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1002.3306
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# Free Fermions Violate the Area Law For Entanglement Entropy
Robert C. Helling 11 Arnold Sommerfeld Center, Ludwig-Maximilians-Universität
München
Wolfgang Spitzer 22 Institut für Theoretische Physik, Universität Erlangen-
Nürnberg
###### Abstract
We show that the entanglement entropy associated to a region grows faster than
the area of its boundary surface. This is done by proving a special case of a
conjecture due to Widom that yields a surprisingly simple expression for the
leading behaviour of the entanglement entropy.
###### keywords:
Entanglement entropy, Area Law, LMU-ASC-08-10
## 1 Background
An interesting model for the area law of Bekenstein’s black-hole entropy is a
local quantum field theory for which one restricts all observations to the
complement of a compact spatial region $\Omega$. One should think of $\Omega$
as the interior of the horizon although one does not require $\Omega$ to have
any gravitational relevance. In fact, we will restrict our attention to a
region in the $n$-dimensional flat Euclidean space ${\mathbb{R}}^{n}$. Even if
the quantum field is globally in a pure state (e.g., the vacuum), the state
restricted to the complement of $\Omega$ will in general be mixed with finite
von Neumann entropy, called the entanglement entropy, $S(\Omega)$ [1, 2, 3].
For many types of quantum field theories, it was observed that in the semi-
classical limit of large $\Omega$, the entanglement entropy $S(\Omega)$ scales
like the area of $\partial\Omega$. In other words, if we rescale
$\Omega\subset{\mathbb{R}}^{n}$ by a factor $R$, the entanglement entropy,
$S(R\Omega)$, should asymptotically scale like $R^{n-1}$ for large $R$.
Recently, it was observed by Gioev and Klich [4, 5], that for free fermions at
zero temperature, however, the entanglement entropy of the ground state scales
like $R^{n-1}\log(R)$ for $R\to\infty$ based on a conjecture by Widom [6, 7].
They provide a lower bound on the entanglement entropy in terms of the trace
of a quadratic function of the restricted state. For the latter, we have
proved in [8] the leading asymptotic behaviour. This establishes indeed that
the entanglement entropy scales at least like $R^{n-1}\log(R)$, which violates
of the beforementioned area law scaling.
For notational clarity we will often write equalities that only hold
asymptotically for large $R$, that is, we will drop subleading terms that are
not central to our argument and only mention it in the text.
General arguments imply that for a pure state, the entanglement entropy of
this state with respect to $\Omega$ is the same as for the complement
${\mathbb{R}}^{n}\setminus\Omega$. For simplicity, we will restrict the state
to a compact region $\Omega$. The ground state, $\rho_{\Gamma}$, of a system
of non-interacting fermions in ${\mathbb{R}}^{n}$ is given in terms of the
Fermi surface at Fermi energy $\epsilon_{F}$: All one-particle states with
momentum
$\mathbf{p}\in\Gamma=\\{\mathbf{p}\in{\mathbb{R}}^{n}|E(\mathbf{p})\leq\epsilon_{F}\\}$
and energy111We will not discuss here assumptions on the dispersion relation
$E(\mathbf{p})$ but one may, of course, think of the example
$E(\mathbf{p})=\mathbf{p}^{2}$. We require that $\Gamma$ and $\Omega$ are
compact sets in ${\mathbb{R}}^{n}$ with sufficiently smooth
boundaries.$E(\mathbf{p})$ are occupied. The ground state $\rho_{\Gamma}$ is
then characterised by the one-particle Fermi projector, $P_{\Gamma}$, defined
in momentum space by the kernel
$P_{\Gamma}(\mathbf{p},\mathbf{p}^{\prime})=\chi_{\Gamma}(\mathbf{p})\delta(\mathbf{p}-\mathbf{p}^{\prime})$.
Here, and in the following, $\chi_{A}$ denotes the indicator function of a set
$A$. In position space, the kernel
$P_{\Gamma}(\mathbf{x},\mathbf{x}^{\prime})=\widehat{\chi_{\Gamma}}(\mathbf{x}-\mathbf{x}^{\prime})$
with
$\widehat{\chi_{\Gamma}}(\mathbf{x})=(2\pi)^{-n}\int_{\Gamma}d\mathbf{p}\,e^{i\mathbf{x}\cdot\mathbf{p}}$
being the inverse Fourier transform of $\chi_{\Gamma}$.
In order to restrict the state $\rho_{\Gamma}$ to the region $\Omega$ we
project the Fermi projector $P_{\Gamma}$ onto $L^{2}(\Omega)$ with
$Q_{\Omega}=\chi_{\Omega}$. This gives the reduced one-particle density matrix
$\varrho_{\Omega,\Gamma}=Q_{\Omega}P_{\Gamma}Q_{\Omega}$. The entanglement
entropy, $S(\Omega,\Gamma)$, of the many particle system in the ground state
$\rho_{\Gamma}$ restricted to the region $\Omega$ is then defined as the grand
canonical entropy of $\varrho_{\Omega,\Gamma}$, that is,
$S(\Omega,\Gamma)=\operatorname{{\mathrm{t}r}}\eta(\varrho_{\Omega,\Gamma})$
with $\eta(t)=-t\log{(t)}-(1-t)\log{(1-t)}$ for $0<t<1$. For details see [8,
Section 4]. We are interested here in the behaviour of this entropy for fixed
$\Gamma$ but large $\Omega$. To this end, we also fix $\Omega$ and study the
asymptotic behaviour of $S(R\Omega,\Gamma)$ as $R\to\infty$.
Our main result is the asymptotic computation of
$\operatorname{{\mathrm{t}r}}[\varrho_{R\Omega,\Gamma}(1-\varrho_{R\Omega,\Gamma})]$
as $R\to\infty$:
$\displaystyle\operatorname{{\mathrm{t}r}}[\varrho_{R\Omega,\Gamma}(1-\varrho_{R\Omega,\Gamma})]=\left(\frac{R}{2\pi}\right)^{n-1}\frac{\log(R)}{4\pi^{2}}\int_{\partial\Omega\times\partial\Gamma}dA(\mathbf{x})dA(\mathbf{p})\,|\mathbf{n}_{\mathbf{x}}\cdot\mathbf{n}_{\mathbf{p}}|,$
(1)
up to terms that grow slower in $R$. Here, $\mathbf{n}_{\mathbf{x}}$ denotes
the unit normal vector at $\mathbf{x}\in\partial\Omega$, $dA(\mathbf{x})$ is
the surface measure on $\partial\Omega$, and similarly for
$\mathbf{n}_{\mathbf{p}}$ and $dA(\mathbf{p})$.
Since $\eta(t)\geq\log(2)4t(1-t)$ for $0<t<1$, we obtain the asymptotic lower
bound on the entanglement entropy,
$S(R\Omega,\Gamma)\geq\frac{\log(2)}{\pi^{2}}\left(\frac{R}{2\pi}\right)^{n-1}\log(R)\int_{\partial\Omega\times\partial\Gamma}dA(\mathbf{x})dA(\mathbf{p})\,|\mathbf{n}_{\mathbf{x}}\cdot\mathbf{n}_{\mathbf{p}}|.$
(2)
Gioev and Klich conjectured in [5] that the exact scaling of
$S(R\Omega,\Gamma)$ is obtained if we replace the factor $\log{(2)}/\pi^{2}$
in (2) by $\frac{1}{12}$. This remains an open problem.
## 2 Computation of
$\operatorname{{\mathrm{t}r}}[\varrho_{R\Omega,\Gamma}(1-\varrho_{R\Omega,\Gamma})]$
The trace of $\varrho_{R\Omega,\Gamma}$ is simply equal to
$\left(\frac{R}{2\pi}\right)^{n}|\Omega||\Gamma|$, where $|\cdot|$ denotes the
$n$-dimensional Lebesgue volume. The trace of $\varrho_{R\Omega,\Gamma}^{2}$
equals
$\displaystyle\operatorname{{\mathrm{t}r}}(Q_{R\Omega}P_{\Gamma}Q_{R\Omega}P_{\Gamma})\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\int_{R\Omega}\\!\\!\\!d\mathbf{x}\int_{R\Omega}\\!\\!\\!d\mathbf{x}^{\prime}\,|\widehat{\chi_{\Gamma}}(\mathbf{x}-\mathbf{x}^{\prime})|^{2}=\int_{R(\Omega-\Omega)}\\!\\!\\!d\mathbf{v}\,|\widehat{\chi_{\Gamma}}(\mathbf{v})|^{2}\,|R\Omega\cap(R\Omega-\mathbf{v})|\,.$
(3)
In the last line we have changed the variables $\mathbf{x}$ and
$\mathbf{x}^{\prime}$ to $\mathbf{u}=\mathbf{x}$ and
$\mathbf{v}=\mathbf{x}-\mathbf{x}^{\prime}$. Then we expand the volume
$|R\Omega\cap(R\Omega-\mathbf{v})|$ to first order in $\mathbf{v}$ (cf. [9,
Theorem 2.1]),
$|R\Omega\cap(R\Omega-\mathbf{v})|=R^{n}|\Omega|+R^{n-1}\\!\\!\\!\int_{\partial\Omega}\\!\\!\\!dA(\mathbf{x})\,\max(0,\mathbf{v}\cdot\mathbf{n}_{\mathbf{x}})+R^{n-2}O(|\mathbf{v}|^{2}).$
(4)
Let us first look at the contribution of $R^{n}|\Omega|$ to the trace of
$\varrho_{R\Omega,\Gamma}^{2}$. The function
$\mathbf{v}\mapsto\widehat{\chi_{\Gamma}}(\mathbf{v})$ decays like
$|\mathbf{v}|^{-(n+1)/2}$ for large $|\mathbf{v}|$, see (7). At the cost of an
order $R^{n-1}$-term we may therefore extend the $\mathbf{v}$-integration to
all of ${\mathbb{R}}^{n}$. By the Plancherel formula this integral gives
$(2\pi)^{-n}|\Gamma|$ and cancels with
$\operatorname{{\mathrm{t}r}}\varrho_{R\Omega,\Gamma}$. When integrated over
$\mathbf{v}$, the remainder term $R^{n-2}O(|\mathbf{v}|^{2})$ is also seen to
yield a term of the order $R^{n-1}$ by using again the above mentioned decay
of $\widehat{\chi_{\Gamma}}$.
Thus, our bound on the entanglement entropy will come from the second term in
(4). Here, we write
$\max(0,\mathbf{v}\cdot\mathbf{n}_{\mathbf{x}})=\chi_{[0,\infty)}(\mathbf{v}\cdot\mathbf{n}_{\mathbf{x}})\,\mathbf{v}\cdot\mathbf{n}_{\mathbf{x}}$
and use the Gauß Theorem so that
$(2\pi)^{n}\,\mathbf{v}\,\widehat{\chi_{\Gamma}}(\mathbf{v})=-i\int_{\partial\Gamma}dA(\mathbf{p})\,\mathbf{n}_{\mathbf{p}}\,e^{i\mathbf{v}\cdot\mathbf{p}}.$
(5)
So it remains to show that for $\mathbf{p}\in\partial\Gamma$,
$\Big{|}\int_{R(\Omega-\Omega)}d\mathbf{v}\,\chi_{[0,\infty)}(\mathbf{v}\cdot\mathbf{n}_{\mathbf{x}})\,\widehat{\chi_{\Gamma}}(-\mathbf{v})\,e^{i\mathbf{v}\cdot\mathbf{p}}+(2\pi
i)^{-1}\operatorname{{\mathrm{s}gn}}({\bm{n}}_{\mathbf{x}}\cdot{\bm{n}}_{\mathbf{p}})\log{(R)}\Big{|}=o(R)\,.$
(6)
Let us now consider the function
$\mathbf{v}\mapsto\widehat{\chi_{\Gamma}}(-\mathbf{v})$ in detail. We use the
representation from (5), that is,
$(2\pi)^{-n}\widehat{\chi_{\Gamma}}(-\mathbf{v})=\frac{i\mathbf{v}}{|\mathbf{v}|^{2}}\cdot\int_{\partial\Gamma}dA(\mathbf{p}^{\prime})\,\mathbf{p}^{\prime}\,e^{-i\mathbf{v}\cdot\mathbf{p}^{\prime}}$.
Then we introduce a coordinate system where $\mathbf{v}=(0,\ldots,0,v)$ and
where the boundary $\partial\Gamma$ is locally written as the graph of a
function $f\colon U\subset{\mathbb{R}}^{n-1}\to{\mathbb{R}}$, that is,
$\mathbf{p}^{\prime}=(\mathbf{t},f(\mathbf{t}))$ and
$dA(\mathbf{p}^{\prime})=\sqrt{1+|\nabla f|^{2}}d\mathbf{t}$. The unit normal
vector is
${\bm{n}}_{\bm{p}^{\prime}}=\operatorname{{\mathrm{s}gn}}(\mathbf{v}\cdot\mathbf{p}^{\prime})(-\nabla
f,1)/{\sqrt{1+|\nabla f|}}$. Then,
$\int_{f(U)}dA(\mathbf{p}^{\prime})\,({\bm{n}}_{\bm{p}^{\prime}})_{n}\,e^{-i\mathbf{v}\cdot\mathbf{p}}=-\frac{1}{v}\int_{U}d\mathbf{t}\,\operatorname{{\mathrm{s}gn}}(f(\mathbf{t}))\,e^{-ivf(\mathbf{t})}.$
In order to find the leading asymptotic behaviour of this
$\mathbf{t}$-integral for large $v$ we apply the method of stationary phase.
Let
${\mathbf{k}_{a}}={\mathbf{k}_{a}}(\mathbf{v})=(\mathbf{t}_{a}(\mathbf{v}),f(\mathbf{t}_{a}(\mathbf{v})))$
be the collection of all stationary points of such local functions $f$, that
is, $\nabla f(\mathbf{t}_{a})=0$; in other words, the points
${\mathbf{k}_{a}}\in\partial\Gamma$ are such that the unit normal vector
$\bm{n}_{{\mathbf{k}_{a}}}$ at ${\mathbf{k}_{a}}$ is parallel to $\mathbf{v}$.
Thus,
$\displaystyle\widehat{\chi_{\Gamma}}(-\mathbf{v})=-i(2\pi
v)^{-(n+1)/2}\sum_{{\mathbf{k}_{a}}}\frac{\operatorname{{\mathrm{s}gn}}(\mathbf{v}\cdot{\mathbf{k}_{a}})}{\sqrt{|\det
f_{ij}(\mathbf{t}_{a})}|}\,e^{-i\mathbf{v}\cdot{\mathbf{k}_{a}}-i\frac{\pi}{4}\operatorname{{\mathrm{s}gn}}(f_{ij}(\mathbf{t}_{a}))}+o(v^{-(n+1)/2}).$
(7)
Here, $f_{ij}(\mathbf{t}_{a})$ denotes the Hessian of $f$ at $\mathbf{t}_{a}$,
$\operatorname{{\mathrm{s}gn}}(f_{ij}(\mathbf{t}_{a}))$ the signum of this
Hessian. The determinant, $\det f_{ij}(\mathbf{t}_{a})$, equals the Gaußian
curvature of $\partial\Gamma$ at ${\mathbf{k}_{a}}$.
Using (7) we return to the oscillatory integral in (6), and employ once more
the method of stationary phase. The composite phase from (5) and (7) is equal
to $\mathbf{v}\cdot(\mathbf{p}-{\mathbf{k}_{a}}(\mathbf{v}))$. Next, we
introduce generalised spherical coordinates for $\mathbf{v}$ as
$\mathbf{v}=\rho(\mathbf{u},h(\mathbf{u}))$, where the map $h\colon
V\subset{\mathbb{R}}^{n-1}\to{\mathbb{R}}$ locally parametrises the boundary
$\partial(\Omega-\Omega)\ni(\mathbf{u},h(\mathbf{u}))$ and $\rho\in[0,R]$ is a
radial coordinate. The stationary point ${\mathbf{k}_{a}}$ is now a function
of $\mathbf{u}$. We have the freedom to assume that
$\mathbf{n}_{\mathbf{p}}=(0,\ldots,0,1)$ such that
${\mathbf{k}_{a}}((0,h(0)))=\mathbf{p}$ for one index $a$ and that $h$ has its
extremum at the origin $\mathbf{u}=0$. Instead of integrating $\rho$ from 0 to
$R$ we will actually integrate $\rho$ only over $[1,R]$ thereby making an
irrelevant error independent of $R$.
To express ${\mathbf{k}_{a}}(\mathbf{v})$ as a function of $\mathbf{u}$ we
equate $\mathbf{v}/v=\bm{n}_{{\bm{k}}_{a}(\mathbf{v})}$, that is,
$\frac{(\mathbf{u},h(\mathbf{u}))}{\sqrt{\mathbf{u}^{2}+h(\mathbf{u})^{2}}}=\operatorname{{\mathrm{s}gn}}(\mathbf{v}\cdot\mathbf{n}_{\mathbf{k}_{a}})\frac{(-\nabla_{\mathbf{t}}f(\mathbf{t}),1)}{\sqrt{1+|\nabla_{\mathbf{t}}f(\mathbf{t})|^{2}}}.$
(8)
Taking derivatives and evaluating at $\mathbf{u}=0$, we find $\frac{\partial
t_{j}}{\partial u_{i}}(0)=-f_{ij}^{-1}({\mathbf{k}_{a}}(0))/h(0)$, where
${\mathbf{k}_{a}}(0)$ is short for ${\mathbf{k}_{a}}(0,h(0))$ and
$f_{ij}^{-1}$ is the matrix inverse of the Hessian of $f$. With this, we can
expand the phase of the remaining $\mathbf{v}$-integral to second order as
$\mathbf{v}\cdot(\mathbf{p}-{\mathbf{k}_{a}}(\mathbf{v}))=\rho
h(0)(\mathbf{p}-{\mathbf{k}_{a}}(0))_{n}+\rho\frac{f_{ij}^{-1}({\mathbf{k}_{a}}(0))}{2h(0)}u_{i}u_{j},$
(9)
The volume element is given by $d\mathbf{v}=\rho^{n-1}h(\mathbf{u})d\rho
d\mathbf{u}$. The stationary phase integral over the $n-1$ coordinates
$\mathbf{u}$ yields $(2h(0)/\rho)^{(n-1)/2}\sqrt{|\det f_{ij}^{-1}|}\exp(i\rho
h(0)(\mathbf{p}-{\mathbf{k}_{a}}(0))_{n})+i\frac{\pi}{4}\operatorname{{\mathrm{s}gn}}(f_{ij}(\mathbf{t}_{a}))$.
This, surprisingly, reduces the remaining $\rho$-integral to
$\int_{1}^{R}d\rho\;e^{i\rho h(0)(\mathbf{p}-{\mathbf{k}_{a}}(0))_{n}}/\rho$.
As noted above, for one index $a$, the exponent vanishes as
$\mathbf{p}={\mathbf{k}_{a}}(0)$ and the integral is the desired $\log(R)$.
If, however, $\mathbf{p}\neq{\mathbf{k}_{a}}(0)$ the integral is bounded for
large $R$ and does not contribute to leading order.
Collecting all terms we have thus proved the lower bound (1) which grows
faster than the area law scaling $R^{n-1}$ by a factor of $\log(R)$. Note that
the stationary phase integrals localises $\mathbf{v}$ and
$\mathbf{p}^{\prime}$ such that there are only contributions from
$\mathbf{p}$, $\mathbf{n}_{\mathbf{v}}$, ${\bm{n}}_{\bm{p}^{\prime}}$ all
being parallel.
## 3 Discussion
Above, we presented a lower bound for the entanglement entropy which violates
an $R^{n-1}$ area law scaling. The expression (2) is in terms of integrals
over the boundaries $\partial\Omega$ and $\partial\Gamma$ and is thus still a
“boundary effect”.
We find it most curious that although we have used stationary phase methods,
which in general depend on (the existence of) second order derivatives at the
stationary points, all these curvature terms involving $f_{ij}$ and $h_{ij}$
eventually cancel out and the integrand in the bound does not contain
derivatives. However, one expects [4, 10] that a fractal boundary
$\partial\Omega$ of dimension $n-1+\alpha$ with $0<\alpha<1$ leads to a
scaling of the entanglement entropy of at least of the order $R^{n-1+\alpha}$
and, presumably, without a $\log{(R)}$ correction.
For $n=1$, if $\Omega$ and $\Gamma$ are a disjoint union of $k$ and $\ell$
compact intervals of finite length respectively, then our lower bound (2) for
the entanglement entropy gives $\log(R)\log(2)4k\ell/\pi^{2}$. In this one-
dimensional case, the precise scaling has been proved, namely,
$\log(R)k\ell/3$. If we then consider the hypercubes, say
$\Omega=\Gamma=[0,1]^{n}$, it is not difficult to derive the exact asymptotic
scaling of $S(R\Omega,\Gamma)$ to be $(R/(2\pi))^{n-1}\log{(R)}n^{2}/3$, which
is in agreement with the conjecture by Gioev and Klich mentioned at the end of
Section 1. Our method of proof requires that the surfaces $\partial\Omega$ and
$\partial\Gamma$ are $C^{3}$. Hence, hypercubes are not included. For a
$C^{3}$ surface, $\partial\Gamma$, it was crucial that
$\widehat{\chi_{\Gamma}}(\mathbf{v})$ behaves like $|\mathbf{v}|^{-(n+1)/2}$
for large $|\mathbf{v}|$. This is not the case for a non-smooth surface such
as the hypercube, where
$\widehat{\chi_{[0,1]^{n}}}(\mathbf{v})=\prod_{i=1}^{n}\sin(v_{i})/(\pi
v_{i})$.
It should be noted as well that the discontinuity of $\chi_{\Gamma}$ is
crucial for the decay of its Fourier transform. For example, for the
equilibrium state at positive temperature, the entanglement entropy scales
like the volume $R^{n}$, see [5]. We did not have to introduce an ultraviolet
regulator since momentum integrations are limited to the compact region
$\Gamma$.
## Acknowledgments
We are grateful to Hajo Leschke with whom the work in [8] was performed on
which is text is based. Furthermore, RCH would like to thank the Elitenetwork
of Bavaria for financial support and Jacobs University Bremen, where this work
was started. We also thank Urs Frauenfelder for discussions.
## References
* [1] M. Srednicki, Phys. Rev. Lett. 71(5), 666–669 (1993).
* [2] L. Bombelli, R. K. Koul, J. Lee, and R. D. Sorkin, Phys. Rev. D 34(2), 373–383 (1986).
* [3] J. Eisert, M. Cramer, and M. B. Plenio, Area laws for the entanglement entropy - a review, 2008, arXiv.org:0808.3773.
* [4] D. Gioev, Int. Math. Res. Not. 2006(O95181), 95181–23 (2006).
* [5] D. Gioev and I. Klich, Phys. Rev. Lett. 96(10), 100503 (2006).
* [6] H. Widom, Trans. AMS 94(1), 170–180 (1960).
* [7] H. Widom, J. Funct. Anal. 88, 166–193 (1990).
* [8] R. C. Helling, H. Leschke, and W. L. Spitzer, A special case of a conjecture by Widom with implications to fermionic entanglement entropy, 2009, arXiv.org:0906.4946.
* [9] R. Roccaforte, Trans. AMS 285(2), 581–602 (1984).
* [10] M. Fannes, B. Haegeman, and M. Mosonyi, J. Math. Phys. 44(12), 6005–6019 (2003).
|
arxiv-papers
| 2010-02-17T16:33:21 |
2024-09-04T02:49:08.441897
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Robert C. Helling (Arnold Sommerfeld Center, LMU Munich), Wolfgang\n Spitzer (U Erlangen-Nuernberg)",
"submitter": "Robert C. Helling",
"url": "https://arxiv.org/abs/1002.3306"
}
|
1002.3389
|
# Deformations of quantum field theories and Connes–Marcolli’s renormalization
group in Epstein–Glaser scheme
Özgür Ceyhan Ö. Ceyhan: Korteweg-de Vries Institute for Mathematics,
University of Amsterdam P. O. Box 94248, 1090 GE Amsterdam, Netherlands
o.ceyhan@uva.nl
## 1\. introduction
Over the decades, physicists have developed a number of state of the art
techniques to produce quantities of great physical relevance out of
mathematically ill-defined quantum field theories (QFTs). The general strategy
is to define physical quantities in families in such a way that they are well-
defined away from the certain limits of the parameters, and then to extract
finite limits of these quantities by using regularization techniques.
The momentum space renormalizations have been widely used in QFTs. The
combinatoric of divergencies and regularizations are beautifully encoded by
Hopf algebras of Feynmann diagrams and Birkhoff decomposition of loops in
these Hopf algebras (see [4, 5]). The complexified dimension plays the role of
deformation parameter in Connes–Kreimer picture. In [6, 7], Connes & Marcolli
have constructed a Riemann–Hilbert correspondence associated to perturbative
renormalization based on Connes–Kreimer’s approach. They have observed an
action of a pro-unipotent affine group scheme ${\mathbb{U}}^{*}$, universal
with respect to the physical theories, and pointed out its connection to the
motivic Galois group of the scheme of 4-cyclotomic integers
${\mathbb{Z}}[i][\frac{1}{2}]$.
On the other hand, Epstein–Glaser renormalization distinguishes itself among
others: It produces finite QFTs from the very definition by choosing the
domain of physical parameters suitably. The Epstein–Glaser’s approach is based
on Dyson series
(1.1) $\displaystyle
S=1+\sum_{n=1}^{\infty}\frac{(i)^{n}}{n!}\int_{M^{n}}dx_{1}\cdots dx_{n}\
T_{n}({\mathcal{L}}_{I}(x_{1}),\dots,{\mathcal{L}}_{I}(x_{n}))$
of scattering operator (S-matrix) for a given potential term
${\mathcal{L}}_{I}$ of a Lagrangian. The problem in Epstein–Glaser setting is
formulated as the problem of extensions of distributions
$T_{n}({\mathcal{L}}_{I}(x_{1}),\dots,{\mathcal{L}}_{I}(x_{n}))$ defined on
the configuration space $M^{n}\setminus\Delta$ of points on the spacetime $M$
to the diagonals $\Delta$.
Contrary to the common perception, that points at divergencies as sources of
ambiguities in QFTs, ambiguities are still present in finite QFTs and are
determined by distributions supported on the diagonals in Epstein–Glaser
setting. This short note aims to describe the deformations of QFTs in terms of
the distributions supported on the diagonals and then give an action of the
pro-algebraic group ${\mathbb{U}}^{*}$ which appears in Connes–Marcolli’s
setting, on the finite QFTs constructed by Epstein-Glaser renormalization
scheme.
This paper is organized as follows: In Section 2, we review some basic facts
on Epstein–Glaser constructions of time ordered products. In the following
section, we describe the deformation space of QFTs. In Section 4, we give an
action of the pro-algebraic group ${\mathbb{U}}^{*}$ on the space of QFTs.
Finally, in Section 5, we discuss a number of corollaries of our constructions
in §3 and §4, and their connections to some other renormalization related
problems.
## 2\. Epstein-Glaser renormalization in a nutshell
Let spacetime $M$ be Euclidian space ${\mathbb{R}}^{d}$, and
${\mathcal{D}}(M)$ be the space of test functions on $M$ with the usual
topology. Let ${\mathcal{H}}$ denote the Hilbert space of the free fields, $D$
a suitable dense subspace and $\Omega$ be the vacuum state.
### 2.1. Time ordered products
Time ordered products form a collection of operator valued distributions
(2.1) $\displaystyle\\{T_{N}:{\mathcal{D}}(M^{N})\to End(D)\mid
N:=\\{1,\dots,n\\}\\},$
and, in Epstein-Glaser renormalization scheme, they are expected to satisfy a
set of basic properties:
#### 2.1.1. Symmetry
$T_{N}$’s are symmetric under permutations of indices, i.e.,
$T_{N}(f_{1}\otimes\dots\otimes f_{n})=T_{N}(f_{\sigma(1)}\otimes\dots\otimes
f_{\sigma(n)})$
for all $\sigma$ in the symmetric group of index set $N$.
#### 2.1.2. Causality
$T_{N}$ factorizes casually, i.e., if $I,I^{c}\neq\emptyset$ is a partition of
$N$, and if $supp(f_{i})\cap supp(f_{j})=\emptyset$ for all $i\in I$ and $j\in
I^{c}$, then
(2.2) $\displaystyle T_{N}(f_{1}\otimes\dots\otimes
f_{n})=T_{I}(\bigotimes_{i\in I}f_{i})\cdot T_{N\setminus I}(\bigotimes_{j\in
I^{c}}f_{j}).$
#### 2.1.3. Translation invariance
$T_{N}$ is invariant under translations:
$T_{N}(f_{1}(x_{1})\otimes\dots\otimes
f_{n}(x_{n}))=T_{N}(f_{1}(x_{1}-a)\otimes\dots\otimes f_{n}(x_{n}-a)).$
Epstein & Glaser constructed time ordered products essentially by using the
causality in [9].
###### Theorem 2.1 (Epstein & Glaser, [9]).
Time-ordered products exist.
### 2.2. Wick expansions of time ordered products
The extension problem of operator valued distributions is reduced to an
extension problem for numerical distributions by expanding time ordered
products in terms of the Wick expansions.
###### Theorem 2.2.
Let $:\phi^{k_{1}}(x_{1}):,\dots,:\phi^{k_{n}}(x_{n}):$ Wick mononomials for
noncoinciding points $x_{1},\dots,x_{n}$ in $M$. Then
$\displaystyle
T_{N}(:\phi^{k_{1}}(x_{1}):\cdots:\phi^{k_{n}}(x_{n}):)=\sum_{\mathbf{J}=(i_{1},\dots,i_{n})=0}^{(k_{1},\dots,k_{n})}t_{\mathbf{J}}(x_{1},\dots,x_{n})\times\frac{:\phi^{i_{1}}\cdots\phi^{i_{n}}:}{i_{1}!\cdots
i_{n}!}$
where the numerical distribution $t_{\mathbf{J}}(x_{1},\dots,x_{n})$ is
$\langle\Omega,T_{N}(:\phi^{k_{1}-i_{1}}(x_{1}):\cdots:\phi^{k_{n}-i_{n}}(x_{n}):)\
\Omega\rangle$
(for instance, see Theorem 2.4 in [2]).
## 3\. Deformations of QFTs in Epstein–Glaser scheme
One of the main consequences of Epstein–Glaser construction is that the space
which parameterizes the collection of time ordered products can be observed
explicitly:
###### Lemma 3.1.
Let $T_{N},\widehat{T}_{N}$ be two different time ordered products. Then, the
difference $T_{N}-\widehat{T}_{N}$ is an operator valued distribution
supported on the diagonals $\Delta\subset M^{N}$.
The proof of this lemma is straightforward and can be found in [18] and [14]
for Minkowski and Euclidean cases respectively.
### 3.1. The space of QFTs
We can reformulate Lemma 3.1 on the level of numerical distributions and give
the space of QFTs as follows: First, we use the translation invariance and set
one of the points, say $x_{1}$, to $0$, so that
$t_{\mathbf{J}}\in{\mathcal{D}}(M^{|N|-1})$ for $n\geq 2$. Due to Lemma 3.1,
we obtain a new distribution by adding another numerical distribution
supported on the union of diagonals $\Delta=\bigcup_{I\subset N}\Delta_{I}$
where $\Delta_{I}:=\\{(0,x_{2},\dots,x_{n})\mid x_{i}=x_{i}\ \text{iff}\
i,j\in I\\}\subset M^{|N|-1}$, i.e.,
(3.1) $t_{\mathbf{J}}\mapsto t_{\mathbf{J}}+d_{\mathbf{J}},\ \ \ \text{where}\
\ \ d_{\mathbf{J}}=\sum_{I\subset N}d_{\mathbf{J},I},$
and $d_{\mathbf{J},I}$’s are numerical distributions supported on the
corresponding diagonals $\Delta_{I}\subset M^{|N|-1}$.
Due to the well known fact that distributions supported at one point are
finite linear combinations of the $\delta$ distribution and its derivatives,
the summand supported on the deepest diagonal $\Delta_{N}=\\{0\\}$ in (3.1) is
given by
$d_{\mathbf{J},N}=\sum_{{\alpha=(\alpha_{1},\dots,\alpha_{nd})}\atop{\sum\alpha_{*}\leq
sd(t_{\mathbf{J}})}}b_{\mathbf{J},N}^{\alpha}\cdot\frac{\partial^{\alpha_{2}}}{\partial
x_{2}^{\alpha_{2}}}\cdots\frac{\partial^{\alpha_{nd}}}{\partial
x_{nd}^{\alpha_{nd}}}\delta_{N}$
where $\delta_{N}$ is the delta function supported on $\\{0\\}\subset
M^{|N|-1}$. The degree is bounded by the generalized degree of homogeneity,
called scaling degree
$sd(t_{\mathbf{J}}):=\inf\\{s\mid\lim_{\lambda\to 0}\lambda^{s}\cdot\int\
t_{\mathbf{J}}(\lambda x)\omega(x)dx\\}.$
The other $d_{\mathbf{J},I}$’s in (3.1) can be given as above case;
(3.2)
$d_{\mathbf{J},I}=\sum_{{\alpha=(\alpha_{1},\dots,\alpha_{nd})}\atop{\sum\alpha_{*}\leq
sd(t_{\mathbf{J}})}}b_{\mathbf{J},I}^{\alpha}\cdot\frac{\partial^{\alpha_{2}}}{\partial
x_{2}^{\alpha_{2}}}\cdots\frac{\partial^{\alpha_{nd}}}{\partial
x_{nd}^{\alpha_{nd}}}\delta_{I}$
where $\delta_{I}$ is the delta function supported on $\Delta_{I}\subset
M^{|N|-1}$.
Hence, we can rephrase Lemma 3.1 as a deformation theory for QFTs in
Epstein–Glaser setting: Let $\text{Def}({\mathcal{Q}})$ be the space of QFTs
around a given QFT determined by the set of numerical distributions
${\mathcal{Q}}=\\{t_{\mathbf{J}}\\}$.
###### Theorem–Definition 3.1.
$\text{Def}({\mathcal{Q}})$ is an infinite dimensional Euclidean space whose
coordinate ring ${\mathcal{H}}$ is ${\mathbb{C}}[b_{\mathbf{J},I}^{\alpha}]$
where $|\mathbf{J}|>2,|\alpha|\leq sd((t_{\mathbf{J}}))$ and $I\subset N$.
It is important to note that $\text{Def}({\mathcal{Q}})$ is unobstructed since
the coordinate ring is ${\mathbb{C}}[b_{\mathbf{J},I}^{\alpha}]$, and
therefore all $k$-th order deformations extends to the next order for all $k$.
### 3.2. Filtration of $\text{Def}({\mathcal{Q}})$
$\text{Def}({\mathcal{Q}})$ is filtered
$\emptyset\subset\text{Def}^{(1)}({\mathcal{Q}})\subset\dots\subset\text{Def}^{(n)}({\mathcal{Q}})\subset\text{Def}^{(n+1)}({\mathcal{Q}})\subset\cdots$
according to the cardinality of index set $\mathbf{J}=(j_{1},\dots,j_{n})$:
$\text{Def}^{(n)}({\mathcal{Q}})=\\{d_{\mathbf{J}}\mid|\mathbf{J}|=n+1\\}.$
for all $n=1,2,3,\dots$ The inclusions $\iota:I\hookrightarrow N$ of index
sets induce imbeddings
$\iota_{\\#}:\text{Def}^{(|I|-1)}({\mathcal{Q}})\hookrightarrow\text{Def}^{(|N|-1)}({\mathcal{Q}})$.
## 4\. Renormalization group in Epstein–Glaser scheme
### 4.1. Symmetries acting on the space of QFTs
The most general form of symmetries of $\text{Def}({\mathcal{Q}})$ are given
by the pseudo-group of all formal (local) diffeomorphisms
$\mu:\text{Def}({\mathcal{Q}})\to\text{Def}({\mathcal{Q}})$. More elaborate
symmetries form a Lie pseudo-group. They are prescribed by systems of
nonlinear equations on jet bundles that are satisfying formal integrability
and local solvability conditions. The remarkable fact is that the
Maurer–Cartan form produces an explicit form of the pseudo-group structure
equations, see [17].
A version of such a symmetry group, called the group of diffeographisms, is
introduced as the group of formal diffeomorphisms tangent to the identity of
the space of coupling constants of the theory by Connes & Kreimer in [5].
### 4.2. Connes–Marcolli’s renormalization group in Epstein–Glaser setting
From physics perspective, the subgroup of symmetries of
$\text{Def}({\mathcal{Q}})$ generated by the scaling transformations is of
particular interest since it essentially gives the renormalization group.
In this paragraph, we present the action of a subgroup of scalings on the
space of QFTs. Namely, we consider a pro-algebraic group of the form
${\mathbb{U}}^{*}={\mathbb{U}}\rtimes{\mathbb{G}}_{m}$ whose unipotent part
${\mathbb{U}}$ is generated by scaling transformations and is associated to
the free graded Lie algebra ${\mathcal{F}}(1,2,\cdots)_{\bullet}$ with one
generator $e_{n}$ in each degree $n>0$. The semi-direct product is given by
the grading of ${\mathbb{U}}$.
#### 4.2.1. Pro-unipotent part ${\mathbb{U}}$
Consider the scaling transformations
(4.1) $b_{\mathbf{J},I}^{\alpha}\mapsto\sum_{K\subseteq
I}\epsilon(\alpha,K,\mathbf{J})\ b_{\mathbf{J},K}^{\alpha}$
where
(4.2)
$\epsilon(\alpha,K,\mathbf{J})=\left\\{\begin{array}[]{ccc}\lambda^{|K|}&\text{if}&j_{1}>j_{2}\\\
1&\text{if}&j_{1}=j_{2}\\\ 0&\text{if}&j_{1}<j_{2}.\end{array}\right.$
They act upon the degree $n$ piece $\text{Def}^{(n)}({\mathcal{Q}})$ of
$\text{Def}({\mathcal{Q}})$. Note that, the definition of $\epsilon$
guarantees that the matrix in (4.1) is upper triangular and therefore the
transformation in (4.1) is pro-unipotent.
The infinitesimal generator of (4.1) is given by the following vector field
$\mathbf{e}_{n}=\sum_{\mathbf{J}:j_{1}>j_{2}}\sum_{I\subset
N,\alpha}\left(\sum_{K\subseteq
I}|K|b_{\mathbf{J},K}^{\alpha}\frac{\partial}{\partial
b_{\mathbf{J},K}^{\alpha}}\right)$
in $T^{*}\text{Def}({\mathcal{Q}})$. The pro-unipotent part ${\mathbb{U}}$ is
associated to free graded Lie algebra ${\mathcal{F}}(1,2,\cdots)_{\bullet}$
which is generated by the elements $e_{n}$ at each positive degree $n$.
#### 4.2.2. Multiplicative group ${\mathbb{G}}_{m}$ and semi-direct product
Consider the 1-parameter group of automorphisms
$\theta_{z}:d_{\mathbf{J}}\mapsto e^{nz}\cdot d_{\mathbf{J}},\ \ \ \forall
z\in{\mathbb{C}}$
implementing the grading. Its infinitesimal generator is given by the grading
operator
$Y(d_{\mathbf{J}}):=\frac{d}{dz}(\theta_{z}d_{\mathbf{J}})\mid_{z=0}=n\cdot
d_{\mathbf{J}}.$
Finally, we define, for all $u\in{\mathbb{G}}_{m}$, an action $u^{Y}$ on
${\mathbb{U}}$ by
$u^{Y}(X)=u^{n}X,\ \forall\ X\ \text{of degree}\ n.$
We can then form the semi-direct product
${\mathbb{U}}^{*}={\mathbb{U}}\rtimes{\mathbb{G}}_{m}$
and this shows that
###### Theorem 4.1.
The pro-algebraic group ${\mathbb{U}}^{*}$ acts upon the space
$\text{Def}({\mathcal{Q}})$.
${\mathbb{U}}^{*}$ is universal with respect to the set of physical theories,
in the sense that it is canonically defined and independent of the physical
theory.
###### Remark 4.2.
In their seminal paper [6], Connes & Marcolli considered the same
${\mathbb{U}}^{*}$ as renormalization group. Their motivation is to identify
the renormalization group as a motivic Galois group as Cartier suggested in
[3]. In their approach, the pro-unipotent part is the graded dual of the
universal enveloping algebra
${\mathcal{U}}({\mathcal{F}}(1,2,\cdots)_{\bullet})^{\vee}$
as a Hopf algebra. Then, they showed that the Tannakian category of flat
equisingular connections which they have obtained from the differential system
of counterterms is equivalent to a category of representations of the affine
groups scheme ${\mathbb{U}}^{*}$.
## 5\. Remarks and further directions
There are numerous connections between Epstein–Glaser and other approaches to
QFTs. Below, we summarize a few direct corollaries of the discussions of the
previous section and speculate on a few possible applications in related
fields.
#### 5.0.1. Spacetime other than Euclidean spaces
In §4.2, we have presented an action of pro-algebraic group ${\mathbb{U}}^{*}$
on the space $\text{Def}({\mathcal{Q}})$ of $S$-matrices on Euclidean
spacetime. However, the basics of this approach can be directly adapted for
any spacetime manifold $M$. Once the time ordered products are given in terms
of numerical distributions (as in [2], for instance), one can define a
representation of ${\mathbb{U}}^{*}$ by considering scaling properties of
distributions supported on the diagonals $\Delta\subset M^{n}$. A construction
of time ordered products for curved space-time along with a discussion of
renormalization group which is very close to our desciption here can be found
in [12].
#### 5.0.2. Causal treatment of gauge theories
In their papers [15] and [20, 21], Kreimer and van Suijlekom extended the
results of Connes & Marcolli to gauge field theories by discussing the
Slavnov-Taylor identities for the couplings at the Hopf algebra level. Van
Suijlekom showed that the Slavnov-Taylor identities generate a Hopf ideal of
Hopf algebra of Feynman diagrams. Hence, these identities are compatible with
renormalization, and the affine group scheme ${\mathbb{U}}^{*}$ remains as a
part of the renormalization picture.
On the other hand, using Epstein–Glaser in gauge field theories is not new to
physics literature and has been studied extensively in both abelian and non-
abelian gauge theories (for instance, see [8, 11, 13, 10, 19]). The gauge
invariance condition in the causal approach is expressed in every order of
perturbation theory separately by a relation of the $n$-point distributions
$T_{N}$ with the charge $Q$, the generator of the free operator gauge
transformations
$[Q,T_{N}]=d_{Q}T_{N},$
and this equation essentially encodes Slavnov-Taylor identities. By using the
gauge invariant distributions supported on the diagonals, one gives the role
of symmetry group of perturbative gauge theories to same ${\mathbb{U}}^{*}$.
The essential tool for casual approach in gauge theories is time ordered
products in Grassmann variables and it can be found in [19]. A very detailed
exposition for Yang-Mills theoy can be found in [11].
#### 5.0.3. Epstein-Glaser vs. dimensional regularization
Even though, our main theorem states an action of the same pro-algebraic group
${\mathbb{U}}^{*}$ on perturbative QFT’s as in Connes–Marcolli’s case, the
action is quite different in nature. The main distinction is that
${\mathbb{U}}^{*}$ acts upon the counterterms in their case. By contrast, the
representation in §4.2 is given by an action on the renormalized values.
Moreover, in Connes–Marcolli’s construction, the affine group scheme
${\mathbb{U}}^{*}$ appears as a motivic Galois group. However, it is unclear
to us whether ${\mathbb{U}}^{*}$ has any direct motivic role in Epstein–Glaser
renormalization in the form discussed above. This question simply arises from
the fact that the integrals in (1.1) contain distributions not rational
functions, and therefore they are not periods.
#### 5.0.4. Feynmann motives and motives of configuration spaces
There is an ongoing search for motivic origins of Feynman amplitudes (for an
extensive account, see [16] and reference therein). This is a program
initiated by Kontsevich’s suggestion that these numbers should be related to
mixed Tate motives. There are several positive and negative results in this
direction.
Epstein–Glaser approach hints a motivic treatment of Feynman integrals:
Feynman rules associates a distribution to each Feynman graph on a
configuration space which is also determined by the same graph. The
divergencies of these integrals can be treated by the techniques that we have
used for time ordered products. Alternatively, one can use Fulton–MacPherson
type of compactifications of these configuration spaces and try to obtain
regularized integrals on them. Fulton–MacPherson compactifications of these
configuration spaces are mixed Tate motives when the spacetime is itself mixed
Tate. This observation is quite intriguing and we are planning to discuss this
approach in a subsequent paper.
#### Acknowledgements
I take this opportunity to express my deep gratitude to K. Kremnizer and M.
Marcolli. This paper essentially contains what I have learned from them. I
also wish to thank to S. Agarwala and W. van Suijlekom for their interest and
suggestions which have been invaluable for me.
Part of this work was carried out during the author’s stay at Caltech, which
he thanks for the hospitality and support. The author is partially supported
by a NWO grant.
## References
* [1] C. Bergbauer, D. Kreimer, The Hopf algebra of rooted trees in Epstein-Glaser renormalization. Ann. Henri Poincaré 6 (2005), no. 2, 343 367.
* [2] R. Brunetti, K. Fredenhagen, Microlocal analysis and interacting quantum field theories: renormalization on physical backgrounds. Comm. Math. Phys. 208 (2000), no. 3, 623–661.
* [3] P. Cartier, A mad day s work: from Grothendieck to Connes and Kontsevich. The evolution of concepts of space and symmetry. Bull. Amer. Math. Soc. (N.S.) 38 (2001), no. 4, 389 408.
* [4] A. Connes, D. 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 (2000), no. 1, 249–273.
* [5] A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann–Hilbert problem. II. The $\beta$-function, diffeomorphisms and the renormalization group. Comm. Math. Phys. 216 (2001), no. 1, 215–241.
* [6] A. Connes, M. Marcolli, From physics to number theory via noncommutative geometry II: Renormalization, the Riemann-Hilbert correspondence, and motivic Galois theory. in ‘Frontiers in Number Theory, Physics, and Geometry, II’ pp.617–713, Springer Verlag, 2006.
* [7] A. Connes, M. Marcolli, Noncommutative Geometry, Quantum Fields and Motives,Colloquium Publications, Vol.55, American Mathematical Society, 2008.
* [8] M. Dütsch, T. Hurth, F. Krahe, G. Scharf, Causal construction of Yang-Mills theories. I. Nuovo Cimento A (11) 106 (1993), no. 8, 1029–1041.
* [9] H. Epstein, V. Glaser, The role of locality in perturbation theory. Ann. Inst. Henri Poincaré, Section A, Vol. XIX, n. 3 (1973) 211.
* [10] D.R. Grigore, The standard model and its generalizations in the Epstein–Glaser approach to renormalization theory. J. Phys. A 33 (2000), no. 47, 8443–8476.
* [11] S. Hollands, Renormalized quantum Yang-Mills fields in curved spacetime. Rev. Math. Phys. 20 (2008), no. 9, 1033–1172.
* [12] S. Hollands, R.M. Wald, On the renormalization group in curved spacetime. Comm. Math. Phys. 237 (2003), no. 1-2, 123–160.
* [13] T. Hurth, Nonabelian gauge theories: the causal approach. Ann. Physics 244 (1995), no. 2, 340–425.
* [14] K.J. Keller Euclidean Epstein-Glaser Renormalization, arXiv:0902.4789
* [15] D. Kreimer, Anatomy of a gauge theory. Ann. Physics 321 (2006), no. 12, 2757–2781.
* [16] M. Marcolli, Feynman motives, World Scientific, 2010.
* [17] P.J. Olver, J. Pohjanpelto, Maurer-Cartan forms and the structure of Lie pseudo-groups. Selecta Math. 11 (2005) 99-126.
* [18] G. Pinter, Finite renormalizations in the Epstein Glaser framework and renormalization of the $S$-matrix of $\Phi^{4}$-theory. Ann. Phys. (8) 10 (2001), no. 4, 333–363.
* [19] G. Scharf, Finite Quantum Electrodynamics: The Causal Approach, Springer; 2nd edition, 1995.
* [20] W.D. Van Suijlekom, The Hopf algebra of Feynman graphs in quantum electrodynamics. Lett. Math. Phys. 77 (2006), no. 3, 265–281.
* [21] W.D. Van Suijlekom, Renormalization of gauge fields: a Hopf algebra approach. Comm. Math. Phys. 276 (2007), no. 3, 773–798.
|
arxiv-papers
| 2010-02-17T22:08:01 |
2024-09-04T02:49:08.446831
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ozgur Ceyhan",
"submitter": "Ozgur Ceyhan",
"url": "https://arxiv.org/abs/1002.3389"
}
|
1002.3433
|
11institutetext: Indian Institute of Astrophysics, Bangalore 560034, India.
nkrao@iiap.res.in 22institutetext: The W.J. McDonald Observatory, The
University of Texas, Austin, TX 78712-1083, USA
dll@astro.as.utexas.edu
# High-resolution spectroscopy of the R Coronae Borealis and Other Hydrogen
Deficient Stars
N. Kameswara Rao 11 David L. Lambert 22
###### Abstract
High-resolution spectroscopy is a very important tool for studying stellar
physics, perhaps, particularly so for such enigmatic objects like the R
Coronae Borealis and related Hydrogen deficient stars that produce carbon dust
in addition to their peculiar abundances. Examples of how high-resolution
spectroscopy is used in the study of these stars to address the two major
puzzles are presented: (i) How are such rare H-deficient stars created? and
(ii) How and where are the obscuring soot clouds produced around the R Coronae
Borealis stars?
## 1 Introduction
We congratulate the organizers for arranging and conducting this conference,
celebrating the 150 years of the discovery by (Gustav Robert) Kirchoff and
(Robert Wilhelm Eberhard) Bunsen that various gases can be easily and
positively identified by a detailed study of the light they emit and absorb.
As Paul Merrill (Merrill et al. (1963)) noted, a new era in astronomy began
when Bunsen saw ‘in yellow flame of an ordinary alcohol lamp whose wick was
sprinkled with salt, and possibilities of the chemical analysis of the most
distant stars’. Thus began the era of astronomical spectroscopy. Merrill’s own
contributions to our discipline are legendary with his discovery of Tc i lines
in the spectra of S stars signaling that element synthesis in stars was a
continuing phenomena (Merrill (1952)).
A few historical remarks might be appropriate. Astronomical spectroscopy in
India started during the famous total solar eclipse that occurred on 1868
August 18 in which a spectroscope was first used to study the nature of solar
prominences leading to the discovery of helium. Madras observatory, from which
the present host Institution evolved, also played a role. Norman Robert
Pogson, then director of the observatory, used a visual spectroscope to look
at the prominences and noted a bright line in the yellow close to but not
coincident with sodium D lines. This was the D3 line - the same line as found
by Janssen during the 1868 eclipse but by Lockyer without the aid of eclipse.
The first stellar spectrum recorded in India was obtained at Madras (now
Chennai); this was a spectrum of the Wolf-Rayet star $\gamma^{2}$ Velorum
observed in 1871.
Spectroscopic analysis of some of the most enigmatic stars yet discovered is
the theme of our contribution. The chosen stars are the R Coronae Borealis
stars and their putative cooler relatives, the hydrogen-deficient cool carbon
stars (HdCs). The enigma presented by these stars has two principal parts: (i)
How did these stars become so H-deficient? and (ii) how do the RCBs but not
the HdCs develop thick clouds of soot that obscure the stellar photosphere
from view? Answers to these questions are emerging. Here, we highlight
particular contributions to their solution from analysis of high-resolution
optical and infrared spectra. Disney (2000) has lamented (tongue in cheek, we
suppose) that ‘the tragedy of Astronomy is that most information lies in
spectra’. We would counter by substituting ‘allure’ for ‘tragedy’ and add that
one should expect to apply all available observational tools in seeking
answers to our questions.
R Coronae Borealis stars are a rare class of peculiar variable stars. The two
defining characteristics of RCBs are (i) a propensity to fade at unpredictable
times by up to about 8 magnitudes as a result of obscuration by clouds of
soot, and (ii) a supergiant-like atmosphere that is very H-deficient and He-
rich. The $T_{\rm eff}$ of the class ranges from 8000 to 4000 K and $\log g$
from 0.5 to 1.5 (cgs units) except for the two hotter stars DY Cen (19000 K)
and MV Sgr (14000 K). The MV ranges from -5 to -2.5 and $\log L$ from 4.0 to
3.2 L⊙. Presently, there are 52 known RCBs in the Galaxy (Tisserand et al.
(2008); Clayton et al. (2009)), 21 known members in the Large Magellanic Cloud
and 6 in the Small Magellanic Cloud. The total number of RCBs in the Galaxy
may be about 3200 (Alcock et al. (2001)) There are only 5 known HdC stars in
the Galaxy. Their detection may be hampered because they do not show RCB-like
light variations, an infrared excess or a stellar wind (Geballe, Rao & Clayton
(2009)). They overlap the RCBs in Teff with a spread from about 6500 to 5000
K. At the high $T_{\rm eff}$ end of the RCBs sit the Extreme helium stars
(EHes) of which 21 are known in the Galaxy with $T_{\rm eff}$ from 32000 to
9000 K and $\log g$ from 0.75 to 4.0. Their $\log L$ ranges from 4.4 to 3.0
L⊙. Here, we give no more than passing attention to the EHes and other, even
hotter H-deficient stars that may form an evolutionary sequence with the HdCs
and RCBs.
A fascinating aspect of RCBs is that they present several faces to observers,
like the Hindu mythological god Karthikeya (son of Kritthikas -Pleiades) with
his six faces. The faces presented to our view are (i) a chemically peculiar
supergiant, (ii) a variable star obscured without warning from our view by
clouds of soot, (iii) a small amplitude pulsation both in light and radial
velocity with a period around 40 days, (iv) an infrared source producing dust
and blowing it away after several days, weeks, months or even years as clouds,
(v) a hot stellar wind, and (vi) and may even be a central star of a low
density nebula as evidenced by the presence of nebular lines of [O II], [N II]
(Fig. 1) and [S II]. They also have putative relatives of higher and lower
temperatures, as intriguing as Karthikeya’s relatives.
Our emphasis is on insights obtainable from high-resolution spectra with the
assumption that relevant atomic and molecular data of adequate precision and
completeness are to be provided from laboratory and theoretical spectroscopic
studies. This assumption is patently false, as is no surprise to practicing
astronomical spectroscopists. On the chance that a reader of this paper may
wish to take up a challenge, we mention two aspects of the incompleteness of
knowledge of the spectrum of neutral carbon that compromise some analyses of
the spectra of RCBs.
Spectra of RCBs are crossed by many lines of C i, as noted by Keenan &
Greenstein (1963). One might refer to the spectrum of a RCB as arising from a
column of somewhat contaminated high-temperature carbon vapour. A great step
forward in identifying the C i lines was made when Johansson (1965) extended
the laboratory spectrum and knowledge of the C i term system. With modern
spectra of RCBs, many suspected C i lines remain unidentified and cry out for
a new laboratory investigation of the C i spectrum. Today, astronomical
spectroscopists often require more than a set of wavelengths and a table of
energy levels. This is certainly true for the pursuit of an abundance analysis
leading to the chemical composition of a RCB. Abundance analysis at a minimum
requires knowledge of the $gf$-values for identified lines including the C i
lines. Asplund et al. (2000) report on an extensive abundance analysis of RCBs
and their uncovering of a ‘carbon problem’ which may arise because the adopted
theoretical $gf$-values or the photoionization cross-sections for neutral
carbon were too large by a factor of about four. (Asplund et al. also
suggested possible ways to reduce the carbon problem without appealing to
deficiencies in atomic physics.) Where are the physicists who will attack the
spectrum of the carbon atom with the precision and thoroughness that we seek?
Figure 1: Low resolution spectra obtained on 2008 May 6 and 2008 June 19 with
the Himalayan Chandra Telescope (HCT) during the current minimum of R CrB
which began in 2007. Emission lines are identified. A variety of emission
lines both broad and sharp are present during deep minimum. Several forbidden
lines of [O II], [N II] representing low density nebular gas are also present
(see text).
## 2 Major Puzzles
As stated in the Introduction, the two principal puzzles presented by the RCBs
concern their origins and their ability to produce soot clouds. The HdCs share
the puzzle about their origins but not that of soot production. Of course, the
‘dust’ puzzle might incorporate the HdCs with the RCBs by asking ‘Why is it
that RCBs but not the HdCs, which are on average cooler than the RCBs, are
prone to soot production?’
### 2.1 Origins
The basic problem is to understand how a H-deficient star evolves from a
H-rich single or binary star; H-poor He-rich gas clouds capable of sustaining
stellar nurseries have not been found. Complete resolution of the problem
would show how the classes of H-deficient stars – HdCs - RCB - EHe - beyond –
are or are not related.
Presently, there are two principal hypotheses for the origins:(i) the double-
degenerate (DD) scenario and (ii) a late thermal pulse or final flash (the FF
scenario) in a luminous star on the white-dwarf cooling track.
In the DD scenario, a He white dwarf is accreted by a C-O white dwarf as loss
of orbital energy drives the white dwarfs to merge. There are rather specific
ranges of initial masses and separations for normal (H-rich) stars in a binary
to merge and produce a H-deficient He-rich star. (Other combinations produce
other odd stars such as a single white dwarf with a mass exceeding
Chandrasekhar’s limit that must go bang in the night or day.) The post-merger
product has the greatly extended envelope required for a supergiant RCB and
HdC star. Subsequent evolution follows the canonical post-AGB contraction to
the white dwarf track (Webbink (1984); Iben & Tutukov (1984); Saio & Jeffery
(2002)).
In the FF scenario, H-rich post-AGB single stars that reach the white dwarf
cooling track with the potential to experience a final thermal pulse in their
He-shell. Under particular conditions, the thin surviving H-envelope may
experience H-burning. Onset of the pulse leads to a return of the star to a
supergiant-like form but, if conditions are ripe, this supergiant will be
H-poor and He-rich. Subsequent evolution takes the supergiant across the HR-
diagram to the white dwarf cooling track. Schonberner (1979), Iben et al.
(1983), Renzini et al. (1990), and Herwig (2001) among others discuss aspects
of the FF scenario. Enigmatic stars such as FG Sge and Sakurai’s object (V4334
Sgr) are commonly identified as FF scenario products.
Quantitative comparison of predictions of the FF and DD scenario and the
observations of the RCB and HdC stars involves several dimensions. Our focus
here is on the chemical composition obtainable only from examination of the
spectrum of RCB and HdC stars. The (preferably) high-resolution spectra from
the ultraviolet, optical and near-infrared are analyzed with model atmospheres
computed for the appropriate composition (Asplund et al. (2000); Pandey et al.
(2006); García-Hernándezet al. (2009)). Recent elemental abundance studies of
C, N, O in particular seem to favour the DD scenario for the RCBs and HdCs
(Asplund et al. 2000; Saio & Jeffery 2002; Pandey et al. 2006).
Two remarkable pieces of observational evidence seems to tilt the balance in
favour of the DD scenario. First, Clayton et al. (2005, 2007) made the
dramatic discovery that 18O is more abundant than 16O from moderate resolution
spectra of HdC and RCB stars showing the first-overtone CO vibration-rotation
bands at about 2.3 microns. Second, Pandey (2006) for EHes and Pandey, Lambert
& Rao (2008) for RCBs showed that 19F was drastically overabundant. The 18O
and F results show that the DD scenario was not a quiescent mixing process but
either involved nucleosynthesis during the merger and/or subsequently during
the evolution of the merged product to and from its supergiant phase.
It is amusing to see how this discovery of 18O was made in HD137613, as told
us by Tom Geballe. He got an email from Geoffrey Clayton at 5:30 p.m on 2004
November 10: ‘Hi Tom. I have this spectrum of the first overtone CO bands in
the HdC star, HD 137613. I took a look at it and the bands were split, so I
said to myself, oh well there’s 13CO, but the extra components aren’t at the
right wavelengths. So I’m a bit mystified. Take a look at the attached plot
and see if you have an idea of what this is’. To which Tom replied: ‘ That is
amazing!! I happen to have a book of CO wavelengths that I calculated when I
was a grad student in the 70s. All dv=1 and dv=2 lines of all isotopic
species. These bandheads match 12C18O!! The first three bandheads (2-0, 3-1,
4-2) of 12C18O are at 2.349, 2.378, and 2.408. I also looked 12C17O and it
doesn’t match. How could a star have as much 18O as 16O? What kind of star is
this? Cheers and congrats, Tom (Geballe 2008).
High resolution spectroscopy is essential for properly estimating the isotope
ratios 16O to 18O, 16O to 17O (as well as detection of the weak lines of 19F).
García-Hernández et al. (2009) used a spectral resolution of 50000 in their
analysis of the 2.3 micron CO bands. They confirmed Clayton et al.’s result of
a high 18O overabundance for HdCs and the RCB S Aps obtaining a ratio of 16O
to 18O of 0.5 in HdCs and the ratio 16 for S Aps (Fig.2). The high resolution
optical spectra obtained with the Harlan J. Smith telescope at the McDonald
Observatory and with the Vainu Bappu Telescope at Kavalur led to the discovery
that 19F is enhanced by a factor of several hundreds to a 1000 times solar in
these stars. In passing we note that Harlan J. Smith and M.K. Vainu Bappu were
great friends.
Figure 2: High resolution spectra of two HdC stars and four RCBs in the region
of 2-0 band of 12C18O at 2.3485 microns. The top plots show synthetic spectra
computed for the HdC star HD137613 for two CO isotopologues and CN. This
figure is from García-Hernández et al. (2009).
There can be no doubt that 18O and 19F were synthesized in or subsequent to
the creation of the RCBs and HdCs. Before full acceptance of the DD scenario
is made, the scenario has to account for the nucleosynthesis of these and
other observable elemental and isotopic abundances. In principle, as first
noted by Warner (1967), 18O may be synthesized by $\alpha$-capture from
abundant 14N. This may occur during the merger in the DD scenario but not with
ease in the FF scenario. The reservoir of 14N is finite and probably not
easily restored, thus, one wonders what cuts off the high temperatures at the
point that there remains an observable amount of nitrogen. Other complexities
remain. In particular, some RCBs and one HdC show an appreciable amount of
lithium, presumably 7Li. Discussions of the nucleosynthesis in and following
the merger are provided by Clayton et al. (2007), Pandey, Lambert & Rao
(2008), and García-Hernández et al. (2009). Additional high-resolution
spectroscopy, especially in the infrared, is desirable to determine the O
isotopic abundances across the RCB sample.
The FF scenario does plausibly account for other stars. Most notable among the
FF candidates are FG Sge, V4334 Sgr and V605 Aql (Clayton et al. 2005). It is
not impossible that RCBs from the FF scenario lurk among the known Galactic,
LMC and SMC examples.
### 2.2 Dust Production by RCBs
Historically, RCB stars were the first stars in which circumstellar dust
production was invoked to explain brightness variations, i.e., the dramatic
declines of several magnitudes (Loreta (1934); O’Keefe (1939)). It is now well
established that soot (carbon) clouds form close to the star and obstruct the
star light reaching us. Such clouds have now been imaged in the infrared
around RY Sgr at distances of about 1000 stellar radii (de Laverny & Mekarnia
(2004)). These clouds were formed presumably closer to the star; observational
constraints prohibited their detection close to the star.
Questions abound concerning the dust. What provides the trigger for dust
production in or around the star? How may dust condense in or around such hot
stars as RCBs? The trigger appears to be related to the atmospheric pulsations
which are revealed by light and radial velocity variations. Pugach (1977)
noted a correlation between the beginning of a light decline and pulsation
phase in RY Sgr. It has now been shown by Crause, Lawson & Henden (2007) for
four stars, by longterm monitoring of the brightness, that when a decline
happens it does so at a particular phase although not every pulsation results
in a decline. Woitke et al. (1996) developed a model for dust formation
following a pulsation-induced shock in the upper atmosphere. When the
pulsation amplitudes are large, the temperatures and densities in the post-
shock gas are predicted to be conducive for nucleation to occur. The question
intriguing spectroscopists is – how may we observe a strong shock develop from
a pulsation? What evidence may we offer that dust condensation actually takes
place in the atmosphere of the star? Can we detect the presence of cool gas
from which dust could condense? High-resolution spectroscopic observations
prior to the onset of the minimum are crucial to studying such questions.
Recently, R CrB has entered a prolonged light decline that commenced on July
2007 and has not yet ended (July 2009). This is one of the deepest minima the
star has had in several decades, reaching more than 9 magnitudes. At minimum
the star shows a variety of emission lines including broad (fwhm of 250 km
s-1) emission lines of He i, H & K lines of Ca ii, Na i D lines (Fig.1) and
sharp (fwhm $\leq$ 15 km s-1) emission lines due to Fe ii, Ti ii etc. (Rao et
al. (1999)). We postpone the discussion of this intriguing emission spectrum
for a later occasion. We were fortunate to catch the star just on the verge of
the decline. Our optical spectra obtained at the McDonald Observatory and at
Vainu Bappu Observatory have a spectral resolution over 60000. Fig. 3 shows
the light curve with the dates of our spectroscopic observations marked. The
spectral changes that R CrB underwent even prior to light minimum may provide
important clues to the phenomenon of dust production.
Figure 3: The visual (red dots) light curve of R CrB from 2007 March to 2008
October. Blue dots refer to V magnitudes, blue circles are either no-filter or
unknown filter measurements, red dots are daily averages of the visual
estimates. All the data are obtained from AAVSO data base. The minimum started
around 2007 July 10. Dates on which high-resolution spectra were obtained are
indicated by the dashed lines at the top. The dates on which low-resolution
spectra from the HCT are shown as short lines.
## 3 Prelude to Minimum: The Spectrum at Maximum Light
The present series of spectra that precede the light minimum starts on 2007
March 4. Spectra at maximum light are dominated by lines of C i, O i, N i, Si
i and S i in addition to the many lines of neutral and ionized metals seen in
spectra of normal F-G supergiants.
The star remained at maximum until 2007 July 10 when the decline in light
started. In the following account, we discuss briefly the spectral changes
that took place during maximum light. Even four months prior to the onset of
the minimum, the star appears to be disturbed with emission in some line cores
suggesting a component with an inverse P-cygni profile. Line doubling of the
absorption lines is seen two months prior to the onset. The blue and red
components of the doubled lines showed a different level of excitation for the
rising gas ($T_{\rm exc}$ of 4300 K) from the infalling gas ($T_{\rm exc}$ of
5600 K).
Some lines, like Ti ii 5154 $\AA$, are not doubled but show emission in the
line core. Even the cores of the Na i D lines show emission compared to the
reference spectrum at ‘normal’ maximum light. Moreover, the radial velocity of
the emissions is same as that of the star.
The absorption lines also broadened without altering the equivalent widths, a
month and half before the onset of minimum as though the starlight was passing
through a scattering medium (see Fig. 4). Study of such line broadening
requires high spectral resolution.
Some aspects of the spectroscopic changes months ahead of the decline
beginning in 2007 July are not unusual. For example, line doubling appears to
be a regular feature of stars such as R CrB and RY Sgr where the atmospheric
pulsation is seen clearly. It will be of interest to see if these changes
occur at the same or similar phases in the pulsation. Perhaps, there is a
quasi-steady amplification of the changes at the particular phase lasting a
sequence of several pulsations and culminating in the physical conditions
amounting to the trigger for a decline. Examination of this idea will call for
routine observations of a RCB such as R CrB, RY Sgr or V854 Cen, a star prone
to frequent declines. Considerable observing time may be wasted in that the
declines are unpredictable and irritation of Telescope Allocation Committees
seems assured. But these datasets are most likely to reveal new phenomena that
may tighten the link between pulsation and the trigger for dust production.
For example, the spectroscopic variations – line asymmetry, doubling, radial
velocity and strength – may be more extreme at particular phases.
More significantly, major disturbances were present in the spectrum of R CrB
three days prior to the descent. Comparison of pre-maximum and 2007 July 7
spectra shows that most lines in 2007 July 7 spectrum have shifted-absorption
components (Fig. 5) in addition to the usual photospheric lines. Fig. 5 shows
a comparison of the spectrum on 2007 July 7 with respect to an undisturbed
spectrum obtained on 1995 August 17 illustrating the additional absorption
components to several lines that appear in the 2007 July 7 spectrum. However,
the C i lines do not show such additional absorption components. Note that,
while the neutral metal lines (e.g., Fe i, and Cr i) show strong blue-shifted
components, the ionized metal lines (e.g., Fe ii, and Cr ii) show red-shifted
components on 2007 July 7. Such differences suggests that rising and falling
gases have different levels of ionization. As further shown in Fig. 6 during
the descent to minimum the blue (ionized) and redshifted (neutral) components
are separated by 53 km s-1 probably as a result of passage of a strong shock.
Of seemingly particular relevance to dust production, the 2007 July 7 spectrum
also displays lines of the C2 Phillips system showing the presence of gas at
temperatures (1000 to 800 K, Figs. 7 and 8) suggestive of a site for dust
nucleation. One supposes these unusual line profiles and the appearance of
cool gas are manifestations of the shock induced by the pulsation that
triggered the decline.
Figure 4: Figure illustrates the broadening of stellar spectrum prior to the
onset of the light minimum. (a) The spectrum on 2007 May 4 (blue) is similar
to a typical maximum one (green - 1995 August 17). (b) shows the spectral
lines on 2007 May 23 (red) are broader than in the spectrum obtained 19 days
earlier (blue). In (c) the 2007 May 4 spectrum is artificially brodened
without changing the equivalent widths to match the 2007 May 23 spectrum. Only
the line shapes got altered.
### 3.1 2007 July 7-10 –At the edge of the descent
Figure 5: Comparison of the spectral lines of C i, Fe i, Fe ii, Cr i, and Cr
ii in the disturbed spectrum on 2007 July 7(red), prior to the descent in
light, with a normal undisturbed spectrum on 1995 August 17 (blue). Note
various additional absorption components to both ionized and neutral lines in
2007 July 7. Figure 6: Typical velocities of the extra absorption components
to neutral (Fe i-blue-shifted) and ionized (Fe ii\- red-shifted) lines on 2007
July 7. The arrow denotes the radial velocity of the star measured from high
excitation lines without components on 2007 July 7. Note the infalling and
expanding layers have a velocity difference over 50 km s-1 that represents the
atmospheric shock velocity.
#### Absorption lines of C2 Phillips system
The 12C2 Swan bands are fairly strong in the spectrum of R CrB. Bandheads of
the 0-0 band at 5165 Å, the 1-0 band at 4737 Å, and the 0-1 band at 6174 Å are
readily identified. More interestingly, the spectrum of 2007 July 7 shows weak
bands of the Phillips system of C2 in absorption – see Fig. 7 for examples of
lines from the 2-0 band. These C2 lines not present in the regular stellar
spectrum are shifted in velocity from the stellar lines and present with a red
and a blue component. Our spectra show that the Phillips lines were not
present on 2007 June 6 spectrum, a month before the onset of the decline.
Figure 7: The spectrum of R CrB on 2007 July 7 (red) in the region 8780 $\AA$
superposed on the reference spectrum obtained on 1995 August 17 (blue).
Absorption lines of the 12C2 Phillips 2-0 band are seen as extra absorptions
in the 2007 July 7 spectrum but are absent from the 1995 August 17 spectrum.
Line doubling in the C2 lines can also be seen. Figure 8: Boltzmann plots of
the red and blue absorption components of C2 2-0 lines of the Phillips system
from the spectrum of R CrB obtained on 2007 July 7. Blue filled circle, open
circle and red filled circle symbols refer to Q, R and P branch lines,
respectively.
Rotational temperatures of 1048$\pm$30 K and 784$\pm$30 K were obtained (Fig.
14) for the blue component at $-$14.5 km s-1 and the red component at $+$5 km
s-1, respectively, using the molecular data given by Bakker et al. (1997). The
corresponding molecular column densities are 7.7$\times$1014 cm-2 and
4.5$\times$1014 cm-2. The radial velocities of C2 molecules of $-$14.5 km s-1
and $+$5 km s-1 suggest cool gas rising relative to the star with stellar
lines giving the radial velocity of 21 km s-1. The cooler gas is less rapidly
rising than the warmer molecular gas.
These rotational temperatures show that there was gas at the condensation
temperatures conducive to formation of carbon soot. Similar detections of cool
C2 molecules have been reported by us for V854 Cen (Rao & Lambert (2000)), R
CrB (Rao, Lambert & Shetrone (2006)) , and V CrA (Rao & Lambert (2008)). But
these detections were during minimum light. In contrast, the present detection
of cool C2 molecules at maximum light but on the verge of a minimum is
significant and leads to the conclusion that the light minima are indeed
caused by formation of dust grains. Woitke et al’s (1996) model seems to be
consistent with observations! Intense spectroscopic coverage of the days
leading up to a descent to a minimum is an important unfulfilled challenge
made more by difficult by the unpredictable onset of the descent.
## 4 The stellar wind
A mark that distinguishes RCBs from HdCs is the absence from the latter’s
defining characteristics, the declines – deep or even shallow – that are such
a strong feature of the RCBs. Absence of declines accounts for the lack of an
infrared excess from the HdCs. Another distinguishing mark between RCBs and
HdCs is the presence of a stellar wind from RCBs but not from HdCs. Are these
two marks different sides of one difference between the two groups of
H-deficient stars? Clayton, Geballe & Bianchi (2003) reported the presence of
a P-Cygni profile for the He i 10830 Å line at maximum light for several RCBs.
These observations show that the RCBs possess a stellar wind with an outflow
velocity of 200 to 300 km s-1. Then, Rao, Lambert & Shetrone (2006) showed
that in R CrB the strong photospheric lines, particularly the O i 7771 Å line,
had a pronounced blue wing suggesting a component with an expansion velocity
of 120 km s-1. Examination of profiles of other lines showed a component
associated with the stellar wind with the velocity proportional to the
excitation energy of the lines, i.e., the range of lines sample the region in
the upper atmosphere where the wind is accelerating to the (possibly terminal)
velocity measured from the He i 10830 Å line. Our high-resolution spectra now
show that a stellar wind may be common, perhaps ubiquitous, among RCBs.
The temporal dependence of the RCB stellar wind is not yet fully known. There
is no indication that it depends greatly or at all on the phase of the
pulsation cycle. Long-term behaviour is also not known. Observations at the
1995–1996 minimum of R CrB suggest that the wind-affected wings of the O i
lines are undisturbed even though the core is affected by transient emission
in the early decline (Rao et al. 1999). What drives this wind? Is there any
connection with magnetic fields? In the R CrB wind, low excitation lines
(e.g., the Al i resonance lines) show variability on a short time scale but
there is no change in the high excitation lines. Examination of magnetically-
sensitive and magnetically-insensitive Fe i lines of comparable strength and
excitation potential from a series of high-resolution spectra of R CrB taken
at maximum light suggests the variability in the former, but not in the latter
(Rao 2008). This result suggests a connection between surface magnetic field
and the wind. The relation between the (apparently) permanent and (seemingly)
pulsation phase-independent wind and trigger for a decline is as yet unknown.
In contrast, Geballe, Rao & Clayton (2009) examined the five known HdCs at the
He i 10830 Å line on high-resolution spectra and found no evidence of a
stellar wind. The absence of a wind and the lack of deep declines among the
HdCs and the presence of a wind among the RCBs with their deep declines is
suggestive that the wind or the presence of an extended atmosphere as a result
of the wind is a necessary condition for occurrence of deep declines.
Figure 9: Profiles of O i 7771 Å (red) and Al i 3944 Å (blue) in R CrB
obtained at different times and adjusted to the stellar velocity. Note the
variability in the low excitation Al i profiles and the unchanged high
excitation O i line profile. The wind velocity suggested by these lines 50 km
s-1 for Al and 120 km s-1 for O. O i line profile of a normal supergiant
$\gamma$ Cyg (black) is shown for comaprison.
## 5 Concluding remarks
Answers are slowly emerging for the important questions posed by RCB and HdC
stars concerning the origin of these H-deficient stars, and the mechanisms by
which a cloud of soot forms and obscures the star. Evidence suggests that the
DD rather than the FF scenario provides a superior accounting for the
elemental abundances of C, N and O for RCB stars and their likely relatives,
the extreme Helium (EHe) stars and cool hydrogen deficient (HdC) stars. The FF
scenario does plausibly account for other stars. Most notable among the FF
candidates are FG Sge and V4334 Sgr, also known as Sakurai’s object and V605
Aql (Clayton et al. (2006)). It is not impossible that RCBs from the FF
scenario lurk among the analysed sample attributed in the main to the DD
scenario.
The process of soot formation is most likely associated with the atmospheric
pulsation which, on occasions at a certain phase of the pulsation, leads to a
stronger than usual shock in the atmosphere such that the physical conditions
are conducive to molecule formation and dust nucleation, as envisaged by
Woitke et al.(1996). How the dust grains grow and soot clouds form represent
questions for future study.
As our essay has hopefully shown, high-resolution optical and infrared spectra
have and will prove crucial to addressing the leading questions about the
H-deficient RCB and HdC stars. In preparing the paper, we came across a
collection of articles on modern high-resolution spectroscopic techniques.
There, the leading chapter by Sir Harry Kroto carried the title ‘Old
spectroscopists forget a lot but they do remember their lines’ (Kroto (2009)).
This pair of old spectroscopists remember at least the important lines.
###### Acknowledgements.
We acknowledge with thanks the variable star observations from the AAVSO
database. This research has made use of the SIMBAD database, operated at CDS,
Strasbourg, France. Our sincere thanks to the editors of this volume for
accepting with admirable patience our delay in submitting this article. This
research has been supported in part by a grant (F-634) to DLL from the Robert
A. Welch Foundation of Houston, Texas.
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|
arxiv-papers
| 2010-02-18T06:33:14 |
2024-09-04T02:49:08.451981
|
{
"license": "Public Domain",
"authors": "N. Kameswara Rao (1), David L. Lambert (2) ((1) Indian Institute of\n Astrophysics, Bangalore, India, (2) The W.J. McDonald Observatory, The\n University of Texas, Austin, USA)",
"submitter": "Kameswara Rao",
"url": "https://arxiv.org/abs/1002.3433"
}
|
1002.3446
|
# Thread-Scalable Evaluation of Multi-Jet Observables
Walter T. Giele111email: giele@fnal.gov, Gerben C. Stavenga222email:
stavenga@fnal.gov and Jan Winter333email: jwinter@fnal.gov
Fermi National Accelerator Laboratory, Batavia, IL 60510, USA
###### Abstract:
A leading-order, leading-color parton-level event generator is developed for
use on a multi-threaded GPU. Speed-up factors between 150 and 300 are obtained
compared to an unoptimized CPU-based implementation of the event generator. In
this first paper we study the feasibility of a GPU-based event generator with
an emphasis on the constraints imposed by the hardware. Some studies of Monte
Carlo convergence and accuracy are presented for $PP\rightarrow 2,\ldots,10$
jet observables using of the order of $10^{11}$ events.
QCD, LO Computations, Jets, Hadronic Colliders
††preprint: FERMILAB-PUB-10-025-T
## 1 Introduction
Leading order (LO) parton-level Monte Carlos (MCs) play a prominent role in
collider phenomenology [1, 2, 3, 4, 5, 6]. As one needs to average the
calculation of the observable over many events, the evaluation time for the
event generation is a crucial issue in the development of LO parton level MCs.
Furthermore, to make full use of the recent progress in the calculation of
virtual corrections [7, 8, 9], fast tree-level event generators are needed for
the calculation of the radiative contributions in a next-to-leading order MC.
One can use large-scale grids for the generation of the tree-level events.
Such grids are expensive and need a large infrastructure. A more preferable
solution would be to run the MC on a single, affordable workstation. As we
will show this is possible using a massively parallel GPU. The NVIDIA Tesla
chip is designed for numerical applications [10] and the CUDA C compiler [11]
provides a familiar development environment. We will use the Tesla GPU
throughout the paper.444We thank the LQCD collaboration for giving us access
to the Tesla GPU processors.
In this paper we will execute all steps needed for event generation on the
GPU. These steps include the implementation of the unit-weight phase-space
generator Rambo [12], the evaluation of the strong coupling and parton density
function using LHAPDF [13], the evaluation of the leading-color $gg\rightarrow
2,\ldots,10$ gluon matrix elements at LO and the calculation of some
observables. The CPU is tasked with calculating the distributions using the
event weight and observables provided by the GPU. By utilizing memory with a
fast access time only, considerable speed-ups are obtained in the event
generation time. This memory is limited in size, requiring some coding effort.
As the GPU chips are developing fast, we can enhance the capabilities of our
parton-level generator in accordance.
In Refs. [14, 15] methods have been developed to evaluate multi-jet cross
sections on GPUs within the framework of the Helas matrix-element evaluator
[16], which forms the basis of the Madgraph event generator [1]. The method is
based on individual Feynman diagram evaluations. As such the scaling with the
number of external particles of the scattering process is faster than
factorial. Such an algorithm will have limited scalability properties, which
cannot be compensated by deploying a large number of threads. Instead, an
algorithm of polynomial complexity will have excellent scaling properties; its
only limitation is the available fast-access memory size. Polynomial
algorithms for the evaluation of ordered LO multi-parton matrix elements have
been formulated in the form of Berends–Giele (BG) recursion relations [17].
For a leading-color generator, any Standard Model matrix element can be
evaluated with an algorithm of polynomial complexity of degree 4 [18] or, by
using more memory storage, of degree 3 [19]. For any fixed color expansion,
the complexity remains polynomial. Therefore, we will use ordered recursive
evaluations of the matrix elements instead of Feynman diagram evaluations.
In this paper we present a GPU-based implementation of all basic tools needed
for a LO generator. In Section 2 we discuss the GPU and its hardware
limitations. According to these limitations, we will determine the optimal
running configuration as a function of the number of gluons. The algorithmic
implementation of the recursion algorithm and other tools such as phase-space
generation, experimental cuts and parton density functions are discussed in
Section 3. Finally, in Section 4 we put all pieces together and construct the
leading-color LO parton-level generator capable of generating up to $PP\to 10$
jets with sufficient statistics for serious phenomenology. The conclusions and
outlook are given in Section 5.
## 2 Thread-Scalable Algorithms for Event Generators
Monte Carlo algorithms belong to a class of algorithms, which can be trivially
parallelized, by dividing the events over the threads. Optimized for graphics
processing, the GPU works by having many threads executing essentially the
same instructions over different data. For a given class of events, e.g.
$n$-gluon scattering, the only difference between the events is due to the
external sources, i.e. the momentum and polarization four-vectors of each
gluon defining the state of the external gluon. The recursive algorithm acts
on these input sources in an identical manner. That is, each thread can
execute the same processor instructions to calculate the matrix-element
weight.
However, because of hardware constraints such a straightforward approach is
limited by the amount of available fast access memory. The GPU memory is
independent from the CPU memory and divided into the off-chip global memory
and the on-chip memory. This distinction is important as the off-chip memory
is large (of the order of gigabytes) but slow to access by the threads.
Therefore, we want to limit the access to the global memory by using it only
for the transfer of results to the CPU memory. The on-chip memory is fast to
access, but limited in size (of the order of tens of kilobytes). The first on-
chip memory structures are the registers. Each thread has its own registers,
which cannot be accessed by other threads. These registers are used within the
algorithm for variable storage, function evaluations, etc. The other on-chip
memory structure is shared memory, which is accessible to all the threads on a
multi-processor (MP). The current GPUs are not yet optimize-able to one event
per thread due to these shared memory and register constraints. With the next
generation of GPUs the shared memory will increase significantly, and we will
reach the point at which we can evaluate one event per thread up to large
multiplicities of gluons.
From this discussion the limitations are clear as each event requires a
certain amount of the limited register and shared memory. For the optimal
solution, we put the maximum number of events on one MP, such that the
evaluation does not exceed the available on-chip memory. The resulting
multiple threads per event can be used to “unroll” do-loops etc., thereby help
speed up the evaluation. This optimal solution is dependent on the rapidly
evolving hardware structure of the GPU chips.
By lowering the number of events per MP below the optimal solution, the number
of available threads per event increases. However, this will not lead to an
effective speed-up of the overall event generation as the total number of
threads per GPU is fixed. Once the number of events to be used per MP has been
determined, the GPU evaluation becomes scalable. The MC generator now simply
scales with the number of available MPs on the GPU.
$n$ | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
---|---|---|---|---|---|---|---|---|---
events/MP | 102 | 68 | 48 | 36 | 28 | 22 | 18 | 15 | 13
threads/event | 10 | 15 | 21 | 28 | 36 | 45 | 55 | 66 | 78
Table 1: The number of $n$-gluon events, which can be simultaneously executed
on one MP (and is equal to $2048/[n\times(n+1)]$) and the number of available
threads per event (equal to $n\times(n+1)/2$). The total number of events
evaluated in parallel on the Tesla chip is 30$\times$(events/MP).
We will use the current NVIDIA Tesla chip for the numerical evaluations. This
chip consists of 30 MPs each capable of running up to 1024 threads. Each MP
has 16,384 32-bit registers and an internal shared memory of 16,384 bytes.
Each thread is assigned its own registers from the pool. Compiling the current
MC implementation indicates that 35 registers per thread are needed. This
gives us an upper maximum based solely on the use of registers of
$16,384/35=468$ threads per MP (each of which could potentially be used to
evaluate one event). The momenta and current storage is of more concern. As we
will see in the next section, for the evaluation of the $n$-gluon matrix
element, we need to store $n\times(n+1)/2$ four-vectors in single (float)
precision. This requires $8\times n\times(n+1)$ bytes of shared memory per
event on the MP.555A bit of calculus shows that when we need to store
$n\times(n+1)/2$ real-valued four-vectors in single precision we require
$4\times 4\times n\times(n+1)/2$ bytes of shared memory. The resulting maximum
number of events per MP as a function of the number of gluons is given in
Table 1. Note that up to 44-gluon scattering can be evaluated on the MP
(albeit with only one event per MP). Beyond 44 gluons the shared memory is too
small to store all the required four-vectors.
## 3 The Implementation of the Thread-Scalable Algorithm
Now that we have determined the optimal running configuration, i.e. the number
of events per MP, we can implement the algorithm. We will describe the
implementation of the Threaded EventS Simulator MC, which we name Tess MC, for
the NVIDIA Tesla chip.666The Tess MC code can be downloaded from the website:
http://vircol.fnal.gov/TESS.html. As we have many threads available per event,
we will use these threads to speed up the MC. In Figure 1 we show the thread
usage during different stages of the event generator for a $2\rightarrow n$
gluon process. The fraction of the evaluation time spent in each stage depends
on the gluon multiplicity. For 4 gluons, we get 20%, 20%, 50% and 0% for the
Rambo, PS-weight, ME-weight and the epilogue phases, respectively. For 12
gluons, the time consumption divides up into 2%, 18%, 75% and 4% for the four
phases.
Figure 1: The thread usage for an $n$-gluon event as the algorithm progresses
through the stages of the event generation: flat phase-space generation,
phase-space weight evaluation (including parton density functions and
$\alpha_{S}$), matrix-element evaluation and finalization phase.
The initialization phase (not shown in Figure 1) consists of starting up the
kernel on the GPU. This is taken care of by the CUDA runtime code and does
essentially not depend on the number of threads it has to spawn. However it is
a significant part when the total kernel time is small, as for the 4-gluon
case.
The kernel starts initiating the unit-weight phase-space generator Rambo. On
the CPU this algorithm grows linearly with $n$ as we have to construct the
$n-2$ outgoing momenta. On the GPU we can employ $n-2$ threads to
simultaneously generate the outgoing momenta, making the Rambo code in
practice independent of $n$.777The Rambo algorithm has some summation
operations, which grow linearly with $n$, but this time scaling is very small
compared to the overall evaluation time of the Rambo algorithm.
After the momenta are generated, we have to calculate the strong coupling
constant, the parton density functions and the observables. We also determine,
if the event passes the canonical cuts. If the event fails the cuts, it is
only flagged as such; the matrix-element weight will still be evaluated as
this has no effect on the overall evaluation time. This means one can deviate
from the chosen canonical cuts on the CPU during the histogramming phase if so
desired. Note that we could in principle generate more events, which pass the
cut before starting the calculation of the matrix-element weights. This should
increase the performance of the Monte Carlo, at the cost of additional
bookkeeping.
The evaluation of the strong coupling constant and parton density functions
requires special attention. As we have used all shared memory for the four-
vector storage of the gluon currents and momenta, we have to use the off-chip
global memory to store the parton density and strong coupling constant
information in the form of grids. Furthermore, interpolation is required
between the grid points. To facilitate this, we use a special type of memory,
the so-called texture memory. This off-chip memory was designed for graphics
applications and performs hardware interpolations of the grid. Specifically,
we set up a 1-dimensional grid for the strong coupling constant. The value of
the strong coupling constant is stored as a function of the renormalization
scale at integer values of the grid. For the 2-dimensional grid used by the
parton density functions, the two dimensions are given by the factorization
scale and the parton fractions. This parton grid is directly obtained from
LHAPDF [13]. After the grid initialization, the texture memory can be accessed
by the GPU and its hardware will perform the appropriate linear interpolation
between the grid points when accessing the grid using non-integer values. This
way we have a very fast evaluation of the strong coupling and parton density
functions taking only about $6$% and $0.6$% of the total GPU time for 4-gluon
and 12-gluon processes, respectively.
The four-momenta are generated and the phase-space weight is determined, hence
we have to evaluate the matrix-element weight next. This happens at the core
of the event generator where we use recursion relations to compute these
weights. For this proof-of-concept program, we decided to use the recursion
relation of Ref. [17] and restricted ourselves to the case of pure gluonic
cross sections; quarks can be easily added at a later stage without changing
the event generator in a fundamental way. The recursion relations we employ
are given by
$J_{\mu}[m,\ldots,n]\;=\;\frac{1}{K[m,\ldots,n]^{2}}\;\Bigg{(}\,\sum_{i=m}^{n-1}\Big{[}J[m,\ldots,i],J[i+1,\ldots,n]\Big{]}_{\mu}\\\
+\;\sum_{i=m}^{n-2}\sum_{j=i+1}^{n-1}\Big{\\{}J[m,\ldots,i],J[i+1,\ldots,j],J[j+1,\ldots,n]\Big{\\}}_{\mu}\Bigg{)}\
,$ (1)
where $J_{\mu}[m,\ldots,n]$ is a conserved four-vector current depending on
the external gluons $\\{m,\ldots,n\\}$. Furthermore, we have used the
shorthand notations
$\begin{split}K^{\mu}[m,\ldots,n]\;&=\;\sum_{i=m}^{n}k_{i}^{\mu}\
,\\\\[8.53581pt]
\Big{[}J[\\{a\\}],J[\\{b\\}]\Big{]}_{\mu}\,&=\;2\,\Big{(}J[\\{a\\}]\cdot
K[\\{b\\}]\Big{)}\,J_{\mu}[\\{b\\}]-2\,\Big{(}K[\\{a\\}]\cdot
J[\\{b\\}]\Big{)}\,J_{\mu}[\\{a\\}]\\\ &+\Big{(}J[\\{a\\}]\cdot
J[\\{b\\}]\Big{)}\,\Big{(}K_{\mu}[\\{a\\}]-K_{\mu}[\\{b\\}]\Big{)}\
,\\\\[8.53581pt]
\Big{\\{}J[\\{a\\}],J[\\{b\\}],J[\\{c\\}]\Big{\\}}_{\mu}\,&=\;2\,\Big{(}J[\\{a\\}]\cdot
J[\\{c\\}]\Big{)}J_{\mu}[\\{b\\}]\\\ &-\Big{(}J[\\{a\\}]\cdot
J[\\{b\\}]\Big{)}\,J_{\mu}[\\{c\\}]-\Big{(}J[\\{c\\}]\cdot
J[\\{b\\}]\Big{)}\,J_{\mu}[\\{a\\}]\ ,\end{split}$ (2)
where the external gluon labeled $i$ has momentum $k_{i}^{\mu}$ and
polarization state $J^{\mu}[i]$. These four-vectors form the initial
conditions for the recursion relation. In addition to the $n$ momenta, the
recursion relation requires $n\times(n-1)/2$ four-vector currents to be stored
giving a total storage of $n\times(n+1)/2$ four-currents per event.
The recursion relations have a polynomial complexity of order $n^{4}$ for
calculating the currents [18]. By exploiting the available threads for each
event, we can reduce the algorithmic complexity of the BG recursion relation.
The relation is easily thread-able, which enables us to lower the polynomial
scaling of the evaluation time of the recursion relation to $n^{3}$. A full
recursion for an $n$-gluon process is completed in $n-1$ steps. In the first
step, we use $n-1$ threads to calculate the polarization vectors
$\\{J[2],\ldots,J[n]\\}$ needed as a starting point in the recursion relation.
We choose each polarization vector as a random unit vector orthogonal to the
respective gluon momentum. By doing this, instead of employing the
conventional helicity vectors, we obtain real-valued currents. This avoids
complex multiplications and reduces the shared-memory usage, resulting in a
significant time gain.
After the 1-currents have been determined, we use $n-2$ threads to calculate
the 2-currents $\\{J[2,3],J[3,4],\ldots,J[n-1,n]\\}$. We continue with the
$n-1$ steps until we have determined $J[2,3,\ldots,n]$ at which point we can
calculate the ordered amplitude and, hence, the matrix-element weight. Note
that because we make use of the multiple threads we have reduced the
computational effort from ${\cal O}(n^{4})$ to ${\cal O}(n^{3})$ complexity.
In principle we may be able to improve even further. The initial ${\cal
O}(n^{4})$ growth of the one-threaded recursion relation to calculate the
$J[2,3,\ldots,n]$ current can be reduced by rewriting the recursion relation
as
$J_{\mu}[m,\ldots,n]\;=\;\sum_{i=m}^{n-1}\left[\Big{(}W[i+1,\ldots,n]\cdot
J[m,\ldots,i]\Big{)}_{\mu}-\Big{(}W[m,\ldots,i]\cdot
J[i+1,\ldots,n]\Big{)}_{\mu}\right]\ ,$ (3)
where the tensor $W_{\mu\nu}$ is defined as
$\begin{split}W_{\mu\nu}[m,\ldots,n]\;&=\;2\,J_{\mu}[m,\ldots,n]\,K_{\nu}[m,\ldots,n]-K_{\mu}[m,\ldots,n]\,J_{\nu}[m,\ldots,n]\\\
&+\sum_{i=m}^{n-1}\Big{(}J_{\mu}[m,\ldots,i]\,J_{\nu}[i+1,\ldots,n]-J_{\mu}[i+1,\ldots,n]\,J_{\nu}[m,\ldots,i]\Big{)}\
.\end{split}$ (4)
By undoing the nested summations in the second term of Eq. 1 we have lowered
the complexity of the algorithm to ${\cal O}(n^{3})$. However, this is only
achieved at the cost of using significantly more storage. For each event, one
would have to store $n\times(n-1)/2$ $4\times 4$-tensors in addition to the
$n\times(n+1)/2$ momenta and current four-vectors. Up to $n\approx 10$ the
extra work of doing matrix multiplications together with the fact that the
relative pre-factor of the $n^{4}$-algorithm is small, $1/4$, compared to the
$n^{3}$-algorithm actually make the $n^{3}$-algorithm slower than the
$n^{4}$-algorithm. Moreover, the extra storage demand does not make the
$n^{3}$-algorithm attractive for our GPU implementation.
From the current for $n-1$ gluons we then obtain the amplitude for the
$n$-gluon matrix element by putting the off-shell leg on-shell, contracting in
with the final polarization vector and symmetrizing over the gluons in the
current. Specifically,
$\mathcal{A}(1,\ldots,n)\;=\;\sum_{\pi}^{(n-1)!}\mbox{Tr}\Big{[}T^{a_{\pi_{1}}}\ldots
T^{a_{\pi_{n-1}}}T^{a_{n}}\Big{]}\,m(\pi_{1},\ldots,\pi_{n-1},n)\ ,$ (5)
with
$m(\pi_{1},\ldots,\pi_{n-1},n)\;=\;\left.\Big{(}J[\pi_{1},\ldots,\pi_{n-1}]\cdot
J[n]\Big{)}\times K^{2}[1,\ldots,n-1]\right\rfloor_{K[1,\ldots,n]=0}\ .$ (6)
Notice that for a given phase-space point, we have to perform the permutation
sum requiring $(n-1)!$ steps to arrive at the full amplitude. This would
immediately lead to a factorial growth in the computer time. We can circumvent
the super-exponential sum over permutations in Eq. 5. In the leading-color
approximation this is easily accomplished and the color-summed, squared
amplitude is given by
$\left|\mathcal{A}(1,\ldots,n)\right|^{2}\;\sim\;N_{\mathrm{C}}^{n-2}\left(N_{\mathrm{C}}^{2}-1\right)\left(\sum_{\pi}^{(n-1)!}\left|m(\pi_{1},\ldots,\pi_{n-1},n)\right|^{2}+{\cal
O}\left(\frac{1}{N_{\mathrm{C}}^{2}}\right)\right)\ .$ (7)
As we will use this matrix element in a $2\rightarrow n-2$ gluon-scattering
phase-space integration, we can use the symmetry of the final state to remove
the permutation sum over the ordered amplitudes. In detail,
$\displaystyle d\,\sigma(PP\rightarrow n-2\ \mbox{jets})$ (8) $\displaystyle=$
$\displaystyle\int d\,x_{1}\ d\,x_{2}\
\frac{F_{g}(x_{1})F_{g}(x_{2})}{4\,p_{1}\cdot p_{2}}\frac{1}{(n-2)!}\int
d\,\Phi(p_{1}p_{2}\rightarrow p_{3}\cdots
p_{n})\sum_{\pi}^{(n-1)!}\left|m(\pi_{1},\ldots,\pi_{n-1},n)\right|^{2}$
$\displaystyle=$ $\displaystyle\int d\,x_{1}\ d\,x_{2}\
\frac{F_{g}(x_{1})F_{g}(x_{2})}{4\,p_{1}\cdot p_{2}}\,(n-1)\int
d\,\Phi(p_{1}p_{2}\rightarrow p_{3}\cdots
p_{n})\left|m(1,\sigma_{2},\ldots,\sigma_{n})\right|^{2}\ ,$
where $p_{1}=x_{1}\,P_{1}$, $p_{2}=x_{2}\,P_{2}$, the parton density function
is given by $F_{g}(x)$, $d\,\Phi$ is the phase-space integration measure and
$\\{\sigma_{2},\ldots,\sigma_{n}\\}$ is a permutation of the list
$\\{2,\ldots,n\\}$ assigned randomly for each MC phase-space point evaluation.
Eventually, in the very last step of our threaded event simulation all the
results are put together and returned to the CPU for processing.
By using the Tess MC, we can evaluate the differential $n$-jet cross sections
in the leading-color approximation. The algorithm is of polynomial complexity
and scales as $n^{3}$ with the number $n$ of gluons.
## 4 A Numerical Study of the Threaded Events Simulator
Figure 2: The horizontal axis is the number of events per MP in a sweep,
giving a total number of 30$\times$(events per MP) evaluated events per sweep.
The red curves used together with the vertical axes on the right indicate the
total GPU time in seconds for 1,000,000 sweeps. The blue curves depict the
evaluation time of one event in seconds as labeled by the vertical axes on the
left.
The first issue to study is the timing behavior of the Tess MC. We show our
results in Figure 2 where we plot the GPU timing as a function of events per
MP for several gluon multiplicities. Each MP will evaluate in a sweep a number
of events in parallel using the threads. In principle the sweep time should be
independent of the number of events evaluated by each MP as long as the
shared-memory constraints are not exceeded, cf. Table 1. However, we have to
execute a substantial amount of transcendental function calls per event, which
induces some queuing at the special function units each MP uses for evaluating
these functions. This queuing effect will increase as the number of events per
MP rises and, hence, lead to a slower execution of the sweep. In Figure 2, one
can see this complicated timing behavior, which is controlled by the GPU
hardware. As discussed the overall evaluation time increases with the number
of events per MP, see the red curves in the plots. In fact, the increase of
the overall evaluation time is overcome by the gain we achieve in evaluating
more events per MP. The more relevant quantity therefore is the evaluation
time per event, defined as the GPU evaluation time divided by the total number
of generated events. As clearly indicated by the blue curves in Figure 2, the
time consumption per event steadily decreases as the number of events per MP
increases. The best performance will be achieved by using the maximal number
of events available per MP.
$n$ | $T_{n}^{\mathrm{GPU}}$ (seconds) | $P_{n}(3)$ | $T_{n}^{\mathrm{CPU}}$ (seconds) | $P_{n}(4)$ | $G_{n}$
---|---|---|---|---|---
4 | $2.975\times 10^{-8}$ | | $8.753\times 10^{-6}$ | | 294
5 | $4.438\times 10^{-8}$ | 0.91 | $1.247\times 10^{-5}$ | 0.87 | 281
6 | $8.551\times 10^{-8}$ | 1.03 | $1.966\times 10^{-5}$ | 0.93 | 230
7 | $2.304\times 10^{-7}$ | 1.19 | $3.047\times 10^{-5}$ | 0.96 | 132
8 | $3.546\times 10^{-7}$ | 1.01 | $4.736\times 10^{-5}$ | 0.98 | 133
9 | $4.274\times 10^{-7}$ | 0.94 | $7.263\times 10^{-5}$ | 0.99 | 170
10 | $6.817\times 10^{-7}$ | 1.05 | $1.044\times 10^{-4}$ | 0.99 | 153
11 | $9.750\times 10^{-7}$ | 1.02 | $1.529\times 10^{-4}$ | 1.00 | 157
12 | $1.356\times 10^{-6}$ | 1.02 | $2.129\times 10^{-4}$ | 1.00 | 158
Table 2: The GPU and CPU evaluation times per event, $T_{n}^{\mathrm{GPU}}$
and $T_{n}^{\mathrm{CPU}}$, given as a function of the number $n$ of gluons
for $gg\to(n-2)\,g$ processes. The polynomial scaling measures are also shown,
for the GPU, $P_{n}(3)$, and for the CPU, $P_{n}(4)$. The $P_{n}(m)$ are
defined as $P_{n}(m)=[(n-1)/n]\times\sqrt[m]{T_{n}/T_{n-1}\;}$. The rightmost
column finally displays the gain
$G=T_{n}^{\mathrm{CPU}}/T_{n}^{\mathrm{GPU}}$.
Now that we have determined the optimal running conditions, we give in Table 2
the evaluation time per event on the GPU compared to the evaluation time of
the same algorithm when executed on the CPU. As can be seen the speed-up in
evaluation time is substantial, ranging from almost a factor of 300 for
4-gluon processes to a factor of around 150 for 12-gluon processes. Note that
the speed-up is completely due to the fact that we evaluate in parallel 3060
and 390 events for the 4- and 12-gluon case, respectively. We have also tested
that running the events sequentially on the GPU using only one event per sweep
results in an event evaluation time, which is slower than the CPU evaluation
time. In particular, we found factors of 10 and 2 for the 4- and 12-gluon
computations, respectively. Because of the substantial time gains, a single
GPU can replace a large grid of hundreds of CPUs.
Also of interest is the scaling behavior of the algorithm. As expected, on the
CPU it is simply polynomial scaling with a factor of 4 in the limit of a large
number of gluons. We see from the table that this scaling is setting in
quickly. The GPU algorithm scales with a factor of 3 as discussed in Section
3. However, as the number of gluons increases, the number of events per MP
decreases. This makes the timing more dependent on specific hardware issues.
As can be seen from Table 2 the polynomial scaling is trending towards a
factor of 3.
Figure 3: The number of sweeps versus several $gg\to(n-2)\,g$ cross sections
normalized to their respective best cross section estimates as given in Table
3. The error is the mean standard deviation.
Given the fast evaluation of events, we can easily generate ${\cal
O}(10^{11})$ events for the calculation of the LO cross sections. With these
large numbers of generated events, one has to carefully consider the
performance of the random number generators. In our case this should cause no
issues, since the number of generated random numbers is of the order of the
square root of the generator’s cycle length. However, as we average over
${\cal O}(10^{11})$ numbers, care has to be taken concerning the loss of
precision, which would result in a systematic underestimation of the cross
section. This is demonstrated in the first graph of Figure 3 where we have
used a single-precision summation to calculate the 4-gluon cross section. As
can be seen the effect becomes dramatic as the number of sweeps is rising and
we end up with a totally wrong determination of the cross section. We avoid
this problem by using the Kahan summation algorithm [20]. All other graphs of
the figure are produced by following this procedure. These additional graphs
display the convergence of the cross section estimate including its mean
standard deviation as a function of the number of GPU sweeps. The vertical
axis has been normalized to the respective best estimate of the cross section;
all of which are listed in Table 3. For this study, we have used Rambo as the
momenta generator, therefore, a severe under-sampling of small phase-space
regions with large weights may occur especially for larger gluon
multiplicities. Because the Rambo phase-space generation is flat and does not
reflect the scattering amplitudes’ strong dipole structure, such under-
sampling effects are expected and cause the peaking behavior of our cross
section estimates. Even with ${\cal O}(10^{10})$ phase-space points an
estimate of the 12-gluon cross section using the Rambo event generator is
quite unreliable and the mean standard deviation error estimate does not fully
reflect the true uncertainty. In a further development step, one may implement
a phase-space generator like Sarge [21], which is capable of adapting to the
QCD antenna structures as occurring in the matrix elements. As pointed out in
Ref. [6], this would resolve the phase-space integration issues we have seen
here.
$n$ | $\sigma_{n}$ (pb) | $N_{\mathrm{generated}}$ | $N_{\mathrm{accepted}}$ | $\sigma_{n}^{\textsc{Comix}}$ (pb)
---|---|---|---|---
4 | $(2.32421\pm 0.00047)\times 10^{8}$ | $3.06\times 10^{11}$ | $1.96848\times 10^{11}$ | $(2.3283\pm 0.0023)\times 10^{8}$
5 | $(1.4353\pm 0.0011)\times 10^{7}$ | $2.04\times 10^{11}$ | $1.12939\times 10^{11}$ | $(1.4355\pm 0.0014)\times 10^{7}$
6 | $(2.84780\pm 0.00096)\times 10^{6}$ | $1.44\times 10^{11}$ | $6.98918\times 10^{10}$ | $(2.8560\pm 0.0030)\times 10^{6}$
7 | $(6.356\pm 0.012)\times 10^{5}$ | $1.08\times 10^{11}$ | $4.49985\times 10^{10}$ | $(6.408\pm 0.015)\times 10^{5}$
8 | $(1.608\pm 0.011)\times 10^{5}$ | $8.40\times 10^{10}$ | $2.93316\times 10^{10}$ |
9 | $(4.38\pm 0.11)\times 10^{4}$ | $6.60\times 10^{10}$ | $1.88182\times 10^{10}$ |
10 | $(1.193\pm 0.024)\times 10^{4}$ | $5.40\times 10^{10}$ | $1.22356\times 10^{10}$ |
11 | $(3.550\pm 0.020)\times 10^{4}$ | $4.50\times 10^{10}$ | $7.88017\times 10^{9}$ |
12 | $(9.64361\pm 0.74)\times 10^{3}$ | $3.90\times 10^{10}$ | $5.13041\times 10^{9}$ |
Table 3: The cross sections $\sigma_{n}$ for $gg\to(n-2)\,g$ and their mean
standard deviations in pb as calculated by the Tess MC using $10^{9}$ sweeps.
The two center columns show the total number of generated events and the
number of events passing the jet cuts. For comparison, the cross sections
$\sigma_{n}^{\textsc{Comix}}$ in pb as computed by Comix considering the full-
color dependence are also given.
Figure 4: Left panels: the profile plots of the relative gauge invariance as
a function of the decimal logarithm of the matrix-element weight,
$\log_{10}W_{\mathrm{ME}}$; center panels: the normalized $H_{T}$
distributions and right panels: the normalized minimum $R$-separations between
pairs of jets. All of which is shown for $gg\to(n-2)\,g$ scatterings at a 14
TeV center-of-mass energy for $n=5,7,9,11$.
The convergence issues reflected in Figure 3 should be taken into account when
interpreting the uncertainties of our best cross section estimates, which are
listed in Table 3. For these cross section calculations of $gg\to(n-2)\,g$
scattering processes at a center-of-mass energy of 14 TeV, we have used the
CTEQ6L1 parton density function set [22] as implemented in LHAPDF [13] with a
fixed renormalization and factorization scale taken at $M_{Z}=91.188$ GeV. For
the jet cuts, we have chosen $p_{T}^{\mathrm{jet}}>20$ GeV,
$|\eta^{\mathrm{jet}}|<2.5$ and $\Delta R_{\textrm{jet--jet}}>0.4$. The cut
efficiencies for different numbers $n$ of gluons can be read off Table 3.
Employing these cuts we were also able to verify the jet production cross
sections that we have produced using Comix with the results reported in Ref.
[6]. To have a stringent comparison, we ran Comix for pure gluon scatterings
yielding cross sections that take the full-color dependence into account.
These results are also listed in the table; for the 4-gluon and 5-gluon
processes, they can be directly compared to the cross section estimates
obtained with the Tess MC on the GPU, since the leading-color approximation
already gives the exact result. The agreement is found to be satisfactory.
We show differential distributions in Figure 4. To obtain them we again used
$10^{9}$ sweeps where, for a certain gluon multiplicity, the total number of
generated events can be read off Table 3. We kept most of the input parameters
unaltered except for the jet cuts, which we changed to
$p_{T}^{\mathrm{jet}}>60$ GeV, $|\eta^{\mathrm{jet}}|<2.0$ and $\Delta
R_{\textrm{jet--jet}}>0.4$, and the choice of the renormalization and
factorization scales, which we decided to set dynamically using $H_{T}$ as a
scale. On the right hand side of Figure 4 we show for 3, 5, 7, 9 gluon jets in
the final state the normalized distributions for the $H_{T}$ observable and
the minimum $R$-separation, $R_{\mathrm{min}}$, which we define through the
jet–jet pair being closest in $R$-space, $R_{\mathrm{min}}=\min\\{\Delta
R_{ij}\\}$. As can be seen smooth distributions are easily obtained using the
Rambo phase-space generator. They are normalized to the total cross sections,
which have been calculated by Tess as $\sigma_{5}=(6.97838\pm 0.00044)\times
10^{4}$ pb, $\sigma_{7}=(4.9761\pm 0.0043)\times 10^{2}$ pb,
$\sigma_{9}=(4.532\pm 0.044)$ pb and $\sigma_{11}=(4.51\pm 0.19)\times
10^{-2}$ pb. On the left hand side of Figure 4 we have added profile plots
displaying the relative gauge invariance versus the decimal logarithm of the
matrix-element weight. Specifically, we show the average
$\left|K[1]\cdot\,J[2,\ldots,n]\right|^{2}/\left|J[1]\cdot\,J[2,\ldots,n]\right|^{2}$
and its mean standard deviation as a function of the matrix-element weight
$W_{\mathrm{ME}}=\left|m(1,2,\ldots,n)\right|^{2}=\left|(J[1]\cdot\,J[2,\ldots,n])\times
K^{2}[2,\ldots,n]\right|^{2}$. The behavior is as expected; for large weights,
we see gauge cancellations up to float precision. For small weights, the gauge
cancellations are less precise. However, these small-weight events are not
important since they do not contribute to the calculation of the observables.
Figure 5: The ratio $(\sigma_{5}^{\textsc{Tess}}\times
d\sigma_{5}^{\textsc{Comix}}/dX)/(\sigma_{5}^{\textsc{Comix}}\times
d\sigma_{5}^{\textsc{Tess}}/dX)-1$ for the 5-gluon $X=H_{T}$ (left panel) and
$X=R_{\mathrm{min}}$ distributions, see the text. The mean standard deviation
error bars of the Comix calculation are also shown.
Finally, in Figure 5 we compare our results to the results obtained with the
Comix event generator [6]. For this comparison, we use both the $H_{T}$ and
$R_{\mathrm{min}}$ 5-gluon distributions and we fix the renormalization and
factorization scale through $M_{Z}=91.118$ GeV to avoid any issues resulting
from slight differences in the evolution codes for running scales between the
two MCs. Furthermore, to have a sole shape comparison, we plot the ratio
$(\sigma_{5}^{\textsc{Tess}}\times
d\sigma_{5}^{\textsc{Comix}}/dX)/(\sigma_{5}^{\textsc{Comix}}\times
d\sigma_{5}^{\textsc{Tess}}/dX)-1$ with the results shown in Figure 5 and $X$
being the observable in consideration. Note that for the minimum
$R$-separation distribution, we have excellent agreement with Comix. For the
$H_{T}$ distribution, we have to realize that the cross section spans 28
orders of magnitude. As Comix relies on importance sampling, it only sparsely
populates the tail of the distribution. This leads to large uncertainties at
large values of $H_{T}$ and, in these regions, Comix will hence tend to
underestimate the value for the cross section.
## 5 Conclusions and Outlook
In our first exploration of the potential of using multi-threaded GPU-based
workstations for Monte Carlo programs, we obtained very encouraging results.
We implemented the entire Tess Monte Carlo on the GPU chip; the only off-chip
usage occurs through utilizing the texture memory for the evaluation of the
parton density function and the strong coupling constant. The GPU global
memory is solely used for transferring the Monte Carlo results to the CPU
memory. At this exploratory phase of the project, we limited ourselves to the
calculation of leading-color leading-order $n$-gluon matrix elements. With
respect to the CPU-based implementation of our Monte Carlo we have found
impressive speed-ups in the computations reaching from ${\cal O}(300)$ for
$PP\to 2$ jets to ${\cal O}(150)$ for $PP\to 10$ jets.
Given these results we are encouraged to further develop the Tess Monte Carlo
by including quarks, vector bosons and subleading color contributions. We are
also planning to implement on the GPU a dipole-based phase-space generator
like Sarge as an alternative to the unit-weight phase-space generator Rambo.
This will avoid the under-sampling issues in high jet-multiplicity final
states. These improvements will result in a full leading-order parton-level
event generator, which will be at least two orders of magnitude faster than
existing leading-order parton-level generators.
More importantly, a GPU-based Monte Carlo can be used as the generator for the
real corrections in an automated next-to-leading order parton-level MC
generator. The virtual corrections can be calculated by using a generalized-
unitarity based method. These methods themselves spend about 90% of their
evaluation time calculating leading-order matrix elements. By using the GPU-
based matrix-element evaluator of the Tess Monte Carlo, one can safely
estimate a speed-up factor of ${\cal O}(10)$ in the evaluation time of the
virtual corrections. This means that the GPU-based automated NLO parton-level
generator can be successfully implemented on the GPU-based workstation and
obtain accurate results on a timescale of a day without resorting to a large-
scale PC farm.
Finally, GPU chips for numerical evaluations are still evolving rapidly. This
will lead to additional significant speed-ups over CPU-based Monte Carlos in
the coming years.
###### Acknowledgments.
We want to thank Jim Simone for suggesting the Tesla GPU chips for use in
event generators. We thank the High-Performance Computing Department at
Fermilab for giving us support and access to the LQCD Tesla-based
workstations. We would also like to thank Patrick Fox, who came up with the
name “Tess” and suggested it to us as the name for our Monte Carlo program.
Fermilab is operated by Fermi Research Alliance, LLC, under contract DE-
AC02-07CH11359 with the United States Department of Energy.
## References
* [1] T. Stelzer and W. F. Long, Comput. Phys. Commun. 81 (1994) 357 [arXiv:hep-ph/9401258].
* [2] F. Krauss, R. Kuhn and G. Soff, JHEP 0202 (2002) 044 [arXiv:hep-ph/0109036].
* [3] P. D. Draggiotis, R. H. P. Kleiss and C. G. Papadopoulos, Eur. Phys. J. C 24 (2002) 447 [arXiv:hep-ph/0202201].
* [4] M. L. Mangano, M. Moretti, F. Piccinini, R. Pittau and A. D. Polosa, JHEP 0307 (2003) 001 [arXiv:hep-ph/0206293].
* [5] E. Boos et al. [CompHEP Collaboration], Nucl. Instrum. Meth. A 534 (2004) 250 [arXiv:hep-ph/0403113].
* [6] T. Gleisberg and S. Höche, JHEP 0812 (2008) 039 [arXiv:0808.3674 [hep-ph]].
* [7] W. T. Giele, Z. Kunszt and K. Melnikov, JHEP 0804, 049 (2008) [arXiv:0801.2237 [hep-ph]].
* [8] C. F. Berger et al., Phys. Rev. D 78, 036003 (2008) [arXiv:0803.4180 [hep-ph]].
* [9] A. van Hameren, C. G. Papadopoulos and R. Pittau, arXiv:0903.4665 [hep-ph].
* [10] See the website “http://www.nvidia.com/object/product_tesla_c1060_us.html” for further information.
* [11] See the website “http://www.nvidia.com/object/cuda_home.html” for further information.
* [12] R. Kleiss, W. J. Stirling and S. D. Ellis, Comput. Phys. Commun. 40 (1986) 359.
* [13] M. R. Whalley, D. Bourilkov and R. C. Group, arXiv:hep-ph/0508110.
* [14] K. Hagiwara, J. Kanzaki, N. Okamura, D. Rainwater and T. Stelzer, arXiv:0908.4403 [physics.comp-ph].
* [15] K. Hagiwara, J. Kanzaki, N. Okamura, D. Rainwater and T. Stelzer, arXiv:0909.5257 [hep-ph].
* [16] H. Murayama, I. Watanabe and K. Hagiwara, “HELAS: HELicity Amplitude Subroutines for Feynman diagram evaluations”, KEK-91-11
* [17] F. A. Berends and W. T. Giele, Nucl. Phys. B 306, 759 (1988).
* [18] R. Kleiss and H. Kuijf, Nucl. Phys. B 312 (1989) 616.
* [19] P. Draggiotis, R. H. P. Kleiss and C. G. Papadopoulos, Phys. Lett. B 439 (1998) 157 [arXiv:hep-ph/9807207].
* [20] W. Kahan, ”Further remarks on reducing truncation errors”, Communications of the ACM 8 (1): 40 (1965)
* [21] P. D. Draggiotis, A. van Hameren and R. Kleiss, Phys. Lett. B 483 (2000) 124 [arXiv:hep-ph/0004047].
* [22] J. Pumplin, D. R. Stump, J. Huston, H. L. Lai, P. M. Nadolsky and W. K. Tung, JHEP 0207 (2002) 012 [arXiv:hep-ph/0201195].
|
arxiv-papers
| 2010-02-18T18:06:43 |
2024-09-04T02:49:08.458004
|
{
"license": "Public Domain",
"authors": "Walter Giele, Gerben Stavenga, Jan-Christopher Winter",
"submitter": "Jan Winter",
"url": "https://arxiv.org/abs/1002.3446"
}
|
1002.3559
|
Common dynamics of two Pisot substitutions with the same incidence matrix
Tarek Sellami111sellami@iml.univ-mrs.fr
Institut de Mathématiques de Luminy, CNRS U.M.R. 6206, 163, Avenue de Luminy,
Case 907, 13288 Marseille Cedex 09 France.
Département de Mathématiques, Faculté des Sciences de Sfax, BP 802, 3018 Sfax,
Tunisie.
Abstract. The matrix of a substitution is not sufficient to completely
determine the dynamics associated, even in simplest cases since there are many
words with the same abelianization.
In this paper we study the common points of the canonical broken lines
associated to two different Pisot irreducible substitutions $\sigma_{1}$ and
$\sigma_{2}$ having the same incidence matrix. We prove that if $0$ is inner
point to the Rauzy fractal associated to $\sigma_{1}$ these common points can
be generated with a substitution on an alphabet of so-called "balanced
blocks".
Résumé: On sait que la matrice d’une substitution ne suffit pas à déterminer
complètement le système dynamique associé; même dans les cas les plus simples,
il existe de nombreuses substitutions associées à une matrice : il existe de
nombreux mots ayant le même abélianisé.
Dans ce papier, on étudie les points commun de deux lignes brisées associé aux
deux substitutions $\sigma_{1}$ et $\sigma_{2}$ irreducible de type Pisot qui
ont la même matrice d’incidence. On montre que si $0$ est un point intérieur à
l’un des deux fractals de Rauzy associé à $\sigma_{1}$ ou $\sigma_{2}$ alors
ces points communs peut être générés par une substitution définit sur un
alphabet appelé "block balncé".
## 1 Introduction
Let $\sigma_{1}$ and $\sigma_{2}$ be two different Pisot substitutions having
the same incidence matrix. Although the fixed points of each substitution have
the same letter frequencies, they usually show different dynamical and
geometrical properties, e.g., their Rauzy fractals have different properties.
(The Rauzy fractals can give a geometric model of the dynamical system defined
by the substitution, for more detail see section $2$).
A classic example is given by the Tribonacci substitution and the flipped
Tribonacci substitution, i.e.,
$\sigma_{1}:\left\\{\begin{array}[]{ll}a\rightarrow ab\\\ b\rightarrow ac\\\
c\rightarrow a\end{array}\right.$ and
$\sigma_{2}:\left\\{\begin{array}[]{ll}a\rightarrow ab\\\ b\rightarrow ca\\\
c\rightarrow a\end{array}\right.$
The incidence matrix of $\sigma_{1}$ and $\sigma_{2}$ is
$\begin{pmatrix}1&1&1\\\ 1&0&0\\\ 0&1&0\\\ \end{pmatrix}$. The dominant
eigenvalue satisfies the relation $X^{3}-X^{2}-X-1=0$, hence the name
Tribonacci for the substitution.
The Rauzy fractal of the first substitution is a topological disc [1], simply
connected , while it is a well known fact that the second fractal is not
simply connected, compare Figure[1].
Figure 1: The Rauzy fractals of $\sigma_{1}$ and $\sigma_{2}$
We consider another simple example of substitutions $\tau_{1}$ and $\tau_{2}$,
i.e.,
$\tau_{1}:\left\\{\begin{array}[]{ll}a\rightarrow aba\\\ b\rightarrow ab\\\
\end{array}\right.$ and $\tau_{2}:\left\\{\begin{array}[]{ll}a\rightarrow
aab\\\ b\rightarrow ba\\\ \end{array}\right.$
The Rauzy fractal of $\tau_{2}$ is the closure of a countable union of
disjoint intervals and the Rauzy fractal of $\tau_{1}$ is an interval, see [9]
and Figure[6].
We can deduce from one matrix we can obtain many different substitutions, so
many different Rauzy fractals. We are interested to studies commons dynamics
of these Rauzy fractals, we are interested to characterize their intersection,
for this we need to define a new object. prove that we can consider their
intersection as a substitutive set.
###### Definition 1.1.
A substitutive set is the closure of the projection of a canonical stepped
line associated to a primitive substitution on a contracting space associated
to the restriction of a positive integer matrix. For more detail see section
2.
The main result of this paper is the following:
###### Theorem 1.1.
Let $\sigma_{1}$ and $\sigma_{2}$ be two irreducible unimodular Pisot
substitutions with the same incidence matrix. Let $X_{\sigma_{1}}$ and
$X_{\sigma_{2}}$ the two associated Rauzy fractals; suppose that $0$ is inner
point to $X_{\sigma_{1}}$ .
Then the intersection of $X_{\sigma_{1}}$ and $X_{\sigma_{2}}$ has non-empty
interior, and it is substitutive.There is an algorithm to obtain the
substitution for intersection.
## 2 Substitutions and Rauzy fractals
### 2.1 General setting
Let $\mathcal{A}:=\\{a_{1},...,a_{d}\\}$ be a finite set of cardinal $d$
called alphabet. The free monoid $\mathcal{A}^{*}$ on the alphabet
$\mathcal{A}$ with empty word $\varepsilon$ is defined as the set of finite
words on the alphabet $\mathcal{A}$, this is
$\mathcal{A}^{*}:=\bigcup_{k\in\mathbb{N}}\mathcal{A}^{k}$, endowed with the
concatenation map. We denote by $\mathcal{A}^{\mathbb{N}}$ and
$\mathcal{A}^{\mathbb{Z}}$ the set of one and two-sided sequences on
$\mathcal{A}$, respectively. The topology of $\mathcal{A}^{\mathbb{N}}$ and
$\mathcal{A}^{\mathbb{Z}}$ is the product topology of discrete topology on
each copy of $\mathcal{A}$. Both spaces are metrizable.
The length of a word $w\in\mathcal{A}^{n}$ with $n\in\mathbb{N}$ is defined as
$|w|=n$. For any letter $a\in\mathcal{A}$, we define the number of occurrences
of $a$ in $w=w_{1}w_{2}\ldots w_{n-1}w_{n}$ by
$|w|_{a}=\sharp\\{i|w_{i}=a\\}$.
Let
$l:\mathcal{A}^{*}\mapsto\mathbb{Z}^{d}:w\mapsto(|w|_{a})_{a\in\mathcal{A}}\in\mathbb{N}^{d}$
be the natural homomorphism obtained by abelianization of the free monoid,
called the abelianization map.
A substitution over the alphabet $\mathcal{A}$ is an endomorphism of the free
monoid $\mathcal{A}^{*}$ such that the image of each letter of $\mathcal{A}$
is a nonempty word.
A substitution $\sigma$ is primitive if there exists an integer $k$ such that,
for each pair $(a,b)\in\mathcal{A}^{2}$, $|\sigma^{k}(a)|_{b}>0$. We will
always suppose that the substitution is primitive, this implies that for all
letter $j\in\mathcal{A}$ the length of the successive iterations
$\sigma^{k}(j)$ tends to infinity.
A substitution naturally extends to the set of two sided sequences
$\mathcal{A}^{\mathbb{Z}}$. We associate to every substitution $\sigma$ its
incidence matrix $M$ which is the $n\times n$ matrix obtained by
abelianization, i.e. $M_{i,j}=|\sigma(j)|_{i}$. It holds that
$l(\sigma(w))=Ml(w)$ for all $w\in\mathcal{A}^{*}$.
Remark. The incidence matrix of a primitive substitution is a primitive
matrix, so with the Perron-Frobenius theorem, it has a simple real positive
dominant eigenvalue $\beta$.
### 2.2 Rauzy fractals
###### Definition 2.1.
A Pisot number is an algebric integer $\beta>1$ such that each Galois
conjugate $\beta^{(i)}$ of $\beta$ satisfies $\mid\beta^{(i)}\mid<1$.
From now, we will suppose that all the substitutions that we consider are
irreducible of Pisot type and unimodular. This mean that the characteristic
polynomial of its incidence matrix is irreducible, its determinant is equal to
$\pm 1$ and its dominant eigenvalues is a Pisot number. We can prove that any
irreducible Pisot substitution is primitive (see [8]).
Remark. Note that there exist substitution whose largest eigenvalue is Pisot
but whose incidence matrix has eigenvalues that are not conjugate to the
dominant eigenvalue. Example is 1$\rightarrow$ 12, 2$\rightarrow$ 3,
3$\rightarrow$ 4, 4$\rightarrow$ 5, 5$\rightarrow$1\. The characteristic
polynomial is reducible. Such substitutions are called Pisot reducible.
###### Definition 2.2.
Let $\sigma$ a substitution and $u\in\mathcal{A}^{\mathbb{N}}$, $u$ is a fixed
point of $\sigma$ if $\sigma(u)=u$. The infinite word $u$ is a periodic point
of $\sigma$ if there exist $k\in\mathbb{N}$ such that $\sigma^{k}(u)=u$.
Let $\sigma$ be a primitive substitution, then there exist a finite number of
periodic points (see [7]). We associate to the fixed point $u$ of the
substitution a symbolic dynamical system $(\Omega_{u},S)$ where $S$ is the
shift map on $\mathcal{A}^{\mathbb{N}}$ given by
$S(a_{0}a_{1}...)=a_{1}a_{2}...$ and $\Omega_{u}$ is the closure of
$\\{S^{m}(u):m\geq 0\\}$ in $\mathcal{A}^{\mathbb{N}}$.
Remark. If $\sigma$ is a primitive substitution then the symbolic dynamical
system $(\Omega_{u},S)$ does not depend on $u$; we denote it by
$(\Omega_{\sigma},S$).
We say that a dynamical system $(X,f)$ is semiconjugate to another dynamical
system $(Y,g)$ if there exists a continuous surjective map
$\Theta:X\rightarrow Y$ such that $\Theta\circ f=g\circ\Theta$. An important
question is whether and how the symbolic dynamical system $(\Omega_{u},S)$
admit a geometric model. By geometrically realizable we mean there exists a
dynamical system $(X,f)$ defined on a geometrical structure, such that
$(\Omega_{u},S)$ is semiconjugate to $(X,f)$.
In [10], G.Rauzy proves that the dynamical system generated by the
substitution $\sigma(1)=12$, $\sigma(2)=13$, $\sigma(3)=1$, is measure-
theoretically conjugate to an exchange of domains in a compact set
$\mathcal{R}$ of the complex plane. This compact subset has a self-similar
structure : using method introduced by F.M.Dekking in [6], S.Ito and Pierre
Arnoux obtain in [1] an alternative construction of $\mathcal{R}$ and prove
that each of exchanged domains has fractal boundary. We will use the
projection method to obtain the Rauzy fractal.
###### Definition 2.3.
A stepped line $L=(x_{n})$ in $\mathbb{R}^{d}$ is a sequence (it could be
finite or infinite) of points in $\mathbb{R}^{d}$ such that $x_{n+1}-x_{n}$
belong to a finite set.
A canonical stepped line is a stepped line such that $x_{0}=0$ and for all
$n\geq 0$, $x_{n+1}-x_{n}$ belong to the canonical basis of $\mathbb{R}^{d}$.
Using the abelianization map, to any finite or infinite word $W$, we can
associate a canonical stepped line in $\mathbb{R}^{d}$ as a sequence
$(l(P_{n}))$, where $P_{n}$ are the prefix of length $n$ of $W$.
An interesting property of the canonical stepped line associated to a fixed
point of primitive Pisot substitution is that it remains within bounded
distance from the expanding direction (given by the right eigenvector of
Perron-Frobenius of $M_{\sigma}$). We denote by $E_{s}$ the stable space (or
contracting space) and $E_{u}$ the unstable space (or expanding direction). We
denote by $\pi_{s}$ the linear projection in the contracting plane, parallel
to the expanding direction and $\pi_{u}$ the projection in the expanding
direction parallel to the contracting plane. We will project the stepped line
on the contracting space in the direction of the right Perron-Frobenius
eigenvector. We obtain a bounded set in $(d-1)$-dimensional vector space.
###### Definition 2.4.
Let $\sigma$ an irreducible Pisot substitution. The Rauzy fractal associated
to $\sigma$ is the closure of the projection of the canonical stepped line
associated to any fixed point of $\sigma$ in the contracting plane parallel to
the expanding direction.
We note the projection $\pi$ of the orbit of the fixed point associated to a
Pisot irreducible substitution $\sigma$ on a contracting space associated to
its incidence matrix.
###### Proposition 2.1.
The projection $\pi$ of the symbolic dynamical system $\Omega_{\sigma}$
associated to a Pisot irreducible substitution $\sigma$ to the Rauzy fractal
is a continuous map.
###### Proof.
The proof is given in [7]. ∎
We denote by $X_{\sigma}$ the Rauzy fractal (Central tile) associated to
$\sigma$ : $X_{\sigma}:=\overline{\\{\pi_{s}(l(u_{0}\ldots
u_{k-1}),k\in{\mathbb{N}}\\}}$. with $u_{0}\ldots u_{k-1}$ is a prefix of the
fixed point of length $k$. Subtiles of the central tile $X_{\sigma}$ are
naturally defined, depending on the letter associated to the vertex of the
stepped line that is projected. On thus sets for $i\in\mathcal{A}$ :
$X_{\sigma}(i):=\overline{\\{\pi_{s}(l(u_{0}\ldots
u_{k-1}),k\in{\mathbb{N}},u_{k}=i\\}}$.
###### Proposition 2.2.
Let $\sigma$ a Pisot substitution and $X_{\sigma}$ its associated Rauzy
fractal. The boundary of $X_{\sigma}$ has zero measure.
###### Proof.
See [4] and [16]. ∎
Figure 2: The projection method to get the Rauzy fractal.
### 2.3 Central tiles viewed as a graph directed iterated function
The tiles $X_{\sigma}(i)$ can be written as a so-called graph iterated
function system (GIFS).
###### Definition 2.5.
(GIFS)
Let G be a finite directed graph with set of vertices $\\{1,\ldots,q\\}$ and
set of edges $E$. Denote the set of edges leading from $i$ to $j$ by $E_{ij}$.
To each $e\in E$ associated a contractive mapping
$\tau_{e}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$. If for each $i$ there is
some outgoing edge we call $(G,\\{\tau_{e}\\})$ a GIFS.
###### Definition 2.6.
(Prefix-suffix automaton)
Let $\sigma$ be a substitution over the alphabet $\mathcal{A}$ and let
$\mathcal{P}$ be the finite set
${P}:=\\{(p,a,s)\in\mathcal{A}^{\ast}\times\mathcal{A}\times\mathcal{A}^{\ast};\exists
b\in\mathcal{A},\sigma(b)=pas\\}$.
The prefix-suffix automaton of sigma has $\mathcal{A}$ as a set of vertices
and $\mathcal{P}$ as a set of label edges : there is an edge labeled by
$(p,a,s)$ from $a$ to $b$ if and only if $pas=\sigma(b)$.
Example. For the Fibonacci substitution $1\mapsto 12$ and $2\mapsto 1$, one
gets:
${P}=\\{(e,1,2),(1,2,e),(e,1,e)\\}$.
The prefix-suffix automaton of the Fibonacci substitution is :
Figure 3: Prefix-suffix automaton for the fibonacci substitution.
###### Theorem 2.1.
Let $\sigma$ be a primitive unit Pisot substitution over the alphabet
$\mathcal{A}$. The central tile $X_{\sigma}$ is a compact subset with nonempty
interior. Each subtile is the closure of its interior. The subtiles of
$X_{\sigma}$ are solution of the $GIFS$
$\forall
i\in\mathcal{A},X_{\sigma}(i)=\bigcup_{j\in\mathcal{A},i\xrightarrow[\text{}]{\text{(p,i,s)}}j}$
$MX_{\sigma}(j)+\pi_{s}l(p)$.
###### Proof.
The proof is given in [13].
∎
### 2.4 Disjointness of the subtiles of the central tile
To ensure that the subtiles are disjoint, we introduce the following
combinatorial condition in substitutions.
###### Definition 2.7.
(Strong coincidence condition).
A substitution $\sigma$ over the alphabet $\mathcal{A}$ satisfies the strong
coincidence condition if for every pair $(b_{1},b_{2})\in\mathcal{A}^{2}$,
there exist $k\in\mathbb{N}$ and $a\in\mathcal{A}$ such that
$\sigma^{k}(b_{1})=p_{1}as_{1}$ and $\sigma^{k}(b_{2})=p_{2}as_{2}$ with
$l(p_{1})=l(p_{2})$ or $l(s_{1})=l(s_{2})$.
Remark. The strong coincidence condition is satisfied by every unit Pisot
substitution over two letter alphabet [3]. It is conjectured that every
substitution of Pisot type satisfies the strong coincidence condition.
###### Theorem 2.2.
Let $\sigma$ be a primitive unit Pisot substitution. If $\sigma$ satisfies the
strong coincidence condition, then the subtiles of the central tiles have
disjoint interiors.
###### Proof.
The proof for the disjointness is given in [1]. ∎
Remark. If $0$ is inner point to the Rauzy fractal associated to a Pisot
substitution then the subtiles of the central tiles have disjoint interiors
(see [13]).
### 2.5 Substitutive sets
A substitutive set is the closure of the projection of a canonical stepped
line associated to substitution on a contracting space of a restriction of a
positive integer matrix. In particular a Rauzy fractal is a substitutive set
since it is the projection of canonical stepped line associated to a fixed
point on the contracting space associated to the matrix of substitution. So we
can expand the definition of Rauzy fractal to substitutive set. In particular
a substitutive set can be expressed as the attractor of some graph directed
iterated function system (IFS). See [2]
## 3 Intersection of Rauzy fractals
Let $\sigma_{1}$ and $\sigma_{2}$ two Pisot irreducible substitutions with the
same incidence matrix, we consider $X_{\sigma_{1}}$ and $X_{\sigma_{2}}$ their
associated Rauzy fractals respectively.
The intersection of $X_{\sigma_{1}}$ and $X_{\sigma_{2}}$ is non-empty since
it contains $0$, and it is a compact set (intersection of two compacts).
###### Proposition 3.1.
Let $\sigma_{1}$ and $\sigma_{2}$ be two Pisot irreducible substitutions with
the same incidence matrix. We consider $L_{1}$ and $L_{2}$ the canonical
broken lines associated to a fixed point of $\sigma_{1}$ and $\sigma_{2}$
respectively, let $P_{1}$ and $P_{2}$ two points from $L_{1}$ and $L_{2}$
respectively. Then $\pi_{s}(P_{1})=\pi_{s}(P_{2})$ implies $P_{1}=P_{2}$.
###### Proof.
The Perron Frobenius eigenvalues is irrational in the irreducible case. We
project $P_{1}$ and $P_{2}$ in the contracting space parallel to the expanding
space (the direction of the Perron Frobenius eigenvectors). If
$\pi_{s}(P_{1})=\pi_{s}(P_{2})$ then $(P_{1}P_{2})$ is parallel to the
expanding direction. This implies that expanding direction is rational.
∎
###### Proposition 3.2.
Let $\sigma_{1}$ and $\sigma_{2}$ be two substitutions with the same incidence
matrix, we consider $X_{\sigma_{1}}$ (resp. $X_{\sigma_{2}}$) the Rauzy
fractal associated to $\sigma_{1}$ (resp. $\sigma_{2}$) and $X_{\sigma}$ the
common point of $X_{\sigma_{1}}$ and $X_{\sigma_{2}}$.
Then the boundary of $X_{\sigma}$ is included in the union boundary of
$X_{\sigma_{1}}$ and $X_{\sigma_{2}}$ and has zero measure.
###### Proof.
Let $x$ a point from the boundary of $X_{\sigma}$. We suppose that $x$ is not
on the boundary of $X_{\sigma_{1}}$. Then there exist $r_{1}>0$ such that
$B(x,r_{1})\subset X_{\sigma_{1}}$. If $x$ is not in the boundary of
$X_{\sigma_{2}}$ then there exist $r_{2}>0$ such that $B(x,r_{2})\subset
X_{\sigma_{2}}$. Then there exist $r=\min(r_{1},r_{2})$ such that
$B(x,r)\subset X_{\sigma_{1}}\cap X_{\sigma_{2}}$, $x$ is in the boundary of
$X_{\sigma}$ then $x$ is in the boundary of $X_{\sigma_{2}}$. Then $\partial
X_{\sigma}\subset\partial X_{\sigma_{1}}\cup\partial X_{\sigma_{2}}$.
Since $\partial X_{\sigma_{1}}$ and $\partial X_{\sigma_{2}}$ have zero
measure then $\partial X_{\sigma}$ has zero measure. ∎
Figure 4: Sets of common points of the fractals of Tribonacci and the flipped
Tribonacci .
### 3.1 The main result: Morphism generating the common points of two Pisot
substitutions with the same incidence matrix
In this section we consider $\sigma_{1}$ and $\sigma_{2}$ be two unimodular
irreducible Pisot substitutions with the same incidence matrix. We denote
$X_{\sigma_{1}}$ and $X_{\sigma_{2}}$ their associated Rauzy fractals
respectively. We suppose that $0$ is an inner point to $X_{\sigma_{1}}$. We
note $X_{\sigma}$ the closure of the intersection of the interior of
$X_{\sigma_{1}}$ and the interior of $X_{\sigma_{2}}$. Let
$(\Omega_{\sigma_{1}},S)$ and $(\Omega_{\sigma_{2}},S)$ the symbolic dynamical
systems associated to $\sigma_{1}$ and $\sigma_{2}$ respectively. We consider
$\pi_{1}$ (resp. $\pi_{2}$) the projection map from the symbolic dynamical
system $(X_{\sigma_{1}},S)$ into the Rauzy fractal (rep.$\pi_{2}$).
We will prove that $X_{\sigma}$ is a substitutive set, and it can be generated
by a substitution obtained with algorithm generating the common point of the
interior of $X_{\sigma_{1}}$ and $X_{\sigma_{2}}$.
###### Definition 3.1.
For a dynamical system $(X,T)$ if $A$ is a subset of $X$, and $x\in A$, we
define the first return time of $x$ as
$n_{x}=\inf\\{n\in\mathbb{N}^{*}|T^{n}(x)\in A\\}$ (it is infinite if the
orbit of $x$ does not come back to $A$). If the first return time is finite
for all $x\in A$, we define the induced map of $T$ on $A$ (or first return
map) as the map $x\mapsto T^{n_{k}}(x)$, and we denote this map by $T_{A}$.
###### Definition 3.2.
A sequence $u=(u_{n})$ is minimal (or uniformly recurrent) if every word
occurring in $u$ occurs in an infinite number of positions with bounded gaps,
that is, if for every factor $W$, there exist $s$ such that for every $n$, $W$
is a factor of $u_{n}\ldots u_{n+s-1}$.
###### Lemma 3.1.
The closure of the intersection $X_{\sigma}$ has non empty interior and non-
zero Lebesque measure.
###### Proof.
We suppose that $0$ is an inner point to $X_{\sigma_{1}}$. Then there exist an
open set $U$ such that $0\in U\subset X_{\sigma_{1}}$. The Rauzy fractal is
the closure of its interior and $0$ is a point of $X_{\sigma_{2}}$, hence
there exist a sequence of points $(x_{n})_{n\in\mathbb{N}}$ from the interior
of $X_{\sigma_{2}}$ which converges to $0$. Then there exist open sets $V_{n}$
such that $x_{n}\in V_{n}\subset X_{\sigma_{2}}$. Since $(x_{n})$ converge to
$0$, there exists $N\in\mathbb{N}$ such that $x_{N}\in U$.
We denote by $W$ the open set $W=U\cap V_{N}$,$W$ is non-empty and $W\subset
X_{\sigma_{1}}\cap X_{\sigma_{2}}$. The intersection of $X_{\sigma_{1}}$ and
$X_{\sigma_{2}}$ contains a non empty open set, hence it has non-zero Lebesque
mesure. ∎
We define the subgroup $\Gamma$ of $\mathbb{Z}^{d}$ as :
$\Gamma=\\{\sum_{i=1}^{d}n_{i}e_{i}/\sum_{i=1}^{d}n_{i}=0,n_{i}\in\mathbb{Z}\\}$
with $e_{i}$ is the canonical bases of $\mathbb{R}^{d}.$
###### Lemma 3.2.
Let $\sigma$ be an irreducible Pisot substitution, and $X_{\sigma}$ its
associated Rauzy fractal. If $0$ is inner point to $X_{\sigma}$ then
$X_{\sigma}$ is a fundamental domain of $E_{s}$ for the projection of $\Gamma$
on the stable space.
###### Proof.
The proof is given in [13].
∎
###### Lemma 3.3.
Let $W$ be a non-empty open set in $X_{\sigma}$. Let $V_{1}=\pi_{1}^{-1}(W)$
and $V_{2}=\pi_{2}^{-1}(W)$ from $\Omega_{\sigma_{1}}$ and
$\Omega_{\sigma_{2}}$ respectively. If $n$ is a first return time in $V_{2}$
then $n$ is a return time in $V_{1}$.
###### Proof.
We consider $v_{1}\in V_{1}$ and $v_{2}\in V_{2}$ such that
$\pi_{1}(v_{1})=\pi_{2}(v_{2})$. Let $n$ the first return time of $v_{2}$ in
$V_{2}$. Then there exist $w\in\Gamma$ such that
$\pi_{1}(S^{n}(v_{1}))=\pi_{2}(S^{n}(v_{2}))+\pi_{2}(w)$.
$\pi_{2}(S^{n}(v_{2}))$ is a point from the interior of $X_{\sigma}$, then it
is a point from the interior of $X_{\sigma_{1}}$. And we have
$\pi_{1}(S^{n}(v_{1}))$ is a point from $X_{\sigma_{1}}$.
Since $0$ is an inner point to $X_{\sigma_{1}}$, from lemma $3.3$,
$X_{\sigma_{1}}$ is a fundamental domain. Then we obtain $\pi_{2}(w)=0$. So we
have $\pi_{1}(S^{n}(v_{1}))=\pi_{2}(S^{n}(v_{2}))$. Then if $n$ is a return
time in $V_{2}$ we deduce that $n$ is a return time in $V_{1}$. ∎
###### Definition 3.3.
Let $U$ and $V$ two finite words, we say that $\begin{pmatrix}U\\\ V\\\
\end{pmatrix}$ is balanced block if $l(U)=l(V)$, where $l$ is the
abelianization map from $\mathcal{A}^{*}$ in $\mathbb{Z}^{d}$.
###### Definition 3.4.
A minimal balanced block is a balanced block, such for every strict prefix
$U_{k}$, $V_{k}$ of $U$ and $V$ respectively of length $k$, $l(U_{k})\neq
l(V_{k})$.
###### Lemma 3.4.
Let $u$ and $v$ be tow fixed points of $\sigma_{1}$ and $\sigma_{2}$
respectively, then we can decompose $u$ and $v$ on a finite minimal balanced
blokcs.
###### Proof.
Let $u$ and $v$ be tow fixed points of $\sigma_{1}$ and $\sigma_{2}$
respectively. We have $0\in X_{\sigma}$ then there exist $v_{1}$ and $v_{2}$
two prefix of $u$ and $v$ respectively such that $x=\pi(v_{1})=\pi(v_{2})$ and
$l(v_{1})=l(v_{2})$. We obtain a balanced block: $\begin{pmatrix}v_{1}\\\
v_{2}\\\ \end{pmatrix}$, we can decompose it with minimal balanced blocks and
we consider the image of each new minimal balanced block with $\sigma_{1}$ and
$\sigma_{2}$. Then there exist new minimal balanced blocks which appear, we
consider the image of each new blocks by $\sigma_{1}$ and $\sigma_{2}$. Since
every word appears with a bounded distance, all the minimal balanced blocks
will appear after a finite time. Then we can obtain a decomposition of $u$ and
$v$ with a finite number of minimal balanced blocks. A simple case appears
when $u$ and $v$ begin with the same letter $i$, then the first minimal
balanced block is $\begin{pmatrix}i\\\ i\\\ \end{pmatrix}$.
∎
###### Theorem 3.1.
$X_{\sigma}$ is a substitutive set.
###### Proof.
We have $X_{\sigma}$ is the closure of the projection of points associated to
balanced blocks, from the two stepped lines associated to the fixed points of
$\sigma_{1}$ and $\sigma_{2}$. These common points can be obtained as a fixed
point of a new substitution defined on the set of the minimal balanced blocks.
There exist an algorithm to obtain this morphism (or substitution). Since $0$
is a point from $X_{\sigma}$ there exist two minimal initial word $v_{1}$ and
$v_{2}$ from the language of $\sigma_{1}$ and $\sigma_{2}$ respectively such
that $l(v_{1})=l(v_{2})$
We denote the block $\begin{pmatrix}v_{1}\\\ v_{2}\\\ \end{pmatrix}$ and we
consider $\sigma_{1}(v_{1})$ and $\sigma_{2}(v_{2})$ we obtain a second block
$\begin{pmatrix}\sigma_{1}(v_{1})\\\ \sigma_{2}(v_{2})\\\ \end{pmatrix}$ with
the property $l(\sigma_{1}(v_{1}))=l(\sigma_{2}(v_{2}))$ because $\sigma_{1}$
and $\sigma_{2}$ have the same matrix. These blocks have a finite length,
because the return time in $X_{\sigma}$ is bounded. We consider the
decomposition of this balanced block $\begin{pmatrix}\sigma_{1}(v_{1})\\\
\sigma_{2}(v_{2})\\\ \end{pmatrix}$ with minimal balanced blocks.
This mean we can write $\begin{pmatrix}\sigma_{1}(v_{1})\\\
\sigma_{2}(v_{2})\\\ \end{pmatrix}=\begin{pmatrix}u_{1}\ldots u_{k}\\\
w_{1}\ldots w_{k}\\\ \end{pmatrix}$ with the property $l(u_{1})=l(w_{1})$,
…,$l(u_{n})=l(v_{n})$.
With this method we obtain a finite numbers of blocks with the same
abelianization. We consider this set of blocks and we consider the image of
each block with the two substitutions $\sigma_{1}$ and $\sigma_{2}$ and we
obtain a morphism witch generate all the common points of the stepped lines.
∎
## 4 Examples
### 4.1 Algorithm to obtain the morphism of the common points of two Rauzy
fractals
#### 4.1.1 Example 1
I will take the example of $\tau_{1}$ and $\tau_{2}$ to show how the algorithm
is working. In this example the first minimal balanced block that we consider
is the beginning of the two fixed points associated to $\tau_{1}$ and
$\tau_{2}$ it will be $\begin{pmatrix}a\\\ a\\\ \end{pmatrix}$.
And we consider the image of the first element of this block by $\tau_{1}$ and
the second one by $\tau_{2}$ so we obtain : $\begin{pmatrix}a\\\ a\\\
\end{pmatrix}\overset{\tau_{1},\tau_{2}}{\longrightarrow}\begin{pmatrix}aba\\\
aab\\\ \end{pmatrix}$.
We denote by $A$ the minimal balanced block $\begin{pmatrix}a\\\ a\\\
\end{pmatrix}$ and by $B$ the minimal balanced block $\begin{pmatrix}ba\\\
ab\\\ \end{pmatrix}$.
So we obtain $A\rightarrow AB.$
The second step is to consider the same thing with the new block
$\begin{pmatrix}ba\\\ ab\\\ \end{pmatrix}$.
We consider the image of this block with the two substitution $\tau_{1}$ and
$\tau_{2}$, and we obtain :
$\begin{pmatrix}ba\\\ ab\\\
\end{pmatrix}\overset{\tau_{1},\tau_{2}}{\longrightarrow}\begin{pmatrix}ababa\\\
aabba\\\ \end{pmatrix}$.
We obtain an other block $\begin{pmatrix}b\\\ b\\\ \end{pmatrix}$ and we
denote by $C$ the projection over this new block and we obtain the image of
$B$ is $ABCA$. We continuous with this algorithm and we obtain the image of
the block $\begin{pmatrix}b\\\ b\\\ \end{pmatrix}$ is the new block
$\begin{pmatrix}ab\\\ ba\\\ \end{pmatrix}$. So we obtain the image of the
letter $C$ is a new letter $D$. Finally the image of the letter $D$ is $DAAC$.
So, we obtain an alphabet $\mathcal{B}$ in $4$ letters and we can define the
morphism $\phi$ as :
$\phi:\left\\{\begin{array}[]{ll}A\rightarrow AB\\\ B\rightarrow ABCA\\\
C\rightarrow D\\\ D\rightarrow DAAC\\\ \end{array}\right.$
And we consider the projection $\pi$ of the letters $A,B,C,D$ in the sets of
blocks $\begin{pmatrix}a\\\ a\\\ \end{pmatrix}$, $\begin{pmatrix}ba\\\ ab\\\
\end{pmatrix}$, $\begin{pmatrix}b\\\ b\\\ \end{pmatrix}$ et
$\begin{pmatrix}ab\\\ ba\\\ \end{pmatrix}$
Then we have : $\begin{pmatrix}\tau_{1}^{n}(a)\\\ \tau_{2}^{n}(a)\\\
\end{pmatrix}$ $=\pi(\phi^{n}(A))$.
The morphism $\phi$ generate all the common points of the two Rauzy fractals
associated to $\tau_{1}$ and $\tau_{2}$.
Figure 5: The Rauzy fractals of $\tau_{1}$ and $\tau_{2}$.
Figure 6: Common points of $\tau_{1}$ and $\tau_{2}$ with distinction of
blocs defined with $\phi$
#### 4.1.2 Example 2
For the two substitutions of Tribonacci and the flipped Tribonacci it is more
complicated see Figure[7], we can define the morphism $\phi$ which generate
all the common points as follows:
$\phi:\left\\{\begin{array}[]{ll}A\rightarrow AB\\\ B\rightarrow C\\\
C\rightarrow AD\\\ D\rightarrow AE\\\ E\rightarrow F\\\ F\rightarrow ADDGA\\\
G\rightarrow AH\\\ H\rightarrow ID\\\ I\rightarrow ADJ\\\ J\rightarrow AHK\\\
K\rightarrow IDGA\end{array}\right.$
and the projection map $\pi$ :
$\pi:\left\\{\begin{array}[]{ll}A\rightarrow\begin{pmatrix}a\\\ a\\\
\end{pmatrix}\\\ B\rightarrow\begin{pmatrix}b\\\ b\\\ \end{pmatrix}\\\
C\rightarrow\begin{pmatrix}ac\\\ ca\\\ \end{pmatrix}\\\
D\rightarrow\begin{pmatrix}ba\\\ ab\\\ \end{pmatrix}\\\
E\rightarrow\begin{pmatrix}cab\\\ bca\\\ \end{pmatrix}\\\
F\rightarrow\begin{pmatrix}aabac\\\ caaab\\\ \end{pmatrix}\\\
G\rightarrow\begin{pmatrix}cab\\\ abc\\\ \end{pmatrix}\\\
H\rightarrow\begin{pmatrix}abac\\\ bcaa\\\ \end{pmatrix}\\\
I\rightarrow\begin{pmatrix}abaca\\\ caaab\\\ \end{pmatrix}\\\
J\rightarrow\begin{pmatrix}cabaab\\\ ababca\\\ \end{pmatrix}\\\
K\rightarrow\begin{pmatrix}ababac\\\ bcaaab\\\
\end{pmatrix}\end{array}\right.$
Figure 7: Sets of common points of the Tribonacci substitution and the flipped
substitution. Each color stands for a different letter of $\mathcal{B}$ and
shows the dynamics of the morphism $\phi$.
###### Corollaire 4.1.
Let $\sigma_{1}$ and $\sigma_{2}$ be the two substitution Tribonacci and the
flipped Tribonacci defined as follows:
$\sigma_{1}:\left\\{\begin{array}[]{ll}a\rightarrow ab\\\ b\rightarrow ac\\\
c\rightarrow a\end{array}\right.$ et
$\sigma_{2}:\left\\{\begin{array}[]{ll}a\rightarrow ab\\\ b\rightarrow ca\\\
c\rightarrow a\end{array}\right.$
We consider $U$ and $V$ their two fixed points, then the letter $c$ doesn’t
occur in the same position in $U$ and $V$.
###### Proof.
Minimal balanced blocks represents a decomposition of the two fixed points
$U$and $V$. We remark that in these finite minimal blocks there is no $c$
which appears in the same position. One can then deduce that the letter $c$
does not appear in the same position in two fixed points $U$ and $V$. ∎
#### 4.1.3 Example 3
Now we will consider more general example defined as follows :
$\delta_{i}^{1}:\left\\{\begin{array}[]{ll}a\rightarrow a^{i}b\\\ b\rightarrow
a^{i-1}c\\\ c\rightarrow a\end{array}\right.$ and
$\delta_{i}^{2}:\left\\{\begin{array}[]{ll}a\rightarrow aba^{i-1}\\\
b\rightarrow aca^{i-2}\\\ c\rightarrow a\end{array}\right.$
$\delta_{i}^{1}$ and $\delta_{i}^{2}$ have the same incidence matrix. We can
define the morphism of their common points for all $i\geq 3$ as :
Figure 8: The Rauzy fractals of $\delta_{3}^{1}$ and $\delta_{3}^{2}$.
$\phi_{i}:\left\\{\begin{array}[]{ll}A\rightarrow AB\\\ B\rightarrow AC\\\
C\rightarrow(AAD)^{i-1}[AAE(AAD)^{i}]^{i-2}AAE(AAD)^{i-1}A\\\ D\rightarrow
AF\\\ E\rightarrow(AAD)^{i-3}A\\\
F\rightarrow(AAD)^{i-1}[AAE(AAD)^{i}]^{i-3}AAE(AAD)^{i-1}A.\end{array}\right.$
$\pi_{i}:\left\\{\begin{array}[]{ll}A\rightarrow\begin{pmatrix}a\\\ a\\\
\end{pmatrix}\\\ B\rightarrow\begin{pmatrix}a^{i-1}b\\\ ba^{i-1}\\\
\end{pmatrix}\\\ C\rightarrow\begin{pmatrix}a^{i-1}b(a^{i}b)^{i-2}a^{i-1}c\\\
ca^{i-1}(ba^{i})^{i-2}ba^{i-1}\\\ \end{pmatrix}\\\
D\rightarrow\begin{pmatrix}a^{i-2}b\\\ ba^{i-2}\\\ \end{pmatrix}\\\
E\rightarrow\begin{pmatrix}a^{i-3}c\\\ ca^{i-3}\\\ \end{pmatrix}\\\
F\rightarrow\begin{pmatrix}a^{i-1}b(a^{i}b)^{i-3}a^{i-1}c\\\
ca^{i-1}(ba^{i})^{i-3}ba^{i-1}\\\ \end{pmatrix}\\\ \end{array}\right.$
Figure 9: Sets of common points of $\delta_{3}^{1}$ and $\delta_{3}^{2}$.
Remark. The property $0$ is inner point is sufficient, and we have this
example of substitutions with the same incidence matrix but the intersection
is reduced to the origin.
We can give an example where the intersection is empty. We consider the two
substitutions $\chi_{1}$ and $\chi_{2}$ defined as follows :
$\chi_{1}:\left\\{\begin{array}[]{ll}a\rightarrow aab\\\ b\rightarrow ab\\\
\end{array}\right.$ and $\chi_{2}:\left\\{\begin{array}[]{ll}a\rightarrow
baa\\\ b\rightarrow ba\\\ \end{array}\right.$
Figure 10: The Rauzy fractals of $\chi_{1}$ and $\chi_{2}$.
###### Proof.
We consider $u_{1}$ and $u_{2}$ the two fixed points associated to $\chi_{1}$
and $\chi_{2}$ respectively. If $a.x$ is a prefix of $u_{1}$ then $b.x$ is a
prefix of $u_{2}$.
We will reason by induction : for $x=a$ it is so verified for $n=1$.
We suppose now that $a.x$ is prefix of $u_{1}$ and $b.x$ is a prefix of
$u_{2}$ with $|x|=n$.
$\chi_{1}(a.x)=aab\chi_{1}(x)$ is prefix of $u_{1}$,
$\chi_{2}(b.x).b=ba\chi_{2}(x)b$ is a prefix of $u_{2}$ if and only if
$\chi_{1}(x)=\chi_{2}(x)b$.
We have for the two letter $a$ and $b$:
* •
$x=a$ : $b.\chi_{1}(a)=baab=\chi_{2}(a)b.$
* •
$x=b$ : $b.\chi_{1}(b)=bab=\chi_{2}(b)b.$
We consider now $x=x_{1}x_{2}\ldots x_{n}$ with $x_{i}\in\\{a,b\\}$
$b.\chi_{1}(x)=b.\chi_{1}(x_{1}x_{2}\ldots
x_{n})=b.\chi_{1}(x_{1})\ldots\chi_{1}(x_{n})$
$=\chi_{2}(x_{1}).b.\chi_{1}(x_{2})\ldots\chi_{1}(x_{n})$
⋮
$=\chi_{2}(x_{1})\chi_{2}(x_{2})\ldots\chi_{2}(x_{n}).b$
So we prove that there exist an infinite word $u$ such that $u_{1}=a.u$ and
$u_{2}=b.u$. ∎
## References
* [1] P. Arnoux and S. Ito "Pisot substitutions and Rauzy fractals" Bull. Belg. Math. Soc. Simon Stevin 8: 181–207 (2001).
* [2] Pierre Arnoux, Julian Bernat and Xavier Bressaud. "Geometrical model for substitutions".
* [3] M.Barge and B.Diamond. "Coincidence for substitutions of Pisot Type". Bell.Soc.Math. France, 130:619-626, 2002.
* [4] Marcy Barge and Jarek Kwapisz, "Geometric theory of unimodular Pisot substitutions". American journal of mathematics(Print) 128:55, 1219-1282.
* [5] Vicent Canterini and Anne Siegel : "Automate des préfixes-suffixes associé une substitution primitive".J. Théor. Nombres Bordeaux 13, no. 2, 353–369, 2001.
* [6] F.M.Dekking, recurrent sets, Adv. in Math. 44 (1982), no.1, 78-104. MR 84:52023
* [7] P.Fogg, "Substitutions in dynamics, arithmetics and combinatorics" (Lecture Notes in Mathematics, Vol.1794).
* [8] J.M.Luck, C.Godrèche and A. Janner and T.Janssen, "The nature of the atomic surfaces of quasiperiodic self similar structures", J.Phys.A: Math. Gen.26: 1951-1999 (1993).
* [9] A.Messaoudi: "Propriétés arithmétiques et dynamiques du fractal de Rauzy", journal de Théorie des nombres de bordeaux, 10, 1998, 135-162.
* [10] G.Rauzy : "Nombre algébrique et substitution", Bull.Soc.Math. France 110 (1982), 147-178.
* [11] M.Queffelec, "Substitution dynamical system", Spectra analysis, lecture note in mathematics, 1294, Springer-Verlag, Berlin, 1987.
* [12] Anne Siegel : "Autour des fractals de Rauzy", Journées Femmes et Mathématiques, Paris (03/2002).
* [13] Anne Siegel and Jorg M.Thuswaldner: "Topological proprieties of Rauzy fractal", Mémoires de la SMF
* [14] Victor F.Sirvent : "The common dynamics of the tribonacci substitutions", Bull. Belg .Math. Soc. 7 (2000), 571-582.
* [15] B.Sing and V.Sirvent : "Geometry of the common dynamics of flipped Pisot substitution", Monatshefte für Mathematik, 155 (2008), 431-448.
* [16] Victor Sirvent and Yang Sirvent, "Self affine tiling via substitution dynamical systems and Rauzy fractal". Pacific journal of mathematics Vol 206, N 2, 2002.
|
arxiv-papers
| 2010-02-18T16:16:28 |
2024-09-04T02:49:08.464806
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tarek Sellami",
"submitter": "Sellami Tarek",
"url": "https://arxiv.org/abs/1002.3559"
}
|
1002.3603
|
# The Spectrum of the Isotropic Diffuse Gamma-Ray Emission Derived From First-
Year Fermi Large Area Telescope Data
A. A. Abdo Space Science Division, Naval Research Laboratory, Washington, DC
20375, USA National Research Council Research Associate, National Academy of
Sciences, Washington, DC 20001, USA M. Ackermann markusa@slac.stanford.edu
W. W. Hansen Experimental Physics Laboratory, Kavli Institute for Particle
Astrophysics and Cosmology, Department of Physics and SLAC National
Accelerator Laboratory, Stanford University, Stanford, CA 94305, USA M.
Ajello W. W. Hansen Experimental Physics Laboratory, Kavli Institute for
Particle Astrophysics and Cosmology, Department of Physics and SLAC National
Accelerator Laboratory, Stanford University, Stanford, CA 94305, USA W. B.
Atwood Santa Cruz Institute for Particle Physics, Department of Physics and
Department of Astronomy and Astrophysics, University of California at Santa
Cruz, Santa Cruz, CA 95064, USA L. Baldini Istituto Nazionale di Fisica
Nucleare, Sezione di Pisa, I-56127 Pisa, Italy J. Ballet Laboratoire AIM,
CEA-IRFU/CNRS/Université Paris Diderot, Service d’Astrophysique, CEA Saclay,
91191 Gif sur Yvette, France G. Barbiellini Istituto Nazionale di Fisica
Nucleare, Sezione di Trieste, I-34127 Trieste, Italy Dipartimento di Fisica,
Università di Trieste, I-34127 Trieste, Italy D. Bastieri Istituto Nazionale
di Fisica Nucleare, Sezione di Padova, I-35131 Padova, Italy Dipartimento di
Fisica “G. Galilei”, Università di Padova, I-35131 Padova, Italy B. M.
Baughman Department of Physics, Center for Cosmology and Astro-Particle
Physics, The Ohio State University, Columbus, OH 43210, USA K. Bechtol W. W.
Hansen Experimental Physics Laboratory, Kavli Institute for Particle
Astrophysics and Cosmology, Department of Physics and SLAC National
Accelerator Laboratory, Stanford University, Stanford, CA 94305, USA R.
Bellazzini Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, I-56127
Pisa, Italy B. Berenji W. W. Hansen Experimental Physics Laboratory, Kavli
Institute for Particle Astrophysics and Cosmology, Department of Physics and
SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94305,
USA R. D. Blandford W. W. Hansen Experimental Physics Laboratory, Kavli
Institute for Particle Astrophysics and Cosmology, Department of Physics and
SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94305,
USA E. D. Bloom W. W. Hansen Experimental Physics Laboratory, Kavli
Institute for Particle Astrophysics and Cosmology, Department of Physics and
SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94305,
USA E. Bonamente Istituto Nazionale di Fisica Nucleare, Sezione di Perugia,
I-06123 Perugia, Italy Dipartimento di Fisica, Università degli Studi di
Perugia, I-06123 Perugia, Italy A. W. Borgland W. W. Hansen Experimental
Physics Laboratory, Kavli Institute for Particle Astrophysics and Cosmology,
Department of Physics and SLAC National Accelerator Laboratory, Stanford
University, Stanford, CA 94305, USA J. Bregeon Istituto Nazionale di Fisica
Nucleare, Sezione di Pisa, I-56127 Pisa, Italy A. Brez Istituto Nazionale di
Fisica Nucleare, Sezione di Pisa, I-56127 Pisa, Italy M. Brigida
Dipartimento di Fisica “M. Merlin” dell’Università e del Politecnico di Bari,
I-70126 Bari, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Bari,
70126 Bari, Italy P. Bruel Laboratoire Leprince-Ringuet, École
polytechnique, CNRS/IN2P3, Palaiseau, France T. H. Burnett Department of
Physics, University of Washington, Seattle, WA 98195-1560, USA S. Buson
Dipartimento di Fisica “G. Galilei”, Università di Padova, I-35131 Padova,
Italy G. A. Caliandro Institut de Ciencies de l’Espai (IEEC-CSIC), Campus
UAB, 08193 Barcelona, Spain R. A. Cameron W. W. Hansen Experimental Physics
Laboratory, Kavli Institute for Particle Astrophysics and Cosmology,
Department of Physics and SLAC National Accelerator Laboratory, Stanford
University, Stanford, CA 94305, USA P. A. Caraveo INAF-Istituto di
Astrofisica Spaziale e Fisica Cosmica, I-20133 Milano, Italy J. M. Casandjian
Laboratoire AIM, CEA-IRFU/CNRS/Université Paris Diderot, Service
d’Astrophysique, CEA Saclay, 91191 Gif sur Yvette, France E. Cavazzuti
Agenzia Spaziale Italiana (ASI) Science Data Center, I-00044 Frascati (Roma),
Italy C. Cecchi Istituto Nazionale di Fisica Nucleare, Sezione di Perugia,
I-06123 Perugia, Italy Dipartimento di Fisica, Università degli Studi di
Perugia, I-06123 Perugia, Italy Ö. Çelik NASA Goddard Space Flight Center,
Greenbelt, MD 20771, USA Center for Research and Exploration in Space Science
and Technology (CRESST) and NASA Goddard Space Flight Center, Greenbelt, MD
20771, USA Department of Physics and Center for Space Sciences and
Technology, University of Maryland Baltimore County, Baltimore, MD 21250, USA
E. Charles W. W. Hansen Experimental Physics Laboratory, Kavli Institute for
Particle Astrophysics and Cosmology, Department of Physics and SLAC National
Accelerator Laboratory, Stanford University, Stanford, CA 94305, USA A.
Chekhtman Space Science Division, Naval Research Laboratory, Washington, DC
20375, USA George Mason University, Fairfax, VA 22030, USA C. C. Cheung
Space Science Division, Naval Research Laboratory, Washington, DC 20375, USA
National Research Council Research Associate, National Academy of Sciences,
Washington, DC 20001, USA J. Chiang W. W. Hansen Experimental Physics
Laboratory, Kavli Institute for Particle Astrophysics and Cosmology,
Department of Physics and SLAC National Accelerator Laboratory, Stanford
University, Stanford, CA 94305, USA S. Ciprini Dipartimento di Fisica,
Università degli Studi di Perugia, I-06123 Perugia, Italy R. Claus W. W.
Hansen Experimental Physics Laboratory, Kavli Institute for Particle
Astrophysics and Cosmology, Department of Physics and SLAC National
Accelerator Laboratory, Stanford University, Stanford, CA 94305, USA J.
Cohen-Tanugi Laboratoire de Physique Théorique et Astroparticules, Université
Montpellier 2, CNRS/IN2P3, Montpellier, France L. R. Cominsky Department of
Physics and Astronomy, Sonoma State University, Rohnert Park, CA 94928-3609,
USA J. Conrad Department of Physics, Stockholm University, AlbaNova, SE-106
91 Stockholm, Sweden The Oskar Klein Centre for Cosmoparticle Physics,
AlbaNova, SE-106 91 Stockholm, Sweden Royal Swedish Academy of Sciences
Research Fellow, funded by a grant from the K. A. Wallenberg Foundation S.
Cutini Agenzia Spaziale Italiana (ASI) Science Data Center, I-00044 Frascati
(Roma), Italy C. D. Dermer Space Science Division, Naval Research
Laboratory, Washington, DC 20375, USA A. de Angelis Dipartimento di Fisica,
Università di Udine and Istituto Nazionale di Fisica Nucleare, Sezione di
Trieste, Gruppo Collegato di Udine, I-33100 Udine, Italy F. de Palma
Dipartimento di Fisica “M. Merlin” dell’Università e del Politecnico di Bari,
I-70126 Bari, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Bari,
70126 Bari, Italy S. W. Digel W. W. Hansen Experimental Physics Laboratory,
Kavli Institute for Particle Astrophysics and Cosmology, Department of Physics
and SLAC National Accelerator Laboratory, Stanford University, Stanford, CA
94305, USA G. Di Bernardo Istituto Nazionale di Fisica Nucleare, Sezione di
Pisa, I-56127 Pisa, Italy E. do Couto e Silva W. W. Hansen Experimental
Physics Laboratory, Kavli Institute for Particle Astrophysics and Cosmology,
Department of Physics and SLAC National Accelerator Laboratory, Stanford
University, Stanford, CA 94305, USA P. S. Drell W. W. Hansen Experimental
Physics Laboratory, Kavli Institute for Particle Astrophysics and Cosmology,
Department of Physics and SLAC National Accelerator Laboratory, Stanford
University, Stanford, CA 94305, USA A. Drlica-Wagner W. W. Hansen
Experimental Physics Laboratory, Kavli Institute for Particle Astrophysics and
Cosmology, Department of Physics and SLAC National Accelerator Laboratory,
Stanford University, Stanford, CA 94305, USA R. Dubois W. W. Hansen
Experimental Physics Laboratory, Kavli Institute for Particle Astrophysics and
Cosmology, Department of Physics and SLAC National Accelerator Laboratory,
Stanford University, Stanford, CA 94305, USA D. Dumora Université de
Bordeaux, Centre d’Études Nucléaires Bordeaux Gradignan, UMR 5797, Gradignan,
33175, France CNRS/IN2P3, Centre d’Études Nucléaires Bordeaux Gradignan, UMR
5797, Gradignan, 33175, France C. Farnier Laboratoire de Physique Théorique
et Astroparticules, Université Montpellier 2, CNRS/IN2P3, Montpellier, France
C. Favuzzi Dipartimento di Fisica “M. Merlin” dell’Università e del
Politecnico di Bari, I-70126 Bari, Italy Istituto Nazionale di Fisica
Nucleare, Sezione di Bari, 70126 Bari, Italy S. J. Fegan Laboratoire
Leprince-Ringuet, École polytechnique, CNRS/IN2P3, Palaiseau, France W. B.
Focke W. W. Hansen Experimental Physics Laboratory, Kavli Institute for
Particle Astrophysics and Cosmology, Department of Physics and SLAC National
Accelerator Laboratory, Stanford University, Stanford, CA 94305, USA P.
Fortin Laboratoire Leprince-Ringuet, École polytechnique, CNRS/IN2P3,
Palaiseau, France M. Frailis Dipartimento di Fisica, Università di Udine and
Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, Gruppo Collegato di
Udine, I-33100 Udine, Italy Y. Fukazawa Department of Physical Sciences,
Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8526, Japan S. Funk
W. W. Hansen Experimental Physics Laboratory, Kavli Institute for Particle
Astrophysics and Cosmology, Department of Physics and SLAC National
Accelerator Laboratory, Stanford University, Stanford, CA 94305, USA P. Fusco
Dipartimento di Fisica “M. Merlin” dell’Università e del Politecnico di Bari,
I-70126 Bari, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Bari,
70126 Bari, Italy D. Gaggero Istituto Nazionale di Fisica Nucleare, Sezione
di Pisa, I-56127 Pisa, Italy F. Gargano Istituto Nazionale di Fisica
Nucleare, Sezione di Bari, 70126 Bari, Italy D. Gasparrini Agenzia Spaziale
Italiana (ASI) Science Data Center, I-00044 Frascati (Roma), Italy N. Gehrels
NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA Department of
Astronomy and Astrophysics, Pennsylvania State University, University Park, PA
16802, USA Department of Physics and Department of Astronomy, University of
Maryland, College Park, MD 20742, USA S. Germani Istituto Nazionale di
Fisica Nucleare, Sezione di Perugia, I-06123 Perugia, Italy Dipartimento di
Fisica, Università degli Studi di Perugia, I-06123 Perugia, Italy B. Giebels
Laboratoire Leprince-Ringuet, École polytechnique, CNRS/IN2P3, Palaiseau,
France N. Giglietto Dipartimento di Fisica “M. Merlin” dell’Università e del
Politecnico di Bari, I-70126 Bari, Italy Istituto Nazionale di Fisica
Nucleare, Sezione di Bari, 70126 Bari, Italy P. Giommi Agenzia Spaziale
Italiana (ASI) Science Data Center, I-00044 Frascati (Roma), Italy F.
Giordano Dipartimento di Fisica “M. Merlin” dell’Università e del Politecnico
di Bari, I-70126 Bari, Italy Istituto Nazionale di Fisica Nucleare, Sezione
di Bari, 70126 Bari, Italy T. Glanzman W. W. Hansen Experimental Physics
Laboratory, Kavli Institute for Particle Astrophysics and Cosmology,
Department of Physics and SLAC National Accelerator Laboratory, Stanford
University, Stanford, CA 94305, USA G. Godfrey W. W. Hansen Experimental
Physics Laboratory, Kavli Institute for Particle Astrophysics and Cosmology,
Department of Physics and SLAC National Accelerator Laboratory, Stanford
University, Stanford, CA 94305, USA I. A. Grenier Laboratoire AIM, CEA-
IRFU/CNRS/Université Paris Diderot, Service d’Astrophysique, CEA Saclay, 91191
Gif sur Yvette, France M.-H. Grondin Université de Bordeaux, Centre d’Études
Nucléaires Bordeaux Gradignan, UMR 5797, Gradignan, 33175, France CNRS/IN2P3,
Centre d’Études Nucléaires Bordeaux Gradignan, UMR 5797, Gradignan, 33175,
France J. E. Grove Space Science Division, Naval Research Laboratory,
Washington, DC 20375, USA L. Guillemot Max-Planck-Institut für
Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany S. Guiriec Center for
Space Plasma and Aeronomic Research (CSPAR), University of Alabama in
Huntsville, Huntsville, AL 35899, USA M. Gustafsson Dipartimento di Fisica
“G. Galilei”, Università di Padova, I-35131 Padova, Italy Istituto Nazionale
di Fisica Nucleare, Sezione di Padova, I-35131 Padova, Italy Y. Hanabata
Department of Physical Sciences, Hiroshima University, Higashi-Hiroshima,
Hiroshima 739-8526, Japan A. K. Harding NASA Goddard Space Flight Center,
Greenbelt, MD 20771, USA M. Hayashida W. W. Hansen Experimental Physics
Laboratory, Kavli Institute for Particle Astrophysics and Cosmology,
Department of Physics and SLAC National Accelerator Laboratory, Stanford
University, Stanford, CA 94305, USA R. E. Hughes Department of Physics,
Center for Cosmology and Astro-Particle Physics, The Ohio State University,
Columbus, OH 43210, USA R. Itoh Department of Physical Sciences, Hiroshima
University, Higashi-Hiroshima, Hiroshima 739-8526, Japan M. S. Jackson The
Oskar Klein Centre for Cosmoparticle Physics, AlbaNova, SE-106 91 Stockholm,
Sweden Department of Physics, Royal Institute of Technology (KTH), AlbaNova,
SE-106 91 Stockholm, Sweden G. Jóhannesson W. W. Hansen Experimental Physics
Laboratory, Kavli Institute for Particle Astrophysics and Cosmology,
Department of Physics and SLAC National Accelerator Laboratory, Stanford
University, Stanford, CA 94305, USA A. S. Johnson W. W. Hansen Experimental
Physics Laboratory, Kavli Institute for Particle Astrophysics and Cosmology,
Department of Physics and SLAC National Accelerator Laboratory, Stanford
University, Stanford, CA 94305, USA R. P. Johnson Santa Cruz Institute for
Particle Physics, Department of Physics and Department of Astronomy and
Astrophysics, University of California at Santa Cruz, Santa Cruz, CA 95064,
USA T. J. Johnson NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
Department of Physics and Department of Astronomy, University of Maryland,
College Park, MD 20742, USA W. N. Johnson Space Science Division, Naval
Research Laboratory, Washington, DC 20375, USA T. Kamae W. W. Hansen
Experimental Physics Laboratory, Kavli Institute for Particle Astrophysics and
Cosmology, Department of Physics and SLAC National Accelerator Laboratory,
Stanford University, Stanford, CA 94305, USA H. Katagiri Department of
Physical Sciences, Hiroshima University, Higashi-Hiroshima, Hiroshima
739-8526, Japan J. Kataoka Waseda University, 1-104 Totsukamachi, Shinjuku-
ku, Tokyo, 169-8050, Japan N. Kawai Department of Physics, Tokyo Institute
of Technology, Meguro City, Tokyo 152-8551, Japan Cosmic Radiation
Laboratory, Institute of Physical and Chemical Research (RIKEN), Wako, Saitama
351-0198, Japan M. Kerr Department of Physics, University of Washington,
Seattle, WA 98195-1560, USA J. Knödlseder Centre d’Étude Spatiale des
Rayonnements, CNRS/UPS, BP 44346, F-30128 Toulouse Cedex 4, France M. L.
Kocian W. W. Hansen Experimental Physics Laboratory, Kavli Institute for
Particle Astrophysics and Cosmology, Department of Physics and SLAC National
Accelerator Laboratory, Stanford University, Stanford, CA 94305, USA F. Kuehn
Department of Physics, Center for Cosmology and Astro-Particle Physics, The
Ohio State University, Columbus, OH 43210, USA M. Kuss Istituto Nazionale di
Fisica Nucleare, Sezione di Pisa, I-56127 Pisa, Italy J. Lande W. W. Hansen
Experimental Physics Laboratory, Kavli Institute for Particle Astrophysics and
Cosmology, Department of Physics and SLAC National Accelerator Laboratory,
Stanford University, Stanford, CA 94305, USA L. Latronico Istituto Nazionale
di Fisica Nucleare, Sezione di Pisa, I-56127 Pisa, Italy M. Lemoine-Goumard
Université de Bordeaux, Centre d’Études Nucléaires Bordeaux Gradignan, UMR
5797, Gradignan, 33175, France CNRS/IN2P3, Centre d’Études Nucléaires
Bordeaux Gradignan, UMR 5797, Gradignan, 33175, France F. Longo Istituto
Nazionale di Fisica Nucleare, Sezione di Trieste, I-34127 Trieste, Italy
Dipartimento di Fisica, Università di Trieste, I-34127 Trieste, Italy F.
Loparco Dipartimento di Fisica “M. Merlin” dell’Università e del Politecnico
di Bari, I-70126 Bari, Italy Istituto Nazionale di Fisica Nucleare, Sezione
di Bari, 70126 Bari, Italy B. Lott Université de Bordeaux, Centre d’Études
Nucléaires Bordeaux Gradignan, UMR 5797, Gradignan, 33175, France CNRS/IN2P3,
Centre d’Études Nucléaires Bordeaux Gradignan, UMR 5797, Gradignan, 33175,
France M. N. Lovellette Space Science Division, Naval Research Laboratory,
Washington, DC 20375, USA P. Lubrano Istituto Nazionale di Fisica Nucleare,
Sezione di Perugia, I-06123 Perugia, Italy Dipartimento di Fisica, Università
degli Studi di Perugia, I-06123 Perugia, Italy G. M. Madejski W. W. Hansen
Experimental Physics Laboratory, Kavli Institute for Particle Astrophysics and
Cosmology, Department of Physics and SLAC National Accelerator Laboratory,
Stanford University, Stanford, CA 94305, USA A. Makeev Space Science
Division, Naval Research Laboratory, Washington, DC 20375, USA George Mason
University, Fairfax, VA 22030, USA M. N. Mazziotta Istituto Nazionale di
Fisica Nucleare, Sezione di Bari, 70126 Bari, Italy W. McConville NASA
Goddard Space Flight Center, Greenbelt, MD 20771, USA Department of Physics
and Department of Astronomy, University of Maryland, College Park, MD 20742,
USA J. E. McEnery NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
Department of Physics and Department of Astronomy, University of Maryland,
College Park, MD 20742, USA C. Meurer Department of Physics, Stockholm
University, AlbaNova, SE-106 91 Stockholm, Sweden The Oskar Klein Centre for
Cosmoparticle Physics, AlbaNova, SE-106 91 Stockholm, Sweden P. F. Michelson
W. W. Hansen Experimental Physics Laboratory, Kavli Institute for Particle
Astrophysics and Cosmology, Department of Physics and SLAC National
Accelerator Laboratory, Stanford University, Stanford, CA 94305, USA W.
Mitthumsiri W. W. Hansen Experimental Physics Laboratory, Kavli Institute for
Particle Astrophysics and Cosmology, Department of Physics and SLAC National
Accelerator Laboratory, Stanford University, Stanford, CA 94305, USA T.
Mizuno Department of Physical Sciences, Hiroshima University, Higashi-
Hiroshima, Hiroshima 739-8526, Japan A. A. Moiseev Center for Research and
Exploration in Space Science and Technology (CRESST) and NASA Goddard Space
Flight Center, Greenbelt, MD 20771, USA Department of Physics and Department
of Astronomy, University of Maryland, College Park, MD 20742, USA C. Monte
Dipartimento di Fisica “M. Merlin” dell’Università e del Politecnico di Bari,
I-70126 Bari, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Bari,
70126 Bari, Italy M. E. Monzani W. W. Hansen Experimental Physics
Laboratory, Kavli Institute for Particle Astrophysics and Cosmology,
Department of Physics and SLAC National Accelerator Laboratory, Stanford
University, Stanford, CA 94305, USA A. Morselli Istituto Nazionale di Fisica
Nucleare, Sezione di Roma “Tor Vergata”, I-00133 Roma, Italy I. V. Moskalenko
W. W. Hansen Experimental Physics Laboratory, Kavli Institute for Particle
Astrophysics and Cosmology, Department of Physics and SLAC National
Accelerator Laboratory, Stanford University, Stanford, CA 94305, USA S.
Murgia W. W. Hansen Experimental Physics Laboratory, Kavli Institute for
Particle Astrophysics and Cosmology, Department of Physics and SLAC National
Accelerator Laboratory, Stanford University, Stanford, CA 94305, USA P. L.
Nolan W. W. Hansen Experimental Physics Laboratory, Kavli Institute for
Particle Astrophysics and Cosmology, Department of Physics and SLAC National
Accelerator Laboratory, Stanford University, Stanford, CA 94305, USA J. P.
Norris Department of Physics and Astronomy, University of Denver, Denver, CO
80208, USA E. Nuss Laboratoire de Physique Théorique et Astroparticules,
Université Montpellier 2, CNRS/IN2P3, Montpellier, France T. Ohsugi
Department of Physical Sciences, Hiroshima University, Higashi-Hiroshima,
Hiroshima 739-8526, Japan N. Omodei Istituto Nazionale di Fisica Nucleare,
Sezione di Pisa, I-56127 Pisa, Italy E. Orlando Max-Planck Institut für
extraterrestrische Physik, 85748 Garching, Germany J. F. Ormes Department of
Physics and Astronomy, University of Denver, Denver, CO 80208, USA D. Paneque
W. W. Hansen Experimental Physics Laboratory, Kavli Institute for Particle
Astrophysics and Cosmology, Department of Physics and SLAC National
Accelerator Laboratory, Stanford University, Stanford, CA 94305, USA J. H.
Panetta W. W. Hansen Experimental Physics Laboratory, Kavli Institute for
Particle Astrophysics and Cosmology, Department of Physics and SLAC National
Accelerator Laboratory, Stanford University, Stanford, CA 94305, USA D.
Parent Université de Bordeaux, Centre d’Études Nucléaires Bordeaux Gradignan,
UMR 5797, Gradignan, 33175, France CNRS/IN2P3, Centre d’Études Nucléaires
Bordeaux Gradignan, UMR 5797, Gradignan, 33175, France V. Pelassa
Laboratoire de Physique Théorique et Astroparticules, Université Montpellier
2, CNRS/IN2P3, Montpellier, France M. Pepe Istituto Nazionale di Fisica
Nucleare, Sezione di Perugia, I-06123 Perugia, Italy Dipartimento di Fisica,
Università degli Studi di Perugia, I-06123 Perugia, Italy M. Pesce-Rollins
Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, I-56127 Pisa, Italy
F. Piron Laboratoire de Physique Théorique et Astroparticules, Université
Montpellier 2, CNRS/IN2P3, Montpellier, France T. A. Porter
tporter@scipp.ucsc.edu Santa Cruz Institute for Particle Physics, Department
of Physics and Department of Astronomy and Astrophysics, University of
California at Santa Cruz, Santa Cruz, CA 95064, USA S. Rainò Dipartimento di
Fisica “M. Merlin” dell’Università e del Politecnico di Bari, I-70126 Bari,
Italy Istituto Nazionale di Fisica Nucleare, Sezione di Bari, 70126 Bari,
Italy R. Rando Istituto Nazionale di Fisica Nucleare, Sezione di Padova,
I-35131 Padova, Italy Dipartimento di Fisica “G. Galilei”, Università di
Padova, I-35131 Padova, Italy M. Razzano Istituto Nazionale di Fisica
Nucleare, Sezione di Pisa, I-56127 Pisa, Italy A. Reimer Institut für Astro-
und Teilchenphysik and Institut für Theoretische Physik, Leopold-Franzens-
Universität Innsbruck, A-6020 Innsbruck, Austria W. W. Hansen Experimental
Physics Laboratory, Kavli Institute for Particle Astrophysics and Cosmology,
Department of Physics and SLAC National Accelerator Laboratory, Stanford
University, Stanford, CA 94305, USA O. Reimer Institut für Astro- und
Teilchenphysik and Institut für Theoretische Physik, Leopold-Franzens-
Universität Innsbruck, A-6020 Innsbruck, Austria W. W. Hansen Experimental
Physics Laboratory, Kavli Institute for Particle Astrophysics and Cosmology,
Department of Physics and SLAC National Accelerator Laboratory, Stanford
University, Stanford, CA 94305, USA T. Reposeur Université de Bordeaux,
Centre d’Études Nucléaires Bordeaux Gradignan, UMR 5797, Gradignan, 33175,
France CNRS/IN2P3, Centre d’Études Nucléaires Bordeaux Gradignan, UMR 5797,
Gradignan, 33175, France S. Ritz Santa Cruz Institute for Particle Physics,
Department of Physics and Department of Astronomy and Astrophysics, University
of California at Santa Cruz, Santa Cruz, CA 95064, USA L. S. Rochester W. W.
Hansen Experimental Physics Laboratory, Kavli Institute for Particle
Astrophysics and Cosmology, Department of Physics and SLAC National
Accelerator Laboratory, Stanford University, Stanford, CA 94305, USA A. Y.
Rodriguez Institut de Ciencies de l’Espai (IEEC-CSIC), Campus UAB, 08193
Barcelona, Spain M. Roth Department of Physics, University of Washington,
Seattle, WA 98195-1560, USA F. Ryde Department of Physics, Royal Institute
of Technology (KTH), AlbaNova, SE-106 91 Stockholm, Sweden The Oskar Klein
Centre for Cosmoparticle Physics, AlbaNova, SE-106 91 Stockholm, Sweden H.
F.-W. Sadrozinski Santa Cruz Institute for Particle Physics, Department of
Physics and Department of Astronomy and Astrophysics, University of California
at Santa Cruz, Santa Cruz, CA 95064, USA D. Sanchez Laboratoire Leprince-
Ringuet, École polytechnique, CNRS/IN2P3, Palaiseau, France A. Sander
Department of Physics, Center for Cosmology and Astro-Particle Physics, The
Ohio State University, Columbus, OH 43210, USA P. M. Saz Parkinson Santa
Cruz Institute for Particle Physics, Department of Physics and Department of
Astronomy and Astrophysics, University of California at Santa Cruz, Santa
Cruz, CA 95064, USA J. D. Scargle Space Sciences Division, NASA Ames
Research Center, Moffett Field, CA 94035-1000, USA A. Sellerholm
sellerholm@physto.se Department of Physics, Stockholm University, AlbaNova,
SE-106 91 Stockholm, Sweden The Oskar Klein Centre for Cosmoparticle Physics,
AlbaNova, SE-106 91 Stockholm, Sweden C. Sgrò Istituto Nazionale di Fisica
Nucleare, Sezione di Pisa, I-56127 Pisa, Italy M. S. Shaw W. W. Hansen
Experimental Physics Laboratory, Kavli Institute for Particle Astrophysics and
Cosmology, Department of Physics and SLAC National Accelerator Laboratory,
Stanford University, Stanford, CA 94305, USA E. J. Siskind NYCB Real-Time
Computing Inc., Lattingtown, NY 11560-1025, USA D. A. Smith Université de
Bordeaux, Centre d’Études Nucléaires Bordeaux Gradignan, UMR 5797, Gradignan,
33175, France CNRS/IN2P3, Centre d’Études Nucléaires Bordeaux Gradignan, UMR
5797, Gradignan, 33175, France P. D. Smith Department of Physics, Center for
Cosmology and Astro-Particle Physics, The Ohio State University, Columbus, OH
43210, USA G. Spandre Istituto Nazionale di Fisica Nucleare, Sezione di
Pisa, I-56127 Pisa, Italy P. Spinelli Dipartimento di Fisica “M. Merlin”
dell’Università e del Politecnico di Bari, I-70126 Bari, Italy Istituto
Nazionale di Fisica Nucleare, Sezione di Bari, 70126 Bari, Italy J.-L. Starck
Laboratoire AIM, CEA-IRFU/CNRS/Université Paris Diderot, Service
d’Astrophysique, CEA Saclay, 91191 Gif sur Yvette, France M. S. Strickman
Space Science Division, Naval Research Laboratory, Washington, DC 20375, USA
A. W. Strong Max-Planck Institut für extraterrestrische Physik, 85748
Garching, Germany D. J. Suson Department of Chemistry and Physics, Purdue
University Calumet, Hammond, IN 46323-2094, USA H. Tajima W. W. Hansen
Experimental Physics Laboratory, Kavli Institute for Particle Astrophysics and
Cosmology, Department of Physics and SLAC National Accelerator Laboratory,
Stanford University, Stanford, CA 94305, USA H. Takahashi Department of
Physical Sciences, Hiroshima University, Higashi-Hiroshima, Hiroshima
739-8526, Japan T. Takahashi Institute of Space and Astronautical Science,
JAXA, 3-1-1 Yoshinodai, Sagamihara, Kanagawa 229-8510, Japan T. Tanaka W. W.
Hansen Experimental Physics Laboratory, Kavli Institute for Particle
Astrophysics and Cosmology, Department of Physics and SLAC National
Accelerator Laboratory, Stanford University, Stanford, CA 94305, USA J. B.
Thayer W. W. Hansen Experimental Physics Laboratory, Kavli Institute for
Particle Astrophysics and Cosmology, Department of Physics and SLAC National
Accelerator Laboratory, Stanford University, Stanford, CA 94305, USA J. G.
Thayer W. W. Hansen Experimental Physics Laboratory, Kavli Institute for
Particle Astrophysics and Cosmology, Department of Physics and SLAC National
Accelerator Laboratory, Stanford University, Stanford, CA 94305, USA D. J.
Thompson NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA L.
Tibaldo Istituto Nazionale di Fisica Nucleare, Sezione di Padova, I-35131
Padova, Italy Dipartimento di Fisica “G. Galilei”, Università di Padova,
I-35131 Padova, Italy Laboratoire AIM, CEA-IRFU/CNRS/Université Paris
Diderot, Service d’Astrophysique, CEA Saclay, 91191 Gif sur Yvette, France D.
F. Torres Institució Catalana de Recerca i Estudis Avançats (ICREA),
Barcelona, Spain Institut de Ciencies de l’Espai (IEEC-CSIC), Campus UAB,
08193 Barcelona, Spain G. Tosti Istituto Nazionale di Fisica Nucleare,
Sezione di Perugia, I-06123 Perugia, Italy Dipartimento di Fisica, Università
degli Studi di Perugia, I-06123 Perugia, Italy A. Tramacere W. W. Hansen
Experimental Physics Laboratory, Kavli Institute for Particle Astrophysics and
Cosmology, Department of Physics and SLAC National Accelerator Laboratory,
Stanford University, Stanford, CA 94305, USA Consorzio Interuniversitario per
la Fisica Spaziale (CIFS), I-10133 Torino, Italy Y. Uchiyama W. W. Hansen
Experimental Physics Laboratory, Kavli Institute for Particle Astrophysics and
Cosmology, Department of Physics and SLAC National Accelerator Laboratory,
Stanford University, Stanford, CA 94305, USA T. L. Usher W. W. Hansen
Experimental Physics Laboratory, Kavli Institute for Particle Astrophysics and
Cosmology, Department of Physics and SLAC National Accelerator Laboratory,
Stanford University, Stanford, CA 94305, USA V. Vasileiou Center for
Research and Exploration in Space Science and Technology (CRESST) and NASA
Goddard Space Flight Center, Greenbelt, MD 20771, USA Department of Physics
and Center for Space Sciences and Technology, University of Maryland Baltimore
County, Baltimore, MD 21250, USA N. Vilchez Centre d’Étude Spatiale des
Rayonnements, CNRS/UPS, BP 44346, F-30128 Toulouse Cedex 4, France V. Vitale
Istituto Nazionale di Fisica Nucleare, Sezione di Roma “Tor Vergata”, I-00133
Roma, Italy Dipartimento di Fisica, Università di Roma “Tor Vergata”, I-00133
Roma, Italy A. P. Waite W. W. Hansen Experimental Physics Laboratory, Kavli
Institute for Particle Astrophysics and Cosmology, Department of Physics and
SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94305,
USA P. Wang W. W. Hansen Experimental Physics Laboratory, Kavli Institute
for Particle Astrophysics and Cosmology, Department of Physics and SLAC
National Accelerator Laboratory, Stanford University, Stanford, CA 94305, USA
B. L. Winer Department of Physics, Center for Cosmology and Astro-Particle
Physics, The Ohio State University, Columbus, OH 43210, USA K. S. Wood Space
Science Division, Naval Research Laboratory, Washington, DC 20375, USA T.
Ylinen Department of Physics, Royal Institute of Technology (KTH), AlbaNova,
SE-106 91 Stockholm, Sweden School of Pure and Applied Natural Sciences,
University of Kalmar, SE-391 82 Kalmar, Sweden The Oskar Klein Centre for
Cosmoparticle Physics, AlbaNova, SE-106 91 Stockholm, Sweden M. Ziegler
Santa Cruz Institute for Particle Physics, Department of Physics and
Department of Astronomy and Astrophysics, University of California at Santa
Cruz, Santa Cruz, CA 95064, USA
###### Abstract
We report on the first Fermi Large Area Telescope (LAT) measurements of the
so-called “extragalactic” diffuse $\gamma$-ray emission (EGB). This component
of the diffuse $\gamma$-ray emission is generally considered to have an
isotropic or nearly isotropic distribution on the sky with diverse
contributions discussed in the literature. The derivation of the EGB is based
on detailed modelling of the bright foreground diffuse Galactic $\gamma$-ray
emission (DGE), the detected LAT sources and the solar $\gamma$-ray emission.
We find the spectrum of the EGB is consistent with a power law with
differential spectral index $\gamma=2.41\pm 0.05$ and intensity, $I(>100\,{\rm
MeV})=(1.03\pm 0.17)\times 10^{-5}$ cm-2 s-1 sr-1, where the error is
systematics dominated. Our EGB spectrum is featureless, less intense, and
softer than that derived from EGRET data.
###### pacs:
95.30.Cq,95.55.Ka,95.85.Pw,96.50.sb,98.70.Sa
Introduction: The high-energy diffuse $\gamma$-ray emission is dominated by
$\gamma$-rays produced by cosmic rays (CR) interacting with the Galactic
interstellar gas and radiation fields, the so-called diffuse Galactic emission
(DGE). A much fainter component, commonly designated as “extragalactic
$\gamma$-ray background” (EGB), was first detected against the bright DGE
foreground by the SAS-2 satellite Fichtel et al. (1978) and later confirmed by
analysis of the EGRET data Sreekumar et al. (1998). The EGB by definition has
an isotropic sky distribution and is considered by many to be the
superposition of contributions from unresolved extragalactic sources including
active galactic nuclei, starburst galaxies and $\gamma$-ray bursts (Dermer
(2007) and references therein) and truly-diffuse emission processes. These
diffuse processes include the possible signatures of large-scale structure
formation Loeb and Waxman (2000), emission produced by the interactions of
ultra-high-energy CRs with relic photons Kalashev et al. (2009), the
annihilation or decay of dark matter, and many other processes (e.g., Dermer
(2007) and references therein). However, the diffuse $\gamma$-ray emission
from inverse Compton (IC) scattering by an extended Galactic halo of CR
electrons could also be attributed to such a component if the size of the halo
is large enough (i.e., $\sim 25$ kpc) Keshet et al. (2004). In addition,
$\gamma$-ray emission from CRs interacting in populations of small solar
system bodies Moskalenko and Porter (2009) and the all-sky contribution of IC
scattering of solar photons with local CRs can provide contributions
Moskalenko et al. (2006, 2007); Orlando and Strong (2008). Hence, an
extragalactic origin for such a component is not clear, even though we will
use the abbreviation ‘EGB’ throughout this paper.
In this paper, we present analysis and first results for the EGB derived from
the Fermi Large Area Telescope (LAT) Atwood et al. (2009) data. Our analysis
uses data from the initial 10 months of the science phase of the mission.
Essential to this study is an event-level data selection with a higher level
of background rejection than the standard LAT data selections, and
improvements to the instrument simulation. These have been made following
extensive on-orbit studies of the LAT performance and of charged particle
backgrounds. Together, these improvements over the pre-launch modelling and
background rejection allow a robust derivation of the spectrum of the EGB that
is not possible with the standard low-background event selection.
Data selection: The LAT is a pair-conversion telescope with a precision
tracker and segmented calorimeter, each consisting of a $4\times 4$ array of
16 modules, a segmented anti-coincidence detector (ACD) that covers the
tracker array, and a programmable trigger and data acquisition system. Details
of the on-board and ground data processing are given in Atwood et al. (2009).
The LAT ground processing makes use of the pre-launch background rejection
scheme described in Atwood et al. (2009). The standard low-background event
selection resulting from this multivariate analysis, termed “diffuse” class,
has a Monte Carlo predicted background rate of $\sim 0.1$ Hz when integrated
over the full instrument acceptance $>100$ MeV. On-orbit investigations of the
residual background of misclassified particles in the diffuse event selection
indicated a higher level than predicted from pre-launch modelling. To reduce
the residual particle background further, we developed an event selection
comprised of the following four criteria in addition to the standard diffuse
event classification: 1) events are required to have a multivariate-analysis
assigned $\gamma$-ray probability that is higher than the standard diffuse
selection, with the required probability an increasing function with energy
instead of a constant value as for diffuse class events; 2) the distance of
extrapolated reconstructed particle tracks from the corners of the ACD must be
higher than a set minimum value to remove particles that enter the LAT in a
region where the ACD has a lower than average efficiency; 3) the average
charge deposit in the silicon layers of the tracker is required to be small;
4) the reconstructed transverse shower size of events in the calorimeter is
within a size range expected for electromagnetic showers. The first two
criteria assist in reducing the overall level of CR background. The second two
criteria provide an additional veto against hadronic showers and heavy ions
that leak through the standard diffuse event classification. In addition to
these analysis cuts the particle background modelling has been updated to be
closer to the observed on-orbit charged particle rates. Furthermore, the
instrument simulation now takes into account pile-up and accidental
coincidence effects in the detector subsystems that were not considered in the
definition of the pre-launch instrument response functions (IRFs) Rando et al.
(2009).
Figure 1: Comparison of (a) LAT on-axis effective area and (b) orbit-averaged
CR background rate integrated over the FOV between the enhanced low-background
event selection and the standard “diffuse” event selection. Figure 2:
Comparison of expected and measured orbit-averaged event rates for two CR
dominated data samples.
Figure 1a shows the on-axis effective area (${\rm A_{eff}}$) for our enhanced
low-background and standard diffuse selections, respectively. The ${\rm
A_{eff}}$ for the enhanced selection is reduced for energies $>300$ MeV with a
peak value $\sim 0.74$ m2 compared to $\sim 0.84$ m2 for diffuse class events.
The ${\rm A_{eff}}$ systematic uncertainties for our enhanced low-background
selection are of the same magnitude as those for the diffuse class events,
evaluated by comparing the efficiencies of analysis cuts for data and
simulation of observations of Vela: 10% below 100 MeV, decreasing to 5% at 560
MeV, and increasing to 20% at 10 GeV and above. Figure 1b shows the orbit-
averaged residual background rate of our enhanced low-background and standard
diffuse selection, respectively, determined from our improved simulation. With
our enhanced event selection, the predicted background rejection is improved
by a factor $1.3$–$10$.
We estimate the uncertainty of the residual CR background in our simulation by
comparing two CR background dominated LAT data samples to the predictions. The
first data sample (A) contains all events passing the on-board filters,
corresponding to a minimal background rejection level. The second data sample
(B) contains events that pass the “source” classification in the standard
analysis Atwood et al. (2009) but fail to pass the more stringent diffuse
selection. This second sample corresponds to a very high level of background
rejection but is still dominated by charged particles compared to the standard
diffuse selection, particularly at intermediate and high Galactic latitudes.
These particles are from the extreme tails of the CR distributions that are
difficult to reject. To both samples we apply the selection cuts 3) and 4),
described above, to remove the heavy ion and hadronic shower backgrounds
observed on-orbit that are not modeled with sufficient accuracy in the
simulation. In addition, to reduce the $\gamma$-ray fraction in both samples
only events from Galactic latitudes $|b|\geq 45^{\circ}$ are used. The
remaining $\gamma$-ray “contamination” is negligible for sample A due to the
overwhelming CR rate. For sample B, the contamination is less than 10% above 1
GeV, but is almost 30% at 200 MeV, even at high Galactic latitudes. To remove
this $\gamma$-ray contamination from sample B we use the intensity maps from
our fits to the data (see below), combined with the IRFs corresponding to
sample B, to determine the expected $\gamma$-ray rate. This is subtracted from
the observed rate of sample B events.
Figure 2 compares the orbit-averaged event rates measured by the LAT and
predicted by our simulation for datasets A and B. At the minimal background
rejection level represented by sample A we find agreement within $\pm$ 20%.
This shows that the bulk of the remaining CR background is well described by
the simulation after removing the particular class of heavy ion and hadronic
shower events mentioned above. For sample B, the agreement is within
$+50$%/$-30$%, indicating the uncertainty in the description of the extreme
tails of the CR distributions. As these tails are responsible for the limiting
background in the present analysis, we adopt the results for sample B as
representative of the uncertainty on the expected residual CR background.
Analysis: We use data taken in the nominal “scanning” mode from the
commencement of scientific operations in mid-August 2008 to mid-June 2009. The
data were prepared using the LAT Science Tools package, which is available
from the Fermi Science Support Center111http://fermi.gsfc.nasa.gov/ssc/.
Events satisfying our enhanced low-background event selection, coming from
zenith angles $<100^{\circ}$ (to greatly reduce the contribution by Earth
albedo $\gamma$-rays) and incidence angles within $65^{\circ}$ of the LAT
z-axis (the LAT field-of-view) were used. This leaves 19 Ms of total
observation time in the data set. The energy-dependent exposure was calculated
using the IRFs corresponding to our enhanced low-background event selection
described above.
The photon counts and exposure were further processed using the GaDGET
package, part of a suite of tools we have developed to analyse the DGE
Ackermann et al. (2009). Gamma-ray skymaps were generated using a HEALPix
Górski et al. (2005) isopixelisation scheme at order 6 with 9 independent
energy bins from 200 MeV to 102 GeV with GaDGET used to simultaneously fit a
DGE model, solar $\gamma$-ray emission, and sources (described below) to the
resulting skymaps. We only consider the Galactic latitude range
$|b|>10^{\circ}$ in this analysis where the DGE is more than an order of
magnitude weaker than in the Galactic plane.
The model used for the large-scale DGE is based on the GALPROP
code222http://galprop.stanford.edu, model id 77XvMM7A.. Recent improvements
include use of the formalism and corresponding code for pion production in
$pp$-interactions by Kamae et al. (2006); Kelner et al. (2006), a complete
recalculation of the ISRF Porter et al. (2008), updated Hi and H2 gas maps,
including corrections to the total gas column density derived from dust
reddening maps Grenier et al. (2005) an improved line-of-sight integration
routine, and the addition of information from our ongoing studies of the DGE
with the LAT Abdo et al. (2009a, b). Cosmic-ray intensities and spectra are
calculated using a diffusive reacceleration CR transport model for a nominal
halo size of 4 kpc, with a rigidity dependent diffusion coefficient that is
consistent with available CR data for the B/C and 10Be/9Be ratios,
respectively. We also consider bounding halo sizes 2 kpc and 10 kpc, with
corresponding self-consistently derived diffusion coefficients, since the size
of the CR halo is one of the principal uncertainties in the DGE foreground.
The injection spectra for CR protons and primary electrons are chosen to
reproduce after propagation the locally measured spectra, including the
recently reported Fermi LAT CR electron spectrum Abdo et al. (2009). Gamma-ray
emissivities are calculated using the propagated CR spectra and intensities
folded with the appropriate target distributions included in the GALPROP code:
Hi, H2, and Hii gas distributions for $\pi^{0}$-decay and bremsstrahlung, and
the ISRF for IC scattering. Gamma-ray intensity skymaps are obtained by direct
line-of-sight integration of the calculated $\gamma$-ray emissivities.
For the dominant high latitude components, bremssstrahlung and $\pi^{0}$-decay
emission from Hi and Hii in the local Galaxy ($7.5\,{\rm kpc}<R<9.5\,{\rm
kpc}$) and IC emission, the intensities are fit to the LAT data via scale
factors. We use the GALPROP skymaps as templates with the component
normalisations per energy bin as fit parameters. The sub-dominant high-
latitude DGE components, bremsstrahlung and $\pi^{0}$-decay from H2, as well
as Hi and Hii outside the local region defined above, are taken from GALPROP
predictions and do not vary in the fit. All sources with test statistic above
200 (i.e., larger than $\sim 14\sigma$) found in the internal LAT 9-month
source list are included with the flux per energy band per source as a fit
parameter. Weaker sources are included with fluxes derived from the LAT
catalogue analysis. In addition templates for the intensity of the
$\gamma$-ray emission from CRs interacting in the solar disk and radiation
field Moskalenko et al. (2006, 2007); Orlando and Strong (2008) that take into
account the relative exposure as the Sun transits the celestial sphere are
included with their normalisations as fit parameters.
Figure 3: LAT measured $\gamma$-ray intensity with fit results for $|b|\geq 10^{\circ}$ including statistical and systematic errors. Fit results by component are given in Table 1. Note LAT data are dominated by systematic uncertainties for the energy range shown in the figure. Table 1: Fit results and uncertainties for the EGB and other components for $|b|\geq 10^{\circ}$. | Intensity integrated over energy band (cm-2 s-1 sr-1)
---|---
Energy in GeV | 0.2–0.4 | 0.4–0.8 | 0.8–1.6 | 1.6–3.2 | 3.2–6.4 | 6.4–12.8 | 12.8–25.6 | 25.6–51.2 | 51.2–102.4
Scale factor | $\times\;10^{-6}$ | $\times\;10^{-7}$ | $\times\;10^{-7}$ | $\times\;10^{-8}$ | $\times\;10^{-8}$ | $\times\;10^{-9}$ | $\times\;10^{-9}$ | $\times\;10^{-9}$ | $\times\;10^{-10}$
EGB | $2.4\pm 0.6$ | $9.3\pm 1.8$ | $3.5\pm 0.6$ | $12.7\pm 2.1$ | $5.0\pm 1.0$ | $14.3\pm 4.0$ | $6.3\pm 1.5$ | $2.6\pm 0.7$ | $11.1\pm 2.9$
Galactic diffuse (fit) | $4.9\pm 0.4$ | $25.9\pm 1.8$ | $12.6\pm 1.3$ | $50.7\pm 7.2$ | $17.0\pm 3.0$ | $50.0\pm 10$ | $17.1\pm 3.6$ | $6.1\pm 1.4$ | $19.1\pm 5.2$
Galactic diffuse (model) | 5.0 | 26.0 | 11.5 | 43.3 | 14.7 | 47.9 | 15.7 | 5.2 | 17.0
IC (fit) | $1.5\pm 0.1$ | $6.8\pm 0.5$ | $3.5\pm 0.4$ | $16.1\pm 2.3$ | $6.6\pm 1.2$ | $23.3\pm 4.9$ | $9.3\pm 2.1$ | $3.9\pm 1.0$ | $10.6\pm 3.7$
IC (model) | 1.2 | 5.3 | 2.3 | 9.7 | 4.0 | 16.2 | 6.3 | 2.4 | 8.7
local Hi (fit) | $2.7\pm 0.2$ | $15.4\pm 1.1$ | $7.4\pm 0.8$ | $28.3\pm 4.0$ | $8.3\pm 1.5$ | $20.6\pm 4.2$ | $5.9\pm 1.2$ | $1.6\pm 0.4$ | $7.0\pm 2.2$
local Hi (model) | 3.1 | 17.0 | 7.6 | 27.6 | 8.7 | 26.0 | 7.7 | 2.3 | 6.8
Sources | $0.8\pm 0.1$ | $3.8\pm 0.2$ | $1.7\pm 0.1$ | $7.2\pm 0.8$ | $2.7\pm 0.4$ | $9.0\pm 1.3$ | $3.4\pm 0.5$ | $1.5\pm 0.2$ | $6.3\pm 1.0$
CR background | $1.4\pm 0.6$ | $4.2\pm 1.7$ | $1.0\pm 0.4$ | $2.8\pm 1.2$ | $0.8\pm 0.4$ | $6.3\pm 3.0$ | $1.4\pm 0.8$ | $0.6\pm 0.4$ | $0.9\pm 0.9$
Solar | $0.1\pm 0.01$ | $0.4\pm 0.04$ | $0.2\pm 0.02$ | $1.0\pm 0.2$ | $0.4\pm 0.2$ | $1.7\pm 0.4$ | $0.7\pm 1.6$ | $0.1\pm 0.04$ | $0.8\pm 0.5$
LAT | $9.6\pm 0.8$ | $44.0\pm 3.0$ | $18.8\pm 2.0$ | $72.9\pm 10$ | $25.3\pm 4.5$ | $81.3\pm 16$ | $28.3\pm 5.7$ | $10.6\pm 2.1$ | $37.9\pm 7.7$
| Foreground modeling related uncertainty in cm-2 s-1 sr-1
Hi column density | $+0.1$/$-0.3$ | $+0.1$/$-1.7$ | $+0.1$/$-0.9$ | $+0.1$/$-3.6$ | $+0.1$/$-1.1$ | $+0.1$/$-2.4$ | $+0.1$/$-0.9$ | $+0.1$/$-0.2$ | $+0.1$/$-1.1$
IC + halo size | $+0.1$/$-0.2$ | $+0.1$/$-0.8$ | $+0.1$/$-0.5$ | $+0.1$/$-1.8$ | $+0.1$/$-0.5$ | $+0.1$/$-0.7$ | $+0.3$/$-0.3$ | $+0.4$/$-0.1$ | $+2.9$/$-0.5$
CR propagation model | $+0.1$/$-0.3$ | $+0.1$/$-1.1$ | $+0.1$/$-0.6$ | $+0.1$/$-0.8$ | $+0.1$/$-0.3$ | $+0.1$/$-1.2$ | $+1.4$/$-0.1$ | $+0.4$/$-0.1$ | $+3.0$/$-0.1$
Subregions of $|b|>10^{\circ}$ sky | $+0.2$/$-0.3$ | $+0.8$/$-1.5$ | $+0.4$/$-0.9$ | $+1.9$/$-2.1$ | $+0.7$/$-0.5$ | $+2.5$/$-1.9$ | $+1.0$/$-1.5$ | $+0.5$/$-0.3$ | $+2.7$/$-0.9$
Results: Figure 3 shows the $\gamma$-ray intensity measured by the LAT and the
fit results for the Galactic latitude range $|b|\geq 10^{\circ}$. Table 1
summarises the numerical values and uncertainties, including the intensity
values for the individually fitted DGE components that are not distinguished
in figure 3 for clarity. The residual intensity obtained after fitting the DGE
model components, solar emission, and sources is the sum of CR background and
EGB. The simulation is used to estimate the CR background and uncertainty, as
described earlier. The CR background is isotropic when averaged over the data
taking period in this paper and is subtracted to obtain the EGB intensity.
Additional figures for different latitude bands and regions of the sky can be
found online epa .
Our formal uncertainty on the EGB comes from the fit using the nominal model.
However, the RMS of the residual count fraction between LAT data and our model
for energies above 200 MeV is 8.2%, when averaged over regions of $13.4$ deg2
to ensure sufficient statistics. This is larger than the 3.3% value expected
solely from statistical fluctuations. We also see correlation of the residual
count fraction with structures in the DGE model skymaps. This suggests a
limitation in the accuracy of the description of the DGE model. We
investigated the uncertainty on the EGB flux related to the DGE components by
varying the relevant parameters in the model and re-evaluating the fits for
$|b|>10^{\circ}$. At high latitudes, the model parameters principally
affecting the DGE are: the change of the IC emission with different halo sizes
and the calculation of the IC emission using the anisotropic/isotropic
formalism Moskalenko and Strong (2000) (IC + halo in Table 1), variations of
the CR source distribution and $X_{\rm CO}$ gradient (CR propagation model),
and how assumptions used to derive Hi column densities from radio data and
dust reddening measurements affect the distribution of Hi in the local region
(Hi column density). To quantify the uncertainty connected to the residual
count fraction, we used the nominal model and examined the variation of the
derived EGB when different subregions of the $|b|>10^{\circ}$ sky are fitted
(Subregions of $|b|>10^{\circ}$ sky). No single component dominates the
uncertainties shown in the lower half of Table 1. We caution that the
uncertainties for the model components cannot be assumed to be independent.
Hence, there is no simple relationship between the combination of individual
components and the total formal uncertainty.
The large statistics allow sub-samples of the total data set to be used as a
cross check. We repeated our analysis for events passing our enhanced
selection with 1) different onboard trigger rates and 2) conversions in the
thin or thick sections of the tracker Atwood et al. (2009). The first sub-
sample ensures that we have properly estimated the residual CR background,
while the second checks that the small fraction of misreconstructed Earth
albedo events that enter the LAT in the back section do not affect the result.
The derived EGB spectrum for these sub-samples is completely consistent with
that derived from the full data set using the same analysis procedure.
Figure 4: EGB intensity derived in this work compared with EGRET-derived
intensities taken from table 1 in Sreekumar et al. (1998) and table 3 in
Strong et al. (2004). Our derived spectrum is compatible with a simple power-
law with index $\gamma=2.41\pm 0.05$ and intensity $I(>100\,{\rm
MeV})=(1.03\pm 0.17)\times 10^{-5}$ cm-2 s-1 sr-1 where the uncertainties are
systematics dominated.
Finally, we note that our analysis also indicates a significant detection of
the combined solar disk and extended solar IC emission. This finding will be
explored in more detail in a separate study.
Discussion: Figure 4 shows the spectrum of the EGB above 200 MeV derived in
the present analysis, and from EGRET data Sreekumar et al. (1998); Strong et
al. (2004). Our intensity extrapolated to 100 MeV based on the power-law fit,
$I(>100\,{\rm MeV})=(1.03\pm 0.17)\times 10^{-5}$ cm-2 s-1 sr-1, is
significantly lower than that obtained from EGRET data: $I_{\rm
EGRET}(>100\,{\rm MeV})=(1.45\pm 0.05)\times 10^{-5}$ cm-2 s-1 sr-1 Sreekumar
et al. (1998). Furthermore, our spectrum is compatible with a featureless
power law with index $\gamma=2.41\pm 0.05$. This is significantly softer than
the EGRET spectrum with index $\gamma_{\rm EGRET}=2.13\pm 0.03$ Sreekumar et
al. (1998). To check that the different spectra are not due to the
instrumental point-source sensitivities, we adopt $F(>100\,{\rm MeV})=10^{-7}$
cm-2 s-1, comparable to the average EGRET sensitivity, and attribute the flux
of all detected LAT sources below this threshold to the EGB. We obtain an
intensity $I_{res}(>100\,{\rm MeV})=(1.19\pm 0.18)\times 10^{-5}$ cm-2 s-1
sr-1 and a spectrum compatible with a power-law with index
$\gamma_{res}=2.37\pm 0.05$. Therefore, the discrepancy cannot be attributed
to a lower threshold for resolving point sources. Our EGB intensity is
comparable to that obtained in the EGRET re-analysis by Strong et al. (2004)
with an updated DGE model, $I_{\rm SMR}(>100\,{\rm MeV})=(1.11\pm 0.1)\times
10^{-5}$ cm-2 s-1 sr-1. However, our EGB spectrum does not show the
distinctive harder spectrum above $\gtrsim 1$ GeV and peak at $\sim 3$ GeV
found in the same EGRET reanalysis. We note that the LAT-measured spectra are
softer above $\gtrsim 1$ GeV than those measured by EGRET also for the DGE at
intermediate latitudes Abdo et al. (2009b) and for the Vela Pulsar Abdo et al.
(2009).
Acknowledgements: The Fermi LAT Collaboration acknowledges support from a
number of agencies and institutes for both development and the operation of
the LAT as well as scientific data analysis. These include NASA and DOE in the
United States, CEA/Irfu and IN2P3/CNRS in France, ASI and INFN in Italy, MEXT,
KEK, and JAXA in Japan, and the K. A. Wallenberg Foundation, the Swedish
Research Council and the National Space Board in Sweden. Additional support
from INAF in Italy and CNES in France for science analysis during the
operations phase is also gratefully acknowledged. GALPROP development is
partially funded via NASA grant NNX09AC15G. Some of the results in this paper
have been derived using the HEALPix Górski et al. (2005) package.
## References
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* (22) See EPAPS Document xxx for additional figures. For more information on EPAPS, see http://www.aip.org/pubservs/epaps.html.
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## I The Spectrum of the Isotropic Diffuse Gamma-Ray Emission Derived From
First-Year Fermi Large Area Telescope Data. Supplementary online material.
### I.1 Abstract
Supplementary material concerning the analysis presented in ”The Spectrum of
the Isotropic Diffuse Gamma-Ray Emission Derived From First-Year Fermi Large
Area Telescope Data” is presented here.
### I.2 Galactic diffuse model
Figure 5 displays the individually fitted contributions to the galactic
diffuse emission arising from inverse Compton emission and interaction of CRs
with atomic hydrogen in the local Galaxy (7.5 kpc $\leq$ r $\leq$ 9.5 kpc) via
bremsstrahlung and pion decay. These contributions are omitted in figure 3 of
the published article for visual clarity, but numerically available in table
1. The intensity is averaged over galactic latitudes $|b|\geq 10^{\circ}$. The
cosmic-ray contamination is subtracted from the EGB component shown here. Its
contribution can be found in figure 3 of the published article. The errors
shown in the graphs are the quadratic sum of statistical and systematic errors
due to uncertainties in the LAT effective area and CR background subtraction.
Figure 5: LAT measured $\gamma$-ray intensity with fit results for $|b|\geq
10^{\circ}$ including statistical and systematic errors. Independently fitted
components of the galactic diffuse emission are shown individually. “Other
Galactic diffuse” denotes the sum of the H2 and non-local Hi and Hii
contributions which are included using the model intensities (i.e., no fit).
Note LAT data are dominated by systematic uncertainties for the energy range
shown in the figure. Ths intensity of the solar emission considered in the
model is below the range shown in the figure.
### I.3 Comparisons for different sky regions
Figure 3 in the published article compares the modeled with the measured
$\gamma$-ray intensity averaged over all galactic latitudes $|b|\geq
10^{\circ}$. To illustrate the good agreement of the used $\gamma$-ray
emission model, i.e. galactic diffuse emission, EGB, point sources and solar
emission, over the whole fitted region, it is interesting to display this
comparison for different sub-regions of the sky.
Figure 6 shows the intensities of different components in the model when
averaged over independent Galactic latitude ranges covering low, mid and high
galactic latitudes, $10^{\circ}\leq|b|\leq 20^{\circ}$, $20^{\circ}\leq|b|\leq
60^{\circ}$ and $|b|\geq 60^{\circ}$. Figure 7 shows the intensities of
different components in the model when averaged over different hemispheres.
The hemispheres considered are centered at the North Galactic pole ($b\geq
0^{\circ}$), the South Galactic pole ($b\leq 0^{\circ}$), the Galactic center
($270^{\circ}\leq l\leq 90^{\circ}$) and anticenter ($90^{\circ}\leq l\leq
270^{\circ}$). Furthermore, Galactic latitudes $|b|<10^{\circ}$ are excluded
from all hemispheres.
We emphasize that the intensities of the components shown in the figures are
from the single gamma-ray emission model used in the analysis. In particular,
the EGB component in each figure is identical because it is isotropic by
construction and thus its average intensity does not vary across the sky. The
accuracy of the model with respect to the different sub-regions of the sky
shown can be judged by comparing the total predicted model intensity (black
line in figure 6) to the measured gamma-ray intensities per energy band
represented by the data points.
Figure 6: LAT measured intensity compared to the $\gamma$-ray emission model
used in the derivation of the EGB averaged over different ranges in Galactic
latitude. The regions shown are $10^{\circ}\leq|b|\leq 20^{\circ}$ (upper
left), $20^{\circ}\leq|b|\leq 60^{\circ}$ (upper right) and $|b|\geq
60^{\circ}$ (lower center). “Other Galactic diffuse” denotes the sum of the H2
and non-local Hi and Hii contributions which are included using the model
intensities (i.e., no fit). Errors include statistical and systematic errors.
LAT data are dominated by systematic uncertainties for the energy range
displayed in the figure.
Figure 7: LAT measured intensity compared to the $\gamma$-ray emission model
used in the derivation of the EGB averaged over different hemispheres on the
sky for Galactic latitudes $|b|\geq 10^{\circ}$. The hemispheres shown are
centered at the North Galactic pole (upper left), the South Galactic pole
(upper right), the Galactic center (lower left) and anti-center (lower right).
“Other Galactic diffuse” denotes the sum of the H2 and non-local Hi and Hii
contributions which are included using the model intensities (i.e., no fit).
Errors include statistical and systematic errors. LAT data are dominated by
systematic uncertainties for the energy range displayed in the figure.
|
arxiv-papers
| 2010-02-18T20:39:26 |
2024-09-04T02:49:08.470691
|
{
"license": "Public Domain",
"authors": "The Fermi-LAT collaboration",
"submitter": "Markus Ackermann",
"url": "https://arxiv.org/abs/1002.3603"
}
|
1002.3741
|
# New dissipated energy for the unstable thin film equation
Marina Chugunova, Roman M. Taranets
###### Abstract
The fluid thin film equation
$h_{t}=-(h^{n}h_{xxx})_{x}-a_{1}\,(h^{m}h_{x})_{x}$ is known to conserve mass
$\int\,h\,dx$, and in the case of $a_{1}\leq~{}0$, to dissipate entropy
$\int\,h^{3/2-n}\,dx$ (see [8]) and the $L^{2}$-norm of the gradient
$\int\,h_{x}^{2}\,dx$ (see [3]). For the special case of $a_{1}=0$ a new
dissipated quantity $\int\,h^{\alpha}\,h_{x}^{2}\,dx$ was recently discovered
for positive classical solutions by Laugesen (see [15]). We extend it in two
ways. First, we prove that Laugesen’s functional dissipates strong nonnegative
generalized solutions. Second, we prove the full $\alpha$-energy
$\int\,\bigl{(}\tfrac{1}{2}\,h^{\alpha}\,h_{x}^{2}\,-$
$\tfrac{a_{1}\,h^{\alpha+m-n+2}}{(\alpha+m-n+1)(\alpha+m-n+2)}\bigr{)}\,dx$
dissipation for strong nonnegative generalized solutions in the case of the
unstable porous media perturbation $a_{1}>0$ and the critical exponent
$m=n+2$.
2000 MSC: 35K55, 35K35, 35Q35, 76D08
keywords: fourth-order degenerate parabolic equations, thin liquid films,
energy, entropy
## 1 Introduction
It is well known that analysis of the existence, uniqueness and regularity of
weak solutions for nonlinear evolution equations relies heavily on a priori
estimates. Often, the physical energy or entropy which originate from the
related model can provide non-increasing in time quantities. Unfortunately, it
is far from obvious how to construct new non-increasing Lyapunov type
functionals. A general algebraic approach to the construction of entropies in
higher-order nonlinear PDEs can be found in [14] and can be applied to analyse
thin film equations with stabilizing porus media type perturbations. In this
paper, inspired by Laugesen’s result [15] on dissipation, we prove that the
energy functional introduced in [15] dissipates strong nonnegative generalized
solutions. However, our method of the proof is only applicable to some subset
of the Laugsen’s dissipation region [15] (see the shaded area on Figure 1).
We study the longwave-unstable generalized thin film equation
$h_{t}=-(h^{n}\,h_{xxx})_{x}-a_{1}\,(h^{m}\,h_{x})_{x},$ (1.1)
where $h(x,t)$ gives the height of the evolving free-surface. The exponent $n$
plays a stabilizing role due to fourth-order forward diffusion term and the
exponent $m$ plays a destabilizing role due to backward second-order diffusion
term for the case when $a_{1}>0$. This class of equations originates from many
physical/industrial applications involving air-fluid interface. For example:
the case $n=1,\quad m=1$ describes a thin jet in a Hele-Shaw cell [10], the
case $n=3,\quad m=-1$ describes Van der Waals driven rupture of thin films
[19], the case $m=n=3$ describes shape of fluid droplets hanging from a
ceiling [11], and the case $n=0,\quad m=1$ describes solidification of a
hyper-cooled melt (this is a modified Kuramoto-Sivashinsky equation) [4].
To prove that the nonnegativity property is preserved in nonlinear thin film
equation $h_{t}=-(h^{n}\,h_{xxx})_{x}$ for $n\geq 1$ (case $a_{1}=0$) Bernis
and Friedman [3] used set of dissipated and conserved quantities: mass
conservation $\int\,h\,dx=M$, surface energy dissipation
$\frac{d}{dt}\int{h_{x}^{2}\,dx}\leq 0$, and entropy dissipation
$\frac{d}{dt}\int{h^{2-n}\,dx}\leq 0$. The new so-called $\beta$-entropy
$\int\,h^{2-n+\beta}\,dx$ was introduced by Bertozzi and Pugh [5] and
independently and simultaneously by Beretta, Bertsch, Dal Passo [1] to extend
this result to $n>0$. They also successfully used this new entropy to obtain
exponential with respect to the $L^{\infty}$-norm convergence toward the mean
value steady state solution. To analyse this convergence rate in $H^{1}$-norm
for the special case $n=1,\ a_{1}=0$ Carlen and Ulusoy [9] used the dissipated
energy $\int{h^{\alpha}\,h_{x}^{2}\,dx}$ constructed by Laugesen [15] for
classical positive solutions. Exponential asymptotic convergence toward the
mean value was also studied by Tudorascu in [18]. This list of connections
between new properties of solutions in thin film PDEs proved by means of newly
discovered dissipated quantities is far from complete.
In this paper we prove that there exists a subinterval $I$ of $-1<\alpha<~{}1$
($I$ depends on $n$ only) and a nonnegative strong generalized solution such
that for any $\alpha\in I$ the full $\alpha$-energy
$\mathcal{E}_{0}^{(\alpha)}(t)=\int\limits_{\Omega}{\left(\tfrac{1}{2}\,h^{\alpha}\,h_{x}^{2}\,-\tfrac{a_{1}\,h^{\alpha+m-n+2}}{(\alpha+m-n+1)(\alpha+m-n+2)}\right)\,dx}$
dissipates. For the unstable porus media perturbation case $a_{1}>0$ this
dissipation is proven under the assumptions that the total mass of the
solution is less than or equal to the critical one, $m=n+2$ and domain
$\Omega$ is unbounded or $h$ is compactly supported. For the stable case
$a_{1}\leq 0$ no such assumptions are needed.
We proceed as follows. First, we show the dissipation for the classical
solutions of the regularized problem and then we take this dissipation to the
limit. We prove dissipation of the full $\alpha$-energy for positive classical
solutions of the regularized problem for any value of the coefficient $a_{1}$
and without any additional assumptions about the total mass of the solution or
its support. However our method of taking the dissipation to the limit due to
the Bernis-Friedman method of regularization requires additional conditions
for the case $a_{1}>0$.
## 2 Auxiliary results to generalized weak solutions
We consider nonnegative weak solutions to the following initial–boundary
problem:
$\displaystyle(\textup{P})$ $\displaystyle
h_{t}+\left({h^{n}h_{xxx}+a_{1}h^{m}h_{x}}\right)_{x}=0\text{ in
}Q_{T},\hfill$ (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 $h=h(t,x)$, $\Omega=(-a,a)$, $Q_{T}=(0,T)\times\Omega$, $n>0$, $m>0$,
and $a_{1}\in\mathbb{R}^{1}$. We define a generalized weak solution in the
Bernis-Friedman sense (see, e. g. [1, 3]).
###### Definition 2.1 (generalized weak solution).
Let $n>0$, $m>0$, and $a_{1}\in\mathbb{R}^{1}$. A generalized weak solution of
problem $($P$)$ is a function $h$ satisfying
$\displaystyle h\in C^{1/2,1/8}_{x,t}(\overline{Q}_{T})\cap
L^{\infty}(0,T;H^{1}(\Omega)),$ (2.4) $\displaystyle h_{t}\in
L^{2}(0,T;(H^{1}(\Omega))^{\prime}),$ (2.5) $\displaystyle h\in
C^{4,1}_{x,t}(\mathcal{P}),\,\,\,h^{\frac{n}{2}}(h_{xxx}+a_{1}h^{m-n}h_{x})\in
L^{2}(\mathcal{P}),\,\,$ (2.6)
where $\mathcal{P}=\overline{Q}_{T}\setminus(\\{h=0\\}\cup\\{t=0\\})$ and $h$
satisfies (2.1) in the following sense:
$\int\limits_{0}^{T}\langle
h_{t}(\cdot,t),\phi\rangle\;dt-\iint\limits_{\mathcal{P}}{h^{n}(h_{xxx}+a_{1}h^{m-n}h_{x})\phi_{x}\,dxdt}\
=0$ (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 [3]; there, the first
term was the integral of $h\phi_{t}$. The proof of the existence of
generalized weak solutions follows the ideas of [3, 1, 5, 6, 7, 17].
Let
$G_{0}^{(\beta)}(z):=\left\\{\begin{gathered}\tfrac{z^{\beta-n+2}}{(\beta-n+2)(\beta-n+1)}\text{
if }\beta-n\neq\\{-1,-2\\},\\\ z\ln z-z\text{ if }\beta-n=-1,\\\ -\ln z\text{
if }\beta-n=-2,\end{gathered}\right.$ (2.10)
$(G^{(\beta)}_{0}(z))^{\prime\prime}=z^{\beta-n}$, and
$G_{0}(z):=G_{0}^{(0)}(z)$.
###### Theorem 1.
Let $a_{1}\in\mathbb{R}^{1}$, $n>0$; $m\geqslant n/2$ for $a_{1}>0$, and $m>0$
for $a_{1}\leq 0$.
(a) [Existence.] Let the nonnegative initial data $h_{0}\in H^{1}(\Omega)$
satisfy
$\int\limits_{\Omega}{G_{0}(h_{0}(x))\,dx}<\infty,$ (2.11)
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,
$h\in L^{2}(0,T_{loc};H^{2}(\Omega)).$ (2.12)
Let
$\mathcal{E}_{0}(T):=\int\limits_{\Omega}{\\{\tfrac{1}{2}h_{x}^{2}(x,T)-a_{1}D_{0}(h(x,T))\\}\,dx},$
(2.13)
where $D_{0}(z):=\frac{z^{m-n+2}}{(m-n+1)(m-n+2)}$. Then the weak solution
satisfies
$\mathcal{E}_{0}(T)+\iint\limits_{\\{h>0\\}}{h^{n}(h_{xxx}+a_{1}h^{m-n}h_{x})^{2}\,dxdt}\leqslant\mathcal{E}_{0}(0),$
(2.14)
$\iint\limits_{\\{h>0\\}}{h^{n}h^{2}_{xxx}\,dxdt}\leqslant\,\,const<\infty.$
(2.15)
for all $T\leq T_{loc}$. The time of existence, $T_{loc}$, is determined by
$a_{1}$, $|\Omega|$, $\int h_{0}$, $\|h_{0x}\|_{2}$, and $\int G_{0}(h_{0})$.
Moreover, $T_{loc}=+\infty$ for $a_{1}\leq 0$.
(b) [Regularity.] If the initial data from (a) also satisfies
$\int\limits_{\Omega}{G^{(\beta)}_{0}(h_{0}(x))\,dx}<\infty$
for some $-1/2<\beta<1,\ \beta\neq 0$ then there exists
$0<T_{loc}^{(\beta)}\leq T_{loc}$ such that the nonnegative generalized weak
solution has the extra regularity
$h^{\tfrac{\beta+2}{2}}\in L^{2}(0,T_{loc}^{(\beta)};H^{2}(\Omega))\text{ and
}h^{\tfrac{\beta+2}{4}}\in L^{2}(0,T_{loc}^{(\beta)};W^{1}_{4}(\Omega)).$
(2.16)
The time of existence, $T_{loc}^{(\beta)}$, is determined by $a_{1}$,
$|\Omega|$, $\int h_{0}$, $\|h_{0x}\|_{2}$, and $\int G^{(\beta)}_{0}(h_{0})$.
Moreover, $T_{loc}^{(\beta)}=+\infty$ for $a_{1}\leq 0$.
There is nothing special about the time $T_{loc}$ in the Theorem 1. In the
case $a_{1}>0$ and $n/2\leq m<n+2$ (or $m=n+2$ and $M\leq M_{c}$), 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 [3] also holds: there exists a nonnegative weak
solution with the integral representation
$\displaystyle\int\limits_{0}^{T}\langle
h_{t}(\cdot,t),\phi\rangle\;dt+\iint\limits_{Q_{T}}(nh^{n-1}h_{x}h_{xx}\phi_{x}+h^{n}h_{xx}\phi_{xx})\;dxdt$
(2.17) $\displaystyle\hskip 108.405pt-
a_{1}\iint\limits_{Q_{T}}{h^{m}h_{x}\phi_{x}\;dxdt}=0.$
## 3 Dissipation of energy for nonnegative weak solutions
The main result of the present paper is the following
###### Theorem 2.
Let $a_{1}\in\mathbb{R}^{1}$, $1/2<n<3$; $m\geqslant n/2$ for $a_{1}>0$, and
$m>0$ for $a_{1}\leq 0$, and
$\mathcal{E}_{0}^{(\alpha)}(T):=\int\limits_{\Omega}{\\{\tfrac{1}{2}h^{\alpha}h_{x}^{2}(x,T)-a_{1}\tilde{D}_{0}(h(x,T))\\}\,dx},$
(3.1)
where
$\tilde{D}_{0}(z):=\frac{z^{\alpha+m-n+2}}{(\alpha+m-n+1)(\alpha+m-n+2)}$, and
$\mathcal{E}_{0}^{(0)}(T)=\mathcal{E}_{0}(T)$. Then there exists a non-empty
subinterval $I$ (see [15] for the explicit form of the $I$) of $0\leq\alpha<1$
for $\frac{1}{2}<n<3$, and of $\frac{3}{2}-n<\alpha<0$ for $\frac{3}{2}<n<3$
such that for any $\alpha\in I$ the nonnegative weak solution from Theorem 1
satisfies the following estimates:
(i) if $a_{1}\leqslant 0$ then
$\mathcal{E}^{(\alpha)}_{0}(T)\leqslant\mathcal{E}^{(\alpha)}_{0}(0);$ (3.2)
(ii) if $a_{1}>0$ then
$\mathcal{E}^{(\alpha)}_{0}(T)\leqslant\mathcal{E}^{(\alpha)}_{0}(0)+C_{1}\iint\limits_{Q_{T}}{h^{\alpha+3m-2n+2}dxdt}\text{
for }m>n+2;$ (3.3)
$\mathcal{E}^{(\alpha)}_{0}(T)\leqslant\mathcal{E}^{(\alpha)}_{0}(0)+T(C_{1}M^{\frac{2\alpha+5m-3n+4}{n+2-m}}+C_{2}M^{\alpha+3m-2n+2})$
(3.4)
for $m<n+2$ and $\alpha>2n-3m-1$;
$\mathcal{E}^{(\alpha)}_{0}(T)\leqslant\mathcal{E}^{(\alpha)}_{0}(0)+C_{3}T\,M^{\alpha+3m-2n+2}$
(3.5)
for $m<n+2$ and $2n-3m-2<\alpha\leqslant 2n-3m-1$;
$\mathcal{E}^{(\alpha)}_{0}(T)\leqslant\mathcal{E}^{(\alpha)}_{0}(0)+C_{2}T\,M^{\alpha+n+8}\text{
for }m=n+2\text{ and }0<M\leqslant M_{c}.$ (3.6)
Here $C_{2}=0$ if $\Omega$ is unbounded or $h$ has compact support.
###### Remark 3.1 (Extra Regularity).
In particular, the extra regularity $h^{\frac{\alpha+2}{2}}\in
L^{\infty}(0,T;H^{1}(\Omega))$ follows directly from Theorem 2. Hence,
$h^{\frac{\alpha+2}{2}}(\cdot,T)\in H^{1}(\Omega)$ for almost all
$T\in[0,T_{loc}^{(\beta)}]$ and therefore $h^{\frac{\alpha+2}{2}}(\cdot,T)\in
C^{1/2}(\overline{\Omega})$ for almost all $T\in[0,T_{loc}^{(\beta)}]$. Assume
that $T_{0}$ is chosen such that $h^{\frac{\alpha+2}{2}}(\cdot,T_{0})\in
C^{1/2}(\overline{\Omega})$ and $h(x_{0},T_{0})=0$ at some
$x_{0}\in\bar{\Omega}$. Then there exists a constant $L$ such that
$h^{\frac{\alpha+2}{2}}(x,T_{0})=|h^{\frac{\alpha+2}{2}}(x,T_{0})-h^{\frac{\alpha+2}{2}}(x_{0},T_{0})|\leq
L|x-x_{0}|^{1/2}.$
Hence $h(x,T_{0})\leq L^{\frac{2}{\alpha+2}}|x-x_{0}|^{\frac{1}{\alpha+2}}$,
i. e. $h(.,T)\in C^{\frac{1}{\alpha+2}}(\bar{\Omega})$ for almost every
$T\in[0,T_{loc}^{(\beta)}]$.
###### Remark 3.2 (Rate of decrease).
For $a_{1}\leq 0$ and $1/2<n<3$ we can generalize the results from [9, Theorem
1.1] in the following way:
$\int\limits_{\Omega}{h^{\alpha}h_{x}^{2}(x,t)\,dx}\leq
C(1+t)^{-\frac{1}{2}}\text{ for }\tfrac{n-4}{2}\leq\alpha<0,$
whence $\|h-\bar{h}\|_{\infty}\leq C(1+t)^{-\frac{1}{4}}$ for any nonnegative
strong solution $h$. Here
$C=C(a_{1},\alpha,n,\bar{h},\mathcal{E}^{(\alpha)}_{0}(0))$, and
$\bar{h}=\frac{1}{|\Omega|}\|h_{0}\|_{1}$. The proof is similar to [9].
### 3.1 Regularized Problem
Given $\delta,\varepsilon>0$, a regularized parabolic problem, similar to that
of Bernis and Friedman [3], is considered:
$\displaystyle(\textup{P}_{\delta,\varepsilon})$ $\displaystyle
h_{t}+\left({f_{\delta\varepsilon}(h)(h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})}\right)_{x}=0,\qquad\hfill$
(3.7) $\displaystyle(\textup{P}_{\delta,\varepsilon})$
$\displaystyle\tfrac{\partial^{i}h}{\partial
x^{i}}(-a,t)=\tfrac{\partial^{i}h}{\partial x^{i}}(a,t)\text{ for
}t>0,\,i=\overline{0,3},\hfill$ (3.8)
$\displaystyle(\textup{P}_{\delta,\varepsilon})$ $\displaystyle\qquad\qquad
h(x,0)=h_{0,\delta\varepsilon}(x),\hfill$ (3.9)
where
$f_{\delta\varepsilon}(z):=f_{\varepsilon}(z)+\delta=\tfrac{|z|^{s+n}}{|z|^{s}+\varepsilon|z|^{n}}+\delta,\
D^{\prime\prime}_{\varepsilon}(z):=\tfrac{|z|^{m-n}}{1+\varepsilon|z|^{m-n}}$
(3.10)
$\forall\,z\in\mathbb{R}^{1},\ \varepsilon>0,\ s\geqslant 4$. The $\delta>0$
in (3.10) makes the problem (3.7) 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 $f_{\varepsilon}(h)\sim
h^{s}$. The nonnegative initial data, $h_{0}$, is approximated via
$\begin{gathered}h_{0,\delta\varepsilon}\in C^{4+\gamma}(\Omega),\
h_{0,\delta\varepsilon}\geqslant h_{0,\delta}+\varepsilon^{\theta}\text{ for
some }0<\theta<\tfrac{2}{2s-3},\\\
\tfrac{\partial^{i}h_{0,\delta\varepsilon}}{\partial
x^{i}}(-a)=\tfrac{\partial^{i}h_{0,\delta\varepsilon}}{\partial
x^{i}}(a)\text{ for }i=\overline{0,3},\\\ h_{0,\delta\varepsilon}\to
h_{0}\text{ strongly in }H^{1}(\Omega)\text{ as }\delta,\varepsilon\to
0.\end{gathered}$ (3.11)
The $\varepsilon$ term in (3.11) ‘‘lifts’’ the initial data so that it will be
positive even if $\delta=0$ and the $\delta$ is involved in smoothing the
initial data from $H^{1}(\Omega)$ to $C^{4+\gamma}(\Omega)$.
_Sketch of Proof_ : By Eĭdelman [12, Theorem 6.3, p.302], the regularized
problem has the 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 values of $\delta$ and
$\varepsilon$, by Eĭdelman [12, Theorem 9.3, p.316] if one can prove a uniform
in time an a priori bound $|h_{\delta\varepsilon}(x,t)|\leq
A_{\delta\varepsilon}<\infty$ 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 [12, 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 1) $T_{loc,\delta\varepsilon}\to
T_{loc}>0$, 2) the solutions $h_{\delta\varepsilon}$ converge to a
(nonnegative) limit, $h$, which is a generalized weak solution, and 3) $h$
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 $\varepsilon\to 0$ will then result in the desired
generalized weak solution $h$. This last step is where the possibility of non-
unique weak solutions arise; see [1] for simple examples of how such
constructions applied to $h_{t}=-(|h|^{n}h_{xxx})_{x}$ can result in two
different solutions arising from the same initial data.
### 3.2 Dissipation of energy for positive solutions
Figure 1: The dissipation region computed numerically by Matlab: $\alpha$
versus $n$. The dashed line corresponds to $\alpha=3/2-n$.
###### Lemma 3.1.
Let $\alpha$ belong to the full domain shown on Figure 1, and
$\mathcal{E}_{\varepsilon}^{(\alpha)}(T):=\int\limits_{\Omega}{\\{\tfrac{1}{2}h^{\alpha}h_{x}^{2}(x,T)-a_{1}\tilde{D}_{\varepsilon}(h(x,T))\\}\,dx},$
(3.12)
where
$\tilde{D}^{\prime\prime}_{\varepsilon}(z):=z^{\alpha}D^{\prime\prime}_{\varepsilon}(z)$.
Then the unique positive classical solution $h_{\varepsilon}$ of the problem
($P_{0,\varepsilon}$) satisfies
$\mathcal{E}^{(\alpha)}_{\varepsilon}(T)\leqslant\mathcal{E}^{(\alpha)}_{\varepsilon}(0)+\mu\iint\limits_{Q_{T}}{h^{\alpha-2}f_{\varepsilon}(h)D^{\prime\prime}_{\varepsilon}(h)h_{x}^{4}dxdt}+\\\
\varepsilon\,k_{1}\int\limits_{\Omega}{h^{\alpha-s-4}f_{\varepsilon}^{2}(h)h_{x}^{6}\,dx}+\varepsilon^{2}k_{2}\int\limits_{\Omega}{h^{\alpha-2s-4}f_{\varepsilon}^{3}(h)h_{x}^{6}\,dx},$
(3.13)
where $k_{i}=k_{i}(\alpha,n,s)$ are constants, and $\mu=\mu(\alpha,a_{1})$
such that $\mu(0,a_{1})=0$ and $\mu(\alpha,a_{1})\leqslant 0$ for
$a_{1}\leqslant 0$.
Note that, although we use the same convenient notations introduced in [15],
the proof of Lemma 3.1 has essential differences from the proof of Theorem 1
of [15]. Indeed, we introduce new ideas in order to estimate the lower-order
term in the equation (3.7). In particular, the new quantity $N$ is introduced,
the quantity $R$ is modified, and so are the terms involving the
regularization parameter $\varepsilon$ in (3.19).
###### Proof of Lemma 3.1.
To prove the bound (3.13), multiply (3.7) with $\delta=0$ by
$-\frac{\alpha}{2}h^{\alpha-1}h_{x}^{2}-h^{\alpha}h_{xx}-a_{1}\tilde{D}^{\prime}_{\varepsilon}(h)$,
integrate over $\Omega$, use integration by parts, apply the periodic boundary
conditions (3.8), to find
$\tfrac{d}{dt}\mathcal{E}^{(\alpha)}_{\varepsilon}(t)=-\int\limits_{\Omega}{h^{\alpha}f_{\varepsilon}(h)(h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})^{2}\,dx}-\\\
\tfrac{\alpha}{2}(\alpha-1)\int\limits_{\Omega}{h^{\alpha-2}h_{x}^{3}f_{\varepsilon}(h)(h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})\,dx}-\\\
2\alpha\int\limits_{\Omega}{h^{\alpha-1}h_{x}h_{xx}f_{\varepsilon}(h)(h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})\,dx}.$
(3.14)
The equality (3.14) can be rewritten as
$\tfrac{d}{dt}\mathcal{E}^{(\alpha)}_{\varepsilon}(t)=-R^{2}-2\alpha\,RS-\tfrac{\alpha}{2}(\alpha-1)\,RL,$
(3.15)
where the quantities
$R:=\langle(h^{\alpha}f_{\varepsilon}(h))^{1/2}(h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})|=|(h^{\alpha}f_{\varepsilon}(h))^{1/2}(h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})\rangle,$
$S:=\langle(h^{\alpha-2}f_{\varepsilon}(h))^{1/2}h_{x}h_{xx}|=|(h^{\alpha-2}f_{\varepsilon}(h))^{1/2}h_{x}h_{xx}\rangle,$
$L:=\langle(h^{\alpha-4}f_{\varepsilon}(h))^{1/2}h_{x}^{3}|=|(h^{\alpha-4}f_{\varepsilon}(h))^{1/2}h_{x}^{3}\rangle,$
$N:=\langle(h^{\alpha-2}f_{\varepsilon}(h)D^{\prime\prime}_{\varepsilon}(h))^{1/2}h_{x}^{2}|=|(h^{\alpha-2}f_{\varepsilon}(h)D^{\prime\prime}_{\varepsilon}(h))^{1/2}h_{x}^{2}\rangle,$
each represent half of an inner product in $L^{2}(\Omega)$. We will need the
following integration by parts formulas
$SL=-\tfrac{1}{5}(\alpha-3)\int\limits_{\Omega}{h^{\alpha-4}f_{\varepsilon}(h)h_{x}^{6}\,dx}-\tfrac{1}{5}\int\limits_{\Omega}{h^{\alpha-3}f^{\prime}_{\varepsilon}(h)h_{x}^{6}\,dx}=\\\
-\tfrac{1}{5}(\alpha+n-3)L^{2}-\tfrac{1}{5}\varepsilon(s-n)\int\limits_{\Omega}{h^{\alpha-s-4}f_{\varepsilon}^{2}(h)h_{x}^{6}\,dx},$
(3.16)
$RL=-(\alpha+n-2)SL-\varepsilon(s-n)\int\limits_{\Omega}{h^{\alpha-s-3}f_{\varepsilon}^{2}(h)h_{x}^{4}h_{xx}\,dx}-3S^{2}+a_{1}N^{2}=\\\
\tfrac{1}{5}(\alpha+n-2)(\alpha+n-3)L^{2}-3S^{2}+a_{1}N^{2}-\varepsilon(s-n)\int\limits_{\Omega}{h^{\alpha-s-3}f_{\varepsilon}^{2}(h)h_{x}^{4}h_{xx}\,dx}+\\\
\tfrac{1}{5}\varepsilon(s-n)(\alpha+n-2)\int\limits_{\Omega}{h^{\alpha-s-4}f_{\varepsilon}^{2}(h)h_{x}^{6}\,dx}=\tfrac{1}{5}(\alpha+n-2)(\alpha+n-3)L^{2}-3S^{2}+\\\
a_{1}N^{2}+\tfrac{1}{5}\varepsilon(s-n)(2\alpha+3n-s-5)\int\limits_{\Omega}{h^{\alpha-s-4}f_{\varepsilon}^{2}(h)h_{x}^{6}\,dx}+\\\
\tfrac{2}{5}\varepsilon^{2}(s-n)^{2}\int\limits_{\Omega}{h^{\alpha-2s-4}f_{\varepsilon}^{3}(h)h_{x}^{6}\,dx}.$
(3.17)
Here we use the auxiliary equality
$f^{\prime}_{\varepsilon}(z)=nz^{-1}f_{\varepsilon}(z)+\varepsilon(s-n)z^{-(s+1)}f_{\varepsilon}^{2}(z)$.
Thus, from (3.15) we have
$\tfrac{d}{dt}\mathcal{E}^{(\alpha)}_{\varepsilon}(t)+\tfrac{\varepsilon^{2}}{5}\alpha(\alpha-1)(s-n)^{2}\int\limits_{\Omega}{h^{\alpha-2s-4}f_{\varepsilon}^{3}(h)h_{x}^{6}\,dx}+\tfrac{a_{1}}{2}\alpha(\alpha-1)N^{2}=\\\
-R^{2}-2\alpha\,RS-\tfrac{\alpha}{10}(\alpha-1)(\alpha+n-2)(\alpha+n-3)L^{2}+\tfrac{3\alpha}{2}(\alpha-1)S^{2}+\\\
\tfrac{\varepsilon}{10}\alpha(\alpha-1)(s-n)(s-2\alpha-3n+5)\int\limits_{\Omega}{h^{\alpha-s-4}f_{\varepsilon}^{2}(h)h_{x}^{6}\,dx}.$
(3.18)
Our next step is to express (3.18) as the negative of a sum of squares to
obtain the energy dissipation. To achieve this, we use (3.16) and (3.17) to
deduce that for all $\kappa\in\mathbb{R}^{1}$,
$\tfrac{d}{dt}\mathcal{E}^{(\alpha)}_{\varepsilon}(t)=-(R+\alpha\,S+\kappa\,L)^{2}+\beta(S+\tfrac{1}{5}(\alpha+n-3)\,L)^{2}+\gamma\,L^{2}+\mu\,N^{2}+\\\
\varepsilon\,k_{1}\int\limits_{\Omega}{h^{\alpha-s-4}f_{\varepsilon}^{2}(h)h_{x}^{6}\,dx}+\varepsilon^{2}k_{2}\int\limits_{\Omega}{h^{\alpha-2s-4}f_{\varepsilon}^{3}(h)h_{x}^{6}\,dx},$
(3.19)
where
$k_{1}=\tfrac{2}{25}(s-n)\biggl{(}5\kappa(\alpha-s+3n-5)+\tfrac{5\alpha}{4}(\alpha-1)(s-2\alpha-3n+5)+\\\
(\alpha+n-3)(\tfrac{\alpha}{2}(5\alpha-3)-6\kappa)\biggr{)},\
k_{2}=\tfrac{1}{5}(s-n)^{2}\biggl{(}4\kappa-\alpha(\alpha-1)\biggr{)},$
$\beta=\tfrac{\alpha}{2}(5\alpha-3)-6\kappa,\,\
\mu=a_{1}\biggl{(}2\kappa-\tfrac{\alpha}{2}(\alpha-1)\biggr{)},$
$\gamma=\kappa^{2}-\tfrac{2}{25}\kappa(\alpha+n-3)\bigl{(}5(2-n)+3(\alpha+n-3)\bigr{)}-\\\
\tfrac{\alpha}{50}(\alpha+n-3)\bigl{(}5(\alpha-1)(\alpha+n-2)-(5\alpha-3)(\alpha+n-3)\bigr{)}=\\\
\kappa^{2}-\tfrac{6}{25}\kappa(\alpha+n-3)\bigl{(}\alpha-\tfrac{2n-1}{3}\bigr{)}-\tfrac{3}{50}\alpha(\alpha+n-3)\bigl{(}\alpha-\tfrac{2n-1}{3}\bigr{)}.$
Now we have to choose the parameter $\kappa$ in such a way that
$\beta\leqslant 0$ and $\gamma\leqslant 0$. In this case, the parameter
$\mu>0$ for $a_{1}>0$, and $\mu\leqslant 0$ for $a_{1}\leqslant 0$. According
to [15], we can find $\kappa$ such that $\beta\leqslant 0$, and
$\gamma\leqslant 0$ when $1/2<n<3$, see also Figure 1 where this region was
computed numerically by Matlab (see [15] for the explicit form of the domain).
### 3.3 Limit process in (3.13)
Rewrite the integral $\iint\limits_{Q_{T}}{\varepsilon
h_{\varepsilon}^{\alpha-s-4}f_{\varepsilon}^{2}(h_{\varepsilon})h_{\varepsilon,x}^{6}\,dxdt}$
in the form
$\iint\limits_{Q_{T}}{\varepsilon
h_{\varepsilon}^{\alpha-s-4}f_{\varepsilon}^{2}(h_{\varepsilon})h_{\varepsilon,x}^{6}\,dxdt}=\iint\limits_{Q_{T}}{\tfrac{\varepsilon\,h_{\varepsilon}^{\alpha+s-n}}{(h_{\varepsilon}^{s-n}+\varepsilon)^{2}}h_{\varepsilon}^{n-4}h_{\varepsilon,x}^{6}\,dxdt}.$
Using the Young’s inequality
$ab\leqslant\tfrac{a^{p}}{p}+\tfrac{b^{q}}{q}\Rightarrow p\,ab\leqslant
a^{p}+(p-1)b^{q},\ \tfrac{1}{p}+\tfrac{1}{q}=1,$ (3.20)
with $a=z^{\frac{s-n}{p}}$ and
$b=\bigl{(}\frac{\varepsilon}{p-1}\bigr{)}^{\frac{1}{q}}$, we deduce
$\varepsilon\tfrac{z^{\alpha+s-n}}{(z^{s-n}+\varepsilon)^{2}}\leqslant\varepsilon\tfrac{z^{\alpha}}{z^{s-n}+\varepsilon}\leqslant\tfrac{(p-1)^{\frac{1}{q}}}{p}\tfrac{\varepsilon\,z^{\alpha}}{z^{\frac{s-n}{p}}\varepsilon^{\frac{1}{q}}}=\tfrac{(p-1)^{\frac{1}{q}}}{p}\varepsilon^{\frac{q-1}{q}}z^{\frac{p\alpha-s+n}{p}},$
choosing $p=\frac{s-n}{\alpha}>1$ and $q=\frac{s-n}{s-n-\alpha}>1$
($\Rightarrow 0<\alpha<s-n$), we find
$\varepsilon\tfrac{z^{\alpha+s-n}}{(z^{s-n}+\varepsilon)^{2}}\leqslant\tfrac{\alpha}{s-n}\bigl{(}\tfrac{s-n-\alpha}{\alpha}\bigr{)}^{\frac{s-n-\alpha}{s-n}}\varepsilon^{\frac{\alpha}{s-n}}.$
Similarly, we deal with the integral
$\varepsilon^{2}\int\limits_{\Omega}{h_{\varepsilon}^{\alpha-2s-4}f_{\varepsilon}^{3}(h)h_{\varepsilon,x}^{6}\,dx}$.
Due to Lemma A.1 and (2.15),
$\iint\limits_{Q_{T}}{h_{\varepsilon}^{n-4}h_{\varepsilon,x}^{6}\,dxdt}$ is
uniformly bounded then
$\Bigl{|}\iint\limits_{Q_{T}}{(k_{1}\varepsilon
h_{\varepsilon}^{\alpha-s-4}f_{\varepsilon}^{2}(h_{\varepsilon})+k_{2}\varepsilon^{2}h_{\varepsilon}^{\alpha-2s-4}f_{\varepsilon}^{3}(h_{\varepsilon}))h_{\varepsilon,x}^{6}\,dxdt}\Bigr{|}\leqslant\\\
C\,\varepsilon^{\frac{\alpha}{s-n}}\iint\limits_{Q_{T}}{h_{\varepsilon}^{n-4}h_{\varepsilon,x}^{6}\,dxdt}\leqslant
C\,\varepsilon^{\frac{\alpha}{s-n}},$ (3.21)
where the positive constant $C$ is independent of $\varepsilon$. Letting
$\varepsilon\to 0$, from (3.21) we obtain
$(k_{1}\varepsilon
h_{\varepsilon}^{\alpha-s-4}f_{\varepsilon}^{2}(h_{\varepsilon})+k_{2}\varepsilon^{2}h_{\varepsilon}^{\alpha-2s-4}f_{\varepsilon}^{3}(h_{\varepsilon}))h_{\varepsilon,x}^{6}\to
0\text{ in }L^{1}(Q_{T})$ (3.22)
for $0<\alpha<s-n$.
Now, we show (3.22) for the case of $\alpha<0$. Rewrite the integral
$\iint\limits_{Q_{T}}{\varepsilon
h_{\varepsilon}^{\alpha-s-4}f_{\varepsilon}^{2}(h_{\varepsilon})h_{\varepsilon,x}^{6}\,dxdt}$
in the form
$\iint\limits_{Q_{T}}{\varepsilon
h_{\varepsilon}^{\alpha-s-4}f_{\varepsilon}^{2}(h_{\varepsilon})h_{\varepsilon,x}^{6}\,dxdt}=\iint\limits_{Q_{T}}{\tfrac{\varepsilon\,h_{\varepsilon}^{s-2}}{(h_{\varepsilon}^{s-n}+\varepsilon)^{2}}h_{\varepsilon}^{\alpha-2}h_{\varepsilon,x}^{6}\,dxdt}.$
Using the inequality (3.20) with $a=z^{\frac{s-n}{p}}$ and
$b=\bigl{(}\frac{\varepsilon}{p-1}\bigr{)}^{\frac{1}{q}}$, we obtain
$\varepsilon\tfrac{z^{s-2}}{(z^{s-n}+\varepsilon)^{2}}\leqslant\tfrac{(p-1)^{\frac{2}{q}}}{p^{2}}\tfrac{\varepsilon\,z^{s-2}}{z^{\frac{2(s-n)}{p}}\varepsilon^{\frac{2}{q}}}=\tfrac{(p-1)^{\frac{2}{q}}}{p^{2}}\varepsilon^{\frac{q-2}{q}}z^{\frac{p(s-2)-2(s-n)}{p}},$
choosing $p=\frac{2(s-n)}{s-2+\alpha-\beta}<2$ and
$q=\frac{2(s-n)}{2(s-n)-s+2-\alpha+\beta}>2$ ($\Rightarrow n>2-\alpha+\beta$),
we find
$\varepsilon\tfrac{z^{s-2}}{(z^{s-n}+\varepsilon)^{2}}\leqslant\bigl{(}\tfrac{s-2+\alpha-\beta}{2(s-n)}\bigr{)}^{2}\bigl{(}\tfrac{s-2(n-1)-\alpha+\beta}{s-2+\alpha-\beta}\bigr{)}^{\frac{s-2(n-1)-\alpha+\beta}{s-n}}\varepsilon^{\frac{n-2+\alpha-\beta}{s-n}}z^{\beta-\alpha},$
where $\beta\in(-1/2,1)$ follows from (2.16). Similarly, we deal with the
integral
$\varepsilon^{2}\int\limits_{\Omega}{h_{\varepsilon}^{\alpha-2s-4}f_{\varepsilon}^{3}(h)h_{\varepsilon,x}^{6}\,dx}$.
Due to $h\in L^{\infty}(0,T;H^{1}(\Omega))$ and (2.16),
$\iint\limits_{Q_{T}}{h_{\varepsilon}^{\alpha-2}h_{\varepsilon,x}^{6}\,dxdt}$
is uniformly bounded then
$\Bigl{|}\iint\limits_{Q_{T}}{(k_{1}\varepsilon
h_{\varepsilon}^{\alpha-s-4}f_{\varepsilon}^{2}(h_{\varepsilon})+k_{2}\varepsilon^{2}h_{\varepsilon}^{\alpha-2s-4}f_{\varepsilon}^{3}(h_{\varepsilon}))h_{\varepsilon,x}^{6}\,dxdt}\Bigr{|}\leqslant\\\
C\,\varepsilon^{\frac{n-2+\alpha-\beta}{s-n}}\iint\limits_{Q_{T}}{h_{\varepsilon}^{\beta-2}h_{\varepsilon,x}^{6}\,dxdt}\leqslant
C\,\varepsilon^{\frac{n-2+\alpha-\beta}{s-n}}\iint\limits_{Q_{T}}{h_{\varepsilon}^{\beta-2}h_{\varepsilon,x}^{4}\,dxdt}\leqslant\\\
C\,\varepsilon^{\frac{n-2+\alpha-\beta}{s-n}},$ (3.23)
where the positive constant $C$ is independent of $\varepsilon$. Letting
$\varepsilon\to 0$, we obtain (3.22) for $\frac{3}{2}-n<\alpha<0$ and
$\frac{3}{2}<n<3$. In view of the Lebesgue’s theorem, we have
$\iint\limits_{Q_{T}}{h_{\varepsilon}^{\alpha-2}f_{\varepsilon}(h_{\varepsilon})D^{\prime\prime}_{\varepsilon}(h_{\varepsilon})h_{\varepsilon,x}^{4}\,dxdt}\to\iint\limits_{Q_{T}}{h^{\alpha+m-2}h_{x}^{4}\,dxdt}$
(3.24)
if $m>0$ and $\alpha>-\frac{1}{2}-m$, due to $h\in
L^{\infty}(0,T;H^{1}(\Omega))$ and (2.16).
Integrating (3.19) over the time interval, and letting $\varepsilon\to 0$, in
view of (3.22) and (3.24), we obtain (3.13) for some subinterval $I$ for
$0\leq\alpha<1$ and $\frac{1}{2}<n<3$ or for $-1<\alpha<0$ and
$\frac{3}{2}<n<3$. Note that, the convergence on the left-hand side follows
from Fatou’s lemma and from the corresponding a priori estimate (see, for
example, [3, 5, 7, 17]). ∎
### 3.4 Proof of Theorem 2
Taking the limit $\varepsilon\to 0$ we obtain
$\mathcal{E}^{(\alpha)}_{0}(T)+\gamma\iint\limits_{Q_{T}}{h^{\alpha+n-4}h_{x}^{6}dxdt}\leqslant\mathcal{E}^{(\alpha)}_{0}(0)+\mu\iint\limits_{Q_{T}}{h^{\alpha+m-2}h_{x}^{4}dxdt}.$
(3.25)
Now, we estimate $\iint\limits_{Q_{T}}{h^{\alpha+m-2}h_{x}^{4}dxdt}$. Using
the Hölder inequality, we obtain
$\iint\limits_{Q_{T}}{h^{\alpha+m-2}h_{x}^{4}dxdt}\leqslant\int\limits_{0}^{T}{\Bigl{(}\int\limits_{\Omega}{h^{\alpha+n-4}h_{x}^{6}dx}\Bigr{)}^{\frac{2}{3}}\Bigl{(}\int\limits_{\Omega}{h^{\alpha+3m-2n+2}dx}\Bigr{)}^{\frac{1}{3}}dt}.$
(3.26)
Applying Lemma A.2 to $v=h^{\frac{\alpha+n+2}{6}}$ with
$a=\frac{6(\alpha+3m-2n+2)}{\alpha+n+2}$, $d=6$, $b=\frac{6}{\alpha+n+2}<a\
(\Rightarrow\alpha>2n-3m-1)$, $i=0$, and $j=1$, we deduce
$\int\limits_{\Omega}{h^{\alpha+3m-2n+2}dx}\leqslant
d_{1}\Bigl{(}\int\limits_{\Omega}{v_{x}^{6}dx}\Bigr{)}^{\frac{\alpha+3m-2n+1}{\alpha+n+7}}\Bigl{(}\int\limits_{\Omega}{h\,dx}\Bigr{)}^{\frac{3(2\alpha+5m-3n+4)}{\alpha+n+7}}+\\\
d_{2}\Bigl{(}\int\limits_{\Omega}{h\,dx}\Bigr{)}^{\alpha+3m-2n+2}\leqslant
c_{1}M^{\frac{3(2\alpha+5m-3n+4)}{\alpha+n+7}}\Bigl{(}\int\limits_{\Omega}{h^{\alpha+n-4}h_{x}^{6}dx}\Bigr{)}^{\frac{\alpha+3m-2n+1}{\alpha+n+7}}+\\\
c_{2}M^{\alpha+3m-2n+2}.$ (3.27)
Substituting (3.27) in (3.26), we find
$\iint\limits_{Q_{T}}{h^{\alpha+m-2}h_{x}^{4}dxdt}\leqslant
c_{1}M^{\frac{2\alpha+5m-3n+4}{\alpha+n+7}}\int\limits_{0}^{T}{\Bigl{(}\int\limits_{\Omega}{h^{\alpha+n-4}h_{x}^{6}dx}\Bigr{)}^{\frac{\alpha+m+5}{\alpha+n+7}}dt}+\\\
c_{2}M^{\frac{\alpha+3m-2n+2}{3}}\int\limits_{0}^{T}{\Bigl{(}\int\limits_{\Omega}{h^{\alpha+n-4}h_{x}^{6}dx}\Bigr{)}^{\frac{2}{3}}dt}.$
(3.28)
If $m<n+2$ then, using Young’s inequality, from (3.28) we arrive at
$\iint\limits_{Q_{T}}{h^{\alpha+m-2}h_{x}^{4}dxdt}\leqslant\epsilon\iint\limits_{Q_{T}}{h^{\alpha+n-4}h_{x}^{6}dxdt}+\\\
C(\epsilon)T(c_{1}M^{\frac{2\alpha+5m-3n+4}{n+2-m}}+c_{2}M^{\alpha+3m-2n+2}).$
(3.29)
Substituting (3.29) in (3.25), and choosing $\epsilon$ small enough, we obtain
$\mathcal{E}^{(\alpha)}_{0}(T)\leqslant\mathcal{E}^{(\alpha)}_{0}(0)+T(C_{1}M^{\frac{2\alpha+5m-3n+4}{n+2-m}}+C_{2}M^{\alpha+3m-2n+2})$
(3.30)
for $\alpha>2n-3m-1$ and $m<n+2$. Here $C_{2}=0$ if $\Omega$ is unbounded or
$h$ is compactly supported. In particular, if $0<\alpha+3m-2n+2\leqslant 1$,
i. e. $2n-3m-2<\alpha\leqslant 2n-3m-1$ then, using the Hölder inequality and
applying Young’s inequality, from (3.26) we obtain
$\iint\limits_{Q_{T}}{h^{\alpha+m-2}h_{x}^{4}dxdt}\leqslant\epsilon\iint\limits_{Q_{T}}{h^{\alpha+n-4}h_{x}^{6}dxdt}+\\\
C(\epsilon)|\Omega|^{2n-3m-1-\alpha}T\,M^{\alpha+3m-2n+2}$ (3.31)
for $2n-3m-2<\alpha\leqslant 2n-3m-1$. Substituting (3.31) in (3.25), and
choosing $\epsilon$ small enough, we obtain
$\mathcal{E}^{(\alpha)}_{0}(T)\leqslant\mathcal{E}^{(\alpha)}_{0}(0)+C_{3}T\,M^{\alpha+3m-2n+2}.$
(3.32)
If $m=n+2$ then, using Young’s inequality, from (3.28) we deduce
$\iint\limits_{Q_{T}}{h^{\alpha+m-2}h_{x}^{4}dxdt}\leqslant
c_{1}M^{2}\iint\limits_{Q_{T}}{h^{\alpha+n-4}h_{x}^{6}dxdt}+\\\
\epsilon\iint\limits_{Q_{T}}{h^{\alpha+n-4}h_{x}^{6}dxdt}+C(\epsilon)T\,M^{\alpha+n+8}.$
(3.33)
Substituting (3.33) in (3.25), and choosing $\epsilon$ enough small, we obtain
$\mathcal{E}^{(\alpha)}_{0}(T)\leqslant\mathcal{E}^{(\alpha)}_{0}(0)+C_{2}T\,M^{\alpha+n+8}$
(3.34)
for $\alpha>-n-7$, $m=n+2$ and $M\leqslant M_{c}$. Here $C_{2}=0$ if $\Omega$
is unbounded or $h$ is compactly supported.
Acknowledgement. The authors thank R. S. Laugesen and A. Burchard for useful
comments and discussions. The research of M. Chugunova is supported by the
NSERC Postdoctoral Fellowship. R. M. Taranets would like to thank M. C. Pugh
for the hospitality of the University of Toronto.
## Appendix A
###### Lemma A.1.
([13, 2]) 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
$\frac{1}{2}<n<3$ for $N=1$. Then the following estimate holds for any
positive functions $v\in H^{2}(\Omega)$ such that $\nabla v\cdot\vec{n}=0$ on
$\partial\Omega$ and $\int\limits_{\Omega}{v^{n}|\nabla\Delta v|^{2}}<\infty$:
$\int\limits_{\Omega}{\varphi^{6}\\{v^{n-4}|\nabla
v|^{6}+v^{n-2}|D^{2}v|^{2}|\nabla v|^{2}\\}}\leqslant\\\
c\Bigl{\\{}\int\limits_{\Omega}{\varphi^{6}v^{n}|\nabla\Delta
v|^{2}}+\int\limits_{\\{\varphi>0\\}}{v^{n+2}|\nabla\varphi|^{6}}\Bigr{\\}},$
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 A.2.
([16]) 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 $v(x)\in
W^{j,d}(\Omega)\cap L^{b}(\Omega)$:
$\left\|{D^{i}v}\right\|_{L^{a}(\Omega)}\leqslant
d_{1}\left\|{D^{j}v}\right\|_{L^{d}(\Omega)}^{\theta}\left\|v\right\|_{L^{b}(\Omega)}^{1-\theta}+d_{2}\left\|v\right\|_{L^{b}(\Omega)},\
\theta=\frac{{\tfrac{1}{b}+\tfrac{i}{N}-\tfrac{1}{a}}}{{\tfrac{1}{b}+\tfrac{j}{N}-\tfrac{1}{d}}}\in\left[{\tfrac{i}{j},1}\right)\\!\\!.$
## References
* [1] 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.
* [2] 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\.
* [3] Francisco Bernis and Avner Friedman. Higher order nonlinear degenerate parabolic equations. J. Differential Equations, 83(1):179–206, 1990.
* [4] 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.
* [5] 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.
* [6] 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.
* [7] 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.
* [8] 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.
* [9] E. Carlen, S. Ulusoy. An entropy dissipation-entropy estimate for a thin film type equation. Comm. Math. Sci., 3(2):171–178, 2005.
* [10] P. Constantin, T. F. Dupont, R. E. Goldstein, Leo P. Kadanoff, M. J. Shelley, and S. M. Zhou. Droplet breakup in a model of the Hele-Shaw cell. Physical review E, 47(6):4169–4181, june 1993.
* [11] P. Ehrhard. The spreading of hanging drops. Journal of colloid and interface science, 168(1):242–246, nov 1994.
* [12] S. D. Èĭdel′man. Parabolic systems. Translated from the Russian by Scripta Technica, London. North-Holland Publishing Co., Amsterdam, 1969.
* [13] 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.
* [14] Ansgar Jungel, and Danial Matthes. An algorithmic construction of entropies in higher-order nonlinear PDEs. Nonlinearity, 19:633–659, 2006.
* [15] R. S. Laugesen. New dissipated energies for the thin fluid film equation. Commun. Pure Appl. Anal., 4 (3): 613–634, 2005.
* [16] L. Nirenberg. An extended interpolation inequality. Ann. Scuola Norm. Sup. Pisa (3), 20:733–737, 1966.
* [17] 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.
* [18] A. Tudorascu. Lubrication approximation for thin viscous films: asymptotic behavior of nonnegative solutions. Communications in PDE, 32:1147–1172, 2007.
* [19] 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
| 2010-02-19T14:16:51 |
2024-09-04T02:49:08.477583
|
{
"license": "Public Domain",
"authors": "Marina Chugunova, Roman M. Taranets",
"submitter": "Roman Taranets",
"url": "https://arxiv.org/abs/1002.3741"
}
|
1002.3829
|
020001 2010 S. A. Cannas P. Netz, Inst. de Química, Univ. Federal do Rio
Grande do Sul, Brazil 020001
I have investigated the structural and dynamic properties of water by
performing a series of molecular dynamic simulations in the range of
temperatures from 213 K to 360 K, using the Simple Point Charge-Extended
(SPC/E) model. I performed isobaric-isothermal simulations (1 bar) of 1185
water molecules using the GROMACS package. I quantified the structural
properties using the oxygen-oxygen radial distribution functions, order
parameters, and the hydrogen bond distribution functions, whereas, to analyze
the dynamic properties I studied the behavior of the history-dependent bond
correlation functions and the non-Gaussian parameter $\alpha_{2}(t)$ of the
mean square displacement of water molecules. When the temperature decreases,
the translational ($\tau$) and orientational (Q) order parameters are linearly
correlated, and both increase indicating an increasing structural order in the
systems. The probability of occurrence of four hydrogen bonds and Q both have
a reciprocal dependence with $T$, though the analysis of the hydrogen bond
distributions permits to describe the changes in the dynamics and structure of
water more reliably. Thus, an increase on the caging effect and the occurrence
of long-time hydrogen bonds occur below $\sim$ 293 K, in the range of
temperatures in which predominates a four hydrogen bond structure in the
system.
# Structural and dynamic properties of SPC/E water
M. G. Campo [inst1] E-mail: mario@exactas.unlpam.edu.ar
(13 July 2009, Revised: 29 December 2009; 7 February 2010)
††volume: 2
99 inst1 Universidad Nacional de La Pampa, Facultad de Ciencias Exactas y
Naturales, Uruguay 151, 6300 Santa Rosa, La Pampa, Argentina.
## 1 Introduction
Water is the subject of numerous studies due to its biological significance
and its universal presence [1–3]. The thermodynamic behavior of water presents
important differences compared with those of the other substances, and many of
the characteristics of such behavior are often attributed to the existence of
hydrogen bonds between water molecules. Scientists have found that the water
structure produced by the hydrogen bonds is peculiar as compared to that of
other liquids. Then, the advances in the knowledge of hydrogen bond behavior
are crucial to understanding water properties.
The method of molecular dynamics (MD) allows to analyze the structure and
dynamics of water at the microscopic level and hence to complement
experimental techniques in which these properties can be interpreted only in a
qualitative way (infra-red absorption and Raman scattering [4], depolarized
light scattering [5, 6], neutron scattering [7], femtosecond spectroscopy
[8–11] and other techniques [12–14].
Among the usual methods to study the short range order in MD simulations of
water are the calculus of radial distribution functions, hydrogen bond
distributions and order parameters. The orientational order parameter Q
measures the tendency of the system to adopt a tetrahedral configuration
considering the water oxygen atom as vertices of a tetrahedron, whereas the
translational order parameter $\tau$ quantifies the deviation of the pair
correlation function from the uniform value of unity seen in an ideal gas [15,
16]. The order parameters are used to construct an order map, in which
different states of a system are mapped onto a plane $\tau$-Q. The order
parameters are, in general, independent, but they are linearly correlated in
the region in which the water behaves anomalously [17].
The dynamics of water can by characterized by the bond lifetime, $\tau_{HB}$,
associated to the process of rupturing and forming of hydrogen bonds between
water molecules which occurs at very short time scale [9, 18, 19, 21–23].
$\tau_{HB}$ is obtained in MD using the history-dependent bond correlation
function $P(t)$, which represents the probability that an hydrogen bond formed
at time $t=0$ remained continuously unbroken and breaks at time $t$ [24, 25].
Also, the dynamics of water can be studied by analyzing the mean-square
displacement time series $M(t)$. In addition to the diffusion coefficient
calculation at long times in which $M(t)\propto t$, in the supercooled region
of temperatures and at intermediate times $M(t)\propto t^{\alpha}$
($0<\alpha<1$). This behavior of $M(t)$ is associated to the subdiffusive
movement of the water molecules, caused by the caging effect in which a water
molecule is temporarily trapped by its neighbors and then moves in short
bursts due to nearby cooperative motion. A time $t^{\ast}$ characterizes this
caging effect (see Sec. II for more details) [26, 27].
In a previous work, we found a $q$-exponential behavior in $P(t)$, in which
$q$ increases with $T^{-1}$ approximately below 300 K. $q(T)$ is also
correlated with the probability of occurrence of four hydrogen bonds, and the
subdiffusive motion of the water molecules [28].
The relationship between dynamics and structural properties of water has not
been clearly established to date. In this paper, I explore whether the effect
that temperature has on the water dynamics reflects a more general connection
between the structure and the dynamics of this substance.
## 2 Theory and method
I have performed molecular dynamic simulations of SPC/E water model using the
GROMACS package [29, 30], simulating fourteen similar systems of 1185
molecules at 1 bar of pressure in a range of temperatures from 213 K to 360 K.
I initialized the system at 360 K using an aleatory configuration of water
molecules, assigning velocities to the molecules according to a Boltzmann’s
distribution at this temperature. For stabilization, I applied Berendsen’s
thermal and hydrostatic baths at the same temperature and 1 bar of pressure
[31]. Then, I ran an additional MD obtaining an isobaric-isothermal ensemble.
I obtained the other systems in a similar procedure, but using as initial
configuration that of the system of the preceding higher temperature and
cooling it at the slow rate of 30 K ns-1 [17]. Stabilization and sampling
periods for the systems at different temperatures are indicated in Table 1.
Simulation and sampling time steps were 2 fs and 10 fs, respectively. The
sampling time step was shorter than the typical time during which a hydrogen
bond can be destroyed by libration movements.
Table 1: Details of the simulation procedure. Duration of the stabilization period ($t_{est}$) and the MD sampling ($t_{MD}$) in the different ranges of temperatures Temp. range (K) | $t_{est}$ (ns) | $t_{MD}$ (ns)
---|---|---
213 - 243 | 20.0 | 10.0
253 - 273 | 16.0 | 10.0
283 - 360 | 16.0 | 8.0
Figure 1: Hydrogen bonds distribution functions $f(n)$ ($n=0,...,5$) versus
$T$. The zones A, B and C correspond to ranges of temperatures in which occur
different relationships between $f(4),f(3)$ and $f(2)$. Note the reciprocal
scale for the temperatures. See the text for details.
I calculated the hydrogen bond distribution functions $f(n)$ ($n=0,1,...,5$),
which is the probability of occurrence of $n$ hydrogen bonds by molecule,
considering a geometric definition of hydrogen bond [20]. As parameters for
this calculation, I used a maximum distance between oxygen atoms of 3.5
$\mathring{A}$ and a minimum angle between the atoms Odonor–H–Oacceptor of
145∘.
The radial distribution function (RDF) is a standard tool used in experiments,
theories, and simulations to characterize the structure of condensed matter.
Using RDFs, I obtained the average number, $N$, of water molecules in the
first hydration layer (the hydration number)
$N=4\pi\rho\int_{0}^{r_{min}}g(r)r^{2}dr$ (1)
where $\rho$ is the number density.
Figure 2: Oxygen-oxygen radial distribution functions for the systems at 213 K
(continuous line), 293 K (dashed line), and 360 K (dotted line). Inset: The
hydration number $N$ vs. $T^{5}$.
The translational order parameter, $\tau$, is defined in Ref. [16] as
$\tau\equiv\int_{0}^{S_{c}}|g(s)-1|ds$ (2)
where the dimensionless variable $s\equiv rn^{1/3}$ is the radial distance r
scaled by the mean intermolecular distance $n^{1/3}$, and $S_{c}$ corresponds
to half of the simulation box size.
The orientational order parameter $Q$ is defined as [15]
$Q=\left\langle
1-\frac{3}{8}\sum_{i=1}^{N}\sum_{j=1}^{4}\sum_{l=j+1}^{4}\left[cos\theta_{jik}+\frac{1}{3}\right]^{2}\right\rangle$
(3)
where $\theta_{jik}$ is the angle formed by the atoms Oj–Oi–Ok. Here, Oi is
the reference oxygen atom, and Oj and Ok are two of its four nearest
neighbors. $Q$=1 in an ideal configuration in which the oxygen atoms would be
located in the vertices of a tetrahedron.
I obtained the bond correlation function $P(t)$ from the simulations by
building a histogram of the hydrogen bonds lifetimes for each configuration.
Then, I fitted this function with a Tsallis distribution of the form
$\exp_{q}(t)=\left[1+\left(1-q\right)t\right]^{1/\left(1-q\right)}$ (4)
being $t$ the hydrogen bond lifetime and $q$ the nonextensivity parameter [28,
32]. If $q=1$, Eq. (4) reduces to an exponential, whereas if $q>1$, $P(t)$
decays more slowly than an exponential. This last behavior occurs when long
lasting hydrogen bonds increase their frequency of occurrence.
The subdiffusive movement of water occurs when the displacement of the
molecules obeys a non-Gaussian statistics. This behavior is characterized by
$t^{*}$, the time in which the non-Gaussian parameter $\alpha_{2}(t)$ reaches
a maximum [see Eq. (5)]. Then, $t^{*}$ is the parameter associated to the
average time during which a water molecule is trapped by its environment
(caging effect), and this prevents it from reaching the diffusive state [26,
27].
$\alpha_{2}(t)=\frac{3\langle r^{4}(t)\rangle}{5\langle r^{2}(t)\rangle}-1$
(5)
## 3 Results and discussion
Three zones or ranges of temperatures can be distinguished in the graph of the
hydrogen bond distributions $f(n)$ vs. $T$ (see Fig. 1). Zone A ($T$ $>$ 350
K) in which $f(3)>f(2)>f(4)$, zone B (293 K $>T>$ 350 K) in which
$f(3)>f(4)>f(2)$, and zone C ($T$ $<$ 293 K) in which $f(4)>f(3)>f(2)$. These
results indicate a predominant structure of three and two hydrogen bonds
(3HB-2HB) in zone A, 3HB-4HB in zone B, and 4HB-3HB in zone C, respectively.
$f(4)\propto T^{-1}$ in all ranges of temperatures, showing that the
tetrahedral structure of water decreases with the increase of this
variable.$f(3)$ increases with $T$ up to 293 K, and then remains approximately
constant ($\sim 0.4$) up to 360 K. $f(2)$ also increases with ${T}$ in all
range of temperatures, but only overcomes $f(4)$ at $T>$ 350 K.
Figure 3: Position of the first minimum of the oxygen-oxygen radial
distribution function vs $T^{4}$, associated to the size of the first
hydration layer.
Fig. 2 shows the oxygen-oxygen RDFs corresponding to the systems at 213 K, 293
K and 360 K. When the temperature decreases, the minimum and maximum tend to
be more defined. This being associated with an increasing order in the system.
The position of the first minimum moves closer to the origin decreasing the
size of the first hydration layer ($\propto T^{4}$ see Fig. 3). Both facts can
be associated to the decrease of the hydration number from $N\sim$ 5 to
$N\sim$ 4 (see inset, Fig. 2).
Figure 4: Order map with the values of the order parameters corresponding to
the simulated systems. Note the change in the slope of the line at $T\sim$ 273
K
The simultaneous behavior of Q and $\tau$ is shown in the order map of Fig. 4,
in which the location of the values corresponding to 293 K are indicated by an
arrow. The order parameters present similar behaviors with the temperature.
Upon cooling, these parameters are linearly correlated and move in the order
map along a line of increasing values, up to reaching maximum values at 213 K.
The slope of the line increases a little for $T>$ 293 K, indicating that
$\tau$ has a response to the increase of $T$ slightly higher than Q in this
range of temperatures. The positive values of the slopes indicate an
increasing order of the system when the temperature decrease.
The $f(n)$ functions allow to obtain a more detailed picture of the structural
orientational changes at shorter ranges between water molecules than the
orientational order parameter. While a small change at 293 K occurs in the
order map, the structures of two, three and four hydrogen bonds are alternated
in importance when the temperature changes. The ability of $f(n)$ to more
reliably describe the structure of the water occurs because the calculation of
the hydrogen bond distributions includes the location of the hydrogen atoms,
whereas Q only quantifies the changes in the average angle between neighbor
oxygen atoms. Although the behavior of $f(4)$ and Q are correlated (see Fig.
5), $f(4)$ shows a greater response to the temperature than Q, indicating that
the main change in the tetrahedral structure with the decrease of the
temperature occurs mainly in the orientation of the bonds between water
molecules. The approximately linear correlation between both variables also
indicates a similar dependence with the temperature ($\propto T^{-1}$).
Figure 5: Q vs $f(4)$. The change in $f(4)$ is higher than that of Q in the
range of temperatures studied. See the text for details. Figure 6: (a) Semilog
plot of $t^{*}$ vs. Q. (b) $q$ vs. Q. See the text for details.
Figure 6 shows the behavior of the dynamical parameters $t^{*}$ and $q$ with
Q. The characteristic time $t^{*}$ has an exponential response to $Q\geq 0.58$
(T $\leq$ 360 K), but the slope of the semilog plot of $t^{*}$ vs. Q increases
significantly for $Q\geq 0.67$. A similar change occurs for $Q\geq 0.67$ in
the linear correlation between $q$ and $Q$. Then, the values $Q\approx 0.67$
and $\tau\approx 1.1$ of the order map can be associated to changes in the
dynamics of the system. The transition of $q\approx 1$ to $q>1$ indicates the
increase of the probability of two water molecules remaining bonded by a
hydrogen bond during an unusual long time, whereas the increase of $t^{*}$ is
associated to the increase of the time during which the molecules remain in a
subdiffusive regime.
However, only the analysis of the $f(n)$ functions reveals the structural
modification that explains the structural and dynamic changes in the system.
The changes in the increase of the order map, $t^{*}(Q)$ and $q(Q)$ occur
below 293 K, in the range of temperatures in which prevail a structure of four
hydrogen bonds in the system.
## 4 Conclusions
The molecular dynamic method allows to study the structure and dynamics of the
SPC/E model of water in the range of temperatures from 213 K to 360 K.
Lowering the temperature of the system from 360 K to 213 K, the number of
water molecules in the first hydration layer decreases from $N\sim 5$ to
$N\sim 4$, along with a decrease in size. The increase of the tetrahedral
structure of the system is also characterized by a growth of the percentage of
occurrence of four hydrogen bonds and the orientational order parameter $Q$.
However, only the analysis of the behavior of the hydrogen bond distribution
allows to deduce that, when a tetrahedral structure associated to the
percentage of four hydrogen bonds predominates, the behavior of the dynamical
variables $P(t)$ and $t^{*}$ show the occurrence of long lasting hydrogen
bonds and caging effect between the molecules of the system.
###### Acknowledgements.
I am grateful for the financial support by PICTO UNLPAM 2005 30807 and
Facultad de Ciencias Exactas y Naturales (UNLPam).
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|
arxiv-papers
| 2010-02-19T21:38:32 |
2024-09-04T02:49:08.483703
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. G. Campo",
"submitter": "Luis Ariel Pugnaloni",
"url": "https://arxiv.org/abs/1002.3829"
}
|
1002.3897
|
# Sphere-foliated minimal and constant mean curvature hypersurfaces in product
spaces
Keomkyo Seo
###### Abstract.
In this paper, we prove that minimal hypersurfaces when $n\geq 3$ and nonzero
constant mean curvature hypersurfaces when $n\geq 2$ foliated by spheres in
parallel horizontal hyperplanes in ${\mathbb{H}}^{n}\times\mathbb{R}$ must be
rotationally symmetric.
Mathematics Subject Classification(2000) : 53A10, 53C42.
Key words and phrases : foliation, constant mean curvature, rotationally
symmetric hypersurface, product space.
## 1\. Introduction
In $\mathbb{R}^{3}$, catenoids and Riemann’s examples are the only minimal
surfaces foliated by circles. However, in higher-dimensional Euclidean space,
there are no examples of non-rotationally symmetric minimal hypersurfaces such
as Riemann’s examples in $\mathbb{R}^{3}$. In 1991, Jagy [6] proved that if
$M^{n}$ is a minimal hypersurface in $\mathbb{R}^{n+1}$ $(n\geq 3)$ and
foliated by $(n-1)$-dimensional spheres in parallel hyperplanes, then $M^{n}$
is rotationally symmetric about the axis containing the centers of all the
spheres. This result has been generalized to other spaces forms: sphere, the
hyperbolic and Lorentz-Minkowski space.(See [7], [8], and [11].)
In $\mathbb{H}^{2}\times\mathbb{R}$, Nelli and Rosenberg [9] found a
rotationally symmetric minimal surface which is called a $catenoid$ in
$\mathbb{H}^{2}\times\mathbb{R}$. In [4], Hauswirth provided several examples
of minimal surfaces foliated by horizontal curves of constant curvature in
$\mathbb{H}^{2}\times\mathbb{R}$. In particular, he constructed a two-
parameter family of Riemann type surfaces. Recently, Bérard and Sa Earp [2]
obtained some results on total curvature and index of higher-dimensional
catenoids in ${\mathbb{H}}^{n}\times\mathbb{R}$. On the other hand, Nelli et
al. [10] described the geometric behavior of rotationally symmetric constant
mean curvature surfaces in ${\mathbb{H}}^{2}\times\mathbb{R}$. They showed
that for $|H|>1/2$, the properties of rotationally symmetric constant mean
curvature surfaces in ${\mathbb{H}}^{2}\times\mathbb{R}$ are analogous to
those of the Delaunay surfaces in $\mathbb{R}^{3}$. Rotationally symmetric
constant mean curvature surfaces in ${\mathbb{H}}^{2}\times\mathbb{R}$ have
been studied in [1, 3, 5, 12].
Throughout this paper, we consider the upper half-space model of hyperbolic
space
$\displaystyle\mathbb{H}^{n}=\\{(x_{1},\cdots,x_{n})\in\mathbb{R}^{n}:x_{n}>0\\}$
equipped with the metric
$\displaystyle{ds^{2}=\frac{(dx_{1})^{2}+\cdots+(dx_{n})^{2}}{{x_{n}}^{2}}}$.
For a product space $\mathbb{H}^{n}\times\mathbb{R}$, we fix the metric
$ds^{2}+\varepsilon dt^{2}\,(\varepsilon=\pm 1)$. This metric is called
$Riemannian$ if $\varepsilon=1$ and $Lorentzian$ if $\varepsilon=-1$.
In this paper, we study hypersurfaces foliated by $(n-1)$-dimensional spheres
lying in parallel hyperplanes in some Riemannian and Lorentzian product
spaces. In Section 2, we shall prove that minimal hypersurfaces $(n\geq 3)$
and non-zero constant mean curvature hypersurfaces $(n\geq 2)$ foliated by
$(n-1)$-dimensional spheres in parallel horizontal hyperplanes in the
Riemannian product ${\mathbb{H}}^{n}\times\mathbb{R}$ should be rotationally
symmetric.(Theorem 2.1) As a consquence, one can see that there is no Riemann
type minimal hypersurfaces foliated by $(n-1)$-dimensional spheres in
${\mathbb{H}}^{n}\times\mathbb{R}$ for $n\geq 3$. We shall use Jagy’s idea [6]
to prove this result. (See also [8].) A key ingredient of the proof is the
following. We describe a hypersurface $M$ in
${\mathbb{H}}^{n}\times\mathbb{R}$ locally as the level set for a smooth
function $f$. If we orient $M$ by the unit normal vector field
$\displaystyle{N=-\frac{\nabla f}{|\nabla f|}}$, then the mean curvature $H$
is given by
(1.1) $\displaystyle nH=-{\rm div}\frac{\nabla f}{|\nabla f|},$
where $\nabla$ and ${\rm div}$ denote the gradient and divergence in
${\mathbb{H}}^{n}\times\mathbb{R}$, respectively. A straightforward
computation using the fact that $M$ is foliated by spheres in parallel
horizontal hyperplanes gives us the conclusion. In Section 3, applying the
similar arguments as in Section 2, we prove an analogue in the Lorentzian
product ${\mathbb{H}}^{n}\times\mathbb{R}$.
The author would like to thank the referee for useful suggestions on improving
the presentation of this paper.
## 2\. Sphere-foliated hypersurfaces in the Riemannian product
$\mathbb{H}^{n}\times\mathbb{R}$
In $\mathbb{H}^{n}\times\mathbb{R}$, a one-parameter family of hyperplanes
$\mathbb{H}^{n}\times\\{t\\}$ for $t\in\mathbb{R}$ are called $parallel$
$horizontal$ hyperplanes. We will deal with hypersurfaces foliated by spheres
in parallel horizontal hyperplanes in the Riemannian product space
$\mathbb{H}^{n}\times\mathbb{R}$.
###### Theorem 2.1.
Let $M$ be an $n$-dimensional hypersurface with constant mean curvature $H$ in
the Riemannian product $\mathbb{H}^{n}\times\mathbb{R}$ and foliated by
spheres in parallel horizontal hyperplanes. If $H\neq 0$ or $H=0$ and $n\geq
3$, then $M$ is a rotationally symmetric hypersurface.
Before proving the above theorem, we need the following well-known fact.
###### Lemma 2.2 ([13], p.81-82).
If an $(n-1)$-dimensional sphere has Euclidean center
$(0,\ldots,0,k)\in\mathbb{R}^{n}_{+}:=\\{(x_{1},\cdots,x_{n}):x_{n}>0\\}$ and
a Euclidean radius $r$, then it has the hyperbolic center
$(0,\ldots,0,K)\in\mathbb{R}^{n}_{+}$ and the hyperbolic radius $R$, where
$\displaystyle K=\sqrt{k^{2}-r^{2}}\,\,\mbox{and
}\,\,R=\frac{1}{2}\ln\frac{k+r}{k-r}.$
Proof of Theorem 2.1. Let $P_{t_{1}}=\mathbb{H}^{n}\times\\{t_{1}\\}$ and
$P_{t_{2}}=\mathbb{H}^{n}\times\\{t_{2}\\}$ be two horizontal hyperplanes of
the foliation for $t_{1}<t_{2}$. Let $M^{*}$ be the piece of $M$ between
$P_{t_{1}}$ and $P_{t_{2}}$. The boundary $\partial M^{*}$ of $M^{*}$ consists
of two $(n-1)$-dimensional spheres $(M^{*}\cap P_{t_{1}})\cup(M^{*}\cap
P_{t_{2}})$. After an isometric transformation in
$\mathbb{H}^{n}\times\mathbb{R}$, we may assume that the hyperbolic centers of
the two boundary spheres are given by $(0,\cdots,0,k_{1},t_{1})$ and
$(0,\cdots,0,k_{2},t_{2})$ in $\mathbb{R}^{n}_{+}\times\mathbb{R}$ for some
constants $k_{1},k_{2}>0$, respectively. Note that these two boundary spheres
are symmetric to the hyperplanes $\\{x_{1}=0\\},\cdots,\\{x_{n-1}=0\\}$. The
well-known Aleksandrov reflection principle shows that $M^{*}$ inherits the
symmetries of its boundary $\partial M^{*}$. Therefore, for each $t_{1}\leq
t\leq t_{2}$, the hyperbolic center of each level $M\cap\\{x_{n+1}=t\\}$ is
symmetric to the hyperplanes $\\{x_{1}=0\\},\cdots,\\{x_{n-1}=0\\}$. Hence it
follows that the hyperbolic center of each level lies in the $2$-plane
$\\{x_{1}=\cdots=x_{n-1}=0\\}$.
Using Lemma 2.2, we parametrize the hyperbolic centers of the
$(n-1)$-dimensional spheres by
$t\longmapsto(0,\ldots,0,K(t),t)\in\mathbb{H}^{n}\times\mathbb{R}$ for
$t\in[t_{1},t_{2}]$, and hence the Euclidean centers of the spheres by
$t\longmapsto(0,\ldots,0,k(t),t)\in\mathbb{H}^{n}\times\mathbb{R}$. Then it
follows from Lemma 2.2 that
$\displaystyle K(t)=\sqrt{k(t)^{2}-r(t)^{2}},$
where $r(t)$ is the Euclidean radius of $M\cap\mathbb{H}^{n}\times\\{t\\}$.
Note that $K(t)>0$ for $t\in[t_{1},t_{2}]$. Moreover, we see that $M^{*}$ is
the level set $\\{f=0\\}$ of a function $f$ given by
(2.1) $\displaystyle
f(x_{1},\ldots,x_{n},t)=\sum_{i=1}^{n-1}x_{i}^{2}+(x_{n}-k(t))^{2}-r(t)^{2}.$
To prove the theorem, it is sufficient to show that
$\displaystyle\frac{dK(t)}{dt}=0,$
which means that $M$ is a rotationally symmetric hypersurface whose rotation
axis is the geodesic $\gamma(t)=\\{(0,\ldots,0,K,t)\\}$ for some constant $K$.
Note that the metric on $\mathbb{H}^{n}\times\mathbb{R}$ is given by
$\displaystyle\sum_{i,j}g_{ij}dx_{i}\otimes
dx_{j}=\frac{1}{x_{n}^{2}}dx_{1}^{2}+\cdots+\frac{1}{x_{n}^{2}}dx_{n}^{2}+dt^{2}.$
Since
$\displaystyle\nabla f$ $\displaystyle=$
$\displaystyle\sum_{i,j}g^{ij}\frac{\partial f}{\partial
x_{i}}\frac{\partial}{\partial x_{j}}$ $\displaystyle=$
$\displaystyle(2x_{n}^{2}x_{1},\cdots,2x_{n}^{2}x_{n-1},2x_{n}^{2}(x_{n}-k),-2(x_{n}-k)k^{\prime}-2rr^{\prime}),$
we have
$\displaystyle|\nabla
f|^{2}=4\Big{(}x_{n}^{2}x_{1}^{2}+\cdots+x_{n}^{2}x_{n-1}^{2}+x_{n}^{2}(x_{n}-k)^{2}+((x_{n}-k)k^{\prime}+rr^{\prime})^{2}\Big{)}.$
Now we compute the mean curvature of $M^{*}$ using the equation (1.1).
$\displaystyle-nH={\rm div}\frac{\nabla f}{|\nabla f|}$ $\displaystyle=$
$\displaystyle\sum_{i,j}\frac{1}{\sqrt{g}}\frac{\partial}{\partial
x_{j}}\Big{(}\sqrt{g}\frac{g^{ij}}{|\nabla f|}\frac{\partial f}{\partial
x_{i}}\Big{)}$ $\displaystyle=$ $\displaystyle\sum_{j}\frac{\partial
Z^{j}}{\partial
x_{j}}+\sum_{j}\frac{1}{\sqrt{g}}\Big{(}\frac{\partial\sqrt{g}}{\partial
x_{j}}\Big{)}Z^{j},$
where $\displaystyle{Z^{j}=\sum_{i}\frac{g^{ij}}{|\nabla f|}\frac{\partial
f}{\partial x_{i}}}$ and $g=\det(g_{ij})=x_{n}^{-2n}$. Then we have
$\displaystyle-nH$
$\displaystyle=\frac{x_{n}^{2}}{S}-\frac{(x_{n}^{2}x_{1})^{2}}{S^{3}}+\cdots+\frac{x_{n}^{2}}{S}-\frac{(x_{n}^{2}x_{n-1})^{2}}{S^{3}}$
$\displaystyle\quad+\frac{x_{n}^{2}+2x_{n}(x_{n}-k)}{S}-\frac{x_{n}^{2}(x_{n}-k)\\{x_{n}r^{2}+x_{n}^{2}(x_{n}-k)+Ak^{\prime}\\}}{S^{3}}$
$\displaystyle\quad+\frac{B}{S}-\frac{A\\{x_{n}^{2}(x_{n}-k)k^{\prime}+AB\\}}{S^{3}}$
$\displaystyle\quad+x_{n}^{n}(-nx_{n}^{-n-1})\frac{x_{n}^{2}(x_{n}-k)}{S},$
where $A=(x_{n}-k)k^{\prime}+rr^{\prime},$ $B=k^{\prime
2}-(x_{n}-k)k^{\prime\prime}-r^{\prime 2}-rr^{\prime\prime},$ and
$\displaystyle S=\frac{|\nabla f|}{2}$ $\displaystyle=$
$\displaystyle\sqrt{x_{n}^{2}x_{1}^{2}+\cdots+x_{n}^{2}x_{n-1}^{2}+x_{n}^{2}(x_{n}-k)^{2}+((x_{n}-k)k^{\prime}+rr^{\prime})^{2}}$
$\displaystyle=$ $\displaystyle\sqrt{x_{n}^{2}r^{2}+A^{2}}.$
Thus we have
$\displaystyle-
nHS^{3}=2A^{2}x_{n}^{2}+(n-1)kr^{2}x_{n}^{3}+(n-2)kx_{n}A^{2}+r^{2}x_{n}^{2}B-2x_{n}^{3}k^{\prime}A+2kk^{\prime}x_{n}^{2}A.$
Squaring the above equation, we obtain
$\displaystyle n^{2}H^{2}S^{6}$
$\displaystyle=n^{2}H^{2}\\{x_{n}^{2}r^{2}+(k^{\prime}x_{n}+rr^{\prime}-kk^{\prime})^{2}\\}^{3}$
(2.2) $\displaystyle=\Big{[}\\{2rr^{\prime}k^{\prime}+(n-2)kk^{\prime
2}+(n-1)kr^{2}-r^{2}k^{\prime\prime}\\}x_{n}^{3}$
$\displaystyle\quad+\\{(k^{\prime 2}+r^{\prime
2}-rr^{\prime\prime}+kk^{\prime\prime})r^{2}+2(n-3)kk^{\prime}rr^{\prime}-2(n-2)k^{\prime
2}k^{2}\\}x_{n}^{2}$
$\displaystyle\quad+k(n-2)(rr^{\prime}-kk^{\prime})^{2}x_{n}\Big{]}^{2}$
Suppose that $H\neq 0$. Let us fix a section $t$. Since $x_{n}$ is varied, we
regard (2) as an equation on $x_{n}$ where the coefficients are functions of
the independent variable $t$. Comparing the degree $0$-terms in both sides of
(2), we get
$\displaystyle n^{2}H^{2}(rr^{\prime}-kk^{\prime})^{6}=0.$
Therefore it follows that
$\displaystyle\frac{dK(t)}{dt}=\frac{d}{dt}\sqrt{k(t)^{2}-r(t)^{2}}=\frac{kk^{\prime}-rr^{\prime}}{\sqrt{k^{2}-r^{2}}}=0.$
Now suppose that $H=0$ and $n\geq 3$. Comparing the coefficients of the degree
$2$-terms in both sides of (2), we have
$\displaystyle k(n-2)(rr^{\prime}-kk^{\prime})^{2}=0.$
Therefore $rr^{\prime}-kk^{\prime}=0$, which also implies that
$\displaystyle\frac{dK(t)}{dt}=0.$
Hence we can conclude that $M$ is a rotationally symmetric hypersurface in
both cases.
∎
Remark. In $\mathbb{H}^{2}\times\mathbb{R}$, Hauswirth [4] constructed several
Riemann type minimal surfaces foliated by circles. However, as mentioned in
the introduction, it follows from the above theorem that there is no Riemann
type minimal hypersurface which is not rotationally symmetric and foliated by
$(n-1)$-dimensional spheres lying in parallel horizontal hyperplanes in
$\mathbb{H}^{n}\times\mathbb{R}$ when $n\geq 3$.
## 3\. Sphere-foliated hypersurfaces in the Lorentzian product
$\mathbb{H}^{n}\times\mathbb{R}$
An immersed hypersurface $M$ in the Lorentz product space
$\mathbb{H}^{n}\times\mathbb{R}$ endowed with the Lorentzian metric
$\displaystyle
ds^{2}=\frac{(dx_{1})^{2}+\cdots+(dx_{n})^{2}}{{x_{n}}^{2}}-(dt)^{2}$
is called $spacelike$ if the induced metric on $M$ is a Riemannian metric. If
the hypersurface is locally the level set of a smooth function $f$, the fact
that $M$ is spacelike means that $\nabla f$ is a $timelike$ vector:
$\displaystyle\langle\nabla f,\nabla f\rangle<0.$
If we orient $M$ by the unit normal vector field
$\displaystyle{N=-\frac{\nabla f}{|\nabla f|}}$, then the mean curvature $H$
is given by
(3.1) $\displaystyle nH=-{\rm div}\frac{\nabla f}{|\nabla f|},$
where $|\nabla f|=\sqrt{-\langle\nabla f,\nabla f\rangle}$ and ${\rm div}$
denotes the divergence with respect to the Lorentzian metric on the product
space $\mathbb{H}^{n}\times\mathbb{R}$.
As in the proof of Theorem 2.1, consider two horizontal hyperplanes of the
foliation $P_{t_{1}}=\mathbb{H}^{n}\times\\{t_{1}\\}$ and
$P_{t_{2}}=\mathbb{H}^{n}\times\\{t_{2}\\}$ for $t_{1}<t_{2}$. Applying the
Aleksandrov reflection principle in ${\mathbb{H}}^{n}\times\mathbb{R}$, we see
that the piece $M^{*}$ between $P_{t_{1}}$ and $P_{t_{2}}$ has the symmetries
of its boundary $\partial M^{*}=(M^{*}\cap P_{t_{1}})\cup(M^{*}\cap
P_{t_{2}})$. Therefore, for each $t_{1}\leq t\leq t_{2}$, the hyperbolic
center of each level $M\cap\\{x_{n+1}=t\\}$ lies in the same $2$-plane. After
a translation in ${\mathbb{H}}^{n}\times\mathbb{R}$, we may assume that this
$2$-plane is defined by $x_{1}=\cdots=x_{n-1}=0$.
Using Lemma 2.2 again, we can parametrize the hyperbolic centers of the
$(n-1)$-dimensional spheres by
$t\longmapsto(0,\ldots,0,K(t),t)\in\mathbb{H}^{n}\times\mathbb{R}$ for
$t\in[t_{1},t_{2}]$, and the Euclidean centers of the spheres by
$t\longmapsto(0,\ldots,0,k(t),t)\in\mathbb{H}^{n}\times\mathbb{R}$, where
$K(t)=\sqrt{k(t)^{2}-r(t)^{2}}$ and $r(t)$ is the Euclidean radius of
$M\cap\mathbb{H}^{n}\times\\{t\\}$. Then $M^{*}$ is the level set $\\{f=0\\}$
of a function $f$ defined as in (2.1). Note that the metric on the Lorentzian
product $\mathbb{H}^{n}\times\mathbb{R}$ is given by
$\displaystyle\sum_{i,j}g_{ij}dx_{i}\otimes
dx_{j}=\frac{1}{x_{n}^{2}}dx_{1}^{2}+\cdots+\frac{1}{x_{n}^{2}}dx_{n}^{2}-dt^{2}.$
Since
$\displaystyle\nabla f$ $\displaystyle=$
$\displaystyle\sum_{i,j}g^{ij}\frac{\partial f}{\partial
x_{i}}\frac{\partial}{\partial x_{j}}$ $\displaystyle=$
$\displaystyle\Big{(}2x_{n}^{2}x_{1},\cdots,2x_{n}^{2}x_{n-1},2x_{n}^{2}(x_{n}-k),2((x_{n}-k)k^{\prime}+rr^{\prime})\Big{)},$
we get
$\displaystyle-\langle\nabla f,\nabla
f\rangle=4\Big{(}-x_{n}^{2}x_{1}^{2}-\cdots-
x_{n}^{2}x_{n-1}^{2}-x_{n}^{2}(x_{n}-k)^{2}+((x_{n}-k)k^{\prime}+rr^{\prime})^{2}\Big{)}.$
Using the mean curvature equation (3.1), we have
$\displaystyle-nH={\rm div}\frac{\nabla f}{|\nabla f|}$ $\displaystyle=$
$\displaystyle\sum_{i,j}\frac{1}{\sqrt{|g|}}\frac{\partial}{\partial
x_{j}}\Big{(}\sqrt{|g|}\frac{g^{ij}}{|\nabla f|}\frac{\partial f}{\partial
x_{i}}\Big{)},$
where $|\nabla f|=\sqrt{-\langle\nabla f,\nabla f\rangle}$ and
$|g|=|\det(g_{ij})|=x_{n}^{-2n}$.
A similar computation as in the proof of Theorem 2.1 shows that
$\displaystyle n^{2}H^{2}$
$\displaystyle\\{-x_{n}^{2}r^{2}+(k^{\prime}x_{n}+rr^{\prime}-kk^{\prime})^{2}\\}^{3}$
(3.2) $\displaystyle=\Big{[}\\{2rr^{\prime}k^{\prime}-(n-2)kk^{\prime
2}-(n-1)kr^{2}+r^{2}k^{\prime\prime}\\}x_{n}^{3}$
$\displaystyle\quad+\\{(k^{\prime 2}+r^{\prime
2}-rr^{\prime\prime}+kk^{\prime\prime})r^{2}-2(n-1)kk^{\prime}rr^{\prime}+2(n-2)k^{\prime
2}k^{2}\\}x_{n}^{2}$
$\displaystyle\quad-k(n-2)(rr^{\prime}-kk^{\prime})^{2}x_{n}\Big{]}^{2}$
Suppose that $H\neq 0$. Comparing the degree $0$-terms in both sides of the
equation (3) of variable $x_{n}$, we obtain
$\displaystyle n^{2}H^{2}(rr^{\prime}-kk^{\prime})^{6}=0.$
Therefore it follows that
(3.3)
$\displaystyle\frac{dK(t)}{dt}=\frac{d}{dt}\sqrt{k(t)^{2}-r(t)^{2}}=\frac{kk^{\prime}-rr^{\prime}}{\sqrt{k^{2}-r^{2}}}=0.$
Now suppose that $H=0$ and $n\geq 3$. Comparing the coefficients of the degree
$2$-terms in both sides of (3), we have
$\displaystyle k(n-2)(rr^{\prime}-kk^{\prime})^{2}=0.$
So we have $rr^{\prime}-kk^{\prime}=0$, which also implies that
(3.4) $\displaystyle\frac{dK(t)}{dt}=0.$
From (3.3) and (3.4), it follows that the hyperbolic center of each
hypersphere in parallel horizontal hyperplane lies in a vertical geodesic line
of the Lorentzian product $\mathbb{H}^{n}\times\mathbb{R}$. Therefore we
obtain the following.
###### Theorem 3.1.
Let $M$ be an $n$-dimensional spacelike hypersurface with constant mean
curvature $H$ in the Lorentzian product $\mathbb{H}^{n}\times\mathbb{R}$ and
foliated by spheres in parallel horizontal hyperplanes. If $H\neq 0$ or $H=0$
and $n\geq 3$, then $M$ is rotationally symmetric.
## References
* [1] U. Abresch and H. Rosenberg, A Hopf differential for constant mean curvature surfaces in $\mathbb{S}^{2}\times\mathbb{R}$ and $\mathbb{H}^{2}\times\mathbb{R}$, Acta Math. 193 (2004), no. 2, 141-174.
* [2] P. Bérard and R. Sa Earp, Minimal hypersurfaces in ${\mathbb{H}}^{n}\times\mathbb{R}$, total curvature and index, arXiv: 0808.3838v1.
* [3] M. Cavalcante and J. de Lira, Examples and structure of CMC surfaces in some Riemannian and Lorentzian homogeneous spaces, Michigan Math. J. 55 (2007), no. 1, 163-181.
* [4] L. Hauswirth, Minimal surfaces of Riemann type in three-dimensional product manifolds, Pacific J. Math. 224 (2006), no. 1, 91-117.
* [5] W.-T. Hsiang and W.-Y. Hsiang, On the uniqueness of isoperimetric solutions and imbedded soap bubbles in noncompact symmetric spaces. I, Invent. Math. 98 (1989), no. 1, 39-58.
* [6] W. Jagy, Minimal hypersurfaces foliated by spheres, Michigan Math. J. 38 (1991), no. 2, 255-270.
* [7] W. Jagy, Sphere-foliated constant mean curvature submanifolds, Rocky Mountain J. Math. 28 (1998), no. 3, 983-1015.
* [8] R. López, Constant mean curvature hypersurfaces foliated by spheres, Differential Geom. Appl. 11 (1999), no. 3, 245-256.
* [9] B. Nelli and H. Rosenberg, Minimal surfaces in ${\mathbb{H}}^{2}\times\mathbb{R}$, Bull. Braz. Math. Soc. (N.S.) 33 (2002), no. 2, 263-292.
* [10] B. Nelli, R. Sa Earp, W. Santos, and E. Toubiana, Uniqueness of $H$-surfaces in ${\mathbb{H}}^{2}\times\mathbb{R},|H|\leq 1/2$, with boundary one or two parallel horizontal circles, Ann. Global Anal. Geom. 33 (2008), no. 4, 307-321.
* [11] S.-H. Park, Sphere-foliated minimal and constant mean curvature hypersurfaces in space forms and Lorentz-Minkowski space, Rocky Mountain J. Math. 32 (2002), no. 3, 1019-1044.
* [12] R. Sa Earp and E. Toubiana, Screw motion surfaces in $\mathbb{H}^{2}\times\mathbb{R}$ and $\mathbb{S}^{2}\times\mathbb{R}$, Illinois J. Math. 49 (2005), no. 4, 1323-1362.
* [13] S. Stahl, The Poincare half-plane. A gateway to modern geometry. Jones and Bartlett Publishers, Boston, MA, 1993.
Department of Mathematics
Sookmyung Women’s University
Hyochangwongil 52, Yongsan-ku
Seoul, 140-742, Korea
e-mail : kseo@sookmyung.ac.kr
|
arxiv-papers
| 2010-02-20T14:44:26 |
2024-09-04T02:49:08.489509
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Keomkyo Seo",
"submitter": "Keomkyo Seo",
"url": "https://arxiv.org/abs/1002.3897"
}
|
1002.4030
|
11institutetext: Key Laboratory for Particle Astrophysics, Institute of High
Energy Physics, Beijing 100049, China 22institutetext: Max-Planck-Institut für
Extraterrestrische Physik, P.O. Box 1603, 85740 Garching, Germany
33institutetext: ICREA & Institut de Ciències de l’Espai (IEEC-CSIC), Campus
UAB, Facultat de Ciències, Torre C5-parell, 2a planta, 08193 Barcelona, Spain
44institutetext: Theoretical Physics Center for Science Facilities (TPCSF),
CAS
# INTEGRAL and Swift/XRT observations
of the source PKS 0208-512
Shu Zhang 11 Werner Collmar 22 Diego F. Torres 33 Jian-Min Wang 1144 Michael
Lang 22 Shuang-Nan Zhang 11 szhang@mail.ihep.ac.cn
(Received / Accepted )
###### Abstract
Aims. The active galaxy PKS 0208-512, detected at lower energies by COMPTEL,
has been claimed to be a MeV blazar from EGRET. We report on the most recent
_INTEGRAL_ observations of the blazar PKS 0208-512, which are supplemented by
_Swift_ ToO observations.
Methods. The high energy X-ray and $\gamma$-ray emission of PKS 0208-512
during August - December 2008 has been studied using 682 ks of INTEGRAL guest
observer time and $\sim$ 56 ks of _Swift_ /XRT observations. These data were
collected during the decay of a $\gamma$-ray flare observed by _Fermi_ /LAT.
Results. At X-ray energies (0.2 – 10 keV) PKS 0208-512 is significantly
detected by _Swift_ /XRT, showing a power-law spectrum with a photon index of
$\sim$ 1.64. Its X-ray luminosity varied by roughly 30$\%$ during one month.
At hard X-/soft $\gamma$-ray energies PKS 0208-512 shows a marginally
significant ($\sim$ 3.2 $\sigma$) emission in the 0.5-1 MeV band when
combining all _INTEGRAL_ /SPI data. Non-detections at energies below and above
this band by _INTEGRAL_ /SPI may indicate intrinsic excess emission. If this
possible excess is produced by the blazar, one possible explanation could be
that its jet consists of an abundant electron-positron plasma, which may lead
to the emission of an annihilation radiation feature. Assuming this scenario,
we estimate physical parameters of the jet of PKS 0208-512.
###### Key Words.:
X-rays: individual: PKS 0208-512
††offprints: Shu Zhang
## 1 Introduction
It has been suggested that relativistic jets of Active Galactic Nuclei (AGN)
may contain electron-positron pair plasmas (Begelman et al. 1984). The
detection of the high energy $\gamma$-ray emission from a very compact region
around the central supermassive black hole in M87 (Aharonian et al. 2006,
Acciari et al. 2009) suggested that electron-positron pair plasmas can be
generated near black holes, thus setting up favorable conditions for launching
electron-positron pair plasma jets. This is further supported by several AGN
observations, like those of PKS 2155-304 and Mkn 501, which have been detected
to process much shorter (3-5 minutes) variability (Aharonian et al. 2007;
Albert et al. 2007). Reynolds et al. (1996) argued for the dominance of an
electron-positron pair plasma in the jet of M87 and thus in other AGNs, and
e.g. Wardle et al. 1998, Hirotani et al. 2000, Lobanov & Zensus 2001, Kino &
Takahara 2004, Dunn et al. 2006 studied different aspects of this eventuality.
Other recent studies propose that extragalactic jets can also be dynamically
dominated by cold protons (e.g., Celotti & Ghisellini 2008). The determination
of the jet content will certainly have important implications for
understanding the mechanisms of jet launching, acceleration and collimation.
The annihilation radiation in AGN jets may reveal itself in two different
manners. Jets containing positrons hit a dense ambient medium, then the
positrons can be thermalized and annihilate with the ambient electrons,
thereby forming a narrow peak around 511 keV. Marscher et al. (2007) searched
for such a narrow ($\leq$ 3 keV) emission feature in the spectrum of the radio
galaxy 3C 120, in which the jet strongly interacts with interstellar clouds.
They did not find a signal, and their upper limit did not constrain the
positron-to-proton ratio in the jet. Alternatively, electrons and positrons
may be already thermalized in the jets, and thus a broadened and blue-shifted
annihilation component could show up on top of the spectral continuum in the
MeV energy range (e.g., Böttcher & Schlickeiser 1996; Skibo et al. 1997).
During the Compton Gamma Ray Observatory (_CGRO_) era, several AGNs were
reported to show such broad bumps at MeV energies (e.g., Bloemen et al. 1995;
Blom et al. 1995), which were called “MeV blazars” due to their excess
emission at MeV energies. However, none of them passed subsequent data
analysis checks (Blom et al. 1995; Williams et al. 2001; Stacy et al. 2003),
and so these reports were discussed quite controversially. In fact, the first
convincing signal of electron-positron annihilation radiation from AGN jets is
yet to be found.
The source PKS 0208-512 is one such _CGRO_ -detected $\gamma$-ray source
claimed to be a MeV blazar. Two COMPTEL observations showed excess MeV-
emission (Blom et al. 1995) compared to the extrapolation of the EGRET
spectrum, measured at energies above 100 MeV (Bertsch et al. 1993; Skibo et
al. 1997). The variability of the source above 100 MeV was studied by von
Montigny et al. (1995). They found a characteristic variability timescale of
eight days, which suggested a $\gamma$-ray emission region on the order of
several Schwarzschild radii for a black hole of 1010 M⊙. Further $\gamma$-ray
observations and data analyses of this blazar led to the detection of
persistent low-energy ($<$3 MeV) MeV emission at a significance level of about
4 $\sigma$ for the period 1991-1998 (Williams et al. 2001). A comparison to
the contemporary EGRET spectrum ($>$100 MeV) (Hartman et al. 1999) indicated a
MeV excess. The strength of this excess however is uncertain, because it
varied with COMPTEL event selections (Williams et al. 2001). Furthermore, the
statistical significance of this excess was challenged by analyses using all
COMPTEL data (Stacy et al. 2003). So, even today there is still no unambiguous
evidence for a MeV excess in the spectrum of PKS 0208-512, or any other
blazar.
The satellite BeppoSAX observed PKS 0208-512 on 14th January 2001 and measured
a spectral index of 1.64$\pm$0.10 at soft X-rays (Donato et al. 2005). The
Chandra observation (Schwartz 2006, 2007) with an exposure of $\sim$ 5 ks
showed a complex X-ray, which can be resolved into at least four regions: the
dominant core, two regions along the jet that may correspond to a hot spot and
an extended lobe, and a possible fourth region outside the jet. By adopting a
CMB (cosmic microwave background) model, where the X-rays are assumed to be
produced via Comptonization of the cosmic microwave background photons by jet
blob electrons, the jet Doppler factor was estimated as 5.7-7.3. The fit of
the Spectral Energy Distribution (SED), containing the 2008 _Fermi_ data at
energies above 200 MeV, resulted in a Lorentz factor of about 10 (Ghisellini
et al. 2009).
We conducted _INTEGRAL_ observations of PKS 0208-512 between August and
December 2008 for a total of $\sim$ 682 ks. Since August 2008 the _Fermi_
observatory (Atwood et al. 2009) surveys the sky at energies above 30 MeV with
its _Large Area Telescope_ (LAT), thereby also monitoring PKS 0208-512 on a
daily basis. The source showed a flare in 2008. The long-term monitoring in
optical and near-IR bands revealed a significant brightening of PKS 0208-512,
with an increase of 1.3 mag in the B-band and 1.4 mag in the R-band between
September 11 and 30, 2008 (Buxton et al. 2008). A contemporaneous increase in
$\gamma$-rays was detected by _Fermi_ (Tosti et al. 2008). The publicly
available $\gamma$-ray lightcurve111at
http://fermi.gsfc.nasa.gov/ssc/data/access/lat/msl$\\_$lc shows a variable
flux, up to about a factor of 6 on timescales of weeks. Our _INTEGRAL_
observations were carried out during the decay of this flaring event.
Supplementary to our _INTEGRAL_ /SPI observations, we were granted two
contemporaneous _Swift_ ToO observations, providing the blazar state at softer
X-ray energies. In this paper, we report on our findings.
## 2 Observations and data analysis
The ESA scientific mission _INTEGRAL_ is dedicated to high-resolution
spectroscopy ($E/\Delta E\simeq 500$; SPI see Vedrenne et al. 2003) and
imaging (angular resolution: $12^{\prime}$ FWHM, point source location
accuracy: $\simeq 1^{\prime}-3^{\prime}$; IBIS/ISGRI, see Ubertini et al. 2003
and Lebrun et al. 2003) of celestial sources in the energy range $15$ keV to
$10$ MeV. _INTEGRAL_ also provides simultaneous monitoring at X-rays ($3-35$
keV, angular resolution: $3^{\prime}$; JEM-X, see Lund et al. 2003) and
optical wavelengths (Johnson V-filter, $550$ nm; OMC, see Mas-Hesse et al.
2003). All _INTEGRAL_ instruments, except OMC, work with coded masks.
Observations of 682 ks in total were carried out between August and December
2008 by applying the $5\times 5$ dithering mode centered on PKS 0208-512.
These data comprise 197 science windows (scw), each lasting typically for 3
ks. _INTEGRAL_ /SPI was operational for about 510 ks during these
observations. Table 1 gives the details of the _INTEGRAL_ observations. The
data are analyzed with the _INTEGRAL_ Offline Scientific Analysis (OSA)
version 7.0. For the spectral analysis of the SPI data we used SPIMODFIT
version 3.0, which is now also available in OSA version 8.0.
The $\gamma$-ray burst explorer _Swift_ was launched on November 20, 2004. It
carries three co-aligned detectors (Gehrels et al. 2004), namely the Burst
Alert Telescope (BAT, Barthelmy et al. 2005), the X-Ray Telescope (XRT,
Burrows, et al. 2005) and the Ultraviolet/Optical Telescope (UVOT, Roming et
al. 2005). Between 2005 and 2008 _Swift_ /XRT observed PKS 0208-512 13 times,
exposing longer than 2 ks (Table 2). This yielded a total exposure time of
$\sim$56 ks. The 2008 observations were – by our ToO request – contemporaneous
to the _INTEGRAL_ observations. We selected all these observations for our
studies, and analyzed the XRT data with Heasoft v. 6.2. For the spectral
fitting we applied XSPEC v 12.3.1 and estimated the model parameters at the
$90$% confidence level.
## 3 Results
The source PKS 0208-512 was not detected by the _INTEGRAL_ instruments
IBIS/ISGRI and JEM-X, neither in individual scw nor in the sum of all data.
However, the image of _INTEGRAL_ /SPI, combining the whole data set ($\sim$
510 ks), reveals evidence for a detection at the 3.2 $\sigma$ level in the
0.5–1 MeV band (Fig. 1). A single source, exactly coincident in location with
PKS 0208-512, is visible in the SPI map, which is otherwise rather empty and
clean, at a flux level of (1.50$\pm$0.47)$\times$10-3 ph cm-2 s-1. The source
is not visible with SPI in other energy ranges, i.e. below 0.5 MeV and above 1
MeV. Maps in the 0.3–0.5 MeV band and above 1.0 MeV remain empty, i.e.,
yielding only upper flux limits. The SPI energy bands 0.3-0.5 MeV and 0.5-1.0
MeV adopted here are commonly used in SPI analyses. For example, Knödlseder et
al. (2007) used 0.3-0.5 MeV and 0.514-1.0 MeV for producing SPI all-sky maps;
Bouchet et al. (2005, 2008) used 0.3-0.6 MeV and 0.6-1.0 MeV bands, and Petry
et al. (2009) chose 0.278-0.502 MeV and 0.502-1.0 MeV for imaging SPI sources.
We selected our bands before the analysis, based on these typical SPI
choices.222 Above 1.0 MeV we took the high-energy band limit as 1.4 MeV
because the sensitivity above $\sim$ 1.4 MeV worsens (see the SPI user manual,
Dubath & Kreykenbohm 2007). We also notice that the presentation of the
imaging results in an alternative binning, for instance using a constant ratio
of the high to low energies defining the band, $E_{high}/E_{low}$=2, (i.e.,
bands as 0.25–0.5 MeV, 0.5–1.0 MeV, 1.0–2.0 MeV) differs little, if at all,
from those reported in Table 1.
In order to test for a possible flare or for a temporary instrumental effect,
we also analyzed the SPI data in sub-intervals. In each time interval, we
found hints for an emission of the blazar, though at lower statistical
significance (Table 1), indicating a likely stable 0.5–1 MeV emission of PKS
0208-512 across the SPI observations.
For the spectral analysis we used the cataloged sky position of PKS 0208-512
and generated SPI spectra by running SPIMODFIT for narrow energy bands. The
program SPIMODFIT was developed at the Max-Planck-institut für
extraterrestrische Physik (MPE) for the analysis of spectra of point sources.
In order to test for possible instrumental effects, we generated the spectra
for different event selections, i.e., using only the SPI “singles” (events
detected only in one SPI detector), using only the SPI “multiples” (events
that scattered once or twice inside SPI and so are “seen” by two or three
detectors), and also using both event types combined. In the first step we
generated broadband spectra by similar energy cuts as in the imaging analysis
for the different event types. The spectra of the “single” and “multiple”
events are statistically consistent with each other, and so independently
indicate a source flux at a level of $\sim$1.5$\times$10-3 ph cm-2 s-1 in the
0.5 to 1 MeV band. Both spectra are also consistent in flux with the results
of the imaging analysis. Since both spectra show the same behavior, we then
combined both event types for additional analyses. The resulting broadband
spectrum is shown in Fig. 2 (left): It shows a flux level of
(1.46$\pm$0.45)$\times$10-3 ph cm-2 s-1 in the 0.5–1 MeV band, and fluxes
consistent with zero at energy bands above and below the latter. We repeated
this analysis with a finer spectral binning to check for a possible narrow
line emission. The spectrum with a binning of 100 keV between 0.3 and 1.4 MeV
is given in Fig. 2 (right). It shows a generally higher flux level in the 0.5
to 1.0 MeV band, which is consistent with the imaging analysis, but shows no
obvious narrow band line emission. A subsequent analysis with an even finer
binning of 50 keV resulted in the same picture. The SPI spectrum (Fig. 2)
suggests a weak emission excess in the 0.5-1 MeV band with respect to the
neighboring energies. The width of this excess is constrained to $\sim$0.5 MeV
by the SPI flux measurements at energies above and below the latter band. The
results of the _INTEGRAL_ data analysis are given in Table 1.
The _Swift_ /XRT data are well represented by a simple power law during the
time period between 2005 and 2008. The spectral index remained always at
$\sim$ 1.6, but the flux dropped by $\sim$ 30$\%$ in the December 2008
observations compared to the previous ones. Table 2 gives the details on the
_Swift_ observations and on the spectral results. The combined 2008 XRT
spectrum is added to the SED (Table 2, Fig. 3) to provide the contemporaneous
X-ray spectrum.
## 4 Discussion and summary
In an exposure of $\sim$ 510 ks SPI found hints at a $\sim$ 3.2-$\sigma$
level, for emission from PKS 0208-512 between 0.5 and 1 MeV at a flux of
$\sim$1.5$\times$10-3 ph cm-2s-1, without recognizable emission at adjacent
energies. A $\chi^{2}$ test on this excess gives 1.1-$\sigma$ and 2.8-$\sigma$
deviations from a linear fit for the right and the left panels of the Fig. 2,
respectively, consistent with the low statistics of the flux. Far from being
proven, the excess remains as an interesting possibility, whose consequences
warrant analysis.
To further investigate this possible emission, we analyzed the contemporaneous
_Fermi_ /LAT data (collected during the SPI revolutions 746 to 757), and
generated the simultaneous energy spectrum at energies above 200 MeV. This
spectrum is added to the SED (Fig. 3). The direct extrapolation of this
_Fermi_ /LAT spectrum to soft $\gamma$-rays falls short compared to the
measured SPI emission in the 0.5-1 MeV band. However, the unmeasured part of
the high-energy spectra covers more than two decades, which may be misleading
for a direct comparison of the two measurements. Anyway, if we take a flux of
$\sim$2$\times$10-7ph cm-2s-1, as was measured contemporaneously by _Fermi_
/LAT, and connect it to the SPI flux in the 0.5–1 MeV band with a power-law
shape, then the $\gamma$-ray photon index has to be steeper than three. On the
other hand, the SPI upper limit at 0.3–0.5 MeV and the flux at 0.5–1 MeV
require a photon index at hard X-ray energies harder than 0.6. Therefore, if
the measured SPI flux at 0.5–1 MeV is canonical inverse-Compton emission, the
change in photon index from hard X-rays to $\gamma$-rays has to be larger than
2.4. This is hard to account for in the current External Comptonization (EC)
or Synchrotron Self-Comptonization (SSC) models, unless a very unusual
electron energy spectrum is assumed. Consequently this might be indicative of
an additional spectral component at 0.5–1 MeV in its SED.
A similar trend of excess emission at soft $\gamma$-rays was indicated already
about ten years ago, when COMPTEL and EGRET data were combined (e.g., Blom et
al. 1995; Williams et al 2001). In Fig. 3 we include the COMPTEL spectrum
(Williams et al. 2001), where the flux at the lowest COMPTEL energies (0.75-1
MeV) is about a factor of 3 higher than the extrapolation of the EGRET
spectrum. The results from COMPTEL/EGRET and _INTEGRAL_ /SPI/_Fermi_ are
independently derived and may mutually strengthen each other.
The broad emission feature between 0.5 and 1 MeV, if emitted by the blazar,
could be understood as a broadened and Doppler blue-shifted pair annihilation
radiation, emitted by a jet containing an electron-positron pair plasma.
Assuming such a scenario, we derive by the arguments below the following
estimates on the blazar jet of PKS 0208-512 and compare these to measured
parameters using data from other wavelengths.
The central energy $E$ of this emission feature is related to the kinematics
of the jet, $E=\gamma_{\rm min}D/(1+z)\times$ 511 keV, where $\gamma_{\rm
min}$ is the minimum Lorentz factor of the electrons and positrons in the jet,
and $D$ is the Doppler boosting factor (Böttcher & Schlickeiser 1996). To
estimate $E$, we fitted the right panel of Fig. 2 with a Gaussian shape. The
central energy was derived as 803${}^{+233}_{-291}$ keV, well within the
energy range of 0.5–1.0 MeV, and the width was $\sim$ 200 keV without a well-
constrained error. The reduced $\chi^{2}$ for the fit was 1.1 for 7 degrees of
freedom. The apparent speed of the relativistic bulk motion of PKS 0208-512
was measured with VLBI as $\sim$ (2.4$\pm$3.1)$c$ (Tingay et al. 2002). By
taking $\gamma_{\rm min}=1$ and $D\sim$ 3, the bulk Lorentz factor $\Gamma$
could then be estimated to $\sim$ $2.6^{+3.0}_{-2.6}$. Subsequently, the
offset angle $\theta$ between our line of sight and the jet became $\sim$
10∘-19∘. We note that a caveat on the estimation of this offset angle is that
the VLBI measurement of the bulk motion was not contemporary.
The density of a relativistic electron-positron pair plasma can be up to
$\sim$ $10^{10}$$(L_{e^{\pm}}D^{-4}/V_{b})$1/2 cm-3 (Roland & Hermsen 1995).
Here the beam was assumed to consist of $e^{\pm}$ and the annihilation
emission could dominate in a volume $V_{b}$, which was inferred from the time
variability in $\gamma$-rays. The relativistic $e^{\pm}$ in the beam were
supposed to have a power law distribution with an index $\sim$ 3 (Roland &
Hermsen 1995). The annihilation luminosity could then be estimated under an
approximation of the annihilation rate (Coppi & Blandford 1990). If we take
$L_{e^{\pm}}$ as the measured luminosity at the 0.5–1 MeV band, $D$=3, and a
lower limit of $V_{b}=\pi R_{b}^{2}L_{b}\sim 5\times 10^{46}$ cm3, we derive
an upper limit of $\sim$ $10^{9}$ cm-3 for the density of electron-positron
population in the jet. Here we took the radius of the beam as $R_{b}$ $\sim$
$20GM/c^{2}$ (Marcowith et al. 1995), the beam length $L_{b}$ of $\sim$ 100
light days due to the variability time scale of the $\gamma$-rays flare
observed by _Fermi_ in 2008, and the central black hole mass as $M$ $\sim$
$10^{8}M_{\odot}$, which is typical for a blazar. The $L_{e^{\pm}}$ is about
6.3 $\times 10^{48}$ erg/s, given a redshift $z$=1.003 (we used $H_{0}$=75 km
s-1Mpc-1, $q_{0}$=0.5). By taking the Eddington limit of 1.3 $\times 10^{38}$
M/M⊙ erg/s, the central black hole mass was estimated to be larger than
6$\times$108 M⊙. Using this mass, the upper limit for the density of the
electron-positron population in the jet decreases to $\sim$ $10^{8}$ cm-3. A
significant number of cold leptons, responsible for the annihilation line, may
generate the bulk spectral feature in the soft X-ray energy range via
Comptonization off the surrounding UV photons, as discussed in e.g. Sikora et
al. (1997), where the emission peak is estimated at $\sim$ $(\Gamma/10)^{2}$
keV. By taking $\Gamma$ $\sim$ 2.6, the emission peak is about 0.07 keV, well
below the Swift/XRT energy domain. We notice that the XRT spectrum can be well
fitted by a simple power-law shape, without any further components.
The gamma-ray flux above 100 MeV, averaged from August to October 2008, was a
factor of $\sim$ 3.5 lower than roughly ten years (1991-2000) ago during the
_CGRO_ era (Hartman et al. 1999; Abdo et al. 2009). The _Swift_ /XRT
observations show that the X-ray flux (2–10 keV) of PKS 0208-512 also
decreased by more than 50% in comparison to the BeppoSAX observation in 2001
(Tavecchio et al. 2002; Donato et al. 2005). However, if the the soft
$\gamma$-ray excess observed by SPI at 0.5-1 MeV is emitted by the blazar,
this component is even brighter now than during the CGRO times. A possible
explanation could be that the $\gamma$-rays generated by inverse-Compton
emission of the non-thermal pairs are more dependent on the Doppler factor
than the thermal annihilation radiation in the jet (Skibo et al. 1997).
Therefore, blazars, viewed at a moderate jet offset-angle, can show a
significant blue-shifted annihilation radiation, which outshines the continuum
emission (Skibo et al. 1997). This may actually have happened in PKS 0208-512:
a viewing angle of $\sim$ 10∘-19∘ matches the prediction for this viewing-
angle scenario. The moderate distance at a redshift $z\sim$1 makes PKS
0208-512 detectable in gamma-rays, despite its relatively large jet offset-
angle.
Further monitoring of this source by _INTEGRAL_ , _Fermi_ /LAT and _Swift_ ,
as well as by the planned NeXT high energy astronomy mission with a more
sensitive Compton telescope below 600 keV and a broad band capability down to
about 0.3 keV should clarify this picture. If confirmed, the monitoring will
be able to constrain the pair density in the PKS 0208-512 jet directly from
observing the annihilation radiation of the pair plasma.
###### Acknowledgements.
We are grateful to the anonymous referee for his/her comments which were of
great helpful to polish the paper. This work was subsidized by the National
Natural Science Foundation of China, and the CAS key Project KJCX2-YW-T03 and
the 973 Program 2009CB824800. DFT has been supported by grants AYA2009-07391
and SGR2009- 811. J.-M. Wang and S.-N. Zhang thank the Natural Science
Foundation of China for support via NSFC-10325313, 10733010, 10725313 and
10821061. Shu Zhang would like to thank _INTEGRAL_ and _Swift_ for approving
_INTEGRAL_ proposal No. 0620052 and two SWFIT ToO (Target of Opportunity)
proposals (ID No. 35002), and for subsequently carrying out the observations
of roughly 700 ks to support this research.
Figure 1: SPI significance map of the sky region centered on PKS 0208-512 in
the 0.5–1 MeV band is shown for an exposure of $\sim$ 510 ks. A source at the
position of PKS 0208-512, marked by a cross, is clearly visible. The
equatorial coordinates are overlaid, and the color bar at the bottom gives the
detection significance scale in $\sigma$.
Figure 2: Broadband _INTEGRAL_ /SPI spectrum from the cataloged sky position
of PKS 0208-512 with the energy bins of the imaging analysis (left panel) and
a resolution of 100 keV (right panel). The spectral analysis is consistent in
behavior and flux with the results of the imaging analysis, showing a weak
($\sim$ 3.2 $\sigma$) emission between 0.5 and 1 MeV, and no evidence for the
source at energies below and above. Over-plotted in the right panel is a
Gaussian fit of the emission excess.
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Figure 3: Contemporaneous high-energy SED of PKS 0208-512 for two different
epochs, the _CGRO_ era (COMPTEL, EGRET; 1991-2000) and recent 2008
measurements (_Swift_ , _INTEGRAL_ , _Fermi_). The COMPTEL data (open
diamonds) are from Williams et al. (2001) covering the time span 1991 to 1998,
and the EGRET spectral shape is from Hartman et al. (1999), measured between
1991 and 1994. The rest is from observations in 2008 by _Swift_ /XRT (spectral
shape; 0.2-10 keV), _INTEGRAL_ (JEM-X, IBIS/ISGRI, SPI; open squares; 6 keV-8
MeV), and _Fermi_ /LAT (spectral shape; above 200 MeV) between November 22 and
December 27. The dashed lines indicate the spectral extrapolations of the
measured spectral shapes. The error bars are 1 $\sigma$ and the upper limits
are 2 $\sigma$. Table 1: _INTEGRAL_ observations of PKS 0208-512.
Rev. ID | Date | MJD | SCWs | Exposure | Flux(0.5–1 MeV)
---|---|---|---|---|---
| | (day) | | (ks) | ($\times$10-3 ph cm-2s-1)
714 | 2008 Aug 18-20 | 54696.26-54698.44 | 50 | 172 |
746 | 2008 Nov 22-23 | 54792.75-54793.75 | 24 | 82 | 1.5$\pm$1.2
754 | 2008 Dec 16-17 | 54816.69-54817.77 | 25 | 85 | 1.1$\pm$1.1
755 | 2008 Dec 19-21 | 54819.92-54820.28 | 31 | 106 | 2.5$\pm$1.0
756 | 2008 Dec 22-23 | 54822.92-54823.74 | 19 | 69 | 1.6$\pm$1.3
757 | 2008 Dec 24-26 | 54824.69-54826.78 | 48 | 168 | 1.2$\pm$0.8
Exposure | $E$-band | Flux | INTEGRAL
(ks) | (MeV) | ($\times$10-4ph cm-2s-1) | (instrument)
510 | 0.3–0.5 | $<$6.4 |
| 0.5–1.0 | 15.0$\pm$4.7 | SPI(imaging)
| 1.0–1.4 | $<$8.5 |
| 1.4–8.0 | $<$24.2 |
510 | 0.3–0.5 | $<$6.0 |
| 0.5–1.0 | 14.6$\pm$4.5 | SPI(spectral)
| 1.0–1.4 | $<$9.3 |
682 | 0.02–0.04 | $<$0.7 |
| 0.04–0.1 | $<$0.7 | ISGRI
| 0.1–0.3 | $<$1.6 |
682 | 0.006–0.01 | $<$17.0 | JEM-X
| 0.01–0.02 | $<$17.6 |
* Note:
The revolution ID, calendar date, time in MJD, number of science windows,
exposure, energy band and flux are given. The errors are 1 $\sigma$ and upper
limits 2 $\sigma$.
Table 2: _Swift_ /XRT observations of PKS 0208-512.
Obs. ID | Date | MJD | Offset | Expo. | Index | N | L | $\chi^{2}$/dof
---|---|---|---|---|---|---|---|---
(Swift/XRT) | | (day) | (arcsecond) | (ks) | | | |
00035002001 | 2005 Apr 23 | 53483.67 | 0.8 | 12.4 | | | |
00035002003 | 2005 May 04 | 53495.00 | 1.1 | 2.1 | | | |
00035002004 | 2005 May 12 | 53502.02 | 1.1 | 2.1 | | | |
00035002005 | 2005 May 10 | 53500.01 | 1.3 | 4.2 | | | |
combined | | | | 20.8 | 1.64$\pm$0.06 | 4.4$\pm$0.2 | 3.4 | 0.9/46
00035002014 | 2008 Oct 23 | 54762.56 | 1.5 | 3.7 | | | |
00035002017 | 2008 Nov 10 | 54780.65 | 2.3 | 3.8 | | | |
00035002021 | 2008 Nov 25 | 54795.43 | 1.5 | 2.2 | | | |
combined | | | | 9.7 | 1.63$\pm$0.11 | 4.3$\pm$0.3 | 3.4 | 1.28/19
00035002026 | 2008 Dec 17 | 54817.31 | 4.1 | 4.2 | | | |
00035002027 | 2008 Dec 20 | 54820.07 | 0.8 | 4.2 | | | |
00035002028 | 2008 Dec 20 | 54820.73 | 0.4 | 4.4 | | | |
00035002029 | 2008 Dec 23 | 54823.07 | 1.1 | 4.4 | | | |
00035002030 | 2008 Dec 24 | 54824.81 | 0.2 | 4.2 | | | |
00035002031 | 2008 Dec 26 | 54826.10 | 0.8 | 4.6 | | | |
combined | | | | 26 | 1.64$\pm$0.07 | 3.0$\pm$0.2 | 2.3 | 1.2/38
combined all | | | | 56.5 | 1.62$\pm$0.04 | 3.9$\pm$0.1 | 3.1 | 1.25/100
combined 2008 | | | | 35.7 | 1.60$\pm$0.05 | 3.3$\pm$0.1 | 2.7 | 1.1/59
* Note:
The observation ID, time in calendar date and MJD, the pointing offset angle,
exposure, spectral index, normalization (in units of 10-4 ph cm-2s-1keV-1),
luminosity (in units of 10-12 ergs cm-2s-1, at 0.2–10 keV), and the reduced
$\chi^{2}$ and degree of freedom (dof) of the power-law fits are given.
|
arxiv-papers
| 2010-02-22T00:43:46 |
2024-09-04T02:49:08.494905
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shu Zhang, Werner Collmar, Diego F. Torres, Jian-Min Wang, Michael\n Lang, Shuang-Nan Zhang",
"submitter": "Shu Zhang",
"url": "https://arxiv.org/abs/1002.4030"
}
|
1002.4228
|
11institutetext: Department of Combinatorics and Optimization
University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
11email: {djao,vsoukhar}@math.uwaterloo.ca
# A Subexponential Algorithm for Evaluating Large Degree Isogenies
David Jao Vladimir Soukharev
###### Abstract
An isogeny between elliptic curves is an algebraic morphism which is a group
homomorphism. Many applications in cryptography require evaluating large
degree isogenies between elliptic curves efficiently. For ordinary curves of
the same endomorphism ring, the previous best known algorithm has a worst case
running time which is exponential in the length of the input. In this paper we
show this problem can be solved in subexponential time under reasonable
heuristics. Our approach is based on factoring the ideal corresponding to the
kernel of the isogeny, modulo principal ideals, into a product of smaller
prime ideals for which the isogenies can be computed directly. Combined with
previous work of Bostan et al., our algorithm yields equations for large
degree isogenies in quasi-optimal time given only the starting curve and the
kernel.
## 1 Introduction
A well known theorem of Tate [29] states that two elliptic curves defined over
the same finite field $\mathbb{F}_{q}$ are isogenous (i.e. admit an isogeny
between them) if and only if they have the same number of points over
$\mathbb{F}_{q}$. Using fast point counting algorithms such as Schoof’s
algorithm and others [9, 25], it is very easy to check whether this condition
holds, and thus whether or not the curves are isogenous. However, constructing
the actual isogeny itself is believed to be a hard problem due to the
nonconstructive nature of Tate’s theorem. Indeed, given an ordinary curve
$E/\mathbb{F}_{q}$ and an ideal of norm $n$ in the endomorphism ring, the
fastest previously known algorithm for constructing the unique (up to
isomorphism) isogeny having this ideal as kernel has a running time of
$O(n^{3+\varepsilon})$, except in a certain very small number of special cases
[4, 16, 17]. In this paper, we present a new probabilistic algorithm for
evaluating such isogenies, which in the vast majority of cases runs
(heuristically) in subexponential time. Specifically, we show that for
ordinary curves, one can evaluate isogenies of degree $n$ between curves of
nearly equal endomorphism ring over $\mathbb{F}_{q}$ in time less than
$L_{q}(\frac{1}{2},\frac{\sqrt{3}}{2})\log(n)$, provided $n$ has no large
prime divisors in common with the endomorphism ring discriminant. Although
this running time is not polynomial in the input length, our algorithm is
still much faster than the (exponential) previous best known algorithm, and in
practice allows for the evaluation of isogenies of cryptographically sized
degrees, some examples of which we present here. We emphasize that, in
contrast with the previous results of Bröker et al. [4], our algorithm is not
limited to special curves such as pairing friendly curves with small
discriminant.
If an explicit equation for the isogeny as a rational function is desired, our
approach in combination with the algorithm of Bostan et al. [3] can produce
the equation in time $O(n^{1+\varepsilon})$ given $E$ and an ideal of norm
$n$, which is quasi-optimal in the sense that (up to log factors) it is equal
to the size of the output. To our knowledge, this method is the only known
algorithm for computing rational function expressions of large degree
isogenies in quasi-optimal time in the general case, given only the starting
curve and the kernel.
Apart from playing a central role in the implementation of the point counting
algorithms mentioned above, isogenies have been used in cryptography to
transfer the discrete logarithm problem from one elliptic curve to another [9,
16, 17, 20, 23, 30]. In many of these applications, our algorithm cannot be
used directly, since in cryptography one is usually given two isogenous
curves, rather than one curve together with the isogeny degree. However,
earlier results [16, 17, 20] have shown that the problem of computing
isogenies between a given pair of curves can be reduced to the problem of
computing isogenies of prime degree starting from a given curve. It is
therefore likely that the previous best isogeny construction algorithms in the
cryptographic setting can be improved or extended in light of the work that we
present here.
## 2 Background
Let $E$ and $E^{\prime}$ be elliptic curves defined over a finite field
$\mathbb{F}_{q}$ of characteristic $p$. An isogeny $\phi\colon E\to
E^{\prime}$ defined over $\mathbb{F}_{q}$ is a non-constant rational map
defined over $\mathbb{F}_{q}$ which is also a group homomorphism from
$E(\mathbb{F}_{q})$ to $E^{\prime}(\mathbb{F}_{q})$. This definition differs
slightly from the standard definition in that it excludes constant maps [27,
§III.4]. The degree of an isogeny is its degree as a rational map, and an
isogeny of degree $\ell$ is called an $\ell$-isogeny. Every isogeny of degree
greater than 1 can be factored into a composition of isogenies of prime degree
defined over $\bar{\mathbb{F}}_{q}$ [11].
For any elliptic curve $E\colon
y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}$ defined over
$\mathbb{F}_{q}$, the Frobenius endomorphism is the isogeny $\pi_{q}\colon
E\to E$ of degree $q$ given by the equation $\pi_{q}(x,y)=(x^{q},y^{q})$. The
characteristic polynomial of $\pi_{q}$ is $X^{2}-tX+q$ where
$t=q+1-\\#E(\mathbb{F}_{q})$ is the trace of $E$.
An endomorphism of $E$ is an isogeny $E\to E$ defined over the algebraic
closure $\bar{\mathbb{F}}_{q}$ of $\mathbb{F}_{q}$. The set of endomorphisms
of $E$ together with the zero map forms a ring under the operations of
pointwise addition and composition; this ring is called the endomorphism ring
of $E$ and denoted $\operatorname{End}(E)$. The ring $\operatorname{End}(E)$
is isomorphic either to an order in a quaternion algebra or to an order in an
imaginary quadratic field [27, V.3.1]; in the first case we say $E$ is
supersingular and in the second case we say $E$ is ordinary.
Two elliptic curves $E$ and $E^{\prime}$ defined over $\mathbb{F}_{q}$ are
said to be isogenous over $\mathbb{F}_{q}$ if there exists an isogeny
$\phi\colon E\to E^{\prime}$ defined over $\mathbb{F}_{q}$. A theorem of Tate
states that two curves $E$ and $E^{\prime}$ are isogenous over
$\mathbb{F}_{q}$ if and only if
$\\#E(\mathbb{F}_{q})=\\#E^{\prime}(\mathbb{F}_{q})$ [29, §3]. Since every
isogeny has a dual isogeny [27, III.6.1], the property of being isogenous over
$\mathbb{F}_{q}$ is an equivalence relation on the finite set of
$\bar{\mathbb{F}}_{q}$-isomorphism classes of elliptic curves defined over
$\mathbb{F}_{q}$. Moreover, isomorphisms between elliptic curves can be
classified completely and computed efficiently in all cases [16]. Accordingly,
we define an isogeny class to be an equivalence class of elliptic curves,
taken up to $\bar{\mathbb{F}}_{q}$-isomorphism, under this equivalence
relation.
Curves in the same isogeny class are either all supersingular or all ordinary.
The vast majority of curves are ordinary, and indeed the number of isomorphism
classes of supersingular curves is finite for each characteristic. Also,
ordinary curves form the majority of the curves of interest in applications
such as cryptography. Hence, we assume for the remainder of this paper that we
are in the _ordinary case_.
Let $K$ denote the imaginary quadratic field containing
$\operatorname{End}(E)$, with maximal order $\mathcal{O}_{K}$. For any order
$\mathcal{O}\subseteq\mathcal{O}_{K}$, the conductor of $\mathcal{O}$ is
defined to be the integer $[\mathcal{O}_{K}:\mathcal{O}]$. The field $K$ is
called the CM field of $E$. We write $c_{E}$ for the conductor of
$\operatorname{End}(E)$ and $c_{\pi}$ for the conductor of
$\mathbb{Z}[\pi_{q}]$. It follows from [12, §7] that
$\operatorname{End}(E)=\mathbb{Z}+c_{E}\mathcal{O}_{K}$ and
$\Delta=c_{E}^{2}\Delta_{K},$ where $\Delta$ (respectively, $\Delta_{K}$) is
the discriminant of the imaginary quadratic order $\operatorname{End}(E)$
(respectively, $\mathcal{O}_{K}$). Furthermore, the characteristic polynomial
has discriminant
$\Delta_{\pi}=t^{2}-4q=\operatorname{disc}(\mathbb{Z}[\pi_{q}])=c_{\pi}^{2}\Delta_{K}$,
with $c_{\pi}=c_{E}\cdot[\operatorname{End}(E):\mathbb{Z}[\pi_{q}]]$.
Following [14] and [16], we say that an isogeny $\phi\colon E\to E^{\prime}$
of prime degree $\ell$ defined over $\mathbb{F}_{q}$ is “down” if
$[\operatorname{End}(E):\operatorname{End}(E^{\prime})]=\ell$, “up” if
$[\operatorname{End}(E^{\prime}):\operatorname{End}(E)]=\ell$, and
“horizontal” if $\operatorname{End}(E)=\operatorname{End}(E)$. Two curves in
an isogeny class are said to “have the same level” if their endomorphism rings
are equal. Within each isogeny class, the property of having the same level is
an equivalence relation. A horizontal isogeny always goes between two curves
of the same level; likewise, an up isogeny enlarges the endomorphism ring and
a down isogeny reduces it. Since there are fewer elliptic curves at higher
levels than at lower levels, the collection of elliptic curves in an isogeny
class visually resembles a “pyramid” or a “volcano” [14], with up isogenies
ascending the structure and down isogenies descending. If we restrict to the
graph of $\ell$-isogenies for a single $\ell$, then in general the
$\ell$-isogeny graph is disconnected, having one $\ell$-volcano for each
intermediate order
$\mathbb{Z}[\pi_{q}]\subset\mathcal{O}\subset\mathcal{O}_{K}$ such that
$\mathcal{O}$ is maximal at $\ell$ (meaning
$\ell\nmid[\mathcal{O}_{K}:\mathcal{O}]$). The “top level” of the class
consists of curves $E$ with $\operatorname{End}(E)=\mathcal{O}_{K}$, and the
“bottom level” consists of curves with
$\operatorname{End}(E)=\mathbb{Z}[\pi_{q}]$.
We say that $\ell$ is an _Elkies prime_ [2, p. 119] if $\ell\nmid c_{E}$ and
${\genfrac{(}{)}{}{}{\Delta}{\ell}}\neq-1$, or equivalently if and only if $E$
admits a horizontal isogeny of degree $\ell$. The number of $\ell$-isogenies
of each type can easily be determined explicitly [14, 16, 21]. In particular,
for all but the finitely many primes $\ell$ dividing
$[\mathcal{O}_{K}:\mathbb{Z}[\pi_{q}]]$, we have that every rational
$\ell$-isogeny admitted by $E$ is horizontal.
## 3 The Bröker-Charles-Lauter algorithm
Our algorithm is an extension of the algorithm developed by Bröker, Charles,
and Lauter [4] to evaluate large degree isogenies over ordinary elliptic
curves with endomorphism rings of small class number, such as pairing-friendly
curves [15]. In this section we provide a summary of their results.
The following notation corresponds to that of [4]. Let $E/\mathbb{F}_{q}$ be
an ordinary elliptic curve with endomorphism ring $\operatorname{End}(E)$
isomorphic to an imaginary quadratic order $\mathcal{O}_{\Delta}$ of
discriminant $\Delta<0$. Identify $\operatorname{End}(E)$ with
$\mathcal{O}_{\Delta}$ via the unique isomorphism $\iota$ such that
$\iota^{*}(x)\omega=x\omega$ for all invariant differentials $\omega$ and all
$x\in\mathcal{O}_{\Delta}$. Then every horizontal separable isogeny on $E$ of
prime degree $\ell$ corresponds (up to isomorphism) to a unique prime ideal
$\mathfrak{L}\subset\mathcal{O}_{\Delta}$ of norm $\ell$ for some Elkies prime
$\ell$. We denote the kernel of this isogeny by $E[\mathfrak{L}]$. Any two
distinct isomorphic horizontal isogenies (i.e., pairs of isogenies where one
is equal to the composition of the other with an isomorphism) induce different
maps on the space of differentials of $E$, and a separable isogeny is uniquely
determined by the combination of its kernel and the induced map on the space
of differentials. A _normalized_ isogeny is an isogeny $\phi\colon E\to
E^{\prime}$ for which $\phi^{*}(\omega_{E^{\prime}})=\omega_{E}$ where
$\omega_{E}$ denotes the invariant differential of $E$. Algorithm 1 (identical
to Algorithm 4.1 in [4]) evaluates, up to automorphisms of $E$, the unique
normalized horizontal isogeny of degree $\ell$ corresponding to a given kernel
ideal $\mathfrak{L}\subset\mathcal{O}_{\Delta}$.
The following theorem, taken verbatim from [4], shows that the running time of
Algorithm 1 is polynomial in the quantities $\log(\ell)$, $\log(q)$, $n$, and
$|\Delta|$.
###### Theorem 3.1
Let $E/\mathbb{F}_{q}$ be an ordinary elliptic curve with Frobenius $\pi_{q}$,
given by a Weierstrass equation, and let $P\in E(\mathbb{F}_{q^{n}})$ be a
point on $E$. Let $\Delta=\operatorname{disc}(\operatorname{End}(E))$ be
given. Assume that $[\operatorname{End}(E):\mathbb{Z}[\pi_{q}]]$ and
$\\#E(\mathbb{F}_{q^{n}})$ are coprime, and let
$\mathfrak{L}=(\ell,c+d\pi_{q})$ be an $\operatorname{End}(E)$-ideal of prime
norm $\ell\neq\operatorname{char}(\mathbb{F}_{q})$ not dividing the index
$[\operatorname{End}(E):\mathbb{Z}[\pi_{q}]]$. Algorithm 1 computes the unique
elliptic curve $E^{\prime}$ such that there exists a normalized isogeny
$\phi\colon E\to E^{\prime}$ with kernel $E[\mathfrak{L}]$. Furthermore, it
computes the $x$-coordinate of $\phi(P)$ if $\operatorname{End}(E)$ does not
equal $\mathbb{Z}[i]$ or $\mathbb{Z}[\zeta_{3}]$ and the square, respectively
cube, of the $x$-coordinate of $\phi(P)$ otherwise. The running time of the
algorithm is polynomial in $\log(\ell)$, $\log(q)$, $n$ and $|\Delta|$.
Algorithm 1 The Bröker-Charles-Lauter algorithm
0: A discriminant $\Delta$, an elliptic curve $E/\mathbb{F}_{q}$ with
$\operatorname{End}(E)=\mathcal{O}_{\Delta}$ and a point $P\in
E(\mathbb{F}_{q^{n}})$ such that $[\operatorname{End}(E):\mathbb{Z}[\pi_{q}]]$
and $\\#E(\mathbb{F}_{q^{n}})$ are coprime, and an
$\operatorname{End}(E)$-ideal $\mathfrak{L}=(\ell,c+d\pi_{q})$ of prime norm
$\ell\neq\operatorname{char}(\mathbb{F}_{q})$ not dividing the index
$[\operatorname{End}(E):\mathbb{Z}[\pi_{q}]]$.
0: The unique elliptic curve $E^{\prime}$ admitting a normalized isogeny
$\phi\colon E\to E^{\prime}$ with kernel $E[\mathfrak{L}]$, and the
$x$-coordinate of $\phi(P)$ for $\Delta\neq-3,-4$ and the square (resp. cube)
of the $x$-coordinate otherwise.
1: Compute the direct sum decomposition
$\operatorname{Pic}(\mathcal{O}_{\Delta})=\bigotimes\langle[I_{i}]\rangle$ of
$\operatorname{Pic}(\mathcal{O}_{\Delta})$ into cyclic groups generated by the
degree 1 prime ideals $I_{i}$ of smallest norm that are coprime to the product
$p\cdot\\#E(\mathbb{F}_{q^{n}})\cdot[\operatorname{End}(E):\mathbb{Z}[\pi_{q}]]$.
2: Using brute force111Bröker, Charles, and Lauter mention that this
computation can be done in “various ways” [4, p. 107], but the only explicit
method given in [4] is brute force. The use of brute force limits the
algorithm to elliptic curves for which $|\Delta|$ is small, such as pairing-
friendly curves., find $e_{1},e_{2},\ldots,e_{k}$ such that
$[\mathfrak{L}]=[I_{1}^{e_{1}}]\cdot[I_{2}^{e_{2}}]\cdots[I_{k}^{e_{k}}]$.
3: Find $\alpha$ (using Cornacchia’s algorithm) and express
$\mathfrak{L}=I_{1}^{e_{1}}\cdot I_{2}^{e_{2}}\cdots
I_{k}^{e_{k}}\cdot(\alpha)$.
4: Compute a sequence of isogenies $(\phi_{1},\ldots,\phi_{s})$ such that the
composition $\phi_{c}:E\rightarrow E_{c}$ has kernel $E[I_{1}^{e_{1}}\cdot
I_{2}^{e_{2}}\cdots I_{k}^{e_{k}}]$ using the method of [4, § 3].
5: Evaluate $\phi_{c}(P)\in E_{c}(\mathbb{F}_{q^{n}})$.
6: Write $\alpha=(u+v\pi_{q})/(zm)$. Compute the isomorphism $\eta\colon
E_{c}\stackrel{{\scriptstyle\sim}}{{\to}}E^{\prime}$ with
$\eta^{*}(\omega_{E^{\prime}})=(u/zm)\omega_{E_{c}}$. Compute
$Q=\eta(\phi_{c}(P))$.
7: Compute $(zm)^{-1}\bmod\\#E(\mathbb{F}_{q^{n}})$, and compute
$R=((zm)^{-1}(u+v\pi_{q}))(Q)$.
8: Put $r=x(R)^{|\mathcal{O}_{\Delta}|^{*}/2}$ and return $(E^{\prime},r)$.
## 4 A subexponential algorithm for evaluating horizontal isogenies
As was shown in Sections 2 and 3, any horizontal isogeny can be expressed as a
composition of prime degree isogenies, one for each prime factor of the
kernel, and any prime degree isogeny is a composition of a normalized isogeny
and an isomorphism. Therefore, to evaluate a horizontal isogeny given its
kernel, it suffices to treat the case of horizontal normalized prime degree
isogenies.
Our objective is to evaluate the unique horizontal normalized isogeny on a
given elliptic curve $E/\mathbb{F}_{q}$ whose kernel ideal is given as
$\mathfrak{L}=(\ell,c+d\pi_{q})$, at a given point $P\in
E(\mathbb{F}_{q^{n}})$, where $\ell$ is an Elkies prime. As in [4], we must
also impose the additional restriction that
$\ell\nmid[\operatorname{End}(E):\mathbb{Z}[\pi_{q}]]$; for Elkies primes, an
equivalent restriction is that
$\ell\nmid[\mathcal{O}_{K}:\mathbb{Z}[\pi_{q}]]$, but we retain the original
formulation for consistency with [4].
In practice, one is typically given $\ell$ instead of $\mathfrak{L}$, but
since it is easy to calculate the list of (at most two) possible primes
$\mathfrak{L}$ lying over $\ell$ (cf. [6]), these two interpretations are for
all practical purposes equivalent, and we switch freely between them when
convenient. When $\ell$ is small, one can use modular polynomial based
techniques [4, §3.1], which have running time
$O(\ell^{3}\log(\ell)^{4+\varepsilon})$ [13]. However, for isogeny degrees of
cryptographic size (e.g. $2^{160}$), this approach is impractical. The Bröker-
Charles-Lauter algorithm sidesteps this problem, by using an alternative
factorization of $\mathfrak{L}$. However, the running time of Bröker-Charles-
Lauter is polynomial in $|\Delta|$, and therefore even this method only works
for small values of $|\Delta|$. In this section we present a modified version
of the Bröker-Charles-Lauter algorithm which is suitable for large values of
$|\Delta|$.
We begin by giving an overview of our approach. In order to handle large
values of $|\Delta|$, there are two main problems to overcome. One problem is
that we need a fast way to produce a factorization
$\mathfrak{L}=I_{1}^{e_{1}}I_{2}^{e_{2}}\cdots I_{k}^{e_{k}}\cdot(\alpha)$ (1)
as in lines 2 and 3 of Algorithm 1. The other problem is that the exponents
$e_{i}$ in Equation (1) need to be kept small, since the running times of
lines 3 and 4 of Algorithm 1 are proportional to
$\sum_{i}|e_{i}|\operatorname{Norm}(I_{i})^{2}$. The first problem, that of
finding a factorization of $\mathfrak{L}$, can be solved in subexponential
time using the index calculus algorithm of Hafner and McCurley [18] (see also
[6, Chap. 11]). To resolve the second problem, we turn to an idea which was
first introduced by Galbraith et. al [17], and recently further refined by
Bisson and Sutherland [1]. The idea is that, in the process of sieving for
smooth norms, one can arbitrarily restrict the input exponent vectors to
sparse vectors $(e_{1},e_{2},...,e_{k})$ such that
$\sum_{i}|e_{i}|N(I_{i})^{2}$ is kept small. This restriction is implemented
in line 6 of Algorithm 3. As in [1], one then assumes heuristically that the
imposition of this restriction does not affect the eventual probability of
obtaining a smooth norm in the Hafner and McCurley algorithm. Note that,
unlike the input exponents, the exponents appearing in the factorizations of
the ensuing smooth norms (that is, the values of $y_{i}$ in Algorithm 3) are
always small, since the norm in question is derived from a reduced quadratic
form.
We now describe the individual components of our algorithm in detail.
### 4.1 Finding a factor base
Let $\operatorname{Cl}(\mathcal{O}_{\Delta})$ denote the ideal class group of
$\mathcal{O}_{\Delta}$. Algorithm 2 produces a factor base consisting of split
primes in $\mathcal{O}_{\Delta}$ of norm less than some bound $N$. The optimal
value of $N$ will be determined in Section 4.4.
Algorithm 2 Computing a factor base
0: A discriminant $\Delta$, a bound $N$.
0: The set $\mathcal{I}$ consisting of split prime ideals of norm less than
$N$, together with the corresponding set $\mathcal{F}$ of quadratic forms.
1: Set $\mathcal{F}\leftarrow\emptyset$.
2: Set $\mathcal{I}\leftarrow\emptyset$.
3: Find all primes $p<N$ such that $(\frac{\Delta}{p})=1$. Call this set $P$.
Let $k=|P|$.
4: For each prime $p_{i}\in P$, find an ideal $\mathfrak{p}_{i}$ of norm
$p_{i}$ (using Cornacchia’s algorithm).
5: For each $i$, find a quadratic form $f_{i}=[(p_{i},b_{i},c_{i})]$
corresponding to $\mathfrak{p}_{i}$ in
$\operatorname{Cl}(\mathcal{O}_{\Delta})$, using the technique of [26, §3].
6: Output
$\mathcal{I}=\\{\mathfrak{p}_{1},\mathfrak{p}_{2},\ldots,\mathfrak{p}_{k}\\}$
and $\mathcal{F}=\\{f_{1},f_{2},\ldots,f_{k}\\}$.
Algorithm 3 “Factoring” a prime ideal
0: A discriminant $\Delta$, an elliptic curve $E/\mathbb{F}_{q}$ with
$\operatorname{End}(E)=\mathcal{O}_{\Delta}$, a smoothness bound $N$, a prime
ideal $\mathfrak{L}$ of norm $\ell$ in $\mathcal{O}_{\Delta}$, an extension
degree $n$.
0: Relation of the form
$\mathfrak{L}=(\alpha)\cdot\prod_{i=1}^{k}{I_{i}^{e_{i}}}$, where $(\alpha)$
is a fractional ideal, $I_{i}$ are as in Algorithm 1, and $e_{i}>0$ are small
and sparse.
1: Run Algorithm 2 on input $\Delta$ and $N$ to obtain
$\mathcal{I}=\\{\mathfrak{p}_{1},\mathfrak{p}_{2},\ldots,\mathfrak{p}_{k}\\}$
and $\mathcal{F}=\\{f_{1},f_{2},\ldots,f_{k}\\}$. Discard any primes dividing
$p\cdot\\#E(\mathbb{F}_{q^{n}})\cdot[\operatorname{End}(E):\mathbb{Z}[\pi_{q}]]$.
2: Set $p_{i}\leftarrow\operatorname{Norm}(\mathfrak{p}_{i})$. (These values
are also calculated in Algorithm 2.)
3: Obtain the reduced quadratic form $[\mathfrak{L}]$ corresponding to the
ideal class of $\mathfrak{L}$.
4: repeat
5: for $i=1,\ldots,k$ do
6: Pick exponents $x_{i}$ in the range $[0,(N/p_{i})^{2}]$ such that at most
$k_{0}$ are nonzero, where $k_{0}$ is a global absolute constant (in practice,
$k_{0}=3$ suffices).
7: end for
8: Compute the reduced quadratic form $\mathfrak{a}=(a,b,c)$ for which the
ideal class $[\mathfrak{a}]$ is equivalent to
$[\mathfrak{L}]\cdot\prod_{i=1}^{k}{f_{i}^{x_{i}}}$.
9: until The integer $a$ factors completely into the primes $p_{i}$, and the
relation derived from
$[\mathfrak{a}]=[\mathfrak{L}]\cdot\prod_{i=1}^{k}{f_{i}^{x_{i}}}$ contains
fewer than $\sqrt{\log(|\Delta|/3)}/z$ nonzero exponents.
10: Write $a=\prod_{i=1}^{k}{p_{i}^{u_{i}}}$.
11: for i=$1,\ldots,k$ do
12: Using the technique of Seysen ([26, Theorem 3.1]), determine the signs of
the exponents $y_{i}=\pm u_{i}$ for which
$\mathfrak{a}=\prod_{i=1}^{k}{f_{i}^{y_{i}}}$.
13: Let $e_{i}=y_{i}-x_{i}$. (These exponents satisfy
$[\mathfrak{L}]=\prod_{i=1}^{k}{f_{i}^{e_{i}}}$.)
14: if $e_{i}\geq 0$ then
15: Set $I_{i}\leftarrow\bar{\mathfrak{p}}_{i}$
16: else
17: Set $I_{i}\leftarrow\mathfrak{p}_{i}$
18: end if
19: end for
20: Compute the principal ideal
$I=\mathfrak{L}\cdot\prod_{i=1}^{k}{I_{i}^{|e_{i}|}}$.
21: Using Cornacchia’s algorithm, find a generator
$\beta\in\mathcal{O}_{\Delta}$ of $I$.
22: Set $m\leftarrow\prod_{i=1}^{k}{p_{i}^{|e_{i}|}}$ and
$\alpha\leftarrow\frac{\beta}{m}$.
23: Output
$\mathfrak{L}=(\alpha)\cdot\bar{I}_{1}^{|e_{1}|}\cdot\bar{I}_{2}^{|e_{2}|}\cdots\bar{I}_{k}^{|e_{k}|}$.
### 4.2 “Factoring” large prime degree ideals
Algorithm 3, based on the algorithm of Hafner and McCurley, takes as input a
discriminant $\Delta$, a curve $E$, a prime ideal $\mathfrak{L}$ of prime norm
$\ell$ in $\mathcal{O}_{\Delta}$, a smoothness bound $N$, and an extension
degree $n$. It outputs a factorization
$\mathfrak{L}=I_{1}^{e_{1}}I_{2}^{e_{2}}\cdots I_{k}^{e_{k}}\cdot(\alpha)$
as in Equation 1, where the $I_{i}$’s are as in Algorithm 1, the exponents
$e_{i}$ are positive, sparse, and small (i.e., polynomial in $N$), and the
ideal $(\alpha)$ is a principal fractional ideal generated by $\alpha$.
### 4.3 Algorithm for evaluating prime degree isogenies
The overall algorithm for evaluating prime degree isogenies is given in
Algorithm 4. This algorithm is identical to Algorithm 1, except that the
factorization of $\mathfrak{L}$ is performed using Algorithm 3. To maintain
consistency with [4], we have included the quantities $\Delta$ and
$\operatorname{End}(E)$ as part of the input to the algorithm. However, we
remark that these quantities can be computed from $E/\mathbb{F}_{q}$ in
$L_{q}(\frac{1}{2},\frac{\sqrt{3}}{2})$ operations using the algorithm of
Bisson and Sutherland [1], even if they are not provided as input.
Algorithm 4 Evaluating prime degree isogenies
0: A discriminant $\Delta$, an elliptic curve $E/\mathbb{F}_{q}$ with
$\operatorname{End}(E)=\mathcal{O}_{\Delta}$ and a point $P\in
E(\mathbb{F}_{q^{n}})$ such that $[\operatorname{End}(E):\mathbb{Z}[\pi_{q}]]$
and $\\#E(\mathbb{F}_{q^{n}})$ are coprime, and an
$\operatorname{End}(E)$-ideal $\mathfrak{L}=(\ell,c+d\pi_{q})$ of prime norm
$\ell\neq\operatorname{char}(\mathbb{F}_{q})$ not dividing the index
$[\operatorname{End}(E):\mathbb{Z}[\pi_{q}]]$.
0: The unique elliptic curve $E^{\prime}$ admitting a normalized isogeny
$\phi\colon E\to E^{\prime}$ with kernel $E[\mathfrak{L}]$, and the
$x$-coordinate of $\phi(P)$ for $\Delta\neq-3,-4$ and the square (resp. cube)
of the $x$-coordinate otherwise.
1: Choose a smoothness bound $N$ (see Section 4.4).
2: Using Algorithm 3 on input $(\Delta,E,N,\mathfrak{L},n)$, obtain a
factorization of the form $\mathfrak{L}=I_{1}^{e_{1}}\cdot I_{2}^{e_{2}}\cdots
I_{k}^{e_{k}}\cdot(\alpha)$.
3: Compute a sequence of isogenies $(\phi_{1},\ldots,\phi_{s})$ such that the
composition $\phi_{c}:E\to E_{c}$ has kernel $E[I_{1}^{e_{1}}\cdot
I_{2}^{e_{2}}\cdots I_{k}^{e_{k}}]$ using the method of [4, § 3].
4: Evaluate $\phi_{c}(P)\in E_{c}(\mathbb{F}_{q^{n}})$.
5: Write $\alpha=(u+v\pi_{q})/(zm)$. Compute the isomorphism $\eta\colon
E_{c}\stackrel{{\scriptstyle\sim}}{{\to}}E^{\prime}$ with
$\eta^{*}(\omega_{E^{\prime}})=(u/zm)\omega_{E_{c}}$. Compute
$Q=\eta(\phi_{c}(P))$.
6: Compute $(zm)^{-1}\bmod\\#E(\mathbb{F}_{q^{n}})$, and compute
$R=((zm)^{-1}(u+v\pi_{q}))(Q)$.
7: Put $r=x(R)^{|\mathcal{O}_{\Delta}|^{*}/2}$ and return $(E^{\prime},r)$.
### 4.4 Running time analysis
In this section, we determine the theoretical running time of Algorithm 4, as
well as the optimal value of the smoothness bound $N$ to use in line 1 of the
algorithm. As is typical for subexponential time factorization algorithms
involving a factor base, these two quantities depend on each other, and hence
both are calculated simultaneously.
As in [9], we define222The definition of $L_{n}(\alpha,c)$ in [6] differs from
that of [9] in the $o(1)$ term. We account for this discrepancy in our text.
$L_{n}(\alpha,c)$ by
$L_{n}(\alpha,c)=O(\exp((c+o(1))(\log(n))^{\alpha}(\log(\log(n)))^{1-\alpha})).$
The quantity $L_{n}(\alpha,c)$ interpolates between polynomial and exponential
size as $\alpha$ ranges from $0$ to $1$. We set
$N=L_{|\Delta|}(\frac{1}{2},z)$ for an unspecified value of $z$, and in the
following paragraphs we determine the optimal value of $z$ which minimizes the
running time of Algorithm 4. (The fact that $\alpha=\frac{1}{2}$ is optimal is
clear from the below analysis, as well as from prior experience with integer
factorization algorithms.) For convenience, we will abbreviate
$L_{|\Delta|}(\alpha,c)$ to $L(\alpha,c)$ throughout.
Line 2 of Algorithm 4 involves running Algorithm 3, which in turn calls
Algorithm 2. As it turns out, Algorithm 2 is almost the same as Algorithm 11.1
from [6], which requires $L(\frac{1}{2},z)$ time, as shown in [6]. The only
difference is that we add an additional step where we obtain the quadratic
form corresponding to each prime ideal in the factor base. This extra step
requires $O(\log(\operatorname{Norm}(I))^{1+\varepsilon})$ time for a prime
ideal $I$, using Cornacchia’s Algorithm [19]. Thus, the overall running time
for Algorithm 2 is bounded above by
$\textstyle
L(\frac{1}{2},z)\cdot\log(L(\frac{1}{2},z))^{1+\varepsilon}=L(\frac{1}{2},z).$
Line 2 of Algorithm 3 takes $\log(\ell)$ time using standard algorithms [12].
The loop in lines 4–9 of Algorithm 3 is very similar to the FindRelation
algorithm in [1], except that we only use one discriminant, and we omit the
requirement that $\\#R/D_{1}>\\#R/D_{2}$ (which in any case is meaningless
when there is only one discriminant). Needless to say, this change can only
speed up the algorithm. Taking $\mu=\sqrt{2}z$ in [1, Prop. 6], we find that
the (heuristic) expected running time of the loop in lines 4–9 of Algorithm 3
is $L(\frac{1}{2},\frac{1}{4z})$.
The next step in Algorithm 3 having nontrivial running time is the computation
of the ideal product in line 20. To exponentiate an element of an arbitrary
semigroup to a power $e$ requires $O(\log e)$ semigroup multiplication
operations [10, §1.2]. To multiply two ideals $I$ and $J$ in an imaginary
quadratic order (via composition of quadratic forms) requires
$O(\max(\log(\operatorname{Norm}(I)),\log(\operatorname{Norm}(J)))^{1+\varepsilon})$
bit operations using fast multiplication [24, §6]. Each of the expressions
$|I_{i}|^{|e_{i}|}$ therefore requires $O(\log|e_{i}|)$ ideal multiplication
operations to compute, with each individual multiplication requiring
$\displaystyle
O((|e_{i}|\log(\operatorname{Norm}(I_{i})))^{1+\varepsilon})=O\left(\left(\left(\frac{N}{p_{i}}\right)^{2}\log(p_{i})\right)^{1+\varepsilon}\right)=O(N^{2+\varepsilon})$
bit operations, for a total running time of $(\log
e_{i})O(N^{2+\varepsilon})=L(\frac{1}{2},2z)$ for each $i$. This calculation
must be performed once for each nonzero exponent $e_{i}$. By line 9, the
number of nonzero exponents appearing in the relation is at most
$\sqrt{\log(|\Delta|/3)}/z$, so the amount of time required to compute all of
the $|I_{i}|^{|e_{i}|}$ for all $i$ is
$(\sqrt{\log(|\Delta|/3)}/z)L(\frac{1}{2},2z)=L(\frac{1}{2},2z)$. Afterward,
the values $|I_{i}|^{|e_{i}|}$ must all be multiplied together, a calculation
which entails at most $\sqrt{\log(|\Delta|/3)}/z$ ideal multiplications where
the log-norms of the input multiplicands are bounded above by
$\log\operatorname{Norm}(I_{i}^{|e_{i}|})=|e_{i}|\log\operatorname{Norm}(I_{i})\leq\left(\frac{N}{p_{i}}\right)^{2}\log
p_{i}\leq N^{2}=\textstyle{L(\frac{1}{2},2z)},$
and thus each of the (at most) $\sqrt{\log(|\Delta|/3)}/z$ multiplications in
the ensuing product can be completed in time at most
$(\sqrt{\log(|\Delta|/3)}/z)L(\frac{1}{2},2z)=L(\frac{1}{2},2z)$. Finally, we
must multiply this end result by $\mathfrak{L}$, an operation which requires
$O(\max(\log\ell,L(\frac{1}{2},2z))^{1+\varepsilon})$ time. All together, the
running time of step 20 is
$L(\frac{1}{2},2z)+O(\max(\log\ell,L(\frac{1}{2},2z))^{1+\varepsilon})=\max((\log\ell)^{1+\varepsilon},L(\frac{1}{2},2z))$,
and the norm of the resulting ideal $I$ is bounded above by
$\ell\cdot\exp(L(\frac{1}{2},2z))$.
Obtaining the generator $\beta$ of $I$ in line 21 of Algorithm 3 using
Cornacchia’s algorithm requires
$\textstyle
O(\log(\operatorname{Norm}(I))^{1+\varepsilon})=(\log\ell+L(\frac{1}{2},2z))^{1+\varepsilon}$
time. We remark that finding $\beta$ given $I$ is substantially easier than
the usual Cornacchia’s algorithm, which entails finding $\beta$ given only
$\operatorname{Norm}(I)$. The usual algorithm requires finding _all_ the
square roots of $\Delta$ modulo $\operatorname{Norm}(I)$, which is very slow
when $\operatorname{Norm}(I)$ has a large number of prime divisors. This time-
consuming step is unnecessary when the ideal $I$ itself is given, since the
embedding of the ideal $I$ in $\operatorname{End}(E)$ already provides (up to
sign) the correct square root of $\Delta$ mod $I$. A detailed description of
this portion of Cornacchia’s algorithm in the context of the full algorithm,
together with running time figures specific to each sub-step, is given by
Hardy et al. [19]; for our purposes, the running time of a single iteration of
Step 6 in [19, §4] is the relevant figure. This concludes our analysis of
Algorithm 3.
Returning to Algorithm 4, we find that (as in [4]) the computation of the
individual isogenies $\phi_{i}$ in line 3 of Algorithm 4 is limited by the
time required to compute the modular polynomials $\Phi_{n}(x,y)$. Using the
Chinese remainder theorem-based method of Bröker et al. [5], these polynomials
can be computed mod $q$ in time $O(n^{3}\log^{3+\varepsilon}(n))$, and the
resulting polynomials require $O(n^{2}(\log^{2}n+\log q))$ space. For each
ideal $I_{i}$, the corresponding modular polynomial of level $p_{i}$ only
needs to be computed once, but the polynomial once computed must be evaluated,
differentiated, and otherwise manipulated $e_{i}$ times, at a cost of
$O(p_{i}^{2+\varepsilon})$ field operations in $\mathbb{F}_{q}$ per
manipulation, or $O(p_{i}^{2+\varepsilon})(\log q)^{1+\varepsilon}$ bit
operations using fast multiplication. The total running time of line 3 is
therefore
$\displaystyle
O(p_{i}^{3+\varepsilon})+\sum_{i}|e_{i}|p_{i}^{2+\varepsilon}(\log
q)^{1+\varepsilon}\leq
O(N^{3+\varepsilon})+\sum_{i}\left(\left(\frac{N}{p_{i}}\right)^{2}\right)p_{i}^{2+\varepsilon}(\log
q)^{1+\varepsilon}$ $\displaystyle\leq
O(N^{3+\varepsilon})+\frac{\sqrt{\log(|\Delta|/3)}}{z}N^{2+\varepsilon}(\log
q)^{1+\varepsilon}=\textstyle{L(\frac{1}{2},3z)+L(\frac{1}{2},2z)(\log
q)^{1+\varepsilon}}.$
Similarly, the evaluation of $\phi_{c}$ in line 4 requires
$\sum_{i}|e_{i}|p_{i}^{2+\varepsilon}=\textstyle{L(\frac{1}{2},2z)}$
field operations in $\mathbb{F}_{q^{n}}$, which corresponds to
$L(\frac{1}{2},2z)(\log q^{n})^{1+\varepsilon}$ bit operations using fast
multiplication.
Combining all the above quantities, we obtain a total running time of
$\displaystyle\textstyle L(\frac{1}{2},z)$ (algorithm 2)
$\displaystyle\textstyle\quad{}+L(\frac{1}{2},\frac{1}{4z})$ (lines 4–9,
algorithm 3)
$\displaystyle\textstyle\quad{}+\max((\log\ell)^{1+\varepsilon},L(\frac{1}{2},2z))$
(line 20, algorithm 3)
$\displaystyle\textstyle\quad{}+(\log\ell+L(\frac{1}{2},2z))^{1+\varepsilon}$
(line 21, algorithm 3)
$\displaystyle\textstyle\quad{}+L(\frac{1}{2},3z)+L(\frac{1}{2},2z)(\log
q)^{1+\varepsilon}$ (line 3, algorithm 4)
$\displaystyle\textstyle\quad{}+L(\frac{1}{2},2z)(\log q^{n})^{1+\varepsilon}$
(line 4, algorithm 4) $\displaystyle=\textstyle
L(\frac{1}{2},\frac{1}{4z})+(\log\ell+L(\frac{1}{2},2z))^{1+\varepsilon}+L(\frac{1}{2},3z)+L(\frac{1}{2},2z)(\log
q^{n})^{1+\varepsilon}.$
When $|\Delta|$ is large, we may impose the reasonable assumption that
$\log(\ell)\ll L(\frac{1}{2},z)$ and $\log(q^{n})\ll L(\frac{1}{2},z)$. In
this case, the running time of Algorithm 4 is dominated by the expression
$L(\frac{1}{2},\frac{1}{4z})+L(\frac{1}{2},3z),$ which attains a minimum at
$z=\frac{1}{2\sqrt{3}}$. Taking this value of $z$, we find that the running
time of Algorithm 4 is equal to
$L_{|\Delta|}(\frac{1}{2},\frac{\sqrt{3}}{2})$. Since the maximum value of
$|\Delta|\leq|\Delta_{\pi}|=4q-t^{2}$ is $4q$, we can alternatively express
this running time as simply $L_{q}(\frac{1}{2},\frac{\sqrt{3}}{2})$.
In the general case, $\log(\ell)$ and $\log(q^{n})$ might be non-negligible
compared to $L(\frac{1}{2},z)$. This can happen in one of two ways: either
$|\Delta|$ is small, or (less likely) $\ell$ is very large and/or $n$ is
large. When this happens, we can still bound the running time of Algorithm 4
by taking $z=\frac{1}{2\sqrt{3}}$ in the foregoing calculation, although such
a choice may fail to be optimal. We then find that the running time of
Algorithm 4 is bounded above by
$\textstyle(\log(\ell)+L(\frac{1}{2},\frac{1}{\sqrt{3}}))^{1+\varepsilon}+L(\frac{1}{2},\frac{\sqrt{3}}{2})+L(\frac{1}{2},\frac{1}{\sqrt{3}})(\log
q^{n})^{1+\varepsilon}.$
We summarize our results in the following theorem.
###### Theorem 4.1
Let $E/\mathbb{F}_{q}$ be an ordinary elliptic curve with Frobenius $\pi_{q}$,
given by a Weierstrass equation, and let $P\in E(\mathbb{F}_{q^{n}})$ be a
point on $E$. Let $\Delta=\operatorname{disc}(\operatorname{End}(E))$ be
given. Assume that $[\operatorname{End}(E):\mathbb{Z}[\pi_{q}]]$ and
$\\#E(\mathbb{F}_{q^{n}})$ are coprime, and let
$\mathfrak{L}=(\ell,c+d\pi_{q})$ be an $\operatorname{End}(E)$-ideal of prime
norm $\ell\neq\operatorname{char}(\mathbb{F}_{q})$ not dividing the index
$[\operatorname{End}(E):\mathbb{Z}[\pi_{q}]]$. Under the heuristics of [1,
§4], Algorithm 4 computes the unique elliptic curve $E^{\prime}$ such that
there exists a normalized isogeny $\phi\colon E\to E^{\prime}$ with kernel
$E[\mathfrak{L}]$. Furthermore, it computes the $x$-coordinate of $\phi(P)$ if
$\operatorname{End}(E)$ does not equal $\mathbb{Z}[i]$ or
$\mathbb{Z}[\zeta_{3}]$ and the square, respectively cube, of the
$x$-coordinate of $\phi(P)$ otherwise. The running time of the algorithm is
bounded above by
$\textstyle(\log(\ell)+L(\frac{1}{2},\frac{1}{\sqrt{3}}))^{1+\varepsilon}+L(\frac{1}{2},\frac{\sqrt{3}}{2})+L(\frac{1}{2},\frac{1}{\sqrt{3}})(\log
q^{n})^{1+\varepsilon}.$
The running time of the algorithm is subexponential in $\log|\Delta|$, and
polynomial in $\log(\ell)$, $\log(q)$, and $n$.
## 5 Examples
### 5.1 Small example
Let $p=10^{10}+19$ and let $E/\mathbb{F}_{p}$ be the curve
$y^{2}=x^{3}+15x+129$. Then $E(\mathbb{F}_{p})$ has cardinality
$10000036491=3\cdot 3333345497$ and trace $t=-36471$. To avoid any bias in the
selection of the prime $\ell$, we set $\ell$ to be the smallest Elkies prime
of $E$ larger than $p/2$, namely $\ell=5000000029$. We will evaluate the
$x$-coordinate of $\phi(P)$, where $\phi$ is an isogeny of degree $\ell$, and
$P$ is chosen arbitrarily to be the point $(5940782169,2162385016)\in
E(\mathbb{F}_{p})$. We remark that, although this example is designed to be
artificially small for illustration purposes, the evaluation of this isogeny
would already be infeasible if we were using prior techniques based on modular
functions of level $\ell$.
The discriminant $\Delta$ of $E$ is $\Delta=t^{2}-4p=-38669866235$. Set
$w=\frac{1+\sqrt{\Delta}}{2}$ and $\mathcal{O}=\mathcal{O}_{\Delta}$. The
quadratic form $(5000000029,-2326859861,270713841)$ represents a prime ideal
$\mathfrak{L}$ of norm $\ell$, and we show how to calculate the isogeny $\phi$
having kernel corresponding to $E[\mathfrak{L}]$. Using an implementation of
Algorithm 3 in MAGMA [22], we find immediately the relation
$\mathfrak{L}=(\frac{\beta}{m})\cdot\mathfrak{p}_{19}\cdot\mathfrak{p}_{31}^{24}$
where $\beta=588048307603210005w-235788727470005542279904$, $m=19\cdot
31^{24}$, $\mathfrak{p}_{19}=(19,2w+7)$, and $\mathfrak{p}_{31}=(31,2w+5)$.
Using this factorization, we can then evaluate $\phi\colon E\to E^{\prime}$
using the latter portion of Algorithm 4. We find that $E^{\prime}$ is the
curve with Weierstrass equation $y^{2}=x^{3}+3565469415x+7170659769$, and
$\phi(P)=(7889337683,\pm 3662693258)$. We omit the details of these steps,
since this portion of the algorithm is identical to the algorithm of Bröker,
Charles and Lauter, and the necessary steps are already extensively detailed
in their article [4].
We can check our computations for consistency by performing a second
computation, starting from the curve
$E^{\prime}:y^{2}=x^{3}+3565469415x+7170659769$, the point
$P^{\prime}=(7889337683,3662693258)\in E^{\prime}(\mathbb{F}_{p})$, and the
conjugate ideal $\bar{\mathfrak{L}}$, which is represented by the quadratic
form $(5000000029,2326859861,270713841)$. Let $\bar{\phi}\colon E^{\prime}\to
E^{\prime\prime}$ denote the unique normalized isogeny with kernel
$E^{\prime}[\bar{\mathfrak{L}}]$. Up to a normalization isomorphism
$\iota\colon E\to E^{\prime\prime}$, the isogeny $\bar{\phi}$ should equal the
dual isogeny $\hat{\phi}$ of $\phi$, and the composition $\bar{\phi}(\phi(P))$
should yield $\iota(\ell P)$. Indeed, upon performing the computation, we find
that $E^{\prime\prime}$ has equation
$y^{2}=x^{3}+(15/\ell^{4})x+(129/\ell^{6}),$
which is isomorphic to $E$ via the isomorphism $\iota\colon E\to
E^{\prime\prime}$ defined by $\iota(x,y)=(x/\ell^{2},y/\ell^{3})$, and
$\bar{\phi}(\phi(P))=(3163843645,8210361642)=(5551543736/\ell^{2},6305164567/\ell^{3}),$
in agreement with the value of $\ell P$, which is $(5551543736,6305164567)$.
### 5.2 Medium example
Let $E$ be the ECCp-109 curve [8] from the Certicom ECC Challenge [7], with
equation $y^{2}=x^{3}+ax+b$ over $\mathbb{F}_{p}$ where
$\displaystyle p$ $\displaystyle=564538252084441556247016902735257$
$\displaystyle a$ $\displaystyle=321094768129147601892514872825668$
$\displaystyle b$ $\displaystyle=430782315140218274262276694323197$
As before, to avoid any bias in the choice of $\ell$, we set $\ell$ to be the
least Elkies prime greater than $p/2$, and we define
$w=\frac{1+\sqrt{\Delta}}{2}$ where
$\Delta=\operatorname{disc}(\operatorname{End}(E))$. Let $\mathfrak{L}$ be the
prime ideal of norm $\ell$ in $\operatorname{End}(E)$ corresponding to the
reduced quadratic form $(\ell,b,c)$ of discriminant $\Delta$, where
$b=-105137660734123120905310489472471$. For each Elkies prime $p$, let
$\mathfrak{p}_{p}$ denote the unique prime ideal corresponding to the reduced
quadratic form $(p,b,c)$ where $b\geq 0$. Our smoothness bound in this case is
$N=L(\frac{1}{2},\frac{1}{2\sqrt{3}})\approx 200$. Using Sutherland’s
smoothrelation package [28], which implements the FindRelation algorithm of
[1], one finds in a few seconds (using an initial seed of 0) the relation
$\mathfrak{L}=\left(\frac{\beta}{m}\right)\mathfrak{I}$, where
$\displaystyle\mathfrak{I}$
$\displaystyle=\bar{\mathfrak{p}}_{7}^{72}\bar{\mathfrak{p}}_{13}^{100}\bar{\mathfrak{p}}_{23}^{14}\bar{\mathfrak{p}}_{47}^{2}\bar{\mathfrak{p}}_{73}^{2}\bar{\mathfrak{p}}_{103}\mathfrak{p}_{179}\mathfrak{p}_{191}$
$\displaystyle m$
$\displaystyle=7^{72}13^{100}23^{14}47^{2}73^{2}103^{1}179^{1}191^{1}$
and
$\displaystyle\beta$
$\displaystyle=3383947601020121267815309931891893555677440374614137047492987151\backslash$
$\displaystyle 2226041731462264847144426019711849448354422205800884837$
$\displaystyle{}-1713152334033312180094376774440754045496152167352278262491589014\backslash$
$\displaystyle 097167238827239427644476075704890979685\cdot w$
We find that the codomain $E^{\prime}$ of the normalized isogeny $\phi\colon
E\to E^{\prime}$ of kernel $E[\mathfrak{L}]$ has equation
$y^{2}=x^{3}+a^{\prime}x+b^{\prime}$ where
$\displaystyle a^{\prime}$ $\displaystyle=84081262962164770032033494307976$
$\displaystyle b^{\prime}$ $\displaystyle=506928585427238387307510041944828$
and that the base point
$\displaystyle
P=(97339010987059066523156133908935,149670372846169285760682371978898)$
of $E$ given in the Certicom ECC challenge has image
$\displaystyle(450689656718652268803536868496211,\pm
345608697871189839292674734567941).$
under $\phi$. As with the first example, we checked the computation for
consistency by using the conjugate ideal.
### 5.3 Large example
Let $E$ be the ECCp-239 curve [8] from the Certicom ECC Challenge [7]. Then
$E$ has equation $y^{2}=x^{3}+ax+b$ over $\mathbb{F}_{p}$ where
$\displaystyle p\\!$
$\displaystyle=\\!862591559561497151050143615844796924047865589835498401307522524859467869$
$\displaystyle a\\!$
$\displaystyle=\\!820125117492400602839381236756362453725976037283079104527317913759073622$
$\displaystyle b\\!$
$\displaystyle=\\!545482459632327583111433582031095022426858572446976004219654298705912499$
Let $\mathfrak{L}$ be the prime ideal whose norm is the least Elkies prime
greater than $p/2$ and whose ideal class is represented by the quadratic form
$(\ell,b,c)$ with $b\geq 0$. We have
$N=L(\frac{1}{2},\frac{1}{2\sqrt{3}})\approx 5000$, and one finds in a few
hours using smoothrelation [28] that $\mathfrak{L}$ is equivalent to
$\mathfrak{I}=\bar{\mathfrak{p}}_{7}^{2}\mathfrak{p}_{11}\mathfrak{p}_{19}\mathfrak{p}_{37}^{2}\bar{\mathfrak{p}}_{71}^{2}\bar{\mathfrak{p}}_{131}\mathfrak{p}_{211}\bar{\mathfrak{p}}_{389}\bar{\mathfrak{p}}_{433}\bar{\mathfrak{p}}_{467}\bar{\mathfrak{p}}_{859}^{18}\mathfrak{p}_{863}\bar{\mathfrak{p}}_{1019}\bar{\mathfrak{p}}_{1151}\bar{\mathfrak{p}}_{1597}\bar{\mathfrak{p}}_{2143}^{6}\bar{\mathfrak{p}}_{2207}^{5}\bar{\mathfrak{p}}_{3359}$
where each ideal $\mathfrak{p}_{p}$ is represented by the reduced quadratic
form $(p,b,c)$ having $b\geq 0$ (this computation can be reconstructed with
[28] using the seed $7$). The quotient $\mathfrak{L}/\mathfrak{I}$ is
generated by $\beta/m$ where $m=\operatorname{Norm}(\mathfrak{I})$ and $\beta$
is
$\displaystyle\\!-\\!923525986803059652225406070265439117913488592374741428959120914067053307\backslash$
$\displaystyle
4585317-917552768623818156695534742084359293432646189962935478129227909w.$
Given this relation, evaluating isogenies of degree $\ell$ is a tedious but
routine computation using Elkies-Atkin techniques [4, §3.1]. Although we do
not complete it here, the computation is well within the reach of present
technology; indeed, Bröker et al. [5] have computed classical modular
polynomials mod $p$ of level up to $20000$, well beyond the largest prime of
$3389$ appearing in our relation.
## 6 Related work
Bisson and Sutherland [1] have developed an algorithm to compute the
endomorphism ring of an elliptic curve in subexponential time, using relation-
finding techniques which largely overlap with ours. Although our main results
were obtained independently, we have incorporated their ideas into our
algorithm in several places, resulting in a simpler presentation as well as a
large speedup compared to the original version of our work.
Given two elliptic curves $E$ and $E^{\prime}$ over $\mathbb{F}_{q}$ admitting
a normalized isogeny $\phi\colon E\to E^{\prime}$ of degree $\ell$, the
equation of $\phi$ as a rational function contains $O(\ell)$ coefficients.
Bostan et al. [3] have published an algorithm which produces this equation,
given $E$, $E^{\prime}$, and $\ell$. Their algorithm has running time
$O(\ell^{1+\varepsilon})$, which is quasi-optimal given the size of the
output. Using our algorithm, it is possible to compute $E^{\prime}$ from $E$
and $\ell$ in time $\log(\ell)L_{|\Delta|}(\frac{1}{2},\frac{\sqrt{3}}{2})$
for large $\ell$. Hence the combination of the two algorithms can produce the
equation of $\phi$ within a quasi-optimal running time of
$O(\ell^{1+\varepsilon})$, given only $E$ and $\ell$ (or $E$ and
$\mathfrak{L}$), without the need to provide $E^{\prime}$ in the input.
## 7 Acknowledgments
We thank the anonymous referees for numerous suggestions which led to
substantial improvements in our main result.
## References
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* [5] R. Bröker, K. Lauter, and A. Sutherland. Modular polynomials via isogeny volcanoes, 2010.
* [6] J. Buchmann and U. Vollmer. Binary quadratic forms, volume 20 of Algorithms and Computation in Mathematics. Springer, Berlin, 2007. An algorithmic approach.
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|
arxiv-papers
| 2010-02-23T00:39:16 |
2024-09-04T02:49:08.501960
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "David Jao and Vladimir Soukharev",
"submitter": "David Jao",
"url": "https://arxiv.org/abs/1002.4228"
}
|
1002.4286
|
6 (2:3) 2010 1–33 Jan. 13, 2009 Jun. 22, 2010
# Redundancy, Deduction Schemes, and Minimum-Size Bases for Association Rules
José L. Balcázar Dep. de Matemáticas, Estadística y Computación
Universidad de Cantabria
Santander, Spain joseluis.balcazar@unican.es
###### Abstract.
Association rules are among the most widely employed data analysis methods in
the field of Data Mining. An association rule is a form of partial implication
between two sets of binary variables. In the most common approach, association
rules are parametrized by a lower bound on their confidence, which is the
empirical conditional probability of their consequent given the antecedent,
and/or by some other parameter bounds such as “support” or deviation from
independence. We study here notions of redundancy among association rules from
a fundamental perspective. We see each transaction in a dataset as an
interpretation (or model) in the propositional logic sense, and consider
existing notions of redundancy, that is, of logical entailment, among
association rules, of the form “any dataset in which this first rule holds
must obey also that second rule, therefore the second is redundant”. We
discuss several existing alternative definitions of redundancy between
association rules and provide new characterizations and relationships among
them. We show that the main alternatives we discuss correspond actually to
just two variants, which differ in the treatment of full-confidence
implications. For each of these two notions of redundancy, we provide a sound
and complete deduction calculus, and we show how to construct complete bases
(that is, axiomatizations) of absolutely minimum size in terms of the number
of rules. We explore finally an approach to redundancy with respect to several
association rules, and fully characterize its simplest case of two partial
premises.
###### Key words and phrases:
Data mining, association rules, implications, redundancy, deductive calculus,
optimum bases
###### 1991 Mathematics Subject Classification:
I.2.3, H.2.8, I.2.4, G.2.3, F.4.1
This work is supported in part by project TIN2007-66523 (FORMALISM) of
Programa Nacional de Investigación, Ministerio de Ciencia e Innovación
(MICINN), Spain, and by the PASCAL2 Network of Excellence of the European
Union.
## 1\. Introduction
The relatively recent discipline of Data Mining involves a wide spectrum of
techniques, inherited from different origins such as Statistics, Databases, or
Machine Learning. Among them, Association Rule Mining is a prominent
conceptual tool and, possibly, a cornerstone notion of the field, if there is
one. Currently, the amount of available knowledge regarding association rules
has grown to the extent that the tasks of creating complete surveys and
websites that maintain pointers to related literature become daunting. A
survey, with plenty of references, is [CegRod], and additional materials are
available in [HahslerWeb]; see also [AIS], [AMSTV], [Freitas], [PasBas],
[Zaki], [ZO], and the references and discussions in their introductory
sections.
Given an agreed general set of “items”, association rules are defined with
respect to a dataset that consists of “transactions”, each of which is,
essentially, a set of items. Association rules are customarily written as
$X\to Y$, for sets of items $X$ and $Y$, and they hold in the given dataset
with a specific “confidence” quantifying how often $Y$ appears among the
transactions in which $X$ appears.
A close relative of the notion of association rule, namely, that of exact
implication in the standard propositional logic framework, or, equivalently,
association rule that holds in 100% of the cases, has been studied in several
guises. Exact implications are equivalent to conjunctions of definite Horn
clauses: the fact, well-known in logic and knowledge representation, that Horn
theories are exactly those closed under bitwise intersection of propositional
models leads to a strong connection with Closure Spaces, which are
characterized by closure under intersection (see the discussions in [DP] or
[KR]). Implications are also very closely related to functional dependencies
in databases. Indeed, implications, as well as functional dependencies, enjoy
analogous, clear, robust, hardly disputable notions of redundancy that can be
defined equivalently both in semantic terms and through the same syntactic
calculus. Specifically, for the semantic notion of entailment, an implication
$X\to Y$ is entailed from a set of implications $\mathcal{R}$ if every dataset
in which all the implications of $\mathcal{R}$ hold must also satisfy $X\to
Y$; and, syntactically, it is known that this happens if and only if $X\to Y$
is derivable from $\mathcal{R}$ via the Armstrong axiom schemes, namely,
Reflexivity ($X\to Y$ for $Y\subseteq X$), Augmentation (if $X\to Y$ and
$X^{\prime}\to Y^{\prime}$ then $XX^{\prime}\to YY^{\prime}$, where
juxtaposition denotes union) and Transitivity (if $X\to Y$ and $Y\to Z$ then
$X\to Z$).
Also, such studies have provided a number of ways to find implications (or
functional dependencies) that hold in a given dataset, and to construct small
subsets of a large set of implications, or of functional dependencies, from
which the whole set can be derived; in Closure Spaces and in Data Mining these
small sets are usually called “bases”, whereas in Dependency Theory they are
called “covers”, and they are closely related to deep topics such as
hypergraph theory. Associated natural notions of minimality (when no
implication can be removed), minimum size, and canonicity of a cover or basis
do exist; again it is inappropriate to try to give a complete set of
references here, but see, for instance, [DP], [EiterG], [GW], [GD],
[GunoEtAl], [KR], [PT], [Wild], [ZO], and the references therein.
However, the fact has been long acknowledged (e.g. already in [Lux]) that,
often, it is inappropriate to search only for absolute implications in the
analysis of real world datasets. Partial rules are defined in relation to
their “confidence”: for a given rule $X\to Y$, the ratio of how often $X$ and
$Y$ are seen together to how often $X$ is seen. Many other alternative
measures of intensity of implication exist [Garriga], [GH]; we keep our focus
on confidence because, besides being among the most common ones, it has a
natural interpretation for educated users through its correspondence with the
observed conditional probability.
The idea of restricting the exploration for association rules to frequent
itemsets, with respect to a support threshold, gave rise to the most widely
discussed and applied algorithm, called Apriori [AMSTV], and to an intense
research activity. Already with full-confidence implications, the output of an
association mining process often consists of large sets of rules, and a well-
known difficulty in applied association rule mining lies in that, on large
datasets, and for sensible settings of the confidence and support thresholds
and other parameters, huge amounts of association rules are often obtained.
Therefore, besides the interesting progress in the topic of how to organize
and query the rules discovered (see [LiuHsuMa], [LiuHuHsu], [TuLiu]), one
research topic that has been worthy of attention is the identification of
patterns that indicate redundancy of rules, and ways to avoid that redundancy;
and each proposed notion of redundancy opens up a major research problem,
namely, to provide a general method for constructing bases of minimum size
with respect to that notion of redundancy.
For partial rules, the Armstrong schemes are not valid anymore. Reflexivity
does hold, but Transitivity takes a different form that affects the confidence
of the rules: if the rule $A\to B$ (or $A\to AB$, which is equivalent) and the
rule $B\to C$ both hold with confidence at least $\gamma$, we still know
nothing about the confidence of $A\to C$; even the fact that both $A\to AB$
and $AB\to C$ hold with confidence at least $\gamma$ only gives us a
confidence lower bound of $\gamma^{2}<\gamma$ for $A\to C$ (assuming
$\gamma<1$). Augmentation does not hold at all; indeed, enlarging the
antecedent of a rule of confidence at least $\gamma$ may give a rule with much
smaller confidence, even zero: think of a case where most of the times $X$
appears it comes with $Z$, but it only comes with $Y$ when $Z$ is not present;
then the confidence of $X\to Z$ may be high whereas the confidence of $XY\to
Z$ may be null. Similarly, if the confidence of $X\to YZ$ is high, it means
that $Y$ and $Z$ appear together in most of the transactions having $X$,
whence the confidences of $X\to Y$ and $X\to Z$ are also high; but, with
respect to the converse, the fact that both $Y$ and $Z$ appear in fractions at
least $\gamma$ of the transactions having $X$ does not inform us that they
show up together at a similar ratio of these transactions: only a ratio of
$2\gamma-1<\gamma$ is guaranteed as a lower bound. In fact, if we look only
for association rules with singletons as consequents (as in some of the
analyses in [AgYu], or in the “basic association rules” of [LiHa], or even in
the traditional approach to association rules [AIS] and the useful apriori
implementation of Borgelt available on the web [BorgeltApriori]) we are almost
certain to lose information. As a consequence of these failures of the
Armstrong schemes, the canonical and minimum-size cover construction methods
available for implications or functional dependencies are not appropriate for
partial association rules.
On the semantic side, a number of formalizations of the intuition of
redundancy among association rules exist in the literature, often with
proposals for defining irredundant bases (see [AgYu], [CrisSim], [KryszPAKDD],
[Lux], [PasBas], [PhanLuongICDM], [Zaki], the survey [Krysz], and section 6 of
the survey [CegRod]). All of these are weaker than the notion that we would
consider natural by comparison with implications (of which we start the study
in the last section of this paper). We observe here that one may wish to
fulfill two different roles with a basis, and that both appear (somewhat
mixed) in the literature: as a computer-supported data structure from which
confidences and supports of rules are computed (a role for which we use the
closures lattice instead) or, in our choice, as a means of providing the user
with a smallish set of association rules for examination and, if convenient,
posterior enumeration of the rules that follow from each rule in the basis.
That is, we will not assume to have available, nor to wish to compute, exact
values for the confidence, but only discern whether it stays above a certain
user-defined threshold. We compute actual confidences out of the closure
lattice only at the time of writing out rules for the user.
This paper focuses mainly on several such notions of redundancy, defined in a
rather general way, by resorting to confidence and support inequalities:
essentially, a rule is redundant with respect to another if it has at least
the same confidence and support of the latter for every dataset. We also
discuss variants of this proposal and other existing definitions given in set-
theoretic terms. For the most basic notion of redundancy, we provide formal
proofs of the so far unstated equivalence among several published proposals,
including a syntactic calculus and a formal proof of the fact, also previously
unknown, that the existing basis known as the Essential Rules or the
Representative Rules ([AgYu], [KryszPAKDD], [PhanLuongICDM]) is of absolutely
minimum size.
It is natural to wish further progress in reducing the size of the basis. Our
theorems indicate that, in order to reduce further the size without losing
information, more powerful notions or redundancy must be deployed. We consider
for this role the proposal of handling separately, to a given extent, full-
confidence implications from lower-than-1-confidence rules, in order to profit
from their very different combinatorics. This separation is present in many
constructions of bases for association rules [Lux], [PasBas], [Zaki]. We
discuss corresponding notions of redundancy and completeness, and prove new
properties of these notions; we give a sound and complete deductive calculus
for this redundancy; and we refine the existing basis constructions up to a
point where we can prove again that we attain the limit of the redundancy
notion.
Next, we discuss yet another potential for strengthening the notion of
redundancy. So far, all the notions have just related one partial rule to
another, possibly in the presence of full implications. Is it possible to
combine two partial rules, of confidence at least $\gamma$, and still obtain a
partial rule obeying that confidence level? Whereas the intuition is that
these confidences will combine together to yield a confidence lower than
$\gamma$, we prove that there is a specific case where a rule of confidence at
least $\gamma$ is nontrivially entailed by two of them. We fully characterize
this case and obtain from the caracterization yet another deduction scheme. We
hope that further progress along the notion of a set of partial rules
entailing a partial rule will be made along the coming years.
Preliminary versions of the results in sections LABEL:dedplain,
LABEL:redundcalculus, LABEL:clocalcsoundcompl, and LABEL:closbasedent have
been presented at Discovery Science 2008 [Bal08b]; preliminary versions of the
remaining results (except those in section LABEL:suppbound, which are newer
and unpublished) have been presented at ECMLPKDD 2008 [Bal08].
## 2\. Preliminaries
Our notation and terminology are quite standard in the Data Mining literature.
All our developments take place in the presence of a “universe” set
$\mathcal{U}$ of atomic elements called items; their absence or presence in
sets or items plays the same role as binary-valued attributes of a relational
table. Subsets of $\mathcal{U}$ are called itemsets. A dataset $\mathcal{D}$
is assumed to be given; it consists of transactions, each of which is an
itemset labeled by a unique transaction identifier. The identifiers allow us
to distinguish among transactions even if they share the same itemset. Upper-
case, often subscripted letters from the end of the alphabet, like $X_{1}$ or
$Y_{0}$, denote itemsets. Juxtaposition denotes union of itemsets, as in $XY$;
and $Z\subset X$ denotes proper subsets, whereas $Z\subseteq X$ is used for
the usual subset relationship with potential equality.
For a transaction $t$, we denote $t\models X$ the fact that $X$ is a subset of
the itemset corresponding to $t$, that is, the transaction satisfies the
minterm corresponding to $X$ in the propositional logic sense.
From the given dataset we obtain a notion of support of an itemset:
$s_{\mathcal{D}}(X)$ is the cardinality of the set of transactions that
include it, $\\{t\in\mathcal{D}\bigm{|}t\models X\\}$; sometimes, abusing
language slightly, we also refer to that set of transactions itself as
support. Whenever $\mathcal{D}$ is clear, we drop the subindex: $s(X)$.
Observe that $s(X)\geq s(Y)$ whenever $X\subseteq Y$; this is immediate from
the definition. Note that many references resort to a normalized notion of
support by dividing by the dataset size. We chose not to, but there is no
essential issue here. Often, research work in Data Mining assumes that a
threshold on the support has been provided and that only sets whose support is
above the threshold (then called “frequent”) are to be considered. We will
require this additional constraint occassionally for the sake of discussing
the applicability of our developments.
We immediately obtain by standard means (see, for instance, [GW] or [Zaki]) a
notion of closed itemsets, namely, those that cannot be enlarged while
maintaining the same support. The function that maps each itemset to the
smallest closed set that contains it is known to be monotonic, extensive, and
idempotent, that is, it is a closure operator. This notion will be reviewed in
more detail later on. Closed sets whose support is above the support
threshold, if given, are usually termed closed frequent sets.
Association rules are pairs of itemsets, denoted as $X\to Y$ for itemsets $X$
and $Y$. Intuitively, they suggest the fact that $Y$ occurs particularly often
among the transactions in which $X$ occurs. More precisely, each such rule has
a confidence associated: the confidence $c_{\mathcal{D}}(X\to Y)$ of an
association rule $X\to Y$ in a dataset $\mathcal{D}$ is $\frac{s(XY)}{s(X)}$.
As with support, often we drop the subindex $\mathcal{D}$. The support in
$\mathcal{D}$ of the association rule $X\to Y$ is $s_{\mathcal{D}}(X\to
Y)=s_{\mathcal{D}}(XY)$.
We can switch rather freely between right-hand sides that include the left-
hand side and right-hand sides that don’t:
Rules $X_{0}\to Y_{0}$ and $X_{1}\to Y_{1}$ are equivalent by reflexivity if
$X_{0}=X_{1}$ and $X_{0}Y_{0}=X_{1}Y_{1}$.
Clearly, $c_{\mathcal{D}}(X\to Y)=c_{\mathcal{D}}(X\to
XY)=c_{\mathcal{D}}(X\to X^{\prime}Y)$ and, likewise, $s_{\mathcal{D}}(X\to
Y)=s_{\mathcal{D}}(X\to XY)=s_{\mathcal{D}}(X\to X^{\prime}Y)$ for any
$X^{\prime}\subseteq X$; that is, the support and confidence of rules that are
equivalent by reflexivity always coincide. A minor notational issue that we
must point out is that, in some references, the left-hand side of a rule is
required to be a subset of the right-hand side, as in [Lux] or
[PhanLuongICDM], whereas many others require the left- and right-hand sides of
an association rule to be disjoint, such as [Krysz] or the original [AIS].
Both the rules whose left-hand side is a subset of the right-hand side, and
the rules that have disjoint sides, may act as canonical representatives for
the rules equivalent to them by reflexivity. We state explicitly one version
of this immediate fact for later reference:
###### Proposition 1.
If rules $X_{0}\to Y_{0}$ and $X_{1}\to Y_{1}$ are equivalent by reflexivity,
$X_{0}\cap Y_{0}=\emptyset$, and $X_{1}\cap Y_{1}=\emptyset$, then they are
the same rule: $X_{0}=X_{1}$ and $Y_{0}=Y_{1}$.
In general, we do allow, along our development, rules where the left-hand
side, or a part of it, appears also at the right-hand side, because by doing
so we will be able to simplify the mathematical arguments. We will assume here
that, at the time of printing out the rules found, that is, for user-oriented
output, the items in the left-hand side are removed from the right-hand side;
accordingly, we write our rules sometimes as $X\to Y-X$ to recall this
convention.
Also, many references require the right-hand side of an association rule to be
nonempty, or even both sides. However, empty sets can be handled with no
difficulty and do give meaningful, albeit uninteresting, rules. A partial rule
$X\to\emptyset$ with an empty right-hand side is equivalent by reflexivity to
$X\to X$, or to $X\to X^{\prime}$ for any $X^{\prime}\subseteq X$, and all of
these rules have always confidence 1. A partial rule with empty left-hand
side, as employed, for instance, in [Krysz], actually gives the normalized
support of the right-hand side as confidence value:
###### Fact 2.
In a dataset $\mathcal{D}$ of $n$ transactions, $c(\emptyset\to Y)=s(Y)/n$.
Again, these sorts of rules could be omitted from user-oriented output, but
considering them conceptually valid simplifies the mathematical development.
We also resort to the convention that, if $s(X)=0$ (which implies that
$s(XY)=0$ as well) we redefine the undefined confidence $c(X\to Y)$ as 1,
since the intuitive expression “all transactions having $X$ do have also $Y$”
becomes vacuously true. This convention is irrespective of whether
$Y\neq\emptyset$.
Throughout the paper, “implications” are association rules of confidence 1,
whereas “partial rules” are those having a confidence below 1. When the
confidence could be 1 or could be less, we say simply “rule”.
## 3\. Redundancy Notions
We start our analysis from one of the notions of redundancy defined formally
in [AgYu]. The notion is employed also, generally with no formal definition,
in several papers on association rules, which subsequently formalize and study
just some particular cases of redundancy (e.g. [KryszPAKDD], [SaquerDeogun]);
thus, we have chosen to qualify this redundancy as “standard”. We propose also
a small variation, seemingly less restrictive; we have not found that variant
explicitly defined in the literature, but it is quite natural.
1. (1)
[AgYu] $X_{0}\to Y_{0}$ has standard redundancy with respect to $X_{1}\to
Y_{1}$ if the confidence and support of $X_{0}\to Y_{0}$ are larger than or
equal to those of $X_{1}\to Y_{1}$, in all datasets.
2. (2)
$X_{0}\to Y_{0}$ has plain redundancy with respect to $X_{1}\to Y_{1}$ if the
confidence of $X_{0}\to Y_{0}$ is larger than or equal to the confidence of
$X_{1}\to Y_{1}$, in all datasets.
Generally, we will be interested in applying these definitions only to rules
$X_{0}\to Y_{0}$ where $Y_{0}\not\subseteq X_{0}$ since, otherwise,
$c(X_{0}\to Y_{0})=1$ for all datasets and the rule is trivially redundant. We
state and prove separately, for later use, the following new technical claim:
###### Lemma 3.
Assume that rule $X_{0}\to Y_{0}$ is plainly redundant with respect to rule
$X_{1}\to Y_{1}$, and that $Y_{0}\not\subseteq X_{0}$. Then
$X_{0}Y_{0}\subseteq X_{1}Y_{1}$.
###### Proof 3.1.
Assume $X_{0}Y_{0}\not\subseteq X_{1}Y_{1}$, to argue the contrapositive.
Then, we can consider a dataset consisting of one transaction $X_{0}$ and,
say, $m$ transactions $X_{1}Y_{1}$. No transaction includes $X_{0}Y_{0}$,
therefore $c(X_{0}\to Y_{0})=0$; however, $c(X_{1}\to Y_{1})$ is either 1 or
$m/(m+1)$, which can be pushed up as much as desired by simply increasing $m$.
Then, plain redundancy does not hold, because it requires $c(X_{0}\to
Y_{0})\geq c(X_{1}\to Y_{1})$ to hold for all datasets whereas, for this
particular dataset, the inequality fails.∎
The first use of this lemma is to show that plain redundancy is not, actually,
weaker than standard redundancy.
###### Theorem 4.
Consider any two rules $X_{0}\to Y_{0}$ and $X_{1}\to Y_{1}$ where
$Y_{0}\not\subseteq X_{0}$. Then $X_{0}\to Y_{0}$ has standard redundancy with
respect to $X_{1}\to Y_{1}$ if and only if $X_{0}\to Y_{0}$ has plain
redundancy with respect to $X_{1}\to Y_{1}$.
|
arxiv-papers
| 2010-02-23T10:02:24 |
2024-09-04T02:49:08.508165
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jose L. Balcazar",
"submitter": "Jos\\'e L Balc\\'azar",
"url": "https://arxiv.org/abs/1002.4286"
}
|
1002.4380
|
# $k^{-3}$ superfluid spectrum of highly curved interacting quantum vortices
Jeffrey Yepez Air Force Research Laboratory, Hanscom Air Force Base,
Massachusetts 01731, USA
###### Abstract
Presented is a prediction, based on the Frenet-Serret differential geometry of
space curves, that the wave number dependence of the average kinetic energy
per unit length of two mutually interacting highly curved quantum vortex
scales as $k^{-3}$. The interacting quantum vortices are helical in shape,
supporting circularly polarized counter-propagating waves, with arbitrary
curvature and torsion. This power-law spectrum agrees with the high-$k$
spectrum found in precise quantum simulations of turbulent superfluidity with
tangle of highly curved and excited quantum vortices.
BEC superfluid, Frenet-Serret formulas, mutually interacting vortices,
$k^{-3}$ helical wave spectrum
###### pacs:
67.25.dk,67.25.dt,67.85.De
Superfluid turbulence is an intriguing low-temperature phenomenon with power-
law energy cascades that are undergoing active investigation. In this Letter
we consider the origin of a $k^{-3}$ power-law in the kinetic energy spectrum
at high-$k$ wave numbers ($\gtrapprox$ the inverse coherence length)
associated with highly curved quantum vortices. The simplest theory for a
superfluid condensate in the zero-temperature limit is
${\cal L}_{\text{\tiny
BEC}}=i\hbar\,\varphi^{\ast}\partial_{t}\varphi+\frac{\hbar^{2}}{2m}(\nabla\varphi^{\ast})\cdot\nabla\varphi+\mu\,\varphi^{\ast}\varphi-\varphi^{\ast}V_{\text{\tiny
H}}\,\varphi,\qquad$ (1)
where $\varphi(x)$ is a complex scalar field for the Bose-Einstein condensate
(BEC), $V_{\text{\tiny H}}$ is a local self-consistent Hartree potential, and
$\mu$ is the chemical potential. Minimizing the action $\int d^{4}x\,{\cal
L}_{\text{\tiny BEC}}$ leads to the Euler-Lagrange equation, a nonlinear
Schroedinger equation when $V_{\text{\tiny
H}}(|\varphi|^{2})=\frac{1}{2}g|\varphi|^{2}$, where $g$ is the real-valued
coupling strength of the nonlinear interaction, known as the Gross-Pitaevskii
equation (GPE)Gross (1963); Pitaevskii (1961)
$i\hbar\partial_{t}\varphi=-\frac{\hbar^{2}}{2m}\nabla^{2}\varphi+(g|\varphi|^{2}-\mu)\,\varphi.$
It captures complex vortex-vortex interplay (nucleation, emission and
absorption of vortex rings, and reconnection) as well as quantum Kelvin wave
mode (kelvon) production on the vortices and sound modes (phonons) that escape
into the bulk region of the quantum fluid.
To determine a steady-state solution of the quantum vortex in a superfluid,
with a BEC wave function denoted by $\varphi_{\text{\tiny v}}$, one solves the
time-independent GPE: $-\xi^{2}\nabla^{2}\varphi_{\text{\tiny
v}}+\left(\frac{g}{\mu}\,|\varphi_{\text{\tiny
v}}|^{2}-1\right)\varphi_{\text{\tiny v}}=0$ with healing length
$\xi\equiv\hbar/\sqrt{2m\mu}$. A solution for the background condensate wave
function of a single rectilinear quantum vortex (with vorticity along
$\hat{\bm{z}}$) is found by separation of variables in polar coordinates.
Inserting $\varphi_{\text{\tiny v}}(r,\vartheta,z)=\phi_{\text{\tiny
v}}(r)Z_{\text{\tiny v}}(z)e^{in\vartheta}$ into time-independent GPE with
$g/\mu=\xi^{2}$ gives the following equations with a separation constant
$k_{\parallel}^{2}$:
$\frac{d^{2}Z_{\text{\tiny v}}(z)}{dx^{2}}+k_{\parallel}^{2}Z_{\text{\tiny
v}}(z)=0,\vspace{-.10in}$ (2a) $\frac{d^{2}\phi_{\text{\tiny
v}}(r)}{dr^{2}}+\frac{1}{r}\frac{d\phi_{\text{\tiny
v}}(r)}{dr}-\frac{n^{2}}{r^{2}}\phi_{\text{\tiny
v}}(r)+\left(a-\phi_{\text{\tiny v}}(r)^{2}\right)\phi_{\text{\tiny v}}(r)=0,$
(2b)
where $a\equiv\xi^{-2}-k_{\parallel}^{2}$. Equation (2a) admits sinusoidal
solutions and (2b) can be solved for integer winding number $n$. For the
simplest $n=1$ case, the Padé approximant $\phi_{\text{\tiny
v}}(r)=\sqrt{a}\sqrt{\frac{11ar^{2}(12+ar^{2})}{384+ar^{2}(128+11ar^{2})}}$
solves the time-independent GPE with errors at ${\cal
O}\left[(r\sqrt{a})^{7}\right]$. Notice that $\phi_{\text{\tiny
v}}(r)\rightarrow\sqrt{a}$ and $r\rightarrow\infty$, and thus the nonlinear
term in (2b) vanishes in the bulk.
The rectilinear quantum vortex solution of the GPE is $\varphi_{\text{\tiny
v}}(x)=\phi_{\text{\tiny
v}}(r)\left[Z_{\circ}^{+}e^{i(\omega_{\parallel}t+k_{\parallel}z)}+Z_{\circ}^{-}e^{i(\omega_{\parallel}t-k_{\parallel}z)}\right],$
with a parabolic dispersion relation
$\hbar\omega_{\parallel}=\frac{\hbar^{2}k_{\parallel}^{2}}{2m}$. For a vortex
line along the $\hat{\bm{z}}$-direction in cylindrical coordinates with unit
winding number, the irrotational part of the superfluid velocity has a
divergent (perpendicular) part characteristic of inviscid flow and an
advective (parallel) part characteristic of rigid translation
$\displaystyle\bm{v}\equiv\frac{\hbar}{m}\nabla\left(\vartheta\pm
k_{\parallel}z\right)=\frac{\hbar}{mr}\hat{\bm{\vartheta}}\pm\frac{\hbar
k_{\parallel}}{m}\hat{\bm{z}}.$ (3)
From this velocity field, we know that the circulation is quantized,
$\kappa\equiv\oint d\bm{l}\cdot\bm{v}=\frac{h}{m}.$ From Stokes’ theorem, we
have $\int d\bm{S}\cdot\nabla\times\bm{v}=\int
d\bm{S}\cdot\bm{\omega}=\frac{h}{m}.$ For a rectilinear
$\hat{\bm{z}}$-directed quantum vortex, the real part of the vorticity is
pinned at the vortex center
$\bm{\omega}=\frac{h}{m}\delta^{(2)}(r)\hat{\bm{z}}.$ For a vortex filament of
any shape, say a curve ${\cal C}$, the velocity field in general may be
written as
$\bm{v}(\bm{r})=\frac{\hbar}{2m}\oint_{\cal
C}\frac{d\bm{s}^{\prime}\times(\bm{r}-\bm{s}^{\prime})}{|\bm{r}-\bm{s}^{\prime}|^{3}},$
(4)
where $d\bm{s}^{\prime}$ is the differential length along the vortex filament,
$\bm{s}^{\prime}$ is the parametrization of ${\cal C}$, $\bm{r}$ is the field
point, and $d\bm{s}^{\prime}$ is the differential line element at the vortex
center and parallel to the vorticitySchwarz (1985). The Biot-Savart formula
(4) reduces to $\bm{v}=\hat{\bm{\vartheta}}\,{\hbar}/{(mr)}$ for the case of
an infinite rectilinear quantum vortex positioned along the center of a
cylindrical coordinate system with $\bm{s}^{\prime}={\bm{z}}$,
$d\bm{s}^{\prime}=dz\,\hat{\bm{z}}$, and
$d\bm{s}^{\prime}\times(\hat{\bm{r}}-\hat{\bm{s}}^{\prime})=\hat{\bm{\vartheta}}\,dz\,|z|/|\bm{r}-\bm{s}^{\prime}|$.
Using the Madelung transformationMadelung (1927)
$\varphi=\sqrt{\rho}\,e^{iS/\hbar}$, (1) can be written in terms of the
conjugate fluid variables (the action $S$ and number density $\rho$) as
follows: ${\cal L}_{\text{\tiny
BEC}}=-\rho\,\partial_{t}S-\rho\left[\frac{\left(\nabla
S\right)^{2}}{2m}+\frac{\hbar^{2}}{2m}\left(\frac{\nabla\rho}{2\rho}\right)^{2}-\mu+V_{\text{\tiny
H}}(\rho)\right].$ The quantity in square brackets is identified with a
Hamiltonian energy functional. Bohm originally made this identification while
considering quantum flow in a spatially-dependent linear external potential
$V(\bm{x})$ Bohm (1952). For a BEC, the nonlinear internal potential energy is
$E_{\text{\tiny int}}(\rho)\equiv V_{\text{\tiny H}}(\rho)-\mu.$ A
semiclassical energy functional is thus $H_{\text{\tiny
BEC}}=\frac{\left(\nabla
S\right)^{2}}{2m}+\frac{\hbar^{2}}{2m}\left(\frac{\nabla\rho}{2\rho}\right)^{2}+E_{\text{\tiny
int}}(\rho).$ The average energy is a statistical volume integral
$\overline{E}=\int d^{3}x\,\rho\,H_{\text{\tiny BEC}}$ with $\rho$ taken as
the effective probability distribution $\overline{E}=\int
d^{3}x\,\rho\left[\frac{\left(\nabla
S\right)^{2}}{2m}+\frac{\hbar^{2}}{2m}\left(\frac{\nabla\rho}{2\rho}\right)^{2}+E_{\text{\tiny
int}}(\rho)\right].$ $\overline{E}$ is a conserved quantity. Here we are
interested in the first term $\overline{E}_{\text{kin}}^{\text{cl}}=\int
d^{3}x\,\rho\frac{\left(\nabla S\right)^{2}}{2m}=\int
d^{3}x\,\rho(x)\,\left(\frac{1}{2}m\bm{v}(x)^{2}\right)$, where the de Broglie
relation $m\bm{v}=\nabla S$ is used. The average classical kinetic energy per
unit length $L$ of a single linear quantum vortex is ${\cal
E}\equiv\overline{E}_{\text{kin}}^{\text{cl}}/L$:
${\cal E}=m\rho_{\circ}\int_{r_{c}}^{r_{b}}2\pi
rdr\,\frac{v_{\vartheta}^{2}}{2}\stackrel{{\scriptstyle(\ref{irrotational_part_of_superfluid_velocity_field_c})}}{{=}}\frac{\rho_{\circ}h^{2}}{4\pi
m}\int_{r_{c}}^{r_{b}}\frac{dr}{r}=m\rho_{\circ}\frac{\kappa_{\circ}^{2}}{4\pi}\log\frac{r_{b}}{r_{c}},$
(5)
where $\kappa_{\circ}\equiv h/m$ is the quantum of circulation and
$\rho_{\circ}$ is the constant background number density of the condensate,
$r_{b}$ is a regularizing parameter associated with the size of the vessel
containing a single quantum vortex and $r_{c}$ is an effective cutoff
parameter to the divergent angular velocity field.111$r_{c}$ may be chosen so
that the integral for ${\cal E}$, with a cutoff that avoids the singularity at
the origin, is equivalent to the original nonsingular integral with no cutoff.
So, $r_{c}$ is technically not a cutoff parameter per se because the original
integral is nonsingular. Instead, $r_{c}$ is merely a matching parameter
useful for replacing a nonsingular but difficult integrand with an
analytically simpler one. All the expressions of the average classical kinetic
energy per unit length in (5) are equivalent and apply to a nearly straight
(high-tension and low-curvature) vortex.
Consider the original treatment by Fetter for nearly-rectilinear vortices
Fetter (1967). The initial (single vortex) equilibrium states are parallel
linear filaments with an arc length parametrization given by the following
vectors $\bm{R}_{1}^{(0)}=(\bm{r}_{1},z_{1})$ and
$\bm{R}_{2}^{(0)}=(\bm{r}_{2},z_{2}).$ The consequent deformed state due to
the mutual interaction of the vortices is parametrized by
$\bm{R}_{1}=(\bm{r}_{1}+\bm{u}_{1}(z_{1}),z_{1})\qquad\qquad\bm{R}_{2}=(\bm{r}_{2}+\bm{u}_{2}(z_{2}),z_{2}),$
(6)
where $\bm{u}_{1}$ and $\bm{u}_{2}$ are treated as small amplitude
perturbations in the radial directions with respect to the initial unperturbed
filamentary lines. Each vortex filament is a stretched helix, approximating a
nearly straight line parallel to the $\hat{\bm{z}}$-axis.
The first step is to calculate the fluctuation in position of a vortex element
at the first vortex due to the presence of the second vortex. The fluid
velocity at the first vortex located at $\bm{R}_{1}$ caused by the second
vortex at $\bm{R}_{2}$ is given by the Biot-Savart law (4)
$\bm{v}(\bm{R}_{1})=\frac{\kappa_{\circ}}{4\pi}\oint_{{\cal
C}_{2}}\frac{d\bm{s}_{2}\times(\bm{R}_{1}-\bm{R}_{2})}{|\bm{R}_{1}-\bm{R}_{2}|^{3}}.$
(7)
With $i=1,2$ denoting the vortices, the differential arc length is
$d\bm{s}_{i}=\frac{d\bm{s}_{i}}{dz_{i}}dz_{i}=dz_{i}\left(\hat{\bm{z}}+\frac{d\bm{u}_{i}(z_{i})}{dz_{i}}\right).$
Then, according to (7), the fluctuation of the position of the vortex element
originally at $\bm{R}^{(0)}_{1}$ is
$\displaystyle\frac{d\bm{u}_{1}}{dt}\\!\\!\\!$
$\displaystyle\stackrel{{\scriptstyle(\ref{deformed_R_s})}}{{=}}$
$\displaystyle\\!\\!\\!\frac{\kappa_{\circ}}{4\pi}\int
dz_{2}\frac{\left(\hat{\bm{z}}+\frac{d\bm{u}_{2}}{dz_{2}}\right)\times\left(\bm{r}_{12}+\hat{\bm{z}}z_{12}+\bm{u}_{12}\right)}{|\bm{r}_{12}+\hat{\bm{z}}z_{12}+\bm{u}_{12}|^{3}},$
where $\bm{r}_{12}\equiv\bm{r}_{1}-\bm{r}_{2}$, $z_{12}\equiv z_{1}-z_{2}$,
and $\bm{u}_{12}\equiv\bm{u}_{1}-\bm{u}_{2}$. Also, defining
$\bm{R}_{12}^{(0)}\equiv\bm{r}_{12}+\hat{\bm{z}}z_{12}$ and making use of the
Taylor expansion
$\frac{1}{|\bm{R}_{12}^{(0)}+\bm{u}_{12}|^{3}}=\frac{1}{|\bm{R}_{12}^{(0)}|^{3}}-\frac{3\bm{u}_{12}\cdot\bm{R}_{12}^{(0)}}{|\bm{R}_{12}^{(0)}|^{5}}+\cdots,$
we can write a leading order expansion of the position fluctuation (velocity
of the first vortex element at $z_{1}$ due to the presence of the second
vortex) $\displaystyle\frac{d\bm{u}_{1}}{dt}\\!\\!\\!$
$\displaystyle\stackrel{{\scriptstyle(\ref{velocity_fluctuation_c})}}{{=}}$
$\displaystyle\\!\\!\\!\frac{\kappa_{\circ}}{4\pi}\hat{\bm{z}}\times\int
dz_{2}\Big{[}\frac{\bm{r}_{12}+\bm{u}_{12}-z_{12}\frac{d\bm{u}_{2}}{dz_{2}}}{|\bm{R}_{12}^{(0)}|^{3}}$
(8b) $\displaystyle\hskip
54.2025pt-\frac{3\,\bm{r}_{12}\cdot\bm{u}_{12}}{|\bm{R}_{12}^{(0)}|^{5}}\bm{r}_{12}+\cdots\Big{]}.$
Making the analogy to the mutual inductance of two line currents, the Neumann
formula can be used to calculate the interaction energy of two vortices
$E_{12}=m\rho_{\circ}\int_{r_{c}}d^{3}x\,\frac{\bm{v}(x)^{2}}{2}\stackrel{{\scriptstyle(\ref{quantum_vortex_Biot_Savart_formula})}}{{=}}\frac{m\rho\kappa_{\circ}^{2}}{4\pi}\oint_{{\cal
C}_{1}}\oint_{{\cal C}_{2}}\frac{d\bm{s}_{1}\cdot
d\bm{s}_{2}}{|\bm{R}_{1}-\bm{R}_{2}|}.$ (9)
The next step is to develop an expansion for the interaction energy
$E_{12}=\frac{m\rho\kappa_{\circ}^{2}}{4\pi}\int\int
dz_{1}dz_{2}\frac{\left(1+\frac{d\bm{u}_{1}}{dz_{1}}\cdot\frac{d\bm{u}_{2}}{dz_{2}}+\cdots\right)}{|\bm{R}_{12}^{(0)}+\bm{u}_{12}|},\qquad$
where the cross-terms vanish because, in the reference frame at the original
center of the unperturbed $i$th vortex line along $\hat{\bm{z}}$, the motion
of the perturbed filament is in the polar direction
$\frac{d\bm{u}_{i}}{dz_{i}}/|\frac{d\bm{u}_{1}}{dz_{1}}|\approx\pm\hat{\bm{\vartheta}}$.
Employing Taylor’s theorem, the denominator is expanded to second order
$\frac{1}{|\bm{R}_{12}^{(0)}+\bm{u}_{12}|}=\frac{1}{|\bm{R}_{12}^{(0)}|}-\frac{\bm{r}_{12}\cdot\bm{u}_{12}}{|\bm{R}_{12}^{(0)}|^{3}}+\frac{3(\bm{r}_{12}\cdot\bm{u}_{12})^{2}}{2|\bm{R}_{12}^{(0)}|^{5}}-\frac{(\bm{u}_{12})^{2}}{2|\bm{R}_{12}^{(0)}|^{3}}+\cdots,$
so in turn the interaction energy expansion becomes
$\displaystyle E_{12}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{m\rho\kappa_{\circ}^{2}}{4\pi}\int\int
dz_{1}dz_{2}\Big{[}\frac{1}{|\bm{R}_{12}^{(0)}|}-\frac{\bm{r}_{12}\cdot\bm{u}_{12}-\frac{1}{2}(\bm{u}_{12})^{2}}{|\bm{R}_{12}^{(0)}|^{3}}$
(10)
$\displaystyle+\frac{z_{12}}{|\bm{R}_{12}^{(0)}|^{3}}\frac{d\bm{u}_{2}}{dz_{2}}\cdot\bm{u}_{1}+\frac{\frac{3}{2}(\bm{r}_{12}\cdot\bm{u}_{12})^{2}}{|\bm{R}_{12}^{(0)}|^{5}}+\cdots\Big{]},$
where in (10) the term obtained by integrated by parts was rewritten as
$-\frac{d}{dz_{1}}\frac{1}{|\bm{R}_{12}^{(0)}|}=\frac{\hat{\bm{z}}\cdot\bm{R}_{12}^{(0)}}{|\bm{R}_{12}^{(0)}|^{3}}=\frac{z_{12}}{|\bm{R}_{12}^{(0)}|^{3}}$.
Hence, it is straightforward to calculate the variation of the mutual
interaction energy with respect to a fluctuation at $z_{1}$ of the center of
the first vortex line
$\begin{split}\frac{\delta
E_{12}}{\delta\bm{u}_{1}(z_{1})}=&\frac{m\rho\kappa_{\circ}^{2}}{4\pi}\int
dz_{2}\Big{[}-\frac{\bm{r}_{12}+\bm{u}_{12}-z_{12}\frac{d\bm{u}_{2}}{dz_{2}}}{|\bm{R}_{12}^{(0)}|^{3}}\\\
&+\frac{3\,\bm{r}_{12}\cdot\bm{u}_{12}}{|\bm{R}_{12}^{(0)}|^{5}}\bm{r}_{12}+\cdots\Big{]}.\end{split}$
(11)
Comparing this result with the previous result (8b) yields the useful relation
$m\rho\kappa_{\circ}\frac{d\bm{u}_{1}(z_{1})}{dt}=-\hat{\bm{z}}\times\frac{\delta
E_{12}}{\delta\bm{u}_{1}(z_{1})}.$ (12)
The mutual interaction energy part of the condensate energy arising from a
perturbed quantum vortex of length $L=\oint_{\cal C}\sqrt{dz^{2}+d\bm{u}^{2}}$
is primarily due to its bending, assuming a sufficient separation distance
exists between the vortices so that vortex-vortex straining has no low-order
effect. Therefore, we have
$\displaystyle\frac{\delta E_{12}}{\delta\bm{u}_{1}(z_{1})}$
$\displaystyle\stackrel{{\scriptstyle(\ref{Neumann_mutual_interaction_formula})}}{{\stackrel{{\scriptstyle(\ref{average_classical_kinetic_energy_per_unit_length})}}{{=}}}}$
$\displaystyle{\cal E}\frac{\delta L}{\delta\bm{u}_{1}(z_{1})}=-{\cal
E}\,\frac{d^{2}\bm{u}_{1}(z_{1})}{dz_{1}^{2}}+\cdots$ (13a)
Inserting this result into (12), yields
$m\rho\kappa_{\circ}\frac{d\bm{u}_{1}(z_{1})}{dt}={\cal
E}\hat{\bm{z}}\times\frac{d^{2}\bm{u}_{1}(z_{1})}{dz_{1}^{2}}.$ (14)
Having completed our review of Fetter’s treatment, let us consider a high-
curvature generalization of (14).
The Frenet-Serret formulas of multivariable calculus concerning the geometry
of curves describe the kinematic properties of a particle at position $\bm{R}$
moving along a continuous and differentiable curve $\cal C$ (the particle’s
trajectory or world line) embedded in three-dimensional Euclidean space
$\mathbb{R}^{3}$
$\displaystyle\frac{d\hat{\bm{t}}}{ds}$ $\displaystyle=$
$\displaystyle\kappa\hat{\bm{n}}\qquad\quad\frac{d\hat{\bm{n}}}{ds}=-\kappa\hat{\bm{t}}+\tau\hat{\bm{b}}\qquad\quad\frac{d\hat{\bm{b}}}{ds}=-\tau\hat{\bm{n}},\qquad$
(15)
where $s$ is the arc length parameter along $\cal C$, $\hat{\bm{t}}$ is the
unit tangent to $\cal C$ at the point $\bm{R}$, $\hat{\bm{n}}$ is the unit
normal perpendicular to $\hat{\bm{t}}$ at $\bm{R}$, $\hat{\bm{b}}$ is the unit
bi-normal perpendicular to both $\hat{\bm{t}}$ and $\hat{\bm{n}}$, and
$\kappa$ is the curvature and $\tau$ is the torsion of $\cal C$ at $\bm{R}$.
Given a fixed curve $\cal C$, one constructs a local Frenet-Serret frame as
follows
$\displaystyle\hat{\bm{t}}\equiv\frac{\bm{R}^{\prime}(s)}{|\bm{R}^{\prime}(s)|}\qquad\hat{\bm{n}}\equiv\frac{\hat{\bm{t}}^{\prime}(s)}{|\hat{\bm{t}}^{\prime}(s)|}\qquad\hat{\bm{b}}\equiv\hat{\bm{t}}\times\hat{\bm{n}},$
(16)
where the prime indicates differentiation with respect to $s$. So
$\hat{\bm{n}}$ points along the direction of the derivative of $\hat{\bm{t}}$
with respect to the arc length parameter of the curve and equating (15) with
(16), the curvature is $\kappa=|\hat{\bm{t}}^{\prime}(s)|.$ Therefore, the
unit vectors $\hat{\bm{t}}$, $\hat{\bm{n}}$, and $\hat{\bm{b}}$ serve as an
orthogonal coordinate system centered at $\bm{R}$, a local reference frame
that moves with the particle. For example, specifying the points in
$\mathbb{R}^{3}$ with the cylindrical coordinates $(r,\vartheta,z)$, if $\cal
C$ is a helix with its axis along $r=0$
$\displaystyle\bm{R}(s)$ $\displaystyle=$ $\displaystyle\left(\mathfrak{r}\cos
kz,\mathfrak{r}\sin kz,0\middle)+\middle(0,0,\mathfrak{h}kz\right)$ (17a)
$\displaystyle=$ $\displaystyle\bm{u}(kz)+\begin{cases}0+{\cal
O}\left(\frac{\mathfrak{h}}{\mathfrak{r}}\right)&\mathfrak{h}\ll\mathfrak{r},\\\
s\hat{\bm{z}}-{\cal
O}\left(\frac{\mathfrak{r}^{2}}{\mathfrak{h}^{2}}\right)&\mathfrak{r}\ll\mathfrak{h}\end{cases}$
(17b)
with $kz\equiv s/\sqrt{\mathfrak{r}^{2}+\mathfrak{h}^{2}}$, then the Frenet-
Serret frame is
$\displaystyle\hat{\bm{t}}$ $\displaystyle=$
$\displaystyle\frac{\mathfrak{r}\hat{\bm{\vartheta}}+\mathfrak{h}\hat{\bm{z}}}{\sqrt{\mathfrak{r}^{2}+\mathfrak{h}^{2}}}\qquad\hat{\bm{n}}=-\hat{\bm{r}}\qquad\hat{\bm{b}}=\frac{-\mathfrak{h}\hat{\bm{\vartheta}}+\mathfrak{r}\hat{\bm{z}}}{\sqrt{\mathfrak{r}^{2}+\mathfrak{h}^{2}}}$
(18)
and the local curvature and torsion of ${\cal C}$ are
$\displaystyle\kappa=\frac{\mathfrak{r}}{\mathfrak{r}^{2}+\mathfrak{h}^{2}}\qquad\qquad\tau=\frac{\mathfrak{h}}{\mathfrak{r}^{2}+\mathfrak{h}^{2}}.$
(19)
In the limit $\mathfrak{h}\ll\mathfrak{r}$, the curve is a compressed helix
where each cycle approximates a circle with curvature $1/\mathfrak{r}$. In the
opposite limit $\mathfrak{h}\gg\mathfrak{r}$, the curve is a stretched helix
(approximating a straight line) with torsion $1/\mathfrak{h}$.
We may reconsider the case of a perturbation to a rectilinear quantum vortex:
a quantum vortex supporting a small amplitude circularly-polarized plane
Kelvin wave counterrotating in a sense opposite to the vorticity direction of
the unperturbed vortex line. The solution we seek is based on the general
Frenet-Serret formulas.222 To make additional contact with the previous
literature, consider the limit of small curvature with the perturbed vortex is
a stretched helix. The local frame, which we may choose to fix at $z_{1}$
centered on the first quantum vortex, is
$\hat{\bm{t}}\equiv\frac{\bm{u}_{1}^{\prime}(z_{1})}{|\bm{u}_{1}^{\prime}(z_{1})|}=\hat{\bm{z}}+\cdots$,
$\hat{\bm{n}}\equiv\frac{\bm{u}_{1}^{\prime\prime}(z_{1})}{|\bm{u}_{1}^{\prime\prime}(z_{1})|}=-\hat{\bm{r}}+\cdots,$
and
$\hat{\bm{b}}\equiv\hat{\bm{t}}\times\hat{\bm{n}}=-\hat{\bm{\vartheta}}+\cdots$,
where the arc length is parametrized by $s\approx z_{1}$. The mutual
interaction of the vortices causes them to bend into the filamentary shape of
a rotating helix—a circularly polarized Kelvin wave. Equation (14) may be
rewritten as
$\dot{\bm{u}}\stackrel{{\scriptstyle(\ref{Frenet_Serret_construction_a})}}{{=}}\frac{{\cal
E}}{m\rho\kappa_{\circ}}\frac{\bm{u}^{\prime}}{|\bm{u}^{\prime}|}\times\bm{u}^{\prime\prime},$
In a parametrization with $|\bm{u}^{\prime}|=1$, this is the local induction
approximation (LIA) used for quantum turbulence simulations Schwarz (1985).
Since the second derivative of the radial perturbation of the quantum vortex
center is
$\bm{u}_{1}^{\prime\prime}\stackrel{{\scriptstyle(\ref{particle_trajectory})}}{{=}}-k^{2}\bm{u}_{1},$
(14) takes the form of an undamped Bloch equation
$\frac{d\bm{u}_{1}(z_{1})}{dt}=\frac{{\cal
E}k^{2}}{m\rho\kappa_{\circ}}\,\bm{u}_{1}(z_{1})\times\hat{\bm{s}}_{2}.$ (20)
The radial displacement $\bm{u}_{1}$ behaves like the magnetization vector of
a nuclear spin precessing about a background magnetic field along
$\hat{\bm{s}}_{2}\approx\hat{z}$. Thus, two segments of the mutually
interacting quantum vortices behave like coupled nuclear spins.
The helix rotates in time as a sinusoidal perturbation
$\bm{u}_{1}\sim\mathfrak{r}_{1}\,e^{i(kz-\omega t)}$, so (14) may be written
as
$\displaystyle i\omega\bm{u}_{1}(z_{1})$ $\displaystyle=$
$\displaystyle\frac{{\cal
E}(k)k^{2}}{m\rho\kappa_{\circ}}\hat{\bm{z}}\times\bm{u}_{1}(z_{1}).$ (21)
Then with $\bm{u}_{1}=(x,y)$, we have $i\omega(x,y)=\frac{{\cal
E}(k)k^{2}}{m\rho\kappa_{\circ}}(-y,x),$ or in matrix form
$\scriptsize\begin{pmatrix}i\omega&\frac{{\cal
E}k^{2}}{m\rho\kappa_{\circ}}\\\ -\frac{{\cal
E}k^{2}}{m\rho\kappa_{\circ}}&i\omega\end{pmatrix}\begin{pmatrix}x\\\
y\end{pmatrix}=\begin{pmatrix}0\\\ 0\end{pmatrix}.$ The well known solution by
setting the determinant to zero, for $r_{b}=1/k$, is the Kelvin wave
dispersion relation
$\displaystyle\omega_{\text{\tiny K}}$ $\displaystyle=$
$\displaystyle\frac{{\cal
E}(k)k^{2}}{m\rho\kappa_{\circ}}\stackrel{{\scriptstyle(\ref{average_classical_kinetic_energy_per_unit_length})}}{{=}}\frac{\kappa_{\circ}k^{2}}{4\pi}\log\frac{1}{r_{c}k},$
(22)
valid in the limit where $\mathfrak{r}/\mathfrak{h}$ is a small quantity with
${\cal C}$ nearly a line. Now, with a locally helical-shaped quantum vortex
parametrized by (17), we need not restrict $\mathfrak{r}/\mathfrak{h}$ to be a
small parameter. Instead the only generic constraint imposed upon the shape of
the perturbed quantum vortex is that it be locally parametrized by (17a). We
will not loose the generality of two interacting quantum vortices of arbitrary
shape by restricting their mutually interacting segments to helices.
Since $|\bm{R}^{\prime}(s)|=1$,
$\hat{\bm{t}}\stackrel{{\scriptstyle(\ref{Frenet_Serret_construction_a})}}{{=}}\bm{R}^{\prime}(s)$
and in turn (15) is
$\frac{d^{2}\bm{R}(s)}{ds^{2}}\stackrel{{\scriptstyle(\ref{curvature_and_torsion_of_helix})}}{{=}}\frac{\mathfrak{r}}{\mathfrak{r}^{2}+\mathfrak{h}^{2}}\hat{\bm{n}}.$
(23)
So, allowing for high curvature, (16) can be written as
$\displaystyle\hat{\bm{b}}$
$\displaystyle\stackrel{{\scriptstyle(\ref{curvature_equation})}}{{=}}$
$\displaystyle\frac{\mathfrak{r}^{2}+\mathfrak{h}^{2}}{\mathfrak{r}}\hat{\bm{t}}\times\frac{d^{2}\bm{R}(s)}{ds^{2}}$
(24a)
$\displaystyle\stackrel{{\scriptstyle(\ref{particle_trajectory_a})}}{{=}}$
$\displaystyle\frac{1}{\mathfrak{r}}\,\hat{\bm{t}}\times\bm{R}(s)$ (24b)
$\displaystyle-\mathfrak{h}\hat{\bm{\vartheta}}+\mathfrak{r}\hat{\bm{z}}$
$\displaystyle\stackrel{{\scriptstyle(\ref{Frenet_Serret_frame})}}{{=}}$
$\displaystyle\left(\hat{\bm{\vartheta}}+\frac{\mathfrak{h}}{\mathfrak{r}}\hat{\bm{z}}\right)\times\bm{R}(s).$
(24c)
We are now considering a pair of parallel vortex segments, arbitrary helices
equal in magnitude and spin, circling around each other whereby each segment
moves by the influence of the velocity field of the other. Since the velocity
field of a vortex segment (with a circularly-polarized radial perturbation) is
anti-parallel to the bi-normal unit vector of a helical curve (i.e.,
$\bm{v}\propto-\hat{\bm{b}}$), this velocity field may be generally expressed
in the Frenet-Serret frame as
$\displaystyle\bm{v}$
$\displaystyle\stackrel{{\scriptstyle(\ref{Frenet_Serret_frame})}}{{=}}$
$\displaystyle\omega_{\circ}(\mathfrak{h}\hat{\bm{\vartheta}}-\mathfrak{r}\hat{\bm{z}}).$
(25)
Next, let us determine the radius $\mathfrak{r}$ and the stretching
$\mathfrak{h}$ of the helix directly in terms of the available physical
parameters describing the mutually interacting quantum vortices, such as the
separation distance between the vortices $2R_{12}^{(0)}$. Let us denote the
orbital radius at $R_{\circ}\equiv R_{12}^{(0)}$. Thus, the magnitude of the
polar velocity component is $v_{\vartheta}=\frac{\hbar}{m(2R_{\circ})}.$ If
$\bm{v}_{\vartheta}=\bm{\omega}_{\circ}\times\bm{R}_{\circ}$, where
$\bm{\omega}_{\circ}=\omega_{\circ}\hat{\bm{z}}$ is the effective orbital
“cyclotron” spin vector, then the orbital angular frequency of the vortex pair
is
$\omega_{\circ}=\frac{v_{\vartheta}}{R_{\circ}}=\frac{\hbar}{2mR_{\circ}^{2}}=\frac{\kappa_{\circ}}{4\pi
R_{\circ}^{2}}.$ The quantum vortex solution (3) with a plane-wave phonon mode
counter-propagating to $\bm{\omega}_{\circ}$ along the $-\hat{\bm{z}}$
direction with wave number $k=k_{\parallel}$ is concomitant to the quantum
vortex kelvon mode
$\displaystyle\bm{v}$
$\displaystyle\stackrel{{\scriptstyle(\ref{v_field})}}{{=}}$
$\displaystyle\frac{\hbar}{2mR_{\circ}}\hat{\bm{\vartheta}}-\frac{\hbar
k}{m}\hat{\bm{z}}=\omega_{\circ}\left(R_{\circ}\hat{\bm{\vartheta}}-2R_{\circ}^{2}k\hat{\bm{z}}\right).$
(26)
Equating (26) to (25), the helical parameters $\mathfrak{h}$ and
$\mathfrak{r}$ are thus analytically determined to be $\mathfrak{h}=R_{\circ}$
and $\mathfrak{r}=2R_{\circ}^{2}k.$ In turn, the curvature and torsion of the
helix are
$\kappa\stackrel{{\scriptstyle(\ref{curvature_and_torsion_of_helix})}}{{=}}\frac{2k_{\parallel}}{1+4k^{2}R_{\circ}^{2}}$
and
$\tau\stackrel{{\scriptstyle(\ref{curvature_and_torsion_of_helix})}}{{=}}\frac{1}{R_{\circ}}\,\frac{1}{1+4k^{2}R_{\circ}^{2}}.$
One observes that the maximum curvature $\kappa_{\text{max}}=k$ occurs when
$k=1/(2R_{\circ})$ and the maximum torsion $\tau_{\text{max}}=1/R_{\circ}$
occurs when $k=0$. Both the curvature and torsion vanish in the limit of
infinitely separated straight-line vortices. Inserting (25) into (24c), one
finds a precise self-consistent equation governing the dynamics of a quantum
vortex with large-amplitude helical wave with arbitrary wave number $k$ in
mutual interaction with another quantum vortex
$\dot{\bm{R}}=-\omega_{\circ}\left(\hat{\bm{\vartheta}}+\frac{\hat{\bm{z}}}{2R_{\circ}k}\right)\times\bm{R}(s).$
(27a) For analytical continuation to Fetter’s treatment in the high-tension
limit $\mathfrak{h}\gg\mathfrak{r}$, the fluctuation of (27a) obeys
$\dot{\bm{u}}\stackrel{{\scriptstyle(\ref{particle_trajectory_c})}}{{=}}-\frac{\omega_{\circ}}{2R_{\circ}k}\hat{\bm{z}}\times\bm{u}.$
(27b)
Equating (27b) to the Bloch equation (20) provides a way to determine the wave
number dependence ${\cal E}={\cal E}(k)$:
${\cal
E}(k)=\frac{m\rho\kappa_{\circ}\omega_{\circ}}{2R_{\circ}k^{3}}=\left(\frac{m\rho\kappa_{\circ}^{2}}{8\pi
R_{\circ}^{3}}\right)\,k^{-3}.$ (28)
A helical wave triggered by a $k$-mode axial phonon counter-propagates along
the segment undergoing mutual interaction. Only the inital axial phonon
dispersion relation $\omega=\frac{\kappa_{\circ}}{4\pi}k^{2}$ is consonant
with the dispersion relation for $\omega_{\text{\tiny K}}$ given by (22) for a
semiclassical Kelvin wave.
Equation (28) is an analytical prediction, based on the Frenet-Serret
differential geometry of space curves, that the helical wave spectrum scales
as $k^{-3}$ for highly curved quantum vortices. This power-law was found in
precise quantum simulations of superfluid turbulence supporting highly curved
vorticesYepez et al. (2009). A $k^{-3}$ spectrum also arises from a single
rectilinear quantum vortex Nore et al. (1997), and so it has been recently
suggested that this straight vortex power-law underlies the high-$k$ part of a
turbulent superfluid spectrum Krstulovic and Brachet (2009). The analysis
presented here suggests that (28), not (5) originally discovered by Lord
Kelvin in 1880 nor the Fourier spectrum of a single rectilinear quantum
vortex, may be responsible for the high-$k$ spectrum because of the effect of
mutually interacting highly-curved quantum vortices characteristic of
superfluid turbulence.
## References
* Gross (1963) E. P. Gross, J.Math.Phys. 4, 195 (1963).
* Pitaevskii (1961) L. P. Pitaevskii, Soviet Phys. JETP 13, 451 (1961).
* Schwarz (1985) K. W. Schwarz, Phys. Rev. B 31, 5782 (1985).
* Madelung (1927) E. Madelung, Zeit. F. Phys. 40, 322 (1927).
* Bohm (1952) D. Bohm, Phys. Rev. 85, 166 (1952).
* Fetter (1967) A. Fetter, Physical Review 162, 143 (1967).
* Yepez et al. (2009) J. Yepez, G. Vahala, L. Vahala, and M. Soe, Phys. Rev. Lett. 103, (2009).
* Nore et al. (1997) C. Nore, M. Abid, and M. E. Brachet, Phys. Fluids 9, 2644 (1997).
* Krstulovic and Brachet (2009) G. Krstulovic and M. Brachet, arXiv:0911.1749 (2009).
|
arxiv-papers
| 2010-02-23T17:53:26 |
2024-09-04T02:49:08.513638
|
{
"license": "Public Domain",
"authors": "Jeffrey Yepez",
"submitter": "Jeffrey Yepez",
"url": "https://arxiv.org/abs/1002.4380"
}
|
1002.4395
|
RECAPP-HRI-2010-005
Reconstruction of the left-chiral tau-sneutrino in supersymmetry with a right-
sneutrino as the lightest supersymmetric particle
Sanjoy Biswas†
Regional Centre for Accelerator-based Particle Physics
Harish-Chandra Research Institute
Chhatnag Road, Jhunsi, Allahabad - 211 019, India
###### Abstract
We have considered a supersymmetric scenario in which the minimal
supersymmetric standard model is augmented with a right-chiral neutrino
superfield for each generation. Such a scenario can have a lightest
supersymmetric particle (LSP) dominated by the right-chiral sneutrino state
and the lighter stau as the next-to-lightest supersymmetric particle (NLSP).
This can easily be motivated by assuming a high scale framework of
supersymmetry breaking like minimal supergravity (mSUGRA). Due to the
extremely small neutrino Yukawa coupling, the decay of the NLSP to the LSP is
suppressed and consequently the NLSP, here the lighter stau mass eigenstate,
becomes stable at the length scale of the detector. The collider signal in
this case consists of charged tracks of massive stable particles in the muon
chamber. Following up on our earlier studies on neutralino and chargino
reconstruction in such a scenario, we have shown the kinematical information
obtained from these charged tracks allows us to reconstruct the left-chiral
tau-sneutrino as well over a significant region of the mSUGRA parameter space.
Two methods for reconstruction are suggested and their relative merits are
discussed.
† e-mail: sbiswas@hri.res.in
## 1 Introduction
Physics beyond the Standard Model (SM), often comes with a proliferation of
new particles, and supersymmetry (SUSY) [1, 2] is not an exception to this.
Therefore, the determination of masses and various other properties of these
new particles in high-energy collider experiments is of paramount importance.
It enables one not only to discriminate among the multitude of existing
models, but also to unveil the fundamental parameters in any given scenario.
As the Large Hadron Collider (LHC) has already started running, we are closer
than ever to the fulfilment of this objective.
However, in most SUSY models with conserved $R$-parity (defined as
$R=(-)^{3B+L+2S}$) the lightest supersymmetric particle (LSP), being neutral
and weakly interacting, goes without being recorded at the detector. This
gives rise to large amount of missing transverse energy ($/\\!\\!\\!E_{T}$)
which makes the reconstruction of the masses of new particles quite difficult.
Although $/\\!\\!\\!E_{T}$ is a canonical signature of most of the
supersymmetric theories, one should remember that the above possibility is not
unique. One can have a charged particle as the next-to-lightest supersymmetric
particle (NLSP) and the decay of it into the LSP may be suppressed, so that
the NLSP becomes stable on the scale of the detector, leaving the charged
track of a massive particle in the muon chamber in a collider event [3, 4, 5,
6, 7]. From the kinematic information of these charged tracks one is able to
reconstruct the masses of the superparticles in these scenarios.
This paper is one in a series where we have considered a scenario in which a
right-chiral sneutrino is the LSP and the NLSP is the superpartner of tau. In
the minimal supersymmetric standard model (MSSM) [8, 9] one can achieve this
by the mere addition of a right-handed neutrino superfield for each generation
in the MSSM spectrum, assuming the neutrinos to be of Dirac type. The
superpotential of the MSSM in this case is given (suppressing family indices)
by
$W_{MSSM}=y_{l}\hat{L}\hat{H_{d}}\hat{E^{c}}+y_{d}\hat{Q}\hat{H_{d}}\hat{D^{c}}+y_{u}\hat{Q}\hat{H_{u}}\hat{U^{c}}+\mu\hat{H_{d}}\hat{H_{u}}+y_{\nu}\hat{L}\hat{H}_{u}\hat{\nu}^{c}_{R}$
(1)
where $\hat{H_{d}}$ and $\hat{H_{u}}$ respectively are the Higgs superfields
that give mass respectively to the $T_{3}=-1/2$ and $T_{3}=+1/2$ fermions.
$y^{\prime}s$ are the strengths of Yukawa interactions. $\hat{L}$ and
$\hat{Q}$ are the left-handed lepton and quark superfields respectively,
whereas $\hat{E^{c}}$, $\hat{D^{c}}$ and $\hat{U^{c}}$, in that order, are the
right handed gauge singlet charged lepton, down-type and up-type quark
superfields. $\mu$ is the Higgsino mass parameter.
Such an LSP interacts only through neutrino Yukawa coupling ($y_{\nu}$) which
is $\sim 10^{-13}$ to account for the tiny neutrino masses. The decays of all
other sparticles including the NLSP into the LSP, is controlled by this Yukawa
coupling. Thus the stau becomes long-lived and appears to be stable at the
detector, drastically changing the signal of SUSY.
It is possible to accommodate such a scenario in a high-scale frame work of
SUSY breaking, like minimal supergravity (mSUGRA) [10] where all the scalar
and gaugino masses at low energy are driven by the renormalisation group
evolution (RGE) of the universal scalar ($m_{0}$) and gaugino mass ($M_{1/2}$)
parameters from high scale respectively. All one has to do in this framework
is to specify at high scale the values of ($m_{0},M_{1/2},A_{0}$,
$sign(\mu)~{}{\rm and}~{}\tan\beta~{}=~{}\langle H_{u}\rangle/\langle
H_{d}\rangle$) where, $A_{0}$ is universal trilinear scalar coupling, $\mu$ is
the Higgsino mass parameter and $\tan\beta$ is the ratio of the vacuum
expectation values of the two Higgs doublets that give masses to the up-and
down-type quarks. The masses of the right-chiral sneutrinos are also assumed
to be evolved from the same universal scalar mass parameter $m_{0}$. At one-
loop level, the RGE of the right-chiral sneutrino mass parameter is given by
[11, 12]
$\displaystyle\frac{dM^{2}_{\tilde{\nu}_{R}}}{dt}=\frac{2}{16\pi^{2}}y^{2}_{\nu}~{}A^{2}_{\nu}$
(2)
where $A_{\nu}$ is obtained by the running of the trilinear soft SUSY breaking
term $A_{0}$ and is responsible for left-right mixing in the sneutrino mass
matrix. Due to the small value of $y_{\nu}$, the value of
$M_{\tilde{\nu}_{R}}$ essentially remains anchored at $m_{0}$, whereas the
other sfermion masses are enhanced at the electroweak scale. Thus, for an
ample range of values of the gaugino masses, one naturally ends up with a
sneutrino LSP ($\tilde{\nu}_{1}$) which is substantially dominated by the
right-chiral state, because of the fact that the mixing angle is also
controlled by the neutrino Yukawa couplings:
$\displaystyle\tilde{\nu}_{1}=-\tilde{\nu}_{L}\sin\theta+\tilde{\nu}_{R}\cos\theta$
(3)
where the mixing angle $\theta$ is given by,
$\displaystyle\tan 2\theta=\frac{2y_{\nu}v\sin\beta|\mu\cot\beta-
A_{\nu}|}{m^{2}_{\tilde{\nu}_{L}}-m^{2}_{\tilde{\nu}_{R}}}$ (4)
It is the NLSP which determines the nature of collider events. The third
generation charged slepton often appears as the NLSP in this scenario, due to
its larger Yukawa coupling. Since the decay width of the NLSP is proportional
to $y^{2}_{\nu}$, all supersymmetric cascades in collider events culminate in
the pair production of the NLSP which decays outside the detector, leaving
behind two charged tracks of massive particles in the muon chamber.
The present work is a follow-up of our earlier work on neutralino and chargino
reconstruction in supersymmetry with long lived stau scenario [13, 14]. In
this work we have concentrated on the mass reconstruction of the heavier tau-
sneutrino whose dominant constituent is the left-chiral state. It is produced
in cascade decay of squarks and/or gluinos via chargino and heavier neutralino
decay. The $\tilde{\nu}_{\tau_{L}}$ thus produced has a substantial branching
fraction of decaying into a $W\tilde{\tau}$ -pair. Hence, reconstructing the
four-momenta of the $W$’s in its hadronic decay mode, it is possible to
reconstruct the heavier mass eigenstate of the tau-sneutrino. Though one can
have a $W$ in the final state in association with two charged tracks in this
long-lived stau scenario, our emphasis nonetheless is on the fact that the
$W$, paired with a particular track giving an invariant mass peak offers a
definite signature of $\tilde{\nu}_{\tau_{L}}$ production in the SUSY cascade.
This paper is assembled as follows: in the following section we have tried to
motivate the choice of benchmark points in the context of a mSUGRA framework.
In section 3 we have discussed the signal under study and its possible
backgrounds. The event selection criteria and reconstruction strategy for
determining the mass of the heavier $\tau$-sneutrino have also been discussed
there. The numerical results which comprise a scan over the region of
$m_{0}$-$M_{1/2}$ plane are presented in section 4. We conclude in section 5.
## 2 The choice of benchmark points
| BP-1 | BP-2 | BP-3 | BP-4 | BP-5 | BP-6
---|---|---|---|---|---|---
mSUGRA | $m_{0}=100$ | $m_{0}=100$ | $m_{0}=100$ | $m_{0}=100$ | $m_{0}=100$ | $m_{0}=100$
input | $m_{1/2}=600$ | $m_{1/2}=500$ | $m_{1/2}=400$ | $m_{1/2}=350$ | $m_{1/2}=325$ | $m_{1/2}=325$
| $\tan\beta=30$ | $\tan\beta=30$ | $\tan\beta=30$ | $\tan\beta=30$ | $\tan\beta=30$ | $\tan\beta=25$
$m_{\tilde{e_{L}}},m_{\tilde{\mu}_{L}}$ | 418 | 355 | 292 | 262 | 247 | 247
$m_{\tilde{e_{R}}},m_{\tilde{\mu}_{R}}$ | 246 | 214 | 183 | 169 | 162 | 162
$m_{\tilde{\nu}_{e_{L}}},m_{\tilde{\nu}_{\mu_{L}}}$ | 408 | 343 | 279 | 247 | 232 | 232
$m_{\tilde{\nu}_{\tau_{L}}}$ | 395 | 333 | 270 | 239 | 224 | 226
$m_{\tilde{\nu}_{i_{R}}}$ | 100 | 100 | 100 | 100 | 100 | 100
$m_{\tilde{\tau}_{1}}$ | 189 | 158 | 127 | 112 | 106 | 124
$m_{\tilde{\tau}_{2}}$ | 419 | 359 | 301 | 273 | 259 | 255
$m_{\chi^{0}_{1}}$ | 248 | 204 | 161 | 140 | 129 | 129
$m_{\chi^{0}_{2}}$ | 469 | 386 | 303 | 261 | 241 | 240
$m_{\chi^{\pm}_{1}}$ | 470 | 387 | 303 | 262 | 241 | 241
$m_{\tilde{g}}$ | 1362 | 1151 | 937 | 829 | 774 | 774
$m_{\tilde{t}_{1}}$ | 969 | 816 | 772 | 582 | 634 | 543
$m_{\tilde{t}_{2}}$ | 1179 | 1008 | 818 | 750 | 683 | 709
$m_{h^{0}}$ | 115 | 114 | 112 | 111 | 111 | 111
Table 1: Proposed benchmark points (BP) for the study of the stau-NLSP
scenario in SUGRA with right-sneutrino LSP. The values of $m_{0}$ and
$M_{1/2}$ are given in GeV. We have also set $A_{0}=100~{}GeV$ and
$sgn(\mu)=+$ for benchmark points under study.
In Table 1, we present the benchmark points used in our earlier studies as
well as in this work. The mass spectrum is obtained using the spectrum
generator ISAJET 7.78 [15]. We have identified those regions of
$m_{0}$-$M_{1/2}$ plane, where one normally has a $\tilde{\tau}$ LSP, in the
absence of a right-chiral sneutrino superfield in an mSUGRA scenario. It
should be noted, however, that the reconstruction technique we have adopted is
not limited to such a scenario. It can be advocated in all those cases where a
stau is the NLSP and its decay length is large compared to the detector scale
[16, 17, 18]. The dominantly right-chiral sneutrino state in this scenario
turns out be a possible dark matter (DM) candidate [19], as it can evade
direct DM searches due to its minuscule Yukawa coupling. The benchmark points
we have chosen are not only compatible with the WMAP data [20], but also
consistent with other experimental constraints such as $b\rightarrow s\gamma$,
correction to the $\rho$-parameter and muon ($g-2$) [21, 22]. The different
points in the parameter space are similar in mass ordering and thus lead to
qualitatively similar collider signals. However, they correspond to different
mass splittings between NLSP and other particles, giving rise to different
kinematics of the final state, which in turn control the reconstructability of
various particles. For all our benchmark points, the mass hierarchy
$m_{\tilde{\tau}}<m_{\chi^{0}_{1}}<m_{\tilde{\nu}_{\tau_{L}}}<m_{\chi^{0}_{2}}\approx
m_{\chi^{\pm}_{1}}$ is satisfied, which affirms the production of
$\tilde{\nu}_{\tau_{L}}$ in cascade decays of squarks and/or gluinos produced
in the initial hard scattering.
## 3 Signal and backgrounds
In order to illustrate that it is possible to reconstruct the left-chiral
$\tau$-sneutrino in this scenario we have considered the following final
state:
* •
$\tau_{j}+W+2\tilde{\tau}+E_{T}/~{}~{}+X$
where, $\tau_{j}$ represents a tau jet, the missing transverse energy is
denoted by $E_{T}/~{}~{}$, $W$ symbolizes a $W$-boson that has been identified
in its hadronic decay mode111We have not considered the leptonic decay of $W$
since it is difficult to reconstruct the four momenta of the $W$ in its
leptonic decay mode due to the presence of the invisible neutrino . and all
other jets coming from cascade decays are included in X. The collider
simulation has been done with a centre of mass energy $E_{cm}$=14 TeV, at two
different integrated luminosities of $30fb^{-1}$ and $100fb^{-1}$ using the
event generator PYTHIA 6.4.16 [23]. A simulation for the early LHC run at
$E_{cm}$=10 TeV and integrated luminosity of $3fb^{-1}$ has also been
predicted. We have used the parton distribution function CTEQ5L [24] with the
factorisation ($\mu_{R}$) and renormalisation ($\mu_{F}$) scale set at
$\mu_{R}=\mu_{F}=$average mass of the final state particles. Following are the
numerical values of various parameters, used in our calculation [21]:
$M_{Z}=91.187$ GeV, $M_{W}=80.398$ GeV, $M_{t}=171.4$ GeV
$M_{H}=120$ GeV, $\alpha^{-1}_{em}(M_{Z})=127.9$, $\alpha_{s}(M_{Z})=0.118$
### 3.1 Basic idea
The $\tilde{\nu}_{\tau_{L}}$ is produced in SUSY cascade predominantly via the
decay of lightest chargino
($\chi^{\pm}_{1}\rightarrow\tilde{\nu}_{\tau_{L}}\tau$) or second lightest
neutralino
($\chi^{0}_{2}\rightarrow\tilde{\nu}_{\tau_{L}}{\bar{\nu}_{\tau}}$). The
corresponding decay branching fraction is given in Table-2. The momentum
information of the charged track enables us to reconstruct the
$\tilde{\nu}_{\tau_{L}}$ mass in this scenario. This closely follows our
earlier studies on neutralino and chargino reconstruction [13, 14]. A
procedure for reconstructing the mass of a left-chiral tau-sneutrino, making
occasional use of the earlier results, is outlined here.
BP-1 | BP-2 | BP-3
---|---|---
$\chi^{0}_{2}\rightarrow\tilde{\nu}_{\tau_{L}}{\bar{\nu}_{\tau}}$(+cc) | $\chi^{\pm}_{1}\rightarrow\tilde{\nu}_{\tau_{L}}\tau^{\pm}$ | $\chi^{0}_{2}\rightarrow\tilde{\nu}_{\tau_{L}}{\bar{\nu}_{\tau}}$(+cc) | $\chi^{\pm}_{1}\rightarrow\tilde{\nu}_{\tau_{L}}\tau^{\pm}$ | $\chi^{0}_{2}\rightarrow\tilde{\nu}_{\tau_{L}}{\bar{\nu}_{\tau}}$(+cc) | $\chi^{\pm}_{1}\rightarrow\tilde{\nu}_{\tau_{L}}\tau^{\pm}$
23% | 24.7% | 25% | 27.6% | 26.6% | 32.5%
BP-4 | BP-5 | BP-6
$\chi^{0}_{2}\rightarrow\tilde{\nu}_{\tau_{L}}{\bar{\nu}_{\tau}}$(+cc) | $\chi^{\pm}_{1}\rightarrow\tilde{\nu}_{\tau_{L}}\tau^{\pm}$ | $\chi^{0}_{2}\rightarrow\tilde{\nu}_{\tau_{L}}{\bar{\nu}_{\tau}}$(+cc) | $\chi^{\pm}_{1}\rightarrow\tilde{\nu}_{\tau_{L}}\tau^{\pm}$ | $\chi^{0}_{2}\rightarrow\tilde{\nu}_{\tau_{L}}{\bar{\nu}_{\tau}}$(+cc) | $\chi^{\pm}_{1}\rightarrow\tilde{\nu}_{\tau_{L}}\tau^{\pm}$
22% | 31.7% | 17.8% | 28.2% | 17.8% | 26.4%
Table 2: The branching fractions for
$\chi^{0}_{2}\rightarrow\tilde{\nu}_{\tau_{L}}{\bar{\nu}_{\tau}}$(+cc) and
$\chi^{\pm}_{1}\rightarrow\tilde{\nu}_{\tau_{L}}\tau^{\pm}$ for respective
benchmark points.
The $\tilde{\nu}_{\tau_{L}}$ has a sizeable decay branching fraction into a
$W^{\pm}\tilde{\tau}^{\mp}_{1}$-pair (ranging from $\approx 34\%$ to $84\%$).
Since the $W$ will always be produced in association with two staus, it is
crucial to identify the correct $W^{\pm}\tilde{\tau}^{\mp}_{1}$ -pair and thus
avoid the combinatorial background. Since charge identification of a $W$ in
its hadronic decay mode is difficult, we have adopted the following methods
for finding the correct pair:
1. 1.
Using opposite sign charged tracks (OSCT): To determine the correct
$W^{\pm}\tilde{\tau}^{\mp}_{1}$-pair we have make use of two opposite sign
charged tracks. For the signal, one has
$\tau_{j}+W+2\tilde{\tau}+E_{T}/~{}~{}+X$ with one tau in the final state,
where the $\tilde{\tau}_{1}^{\pm}\tau^{\mp}$ pair has originated in the decay
of a neutralino $\chi^{0}_{1}$ (or $\chi^{0}_{2}$) and
$\tilde{\nu}_{\tau_{L}}$ has decayed into a $W^{\pm}\tilde{\tau}^{\mp}_{1}$
pair. The stau-track produced in a $\tilde{\nu}_{\tau_{L}}$ decay always has
the same charge as that of the tau in this cases. Hence combining the four
momenta of this track with that of $W$ one can obtain a
$W^{\pm}\tilde{\tau}^{\mp}_{1}$ invariant mass distribution peaking at the
$\tilde{\nu}_{\tau_{L}}$ mass ($m_{\tilde{\nu}_{\tau_{L}}}$).This requires
identification of the charge of a jet out of a tau decay, the efficiency of
which has been assumed to be 100%. The events in which the tau has originated
from the decay of a chargino
($\chi^{\pm}_{1}\rightarrow\tilde{\nu}_{\tau_{L}}\tau^{\pm}$), and the tau out
of a neutralino decay goes unidentified, will contribute to the background
events in this method, which we will discuss in the next subsection.
Nevertheless, this method works very well in determining the correct
$W^{\pm}\tilde{\tau}^{\mp}_{1}$ pair and one obtains the invariant mass peak
at the correct value of the corresponding mass of the left-chiral sneutrino
($m_{\tilde{\nu}_{\tau_{L}}}$). Furthermore, this method can be used
irrespective of the possibility of reconstruction of the neutralino and
chargino mass222It should be mentioned here, that the mass associated with the
charge track, which can be found following our earlier work [13], is an
inevitable input for both the methods..
2. 2.
Using chargino-neutralino mass information (CNMI): Though the method based on
OSCT does not depend on the reconstructability of other superparticle masses,
its main disadvantage is that it not only reduces half of the signal event but
also includes many background events. The correct
$W^{\pm}\tilde{\tau}^{\mp}_{1}$ pair can also be determined if one uses the
information of the chargino or/and neutralino mass, by looking at the end
point in the invariant mass distribution of the track and tau-jet pair. The
right combination will have the end point at the corresponding neutralino mass
($m_{\chi^{0}_{i}},~{}i=1,2$). In cases where the tau out of a neutralino
decay goes undetected and the tau out of a chargino decay gets identified,
then the $\tilde{\tau}-W-\tau_{j}$ pair invariant mass distribution will have
a end point at the corresponding chargino mass ($m_{\chi^{\pm}_{1}}$). We have
combined the $W^{\pm}$ with the corresponding track ($\tilde{\tau}^{\mp}_{1}$)
when either of the $m_{\chi^{0}_{i}}-M_{\tilde{\tau}\tau_{j}}\leq 20~{}GeV$ or
$m_{\chi^{\pm}_{i}}-M_{\tilde{\tau}W\tau_{j}}\leq 20~{}GeV$ criterion is
satisfied.
### 3.2 Backgrounds and event selection criteria
In this subsection we discuss the possible SM backgrounds that can fake our
signal, namely, $W+\tau_{j}+2\tilde{\tau}+E_{T}/~{}~{}+X$ and prescribed the
requisite cuts to minimise them. First of all, we have advocated the following
basic cuts for each event to validate our desired final state:
* •
$p_{T}^{lep},~{}p_{T}^{track}>10$ GeV
* •
$p_{T}^{hardest-jet}>75$ GeV
* •
$p_{T}^{other-jets}>30$ GeV
* •
$/\\!\\!\\!E_{T}>40$ GeV
* •
$|\eta|<2.5$ for leptons, jets and stau
* •
$\Delta R_{ll}>0.2,~{}\Delta R_{lj}>0.4$, where $\Delta
R^{2}=\Delta\eta^{2}+\Delta\phi^{2}$
* •
$\Delta R_{\tilde{\tau}l}>0.2,~{}\Delta R_{\tilde{\tau}j}>0.4$
* •
$\Delta R_{jj}>0.7$
Figure 1: $p_{T}$ of the harder muonlike track (left) and
$\Sigma|\vec{p_{T}}|$ (right) distribution (normalised to unity) for the
signal (BP5) and the background, with $E_{cm}$=10 TeV.
Figure 2: $p_{T}$ of the harder muonlike track (left) and
$\Sigma|\vec{p_{T}}|$ (right) distribution (normalised to unity) for the
signal (BP1, BP2 and BP3) and the background, with $E_{cm}$=14 TeV.
Figure 3: $p_{T}$ of the harder muonlike track (left )and
$\Sigma|\vec{p_{T}}|$ (right) distribution (normalised to unity) for the
signal (BP4, BP5 and BP6) and the background, with $E_{cm}$=14 TeV.
The above cuts are applied for simulation with both the center-of-mass
energies of 10 and 14 TeV. Keeping detector resolutions of the momenta of each
particle in mind, different Gaussian resolution functions have been used for
electrons, muons/staus and jets. These exactly follow the path prescribed in
our earlier work [13]. We have assumed a tau identification efficiency of 50%
for one prong decay of a tau lepton, whereas a rejection factor of 100 has
been used for the non-taujets [25, 26, 27]. To identify the $W$ in its
hadronic decay mode, we have used the following criteria:
* •
the invariant mass of any two jets should lie within
$M_{W}-20<M_{jj}<M_{W}+20$
* •
the separation between the stau and the direction formed out of the vector sum
of the momenta of the two jets (produced in $W$ decay) should lie within
$\Delta R=0.8$
The charged track of a massive particle in the muon chamber can be faked by
the muon itself. In our earlier study, we have shown that the SM backgrounds
like $t\bar{t}$, $ZZ$, $ZW$, $ZH$ can contribute to the
$\tau_{j}+2\tilde{\tau}+E_{T}/~{}~{}+X$ final state. In the present study we
have an additional $W$ in the final state. This requirement further reduces
the contribution from SM processes. Above all, demanding an added degree of
hardness of the charged tracks and a minimum value of the scalar sum of
transverse momenta ($\Sigma|\vec{p_{T}}|$) of all the visible particle in the
final state, the contribution from above SM processes to the
$W+\tau_{j}+2\tilde{\tau}+E_{T}/~{}~{}+X$ final state can be suppressed
considerably. For the simulation at $E_{cm}=10~{}TeV$, we have adopted the
following cuts on the $p_{T}^{\tilde{\tau}}$ and $\Sigma|\vec{p_{T}}|$
variables (See Figure 1):
* •
$p_{T}^{track}>75$ GeV
* •
$\Sigma|\vec{p_{T}}|>700$ GeV
where as, for the simulation at a higher center of mass energy
$E_{cm}=14~{}TeV$ the degree of hardness raised to(See Figure 2 and 3):
* •
$p_{T}^{track}>100$ GeV
* •
$\Sigma|\vec{p_{T}}|>1$ TeV
## 4 Results and discussions
We present the numerical results of our study in this section. In Table 3 we
have presented the results of simulation at $E_{cm}=10~{}TeV$ for BP5, to
illustrate that it is possible to reconstruct the left-chiral tau-sneutrino
even at the early phase of the LHC run at an integrated luminosity of
$3~{}fb^{-1}$ (Figure 4). The results of simulation at $E_{cm}=14~{}TeV$ for
two different luminosities have also been shown in Table 4 and 5. Depending on
the luminosity and available center of mass energy, it is possible to probe
some or all of the benchmark points we have studied, at the LHC.
The numerical results enable us to assess the relative merits of the OSCT and
CNMI methods. It is obvious from Table 3-5 that we have larger number of
events when the chargino-neutrlino mass information (CNMI) rather than the
opposite sign charged tracks (OSCT) has been used to reconstruct the mass of
the $\tilde{\nu}_{\tau_{L}}$, as one would expect the number of events to be
less if one is restricted to opposite sign charged tracks. Also, the chance of
including the wrong $W^{\pm}\tilde{\tau}^{\mp}_{1}$-pair is more in the OSCT
method. For example, as has already been mentioned,
$\chi^{\pm}_{1}\chi^{0}_{1}$ produced in cascades can give rise to
$W+\tau_{j}+2\tilde{\tau}+E_{T}/~{}~{}+X$ final state, where the tau out of a
chargino decay ($\chi^{\pm}_{1}\rightarrow\tilde{\nu}_{\tau_{L}}\tau^{\pm}$)
has been identified and not the one originated from a neutralino decay. In
this situation one ends up with a wrong combination of
$W^{\pm}\tilde{\tau}^{\mp}_{1}$ pair if one is using OSCT. The contribution of
such background, however, is not counted when CNMI is used. In spite of this,
the fact that no information on chargino and neutralino masses is used in
OSCT, is of advantage in independently confirming the nature of the spectrum.
Both the methods discussed above are prone to background contamination within
the model itself, such as misidentification of $W$ and the decay like
$\chi^{\pm}_{1/2}\rightarrow\chi^{0}_{1/2}W^{\pm}\rightarrow\tilde{\tau}_{1}W^{\pm}\tau$,
which smear the peak, as is visible from Figure 5 and 6.
From the numerical results, it is seen that the number of events start getting
increased as one moves from BP1 to BP5, due to the fact that the production
cross section of $\tilde{\nu}_{\tau_{L}}$ in cascade decay of squarks and/or
gluinos get enhanced. However, at BP6 one has less number of events. The
reason is twofold: First, the decay branching fraction of
$\tilde{\nu}_{\tau_{L}}\rightarrow\tilde{\tau}_{1}W$ reduces from 84% for BP1
to 60% for BP5. The enhanced production cross-section thus has dominated
effect. However, for BP6 the branching ratio falls to 34.5%, thus affecting
the event rates adversely. Secondly, the mass difference
$m_{\tilde{\nu}_{\tau_{L}}}-(m_{\tilde{\tau}_{1}}+m_{W})$ is of the order of
20 GeV, which restricts the stau track from passing the requisite hardness cut
for a sizeable number of events. Also, the $\tilde{\tau}_{1}W$ invariant mass
peak is badly affected at BP6, as the contribution from the decay
$\chi^{\pm}_{1/2}\rightarrow\chi^{0}_{1/2}W^{\pm}$ is maximum due to the
increase in the branching ratio. The number of events within a bin of $\pm
20~{}GeV$ around the peak obtained using CNMI is comparable to that obtained
using OSCT at this benchmark point. This is due to the fact that in case of
mass reconstruction using CNMI, the information about the lightest neutralino
mass is not available at BP6 [13].
We have also explored the mSUGRA parameter space to study the feasibility of
sneutrino reconstruction. A thorough scan over the $m_{0}-M_{1/2}$ plane has
been done using the spectrum generator SuSpect v2.34 [28], which leads to a
$\tilde{\tau}$ LSP in a usual mSUGRA scenario without the right handed
sneutrino and identified the region where it is possible to reconstruct the
left-chiral stau neutrino with more than 25 events within the vicinity of the
$\tilde{\nu}_{\tau_{L}}$ mass peak at an integrated luminosity of
$30~{}fb^{-1}$. The corresponding plots are depicted in Figure 7.
The regions where reconstruction is possible have been determined using the
following criteria:
| basic cuts | $p_{T}$+$\Sigma{|p_{T}|}$ | $|M_{peak}-M_{\tilde{\tau}W}|\leq 20$
---|---|---|---
BP5 | CNMI | OSCT | CNMI | OSCT | CNMI | OSCT
| 132 | 84 | 107 | 66 | 23 | 15
Table 3: Number of signal events for the $W+\tau_{j}+2\tilde{\tau}$ (charged-
track)+$E_{T}/~{}~{}+X$ final state, considering all SUSY processes for BP5
with $E_{cm}$=10 TeV at an integrated luminosity of 3 $fb^{-1}$ assuming tau
identification efficiency $\epsilon_{\tau}=50\%$. In Table CNMI stands for
reconstruction of $m_{\tilde{\nu}_{\tau_{L}}}$ using Chargino-Neutralino Mass
Information whereas OSCT implies reconstruction of the same using Opposite
Sign Charged Tracks.
Figure 4: The invariant mass ($M_{\tilde{\tau}W}$) distribution in $W+\tau_{j}+2\tilde{\tau}$ (charged-track)+$E_{T}/~{}~{}+X$ final state for BP5 assuming tau identification efficiency $(\epsilon_{\tau})=50\%$ at an integrated luminosity of 3 $fb^{-1}$ and with $E_{cm}$=10 TeV. In Figure CNMI stands for reconstruction of $m_{\tilde{\nu}_{\tau_{L}}}$ using Chargino- Neutralino Mass Information whereas OSCT implies reconstruction of the same using Opposite Sign Charged Tracks. | BP1 | BP2 | BP3
---|---|---|---
| CNMI | OSCT | CNMI | OSCT | CNMI | OSCT
basic cuts | 318 | 248 | 984 | 715 | 4060 | 2550
$p_{T}$+$\Sigma{|p_{T}|}$ cut | 250 | 205 | 775 | 574 | 3030 | 1904
$|M_{peak}-M_{\tilde{\tau}W}|\leq 20$ | 62 | 38 | 187 | 119 | 634 | 364
Table 4: Number of signal events for the $W+\tau_{j}+2\tilde{\tau}$ (charged-track)+$E_{T}/~{}~{}+X$ final state, considering all SUSY processes, for BP1, BP2 and BP3 with $E_{cm}$=14 TeV at an integrated luminosity of 100 $fb^{-1}$ assuming tau identification efficiency $\epsilon_{\tau}=50\%$. In Table CNMI and OSCT stand for the same as in Table 3. | BP4 | BP5 | BP6
---|---|---|---
| CNMI | OSCT | CNMI | OSCT | CNMI | OSCT
basic cuts | 8634 | 5450 | 12519 | 8014 | 4013 | 3995
$p_{T}$+$\Sigma{|p_{T}|}$ cut | 6145 | 3709 | 8654 | 5363 | 2560 | 2537
$|M_{peak}-M_{\tilde{\tau}W}|\leq 20$ | 1218 | 689 | 1471 | 866 | 385 | 349
Table 5: Number of signal events for the $W+\tau_{j}+2\tilde{\tau}$ (charged-
track)+$E_{T}/~{}~{}+X$ final state, considering all SUSY processes, for BP4,
BP5 and BP6 with $E_{cm}$=14 TeV at an integrated luminosity of 100 $fb^{-1}$
assuming tau identification efficiency $\epsilon_{\tau}=50\%$. In Table CNMI
and OSCT stand for the same as in Table 3.
Figure 5: The invariant mass ($M_{\tilde{\tau}W}$) distribution in
$W+\tau_{j}+2\tilde{\tau}$ (charged-track)+$E_{T}/~{}~{}+X$ final state for
four of our proposed benchmark points assuming tau identification efficiency
$(\epsilon_{\tau})=50\%$ at an integrated luminosity of 30 $fb^{-1}$ and
centre of mass energy 14 $TeV$. In Figure CNMI and OSCT stand for the same as
in Figure 4.
Figure 6: The invariant mass ($M_{\tilde{\tau}W}$) distribution in
$W+\tau_{j}+2\tilde{\tau}$ (charged-track)+$E_{T}/~{}~{}+X$ final state for
all the benchmark points assuming tau identification efficiency
$(\epsilon_{\tau})=50\%$ at an integrated luminosity of 100 $fb^{-1}$ and
centre of mass energy 14 $TeV$. In Figure CNMI and OSCT stand for the same as
in Figure 4.
* •
In the parameter space, we have not gone into regions where the gluino mass
exceeds $\approx 2~{}TeV$.
* •
The number of events within a bin of $\pm 20~{}GeV$ around the peak must be
greater than a specific number for the corresponding luminosity.
* •
Also it is possible to reconstruct at least one of the neutralinos at the
region of $m_{0}-M_{1/2}$ under consideration, when the mass information is
used, as a criterion of finding correct $W^{\pm}\tilde{\tau}^{\mp}_{1}$. We
should mention here that, for this study in identifying the region of mSUGRA
parameter space, we have used neutralino mass information alone and not the
information on chargino mass.
Figure 7: The region in the $m_{0}-M_{1/2}$ plane (with $tan\beta=30$ and
$A_{0}=100$), where it is possible to reconstruct the left-chiral sneutrino at
an integrated luminosity of 30 $fb^{-1}$ and center-of-mass energy 14 $TeV$
with more than 25 events in the vicinity of the peak. In the Figure blue (dark
shade) region represents reconstruction of $m_{\tilde{\nu}_{\tau_{L}}}$ using
only Neutralino Mass Information (NMI), while the pink region (light shade)
stands for reconstruction of the same using Opposite Sign Charged Tracks
(OSCT) for the $W+\tau_{j}+2\tilde{\tau}$ (charged-track)+$E_{T}/~{}~{}+X$
final state. The entire region above the dashed line indicates the scenario
where one has a $\tilde{\nu_{R}}$-LSP and a $\tilde{\tau}$-NLSP.
## 5 Summary and conclusions
We have considered a SUSY scenario where the LSP is dominated by a right-
sneutrino state, while a dominantly right-chiral stau is the NLSP. The stau,
being stable on the length scale of collider detectors, gives rise to charged
track of massive particle at the muon chamber. It is also shown that such a
scenario follows naturally from a high-scale scenario of universal scalar and
gaugino masses.
We have investigated the possibility of reconstruction of the left-chiral tau
sneutrino in such a scenario. For that we have studied the final state
consisting of $W+\tau_{j}+2\tilde{\tau}+E_{T}/~{}~{}+X$. We have also
prescribed two different strategies for the reconstruction of the mass of the
$\tilde{\nu}_{\tau_{L}}$. One is independent of the reconstructability of
other particles as it does not uses the mass information and the other one
does depend on the reconstructability of the chargino and neutralino masses.
The cuts imposed on the kinematic variables to eliminate the SM backgrounds
are motivated by our recent studies on chargino and neutralino reconstruction
under similar circumstances. We have demonstrated the feasibility of
reconstructing the mass of the $\tilde{\nu}_{\tau_{L}}$ even at the early
phase of LHC run with $E_{cm}=10~{}TeV$ and at an integrated luminosity of
$3~{}fb^{-1}$ for a particular benchmark point (BP5) for illustration. The
results for simulation with higher LHC center-of-mass energy $E_{cm}=14~{}TeV$
at two different luminosities have also been shown.
A thorough scan over the $m_{0}-M_{1/2}$ plane has been performed in this
study, which shows that a significant region of the mSUGRA parameter space can
be probed at the LHC with sufficient number of events at an integrated
luminosity of $30~{}fb^{-1}$ with $E_{cm}=14~{}TeV$ using our prescribed
methods.
To conclude, the MSSM with a right-chiral sneutrino superfield for each
generation is a worthful possibility to look at the LHC. It not only offers a
distinct SUSY signal in the form of a pair of charged tracks of massive
particles but also opens a new vista in the reconstruction of the
superparticle masses.
Acknowledgement: The author thanks Biswarup Mukhopadhyaya for helping in the
preparation of this manuscript and giving valuable suggestions. This work is
partially supported by funding available from the Department of Atomic Energy,
Government of India for the Regional Center for Accelerator-based Particle
Physics, Harish-Chandra Research Institute. Computational work for this study
was partially carried out at the cluster computing facility of Harish-Chandra
Research Institute (http://cluster.mri.ernet.in).
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|
arxiv-papers
| 2010-02-23T19:30:40 |
2024-09-04T02:49:08.520057
|
{
"license": "Public Domain",
"authors": "Sanjoy Biswas",
"submitter": "Sanjoy Biswas",
"url": "https://arxiv.org/abs/1002.4395"
}
|
1002.4435
|
# Recognizing Graph Theoretic
Properties with Polynomial Ideals
J.A. De Loera, C. Hillar111The second author is partially supported by an NSA
Young Investigator Grant and an NSF All-Institutes Postdoctoral Fellowship
administered by the Mathematical Sciences Research Institute through its core
grant DMS-0441170., P.N. Malkin, M. Omar 222The fourth author is partially
supported by NSERC Postgraduate Scholarship 281174. 333All other authors are
partially supported by NSF grant DMS-0914107 and an IBM OCR award
###### Abstract
Many hard combinatorial problems can be modeled by a system of polynomial
equations. N. Alon coined the term polynomial method to describe the use of
nonlinear polynomials when solving combinatorial problems. We continue the
exploration of the polynomial method and show how the algorithmic theory of
polynomial ideals can be used to detect $k$-colorability, unique
Hamiltonicity, and automorphism rigidity of graphs. Our techniques are diverse
and involve Nullstellensatz certificates, linear algebra over finite fields,
Gröbner bases, toric algebra, convex programming, and real algebraic geometry.
## 1 Introduction
In his well-known survey [1], Noga Alon used the term polynomial method to
refer to the use of nonlinear polynomials when solving combinatorial problems.
Although the polynomial method is not yet as widely used as its linear
counterpart, increasing numbers of researchers are using the algebra of
multivariate polynomials to solve interesting problems (see for example [2,
12, 13, 17, 19, 23, 24, 32, 31, 35, 36, 38, 43] and references therein). In
the concluding remarks of [1], Alon asked whether it is possible to modify
algebraic proofs to yield efficient algorithmic solutions to combinatorial
problems. In this paper, we explore this question further. We use polynomial
ideals and zero-dimensional varieties to study three hard recognition problems
in graph theory. We show that this approach can be fruitful both theoretically
and computationally, and in some cases, result in efficient recognition
strategies.
Roughly speaking, our approach is to associate to a combinatorial question
(e.g., is a graph $3$-colorable?) a system of polynomial equations $J$ such
that the combinatorial problem has a positive answer if and only if system $J$
has a solution. These highly structured systems of equations (see Propositions
1.1, 1.3, and 1.4), which we refer to as _combinatorial systems of equations_
, are then solved using various methods including linear algebra over finite
fields, Gröbner bases, or semidefinite programming. As we shall see below this
methodology is applicable in a wide range of contexts.
In what follows, $G=(V,E)$ denotes an undirected simple graph on vertex set
$V=\left\\{1,\ldots,n\right\\}$ and edges $E$. Similarly, by $G=(V,A)$ we mean
that $G$ is a _directed_ graph with arcs $A$. When $G$ is undirected, we let
$Arcs(G)=\\{(i,j):i,j\in V,\ \text{and}\ \\{i,j\\}\in E\\}$
consist of all possible arcs for each edge in $G$. We study three classical
graph problems.
First, in Section 2, we explore $k$-colorability using techniques from
commutative algebra and algebraic geometry. The following polynomial
formulation of $k$-colorability is well-known [5].
###### Proposition 1.1.
Let $G=(V,E)$ be an undirected simple graph on vertices $V=\\{1,\ldots,n\\}$.
Fix a positive integer $k$, and let $\mathbb{K}$ be a field with
characteristic relatively prime to $k$. The polynomial system
$J_{G}=\\{x_{i}^{k}-1=0,\ x_{i}^{k-1}+x_{i}^{k-2}x_{j}+\cdots+x_{j}^{k-1}=0:\
i\in V,\ \\{i,j\\}\in E\\}$
has a common zero over $\overline{\mathbb{K}}$ (the algebraic closure of
$\mathbb{K}$) if and only if the graph $G$ is $k$-colorable.
###### Remark 1.2.
Depending on the context, the fields $\mathbb{K}$ we use in this paper will be
the rationals $\mathbb{Q}$, the reals ${\mathbb{R}}$, the complex numbers
$\mathbb{C}$, or finite fields $\mathbb{F}_{p}$ with $p$ a prime number.
Hilbert’s Nullstellensatz [11, Theorem 2, Chapter 4] states that a system of
polynomial equations $\\{f_{1}(x)=0,\dots,f_{r}(x)=0\\}$ with coefficients in
$\mathbb{K}$ has no solution with entries in its algebraic closure
${\overline{\mathbb{K}}}$ if and only if
$1=\sum_{i=1}^{r}\beta_{i}f_{i},\ \ \text{ for some polynomials
$\beta_{1},\ldots,\beta_{r}\in{\mathbb{K}}[x_{1},\dots,x_{n}]$}.$
Thus, if the system has no solution, there is a Nullstellensatz certificate
that the associated combinatorial problem is infeasible. We can find a
Nullstellensatz certificate $1=\sum_{i=1}^{r}\beta_{i}f_{i}$ of a given degree
$D:=\max_{1\leq i\leq r}\\{\deg(\beta_{i})\\}$ or determine that no such
certificate exists by solving a system of _linear equations_ whose variables
are in bijection with the coefficients of the monomials of
$\beta_{1},\ldots,\beta_{r}$ (see [15] and the many references therein). The
number of variables in this linear system grows with the number ${n+D\choose
D}$ of monomials of degree at most $D$. Crucially, the linear system, which
can be thought of as a $D$-th order linear relaxation of the polynomial
system, can be solved in time that is polynomial in the input size for fixed
degree $D$ (see [34, Theorem 4.1.3] or the survey [15]). The degree $D$ of a
Nullstellensatz certificate of an infeasible polynomial system cannot be more
than known bounds [26], and thus, by searching for certificates of increasing
degrees, we obtain a finite (but potentially long) procedure to decide whether
a system is feasible or not (this is the NulLA algorithm in [34, 14, 13]). The
philosophy of “linearizing” a system of arbitrary polynomials has also been
applied in other contexts besides combinatorics, including computer algebra
[18, 25, 37, 44], logic and complexity [9], cryptography [10], and
optimization [30, 28, 29, 39, 40, 41].
As the complexity of solving a combinatorial system with this strategy depends
on its certificate degree, it is important to understand the class of problems
having small degrees $D$. In Theorem 2.1, we give a combinatorial
characterization of non-3-colorable graphs whose polynomial system encoding
has a degree one Nullstellensatz certificate of infeasibility. Essentially, a
graph has a degree one certificate if there is an edge covering of the graph
by three and four cycles obeying some parity conditions on the number of times
an edge is covered. This result is reminiscent of the cycle double cover
conjecture of Szekeres (1973) [47] and Seymour (1979) [42]. The class of
non-3-colorable graphs with degree one certificates is far from trivial; it
includes graphs that contain an odd-wheel or a 4-clique [34] and
experimentally it has been shown to include more complicated graphs (see [34,
13, 15]).
In our second application of the polynomial method, we use tools from the
theory of Gröbner bases to investigate (in Section 3) the detection of
Hamiltonian cycles of a directed graph $G$. The following ideals algebraically
encode Hamiltonian cycles (see Lemma 3.8 for a proof).
###### Proposition 1.3.
Let $G=(V,A)$ be a simple directed graph on vertices $V=\\{1,\ldots,n\\}$.
Assume that the characteristic of $\mathbb{K}$ is relatively prime to $n$ and
that $\omega\in\mathbb{K}$ is a primitive $n$-th root of unity. Consider the
following system in $\mathbb{K}[x_{1},\ldots,x_{n}]$:
$H_{G}=\\{x_{i}^{n}-1=0,\ \prod_{j\in\delta^{+}(i)}(\omega x_{i}-x_{j})=0:\
i\in V\\}.$
Here, $\delta^{+}(i)$ denotes those vertices $j$ which are connected to $i$ by
an arc going from $i$ to $j$ in $G$. The system $H$ has a solution over
$\overline{\mathbb{K}}$ if and only if $G$ has a Hamiltonian cycle.
We prove a decomposition theorem for the ideal $H_{G}$ generated by the above
polynomials, and based on this structure, we give an algebraic
characterization of _uniquely Hamiltonian graphs_ (reminiscent of the one for
$k$-colorability in [24]). Our results also provide an algorithm to decide
this property. These findings are related to a well-known theorem of Smith
[50] which states that if a $3$-regular graph has one Hamiltonian cycle then
it has at least three. It is still an open question to decide the complexity
of finding a second Hamiltonian cycle knowing that it exists [6].
Finally, in Section 4 we explore the problem of determining the automorphisms
$Aut(G)$ of an undirected graph $G$. Recall that the elements of $Aut(G)$ are
those permutations of the vertices of $G$ which preserve edge adjacency. Of
particular interest for us in that section is when graphs are rigid; that is,
$|Aut(G)|=1$. The complexity of this decision problem is still wide open [7].
The combinatorial object $Aut(G)$ will be viewed as an algebraic variety in
${\mathbb{R}}^{n\times n}$ as follows.
###### Proposition 1.4.
Let $G$ be a simple undirected graph and $A_{G}$ its adjacency matrix. Then
$Aut(G)$ is the group of permutation matrices $P=[P_{i,j}]_{i,j=1}^{n}$ given
by the zeroes of the ideal $I_{G}\subseteq{\mathbb{R}}[x_{1},\ldots,x_{n}]$
generated from the equations:
$\begin{split}{(PA_{G}-A_{G}P)}_{i,j}=0,&\ \ 1\leq i,j\leq n;\ \ \
\sum_{i=1}^{n}P_{i,j}=1,\ \ 1\leq j\leq n;\\\ \sum_{j=1}^{n}P_{i,j}=1,&\ \
1\leq i\leq n;\ \ \ {P}^{2}_{i,j}-P_{i,j}=0,\ \ 1\leq i,j\leq n.\\\
\end{split}$ (1)
###### Proof.
The last three sets of equations say that $P$ is a permutation matrix, while
the first one ensures that this permutation preserves adjacency of edges
($PA_{G}P^{\top}=A_{G}$). ∎
In what follows, we shall interchangeably refer to $Aut(G)$ as a group or the
variety of Proposition 1.4. This real variety can be studied from the
perspective of convexity. Indeed, from Proposition 1.4, $Aut(G)$ consists of
the integer vertices of the polytope of doubly stochastic matrices commuting
with $A_{G}$. By replacing the equations ${P}^{2}_{i,j}-P_{i,j}=0$ in (1) with
the linear inequalities $P_{ij}\geq 0$, we obtain a polyhedron $P_{G}$ which
is a convex relaxation of the automorphism group of the graph. This polytope
and its integer hull have been investigated by Friedland and Tinhofer [48,
20], where they gave conditions for it to be integral. Here, we uncover more
properties of the polyhedron $P_{G}$ and its integer vertices $Aut(G)$.
Our first result is that $P_{G}$ is _quasi-integral_ ; that is, the graph
induced by the integer points in the 1-skeleton of $P_{G}$ is connected (see
Definition 7.1 in Chapter 4 of [27]). It follows that one can decide rigidity
of graphs by inspecting the vertex neighbors of the identity permutation.
Another application of this result is an output-sensitive algorithm for
enumerating all automorphisms of any graph [3]. The problem of determining the
triviality of the automorphism group of a graph can be solved efficiently when
$P_{G}$ is integral. Such graphs have been called _compact_ and a fair amount
of research has been dedicated to them (see [8, 48] and references therein).
Next, we use the theory of Gouveia, Parrilo, and Thomas [21], applied to the
ideal $I_{G}$ of Proposition 1.4, to approximate the integer hull of $P_{G}$
by projections of semidefinite programs (the so-called _theta bodies_). In
their work, the authors of [21] generalize the Lovász theta body for $0/1$
polyhedra to generate a sequence of semidefinite programming relaxations
computing the convex hull of the zeroes of a set of real polynomials [33, 32].
The paper [21] provides some applications to finding maximum stable sets [33]
and maximum cuts [21]. We study the theta bodies of the variety of
automorphisms of a graph. In particular, we give sufficient conditions on
$Aut(G)$ for which the first theta body is already equal to $P_{G}$ (in much
the same way that stable sets of perfect graphs are theta-1 exact [21, 33]).
Such graphs will be called _exact_. Establishing these conditions for
exactness requires an interesting generalization of properties of the
symmetric group (see Theorem 4.6 for details). In addition, we prove that
compact graphs are a proper subset of exact graphs (see Theorem 4.4). This is
interesting because we do not know of an example of a graph that is not exact,
and the connection with semidefinite programming may open interesting
approaches to understanding the complexity of the graph automorphism problem.
Below, we assume the reader is familiar with the basic properties of
polynomial ideals and commutative algebra as introduced in the elementary text
[11]. A quick, self-contained review can also be found in Section 2 of [24].
## 2 Recognizing Non-3-colorable Graphs
In this section, we give a complete combinatorial characterization of the
class of non-3-colorable simple undirected graphs $G=(V,E)$ with a degree one
Nullstellensatz certificate of infeasibility for the following system (with
$\mathbb{K}=\mathbb{F}_{2}$) from Proposition 1.1:
$J_{G}=\\{x_{i}^{3}+1=0,\ x_{i}^{2}+x_{i}x_{j}+x_{j}^{2}=0:\ i\in V,\
\\{i,j\\}\in E\\}.$ (2)
This polynomial system has a degree one ($D=1$) Nullstellensatz certificate of
infeasibility if and only if there exist coefficients
$a_{i},a_{ij},b_{ij},b_{ijk}\in\mathbb{F}_{2}$ such that
$\sum_{i\in V}(a_{i}+\sum_{j\in V}a_{ij}x_{j})(x_{i}^{3}+1)+\sum_{\\{i,j\\}\in
E}(b_{ij}+\sum_{k\in V}b_{ijk}x_{k})(x_{i}^{2}+x_{i}x_{j}+x_{j}^{2})=1.$ (3)
Our characterization involves two types of substructures on the graph $G$ (see
Figure 1). The first of these are _oriented partial-3-cycles_ , which are
pairs of arcs $\\{(i,j),(j,k)\\}\subseteq Arcs(G)$, also denoted $(i,j,k)$, in
which $(k,i)\in Arcs(G)$ (the vertices ${i,j,k}$ induce a 3-cycle in $G$). The
second are _oriented chordless 4-cycles_ , which are sets of four arcs
$\\{(i,j),(j,k),(k,l),(l,i)\\}\subseteq Arcs(G)$, denoted $(i,j,k,l)$, with
$(i,k),(j,l)\not\in Arcs(G)$ (the vertices ${i,j,k,l}$ induce a chordless
4-cycle).
(ii)$j$$i$$l$$k$(i)$k$$i$$j$
Figure 1: (i) partial 3-cycle, (ii) chordless 4-cycle
###### Theorem 2.1.
For a given simple undirected graph $G=(V,E)$, the polynomial system over
$\mathbb{F}_{2}$ encoding the $3$-colorability of $G$
$J_{G}=\\{x_{i}^{3}+1=0,\ x_{i}^{2}+x_{i}x_{j}+x_{j}^{2}=0:\ i\in V,\
\\{i,j\\}\in E\\}$
has a degree one Nullstellensatz certificate of infeasibility if and only if
there exists a set $C$ of oriented partial $3$-cycles and oriented chordless
$4$-cycles from $Arcs(G)$ such that
1. 1.
$|C_{(i,j)}|+|C_{(j,i)}|\equiv 0\pmod{2}$ for all $\\{i,j\\}\in E$ and
2. 2.
$\sum_{(i,j)\in Arcs(G),i<j}|C_{(i,j)}|\equiv 1\pmod{2}$,
where $C_{(i,j)}$ denotes the set of cycles in $C$ in which the arc $(i,j)\in
Arcs(G)$ appears. Moreover, the class of non-3-colorable graphs whose
encodings have degree one Nullstellensatz infeasibility certificates can be
recognized in polynomial time.
We can consider the set $C$ in Theorem 2.1 as a covering of $E$ by directed
edges. From this perspective, Condition 1 in Theorem 2.1 means that every edge
of $G$ is covered by an even number of arcs from cycles in $C$. On the other
hand, Condition 2 says that if $\hat{G}$ is the directed graph obtained from
$G$ by the orientation induced by the total ordering on the vertices
$1<2<\cdots<n$, then when summing the number of times each arc in $\hat{G}$
appears in the cycles of $C$, the total is odd.
Note that the 3-cycles and 4-cycles in $G$ that correspond to the partial
3-cycles and chordless 4-cycles in $C$ give an edge-covering of a
non-3-colorable subgraph of $G$. Also, note that if a graph $G$ has a
non-3-colorable subgraph whose polynomial encoding has a degree one
infeasibility certificate, then the encoding of $G$ will also have a degree
one infeasibility certificate.
The class of graphs with encodings that have degree one infeasibility
certificates includes all graphs containing odd wheels as subgraphs (e.g., a
$4$-clique) [34].
###### Corollary 2.2.
If a graph $G=(V,E)$ contains an odd wheel, then the encoding of
$3$-colorability of $G$ from Theorem 2.1 has a degree one Nullstellensatz
certificate of infeasibility.
###### Proof.
Assume $G$ contains an odd wheel with vertices labelled as in Figure 2 below.
Let
$C:=\\{(i,1,i+1):2\leq i\leq n-1\\}\cup\\{(n,1,2)\\}.$
Figure 2: Odd wheel
Figure 2 illustrates the arc directions for the oriented partial 3-cycles of
$C$. Each edge of $G$ is covered by exactly zero or two partial 3-cycles, so
$C$ satisfies Condition 1 of Theorem 2.1. Furthermore, each arc $(1,i)\in
Arcs(G)$ is covered exactly once by a partial 3-cycle in $C$, and there is an
odd number of such arcs. Thus, $C$ also satisfies Condition 2 of Theorem 2.1.
∎
A non-trivial example of a non-3-colorable graph with a degree one
Nullstellensatz certicate is the Grötzsch graph.
###### Example 2.3.
Consider the Grötzsch graph in Figure 3, which has no 3-cycles. The following
set of oriented chordless 4-cycles gives a certificate of non-3-colorability
by Theorem 2.1:
$\displaystyle C:=\\{$
$\displaystyle(1,2,3,7),(2,3,4,8),(3,4,5,9),(4,5,1,10),(1,10,11,7),$
$\displaystyle(2,6,11,8),(3,7,11,9),(4,8,11,10),(5,9,11,6)\\}.$
Figure 3 illustrates the arc directions for the 4-cycles of $C$. Each edge of
the graph is covered by exactly two 4-cycles, so $C$ satisfies Condition 1 of
Theorem 2.1. Moreover, one can check that Condition 2 is also satisfied. It
follows that the graph has no proper 3-coloring. ∎
Figure 3: Grötzsch graph.
We now prove Theorem 2.1 using ideas from polynomial algebra. First, notice
that we can simplify a degree one certificate as follows: Expanding the left-
hand side of (3) and collecting terms, the only coefficient of
$x_{j}x_{i}^{3}$ is $a_{ij}$ and thus $a_{ij}=0$ for all $i,j\in V$.
Similarly, the only coefficient of $x_{i}x_{j}$ is $b_{ij}$, and so $b_{ij}=0$
for all $\\{i,j\\}\in E$. We thus arrive at the following simplified
expression:
$\sum_{i\in V}a_{i}(x_{i}^{3}+1)+\sum_{\\{i,j\\}\in E}(\sum_{k\in
V}b_{ijk}x_{k})(x_{i}^{2}+x_{i}x_{j}+x_{j}^{2})=1.$ (4)
Now, consider the following set $F$ of polynomials:
$\displaystyle x_{i}^{3}+1$ $\displaystyle\qquad\forall i\in V,$ (5)
$\displaystyle x_{k}(x_{i}^{2}+x_{i}x_{j}+x_{j}^{2})$
$\displaystyle\qquad\forall\\{i,j\\}\in E,\ k\in V.$ (6)
The elements of $F$ are those polynomials that can appear in a degree one
certificate of infeasibility. Thus, there exists a degree one certificate if
and only if the constant polynomial 1 is in the linear span of $F$; that is,
$1\in\langle F\rangle_{\mathbb{F}_{2}}$, where $\langle
F\rangle_{\mathbb{F}_{2}}$ is the vector space over $\mathbb{F}_{2}$ generated
by the polynomials in $F$.
We next simplify the set $F$. Let $H$ be the following set of polynomials:
$\displaystyle x_{i}^{2}x_{j}+x_{i}x_{j}^{2}+1$
$\displaystyle\;\forall\\{i,j\\}\in E,$ (7) $\displaystyle
x_{i}x_{j}^{2}+x_{j}x_{k}^{2}$ $\displaystyle\;\forall(i,j),(j,k),(k,i)\in
Arcs(G),$ (8) $\displaystyle
x_{i}x_{j}^{2}+x_{j}x_{k}^{2}+x_{k}x_{l}^{2}+x_{l}x_{i}^{2}$
$\displaystyle\;\forall(i,j),(j,k),(k,l),(l,i)\in Arcs(G),(i,k),(j,l)\not\in
Arcs(G).$ (9)
If we identify the monomials $x_{i}x_{j}^{2}$ as the arcs $(i,j)$, then the
polynomials (2) correspond to oriented partial 3-cycles and the polynomials
(9) correspond to oriented chordless 4-cycles. The following lemma says that
we can use $H$ instead of $F$ to find a degree one certificate.
###### Lemma 2.4.
We have $1\in\langle F\rangle_{\mathbb{F}_{2}}$ if and only if $1\in\langle
H\rangle_{\mathbb{F}_{2}}$.
###### Proof.
The polynomials (6) above can be split into two classes of equations: (i)
$k=i$ or $k=j$ and (ii) $k\neq i$ and $k\neq j$. Thus, the set $F$ consists of
$\displaystyle x_{i}^{3}+1\qquad$ $\displaystyle\forall i\in V,$ (10)
$\displaystyle
x_{i}(x_{i}^{2}+x_{i}x_{j}+x_{j}^{2})=x_{i}^{3}+x_{i}^{2}x_{j}+x_{i}x_{j}^{2}$
$\displaystyle\qquad\forall\\{i,j\\}\in E,$ (11) $\displaystyle
x_{k}(x_{i}^{2}+x_{i}x_{j}+x_{j}^{2})=x_{i}^{2}x_{k}+x_{i}x_{j}x_{k}+x_{j}^{2}x_{k}$
$\displaystyle\qquad\forall\\{i,j\\}\in E,\ k\in V,i\neq k\neq j.$ (12)
Using polynomials (10) to eliminate the $x_{i}^{3}$ terms from (11), we arrive
at the following set of polynomials, which we label $F^{\prime}$:
$\displaystyle x_{i}^{3}+1\qquad$ $\displaystyle\forall i\in V,$ (13)
$\displaystyle
x_{i}^{2}x_{j}+x_{i}x_{j}^{2}+1=(x_{i}^{3}+x_{i}^{2}x_{j}+x_{i}x_{j}^{2})+(x_{i}^{3}+1)$
$\displaystyle\qquad\forall\\{i,j\\}\in E,$ (14) $\displaystyle
x_{i}^{2}x_{k}+x_{i}x_{j}x_{k}+x_{j}^{2}x_{k}$
$\displaystyle\qquad\forall\\{i,j\\}\in E,\ k\in V,i\neq k\neq j.$ (15)
Observe that $\langle F\rangle_{\mathbb{F}_{2}}=\langle
F^{\prime}\rangle_{\mathbb{F}_{2}}$. We can eliminate the polynomials (13) as
follows. For every $i\in V$, $(x_{i}^{3}+1)$ is the only polynomial in
$F^{\prime}$ containing the monomial $x_{i}^{3}$ and thus the polynomial
$(x_{i}^{3}+1)$ cannot be present in any nonzero linear combination of the
polynomials in $F^{\prime}$ that equals 1. We arrive at the following smaller
set of polynomials, which we label $F^{\prime\prime}$.
$\displaystyle x_{i}^{2}x_{j}+x_{i}x_{j}^{2}+1$
$\displaystyle\qquad\forall\\{i,j\\}\in E,$ (16) $\displaystyle
x_{i}^{2}x_{k}+x_{i}x_{j}x_{k}+x_{j}^{2}x_{k}$
$\displaystyle\qquad\forall\\{i,j\\}\in E,k\in V,i\neq k\neq j.$ (17)
So far, we have shown $1\in\langle F\rangle_{\mathbb{F}_{2}}=\langle
F^{\prime}\rangle_{\mathbb{F}_{2}}$ if and only if $1\in\langle
F^{\prime\prime}\rangle_{\mathbb{F}_{2}}$.
Next, we eliminate monomials of the form $x_{i}x_{j}x_{k}$. There are 3 cases
to consider.
Case 1: $\\{i,j\\}\in E$ but $\\{i,k\\}\not\in E$ and $\\{j,k\\}\not\in E$. In
this case, the monomial $x_{i}x_{j}x_{k}$ appears in only one polynomial,
$x_{k}(x_{i}^{2}+x_{i}x_{j}+x_{j}^{2})=x_{i}^{2}x_{k}+x_{i}x_{j}x_{k}+x_{j}^{2}x_{k}$,
so we can eliminate all such polynomials.
Case 2: $i,j,k\in V$, $(i,j),(j,k),(k,i)\in Arcs(G)$. Graphically, this
represents a 3-cycle in the graph. In this case, the monomial
$x_{i}x_{j}x_{k}$ appears in three polynomials:
$\displaystyle
x_{k}(x_{i}^{2}+x_{i}x_{j}+x_{j}^{2})=x_{i}^{2}x_{k}+x_{i}x_{j}x_{k}+x_{j}^{2}x_{k},$
(18) $\displaystyle
x_{j}(x_{i}^{2}+x_{i}x_{k}+x_{k}^{2})=x_{i}^{2}x_{j}+x_{i}x_{j}x_{k}+x_{j}x_{k}^{2},$
(19) $\displaystyle
x_{i}(x_{j}^{2}+x_{j}x_{k}+x_{k}^{2})=x_{i}x_{j}^{2}+x_{i}x_{j}x_{k}+x_{i}x_{k}^{2}.$
(20)
Using the first polynomial, we can eliminate $x_{i}x_{j}x_{k}$ from the other
two:
$\displaystyle
x_{i}^{2}x_{j}+x_{j}x_{k}^{2}+x_{i}^{2}x_{k}+x_{j}^{2}x_{k}=(x_{i}^{2}x_{j}+x_{i}x_{j}x_{k}+x_{j}x_{k}^{2})+(x_{i}^{2}x_{k}+x_{i}x_{j}x_{k}+x_{j}^{2}x_{k}),$
$\displaystyle
x_{i}x_{j}^{2}+x_{i}x_{k}^{2}+x_{i}^{2}x_{k}+x_{j}^{2}x_{k}=(x_{i}x_{j}^{2}+x_{i}x_{j}x_{k}+x_{i}x_{k}^{2})+(x_{i}^{2}x_{k}+x_{i}x_{j}x_{k}+x_{j}^{2}x_{k}).$
We can now eliminate the polynomial (18). Moreover, we can use the polynomials
(16) to rewrite the above two polynomials as follows.
$\displaystyle
x_{k}x_{i}^{2}+x_{i}x_{j}^{2}=(x_{i}^{2}x_{j}+x_{j}x_{k}^{2}+x_{i}^{2}x_{k}+x_{j}^{2}x_{k})+(x_{j}x_{k}^{2}+x_{j}^{2}x_{k}+1)+(x_{i}x_{j}^{2}+x_{i}^{2}x_{j}+1),$
$\displaystyle
x_{i}x_{j}^{2}+x_{j}x_{k}^{2}=(x_{i}x_{j}^{2}+x_{i}x_{k}^{2}+x_{i}^{2}x_{k}+x_{j}^{2}x_{k})+(x_{i}x_{k}^{2}+x_{i}^{2}x_{k}+1)+(x_{j}x_{k}^{2}+x_{j}^{2}x_{k}+1).$
Note that both of these polynomials correspond to two of the arcs of the
$3$-cycle $(i,j),(j,k),(k,i)\in Arcs(G)$.
Case 3: $i,j,k\in V$, $(i,j),(j,k)\in Arcs(G)$ and $(k,i)\not\in Arcs(G)$. We
have
$\displaystyle
x_{k}(x_{i}^{2}+x_{i}x_{j}+x_{j}^{2})=x_{i}^{2}x_{k}+x_{i}x_{j}x_{k}+x_{j}^{2}x_{k},$
(21) $\displaystyle
x_{i}(x_{j}^{2}+x_{j}x_{k}+x_{k}^{2})=x_{i}x_{j}^{2}+x_{i}x_{j}x_{k}+x_{i}x_{k}^{2}.$
(22)
As before we use the first polynomial to eliminate the monomial
$x_{i}x_{j}x_{k}$ from the second:
$\displaystyle
x_{i}x_{j}^{2}+x_{j}x_{k}^{2}+(x^{2}_{i}x_{k}+x_{i}x_{k}^{2}+1)=\ $
$\displaystyle(x_{i}x_{j}^{2}+x_{i}x_{j}x_{k}+x_{i}x_{k}^{2})+(x_{i}^{2}x_{k}+x_{i}x_{j}x_{k}+x_{j}^{2}x_{k})$
$\displaystyle+(x_{j}x_{k}^{2}+x_{j}^{2}x_{k}+1).$
We can now eliminate (21); thus, the original system has been reduced to the
following one, which we label as $F^{\prime\prime\prime}$:
$\displaystyle x_{i}^{2}x_{j}+x_{i}x_{j}^{2}+1$
$\displaystyle\;\forall\\{i,j\\}\in E,$ (23) $\displaystyle
x_{i}x_{j}^{2}+x_{j}x_{k}^{2}$ $\displaystyle\;\forall(i,j),(i,k),(j,k)\in
Arcs(G),$ (24) $\displaystyle
x_{i}x_{j}^{2}+x_{j}x_{k}^{2}+(x^{2}_{i}x_{k}+x_{i}x_{k}^{2}+1)$
$\displaystyle\;\forall(i,j),(j,k)\in Arcs(G),(k,i)\not\in Arcs(G).$ (25)
Note that $1\in\langle F\rangle_{\mathbb{F}_{2}}$ if and only if $1\in\langle
F^{\prime\prime\prime}\rangle_{\mathbb{F}_{2}}$.
The monomials $x^{2}_{i}x_{k}$ and $x_{i}x_{k}^{2}$ with $(k,i)\not\in
Arcs(G)$ always appear together and only in the polynomials (25) in the
expression $(x^{2}_{i}x_{k}+x_{i}x_{k}^{2}+1)$. Thus, we can eliminate the
monomials $x^{2}_{i}x_{k}$ and $x_{i}x_{k}^{2}$ with $(k,i)\not\in Arcs(G)$ by
choosing one of the polynomials (25) and using it to eliminate the expression
$(x^{2}_{i}x_{k}+x_{i}x_{k}^{2}+1)$ from all other polynomials in which it
appears. Let $i,j,k,l\in V$ be such that $(i,j),(j,k),(k,l),(l,i)\in Arcs(G)$
and $(k,i),(i,k)\not\in Arcs(G)$. We can then eliminate the monomials
$x^{2}_{i}x_{k}$ and $x_{i}x_{k}^{2}$ as follows:
$\displaystyle x_{i}x_{j}^{2}+x_{j}x_{k}^{2}+x_{k}x_{l}^{2}+x_{l}x_{i}^{2}=$
$\displaystyle\
(x_{i}x_{j}^{2}+x_{j}x_{k}^{2}+x^{2}_{i}x_{k}+x_{i}x_{k}^{2}+1)$
$\displaystyle+(x_{k}x_{l}^{2}+x_{l}x_{i}^{2}+x^{2}_{i}x_{k}+x_{i}x_{k}^{2}+1).$
Finally, after eliminating the polynomials (25), we have system $H$
(polynomials (7), (2), and (9)):
$\displaystyle x_{i}^{2}x_{j}+x_{i}x_{j}^{2}+1$
$\displaystyle\;\forall\\{i,j\\}\in E,$ $\displaystyle
x_{i}x_{j}^{2}+x_{j}x_{k}^{2}$ $\displaystyle\;\forall(i,j),(j,k),(k,i)\in
Arcs(G),$ $\displaystyle
x_{i}x_{j}^{2}+x_{j}x_{k}^{2}+x_{k}x_{l}^{2}+x_{l}x_{i}^{2}$
$\displaystyle\;\forall(i,j),(j,k),(k,l),(l,i)\in Arcs(G),(i,k),(j,l)\not\in
Arcs(G).$
The system $H$ has the property that $1\in\langle
F^{\prime\prime\prime}\rangle_{\mathbb{F}_{2}}$ if and only if $1\in\langle
H\rangle_{\mathbb{F}_{2}}$, and thus, $1\in\langle F\rangle_{\mathbb{F}_{2}}$
if and only if $1\in\langle H\rangle_{\mathbb{F}_{2}}$ as required ∎
We now establish that the sufficient condition for infeasibility $1\in\langle
H\rangle_{\mathbb{F}_{2}}$ is equivalent to the combinatorial parity
conditions in Theorem 2.1.
###### Lemma 2.5.
There exists a set $C$ of oriented partial 3-cycles and oriented chordless
4-cycles satisfying Conditions 1. and 2. of Theorem 2.1 if and only if
$1\in\langle H\rangle_{\mathbb{F}_{2}}$.
###### Proof.
Assume that $1\in\langle H\rangle_{\mathbb{F}_{2}}$. Then there exist
coefficients $c_{h}\in\mathbb{F}_{2}$ such that $\sum_{h\in H}c_{h}h=1$. Let
$H^{\prime}:=\\{h\in H:c_{h}=1\\}$; then, $\sum_{h\in H^{\prime}}h=1$. Let $C$
be the set of oriented partial 3-cycles $(i,j,k)$ where
$x_{i}x_{j}^{2}+x_{j}x_{k}^{2}\in H^{\prime}$ together with the set of
oriented chordless 4-cycles $(i,j,l,k)$ where
$x_{i}x_{j}^{2}+x_{j}x_{l}^{2}+x_{l}x_{k}^{2}+x_{k}x_{i}^{2}\in H^{\prime}$.
Now, $|C_{(i,j)}|$ is the number of polynomials in $H^{\prime}$ of the form
(2) or (9) in which the monomial $x_{i}x_{j}^{2}$ appears, and similarly,
$|C_{(j,i)}|$ is the number of polynomials in $H^{\prime}$ of the form (2) or
(9) in which the monomial $x_{j}x_{i}^{2}$ appears. Thus, $\sum_{h\in
H^{\prime}}h=1$ implies that, for every pair $x_{i}x_{j}^{2}$ and
$x_{j}x_{i}^{2}$, either
1. 1.
$|C_{(i,j)}|\equiv 0\pmod{2}$, $|C_{(j,i)}|\equiv 0\pmod{2}$, and
$x_{i}^{2}x_{j}+x_{i}x_{j}^{2}+1\not\in H^{\prime}$ or
2. 2.
$|C_{(i,j)}|\equiv 1\pmod{2}$, $|C_{(j,i)}|\equiv 1\pmod{2}$, and
$x_{i}^{2}x_{j}+x_{i}x_{j}^{2}+1\in H^{\prime}$.
In either case, we have $|C_{(i,j)}|+|C_{(j,i)}|\equiv 0\pmod{2}$. Moreover,
since $\sum_{h\in H^{\prime}}h=1$, there must be an odd number of the
polynomials of the form $x_{i}^{2}x_{j}+x_{i}x_{j}^{2}+1$ in $H^{\prime}$.
That is, case 2 above occurs an odd number of times and therefore,
$\sum_{(i,j)\in Arcs(G),i<j}|C_{(i,j)}|\equiv 1\pmod{2}$ as required.
Conversely, assume that there exists a set $C$ of oriented partial 3-cycles
and oriented chordless 4-cycles satisfying the conditions of Theorem 2.1. Let
$H^{\prime}$ be the set of polynomials $x_{i}x_{j}^{2}+x_{j}x_{k}^{2}$ where
$(i,j,k)\in C$ and the set of polynomials
$x_{i}x_{j}^{2}+x_{j}x_{l}^{2}+x_{l}x_{k}^{2}+x_{k}x_{i}^{2}$ where
$(i,j,l,k)\in C$ together with the set of polynomials
$x_{i}^{2}x_{j}+x_{i}x_{j}^{2}+1\in H$ where $|C_{(i,j)}|\equiv 1$. Then,
$|C_{(i,j)}|+|C_{(j,i)}|\equiv 0\pmod{2}$ implies that every monomial
$x_{i}x_{j}^{2}$ appears in an even number polynomials of $H^{\prime}$.
Moreover, since $\sum_{(i,j)\in Arcs(G),i<j}|C_{(i,j)}|\equiv 1\pmod{2}$,
there are an odd number of polynomials $x_{i}^{2}x_{j}+x_{i}x_{j}^{2}+1$
appearing in $H^{\prime}$. Hence, $\sum_{h\in H^{\prime}}h=1$ and $1\in\langle
H\rangle_{\mathbb{F}_{2}}$. ∎
Combining Lemmas 2.4 and 2.5, we arrive at the characterization stated in
Theorem 2.1. That such graphs can be decided in polynomial time follows from
the fact that the existence of a certificate of any fixed degree can be
decided in polynomial time (as is well known and follows since there are
polynomially many monomials up to any fixed degree; see also [34, Theorem
4.1.3]).
Finally, we pose as open problems the construction of a variant of Theorem 2.1
for general $k$-colorability and also combinatorial characterizations for
larger certificate degrees $D$.
###### Problem 2.6.
Characterize those graphs with a given $k$-colorability Nullstellensatz
certificate of degree $D$.
## 3 Recognizing Uniquely Hamiltonian Graphs
Throughout this section we work over an arbitrary algebraically closed field
$\mathbb{K}={\overline{\mathbb{K}}}$, although in some cases, we will need to
restrict its characteristic. Let us denote by $H_{G}$ the Hamiltonian ideal
generated by the polynomials from Proposition 1.3. A connected, directed graph
$G$ with $n$ vertices has a Hamiltonian cycle if and only if the equations
defined by $H_{G}$ have a solution over $\mathbb{K}$ (or, in other words, if
and only if $V(H_{G})\neq\emptyset$ for the algebraic variety $V(H_{G})$
associated to the ideal $H_{G}$). In a precise sense to be made clear below,
the ideal $H_{G}$ actually encodes all Hamiltonian cycles of $G$. However, we
need to be somewhat careful about how to count cycles (see Lemma 3.8). In
practice $\omega$ can be treated as a variable and not as a fixed primitive
$n$-th root of unity. A set of equations ensuring that $\omega$ only takes on
the value of a _primitive_ $n$-th root of unity is the following:
$\\{\omega^{i(n-1)}+\omega^{i(n-2)}+\cdots+\omega^{i}+1=0:\ 1\leq i\leq n\\}.$
We can also use the cyclotomic polynomial $\Phi_{n}(\omega)$ [16], which is
the polynomial whose zeroes are the primitive $n$-th roots of unity.
We shall utilize the theory of Gröbner bases to show that $H_{G}$ has a
special (algebraic) decomposition structure in terms of the different
Hamiltonian cycles of $G$ (this is Theorem 3.9 below). In the particular case
when $G$ has a unique Hamiltonian cycle, we get a specific algebraic criterion
which can be algorithmically verified. These results are Hamiltonian analogues
to the algebraic $k$-colorability characterizations of [24]. We first turn our
attention more generally to cycle ideals of a simple directed graph $G$. These
will be the basic elements in our decomposition of the Hamiltonian ideal
$H_{G}$, as they algebraically encode single cycles $C$ (up to symmetry).
When $G$ has the property that each pair of vertices connected by an arc is
also connected by an arc in the opposite direction, then we call $G$ _doubly
covered_. When $G=(V,E)$ is presented as an undirected graph, we shall always
view it as the doubly covered directed graph on vertices $V$ with arcs
$Arcs(G)$.
Let $C$ be a cycle of length $k>2$ in $G$, expressed as a sequence of arcs,
$C=\\{(v_{1},v_{2}),(v_{2},v_{3}),\ldots,(v_{k},v_{1})\\}.$
For the purpose of this work, we call $C$ a _doubly covered cycle_ if
consecutive vertices in the cycle are connected by arcs in both directions;
otherwise, $C$ is simply called _directed_. In particular, each cycle in a
doubly covered graph is a doubly covered cycle. These definitions allow us to
work with both undirected and directed graphs in the same framework.
###### Definition 3.1 (Cycle encodings).
Let $\omega$ be a fixed primitive $k$-th root of unity and let $\mathbb{K}$ be
a field with characteristic not dividing $k$. If $C$ is a doubly covered cycle
of length $k$ and the vertices in $C$ are $\\{v_{1},\ldots,v_{k}\\}$, then the
cycle encoding of $C$ is the following set of $k$ polynomials in
$\mathbb{K}[x_{v_{1}},\ldots,x_{v_{k}}]$:
$g_{i}=\begin{cases}\
x_{v_{i}}+\frac{(\omega^{2+i}-\omega^{2-i})}{(\omega^{3}-\omega)}x_{v_{k-1}}+\frac{(\omega^{1-i}-\omega^{3+i})}{(\omega^{3}-\omega)}x_{v_{k}}&\
\ \text{$i=1,\ldots,k-2$},\\\ \ (x_{v_{k-1}}-\omega
x_{v_{k}})(x_{v_{k-1}}-\omega^{-1}x_{v_{k}})&\ \ \text{$i=k-1$},\\\ \
x_{v_{k}}^{k}-1&\ \ \text{$i=k$}.\end{cases}$ (26)
If $C$ is a directed cycle of length $k$ in a directed graph, with vertex set
$\\{v_{1},\ldots,v_{k}\\}$, the cycle encoding of $C$ is the following set of
$k$ polynomials:
$g_{i}=\begin{cases}\ x_{v_{k-i}}-\omega^{k-i}x_{v_{k}}&\ \
\text{$i=1,\ldots,k-1$},\\\ \ x_{v_{k}}^{k}-1&\ \ \text{$i=k$}.\end{cases}$
(27)
###### Definition 3.2 (Cycle Ideals).
The cycle ideal associated to a cycle $C$ is $H_{G,C}=\langle
g_{1},\ldots,g_{k}\rangle\subseteq\mathbb{K}[x_{v_{1}},\ldots,x_{v_{k}}]$,
where the $g_{i}$s are the cycle encoding of $C$ given by (26) or (27).
The polynomials $g_{i}$ are computationally useful generators for cycle
ideals. (Once again, see [11] for the relevant background on Gröbner bases and
term orders.)
###### Lemma 3.3.
The cycle encoding polynomials $F=\\{g_{1},\ldots,g_{k}\\}$ are a reduced
Gröbner basis for the cycle ideal $H_{G,C}$ with respect to any term order
$\prec$ with $x_{v_{k}}\prec\cdots\prec x_{v_{1}}$.
###### Proof.
Since the leading monomials in a cycle encoding:
$\\{x_{v_{1}},\ldots,x_{v_{k-2}},x_{v_{k-1}}^{2},x_{v_{k}}^{k}\\}\ \textit{
\rm or }\ \\{x_{v_{1}},\ldots,x_{v_{k-2}},x_{v_{k-1}},x_{v_{k}}^{k}\\}$ (28)
are relatively prime, the polynomials $g_{i}$ form a Gröbner basis for
$H_{G,C}$ (see Theorem 3 and Proposition 4 in [11, Section 2]). That $F$ is
reduced follows from inspection of (26) and (27). ∎
###### Remark 3.4.
In particular, since reduced Gröbner bases (with respect to a fixed term
order) are unique, it follows that cycle encodings are canonical ways of
generating cycle ideals (and thus of representing cycles by Lemma 3.6).
Having explicit Gröbner bases for these ideals allows us to compute their
Hilbert series easily.
###### Corollary 3.5.
The Hilbert series of $\mathbb{K}[x_{v_{1}},\ldots,x_{v_{k}}]/H_{G,C}$ for a
doubly covered cycle or a directed cycle is equal to (respectively)
$\frac{(1-t^{2})(1-t^{k})}{(1-t)^{2}}\text{ or }\ \frac{(1-t^{k})}{(1-t)}.$
###### Proof.
If $\prec$ is a graded term order, then the (affine) Hilbert function of an
ideal and of its ideal of leading terms are the same [11, Chapter 9, §3]. The
form of the Hilbert series is now immediate from (28). ∎
The naming of these ideals is motivated by the following result; in words, it
says that the cycle $C$ is encoded as a complete intersection by the ideal
$H_{G,C}$.
###### Lemma 3.6.
The following hold for the ideal $H_{G,C}$.
1. 1.
$H_{G,C}$ is radical,
2. 2.
$|V(H_{G,C})|=k$ if $C$ is directed, and $|V(H_{G,C})|=2k$ if $C$ is doubly
covered undirected.
###### Proof.
Without loss of generality, we suppose that $v_{i}=i$ for $i=1,\ldots,k$. Let
$\prec$ be any term order in which $x_{k}\prec\cdots\prec x_{1}$. From Lemma
3.3, the set of $g_{i}$ form a Gröbner basis for $H_{G,C}$. It follows that
the number of standard monomials of $H_{G,C}$ is $2k$ if $C$ is doubly covered
undirected (resp. $k$ if it is directed). Therefore by [24, Lemma 2.1], if we
can prove that $|V(H_{G,C})|\geq k$ (resp. $|V(H_{G,C})|\geq 2k$), then both
statements 1. and 2. follow.
When $C$ is directed, this follows easily from the form of (27), so we shall
assume that $C$ is doubly covered undirected. We claim that the $k$ cyclic
permutations of the two points:
$(\omega,\omega^{2},\ldots,\omega^{k}),(\omega^{k},\omega^{k-1},\ldots,\omega)$
are zeroes of $g_{i}$, $i=1,\ldots,k$. Since cyclic permutation is
multiplication by a power of $\omega$, it is clear that we need only verify
this claim for the two points above. In the fist case, when
$x_{i}=\omega^{i}$, we compute that for $i=1,\ldots,k-2$:
$\begin{split}(\omega^{3}-\omega)g_{i}(\omega,\ldots,\omega^{k})=\
&(\omega^{3}-\omega)\omega^{i}+(\omega^{2+i}-\omega^{2-i})\omega^{k-1}+(\omega^{1-i}-\omega^{3+i})\omega^{k}\\\
=\
&\omega^{3+i}-\omega^{1+i}+\omega^{1+i+k}-\omega^{1-i+k}+\omega^{1-i+k}-\omega^{3+i+k}\\\
=\ &0,\end{split}$
since $\omega^{k}=1$. In the second case, when $x_{i}=\omega^{1-i}$, we again
compute that for $i=1,\ldots,k-2$:
$\begin{split}(\omega^{3}-\omega)g_{i}(\omega^{k},\ldots,\omega)=\
&(\omega^{3}-\omega)\omega^{1-i}+(\omega^{2+i}-\omega^{2-i})\omega^{2}+(\omega^{1-i}-\omega^{3+i})\omega\\\
=\
&\omega^{4-i}-\omega^{2-i}+\omega^{4+i}-\omega^{4-i}+\omega^{2-i}-\omega^{4+i}\\\
=\ &0.\end{split}$
Finally, it is obvious that the two points zero $g_{k-1}$ and $g_{k}$, and
this completes the proof. ∎
###### Remark 3.7.
Conversely, it is easy to see that points in $V(H_{G,C})$ correspond to cycles
of length $k$ in $G$. That this variety contains $k$ or $2k$ points
corresponds to there being $k$ or $2k$ ways of writing down the cycle since we
may cyclically permute it and also reverse its orientation (if each arc in the
path is bidirectional).
Before stating our decomposition theorem (Theorem 3.9), we need to explain how
the Hamiltonian ideal encodes all Hamiltonian cycles of the graph $G$.
###### Lemma 3.8.
Let $G$ be a connected directed graph on $n$ vertices. Then,
$V(H_{G})=\bigcup_{C}V(H_{G,C}),$
where the union is over all Hamiltonian cycles $C$ in $G$.
###### Proof.
We only need to verify that points in $V(H_{G})$ correspond to cycles of
length $n$. Suppose there exists a Hamiltonian cycle in the graph $G$. Label
vertex $1$ in the cycle with the number $x_{1}=\omega^{0}=1$ and then
successively label vertices along the cycle with one higher power of $\omega$.
It is clear that these labels $x_{i}$ associated to vertices $i$ zero all of
the equations generating $H_{G}$.
Conversely, let $\textbf{v}=(x_{1},\ldots,x_{n})$ be a point in the variety
$V(H_{G})$ associated to $H_{G}$; we claim that v encodes a Hamiltonian cycle.
From the edge equations, each vertex must be adjacent to one labeled with the
next highest power of $\omega$. Fixing a starting vertex $i$, it follows that
there is a cycle $C$ labeled with (consecutively) increasing powers of
$\omega$. Since $\omega$ is a primitive $n$th root of unity, this cycle must
have length $n$, and thus is Hamiltonian. ∎
Combining all of these ideas, we can prove the following result.
###### Theorem 3.9.
Let $G$ be a connected directed graph with $n$ vertices. Then,
$H_{G}=\bigcap_{C}H_{G,C},$
where $C$ ranges over all Hamiltonian cycles of the graph $G$.
###### Proof.
Since $H_{G}$ contains a square-free univariate polynomial in each
indeterminate, it is radical (see for instance [24, Lemma 2.1]). It follows
that
$\begin{split}H_{G}=\ &I(V(H_{G}))\\\ =\
&I\left(\bigcup_{C}V(H_{G,C})\right)\\\ =\ &\bigcap_{C}I(V(H_{G,C}))\\\ =\
&\bigcap_{C}H_{G,C},\\\ \end{split}$ (29)
where the second inequality comes from Lemma 3.8 and the last one from
$H_{G,C}$ being a radical ideal (Lemma 3.6). ∎
We call a directed graph (resp. doubly covered graph) uniquely Hamiltonian if
it contains $n$ cycles of length $n$ (resp. $2n$ cycles of length $n$).
###### Corollary 3.10.
The graph $G$ is uniquely Hamiltonian if and only if the Hamiltonian ideal
$H_{G}$ is of the form $H_{G,C}$ for some length $n$ cycle $C$.
This corollary provides an algorithm to check whether a graph is uniquely
Hamiltonian. We simply compute a unique reduced Gröbner basis of $H_{G}$ and
then check that it has the same form as that of an ideal $H_{G,C}$. Another
approach is to count the number of standard monomials of any Gröbner bases for
$H_{G}$ and compare with $n$ or $2n$ (since $H_{G}$ is radical). We remark,
however, that it is well-known that computing a Gröbner basis in general
cannot be done in polynomial time [51, p. 400].
We close this section with a directed and an undirected example of Theorem
3.9.
###### Example 3.11.
Let $G$ be the directed graph with vertex set $V=\\{1,2,3,4,5\\}$ and arcs
$A=\\{(1,2),(2,3),(3,4),(4,5),(5,1),(1,3),(1,4)\\}$. Moreover, let $\omega$ be
a primitive $5$-th root of unity. The ideal
$H_{G}\subset\mathbb{K}[x_{1},x_{2},x_{3},x_{4},x_{5}]$ is generated by the
polynomials,
$\\{x_{i}^{5}-1:1\leq i\leq 5\\}\cup\\{(\omega x_{1}-x_{2})(\omega
x_{1}-x_{3})(\omega x_{1}-x_{4}),\omega x_{2}-x_{3},\omega x_{3}-x_{4},\omega
x_{4}-x_{5},\omega x_{5}-x_{1}\\}.$
A reduced Gröbner basis for $H_{G}$ with respect to the ordering $x_{5}\prec
x_{4}\prec x_{3}\prec x_{2}\prec x_{1}$ is
$\\{x_{5}^{5}-1,x_{4}-\omega^{4}x_{5},x_{3}-\omega^{3}x_{5},x_{2}-\omega^{2}x_{5},x_{1}-\omega
x_{5}\\},$
which is a generating set for $H_{G,C}$ with
$C=\\{(1,2),(2,3),(3,4),(4,5),(5,1)\\}$. ∎
Let $G$ be an undirected graph with vertex set $V$ and edge set $E$, and
consider the auxiliary directed graph $\tilde{G}$ with vertices $V$ and arcs
$Arcs(G)$. Notice that $\tilde{G}$ is doubly covered, and hence each of its
cycles are doubly covered. We apply Theorem 3.9 to $H_{\tilde{G}}$ to
determine and count Hamiltonian cycles in $G$. In particular, the cycle
$C=\\{v_{1},v_{2},\ldots,v_{n}\\}$ of $G$ is Hamiltonian if and only if
$\\{(v_{1},v_{2}),(v_{2},v_{3}),\ldots,(v_{n-1},v_{n}),(v_{n},v_{1})\\}$ and
$\\{(v_{2},v_{1}),(v_{3},v_{2}),\ldots,(v_{n},v_{n-1}),(v_{1},v_{n})\\}$ are
Hamiltonian cycles of $\tilde{G}$.
###### Example 3.12.
Let $G$ be the undirected complete graph on the vertex set $V=\\{1,2,3,4\\}$.
Let $\tilde{G}$ be the doubly covered graph with vertex set $V$ and arcs
$Arcs(G)$. Notice that $\tilde{G}$ has twelve Hamiltonian cycles:
$\displaystyle C_{1}=$ $\displaystyle\\{(1,2),(2,3),(3,4),(4,1)\\},$
$\displaystyle\ \ C_{2}=$ $\displaystyle\\{(2,1),(3,2),(4,3),(1,4)\\},$
$\displaystyle C_{3}=$ $\displaystyle\\{(1,2),(2,4),(4,3),(3,1)\\},$
$\displaystyle\ \ C_{4}=$ $\displaystyle\\{(2,1),(4,2),(3,4),(1,3)\\},$
$\displaystyle C_{5}=$ $\displaystyle\\{(1,3),(3,2),(2,4),(4,1)\\},$
$\displaystyle\ \ C_{6}=$ $\displaystyle\\{(3,1),(2,3),(4,2),(1,4)\\},$
$\displaystyle C_{7}=$ $\displaystyle\\{(1,3),(3,4),(4,2),(2,1)\\},$
$\displaystyle\ \ C_{8}=$ $\displaystyle\\{(3,1),(4,3),(2,4),(1,2)\\},$
$\displaystyle C_{9}=$ $\displaystyle\\{(1,4),(4,2),(2,3),(3,1)\\},$
$\displaystyle\ \ C_{10}=$ $\displaystyle\\{(4,1),(2,4),(3,2),(1,3)\\},$
$\displaystyle C_{11}=$ $\displaystyle\\{(1,4),(4,3),(3,2),(2,1)\\},$
$\displaystyle\ \ C_{12}=$ $\displaystyle\\{(4,1),(3,4),(2,3),(1,2)\\}.$
One can check in a symbolic algebra system such as SINGULAR or Macaulay 2 that
the ideal $H_{\tilde{G}}$ is the intersection of the cycle ideals
$H_{\tilde{G},C_{i}}$ for $i=1,\ldots,12$.
## 4 Permutation Groups as Algebraic Varieties and their Convex
Approximations
In this section, we study convex hulls of permutations groups viewed as
permutation matrices. We begin by studying the convex hull of automorphism
groups of undirected simple graphs; these have a natural polynomial
presentation using Proposition 1.4 from the introduction. For background
material on graph automorphism groups see [7, 8].
We write $Aut(G)$ for the automorphism group of a graph $G=(V,E)$. Elements of
$Aut(G)$ are naturally represented as $|V|\times|V|$ permutation matrices;
they are the _integer_ vertices of the rational polytope $P_{G}$ defined in
the discussion following Proposition 1.4. The polytope $P_{G}$ was first
introduced by Tinhofer [48]. Since we are primarily interested in the integer
vertices of $P_{G}$, we investigate $IP_{G}$, the integer hull of $P_{G}$
(i.e. $IP_{G}=conv(P_{G}\cap\mathbb{Z}^{n\times n})$). In the fortunate case
that $P_{G}$ is already integral ($P_{G}$ = $IP_{G}$), we say that the graph
$G$ is _compact_ , a term coined in [48]. This occurs, for example, in the
special case that $G$ is an independent set on $n$ vertices. In this case
$Aut(G)=S_{n}$ and $P_{G}$ is the well-studied Birkhoff polytope, the convex
hull of all doubly-stochastic matrices (see Chapter 5 of [27]). One can
therefore view $P_{G}$ as a generalization of the Birkhoff polytope to
arbitrary graphs. Unfortunately, the polytope $P_{G}$ is not always integral.
For instance, $P_{G}$ is not integral when $G$ is the Petersen graph.
Nevertheless, we can prove the following related result.
###### Proposition 4.1.
The polytope $P_{G}$ is quasi-integral. That is, the induced subgraph of the
integer points of the 1-skeleton of $P_{G}$ is connected.
###### Proof.
We claim that there exists a $0/1$ matrix $A$ such that $P_{G}$ is the set of
points $\\{x\in{\mathbb{R}}^{n\times n}\ :\ Ax=\textbf{1},x\geq 0\\}$ (where 1
is the all 1s vector). By the main theorem of Trubin [49] and independently
[4], polytopes given by such systems are quasi-integral (see also Theorem 7.2
in Chapter 4 of [27]). Therefore, we need to rewrite the defining equations
presented in Proposition 1.4 to fit this desired shape. Fix indices $1\leq
i,j\leq n$ and consider the row of $P_{G}$ defined by the equation
$\sum_{r\in\delta(j)}P_{ir}-\sum_{k\in\delta(i)}P_{kj}=0.$
Here $\delta(i)$ denotes those vertices $j$ which are connected to $i$. Adding
the equation $\sum_{r=1}^{n}P_{rj}=1$ to both sides of this expression yields
$\sum_{r\in\delta(j)}P_{ir}+\sum_{k\notin\delta(i)}P_{kj}=1.$ (30)
We can therefore replace the original $n^{2}$ equations defining $P_{G}$ by
(30) over all $1\leq i,j\leq n$. The result now follows provided that no
summand in each of these equations repeats. However, this is clear since if
summands $P_{kj}$ and $P_{ir}$ are the same, then $r=j$, which is impossible
since $r\in\delta(j)$. ∎
We would still like to find a tighter description of $IP_{G}$ in terms of
inequalities. For this purpose, recall the radical polynomial ideal $I_{G}$ in
Proposition 1.4 and its real variety $V_{{\mathbb{R}}}(I_{G})$. We approximate
a tighter description of $IP_{G}$ using a hierarchy of projected semidefinite
relaxations of $conv(V_{{\mathbb{R}}}(I_{G}))$. When these relaxations are
tight, we obtain a full description of $IP_{G}$ that allows us to optimize and
determine feasibility via semidefinite programming.
We begin with some preliminary definitions from [21] and motivated by Lovász &
Schrijver [33]. Let $I\subset\mathbb{R}[x_{1},\ldots,x_{n}]$ be a real radical
ideal ($I=\mathcal{I}(V_{{\mathbb{R}}}(I))$). A polynomial $f$ is said to be
nonnegative mod $I$ (written $f\geq 0$ (mod $I$)) if $f(p)\geq 0$ for all
$p\in V_{\mathbb{R}}(I)$. Similarly, a polynomial $f$ is said to be a sum of
squares mod $I$ if there exist
$h_{1},\ldots,h_{m}\in\mathbb{R}[x_{1},\ldots,x_{n}]$ such that
$f-\sum_{i=1}^{m}h_{i}^{2}\in I$. If the degrees of the $h_{1},\ldots,h_{m}$
are bounded by some positive integer $k$, we say $f$ is $k$-sos mod $I$.
The _$k$ -th theta body_ of $I$, denoted $TH_{k}(I)$, is the subset of
${\mathbb{R}}^{n}$ that is nonnegative on each $f\in I$ that is $k$-sos mod
$I$. We say that a real variety $V_{\mathbb{R}}(I)$ is theta $k$-exact if
$\overline{conv(V_{\mathbb{R}}(I))}=TH_{k}(I)$. When the ideal $I$ is real
radical, theta bodies provide a hierarchy of semidefinite relaxations of
$\overline{conv(V_{{\mathbb{R}}}(I))}$:
$TH_{1}(I)\supseteq
TH_{2}(I)\supseteq\cdots\supseteq\overline{conv(V_{{\mathbb{R}}}(I))}$
because in this case theta bodies can be expressed as projections of feasible
regions of semidefinite programs (such regions are called _spectrahedra_). In
order to exploit this theory, we must establish that $I_{G}$ is indeed real
radical.
###### Lemma 4.2.
The ideal $I_{G}\subseteq{\mathbb{R}}[x_{1},\ldots,x_{n}]$ is real radical.
###### Proof.
Let $J_{G}$ be the ideal in $\mathbb{C}[x_{1},\ldots,x_{n}]$ generated by the
same polynomials that generate $I_{G}$, and $\sqrt[{\mathbb{R}}]{I_{G}}$ be
the real radical of $I_{G}$. Since the polynomial $x_{i}^{2}-x_{i}\in J_{G}$
for each $1\leq i\leq n$, Lemma 2.1 of [24] implies $J_{G}=\sqrt{J_{G}}$
(where $\sqrt{J_{G}}$ is the radical of $J_{G}$). Together with the fact that
$V_{\mathbb{C}}(J_{G})=V_{{\mathbb{R}}}(I_{G})$, this implies
$J_{G}\supseteq\sqrt[{\mathbb{R}}]{I_{G}}$. Since
$I_{G}=J_{G}\cap{\mathbb{R}}[x_{1},\ldots,x_{n}]$, we conclude
$I_{G}\supseteq\sqrt[{\mathbb{R}}]{I_{G}}$. The result follows since
trivially, $I_{G}\subseteq\sqrt[{\mathbb{R}}]{I_{G}}$. ∎
From Lemma 4.2, we conclude that if $I_{G}$ is theta $k$-exact, linear
optimization over the automorphisms can be performed using semidefinite
programming provided that one first computes a basis for the quotient ring
${\mathbb{R}}[P_{11},P_{12},\dots,P_{nn}]/I_{G}$ (e.g., obtained from the
standard monomials of a Gröbner basis). Using such a basis one can set up the
necessary semidefinite programs (see Section 2 of [21] for details). In fact,
for $k$-exact ideals, one only needs those elements of the basis up to degree
$2k$. This motivates the need for characterizing those graphs for which
$I_{G}$ is $k$-exact.
In this section we focus on finding graphs $G$ such that $I_{G}$ is 1-exact;
we shall call such graphs _exact_ in what follows. The key to finding exact
graphs is the following combinatorial-geometric characterization.
###### Theorem 4.3.
[21] Let $V_{\mathbb{R}}(I)\subset\mathbb{R}^{n}$ be a finite real variety.
Then $V_{\mathbb{R}}(I)$ is exact if and only if there is a finite linear
inequality description of $conv(V_{{\mathbb{R}}}(I))$ such that for every
inequality $g(x)\geq 0$, there is a hyperplane $g(x)=\alpha$ such that every
point in $V_{\mathbb{R}}(I)$ lies either on the hyperplane $g(x)=0$ or the
hyperplane $g(x)=\alpha$.
A result of Sullivant (see Theorem 2.4 in [46]) directly implies that when the
polytope $P=conv(V_{{\mathbb{R}}}(I))$ is lattice isomorphic to an integral
polytope of the form $[0,1]^{n}\cap L$ where $L$ is an affine subspace, then
$P$ satisfies the condition of Theorem 4.3. Putting these ideas together we
can prove compactness implies exactness. Furthermore, the class of exact
graphs properly extends the class of compact graphs. The proof of this latter
fact is an extension of a result in [48].
###### Theorem 4.4.
The class of exact graphs strictly contains the class of compact graphs. More
precisely:
1. 1.
If $G$ is a compact graph, then $G$ is also exact.
2. 2.
Let $G_{1},\ldots,G_{m}$ be $k$-regular connected compact graphs, and let
$G=\bigsqcup_{i=1}^{m}G_{i}$ be the graph that is the disjoint union of
$G_{1},\ldots,G_{m}$. Then $G$ is always exact, but $G$ may not be compact.
Indeed, $G$ is compact if and only if $G_{i}\cong G_{j}$ for all $1\leq
i,j\leq n$.
###### Proof.
If $G$ is compact, then the integer hull of $P_{G}$ is precisely the affine
space
$\\{P\in{\mathbb{R}}^{n\times n}\ :\ PA_{G}=A_{G}P,\
\sum_{i=1}^{n}P_{ij}=\sum_{j=1}^{n}P_{ij}=1,\ 1\leq i,j\leq n\\}$
intersected with the cube ${[0,1]}^{n\times n}$. That $G$ is exact follows
from Theorem 2.4 of [46].
We now prove Statement 2. If $G_{i}\not\cong G_{j}$ for some pair $(i,j)$,
then $G$ was shown to be non-compact by Tinhofer (see [48, Lemma 2]).
Nevertheless, $G$ is exact. We prove this for $m=2$, and the result will
follow by induction. We claim that if $G=G_{1}\sqcup G_{2}$ with
$G_{1}\not\cong G_{2}$, then the integer hull $IP_{G}$ is the solution set to
the following system (which we denote by $\tilde{IP}_{G}$):
$\displaystyle~{}{(PA_{G}-A_{G}P)}_{i,j}=0$ $\displaystyle 1\leq i,j\leq n,$
$\displaystyle\sum_{i=1}^{n}P_{i,j}=1$ $\displaystyle 1\leq j\leq n,$
$\displaystyle\sum_{j=1}^{n}P_{i,j}=1$ $\displaystyle 1\leq i\leq n,$
$\displaystyle\sum_{i=1}^{n_{1}}\sum_{j=n_{1}+1}^{n_{1}+n_{2}}P_{i,j}=0,$
$\displaystyle 0\leq P_{i,j}\leq 1,$
where $n_{i}=|V(G_{i})|$ with $n_{1}\leq n_{2}$. Statement 2 then follows
again from Theorem 2.4 of [46].
We now prove the claim. Let $A_{G_{i}}$ be the adjacency matrix of $G_{i}$.
Index the adjacency matrix of $G=G_{1}\sqcup G_{2}$ so that the first $n_{1}$
rows (and hence first $n_{1}$ columns) index the vertices of $G_{1}$. Any
feasible $P$ of $P_{G}$ can be written as a block matrix
$P=\begin{pmatrix}A_{P}&B_{P}\\\ C_{P}&D_{P}\end{pmatrix},$
in which $A_{P}$ is $n_{1}\times n_{1}$. Since $G_{1}$ and $G_{2}$ are not
isomorphic, the only integer vertices of $P_{G}$ are of the form
$\begin{pmatrix}P_{1}&0\\\ 0&P_{2}\end{pmatrix}$ where $P_{i}$ is an
automorphism of $G_{i}$.
Now let $P$ be any non-integer vertex of $P_{G}$. We claim that the row sums
of $B_{P}$ must be 1. This will establish that $IP_{G}$ is described by the
system $\tilde{IP}_{G}$. To see this, observe that if $Q$ is any point in
$P_{G}$ not in $IP_{G}$, it is a convex combination of points in $P_{G}$, one
of which (say $P$) is non-integer. If the row sums of $B_{P}$ are 1, then $Q$
violates the system $\tilde{IP}_{G}$.
We now prove that if $P$ is a non-integer vertex of $P_{G}$, then the row sums
of $B_{P}$ must be 1. Since $P$ commutes with the adjacency matrix $A_{G}$ of
$G$, we must have
$A_{P}A_{G_{1}}=A_{G_{1}}A_{P},\ \ B_{P}A_{G_{2}}=A_{G_{1}}B_{P},\ \
C_{P}A_{G_{2}}=A_{G_{1}}C_{P},\ \ D_{P}A_{G_{2}}=A_{G_{2}}D_{P}.$
Let $\\{b_{1},\ldots,b_{n_{2}}\\}$ be the column sums of $B_{P}$. We shall
calculate the sum of the entries in each column of
$B_{P}A_{G_{2}}=A_{G_{1}}B_{P}$ in two ways. First, consider $A_{G_{1}}B_{P}$.
Since $G_{1}$ is $k$-regular, each entry of the $i$-th column of $B_{P}$ will
contribute exactly $k$ times to the sum of the entries of the $i$-th column of
$A_{G_{1}}B_{P}$. Thus, the sum of the entries of the $i$-th column of
$A_{G_{1}}B_{P}$ is $kb_{i}$.
Second, consider $B_{P}A_{G_{2}}$. The sum of the entries in its $i$-th column
is the sum of the entries of the columns of $B_{P}$ indexed by the neighbors
of $i$ in $G_{2}$. Thus, the sum of the entries in the $i$-th column of
$B_{P}A_{G_{2}}$ is $\sum_{l\in\delta_{G_{2}}(i)}b_{l}$. It follows that
$kb_{i}=\sum_{l\in\delta_{G_{2}}(i)}b_{l}$ for each $1\leq i\leq n$. This
equality can be written concisely as:
$\begin{pmatrix}kI_{n_{2}\times
n_{2}}-A_{G_{2}}\end{pmatrix}\begin{pmatrix}b_{1}\\\ \vdots\\\
b_{n_{2}}\end{pmatrix}=0.$
The matrix $kI_{n_{2}\times n_{2}}-A_{G_{2}}$ is the Laplacian of $G_{2}$. It
is well known that the kernel of the Laplacian of a connected graph is one
dimensional (see [8], Lemma 13.1.1). Since $G_{2}$ is regular, the kernel
contains the all ones vector. It follows that $b_{1}=\cdots=b_{n_{2}}$. By a
similar argument, the row sums of $C_{P}$ are all the same. Since all row sums
and column sums of $P$ are 1, and the row sums and column sums of $A_{G_{1}}$
are the same, it follows that the row sums of $B_{P}$ are equal and are the
same as the column sums of $C_{P}$.
Now assume for contradiction that the row sums of $B_{P}$ are not 1. If the
row sums are 0, then $B_{P}$ and $C_{P}$ would be 0 matrices. Since $G_{1}$
and $G_{2}$ are compact this would imply $A_{P}$ and $D_{P}$ are permutation
matrices, contradicting that $P$ is not integral. Thus the sum of each row of
$B_{P}$ is $\lambda$ with $0<\lambda<1.$ This implies the sum of the rows of
$A_{P}$ is $1-\lambda$ and that $\frac{1}{1-\lambda}A_{P}$ is a feasible
solution to $P_{G_{1}}$. By compactness of $G_{1}$, the matrix
$\frac{1}{1-\lambda}A_{P}$ is a convex combination
$\sum_{i=1}^{k}\mu_{k}Q_{k}$ of permutations $Q_{k}$ of $G_{1}$. This implies
that
$P=\sum_{i=1}^{k}\mu_{i}\begin{pmatrix}(1-\lambda)Q_{k}&B_{P}\\\
C_{P}&D_{P}\end{pmatrix},$
which is a convex combination of feasible solutions to $P_{G}$, contradicting
$P$ being a vertex. It follows that the row sums of $B_{P}$ must be 1. ∎
Exact graphs are then more abundant than compact graphs and the convex hull of
automorphisms of an exact graph has a description in terms of semidefinite
programming. It is thus desirable to find nice classes of graphs that are
exact. Notice that exactness is really a property of the set of permutation
matrices representing an automorphism group. This discussion motivates the
following question.
###### Question 4.5.
Which permutation subgroups of $S_{n}$ are exact?
Here we view a permutation subgroup of $S_{n}$ through its natural permutation
representation in ${\mathbb{R}}^{n\times n}$. In this light, a permutation
subgroup can be considered as a variety, and we say the permutation subgroup
is exact if this variety is exact. As an example, consider the alternating
group $A_{n}$ as a subgroup of $S_{n}$. It is known (see [7]) that $A_{n}$ is
never the automorphism group of a graph on $n$ vertices, so it cannot be
presented as the integer points of a polytope of the form $P_{G}$ with
$|V(G)|=n$. However, there is a description of $A_{n}$ as a variety whose
points are vertices of the $n\times n$ Birkhoff polytope:
$\begin{split}\sum_{j=1}^{n}P_{i,j}&=1,\ \ 1\leq i\leq n;\ \ \
\sum_{i=1}^{n}P_{i,j}=1,\ \ 1\leq j\leq n;\\\ \det(P)&=1;\ \ \ \ \ \ \ \ \ \ \
\ \ \ \ {P}^{2}_{i,j}-P_{i,j}=0,\ \ 1\leq i,j\leq n.\\\ \end{split}$
More generally, when a finite permutation group has a description as a
variety, we can apply the theory of theta bodies to obtain descriptions of
convex hulls. Using the algebraic-geometric ideas outlined in [45] we give a
sufficient condition for exactness of permutation groups.
Let $A=\\{\sigma_{1},\ldots,\sigma_{d}\\}$ be a subgroup of $S_{n}$. We
consider $A$ as the set of matrices
$\\{P_{\sigma_{1}},\ldots,P_{\sigma_{d}}\\}$ $\subseteq\mathbb{Z}^{n\times
n}$, where $P_{\sigma_{i}}$ is the permutation matrix corresponding to
$\sigma_{i}$. Let
$\mathbb{C}[\textbf{x}]:=\mathbb{C}[x_{\sigma_{1}},\ldots,x_{\sigma_{d}}]$ be
the polynomial ring in $d$ indeterminates indexed by permutations in $A$, and
let $\mathbb{C}[\textbf{t}]:=\mathbb{C}[t_{ij}:1\leq i,j\leq n]$.
The algebra homomorphism induced by the map
$\pi:\mathbb{C}[\textbf{x}]\to\mathbb{C}[\textbf{t}],\
\pi(x_{\sigma_{i}})=\prod_{1\leq j,k\leq n}t_{jk}^{(P_{\sigma_{i}})_{jk}}$
(31)
has kernel $I_{A}$, which is a prime toric ideal [45]. By Theorem 4.3,
Corollary 8.9 in [45], and Corollary 2.5 in [46], the group $A$ is exact if
and only if for every reverse lexicographic term ordering $\prec$ on
$\mathbb{C}[x]$, the initial ideal $in_{\prec}(I_{A})$ is generated by square-
free monomials. We now describe a family of permutation groups that are exact.
Let $A\subseteq\mathbb{Z}^{n\times n}$ be a subgroup of $S_{n}$. We say that
$A$ is permutation summable if for any permutations $P_{1},\ldots,P_{m}\in A$
satisfying the inequality $\sum_{i=1}^{m}P_{i}-I\geq 0$ (entry-wise), we have
that $\sum_{i=1}^{m}P_{i}-I$ is also a sum of permutation matrices in $A$. For
example, Birkhoff’s Theorem (see e.g., Theorem 1.1 in Chapter 5 of [27])
implies $S_{n}$ is permutation summable. Note that in this case $P_{S_{n}}$ is
the Birkhoff polytope which is known to be exact by the results in [21]. We
prove the following result.
###### Theorem 4.6.
Let $A=\\{\sigma_{1},\ldots,\sigma_{d}\\}$ be a permutation group that is a
subgroup of $S_{n}$.
(1) If $A$ is permutation summable, then $A$ is exact.
(2) Suppose $I_{A}$, the toric ideal associated to $A$, has a quadratically
generated Gröbner basis with respect to any reverse lexicographic ordering
$\prec$, then $A$ is exact.
###### Proof.
Let $I_{A}$ be the kernel of the algebra homomorphism induced by (31). We
shall abbreviate the action of $\pi$ on $x_{\sigma}$ by
$\pi(x_{\sigma})=t^{P_{\sigma}}$ for any $\sigma\in A$.
Let $\mathfrak{G}$ be a reduced Gröbner basis for $I_{A}$ with respect to some
reverse lexicographic order $\prec$ on
$\\{x_{\sigma_{1}},\ldots,x_{\sigma_{d}}\\}$. Let $x^{u}-x^{v}\in\mathfrak{G}$
with leading term $x^{u}$. By Theorem 4.3, Corollary 8.9 in [45] and Corollary
2.5 in [46], Statement (1) follows if we can find a square-free monomial
$x^{u^{\prime}}\in in_{\prec}(I_{A})$ such that $x^{u^{\prime}}$ divides
$x^{u}$.
Let $x_{\tau}$ be the smallest variable dividing $x^{v}$ with respect to
$\prec$. Then $x_{\tau}$ is smaller than any variable appearing in $x^{u}$ by
the choice of a reverse lexicographic ordering. Since
$x^{u}-x^{v}\in\mathfrak{G}$, we have $\pi(x^{u})=\pi(x^{v})$. It follows that
$\pi(x_{\tau})$ divides $\pi(x^{u})$, so letting
$x^{u}=x_{\sigma_{i_{1}}}\cdots x_{\sigma_{i_{k}}}$ for some
$\\{\sigma_{i_{1}},\ldots,\sigma_{i_{k}}\\}\subseteq A$, we have
$\dfrac{\pi(x^{u})}{\pi(x_{\tau})}=t^{P_{\sigma_{i_{1}}}+\cdots+P_{\sigma_{i_{k}}}}t^{-P_{\tau}},$
in which $\sum_{j=1}^{k}P_{\sigma_{i_{j}}}-P_{\tau}$ is a matrix with
nonnegative integer entries. Choose a subset $\\{\rho_{1},\ldots,\rho_{r}\\}$
$\subset\\{\sigma_{i_{1}},\ldots,\sigma_{i_{k}}\\}$ such that
$\\{P_{\rho_{1}},\ldots,P_{\rho_{r}}\\}$ minimally supports $P_{\tau}$ with
$P_{\rho_{i}}\neq P_{\rho_{j}}$ for all $i,j$, and let
$x^{u^{\prime}}=x_{\rho_{1}}\cdots x_{\rho_{r}}$. We claim that
$x^{u^{\prime}}$ is a square-free monomial that divides $x^{u}$ and lies in
$in_{\prec}(I_{A})$, which will prove Statement (1).
By construction, all indeterminates $x_{\rho_{1}},\ldots,x_{\rho_{r}}$ are
distinct, so $x^{u^{\prime}}$ is square-free. Moreover, since
$\\{\rho_{1},\ldots,\rho_{r}\\}\subset\\{\sigma_{i_{1}},\ldots,\sigma_{i_{k}}\\}$,
we have that $x^{u^{\prime}}$ divides $x^{u}$. It remains to show that
$x^{u^{\prime}}$ lies in $in_{\prec}(I_{A})$. To see this, note that
$\sum_{i=1}^{r}P_{\rho_{i}}-P_{\tau}$ has nonnegative integer entries, and
hence so does
$M=\sum_{i=1}^{r}{(P_{\tau})}^{-1}P_{\rho_{i}}-I$
(multiplying by $P_{\tau}^{-1}$ permutes matrix entries, and therefore does
not effect nonnegativity). Since $A$ is permutation summable, the matrix $M$
is a sum of matrices in $A$, and hence so is $P_{\tau}M$ =
$\sum_{i=1}^{r}P_{\rho_{i}}-P_{\tau}$. It follows that
$\sum_{i=1}^{r}P_{\rho_{i}}-P_{\tau}=\sum_{j=1}^{r-1}P_{\sigma_{l_{j}}}$
for some $\\{\sigma_{l_{1}},\ldots,\sigma_{l_{r-1}}\\}\subset A$. In
particular, $\pi(x^{u^{\prime}})=\pi(x_{\tau})\cdot\pi(x^{v^{\prime}})$ and so
$x^{u^{\prime}}-x_{\tau}x^{v^{\prime}}\in I_{A}$. Since $x_{\tau}$ is smaller
than any term in $x^{u^{\prime}}$ (the monomial $x^{u^{\prime}}$ divides
$x^{u}$ and the same holds for $x^{u}$), the leading term of
$x^{u^{\prime}}-x_{\tau}x^{v^{\prime}}$ is $x^{u^{\prime}}$; hence,
$x^{u^{\prime}}\in in_{\prec}(I_{A})$. This proves Statement (1).
For Statement (2), since any Gröbner basis is quadratically generated, by part
(1) it suffices to show that if $P_{1},P_{2},Q\in A$ with all entries of
$P_{1}+P_{2}-Q$ nonnegative, then $P_{1}+P_{2}-Q$ is a permutation matrix.
Since $supp(Q)\subset supp(P_{1})\cup supp(P_{2})$, the permutation $Q$ is a
vertex of a face containing $P_{1}$ and $P_{2}$. By Theorem 3.5 of [22], $Q$
is on the smallest face containing $P_{1}$ and $P_{2}$, and this face is
centrally symmetric. Thus, there is a vertex $R$ such that $Q+R=P_{1}+P_{2}$,
and the result follows.
∎
In light of Theorem 4.6, we would like to find permutation groups $A$ that are
permutation summable. As we have seen, Birkhoff’s Theorem (see [45]) implies
that $S_{n}$ is permutation summable. We can use this fact to construct more
permutation summable groups. For instance, $S_{n_{1}}\times\cdots\times
S_{n_{m}}$ is permutation summable, simply by applying the permutation
summability condition on each $S_{n_{i}}$ and taking direct sums. More
generally, if $H_{1},\ldots,H_{m}$ are permutation summable, then so is
$H_{1}\times\cdots\times H_{m}$. We present another class of permutation
summable groups that contains familiar groups.
###### Definition 4.7.
Let $A$ be a permutation subgroup of $S_{n}$. We say $A$ is strongly fixed-
point free if for every $\sigma\in A\backslash\\{1\\}$, we have $\sigma(i)\neq
i$ for any $i\in\\{1,\ldots,n\\}$.
###### Corollary 4.8.
Let $A$ be a strongly fixed-point free subgroup of $S_{n}$. Then $A$ is exact.
###### Proof.
Let $A$ be strongly fixed-point free. Consider any subset
$\\{P_{\sigma_{1}},\ldots,P_{\sigma_{k}}\\}$ of $A$ and assume
$\sum_{i=1}^{k}P_{\sigma_{i}}-I$ is a matrix with nonnegative entries. Then
one of the matrices in $A$ contains a fixed point. Without loss of generality,
assume $P_{\sigma_{1}}$ is one such matrix. Since $A$ is strongly fixed-point
free, we have $P_{\sigma_{1}}=I$. Hence,
$\sum_{i=1}^{k}P_{\sigma_{i}}-I=\sum_{i=2}^{k}P_{\sigma_{i}},$
and thus $A$ is permutation summable. The result now follows from Theorem 4.6.
∎
There are many well-known families of permutation groups that are strongly
fixed-point free, and hence exact. These include the group generated by any
$n$ cycle in $S_{n}$, and even dihedral groups (dihedral groups of order $4n$
as subgroups of $S_{2n}$).
## 5 Acknowledgements
We would like to thank the referee for his or her valuable suggestions and
corrections.
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|
arxiv-papers
| 2010-02-23T23:14:46 |
2024-09-04T02:49:08.526114
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J.A. De Loera, C. Hillar, P.N. Malkin, M. Omar",
"submitter": "Peter Malkin",
"url": "https://arxiv.org/abs/1002.4435"
}
|
1002.4667
|
# Outflows from AGN: their Impact on Spectra and the Environment
Daniel Proga and Ryuichi Kurosawa∗
###### Abstract
We present a brief summary of the main results from our multi-dimensional,
time-dependent simulations of gas dynamics in AGN. We focus on two types of
outflows powered by radiation emitted from the AGN: disk winds and winds
driven from large-scale inflows. We show spectra predicted by the simulations
and discuss their relevance to observations of broad- and narrow-line regions
of the AGN. We finish with a few remarks on whether these outflows can have a
significant impact on their environment and host galaxy.
Department of Physics and Astronomy, University of Nevada Las Vegas, NV 89154,
USA
∗Current Address: Department of Astronomy, Cornell University, Ithaca, NY 1
4853, USA
## 1\. Introduction
While AGN are very strong sources of electromagnetic radiation, they are also
sources of outflows of matter and magnetic energy. There is a growing body of
evidence that outflows are quite common and are an integral part of AGN (e.g.,
Crenshaw et al. 2002; Richards & Hall 2004). Moreover, radiation and mass
outflows in luminous sources are physically coupled as matter can absorber,
scatter and emit radiation and radiation can affect dynamics and ionization of
matter. For example, powerful mass outflows in quasars are very likely winds
driven by radiation from accretion disks (e.g., reviews by K$\ddot{\rm o}$nigl
2006; Proga 2007a). In addition, as we will illustrate here, the same
radiation can drive outflows from large scale inflows.
In this paper, we briefly summarize results from numerical simulations of disk
winds and large-scale outflows. We review the properties of these outflows and
compare them with those observed in AGN. This will allow us to assess the
level of our understanding of AGN outflows and their potential role in the AGN
feedback.
## 2\. Radiation-Driven Outflows from Accretion Disks
The successes of modelling outflows driven by radiation pressure on spectra
lines (line driving) from OB stars (e.g., Castor et al. 1975; Puls et al.
2008) and from accretion disks in cataclysmic variables (CVs; e.g., Proga
2005) motivate applications of a similar physics to model outflows in AGN.
Figure 1.: A map of the poloidal velocity field of the radiation-driven disk
wind model described in the text. The rotation axis of the disk is along the
left hand vertical frame, while the midplane of the disk is along the lower
horizontal frame. The figure is from Proga & Kallman (2004).
In Proga et al. (2000) and Proga & Kallman (2004), we adopted the approach
from Proga et al. (1998, 1999) to calculate axisymmetric time-dependent
hydrodynamical models of line-driven winds from accretion disks in AGN. To
apply the line-driven disk wind model developed for CVs to AGN, we took into
account, in an approximated way, the difference in the spectral energy
distribution between AGN and CVs. In particular, we introduced three major
modifications to our previous approach: 1) calculation of the parameters of
the line force based on the wind properties, 2) effects of optical depth on
the continuum photons, and 3) radiative heating and cooling of the gas.
Our AGN wind calculations follow (i) a hot and low density flow with negative
radial velocity in the polar region, (ii) a dense, warm and fast equatorial
outflow from the disk, and (iii) a transitional zone in which the disk outflow
is hot and struggles to escape the system. Fig. 1 shows an example of a
velocity field of a AGN disk wind model.
To make a direct comparison with observations, we calculated synthetic line
profiles based on our model for the C IV $\lambda$1549 Å line (Proga & Kallman
2004). The synthetic line profiles show a strong dependence on inclination
angle: the absorption forms only when an observer looks at the CE through the
fast wind (i.e., $i\geq 60^{\circ}$ as shown in Fig. 3 of Proga & Kallman
2004; see also Fig. 2 here). This $i$ dependence can explain why only 10% of
QSO have BALs.
Figure 2.: Theoretical profiles of Civ 1549Å based on the wind model of Proga
& Kallman (2004) as a function of inclination angle, $i$ (see bottom right
corner of each panel for the value of $i$). The solid lines show the profiles
due to the absorption only whereas the dashed lines show the profiles due to
the absorption and emission (the line source function includes resonance
scattering and thermal emission). Note the sensitivity of the lines on the
inclination angle. The zero velocity corresponding to the line center is
indicated by the vertical line.
The model also predicts high column densities for the inclination angles at
which strong absorption lines form. Schurch et al. (2009) computed broad band
spectra for various inclination angles using the simulation from Proga &
Kallman (2004). Fig. 3 presents some examples and shows that the model is
consistent with the observations, i.e., BAL QSO are underluminous in X-rays
compared to their non-BAL QSO counterparts (e.g., Brandt et al. 2000; Giustini
et al. 2008).
## 3\. Radiation-Driven Outflows from Large-Scale Inflows
In the previous section, we showed that the powerful radiation from an AGN can
drive a strong wind from an accretion disk, the place where the radiation is
produced. In turn, this wind has a significant impact on the radiation.
However, the same radiation can also change the dynamics of the material at
large distances from the radiation source.
Figure 3.: The 0.3-13 keV transmitted X-ray spectra ($E\,L_{E}$) from five
different lines-of-sight, with different inclinations, through the
hydrodynamic simulation of an AGN outflow presented in Proga & Kallman (2004).
The spectra were calculated with XSCORT (v5.18) using snapshot 800 of the
Proga & Kallman simulation to provide self-consistent physical outflow
properties. The input ionizing power-law spectrum is shown in black. The
spectra correspond to inclinations of $\theta$ = 50, 57, 62, 65, & 67∘, (top
to bottom). These l.o.s were chosen to highlight the range of spectral shapes
that result from the different physical properties throughout the simulated
wind. The figure is from Schurch et al. (2009).
We have begun studying gas dynamics in AGNs on subparsec and parsec scales
(Proga 2007b; Proga et al. 2008; Kurosawa & Proga 2008, 2009a, 2009b; Kurosawa
et al. 2009). In the following, we briefly summarize the main results of this
work.
In Proga (2007b), we calculated a series of models for non-rotating flows that
are under the influence of super massive BH gravity and radiation from an
accretion disk surrounding the BH. Generally, we used the numerical methods
developed by Proga et al. (2000). Our numerical approach allows for the self-
consistent determination of whether the flow is gravitationally captured by
the BH or driven away by thermal expansion or radiation pressure.
For a $10^{8}~{}\rm M_{\odot}$ BH with an accretion luminosity of 0.6 of
$L_{\rm Edd}$ (the same parameters as in the disk wind simulations), we found
that a non-rotating flow settles quickly into a steady state and has two
components (1) an equatorial inflow and (2) a bipolar inflow/outflow with the
outflow leaving the system along the pole. The first component is a
realization of Bondi-like accretion flow. The second component is an example
of a non-radial accretion flow becoming an outflow once it is pushed close to
the rotational axis of the disk where thermal expansion and radiation pressure
can accelerate the flow outward. The main result of these simplified
calculations is that the existence of the above two flow components is robust
yet their properties are sensitive to the geometry, SED of the radiation
field, and the outer boundaries. In particular, the outflow power and the
degree of collimation are higher for the model with radiation dominated by
UV/disk emission than for the model with radiation dominated by X-ray/central
engine emission. This sensitivity is related to the fact that thermal
expansion drives a weaker and wider outflow, compared to the radiation
pressure.
Figure 4.: Three-dimensional hydrodynamical simulations of outflow formation
via redirection of accreting gas by the strong radiation from an accretion
disk around a super massive black hole with its mass $M_{\rm{BH}}=10^{8}\rm
M_{\odot}$. The infalling gas is weakly rotating (sub-Keplerian), and the
Eddington ratio of the system is 0.6. The volume rendering representation of
the density distributions is shown. The outflow morphology is bi-conical, but
the flow contains relatively cold and dense cloud-like structures which
resembles those observed in the NRLs of Seyfert galaxies. The figure is from
Kurosawa & Proga (2009b).
Rotation of the inflowing gas changes the geometry of the flow because the
centrifugal force prevents gas from reaching the rotational axis (see Fig. 1
in Proga et al. 2008). This, in turn, reduces the mass outflow rate because
less gas is pushed toward the polar region. We also found that rotation can
lead to fragmentation and time variability of the outflow. As the flow
fragments, cold and dense clouds form (Figs. 4 and 12 in Proga et al. 2008).
Three-dimensional effects are also important. In Kurosawa & Proga (2008), we
considered effects of radiation due to a precessing accretion disk on a
spherical cloud of gas around the disk. On the other hand, in Kurosawa & Proga
(2009a), we recalculated some models from papers Proga (2007b) and Proga et
al. (2008) in full 3-D. Our 3-D simulations of a nonrotating gas show small
yet noticeable nonaxisymmetric small-scale features inside the outflow.
However, the outflow as a whole and the inflow do not seem to suffer from any
large-scale instability. In the rotating case, the nonaxisymmetric features
are very prominent, especially in the outflow which consists of many cold
dense clouds entrained in a smoother hot flow (e.g., see Figs. 4 and 5). The
3-D outflow is nonaxisymmetric due to the shear and thermal instabilities.
Effects of gas rotations are similar in 2-D and 3-D. In particular, gas
rotation increases the outflow thermal energy flux, but reduces the outflow
mass and kinetic energy fluxes. In addition, rotation leads to time
variability and fragmentation of the outflow in the radial and latitudinal
directions. The collimation of the outflow is reduced in the models with gas
rotation. The main difference between the models with rotation in 3-D and 2-D
is that the time variability in the mass and energy fluxes is reduced in the
3-D case because of the outflow fragmentation in the azimuthal direction.
Figure 5.: Spatial distributions of the “cold clouds” in the model shown in
Fig. 4. The grayscale image shows the density map of the cold clouds in
logarithmic scale (in cgs unit) on the $z$–$x$ plane. The cold clouds here are
defined as the gas with its density higher than $\rho_{\mathrm{min}}=1.6\times
10^{-20}\,\mathrm{g\,cm^{-3}}$ and its temperature less than
$T_{\mathrm{max}}=1.6\times 10^{5}\,\mathrm{K}$. The clouds are not
spherically distributed, but located near the bi-conic surface (which appears
as an X-shaped pattern here) defined by the outflowing gas. Note that the
length scale are in units of pc. The figure is from Kurosawa & Proga (2009b).
To be able to compare these new simulations with observations, we are in a
process of computing synthetic line profiles, broad band spectra and maps.
However, even without these diagnostics we can check if the models are
consistent with the data. For example, we can compare the kinematics study of
NGC 4151 Das et al. (2005) with the velocity of the cold clouds (cf. Fig. 5)
projected ($v_{\mathrm{proj}}$) toward an observer at the inclination angle
$i=45^{\circ}$ , which is the inclination of NGC 4151. Das et al. (2005) used
the kinematics model of the outflows with a bi-conic radial velocity law, and
found a good fit to their observations when the opening angle of the cone is
$\sim 33^{\circ}$. Interestingly, we find the opening angle of the outflows in
our is also about $30^{\circ}$ (cf. Figs. 5).
Figure 6 shows $v_{\mathrm{proj}}$ of the clouds plotted as a function of the
projected vertical distance, which is the distance along the $z$-axis in Fig.
5 projected onto the plane of the sky for an observer viewing the system with
$i=45^{\circ}$. The figure shows that the clouds are accelerated up to
$250\,\mathrm{km\,s^{-1}}$ until the projected distance reaches $\sim 4$ pc,
but the velocity curve starts to flatten beyond this point. Towards the outer
edges (near the outer boundaries), the curve begins to show a sign of
deceleration, but not so clearly. We note that the hot outflowing gas, on the
other hand, does show deceleration at the larger radii in our models (cf. Fig.
9 in Kurosawa & Proga 2009a). Although the physical size of the long slit
observation of NGC 4151 by Das et al. (2005) is in much lager scale ($\sim 50$
times larger) than that of our model, their radial velocities as a function of
the position along the slit (see their Figs. 5 and 6) show a similar pattern
as in our model (Fig. 6). The range of $v_{\mathrm{proj}}$ in our model is
about $-250$ to $300\,\mathrm{km\,\ s^{-1}}$ while the range of the observed
radial velocities in Das et al. (2005) is about $-800$ to
$800\,\mathrm{km\,s^{-1}}$, which is comparable to ours.
Figure 6.: The velocities of the cold cloud elements (as in Fig. 5) projected
toward an observer located at the inclination angle $i=45^{\circ}$ are shown
as a function of the projected vertical distance (the distance along the
$z$-axis in Fig. 5, but projected on to the plane of the sky for the observer
viewing the system with $i=45^{\circ}$). The negative projected distance
indicates the clouds are found in the lower half of the projection plane. The
figure is from Kurosawa & Proga (2009b).
## 4\. Concluding Remarks
The models presented here, which numerically simulate the outflows driven by
radiation from AGN, are in many respects consistent with observations. Clearly
more work is needed to test the models and improve them. However, the first
few steps toward the development a self-consistent physical model of the AGN
outflows has been taken.
Applications of the model are not limited to AGN because other astrophysical
objects – such as X-ray binaries, in particular micro-quasars – have a similar
geometry and can be understood within a similar physical framework. In
addition, having a physical model of the AGN outflows we can apply it to the
so-called AGN feedback problem.
Some results from the outflow models have been already incorporated to galaxy
evolution calculations. In particular, Ciotti et al. (2009), improved and
extended the accretion and feedback physics explored in their previous papers
(e.g., Ciotti & Ostriker 1997, 2001, 2007). By using a high-resolution one-
dimensional hydrodynamical code, Ciotti et al. (2009) studied, the evolution
of an isolated elliptical galaxy, where the cooling and heating functions
include photoionization and Compton effects, and restricting to models which
include only radiative or only mechanical feedback (the latter, in the form of
disk winds properties of which were adopted from Proga et al. 2000; Proga &
Kallman 2004).
These recent calculations confirmed that for Eddington ratios above 0.01, both
accretion and radiative outputs are forced by feedback effects to be in burst
mode, so that strong intermittencies are expected at early times, while at low
redshift the explored models are characterized by smooth, very sub-Eddington
mass accretion rates punctuated by rare outbursts. However, the explored
models always fail some observational tests. For the high mechanical
efficiency of $10^{-2.3}$ as adopted by some investigators, it was found that
most of the gas is ejected from the galaxy, the resulting X-ray luminosity is
less than a value typically observed and little super massive black hole
growth occurs. However, models with low enough mechanical efficiency to
accommodate satisfactory black hole growth tend to allow too strong cooling
flows and leave galaxies at z = 0 with E+A spectra more frequently than is
observed. Surprisingly, it was also found that both types of feedback are
required. Radiative heating over the inner few kilo parsecs is needed to
prevent calamitous cooling flows, and mechanical feedback from active galactic
nucleus winds, which affects primarily the inner few hundred parsecs, is
needed to moderate the luminosity and growth of the central black hole. Models
with combined feedback pass more observational tests (Ciotti et al., in
preparation). Thus it emerges from these simulations that to solve the AGN
feedback problem/explain all aspects of the galaxy evolution, more than one
form of feedback is needed.
### Acknowledgments.
This work was supported by NASA through grant HST-AR-11276 from the Space
Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.
DP acknowledges also support from NSF (grant AST-0807491).
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|
arxiv-papers
| 2010-02-25T00:18:53 |
2024-09-04T02:49:08.536672
|
{
"license": "Public Domain",
"authors": "Daniel Proga (UNLV), Ryuichi Kurosawa (UNLV, Cornell Univ.)",
"submitter": "Daniel Proga",
"url": "https://arxiv.org/abs/1002.4667"
}
|
1002.4772
|
Exotic Galilean Symmetry and Non-Commutative Mechanics
Exotic Galilean Symmetry
and Non-Commutative Mechanics\star\star$\star$This paper is a contribution to
the Special Issue “Noncommutative Spaces and Fields”. The full collection is
available at http://www.emis.de/journals/SIGMA/noncommutative.html
Peter A. HORVÁTHY †, Luigi MARTINA ‡ and Peter C. STICHEL §
P.A. Horváthy, L. Martina and P.C. Stichel
† Laboratoire de Mathématiques et de Physique Théorique, Université de Tours,
† Parc de Grandmont, F-37200 Tours, France horvathy@lmpt.univ-tours.fr
‡ Dipartimento di Fisica - Università del Salento and Sezione INFN di Lecce,
‡ via Arnesano, CP. 193, I-73100 Lecce, Italy Luigi.Martina@le.infn.it
§ An der Krebskuhle 21 D-33 619 Bielefeld, Germany peter@physik.uni-
bielefeld.de
Received March 23, 2010, in final form July 19, 2010; Published online July
26, 2010
Some aspects of the “exotic” particle, associated with the two-parameter
central extension of the planar Galilei group are reviewed. A fundamental
property is that it has non-commuting position coordinates. Other and
generalized non-commutative models are also discussed. Minimal as well as
anomalous coupling to an external electromagnetic field is presented.
Supersymmetric extension is also considered. Exotic Galilean symmetry is also
found in Moyal field theory. Similar equations arise for a semiclassical Bloch
electron, used to explain the anomalous/spin/optical Hall effects.
noncommutative spaces; Galilean symmetry; dynamical systems; quantum field
theory
46L55; 37K65; 37L20; 83C65; 81T70
## 1 Introduction: “exotic” Galilean symmetry
A curious property of massive non-relativistic quantum systems is that
Galilean boosts only act up-to phase, so that only the $1$-parameter central
extension of the Galilei group acts unitarily [2]. True representations only
arise for massless particles.
Do further extension parameters exist? It is well-known that central
extensions are associated with non-trivial Lie algebra cohomology [3, 4], and
Bargmann [5] has shown that, in $d\geq 3$ space dimensions, the Galilei group
only admits a $1$-parameter central extension, identified with the physical
mass, $m$. Lévy-Leblond [6] has recognized, however, that, owing to the
Abelian nature of planar rotations, the _planar_ Galilei group admits a second
central extension. The cohomology is two-dimensional, and is parametrized by
two constants, namely by the mass and a second, “exotic”, parameter $\kappa$.
The second extension is highlighted by the non-commutativity of Galilean boost
generators,
$\displaystyle[K_{1},K_{2}]=i\kappa.$
This fact has long been considered, however, a mere mathematical curiosity, as
planar physics itself has been viewed as a toy.
Around 1995 the situation started to change, though, with the construction of
physical models carrying such an “exotic” structure [7, 8, 9]. [9] uses an
acceleration-dependent Lagrangian, while that in [10], is obtained following
Souriau’s method [4].
These models have the distinctive feature that the Poisson bracket of the
planar coordinates does not vanish,
$\displaystyle\\{x_{1},x_{2}\\}=-\frac{\kappa}{m^{2}}\equiv\theta,$ (1.1)
and provide us with two-dimensional examples of _non-commutative mechanics_
[11, 12, 13, 14, 15, 16]111Our conventions are as follows. Symbols with arrows
denote vectors, and those in boldface are tensors. The position vector
${\vec{r}}$ has coordinates $x_{i}$..
What is the physical origin of exotic mechanics? _What is the quantum
mechanical counterpart?_ An answer has been found soon after its introduction:
it is a sort of “non-relativistic shadow” of (fractional) spin [17, 18, 19].
Our particles can be interpreted therefore, as nonrelativistic anyons [20,
21].
The supersymmetric extension of the theory is outlined in Section 9.
All these examples have been taken from one-particle mechanics; exotic
Galilean symmetry can be found, however, also in field theory [22], as
explained in Section 10.
Remarkably, similar structures were considered, independently and around the
same time, in condensed matter physics, namely for the Bloch electron [23]: it
was argued that the semiclassical dynamics should involve a “Berry term”,
which induces “anomalous” velocity similar to the one in the “exotic” model
[10].
These, 3-space dimensional, models are also non-commutative, but the parameter
$\theta$ is now promoted to a vector-valued function of the quasi-momentum:
${{\vec{\Theta}}}={{\vec{\Theta}}}({\vec{k}})$. Exotic Galilean symmetry,
strictly linked to two space dimensions, is lost. However, a rich Poisson
structure and an intricate interplay with external magnetic fields can be
studied. Further developments include the Anomalous [24], the Spin [25] and
the Optical [26, 27, 28, 29] Hall effects.
This review completes and extends those in [30, 31, 32].
## 2 Exotic model, constructed by Souriau’s method
Models associated with a given symmetry group can be conveniently constructed
using Souriau’s method: the classical phase spaces of “elementary systems”
correspond to coadjoint orbits of their symmetry groups [4]. This was
precisely the way followed in [7, 10] to produce an “elementary” classical
system carrying Lévy-Leblond’s “exotic” Galilean symmetry. Translated from
Souriau’s to more standard terms, the model has an “exotic” symplectic form
and a free Hamiltonian,
$\displaystyle\Omega_{0}=dp_{i}\wedge
dx_{i}+\displaystyle\frac{1}{2}\theta\epsilon_{ij}dp_{i}\wedge{}dp_{j},$
$\displaystyle H_{0}=\frac{\vec{p}{\,}^{2}}{2m}.$
The associated free motions follow the usual straight lines, described by the
equations
$m\dot{x}_{i}=p_{i}-m\theta\epsilon_{ij}\dot{p}_{j},\qquad\dot{p}_{i}=0.$
The “exotic” structure only enters the conserved quantities, namely the boost
and the angular momentum,
$\displaystyle
j=\epsilon_{ij}x_{i}p_{j}+\frac{\theta}{2}{\vec{p}}\,{}^{2},\qquad
K_{i}=-mx_{i}+p_{i}t-m\theta\epsilon_{ij}p_{j}.$ (2.1)
The “exotic” structure behaves hence roughly as spin: it contributes to some
conserved quantities, but the new terms are separately conserved. The new
structure does not seem to lead to any new physics.
The situation changes dramatically, though, if the particle is coupled to a
gauge field. Applying Souriau’s coupling prescription [4] yields indeed
$\displaystyle\Omega=\Omega_{0}+eB\,dq_{1}\wedge dq_{2},\qquad H=H_{0}+eV,$
(2.2)
where $B$ is the magnetic field assumed to be perpendicular to the plane, and
$V$ is the electric potential. For simplicity, both $B$ and $V$ are assumed to
be time-independent.
The associated Poisson bracket then automatically satisfies the Jacobi
identity. The resulting equations of motion read
$\displaystyle
m^{*}\dot{x}_{i}=p_{i}-em\theta\,\epsilon_{ij}E_{j},\qquad\dot{p}_{i}=eE_{i}+eB\,\epsilon_{ij}\dot{x}_{j},$
(2.3)
where the parameter $\theta=k/m^{2}$ measures the non-commutativity of $x_{1}$
and $x_{2}$, and we have introduced the effective mass
$\displaystyle m^{*}=m(1-e\theta B).$ (2.4)
The novel features, crucial for physical applications, are twofold: Firstly,
the relation between velocity and momentum contains an “anomalous velocity
term” $-em\theta\epsilon_{ij}E_{j}$, so that $\dot{x}_{i}$ and $p_{i}$ are not
in general parallel. The second one is the interplay between the exotic
structure and the magnetic field, yielding the effective mass $m^{*}$ in
(2.4).
Equations (2.3) do not derive from a configuration-space Lagrangian (but see
Section 3). The $1$-order “phase” (in fact “evolution space” [4]) formalism
works, however, so that the equations of motion (2.3) come from varying the
action defined by integrating the “Cartan” 1-form [4],
$\displaystyle\lambda=({p_{i}}-{A_{i}})d{x_{i}}-\frac{\vec{p}{\,}^{2}}{2m}dt+\frac{\theta}{2}\epsilon_{ij}{p_{i}}d{p_{j}}$
along the lifted world-line $\widetilde{\gamma}$ in evolution space
$T^{*}{{\mathbb{R}}}^{2}\times{{\mathbb{R}}}$. The exterior derivative,
$\sigma=d\lambda$, of the Cartan form $\lambda$ provides us with a closed
“Lagrange–Souriau” 2-form, which, however, cannot be separated canonically
into a “symplectic” and a “Hamiltonian part” [4]. Thus more general procedures
have to be adopted to build such a system and clarify their Hamiltonian
structure. These aspects will be discussed in detail in the following Sections
3, 4 and 5. Here we limit ourselves to notice that, in fact, when $m^{*}\neq
0$, (2.3) is a Hamiltonian system,
$\displaystyle\dot{\xi}=\\{H,\xi^{\alpha}\\},\qquad\xi=(x_{j},p_{i}),$
with Poisson brackets
$\displaystyle\\{x_{1},x_{2}\\}=\frac{m}{m^{*}}\theta,\qquad\\{x_{i},p_{j}\\}=\frac{m}{m^{*}}\delta_{ij},\qquad\\{p_{1},p_{2}\\}=\frac{m}{m^{*}}eB.$
(2.5)
A remarkable property is that for vanishing effective mass $m^{*}=0$, i.e.,
when the magnetic field takes the critical value
$\displaystyle B=\frac{1}{e\theta},$
the system becomes singular. Then “Faddeev–Jackiw” (alias symplectic)
reduction yields an essentially two-dimensional, simple system, reminiscent of
“Chern–Simons mechanics” [33]. The symplectic plane plays, simultaneously, the
role of both configuration and phase space. The only motions are those which
follow a generalized Hall law.
Quantization of the reduced system yields, moreover, the “Laughlin” wave
functions [34], which are the ground states in the Fractional Quantum Hall
Effect (FQHE).
The relations (2.5) diverge as $m^{*}\to 0$, but after reduction we get, cf.
(1.1),
$\displaystyle\\{x_{1},x_{2}\\}=\frac{1}{eB}=\theta.$
The coordinates are, hence, non-commuting, and their commutator is determined
by the magnetic field, fine-tuned to the parameter $\theta$.
Relation to another non-commutative mechanics. The exotic relations (2.5) are
similar to those proposed (later) in [11],
$\displaystyle\\{x_{i},x_{j}\\}=\theta\epsilon_{ij},\qquad\\{x_{i},p_{j}\\}=\delta_{ij},\qquad\\{p_{1},p_{2}\\}=eB,$
(2.6)
which look indeed simpler. Using the standard Hamiltonian
$H=\frac{p^{2}}{2m}+eV(x)$, the associated equations of motion read
$\displaystyle mx_{i}^{\prime}=p_{i}-em\theta\epsilon_{ij}E_{j},\qquad
p_{i}^{\prime}=eB\epsilon_{ij}\frac{p_{j}}{m}+eE_{i},$ (2.7)
where we noted “time” by $T$ and $(\cdot)^{\prime}=\frac{d}{dT}$.
A short calculation reveals, however, that
$\displaystyle\big{\\{}x_{i},\\{p_{1},p_{2}\\}\big{\\}}_{\rm
cycl}=e\theta\,\epsilon_{ij}{\partial}_{j}B,$
so that _the Jacobi identity is only satisfied if $B$ is a constant_. In other
words, the system (2.6) is _only consistent_ for a constant magnetic field –
which is an _unphysical condition_ in general222For another role of the Jacobi
identity in non-commutative mechanics, see [35]..
The model (2.6) has another strange feature. Let us indeed assume that the
magnetic field is radially symmetric, $B=B(r)$. One would then expect to have
conserved angular momentum. For constant $B$, applying Noether’s theorem to an
infinitesimal rotation $\delta\xi_{i}=\epsilon_{ij}\xi_{j}$ yields indeed
$\delta\xi_{i}=-\\{j^{\rm NP},\xi_{i}\\},$ with
$\displaystyle j^{\rm NP}=\frac{1}{1-e\theta
B}\underbrace{\left({\vec{x}}\times\vec{p}+\frac{\theta}{2}\vec{p}\,{}^{2}+\frac{eB}{2}{\vec{x}}\,{}^{2}\right)}_{j}.$
(2.8)
This differs from the standard expression by the pre-factor $(1-e\theta
B)^{-1}.$ But what is even worse is that, for a radial but non constant
magnetic field, (2.8) is _not conserved_ :
$\frac{dj^{\rm NP}}{dT}=\frac{e\theta j}{(1-e\theta
B)^{2}}\partial_{i}Bx_{i}^{\prime},$
while $j$ in (2.1) is still conserved as it should.
Can the theory defined by (2.6) be extended to an arbitrary $B$?
Let us first assume that $B=$ const. s.t. $m^{*}\neq 0$, and let us redefine
the time333This was suggested to us by G. Marmo (private communication)., as
$\displaystyle T\to t=(1-e\theta B)T\quad\Rightarrow\quad\frac{\
d}{dT}=(1-e\theta B)\frac{\ d}{dt}.$ (2.9)
Then equations (2.7) are carried into the exotic equations, (2.3). When
$B={\rm const}$. s.t. $m^{*}\neq 0$, the two theories are therefore
equivalent.
Remarkably, the _time redefinition ( 2.9)_ actually _extends the previous
theory_ , since it carries it into the “exotic model”, for which _the Jacobi
identity holds for any, not necessarily constant_ $B$. Thus, the
transformation (2.9) (which is singular for $eB\theta=1$), removes the
unphysical restriction to constant magnetic fields: (2.9) regularizes the
system (2.6).
## 3 Acceleration-dependent Lagrangian in configuration space
An independent and rather different approach was followed in [9].
We start again with a particle characterized by the two central charges
$m\qquad\hbox{and}\qquad\kappa=-m^{2}\theta$
of the exotic Galilei group. These charges appear in the following Lie-
brackets (represented by Poisson-brackets (PBs)) between the translation
generators $P_{i}$ and the boost generators $K_{i}$
$\displaystyle\\{P_{i},K_{j}\\}=m\delta_{ij},\qquad\\{K_{i},K_{j}\\}=-m^{2}\theta\epsilon_{ij}.$
(3.1)
In order to find a configuration space Lagrangian whose Noether charges for
boosts satisfy (3.1) we must add the second time derivative of the coordinates
$\ddot{x}_{i}$ to the usual variables $x_{i}$ and $\dot{x}_{i}$. As shown in
[9] the most general one-particle Lagrangian, which is at most linearly
dependent on $\ddot{x}_{i}$, leading to the Euler–Lagrange equations of motion
which are covariant w.r.t. the planar Galilei group, is given, up to gauge
transformations, by
$\displaystyle{\cal
L}=\frac{m}{2}\dot{x}_{i}^{2}+\frac{m^{2}\theta}{2}\epsilon_{ij}\dot{x}_{i}\ddot{x}_{j}.$
(3.2)
Introducing the Lagrange multipliers $p_{i}$ and adding
$p_{i}(\dot{x}_{i}-y_{i})$ to (3.2) yields
$\displaystyle{\cal
L}=p_{i}\dot{x}_{i}+\frac{m^{2}\theta}{2}\epsilon_{ij}y_{i}\dot{y}_{j}-H({{\vec{y}}},{{\vec{p}}})$
(3.3)
with
$\displaystyle H({{\vec{y}}},{{\vec{p}}})=y_{i}p_{i}-\frac{m}{2}y_{i}^{2}.$
(3.3) describes a constrained system, because we have
$\displaystyle\frac{\partial{\cal
L}}{\partial\dot{y}_{i}}=-\frac{m^{2}\theta}{2}\epsilon_{ij}y_{j}.$
Therefore the PBs, obtained by means of the Faddeev–Jackiw procedure, take a
non-standard form
$\displaystyle\\{x_{i},p_{j}\\}=\delta_{ij},\qquad\\{y_{i},y_{j}\\}=-\frac{1}{m^{2}\theta}\epsilon_{ij}.$
(3.4)
All other PBs vanish.
For the conserved boost generator we obtain
$\displaystyle K_{i}=-mx_{i}+p_{i}t-m^{2}\theta\epsilon_{ij}y_{j}$
and therefore, due to (3.4), the PB resp. commutator of two boosts is
nonvanishing
$\displaystyle\\{K_{i},K_{j}\\}=-m^{2}\theta\epsilon_{ij}.$ (3.5)
The Lagrangian (3.3) shows that the phase space is 6-dimensional. In order to
split off two internal degrees of freedom, we have to look for a Galilean
invariant decomposition of the 6-dim phase space into two dynamically
independent parts: a 4-dim external and a 2-dim internal part. This
decomposition is achieved by the transformation [9, 20]
$({\vec{x}},{\vec{p}},{\vec{y}})\to({\vec{X}},{\vec{p}},{\vec{Q}})$ with
$\displaystyle
y_{i}=\frac{p_{i}}{m}+\frac{Q_{i}}{m\theta}\qquad\hbox{and}\qquad
x_{i}=X_{i}-\epsilon_{ij}Q_{j},$ (3.6)
leading to the following decomposition of the Lagrangian (3.3)
${\cal L}={\cal L}_{\rm ext}+{\cal L}_{\rm int}$
with
$\displaystyle{\cal L}_{\rm
ext}=p_{i}\dot{X}_{i}+\frac{\theta}{2}\epsilon_{ij}p_{i}\dot{p}_{j}-\frac{p_{i}^{2}}{2m}\qquad\hbox{and}\qquad{\cal
L}_{\rm
int}=\frac{1}{2m\theta^{2}}Q^{2}_{i}+\frac{1}{2\theta}\epsilon_{ij}Q_{i}\dot{Q}_{j}.$
(3.7)
From (3.6) and the PBs (3.4) it now follows that the new coordinates $X_{i}$
are noncommutative
$\displaystyle\\{X_{i},X_{j}\\}=\theta\epsilon_{ij}.$ (3.8)
The remaining nonvanishing PBs are
$\displaystyle\\{X_{i},p_{j}\\}=\delta_{ij},\qquad\\{Q_{i},Q_{j}\\}=-\theta\epsilon_{ij}.$
Conclusion. The particle Lagrangian (3.2) containing $\ddot{x}_{i}$ leads to a
nonvanishing commutator of two boosts. But in order to obtain noncommutative
coordinates we are forced to decompose the 6-dim phase space in a Galilean
invariant manner into two dynamically independent 4-dim external and 2-dim
internal phase spaces.
Relation of the DH and LSZ models. The relation of the “minimal” and the
configuration-space models of DH [10] and of L.S.Z. [9] respectively, has been
studied in [20]. Introducing the coordinates $X_{i}$, $Q_{i}$, and $p_{i}$ on
$6$-dimensional phase space according to (3.6) allows us to present the
symplectic structure and the Hamiltonian associated with (3.2) as
$\displaystyle\Omega=dp_{i}\wedge
dX_{i}+\frac{\theta}{2}\varepsilon_{ij}dp_{i}\wedge
dp_{j}+\frac{1}{2\theta}\varepsilon_{ij}dQ_{i}\wedge dQ_{j},$ $\displaystyle
H=\frac{{\vec{p}}\,{}^{2}}{2m}-\frac{1}{2m\theta^{2}}{\vec{Q}}^{2}.$
Thus, the model of L.S.Z. is decomposed into the DH theory, augmented with a
two-dimensional internal space, and a negative zero point of the Hamiltonian.
Note that the “external” and “internal” phase spaces are “almost” independent:
the only effect of internal motion is indeed through the length of the
internal vector, $|{\vec{Q}}|$.
Generalization of (3.2). If we add to (3.2) a term $f(\ddot{x}_{i}^{2})$, the
obtained Lagrangian is the most general one involving, in a Galilean quasi-
invariant manner, the variables $x_{i}$, $\dot{x}_{i}$ and $\ddot{x}_{i}$.
Then one can show
i) the PB of the two boosts (3.5) will not change,
ii) the new 8-dim phase space may be decomposed again in a Galilean invariant
manner into two dynamically independent 4-dim parts, an external and an
internal one.
Commutative – versus noncommutative plane.
The subalgebra of the Galilean algebra containing only translations and boosts
is given in the cases of, respectively, their one- or two-fold central
extensions by
one-fold centrally extended two-fold centrally extended
$\displaystyle\\{p_{i},K_{j}\\}=m\delta_{ij},\qquad$
$\displaystyle\\{p_{i}^{\prime},K_{j}^{\prime}\\}=m\delta_{ij},$
$\displaystyle\\{p_{i},p_{j}\\}=0,$
$\displaystyle\\{p_{i}^{\prime},p^{\prime}_{j}\\}=0,$
$\displaystyle\\{K_{i},K_{j}\\}=0,$
$\displaystyle\\{K_{i}^{\prime},K_{j}^{\prime}\\}=-m^{2}\theta\epsilon_{ij}.$
Obviously both are related by the transformations
$\displaystyle K^{\prime}_{i}=K_{i}-\frac{m\theta}{2}\epsilon_{ij}p_{j},\qquad
p_{i}^{\prime}=p_{i}.$
To this corresponds the following point transformation between noncommutative
coordinates $X_{i}$ and commutative ones $q_{i}$
$\displaystyle X_{i}=q_{i}-\frac{\theta}{2}\epsilon_{ij}p_{j}$
as can be read off immediately from the form of ${\cal L}_{\rm ext}$ in (3.7).
Now the question arises: What to use in physics, the commutative or the non-
commutative plane?
Answer. For free particles both possibilities are equivalent. But in the case
of a nontrivial interaction one has to use the commutative (noncommutative)
plane, if a local potential or gauge interaction is given in terms of $q_{i}$
$(X_{i})$.
It is worth mentioning that the acceleration-dependent model presented in this
Section can be related to radiation damping [36].
## 4 General form of noncommutative mechanics
Up to now noncommutativity has been described by a constant $\theta$ in the PB
(3.8). But it is possible to get $\theta$ as a function of ${{\vec{X}}}$ and
${{\vec{p}}}$ if one considers external Lagrangians more general than (3.7).
To do this consider a very general class of Lagrangians given by
$\displaystyle{\cal
L}=p_{i}\dot{X}_{i}+\tilde{A}_{i}({{\vec{X}}},{{\vec{p}}})\dot{p}_{i}-H({{\vec{p}}},{{\vec{X}}})$
(4.1)
leading to the PBs
$\displaystyle\\{X_{i},X_{j}\\}\sim\epsilon_{ij}\tilde{B},\qquad\tilde{B}=\epsilon_{k\ell}\partial_{p_{k}}\tilde{A}_{\ell}({{\vec{X}}},{{\vec{p}}})$
with
$\\{P_{i},P_{j}\\}=0.$
We dispense with the reproduction of the more complicated form of the PBs for
the phase space coordinates $\left(X_{i},p_{j}\right)$. Again by the point
transformation
$\displaystyle X_{i}\to q_{i}=X_{i}-\tilde{A}_{i}({{\vec{X}}},{{\vec{p}}})$
we obtain commuting coordinates $q_{i}$ as follows from
$p_{i}\dot{X}_{i}+\tilde{A}_{i}\dot{p}_{i}=p_{i}\dot{q}_{i}+\frac{d}{dt}(\tilde{A}_{i}p_{i}).$
Examples:
i)
$\displaystyle\tilde{A}_{i}=f(p^{2})({{\vec{X}}}\cdot{{\vec{p}}}\,)p_{i}$
(4.2)
leading to the PBs of the phase space variables
$\displaystyle\\{X_{i},X_{j}\\}=\frac{f(p^{2})}{1-p^{2}f(p^{2})}\epsilon_{ij}L,\qquad
L=\epsilon_{k\ell}X_{k}p_{\ell},$ (4.3)
$\displaystyle\\{X_{i},p_{j}\\}=\delta_{ij}+\frac{f(p^{2})}{1-p^{2}f(p^{2})}p_{i}p_{j}.$
(4.4)
A particular example is given by
$f(p^{2})=\frac{\theta}{1+p^{2}\theta}\qquad\hbox{ and
therefore}\qquad\frac{f}{1-p^{2}f}=\theta.$
This gives exactly Snyder’s NC-algebra, presented in 1947 [37].
Another case, defined by
$\displaystyle f(p^{2})=\frac{2}{p^{2}},$ (4.5)
can be related to a deformed Galilei algebra (to be discussed in the next
section).
ii)
$\displaystyle\tilde{A}_{i}=\tilde{A}_{i}({{\vec{p}}})$
leading to the PBs
$\displaystyle\\{X_{i},X_{j}\\}=\epsilon_{ij}\tilde{B}({\vec{p}}),\qquad\\{X_{i},p_{j}\\}=\delta_{ij}.$
$\tilde{B}$ is the Berry curvature for the semiclassical dynamics of electrons
in condensed matter to be discussed in Section 11.
We may generalize (4.1) to the most general 1st-order Lagrangian
$\displaystyle{\cal
L}=(p_{i}+A_{i}({{\vec{X}}},{{\vec{p}}}))\dot{X}_{i}+\tilde{A}_{i}({{\vec{X}}},{{\vec{p}}})\dot{p}_{i}-H({\vec{p}},{{\vec{X}}}).$
Here $A_{i}({{\vec{X}}})$ describes standard electromagnetic interaction (cp.
Section 6, and Section 11 for the 3 dimensional case, respectively). A
particular case of a ${{\vec{p}}}$-dependent $A_{i}$ has been considered in
[38].
## 5 Lagrangian realization of the $\boldsymbol{\tilde{k}}$-deformed Galilei
algebra
as a symmetry algebra
In 1991 Lukierski, Nowicki, Ruegg and Tolstoy invented the $k$-deformed
Poincaré algebra [39], which later found applications, e.g., in Quantum
Gravity [40]. By rescaling the Poincaré generators and the deformation
parameter $k$, the corresponding nonrelativisitic limit, the
$\tilde{k}$-deformed Galilei algebra, has been derived by Giller et al. [41]
and, in a different basis, by Azcarraga et al. [42]. In this Section we will
describe a Lagrangian realization of the latter.
Again we look at the classical Lagrangian (4.1) specified by (4.2) and (4.5)
together with the following choice of the Hamiltonian
$\displaystyle H=\tilde{k}\ln(p^{2}/2).$ (5.1)
According to (4.3), (4.4) we obtain the PBs
$\displaystyle\\{X_{i},X_{j}\\}=-\frac{2}{p^{2}}\epsilon_{ij}L\qquad\mbox{and}\qquad\\{X_{i},p_{j}\\}=\delta_{ij}-\frac{2}{p^{2}}p_{i}p_{j}$
which lead, together with the Hamiltonian (5.1), to the equations of motion
$\displaystyle\dot{p}_{i}=0\qquad\mbox{and}\qquad\dot{X}_{i}=-\frac{2\tilde{k}}{p^{2}}p_{i}.$
Then, we may define the “pseudo-boosts” $K_{i}$
$\displaystyle K_{i}=p_{i}t+\frac{p^{2}}{2\tilde{k}}X_{i}$
which are conserved. They satisfy, together with $p_{i}$ and $H$, the PB-
algebra
$\displaystyle\\{K_{i},p_{j}\\}=\frac{\delta_{ij}}{2\tilde{k}}p^{2}-\frac{p_{i}p_{j}}{\tilde{k}},\qquad\\{K_{i},H\\}=-p_{i},\qquad\\{K_{i},K_{j}\\}=0.$
(5.2)
Together with the standard algebra of translations (represented by $p_{i}$ and
$H$) and rotations (represented by $L$) the relations (5.2), build the
$\tilde{k}$-deformed Galilei algebra derived in [42].
The limit $\tilde{k}\to\infty$ leads to a divergent Hamiltonian (5.1).
Therefore, the $\tilde{k}$-deformation does not have a standard “no-
deformation limit”.
## 6 Physical origin of the exotic structure
A free relativistic “elementary” particle in the plane corresponds to a
unitary representation of the planar Lorentz group $O(2,1)$ [18]. These
representations are in turn associated with the coadjoint orbits of $SO(2,1)$,
endowed with their canonical symplectic structures, interpreted by Souriau as
classical phase spaces [4]. Applied to the planar Lorentz group, the procedure
yields the _relativistic_ model [18, 19]
$\displaystyle\Omega_{\rm rel}=dp_{\alpha}\wedge
dx^{\alpha}+\frac{s}{2}\epsilon^{\alpha\beta\gamma}\frac{p_{\alpha}dp_{\beta}\wedge
dp_{\gamma}}{(p^{2})^{3/2}},$ $\displaystyle H_{\rm
rel}=\frac{1}{2m}\big{(}p^{2}-m^{2}c^{2}\big{)}.$
The $p$-dependent contribution looks like a “magnetic monopole in momentum
space” (cf. (11.10) below).
As pointed out by Jackiw and Nair [17], the free exotic model can be recovered
considering a tricky non-relativistic limit , namely
$\displaystyle s/c^{2}\to\kappa=m^{2}\theta.$
The two-form $\Omega_{\rm rel}\big{|}_{H_{\rm rel}=0}$ goes indeed over into
the exotic symplectic form. Intuitively, the exotic structure can be viewed as
a _“non-relativistic shadow” of relativistic spin_.
The exotic Galilei group can itself be derived from the planar Poincaré group
by “Jackiw–Nair” contraction [17]. One starts with the planar Lorentz
generators,
$\displaystyle\\{J^{\alpha},J^{\beta}\\}=\epsilon^{\alpha\beta\gamma}J_{\gamma}.$
For the classical system
$\displaystyle
J_{\mu}=\epsilon_{\mu\nu\rho}x^{\nu}p^{\rho}+s\frac{p_{\mu}}{\sqrt{p^{2}}}.$
A non-relativistic boost is the “JN” limit of a suitable Lorentz
transformation,
$\displaystyle\frac{1}{c}\epsilon_{ij}J^{j}\to
mx_{i}-p_{i}t+m\theta\epsilon_{ij}p_{j}=-K_{i},$
and the exotic relation is recovered,
$\displaystyle\\{K_{1},K_{2}\\}=J_{0}/c^{2}\to\frac{s}{c^{2}}=\kappa.$
The angular momentum is in turn
$\displaystyle
J_{0}=\vec{x}\times\vec{p}+s+\frac{s}{m^{2}c^{2}}\vec{p}\,{}^{2}\to\vec{x}\times\vec{p}+\frac{\theta}{2}\vec{p}\,{}^{2}=j.$
whereas the divergent term $s=\kappa c^{2}$ has to be removed by hand.
It is worth mentioning that the “Jackiw–Nair limit” of a relativistic particle
with torsion [43] provides us with the L.S.Z. model [20], and a similar
procedure yields the so-called “Newton–Hooke” system [44]. Applied to the
infinite-component Majorana-type anyon equations [18, 19] yields, furthermore,
a first-order infinite-component “Lévy-Leblond type” system with exotic
Galilean symmetry [21].
About anyons constructed from orbits, see also [45].
## 7 Anomalous coupling of anyons
It has been suggested [46] that a classical, _relativistic_ anyon in an
electromagnetic field should be described by the equations
$\displaystyle m\frac{dx^{\alpha}}{d\tau}=p^{\alpha}\qquad$
$\displaystyle\hbox{(velocity-momentum)},$
$\displaystyle\frac{dp^{\alpha}}{d\tau}=\frac{e}{m}F^{\alpha\beta}p_{\beta}\qquad$
$\displaystyle\hbox{(Lorentz equation)},$ (7.1)
$\alpha,\beta,\ldots=0,1,2$ and $\tau$ denotes the proper time. These
equations are Hamiltonian, with symplectic form and Hamilton’s function
$\displaystyle\Omega=\Omega_{\rm
rel}+\frac{1}{2}eF_{\alpha\beta}dx^{\alpha}\wedge dx^{\beta},$ $\displaystyle
H=H_{\rm
rel}+\frac{es}{2m\sqrt{p^{2}}}\epsilon_{\alpha\beta\gamma}F^{\alpha\beta}p^{\gamma},$
(7.2)
respectively. Let us observe that the second, non-minimal term in the
Hamiltonian is _dictated by the required form of the velocity relation_ in
(7.1).
The model of [46] has gyromagnetic ratio $g=2$, and some theoreticians have
long believed [46, 47] that this is indeed the “correct” $g$ value of anyons.
Experimental evidence shows, however, that in the Fractional Quantum Hall
Effect, for example, the measured value of $g$ is approximately _zero_ [48].
Is it possible to construct an “anomalous” model with $g\neq 2$? The answer is
affirmative [49], as we now explain.
Planar spin has to satisfy the relation $S_{\alpha\beta}p^{\beta}=0$. The spin
tensor has, therefore, the form
$\displaystyle
S_{\alpha\beta}=\frac{s}{\sqrt{p^{2}}}\epsilon_{\alpha\beta\gamma}p^{\gamma}.$
Introducing the shorthand $-F_{\alpha\beta}S^{\alpha\beta}=F\cdot S$, the
Hamiltonian (7.2) is presented as
$\displaystyle H^{\rm
CNP}=\frac{1}{2m}\big{(}p^{2}-M^{2}c^{2}\big{)}\qquad\hbox{where}\qquad
M^{2}=m^{2}+\frac{e}{c^{2}}F\cdot S.$ (7.3)
Let us observe that the “mass” $M$ depends here on spin-field coupling. Our
clue for generalizing this model has been the formula put forward by Duval
[50, 51]: let us posit, instead of (7.3), the mass formula
$\displaystyle M^{2}=m^{2}+\frac{g}{2}\frac{e}{c^{2}}F\cdot S,$
where $g$ is an arbitrary real constant. Then consistent equations of motion
are obtained for any $g$, namely
$\displaystyle
D\frac{dx^{\alpha}}{d\tau}=G\frac{p^{\alpha}}{M}+(g-2)\frac{es}{4M^{2}}\epsilon^{\alpha\beta\gamma}F_{\beta\gamma},$
(7.4)
$\displaystyle\frac{dp^{\alpha}}{d\tau}=\frac{e}{m}F^{\alpha\beta}p_{\beta},$
(7.5)
where the coefficients denote the complicated, field-dependent expressions
$\displaystyle D=1+\frac{eF\cdot S}{2M^{2}c^{2}},\qquad
G=1+\frac{g}{2}\frac{eF\cdot S}{2M^{2}c^{2}}.$
_Choosing_ $g=2$, the generalized model plainly reduces to equation (7.1),
proposed in [46]. We stress, however, that _no physical principle requires
that the second, “anomalous” term should vanish_ in (7.4). $g=2$ is _not_ a
physical necessity therefore: a perfectly consistent model is obtained for any
$g$, as it has been advocated a long time ago [50, 51].
Non-relativistic anyon with anomalous coupling. We can now consider the
“Jackiw–Nair” non-relativistic limit of the above relativistic model. This
provides us, for any $g$, with the Lorentz equation (7.5), supplemented with
$\displaystyle(M_{g}D)\dot{x}_{i}=Gp_{i}-\left(1-\frac{g}{2}\right)eM_{g}\theta\epsilon_{ij}E_{j},$
where
$\displaystyle M_{g}=m(\sqrt{1-g\theta eB}),\qquad D=\big{(}1-(g+1)\theta
eB\big{)},\qquad G=\big{(}1-(3g/2))\theta eB\big{)}.$
$\bullet$ It is a most important fact that, for any $g\neq 2$, the only
consistent motions follow a generalized Hall law, whenever the field takes
either of the critical values
$\displaystyle
B=\frac{1}{1+g}\frac{1}{e\theta}\qquad\hbox{or}\qquad\frac{2}{3g}\frac{1}{e\theta}.$
One can indeed show that, for any $g\neq 2$, the models can be transformed
into each other by a suitable redefinition. For $g=0$ the equations become
identically satisfied. See [49] for details.
$\bullet$ In particular, for $g=0$ the minimal exotic model of [10] is
recovered. The latter is, hence, not the NR limit of the model of [46] (7.1)
[which has $g=2$, as said]. _The experimental evidence _[_48] is, hence, a
strong argument in favor of the minimal model of _[_10]_.
$\bullet$ $g=2$ is the only case when the velocity and the momentum are
parallel. This is, however, not required by any first principle.
Having an anomalous velocity relation seems to be unusual in high-energy
physics; it is, however, a well accepted requirement in condensed matter
physics, as explained in Section 11.
Let us mention that relativistic anyons can be described, at the field
theoretical level, by infinite-component fields of the Majorana–Dirac type
[19]. Coupling them to an external gauge field is a major unsolved problem.
Partial results can be obtained in the non-relativistic case [52].
## 8 Two ways of introducing electromagnetic interactions
In this section we will show that Souriau’s coupling prescription (2.2) is not
the only possibility to introduce electromagnetic (e.m.) interaction into the
Lagrangian ${\cal L}_{\rm ext}$ (3.7).
In the commutative case we have the principle of minimal e.m. coupling
$\displaystyle
p_{i}\dot{X}_{i}-\frac{p^{2}_{i}}{2m}\to(p_{i}+eA_{i}({{\vec{X}}},t))\dot{X}_{i}-\frac{p_{i}^{2}}{2m}+eA_{0}({{\vec{X}}},t),$
called the minimal additon rule, which is equivalent, due to the point
transformation $p_{i}\to p_{i}-eA_{i}$, to the minimal substitution rule [53],
$\displaystyle p_{i}\dot{X}_{i}-\frac{p_{i}^{2}}{2m}\to
p_{i}\dot{X}_{i}-\frac{(p_{i}-eA_{i})^{2}}{2m}+eA_{0}({{\vec{X}}},t).$
In the noncommutative case the equivalence of minimal addition and minimal
substitution rule is not valid. Therefore we have to consider two different
ways of introducing the minimal e.m. coupling:
Minimal addition (Duval–Horvathy [10], called DH-model)
$\displaystyle{\cal L}\to{\cal L}_{\rm e.m.}={\cal
L}+e(A_{i}\dot{X}_{i}+A_{0}),$ (8.1)
which, as usual, is quasi-invariant w.r.t. standard gauge transformations
$\displaystyle A_{\mu}({{\vec{X}}},t)\to
A_{\mu}({{\vec{X}}},t)+\partial_{\mu}\Lambda({{\vec{X}}},t).$
Obviously the minimal addition rule (8.1) is equivalent to Souriau’s
prescription (2.2).
Minimal substitution (Lukierski–Stichel–Zakrzewski [53], called L.S.Z.
model)444 In this model the gauge fields carry a “hat” in order to distinguish
them from the corresponding quantities in the DH-model.
$\displaystyle H=\frac{p^{2}_{i}}{2m}\to H_{\rm
e.m.}=\frac{(p_{i}-e\hat{A}_{i})^{2}}{2m}-e\hat{A}_{0}.$
The corresponding Lagrangian is quasi-invariant w.r.t. generalized gauge
transformations, given in infinitesimal form by
$\displaystyle\delta\hat{A}_{\mu}({{\vec{X}}},t)=\hat{A}_{\mu}^{\prime}({{\vec{X}}}+\delta{{\vec{X}}},t)-\hat{A}_{\mu}({{\vec{X}}},t)=\partial_{\mu}\Lambda({{\vec{X}}},t),$
with
$\displaystyle\delta X_{i}=-e\theta\epsilon_{ij}\partial_{j}\Lambda$ (8.2)
and supplemented by
$\displaystyle\delta p_{i}=e\partial_{i}\Lambda.$
Note that the coordinate transformations (8.2) are area preserving.
It turns out that both models are related to each other by a noncanonical
transformation of phase space variables supplemented by a classical
Seiberg–Witten transformation of the corresponding gauge potentials:
If we denote the phase space variables and potentials for
1. –
the DH-model by $(\vec{\eta},\vec{{\cal P}},A_{\mu})$,
2. –
the L.S.Z.-model by $({{\vec{X}}},{{\vec{p}}},\hat{A}_{\mu})$,
then we find the relations
$\displaystyle\eta_{i}({{\vec{X}}},t)=X_{i}+e\theta\epsilon_{ij}\hat{A}_{j}({{\vec{X}}},t),$
$\displaystyle{\cal P}_{i}=p_{i}-e\hat{A}_{i}({{\vec{X}}},t)$
with the corresponding field strengths related by
$\displaystyle\hat{F}_{\mu\nu}({{\vec{X}}},t)=\frac{F_{\mu\nu}(\vec{\eta},t)}{1-e\theta
B(\vec{\eta},t)}.$ (8.3)
The Seiberg–Witten transformation between the resp. gauge fields is more
involved, and will not be reproduced here (for details cp. [53]).
These results lead to an interesting by-product: Consider the PBs of
coordinates in both models, given by
$\displaystyle\\{\eta_{i},\eta_{j}\\}=\frac{\theta\ \epsilon_{ij}}{1-e\theta
B(\vec{\eta},t)}\qquad\mbox{and}\qquad\\{X_{i},X_{j}\\}=\theta\,\epsilon_{ij}.$
(8.4)
Then the foregoing results implicitly give the coordinate transformation
between a model with a constant noncommutativity parameter $m^{2}\theta$ and
one with arbitrary coordinate-dependent noncommutativity function
$m^{2}\theta({{\vec{X}}},t)$ (this result has been rediscovered in [54]).
Now the question arises, which of both models has to be used for physical
applications? Let us look at one example, the Quantum Hall effect. As already
shown in Section 2 in the case of the DH-model [10] the Hall law,
$\displaystyle\dot{X}_{i}=\epsilon_{ij}\frac{E_{j}}{B},$ (8.5)
is valid at the critical magnetic field
$\displaystyle B_{\rm crit}=\big{(}e\theta\big{)}^{-1}.$
Then it follows from the field transformation law (8.3) that, for the
L.S.Z.-model, the Hall law is valid in the limit of large e.m. fields. In
order to see this in more detail we have to consider the equations of motion
for the L.S.Z.-model formulated in terms of the gauge-invariant phase space
variables $\vec{\eta}$ and $\vec{\cal P}$. For that, we use the equations of
motion (2.3) for the DH model written in terms of $\eta_{i}$ and ${\cal
P}_{i}$, transform the e.m. fields according to (8.3) and we obtain $(e=1$,
$m=1)$
$\displaystyle\dot{\eta}_{i}=(1+\theta\hat{B}){\cal
P}_{i}-\theta\epsilon_{ij}\hat{E}_{j},$ $\displaystyle\dot{{\cal
P}}_{i}=\hat{B}\epsilon_{ij}{\cal P}_{j}+\hat{E}_{i}.$ (8.6)
For the particular case of homogeneous e.m. fields we obtain finally
$\displaystyle\ddot{\eta}_{i}=\hat{B}\epsilon_{ij}\dot{\eta}_{j}+\hat{E}_{i}$
(8.7)
leading to the Hall law (8.5) in the high field limit.
Note that (8.7) has the same functional form as in the commutative case.
Another point of view is presented in [55].
## 9 Supersymmetry
In the following, we supersymmetrize the e.m. coupling models treated in the
last section. To do that we follow the treatment in Section 3 of [56]. For
that, we consider standard $N=2$ SUSY characterized by
$\displaystyle H=\frac{i}{2}\\{Q,\bar{Q}\\}$ (9.1)
and
$\displaystyle\\{Q,Q\\}=\\{\bar{Q},\bar{Q}\\}=0.$ (9.2)
In order to construct the supercharge $Q$, satisfying (9.1), we start with the
common structure of the bosonic Hamiltonian $H_{b}$ for both models $(e=1$,
$m=1)$
$\displaystyle H_{b}=\frac{1}{2}\big{(}{\cal
P}^{2}_{i}+W_{i}^{2}({{\vec{X}}})\big{)}$ (9.3)
with
${\cal P}_{i}=p_{i}\qquad\mbox{for the DH-model}$
and
${\cal P}_{i}=p_{i}-A_{i}\qquad\mbox{for the L.S.Z.-model}.$
Note that, in accordance with the quantized form of (9.1), the potential term
in (9.3) is chosen to be positive
$\displaystyle A_{0}=-\frac{1}{2}W^{2}_{i}.$ (9.4)
In order to add to (9.3) its fermionic superpartner, we supplement the bosonic
phase space variables with fermionic coordinates $\psi_{i}(\bar{\psi}_{i})$
satisfying canonical PBs
$\displaystyle\\{\psi_{i},\bar{\psi}_{j}\\}=-i\delta_{ij}.$
Now we assume
$\displaystyle Q=i({\cal P}_{i}+iW_{i})\psi_{i}$
such that (9.3) is valid. But now the relations (9.2) are fulfilled only if
the following two conditions are satisfied:
$\displaystyle\\{{\cal P}_{i},{\cal P}_{j}\\}=\\{W_{i},W_{j}\\}$ (9.5)
and
$\displaystyle\\{{\cal P}_{i},W_{j}\\}=\\{{\cal P}_{j},W_{i}\\}.$ (9.6)
It can be shown that (9.6) is satisfied automatically in both models, whereas
(9.5) fixes the magnetic field in terms of $W_{i}$ (same form for both
models):
$\displaystyle
B=\frac{\theta}{2}\epsilon_{ij}\epsilon_{k\ell}\partial_{k}W_{i}\partial_{\ell}W_{j}.$
(9.7)
The connection between B-field (9.7) and electric potential $A_{0}$ (9.4)
takes a simple form in the case of rotational invariance. From
$\displaystyle W_{i}({{\vec{X}}})=\partial_{i}W(r)$
we obtain
$\displaystyle A_{0}(r)=-\frac{1}{2}(W^{\prime}(r))^{2}$
and
$\displaystyle B(r)=-\frac{\theta}{r}A^{\prime}_{0}(r).$
As an example, consider the harmonic oscillator. Then
$\displaystyle A_{0}=-\frac{\omega^{2}}{2}r^{2}$
and we obtain a homogeneous $B$-field of strength
$\displaystyle B=\theta\omega^{2}.$
The supersymmetric extension of the DH model, and of anyons, have also been
studied in [57] and in [58], respectively.
## 10 Galilean symmetry in Moyal field theory
As we mentioned already, the physical explanation of the Fractional Quantum
Hall Effect (FQHE) relies on the dynamics of quasiparticles which carry both
an electric and a magnetic charge [34]. In the field theory context, these
quasiparticles arise as charged vortex solutions of the coupled field
equations. The phenomenologically preferred theory of Zhang et al. [59] is
Galilei invariant; the Galilean boost commute for these models, though. Does
there exist a field theoretical model with “exotic” Galilean symmetry? The
answer is yes, if we consider Moyal field theory [15, 60]. Here one considers
the usually-looking Lagrangian
$\displaystyle
L=i\bar{\psi}D_{t}\psi-\frac{1}{2}\big{|}{\vec{D}\psi}\big{|}^{2}+\kappa\left(\frac{1}{2}\epsilon_{ij}\partial_{t}A_{i}A_{j}+A_{t}B\right)$
but where the covariant derivative and the field strength,
$\displaystyle D_{\mu}\psi={\partial}_{\mu}\psi-ieA_{\mu}\star\psi,$
$\displaystyle
F_{\mu\nu}={\partial}_{\mu}A_{\nu}-{\partial}_{\nu}A_{\mu}-ie\big{(}A_{\mu}\star
A_{\nu}-A_{\nu}\star A_{\mu}\big{)},$
respectively, involve the Moyal “star” product, associated with the parameter
$\theta$,
$\displaystyle\big{(}f\star
g\big{)}(x_{1},x_{2})=\exp\left(i\frac{\theta}{2}\big{(}{\partial}_{x_{1}}{\partial}_{y_{2}}-{\partial}_{x_{2}}{\partial}_{y_{1}}\big{)}\right)f(x_{1},x_{2})g(y_{1},y_{2})\Big{|}_{{\vec{x}}=\vec{y}}.$
Here the matter field $\psi$ is in the fundamental representation of the gauge
group $U(1)_{*}$ i.e., $A_{\mu}$ acts from the left. The associated field
equations look formally as in the commutative case,
$\displaystyle iD_{t}\psi+\frac{1}{2}\vec{D}^{2}\psi=0,$ $\displaystyle\kappa
E_{i}-{e}\epsilon_{ik}j^{l}_{\ k}=0,$ $\displaystyle\kappa B+e\rho^{l}=0,$
(10.1)
where $B=\epsilon_{ij}F_{ij}$, $E_{i}=F_{i0}$. Note, however, that $\rho^{l}$
and $\vec{j}\,{}^{l}$ denote here the left density and left current,
respectively,
$\displaystyle\rho^{l}=\psi\star\bar{\psi},\qquad\vec{j}\,{}^{l}=\frac{1}{2i}\left(\vec{D}\psi\star\bar{\psi}-\psi\star(\overline{\vec{D}\psi})\right).$
These theories admit static, finite-energy vortex solutions [60] which
generalize those found before in ordinary CS theory [59, 61].
Are these theories Galilean invariant? At first sight, the answer seems to be
negative, and it has been indeed a widely shared view that Moyal field theory
is inconsistent with Galilean symmetry. The situation is more subtle, however.
The conventional infinitesimal implementation of a Galilean boost,
$\displaystyle\delta^{0}B=-t\vec{b}\cdot\vec{\nabla}B\qquad\hbox{but}\qquad\delta^{0}\rho^{l}=-\frac{\theta}{2}\vec{b}\times\vec{\nabla}\rho^{l}-\vec{b}\cdot\vec{\nabla}\rho^{l}.$
is indeed broken, as the Gauss constraint (10.1) is not preserved. Galilean
symmetry can be restored taking into account the Moyal structure [22], namely
considering the antifundamental representation
$\displaystyle\delta^{r}\psi=\psi\star(i{\vec{b}}\cdot{\vec{x}})-t{\vec{b}}\cdot{\vec{\nabla}}\psi=(i{\vec{b}}\cdot{\vec{x}})\psi+\frac{\theta}{2}{\vec{b}}\times{\vec{\nabla}}\psi-t{\vec{b}}\cdot{\vec{\nabla}}\psi.$
Observing that
$\displaystyle\delta^{r}\psi=\delta^{0}\psi+\frac{\theta}{2}{\vec{b}}\times{\vec{\nabla}}\psi$
we find that the $\theta$-terms cancel in $\delta^{r}\rho^{l}$, leaving us
with the homogeneous transformation law
$\displaystyle\delta^{r}\rho^{l}=-t{\vec{b}}\cdot{\vec{\nabla}}\rho^{l}.$
Putting $\delta^{r}A_{\mu}=\delta^{0}A_{\mu},$ so that
$\delta^{r}B=\delta^{0}B$, the Gauss constraint (10.1) is right-invariant, as
are all the remaining equations. The associated boost generator, calculated
using the Noether theorem, reads
$\displaystyle\vec{K}^{r}=t{\vec{P}}-\int{\vec{x}}\rho^{r}d^{2}{\vec{x}},$
(10.2)
where
$\displaystyle
P_{i}=\int\frac{1}{2i}\big{(}\bar{\psi}{\partial}_{i}\psi-(\overline{{\partial}_{i}\psi})\psi\big{)}d^{2}{\vec{x}}-\frac{\kappa}{2}\int\epsilon_{jk}A_{k}{\partial}_{i}A_{j}d^{2}{\vec{x}}$
is the conserved momentum. The conservation of (10.2) can also be checked
directly, using the continuity equation satisfied by the right density,
$\rho^{r}=\bar{\psi}\star\psi$. At last, the boost components have the exotic
commutation relation
$\displaystyle\big{\\{}K_{i},K_{j}\big{\\}}=\epsilon_{ij}k,\qquad
k\equiv-\theta\int|\psi|^{2}d^{2}x.$
Let us note, in conclusion, that Galilean symmetry as established here makes
it possible to produce moving vortices by boosting the static solutions
constructed in [60], see [62].
## 11 Noncommutativity in 3 dimensions:
the semiclassical Bloch electron
### 11.1 The semiclassical model
Around the same time and with no relation to the above developments, a very
similar theory has arisen in condensed matter physics. For instance, applying
a Berry-phase argument to a Bloch electron in a lattice, the standard
semiclassical equations [63] are modified by new terms [23], generating purely
quantum effects on the mean values of the electron’s position and quasi-
momentum ${{\vec{r}}}$ and ${{\vec{p}}}$, respectively, which add to the force
due to the momentum gradient of the energy band dispersion relation
$\epsilon_{n}({{\vec{p}}})$ and to the external (for instance, Lorentz)
forces. The semiclassical approach allows several applications and
generalizations, both from the physical [23, 24, 25, 26, 14], and the
mathematical [30, 64, 65, 66] side.
The clue is that the semiclassical model fits perfectly into Souriau’s general
framework [4] presented above. One starts with a “microscopic” Hamiltonian
operator
${\hat{H}}\big{[}{\hat{\vec{r}}},{\hat{\vec{p}}},f({\hat{\vec{r}}},t)\big{]}$
for a particle (electron) for a periodic potential, which is adiabatically (in
space-time) modified by a perturbation $f$ (possibly an external field). The
position/momentum operators $\hat{\vec{r}}$ and ${\hat{\vec{p}}}$ satisfy the
Heisenberg algebra, as usual. Moreover, for any constant $f$, the Hamiltonian
${\hat{H}}$ reduces to the usual one for a periodic crystal lattice.
The adiabatic features of the perturbation $f$ are expressed by the
inequalities $l_{\rm latt}\ll l_{wp}\ll l_{\rm mod}$, among the lattice
constant length $l_{\rm latt}$, the wave-packet dispersion length $l_{wp}$ and
the modulation wave-length $l_{\rm mod}$. Furthermore, the characteristic time
scale $\hbar/\Delta E_{\rm gap}$ must be much smaller than the typical time-
scale of variations of $f$.
The first order truncation of the Hamiltonian around the instantaneous mean
position ${\vec{r}}_{c}$,
$\displaystyle{\hat{H}}\big{[}{\hat{\vec{r}}},{\hat{\vec{p}}},f({\hat{\vec{r}}},t)\big{]}={\hat{H}}_{({\vec{r}}_{c},t)}+{\hat{W}}_{({\vec{r}}_{c},t)},$
$\displaystyle{\hat{W}}_{({\vec{r}}_{c},t)}=\frac{1}{2}\big{[}{\partial}_{f}{\hat{H}}\,\nabla_{{\vec{r}}_{c}}f({{\vec{r}}_{c}},t)\cdot({\hat{\vec{r}}}-{\vec{r}}_{c})+{\rm
h.c.}\big{]},$
defines a quasi-static Hamiltonian ${\hat{H}}_{\left({\vec{r}}_{c},t\right)}$,
depending on the “slow” parameters $c=\left({{\vec{r}}_{c}},t\right)$.
${\hat{H}}_{\left({\vec{r}}_{c},t\right)}$ is periodic under ${\vec{a}}$ –
translations, and its eigenstates are Bloch the waves. The latter are defined,
for any fixed time $t$ and ${\vec{r}}_{c}$, by
$\displaystyle{\hat{H}}_{\left({\vec{r}}_{c},t\right)}|\psi_{\left({\vec{r}}_{c},t\right)}^{n,{\vec{q}}}\rangle=E_{\left({\vec{r}}_{c},t\right)}^{n,{\vec{q}}}|\psi_{\left({\vec{r}}_{c},t\right)}^{n,{\vec{q}}}\rangle,\qquad$
$\displaystyle\langle\psi_{\left({\vec{r}}_{c},t\right)}^{n,{\vec{q}}}|\psi_{\left({\vec{r}}_{c},t\right)}^{n^{\prime},{\vec{q}}\,^{\prime}}\rangle=\delta_{n,n^{\prime}}\delta\left({\vec{q}}-{\vec{q}}\,^{\prime}\right),$
$\displaystyle\langle{\vec{r}}|\psi_{\left({\vec{r}}_{c},t\right)}^{n,{\vec{q}}}\rangle=e^{i{\vec{q}}\cdot{\vec{r}}}u_{\left({\vec{r}}_{c},t\right)}^{n,{\vec{q}}}\left({\vec{r}}\right),\qquad$
$\displaystyle
u_{\left({\vec{r}}_{c},t\right)}^{n,{\vec{q}}}\left({\vec{r}}+{\vec{a}}\right)=u_{\left({\vec{r}}_{c},t\right)}^{n,{\vec{q}}}\left({\vec{r}}\right),$
where the energy eigenvalues $E_{\left({\vec{r}}_{c},t\right)}^{n,{\vec{q}}}$
are labeled by the band index, $n$, and by the quasi-momentum ${\vec{q}}$,
restricted to the first Brillouin zone (IBZ). We assume that the time
evolution of ${{\vec{r}}_{c}}$ closely follows the one obtained by the exact
integration of the Schrödinger equation and that the eigenvalues
$E_{\left({\vec{r}}_{c},t\right)}^{n,{\vec{q}}}$ form well separated bands,
and that band jumping is forbidden. The label $n$ will be dropped in what
follows.
A classical result by Karplus and Luttinger [67] says that
$\displaystyle\langle\psi_{\left({\vec{r}}_{c},t\right)}^{{\vec{q}}}|\,{\hat{\vec{r}}}\,|\psi_{\left({\vec{r}}_{c},t\right)}^{{\vec{q}}^{\prime}}\rangle=\big{[}i\nabla_{{\vec{q}}}+\,\langle
u_{\left({\vec{r}}_{c},t\right)}^{{\vec{q}}}\left({\vec{r}}\right)|i\nabla_{{\vec{q}}}\,u_{\left({\vec{r}}_{c},t\right)}^{{\vec{q}}\,}\left({\vec{r}}\right)\rangle_{\rm
cell}\big{]}\delta\left({\vec{q}}\,^{\prime}-{\vec{q}}\right).$ (11.1)
That is, the momentum representation of ${\hat{\vec{r}}}$ is
$\displaystyle{\hat{\vec{r}}}=i\nabla_{{\vec{q}}}+\vec{{\cal
A}}\left({\vec{r}}_{c},{\vec{q}},t\right),\qquad\vec{{\cal A}}=\langle
u_{\left({\vec{r}}_{c},t\right)}^{{\vec{q}}}|i\nabla_{{\vec{q}}}\,u_{\left({\vec{r}}_{c},t\right)}^{{\vec{q}}}\rangle_{\rm
cell},$
where $\langle\cdot|\cdot\rangle_{\rm cell}$ is the restriction of the scalar
product to the unit cell with periodic boundary conditions, and with
normalization factor $\left(2\pi\right)^{3}/V_{\rm cell}$. Then the quantity
$\vec{{\cal A}}\,{\left({\vec{r}}_{c},{\vec{q}},t\right)}$ is interpreted as a
$U(1)$ _Berry connection_ , whose curvature appears in the commutation
relations for the position operator components,
$\displaystyle\left[{\hat{r}}_{j},{\hat{r}}_{l}\right]=i\,\epsilon_{jl}\,{\partial}_{q_{j}}{{{\cal
A}}}\,_{l}{\left({\vec{r}}_{c},{\vec{q}},t\right)}=\Theta_{jl}\left({\vec{r}}_{c},{\vec{q}},t\right),$
(11.2)
which converts the dynamics of an ordinary particle in a periodic background
potential into a quantum mechanical system in a _non-commutative
configuration space_ [15]. The antisymmetric tensor
${\mathbf{\Theta}}=(\Theta_{ij})$ generalizes in fact the scalar parameter
$\theta$ of the planar non-commutative theory.
Its effects cannot be disregarded for the semiclassical motion of a wave-
packet
$\displaystyle|\widetilde{\Psi}\left[{\vec{r}}_{c}\left(t\right),{\vec{q}}_{c}\left(t\right)\right]\rangle=\int_{\rm
IBZ}\Phi\left({\vec{q}},t\right)|\psi_{\left({\vec{r}}_{c},t\right)}^{{\vec{q}}}\rangle\;d{\vec{q}},$
built by superimposing one-band Bloch waves with a normalized amplitude
$\Phi\left({\vec{q}},t\right)$. In fact, under the assumptions of small
momentum dispersion, $\Delta_{q}\ll 2\pi/l_{\rm latt}$, it can be proved that
the mean packet-position is
$\displaystyle{\vec{r}}_{c}\left(t\right)=\langle\widetilde{\Psi}|\,{\hat{{\vec{r}}}}\,|\widetilde{\Psi}\rangle\approx-\nabla_{{\vec{q}}_{c}}\,{\rm
arg}\left[\Phi\left({\vec{q}}_{c},t\right)\right]+{\vec{{\cal
A}}}\,{\left({\vec{r}}_{c},{\vec{q}}_{c},t\right)},$
where the mean quasi-momentum,
$\displaystyle{\vec{q}}_{c}\left(t\right)=\int_{{}_{IBZ}}{\vec{q}}\;|\Phi\left({\vec{q}},t\right)|^{2}\;d\,{\vec{q}},$
(11.3)
has been introduced. Then, the semiclassical description of the wave-packet is
reduced to that of a particle – like system in the
$\left({\vec{r}}_{c},{\vec{q}}_{c}\right)$ “phase space”, the dynamics of
which is obtained by minimizing the Schrödinger field action
$\displaystyle
S=\int_{t_{1}}^{t_{2}}\left\\{\frac{i}{2}\frac{\langle\Psi|\frac{d\Psi}{dt}\rangle-\langle\frac{d\Psi}{dt}|\Psi\rangle}{\langle\Psi|\Psi\rangle}-\frac{\langle\Psi|{\hat{H}}|\Psi\rangle}{\langle\Psi|\Psi\rangle}\right\\}dt,$
where $\left({\vec{r}}_{c}\left(t\right),{\vec{q}}_{c}\left(t\right)\right)$
parametrize the wave-function [68]. This leads to an “approximate Lagrangian”
for a point-like classical particle of the form (4.1), namely to
$\displaystyle L_{\rm
app}=\dot{\vec{r}}{}_{c}\cdot\left({\vec{q}}_{c}+\overrightarrow{{\cal
R}}\left({\vec{r}}_{c},{\vec{q}}_{c},t\right)\right)+\dot{{\vec{q}}}_{c}\cdot{\vec{{\cal
A}}}\,{\left({\vec{r}}_{c},{\vec{q}}_{c},t\right)}+{\cal
T}\left({\vec{r}}_{c},{\vec{q}}_{c},t\right)$ $\displaystyle\hphantom{L_{\rm
app}=}{}-{{\cal E}}\left({\vec{r}}_{c},{\vec{q}}_{c},t\right)-\Delta{{\cal
E}}\left({\vec{r}}_{c},{\vec{q}}_{c},t\right),$ (11.4)
where
$\displaystyle{\cal T}\left({\vec{r}}_{c},{\vec{q}}_{c},t\right)=\langle
u_{\left({\vec{r}}_{c},t\right)}^{{\vec{q}}_{c}}|i{\partial}_{t}\,u_{\left({\vec{r}}_{c},t\right)}^{{\vec{q}}_{c}}\rangle_{\rm
cell},$ $\displaystyle\overrightarrow{{\cal
R}}\left({\vec{r}}_{c},{\vec{q}}_{c},t\right)=\langle
u_{\left({\vec{r}}_{c},t\right)}^{{\vec{q}}_{c}}|i\nabla_{{\vec{r}}_{c}}\,u_{\left({\vec{r}}_{c},t\right)}^{{\vec{q}}_{c}}\rangle_{\rm
cell},$ $\displaystyle{{\cal
E}}=\langle\widetilde{\Psi}|{\hat{H}}_{\left({\vec{r}}_{c},t\right)}|\widetilde{\Psi}\rangle,\qquad\Delta{{\cal
E}}=\langle\widetilde{\Psi}|{\hat{W}}_{\left({\vec{r}}_{c},t\right)}|\widetilde{\Psi}\rangle.$
Together with $\vec{{\cal A}}$, the scalar ${\cal T}$ and the vector field
$\overrightarrow{{\cal R}}$ provide us with the complete Berry connection on
the entire “environmental parameter space”
$\left({\vec{r}}_{c},{\vec{q}}_{c},t\right)$. The quantity ${{\cal E}}$
expresses the potential energy felt by the wave packet in the periodic
environment and $\Delta{{\cal E}}$ comes from the adiabatic perturbations.
For slowly changing electromagnetic potentials
$\big{(}{\vec{A}}\left({\vec{r}},t\right),V_{el}\left({\vec{r}},t\right)\big{)}$,
the rather involved expressions above take an elegant form [23], – but the
Bloch eigenfunctions get a gauge-dependent phase modification $\approx
e{\vec{A}}({\vec{r}}_{c},t)\cdot{\vec{r}}$. In fact, a change of phase has no
influence on the Berry connection because of (11.1), so one can introduce the
gauge invariant kinetic momentum
$\displaystyle{\vec{k}}_{c}={\vec{q}}_{c}-e{\vec{A}}({\vec{r}}_{c},t)$
and set again
$\displaystyle{\vec{{\cal
A}}}\,{\left({\vec{r}}_{c},{\vec{q}}_{c},t\right)}={\vec{{\cal
A}}}\,({\vec{k}}_{c}),$ $\displaystyle\overrightarrow{{\cal
R}}\simeq-e\,\nabla_{{\vec{r}}_{c}}\big{(}{{\vec{A}}}\,({\vec{r}}_{c},t)\cdot{\vec{r}}\big{)}\Big{|}_{{\vec{r}}={\vec{r}}_{c}},\qquad{\cal
T}\simeq-e\,{\partial}_{t}{{{\vec{A}}}}\,{({\vec{r}}_{c},t)\cdot{\vec{r}}_{c}},$
$\displaystyle{{\cal E}}={{\cal
E}}_{0}({\vec{k}}_{c})+e\,V_{el}({\vec{r}}_{c},t),\qquad\Delta{{\cal
E}}=-{\vec{M}}({\vec{k}}_{c},t)\cdot{\vec{B}}({\vec{r}}_{c},t),$
where
${\vec{M}}({\vec{k}}_{c},t)=-\frac{e}{2m}\langle\widetilde{\Psi}|{\hat{\vec{L}}}|\widetilde{\Psi}\rangle$
is the mean magnetic moment of the wave-packet. ${\vec{B}}({\vec{r}}_{c},t)$
and ${\vec{E}}({\vec{r}}_{c},t)$ are defined as usual from the mean values of
the potentials. Dropping the label $c$, and putting
$\displaystyle\Theta_{i}={\scriptstyle{\frac{1}{2}}}\epsilon_{ijk}\Theta_{jk},$
the generalized semiclassical equations of motion are
$\displaystyle\dot{{\vec{r}}}=\nabla_{{\vec{k}}}\big{[}{\cal
E}_{0}({\vec{k}})-{\vec{M}}({\vec{k}},t)\cdot{\vec{B}}({\vec{r}},t)\big{]}-{\dot{{\vec{k}}}}\times{{\vec{\Theta}}}({\vec{k}}),$
$\displaystyle\dot{{\vec{k}}}=-e\big{(}{\dot{{\vec{r}}}}\times{{\vec{B}}}({\vec{r}},t)+{\vec{E}}({\vec{r}},t)\big{)}+\nabla_{{{\vec{r}}}}\big{(}{\vec{M}}({\vec{k}},t)\cdot{{\vec{B}}}({\vec{r}},t)\big{)},$
(11.5)
further confirming the idea of the non-commutativity parameter, now a function
of momentum-space variables, is in fact a Berry phase effect.
Notice, here that the semiclassical procedure has consistently “averaged” on
the gauge degrees of freedom at local scales of order $\sim l_{wp}$, but the
final model still possesses the same gauge invariant character as a point-like
particle interacting with an external e.m. field.
Then, for the electronic wave-packet semiclassically described by (11.5), one
can adapt the symplectic techniques described in the previous sections and it
can be used for a Hamiltonian formulation.
### 11.2 Hamiltonian structure
Comparing the system (11.5) with the previous ones in (2.3) or (8.6), one
recognizes a general common structure. The nice group-geometrical symmetry
properties of the 2D Galilei group, which partially motivated the present
research, are broken in general. However, the unifying framework for such
differential systems is provided by the same ideology adopted in Sections 2
and 6, i.e. writing them as the kernel of a postulated anti-symmetric, closed,
constant-rank Lagrange–Souriau 2-form $\sigma$ of the form
$\displaystyle\sigma=\left[\left(1-Q_{i}\right)dq_{i}-eE_{i}dt\right]\wedge\left(dr_{i}-g_{i}dt\right)$
$\displaystyle\phantom{\sigma=}{}+\frac{1}{2}e\epsilon_{ijk}B_{k}dr_{i}\wedge
dr_{j}+\frac{1}{2}\epsilon_{ijk}\Theta_{k}dq_{i}\wedge
dq_{j}+Q_{0}\epsilon_{ij}dr_{i}\wedge dq_{j},$ (11.6)
where the Souriau’s prescription to explicitly include the electromagnetic
contributions has been used. The vector fields ${\vec{g}}$,
${{\vec{\Theta}}}$, ${\vec{Q}}$ and the scalar functions $Q_{0}$ may depend on
all independent variables $\left({\vec{r}},{\vec{q}},t\right)$. Notice that
(11.6) only contains “forces”, i.e. gauge invariant quantities. Moreover, the
so called “Maxwell principle” [4], i.e. the closure relation $d\sigma=0$,
implies a set of integrability conditions for functions involved, which reduce
to the usual Maxwell equations for $\big{(}{\vec{E}},{\vec{B}}\big{)}$, when
the new extra fields are set to constants. Even in this case, and in 2 space
dimensions, the resulting equations are non-trivial, coinciding for instance
with (2.3) after the identifications $r_{i}\rightarrow x_{i}$,
$g_{i}\rightarrow q_{i}/m$, $\theta_{3}\rightarrow\theta$ and $Q_{i}\equiv 0$.
We note (like in Section 2) that a model defined by the 2-form (11.6) may not
possess a globally defined configuration space Lagrangian. This makes the
value of the semiclassical Lagrangian (11.4) questionable. If it is assumed
valid at least locally, the physical meaning of the coefficients appearing in
(11.6) can be deduced, via exterior derivative, from the Cartan 1-form
$\displaystyle\lambda=\big{(}{\vec{q}}+\overrightarrow{{\cal R}}\big{)}\cdot
d{\vec{r}}+{\vec{{\cal A}}}\cdot d{\vec{q}}+\left({\cal T}-{{\cal
E}}-\Delta{{\cal E}}\right)dt.$
Thus, the most general equations of motion deriving from (11.4) (or
equivalently from (11.6)) are
$\displaystyle\left(1+{\mathbf{\Xi}}\right)\dot{{\vec{r}}}+\Theta{\dot{{\vec{q}}}}=\nabla_{{\vec{q}}}\left[{\cal
E}+\Delta{{\cal E}}-{\cal T}\right]+{\partial}_{t}{\vec{\cal A}},$
$\displaystyle
X{\dot{{\vec{r}}}}+\left(1+{\mathbf{\Xi}}\right)\dot{{\vec{q}}}=-\nabla_{{\vec{r}}}\left[{\cal
E}+\Delta{{\cal E}}-{\cal T}\right]-{\partial}_{t}{\vec{\cal R}},$ (11.7)
where the antisymmetric matrices ${\mathbf{\Xi}}=(\Xi_{ij})$ and
${\mathbf{X}}=(X_{ij})$ have elements
$\displaystyle{\mathbf{\Xi}}_{ij}={\partial}_{r_{i}}{\cal
A}_{j}-{\partial}_{q_{j}}{\cal R}_{i},\qquad X_{ij}={\partial}_{r_{i}}{\cal
R}_{j}-{\partial}_{r_{j}}{\cal R}_{i}.$ (11.8)
The dynamical system (11.7) is defined on the tangent manifold of the
configuration space, endowed with generalized coordinates
$\vec{\xi}=\left({\vec{r}},{\vec{q}}\right)$. But, when
${\partial}_{t}\vec{{\cal A}}={\partial}_{t}\vec{{\cal R}}\equiv 0$, the
rearrangement $\sigma=\omega-dH\wedge dt$ of the terms in (11.6) is possible,
introducing the symplectic 2-form
$\displaystyle\omega=\left(\delta_{i,j}+\Xi_{ij}\right)dr_{i}\wedge
dq_{j}+\frac{1}{2}\left[X_{ij}dq_{i}\wedge dq_{j}-\Theta_{ij}dr_{i}\wedge
dr_{j}\right]$
and the Hamiltonian function ${\cal H}={\cal E}+\Delta{{\cal E}}-{\cal T}$.
Actually, the closure of $\sigma$ implies that, $d\omega=0$, for $\omega$.
Equivalently, the set of differential constraints
$\displaystyle\varepsilon_{ijk}{\partial}_{q_{i}}\Theta_{jk}=0,\qquad$
$\displaystyle\varepsilon_{ijk}{\partial}_{r_{i}}X_{jk}=0,$
$\displaystyle{\partial}_{q_{j}}\Xi_{ij}=-{\partial}_{r_{j}}\Theta_{ij},\qquad$
$\displaystyle{\partial}_{r_{j}}\Xi_{ij}={\partial}_{q_{j}}X_{ij},$ (11.9)
$\displaystyle\left(1-\delta_{hk}\right)\varepsilon_{kij}{\partial}_{q_{k}}\Xi_{ij}=\varepsilon_{hij}{\partial}_{r_{h}}\Theta_{ij},\qquad$
$\displaystyle\left(1-\delta_{hk}\right)\varepsilon_{kij}{\partial}_{r_{k}}\Xi_{ij}=-\varepsilon_{hij}{\partial}_{q_{h}}X_{ij},$
which, however, are automatically satisfied, because of the antisymmetry and
the differentiability properties of the tensors ${\mathbf{\Theta}}$,
${\mathbf{\Xi}}$ and ${\mathbf{X}}$ defined in (11.2) and (11.8). Thus, for
non degenerate $\omega=\omega_{\alpha\beta}d\xi_{\alpha}\wedge d\xi_{\beta}$,
Poisson brackets,
$\displaystyle\left\\{f,g\right\\}=\omega^{\alpha\beta}{\partial}_{\alpha}f{\partial}_{\beta}g$
can be defined for any pair of functions $f(\vec{\xi})$ and $g(\vec{\xi})$,
where $\omega^{\alpha\gamma}\omega_{\gamma\beta}=\delta^{\alpha}_{\beta}$ is
the inverse of the symplectic matrix [4, 69]. Thus, the equations (11.7) take
the usual Hamiltonian form $\dot{\xi}_{\alpha}=\\{\xi_{\alpha},{\cal H}\\}.$
In the present case $\left(\omega_{\alpha\beta}\right)$ is a real symplectic
$6\times 6$ matrix, which is non degenerate when
$\displaystyle\sqrt{\det\left(\omega_{\alpha\beta}\right)}=1-\frac{1}{2}{\rm
Tr}\left({\mathbf{\Xi}}^{2}+{\mathbf{X}}\left({\bf
1}+2{\mathbf{\Xi}}\right){\mathbf{\Theta}}\right)\neq 0.$
Such a factor generalizes the denominators present in the Poisson brackets
(2.5), (4.3) or (8.4). Moreover, it crucially appears in the expression of the
invariant phase-space volume, ensuring the validity of the Liouville theorem
[70, 65].
As special example, we deal with only momentum (gauge invariant) dependent
Berry curvature ${{\vec{\Theta}}}\left({\vec{q}}\right)$ which is to be
divergence-free according to the first equation in (11.9). That condition can
be satisfied, except in one point, e.g., by a monopole in ${{\vec{q}}}$-space,
$\displaystyle{{\vec{\Theta}}}=g\frac{{\vec{q}}}{q^{3}},$ (11.10)
which is indeed the only possibility consistent with the spherical symmetry
and the canonical relations $\\{x_{i},q_{j}\\}=\delta_{ij}$ [66]. The
expression (11.10) appears to be consistent, at least qualitatively, with the
data reported in [24] and in Spin Hall Effects [25].
In absence of a magnetic field and taking, for simplicity, the energy band
$\epsilon_{n}({{\vec{q}}})$ to be parabolic, the equations (11.5) for become
$\displaystyle\dot{{\vec{r}}}={{\vec{q}}}+\frac{eg}{q^{3}}{\vec{E}}\times{{\vec{q}}},\qquad\dot{{\vec{q}}}=-e{\vec{E}}.$
The anomalous term shifts the velocity and deviates, hence, the particle’s
trajectory perpendicularly to the electric field, just like in the anomalous
Hall effect, see [24].
A similar pattern arises in optics [26, 27, 28, 29]: to first order in the
gradient of the refractive index $n$, spinning light is approximately
described by the equations
$\displaystyle\dot{{\vec{r}}}\approx{{\vec{p}}}-\frac{s}{\omega}\,{\rm
grad}\left(\frac{1}{n}\right)\times{{\vec{p}}},\qquad\dot{{\vec{p}}}\approx-n^{3}\omega^{2}{\rm
grad}\left(\frac{1}{n}\right),$
where $s$ denotes the photon’s spin. In the first relation we recognize, once
again, an anomalous velocity relation of the type (11.5). The new term makes
the light’s trajectory deviate from that predicted in ordinary geometrical
optics, giving rise to the “optical Magnus effect” [26]. A manifestation of
this is the displacement of the light ray perpendicularly to the plane of
incidence at the interface of two media with different refraction index: this
is the “Optical Hall Effect” [27, 28, 29].
Another nice illustration is provided by the non-commutative Kepler problem
[14]. Choosing the non-commutative vector ${\vec{\Theta}}$ in the vertical
direction,
$\displaystyle\Theta_{i}=\theta\delta_{iz}$
the 3D problem reduces to the “exotic” model presented in Section 2. Then the
authors of [14] show that, for the Kepler potential $V\propto r^{-1}$ the
perturbation due to non-commutativity induces the precession of the perihelion
point of planetary orbit.
As yet another example, we would like to mention the recent work [71], in
which it is shown that a particle with “monopole-type” noncommutativity
(11.10) admits a conserved Runge–Lenz vector, namely
$\displaystyle\vec{K}={\vec{r}}\times\vec{J}-\alpha\frac{{\vec{q}}}{q},$
provided the Hamiltonian is
$\displaystyle
H=\frac{{\vec{r}}\,{}^{2}}{2}+\frac{g^{2}}{2q^{2}}+\frac{\alpha}{q}.$
Note that ${\vec{q}}$ here is the momentum: the “monopole” is in “dual space”.
Let us observe that this expression is reminiscent of the Chern–Simons
mechanics” [33] in that it has no mass term. The associated the equations of
motion read,
$\displaystyle\dot{{\vec{r}}}=-\left(\frac{g^{2}}{q^{4}}+\frac{\alpha}{q^{3}}\right){\vec{q}}+g\frac{{\vec{q}}\times{{\vec{r}}}}{q^{3}},\qquad\dot{{\vec{q}}}=-{{\vec{r}}}.$
The Kepler-type dynamical symmetry then allows one to show that the classical
motions follow (arcs of) oblique ellipses [71].
### Acknowledgments
This review also includes results obtained jointly with C. Duval, Z. Horváth,
J. Lukierski, M. Plyushchay and W.J. Zakrzewski, to whom we express our
indebtedness. L.M. thanks the INFN - Sezione of Lecce for partial financial
support under the project LE41.
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|
arxiv-papers
| 2010-02-25T12:27:07 |
2024-09-04T02:49:08.542654
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "P. A. Horvathy, L. Martina, P.C. Stichel",
"submitter": "Peter Horvathy",
"url": "https://arxiv.org/abs/1002.4772"
}
|
1002.4853
|
# On the connectivity of the Sylow graph of a finite group
Francesco G. Russo University of Palermo
Department of Mathematics
via Archirafi 34, 90123, Palermo, Italy francescog.russo@yahoo.com
###### Abstract.
The Sylow graph $\Gamma(G)$ of a finite group $G$ originated from recent
investigations on the so–called $\mathbf{N}$–closed classes of groups. The
connectivity of $\Gamma(G)$ was proved only few years ago, involving the
classification of finite simple groups, and the structure of $G$ may be
strongly restricted, once information on $\Gamma(G)$ are given. The first
result of the present paper deals with a condition on $\mathbf{N}$–closed
classes of groups. The second result deals with a computational criterion,
related to the connectivity of $\Gamma(G)$.
###### Key words and phrases:
Sylow graph of a finite group; normalizers; almost–simple groups; formations;
symmetric coverings
###### 2010 Mathematics Subject Classification:
20E32, 20D20, 20F17.
## 1\. Sylow graph and formations of groups
All the groups, which are considered in the present paper, are finite. If $G$
is a group in which every Sylow subgroup is self–normalizing, then [12,
Theorem 1] shows that $G$ is a $p$–group for some prime $p$. [12, Theorem 1]
is a classical result and belongs to a long standing line of research which
investigates the structural properties of a group, once restrictions on its
normalizers are given (see [10, Chapter 5]). As noted in [3], we may introduce
the group class operator $\mathbf{N}$:
(1.1) $\mathbf{N}(\mathfrak{X})=(G:N_{G}(G_{p})\in\mathfrak{X},\forall
p\in\pi(G)),$
where $\mathfrak{X}$ is a class of groups, $\pi(G)$ is the set of prime
divisors of the order of $G$ and $G_{p}$ denotes the Sylow $p$– subgroup of
$G$, briefly $G_{p}\in\mathrm{Syl}_{p}(G)$. $\mathbf{N}$ was largely studied
in [1, 2, 3, 5, 6, 7, 8, 9, 13]. Originally, [12, Corollary 2] shows that the
class $\mathfrak{E}_{p}$ of all $p$–groups is $\mathbf{N}$–closed. More
generally, [3, Theorem 2] shows that the class $\mathfrak{N}$ of all nilpotent
groups is $\mathbf{N}$–closed. A further improvement is [1, Theorem 2], where
it is considered the class of all $p$–nilpotent groups. In the class
$\mathfrak{S}$ of all solvable groups, relations among $\mathbf{N}$ and
classes of groups which are closed with respect to forming subgroups (briefly,
$\mathbf{S}$–closed), were investigated in [5, 6]. These researches continued
in [7, 8, 9], introducing some technical notions, which refer to ideas and
techniques in the theory of formations of groups due to Gaschütz, Lubesender
and Shemetkov. The terminology is standard in literature and can be found in
[10]. A significant notion is the following.
###### Definition 1.1 (See [6]).
Assume that $\pi$ is a set of primes. A covering
$\mathcal{R}=(\pi(p))_{p\in\mathbb{P}}$ is called $symmetric$ in $\pi$, if the
following properties hold:
* (i)
$\pi={\underset{p\in\pi}{\bigcup}}\pi(p),$
* (ii)
$p\in\pi(p)$, for each $p\in\pi$,
* (iii)
if $q\in\pi(p)$, then $p\in\pi(q)$.
Let $f_{\mathcal{R}}$ be the formation function, defined by
(1.2)
$f_{\mathcal{R}}(p)=\left\\{\begin{array}[]{lcl}\mathfrak{E}_{\pi(p)},\,\,\mathrm{if}\,\,p\in\pi\\\
\emptyset,\,\,\,\,\,\,\,\,\,\,\mathrm{if}\,\,p\not\in\pi.\end{array}\right.$
The saturated formation $LF(f_{\mathcal{R}})=\mathfrak{E}_{\mathcal{R}},$
locally defined by $f_{\mathcal{R}}$, is said to be a $covering$–$formation$,
associated to $\mathcal{R}$. The intersection
$\mathfrak{E}_{\mathcal{R}}\cap\mathfrak{S}$ is said to be a
covering–formation of solvable groups, associated to $\mathcal{R}$. If
$\mathcal{R}$ is a partition of $\pi$, $\mathfrak{E}_{\mathcal{R}}$ is called
$lattice$–$formation$.
In [1, 2, 3, 5, 6, 7, 8, 9] the covering–formations and the lattice–formations
originated the following graph, which appears in [9] for the first time.
###### Definition 1.2 (See [9]).
Let $G$ be a group and $p,q$ two distinct primes in $\pi(G)$. Then there is a
sequence $p_{1}=p,p_{2},\ldots,p_{n}=q$ in $\pi(G)$ such that either $p_{i}$
divides $|N_{G}(G_{p_{i+1}}):C_{G}(G_{p_{i+1}})|$ or $p_{i+1}$ divides
$|N_{G}(G_{p_{i}}):C_{G}(G_{p_{i}})|,$ for each $i=1,2,\ldots,n$. When this
happen, we write briefly $p\approx q$. $\Delta(G)$ denotes the graph with set
of vertices $\pi(G)$ and edges given by the relation $\approx$.
The connectivity of $\Delta(G)$ was hard to investigate in the form of
Definition 1.2. In fact it was reformulated under a different prospective by
Kazarin and others in [13]. These authors study the following graph, known as
the $Sylow$ $graph$ $of$ $G$.
###### Definition 1.3 (See [13]).
In a group $G$ define the automiser $A_{p}(G)$ to be the group
$N_{G}(G_{p})/G_{p}C_{G}(G_{p})$. The Sylow graph $\Gamma(G)$ of $G$, with set
of vertices $\pi(G)$, is given by the following rules: Two vertices
$p,q\in\pi(G)$ form an edge of $\Gamma_{A}(G)$ if either $q\in\pi(A_{p}(G))$
or $p\in\pi(A_{q}(G))$. Briefly, we write $p\rightarrow q$ if
$q\in\pi(A_{p}(G))$.
It is easy to see that $p\rightarrow q$ implies $p\approx q$, since
$|N_{G}(G_{p}):G_{p}N_{G}(G_{p})|\leq|N_{G}(G_{p}):C_{G}(G_{p})|$. In
particular, if $G_{p}$ is abelian, then $G_{p}\subseteq C_{G}(G_{p})$ and we
conclude that $\Delta(G)=\Gamma(G)$. Therefore the two graphs are related and
we can summarize all in the following observation.
###### Remark 1.4.
While $\Delta(G)$ admits loops, $\Gamma(G)$ does not admit loops. This fact
does not affect the connectivity of $\Delta(G)$ and $\Gamma(G)$ so that
$\Delta(G)$ is connected if, and only if, $\Gamma(G)$ is connected.
The connectivity of $\Gamma(G)$ was proved in [13, Main Theorem] and
influenced heavily the first versions of the present paper. An immediate
consequence, conjectured in [9], is that the lattice–formations are
$\mathbf{N}$–closed. Therefore $\Gamma(G)$ allows us to decide whether a class
of groups is $\mathbf{N}$–closed or not. In Section 2 we describe some
abstract conditions, which are useful for the same scope, without using the
methods of the graph theory. Since the properties of $\Gamma(G)$ are related
to those of the Sylow normalizers of $G$, it is used the classification of
finite simple groups in [13]. In Section 3 we illustrate a criterion for the
connectivity of $\Gamma(G)$, dealing with a computational method. In this way,
we avoid the classification of finite simple groups.
## 2\. Some abstract conditions
We begin with some examples to become confident with $\mathbf{N}$–closed
classes.
###### Example 2.1 (See [13], Examples 2,3 and Remark in Section 5).
Consider $\pi=\\{2,3,5\\},$ $\pi(2)=\\{2,3,5\\},$ $\pi(3)=\\{2,3\\},$
$\pi(5)=\\{2,5\\}$ and $\mathcal{R}=\pi(2)\cup\pi(3)\cup\pi(5)$. Definition
1.1 is satisfied and $\mathcal{R}$ is symmetric in $\pi$. Note that the
alternating groups on 5 elements belong to
$\mathbf{N}(\mathfrak{E}_{\mathcal{R}})\setminus\mathfrak{E}_{\mathcal{R}}$.
This suggests that there is not a characterization of covering–formations in
the universe $\mathfrak{E}$ of all finite groups in terms of $\mathbf{N}$.
Example 2.1 shows the following fact. If $E$ is a simple group, belonging to a
covering–formation, then $E$ is a $\pi(p)$–group, for each $p\in\pi(E)$,
because $E$ is clearly the only chief factor of itself.
###### Remark 2.2.
Still from Definition 1.1, the lattice–formation, related to the minimal
partition of $\pi$, is the formation $\mathfrak{N}_{\pi}$ of all nilpotent
$\pi$–groups. In particular, $\mathfrak{N}_{\mathbb{P}}=\mathfrak{N}$. The
lattice–formation, related to the maximal partition of $\pi$, is
$\mathfrak{E}_{\pi}$. In particular, $\mathfrak{E}_{\mathbb{P}}=\mathfrak{E}$.
From [2], the lattice–formation, related to a partition $(\pi_{i})_{i\in I}$
of $\pi\subseteq\mathbb{P}$, is the class of the groups which are direct
product of $\pi_{i}$–groups.
###### Example 2.3.
Still from Definition 1.1, the class $\mathfrak{D}$ of all groups which have
odd order is both a lattice–formation and a covering–formation of solvable
groups, thanks to a well–known result of J. Thompson and W. Feit. More
precisely, $\mathfrak{D}=LF(f_{\mathcal{R}})$, where
$f_{\mathcal{R}}(2)=\emptyset$ and
$f_{\mathcal{R}}(p)=\mathfrak{S}_{2^{\prime}}$ if $p\not=2.$ Note that
$\mathfrak{D}=\mathfrak{S}_{2^{\prime}}$ and the characteristic of the
formation $\mathfrak{D}$ (see [10] for the terminology) is equal to
$2^{\prime}=\mathbb{P}\setminus\\{2\\}$.
The above considerations show that saturated formations, which are
$\mathbf{S}$–closed and $\mathbf{N}$–closed, could not be covering–formations.
This was already noted in [8, Section 3]. The following condition is more easy
to check in a computational way.
###### Hypothesis.
Assume that in a group $G$, if $p\in\pi(G)$ and $p$ is odd, then there exists
a prime $q$ such that $q<p$ and $q$ divides $|N_{G}(G_{p}):C_{G}(G_{p})|$ for
each $G_{p}\in\mathrm{Syl}_{p}(G)$.
If Hypothesis is true, then $\Delta(G)$ is connected and, consequently,
$\Gamma(G)$ is connected. Unfortunately, the converse is false, as shown
below.
###### Example 2.4.
Let $H=PSL_{2}(3^{3^{k}})$ for any integer $k\geq 1$ and consider
$G=H\langle\sigma\rangle$, where
(2.1) $\sigma:\left(\begin{array}[]{ccccccc}a&b\\\ c&d\\\
\end{array}\right)\in
PSL_{2}(3^{3^{k}})\longmapsto\left(\begin{array}[]{ccccccc}a^{3}&b^{3}\\\
c^{3}&d^{3}\\\ \end{array}\right)\in PSL_{2}(3^{3^{k}}).$
$G$ is an almost–simple group, i.e.: $\mathrm{Inn}(S)\leq
G\leq\mathrm{Aut}(S)$ for some non–abelian simple group $S$, and
$\\{2,3\\}\subseteq\pi(G)$. Now $G$ does not satisfies Hypothesis, because $2$
does not divides $|N_{G}(G_{3}):C_{G}(G_{3})|$. At the same time, one can see
that there is a prime $l\in\pi(G)\setminus\\{2,3\\}$ such that $2$ divides
$|N_{G}(G_{l}):C_{G}(G_{l})|$ and $3$ divides $|N_{G}(G_{l}):C_{G}(G_{l})|$ so
that $2\approx 3$.
Saturated formations, which are both $\mathbf{S}$–closed and
$\mathbf{N}$–closed, are characterized in [8, Theorem 2] and yields to
interesting constructions as in [8, Section 3] or in [13, Section 5]. We
recall just one case.
###### Example 2.5.
Consider a set of primes
$\pi=\\{p\\}\cup(\pi\setminus\\{p\\})=\\{p\\}\cup\omega$ such that there exist
a non–abelian simple $\omega$–group. Then the formation function $f$, defined
by $f(q)=\emptyset$, if $q\not\in\pi$; $f(q)=\mathfrak{S}_{\pi}$, if $q=p$;
$f(q)=\mathfrak{E}_{\pi}$, if $q\in\omega$, determines the saturated formation
$\mathfrak{F}=LF(f)$, locally defined by $f$, which is in many situations both
$\mathbf{S}$–closed and $\mathbf{N}$–closed but not a covering–formation. See
[8, Section 3] for details.
We may extend Example 2.5 in a more appropriate way. Recall that $G^{\infty}$
denotes the largest perfect subgroup of a group $G$. Recall also that
$\mathfrak{F}$ in Example 2.5 is the class of all finite groups $G$ such that
whenever $H/K$ is a chief–factor of $G$ and $q$ is a prime dividing $|H/K|$,
then $G/C_{G}(H/K)\in f(q)$.
###### Lemma 2.6.
$G\in\mathfrak{F}$ if and only if $G$ is a $\pi$–group and $G^{\infty}$ is a
$p^{\prime}$–group.
###### Proof.
Suppose $G\in\mathfrak{F}$. If $q$ is a prime divisor of $|G|$ then there
exists a chief factor $H/K$ of $G$ such that $q$ divides $|H/K|$. Then
$G/C_{G}(H/K)\in f(q)$. Thus $f(q)$ is not empty and so $q\in\pi$. So $G$ is a
$\pi$–group. Suppose for a contradiction that $G^{\infty}$ is not a
$p^{\prime}$–group. Then there exists a chief factor $H/K$ of $G$ such that
$H\leq G^{\infty}$, $p$ divides $|H/K|$ and $G^{\infty}/H$ is a
$p^{\prime}$–group. It follows that $G/C_{G}(H/K)\in f(p)$ and so
$G/C_{G}(H/K)$ is solvable. Thus $G^{\infty}\leq C_{G}(H/K)$ and so $H/K\leq
Z(G^{\infty}/K)$. Since $p$ divides $|H/K|$, $H/K$ is a $p$–elementary abelian
group. Since $G^{\infty}/K$ is $p^{\prime}$–group, we conclude that
$G^{\infty}/K=H/K\times L/K$ for some subgroup $L/K$ of $G^{\infty}/K$. This
is a contradiction, since $G^{\infty}/K$ is perfect and $H/K$ is abelian.
Suppose next that $G$ is a $\pi$–group and $G^{\infty}$ is a
$p^{\prime}$–group. Let $H/K$ be a chief factor of $G$ and $q$ a prime
dividing $|H/K|$. Since $G$ is a $\pi$–group, $q\in\pi$ and $G/C_{G}(H/K)$ is
a $\pi$–group. So if $q\neq p$ we have $G/C_{G}(H/K)\in f(q)$. Suppose $q=p$.
Since $[H/K,G^{\prime}]=[H,G^{\prime}]K/K$ is a $G$–invariant
$p^{\prime}$–subgroup of $H/K$, we conclude that $G^{\prime}\leq C_{G}(H/K)$
and so $G/C_{G}(H/K)$ is solvable. Hence again $G/C_{G}(H/K)\in f(q)$. So
$G\in\mathfrak{F}$. ∎
###### Corollary 2.7.
$\mathfrak{F}$ is $\mathbf{S}$–closed.
###### Proof.
This follows immediately from Lemma 2.6. ∎
In the next results we will use the $generalized$ $Fitting$ $subgroup$
$F^{*}(G)$ of a group $G$ and the $components$ $of$ $G$. These are well–known
notions, which can be found in [10, p.580].
###### Lemma 2.8.
Let $H/K$ be a non–abelian chief–factor of a group $G$, $L$ a component of
$H/K$, $\overline{U}=N_{G/K}(L)/C_{G/K}(L)$, $q$ a prime,
$T\in\mathrm{Syl}_{q}(\overline{U})$ and $S\in\mathrm{Syl}_{q}(G)$. Then
$F^{*}(\overline{U})=\overline{L}\cong L$ and
$N_{\overline{L}}(T)/\overline{L}\cap T$ is isomorphic to a subgroup of
$N_{H}(S)/(S\cap H)K$.
###### Proof.
The first statement is obvious. To simplify the notation we may replace $G$ by
$G/K$ and assume that $K=1$. Put $R=N_{S}(L)$. We also may assume that
$T=\overline{R}$. Then $N_{\overline{L}}(T)/\overline{L}\cap T\cong
N_{L}(R)/L\cap R$. Let $s_{1},\ldots s_{k}$ be a transversal to $R$ in $S$
with $s_{1}=1$ and put $L_{i}=L^{s_{i}}$. Then $M=\langle
L^{S}\rangle=L_{1}\times L_{2}\times\ldots L_{k}$ and it is now easy to see
that $N_{M}(S)/S\cap M\cong N_{L}(R)/L\cap R$.∎
###### Corollary 2.9.
Let $G$, $L$ and $\overline{U}$ be as in Lemma 2.8. If
$G\in\mathbf{N}(\mathfrak{F})$, then
$\overline{U}\in\mathbf{N}(\mathfrak{F})$.
###### Proof.
Since $\mathfrak{F}$ is closed under sections, this follows from Lemma 2.8. ∎
Note that a group $U$ is almost–simple if and only if $F^{*}(U)$ is simple.
###### Theorem 2.10.
$\mathfrak{F}=\mathbf{N}(\mathfrak{F})$ if and only the following statements
are true:
* (a)
If $U$ is a non–solvable almost–simple group in $\mathbf{N}(\mathfrak{F})$,
then $F^{*}(U)$ is a $p^{\prime}$–group.
* (b)
If $L$ is a non–abelian finite simple $(\pi\setminus\\{p\\})$–group and
$S\in\mathrm{Syl}_{p}(\mathrm{Aut}(L))$, then $C_{L}(S)$ is non–solvable.
###### Proof.
Suppose first that $\mathfrak{F}=\mathbf{N}(\mathfrak{F})$. Let $U$ be an
almost–simple group in $\mathbf{N}(\mathfrak{F})$. Since
$\mathfrak{F}=\mathbf{N}(\mathfrak{F})$, we conclude that $L\in\mathfrak{F}$
and so $U^{\infty}$ is a $p^{\prime}$–group. Since $U$ is non–solvable,
$F^{*}(U)$ is perfect and so $F^{*}(U)\leq U^{\infty}$ and $F^{*}(U)$ is a
$p^{\prime}$–group. Let $L$ be a finite simple $(\pi\setminus\\{p\\})$–group
and $S\in\mathrm{Syl}_{p}(\mathrm{Aut}(L))$. Let $\mathbb{F}$ be a finite
field of characteristic $p$ such that $q$ divides $|\mathbb{F}^{\sharp}|$ for
all $q\in\pi$ with $q\neq p$ ($\mathbb{F}^{\sharp}$ denotes the multiplicative
group of $\mathbb{F}$). Let $D$ be the $(\pi\setminus\\{p\\})$–Hall subgroup
of $\mathbb{F}^{\sharp}$. Let $V$ be a faithful $\mathbb{F}LS$–module. Then
$V$ is also a module for $D\times LS$ and we can consider the semidirect
product $H=V(D\times LS)$. Note that $H$ is a $\pi$–group. Also $[V,L]\leq
H^{\infty}$ and so $H^{\infty}$ is not a $p^{\prime}$–group. So
$H\not\in\mathfrak{F}$. Since $\mathfrak{F}=\mathbf{N}(\mathfrak{F})$, this
gives $H\not\in\mathbf{N}(\mathfrak{F})$. Hence there exists $q\in\pi$ and
$T\in\mathrm{Syl}_{q}(H)$ such that $N_{H}(T)\not\in\mathfrak{F}$. Since
$N_{H}(R)$ is a $\pi$–group, this means that $N_{H}(T)^{\infty}$ is not a
$p^{\prime}$–group. Since $N_{H}(T)^{\infty}\leq H^{\infty}\leq VL$ and $L$ is
a $p^{\prime}$–group we conclude that $N_{H}(T)\cap V\neq 1$. Assume that
$q=p$. Then $T\cap V=1$ and $1\neq N_{H}(T)\cap V=C_{V}(T)=C_{V}(VT)$. Since
$T$ is a Sylow $q$–subgroup of $H$ and $q$ divides $|D|$, we get $VT\cap D\neq
1$. But each non–trivial element of $D$ acts fixed point freely on $V$ and so
$C_{V}(VT\cap D)=1$, a contradiction to $C_{V}(VT)\neq 1$. Thus $q=p$ and we
may assume that $T=VS$. Then $N_{H}(T)=VDN_{L}(S)$ and, since $N_{H}(T)$ is
non–solvable, we conclude that $N_{L}(S)$ is non–solvable. Suppose next that
(a) and (b) hold. We will first show that
(2.2) $\mathrm{Any}\ \mathrm{non}-\mathrm{abelian}\ \mathrm{composition}\
\mathrm{factor}\ L\ \mathrm{of}\ G\in\mathbf{N}(\mathfrak{F})\ \mathrm{is}\
\mathrm{a}\ p^{\prime}-\mathrm{group}.$
Indeed by Lemma 2.8 and Corollary 2.9, $L\cong F^{*}(\overline{U})$ for the
simple group $\overline{U}\in\mathbf{N}(\mathfrak{F})$. Thus by (a), $L$ is a
$p^{\prime}$–group. So (2.2) holds.
Suppose for a contradiction that there exists
$G\in\mathbf{N}(\mathfrak{F})\setminus\mathfrak{F}$. Then there exist a
chief–factor $H/K$ and a prime $q$ dividing $|H/K|$ such that
$G/C_{G}(H/K)\not\in f(q)$. Since $G$ is a $\pi$–group, this implies that
$q=p$ and $G/C_{G}(H/K)$ is non–solvable. Thus $G^{\infty}\nleq C_{G}(H/K)$
and there exists a normal subgroup $M$ of $G$ maximal with $C_{G}(H/K)\leq
M<G$. Observe that $G^{\infty}/M$ is a non–abelian chief factor of $G$. Let
$L$ be a composition factor of $G^{\infty}/M$. By (2.2), $L$ is a
$p^{\prime}$–group. From (b) and Lemma 2.8 we conclude that
$N_{G^{\infty}}(S)M/M$ is non–solvable and so
(2.3) $N_{G}(S)^{\infty}\nleq C_{G}(H/K).$
Since $G\in\mathbf{N}(\mathfrak{F})$, $N_{G}(S)^{\infty}$ is a
$p^{\prime}$–group. Since $[N_{G}(S)^{\infty},S]\leq N_{G}(S)^{\infty}\cap S,$
we conclude that $N_{G}(S^{\infty})\leq C_{G}(S)$. By (2.2), $H/K$ is a
$p$–group. Thus $H=(S\cap H)K$ and so
(2.4) $[N_{G}(S)^{\infty},H]\leq[N_{G}(S^{\infty}),S]K=K,$
which is in contradiction with (2.3). ∎
###### Remark 2.11.
(a) and (b) in Theorem 2.10 follow from: (c) If $L$ is a finite simple
$\pi$–group, then $\mathrm{Aut}(L)$ is a $p^{\prime}$–group.
## 3\. A computational method
In the present section we illustrate an algorithm in GAP (see [14]), which
allows us to decide whether $\Delta(G)$ is connected or not in an empiric way.
The times of answer of the oracle are quite long, but, in case we use
Hypothesis, they are significatively reduced. The present algorithm has been
called ROSN1, achronymus of ”Restrictions On Sylow Normalizers”. In Appendix
the reader may find the description of $\Delta(G)$ for all the sporadic simple
groups ($\textrm{Div}(k)$ denotes the set of divisors of the integer $k\geq
1$). These results are deduced by looking at [4] and confirm the empiric
results which we get on ROSN1.
TestGroup:=function(G)
local pi,p,S,x,N,C,Q,R,i,res;
pi:=AsSet(FactorsInt(Size(G)));
S:=List(pi,p->SylowSubgroup(G,p));
N:=List(S,x->Normalizer(G,x));
C:=List(S,x->Centralizer(G,x));
Q:=List([1..Size(pi)],i->Size(N[i])/Size(C[i]));
R:=List([1..Size(pi)],i->ForAny(FactorsInt(Q[i]),p->p<pi[i]));
if pi[1]=2 then
res:=ForAll(R{[2..Size(pi)]},x->x);
else res:=ForAll(R,x->x);
fi;
return res;
end;
## acknowledgement
The present work was supported by GNSAGA (Indam, Florence, Italy) in the years
2008–09. This allowed me to visit Michigan State University (USA), TU of
Darmstadt (Germany), Free University of Berlin (Germany) and University
Technology of Malaysia (Malaysia), finding the inspiration for the present
paper.
## References
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* [2] Ballester–Bolinches, A., Martinez–Pastor, A., Pedraza–Aguilera, M. C. and Perez–Ramos, M. D., On nilpotent–like Fitting formations, in: Groups St. Andrews 2001 in Oxford, Vol. I, LMS Lecture Notes Ser., 304, Cambridge University Press, Cambridge, 2003, 31–38
* [3] Bianchi, M., Gillio Berta Mauri, A. and Hauck, P., On finite groups with nilpotent Sylow–normalizers, Arch. Math. (Basel), 47 (1986), 193–197
* [4] Conway, J., Curtis, R., Norton, S., Parker, R. and Wilson, R., Atlas of Finite Groups, Clarendon Press, Oxford, 1985
* [5] D’Aniello, A., De Vivo, C. and Giordano, G., Finite groups with primitive Sylow normalizers, Boll. Unione Mat. Ital. Sez. B, 5 (2002), 235–245
* [6] D’Aniello, A., De Vivo, C. and Giordano, G., Saturated formations and Sylow normalisers, Bull. Austral. Math. Soc., 69 (2004), 25–33
* [7] D’Aniello, A., De Vivo, C., Giordano, G. and Perez–Ramos, M. D., Saturated formations closed under Sylow normalizers, Comm. Algebra, 33 (2005), 2801–2808
* [8] D’Aniello, A., De Vivo, C. and Giordano, G., On certain saturated formations of finite groups, in: Proceedings of Ischia Group Theory 2006, World Scientific Publishing, Singapore, 2006, 12-31
* [9] D’Aniello, A., De Vivo, C. and Giordano, G., Lattice formations and Sylow normalizers: a conjecture, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia, 55 (2007), 107–112.
* [10] Doerk, K. and Hawkes,T., Finite Soluble Groups, de Gruyter, Berlin, 1992
* [11] Glauberman, G., Prime–power factor groups of finite groups, Math. Z., 107 (1968), 159–172
* [12] Glauberman, G., Prime–power factor groups of finite groups II, Math. Z., 117 (1970), 46–56
* [13] Kazarin, L., Martinez–Pastor, A. and Perez–Ramos, M. D., On the Sylow graph of a group and Sylow normalizers, e-print, Cornell University, arxiv:0912.2839v1.
* [14] The GAP Group, GAP–Groups, Algorithms and Programming, version 4.4, available online at http://www.gap-system.org, 2005.
## 4\. Appendix
* (1)
For $G=M_{11}$, $\pi(M_{11})=\\{2,3,5,11\\}$.
$2\in\textrm{Div}(|N_{M_{11}}(G_{3}):C_{M_{11}}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|)\cap\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|)$. These data are enough.
* (2)
For $G=M_{12}$, $\pi(M_{12})=\\{2,3,5,11\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|)\cap\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|),$ These data are enough.
* (3)
For $G=J_{1}$, $\pi(J_{1})=\\{2,3,5,7,11,19\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|)\cap\textrm{Div}(|N_{G}(G_{19}):C_{G}(G_{19})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|)$. These data are enough.
* (4)
For $G=M_{22}$, $\pi(M_{22})=\\{2,3,5,7,11\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|).$ These data are enough.
* (5)
For $G=J_{2}$, $\pi(J_{2})=\\{2,3,5,7\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|).$
These data are enough.
* (6)
For $G=M_{23}$, $\pi(M_{23})=\\{2,3,5,7,11,23\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|),$
$11\in\textrm{Div}(|N_{G}(G_{23}):C_{G}(G_{23})|)$. These data are enough.
* (7)
For $G=H_{5}$, $\pi(H_{5})=\\{2,3,5,7,11\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|).$ These data are enough.
* (8)
For $G=J_{3}$, $\pi(J_{3})=\\{2,3,5,17,19\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|)\cap\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|),$
$3\in\textrm{Div}(|N_{G}(G_{19}):C_{G}(G_{19})|).$ These data are enough.
* (9)
For $G=M_{24}$, $\pi(M_{24})=\\{2,3,5,17,19\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|)\cap\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|),$
$3\in\textrm{Div}(|N_{G}(G_{19}):C_{G}(G_{19})|).$ These data are enough.
* (10)
For $G=McL$,
$\pi(McL)=\\{2,3,5,7,11\\}$.$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|).$ These data are enough.
* (11)
For $G=He$, $\pi(He)=\\{2,3,5,7,17\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|)\cap\textrm{Div}(|N_{G}(G_{17}):C_{G}(G_{17})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|).$ These data are enough.
* (12)
For $G=Ru$, $\pi(Ru)=\\{2,3,5,7,13,29\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|)\cap\textrm{Div}(|N_{G}(G_{13}):C_{G}(G_{13})|),$
$7\in\textrm{Div}(|N_{G}(G_{29}):C_{G}(G_{29})|).$ These data are enough.
* (13)
For $G=Suz$, $\pi(Suz)=\\{2,3,5,7,11,13\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|)\cap\textrm{Div}(|N_{G}(G_{13}):C_{G}(G_{13})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|).$ These data are enough.
* (14)
For $G=O^{\prime}N$, $\pi(O^{\prime}N)=\\{2,3,5,7,11,19,31\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|)\cap\textrm{Div}(|N_{G}(G_{19}):C_{G}(G_{19})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|)\cap\textrm{Div}(|N_{G}(G_{31}):C_{G}(G_{31})|).$
These data are enough.
* (15)
For $G=Co3$, $\pi(Co3)=\\{2,3,5,7,11,23\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|),$
$11\in\textrm{Div}(|N_{G}(G_{23}):C_{G}(G_{23})|).$ These data are enough.
* (16)
For $G=Co2$, $\pi(Co2)=\\{2,3,5,7,11,23\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|),$
$11\in\textrm{Div}(|N_{G}(G_{23}):C_{G}(G_{23})|).$ These data are enough.
* (17)
For $G=Co1$, $\pi(Co1)=\\{2,3,5,7,11,13,23\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|)\cap\textrm{Div}(|N_{G}(G_{13}):C_{G}(G_{13})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|),$
$11\in\textrm{Div}(|N_{G}(G_{23}):C_{G}(G_{23})|).$ These data are enough.
* (18)
For $G=Fi_{22}$, $\pi(Fi_{22})=\\{2,3,5,7,11,13\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|)\cap\textrm{Div}(|N_{G}(G_{13}):C_{G}(G_{13})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|).$ These data are enough.
* (19)
For$G=Fi_{24}$, $\pi(Fi_{24})=\\{2,3,5,7,11,13,17,23,29\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|)\cap\textrm{Div}(|N_{G}(G_{17}):C_{G}(G_{17})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|)\cap\textrm{Div}(|N_{G}(G_{13}):C_{G}(G_{13})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|),$
$7\in\textrm{Div}(|N_{G}(G_{29}):C_{G}(G_{29})|),$
$11\in\textrm{Div}(|N_{G}(G_{23}):C_{G}(G_{23})|),$ These data are enough.
* (20)
For $G=HN$, $\pi(HN)=\\{2,3,5,7,11,19\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|)\cap\textrm{Div}(|N_{G}(G_{19}):C_{G}(G_{19})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|).$ These data are enough.
* (21)
For $G=Ly$, $\pi(Ly)=\\{2,3,5,7,11,31,37,67\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|)\cap\textrm{Div}(|N_{G}(G_{37}):C_{G}(G_{37})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|)\cap\textrm{Div}(|N_{G}(G_{31}):C_{G}(G_{31})|)l,$
$11\in\textrm{Div}(|N_{G}(G_{37}):C_{G}(G_{37})|).$ These data are enough.
* (22)
For $G=Fi_{23}$, $\pi(Fi_{24})=\\{2,3,5,7,11,13,17,23\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|)\cap\textrm{Div}(|N_{G}(G_{17}):C_{G}(G_{17})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|)\cap\textrm{Div}(|N_{G}(G_{13}):C_{G}(G_{13})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|),$
$11\in\textrm{Div}(|N_{G}(G_{23}):C_{G}(G_{23})|).$ These data are enough.
* (23)
For $G=Th$, $\pi(Th)=\\{2,3,5,7,11,13,19,31\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|)\cap\textrm{Div}(|N_{G}(G_{13}):C_{G}(G_{13})|)\cap\textrm{Div}(|N_{G}(G_{19}):C_{G}(G_{19})|),$
$5\in\textrm{Div}(|N_{G}(G_{31}):C_{G}(G_{31})|).$ These data are enough.
* (24)
For $G=J_{4}$, $\pi(J_{4})=\\{2,3,5,7,11,23,29,31,37,43\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|)\cap\textrm{Div}(|N_{G}(G_{23}):C_{G}(G_{23})|)\cap\textrm{Div}(|N_{G}(G_{37}):C_{G}(G_{37})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|)\cap\textrm{Div}(|N_{G}(G_{31}):C_{G}(G_{31})|),$
$7\in\textrm{Div}(|N_{G}(G_{29}):C_{G}(G_{29})|)\cap\textrm{Div}(|N_{G}(G_{43}):C_{G}(G_{43})|).$
These data are enough.
* (25)
For $G=B$, $\pi(B)=\\{2,3,5,7,11,13,17,19,23,31,47\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|)\cap\textrm{Div}(|N_{G}(G_{17}):C_{G}(G_{17})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|)\cap\textrm{Div}(|N_{G}(G_{13}):C_{G}(G_{13})|)$$\cap\textrm{Div}(|N_{G}(G_{19}):C_{G}(G_{19})|)\cap\textrm{Div}(|N_{G}(G_{37}):C_{G}(G_{37})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|)\cap\textrm{Div}(|N_{G}(G_{23}):C_{G}(G_{23})|)\cap\textrm{Div}(|N_{G}(G_{31}):C_{G}(G_{31})|),$
$23\in\textrm{Div}(|N_{G}(G_{47}):C_{G}(G_{47})|).$ These data are enough.
* (26)
For $G=M$, $\pi(M)=\\{2,3,5,7,11,13,17,19,23,29,31,41,47,59,71\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|)\cap\textrm{Div}(|N_{G}(G_{17}):C_{G}(G_{17})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|)\cap\textrm{Div}(|N_{G}(G_{13}):C_{G}(G_{13})|)$$\cap\textrm{Div}(|N_{G}(G_{19}):C_{G}(G_{19})|)\cap\textrm{Div}(|N_{G}(G_{37}):C_{G}(G_{37})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|)\cap\textrm{Div}(|N_{G}(G_{23}):C_{G}(G_{23})|)$$\cap\textrm{Div}(|N_{G}(G_{31}):C_{G}(G_{31})|)\cap\textrm{Div}(|N_{G}(G_{41}):C_{G}(G_{41})|),$$7\in\textrm{Div}(|N_{G}(G_{71}):C_{G}(G_{71})|),$
$23\in\textrm{Div}(|N_{G}(G_{47}):C_{G}(G_{47})|),$
$29\in\textrm{Div}(|N_{G}(G_{59}):C_{G}(G_{59})|).$ These data are enough.
|
arxiv-papers
| 2010-02-25T19:34:27 |
2024-09-04T02:49:08.551101
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Francesco G. Russo (Universita' degli Studi di Palermo, Palermo,\n Italy)",
"submitter": "Francesco G. Russo",
"url": "https://arxiv.org/abs/1002.4853"
}
|
1002.4855
|
# On the Schwartz space isomorphism theorem for the Riemannian symmetric
spaces
Joydip Jana Syamaprasad College
Department of Mathematics
92, S. P. Mukherjee Road
Kolkata-700 026, India
E-mail : joydipjana@gmail.com
###### Abstract.
We deduce a proof of the isomorphism theorem for certain closed subspace
$\mathcal{S}^{p}_{\Gamma}(X)$ of the $L^{p}$-Schwartz class functions
$(0<p\leq 2)$ on a Riemannian symmetric space $X$ where $\Gamma$ is a finite
subset of $\widehat{K}_{M}$. The Fourier transform considered is the Helgason
Fourier transform. Our proof relies only on the Paley-Wiener theorem for the
corresponding class of functions and hence it does not use the complicated
higher asymptotics of the elementary spherical functions.
###### Key words and phrases:
Schwartz spaces, $\delta$-spherical transform, Helgason Fourier transform
_Mathematical Subject Classification_ : 43A80, 43A85, 43A90
## Introduction
Let $X$ be a Riemannian symmetric space realized as $G/K$, where $G$ is a
connected, noncompact semisimple Lie group with finite center. Let us fix $K$
is a maximal compact subgroup of $G$. The $L^{p}$-Schwartz space isomorphism
theorem for bi-$K$-invariant functions on the group $G$ under the spherical
transform was first proved by Harish-Chandra [HC58a, HC58b, HC66] (for $p=2$),
Trombi and Varadarajan [TV71] (for $0<p<2$). Recently Anker [Ank91] gave a
remarkable short and elegant proof of the above theorem for $0<p\leq 2$.
Anker’s work does not involve the asymptotic expansion of the elementary
spherical functions which has a crucial role in the earlier works. The aim of
this paper is to extend Anker’s technique for the $L^{p}$-Schwartz class
functions (for $0<p\leq 2$) on $X$ and to establish an isomorphism theorem
under the Helgason fourier transform (HFT).
The main result of this paper is developed in two theorems Theorem 3.8 and
Theorem 5.3. In Theorem 3.8 the basic $L^{p}$-Schwartz space
$\mathcal{S}^{p}_{\delta}(X)$ is the space of operator valued
left-$\delta$-type $(\delta\in\widehat{K}_{M})$, smooth functions on $X$. HFT
when restricted to $\mathcal{S}^{p}_{\delta}(X)$ is identified with the
‘$\delta$-spherical transform’. In Theorem 3.8 we establish an isomorphism
between the Schwartz spaces $\mathcal{S}^{p}_{\delta}(X)$ and
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ under the
$\delta$-spherical transform. The image
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ is a space of matrix
valued functions with certain decay defined on a closed complex tube
$\mathfrak{a}^{*}_{\varepsilon}=\mathfrak{a}^{*}+iC^{\varepsilon\rho}$.
Explicit definition of the space
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ and the domain
$\mathfrak{a}^{*}_{\varepsilon}$ will be given in the next section.
Restriction of Theorem 3.8 to the rank-one case is a part of the result of
Eguchi and Kowata [EK76]. Theorem 3.8 also relaxes the rank restriction of the
similar result obtained in [JS07].
Theorem 5.3 further extends the isomorphism obtained in Theorem 3.8 to the
space $\mathcal{S}^{p}_{\Gamma}(X)$ of scalar valued $K$-finite Schwartz class
functions on $X$ for which the left $K$-types lie in a fixed finite subset
$\Gamma\subset\widehat{K}_{M}$. The transform considered in Theorem 5.3 is the
HFT.
We shall closely follow the notations of [Hel, Hel94]. Some basic definitions
and results used in this paper are given in the following section.
## 1\. Notation and Preliminaries
Let $G$ be a connected, noncompact semisimple Lie group with finite center and
$K$ be a maximal compact subgroup of $G$. Let $\theta$ be the Cartan
involution corresponding to $K$. Let $X$ be a Riemannian symmetric space
realized as $G/K$. Let $\mathfrak{g}$ be the Lie algebra of $G$ and
$\mathfrak{g}_{\mathbb{C}}$ be its complexification. The Hermitian norm of
both $\mathfrak{g}$ and $\mathfrak{g}_{\mathbb{C}}$ will be denoted by the
notation $\|\cdot\|$. We denote $\mathfrak{k}$ for the Lie algebra of the
maximal compact subgroup $K$ of $G$. Let
$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{s}$ be the Cartan decomposition of
the Lie algebra. We fix a maximal abelian subspace $\mathfrak{a}$ in
$\mathfrak{s}$. Let $A$ be analytic subgroup of $G$ with the Lie algebra
$\mathfrak{a}$. Let $\mathfrak{a}^{*}$ and $\mathfrak{a}^{*}_{\mathbb{C}}$
respectively be the real dual of $\mathfrak{a}$ and its complexification. The
Killing form induces a scalar product on $\mathfrak{a}$ and hence on
$\mathfrak{a}^{*}$. We shall denote $\langle\cdot,\cdot\rangle_{1}$ for the
$\mathbb{C}$-bilinear extension of that scalar product to
$\mathfrak{a}^{*}_{\mathbb{C}}$. A semisimple Lie group $G$ is said to be of
‘real rank-$n$’ if $dim\mathfrak{a}=n$ and the corresponding symmetric spaces
$X$ realized as $X=G/K$ are called ‘rank-$n$’ symmetric spaces.
Let $\Sigma$ be the root system associated with the pair
$(\mathfrak{g},\mathfrak{a})$. For each $\alpha\in\Sigma$ we write
$\mathfrak{g}_{\alpha}$ for the corresponding root space. Let $M^{\prime}$ and
$M$ respectively be the normalizer and the centralizer of $A$ in $K$. The
quotient $W=M^{\prime}/M$ be the Weyl group associated with the root system
$\Sigma$.
Let us choose and fix a system of positive roots which we denote by
$\Sigma^{+}$. Let $\mathfrak{a}^{+}$ be the corresponding positive Weyl
chamber and $\overline{\mathfrak{a}^{+}}$ be its closer. We denote
$\mathfrak{a}^{*+}$ and $\overline{\mathfrak{a}^{*+}}$ for the similar
chambers in $\mathfrak{a}^{*}$. We put $A^{+}=\exp\mathfrak{a}^{+}$ and
$\overline{A^{+}}=\exp\overline{\mathfrak{a}^{+}}$. The element
$\rho\in\mathfrak{a}^{*}$ is denoted by
(1.1)
$\rho(H)=\frac{1}{2}\sum_{\alpha\in\Sigma^{+}}m_{\alpha}\alpha(H),~{}~{}\mbox{
where}~{}m_{\alpha}=dim\mathfrak{g}_{\alpha}~{}\mbox{and}~{}H\in\mathfrak{a}.$
Let $\Sigma_{0}\subset\Sigma$ be the set of all indivisible roots and
$\Sigma_{0}^{+}=\Sigma_{0}\cap\Sigma^{+}$.
$\mathfrak{n}=\oplus_{\alpha\in\Sigma^{+}}\mathfrak{g}_{\alpha}$ is a
nilpotent subalgebra of $\mathfrak{g}$. Let $N$ be the nilpotent subgroup of
$G$ with the Lie algebra $\mathfrak{n}$. The Iwasawa decomposition of the
group $G$ is given as $G=KAN$. The map $(k,a,n)\mapsto kan$ is a
diffeomorphism from $K\times A\times N$ onto $G$. Let
$\mathcal{H}:G\rightarrow\mathfrak{a}$ and
$\mathcal{A}:G\rightarrow\mathfrak{a}$ are the $\mathfrak{a}$ projections of
$g\in G$ in Iwasawa $KAN$ and $NAK$ decompositions respectively. These two
projections are related by $\mathcal{A}(g)=-\mathcal{H}(g^{-1})$ for each
$g\in G$. Any element $g\in G$ can therefore be written as
$g=k\exp{\mathcal{H}(g)}n=n_{1}\exp{\mathcal{A}(g)}k_{1}$. In the Iwasawa
$KAN$ decomposition the Haar measure of the group $G$ is given by
(1.2)
$\int_{G}f(g)dg=const.\int_{K}dk\int_{\mathfrak{a}^{+}}e^{2\rho(\mathcal{H}(g))}d\mathcal{H}(g)\int_{N}dn~{}f(k\exp{\mathcal{H}(g)}n),$
where $const.$ is a normalizing constant. The ‘Cartan decomposition’ of the
group is $G=K\overline{A^{+}}K=K\exp\overline{\mathfrak{a}^{+}}K$. Let $g^{+}$
denote the $\mathfrak{a}^{+}$ component of $g\in G$, and we denote
$|g|=\|g^{+}\|$. We have a basic estimate: for some constant $c>0$
(1.3) $\|\mathcal{H}(g)\|\leq c|g|,\mbox{~{}for each~{} }g\in G.$
The Haar measure for the Cartan decomposition is given by
(1.4)
$\int_{G}f(g)dg=const.\int_{K}dk\int_{\mathfrak{a}^{+}}\Delta(H)dh\int_{K}dk^{\prime}f(k\exp
Hk^{\prime}),~{}~{}(H\in\overline{\mathfrak{a}^{+}})$
where $\Delta(H)=\prod_{\alpha\in\Sigma^{+}}\sinh^{m_{\alpha}}H$ and $const.$
is a positive normalizing constant. We shall be using the following estimate
for the density $\Delta(H)$:
(1.5) $0\leq\Delta(H)\leq ce^{2\rho(H)},\mbox{~{}for~{}}H\in\mathfrak{a}^{+}.$
A function $f$ on $G$ is said to be ‘bi-$K$-invariant’ if
$f(k_{1}gk_{2})=f(g)$ for all $k_{1},k_{2}\in K$ and $g\in G$. We refer a
function a function as ‘right-$K$-invariant’ if invariant under the right $K$
action on $G$ that is $f(gk)=f(g)$ for all $k\in K,g\in G$. Althrough in this
paper we shall consider a function on the symmetric space $X=G/K$ as a
right-$K$-invariant function on the group $G$. For any function space
$\mathfrak{F}(G)$ on $G$ or $\mathfrak{F}(G/K)$ on $X$, we shall denote
$\mathfrak{F}(G//K)$ for the corresponding subspace of bi-$K$-invariant
functions.
We denote $\mathcal{C}^{\infty}(G)$ for the set of all smooth functions on
$G$. We fix a basis $\\{X_{j}\\}$ for the Lie algebra $\mathfrak{g}$. Let
$\mathcal{U}(\mathfrak{g})$ be the ‘universal enveloping algebra’ over
$\mathfrak{g}$. Let $D_{1}\cdots D_{m},E_{1}\cdots
E_{n}\in\mathcal{U}(\mathfrak{g})$, then the action of
$\mathcal{U}(\mathfrak{g})$ on a function $f\in\mathcal{C}^{\infty}(G)$ is
defined as follows:
$\displaystyle f(D_{1}\cdots D_{m},x,E_{1}\cdots E_{n})=$
$\displaystyle\frac{d}{dt_{1}}\mid_{t_{1}=0}\cdots\frac{d}{dt_{m}}\mid_{t_{m}=0}~{}\frac{d}{ds_{1}}\mid_{s_{1}=0}\cdots\frac{d}{ds_{n}}\mid_{s_{n}=0}f(\exp{(t_{1}D_{1})}\cdots\exp{(t_{m}D_{m})}x\exp{(s_{1}E_{1})}\cdots\exp{(s_{n}E_{n})}).$
Let $b_{ij}=\mathfrak{B}(X_{i},X_{j})$ and $(b^{ij})$ be the inverse of the
matrix $(b_{ij})$. We now define a distinguished element, called the ‘Casimir
element’, of $\mathcal{U}(\mathfrak{g})$ by
$\Omega=\sum_{i,j}b^{ij}X_{i}X_{j}$. The differential operator $\Omega$ lies
in the center of $\mathcal{U}(\mathfrak{g})$. The action of the ‘Laplace-
Beltrami operator’ $\mathbf{L}$ on $X$ is defined by the action of $\Omega$:
(1.6) $\mathbf{L}f(xK)=f(x,\Omega),\hskip 36.135ptx\in G.$
Let us briefly describe the method of construction of a family of rank-one
symmetric spaces, the rank-one reductions, which are totally geodesic
submanifolds of the general rank symmetric space $G/K$ ( see [Hel01, Ch. IX,
§2], [Hel, Ch. IV, §6], [Kna03] ). Let $\beta$ be an indivisible root of the
system $\Sigma$ and $\mathfrak{g}_{(\beta)}$ be the Lie subalgebra of
$\mathfrak{g}$ generated by the root spaces
$\mathfrak{g}_{\beta},~{}\mathfrak{g}_{2\beta},~{}\mathfrak{g}_{-\beta}$ and
$\mathfrak{g}_{-2\beta}$. The subalgebra $\mathfrak{g}_{(\beta)}$ is stable
under the Cartan involution and it is simple. Let $G_{(\beta)}$ be the
analytic subgroup of $G$ corresponding to the Lie subalgebra
$\mathfrak{g}_{(\beta)}$. The Iwasawa decomposition of $G_{(\beta)}$ be
$G_{(\beta)}=K_{(\beta)}A_{(\beta)}N_{(\beta)}$ where $K_{(\beta)}=K\cap
G_{(\beta)}$, $A_{(\beta)}=A\cap G_{(\beta)}$ and $N_{(\beta)}=N\cap
G_{(\beta)}$. Also the centralizer of $A_{(\beta)}$ in $K_{(\beta)}$ is
$M_{(\beta)}=M\cap G_{(\beta)}$. The abelian subgroup $A_{(\beta)}$ is one-
dimensional and its Lie algebra $\mathfrak{a}_{(\beta)}$ is generated by the
element $H_{\beta}\in\mathfrak{a}$ determined by
$\lambda(H_{\beta})=\langle\lambda,\beta\rangle$. Hence $G_{(\beta)}$ is of
real-rank-one and consequently $G_{(\beta)}/K_{(\beta)}$ is a rank-one
symmetric space.
The restricted roots of $G_{(\beta)}$ are $\\{\beta,2\beta,-\beta,-2\beta\\}$
or $\\{\beta,-\beta\\}$ according as $2\beta\in\Sigma$ or not. Let us further
consider $\beta$ to be a positive root of $G_{(\beta)}$ thus the Lie algebra
$\mathfrak{n}_{\beta}$ of $N_{(\beta)}$ is the sum of the root spaces
$\mathfrak{g}_{\beta}$ and $\mathfrak{g}_{2\beta}$. We write $\rho_{(\beta)}$
for the $\rho$-function of $\mathfrak{g}_{(\beta)}$. It can be shown that
[Hel, B.3, page: 483] $\rho(H_{\beta})\geq\rho_{(\beta)}(H_{\beta})$ for all
$\beta\in\Sigma_{0}^{+}$. The equality holds only when $\beta$ is simple and
in that case $\rho_{(\beta)}$ is exactly the restriction of $\rho$ to
$\mathfrak{g}_{(\beta)}$. For each $\beta\in\Sigma_{0}$ the restriction
$\lambda_{\beta}$ of $\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$ to
$\mathfrak{a}_{(\beta)}$ is given by the expression
$\lambda_{\beta}=\frac{{\langle\lambda,\beta\rangle}_{1}}{\langle\beta,\beta\rangle}\beta$.
Let $\pi_{\lambda}$ ($\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$) be the
spherical principal series representations of $G$ realized on the Hilbert
space $L^{2}(K/M)$ and given by the following:
(1.7)
$\left\\{\pi_{\lambda}(g)\zeta\right\\}(kM)=e^{-(i\lambda+\rho)\mathcal{H}(g^{-1}k)}\zeta(\mathcal{K}(g^{-1}k)M),$
where $\lambda\in\mathfrak{a}^{*}_{\mathbb{C}},~{}\zeta\in L^{2}(K/M)$ and
$\mathcal{K}(g^{-1}k)$ denotes the $K$ part of $g^{-1}k$ in the Iwasawa $KAN$
decomposition. The spherical functions $\varphi_{\lambda}(\cdot)$, given by
the following formula
(1.8)
$\varphi_{\lambda}(g)=\int_{K}e^{(i\lambda-\rho)\mathcal{H}(g^{-1}k)}dk,~{}\mbox{where}~{}g\in
G,~{}\lambda\in\mathfrak{a}^{*}_{\mathbb{C}},$
are the matrix coefficient of the principal series representations. For each
$g\in G$ and $\omega\in W$ $\varphi_{\omega\lambda}(g)=\varphi_{\lambda}(g)$.
We shall use the following basic estimates for the elementary spherical
functions for our purpose.
1. (i)
For each $H\in\overline{\mathfrak{a}^{+}}$ and
$\lambda\in\overline{{\mathfrak{a}^{*}}^{+}}$, we have
(1.9) $0<|\varphi_{-i\lambda}(\exp H)|\leq e^{\lambda(H)}\varphi_{0}(\exp H),$
here, $\varphi_{0}(\cdot)$ is the spherical function corresponding to
$\lambda=0$. For a proof of the above estimate see [GV88], Proposition 4.6.1.
2. (ii)
For all $g\in G$, $0<\varphi_{0}(g)\leq 1$ [GV88, Proposition 4.6.3] . Also
for $\overline{\mathfrak{a}^{+}}$ we have the following estimate
(1.10) $\displaystyle e^{\rho(H)}\leq\varphi_{0}(\exp
H)\leq~{}{\ss}(1+\|H\|)^{c_{\mathfrak{a}}}~{}e^{\rho(H)},$
where, ${\ss},c_{\mathfrak{a}}>0$ are group dependent constants. This is an
work of Harish-Chandra, the above optimal estimate was obtained by Anker
[Ank87].
Let $\mathcal{D}(G)$ be the subspace of $\mathcal{C}^{\infty}(G)$ generated by
the compactly supported scalar valued smooth functions on $G$. For each
$f\in\mathcal{D}(G//K)$ the ‘spherical Fourier transform’ $\mathcal{S}f$ is
defined by
(1.11)
$\mathcal{S}f(\lambda)=\int_{G}f(x)\varphi_{-\lambda}(x)dx,~{}\mbox{where}~{}\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}.$
The inversion of the spherical transform is given by
(1.12)
$f(x)=\frac{1}{|W|}\int_{\mathfrak{a}^{*}_{\mathbb{C}}}\varphi_{\lambda}(x)~{}\mathcal{S}f(\lambda)~{}|\mathbf{c}(\lambda)|^{-2}d\lambda,$
where $|W|$ is the cardinality of the group $W$ and $\mathbf{c}(\lambda)$ is
the ‘Harish-Chandra $\mathbf{c}$-function’. For our purpose we shall use the
following estimate [Ank92]: there exists constants $a,b>0$ such that
(1.13) $|\mathbf{c}(\lambda)|^{-2}\leq~{}a(\|\lambda\|+1)^{b},\mbox{ ~{}for
all~{}}\lambda\in\mathfrak{a}^{*}.$
For $f\in\mathcal{D}(X)$, the Helgason Fourier transform (HFT) $\mathcal{F}f$
is a function on $\mathfrak{a}^{*}_{\mathbb{C}}\times K/M$ and it is defined
by ( [Hel94], Ch. III, $\S$ 1)
(1.14)
$\mathcal{F}f(\lambda,kM)=\int_{G}f(x)~{}e^{(i\lambda-\rho)(H(x,kM))}dx$
where the function $H:G\times K/M\mapsto\mathfrak{a}$ is given by
$H(x,kM)=\mathcal{H}(x^{-1}k)$. For the sake of simplicity we fix the
notational convention $\mathcal{F}f(\lambda,kM)=\mathcal{F}f(\lambda,k)$. We
should note that, for a bi-$K$-invariant function (that is a
left-$K$-invariant function) $g$ on $X$, the HFT reduces to the spherical
transform:
$\mathcal{F}g(\lambda,k)=\mathcal{F}g(\lambda,e)=\mathcal{S}g(\lambda)$.
The inversion formula for HFT [Hel94, Ch.-III, Theorem 1.3] for
$f\in\mathcal{D}(X)$ is as follows:
(1.15)
$f(x)=\frac{1}{|W|}\int_{\mathfrak{a}^{*}}\int_{K}\mathcal{F}f(\lambda,k)~{}e^{-(i\lambda+\rho)(\mathcal{H}(x^{-1}k))}|\mathbf{c}(\lambda)|^{-2}d\lambda~{}dk.$
Let $\delta$ be a unitary irreducible representation of K realized on a finite
dimensional vector space $V_{\delta}$ with an inner product
$\langle\cdot,\cdot\rangle$. Let us denote $dimV_{\delta}=d_{\delta}$. We
denote by $\widehat{K}$ the set of equivalence classes of unitary irreducible
representations of $K$ and by customary abuse of notation regard each element
of $\widehat{K}$ as a representation from its equivalence class. For each
$\delta\in\widehat{K}$, let $\chi_{\delta}$ stand for the character of the
representation $\delta$ and $V_{\delta}^{M}=\\{v\in
V_{\delta}~{}|~{}\delta(m)v=v\mbox{~{}for all~{}}m\in M\\}$ is the subspace of
$V_{\delta}$ fixed under $\delta|_{M}$. Let $\widehat{K}_{M}$ stands for the
subset of $\widehat{K}$ consisting of $\delta$ for which
$V_{\delta}^{M}\neq\\{0\\}$ and we will mostly be interested in
representations $\delta\in\widehat{K}_{M}$. We set an orthogonal basis
$\\{v_{j}\\}_{1\leq j\leq d_{\delta}}$ of $V_{\delta}$ and we assume that
$\\{v_{1},\cdots v_{\ell_{\delta}}\\}$ generates $V_{\delta}^{M}$ where
$dimV_{\delta}^{M}=\ell_{\delta}$.
We also define a norm for each unitary irreducible representation of $K$. Let
$\Theta$ be the restriction of the Cartan-Killing form $\mathfrak{B}$ to
$\mathfrak{k}\times\mathfrak{k}$. Let $\mathcal{K}_{1},...,\mathcal{K}_{r}$ be
a basis for $\mathfrak{k}$ over $\mathbb{R}$ orthonormal with respect to
$\Theta$. Let
$\omega_{\mathfrak{k}}=-(\mathcal{K}_{1}^{2}+...+\mathcal{K}_{r}^{2})$ be the
Casimir element of $K$. Clearly $\omega_{\mathfrak{k}}$ is a differential
operator which commutes with both left and right translations of $K$. Thus
$\delta(\omega_{\mathfrak{k}})$ commutes with $\delta(k)$ for all $k\in K$.
Hence by Schur’s lemma [Sug90, Ch.I, Theorem 2.1]:
$\delta(\omega_{\mathfrak{k}})=c(\delta)\delta(e)$ where
$c(\delta)\in\mathbb{C}$. As $\delta(\mathcal{K}_{i})$ $(1\leq i\leq r)$ are
skew-adjoint operators, $c(\delta)$ is real and $c(\delta)\geq 0$. We define
$|\delta|^{2}=c(\delta)$, for $\delta\in\widehat{K}_{M}$. As,
$\delta\in\widehat{K}_{M}$, $\delta(k)$ is a unitary matrix of order
$d_{\delta}\times d_{\delta}$. So
$\|\delta(k)\|_{\mathbf{2}}=\sqrt{d_{\delta}}$ where $\|\cdot\|_{\mathbf{2}}$
denotes the Hilbert Schmidt norm. Also, from Weyl’s dimension formula we can
choose an $r\in\mathbb{Z}^{+}$ and a positive constant $c$ independent of
$\delta$ such that $\|\delta(k)\|_{\mathbf{2}}\leq c(1+|\delta|)^{r}$ for all
$k\in K$. Thus, $d_{\delta}\leq c^{\prime}(1+|\delta|)^{2r}$ with
$c^{\prime}>0$ independent of $\delta$.
For any $f\in\mathcal{C}^{\infty}(X)$ we put:
(1.16) $f^{\delta}(x)=d_{\delta}\int_{K}f(kx)\delta(k^{-1})dk.$
Clearly, $f^{\delta}$ is a $\mathcal{C}^{\infty}$ map from $X$ to
$Hom(V_{\delta},V_{\delta})$ satisfying
(1.17) $f^{\delta}(kx)=\delta(k)f^{\delta}(x),~{}~{}\mbox{for all~{}}x\in
X,k\in K.$
Any function satisfying the property (1.17) will be referred to as (a
$d_{\delta}\times d_{\delta}$ matrix valued) left $\delta$-type function. For
any function space $\mathcal{E}(X)\subseteq\mathcal{C}^{\infty}(X)$, we write
$\mathcal{E}_{\delta}(X)=\\{f^{\delta}~{}|~{}f\in\mathcal{E}(X)\\}$. We shall
denote by $\check{\delta}$ the contragradient representation of the
representation $\delta\in\widehat{K}_{M}$. and a function $f$ will be called a
scalar valued left $\check{\delta}$-type function if $f\equiv
d_{\delta}\chi_{\delta}\ast f$, where the operation $\ast$ is the convolution
over $K$. For any class of scalar valued functions $\mathcal{G}(X)$ we shall
denote
$\mathcal{G}(\check{\delta},X)=\\{g\in\mathcal{G}(X)~{}|~{}g\equiv
d_{\delta}\chi_{\delta}\ast g\\}.$
The following theorem, due to Helgason, identifies the two classes
$\mathcal{D}_{\delta}(X)$ and $\mathcal{D}(\check{\delta},X)$ corresponding to
each $\delta\in\widehat{K}_{M}$.
###### Proposition 1.1.
_[Helgason[Hel94, Ch.III, Proposition 5.10]]_
The map $\mathcal{Q}:f\mapsto g$, $g(x)=tr\left(f(x)\right)$ $(x\in X)$ is a
homeomorphism from $\mathcal{D}_{\delta}(X)$ onto
$\mathcal{D}(\check{\delta},X)$ and its inverse is given by $g\mapsto
f=g^{\delta}$.
###### Remark 1.2.
For each $\delta\in\widehat{K}_{M}$, the space
$\mathcal{D}(X,Hom(V_{\delta},V_{\delta}))$ of $\mathcal{C}^{\infty}$
functions on $X$ taking values in $Hom(V_{\delta},V_{\delta})$, carries the
inductive limit topology of the Fréchet spaces
$\mathcal{D}^{R}(X,Hom(V_{\delta},V_{\delta}))=\\{F\in\mathcal{D}(X,Hom(V_{\delta},V_{\delta}))~{}|~{}suppF\subseteq\overline{B^{R}(0)}\\},$
for $R=0,1,2,\cdots.$ As $\mathcal{D}(\check{\delta},X)\subset\mathcal{D}(X)$,
so the natural topology of $\mathcal{D}(\check{\delta},X)$ is the inherited
subspace topology.
A consequence of the Peter-Weyl theorem can be stated [Hel, Ch.IV, Corollary
3.4] in the form that any $f\in\mathcal{C}^{\infty}(X)$ has the decomposition
(1.18) $f(x)=\sum_{\delta\in\widehat{K}_{M}}tr(f^{\delta}(x)).$
A function $f\in\mathcal{C}^{\infty}(X)$ is said to be ‘left-$K$ finite’ if
there exists a finite subset $\Gamma(f)\subset\widehat{K}_{M}$ (depending on
the function $f$) such that $tr(f^{\gamma}(\cdot))\equiv 0$ for all
$\gamma\in\widehat{K}_{M}\setminus\Gamma(f)$. For any class
$\mathfrak{H}(X)\subseteq\mathcal{C}^{\infty}(X)$ of function we shall denote
$\mathfrak{H}(X)_{K}$ for its left $K$ finite subclass. Let $\Gamma$ be a
fixed subset (finite or infinite) of $\widehat{K}_{M}$. Then we shall use the
notation $\mathfrak{H}_{\Gamma}(X)$ for the subclass of $\mathfrak{H}(X)$
(1.19)
$\mathfrak{H}_{\Gamma}(X)=\\{g\in\mathfrak{H}(X)~{}|~{}g^{\delta}(\cdot)\equiv
0,\mbox{~{}for all~{}}\delta\in\widehat{K}_{M}\setminus\Gamma\\}.$
For each $f\in\mathcal{D}(X)$ and $\delta\in\widehat{K}_{M}$, we define the
$\delta$ projection of it’s HFT $\mathcal{F}f$ as follows:
(1.20)
$\left(\mathcal{F}f\right)^{\delta}(\lambda,k)=d_{\delta}\int_{K}\mathcal{F}f(\lambda,k_{1}k)\delta(k_{1}^{-1})dk_{1},~{}~{}\mbox{where}~{}\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}~{}\mbox{and}~{}k\in
K.$
The HFT $\mathcal{F}(f^{\delta})$ of $f^{\delta}$ is also defined by the
formula (1.14), in this case the integration is taken over each matrix entry.
###### Lemma 1.3.
For each $f\in\mathcal{D}(X)$ and $\delta\in\widehat{K}_{M}$ the following are
true
1. (i)
$\left(\mathcal{F}f\right)^{\delta}(\lambda,k)=\delta(k)\left(\mathcal{F}f\right)^{\delta}(\lambda,e),$
2. (ii)
$\mathcal{F}(f^{\delta})(\lambda,k)=\left(\mathcal{F}f\right)^{\delta}(\lambda,k),\mbox{~{}for
all}~{}\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}~{}\mbox{and}~{}k\in K.$
###### Proof.
Part (i) of the Lemma follows trivially from (1.20). Part (ii) can be deduced
from the following
(1.21) $\displaystyle\mathcal{F}(f^{\delta})(\lambda,kM)$ $\displaystyle=$
$\displaystyle\int_{X}f^{\delta}(x)e^{(i\lambda-1)\mathcal{H}(x^{-1}k)}dx,$
$\displaystyle=$ $\displaystyle
d_{\delta}\int_{X}\left\\{\int_{K}f(k_{1}x)\delta(k_{1}^{-1})dk_{1}\right\\}e^{(i\lambda-1)\mathcal{H}(x^{-1}k)}dx$
Now the desired result follows from (1.21) by a simple application of the
Fubini’s theorem.
## 2\. The $\delta$-spherical transform
Let us now define the ‘$\delta$-spherical transform’ on
$\mathcal{D}_{\delta}(X)$. Most of the basic analysis was done by Helgason
[Hel94] on $\mathcal{D}({\check{\delta}},X)$, we shall follow those results
closely and prove them on $\mathcal{D}_{\delta}(X)$ using the homeomorphism
$\mathcal{Q}$, defined in Proposition 1.1.
###### Definition 2.1.
For $f\in\mathcal{D}_{\delta}(X)$ the $\delta$-spherical transform
$\widetilde{f}$ is an operator valued function on
$\mathfrak{a}^{*}_{\mathbb{C}}$ and is given by
(2.1)
$\widetilde{f}(\lambda)=d_{\delta}\int_{G}trf(x)\Phi_{\overline{\lambda},\delta}^{*}(x)dx$
where for each $\delta\in\widehat{K}_{M}$ and
$\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$, the function
(2.2)
$\Phi_{\lambda,\delta}(x)=\int_{K}e^{-(i\lambda+1)H(x^{-1}k)}\delta(k)dk,\hskip
14.45377ptx\in G,$
is called the ‘generalized spherical function’ of class $\delta$. For each
$x\in G$, $\Phi_{\lambda,\delta}(x)$ is an operator in
$Hom(V_{\delta},V_{\delta})$. Taking point-wise adjoint leads to the
expression
(2.3)
$\Phi_{\overline{\lambda},\delta}^{*}(x):=\Phi_{\overline{\lambda},\delta}^{*}(x)=\int_{K}e^{(i\lambda-1)H(x^{-1}k)}\delta(k^{-1})dk,\hskip
14.45377ptx\in G.$
###### Remark 2.2.
From the Iwasawa decomposition, if $x\in G$ and $\tau\in K$, $\mathcal{H}(\tau
x)=\mathcal{H}(x)$. Hence, the expressions (2.2) and (2.3) show that both
$\Phi_{\lambda,\delta}$ and $\Phi_{\overline{\lambda},\delta}^{*}$ can be
considered as functions on the space $X=G/K$.
In the following proposition we list out some basic properties of the
generalized spherical functions that we will be using.
###### Proposition 2.3.
1. (i)
For each $x\in X$, the function $\lambda\mapsto\Phi_{\lambda,\delta}(x)$ is
holomorphic on $\mathfrak{a}^{*}_{\mathbb{C}}$.
2. (ii)
Let $\delta\in\widehat{K}_{M}$ and $\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$.
Then for each $x\in X$ and $k\in K$ we have
(2.4)
$\Phi_{{\lambda},\delta}(kx)=\delta(k)\Phi_{{\lambda},\delta}(x)\mbox{~{}~{}and~{}~{}}\Phi_{\overline{\lambda},\delta}^{*}(kx)=\Phi_{\overline{\lambda},\delta}^{*}(x)\delta(k^{-1}).$
Let $v\in V_{\delta}$ and $m\in M$ then
(2.5)
$\delta(m)\left(\Phi_{\overline{\lambda},\delta}^{*}(x)v\right)=\Phi_{\overline{\lambda},\delta}^{*}(x)v.$
3. (iii)
_[Helgason[Hel94, Ch.III, Theorem 5.15 ]]_ For each
$\delta\in\widehat{K}_{M}$, $\omega\in W$ and for all
$\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$, the restrictions
$\Phi_{\lambda,\delta}|_{A}$ and $\Phi_{\overline{\lambda},\delta}^{*}|_{A}$
satisfy the relations
(2.6) $\displaystyle\Phi_{\lambda,\delta}|_{A}Q^{\delta}(\lambda)$
$\displaystyle=\Phi_{\omega\lambda,\delta}|_{A}Q^{\delta}(\omega\lambda),$
(2.7) $\displaystyle
Q^{\delta}(\lambda)^{-1}\Phi_{\overline{\lambda},\delta}^{*}|_{A}$
$\displaystyle=Q^{\delta}(\omega\lambda)^{-1}\Phi_{-\overline{\omega\lambda},\delta}^{*}|_{A}$
where $Q^{\delta}(\lambda)$ is a $(\ell_{\delta}\times\ell_{\delta})$ matrix
whose entries are certain constant coefficient polynomials in
$\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$ (see [Hel94, Ch. III, §2] for
details ). Furthermore, both sides of (2.7) are holomorphic for all
$\lambda\in\mathbb{C}$, implying that
$\Phi_{\overline{\lambda},\delta}^{*}|_{A}$ is divisible by
$Q^{\delta}(\lambda)$ in the ring of entire functions.
4. (iv)
For each fixed $\lambda$ and $\delta$, the function
$\Phi_{{\lambda},\delta}(x)$ and its adjoint are both joint eigenfunctions of
all $G$-invariant differential operators of $X$. Particularly, for the
Laplace-Beltrami operator $\mathbf{L}$, the eigenvalues are as follows:
(2.8)
$\left(\mathbf{L}\Phi_{\lambda,\delta}\right)(x)=-\left(\langle\lambda,\lambda\rangle_{1}+\|\rho\|^{2}\right)\Phi_{\lambda,\delta}(x),~{}~{}x\in
X.$
5. (v)
For each $\delta\in\widehat{K}_{M}$, the generalized spherical function
corresponding to $\delta$ is related with the elementary spherical function by
the following differential equation [Hel94, Ch.III, §5, Corollary 5.17]
(2.9)
$\Phi_{\lambda,\delta}(gK)|_{V_{\delta}^{M}}=\left(\mathbf{D}^{\delta}\varphi_{\lambda}\right)(g)Q^{\delta}(\lambda)^{-1}$
where $\mathbf{D}^{\delta}$ is a differential operator matrix of order
$(d_{\delta}\times\ell_{\delta})$. Individual matrix entries of
$\mathbf{D}^{\delta}$ are certain constant coefficient differential operators
on $G$.
6. (vi)
For any
$\textbf{g}_{1},\textbf{g}_{2}\in\mathcal{U}(\mathfrak{g}_{\mathbb{C}})$ there
exist constants $c=c(\textbf{g}_{1},\textbf{g}_{2})$,
$b=b(\textbf{g}_{1},\textbf{g}_{2})$ and $c_{0}>0$ so that
(2.10)
$\|\Phi_{\lambda,\delta}(\textbf{g}_{1},x,\textbf{g}_{2})\|_{\mathbf{2}}\leq
c(1+|\delta|)^{b}(1+\|\lambda\|)^{b}\varphi_{0}(x)e^{c_{0}\|\Im{\lambda}\|(1+|x|)},$
for all $x\in X$ and $\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$.
###### Proof.
Property (2.4) follows trivially from the definition of the generalized
spherical function. (2.5) also follows from (2.2) as below:
$\displaystyle\delta(m)\left(\Phi_{\overline{\lambda},\delta}^{*}(x)v\right)$
$\displaystyle=\left\\{\int_{K}e^{(i\lambda-1)\mathcal{H}(x^{-1}k)}\delta(mk^{-1})dk\right\\}v$
$\displaystyle=\left\\{\int_{K}e^{(i\lambda-1)\mathcal{H}(x^{-1}k^{\prime}m)}\delta(k^{\prime-1})dk^{\prime}\right\\}v.$
The last line follows by a simple change of variable $mk^{-1}$ to
$k^{\prime-1}$. In the last expression above, let
$x^{-1}k^{\prime}=\mathcal{K}(x^{-1}k^{\prime})(\exp{\mathcal{H}(x^{-1}k^{\prime})})n^{\prime}$
for some $n^{\prime}\in N$. As $M$ normalizes $N$ and centralizes $A$ we have
$x^{-1}k^{\prime}m=\mathcal{K}(x^{-1}k^{\prime})m(\exp{\mathcal{H}(x^{-1}k^{\prime})})\mathcal{N}(x^{-1}k^{\prime}).$
This shows that
$\mathcal{H}(x^{-1}k^{\prime})=\mathcal{H}(x^{-1}k^{\prime}m)$. Thus
$\displaystyle\delta(m)\left(\Phi_{\overline{\lambda},\delta}^{*}(x)v\right)$
$\displaystyle=\left\\{\int_{K}e^{(i\lambda-1)\mathcal{H}(x^{-1}k^{\prime})}\delta(k^{\prime-1})dk^{\prime}\right\\}v=\Phi_{\overline{\lambda},\delta}^{*}(x)v.$
A proof of property (ii) may be found in [Hel94, Ch.III, §1 (6)] and [Hel,
Ch.II, Corollary 5.20]. The estimate (2.10) is a work of Arthur [Art79].
###### Remark 2.4.
The property (2.5) clearly shows that for each $x\in X$ the operator
$\Phi_{\overline{\lambda},\delta}^{*}(x)$ maps $V_{\delta}$ to
$V_{\delta}^{M}$. Hence $\Phi_{\overline{\lambda},\delta}^{*}(x)$ is a
$d_{\delta}\times d_{\delta}$ matrix whose only the first $\ell_{\delta}$ rows
can nonzero. Consequently, for each $x\in X$, $\Phi_{\lambda,\delta}(x)$ is a
$d_{\delta}\times d_{\delta}$ matrix of which only the first $\ell_{\delta}$
columns can be nonzero. In other words, the operator
$\Phi_{\lambda,\delta}(x)$ vanishes identically on the orthogonal complement
of the subspace $V_{\delta}^{M}$.
###### Remark 2.5.
In the case of the rank-one symmetric spaces the Kostant matrix $Q^{\delta}$
reduces to a constant coefficient polynomial (Kostant polynomial) [Hel94,
Theorem 11.2,S11, Ch.III]. The degree of the Kostant polynomials depends on
the choice of $\delta\in\widehat{K}_{M}$. Furthermore all the zeros of the
Kostant polynomials lie on the open lower half of the imaginary axis.
The general rank analogue of the above result states that $\det
Q^{\delta}(\lambda)\neq 0$ for all
$\lambda\in\mathfrak{a}^{*}+i\overline{\mathfrak{a}^{*+}}$. This is an easy
consequence of Lemma 2.11 and Proposition 4.1 of [Hel94, Ch.III].
###### Lemma 2.6.
If $f\in\mathcal{D}_{\delta}(X)$, where $\delta\in\widehat{K}_{M}$, then
$\mathcal{F}f(\lambda,e)=\widetilde{f}(\lambda)$ for all
$\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$.
###### Proof.
For any $f\in\mathcal{D}_{\delta}(X)$, using the topological isomorphism
$\mathcal{Q}$ as described in Proposition 1.1, we get
$trf(\cdot)\in\mathcal{D}(\check{\delta},X)$ and also
$f(x)=d_{\delta}\int_{K}trf(kx)\delta(k^{-1})dk$. Now from the definition of
HFT (1.14) we get:
$\displaystyle\mathcal{F}f(\lambda,e)$
$\displaystyle=\int_{G}f(x)e^{(i\lambda-\rho)\mathcal{H}(x^{-1}e)}dx,$ (2.11)
$\displaystyle=d_{\delta}\int_{G}\int_{K}trf(kx)\delta(k^{-1})dke^{(i\lambda-\rho)\mathcal{H}(x^{-1}e)}dx.$
A simple application of the Fubini theorem and a substitution $kx=y$ in the
integrand of (2) gives the following.
$\displaystyle\mathcal{F}f(\lambda,e)$
$\displaystyle=d_{\delta}\int_{G}trf(y)\left\\{\int_{K}e^{(i\lambda-\rho)\mathcal{H}(y^{-1}k)}\delta(k^{-1})dk\right\\}dy,$
$\displaystyle=d_{\delta}\int_{G}trf(y)\Phi_{\overline{\lambda},\delta}^{*}(y)dy=\widetilde{f}(\lambda).$
###### Lemma 2.7.
Let $f\in\mathcal{D}_{\delta}(X)$, then the inversion formula for the
$\delta$-spherical transform (Definition 2.1) is given by:
(2.12)
$f(x)=\frac{1}{|W|}\int_{\mathfrak{a}^{*}}\Phi_{\lambda,\delta}(x)\widetilde{f}(\lambda)|\mathbf{c}(\lambda)|^{-2}d\lambda.$
Furthermore we get:
$\int_{G}{\|f(x)\|_{\mathbf{2}}}^{2}dx=\frac{1}{|W|}\int_{\mathfrak{a}^{*}}{\|\widetilde{f}(\lambda)\|_{\mathbf{2}}}^{2}~{}|\mathbf{c}(\lambda)|^{-2}d\lambda.$
###### Proof.
The formula (2.12) is derived from the inversion formula (1.15) of the HFT.
$\displaystyle f(x)$
$\displaystyle=\frac{1}{|W|}\int_{\mathfrak{a}^{*}}\int_{K}\mathcal{F}f(\lambda,k)e^{-(i\lambda+\rho)(\mathcal{H}(x^{-1}k))}|\mathbf{c}(\lambda)|^{-2}dk~{}d\lambda,$
$\displaystyle=\frac{1}{|W|}\int_{\mathfrak{a}^{*}}\int_{K}\delta(k)\mathcal{F}f(\lambda,e)e^{-(i\lambda+\rho)(\mathcal{H}(x^{-1}k))}|\mathbf{c}(\lambda)|^{-2}dk~{}d\lambda,$
$\displaystyle=\frac{1}{|W|}\int_{\mathfrak{a}^{*}}\left\\{\int_{K}e^{-(i\lambda+\rho)(\mathcal{H}(x^{-1}k))}\delta(k)dk\right\\}\widetilde{f}(\lambda)|\mathbf{c}(\lambda)|^{-2}d\lambda,$
$\displaystyle=\frac{1}{|W|}\int_{\mathfrak{a}^{*}}\Phi_{\lambda,\delta}(x)\widetilde{f}(\lambda)|\mathbf{c}(\lambda)|^{-2}d\lambda.$
The second and the third line of the above deduction are consequences of Lemma
1.3 and Lemma 2.6 respectively.
The second relation of this Lemma also follows from the Plancherel formula of
the HFT ([Hel94], Ch.-III, $\S$1, Theorem 1.5 ). For
$f\in\mathcal{D}_{\delta}(X)$, the HFT is defined matrix entry wise, hence in
this case the Plancherel formula is given by:
(2.13)
$\int_{G}{\|f(x)\|_{\mathbf{2}}}^{2}dx=\frac{1}{|W|}\int_{\mathfrak{a}^{*}}\int_{K}{\|\mathcal{F}f(\lambda,k)\|_{\mathbf{2}}}^{2}|\mathbf{c}(\lambda)|^{-2}dk~{}d\lambda,$
which follows easily from the classical Plancherel formula given in [Hel94].
The required Plancherel formula for the $\delta$-spherical transform follows
from (2.13) by using relation (i) of Lemma 1.3 and Schur’s Orthogonality
relation ([Sug90], Theorem 3.2).
We shall next deduce the deduce a topological Paley-Wiener (P-W) theorem for
the $\delta$-spherical transform (2.1).
A holomorphic function $\psi:\mathfrak{a}^{*}_{\mathbb{C}}\longrightarrow
Hom(V_{\delta},V_{\delta}^{M})$ is said to be of ‘exponential type-$R$’
($R\in\mathbb{R}^{+}$) if
$\sup_{\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}}e^{-R\|\Im\lambda\|}(1+\|\lambda\|)^{N}\|\psi(\lambda)\|_{\mathbf{2}}<+\infty\mbox{\hskip
36.135ptfor each~{} }N\in\mathbb{Z}^{+}.$
Let $\mathcal{H}_{\delta}^{R}(\mathfrak{a}^{*}_{\mathbb{C}})$ be the class of
all $Hom(V_{\delta},V_{\delta}^{M})$ valued exponential type-$R$ functions on
$\mathfrak{a}^{*}_{\mathbb{C}}$ and further let
$\mathcal{H}_{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})=\bigcup_{R>0}\mathcal{H}^{R}_{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$.
###### Theorem 2.8.
For each fixed $\delta\in\widehat{K}_{M}$, the $\delta$-spherical transform
given by (2.1) is a homeomorphism between the spaces $\mathcal{D}_{\delta}(X)$
and $\mathcal{P}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$, where
(2.14)
$\mathcal{P}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})=\left\\{\xi\in\mathcal{H}_{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})~{}|~{}\lambda\mapsto
Q^{\delta}(\lambda)^{-1}\xi(\lambda)~{}\mbox{is an}~{}W\mbox{-invariant entire
function}\right\\}.$
Here $Q^{\delta}(\lambda)$ is the matrix of constant coefficient polynomials
appeared in the expressions (2.6) and (2.7).
###### Proof.
Our proof solely relies on the proof of the topological Paley-Wiener theorem
given by Helgason ([Hel94], Ch.-III, Theorem 5.11), where he characterized the
image of the space $\mathcal{D}(\check{\delta},X)$ under the transform
$f\mapsto\widehat{f}$, where
(2.15)
$\widehat{f}(\lambda)=d_{\delta}\int_{G}f(x)~{}\Phi_{\overline{\lambda},\delta}^{*}(x)dx,~{}~{}~{}(\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}).$
Helgason proved that the above transform is a topological isomorphism between
the spaces $\mathcal{D}(\check{\delta},X)$ and
$\mathcal{P}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$. From the Proposition
1.1 and the definition (2.1) of the $\delta$-spherical transform we get: for
each $f\in\mathcal{D}_{\delta}(X)$,
$\widehat{(\mathcal{Q}f)}(\lambda)=\widetilde{f}(\lambda)$,
$(\forall\lambda\in\mathfrak{a}^{*}_{\mathbb{C}})$. The proposition now
follows from the fact that both the maps $\mathcal{Q}$ and
$f\mapsto\widehat{f}$ are homeomorphisms.
Let us now consider the function space
$\mathcal{P}_{0}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})=\left\\{h\in\mathcal{H}(\mathfrak{a}^{*}_{\mathbb{C}})|~{}h~{}\mbox{is
}~{}W\mbox{-invariant}\right\\}$ with the relative topology. A function
$h\in\mathcal{P}^{\delta}_{0}(\mathfrak{a}^{*}_{\mathbb{C}})$ can be written
as $h\equiv\left(h_{ij}\right)_{{\ell_{\delta}}\times d_{\delta}}$ where each
of the scalar valued component function $h_{ij}$ is entire, $W$-invariant and
of exponential type. Let $\mathcal{D}(G//K)$ and
$\mathcal{D}(G//K,Hom(V_{\delta},V_{\delta}^{M}))$ are respectively be the
spaces of scalar valued and $Hom(V_{\delta},V_{\delta}^{M})$ valued
bi-$K$-invariant, compactly supported, $\mathcal{C}^{\infty}$ functions on
$G$. The Paley-Wiener theorem for the bi-$K$-invariant functions gives an
unique $f_{ij}\in\mathcal{D}(G//K)$ such that $\mathcal{S}f_{ij}=h_{ij}$. We
set $f\equiv\left(f_{ij}\right)_{{\ell_{\delta}}\times
d_{\delta}}\in\mathcal{D}(G//K,Hom(V_{\delta},V_{\delta}^{M}))$ and define the
spherical transform $\mathcal{S}f$ matrix entry wise to get $\mathcal{S}f=h$.
Moreover by suitably modifying the proof of the P-W theorem for the
bi-$K$-invariant functions one can show that $\mathcal{S}$ is infact a
homeomorphism between the spaces
$\mathcal{D}(G//K,Hom(V_{\delta},V_{\delta}^{M}))$ and
$\mathcal{P}^{\delta}_{0}(\mathfrak{a}^{*}_{\mathbb{C}})$. The following Lemma
identifies the two PW-spaces
$\mathcal{P}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$ and
$\mathcal{P}^{\delta}_{0}(\mathfrak{a}^{*}_{\mathbb{C}})$.
###### Lemma 2.9.
_[Helgason[Hel94, Ch.-III, $\S$5, Lemma 5.12]]_
The mapping $\psi(\lambda)\mapsto Q^{\delta}(\lambda)\psi(\lambda)$
($\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$) is a homeomorphism from
$\mathcal{P}^{\delta}_{0}(\mathfrak{a}^{*}_{\mathbb{C}})$ onto
$\mathcal{P}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$.
###### Lemma 2.10.
Any function $f\in\mathcal{D}_{\delta}(X)$ can be written as
$f(x)=\mathbf{D}^{\delta}\phi(x)$ ($\forall x\in G$), where
$\phi\in\mathcal{D}(G//K,Hom(V_{\delta},V_{\delta}^{M}))$ and
$\mathbf{D}^{\delta}$ is the ${d_{\delta}}\times{\ell_{\delta}}$ matrix of
constant coefficient differential operators as mentioned in (2.9).
###### Proof.
Let $f\in\mathcal{D}_{\delta}(X)$, then by Theorem 2.8, its $\delta$-spherical
transform
$\widetilde{f}\in\mathcal{P}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$. Using
the homeomorphism given in Lemma 2.9, we get an unique function
$\lambda\mapsto\Phi(\lambda)=Q^{\delta}(\lambda)^{-1}\widetilde{f}(\lambda)$
in $\mathcal{P}^{\delta}_{0}(\mathfrak{a}^{*}_{\mathbb{C}})$. By the PW-
theorem for the bi-$K$-invariant functions we get a function
$\phi\in\mathcal{D}(G//K,Hom(V_{\delta},V_{\delta}^{M}))$ such that:
(2.16)
$\phi(x)=\frac{1}{|W|}\int_{\mathfrak{a}^{*}}\varphi_{\lambda}(x)\Phi(\lambda)|\mathbf{c}(\lambda)|^{-2}d\lambda.$
Now by applying the differential operator $\mathbf{D}^{\delta}$ on the both
sides of (2.16) and by using the absolute convergence of the integral in
(2.16) we get:
$\displaystyle\left(\mathbf{D}^{\delta}\phi\right)(x)$
$\displaystyle=\frac{1}{|W|}\int_{\mathfrak{a}^{*}}\left(\mathbf{D}^{\delta}\varphi_{\lambda}(x)\right)\Phi(\lambda)|\mathbf{c}(\lambda)|^{-2}d\lambda,$
$\displaystyle=\frac{1}{|W|}\int_{\mathfrak{a}^{*}}\Phi_{\lambda,\delta}(x)Q^{\delta}(\lambda)\Phi(\lambda)|\mathbf{c}(\lambda)|^{-2}d\lambda,$
$\displaystyle=\frac{1}{|W|}\int_{\mathfrak{a}^{*}}\Phi_{\lambda,\delta}(x)\widetilde{f}(\lambda)|\mathbf{c}(\lambda)|^{-2}d\lambda=f(x).$
The second line of the above calculation is a consequence of (2.9) and the
last line is merely the inversion formula (2.12).
We conclude this section with a ‘product formula’, due to Helgason, for the
polynomial $\det Q^{\delta}(\lambda)$ with
$\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$. First we shall characterize the set
$\widehat{K_{\beta}}_{M_{\beta}}$ ( $\beta\in\Sigma^{+}_{0}$) of equivalence
classes of representations of $K_{\beta}$ as certain restrictions of the
representations $\delta\in\widehat{K}_{M}$. Let $\delta\in\widehat{K}_{M}$ and
$V_{\delta}$ and $V_{\delta}^{M}$ be as explained earlier. Let $V$ denote the
$K_{\beta}M$-invariant subspace of $V_{\delta}$ generated by $V_{\delta}^{M}$.
Then $V$ decomposes into $K_{\beta}M$-irreducible subspaces as
$V=\bigoplus_{i=1}^{\ell_{\delta}}V_{i}.$ Let $\delta(i,\beta)$ be the
representation of $K_{\beta}M$ on $V_{i}$ given by $\delta$. Then each
$\delta(i,\beta)$ is irreducible except when $dimK_{\beta}=1$ and
$mkm^{-1}=k^{-1}$ for all $k\in K_{\beta}$ and some $m\in M$ [Hel94, Ch. III,
Lemma 3.9 - 3.11 ]. In this case $\delta(i,\beta)$ brakes up into two
irreducible one dimensional representations as
$\delta(i,\beta)=\delta(i,\beta)_{0}\oplus\delta(i,\beta)_{0}\check{}$ where
$\delta(i,\beta)_{0}\check{}$ is the contragradient representation of
$\delta(i,\beta)_{0}$ and in this particular case we choose
$\delta(i,\beta)_{0}\check{}$ as $\delta(i,\beta)$.
###### Lemma 2.11.
_Helgason[Hel94, Ch. III, Prposition 4.3]_ For each $\beta\in\Sigma_{0}^{+}$,
as $\delta$ runs through $\widehat{K}_{M}$, the representations
$\delta(i,\beta)$ ($1\leq i\leq\ell$) runs through all of
$\widehat{K_{\beta}}_{M_{\beta}}$.
Let $Q^{\delta(i,\beta)}_{\beta}(\lambda_{\beta})$ be the Kostant polynomial
for the rank-one symmetric space $G_{\beta}/K_{\beta}$ corresponding to the
representation $\delta(i,\beta)\in\widehat{K_{\beta}}_{M_{\beta}}$. Then the
determinant of the Kostant matrix $Q^{\delta}(\lambda)$
($\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$) can be represented by a product
formula [Hel94, Ch. III, §3, (50) and §4, Theorem 4.2 ]
(2.17) $\det
Q^{\delta}(\lambda)=\mathcal{C}_{\delta}\prod_{\beta\in\Sigma_{0}^{+},~{}1\leq
i\leq\ell_{\delta}}Q^{\delta(i,\beta)}_{\beta}(\lambda_{\beta}),\mbox{~{}for
all~{}}\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$
where $\mathcal{C}_{\delta}$ is a nonzero constant depending on
$\delta\in\widehat{K}_{M}$.
## 3\. $L^{p}$-Schwartz spaces
###### Definition 3.1.
_[Classical $L^{p}$ Schwartz space]_
A $\mathcal{C}^{\infty}$ function $f$ on $X$ is said to be in the
$L^{p}$-Schwartz space ( $0<p\leq 2$ ) $\mathcal{S}^{p}(X)$ if for each
nonnegative integer $n$ and $D,E\in\mathcal{U}(\mathfrak{g}_{\mathbb{C}})$ the
function $f$ satisfies the following decay condition:
(3.1) $\mu_{D,E,n}(f)=\sup_{x\in
G}|f(D,x,E)|~{}\varphi_{0}^{-\frac{2}{p}}(x)~{}(1+|x|)^{n}~{}<+\infty.$
The topology induced by the countable family of seminorms $\mu_{D,E,n}(\cdot)$
makes $\mathcal{S}^{p}(X)$ a Fréchet space. It can be shown that
$\mathcal{D}(X)$ is a dense subspace of $\mathcal{S}^{p}(X)$ for $0<p\leq 2$.
Let $f\in\mathcal{S}^{p}(X)$, we take its $\delta$-projection $f^{\delta}$ as
defined in (1.16). Then $f^{\delta}$ is a left $\delta$-type
$Hom(V_{\delta},V_{\delta})$ valued function with a decay
(3.2) $\mu_{D,E,n}(f^{\delta})=\sup_{x\in
G}\|f^{\delta}(D,x,E)\|_{\mathbf{2}}~{}\varphi_{0}^{-\frac{2}{p}}(x)~{}(1+|x|)^{n}~{}<+\infty,$
where $\|\cdot\|_{\mathbf{2}}$ denotes the Hilbert Schmidt norm. Let us denote
$\mathcal{S}^{p}_{\delta}(X)$ for the class of left $\delta$-type
$Hom(V_{\delta},V_{\delta})$ valued functions with the decay (3.2). For each
$h\in\mathcal{S}^{p}_{\delta}(X)$, the scalar valued function $trh(\cdot)$
satisfies $trh\equiv d_{\check{\delta}}(\chi_{\check{\delta}}\ast trh)$ and
the decay (3.1). Let
$\mathcal{S}^{p}(\check{\delta},X)\subset\mathcal{S}^{p}(X)$ be the class of
all scalar valued left-$\check{\delta}$ type Schwartz class functions. Both
the spaces $\mathcal{S}^{p}_{\delta}(X)$ and
$\mathcal{S}^{p}(\check{\delta},X)$ becomes Frèchet spaces with the topologies
induced by the family of seminorms given in (3.2) and (3.1) respectively.
Moreover, $\mathcal{D}_{\delta}(X)$ and $\mathcal{D}(\check{\delta},X)$ are
respectively dense subspaces of $\mathcal{S}^{p}_{\delta}(X)$ and
$\mathcal{S}^{p}(\check{\delta},X)$ in the respective Schwartz space
topologies. The topological isomorphism $\mathcal{Q}$ in Proposition 1.1 can
be extended between the Schwartz spaces $\mathcal{S}^{p}_{\delta}(X)$ and
$\mathcal{S}^{p}(\check{\delta},X)$.
Next we shall to extend the definition of the $\delta$-spherical transform to
the Schwartz space $\mathcal{S}^{p}_{\delta}(X)$ ($0<p\leq 2$). The spherical
transform (1.11) defined on $\mathcal{D}(G//K)$ can be extended to the
$L^{p}$-Schwartz spaces $\mathcal{S}^{p}(G//K)$ of bi-$K$-invariant functions
on the group $G$. The image $\mathcal{S}(\mathfrak{a}^{*}_{\varepsilon})$
(defined below) of $\mathcal{S}^{p}(G//K)$ under the spherical transform is
again a Schwartz class of functions defined o the complex tube
$\mathfrak{a}^{*}_{\varepsilon}=\mathfrak{a}^{*}+iC^{\varepsilon\rho}$ where
$\varepsilon=\left(\frac{2}{p}-1\right)$ and
$C^{\varepsilon\rho}=\\{\lambda\in\mathfrak{a}^{*}~{}|~{}\omega\lambda(H)\leq\varepsilon\rho(H)\mbox{~{}for
all~{}}H\in\overline{\mathfrak{a}^{+}}\mbox{~{}and~{}}\omega\in W\\}.$
###### Definition 3.2.
The space $\mathcal{S}(\mathfrak{a}^{*}_{\varepsilon})$ consists of the
complex valued functions $h$ on $\mathfrak{a}^{*}_{\varepsilon}$ such that:
1. (i)
$h$ is holomorphic in the interior of the tube
$\mathfrak{a}^{*}_{\varepsilon}$ and continues on the closed tube,
2. (ii)
$h$ is $W$-invariant,
3. (iii)
for any polynomial $P$ in the algebra $S(\mathfrak{a})$ of the symmetric
polynomials on $\mathfrak{a}^{*}$ and any positive integer $r$
(3.3) $\tau_{P,r}(h)=\sup_{\lambda\in
Int\mathfrak{a}^{*}_{\varepsilon}}(1+\|\lambda\|)^{r}\left|P\left(\frac{\partial}{\partial\lambda}\right)h(\lambda)\right|<+\infty.$
The countable family $\left\\{\tau_{P,r}~{}|~{}P\in
S(\mathfrak{a}),r\in\mathbb{Z}^{+}\cup\\{0\\}\right\\}$ gives a Frèchet norm
on the space $S(\mathfrak{a}^{*}_{\varepsilon})$.
###### Remark 3.3.
The topology of $\mathcal{S}(\mathfrak{a}^{*}_{\varepsilon})$ can also be
given by the following equivalent family of seminorms.
(3.4) $\tau_{P,r}^{+}(h)=\sup_{\lambda\in
Int\mathfrak{a}^{*}_{\varepsilon}\cap\left(\mathfrak{a}^{*}+i\overline{\mathfrak{a}^{*+}}\right)}(1+\|\lambda\|)^{r}\left|P\left(\frac{\partial}{\partial\lambda}\right)h(\lambda)\right|<+\infty.$
This is an easy consequence of (ii) of Definition 3.2 and the fact
$\|\omega\lambda\|=\|\lambda\|$ for all
$\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$ and $\omega\in W$.
We now state the $L^{p}$-Schwartz space isomorphism theorem for the
bi-$K$-invariant functions.
###### Theorem 3.4.
The spherical transform $f\mapsto\mathcal{S}f$ defined in (1.11) is
topological isomorphism between the spaces $\mathcal{S}^{p}(G//K)$ ($0<p\leq
2$) and $\mathcal{S}(\mathfrak{a}^{*}_{\varepsilon})$.
For a proof of this theorem one can see [GV88]. We are mainly interested in
Anker’s proof [Ank91] of the above theorem, which relies on the PW theorem for
the spherical transform.
Let $\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$ be the space of all
$Hom(V_{\delta},V_{\delta}^{M})$ valued functions $h$ on
$\mathfrak{a}^{*}_{\varepsilon}$ satisfying (i), (ii) of Definition 3.2 along
with the decay: for each polynomial $P\in S(\mathfrak{a})$ and integer $n\geq
0$
(3.5) $\tau_{P,n}^{+}(h)=\sup_{\lambda\in
Int\mathfrak{a}^{*}_{\varepsilon}\cap\left(\mathfrak{a}^{*}+i\overline{\mathfrak{a}^{*+}}\right)}(1+\|\lambda\|)^{n}\left\|P\left(\frac{\partial}{\partial\lambda}\right)h(\lambda)\right\|_{\mathbf{2}}<+\infty.$
The countable family $\\{\tau_{P,n}^{+}\\}$ induces a a Frèchet structure on
$\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$. The space
$\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$ can also be viewed as a
space of $Hom(V_{\delta},V_{\delta}^{M})$ valued functions with each of its
matrix entry function in $\mathcal{S}(\mathfrak{a}^{*}_{\varepsilon})$. For
our purpose we shall use the following equivalent (inducing the same topology)
family of seminorms on $\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$:
(3.6) $\tau^{+*}_{P,r}(h)=\sup_{\lambda\in
Int(\mathfrak{a}^{*}_{\varepsilon}\cap(\mathfrak{a}^{*}+i\overline{\mathfrak{a}^{*+}}))}\left\|P\left(\frac{\partial}{\partial\lambda}\right)\left\\{h(\lambda)(\|\rho\|^{2}+\langle\lambda,\lambda\rangle_{1})^{r}\right\\}\right\|_{\mathbf{2}},~{}P\in
S(\mathfrak{a}),r\in\mathbb{Z}^{+}\cup\\{0\\}.$
Let $f=(f_{ij})\in\mathcal{S}^{p}(G//K,Hom(V_{\delta},V_{\delta}^{M}))$. Then
by defining $\mathcal{S}f=(\mathcal{S}f_{ij})$ we can the following extension
of the isomorphism of Theorem 3.4.
###### Lemma 3.5.
The spherical transform is a topological isomorphism between the spaces
$\mathcal{S}^{p}(G//K,Hom(V_{\delta},V_{\delta}^{M}))$ and
$\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$.
This Lemma can be proved easily by using the conclusion of the Theorem 3.4 for
each matrix entry of the functions of
$\mathcal{S}^{p}(G//K,Hom(V_{\delta},V_{\delta}^{M}))$.
We now define the ambient space for the image of the Schwartz space
$\mathcal{S}^{p}_{\delta}(X)$ under the $\delta$-spherical transform.
###### Definition 3.6.
Let $\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ be the set of all
$Hom(V_{\delta},V_{\delta}^{M})$ valued functions $h$ on the complex tube
$\mathfrak{a}^{*}_{\varepsilon}$ such that
1. (i)
h is holomorphic in the interior of the tube $\mathfrak{a}^{*}_{\varepsilon}$
and extends as a continuous function to the closed tube.
2. (ii)
$\lambda\mapsto Q^{\delta}(\lambda)^{-1}~{}h(\lambda)$ is $W$-invariant
holomorphic function the interior of the complex tube
$\mathfrak{a}^{*}_{\varepsilon}$.
3. (iii)
for each polynomial $P\in S(\mathfrak{a})$ and integer $n\geq 0$
(3.7) $\nu_{P,n}(h)=\sup_{\lambda\in
Int\mathfrak{a}^{*}_{\varepsilon}}(1+\|\lambda\|)\left\|P\left(\frac{\partial}{\partial\lambda}\right)h(\lambda)\right\|_{\mathbf{2}}<+\infty.$
Clearly, $\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ becomes a
Frèchet space with the topology induced by the countable family
$\left\\{\nu_{P,n}\right\\}$ of seminorms.
###### Remark 3.7.
It is easy to observe that the topology of the space
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ can also be determined
by the following equivalent family of seminorms: for each polynomial $P\in
S(\mathfrak{a})$ and each nonnegative integer $n$
(3.8) $h\mapsto{\nu}^{\\#}_{P,n}(h)=\sup_{\lambda\in
Int\mathfrak{a}^{*}_{\varepsilon}}\left|P\left(\frac{\partial}{\partial\lambda}\right)\left\\{(\langle\lambda,\lambda\rangle_{1}+\|\rho\|^{2})^{n}h(\lambda)\right\\}\right|.$
Let us now state the first main theorem of this paper.
###### Theorem 3.8.
The $\delta$-spherical transform (2.1) is a topological isomorphism between
the Schwartz spaces $\mathcal{S}^{p}_{\delta}(X)$ and
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ with $0<p\leq 2$ and
$\varepsilon=\left(\frac{2}{p}-1\right)$.
The following proposition is a key step to prove Theorem 3.8.
###### Proposition 3.9.
For each $\delta\in\widehat{K}_{M}$, there exists a $\delta$-dependent
constant $c_{\delta}>0$ such that
(3.9) $\inf_{\lambda\in(\mathfrak{a}^{*}+i\overline{\mathfrak{a}^{*+}})}|\det
Q^{\delta}(\lambda)|\geq c_{\delta}.$
###### Proof.
To prove this we use the product formula (2.17) of $\det Q^{\delta}(\lambda)$.
Each of the factors $Q^{\delta(i,\beta)}_{\beta}$ is the Kostant polynomials
of he rank-one restrictions $G_{\beta}/K_{\beta}$ and hence it is a polynomial
in one complex variable
$\lambda_{\beta}\in{\mathfrak{a}^{*}_{(\beta)}}_{\mathbb{C}}\cong\mathbb{C}$.
It is easy to check that
$\lambda\in\mathfrak{a}^{*}+i\overline{\mathfrak{a}^{*+}}$ if and only if
$\Re{\langle i\lambda,\alpha\rangle}\leq 0$ for all $\alpha\in\Sigma^{+}$.
Using the definition of the restriction
$\lambda_{\beta}=\frac{\langle\lambda,\beta\rangle_{1}}{\langle\beta,\beta\rangle}\beta$
we get
(3.10) $\displaystyle\langle i\lambda_{\beta},\beta\rangle_{\beta}$
$\displaystyle=\left\langle
i\frac{\langle\lambda,\beta\rangle_{1}}{\langle\beta,\beta\rangle}\beta,\beta\right\rangle_{\beta}=\langle
i\lambda,\beta\rangle$
where $\langle\cdot,\cdot\rangle_{\beta}$ denotes the inner product on
$\mathfrak{a}^{*}_{(\beta)}$ as well as its $\mathbb{C}$-bilinear extension to
the complexification ${\mathfrak{a}^{*}_{(\beta)}}_{\mathbb{C}}$. Now (3.10)
clearly suggests that if
$\lambda\in\mathfrak{a}^{*}+i\overline{\mathfrak{a}^{*+}}$ then for each
$\beta\in\Sigma_{0}^{+}$ the restriction
$\lambda_{\beta}\in\mathfrak{a}_{(\beta)}^{*}+i\overline{\mathfrak{a}_{(\beta)}^{*+}}$.
As the polynomial $Q^{\delta(i,\beta)}_{\beta}(\lambda_{\beta})\neq 0$ for all
$\lambda_{\beta}\in\mathfrak{a}_{(\beta)}^{*}+i\overline{\mathfrak{a}_{(\beta)}^{*+}}$
and $\mathfrak{a}_{(\beta)}^{*}+i\overline{\mathfrak{a}_{(\beta)}^{*+}}$ being
a closed subset of $\mathbb{C}$ so we get a positive constant
$c_{\delta(i,\beta)}$ such that
$\inf_{\lambda_{\beta}\in\mathfrak{a}_{(\beta)}^{*}+i\overline{\mathfrak{a}_{(\beta)}^{*+}}}|Q^{\delta(i,\beta)}_{\beta}(\lambda_{\beta})|\geq
c_{\delta(i,\beta)}.$
Since corresponding to each $\delta\in\widehat{K}_{M}$ the product formula
(2.17) of $\det Q^{\delta}(\lambda)$ has only finitely many factors hence we
get the desired conclusion of the proposition.
The next lemma extends the homeomorphism given in Lemma 2.9 between the P-W
spaces to the corresponding Schwartz classes.
###### Lemma 3.10.
The map
(3.11) $g(\lambda)\mapsto Q^{\delta}(\lambda)g(\lambda),~{}~{}\mbox{~{}~{}for
all ~{}}\lambda\in\mathfrak{a}^{*}_{\varepsilon},$
is a homeomorphism from the space
$\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$ onto
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$.
###### Proof.
Let us first take $g\in\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$. We
denote $h(\cdot)=Q^{\delta}(\cdot)g(\cdot)$. We shall show that
$h\in\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$. As
$Q^{\delta}(\lambda)$ is an ${\ell_{\delta}}\times{\ell_{\delta}}$ matrix of
polynomials in $\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$ so $\lambda\mapsto
Q^{\delta}(\lambda)$ is a holomorphic function on
$\mathfrak{a}^{*}_{\mathbb{C}}$. Hence the function $h$ satisfies condition
(i) and (ii) of Definition 3.6 which easily follows from the similar
properties of $g\in\mathcal{S}_{0}(\mathfrak{a}_{\varepsilon})$ and the
construction (3.11) of the function $h$.
To establish the decay condition (3.7) for $h$ let us take a polynomial $P\in
S(\mathfrak{a})$ and $m\in\mathbb{Z}^{+}\cup\\{0\\}$. Then
$\displaystyle\sup_{\lambda\in Int\mathfrak{a}^{*}_{\varepsilon}}$
$\displaystyle\left\|P\left(\frac{\partial}{\partial\lambda}\right)h(\lambda)\right\|_{\mathbf{2}}(1+\|\lambda\|)^{m}$
$\displaystyle\leq\sup_{\lambda\in
Int\mathfrak{a}^{*}_{\varepsilon}}\sum_{\kappa}c_{\kappa}\left\|\left\\{P^{\prime}_{\kappa}\left(\frac{\partial}{\partial\lambda}\right)Q^{\delta}(\lambda)\right\\}\left\\{P_{\kappa}\left(\frac{\partial}{\partial\lambda}\right)g(\lambda)\right\\}\right\|_{\mathbf{2}}(1+\|\lambda\|)^{m}$
$\displaystyle\leq\sup_{\lambda\in
Int\mathfrak{a}^{*}_{\varepsilon}}\sum_{\kappa}c_{\kappa}\left\|\left\\{P^{\prime}_{\kappa}\left(\frac{\partial}{\partial\lambda}\right)Q^{\delta}(\lambda)\right\\}\right\|_{\mathbf{2}}\left\|\left\\{P_{\kappa}\left(\frac{\partial}{\partial\lambda}\right)g(\lambda)\right\\}\right\|_{\mathbf{2}}(1+\|\lambda\|)^{m}$
(3.12) $\displaystyle\leq\sup_{\lambda\in
Int\mathfrak{a}^{*}_{\varepsilon}}\sum_{\kappa}c_{\kappa}^{\delta}\left\|\left\\{P_{\kappa}\left(\frac{\partial}{\partial\lambda}\right)g\right\\}(\lambda)\right\|_{\mathbf{2}}(1+\|\lambda\|)^{m_{\kappa}^{\delta}}$
where, $m_{\kappa}^{\delta}$ are nonnegative integers and
$c_{\kappa}^{\delta}$ are positive constants both depending on
$\delta\in\widehat{K}_{M}$. As
$g\in\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$, the right hand side of
(3) is clearly finite. Moreover, (3) shows that the map (3.11) is a continuous
function from $\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$ into
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$.
Now let $\psi\in\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ and
define $g(\cdot):=Q^{\delta}(\cdot)^{-1}\psi(\cdot)$. As, $\psi\in
S_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$, by Definition 3.6 the function
$\lambda\mapsto g(\lambda)=\frac{1}{\det
Q^{\delta}(\lambda)}Q^{\delta}_{c}(\lambda)\psi(\lambda)$
(here, $Q^{\delta}_{c}(\lambda)$ is the cofactor matrix of
$Q^{\delta}(\lambda)$) is $W$-invariant and it is holomorphic in the interior
of the tube $\mathfrak{a}^{*}_{\varepsilon}$. To infer
$g\in\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$ all we have to show is
that the function $g$ has certain decay. Let $P\in S(\mathfrak{a})$ and $t$ be
any nonnegative integer, then
$\displaystyle\left\\{P\left(\frac{\partial}{\partial\lambda}\right)g(\lambda)\right\\}$
$\displaystyle=\left\\{P\left(\frac{\partial}{\partial\lambda}\right)\frac{1}{\det
Q^{\delta}(\lambda)}Q^{\delta}_{c}(\lambda)\psi(\lambda)\right\\}$ (3.13)
$\displaystyle=\sum_{\kappa}\frac{P_{\kappa}\left(\frac{\partial}{\partial\lambda}\right)Q^{\delta}_{c}(\lambda)P^{\prime}_{\kappa}\left(\frac{\partial}{\partial\lambda}\right)\psi(\lambda)}{\left(\det
Q^{\delta}(\lambda)\right)^{m_{\kappa}}}.$
The last line of (3) follows by an easy application of the Leibniz rule. Here
$P_{\kappa},P^{\prime}_{\kappa}$ are finite degree polynomials, $m_{\kappa}$
is a positive integer depending on $\kappa$ and the sum is over a finite set.
From (3) we get:
$\displaystyle\left\|P\left(\frac{\partial}{\partial\lambda}\right)g(\lambda)\right\|_{\mathbf{2}}$
$\displaystyle\leq\sum_{\kappa}\frac{\|P_{\kappa}\left(\frac{\partial}{\partial\lambda}\right)Q^{\delta}_{c}(\lambda)\|_{\mathbf{2}}\|P^{\prime}_{\kappa}\left(\frac{\partial}{\partial\lambda}\right)\psi(\lambda)\|_{\mathbf{2}}}{|\det
Q^{\delta}(\lambda)|^{m_{\kappa}}}$ (3.14) $\displaystyle\leq
c(\delta)\sum_{\kappa}\frac{\|P^{\prime}_{\kappa}\left(\frac{\partial}{\partial\lambda}\right)\psi(\lambda)\|_{\mathbf{2}}}{|\det
Q^{\delta}(\lambda)|^{m_{\kappa}}}(1+\|\lambda\|)^{n_{\kappa}}.$
The above inequality is obtained by using the fact that:
$\|P_{\kappa}\left(\frac{\partial}{\partial\lambda}\right)Q^{\delta}_{c}(\lambda)\|_{\mathbf{2}}\leq
c(\delta)(1+\|\lambda\|)^{n_{\kappa}}$ where $c(\delta)>0$ is a
$\delta$-dependent constant and $n_{\kappa}$ is a positive integer depending
on the degree of $P_{\kappa}$ (it may also depend on $\delta$). Now from (3)
we get the following inequality for any nonnegative integer $t$.
$\displaystyle\sup_{Int(\mathfrak{a}^{*}_{\varepsilon}\cap(\mathfrak{a}^{*}+i\overline{\mathfrak{a}^{*+}}))}\left\|P\left(\frac{\partial}{\partial\lambda}\right)g(\lambda)\right\|_{\mathbf{2}}(1+\|\lambda\|)^{t}$
$\displaystyle\leq\sum_{\kappa}\sup_{Int(\mathfrak{a}^{*}_{\varepsilon}\cap(\mathfrak{a}^{*}+i\overline{\mathfrak{a}^{*+}}))}\frac{\left\|P^{\prime}_{\kappa}\left(\frac{\partial}{\partial\lambda}\right)\psi(\lambda)\right\|_{\mathbf{2}}(1+\|\lambda\|)^{t+n_{\kappa}}}{|\det
Q^{\delta}(\lambda)|^{m_{\kappa}}}$
$\displaystyle\leq\sum_{\kappa}\frac{\sup_{\lambda\in\mathfrak{a}^{*}_{\varepsilon}}\left\|P^{\prime}_{\kappa}\left(\frac{\partial}{\partial\lambda}\right)\psi(\lambda)\right\|_{\mathbf{2}}(1+\|\lambda\|)^{t+n_{\kappa}}}{\inf_{\lambda\in
Int(\mathfrak{a}^{*}_{\varepsilon}\cap(\mathfrak{a}^{*}+i\overline{\mathfrak{a}^{*+}}))}|\det
Q^{\delta}(\lambda)|^{m_{\kappa}}}$ (3.15)
$\displaystyle\leq\sum_{\kappa}\frac{1}{c_{\delta}^{\kappa}}\sup_{\lambda\in\mathfrak{a}^{*}_{\varepsilon}}\left\|P^{\prime}_{\kappa}\left(\frac{\partial}{\partial\lambda}\right)\psi(\lambda)\right\|_{\mathbf{2}}(1+\|\lambda\|)^{t+n_{\kappa}}.$
The last line of the above successive inequalities is a consequence of
Proposition 3.9. As
$\psi\in\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ so each term of
the finite summation on the right hand side of (3) is finite. Hence we
conclude that $g$ satisfies the decay (3.5) of the space
$\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$. The inequality (3) further
concludes that the continuous map (3.11) from
$\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$ onto
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ is injective and also
its inverse map is continuous. As both the spaces
$\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$ and
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ are Frèchet spaces so
the map (3.11) is a homeomorphism.
###### Lemma 3.11.
The Paley-Wiener space $\mathcal{P}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$,
defined in (2.14), is a dense subspace of the Schwartz space
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$.
###### Proof.
Let us take any $H\in\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$, it
is enough to show that, there is a sequence $\\{G_{n}\\}$
($G_{n}\in\mathcal{P}(\mathfrak{a}^{*}_{\mathbb{C}})$) converging to $H$ in
the topology of the space
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$. Let
$H(\lambda)=\left(H_{ij}(\lambda)\right)_{{\ell_{\delta}}\times d_{\delta}}$.
By the isomorphism obtained in Lemma 3.10, we get one unique
$G\in\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$ such that
$\displaystyle H(\lambda)=\left(H_{ij}(\lambda)\right)_{{\ell_{\delta}}\times
d_{\delta}}$ $\displaystyle=Q^{\delta}(\lambda)G(\lambda),$
$\displaystyle=Q^{\delta}(\lambda)\left(G_{ij}(\lambda)\right)_{{\ell_{\delta}}\times
d_{\delta}},$ (3.16)
$\displaystyle=\left(\sum_{k=1}^{{\ell_{\delta}}}Q^{\delta}(\lambda)_{ik}G_{kj}(\lambda)\right)_{{\ell_{\delta}}\times
d_{\delta}}.$
As $G\in\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$, so from the
definition of the Schwartz space
$\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$ it follows that the matrix
entry functions $G_{ij}\in\mathcal{S}(\mathfrak{a}^{*}_{\varepsilon})$ for
each $1\leq i\leq{\ell_{\delta}}$ and $1\leq j\leq d_{\delta}$. We know that
the Paley-Wiener space $\mathcal{P}(\mathfrak{a}^{*}_{\mathbb{C}})$ under the
spherical transform is dense in the Schwartz class
$\mathcal{S}(\mathfrak{a}^{*}_{\varepsilon})$ [GV88]. Therefore we can get a
sequence
$\left\\{{g_{ij}}_{n}\right\\}_{n}\subset\mathcal{P}(\mathfrak{a}^{*}_{\mathbb{C}})$
converging to $G_{ij}$ in $\mathcal{S}(\mathfrak{a}^{*}_{\varepsilon})$. As,
each $Q^{\delta}_{ik}(\lambda)$ ($1\leq i\leq{\ell_{\delta}},1\leq k\leq
d_{\delta}$) is a polynomial in $\lambda$, so the sequence
$\left\\{\sum_{k=1}^{\ell_{\delta}}Q^{\delta}_{ik}(\lambda){g_{ij}}_{n}(\lambda)\right\\}_{n}$
converges to
$\sum_{k=1}^{\ell_{\delta}}Q^{\delta}_{ik}(\lambda)G_{kj}(\lambda)$ (for each
$\lambda$) in $\mathcal{S}(\mathfrak{a}^{*}_{\varepsilon})$.
Let $g_{n}(\lambda)=\left({g_{ij}}_{n}(\lambda)\right)_{{\ell_{\delta}}\times
d_{\delta}}$. As each
${g_{ij}}_{n}\in\mathcal{P}(\mathfrak{a}^{*}_{\mathbb{C}})$, so from the
definition it follows that, the matrix valued function
$g_{n}\in\mathcal{P}^{\delta}_{0}(\mathfrak{a}^{*}_{\mathbb{C}})$. Clearly by
Lemma 2.9 for each natural number $n$,
$Q^{\delta}(\cdot)g_{n}(\cdot)\in\mathcal{P}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$.
Let $P$ be any polynomial in $S(\mathfrak{a})$ and $t$ be any nonnegative
integer then:
$\displaystyle\tau_{P,t}\left(Q^{\delta}(\cdot)g_{n}(\cdot)-Q^{\delta}(\cdot)G(\cdot)\right)$
$\displaystyle=\sup_{\lambda\in
Int\mathfrak{a}^{*}_{\varepsilon}}\left\|P\left(\frac{\partial}{\partial\lambda}\right)\left\\{Q^{\delta}(\lambda)g_{n}(\lambda)-Q^{\delta}(\lambda)G(\lambda)\right\\}\right\|_{\mathbf{2}}(1+\|\lambda\|)^{t},$
(3.17) $\displaystyle=\sup_{\lambda\in
Int\mathfrak{a}^{*}_{\varepsilon}}\sum_{i=1}^{\ell_{\delta}}\sum_{j=1}^{d_{\delta}}\left\|P\left(\frac{\partial}{\partial\lambda}\right)\sum_{k=1}^{{{\ell_{\delta}}}}\left\\{Q^{\delta}(\lambda)_{ik}{g_{ik}}_{n}(\lambda)-Q^{\delta}(\lambda)_{ik}G_{ik}(\lambda)\right\\}\right\|^{2}(1+\|\lambda\|)^{t}.$
A suitable choice of $n$ can made the right hand side of (3) arbitrarily
small. Hence we get the sequence $\left\\{Q^{\delta}(\cdot)g_{n}\right\\}_{n}$
in $\mathcal{P}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$ converging to $H$ in
the topology of $\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$. This
completes the proof of the Lemma.
Next we shall try to extend the definition of the $\delta$-spherical transform
(2.1) to the Schwartz class $\mathcal{S}^{p}_{\delta}(X)$ where $0<p\leq 2$.
###### Lemma 3.12.
For each $f\in\mathcal{S}^{p}_{\delta}(X)$, the function
$\lambda\mapsto\widetilde{f}(\lambda)$, where $\widetilde{f}$ is given by
(2.1), is a holomorphic function in the interior of the complex tube
$\mathfrak{a}^{*}_{\varepsilon}$.
###### Proof.
For each $f\in\mathcal{S}^{p}_{\delta}(X)$ the function $x\mapsto trf(x)$ has
the following decay: for each $D,E\in\mathcal{U}(\mathfrak{g}_{\mathbb{C}})$
and integer $n\geq 0$
(3.18) $\sup_{x\in
G}|trf(D,x,E)|(1+|x|)^{n}\varphi_{0}^{-\frac{2}{p}}(x)<+\infty,$
which follows easily from (3.2) and the fact that
$|trf(x)|\leq\|f(x)\|_{\mathbf{2}}$ ( $x\in X$). Using (3.18) and the estimate
(2.10) of the generalized spherical functions one can show, by following a
standard argument (see [GV88, $\S$6.2], [EK76]), that the $\delta$-spherical
transform, defined by the integral (2.1), exists for
$\lambda\in\mathfrak{a}^{*}_{\varepsilon}$.
Let $\gamma$ be a closed curve in the interior of the tube
$\mathfrak{a}^{*}_{\varepsilon}$. Then for $f\in\mathcal{S}^{p}_{\delta}(X)$
we get
$\displaystyle\int_{\gamma}\widetilde{f}(\lambda)d\lambda$
$\displaystyle=d_{\delta}\int_{\gamma}\left\\{\int_{G}trf(x)\Phi_{\overline{\lambda},\delta}(x)dx\right\\}d\lambda.$
As the integral within braces exists absolutely for
$\lambda\in\mathfrak{a}^{*}_{\varepsilon}$, so we apply Fubini’s theorem to
get:
$\int_{\gamma}\widetilde{f}(\lambda)d\lambda=d_{\delta}\int_{G}trf(x)\left\\{\int_{\gamma}\Phi_{\overline{\lambda},\delta}(x)d\lambda\right\\}dx$.
We also recall that the functions
$\lambda\mapsto\Phi_{\overline{\lambda},\delta}(\cdot)$ are entire. Hence by
an application of Morera’s theorem the desired conclusion of the lemma
follows.
## 4\. Proof of Theorem 3.8
To show that the $\delta$-spherical transform is a topological isomorphism it
is enough to show that it is a continuous surjection from
$\mathcal{S}^{p}_{\delta}(X)$ onto
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$.
###### Lemma 4.1.
The $\delta$-spherical transform $f\mapsto\widetilde{f}$ is a continuous map
from the Schwartz space $\mathcal{S}^{p}_{\delta}(X)$ into
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$.
###### Proof.
It is enough to show that given any seminorm $\nu$ (or equivalently
$\nu^{\\#}$) on $\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ one can
find a seminorm $\mu$ on $\mathcal{S}^{p}_{\delta}(X)$ such that
$\nu(\widetilde{f})\leq c\mu(f)\mbox{~{}for
all~{}}f\in\mathcal{S}^{p}_{\delta}(X).$
With $\texttt{P}\in S(\mathfrak{a})$ and $t\in\mathbb{N}\cup\\{0\\}$ we get
the following by using the integral expression (2.1) of the $\delta$-spherical
transform.
$\displaystyle\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)\left\\{\left(\langle\lambda,\lambda\rangle_{1}+\|\rho\|^{2}\right)^{t}\widetilde{f}(\lambda)\right\\}$
$\displaystyle=\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)\int_{G}trf(x)~{}\left(\langle\lambda,\lambda\rangle_{1}+\|\rho\|^{2}\right)^{t}\Phi_{\overline{\lambda},\delta}^{*}(x)dx$
(4.1)
$\displaystyle=(-1)^{n}\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)\int_{G}trf(x)~{}~{}\mathbf{L}^{t}\Phi_{\overline{\lambda},\delta}^{*}(x)dx.$
The last equality is a consequence of the property (2.8) of the generalized
spherical function. Now a simple application of integration by parts gives:
$\displaystyle(\ref{ali:10})$
$\displaystyle=(-1)^{n}\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)\int_{G}\mathbf{L}^{t}trf(x)~{}~{}\Phi_{\overline{\lambda},\delta}^{*}(x)dx$
$\displaystyle=(-1)^{n}\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)\int_{G}tr~{}\mathbf{L}^{t}f(x)~{}~{}\Phi_{\overline{\lambda},\delta}^{*}(x)dx$
(4.2)
$\displaystyle=(-1)^{n}\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)\int_{G}tr~{}\mathbf{L}^{t}f(x)\left\\{\int_{K}e^{(i\lambda-\rho)(\mathcal{H}(x^{-1}k))}\delta(k^{-1})dk\right\\}dx.$
The second line in the above chain of equalities uses the fact
$\mathbf{L}trf(\cdot)=tr\mathbf{L}f(\cdot)$ which is clear as the differential
operator $\mathbf{L}$ acts entry-wise to the operator valued function $f$.
As $f\in\mathcal{S}^{p}_{\delta}(X)$ so it can also be considered as a
right-$K$-invariant function on the group $G$. The action of the Laplace
Beltrami operator $\mathbf{L}$ on $f$ is the same as the action of the Casimir
operator on $f$ considering as a function on $G$. Therefore, by the property
of the Casimir operator, the action of $\mathbf{L}$ does not change the
left-$K$-type of the function $f$, i. e the function $\mathbf{L}^{t}f(\cdot)$
is again of left-$\delta$-type. Moreover, for each nonnegative integer $n$ the
function $\mathbf{L}^{t}f(\cdot)\in\mathcal{S}^{p}_{\delta}(X)$. Hence by
Lemma 3.12 the integral on the right hand side of (4) exists absolutely. We
apply Fubini’s theorem to interchange the integrals and then we put
$x^{-1}k=y^{-1}$ to get:
$\displaystyle(\ref{ali:101})$
$\displaystyle=(-1)^{n}\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)\int_{K}\int_{G}tr\mathbf{L}^{t}f(ky)~{}~{}e^{(i\lambda-\rho)\mathcal{H}(y^{-1})}\delta(k^{-1})dydk$
$\displaystyle=(-1)^{n}\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)\int_{G}\left\\{\int_{K}tr\mathbf{L}^{t}f(ky)~{}~{}\delta(k^{-1})dk\right\\}~{}~{}e^{(i\lambda-\rho)\mathcal{H}(y^{-1})}dy$
$\displaystyle=\frac{(-1)^{t}}{d_{\delta}}\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)\int_{G}\mathbf{L}^{t}f(y)~{}~{}e^{(i\lambda-\rho)\mathcal{H}(y^{-1})}dy$
(4.3)
$\displaystyle=\frac{(-1)^{t}}{d_{\delta}}\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)\int_{G}\mathbf{L}^{t}f(y^{-1})~{}~{}e^{(i\lambda-\rho)\mathcal{H}(y)}dy$
The third lie follows by using the Schwartz space extension of the isomorphism
$\mathcal{Q}$ of Proposition 1.1 and the last line uses the invariance of the
Haar measure under the transformation $g\mapsto g^{-1}$. Let us now break up
the group $G$ as well as the Haar measure using the Iwasawa decomposition
$KAN$ decomposition to get.
$\displaystyle(\ref{ali:102})=$
$\displaystyle\frac{(-1)^{t}}{d_{\delta}}\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)\int_{K}\int_{\mathfrak{a}^{+}}\int_{N}\mathbf{L}^{t}f(n^{-1}(\exp
H)^{-1}k^{-1})e^{(i\lambda-\rho)(\mathcal{H}(k(\exp H)n)}dke^{2\rho(H)}dHdn$
(4.4)
$\displaystyle=\frac{(-1)^{t}}{d_{\delta}}\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)\int_{\mathfrak{a}^{+}}\int_{N}\mathbf{L}^{t}f(n^{-1}(\exp
H)^{-1})e^{(i\lambda+\rho)(H)}dHdn.$
Let $\alpha_{1},\alpha_{2},\cdots,\alpha_{r}$ be the set of all positive
restricted roots. Let $\varepsilon_{i}\in\mathfrak{a}^{*}$ ($1\leq i\leq r$)
be such that $\langle\alpha_{i},\varepsilon_{j}\rangle=\delta_{ij}$. Then
clearly $\\{\varepsilon_{i}\\}_{1\leq i\leq r}$ forms a basis of
$\mathfrak{a}^{*}$ and thus we introduce a global coordinate on
$\mathfrak{a}^{*}$ by $\lambda=\sum_{i=1}^{r}\lambda_{i}\varepsilon_{i}$
($\forall\lambda\in\mathfrak{a}^{*}$).
Let
$\texttt{P}(\lambda)=\sum_{\theta=0}^{\beta}\sum_{\beta_{1}+\cdots+\beta_{r}=\theta}\alpha_{\beta_{1}+\cdots+\beta_{r}}\lambda_{1}^{\beta_{1}}\lambda_{2}^{\beta_{2}}\cdots\lambda_{r}^{\beta_{r}}$.
Thus,
$\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)=\sum_{\theta=0}^{\beta}\sum_{\beta_{1}+\cdots+\beta_{r}=\theta}\alpha_{\beta_{1}+\cdots+\beta_{r}}\left(\frac{\partial}{\partial\lambda_{1}}\right)^{\beta_{1}}\left(\frac{\partial}{\partial\lambda_{2}}\right)^{\beta_{2}}\cdots\left(\frac{\partial}{\partial\lambda_{r}}\right)^{\beta_{r}}.$
There fore it is easy to check that:
(4.5)
$\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)e^{(i\lambda+\rho)(H)}=\texttt{P}(H)~{}e^{(i\lambda+\rho)(H)},$
where
$\texttt{P}(H)=\sum_{\theta=0}^{\beta}\sum_{\beta_{1}+\cdots+\beta_{r}=\theta}\alpha_{\beta_{1}+\cdots+\beta_{r}}\varepsilon_{1}^{\beta_{1}}(iH))\cdots\varepsilon_{r}^{\beta_{r}}(iH)$.
From (4) it follows that:
$\displaystyle\left\|\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)\left\\{\left(\langle\lambda,\lambda\rangle_{1}+\|\rho\|^{2}\right)^{n}\widetilde{f}(\lambda)\right\\}\right\|_{\mathbf{2}}$
$\displaystyle\hskip
72.26999pt\leq\frac{1}{d_{\delta}}\int_{\mathfrak{a}^{+}}\int_{N}\|\mathbf{L}^{t}f(n^{-1}(\exp
H)^{-1})\|_{\mathbf{2}}~{}\|\texttt{P}(H)\|\left|e^{(i\lambda+\rho)(H)}\right|dHdn$
(4.6) $\displaystyle\hskip
72.26999pt\leq\frac{1}{d_{\delta}}\int_{\mathfrak{a}^{+}}\int_{A}\|\mathbf{L}^{t}f(n^{-1}(\exp
H)^{-1})\|_{\mathbf{2}}~{}\|\texttt{P}(H)\|~{}e^{(|\Im\lambda|+\rho)(H)}dHdn.$
Some basic estimates gives the following:
$\displaystyle\|\varepsilon_{j}^{\beta_{j}}(iH)\|$
$\displaystyle\leq\|\varepsilon_{j}\|^{\beta_{j}}\|H\|$ $\displaystyle\leq
c\|\varepsilon_{j}\|^{\beta_{j}}|(\exp H)n|$ (4.7) $\displaystyle\leq
c\|\varepsilon_{i}\|^{\beta_{i}}(1+|(\exp H)n|).$
The above estimate is a consequence of (1.3) and it is true for all $n\in N$.
Using (4) one can find $d_{\texttt{P}}\in\mathbb{Z}^{+}$ such that
(4.8) $\|\texttt{P}(H)\|\leq c_{1}(1+|(\exp H)n|)^{d_{\texttt{P}}}.$
As $f\in\mathcal{S}^{p}_{\delta}(X)$ so for each $m\in\mathbb{Z}^{+}$ we have:
(4.9) $\|\mathbf{L}^{t}f(n^{-1}(\exp
H)^{-1})\|_{\mathbf{2}}\leq\mu_{\mathbf{L}^{t},m}(f)~{}(1+|(\exp
H)n|)^{-m}~{}\varphi_{0}^{\frac{2}{p}}(n^{-1}(\exp H)^{-1}).$
The above inequality also uses the fact that $|g|=|g^{-1}|$ for all $g\in G$.
The estimates (4) and (4.9) reduce the inequation (4) to the following:
$\displaystyle\left\|\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)\left\\{\left(\langle\lambda,\lambda\rangle_{1}+\|\rho\|^{2}\right)^{t}\widetilde{f}(\lambda)\right\\}\right\|_{\mathbf{2}}$
$\displaystyle\leq
c_{1}\frac{\mu_{\mathbf{L}^{t},m}(f)}{d_{\delta}}\int_{\mathfrak{a}^{+}}\int_{N}\varphi_{0}^{\frac{2}{p}}(n^{-1}(\exp
H)^{-1})~{}(1+|(\exp
H)n|)^{-m+d_{\texttt{P}}}~{}e^{(|\Im\lambda|+\rho)(H)}dHdn$
$\displaystyle=c_{1}\frac{\mu_{\mathbf{L}^{t},m}(f)}{d_{\delta}}\int_{K}\int_{\mathfrak{a}^{+}}\int_{N}\varphi_{0}^{\frac{2}{p}}(n^{-1}(\exp
H)^{-1}k^{-1})(1+|k(\exp H)n|)^{-m+d_{\texttt{P}}}$ $\displaystyle\hskip
224.03743pte^{(|\Im\lambda|-\rho)(\mathcal{H}(k(\exp H)n))}dke^{2\rho(H)}dHdn$
(4.10)
$\displaystyle=c_{\delta}\mu_{\mathbf{L}^{t},m}(f)\int_{G}\varphi_{0}^{\frac{2}{p}}(g^{-1})(1+|g|)^{-m+d_{\texttt{P}}}e^{(|\Im\lambda|-\rho)(\mathcal{H}(g))}dg$
where $c_{\delta}=c_{1}\frac{1}{d_{\delta}}$. Now we use the Cartan
decomposition i.e $g=k_{1}\exp{|g|}k_{2}$ and appropriate form of the Haar
measure (1.4) to get:
$\displaystyle(\ref{ali:14})$
$\displaystyle=c_{\delta}\mu_{\mathbf{L}^{t},m}(f)\int_{\mathfrak{a}^{+}}\int_{K}\varphi_{0}^{\frac{2}{p}}(\exp{|g^{-1}|})(1+|g|)^{-m+d_{\texttt{P}}}e^{(|\Im\lambda|-\rho)(\mathcal{H}(\exp{|g|}k_{2}))}\Delta(|g|)d|g|~{}dk_{2},$
$\displaystyle=c_{\delta}\mu_{\mathbf{L}^{t},m}(f)\int_{\mathfrak{a}^{+}}\varphi_{0}^{\frac{2}{p}}(\exp{|g|})(1+|g|)^{-m+d_{\texttt{P}}}\left\\{\int_{K}e^{(i(-i|\Im\lambda|)-\rho)(\mathcal{H}(\exp{|g|}k_{2}))}dk_{2}\right\\}$
$\displaystyle\hskip 325.215pt\Delta(|g|)d|g|$
$\displaystyle=c_{\delta}\mu_{\mathbf{L}^{t},m}(f)\int_{\mathfrak{a}^{+}}\varphi_{0}^{\frac{2}{p}}(\exp{|g|})(1+|g|)^{-m+d_{\texttt{P}}}~{}\varphi_{-i|\Im\lambda|}(\exp|g^{-1}|)\Delta(|g|)d|g|$
(4.11) $\displaystyle\leq
c_{\delta}\mu_{\mathbf{L}^{t},m}(f)\int_{\mathfrak{a}^{+}}\varphi_{0}^{\frac{2}{p}+1}(\exp{|g|})(1+|g|)^{-m+d_{\texttt{P}}}~{}e^{|\Im\lambda(|g|)|}\Delta(|g|)d|g|$
where the last inequality in this chain follows by using the estimate (1.9) of
the elementary spherical function. We take
$\lambda\in\mathfrak{a}^{*}_{\varepsilon}$, therefore
$|\Im\lambda(|g|)|\leq\varepsilon\rho(|g|)$ where
$\varepsilon=\left(\frac{2}{p}-1\right)$. Now by using the another fundamental
estimate (1.10) we further reduce the inequality (4) to the following
$\displaystyle\left\|\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)\left\\{\left(\langle\lambda,\lambda\rangle_{1}+\|\rho\|^{2}\right)^{t}\widetilde{f}(\lambda)\right\\}\right\|_{\mathbf{2}}$
$\displaystyle\hskip 108.405pt\leq
c_{\delta}\mu_{\mathbf{L}^{t},m}(f)\int_{\mathfrak{a}^{+}}\varphi_{0}^{2}(\exp{|g|})(1+|g|)^{-m+d_{\texttt{P}}+\varepsilon\theta}\Delta(|g|)~{}d|g|,$
(4.12) $\displaystyle\hskip
108.405pt=c_{\delta}\mu_{\mathbf{L}^{t},m}(f)\int_{G}\varphi_{0}^{2}(g)~{}(1+|g|)^{-m+d_{\texttt{P}}+\varepsilon\theta}~{}dg.$
We choose a suitably large $m\in\mathbb{Z}^{+}$ so the integral (4) converges
([HC66, Lemma 7]). This completes the proof of the Lemma.
We now take up the extension of the inversion formula (2.12) of the
$\delta$-spherical transform.
###### Lemma 4.2.
For each $h\in\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ the
following integral
(4.13)
$\frac{1}{\omega}\int_{\mathfrak{a}^{*}}\Phi_{\lambda,\delta}(x)~{}h(\lambda)~{}|\mathbf{c}(\lambda)|^{-2}d\lambda,~{}~{}\mbox{~{}~{}~{}~{}for
}x\in X,$
gives a $Hom(V_{\delta},V_{\delta})$ valued, left-$\delta$-type
$\mathcal{C}^{\infty}$ function on $X$. (From now on we shall denote this
function by $\mathcal{I}h(\cdot)$.)
###### Proof.
Let us take any $D\in\mathcal{U}(\mathfrak{g}_{\mathbb{C}})$. Then,
(4.14)
$\displaystyle\frac{1}{\omega}\int_{\mathfrak{a}^{*}}\left\|\Phi_{\lambda,\delta}(D,x)\right\|_{\mathbf{2}}$
$\displaystyle\|h(\lambda)\|_{\mathbf{2}}|\mathbf{c}(\lambda)|^{-2}d\lambda\leq
c_{\delta}~{}\varphi_{0}(x)\int_{\mathfrak{a}^{*}}(1+\|\lambda\|)^{b_{\textbf{D}}+b-n}d\lambda.$
The above inequality follows from the fact that
$h\in\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ and by using the
decay (3.7), the estimate (1.13) and the estimate (2.10) for the generalized
spherical functions. One can choose a suitably large $n$ so that the integral
in the right hand side of (4.14) converges. This proves $\mathcal{I}h$ is a
function on $X$ and $\textbf{D}\mathcal{I}h$ exists for all
$\textbf{D}\in\mathcal{U}(\mathfrak{g}_{\mathbb{C}})$. Hence
$\mathcal{I}h\in\mathcal{C}^{\infty}(X,Hom(V_{\delta},V_{\delta}))$. As,
$\Phi_{\lambda,\delta}(\cdot)$ is of left-$\delta$-type ( (2.4) of Proposition
2.3), so is $\mathcal{I}h$.
###### Lemma 4.3.
If $h\in\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ then the inverse
$\mathcal{I}h\in\mathcal{S}^{p}_{\delta}(X)$.
###### Proof.
To prove this Lemma we shall first consider the spaces
$\mathcal{P}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$ and
$\mathcal{D}_{\delta}(X)$ equipped with the topologies of the respective
Schwartz spaces containing them. We have already noticed that
$\mathcal{P}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$ and
$\mathcal{D}_{\delta}(X)$ are dense subspaces of
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ and
$\mathcal{S}^{p}_{\delta}(X)$ respectively.
We shall show that $\mathcal{I}$ is a continuous map from
$\mathcal{P}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$ onto (by Theorem 2.8)
$\mathcal{D}_{\delta}(X)$ with respective to the Schwartz space topologies.
That is for $h\in\mathcal{P}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$ and for
each seminorm $\mu$ on $\mathcal{D}_{\delta}(X)$ , there exists a seminorm
$\nu$ on $\mathcal{P}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$ such that
$\mu(f)\leq c_{\delta}\nu(h)$, where
$f=\mathcal{I}h\in\mathcal{D}_{\delta}(X)$ and $c_{\delta}$ is a positive
constant depending on $\delta\in\widehat{K}_{M}$.
As, $f\in\mathcal{D}_{\delta}(X)$, by Lemma 2.10, we get a function
$\phi\in\mathcal{D}(G//K,Hom(V_{\delta},V_{\delta}))$ such that
$f\equiv\textbf{D}^{\delta}\phi$. If $\Phi$ be the image of $\phi$ under the
spherical transform then it follows easily that $h=Q^{\delta}\Phi$. Let
$\textbf{D},\textbf{E}\in\mathcal{U}(\mathfrak{g}_{\mathbb{C}})$ and $n$ be
any nonnegative integer, then
$\displaystyle\mu_{\textbf{D},\textbf{E},n}(f)$ $\displaystyle=\sup_{x\in
G}\|f(\textbf{D},x,\textbf{E})\|_{\mathbf{2}}(1+|x|)^{n}\varphi^{-\frac{2}{p}}_{0}(x),$
$\displaystyle=\sup_{x\in
G}\|\textbf{D}^{\delta}\phi(\textbf{D},x,\textbf{E})\|_{\mathbf{2}}(1+|x|)^{n}\varphi^{-\frac{2}{p}}_{0}(x),$
(4.15)
$\displaystyle={\mu_{0}}_{\textbf{D}^{\delta}\textbf{D},\textbf{E},n}(\phi).$
( _Here $\mu_{0}$ denote the seminorms on the Fréchet space
$\mathcal{S}^{p}(G//K,Hom(V_{\delta},V_{\delta}))$_. ) At this point we use
Anker’s [Ank91] proof of the Schwartz space isomorphism theorem for
bi-$K$-invariant functions. For each
$\textbf{D},\textbf{E}\in\mathcal{U}(\mathfrak{g}_{\mathbb{C}})$ and
$n\in\mathbb{Z}^{+}$ one can find a polynomial $P\in S(\mathfrak{a})$ and
$m_{\delta}\in\mathbb{Z}^{+}$ (depending on $d_{\delta}$) such that,
$\displaystyle{\mu_{0}}_{\textbf{D}^{\delta}\textbf{D},\textbf{E},n}(\phi)$
$\displaystyle\leq c_{\delta}\sup_{\lambda\in
Int\mathfrak{a}^{*}_{\varepsilon}}\left\|P\left(\frac{\partial}{\partial\lambda}\right)\Phi(\lambda)\right\|_{\mathbf{2}}(1+\|\lambda\|)^{m_{\delta}},$
(4.16) $\displaystyle\leq c_{\delta}\sup_{\lambda\in
Int\mathfrak{a}^{*}_{\varepsilon}}\left\|P_{1}\left(\frac{\partial}{\partial\lambda}\right)h(\lambda)\right\|_{\mathbf{2}}(1+\|\lambda\|)^{m^{\prime}_{\delta}}.$
The last line in (4) follows by using the isomorphism, proved in Lemma 3.10,
between the Schwartz spaces $S_{0}(\mathfrak{a}^{*}_{\varepsilon})$ and
$S_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$. Hence (4) and (4) togather gives
$\mu_{\textbf{D},\textbf{E},n}(f)\leq
c_{\delta}\nu_{P_{1},m^{\prime}_{\delta}}(h)$. As we have started with an
$h\in\mathcal{P}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})\subset\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$,
the right hand side of (4) is clearly finite. Hence
$\mathcal{I}h=f\in\mathcal{S}^{p}_{\delta}(X)$.
Now we apply the density argument to conclude the Lemma. Let us now take
$h\in\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$. As,
$\mathcal{P}_{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$ is dense in
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$, there exists a Cauchy
sequence
$\left\\{h_{n}\right\\}\subset\mathcal{P}_{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$
converging to $h$. Then, by what we have proved above, we can get a Cauchy
sequence $\left\\{f_{n}\right\\}\subset\mathcal{D}_{\delta}(X)$ such that
$\widetilde{f_{n}}=h_{n}$. As $\mathcal{S}^{p}_{\delta}(X)$ is a Fréchet space
the sequence must converge to some $f\in\mathcal{S}^{p}_{\delta}(X)$. Clearly,
$f=\mathcal{I}h$. This completes the proof of the Lemma.
We note that, the Lemma 4.3 also implies the fact that the $\delta$-spherical
transform is an injection in the corresponding Schwartz space level.
Finally, Lemma 4.1 and Lemma 4.3 together shows that the $\delta$-spherical
transform is a continuous surjection of $\mathcal{S}_{\delta}^{p}(X)$ onto
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ for $(0<p\leq 2)$. A
simple application of the open mapping theorem concludes that the
$\delta$-spherical transform is a topological isomorphism between the
corresponding Schwartz spaces. This proves the Theorem 3.8.
In the next section we shall extend this result to a slightly larger class of
functions.
## 5\. Finite $K$-type functions
Let choose and fix a finite subset $\Gamma\subset\widehat{K}_{M}$. We denote
$\mathcal{D}_{\Gamma}(X)$ for the space of all compactly supported
$\mathcal{C}^{\infty}$ functions on $X$ with the property that: for
$f\in\mathcal{D}_{\Gamma}(X)$ $f^{\delta}\equiv 0$ for $\delta\notin\Gamma$.
We take the subclass $\mathcal{S}_{\Gamma}^{p}(X)$ of the Schwartz class
$\mathcal{S}^{p}(X)$ (for $0<p\leq 2$) defined by
(5.1)
$\mathcal{S}^{p}_{\Gamma}(X)=\\{f\in\mathcal{S}^{p}(X)~{}|~{}f(X)=\sum_{\delta\in\Gamma}trf^{\delta}(x)~{}\mbox{~{}for
all~{}}x\in X\\}.$
The seminorms on $\mathcal{S}^{p}_{\Gamma}(X)$ are as follows: for each
$\textbf{D},\textbf{E}\in\mathcal{U}(\mathfrak{g}_{\mathbb{C}})$ and
$n\in\mathbb{Z}^{+}$,
(5.2) ${\mu_{\Gamma}}_{\textbf{D},\textbf{E},n}(f)=\sup_{\delta\in\Gamma,x\in
X}\left\|f^{\delta}(\textbf{D},x,\textbf{E})\right\|_{\mathbf{2}}~{}(1+|x|)^{n}~{}\varphi_{0}^{-\frac{2}{p}}(x)~{}<+\infty.$
Clearly $\mathcal{D}_{\Gamma}(X)$ is a dense subset of the Schwartz space
$\mathcal{S}^{p}_{\Gamma}(X)$ with respect to the Fréchet topology induced by
the countable family of seminorms
$\\{{\mu_{\Gamma}}_{\textbf{D},\textbf{E},n}\\}$. It also follows easily from
the definition 3.1 and (3.2 ) that, if $f\in\mathcal{S}^{p}_{\Gamma}(X)$ then
for each $\delta\in\Gamma$ the projection
$f^{\delta}\in\mathcal{S}^{p}_{\delta}(X)$. For these classes of functions the
transform we shall mainly consider is the Helgason Fourier transform.
Let us now define the Schwartz class functions on the domain
$\mathfrak{a}^{*}_{\varepsilon}\times K/M$.
###### Definition 5.1.
Let $\mathcal{S}_{\Gamma}(\mathfrak{a}^{*}_{\varepsilon}\times K/M)$ denotes
the class of functions $h$ on $\mathfrak{a}^{*}_{\varepsilon}\times K/M$
satisfying the following properties:
1. (i)
For each $kM\in K/M$, the function $\lambda\mapsto h(\lambda,kM)$ is
holomorphic on $Int\mathfrak{a}^{*}_{\varepsilon}$, and it extends as a
continuous function on the closed complex tube
$\mathfrak{a}^{*}_{\varepsilon}$. The function $h$ is a smooth function in the
$k\in K/M$ variable.
2. (ii)
For all $\lambda\in\mathfrak{a}^{*}_{\varepsilon}$, $\omega\in W$ and $x\in G$
(5.3) $\check{h}(\lambda,x)=\check{h}(\omega\lambda,x),$
where
$\check{h}(\lambda,x)=\int_{K}h(\lambda,k)e^{-(i\lambda+\rho)H(x^{-1}k)}dk$.
3. (iii)
For each $P\in S(\mathfrak{a})$ and for integers $n,m>0$ the function $h$
satisfies the following decay condition
(5.4) $\sup_{(\lambda,k)\in Int\mathfrak{a}^{*}_{\varepsilon}\times
K/M}\left|P\left(\frac{d}{d\lambda}\right)h(\lambda,k,\omega^{m}_{\mathfrak{k}})\right|~{}(1+|\lambda|)^{n}<~{}+\infty.$
4. (iv)
For each $\delta\in\widehat{K}_{M}\setminus\Gamma$ the
left-$\delta$-projection $h^{\delta}$ defined by
(5.5)
$h^{\delta}(\lambda,k)=d_{\delta}\int_{K}h(\lambda,k_{1}k)\delta(k_{1}^{-1})dk_{1},$
is identically a zero function on $\mathfrak{a}^{*}_{\varepsilon}\times K/M$.
The space $\mathcal{S}_{\Gamma}(\mathfrak{a}^{*}_{\varepsilon}\times K/M)$ is
a Fréchet space with the topology induced by the seminorms (5.4). By the
theory of smooth functions on compact groups [Sug71], the topology of the
space $\mathcal{S}_{\Gamma}(\mathfrak{a}^{*}_{\varepsilon}\times K/M)$ can be
given by the following equivalent family of seminorms, for each $P\in
S(\mathfrak{a})$ and $m\in\mathbb{Z}^{+}$ we have
(5.6) $\sup_{\lambda\in Int\mathfrak{a}^{*}_{\varepsilon},\delta\in
F}\left\|P\left(\frac{d}{d\lambda}\right)h^{\delta}(\lambda,eM)\right\|_{\textbf{2}}(1+|\lambda|)^{m}<+\infty,\mbox{~{}for~{}}h\in\mathcal{S}_{\Gamma}(\mathfrak{a}^{*}_{\varepsilon}\times
K/M).$
We denote by $\mathcal{S}(\mathfrak{a}^{*}_{\varepsilon}\times K/M)$ the
Fréchet space satisfying all the conditions of the Definition 5.1 except
condition (iv). The space
$\mathcal{S}_{\Gamma}(\mathfrak{a}^{*}_{\varepsilon}\times K/M)$ is a closed
subspace of $\mathcal{S}(\mathfrak{a}^{*}_{\varepsilon}\times K/M)$. We know
that the HFT can extended to the Schwartz class $\mathcal{S}^{p}(X)$ [EK76],
furthermore the HFT is a continuous map from $\mathcal{S}^{p}(X)$ into
$\mathcal{S}(\mathfrak{a}^{*}_{\varepsilon}\times K/M)$. Hence the HFT is a
continuous map from $\mathcal{S}^{p}(F,X)$ into
$\mathcal{S}_{\Gamma}(\mathfrak{a}^{*}_{\varepsilon}\times K/M)$.
###### Lemma 5.2.
Let $h\in\mathcal{S}_{\Gamma}(\mathfrak{a}^{*}_{\varepsilon}\times K/M)$, then
for each $\delta\in F$, the left-$\delta$-projection
$h^{\delta}\in\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$.
###### Proof.
The function $\lambda\mapsto h^{\delta}(\lambda,eM)$ trivially satisfies
condition (i) of Definition 3.6. The required decay (3.3) is also an easy
consequence of (5.4). It can be shown that the $\delta$-projection
$h^{\delta}$ also satisfies the condition
$\check{(h^{\delta})}(\lambda,x)=\check{(h^{\delta})}(\omega\lambda,x)\mbox{~{}for
all~{}}\omega\in W.$
It is easy to check that
$h^{\delta}(\lambda,kM)=\delta(k)h^{\delta}(\lambda,eM)$, hence for each
$(\lambda,a)\in\mathfrak{a}^{*}_{\varepsilon}\times A$ we write
$\check{(h^{\delta})}(\lambda,a)$ as follows
$\check{(h^{\delta})}(\lambda,a)=\Phi_{\lambda,\delta}(a)h^{\delta}(\lambda,eM).$
By the property (2.6) of the generalized spherical functions it follows that
the function $\lambda\mapsto Q^{\delta}(\lambda)^{-1}h^{\delta}(\lambda,eM)$
is $W$-invariant. Hence we conclude that
$h^{\delta}(\cdot,eM)\in\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$.
By using Theorem 3.8, for each
$h\in\mathcal{S}_{\Gamma}(\mathfrak{a}^{*}_{\varepsilon}\times K/M)$ we get an
unique finite sequence $\\{f^{\delta}\\}_{\delta\in\Gamma}$ of
$\mathcal{C}^{\infty}$ functions on $X$ such that each member
$f^{\delta}\in\mathcal{S}^{p}_{\delta}(X)$. We consider the following scalar
valued function
(5.7) $f(x)=\sum_{\delta\in F}trf^{\delta}(x),~{}~{}~{}x\in X.$
For each $\delta\in\Gamma$,
$\left(\mathcal{F}^{-1}h\right)^{\delta}(x)=\mathcal{I}(h^{\delta})(x)=f^{\delta}(x)$.
Hence, we get $\mathcal{F}^{-1}h(x)=f(x)$ for all $x\in X$. The function
$f\in\mathcal{S}^{p}_{\Gamma}(X)$. Furthermore for each
$D,E\in\mathcal{U}(\mathfrak{g}_{\mathbb{C}})$ and $n\in\mathbb{Z}^{+}$ we
have
$\displaystyle\sup_{x\in G}|f(D,x,E)|$
$\displaystyle(1+|x|)^{n}\varphi_{0}^{-\frac{2}{p}}(x)$ $\displaystyle\leq
c\sup_{x\in
G,\delta\in\Gamma}\|f^{\delta}(D,x,E)\|_{\textbf{2}}(1+|x|)^{n}\varphi_{0}^{-\frac{2}{p}}(x)$
$\displaystyle\leq c_{1}\sup_{\lambda\in
Int\mathfrak{a}^{*}_{\varepsilon},\delta\in\Gamma}\left\|P\left(\frac{d}{d\lambda}\right)h^{\delta}(\lambda,eM)\right\|_{\textbf{2}}(1+|\lambda|)^{m}$
$\displaystyle\leq c_{2}\sup_{\lambda\in
Int\mathfrak{a}^{*}_{\varepsilon},k\in
K}\left|P_{1}\left(\frac{d}{d\lambda}\right)h(\lambda,k,\omega_{\mathfrak{k}}^{r})\right|(1+|\lambda|)^{m_{1}},$
for some $P_{1}\in\mathcal{S}(\mathfrak{a}^{*})$ and
$r,m_{1}\in\mathbb{Z}^{+}$. Thus, the HFT is a bijective map from
$\mathcal{S}^{p}_{\Gamma}(X)$ to
$\mathcal{S}_{\Gamma}(\mathfrak{a}^{*}_{\varepsilon}\times K/M)$. Once again,
by the open mapping theorem we conclude the following.
###### Theorem 5.3.
Let $\Gamma$ be a finite subset of $\widehat{K}_{M}$, then the HFT is a
topological isomorphism of the space $\mathcal{S}^{p}_{\Gamma}(X)$ onto the
Fréchet space $\mathcal{S}_{\Gamma}(\mathfrak{a}^{*}_{\varepsilon}\times
K/M)$.
Acknowledgments: The author is thankful to Prof. Angela Pasquale of Université
de Metz, France for her valuable suggestions.
## References
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* [Ank91] by same author, _The spherical Fourier transform of rapidly decreasing functions. A simple proof of a characterization due to Harish-Chandra, Helgason, Trombi, and Varadarajan_ , J. Funct. Anal. 96 (1991), no. 2, 331–349.
* [Ank92] by same author, _Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces_ , Duke Math. J. 65 (1992), no. 2, 257–297. MR MR1150587 (93b:43007)
* [Art79] James Arthur, _Eisenstein series and the trace formula_ , Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 253–274. MR MR546601 (81b:10020)
* [EK76] Masaaki Eguchi and Atsutaka Kowata, _On the Fourier transform of rapidly decreasing functions of $L^{p}$ type on a symmetric space_, Hiroshima Math. J. 6 (1976), no. 1, 143–158.
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* [HC58a] Harish-Chandra, _Spherical functions on a semisimple Lie group. I_ , Amer. J. Math. 80 (1958), 241–310.
* [HC58b] by same author, _Spherical functions on a semisimple Lie group. II_ , Amer. J. Math. 80 (1958), 553–613.
* [HC66] by same author, _Discrete series for semisimple Lie groups. II. Explicit determination of the characters_ , Acta Math. 116 (1966), 1–111.
* [Hel] Sigurdur Helgason, _Groups and geometric analysis_ , Mathematical Surveys and Monographs, vol. 83.
* [Hel94] by same author, _Geometric analysis on symmetric spaces_ , Mathematical Surveys and Monographs, vol. 39, American Mathematical Society, Providence, RI, 1994.
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|
arxiv-papers
| 2010-02-25T19:40:55 |
2024-09-04T02:49:08.556793
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Joydip Jana",
"submitter": "Joydip Jana",
"url": "https://arxiv.org/abs/1002.4855"
}
|
1002.4856
|
# The High-Density Ionized Gas in the Central Parsecs of the Galaxy
Jun-Hui Zhao, Ray Blundell, James M. Moran D. Downes, Karl F. Schuster Dan
Marrone
###### Abstract
We report the results from observations of H30$\alpha$ line emission in Sgr A
West with the Submillimeter Array at a resolution of 2′′ and a field of view
of about 40′′. The H30$\alpha$ line is sensitive to the high-density ionized
gas in the minispiral structure. We compare the velocity field obtained from
H30$\alpha$ line emission to a Keplerian model, and our results suggest that
the supermassive black hole at Sgr A* dominates the dynamics of the ionized
gas. However, we also detect significant deviations from the Keplerian motion,
which show that the impact of strong stellar winds from the massive stars
along the ionized flows and the interaction between Northern and Eastern arms
play significant roles in the local gas dynamics.
Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, MS 78,
Cambridge, MA 02138
Institut de Radio Astronomie Millimétrique, 38406 Saint Martin d’Hères, France
Department of Astronomy, University of Chicago, Chicago, IL 60637
## 1\. Introduction
The Galactic Center harbors a supermassive black hole (SMBH) with a mass of
$4.2\times 10^{6}$ M⊙ at the position of Sgr A* (Ghez et al. 2008; Gillessen
et al. 2009). Previous radio recombination line studies (Schwarz, Bregman, &
van Gorkom 1989; Roberts & Goss 1993; Roberts, Yusef-Zadeh, & Goss 1996)
revealed complex structures of ionized gas around Sgr A*, including the
“minispiral” and “Bar.” Parts of these structures have been modeled as gas
orbiting Sgr A* (Schwarz, Bregman, & van Gorkom 1989; Sanders 1998; Paumard,
Maillard, & Morris 2004; Vollmer & Duschl 2000; Liszt 2003; Zhao et al. 2009)
while other parts of the structures may be gravitationally unbound (Yusef-
Zadeh et al. 1998). In particular, the strong stellar winds from the clusters
of massive stars in the vicinity of Sgr A*, with the help of gravitational
focusing, create large-scale, unbound features (Lutz, Krabbe, & Genzel 1993).
There are a number of difficulties in existing line observations of these
structures. In the cm–radio maps from the Very Large Array (VLA), the
recombination line strength is low, the line-to-continuum ratio is low, the
bandwidth coverage is limited or irregular, and it is especially difficult to
detect high-velocity line wings. In the IR, the picture given by the emission
lines from the ionized gas is distorted, because of high obscuration by dust.
In this paper, we summarize the main results from our observations of the
H30$\alpha$ line at 1.3 mm with the SMA from three array configurations, the
combination of which provides an angular resolution of 2′′.
## 2\. Distribution of the High-Density Ionized Gas
Figure 1 presents the image integrated H30$\alpha$ line emission observed in
the central 1.5 parsecs, showing the Northern and Eastern arms of the
minispiral. The Western Arc does not shown well in our image in the
H30$\alpha$ line since most of the line emission is below the 4$\sigma$
sensitivity cutoff. The H30$\alpha$ line spectra are shown toward selected
regions with IR sources. A detailed comparison of the H30$\alpha$ line spectra
and the H92$\alpha$ line spectra observed with the VLA yields line ratios of
H30$\alpha$/H92$\alpha$ in most of the regions that are close to the frequency
ratio $\nu_{H30\alpha}/\nu_{H92\alpha}\sim 28$, suggesting that the ionized
gas in the central parsecs is optically thin and under LTE conditions. In some
regions, however (e.g., IRS 1W and 10W), the line ratios are significantly
smaller than the LTE value.
Figure 1.: The image of the H30$\alpha$ line emission around Sgr A* (plus-dot)
in the central 1.5-parsec region observed with the SMA with a resolution of
2′′. The H30$\alpha$ line spectra of the HII sources within the minispiral are
plotted and labelled with the names of the corresponding IRS sources.
To assess the physical conditions of the ionized gas, we have also fit a model
to the observed lines based on the assumption that the HII sources are
isothermal and homogeneous. The derived typical electron densities in the HII
clumps are 5–10$\times 10^{4}$ cm-3, consistent with the values ($10^{4}$
cm-3) derived from Pa$\alpha$ line emission at 1.87 $\mu$m (Scoville et al.
2003) for a given uncertainty in extinction corrections in the near IR. The
typical electron temperature in the minispiral arms is 5000–7000 K, in good
agreement with the previous determination from H92$\alpha$ (Roberts & Goss
1993). The highest density of $2\times 10^{5}$ cm-3 occurs in the IRS 13E
region while the highest temperatures of up to 13000 K were found in the Bar.
Both the H30$\alpha$ and H92$\alpha$ line profiles are broadened due to the
large velocity gradients along the line of sight and across the 2′′ beam
produced by the large dynamical motions near Sgr A*. For the H92$\alpha$ line,
the line width due to pressure broadening appears to be comparable to that of
thermal Doppler broadening while for the H30$\alpha$ lines, the pressure
broadening is negligible.
## 3\. Kinematics and Dynamics
Figure 2a shows the image of radial velocities determined from fitting the
peak velocities of the H30$\alpha$ line in the central 40′′, showing large
velocity gradients along the Northern and Eastern arms. The observed radial
velocity field is, in general, consistent with that predicted from a Keplerian
model (Zhao et al. 2009), suggesting that the SMBH at Sgr A* governs the
dynamics of the ionized gas. However, in several regions, the observed
kinematics deviate significantly from those predicted from the Keplerian
model, as described below.
Figure 2.: The observed radial velocity field (a) around Sgr A* (plus-dot)
from the SMA observations of the H30$\alpha$ line is compared with that
computed from the Keplerian model (b). The velocity wedge is in units of km
s-1. The inset (c) is the H30$\alpha$ spectrum toward the IRS 34 region
outlined with the black circle NW of Sgr A* in which five massive stars 7SW
(WN8), 34W (Ofpe/WN9), 34E (O9-9.5), 34NW (WN7), and 3E (WC5/6) are located
(shown with “+” symbols, Paumard et al. 2006). The arrow indicates the
direction of the averaged transverse velocity of the four stars that have
proper-motion information. The red circle indicates the region with high
electron temperature where the interaction between the Northern and Eastern
arms occurs. The arrow SE of Sgr A* indicates the transverse velocity of the
IRS 9W star (WN8).
IRS 34: In the northwest end of the Eastern Arm, a kinematic feature with
radial velocity of up to $-200$ km s-1 shows a large deviation from the value
of $-100$ km s-1 predicted from the Keplerian model (see Figure 2b). Near this
region, there is a group of Wolf-Rayet (WR) stars, evolved from early-type O
stars with a mass loss rate of 10-5 M⊙ yr-1. The suspected strong stellar
winds from these massive stars may play a considerable role in the local
kinematics.
IRS 9W: In the “tip” region (Paumard, Maillard, & Morris 2004), a peculiar
radial velocity of +300 km s-1 was detected from our H30$\alpha$ observations,
showing a large deviation from the radial velocity of +150 km s-1 predicted
from the Keplerian model. The impact of the strong stellar wind from the IRS
9W star (WN8) on the orbiting ionized flow in the Eastern Arm is likely
responsible for the high velocity of the tip.
The Bar: Deviations from Keplerian motion are also observed in the Bar region.
These non-Keplerian kinematics could be attributed to the interaction between
the ionized flows in both Northern and Eastern arms and the interaction of
ionized flows with the strong winds from the massive star clusters, e.g., IRS
16 and 13.
Finally, in the locations with large offsets from Sgr A*, we noticed that the
deviations in radial velocity from the Keplerian motions become significant
for the Northern Arm ($>$ 50 km s-1) while for the Eastern Arm the deviations
are less significant. The difference in the deviation of the observed radial
velocities from the Keplerian velocities between the Northern and Eastern arms
might suggest that the distribution of mass has a preference for the plane of
the circumnuclear disk, which is coplanar with the orbit of the Northern Arm
flow.
## References
* Ghez et al. (2008) Ghez, A. M., et al., 2008, ApJ, 689, 1044
* Gillessen et al. (2009) Gillessen, S., et al., 2009, ApJ, 692, 1075
* Liszt (2003) Liszt, H. S., 2003, A&A, 408, 1009
* Lutz, Krabbe, & Genzel (1993) Lutz, D., Krabbe, A., & Genzel, R., 1993, ApJ, 418, L244
* Paumard et al. (2006) Paumard, T., et al., 2006, ApJ, 643, 1011
* Paumard, Maillard, & Morris (2004) Paumard, T., Maillard, J. P., & Morris, M., 2004, A&A, 426, 81
* Roberts & Goss (1993) Roberts, D. A. & Goss, W. M., 1993, ApJS, 86, 133
* Roberts, Yusef-Zadeh, & Goss (1996) Roberts, D. A., Yusef-Zadeh, F., & Goss, W. M., 1996, ApJ, 459, 627
* Sanders (1998) Sanders, R. H., 1998, MNRAS, 294, 35
* Schwarz, Bregman, & van Gorkom (1989) Schwarz, U. J., Bregman, J. D., & van Gorkom, J. H., 1989, A&A, 215, 33
* Scoville et al. (2003) Scoville, N. Z., et al., 2003, ApJ, 594, 294
* Vollmer & Duschl (2000) Vollmer, B. & Duschl, W.J., 2000, New Astronomy, 4, 581
* Yusef-Zadeh et al. (1998) Yusef-Zadeh, F., Roberts, D. A., Biretta, J., 1998, ApJL, 499, L159
* Zhao et al. (2009) Zhao, J.-H., Morris, M. R., Goss, W. M., & An, T., 2009, ApJ, 699, 186
|
arxiv-papers
| 2010-02-25T19:56:22 |
2024-09-04T02:49:08.564307
|
{
"license": "Public Domain",
"authors": "Jun-Hui Zhao, Ray Blundell, James M. Moran, D. Downes, Karl F.\n Schuster, Dan Marrone",
"submitter": "Jun-Hui Zhao",
"url": "https://arxiv.org/abs/1002.4856"
}
|
1003.0022
|
18D10 57M27 17B10 81R05 57R56
# Knot polynomial identities and quantum group coincidences
Scott Morrison scott@tqft.net Emily Peters Supported in part by NSF Grant
DMS0401734 eep@euclid.unh.edu Noah Snyder Supported in part by RTG grant
DMS-0354321 and an NSF Postdoctoral fellowship nsnyder@math.columbia.edu
URLs:http://tqft.net/ http://euclid.unh.edu/~eep
and http://math.columbia.edu/~nsnyder
( First edition: the mysterious future This edition: . )
###### Abstract
We construct link invariants using the $\mathcal{D}_{2n}$ subfactor planar
algebras, and use these to prove new identities relating certain
specializations of colored Jones polynomials to specializations of other
quantum knot polynomials. These identities can also be explained by
coincidences between small modular categories involving the even parts of the
$\mathcal{D}_{2n}$ planar algebras. We discuss the origins of these
coincidences, explaining the role of $SO$ level-rank duality, Kirby-Melvin
symmetry, and properties of small Dynkin diagrams. One of these coincidences
involves $G_{2}$ and does not appear to be related to level-rank duality.
###### keywords:
Planar Algebras, Quantum Groups, Fusion Categories, Knot Theory, Link
Invariants
## 1 Introduction and background
The goal of this paper is to construct knot and link invariants from the
$\mathcal{D}_{2n}$ subfactor planar algebras, and to use these invariants to
prove new identities between quantum group knot polynomials. These identities
relate certain specializations of colored Jones polynomials to specializations
of other knot polynomials. In particular we prove that for any knot $K$ (but
not for a link!),
$\displaystyle\mathcal{J}_{\mathfrak{sl}(2),(2)}(K){}_{\mid q=\exp(\frac{2\pi
i}{12})}{}$ $\displaystyle=2\mathcal{J}_{\mathcal{D}_{4},P}(K)$
$\displaystyle=2,$ (Theorem 3.3)
$\displaystyle\mathcal{J}_{\mathfrak{sl}(2),(4)}(K){}_{\mid q=\exp(\frac{2\pi
i}{20})}{}$ $\displaystyle=2\mathcal{J}_{\mathcal{D}_{6},P}(K)$
$\displaystyle=2\mathcal{J}_{\mathfrak{sl}(2),(1)}(K){}_{\mid
q=\exp(-\frac{2\pi i}{10})}{},$ (Theorem 3.4)
$\displaystyle\mathcal{J}_{\mathfrak{sl}(2),(6)}(K){}_{\mid q=\exp(\frac{2\pi
i}{28})}{}$ $\displaystyle=2\mathcal{J}_{\mathcal{D}_{8},P}(K)$
$\displaystyle=2\operatorname{HOMFLYPT}(K)(\exp(2\pi
i\frac{5}{7}),\exp(-\frac{2\pi i}{14})-\exp(\frac{2\pi i}{14})),$ (Theorem
3.5) $\displaystyle\mathcal{J}_{\mathfrak{sl}(2),(8)}(K){}_{\mid
q=\exp(\frac{2\pi i}{36})}{}$
$\displaystyle=2\mathcal{J}_{\mathcal{D}_{10},P}(K)$
$\displaystyle=2\operatorname{Kauffman}(K)(\exp(2\pi i\frac{31}{36}),\exp(2\pi
i\frac{25}{36})+\exp(2\pi i\frac{11}{36}))$
$\displaystyle=2\operatorname{Kauffman}(K)(-iq^{7},i(q-q^{-1})){}_{\mid
q=-\exp(\frac{-2\pi i}{18})}{}$ (Theorem 1) and
$\displaystyle\mathcal{J}_{\mathfrak{sl}(2),(12)}(K){}_{\mid q=\exp(\frac{2\pi
i}{52})}{}$ $\displaystyle=2\mathcal{J}_{\mathcal{D}_{14},P}(K)$
$\displaystyle=2\mathcal{J}_{G_{2},V_{(10)}}(K){}_{\mid q=\exp{2\pi
i\frac{23}{26}}}{},$ (Theorem 7)
where $\mathcal{J}_{\mathfrak{sl}(2),(k)}$ denotes the $k^{\text{th}}$ colored
Jones polynomial, $\mathcal{J}_{G_{2},V_{(1,0)}}$ denotes the knot invariant
associated to the $7$-dimensional representation of $G_{2}$ and
$\mathcal{J}_{\mathcal{D}_{2n},P}$ is the $\mathcal{D}_{2n}$ link invariant
for which we give a skein-theoretic construction.111Beware, the
$\mathcal{D}_{2n}$ planar algebra is not related to the lie algebra
$\mathfrak{so}_{4n}$ with Dynkin diagram $D_{2n}$ but is instead a quantum
subgroup of $U_{q}(\mathfrak{su}_{2})$. (For our conventions for these
polynomials, in particular their normalizations, see Section 1.1.3.)
These formulas should appear somewhat mysterious, and much of this paper is
concerned with discovering the explanations for them. It turns out that each
of these knot invariant identities comes from a coincidence of small modular
categories involving the even part of one of the $\mathcal{D}_{2n}$. Just as
families of finite groups have coincidences for small values (for example, the
isomorphism between the finite groups $\text{Alt}_{5}$ and
$\mathbf{PSL}_{2}(\mathbb{F}_{5})$ or the outer automorphism of $S_{6}$),
modular categories also have small coincidences. Explicitly, we prove the
following coincidences, where $\frac{1}{2}\mathcal{D}_{2n}$ denotes the even
part of $\mathcal{D}_{2n}$ (the first of these coincidences is well-known).
* •
$\frac{1}{2}\mathcal{D}_{4}\cong\operatorname{Rep}{\mathbb{Z}/3}$, sending $P$
to $\chi_{\exp({\frac{2\pi i}{3}})}$ (but an unusual braiding on
$\operatorname{Rep}{\mathbb{Z}/3}$!).
* •
$\frac{1}{2}\mathcal{D}_{6}\cong\operatorname{Rep}^{\text{uni}}{U_{s=\exp({2\pi
i\frac{7}{10}})}(\mathfrak{sl}_{2}\oplus\mathfrak{sl}_{2})}^{modularize}$,
sending $P$ to $V_{1}\boxtimes V_{0}$. (See Theorem 8, and §3.4.)
* •
$\frac{1}{2}\mathcal{D}_{8}\cong\operatorname{Rep}^{\text{uni}}{U_{s=\exp({2\pi
i\frac{5}{14}})}(\mathfrak{sl}_{4})}^{modularize}$, sending $P$ to
$V_{(100)}$. (See Theorem 8.)
* •
$\frac{1}{2}\mathcal{D}_{10}$ has an order $3$ automorphism:
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(See Theorem 8.)
* •
$\frac{1}{2}\mathcal{D}_{14}\cong\operatorname{Rep}{U_{\exp({2\pi
i\frac{23}{26}})}(\mathfrak{g}_{2})}$ sending $P$ to $V_{(10)}$. (See Theorem
8.)
To interpret the right hand sides of these equivalences, recall that the
definition of the braiding (although not of the quantum group itself) depends
on a choice of $s=q^{\frac{1}{L}}$, where $L$ is the index of the root lattice
in the weight lattice. Furthermore, the ribbon structure on the category of
representations depends on a choice of a certain square root. In particular,
besides the usual pivotal structure there’s also another pivotal structure,
which is called “unimodal” by Turaev [42] and discussed in §1.1.4 below. By
“modularize” we mean take the modular quotient described by Bruguières [6] and
Müger [31] and recalled in §1.1.8 below.
We first prove the knot polynomial identities directly, and later we give more
conceptual explanations of the coincidences using
* •
coincidences of small Dynkin diagrams,
* •
level-rank duality, and
* •
Kirby-Melvin symmetry.
These conceptual explanations do not suffice for the equivalence between the
even part of $\mathcal{D}_{14}$ and $\operatorname{Rep}{U_{\exp({2\pi
i\frac{\ell}{26}})}(\mathfrak{g}_{2})}$, $\ell=-3$ or $10$, which deserves
further exploration. Nonetheless we can prove this equivalence using direct
methods (see Section 3.5), and it answers a conjecture of Rowell’s [34]
concerning the unitarity of $(G_{2})_{\frac{1}{3}}$.
We illustrate each of these coincidences of tensor categories with diagrams of
the appropriate quantum group Weyl alcoves; see in particular Figures 9, 10,
11 and 12 at the end of the paper. An ambitious reader might jump to those
diagrams and try to understand them and their captions, then work back through
the paper to pick up needed background or details.
In more detail the outline of the paper is as follows. In the background
section we recall some important facts about planar algebras, tensor
categories, quantum groups, knot invariants and their relationships. We fix
our conventions for knot polynomials. We also briefly recall several key
concepts like semisimplification, deequivariantization, and modularization.
In Section 2 we use the skein theoretic description of $\mathcal{D}_{2n}$ to
show that the Kauffman bracket gives a braiding up to sign for
$\mathcal{D}_{2n}$, and in particular gives a braiding on the even part (this
was already known; see for example the description of $\mathrm{Rep}^{0}A$ in
[22, p. 33]). Using this, we define and discuss some new invariants of links
which are the $\mathcal{D}_{2n}$ analogues of the colored Jones polynomials.
We also define some refinements of these invariants for multi-component links.
In Section 3, we discuss some identities relating the $\mathcal{D}_{2n}$ link
invariants at small values of $n$ to other link polynomials. This allows us to
prove the above identities between quantum group invariants of knots. The main
technique is to apply the following schema to an object $X$ in a ribbon
category (where $A$ and $B$ always denote simple objects),
* •
if $X\otimes X=A$ then the knot invariant coming from $X$ is trivial,
* •
if $X\otimes X=1\oplus A$ then the knot invariant coming from $X$ is a
specialization of the Jones polynomial,
* •
if $X\otimes X=A\oplus B$ then the knot invariant coming from $X$ is a
specialization of the HOMFLYPT polynomial,
* •
if $X\otimes X=1\oplus A\oplus B$ then the knot invariant coming from $X$ is a
specialization of the Kauffman polynomial or the Dubrovnik polynomial,
Furthermore we give formulas that identify which specialization occurs. This
technique is due to Kauffman, Kazhdan, Tuba, Wenzl, and others [40, 20, 17],
and is well-known to experts. We also use a result of Kuperberg’s which gives
a similar condition for specializations of the $G_{2}$ knot polynomial.
In Section 4, we reprove the results of the previous section using
coincidences of Dynkin diagrams, generalized Kirby-Melvin symmetry, and level-
rank duality. In particular, we give a new simple proof of Kirby-Melvin
symmetry which applies very generally, and we use a result of Wenzl and Tuba
to strengthen Beliakova and Blanchet’s statement of $SO$ level-rank duality.
We’d like to thank Stephen Bigelow, Vaughan Jones, Greg Kuperberg, Nicolai
Reshetikhin, Kevin Walker, and Feng Xu for helpful conversations. During our
work on this paper, Scott Morrison was at Microsoft Station Q, Emily Peters
was supported in part by NSF grant DMS0401734 and Noah Snyder was supported in
part by RTG grant DMS-0354321 and by an NSF postdoctoral grant.
### 1.1 Background and Conventions
The subject of quantum groups and quantum knot invariants suffers from a
plethora of inconsistent conventions. In this section we quickly recall
important notions, specify our conventions, and give citations. The reader is
encouraged to skip this section and refer back to it when necessary. In
particular, most of Sections 2 and 3 involve only diagram categories and do
not require understanding quantum group constructions or their notation.
#### 1.1.1 General conventions
###### Definition 1.1.
The $n^{th}$ quantum number $[n]_{q}$ is defined as
$\frac{q^{n}-q^{-n}}{q-q^{-1}}=q^{n-1}+q^{n-3}+\cdots+q^{-n+1}.$
Following [36] the symbol $s$ will always denote a certain root of $q$ which
will be specified as appropriate.
#### 1.1.2 Ribbon categories, diagrams, and knot invariants
A ribbon category is a braided pivotal monoidal category satisfying a
compatibility relation between the pivotal structure and the braiding. See
[37] for details (warning: that reference uses the word tortile in the place
of ribbon). We use the optimistic convention where diagrams are read upward.
The key property of ribbon categories is that if $\mathcal{C}$ is a ribbon
category there is a functor $\mathcal{F}$ from the category of ribbons
labelled by objects of $\mathcal{C}$ with coupons labelled by morphisms in
$\mathcal{C}$ to the category $\mathcal{C}$ (see [33, 42, 37]). In particular,
if $V$ is an object in $\mathcal{C}$ and $L_{V}$ denotes a framed oriented
link $L$ labelled by $V$, then
$\tilde{\mathcal{J}}_{\mathcal{C},V}:L\mapsto\mathcal{F}(L_{V})\in\operatorname{End}\left(\boldsymbol{1}\right)$
is an invariant of oriented framed links (due to Reshetikhin-Turaev [33]).
Whenever $V$ is a simple object, the invariant depends on the framing through
a ‘twist factor’. That is, two links $L$ and $L^{\prime}$ which are the same
except that $w(L)=w(L^{\prime})+1$, where $w$ denotes the writhe, have
invariants satisfying
$\tilde{\mathcal{J}}_{\mathcal{C},V}(L)=\theta_{V}\tilde{\mathcal{J}}_{\mathcal{C},V}(L^{\prime})$
for some $\theta_{V}$ in the ground field (not depending on $L$). Thus
$\tilde{\mathcal{J}}_{\mathcal{C},V}$ can be modified to give an invariant
which does not depend on framing.
###### Theorem 1.2.
Let
$\mathcal{J}_{\mathcal{C},V}(L)=\theta_{V}^{-w(L)}\tilde{\mathcal{J}}_{\mathcal{C},V}(L)$.
Then $\mathcal{J}_{\mathcal{C},V}(L)$ is an invariant of links.
Given any pivotal tensor category $\mathcal{C}$ (in particular any ribbon
category) and a chosen object $X\in\mathcal{C}$, one can consider the full
subcategory whose objects are tensor products of $X$ and $X^{*}$. This
subcategory is more convenient from the diagrammatic perspective because one
can drop the labeling of strands by objects and instead assume that all
strands are labelled by $X$ (here $X^{*}$ appears as the downward oriented
strand). Thus this full subcategory becomes a spider [26], which is an
oriented version of Jones’s planar algebras [13]. If $X$ is symmetrically
self-dual then this full subcategory is an unoriented unshaded planar algebra
in the sense of [30].
Often one only describes the full subcategory (via diagrams) but wishes to
recover the whole category. If the original category was semisimple and $X$ is
a tensor generator, then this can be acheived via the idempotent completion.
This is explained in detail in [30, 41, 26]. The simple objects in the
idempotent completion are the minimal projections in the full subcategory.
#### 1.1.3 Conventions for knot polynomials and their diagram categories
In this subsection we give our conventions for the following knot polynomials:
the Jones polynomial, the colored Jones polynomials, the HOMFLYPT polynomial,
the Kauffman polynomial, and the Dubrovnik polynomial. Each of these comes in
a framed version as well as an unframed version. The framed versions of these
polynomials (other than the colored Jones polynomial) are given by simple
skein relations. These skein relations can be thought of as defining a ribbon
category whose objects are collections of points (possibly with orientations)
and whose morphisms are tangles modulo the skein relations and modulo all
negligible morphisms (see §1.1.6).
We will often use the same name to refer to the knot polynomial and the
category. This is very convenient for keeping track of conventions. The
HOMFLYPT skein category and the Dubrovnik skein category are more commonly
known as the Hecke category and the BMW category.
Contrary to historical practice, we normalize the polynomials so they are
multiplicative for disjoint union. In particular, the invariant of the empty
link is $1$, while the invariant of the unknot is typically nontrivial.
##### The Temperley-Lieb category
We first fix our conventions for the Temperley-Lieb ribbon category
$\mathcal{TL}(s)$. Let $s$ be a complex number with $q=s^{2}$.
The objects in Temperley-Lieb are natural numbers (thought of as disjoint
unions of points). The morphism space $\operatorname{Hom}\left(a,b\right)$
consists of linear combinations of planar tangles with $a$ boundary points on
the bottom and $b$ boundary points on the top, modulo the relation that each
closed circle can be replaced by a multiplicative factor of
$[2]_{q}=q+q^{-1}$. The endomorphism space of the object consisting of $k$
points will be called $\mathcal{TL}_{2k}$.
The braiding (which depends on the choice of $s=q^{\frac{1}{2}}$) is given by
$\leavevmode\hbox to28.85pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip
14.42638pt\lower-14.42638pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}{{}{}}{}}{}{{}}{}
{}{}{}\pgfsys@moveto{3.533pt}{3.53333pt}\pgfsys@lineto{14.22638pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{{}{}}{}}{}{{}}{}
{}{}{}\pgfsys@moveto{-3.533pt}{3.53333pt}\pgfsys@lineto{-14.22638pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{{}{}}{}}{}{{}}{}
{}{}{}\pgfsys@moveto{3.533pt}{-3.53333pt}\pgfsys@lineto{14.22638pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{{}{}}{}}{}{{}}{}
{}{}{}\pgfsys@moveto{3.533pt}{3.53333pt}\pgfsys@lineto{-14.22638pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=is\leavevmode\hbox
to28.85pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip-42.47914pt\lower-14.42638pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{42.67914pt}{14.22638pt}\pgfsys@curveto{56.90552pt}{0.0pt}{56.90552pt}{0.0pt}{42.67914pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{71.1319pt}{14.22638pt}\pgfsys@curveto{56.90552pt}{0.0pt}{56.90552pt}{0.0pt}{71.1319pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-is^{-1}\leavevmode\hbox
to28.85pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip-99.38466pt\lower-14.42638pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{99.58466pt}{14.22638pt}\pgfsys@curveto{113.81104pt}{0.0pt}{113.81104pt}{0.0pt}{128.03741pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{99.58466pt}{-14.22638pt}\pgfsys@curveto{113.81104pt}{0.0pt}{113.81104pt}{0.0pt}{128.03741pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}.$
We also use the following important diagrams.
* •
The Jones projections in $\mathcal{TL}_{2n}$:
$e_{i}=[2]_{q}^{-1}\leavevmode\hbox to60.15pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip-5.49046pt\lower-14.42638pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{5.69046pt}{-14.22638pt}\pgfsys@lineto{5.69046pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{13.32182pt}{-2.5pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\cdots$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{}\pgfsys@moveto{28.45276pt}{-14.22638pt}\pgfsys@lineto{28.45276pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{34.14322pt}{-14.22638pt}\pgfsys@curveto{34.14322pt}{-12.65497pt}{35.41708pt}{-11.3811pt}{36.9885pt}{-11.3811pt}\pgfsys@curveto{38.5599pt}{-11.3811pt}{39.83377pt}{-12.65497pt}{39.83377pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{{}{}{}{}}} {{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{34.14322pt}{14.22638pt}\pgfsys@curveto{34.14322pt}{12.65497pt}{35.41708pt}{11.3811pt}{36.9885pt}{11.3811pt}\pgfsys@curveto{38.5599pt}{11.3811pt}{39.83377pt}{12.65497pt}{39.83377pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{45.52458pt}{-14.22638pt}\pgfsys@lineto{45.52458pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{53.15552pt}{-2.5pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\cdots$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{}\pgfsys@moveto{65.44142pt}{-14.22638pt}\pgfsys@lineto{65.44142pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},\,i\in\\{1,\ldots,n-1\\};$
* •
The Jones-Wenzl projection $f^{(n)}$ in $\mathcal{TL}_{2n}$ [44] which is the
unique projection with the property
$f^{(n)}e_{i}=e_{i}f^{(n)}=0,\,\text{for each $i\in\\{1,\ldots,n-1\\}$}.$
##### The Jones polynomial
The framed Jones polynomial $\widetilde{J}$ (or Kauffman bracket) is the
invariant coming from the ribbon category $\mathcal{TL}(s)$. In particular, it
is defined for unoriented framed links by
$\displaystyle\bigcirc$ $\displaystyle=q+q^{-1}$ and
$\displaystyle=is\leavevmode\hbox to28.85pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip-42.47914pt\lower-14.42638pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{42.67914pt}{14.22638pt}\pgfsys@curveto{56.90552pt}{0.0pt}{56.90552pt}{0.0pt}{42.67914pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{71.1319pt}{14.22638pt}\pgfsys@curveto{56.90552pt}{0.0pt}{56.90552pt}{0.0pt}{71.1319pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-is^{-1}\leavevmode\hbox
to28.85pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip-99.38466pt\lower-14.42638pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{99.58466pt}{14.22638pt}\pgfsys@curveto{113.81104pt}{0.0pt}{113.81104pt}{0.0pt}{128.03741pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{99.58466pt}{-14.22638pt}\pgfsys@curveto{113.81104pt}{0.0pt}{113.81104pt}{0.0pt}{128.03741pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}.$ (1.1)
This implies that
$\leavevmode\hbox to31.7pt{\vbox
to34.54pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-3.04544pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{} {}{{}}{}
{}{} {}{}
{}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{27.27438pt}{27.27438pt}\pgfsys@curveto{29.4967pt}{29.4967pt}{31.2982pt}{28.75049pt}{31.2982pt}{25.60765pt}\pgfsys@lineto{31.2982pt}{2.84511pt}\pgfsys@curveto{31.2982pt}{-0.29773pt}{29.4967pt}{-1.04395pt}{27.27438pt}{1.17838pt}\pgfsys@lineto{17.07182pt}{11.38092pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{11.38092pt}{17.07182pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=is^{3}\leavevmode\hbox
to14.63pt{\vbox to28.85pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{} {}{{}}{}
{}{}
{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{10.20256pt}{10.20256pt}\pgfsys@curveto{12.42488pt}{12.42488pt}{12.42488pt}{16.02788pt}{10.20256pt}{18.2502pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\qquad\text{and}\qquad\leavevmode\hbox
to31.7pt{\vbox to34.54pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-31.4982pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}
{}{{}}{} {}{} {}{}
{}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{27.27438pt}{-27.27438pt}\pgfsys@curveto{29.4967pt}{-29.4967pt}{31.2982pt}{-28.75049pt}{31.2982pt}{-25.60765pt}\pgfsys@lineto{31.2982pt}{-2.84511pt}\pgfsys@curveto{31.2982pt}{0.29773pt}{29.4967pt}{1.04395pt}{27.27438pt}{-1.17838pt}\pgfsys@lineto{17.07182pt}{-11.38092pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{11.38092pt}{-17.07182pt}\pgfsys@lineto{0.0pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=-is^{-3}\leavevmode\hbox
to14.63pt{\vbox to28.85pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{} {}{{}}{}
{}{}
{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{10.20256pt}{10.20256pt}\pgfsys@curveto{12.42488pt}{12.42488pt}{12.42488pt}{16.02788pt}{10.20256pt}{18.2502pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$
so the twist factor is $is^{3}$.
The framing-independent Jones polynomial is defined by
$J(L)=(-is^{-3})^{\operatorname{writhe}(L)}\widetilde{J}(L)$. It satisfies the
following version of the Jones skein relation
$\displaystyle q^{2}\leavevmode\hbox to28.85pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip
14.42638pt\lower-14.42638pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}{{}{}}{}}{}{{}}{}
{}{}{}{{
{\pgfsys@beginscope\pgfsys@setlinewidth{0.32pt}\pgfsys@setdash{}{0.0pt}\pgfsys@roundcap\pgfsys@roundjoin{}
{}{}{} {}{}{}
\pgfsys@moveto{-1.19998pt}{1.59998pt}\pgfsys@curveto{-1.09998pt}{0.99998pt}{0.0pt}{0.09999pt}{0.29999pt}{0.0pt}\pgfsys@curveto{0.0pt}{-0.09999pt}{-1.09998pt}{-0.99998pt}{-1.19998pt}{-1.59998pt}\pgfsys@stroke\pgfsys@endscope}}
}{}{}{{}}\pgfsys@moveto{3.533pt}{3.53333pt}\pgfsys@lineto{13.90112pt}{13.90112pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.70712}{0.7071}{-0.7071}{0.70712}{13.90111pt}{13.90112pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {{}{{}{}}{}}{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-3.533pt}{3.53333pt}\pgfsys@lineto{-13.90112pt}{13.90112pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.70712}{0.7071}{-0.7071}{-0.70712}{-13.90111pt}{13.90112pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {{}{{}{}}{}}{}{{}}{}
{}{}{}\pgfsys@moveto{3.533pt}{-3.53333pt}\pgfsys@lineto{14.22638pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{{}{}}{}}{}{{}}{}
{}{}{}\pgfsys@moveto{3.533pt}{3.53333pt}\pgfsys@lineto{-14.22638pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-q^{-2}\leavevmode\hbox
to28.85pt{\vbox to28.85pt{\pgfpicture\makeatletter\hbox{\hskip
14.42638pt\lower-14.42638pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{14.22638pt}{-14.22638pt}\pgfsys@lineto{-13.90112pt}{13.90112pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.7071}{0.7071}{-0.7071}{-0.7071}{-13.90112pt}{13.90112pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{2.84544pt}{2.84544pt}\pgfsys@lineto{13.90112pt}{13.90112pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.7071}{0.7071}{-0.7071}{0.7071}{13.90112pt}{13.90112pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{}
{}{}{}\pgfsys@moveto{-2.84544pt}{-2.84544pt}\pgfsys@lineto{-14.22638pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$
$\displaystyle=-is\leavevmode\hbox to28.85pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip
14.42638pt\lower-14.42638pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}{{}{}}{}}{}{{}}{}
{}{}{}\pgfsys@moveto{3.533pt}{3.53333pt}\pgfsys@lineto{14.22638pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{{}{}}{}}{}{{}}{}
{}{}{}\pgfsys@moveto{-3.533pt}{3.53333pt}\pgfsys@lineto{-14.22638pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{{}{}}{}}{}{{}}{}
{}{}{}\pgfsys@moveto{3.533pt}{-3.53333pt}\pgfsys@lineto{14.22638pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{{}{}}{}}{}{{}}{}
{}{}{}\pgfsys@moveto{3.533pt}{3.53333pt}\pgfsys@lineto{-14.22638pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-is^{-1}\leavevmode\hbox
to28.85pt{\vbox to28.85pt{\pgfpicture\makeatletter\hbox{\hskip
14.42638pt\lower-14.42638pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{}
{}{}{}\pgfsys@moveto{14.22638pt}{-14.22638pt}\pgfsys@lineto{-14.22638pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{2.84544pt}{2.84544pt}\pgfsys@lineto{14.22638pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-2.84544pt}{-2.84544pt}\pgfsys@lineto{-14.22638pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$
$\displaystyle=q\leavevmode\hbox to28.85pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip-42.47914pt\lower-14.42638pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{42.67914pt}{14.22638pt}\pgfsys@curveto{56.90552pt}{0.0pt}{56.90552pt}{0.0pt}{42.67914pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{71.1319pt}{14.22638pt}\pgfsys@curveto{56.90552pt}{0.0pt}{56.90552pt}{0.0pt}{71.1319pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-q^{-1}\leavevmode\hbox
to28.85pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip-42.47914pt\lower-14.42638pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{42.67914pt}{14.22638pt}\pgfsys@curveto{56.90552pt}{0.0pt}{56.90552pt}{0.0pt}{42.67914pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{71.1319pt}{14.22638pt}\pgfsys@curveto{56.90552pt}{0.0pt}{56.90552pt}{0.0pt}{71.1319pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$
$\displaystyle=(q-q^{-1})\leavevmode\hbox to28.85pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip-42.47914pt\lower-14.42638pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}{{}}{}{{}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{43.0044pt}{13.90112pt}\pgfsys@curveto{56.90552pt}{0.0pt}{56.90552pt}{0.0pt}{42.67914pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.7071}{0.7071}{-0.7071}{-0.7071}{43.0044pt}{13.90112pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}
{}{{}}{}{{}}{}{{}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{70.80664pt}{13.90112pt}\pgfsys@curveto{56.90552pt}{0.0pt}{56.90552pt}{0.0pt}{71.1319pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.7071}{0.7071}{-0.7071}{0.7071}{70.80664pt}{13.90112pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}.$
##### The colored Jones polynomial
The framed colored Jones polynomial
$\tilde{\mathcal{J}}_{\mathfrak{sl}(2),(k)}(K)$ is the invariant coming from
the simple projection $f^{(k)}$ in $\mathcal{TL}(s)$. The twist factor is
$i^{k^{2}}s^{k^{2}+2k}$.
##### The HOMFLYPT polynomial
The framed HOMFLYPT polynomial is given by the following skein relation.
$w\leavevmode\hbox to28.85pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip
14.42638pt\lower-14.42638pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}{{}{}}{}}{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{3.533pt}{3.53333pt}\pgfsys@lineto{13.90112pt}{13.90112pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.70712}{0.7071}{-0.7071}{0.70712}{13.90111pt}{13.90112pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {{}{{}{}}{}}{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-3.533pt}{3.53333pt}\pgfsys@lineto{-13.90112pt}{13.90112pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.70712}{0.7071}{-0.7071}{-0.70712}{-13.90111pt}{13.90112pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {{}{{}{}}{}}{}{{}}{}
{}{}{}\pgfsys@moveto{3.533pt}{-3.53333pt}\pgfsys@lineto{14.22638pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{{}{}}{}}{}{{}}{}
{}{}{}\pgfsys@moveto{3.533pt}{3.53333pt}\pgfsys@lineto{-14.22638pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-w^{-1}\leavevmode\hbox
to28.85pt{\vbox to28.85pt{\pgfpicture\makeatletter\hbox{\hskip
14.42638pt\lower-14.42638pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{14.22638pt}{-14.22638pt}\pgfsys@lineto{-13.90112pt}{13.90112pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.7071}{0.7071}{-0.7071}{-0.7071}{-13.90112pt}{13.90112pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{2.84544pt}{2.84544pt}\pgfsys@lineto{13.90112pt}{13.90112pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.7071}{0.7071}{-0.7071}{0.7071}{13.90112pt}{13.90112pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{}
{}{}{}\pgfsys@moveto{-2.84544pt}{-2.84544pt}\pgfsys@lineto{-14.22638pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=z\leavevmode\hbox
to28.85pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip-42.47914pt\lower-14.42638pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}{{}}{}{{}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{43.0044pt}{13.90112pt}\pgfsys@curveto{56.90552pt}{0.0pt}{56.90552pt}{0.0pt}{42.67914pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.7071}{0.7071}{-0.7071}{-0.7071}{43.0044pt}{13.90112pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}
{}{{}}{}{{}}{}{{}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{70.80664pt}{13.90112pt}\pgfsys@curveto{56.90552pt}{0.0pt}{56.90552pt}{0.0pt}{71.1319pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.7071}{0.7071}{-0.7071}{0.7071}{70.80664pt}{13.90112pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (1.2)
$\displaystyle=w^{-1}a\leavevmode\hbox to14.63pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{} {}{{}}{}
{}{}
{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{10.20256pt}{10.20256pt}\pgfsys@curveto{12.42488pt}{12.42488pt}{12.42488pt}{16.02788pt}{10.20256pt}{18.2502pt}\pgfsys@lineto{0.32526pt}{28.1275pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.7071}{0.7071}{-0.7071}{-0.7071}{0.32526pt}{28.1275pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$
$\displaystyle=aw^{-1}\leavevmode\hbox to14.63pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{} {}{{}}{}
{}{}
{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{10.20256pt}{10.20256pt}\pgfsys@curveto{12.42488pt}{12.42488pt}{12.42488pt}{16.02788pt}{10.20256pt}{18.2502pt}\pgfsys@lineto{0.32526pt}{28.1275pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.7071}{0.7071}{-0.7071}{-0.7071}{0.32526pt}{28.1275pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (1.3)
The twist factor is just $w^{-1}a$.
Thus, the framing-independent HOMFLYPT polynomial is given by the following
skein relation.
$a\leavevmode\hbox to28.85pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip
14.42638pt\lower-14.42638pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}{{}{}}{}}{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{3.533pt}{3.53333pt}\pgfsys@lineto{13.90112pt}{13.90112pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.70712}{0.7071}{-0.7071}{0.70712}{13.90111pt}{13.90112pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {{}{{}{}}{}}{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-3.533pt}{3.53333pt}\pgfsys@lineto{-13.90112pt}{13.90112pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.70712}{0.7071}{-0.7071}{-0.70712}{-13.90111pt}{13.90112pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {{}{{}{}}{}}{}{{}}{}
{}{}{}\pgfsys@moveto{3.533pt}{-3.53333pt}\pgfsys@lineto{14.22638pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{{}{}}{}}{}{{}}{}
{}{}{}\pgfsys@moveto{3.533pt}{3.53333pt}\pgfsys@lineto{-14.22638pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-a^{-1}\leavevmode\hbox
to28.85pt{\vbox to28.85pt{\pgfpicture\makeatletter\hbox{\hskip
14.42638pt\lower-14.42638pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{14.22638pt}{-14.22638pt}\pgfsys@lineto{-13.90112pt}{13.90112pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.7071}{0.7071}{-0.7071}{-0.7071}{-13.90112pt}{13.90112pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{2.84544pt}{2.84544pt}\pgfsys@lineto{13.90112pt}{13.90112pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.7071}{0.7071}{-0.7071}{0.7071}{13.90112pt}{13.90112pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{}
{}{}{}\pgfsys@moveto{-2.84544pt}{-2.84544pt}\pgfsys@lineto{-14.22638pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=z\leavevmode\hbox
to28.85pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip-42.47914pt\lower-14.42638pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}{{}}{}{{}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{43.0044pt}{13.90112pt}\pgfsys@curveto{56.90552pt}{0.0pt}{56.90552pt}{0.0pt}{42.67914pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.7071}{0.7071}{-0.7071}{-0.7071}{43.0044pt}{13.90112pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}
{}{{}}{}{{}}{}{{}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{70.80664pt}{13.90112pt}\pgfsys@curveto{56.90552pt}{0.0pt}{56.90552pt}{0.0pt}{71.1319pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.7071}{0.7071}{-0.7071}{0.7071}{70.80664pt}{13.90112pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (1.4)
##### The Kauffman polynomial
The Kauffman polynomial comes in two closely related versions, known as the
Kauffman and Dubrovnik normalizations. Both are invariants of unoriented
framed links. The framed Kauffman polynomial
$\widetilde{\operatorname{Kauffman}}$ is defined by
$\leavevmode\hbox to28.85pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip
14.42638pt\lower-14.42638pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}{{}{}}{}}{}{{}}{}
{}{}{}\pgfsys@moveto{3.533pt}{3.53333pt}\pgfsys@lineto{14.22638pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{{}{}}{}}{}{{}}{}
{}{}{}\pgfsys@moveto{-3.533pt}{3.53333pt}\pgfsys@lineto{-14.22638pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{{}{}}{}}{}{{}}{}
{}{}{}\pgfsys@moveto{3.533pt}{-3.53333pt}\pgfsys@lineto{14.22638pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{{}{}}{}}{}{{}}{}
{}{}{}\pgfsys@moveto{3.533pt}{3.53333pt}\pgfsys@lineto{-14.22638pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\leavevmode\hbox
to28.85pt{\vbox to28.85pt{\pgfpicture\makeatletter\hbox{\hskip
14.42638pt\lower-14.42638pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{}
{}{}{}\pgfsys@moveto{14.22638pt}{-14.22638pt}\pgfsys@lineto{-14.22638pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{2.84544pt}{2.84544pt}\pgfsys@lineto{14.22638pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-2.84544pt}{-2.84544pt}\pgfsys@lineto{-14.22638pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=z\left(\leavevmode\hbox
to28.85pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip-42.47914pt\lower-14.42638pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{42.67914pt}{14.22638pt}\pgfsys@curveto{56.90552pt}{0.0pt}{56.90552pt}{0.0pt}{42.67914pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{71.1319pt}{14.22638pt}\pgfsys@curveto{56.90552pt}{0.0pt}{56.90552pt}{0.0pt}{71.1319pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\leavevmode\hbox
to28.85pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip-99.38466pt\lower-14.42638pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{99.58466pt}{14.22638pt}\pgfsys@curveto{113.81104pt}{0.0pt}{113.81104pt}{0.0pt}{128.03741pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{99.58466pt}{-14.22638pt}\pgfsys@curveto{113.81104pt}{0.0pt}{113.81104pt}{0.0pt}{128.03741pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right)$ (1.5)
$\displaystyle=a\leavevmode\hbox to14.63pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{} {}{{}}{}
{}{}
{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{10.20256pt}{10.20256pt}\pgfsys@curveto{12.42488pt}{12.42488pt}{12.42488pt}{16.02788pt}{10.20256pt}{18.2502pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$
$\displaystyle=a^{-1}\leavevmode\hbox to14.63pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{} {}{{}}{}
{}{}
{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{10.20256pt}{10.20256pt}\pgfsys@curveto{12.42488pt}{12.42488pt}{12.42488pt}{16.02788pt}{10.20256pt}{18.2502pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}.$ (1.6)
Here the value of the unknot is $\frac{a+a^{-1}}{z}-1$.
The framed Dubrovnik polynomial $\widetilde{\operatorname{Dubrovnik}}$ is
defined by
$\leavevmode\hbox to28.85pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip
14.42638pt\lower-14.42638pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}{{}{}}{}}{}{{}}{}
{}{}{}\pgfsys@moveto{3.533pt}{3.53333pt}\pgfsys@lineto{14.22638pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{{}{}}{}}{}{{}}{}
{}{}{}\pgfsys@moveto{-3.533pt}{3.53333pt}\pgfsys@lineto{-14.22638pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{{}{}}{}}{}{{}}{}
{}{}{}\pgfsys@moveto{3.533pt}{-3.53333pt}\pgfsys@lineto{14.22638pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{{}{}}{}}{}{{}}{}
{}{}{}\pgfsys@moveto{3.533pt}{3.53333pt}\pgfsys@lineto{-14.22638pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-\leavevmode\hbox
to28.85pt{\vbox to28.85pt{\pgfpicture\makeatletter\hbox{\hskip
14.42638pt\lower-14.42638pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{}
{}{}{}\pgfsys@moveto{14.22638pt}{-14.22638pt}\pgfsys@lineto{-14.22638pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{2.84544pt}{2.84544pt}\pgfsys@lineto{14.22638pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-2.84544pt}{-2.84544pt}\pgfsys@lineto{-14.22638pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=z\left(\leavevmode\hbox
to28.85pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip-42.47914pt\lower-14.42638pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{42.67914pt}{14.22638pt}\pgfsys@curveto{56.90552pt}{0.0pt}{56.90552pt}{0.0pt}{42.67914pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{71.1319pt}{14.22638pt}\pgfsys@curveto{56.90552pt}{0.0pt}{56.90552pt}{0.0pt}{71.1319pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-\leavevmode\hbox
to28.85pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip-99.38466pt\lower-14.42638pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{99.58466pt}{14.22638pt}\pgfsys@curveto{113.81104pt}{0.0pt}{113.81104pt}{0.0pt}{128.03741pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{99.58466pt}{-14.22638pt}\pgfsys@curveto{113.81104pt}{0.0pt}{113.81104pt}{0.0pt}{128.03741pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right)$ (1.7)
$\displaystyle=a\leavevmode\hbox to14.63pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{} {}{{}}{}
{}{}
{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{10.20256pt}{10.20256pt}\pgfsys@curveto{12.42488pt}{12.42488pt}{12.42488pt}{16.02788pt}{10.20256pt}{18.2502pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$
$\displaystyle=a^{-1}\leavevmode\hbox to14.63pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{} {}{{}}{}
{}{}
{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{10.20256pt}{10.20256pt}\pgfsys@curveto{12.42488pt}{12.42488pt}{12.42488pt}{16.02788pt}{10.20256pt}{18.2502pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}.$ (1.8)
Here the value of the unknot is $\frac{a-a^{-1}}{z}+1$.
In both cases, the twist factor is $a$. The unframed Kauffman and Dubrovnik
polynomials do not satisfy any conveniently stated skein relations. The
Kauffman and Dubrovnik polynomials are closely related to each other by
$\widetilde{\operatorname{Dubrovnik}}(L)(a,z)=i^{-w(L)}(-1)^{\\#L}\widetilde{\operatorname{Kauffman}}(L)(ia,-iz),$
where $\\#L$ is the number of components of the link and $w(L)$ is the writhe
of any choice of orientation for $L$ (which turns out not to depend, modulo
$4$, on the choice of orientation). This is due to Lickorish [29] [17, p.
466].
##### Kuperberg’s $G_{2}$ Spider
We recall Kuperberg’s skein theoretic description of the quantum $G_{2}$ knot
invariant [26, 25] (warning, there is a sign error in the former source).
Kuperberg’s $q$ is our $q^{2}$ (which agrees with the usual quantum group
conventions). The quantum $G_{2}$ invariant is defined by
$\leavevmode\hbox to28.85pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip
14.42638pt\lower-14.42638pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}{{}{}}{}}{}{{}}{}
{}{}{}\pgfsys@moveto{3.533pt}{3.53333pt}\pgfsys@lineto{14.22638pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{{}{}}{}}{}{{}}{}
{}{}{}\pgfsys@moveto{-3.533pt}{3.53333pt}\pgfsys@lineto{-14.22638pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{{}{}}{}}{}{{}}{}
{}{}{}\pgfsys@moveto{3.533pt}{-3.53333pt}\pgfsys@lineto{14.22638pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{{}{}}{}}{}{{}}{}
{}{}{}\pgfsys@moveto{3.533pt}{3.53333pt}\pgfsys@lineto{-14.22638pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\frac{1}{1+q^{-2}}\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=15.89673pt]{./diagrams/eps/G2/I.pdf}}\end{array}+\frac{1}{1+q^{2}}\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=31.79951pt]{./diagrams/eps/G2/H.pdf}}\end{array}+\frac{1}{q^{2}+q^{4}}\leavevmode\hbox
to28.85pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip-99.38466pt\lower-14.42638pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{99.58466pt}{14.22638pt}\pgfsys@curveto{113.81104pt}{0.0pt}{113.81104pt}{0.0pt}{128.03741pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{99.58466pt}{-14.22638pt}\pgfsys@curveto{113.81104pt}{0.0pt}{113.81104pt}{0.0pt}{128.03741pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\frac{1}{q^{-2}+q^{-4}}\leavevmode\hbox
to28.85pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip-42.47914pt\lower-14.42638pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{42.67914pt}{14.22638pt}\pgfsys@curveto{56.90552pt}{0.0pt}{56.90552pt}{0.0pt}{42.67914pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{71.1319pt}{14.22638pt}\pgfsys@curveto{56.90552pt}{0.0pt}{56.90552pt}{0.0pt}{71.1319pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$
where the trivalent vertex satisfies the following relations
$\displaystyle\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=15.89673pt]{./diagrams/eps/G2/loop.pdf}}\end{array}$
$\displaystyle=q^{10}+q^{8}+q^{2}+1+q^{-2}+q^{-8}+q^{-10}$
$\displaystyle\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=15.89673pt]{./diagrams/eps/G2/lollipop.pdf}}\end{array}$
$\displaystyle=0$
$\displaystyle\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=15.89673pt]{./diagrams/eps/G2/bigon.pdf}}\end{array}$
$\displaystyle=-\left(q^{6}+q^{4}+q^{2}+q^{-2}+q^{-4}+q^{-6}\right)\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=11.92406pt]{./diagrams/eps/G2/strand.pdf}}\end{array}$
$\displaystyle\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=23.84811pt]{./diagrams/eps/G2/triangle.pdf}}\end{array}$
$\displaystyle=\left(q^{4}+1+q^{-4}\right)\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=23.84811pt]{./diagrams/eps/G2/vertex.pdf}}\end{array}$
$\displaystyle\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=23.84811pt]{./diagrams/eps/G2/square.pdf}}\end{array}$
$\displaystyle=-\left(q^{2}+q^{-2}\right)\left(\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=15.89673pt]{./diagrams/eps/G2/I.pdf}}\end{array}+\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=31.79951pt]{./diagrams/eps/G2/H.pdf}}\end{array}\right)+\left(q^{2}+1+q^{-2}\right)\left(\leavevmode\hbox
to28.85pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip-99.38466pt\lower-14.42638pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{99.58466pt}{14.22638pt}\pgfsys@curveto{113.81104pt}{0.0pt}{113.81104pt}{0.0pt}{128.03741pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{99.58466pt}{-14.22638pt}\pgfsys@curveto{113.81104pt}{0.0pt}{113.81104pt}{0.0pt}{128.03741pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\leavevmode\hbox
to28.85pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip-42.47914pt\lower-14.42638pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{42.67914pt}{14.22638pt}\pgfsys@curveto{56.90552pt}{0.0pt}{56.90552pt}{0.0pt}{42.67914pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{71.1319pt}{14.22638pt}\pgfsys@curveto{56.90552pt}{0.0pt}{56.90552pt}{0.0pt}{71.1319pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right)$
$\displaystyle\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=23.84811pt]{./diagrams/eps/G2/pentagon.pdf}}\end{array}$
$\displaystyle=\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=23.84811pt]{./diagrams/eps/G2/tree1.pdf}}\end{array}+\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=23.84811pt]{./diagrams/eps/G2/tree2.pdf}}\end{array}+\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=23.84811pt]{./diagrams/eps/G2/tree3.pdf}}\end{array}+\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=23.84811pt]{./diagrams/eps/G2/tree4.pdf}}\end{array}+\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=23.84811pt]{./diagrams/eps/G2/tree5.pdf}}\end{array}$
$\displaystyle\qquad-\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=23.84811pt]{./diagrams/eps/G2/forest1.pdf}}\end{array}-\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=23.84811pt]{./diagrams/eps/G2/forest2.pdf}}\end{array}-\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=23.84811pt]{./diagrams/eps/G2/forest3.pdf}}\end{array}-\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=23.84811pt]{./diagrams/eps/G2/forest4.pdf}}\end{array}-\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=23.84811pt]{./diagrams/eps/G2/forest5.pdf}}\end{array}.$
#### 1.1.4 Quantum groups
Quantum groups are key sources of ribbon categories. If $\mathfrak{g}$ is a
complex semisimple Lie algebra, let $U_{s}(\mathfrak{g})$ denote the
Drinfel’d-Jimbo quantum group, and let
$\operatorname{Rep}{U_{s}(\mathfrak{g})}$ denote its category of
representations. This category is a ribbon category and hence given a quantum
group and any representation the Reshetikhin-Turaev procedure gives a knot
invariant.
We follow the conventions from [36]. See [36, p. 2] for a comprehensive
summary of how his conventions line up with those in other sources. (In
particular, our $q$ is the same as both Sawin’s $q$ and Lusztig’s $v$.) We
make one significant change: we only require that the underlying Lie algebra
$\mathfrak{g}$ be semi-simple rather than simple. This does not cause any
complications because $U_{s}(\mathfrak{g_{1}}\oplus\mathfrak{g_{2}})\cong
U_{s}(\mathfrak{g_{1}})\boxtimes U_{s}(\mathfrak{g_{2}})$.
In particular, following [36], we have variables $s$ and $q$ and the relation
$s^{L}=q$ where $L$ is the smallest integer such that $L$ times any inner
product of weights is an integer. The values of $L$ for each simple Lie
algebra appear in Figure 7. The quantum group itself and its representation
theory only depend on $q$, while the braiding and the ribbon category depend
on the additional choice of $s$.
For the quantum groups $U_{s}(\mathfrak{so}(n))$ we denote by
$\operatorname{Rep}^{\text{vector}}(U_{q}(\mathfrak{so}(n)))$ the collection
of representations whose highest weight corresponds to a vector representation
of $\mathfrak{so}(n)$ (that is, a representation of $\mathfrak{so}(n)$ which
lifts to the non-simply connected Lie group $SO(n)$). Note that the braiding
on the vector representations does not depend on $s$ so we use $q$ as our
subscript here instead.
Often the pivotal structure on a tensor category is not unique, and indeed for
the representation theory of a quantum group the pivotal structures are a
torsor over the group of maps from the weight lattice modulo the root lattice
to $\pm 1$. In general there is no ‘standard’ pivotal structure, but for the
representation theory of a quantum group there is both the usual one defined
by the Hopf algebra structure of the Drinfel’d-Jimbo quantum group, and
Turaev’s ‘unimodal’ pivotal structure,
$\operatorname{Rep}^{\text{uni}}{U_{s}(\mathfrak{g})}$. Changing the pivotal
structure by $\chi$, a map from the weight lattice modulo the root lattice to
$\pm 1$, has two major effects: it changes both the dimension of an object and
its twist factor by multiplying by $\chi(V)$. The unimodal pivotal structure
is characterized by the condition that every self-dual object is symmetrically
self-dual. One important particular case is that
$\operatorname{Rep}^{\text{uni}}{U_{s}(\mathfrak{sl}(2))}\cong\mathcal{TL}(-is)$
[38, 39].
The twist factor for an irreducible representation $V$ is determined by the
action of the ribbon element, giving (for the standard pivotal structure)
$q^{\langle\lambda,\lambda+2\rho\rangle}$ where $\lambda$ is the highest
weight of $V$. Note that since
$\langle\lambda,\lambda+2\rho\rangle\in\frac{1}{L}\mathbb{Z}$ (where $L$ is
the exponent of the weight lattice mod the root lattice), the twist factor in
general depends on a choice of $s=q^{\frac{1}{L}}$.
For $V=V_{(k)}$, the representation of $U_{s}(\mathfrak{sl}(2))$ with highest
weight $k$, the twist factor is $s^{k^{2}+2k}$ (notice this is the same as the
$k$-colored Jones polynomial for $k$ even; for $k$ odd the twist factors
differ by a sign as predicted by
$\operatorname{Rep}^{\text{uni}}{U_{s}(\mathfrak{sl}(2))}\cong\mathcal{TL}(-is)$).
For $V=V^{\natural}$, the standard representation of
$U_{s}(\mathfrak{sl}(n))$, the twist factor is $s^{n-1}$. For the standard
representations of $\mathfrak{so}(2n+1)$, $\mathfrak{sp}(2n)$ and
$\mathfrak{so}(2n)$ the twist factors are $q^{4n}$, $q^{2n+1}$ and $q^{2n-1}$
respectively. The twist factor for the representation $V_{ke_{1}}$ of
$\mathfrak{so}(2n+1)$ is $q^{2k^{2}+(4n-2)k}$. Note the the representation
$V_{ke_{1}}$ of $\mathfrak{so}(3)$ is the representation $V_{(2k)}$ of
$\mathfrak{sl}(2)$ and in this case the twist factor agrees with the first one
given in this paragraph. The twist factor for the representation $V_{ke_{1}}$
of $\mathfrak{so}(2n)$ is $q^{k^{2}+2(n-1)k}$. Later we will need the twist
factors for the representations $V_{3e_{1}}$, $V_{e_{n-1}}$ and $V_{3e_{n-1}}$
of $\mathfrak{so}(2n)$. These are $q^{6n+3}$, $q^{\frac{1}{4}n(2n-1)}$ and
$q^{\frac{3}{4}n(2n+1)}$. The twist factor for the $7$-dimensional
representation of $G_{2}$ is $q^{12}$.
The invariants of the unknot are just the quantum dimensions. For the standard
representations of $\mathfrak{sl}(n)$, $\mathfrak{so}(2n+1)$,
$\mathfrak{sp}(2n)$ and $\mathfrak{so}(2n)$ these are
$[n]_{q},[2n]_{q^{2}}+1,[2n+1]_{q}-1$ and $[2n-1]_{q}+1$ respectively.
The invariants of the standard representations are specializations of the
HOMFLYPT or Dubrovnik polynomials. Written in terms of the framing-independent
invariants, we have
$\displaystyle\text{HOMFLYPT}(L)(q^{n},q-q^{-1})$
$\displaystyle=\mathcal{J}_{\mathfrak{sl}(n),V^{\natural}}(L)(q),$ (1.9)
$\displaystyle\text{Dubrovnik}(L)(q^{4n},q^{2}-q^{-2})$
$\displaystyle=\mathcal{J}_{\mathfrak{so}(2n+1),V^{\natural}}(L)(q),$ (1.10)
$\displaystyle\text{Dubrovnik}(L)(-q^{2n+1},q-q^{-1})$
$\displaystyle=(-1)^{\sharp
L}\mathcal{J}_{\mathfrak{sp}(2n),V^{\natural}}(L)(q),$ (1.11) and
$\displaystyle\text{Dubrovnik}(L)(q^{2n-1},q-q^{-1})$
$\displaystyle=\mathcal{J}_{\mathfrak{so}(2n),V^{\natural}}(L)(q).$ (1.12)
These identities are ‘well-known’, but it’s surprisingly hard to find precise
statements in the literature, and we include these mostly for reference. The
identities follow immediately from Theorem 3.2 below, and the fact that the
eigenvalues of the braiding on the natural representations of
$\mathfrak{sl}(n),\mathfrak{so}(2n+1),\mathfrak{sp}(2n)$ and
$\mathfrak{so}(2n)$ are
$(-s^{-n-1},s^{n-1}),(q^{-4n},q^{2},-q^{-2}),(-q^{-2n-1},q,-q^{-1})$ and
$(q^{-2n+1},q,-q^{-1})$ respectively. The sign in Equation (1.11) appears
because Theorem 3.2 does not apply immediately to the natural representation
of $\mathfrak{sp}(2n)$, which is antisymmetrically self-dual. Changing to the
unimodal pivotal structure fixes this, introduces the sign in the knot
invariant, and explains the discrepancy between the value of $a$ in the
specialization of the Dubrovnik polynomial and the twist factor for the
natural representation of $\mathfrak{sp}(2n)$.
We’ll show using techniques inspired by [4, 5, 40] that several of these
identities between knot polynomials come from functors between the
corresponding categories.
#### 1.1.5 Comparison with the KnotTheory‘ package
The HOMFLYPT and Kauffman polynomials defined here agree with those available
in the Mathematica package KnotTheory‘ (available at the Knot Atlas [16]),
except that in the package the invariants are normalized so that their value
on the unknot is $1$. The Jones polynomial in the package uses ‘bad’
conventions from the point of view of quantum groups. You’ll need to
substitute $q\mapsto q^{-2}$, and then multiply by $q+q^{-1}$ to get from the
invariant implemented in KnotTheory‘ to the one described here. The $G_{2}$
spider invariant described here agrees with that calculated using the
QuantumKnotInvariant function in the package. The function
QuantumKnotInvariant in the package calculates the framing-independent
invariants from quantum groups described here.
#### 1.1.6 Semisimplification
Suppose that $\mathcal{C}$ is a spherical tensor category which is
$\mathbb{C}$-linear and which is idempotent complete (every projection has a
kernel and an image). Let $N$ be the collection of negligible morphisms ($f$
is negligible if $\text{tr}(fg)=0$ for all $g$). Call a collection of
morphisms $I\in\mathcal{C}$ an ideal if $I$ is closed under composition and
tensor product with arbitrary morphisms in $\mathcal{C}$. We recall the
following facts:
* •
$N$ is an ideal.
* •
Any ideal in $\mathcal{C}$ is contained in $N$.
* •
If $\mathcal{C}$ semisimple then $N=0$.
* •
If $\mathcal{C}$ is abelian, then $\mathcal{C}/N=\mathcal{C}^{ss}$ is
semisimple.
* •
If $\mathcal{D}$ is psuedo-unitary (pivotal, and all quantum dimensions are
positive, up to a fixed Galois conjugacy) and
$\mathcal{F}:\mathcal{C}\rightarrow\mathcal{D}$ is a functor of pivotal
categories, then $\mathcal{F}$ is trivial on $N$.
There are some technical issues which, while not immediately relevant to this
paper, are important to keep in mind when dealing with semisimplifications.
First, $\mathcal{C}/N$ may not always be semisimple. Furthermore, if
$\mathcal{D}$ is semisimple but not psuedo-unitary there may be a functor
$\mathcal{F}:\mathcal{C}\rightarrow\mathcal{D}$ which does not factor through
$\mathcal{C}/N$.
###### Example 1.3.
If $q$ is a root of unity, and $a$ is not an integer power of $q$, then the
quotient of the Dubrovnik category at $(a,z=q-q^{-1})$ by negligibles is not
semisimple [40, Cor. 7.9].
###### Example 1.4.
This example is adapted from [8, Remark 8.26]. Let
$\mathcal{E}=\operatorname{Rep}^{\text{vector}}U_{q=\exp(\frac{2\pi
i}{10})}(\mathfrak{so}(3))$
be the Yang-Lee category. This fusion category has two objects, $1$ and $X$,
satisfying $X\otimes X\cong X\oplus 1$. The object $X$ has dimension the
golden ratio. Let $\mathcal{E}^{\prime}$ be a Galois conjugate of
$\mathcal{E}$ where $X$ has dimension the conjugate of the golden ratio. Let
$\mathcal{D}=\mathcal{E}\boxtimes\mathcal{E}^{\prime}$; this is a non-pseudo-
unitary semisimple category. Note that $X\boxtimes X$ is a symmetrically self-
dual object with dimension $-1$. Hence there is a functor from
$\mathcal{C}=TL_{d=-1}\rightarrow\mathcal{D}$ sending the single strand to
$X\boxtimes X$ (see §3.4 for more details). The second Jones-Wenzl idempotent
is negligible in $TL_{d=-1}$ but it is not killed by this functor.
For further details see [3] [41] and [7, Proposition 5.7].
#### 1.1.7 Quantum groups at roots of unity
When $s$ is a root of unity, by $\operatorname{Rep}{U_{s}(\mathfrak{g})}$ we
mean the semisimplified category of tilting modules of the Lusztig integral
form. We only ever consider cases where $q$ is a primitive $\ell$th root of
unity with $\ell$ large enough in the sense of [36, Theorem 2]. The key facts
about this category are described in full generality in [36] (based on earlier
work by Andersen, Lusztig, and others):
* •
The isomorphism classes of simple objects correspond to weights in the
fundamental alcove. (Be careful, as when the Lie algebra is not simply laced
the shape of the fundamental alcove depends on the factorization of the order
of the root of unity [36, Lemma 1].
* •
The dimensions and twist factors for these simple objects are given by
specializing the formulas for dimensions and twist factors from generic $q$.
* •
The tensor product rule is given by the quantum Racah rule [36, §5].
#### 1.1.8 Modularization
We review the theory of modularization or deequivariantization developed by
Müger [31] and Bruguières [6]. Suppose that $\mathcal{C}$ is a ribbon category
and that $\mathcal{G}$ is a collection of invertible ($X\otimes
X^{*}\cong\mathbf{1}$, or equivalently $\dim X=\pm 1$) simple objects in
$\mathcal{C}$ which satisfy four conditions:
* •
$\mathcal{G}$ is closed under tensor product.
* •
Every $V\in\mathcal{G}$ is transparent (that is, the positive and negative
braidings with any object $W\in\mathcal{C}$ are equal).
* •
$\dim V=1$ for every $V\in\mathcal{G}$.
* •
The twist factor $\theta_{V}$ is $1$ for every $V\in\mathcal{G}$.
Let $\mathcal{C}//\mathcal{G}$ denote the Müger-Bruguières
deequivariantization. There is a faithful essentially surjective functor
$\mathcal{C}\rightarrow\mathcal{C}//\mathcal{G}$. This functor is not full
because in the deequivariantization there are more maps: in
$\mathcal{C}//\mathcal{G}$ every object in the image of $\mathcal{G}$ becomes
isomorphic to the trivial object.
We’ll often write $\mathcal{C}//X$ to denote the deequivariantization by the
collection of tensor powers of some object $X$.
A ribbon functor between premodular (that is, ribbon and fusion) categories
$\mathcal{F}:\mathcal{C}\rightarrow\mathcal{C}^{\prime}$ is called a
modularization if it is dominant (every simple object in
$\mathcal{C}^{\prime}$ appears as a summand of an object in the image of
$\mathcal{F}$) and if $\mathcal{C}^{\prime}$ is modular.
###### Theorem 1.5.
Suppose that $\mathcal{C}$ is a premodular category whose global dimension is
nonzero. Any modularization is naturally isomorphic to
$\mathcal{F}:\mathcal{C}\rightarrow\mathcal{C}//\mathcal{G}$ where
$\mathcal{G}$ is the set of all transparent objects. A modularization exists
if and only if every transparent object has dimension $1$ and twist factor
$1$.
In Section 4.2, we compute modularizations using the following lemma.
###### Lemma 1.6.
Suppose $\mathcal{M}$ is a modular $\otimes$-category, which is a full
subcategory of a tensor category $\mathcal{C}$. Denote by $\mathcal{I}$ the
subcategory of invertible objects in $\mathcal{C}$, and assume they are all
transparent, and that the group of objects $\mathcal{I}$ acts (by tensor
product) freely on $\mathcal{C}$. Then the orbits of $\mathcal{I}$ each
contain exactly one object from $\mathcal{M}$, and the modularization
$\mathcal{C}//\mathcal{I}$ is equivalent to $\mathcal{M}$.
For further detail, see [4, §1.3-1.4]. Related notions appear in the physics
literature, for example [2].
## 2 Link invariants from $\mathcal{D}_{2n}$
### 2.1 The $\mathcal{D}_{2n}$ planar algebras
The $\mathcal{D}_{2n}$ subfactors were first constructed in [19] using an
automorphism of the subfactor $A_{4n-3}$. Since then several papers have
offered alternative constructions; via planar algebras in [15], and as a
module category over an algebra object in
$\operatorname{Rep}{U_{s=\exp({\frac{2\pi i}{16n-8}})}(\mathfrak{sl}(2))}$ in
[22]. Our main tool is our skein theoretic construction from [30]. That paper
is also intended as a quick introduction to planar algebras.
###### Definition 2.1.
Fix $q=\exp(\frac{2\pi i}{8n-4})$. Let $\mathcal{PA}(S)$ be the planar algebra
generated by a single “box” $S$ with $4n-4$ strands, modulo the following
relations.
1. 1.
A closed circle is equal to $[2]_{q}=(q+q^{-1})=2\cos(\frac{2\pi}{8n-4})$
times the empty diagram.
2. 2.
Rotation relation: $\leavevmode\hbox to34.14pt{\vbox
to71.33pt{\pgfpicture\makeatletter\hbox{\hskip
17.07182pt\lower-28.45276pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@moveto{-17.07182pt}{-28.45276pt}\pgfsys@moveto{-17.07182pt}{-28.45276pt}\pgfsys@lineto{-17.07182pt}{28.45276pt}\pgfsys@lineto{17.07182pt}{28.45276pt}\pgfsys@lineto{17.07182pt}{-28.45276pt}\pgfsys@closepath\pgfsys@moveto{17.07182pt}{28.45276pt}\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}\par{}{{}}{} {{}{}}{}{{{{}{}{}{}}} {{}{}{}{}}}{} {} {}{}
{{}{}}{}{{{{}{}{}{}}} {{}{}{}{}}}{} {} {}{}
{{}{}}{}{}\pgfsys@moveto{-9.10509pt}{0.0pt}\pgfsys@lineto{-9.10509pt}{8.53583pt}\pgfsys@curveto{-9.10509pt}{10.10724pt}{-10.37895pt}{11.3811pt}{-11.95036pt}{11.3811pt}\pgfsys@curveto{-13.52177pt}{11.3811pt}{-14.79564pt}{10.10724pt}{-14.79564pt}{8.53583pt}\pgfsys@lineto{-14.79564pt}{-1.42264pt}\pgfsys@curveto{-14.79564pt}{-9.75117pt}{-8.04422pt}{-16.5026pt}{0.28432pt}{-16.5026pt}\pgfsys@curveto{8.61285pt}{-16.5026pt}{15.36427pt}{-9.75117pt}{15.36427pt}{-1.42264pt}\pgfsys@lineto{15.36427pt}{41.25648pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{{}{}}{}{}\pgfsys@moveto{9.10509pt}{0.0pt}\pgfsys@lineto{9.10509pt}{42.67912pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{{}{}}{}{}\pgfsys@moveto{-4.26773pt}{0.0pt}\pgfsys@lineto{-4.26773pt}{42.67912pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{}}{}\pgfsys@moveto{1.9919pt}{19.91684pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.17477pt}{19.38907pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{...}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\par{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{9.4939pt}{0.0pt}\pgfsys@curveto{9.4939pt}{5.2434pt}{5.2434pt}{9.4939pt}{0.0pt}{9.4939pt}\pgfsys@curveto{-5.2434pt}{9.4939pt}{-9.4939pt}{5.2434pt}{-9.4939pt}{0.0pt}\pgfsys@curveto{-9.4939pt}{-5.2434pt}{-5.2434pt}{-9.4939pt}{0.0pt}{-9.4939pt}\pgfsys@curveto{5.2434pt}{-9.4939pt}{9.4939pt}{-5.2434pt}{9.4939pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.35416pt}{-3.41666pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$S$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}{{}}{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{-9.6939pt}{0.0pt}\pgfsys@moveto{-8.27126pt}{0.0pt}\pgfsys@curveto{-8.27126pt}{0.7857pt}{-8.90819pt}{1.42264pt}{-9.6939pt}{1.42264pt}\pgfsys@curveto{-10.4796pt}{1.42264pt}{-11.11653pt}{0.7857pt}{-11.11653pt}{0.0pt}\pgfsys@curveto{-11.11653pt}{-0.7857pt}{-10.4796pt}{-1.42264pt}{-9.6939pt}{-1.42264pt}\pgfsys@curveto{-8.90819pt}{-1.42264pt}{-8.27126pt}{-0.7857pt}{-8.27126pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-9.6939pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=i\cdot\leavevmode\hbox
to34.14pt{\vbox to71.33pt{\pgfpicture\makeatletter\hbox{\hskip
17.07182pt\lower-28.45276pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@moveto{-17.07182pt}{-28.45276pt}\pgfsys@moveto{-17.07182pt}{-28.45276pt}\pgfsys@lineto{-17.07182pt}{28.45276pt}\pgfsys@lineto{17.07182pt}{28.45276pt}\pgfsys@lineto{17.07182pt}{-28.45276pt}\pgfsys@closepath\pgfsys@moveto{17.07182pt}{28.45276pt}\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}\par{}{{}}{}
{{}{}}{}{}\pgfsys@moveto{-9.10509pt}{0.0pt}\pgfsys@lineto{-9.10509pt}{42.67912pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{{}{}}{}{}\pgfsys@moveto{9.10509pt}{0.0pt}\pgfsys@lineto{9.10509pt}{42.67912pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{}}{}\pgfsys@moveto{-1.9919pt}{19.91684pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.15857pt}{19.38907pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{...}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{{}{}}{}{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@lineto{4.26773pt}{42.67912pt}\pgfsys@stroke\pgfsys@invoke{
} \par{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{9.4939pt}{0.0pt}\pgfsys@curveto{9.4939pt}{5.2434pt}{5.2434pt}{9.4939pt}{0.0pt}{9.4939pt}\pgfsys@curveto{-5.2434pt}{9.4939pt}{-9.4939pt}{5.2434pt}{-9.4939pt}{0.0pt}\pgfsys@curveto{-9.4939pt}{-5.2434pt}{-5.2434pt}{-9.4939pt}{0.0pt}{-9.4939pt}\pgfsys@curveto{5.2434pt}{-9.4939pt}{9.4939pt}{-5.2434pt}{9.4939pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.35416pt}{-3.41666pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$S$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}{{}}{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{-9.6939pt}{0.0pt}\pgfsys@moveto{-8.27126pt}{0.0pt}\pgfsys@curveto{-8.27126pt}{0.7857pt}{-8.90819pt}{1.42264pt}{-9.6939pt}{1.42264pt}\pgfsys@curveto{-10.4796pt}{1.42264pt}{-11.11653pt}{0.7857pt}{-11.11653pt}{0.0pt}\pgfsys@curveto{-11.11653pt}{-0.7857pt}{-10.4796pt}{-1.42264pt}{-9.6939pt}{-1.42264pt}\pgfsys@curveto{-8.90819pt}{-1.42264pt}{-8.27126pt}{-0.7857pt}{-8.27126pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-9.6939pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$
3. 3.
Capping relation: $\leavevmode\hbox to36.86pt{\vbox
to39.43pt{\pgfpicture\makeatletter\hbox{\hskip
18.42972pt\lower-21.00433pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{9.4939pt}{0.0pt}\pgfsys@curveto{9.4939pt}{5.2434pt}{5.2434pt}{9.4939pt}{0.0pt}{9.4939pt}\pgfsys@curveto{-5.2434pt}{9.4939pt}{-9.4939pt}{5.2434pt}{-9.4939pt}{0.0pt}\pgfsys@curveto{-9.4939pt}{-5.2434pt}{-5.2434pt}{-9.4939pt}{0.0pt}{-9.4939pt}\pgfsys@curveto{5.2434pt}{-9.4939pt}{9.4939pt}{-5.2434pt}{9.4939pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.35416pt}{-3.41666pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$S$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}{{}}{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{-8.92323pt}{3.7877pt}\pgfsys@moveto{-7.5006pt}{3.7877pt}\pgfsys@curveto{-7.5006pt}{4.57341pt}{-8.13753pt}{5.21034pt}{-8.92323pt}{5.21034pt}\pgfsys@curveto{-9.70894pt}{5.21034pt}{-10.34587pt}{4.57341pt}{-10.34587pt}{3.7877pt}\pgfsys@curveto{-10.34587pt}{3.002pt}{-9.70894pt}{2.36507pt}{-8.92323pt}{2.36507pt}\pgfsys@curveto{-8.13753pt}{2.36507pt}{-7.5006pt}{3.002pt}{-7.5006pt}{3.7877pt}\pgfsys@closepath\pgfsys@moveto{-8.92323pt}{3.7877pt}\pgfsys@fillstroke\pgfsys@invoke{
}
{{}{}{{}}{}}{}{{}}{}{{{}{}}}{{}{}{{}}{}}{}{{{}{}}}{}{}{}{}\pgfsys@moveto{-6.85461pt}{6.85461pt}\pgfsys@curveto{-12.89035pt}{12.89035pt}{0.0pt}{18.22972pt}{0.0pt}{9.6939pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{}{{}}{}}{}{{}}{}
{{}{}}{}{}\pgfsys@moveto{6.85461pt}{6.85461pt}\pgfsys@lineto{12.89035pt}{12.89035pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{}{{}}{}}{}{{}}{}
{{}{}}{}{}\pgfsys@moveto{9.6939pt}{0.0pt}\pgfsys@lineto{18.22972pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}
{{}{}{{}}{}}{}{{}}{}{{}}{}\pgfsys@moveto{6.85461pt}{-6.85461pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.46573pt}{-14.83205pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\cdot$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}{{}}{}}{}{{}}{}{{}}{}\pgfsys@moveto{0.0pt}{-9.6939pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.38889pt}{-17.67133pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\cdot$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}{{}}{}}{}{{}}{}{{}}{}\pgfsys@moveto{-6.85461pt}{-6.85461pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-8.2435pt}{-14.83205pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\cdot$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}{}{{}}{}}{}{{}}{}
{{}{}}{}{}\pgfsys@moveto{-9.6939pt}{0.0pt}\pgfsys@lineto{-18.22972pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=0$
4. 4.
Two-$S$ relation: $\leavevmode\hbox to30.2pt{\vbox
to90.95pt{\pgfpicture\makeatletter\hbox{\hskip
15.10226pt\lower-45.47675pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{9.4939pt}{19.91684pt}\pgfsys@curveto{9.4939pt}{25.16023pt}{5.2434pt}{29.41074pt}{0.0pt}{29.41074pt}\pgfsys@curveto{-5.2434pt}{29.41074pt}{-9.4939pt}{25.16023pt}{-9.4939pt}{19.91684pt}\pgfsys@curveto{-9.4939pt}{14.67345pt}{-5.2434pt}{10.42294pt}{0.0pt}{10.42294pt}\pgfsys@curveto{5.2434pt}{10.42294pt}{9.4939pt}{14.67345pt}{9.4939pt}{19.91684pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{19.91684pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.35416pt}{16.50018pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$S$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}{{}}{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{-9.6939pt}{19.91684pt}\pgfsys@moveto{-8.27126pt}{19.91684pt}\pgfsys@curveto{-8.27126pt}{20.70255pt}{-8.90819pt}{21.33948pt}{-9.6939pt}{21.33948pt}\pgfsys@curveto{-10.4796pt}{21.33948pt}{-11.11653pt}{20.70255pt}{-11.11653pt}{19.91684pt}\pgfsys@curveto{-11.11653pt}{19.13113pt}{-10.4796pt}{18.4942pt}{-9.6939pt}{18.4942pt}\pgfsys@curveto{-8.90819pt}{18.4942pt}{-8.27126pt}{19.13113pt}{-8.27126pt}{19.91684pt}\pgfsys@closepath\pgfsys@moveto{-9.6939pt}{19.91684pt}\pgfsys@fillstroke\pgfsys@invoke{
} \par{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{9.4939pt}{-19.91684pt}\pgfsys@curveto{9.4939pt}{-14.67345pt}{5.2434pt}{-10.42294pt}{0.0pt}{-10.42294pt}\pgfsys@curveto{-5.2434pt}{-10.42294pt}{-9.4939pt}{-14.67345pt}{-9.4939pt}{-19.91684pt}\pgfsys@curveto{-9.4939pt}{-25.16023pt}{-5.2434pt}{-29.41074pt}{0.0pt}{-29.41074pt}\pgfsys@curveto{5.2434pt}{-29.41074pt}{9.4939pt}{-25.16023pt}{9.4939pt}{-19.91684pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-19.91684pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.35416pt}{-23.3335pt}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}{{}}{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{-9.6939pt}{-19.91684pt}\pgfsys@moveto{-8.27126pt}{-19.91684pt}\pgfsys@curveto{-8.27126pt}{-19.13113pt}{-8.90819pt}{-18.4942pt}{-9.6939pt}{-18.4942pt}\pgfsys@curveto{-10.4796pt}{-18.4942pt}{-11.11653pt}{-19.13113pt}{-11.11653pt}{-19.91684pt}\pgfsys@curveto{-11.11653pt}{-20.70255pt}{-10.4796pt}{-21.33948pt}{-9.6939pt}{-21.33948pt}\pgfsys@curveto{-8.90819pt}{-21.33948pt}{-8.27126pt}{-20.70255pt}{-8.27126pt}{-19.91684pt}\pgfsys@closepath\pgfsys@moveto{-9.6939pt}{-19.91684pt}\pgfsys@fillstroke\pgfsys@invoke{
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{{}{}}{}{}\pgfsys@moveto{-6.85461pt}{26.77145pt}\pgfsys@lineto{-14.90227pt}{34.8191pt}\pgfsys@stroke\pgfsys@invoke{
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{{}{}}{}{}\pgfsys@moveto{-6.85461pt}{-26.77145pt}\pgfsys@lineto{-14.90227pt}{-34.8191pt}\pgfsys@stroke\pgfsys@invoke{
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{{}{}}{}{}\pgfsys@moveto{6.85461pt}{-26.77145pt}\pgfsys@lineto{14.90227pt}{-34.8191pt}\pgfsys@stroke\pgfsys@invoke{
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}}
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{{}{{}{}}{}}{}{{}}{}{}\pgfsys@moveto{-14.1109pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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In [30], our main theorem included the following statements:
###### Theorem 2.2.
$\mathcal{PA}(S)$ is the $\mathcal{D}_{2n}$ subfactor planar algebra:
1. 1.
the space of closed diagrams is $1$-dimensional (in particular, the relations
are consistent),
2. 2.
it is spherical,
3. 3.
It is unitary, and hence pseudo-unitary and semisimple.
4. 4.
the principal graph of $\mathcal{PA}(S)$ is the Dynkin diagram $D_{2n}$.
In order to prove this theorem, we made liberal use of the following “half-
braided” relation:
###### Theorem 2.3.
You can isotope a strand above an $S$-box, but isotoping a strand below an
$S$-box introduces a factor of $-1$.
1. 1.
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{{}{}}{}{}\pgfsys@moveto{6.85461pt}{6.85461pt}\pgfsys@lineto{6.85461pt}{49.53374pt}\pgfsys@stroke\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}{{}}{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{-9.6939pt}{0.0pt}\pgfsys@moveto{-8.27126pt}{0.0pt}\pgfsys@curveto{-8.27126pt}{0.7857pt}{-8.90819pt}{1.42264pt}{-9.6939pt}{1.42264pt}\pgfsys@curveto{-10.4796pt}{1.42264pt}{-11.11653pt}{0.7857pt}{-11.11653pt}{0.0pt}\pgfsys@curveto{-11.11653pt}{-0.7857pt}{-10.4796pt}{-1.42264pt}{-9.6939pt}{-1.42264pt}\pgfsys@curveto{-8.90819pt}{-1.42264pt}{-8.27126pt}{-0.7857pt}{-8.27126pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-9.6939pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{{}{}{{}}{}}{}{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \par{}{}{{}}{} {}{} {}{}
{}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}\pgfsys@moveto{-28.45276pt}{42.67914pt}\pgfsys@lineto{-28.45276pt}{34.14331pt}\pgfsys@curveto{-28.45276pt}{31.00047pt}{-25.90504pt}{28.45276pt}{-22.7622pt}{28.45276pt}\pgfsys@lineto{22.7622pt}{28.45276pt}\pgfsys@curveto{25.90504pt}{28.45276pt}{28.45276pt}{31.00047pt}{28.45276pt}{34.14331pt}\pgfsys@lineto{28.45276pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{-6.85461pt}{6.85461pt}\pgfsys@lineto{-6.85461pt}{24.91975pt}\pgfsys@stroke\pgfsys@invoke{
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{{}{}}{}{}\pgfsys@moveto{-6.85461pt}{31.98576pt}\pgfsys@lineto{-6.85461pt}{46.21214pt}\pgfsys@stroke\pgfsys@invoke{
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{{}{}{{}}{}}{}{{}}{}{{}}{}\pgfsys@moveto{0.0pt}{9.6939pt}\pgfsys@stroke\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}{{}{}}{}}{}{{}}{}
{{}{}}{}{}\pgfsys@moveto{6.85461pt}{31.98576pt}\pgfsys@lineto{6.85461pt}{46.21214pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{6.85461pt}{6.85461pt}\pgfsys@lineto{6.85461pt}{24.91975pt}\pgfsys@stroke\pgfsys@invoke{
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2. 2.
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In [30] we gave a skein theoretic description of each isomorphism class of
simple projections in $\mathcal{D}_{2n}$. These are $f^{(i)}$ for $0\leq i\leq
2n-3$, the projection $P=\frac{1}{2}(f^{(2n-2)}+S)$, and the projection
$Q=\frac{1}{2}(f^{(2n-2)}-S)$. We also gave a complete description of the
tensor product rules for these projections (most of which appear in [11, 22]).
By the even part of $\mathcal{D}_{2n}$, which we’ll denote
$\frac{1}{2}\mathcal{D}_{2n}$, we mean the full subcategory whose objects
consist of collections of an even number of points. The simple projections in
the even part of $\mathcal{D}_{2n}$ are
$f^{(0)},f^{(2)},\ldots,f^{(2n-4)},P,Q$.
###### Proposition 2.4.
$\frac{1}{2}\mathcal{D}_{2n}\cong\operatorname{Rep}^{\text{vector}}{U_{q=\exp(\frac{2\pi
i}{8n-4})}(\mathfrak{so}(3))}^{modularize}$.
###### Proof.
To see this we observe that
$\operatorname{Rep}^{\text{vector}}{U_{q=\exp(\frac{2\pi
i}{8n-4})}(\mathfrak{so}(3))}$ is the even part of
$\operatorname{Rep}{U_{q=\exp(\frac{2\pi i}{8n-4})}(\mathfrak{sl}(2))}$, and
the even parts of $\operatorname{Rep}{U_{q=\exp(\frac{2\pi
i}{8n-4})}(\mathfrak{sl}(2))}$ and $\mathcal{TL}$ are the same at that value
of $q$ (the change in pivotal structure does not affect the even part). Hence
there is a functor
$\operatorname{Rep}^{\text{vector}}{U_{q=\exp(\frac{2\pi
i}{8n-4})}(\mathfrak{so}(3))}\rightarrow\frac{1}{2}\mathcal{D}_{2n}$
The description of simple objects in $\mathcal{D}_{2n}$ shows that this
functor is dominant (as $P+Q=f^{(2n-2)}$), and a simple calculation shows that
$\frac{1}{2}\mathcal{D}_{2n}$ has no transparent objects and so is modular.
Hence, the claim follows by the uniqueness of modularization. ∎
### 2.2 Invariants from $\mathcal{D}_{2n}$
Although $\mathcal{D}_{2n}$ is not a ribbon category, its even part is ribbon.
This is in [22, p. 33]; we prefer to give a skein theoretic explanation.
Define the braiding using the Kauffman bracket formula. This braiding clearly
satisfies Reidemeister moves $2$ and $3$, as well as the additional ribbon
axiom: all of these equalities of diagrams only involve diagrams in Temperley-
Lieb, which is a ribbon category. The only additional thing to check is
naturality, which means that any diagram can pass over or under a crossing
without changing. This follows immediately from the “half-braiding” relation,
because all crossings involve an even number of strands.
Since the even part of $\mathcal{D}_{2n}$ is a ribbon category, any simple
object in it defines a link invariant. For the simple objects $f^{(2k)}$ this
invariant is just a colored Jones polynomial. So we concentrate on invariants
involving $P$ and $Q$. Given an oriented framed link $L$, to get the framed
$P$-invariant, we first $2n-2$ cable it and place a $P$ (going in the
direction of the orientation) on each component. See Figure 1 for an example.
Then we evaluate this new picture in the $\mathcal{D}_{2n}$ planar algebra
(using the Kauffman resolution of crossings).
In the usual way, we can make it into an invariant of unframed links, which we
will call $\mathcal{J}_{\mathcal{D}_{2n},P}(L)$. Since $P=Pf^{(2n-2)}$, the
twist factor is the same as that for $f^{(2n-2)}$, namely $q^{2n(n-1)}$.
###### Theorem 2.5.
For a knot $K$ (but not for a link!),
$\mathcal{J}_{\mathcal{D}_{2n},P}(K)=\frac{1}{2}\mathcal{J}_{\mathfrak{sl}(2),(2n-2)}(K)=\mathcal{J}_{\mathcal{D}_{2n},Q}(K)$
###### Proof.
To compute $\tilde{\mathcal{J}}_{\mathcal{D}_{2n},P}(K)$ we $(2n-2)$-cable
$K$, and insert one $P=\frac{1}{2}(f^{(2n-2)}+S)$ somewhere. When we split
this into the sum of two diagrams, the diagram with the $S$ in it is zero,
since in every resolution the $S$ connects back up with itself. Meanwhile, the
diagram with the $f^{(2n-2)}$ in it is the colored Jones polynomial. Thus
$\tilde{\mathcal{J}}_{\mathcal{D}_{2n},P}(K)=\frac{1}{2}\tilde{\mathcal{J}}_{\mathfrak{sl}(2),(2n-2)}(K)$.
Exactly the same argument holds for $Q$. Furthermore, the twist factors for
$P$, $Q$ and $f^{(2n-2)}$ are all equal as computed above. ∎
$\mapsto$ $\;P\;$$\;P\;$
Figure 1: Computing the framed $\mathcal{D}_{10}$ invariant of the Hopf link.
### 2.3 A refined invariant
Although this section isn’t necessary for the rest of this paper, it may
nevertheless be of interest. We can slightly modify this construction to
produce a more refined invariant for links. Instead of labeling every
component with $P$ or every component with $Q$ we could instead label some
components with $P$ and others with $Q$. This would not be an invariant of
links, but if you fix which number of links to label with $P$ and sum over all
choices of components this is a link invariant. Notice that since the twist
factors for $P$ and $Q$ are the same, this definition makes sense either for
framed or unframed versions of the invariant.
###### Definition 2.6.
For $a$ a positive integer, let $\mathcal{J}_{\mathcal{D}_{2n},P/Q}^{a}(L)$ be
the sum over all ways of labelling $a$ components of $L$ with $P$ and the
remaining components with $Q$.
Since $P=\frac{1}{2}(f^{(2n-2)}+S)$ and $Q=\frac{1}{2}(f^{(2n-2)}-S)$, these
invariants can be written in terms of simpler-to-compute invariants.
###### Definition 2.7.
Let $\mathcal{J}^{k}_{\mathcal{D}_{2n},S/f}(L)$ be $2^{-\ell}$ times the sum
of all the ways to put an $S$ on $k$ of the link components and an $f^{(n)}$
on the rest of the components. We call this the $k$-refined
$(\mathcal{D}_{2n},P)$-invariant of an $\ell$-component link $L$.
This is a refinement of the $(\mathcal{D}_{2n},P)$ link invariant in that
$\sum_{k=0}^{k=\ell}\mathcal{J}^{k}_{\mathcal{D}_{2n},S/f}(L)=\mathcal{J}_{\mathcal{D}_{2n},P}$
More precisely we have the following lemma.
###### Lemma 2.8.
$\displaystyle\mathcal{J}_{\mathcal{D}_{2n},P/Q}^{a}(L)$
$\displaystyle=\sum_{i=0}^{a}\sum_{j=0}^{\ell-a}(-1)^{\ell-
a-j}\begin{pmatrix}i+j\\\ i\end{pmatrix}\begin{pmatrix}\ell-(i+j)\\\
a-i\end{pmatrix}\mathcal{J}_{\mathcal{D}_{2n},S/f}^{i+j}(L)$
$\displaystyle=\sum_{k=0}^{\ell}(-1)^{\ell-
a-k}\mathcal{J}_{\mathcal{D}_{2n},S/f}^{k}(L)\sum_{i=0}^{\text{min}(k,a)}(-1)^{i}\begin{pmatrix}k\\\
i\end{pmatrix}\begin{pmatrix}\ell-k\\\ a-i\end{pmatrix}.$
These refined invariants can detect more information than the ordinary
invariant. For example, although we will show in the next section that the
$\mathcal{D}_{4}$ invariant is trivial, it is not difficult to see using the
methods of the next section that its refined invariants detect linking number
mod $3$.
## 3 Knot polynomial identities
The theorems of this section describe how to identify an invariant coming from
an object in a ribbon category as a specialization of the Jones, HOMFLYPT or
Kauffman polynomials. These theorems are well-known to the experts, and
versions of them can be found in [17, 20, 40]. Since we need explicit formulas
for which specializations appear we collect the proofs here.
We then identify cases in which these theorems apply, namely
$\mathcal{D}_{2n}$ for $n=2,3,4$ and $5$, and explain exactly which
specializations occur.
There is a similar procedure, due to Kuperberg, for recognizing knot
invariants which are specializations of the $G_{2}$ knot polynomial. We apply
this technique to $\mathcal{D}_{14}$.
The identities in this section do not follow from the knot polynomial
identities in [14] [28, §6, Table 2]. (But most of those identities follow
from the technique outlined in this section.)
### 3.1 Recognizing a specialization of Jones, HOMFLYPT, or Kauffman
Identifying a knot invariant as a specialization of a classical knot
polynomial happens in two steps. Let’s say you’re looking at the knot
invariant coming from an object $V$ in a ribbon category. First, you look at
the direct sum decomposition of $V\otimes V$, and hope that you don’t see too
many summands. Theorem 3.1 below describes how to interpret this
decomposition, hopefully guaranteeing that the invariant is either trivial, or
a specialization of Jones, HOMFLYPT, or Kauffman. If this proves successful,
you next look at the eigenvalues of the braiding on the summands of $V\otimes
V$. Theorem 3.2 then tells you exactly which specialization you have.
###### Theorem 3.1.
Suppose that $V$ is a simple object in a ribbon category $\mathcal{C}$ and
that if $V$ is self-dual then it is symmetrically self-dual.
1. 1.
If $V\otimes V$ is simple, then $\dim{V}=\pm 1$ and the link invariant
$\mathcal{J}_{\mathcal{C},V}=(\dim V)^{\\#}$ where $\\#$ is the number of
components of the link.
2. 2.
If $V\otimes V=\boldsymbol{1}\oplus L$ for some simple object $L$, then the
link invariant $\mathcal{J}_{\mathcal{C},V}$ is a specialization of the Jones
polynomial.
3. 3.
If $V\otimes V=L\oplus M$ for some simple objects $L$ and $M$, then the link
invariant $\mathcal{J}_{\mathcal{C},V}$ is a specialization of HOMFLYPT.
4. 4.
If $V\otimes V=\boldsymbol{1}\oplus L\oplus M$ for some simple objects $L$ and
$M$, then the link invariant $\mathcal{J}_{\mathcal{C},V}$ is a specialization
of either the Kauffman polynomial or the Dubrovnik polynomial.
###### Proof.
1. 1.
Trivial case
Since the category is spherical and braided, $\operatorname{End}\left(V\otimes
V\right)\cong\operatorname{End}\left(V\otimes V^{*}\right)$. Hence if
$V\otimes V$ is simple we must have $V\otimes V^{*}=\boldsymbol{1}$, so $\dim
V=\pm 1$. Also by simplicity, $\operatorname{End}\left(V\otimes V\right)$ is
one dimensional, and so, up to constants, a crossing is equal to the identity
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{}{}{}\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@curveto{56.90552pt}{31.42845pt}{31.42845pt}{56.90552pt}{0.0pt}{56.90552pt}\pgfsys@curveto{-31.42845pt}{56.90552pt}{-56.90552pt}{31.42845pt}{-56.90552pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{68.28644pt}{0.0pt}\pgfsys@curveto{68.28644pt}{37.71413pt}{37.71396pt}{68.2866pt}{-0.00017pt}{68.2866pt}\pgfsys@curveto{-37.7143pt}{68.2866pt}{-68.28677pt}{37.71413pt}{-68.28677pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@curveto{56.90552pt}{-31.42845pt}{31.42845pt}{-56.90552pt}{0.0pt}{-56.90552pt}\pgfsys@curveto{-31.42845pt}{-56.90552pt}{-56.90552pt}{-31.42845pt}{-56.90552pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{51.21504pt}{0.0pt}\pgfsys@curveto{51.21504pt}{-28.2856pt}{28.28568pt}{-51.21497pt}{0.00008pt}{-51.21497pt}\pgfsys@curveto{-28.28552pt}{-51.21497pt}{-51.21489pt}{-28.2856pt}{-51.21489pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{62.59598pt}{0.0pt}\pgfsys@curveto{62.59598pt}{-34.57129pt}{34.57121pt}{-62.59605pt}{-0.00008pt}{-62.59605pt}\pgfsys@curveto{-34.57137pt}{-62.59605pt}{-62.59613pt}{-34.57129pt}{-62.59613pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}\pgfsys@beginscope\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{136.57332pt}{0.0pt}\pgfsys@curveto{136.57332pt}{-28.2856pt}{113.64395pt}{-51.21497pt}{85.35835pt}{-51.21497pt}\pgfsys@curveto{57.07275pt}{-51.21497pt}{34.14339pt}{-28.2856pt}{34.14339pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{147.95425pt}{0.0pt}\pgfsys@curveto{147.95425pt}{-34.57129pt}{119.92949pt}{-62.59605pt}{85.3582pt}{-62.59605pt}\pgfsys@curveto{50.78691pt}{-62.59605pt}{22.76215pt}{-34.57129pt}{22.76215pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{153.64471pt}{0.0pt}\pgfsys@curveto{153.64471pt}{-37.71413pt}{123.07224pt}{-68.2866pt}{85.35811pt}{-68.2866pt}\pgfsys@curveto{47.64398pt}{-68.2866pt}{17.0715pt}{-37.71413pt}{17.0715pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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two tells us that $\leavevmode\hbox to28.85pt{\vbox
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{}{}{}\pgfsys@moveto{136.57332pt}{0.0pt}\pgfsys@curveto{136.57332pt}{28.2856pt}{113.64395pt}{51.21497pt}{85.35835pt}{51.21497pt}\pgfsys@curveto{57.07275pt}{51.21497pt}{34.14339pt}{28.2856pt}{34.14339pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{147.95425pt}{0.0pt}\pgfsys@curveto{147.95425pt}{34.57129pt}{119.92949pt}{62.59605pt}{85.3582pt}{62.59605pt}\pgfsys@curveto{50.78691pt}{62.59605pt}{22.76215pt}{34.57129pt}{22.76215pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}\pgfsys@beginscope\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{51.21504pt}{0.0pt}\pgfsys@curveto{51.21504pt}{-28.2856pt}{28.28568pt}{-51.21497pt}{0.00008pt}{-51.21497pt}\pgfsys@curveto{-28.28552pt}{-51.21497pt}{-51.21489pt}{-28.2856pt}{-51.21489pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{62.59598pt}{0.0pt}\pgfsys@curveto{62.59598pt}{-34.57129pt}{34.57121pt}{-62.59605pt}{-0.00008pt}{-62.59605pt}\pgfsys@curveto{-34.57137pt}{-62.59605pt}{-62.59613pt}{-34.57129pt}{-62.59613pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{68.28644pt}{0.0pt}\pgfsys@curveto{68.28644pt}{-37.71413pt}{37.71396pt}{-68.2866pt}{-0.00017pt}{-68.2866pt}\pgfsys@curveto{-37.7143pt}{-68.2866pt}{-68.28677pt}{-37.71413pt}{-68.28677pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}{}{}{}}}{} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}{}{}{}}}{} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}{}{}{}}}{} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{4.0pt}\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
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{}{}{}\pgfsys@moveto{142.2638pt}{0.0pt}\pgfsys@curveto{142.2638pt}{-31.42845pt}{116.78673pt}{-56.90552pt}{85.35828pt}{-56.90552pt}\pgfsys@curveto{53.92982pt}{-56.90552pt}{28.45276pt}{-31.42845pt}{28.45276pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{136.57332pt}{0.0pt}\pgfsys@curveto{136.57332pt}{-28.2856pt}{113.64395pt}{-51.21497pt}{85.35835pt}{-51.21497pt}\pgfsys@curveto{57.07275pt}{-51.21497pt}{34.14339pt}{-28.2856pt}{34.14339pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{147.95425pt}{0.0pt}\pgfsys@curveto{147.95425pt}{-34.57129pt}{119.92949pt}{-62.59605pt}{85.3582pt}{-62.59605pt}\pgfsys@curveto{50.78691pt}{-62.59605pt}{22.76215pt}{-34.57129pt}{22.76215pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{153.64471pt}{0.0pt}\pgfsys@curveto{153.64471pt}{-37.71413pt}{123.07224pt}{-68.2866pt}{85.35811pt}{-68.2866pt}\pgfsys@curveto{47.64398pt}{-68.2866pt}{17.0715pt}{-37.71413pt}{17.0715pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
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}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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{}{{}}{}{{}}{}{{}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{70.80664pt}{13.90112pt}\pgfsys@curveto{56.90552pt}{0.0pt}{56.90552pt}{0.0pt}{71.1319pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$. Capping this off
shows that the twist factor is $\alpha\dim V$. Thus the framing corrected
skein relation is
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{{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{39.83368pt}{0.0pt}\pgfsys@curveto{39.83368pt}{21.99982pt}{21.99982pt}{39.83368pt}{0.0pt}{39.83368pt}\pgfsys@curveto{-21.99982pt}{39.83368pt}{-39.83368pt}{21.99982pt}{-39.83368pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}{}{}{}{{}}\pgfsys@moveto{39.83368pt}{-0.45999pt}\pgfsys@curveto{39.83368pt}{-21.99982pt}{21.99982pt}{-39.83368pt}{0.0pt}{-39.83368pt}\pgfsys@curveto{-21.99982pt}{-39.83368pt}{-39.83368pt}{-21.99982pt}{-39.83368pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{99.58421pt}{0.0pt}\pgfsys@curveto{99.58421pt}{-21.99982pt}{81.75035pt}{-39.83368pt}{59.75053pt}{-39.83368pt}\pgfsys@curveto{37.75072pt}{-39.83368pt}{19.91685pt}{-21.99982pt}{19.91685pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{39.83368pt}{0.0pt}\pgfsys@lineto{43.81699pt}{-3.98329pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{39.83368pt}{0.0pt}\pgfsys@lineto{35.85037pt}{-3.98329pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{19.91684pt}{0.0pt}\pgfsys@lineto{23.90015pt}{-3.98329pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{142.2638pt}{0.0pt}\pgfsys@curveto{142.2638pt}{31.42845pt}{116.78673pt}{56.90552pt}{85.35828pt}{56.90552pt}\pgfsys@curveto{53.92982pt}{56.90552pt}{28.45276pt}{31.42845pt}{28.45276pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{136.57332pt}{0.0pt}\pgfsys@curveto{136.57332pt}{28.2856pt}{113.64395pt}{51.21497pt}{85.35835pt}{51.21497pt}\pgfsys@curveto{57.07275pt}{51.21497pt}{34.14339pt}{28.2856pt}{34.14339pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{147.95425pt}{0.0pt}\pgfsys@curveto{147.95425pt}{34.57129pt}{119.92949pt}{62.59605pt}{85.3582pt}{62.59605pt}\pgfsys@curveto{50.78691pt}{62.59605pt}{22.76215pt}{34.57129pt}{22.76215pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}{}{}{}}}{} {}
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}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@stroke@opacity{1}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}{}{}{}}}{} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@curveto{56.90552pt}{31.42845pt}{31.42845pt}{56.90552pt}{0.0pt}{56.90552pt}\pgfsys@curveto{-31.42845pt}{56.90552pt}{-56.90552pt}{31.42845pt}{-56.90552pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{51.21504pt}{0.0pt}\pgfsys@curveto{51.21504pt}{28.2856pt}{28.28568pt}{51.21497pt}{0.00008pt}{51.21497pt}\pgfsys@curveto{-28.28552pt}{51.21497pt}{-51.21489pt}{28.2856pt}{-51.21489pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{62.59598pt}{0.0pt}\pgfsys@curveto{62.59598pt}{34.57129pt}{34.57121pt}{62.59605pt}{-0.00008pt}{62.59605pt}\pgfsys@curveto{-34.57137pt}{62.59605pt}{-62.59613pt}{34.57129pt}{-62.59613pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{68.28644pt}{0.0pt}\pgfsys@curveto{68.28644pt}{37.71413pt}{37.71396pt}{68.2866pt}{-0.00017pt}{68.2866pt}\pgfsys@curveto{-37.7143pt}{68.2866pt}{-68.28677pt}{37.71413pt}{-68.28677pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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} {}{{}}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{51.21504pt}{0.0pt}\pgfsys@curveto{51.21504pt}{-28.2856pt}{28.28568pt}{-51.21497pt}{0.00008pt}{-51.21497pt}\pgfsys@curveto{-28.28552pt}{-51.21497pt}{-51.21489pt}{-28.2856pt}{-51.21489pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{62.59598pt}{0.0pt}\pgfsys@curveto{62.59598pt}{-34.57129pt}{34.57121pt}{-62.59605pt}{-0.00008pt}{-62.59605pt}\pgfsys@curveto{-34.57137pt}{-62.59605pt}{-62.59613pt}{-34.57129pt}{-62.59613pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{68.28644pt}{0.0pt}\pgfsys@curveto{68.28644pt}{-37.71413pt}{37.71396pt}{-68.2866pt}{-0.00017pt}{-68.2866pt}\pgfsys@curveto{-37.7143pt}{-68.2866pt}{-68.28677pt}{-37.71413pt}{-68.28677pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{-2.84544pt}{-2.84544pt}\pgfsys@lineto{-14.22638pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
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The equality of the two crossings lets us unlink any link, showing that the
framing corrected invariant is $(\dim V)^{\\#}$, where $\\#$ is the number of
components.
2. 2.
Jones polynomial case
Since $\operatorname{End}\left(V\otimes V\right)$ is $2$-dimensional there
must be a linear dependence between the crossing and the two basis diagrams of
Temperley-Lieb. (If these two Temperley-Lieb diagrams were linearly dependent,
then $V\otimes V\cong\boldsymbol{1}$, contradicting the assumption). Hence we
must have a relation of the form
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Following Kauffman, rotate this equation, glue them together and apply
Reidemeister $2$ to see that $B=A^{-1}$ and $A^{2}+A^{-2}=\dim V$. Hence this
invariant is given by the Kauffman bracket.
3. 3.
HOMFLYPT case
Since $\operatorname{End}\left(V\otimes V\right)$ is $2$-dimensional there
must be a linear dependence between the two crossings and the identity (we
can’t use the cup-cap diagram here because $V$ is not self dual). Hence, we
have that
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for some $\alpha$, $\beta$, and $\gamma$. If $\alpha$ or $\beta$ were zero,
$\operatorname{End}\left(V\otimes V\right)$ would be $1$-dimensional, so we
must have that $\alpha$ and $\beta$ are nonzero. Hence we can rescale the
relation so that $\alpha=w$, $\beta=-w^{-1}$, and $\gamma=z$. Since the twist
is some multiple of the single strand we can define $a$ such that the twist
factor is $w^{-1}a$. Thus we’ve recovered the framed HOMFLYPT skein relations.
4. 4.
Kauffman case
Since $V\otimes V$ has three simple summands, its endomorphism space is $3$
dimensional. Moreover, since one of the summands is the trivial
representation, one such endomorphism is the ‘cup-cap’ diagram . There must be
some linear relation of the form
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The space of such relations is invariant under a $\pi/2$ rotation, and fixed
under a $\pi$ rotation, so there must be a linear relation which is either a
$(+1)$\- or $(-1)$-eigenvector of the $\pi/2$ rotation. That is, there must be
a relation of the form
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If $A$ were zero, this would be a linear relation between and , which would
imply that $V\otimes V\cong\boldsymbol{1}$. Thus we can divide by $A$, and
obtain either the Kauffman polynomial (Equation (1.5)) or Dubrovnik polynomial
(Equation (1.7)) skein relation with $z=B/A$.
∎
This argument for the Dubrovnik polynomial is similar in spirit to Kauffman’s
original description in [17], and the argument for HOMFLYPT polynomial is
similar to [12, §4]. Similar results were also obtained in [40, 20].
We’ll now need some notation for eigenvalues. Suppose $N$ appears once as a
summand of $V\otimes V$, and consider the braiding as an endomorphism acting
by composition on $\operatorname{End}\left(V\otimes V\right)$. Then the
idempotent projecting onto $N\subset V\otimes V$ is an eigenvector for the
braiding, and we’ll write $R_{N\subset V\otimes V}$ for the corresponding
eigenvalue. The following is well-known (for example the HOMFLYPT case is
essentially [12, §4]).
###### Theorem 3.2.
If one of conditions (2)-(4) of Theorem 3.1 holds, then we can find which
specialization occurs by computing eigenvalues.
1. (2)
If $R_{L\subset V\otimes V}=\lambda$, then $R_{\boldsymbol{1}\subset V\otimes
V}=-\lambda^{-3}$ and
$\mathcal{J}_{\mathcal{C},V}=\mathcal{J}_{\mathfrak{sl}(2),(1)}(a)$
with $a=-\lambda^{2}$.
2. (3)
If $R_{L\subset V\otimes V}=\lambda$, $R_{M\subset V\otimes V}=\mu$, and
$\theta$ is the twist factor, then
$\mathcal{J}_{\mathcal{C},V}=\operatorname{HOMFLYPT}(a,z)$
with $a=\frac{\theta}{\sqrt{-\lambda\mu}}$ and
$z=\frac{\lambda+\mu}{\sqrt{-\lambda\mu}}$.
3. (4)
If $R_{L\subset V\otimes V}=\lambda$ and $R_{M\subset V\otimes V}=\mu$, then
$\lambda\mu=\pm 1$.
1. (a)
If $\lambda\mu=-1$ then
$\mathcal{J}_{\mathcal{C},V}=\operatorname{Dubrovnik}(a,z)$
with $a=R_{1\subset V\otimes V}^{-1}$ and $z=\lambda+\mu$.
2. (b)
If $\lambda\mu=1$ then
$\mathcal{J}_{\mathcal{C},V}=\operatorname{Kauffman}(a,z)$
with $a=R_{\boldsymbol{1}\subset V\otimes V}^{-1}$ and $z=\lambda+\mu$.
###### Proof.
These proofs all follow the same outline. We consider the operator $X$ which
acts on tangles with four boundary points by multiplication with a positive
crossing. We find the eigenvalues of $X$ in terms of the parameters ($a$
and/or $z$) and then solve for the parameters in terms of the eigenvalues.
1. (2)
The Jones skein relation for unoriented framed links is
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if closed circles count for $[2]_{a}=(a+a^{-1})$.
The eigenvectors for $X$, multiplication by the positive crossing, are
$f^{(2)}$ and
which have eigenvalues $ia^{\frac{1}{2}}$ and $-ia^{\frac{-3}{2}}$.
The cup-cap picture must correspond to the summand $\boldsymbol{1}$, and so we
see that if $R_{L\subset V\otimes V}=\lambda$, then $a=-\lambda^{2}$ and
$R_{\boldsymbol{1}\subset V\otimes V}=-\lambda^{-3}$.
2. (3)
The HOMFLYPT skein relation is for oriented framed links:
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}{}\pgfsys@moveto{39.83368pt}{0.0pt}\pgfsys@curveto{39.83368pt}{21.99982pt}{21.99982pt}{39.83368pt}{0.0pt}{39.83368pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\par{}{{}}{}{{{{}{}{}{}}}
{{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{39.83368pt}{0.0pt}\pgfsys@curveto{39.83368pt}{21.99982pt}{21.99982pt}{39.83368pt}{0.0pt}{39.83368pt}\pgfsys@curveto{-21.99982pt}{39.83368pt}{-39.83368pt}{21.99982pt}{-39.83368pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}{}{}{}{{}}\pgfsys@moveto{39.83368pt}{-0.45999pt}\pgfsys@curveto{39.83368pt}{-21.99982pt}{21.99982pt}{-39.83368pt}{0.0pt}{-39.83368pt}\pgfsys@curveto{-21.99982pt}{-39.83368pt}{-39.83368pt}{-21.99982pt}{-39.83368pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \par{}{{}}{}{{{}{}{}{}}}{} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{4.0pt}\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
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}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@stroke@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{19.91684pt}{0.0pt}\pgfsys@curveto{19.91684pt}{-21.99982pt}{37.7507pt}{-39.83368pt}{59.75052pt}{-39.83368pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
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{}{}{}\pgfsys@moveto{99.58421pt}{0.0pt}\pgfsys@curveto{99.58421pt}{-21.99982pt}{81.75035pt}{-39.83368pt}{59.75053pt}{-39.83368pt}\pgfsys@curveto{37.75072pt}{-39.83368pt}{19.91685pt}{-21.99982pt}{19.91685pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \par{}{{}}{}
{}{}{}\pgfsys@moveto{39.83368pt}{0.0pt}\pgfsys@lineto{43.81699pt}{-3.98329pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{39.83368pt}{0.0pt}\pgfsys@lineto{35.85037pt}{-3.98329pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{19.91684pt}{0.0pt}\pgfsys@lineto{23.90015pt}{-3.98329pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{19.91684pt}{0.0pt}\pgfsys@lineto{15.93352pt}{-3.98329pt}\pgfsys@stroke\pgfsys@invoke{
} \par\par\par\hss} {}{{}}{}{{{{}{}{}{}}} {{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{142.2638pt}{0.0pt}\pgfsys@curveto{142.2638pt}{31.42845pt}{116.78673pt}{56.90552pt}{85.35828pt}{56.90552pt}\pgfsys@curveto{53.92982pt}{56.90552pt}{28.45276pt}{31.42845pt}{28.45276pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{{}{}{}{}}} {{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{136.57332pt}{0.0pt}\pgfsys@curveto{136.57332pt}{28.2856pt}{113.64395pt}{51.21497pt}{85.35835pt}{51.21497pt}\pgfsys@curveto{57.07275pt}{51.21497pt}{34.14339pt}{28.2856pt}{34.14339pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{{}{}{}{}}} {{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{147.95425pt}{0.0pt}\pgfsys@curveto{147.95425pt}{34.57129pt}{119.92949pt}{62.59605pt}{85.3582pt}{62.59605pt}\pgfsys@curveto{50.78691pt}{62.59605pt}{22.76215pt}{34.57129pt}{22.76215pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{{}{}{}{}}} {{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{153.64471pt}{0.0pt}\pgfsys@curveto{153.64471pt}{37.71413pt}{123.07224pt}{68.2866pt}{85.35811pt}{68.2866pt}\pgfsys@curveto{47.64398pt}{68.2866pt}{17.0715pt}{37.71413pt}{17.0715pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \par{}{{}}{}{{{}{}{}{}}}{} {} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{4.0pt}\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@stroke@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@curveto{56.90552pt}{31.42845pt}{31.42845pt}{56.90552pt}{0.0pt}{56.90552pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}{}{}{}}}{} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{4.0pt}\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@stroke@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{51.21504pt}{0.0pt}\pgfsys@curveto{51.21504pt}{28.2856pt}{28.28568pt}{51.21497pt}{0.00008pt}{51.21497pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}{}{}{}}}{} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{4.0pt}\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@stroke@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{62.59598pt}{0.0pt}\pgfsys@curveto{62.59598pt}{34.57129pt}{34.57121pt}{62.59605pt}{-0.00008pt}{62.59605pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}{}{}{}}}{} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{4.0pt}\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@stroke@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{68.28644pt}{0.0pt}\pgfsys@curveto{68.28644pt}{37.71413pt}{37.71396pt}{68.2866pt}{-0.00017pt}{68.2866pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\par{}{{}}{}{{{{}{}{}{}}}
{{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@curveto{56.90552pt}{31.42845pt}{31.42845pt}{56.90552pt}{0.0pt}{56.90552pt}\pgfsys@curveto{-31.42845pt}{56.90552pt}{-56.90552pt}{31.42845pt}{-56.90552pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{{}{}{}{}}} {{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{51.21504pt}{0.0pt}\pgfsys@curveto{51.21504pt}{28.2856pt}{28.28568pt}{51.21497pt}{0.00008pt}{51.21497pt}\pgfsys@curveto{-28.28552pt}{51.21497pt}{-51.21489pt}{28.2856pt}{-51.21489pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{{}{}{}{}}} {{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{62.59598pt}{0.0pt}\pgfsys@curveto{62.59598pt}{34.57129pt}{34.57121pt}{62.59605pt}{-0.00008pt}{62.59605pt}\pgfsys@curveto{-34.57137pt}{62.59605pt}{-62.59613pt}{34.57129pt}{-62.59613pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{{}{}{}{}}} {{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{68.28644pt}{0.0pt}\pgfsys@curveto{68.28644pt}{37.71413pt}{37.71396pt}{68.2866pt}{-0.00017pt}{68.2866pt}\pgfsys@curveto{-37.7143pt}{68.2866pt}{-68.28677pt}{37.71413pt}{-68.28677pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \par{}{{}}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@curveto{56.90552pt}{-31.42845pt}{31.42845pt}{-56.90552pt}{0.0pt}{-56.90552pt}\pgfsys@curveto{-31.42845pt}{-56.90552pt}{-56.90552pt}{-31.42845pt}{-56.90552pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{51.21504pt}{0.0pt}\pgfsys@curveto{51.21504pt}{-28.2856pt}{28.28568pt}{-51.21497pt}{0.00008pt}{-51.21497pt}\pgfsys@curveto{-28.28552pt}{-51.21497pt}{-51.21489pt}{-28.2856pt}{-51.21489pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{62.59598pt}{0.0pt}\pgfsys@curveto{62.59598pt}{-34.57129pt}{34.57121pt}{-62.59605pt}{-0.00008pt}{-62.59605pt}\pgfsys@curveto{-34.57137pt}{-62.59605pt}{-62.59613pt}{-34.57129pt}{-62.59613pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{68.28644pt}{0.0pt}\pgfsys@curveto{68.28644pt}{-37.71413pt}{37.71396pt}{-68.2866pt}{-0.00017pt}{-68.2866pt}\pgfsys@curveto{-37.7143pt}{-68.2866pt}{-68.28677pt}{-37.71413pt}{-68.28677pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \par{}{{}}{}{{{}{}{}{}}}{} {} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{4.0pt}\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@stroke@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@curveto{28.45276pt}{-31.42845pt}{53.92982pt}{-56.90552pt}{85.35828pt}{-56.90552pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}{}{}{}}}{} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{4.0pt}\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@stroke@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@curveto{34.14322pt}{-28.2856pt}{57.07259pt}{-51.21497pt}{85.35818pt}{-51.21497pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}{}{}{}}}{} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{4.0pt}\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@stroke@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{22.76228pt}{0.0pt}\pgfsys@curveto{22.76228pt}{-34.57129pt}{50.78705pt}{-62.59605pt}{85.35834pt}{-62.59605pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}{}{}{}}}{} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{4.0pt}\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@stroke@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{17.07182pt}{0.0pt}\pgfsys@curveto{17.07182pt}{-37.71413pt}{47.6443pt}{-68.2866pt}{85.35843pt}{-68.2866pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\par{}{{}}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{142.2638pt}{0.0pt}\pgfsys@curveto{142.2638pt}{-31.42845pt}{116.78673pt}{-56.90552pt}{85.35828pt}{-56.90552pt}\pgfsys@curveto{53.92982pt}{-56.90552pt}{28.45276pt}{-31.42845pt}{28.45276pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{136.57332pt}{0.0pt}\pgfsys@curveto{136.57332pt}{-28.2856pt}{113.64395pt}{-51.21497pt}{85.35835pt}{-51.21497pt}\pgfsys@curveto{57.07275pt}{-51.21497pt}{34.14339pt}{-28.2856pt}{34.14339pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{147.95425pt}{0.0pt}\pgfsys@curveto{147.95425pt}{-34.57129pt}{119.92949pt}{-62.59605pt}{85.3582pt}{-62.59605pt}\pgfsys@curveto{50.78691pt}{-62.59605pt}{22.76215pt}{-34.57129pt}{22.76215pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{153.64471pt}{0.0pt}\pgfsys@curveto{153.64471pt}{-37.71413pt}{123.07224pt}{-68.2866pt}{85.35811pt}{-68.2866pt}\pgfsys@curveto{47.64398pt}{-68.2866pt}{17.0715pt}{-37.71413pt}{17.0715pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \par\hss}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}{{}}{}{{}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{43.0044pt}{13.90112pt}\pgfsys@curveto{56.90552pt}{0.0pt}{56.90552pt}{0.0pt}{42.67914pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.7071}{0.7071}{-0.7071}{-0.7071}{43.0044pt}{13.90112pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}
{}{{}}{}{{}}{}{{}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{70.80664pt}{13.90112pt}\pgfsys@curveto{56.90552pt}{0.0pt}{56.90552pt}{0.0pt}{71.1319pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.7071}{0.7071}{-0.7071}{0.7071}{70.80664pt}{13.90112pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$ (3.1)
and the characteristic equation for the operator $X$ which multiplies by the
positive crossing is
$wx-w^{-1}x^{-1}=z\quad\Longleftrightarrow\quad
x^{2}-\frac{z}{w}x-\frac{1}{w^{2}}=0.$
So if $\lambda$ and $\mu$ are the eigenvalues of $X$, we have
$\lambda\mu=-w^{-2}$ and $\lambda+\mu=\frac{z}{w}$, so that
$w=\frac{1}{\sqrt{-\lambda\mu}}\text{\quad and
\quad}z=\frac{\lambda+\mu}{\sqrt{-\lambda\mu}}$
To recover $a$ we note that the twist factor is $aw^{-1}$, hence $a=w\theta$.
3. (4)
For the Dubrovnik or Kauffman skein relation we have
$\leavevmode\hbox to28.85pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip
14.42638pt\lower-14.42638pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\hbox to0.0pt{\hbox to0.0pt{
{}{{}}{}{{{{}{}{}{}}} {{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{99.58421pt}{0.0pt}\pgfsys@curveto{99.58421pt}{21.99982pt}{81.75035pt}{39.83368pt}{59.75053pt}{39.83368pt}\pgfsys@curveto{37.75072pt}{39.83368pt}{19.91685pt}{21.99982pt}{19.91685pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \par{}{{}}{}{{{}{}{}{}}}{} {} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{4.0pt}\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@stroke@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{39.83368pt}{0.0pt}\pgfsys@curveto{39.83368pt}{21.99982pt}{21.99982pt}{39.83368pt}{0.0pt}{39.83368pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\par{}{{}}{}{{{{}{}{}{}}}
{{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{39.83368pt}{0.0pt}\pgfsys@curveto{39.83368pt}{21.99982pt}{21.99982pt}{39.83368pt}{0.0pt}{39.83368pt}\pgfsys@curveto{-21.99982pt}{39.83368pt}{-39.83368pt}{21.99982pt}{-39.83368pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \par\par{}{{}}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {}
{}{}{}{}{}{}{{}}\pgfsys@moveto{39.83368pt}{-0.45999pt}\pgfsys@curveto{39.83368pt}{-21.99982pt}{21.99982pt}{-39.83368pt}{0.0pt}{-39.83368pt}\pgfsys@curveto{-21.99982pt}{-39.83368pt}{-39.83368pt}{-21.99982pt}{-39.83368pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{39.83368pt}{-0.45999pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \par{}{{}}{}{{{}{}{}{}}}{} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{4.0pt}\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@stroke@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{19.91684pt}{0.0pt}\pgfsys@curveto{19.91684pt}{-21.99982pt}{37.7507pt}{-39.83368pt}{59.75052pt}{-39.83368pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\par{}{{}}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{99.58421pt}{0.0pt}\pgfsys@curveto{99.58421pt}{-21.99982pt}{81.75035pt}{-39.83368pt}{59.75053pt}{-39.83368pt}\pgfsys@curveto{37.75072pt}{-39.83368pt}{19.91685pt}{-21.99982pt}{19.91685pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \par{}{{}}{}
{}{}{}\pgfsys@moveto{39.83368pt}{0.0pt}\pgfsys@lineto{43.81699pt}{-3.98329pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
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{}{}{}\pgfsys@moveto{142.2638pt}{0.0pt}\pgfsys@curveto{142.2638pt}{-31.42845pt}{116.78673pt}{-56.90552pt}{85.35828pt}{-56.90552pt}\pgfsys@curveto{53.92982pt}{-56.90552pt}{28.45276pt}{-31.42845pt}{28.45276pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{136.57332pt}{0.0pt}\pgfsys@curveto{136.57332pt}{-28.2856pt}{113.64395pt}{-51.21497pt}{85.35835pt}{-51.21497pt}\pgfsys@curveto{57.07275pt}{-51.21497pt}{34.14339pt}{-28.2856pt}{34.14339pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{147.95425pt}{0.0pt}\pgfsys@curveto{147.95425pt}{-34.57129pt}{119.92949pt}{-62.59605pt}{85.3582pt}{-62.59605pt}\pgfsys@curveto{50.78691pt}{-62.59605pt}{22.76215pt}{-34.57129pt}{22.76215pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{153.64471pt}{0.0pt}\pgfsys@curveto{153.64471pt}{-37.71413pt}{123.07224pt}{-68.2866pt}{85.35811pt}{-68.2866pt}\pgfsys@curveto{47.64398pt}{-68.2866pt}{17.0715pt}{-37.71413pt}{17.0715pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{99.58466pt}{14.22638pt}\pgfsys@curveto{113.81104pt}{0.0pt}{113.81104pt}{0.0pt}{128.03741pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
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{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{99.58466pt}{-14.22638pt}\pgfsys@curveto{113.81104pt}{0.0pt}{113.81104pt}{0.0pt}{128.03741pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
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Multiplying by the crossing we see that,
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{}{}{}\pgfsys@moveto{51.21504pt}{0.0pt}\pgfsys@curveto{51.21504pt}{28.2856pt}{28.28568pt}{51.21497pt}{0.00008pt}{51.21497pt}\pgfsys@curveto{-28.28552pt}{51.21497pt}{-51.21489pt}{28.2856pt}{-51.21489pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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Subtracting $a^{-1}$ times the first equation from the second and rearranging
slightly we see that the characteristic equation for the crossing operator is
$(x-a^{-1})(x^{2}-zx\pm 1)=0$. Hence the eigenvalues are $a^{-1}$, $\lambda$,
and $\mu$, where $\lambda+\mu=z$ and $\lambda\mu=\pm 1$. Since $a$ is the
twist factor it is the inverse of the eigenvalue corresponding to
$\boldsymbol{1}$ (compare with case (2)).
∎
### 3.2 Knot polynomial identities for $\mathcal{D}_{4}$, $\mathcal{D}_{6}$,
$\mathcal{D}_{8}$ and $\mathcal{D}_{10}$
We state four theorems, give two lemmas, and then give rather pedestrian
proofs of the theorems. Snazzier proofs appear in Section 4, as special cases
of Theorem 18. In each of these theorems, we relate two quantum knot
invariants via an intermediate knot invariant coming from $\mathcal{D}_{2n}$.
You can think of these results as purely about quantum knot invariants,
although the proofs certainly use $\mathcal{D}_{2n}$.
###### Theorem 3.3 (Identities for $n=2$).
$\displaystyle\mathcal{J}_{\mathfrak{sl}(2),(2)}(K){}_{\mid q=\exp(\frac{2\pi
i}{12})}{}$ $\displaystyle=2\mathcal{J}_{\mathcal{D}_{4},P}(K)$
$\displaystyle=2$
###### Theorem 3.4 (Identities for $n=3$).
$\displaystyle\mathcal{J}_{\mathfrak{sl}(2),(4)}(K){}_{\mid q=\exp(\frac{2\pi
i}{20})}{}$ $\displaystyle=2\mathcal{J}_{\mathcal{D}_{6},P}(K)$
$\displaystyle=2\mathcal{J}_{\mathfrak{sl}(2),(1)}(K){}_{\mid
q=\exp(-\frac{2\pi i}{10})}{}$
###### Theorem 3.5 (Identities for $n=4$).
$\displaystyle\mathcal{J}_{\mathfrak{sl}(2),(6)}(K){}_{\mid q=\exp(\frac{2\pi
i}{28})}{}$ $\displaystyle=2\mathcal{J}_{\mathcal{D}_{8},P}(K)$
$\displaystyle=2\operatorname{HOMFLYPT}(K)(\exp(2\pi
i\frac{3}{14}),\exp(\frac{2\pi i}{14})-\exp(-\frac{2\pi i}{14}))$
$\displaystyle=2\operatorname{HOMFLYPT}(K)(\exp(2\pi
i\frac{5}{7}),\exp(-\frac{2\pi i}{14})-\exp(\frac{2\pi i}{14}))$
###### Remark 1.
This isn’t just any specialization of the HOMFLYPT polynomial:
$\displaystyle\operatorname{HOMFLYPT}(K)$ $\displaystyle(\exp(2\pi
i\frac{3}{14}),\exp(\frac{2\pi i}{14})-\exp(-\frac{2\pi i}{14}))$
$\displaystyle=\operatorname{HOMFLYPT}(K)(q^{3},q-q^{-1}){}_{\mid
q=\exp(\frac{2\pi i}{14})}{}$
$\displaystyle=\mathcal{J}_{\mathfrak{sl}(3),(1,0)}(K){}_{\mid
q=\exp(\frac{2\pi i}{14})}{}$ and $\displaystyle\operatorname{HOMFLYPT}(K)$
$\displaystyle(\exp(2\pi i\frac{5}{7}),\exp(-\frac{2\pi
i}{14})-\exp(\frac{2\pi i}{14}))$
$\displaystyle=\operatorname{HOMFLYPT}(K)(q^{4},q-q^{-1}){}_{\mid
q=\exp(-\frac{2\pi i}{14})}{}$
$\displaystyle=\mathcal{J}_{\mathfrak{sl}(4),(1,0,0)}(K){}_{\mid
q=\exp(-\frac{2\pi i}{14})}{}$
$\displaystyle=-\mathcal{J}_{\mathfrak{sl}(4),(1,0,0)}(K){}_{\mid
q=-\exp(-\frac{2\pi i}{14})}{}$
(The last identity here follows from the fact that every exponent of $q$ in
$\mathcal{J}_{\mathfrak{sl}(4),(1,0,0)}(K)$ is odd. We’ve included this form
here to foreshadow §4.4 where we’ll give an independent proof of this theorem,
and in which this particular value of $q=-\exp(-\frac{2\pi i}{14})$ will
spontaneously appear.)
###### Theorem 1 (Identities for $n=5$).
$\displaystyle\mathcal{J}_{\mathfrak{sl}(2),(8)}(K){}_{\mid q=\exp(\frac{2\pi
i}{36})}{}$ $\displaystyle=2\mathcal{J}_{\mathcal{D}_{10},P}(K)$
$\displaystyle=2\operatorname{Dubrovnik}(K)(\exp(2\pi i\frac{4}{36}),\exp(2\pi
i\frac{2}{36})+\exp(2\pi i\frac{16}{36}))$
###### Remark 2.
Again, this isn’t just any specialization of the Dubrovnik polynomial:
$\displaystyle\operatorname{Dubrovnik}(K)(\exp(2\pi i\frac{4}{36}),$
$\displaystyle\exp(2\pi i\frac{2}{36})+\exp(2\pi i\frac{16}{36}))$
$\displaystyle=\operatorname{Dubrovnik}(K)(q^{7},q-q^{-1}){}_{\mid
q=-\exp(\frac{-2\pi i}{18})}{}$
$\displaystyle=\mathcal{J}_{\mathfrak{so}(8),(1,0,0,0)}(K){}_{\mid
q=-\exp(\frac{-2\pi i}{18})}{}.$
For the proofs of these statements, we’ll need to know how $P\otimes P$
decomposes in each $\mathcal{D}_{2n}$. The following formula was proved in
[11].
$P\otimes
P\cong\begin{cases}Q\oplus\bigoplus_{l=0}^{\frac{n-4}{2}}f^{(4l+2)}&\text{when
$n$ is even}\\\ P\oplus\bigoplus_{l=0}^{\frac{n-3}{2}}f^{(4l)}&\text{when $n$
is odd}\end{cases}$ (3.2)
In particular,
$\displaystyle P\otimes P$ $\displaystyle\cong Q$ in $\mathcal{D}_{4}$,
$\displaystyle P\otimes P$ $\displaystyle\cong P\oplus f^{(0)}$ in
$\mathcal{D}_{6}$, $\displaystyle P\otimes P$ $\displaystyle\cong Q\oplus
f^{(2)}$ in $\mathcal{D}_{8}$, and $\displaystyle P\otimes P$
$\displaystyle\cong P\oplus f^{(0)}\oplus f^{(4)}$ in $\mathcal{D}_{10}$.
Further, we’ll need a lemma calculating the eigenvalues of the braiding.
###### Lemma 3.
Suppose $X$ is an idempotent in the set
$\\{f^{(2)},f^{(6)},\ldots,f^{(2n-6)},Q\\}$ if $n$ is even, or
$X\in\\{f^{(0)},f^{(4)},\ldots,f^{(2n-6)},P\\}$ if $n$ is odd. Then the
eigenvalues for the braiding in $\mathcal{D}_{2n}$ are
$R_{X\subset P\otimes P}=(-1)^{k}q^{k(k+1)-2n(n-1)}$
where $2k$ is the number of strands in the idempotent $X$.
###### Proof 3.1.
The endomorphism space for $P\otimes P$ is spanned by the projections onto the
direct summands described above in Equation (3.2), and thus by the diagrams
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}{{}}\pgfsys@beginscope\pgfsys@invoke{
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{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor[named]{.}{rgb}{1,.5,0}\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}$2n-2-k$}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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We calculate
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{}{{}}{}{{{{}{}{}{}}} {{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{99.58421pt}{0.0pt}\pgfsys@curveto{99.58421pt}{21.99982pt}{81.75035pt}{39.83368pt}{59.75053pt}{39.83368pt}\pgfsys@curveto{37.75072pt}{39.83368pt}{19.91685pt}{21.99982pt}{19.91685pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@stroke@opacity{1}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\par{}{{}}{}{{{{}{}{}{}}}
{{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{39.83368pt}{0.0pt}\pgfsys@curveto{39.83368pt}{21.99982pt}{21.99982pt}{39.83368pt}{0.0pt}{39.83368pt}\pgfsys@curveto{-21.99982pt}{39.83368pt}{-39.83368pt}{21.99982pt}{-39.83368pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}{}{}{}{{}}\pgfsys@moveto{39.83368pt}{-0.45999pt}\pgfsys@curveto{39.83368pt}{-21.99982pt}{21.99982pt}{-39.83368pt}{0.0pt}{-39.83368pt}\pgfsys@curveto{-21.99982pt}{-39.83368pt}{-39.83368pt}{-21.99982pt}{-39.83368pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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{}{}\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
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{}{}{}\pgfsys@moveto{99.58421pt}{0.0pt}\pgfsys@curveto{99.58421pt}{-21.99982pt}{81.75035pt}{-39.83368pt}{59.75053pt}{-39.83368pt}\pgfsys@curveto{37.75072pt}{-39.83368pt}{19.91685pt}{-21.99982pt}{19.91685pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \par{}{{}}{}
{}{}{}\pgfsys@moveto{39.83368pt}{0.0pt}\pgfsys@lineto{43.81699pt}{-3.98329pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{39.83368pt}{0.0pt}\pgfsys@lineto{35.85037pt}{-3.98329pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{19.91684pt}{0.0pt}\pgfsys@lineto{23.90015pt}{-3.98329pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{19.91684pt}{0.0pt}\pgfsys@lineto{15.93352pt}{-3.98329pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{142.2638pt}{0.0pt}\pgfsys@curveto{142.2638pt}{31.42845pt}{116.78673pt}{56.90552pt}{85.35828pt}{56.90552pt}\pgfsys@curveto{53.92982pt}{56.90552pt}{28.45276pt}{31.42845pt}{28.45276pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{{}{}{}{}}} {{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{136.57332pt}{0.0pt}\pgfsys@curveto{136.57332pt}{28.2856pt}{113.64395pt}{51.21497pt}{85.35835pt}{51.21497pt}\pgfsys@curveto{57.07275pt}{51.21497pt}{34.14339pt}{28.2856pt}{34.14339pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{{}{}{}{}}} {{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{147.95425pt}{0.0pt}\pgfsys@curveto{147.95425pt}{34.57129pt}{119.92949pt}{62.59605pt}{85.3582pt}{62.59605pt}\pgfsys@curveto{50.78691pt}{62.59605pt}{22.76215pt}{34.57129pt}{22.76215pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{153.64471pt}{0.0pt}\pgfsys@curveto{153.64471pt}{37.71413pt}{123.07224pt}{68.2866pt}{85.35811pt}{68.2866pt}\pgfsys@curveto{47.64398pt}{68.2866pt}{17.0715pt}{37.71413pt}{17.0715pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}{}{}{}}}{} {}
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{}{}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@stroke@opacity{1}\pgfsys@invoke{
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}{}\pgfsys@moveto{62.59598pt}{0.0pt}\pgfsys@curveto{62.59598pt}{34.57129pt}{34.57121pt}{62.59605pt}{-0.00008pt}{62.59605pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}{}{}{}}}{} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{4.0pt}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\par{}{{}}{}{{{{}{}{}{}}}
{{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@curveto{56.90552pt}{31.42845pt}{31.42845pt}{56.90552pt}{0.0pt}{56.90552pt}\pgfsys@curveto{-31.42845pt}{56.90552pt}{-56.90552pt}{31.42845pt}{-56.90552pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{{}{}{}{}}} {{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{51.21504pt}{0.0pt}\pgfsys@curveto{51.21504pt}{28.2856pt}{28.28568pt}{51.21497pt}{0.00008pt}{51.21497pt}\pgfsys@curveto{-28.28552pt}{51.21497pt}{-51.21489pt}{28.2856pt}{-51.21489pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{{}{}{}{}}} {{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{62.59598pt}{0.0pt}\pgfsys@curveto{62.59598pt}{34.57129pt}{34.57121pt}{62.59605pt}{-0.00008pt}{62.59605pt}\pgfsys@curveto{-34.57137pt}{62.59605pt}{-62.59613pt}{34.57129pt}{-62.59613pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{{}{}{}{}}} {{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{68.28644pt}{0.0pt}\pgfsys@curveto{68.28644pt}{37.71413pt}{37.71396pt}{68.2866pt}{-0.00017pt}{68.2866pt}\pgfsys@curveto{-37.7143pt}{68.2866pt}{-68.28677pt}{37.71413pt}{-68.28677pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \par{}{{}}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@curveto{56.90552pt}{-31.42845pt}{31.42845pt}{-56.90552pt}{0.0pt}{-56.90552pt}\pgfsys@curveto{-31.42845pt}{-56.90552pt}{-56.90552pt}{-31.42845pt}{-56.90552pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{51.21504pt}{0.0pt}\pgfsys@curveto{51.21504pt}{-28.2856pt}{28.28568pt}{-51.21497pt}{0.00008pt}{-51.21497pt}\pgfsys@curveto{-28.28552pt}{-51.21497pt}{-51.21489pt}{-28.2856pt}{-51.21489pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{62.59598pt}{0.0pt}\pgfsys@curveto{62.59598pt}{-34.57129pt}{34.57121pt}{-62.59605pt}{-0.00008pt}{-62.59605pt}\pgfsys@curveto{-34.57137pt}{-62.59605pt}{-62.59613pt}{-34.57129pt}{-62.59613pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{68.28644pt}{0.0pt}\pgfsys@curveto{68.28644pt}{-37.71413pt}{37.71396pt}{-68.2866pt}{-0.00017pt}{-68.2866pt}\pgfsys@curveto{-37.7143pt}{-68.2866pt}{-68.28677pt}{-37.71413pt}{-68.28677pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \par{}{{}}{}{{{}{}{}{}}}{} {} {}{}\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}{}{}{}}}{} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}{}{}{}}}{} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \par
\pgfsys@invoke{\lxSVG@closescope
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Here there are negative half-twists on $2n-2$ strands below the top $P$s, and
a positive half-twist on $2k$ strands above $X$. The $2n-2-k$ strands
connecting the two $P$s each have a negative kink.
A positive half-twist on $\ell$ strands adjacent to an “uncappable” element,
such as a minimal projection, gives a factor of $(is)^{\ell(\ell-1)/2}$, a
negative half-twist on $\ell$ strands adjacent to an uncappable element gives
a factor of $(-is^{-1})^{\ell(\ell-1)/2}$, and a negative kink gives a factor
of $-is^{-3}$. Remembering $q=s^{2}$, this shows that
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Thus
$R_{X\subset P\otimes P}=(-1)^{k}q^{k(k+1)-2n(n-1)}.$
###### Proof 3.2 (Proof of Theorem 3.3).
In $\mathcal{D}_{4}$, $P\otimes P\cong Q$, so part one of Theorem 3.1 applies.
Furthermore $\dim P=1$, so the unframed invariant for the object $P$ in
$\mathcal{D}_{4}$ is trivial. The first equation is just Theorem 2.5.
The same argument yields a previously known identity [14]. Consider Temperley-
Lieb at $q=\exp({\frac{2\pi i}{6}})$, and notice that $f^{(1)}\otimes
f^{(1)}\cong f^{(0)}$ and $\dim f^{(1)}=1$. Thus
$\mathcal{J}_{\mathfrak{sl}(2),(1)}(K){}_{\mid q=\exp(\frac{2\pi i}{6})}{}=1$.
###### Proof 3.3 (Proof of Theorem 3.4).
In $\mathcal{D}_{6}$, we have that $P\otimes P\cong P\oplus f^{(0)}$, so part
two of Theorem 3.1 applies, and we know $\mathcal{J}_{\mathcal{D}_{6},P}(K)$
is some specialization of the Jones polynomial. Using Lemma 3, we compute the
two eigenvalues as
$R_{f^{(0)}\subset P\otimes P}=\exp({\frac{2\pi i}{20}})^{-12}=-(\exp({2\pi
i\frac{-3}{10}}))^{-3}$
and
$R_{P\subset P\otimes P}=\exp({\frac{2\pi i}{20}})^{-6}=\exp({2\pi
i\frac{-3}{10}})$
which is consistent with $R_{P\subset P\otimes P}=\lambda$ and
$R_{f^{(0)}\subset P\otimes P}=-\lambda^{-3}$. So, we conclude that
$a=-\lambda^{2}=-\exp({-6\frac{2\pi i}{10}})=\exp({-\frac{2\pi i}{10}})$.
###### Proof 3.4 (Proof of Theorem 3.5).
In much the same way, for $\mathcal{D}_{8}$ we have $P\otimes P\cong Q\oplus
f^{(2)}$, so $\mathcal{J}_{\mathcal{D}_{8},P}(K)$ is some specialization of
the HOMFLYPT polynomial. The eigenvalues are
$\displaystyle\lambda=R_{f^{(2)}\subset P\otimes P}$ $\displaystyle=\exp(2\pi
i\frac{10}{14})$ and $\displaystyle\mu=R_{Q\subset P\otimes P}$
$\displaystyle=\exp(2\pi i\frac{1}{14}),$
so $\frac{1}{\sqrt{-\lambda\mu}}=\pm\exp\left(2\pi i\frac{-2}{14}\right)$. The
twist factor is $\theta=\exp\left(2\pi i\frac{-2}{14}\right)$, and so we get
$\mathcal{J}_{\mathcal{D}_{8},P}(K)=\operatorname{HOMFLYPT}(K)(a,z)$ with
either
$a=\exp(2\pi i\frac{5}{7}),\,z=\exp(-2\pi i\frac{1}{14})-\exp(2\pi
i\frac{1}{14})$
(taking the ‘positive’ square root) or
$a=\exp(2\pi i\frac{3}{14}),\,z=\exp(2\pi i\frac{1}{14})-\exp(-2\pi
i\frac{1}{14})$
(taking the other).
###### Proof 3.5 (Proof of Theorem 1).
Again, in $\mathcal{D}_{10}$ we have $P\otimes P\cong P\oplus f^{(0)}\oplus
f^{(4)}$, so $\mathcal{J}_{\mathcal{D}_{10},P}(K)$ is a specialization of
either the Kauffman polynomial or the Dubrovnik polynomial. The eigenvalues
are
$\displaystyle a^{-1}$ $\displaystyle=R_{f^{(0)}\subset P\otimes P}=\exp(2\pi
i\frac{-1}{9}),$ $\displaystyle\lambda$ $\displaystyle=R_{f^{(4)}\subset
P\otimes P}=\exp(2\pi i\frac{1}{18})$ and $\displaystyle\mu$
$\displaystyle=R_{P\subset P\otimes P}=\exp(2\pi i\frac{4}{9})$
Now we apply Theorem 3.2 (4) to these; we see that $R_{f^{(4)}\subset P\otimes
P}R_{P\subset P\otimes P}=-1$, so we’re in the Dubrovnik case. We read off
$z=\exp(2\pi i\frac{1}{18})+\exp(2\pi i\frac{4}{9})$.
We’ve now shown that
$\displaystyle\mathcal{J}_{\mathfrak{sl}(2),(8)}(K){}_{\mid q=\exp(\frac{2\pi
i}{36})}{}$ $\displaystyle=2\mathcal{J}_{\mathcal{D}_{10},P}(K)$
$\displaystyle=2\operatorname{Dubrovnik}(K)(\exp(2\pi i\frac{4}{36}),\exp(2\pi
i\frac{2}{36})+\exp(2\pi i\frac{16}{36})).$
To get the last identity, we note that
$(q^{7},q-q^{-1}){}_{\mid q=-\exp(\frac{-2\pi i}{18})}{}=(\exp(2\pi
i\frac{4}{36}),\exp(2\pi i\frac{16}{36})+\exp(2\pi i\frac{2}{36}))$
and use the specialization appearing in Equation (1.12).
We remark that when $2n\geq 12$, Equation (3.2) shows that $P\otimes P$ has at
least three summands which are not isomorphic to $f^{(0)}$, and thus Theorem
3.1 does not apply.
### 3.3 Recognizing specializations of the $G_{2}$ knot invariant
If $V$ is an object in a ribbon category such that $V\otimes
V\cong\boldsymbol{1}\oplus V\oplus A\oplus B$ then it is reasonable to guess
that the knot invariants coming from $V$ are specializations of the $G_{2}$
knot polynomial. In particular $\mathcal{D}_{14}$ might be related to $G_{2}$,
since in $\mathcal{D}_{14}$ we have $P\otimes P\cong f^{(0)}\oplus
f^{(4)}\oplus f^{(8)}\oplus P$ by Equation (3.2). In this section we prove
such a relationship using results of Kuperberg [25]. Applying Kuperberg’s
theorem requires some direct but tedious calculations.
In work in progress, Snyder has shown that, outside of a few small exceptions,
all nontrivial knot invariants coming from tensor categories with $V\otimes
V\cong\boldsymbol{1}\oplus V\oplus A\oplus B$ come from the $G_{2}$ link
invariant, which would obviate the need for these calculations. (The
“nontrivial” assumption in the last sentence is crucial as the standard
representation of the symmetric group $S_{n}$, or more generally the standard
object in Deligne’s category $S_{t}$, also satisfies $V\otimes
V\cong\boldsymbol{1}\oplus V\oplus A\oplus B$.)
In the following, by a trivalent vertex we mean a rotationally invariant map
$V\otimes V\to V$ for some symmetrically self-dual object $V$. By a tree we
mean a trivalent graph without cycles (allowing disjoint components).
###### Theorem 3 ([25, Theorem 2.1]).
Suppose we have a symmetrically self-dual object $V$ and a trivalent vertex in
a ribbon category $\mathcal{C}$, such that trees with $5$ or fewer boundary
points form a basis for the spaces
$\operatorname{Inv}_{\mathcal{C}}\left(V^{\otimes k}\right)$ for $k\leq 5$.
Then the link invariant $\mathcal{J}_{\mathcal{C},V}$ is a specialization of
the $G_{2}$ link invariant for some $q$.
###### Remark 4.
The trivalent vertex in $\mathcal{C}$ is some scalar multiple of the trivalent
vertex in the $G_{2}$ spider. Note that the $G_{2}$ link invariant is the same
at $q$ and $-q$ since all the relations only depend on $q^{2}$.
###### Lemma 5.
Suppose that $\mathcal{C}$ is a pivotal tensor category with a trivalent
vertex such that trees form a basis of $\operatorname{Inv}\left(V^{\otimes
k}\right)$ for $k\leq 3$. Then
1. 1.
trees are linearly independent in $\operatorname{Inv}\left(V^{\otimes
4}\right)$ if and only if
$-2b^{4}d^{5}+b^{4}d^{6}-2b^{3}d^{4}t+(b^{2}d^{4}-b^{2}d^{6})t^{2}\neq 0,$
(3.3)
2. 2.
trees are linearly independent in $\operatorname{Inv}\left(V^{\otimes
5}\right)$ if and only if
$\displaystyle b^{20}\left(d^{15}-10d^{13}-5d^{12}+65d^{11}-62d^{10}\right)$
$\displaystyle+5b^{19}t\left(d^{14}+d^{13}-7d^{12}-d^{11}+10d^{10}\right)$
$\displaystyle-5b^{18}t^{2}\left(d^{15}-10d^{13}-3d^{12}+55d^{11}-61d^{10}\right)$
$\displaystyle-5b^{17}t^{3}\left(6d^{14}+7d^{13}-40d^{12}-41d^{11}+83d^{10}\right)$
$\displaystyle+5b^{16}t^{4}\left(2d^{15}+3d^{14}-15d^{13}-17d^{12}+72d^{11}-68d^{10}\right)$
$\displaystyle+b^{15}t^{5}\left(2d^{15}+60d^{14}+60d^{13}-405d^{12}-485d^{11}+930d^{10}\right)$
$\displaystyle-5b^{14}t^{6}\left(3d^{15}+12d^{14}-8d^{13}-64d^{12}+3d^{11}+71d^{10}\right)$
$\displaystyle-5b^{13}t^{7}\left(5d^{14}+5d^{13}-44d^{12}-50d^{11}+96d^{10}\right)$
$\displaystyle+5b^{12}t^{8}\left(3d^{15}+12d^{14}-6d^{13}-70d^{12}-17d^{11}+112d^{10}\right)$
$\displaystyle-5b^{11}t^{8}\left(2d^{15}+6d^{14}-5d^{13}-29d^{12}+4d^{11}+45d^{10}\right)$
$\displaystyle+b^{10}t^{10}\left(2d^{15}+5d^{14}-5d^{13}-20d^{12}+10d^{11}+33d^{10}\right)$
$\displaystyle\neq 0$
Where $d$, $b$ and $t$ are defined by
$\displaystyle\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=15.89673pt]{./diagrams/eps/G2/loop}}\end{array}$
$\displaystyle=d,$
$\displaystyle\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=15.89673pt]{./diagrams/eps/G2/bigon}}\end{array}$
$\displaystyle=b\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=15.89673pt]{./diagrams/eps/G2/strand}}\end{array},$
$\displaystyle\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=39.75092pt]{./diagrams/eps/G2/triangle}}\end{array}$
$\displaystyle=t\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=31.79951pt]{./diagrams/eps/G2/vertex}}\end{array}.$
###### Proof 3.6.
Compute the matrix of inner products between trees. Each of these inner
products can be calculated using only the relations for removing circles,
bigons, and triangles. If the determinant of this matrix is nonzero then the
trees are linearly independent.
For $\mathcal{D}_{2n}$ the single strand corresponds to $P$, and the trivalent
vertex is
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{{}{}}{}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@lineto{-14.22638pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
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{{}{}}{}{}\pgfsys@moveto{-2.84523pt}{0.0pt}\pgfsys@lineto{-2.84523pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
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{{}{}}{}{}\pgfsys@moveto{2.84523pt}{0.0pt}\pgfsys@lineto{2.84523pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
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{{}{}}{}{}\pgfsys@moveto{8.5359pt}{0.0pt}\pgfsys@lineto{8.5359pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
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{{}{}}{}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@lineto{14.22638pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\pgfsys@color@rgb@stroke{1}{.5}{0}\pgfsys@invoke{
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}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{68.68652pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\pgfsys@color@rgb@stroke{1}{.5}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{.5}{0}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{1,.5,0}\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}$6$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$
which is rotationally invariant, because $P$ is invariant under $180$-degree
rotation.
In order to apply Lemma 5 we must compute the values of $b$ and $t$ in
$\frac{1}{2}\mathcal{D}_{14}$. In order to do so we simplify the expression
for the trivalent vertex.
###### Lemma 6.
$\leavevmode\hbox to144.51pt{\vbox
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\hbox to0.0pt{\hbox to0.0pt{
{}{{}}{}{{{{}{}{}{}}} {{}{}{}{}}}{} {}
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{{}{}{}{}}}{} {}
{}{}{}\pgfsys@moveto{39.83368pt}{0.0pt}\pgfsys@curveto{39.83368pt}{21.99982pt}{21.99982pt}{39.83368pt}{0.0pt}{39.83368pt}\pgfsys@curveto{-21.99982pt}{39.83368pt}{-39.83368pt}{21.99982pt}{-39.83368pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{99.58421pt}{0.0pt}\pgfsys@curveto{99.58421pt}{-21.99982pt}{81.75035pt}{-39.83368pt}{59.75053pt}{-39.83368pt}\pgfsys@curveto{37.75072pt}{-39.83368pt}{19.91685pt}{-21.99982pt}{19.91685pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{{}{}}{}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@lineto{14.22638pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$
###### Proof 3.7.
Expand $f^{(12)}=P+Q$ and use the fact that $P\otimes Q$, $Q\otimes P$, and
$Q\otimes Q$ do not have nonzero maps to $P$.
###### Lemma 7.
In $\frac{1}{2}\mathcal{D}_{14}$, using the above trivalent vertex, we have
that $d$ is the root of $x^{6}-3x^{5}-6x^{4}+4x^{3}+5x^{2}-x-1$ which is
approximately $4.14811$, $b$ is the root of
$x^{6}-12x^{5}-499x^{4}-2760x^{3}-397x^{2}+276x-1$ which is approximately
$0.00364276$ and $t$ is the root of
$x^{6}+136x^{5}+5072x^{4}+53866x^{3}+13132x^{2}+721x+1$ which is approximately
$-0.00142366$.
###### Proof 3.8.
The formula for $d$ is just the dimension of $P$.
We use the alternate description of the trivalent vertex to reduce the
calculation of $b$ and $t$ to a calculation in Temperley-Lieb which we do
using the formulas of [18].
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{{}{}}{}{}\pgfsys@moveto{6.82861pt}{68.28557pt}\pgfsys@lineto{6.82861pt}{113.8093pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{{}{}}{}{}\pgfsys@moveto{11.38092pt}{68.28557pt}\pgfsys@lineto{11.38092pt}{113.8093pt}\pgfsys@stroke\pgfsys@invoke{
} \par{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}
{\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@rect{-19.20988pt}{-79.47057pt}{38.41977pt}{22.37pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-13.51933pt}{-71.83557pt}\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@rect{-15.15048pt}{-105.84511pt}{30.30096pt}{18.21442pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{}{}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\pgfsys@color@rgb@stroke{1}{.5}{0}\pgfsys@invoke{
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}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-12.119pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\pgfsys@color@rgb@stroke{1}{.5}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{.5}{0}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{1,.5,0}\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}$6$}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=b^{\prime}P$
where $b^{\prime}$ is the coefficient for removing bigons labelled with
$f^{(12)}$ in Temperley-Lieb.
The calculation for $t$ is similar: we replace each trivalent vertex with a
trivalent vertex with a $P$ on the outside and $f^{(12)}$s in the middle. Then
we reduce the inner triangle in Temperley-Lieb.
###### Theorem 7.
For $\ell=-3$ or $10$,
$\displaystyle\mathcal{J}_{\mathfrak{sl}(2),(12)}(K){}_{\mid q=\exp(\frac{2\pi
i}{52})}{}$ $\displaystyle=2\mathcal{J}_{\mathcal{D}_{14},P}(K)$
$\displaystyle=2\mathcal{J}_{G_{2},V_{(10)}}(K){}_{\mid q=\exp{2\pi
i\frac{\ell}{26}}}{}$
###### Proof 3.9.
Since $\dim\operatorname{Inv}\left(P^{\otimes
0}\right)=\dim\operatorname{Inv}\left(P^{\otimes
2}\right)=\dim\operatorname{Inv}\left(P^{\otimes 3}\right)=1$ and
$\dim\operatorname{Inv}\left(P\right)=0$, trees form a basis of
$\operatorname{Inv}\left(P^{\otimes k}\right)$ for $k\leq 3$. By Lemma 5 we
see that trees are linearly independent in $\operatorname{Inv}\left(P^{\otimes
4}\right)$ and $\operatorname{Inv}\left(P^{\otimes 5}\right)$. A dimension
count shows that trees form a basis for these spaces. Now we apply Theorem 3
to see that the theorem holds for some $q$. We need to normalize the
$\mathcal{D}_{14}$ trivalent vertex for $P$ before it satisfies the $G_{2}$
relations, specifically multiplying it by the largest real root of
$x^{12}-645x^{10}-10928x^{8}-32454x^{6}-4752x^{4}+2x^{2}+1$. The quantities
$b$ and $t$ are both homogeneous of degree $2$ with respect to scaling the
trivalent vertex, so they are both multiplied by the square of this quantity.
We now solve the equations
$\displaystyle d$ $\displaystyle=q^{10}+q^{8}+q^{2}+1+q^{-2}+q^{-8}+q^{-10}$
$\displaystyle b$
$\displaystyle=-\left(q^{6}+q^{4}+q^{2}+q^{-2}+q^{-4}+q^{-6}\right)$
$\displaystyle t$ $\displaystyle=q^{4}+1+q^{-4}$
and find that they have a four solutions, $q=\exp{2\pi i\frac{\ell}{26}}$ with
$\ell=\pm 3,\pm 10$. Not all of these give the correct twist factor, however.
The twist factor for $P$ is $\exp(2\pi i\frac{-10}{26})$, while the twist
factor for the representation $V_{(10)}$ of $G_{2}$ is $q^{12}$; these only
agree for $\ell=-3$ or $10$. Since the identity holds for some $q$, and the
knot invariant only depends on $q^{2}$, the identity must hold for each of
these values.
### 3.4 Ribbon functors
The proofs of Theorems 3.1 , 3.2, and 3 actually construct ribbon functors
from a certain diagrammatic category to the ribbon category $\mathcal{C}$.
Combining this functor with the description of quantum group categories by
diagrams in [5, 4] and [1] one could prove the coincidences described in the
introduction (that is, Theorems 8, 8 and 8). To do this we need the following
lemma.
###### Lemma 8.
Suppose that $\mathcal{C}$ is a ribbon category such that $\mathcal{C}^{ss}$
is premodular, that $\mathcal{D}$ is a pseudo-unitary modular category, and
that $\mathcal{F}$ is a dominant ribbon functor
$\mathcal{C}\rightarrow\mathcal{D}$. Then $\mathcal{D}\cong\mathcal{C}^{ss\
modularize}$.
###### Proof 3.10.
Since $\mathcal{D}$ is pseudo-unitary the functor must factor through the
semisimplification, and thus the result follows from the uniqueness of
modularization.
In our cases $\mathcal{D}_{2n}$ is the target category, and is certainly
unitary and modular. The source category is a category of diagrams (coming
from Temperley-Lieb, Kauffman/Dubrovnik, HOMFLYPT, or the $G_{2}$ spider).
Dominance of the functor is a simple calculation in the fusion ring of
$\mathcal{D}_{2n}$. If $q$ is a large enough root of unity, then the
semisimplification of that diagram category has been proven to be pre-modular
for each of these cases [4, 40, 5] except the $G_{2}$ spider. Hence the
argument of the last subsection does not yet give a proof of the $G_{2}$
coincidence. We give a completely different proof in the next subsection.
### 3.5 Recognizing $\mathcal{D}_{2n}$ modular categories
Earlier in this section we found knot polynomial identities and coincidences
of modular tensor categories by observing that $P^{\otimes 2}$ broke up in
some particular way. In this section we work in the reverse direction. The
category $\frac{1}{2}\mathcal{D}_{2n}$ has a small object $f^{(2)}$ and
$f^{(2)}\otimes f^{(2)}\cong\boldsymbol{1}\oplus f^{(2)}\oplus f^{(4)}$. If we
are to have a coincidence of modular tensor categories
$\mathcal{D}_{2n}\cong\mathcal{C}$ then there must be an object in
$\mathcal{C}$ which breaks up the same way. Using the characterization of the
Kauffman and Dubrovnik categories above we can prove that
$\mathcal{D}_{2n}\cong\mathcal{C}$ by producing this object. In the following
theorem, we use this technique to show
$\frac{1}{2}\mathcal{D}_{14}\cong\operatorname{Rep}{U_{\exp({2\pi
i\frac{\ell}{26}})}(\mathfrak{g}_{2})}$, for $\ell=-3$ or $10$, sending
$P\mapsto V_{(10)}$. It’s also possible to prove Theorems 8, 8 and 8 by this
technique, although we don’t do this.
###### Theorem 8.
There is an equivalence of modular tensor categories
$\operatorname{Rep}{U_{\exp({2\pi
i\frac{\ell}{26}})}(\mathfrak{g}_{2})}\cong\frac{1}{2}\mathcal{D}_{14},$
where $\ell=-3$ or $10$, sending $f^{(2)}\mapsto V_{(02)}$. Under this
equivalence we also have $P\mapsto V_{(10)}$.
###### Proof 3.11.
Using the Racah rule for tensor products in $\operatorname{Rep}{U_{\exp({2\pi
i\frac{\ell}{26}})}(\mathfrak{g}_{2})}$ we see that $V_{(02)}^{\otimes
2}\cong\boldsymbol{1}\oplus V_{(01)}\oplus V_{(02)}$.
The eigenvalues for the square of a crossing can be read off from twist
factors:
$R_{X\subset Y\otimes Y}^{2}=\theta_{X}\theta_{Y}^{-2}.$
The twist factors for the representations $V_{(00)},V_{(01)}$ and $V_{(02)}$
are $1,q^{24}$ and $q^{60}$ respectively, so the corresponding eigenvalues for
the crossing are $q^{-60},\sigma_{1}q^{-48}$ and $\sigma_{2}q^{-30}$ for some
signs $\sigma_{1}$ and $\sigma_{2}$. We thus compute, whether we are in the
Kauffman or Dubrovnik settings, that $a=q^{60}$ and
$z=\sigma_{1}q^{-48}+\sigma_{2}q^{-30}$.
If we are in the Kauffman setting, we must have
$\sigma_{1}\sigma_{2}q^{-78}=1$, so $\sigma_{1}=\sigma_{2}$. We now see the
dimension formula $d=\frac{a+a^{-1}}{z}-1$ can not be equal to
$\dim(f^{(2)})=[3]_{q=\exp(\frac{2\pi i}{52})}$ for any choice of
$\sigma_{1},\sigma_{2}$.
Hence we must be in the Dubrovnik setting where we have
$\sigma_{1}=-\sigma_{2}$ and $d=\frac{a-a^{-1}}{z}+1$. Now the dimensions
match up exactly when $\sigma_{1}=-1$ and $\sigma_{2}=1$.
By §3.4 and Theorems 3.1 and 3.2 we have a functor from the Dubrovnik category
with $a=\exp(2\pi i\frac{1}{13})$ and $z=\exp(2\pi i\frac{1}{26})-\exp(2\pi
i\frac{-1}{26})$ to $\operatorname{Rep}{U_{\exp({2\pi
i\frac{\ell}{26}})}(\mathfrak{g}_{2})}$. Since the target category is pseudo-
unitary [34], this functor factors through the semisimplification of the
diagram category, which is the premodular category
$\operatorname{Rep}{U_{q=\exp(2\pi i\frac{1}{52})}(\mathfrak{so}(3))}$. Since
the target is modular [36] and the functor is dominant (a straightforward
calculation via the Racah rule in the Grothendieck group of
$\operatorname{Rep}{U_{\exp({2\pi i\frac{\ell}{26}})}(\mathfrak{g}_{2})}$)
this functor induces an equivalence between the modularization of
$\operatorname{Rep}{U_{q=\exp(2\pi i\frac{1}{52})}(\mathfrak{so}(3))}$, which
is nothing but $\frac{1}{2}\mathcal{D}_{14}$, and
$\operatorname{Rep}{U_{\exp({2\pi i\frac{\ell}{26}})}(\mathfrak{g}_{2})}$.
The correspondence between simples shown in Figure 2, can be computed
inductively. Begin with the observation that $f^{(2)}$ is sent to $V_{(02)}$
by construction; after that, everything else is determined by working out the
tensor product rules in both categories.
$\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=119.2467pt]{./diagrams/eps/coincidences/g2-weyl-
chamber}}\end{array}$
Figure 2: The positive Weyl chamber for $G_{2}$, showing the surviving
irreducible representations in the semisimple quotient at $q=\pm\exp({2\pi
i\frac{-3}{26}})$, and the correspondence with the even vertices of
$\mathcal{D}_{14}$.
Note that $q=\pm\exp({2\pi i\frac{-3}{26}})$ corresponds to the fractional
level $\frac{1}{3}$ of $G_{2}$ (see [36]), which has previously been
conjectured to be unitary [34]. This theorem proves that conjecture.
Finally, we note that the same method gives an equivalence between
$\operatorname{Rep}{U_{\pm\exp({2\pi i\frac{-3}{28}})}(\mathfrak{g}_{2})}$ and
the subcategory of $\operatorname{Rep}{U_{\exp({2\pi
i\frac{1}{28}})}(\mathfrak{sp}(6))}$ generated by the representation
$V_{(012)}$, sending the representation $V_{(12)}$ of $\mathfrak{g}_{2}$ to
$V_{(012)}$. On the $\mathfrak{g}_{2}$ side, we have $V_{(12)}^{\otimes
2}\cong V_{(00)}\oplus V_{(01)}\oplus V_{(02)}$ with corresponding eigenvalues
$1,\exp(2\pi i\frac{3}{14})$ and $-\exp(2\pi i\frac{4}{14})$. On the
$\mathfrak{sp}(6)$ side we have $V_{(012)}^{\otimes 2}\cong V_{(000)}\oplus
V_{(010)}\oplus V_{(200)}$ with corresponding eigenvalues $1,\exp(2\pi
i\frac{3}{14})$ and $-\exp(2\pi i\frac{4}{14})$. Thus both categories, which
are each modular, are the modularization of the semisimplification of the
Kauffman category at $a=1,z=\exp(2\pi i\frac{3}{14})-\exp(2\pi
i\frac{4}{14})$. This proves the conjecture of [34] that $G_{2}$ at level
$\frac{2}{3}$ is also unitary.
## 4 Coincidences of tensor categories
In the previous section we found identities between knot polynomials coming
from (a priori) different ribbon categories. In Section 3.4 we showed that
these identities must come from unexpected functors between these ribbon
categories. In this section we explain how these coincidences of tensor
categories follow from general theory. One should think of the results of this
section as quantum analogs of small coincidences in group theory, such as
$\operatorname{Alt}_{5}\cong\mathbf{PSL}_{2}(\mathbb{F}_{5})$.
There are three important sources of unexpected equivalences (or
autoequivalences) between ribbon categories coming from quantum groups:
coincidences of small Dynkin diagrams, (deequivariantization related to)
generalized Kirby-Melvin symmetry, and level-rank duality.
There are sometimes coincidences between Dynkin diagrams in different
families. For instance, the Dynkin diagrams $A_{3}$ and $D_{3}$ are equal,
from which it follows that $\mathfrak{sl}(4)\cong\mathfrak{so}(6)$ and the
associated categories of representations of quantum groups are equivalent too.
Kirby-Melvin symmetry relates link invariants coming from different objects in
the same category, when that category has an invertible object. Under certain
auspicious conditions, one can go further and deequivariantize by the
invertible object.
Level-rank duality is a collection of equivalences relating $SU(n)_{k}$ with
$SU(k)_{n}$, and relating $SO(n)_{k}$ with $SO(k)_{n}$, where $SU(n)_{k}$ or
$SO(n)_{k}$ refers to the semisimplified representation category of the
rank-$n$ quantum group, at a carefully chosen root of unity which depends on
the ‘level’ $k$. In some sense, level-rank duality is more natural in the
context of $U(n)$ and $O(n)$, and new difficulties arise formulating level-
rank duality for the quantum groups $SU(n)$ and $SO(n)$. We give, in Theorem
18, a precise statement for $SO$ level-rank duality with $n=3$ and $k$ even.
We will discuss each of these three sources of unexpected equivalences in the
following sections, and then use them to prove the following results:
###### Theorem 8.
There is an equivalence of modular tensor categories
$\frac{1}{2}\mathcal{D}_{6}\cong\operatorname{Rep}^{\text{uni}}{U_{s=\exp\left(\frac{7}{10}2\pi
i\right)}(\mathfrak{sl}(2)\oplus\mathfrak{sl}(2)})^{modularize},$
sending $P\mapsto V_{(1)}\boxtimes V_{(0)}$.
###### Theorem 8.
There is an equivalence of modular tensor categories
$\frac{1}{2}\mathcal{D}_{8}\cong\operatorname{Rep}^{\text{uni}}{U_{s=\exp\left(\frac{5}{14}2\pi
i\right)}(\mathfrak{sl}(4))}^{modularize},$
sending $P\mapsto V_{(100)}$.
###### Theorem 8.
The modular tensor category $\frac{1}{2}\mathcal{D}_{10}$ has an order $3$
automorphism, fixing $f^{(0)},f^{(4)}$ and $f^{(6)}$, and permuting
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Finally we note that there are other coincidences of small tensor categories
that do not follow from these general techniques. In particular it would be
very interesting to better explain the coincidences involving $G_{2}$.
### 4.1 Dynkin diagram coincidences and quantum groups
The definition of the quantum group and its ribbon category of representations
depend only on the Dynkin diagram itself. For the quantum group and its tensor
category this is obvious from the presentation by generators and relations.
For the braiding and the ribbon structure this follows from the independence
of choice of decomposition of the longest word in the Weyl group in the
multiplicative formula for the R-matrix.
In particular, every coincidence between Dynkin diagrams lifts to a statement
about the quantum groups. We will use that $D_{2}=A_{1}\times A_{1}$, that
$D_{3}=A_{3}$, and that $D_{4}$ has triality symmetry.
The reason these coincidences are useful is that they give two different
diagrammatic presentations of the same ribbon category. For example, the fact
that $B_{1}=A_{1}$ tells you that the even part of Temperley-Lieb can be
described using the Dubrovnik category, which we used implicity in Section
3.5. The only coincidence we don’t use is $B_{2}=C_{2}$. Since $B_{2}$ is the
Dynkin diagram for $\mathfrak{so}(5)$, there is no relationship via level-rank
duality with the $\mathcal{D}_{2n}$ planar algebras.
### 4.2 Kirby-Melvin symmetry
Kirby-Melvin symmetry relates link invariants from one representation of a
quantum group to link invariants coming from another representation which is
symmetric to it under a symmetry of the affine Weyl chamber. This symmetry
principle was proved in type $A_{1}$ by Kirby and Melvin [21], in type $A_{n}$
by Kohno and Takata [23], and for a general quantum group by Le [27]. We give
a diagrammatic proof which generalizes this result to tensor categories which
might not come from quantum groups.
Suppose that $\mathcal{C}$ is a semi-simple ribbon category and that $X$ is an
object which is invertible in the sense that $X\otimes
X^{*}\cong\boldsymbol{1}$. Kirby-Melvin symmetry relates link invariants
coming from a simple object $A$ to invariants coming from the (automatically
simple) object $A\otimes X$.
The key observation is that, for any simple $A$, the objects $A\otimes X$ and
$X\otimes A$ are simple (since $\operatorname{Hom}\left(A\otimes X,A\otimes
X\right)=\operatorname{Hom}\left(A\otimes X\otimes X^{*},A\right)$), so the
Hom space between them is one dimensional. Thus the over-crossing and under-
crossing must be scalar multiples. Define $c_{A}$ by the following formula,
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope
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Note that $c_{A}^{-1}\dim A\dim X=S_{XA}$ where $S$ is the $S$-matrix. Using
the formula for the square of the crossing in terms of the ribbon element, we
see that $c_{A}=\frac{\theta_{A}\theta_{X}}{\theta_{A\otimes X}}$.
###### Theorem 8.
Let $\mathcal{C}$ be a semi-simple ribbon category, $A$ be a simple object in
$\mathcal{C}$, $X$ be a simple invertible object, and $L$ a link with $\\#L$
components. Then,
$\mathcal{J}_{\mathcal{C},A\otimes
X}(L)=\mathcal{J}_{\mathcal{C},A}(L)\mathcal{J}_{\mathcal{C},X}(L)=(\dim
X)^{\\#L}\mathcal{J}_{\mathcal{C},A}(L).$
###### Proof 4.1.
First look at the framed version of the knot invariants. The framed $A\otimes
X$ invariant comes from cabling $L$ and labeling one of the two cables $A$ and
the other one $X$. We unlink the link labeled $A$ from the link labeled $X$ by
successively changing crossings where $X$ goes under $A$ to crossings where
$X$ goes over $A$. Each crossing in the original link gives rise to two
crossings between the $X$-labelled link and the $A$-labelled link, and exactly
one of these crossings needs to be switched. Furthermore, the sign of the
crossing that needs to be switched is the same as the sign of the original
crossing. See the following diagram for what happens at each positive
crossing.
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Hence, unlinking the $X$-labelled link from the $A$-labelled link picks up a
factor of $c_{A}^{-writhe}$. At this point, the link labelled by $A$ lies
completely behind the link labelled by $X$, and we can compute their
invariants separately. Thus,
$\theta_{A\otimes X}^{writhe}\mathcal{J}_{\mathcal{C},A\otimes
X}(L)=c_{A}^{-writhe}\theta_{A}^{writhe}\mathcal{J}_{\mathcal{C},A}(L)\theta_{X}^{writhe}\mathcal{J}_{\mathcal{C},X}(L).$
Rearranging terms and writing $c_{A}$ in terms of twist factors, we see that
$\mathcal{J}_{\mathcal{C},A\otimes
X}(L)=\mathcal{J}_{\mathcal{C},A}(L)\mathcal{J}_{\mathcal{C},X}(L)$. The final
equation follows from Theorem 3.1.
Note that $\dim{X}$ above has to be $1$ or $-1$, since $\dim{X}=\dim{X^{*}}$
and $X\otimes X^{*}=1$.
Suppose that you have a finite ribbon category whose fusion graph is
symmetric. Take $X$ to be any projection which is symmetric in the fusion
graph with $\boldsymbol{1}$. Then it is easy to see that its Frobenius-Perron
dimension $\dim_{FP}(X)=1$, and thus that $X$ is invertible. Hence, any time
the fusion graph has a symmetry so do the knot invariants.
If $X$ gives a Kirby-Melvin symmetry, then if you’re lucky you can set
$X\cong\boldsymbol{1}$ using the deequivariantization procedure. Furthermore,
even if you can’t deequivariantize immediately (for example, if $\dim X\neq
1$) you might still be able to modify the category $\mathcal{C}$ is some mild
way (changing the braiding or changing the pivotal structure, neither of which
changes the link invariant significantly) and then be able to
deequivariantize. We give three examples of this:
Consider $\operatorname{Rep}{U_{q=-\exp\left(-2\pi
i\frac{1}{10}\right)}(\mathfrak{sl}(2))}$. The representation $V_{3}$ is
invertible and thus gives a Kirby-Melvin symmetry. We can make this monoidal
category into a ribbon category in many ways: first we can choose
$s=q^{\frac{1}{2}}$ in two different ways; second we can choose either the
usual pivotal structure or the unimodal one. For each of these four choices we
check each of the conditions needed to define the deequivariantization
$\mathcal{C}//V_{3}$ (transparency, dimension $1$, and twist factor $1$).
| $V_{3}$ transparent | $\dim V_{3}$ | $\theta_{V_{3}}$
---|---|---|---
$\operatorname{Rep}{U_{s=\exp\left(2\pi i\frac{1}{5}\right)}(\mathfrak{sl}(2))}$ | Yes | -1 | 1
$\operatorname{Rep}{U_{s=\exp\left(2\pi i\frac{7}{10}\right)}(\mathfrak{sl}(2))}$ | No | -1 | -1
$\operatorname{Rep}^{\text{uni}}{U_{s=\exp\left(2\pi i\frac{1}{5}\right)}(\mathfrak{sl}(2))}$ | No | 1 | -1
$\operatorname{Rep}^{\text{uni}}{U_{s=\exp\left(2\pi i\frac{7}{10}\right)}(\mathfrak{sl}(2))}$ | Yes | 1 | 1
Let $\operatorname{Rep}^{\text{root}}U_{q}(\mathfrak{g})$ denote the full
subcategory of representations whose highest weights are in the root lattice.
(Notice that this ribbon category only depends on $q$, not on a choice of
$s=q^{\frac{1}{L}}$. Furthermore, it does not depend on the choice of ribbon
element.)
###### Lemma 9.
$\operatorname{Rep}^{\text{root}}U_{q=-\exp\left(-2\pi
i\frac{1}{10}\right)}(\mathfrak{sl}(2))\cong\operatorname{Rep}^{\text{uni}}{U_{s=\exp\left(2\pi
i\frac{7}{10}\right)}(\mathfrak{sl}(2))^{modularize}}$.
###### Proof 4.2.
We restrict the deequivariantization
$\mathcal{F}:\operatorname{Rep}^{\text{uni}}{U_{s=\exp\left(2\pi
i\frac{7}{10}\right)}(\mathfrak{sl}(2))}\rightarrow\operatorname{Rep}^{\text{uni}}{U_{s=\exp\left(2\pi
i\frac{7}{10}\right)}(\mathfrak{sl}(2))}//V_{3}$
to $\operatorname{Rep}^{\text{root}}$. Since $\otimes V_{3}$ acts freely on
the isomorphism classes of simple objects and since every orbit contains
exactly one object in $\operatorname{Rep}^{\text{root}}$ the restriction of
this functor is an equivalence by Lemma 1.6.
We will need two similar results, for $\operatorname{Rep}U_{q=-\exp\left(-2\pi
i\frac{1}{10}\right)}(\mathfrak{sl}(2)\oplus\mathfrak{sl}(2))$ and for
$\operatorname{Rep}{U_{q=-\exp\left(-\frac{2\pi
i}{14}\right)}(\mathfrak{sl}(4))}$.
In $\operatorname{Rep}U_{q=-\exp\left(-2\pi
i\frac{1}{10}\right)}(\mathfrak{sl}(2)\oplus\mathfrak{sl}(2))$ we can consider
the root representations, those of the form $V_{a}\boxtimes V_{b}$ with both
$a$ and $b$ even, as well as the vector representations, those $V_{a}\boxtimes
V_{b}$ with $a+b$ even. We call these the vector representations because they
become the vector representations under the identification
$\mathfrak{sl}(2)\oplus\mathfrak{sl}(2)\cong\mathfrak{so}(4)$.
###### Lemma 10.
$\displaystyle\operatorname{Rep}^{\text{root}}U_{q=-\exp\left(-2\pi
i\frac{1}{10}\right)}(\mathfrak{sl}(2)\oplus\mathfrak{sl}(2))$
$\displaystyle\qquad\cong\operatorname{Rep}^{\text{vector}}U_{q=-\exp\left(-2\pi
i\frac{1}{10}\right)}(\mathfrak{sl}(2)\oplus\mathfrak{sl}(2))//V_{3}\boxtimes
V_{3}$
$\displaystyle\qquad\cong\operatorname{Rep}^{\text{uni}}{U_{s=\exp\left(2\pi
i\frac{7}{10}\right)}(\mathfrak{sl}(2)\oplus\mathfrak{sl}(2))}^{modularize}.$
###### Proof 4.3.
We make the abbreviations
$\displaystyle\mathcal{R}$
$\displaystyle=\operatorname{Rep}^{\text{root}}U_{q=-\exp\left(-2\pi
i\frac{1}{10}\right)}(\mathfrak{sl}(2)\oplus\mathfrak{sl}(2))$
$\displaystyle\mathcal{V}$
$\displaystyle=\operatorname{Rep}^{\text{vector}}U_{q=-\exp\left(-2\pi
i\frac{1}{10}\right)}(\mathfrak{sl}(2)\oplus\mathfrak{sl}(2))$
$\displaystyle\mathcal{U}$
$\displaystyle=\operatorname{Rep}^{\text{uni}}{U_{s=\exp\left(2\pi
i\frac{7}{10}\right)}(\mathfrak{sl}(2)\oplus\mathfrak{sl}(2))}.$
It is easy to check that $\mathcal{R}$ and $\mathcal{V}$ are not affected by
either the choice of $s$ (recall in this situation $s$ is a square root of
$q$, required for the definition of the braiding), or changing between the
usual and the unimodal pivotal structures. Thus we have inclusions
$\mathcal{R}\subset\mathcal{V}\subset\mathcal{U}.$
The invertible objects in $\mathcal{U}$ are the representations
$V_{0}\boxtimes V_{0},V_{0}\boxtimes V_{3},V_{3}\boxtimes V_{0}$ and
$V_{3}\boxtimes V_{3}$. For any choice of $s$ and either pivotal structure,
$V_{3}\boxtimes V_{3}$ is transparent. The representations $V_{0}\boxtimes
V_{3}$ and $V_{3}\boxtimes V_{0}$ are transparent only with $s=\exp\left(2\pi
i\frac{7}{10}\right)$ and the unimodal pivotal structure. Under tensor
product, the invertible objects form the group
$\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$. The invertible objects
in $\mathcal{V}$ are $V_{0}\boxtimes V_{0}$ and $V_{3}\boxtimes V_{3}$,
forming the group $\mathbb{Z}/2\mathbb{Z}$.
We have
$\displaystyle(V_{a}\boxtimes V_{b})\otimes(V_{0}\boxtimes V_{3})$
$\displaystyle\cong V_{a}\boxtimes V_{3-b}$ $\displaystyle(V_{a}\boxtimes
V_{b})\otimes(V_{3}\boxtimes V_{0})$ $\displaystyle\cong V_{3-a}\boxtimes
V_{b}$ $\displaystyle(V_{a}\boxtimes V_{b})\otimes(V_{3}\boxtimes V_{3})$
$\displaystyle\cong V_{3-a}\boxtimes V_{3-b},$
and so see that the action of the group of invertible objects is free. Each
$\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ orbit on $\mathcal{U}$
contains exactly one object from $\mathcal{R}$, and each
$\mathbb{Z}/2\mathbb{Z}$ orbit on $\mathcal{V}$ contains exactly one object
from $\mathcal{R}$. See Figure 3.
$\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=139.12216pt]{./diagrams/eps/coincidences/so4-4fold-quotient}}\end{array}$ | $\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=139.12216pt]{./diagrams/eps/coincidences/so4-vector-quotient}}\end{array}$
---|---
(a) | (b)
Figure 3: (a) The $4$-fold quotient of the Weyl alcove and (b) the $2$-fold
quotient of the Weyl alcove, with vector representations marked. Lemma 10
identifies the two resulting $4$-object categories.
Thus both equivalences in this lemma are deequivariantizations, by applying
Lemma 1.6 to the inclusions $\mathcal{R}\subset\mathcal{U}$ and
$\mathcal{V}\subset\mathcal{U}$.
In $\operatorname{Rep}{U_{q=-\exp\left(-\frac{2\pi
i}{14}\right)}(\mathfrak{sl}(4))}$ we can again consider two subcategories,
the root representations and the vector representations. The root
representations of $\mathfrak{sl}(4)$ are those whose highest weight is an
$\mathbb{N}$-linear combination of $(2,-1,0),(-1,2,-1),(0,-1,2)$ in
$\mathbb{N}^{3}$. They form an index $4$ sublattice of the weight lattice. The
Weyl alcove for $\mathfrak{sl}(4)$ at a $14$-th root of unity consists of
those weights $(a,b,c)\in\mathbb{N}^{3}$ with $a+b+c\leq 3$, and so the
relevant root representations are $V_{(000)},V_{(101)},V_{(210)},V_{(012)}$
and $V_{(020)}$. The vector representations
$\operatorname{Rep}^{\text{vector}}U_{q=-\exp\left(-\frac{2\pi
i}{14}\right)}(\mathfrak{sl}(4))$ are those that become vector representations
under the identification $\mathfrak{sl}(4)\cong\mathfrak{so}(6)$ (this is
$A_{3}=D_{3}$), namely those $V_{(abc)}$ with $a+c$ even. These form an index
$2$ sublattice of the weight lattice, containing the root lattice. Both
sublattices are illustrated in Figure 4; hopefully having these diagrams in
mind will ease later arguments.
$\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=178.86702pt]{./diagrams/eps/coincidences/sl4-vector-labels}}\end{array}$ | $\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=178.86702pt]{./diagrams/eps/coincidences/sl4-root-labels}}\end{array}$
---|---
(a) | (b)
Figure 4: The $\mathfrak{sl}(4)$ Weyl alcove at a $14$-th root of unity,
showing (a) the vector representation sublattice and (b) the root
representation sublattice.
###### Lemma 11.
$\displaystyle\operatorname{Rep}^{\text{root}}U_{q=-\exp\left(-\frac{2\pi
i}{14}\right)}(\mathfrak{sl}(4))$
$\displaystyle\cong\operatorname{Rep}^{\text{vector}}U_{q=-\exp\left(-\frac{2\pi
i}{14}\right)}(\mathfrak{sl}(4))//V_{(030)}$
$\displaystyle\cong\operatorname{Rep}^{\text{uni}}{U_{s=\exp\left(2\pi
i\frac{5}{14}\right)}(\mathfrak{sl}(4))}^{modularize}.$
###### Proof 4.4.
We make the abbreviations
$\displaystyle\mathcal{R}$
$\displaystyle=\operatorname{Rep}^{\text{root}}U_{q=-\exp\left(-\frac{2\pi
i}{14}\right)}(\mathfrak{sl}(4))$ $\displaystyle\mathcal{V}$
$\displaystyle=\operatorname{Rep}^{\text{vector}}U_{q=-\exp\left(-\frac{2\pi
i}{14}\right)}(\mathfrak{sl}(4))$ $\displaystyle\mathcal{U}$
$\displaystyle=\operatorname{Rep}^{\text{uni}}{U_{s=\exp\left(2\pi
i\frac{5}{14}\right)}(\mathfrak{sl}(4))}.$
It is easy to check that $\mathcal{R}$ and $\mathcal{V}$ are not affected by
either the choice of $s$ (recall in this situation $s$ is a $4$-th root of
$q$, required for the definition of the braiding), or any variation of pivotal
structure. Thus we have inclusions
$\mathcal{R}\subset\mathcal{V}\subset\mathcal{U}.$
The invertible objects in $\mathcal{U}$ are the representations
$V_{(000)},V_{(300)},V_{(030)}$ and $V_{(003)}$. For any choice of $s$ and
pivotal structure, $V_{(030)}$ is transparent. The representations $V_{(300)}$
and $V_{(003)}$ are transparent only with $s=\exp\left(2\pi
i\frac{5}{14}\right)$ and the unimodal pivotal structure. Under tensor
product, the invertible objects form the group $\mathbb{Z}/4\mathbb{Z}$. The
invertible objects in $\mathcal{V}$ are $V_{000}$ and $V_{030}$, forming the
group $\mathbb{Z}/2\mathbb{Z}$.
The action of the group of invertible objects is free, and shown in Figure 5.
Each $\mathbb{Z}/4\mathbb{Z}$ orbit on $\mathcal{U}$ contains exactly one
object from $\mathcal{R}$, and each $\mathbb{Z}/2\mathbb{Z}$ orbit on
$\mathcal{V}$ contains exactly one object from $\mathcal{R}$. See Figure 6.
$\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=119.2467pt]{./diagrams/eps/coincidences/sl4-300}}\end{array}$ | $\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=119.2467pt]{./diagrams/eps/coincidences/sl4-030}}\end{array}$ | $\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=119.2467pt]{./diagrams/eps/coincidences/sl4-003}}\end{array}$
---|---|---
(a) | (b) | (c)
Figure 5: The action of tensor product with an invertible object. (a) $-\otimes V_{(300)}$ and (c) $-\otimes V_{(003)}$ act by orientation reversing isometries, while (b) $-\otimes V_{(030)}$ acts by a $\pi$ rotation. $\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=178.86702pt]{./diagrams/eps/coincidences/sl4-4fold-quotient}}\end{array}$ | $\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=178.86702pt]{./diagrams/eps/coincidences/sl4-vector-quotient}}\end{array}$
---|---
(a) | (b)
Figure 6: (a) The $4$-fold quotient of the Weyl alcove and (b) the $2$-fold
quotient of the Weyl alcove, with vector representations marked. Lemma 11
identifies the two resulting $5$-object categories.
Thus both equivalences in the Lemma are de-equivariantizations, by applying
Lemma 1.6 to the inclusions $\mathcal{R}\subset\mathcal{U}$ and
$\mathcal{V}\subset\mathcal{U}$.
Finally, the usual statement in the literature of generalized Kirby-Melvin
symmetry involves changing the label of only one component on the link. This
can be proved in a completely analogous way to the result above. We recall the
statement here.
###### Theorem 11.
Let $J_{A_{1},\ldots,A_{k}}(L)$ be the value of a framed link $L$ (with
components $L_{1},\ldots,L_{k}$), labeled by simple objects
$A_{1},\ldots,A_{k}$. Suppose now that $A_{1}$ is replaced by $A_{1}\otimes X$
(with $X$ invertible). Then
$J_{A_{1}\otimes X,A_{2},\ldots,A_{k}}(L)=\dim{X}\cdot
c_{X}^{\operatorname{writhe}(L_{1})}\cdot\prod_{i=1,\ldots,k}c_{A_{i}}^{\operatorname{linking}(L_{1}^{\prime},L_{i})}\cdot
J_{A_{1},\ldots,A_{k}}(L)$
where $L_{1}^{\prime}$ is a copy of $L_{1}$ running parallel to $L_{1}$ in the
blackboard framing.
### 4.3 Level-rank duality
Level-rank duality is a collection of ideas saying that the semisimplified
representation theory of a quantum group at a certain root of unity is related
to that of a different quantum group, at a (potentially) different root of
unity. The rank of a quantum group in this setting is dimension of its natural
representation (i.e. the $n$ in $\mathfrak{so}(n)$ or $\mathfrak{sl}(n)$). The
level describes the root of unity. The name “level” comes from the connection
between quantum groups at roots of unity and projective representations of
loop groups at a fixed level. Here the relationship between the root of unity
and the level is given by the formula
$k=\frac{l}{2D}-\check{h}$
where $l$ is the order of the root of unity, $D$ is the lacing number of the
quantum group, and $\check{h}$ is the dual Coxeter number. See Figure 7 for
the values for each simple Lie algebra. Notice that not all roots of unity
come from loop groups under this correspondence.
type | Lie group | rank | lacing number $D$ | dual Coxeter number $\check{h}$ | L
---|---|---|---|---|---
$A_{n}$ | $\mathfrak{sl}(n+1)$ | $n$ | $1$ | $n+1$ | $n+1$
$B_{\text{$n$ even}}$ | $\mathfrak{so}(2n+1)$ | $n$ | $2$ | $2n-1$ | 1
$B_{\text{$n$ odd}}$ | $\mathfrak{so}(2n+1)$ | $n$ | $2$ | $2n-1$ | 2
$C_{n}$ | $\mathfrak{sp}(2n)$ | $n$ | $2$ | $n+1$ | $1$
$D_{\text{$n$ even}}$ | $\mathfrak{so}(2n)$ | $n$ | $1$ | $2n-2$ | 2
$D_{\text{$n$ odd}}$ | $\mathfrak{so}(2n)$ | $n$ | $1$ | $2n-2$ | 4
$E_{n}$ | $E_{6|7|8}$ | $6,7,8$ | $1$ | $12,18,30$ | $3,2,1$
$F_{4}$ | $F_{4}$ | $4$ | $2$ | $9$ | $1$
$G_{2}$ | $G_{2}$ | $2$ | $3$ | $4$ | $1$
Figure 7: Combinatorial data for the simple Lie algebras.
Nonetheless there are versions of level-rank duality for quantum groups at
roots of unity not corresponding to loop groups. In this context what the
“level” measures is which quantum symmetrizers vanish, while the rank measures
which quantum antisymmetrizers vanish. At the level of combinatorics, the rank
gives the bound on the number of rows in Young diagrams, while the level gives
a bound on the number of columns, and duality is realized by reflecting Young
diagrams thus interchanging the roles of rank and level.
We want statements of level-rank duality that give equivalences of braided
tensor categories. In order to get such precise statements several
technicalities appear. First, level-rank duality concerns $SO$, not $Spin$, so
we only look at the vector representations. Second, there is a subtle
relationship between the roots of unity you need to pick on each side of the
equivalence. In particular, if the root of unity on the left side is of the
form $\exp(\frac{2\pi i}{m})$ then the root of unity on the right side
typically will not be of that form. Finally, level-rank duality is most
natural as a statement about $U$ and $O$, not about $SU$ and $SO$. Getting
statements about $SU$ and $SO$ requires considering modularizations. (It may
seem surprising that this is even possible, since we know that
$\operatorname{Rep}(U_{\zeta(\ell)}(\mathfrak{so}(n)))$ is already a modular
tensor category [36, Theorem 6]. When we restrict to the subcategory of vector
representations, however, we lose modularity.)
We found the papers [5] (on the $SU$ case) and [4] (on the $SO$ and $Sp$
cases) to be exceedingly useful, and we’ll give statements and proofs that
closely follow their methods. Level-rank duality for $SO(3)-SO(4)$ appears in
the paper [9], where it is used to prove Tutte’s golden identity for the
chromatic polynomial. For our particular case of level-rank duality involving
$\mathfrak{so}_{3}$ and the $\mathcal{D}_{2m}$ subfactor planar algebra, see
the more physically minded [32]. For some more background on level-rank
duality, see [35, 10, 5] for the $SU$ cases, [24] for level-rank duality at
the level of $3$-manifold invariants and [45] for loop groups.
As explained by Beliakova and Blanchet, level-rank duality is easiest to
understand in a diagrammatic setting, where it says that $U(n)_{k}\cong
U(k)_{n}$ and $O(n)_{k}\cong O(k)_{n}$, with $U$ and $O$ being interpreted as
categories of tangles modulo either the HOMFLYPT or Dubrovnik relations. The
equivalences come from almost trivial symmetries of the relations. The reason
this modularization is necessary is that to recover $SO$ from $O$, we need to
quotient out by the determinant representation. Thus, to translate an
equivalence $O(n)_{k}\cong O(k)_{n}$ into something like $SO(n)_{k}\cong
SO(k)_{n}$, we find that in each category there is the ‘shadow’ of the
determinant representation in the other category, which we still need to
quotient out. See Figure 8 for a schematic diagram illustrating this.
$\textstyle{O(n)_{k}=O(k)_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{det}_{n}\cong\boldsymbol{1}}$$\scriptstyle{\mathrm{det}_{k}\cong\boldsymbol{1}}$$\textstyle{SO(n)_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{det}_{k}\cong\boldsymbol{1}}$$\textstyle{SO(k)_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{det}_{n}\cong\boldsymbol{1}}$$\textstyle{SO(n)_{k}^{\operatorname{mod}}\cong
SO(k)_{n}^{\operatorname{mod}}}$ Figure 8: A schematic description of $SO$
level-rank duality, suppressing the details of the actual roots of unity
appearing.
Here is the precise statement of level-rank duality which we will be using.
Define
$\ell_{n,k}=\begin{cases}2(n+k-2)&\text{if $n$ and $k$ are even}\\\
4(n+k-2)&\text{if $n$ is odd and $k$ is even}\\\ n+k-2&\text{if $n$ is even
and $k$ is odd}\end{cases}$
###### Theorem 11 (SO level-rank duality).
Suppose $n,k\geq 3$ are not both odd. Suppose $q_{1}$ is a primitive root of
unity with order $\ell_{n,k}$. Choose $q_{2}$ so that
$-1=\begin{cases}q_{1}q_{2}&\text{if $n$ and $k$ are both even}\\\
q_{1}^{2}q_{2}&\text{if $n$ is odd and $k$ is even}\\\ q_{1}q_{2}^{2}&\text{if
$n$ is even and $k$ is odd}\\\ \end{cases}$
As ribbon categories, there is an equivalence
$\operatorname{Rep}^{\text{vector}}(U_{q=q_{1}}(\mathfrak{so}(n)))//V_{ke_{1}}\cong\operatorname{Rep}^{\text{vector}}(U_{q=q_{2}}(\mathfrak{so}(k)))//V_{ne_{1}}.$
###### Remark 12.
When both $n$ and $k$ are odd, there is some form of level-rank duality in
terms of the Dubrovnik skein relation, pursued in [4] where it is called the
$B^{n,-k}$ case. However it does not seem possible to express this case purely
in terms of quantum groups.
###### Remark 13.
Notice that the order of $q_{2}$ is always $\ell_{k,n}$. When $n$ and $k$ are
both even then the roots of unity on both sides come from loop groups.
However, when $n$ or $k$ is odd the roots of unity are not the ones coming
from loop groups.
###### Proof 4.5.
We begin by defining a diagrammatic category $\mathcal{O}(t,w)$, and then
seeing that a certain $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$
quotient can be realised via two steps of deequivariantization in two
different ways. In the first way, after the initial deequivariantization we
obtain a category equivalent to
$\operatorname{Rep}^{\text{vector}}(U_{q_{1}}(\mathfrak{so}(n)))$, while in
the second way we obtain a category equivalent to
$\operatorname{Rep}^{\text{vector}}(U_{q_{2}}(\mathfrak{so}(k)))$ instead. The
second steps of deequivariantizations give the categories in the statement
above; since both are the modular quotient of $\mathcal{O}(t,w)$ for a certain
$t$ and $w$, they are equivalent.
###### Definition 14.
The category $\widetilde{\mathcal{O}}(t,w)$ is the idempotent completion of
the BMW category (the quotient of the tangle category by the Dubrovnik skein
relations) with
$\displaystyle a$ $\displaystyle=w^{t-1}$ $\displaystyle z$
$\displaystyle=w-w^{-1}.$
The category $\mathcal{O}(t,w)$ is the quotient of
$\widetilde{\mathcal{O}}(t,w)$ by all negligible morphisms.
Now define $w_{n,k}$ by
$w_{n,k}=\begin{cases}q_{1}&\text{if $n$ is even}\\\ q_{1}^{2}&\text{if $n$ is
odd.}\end{cases}$
Note the $w_{n,k}$ is a root of unity of order $2(n+k-2)$ when $k$ is even and
of order $n+k-2$ with $k$ is odd. The hypotheses of the theorem then ensure
that
$-w_{n,k}^{-1}=\begin{cases}q_{2}&\text{if $k$ is even}\\\ q_{2}^{2}&\text{if
$k$ is odd.}\end{cases}$
###### Lemma 15.
For $n,k\in\mathbb{N}$ and not both odd, the categories
$\mathcal{O}(n,w_{n,k})$ and $\mathcal{O}(k,-w_{n,k}^{-1})$ are equivalent.
###### Proof 4.6.
In $\mathcal{O}(k,-w_{n,k}^{-1})$, $z=-w_{n,k}^{-1}+w_{n,k}$, which is the
same value of $z$ as appears in $\mathcal{O}(n,w_{n,k})$. Similarly, in
$\mathcal{O}(k,-w_{n,k}^{-1})$, we have
$\displaystyle a$ $\displaystyle=(-w_{n,k}^{-1})^{k-1}$
$\displaystyle=\begin{cases}-w_{n,k}^{1-k}=w_{n,k}^{n+k-2+1-k}&\text{if $k$ is
even}\\\ w_{n,k}^{1-k}=w_{n,k}^{n+k-2+1-k}&\text{if $k$ is odd}\end{cases}$
$\displaystyle=w_{n,k}^{n-1}$
and so the same values of $a$ appear in both categories; thus they actually
have exactly the same definition.
###### Lemma 16.
When $n\in\mathbb{N}$, the category $\mathcal{O}(n,w)$ has a transparent
object with quantum dimension $1$, which we’ll call $\det{}_{n}$. Further, if
$w=w_{n,k}$, there is another such object $\det{}_{k}$ coming from
$\mathcal{O}(k,-w^{-1})$ via the equivalence of the previous lemma. These
transparent objects form the group
$\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}=\\{\boldsymbol{1},\det{}_{n},\det{}_{k},\det{}_{n}\otimes\det{}_{k}\\}$
under tensor product.
###### Proof 4.7.
See [4, Lemmas 4.1.ii and 4.3].
Write $\ell(q)$ for the order of a root of unity $q$, and define
$\ell^{\prime}(q)=\begin{cases}\ell(q)&\text{if $2\nmid\ell(q)$}\\\
\ell(q)/2&\text{if $2\mid\ell(q)$.}\end{cases}$
###### Lemma 17.
We can identify the deequivariantizations as
$\displaystyle\operatorname{Rep}^{\text{vector}}(U_{q}(\mathfrak{so}(n)))$
$\displaystyle\cong\begin{cases}\mathcal{O}(n,q)//\det{}_{n}&\text{if $n$ is
even}\\\ \mathcal{O}(n,q^{2})//\det{}_{n}&\text{if $n$ is odd}\end{cases}$
for any $q$, as long as if $q$ is a root of unity, when $n$ is even,
$\ell^{\prime}(q)\geq n-2$, and when $n$ is odd, $\ell^{\prime}(q)\geq 2(n-2)$
when $2\mid\ell^{\prime}(q)$ and $\ell^{\prime}(q)>n-1$ when
$2\nmid\ell^{\prime}(q)$.
In particular when $q=q_{1}$ we obtain
$\displaystyle\mathcal{O}(n,w_{n,k})//\det{}_{n}$
$\displaystyle\cong\operatorname{Rep}^{\text{vector}}(U_{q_{1}}(\mathfrak{so}(n)))$
and further, $\displaystyle\mathcal{O}(n,w_{n,k})//\det{}_{k}$
$\displaystyle\cong\operatorname{Rep}^{\text{vector}}(U_{q_{2}}(\mathfrak{so}(k)))$
Moreover, in $\mathcal{O}(n,w_{n,k})//\det{}_{n}$, we have $\det{}_{k}\cong
V_{ke_{1}}$ and in $\mathcal{O}(n,w_{n,k})//\det{}_{k}$, we have
$\det{}_{n}\cong V_{ne_{1}}$.
###### Proof 4.8.
The first equivalence follows from the main results of [40]. We give a quick
sketch of their argument. The fact that the eigenvalues of the $R$-matrix
acting on the standard representation of $\mathfrak{so}(n)$ are
$q^{-2n+2},-q^{-2}$ and $q^{2}$ when $n$ is odd, or $q^{-n+1},-q^{-1}$ and $q$
when $n$ is even ensures that this is a functor from
$\widetilde{\mathcal{O}}(n,q^{2})$ or $\widetilde{\mathcal{O}}(n,q)$, by
Theorems 3.1, 3.2 and §3.4. One then checks that the functor factors through
the deequivariantization. Finally, by computing dimensions of Hom-spaces one
concludes that the functor must kill all negligibles and must be surjective.
One can check that $\ell^{\prime}(q_{1})=n+k-2$ when $n$ is even or $2(n+k-2)$
when $n$ is odd, and so the required inequalities always hold for
$\mathfrak{so}(n)$.
The last equivalence follows from the first and Lemma 15,:
$\displaystyle\mathcal{O}(n,w_{n,k})//\det{}_{k}$
$\displaystyle\cong\mathcal{O}(k,-w_{n,k}^{-1})//\det{}_{k}$
$\displaystyle\cong\operatorname{Rep}^{\text{vector}}(U_{q_{2}}(\mathfrak{so}(k)))$
Here we check that $\ell^{\prime}(q_{2})=n+k-2$ when $k$ is even or $2(n+k-2)$
when $k$ is odd, satisfying the inequalities for $\mathfrak{so}(k)$.
The proof of the theorem is now immediate; we write
$\mathcal{O}(n,w_{n,k})//\\{\det{}_{n},\det{}_{k}\\}$ in two different ways,
obtaining
$\displaystyle\mathcal{O}(n,w_{n,k})//\\{\det{}_{n},\det{}_{k}\\}$
$\displaystyle=\mathcal{O}(n,w_{n,k})//\det{}_{n}//\det{}_{k}$
$\displaystyle\cong\operatorname{Rep}^{\text{vector}}(U_{q_{1}}(\mathfrak{so}(n)))//V_{ke_{1}}$
and $\displaystyle\mathcal{O}(n,w_{n,k})//\\{\det{}_{n},\det{}_{k}\\}$
$\displaystyle=\mathcal{O}(n,w_{n,k})//\det{}_{k}//\det{}_{n}$
$\displaystyle\cong\operatorname{Rep}^{\text{vector}}(U_{q_{2}}(\mathfrak{so}(k)))//V_{ne_{1}}.$
###### Remark 18.
One can easily verify an essential condition, that the twist factor for
$V_{ke_{1}}$ inside
$\operatorname{Rep}^{\text{vector}}(U_{\zeta(\ell_{n,k})}(\mathfrak{so}(n)))$
is $+1$, from the formulas for the twist factor given in §1.1.4.
Finally, we specialize to the case $n=3$, where the $\mathcal{D}_{2m}$ planar
algebras appear.
###### Theorem 18 ($SO(3)$-$SO(k)$ level-rank duality).
Suppose $k\geq 4$ is even. There is an equivalence of ribbon categories
$\frac{1}{2}\mathcal{D}_{k+2}\cong\operatorname{Rep}^{\text{vector}}(U_{q=-\exp\left(-\frac{2\pi
i}{2k+2}\right)}(\mathfrak{so}(k)))//V_{3e_{1}}$
sending the tensor generator $W_{2}$ of $\frac{1}{2}\mathcal{D}_{k+2}$ to
$V_{2e_{1}}$ and $P$ to $V_{2e_{\frac{k}{2}-1}}$.
This follows immediately, from the description in §2.1 of the even part of
$\mathcal{D}_{2n}$ as
$\frac{1}{2}\mathcal{D}_{2n}\cong\operatorname{Rep}^{\text{vector}}(U_{q=\exp(\frac{2\pi
i}{8n-4})}(\mathfrak{so}(3))^{modularize}$, and the general case of level-rank
duality.
### 4.4 Applications
#### 4.4.1 Knot invariants
Combining $SO(3)$-$SO(k)$ level-rank duality for even $k\geq 8$ with Kirby-
Melvin symmetry, we obtain the following knot polynomial identities:
###### Theorem 18 (Identities for $n\geq 3$).
For all knots $K$,
$\displaystyle\mathcal{J}_{\mathfrak{sl}(2),(2n-2)}(K){}_{\mid
q=\exp(\frac{2\pi i}{8n-4})}{}$
$\displaystyle=2\mathcal{J}_{\mathcal{D}_{2n},P}(K)$
$\displaystyle=2\mathcal{J}_{\mathfrak{so}(2n-2),2e_{n-2}}(K){}_{\mid
q=-\exp(-\frac{2\pi i}{4n-2})}{}$
$\displaystyle=(-1)^{1+\left\lceil\frac{n}{2}\right\rceil}2\mathcal{J}_{\mathfrak{so}(2n-2),e_{n-2}}(K){}_{\mid
q=-\exp(-\frac{2\pi i}{4n-2})}{}$ (4.1) and for all links $L$,
$\displaystyle\mathcal{J}_{\mathfrak{sl}(2),(2)}(L){}_{\mid q=\exp(\frac{2\pi
i}{8n-4})}{}$ $\displaystyle=\mathcal{J}_{\mathcal{D}_{2n},W_{2}}(L)$
$\displaystyle=\mathcal{J}_{\mathfrak{so}(2n-2),2e_{1}}(L){}_{\mid
q=-\exp(-\frac{2\pi i}{4n-2})}{}$
$\displaystyle=\mathcal{J}_{\mathfrak{so}(2n-2),e_{1}}(L){}_{\mid
q=-\exp(-\frac{2\pi i}{4n-2})}{}.$ (4.2)
(The representation of $\mathfrak{so}(2n-2)$ with highest weight $e_{n-1}$ is
one of the spinor representations.)
###### Proof 4.9.
The first two identities are immediate applications of Theorems 2.5 and 18.
For the next identity, we use the statement of Kirby-Melvin symmetry in
Theorem 8, with $A=V_{2e_{n-2}}$ and $X=V_{3e_{n-2}}$. We calculate that
$\dim{X}=(-1)^{1+\left\lceil\frac{n}{2}\right\rceil}$ by the following trick.
At $q=\exp(\frac{2\pi i}{4n-2})$, this dimension must be $+1$, since it is the
dimension of an invertible object in a unitary tensor category. At
$q=\exp(-\frac{2\pi i}{4n-2})$ it is the same, since quantum dimensions are
invariant under $q\mapsto q^{-1}$, and finally we can calculate the sign at
$q=-\exp(-\frac{2\pi i}{4n-2})$ by checking the parity of the exponents in the
Weyl dimension formula. This implies that $X\otimes X^{*}\cong V_{0}$. Using
the Racah rule, we find $A\otimes X=V_{e_{n-2}}$.
Now we do the same computation again with $A=V_{2e_{1}}$ and $X=V_{3e_{1}}$.
This case is simpler since $\dim V_{3e_{1}}=1$.
###### Remark 19.
We found keeping all the details of this theorem straight very difficult, and
we’d encourage you to wonder if we eventually got it right. We had some help,
however, in the form of computer computations. You too can readily check the
details of this theorem on small knots and links, assuming you have access to
Mathematica. Download and install the KnotTheory‘ package from
http://katlas.org. This includes with it the QuantumGroups‘ package written by
Morrison, which, although rather poorly documented, provides the function
QuantumKnotInvariant. This function can in principle compute any knot
invariant coming from an irreducible representation of a quantum group, but in
practice runs into time and memory constraints quickly. The explicit commands
for checking small cases of the above theorem are included as a Mathematica
notebook aux/check.nb with the arXiv source of this paper.
Note that the $n=5$ case of Equation (4.1) in Theorem 18 reproduces the
statement of Theorem 1.
The $n=3$ and $n=4$ cases of Theorem 18 also reproduce previous results. The
Lie algebra $\mathfrak{so}(2n-2)$ has Dynkin diagram $D_{n-1}$, with the
spinor representations corresponding to the two extreme vertices. At $n=4$,
$D_{n-1}$ becomes the Dynkin diagram $A_{3}$, and the spinor representations
become the standard and dual representations; this explains Theorem 3.5. See
Theorem 8 and Figure 10 for a full explanation.
At $n=3$, $D_{n-1}$ becomes $A_{1}\times A_{1}$, and the spinor
representations become $(\text{standard})\boxtimes(\text{trivial})$ and
$(\text{trivial})\boxtimes(\text{standard})$, giving the case described in
Theorem 3.4. See Theorem 8 and Figure 9 for a full explanation.
#### 4.4.2 Coincidences
###### Proof 4.10 (Proof of Theorem 8).
We want to construct an equivalence
$\frac{1}{2}\mathcal{D}_{6}\cong\operatorname{Rep}^{\text{uni}}{U_{s=\exp\left(\frac{7}{10}2\pi
i\right)}(\mathfrak{sl}(2)\oplus\mathfrak{sl}(2))}^{modularize},$
sending $P\mapsto V_{(1)}\boxtimes V_{(0)}$.
First, we recall that the $k=4$ case of $SO(3)$ level-rank duality (Theorem
18) gave us the equivalence
$\displaystyle\frac{1}{2}\mathcal{D}_{6}$
$\displaystyle\cong\operatorname{Rep}^{\text{vector}}U_{q=\exp(\frac{2\pi
i}{20})}(\mathfrak{so}(3))^{modularize}$
$\displaystyle\cong\operatorname{Rep}^{\text{vector}}U_{q=-\exp\left(-\frac{2\pi
i}{10}\right)}(\mathfrak{so}(4))//V_{3e_{1}}$
Since the Dynkin diagrams $D_{2}$ and $A_{1}\times A_{1}$ coincide we can
replace $\mathfrak{so}(4)$ by $\mathfrak{sl}(2)\oplus\mathfrak{sl}(2)$. The
representation $V_{3e_{1}}$ of $\mathfrak{so}(4)$ is sent to $V_{3}\boxtimes
V_{3}$, so we have
$\displaystyle\frac{1}{2}\mathcal{D}_{6}$
$\displaystyle\cong\operatorname{Rep}^{\text{vector}}U_{q=-\exp\left(-\frac{2\pi
i}{10}\right)}(\mathfrak{sl}(2)\oplus\mathfrak{sl}(2))//V_{3}\boxtimes V_{3}$
Next, by Lemma 10 we can replace this $2$-fold quotient of the vector
representations with a $4$-fold quotient of the entire representations
category of $\mathfrak{sl}(2)\oplus\mathfrak{sl}(2)$, as long as we carefully
choose $s$ and the unimodal pivotal structure. Figure 9 shows the
identification between the objects of $\frac{1}{2}\mathcal{D}_{6}$ and the
corresponding objects in the $4$-fold quotient.
$\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=198.7425pt]{./diagrams/eps/coincidences/SO4-chamber}}\end{array}$
Figure 9: The simple objects of $\frac{1}{2}\mathcal{D}_{6}$ may be
identified with the representatives of the vector representations in the
quotient $\operatorname{Rep}U_{q=-\exp\left(-\frac{2\pi
i}{10}\right)}(\mathfrak{so}(4))//V_{3e_{1}}$, via level-rank duality. We can
replace $\mathfrak{so}(4)$ here with $\mathfrak{sl}(2)\oplus\mathfrak{sl}(2)$.
The object $V_{3e_{1}}$ becomes $V_{3}\boxtimes V_{3}$. The circles above
indicate the vector representations, labelled by the corresponding objects of
$\frac{1}{2}\mathcal{D}_{6}$. Next we can apply Lemma 10 to realize
$\frac{1}{2}\mathcal{D}_{6}$ as the modularization of
$\operatorname{Rep}^{\text{uni}}{U_{s=\exp\left(\frac{7}{10}2\pi
i\right)}(\mathfrak{sl}(2)\oplus\mathfrak{sl}(2))}$. In this modularization,
we quotient out the four corner vertices. Note that $P$ is sent to
$V_{1}\boxtimes V_{0}$, and in particular the knot invariant coming from $P$
recovers a specialization of the Jones polynomial.
###### Remark 20.
Theorem 3.4 is now an immediate corollary.
###### Remark 21.
The coincidence of Dynkin diagrams $D_{2}=A_{1}\times A_{1}$ also implies that
the $D_{2}$ specialization of the Dubrovnik polynomial is equal to the square
of the Jones polynomial:
$\operatorname{Dubrovnik}(K)(q^{3},q-q^{-1})=J(K)(q)^{2}.$
This was proved by Lickorish [29, Theorem 3], without using quantum groups.
###### Corollary 22.
Looking at the object $f^{(2)}\in\frac{1}{2}\mathcal{D}_{6}$, we have
$\mathcal{J}_{\mathfrak{sl}(2),(2)}(K){}_{\mid q=\exp(\frac{2\pi
i}{20})}{}=\mathcal{J}_{\mathfrak{sl}(2),(1)}(K)^{2}{}_{\mid
q=\exp(-\frac{2\pi i}{10})}{}.$
This identity is closely related to Tutte’s golden identity, c.f. [9].
###### Proof 4.11 (Proof of Theorem 8).
We want to construct an equivalence
$\displaystyle\frac{1}{2}\mathcal{D}_{8}$
$\displaystyle\cong\operatorname{Rep}^{\text{vector}}U_{q=-\exp\left(-\frac{2\pi
i}{14}\right)}(\mathfrak{so}(6))//(V_{3e_{1}})$
$\displaystyle\cong\operatorname{Rep}^{\text{uni}}{U_{s=\exp\left(\frac{5}{14}2\pi
i\right)}(\mathfrak{sl}(4))}^{modularize},$
sending $P$ to a spinor representation of $\mathfrak{so}(6)$ and to
$V_{(100)}$, the standard representation of $\mathfrak{sl}(4)$.
The first step is the $k=6$ special case of Theorem 18 on level-rank duality.
The second step uses the coincidence of Dynkin diagrams $D_{3}=A_{3}$ to
obtain
$\operatorname{Rep}^{\text{vector}}U_{q=-\exp\left(-\frac{2\pi
i}{14}\right)}(\mathfrak{so}(6)//(V_{3e_{1}})\cong\operatorname{Rep}^{\text{vector}}U_{q=-\exp\left(-\frac{2\pi
i}{14}\right)}(\mathfrak{sl}(4))//V_{(030)}$
after which Lemma 11 gives the desired result. For more details see Figure 10.
###### Remark 23.
Theorem 3.5 is now an immediate corollary.
$\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=317.9892pt]{./diagrams/eps/coincidences/sl4-modularisation-
domain}}\end{array}$ Figure 10: We can realise $\frac{1}{2}\mathcal{D}_{8}$
as the vector representations in the $2$-fold quotient
$\operatorname{Rep}U_{q=-\exp\left(-\frac{2\pi
i}{14}\right)}(\mathfrak{sl}(4))//(V_{(030)})$, via level-rank duality and the
$A_{3}=D_{3}$ coincidence of Dynkin diagrams. The figure shows a fundamental
domain for the $2$-fold quotient. The objects of $\frac{1}{2}\mathcal{D}_{8}$
are shown circled (with fainter circles in the other domain showing their
other representatives). Now we can apply Lemma 11, and instead identify these
vector representations with representations in the $4$-fold quotient
$\operatorname{Rep}^{\text{uni}}{U_{s=\exp\left(2\pi
i\frac{5}{14}\right)}(\mathfrak{sl}(4))}^{modularize}$ of the unimodal
representation theory of $\mathfrak{sl}(4)$, at a particular choice of $s$.
These identifications are shown as arrows. Note that $P$ is sent to
$V_{(100)}$, the standard representation of $\mathfrak{sl}(4)$. In particular,
the knot invariant coming from $P$ matches up with a specialization of the
HOMFLYPT polynomial.
###### Proof 4.12 (Proof of Theorem 8).
We want to show that $\frac{1}{2}\mathcal{D}_{10}$ has an order $3$
automorphism:
$\begin{array}[]{c}\raisebox{-2.5pt}{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.90451pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\\\&&\crcr}}}\ignorespaces{\hbox{\kern-6.90451pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
6.90865pt\raise 3.76483pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@stopper}}}}}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
28.50887pt\raise 3.78268pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern
14.70622pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 28.50793pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
35.53839pt\raise-4.94655pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@stopper}}}}}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
27.26883pt\raise-24.57826pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern-3.0pt\raise-31.3475pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
8.14232pt\raise-31.3475pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{f^{(2)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
8.14467pt\raise-24.5621pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@stopper}}}}}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern-0.12143pt\raise-2.99557pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern
32.46071pt\raise-31.3475pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces}\end{array}.$
Again, we first apply the $k=8$ special case of level-rank duality (Theorem
18) to see there is a functor
$\mathcal{L}:\frac{1}{2}\mathcal{D}_{10}\xrightarrow{\cong}\operatorname{Rep}^{\text{vector}}U_{q=-\exp\left(-\frac{2\pi
i}{18}\right)}(\mathfrak{so}(8))//V_{(3000)}$
with $f^{(2)}$ corresponding to $V_{(1000)}$ and $P$ to $V_{(0002)}$. In an
exactly analogous manner as in Lemmas 10 and 11, we can identify this two-fold
quotient of the vector representations of $\mathfrak{so}(8)$ with a four-fold
quotient of all the representations in the Weyl alcove. That is, there is a
functor
$\mathcal{K}:\operatorname{Rep}^{\text{vector}}U_{q=-\exp\left(-\frac{2\pi
i}{18}\right)}(\mathfrak{so}(8))//V_{(3000)}\xrightarrow{\cong}\\\
\operatorname{Rep}^{\text{uni}}U_{q=-\exp\left(-\frac{2\pi
i}{18}\right)}(\mathfrak{so}(8))//(V_{(3000)},V_{(0030)},V_{(0003)}).$
The triality automorphism of the Dynkin diagram $D_{4}$ gives an automorphism
$T$ of this category. A direct computation shows that $T$ induces a cyclic
permutation of $P$, $Q$, and $f^{(2)}$ in $\frac{1}{2}\mathcal{D}_{10}$. For
example,
$\displaystyle\mathcal{L}^{-1}(\mathcal{K}^{-1}(T(\mathcal{K}(\mathcal{L}(f^{(2)})))))$
$\displaystyle=\mathcal{L}^{-1}(\mathcal{K}^{-1}(T(V_{(1000)})))$
$\displaystyle=\mathcal{L}^{-1}(\mathcal{K}^{-1}(V_{(0001)}))$
$\displaystyle=\mathcal{L}^{-1}(\mathcal{K}^{-1}(V_{(0001)}\otimes
V_{(0003)}))$ $\displaystyle=\mathcal{L}^{-1}(\mathcal{K}^{-1}(V_{(0002)}))$
$\displaystyle=\mathcal{L}^{-1}(V_{(0002)})$ $\displaystyle=P$
See Figures 11 and 12 for more details. It may be that this automorphism of
$\frac{1}{2}\mathcal{D}_{10}$ is related to the exceptional modular invariant
associated to $\mathfrak{sl}(2)$ at level $16$ described in [43].
$\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=317.9892pt]{./diagrams/eps/coincidences/SO8-chamber}}\end{array}$
Figure 11: We can realise $\frac{1}{2}\mathcal{D}_{10}$ as the vector
representations in the $2$-fold quotient
$\operatorname{Rep}^{\text{vector}}U_{q=-\exp\left(-\frac{2\pi
i}{18}\right)}(\mathfrak{so}(8))//(V_{(3000)})$, via level-rank duality. The
Weyl alcove for $\mathfrak{so}(8)$ at $q=-\exp\left(-\frac{2\pi i}{18}\right)$
consists of those $V_{(abcd)}$ such that $a+2b+c+d\leq 3$. In particular,
$b=0$ or $b=1$. So we draw this alcove as two tetrahedra, the $V_{\star
0\star\star}$ tetrahedron, and the $V_{\star 1\star\star}$ tetrahedron. The
vector representations are those $V_{(abcd)}$ with $c+d$ even. We show a
fundamental domain for the modularization involution $\otimes V_{(3000)}$,
which acts on the $V_{\star 0\star\star}$ tetrahedron by $\pi$ rotation about
the line joining $\frac{3}{2}000$ and $00\frac{3}{2}\frac{3}{2}$ and on the
$V_{\star 1\star\star}$ tetrahedron by $\pi$ rotation about the line joining
$\frac{1}{2}100$ and $01\frac{1}{2}\frac{1}{2}$. The tensor category of
$\frac{1}{2}\mathcal{D}_{10}$ is equivalent to the tensor subcategory of this
modularization consisting of images of vector representations, with the
equivalence sending $f^{(0)}\mapsto V_{(0000)}$, $f^{(2)}\mapsto V_{(1000)}$,
$f^{(4)}\mapsto V_{(0011)}$, $f^{(6)}\mapsto V_{(0100)}$, $P\mapsto
V_{(0020)}$ and $Q\mapsto V_{(1020)}$. The blue arrows shows the action of the
triality automorphism $V_{(abcd)}\mapsto V_{(cbda)}$ for $\mathfrak{so}(8)$ on
the image of $\frac{1}{2}\mathcal{D}_{10}$. This action is computed via the
equivalence with the $4$-fold quotient of all representations, shown in Figure
12. Notice that under this automorphism $P$ is sent to the standard
representation. In particular, the knot invariant coming from $P$ matches up
with a specialization of the Kauffman polynomial.
$\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[width=238.49341pt]{./diagrams/eps/coincidences/SO8-4fold-
quotient}}\end{array}$ Figure 12: The action of the $D_{4}$ triality
automorphism on the four-fold quotient
$\operatorname{Rep}U_{q=-\exp\left(-\frac{2\pi
i}{18}\right)}(\mathfrak{so}(8))//(V_{(3000)},V_{(0030)},V_{(0003)}).$
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This paper is available online at arXiv:1003.0022, and at
http://tqft.net/identities.
|
arxiv-papers
| 2010-02-26T21:57:31 |
2024-09-04T02:49:08.588995
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Scott Morrison, Emily Peters, Noah Snyder",
"submitter": "Scott Morrison",
"url": "https://arxiv.org/abs/1003.0022"
}
|
1003.0031
|
###### Abstract
We show that the natural scaling of measurement for a particular problem
defines the most likely probability distribution of observations taken from
that measurement scale. Our approach extends the method of maximum entropy to
use measurement scale as a type of information constraint. We argue that a
very common measurement scale is linear at small magnitudes grading into
logarithmic at large magnitudes, leading to observations that often follow
Student’s probability distribution which has a Gaussian shape for small
fluctuations from the mean and a power law shape for large fluctuations from
the mean. An inverse scaling often arises in which measures naturally grade
from logarithmic to linear as one moves from small to large magnitudes,
leading to observations that often follow a gamma probability distribution. A
gamma distribution has a power law shape for small magnitudes and an
exponential shape for large magnitudes. The two measurement scales are natural
inverses connected by the Laplace integral transform. This inversion connects
the two major scaling patterns commonly found in nature. We also show that
superstatistics is a special case of an integral transform, and thus can be
understood as a particular way in which to change the scale of measurement.
Incorporating information about measurement scale into maximum entropy
provides a general approach to the relations between measurement, information
and probability.
###### keywords:
maximum entropy; information theory; superstatistics; power law; Student’s
distribution; gamma distribution
10.3390/e12030289 12 Received: 18 January 2010; in revised form: 16 February
2010 / Accepted: 23 February 2010 / Published: 26 February 2010 Measurement
Invariance, Entropy, and Probability Steven A. Frank 1,2,⋆ and D. Eric Smith 2
E-mail: safrank@uci.edu.
## 1 Introduction
Suppose you have a ruler that is about the length of your hand. With that
ruler, you can measure the size of all the visible objects in your office.
That scaling of objects in your office with the length of the ruler means that
those objects have a natural linear scaling in relation to your ruler.
Now consider the distances from your office to various galaxies. Your ruler is
of no use, because you cannot distinguish whether a particular galaxy moves
farther away by one ruler unit. Instead, for two galaxies, you can measure the
ratio of distances from your office to each galaxy. You might, for example,
find that one galaxy is twice as far as another, or, in general, that a galaxy
is some percentage farther away than another.
Percentage changes define a ratio scale of measure, which has natural units in
logarithmic measure (Hand04Measurement, ). For example, a doubling of distance
always adds $\log(2)$ to the logarithm of the distance, no matter what the
initial distance.
Measurement naturally grades from linear at local magnitudes to logarithmic at
distant magnitudes when compared to some local reference scale. The transition
between linear and logarithmic varies between problems. Measures from some
phenomena remain primarily in the linear domain, such as measures of height
and weight in humans. Measures for other phenomena remain primarily in the
logarithmic domain, such as cosmological distances. Other phenomena scale
between the linear and logarithmic domains, such as fluctuations in the price
of financial assets (Aparicio01Empirical, ) or the distribution of income and
wealth (dragulescu01exponential, ).
The second section of this article shows how the characteristic scaling of
measurement constrains the most likely probability distribution of
observations. We use the standard method of maximum entropy to find the most
likely probability distribution (Jaynes03Science, ). But, rather than follow
the traditional approach of starting with the information in the summary
statistics of observations, such as the mean or variance, we begin with the
information in the characteristic scale of measurement. We argue that
measurement sets the fundamental nature of information and shapes the
probability distribution of observations. We present a novel extension of the
method of maximum entropy to incorporate information about the scale of
measurement.
The third section emphasizes the naturalness of the measurement scale that
grades from linear at small magnitudes to logarithmic at large magnitudes.
This linear to logarithmic scaling leads to observations that often follow a
linear-log exponential or Student’s probability distribution. A linear-log
exponential distribution is an exponential shape for small magnitudes and a
power law shape for large magnitudes. Student’s distribution is a Gaussian
shape for small fluctuations from the mean and a power law for large
fluctuations from the mean. The shapes correspond to linear scaling at small
magnitudes and logarithmic scaling at large magnitudes. Many naturally
observed patterns follow these distributions. The particular form depends on
whether the measurement scale for a problem is primarily linear, primarily
logarithmic, or grades from linear to logarithmic.
The fourth section inverts the natural linear to logarithmic scaling for
magnitudes. Because magnitudes often scale from linear to logarithmic as one
moves from small to large magnitudes, inverse measures often scale from
logarithmic to linear as one moves from small to large magnitudes. This
logarithmic to linear scaling leads to observations that often follow a gamma
probability distribution. A gamma distribution is a power law shape for small
magnitudes and an exponential shape for large magnitudes, corresponding to
logarithmic scaling at small values and linear scaling at large values. The
gamma distribution includes as special cases the exponential distribution, the
power law distribution, and the chi-square distribution, subsuming many
commonly observed patterns.
The fifth section demonstrates that the Laplace integral transform provides
the formal connection between the inverse measurement scales. The Laplace
transform, like its analytic continuation the Fourier transform, changes a
magnitude with dimension $d$ on one scale into an inverse magnitude with
dimension $1/d$ on the other scale. This inversion explains the close
association between the linear to logarithmic scaling as magnitudes increase
and the inverse scale that grades from logarithmic to linear as magnitudes
increase. We discuss the general role of integral transforms in changing the
scale of measurement. Superstatistics is the averaging of a probability
distribution with a variable parameter over a probability distribution for the
variable parameter (Beck03Superstatistics, ). We show that superstatistics is
a special case of an integral transform, and thus can be understood as a
particular way in which to change the scale of measurement.
In the sixth section, we relate our study of measurement invariance for
continuous variables to previous methods of maximum entropy for discrete
variables. We also distinguish the general definition of measurement scale by
information invariance from our particular argument about the commonness of
linear-log scales.
In the discussion, we contrast our emphasis on the primacy of measurement with
alternative approaches to understanding measurement, randomness, and
probability. One common approach changes the definition of randomness and
entropy to incorporate a change in measurement scale (Tsallis09Introduction,
). We argue that our method makes more sense, because we directly incorporate
the change in measurement scale as a kind of information, rather than alter
the definition of randomness and entropy to match each change in measurement
scale. It is measurement that changes empirically between problems rather than
the abstract meaning of randomness and information. Although we focus on the
duality between linear to logarithmic scaling and its inverse logarithmic to
linear scaling, our general approach applies to any type of measure invariance
and measurement scale.
## 2 Measurement, Information Invariance, and Probability
We derive most likely probability distributions. Our method follows the
maximum entropy approach (Jaynes57Information, ; Jaynes57Informationb, ;
Jaynes03Science, ). That approach assumes that the most likely distribution
has the maximum amount of randomness, or entropy, subject to the constraint
that the distribution must capture all of the information available to us. For
example, if we know the average value of a sample of observations, and we know
that all values from the underlying probability distribution are positive,
then all candidate probability distributions must have only positive values
and have a mean value that agrees with the average of the empirically observed
values. By maximum entropy, the most random distribution constrained to have
positive values and a fixed mean is the exponential distribution.
We express the available information by constraints. Typical constraints
include the average or variance of observations. But we must use all available
information, which may include information about the scale of measurement
itself. Previous studies have discussed how the scale of measurement provides
information. However, that aspect of maximum entropy has not been fully
developed (Jaynes03Science, ; Frank09The-common, ). Our goal is to develop the
central role of measurement scaling in shaping the commonly observed
probability distributions.
In the following sections, we show how to use information about measurement
invariances and associated measurement scales to find most likely probability
distributions.
### 2.1 Maximum entropy
The method of maximum entropy defines the most likely probability distribution
as the distribution that maximizes a measure of entropy (randomness) subject
to various information constraints. We write the quantity to be maximized as
$\Lambda={\cal{E}}-\alpha C_{0}-\sum_{i=1}^{n}\lambda_{i}C_{i}$ (1)
where ${\cal{E}}$ measures entropy, the $C_{i}$ are the constraints to be
satisfied, and $\alpha$ and the $\lambda_{i}$ are the Lagrange multipliers to
be found by satisfying the constraints. Let $C_{0}=\int p_{y}{\hbox{\rm
d}}y-1$ be the constraint that the probabilities must total one, where $p_{y}$
is the probability distribution function of $y$. The other constraints are
usually written as $C_{i}=\int p_{y}f_{i}(y){\hbox{\rm d}}y-\left\langle
f_{i}(y)\right\rangle$, where the $f_{i}(y)$ are various transformed
measurements of $y$. Angle brackets denote mean values. A mean value is either
the average of some function applied to each of a sample of observed values,
or an a priori assumption about the average value of some function with
respect to a candidate set of probability laws. If $f_{i}(y)=y^{i}$, then
$\left\langle y^{i}\right\rangle$ are the moments of the distribution—either
the moments estimated from observations or a priori values of the moments set
by assumption. The moments are often regarded as “normal” constraints,
although from a mathematical point of view, any properly formed constraint can
be used.
Here, we confine ourselves to a single constraint of measurement. We express
that constraint with a more general notation, $C_{1}=\int
p_{y}\textrm{T}[f(y)]{\hbox{\rm
d}}y-\left\langle\textrm{T}[f(y)]\right\rangle$, where $\textrm{T}()$ is a
transformation. We could, of course, express the constraining function for $y$
directly through $f(y)$. However, we wish to distinguish between an initial
function $f(y)$ that can be regarded as a normal measurement, in any sense in
which one chooses to interpret the meaning of normal, and a transformation of
normal measurements denoted by $\textrm{T}()$ that arises from information
about the measurement scale.
The maximum entropy distribution is obtained by solving the set of equations
${{\partial\Lambda}\over{\partial p_{y}}}={{\partial{\cal{E}}}\over{\partial
p_{y}}}-\alpha-\lambda\textrm{T}[f(y)]=0$ (2)
where one checks the candidate solution for a maximum and obtains $\alpha$ and
$\lambda$ by satisfying the constraint on total probability and the constraint
on $\left\langle\textrm{T}[f(y)]\right\rangle$. We assume that we can treat
entropy measures as the continuous limit of the discrete case.
In the standard approach, we define entropy by Shannon information
${\cal{E}}=-\int p_{y}\log(p_{y}){\hbox{\rm d}}y$ (3)
which yields the solution of Equation (2) as
$p_{y}=ke^{-\lambda\textrm{T}[f(y)]}$ (4)
where $k$ and $\lambda$ satisfy the two constraints.
### 2.2 Measurement and transformation
Maximum entropy, in order to be a useful method, must capture all of the
available information about a particular problem. One form of information
concerns transformations to the measurement scale that leave the most likely
probability distribution unchanged. Suppose, for example, that we obtain the
same information from measurements of $x$ and transformed measurements,
$\textrm{G}(x)$. Put another way, if one has access only to measurements on
the $\textrm{G}(x)$ scale, one has the same information that would be obtained
if the measurements were reported on the $x$ scale. We say that the
measurements $x$ and $\textrm{G}(x)$ are equivalent with respect to
information, or that the transformation $x\rightarrow\textrm{G}(x)$ is an
invariance (Hand04Measurement, ; luce08measurement, ; narens08meaningfulness,
).
To capture this information invariance in maximum entropy, we must express our
measurements on a transformed scale. In particular, we must choose the
transformation, $\textrm{T}()$, for expressing measurements so that
$\textrm{T}(x)=\gamma+\delta\textrm{T}[\textrm{G}(x)]$ (5)
for some arbitrary constants $\gamma$ and $\delta$. Putting this definition of
$\textrm{T}(x)$ into Equation (4) shows that we get the same maximum entropy
solution whether we use the direct scale $x$ or the alternative measurement
scale, $\textrm{G}(x)$, because the $k$ and $\lambda$ constants will adjust to
the constants $\gamma$ and $\delta$ so that the distribution remains
unchanged.
Given the transformation $\textrm{T}(x)$, the derivative of that
transformation expresses the information invariance in terms of measurement
invariance. In particular, we have the following invariance of the measurement
scale under a change ${\hbox{\rm d}}x$
${\hbox{\rm d}}\textrm{T}(x)\propto{\hbox{\rm d}}\textrm{T}[\textrm{G}(x)]$
(6)
We may also examine $m_{x}=\textrm{T}^{\prime}(x)={\hbox{\rm
d}}\textrm{T}(x)/{\hbox{\rm d}}x$ to obtain the change in measurement scale
required to preserve the information invariance between $x$ and
$\textrm{G}(x)$.
If we know the measurement invariance, $\textrm{G}(x)$, we can find the
correct transformation from Equation (5). If we know the transformation
$\textrm{T}(x)$, we can find $\textrm{G}(x)$ by inverting Equation (5) to
obtain
$\textrm{G}(x)=\textrm{T}^{-1}\left[{{\textrm{T}(x)-\gamma}\over{\delta}}\right]$
(7)
Alternatively, we may deduce the transformation $\textrm{T}(x)$ by examining
the form of a given probability distribution and using Equation (4) to find
the associated transformation.
In summary, $x$ and $\textrm{G}(x)$ provide invariant information, and the
transformation of measurements $\textrm{T}(x)$ captures that information
invariance in terms of measurement invariance.
### 2.3 Example: ratio and scale invariance
Suppose the information we obtain from positive-valued measurements depends
only on the ratio of measurements, $y_{2}/y_{1}$. In this particular case, all
measurements with the same ratio map to the same value, so we say that the
measurement scale has ratio invariance. Pure ratio measurements also have
scale invariance, because ratios do not depend on the magnitude or scale of
the observations.
We express the invariances that characterize a measurement scale by the
transformations that leave the information in the measurements unchanged
(Hand04Measurement, ; luce08measurement, ; narens08meaningfulness, ). If we
obtain values $x$ and use the measurement scale from the transformation
$\textrm{T}(x)=\log(x)$, the information in $x$ is the same as in
$\textrm{G}(x)=x^{c}$, because $\textrm{T}(x)=\log(x)$ and
$\textrm{T}[\textrm{G}(x)]=c\log(x)$, so in general
$\textrm{T}(x)\propto\textrm{T}[\textrm{G}(x)]$, which means that the
information in the measurement scale given by $\textrm{T}(x)$ is invariant
under the transformation $\textrm{G}(x)$.
We can express the invariance in a way that captures how measurement relates
to information and probability. The transformation $\textrm{T}(x)=\log(x)$
shrinks increments on the uniform scaling of $x$ so that each equally spaced
increment on the original uniform scale shrinks to length $1/x$ on the
transformed scale. We can in general quantify the deformation in incremental
scaling by the derivative of the transformation $\textrm{T}(x)$ with respect
to $x$. In the case of the logarithmic measurement scale with ratio
invariance, the measure invariance in Equation (6) is
${\hbox{\rm d}}\log(x)\propto{\hbox{\rm
d}}\log[\textrm{G}(x)]\Rightarrow{{1}\over{x}}\propto{{c}\over{x}}$
showing in another way that the logarithmic measure $\textrm{T}(x)$ is
invariant under the transformation $\textrm{G}(x)$. With regard to probability
or information, we can think of the logarithmic scale with ratio invariance as
having an expected density of probability per increment in proportion to
$1/x$, so that the expected density of observations at scale $x$ decreases in
proportion to $1/x$. Roughly, we may also say that the information value of an
increment decreases in proportion to $1/x$. For example, the increment length
of our hand is an informative measure for the visible objects near us, but
provides essentially no information on a cosmological scale.
If we have measurements $f(y)=y$, and we transform those measurements in a way
consistent with a ratio and scale invariance of information, then we have the
transformed measures $\textrm{T}[f(y)]=\log(y)$. The constraint for maximum
entropy corresponds to $\left\langle\log(y)\right\rangle$, which is logarithm
of the geometric mean of the observations on the direct scale $y$. Given that
constraint, the maximum entropy distribution is a power law
$p_{y}=ke^{-\lambda\textrm{T}[f(y)]}=ke^{-\lambda\log(y)}=ky^{-\lambda}$
For $y\geq 1$, we can solve for the constants $k$ and $\lambda$, yielding
$p_{y}=\delta y^{-(1+\delta)}$, with
$\delta=1/\left\langle\log(y)\right\rangle$.
## 3 The linear to Logarithmic Measurement Scale
### 3.1 Measurement
In the previous section, we obtained ratio and scale invariance with a measure
$m_{x}=\textrm{T}^{\prime}(x)\propto 1/x$. In this section, we consider the
more general measure
$m_{x}\propto{{1}\over{1+bx}}$
At small values of $x$, the measure becomes linear, $m_{x}\propto 1$, and at
large values of $x$, the measure becomes ratio invariant (logarithmic),
$m_{x}\propto 1/x$. This measure has scale dependence with ratio invariance at
large scales, because the measure changes with the magnitude (scale) of $x$,
becoming ratio invariant at large values of $x$. The parameter $b$ controls
the scale at which the measure grades between linear and logarithmic.
Given $m_{x}=\textrm{T}^{\prime}(x)$, we can integrate this deformation of
measurement to obtain the associated scale of measurement as
$\textrm{T}(x)={{1}\over{a}}\log(1+bx)=\log(1+bx)^{{1}\over{a}}\propto\log(1+bx)$
(8)
where we have expressed the proportionality constant as $1/a$ and we have
dropped the constant of integration. The expression $\log(1+bx)$ is just a
logarithmic measurement scale for positive values in relation to a fixed
origin at $x=0$, because $\log(1)=0$. The standard logarithmic expression,
$\log(x)$, has an implicit origin for positive values at $x=1$, which is only
appropriate for purely ratio invariant problems with no notion of an origin to
set the scale of magnitudes. In most empirical problems, there is some
information about the scaling of magnitudes. Thus, $\log(1+bx)$ is more often
the natural measurement scale.
Next, we seek an expression $\textrm{G}(x)$ to describe the information
invariance in the measurement scale, such that the information in $x$ and in
$\textrm{G}(x)$ is the same. The expression in Equation (6), ${\hbox{\rm
d}}\textrm{T}(x)\propto{\hbox{\rm d}}\textrm{G}[\textrm{T}(x)]$, sets the
condition for information invariance, leading to
$\textrm{G}(x)={{(1+bx)^{{1}\over{a}}-1}\over{b}}$ (9)
On the measurement scale $\textrm{T}(x)$, the information in $x$ is the same
as in $\textrm{G}(x)$, because
${\hbox{\rm d}}\textrm{T}(x)\propto{\hbox{\rm
d}}\textrm{T}[\textrm{G}(x)]\Rightarrow{{b/a}\over{1+bx}}\propto{{b/a^{2}}\over{1+bx}}$
We now use $x=f(y)$ to account for initial normal measures that may be taken
in any way we choose. Typically, we use direct values, $f(y)=y$, or squared
values, $f(y)=y^{2}$, corresponding to initial measures related to the first
and second moments—the average and variance. For now, we use $f(y)$ to hold
the place of whatever direct values we will use. Later, we consider the
interpretations of the first and second moments.
### 3.2 Probability
The constraint for maximum entropy corresponds to
$\left\langle\textrm{T}[f(y)]\right\rangle=\left\langle\log[1+bf(y)]^{{1}\over{a}}\right\rangle$,
a value that approximately corresponds to an interpolation between the linear
mean and the geometric mean of $f(y)$. Given that constraint, the maximum
entropy distribution from Equation (4) is
$p_{y}\propto[1+bf(y)]^{-\alpha}$ (10)
where $\alpha=\lambda/a$ acts as a single parameter chosen to satisfy the
constraint, and $b$ is a parameter derived from the measurement invariance
that expresses the natural scale of measurement for a particular problem.
From Equation (10), we can express simple results when in either the purely
linear or purely logarithmic regime. For small values of $bf(y)$ we can write
$p_{y}\propto e^{-\alpha bf(y)}$. For large values of $bf(y)$ we can write
$p_{y}\propto f(y)^{-\alpha}$, where we absorb $b^{-\alpha}$ into the
proportionality constant. Thus, the probability distribution grades from
exponential in $f(y)$ at small magnitudes to a power law in $f(y)$ at large
magnitudes, corresponding to the grading of the linear to logarithmic
measurement scale.
### 3.3 Transition between linear and logarithmic scales
We mentioned that one can obtain the parameter $\alpha$ in Equation (10)
directly from the constraint $\left\langle\textrm{T}[f(y)]\right\rangle$,
which can be calculated directly from observed values of the process or set by
assumption. What about the parameter $b$ that sets the grading between the
linear and logarithmic regimes?
When we are in the logarithmic regime at large values of $bf(y)$,
probabilities scale as $p_{y}\propto f(y)^{-\alpha}$ independently of $b$.
Thus, with respect to $b$, we only need to know the magnitude of observations
above which ratio invariance and logarithmic scaling become reasonable
descriptions of the measurement scale.
In the linear regime, $p_{y}\propto e^{-\alpha bf(y)}$, thus $b$ only arises
as a constant multiplier of $\alpha$ and so can be subsumed into a single
combined parameter $\beta=\alpha b$ estimated from the single constraint.
However, it is useful to consider the meaning of $b$ in the linear regime to
provide guidance for how to interpret $b$ in the mixed regime in which we need
the full expression in Equation (10).
When $f(y)=y$, the linear regime yields an exponential distribution
$p_{y}\propto e^{-\alpha by}$. In this case, $b$ weights the intensity or rate
of the process $\alpha$ that sets the scaling of the distribution.
When $f(y)=y^{2}$, the linear regime yields a Gaussian distribution
$p_{y}\propto e^{-\alpha by^{2}}$, where $2\alpha b$ is the reciprocal of the
variance that defines the precision of measurements—the amount of information
a measurement provides about the location of the average value. In this case,
$b$ weights the precision of measurement. The greater the value of $b$, the
more information per increment on the measurement scale.
### 3.4 Linear-log exponential distribution
When $f(y)=y$, we obtain from Equation (10) what we will call the linear-log
exponential distribution
$p_{y}\propto[1+by]^{-\alpha}$ (11)
for $y>0$. This distribution is often called the generalized type II Pareto
distribution or the Lomax distribution (Johnson94Distributions, ). Small
values of $by$ lead to an exponential shape, $p_{y}\propto e^{-\alpha by}$.
Large values of $by$ lead to power law tails, $p_{y}\propto y^{-\alpha}$. The
parameter $b$ determines the grading from the exponential to the power law.
Small values of $b$ extend the exponential to higher values of $y$, whereas
large values of $b$ move the extent of the power law shape toward smaller
values of $y$. Many natural phenomena follow a linear-log exponential
distribution (Tsallis09Introduction, ).
### 3.5 Student’s distribution
When $f(y)=y^{2}$, we obtain from Equation (10) Student’s distribution
$p_{y}\propto[1+by^{2}]^{-\alpha}$ (12)
Here, we assume that $y$ expresses deviations from the average. Small
deviations lead to a Gaussian shape around the mean, $p_{y}\propto e^{-\alpha
by^{2}}$. Large deviations lead to power law tails, $p_{y}\propto
f(y)^{-\alpha}$. The parameter $b$ determines the grading from the Gaussian to
the power law. Small values of $b$ expand the Gaussian shape far from the
mean, whereas large values of $b$ move the extent of the power law shape
closer to the central value at the average. Many natural phenomena expressed
as deviations from a central value follow Student’s distribution
(Tsallis09Introduction, ).
The ubiquity of both Student’s distribution and the linear-log exponential
distribution arises from the fact that the grading between linear measurement
scaling at small magnitudes and logarithmic measurement scaling at large
magnitudes is inevitably widespread. Many cases will be primarily in the
linear regime and so be mostly exponential or Gaussian except in the extreme
tails. Many other cases will be primarily in the logarithmic regime and so be
mostly power law except in the regime of small deviations near the origin or
the central location. Other cases will produce measurements across both scales
and their transition.
## 4 The Inverse Logarithmic to Linear Measurement Scale
We have argued that the linear to logarithmic measurement scale is likely to
be common. Magnitudes such as time or distance naturally grade from linear at
small scales to logarithmic at large scales.
Many problems measure inverse dimensions, such as the reciprocals of time or
distance. If magnitudes of time or space naturally grade from linear to
logarithmic as scale increases from small to large, then how do the
reciprocals scale? In this section, we argue that the inverse scale naturally
grades from logarithmic to linear as scale increases from small to large.
We first describe the logarithmic to linear measurement scale and its
consequences for probability. We then show the sense in which the logarithmic
to linear scale is the natural inverse of the linear to logarithmic scale.
### 4.1 Measurement
The transformation
$\textrm{T}(x)=x+b\log(x)$
corresponds to the change in measurement scale $m_{x}=1+b/x$. As $x$ becomes
small, the measurement scaling $m_{x}\rightarrow 1/x$ becomes the ratio-
invariant logarithmic scale. As $x$ increases, the measurement scaling
$m_{x}\rightarrow 1$ becomes the uniform measure associated with the standard
linear scale. Thus, the scaling $m_{x}=1+b/x$ interpolates between logarithmic
and linear measurements, with the weighting of the two scales shifting from
logarithmic to linear as $x$ increases from small to large values.
### 4.2 Probability
The constraint for maximum entropy corresponds to
$\left\langle\textrm{T}[f(y)]\right\rangle=\left\langle
f(y)+b\log[f(y)]\right\rangle$, a value that interpolates between the linear
mean and the geometric mean of $f(y)$. Given that constraint, the maximum
entropy distribution is $p_{y}\propto f(y)^{-\lambda b}e^{-\lambda f(y)}$,
with $\lambda$ chosen to satisfy the constraint.
The direct measure $f(y)=y$ for positive values is the gamma distribution
$p_{y}\propto y^{-\lambda b}e^{-\lambda y}$ (13)
As $y$ becomes small, the distribution approaches a power law form,
$p_{y}\propto y^{-\lambda b}$. As $y$ becomes large, the distribution
approaches an exponential form in the tails, $p_{y}\propto e^{-\lambda y}$.
Thus, the distribution grades from power law at small scales to exponential at
large scales, corresponding to the measurement scale that grades from
logarithmic to linear as magnitude increases. Larger values of $b$ extend the
power law to higher magnitudes by pushing the logarithmic to linear change in
measure to higher magnitudes. The combination of power law and exponential
shapes in the gamma distribution is the direct inverse of the linear-log
exponential distribution given in Equation (11).
The squared values $f(y)=y^{2}$, which we interpret as squared deviations from
the average value, lead to
$p_{y}\propto y^{-\lambda b}e^{-\lambda y^{2}/2}$ (14)
where the exponent of two on the first power law component is subsumed in the
other parameters. This distribution is a power law at small scales with
Gaussian tails at large scales, providing the inverse of Student’s
distribution in Equation (12). This distribution is a form of the generalized
gamma distribution (Johnson94Distributions, ), which we call the gamma-Gauss
distribution. This distribution may, for example, arise as the sum of
truncated power laws or Lévy flights (Frank09The-common, ).
## 5 Integral Transforms and Superstatistics
The previous sections showed that linear to logarithmic scaling has a simple
relation to its inverse of logarithmic to linear scaling. That simple relation
suggests that the two inverse scales can be connected by some sort of
transformation of measure. We will now show the connection.
Suppose we start with a particular measurement scale given by $\textrm{T}(x)$
and its associated probability distribution given by
$p_{x}\propto e^{-\alpha\textrm{T}(x)}$
Consider a second measurement scale $\widetilde{\textrm{T}}(\sigma)$ with
associated probability distribution
$p_{\sigma}\propto e^{-\lambda\widetilde{\textrm{T}}(\sigma)}$
What sort of transformation relates the two measurement scales?
The integral transforms often provide a way to connect two measurement scales.
For example, we could write
$p_{x}\propto\int_{\sigma^{-}}^{\sigma^{+}}p_{\sigma}g_{x|\sigma}{\hbox{\rm
d}}\sigma$ (15)
This expression is called an integral transform of $p_{\sigma}$ with respect
to the transformation kernel $g_{x|\sigma}$. If we interpret $g_{x|\sigma}$ as
a probability distribution of $x$ given a parameter $\sigma$, and $p_{\sigma}$
as a probability distribution over the variable parameter $\sigma$, then the
expression for $p_{x}$ is called a superstatistic: the probability
distribution, $p_{x}$, that arises when one starts with a different
distribution, $g_{x|\sigma}$, and averages that distribution over a variable
parameter with distribution $p_{\sigma}$ (Beck03Superstatistics, ).
It is often useful to think of a superstatistic as an integral transform that
transforms the measurement scale. In particular, we can expand Equation (15)
as
$e^{-\alpha\textrm{T}(x)}\propto\int_{\sigma^{-}}^{\sigma^{+}}e^{-\lambda\widetilde{\textrm{T}}(\sigma)}g_{x|\sigma}{\hbox{\rm
d}}\sigma$ (16)
which shows that the transformation kernel $g_{x|\sigma}$ changes the
measurement scale from $\widetilde{\textrm{T}}(\sigma)$ to $\textrm{T}(x)$. It
is not necessary to think of $g_{x|\sigma}$ as a probability distribution—the
essential role of $g_{x|\sigma}$ concerns a change in measurement scale.
The Laplace transform provides the connection between our inverse linear-
logarithmic measurement scales. To begin, expand the right side of Equation
(16) using the Laplace transform kernel $g_{x|\sigma}=e^{-\sigma x}$, and use
the inverse logarithmic to linear measurement scale,
$\widetilde{\textrm{T}}(\sigma)=\sigma+b\log(\sigma)$. Integrating from zero
to infinity yields
$e^{-\alpha\textrm{T}(x)}\propto(1+x/\lambda)^{b\lambda-1}$
with the requirement that $b\lambda<1$. From this, we have
$\textrm{T}(x)\propto\log(1+x/\lambda)$, which is the linear to logarithmic
scale. Thus, the Laplace transform inverts the logarithmic to linear scale
into the linear to logarithmic scale. The inverse Laplace transform converts
in the other direction.
If we use $x=y$, then the transform relates the linear-log exponential
distribution of Equation (11) to the gamma distribution of Equation (13). If
we use $x=y^{2}$, then the transform relates Student’s distribution of
Equation (12) to the gamma-Gauss distribution of Equation (14).
The Laplace transform inverts the measurement scales. This inversion is
consistent with a common property of Laplace transforms, in which the
transform inverts a measure with dimension $d$ to a measure with dimension
$1/d$. One sometimes interprets the inversion as a change from a direct
measure to a rate or frequency. Here, it is only the inversion of dimension
that is significant. The inversion arises because, in the transformation
kernel $g_{x|\sigma}=e^{-\sigma x}$, the exponent $\sigma x$ is typically non-
dimensional, so that the dimensions of $\sigma$ and $x$ are reciprocals of
each other. The transformation takes a distribution in $\sigma$ given by
$p_{\sigma}$ and returns a distribution in $x$ given by $p_{x}$. Thus, the
transformation typically inverts the dimension.
## 6 Connections and Caveats
### 6.1 Discrete versus continuous variables
We used measure invariance to analyze maximum entropy for continuous
variables. We did not discuss discrete variables, because measure invariance
applied to discrete variables has been widely and correctly used in maximum
entropy (Jaynes03Science, ; Sivia06Tutorial, ; Frank09The-common, ). In the
Discussion, we describe why previous attempts to apply invariance to
continuous variables did not work in general. That failure motivated our
current study.
Here, we briefly review measure invariance in the discrete case for comparison
with our analysis of continuous variables. We use the particular example of
$N$ Bernoulli trials with a sample measure of the number of successes
$y=0,1,\ldots,N$. Frank (Frank09The-common, ) describes the measure invariance
for this case: “How many different ways can we can obtain $y=0$ successes in
$N$ trials? Just one: a series of failures on every trial. How many different
ways can we obtain $y=1$ success? There are $N$ different ways: a success on
the first trial and failures on the others; a success on the second trial, and
failures on the others; and so on. The uniform solution by maximum entropy
tells us that each different combination is equally likely. Because each value
of $y$ maps to a different number of combinations, we must make a correction
for the fact that measurements on $y$ are distinct from measurements on the
equally likely combinations. In particular, we must formulate a measure…that
accounts for how the uniformly distributed basis of combinations translates
into variable values of the number of successes, $y$. Put another way, $y$ is
invariant to changes in the order of outcomes given a fixed number of
successes. That invariance captures a lack of information that must be
included in our analysis.”
In this particular discrete case, transformations of order do not change our
information about the total number of successes. Our measurement scale
expresses that invariance, and that invariance is in turn captured in the
maximum entropy distribution.
The nature of invariance is easy to see in the discrete case by combinatorics.
The difficulty in past work has been in figuring out exactly how to capture
the same notion of invariance in the continuous case. We showed that the
answer is perhaps as simple as it could be: use the transformations that do
not change information in the context of a particular problem. Jaynes
(Jaynes03Science, ) hinted at this approach, but did not develop and apply the
idea in a general way.
### 6.2 General measure invariance versus particular linear-log scales
Our analysis followed two distinct lines of argument. First, we presented the
general expression for invariance as a form of information in maximum entropy.
We developed that expression particularly for the case of continuous
variables. The general expression sets the conditions that define measurement
scales and the relation between measurement and probability. But the general
expression does not tell us what particular measurement scale will arise in
any problem.
Our second line of argument claimed that various types of grading between
linear and logarithmic measures arise very commonly in natural problems. Our
argument for commonness is primarily inductive and partially subjective. On
the inductive side, the associated probability distributions seem to be those
most commonly observed in nature. On the subjective side, the apparently
simplest assumptions about invariance lead to what we called the common
gradings between linear and logarithmic scales. We do not know of any way to
prove commonness or naturalness. For now, we are content that the general
mathematical arguments lead in a simple way to those probability distributions
that appear to arise commonly in nature.
A different view of what is common or what is simple would of course lead to
different information invariances, measurement scales, and probability
distributions. In that case, our general mathematical methods would provide
the tools by which to analyze the alternative view.
## 7 Discussion
We developed four topics. First, we provided a new extension to the method of
maximum entropy in which we use the measurement scale as a primary type of
information constraint. Second, we argued that a measurement scale that grades
from linear to logarithmic as magnitude increases is likely to be very common.
The linear-log exponential and Student’s distributions follow immediately from
this measurement scale. Third, we showed that the inverse measure that grades
from logarithmic at small scales to linear at large scales leads to the gamma
and gamma-Gauss distributions. Fourth, we demonstrated that the two
measurement scales are natural inverses related by the Laplace integral
transform. Superstatistics are a special case of integral transforms and can
be understood as changes in measurement scale.
In this discussion, we focus on measurement invariance, alternative
definitions of entropy, and maximum entropy methods.
Jaynes (Jaynes03Science, ) summarized the problem of incorporating measurement
invariance as a form of information in maximum entropy. The standard
conclusion is that one should use relative entropy to account for measurement
invariance. In our notation, for a measurement scale $\textrm{T}(y)$ with
measure deformation $m_{y}=\textrm{T}^{\prime}(y)$, the form of relative
entropy is the Kullback-Leibler divergence
${\cal{E}}=-\int p_{y}\log\left({{p_{y}}\over{m_{y}}}\right){\hbox{\rm d}}y$
in which the $m_{y}$ is proportional to a prior probability distribution that
incorporates the information from the measurement scale and leads to the
analysis of maximum relative entropy. This approach works in cases where the
measure change, $m_{y}$, is directly related to a change in the probability
measure. Such changes in probability measure typically arise in combinatorial
problems, such as a type of measurement that cannot distinguish between the
order of elements in sets.
For continuous deformations of the measurement scale, using $m_{y}$ as a
relative scaling for probability does not always give the correct answer. In
particular, if one uses the constraint $\left\langle f(y)\right\rangle$ and
the measure $m_{y}$ in the above definition of relative entropy, the maximum
relative entropy gives the probability distribution
$p_{y}\propto m_{y}e^{-\lambda f(y)}$
which is often not the correct result. Instead, the correct result follows
from the method we gave, in which the information from measurement invariance
is incorporated by transforming the constraint as
$\left\langle\textrm{T}[f(y)]\right\rangle$, yielding the maximum entropy
solution
$p_{y}\propto e^{-\lambda\textrm{T}[f(y)]}$
It is possible to change the definition of entropy, such that maximum entropy
applied to the transformed measure of entropy plus the direct constraint
$\left\langle f(y)\right\rangle$ gives the correct answer
(Tsallis09Introduction, ). The resulting probability distributions are of
course the same when transforming the constraint or using an appropriate
matching transformation of the entropy measure. We discuss the mathematical
relation between the alternative transformations in a later paper.
We prefer the transformation of the constraint, because that approach directly
shows how information about measurement alters scale and is incorporated into
maximum entropy. By contrast, changing the definition of entropy requires each
measurement scale to have its own particular definition of entropy and
information. Measurement is inherently an empirical phenomenon that is
particular to each type of problem and so naturally should be changed as the
nature of the problem changes. The abstract notions of entropy and information
are not inherently empirical factors that change among problems, so it seems
perverse to change the definition of randomness with each transformation of
measurement.
The Tsallis and Rényi entropy measures are transformations of Shannon entropy
that incorporate the scaling of measurement from linear to logarithmic as
magnitude increases (Tsallis09Introduction, ). Those forms of entropy
therefore could be used as a common alternative to Shannon entropy whenever
measurements scale linearly to logarithmically. Although mathematically
correct, such entropies change the definition of randomness to hide the
particular underlying transformation of measurement. That approach makes it
very difficult to understand how alternative measurement scales alter the
expected types of probability distributions.
## 8 Conclusions
Linear and logarithmic measurements seem to be the most common natural scales.
However, as magnitudes change, measurement often grades between the linear and
logarithmic scales. That transition between scales is often overlooked. We
showed how a measurement scale that grades from linear to logarithmic as
magnitude increases leads to some of the most common patterns of nature
expressed as the linear-log exponential distribution and Student’s
distribution. Those distributions include the exponential, power law, and
Gaussian distributions as special cases, and also include hybrids between
those distribution that must commonly arise when measurements span the linear
and logarithmic regimes.
We showed that a measure grading from logarithmic to linear as magnitude
increases is a natural inverse scale. That measurement scale leads to the
gamma and gamma-Gauss distributions. Those distributions are also composed of
exponential, power law, and Gaussian components. However, those distributions
have the power law forms at small magnitudes corresponding to the logarithmic
measure at small magnitudes, whereas the inverse scale has the power law
components at large magnitudes corresponding to the logarithmic measure at
large magnitudes.
The two measurement scales are natural inverses connected by the Laplace
transform. That transform inverts the dimension, so that a magnitude of
dimension $d$ on one scale becomes an inverse magnitude of dimension $1/d$ on
the other scale. Inversion connects the two major scaling patterns commonly
found in nature. Our methods of incorporating information about measurement
scale into maximum entropy also apply to other forms of measurement scaling
and invariance, providing a general method to study the relations between
measurement, information, and probability.
## Acknowledgements
SAF is supported by National Science Foundation grant EF-0822399, National
Institute of General Medical Sciences MIDAS Program grant U01-GM-76499, and a
grant from the James S. McDonnell Foundation. DES thanks Insight Venture
Partners for support.
## References
* (1) Hand, D. Measurement Theory and Practice; Arnold: London, UK, 2004.
* (2) Aparicio, F.; Estrada, J. Empirical distributions of stock returns: European securities markets, 1990-95. Eur. J. Financ. 2001, 7, 1–21.
* (3) Dragulescu, A.A.; Yakovenko, V.M. Exponential and power-law probability distributions of wealth and income in the United Kingdom and the United States. Physica A 2001, 299, 213–221.
* (4) Jaynes, E.T. Probability Theory: The Logic of Science; Cambridge University Press: New York, NY, USA, 2003.
* (5) Beck, C.; Cohen, E. Superstatistics. Physica A: Statist. Mech. Appl. 2003, 322, 267–275.
* (6) Tsallis, C. Introduction to Nonextensive Statistical Mechanics; Springer: New York, NY, USA, 2009.
* (7) Jaynes, E.T. Information theory and statistical mechanics. Phys. Rev. 1957, 106, 620–630.
* (8) Jaynes, E.T. Information theory and statistical mechanics, II. Phys. Rev. 1957, 108, 171–190.
* (9) Frank, S.A. The common patterns of nature. J. Evol. Biol. 2009, 22, 1563–1585.
* (10) Luce, R.D.; Narens, L. Measurement, theory of. In The New Palgrave Dictionary of Economics; Durlauf, S.N., Blume, L.E., Eds.; Palgrave Macmillan: Basingstoke, UK, 2008.
* (11) Narens, L.; Luce, R.D. Meaningfulness and invariance. In The New Palgrave Dictionary of Economics; Durlauf, S.N., Blume, L.E., Eds.; Palgrave Macmillan: Basingstoke, UK, 2008.
* (12) Johnson, N.L.; Kotz, S.; Balakrishnan, N. Continuous Univariate Distributions, 2nd ed.; Wiley: New York, NY, USA, 1994.
* (13) Sivia, D.S.; Skilling, J. Data Analysis: A Bayesian Tutorial; Oxford University Press: New York, NY, USA, 2006.
|
arxiv-papers
| 2010-02-26T22:49:24 |
2024-09-04T02:49:08.631872
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Steven A. Frank and D. Eric Smith",
"submitter": "Steven Frank",
"url": "https://arxiv.org/abs/1003.0031"
}
|
1003.0111
|
# Anomalous Valley Magnetic Moment of Graphene
Daqing Liu, Shengli Zhang, Erhu Zhang, Ning Ma, Huawei Chen
Department of Applied Physics, Non-equilibrium Condensed Matter and Quantum
Engineering Laboratory,
Key Laboratory of Ministry of Education, Xi’an Jiaotong University, Xi’an,
710049, China
###### Abstract
Carrier interactions on graphene are studied. The study shows that besides the
well known Coulomb repulsion between carriers, there also exist four-fermion
interactions associated with U-process, one of which attracts carriers in
different valleys. We then calculate the contributions to valley magnetic
moment from vertex correction and from four-fermion corrections explicitly.
The relative contributions are -18% and 3% respectively. At last we point out
that we can mimic heavy quarkonium system by carrier interactions in graphene.
## 1 Introduction
Graphene [1], newly discovered two-dimensional crystals, has attracted more
and more attentions of theorists and experimentalists[2, 3, 4, 5]. In
graphene, there is a typical valley degeneracy, corresponding to the presence
of two different valleys in the band structure. However, as stated in the
reference [6], such degeneracy makes it difficult to observe the intrinsic
physics of a single valley in experiments[7, 8]. How to distinguish carriers
in the two valleys is therefore always a topic attracting literatures[6, 11,
10, 9].
Ref. [9] pointed out that in close analogy with the spin degree, there is an
intrinsic magnetic moment associated with the valley index, which was called
as valley magnetic moment (VMM). At tree level the valley magnetic moment is
about 30 times that of the usual spin magnetic moment, therefore,
”valleytronics” provides a new and much more standard pathway to potential
applications in a broad class of semiconductors as compared with the novel
valley device in graphene nanoribbon[10]. However, since in graphene the
effective coupling $\frac{e^{2}}{\varepsilon\hbar v}\sim 1$, a question to be
posed is to what extent the calculation in ref. [9] is valid.
To answer the question, we first study carrier interactions. The study shows
that, in tight binding approximation, besides the well-known Coulomb repulsion
between electrons, there are also four-fermion interactions associated with
U-process. The four-fermion interactions are type dependent and more
significant, one of them attracts carriers in different valleys. Armed with
the understanding of the interactions, we point out that there are two
corrections to VMM at one-loop level. One is the vertex correction and the
other is the four-fermion correction. The vertex correction is similar to the
anomalous magnetic moment of a particle in quantum electrodynamics (QED)
except that carrier interactions on graphene are not ”Lorentz covariant”.
Therefore, such correction always appears even in a one-valley system.
Meanwhile, the valley degree is similar to the flavor degree in particle
physics or high-energy physics. To compute anomalous magnetic moment of a
particle due to flavor degree, one should also consider the weak interaction,
an interaction between flavor degree. Our Yakawa-like four-fermion
interactions are similar to the lower-energy approximation of the weak
interactions. In this way, the correction to VMM due to valley degree appears
at one-loop level. In contrast, such correction can not occur in QED.
Our study shows that the total correction is about $-15\%$. Furthermore, since
the corrections are independent on the divergence of the loop calculations,
VMM can be used to check the validity of the perturbational calculation.
## 2 Carrier interactions
Here we study carrier interactions. The study shows that besides the well
known Coulomb repulsion, there are also four-fermion interactions between
carriers at different valleys, which are not only short-range but also
contacting ones.
For simplicity, we set $\hbar\equiv 1$ and
$X({\mathbf{r}}-{\mathbf{r}}_{A^{\prime}})$ the normalized orbital $p_{z}$
wave function of electron bound to atom $A^{\prime}$, i.e. it satisfies $\int
d{\mathbf{r}}X({\mathbf{r}}-{\mathbf{r}}_{A^{\prime}})X({\mathbf{r}}-{\mathbf{r}}_{A^{\prime\prime}})=\delta_{A^{\prime}A^{\prime\prime}}$
[12]. A-electron wave function $\psi_{A}({\mathbf{k}})$ in position space is
$\psi_{A}({\mathbf{k}})=\sqrt{\omega}/(2\pi)\sum\limits_{A}\textrm{e}^{\textrm{i}{\mathbf{k}\cdot{\mathbf{r}}_{A}}}X({\mathbf{r}}-{\mathbf{r}}_{A})$,
where $\omega=\sqrt{3}a^{2}/2$ is the area of the hexagonal cell. For
B-electron the case is similar. We then have
$<\psi_{A^{\prime}_{0}}({\mathbf{k}}^{\prime})|\psi_{A_{0}}({\mathbf{k}})>=\delta_{A_{0}A_{0}^{\prime}}\delta({\mathbf{k}}-{\mathbf{k}^{\prime}})$,
where $A_{0},A_{0}^{\prime}=A$ or $B$.
To study carrier interactions, we consider
$V({\bf k})=\frac{(2\pi)^{4}}{N\omega}\psi_{A_{2}^{\prime}}^{*}({\bf
k}_{2}+{\bf k})\psi_{A_{2}}({\bf k}_{2})\hat{V}\psi_{A_{1}^{\prime}}^{*}({\bf
k}_{1}-{\bf k})\psi_{A_{1}}({\bf k}_{1}),$ (1)
where $A_{1},\,A_{1}^{\prime},\,A_{2},\,A_{2}^{\prime}$ equal to $A$ or $B$.
If we ignore interchange interactions, the main contribution to
$X^{*}({\mathbf{r}}^{\prime}-{\mathbf{r}_{A_{2}^{\prime}}})X({\mathbf{r}}^{\prime}-{\mathbf{r}_{A_{2}}})\hat{V}X^{*}({\mathbf{r}}-{\mathbf{r}_{A_{1}^{\prime}}})X({\mathbf{r}}-{\mathbf{r}_{A_{1}}})$
should be at vicinity ${\mathbf{r}}_{A_{1}^{\prime}}={\mathbf{r}}_{A_{1}}$,
${\mathbf{r}}_{A_{2}^{\prime}}={\mathbf{r}}_{A_{2}}$,
${\mathbf{r}}^{\prime}\approx{\mathbf{r}}_{A_{2}}$ and
${\mathbf{r}}\approx{\mathbf{r}}_{A_{1}}$. We get
$\displaystyle V({\bf k})$ $\displaystyle=$
$\displaystyle\delta_{A_{1}A_{1}^{\prime}}\delta_{A_{2}A_{2}^{\prime}}\sum\limits_{A_{1}A_{2}}e^{\textrm{i}{\bf
k}({\mathbf{r}}_{A_{1}}-{\mathbf{r}}_{A_{2}})}\frac{\omega
e^{2}}{N|{\mathbf{r}}_{A_{1}}-{\mathbf{r}}_{A_{2}}|}$ (2) $\displaystyle=$
$\displaystyle\delta_{A_{1}A_{1}^{\prime}}\delta_{A_{2}A_{2}^{\prime}}\sum\limits_{A_{2}}e^{\textrm{i}{\bf
k}({\mathbf{r}}_{A_{1}}-{\mathbf{r}}_{A_{2}})}\frac{\omega
e^{2}}{|{\mathbf{r}}_{A_{1}}-{\mathbf{r}}_{A_{2}}|},$
where we have fixed $A_{1}$ in the last step. We shall ignore the two delta
functions thereinafter.
Figure 1: Graphene hexagonal lattice constructed as a superposition of two
triangular lattices A and B, with bases vectors
${\mathbf{a}}_{1}=a(\frac{1}{2},\frac{\sqrt{3}}{2})$ and
${\mathbf{a}}_{2}=a(\frac{1}{2},-\frac{\sqrt{3}}{2})$, where lattice constant
$a=|{\mathbf{a}}_{1}|=|{\mathbf{a}}_{2}|$. The reciprocal lattice vectors are
${\mathbf{b}}_{1}=\frac{2\pi}{a}(1,\frac{1}{\sqrt{3}})$ and
${\mathbf{b}}_{2}=\frac{2\pi}{a}(1,-\frac{1}{\sqrt{3}})$ respectively.
Without loss of generality, we set $A_{1}=A$. To compute interactions between
carriers, we first mark coordinates of $A$ and $B$ with two integers $n_{1}$
and $n_{2}$. From Fig.2, the coordinate of one atom A is set as $(0,0)$. Then,
for infinitely large graphene, coordinates of atom $A$ are depicted as
$(n_{1}/2,\,\sqrt{3}(n_{1}-2n_{2})/2)a$ and coordinates of atoms $B$
$(n_{1}/2,\,\sqrt{3}(n_{1}-2n_{2}+\frac{2}{3})/2)a$ respectively, where $a$ is
lattice constant and $n_{1}$, $n_{2}$ are arbitrary integers.
We put our focus on the interactions between electrons around $\pm{\bf
K}=\pm(4\pi/3a,0)$. We first study the case where there is no valley-valley
transition during interactions. For this case, we suppose $|{\mathbf{k}}a|\ll
1$. In Eq. (2) the function in the summation is a slow-moving function,
therefore, the summation can be replaced by an integral,
$\displaystyle V_{c}({\bf k})\approx e^{2}\int
d{\mathbf{r}}\frac{e^{\textrm{i}{\bf
k}\cdot{\mathbf{r}}}}{|{\mathbf{r}}|}=\frac{2\pi e^{2}}{\varepsilon k}.$ (3a)
where we have inserted the effective permittivity $\varepsilon$ in the last
equation to include screening effect. We thus obtain the well known Coulomb
interaction. The type of $A_{2}$ does not influence the results, that is, the
Coulomb repulsion works both for carriers in the same valley and for carriers
in the different valleys.
Besides the well known Coulomb interactions, there are other interactions
which is related to valley-valley transition. Such interactions correspond to
a U-process and therefore ${\mathbf{k}}\approx\pm({4\pi/3a},0)$. To deal with
such case, we substitute ${\mathbf{k}}+(\frac{4\pi}{3a},0)$ for ${\mathbf{k}}$
in Eq. (2).
We first consider A-B interactions, that is, $A_{2}=B$ in Eq. (2). We get then
$\displaystyle V_{d}({\mathbf{k}})\approx e^{2}\sum\limits_{n_{1}\neq
0}\frac{a}{2}e^{-\textrm{i}k_{x}{n_{1}a\over 2}-\textrm{i}{2\pi\over
3}n_{1}}2K_{0}(|k_{y}\frac{n_{1}a}{2}|)$
$\displaystyle+e^{2}\frac{a}{2}\sum\limits_{n_{2}}\frac{e^{\textrm{i}k_{y}\sqrt{3}a(n_{2}-\frac{1}{3})}}{|n_{2}-\frac{1}{3}|}\approx
0.1e^{2}a.$ (3b)
Compared to the long-wavelength result in Eq. (3a), the valley-valley
interaction suffers a coefficient suppression due to the large momentum
transfer. However, since such valley-valley interactions are short-range, it
is not needed to consider screening effect. We therefore does not insert the
effective permittivity $\varepsilon$ in the above equation.
Whereas when $A_{2}=A$, one should subtract the contribution from self
interactions, which corresponds to $(n_{1},n_{2})=(0,0)$ in Eq. (2). The
result is then
$\displaystyle V_{s}({\mathbf{k}})\approx-1.55e^{2}a.$ (3c)
Here, the large negative coefficient $-1.55$ is due to the subtraction.
Since Coulomb interaction is long-range, it does not depend on the
distributing detail of the adjoint electrons. Thus, as shown in Eq. (3a), such
interaction is type-independent. However, the four-fermion form of the
U-process implies that such interactions are short-range and they therefore
depend on the distributing detail. Therefore, as shown by (3b) and (3c), such
interactions are type-dependent. Reference [13, 14] also proposed four-fermion
interactions from different aspects. In Ref. [13] the authors add a near-
neighbor interaction term and then, when they carry out momentum integral in
the first Brillouin zone, they adhere the short-range interaction with the
usual Coulomb interaction at
$|{\mathbf{k}}|=\frac{1}{2}\frac{\pi}{\sqrt{3}a}$, where ${\mathbf{k}}$ is the
transfer momentum. In contrast, in our approach, there is no artificial
adhering and the interactions due to the valley transition are shown
explicitly. Furthermore, the results obtained by our approach are suitable to
take the quantum field theory calculations.
We emphasize that, besides the vertex correction, $V_{s}$ also contributes to
VMM. Furthermore, since $V_{s}<0$, it takes attracting force between electrons
in different valleys. The interaction may play crucial role in superconduction
phenomena [15]. Therefore, $V_{s}$ deserves further research.
## 3 The formal development of Lagrangian
We first define two two-component spinors $\varphi$ and $\chi$ as follows:
$\varphi=\left(\begin{array}[]{c}a_{\bf K}(\bf p)\\\ b_{\bf K}(\bf p)\\\
\end{array}\right)$ and $\chi=\left(\begin{array}[]{c}-\textrm{i}b_{\bf-K}(\bf
p)\\\ \textrm{i}a_{\bf-K}(\bf p)\\\ \end{array}\right)$, where ${\bf\pm K}$
are two valleys. To describe the graphene dynamics in field theory language,
we read the Lagrangian,
$\displaystyle\mathcal{L}_{0}$ $\displaystyle=$
$\displaystyle\bar{\varphi}(\textrm{i}\gamma^{0}{\partial_{t}}+\textrm{i}v{\bf\gamma}\cdot\nabla-m)\varphi-e\bar{\varphi}(\gamma^{0}A^{0}-\beta{\mathbf{\gamma}}\cdot{\mathbf{A}})\varphi$
(4)
$\displaystyle+\bar{\chi}(\textrm{i}\gamma^{0}{\partial_{t}}+\textrm{i}v\mbox{{$\gamma$}}\cdot\nabla+m)\chi-e\bar{\chi}(\gamma^{0}A^{0}-\beta{\mathbf{\gamma}}\cdot{\mathbf{A}})\chi$
$\displaystyle-\frac{\lambda_{1}}{2}\bar{\varphi}\gamma^{1}\chi\bar{\chi}\gamma_{1}\varphi-\frac{\lambda_{2}}{2}\bar{\varphi}\gamma^{2}\chi\bar{\chi}\gamma_{2}\varphi,$
where $\beta={v/c}$, $v$ is the Fermi velocity of carriers, $c$ is the
effective light velocity in graphene, $\lambda_{1}=-(V_{s}-V_{d})/2$ and
$\lambda_{2}=-(V_{s}+V_{d})/2$. We also set three gamma matrices as
$\gamma^{0}=\gamma_{0}=\sigma_{3}=\left(\begin{array}[]{cc}1&0\\\ 0&-1\\\
\end{array}\right)$,
$\gamma^{1}=g^{11}\gamma_{1}=-\gamma_{1}=\gamma^{0}\sigma_{1}=\left(\begin{array}[]{cc}0&1\\\
-1&0\\\ \end{array}\right)$ and
$\gamma^{2}=g^{22}\gamma_{2}=-\gamma_{2}=\gamma^{0}\sigma_{2}=\left(\begin{array}[]{cc}0&-i\\\
-i&0\\\ \end{array}\right)$, where $\sigma_{i}$’s are three Pauli matrices and
metric matrix $g^{\mu\nu}=diag\\{1,-1,-1\\}$. Since four-fermion interactions
in Eqs. (3b) and (3c) are contacting ones, it is not necessary to introduce
corresponding gauge field. In the above equation the energy gap $m$ can be
used to improve the use of graphene in making transistors and is therefore the
one of the hot spots of literatures. In Ref. [16], the authors investigate the
energy gap of graphene on a substrate BN, which is generated by the breaking
of the A-B sublattice symmetry. However, such energy gap has not been observed
up to now. In Ref. [17] the authors report that single layer graphene on SiC
has a gap of 0.26eV, but the result is under debate [18, 19].
Utilizing the definitions of $\gamma$ matrix and boost generators
corresponding to ”Lorentz” transformation we find positive solution and
negative solution for $\varphi$ field (or negative solution and positive
solution for $\chi$ field) as
$u({\mathbf{p}})=(\sqrt{p^{0}+m},{v(p^{1}+\textrm{i}p^{2})/\sqrt{p^{0}+m}})^{T}$
and
$v({\mathbf{p}})=({v(p^{1}-\textrm{i}p^{2})/\sqrt{p^{0}+m}},\sqrt{p^{0}+m})^{T}$
respectively. They meet
$\tilde{p}\\!\\!\\!/u(p)=mu(p),\,\bar{v}(p^{\prime})\tilde{p}\\!\\!\\!/^{\prime}=-\bar{v}(p^{\prime})m$,
where $\tilde{p}\\!\\!\\!/=\gamma_{\mu}\tilde{p}^{\mu}$ and
$\tilde{p}=(p^{0},\,v{\mathbf{p}})$.
However, the Lagrangian in Eq. (4) is a bare one and it needs renormalization
to match the observable quantities[20]. Having set
$\varphi=Z^{1/2}_{2}\varphi_{r},\,\chi=Z^{1/2}_{2}\chi_{r}$ and
$A=Z^{1/2}_{3}A_{r}$, where $\varphi_{r}$, $\chi_{r}$ and $A_{r}$ are
renormalized quantities, we split each term of the Lagrangian into two pieces
as follows:
$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\bar{\varphi}_{r}(\textrm{i}\gamma^{0}{\partial_{t}}+\textrm{i}v_{r}{\bf\gamma}\cdot\nabla-
m_{r})\varphi_{r}$ (5) $\displaystyle-
e_{r}\bar{\varphi}_{r}(\gamma^{0}A^{0}_{r}-\beta_{r}{\mathbf{\gamma}}\cdot{\mathbf{A}}_{r})\varphi_{r}$
$\displaystyle+\bar{\chi}_{r}(\textrm{i}\gamma^{0}{\partial_{t}}+\textrm{i}v_{r}\gamma\cdot\nabla+m_{r})\chi_{r}$
$\displaystyle-
e_{r}\bar{\chi}_{r}(\gamma^{0}A^{0}_{r}-\beta_{r}{\mathbf{\gamma}}\cdot{\mathbf{A}}_{r})\chi_{r}$
$\displaystyle-\frac{\lambda_{1r}}{2}\bar{\varphi}_{r}\gamma^{1}\chi_{r}\bar{\chi}_{r}\gamma_{1}\varphi_{r}-\frac{\lambda_{2r}}{2}\bar{\varphi}_{r}\gamma^{2}\chi_{r}\bar{\chi}_{r}\gamma_{2}\varphi_{r}$
$\displaystyle+\bar{\varphi}_{r}(\textrm{i}\delta_{2}\gamma^{0}\partial_{t}+i\delta_{v}v_{r}{\mathbf{\gamma}}\cdot\nabla-\delta_{m})\varphi_{r}$
$\displaystyle+\bar{\chi}_{r}(\textrm{i}\delta_{2}\gamma^{0}\partial_{t}+i\delta_{v}v_{r}{\mathbf{\gamma}}\cdot\nabla+\delta_{m})\chi_{r}$
$\displaystyle-
e_{r}\bar{\varphi}_{r}(\delta_{1}\gamma^{0}A^{0}_{r}-\beta_{r}\delta_{c}\gamma\cdot{\mathbf{A}}_{r})\varphi_{r}$
$\displaystyle-
e_{r}\bar{\chi}_{r}(\delta_{1}\gamma^{0}A^{0}_{r}-\beta_{r}\delta_{c}\gamma\cdot{\mathbf{A}}_{r})\chi_{r}$
$\displaystyle-\frac{\delta_{1\lambda}}{2}\bar{\varphi}_{r}\gamma^{1}\chi_{r}\bar{\chi}_{r}\gamma_{1}\varphi_{r}-\frac{\delta_{2\lambda}}{2}\bar{\varphi}_{r}\gamma^{2}\chi_{r}\bar{\chi}_{r}\gamma_{2}\varphi_{r},$
where counterterm coefficients $\delta_{2}=Z_{2}-1$,
$\delta_{m}=Z_{2}m-m_{r}$, $\delta_{v}=Z_{2}v/v_{r}-1$,
$\delta_{1}=Z_{2}Z_{3}^{1/2}e/e_{r}-1\equiv Z_{1}-1$,
$\delta_{c}=Z_{1}\beta/\beta_{r}-1$,
$\delta_{1\lambda}=Z_{2}^{2}\lambda_{1}-\lambda_{1r}$ and
$\delta_{2\lambda}=Z_{2}^{2}\lambda_{2}-\lambda_{2r}$ are determined by
renormalized conditions.
Since all the quantities in the following are renormalized ones, all the
subscripts $r$ will be dropped out.
## 4 Calculation of VMM
To compute the VMM we first show Feynman rules in Fig. 4 (a)-(e). The
contribution to VMM up to $e^{2}$ is depicted by Fig.4(g), two diagrams in
Fig.4(h), which is denoted by $\delta\Gamma_{l}^{\mu}$, $l=1,\,2$, Fig.4(i),
which is denoted by $\Gamma^{\mu}$, and the counterterm, Fig.4)f). The
scattering amplitude of carrier under external gauge field is
$\textrm{i}\mathcal{M}=-\textrm{i}e^{\prime}\\{\bar{u}(p^{\prime})(1+\delta_{1})\gamma^{\mu}u(p)+\Gamma^{\mu}+\sum\limits_{l=1}^{2}\delta\Gamma_{l}^{\mu}\\}A_{\mu}^{cl}({\mathbf{q}}),$
(6)
where counterterm $\delta_{1}$ plays a similar role as $Z_{int}-1$ and
$Z_{kin}-1$ in Ref. [21], and $A_{\mu}^{cl}({\mathbf{q}})$ is the Fourier
transformation of the external field, $p^{\prime}$ and $p$ are outgoing and
incoming momentums of carrier respectively. Since we are working in lower
energy limit, we ignore the renormalization of Fermi velocity and charge. We
get
$\displaystyle\delta\Gamma_{l}^{\mu}$ $\displaystyle=$
$\displaystyle-\textrm{i}\lambda_{l}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{\bar{u}(p^{\prime})\gamma^{l}(\tilde{k}\\!\\!\\!/-m)\gamma^{\mu}(\tilde{k}\\!\\!\\!/^{\prime}-m)\gamma_{l}u(p)}{[\tilde{k}\\!\\!\\!/^{2}-m^{2}+\textrm{i}\eta][\tilde{k}\\!\\!\\!/^{\prime
2}-m^{2}]}$ $\displaystyle\Gamma^{\mu}$ $\displaystyle=$
$\displaystyle\frac{2\pi\textrm{i}e^{2}}{\varepsilon
v^{4}}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{u^{\dagger}(p^{\prime})(\tilde{k}\\!\\!\\!/^{\prime\prime}+m)\gamma^{\mu}(\tilde{k}\\!\\!\\!/+m)\gamma_{0}u(p)}{[\tilde{k}\\!\\!\\!/^{\prime\prime
2}-m^{2}+\textrm{i}\eta][\tilde{k}\\!\\!\\!/^{2}-m^{2}]|{\mathbf{p}}-{\mathbf{k}}|}$
respectively, where $q=p^{\prime}-p$, $k^{\prime\prime}=k+q$ and
$k^{\prime}=k-q$. In the above equation, we do not sum over the repeated index
$l$ and the terms proportional to $\beta^{2}\sim 10^{-4}$ are neglected.
Figure 2: Feynman rule and Feynman diagram on graphene. a) Propagator of
$\varphi$ field with momentum $p$,
$\textrm{i}\frac{\tilde{p}\\!\\!\\!/+m}{(\tilde{p})^{2}-m^{2}+\textrm{i}\epsilon}$,
where $\epsilon$ is infinitesimal positive. b) Propagator of $\chi$ field,
$\textrm{i}\frac{\tilde{p}\\!\\!\\!/-m}{(\tilde{p})^{2}-m^{2}+\textrm{i}\epsilon}$.
c) Propagator of gauge field,
$\frac{2\pi\textrm{i}}{\varepsilon}\frac{g_{\mu\nu}}{p-\textrm{i}\epsilon}$.
d) Interaction vertex between $\varphi$ field and gauge field,
$-\textrm{i}e^{\prime}\gamma^{\mu}$, where $e^{\prime}=e$ for $\mu=0$ and
$e^{\prime}=\beta e$ for $\mu=1,\,2$. For $\chi$ field the interaction is
similar. e) Two vertices of four-fermion interactions,
$-\textrm{i}\lambda_{1}\gamma^{1}_{i_{1}i_{2}}\gamma_{1j_{1}j_{2}}$ and
$-\textrm{i}\lambda_{2}\gamma^{2}_{i_{1}i_{2}}\gamma_{2j_{1}j_{2}}$.
f)Counterterm vertex, $-\textrm{i}e^{\prime}\delta_{1}\gamma^{\mu}$. Since we
are only concerned about the correction to VMM up to order $e^{2}$, the
renormalization of fermion velocity is ignored. g) Tree level diagram
contributing to VMM. h) Four-fermion corrections. i) Vertex correction.
Since the external field is time-independent, $q^{0}=p^{\prime 0}-p^{0}=0$ in
Eq. (LABEL:gamma). If the electromagnetic field varies very slowly over a
large region, Fourier components of the electromagnetic field will be
concentrated about ${\mathbf{q}}=0$. We can thus take nonrelativistic limit,
${\mathbf{q}}\rightarrow 0$, in $\textrm{i}\mathcal{M}$, which means
$|{\mathbf{p}}|,\,|{\mathbf{p}}^{\prime}|,\,|{\mathbf{q}}|\leq m/v$.
Therefore, we have relations $-(\tilde{q})^{2}=v^{2}{\mathbf{q}}^{2}>0$ and
$\tilde{p}\tilde{p}^{\prime}=(p^{0})^{2}-v^{2}{\mathbf{p}}\cdot{\mathbf{p}}^{\prime}\approx
m^{2}$.
To study the response to external magnetic field, we set time component of
$A_{cl}$ as zero, i.e. $A_{cl}({\mathbf{q}})=(0,{\mathbf{A}}({\mathbf{q}}))$.
We therefore only need to calculate the spatial part in
$\textrm{i}\mathcal{M}$.
Since our theory violets the ”Lorentz covariance”, we should treat the result
carefully. Furthermore, all the integrals in Eq. (LABEL:gamma) are divergent
and therefore the result seems ambiguous. However, we have the good news that
the ambiguity have no effect on VMM. After a lengthy calculation, such as Wick
rotation[22] and the expansion of the result to order
$|{\mathbf{p}}|,\,|{\mathbf{p}}^{\prime}|,\,|{\mathbf{q}}|$, we write
$\Gamma^{i}$ and $\delta\Gamma^{i}_{l}$ as
$\bar{u}(C_{1}\gamma^{i}+C_{2}\textrm{i}\epsilon^{ij}q^{j}\sigma_{3}/2)u$ in
nonrelativistic limit, where $\epsilon^{12}=-\epsilon^{21}=1$,
$\epsilon^{11}=\epsilon^{22}=0$, $C_{1}$ and $C_{2}$ depend on $\Gamma^{\mu}$,
$\delta\Gamma^{i}_{1}$ and $\delta\Gamma^{i}_{2}$. For all the cases, $C_{1}$
is divergent while $C_{2}$ is finite. Together with the tree diagram and
counterterm,
$(1+\delta_{1}+C_{1}(\Gamma^{i})+C_{1}(\delta\Gamma^{i}_{1})+C_{1}(\delta\Gamma^{i}_{2}))\bar{u}\gamma^{i}u$
should be fixed to match renormalization conditions. Comparing with the Born
approximation for scattering from a potential of carrier nearly
${\mathbf{p}},{\mathbf{q}}\rightarrow 0$, we find that
$1+\delta_{1}+C_{1}(\Gamma^{i})+C_{1}(\delta\Gamma^{i}_{1})+C_{1}(\delta\Gamma^{i}_{2})$
is just the electric charge of carrier, in units of $e$. Due to this, we set
the renormalization condition as
$1+\delta_{1}+C_{1}(\Gamma^{i})+C_{1}(\delta\Gamma^{i}_{1})+C_{1}(\delta\Gamma^{i}_{2})=1$
at ${\mathbf{p}},{\mathbf{q}}\rightarrow 0$. This renormalization condition
corresponds with the fact that the carrier at lower energy (${\mathbf{p}}=0$)
possesses unit (renormalized) charge $e$ when scattered under external
potential which varies very slowly.
For finite term $C_{2}$, we have
$C_{2}(\Gamma^{i})=\frac{e^{2}}{4\varepsilon
m},\,~{}~{}C_{2}(\delta\Gamma^{i}_{l})=-\frac{\lambda_{l}}{4\pi v}.$ (8)
Ignoring term proportional to ${\mathbf{p}}+{\mathbf{p}}^{\prime}$, which is
the contribution of the operator
${\mathbf{p}}\cdot{\mathbf{A}}+{\mathbf{A}}\cdot{\mathbf{p}}$ in the standard
kinetic energy term of nonrelativistic quantum mechanics, we rewrite
$\bar{u}\gamma^{i}u$ term as
$\bar{u}(p^{\prime})\gamma^{i}u(p)\rightarrow\frac{-\textrm{i}v\epsilon^{ij}q^{j}}{m}\bar{u}\frac{\sigma^{3}}{2}u.$
(9)
We obtain, then,
$\textrm{i}\mathcal{M}=-\textrm{i}2m\xi^{\dagger}\frac{\sigma_{3}}{2}\xi{ev\beta\over
m}(1-\frac{e^{2}}{4\varepsilon v}+\frac{m}{4\pi
v^{2}}(\lambda_{1}+\lambda_{2}))B^{3},$ (10)
where $B^{3}=-\textrm{i}(q^{1}A_{cl}^{2}-q^{2}A_{cl}^{1})$ is magnetic field
perpendicular to graphene, $\xi=(1,0)^{T}$ is two-component spinor and
$\xi^{\dagger}\frac{\sigma_{3}}{2}\xi=1/2\equiv s_{3}$ indicates that electron
pseudo-spin is 1/2.
We interpret $\mathcal{M}$ as the Born approximation to the scattering of the
electron from a potential. The potential is just that of a magnetic moment
interaction, $V({\mathbf{x}})=-\mu_{e}^{3}({\mathbf{K}})B^{3}$, where
$\mu_{e}^{3}({\mathbf{K}})=\frac{ev\beta}{m}(1-\frac{e^{2}}{4\varepsilon
v}+\frac{m}{4\pi v^{2}}(\lambda_{1}+\lambda_{2}))s_{3}$ (11)
is the carrier VMM parallel to $B^{3}$ at ${\mathbf{K}}$ valley. For the hole,
we get the same value with a necessary minus sign. Similarly, for carrier at
$-{\mathbf{K}}$ valley, VMM is also the same with a minus sign. Such
phenomenon is known as the broken inversion symmetry in Ref. [9].
By recovering $\hbar$, the leading term of VMM is ${e\hbar v\beta/m}$, which
is also obtained by ref. [9]. However, besides the leading term, there are
also other contributions to VMM. The relative contribution to VMM are
$-\frac{e^{2}}{4\varepsilon\hbar
v}-\frac{mV_{s}}{4\pi\hbar^{2}v^{2}}=\alpha(-\frac{1}{4}+\frac{1.55\varepsilon\,m\,a}{4\pi\hbar
v}),$ (12)
where $\alpha=e^{2}/(\varepsilon\hbar v)\approx 0.73$ when $\varepsilon=3$.
Substituting $a=2.46\overset{\circ}{A}$, $v\approx 10^{-8}cm/s$ into Eq. (12),
we find the relative modifications to VMM due to vertex correction and four-
fermion interactions are about $-18\%$ and $3\%$ respectively if we choose
$m=0.26eV$.
It looks strange that it is not $V_{d}$ but $V_{s}$ which contributes to VMM.
Such behavior stems from the definition of $\chi$ field. From the definition
of $\varphi$ field and $\chi$ field $V_{d}$ only relates to interaction
between carriers in different valley with the same pseudo-spin so that it does
not contribute to VMM. On the contrary, $V_{s}$ relates to interaction between
carriers in different valley with the different pseudo-spin. Therefore, only
$V_{s}$ contributes to VMM.
## 5 Discussions
In this paper we have discussed the carrier interactions. The study reveals
that besides the well known Coulomb repulsion between carriers, there are
four-fermion interactions between carriers in different valleys. Since the
interactions are short-range and contacting ones, they depend on the atom
collocation detail. Therefore, the four-fermion interactions are type
dependent. Our study shows that one of the four-fermion interactions attracts
carriers in different valleys, which we believe to be helpful in understanding
the unusual superconduction effect in graphene.
We also compute VMM from the tree level diagram, the vertex correction and the
four-fermion interactions respectively. The contribution from the tree level
diagram agrees with the result obtained in ref. [9]. The other two
contributions counteract each other and therefore the total contribution to
VMM is about $-15\%$ if we choose $m=0.26eV$ and $\varepsilon=3$.
The very high accurate measurement of spin magnetic moment is very important,
both from theoretical viewpoint and from practical one. Similarly, our result
on VMM is also significant to valleytronics in graphene, especially to the
future apparatus design based on valleytronics. Our study also points out
that, in close analogy to the Zeeman split, the contribution to VMM induced by
$V_{s}$ is inherent, since it is independent on the energy gap $m$. In other
words, to measure the magnetic moment induced by $V_{s}$, we may choose the
substrate freely, although different substrate may generate different energy
gap and different effective permittivity.
From Eq. (12), $\alpha$ plays the same role as the fine structure constant in
QED, $\alpha_{e}=\frac{e^{2}}{\hbar c}\approx 1/137$. However, because
$\alpha$ is about 100 times larger than $\alpha_{e}$, it is hard to state that
we mimic QED by carrier interaction in graphene. Meanwhile, when we deal with
problems dominated by quantum chromodynamics(QCD), especially in heavy
quarkonium, such as $c\bar{c}$ system and $b\bar{b}$ system, we always take
$\alpha_{s}=\frac{g^{2}_{s}}{\hbar c}$, where $g_{s}$ is the QCD coupling, as
the estimate of the effectiveness of perturbational expansion. (In many cases
when we deal with such problem we take an approach very similar to QED, up to
an unimportant color factor.) At energy scale $740Mev$,
$\alpha_{s}(740Mev)\approx 0.73\approx\alpha$[22]. Noticing that the energy
scale is close to the soft scale of $c\bar{c}$ and $b\bar{b}$ systems[23], the
dynamics of which is depicted by nonrelativistic QCD, we conclude that we can
mimic the heavy quarkonium system by carrier interactions in graphene.
Therefore, the study on the heavy-quarkonium system can also be carried out in
graphene.
This work is supported by the Cultivation Fund of the Key Scientific and
Technical Innovation Project-Ministry of Education of China (No. 708082).
## References
* [1] K.S. Novoselov, et al., Science, 306, 669(2004); K.S. Novoselov et al., Nature (London) 438, 197 (2005); Y. Zhang et al., Nature (London) 438, 201(2005).
* [2] See, for instance, A.K. Geim and K.S. Novoselov, Nature Mater. 6, 183 (2007); M.I. Katsnelson, Mater. Today 10, Issue 1&2, 20 (2007); M.I. Katsnelson and K.S. Novoselov, cond-mat/0703374; A. Kashuba, arXiv:0802.2261; K. Shizuya, Phys.Rev. B75, 245417; F. Guinea et al, arXiv:0803.1958.
* [3] D.Q. Liu and S.L. Zhang J. Phys.: Condens. Matter 20, 175222 (2008); S.A. Mikhailov, K. Ziegler, J. Phys.: Condens. Matter20, 384204 (2008)
* [4] V.P. Gusynin, S.G. Sharapov and J.P. Carbotte, Phys. Rev. B75, 165407 (2007); V.P. Gusynin1, S.G. Sharapov and J.P. Carbotte, J. Phys.: Condens. Matter 19, 026222 (2007).
* [5] V.P. Gusynin and S.G. Sharapov and J.P. Carbotte, Int.J.Mod.Phys. B21, 4611 (2007).
* [6] P. Recher et al, Phys. Rev. B76, 235404 (2007).
* [7] S.V. Morozov et al., Phys.Rev.Lett. 97, 016801 (2006); A.F. Morpurgo and F. Guinea, Phys. Rev. Lett. 97, 196804 (2006).
* [8] D. Xiao, J. Shi and Q. Niu, Phys.Rev.Lett. 95, 137204 (2005).
* [9] D. Xiao, W. Yao and Q. Niu, Phys.Rev.Lett. 99, 236809 (2007).
* [10] A. Rycerz, J. Tworzydlo and C.W.J. Beenakker, Nature Phys. 3, 172 (2007).
* [11] D.V. Bulaev, B. Trauzettel and D. Loss, Phys.Rev. B77, 235301 (2008).
* [12] P.R. Wallace, Phys. Rev. 71, 622 (1947).
* [13] R. Rold$\acute{\mathrm{a}}$n, M.P. L$\acute{\mathrm{o}}$pez-Sancho and F. Guinea, Phys. Rev. B77, 115410 (2008).
* [14] C. Zhang, L. Chen and Z. Ma, Phys.Rev. B77, 241402 (2008).
* [15] B. Uchoa and CastroNeto,A.H., Phys.Rev.Lett. 98, 146801 (2007).
* [16] G. Giovannetti et al Phys. Rev. B76, 073103 (2007).
* [17] S.Y. Zhou, et al, Nature Mater. 6,770 2007; W. Yao, S.A. Yang, and Q. Niu, Phys.Rev.Lett. 102, 096801 (2009).
* [18] T. Ohta, A. Bostwick, T. Seyller, K. Horn and E. Rotenberg, Science 313, 951 (2006)
* [19] S.Y. Zhou, et al, Nature Mater. 7, 259 (2008)
* [20] C.K. Xu, Phys.Rev. B78, 054432 (2008).
* [21] J. Gonz$\acute{a}$lez, F. Guinea and M.A.H. Vozmediano, Nucl.Phys. B424, 595 (1994).
* [22] D. Liu, Comm.Theor.Phys. V46, 1027 (2006); Y. Koide, hep-ph/9410270.
* [23] N. Brambilla, arXiv:hep-ph/0702105, (2007); D. Liu, Chin.Phys. 16, 962 (2007).
|
arxiv-papers
| 2010-02-27T16:40:19 |
2024-09-04T02:49:08.643992
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Daqing Liu, Shengli Zhang, Erhu Zhang, Ning Ma, Huawei Chen",
"submitter": "Daqing Liu",
"url": "https://arxiv.org/abs/1003.0111"
}
|
1003.0149
|
# $\mathbf{Phase}$ $\mathbf{structure}$ $\mathbf{of}$
$\mathbf{Topologically}$ $\mathbf{massive}$ $\mathbf{gauge}$ $\mathbf{theory}$
$\mathbf{with}$ $\mathbf{fermion}$
Yuichi Hoshino
Kushiro National College of Technology,
Otanoshike-nishi 2-32-1,Kushro 084-0916,Hokkaido,Japan
###### Abstract
Using Bloch-Nordsieck approximation fermion propagator in 3-dimensional gauge
theory with topological mass is studied.Infrared divergence of Chern-Simon
term is soft,which modifes anomalous dimension.In unquenched QCD with
2-component spinor anomalous dimension has fractional value,where order
parmeter is divergent.
## 1 Introduction
The Lagrangeans of Topologically massive gauge theory with fermion are[1]
$L=\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{4}\theta\epsilon^{\mu\nu\rho}F_{\mu\nu}A_{\rho}+\overline{\psi}(i\gamma\cdot(\partial-
ieA)-m)\psi+\frac{1}{2d}(\partial\cdot A)^{2},$ (1) $\displaystyle L$
$\displaystyle=\frac{1}{4g^{2}}tr(F_{\mu\nu}F^{\mu\nu})-\frac{\theta}{4g^{2}}\epsilon^{\mu\nu\rho}tr(F_{\mu\nu}A_{\rho}-\frac{2}{3}A_{\mu}A_{\nu}A\rho)$
$\displaystyle+\overline{\psi}(i\gamma\cdot(\partial-
ieA)-m)\psi+\frac{1}{2d}(\partial\cdot A)^{2}$ (2)
,where $\theta=(g^{2}/4\pi)n,(n=0,\pm 1,\pm 2...)$ with $\gamma$ matrix for
4-comonent fermion [2].In comparison with massless $QED_{3},QCD_{3}$
topological mass term seems to soften the infrared divergence of massive
fermion near its on-shell.In Minkowski metric we have 5 $\gamma$ matrices
$\\{\gamma_{\mu},\gamma_{\nu}\\}=2g_{\mu\nu},(\mu=0,1,2)$
$\displaystyle\gamma_{0}$
$\displaystyle=\left(\begin{array}[]{cc}\sigma_{3}&0\\\
0&-\sigma_{3}\end{array}\right),\gamma_{1,2}=-i\left(\begin{array}[]{cc}\sigma_{1,2}&0\\\
0&-\sigma_{1,2}\end{array}\right),\gamma_{4}=\left(\begin{array}[]{cc}0&I\\\
I&0\end{array}\right),\gamma_{5}=\left(\begin{array}[]{cc}0&-iI\\\
iI&0\end{array}\right),$ (11) $\displaystyle\tau$
$\displaystyle\equiv\frac{-i}{2}[\gamma_{4},\gamma_{5}]=diag\left(I,-I\right),\tau_{\pm}=\frac{1\pm\tau}{2}.$
(12)
There are two redundant matrices $\gamma_{4\text{ }}$and $\gamma_{5}$ which
anticommutes with other three $\gamma$ matrices.There exists two kind of
chiral transformation
$\psi\rightarrow\exp(i\alpha\gamma_{4})\psi,\psi\rightarrow\exp(\alpha\gamma_{5})\psi.$The
matrices $\\{\gamma_{4},\gamma_{5},I_{4},\tau\\}$ generate a $U(2)$ chiral
symmetry containing massless spinor fields.This $U(2)$ symmetry is broken down
to $U(1)\times U(1)$ by a spinor mass term $m_{e}\overline{\psi}\psi$ and
parity violating mass $m_{o}\overline{\psi}\tau\psi,$where
$\overline{\psi}\tau\psi$ is a spin density.Here after we take 4-component
spinors to study chiral symmetry breaking by $m^{e}\overline{\psi}\psi$ in
pure QED${}_{3},$where $\overline{\psi}\tau\psi$ is invariant under chiral
transformation.After that the effects of Chern-Simon term will be studied by
2-component spinors.
## 2 Fermion spectral function
The spectral function of 4-component fermion and photon propagator[3] are
defined as
$S_{F}(x)=S_{F}^{0}(x)\exp(F(x)),S_{F}^{0}(x)=-(i\gamma\cdot\partial+m)\frac{\exp(-m\sqrt{-x^{2}})}{4\pi\sqrt{x^{2}}}.$
(13)
$D_{F}^{0}(k)=-i(\frac{g_{\mu\nu}-k_{\mu}k_{\nu}/k^{2}-i\theta\epsilon_{\mu\nu\rho}k^{\rho}/k^{2}}{k^{2}-\theta^{2}+i\epsilon})+id\frac{k_{\mu}k_{\nu}}{k^{4}},$
(14)
where $F$ is an $O(e^{2})$ matrix element $|T_{1}|^{2}$ for the process
electron $(p+k)\rightarrow$electron $(p)+$photon $(k)$ as
$\displaystyle T_{1}$
$\displaystyle=-ie\frac{\epsilon_{\mu}(k,\lambda)}{\gamma\cdot(p+k)-m}\gamma^{\mu}\exp(i(p+k)\cdot
x)U(p,s),$ (15) $\displaystyle\sum_{\lambda,S}T_{1}\overline{T_{1}}$
$\displaystyle=-\frac{\gamma\cdot p+m}{2m}e^{2}[\frac{m^{2}}{(p\cdot
k)^{2}}+\frac{1}{p\cdot k}+\frac{(d-1)}{k^{2}}]-\frac{\gamma\cdot
p}{m}\frac{e^{2}}{4\theta}\frac{m}{p\cdot k},$ (16)
$F=\int\frac{d^{3}k}{(2\pi)^{2}}\exp(ikx)\theta(k_{0})\delta(k^{2}-\theta^{2})\sum_{\lambda,S}T_{1}\overline{T_{1}}.$
(17)
Here we use the retarded propagator to derive the function $F$
$D_{+}(x)=\int\frac{d^{3}k}{i(2\pi)^{2}}\exp(ik\cdot
x)\theta(k_{0})\delta(k^{2}-\theta^{2})=\frac{\exp(-\theta\sqrt{-x^{2}})}{8\pi
i\sqrt{-x^{2}}}$ (18)
The function $F$ is evaluated by $\alpha$ integration for pure QED3[4, 3].
$\displaystyle F$ $\displaystyle=ie^{2}m^{2}\int_{0}^{\infty}\alpha d\alpha
D_{F}(x+\alpha p)-e^{2}\int_{0}^{\infty}d\alpha D_{F}(x+\alpha
p)-i(d-1)e^{2}\frac{\partial}{\partial\theta^{2}}D_{F}(x,\theta^{2})$
$\displaystyle=\frac{e^{2}}{8\pi}[\frac{\exp(-\theta|x|)-\theta|x|E_{1}(\theta|x|)}{\theta}-\frac{E_{1}(\theta|x|)}{m}+\frac{(d-1)\exp(-\theta\left|x\right|))}{2\theta}].$
(19)
It is well known that function $F$ has linear and logarithmic infrared
divergence with respect to $\theta$ where $\theta$ is a bare photon mass.Here
we notice the followings.(1) $\exp(F)$ includes all infrared divergences.(2)
quenhed propagator has linear and logarithmic infrared divergences.Linear
divegence is absent in a special gauge.In unquenched case $\theta$ dependence
of $\exp(F)$ is modified by dressed boson spectral function to $\int
ds\rho_{\gamma}(s)\exp(F(x,\sqrt{s}),$where
$\pi\rho_{\gamma}(s)=-\Im(s-\Pi(s))^{-1}.$
## 3 Phase strucutures
For short and long distance we have the approximate form of the function $F$
$F_{S}\sim
A-\theta|x|+(D+C|x|)\ln(\theta\left|x\right|))-\frac{(d+1-2\gamma)e^{2}|x|}{16\pi},(\theta|x|\ll
1),F_{L}\sim 0,(1\ll\theta|x|)..$ (20)
From the above formulae we have
$\displaystyle\exp(F)$ $\displaystyle=A(\theta|x|)^{D+C|x|}(\theta|x|\ll
1),A=\exp(\frac{e^{2}(1+d)}{16\pi\theta}+\frac{e^{2}\gamma}{8\pi m}),$
$\displaystyle C$ $\displaystyle=\frac{e^{2}}{8\pi},D=\frac{e^{2}}{8\pi m},$
(21)
where $m$ is the physical mass and $\gamma$ is an Euler’s constant.Here we see
$D$ acts to change the power of $|x|$.For $D=1,S_{F}(0)$ is finite and we have
$\left\langle\overline{\psi}\psi\right\rangle\neq 0$ [4, 6].Thus if we require
$D=1,$we obtain $m=e^{2}/8\pi.$In the same way we add the conrtribution of
Chern-Simon term.For simplicity we consider the 2-component spinors in(7).In
the condensed phase we have the modified anomalous dimension for $\theta>0$
$D=\frac{e^{2}}{8\pi
m}+\frac{e^{2}}{32\pi\theta}=1,m^{{}^{\prime}}=\frac{e^{2}}{8\pi}/(1-\frac{e^{2}}{32\pi\theta}),\Delta
m=\frac{e^{2}}{8\pi}\frac{e^{2}}{32\pi\theta}/(1-\frac{e^{2}}{32\pi\theta}).$
(22)
In unquenched case there are parity even and odd spectral function of gauge
boson by vacuum polarization[1].In this case we separate $D$ into parity even
and odd contribution $D^{e}=e^{2}/8\pi m,D^{O}=e^{2}/32\pi\theta.$For
Topologically Massive QCD,$\theta$ is quantized with $n(0,\pm 1,\pm
2,..).$Thus we have $D=e^{2}/(8\pi m)+1/8n=1$ for quenched case, and
$D^{e}=e^{2}/8\pi m=1,D^{O}=1/(8n)$ for unquenched case,which leads to
$\left\langle\overline{\psi}\psi\right\rangle=\infty$ for any $n\neq 0.$In the
4-component spinor free fermion propagator is decomposed into chiral
representation
$S_{F}(p)=\frac{1}{m_{\epsilon}I+m_{O}\tau-\gamma\cdot p}=\frac{(\gamma\cdot
p+m_{+})\tau_{+}}{p^{2}-m_{+}^{2}+i\epsilon}+\frac{(\gamma\cdot
p+m_{-})\tau_{-}}{p^{2}-m_{-}^{2}+i\epsilon}.$ (23)
The difference in two spectral functions is a opposite sign of each Chern-
Simon contribution for $\tau_{\pm}$.In Toplogically massive gauge theory
dynamical mass is parity even and Chern-Simon term shifts mass of different
chirality with opposite sign.However shifted mass may not strongly depend on
$\theta$ but $\left\langle\overline{\psi}\psi\right\rangle$ is proportional to
$\theta$(12)[5].Our approximation is convenient for unquenched case by the use
of gauge boson spectral function[6].
## References
* [1] S.Deser,R.Jackiw&Templeton,Annals of Physics 281,409-449(2000).
* [2] C.J.Burden,NuclerPhysicsB387(1992)419-446,Kei-ichi Kondo,Int.J.Mod.Phys.$\mathbf{A11}$;777-822,1996.
* [3] R.Jackiw,L.Soloviev,Phys.Rev.173.5(1968)1485.
* [4] Yuichi Hoshino,in CONTINUOUS ADVANCES IN QCD 2008;361-372.
* [5] Toyoki Matsuyama,Hideko Nagahiro,Mod.Phys.Lett.A15 (2000) 2373-2386.
* [6] C.S.Fisher,Reinhart Alkofer,T.Darm,P.Maris,Phys.Rev.D70:073007,2004.
|
arxiv-papers
| 2010-02-28T04:27:00 |
2024-09-04T02:49:08.650696
|
{
"license": "Public Domain",
"authors": "Yuichi Hoshino",
"submitter": "Yuichi Hoshino",
"url": "https://arxiv.org/abs/1003.0149"
}
|
1003.0190
|
# Generating Function For Network Delay
A.M. Sukhov, N.Yu. Kuznetsova, A.K. Pervitsky and A.A. Galtsev Samara State
Aerospace University, Moskovskoe sh., 34, Samara, 443086, Russia
e-mails: amskh@yandex.ru, meneger$\\_$job@mail.ru, pervitskiy.alex@mail.ru,
galaleksey@gmail.com
###### Abstract
In this paper correspondence between experimental data for packet delay and
two theoretical types of distribution is investigated. Statistical tests have
shown that only exponential distribution can be used for the description of
packet delays in global network. Precision experimental data to within
microseconds are gathered by means of the RIPE Test Box. Statistical
verification of hypothesis has shown that distribution parameters remain
constants during 500 second intervals at least. In paper cumulative
distribution function and generating function for packet delay in network are
in an explicit form written down, the algorithm of search of parameters of
distribution is resulted.
## I Introduction
The special area of the control theory, named networked control systems in
which transfers as environment of operating signals were used computer
networks, has arisen in the late nineties of the XX-th century [18].
Originally, as the network environment of control systems local networks [8]
which differ high-speed data transfer and in the minimum percent of packet
loss were used.
The networked control systems in which as the handle environment the global
network Internet is used, is extremely complicated because of random character
of distribution of packet delay and their big absolute values [10, 15, 17].
However till now, results of the advanced network researches are not used in
the control theory and algorithms on their basis are not created. The present
project assumes introduction of new network decisions in networked control
systems.
Except classical problems of the control theory, there is requirement of
management and monitoring of network processes. For example, for video
transfer over networks TCP/IP networks the important parameter is the
available bandwidth between two hosts [4]. For problems of routing it is
necessary to know throughput between routers. Special network emulators are
applied to modeling of the majority of network processes, for example,
INET/OMNET ++ [16].
Now numerous programs which imitate transmission of packets through TCP/IP
protocol network are created. In a basis of operation of all emulators
position that the type of delay distribution is unknown is occupied. The
purpose of our research consists in that not only to define delay type, but
also to find generating function for the traffic emulators.
For the decision of problems of the networked control systems on the basis of
stack TCP/IP it is more convenient to use the process approach to the control
theory, based on idea of existence of some universal functions of control. The
purpose of our research is the finding of this function for network
components. In the modern theory of computer networks there were many
utilities working with a network delay, there is a progress in studying and
modeling of transmitting of packets. Our problem consists in trying to
describe process of a network delay of management packets and to show ways of
practical calculation of all parameters entering into corresponding
distribution functions [9].
By transmission of control signals through TCP/IP network, the separate
packets of the controlling data flow transferring the information are supplied
non-uniformly, and the part of packets in general is lost by transmission on a
network and does not reach a target. For rise of efficiency of control
algorithms it is necessary to reduce to a minimum of packets delay and their
variation, and also percent of packet loss. Similar algorithms are used for
transmission voice and video streams, in grid systems, at control of robust
systems, in network computer games, etc.
At first it would be desirable to result the brightest research on a
distribution type for network delay. To understand, about what there is a
speech in described papers, will give definitions of notations used in them:
* •
Round-trip time (RTT) time is the time required for a packet to travel from
the testing host to a remote computer that receives the packet and retransmits
it back to the source.
* •
The One-Way Delay (OWD) value is calculated between two synchronized points A
and B of an IP network, and it is the time in seconds that a packet spends in
travelling across the IP network from A to B.
In particular, Elteto and Molnar [6] have spent measurements of round-trip
delay in the Ericsson Corporate Network, complex analysis of the received data
has allowed to build the supposition about distribution type for network
delay. The main finding of their research is that the round-trip delay can be
well approximated by a truncated normal distribution.
Konstantina Papagiannaki et al [12] in the research have measured and have
analyzed packet delay between two adjacent routers in the core network. On the
basis of the received measurements, they have made the supposition about the
factors influencing occurrence of delay, and very big delays which cannot be
explained in the way of batch processing in routers on algorithm FIFO have
been noticed.
Recently, fulfilling a series of operations on measurement of an available
bandwidth [13], we have installed that for a type definition of delay
distribution we should research only a variable part of delay while its most
part remains constant. This fact also has served as a starting point of our
operation.
## II Premises for model
In 1999 Downey [5] for the first time has detected linear dependence of the
minimum possible round trip time on the size of transferred packets. In 2004
precise experiments by Choi et al [1], Hohn et al [11] proved that the minimum
fixed delay component $D^{fixed}(W)$ for a packet of size $W$ is a linear (or
precisely, an affine) function of its size,
$D^{fixed}(W)=W\sum_{i=1}^{h}1/C_{i}+\sum_{i=1}^{h}\delta_{i}$ (1)
where $C_{i}$ is each link of capacity of $h$ hops and $\delta_{i}$ is
propagation delay. To validate this assumption, they check the minimum delay
of packets of the same size for three path, and plot the minimum delay against
the packet size.
Let $D(W)$ represents the one way delay (point-to-point delay) of a packet.
Here we refer to it as the minimum path transit time for the given packet size
$W$, denoted by $D^{fixed}(W)=\min D(W)$. With the fixed delay component
$D^{fixed}(W)$ identified, we can now subtract it from the point-to-point
delay of each packet to study the variable delay component $d^{var}$. The
variable delay component of the packet, $d^{var}$, is given by
$D(W)=D^{fixed}(W)+d^{var}$ (2)
Computed minimal delay $D^{fixed}(W)$ is
$D^{fixed}(W)=D_{min}+W/C,$ (3)
where $C=(W_{2}-W_{1})/(D_{2}-D_{1})$ is end-to-end available bandwidth and it
is searched, comparing average time of packet delay of the different sizes
$W_{2}$ and $W_{1}$ [13]. Here
$D_{min}=\lim_{W\rightarrow 0}D^{fixed}(W)$ (4)
The value $D_{min}$ is related to the distance between the sites (i.e.
propagation delay) and per-packet router processing time at each hop along the
path between the sites [2, 3]. This value represents as the minimum delay
$D_{min}$ for which the very small package can be transmitted on a network
from one point in another.
The minimal delay [13] of datagram transmission $D_{min}$ may be calculated as
$D_{min}=\frac{W_{2}D^{fixed}(W_{1})-W_{1}D^{fixed}(W_{2})}{W_{2}-W_{1}}$ (5)
This value as well as the methods of its measurement has a important
significance in applied tasks of control theory [18]. The second significant
question of networking control theory is the distribution type for variable
delay component $d^{var}$ which should be studied in rest of paper. To know
the expression for this parameter we may easy calculate the duration of buffer
for streaming application on receiving side.
## III Experimental search
In order to determine distribution type for a variable delay component
$d^{var}$ we should run considerable quantity of measurements between various
hosts in the Internet. The basic problem of experimental testing is the
precise of delay measurements that is necessary for accurate result. Similar
measurements demand, at least, microsecond precision for delay measurements;
we are reaching such accuracy with help of RIPE Test Box mechanism [7, 14]. In
order to prepare the experiments three Test Boxes have been installed in
Moscow, Samara and Rostov on Don during 2006-2008 years in framework of RFBR
grant 06-07-89074. Each RIPE Test Box represents a server under management of
an FreeBSD operating system with the GPS receiver connected to it.
Characteristic times of investigated processes (a packet delay, jitter) have
the order from 10 $ms$ to 1 $sec$, therefore is enough to use system hours of
a RIPE Test Box for their reliable measurement. At the first stage experiment
between tt01.ripe.net (RIPE NCC at AMS-IX, Amsterdam), tt143.ripe.net (Samara,
SSAU), tt17.ripe.net (Bologna) and tt74.ripe.net (Melbourne) have been made
which include precision measurement of packet delay with accuracy 2-12 $\mu
s$. Testing results are available via telnet to corresponding RIPE Test Box on
port 9142. It is important to come and write down simultaneously the data on
both ends of the investigated connection.
On the basis of the received data set it is easy to construct a cumulative
distribution function for network delay $D$:
$F(D)=P(x\leq D)$ (6)
For initial comparison truncated normal and exponential distributions have
been chosen, expressions for which are written down.
For truncated normal distribution it is possible to select following
approximation:
$F(D)=\left\\{\begin{IEEEeqnarraybox}[]{[}][c]{rl}0,&D<D_{min};\\\
\frac{\sqrt{2/\pi}}{\sigma}\int\limits_{D_{min}}^{D}\exp\left\\{-\frac{(x-D_{min})^{2}}{2\sigma^{2}}\right\\}dx,&D\geq
D_{min}\end{IEEEeqnarraybox}\right.$ (7)
where
$\sigma=D_{av}-D_{min}$ (8)
is the difference between average network delay $D_{av}(W)=\mathbb{E}[D(W)]$
and minimum delay $D_{min}(W)$.
It should be noted that all statistical data has been gathered by us for the
fixed size of a packets $W$. By default for RIPE Test Box it equals to 100
bytes. In Section V we update a cumulative distribution function $F(D,W)$
taking into account the packets size $W$.
The alternative type of allocation which will be checked on correspondence is
an exponential distribution, expression for which is written below.
$F(D)=\left\\{\begin{IEEEeqnarraybox}[]{[}][c]{rl}0,&D<D_{min}\\\
1-\exp\left\\{-\lambda(D-D_{min})\right\\},&D\geq
D_{min}\end{IEEEeqnarraybox}\right.$ (9)
where
$\lambda=1/(D_{av}-D_{min})$ (10)
is reciprocal to the difference between average network delay
$D_{av}(W)=\mathbb{E}[D(W)]$ and minimum delay $D_{min}(W)$.
For initial check of conformity to distribution type two methods will be used:
calculation of Pearson correlation coefficient and a graphic method. We will
designate as $K_{nor}$ correlation coefficient between experimental and normal
distributions then $K_{exp}$ is correlation coefficient between experimental
and exponential distributions.
The obtained data is shown in Table I, where the column host corresponds to a
direction between two RIPE Test Boxes, and the column $W$ specifies in the
size of a testing packet.
N | host | $W$ (bytes) | $K_{nor}$ | $K_{exp}$
---|---|---|---|---
1 | bolonia | | |
| tt01$\Rightarrow$tt17 | 100 | 0.76 | 0.97
2 | samara | | |
| tt01$\Rightarrow$tt143 | 100 | 0.87 | 0.98
3 | samara | | |
| tt01$\Rightarrow$tt143 | 1024 | 0.99 | 0.99
4 | melburn | | |
| tt01$\Rightarrow$tt74 | 100 | 0.66 | 0.97
TABLE I: Precise measurements
Except correlation coefficients it is possible to compare graphics
representation of cumulative distribution functions (CDF), showing all three
functions on one plot. On the uniform graphics (see Figures 1, 2) dash line
selects an experimental curve, dot-dash curve corresponds to normal
allocation. In dot style the exponential distribution is painted.
The plots constructed in Figures 1 and 2, represent dependence of CDF on delay
of a packet on a site from Amsterdam to Samara (tt01$\Rightarrow$tt143). The
first plot describes testing of a network by packages in the 100 bytes size,
the second plot corresponds to packages in the size of 1024 bytes. Time on
axis is measured in milliseconds.
Figure 1: Experimental (dash), normal (dash-dot) and exponential (dot) CDFs,
precise testing. Direction: tt01$\Rightarrow$tt143, $W=$100 bytes Figure 2:
Experimental (dash), normal (dash-dot) and exponential (dot) CDFs, precise
testing. Direction: tt01$\Rightarrow$tt143, $W=$1024 bytes
All experiments resulted above testify that the preferable type of
distribution describing packet delay in a global network is an exponential
distribution. Thus, as have shown our researches, the random variable of
packet delay between two network points is arranged on an exponential low with
the parameter calculated from experimental values under the Equation (10).
However, not each investigator who is engaged in the control theory has the
equipment for the precision measurements, similarly RIPE Test Boxes. Therefore
in this part it would be desirable to present technique which uses the data of
well-known utilities and doesn’t demand the expensive equipment.
For testing it is possible to use the utility ping as it is the most
widespread resource for verification of connection quality in TCP/IP networks.
Let’s mark only that this utility measures round-trip time, instead of one way
delay.
The data received with help of ping has the millisecond precision that is
exact enough to judge delay distribution. The utility ping allows testing
connections between points AIST - New Zealand (tt47.ripe.net), Volgatelekom -
Australia (tt74.ripe.net) and SSAU-Melbourn (tt74.ripe.net). As remote hosts
were used servers of RIPE measurement system, AIST, Volgatelecom (VT),
Infolada and SSAU is local Internet Service Providers from Samara region,
Russia. Processing the obtained data on the above described algorithm, we have
received the results presented in the Table II.
N | host | $W$ (bytes) | $K_{nor}$ | $K_{exp}$
---|---|---|---|---
1 | AIST$\Rightarrow$ | | |
| New Zeland | 32 | 0.94 | 0.95
2 | Volgatelecom$\Rightarrow$ | | |
| Australia | 32 | 0.96 | 0.98
3 | SSAU$\Rightarrow$ | | |
| Melburn | 64 | 0.66 | 0.97
4 | Infolada$\Rightarrow$ | | |
| Athens | 32 | 0.98 | 0.98
TABLE II: ping measurements
The evident illustration is resulted in definition of distribution type in
Figure 3-5.
Figure 3: Experimental (dash), normal (dash-dot) and exponential (dot) CDFs,
precise testing. Direction: Samara$\Rightarrow$Holland, $W=$32 bytes Figure 4:
Experimental (dash), normal (dash-dot) and exponential (dot) CDFs, precise
testing. Direction: Infolada$\Rightarrow$Athens, $W=$32 bytes Figure 5:
Experimental (dash), normal (dash-dot) and exponential (dot) CDFs, precise
testing. Direction: SSAU$\Rightarrow$Australia, $W=$1064 bytes
It should be noted that the utility ping allows finding automatically values
of variables $D_{av}$ and $D_{min}$ (see Eqns. (7) and (9)) which completely
define the distribution form, both normal and exponential types. It is enough
to give sequence from 20 packets to obtain the given values with split-hair
accuracy, sufficient for the description of processes of the control theory.
## IV Statistical hypothesis testing
The verification executed in the previous section about conformity of
distribution type has preliminary character and isn’t strict. In this section
for check of distribution type Pearson’s chi-square test will be used.
For processing some data sets from 2000 to 2500 values the delays collected
with interval in 2 seconds have been used. This data divides into intervals in
50, 100, 250, 500, 1000 and 2000 values and was tested on Pearson. Testing
results could be found in Tables III, IV.
At construction of these Tables following designations were used:
* •
Dimension of observations $N$ (number of measurements)
* •
$n$ is the number of cells. All observations $N$ are divided among $n$ cells
according Sturges’ rule $n=(1+3.22\lg N)+1$
* •
$t$ is the value of the test-statistic
* •
$\chi^{2}_{0.95,n-1}$ is the theoretical value of threshold of hypothesis
acceptance
If $t<\chi^{2}_{0.95,n-1}$ then hypothesis about corresponding type of
distribution is accepted, differently hypothesis is rejected.
In the beginning we will check up on conformity to exponential distribution
the data, reception by means of RIPE Test Box, see Table III.
$N$ | 50 | 100 | 200 | 250 | 500 | 1000 | 2000
---|---|---|---|---|---|---|---
$n$ | 14 | 17 | 19 | 20 | 22 | 24 | 27
$\chi^{2}_{0.95,n-1}$ | 22.36 | 26.30 | 28.87 | 30.14 | 32.67 | 35.17 | 38.89
$t$ | 21.29 | 25.35 | 22.77 | 23.10 | 134.31 | 547.16 | 978.98
Hypothesis acceptance | Yes | Yes | Yes | Yes | No | No | No
TABLE III: Verification of exponential distribution, Samara-Amsterdam,
(tt143$\Rightarrow$tt01), packet size 100 bytes
From Table III follows that within 500 second intervals (250 measurements) the
packet delay is distributed on the exponential law.
The data received with RIPE Test Box, was checked also on conformity to the
truncated normal distribution, see Table IV.
$N$ | 50 | 100 | 200 | 250 | 500 | 1000 | 2000
---|---|---|---|---|---|---|---
$n$ | 14 | 17 | 19 | 20 | 22 | 24 | 27
$\chi^{2}_{0.95,n-1}$ | 22.36 | 26.30 | 28.87 | 30.14 | 32.67 | 35.17 | 38.89
$t$ | 43.32 | 217.46 | 2906.47 | 6077.07 | $\infty$ | $\infty$ | $\infty$
Hypothesis acceptance | No | No | No | No | No | No | No
TABLE IV: Verification of truncated normal distribution, Samara-Amsterdam,
(tt143$\Rightarrow$tt01), packet size 1024 bytes
Pearson’s chi-square test allows rejecting hypothesis about the truncated
normal type of distribution for the description of process of packet delay.
## V Distribution Type for Delay and Generating Function for Traffic Emulator
In real the Internet processes the size of transferred packages can vary,
therefore the cumulative distribution function should be updated. For each
size of a packet $W$ there is the minimum time $D^{fixed}(W)$ defined by the
Eq. (3).
Then, the final cumulative distribution function $F(D,W)$ is
$F(D,W)=\left\\{\begin{IEEEeqnarraybox}[]{[}][c]{rl}0,D<D_{min}+W/C&\\\
1-\exp\left\\{-\lambda(D-D_{min}-W/C)\right\\},&\\\ D\geq
D_{min}+W/C&\end{IEEEeqnarraybox}\right.$ (11)
where
$\lambda=1/(D_{av}-D_{min})$ (12)
is reciprocal to the difference between average network delay
$D_{av}(W)=\mathbb{E}[D(W)]$ and minimum delay $D_{min}(W)$ and $C$ is end-to-
end capacity.
It is important to notice that the data on the delay, received with use of
packages of various length, comprises the additional variable component caused
network jitter $j$ (delay variation) [4]. Therefore, the best the controlling
algorithm will form packages of the identical size. If we use the utility ping
for delay definition in it there is a special key for resizing of a testing
package ($-l$ in Windows, $-s$ in Linux).
Our model has one more wide application except tasks of the control theory.
Results of our operation can be applied at writing of emulators of the traffic
in a global network [16]. Till now it was considered that the type of delay
distribution is unknown, and traffic emulators used own generating functions
for delay generation. On the basis of the type of delay found us (see Eqn.
(11)) it is possible to write generating function
$D=D_{min}+W/C-(1/\lambda)\ln(1-F(D,W))$ (13)
In this equation content distribution function (CDF) $F(D,W)$ can be set the
generator of random numbers. The received numbers will give values of delay
for network packet of the different size. We will notice once again that in
real networks these values can be calculated according to the specilised
utility.
## VI Conclusion
In the present work for the description of process of the packet delay in a
global networks it has been chosen exponential distribution. In comparison
with truncated normal distribution it has shown the best correlation with
experimental results. Unlike the truncated normal distribution it passes check
by statistical methods.
Experimental data were gathered by means of the precision RIPE measuring
system to within microseconds, and also by means of the standard utility ping.
This utility measures round-trip time to within milliseconds. During small
periods about several minutes when it is possible to consider conditions of
transmission on a TCP/IP network invariable, such approach gives correlations
from above 0.99. At change of network conditions the elementary ping testing
by a series from 20 packets will allow to change exponential distribution
parameters instantly.
We write in an explicit form of cumulative distribution functions for normal
and exponential distribution of delay. Generating function for packet delay
has been found in real networks which can be used in emulators of the traffic
of a global network.
In summary we would like to thank Leonid Fridman, the Professor from
University of Mexico for fruitful dialogues in which course the idea of this
article has taken shape. Also it would be desirable to thank all collective of
technical service RIPE ncc and especially Ruben van Staveren and Roman
Kalyakin for constant assistance in comprehension of subtleties of a measuring
infrastructure. We also would like to express the gratitude to the Wolfram
Research corporation, which the first has marked our preprint and has given us
licenses for the right of use of Mathematica
## References
* [1] Choi, B.-Y., Moon, S., Zhang, Z.-L., Papagiannaki, K. and Diot, C.: Analysis of Point-To-Point Packet Delay In an Operational Network. In: Infocom 2004, Hong Kong, pp. 1797-1807 (2004)
* [2] Cottrell, L., Matthews, W. and Logg C.: Tutorial on Internet Monitoring $\&$ PingER at SLAC. http://www.slac.stanford.edu/comp/net/wan-mon/tutorial.html
* [3] Crovella, M.E. and Carter, R.L.: Dynamic Server Selection in the Internet. In: Proc. of the Third IEEE Workshop on the Architecture and Implementation of High Performance Communication Subsystems (1995)
* [4] Dovrolis C., Ramanathan P., and Moore D., Packet-Dispersion Techniques and a Capacity-Estimation Methodology, IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 12, NO. 6, DECEMBER 2004, p. 963-977
* [5] Downey A.B., Using Pathchar to estimate internet link characteristics, in Proc. ACM SICCOMM, Sept. 1999, pp. 222–223.
* [6] Elteto, T., Molnar, S., On the distribution of round-trip delays in TCP/IP networks, in The Proceedings of the Local Computer Networks (LCN 99) Conference, IEEE, 1999, pp. 172–181
* [7] Georgatos, F., Gruber, F., Karrenberg, D., Santcroos, M., Susanj, A., Uijterwaal, H. and Wilhelm R., Providing active measurements as a regular service for ISP’s. In: PAM2001
* [8] Georges J.-P., Divoux T., and Rondeau E., Confronting the performances of a switched ethernet network with industrial constraints by using the network calculus, _International Journal of Communication Systems(IJCS)_ , vol. 18, no. 9, pp. 877–903, 2005
* [9] Fridman E., Seuret A., and Richard J.-P., “Robust sampled-data stabilization of linear systems: an input delay approach,” _Automatica_ , vol. 40, pp. 1441–1446, 2004
* [10] Hespanha J.P., Naghshtabrizi P., and Xu Y., “A survey of recent results in networked control systems,” _Proceedings of the IEEE_ , vol. 95, pp. 138–162, January 2007
* [11] N. Hohn, D. Veitch, K. Papagiannaki and C. Diot, Bridging Router Performance And Queuing Theory, Proc. ACM SIGMETRICS, New York, USA, Jun 2004
* [12] D. Papagiannaki, S. Moon, C. Fraleigh, P. Thiran, F. Tobagi, and C. Diot, Analysis of measured single-hop delay from an operational backbone network, in Proc. IEEE INFOCOM 2002, New York, New York, June 2002
* [13] A.M. Sukhov, T.G. Sultanov, M.V. Strizhov, A.P. Platonov, Throughput metrics and packet delay in TCP/IP networks, RIPE59 Meeting, Lisbon, 2009; arXiv:0907.3710
* [14] Ripe Test Box, http://ripe.net/projects/ttm/
* [15] Y. Tipsuwan and M.-Y. Chow, ”Control methodologies in networked control systems,” Control Engineering Practice, vol. 11, pp. 1099-1111, 2003
* [16] A. Varga, The OMNeT++ distrete event simulation system, Software on-line: http://whale.hit.bme.hu/omnetpp/, 1999
* [17] Zampieri S., “Trends in networked control systems,” _Proceedings of the 17th World Congress, The International federation of Automatic Control_ , July 2008
* [18] W. Zhang and M.S. Branicky and S.M. Phillips, Stability of Networked Control Systems, IEEE Control System Magazine, 21(1):84-99, February 2001
|
arxiv-papers
| 2010-02-28T16:42:36 |
2024-09-04T02:49:08.656005
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A.M. Sukhov, N.Yu. Kuznetsova, A.K. Pervitsky and A.A. Galtsev",
"submitter": "Andrei Sukhov M",
"url": "https://arxiv.org/abs/1003.0190"
}
|
1003.0487
|
# Scalable Large-Margin Mahalanobis Distance Metric Learning
Chunhua Shen, Junae Kim, and Lei Wang Manuscript received April X, 200X;
revised March X, 200X. NICTA is funded by the Australian Government as
represented by the Department of Broadband, Communications and the Digital
Economy and the Australian Research Council through the ICT Center of
Excellence program. C. Shen is with NICTA, Canberra Research Laboratory,
Locked Bag 8001, Canberra, ACT 2601, Australia, and also with the Australian
National University, Canberra, ACT 0200,Australia (e-mail:
chunhua.shen@nicta.com.au). J. Kim and L. Wang are with the Australian
National University, Canberra, ACT 0200, Australia (e-mail: {junae.kim,
lei.wang}@anu.edu.au). Color versions of one or more of the figures in this
brief are available online at http://ieeexplore.ieee.org. Digital Object
Identifier 10.1109/TNN.200X.XXXXXXX
###### Abstract
For many machine learning algorithms such as $k$-Nearest Neighbor ($k$-NN)
classifiers and $k$-means clustering, often their success heavily depends on
the metric used to calculate distances between different data points. An
effective solution for defining such a metric is to learn it from a set of
labeled training samples. In this work, we propose a fast and scalable
algorithm to learn a Mahalanobis distance metric. The Mahalanobis metric can
be viewed as the Euclidean distance metric on the input data that have been
linearly transformed. By employing the principle of margin maximization to
achieve better generalization performances, this algorithm formulates the
metric learning as a convex optimization problem and a positive semidefinite
(p.s.d.) matrix is the unknown variable. Based on an important theorem that a
p.s.d. trace-one matrix can always be represented as a convex combination of
multiple rank-one matrices, our algorithm accommodates any differentiable loss
function and solves the resulting optimization problem using a specialized
gradient descent procedure. During the course of optimization, the proposed
algorithm maintains the positive semidefiniteness of the matrix variable that
is essential for a Mahalanobis metric. Compared with conventional methods like
standard interior-point algorithms [2] or the special solver used in Large
Margin Nearest Neighbor (LMNN) [23], our algorithm is much more efficient and
has a better performance in scalability. Experiments on benchmark data sets
suggest that, compared with state-of-the-art metric learning algorithms, our
algorithm can achieve a comparable classification accuracy with reduced
computational complexity.
###### Index Terms:
Large-margin nearest neighbor, distance metric learning, Mahalanobis distance,
semidefinite optimization.
## I Introduction
In many machine learning problems, the distance metric used over the input
data has critical impact on the success of a learning algorithm. For instance,
$k$-Nearest Neighbor ($k$-NN) classification [4], and clustering algorithms
such as $k$-means rely on if an appropriate distance metric is used to
faithfully model the underlying relationships between the input data points. A
more concrete example is visual object recognition. Many visual recognition
tasks can be viewed as inferring a distance metric that is able to measure the
(dis)similarity of the input visual data, ideally being consistent with human
perception. Typical examples include object categorization [24] and content-
based image retrieval [17], in which a similarity metric is needed to
discriminate different object classes or relevant and irrelevant images
against a given query. As one of the most classic and simplest classifiers,
$k$-NN has been applied to a wide range of vision tasks and it is the
classifier that directly depends on a predefined distance metric. An
appropriate distance metric is usually needed for achieving a promising
accuracy. Previous work (e.g., [25, 26]) has shown that compared to using the
standard Euclidean distance, applying an well-designed distance often can
significantly boost the classification accuracy of a $k$-NN classifier. In
this work, we propose a scalable and fast algorithm to learn a Mahalanobis
distance metric. Mahalanobis metric removes the main limitation of the
Euclidean metric in that it corrects for correlation between the different
features.
Recently, much research effort has been spent on learning a Mahalanobis
distance metric from labeled data [25, 26, 23, 5]. Typically, a convex cost
function is defined such that a global optimum can be achieved in polynomial
time. It has been shown in the statistical learning theory [22] that
increasing the margin between different classes helps to reduce the
generalization error. Inspired by the work of [23], we directly learn the
Mahalanobis matrix from a set of distance comparisons, and optimize it via
margin maximization. The intuition is that such a learned Mahalanobis distance
metric may achieve sufficient separation at the boundaries between different
classes. More importantly, we address the scalability problem of learning the
Mahalanobis distance matrix in the presence of high-dimensional feature
vectors, which is a critical issue of distance metric learning. As indicated
in a theorem in [18], a positive semidefinite trace-one matrix can always be
decomposed as a convex combination of a set of rank-one matrices. This theorem
has inspired us to develop a fast optimization algorithm that works in the
style of gradient descent. At each iteration, it only needs to find the
principal eigenvector of a matrix of size ${D\times D}$ ($D$ is the
dimensionality of the input data) and a simple matrix update. This process
incurs much less computational overhead than the metric learning algorithms in
the literature [23, 2]. Moreover, thanks to the above theorem, this process
automatically preserves the p.s.d. property of the Mahalanobis matrix. To
verify its effectiveness and efficiency, the proposed algorithm is tested on a
few benchmark data sets and is compared with the state-of-the-art distance
metric learning algorithms. As experimentally demonstrated, $k$-NN with the
Mahalanobis distance learned by our algorithms attains comparable (sometimes
slightly better) classification accuracy. Meanwhile, in terms of the
computation time, the proposed algorithm has much better scalability in terms
of the dimensionality of input feature vectors.
We briefly review some related work before we present our work. Given a
classification task, some previous work on learning a distance metric aims to
find a metric that makes the data in the same class close and separates those
in different classes from each other as far as possible. Xing _et al_. [25]
proposed an approach to learn a Mahalanobis distance for supervised
clustering. It minimizes the sum of the distances among data in the same class
while maximizing the sum of the distances among data in different classes.
Their work shows that the learned metric could improve clustering performance
significantly. However, to maintain the p.s.d. property, they have used
projected gradient descent and their approach has to perform a full eigen-
decomposition of the Mahalanobis matrix at each iteration. Its computational
cost rises rapidly when the number of features increases, and this makes it
less efficient in coping with high-dimensional data. Goldberger _et al_. [7]
developed an algorithm termed Neighborhood Component Analysis (NCA), which
learns a Mahalanobis distance by minimizing the leave-one-out cross-validation
error of the $k$-NN classifier on the training set. NCA needs to solve a non-
convex optimization problem, which might have many local optima. Thus it is
critically important to start the search from a reasonable initial point.
Goldberger _et al_. have used the result of linear discriminant analysis as
the initial point. In NCA, the variable to optimize is the projection matrix.
The work closest to ours is Large Margin Nearest Neighbor (LMNN) [23] in the
sense that it also learns a Mahalanobis distance in the large margin
framework. In their approach, the distances between each sample and its
“target neighbors” are minimized while the distances among the data with
different labels are maximized. A convex objective function is obtained and
the resulting problem is a semidefinite program (SDP). Since conventional
interior-point based SDP solvers can only solve problems of up to a few
thousand variables, LMNN has adopted an alternating projection algorithm for
solving the SDP problem. At each iteration, similar to [25], also a full
eigen-decomposition is needed. Our approach is largely inspired by their work.
Our work differs LMNN [23] in the following: (1) LMNN learns the metric from
the pairwise distance information. In contrast, our algorithm uses examples of
proximity comparisons among triples of objects (e.g., example $i$ is closer to
example $j$ than example $k$). In some applications like image retrieval, this
type of information could be easier to obtain than to tag the actual class
label of each training image. Rosales and Fung [16] have used similar ideas on
metric learning; (2) More importantly, we design an optimization method that
has a clear advantage on computational efficiency (we only need to compute the
leading eigenvector at each iteration). The optimization problems of [23] and
[16] are both SDPs, which are computationally heavy. Linear programs (LPs) are
used in [16] to approximate the SDP problem. It remains unclear how well this
approximation is.
The problem of learning a kernel from a set of labeled data shares
similarities with metric learning because the optimization involved has
similar formulations. Lanckriet _et al_. [11] and Kulis _et al_. [10]
considered learning p.s.d. kernels subject to some pre-defined constraints. An
appropriate kernel can often offer algorithmic improvements. It is possible to
apply the proposed gradient descent optimization technique to solve the kernel
learning problems. We leave this topic for future study.
The rest of the paper is organized as follows. Section II presents the convex
formulation of learning a Mahalanobis metric. In Section III, we show how to
efficiently solve the optimization problem by a specialized gradient descent
procedure, which is the main contribution of this work. The performance of our
approach is experimentally demonstrated in Section IV. Finally, we conclude
this work in Section V.
## II Large-Margin Mahalanobis Metric Learning
In this section, we propose our distance metric learning approach as follows.
The intuition is to find a particular distance metric for which the margin of
separation between the classes is maximized. In particular, we are interested
in learning a quadratic Mahalanobis metric.
Let $\mathbf{a}_{i}\in\mathbb{R}^{D}(i=1,2,\cdots,n)$ denote a training sample
where $n$ is the number of training samples and $D$ is the number of features.
To learn a Mahalanobis distance, we create a set $\mathcal{S}$ that contains a
group of training triplets as
$\mathcal{S}=\\{(\mathbf{a}_{i},\mathbf{a}_{j},\mathbf{a}_{k})\\}$, where
$\mathbf{a}_{i}$ and $\mathbf{a}_{j}$ come from the same class and
$\mathbf{a}_{k}$ belongs to different classes. A Mahalanobis distance is
defined as follows. Let $\mathbf{P}\in\mathbb{R}^{D\times d}$ denote a linear
transformation and $\mathbf{dist}$ be the squared Euclidean distance in the
transformed space. The squared distance between the projections of
$\mathbf{a}_{i}$ and $\mathbf{a}_{j}$ writes:
$\mathbf{dist}_{ij}=\|\mathbf{P}^{{\\!\top}}\mathbf{a}_{i}-\mathbf{P}^{{\\!\top}}\mathbf{a}_{j}\|^{2}_{2}=(\mathbf{a}_{i}-\mathbf{a}_{j})^{{\\!\top}}\mathbf{P}\mathbf{P}^{{\\!\top}}(\mathbf{a}_{i}-\mathbf{a}_{j}).$
(1)
According to the class memberships of $\mathbf{a}_{i}$, $\mathbf{a}_{j}$ and
$\mathbf{a}_{k}$, we wish to achieve
$\mathbf{dist}_{ik}\geq\mathbf{dist}_{ij}$ and it can be obtained as
$(\mathbf{a}_{i}-\mathbf{a}_{k})^{{\\!\top}}\mathbf{P}\mathbf{P}^{{\\!\top}}(\mathbf{a}_{i}-\mathbf{a}_{k})\geq(\mathbf{a}_{i}-\mathbf{a}_{j})^{{\\!\top}}\mathbf{P}\mathbf{P}^{{\\!\top}}(\mathbf{a}_{i}-\mathbf{a}_{j}).$
(2)
It is not difficult to see that this inequality is generally not a convex
constrain in $\mathbf{P}$ because the difference of quadratic terms in
$\mathbf{P}$ is involved. In order to make this inequality constrain convex, a
new variable $\mathbf{X}=\mathbf{P}\mathbf{P}^{{\\!\top}}$ is introduced and
used throughout the whole learning process. Learning a Mahalanobis distance is
essentially learning the Mahalanobis matrix $\mathbf{X}$. (2) becomes linear
in $\bf X$. This is a typical technique to convexify a problem in convex
optimization [2].
### II-A Maximization of a soft margin
In our algorithm, a margin is defined as the difference between
$\mathbf{dist}_{ik}$ and $\mathbf{dist}_{ij}$, that is,
$\begin{array}[]{ll}\rho_{r}=(\mathbf{a}_{i}-\mathbf{a}_{k})^{{\\!\top}}{\mathbf{X}}(\mathbf{a}_{i}-\mathbf{a}_{k})-(\mathbf{a}_{i}-\mathbf{a}_{j})^{{\\!\top}}{\mathbf{X}}(\mathbf{a}_{i}-\mathbf{a}_{j}),\\\
\forall(\mathbf{a}_{i},\mathbf{a}_{j},\mathbf{a}_{k})\in\mathcal{S},~{}~{}r=1,2,\cdots,|\mathcal{S}|.\end{array}$
(3)
Similar to the large margin principle that has been widely used in machine
learning algorithms such as support vector machines and boosting, here we
maximize this margin (3) to obtain the optimal Mahalanobis matrix
$\mathbf{X}$. Clearly, the larger is the margin $\rho_{r}$, the better metric
might be achieved. To enable some flexibility, _i.e_., to allow some
inequalities of (2) not to be satisfied, a soft-margin criterion is needed.
Considering these factors, we could define the objective function for learning
$\mathbf{X}$ as
$\begin{array}[]{lll}&\\!\\!\max_{\rho,\mathbf{X},\boldsymbol{\xi}}\quad{\rho-C\sum_{r=1}^{|\mathcal{S}|}\xi_{r}},\,\,\mathop{\mathrm{subject~{}to}}\nolimits\\\
&\mathbf{X}\succcurlyeq 0,\mathbf{Tr}(\mathbf{X})=1,\\\ &\xi_{r}\geq
0,~{}~{}r=1,2,\cdots,|\mathcal{S}|,\\\
&(\mathbf{a}_{i}-\mathbf{a}_{k})^{{\\!\top}}\mathbf{X}(\mathbf{a}_{i}-\mathbf{a}_{k})-(\mathbf{a}_{i}-\mathbf{a}_{j})^{{\\!\top}}\mathbf{X}(\mathbf{a}_{i}-\mathbf{a}_{j})\geq\rho-\xi_{r},\\\
&\forall(\mathbf{a}_{i},\mathbf{a}_{j},\mathbf{a}_{k})\in\mathcal{S},\end{array}$
(4)
where $\mathbf{X}\succcurlyeq 0$ constrains $\mathbf{X}$ to be a p.s.d. matrix
and $\mathbf{Tr}(\mathbf{X})$ denotes the trace of $\mathbf{X}$. $r$ indexes
the training set $\mathcal{S}$ and $|\mathcal{S}|$ denotes the size of
$\mathcal{S}$. $C$ is an algorithmic parameter that balances the violation of
(2) and the margin maximization. $\xi\geq 0$ is the slack variable similar to
that used in support vector machines and it corresponds to the soft-margin
hinge loss. Enforcing $\mathbf{Tr}(\mathbf{X})=1$ removes the scale ambiguity
because the inequality constrains are scale invariant. To simplify exposition,
we define
$\mathbf{A}^{r}=(\mathbf{a}_{i}-\mathbf{a}_{k})(\mathbf{a}_{i}-\mathbf{a}_{k})^{{\\!\top}}-(\mathbf{a}_{i}-\mathbf{a}_{j})(\mathbf{a}_{i}-\mathbf{a}_{j})^{{\\!\top}}.$
(5)
ssss Therefore, the last constraint in (4) can be written as
$\big{<}\mathbf{A}^{r},\mathbf{X}\big{>}\geq\rho-\xi_{r},\quad
r=1,\cdots,|S|.$ (6)
Note that this is a linear constrain on $\mathbf{X}$. Problem (4) is thus a
typical SDP problem since it has a linear objective function and linear
constraints plus a p.s.d. conic constraint. One may solve it using off-the-
shelf SDP solvers like CSDP [1]. However, directly solving the problem (4)
using those standard interior-point SDP solvers would quickly become
computationally intractable with the increasing dimensionality of feature
vectors. We show how to efficiently solve (4) in a fashion of first-order
gradient descent.
### II-B Employment of a differentiable loss function
It is proved in [18] that a p.s.d. matrix can always be decomposed as a linear
convex combination of a set of rank-one matrices. In the context of our
problem, this means that $\mathbf{X}=\sum_{i}\theta_{i}\mathbf{Z}_{i}$, where
$\mathbf{Z}_{i}$ is a rank-one matrix and $\mathbf{Tr}(\mathbf{Z}_{i})=1$.
This important result inspires us to develop a gradient descent based
optimization algorithm. In each iteration, $\mathbf{X}$ can be updated as
$\mathbf{X}_{i+1}=\mathbf{X}_{i}+\alpha(\delta{\mathbf{X}}-{\mathbf{X}}_{i})=\mathbf{X}_{i}+\alpha\mathbf{p}_{i},~{}~{}~{}0\leq\alpha\leq
1,$ (7)
where $\delta\mathbf{X}$ is a rank-one and trace-one matrix. $\mathbf{p}_{i}$
is the search direction. It is straightforward to verify that ${\bf Tr}({\bf
X}_{i+1})=1$, and ${\bf X}_{i+1}\succcurlyeq 0$ hold. This is the starting
point of our gradient descent algorithm. With this update strategy, the trace-
one and positive semidefinteness of $\bf X$ is always retained. We show how to
calculate this search direction in Algorithm LABEL:ALG:2. Although it is
possible to use subgradient methods to optimize non-smooth objective
functions, we use a differentiable objective function instead so that the
optimization procedure is simplified (standard gradient descent can be
applied). So, we need to ensure that the objective function is differentiable
with respect to the variables $\rho$ and $\mathbf{X}$.
Let $f(\cdot)$ denote the objective function and $\lambda(\cdot)$ be a loss
function. Our objective function can be rewritten as
$f(\mathbf{X},\rho)=\rho-C\cdot\sum_{r=1}^{|\mathcal{S}|}{\lambda\left(\big{<}\mathbf{A}^{r},\mathbf{X}\big{>}-\rho\right)}.$
(8)
The above problem (4) adopts the hinge loss function that is defined as
$\lambda(z)=\max(0,-z)$. However, the hinge loss is not differentiable at the
point of $z=0$, and standard gradient-based optimization cam be applied
directly. In order to make standard gradient descent methods applicable, we
propose to use differentiable loss functions, for example, the squared hinge
loss or Huber loss functions as discussed below.
Figure 1: The hinge loss, squared hinge loss and Huber loss.
The squared hinge loss function can be represented as
$\displaystyle\lambda\left(\big{<}\mathbf{A}^{r},\mathbf{X}\big{>}-\rho\right)=$
$\displaystyle\quad\left\\{\begin{array}[]{ll}0,&\mbox{if
}\left(\big{<}\mathbf{A}^{r},\mathbf{X}\big{>}-\rho\right)\geq 0,\\\
\left(\big{<}\mathbf{A}^{r},\mathbf{X}\big{>}-\rho\right)^{2},&\mbox{if
}\left(\big{<}\mathbf{A}^{r},{\mathbf{X}}\big{>}-\rho\right)<0.\end{array}\right.$
(11)
As shown in Fig. 1, this function connects the positive and zero segments
smoothly and it is differentiable everywhere including the point $z=0$. We
also consider the Huber loss function in this work:
$\displaystyle\lambda\left(\big{<}\mathbf{A}^{r},\mathbf{X}\big{>}-\rho\right)=$
$\displaystyle\quad\left\\{\begin{array}[]{ll}0,&\mbox{if
}\left(\big{<}\mathbf{A}^{r},\mathbf{X}\big{>}-\rho\right)\geq h,\\\
\frac{\left(h-\left(\left<\mathbf{A}^{r},\mathbf{X}\right>-\rho\right)\right)^{2}}{4h},&\mbox{if
}{-h}<\left(\big{<}\mathbf{A}^{r},\mathbf{X}\big{>}-\rho\right)<h,\\\ \\\
-(\left<\mathbf{A}^{r},\mathbf{X}\right>-\rho),&\mbox{if
}\left(\big{<}\mathbf{A}^{r},\mathbf{X}\big{>}-\rho\right)\leq-h,\end{array}\right.$
(16)
where $h$ is a parameter whose value is usually between $0.01$ and $0.5$. A
Huber loss function with $h=0.5$ is plotted in Fig. 1. There are three
different parts in the Huber loss function, and they together form a
continuous and differentiable function. This loss function approaches the
hinge loss curve when $h\to 0$. Although the Huber loss is more complicated
than the squared hinge loss, its function value increases linearly with the
value of $\big{<}\mathbf{A}^{r},\mathbf{X}\big{>}-\rho$. Hence, when a
training set contains outliers or samples heavily contaminated by noise, the
Huber loss might give a more reasonable (milder) penalty than the squared
hinge loss does. We discuss both loss functions in our experimental study.
Again, we highlight that by using these two loss functions, the cost function
$f(\mathbf{X},\rho)$ that we are going to optimization becomes differentiable
with respect to both $\mathbf{X}$ and $\rho$.
Algorithm 1 The proposed optimization algorithm.
6
6
6
6
6
6
Algorithm 2 Compute $\mathbf{X}_{k}$ in the proposed algorithm.
4
4
4
4
## III A scalable and fast optimization algorithm
The proposed algorithm maximizes the objective function iteratively, and in
each iteration the two variables $\mathbf{X}$ and $\rho$ are optimized
alternatively. Note that the optimization in this alternative strategy retains
the global optimum because $f(\mathbf{X},\rho)$ is a convex function in both
variables $(\mathbf{X},\rho)$ and $(\mathbf{X},\rho)$ are not coupled
together. We summarize the proposed algorithm in Algorithm LABEL:ALG:0. Note
that $\rho_{k}$ is a scalar and Line 3 in Algorithm LABEL:ALG:0 can be solved
directly by a simple one-dimensional maximization process. However,
$\mathbf{X}$ is a p.s.d. matrix with size of $D\times D$. Recall that $D$ is
the dimensionality of feature vectors. The following section presents how
$\mathbf{X}$ is efficiently optimized in our algorithm.
### III-A Optimizing for the Mahalanobis matrix ${\mathbf{X}}_{k}$
Let $\mathcal{P}=\\{\mathbf{X}\in\mathbb{R}^{D\times D}:\mathbf{X}\succcurlyeq
0,\mathbf{Tr}(\mathbf{X})=1\\}$ be the domain in which a feasible $\mathbf{X}$
lies. Note that $\mathcal{P}$ is a convex set of $\mathbf{X}$. As shown in
Line 4 in Algorithm LABEL:ALG:0, we need to solve the following maximization
problem:
$\underset{\mathbf{X}\in\mathcal{P}}{\max}\,\,\,{f(\mathbf{X},\rho_{k})},$
(17)
where $\rho_{k}$ is the output of Line 3 in Algorithm LABEL:ALG:0. Our
algorithm offers a simple and efficient way for solving this problem by
explicitly maintaining the positive semidefiniteness property of the matrix
$\mathbf{X}$. It needs only compute the largest eigenvalue and the
corresponding eigenvector whereas most previous approaches such as the method
of [23] require a full eigen-decomposition of $\mathbf{X}$. Their
computational complexities are $O(D^{2})$ and $O(D^{3})$, respectively. When
$D$ is large, this computational complexity difference could be significant.
Let $\nabla f(\mathbf{X},\rho_{k})$ be the gradient matrix of $f(\cdot)$ with
respect to $\mathbf{X}$ and $\alpha$ be the step size for updating
$\mathbf{X}$. Recall that we update $\mathbf{X}$ in such a way that
$\mathbf{X}_{i+1}=(1-\alpha)\mathbf{X}_{i+1}+\alpha\delta\mathbf{X}$, where
$\mathbf{rank}(\delta\mathbf{X})=1$ and $\mathbf{Tr}(\delta\mathbf{X})=1$. To
find the $\delta\mathbf{X}$ that satisfies these constraints and in the
meantime can best approximate the gradient matrix $\nabla
f(\mathbf{X},\rho_{k})$, we need to solve the following optimization problem:
$\displaystyle\max_{\delta\bf X}\quad$ $\displaystyle\big{<}\nabla
f(\mathbf{X},\rho_{k}),\delta\mathbf{X}\big{>}$
$\displaystyle\mathop{\mathrm{subject~{}to}}\nolimits\,\,\,$
$\displaystyle\mathbf{rank}(\delta\mathbf{X})=1,\mathbf{Tr}(\delta\mathbf{X})=1.$
(18)
The optimal $\delta{\mathbf{X}}^{\star}$ is exactly
$\mathbf{v}\mathbf{v}^{{\\!\top}}$ where $\mathbf{v}$ is the eigenvector of
$\nabla f(\mathbf{X},\rho_{k})$ that corresponds to the largest eigenvalue.
The constraints says that $\delta{\mathbf{X}}$ is a outer product of a unit
vector: $\delta{\mathbf{X}}={\bf v}{\bf v}^{\\!\top}$ with $||{\bf
v}||_{2}=1$. Here $||\cdot||_{2}$ is the Euclidean norm. Problem (III-A) can
then be written as: $\max_{\bf v}{\bf v}^{\\!\top}[\nabla
f(\mathbf{X},\rho_{k})]\bf v$, subject to $||{\bf v}||_{2}=1$. It is clear now
that an eigen-decomposition gives the solution to the above problem.
Hence, to solve the above optimization, we only need to compute the leading
eigenvector of the matrix $\nabla f(\mathbf{X},\rho_{k})$. Note that
$\mathbf{X}$ still retains the properties of $\mathbf{X}\succcurlyeq
0,\mathbf{Tr}(\mathbf{X})=1$ after applying this process.
Clearly, a key parameter of this optimization process is $\alpha$ which
implicitly decides the total number of iterations. The computational overhead
of our algorithm is proportional to the number of iterations. Hence, to
achieve a fast optimization process, we need to ensure that in each iteration
the $\alpha$ can lead to a sufficient reduction on the value of $f$. This is
discussed in the following part.
### III-B Finding the optimal step size $\alpha$
We employ the backtracking line search algorithm in [15] to identify a
suitable $\alpha$. It reduces the value of $\alpha$ until the Wolfe conditions
are satisfied. As shown in Algorithm LABEL:ALG:2, the search direction is
$\mathbf{p}_{i}=\mathbf{v}_{i}\mathbf{v}_{i}^{{\\!\top}}-\mathbf{X}_{i}$. The
Wolfe conditions that we use are
$f(\mathbf{X}_{i}+\alpha\mathbf{p}_{i},\rho_{i})\leq
f(\mathbf{X}_{i},\rho_{i})+c_{1}\alpha\mathbf{p}_{i}^{{\\!\top}}\nabla
f(\mathbf{X}_{i},\rho_{i}),$ $\big{|}\mathbf{p}_{i}^{{\\!\top}}\nabla
f(\mathbf{X}_{i}+\alpha\mathbf{p}_{i},\rho_{i})\big{|}\leq
c_{2}\big{|}\mathbf{p}_{i}^{{\\!\top}}\nabla
f(\mathbf{X}_{i},\rho_{i})\big{|},$ (19)
where $0<c_{1}<c_{2}<1$. The result of backtracking line search is an
acceptable $\alpha$ which can give rise to sufficient reduction on the
function value of $f(\cdot)$. We show in the experiments that with this
setting our optimization algorithm can achieve higher computational efficiency
than some of the existing solvers.
## IV Experiments
The goal of these experiments is to verify the efficiency of our algorithm in
achieving comparable (or sometimes even better) classification performances
with a reduced computational cost. We perform experiments on $10$ data sets
described in Table I. For some data sets, PCA is performed to remove noises
and reduce the dimensionality. The metric learning algorithms are then run on
the data sets pre-processed by PCA. The Wine, Balance, Vehicle, Breast-Cancer
and Diabetes data sets are obtained from UCI Machine Learning Repository [14],
and USPS, MNIST and Letter are from LibSVM [3] For MNIST, we only use its test
data in our experiment. The ORLface data is from att
research111http://www.uk.research.att.com/facedatabase.html and Twin-Peaks is
downloaded from L. van der Maaten’s website222http://ticc.uvt.nl/lvdrmaaten/.
The Face and Background classes (435 and 520 images respectively) in the image
retrieval experiment are obtained from the Caltech-101 object database [6]. In
order to perform statistics analysis, the ORLface, Twin-Peaks, Wine, Balance,
Vehicle, Diabetes and Face-Background data sets are randomly split as 10 pairs
of train/validation/test subsets and experiments on those data set are
repeated 10 times on each split.
TABLE I: The ten benchmark data sets used in the experiment. Missing entries in “dimension after PCA” indicate no PCA processing. | # training | # validation | # test | dimension | dimension after PCA | # classes | # runs | # triplets for SDPMetric
---|---|---|---|---|---|---|---|---
USPSPCA | 5,833 | 1,458 | 2,007 | 256 | 60 | 10 | 1 | 52,497
USPS | 5,833 | 1,458 | 2,007 | 256 | | 10 | 1 | 5,833
MNISTPCA | 7,000 | 1,500 | 1,500 | 784 | 60 | 10 | 1 | 54,000
MNIST | 7,000 | 1,500 | 1,500 | 784 | | 10 | 1 | 7,000
Letter | 10,500 | 4,500 | 5,000 | 16 | | 26 | 1 | 94,500
ORLface | 280 | 60 | 60 | 2,576 | 42 | 40 | 10 | 280
Twin-Peaks | 14,000 | 3,000 | 3,000 | 3 | | 11 | 10 | 14,000
Wine | 126 | 26 | 26 | 13 | | 3 | 10 | 1,134
Balance | 439 | 93 | 93 | 4 | | 3 | 10 | 3,951
Vehicle | 593 | 127 | 126 | 18 | | 4 | 10 | 5,337
Breast-Cancer | 479 | 102 | 102 | 10 | | 2 | 10 | 4,311
Diabetes | 538 | 115 | 115 | 8 | | 2 | 10 | 4,842
Face-Background | 472 | 101 | 382 | 100 | | 2 | 10 | 4,428
The $k$-NN classifier with the Mahalanobis distance learned by our algorithm
(termed SDPMetric in short) is compared with the $k$-NN classifiers using a
simple Euclidean distance (“Euclidean” in short) and that learned by the Large
Margin Nearest Neighbor in [23] (LMNN333In our experiment, we have used the
implementation of LMNN’s authors. Note that to be consistent with the setting
in [23], LMNN here also uses the “obj=1” option and updates the projection
matrix to speed up its computation. If we update the distance matrix directly
to get global optimum, LMNN would be much more slower due to full eigen-
decomposition at each iteration. in short). Since Weinberger _et al_. [23] has
shown that LMNN obtains the classification performance comparable to support
vector machines on some data sets, we focus on the comparison between our
algorithm and LMNN, which is considered as the state-of-the-art. To prepare
the training triplet set $\mathcal{S}$, we apply the $3$-NN method to these
data sets and generate the training triplets for our algorithm. The training
data sets for LMNN is also generated using $3$-NN, except that the Twin-peaks
and ORLface are applied with the $1$-NN method. Also, the experiment compares
the two variants of our proposed SDPMetric, which use the squared hinge loss
(denoted as SDPMetric-S) and the Huber loss(SDPMetric-H), respectively. We
split each data set into 70/15/15% randomly and refer to those split sets as
training, cross validation and test sets except pre-separated data sets
(Letter and USPS) and Face-Background which was made for image retrieval.
Following [23], LMNN uses 85/15% data for training and testing. The training
data is also split into 70/15% in LMNN for cross validation to be consistent
with our SDPMetric. Since USPS data set has been split into training/test
already, only the training data are divided into 70/15% for training and
validation. The Letter data set is separated according to Hsu and Lin [9].
Same as in [23], PCA is applied to USPS, MNIST and ORLface to reduce the
dimensionality of feature vectors.
The following experimental study demonstrates that our algorithm achieves
slightly better classification accuracy rates with a much less computational
cost than LMNN on most of the tested data sets. The detailed test error rates
and timing results are reported in Tables II and III. As we can see, the test
error rates of SDPMetric-S are comparable to those of LMNN. SDPMetric-H
achieves lower misclassification error rates than LMNN and the Euclidean
distance on most of data sets except Face-Background data (which is treated as
an image retrieval problem) and MNIST, on which SDPMetric-S achieves a lower
error rate. Overall, we can conclude that the proposed SDPMetric either with
squared hinge loss or Huber loss is at least comparable to (or sometimes
slightly better than) the state-of-the-art LMNN method in terms of
classification performance.
Before reporting the timing result on these benchmark data sets, we compared
our algorithm (SDPMetric-H) with two convex optimization solvers, namely,
SeDuMi [20] and SDPT3 [21] which are used as internal solvers in the
disciplined convex programming software CVX [8]. Both SeDuMi and SDPT3 use
interior-point based methods. To perform eigen-decomposition, our SDPMetric
uses ARPACK [19], which is designed to solve large scale eigenvalue problems.
Our SDPMetric is implemented in standard C/C++. Experiments have been
conducted on a standard desktop. We randomly generated $1,000$ training
triplets and gradually increase the dimensionality of feature vectors from
$20$ to $100$. Fig. 2 illustrates computational time of ours, CVX/SeDuMi and
CVX/SDPT3. As shown, the computational load of our algorithm almost keeps
constant as the dimensionality increases. This might be because the proportion
of eigen-decomposition’s CPU time does not dominate with dimensions varying
from $20$ to $100$ in SDPMetric on this data set. In contrast, the
computational loads of CVX/SeDuMi and CVX/SDPT3 increase quickly in this
course. In the case of the dimension of $100$, the difference on CPU time can
be as large as $800\sim 1000$ seconds. This shows the inefficiency and poor
scalability of standard interior-point methods. Secondly, the computational
time of LMNN, SDPMetric-S and SDPMetric-H on these benchmark data sets are
compared in Table III. As shown, LMNN is always slower than the proposed
SDPMetric which converges very fast on these data sets. Especially, on the
Letter and Twin-Peaks data sets, SDPMetric shows significantly improved
computational efficiency.
Figure 2: Computational time versus the dimensionality of feature vectors.
Face-Background data set consists of the two object classes, Face-easy and
Background-Google in [6], as a retrieval problem. The images in the class of
Background-Google are randomly collected from the Internet and they are used
to represent the non-target class. For each image, a number of interest
regions are identified by the Harris-Affine detector [13] and the visual
content in each region is characterized by the SIFT descriptor [12]. A
codebook of size $100$ is created by using $k$-means clustering. Each image is
then represented by a $100$-dimensional histogram vector containing the number
of occurrences of each visual word. We evaluate retrieval accuracy using each
facial image in a test subset as a query. For each compared metric, the
accuracy of the retrieved top $1$ to $20$ images are computed, which is
defined as the ratio of the number of facial images to the total number of
retrieved images. We calculate the average accuracy of each test subset and
then average over the whole $10$ test subsets. Fig. 3 shows the retrieval
accuracies of the Mahalanobis distances learned by Euclidean, LMNN and
SDPMetric. Clearly we can observe that SDPMetric-H and SDPMetric-S
consistently present higher retrieval accuracy values, which again verifies
their advantages over the LMNN method and Euclidean distance.
TABLE II: $3$-Nearest Neighbor misclassification error rates. The standard deviation values are in brackets. The best results are highlighted in bold. | Euclidean | LMNN | SDPMetric-S | SDPMetric-H
---|---|---|---|---
USPSPCA | 5.63 | 5.18 | 5.28 | 5.18
USPS | 4.42 | 3.58 | 4.36 | 3.21
MNISTPCA | 3.15 | 3.15 | 3.00 | 3.35
MNIST | 4.80 | 4.60 | 4.13 | 4.53
Letter | 5.38 | 4.04 | 3.60 | 3.46
ORLface | 6.00 (3.46) | 5.00 (2.36) | 4.75 (2.36) | 4.25 (2.97)
Twin-Peaks | 1.03 (0.21) | 0.90 (0.19) | 1.17 (0.20) | 0.79 (0.19)
Wine | 24.62 (5.83) | 3.85 (2.72) | 3.46 (2.69) | 3.08 (2.31)
Bal | 19.14 (1.59) | 14.19 (4.12) | 9.78 (3.17) | 10.32 (3.44)
Vehicle | 28.41 (2.41) | 21.59 (2.71) | 21.67 (4.00) | 20.87 (2.97)
Breast-Cancer | 4.51 (1.49) | 4.71 (1.61) | 3.33 (1.40) | 2.94 (0.88)
Diabetes | 28.00 (2.84) | 27.65 (3.45) | 28.70 (3.67) | 27.64 (3.71)
Face-Background | 26.41 (2.72) | 14.71 (1.33) | 16.75 (1.72) | 15.86 (1.37)
TABLE III: Computational time for each run. | LMNN | SDPMetric-S | SDPMetric-H
---|---|---|---
USPSPCA | 256s | 111s | 258s
USPS | 1.6h | 16m | 20m
MNISTPCA | 219s | 111s | 99s
MNIST | 9.7h | 1.4h | 37m
Letter | 1036s | 6s | 136s
ORLface | 13s | 4s | 3s
Twin-peakes | 595s | less than 1s | less than 1s
Wine | 9s | 2s | 2s
Bal | 7s | less than 1s | 2s
Vehicle | 19s | 2s | 7s
Breast-Cancer | 4s | 2s | 3s
Diabetes | 10s | less than 1s | 2s
Face-Background | 92s | 5s | 5s
Figure 3: Retrieval performances of SDPMetric-S, SDPMetric-H, LMNN and the
Euclidean distance. The curves of SDPMetric-S and SDPMetric-H are very close.
## V Conclusion
We have proposed a new algorithm to demonstrate how to efficiently learn a
Mahalanobis distance metric with the principle of margin maximization.
Enlightened by the important theorem on p.s.d. matrix decomposition in [18],
we have designed a gradient descent method to update the Mahalanobis matrix
with cheap computational loads and at the same time, the p.s.d. property of
the learned matrix is maintained during the whole optimization process.
Experiments on benchmark data sets and the retrieval problem verify the
superior classification performance and computational efficiency of the
proposed distance metric learning algorithm.
The proposed algorithm may be used to solve more general SDP problems in
machine learning. To look for other applications is one of the future research
directions.
## References
* [1] B. Borchers, “CSDP, a C library for semidefinite programming,” _Optim. Methods and Softw._ , vol. 11, no. 1, pp. 613–623, 1999.
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* [19] D. C. Sorensen, “Implicitly restarted Arnoldi/Lanczos methods for large scale eigenvalue calculations,” Tech. Rep., 1996.
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* [22] V. Vapnik, _Statistical learning theory_. New York: John Wiley and Sons Inc., 1998.
* [23] K. Q. Weinberger, J. Blitzer, and L. K. Saul, “Distance metric learning for large margin nearest neighbor classification,” in _Proc. Adv. Neural Inf. Process. Syst._ , Vancouver, Canada, Dec. 2006, pp. 1475–1482.
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|
arxiv-papers
| 2010-03-02T01:12:34 |
2024-09-04T02:49:08.669719
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chunhua Shen, Junae Kim, and Lei Wang",
"submitter": "Chunhua Shen",
"url": "https://arxiv.org/abs/1003.0487"
}
|
1003.0506
|
# Double occupancies in confined attractive fermions on optical lattices
Ji-Hong Hu Department of Physics, Zhejiang Normal University, Jinhua 340012,
China Gao Xianlong gaoxl@zjnu.edu.cn Department of Physics, Zhejiang Normal
University, Jinhua 340012, China
###### Abstract
We perform a numerical study of a one-dimensional Fermion-Hubbard model in
harmonic traps within the Thomas-Fermi approximation based on the exact Bethe-
ansatz solution. The $\rho-U/t$ phase diagram is shown for the systems of
attractive interactions ($\rho$ is the characteristic density and $U/t$ the
interaction strength scaled in units of the hopping parameter.). We study the
double occupancy, the local central density and their derivatives. Their roles
are discussed in details in detecting the composite phases induced by the
trapping potential.
###### pacs:
05.30.Fk,03.75.Ss,71.10.Pm,71.15.Pd
The recent remarkable experimental progress in cooling and manipulating
ultracold atomic gases in optical lattices provides us an alternative way to
investigate the many-body effects in condensed matter system review paper .
For the optical lattices loaded with atomic gases, the on-site interaction
between different species can be tuned to be very strong by means of a
technique named Feshbach resonance Feshbach or by increasing the lattice
depth to decrease the hopping between neighbor sites. Thus the outstanding
controllability offers the possibility to use these systems to ”quantum
simulate” quantum many-body physics in a well-controlled way. An example is
the experimentally realized Tonks-Girardeau gas in a one-dimensional (1D)
optical lattice Paredes .
In the experiments, the trapping potential used to confine atoms induces
inhomogeneity and complexity in characterizing the quantum phases. Though a
few techniques exist in experiments observing the new quantum phases and
probing various properties in ultracold gases in optical lattices, such as the
time-of-flight imaging of the momentum distribution, noise correlations and
Bragg spectroscopy review paper , there are difficulties in observing the
composite phases induced by the inherent trapping potential, for example, the
incompressible bulk Mott- and band-insulating phases surrounded by the
compressible fluid phases at the edges in the 1D Fermi-Hubbard model under
harmonic confinement. Scarola et al. proposed to use the core compressibility
by filtering out the edge effects, which offers a direct probe of
incompressible phases independent of inhomogeneity Scarola . Duan et al.
suggested to use the Fourier sampling of time-of-flight images to reveal new
correlation functions Duan . Kollath et al. used the double occupancy (DO)
induced by periodic lattice modulations via the creation of molecules to
identify the pair-binding energy and the spin-ordering in a Fermi gas in an
optical lattice Kollath , which was recently measured in the experiments by
molecule formation and used to identify an incompressible Mott-insulator phase
of two hyperfine states of strongly repulsive fermionic 40K atoms Jordens .
The compressibility of the quantum gas in the trap was used recently in
identifying the incompressible Mott-insulating phase at strong interactions by
independent control of external confinement and lattice depth Schneider . Here
in this paper we discuss how to characterize the different composite phases of
1D confined fermions on optical lattices, by investigating observables like
the double occupation sites, the local central density and their derivatives.
We consider a two-component Fermi gas in a tube with $N_{f}$ atoms and $N_{s}$
lattice sites, which can be realized by a strong confinement in the transverse
directions moritz_2005 with an additional periodic potential applied along
the tube and can be described by a one-band inhomogeneous Fermi-Hubbard model
jaksch_98 ,
$\displaystyle\hat{H}_{s}$ $\displaystyle=$
$\displaystyle-t\sum_{i=1,\sigma}^{N_{s}}({\hat{c}}^{\dagger}_{i\sigma}{\hat{c}}_{i+1\sigma}+{H}.{c}.)+U\sum_{i=1}^{N_{s}}\,{\hat{D}}_{i}$
(1) $\displaystyle+V_{2}\sum_{i=1}^{N_{s}}(i-N_{s}/2)^{2}{\hat{n}}_{i}$
where the spin degrees of freedom $\sigma=\uparrow,\downarrow$ are
pseudospin-$1/2$ labels for two internal hyperfine states.
${\hat{c}}_{i\sigma}$ (${\hat{c}}^{\dagger}_{i\sigma}$) are fermionic
operators annihilating (creating) particles with spin $\sigma$ in a Wannier
state at site $i$.
${\hat{n}}_{i}=\sum_{\sigma}{\hat{n}}_{i\sigma}=\sum_{\sigma}{\hat{c}}^{\dagger}_{i\sigma}{\hat{c}}_{i\sigma}$
is the total site occupation operator, $t$ is the tunneling between nearest
neighbors, $U$ is the strength of the on-site attractive interaction,
${\hat{D}}_{i}={\hat{n}}_{i\uparrow}{\hat{n}}_{i\downarrow}$ is the double
occupancy operator at site $i$, and $V_{i}$ describes the strength of the
trapping potential.
The homogeneous 1D Fermi-Hubbard model belongs to the universality class of
Luttinger liquids. At zero temperature, the properties of this model in the
thermodynamic limit ($N_{\sigma},N_{s}\rightarrow\infty$, but with finite
$N_{\sigma}/N_{s}$) are determined by the fillings
$n_{\sigma}=N_{\sigma}/N_{s}$ and by the dimensionless coupling constant
$u=U/t$. According to Lieb and Wu LiebWu , the ground state properties for
different fillings in the thermodynamic limit are described by the coupled
integral equations (details see Refs. KocharianPRB, ; gaoprb78, ).
Figure 1: (Color online) Phase diagram for the one-dimensional inhomogeneous
fermion-Hubbard model as a function of attractive interaction strength $u=U/t$
and characteristic particle density $\rho=N_{f}\sqrt{V_{2}/t}$. For fixed $u$,
the shape of the density profile is completely determined by $\rho$. The band
insulator (BI) phase is determined with the local occupation in the trap
center satisfying $|n_{i}-2|<0.005$. Below the BI phase, it is the Luther-
Emery (LE) liquid phase. The boundary between BI and LE phases is plotted as
open squares ($\square$) with solid lines as a guide for the eye. The solid
circles ($\bullet$) represent the phase boundary determined from the
derivative of the double occupancy, and the solid upper triangles
($\blacktriangle$) represent the one determined from the derivative of the
local central density (See details in the text and in Fig. 4). All of them can
be used as identifying the onset of the BI phase.
Figure 2: (Color online) The ground-state concentration of doubly occupied
sites in the homogeneous system $D^{\rm hom}$ (a) as a function of interaction
strength $u$ for various $n$ and (b) as a function of particle density $n$ for
various interacting strengths $u$. For $0\leq n\leq 2$, $D^{\rm hom}$ is in
the range $0\leq D^{\rm hom}\leq 1$. For fixed $n$, $D^{\rm hom}$ increases
monotonically with increasing $|u|$ and for fixed $u$ it increases
monotonically with increasing $n$. At $U=0$, $D^{\rm hom}=n^{2}/4$ and at
strongly attractive coupling limit $D^{\rm hom}=n/2$. The attractive
interaction favors the formation of pairs between opposite spins. In the inset
of (b), the derivative of the double occupancy with respect to the particle
density $n$ is plotted, which is used to calculate the double occupancy for
the inhomogeneous system in the spirit of local density approximation.
The inhomogeneity of 1D Fermi-Hubbard model caused by the boundary conditions
or the external potential, normally invalidates a reliable analytical method
that usually used in the homogeneous system. Many numerically accurate
schemes, such as the density-matrix renormalization group Machida1 ; gaoprb78
; Molina , quantum-Monto-carlo (QMC) Rigol ; Rigol2004 ; Astrakhardik ; Casula
, exact diagnolization Machida1 ; exact diagnolization ; Nikkarila and
density-functional theory based on the exact Bethe-ansatz solution gaoprb78 ,
are used in obtaining the phase diagram and the collective oscillations of the
atomic mass density. The Thomas-Fermi approximation based on the exact Bethe-
ansatz solution has also been successfully used in characterizing the phase
diagrams Liu . The rich phases induced by the trapping potential or the phase
transitions driven by the system parameters can be identified by the site
occupation, the variance of the local density, the local compressibility
Rigol2004 , or the thermodynamic stiffness (i.e., the inverse of the global
compressibility) gaoprb78 .
Here we study the inhomogeneous Fermi-Hubbard model described by Eq. (1),
which allows for the spatial coexistence of different quantum states Rigol ;
Liu . For attractive interactions, $U<0$, there are two different composite
phases, well classified in the phase diagram defined by a scaled dimensionless
characteristic density $\rho=N_{f}\sqrt{V_{2}/t}$ and $u=U/t$ Rigol : the
Luther-Emery (LE) liquid phase characterized by fermion pairs and atomic
density wave, and band insulator (BI) phase with a plateau of $n_{i}=2$ in the
trap center of tightly bound spin-singlet dimers surrounded by Luther-Emery
layers gaoprb78 .
All these phases have been analyzed in the $\rho-u$ Rigol2004 ; Heiselberg ,
$\mu-u$ ($\mu$ is the chemical potential) Campo , and $n_{0}-\nu$ Liu
($n_{0}$ is the central density and $\nu=2\sqrt{N_{f}}/\pi$ is the
characteristic filling factor) phase diagrams.
Figure 3: (Color online) (a) The ground-state concentration of doubly occupied
sites $D$ as a function of characteristic density $\rho$ in the system under
the presence of trapping potential for various interacting strength $u$. (b)
The local central density $n_{0}$ as a function of characteristic density
$\rho$ for various interacting strength $u$. For comparison, the double
occupancy and the local central density for the noninteracting case ($u=0$)
are also shown in (a) and (b), respectively.
Figure 4: (Color online) (a) The derivative of the ground-state concentration
of doubly occupied sites $D$ as a function of characteristic density $\rho$ in
the system under the presence of trapping potential. In the inset, the local
inverse compressibility $\kappa^{-1}_{i}$ (in units of $t$) is shown as a
function of $i$ with the system parameters determined from the critical
characteristic density on the peaks. (b) The derivative of the local central
density $n_{0}$ as a function of characteristic density $\rho$ for different
attractive interactions. For comparison, the derivatives of the double
occupancy and the local central density for the noninteracting case ($u=0$)
are also shown in (a) and (b), respectively.
The harmonic trapping potential in Eq. (1) induces a position dependence of
the chemical potentials which can be evaluated within the local density
approximation based on the Bethe-ansatz equations. This amounts to determining
the global chemical potential $\mu$ of the system from the local equilibrium
condition,
$\mu=\left.\mu_{\rm hom}(n,u)\right|_{n\rightarrow
n_{i}}+V_{2}(i-N_{s}/2)^{2},$ (2)
while imposing the normalization condition $\sum_{i}n_{i}=N_{f}$. Here
$\mu_{\rm hom}(n,u)$ is the chemical potential of the homogeneous system
calculated from the Bethe-ansatz coupled equations gao_long_2005 ; gaoprb78 .
In Fig. 1, we show the above-mentioned composite phases through the $\rho-u$
phase diagram, which is insensitive to the individual values of $N_{f}$ and
$V_{2}$, that is, for fixed $u$, the shape of the density profile in this
system is completely determined by $\rho$ Rigol . Below the BI phase, it is
the LE liquid phase. We would like to mention that the similar phase diagram
has been obtained by Rigol et al. for the repulsive interaction Rigol and
Heiselberg but with a fitting function for $\mu_{\rm hom}(n,u)$ Heiselberg .
Here we solve the local equilibrium condition with the exact numerical data
for $\mu_{\rm hom}(n,u)$ calculated from the coupled Bethe-ansatz equations.
We conclude that the phase diagram based on the Thomas-Fermi approximation
gives qualitatively correct results when comparing to QMC.
To detect the composite phases induced by the trapping potential, we
investigate firstly the concentration of the doubly occupied sites, which is
defined as the derivative of the ground state energy with respect to $U$ Leo ,
$D=\frac{2}{N_{f}}\sum^{N_{s}}_{i=1}\langle\hat{D}_{i}\rangle=\frac{2}{N_{f}}\frac{\partial
E_{0}}{\partial U}\,.$
Here $E_{0}$ is the ground state energy per site of the system. The DO in the
homogeneous system, $D^{\rm hom}$, can be calculated exactly from the Bethe-
ansatz coupled equations KocharianPRB , which is shown in Fig. 2. Notice, at
$U=0$, we have $D^{\rm hom}=n^{2}/4$ and at strongly attractive coupling limit
$D^{\rm hom}=n/2$. In the presence of trapping potential, we calculate the DO
in the spirit of local density approximation, $D=\frac{2}{N_{f}}\sum_{i}D^{\rm
hom}(n)|_{n\rightarrow n_{i}}$. From Fig. 2, it becomes obvious that the
attractive interaction favors the formation of pairs between the two spin
species while the repulsive interaction suppresses them.
Now we show in Fig. 3 (a) the DO and in Fig. 3 (b) the local central density
$n_{0}$ for the inhomogeneous system described in Eq. (1) as a function of the
characteristic density $\rho$. For large $\rho$, the DO and the central
density saturate at $D\approx 1$ and $n_{0}\approx 2$, respectively,
corresponding to the BI regime, where an incompressible phase of band-
insulating type is realized in the core of the trap. The more attractive of
the interaction is, the sooner for $D$ to reach the value of $1$ and for
$n_{0}$ to reach $2$. These manifest that the attractive interaction favors
pairing, and thus increases DO, different from the case of repulsive
interaction where it suppresses DO and supports Mott-insulating phase.
In the following, we show the derivative of the DO in Fig. 4 (a) and the
derivative of the local central density in Fig. 4 (b), respectively, as probes
for the composite quantum phase transitions and especially as signs for band-
insulating phases. The derivatives of the DO and the local central density are
performed numerically using the central finite difference method. We find that
the singularities in these two quantities provide a further support and clear
signatures of the composite quantum phase transitions accompanying Fig. 3. In
Fig. 4 (a), the phase transition from LE to BI is manifested by an obvious
peak. The onset of the peak gives a critical value of the characteristic
density, which determines the critical particle number for a given $V_{2}/t$.
Based on these parameters, we plot in the inset of Fig. 4 (a) the local
inverse compressibility defined by $\kappa^{-1}_{i}=\partial^{2}E_{0}/\partial
n^{2}|_{n\rightarrow n_{i}}$ as a function of site position. The
incompressible nature of the BI phase is manifested by the disappearance of
$\kappa^{-1}_{i}$, which is compatible with the onset of the peak in
$dD/d\rho$. In Fig. 4 (b), the vanishing of the derivative of the local
central density gives very direct signatures of the incompressible nature of
BI in the center of the trap. The phase boundaries of the onset of the BI
phase are plotted in Fig. 1, which are determined by the appearance of the
peak in the derivative of the DO and the vanishing of the derivative of the
local central density. We find that the phase boundary determined from the DO
slightly overestimates the onset of the BI phase, while the central density
slightly underestimates it.
In conclusion, motivated by earlier theoretical work and by ongoing
experimental efforts, we present a fully numerical study for the ground state
properties of a one-dimensional attractive Fermion-Hubbard model in harmonic
traps within the Thomas-Fermi approximation based on the exact Bethe-ansatz
solution. We investigate the double occupancy, the local central density and
their derivatives. We found they are good indicators of the formation of an
incompressible band-insulating phase in the center of the trap, which can be
detected in the near-future in the experiments in the 1D cold fermionic atomic
gases.
This work was supported by NSF of China under Grant No. 10704066, 10974181,
Qianjiang River Fellow Fund 2008R10029, and Program for Innovative Research
Team in Zhejiang Normal University.
## References
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|
arxiv-papers
| 2010-03-02T05:59:45 |
2024-09-04T02:49:08.677490
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ji-Hong Hu and Gao Xianlong",
"submitter": "Gao Xianlong",
"url": "https://arxiv.org/abs/1003.0506"
}
|
1003.0514
|
# The finite-dimensional Witsenhausen counterexample
Pulkit Grover, Se Yong Park and Anant Sahai
Department of EECS, University of California at Berkeley, CA-94720, USA
{pulkit, sahai, separk}@eecs.berkeley.edu
###### Abstract
Recently, a vector version of Witsenhausen’s counterexample was considered and
it was shown that in that limit of infinite vector length, certain
quantization-based control strategies are provably within a constant factor of
the optimal cost for all possible problem parameters. In this paper, finite
vector lengths are considered with the dimension being viewed as an additional
problem parameter. By applying a large-deviation “sphere-packing” philosophy,
a lower bound to the optimal cost for the finite dimensional case is derived
that uses appropriate shadows of the infinite-length bound. Using the new
lower bound, we show that good lattice-based control strategies achieve within
a constant factor of the optimal cost uniformly over all possible problem
parameters, including the vector length. For Witsenhausen’s original problem —
the scalar case — the gap between regular lattice-based strategies and the
lower bound is numerically never more than a factor of $8$.
## I Introduction
Distributed control problems have long proved challenging for control
engineers. In 1968, Witsenhausen [1] gave a counterexample showing that even a
seemingly simple distributed control problem can be hard to solve. For the
counterexample, Witsenhausen chose a two-stage distributed LQG system and
provided a nonlinear control strategy that outperforms all linear laws. It is
now clear that the non-classical information pattern of Witsenhausen’s problem
makes it quite challenging111In words of Yu-Chi Ho [2], “the simplest problem
becomes the hardest problem.”; the optimal strategy and the optimal costs for
the problem are still unknown — non-convexity makes the search for an optimal
strategy hard [3, 4, 5]. Discrete approximations of the problem [6] are even
NP-complete222More precisely, results in [7] imply that the discrete
counterparts to the Witsenhausen counterexample are NP-complete if the
assumption of Gaussianity of the primitive random variables is relaxed.
Further, it is also shown in [7] that with this relaxation, a polynomial time
solution to the original continuous problem would imply $P=NP$, and thus
conceptually the relaxed continuous problem is also hard. [7].
In the absence of a solution, research on the counterexample has bifurcated
into two different directions. Since there is no known systematic approach to
obtain provably optimal solutions, a body of literature (e.g. [4] [5] [8] and
the references therein) applies search heuristics to explore the space of
possible control actions and obtain intuition into the structure of good
strategies. Work in this direction has also yielded considerable insight into
addressing non-convex problems in general.
In the other direction the emphasis is on understanding the role of implicit
communication in the counterexample. In distributed control, control actions
not only attempt to reduce the immediate control costs, they can also
communicate relevant information to other controllers to help them reduce
costs. Witsenhausen [1, Section 6] and Mitter and Sahai [9] aim at developing
systematic constructions based on implicit communication. Witsenhausen’s two-
point quantization strategy is motivated from the optimal strategy for two-
point symmetric distributions of the initial state [1, Section 5] and it
outperforms linear strategies for certain parameter choices. Mitter and Sahai
[9] propose multipoint-quantization strategies that, depending on the problem
parameters, can outperform linear strategies by an arbitrarily-large factor.
Various modifications to the counterexample investigate if misalignment of
these two goals of control and implicit communication makes the problems hard
[3, 10, 11, 12, 13, 14] (see [15] for a survey of other such modifications).
Of particular interest are two works, those of Rotkowitz and Lall [12], and
Rotkowitz [14]. The first work [12] shows that with extremely fast, infinite-
capacity, and perfectly reliable external channels, the optimal controllers
are linear not just for the Witsenhausen’s counterexample (which is a simple
observation), but for more general problems as well. This suggests that
allowing for an external channel between the two controllers in Witsenhausen’s
counterexample might simplify the problem. However, when the channel is not
perfect, Martins [16] shows that finding optimal solutions can be
hard333Martins shows that nonlinear strategies that do not even use the
external channel can outperform linear ones that do use the channel where the
external channel SNR is high. As is suggested by what David Tse calls the
“deterministic perspective” (along the lines of [17, 18, 19]), linear
strategies do not make good use of the external channel because they only
communicate the “most significant bits” — which can anyway be estimated
reliably at the second controller. So if the uncertainty in the initial state
is large, the external channel is only of limited help and there may be
substantial advantage in having the controllers talk through the plant. A
similar problem is considered by Shoarinejad et al in [20], where noisy side
information of the source is available at the receiver. Since this formulation
is even more constrained than that in [16], it is clear that nonlinear
strategies outperform linear for this problem as well.. A closer inspection of
the problem in [16] reveals that nonlinear strategies can outperform linear
ones by an arbitrarily large factor for any fixed SNR on the external channel.
Even to make good use of the external channel resource, one needs nonlinear
strategies.
The second work [14] shows that if one considers the induced norm instead of
the original expected quadratic cost, linear control laws are optimal and easy
to find. The induced norm formulation is therefore easy to solve, and at the
same time, it makes no assumptions on the state and the noise distributions.
This led Doyle to ask if Witsenhausen’s counterexample (with expected
quadratic cost) is at all relevant [21] — after all, not only is the LQG
formulation more constrained, it is also harder to solve. The question thus
becomes what norm is more appropriate, and the answer must come from what is
relevant in practical situations. In practice, one usually knows the “typical”
amplitude of the noise and the initial state, or at least rough bounds them.
The induced-norm formulation may therefore be quite conservative: since no
assumptions are made on the state and the noise, it requires budgeting for
completely arbitrary behavior of state and noise — they can even collude to
raise the costs for the chosen strategy. To see how conservative the induced-
norm formulation can be, notice the following: even allowing for colluding
state and noise, mere knowledge of a bound on the noise amplitude suffices to
have quantization-based nonlinear strategies outperform linear strategies by
an arbitrarily large factor (with the expected cost replaced by a hard-budget.
The proof is simpler than that in [9], and is left as an exercise to the
interested reader for reasons of limited space). Conceptually, the LQG
formulation is only abstracting some knowledge of noise and initial state
behavior. In practical situations where such knowledge exists, designs based
on an induced norm formulation (and linear strategies) may be needlessly
expensive because they budget for impossible events.
The fact that nonlinear strategies can be arbitrarily better brings us to a
question that has received little attention in the literature — how far are
the proposed nonlinear strategies from the optimal? It is believed that the
strategies of Lee, Lau and Ho [5] are close to optimal. In Section VI, we will
see that these strategies can be viewed as an instance of the “dirty-paper
coding” strategy in information theory, and quantify their advantage over pure
quantization based strategies. Despite their improved performance, there was
no guarantee that these strategies are indeed close to optimal444The search in
[5] is not exhaustive. The authors first find a good quantization-based
solution. Inspired by piecewise linear strategies (from the neural networks
based search of Baglietto et al [4]), each quantization step is broken into
several small sub-steps to approximate a piecewise linear curve. .
Witsenhausen [1, Section 7] derived a lower bound on the costs that is loose
in the interesting regimes of small $k$ and large $\sigma_{0}^{2}$ [15, 22],
and hence is insufficient to obtain any guarantee on the gap from optimality.
Towards obtaining such a guarantee, a strategic simplification of the problem
was introduced in [23, 15] where we consider an asymptotically-long vector
version of the problem. This problem is related to a toy communication problem
that we call “Assisted Interference Suppression” (AIS) which is an extension
of the dirty-paper coding (DPC) [24] model in information theory. There has
been a burst of interest in extensions to DPC in information theory mainly
along two lines of work — multi-antenna Gaussian channels, and the “cognitive-
radio channel.” For multi-antenna Gaussian channels, a problem of much
theoretical and practical interest, DPC turns out to be the optimal strategy
(see [25] and the references therein). The “cognitive radio channel” problem
was formulated by Devroye et al [26]. This inspired much work in asymmetric
cooperation between nodes [27, 28, 29, 30, 31]. In our work [15, 23], we
developed a new lower bound to the optimal performance of the vector
Witsenhausen problem. Using this bound, we show that vector-quantization based
strategies attain within a factor of $4.45$ of the optimal cost for all
problem parameters in the limit of infinite vector length. Further,
combinations of linear and DPC-based strategies attain within a factor $2$ of
the optimal cost. This factor was later improved to $1.3$ in [32] by improving
the lower bound. While a constant-factor result does not establish true
optimality, such results are often helpful in the face of intractable problems
like those that are otherwise NP-hard [33]. This constant-factor spirit has
also been useful in understanding other stochastic control problems [34, 35]
and in the asymptotic analysis of problems in multiuser wireless communication
[36, 17].
While the lower bound in [15] holds for all vector lengths, and hence for the
scalar counterexample as well, the ratio of the costs attained by the
strategies of [9] and the lower bound diverges in the limit $k\rightarrow 0$
and $\sigma_{0}\rightarrow\infty$. This suggests that there is a significant
finite-dimensional aspect of the problem that is being lost in the infinite-
dimensional limit: either quantization-based strategies are bad, or the lower
bound of [15] is very loose. This effect is elucidated in [22] by deriving a
different lower bound showing that quantization-based strategies indeed attain
within a constant555The constant is large in [22], but as this paper shows,
this is an artifact of the proof rather than reality. factor of the optimal
cost for Witsenhausen’s original problem. The bound in [22] is in the spirit
of Witsenhausen’s original lower bound, but is more intricate. It captures the
idea that observation noise can force a second-stage cost to be incurred
unless the first stage cost is large.
In this paper, we revert to the line of attack initiated by the vector
simplification of [15]. In Section II, we formally state the vector version of
the counterexample. For obtaining good control strategies, we observe that the
action of the first controller in the quantization-based strategy of [9] can
be thought of as forcing the state to a point on a one-dimensional lattice.
Extending this idea, in Section III, we provide lattice-based quantization
strategies for finite dimensional spaces and analyze their performance.
Building upon the vector lower bound of [15], a new lower bound is derived in
Section IV which is in the spirit of large-deviations-based information-
theoretic bounds for finite-length communication problems666An alternative
Central Limit Theorem (CLT)-based approach has also been used in the
information-theory literature [37, 39, 38]. In [39, 38], the approach is used
to obtain extremely tight approximations at moderate blocklengths for
Shannon’s noisy communication problem. (e.g. [40, 41, 42, 43]). In particular,
our new bound extends the tools in [43] to a setting with unbounded distortion
measure. In Section V, we combine the lattice-based upper bound (Section III)
and the large-deviations lower bound (Section IV) to show that lattice-based
quantization strategies attain within a constant factor of the optimal cost
for any finite length, uniformly over all problem parameters. For example,
this constant factor is numerically found to be smaller than $8$ for the
original scalar problem. We also provide a constant factor that holds
uniformly over all vector lengths.
To understand the significance of the result, consider the following. At
$k=0.01$ and $\sigma_{0}=500$, the cost attained by the optimal linear scheme
is close to $1$. The cost attained by a quantization-based777The quantization
points are regularly spaced about $9.92$ units apart. This results in a first
stage cost of about $8.2\times 10^{-4}$ and a second stage cost of about
$6.7\times 10^{-5}$. scheme is $8.894\times 10^{-4}$. Our new lower bound on
the cost is $3.170\times 10^{-4}$. Despite the small value of the lower bound,
the ratio of the quantization-based upper bound and the lower bound for this
choice of parameters is less than three!
We conclude in Section VI outlining directions of future research and
speculating on the form of finite-dimensional strategies (following [15]) that
we conjecture might be optimal.
## II Notation and problem statement
Figure 1: Block-diagram for vector version of Witsenhausen’s counterexample of
length $m$.
Vectors are denoted in bold. Upper case tends to be used for random variables,
while lower case symbols represent their realizations.
$W(m,k^{2},\sigma_{0}^{2})$ denotes the vector version of Witsenhausen’s
problem of length $m$, defined as follows (shown in Fig. 1):
* •
The initial state $\mathbf{X}^{m}_{0}$ is Gaussian, distributed
$\mathcal{N}(0,\sigma_{0}^{2}\mathbb{I}_{m})$, where $\mathbb{I}_{m}$ is the
identity matrix of size $m\times m$.
* •
The state transition functions describe the state evolution with time. The
state transitions are linear:
$\displaystyle\mathbf{X}^{m}_{1}$ $\displaystyle=$
$\displaystyle\mathbf{X}^{m}_{0}+\mathbf{U}^{m}_{1},\;\;\;\text{and}$
$\displaystyle\mathbf{X}^{m}_{2}$ $\displaystyle=$
$\displaystyle\mathbf{X}^{m}_{1}-\mathbf{U}^{m}_{2}.$
* •
The outputs observed by the controllers:
$\displaystyle\mathbf{Y}^{m}_{1}$ $\displaystyle=$
$\displaystyle\mathbf{X}^{m}_{0},\;\;\;\text{ and}$
$\displaystyle\mathbf{Y}^{m}_{2}$ $\displaystyle=$
$\displaystyle\mathbf{X}^{m}_{1}+\mathbf{Z}^{m},$ (1)
where $\mathbf{Z}^{m}\sim\mathcal{N}(0,\sigma_{Z}^{2}\mathbb{I}_{m})$ is
Gaussian distributed observation noise.
* •
The control objective is to minimize the expected cost, averaged over the
random realizations of $\mathbf{X}^{m}_{0}$ and $\mathbf{Z}^{m}$. The total
cost is a quadratic function of the state and the input given by the sum of
two terms:
$\displaystyle J_{1}(\mathbf{x}^{m}_{1},\mathbf{u}^{m}_{1})$ $\displaystyle=$
$\displaystyle\frac{1}{m}k^{2}\|\mathbf{u}^{m}_{1}\|^{2},\;\text{and}$
$\displaystyle J_{2}(\mathbf{x}^{m}_{2},\mathbf{u}^{m}_{2})$ $\displaystyle=$
$\displaystyle\frac{1}{m}\|\mathbf{x}^{m}_{2}\|^{2}$
where $\|\cdot\|$ denotes the usual Euclidean 2-norm. The cost expressions are
normalized by the vector-length $m$ to allow for natural comparisons between
different vector-lengths. A control strategy is denoted by
$\gamma=(\gamma_{1},\gamma_{2})$, where $\gamma_{i}$ is the function that maps
the observation $\mathbf{y}^{m}_{i}$ at $\underline{\underline{\text{C}_{i}}}$
to the control input $\mathbf{u}^{m}_{i}$. For a fixed $\gamma$,
$\mathbf{x}^{m}_{1}=\mathbf{x}^{m}_{0}+\gamma_{1}(\mathbf{x}^{m}_{0})$ is a
function of $\mathbf{x}^{m}_{0}$. Thus the first stage cost can instead be
written as a function
$J_{1}^{(\gamma)}(\mathbf{x}^{m}_{0})=J_{1}(\mathbf{x}^{m}_{0}+\gamma_{1}(\mathbf{x}^{m}_{0}),\gamma_{1}(\mathbf{x}^{m}_{0}))$
and the second stage cost can be written as
$J_{2}^{(\gamma)}(\mathbf{x}^{m}_{0},\mathbf{z}^{m})=J_{2}(\mathbf{x}^{m}_{0}+\gamma_{1}(\mathbf{x}^{m}_{0})-\gamma_{2}(\mathbf{x}^{m}_{0}+\gamma_{1}(\mathbf{x}^{m}_{0})+\mathbf{z}^{m}),\gamma_{2}(\mathbf{x}^{m}_{0}+\gamma_{1}(\mathbf{x}^{m}_{0})+\mathbf{z}^{m}))$.
For given $\gamma$, the expected costs (averaged over $\mathbf{x}^{m}_{0}$ and
$\mathbf{z}^{m}$) are denoted by $\bar{J}^{(\gamma)}(m,k^{2},\sigma_{0}^{2})$
and $\bar{J}_{i}^{(\gamma)}(m,k^{2},\sigma_{0}^{2})$ for $i=1,2$. We define
$\bar{J}^{(\gamma)}_{\min}(m,k^{2},\sigma_{0}^{2})$ as follows
$\bar{J}_{\min}(m,k^{2},\sigma_{0}^{2}):=\inf_{\gamma}\bar{J}^{(\gamma)}(m,k^{2},\sigma_{0}^{2}).$
(2)
We note that for the scalar case of $m=1$, the problem is Witsenhausen’s
original counterexample [1].
Observe that scaling $\sigma_{0}$ and $\sigma_{Z}$ by the same factor
essentially does not change the problem — the solution can also be scaled by
the same factor (with the resulting cost scaling quadratically with it). Thus,
without loss of generality, we assume that the variance of the Gaussian
observation noise is $\sigma_{Z}^{2}=1$ (as is also assumed in [1]). The pdf
of the noise $\mathbf{Z}^{m}$ is denoted by $f_{Z}(\cdot{})$. In our proof
techniques, we also consider a hypothetical observation noise
$\mathbf{Z}^{m}_{G}\sim\mathcal{N}(0,\sigma_{G}^{2})$ with the variance
$\sigma_{G}^{2}\geq 1$. The pdf of this test noise is denoted by
$f_{G}(\cdot{})$. We use $\psi(m,r)$ to denote $\Pr(\|\mathbf{Z}^{m}\|\geq r)$
for $\mathbf{Z}^{m}\sim\mathcal{N}(0,\mathbb{I})$.
Subscripts in expectation expressions denote the random variable being
averaged over (e.g.
$\mathbb{E}_{\mathbf{X}^{m}_{0},\mathbf{Z}^{m}_{G}}\left[{\cdot{}}\right]$
denotes averaging over the initial state $\mathbf{X}^{m}_{0}$ and the test
noise $\mathbf{Z}^{m}_{G}$).
## III Lattice-based quantization strategies
Figure 2: Covering and packing for the 2-dimensional hexagonal lattice. The
packing-covering ratio for this lattice is $\xi=\frac{2}{\sqrt{3}}\approx
1.15$ [44, Appendix C]. The first controller forces the initial state
$\mathbf{x}^{m}_{0}$ to the lattice point nearest to it. The second controller
estimates $\mathbf{\widehat{x}}^{m}_{1}$ to be a lattice point at the centre
of the sphere if it falls in one of the packing spheres. Else it essentially
gives up and estimates $\mathbf{\widehat{x}}^{m}_{1}=\mathbf{y}^{m}_{2}$, the
received output itself. A hexagonal lattice-based scheme would perform better
for the 2-D Witsenhausen problem than the square lattice (of
$\xi=\sqrt{2}\approx 1.41$ [44, Appendix C]) because it has a smaller $\xi$.
Lattice-based quantization strategies are the natural generalizations of
scalar quantization-based strategies [9]. An introduction to lattices can be
found in [45, 46]. Relevant definitions are reviewed below. $\mathcal{B}$
denotes the unit ball in $\mathbb{R}^{m}$.
###### Definition 1 (Lattice)
An $m$-dimensional lattice $\Lambda$ is a set of points in $\mathbb{R}^{m}$
such that if $\mathbf{x}^{m},\mathbf{y}^{m}\in\Lambda$, then
$\mathbf{x}^{m}+\mathbf{y}^{m}\in\Lambda$, and if $\mathbf{x}^{m}\in\Lambda$,
then $-\mathbf{x}^{m}\in\Lambda$.
###### Definition 2 (Packing and packing radius)
Given an $m$-dimensional lattice $\Lambda$ and a radius $r$, the set
$\Lambda+r\mathcal{B}$ is a packing of Euclidean $m$-space if for all points
$\mathbf{x}^{m},\mathbf{y}^{m}\in\Lambda$,
$(\mathbf{x}^{m}+r\mathcal{B})\bigcap(\mathbf{y}^{m}+r\mathcal{B})=\emptyset$.
The packing radius $r_{p}$ is defined as
$r_{p}:=\sup\\{r:\Lambda+r\mathcal{B}\;\text{is a packing}\\}$.
###### Definition 3 (Covering and covering radius)
Given an $m$-dimensional lattice $\Lambda$ and a radius $r$, the set
$\Lambda+r\mathcal{B}$ is a covering of Euclidean $m$-space if
$\mathbb{R}^{m}\subseteq\Lambda+r\mathcal{B}$. The covering radius $r_{c}$ is
defined as $r_{c}:=\inf\\{r:\Lambda+r\mathcal{B}\;\text{is a covering}\\}$.
###### Definition 4 (Packing-covering ratio)
The packing-covering ratio (denoted by $\xi$) of a lattice $\Lambda$ is the
ratio of its covering radius to its packing radius, $\xi=\frac{r_{c}}{r_{p}}$.
Because it creates no ambiguity, we do not include the dimension $m$ and the
choice of lattice $\Lambda$ in the notation of $r_{c}$, $r_{p}$ and $\xi$,
though these quantities depend on $m$ and $\Lambda$.
For a given dimension $m$, a natural control strategy that uses a lattice
$\Lambda$ of covering radius $r_{c}$ and packing radius $r_{p}$ is as follows.
The first controller uses the input $\mathbf{u}^{m}_{1}$ to force the state
$\mathbf{x}^{m}_{0}$ to the lattice point nearest to $\mathbf{x}^{m}_{0}$. The
second controller estimates $\mathbf{x}^{m}_{1}$ to be the lattice point
nearest to $\mathbf{y}^{m}_{2}$. For analytical ease, we instead consider an
inferior strategy where the second controller estimates $\mathbf{x}^{m}_{1}$
to be a lattice point only if the lattice point lies within the sphere of
radius $r_{p}$ around $\mathbf{y}^{m}_{2}$. If no lattice point exists in the
sphere, the second controller estimates $\mathbf{x}^{m}_{1}$ to be
$\mathbf{y}^{m}_{2}$, the received vector itself. The actions
$\gamma_{1}(\cdot{})$ of $\underline{\underline{\text{C}_{1}}}$ and
$\gamma_{2}(\cdot{})$ of $\underline{\underline{\text{C}_{2}}}$ are therefore
given by
$\displaystyle\gamma_{1}(\mathbf{x}^{m}_{0})$ $\displaystyle=$
$\displaystyle-\mathbf{x}^{m}_{0}+\underset{{\mathbf{x}^{m}_{1}}\in\Lambda}{\text{arg
min}}\;\|\mathbf{x}^{m}_{1}-\mathbf{x}^{m}_{0}\|^{2},$
$\displaystyle\gamma_{2}(\mathbf{y}^{m}_{2})$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cc}\mathbf{\widetilde{x}}^{m}_{1}&\text{if}\;\exists\;\mathbf{\widetilde{x}}^{m}_{1}\in\Lambda\;\text{s.t.}\;\|\mathbf{y}^{m}_{2}-\mathbf{\widetilde{x}}^{m}_{1}\|^{2}<r_{p}^{2}\\\
\mathbf{y}^{m}_{2}&\text{otherwise}\end{array}\right..$
The event where there exists no such
$\mathbf{\widetilde{x}}^{m}_{1}\in\Lambda$ is referred to as decoding failure.
In the following, we denote $\gamma_{2}(\mathbf{y}^{m}_{2})$ by
$\mathbf{\widehat{x}}^{m}_{1}$, the estimate of $\mathbf{x}^{m}_{1}$.
###### Theorem 1
Using a lattice-based strategy (as described above) for
$W(m,k^{2},\sigma_{0}^{2})$ with $r_{c}$ and $r_{p}$ the covering and the
packing radius for the lattice, the total average cost is upper bounded by
$\displaystyle\bar{J}^{(\gamma)}(m,k^{2},\sigma_{0}^{2})\leq\inf_{P\geq
0}k^{2}P+\left(\sqrt{\psi(m+2,r_{p})}+\sqrt{\frac{P}{\xi^{2}}}\sqrt{\psi(m,r_{p})}\right)^{2},$
where $\xi=\frac{r_{c}}{r_{p}}$ is the packing-covering ratio for the lattice,
and $\psi(m,r)=\Pr(\|\mathbf{Z}^{m}\|\geq r)$. The following looser bound also
holds
$\displaystyle\bar{J}^{(\gamma)}(m,k^{2},\sigma_{0}^{2})\leq\inf_{P>\xi^{2}}k^{2}P+\left(1+\sqrt{\frac{P}{\xi^{2}}}\right)^{2}e^{-\frac{mP}{2\xi^{2}}+\frac{m+2}{2}\left(1+\ln\left(\frac{P}{\xi^{2}}\right)\right)}.$
Remark: The latter loose bound is useful for analytical manipulations when
proving explicit bounds on the ratio of the upper and lower bounds in Section
V.
###### Proof:
Note that because $\Lambda$ has a covering radius of $r_{c}$,
$\|\mathbf{x}^{m}_{1}-\mathbf{x}^{m}_{0}\|^{2}\leq r_{c}^{2}$. Thus the first
stage cost is bounded above by $\frac{1}{m}k^{2}r_{c}^{2}$. A tighter bound
can be provided for a specific lattice and finite $m$ (for example, for $m=1$,
the first stage cost is approximately $k^{2}\frac{r_{c}^{2}}{3}$ if
$r_{c}^{2}\ll\sigma_{0}^{2}$ because the distribution of $\mathbf{x}^{m}_{0}$
conditioned on it lying in any of the quantization bins is approximately
uniform at least for the most likely bins).
For the second stage, observe that
$\displaystyle\mathbb{E}_{\mathbf{X}^{m}_{1},\mathbf{Z}^{m}}\left[{\|\mathbf{X}^{m}_{1}-\mathbf{\widehat{X}}^{m}_{1}\|^{2}}\right]=\mathbb{E}_{\mathbf{X}^{m}_{1}}\left[{\mathbb{E}_{\mathbf{Z}^{m}}\left[{\|\mathbf{X}^{m}_{1}-\mathbf{\widehat{X}}^{m}_{1}\|^{2}|\mathbf{X}^{m}_{1}}\right]}\right].$
(4)
Denote by $\mathcal{E}_{m}$ the event $\\{\|\mathbf{Z}^{m}\|^{2}\geq
r_{p}^{2}\\}$. Observe that under the event $\mathcal{E}_{m}^{c}$,
$\mathbf{\widehat{X}}^{m}_{1}=\mathbf{X}^{m}_{1}$, resulting in a zero second-
stage cost. Thus,
$\displaystyle\mathbb{E}_{\mathbf{Z}^{m}}\left[{\|\mathbf{X}^{m}_{1}-\mathbf{\widehat{X}}^{m}_{1}\|^{2}|\mathbf{X}^{m}_{1}}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}_{\mathbf{Z}^{m}}\left[{\|\mathbf{X}^{m}_{1}-\mathbf{\widehat{X}}^{m}_{1}\|^{2}{1{1}}_{\left\\{\mathcal{E}_{m}\right\\}}|\mathbf{X}^{m}_{1}}\right]+\mathbb{E}_{\mathbf{Z}^{m}}\left[{\|\mathbf{X}^{m}_{1}-\mathbf{\widehat{X}}^{m}_{1}\|^{2}{1{1}}_{\left\\{\mathcal{E}_{m}^{c}\right\\}}|\mathbf{X}^{m}_{1}}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}_{\mathbf{Z}^{m}}\left[{\|\mathbf{X}^{m}_{1}-\mathbf{\widehat{X}}^{m}_{1}\|^{2}{1{1}}_{\left\\{\mathcal{E}_{m}\right\\}}|\mathbf{X}^{m}_{1}}\right].$
We now bound the squared-error under the error event $\mathcal{E}_{m}$, when
either $\mathbf{x}^{m}_{1}$ is decoded erroneously, or there is a decoding
failure. If $\mathbf{x}^{m}_{1}$ is decoded erroneously to a lattice point
$\mathbf{\widetilde{x}}^{m}_{1}\neq\mathbf{x}^{m}_{1}$, the squared-error can
be bounded as follows
$\displaystyle\|\mathbf{x}^{m}_{1}-\mathbf{\widetilde{x}}^{m}_{1}\|^{2}=\|\mathbf{x}^{m}_{1}-\mathbf{y}^{m}_{2}+\mathbf{y}^{m}_{2}-\mathbf{\widetilde{x}}^{m}_{1}\|^{2}\leq\left(\|\mathbf{x}^{m}_{1}-\mathbf{y}^{m}_{2}\|+\|\mathbf{y}^{m}_{2}-\mathbf{\widetilde{x}}^{m}_{1}\|\right)^{2}\leq\left(\|\mathbf{z}^{m}\|+r_{p}\right)^{2}.$
If $\mathbf{x}^{m}_{1}$ is decoded as $\mathbf{y}^{m}_{2}$, the squared-error
is simply $\|\mathbf{z}^{m}\|^{2}$, which we also upper bound by
$\left(\|\mathbf{z}^{m}\|+r_{p}\right)^{2}$. Thus, under event
$\mathcal{E}_{m}$, the squared error
$\|\mathbf{x}^{m}_{1}-\mathbf{\widehat{x}}^{m}_{1}\|^{2}$ is bounded above by
$\left(\|\mathbf{z}^{m}\|+r_{p}\right)^{2}$, and hence
$\displaystyle\mathbb{E}_{\mathbf{Z}^{m}}\left[{\|\mathbf{X}^{m}_{1}-\mathbf{\widehat{X}}^{m}_{1}\|^{2}|\mathbf{X}^{m}_{1}}\right]$
$\displaystyle\leq$
$\displaystyle\mathbb{E}_{\mathbf{Z}^{m}}\left[{\left(\|\mathbf{Z}^{m}\|+r_{p}\right)^{2}{1{1}}_{\left\\{\mathcal{E}_{m}\right\\}}|\mathbf{X}^{m}_{1}}\right]$
(5) $\displaystyle\overset{(a)}{=}$
$\displaystyle\mathbb{E}_{\mathbf{Z}^{m}}\left[{\left(\|\mathbf{Z}^{m}\|+r_{p}\right)^{2}{1{1}}_{\left\\{\mathcal{E}_{m}\right\\}}}\right],$
where $(a)$ uses the fact that the pair
$(\mathbf{Z}^{m},{1{1}}_{\left\\{\mathcal{E}_{m}\right\\}})$ is independent of
$\mathbf{X}^{m}_{1}$. Now, let $P=\frac{r_{c}^{2}}{m}$, so that the first
stage cost is at most $k^{2}P$. The following lemma helps us derive the upper
bound.
###### Lemma 1
For a given lattice with
$r_{p}^{2}=\frac{r_{c}^{2}}{\xi^{2}}=\frac{mP}{\xi^{2}}$, the following bound
holds
$\displaystyle\frac{1}{m}\mathbb{E}_{\mathbf{Z}^{m}}\left[{\left(\|\mathbf{Z}^{m}\|+r_{p}\right)^{2}{1{1}}_{\left\\{\mathcal{E}_{m}\right\\}}}\right]\leq\left(\sqrt{\psi(m+2,r_{p})}+\sqrt{\frac{P}{\xi^{2}}}\sqrt{\psi(m,r_{p})}\right)^{2}.$
The following (looser) bound also holds as long as $P>\xi^{2}$,
$\displaystyle\frac{1}{m}\mathbb{E}_{\mathbf{Z}^{m}}\left[{\left(\|\mathbf{Z}^{m}\|+r_{p}\right)^{2}{1{1}}_{\left\\{\mathcal{E}_{m}\right\\}}}\right]\leq\left(1+\sqrt{\frac{P}{\xi^{2}}}\right)^{2}e^{-\frac{mP}{2\xi^{2}}+\frac{m+2}{2}\left(1+\ln\left(\frac{P}{\xi^{2}}\right)\right)}.$
###### Proof:
See Appendix A. ∎
The theorem now follows from (4), (5) and Lemma 1. ∎
## IV Lower bounds on the cost
Figure 3: A pictorial representation of the proof for the lower bound assuming
$\sigma_{0}^{2}=30$. The solid curves show the vector lower bound of [15] for
various values of observation noise variances, denoted by $\sigma_{G}^{2}$.
Conceptually, multiplying these curves by the probability of that channel
behavior yields the shadow curves for the particular $\sigma_{G}^{2}$, shown
by dashed curves. The scalar lower bound is then obtained by taking the
maximum of these shadow curves. The circles at points along the scalar bound
curve indicate the optimizing value of $\sigma_{G}$ for obtaining that point
on the bound.
Bansal and Basar [3] use information-theoretic techniques related to rate-
distortion and channel capacity to show the optimality of linear strategies in
a modified version of Witsenhausen’s counterexample where the cost function
does not contain a product of two decision variables. Following the same
spirit, in [15] we derive the following lower bound for Witsenhausen’s
counterexample itself.
###### Theorem 2
For $W(m,k^{2},\sigma_{0}^{2})$, if for a strategy $\gamma(\cdot{})$ the
average power
$\frac{1}{m}\mathbb{E}_{\mathbf{X}^{m}_{0}}\left[{\|\mathbf{U}^{m}_{1}\|^{2}}\right]=P$,
the following lower bound holds on the second stage cost
$\bar{J}_{2}^{(\gamma)}(m,k^{2},\sigma_{0}^{2})\geq\left(\left(\sqrt{\kappa(P,\sigma_{0}^{2})}-\sqrt{P}\right)^{+}\right)^{2},$
where $(\cdot{})^{+}$ is shorthand for $\max(\cdot{},0)$ and
$\kappa(P,\sigma_{0}^{2})=\frac{\sigma_{0}^{2}}{\sigma_{0}^{2}+P+2\sigma_{0}\sqrt{P}+1}.$
(6)
The following lower bound thus holds on the total cost
$\bar{J}^{(\gamma)}(m,k^{2},\sigma_{0}^{2})\geq\inf_{P\geq
0}k^{2}P+\left(\left(\sqrt{\kappa(P,\sigma_{0}^{2})}-\sqrt{P}\right)^{+}\right)^{2}.$
###### Proof:
We refer the reader to [15] for the full proof. We outline it here because
these ideas are used in the derivation of the new lower bound in Theorem 3.
Using a triangle inequality argument, we show
$\displaystyle\sqrt{\frac{1}{m}\mathbb{E}_{\mathbf{X}^{m}_{0},\mathbf{Z}^{m}}\left[{\|\mathbf{X}^{m}_{0}-\mathbf{\widehat{X}}^{m}_{1}\|^{2}}\right]}\leq\sqrt{\frac{1}{m}\mathbb{E}_{\mathbf{X}^{m}_{0},\mathbf{Z}^{m}}\left[{\|\mathbf{X}^{m}_{0}-\mathbf{X}^{m}_{1}\|^{2}}\right]}+\sqrt{\frac{1}{m}\mathbb{E}_{\mathbf{X}^{m}_{0},\mathbf{Z}^{m}}\left[{\|\mathbf{X}^{m}_{1}-\mathbf{\widehat{X}}^{m}_{1}\|^{2}}\right]}.$
(7)
The first term on the RHS is $\sqrt{P}$. It therefore suffices to lower bound
the term on the LHS to obtain a lower bound on
$\mathbb{E}_{\mathbf{X}^{m}_{0},\mathbf{Z}^{m}}\left[{\|\mathbf{X}^{m}_{1}-\mathbf{\widehat{X}}^{m}_{1}\|^{2}}\right]$.
To that end, we interpret $\mathbf{\widehat{X}}^{m}_{1}$ as an estimate for
$\mathbf{X}^{m}_{0}$, which is a problem of transmitting a source across a
channel. For an iid Gaussian source to be transmitted across a memoryless
power-constrained additive-noise Gaussian channel (with one channel use per
source symbol), the optimal strategy that minimizes the mean-square error is
merely scaling the source symbol so that the average power constraint is met
[47]. The estimation at the second controller is then merely the linear MMSE
estimation of $\mathbf{X}^{m}_{0}$, and the obtained MMSE is
$\kappa(P,\sigma_{0}^{2})$. The lemma now follows from (7). ∎
Observe that the lower bound expression is the same for all vector lengths. In
the following, large-deviation arguments [48, 49] (called sphere-packing style
arguments for historical reasons) are extended following [41, 42, 43] to a
joint source-channel setting where the distortion measure is unbounded. The
obtained bounds are tighter than those in Theorem 2 and depend explicitly on
the vector length $m$.
###### Theorem 3
For $W(m,k^{2},\sigma_{0}^{2})$, if for a strategy $\gamma(\cdot{})$ the
average power
$\frac{1}{m}\mathbb{E}_{\mathbf{X}^{m}_{0}}\left[{\|\mathbf{U}^{m}_{1}\|^{2}}\right]=P$,
the following lower bound holds on the second stage cost for any choice of
$\sigma_{G}^{2}\geq 1$ and $L>0$
$\bar{J}_{2}^{(\gamma)}(m,k^{2},\sigma_{0}^{2})\geq\eta(P,\sigma_{0}^{2},\sigma_{G}^{2},L).$
where
$\displaystyle\eta(P,\sigma_{0}^{2},\sigma_{G}^{2},L)=\frac{\sigma_{G}^{m}}{c_{m}(L)}\exp\left(-\frac{mL^{2}(\sigma_{G}^{2}-1)}{2}\right)\left(\left(\sqrt{\kappa_{2}(P,\sigma_{0}^{2},\sigma_{G}^{2},L)}-\sqrt{P}\right)^{+}\right)^{2},$
where $\kappa_{2}(P,\sigma_{0}^{2},\sigma_{G}^{2},L):=$
$\displaystyle\frac{\sigma_{0}^{2}\sigma_{G}^{2}}{c_{m}^{\frac{2}{m}}(L)e^{1-d_{m}(L)}\left((\sigma_{0}+\sqrt{P})^{2}+d_{m}(L)\sigma_{G}^{2}\right)},$
$c_{m}(L):=\frac{1}{\Pr(\|\mathbf{Z}^{m}\|^{2}\leq
mL^{2})}=\left(1-\psi(m,L\sqrt{m})\right)^{-1}$,
$d_{m}(L):=\frac{\Pr(\|\mathbf{Z}^{m+2}\|^{2}\leq
mL^{2})}{\Pr(\|\mathbf{Z}^{m}\|^{2}\leq
mL^{2})}=\frac{1-\psi(m+2,L\sqrt{m})}{1-\psi(m,L\sqrt{m})}$,
$0<d_{m}(L)<1$, and $\psi(m,r)=\Pr(\|\mathbf{Z}^{m}\|\geq r)$. Thus the
following lower bound holds on the total cost
$\bar{J}_{\min}(m,k^{2},\sigma_{0}^{2})\geq\inf_{P\geq
0}k^{2}P+\eta(P,\sigma_{0}^{2},\sigma_{G}^{2},L),$ (8)
for any choice of $\sigma_{G}^{2}\geq 1$ and $L>0$ (the choice can depend on
$P$). Further, these bounds are at least as tight as those of Theorem 2 for
all values of $k$ and $\sigma_{0}^{2}$.
###### Proof:
From Theorem 2, for a given $P$, a lower bound on the average second stage
cost is $\left(\left(\sqrt{\kappa}-\sqrt{P}\right)^{+}\right)^{2}$. We derive
another lower bound that is equal to the expression for
$\eta(P,\sigma_{0}^{2},\sigma_{G}^{2},L)$. The high-level intuition behind
this lower bound is presented in Fig. 3.
Define $\mathcal{S}_{L}^{G}:=\\{\mathbf{z}^{m}:\|\mathbf{z}^{m}\|^{2}\leq
mL^{2}\sigma_{G}^{2}\\}$ and use subscripts to denote which probability model
is being used for the second stage observation noise. $Z$ denotes white
Gaussian of variance $1$ while $G$ denotes white Gaussian of variance
$\sigma_{G}^{2}\geq 1$.
$\displaystyle\mathbb{E}_{\mathbf{X}^{m}_{0},\mathbf{Z}^{m}}\left[{J_{2}^{(\gamma)}(\mathbf{X}^{m}_{0},\mathbf{Z}^{m})}\right]$
$\displaystyle=$
$\displaystyle\int_{\mathbf{z}^{m}}\int_{\mathbf{x}^{m}_{0}}J_{2}^{(\gamma)}(\mathbf{x}^{m}_{0},\mathbf{z}^{m})f_{0}(\mathbf{x}^{m}_{0})f_{Z}(\mathbf{z}^{m})d\mathbf{x}^{m}_{0}d\mathbf{z}^{m}$
(9) $\displaystyle\geq$
$\displaystyle\int_{\mathbf{z}^{m}\in\mathcal{S}_{L}^{G}}\left(\int_{\mathbf{x}^{m}_{0}}J_{2}^{(\gamma)}(\mathbf{x}^{m}_{0},\mathbf{z}^{m})f_{0}(\mathbf{x}^{m}_{0})d\mathbf{x}^{m}_{0}\right)f_{Z}(\mathbf{z}^{m})d\mathbf{z}^{m}$
$\displaystyle=$
$\displaystyle\int_{\mathbf{z}^{m}\in\mathcal{S}_{L}^{G}}\left(\int_{\mathbf{x}^{m}_{0}}J_{2}^{(\gamma)}(\mathbf{x}^{m}_{0},\mathbf{z}^{m})f_{0}(\mathbf{x}^{m}_{0})d\mathbf{x}^{m}_{0}\right)\frac{f_{Z}(\mathbf{z}^{m})}{f_{G}(\mathbf{z}^{m})}f_{G}(\mathbf{z}^{m})d\mathbf{z}^{m}.$
The ratio of the two probability density functions is given by
$\displaystyle\frac{f_{Z}(\mathbf{z}^{m})}{f_{G}(\mathbf{z}^{m})}=\frac{e^{-\frac{\|\mathbf{z}^{m}\|^{2}}{2}}}{\left(\sqrt{2\pi}\right)^{m}}\frac{\left(\sqrt{2\pi\sigma_{G}^{2}}\right)^{m}}{e^{-\frac{\|\mathbf{z}^{m}\|^{2}}{2\sigma_{G}^{2}}}}=\sigma_{G}^{m}e^{-\frac{\|\mathbf{z}^{m}\|^{2}}{2}\left(1-\frac{1}{\sigma_{G}^{2}}\right)}.$
Observe that $\mathbf{z}^{m}\in\mathcal{S}_{L}^{G}$,
$\|\mathbf{z}^{m}\|^{2}\leq mL^{2}\sigma_{G}^{2}$. Using $\sigma_{G}^{2}\geq
1$, we obtain
$\frac{f_{Z}(\mathbf{z}^{m})}{f_{G}(\mathbf{z}^{m})}\geq\sigma_{G}^{m}e^{-\frac{mL^{2}\sigma_{G}^{2}}{2}\left(1-\frac{1}{\sigma_{G}^{2}}\right)}=\sigma_{G}^{m}e^{-\frac{mL^{2}(\sigma_{G}^{2}-1)}{2}}.$
(10)
Using (9) and (10),
$\displaystyle\mathbb{E}_{\mathbf{X}^{m}_{0},\mathbf{Z}^{m}}\left[{J_{2}^{(\gamma)}(\mathbf{X}^{m}_{0},\mathbf{Z}^{m})}\right]$
$\displaystyle\geq$
$\displaystyle\sigma_{G}^{m}e^{-\frac{mL^{2}(\sigma_{G}^{2}-1)}{2}}\int_{\mathbf{z}^{m}\in\mathcal{S}_{L}^{G}}\left(\int_{\mathbf{x}^{m}_{0}}J_{2}^{(\gamma)}(\mathbf{x}^{m}_{0},\mathbf{z}^{m})f_{0}(\mathbf{x}^{m}_{0})d\mathbf{x}^{m}_{0}\right)f_{G}(\mathbf{z}^{m})d\mathbf{z}^{m}$
(11) $\displaystyle=$
$\displaystyle\sigma_{G}^{m}e^{-\frac{mL^{2}(\sigma_{G}^{2}-1)}{2}}\mathbb{E}_{\mathbf{X}^{m}_{0},\mathbf{Z}^{m}_{G}}\left[{J_{2}^{(\gamma)}(\mathbf{X}^{m}_{0},\mathbf{Z}^{m}_{G}){1{1}}_{\left\\{\mathbf{Z}^{m}_{G}\in\mathcal{S}_{L}^{G}\right\\}}}\right]$
$\displaystyle=$
$\displaystyle\sigma_{G}^{m}e^{-\frac{mL^{2}(\sigma_{G}^{2}-1)}{2}}\mathbb{E}_{\mathbf{X}^{m}_{0},\mathbf{Z}^{m}_{G}}\left[{J_{2}^{(\gamma)}(\mathbf{X}^{m}_{0},\mathbf{Z}^{m}_{G})|\mathbf{Z}^{m}_{G}\in\mathcal{S}_{L}^{G}}\right]\Pr(\mathbf{Z}^{m}_{G}\in\mathcal{S}_{L}^{G}).$
Analyzing the probability term in (11),
$\displaystyle\Pr(\mathbf{Z}^{m}_{G}\in\mathcal{S}_{L}^{G})$ $\displaystyle=$
$\displaystyle\Pr\left(\|\mathbf{Z}^{m}_{G}\|^{2}\leq
mL^{2}\sigma_{G}^{2}\right)=\Pr\left(\left(\frac{\|\mathbf{Z}^{m}_{G}\|}{\sigma_{G}}\right)^{2}\leq
mL^{2}\right)$ (12) $\displaystyle=$ $\displaystyle
1-\Pr\left(\left(\frac{\|\mathbf{Z}^{m}_{G}\|}{\sigma_{G}}\right)^{2}>mL^{2}\right)=1-\psi(m,L\sqrt{m})=\frac{1}{c_{m}(L)},$
because
$\frac{\mathbf{Z}^{m}_{G}}{\sigma_{G}}\sim\mathcal{N}(0,\mathbb{I}_{m})$. From
(11) and (12),
$\displaystyle\mathbb{E}_{\mathbf{X}^{m}_{0},\mathbf{Z}^{m}}\left[{J_{2}^{(\gamma)}(\mathbf{X}^{m}_{0},\mathbf{Z}^{m})}\right]$
$\displaystyle\geq$
$\displaystyle\sigma_{G}^{m}e^{-\frac{mL^{2}(\sigma_{G}^{2}-1)}{2}}\mathbb{E}_{\mathbf{X}^{m}_{0},\mathbf{Z}^{m}_{G}}\left[{J_{2}^{(\gamma)}(\mathbf{X}^{m}_{0},\mathbf{Z}^{m}_{G})|\mathbf{Z}^{m}_{G}\in\mathcal{S}_{L}^{G}}\right](1-\psi(m,L\sqrt{m}))$
(13) $\displaystyle=$
$\displaystyle\frac{\sigma_{G}^{m}e^{-\frac{mL^{2}(\sigma_{G}^{2}-1)}{2}}}{c_{m}(L)}\mathbb{E}_{\mathbf{X}^{m}_{0},\mathbf{Z}^{m}_{G}}\left[{J_{2}^{(\gamma)}(\mathbf{X}^{m}_{0},\mathbf{Z}^{m}_{G})|\mathbf{Z}^{m}_{G}\in\mathcal{S}_{L}^{G}}\right].$
We now need the following lemma, which connects the new finite-length lower
bound to the infinite-length lower bound of [15].
###### Lemma 2
$\displaystyle\mathbb{E}_{\mathbf{X}^{m}_{0},\mathbf{Z}^{m}_{G}}\left[{J_{2}^{(\gamma)}(\mathbf{X}^{m}_{0},\mathbf{Z}^{m}_{G})|\mathbf{Z}^{m}_{G}\in\mathcal{S}_{L}^{G}}\right]\geq\left(\left(\sqrt{\kappa_{2}(P,\sigma_{0}^{2},\sigma_{G}^{2},L)}-\sqrt{P}\right)^{+}\right)^{2},$
for any $L>0$.
###### Proof:
See Appendix B. ∎
The lower bound on the total average cost now follows from (13) and Lemma 2.
We now verify that $d_{m}(L)\in(0,1)$. That $d_{m}(L)>0$ is clear from
definition. $d_{m}(L)<1$ because
$\\{\mathbf{z}^{m+2}:\|\mathbf{z}^{m+2}\|^{2}\leq
mL^{2}\sigma_{G}^{2}\\}\subset\\{\mathbf{z}^{m+2}:\|\mathbf{z}^{m}\|^{2}\leq
mL^{2}\sigma_{G}^{2}\\}$, _i.e._ , a sphere sits inside a cylinder.
Finally we verify that this new lower bound is at least as tight as the one in
Theorem 2. Choosing $\sigma_{G}^{2}=1$ in the expression for
$\eta(P,\sigma_{0}^{2},\sigma_{G}^{2},L)$,
$\displaystyle\eta(P,\sigma_{0}^{2},\sigma_{G}^{2},L)\geq\sup_{L>0}\frac{1}{c_{m}(L)}\left(\left(\sqrt{\kappa_{2}(P,\sigma_{0}^{2},1,L)}-\sqrt{P}\right)^{+}\right)^{2}.$
Now notice that $c_{m}(L)$ and $d_{m}(L)$ converge to $1$ as
$L\rightarrow\infty$. Thus
$\kappa_{2}(P,\sigma_{0}^{2},1,L)\overset{L\rightarrow\infty}{\longrightarrow}\kappa(P,\sigma_{0}^{2})$
and therefore, $\eta(P,\sigma_{0}^{2},\sigma_{G}^{2},L)$ is lower bounded by
$\left(\left(\sqrt{\kappa}-\sqrt{P}\right)^{+}\right)^{2}$, the lower bound in
Theorem 2.
∎
## V Combination of linear and lattice-based strategies attain within a
constant factor of the optimal cost
###### Theorem 4 (Constant-factor optimality)
The costs for $W(m,k^{2},\sigma_{0}^{2})$ are bounded as follows
$\displaystyle\inf_{P\geq 0}\sup_{\sigma_{G}^{2}\geq
1,L>0}k^{2}P+\eta(P,\sigma_{0}^{2},\sigma_{G}^{2},L)\leq\bar{J}_{min}(m,k^{2},\sigma_{0}^{2})\leq\mu\left(\inf_{P\geq
0}\sup_{\sigma_{G}^{2}\geq
1,L>0}k^{2}P+\eta(P,\sigma_{0}^{2},\sigma_{G}^{2},L)\right),$
where $\mu=100\xi^{2}$, $\xi$ is the packing-covering ratio of any lattice in
$\mathbb{R}^{m}$, and $\eta(\cdot)$ is as defined in Theorem 3. For any $m$,
$\mu<1600$. Further, depending on the $(m,k^{2},\sigma_{0}^{2})$ values, the
upper bound can be attained by lattice-based quantization strategies or linear
strategies. For $m=1$, a numerical calculation (MATLAB code available at [50])
shows that $\mu<8$ (see Fig. 5).
###### Proof:
Figure 4: The ratio of the upper and the lower bounds for the scalar
Witsenhausen problem (top), and the 2-D Witsenhausen problem (bottom, using
hexagonal lattice of $\xi=\frac{2}{\sqrt{3}}$) for a range of values of $k$
and $\sigma_{0}$. The ratio is bounded above by $17$ for the scalar problem,
and by $14.75$ for the 2-D problem. Figure 5: An exact calculation of the
first and second stage costs yields an improved maximum ratio smaller than $8$
for the scalar Witsenhausen problem.
Let $P^{*}$ denote the power $P$ in the lower bound in Theorem 3. We show here
that for any choice of $P^{*}$, the ratio of the upper and the lower bound is
bounded.
Consider the two simple linear strategies of zero-forcing
($\mathbf{u}^{m}_{1}=-\mathbf{x}^{m}_{0}$) and zero-input
($\mathbf{u}^{m}_{1}=0$) followed by LLSE estimation at
$\underline{\underline{\text{C}_{2}}}$. It is easy to see [15] that the
average cost attained using these two strategies is $k^{2}\sigma_{0}^{2}$ and
$\frac{\sigma_{0}^{2}}{\sigma_{0}^{2}+1}<1$ respectively. An upper bound is
obtained using the best amongst the two linear strategies and the lattice-
based quantization strategy.
Case 1: $P^{*}\geq\frac{\sigma_{0}^{2}}{100}$.
The first stage cost is larger than $k^{2}\frac{\sigma_{0}^{2}}{100}$.
Consider the upper bound of $k^{2}\sigma_{0}^{2}$ obtained by zero-forcing.
The ratio of the upper bound and the lower bound is no larger than $100$.
Case 2: $P^{*}<\frac{\sigma_{0}^{2}}{100}$ and $\sigma_{0}^{2}<16$.
Using the bound from Theorem 2 (which is a special case of the bound in
Theorem 3),
$\displaystyle\kappa$ $\displaystyle=$
$\displaystyle\frac{\sigma_{0}^{2}}{(\sigma_{0}+\sqrt{P^{*}})^{2}+1}\overset{\left(P^{*}<\frac{\sigma_{0}^{2}}{100}\right)}{\geq}\frac{\sigma_{0}^{2}}{\sigma_{0}^{2}\left(1+\frac{1}{\sqrt{100}}\right)^{2}+1}$
$\displaystyle\overset{(\sigma_{0}^{2}<16)}{\geq}$
$\displaystyle\frac{\sigma_{0}^{2}}{16\left(1+\frac{1}{\sqrt{100}}\right)^{2}+1}=\frac{\sigma_{0}^{2}}{20.36}\geq\frac{\sigma_{0}^{2}}{21}.$
Thus, for $\sigma_{0}^{2}<16$ and $P^{*}\leq\frac{\sigma_{0}^{2}}{100}$,
$\displaystyle\bar{J}_{min}$ $\displaystyle\geq$
$\displaystyle\left((\sqrt{\kappa}-\sqrt{P^{*}})^{+}\right)^{2}\geq\sigma_{0}^{2}\left(\frac{1}{\sqrt{21}}-\frac{1}{\sqrt{100}}\right)^{2}\approx
0.014\sigma_{0}^{2}\geq\frac{\sigma_{0}^{2}}{72}.$
Using the zero-input upper bound of $\frac{\sigma_{0}^{2}}{\sigma_{0}^{2}+1}$,
the ratio of the upper and lower bounds is at most
$\frac{72}{\sigma_{0}^{2}+1}\leq 72$.
Case 3: $P^{*}\leq\frac{\sigma_{0}^{2}}{100},\sigma_{0}^{2}\geq
16,P^{*}\leq\frac{1}{2}$.
In this case,
$\displaystyle\kappa$ $\displaystyle=$
$\displaystyle\frac{\sigma_{0}^{2}}{(\sigma_{0}+\sqrt{P^{*}})^{2}+1}\overset{(P^{*}\leq\frac{1}{2})}{\geq}\frac{\sigma_{0}^{2}}{(\sigma_{0}+\sqrt{0.5})^{2}+1}$
$\displaystyle\overset{(a)}{\geq}$
$\displaystyle\frac{16}{(\sqrt{16}+\sqrt{0.5})^{2}+1}\approx 0.6909\geq 0.69,$
where $(a)$ uses $\sigma_{0}^{2}\geq 16$ and the observation that
$\frac{x^{2}}{(x+b)^{2}+1}=\frac{1}{\left(1+\frac{b}{x}\right)^{2}+\frac{1}{x^{2}}}$
is an increasing function of $x$ for $x,b>0$. Thus,
$\displaystyle\left((\sqrt{\kappa}-\sqrt{P})^{+}\right)^{2}\geq((\sqrt{0.69}-\sqrt{0.5})^{+})^{2}\approx
0.0153\geq 0.015.$
Using the upper bound of $\frac{\sigma_{0}^{2}}{\sigma_{0}^{2}+1}<1$, the
ratio of the upper and the lower bounds is smaller than $\frac{1}{0.015}<67$.
Case 4: $\sigma_{0}^{2}>16$, $\frac{1}{2}<P^{*}\leq\frac{\sigma_{0}^{2}}{100}$
Using $L=2$ in the lower bound,
$\displaystyle c_{m}(L)$ $\displaystyle=$
$\displaystyle\frac{1}{\Pr(\|\mathbf{Z}^{m}\|^{2}\leq
mL^{2})}=\frac{1}{1-\Pr(\|\mathbf{Z}^{m}\|^{2}>mL^{2})}$
$\displaystyle\overset{\text{(Markov's ineq.)}}{\leq}$
$\displaystyle\frac{1}{1-\frac{m}{mL^{2}}}\overset{(L=2)}{=}\frac{4}{3},$
Similarly,
$\displaystyle d_{m}(2)$ $\displaystyle=$
$\displaystyle\frac{\Pr(\|\mathbf{Z}^{m+2}\|^{2}\leq
mL^{2})}{\Pr(\|\mathbf{Z}^{m}\|^{2}\leq mL^{2})}$ $\displaystyle\geq$
$\displaystyle\Pr(\|\mathbf{Z}^{m+2}\|^{2}\leq
mL^{2})=1-\Pr(\|\mathbf{Z}^{m+2}\|^{2}>mL^{2})$
$\displaystyle\overset{\text{(Markov's ineq.)}}{\geq}$ $\displaystyle
1-\frac{m+2}{mL^{2}}=1-\frac{1+\frac{2}{m}}{4}\overset{(m\geq
1)}{\geq}1-\frac{3}{4}=\frac{1}{4}.$
In the bound, we are free to use any $\sigma_{G}^{2}\geq 1$. Using
$\sigma_{G}^{2}=6P^{*}>1$,
$\displaystyle\kappa_{2}$ $\displaystyle=$
$\displaystyle\frac{\sigma_{G}^{2}\sigma_{0}^{2}}{\left((\sigma_{0}+\sqrt{P^{*}})^{2}+d_{m}(2)\sigma_{G}^{2}\right)c_{m}^{\frac{2}{m}}(2)e^{1-d_{m}(2)}}$
$\displaystyle\overset{(a)}{\geq}$
$\displaystyle\frac{6P^{*}\sigma_{0}^{2}}{\left((\sigma_{0}+\frac{\sigma_{0}}{10})^{2}+\frac{6\sigma_{0}^{2}}{100}\right)\left(\frac{4}{3}\right)^{\frac{2}{m}}e^{\frac{3}{4}}}\overset{(m\geq
1)}{\geq}1.255P^{*}.$
where $(a)$ uses
$\sigma_{G}^{2}=6P^{*},P^{*}<\frac{\sigma_{0}^{2}}{100},c_{m}(2)\leq\frac{4}{3}$
and $1>d_{m}(2)\geq\frac{1}{4}$. Thus,
$\left((\sqrt{\kappa_{2}}-\sqrt{P^{*}})^{+}\right)^{2}\geq
P^{*}(\sqrt{1.255}-1)^{2}\geq\frac{P^{*}}{70}.$ (14)
Now, using the lower bound on the total cost from Theorem 3, and substituting
$L=2$,
$\displaystyle\bar{J}_{min}(m,k^{2},\sigma_{0}^{2})$ $\displaystyle\geq$
$\displaystyle
k^{2}P^{*}+\frac{\sigma_{G}^{m}}{c_{m}(2)}\exp\left(-\frac{mL^{2}(\sigma_{G}^{2}-1)}{2}\right)\left(\left(\sqrt{\kappa_{2}}-\sqrt{P^{*}}\right)^{+}\right)^{2}$
(15) $\displaystyle\overset{(\sigma_{G}^{2}=6P^{*})}{\geq}$ $\displaystyle
k^{2}P^{*}+\frac{(6P^{*})^{m}}{c_{m}(2)}\exp\left(-\frac{4m(6P^{*}-1)}{2}\right)\;\frac{P^{*}}{70}$
$\displaystyle\overset{(a)}{\geq}$ $\displaystyle
k^{2}P^{*}+\frac{3^{m}}{\frac{4}{3}}e^{2m}e^{-12P^{*}m}\;\frac{1}{70\times 2}$
$\displaystyle\overset{(m\geq 1)}{\geq}$ $\displaystyle
k^{2}P^{*}+\frac{3\times 3\times e^{2}}{4\times 70\times 2}e^{-12mP^{*}}$
$\displaystyle>$ $\displaystyle k^{2}P^{*}+\frac{1}{9}e^{-12mP^{*}},$
where $(a)$ uses $c_{m}(2)\leq\frac{4}{3}$ and $P^{*}\geq\frac{1}{2}$. We
loosen the lattice-based upper bound from Theorem 1 and bring it into a form
similar to (15). Here, $P$ is a part of the optimization:
$\displaystyle\bar{J}_{min}(m,k^{2},\sigma_{0}^{2})$ (16) $\displaystyle\leq$
$\displaystyle\inf_{P>\xi^{2}}k^{2}P+\left(1+\sqrt{\frac{P}{\xi^{2}}}\right)^{2}e^{-\frac{mP}{2\xi^{2}}+\frac{m+2}{2}\left(1+\ln\left(\frac{P}{\xi^{2}}\right)\right)}$
$\displaystyle\leq$
$\displaystyle\inf_{P>\xi^{2}}k^{2}P+\frac{1}{9}e^{-\frac{0.5mP}{\xi^{2}}+\frac{m+2}{2}\left(1+\ln\left(\frac{P}{\xi^{2}}\right)\right)+2\ln\left(1+\sqrt{\frac{P}{\xi^{2}}}\right)+\ln\left(9\right)}$
$\displaystyle\leq$
$\displaystyle\inf_{P>\xi^{2}}k^{2}P+\frac{1}{9}e^{-m\left(\frac{0.5P}{\xi^{2}}-\frac{m+2}{2m}\left(1+\ln\left(\frac{P}{\xi^{2}}\right)\right)-\frac{2}{m}\ln\left(1+\sqrt{\frac{P}{\xi^{2}}}\right)-\frac{\ln\left(9\right)}{m}\right)}$
$\displaystyle=$
$\displaystyle\inf_{P>\xi^{2}}k^{2}P+\frac{1}{9}e^{-\frac{0.12mP}{\xi^{2}}}\times
e^{-m\left(\frac{0.38P}{\xi^{2}}-\frac{1+\frac{2}{m}}{2}\left(1+\ln\left(\frac{P}{\xi^{2}}\right)\right)-\frac{2}{m}\ln\left(1+\sqrt{\frac{P}{\xi^{2}}}\right)-\frac{\ln\left(9\right)}{m}\right)}$
$\displaystyle\overset{(m\geq 1)}{\leq}$
$\displaystyle\inf_{P>\xi^{2}}k^{2}P+\frac{1}{9}e^{-\frac{0.12mP}{\xi^{2}}}e^{-m\left(\frac{0.38P}{\xi^{2}}-\frac{3}{2}\left(1+\ln\left(\frac{P}{\xi^{2}}\right)\right)-2\ln\left(1+\sqrt{\frac{P}{\xi^{2}}}\right)-\ln\left(9\right)\right)}$
$\displaystyle\leq$ $\displaystyle\inf_{P\geq
34\xi^{2}}k^{2}P+\frac{1}{9}e^{-\frac{0.12mP}{\xi^{2}}},$
where the last inequality follows from the fact that
$\frac{0.38P}{\xi^{2}}>\frac{3}{2}\left(1+\ln\left(\frac{P}{\xi^{2}}\right)\right)+2\ln\left(1+\sqrt{\frac{P}{\xi^{2}}}\right)+\ln\left(9\right)$
for $\frac{P}{\xi^{2}}>34$. This can be checked easily by plotting it.888It
can also be verified symbolically by examining the expression
$g(b)=0.38b^{2}-\frac{3}{2}(1+\ln b^{2})-2\ln(1+b)-\ln\left(9\right)$, taking
its derivative $g^{\prime}(b)=0.76b-\frac{3}{b}-\frac{2}{1+b}$, and second
derivative $g^{\prime\prime}(b)=0.76+\frac{3}{b^{2}}+\frac{2}{(1+b)^{2}}>0$.
Thus $g(\cdot{})$ is convex-$\cup$. Further, $g^{\prime}(\sqrt{34})\approx
3.62>0$, and $g(\sqrt{34})\approx 0.09$ and so $g(b)>0$ whenever
$b\geq\sqrt{34}$.
Using $P=100\xi^{2}P^{*}\geq 50\xi^{2}>34\xi^{2}$ (since
$P^{*}\geq\frac{1}{2}$) in (16),
$\displaystyle\bar{J}_{min}(m,k^{2},\sigma_{0}^{2})$ $\displaystyle\leq$
$\displaystyle k^{2}100\xi^{2}P^{*}+\frac{1}{9}e^{-m\frac{0.12\times
100\xi^{2}P^{*}}{\xi^{2}}}$ (17) $\displaystyle=$ $\displaystyle
k^{2}100\xi^{2}P^{*}+\frac{1}{9}e^{-12mP^{*}}.$
Using (15) and (17), the ratio of the upper and the lower bounds is bounded
for all $m$ since
$\mu\leq\frac{k^{2}100\xi^{2}P^{*}+\frac{1}{9}e^{-12mP^{*}}}{k^{2}P^{*}+\frac{1}{9}e^{-12mP^{*}}}\leq\frac{k^{2}100\xi^{2}P^{*}}{k^{2}P^{*}}=100\xi^{2}.$
(18)
For $m=1$, $\xi=1$, and thus in the proof the ratio $\mu\leq 100$. For $m$
large, $\xi\approx 2$ [46], and $\mu\lesssim 400$. For arbitrary $m$, using
the recursive construction in [51, Theorem 8.18], $\xi\leq 4$, and thus
$\mu\leq 1600$ regardless of $m$. ∎
Though the proof above succeeds in showing that the ratio is uniformly bounded
by a constant, it is not very insightful and the constant is large. However,
since the underlying vector bound can be tightened (as shown in [32]), it is
not worth improving the proof for increased elegance at this time. The
important thing is that such a uniform constant exists.
A numerical evaluation of the upper and lower bounds (of Theorem 1 and 3
respectively) shows that the ratio is smaller than $17$ for $m=1$ (see Fig.
4). A precise calculation of the cost of the quantization strategy improves
the upper bound to yield a maximum ratio smaller than $8$ (see Fig. 5).
A simple grid lattice has a packing-covering ratio $\xi=\sqrt{m}$. Therefore,
while the grid lattice has the best possible packing-covering ratio of $1$ in
the scalar case, it has a rather large packing covering ratio of
$\sqrt{2}\;(\approx 1.41)$ for $m=2$. On the other hand, a hexagonal lattice
(for $m=2$) has an improved packing-covering ratio of
$\frac{2}{\sqrt{3}}\approx 1.15$. In contrast with $m=1$, where the ratio of
upper and lower bounds of Theorem 1 and 3 is approximately $17$, a hexagonal
lattice yields a ratio smaller than $14.75$, despite having a larger packing-
covering ratio. This is a consequence of the tightening of the sphere-packing
lower bound (Theorem 3) as $m$ gets large999Indeed, in the limit
$m\rightarrow\infty$, the ratio of the asymptotic average costs attained by a
vector-quantization strategy and the vector lower bound of Theorem 2 is
bounded by $4.45$ [15]..
## VI Discussions of numerical explorations and Conclusions
Though lattice-based quantization strategies allow us to get within a constant
factor of the optimal cost for the vector Witsenhausen problem, they are not
optimal. This is known for the scalar [5] and the infinite-length case [15].
It is shown in [15] that the “slopey-quantization” strategy of Lee, Lau and Ho
[5] that is believed to be very close to optimal in the scalar case can be
viewed as an instance of a linear scaling followed by a dirty-paper coding
(DPC) strategy. Such DPC-based strategies are also the best known strategies
in the asymptotic infinite-dimensional case, requiring optimal power $P$ to
attain $0$ asymptotic mean-square error in the estimation of
$\mathbf{x}^{m}_{1}$, and attaining costs within a factor of $1.3$ of the
optimal [32] for all $(k,\sigma_{0}^{2})$. This leads us to conjecture that a
DPC-like strategy might be optimal for finite-vector lengths as well. In the
following, we numerically explore the performance of DPC-like strategies.
Figure 6: Ratio of the achievable costs to the scalar lower bound along
$k\sigma_{0}=10^{-0.5}$ for various strategies. Quantization with MMSE-
estimation at the second controller outperforms quantization with MLE, or even
scaled MLE. For slopey-quantization with heuristic DPC-parameter, the
parameter $\alpha$ in DPC-based scheme is borrowed from the infinite-length
analysis. The figure suggests that along this path ($k\sigma_{0}=\sqrt{10}$),
the difference between optimal-DPC and heuristic DPC is not substantial.
However, Fig. 7 (b) shows that this is not true in general. Figure 7: (a)
shows the ratio of cost attained by linear+quantization (with MMSE decoding)
to DPC with parameter $\alpha$ obtained by brute-force optimization. DPC can
do up to $15\%$ better than the optimal quantization strategy. Also the
maximum is attained along $k\approx 0.6$ which is different from $k=0.2$ of
the benchmark problem [5]. (b) shows the ratio of cost attained by
linear+quantization to DPC with $\alpha$ borrowed from infinite-length
optimization. Heuristic DPC does not outperform linear+quantization (with MMSE
estimation) substantially.
It is natural to ask how much there is to gain using a DPC-based strategy over
a simple quantization strategy. Notice that the DPC-strategy gains not only
from the slopey quantization, but also from the MMSE-estimation at the second
controller. In Fig. 6, we eliminate the latter advantage by considering first
a uniform quantization-based strategy with an appropriate scaling of the MLE
so that it approximates the MMSE-estimation performance, and then the actual
MMSE-estimation strategy for uniform quantization. Along the curve
$k\sigma_{0}=\sqrt{10}$, there is significant gain in using this approximate-
MMSE estimation over MLE, and further gain in using MMSE-estimation itself.
This also shows that there is an interesting tradeoff between the complexity
of the second controller and the system performance.
From Fig. 6, along the curve $k\sigma_{0}=\sqrt{10}$, the DPC-based strategy
performs only negligibly better than a quantization-based strategy with MMSE
estimation. Fig. 7 (a) shows that this is not true in general. A DPC-based
strategy can perform up to $15\%$ better than a simple quantization-based
scheme depending on the problem parameters. Interestingly, the advantage of
using a DPC-based strategy for the case of $k=0.2,\sigma_{0}=5$ (which is used
as the benchmark case in many papers, e.g. [5, 8]) is quite small. The maximum
gain of about $15\%$ is obtained at $k\approx 10^{-0.2}\approx 0.63$, and
$\sigma_{0}=1$ (and indeed, any $\sigma_{0}>1$. In the future, we suggest the
community use the point $(0.63,1)$ as the benchmark case.
Given that there is an advantage in using a DPC-like strategy, an interesting
question is whether the DPC parameter $\alpha$ that optimizes the DPC-based
strategy’s performance at infinite-lengths (in [15]) gives good performance
for the scalar case as well. Fig. 7 (b) answers this question at least
partially in the negative. This heuristic-DPC does only slightly better than a
quantization strategy with MMSE estimation, whereas other values of $\alpha$
do significantly better.
Finally, we observe that while uniform bin-size quantization or DPC-based
strategies are designed for atypical noise behavior, atypical behavior of the
the initial state is better accommodated by using nonuniform bin-sizes (such
as those in [5, 8]). Table I compares the two. Clearly, the advantage in
having nonuniform slopey-quantization is small, but not negligible. It would
be interesting to calibrate the advantage of nonuniform-bin sizes for
$(k,\sigma_{0})=(0.63,1)$, a maximum gain point for uniform-bin size slopey-
quantization strategies.
TABLE I: Costs attained for the benchmark case of $k=0.2$, $\sigma_{0}=5$. | linear+quantization | Slopey-quantization
---|---|---
Lee, Lau and Ho [5] | 0.1713946 | 0.1673132
Li, Marden and Shamma [8] | — | 0.1670790
This paper | 0.1715335 | 0.1673654
There are plenty of open problems that arise naturally. Both the lower and the
upper bounds have room for improvement. The lower bound can be improved by
tightening the vector lower bound of [15] (one such tightening is performed in
[32]) and obtaining corresponding finite-length results using the sphere-
packing tools developed here.
Tightening the upper bound can be performed by using DPC-based techniques over
lattices. Further, an exact analysis of the required first-stage power when
using a lattice would yield an improvement (as pointed out earlier, for $m=1$,
$\frac{1}{m}k^{2}r_{c}^{2}$ overestimates the required first-stage cost),
especially for small $m$. Improved lattice designs with better packing-
covering ratios would also improve the upper bound.
Perhaps a more significant set of open problems are the next steps in
understanding more realistic versions of Witsenhausen’s problem, specifically
those that include costs on all the inputs and all the states [13], with noisy
state evolution and noisy observations at both controllers. The hope is that
solutions to these problems can then be used as the basis for provably-good
nonlinear controller synthesis for larger distributed systems. Further, tools
developed for solving these problems might help address multiuser problems in
information theory, in the spirit of [52, 53].
## Acknowledgments
We gratefully acknowledge the support of the National Science Foundation
(CNS-403427, CNS-093240, CCF-0917212 and CCF-729122), Sumitomo Electric and
Samsung. We thank Amin Gohari, Bobak Nazer and Anand Sarwate for helpful
discussions, and Gireeja Ranade for suggesting improvements in the paper.
## Appendix A Proof of Lemma 1
$\displaystyle\mathbb{E}_{\mathbf{Z}^{m}}\left[{\left(\|\mathbf{Z}^{m}\|+r_{p}\right)^{2}{1{1}}_{\left\\{\mathcal{E}_{m}\right\\}}}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}_{\mathbf{Z}^{m}}\left[{\|\mathbf{Z}^{m}\|^{2}{1{1}}_{\left\\{\mathcal{E}_{m}\right\\}}}\right]+r_{p}^{2}\Pr(\mathcal{E}_{m})+2r_{p}\mathbb{E}_{\mathbf{Z}^{m}}\left[{\left({1{1}}_{\left\\{\mathcal{E}_{m}\right\\}}\right)\left(\|\mathbf{Z}^{m}\|{1{1}}_{\left\\{\mathcal{E}_{m}\right\\}}\right)}\right]$
(19) $\displaystyle\overset{(a)}{\leq}$
$\displaystyle\mathbb{E}_{\mathbf{Z}^{m}}\left[{\|\mathbf{Z}^{m}\|^{2}{1{1}}_{\left\\{\mathcal{E}_{m}\right\\}}}\right]+r_{p}^{2}\Pr(\mathcal{E}_{m})+2r_{p}\sqrt{\mathbb{E}_{\mathbf{Z}^{m}}\left[{{1{1}}_{\left\\{\mathcal{E}_{m}\right\\}}}\right]}\sqrt{\mathbb{E}_{\mathbf{Z}^{m}}\left[{\|\mathbf{Z}^{m}\|^{2}{1{1}}_{\left\\{\mathcal{E}_{m}\right\\}}}\right]}$
$\displaystyle=$
$\displaystyle\left(\sqrt{\mathbb{E}_{\mathbf{Z}^{m}}\left[{\|\mathbf{Z}^{m}\|^{2}{1{1}}_{\left\\{\mathcal{E}_{m}\right\\}}}\right]}+r_{p}\sqrt{\Pr(\mathcal{E}_{m})}\right)^{2},$
where $(a)$ uses the Cauchy-Schwartz inequality [54, Pg. 13].
We wish to express
$\mathbb{E}_{\mathbf{Z}^{m}}\left[{\|\mathbf{Z}^{m}\|^{2}{1{1}}_{\left\\{\mathcal{E}_{m}\right\\}}}\right]$
in terms of $\psi(m,r_{p}):=\Pr(\|\mathbf{Z}^{m}\|\geq
r_{p})=\int_{\|\mathbf{z}^{m}\|\geq
r_{p}}\frac{e^{-\frac{\|\mathbf{z}^{m}\|^{2}}{2}}}{\left(\sqrt{2\pi}\right)^{m}}d\mathbf{z}^{m}$.
Denote by
$\mathcal{A}_{m}(r):=\frac{2\pi^{\frac{m}{2}}r^{m-1}}{\Gamma\left(\frac{m}{2}\right)}$
the surface area of a sphere of radius $r$ in $\mathbb{R}^{m}$ [55, Pg. 458],
where $\Gamma(\cdot{})$ is the Gamma-function satisfying
$\Gamma(m)=(m-1)\Gamma(m-1)$, $\Gamma(1)=1$, and
$\Gamma(\frac{1}{2})=\sqrt{\pi}$. Dividing the space $\mathbb{R}^{m}$ into
shells of thickness $dr$ and radii $r$,
$\displaystyle\mathbb{E}_{\mathbf{Z}^{m}}\left[{\|\mathbf{Z}^{m}\|^{2}{1{1}}_{\left\\{\mathcal{E}_{m}\right\\}}}\right]$
$\displaystyle=$ $\displaystyle\int_{\|\mathbf{z}^{m}\|\geq
r_{p}}\|\mathbf{z}^{m}\|^{2}\frac{e^{-\frac{\|\mathbf{z}^{m}\|^{2}}{2}}}{\left(\sqrt{2\pi}\right)^{m}}d\mathbf{z}^{m}=\int_{r\geq
r_{p}}r^{2}\frac{e^{-\frac{r^{2}}{2}}}{\left(\sqrt{2\pi}\right)^{m}}\mathcal{A}_{m}(r)dr$
(20) $\displaystyle=$ $\displaystyle\int_{r\geq
r_{p}}r^{2}\frac{e^{-\frac{r^{2}}{2}}}{\left(\sqrt{2\pi}\right)^{m}}\frac{2\pi^{\frac{m}{2}}r^{m-1}}{\Gamma\left(\frac{m}{2}\right)}dr$
$\displaystyle=$ $\displaystyle\int_{r\geq
r_{p}}\frac{e^{-\frac{r^{2}}{2}}2\pi}{\left(\sqrt{2\pi}\right)^{m+2}}\frac{2\pi^{\frac{m+2}{2}}r^{m+1}}{\pi\frac{2}{m}\Gamma\left(\frac{m+2}{2}\right)}dr=m\psi(m+2,r_{p}).$
Using (19), (20), and $r_{p}=\sqrt{\frac{mP}{\xi^{2}}}$
$\displaystyle\mathbb{E}_{\mathbf{Z}^{m}}\left[{\left(\|\mathbf{Z}^{m}\|+r_{p}\right)^{2}{1{1}}_{\left\\{\mathcal{E}_{m}\right\\}}}\right]\leq
m\left(\sqrt{\psi(m+2,r_{p})}+\sqrt{\frac{P}{\xi^{2}}}\sqrt{\psi(m,r_{p})}\right)^{2},$
which yields the first part of Lemma 1. To obtain a closed-form upper bound we
consider $P>\xi^{2}$. It suffices to bound $\psi(\cdot{},\cdot{})$.
$\displaystyle\psi(m,r_{p})$ $\displaystyle=$
$\displaystyle\Pr(\|\mathbf{Z}^{m}\|^{2}\geq
r_{p}^{2})=\Pr(\exp(\rho\sum_{i=1}^{m}Z_{i}^{2})\geq\exp(\rho r_{p}^{2}))$
$\displaystyle\overset{(a)}{\leq}$
$\displaystyle\mathbb{E}_{\mathbf{Z}^{m}}\left[{\exp(\rho\sum_{i=1}^{m}Z_{i}^{2})}\right]e^{-\rho
r_{p}^{2}}=\mathbb{E}_{Z_{1}}\left[{\exp(\rho Z_{1}^{2})}\right]^{m}e^{-\rho
r_{p}^{2}}\overset{(\text{for}\;0<\rho<0.5)}{=}\frac{1}{(1-2\rho)^{\frac{m}{2}}}e^{-\rho
r_{p}^{2}},$
where $(a)$ follows from the Markov inequality, and the last inequality
follows from the fact that the moment generating function of a standard
$\chi_{2}^{2}$ random variable is $\frac{1}{(1-2\rho)^{\frac{1}{2}}}$ for
$\rho\in(0,0.5)$ [56, Pg. 375]. Since this bound holds for any
$\rho\in(0,0.5)$, we choose the minimizing
$\rho^{*}=\frac{1}{2}\left(1-\frac{m}{r_{p}^{2}}\right)$. Since
$r_{p}^{2}=\frac{mP}{\xi^{2}}$, $\rho^{*}$ is indeed in $(0,0.5)$ as long as
$P>\xi^{2}$. Thus,
$\displaystyle\psi(m,r_{p})\leq\frac{1}{(1-2\rho^{*})^{\frac{m}{2}}}e^{-\rho^{*}r_{p}^{2}}=\left(\frac{r_{p}^{2}}{m}\right)^{\frac{m}{2}}e^{-\frac{1}{2}\left(1-\frac{m}{r_{p}^{2}}\right)r_{p}^{2}}=e^{-\frac{r_{p}^{2}}{2}+\frac{m}{2}+\frac{m}{2}\ln\left(\frac{r_{p}^{2}}{m}\right)}.$
Using the substitutions $r_{c}^{2}=mP$, $\xi=\frac{r_{c}}{r_{p}}$ and
$r_{p}^{2}=\frac{mP}{\xi^{2}}$,
$\displaystyle\Pr(\mathcal{E}_{m})=\psi(m,r_{p})=\psi\left(m,\sqrt{\frac{mP}{\xi^{2}}}\right)\leq
e^{-\frac{mP}{2\xi^{2}}+\frac{m}{2}+\frac{m}{2}\ln\left(\frac{P}{\xi^{2}}\right)},\;\text{and}$
(21)
$\displaystyle\mathbb{E}_{\mathbf{Z}^{m}}\left[{\|\mathbf{Z}^{m}\|^{2}{1{1}}_{\left\\{\mathcal{E}_{m}\right\\}}}\right]\leq
m\psi\left(m+2,\sqrt{\frac{mP}{\xi^{2}}}\right)\leq
me^{-\frac{mP}{2\xi^{2}}+\frac{m+2}{2}+\frac{m+2}{2}\ln\left(\frac{mP}{(m+2)\xi^{2}}\right)}.$
(22)
From (19), (21) and (22),
$\displaystyle\mathbb{E}_{\mathbf{Z}^{m}}\left[{\left(\|\mathbf{Z}^{m}\|+r_{p}\right)^{2}{1{1}}_{\left\\{\mathcal{E}_{m}\right\\}}}\right]$
$\displaystyle\leq$
$\displaystyle\bigg{(}\sqrt{m}e^{-\frac{mP}{4\xi^{2}}+\frac{m+2}{4}+\frac{m+2}{4}\ln\left(\frac{mP}{(m+2)\xi^{2}}\right)}\sqrt{\frac{mP}{\xi^{2}}}e^{-\frac{mP}{4\xi^{2}}+\frac{m}{4}+\frac{m}{4}\ln\left(\frac{P}{\xi^{2}}\right)}\bigg{)}^{2}$
$\displaystyle\overset{(\text{since}\;P>\xi^{2})}{<}$
$\displaystyle\left(\sqrt{m}\left(1+\sqrt{\frac{P}{\xi^{2}}}\right)e^{-\frac{mP}{4\xi^{2}}+\frac{m+2}{4}+\frac{m+2}{4}\ln\left(\frac{P}{\xi^{2}}\right)}\right)^{2}$
$\displaystyle=$ $\displaystyle
m\left(1+\sqrt{\frac{P}{\xi^{2}}}\right)^{2}e^{-\frac{mP}{2\xi^{2}}+\frac{m+2}{2}+\frac{m+2}{2}\ln\left(\frac{P}{\xi^{2}}\right)}.$
## Appendix B Proof of Lemma 2
The following lemma is taken from [15].
###### Lemma 3
For any three random variables $A$, $B$ and $C$,
$\displaystyle\mathbb{E}\left[{\|B-C\|^{2}}\right]\geq\left(\left(\sqrt{\mathbb{E}\left[{\|A-C\|^{2}}\right]}-\sqrt{\mathbb{E}\left[{\|A-B\|^{2}}\right]}\right)^{+}\right)^{2}.$
###### Proof:
See [15, Appendix II]. ∎
Choosing $A=\mathbf{X}^{m}_{0}$, $B=\mathbf{X}^{m}_{1}$ and
$C=\mathbf{\widehat{X}}^{m}_{1}$,
$\displaystyle\mathbb{E}_{\mathbf{X}^{m}_{0},\mathbf{Z}^{m}_{G}}\left[{J_{2}^{(\gamma)}(\mathbf{X}^{m}_{0},\mathbf{Z}^{m}_{G})|\mathbf{Z}^{m}_{G}\in\mathcal{S}_{L}^{G}}\right]=\frac{1}{m}\mathbb{E}_{\mathbf{X}^{m}_{0},\mathbf{Z}^{m}_{G}}\left[{\|\mathbf{X}^{m}_{1}-\mathbf{\widehat{X}}^{m}_{1}\|^{2}|\mathbf{Z}^{m}_{G}\in\mathcal{S}_{L}^{G}}\right]$
(23) $\displaystyle\geq$
$\displaystyle\bigg{(}\bigg{(}\sqrt{\frac{1}{m}\mathbb{E}_{\mathbf{X}^{m}_{0},\mathbf{Z}^{m}_{G}}\left[{\|\mathbf{X}^{m}_{0}-\mathbf{\widehat{X}}^{m}_{1}\|^{2}|\mathbf{Z}^{m}_{G}\in\mathcal{S}_{L}^{G}}\right]}-\sqrt{\frac{1}{m}\mathbb{E}_{\mathbf{X}^{m}_{0},\mathbf{Z}^{m}_{G}}\left[{\|\mathbf{X}^{m}_{0}-\mathbf{X}^{m}_{1}\|^{2}|\mathbf{Z}^{m}_{G}\in\mathcal{S}_{L}^{G}}\right]}\bigg{)}^{+}\bigg{)}^{2}$
$\displaystyle=$
$\displaystyle\bigg{(}\bigg{(}\sqrt{\frac{1}{m}\mathbb{E}_{\mathbf{X}^{m}_{0},\mathbf{Z}^{m}_{G}}\left[{\|\mathbf{X}^{m}_{0}-\mathbf{\widehat{X}}^{m}_{1}\|^{2}|\mathbf{Z}^{m}_{L}\in\mathcal{S}_{L}^{G}}\right]}-\sqrt{P}\bigg{)}^{+}\bigg{)}^{2},$
since $\mathbf{X}^{m}_{0}-\mathbf{X}^{m}_{1}=\mathbf{U}^{m}_{1}$ is
independent of $\mathbf{Z}^{m}_{G}$ and
$\mathbb{E}\left[{\|\mathbf{U}^{m}_{1}\|^{2}}\right]=mP$. Define
$\mathbf{Y}^{m}_{L}:=\mathbf{X}^{m}_{1}+\mathbf{Z}^{m}_{L}$ to be the output
when the observation noise $\mathbf{Z}^{m}_{L}$ is distributed as a truncated
Gaussian distribution:
$f_{Z_{L}}(\mathbf{z}^{m}_{L})=\left\\{\begin{array}[]{ll}c_{m}(L)\frac{e^{-\frac{\|\mathbf{z}^{m}_{L}\|^{2}}{2\sigma_{G}^{2}}}}{\left(\sqrt{2\pi\sigma_{G}^{2}}\right)^{m}}&\mathbf{z}^{m}_{L}\in\mathcal{S}_{L}^{G}\\\
0&\text{otherwise.}\end{array}\right.$ (24)
Let the estimate at the second controller on observing $\mathbf{y}^{m}_{L}$ be
denoted by $\mathbf{\widehat{X}}^{m}_{L}$. Then, by the definition of
conditional expectations,
$\mathbb{E}_{\mathbf{X}^{m}_{0},\mathbf{Z}^{m}_{G}}\left[{\|\mathbf{X}^{m}_{0}-\mathbf{\widehat{X}}^{m}_{1}\|^{2}|\mathbf{Z}^{m}_{G}\in\mathcal{S}_{L}^{G}}\right]=\mathbb{E}_{\mathbf{X}^{m}_{0},\mathbf{Z}^{m}_{G}}\left[{\|\mathbf{X}^{m}_{0}-\mathbf{\widehat{X}}^{m}_{L}\|^{2}}\right].$
(25)
To get a lower bound, we now allow the controllers to optimize themselves with
the additional knowledge that the observation noise $\mathbf{z}^{m}$ must fall
in $\mathcal{S}_{L}^{G}$. In order to prevent the first controller from
“cheating” and allocating different powers to the two events (_i.e._
$\mathbf{z}^{m}$ falling or not falling in $\mathcal{S}_{L}^{G}$), we enforce
the constraint that the power $P$ must not change with this additional
knowledge. Since the controller’s observation $\mathbf{X}^{m}_{0}$ is
independent of $\mathbf{Z}^{m}$, this constraint is satisfied by the original
controller (without the additional knowledge) as well, and hence the cost for
the system with the additional knowledge is still a valid lower bound to that
of the original system.
The rest of the proof uses ideas from channel coding and the rate-distortion
theorem [57, Ch. 13] from information theory. We view the problem as a problem
of implicit communication from the first controller to the second. Notice that
for a given $\gamma(\cdot{})$, $\mathbf{X}^{m}_{1}$ is a function of
$\mathbf{X}^{m}_{0}$,
$\mathbf{Y}^{m}_{L}=\mathbf{X}^{m}_{1}+\mathbf{Z}^{m}_{L}$ is conditionally
independent of $\mathbf{X}^{m}_{0}$ given $\mathbf{X}^{m}_{1}$ (since the
noise $\mathbf{Z}^{m}_{L}$ is additive and independent of $\mathbf{X}^{m}_{1}$
and $\mathbf{X}^{m}_{0}$). Further, $\mathbf{\widehat{X}}^{m}_{L}$ is a
function of $\mathbf{Y}^{m}_{L}$. Thus
$\mathbf{X}^{m}_{0}-\mathbf{X}^{m}_{1}-\mathbf{Y}^{m}_{L}-\mathbf{\widehat{X}}^{m}_{L}$
form a Markov chain. Using the data-processing inequality [57, Pg. 33],
$I(\mathbf{X}^{m}_{0};\mathbf{\widehat{X}}^{m}_{L})\leq
I(\mathbf{X}^{m}_{1};\mathbf{Y}^{m}_{L}),$ (26)
where $I(A,B)$ is the expression for mutual information expression between two
random variables $A$ and $B$ (see, for example, [57, Pg. 18, Pg. 231]). To
estimate the distortion to which $\mathbf{X}^{m}_{0}$ can be communicated
across this truncated Gaussian channel (which, in turn, helps us lower bound
the MMSE in estimating $\mathbf{X}^{m}_{1}$), we need to upper bound the term
on the RHS of (26).
###### Lemma 4
$\frac{1}{m}I(\mathbf{X}^{m}_{1};\mathbf{Y}^{m}_{L})\leq\frac{1}{2}\log_{2}\left(\frac{e^{1-d_{m}(L)}(\bar{P}+d_{m}(L)\sigma_{G}^{2})c_{m}^{\frac{2}{m}}(L)}{\sigma_{G}^{2}}\right).$
###### Proof:
We first obtain an upper bound to the power of $\mathbf{X}^{m}_{1}$ (this
bound is the same as that used in [15]):
$\displaystyle\mathbb{E}_{\mathbf{X}^{m}_{0}}\left[{\|\mathbf{X}^{m}_{1}\|^{2}}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}_{\mathbf{X}^{m}_{0}}\left[{\|\mathbf{X}^{m}_{0}+\mathbf{U}^{m}_{1}\|^{2}}\right]=\mathbb{E}_{\mathbf{X}^{m}_{0}}\left[{\|\mathbf{X}^{m}_{0}\|^{2}}\right]+\mathbb{E}_{\mathbf{X}^{m}_{0}}\left[{\|\mathbf{U}^{m}_{1}\|^{2}}\right]+2\mathbb{E}_{\mathbf{X}^{m}_{0}}\left[{{\mathbf{X}^{m}_{0}}^{T}\mathbf{U}^{m}_{1}}\right]$
$\displaystyle\overset{(a)}{\leq}$
$\displaystyle\mathbb{E}_{\mathbf{X}^{m}_{0}}\left[{\|\mathbf{X}^{m}_{0}\|^{2}}\right]+\mathbb{E}_{\mathbf{X}^{m}_{0}}\left[{\|\mathbf{U}^{m}_{1}\|^{2}}\right]+2\sqrt{\mathbb{E}_{\mathbf{X}^{m}_{0}}\left[{\|\mathbf{X}^{m}_{0}\|^{2}}\right]}\sqrt{\mathbb{E}_{\mathbf{X}^{m}_{0}}\left[{\|\mathbf{U}^{m}_{1}\|^{2}}\right]}$
$\displaystyle\leq$ $\displaystyle m(\sigma_{0}+\sqrt{P})^{2},$
where $(a)$ follows from the Cauchy-Schwartz inequality. We use the following
definition of differential entropy $h(A)$ of a continuous random variable $A$
[57, Pg. 224]:
$h(A)=-\int_{S}f_{A}(a)\log_{2}\left(f_{A}(a)\right)da,$ (27)
where $f_{A}(a)$ is the pdf of $A$, and $S$ is the support set of $A$.
Conditional differential entropy is defined similarly [57, Pg. 229].
Let $\bar{P}:=(\sigma_{0}+\sqrt{P})^{2}$. Now,
$\mathbb{E}\left[{Y_{L,i}^{2}}\right]=\mathbb{E}\left[{X_{1,i}^{2}}\right]+\mathbb{E}\left[{Z_{L,i}^{2}}\right]$
(since $X_{1,i}$ is independent of $Z_{L,i}$ and by symmetry, $Z_{L,i}$ are
zero mean random variables). Denote
$\bar{P}_{i}=\mathbb{E}\left[{X_{1,i}^{2}}\right]$ and
$\sigma_{G,i}^{2}=\mathbb{E}\left[{Z_{L,i}^{2}}\right]$. In the following, we
derive an upper bound $C_{G,L}^{(m)}$ on
$\frac{1}{m}I(\mathbf{X}^{m}_{1};\mathbf{Y}^{m}_{L})$.
$\displaystyle C_{G,L}^{(m)}$ $\displaystyle:=$
$\displaystyle\sup_{p(\mathbf{X}^{m}_{1}):\mathbb{E}\left[{\|\mathbf{X}^{m}_{1}\|^{2}}\right]\leq
m\bar{P}}\frac{1}{m}I(\mathbf{X}^{m}_{1};\mathbf{Y}^{m}_{L})$ (28)
$\displaystyle\overset{(a)}{=}$
$\displaystyle\sup_{p(\mathbf{X}^{m}_{1}):\mathbb{E}\left[{\|\mathbf{X}^{m}_{1}\|^{2}}\right]\leq
m\bar{P}}\frac{1}{m}h(\mathbf{Y}^{m}_{L})-\frac{1}{m}h(\mathbf{Y}^{m}_{L}|\mathbf{X}^{m}_{1})$
$\displaystyle\overset{}{=}$
$\displaystyle\sup_{p(\mathbf{X}^{m}_{1}):\mathbb{E}\left[{\|\mathbf{X}^{m}_{1}\|^{2}}\right]\leq
m\bar{P}}\frac{1}{m}h(\mathbf{Y}^{m}_{L})-\frac{1}{m}h(\mathbf{X}^{m}_{1}+\mathbf{Z}^{m}_{L}|\mathbf{X}^{m}_{1})$
$\displaystyle\overset{(b)}{=}$
$\displaystyle\sup_{p(\mathbf{X}^{m}_{1}):\mathbb{E}\left[{\|\mathbf{X}^{m}_{1}\|^{2}}\right]\leq
m\bar{P}}\frac{1}{m}h(\mathbf{Y}^{m}_{L})-\frac{1}{m}h(\mathbf{Z}^{m}_{L}|\mathbf{X}^{m}_{1})$
$\displaystyle\overset{(c)}{=}$
$\displaystyle\sup_{p(\mathbf{X}^{m}_{1}):\mathbb{E}\left[{\|\mathbf{X}^{m}_{1}\|^{2}}\right]\leq
m\bar{P}}\frac{1}{m}h(\mathbf{Y}^{m}_{L})-\frac{1}{m}h(\mathbf{Z}^{m}_{L})$
$\displaystyle\overset{(d)}{\leq}$
$\displaystyle\sup_{p(\mathbf{X}^{m}_{1}):\mathbb{E}\left[{\|\mathbf{X}^{m}_{1}\|^{2}}\right]\leq
m\bar{P}}\frac{1}{m}\sum_{i=1}^{m}h(Y_{L,i})-\frac{1}{m}h(\mathbf{Z}^{m}_{L})$
$\displaystyle\overset{(e)}{\leq}$
$\displaystyle\sup_{\bar{P}_{i}:\sum_{i=1}^{m}\bar{P}_{i}\leq
m\bar{P}}\frac{1}{m}\sum_{i=1}^{m}\frac{1}{2}\log_{2}\left(2\pi
e(\bar{P}_{i}+\sigma_{G,i}^{2})\right)-\frac{1}{m}h(\mathbf{Z}^{m}_{L})$
$\displaystyle\overset{(f)}{\leq}$ $\displaystyle\frac{1}{2}\log_{2}\left(2\pi
e(\bar{P}+d_{m}(L)\sigma_{G}^{2})\right)-\frac{1}{m}h(\mathbf{Z}^{m}_{L}).$
Here, $(a)$ follows from the definition of mutual information [57, Pg. 231],
$(b)$ follows from the fact that translation does not change the differential
entropy [57, Pg. 233], $(c)$ uses independence of $\mathbf{Z}^{m}_{L}$ and
$\mathbf{X}^{m}_{1}$, and $(d)$ uses the chain rule for differential entropy
[57, Pg. 232] and the fact that conditioning reduces entropy [57, Pg. 232]. In
$(e)$, we used the fact that Gaussian random variables maximize differential
entropy. The inequality $(f)$ follows from the concavity-$\cap$ of the
$\log(\cdot{})$ function and an application of Jensen’s inequality [57, Pg.
25]. We also use the fact that
$\frac{1}{m}\sum_{i=1}^{m}\sigma_{G,i}^{2}=d_{m}(L)\sigma_{G}^{2}$, which can
be proven as follows
$\displaystyle\frac{1}{m}\mathbb{E}\left[{\sum_{i=1}^{m}Z_{L,i}^{2}}\right]$
$\displaystyle\overset{(\text{using}~{}\eqref{eq:fz})}{=}$
$\displaystyle\frac{\sigma_{G}^{2}}{m}\int_{\mathbf{z}^{m}\in\mathcal{S}_{L}^{G}}\frac{\|\mathbf{z}^{m}\|^{2}}{\sigma_{G}^{2}}c_{m}(L)\frac{\exp\left(-\frac{\|\mathbf{z}^{m}_{G}\|^{2}}{2\sigma_{G}^{2}}\right)}{\left(\sqrt{2\pi\sigma_{G}^{2}}\right)^{m}}d\mathbf{z}^{m}_{G}$
(29) $\displaystyle=$
$\displaystyle\frac{c_{m}(L)\sigma_{G}^{2}}{m}\mathbb{E}\left[{\|\mathbf{Z}^{m}_{G}\|^{2}{1{1}}_{\left\\{\|\mathbf{Z}^{m}_{G}\|\leq\sqrt{mL^{2}\sigma_{G}^{2}}\right\\}}}\right]$
$\displaystyle\overset{(\mathbf{\widetilde{Z}}^{m}:=\frac{\mathbf{Z}^{m}_{G}}{\sigma_{G}})}{=}$
$\displaystyle\frac{c_{m}(L)\sigma_{G}^{2}}{m}\mathbb{E}\left[{\|\mathbf{\widetilde{Z}}^{m}\|^{2}{1{1}}_{\left\\{\|\mathbf{\widetilde{Z}}^{m}\|\leq\sqrt{mL^{2}}\right\\}}}\right]$
$\displaystyle=$
$\displaystyle\frac{c_{m}(L)\sigma_{G}^{2}}{m}\bigg{(}\mathbb{E}\left[{\|\mathbf{\widetilde{Z}}^{m}\|^{2}}\right]-\mathbb{E}\left[{\|\mathbf{\widetilde{Z}}^{m}\|^{2}{1{1}}_{\left\\{\|\mathbf{\widetilde{Z}}^{m}\|>\sqrt{mL^{2}}\right\\}}}\right]\bigg{)}$
$\displaystyle\overset{(\text{using}~{}\eqref{eq:psinplus2})}{=}$
$\displaystyle\frac{c_{m}(L)\sigma_{G}^{2}}{m}\left(m-m\psi(m+2,\sqrt{mL^{2}})\right)$
$\displaystyle=$ $\displaystyle
c_{m}(L)\left(1-\psi(m+2,L\sqrt{m})\right)\sigma_{G}^{2}=d_{m}(L)\sigma_{G}^{2}.$
We now compute $h(\mathbf{Z}^{m}_{L})$
$\displaystyle h(\mathbf{Z}^{m}_{L})$ $\displaystyle=$
$\displaystyle\int_{\mathbf{z}^{m}\in\mathcal{S}_{L}^{G}}f_{Z_{L}}(\mathbf{z}^{m})\log_{2}\left(\frac{1}{f_{Z_{L}}(\mathbf{z}^{m})}\right)d\mathbf{z}^{m}=\int_{\mathbf{z}^{m}\in\mathcal{S}_{L}^{G}}f_{Z_{L}}(\mathbf{z}^{m})\log_{2}\left(\frac{\left(\sqrt{2\pi\sigma_{G}^{2}}\right)^{m}}{c_{m}(L)e^{-\frac{\|\mathbf{z}^{m}\|^{2}}{2\sigma_{G}^{2}}}}\right)d\mathbf{z}^{m}$
(30) $\displaystyle=$
$\displaystyle-\log_{2}\left(c_{m}(L)\right)+\frac{m}{2}\log_{2}\left(2\pi\sigma_{G}^{2}\right)+\int_{\mathbf{z}^{m}\in\mathcal{S}_{L}^{G}}c_{m}(L)f_{G}(\mathbf{z}^{m})\frac{\|\mathbf{z}^{m}\|^{2}}{2\sigma_{G}^{2}}\log_{2}\left(e\right)d\mathbf{z}^{m}.$
Analyzing the last term of (30),
$\displaystyle\int_{\mathbf{z}^{m}\in\mathcal{S}_{L}^{G}}c_{m}(L)f_{G}(\mathbf{z}^{m})\frac{\|\mathbf{z}^{m}\|^{2}}{2\sigma_{G}^{2}}\log_{2}\left(e\right)d\mathbf{z}^{m}$
$\displaystyle=$
$\displaystyle\frac{\log_{2}\left(e\right)}{2\sigma_{G}^{2}}\int_{\mathbf{z}^{m}\in\mathcal{S}_{L}^{G}}c_{m}(L)\frac{e^{-\frac{\|\mathbf{z}^{m}\|^{2}}{2\sigma_{G}^{2}}}}{\left(\sqrt{2\pi\sigma_{G}^{2}}\right)^{m}}\|\mathbf{z}^{m}\|^{2}d\mathbf{z}^{m}$
(31) $\displaystyle=$
$\displaystyle\frac{\log_{2}\left(e\right)}{2\sigma_{G}^{2}}\int_{\mathbf{z}^{m}}f_{Z_{L}}(\mathbf{z}^{m})\|\mathbf{z}^{m}\|^{2}d\mathbf{z}^{m}$
$\displaystyle\overset{(\text{using}~{}\eqref{eq:fz})}{=}$
$\displaystyle\frac{\log_{2}\left(e\right)}{2\sigma_{G}^{2}}\mathbb{E}_{G}\left[{\|\mathbf{Z}^{m}_{L}\|^{2}}\right]=\frac{\log_{2}\left(e\right)}{2\sigma_{G}^{2}}\mathbb{E}_{G}\left[{\sum_{i=1}^{m}Z_{L,i}^{2}}\right]$
$\displaystyle\overset{(\text{using}~{}\eqref{eq:expectzl})}{=}$
$\displaystyle\frac{\log_{2}\left(e\right)}{2\sigma_{G}^{2}}md_{m}(L)\sigma_{G}^{2}=\frac{m\log_{2}\left(e^{d_{m}(L)}\right)}{2}.$
The expression $C_{G,L}^{(m)}$ can now be upper bounded using (28), (30) and
(31) as follows.
$\displaystyle C_{G,L}^{(m)}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log_{2}\left(2\pi
e(\bar{P}+d_{m}(L)\sigma_{G}^{2})\right)+\frac{1}{m}\log_{2}\left(c_{m}(L)\right)-\frac{1}{2}\log_{2}\left(2\pi\sigma_{G}^{2}\right)-\frac{1}{2}\log_{2}\left(e^{d_{m}(L)}\right)$
(32) $\displaystyle=$ $\displaystyle\frac{1}{2}\log_{2}\left(2\pi
e(\bar{P}+d_{m}(L)\sigma_{G}^{2})\right)+\frac{1}{2}\log_{2}\left(c_{m}^{\frac{2}{m}}(L)\right)-\frac{1}{2}\log_{2}\left(2\pi\sigma_{G}^{2}\right)-\frac{1}{2}\log_{2}\left(e^{d_{m}(L)}\right)$
$\displaystyle=$ $\displaystyle\frac{1}{2}\log_{2}\left(\frac{2\pi
e(\bar{P}+d_{m}(L)\sigma_{G}^{2})c_{m}^{\frac{2}{m}}(L)}{2\pi\sigma_{G}^{2}e^{d_{m}(L)}}\right)=\frac{1}{2}\log_{2}\left(\frac{e^{1-d_{m}(L)}(\bar{P}+d_{m}(L)\sigma_{G}^{2})c_{m}^{\frac{2}{m}}(L)}{\sigma_{G}^{2}}\right).$
∎
Now, recall that the rate-distortion function $D_{m}(R)$ for squared error
distortion for source $\mathbf{X}^{m}_{0}$ and reconstruction
$\mathbf{\widehat{X}}^{m}_{L}$ is,
$D_{m}(R):=\inf_{\scriptsize\begin{array}[]{c}p(\mathbf{\widehat{X}}^{m}_{L}|\mathbf{X}^{m}_{0})\\\
\frac{1}{m}I(\mathbf{X}^{m}_{0};\mathbf{\widehat{X}}^{m}_{L})\leq
R\end{array}}\frac{1}{m}\mathbb{E}_{\mathbf{X}^{m}_{0},\mathbf{Z}^{m}_{G}}\left[{\|\mathbf{X}^{m}_{0}-\mathbf{\widehat{X}}^{m}_{L}\|^{2}}\right],$
(33)
which is the dual of the rate-distortion function [57, Pg. 341]. Since
$I(\mathbf{X}^{m}_{0};\mathbf{\widehat{X}}^{m}_{L})\leq mC_{G,L}^{(m)}$, using
the converse to the rate distortion theorem [57, Pg. 349] and the upper bound
on the mutual information represented by $C_{G,L}^{(m)}$,
$\frac{1}{m}\mathbb{E}_{\mathbf{X}^{m}_{0},\mathbf{Z}^{m}_{G}}\left[{\|\mathbf{X}^{m}_{0}-\mathbf{\widehat{X}}^{m}_{L}\|^{2}}\right]\geq
D_{m}(C_{G,L}^{(m)}).$ (34)
Since the Gaussian source is iid, $D_{m}(R)=D(R)$, where
$D(R)=\sigma_{0}^{2}2^{-2R}$ is the distortion-rate function for a Gaussian
source of variance $\sigma_{0}^{2}$ [57, Pg. 346]. Thus, using (23), (25) and
(34),
$\displaystyle\mathbb{E}_{\mathbf{X}^{m}_{0},\mathbf{Z}^{m}_{G}}\left[{J_{2}^{(\gamma)}(\mathbf{X}^{m}_{0},\mathbf{Z}^{m})|\mathbf{Z}^{m}\in\mathcal{S}_{L}^{G}}\right]\geq\left(\left(\sqrt{D(C_{G,L}^{(m)})}-\sqrt{P}\right)^{+}\right)^{2}.$
Substituting the bound on $C_{G,L}^{(m)}$ from (32),
$\displaystyle
D(C_{G,L}^{(m)})=\sigma_{0}^{2}2^{-2C_{G,L}^{(m)}}=\frac{\sigma_{0}^{2}\sigma_{G}^{2}}{c_{m}^{\frac{2}{m}}(L)e^{1-d_{m}(L)}(\bar{P}+d_{m}(L)\sigma_{G}^{2})}.$
Using (23), this completes the proof of the lemma. Notice that
$c_{m}(L)\rightarrow 1$ and $d_{m}(L)\rightarrow 1$ for fixed $m$ as
$L\rightarrow\infty$, as well as for fixed $L>1$ as $m\rightarrow\infty$. So
the lower bound on $D(C_{G,L}^{(m)})$ approaches $\kappa$ of Theorem 2 in both
of these two limits.
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|
arxiv-papers
| 2010-03-02T07:44:14 |
2024-09-04T02:49:08.684067
|
{
"license": "Public Domain",
"authors": "Pulkit Grover, Se Yong Park and Anant Sahai",
"submitter": "Pulkit Grover",
"url": "https://arxiv.org/abs/1003.0514"
}
|
1003.0520
|
# Information Embedding meets Distributed Control
Pulkit $\text{Grover}^{\dagger}$, Aaron B. $\text{Wagner}^{\ddagger}$ and
Anant $\text{Sahai}^{\dagger}$ $\dagger$Wireless Foundations, Department of
EECS, University of California at Berkeley. Email: $\\{$pulkit, sahai$\\}$ @
eecs.berkeley.edu. $\ddagger$ School of Electrical and Computer Engineering,
Cornell University. Email: wagner @ ece.cornell.edu. An abridged version of
this paper will be presented at the 2010 Information Theory Workshop (ITW),
Cairo, Egypt.
###### Abstract
We consider the problem of information embedding where the encoder modifies a
white Gaussian host signal in a power-constrained manner to encode the
message, and the decoder recovers both the embedded message and the modified
host signal. This extends the recent work of Sumszyk and Steinberg to the
continuous-alphabet Gaussian setting. We show that a dirty-paper-coding based
strategy achieves the optimal rate for perfect recovery of the modified host
and the message. We also provide bounds for the extension wherein the modified
host signal is recovered only to within a specified distortion. When
specialized to the zero-rate case, our results provide the tightest known
lower bounds on the asymptotic costs for the vector version of a famous open
problem in distributed control — the Witsenhausen counterexample. Using this
bound, we characterize the asymptotically optimal costs for the vector
Witsenhausen problem numerically to within a factor of $1.3$ for all problem
parameters, improving on the earlier best known bound of $2$.
## I Introduction
The problem of interest in this paper (see Fig. 1) derives its motivation from
an information-theoretic standpoint, as well as from a distributed-control
perspective. Information-theoretically, the problem is an extension of an
information embedding problem recently addressed by Sumszyk and Steinberg [1]
— the encoder ensures that the decoder recovers the modified host signal
$\mathbf{X}^{m}$ perfectly, along with the message. Philosophically, the work
in [1] is directed towards understanding how a communication problem changes
when an additional requirement, that of the encoder being able to produce a
copy of the reconstruction at the decoder, is imposed on the system (in source
coding context, the issue was explored by Steinberg in [2]). The problem is
also closely connected to other information theory problems [3, 4, 5, 6]. We
refer the interested reader to [7], where these connections are discussed in
detail.
Figure 1: The host signal $\mathbf{S}^{m}$ is first modified by the encoder
using a power constrained input $\mathbf{U}^{m}$. The modified host signal
$\mathbf{X}^{m}$ and the message $M$ are then reconstructed at the decoder.
The problem is to find the minimum distortion in reconstruction of
$\mathbf{X}^{m}$ given $P$, the power constraint, and $R$, the rate of
reliable message transmission.
In [1], the authors assume that the host signal $\mathbf{S}^{m}$, the modified
host signal (the channel input) $\mathbf{X}^{m}$ and the channel output
$\mathbf{Y}^{m}$ are all finite-alphabet. In this paper, we consider the
Gaussian version of their problem. The extension is non-trivial [8] because
simple Fano’s inequality-based techniques do not work for the infinite-
alphabet formulation. Experience in infinite-alphabet problems might even
suggest that (asymptotic) perfect reconstruction may be impossible because the
problem is set in continuous space. Intriguingly, asymptotic perfect
reconstruction is possible in our problem because the encoder can ensure that
the modified host signal takes values in a discrete subset of the continuous
space. We provide tight results characterizing the tradeoff between rate and
power for perfect reconstruction. As is more natural in a continuous-alphabet
setting, we relax the assumption of perfect recovery of the host signal by
considering recovery within a specified nonzero distortion, and for this
problem we provide upper and lower bounds on the tradeoff between rate, power
and average distortion.
The nonzero distortion problem is closely related to the vector version of a
famous distributed control problem called the Witsenhausen counterexample [9]
— at zero communication rate, the two problems are the same [7]. The scalar
counterexample is believed to be quite challenging (see [7] for a survey of
prior results showing why it is believed to be so). As a conceptual
simplification, Grover and Sahai [7] considered the long-blocklength limit of
the counterexample. Further, they relaxed the requirement of obtaining a
provably optimal strategy to the weaker objective of obtaining strategies that
attain within a constant factor of the optimal cost. For the weighted sum of
power and average distortion costs (see Section II), they then show that
dirty-paper coding techniques attain within a factor of $2$ of the optimal
cost for all problem parameters (i.e. the weights and the variances of the
random variables). Backing off from the infinite blocklength limit, Grover,
Park and Sahai [10] then showed that similar constant-factor results can also
be obtained for finite vector lengths, including the scalar case. The
achievable strategy, which yields the upper bounds, now uses lattices instead
of random codebooks. The lower bound is obtained by applying sphere-packing
ideas from information theory to the bound of [7].
The lower bound in this paper specialized to rate zero provides an improved
lower bound to the costs of the vector Witsenhausen counterexample in the
long-blocklength limit. Using this improved bound, we show that the ratio of
upper and lower bounds is smaller than $1.3$ regardless of the choice of the
weights and the problem parameters. This is an improvement over the previously
best known maximum ratio of two [7].
Control theory has long wrestled with the Witsenhausen counterexample. Because
it is a canonical problem, a comprehensive distributed-control theory would
necessarily include a good understanding of the counterexample. Information-
theory has had long-standing canonical problems of its own. In a line of
investigation started by Gupta and Kumar [11], the question of the capacity of
a large wireless network is studied. By restricting attention to obtaining
just the scaling of the total capacity, the bar for what might constitute a
reasonable information-theoretic solution was lowered. More recently, the
calculation of channel capacity to within a finite number of bits111Our
constant-factor results on control costs are closely related to results on
bounded gap from capacity. A factor of $2$ approximation in power would be a
slightly stronger result than a $\frac{1}{2}$-bit approximation in the
capacity of a real channel. for canonical information-theory problems (e.g.
the interference channel [12]) has led to significant advances in
understanding capacity for larger network communication problems [13, 14]. The
recent results on Witsenhausen’s counterexample thus raise a parallel hope in
distributed control.
## II Problem Statement
The host signal $\mathbf{S}^{m}$ is distributed
$\mathcal{N}(0,\sigma^{2}\mathbb{I})$, and the message $M$ is independent of
$\mathbf{S}^{m}$ and distributed uniformly over $\\{1,2,\ldots,2^{mR}\\}$. The
encoder $\mathcal{E}_{m}$ maps $(M,\mathbf{S}^{m})$ to $\mathbf{X}^{m}$ by
additively distorting $\mathbf{S}^{m}$ using input $\mathbf{U}^{m}$ of average
power (for each message) at most $P$, i.e.
$\mathbb{E}\left[\|\mathbf{S}^{m}-\mathbf{X}^{m}\|^{2}\right]\leq mP$.
Additive white Gaussian noise
$\mathbf{Z}^{m}\sim\mathcal{N}(0,\sigma_{z}^{2}\mathbb{I})$, where
$\sigma_{z}^{2}=1$, is added to $\mathbf{X}^{m}$ by the channel. The decoder
$\mathcal{D}_{m}$ maps the channel outputs $\mathbf{Y}^{m}$ to both an
estimate $\mathbf{\widehat{X}}^{m}$ of the modified host signal
$\mathbf{X}^{m}$ and an estimate $\widehat{M}$ of the message.
Define the error probability
$\epsilon_{m}(\mathcal{E}_{m},\mathcal{D}_{m})=\Pr(M\neq\widehat{M})$. For the
encoder-decoder sequence
$\\{\mathcal{E}_{m},\mathcal{D}_{m}\\}_{m=1}^{\infty}$, define the minimum
asymptotic distortion $MMSE(P,R)$ as follows
$\displaystyle
MMSE(P,R)=\underset{\\{\mathcal{E}_{m},\mathcal{D}_{m}\\}_{m=1}^{\infty}:\epsilon_{m}(\mathcal{E}_{m},\mathcal{D}_{m})\rightarrow
0}{\inf}\;\underset{m\rightarrow\infty}{\lim\sup}\;\frac{1}{m}\mathbb{E}\left[\|\mathbf{X}^{m}-\mathbf{\widehat{X}}^{m}\|^{2}\right].$
We are interested in the tradeoff between the rate $R$, the power $P$, and
$MMSE(P,R)$.
The conventional control-theoretic weighted cost formulation [9] defines the
total cost to be
$J=\frac{1}{m}k^{2}\|\mathbf{U}^{m}\|^{2}+\frac{1}{m}\|\mathbf{X}^{m}-\mathbf{\widehat{X}}^{m}\|^{2},$
(1)
where $k\in\mathbb{R}^{+}$. The objective is to minimize the average cost,
$\mathbb{E}\left[J\right]$ at rate $R$. The average is taken over the
realizations of the host signal, the channel noise, and the message. At $R=0$,
the problem is the vector Witsenhausen counterexample [7].
## III Main Results
### III-A Lower bounds on $MMSE(P,R)$
###### Theorem 1
For the problem as stated in Section II, for communicating reliably at rate
$R$ with input power $P$, the asymptotic average mean-square error in
recovering $\mathbf{X}^{m}$ is lower bounded as follows. For $P\geq 2^{2R}-1$,
$MMSE(P,R)\geq\inf_{\sigma_{SU}}\sup_{\gamma>0}\frac{1}{\gamma^{2}}\left(\left(\sqrt{\frac{\sigma^{2}2^{2R}}{1+\sigma^{2}+P+2\sigma_{SU}}}-\sqrt{(1-\gamma)^{2}\sigma^{2}+\gamma^{2}P-2\gamma(1-\gamma){\sigma_{SU}}}\right)^{+}\right)^{2},$
(2)
where
$\max\left\\{-\sigma\sqrt{P},\frac{2^{2R}-1-P-\sigma^{2}}{2}\right\\}\leq\sigma_{SU}\leq\sigma\sqrt{P}$.
For $P<2^{2R}-1$, reliable communication at rate $R$ is not possible.
###### Corollary 1
For the vector Witsenhausen problem with
$\mathbb{E}\left[\|\mathbf{U}^{m}\|^{2}\right]\leq mP$, the following is a
lower bound on the $MMSE$ in the estimation of $\mathbf{X}^{m}$.
$MMSE(P,0)\geq\inf_{\sigma_{SU}}\sup_{\gamma>0}\frac{1}{\gamma^{2}}\left(\left(\sqrt{\frac{\sigma^{2}}{1+\sigma^{2}+P+2\sigma_{SU}}}-\sqrt{(1-\gamma)^{2}\sigma^{2}+\gamma^{2}P-2\gamma(1-\gamma){\sigma_{SU}}}\right)^{+}\right)^{2}.$
(3)
where $\sigma_{SU}\in[-\sigma\sqrt{P},\sigma\sqrt{P}]$.
###### Proof:
[Of Theorem 1] For conceptual clarity, we first derive the result for the case
$R=0$ (Corollary 1). The tools developed are then used to derive the lower
bound for $R>0$.
###### Proof:
[Of Corollary 1]
For any chosen pair of encoding map $\mathcal{E}_{m}$ and decoding map
$\mathcal{D}_{m}$, there is a Markov chain
$\mathbf{S}^{m}\rightarrow\mathbf{X}^{m}\rightarrow\mathbf{Y}^{m}\rightarrow\mathbf{\widehat{X}}^{m}$.
Using the data-processing inequality
$I(\mathbf{S}^{m};\mathbf{\widehat{X}}^{m})\leq
I(\mathbf{X}^{m};\mathbf{Y}^{m}).$ (4)
The terms in the inequality can be bounded by single letter expressions as
follows. Define $Q$ as a random variable uniformly distributed over
$\\{1,2,\ldots,m\\}$. Define $S=S_{Q}$, $U=U_{Q}$, $X=X_{Q}$, $Z=Z_{Q}$,
$Y=Y_{Q}$ and $\widehat{X}=\widehat{X}_{Q}$. Then,
$\displaystyle I(\mathbf{X}^{m};\mathbf{Y}^{m})$ $\displaystyle=$
$\displaystyle h(\mathbf{Y}^{m})-h(\mathbf{Y}^{m}|\mathbf{X}^{m})$ (5)
$\displaystyle\overset{(a)}{\leq}$
$\displaystyle\sum_{i}h(Y_{i})-h(\mathbf{Y}^{m}|\mathbf{X}^{m})$
$\displaystyle=$ $\displaystyle\sum_{i}h(Y_{i})-h(Y_{i}|X_{i})$
$\displaystyle=$ $\displaystyle\sum_{i}I(X_{i};Y_{i})$ $\displaystyle=$
$\displaystyle mI(X;Y|Q)$ $\displaystyle=$ $\displaystyle
m\left(h(Y|Q)-h(Y|X,Q)\right)$ $\displaystyle\leq$ $\displaystyle
m\left(h(Y)-h(Y|X,Q)\right)$ $\displaystyle\overset{(b)}{=}$ $\displaystyle
m\left(h(Y)-h(Y|X)\right)=mI(X;Y),$
where $(a)$ follows from an application of the chain-rule for entropy followed
by using the fact that conditioning reduces entropy, and $(b)$ follows from
the observation that the additive noise $Z_{i}$ is iid across time, and
independent of the input $X_{i}$ (thus $Y\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Q|X$). Also,
$\displaystyle I(\mathbf{S}^{m};\mathbf{\widehat{X}}^{m})$ $\displaystyle=$
$\displaystyle h(\mathbf{S}^{m})-h(\mathbf{S}^{m}|\mathbf{\widehat{X}}^{m})$
(6) $\displaystyle=$
$\displaystyle\sum_{i}h(S_{i})-h(\mathbf{S}^{m}|\mathbf{\widehat{X}}^{m})$
$\displaystyle\overset{(a)}{\geq}$
$\displaystyle\sum_{i}\left(h(S_{i})-h(S_{i}|\widehat{X}_{i})\right)$
$\displaystyle=$
$\displaystyle\sum_{i}I(S_{i};\widehat{X}_{i})=mI(S;\widehat{X}|Q)$
$\displaystyle=$ $\displaystyle m\left(h(S|Q)-h(S|\widehat{X},Q)\right)$
$\displaystyle\overset{(b)}{\geq}$ $\displaystyle
m\left(h(S)-h(S|\widehat{X})\right)=mI(S;\widehat{X}),$
where $(a)$ and $(b)$ again follow from the fact that conditioning reduces
entropy, and $(b)$ also uses the observation that since $S_{i}$ are iid, $S$,
$S_{i}$, and $S|Q=q$ are distributed identically.
Now, using (4), (5) and (6),
$mI(S;\widehat{X})\leq I(\mathbf{S}^{m};\mathbf{\widehat{X}}^{m})\leq
I(\mathbf{X}^{m};\mathbf{Y}^{m})\leq mI(X;Y).$ (7)
Also observe that from the definitions of $S$, $X$, $\widehat{X}$ and $Y$,
$\mathbb{E}\left[d(\mathbf{S}^{m},\mathbf{X}^{m})\right]=\mathbb{E}\left[d(S,X)\right]$,
and
$\mathbb{E}\left[d(\mathbf{X}^{m},\mathbf{\widehat{X}}^{m})\right]=\mathbb{E}\left[d(X,\widehat{X})\right]$.
Using the Cauchy-Schwartz inequality, the correlation
$\sigma_{SU}=\mathbb{E}\left[SU\right]$ must satisfy the following constraint,
$|\sigma_{SU}|=|\mathbb{E}\left[SU\right]|\leq\sqrt{\mathbb{E}\left[S^{2}\right]}\sqrt{\mathbb{E}\left[U^{2}\right]}\leq\sigma\sqrt{P}.$
(8)
Also,
$\mathbb{E}\left[X^{2}\right]=\mathbb{E}\left[(S+U)^{2}\right]=\sigma^{2}+P+2\sigma_{SU}.$
(9)
Since $Z=Y-X\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}X$, and a Gaussian input distribution
maximizes the mutual information across an average-power-constrained AWGN
channel,
$I(X;Y)\leq\frac{1}{2}\log_{2}\left(1+\frac{P+\sigma^{2}+2\sigma_{SU}}{1}\right).$
(10) $\displaystyle I(S;\widehat{X})$ $\displaystyle=$ $\displaystyle
h(S)-h(S|\widehat{X})$ (11) $\displaystyle=$ $\displaystyle
h(S)-h(S-\gamma\widehat{X}|\widehat{X})\;\forall\gamma$
$\displaystyle\overset{(a)}{\geq}$ $\displaystyle h(S)-h(S-\gamma\widehat{X})$
$\displaystyle=$ $\displaystyle\frac{1}{2}\log_{2}\left(2\pi
e\sigma^{2}\right)-h(S-\gamma\widehat{X}),$
where $(a)$ follows from the fact that conditioning reduces entropy. Also note
here that the result holds for any $\gamma>0$, and in particular, $\gamma$ can
depend on $\sigma_{SU}$. Now,
$\displaystyle h(S-\gamma\widehat{X})$ $\displaystyle=$ $\displaystyle
h(S-\gamma(\widehat{X}-X)-\gamma X)$ (12) $\displaystyle=$ $\displaystyle
h\left(S-\gamma(\widehat{X}-X)-\gamma S-\gamma U\right)$ $\displaystyle=$
$\displaystyle h\left((1-\gamma)S-\gamma U-\gamma(\widehat{X}-X)\right).$
The second moment of a sum of two random variables $A$ and $B$ can be bounded
as follows
$\displaystyle\mathbb{E}\left[(A+B)^{2}\right]$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[A^{2}\right]+\mathbb{E}\left[B^{2}\right]+2\mathbb{E}\left[AB\right]$
(13) $\displaystyle\overset{\text{Cauchy-Schwartz ineq.}}{\leq}$
$\displaystyle\mathbb{E}\left[A^{2}\right]+\mathbb{E}\left[B^{2}\right]+2\sqrt{\mathbb{E}\left[A^{2}\right]}\sqrt{\mathbb{E}\left[B^{2}\right]}$
$\displaystyle=$
$\displaystyle(\sqrt{\mathbb{E}\left[A^{2}\right]}+\sqrt{\mathbb{E}\left[B^{2}\right]})^{2},$
with equality when $A$ and $B$ are aligned, i.e. $A=\lambda B$ for some
$\lambda\in\mathbb{R}$. For the random variable under consideration in (12),
choosing $A=(1-\gamma)S-\gamma U$, and $B=-\gamma(\widehat{X}-X)$ in (13)
$\displaystyle\mathbb{E}\left[\left((1-\gamma)S-\gamma
U-\gamma(\widehat{X}-X)\right)^{2}\right]\leq\left(\sqrt{(1-\gamma)^{2}\sigma^{2}+\gamma^{2}P-2\gamma(1-\gamma){\sigma_{SU}}}+\gamma\sqrt{\mathbb{E}\left[(\widehat{X}-X)^{2}\right]}\right)^{2}.$
(14)
Equality is obtained by aligning222In general, since
$\mathbf{\widehat{X}}^{m}$ is a function of $\mathbf{Y}^{m}$, this alignment
is not actually possible when the recovery of $\mathbf{X}^{m}$ is not exact.
The derived bound is therefore loose. $X-\widehat{X}$ with $(1-\gamma)S-\gamma
U$. Thus,
$\displaystyle I(S;\widehat{X})$ $\displaystyle\geq$
$\displaystyle\frac{1}{2}\log_{2}\left(2\pi
e\sigma^{2}\right)-h(S-\gamma\widehat{X})$ (15) $\displaystyle\geq$
$\displaystyle\frac{1}{2}\log_{2}\left(\frac{\sigma^{2}}{\left(\sqrt{(1-\gamma)^{2}\sigma^{2}+\gamma^{2}P-2\gamma(1-\gamma){\sigma_{SU}}}+\gamma\sqrt{\mathbb{E}\left[(\widehat{X}-X)^{2}\right]}\right)^{2}}\right).$
Using (7), $I(S;\widehat{X})\leq I(X;Y)$. Using the lower bound on
$I(S;\widehat{X})$ from (15) and the upper bound on $I(X;Y)$ from (10), we get
$\displaystyle\frac{1}{2}\log_{2}\left(\frac{\sigma^{2}}{\left(\sqrt{(1-\gamma)^{2}\sigma^{2}+\gamma^{2}P-2\gamma(1-\gamma){\sigma_{SU}}}+\gamma\sqrt{\mathbb{E}\left[(\widehat{X}-X)^{2}\right]}\right)^{2}}\right)\leq\frac{1}{2}\log_{2}\left(1+\frac{P+\sigma^{2}+2\sigma_{SU}}{1}\right),$
for the choice of $\mathcal{E}_{m}$ and $\mathcal{D}_{m}$. Since
$\log_{2}\left(\cdot{}\right)$ is a monotonically increasing function,
$\displaystyle\frac{\sigma^{2}}{\left(\sqrt{(1-\gamma)^{2}\sigma^{2}+\gamma^{2}P-2\gamma(1-\gamma){\sigma_{SU}}}+\gamma\sqrt{\mathbb{E}\left[(\widehat{X}-X)^{2}\right]}\right)^{2}}\leq
1+P+\sigma^{2}+2\sigma_{SU}$
$\displaystyle\text{i.e.}\;\;\left(\sqrt{(1-\gamma)^{2}\sigma^{2}+\gamma^{2}P-2\gamma(1-\gamma){\sigma_{SU}}}+\gamma\sqrt{\mathbb{E}\left[(\widehat{X}-X)^{2}\right]}\right)^{2}\geq\frac{\sigma^{2}}{1+P+\sigma^{2}+2\sigma_{SU}},$
$\displaystyle\text{Since
$\gamma>0$,}\;\;\gamma\sqrt{\mathbb{E}\left[(\widehat{X}-X)^{2}\right]}\geq\sqrt{\frac{\sigma^{2}}{1+P+\sigma^{2}+2\sigma_{SU}}}-\sqrt{(1-\gamma)^{2}\sigma^{2}+\gamma^{2}P-2\gamma(1-\gamma){\sigma_{SU}}}.$
Because the RHS may not be positive, we take the maximum of zero and the RHS
and obtain the following lower bound for $\mathcal{E}_{m}$ and
$\mathcal{D}_{m}$.
$\mathbb{E}\left[(\widehat{X}-X)^{2}\right]\geq\frac{1}{\gamma^{2}}\left(\left(\sqrt{\frac{\sigma^{2}}{1+P+\sigma^{2}+2\sigma_{SU}}}-\sqrt{(1-\gamma)^{2}\sigma^{2}+\gamma^{2}P-2\gamma(1-\gamma){\sigma_{SU}}}\right)^{+}\right)^{2}.$
(16)
Because the bound holds for every $\gamma>0$,
$\mathbb{E}\left[(\widehat{X}-X)^{2}\right]\geq\sup_{\gamma>0}\frac{1}{\gamma^{2}}\left(\left(\sqrt{\frac{\sigma^{2}}{1+P+\sigma^{2}+2\sigma_{SU}}}-\sqrt{(1-\gamma)^{2}\sigma^{2}+\gamma^{2}P-2\gamma(1-\gamma){\sigma_{SU}}}\right)^{+}\right)^{2},$
(17)
for the chosen $\mathcal{E}_{m}$ and $\mathcal{D}_{m}$. Now, from (8),
$\sigma_{SU}$ can take values in $[-\sigma\sqrt{P},\sigma\sqrt{P}]$. Because
the lower bound depends on $\mathcal{E}_{m}$ and $\mathcal{D}_{m}$ only
through $\sigma_{SU}$, we obtain the following lower bound for all
$\mathcal{E}_{m}$ and $\mathcal{D}_{m}$,
$\mathbb{E}\left[(\widehat{X}-X)^{2}\right]\geq\inf_{|\sigma_{SU}|\leq\sigma\sqrt{P}}\sup_{\gamma>0}\frac{1}{\gamma^{2}}\left(\left(\sqrt{\frac{\sigma^{2}}{1+P+\sigma^{2}+2\sigma_{SU}}}-\sqrt{(1-\gamma)^{2}\sigma^{2}+\gamma^{2}P-2\gamma(1-\gamma){\sigma_{SU}}}\right)^{+}\right)^{2},$
(18)
which proves Corollary 1. Notice that we did not take limits in $m$ anywhere,
and hence the lower bound holds for all values of $m$. ∎
### The case of nonzero rate
To prove Theorem 1, consider now the problem when the encoder wants to also
communicate a message $M$ reliably to the decoder at rate $R$.
Using Fano’s inequality, since $\Pr(M\neq\widehat{M})=\epsilon_{m}\rightarrow
0$ as $m\rightarrow\infty$, $H(M|\widehat{M})\leq m\delta_{m}$ where
$\delta_{m}\rightarrow 0$. Thus,
$\displaystyle I(M;\widehat{M})$ $\displaystyle=$ $\displaystyle
H(M)-H(M|\widehat{M})$ (19) $\displaystyle=$ $\displaystyle
mR-H(M|\widehat{M})$ $\displaystyle\geq$ $\displaystyle
mR-m\delta_{m}=m(R-\delta_{m}).$
As before, we consider a mutual information inequality that follows directly
from the Markov chain
$(M,\mathbf{S}^{m})\rightarrow\mathbf{X}^{m}\rightarrow\mathbf{Y}^{m}\rightarrow(\mathbf{\widehat{X}}^{m},\widehat{M})$
:
$I(M,\mathbf{S}^{m};\widehat{M},\mathbf{\widehat{X}}^{m})\leq
I(\mathbf{X}^{m};\mathbf{Y}^{m}).$ (20)
The RHS can be bounded above as in (5). For the LHS,
$\displaystyle I(M,\mathbf{S}^{m};\widehat{M},\mathbf{\widehat{X}}^{m})$
$\displaystyle=$ $\displaystyle
I(M;\widehat{M},\mathbf{\widehat{X}}^{m})+I(\mathbf{S}^{m};\widehat{M},\mathbf{\widehat{X}}^{m}|M)$
(21) $\displaystyle\geq$ $\displaystyle
I(M;\widehat{M})+I(\mathbf{S}^{m};\widehat{M},\mathbf{\widehat{X}}^{m}|M)$
$\displaystyle=$ $\displaystyle
I(M;\widehat{M})+h(\mathbf{S}^{m}|M)-h(\mathbf{S}^{m}|\widehat{M},\mathbf{\widehat{X}}^{m},M)$
$\displaystyle\overset{\mathbf{S}^{m}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}M}{=}$ $\displaystyle
I(M;\widehat{M})+h(\mathbf{S}^{m})-h(\mathbf{S}^{m}|\widehat{M},\mathbf{\widehat{X}}^{m},M)$
$\displaystyle\geq$ $\displaystyle
I(M;\widehat{M})+h(\mathbf{S}^{m})-h(\mathbf{S}^{m}|\mathbf{\widehat{X}}^{m})$
$\displaystyle\geq$ $\displaystyle
I(M;\widehat{M})+I(\mathbf{S}^{m};\mathbf{\widehat{X}}^{m})$
$\displaystyle\overset{\text{using~{}\eqref{eq:isx}}}{\geq}$ $\displaystyle
I(M;\widehat{M})+mI(S;\widehat{X}).$
From (19), (20) and (21), we obtain
$\displaystyle m(R-\delta_{m})+mI(S;\widehat{X})$
$\displaystyle\overset{\text{using}~{}\eqref{eq:fano}}{\leq}$ $\displaystyle
I(M;\widehat{M})+mI(S;\widehat{X})$ (22)
$\displaystyle\overset{\text{using}~{}\eqref{eq:breaking}}{\leq}$
$\displaystyle I(M,\mathbf{S}^{m};\widehat{M},\mathbf{\widehat{X}}^{m})$
$\displaystyle\overset{\text{using}~{}\eqref{eq:dpi2}}{\leq}$ $\displaystyle
I(\mathbf{X}^{m};\mathbf{Y}^{m})\overset{\text{using}~{}\eqref{eq:ixy}}{\leq}mI(X;Y).$
$I(X;Y)$ and $I(S;\widehat{X})$ can be bounded as before in (10) and (15).
Observing that as $m\rightarrow\infty$, $\delta_{m}\rightarrow 0$, we get the
following lower bound on the $MMSE$ for nonzero rate,
$MMSE(P,R)\geq\inf_{\sigma_{SU}}\sup_{\gamma>0}\frac{1}{\gamma^{2}}\left(\left(\sqrt{\frac{\sigma^{2}2^{2R}}{1+\sigma^{2}+P+2\sigma_{SU}}}-\sqrt{(1-\gamma)^{2}\sigma^{2}+\gamma^{2}P-2\gamma(1-\gamma){\sigma_{SU}}}\right)^{+}\right)^{2}.$
(23)
In the limit $\delta_{m}\rightarrow 0$, we require from (22) that $I(X;Y)\geq
R$. This gives the following constraint on $\sigma_{SU}$,
$\displaystyle\frac{1}{2}\log_{2}\left(1+P+\sigma^{2}+2\sigma_{SU}\right)\geq
R$ $\displaystyle\text{i.e.}\;\sigma_{SU}\geq\frac{2^{2R}-1-P-\sigma^{2}}{2},$
(24)
yielding (in conjunction with (8)) the constraint on $\sigma_{SU}$ in Theorem
1. The constraint on $P$ in the Theorem follows from Costa’s result [3],
because the rate $R$ must be smaller than the capacity over a power
constrained AWGN channel with known interference,
$\frac{1}{2}\log_{2}\left(1+P\right)$. ∎ It is insightful to see how the lower
bound in Corollary 1 is an improvement over that in [7]. The lower bound in
[7] is given by
$MMSE(P,0)\geq\left(\left(\sqrt{\frac{\sigma^{2}}{\sigma^{2}+P+2\sigma\sqrt{P}+1}}-\sqrt{P}\right)^{+}\right)^{2},$
(25)
which again holds for all $m$. Because any $\gamma$ provides a valid lower
bound in Corollary 1, choosing $\gamma=1$ in Corollary 1 provides the
following (loosened) bound,
$MMSE(P,0)\geq\inf_{|\sigma_{SU}|\leq\sigma\sqrt{P}}\left(\left(\sqrt{\frac{\sigma^{2}}{\sigma^{2}+P+2\sigma_{SU}+1}}-\sqrt{P}\right)^{+}\right)^{2},$
(26)
which is minimized for $\sigma_{SU}=\sigma\sqrt{P}$. This immediately yields
the lower bound (25) of [7].
### III-B The upper bound and the tightness at $MMSE=0$
We use the combination of linear and dirty-paper coding strategies of [7],
except that we communicate a message at rate $R$ as well. We summarize the
strategy briefly, and refer the interested reader to [7] for a detailed
description and analysis of the achievability.
The encoder divides its input into two parts $\mathbf{U}^{m}_{lin}$ and
$\mathbf{U}^{m}_{dpc}$ of powers $P_{lin}$ and $P_{dpc}$ respectively, such
that $P=P_{lin}+P_{dpc}$ (by construction, $\mathbf{U}^{m}_{lin}$ and
$\mathbf{U}^{m}_{dpc}$ turn out to be orthogonal in the limit). We refer to
$P_{lin}$ as the linear part of the power, and $P_{dpc}$ the dirty-paper
coding part of the power. The linear part is used to scale the host signal
down by a factor $\beta$ (using $\mathbf{U}^{m}_{lin}=-\beta\mathbf{S}^{m}$)
so that the scaled down host signal has variance
$\widetilde{\sigma}^{2}=\sigma^{2}(1-\beta)^{2}$, where
$\beta^{2}\sigma^{2}=P_{lin}$. Using the remaining $P_{dpc}$ power, the
transmitter dirty-paper codes against the scaled-down host signal
$(1-\beta)\mathbf{S}^{m}$ with the DPC parameter $\alpha$ [3] allowed to be
arbitrary (unlike in [3], where it is eventually chosen to be the MMSE
parameter).
A plain DPC strategy achieves the following rate [3, Eq. (6)]
$R=\frac{1}{2}\log_{2}\left(\frac{P(P+\sigma^{2}+1)}{P\sigma^{2}(1-\alpha)^{2}+P+\alpha^{2}\sigma^{2}}\right),$
(27)
The strategy recovers $\mathbf{U}^{m}+\alpha\mathbf{S}^{m}$ at the decoder
with high probability. Because we also have a linear part here, the achieved
rate is
$R=\frac{1}{2}\log_{2}\left(\frac{P_{dpc}(P_{dpc}+\widetilde{\sigma}^{2}+1)}{P_{dpc}\widetilde{\sigma}^{2}(1-\alpha)^{2}+P_{dpc}+\alpha^{2}\widetilde{\sigma}^{2}}\right).$
(28)
The decoder now decodes the codeword
$\mathbf{U}^{m}_{dpc}+\alpha(1-\beta)\mathbf{S}^{m}$. It then performs an MMSE
estimation for estimating
$\mathbf{X}^{m}=\mathbf{S}^{m}+\mathbf{U}^{m}=(1-\beta)\mathbf{S}^{m}+\mathbf{U}^{m}_{dpc}$
using the channel output
$\mathbf{Y}^{m}=(1-\beta)\mathbf{S}^{m}+\mathbf{U}^{m}_{dpc}+\mathbf{Z}^{m}$
and the decoded codeword $\alpha(1-\beta)\mathbf{S}^{m}+\mathbf{U}^{m}_{dpc}$.
The obtained $MMSE$ can now be minimized over the choice of $\alpha$ and
$\beta$ under the constraint (28).
###### Corollary 2
For a given power $P$, a combination of linear and DPC-based strategies
achieves the maximum rate $C(P)$ in the perfect recovery limit $MMSE(P,R)=0$,
where $C(P)$ is given by
$C(P)=\sup_{\sigma_{SU}\in[-\sigma\sqrt{P},0]}\frac{1}{2}\log_{2}\left(\frac{(P\sigma^{2}-\sigma_{SU}^{2})(1+\sigma^{2}+P+2\sigma_{SU})}{\sigma^{2}(\sigma^{2}+P+2\sigma_{SU})}\right).$
(29)
###### Proof:
The achievability
The combination of linear and DPC-based strategies of [7] recovers
$\mathbf{U}^{m}_{dpc}+\alpha(1-\beta)\mathbf{S}^{m}$ at the decoder with high
probability. In order to perfectly recover
$\mathbf{X}^{m}=(1-\beta)\mathbf{S}^{m}+\mathbf{U}^{m}_{dpc}$, we can use
$\alpha=1$, and hence the strategy would achieve a rate of
$R_{ach}=\sup_{P_{lin},P_{dpc}:P=P_{lin}+P_{dpc}}\frac{1}{2}\log_{2}\left(\frac{P_{dpc}(P_{dpc}+\widetilde{\sigma}^{2}+1)}{P_{dpc}+\widetilde{\sigma}^{2}}\right),$
(30)
where we take a supremum over $P_{lin},P_{dpc}$ such that they sum up to $P$.
Let $\sigma_{SU}=-\sigma\sqrt{P_{lin}}$ (note that as $P_{lin}$ varies from
$0$ to $P$, $\sigma_{SU}$ varies from $0$ to $-\sigma\sqrt{P}$). Then,
$P_{dpc}=P-\frac{\sigma_{SU}^{2}}{\sigma^{2}}$, and
$P_{dpc}+\widetilde{\sigma}^{2}=P_{dpc}+\sigma^{2}+P_{lin}-2\sigma\sqrt{P_{lin}}=P+\sigma^{2}+2\sigma_{SU}$.
Thus,
$\displaystyle
R_{ach}=\sup_{\sigma_{SU}\in[-\sigma\sqrt{P},0]}\frac{1}{2}\log_{2}\left(\frac{\left(P-\frac{\sigma_{SU}^{2}}{\sigma^{2}}\right)(P+\sigma^{2}+2\sigma_{SU}+1)}{P+\sigma^{2}+2\sigma_{SU}}\right).$
(31)
Simple algebra shows that this expression matches that in Corollary 2.
The converse
Since we are free to choose $\gamma$ in Theorem 1, let
$\gamma=\gamma^{*}=\frac{\sigma^{2}+\sigma_{SU}}{\sigma^{2}+P+2\sigma_{SU}}$.
Then, $1-\gamma^{*}=\frac{P+\sigma_{SU}}{\sigma^{2}+P+2\sigma_{SU}}$. Thus, we
get
$0\geq\inf_{\sigma_{SU}}\frac{1}{\gamma^{*^{2}}}\left(\left(\sqrt{\frac{\sigma^{2}2^{2R}}{1+\sigma^{2}+P+2\sigma_{SU}}}-\sqrt{(1-\gamma^{*})^{2}\sigma^{2}+\gamma^{*^{2}}P-2\gamma^{*}(1-\gamma^{*}){\sigma_{SU}}}\right)^{+}\right)^{2}.$
(32)
It has to be the case that the term inside $(\cdot{})^{+}$ is non-positive for
some value of $\sigma_{SU}$. This immediately yields
$\displaystyle 2^{2R}$ $\displaystyle\leq$
$\displaystyle\sup_{\sigma_{SU}}\frac{1}{\sigma^{2}}\left((1-\gamma^{*})^{2}\sigma^{2}+\gamma^{*^{2}}P-2\gamma^{*}(1-\gamma^{*}){\sigma_{SU}}\right)(1+\sigma^{2}+P+2\sigma_{SU})$
$\displaystyle=$
$\displaystyle\sup_{\sigma_{SU}}\frac{1}{\sigma^{2}}\frac{\left((P+\sigma_{SU})^{2}\sigma^{2}+(\sigma^{2}+\sigma_{SU})^{2}P-2(P+\sigma_{SU})(\sigma^{2}+\sigma_{SU})\sigma_{SU}\right)}{(\sigma^{2}+P+2\sigma_{SU})^{2}}(1+\sigma^{2}+P+2\sigma_{SU})$
$\displaystyle=$
$\displaystyle\sup_{\sigma_{SU}}\frac{1}{\sigma^{2}}\frac{\left(P^{2}\sigma^{2}-\sigma_{SU}^{2}\sigma^{2}+2P\sigma_{SU}\sigma^{2}+P\sigma^{4}-P\sigma_{SU}^{2}-2\sigma_{SU}^{3}\right)}{(\sigma^{2}+P+2\sigma_{SU})^{2}}(1+\sigma^{2}+P+2\sigma_{SU})$
$\displaystyle=$
$\displaystyle\sup_{\sigma_{SU}}\frac{1}{\sigma^{2}}\frac{\left((P\sigma^{2}-\sigma_{SU}^{2})(P+\sigma^{2}+2\sigma_{SU})\right)}{(\sigma^{2}+P+2\sigma_{SU})^{2}}(1+\sigma^{2}+P+2\sigma_{SU})$
$\displaystyle=$
$\displaystyle\sup_{\sigma_{SU}}\frac{(P\sigma^{2}-\sigma_{SU}^{2})(1+\sigma^{2}+P+2\sigma_{SU})}{\sigma^{2}(\sigma^{2}+P+2\sigma_{SU})}$
Thus, we get the following upper bound on $C(P)$,
$C(P)\leq\sup_{\sigma_{SU}\in[-\sigma\sqrt{P},\sigma\sqrt{P}]}\frac{1}{2}\log_{2}\left(\frac{(P\sigma^{2}-\sigma_{SU}^{2})(1+\sigma^{2}+P+2\sigma_{SU})}{\sigma^{2}(\sigma^{2}+P+2\sigma_{SU})}\right).$
(33)
The term $(P\sigma^{2}-\sigma_{SU}^{2})$ is oblivious to the sign of
$\sigma_{SU}$. However, the term
$\frac{1+\sigma^{2}+P+2\sigma_{SU}}{\sigma^{2}+P+2\sigma_{SU}}=1+\frac{1}{\sigma^{2}+P+2\sigma_{SU}}$
(34)
is clearly larger for $\sigma_{SU}<0$ if we fix $|\sigma_{SU}|$. Thus the
supremum in (33) is attained at some $\sigma_{SU}<0$, and we get
$C(P)\leq\sup_{\sigma_{SU}\in[-\sigma\sqrt{P},0]}\frac{1}{2}\log_{2}\left(\frac{(P\sigma^{2}-\sigma_{SU}^{2})(1+\sigma^{2}+P+2\sigma_{SU})}{\sigma^{2}(\sigma^{2}+P+2\sigma_{SU})}\right),$
(35)
which matches the expression in (31). Thus for perfect reconstruction
($MMSE=0$), the combination of linear and DPC strategies proposed in [7] is
optimal. ∎
## IV Numerical results
Witsenhausen’s original control theoretic formulation seeks to minimize the
sum of weighted costs $k^{2}P+MMSE$. Fig. 2(b) shows that asymptotically, the
ratio of upper and new lower bounds (from Corollary 1) on the weighted cost is
bounded by $1.3$, an improvement over the ratio of $2$ in [7]. The ridge of
ratio $2$ along $\sigma^{2}=\frac{\sqrt{5}-1}{2}$ present in Fig. 2(a)
(obtained using the old bound from [7]) does not exist with the new lower
bound since this small-$k$ regime corresponds to target $MMSE$s close to zero
– where the new lower bound is tight. This is illustrated in Fig. 3 (top).
Also shown in Fig. 3 (bottom) is the lack of tightness in the bounds at small
$P$. The figure explains how this looseness results in the ridge along
$k\approx 1.67$ still surviving in the new ratio plot.
Fig. 4 shows the ratio of upper and lower bounds on $MMSE(P,0)$ versus $P$ and
$\sigma$. While the ratio with the bound of [7] was unbounded (Fig. 4, top),
the new ratio is bounded by a factor of $1.5$ (Fig. 4, bottom). This is again
a reflection of the tightness of the bound at small $MMSE$. A flipped
perspective is shown in Fig. 5, where we compute the ratio of upper and lower
bounds on required power to attain a specified $MMSE$. As further evidence of
the lack of tightness in the small-$P$ (“high distortion”) regime, the ratio
of upper and lower bounds on required power diverges to infinity along the
path $MMSE=\frac{\sigma^{2}}{\sigma^{2}+1}$.
Fig. 6 shows the upper and the lower bounds for $R=0.5$. Again, the bounds are
not tight in the small-$P$ regime — now the looseness is at the lowest power
$P=1$ at which communication at $R=0.5$ is possible. As shown in Corollary 2,
the bounds are still tight at $MMSE=0$. Fig. 7 shows the upper and lower
bounds on $MMSE$ as a function of the rate $R$ for fixed power $P=1$ and
$\sigma^{2}$ equal to the Golden ratio. The figure demonstrates that beyond
the maximum rate with zero distortion, the price of increasing rate is an
increased distortion in the estimation of $\mathbf{X}^{m}$.
The MATLAB code for these figures can be found in [15].
Figure 2: The ratio of upper and lower bounds on the total asymptotic cost for
the vector Witsenhausen counterexample with the lower bound taken from [7] in
(a) and from Corollary 1 in (b). As compared to the previous best known ratio
of $2$ [7], the ratio here is smaller than $1.3$. Further, an infinitely long
ridge along $\sigma^{2}=\frac{\sqrt{5}-1}{2}$ and small $k$ that is present in
lower bounds of [7] is no longer present here. This is a consequence of the
tightness lower bound at $MMSE=0$, and hence for small $k$. A ridge remains
along $k\approx 1.67$ ($\log_{10}(k)\approx 0.22$) and large $\sigma$, and
this can be understood by observing Fig. 3 for $\sigma=10$.
Figure 3: Upper and lower bounds on asymptotic $MMSE$ vs $P$ for
$\sigma=\sqrt{\frac{\sqrt{5}-1}{2}}$ (square-root of the Golden ratio; Fig.
(a)) and $\sigma=10$ (b) for zero-rate (the vector Witsenhausen
counterexample). Tangents are drawn to evaluate the total cost for
$k=\sqrt{0.1}$ for $\sigma=\sqrt{\frac{\sqrt{5}-1}{2}}$, and for $k=1.67$ for
$\sigma=10$ (slope $=-k^{2}$). The intercept on the $MMSE$ axis of the tangent
provides the respective bound on the total cost. The tangents to the upper
bound and the new lower bound almost coincide for small values of $k$. At
$k\approx 1.67$ and $\sigma=10$, however, our bound is not significantly
better than that in [7] and hence the ridge along $k\approx 1.67$ remains in
the new ratio plot in Fig. 2.
Figure 4: Ratio of upper and lower bounds on $MMSE$ vs $P$ and $\sigma$ at
$R=0$. Whereas the ratio diverges to infinity with the old lower bound of [7]
(top), it is bounded by $1.5$ for the new bound (bottom). This is a
consequence of the improved tightness of the new bound at small $MMSE$. Figure
5: Ratio of upper and lower bounds on $P$ vs $MMSE$ and $\sigma$ at $R=0$.
Interestingly, the ratio increases to infinity as $\sigma\rightarrow\infty$
along the path where $P$ is close to zero (corresponding to “high”
$MMSE=\frac{\sigma^{2}}{\sigma^{2}+1}$). Figure 6: Upper and lower bounds on
$P$ vs $MMSE$ for $\sigma=\sqrt{\frac{\sqrt{5}-1}{2}}$ for $R=0.5$. Though the
bounds match at $MMSE=0$ (by Corollary 2), the bounds do not match at the
minimum power ($P=1$ here) for nonzero rates. Below $P=1$, communication at
$R=0.5$ is not possible. Figure 7: Plot of upper and lower bounds on $MMSE$ vs
rate for fixed power $P=1$ and $\sigma=\sqrt{\frac{\sqrt{5}-1}{2}}$. Higher
rates require higher average distortion in the reconstruction of
$\mathbf{X}^{m}$.
## Acknowledgments
P. Grover and A. Sahai acknowledge the support of the National Science
Foundation (CNS-403427, CNS-093240, CCF-0917212 and CCF-729122) and Sumitomo
Electric. A. B. Wagner acknowledges the support of NSF CSF-06-42925 (CAREER)
grant. We thank Hari Palaiyanur, Se Yong Park and Gireeja Ranade for helpful
discussions.
## References
* [1] O. Sumszyk and Y. Steinberg, “Information embedding with reversible stegotext,” in _Proceedings of the 2009 IEEE Symposium on Information Theory_ , Seoul, Korea, Jun. 2009.
* [2] Y. Steinberg, “Simultaneous transmission of data and state with common knowledge,” in _Proceedings of the 2008 IEEE Symposium on Information Theory_ , Toronto, Canada, Jun. 2008, pp. 935–939.
* [3] M. H. M. Costa, “Writing on dirty paper,” _IEEE Trans. Inform. Theory_ , vol. 29, no. 3, pp. 439–441, May 1983.
* [4] Y.-H. Kim, A. Sutivong, and T. M. Cover, “State amplification,” _IEEE Trans. Inform. Theory_ , vol. 54, no. 5, pp. 1850–1859, May 2008.
* [5] S. P. Kotagiri and J. N. Laneman, “Multiaccess channels with state known to some encoders and independent messages,” _EURASIP Journal on Wireless Communications and Networking_ , no. 450680, 2008.
* [6] N. Merhav and S. Shamai, “Information rates subject to state masking,” _IEEE Trans. Inform. Theory_ , vol. 53, no. 6, pp. 2254–2261, Jun. 2007\.
* [7] P. Grover and A. Sahai, “Vector Witsenhausen counterexample as assisted interference suppression,” _Special issue on Information Processing and Decision Making in Distributed Control Systems of the International Journal on Systems, Control and Communications (IJSCC)_ , vol. 2, pp. 197–237, 2010\.
* [8] Y. Steinberg, personal communication, Jun. 2009.
* [9] H. S. Witsenhausen, “A counterexample in stochastic optimum control,” _SIAM Journal on Control_ , vol. 6, no. 1, pp. 131–147, Jan. 1968.
* [10] P. Grover, A. Sahai, and S. Y. Park, “The finite-dimensional Witsenhausen counterexample,” in _Proceedings of the Workshop on Control over Communication Channels (ConCom), 7th Intl. Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt)_ , Seoul, Korea, Jul. 2009.
* [11] P. Gupta and P. R. Kumar, “The capacity of wireless networks,” _Information Theory, IEEE Transactions on_ , vol. 46, no. 2, pp. 388–404, Mar 2000.
* [12] R. Etkin, D. Tse, and H. Wang, “Gaussian interference channel capacity to within one bit,” _IEEE Trans. Inform. Theory_ , pp. 5534–5562, Dec. 2008\.
* [13] A. S. Avestimehr, S. Diggavi, and D. N. C. Tse, “A deterministic approach to wireless relay networks,” in _Proc. of the Allerton Conference on Communications, Control and Computing_ , October 2007.
* [14] A. S. Avestimehr, “Wireless network information flow: A deterministic approach,” Ph.D. dissertation, UC Berkeley, Berkeley, CA, 2008.
* [15] “Code for ‘Information embedding meets distributed control’.” [Online]. Available: http://www.eecs.berkeley.edu/$\sim$pulkit/InformationEmbedding.htm
|
arxiv-papers
| 2010-03-02T08:36:02 |
2024-09-04T02:49:08.696116
|
{
"license": "Public Domain",
"authors": "Pulkit Grover, Aaron B. Wagner and Anant Sahai",
"submitter": "Pulkit Grover",
"url": "https://arxiv.org/abs/1003.0520"
}
|
1003.0532
|
# Phase separation in optical lattices in a spin-dependent external potential
A-Hai Chen Department of Physics, Zhejiang Normal University, Jinhua 340012,
China Gao Xianlong gaoxl@zjnu.edu.cn Department of Physics, Zhejiang Normal
University, Jinhua 340012, China Kavli Institute for Theoretical Physics
China, CAS, Beijing 100190, China
###### Abstract
We investigate the phase separation in one-dimensional Fermi gases on optical
lattices. The density distributions and the magnetization are calculated by
means of density-matrix renormalization method. The phase separation between
spin-up and spin-down atoms is induced by the interplay of the spin-dependent
harmonic confinement and the strong repulsive interaction between
intercomponent fermions. We find the existence of a critical repulsive
interaction strength above which the phase separation evolves. By increasing
the trap imbalance, the composite phase of Mott-insulating core is changed
into the one of ferromagnetic insulating core, which is incompressible and
originates from the Pauli exclusion principle.
###### pacs:
05.30.Fk,03.75.Ss,71.10.Pm,71.15.Pd
## I Introduction
Ultracold atoms in optical lattices provide a new test bed for interacting
quantum many-body systems review paper . Fermionic atoms in optical lattices
can be used to realize the clean Fermi-Hubbard model, which is free of lattice
defects, impurities, and phonons, in contrast to those in solid-state systems.
Over the past few years, many interesting phenomena were observed in optical
lattices, for example, the Fermi surface of the atoms in the lattice, the
transform from a normal state into a band insulator Kohl , and fermionic
superfluidity of attractively interacting fermions Chin . Two other major
breakthroughs achieved recently in fermionic superfluid are: the BEC-BCS
crossover Regal and imbalanced superfluidity Zwierlein .
The spatial inhomogeneity due to the confinement essential for ultracold
atomic experiments is always present, which leads to a spatially varying local
density distribution and normally invalidates a reliable analytical method
usually used in the homogeneous system. Many numerical schemes such as the
density-matrix renormalization group (DMRG) Machida1 ; gaoprl98 ; Molina ,
quantum Monte Carlo Rigol ; Rigol2004 ; Astrakhardik ; Casula , exact
diagnolization exact diagnolization ; Machida1 ; Nikkarila , and density-
functional theory based on the exact Bethe-ansatz solution gaoprl98 are used
in studying the many interesting quantum effects in spin-balanced or
imbalanced systems. Among them, intriguing properties such as phase separation
in a trap and the transition from superfluidity to a normal state have
attracted a great deal of attention both experimentally and theoretically.
In the experiments, a phase separation was observed between the normal
component and the superfluidity of interacting fermionic atom gases with
imbalanced spin populations Zwierlein . In theory, the mean-field approach
provides a qualitative explanation of the phase separation of imbalanced
fermionic atom gases in a trap mean-field theory . The imbalance of the two
species with $N_{\uparrow}\neq N_{\downarrow}$ can be produced by different
trapping frequencies Recati06 , namely, spin-dependent trapping potentials.
Phase separation can occur in trapped spinor boson gases with a weak
anisotropic spin-spin interaction Hao and in multicomponent Fermi gases with
different values of the scattering lengths and particle number Roth . In two-
dimensional optical lattices, the phase separation due to the imbalanced
mixture, antiferromagnetic order Snoek , and pairing symmetries An is
investigated by the mean-field theory. For a one-dimensional (1D) system of
two-component Fermi gases in a continuous space, it is found that there exists
a critical interaction strength beyond which one atomic component expels
another from the center of the trap Karpiuk . For a 1D attractive Hubbard
model, a phase separation between the condensate and unpaired majority atoms
occurs for a certain range of the interaction and population imbalance. At
$T=0$ beyond a critical spin polarization, the phase separation always exists
no matter how strong the interaction is Tezuka . For a 1D repulsive Hubbard
model, the phase separation due to the different trap frequencies is discussed
within the local magnetization by the spin-dependent density-functional theory
Abedinpour .
In the present work, we are interested in the phase separation between
different fermion species induced by the spin-dependent external potentials.
The interplay between the external spin-dependent potentials and the repulsive
interaction of intercomponent fermions will be explored.
## II The model
We consider a two-component Fermi gas in a tube with $N_{f}$ atoms and $N_{s}$
lattice sites with the unit lattice constant, which can be described by a one-
band inhomogeneous Fermi-Hubbard model jaksch_98
$\displaystyle\hat{H}_{s}$ $\displaystyle=$
$\displaystyle-t\sum_{i,\sigma}({\hat{c}}^{\dagger}_{i\sigma}{\hat{c}}_{i+1\sigma}+{H}.{c}.)+U\sum_{i=1}^{N_{s}}\,{\hat{n}}_{i\uparrow}{\hat{n}}_{i\downarrow}$
(1)
$\displaystyle+\sum_{i,\sigma}V_{\sigma}\left[i-(N_{s}-1)/2\right]^{2}{\hat{n}}_{i}\,,$
where the spin degrees of freedom $\sigma=\uparrow,\downarrow$ are
pseudospin-$1/2$ labels for two internal hyperfine states and
${\hat{c}}_{i\sigma}$ (${\hat{c}}^{\dagger}_{i\sigma}$) are fermionic
operators annihilating (creating) particles with spin $\sigma$ in a Wannier
state at site $i$.
${\hat{n}}_{i}=\sum_{\sigma}{\hat{n}}_{i\sigma}=\sum_{\sigma}{\hat{c}}^{\dagger}_{i\sigma}{\hat{c}}_{i\sigma}$
is the total site occupation operator, $t$ is the tunneling between the
nearest neighbors, $U$ is the strength of the on-site interaction, and
$V_{\sigma}$ describes the strength of the spin-dependent harmonic trapping
potentials $V_{\rm har,\sigma}=V_{\sigma}\left[i-(N_{s}-1)/2\right]^{2}$.
The inhomogeneous Fermi-Hubbard model can be realized by a strong confinement
in transverse directions moritz_2005 with an additional periodic potential
applied along the tube. Concerning the experimental realization of the spin-
dependent external potentials, one can use magnetically trapped Fermi mixtures
of a particular atom in the two different hyperfine states equalmass ; Iskin ,
or two different trapped atoms of unequal masses Blume , where the different
magnetic moments make $V_{\uparrow}\neq V_{\downarrow}$. In the experiment of
two 40K fermion species, the ratio of frequencies
$V_{\uparrow}/V_{\downarrow}=\sqrt{9/7}$ is discussed Jin . In optical
lattices, a spin-dependent optical trap can be realized by asymmetrically
detuning the laser frequencies with respect to the two hyperfine states Iskin
. Experimentally, the atomic density we calculated is the most convenient and
clear observable detectable by electron beams, high-resolution cameras, or
noise interference. Recently, a composite phase of an incompressible Mott-
insulator phase in the core was identified Jordens , where the core is
composed of strongly repulsive fermionic atoms in two hyperfine states. It is
shown how the system evolves by increasing confinement from a compressible
dilute metal into a band-insulating state, which also provides a way to
polarize a spin-balanced system where $N_{\uparrow}=N_{\downarrow}=N_{f}/2$
Jordens .
The homogeneous 1D Fermi-Hubbard model belongs to the universality class of
Luttinger liquids. At zero temperature, the properties of this model in the
thermodynamic limit ($N_{\sigma},N_{s}\rightarrow\infty$, but with finite
$N_{\sigma}/N_{s}$) are determined by the fillings
$n_{\sigma}=N_{\sigma}/N_{s}$ and by the dimensionless coupling constant
$u=U/t$. According to Lieb and Wu LiebWu , the ground state (GS) properties
for different fillings in the thermodynamic limit are described by the coupled
integral equations (for details see Refs. [32,33]).
For the inhomogeneous system described by Eq. (1), the coexisted phases
induced by the external spin-independent trapping potentials
($V_{\uparrow}=V_{\downarrow}$) were well identified by many authors Rigol ;
Rigol2004 ; Heiselberg ; Campo ; Liu . We focus in this work on the spin-
dependent potentials ($V_{\uparrow}\neq V_{\downarrow}$) by applying the DMRG
techniques, performed by using the ALPS libraries Albuquerque . During our
DMRG calculations, the states kept are $500$ to $1000$ so that we can restrict
the cut error to be less than $10^{-11}$.
## III Numerical results
In this section we present our numerical results. In the following discussion
we keep the total number of particles constant ($N_{f}=40$) and vary the
number of spin-up and spin-down atoms in the system. We characterize the
confinement imbalance by defining the ratio between the spin-up and spin-down
dependent external potentials as
$\displaystyle\gamma=\frac{V_{\uparrow}}{V_{\downarrow}}.$ (2)
In Fig. 1 we show schematic plots for the spin-dependent harmonic potentials
and the density distributions of the two fermion species with small or large
confinement imbalances. The effects of $\gamma$ are manifested in that two
atoms coexist for small $\gamma$ where the spin-up and spin-down atom mixture
in the center of the trap forms phase mixing (PM) region and separate with
only spin-up atoms left for large $\gamma$ where the phase separated (PS)
region is formed. The PS region is determined with the local occupation in the
trap center (i.e., $i$=0) satisfying $n_{0}\leq 10^{-3}$. We distinguish in
the following the different phases by showing the atomic density profiles and
the local magnetization for different repulsive interactions and confining
strengths in the system of spin-unpolarized or spin-polarized atoms.
Figure 1: (Color online) Schematic illustrations for spin-dependent harmonic
potentials $V_{\rm har,\sigma}$ (in units of $t$) and the density
distributions $n_{\sigma}$ (in units of the lattice constant) of both spin-up
and spin-down atoms in the presence of interactions. The left panel (a) is for
the system of small trap imbalance, where the spin-up and spin-down atom
mixture in the center forms a PM region and the right panel (b) for the system
of large trap imbalance, where the PS region is formed. Figure 2: (Color
online) Phase diagram as a function of $u$ and the confinement ratio $\gamma$
determined by the DMRG technique. The system consists of $N_{\uparrow}=20$ and
$N_{\downarrow}=20$ fermions. The spin-down trap strength is
$V_{\downarrow}=1.0\times 10^{-3}$. The arrow in the top indicates the
position where $\gamma=1$ and the arrow in the right where $u_{c}=1.64$. The
solid line is a power-law fit $u=u_{c}+\alpha/(\gamma-1)$ to the data with
$\alpha=10.932$. The two phases PM and PS are manifested in Fig. 1 and
explained in the text.
First, we study the phase separation between the two-component fermions
induced by the interplay between the repulsive interaction and the spin-
dependent parabolic potentials in the unpolarized system of an equal number of
spin-up and spin-down atoms ($N_{\uparrow}=N_{\downarrow}=20$) and
$V_{\downarrow}=1.0\times 10^{-3}$. The lattice size chosen here and in the
following is always large enough to make sure that the GS densities smoothly
drop down at the edges. In Fig. 2, the phase diagram is shown as a function of
$u$ and the confinement ratio $\gamma$. Two regions are seen: the PM region
with both spin-up and spin-down mixtures in the center of the trap and the PS
region with only spin-up atoms remaining in the center. A critical interaction
strength $u_{c}=1.64$ is obtained, below which there is no phase separation no
matter how large the confinement ratio. For the system considered here, the
condition in which the phase separation happens can be simply fitted by a
power-law relation $u=u_{c}+\alpha/(\gamma-1)$, with $\alpha=10.932$. We
further illustrate in Fig. 3 an explicit example by choosing $\gamma=3$ and
changing the interaction strength. We confirm that there exists a critical
value of the interaction strength ($u=8$) beyond which the spin-down atoms are
depleted from the center of the trap and repelled into the periphery regions
between $V_{\uparrow}$ and $V_{\downarrow}$. In this case, a phase separation
begins to appear (i.e., the Fermi components tend to stay in different spatial
regions). Thus, upon approaching the phase separation point and beyond, the
local polarization of the atomic gases in the center becomes stronger and
stronger. When the complete phase separation is realized, fermions become
fully polarized due to the strong repulsive interaction. As a result, spin-up
atoms locate in the center and spin-down atoms at the periphery of the trap,
which is clearly seen in Fig. 3(c) for $u=20$. Upon reaching the complete
phase separation, further increasing the repulsive interaction only makes the
spin-up density a little more confined and spin-down density more spread out.
In Fig. 3(d), we plot the local magnetization of the system, which is defined
as $m_{i}=(n_{i\uparrow}-n_{i\downarrow})/2$. For the strong repulsive
interaction where the phase separation begins to evolve $m_{i}$ changes from
negative to positive with a big slope.
Figure 3: (Color online) Ground state density profiles for $n_{i\uparrow}$,
$n_{i\downarrow}$ and local magnetization
$m_{i}=(n_{i\uparrow}-n_{i\downarrow})/2$ as a function of $i$ in the spin-
dependent external potentials of $\gamma=3$. The spin-down trap strength is
$10^{-3}$. The system consists of $N_{\uparrow}=20$ and $N_{\downarrow}=20$
fermions. Three different interaction strengths are shown: (a) $u=1$, (b)
$u=8$, and (c) $u=20$. The local magnetization $m_{i}$ is shown in (d). The
solid line connecting the symbols serves as a guide for the eyes. For
comparison, the GS densities of the noninteracting case ($u=0$) for spin-up
(bold solid line) and spin-down (thin solid line) atoms are included in (a),
and the corresponding local magnetization (thin solid line) is also shown in
(d). We find that the repulsive interaction can induce a complete phase
separation between the two components in the spin-dependent external
potentials.
Now, let us concentrate on the phase separation induced by spin-dependent
parabolic potentials. In Figs. 4 and 5 we study the polarized systems of an
unequal number of spin-up and spin-down particles ($N_{\uparrow}=30$ and
$N_{\downarrow}=10$) with weak ($u=1$) and strong ($u=4$) repulsive
interactions. We illustrate the effects of the confinement ratio on the local
density distributions and the local magnetization.
Figure 4: (Color online) Ground state density profiles for $n_{i\uparrow}$,
$n_{i\downarrow}$ and local magnetization $m_{i}$ as a function of $i$ for the
system of weak repulsive interaction ($u=1$) in the spin-dependent external
potentials. The system consists of $N_{\uparrow}=30$ and $N_{\downarrow}=10$
fermions with $V_{\downarrow}=2.5\times 10^{-4}$. Three different ratios of
confining potentials are shown. (a) $\gamma=1$, (b) $\gamma=3$, and (c)
$\gamma=6$. The local magnetization $m_{i}$ is plotted in (d). From (d), we
can see that upon reaching the phase separation point and beyond, the local
magnetization $m_{i}$ becomes more negative at the periphery and more positive
in the bulk region of the trap signaling that more spin-down fermions are
repelled from the center and more spin-up fermions are constrained there.
Figure 5: (Color online) Same as Fig. 4 but for the system of strong repulsive
interaction ($u=4$). From (b), we notice that a complete phase separation
occurs. For comparison, in (a)-(c), the GS densities of the noninteracting
case ($u=0$) for the spin-up (bold solid line) and spin-down (thin solid line)
atoms are also plotted. In (d) we include the corresponding local
magnetizations for the noninteracting case with $\gamma=1$ (bold solid line),
$3$ (thin solid line), and $6$ (dotted line), respectively.
We increase $\gamma$ by keeping the spin-down external potential
$V_{\downarrow}$ as invariant and increasing $V_{\uparrow}$ (i.e., $\gamma\geq
1$). In Fig. 4, the density profiles for the weak repulsive interaction
($u=1$) are shown with different confinement imbalances ($\gamma=1,3$, and
$6$). While increasing the confinement for the spin-up atoms, the interaction
between the spin-up and spin-down atoms in the center of the trap repels the
spin-down atoms into the edges of the trap. However the repulsive interaction
is not strong enough and only a small amount of phase separation appears. From
Fig. 5(b), we can see that, in the system of strong repulsive interaction
($u=4$) and a large trap imbalance ($\gamma=3$), almost all the spin-down
atoms are repelled from the bulk of the trap and a complete phase separation
is realized. Due to the depletion of the spin-down fermions, fully polarized
gases of spin-up fermions are obtained, as can be seen in Figs. 5(b) and 5(c).
For comparison, the GS density distributions of the spin-up and spin-down
atoms for the noninteracting case ($u=0$) are also included, where no phase
separation is observed. We conclude that the intercomponent interaction is
essential in achieving a phase separation between the two-component fermions
in a spin-dependent trap.
The local magnetization $m_{i}$, in Figs. 4(d) and 5(d), gives another
signature of the phase separation. For small $\gamma$, a flat region of
$m_{i}$ is seen in the center of the trap and two bumps are shown at the edges
with the excess spin-up atoms. The increase of the trap imbalance and the
repulsive interaction strength shows a signature that $m_{i}$ is more negative
at the edges, that is, more and more spin-down atoms are repelled from the
center of the trap and accumulate at the periphery region between
$V_{\uparrow}$ and $V_{\downarrow}$.
Figure 6: (Color online) Density distributions for spin-up and spin-down
fermions together with their sum (the total GS density) and difference (the
local magnetization) plotted against the site with strong repulsive
interaction ($u=6$) in the spin-dependent external potentials. The system
consists of $N_{\uparrow}=20$ and $N_{\downarrow}=20$ fermions with
$V_{\downarrow}=6.0\times 10^{-3}$. Three different ratios of confining
potentials are shown: (a) $\gamma=1$, (b) $\gamma=2.6$, and (c) $\gamma=6$.
Figure 7: (Color online) Same as Fig. 6 but for the spin-polarized system of
$N_{\uparrow}=30$ and $N_{\downarrow}=10$. Three different ratios of confining
potentials are shown: (a) $\gamma=1$, (b) $\gamma=1.45$, and (c) $\gamma=6$.
In the following we study how the spin-dependent potentials influence the
composite phase of the Mott-insulating core in the bulk. In Fig. 6, we show
the GS density distributions of an unpolarized system of
$N_{\uparrow}=N_{\downarrow}=20$ under the influence of the different trap
imbalances with $V_{\downarrow}=6.0\times 10^{-3}$ and a strong repulsive
interaction of $u=6$. For $\gamma=1$, a Mott phase is formed in the bulk
region of the trap. With the increase of the confinement for the spin-up
atoms, the spin-down atoms are repelled from the center of the trap. The Mott
phase induced by the interaction between the locally spin-balanced fermions is
changed into the Mott-like phase induced by both the interaction between the
locally spin-imbalanced fermions and the spin-dependent potentials. At the
critical point of $\gamma=2.6$, the phase separation starts [see Fig. 6(b)]
and the strong confinement for the spin-up atoms forms an insulating core of
fully polarized fermions in the center of the trap, over which the local
occupancy is a unit. This insulating core is regarded as a ferromagnetic
insulating phase since it is incompressible in nature and originates from the
Pauli exclusion principle, which differs from the Mott-insulating phase
induced by the repulsive interaction between fermions MachidaPRB , such as in
Fig. 6(a). The unit core becomes stable for $\gamma>6$ by further increasing
$\gamma$. Upon reaching the phase separation point and beyond, a plateau of
constant $m_{i}=0.5$ is formed in the center of the local magnetization.
In Fig. 7 we show the case of a polarized system of $N_{\uparrow}=30$ and
$N_{\downarrow}=10$ with a strong repulsive interaction of $u=6$. For
$\gamma=1$, the spin-up fermions form Wigner-lattice-type profiles inside the
Mott core Soffing , which occurs at low fillings or equivalently at large $u$
and can be explained by mapping Eq. (1) into the antiferromagnetic Heisenberg
model MachidaPRB . We notice that, compared to the unpolarized system of that
in Fig. 6, the polarized system becomes more easily reaches the phase
separation point ($\gamma=1.45$). That is, the Mott phase in the polarized
system is less robust against the increase of the interaction strength and
confinement.
## IV Conclusions
In this article we perform a theoretical study of a 1D Fermi-Hubbard model in
a spin-dependent harmonic trap within the DMRG techniques. The interplay
between the repulsive interaction and the spin-dependent harmonic trap is
studied for the system of spin-balanced or spin-imbalanced Fermi gases. We
find that, for the system in the spin-dependent external potentials, there
exists a critical interaction strength beyond which a phase separation can
occur with two Fermi components staying in the different spatial regions. For
the system with a weak interaction strength, upon increasing the trap
imbalance, the spin-up atoms are confined more and more in the center of the
trap and a depletion occurs for the spin-down atoms due to the intercomponent
repulsive interactions. However, the weak repulsive interaction below a
critical value is not capable of achieving a full phase separation. For the
system with strong intercomponent repulsive interactions, a complete phase
separation is realized at the strong confinement imbalance where spin-down
atoms are repelled out of the bulk region with only spin-up atoms remaining.
For the system with both strong confinement and strong repulsive interactions,
where a composite phase of the Mott-insulating core is formed in the center,
we show that, upon increasing the trap imbalance, the Mott phase induced by
the interaction between the locally spin-balanced fermions is changed into the
Mott-like phase induced under the interplay between the interaction of the
locally spin-imbalanced fermions and the spin-dependent confining potentials.
Upon reaching the phase separation point and beyond, the ferromagnetic
insulating phase due to the Pauli exclusion principle appears, which is of the
unit core. In the distribution of the local magnetization, a step structure
contributed by spin-up atoms alone is formed with a big slope from a negative
to positive value.
## V Acknowledgements
This work was supported by Qianjiang River Fellow Fund 2008R10029, NSF of
China under Grant No. 10704066, 10974181, and Program for Innovative Research
Team in Zhejiang Normal University. We thank Lin Na for critical reading of
the manuscript. The calculations were performed using the ALPS libraries.
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|
arxiv-papers
| 2010-03-02T09:20:42 |
2024-09-04T02:49:08.703003
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A-Hai Chen and Gao Xianlong",
"submitter": "Gao Xianlong",
"url": "https://arxiv.org/abs/1003.0532"
}
|
1003.0579
|
11footnotetext: 2000 Mathematics Subject Classification: 05D10, 46B03
# More $\ell_{r}$ saturated $\mathcal{L}^{\infty}$ spaces
I. Gasparis Department of Mathematics, Aristotle University of Thessaloniki,
Thessaloniki 54124, Greece ioagaspa@math.auth.gr , M.K. Papadiamantis
National Technical University of Athens, Faculty of Applied Sciences,
Department of Mathematics, Zografou Campus, 157 80, Athens, Greece
mpapadiamantis@yahoo.gr and D.Z. Zisimopoulou National Technical University
of Athens, Faculty of Applied Sciences, Department of Mathematics, Zografou
Campus, 157 80, Athens, Greece dzisimopoulou@hotmail.com
###### Abstract.
Given $r\in(1,\infty)$, we construct a new $\mathcal{L}^{\infty}$ separable
Banach space which is $\ell_{r}$ saturated .
###### Key words and phrases:
Banach theory, $\ell_{p}$ saturated, $\mathcal{L}^{\infty}$ spaces
## 1\. Introduction
The Bourgain-Delbaen spaces [7] are examples of separable
$\mathcal{L}^{\infty}$ spaces containing no isomorphic copy of $c_{0}$. They
have played a key role in the solution of the scalar-plus-compact problem by
Argyros and Haydon [3], where a Hereditarily Indecomposable
$\mathcal{L}^{\infty}$ space is presented with the property that every
operator on the space is a compact perturbation of a scalar multiple of the
identity.
There has recently been an interest in the study $\mathcal{L}^{\infty}$ spaces
of the Bourgain-Delbaen type. Freeman, Odell and Schlumprecht [8] showed that
every Banach space with separable dual is isomorphic to a subspace of a
$\mathcal{L}^{\infty}$ space having a separable dual. The aim of this paper is
to present a method of constructing, for every $1<r<\infty$, a new
$\mathcal{L}^{\infty}$ space which is $\ell_{r}$ saturated. Our approach
shares common features with the Argyros-Haydon work. More precisely we
combine, as in [3], the Bourgain-Delbaen method [7] yielding exotic
$\mathcal{L}^{\infty}$ spaces, with the Tsirelson type norms that are
equivalent to some $\ell_{r}$ norm (see [2], [4], [5]). Recall that in [9],
the original Bourgain-Delbaen spaces $\mathfrak{X}_{a,b}$ with $a<1$,
$b<\frac{1}{2}$ and $a+b>1$ where shown to be $\ell_{p}$ saturated for $p$
determined by the formulas $\frac{1}{p}+\frac{1}{q}=1$ and $a^{q}+b^{q}=1$.
This paper is organized as follows. In the second section, for a given
$r\in(1,\infty)$, we construct a Banach space $\mathfrak{X}_{r}$. To do this,
we first choose $n\in\operatorname{\mathbb{N}}$, $n>1$, and a finite sequence
$\overline{b}=(b_{1},b_{2},\ldots,b_{n})$ of positive real numbers with
$b_{1}<1$, $b_{2},b_{3},\ldots,b_{n}<\frac{1}{2}$ such that
$\sum_{i=1}^{n}b_{i}^{r^{\prime}}=1$ and $\frac{1}{r}+\frac{1}{r^{\prime}}=1$.
The definition of $\mathfrak{X}_{r}$ combines the Bourgain-Delbaen method with
the Tsirelson type space
$\operatorname{\mathcal{T}}(\operatorname{\mathcal{A}}_{n},\overline{b})$
which will be later proved to be isomorphic to $\ell_{r}$. In particular, if
$b_{1}=b_{2}=\ldots=b_{n}=\theta$,
$\operatorname{\mathcal{T}}(\operatorname{\mathcal{A}}_{n},\overline{b})$
coincides with
$\operatorname{\mathcal{T}}(\operatorname{\mathcal{A}}_{n},\theta)$ and the
latter is known to be isomorphic to $\ell_{p}$ for some $p\in(1,\infty)$ (see
[4]). It is worth noticing that for $n=2$ the spaces $\mathfrak{X}_{r}$
essentially coincide with the original Bourgain-Delbaen spaces
$\mathfrak{X}_{a,b}$. Thus, our construction of $\mathcal{L}^{\infty}$ spaces
which are $\ell_{r}$ saturated spaces, can be considered as a generalization
of the Bourgain-Delbaen method. We must point out here that when $n=2$, our
proof of the fact that $\mathfrak{X}_{r}$ is $\ell_{r}$ saturated, differs
from Haydon’s (see [9]) corresponding one for $\mathfrak{X}_{a,b}$. To be more
specific, $\mathfrak{X}_{r}$ has a natural FDD $(M_{k})$. Given a normalized
skipped block basis $(u_{k})$ of $(M_{k})$ with the supports of the $u_{k}$’s
lying far enough apart, then it is not hard to check that $(u_{k})$ dominates
$(e_{k})$, the natural basis of
$\operatorname{\mathcal{T}}(\operatorname{\mathcal{A}}_{n},\overline{b})$. The
same holds for every normalized block basis of $(u_{k})$. To obtain a
normalized block basis of $(u_{k})$ equivalent to $(e_{k})$, we select a
sequence $I_{1}<I_{2}<\dots$ of successive finite subsets of $\mathbb{N}$ such
that $\lim_{k}\|\sum_{i\in I_{k}}u_{i}\|=\infty$. Such a choice is possible by
the domination of $(e_{k})$ by $(u_{k})$. We set $v_{k}=\|\sum_{i\in
I_{k}}u_{i}\|^{-1}\sum_{i\in I_{k}}u_{i}$ and show that some subsequence of
$(v_{k})$ is dominated by $(e_{k})$. To accomplish this we adapt the method of
the analysis of the members of a finite block basis of $(e_{k})$ with respect
to a functional in the natural norming set of
$\operatorname{\mathcal{T}}(\operatorname{\mathcal{A}}_{n},\overline{b})$ (see
[6]), to the context of the present construction. We believe that this
approach yields a more transparent proof than Haydon’s, at least for the upper
$\ell_{r}$ estimate.
The rest of the paper is devoted to the proof of the main property, namely
that $\mathfrak{X}_{r}$ is $\ell_{r}$ saturated. In Section 3, we define the
tree analysis of the functionals $\\{e_{\gamma}^{*}:\gamma\in\Gamma\\}$ which
is a 1-norming subset of the unit ball of $\mathfrak{X}_{r}^{*}$. The tree
analysis is similar to the corresponding one used in the Tsirelson and mixed
Tsirelson spaces [4]. In the following two sections we establish the lower and
upper norm estimates for certain block sequences in the space
$\mathfrak{X}_{r}$.
In the final section we show that every block basis of $(M_{k})$ admits a
further normalized block basis $(x_{k})$ such that every normalized block
basis of $(x_{k})$ is equivalent to the natural basis of the space
$\operatorname{\mathcal{T}}(\operatorname{\mathcal{A}}_{n},\overline{b})$.
Zippin’s theorem [12] yields the desired result.
## 2\. Preliminaries
In this section we define the space $\mathfrak{X}_{r}$ combining the Bourgain-
Delbaen construction [7] and the Tsirelson type constructions [2], [4].
Before proceeding, we recall some notation and terminology from [3]. Let
$n\in\operatorname{\mathbb{N}}$ and $0<b_{1},b_{2},...,b_{n}<1$ with
$\sum_{i=1}^{n}b_{i}>1$ and there exists $r^{\prime}\in(1,\infty)$ such that
$\sum_{i=1}^{n}{b_{i}}^{r^{\prime}}=1$. We may also assume without loss of
generality that $b_{1}>b_{2}>\ldots>b_{n}$. We define
$W[(\operatorname{\mathcal{A}}_{n},\overline{b})]$ to be the smallest subset
$W$ of $c_{00}(\operatorname{\mathbb{N}})$ with the following properties:
1. (1)
$\pm e^{*}_{k}\in W$ for all $k\in\mathbb{N}$,
2. (2)
whenever $f_{i}\in W$ and
$\max\operatorname{supp}f_{i}<\min\operatorname{supp}f_{i+1}$ for all $i$, we
have $\sum_{i\leq a}b_{i}f_{i}\in W$, provided that $a\leq n$,
We say that an element $f$ of
$W[(\operatorname{\mathcal{A}}_{n},\overline{b})]$ is of Type $0$ if $f=\pm
e_{k}^{*}$ for some $k$ and of Type I otherwise; an element of Type I is said
to have weight $b_{a}$ for some $a\leq n$ if $f=\sum_{i=1}^{a}f_{i}$ for a
suitable sequence $(f_{i})$ of successive elements of
$W[\operatorname{\mathcal{A}}_{n},\overline{b}]$.
The Tsirelson space
$\operatorname{\mathcal{T}}(\operatorname{\mathcal{A}}_{n},\overline{b})$ is
defined to be the completion of $c_{00}$ with respect to the norm
$\|x\|=\sup\\{\langle f,x\rangle:f\in
W[\operatorname{\mathcal{A}}_{n},\overline{b}]\\}.$
We may also characterize the norm of this space implicitly as being the
smallest function $x\mapsto\|x\|$ satisfying
$\|x\|=\max\bigg{\\{}\|x\|_{\infty},\sup\sum_{i=1}^{n}b_{i}\|E_{i}x\|\bigg{\\}},$
where the supremum is taken over all sequences of finite subsets
$E_{1}<E_{2}<\cdots<E_{n}$.
We shall now present the fundamental aspects related to the Bourgain-Delbaen
construction.
For the interested readers we mention that the following method can be
characterized as the ”dual” construction of the construction presented in [3].
This characterization is based on the fact that in [3] a particular kind of
basis is given to $\ell_{1}(\Gamma)$ and the Bourgain-Delbaen type space $X$
is seen as the predual of its dual, which is $\ell_{1}(\Gamma)$.
Let $(\Gamma_{q})_{q\in{\operatorname{\mathbb{N}}}}$ be a strictly increasing
sequence of finite sets and denote their union by $\Gamma$;
$\Gamma=\cup_{q\in{\operatorname{\mathbb{N}}}}\Gamma_{q}$.
We set $\Delta_{0}=\Gamma_{0}$ and
$\Delta_{q}=\Gamma_{q}\backslash\Gamma_{q-1}$ for $q=1,2,\ldots$
Assume furthermore that to each $\gamma\in\Delta_{q}$, $q\geq 1$, we have
assigned a linear functional
$c_{\gamma}^{*}:\ell^{\infty}(\Gamma_{q-1})\rightarrow\operatorname{\mathbb{R}}$.
Next, for $n<m$ in $\operatorname{\mathbb{N}}$, we define by induction, a
linear operator
$i_{n,m}:\ell^{\infty}(\Gamma_{n})\rightarrow\ell^{\infty}(\Gamma_{m})$ as
follows:
For $m=n+1$, we define
$i_{n,n+1}:\ell^{\infty}(\Gamma_{n})\rightarrow\ell^{\infty}(\Gamma_{n+1})$ by
the rule
$(i_{n,n+1}(x))(\gamma)=\begin{cases}x(\gamma),\ \text{if
}\gamma\in\Gamma_{n}\\\ c_{\gamma}^{*}(x),\ \text{if
}\gamma\in\Delta_{n+1}\end{cases}$
for every $x\in\ell^{\infty}(\Gamma_{n})$.
Then assuming that $i_{n,m}$ has been defined, we set
$i_{n,m+1}=i_{m,m+1}\circ i_{n,m}$. A direct consequence of the above
definition is that for $n<l<m$ it holds that $i_{n,m}=i_{l,m}\circ i_{n,l}$.
Finally we denote by
$i_{n}:\ell^{\infty}(\Gamma_{n})\rightarrow\operatorname{\mathbb{R}}^{\Gamma}$
the direct limit $i_{n}=\lim_{m\rightarrow\infty}i_{n,m}$.
We assume that there exists a $C>0$ such that for every $n<m$ we have
$\|i_{n,m}\|\leq C$. This implies that $\|i_{n}\|\leq C$ and therefore
$i_{n}:\ell^{\infty}(\Gamma_{n})\to\ell^{\infty}(\Gamma)$ is a bounded linear
map. In particular, setting $X_{n}=i_{n}[\ell^{\infty}(\Gamma_{n})]$, we have
that $X_{n}\stackrel{{\scriptstyle
C}}{{\thickapprox}}\ell^{\infty}(\Gamma_{n})$ and furthermore
$(X_{n})_{n\in\operatorname{\mathbb{N}}}$ is an increasing sequence of
subspaces of $\ell^{\infty}(\Gamma)$. We also set
$\mathfrak{X}_{BD}=\overline{\bigcup\limits_{n\in\operatorname{\mathbb{N}}}X_{n}}\hookrightarrow\ell^{\infty}(\Gamma)$
equipped with the supremum norm. Evidently, $\mathfrak{X}_{BD}$ is an
$\mathcal{L}^{\infty}$ space.
Let us denote by $r_{n}:\ell^{\infty}(\Gamma)\to\ell^{\infty}(\Gamma_{n})$ the
natural restriction map, i.e. $r_{n}(x)=x|_{\Gamma_{n}}$. We will also abuse
notation and denote by
$r_{n}:\ell^{\infty}(\Gamma_{m})\to\ell^{\infty}(\Gamma_{n})$ the restriction
function from $\ell^{\infty}(\Gamma_{m})$ to $\ell^{\infty}(\Gamma_{n})$ for
$n<m$.
###### Notation 2.1.
We denote by $e_{\gamma}^{*}$ the restriction of the unit vector
$e_{\gamma}\in\ell^{1}(\Gamma)$ on the space $\mathfrak{X}_{BD}$.
We also extend the functional
$c_{\gamma}^{*}:\ell^{\infty}(\Gamma_{n})\to\operatorname{\mathbb{R}}$ to a
functional $c_{\gamma}^{*}:\mathfrak{X}_{BD}\to\operatorname{\mathbb{R}}$ by
the rule $c_{\gamma}^{*}(x)=(c_{\gamma}^{*}\circ r_{q-1})(x)$ when
$\gamma\in\Delta_{q}$.
As it is well known from [3] and [7], instead of the Schauder basis of
$\mathfrak{X}_{BD}$, it is more convenient to work with a FDD naturally
defined as follows:
For each $q\in\operatorname{\mathbb{N}}$ we set
$M_{q}=i_{q}[\ell^{\infty}(\Delta_{q})]$.
We briefly establish this fact in the following proposition and then continue
with the details of the construction of $\mathfrak{X}_{r}$.
###### Proposition 2.2.
The sequence $(M_{q})_{q\in\operatorname{\mathbb{N}}}$ is a FDD for
$\mathfrak{X}_{BD}$.
###### Proof.
For $q\geq 0$ we define the maps $P_{\\{q\\}}:\mathfrak{X}_{BD}\to M_{q}$ with
$P_{\\{q\\}}(x)=i_{q}(r_{q}(x))-i_{q-1}(r_{q-1}(x))$
It is easy to check that each $P_{\\{q\\}}$ is a projection onto $M_{q}$ and
that for $q_{1}\neq q_{2}$ and $x\in M_{q_{2}}$ we have
$P_{\\{q_{1}\\}}(x)=0$. Also we have that $\|P_{q}\|\leq 2C$. We point out
that in a similar manner one can define projections on intervals of the form
$I=(p,q]$ so that $P_{I}(x)=\sum_{i=p+1}^{q}P_{\\{i\\}}(x)$ for which we can
readily verify the formula
$P_{I}(x)=i_{q}(r_{q}(x))-i_{p}(r_{p}(x))$
Note that $\|P_{I}\|\leq 2C$. This shows that indeed $(M_{q})_{q}$ is a FDD
generating $\mathfrak{X}_{BD}$. ∎
For $x\in\mathfrak{X}_{BD}$ we denote by $\operatorname{supp}x$ the set
$\operatorname{supp}x=\\{q:P_{\\{q\\}}(x)\neq 0\\}$ and by
$\operatorname{ran}x$ the minimal interval of $\operatorname{\mathbb{N}}$
containing $\operatorname{supp}x$.
###### Definition 2.3.
A block sequence $(x_{i})_{i=1}^{\infty}$ in $\mathfrak{X}_{BD}$ is called
skipped (with respect to $(M_{q})_{q\in\operatorname{\mathbb{N}}}$), if there
is a subsequence $(q_{i})_{i=1}^{\infty}$ of $\operatorname{\mathbb{N}}$ so
that for all $i\in\operatorname{\mathbb{N}}$,
$\operatorname{maxsupp}{x_{i}}<q_{i}<\operatorname{minsupp}{x_{i+1}}$.
In the sequel, when we refer to a skipped block sequence, we consider it to be
with respect to the FDD $(M_{q})_{q\in\operatorname{\mathbb{N}}}$.
Let $q\geq 0$. For all $\gamma\in\Delta_{q}$ we set
$d_{\gamma}^{*}=e_{\gamma}\circ P_{\\{q\\}}.$ Then the family
$(d^{*}_{\gamma})_{\gamma\in\Gamma}$ consists of the biorthogonal functionals
of the FDD $(M_{q})_{q\geq 0}$. Notice that for $\gamma\in\Delta_{q}$,
$\displaystyle d_{\gamma}^{*}(x)$ $\displaystyle=$ $\displaystyle
P_{q}(x)(\gamma)=i_{q}(r_{q}(x))(\gamma)-i_{q-1}(r_{q-1}(x))(\gamma)=$
$\displaystyle=$ $\displaystyle
r_{q}(x)(\gamma)-c_{\gamma}^{*}(r_{q-1}(x))=x(\gamma)-c_{\gamma}^{*}(x)=$
$\displaystyle=$ $\displaystyle e^{*}_{\gamma}(x)-c^{*}_{\gamma}(x).$
The sequences $(\Delta_{q})_{q\in\operatorname{\mathbb{N}}}$ and
$(c_{\gamma}^{*})_{\gamma\in\Gamma}$ are determined as in [3], section 4 and
Theorem 3.5.
We give some useful notation. For fixed $n\in\operatorname{\mathbb{N}}$ and
$\overline{b}=(b_{1},b_{2},\ldots,b_{n})$ with $0<b_{1},b_{2},\ldots,b_{n}<1$,
for each $\gamma\in\Delta_{q}$ we assign
1. (a)
$\operatorname{rank}\gamma=q$
2. (b)
age of $\gamma$ denoted by $a(\gamma)=a$ such that $2\leq a\leq n$
3. (c)
weight of $\gamma$ denoted by $w(\gamma)=b_{a}$
In order to proceed to the construction, we first need to fix a positive
integer $n$ and a descending sequence of positive real numbers
$b_{1},\ldots,b_{n}$ such that $b_{1}<1$, $b_{i}<\frac{1}{2}$, for every
$i=2,\ldots,n$ and $\sum_{i=1}^{n}b_{i}>1$. Let $r\in(1,\infty)$ be such that
$\sum_{i=1}^{n}b_{i}^{r^{\prime}}=1$ and $\frac{1}{r}+\frac{1}{r^{\prime}}=1$.
Now we shall define the space $\mathfrak{X}_{r}$ by using the Bourgain-Delbaen
construction that was presented in the preceding paragraphs.
We set $\Delta_{0}=\emptyset$, $\Delta_{1}=\\{0\\}$ and recursively define for
each $q>1$ the set $\Delta_{q}$.
Assume that $\Delta_{p}$ have been defined for all $p\leq q$. We set
$\displaystyle\Delta_{q+1}$ $\displaystyle=$
$\displaystyle\big{\\{}(q+1,a,p,\eta,\varepsilon e_{\xi}^{*}):\ 2\leq a\leq
n,p\leq q,\ \varepsilon=\pm 1,\ e_{\xi}^{*}\in S_{\ell^{1}(\Gamma_{q})},\
\xi\in\Gamma_{q}\setminus\Gamma_{p},$ $\displaystyle\ \ \eta\in\Gamma_{p},\
b_{a-1}=w(\eta)\big{\\}}$
For $\gamma\in\Delta_{q+1}$ it is clear that the first coordinate is the
$\operatorname{rank}$ of $\gamma$, while the second is the age $a(\gamma)$ of
$\gamma$. The functionals $(c_{\gamma}^{*})_{\gamma\in\Delta_{q+1}}$ are
defined in a way that depends on $\gamma\in\Delta_{q+1}$. Namely, let
$x\in\ell^{\infty}(\Gamma_{q})$.
1. (i)
For $\gamma=(q+1,2,p,\eta,\varepsilon e_{\xi}^{*})$ we set
$c_{\gamma}^{*}(x)=b_{1}x(\eta)+b_{2}\varepsilon
e_{\xi}^{*}\big{(}x-i_{p,q}(r_{p}(x))\big{)}.$
2. (ii)
For $\gamma=(q+1,a,p,\eta,\varepsilon e_{\xi}^{*})$ with $a>2$ we set
$c_{\gamma}^{*}(x)=x(\eta)+b_{a}\varepsilon
e_{\xi}^{*}\big{(}x-i_{p,q}(r_{p}(x))\big{)}.$
We may now define sequences $(i_{q})$, $(\Gamma_{q})$, $(X_{q})$ in a similar
manner as before and set
$\mathfrak{X}_{r}=\overline{\bigcup\limits_{q\in\operatorname{\mathbb{N}}}X_{q}}$.
Assuming that $(i_{q})$ is uniformly bounded by a constant C, we conclude that
the space $\mathfrak{X}_{r}$ is a subspace of $\ell_{\infty}(\Gamma)$. The
constant C is determined as in [3] Theorem 3.4, by taking
$C=\frac{1}{1-2b_{2}}$. Thus, for every $m\in\operatorname{\mathbb{N}}$,
$\|i_{m}\|\leq C$. This implies that $\|P_{I}\|\leq 2C$ for every $I$
interval.
###### Remark 2.4.
In the case of $n=2$, i.e. $\overline{b}=(b_{1},b_{2})$, the space
$\mathfrak{X_{r}}$ essentially coincides with the Bourgain-Delbaen space
$\mathfrak{X}_{b_{1},b_{2}}$, since every $\gamma\in\Gamma$ is of age 2.
###### Remark 2.5.
As it is shown in Proposition 6.2, the choice of r, based on the fixed $n$ and
$\overline{b}$, yields that
$\operatorname{\mathcal{T}}(\mathcal{A}_{n},\overline{b})\cong\ell_{r}$.
Moreover, the ingredients of the ”Tsirelson type spaces” theory that are used
throughout this paper are essentially the same with the corresponding ones in
[3]. The basic difference in our approach is that we use only one family
$\operatorname{\mathcal{T}}(\operatorname{\mathcal{A}}_{n},\overline{b})$ for
some appropriate $n$ and $\overline{b}$.
## 3\. The Tree Analysis of $e_{\gamma}^{*}$ for $\gamma\in\Gamma$
We begin by recalling the analysis of $e_{\gamma}^{*}$ in [3] section 4. The
only difference is that in our case all the functionals $e_{\gamma}^{*}$ have
weight depending on their age which is greater or equal to 2.
### 3.1. The evaluation Analysis of $e_{\gamma}^{*}$ for $\gamma\in\Gamma$
First we point out that for $q\in\operatorname{\mathbb{N}}$ every
$\gamma\in\Delta_{q+1}$ admits a unique analysis as follows:
Let $a(\gamma)=a\leq{n}$. Then using backwards induction we determine a
sequence of sets
$\\{p_{i},q_{i},\varepsilon_{i}e_{\xi_{i}}^{*}\\}_{i=1}^{a}\cup\\{\eta_{i}\\}_{i=2}^{a}$
with the following properties.
1. (i)
$p_{1}<q_{1}<\cdots<p_{a}<q_{a}=q$.
2. (ii)
$\varepsilon_{i}=\pm 1,\ \operatorname{rank}\xi_{i}\in(p_{i},q_{i}]$ for
$1\leq i\leq a$ and $\operatorname{rank}\eta_{i}=q_{i}+1$ for $2\leq i\leq a$.
3. (iii)
$\eta_{a}=\gamma$,
$\eta_{i}=(\operatorname{rank}\eta_{i},i,p_{i},\eta_{i-1},\varepsilon_{i}e_{\xi_{i}}^{*})$
for every $i>2$
$\eta_{2}=(\operatorname{rank}\eta_{2},2,p_{2},\varepsilon_{1}\xi_{1},\varepsilon_{2}e_{\xi_{2}}^{*})$
and $(p_{1},q_{1}]=(1,\operatorname{rank}\xi_{1}]$.
###### Definition 3.1.
Let $q\in\operatorname{\mathbb{N}}$ and $\gamma\in\Gamma_{q}$. Then the
sequence
$\\{p_{i},q_{i},\varepsilon_{i}e_{\xi_{i}}^{*}\\}_{i=1}^{a}\cup\\{\eta_{i}\\}_{i=2}^{a}$
satisfying all the above properties will be called the analysis of $\gamma$.
Moreover, following similar arguments as in [3] Proposition 4.6 it holds that,
$e_{\gamma}^{*}=\sum_{i=2}^{a}d_{\eta_{i}}^{*}+\sum_{i=1}^{a}b_{i}\varepsilon_{i}e_{\xi_{i}}^{*}\circ
P_{(p_{i},q_{i}]}=\sum_{i=2}^{a}e_{\eta_{i}}^{*}\circ
P_{\\{q_{i}+1\\}}+\sum_{i=1}^{a}b_{i}\varepsilon_{i}e_{\xi_{i}}^{*}\circ
P_{(p_{i},q_{i}]}.$
We set $g_{\gamma}=\sum\limits_{i=2}^{a}d_{\eta_{i}}^{*}$ and
$f_{\gamma}=\sum\limits_{i=1}^{a}b_{i}\varepsilon_{i}e_{\xi_{i}}^{*}\circ
P_{(p_{i},q_{i}]}$.
### 3.2. The r-Analysis of the functional $e_{\gamma}^{*}$
Let $r\in{\operatorname{\mathbb{N}}}$ and $\gamma\in\Delta_{q+1}$.Let
$a(\gamma)=a\leq{n}$ and
$\\{p_{i},q_{i},\varepsilon_{i}e_{\xi_{i}}^{*}\\}_{i=1}^{a}\bigcup\\{\eta_{i}\\}_{i=2}^{a}$
the evaluation analysis of $\gamma$. We define the r-analysis of
$e_{\gamma}^{*}$ as follows:
1. (a)
If $r\leq{p_{1}}$, then the r-analysis of $e_{\gamma}^{*}$ coincides with the
evaluation analysis of $e_{\gamma}^{*}$.
2. (b)
If $r\geq{q_{a}}$, then we assign no r-analysis to $e_{\gamma}^{*}$ and we say
that $e_{\gamma}^{*}$ is r-indecomposable.
3. (c)
If $p_{1}<r<q_{a}$, we define $i_{r}=\min\\{i:r<q_{i}\\}$. Note that this is
well-defined. The r-analysis of $e_{\gamma}^{*}$ is the following triplet
$\\{(p_{i},q_{i}]\\}_{i\geq i_{r}},\\{\varepsilon_{i}\xi_{i}\\}_{i\geq
i_{r}},\\{\eta_{i}\\}_{i\geq\max\\{2,i_{r}\\}}.$
where $p_{i_{r}}$ is either the same or $r$ in the case that $r>p_{i_{r}}$.
Next we introduce the tree analysis of $e_{\gamma}^{*}$ which is similar to
the tree analysis of a functional in a Mixed Tsirelson space (see [4] Chapter
II.1). Notice that the evaluation analysis and the r-analysis of
$e_{\gamma}^{*}$ form the first level of the tree analysis that we are about
to present.
We start with some notation. We denote by
$(\operatorname{\mathcal{T}},"\preceq")$ a finite partially ordered set which
is a tree. Its elements are finite sequences of natural numbers ordered by the
initial segment partial order. For every $t\in\operatorname{\mathcal{T}}$,we
denote by $S_{t}$ the immediate successors of $t$
Assume now that $(p_{t},q_{t}]_{t\in\operatorname{\mathcal{T}}}$ is a tree of
intervals of $\operatorname{\mathbb{N}}$ such that $t\preceq s$ iff
$(p_{t},q_{t}]\supset(p_{s},q_{s}]$ and $t,s$ are incomparable iff
$(p_{t},q_{t}]\cap(p_{s},q_{s}]=\emptyset$. For such a family
$(p_{t},q_{t}]_{t\in\operatorname{\mathcal{T}}}$ and $t,s$ incomparable we
shall denote by $t<s$ iff $(p_{t},q_{t}]<(p_{s},q_{s}]$ (i.e. $q_{t}<p_{s}$).
### 3.3. The Tree Analysis of the functional $e_{\gamma}^{*}$
Let $\gamma\in\Delta_{q+1}$ with $a(\gamma)=a\leq{n}$. A family of the form
$\mathcal{F}_{\gamma}=\\{\xi_{t},(p_{t},q_{t}]\\}_{t\in\operatorname{\mathcal{T}}}$
is called the tree analysis of $e_{\gamma}^{*}$ if the following are
satisfied:
1. (1)
$\operatorname{\mathcal{T}}$ is a finite tree with a unique root denoted as
$\emptyset$.
2. (2)
We set $\xi_{\emptyset}=\gamma$,$(p_{\emptyset},q_{\emptyset}]=(1,q]$ and let
$\\{p_{i},q_{i},\varepsilon_{i}e_{\xi_{i}}^{*}\\}_{i=1}^{a}\bigcup\\{\eta_{i}\\}_{i=2}^{a}$
the evaluation analysis of $\xi_{\emptyset}$. Set
$S_{\emptyset}=\\{(1),(2),\ldots,(a)\\}$ and for every $s=(i)\in
S_{\emptyset}$, $\\{\xi_{s},(p_{s},q_{s}]\\}=\\{\xi_{i},(p_{i},q_{i}]\\}$.
3. (3)
Assume that for a $t\in\operatorname{\mathcal{T}}$
$\\{\xi_{t},(p_{t},q_{t}]\\}$ has been defined. There are two cases:
1. (a)
If $e_{\xi_{t}}^{*}$ is $p_{t}$-decomposable, let
$\\{(p_{i},q_{i}]\\}_{i\geq i_{p_{t}}},\\{\varepsilon_{i}\xi_{i}\\}_{i\geq
i_{p_{t}}},\\{\eta_{i}\\}_{i\geq\max\\{2,i_{p_{t}}\\}}$
the $p_{t}$ analysis of $e_{\xi_{t}}^{*}$. We set
$S_{t}=\\{(t^{\smallfrown}i):i\geq i_{p_{t}}\\}$ and
$S_{t}^{p_{t}}=\begin{cases}S_{t},\ \text{if }\eta_{i_{p_{t}}}\
\text{exists}\\\ S_{t}\diagdown\\{(t^{\smallfrown}i_{p_{t}})\\},\
\text{otherwise}\end{cases}$
Then, for every $s=(t^{\smallfrown}i)\in S_{t}$, we set
$\\{\xi_{s},(p_{s},q_{s}]\\}=\\{\xi_{i},(p_{i},q_{i}]\\}$ where
$\\{\varepsilon_{i}\xi_{i},(p_{i},q_{i}]\\}$ is a member of the $p_{t}$
analysis of $e_{\xi_{t}}^{*}$.
2. (b)
$e_{\xi_{t}}^{*}$ is $p_{t}$-indecomposable, then $\xi_{t}$ consists a maximal
node of $\operatorname{\mathcal{F}}_{\gamma}$.
###### Notation 3.2.
For later use we need the following:
For every $t\in\operatorname{\mathcal{T}}$ $e_{\xi_{t}}^{*}=f_{t}+g_{t}$,
where $f_{t}=\sum_{s\in S_{t}}b_{s}\varepsilon_{s}e_{\xi_{s}}^{*}\circ
P_{(p_{s},q_{s}]}$ and $g_{t}=\sum_{s\in S_{t}^{p_{t}}}d_{\eta_{s}}^{*}$ and
for $s=(t^{\smallfrown}i)\in S_{t}^{p_{t}}$,
$\eta_{(t^{\smallfrown}i)}=(\operatorname{rank}\eta_{(t^{\smallfrown}i)},i,p_{(t^{\smallfrown}i)},\eta_{(t^{\smallfrown}i-1)},\varepsilon_{(t^{\smallfrown}i)}e_{\xi_{(t^{\smallfrown}i)}}^{*})$.
In the rest of the paper, we set $f_{t}=f_{\xi_{t}}$ and $g_{t}=g_{t}$.
###### Lemma 3.3.
Let $x\in\mathfrak{X}_{r}$ and $\gamma\in\Gamma$. Then,
$e_{\gamma}^{*}(x)=\prod_{\emptyset\preceq s\preceq
t_{x}}(\varepsilon_{s}b_{s})(f_{t_{x}}+g_{t_{x}})(x),$
where $t_{x}=\max\\{t:\operatorname{ran}x\subseteq(p_{t},q_{t}]\\}$.
###### Proof.
Let
$\operatorname{\mathcal{F}}_{\gamma}=\\{\xi_{t},(p_{t},q_{t}]\\}_{t\in\operatorname{\mathcal{T}}}$
a tree analysis of $\gamma$.
If $\\{t:\operatorname{ran}x\subseteq(p_{t},q_{t}]\\}=\emptyset$, then
$e_{\gamma}^{*}(x)=f_{\emptyset}(x)+g_{\emptyset}(x)$ and the equality holds.
If $\\{t:\operatorname{ran}x\subseteq(p_{t},q_{t}]\\}\neq\emptyset$, we can
find $\\{t_{1}\prec t_{2}\prec\ldots\prec
t_{m}\\}\in\operatorname{\mathcal{T}}$ such that $t_{1}\in S_{\emptyset}$ and
$t_{m}=t_{x}$.
For every $t\in\operatorname{\mathcal{T}}$ with $t\prec t_{x}$, $g_{t}(x)=0$.
Indeed, for every $s\in S_{t}^{p_{t}}$,
$d_{\eta_{s}}^{*}(x)=e_{\eta_{s}}^{*}\circ P_{\\{q_{s}+1\\}}(x)=0$ because
$\operatorname{ran}x\subseteq(p_{t_{x}},q_{t_{x}}]\subseteq(p_{s},q_{s}]$.
So, we have that
$\displaystyle e_{\gamma}^{*}(x)$ $\displaystyle=$ $\displaystyle
f_{\emptyset}(x)=\sum_{s\in
S_{\emptyset}}b_{s}\varepsilon_{s}e_{\xi_{s}}^{*}\circ
P_{(p_{s},q_{s}]}(x)=b_{t_{1}}\varepsilon_{t_{1}}e_{\xi_{t_{1}}}^{*}(x)$
$\displaystyle=$ $\displaystyle
b_{t_{1}}\varepsilon_{t_{1}}f_{t_{1}}(x)=b_{t_{1}}\varepsilon_{t_{1}}b_{t_{2}}\varepsilon_{t_{2}}e_{\xi_{t_{2}}}^{*}\circ
P_{(p_{t_{2}},q_{t_{2}}]}(x)=b_{t_{1}}b_{t_{2}}\varepsilon_{t_{1}}\varepsilon_{t_{2}}e_{\xi_{t_{2}}}^{*}(x)$
$\displaystyle=$ $\displaystyle
b_{t_{1}}b_{t_{2}}\varepsilon_{t_{1}}\varepsilon_{t_{2}}f_{t_{2}}(x)=\ldots=\prod_{\emptyset\preceq
s\preceq t_{x}}(\varepsilon_{s}b_{s})(f_{t_{x}}+g_{t_{x}})(x)$
setting $\varepsilon_{\emptyset}=b_{\emptyset}=1$. ∎
###### Corollary 3.4.
If $(f_{t_{x}},(p_{t_{x}},q_{t_{x}}])$ is a maximal node, then
$e_{\gamma}^{*}(x)=0$.
###### Proof.
Let $(f_{t_{x}},(p_{t_{x}},q_{t_{x}}])$ be a maximal node. Then
$f_{t_{x}}(x)=0$ and $g_{t_{x}}(x)=0$ and from Lemma 3.3 we deduce that
$e_{\gamma}^{*}(x)=0$. ∎
## 4\. The lower estimate
###### Definition 4.1.
An $\phi\in W(\operatorname{\mathcal{A}}_{n},\overline{b})$ is said to be a
proper functional if it admits a tree analysis
$(\phi_{t})_{t\in\operatorname{\mathcal{T}}}$ such that for every non-maximal
node $t\in\operatorname{\mathcal{T}}$ the set $\\{\phi_{s}:s\in S_{t}\\}$ has
at least two non-zero elements.
We denote by $W_{pr}(\operatorname{\mathcal{A}}_{n},\overline{b})$ to be the
subset of $W(\operatorname{\mathcal{A}}_{n},\overline{b})$ consisting of all
proper functionals. For every $t\in\operatorname{\mathcal{T}}$ it holds that
$\phi_{t}=\sum_{s\in S_{t}}b_{s}\phi_{s}$ with $\\{b_{s}\\}_{s\in
S_{t}}\subseteq\\{b_{1},b_{2},\ldots,b_{n}\\}$ and $b_{\emptyset}=1$.
###### Lemma 4.2.
The set $W_{pr}(\operatorname{\mathcal{A}}_{n},\overline{b})$ 1-norms the
space
$\operatorname{\mathcal{T}}(\operatorname{\mathcal{A}}_{n},\overline{b})$.
###### Proof.
We shall show that for every $\phi\in
W(\operatorname{\mathcal{A}}_{n},\overline{b})$ there exists $g\in
W_{pr}(\operatorname{\mathcal{A}}_{n},\overline{b})$ such that $|\phi(m)|\leq
g(m)\ \forall m\in{\operatorname{\mathbb{N}}}$. Since the basis is
1-unconditional the previous statement yields the result.
To this end, let $\phi\in W(\operatorname{\mathcal{A}}_{n},\overline{b})$.
Then using a tree analysis $\\{\phi_{t}\\}_{t\in\operatorname{\mathcal{T}}}$
of $\phi$ we easily see that for every $m\in\operatorname{supp}f$, there
exists a maximal node $t_{m}\in\operatorname{\mathcal{T}}$ with
$\phi_{t_{m}}=\varepsilon_{m}e_{m}^{*}$ and
$\phi(m)=\varepsilon_{m}\prod\limits_{t<t_{m}}b_{t}$.
For every $m\in\operatorname{supp}\phi$ we set
$K_{m}=\\{t\in\operatorname{\mathcal{T}}:t<t_{m}\ \text{and}\ \\#S_{t}>1\\}$.
Then it is easy to see that the functional
$g=\sum\limits_{m\in\operatorname{supp}\phi}(\prod\limits_{t\in
K_{m}}b_{t})e_{m}^{*}$ is a functional belonging to
$W_{pr}(\operatorname{\mathcal{A}}_{n},\overline{b})$. Moreover, since
$b_{t}<1$ for every $t\in\operatorname{\mathcal{T}}$ we get that
$|\phi(m)|\leq g(m)\ \forall m\in{\operatorname{\mathbb{N}}}$. ∎
###### Lemma 4.3.
Let $\phi\in W_{pr}(\operatorname{\mathcal{A}}_{n},\overline{b})$ and
$l\in\operatorname{\mathbb{N}}$. If $\operatorname{maxsupp}\phi=l$, then
$h(\operatorname{\mathcal{T}}_{\phi})\leq l$.
###### Proof.
Let $\theta_{n}$ be the amount of nodes at the $n$ level of
$\operatorname{\mathcal{T}}_{\phi}$. Since $\phi$ is proper, it holds that
$\theta_{n+1}>\theta_{n}$ for every $n\in\operatorname{\mathbb{N}}$. Assume to
the contrary that $h(\operatorname{\mathcal{T}}_{\phi})>l$, i.e.
$h(\operatorname{\mathcal{T}}_{\phi})=l+k$ for some
$k\in\operatorname{\mathbb{N}}$. Then,
$\theta_{1}=1,\ \theta_{2}\geq 2,\ \ldots\ ,\ \theta_{l+k}\geq l+k$
Since, the $l+k$ level of $\operatorname{\mathcal{T}}_{\phi}$ consists of
functionals of the form $e_{i}^{*}$, we deduce that
$\operatorname{maxsupp}\phi\geq l+k>l$, which leads to a contradiction. ∎
###### Proposition 4.4.
Let $(x_{k})_{k\in\operatorname{\mathbb{N}}}$ be a normalized skipped block
sequence in $\mathfrak{X}_{r}$ and $(q_{k})_{k\in\operatorname{\mathbb{N}}}$ a
strictly increasing sequence of integers such that
$\operatorname{supp}x_{k}\subset(q_{k}+k,q_{k+1})$. Then, for every sequence
of positive scalars $(a_{k})_{k\in\operatorname{\mathbb{N}}}$ and for every
$l\in\operatorname{\mathbb{N}}$, it holds that
(1)
$\|\sum_{k=1}^{l}a_{k}e_{k}\|_{\operatorname{\mathcal{T}}(\operatorname{\mathcal{A}}_{n},\overline{b})}\leq
C\|\sum_{k=1}^{l}a_{k}x_{k}\|_{\infty}$
where
$(e_{k})_{k\in\operatorname{\mathbb{N}}}\subseteq\operatorname{\mathcal{T}}(\operatorname{\mathcal{A}}_{n},\overline{b})$
and C is an upper bound for the norms of the operators $i_{m}$ in
$\mathfrak{X}_{r}$.
###### Proof.
Let $\phi\in W(\operatorname{\mathcal{A}}_{n},\overline{b})$. From Lemma 4.2
we may assume that $\phi$ is proper. We will use induction on the height of
the tree $\operatorname{\mathcal{T}}_{\phi}$.
If $h(\operatorname{\mathcal{T}}_{\phi})=0$ (i.e. $f$ is maximal), then $\phi$
is of the form $\phi=\varepsilon_{k}e_{k}^{*}$ with $\varepsilon_{k}=\pm 1$.
We observe that, $|\phi(\sum_{k=1}^{l}a_{k}e_{k})|=|a_{k}|=a_{k}$. From [3]
Proposition 4.8, we can choose
$\gamma\in\Gamma_{q_{k+1}-1}\backslash\Gamma_{q_{k}+k}$ such that
$|x_{k}(\gamma)|\geq\frac{1}{C}\|x_{k}\|=\frac{1}{C}$. Then,
$|\phi(\sum_{k=1}^{l}a_{k}e_{k})|=a_{k}\leq
C|a_{k}||x_{k}(\gamma)|=C|e_{\gamma}^{*}(a_{k}x_{k})|\leq
C|e_{\gamma}^{*}(\sum_{k=1}^{l}a_{k}x_{k})|$.
We assume that for every $\phi\in
W(\operatorname{\mathcal{A}}_{n},\overline{b})$ with
$h(\operatorname{\mathcal{T}}_{\phi})=h>0$ and
$\operatorname{maxsupp}\phi=l_{0}$, there exists $\gamma\in\Gamma$, such that:
1. (1)
$\gamma\in\Gamma_{q_{l_{0}+1}+h}\backslash\Gamma_{q_{l_{0}+1}}$
2. (2)
$h(\operatorname{\mathcal{T}}_{\phi})=h(\operatorname{\mathcal{F}}_{\gamma})\leq
l_{0}$
3. (3)
$|\phi(\sum_{k=1}^{l}a_{k}e_{k})|\leq C|\sum_{k=1}^{l}a_{k}x_{k}(\gamma)|$ for
every $l\geq l_{0}$
Observe that assumption (1) yields
$x_{l_{0}}<\operatorname{rank}\gamma<x_{l_{0}+1}$, while assumption (2) gives
us that
$\operatorname{minsupp}x_{l_{0}+1}-\operatorname{maxsupp}x_{l_{0}}>h(\operatorname{\mathcal{T}}_{\phi})$.
Indeed,
$x_{l_{0}}<q_{l_{0}+1}<\operatorname{rank}\gamma\leq q_{l_{0}+1}+h\leq
q_{l_{0}+1}+l_{0}<q_{l_{0}+1}+(l_{0}+1)<x_{l_{0}+1}$ $\text{and
}\operatorname{minsupp}x_{l_{0}+1}-\operatorname{maxsupp}x_{l_{0}}>l_{0}+1>l_{0}\geq
h(\operatorname{\mathcal{F}}_{\gamma}).$
Let $\phi\in W(\operatorname{\mathcal{A}}_{n},\overline{b})$ with
$h(\operatorname{\mathcal{T}}_{\phi})=h+1$, $l_{0}=\operatorname{maxsupp}\phi$
and let $(\phi_{t})_{t\in\operatorname{\mathcal{T}}}$ the tree analysis of
$\phi$. Then, $\phi$ is of the form $\phi=\sum_{s\in
S_{\emptyset}}b_{s}\phi_{s}$, $\\#S_{\emptyset}\leq n$. We observe that for
every $s\in S_{\emptyset}$, $h(\operatorname{\mathcal{T}}_{\phi_{s}})=h$. We
set $p_{1}=1$, for every $s\in S_{\emptyset}\diagdown\\{1\\}$
$p_{s}=\min\\{q_{k}+k:k\in\operatorname{supp}\phi_{s}\\}$ and for every $s\in
S_{\emptyset}$, $r_{s}=q_{l_{s}+1}+h$ where
$l_{s}=\operatorname{maxsupp}\phi_{s}$.
We next apply the inductive hypothesis to obtain
$\xi_{s}\in\Gamma_{r_{s}}\backslash\Gamma_{q_{l_{s}}+1}$ with
$h(\operatorname{\mathcal{T}}_{\phi_{s}})=h(\operatorname{\mathcal{F}}_{\xi_{s}})$
such that
$\displaystyle|\phi_{s}(\sum_{k=1}^{l}a_{k}e_{k})|$ $\displaystyle=$
$\displaystyle|\phi_{s}(\sum_{k\in\operatorname{supp}\phi_{s}}a_{k}e_{k})|\leq
C\varepsilon_{s}\sum_{k\in\operatorname{supp}\phi_{s}}a_{k}x_{k}(\xi_{s})$
$\displaystyle=$ $\displaystyle
C\varepsilon_{s}e_{\xi_{s}}^{*}(\sum_{k\in\operatorname{supp}\phi_{s}}a_{k}x_{k})=C\varepsilon_{s}e_{\xi_{s}}^{*}\circ
P_{(p_{s},r_{s}]}(\sum_{k=1}^{l}a_{k}x_{k}),$
with $\varepsilon_{s}$ such that
$\varepsilon_{s}e_{\xi_{s}}^{*}(\sum_{k\in\operatorname{supp}\phi_{s}}a_{k}x_{k})=|\sum_{k\in\operatorname{supp}\phi_{s}}a_{k}x_{k}(\xi_{s})|$.
Let $\gamma\in\Gamma$ have analysis
$\\{p_{s},r_{s},\varepsilon_{s}e_{\xi_{s}}^{*}\\}_{s\in
S_{\emptyset}}\bigcup\\{\eta_{s}\\}_{s\in S_{\emptyset}\diagdown\\{1\\}}$
where $\eta_{s}\in\Delta_{r_{s}+1}$. Observe that
$\operatorname{rank}\xi_{s}\in{(q_{l_{s}+1},r_{s}]}\subset{(p_{s},r_{s}]}$. It
is clear that for every $s\in S_{\emptyset}\diagdown\\{1\\}$,
$d_{\eta_{s}}^{*}(\sum_{k=1}^{l}a_{k}x_{k})=0$. Indeed,
$\operatorname{supp}x_{l_{s}}<q_{l_{s}+1}<q_{l_{s}+1}+{(h+1)}=r_{s}+1\leq
q_{l_{s}+1}+(l_{s}+1)<\operatorname{supp}x_{l_{s}+1}.$
Therefore,
$\displaystyle|\phi(\sum_{k=1}^{l}a_{k}e_{k})|$ $\displaystyle\leq$
$\displaystyle\sum_{s\in
S_{\emptyset}}|b_{s}\phi_{s}(\sum_{k\in\operatorname{supp}\phi_{s}}a_{k}e_{k})|\leq
C\sum_{s\in S_{\emptyset}}b_{s}\varepsilon_{s}e_{\xi_{s}}^{*}\circ
P_{(p_{s},r_{s}]}(\sum_{k=1}^{l}a_{k}x_{k})$ $\displaystyle\leq$
$\displaystyle C|\sum_{k=1}^{l}a_{k}x_{k}(\gamma)|$
It is clear that
$h(\operatorname{\mathcal{T}}_{\phi})=h(\operatorname{\mathcal{F}}_{\gamma})\leq
l_{0}$ and $x_{l_{0}}<\operatorname{rank}\gamma<x_{l_{0}+1}$. ∎
###### Corollary 4.5.
For every block sequence in $\mathfrak{X}_{r}$ there exists a further block
sequence satisfying inequality (1).
## 5\. The upper estimate
Let $(y_{l})_{l\in\operatorname{\mathbb{N}}}$ be a normalized skipped block
sequence in $\mathfrak{X}_{r}$. From Corollary 4.5, we can find a further
block sequence of $(y_{l})_{l}$, still denoted by $(y_{l})_{l}$, satisfying
inequality (1).
Therefore, we have that
$\|\sum_{l=1}^{m}y_{l}\|_{\infty}\geq\frac{1}{C}\|\sum_{l=1}^{m}e_{l}\|_{\operatorname{\mathcal{T}}(\operatorname{\mathcal{A}}_{n},\overline{b})}$
For every $j\in\operatorname{\mathbb{N}}$, set $M_{j}=\\{1,2,\ldots,n\\}^{j}$.
It is easily checked, after identifying $M_{j}$ with $\\{1,\dots,n^{j}\\}$ for
every $j$, that the functional $f_{j}=\sum_{s\in
M_{j}}(\prod_{i=1}^{j}b_{s_{i}})e_{s}^{*}$ belongs to
$W(\operatorname{\mathcal{A}}_{n},\overline{b})$ where $s_{i}$ is the $i$-th
coordinate of $s$, for each $i=1,2,\ldots,n$ and $\sum_{s\in
M_{j}}\prod_{i=1}^{j}b_{s_{i}}=(\sum_{i=1}^{n}b_{i})^{j}$. Using the fact that
$\\#M_{j}=n^{j}$, we obtain that
$\|\sum_{l=1}^{n^{j}}e_{l}\|_{\operatorname{\mathcal{T}}(\operatorname{\mathcal{A}}_{n},\overline{b})}=\|\sum_{s\in
M_{j}}e_{s}\|_{\operatorname{\mathcal{T}}(\operatorname{\mathcal{A}}_{n},\overline{b})}\geq
f_{j}(\sum_{l=1}^{n^{j}}e_{l})=(\sum_{i=1}^{n}b_{i})^{j}$.
Also, for every $m\in\operatorname{\mathbb{N}}$ large enough we may find
$j\in\operatorname{\mathbb{N}}$ such that $n^{j+1}>m\geq n^{j}$. From the
above and the unconditionality of the basis of the space
$\operatorname{\mathcal{T}}(\operatorname{\mathcal{A}}_{n},\overline{b})$, it
follows that
$\|\sum_{l=1}^{m}y_{l}\|_{\infty}\geq\frac{1}{C}\|\sum_{l=1}^{m}e_{l}\|_{\operatorname{\mathcal{T}}(\operatorname{\mathcal{A}}_{n},\overline{b})}\geq\frac{1}{C}\|\sum_{l=1}^{n^{j}}e_{l}\|_{\operatorname{\mathcal{T}}(\operatorname{\mathcal{A}}_{n},\overline{b})}=(\sum_{i=1}^{n}b_{i})^{j}$
We conclude that $\|\sum_{l=1}^{m}y_{l}\|_{\infty}\stackrel{{\scriptstyle
m\to\infty}}{{\longrightarrow}}\infty$ as $\sum_{i=1}^{n}b_{i}>1$.
We next choose a further block sequence
$(x_{k})_{k\in\operatorname{\mathbb{N}}}$ of
$(y_{l})_{l\in\operatorname{\mathbb{N}}}$ with some additional properties. Let
$\varepsilon>0$ and choose a descending sequence $(\varepsilon_{k})_{k}$ of
positive reals such that $(\sum_{k=1}^{\infty}\varepsilon_{k})<\varepsilon$.
We can also find an increasing sequence $(n_{k})_{k}$ of positive integers and
a sequence $(F_{k})_{k}$ of succesive subsets of $\operatorname{\mathbb{N}}$
such that the following are satisfied:
1. (1)
For every $k\in\operatorname{\mathbb{N}}$, $\frac{1}{n_{k}}<\varepsilon_{k}$.
2. (2)
For every $k\in\operatorname{\mathbb{N}}$, $\|\sum_{l\in F_{k}}y_{l}\|>n_{k}$.
This is possible, due to the above notation.
We have thus constructed a normalized skipped block sequence
$(x_{k})_{k\in\operatorname{\mathbb{N}}}$ of the form $x_{k}=\sum_{l\in
F_{k}}\lambda_{l}y_{l}$, where $\lambda_{l}=\frac{1}{\|\sum_{l\in
F_{k}}y_{l}\|}$. Notice that $|\lambda_{l}|<\varepsilon_{k}$ for every $l\in
F_{k}$.
Let $\gamma\in\Gamma$ with tree analysis
$\operatorname{\mathcal{F}}_{\gamma}=\\{\xi_{t},(p_{t},q_{t}]\\}_{t\in\operatorname{\mathcal{T}}}$.
For every $k\in\operatorname{\mathbb{N}}$, we set
$t_{k}=\max\\{t:\operatorname{ran}x_{k}\subset(p_{t},q_{t}]\\}$. Notice that
if for a given $x_{k}$, $t_{k}$ is non-maximal, then there exist at least two
immediate successors of $t_{k}$, say $s_{1}$, $s_{2}$ such that the
corresponding intervals $(p_{s_{1}},q_{s_{1}}]$, $(p_{s_{2}},q_{s_{2}}]$
intersect $\operatorname{ran}x_{k}$. For later use we shall denote by
$(p_{s_{0}},q_{s_{0}}]$ the first interval in the natural order of disjoint
segments of the natural numbers that intersects $x_{k}$. Notice that $s_{0}$
is not necessarily the first element of $S_{t}$.
For the pair $\gamma$, $(x_{k})_{k\in\operatorname{\mathbb{N}}}$ and for every
$t\in\operatorname{\mathcal{T}}$ we define the following sets:
$D_{t}=\bigcup_{s\succeq t}\\{k:s=t_{k}\\}$, $K_{t}=D_{t}\backslash\cup_{s\in
S_{t}}D_{s}\ =\\{k:t=t_{k}\\}$ and $E_{t}=\\{s\in
S_{t}:D_{s}\neq\emptyset\\}$.
We now set
$x_{k}=x^{\prime}_{k}+x^{\prime\prime}_{k}+x^{\prime\prime\prime}_{k}$ where,
$x^{\prime}_{k}=x_{k}\mid_{(p_{s_{0}},q_{s_{0}}]},\ \
x^{\prime\prime}_{k}=x_{k}\mid_{\bigcup_{s\in S_{t_{k}},s\neq
s_{0}}(p_{s},q_{s}]}\ \text{and
}x^{\prime\prime\prime}_{k}=x_{k}-x^{\prime}_{k}-x^{\prime\prime}_{k}.$
###### Remark 5.1.
1. (1)
The sets $D_{t}$,$K_{t}$,$E_{t}$ are determined by the chosen pair
$\gamma,(x_{k})_{k}$. For a different pair, these sets may differ as well. For
example, let $k\in K_{t}$, for the pair $\gamma,(x_{k})_{k}$. Then $t=t_{k}$
for $x_{k}$. By the construction of $x^{\prime}_{k}$, there exists $s_{k}\in
S_{t}$ such that $x^{\prime}_{k}=x_{k}\mid_{(p_{s_{k}},q_{s_{k}}]}$. Thus,
taking the pair $\gamma,(x^{\prime}_{k})_{k}$ the same $k$ belongs to
$K_{s_{k}}$.
2. (2)
For every $k\in\operatorname{\mathbb{N}}$, $|g_{t_{k}}(x_{k})|\leq
2Cn\varepsilon_{k}$.
Indeed, from the definition of $(x_{k})_{k\in\operatorname{\mathbb{N}}}$ we
have that
$\displaystyle|g_{t_{k}}(x_{k})|$ $\displaystyle\leq$ $\displaystyle\sum_{s\in
S_{t_{k}}^{p_{t_{k}}}}|d_{\eta_{s}}^{*}(x_{k})|\leq\sum_{s\in
S_{t_{k}}^{p_{t_{k}}}}|e_{\eta_{s}}^{*}\circ P_{\\{q_{s}+1\\}}(\sum_{l\in
F_{k}}\lambda_{l}y_{l})|\leq$ $\displaystyle\leq$ $\displaystyle\sum_{s\in
S_{t_{k}}^{p_{t_{k}}}}\|e_{\eta_{s}}^{*}\|\|P_{\\{q_{s}+1\\}}\||\lambda_{l}^{s}|\|y_{l}^{s}\|\leq\sum_{s\in
S_{t_{k}}^{p_{t_{k}}}}2C\varepsilon_{k}\leq$ $\displaystyle\leq$
$\displaystyle 2C\varepsilon_{k}(\sharp S_{t_{k}})\leq 2Cn\varepsilon_{k}.$
3. (3)
It is obvious that $g_{t_{k}}(x_{k})=g_{t_{k}}(x^{\prime\prime\prime}_{k})$,
$f_{t_{k}}(x^{\prime\prime\prime}_{k})=0$ and for every $t\prec t_{k}$,
$g_{t}(x^{\prime\prime\prime}_{k})=0$.
###### Lemma 5.2.
For the pairs $\gamma,(x^{\prime}_{k})_{k\in\operatorname{\mathbb{N}}}$ and
$\gamma,(x^{\prime\prime}_{k})_{k\in\operatorname{\mathbb{N}}}$ it holds that
$\\#(K_{t}\cup E_{t})\leq n$.
###### Proof.
Let $t\in\operatorname{\mathcal{T}}$ and let $k\in K_{t}$.
We set $s_{k}=\max\\{s\in
S_{t}:(p_{s},q_{s}]\cap\operatorname{ran}x^{\prime}_{k}\neq\emptyset\\}$. From
the definition of $t_{k}$, notice that $\\#S_{t}\geq 2$. It holds that
$s_{k}\not\in E_{t}$.
Indeed, from the definition of $t_{k}$, $s_{k}$ we have that
$(p_{t_{k}},q_{t_{k}}]\cap\operatorname{ran}x^{\prime}_{k}=\operatorname{ran}x^{\prime}_{k}$
and
$(p_{s_{k}},q_{s_{k}}]\cap\operatorname{ran}x^{\prime}_{k}=(p_{s_{k}},q_{s_{k}}]$.
Since $s_{k}\in S_{t_{k}}$,
$(p_{s_{k}},q_{s_{k}}]\subseteq(p_{t_{k}},q_{t_{k}}]$. It follows that
$(p_{s_{k}},q_{s_{k}}]\subseteq\operatorname{ran}x^{\prime}_{k}$.
Therefore, we can define a one-to-one map $G:K_{t}\rightarrow S_{t}\backslash
E_{t}$, hence $\\#K_{t}+\\#E_{t}\leq\\#S_{t}\leq n$.
The proof for the pair
$\gamma,(x^{\prime\prime}_{k})_{k\in\operatorname{\mathbb{N}}}$ is similar. ∎
###### Proposition 5.3.
Let $(x_{k})_{k\in\operatorname{\mathbb{N}}}$ be as above.Then for every
$\gamma\in\Gamma$ there exist $\phi_{1},\phi_{2}\in
W(\operatorname{\mathcal{A}}_{n},\overline{b})$ such that for every sequence
$(a_{k})_{k\in\operatorname{\mathbb{N}}}$ of positive scalars, for every
$l\in\operatorname{\mathbb{N}}$ it holds that,
(2)
$|\sum_{k=1}^{l}a_{k}x_{k}(\gamma)|\leq\frac{1}{b_{n}}(\phi_{1}+\phi_{2})(\sum_{k=1}^{l}a_{k}e_{k})+2Cn\varepsilon(\sum_{k=1}^{l}a_{k}^{r})^{\frac{1}{r}}$
###### Proof.
Let $\gamma\in\Delta_{q+1}witha(\gamma)=a\leq n$. Let
$\operatorname{\mathcal{F}}_{\gamma}=\\{\xi_{t},(p_{t},q_{t}]\\}_{t\in\operatorname{\mathcal{T}}}$,
where $\xi_{\emptyset}=\gamma$, be the tree analysis of $\gamma$. We may
assume that
$\bigcup_{k=1}^{l}\operatorname{ran}x_{k}\subset(p_{\emptyset},q_{\emptyset}]$.
###### Claim.
For the pairs $\gamma,(x^{\prime}_{k})_{k\in\operatorname{\mathbb{N}}}$ and
$\gamma,(x^{\prime\prime}_{k})_{k\in\operatorname{\mathbb{N}}}$ there exist
$\phi_{1},\phi_{2}\in W(\operatorname{\mathcal{A}}_{n},\overline{b})$ such
that for every sequence of positive scalars
$(a_{k})_{k\in\operatorname{\mathbb{N}}}$ and for every
$l\in\operatorname{\mathbb{N}}$, it holds that
(3)
$|f_{\emptyset}(\sum_{k=1}^{l}a_{k}x^{\prime}_{k})|\leq\frac{2C}{b_{n}}\phi_{1}(\sum_{k=1}^{l}a_{k}e_{k})$
(4)
$|f_{\emptyset}(\sum_{k=1}^{l}a_{k}x^{\prime\prime}_{k})|\leq\frac{2C}{b_{n}}\phi_{2}(\sum_{k=1}^{l}a_{k}e_{k})$
###### Proof of the Claim.
We only prove inequality 3. The proof of inequality 4 requires the same
arguments. We recall that $f_{t}=\sum_{s\in
S_{t}}b_{s}\varepsilon_{s}(f_{s}+g_{s})\circ P_{(p_{s},q_{s}]}$ for every
$t\in\operatorname{\mathcal{T}}$ non maximal. From the definition of
$(x^{\prime}_{k})_{k\in\operatorname{\mathbb{N}}}$, we have that $g_{s}\circ
P_{(p_{s},q_{s}]}(x^{\prime}_{k})=0$ for every $s\in S_{t}$.Therefore,
$f_{t}(\sum_{k\in D_{t}}a_{k}x^{\prime}_{k})=(\sum_{s\in
S_{t}}b_{s}\varepsilon_{s}f_{s}\circ P_{(p_{s},q_{s}]})(\sum_{k\in
D_{t}}a_{k}x^{\prime}_{k})$. We will use backwards induction on the levels of
the tree $\operatorname{\mathcal{T}}$, i.e we shall show that for every
$t\in\operatorname{\mathcal{T}}$ there exists $\phi_{1}^{t}\in
W(\operatorname{\mathcal{A}}_{n},\overline{b})$ with
$\operatorname{supp}\phi_{1}^{t}\subseteq D_{t}$ such that
$|f_{t}(\sum_{k\in
D_{t}}a_{k}x^{\prime}_{k})|\leq\frac{2C}{b_{n}}\phi_{1}^{t}(\sum_{k\in
D_{t}}a_{k}e_{k})$
.
Let $0<h\leq\max\\{|t|:t\in\operatorname{\mathcal{T}}\\}$
We assume that the proposition has been proved for all $t$ with $|t|=h$.
Let $t\in\operatorname{\mathcal{T}}$ with $|t|=h-1$.Then we have the following
cases:
1. (1)
If $f_{t}$ is a maximal node, $f_{t}(\sum_{k\in D_{t}}a_{k}x^{\prime}_{k})=0$,
so there is nothing to prove. Indeed, $K=D_{t}$, therefore for every $k\in
D_{t}$, from Corollary 3.4 $f_{t}(x^{\prime}_{k})=0$ since $t=t_{k}$.
2. (2)
If $f_{t}$ is a non-maximal node, then
$f_{t}(\sum_{k\in D_{t}}a_{k}x^{\prime}_{k})=(\sum_{s\in
S_{t}}b_{s}\varepsilon_{s}f_{s}\circ P_{(p_{s},q_{s}]})(\sum_{k\in
D_{t}}a_{k}x^{\prime}_{k})=$
$=\sum_{s\in S_{t}}b_{s}\varepsilon_{s}f_{s}(\sum_{k\in
D_{s}}a_{k}x^{\prime}_{k})+\sum_{k\in K}(\sum_{s\in
S_{t}}b_{s}\varepsilon_{s}f_{s})(a_{k}x^{\prime}_{k})$.
From the fact that, for every $k\in K_{t}$, $g_{t}(x^{\prime}_{k})=0$ we get
that
$|f_{t}(x^{\prime}_{k})|=|x^{\prime}_{k}(\xi_{t})|\leq\|x^{\prime}_{k}\|\leq
2C=2Ce_{k}^{*}(e_{k}).$
Moreover, for $s\in E_{t}$ it holds that $|s|=h-1$. For every $k\in D_{s}$,
from the inductive hypothesis we obtain
$|\sum_{s\in
S_{t}}b_{s}f_{s}(x^{\prime}_{k})|=|b_{s}f_{s}(x^{\prime}_{k})|\leq
b_{s}\frac{2C}{b_{n}}\phi_{1}^{s}(e_{k}).$
with $\phi_{1}^{s}\in W(\operatorname{\mathcal{A}}_{n},\overline{b})$ and
$\operatorname{supp}\phi_{1}^{s}\subseteq D_{s}$.
We set $\phi_{1}^{t}=(\sum_{s\in E_{t}}b_{s}\phi_{1}^{s}+\sum_{k\in
K_{t}}b_{k}e_{k}^{*})$.
From Lemma 5.2, it is easily checked that $\phi_{1}^{t}\in
W(\operatorname{\mathcal{A}}_{n},\overline{b})$ and it holds that,
$|f_{t}(\sum_{k\in
D_{t}}a_{k}x^{\prime}_{k})|\leq\frac{2C}{b_{n}}\phi_{1}^{t}(\sum_{k\in
D_{t}}a_{k}e_{k})$.
∎
Recall that
$e_{\gamma}^{*}(\sum_{k=1}^{l}a_{k}x_{k})=g_{\emptyset}(\sum_{k=1}^{l}a_{k}x_{k})+f_{\emptyset}(\sum_{k=1}^{l}a_{k}x_{k})$.
The fact that
$g_{\emptyset}(\sum_{k=1}^{l}a_{k}x^{\prime}_{k})=g_{\emptyset}(\sum_{k=1}^{l}a_{k}x^{\prime\prime}_{k})=g_{\emptyset}(\sum_{k\in{\\{m:t_{m}\neq\emptyset\\}}}a_{k}x^{\prime\prime\prime}_{k})=f_{\emptyset}(\sum_{k\in{\\{m:t_{m}=\emptyset\\}}}a_{k}x^{\prime\prime\prime}_{k})=0$
implies the following:
$\displaystyle|e_{\gamma}^{*}(\sum_{k=1}^{l}a_{k}x_{k})|$ $\displaystyle\leq$
$\displaystyle|g_{\emptyset}(\sum_{k\in{\\{m:t_{m}=\emptyset\\}}}a_{k}x^{\prime\prime\prime}_{k})|+|f_{\emptyset}(\sum_{k=1}^{l}a_{k}x^{\prime}_{k})|$
$\displaystyle+$
$\displaystyle|f_{\emptyset}(\sum_{k=1}^{l}a_{k}x^{\prime\prime}_{k})|+|f_{\emptyset}(\sum_{k\in{\\{m:t_{m}\neq\emptyset\\}}}a_{k}x^{\prime\prime\prime}_{k})|$
¿From Remark 5.1 we get that,
$|g_{\emptyset}(\sum_{k\in{\\{m:t_{m}=\emptyset\\}}}a_{k}x^{\prime\prime\prime}_{k})|\leq\sum_{k\in{\\{m:t_{m}=\emptyset\\}}}a_{k}|g_{\emptyset}(x^{\prime\prime\prime}_{k})|\leq
2Cn\sum_{k\in{\\{m:t_{m}=\emptyset\\}}}a_{k}\varepsilon_{k}.$
From Lemma 3.3 and Remark 5.1 we have that,
$\displaystyle|f_{\emptyset}(\sum_{k\in{\\{m:t_{m}\neq\emptyset\\}}}a_{k}x^{\prime\prime\prime}_{k})|$
$\displaystyle\leq$
$\displaystyle\sum_{k\in{\\{m:t_{m}\neq\emptyset\\}}}a_{k}(\prod_{t<t_{k}}b_{t})|g_{t_{k}}(x^{\prime\prime\prime}_{k})|\leq$
$\displaystyle\leq$ $\displaystyle
2C\frac{1}{2}n\sum_{k\in{\\{m:t_{m}\neq\emptyset\\}}}a_{k}\varepsilon_{k}\leq
2Cn\sum_{k\in{\\{m:t_{m}\neq\emptyset\\}}}a_{k}\varepsilon_{k}.$
Finally, we conclude that
$\displaystyle|\sum_{k=1}^{l}a_{k}x_{k}(\gamma)|$ $\displaystyle\leq$
$\displaystyle
2Cn\sum_{k\in{\\{m:t_{m}=\emptyset\\}}}a_{k}\varepsilon_{k}+\frac{2C}{b_{n}}\phi_{1}(\sum_{k=1}^{l}a_{k}e_{k})$
$\displaystyle+$
$\displaystyle\frac{2C}{b_{n}}\phi_{2}(\sum_{k=1}^{l}a_{k}e_{k})+2Cn\sum_{k\in{\\{m:t_{m}\neq\emptyset\\}}}a_{k}\varepsilon_{k}$
$\displaystyle\leq$
$\displaystyle\frac{2C}{b_{n}}(\phi_{1}+\phi_{2})(\sum_{k=1}^{l}a_{k}e_{k})+2Cn\sum_{k=1}^{l}a_{k}\varepsilon_{k}$
$\displaystyle\leq$
$\displaystyle\frac{2C}{b_{n}}(\phi_{1}+\phi_{2})(\sum_{k=1}^{l}a_{k}e_{k})+2Cn\max\\{a_{k}:k\in\operatorname{\mathbb{N}}\\}(\sum_{k=1}^{l}\varepsilon_{k})$
$\displaystyle\leq$
$\displaystyle\frac{2C}{b_{n}}(\phi_{1}+\phi_{2})(\sum_{k=1}^{l}a_{k}e_{k})+2Cn\varepsilon(\sum_{k=1}^{l}a_{k}^{r})^{\frac{1}{r}}.$
where in the last inequality we used the fact that the $\ell_{r}$ norm
dominates the $c_{0}$ norm. ∎
###### Remark 5.4.
From [4] Theorem I.4, we know that $\|\sum
a_{k}e_{k}\|_{\operatorname{\mathcal{T}}(\operatorname{\mathcal{A}}_{n},\overline{b})}\geq
M(\sum a_{k}^{r})^{\frac{1}{r}}$. This result and the previous Proposition,
yield that
$|\sum_{k=1}^{l}a_{k}x_{k}(\gamma)|\leq\frac{2C}{b_{n}}(\phi_{1}+\phi_{2})(\sum_{k=1}^{l}a_{k}e_{k})+\frac{2Cn\varepsilon}{M}\|\sum_{k=1}^{l}a_{k}e_{k}\|_{\operatorname{\mathcal{T}}(\operatorname{\mathcal{A}}_{n},\overline{b})}.$
For $\varepsilon=\frac{M}{nb_{n}}$,
$|\sum_{k=1}^{l}a_{k}x_{k}(\gamma)|\leq\frac{6C}{b_{n}}\|\sum_{k=1}^{l}a_{k}e_{k}\|_{\operatorname{\mathcal{T}}(\operatorname{\mathcal{A}}_{n},\overline{b})}.$
Therefore,
(5)
$\|\sum_{k=1}^{l}a_{k}x_{k}\|_{\infty}\leq\frac{6C}{b_{n}}\|\sum_{k=1}^{l}a_{k}e_{k}\|_{\operatorname{\mathcal{T}}(\operatorname{\mathcal{A}}_{n},\overline{b})}.$
###### Corollary 5.5.
For every block sequence in $\mathfrak{X}_{r}$ there exists a further block
sequence satisfying inequality (5).
## 6\. The main result
###### Proposition 6.1.
Let $(x_{k})_{k\in\operatorname{\mathbb{N}}}$ be a skipped block sequence in
$\mathfrak{X}_{r}$ satisfying
$\operatorname{minsupp}x_{k+1}>\operatorname{maxsupp}x_{k}+k$ and the
conditions of Proposition 5.3. Then $(x_{k})_{k\in\operatorname{\mathbb{N}}}$
is equivalent to the basis of the Tsirelson space
$\operatorname{\mathcal{T}}(\operatorname{\mathcal{A}}_{n},\overline{b})$ for
$n$ and $\overline{b}$ determined as before.
###### Proof.
It is an immediate consequence of Propositions 4.4, 5.3 and Remark 5.4. ∎
###### Proposition 6.2.
The space
$\operatorname{\mathcal{T}}(\operatorname{\mathcal{A}}_{n},\overline{b})$ is
isomorphic to $\ell_{p}$ for some $p\in(1,\infty)$.
###### Proof.
In a similar manner as in [4] Theorem I.4, one can see that for every
normalized block sequence $(x_{k})_{k}$ of the basis $(e_{j})_{j}$ and for
every scalar sequence $(a_{k})$ it holds that, $\|\sum
a_{k}x_{k}\|\leq\frac{2}{b_{n}}\|\sum a_{k}e_{k}\|$. Zippin’s Theorem [12]
yields that
$\operatorname{\mathcal{T}}(\operatorname{\mathcal{A}}_{n},\overline{b})$ is
isomorphic to some $\ell_{p}$ for some $p\in(1,\infty)$.∎
###### Remark 6.3.
An alternative proof could also be derived using the Results in Sections 4 and
5. Indeed, let $(y_{l})_{l\in\operatorname{\mathbb{N}}}$ be a skipped block
sequence in $\mathfrak{X}_{r}$. Then, there exists a further block sequence
$(x_{k})_{k\in\operatorname{\mathbb{N}}}$ satisfying simultaneously the
assumptions of Corollaries 4.5 and 5.5. Therefore,
$(x_{k})_{k\in\operatorname{\mathbb{N}}}$ satisfies the assumptions of
Proposition 6.1.
Let’s observe that every further block sequence $(z_{k})_{k}$ of $(x_{k})_{k}$
is also skipped block and satisfies Proposition 6.1, thus it is equivalent to
the basis of the space
$\operatorname{\mathcal{T}}(\operatorname{\mathcal{A}}_{n},\overline{b})$.
Hence, every block sequence $(z_{n})_{n}$ of $(x_{k})_{k}$ is equivalent to
$(x_{k})_{k}$. Zippin’s theorem [12] yields that the space
$\overline{<(x_{k})_{k}>}$ is isomorphic to some $\ell_{p}$. Therefore,
$\operatorname{\mathcal{T}}(\operatorname{\mathcal{A}}_{n},\overline{b})\cong\ell_{p}$
for some $p\in(1,\infty)$.
In order to determine the exact value of $p$, we need the following
Proposition.
###### Proposition 6.4.
The space
$\operatorname{\mathcal{T}}(\operatorname{\mathcal{A}}_{n},\overline{b})$ is
isomorphic to $\ell_{r}$ with $\frac{1}{r}+\frac{1}{r^{\prime}}=1$ and
$\sum_{i=1}^{n}b_{i}^{r^{\prime}}=1$.
###### Proof.
First, let observe that for every $x\in c_{00}$, $\|x\|\leq\|x\|_{r}$. We
shall use induction on the cardinality of $\operatorname{supp}x$. If
$|\operatorname{supp}x|=1$, it is trivial. Assume that it holds for every
$y\in c_{00}$ with $|\operatorname{supp}y|\leq n$ and let $x\in c_{00}$ with
$|\operatorname{supp}x|=n+1$. Then either $\|x\|=\|x\|_{\infty}$ or
$\|x\|=\sum_{i=1}^{n}b_{i}\|E_{i}x\|$ for some appropriate subsets
$E_{1}<E_{2}<\ldots<E_{n}$. In the first case, there is nothing to prove as
for every $p\in[r,\infty)$ $\|x\|_{\infty}\leq\|x\|_{p}$. Therefore we only
need to deal with the second case.
It suffices to observe that for every $i=1,2,\ldots,n$, the cardinality of
$\operatorname{supp}E_{i}x$ is less than $\operatorname{supp}x$ and thus,
using the inductive hypothesis along with $H\ddot{o}lder^{\prime}s$
inequality, we get that
$\|x\|\leq\sum_{i=1}^{n}b_{i}\|E_{i}x\|_{r}\leq(\sum_{i=1}^{n}b_{i}^{r^{\prime}})^{\frac{1}{r^{\prime}}}(\sum_{i=1}^{n}\|E_{i}x\|_{r}^{r})^{\frac{1}{r}}=\|x\|_{r}.$
By combining the preceding argument with Proposition 6.2, we conclude that
$\operatorname{\mathcal{T}}(\operatorname{\mathcal{A}}_{n},\overline{b})$ is
isomorphic to $\ell_{p}$ for some $p\in[r,\infty)$.
For every $l\in\operatorname{\mathbb{N}}$ set $M_{l}=\\{1,2,\ldots,n\\}^{l}$.
We have already mentioned that for every $l\in\operatorname{\mathbb{N}}$ the
functional $f_{l}=\sum_{s\in M_{l}}(\prod_{i=1}^{l}b_{s_{i}})e_{s}^{*}$
belongs to $W(\operatorname{\mathcal{A}}_{n},\overline{b})$ where $s_{i}$ is
the $i$-th coordinate of $s$, for each $i=1,2,\ldots,n$ and $\sum_{s\in
M_{l}}\prod_{i=1}^{l}b_{s_{i}}=(\sum_{i=1}^{n}b_{i})^{l}$. We set
$a_{s}=\prod_{i=1}^{l}b_{s_{i}}$ and $x_{l}=\sum_{s\in
M_{l}}a_{s}^{\frac{r^{\prime}}{r}}e_{s}$. It is easily seen that for every
$l\in\operatorname{\mathbb{N}}$, $\|x_{l}\|=1$. Indeed,
$\|x_{l}\|\leq\|x_{l}\|_{r}=(\sum_{s\in
M_{l}}a_{s}^{r^{\prime}})^{\frac{1}{r}}=(\sum_{i=1}^{n}b_{i}^{r^{\prime}})^{\frac{l}{r}}=1=f_{l}(x_{l})\leq\|x_{l}\|.$
We claim that for $p^{\prime}>r$ and every $\varepsilon>0$ there exists
$l\in\operatorname{\mathbb{N}}$ such that
$\|x_{l}\|_{p^{\prime}}<\varepsilon$. If the claim holds we are done as $p$
coincides with $r$.
Proof of the Claim: Notice that for $p^{\prime}>r$,
$\sum_{i=1}^{n}b_{i}^{\frac{r^{\prime}}{r}p^{\prime}}=\sum_{i=1}^{n}b_{i}^{r^{\prime}(1+\delta)}$
for some $0<\delta<1$. But for every $i=1,2,\ldots,n$ $b_{i}<1$, and therefore
$\sum_{i=1}^{n}b_{i}^{r^{\prime}(1+\delta)}<\sum_{i=1}^{n}b_{i}^{r^{\prime}}=1.$
Thus, there exists $l\in\operatorname{\mathbb{N}}$ such that
$(\sum_{i=1}^{n}b_{i}^{r^{\prime}(1+\delta)})^{l}<\varepsilon^{p^{\prime}}$.
Then for this $l$,
$\|x_{l}\|_{p^{\prime}}=(\sum_{s\in
M_{l}}a_{s}^{\frac{r^{\prime}}{r}p^{\prime}})^{\frac{1}{p^{\prime}}}=(\sum_{s\in
M_{l}}a_{s}^{r^{\prime}(1+\delta)})^{\frac{1}{p^{\prime}}}=(\sum_{i=1}^{n}b_{i}^{r^{\prime}(1+\delta)})^{\frac{l}{p^{\prime}}}<\varepsilon.$
∎
###### Theorem 6.5.
For every $r\in(1,\infty)$ the space $\mathfrak{X}_{r}$ is $\ell_{r}$
saturated.
###### Proof.
As it was mentioned in the above Remark, for every skipped block sequence in
$\mathfrak{X}_{r}$ we can find a further block sequence $(x_{k})_{k}$ such
that the space $\overline{<(x_{k})_{k}>}$ is isomorphic to $\ell_{r}$. ∎
###### Remark 6.6.
From the previous Theorem, we deduce that the space $\mathfrak{X}_{r}$ is a
separable $\mathcal{L}^{\infty}$ space which does not contain $\ell_{1}$.
Therefore, the results of D.Lewis-C.Stegall [10] and A. Pelczyński [11] yields
that $\mathfrak{X}_{r}^{*}$ is isomorphic to $\ell_{1}$. Alternatively, one
can use the corresponding argument of D. Alspach [1] and show directly that
$(M_{q})$ is a shrinking FDD for $\mathfrak{X}_{r}$. It then follows that
$(e_{\gamma}^{*})_{\gamma\in\Gamma}$ is a basis for $\mathfrak{X}_{r}^{*}$,
equivalent to the usual $\ell_{1}$-basis.
## References
* [1] D. Alspach, The dual of the Bourgain-Delbaen space, Israel J. Math. 117 (2000), 239–259.
* [2] S.A. Argyros and I. Deliyanni, Banach spaces of the type of Tsirelson, arXiv (math/9207206v1), (1992).
* [3] S.A. Argyros and R. Haydon, A Hereditarily Indecomposable $\mathcal{L}^{\infty}$-space that solves the scalar-plus-compact problem, (submitted).
* [4] S.A. Argyros and S. Todorčević, Ramsey methods in Analysis, Birkhauser (2005).
* [5] S.F. Bellenot, Tsirelson superspaces and $\ell_{p}$, Journal of Funct. Anal., 69 (1986), no. 2, 207–228.
* [6] J. Bernués and I. Deliyanni, Families of finite subsets of $\operatorname{\mathbb{N}}$ of low complexity and Tsirelson type spaces, Math. Nachr., 222 (2001), 15 -29.
* [7] J. Bourgain and F. Delbaen, A class of special $\mathcal{L}^{\infty}$ spaces, Acta Mathematica, 145 (1980), 155–176.
* [8] D. Freeman, E. Odell and Th. Schlumprecht, The universality of a $\ell_{1}$ as a dual Banach space, preprint.
* [9] R. Haydon, Subspaces of the Bourgain-Delbaen space, Studia Math., 139 (2000), no. 3, 275–293.
* [10] D. Lewis and C. Stegall, Banach spaces whose duals are isomorphic to $\ell^{1}(\Gamma)$, J. Funct. Anal., 12 (1971), 167–177.
* [11] A. Pelczyński, On Banach spaces containing $L_{1}(\mu)$, Studia Math., 30 (1968), 231–246.
* [12] M. Zippin, On perfectly homogeneous bases in Banach spaces, Israel J. of Math., 4 (1966), 265–272.
|
arxiv-papers
| 2010-03-02T13:18:43 |
2024-09-04T02:49:08.710799
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "I. Gasparis, M.K. Papadiamantis and D.Z. Zisimopoulou",
"submitter": "Ioannis Gasparis",
"url": "https://arxiv.org/abs/1003.0579"
}
|
1003.0638
|
Permanent address: ]Georgian Technical University, Kostava str. 77, Tbilisi,
0179, Georgia.
# Beer’s law in semiconductor quantum dots
G. T. Adamashvili [ Max-Planck-Institut für Physik Komplexer Systeme
Nöthnitzer Str. 38 D-01187 Dresden, Germany
email:gadama@parliament.ge A. A. Maradudin Department of Physics and
Astronomy
and Institute for Surface and Interface Science
University of California, Irvine, CA, 92697 USA.
email: aamaradu@uci.edu
###### Abstract
The propagation of a coherent optical linear wave in an ensemble of
semiconductor quantum dots is considered. It is shown that a distribution of
transition dipole moments of the quantum dots changes significantly the
polarization and Beer’s absorption length of the ensemble of quantum dots.
Explicit analytical expressions for these quantities are presented.
###### pacs:
78.67. Hc
## I Introduction
Semiconductor quantum dots (SQD), also referred to as zero-dimensional
systems, are nanostructures that allow confinement of charge carriers in all
three spatial directions, which results in atomic-like discrete energy spectra
and strongly enhanced carrier lifetimes. Such features make quantum dots
similar to atoms in many respects (artificial atoms)[1]. Observation of
optical coherence effects in ensembles of quantum dots is usually spoiled by
the inhomogeneous line broadening due to dot-size fluctuations, with a full
width at half maximum of typically more than several tens of meV. Real quantum
dots often have a base length in the range 50-400 Å. Beside the frequency, the
quantum dot-size fluctuations influence also the transition dipole moments of
SQD. Borri et al.[2] have reported measurements of optical Rabi oscillations
in the excitonic ground-state transition of an inhomogeneously broadened
InGaAs quantum dot ensemble. They found that a distribution with a 20 percent
standard deviation of transition dipole moments results in a strong damping of
the oscillations versus pulse area. In the experiments reported in [2] it was
also found that the period of the Rabi oscillations is changed. These results
show quantitatively how uniformity in dot size and shape is important for any
application based on a coherent light-quantum dot ensemble interaction.
In theoretical investigations of optical coherence effects in ensembles of SQD
it is usually assumed that transition dipole moments of the quantum dots are
independent of the size of the dots [3-8]. It is obvious that for an adequate
description of coherent optical phenomena it is necessary to take into account
a distribution of transition dipole moments in the ensemble of quantum dots.
The purpose of the present work is to investigate theoretically the
propagation and absorption of an optical linear wave when a distribution of
transition dipole moments in the ensemble of SQD is taken into account.
## II The optical Bloch equations in semiconductor quantum dots
We consider the propagation of a circularly polarized optical plane wave pulse
in an ensemble of SQD with a pulse width $T<<T_{1,2}$, frequency
$\omega>>T^{-1}$, wave vector $\vec{k}$, and the electric field $\vec{E}$,
which gives rise to the excitonic ground-state transitions (labeled 0-X).
We use the method of slowly varying envelopes, and define a real envelope
$\hat{E}$ by
$\vec{E}(z,t)=(\sqrt{2})\hat{E}(z,t)[\vec{x}\cos(\omega
t-kz)+\vec{y}\sin(\omega t-kz)],$
which defines an optical pulse propagating along the positive $z$ axis, and
$\vec{x}$ and $\vec{y}$ are unit vectors in the directions of the $x$ and $y$
axes. We shall find it useful to define the complex polarization vectors
$\vec{e}_{+}=\frac{1}{\sqrt{2}}(\vec{x}+i\vec{y}),\;\;\;\;\;\;\;\;\;\;\;\;\;\;\vec{e}_{-}=\frac{1}{\sqrt{2}}(\vec{x}-i\vec{y}),$
in terms of which the electric field has the form
$\vec{E}(z,t)=\hat{E}(z,t)(\vec{e}_{-}e^{i(\omega
t-kz)}+\vec{e}_{+}e^{-i(\omega t-kz)}),$ (1)
where $\hat{E}(z,t)$ is the slowly varying complex envelope of the electric
field and $e^{\pm i(kz-\omega t)}$ contains the rapidly varying phase of the
carrier wave. We will assume that
$|\frac{\partial\hat{E}}{\partial
t}|<<\omega|\hat{E}|,\;\;\;|\frac{\partial\hat{E}}{\partial z}|<<k|\hat{E}|.$
(2)
For the description of SQD in a circularly polarized optical pulse we use the
two-level model (0-X transitions) that can be described by the states $|1>$
and $|2>$ with energies $E_{1}=0$ and $E_{2}=\hbar\omega_{0}$, respectively,
where $|1>$ is the ground state. The Hamiltonian and wave function of this
system are:
$H=H_{0}+\hat{V},$
$|\Psi>=\sum_{n=1,2}c_{n}(t)exp(-\frac{i}{\hbar}E_{n}t)|n>,$
where $H_{0}=\hbar\omega_{0}|2><2|$ is the Hamiltonian of the two-level SQD
with the frequency of excitation $\omega_{21}=\omega_{0}$,
$\hat{V}=-\hat{\mu}\vec{E}=-\mu(\hat{\sigma}_{-}E_{+}+\hat{\sigma}_{+}E_{-})$
is the light-SQD interaction Hamiltonian, $\hbar$ is Planck’s constant,
$\hat{\mu}=\vec{\mu}_{12}\hat{\sigma}_{+}+\vec{\mu}^{*}_{12}\hat{\sigma}_{-}$
is the SQD’s dipole moment operator,
$\vec{\mu}_{12}=\frac{\mu}{\sqrt{2}}(\vec{x}-i\vec{y})$
is the electric dipole matrix element for the corresponding transition,
$\mu=|\vec{\mu}_{12}|,\;\;\;\;\;\;\;\;\vec{\mu}_{21}=\vec{\mu}^{*}_{12},$
${E}_{\pm}=\hat{E}e^{\pm i(\omega t-kz)},$
$\hat{{\sigma}}_{\pm}=\frac{1}{2}(\hat{\sigma}_{x}\pm i\hat{\sigma}_{y}),$
and the $\\{\hat{{\sigma}}_{i}\\}$ are the Pauli matrices, which satisfy
$[\hat{{\sigma}}_{x},\hat{{\sigma}}_{y}]=2i\hat{{\sigma}}_{z}$, and
commutation relations resulting from cyclic permutations of the subscripts.
The probability amplitudes $c_{1}$ and $c_{2}$ are determined by the
Schrödinger equations
$i\hbar\frac{\partial c_{1}(t)}{\partial
t}=-\mu\;{E}_{+}c_{2}(t)e^{-i\omega_{0}t}$ $i\hbar\frac{\partial
c_{2}(t)}{\partial t}=-\mu\;{E}_{-}c_{1}(t)e^{i\omega_{0}t}.$ (3)
The average values of the Pauli operators $\hat{\sigma}_{i}$ for the state
$|\Psi>=c_{1}\;|1>+\;c_{2}\;|2>$, are $s_{i}=<\hat{\sigma}_{i}>\;\;$
$=<\Psi|\hat{\sigma}_{i}|\Psi>$, (where $i=1,2,3)$ and have the form [9]:
$s_{x}=c^{*}_{1}(t)c_{2}(t)e^{-i\omega_{0}t}+c_{1}(t)c^{*}_{2}(t)e^{i\omega_{0}t},$
$s_{y}=ic^{*}_{1}(t)c_{2}(t)e^{-i\omega_{0}t}-ic_{1}(t)c^{*}_{2}(t)e^{i\omega_{0}t},$
$s_{z}=c^{*}_{2}(t)c_{2}(t)-c^{*}_{1}(t)c_{1}(t).$
To introduce explicitly the appropriate rotation matrix for the functions
$s_{i}$ in the rotating frame with components $u,v,w$, from the Schrödinger
equations (3) we obtain the Bloch equations:
$\dot{u}=-\Delta v,$ $\dot{v}=i\Delta u+\kappa w\hat{E},$ $\dot{w}=-\kappa
v\hat{E},$ (4)
where
$\kappa=\frac{2\mu}{\hbar},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\Delta=\omega_{0}-\omega,$
while $u$ and $-v$ are the components, in units of the transition moment
$\mu$, of the SDQ’s dipole moment in-phase and in-quadrature with the field
$\vec{E}$. In other words, $v$ is the absorptive component of the SQD dipole
moment, while $u$ is the dispersive component, and
$\frac{1}{2}\hbar\omega_{0}s_{z}$ is the expectation of the atom’s unperturbed
energy.
In addition to (4), we need a description of the pulse propagation in the
medium. The wave equation for the electric field $\vec{E}(z,t)$ of the optical
pulse in the medium is given by
$\frac{\partial^{2}\vec{E}}{\partial{z}^{2}}-\frac{\varepsilon}{c^{2}}\frac{\partial^{2}\vec{E}}{\partial
t^{2}}=\frac{4\pi}{c^{2}}\frac{\partial^{2}\vec{P}}{\partial t^{2}},$ (5)
where $c$ is the speed of light in vacuum, $\varepsilon=\eta^{2}$ is the
dielectric permittivity, $\eta$ is the refractive index of the medium, and
$\vec{P}$ is the polarization of the ensemble of SQD.
For the determination of the polarization of the ensemble of SQD we have to
take into account that the dipole moment of the SQD depends on the size of the
quantum dot. The polarization of the ensemble of SQD is equal to
$\vec{P}=\sum_{i=1}^{n_{0}}<\vec{\mu}_{i}>,$
where $n_{0}$ is the uniform dot density, and $<\vec{\mu}_{i}>$ is the
expectation value of the i-th dipole moment.
We can rewrite this expression in another form:
$\vec{P}=\sum_{j=1}\;n_{j}\;<\vec{\mu}_{j}>,$ (6)
where $n_{j}$ is the number of dipoles with matrix elements $\mu_{j}$ in the
interval $\Delta\mu_{j}$.
We introduce the function $\tilde{h}(\mu_{i})$ such that the quantity
$\tilde{h}(\mu_{i})\Delta\mu_{i}$ is the the fraction of dipoles with dipole
matrix element $\mu_{i}$ within $\Delta\mu_{i}$. Obviously its normalization
is
$\int_{0}^{\infty}\tilde{h}(\mu)d\mu=1,$
or in another form
$\int_{\mu_{min}}^{\infty}\tilde{h}(\mu-\mu_{min})d\mu=1,$
where $\mu_{min}$ is the minimum value of the dipole matrix element in the
ensemble of SQD. To take into account that when $\mu<\mu_{min}$ the number of
dipoles is equal to zero, $\tilde{h}(\mu<\mu_{min})=0$, we can rewrite the
last equation in the form
$\int_{-\infty}^{\infty}\tilde{h}(\mu-\mu_{min})d\mu=1.$
Because the quantity ${\mu}_{0}-\mu_{min}$ is constant, we can introduce the
function
${h}(\mu-{\mu}_{0})=\tilde{h}(\mu-\mu_{min}),$
and for this function the normalization condition has the form
$\int_{-\infty}^{\infty}{h}(\mu-{\mu}_{0})d\mu=1,$
where
$\vec{\mu}_{0}=\frac{\sum_{i=1}^{n_{0}}\vec{\mu}_{i}}{n_{0}}$
is the main dot dipole matrix element.
By using the function $h(\mu-{\mu}_{0})$, we obtain that the quantity
${n_{j}}\approx{n_{0}}\;{h}(\mu_{j}-\mu_{0})\;\Delta\mu_{j}$. Substituting the
last expression into Eq. (6) we obtain the polarization of the ensemble of SQD
in the form
$\vec{P}\approx\;{n_{0}}\;\sum_{j=1}^{\tilde{n}_{0}}\;{h}(\mu_{j}-\mu_{0})\;\Delta\mu_{j}\;<\vec{\mu_{j}}>.$
After transition to a limit we obtain the exact value of the polarization,
namely
$\vec{P}={n_{0}}\;\int_{-\infty}^{\infty}\;{h}(\mu-\mu_{0})\;\;<\vec{\mu}>\;d\mu.$
(7)
Expressing the value of $<\mu>$ in Eq. (7) in terms of the slowly varying
amplitudes $u$ and $v$, we obtain
$\vec{P}=\frac{{n_{0}}}{\sqrt{2}}\;\int_{-\infty}^{\infty}\;\mu\;{h}(\mu-\mu_{0})\;\\{u\;[\vec{x}\cos(\omega
t-kz)\\\ +\vec{y}\sin(\omega t-kz)]+v\;[-\vec{x}\sin(\omega
t-kz)+\vec{y}\cos(\omega t-kz)]\\}\;d\mu.$
The fluctuations in the size of the SQD also lead to the inhomogeneous
broadening of the spectral line. Upon taking into account the inhomogeneous
broadening of the spectral line, the polarization of the ensemble of SQD is
equal to
$\displaystyle\vec{P}(z,t)=\frac{{n_{0}}}{\sqrt{2}}\;\int_{-\infty}^{\infty}\;\int_{-\infty}^{\infty}\;\mu\;{h}(\mu-\mu_{0})\;g(\Delta)\\{u(z,t;\Delta,\mu)[\vec{x}\cos(\omega
t-kz)+\vec{y}\sin(\omega t-kz)]$
$\displaystyle+v(z,t;\Delta,\mu)[-\vec{x}\sin(\omega t-kz)+\vec{y}\cos(\omega
t-kz)]\\}\;d\mu\;d\Delta,$ (8)
where $g(\Delta)$ is the inhomogeneous broadening function. This equation
generalizes the polarization of the ensemble of SQD to the case of a
distribution of transition dipole moments. In the special case, when all
dipole moments are the same, the distribution function is
$h(\mu-\mu_{0})=\delta(\mu-\mu_{0})$, and we obtain the polarization in the
usual form [9]:
$\displaystyle\vec{P_{h}}(z,t)=\frac{{n_{0}}\mu_{0}}{\sqrt{2}}\;\int_{-\infty}^{\infty}\;g(\Delta)\\{u(z,t;\Delta)[\vec{x}\cos(\omega
t-kz)+\vec{y}\sin(\omega t-kz)]$
$\displaystyle+v(z,t;\Delta)[-\vec{x}\sin(\omega t-kz)+\vec{y}\cos(\omega
t-kz)]\\}\;d\Delta.$ (9)
By substituting equations (1) and (8) into the wave equation (5), and using
the assumption of slowly varying amplitudes (2), we obtain the in-phase and
in-quadrature wave equations in the following forms:
$(k^{2}-\frac{\varepsilon}{c^{2}}\omega^{2})\hat{E}=\frac{2\pi
n_{0}{\omega}^{2}}{c^{2}}\;\int_{-\infty}^{\infty}\;\int_{-\infty}^{\infty}\;\mu\;{h}(\mu-\mu_{0})\;g(\Delta)\;u\;d\mu\;d\Delta,$
(10) $\frac{\partial\hat{E}}{\partial
z}+\frac{\omega\varepsilon}{kc^{2}}\frac{\partial\hat{E}}{\partial
t}=\frac{\pi
n_{0}{\omega}^{2}}{c^{2}k}\;\int_{-\infty}^{\infty}\;\int_{-\infty}^{\infty}\;\mu\;{h}(\mu-\mu_{0})\;g(\Delta)\;v\;d\mu\;d\Delta.$
(11)
Equations (10) and (11) are general equations for the slowly varying
amplitudes by means of which we can consider a quite wide class of coherent
optical phenomena (for instance: Rabi oscillations, photon echo, self-induced
transparency, and others) in an ensemble of SQD in the presence of a
distribution of transition dipole moments in the ensemble. Equations (10) and
(11) generalize the Maxwell equations that have been considered up to now for
atomic systems and for SQD when fluctuations of the dipole moments are
neglected [1]. We have to note that an analytic solution of these equations
are significantly difficult than the solution of the Maxwell equations in the
special case, when all dipole moments are the same, because in Eqs.(10) and
(11) the quantities $v$ and $u$ are the functions also of the variable $\mu$.
Nevertheless analytic solution of the equations (10) and (11) are possible in
some special cases, for instance, for Beer’s law in SQD.
## III Beer’s law in semiconductor quantum dots
From Eq. (10) we can obtain the dispersion relation for the wave and from Eq.
(11) we can determine the explicit form of the envelope of the electric field
strength $\hat{E}$.
Usually $g(\Delta)$ is a very broad and smooth function, so that the integral
$\int_{-\infty}^{\infty}g(\Delta)ud\Delta$ in Eq. (10) is very small and is
usually neglected [9]. Consequently, the dispersion law has the form
$k^{2}=\frac{\varepsilon}{c^{2}}\omega^{2}.$ (12)
After integration of the in-quadrature wave equation (11) from $t=-\infty$ up
to the time $\bar{t}$ that occurs after the pulse has passed the point of
observation $z$ we obtain
$\frac{\partial\vartheta(\bar{t},z)}{\partial
z}=\frac{2\pi^{2}n_{0}{\omega}^{2}g(0)}{c^{2}k}\int_{-\infty}^{\infty}\mu{h}(\mu-\mu_{0})v(\mu,t_{0},z,0)d\mu,$
(13)
where
$\vartheta(t,z)=\int_{-\infty}^{t}\;\hat{E}(t^{\prime},z)dt^{\prime}$ (14)
is proportional to the area of the pulse,
$\theta(t,z)=\frac{2\mu}{\hbar}\vartheta(t,z)$, the time $t_{0}$ marks the end
of the pulse, and at $t_{0}$ and for all later times the electric field
envelope $\hat{E}$ is zero.
For an analytic solution of Eq. (13) we have to determine the explicit form of
the quantity $v$ as a function of the electric field and dipole moment. From
the Bloch equations (4) for the absorptive part of the on-resonance dipole
amplitude, we find that
$v(\mu,t_{0},z,0)=-\sin\frac{2\mu}{\hbar}\vartheta(t_{0},z).$ (15)
We consider Beer’s law of absorption for the electric field of the pulse in a
SQD. For this purpose we consider the limit of weak electric fields, in which
the ”area” $\theta(t_{0},z)$ is small, i.e. $|\theta(t_{0},z)|<<1$, for all
values of $\mu$. Under this condition, and taking into account that for all
$t>t_{0}$, $\vartheta(\bar{t},z)=\vartheta(t_{0},z)$, from Eq.(15) we obtain
$v(\mu,t_{0},z,0)\approx-\frac{2\mu}{\hbar}\vartheta(t_{0},z).$ (16)
On substituting Eq. (16) into Eq. (13), we obtain the relation
$\frac{\partial\vartheta(\bar{t},z)}{\partial
z}=-\frac{\alpha}{2}\;\vartheta(\bar{t},z),$ (17)
where
$\alpha=\frac{4\;\pi^{2}n_{0}{\omega}^{2}g(0)}{c^{2}\hbar\;k}\;\;\int_{-\infty}^{\infty}\;{\mu}^{2}\;{h}(\mu-\mu_{0})\;d\mu$
(18)
is the reciprocal Beer’s absorption length, in which fluctuations of the
dipole moments of the quantum dots have been taken into account.
## IV Conclusion
We note that Beer’s law, Eq.(17), shows that the electric field of the wave
decays exponentially with increasing penetration into the medium. For pulses
with an electric field of arbitrary shape, Beer’s law is usually written for
the dimensionless pulse area $\frac{2\mu_{c}}{\hbar}\vartheta$, where the
dipole moment $\mu_{c}$ is the same for all atoms or SQD.
For SQD with a distribution of their dipole moments, it is impossible to write
the area of a pulse in the usual dimensionless form, and we have to use Beer’s
law in the form of Eq. (17), for the quantity $\vartheta$ defined by Eq. (14).
In the limiting case when the fluctuation of the dipole moments is neglected,
the distribution function ${h}(\mu-\mu_{0})$ transforms to the Dirac
$\delta$-function, and the reciprocal Beer’s absorption length reduces to the
usual form which, in atomic systems with identical dipole moments in solids or
in gases, where $\eta=1$ [9], is
$\alpha_{h}=\frac{4\;\pi^{2}n_{0}\omega g(0)}{c\hbar\;\eta}\;{\mu_{0}}^{2}.$
(19)
On comparing Eqs.(9),(18) and (19) we see that a distribution of the dipole
moments in an SQD ensemble significantly changes the polarization Eq.(8) and
Maxwell’s wave equations for an ensemble of quantum dots Eqs.(10) and (11)
and, as a result, in a special case also the Beer’s absorption length Eq.(18).
## References
* [1] D. Bimberg, M. Grundmann, and N. N. Ledentsov, Quantum Dot Heterostructures (Wiley, Chichester, 1999).
* [2] P. Borri, W. Langbein, S. Schneider, U. Woggon, R.L. Sellin, D. Ouyang, and D. Bimberg , Phys. Rev. Lett. 87, 157401 (2001).
* [3] G. Panzarini, U. Hohenester, and E. Molinari, Phys. Rev. B 65, 165322 (2002).
* [4] J. Forstner, C. Weber, J. Danckwerts, and A. Knorr, Phys.Rev. Lett. 91, 127401 (2003).
* [5] G. T. Adamashvili and A. Knorr, Opt. Lett., 31, 74 (2006).
* [6] G. T. Adamashvili, C. Weber, A. Knorr, and N. T. Adamashvili, Phys. Rev. A 75, 063808 (2007).
* [7] G. T. Adamashvili and A. Knorr, Phys. Lett. A, 367, 220 (2007).
* [8] G. T. Adamashvili, C. Weber, and A. Knorr, The European Physical Journal D 47, 113 (2008).
* [9] L. Allen and J. H. Eberly, Optical resonance and two-level atoms (Dover, New York, 1975).
|
arxiv-papers
| 2010-03-02T17:26:25 |
2024-09-04T02:49:08.720369
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "G. T. Adamashvili and A. A. Maradudin",
"submitter": "Guram Adamashvili",
"url": "https://arxiv.org/abs/1003.0638"
}
|
1003.0641
|
# Modeling of amorphous carbon structures with arbitrary structural
constraints
F. H. Jornada Instituto de Física, Universidade Federal do Rio Grande do Sul,
91501-970 Porto Alegre - RS, Brazil V. Gava Universidade de Caxias do Sul,
95070-560 Caxias do Sul - RS, Brazil A. L. Martinotto Universidade de Caxias
do Sul, 95070-560 Caxias do Sul - RS, Brazil L. A. Cassol Universidade de
Caxias do Sul, 95070-560 Caxias do Sul - RS, Brazil C. A. Perottoni
Universidade de Caxias do Sul, 95070-560 Caxias do Sul - RS, Brazil
###### Abstract
In this paper we describe a method to generate amorphous structures with
arbitrary structural constraints. This method employs the Simulated Annealing
algorithm to minimize a simple yet carefully tailored Cost Function (CF). The
Cost Function is composed of two parts: a simple harmonic approximation for
the energy-related terms and a cost that penalizes configurations that do not
have atoms in the desired coordinations. Using this approach, we generated a
set of amorphous carbon structures spawning nearly all the possible
combinations of $sp$, $sp^{2}$ and $sp^{3}$ hybridizations. The bulk moduli of
this set of amorphous carbons structures was calculated using Brenner’s
potential. The bulk modulus strongly depends on the mean coordination,
following a power law behavior with an exponent $\nu=1.51\pm 0.17$. A modified
Cost Function that segregates carbon with different hybridizations is also
presented, and another set of structures was generated. With this new set of
amorphous materials, the correlation between the bulk modulus and the mean
coordination weakens. This method proposed can be easily modified to explore
the effects on physical properties of the presence hydrogen, dangling bonds,
and structural features such as carbon rings.
amorphous carbon, simulated annealing, bulk modulus
###### pacs:
61.43.Bn, 61.43.-j, 62.20.de
††preprint: ???
## I Introduction
Carbon is an impressively versatile chemical element. As it is found in three
distinct hybridizations, $sp^{3}$, $sp^{2}$ and $sp$, each with a well-defined
local topology, this element can form a variety of different allotropes.
Diamond’s bulk hardness and graphite’s laminar softness, for instance, can
each be tracked down to carbon $sp^{3}$ tetrahedron-like bonding and $sp^{2}$
planar configuration. Although these two materials exhibit quite distinctive
physical and chemical properties, they represent only a small fraction of
possible carbon solids.
Due to their many industrial applications, non-crystalline carbon materials
have lately received attention. This class of material can be cheaply produced
by Chemical Vapor Deposition (CVD) Messina and Santagelo (2006) and deposited
over surfaces as thin, hard films, which exhibit good biocompatibility and
chemical inertness Robertson (24 May 2002). In these films, the amount of
$sp^{3}$, $sp^{2}$ and $sp$ hybridized carbon, along with the presence or
absence of hydrogen, directly influences the coating’s stiffness Fanchini and
Tagliaferro (2006). Considering the immense variety of amorphous carbon films
that can be generated experimentally, many efforts were made to theoretically
address how their properties vary according to changes in composition and
local structure.
One of the first models to describe non-crystalline materials was elaborated
by Zachariasen, who introduced the concept of continuous random network (CRN)
to explain the atomic arrangement in $\mathrm{SiO_{2}}$ glasses Zachariasen
(1932). Despite primarily addressing $\mathrm{SiO_{2}}$ structures,
Zachariasen proposed that most oxide glasses could be considered as a random
disposal of atoms in which there are neither bond defects nor long range
crystallinity. Further advances in this field were made possible by employing
computers to generate continuous random networks. Among the most successful
approaches is the one by Wooten, Winer and Weaire (WWW) Wooten et al. (1985).
Their ingenious bond-switching method consists of randomly swapping bonds from
an originally $100\%$ $sp^{3}$ crystalline structure. In the WWW approach, one
starts with a periodic diamond supercell, and follows a cycle of bond
interchanges and geometry relaxation until a fully tetrahedral amorphous
carbon (ta-C) structure, also referred to as amorphous diamond (a-D), is
obtained. This method is not only computationally fast and straightforward to
implement, but also very successful in reproducing the experimental radial
distribution function of amorphous solids Wooten and Weaire (1987).
As computer performance improved, it became possible to generate a-C by
employing Molecular Dynamics (MD). Using this strategy, it is possible to
simulate either the quenching of a melted carbon liquid or the process of film
deposition, and amorphous carbon structures containing $sp$, $sp^{2}$ and
$sp^{3}$ hybridizations may be obtained Kaukonen and Nieminen (1992); Wang and
Ho (1993); Kelires (1994); Marks (2000); Marks et al. (2002); Mathioudakis et
al. (2004). Another approach, analogous to MD, for the computer generation of
amorphous carbon structures is the activation-relaxation technique, proposed
by Barkema and Mousseau Barkema and Mousseau (1996). This method also allows
one to obtain a CRN containing carbon atoms in different coordination, and it
is an efficient means to overcome energy barriers between metastable minima.
Many previous works have pointed out a general trend of a-C structures to
become denser and stiffer with the increase of the mean atomic coordination
Djordjevic and Thorpe (1997); Mathioudakis et al. (2004). These findings are
supported by experimental data Ferrari et al. (1999) and by the percolation
model of Phillips Phillips (1979) and Thorpe Thorpe (1983); He and Thorpe
(1985). The latter work may be seen as the theoretical basis that explains the
strong dependence of elastic properties both on the mean coordination and on
the presence of small rings, and it can also successfully explain CRNs’ bulk
modulus vanishing as the mean atomic coordination $\overline{z}$ approaches
$2.4$.
Computer methods to generate a-C have recently included the use of
increasingly sophisticated Hamiltonians (such as ab initio) to more accurately
simulate the dynamics of carbon atoms Galli et al. (1989); Marks et al.
(1996); McCulloch et al. (2000); Han et al. (2007). Likewise, many of these
algorithms focus on simulating the experimental process that originates a
particular amorphous system. This may be seen as a top-down strategy, as for
each new a-C material produced, it is necessary to perform an MD that
resembles the corresponding experimental conditions. Only after that can the
material’s properties be estimated. For instance, if one were to answer the
effect of a particular local structure on the amorphous carbon properties,
various MDs might possibly have to be performed until a certain simulation
yielded the desired microscopic structure.
Later, with the discovery of new forms of a-C, the influence of some local
scale features on the properties of amorphous systems was pointed out Lyapin
et al. (2000); Hong et al. (2009). The intrinsic complexity of dealing with
these features, such as the presence of rings, the proportion of $sp$,
$sp^{2}$ and $sp^{3}$ hybridized atoms, and the size of clusters, may explain
the difficulty to obtain an universal relationship among density, bulk modulus
and mean coordination Lyapin et al. (2000). In order to efficiently study
these aspects, it is important to develop a method to generate amorphous
carbon structures, as neither WWW’s algorithm nor MD can easily generate a-C
with generic structural constraints. The former because it is limited to
$sp^{3}$ hybridized carbon, and the latter because it follows a top-down
approach and offers no direct way to control the presence of those features.
In order to tackle these difficulties, we introduce a scheme for the computer
generation of a-C structures which follows a bottom-up strategy. Relying on
the algorithm of Simulated Annealing (SA) Kirkpatrick et al. (1983), we
propose a Cost Function (CF) that aims not at simulating realistic atomic
dynamics – as done in Blaudeck and Hoffmann (2003) – but rather at exploring
several metastable configurations that meet some desired criteria. Our aim is
to introduce something of a “theoretical workbench” to flexibly simulate a-C.
One of our main interests is to employ this technique to calculate the bulk
modulus’ dependence on the fraction of $sp$, $sp^{2}$ and $sp^{3}$ hybridized
carbon, and to perhaps find an amorphous structure more incompressible than
diamond.
The paper is organized as follows: First, in the computational strategy
section, we detail the proposed Cost Function. Next, we present the results of
simulations with 512 carbon atoms, using periodic boundary conditions, in
which we calculate the bulk modulus of several amorphous structures spawning
several possible combinations of $sp^{3}$, $sp^{2}$ and $sp$ carbon and also
compare the calculated radial distribution functions with the literature. We
then show that we can modify our CF to force the creation of clusters of atoms
with the same hybridization, and thus explore their effect on the bulk
modulus. We conclude by discussing the advantages of the approach described in
this paper, as well as possible extensions that can be proposed to deal with
other problems.
## II Computational Strategy
Following the idea of the SA technique, we have developed a fast and
customizable CF which guides the exploration of different atomic
configurations. Our Cost Function is composed of two parts, the first a
computationally simple potential, and the second depending on the desired
constraints for the CRN. The CF is required neither to be a continuous
function nor to yield a realistic value for structures far from a metastable
situation; its only requirement is to have an arbitrary low value for
geometrically stable configurations. Thus, the problem of finding a certain
amorphous material that meets some arbitrary constraints can be transformed to
the problem of finding a reasonably low minimum of this Cost Function using
SA.
Considering the potential part of the CF, a simple harmonic-like approximation
was employed. As we wished our model to be as computationally fast as
possible, carbon atoms were considered to be either bonded or non-bonded, with
a cut-off distance $r_{c}$ of $2.0$ Å. This value is somewhat arbitrary, as
long as it is smaller than the typical second-neighbor distance, but we found
that shrinking it too much allows non-bonded atoms to remain too close. Thus,
the potential term of the Cost Function is written as
$\displaystyle\phi_{V}=v_{r}\sum_{r_{ij}}(r_{ij}-r^{\ast}_{c(i)c(j)})^{2}+v_{a}\sum_{\theta_{ijk}}(\theta_{ijk}-\theta^{\ast}_{c(j)})^{2}$
$\displaystyle+v_{t}\sum_{\mathbf{u}_{i},\mathbf{u}_{j}}[1-(\mathbf{u}_{i}\cdot\mathbf{u}_{j})^{2}]$
(1)
The first sum is over all bonds $r_{ij}$ and expresses the stretching energy
relative to the equilibrium distance $r^{\ast}_{c(i)c(j)}$ between atoms $i$
and $j$ (having coordination $c(i)$ and $c(j)$ respectively). The second sum
comprises all $\theta_{ijk}$ angles having a common $j$ center, where
$\theta^{\ast}_{c(j)}$ denotes the equilibrium angle, which in our
approximation depends only on the hybridization $c(j)$ of atom $j$. The last
sum considers only connected $sp^{2}$ centers, and it constitutes the
torsional energy of having two $sp^{2}$ planes, with normal vectors
$\mathbf{u}_{i}$ and $\mathbf{u}_{j}$, nonparallel. In all summations,
repeated terms are not counted. Also, the constants $v_{r}$, $v_{a}$ and
$v_{t}$ merely set the relative strength of the radial, angular and torsion
terms for the total $\phi_{V}$.
The equilibrium quantities do not need to be found with great precision. Since
our objective is simply to obtain a structure obeying a given set of
constraints, there can be small geometrical distortions, all of which can be
removed in a further relaxation by using a more realistic Hamiltonian. Thus,
the following approximations were made: if two bonded atoms are both $sp^{3}$,
their equilibrium distance $r^{\ast}_{44}$ is the same found in diamond; for
$sp^{2}$-$sp^{2}$ bonds, equilibrium interatomic distance $r^{\ast}_{33}$ is
that found in graphite and, in the case of $sp$-$sp$ bonds, one takes the
equilibrium distance of the triple bond in 2-butyne for $r^{\ast}_{22}$ Kynoch
(1962). For a pair of bonded carbon atoms with different hybridizations, say,
$c^{\prime}$ and $c^{\prime\prime}$, the equilibrium distance
$r^{\ast}_{c^{\prime}c^{\prime\prime}}$ is simply is the average
$(r^{\ast}_{c^{\prime}c^{\prime}}+r^{\ast}_{c^{\prime\prime}c^{\prime\prime}})/2$.
Finally, if an atom has a coordination $c^{\prime}>4$, the same value as for
four-fold atoms is assumed ($r^{\ast}_{c^{\prime}c^{\prime}}=r^{\ast}_{44}$).
Likewise, the distance $r^{\ast}_{11}$ between singly-bonded atoms is the same
as for $sp$-$sp$.
The way we presented $\phi_{V}$ alone will not yield any bondings, as linked
atoms will always increase the system’s energy. In order to correct this and
to control the material hybridizations, we introduce another term in the Cost
Function. This term, here referred to as Coordination Cost, allows one to set
how many atoms should be $sp^{3}$, $sp^{2}$ and $sp$:
$\phi_{C}=\sum_{c^{\prime}}\epsilon_{c^{\prime}}|n_{c^{\prime}}-n^{\ast}_{c^{\prime}}|$
(2)
The sum is over all possible $c^{\prime}$ coordinations, where
$n_{c^{\prime}}$ is the number of $c^{\prime}$-coordinated atoms, and
$n^{\ast}_{c^{\prime}}$ is a parameter that sets how many atoms should be
$c^{\prime}$-coordinated. This way, each constant $\epsilon_{c^{\prime}}$ sets
a cost for a configuration having a wrong number
$|n_{c^{\prime}}-n^{\ast}_{c^{\prime}}|$ of $c^{\prime}$-coordinated centers.
Clearly, we must have
$\sum_{c^{\prime}}n^{\ast}_{c^{\prime}}=\sum_{c^{\prime}}n_{c^{\prime}}=N$,
where $N$ is the total number of atoms. Taking the absolute value of
$n_{c^{\prime}}-n^{\ast}_{c^{\prime}}$ rather than squaring it has shown some
advantages, such as the Cost Function exhibiting a sharper minimum. Moreover,
it naturally makes the CF an extensive function, so that constants do not have
to be altered as the number of atoms in the simulation box changes.
Finally, the Cost Function $\Phi$ is simply defined as a linear combination of
the previous terms:
$\Phi=\lambda_{V}\phi_{V}+\lambda_{C}\phi_{C},$ (3)
where $\lambda_{V}$ and $\lambda_{C}$ are constants. With the former
definitions, bonded atoms are stable, provided that their linking decreases
the number of atoms with wrong coordinations. By putting $N$ atoms in a cubic
cell with periodic boundaries and setting how many should be $sp^{3}$,
$sp^{2}$ and $sp$ hybridized (i.e., fixing $n^{\ast}_{4}$, $n^{\ast}_{3}$ and
$n^{\ast}_{2}$), a CRN can be obtained as the set of atomic positions which
minimizes $\Phi$. We expect our CF to possess many metastable minima, and we
employed the Simulated Annealing (SA) algorithm to optimize $\Phi$.
Despite being SA a technique aimed at finding the global minimum of complex
hypersurfaces, we are not necessarily interested in using this technique to
its full extent. Consider, for instance, the generation of fully $sp^{3}$
carbon CRNs. The global minimum of the CF in this case is crystalline diamond,
which is clearly not the solution we are looking for. Accordingly, we propose
the use of the SA algorithm to find deep minima of the Cost Function that
comply with a certain set of constraints. Whether or not this minimum is the
global minimum of the Cost Function is not our concern.
We devised the optimization algorithm the following way: Each step in the SA
scheme constitutes randomly displacing one atom inside a cube of side length
$0.4$ Å, changing the CF by $\Delta\Phi$. Even though system-wide movements
could be implemented, individual movements have shown a great advantage, as
$\Phi$ only changes locally in this case. So, the recalculation of $\phi_{V}$,
which is the most numerically intensive task in our CF, only has to be
evaluated in a small radius around the displaced atom. This way, the
computational complexity of $\Delta\Phi$ approximately does not scale with the
system size. As in classical SA, each movement is accepted stochastically
according to the weighing factor $e^{-\beta\,{\Delta\Phi}}$, where $\beta$ is
the inverse of the fictitious temperature $T$. In addition to atomic
displacements, the system periodically undergoes a random expansion or
contraction, so that one does not need to set the final density a priori.
Although $\Delta\Phi$ can not be evaluated locally in this case, the number of
atomic displacements between full system scalings are such that these
resizings do not compromise the algorithm speed.
A key component to successfully finding a minimum for $\Phi$ is the
determination of the optimal annealing scheme Johnson et al. (1989). Following
Christoph and Hoffmann Christoph and Hoffmann (1993), we decreased $T$ using a
power law. In order to further control the annealing, we separated it in three
regions, and following Johnson et al. Johnson et al. (1989), we fixed the
initial and final acceptance rates instead of the temperatures, as the former
proved to be less sensitive to changes in the CF. The initial and final
acceptance rates and number of steps assigned to each annealing region were
found by minimizing the width of the angular distribution function for a-D,
and results are summarized in Table 1.
Table 1: Parameters used in the three-regions annealing scheme: initial ($P_{i}$) and final ($P_{f}$) acceptance rate and the fraction of the total steps assigned to each region. Region | Steps | $P_{i}$ | $P_{f}$
---|---|---|---
$1$ | $10\%$ | $95\%$ | $70\%$
$2$ | $80\%$ | $70\%$ | $30\%$
$3$ | $10\%$ | $30\%$ | $5\%$
The constants in Eqs. (1)-(5) were determined as follows. Medium-sized (256
atoms) mixed-coordination amorphous carbon networks were generated, and the
resulting structures were further relaxed using Molecular Dynamics and
Brenner’s interatomic potential Brenner et al. (2002) using GULP Gale and Rohl
(2003). Parameters were thus varied to minimize the final structures’ energies
and coordination errors (i.e., the numbers $n_{c^{\prime}}$ of
$c^{\prime}$-coordinated atoms should be as close as possible to the defined
number $n_{c^{\prime}}*$). The optimal constants are shown in Table 2. Also,
we found $\lambda_{V}/\lambda_{C}\approx 0.75$ to be appropriate for most
simulations. One should note that these constants could not be determined with
great precision due to the large statistical fluctuations in the methodology.
Fortunately, precise values are not sought, since small stresses in the CRNs
could be eliminated a posteriori by a supplemental Molecular Dynamics using
more sophisticated interatomic potentials, such as Brenner’s potential Brenner
et al. (2002), or even ab initio calculations.
Table 2: Parameters for Eqs. (1)-(5). Constant | $\lambda_{V}$ | $\lambda_{C}$ | $v_{r}$ | $v_{a}$ | $v_{t}$ | |
---|---|---|---|---|---|---|---
Value | $1.0$ | $2.5$ | $5.0$ | $3.0$ | $1.5$ | |
Constant | $\epsilon_{0}$ | $\epsilon_{1}$ | $\epsilon_{2}$ | $\epsilon_{3}$ | $\epsilon_{4}$ | $\epsilon_{5}$ | $\epsilon_{j(j\geq 5)}$
Value | $10.0$ | $5.0$ | $2.0$ | $1.5$ | $1.0$ | $10.0$ | $10^{j}$
Constant | $r_{11}^{\ast}$ | $r_{22}^{\ast}$ | $r_{33}^{\ast}$ | $r_{44}^{\ast}$ | $r_{jj(j\geq 4)}^{\ast}$ | |
Value $(\AA)$ | $1.2$ | $1.2$ | $1.42$ | $1.54$ | $1.54$ | |
Constant | $\theta_{2}^{\ast}$ | $\theta_{3}^{\ast}$ | $\theta_{4}^{\ast}$ | $\theta_{j(j\geq 4)}^{\ast}$ | | |
Value | $180^{\circ}$ | $120^{\circ}$ | $109.4^{\circ}$ | $109.4^{\circ}$ | | |
As a next step, in order to validate our CF, we first generated a 64-atom
$100\%$ $sp^{3}$ CRN. We used this CRN and a diamond structure as references
to build 118 other CRNs, each of them constructed as a linear combination of
the two reference structures. More specifically, denoting $\mathbf{r}_{i}^{d}$
and $\mathbf{r}_{i}^{a}$ the positions of the $i$th atom of the diamond and
the amorphous structure respectively, each interpolated CRN was defined
according to
$\mathbf{r}_{i}(u)=(1-u)\;\mathbf{r}_{i}^{d}+u\;\mathbf{r}_{i}^{a}$, where $u$
is an interpolation parameter.
For each CRN, we calculated the energy using our Cost Function and Brenner’s
potential, and plotted the results in Fig. 1. It is clear that both models
should yield high values for unstable materials (i.e., for $u$ far from $0$ or
$1$), and that our potential should only partially reproduce the true energy
surface. Nevertheless, our simplified Cost Function very much resembles the
computationally more expensive Brenner’s potential and, particularly, it
agrees with it on the position of the two minima, which is the key feature for
the Simulated Annealing scheme to identify a metastable CRN.
Figure 1: (Color online). Comparison of the energy calculated using our Cost
Function (top red curve) and Brenner’s Brenner et al. (2002) potential (bottom
blue) for a set of structures. Top curve has been displaced vertically for
clarity. The left minimum at $u=0$ corresponds to a diamond structure, and the
right one at $u=1$ to a $sp^{3}$ amorphous material.
## III Results and Discussion
The procedure we are proposing for the generation of customized continuous
random networks was first applied to map the bulk modulus’ dependence on the
fraction of $sp$, $sp^{2}$ and $sp^{3}$ hybridized carbon. To do so, we
generated $45$ amorphous structures, each containing $512$ atoms and having a
different proportion of the possible atomic coordinations. These proportions
were chosen so as to homogeneously cover a ternary graph mapping all possible
coordinations. It took $2\times 10^{8}$ iterations ($4$ hours111Simulations
were carried out in a 32-bits, single core Intel©Celeron©CPU with 2.66GHz
clock and 1Gb RAM.) for each SA simulation. After these structures were
generated, they were submitted to Molecular Dynamics simulations using
Brenner’s potential to minimize non-physical features, such as distorted
angles and distances. The MD also ensures that each structure stays in a
relatively low energy metastable configuration.
Each MD simulation was carried out at a low temperature of $50$ K in order to
preserve the main features of the CRN generated by SA. The structures that
were modified the most by the Molecular Dynamics process had roughly $50\%$
$sp$/$sp^{3}$ carbon atoms, but little or no $sp^{2}$ centers. In the worst
case, a structure with $\overline{z}=2.87$ had $14.65\%$ of its $sp^{3}$ and
$2.15\%$ of its $sp$ atoms turned into the $sp^{2}$ form. On the other hand,
the most $sp^{3}$-rich structure, having final mean coordination
$\overline{z}=3.98$, had less than $1.5\%$ change in its hybridization due to
the MD relaxation.
Due to the high computational cost required to perform a full MD, only the
equilibration phase was performed. Each MD isobaric simulation was performed
for $5$ ps using a $0.1$ fs time step. Afterwards, each structure was
submitted to a full Hessian-driven geometry relaxation, so that the elastic
moduli could be calculated. We could have also estimated the the elastic
moduli directly using the volume fluctuations of the MD, but it would have
required a much longer time period. Both the MD simulations and the Hessian-
driven geometry relaxations were performed using GULP Gale and Rohl (2003).
Fig. 2 shows some CRNs generated, including a $sp$ rich carbon network, which
may be quite difficult to obtain using Molecular Dynamics under conventional
approaches. One should note, however, that Brenner’s potential does not
includes van der Waals forces, which should be quite important in determining
the geometries and elastic properties of these low-density $sp$ carbon
structures.
(a)
(b)
(c)
Figure 2: (Color online). Example of some generated structures. The following
color scheme was used: $sp^{3}$ atoms are shown in green, $sp^{2}$ ones in
blue and $sp$ in red. (a) A $sp$ rich amorphous carbon network. (b) A mixed
$sp^{2}$/$sp^{3}$ structure with $\lambda_{H}=1.5$. Note that the CRN
segregates $sp^{2}$ and $sp^{3}$ atoms in two phases due to the heterogeneity
cost. (c) Another mixed $sp^{2}$/$sp^{3}$ structure but with $\lambda_{H}=0$.
There are no visibly distinct phases in this case.
As an example of the flexibility of our approach to generate a-C, we also
proposed another term to the Cost Function to penalize binded atoms with
different coordinations. This term was proposed to segregate carbon atoms
having different hybridizations for main reasons. The first is purely
theoretical, as, at least in principle, $sp^{2}$ centers embedded in a rigid
$sp^{3}$ matrix should not influence the bulk modulus much, whereas if these
threefold atoms form a segregate phase they might make the material
considerably more compressible. This $sp^{2}$ segregation is also motivated by
the existence of experimental carbon materials containing $sp^{3}$-rich phases
and graphite-like $sp^{2}$-rich regions Lau et al. (2008). The second reason
is that it has been pointed out that the microstructure of hydrogenated a-C,
mainly the size and shape of $sp^{2}$ clusters, plays an important role in
determining the electronic properties of these materials Theye and Paret
(2002). So, we proposed an heterogeneity term $\phi_{H}$,
$\phi_{H}=\sum_{r_{ij}}(1-\delta_{c(i),c(j)})$ (4)
where we are again summing over all possible bonds, and $c(i)$ is the
coordination of the center $i$, and $\delta$ is the Kronecker delta function.
With the addition of this term, the cost function becomes:
$\Phi=\lambda_{V}\phi_{V}+\lambda_{C}\phi_{C}+\lambda_{H}\phi_{H}$ (5)
In our simulations, we used $\lambda_{H}$ = $1.5$, which was the smaller
number for which atoms with different coordinations would be visibly
segregated. This by no means precluded that a small number of atoms with other
coordinations were found in the segregated phases. One may control how pure in
terms of hybridization these regions are by varying $\lambda_{H}$. Using this
new $\Phi$, we generated another set of $45$ structures, later employing
Brenner’s potential as well. In order to distinguish from the other set of
structures, we shall call the latter a-Cs (generated with $\lambda_{H}>0$)
heterogeneous structures, and the former ones (with $\lambda_{H}=0$)
homogeneous structures. Fig. 2 shows the effect of including $\phi_{H}$ into
the Cost Function by showing two CRNs generated with different values of
$\lambda_{H}$ but having similar amounts of $sp^{2}$ and $sp^{3}$ centers.
The possibility of creating homogeneous and heterogeneous structures
exemplifies the flexibility of our method to generate amorphous structures. By
adding a simple and intuitive term to $\Phi$, it is possible to generate CRNs
with quite different characteristics. This same approach can be employed to
generate CRNs having other microscopic features. For instance, a term may be
added to increase the energy of a CRN if $n$-fold carbon rings are present.
Also, the constant $n_{1}^{\ast}$ may have a non-zero value, and centers with
only one bond may be readily mapped into hydrogen atoms or dangling bonds.
One important question that can be readily answered using our algorithm is how
the bulk modulus varies with the atomic coordinations. Although it has been
pointed out that the bulk modulus should depend mainly on the mean
coordination, no previous method exists to generate a-C with predetermined
fractions of $sp^{3}$, $sp^{2}$ and $sp$ carbon, so that this aspect has not
yet been fully studied. We show in Fig. 3 the resulting bulk modulus as a
function of the fraction of $sp^{3}$, $sp^{2}$ and $sp$ carbon, for both the
cases of homogeneous and heterogeneous structures222The graphics were
generated using a custom-made module for pylab, and may be obtained free of
charge from fjornada@if.ufrgs.br.. We also plot the bulk modulus versus the
mean coordination for the homogeneous case in Fig. 4. The largest bulk modulus
we found was $362$ GPa, which is lower than that calculated for crystalline
diamond using Brenner’s potential ($442$ GPa, the same as the diamond’s
experimental bulk modulus Kittel (1968)).
Figure 3: (Color online). Bulk modulus dependency on carbon hybridization. In
each triangle, the lower left vertex represents a $100\%$ $sp$ structure
(having a mean coordination $\overline{z}=2$), the top vertex a $100\%$
$sp^{2}$ structure (with $\overline{z}=3$), and the lower right vertex a
$100\%$ $sp^{3}$ material (with $\overline{z}=4$). Points lying on the same
vertical line have the same mean coordination. Top: No constraint was imposed
on the heterogeneity ($\lambda_{H}=0$). The bulk modulus varies little along
any vertical line, suggesting that it may be well described by the mean
coordination. Bottom: Heterogeneous structures generated with
$\lambda_{H}=1.5$. The mean coordination does not dictate the bulk modulus as
well as in the previous case, since $B$ varies along vertical lines. Figure 4:
(Color online). Variation of the bulk modulus as a function of the mean
coordination. Light diamonds (dark circles) represent data from CRNs generated
by SA without (with) heterogeneity cost. For comparison, crosses show tight-
binding results from Mathioudakis et al. (2004). Inset: solid line
representing the power law fit to data for homogeneous CRNs.
For the homogeneous case, the bulk modulus depended mainly on the mean atomic
coordination, with a Spearman’s rank correlation coefficient Spearman (1904)
$\rho=0.98$ – supporting previous studies also pointing out this trend
Phillips (1979); Thorpe (1983); He and Thorpe (1985); Mathioudakis et al.
(2004). For heterogeneous networks, the dependence on the mean correlation
diminished a little, with $\rho=0.9$. However, considering only the region
with mean coordination $2.5<\overline{z}<3.5$, both correlations drop to
$\rho=0.94$ and $\rho=0.83$, respectively. We explain the larger decrease of
$\rho$ for heterogeneous structures structures this way: since $sp$ hybridized
atoms form floppy Thorpe (1983) phases with null bulk modulus, some
heterogeneous CRNs, such as one made of $50\%$ $sp^{3}$ and $50\%$ $sp$
carbon, will have a very small bulk modulus due to the large floppy region.
Such small bulk modulus will not be observed in a $100\%$ $sp^{2}$ network,
even though both structures have the same mean coordination. If we put
$\lambda_{H}=0$ and let homogeneous structures form, the $sp$ carbon will not
segregate, but it will be incorporated between $sp^{3}$ centers. Thus, there
will be no large floppy regions.
The bulk modulus is also plotted as a function of the mean coordination (Fig.
4). Following previous studies He and Thorpe (1985); Djordjevic and Thorpe
(1997); Mathioudakis et al. (2004), we fitted a power law to the bulk modulus
data for the set of homogeneous CRNs,
$B(\overline{z})=B_{0}\left(\overline{z}-\overline{z}_{f}\right)^{\nu}$ (6)
We found the phase transition from rigid to floppy states Thorpe (1983) to
occur at mean coordination $\overline{z}_{f}=2.10\pm 0.11$, with $B_{0}=140\pm
26$ GPa and $\nu=1.51\pm 0.17$. These results, particularly the exponent, are
close to those reported by He and Thorpe (1985); Djordjevic and Thorpe (1997);
Mathioudakis et al. (2004), as compared in Table 3. The slight deviation for
$\overline{z}_{f}$ can be explained by the size of the simulation cell: Even
for relatively large cells containing $512$ atoms, there is a chance that a
non-floppy carbon chain of $sp^{2}$ or $sp^{3}$ atoms will percolate the
periodic cell. This was not observed by Mathioudkis et al. – whose results
were extrapolated for mean coordination bellow $\overline{z}=2.68$ – nor by He
and Thorpe He and Thorpe (1985) and Djordjevic and Thorpe Djordjevic and
Thorpe (1997), because of a limitation of the bond depleting method which
causes the simulation cell to collapse for small $\overline{z}$.
Table 3: Comparison of the fitted parameters for Eq. (6). Reference | $\overline{z}_{f}$ | $\nu$
---|---|---
He and Thorpe He and Thorpe (1985) | $2.4$ | $1.5\pm 0.2$
Djordjevic and Thorpe Djordjevic and Thorpe (1997) | $2.4$ | $1.4$
Mathioudakis et al. Mathioudakis et al. (2004) | $2.33$ | $1.5\pm 0.1$
This Work | $2.10\pm 0.11$ | $1.51\pm 0.17$
It must be pointed out that long-range effects may be taken into account after
the amorphous structures are generated. In our case, we employed the Brenner
potential, which does not include such interactions, for the relaxation and
calculation of the bulk moduli. It is reasonable to assume that the long $sp$
chains are weakly binded by dispersive forces, so that the bulk modulus of
floppy networks do not vanish completely. So, it is quite possible that using
a potential model for the calculation of the elastic properties that includes
van der Waals interactions would yield higher bulk moduli for low
$\overline{z}$.
After generating the CRNs, the set of $90$ structures may be seen as the
initial set of an expanding database that may be used, for instance, to
extract structural information from experimental results. As an example of
this application, we compared the calculated radial distribution functions
(RDF) for the set of CRNs with results from the literature in Fig. 5. Using
all our available structures, we searched for CRNs that would best reproduce
the RDF of some experimental materials: one sputtered a-C Li and Lannin (1990)
and one ta-C Gilkes et al. (1995). The first experimental a-C was prepared by
rf sputtering, while the ta-C was grown using filtered cathodic arc. Using
least-squares fitting, we found that a $88\%$ $sp^{2}$ and $12\%$ $sp$
homogeneous structure best reproduced the RDF of the sputtered a-C, while a
heterogeneous $50\%$ $sp^{3}$/$sp^{2}$ structure best described the ta-C one.
Furthermore, by adding an additional degree of freedom to scale the $r$
variable, the structure that best described the ta-C one was CRN containing
$80\%$ $sp^{3}$, $10\%$ $sp^{2}$ and $10\%$ $sp$ carbon atoms.
The fit is not optimal: The mean coordinations of the experimental structures
were reported to be $3.34$ and $3.9$, while the fitted mean coordination were
$2.9$ and $3.5$ ($3.74$ if we add the additional degree of freedom). This
error could be related to the potential used in the relaxation process, or to
the number of iterations used to generate the structures, which in turn
control their angular and bond distribution widths. However, even though the
CRNs were not generated for this purpose, the comparison between experimental
and theoretical RDFs was performed to show that the generated CRNs indeed
present similarities with the experimental structures, to such an extent that
they are able to reproduce experimental RDFs. It should be noted that it has
never been in the scope of this work to present a method to extract structural
information or create structures based on given experimental RDFs. There are
specialized methods for this tasks, such as Reverse Monte Carlo McGreevy and
Pusztai (1988) and Hybrid Reverse Monte Carlo Opletal et al. (2002), which are
much more efficient to extract information from experimental RDFs, but not to
generate CRNs having particular coordination or structural constraints.
Figure 5: (Color online). Reduced Radial Distribution Function $G(r)$. Top
curves: sputtered a-C Li and Lannin (1990) (dotted line) and best fit ($100\%$
$sp^{2}$ structure, green solid line). Middle curves: ta-C Gilkes et al.
(1995) (dotted line) and best fit (heterogeneous $50\%$ $sp^{2}$ and $50\%$
$sp^{3}$ structure, red solid line). Bottom curves: the same experimental
curve was used Gilkes et al. (1995), but the theoretical curve could also
scale in the $r$ axis. The yellow curve represents a CRN containing $90\%$
$sp^{3}$, $10\%$ $sp^{2}$ and $10\%$ $sp$ carbon atoms.
Finally, we generated a set of structures to test the performance of the
algorithm to generate $100\%$ $sp^{3}$ structures. CRNs having 64, 128, 256
and 512 atoms were generated, and two structures were created for a given
number of atoms. After relaxation using the same strategy as before (molecular
dynamics followed by Hessian-driven relaxation), the calculated angular widths
ranged from $11.8^{\circ}$ to $13.9^{\circ}$, with a mean value of
$13.2^{\circ}$. We did not observe a significant correlation between the
angular width and the number of atoms, and the angular distribution was
relatively symmetric with an average of $109.07^{\circ}$. For comparison,
high-quality tetrahedral networks having an angular width of $9.19^{\circ}$
have been assembled using a modified version of the WWW algorithm by Barkema
and Mousseau Barkema and Mousseau (2000). Also, a reduction of the angular
distribution width may be possible by increasing the number of steps during
the SA or by extending the relaxation process after the structure is
generated. So, although our strategy is not optimized for a-D as other
methods, it is flexible enough to generate both a-D and a-C with various
hybridizations and structural constraints.
## IV Conclusion
In this paper, we described the computational creation of carbon CRNs
employing the Simulated Annealing algorithm. We proposed a numerically simple
Cost Function able to yield extremely different amorphous materials. As an
example of the capabilities of our algorithm, we generated amorphous
structures spawning nearly all possible combinations of $sp^{3}$, $sp^{2}$ and
$sp$ hybridized centers, and then calculated their bulk moduli using Brenner’s
potential Brenner et al. (2002). We were also able to easily modify our CF to
create heterogeneous materials, in which atoms with the same hybridization
tend to segregate. With the set of homogeneous structures, we observed a phase
transition from floppy to rigid networks, and a power-law fitting of the bulk
modulus dependency on mean atomic coordination was in close agreement with the
literature. However, we noticed that the mean coordination $\overline{z}$ did
not correlate with the bulk modulus of heterogeneous networks as well as it
did for homogeneous ones. This indicates that the heterogeneity may play a
very important role in dictating the elastic properties of a-C.
The strategy we described is completely universal and customizable, and
modifications can be easily made to include other chemical elements, such as
hydrogen, and to control the presence of other features, such as rings and
dangling bonds. Once CRNs with particular features are generated, their
physical properties can be estimated using more sophisticated Hamiltonians,
including ab initio calculations, whenever it is computationally feasible.
Further extension of this approach to include microstructural constraints in
the process of generating CRNs (such as those possibly responsible for
ultrahigh hardness in N. Dubrovinskaia and Liebske (2005)) depends only on the
availability of suitable computational resources.
## V Acknowledgements
The authors thank Ricardo Vargas Dorneles and Gunther Johannes Lewczuk
Gerhardt, who granted access to the computer clusters where most of the
calculations presented in this paper were performed, and John Muller and
Veronica Gouvea for reviewing this manuscript. This work was partially
supported by the Brazilian agency CNPq (Conselho Nacional de Desenvolvimento
Científico e Tecnológico). We would also like to thank Universidade de Caixas
do Sul for the computer resources (GridUCS).
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|
arxiv-papers
| 2010-03-02T17:37:59 |
2024-09-04T02:49:08.726465
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "F. H. Jornada, V. Gava, A. L. Martinotto, L. A. Cassol, C. A.\n Perottoni",
"submitter": "Felipe Jornada",
"url": "https://arxiv.org/abs/1003.0641"
}
|
1003.0684
|
# Is it possible to relate MOND with Hořava Gravity?
Juan M. Romero, R. Bernal-Jaquez, O. González-Gaxiola,
Departamento de Matemáticas Aplicadas y Sistemas,
Universidad Autónoma Metropolitana-Cuajimalpa
México, D.F 01120, México
jromero@correo.cua.uam.mxrbernal@correo.cua.uam.mxogonzalez@correo.cua.uam.mx
###### Abstract
In this work we present a scalar field theory invariant under space-time
anisotropic transformations with a dynamic exponent $z$. It is shown that this
theory possess symmetries similar to Hořava gravity and that in the limit
$z=0$ the equations of motion of the non-relativistic MOND theory are
obtained. This result allow us to conjecture the existence of a Hořava type
gravity that in the limit $z=0$ is consistent with MOND.
Keyswords: Gravity; Mond; Horava Gravity
PACS: 04.50.Kd, 04.50.-h
## 1 Introduction
In recent years, various modifications have been proposed to general
relativity. One of the most notorious at a phenomenological level, is the
relativistic version of the so called Modified Newtonian Dynamics (MOND) [1].
Notoriously, without making use of dark matter, MOND successfully explains the
anomalous dynamics of different astrophysical objects. For example, it
explains the rotation curves of different galaxies and the Tully-Fisher
relation [2]. Besides, MOND has an explanation for the so called Pioneer
anomaly [3]. We also have to mention that, MOND presents problems to predict
the cluster’s of galaxies dynamics, in this case however, the baryonic mass
has not been measured with certainty [4].
MOND’s starting point is to assume that for small accelerations, about
$a_{0}\approx 10^{-8}cm/s^{2}$, Newton’s second law takes on the form [1]
$\displaystyle\mu\left(\frac{|\vec{a}|}{a_{0}}\right)m\vec{a}=\vec{F},$ (1)
with $\mu(u)$ defined as a function that satisfies
$\displaystyle\mu(u)=\left\\{\begin{array}[]{ll}1&\quad{\rm if}\quad u>>1,\\\
u&\quad{\rm if}\quad u<<1.\end{array}\right.$ (4)
The Newtonian regime is obtained if $u>>1$, meanwhile the MOND’s regime is
obtained if $u<<1$. In spite of is phenomenological success, MOND has problems
at a theoretical level, for example, the energy for one particle is not
conserved although it is conserved in several modified versions of the theory
[5].
In the non-relativistic gravitational field regime, MOND is consistent with
[6]
$\displaystyle\vec{\nabla}\cdot\left(\mu\left(\frac{|\vec{\nabla}\phi|}{a_{0}}\right)\vec{\nabla}\phi\right)=4\pi
G\rho,$ (5)
that in the MOND limit takes on the form
$\displaystyle\vec{\nabla}\cdot\left(\frac{|\vec{\nabla}\phi|}{a_{0}}\vec{\nabla}\phi\right)=4\pi
G\rho.$ (6)
This model gives a non-relativistic gravitational field description and does
not present problems with conserved quantities. However, using this model it
is not possible to attack relativistic problems, such as gravitational lenses
or cosmological problems. To analyse these problems is necessary a
relativistic MOND. Though there are several relativistic versions of MOND
theory, the most complete is the so called $TeVeS$ theory [7], see also [8].
This version is compatible with Eq. (6) in the non-relativistic limit and has
been successful in explaining several phenomenological facts. It is worthy to
mention that, with the present WMAP measurements, this theory cannot be
discarded [9, 10]. With finer WMAP experimental measurements, this theory may
be confirmed, modify or even discarded.
However, at a theoretical level, $TeVeS$ still has some inconsistencies, for
example, it has some dynamical problems as instabilities may appear [11, 12].
Therefore cannot be considered as a finished theory. In fact, modifications to
this theory has been recently proposed [11] and also a new relativistic
version of MOND has been recently proposed [13].
In the other hand, Hořava has recently proposed a modified version of general
relativity that in principle is renormalizable and free of ghosts [14]. This
gravity assumes that space-time is compatible with the anisotropic
transformations
$\displaystyle t\to b^{z}t,\qquad\vec{x}\to b\vec{x},$ (7)
where $z$ is a dynamic exponent. As a consequence of the transformation given
in Eq. (7), the usual dispersion relation is substituted by
$\displaystyle
P^{2}_{0}-\tilde{G}\left(\vec{P}^{2}\right)^{z}=0,\qquad\tilde{G}={\rm
constant}.$ (8)
A remarkable point about this dispersion relation is that, it is not obtained
from a geodesic equation [15, 16, 17, 18]. Hořava gravity is not compatible
with Lorentz’s transformations neither invariant under all the
diffeomorphisms, nevertheless for long distances the usual relativity theory
is regained. This theory has some dynamical problems [19, 20, 21], and cannot
be considered as complete. In fact, some recent proposals have been made in
order to improve it [22].
In this work, we will show a scalar field model compatible with the
transformations stated in Eq. (7) whose dynamics in the limit $z=0$ reduces to
MOND Eq. (6). It is shown that this theory possess Weyl’s symmetries similar
to Hořava gravity. This allow us to conjecture that an anisotropic gravity
theory in space-time could exist such that, at $z=0$ reduces to a MOND type
gravity; at $z=1$ standard gravity is regained and at $z=3$ we regain a Hořava
type gravity compatible with quantum mechanics. It worthy to mention that, the
anisotropic transformations in Eq. (7) are of importance for the $AdS/CFT$
non-relativistic duality by means of which efforts are made in order to relate
condensed matter phenomena with string theory [23, 24]. This would make
possible the existence of $AdS/CFT$ duality with a MOND type theory.
This manuscript is organized as follows: In section 2 we present our system
and its equations of motion are studied. Section 3 is devoted to study the
conserved quantities the system, in Section 4 we study the algebra of the
conserved quantities and in Section 5 we present a summary of our results.
## 2 Action
Consider the Action invariant under the transformation Eq. (7)
$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
dx^{d}dt\left[a\left(\frac{\partial\phi(\vec{x},t)}{\partial
t}\right)^{\frac{z+d}{z}}+\gamma\left(\frac{\partial\phi(\vec{x},t)}{\partial
x^{i}}\frac{\partial\phi(\vec{x},t)}{\partial
x^{i}}\right)^{\frac{z+d}{2}}\right]$ (9) $\displaystyle=$ $\displaystyle\int
dx^{d}dt\left[a\left(\frac{\partial\phi(\vec{x},t)}{\partial
t}\right)^{\frac{z+d}{z}}+\gamma\left(\vec{\nabla}\phi\cdot\vec{\nabla}\phi\right)^{\frac{z+d}{2}}\right].$
Note that, if we define the non-zero elements of the metric $g_{\mu\nu}$ as
$g_{00}=1,g_{ij}=\delta_{ij}$, we have $g={\rm det}g_{\mu\nu}$ and therefore
the Action Eq. (9) can be written as
$\displaystyle S=\int
dx^{d}dt\sqrt{g}\left[a\left(g^{00}\frac{\partial\phi(\vec{x},t)}{\partial
t}\frac{\partial\phi(\vec{x},t)}{\partial
t}\right)^{\frac{z+d}{2z}}+\gamma\left(g^{ij}\frac{\partial\phi(\vec{x},t)}{\partial
x^{i}}\frac{\partial\phi(\vec{x},t)}{\partial
x^{j}}\right)^{\frac{z+d}{2}}\right],$ (10)
this expression is invariant under Weyl’s anisotropic transformations
$\displaystyle g_{00}\to\Omega^{2z}(\vec{x},t)g_{00},\qquad g_{ij}\to
g_{ij}\Omega^{2}(\vec{x},t).$ (11)
This kind of symmetry is similar to the one present at Hořava gravity [14].
The equation of motion for Eq. (9) is
$\displaystyle a\left(\frac{z+d}{z}\right)\frac{\partial}{\partial
t}\left(\frac{\partial\phi}{\partial
t}\right)^{\frac{d}{z}}+\gamma\left(z+d\right)\frac{\partial}{\partial
x_{i}}\left(\left(\frac{\partial\phi}{\partial x^{j}}\frac{\partial}{\partial
x^{j}}\right)^{\frac{z+d-2}{2}}\frac{\partial\phi}{\partial x^{i}}\right)=0,$
that can be written as
$\displaystyle a\left(\frac{z+d}{z}\right)\frac{\partial}{\partial
t}\left(\frac{\partial\phi}{\partial
t}\right)^{\frac{d}{z}}+\gamma\left(z+d\right)\vec{\nabla}\cdot\left(|\vec{\nabla}\phi|^{d+z-2}\vec{\nabla}\phi\right)=0.$
(12)
If a source $\rho$ is consider, the Action Eq. (9) is now given by
$\displaystyle S=\int dx^{d}dt\left[a\left(\frac{\partial\phi}{\partial
t}\right)^{\frac{z+d}{z}}+\gamma\left(\vec{\nabla}\phi\cdot\vec{\nabla}\phi\right)^{\frac{z+d}{2}}+\phi\rho\right].$
(13)
If under scaling the source is transformed as $\rho\to\Omega^{-(z+d)}\rho,$
then Eq. (13) is invariant under Weyl’s symmetry, Eq. (11). The equation of
motion given by the Action Eq. (13) is
$\displaystyle a\left(\frac{z+d}{z}\right)\frac{\partial}{\partial
t}\left(\frac{\partial\phi}{\partial
t}\right)^{\frac{d}{z}}+\gamma\left(z+d\right)\vec{\nabla}\cdot\left(|\vec{\nabla}\phi|^{d+z-2}\vec{\nabla}\phi\right)=-\rho.$
(14)
It can be noticed that, at the limit $z\to 0$ the first term of the Action in
Eq. (13) is constant and we can obtain the effective Action as
$\displaystyle S=\int
dx^{d}\left[\gamma\left(\vec{\nabla}\phi\cdot\vec{\nabla}\phi\right)^{\frac{d}{2}}+\phi\rho\right],$
(15)
whose equation of motion is
$\displaystyle\gamma
d\vec{\nabla}\cdot\left(|\vec{\nabla}\phi|^{d-2}\vec{\nabla}\phi\right)=-\rho.$
(16)
If $d=3$ and $\gamma=-1/(12\pi Ga_{0})$ we obtain the MOND non-relativistic
equation of motion Eq. (6), [6].
Therefore the system under study contains the non-relativistic MOND’s theory
and coincides with Hořava gravity symmetries. This fact, make it possible to
conjecture the existence of a Hořava type gravity that, in $z=0$ limit reduces
to MOND. Note that, as the Hořava gravity must be valid in the quantum regime,
the fundamental constant is Planck’s mass $M_{P}$. This constant seems to be
non-related with MOND’s fundamental constant $a_{0}$. However, $a_{0}$ can be
written as $a_{0}\approx m_{N}c\left(6M_{P}^{3}t_{p}\right)^{-1},$ where
$m_{N}$ is the proton mass. It is possible that in this conjectured gravity
this type of relations could arise in a natural way. It can be noticed that,
in this relation we have a collection of apparently dissimilar quantities,
however, by means of the Holographic principle, this type of relations appear
[25].
## 3 Noether’s theorem
In this section, we will find out the conserved quantities of the Action Eq.
(9). First, note that the canonical momentum is given by
$\displaystyle\Pi=\frac{\partial{\cal
L}}{\partial\dot{\phi}}=a\frac{z+d}{z}\left(\dot{\phi}\right)^{\frac{d}{z}},$
(17)
therefore, the equation of motion can be written as
$\displaystyle\frac{\partial\Pi}{\partial
t}+\gamma(z+d)\vec{\nabla}\cdot\left(|\vec{\nabla}\phi|^{d+z-2}\vec{\nabla}\phi\right)=-\rho.$
(18)
Considering Noether’s theorem, we know that the temporal part of
$\displaystyle\int d^{d}J_{\mu}=\int dx^{d}\left(\frac{\partial{\cal
L}}{\partial(\partial^{\mu}\phi)}\frac{\partial\phi}{\partial
x^{\nu}}-g_{\mu\nu}{\cal L}\right)\delta x^{\nu}$ (19)
is conserved. Taking into account that the Action Eq. (9) is invariant under
temporal translations, we can conclude that the Hamiltonian is conserved
$\displaystyle H=\int dx^{d}{\cal H}=\int
dx^{d}\left(\frac{ad}{z}\left(\frac{z}{a(z+d)}\right)^{\frac{d+z}{d}}\Pi^{\frac{z+d}{d}}-\gamma|\vec{\nabla}\phi|^{z+d}\right).$
(20)
Besides, the momentum is conserved
$\displaystyle P_{i}=\int dx^{d}p_{i}=\int
dx^{d}\Pi\frac{\partial\phi}{\partial x^{i}}=\int
dx^{d}a\frac{z+d}{z}\left(\dot{\phi}\right)^{\frac{d}{z}}\frac{\partial\phi}{\partial
x^{i}}$ (21)
as well as the angular momentum
$\displaystyle L_{i}=-\int
dx^{d}\Pi\epsilon_{ijk}x_{j}\frac{\partial\phi}{\partial x^{k}}=-\int
dx^{d}a\frac{z+d}{z}\left(\dot{\phi}\right)^{\frac{d}{z}}\epsilon_{ijk}x_{j}\frac{\partial\phi}{\partial
x^{k}}.$ (22)
The scaling generator
$\displaystyle D=\int dx^{d}\left(zt{\cal H}+p_{i}x^{i}\right)$ (23)
is also conserved. This quantities form the algebra
$\displaystyle\\{H,P_{i}\\}$ $\displaystyle=$ $\displaystyle 0,$ (24)
$\displaystyle\\{H,L_{kl}\\}$ $\displaystyle=$ $\displaystyle 0,$ (25)
$\displaystyle\\{H,D\\}$ $\displaystyle=$ $\displaystyle zH,$ (26)
$\displaystyle\\{D,P_{i}\\}$ $\displaystyle=$ $\displaystyle-P_{i},$ (27)
$\displaystyle\\{D,L_{i}\\}$ $\displaystyle=$ $\displaystyle 0,$ (28)
$\displaystyle\\{P_{i},P_{j}\\}$ $\displaystyle=$ $\displaystyle 0,$ (29)
$\displaystyle\\{P_{i},L_{j}\\}$ $\displaystyle=$
$\displaystyle\epsilon_{ijm}P_{m},$ (30) $\displaystyle\\{L_{i},L_{j}\\}$
$\displaystyle=$ $\displaystyle\epsilon_{ijk}L_{k}.$ (31)
This type of algebraic relations are characteristic of anisotropic scale
invariant systems with dynamic exponent $z$.
## 4 Summary
In this work we have presented a scalar field theory invariant under space-
time anisotropic transformations with a dynamic exponent $z$. It is shown that
this theory possess symmetries similar to Hořava gravity, in particular Weyl’s
symmetries. Also, it is shown that in the limit $z=0$ the equations of motion
of the non-relativistic MOND theory are obtained. This result make it possible
to conjecture the existence of a Hořava type gravity that in the limit $z=0$
is consistent with MOND. Also, conserved quantities and their algebraic
relations have been studied.
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* [21] C. Bogdanos, E. N. Saridakis Perturbative instabilities in Horava gravity, Class. Quant. Grav. 27 075005 (2010).
* [22] D. Blas, O. Pujolas, S. Sibiryakov, A healthy extension of Hořava gravity, arXiv:0909.3525
* [23] D. T. Son, Toward an AdS/cold atoms correspondence: a geometric realization of the Schroedinger symmetry, Phys. Rev. D 78, 046003 (2008).
* [24] K. Balasubramanian, J. McGreevy, Gravity duals for non-relativistic CFTs Phys. Rev. Lett. 101, 061601 (2008).
* [25] G. A. Mena Marugan, S. Carneiro, Holography and the large number hypothesis, Phys. Rev. D 65 (2002) 087303.
|
arxiv-papers
| 2010-03-02T21:14:20 |
2024-09-04T02:49:08.735525
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Juan M. Romero, R. Bernal-Jaquez, O. Gonzalez-Gaxiola",
"submitter": "Juan Manuel Romero",
"url": "https://arxiv.org/abs/1003.0684"
}
|
1003.0921
|
# Experimental Setup for the Measurement of the Thermoelectric Power in Zero
and Applied Magnetic Field
Eundeok Mun1, Sergey L. Bud’ko1, Milton S. Torikachvili2, Paul C. Canfield1
1Ames Laboratory US DOE and Department of Physics and Astronomy, Iowa State
University, Ames, IA 50011, USA 2Department of Physics, San Diego State
University, San Diego, California 92182-1233 USA canfield@ameslab.gov
###### Abstract
An experimental setup was developed for the measurement of the thermoelectric
power (TEP, Seebeck coefficient) in the temperature range from 2 to 350 K and
magnetic fields up to 140 kOe. The system was built to fit in a commercial
cryostat and is versatile, accurate and automated; using two heaters and two
thermometers increases the accuracy of the TEP measurement. High density data
of temperature sweeps from 2 to 350 K can be acquired in under 16 hours and
high density data of isothermal field sweeps from 0 to 140 kOe can be obtained
in under 2 hours. Calibrations for the system have been performed on a
platinum wire and Bi2Sr2CaCu2O8+δ high $T_{c}$ superconductors. The measured
TEP of phosphor-bronze (voltage lead wire) turns to be very small, where the
absolute TEP value of phosphor-bronze wire is much less than 0.5 $\mu$V/K
below 80 K. For copper and platinum wires measured against to the phosphor-
bronze wire, the agreement between measured results and the literature data is
good. To demonstrate the applied magnetic field response of the system, we
report measurements of the TEP on single crystal samples of LaAgSb2 and
CeAgSb2 in fields up to 140 kOe.
###### pacs:
06.60.Ei, 07.20.Mc, 72.15.Jf
Keywords: thermoelectric power, measurement setup, calibration
## 1 Introduction
Since its discovery in 1821 by Thomas Johann Seebeck, relatively few studies
of the magnetic field dependence of the thermoelectric power (TEP) were
carried out, mostly in pure metals [1]. However, over the past few decades,
the magnetic field-dependent TEP studies of many materials ranging from
magnetic multilayers [2] to high $T_{c}$ superconductors [3], to the electron-
topological transition [4] and to strongly correlated electron systems [5, 6,
7] have been provided useful information. Intensive efforts also have been
made in the search for highly efficient thermoelectric materials. This being
said, the measurement of the intrinsic TEP is particularly difficult even in
simple metals such as copper or gold. This is due to the small magnitude of
TEP at low temperatures and its sensitivity to the presence of small
concentrations of impurities, where magnetic impurities can enhance the TEP
below certain temperatures by means of the Kondo effect [1].
Few experimental details have been given in the literature concerning the
measurement setups and the procedure for calibration of lead (as in contacting
the sample, not Pb) wires [8, 9, 10]. Detailed descriptions of the measurement
techniques at low temperatures and high magnetic fields can be found in Refs.
[9, 10]. In this article, we describe the development of an experimental setup
for TEP measurement in a Quantum Design (QD), Physical Property Measurement
System (PPMS). The PPMS sample puck provides both thermal and electrical
contacts to the sample. The merits of this technique are (i) it is easy to
implement using two commercial, Cernox thin-film, resistance cryogenic
temperature sensors and two strain gauge heaters and (ii) it is easy to
control the temperature and magnetic field of the system using the PPMS
platform. Using the PPMS temperature-magnetic field ($T-H$) environment and
the two heaters and two thermometers, an alternating heating method allows for
measurements of the TEP of materials over a temperature range from 2 to 350 K
and magnetic fields up to 140 kOe. The alternating heating method we use
improves the resolution by a factor of two and provides a reliable temperature
gradient. For the measurement, the sample is mounted directly between the two
Cernox thermometers each of which is heated by a strain gauge heater with
constant DC current. An important component of this technique involves the use
of phosphor-bronze lead wires to reduce the background TEP and magneto-
thermoelectric power (MTEP) associated with the lead wires.
## 2 Experimental Setup
In this section we will describe our specific sample holder (sample stage) and
explain data acquisition process. This measurement setup was designed to fit
PPMS cryostat used to control the temperature and magnetic field of the
system. All instruments (current sources, voltmeters, switch system and PPMS)
were controlled by National Instruments LabVIEW software. The sample holder
can be easily modified and adapted to other cryogenic systems, including those
with higher magnetic fields and lower temperatures.
### 2.1 Sample Holder
Figures 1 (a) and (b) show a schematic diagram of the sample stage built on
the PPMS sample puck and a photograph of actual sample stage. The magnetic
field is applied perpendicular to the plane of the heaters, thermometers and
puck platform. Two sample stages are attached to a circular copper heat sink
positioned on the 23 mm diameter PPMS sample puck that, when in use, is
shielded by a gold plated copper cap (not shown). We use Cernox sensors
(CX-1050-SD package) as thermometers that provide high sensitivity at low
temperatures, good sensitivity over a broad range and low magnetic field-
induced errors. The dimensions of this package (1.9$\times$1.1$\times$3.2 mm3)
are large enough to attach a heater and sample simultaneously to the package
surface. Strain gauges (heaters), 0.2 $\times$ 1.4 mm2 and typically $R\sim$
120 $\Omega$, are glued to the top of the Cernox thermometers using Stycast
1266 epoxy. In order to insure thermal isolation, the heat sink (PPMS puck)
and the sample stage was separated by a thin (1 mm thickness) G-10 plate. This
G-10 plate was glued on the bottom of the Cernox thermometer using the Stycast
1266 epoxy. From several test runs we observed that the two Cernox wires and
two heater wires provided enough cooling power to the sample stage since the
strain gauge and Cernox each have low thermal mass. Each sample stage
including heater, thermometer and G-10 plate was glued to the copper heat sink
with GE 7301 varnish, so that it was easy to remove by dissolving the GE-
varnish with ethanol. Because of the constraint of the PPMS sample puck, the
distance between two stages can be varied from $\sim$1.5 mm to $\sim$ 6 mm.
Large flexibility with respect to the sample size can therefore be gained
since the precise configuration of the thermal stage can be easily adjusted.
If the sample length is smaller than 1.5 mm, it is hard to establish a
temperature difference ($\Delta T$) because both thermal stages are isolated
from the heat sink. Typically, samples with length varying from 2 to 7 mm can
be measured. All wires on the measurement cell are thermally anchored to the
heat sink. The TEP measurement was made with the PPMS operating in the high
vacuum mode with pressure $\sim 10^{-5}$ torr.
For mounting the sample, and measuring the voltage, two different
configurations were tested (Fig. 1 (c)). First, samples were mounted on the
two sample stages with GE-varnish. The voltage difference $\Delta V$ is
measured using 25 $\mu$m diameter copper wire or phosphor-bronze wire attached
to the sample using silver epoxy as shown in the top of Fig. 1 (c).
Alternatively, samples were directly mounted to the sample stages using DuPont
4929N silver paste. The silver paste provides good thermal and electrical
contact between the sample and the gold plated layer on the surface of the
Cernox package (bottom of Fig. 1 (c)). The copper wire or phosphor-bronze lead
wire is soldered to this gold plated layer. In this case the voltage
difference is obtained by measuring the voltage difference between two sample
stages. Since the data was taken in a steady state, by assuming the
temperature of the gold layer is the same as silver paste, the TEP
contribution of the sample stage can be ignored. Since the silver paste can be
dissolved in hexyl acetate, the sample can be easily detached by carefully
adding small amount of this solvent without degrading Stycast or GE-varnish.
We ran several test measurements to compare thermal coupling between sample
and thermometers by using silver paste and GE-varnish. We found it to be
essentially the same for both cases. In general the TEP measurement was
performed with the later, silver paste configuration, because the sample
mounting and removal were easier than GE-varnish. The GE-varnish configuration
is preferred mainly when good electrical contact between the sample and the
gold layer of the thermometer with silver paste can not be established. For
example, when we measure the TEP of the Bi2Sr2CaCu2O8+δ (Bi2212) high $T_{c}$
samples for calibration it was hard to get good electrical contact (see next
section).
### 2.2 Determination of $\Delta T$, $\Delta V$, $T_{av}$ and $S$
A block diagram of the TEP measurement is shown in Fig. 1 (d). Since the PPMS
sample puck provides only 12 wires, they had to be used frugally: Six wires
total were used for the two Cernox sensors, which were connected in series,
four wires were used for the heaters (2 each), and two wires were used for the
TEP voltage. The resistance of each Cernox is measured with a Hewlett Packard
34420A nanovoltmeter via a Keithley 7001 switch system with a Keithley 7059
low voltage scanner card. The current was supplied to the Cernox thermometers
by a Keithley 220 programmable current source. A temperature difference
($\Delta T$) across the sample was established by applying a DC current with
two Keithley 220 programmable current source alternately through one of the
strain gauges at a time, while the voltage difference ($\Delta V$) across the
sample was monitored independently with a Hewlett Packard 34420A
nanovoltmeter.
When we apply a small temperature difference across the sample, the
temperatures ($T_{1}(t)$, $T_{2}(t)$) and a voltage ($V(t)$) are recorded as a
function of time, as illustrated in Fig. 2. $T_{1}$ and $T_{2}$ are the
temperatures of the two Cernox thermometers that the sample spans. $t_{i}$
represents the time just before alternating power to the heaters (e.g. #1 on
and #2 off) and $t_{f}$ indicates the time just before the next power switch
(e.g. #1 off and #2 on). As shown in Figs. 2 (c) and 2 (d) in particular, from
a linear fit of the measured voltage and temperature as a function of time,
$\Delta T$ and $\Delta V$, respectively, the sample temperature $T_{av}$ and
the TEP ($S=-\Delta V/\Delta T$) are calculated using the following equations.
$\displaystyle{}2\Delta T$ $\displaystyle=$
$\displaystyle(T_{2f}-T_{1f})+(T_{1i}-T_{2i})$ $\displaystyle 2\Delta V$
$\displaystyle=$ $\displaystyle V_{f}-V_{i}$ $\displaystyle T_{av}$
$\displaystyle=$ $\displaystyle\frac{(T_{2f}+T_{1f})+(T_{2i}+T_{1i})}{4}$
Since the temperature difference is generated by alternately applying power to
one of the heaters, the measured voltage corresponds to 2$\Delta V$. Thus, the
TEP of sample is calculated by $S=-2\Delta V/2\Delta T$. Figure 2 shows the
data corresponding to a measurement performed near 55 K on a platinum (Pt)
wire sample, using phosphor-bronze lead wires. The puck temperature was ramped
at the rate of 0.1 K/min. A complete cycle, used to determine $\Delta T$ and
$\Delta V$, took a time period ($\tau$) of 50 sec. The parameters ($T_{1i}$,
$T_{1f}$, $T_{2i}$, $T_{2f}$, $V_{i}$ and $V_{f}$) were determined by a linear
fit of the data as a function of time as shown in Fig. 2 (c) and (d).
The heater current ($I$) and time period ($\tau$), needed to generate given
$\Delta T$, are not easy to estimate a prior, because of the temperature
dependence of multiple parameters, such as the thermal conductivity and heat
capacity of the sample, sample stage and all electrical wiring of the
apparatus. Therefore, the current and measurement time for given $\Delta T$
were determined empirically at several temperatures by applying constant power
to one of the heaters. For determining the final temperature and voltage,
after switching heater from one to the other, the number of data point for
linear fit was selected within constant temperature and voltage region as a
function of time. Although it depends on the sample under investigation,
typical values of $\tau\sim$ 45 sec at 2 K and $\tau\sim$ 150 sec at 300 K for
this setup allowed an accurate determination of the final values of $T_{f}$
and $V_{f}$. Typical values of the heater current were $I\sim$ 0.8 mA to
generate $\Delta T\sim$ 0.2 K at 2 K, and $I\sim$ 5 mA to generate $\Delta
T\sim$ 1.0 K at 300 K.
By utilizing two heaters and an alternating gradient $\Delta T$, we avoid
problems associated with offset voltages. $V_{i}$ and $V_{f}$ represent the
thermal voltages in the circuit, which include spurious voltages and the TEP
of lead wires. In fact, for very low values of the TEP, it is often necessary
to consider an offset voltage ($V_{off}$) in the system and circuit. A common
source of spurious voltage, for example, is the wiring of the system from the
voltmeter to the sample space since there is a thermal gradient and several
soldering points between various wires. We found that the value of $V_{off}$
for this setup depended on temperature; it was $\sim$0.5 $\mu$V around 300 K
and $\sim$ -1.5 $\mu$V around 10 K. If we suppose that $V_{off}$ is
independent of the small $\Delta T$ across the sample and has a small
temperature dependence as a function of time (adiabatic approximation)
$V_{off}$ can be easily canceled out using two heaters as shown in Fig. 2 (d).
In the early stage of testing this measurement setup, the process of
collecting data was checked by measuring the constantan wire (100 $\mu$m
diameter) against copper wire ($\sim$ 20 $\mu$m diameter). Since constantan
wire has been known to have large TEP value compared to copper wire, the
system can be tested without correcting the contribution of copper wire as
shown in Fig. 3. In this test running, we used the following two protocols.
Firstly, a stable temperature method was applied; in this measurement the
sample puck was held at a constant temperature and the TEP of the constantan
wire using either one heater or two heaters was measured and found to be
basically same within error bar of this measurement setup. However, the TEP
data for the constantan wire showed a small hysteresis upon cooling and
warming between 50 and 260 K with a maximum difference of about 2 %. The
origin of this hysteresis is not clear, we expect this that it is based on
different relaxation times to stablize the temperatures of the system.
Secondly we adopted an alternate method which was to measure the TEP while
slowly warming the system temperature with the ramp rate of 0.1 K/min below 10
K and of 0.45 K/min above 100 K (shown for a measurement of Pt wire in Fig. 2
(a) for $T\sim$55 K). As temperature increases higher than 10 K, the ramp rate
was increased for certain temperature range, for instance 0.2 K/min up to 20 K
and 0.3 K/min up to 100 K. It is worth noting that if the system temperature
is slowly warming, it is necessary to carefully consider the time dependence
of the sample temperatures and voltages. In this case we calculated $\Delta T$
and $\Delta V$ from a linear fit of the data. Continuous measurements while
ramping temperature provide a high density of data and reduce the measurement
time. In general it takes 16 hours to run from 2 to 350 K. This is in contrast
to our finding that the relaxation time to stablize a sample stage completely
under high vacuum at a single temperature is longer than one hour. Figure 3
shows the TEP of Constantan wire based on these two protocols. In this test
run the agreement between measured results and the reference data [13] is
reasonable. The TEP extracted by the second protocol (slow drift of the system
temperature) lies between the data taken on warming and cooling using the
stable temperature method.
## 3 System Calibration and Sample TEP
Since the wires attached to the sample are either copper or phosphor-bronze, a
second thermal voltage is also generated. The measured TEP is then
$\displaystyle{}S_{measured}=S_{sample}-S_{wire}$ (1)
Here $S_{wire}$ represents the sum of the wire and all system contributions.
When measuring an unknown sample the TEP is then the sum of $S_{wire}$ and
$S_{measured}$.
The TEP of copper is strongly dependent on magnetic impurities below 100 K due
to the Kondo effect [1] and therefore no reliable (or universal) reference
data set is available for low temperatures. On the other hand, a
superconducting material is a suitable reference because $S$ = 0 in
superconducting state. In the present study Pt-wire and Bi2212 high $T_{c}$
superconductors were each, separately, mounted between the two sample stages
and calibration measurements were performed. These were sufficient for
determining the lead wire contribution $S_{wire}$. For the high temperature
region pure Pt-wire ($\sim$50 $\mu$m diameter) was used as a reference. Figure
4 shows the TEP of the Pt-wire versus copper wire and Pt-wire versus phosphor-
bronze wire. The result of Pt-wire versus phosphor-bronze wire is in good
agreement with the absolute TEP value of Pt [1] which implies that the
absolute TEP value of phosphor-bronze wire is negligible. Note that below 100
K the Pt-wire manifests slightly different TEP responses depending on the heat
treatment (annealing) of wire. At low temperatures we employed two
superconducting Bi2212 compounds with $T_{c}$ about $\sim$82 K and $\sim$92 K,
where the different $T_{c}$ values may be due to the heating of sample in air.
The results of the TEP measurement for Bi2212 against copper and phosphor-
bronze wire are shown in Fig. 5. In this calibration measurement samples were
mounted on the two sample stages with GE-varnish. The copper and phosphor-
bronze wire were attached to the sample using silver epoxy (top configuration
of Fig. 1 (c)). Here we used Bright Brushing Gold to attach the wire to the
Bi2212 because using only silver epoxy provided a poor electrical contact,
usually on the order of $10^{3}$ $\Omega$. After painting on the Bright
Brushing Gold, the sample was heated up to 400 oC quickly, held for 5 min and
air quenched to room temperature, where the contact resistance was reduced to
below 100 $\Omega$.
The absolute TEP of copper and phosphor-bronze wire we measured and of copper,
from the literature, is shown in Fig. 6. Because $S$ = 0 in the
superconducting state, the observed TEP is the absolute TEP of copper and
phosphor-bronze wire. From Fig. 6 (a) it is dramatically clear that the
absolute TEP value of phosphor-bronze wire is very small, $S\ll$ 0.5 $\mu$V/K,
up to 80 K. For copper wire the agreement between measured results and the
literature data is reasonable. The inset of Fig. 6 (b) shows the low
temperature TEP of copper wire. For the copper wire measured against phosphor-
bronze, no correction was added. These data indicate a fairly good agreement
with the data taken from Fig. 5 (b). The estimated uncertainty for the copper
wire is about 0.3 $\mu V$/K. In addition to the subtraction errors, we believe
that this disagreement is due to a difference in quality of the copper wire in
Ref. [1] and that used in this measurement.
As an aside, it should be noted that the low temperature, oscillatory behavior
of the Bi2212 sample for $H>$0 (Fig. 5) is reproducible. Although similar
behavior was observed in the Nernst signal and associated with the plastic
flow of the vortices [11], the origin of this phenomena is still somewhat
unclear.
Previous TEP measurements at low temperatures and in high magnetic fields have
had to take into account the significant contribution of background voltage.
By using well-known elemental metal wires of copper or gold and
superconducting materials, these background contributions can be accounted
for, correcting the background contribution. For small single crystals an
alternating AC current technique, utilizing a thermocouple, has been used to
measure TEP under high magnetic fields for a wide range of temperatures [9,
10]. Although the thermocouple wire provides a good sensitivity for relative
temperatures, an accurate determination of $\Delta T$ in high magnetic fields
becomes difficult and large efforts are needed to calibrate the field
dependence of the thermocouple wire.
In order to exclude the difficulties due to the magneto-thermoelectric power
(MTEP) measurement based primarily on the field dependence of $S_{wire}$ and
thermometer calibrations, we selected phosphor-bronze wire and Cernox. Whereas
the TEP of copper (Cu) wire is not small and shows a field dependence,
phosphor-bronze wire provides essentially zero TEP over wide temperature range
and is almost temperature and field independent [12] as shown in Figs. 5 and
6. Therefore, in this measurement setup the magnetic field dependence of TEP
of samples, including the quantum oscillation (de Haas-van Alphen oscillation)
at low temperatures, can be reliably measured.
To demonstrate the versatility and reliability of this technique two research
samples (as opposed to wires of Cu or Pt) were measured, the TEP data are
shown in Fig. 7 as a function of temperature and Fig. 8 as a function of
applied magnetic field. LaAgSb2 has been observed to have a charge density
wave order at $\sim$210 K and $\sim$185 K [14, 15], and CeAgSb2 was
characterized as a ferromagnetic Kondo lattice compound with Curie temperature
$T_{c}$=9.8 K [14]. In both compounds de Haas-van Alphen (dHvA) oscillations
at low temperatures have been observed [16]. These single crystals were grown
by excess Sb flux [14]. Samples were prepared with dimensions about
0.8$\times$0.2$\times$2.5 mm3 for LaAgSb2 and 0.8$\times$0.2$\times$3 mm3 for
CeAgSb2. Zero field measurement of resistivity and TEP of both materials are
presented in Fig. 7. The resistivity data are consistent with earlier study
and the TEP has clear features at the same transition temperatures. For
$H\parallel c$ at 2.3 K with $\triangle T$=0.2 K, for both materials, dHvA
type oscillations were observed in TEP as a function of field, $S(H)$, shown
in Fig. 8. Fourier analysis (fast Fourier transform) of the $S(H)$ data
reveals peaks in the spectrum. The observed frequencies match the frequencies
obtained from resistivity and magnetization [16]. The detailed data analysis
will be published elsewhere [17]. So as to provide a clear sense of how
readily $S(T,H)$ data can be aquired using this technique, it should be noted
that the temperature dependence of TEP was taken over $\sim$14 hours and the
field dependence was taken with the ramp rate of 25 Oe/sec ($\sim$2 hours).
The accuracy of this technique was estimated by using the measurement of Pt
and Cu wire. The estimated uncertainty of this system over all temperature
ranges falls within a maximum $\pm$1 $\mu$V/K, and the relative accuracy is
within a maximum of 10 %. In the high temperature region, roughly above 100 K,
the main uncertainty originates from inaccurate determination of the $\Delta
T$ due to the relatively low sensitivity of the Cernox. The absolute and
relative temperature of Cernox was observed within a resolution of 4 mK at low
temperatures, the relative error at high temperatures falls within $\sim$ 200
mK. For materials having low thermal conductivity, the error may be larger due
to the temperature difference between sample and thermometer. For materials
having small TEP, less than 0.5 $\mu V$/K, the error can also be larger due to
noise. More contributions to the error need to be considered for TEP
measurements in the magnetic field. For instance, due to the heat conducting
environment which is mainly caused by induced current by applying magnetic
fields ($d\Phi/dt$), it is very important to make sure that the ramp rate of
magnetic field should be slow enough to avoid additional heating and reduce
the induced voltage due to the open loop. Alternatively, the TEP can be
measured stepping the magnetic field with the magnet in persistent mode for
each value of the field.
## 4 Summary of Technical Parameters and Reference Information
* •
Operation range: temperature range from 2 to 350 K and magnetic fields up to
140 kOe.
* •
Limit of sample dimension: the length of sample is longer than 1.5 mm (smaller
than this length has not been tested).
* •
$\Delta T$: from 0.1 to 2.5 K, depending on the temperature and the absolute
TEP value of sample.
* •
Ramp rate of system temperature: it can be varied up to 1 K/min. For example,
in the calibration measurement, it was selected 0.1 K/min up to 10 K, 0.35
K/min up to 100 K and 0.45 K/min above 100 K.
* •
Estimated accuracy: maximum of $\pm$ 1 $\mu$V/K and 10% depending on the
temperature and sample. The limit of accuracy is mainly due to the uncertainty
of the thermometer and the thermal contact between the sample and the thermal
stage. If the absolute TEP of the sample is smaller than 0.5 $\mu$V/K the
fluctuation of the sample voltage was observed.
* •
Copper wire: 0.025 mm diameter, Puratronic, 99.995% (metals basis), Alfa
Aesar. Detected impurity elements are Fe, Ag, O, S (as provided by supplier).
* •
Phosphor-Bronze wire: Cu0.94Sn0.06 alloy, 0.025 mm dia, GoodFellow.
* •
Platinum wire: 0.05 mm diameter, 99.95% (metals basis), Alfa Aesar.
* •
Silver epoxy: H20E, Epotek.
* •
Strain gauge : FLG-02-23, 0.2$\times$1.4 mm2 grid made by Cu-Ni alloy and
3.5$\times$2.5 mm2 thin epoxy backing, Tokyo Sokki Kenkyujo Co., Ltd.
* •
Silver paste: DuPont 4929N silver paint, DuPont, Inc.
* •
Stycast 1266: Emerson & Cuming, Inc.
We would like to thank A. Kaminski for providing Bi2212 samples, J. Frederick
and S. A. Law for preparing samples RAgSb2 and M. E. Tillman, A. Kreyssig, M.
D. Vannette, C. Martin and M. A. Tanatar for valuable discussion to this
project. C8H10N4O2 for this work was provided, in part, by C. Petrovic. Work
at Ames Laboratory was supported by the Basic Energy Sciences, U.S. Department
of Energy under Contract No. DE-AC02-07CH11358. Milton S. Torikachvili
gratefully acknowledges support of the National Science Foundation under
DMR-0805335.
## References
## References
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* [2] Sakurai J, Horie M, Araki S, Yamamoto H and Shinjo T 1991 J. Phys. Soc. Jpn. 60 2522
* [3] Wang Y, Xu Z A, Kakeshita T, Uchida S, Ono S, Ando Y and Ong N P 2001 Phys. Rev. B 64 224519
* [4] Bud’ko S L, Gapotchenko A G and Itskevich E S 1988 Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki 47 106
* [5] Sakurai J, Takamatsu Y, Kuwai T, Isikawa Y, Mori K, Fukuhara T and Maezawa K 1995 Physica B 206 834
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* [12] Wang Y, Rogado N S, Cava R J and Ong N P 2003 Nature 423 425 Supplimentary Information
* [13] The reference data came from MMR Technologies with constantan wire as a standard.
* [14] Myers K D, Bud’ko S L, Fisher I R, Islam Z, Kleinke H, Lacerda A H and Canfield P C 1999 J. Magn. Magn. Mater. 205 27
* [15] Song C, Park J, Koo J, Lee K B, Rhee J Y, Bud’ko S L, Canfield P C, Harmon B N and Goldman A I 2003 Phys. Rev. B 68 035113
* [16] Myers K D, Bud’ko S L, Antropov V P, Harmon B N, Canfield P C and Lacerda A H 1999 Phys. Rev. B 60 13371
* [17] Mun E $et$ $al$., in preparation.
Figure 1: (a) Schematic diagram of sample stages. A: Strain gauges for heater,
B: Thermometers (Cernox), C: G-10 for thermal insulation from heat sink, D:
Voltage probe wires, E: Sample. (b) A photo of the measurement cell. (c)
Sample mounting method using GE-varnish (top) and silver paste (bottom). (d)
Block diagram of measurement system. The system temperature and magnetic field
is controlled by PPMS. All instruments shown in the block diagram including
PPMS is operated by LabVIEW software. The details of the use of the
instruments are explained in the text.
Figure 2: Measurement procedure to extract the TEP from data corresponding to
the measurement performed near 55 K on Pt-wire versus phosphor-bronze wire.
Actual time period ($\tau$) between subsequent cycles, used to calculate TEP,
was 50 sec. (a) Measured temperatures of both thermometers ($T_{1}$ and
$T_{2}$) and (b) sample voltage ($V$) as a function of time. Note small
($\sim$0.1 K/min) drift superimposed on data. (c) (d) One cycle of measurement
to determine parameters $\Delta T$, $\Delta V$: initial temperature $T_{i}$,
final temperature $T_{f}$, initial voltage $V_{i}$, final voltage $V_{f}$ and
offset voltage $V_{off}$. The solid lines represent the linear fit to the
measurement data. The temperature difference for $T_{1}$ ($T_{2}$) is
determined by $\Delta T_{1}$=$T_{1i}$-$T_{2i}$ ($\Delta
T_{2}$=$T_{2f}$-$T_{1f}$) so that 2$\Delta T$=$\Delta T_{1}$+$\Delta T_{2}$.
The voltage difference is calculated 2$\Delta V$=$V_{f}$-$V_{i}$ (see text).
Figure 3: TEP of constantan wire versus copper wire. Warming up and cooling
down indicate the measurement data using the stable temperature method. The
solid line shows the TEP values using the alternating heating method by slowly
drifting system temperature. The detailed explanations are in the text. We
used the reference data provided from MMR Technologies with constantan as a
standard.
Figure 4: TEP of Pt-wire versus phosphor-bronze wire and Pt-wire versus copper
wire. Circles and solid line represent the measured data from this work
without any corrections. Both reference 1 (open squares) and reference 2
(solid triangles) data are from Ref. [1].
Figure 5: Calibration measurements of lead wires. (a) TEP of Bi2212 versus
phosphor-bronze wire and (b) Bi2212 versus copper wire as a function of
temperature at several constant magnetic fields.
Figure 6: (a) Absolute TEP of copper and phosphor-bronze wire below 80K. The
data are taken from Fig. 5. (b) Calibrated TEP curve of copper wire at $H$=0
(open square) and 140 kOe (open circle). Both closed circles (reference 3) and
stars (reference 4) were taken from Ref. [1]. Inset: expanded view for low
temperature range. The symbols present the measured TEP of copper wire against
to phosphor-bronze wire. No correction was added. Solid lines are taken from
Fig. 5 (b).
Figure 7: TEP (left axis) and electrical resistivity (right axis) as a
function of temperature between 2 and 300 K for (a) CeAgSb2 and (b) LaAgSb2.
Both the temperature gradient and electrical current are applied in the
tetragonal $ab$-plane.
Figure 8: Magneto-thermoelectric power of LaAgSb2 and CeAgSb2 at 2.3 K for
$H\parallel c$. Labels for frequencies shown in the inset of (b) are taken
from Ref. [16].
|
arxiv-papers
| 2010-03-03T21:55:53 |
2024-09-04T02:49:08.745367
|
{
"license": "Public Domain",
"authors": "Eundeok Mun, Sergey L. Bud'ko, Milton S. Torikachvili, Paul C.\n Canfield",
"submitter": "Eundeok Mun",
"url": "https://arxiv.org/abs/1003.0921"
}
|
1003.1109
|
# Interacting new agegraphic viscous dark energy with varying $G$
A. Sheykhi a,c , M. R. Setare b,c
a Department of Physics, Shahid Bahonar University,
P.O. Box 76175, Kerman, Iran
b Department of Science, Payame Noor University, Bijar, Iran
c Research Institute for Astronomy and Astrophysics of Maragha
(RIAAM), Maragha, Iran
Email: sheykhi@mail.uk.ac.irEmail: rezakord@ipm.ir
###### Abstract
We consider the new agegraphic model of dark energy with a varying
gravitational constant, $G$, in a non-flat universe. We obtain the equation of
state and the deceleration parameters for both interacting and noninteracting
new agegraphic dark energy. We also present the equation of motion determining
the evolution behavior of the dark energy density with a time variable
gravitational constant. Finally, we generalize our study to the case of
viscous new agegraphic dark energy in the presence of an interaction term
between both dark components.
Keywords: agegraphic; dark energy; viscous.
## 1 Introduction
Many cosmological observations, such as SNe Ia [1], WMAP [2], SDSS [3],
Chandra X-ray observatory [4], etc., reveal that our universe is undergoing an
accelerating expansion. To explain this cosmic positive acceleration,
mysterious dark energy has been proposed. There are several dark energy models
which can be distinguished by, for instance, their equation of state (EoS)
$(w=\frac{P_{de}}{\rho_{de}})$ during the evolution of the universe. Although
the simplest way to explain this behavior is the consideration of a
cosmological constant [5], the known fine-tuning problem [6] led to the dark
energy paradigm. The dynamical nature of dark energy, at least in an effective
level, can originate from various fields, such is a canonical scalar field
(quintessence) [7], a phantom field, that is a scalar field with a negative
sign of the kinetic term [8], or the combination of quintessence and phantom
in a unified model named quintom [9].
An approach to the problem of DE arises from the holographic principle that
states that the number of degrees of freedom related directly to entropy
scales with the enclosing area of the system. It was shown by ’tHooft and
Susskind [10] that effective local quantum field theories greatly overcount
degrees of freedom because the entropy scales extensively for an effective
quantum field theory in a box of size $L$ with UV cut-off $\Lambda$.
Attempting to solve this problem, Cohen et al. showed [11] that in quantum
field theory, short distance cut-off $\Lambda$ is related to long distance
cut-off $L$ due to the limit set by forming a black hole. In other words the
total energy of the system with size $L$ should not exceed the mass of the
same size black hole i.e. $L^{3}\rho_{\Lambda}\leq LM_{p}^{2}$ where
$\rho_{\Lambda}$ is the quantum zero-point energy density caused by UV cutoff
$\Lambda$ and $M_{P}$ denotes Planck mass ( $M_{p}^{2}=1/{8\pi G})$. The
largest $L$ is required to saturate this inequality. Then its holographic
energy density is given by $\rho_{\Lambda}=3c^{2}M_{p}^{2}/L^{2}$ in which $c$
is free dimensionless parameter and coefficient $3$ is for convenience. More
recently a new dark energy model, dubbed agegraphic dark energy has been
proposed [12] (see also [13]), which takes into account the Heisenberg
uncertainty relation of quantum mechanics together with the gravitational
effect in general relativity. Following the line of quantum fluctuations of
spacetime, Karolyhazy [14] proposed that the distance $t$ in Minkowski
spacetime cannot be known to a better accuracy than $\delta t=\beta
t_{p}^{2/3}t^{1/3}$, where $\lambda$ is a dimensionless constant of order
unity. Based on Karolyhazy relation, Maziashvili proposed that the energy
density of metric fluctuations of Minkowski spacetime is given by [15]
$\rho_{\Lambda}\sim\frac{1}{t_{p}^{2}t^{2}}\sim\frac{M_{p}^{2}}{t^{2}},$ (1)
where $t_{p}$ is the reduced Planck time, and $M_{p}$ is the Planck mass. The
agegraphic models of dark energy have been examined and constrained by various
astronomical observations [16, 17, 18, 19, 20].
Since we know neither the nature of dark energy nor the nature of dark matter,
a microphysical interaction model is not available either. However,
pressureless dark matter in interaction with holographic dark energy is more
than just another model to describe an accelerated expansion of the universe.
Understanding dark energy is one of the biggest challenges to the particle
physics of this century. Studying the interaction between the dark energy and
ordinary matter will open a possibility of detecting the dark energy. It
should be pointed out that evidence was recently provided by the Abell Cluster
A586 in support of the interaction between dark energy and dark matter [21].
However, despite the fact that numerous works have been performed till now,
there are no strong observational bounds on the strength of this interaction
[22]. This weakness to set stringent (observational or theoretical)
constraints on the strength of the coupling between dark energy and dark
matter stems from our unawareness of the nature and origin of dark components
of the Universe. It is therefore more than obvious that further work is needed
to this direction.
Previously, it has been shown that the interacting new agegraphic model of
dark energy can cross the phantom divide [19]. The phantom energy, if it
exists, can cause some peculiar phenomena e.g. violates the strong energy
condition, $\rho+3p\geq 0$. This leads us to consider phantom energy as an
imperfect fluid, implying that the phantom fluid could contain non-zero bulk
and shear viscosities [23]. The bulk viscosities are negligible for non-
relativistic and ultra-relativistic fluids but are important for the
intermediate cases. In viscous cosmology, shear viscosities arise in relation
to space anisotropy while the bulk viscosity accounts for the space isotropy
[24]. Generally, shear viscosities are ignored (as the CMBR does not indicate
significant anisotropies) and only bulk viscosities are taken into account for
the fluids in the cosmological context. Moreover, bulk viscosity related to a
grand unified theory phase transition may lead to an explanation of the
accelerated cosmic expansion [25].
Although the holographic dark energy model with varying gravitational constant
has been studied in [26, 27], however, until now, in all the investigated
agegraphic dark energy models a constant Newton s constant G has been
considered. The role of $G$-variation will be expressed through the pure
number $G^{\prime}/G\equiv\Delta_{G}$, which will be extracted from
observations. In particular, observations of Hulse-Taylor binary pulsar
B$1913+16$ lead to the estimation $\dot{G}/G\sim 2\pm 4\times
10^{-12}{yr}^{-1}$ [41, 42], while helio-seismological data provide the bound
$-1.6\times 10^{-12}{yr}^{-1}<\dot{G}/G<0$ [43]. Similarly, Type Ia supernova
observations [1] give the best upper bound of the variation of $G$ as
$-10^{-11}yr^{-1}\leq\frac{\dot{G}}{G}<0$ at redshifts $z\simeq 0.5$ [44],
while astereoseismological data from the pulsating white dwarf star G117-B15A
lead to $\left|\frac{\dot{G}}{G}\right|\leq 4.10\times 10^{-11}yr^{-1}$ [45].
See also [46] for various bounds on $\dot{G}/G$ from observational data,
noting that all these measurements are valid at relatively low redshifts, i.e
$z<3.5$. Since the limits in $G$-variation are given for $\dot{G}/G$ in units
$yr^{-1}$, and since $\dot{G}/G=HG^{\prime}/G$, we can estimate $\Delta_{G}$
substituting the value of $H$ in $yr^{-1}$ [26].
Besides, there have been many proposals in the literature attempting to
theoretically justified a varying gravitational constant. For example in
Brans-Dicke theory the gravitational constant is replaced by a scalar field
coupling to gravity through a new parameter, and it has been generalized to
various forms of scalar-tensor theories [47], leading to a considerably
broader range of variable-G theories. In addition, justification of a varying
Newton’s constant has been established with the use of conformal invariance
and its induced local transformations [48]. Finally, a varying G can arise
perturbatively through a semiclassical treatment of Hilbert-Einstein action
[49], non-perturbatively through quantum gravitational approaches within the
“Hilbert-Einstein truncation” [50], or through gravitational holography [51,
52].
In the light of all mentioned above, it becomes obvious that the investigation
on the interacting new agegraphic dark energy models with varying
gravitational constant is well motivated. In this paper, we would like to
generalize, following [28], the new agegraphic viscous dark energy models to
the universe with spacial curvature in the presence of interaction between the
dark matter and dark energy with a varying gravitational constant, $G$.
## 2 New ADE with varying gravitational constant
Soon after the original ADE model was introduced [12], an alternative model
dubbed “ new agegraphic dark energy” was proposed by Wei and Cai [29], while
the time scale is chosen to be the conformal time $\eta$ instead of the age of
the universe, which is defined by $dt=ad\eta$, where $t$ is the cosmic time.
It is worth noting that the Karolyhazy relation $\delta{t}=\beta
t_{p}^{2/3}t^{1/3}$ was derived for Minkowski spacetime
$ds^{2}=dt^{2}-d\mathrm{x^{2}}$ [14, 15]. In case of the FRW universe, we have
$ds^{2}=dt^{2}-a^{2}d\mathrm{x^{2}}=a^{2}(d\eta^{2}-d\mathrm{x^{2}})$. Thus,
it might be more reasonable to choose the time scale to be the conformal time
$\eta$ since it is the causal time in the Penrose diagram of the FRW universe.
The new ADE contains some new features different from the original ADE and
overcome some unsatisfactory points. For instance, the original ADE suffers
from the difficulty to describe the matter-dominated epoch while the new
agegraphic dark energy resolved this issue [29]. The energy density of the new
ADE can be written
$\rho_{D}=\frac{3n^{2}}{8\pi G\eta^{2}},$ (2)
where the conformal time is given by
$\eta=\int{\frac{dt}{a}}=\int_{0}^{a}{\frac{da}{Ha^{2}}}.$ (3)
If we write $\eta$ to be a definite integral, there will be an integral
constant in addition. Thus, we have $\dot{\eta}=1/a$. Let us again consider a
FRW universe with spatial curvature containing the new agegraphic dark energy
and pressureless matter. The Friedmann equation can be written
$\displaystyle H^{2}+\frac{k}{a^{2}}=\frac{8\pi
G}{3}\left(\rho_{m}+\rho_{D}\right),$ (4)
where $k$ is the curvature parameter with $k=-1,0,1$ corresponding to open,
flat, and closed universes, respectively. A closed universe with a small
positive curvature ($\Omega_{k}\simeq 0.01$) is compatible with observations
[30]. If we introduce, as usual, the fractional energy densities such as
$\displaystyle\Omega_{m}=\frac{8\pi G\rho_{m}}{3H^{2}},\hskip
14.22636pt\Omega_{D}=\frac{8\pi G\rho_{D}}{3H^{2}},\hskip
14.22636pt\Omega_{k}=\frac{k}{H^{2}a^{2}},$ (5)
then the Friedmann equation can be written
$\displaystyle\Omega_{m}+\Omega_{D}=1+\Omega_{k}.$ (6)
The fractional energy density of the new agegraphic dark energy can also be
written
$\displaystyle\Omega_{D}=\frac{n^{2}}{H^{2}\eta^{2}}.$ (7)
We consider the FRW universe filled with dark energy and dust (dark matter)
which evolves according to their conservation laws
$\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+w_{D})=0,$ (8)
$\displaystyle\dot{\rho}_{m}+3H\rho_{m}=0,$ (9)
where $w_{D}=p_{D}/\rho_{D}$ is the equation of state parameter of new ADE.
Taking the derivative of Eq. (2) with respect to the cosmic time and using Eq.
(7) we have
$\displaystyle\dot{\rho}_{D}=-H\rho_{D}\left(2\frac{\sqrt{\Omega_{D}}}{na}+\frac{G^{\prime}}{G}\right).$
(10)
where the prime stands for the derivative with respect to $x=\ln{a}$.
Inserting this equation in the conservation law (8), we obtain the equation of
state parameter
$\displaystyle w_{D}=-1+\frac{2}{3na}\sqrt{\Omega_{D}}+\frac{G^{\prime}}{3G}.$
(11)
The equation of motion for $\Omega_{D}$ can be obtained by taking the
derivative of Eq. (7). The result is
$\displaystyle{\Omega^{\prime}_{D}}=-\Omega_{D}\left(2\frac{\dot{H}}{H^{2}}+2\frac{\sqrt{\Omega_{D}}}{na}\right).$
(12)
where the dot is the derivative with respect to the time. The next step is to
calculate $\frac{\dot{H}}{H^{2}}$. Taking the derivative of both side of the
Friedman equation (4) with respect to the cosmic time $t$, and using Eqs.
(5)-(9) and (11), we obtain
$\displaystyle
2\frac{\dot{H}}{H^{2}}=3(\Omega_{D}-1)-\frac{2}{na}\Omega^{3/2}_{D}-\Omega_{k}+\frac{G^{\prime}}{G}(1+\Omega_{k}-\Omega_{D})$
(13)
Substituting this relation into Eq. (23), we obtain the evolution behavior of
the new agegraphic dark energy
$\displaystyle{\Omega^{\prime}_{D}}$ $\displaystyle=$
$\displaystyle\Omega_{D}\left[(1-\Omega_{D})\left(3-\frac{2}{na}\sqrt{\Omega_{D}}\right)+\Omega_{k}-\frac{G^{\prime}}{G}(1+\Omega_{k}-\Omega_{D})\right].$
(14)
For completeness, we give the deceleration parameter
$\displaystyle q=-\frac{\ddot{a}}{aH^{2}}=-1-\frac{\dot{H}}{H^{2}},$ (15)
which combined with the Hubble parameter and the dimensionless density
parameters form a set of useful parameters for the description of the
astrophysical observations. Substituting Eq. (13) in Eq. (15) we get
$\displaystyle q$ $\displaystyle=$
$\displaystyle\frac{1}{2}-\frac{3}{2}{\Omega_{D}}+\frac{\Omega^{3/2}_{D}}{na}+\frac{\Omega_{k}}{2}-\frac{G^{\prime}}{2G}(1+\Omega_{k}-\Omega_{D}).$
(16)
## 3 Interacting new ADE with varying gravitational constant
Next we consider the case where the pressureless dark matter and the new ADE
do not conserve separately but interact with each other. Given the unknown
nature of both dark matter and dark energy there is nothing in principle
against their mutual interaction and it seems very special that these two
major components in the universe are entirely independent. Indeed, this
possibility is receiving growing attention in the literature [31, 32, 33] and
appears to be compatible with SNIa and CMB data [34]. The total energy density
satisfies a conservation law
$\dot{\rho}+3H(\rho+p)=0,$ (17)
where $\rho=\rho_{m}+\rho_{D}$ and $p=p_{D}$. However, as stated above, both
components- the pressureless dark matter and the new ADE- are assumed to
interact with each other; thus, one may grow at the expense of the other. The
conservation equations for them read
$\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+w_{D})=-Q,$ (18)
$\displaystyle\dot{\rho}_{m}+3H\rho_{m}=Q,$ (19)
where $Q$ stands for the interaction term. Following [35] we shall assume for
the latter the ansatz $Q=\Gamma\rho_{D}$ with $\Gamma>0$ which means that
there is an energy transfer from the dark energy to dark matter. This
expression for the interaction term was first introduced in the study of the
suitable coupling between a quintessence scalar field and a pressureless cold
dark matter field [31, 32]. We also assume $\Gamma=3b^{2}(1+r)H$ where
$r={\rho_{m}}/{\rho_{D}}$ and $b^{2}$ is a coupling constant. Therefore, the
interaction term $Q$ can be expressed as
$\displaystyle Q=3b^{2}H\rho_{D}(1+r),$ (20)
where
$\displaystyle r$ $\displaystyle=$
$\displaystyle\frac{\Omega_{m}}{\Omega_{D}}=-1+\frac{1+\Omega_{k}}{\Omega_{D}}.$
(21)
Combining Eqs. (10), (20) and (21) with Eq. (18) we obtain the equation of
state parameter
$\displaystyle
w_{D}=-1+\frac{2}{3na}\sqrt{\Omega_{D}}-b^{2}\frac{(1+\Omega_{k})}{\Omega_{D}}+\frac{G^{\prime}}{3G}.$
(22)
If we take following [26] $0<{G^{\prime}}/{G}\leq 0.07$ and assuming
$\Omega_{D}=0.73$ and $\Omega_{k}\approx 0.01$ for the present time and $n=4$,
$b=0.1$, we obtain $-0.87<w_{D}-\leq 0.85$ which is consistent with recent
observations. We can also find the equation of motion for $\Omega_{D}$ by
taking the derivative of Eq. (7). The result is
$\displaystyle{\Omega^{\prime}_{D}}=\Omega_{D}\left(-2\frac{\dot{H}}{H^{2}}-\frac{2}{na}\sqrt{\Omega_{D}}\right).$
(23)
where
$\displaystyle
2\frac{\dot{H}}{H^{2}}=3(\Omega_{D}-1)-\frac{2}{na}\Omega^{3/2}_{D}-\Omega_{k}-b^{2}\frac{(1+\Omega_{k})}{\Omega_{D}}+\frac{G^{\prime}}{G}(1+\Omega_{k}-\Omega_{D})$
(24)
Substituting this relation into Eq. (23), we obtain the evolution behavior of
the interacting new agegraphic dark energy with variable gravitational
constant
$\displaystyle{\Omega^{\prime}_{D}}$ $\displaystyle=$
$\displaystyle\Omega_{D}\left[(1-\Omega_{D})\left(3-\frac{2}{na}\sqrt{\Omega_{D}}\right)+\Omega_{k}-3b^{2}(1+\Omega_{k})-\frac{G^{\prime}}{G}(1+\Omega_{k}-\Omega_{D})\right].$
(25)
The deceleration parameter is now given by
$\displaystyle q$ $\displaystyle=$
$\displaystyle\frac{1}{2}-\frac{3}{2}{\Omega_{D}}+\frac{\Omega^{3/2}_{D}}{na}+\frac{\Omega_{k}}{2}-\frac{3}{2}b^{2}(1+\Omega_{k})-\frac{G^{\prime}}{2G}(1+\Omega_{k}-\Omega_{D}).$
(26)
Again takeing $0<{G^{\prime}}/{G}\leq 0.07$ and assuming $\Omega_{D}=0.73$ and
$\Omega_{k}\approx 0.01$ for the present time and $n=4$, $b=0.1$, we obtain
$-0.46<q-\leq 0.45$ which is again compatible with recent observational data
[53].
## 4 Interacting viscous new ADE with varying $G$
In this section we would like to generalize our study to the interacting
viscous new agegraphic dark energy model. In an isotropic and homogeneous FRW
universe, the dissipative effects arise due to the presence of bulk viscosity
in cosmic fluids. The theory of bulk viscosity was initially investigated by
Eckart [36] and later on pursued by Landau and Lifshitz [37]. Dark energy with
bulk viscosity has a peculiar property to cause accelerated expansion of
phantom type in the late evolution of the universe [24, 38, 39]. It can also
alleviate several cosmological puzzles like age problem, coincidence problem
and phantom crossing. The energy-momentum tensor of the viscous fluid is
$T_{\mu\nu}=\rho_{D}u_{\mu}u_{\nu}+\tilde{p}_{D}(g_{\mu\nu}+u_{\mu}u_{\nu}),$
(27)
where $u_{\mu}$ is the four-velocity vector and
$\tilde{p}_{D}={p}_{D}-3H\xi,$ (28)
is the effective pressure of dark energy and $\xi$ is the bulk viscosity
coefficient. We require $\xi>0$ to get positive entropy production in
conformity with second law of thermodynamics [40]. The energy conservation
equation for interacting viscous dark energy is now given by
$\displaystyle\dot{\rho}_{D}+3H(\rho_{D}+\tilde{p}_{D})=-Q,$ (29)
which can be written
$\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+w_{D})=9H^{2}\xi-Q,$ (30)
Combining Eqs. (10), (20) and (21) with Eq. (30) we obtain the equation of
state parameter
$\displaystyle
w_{D}=-1+\frac{2}{3na}\sqrt{\Omega_{D}}-b^{2}\frac{(1+\Omega_{k})}{\Omega_{D}}+\frac{G^{\prime}}{3G}+\frac{3H\xi}{\rho_{D}}.$
(31)
If we assume $\xi=\alpha H^{-1}\rho_{D}$, where $\alpha$ is a constant
parameter, then we get
$\displaystyle
w_{D}=3\alpha-1+\frac{2}{3na}\sqrt{\Omega_{D}}-b^{2}\frac{(1+\Omega_{k})}{\Omega_{D}}+\frac{G^{\prime}}{3G}.$
(32)
The equation of motion for viscous ADE is obtained as
$\displaystyle{\Omega^{\prime}_{D}}$ $\displaystyle=$
$\displaystyle\Omega_{D}\left[(1-\Omega_{D})\left(3-\frac{2}{na}\sqrt{\Omega_{D}}\right)+\Omega_{k}-3b^{2}(1+\Omega_{k})-\frac{G^{\prime}}{G}(1+\Omega_{k}-\Omega_{D})+9\alpha\Omega_{D}\right].$
(33)
## 5 Conclusions
In this work we have investigated the interacting new agegraphic viscous dark
energy scenario with a varying gravitational constant. We have obtained the
equation of state and the deceleration parameters for both interacting and
noninteracting new agegraphic dark energy. By consideing non-interacting and
interacting cases we have extracted the exact differential equations that
determine the evolution of the dark energy density-parameter, where the
$G$-variation appears as a coefficient in additional terms.
## 6 Acknowledgment
This work has been supported by Research Institute for Astronomy and
Astrophysics of Maragha, Iran.
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|
arxiv-papers
| 2010-03-04T18:39:10 |
2024-09-04T02:49:08.756724
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Sheykhi, M. R. Setare",
"submitter": "Mohammad Reza Setare",
"url": "https://arxiv.org/abs/1003.1109"
}
|
1003.1187
|
# Flexible Sampling of Discrete Scale Invariant Markov Processes: Covariance
and Spectrum
N. Modarresi and S. Rezakhah Faculty of Mathematics and Computer Science,
Amirkabir University of Technology, 424 Hafez Avenue, Tehran 15914, Iran.
E-mail: namomath@aut.ac.ir (N. Modarresi), rezakhah@aut.ac.ir (S. Rezakhah).
###### Abstract
In this paper we consider some flexible discrete sampling of a discrete scale
invariant process $\\{X(t),t\in{\bf R^{+}}\\}$ with scale $l>1$. By this
method we plan to have $q$ samples at arbitrary points ${\bf s}_{0},{\bf
s}_{1},\ldots,{\bf s}_{q-1}$ in interval $[1,l)$ and proceed our sampling in
the intervals $[l^{n},l^{n+1})$ at points $l^{n}{\bf s}_{0},l^{n}{\bf
s}_{1},\ldots,l^{n}{\bf s}_{q-1}$, $n\in{\bf Z}$. Thus we have a discrete time
scale invariant (DT-SI) process and introduce an embedded DT-SI process as
$W(nq+k)=X(l^{n}{\bf s}_{k})$, $q\in{\bf N}$, $k=0,\ldots,q-1$. We also
consider $V(n)=\big{(}V^{0}(n),\ldots,V^{q-1}(n)\big{)}$ where
$V^{k}(n)=W(nq+k)$, as an embedded $q$-dimensional discrete time self-similar
(DT-SS) process. By introducing quasi Lamperti transformation, we find
spectral representation of such process and its spectral density matrix is
given. Finally by imposing wide sense Markov property for $W(\cdot)$ and
$V(\cdot)$, we show that the spectral density matrix of $V(\cdot)$ and
spectral density function of $W(\cdot)$ can be characterized by
$\\{R_{j}(1),R_{j}(0),j=0,\ldots,q-1\\}$ where $R_{j}(k)=E[W(j+k)W(j)]$.
AMS 2000 Subject Classification: 60G18, 62M15.
Keywords: Discrete scale invariance; Wide sense Markov; Multi-dimensional
self-similar processes.
## 1 Introduction
The concept of stationarity and self-similarity are used as a fundamental
property to handle many natural phenomena. Lamperti transformation defines a
one to one correspondence between stationary and self-similar processes.
Discrete scale invariance (DSI) process can be defined as the Lamperti
transform of periodically correlated (PC) process. Many critical systems, like
statistical physics, textures in geophysics, network traffic and image
processing can be interpreted by these processes [1]. Fourier transform is
known as a suited representation for stationarity, but not for self-
similarity. A harmonic like representation of self-similar process is
introduced by using Mellin transform [4].
A process which is Markov and self-similar, is called self-similar Markov
process. These processes are involved in various parts of probability theory,
such as branching processes and fragmentation theory [2].
Current authors considered DSI processes in the wide sense with some scale
$l>1$. They proposed to have some fixed number of samples, say $T$, in each
scale at points $\alpha^{k}$, $k\in{\bf Z}$ where $l=\alpha^{T}$, $T\in{\bf
N}$. By such sampling they provided a discrete time scale invariant process in
the wide sense and found a closed formula for its covariance function [6]. In
this paper we consider $X(\cdot)$ as DSI process with scale $l>1$, and
sampling at arbitrary points $1\leqslant{\bf s}_{0}<{\bf s}_{1}\ldots<{\bf
s}_{q-1}<l$ in the interval $[1,l)$. We also take our samples at points
$l^{n}{\bf s}_{0},l^{n}{\bf s}_{1},\ldots,l^{n}{\bf s}_{q-1}$, $n\in{\bf Z}$
in the intervals $[l^{n},l^{n+1})$. Then we introduce some discrete time
embedded scale invariant (DT-ESI) process $W(nq+k)=X(l^{n}{\bf s}_{k})$,
$q\in{\bf N}$, $k=0,\ldots,q-1$ and corresponding multi-dimensional discrete
time embedded self-similar (DT-ESS) process as
$V(n)=\big{(}V^{0}(n),\ldots,V^{q-1}(n)\big{)}$ where $V^{k}(n)=W(nq+k)$. We
investigate properties of these processes when they are Markov in the wide
sense.
This paper is organized as follows. In section 2, we present a review of
multi-dimensional stationary, periodically correlated, self-similar and
discrete scale invariant processes. Then we define discrete time self-similar
(DT-SS) and scale invariant (DT-SI) processes. We also introduce quasi
Lamperti transformation in this section. Section 3 is devoted to the structure
of the multi-dimensional DT-SS process resulting from the above method of
sampling. We define DT-ESI process and corresponding multi-dimensional DT-ESS
process and characterize the spectral density matrix of it in this section.
Finally covariance function and spectral density matrix of the discrete time
embedded scale invariant Markov (DT-ESIM) processes and corresponding multi-
dimensional discrete time embedded self-similar Markov (DT-ESSM) are obtained
in section 4.
## 2 Theoretical framework
This section is organized in tree subsections. First we review the structure
of the covariance function and spectral distribution matrix of multi-
dimensional stationary processes. We present definitions of DT-SS, DT-SI, wide
sense self-similar and scale invariant processes in subsection 2.2. We define
quasi Lamperti transformation and present its properties which provide a one
to one correspondence between DT-SS and discrete time stationary processes and
also between DT-SI and DT-PC processes.
### 2.1 Stationary and multi-dimensional stationary processes
###### Definition 2.1
A process $\\{Y(t),t\in{\bf R}\\}$ is said to be stationary, if for any
$t,\tau\in{\bf R}$
$\\{Y(t+\tau)\\}\stackrel{{\scriptstyle d}}{{=}}\\{Y(t)\\}$ (2.1)
where $\stackrel{{\scriptstyle d}}{{=}}$ is the equality of all finite-
dimensional distributions.
If $(2.1)$ holds for some $\tau\in{\bf R}$, the process is said to be
periodically correlated. The smallest of such $\tau$ is called period of the
process.
By Rozanov [8], if $Y(t)=\\{Y^{k}(t)\\}_{k=1,\ldots,n}$ be an $n$-dimensional
stationary process, then
$Y(t)=\int e^{i\lambda t}\phi(d\lambda)$ (2.2)
is its spectral representation, where $\phi=\\{\varphi_{k}\\}_{k=1,\ldots,n}$
and $\varphi_{k}$ is the random spectral measure associated with the $k$th
component $Y^{k}$ of the $n$-dimensional process $Y$. Let
$B_{kr}(\tau)=E[Y^{k}(\tau+t)\overline{Y^{r}(t)}],\hskip
19.91692ptk,r=1,\ldots,n$
and $B(\tau)=[B_{kr}(\tau)]_{k,r=1,\ldots,n}$ be the correlation matrix of
$Y$. The components of the correlation matrix of the process $Y$ can be
represented as
$B_{kr}(\tau)=\int e^{i\lambda\tau}F_{kr}(d\lambda),\hskip
19.91692ptk,r=1,\ldots,n$ (2.3)
where for any Borel set $\Delta$,
$F_{kr}(\Delta)=E[\varphi_{k}(\Delta)\overline{\varphi_{r}(\Delta)}]$ are the
complex valued set functions which are $\sigma$-additive and have bounded
variation. For any $k,r=1,\ldots,n$, if the sets $\Delta$ and
$\Delta^{\prime}$ do not intersect,
$E[\varphi_{k}(\Delta)\overline{\varphi_{r}(\Delta^{\prime})}]=0$. For any
interval $\Delta=(\lambda_{1},\lambda_{2})$ when
$F_{kr}(\\{\lambda_{1}\\})=F_{kr}(\\{\lambda_{2}\\})=0$ the following relation
holds
$F_{kr}(\Delta)=\frac{1}{2\pi}\int_{\Delta}\sum_{\tau=-\infty}^{\infty}B_{kr}(\tau)e^{-i\lambda\tau}d\lambda$
(2.4)
$=\frac{1}{2\pi}B_{kr}(0)[\lambda_{2}-\lambda_{1}]+\lim_{T\rightarrow\infty}\frac{1}{2\pi}\sum_{0<|\tau|\leqslant
T}B_{kr}(\tau)\frac{e^{-i\lambda_{2}\tau}-e^{-i\lambda_{1}\tau}}{-i\tau}$
in the discrete parameter case, and
$F_{kr}(\Delta)=\lim_{a\rightarrow\infty}\frac{1}{2\pi}\int_{-a}^{a}\frac{e^{-i\lambda_{2}\tau}-e^{-i\lambda_{1}\tau}}{-i\tau}B_{kr}(\tau)d\tau$
in the continuous parameter case.
### 2.2 Discrete time scale invariant processes
###### Definition 2.2
A process $\\{X(t),t\in{\bf R^{+}}\\}$ is said to be self-similar of index
$H>0$, if for any ${\lambda}>0$
$\\{\lambda^{-H}X(\lambda t)\\}\stackrel{{\scriptstyle d}}{{=}}\\{X(t)\\}.$
(2.5)
The process is said to be DSI of index $H$ and scaling factor
${\lambda}_{0}>0$ or (H,${\lambda}_{0}$)-DSI, if $(2.5)$ holds for
$\lambda=\lambda_{0}$.
As an intuition, self-similarity refers to an invariance with respect to any
dilation factor. However, this may be a too strong requirement for capturing
in situations that scaling properties are only observed for some preferred
dilation factors.
###### Definition 2.3
A process $\\{X(k),k\in{\bf\check{T}}\\}$ is called discrete time self-similar
(DT-SS) process with parameter space $\check{T}$, where $\check{T}$ is any
subset of countable distinct points of positive real numbers, if for any
$k_{1},k_{2}\in\check{T}$
$\\{X(k_{2})\\}\stackrel{{\scriptstyle
d}}{{=}}(\frac{k_{2}}{k_{1}})^{H}\\{X(k_{1})\\}.$ (2.6)
The process $X(\cdot)$ is called discrete time scale invariance (DT-SI) with
scale $l>0$ and parameter space $\check{T}$, if for any
$k_{1},k_{2}=lk_{1}\in\check{T}$, $(2.6)$ holds.
###### Remark 2.1
If the process $\\{X(t),t\in{\bf R^{+}}\\}$ is DSI with scale $l=\alpha^{T}$
for fixed $T\in{\bf N}$ and $\alpha>1$, then by sampling of the process at
points $\alpha^{k},k\in{\bf Z}$, we have $X(\cdot)$ as a DT-SI process with
parameter space $\check{T}=\\{\alpha^{k},k\in{\bf Z}\\}$ and scale
$l=\alpha^{T}$. If we consider sampling of $X(\cdot)$ at points
$\alpha^{nT+k},n\in{\bf Z},\mbox{for fixed}\ k=0,1,\ldots,T-1$, then
$X(\cdot)$ is a DT-SS process with parameter space
$\check{T}=\\{\alpha^{nT+k},n\in{\bf Z}\\}$.
Yazici et.al. [9] introduced wide sense self-similar processes as the
following definition, which can be obtained by applying the Lamperti
transformation ${\cal L}_{H}$ to the class of wide-sense stationary processes.
This class encompasses all strictly self-similar processes with finite
variance, including Gaussian processes such as fractional Brownian motion but
no other alpha-stable processes.
###### Definition 2.4
A random process $\\{X(t),t\in{\bf R^{+}}\\}$ is said to be wide sense self-
similar with index H, for some $H>0$ if the following properties are satisfied
for each $c>0$, $t,t_{1},t_{2}>0$
$(i)\,\,\ E[X^{2}(t)]<\infty$,
$(ii)\,\,E[X(ct)]=c^{H}E[X(t)]$,
$(iii)\,\,E[X(ct_{1})X(ct_{2})]=c^{2H}E[X(t_{1})X(t_{2})]$.
This process is called wide sense DSI of index $H$ and scaling factor
$c_{0}>0$, if the above conditions hold for some $c=c_{0}$.
###### Definition 2.5
A random process $\\{X(k),k\in\check{T}\\}$ is called DT-SS in the wide sense
with index $H>0$ and with parameter space $\check{T}$, where $\check{T}$ is
any subset of distinct countable points of positive real numbers, if for all
$k,k_{1}\in\check{T}$ and all $c>0$, where $ck,ck_{1}\in\check{T}:$
$(i)\,\,\ E[X^{2}(k)]<\infty$,
$(ii)\,\,E[X(ck)]=c^{H}E[X(k)]$,
$(iii)\,\,E[X(ck)X(ck_{1})]=c^{2H}E[X(k)X(k_{1})]$.
If the above conditions hold for some fixed $c=c_{0}$, then the process is
called DT-SI in the wide sense with scale $c_{0}$.
###### Remark 2.2
Let $\\{X(t),t\in{\bf R^{+}}\\}$ in Remark $2.1$ be DSI in the wide sense,
with the same scale $l=\alpha^{T}$. Then $X(\cdot)$ with parameter space
$\check{T}=\\{\alpha^{k},k\in{\bf Z}\\}$ for $\alpha>1$ is DT-SI in the wide
sense and $X(\cdot)$ with parameter space $\check{T}=\\{\alpha^{nT+k},n\in{\bf
Z}\\}$ for fixed $T\in{\bf N}$, $\alpha>1$ is DT-SS for $k=0,\ldots,T-1$ in
the wide sense.
Through this paper we are dealt with wide sense self-similar and wide sense
scale invariant process, and for simplicity we omit the term ”in the wide
sense” hereafter.
### 2.3 Quasi Lamperti transformation
We introduce the quasi Lamperti transformation and its properties by
followings.
###### Definition 2.6
The quasi Lamperti transform with positive index $H$ and $\alpha>1$, denoted
by ${\cal L}_{H,\alpha}$ operates on a random process $\\{Y(t),t\in{\bf R}\\}$
as
${\cal L}_{H,\alpha}Y(t)=t^{H}Y(\log_{\alpha}t)$ (2.7)
and the corresponding inverse quasi Lamperti transform ${\cal
L}^{-1}_{H,\alpha}$ on process $\\{X(t),t\in{\bf R^{+}}\\}$ acts as
${\cal L}^{-1}_{H,\alpha}X(t)={\alpha}^{-tH}X(\alpha^{t}).$ (2.8)
###### Corollary 2.1
If $\\{Y(t),t\in{\bf R}\\}$ is stationary process, its quasi Lamperti
transform $\\{{\cal L}_{H,\alpha}Y(t),t\in{\bf R^{+}}\\}$ is self-similar.
Conversely if $\\{X(t),t\in{\bf R^{+}}\\}$ is self-similar process, its
inverse quasi Lamperti transform $\\{{\cal L}^{-1}_{H,\alpha}X(t),t\in{\bf
R}\\}$ is stationary.
###### Corollary 2.2
If $\\{X(t),t\in{\bf R^{+}}\\}$ is $(H,{\alpha}^{T}$)-DSI then ${\cal
L}^{-1}_{H,\alpha}X(t)=Y(t)$ is PC with period $T>0$. Conversely if
$\\{Y(t),t\in{\bf R}\\}$ is PC with period $T$ then ${\cal
L}_{H,\alpha}Y(t)=X(t)$ is $(H,{\alpha}^{T}$)-DSI.
###### Remark 2.3
If $X(\cdot)$ is a DT-SS process with parameter space
$\check{T}=\\{l^{n},n\in{\bf Z}\\}$, then its stationary counterpart
$Y(\cdot)$ has parameter space $\check{T}=\\{nT,n\in{\bf Z}\\}$
$X(l^{n})={\cal
L}_{H,\alpha}Y(l^{n})=l^{nH}Y(\log_{\alpha}{\alpha^{nT}})=\alpha^{nTH}Y(nT).$
Also it is clear by the following relation that if $X(\cdot)$ is a DT-SI
process with scale $l=\alpha^{T}$, $T\in{\bf N}$ and parameter space
$\check{T}=\\{\alpha^{n},n\in{\bf Z}\\}$, then $Y(\cdot)$ is a discrete time
periodically correlated (DT-PC) process with period $T$ and parameter space
$\check{T}=\\{n,n\in{\bf Z}\\}$
$Y(n)={\cal L}^{-1}_{H,\alpha}X(n)=\alpha^{-nH}X(\alpha^{n}).$
## 3 Structure of the process
In this section we define a multi-dimensional DT-SS process in the wide sense.
We also introduce a new method for sampling of a DSI process with scale $l>1$,
which provide sampling at arbitrary points in the interval $[1,l)$ and at
multiple $l^{n}$ of such points in the intervals $[l^{n},l^{n+1})$, $n\in{\bf
N}$. We introduce DT-ESI process corresponding to the multi-dimensional DT-ESS
process. Finally in Theorem 3.1 we find harmonic like representation and
spectral density matrix of the multi-dimensional DT-ESS process.
###### Definition 3.1
The process $U(t)=(U^{0}(t),U^{1}(t),\ldots,U^{q-1}(t))$ with parameter space
$\check{T}=\\{l^{n},n\in{\bf Z}\\}$, $l=\alpha^{T}$, $\alpha>1$ and $T\in{\bf
N}$ is a q-dimensional discrete time self similar process in the wide sense,
where
$\bf(a)$ $\\{U^{j}(\cdot)\\}$ for all $j=0,1,\cdots,q-1$ is DT-SS process with
parameter space
$\check{T}^{j}=\\{l^{n},n\in{\bf Z}\\}$.
$\bf(b)$ For every $n,\tau\in{\bf Z},\,\ j,k=0,1,\cdots,q-1$
$\mathrm{Cov}\big{(}U^{j}(l^{n+\tau}),U^{k}(l^{n})\big{)}=l^{2nH}\mathrm{Cov}\big{(}U^{j}(l^{\tau}),U^{k}(1)\big{)}.$
Our method of sampling is to provide enough flexibility to choose arbitrary
sample points of a discrete time scale invariant process. So, one could decide
to have $T$ partitions in each scale interval $I_{n}=[l^{n},l^{n+1})$,
$n\in{\bf Z}$ of a continuous time DSI process $X(\cdot)$ with scale $l>1$ and
find $\alpha$ by $l=\alpha^{T}$. Then our partitions in scale interval $I_{n}$
are
$[\alpha^{nT},\alpha^{nT+1}),[\alpha^{nT+1},\alpha^{nT+2}),\ldots,[\alpha^{nT+T-1},\alpha^{(n+1)T}).$
So we consider to have $n_{k}$ samples in partition
$[\alpha^{nT+k},\alpha^{nT+k+1})$ at points
$\alpha^{nT+k}s_{k_{1}},\alpha^{nT+k}s_{k_{2}},\ldots,\alpha^{nT+k}s_{k_{n_{k}}}$
where $1\leqslant s_{k_{1}}<s_{k_{2}}<\ldots<s_{k_{n_{k}}}<\alpha$,
$k=0,\ldots,T-1$ and $q=\sum_{i=0}^{T-1}n_{i}$. Now we can state the following
remark.
###### Remark 3.1
Let $U^{k}(l^{n})=X(l^{n}{\bf s}_{u})$ in Definition $3.1$, where ${\bf
s}_{u}=\alpha^{k}s_{k_{x}}$ in which $\sum_{i=-1}^{k-1}n_{i}\leqslant
u<\sum_{i=-1}^{k}n_{i}$, $n_{-1}=0$ and $u=x+\sum_{i=0}^{k-1}n_{i}$,
$x=1,\ldots,n_{k}$. Thus $X(l^{n}{\bf s}_{u})$ for $u=0,\ldots,q-1$ is a DT-SS
process and $U(l^{n})=\big{(}X(l^{n}{\bf s}_{0}),\ldots,X(l^{n}{\bf
s}_{q-1})\big{)}$ is a $q$-dimensional DT-SS process.
By such method of sampling at discrete points we provide a $q$-dimensional DT-
ESS process $V(n)$ as
$V(n)=\big{(}V^{0}(n),V^{1}(n),\ldots,V^{q-1}(n)\big{)},\hskip
28.45274ptn\in{\bf Z}$
where $q=\sum_{i=0}^{T-1}n_{i}$ and
$V^{u}(n):=X(\alpha^{nT}{\bf s}_{u})$ (3.1)
$\sum_{i=-1}^{k-1}n_{i}\leqslant u<\sum_{i=-1}^{k}n_{i}$, $n_{-1}=0$, ${\bf
s}_{u}=\alpha^{k}s_{k_{x}}$ and $x=u-\sum_{i=0}^{k-1}n_{i}$, $u=0,\ldots,q-1$.
###### Remark 3.2
Corresponding to the $q$-dimensional DT-ESS process $V(n)$ there exist a DT-
ESI process $W(\kappa)$ with scale $l=\alpha^{T}$ as
$W(\kappa):=V^{u}(n)=X(\alpha^{nT}{\bf s}_{u})\hskip 28.45274pt\kappa\in{\bf
Z}$ (3.2)
where $u=\kappa-q[\frac{\kappa}{q}]$, $n=[\frac{\kappa}{q}]$ and
$\kappa=nq+u$, since by $(3.1)$ and $(3.2)$
$W(\kappa+q)=X(\alpha^{(n+1)T}{\bf s}_{u})\stackrel{{\scriptstyle
d}}{{=}}\alpha^{TH}X(\alpha^{nT}{\bf s}_{u})=l^{H}W(\kappa).$
By the following theorem, the spectral density matrix of the $q$-dimensional
DT-ESS process and harmonic like representation of each column is obtained.
###### Theorem 3.1
Let $X(\cdot)$ be a DSI process with scale $l=\alpha^{T}$ and $1\leqslant{\bf
s}_{0}<{\bf s}_{1}<\ldots<{\bf s}_{q-1}<\alpha^{T}$, then
$V(n)=\big{(}V^{0}(n),\ldots,V^{q-1}(n)\big{)}$, where
$V^{u}(n)=X(\alpha^{nT}{\bf s}_{u})$, $n\in{\bf Z}$ and $u=0,\ldots,q-1$ is a
$q$-dimensional DT-ESS process and
(i) The harmonic like representation of $V^{u}(n)$ is
$V^{u}(n)=(\alpha^{nT}{\bf s}_{u})^{H}\int_{0}^{2\pi}e^{i\omega
n}d\phi_{u}(\omega)$ (3.3)
where $\phi_{u}(\omega)$ is an orthogonal spectral measure, that is
$E[d\phi_{u}(\omega)\overline{d\phi_{\nu}(\omega^{\prime})}]=0$,
$u,\nu=0,\ldots,q-1$ when $\omega\neq\omega^{\prime}$.
(ii) The corresponding spectral density matrix of $V(n)$ is
$g^{H}(\omega)=[g_{u,\nu}^{H}(\omega)]_{u,\nu=0,\ldots,q-1}$, where
$g_{u,\nu}^{H}(\omega)=\frac{({\bf s}_{u}{\bf
s}_{\nu})^{-H}}{2\pi}\sum_{\tau=-\infty}^{\infty}\alpha^{-TH\tau}e^{-i\omega\tau}Q^{H}_{u,\nu}(\tau)$
(3.4)
$\tau\in{\bf N}$ and $Q^{H}_{u,\nu}(\tau)$ is the covariance function of
$V^{u}(\tau)$ and $V^{\nu}(0)$.
Proof of (i): Remark $2.3$ implies that
$V^{u}(n)=X(\alpha^{nT}{\bf s}_{u})={\cal L}_{H,\alpha}Y(\alpha^{nT}{\bf
s}_{u})=(\alpha^{nT}{\bf s}_{u})^{H}\eta^{u}(n)$
where $\eta^{u}(n)=Y(nT+\log_{\alpha}{\bf s}_{u})$. Thus $V^{u}(n)$ for every
$u=0,1,\ldots,q-1$ is a DT-ESS process in $n$, where its discrete time
stationary counterpart $\eta^{u}(n)$ for fixed $u=0,1,\ldots,q-1$ has spectral
representation $\eta^{u}(n)=\int_{0}^{2\pi}e^{i\omega n}d\phi_{u}(\omega)$.
Proof of (ii): The covariance matrix of $V(n)$ is denoted by
$Q^{H}(n,\tau)=[Q^{H}_{u,\nu}(n,\tau)]_{u,\nu=0,\ldots,q-1}$ where
$Q^{H}_{u,\nu}(n,\tau)=E[V^{u}(n+\tau)V^{\nu}(n)]=E[X(\alpha^{(n+\tau)T}{\bf
s}_{u})X(\alpha^{nT}{\bf s}_{\nu})]$
By the scale invariant property of the process $X(\cdot)$ we have that
$Q^{H}_{u,\nu}(n,\tau)=\alpha^{2nTH}E[X(\alpha^{\tau T}{\bf s}_{u})X({\bf
s}_{\nu})]=\alpha^{2nTH}Q^{H}_{u,\nu}(\tau)$ (3.5)
where $Q^{H}_{u,\nu}(\tau)=Q^{H}_{u,\nu}(0,\tau)=E[V^{u}(\tau)V^{\nu}(0)]$,
then by (3.3)
$Q^{H}_{u,\nu}(\tau)=E[(\alpha^{\tau T}{\bf s}_{u})^{H}({\bf
s}_{\nu})^{H}\int_{0}^{2\pi}e^{i\omega\tau}d\phi_{u}(\omega)\int_{0}^{2\pi}\overline{d\phi_{v}(\omega^{\prime})}]$
$=\alpha^{\tau TH}({\bf s}_{u}{\bf
s}_{\nu})^{H}\int_{0}^{2\pi}e^{i\omega\tau}dG^{H}_{u,\nu}(\omega)$ (3.6)
where
$E[d\phi_{u}(\omega)\overline{d\phi_{\nu}(\omega^{\prime})}]=dG^{H}_{u,\nu}(\omega)$
when $\omega=\omega^{\prime}$ and is $0$ when $\omega\neq\omega^{\prime}$.
On the other hand, by the definition of $\eta^{u}(n)$ in the proof of part
$(i)$
$Q^{H}_{u,\nu}(\tau)=E[X(\alpha^{\tau T}{\bf s}_{u})X({\bf s}_{\nu})]=E[{\cal
L}_{H,\alpha}Y(\alpha^{\tau T}{\bf s}_{u}){\cal L}_{H,\alpha}Y({\bf
s}_{\nu})]$ $=(\alpha^{\tau T}{\bf s}_{u}{\bf s}_{\nu})^{H}E[Y(\tau
T+\log_{\alpha}{\bf s}_{u})Y(\log_{\alpha}{\bf s}_{\nu})]$ $=(\alpha^{\tau
T}{\bf s}_{u}{\bf s}_{\nu})^{H}E[\eta^{u}(\tau)\eta^{\nu}(0)]=(\alpha^{\tau
T}{\bf s}_{u}{\bf s}_{\nu})^{H}B_{u,\nu}(\tau).$
Then by (3.6)
$B_{u,\nu}(\tau)=\int_{0}^{2\pi}e^{i\omega\tau}dG^{H}_{u,\nu}(\omega),\hskip
14.22636ptu,\nu=0,\ldots,q-1$
Now by (2.3) and (2.4) for $u,\nu=0,\ldots,q-1$ we have that
$G^{H}_{u,\nu}(A)=\frac{1}{2\pi}\int_{A}\sum_{\tau=-\infty}^{\infty}B_{u,\nu}(\tau)e^{-i\lambda\tau}d\lambda.$
By substituting $B_{u,\nu}(\tau)=(\alpha^{\tau T}{\bf s}_{u}{\bf
s}_{\nu})^{-H}Q^{H}_{u,\nu}(\tau)$, the elements of the spectral distribution
function, $G^{H}_{u,\nu}(\cdot)$ has the following representation
$G^{H}_{u,\nu}(A)=\frac{({\bf s}_{u}{\bf
s}_{\nu})^{-H}}{2\pi}\int_{A}\sum_{\tau=-\infty}^{\infty}\alpha^{-TH\tau}e^{-i\lambda\tau}Q^{H}_{u,\nu}(\tau)d\lambda.$
(3.7)
Let $A=(\omega,\omega+d\omega]$, then the elements of the spectral density
matrix, $g_{u,\nu}^{H}(\omega)$ are
$g_{u,\nu}^{H}(\omega):=\frac{G^{H}_{u,\nu}(d\omega)}{d\omega}=\frac{({\bf
s}_{u}{\bf
s}_{\nu})^{-H}}{2\pi}\sum_{\tau=-\infty}^{\infty}\alpha^{-TH\tau}\big{(}\frac{1}{d\omega}\int_{\omega}^{\omega+d\omega}e^{-i\lambda\tau}d\lambda\big{)}Q^{H}_{u,\nu}(\tau)$
$=\frac{({\bf s}_{u}{\bf
s}_{\nu})^{-H}}{2\pi}\sum_{\tau=-\infty}^{\infty}\alpha^{-TH\tau}\big{(}\frac{1}{-i\tau}\lim_{d\omega\rightarrow
0}\frac{e^{-i{(\omega+d\omega)\tau}}-e^{-i\omega\tau}}{d\omega}\big{)}Q^{H}_{u,\nu}(\tau)$
$=\frac{({\bf s}_{u}{\bf
s}_{\nu})^{-H}}{2\pi}\sum_{\tau=-\infty}^{\infty}\alpha^{-TH\tau}\big{(}(\frac{1}{-i\tau})(-i\tau)e^{-i\omega\tau}\big{)}Q^{H}_{u,\nu}(\tau).$
Thus we get to the assertion of part (ii) of the theorem.$\square$
## 4 Multi-dimensional DT-ESSM process
Using our method of sampling in section 3, we find the covariance function of
the DT-ESI process $W(\cdot)$, which is defined in (3.2) and its corresponding
multi-dimensional DT-ESS process $V(\cdot)$, defined in (3.1) for the case
that they are Markov in the wide sense as well, which we call them DT-ESIM and
DT-ESSM respectively in subsection 4.1. We find the spectral density matrix of
these processes in subsection 4.2.
### 4.1 Covariance function of DT-ESIM
Here we characterize the covariance function of the DT-ESIM process
$\\{W(\kappa),\kappa\in{\bf Z}\\}$ in Theorem 4.1 and the covariance function
of the associated $q$-dimensional DT-ESSM process in Theorem 4.2.
###### Theorem 4.1
Let $\\{W(\kappa),\kappa\in{\bf Z}\\}$, defined in $(3.2)$, be DT-ESI and
Markov in the wide sense DT-ESIM, with scale $\alpha^{T}$. Then for
$\tau\in{\bf W}=\\{0,1,\ldots\\}$, $\kappa=nq+\nu$, $\kappa+\tau=mq+u$,
$u,\nu=0,\ldots,q-1$ and $n,m\in{\bf Z}$, the covariance function
$R_{\kappa}(\tau):=E[W(\kappa+\tau)W(\kappa)]=E[X(\alpha^{mT}{\bf
s}_{u})X(\alpha^{nT}{\bf s}_{\nu})]$ (4.1)
can be characterized as
$R_{\kappa}(tq+s)=[\tilde{f}(q-1)]^{t}\tilde{f}(\kappa+s-1)[\tilde{f}(\kappa-1)]^{-1}R_{\kappa}(0)$
(4.2) $R_{\kappa}(-tq+s)=\alpha^{-2tqH}R_{\kappa+s}((t-1)q+q-s)$
where $1\leqslant{\bf s}_{0}<{\bf s}_{1}<\ldots<{\bf s}_{q-1}<\alpha^{T}$,
$t\in{\bf Z}$, $s=0,\ldots,q-1$
$\tilde{f}(r)=\prod_{j=0}^{r}f(j)=\prod_{j=0}^{r}R_{j}(1)/R_{j}(0),\hskip
19.91692ptr\in{\bf Z}$ (4.3)
and $\tilde{f}(-1)=1$.
Before proceeding to the proof of the theorem we present the main property of
covariance function of the wide sense Markov process.
Let $\\{X(n),n\in{\bf Z}\\}$ be a second order process of centered random
variables, $E[X(n)]=0$ and $E[|X(n)|^{2}]<\infty$, $n\in{\bf Z}$. Following
Doob [3], the real valued second order process $X(\cdot)$ is Markov in the
wide sense if
$R(n_{1},n_{2})=G\big{(}\min(n_{1},n_{2})\big{)}H\big{(}\max(n_{1},n_{2})\big{)}$
(4.4)
where $R(n_{1},n_{2}):=E[X(n_{1})X(n_{2})]$ is the covariance function of
$X(\cdot)$ and $G$ and $H$ are defined uniquely up to a constant multiple and
the ratio $G/H$ is a positive nondecreasing function.
Proof of the theorem: As $\\{W(\kappa),\kappa\in{\bf Z}\\}$ is DT-ESI with
scale $\alpha^{T}$, this theorem fully characterize the covariance function of
the DT-ESIM process. From the Markov property (4.4), $R_{\kappa}(\tau)$
defined in (4.1), satisfies
$R_{\kappa}(\tau)=G(\alpha^{nT}{\bf s}_{\nu})H(\alpha^{mT}{\bf s}_{u}),\hskip
28.45274pt\tau\in{\bf Z},\alpha>1$ (4.5)
By substituting $\tau=0$ in the above relation we have $m=n$, then
$G(\alpha^{nT}{\bf s}_{\nu})=\frac{R_{\kappa}(0)}{H(\alpha^{nT}{\bf
s}_{\nu})}.$
Therefore
$R_{\kappa}(\tau)=\frac{H(\alpha^{mT}{\bf s}_{u})}{H(\alpha^{nT}{\bf
s}_{\nu})}R_{\kappa}(0),\hskip 28.45274pt\tau\in{\bf Z}$ (4.6)
Thus
$H(\alpha^{mT}{\bf
s}_{u})=\frac{R_{\kappa}(\tau)}{R_{\kappa}(0)}H(\alpha^{nT}{\bf s}_{\nu}).$
So
${\bf H}(\kappa+\tau)=\frac{R_{\kappa}(\tau)}{R_{\kappa}(0)}{\bf H}(\kappa)$
where ${\bf H}(\kappa)=H(\alpha^{nT}{\bf s}_{\nu})$ and ${\bf
H}(\kappa+\tau)=H(\alpha^{mT}{\bf s}_{u})$. Therefore
$R_{\kappa}(\tau)=\frac{{\bf H}(\kappa+\tau)}{{\bf H}(\kappa)}R_{\kappa}(0).$
(4.7)
For $\tau=1$, we have
${\bf H}(\kappa+1)=\frac{R_{\kappa}(1)}{R_{\kappa}(0)}{\bf H}(\kappa).$
By the recursive relation, it follows that
${\bf
H}(\kappa+1)=\frac{R_{\kappa}(1)}{R_{\kappa}(0)}\frac{R_{\kappa-1}(1)}{R_{\kappa-1}(0)}\ldots\frac{R_{0}(1)}{R_{0}(0)}{\bf
H}(0)={\bf H}(0)\prod_{j=0}^{\kappa}f(j)$
and
${\bf H}(\kappa)={\bf H}(0)\prod_{j=0}^{\kappa-1}f(j)$
where $f(j)=R_{j}(1)/R_{j}(0)$. By the assumptions $n=[\frac{\kappa}{q}]$,
$\nu=\kappa-q[\frac{\kappa}{q}]$ we have $\kappa=nq+\nu$, then
${\bf H}(nq+\nu)={\bf H}(0)\prod_{j=0}^{nq+\nu-1}f(j).$ (4.8)
As mentioned in Remark 3.2, $\\{W(\kappa),\kappa\in{\bf Z}\\}$ is DT-ESI with
scale $l$, then
$f(\kappa+q)=\frac{R_{\kappa+q}(1)}{R_{\kappa+q}(0)}=\frac{E[W(\kappa+q+1)W(\kappa+q)]}{E[W(\kappa+q)W(\kappa+q)]}$
$=\frac{\alpha^{2TH}E[W(\kappa+1)W(\kappa)]}{\alpha^{2TH}E[W(\kappa)W(\kappa)]}=\frac{R_{\kappa}(1)}{R_{\kappa}(0)}=f(\kappa).$
Hence by (4.8)
${\bf H}(nq+\nu)={\bf
H}(0)\big{[}\prod_{j=0}^{q-1}f(j)\big{]}^{n}\prod_{j=0}^{\nu-1}f(j),\hskip
19.91692pt\nu\geqslant 1$
By the definition of $\tilde{f}$ in (4.3)
${\bf H}(nq+\nu)={\bf H}(0)[\tilde{f}(q-1)]^{n}\tilde{f}(\nu-1).$ (4.9)
By a similar method one can verify that
${\bf H}(-nq+\nu)={\bf H}(0)[\tilde{f}(q-1)]^{-n}\tilde{f}(\nu-1).$
Let $\tau=tq+s$ in (4.7), $t\in{\bf W}$ and $s=0,1,\ldots,q-1$, then it
follows from (4.9) that
$R_{\kappa}(tq+s)=\frac{{\bf H}(\kappa+tq+s)}{{\bf
H}(\kappa)}R_{\kappa}(0)=\frac{{\bf
H}(0)[\tilde{f}(q-1)]^{t}\tilde{f}(\kappa+s-1)}{{\bf
H}(0)\tilde{f}(\kappa-1)}R_{\kappa}(0).$
For $\tau=-tq+s$ we have that
$R_{\kappa}(-tq+s)=E[X(\alpha^{-tq+\kappa+s})X(\alpha^{\kappa})]=\alpha^{-2tqH}E[X(\alpha^{\kappa+s})X(\alpha^{tq+\kappa})]$
$=\alpha^{-2tqH}R_{\kappa+s}(tq-s)=\alpha^{-2tqH}R_{\kappa+s}((t-1)q+q-s).\square$
Now we can use this theorem to prove the next result as follows, for
$q$-dimensional DT-ESSM process.
###### Theorem 4.2
Let $\\{W(\kappa),\kappa\in{\bf Z}\\}$ be a DT-ESIM process, and
$\\{V(n),n\in{\bf Z}\\}$ be its associated $q$-dimensional DT-ESSM process
with covariance matrix $Q^{H}(n,\tau)$ which is defined by $(3.5)$. Then
$Q^{H}(n,\tau)=\alpha^{2nTH}[\tilde{f}(q-1)]^{\tau}CR,\hskip
19.91692pt\tau\in{\bf Z}$ (4.10)
where $\tilde{f}(\cdot)$ is defined in $(4.3)$ and the matrices $C$ and $R$
are given by $C=[C_{u,\nu}]_{u,\nu=0,\ldots,q-1}$, where
$C_{u,\nu}=\tilde{f}(u-1)[\tilde{f}(\nu-1)]^{-1}$, and $R$ is a diagonal
matrix with diagonal elements $R_{\nu}(0)$, $\nu=0,1,\ldots,q-1$, which is
defined in $(4.1)$.
Proof: As $W(\cdot)$ is DT-ESI with scale $l$, (3.2) and (3.5) indicate that
$Q^{H}_{u,\nu}(n,\tau)=\alpha^{2nTH}Q^{H}_{u,\nu}(\tau)$. Now by the
assumption $\kappa=nq+\nu$ and $\kappa+\tau=mq+u$ where $m,n\in{\bf Z}$,
$\tau\in{\bf W}$, we have $\tau=(m-n)q+u-\nu$ and therefore
$R_{\kappa}(\tau)=R_{nq+\nu}((m-n)q+u-\nu)=E[W(mq+u)W(nq+\nu)]$
$=E[X(\alpha^{mT}{\bf s}_{u})X(\alpha^{nT}{\bf s}_{\nu})].$
Hence
$Q^{H}_{u,\nu}(\tau)=E[X(\alpha^{\tau T}{\bf s}_{u})X({\bf
s}_{\nu})]=R_{\nu}(\tau q+u-\nu)$ (4.11)
and by the Markov property of $W(\cdot)$ from (4.2) we have
$R_{\nu}(\tau
q+u-\nu)=[\tilde{f}(q-1)]^{\tau}\tilde{f}(u-1)[\tilde{f}(\nu-1)]^{-1}R_{\nu}(0)$
for $u,\nu=0,\ldots,q-1$. Let
$C_{u,\nu}=\tilde{f}(u-1)[\tilde{f}(\nu-1)]^{-1}$, so
$Q^{H}_{u,\nu}(\tau)=[\tilde{f}(q-1)]^{\tau}C_{u,\nu}R_{\nu}(0).$ (4.12)
Thus we can represent the elements of the covariance matrix of $q$-dimensional
DT-ESSM process as
$Q^{H}_{u,\nu}(n,\tau)=\alpha^{2nTH}[\tilde{f}(q-1)]^{\tau}C_{u,\nu}R_{\nu}(0).\square$
### 4.2 Spectral representation of the process
The spectral density matrix of the $q$-dimensional DT-ESSM process is
characterized by the following lemma which is proved in [7].
###### Lemma 4.1
The spectral density matrix
$g^{H}(\omega)=[g^{H}_{u,\nu}(\omega)]_{u,\nu=0,\ldots,q-1}$ of the
$q$-dimensional DT-ESSM process $V(n)$ is specified by
$g_{u,\nu}^{H}(\omega)=\frac{({\bf s}_{u}{\bf
s}_{\nu})^{-H}}{2\pi}\left[\frac{\tilde{f}(u-1)R_{\nu}(0)}{\tilde{f}(\nu-1)\big{(}1-e^{-i\omega}\alpha^{-HT}\tilde{f}(q-1)\big{)}}-\frac{\tilde{f}(\nu-1)R_{u}(0)}{\tilde{f}(u-1)\big{(}1-e^{-i\omega}\alpha^{HT}\tilde{f}^{-1}(q-1)\big{)}}\right]$
where $R_{k}(0)$ is the variance of $W(k)$ and $\tilde{f}(\cdot)$ is defined
by $(4.3)$.
Proof: By applying (3.4) and (4.12), the spectral density matrix of the
process $\\{V(n),n\in{\bf Z}\\}$ which is denoted by
$g^{H}(\omega)=[g^{H}_{u,\nu}(\omega)]_{u,\nu=0,\ldots,q-1}$ can be written as
$g_{u,\nu}^{H}(\omega)=\frac{({\bf s}_{u}{\bf
s}_{\nu})^{-H}}{2\pi}\Big{[}\sum_{\tau=0}^{\infty}\alpha^{-TH\tau}e^{-i\omega\tau}Q^{H}_{u,\nu}(\tau)$
$+\sum_{\tau=-\infty}^{-1}\alpha^{-TH\tau}e^{-i\omega\tau}Q^{H}_{u,\nu}(\tau)\Big{]}=g_{u,\nu,1}^{H}(\omega)+g_{u,\nu,2}^{H}(\omega)$
where
$g_{u,\nu,1}^{H}(\omega)=\frac{({\bf s}_{u}{\bf
s}_{\nu})^{-H}}{2\pi}\sum_{\tau=0}^{\infty}\alpha^{-TH\tau}e^{-i\omega\tau}[\tilde{f}(q-1)]^{\tau}\tilde{f}(u-1)[\tilde{f}(\nu-1)]^{-1}R_{\nu}(0)$
$=\frac{({\bf s}_{u}{\bf
s}_{\nu})^{-H}\tilde{f}(u-1)R_{\nu}(0)}{2\pi\tilde{f}(\nu-1)}\sum_{\tau=0}^{\infty}\big{(}\alpha^{-TH}e^{-i\omega}\tilde{f}(q-1)\big{)}^{\tau}.$
(4.13)
By Remark 3.2, the scale invariant property of $W(\kappa)$ and the assumption,
that at least one of the $\text{Corr}[W(j)W(j+1)]$ be smaller than one, we
have that $|\tilde{f}(q-1)|<\alpha^{TH}$ for $j=0,\ldots,q-1$. Thus
$|e^{-i\omega}\alpha^{-TH}\tilde{f}(q-1)|=|\alpha^{-TH}\tilde{f}(q-1)|<1,$
and (4.13) for $\tau\in{\bf W}$ is convergent. By the equality
$Q_{u,\nu}(-\tau)=E[X(\alpha^{-\tau T}{\bf s}_{u})X({\bf
s}_{\nu})]=\alpha^{-2\tau TH}E[X(\alpha^{\tau T}{\bf s}_{\nu})X({\bf
s}_{u})]=\alpha^{-2\tau TH}Q_{\nu,u}(\tau),$
convergence of $g_{u,\nu,2}^{H}(\omega)$ follows by a similar method.
Therefore
$g_{u,\nu}^{H}(\omega)=\frac{({\bf s}_{u}{\bf
s}_{\nu})^{-H}}{2\pi}\Big{[}\frac{R_{\nu}(0)\tilde{f}(u-1)}{\tilde{f}(\nu-1)}\sum_{\tau=0}^{\infty}\big{(}\alpha^{-TH}e^{-i\omega}\tilde{f}(q-1)\big{)}^{\tau}$
$+\frac{R_{u}(0)\tilde{f}(\nu-1)}{\tilde{f}(u-1)}\sum_{\tau=1}^{\infty}\big{(}\alpha^{-TH}e^{i\omega}\tilde{f}(q-1)\big{)}^{\tau}\Big{]}$
$=\frac{({\bf s}_{u}{\bf
s}_{\nu})^{-H}}{2\pi}\Big{[}\frac{R_{\nu}(0)\tilde{f}(u-1)}{\tilde{f}(\nu-1)\big{(}1-\alpha^{-TH}e^{-i\omega}\tilde{f}(q-1)\big{)}}+\frac{R_{u}(0)\tilde{f}(\nu-1)\alpha^{-TH}e^{i\omega}\tilde{f}(q-1)}{\tilde{f}(u-1)\big{(}1-\alpha^{-TH}e^{i\omega}\tilde{f}(q-1)\big{)}}\Big{]},$
so we arrive at the assertion of the lemma.$\square$
###### Example 4.1
Let
$X(t)=\sum_{n=1}^{\infty}\lambda^{n(H-\frac{1}{2})}I_{[\lambda^{n-1},\lambda^{n})}(t)B(t)$
where $B(\cdot)$ is the standard Brownian motion, $I(\cdot)$ indicator
function, $H>0$ and $\lambda>1$. We call this process Simple Brownian Motion.
We showed in [7] that $\\{X(t),t\in R^{+}\\}$ is DSI and Markov with Hurst
index $H$ and scale $\lambda$. By sampling of this process at points
$\alpha^{nT}{\bf s}_{u}$, $n\in{\bf W}$, where $1\leq{\bf s}_{0}\leq{\bf
s}_{1},\cdots,{\bf s}_{q-1}<\alpha^{T}$, and by assuming $\lambda=\alpha^{T}$,
$W(\kappa):=X(\alpha^{nT}{\bf s}_{u}),$
is a DT-ESIM process, and $V(n)=\big{(}V^{0}(n),\ldots,V_{q-1}(n)\big{)}$
where $V^{u}(n)=W(\kappa)$ is the associated $q$-dimensional DT-ESSM process
where $u=\kappa-q[\frac{\kappa}{q}]$, $n=[\frac{\kappa}{q}]$. By (4.1) we have
that $R_{j}^{H}(0)=R_{j}^{H}(1)=\alpha^{2TH^{\prime}}{\bf s}_{j}$ for
$j=0,\cdots,q-2$ and
$R_{q-1}^{H}(1)=\alpha^{TH^{\prime}}R_{q-1}^{H}(0)=\alpha^{3TH^{\prime}}{\bf
s}_{q-1}$, where $H^{\prime}=H-\frac{1}{2}$. So
$R_{u}(0)=\alpha^{2TH^{\prime}}{\bf s}_{u}$,
$\;R_{\nu}(0)=\alpha^{2TH^{\prime}}{\bf s}_{\nu}$. Also (4.3) implies that
$\tilde{f}(u-1)=\tilde{f}(\nu-1)=1$, $\tilde{f}(q-1)=\alpha^{TH^{\prime}}$.
Thus By Lemma 4.1, the spectral density matrix of $V(n)$ is
$g_{u,\nu}^{H}(\omega)=\frac{({\bf s}_{u}{\bf
s}_{\nu})^{-H}\alpha^{2TH^{\prime}}}{2\pi}\left[\frac{{\bf
s}_{\nu}}{1-e^{-i\omega}\alpha^{-T/2}}-\frac{{\bf
s}_{u}}{1-e^{-i\omega}\alpha^{T/2}}\right].$
## References
* [1] P. Borgnat, P.O. Amblard, P. Flandrin, 2005, ”Scale invariances and Lamperti transformations for stochastic processes”, Journal of Physics A: Mathematical and General, Vol.38, pp.2081 2101.
* [2] M.E. Caballero, L. Chaumont, 2006, ”Weak convergence of positive self-similar Markov processes and overshoots of Levy processes”, The annals of probability, Vol.34, No.3, pp.1012-1034.
* [3] J.L. Doob, ”Stochastic Processes”, Wiley, New York 1953.
* [4] P. Flandrin, P. Borgnat, P.O. Amblard, 2002, ”From stationarity to selfsimilarity, and back : Variations on the Lamperti transformation”, appear in Processes with Long-Range Correlations, pp.88-117.
* [5] E.G. Gladyshev, 1961, ”Periodically correlated random sequences”, Soviet Math. Dokl., No.2, pp.385-388.
* [6] N. Modarresi, S. Rezakhah, 2009, ”Discrete time scale invariant Markov processes”, arxiv/pdf/0905/0905.3959v3.pdf.
* [7] N. Modarresi, S. Rezakhah, 2010, ”Spectral analysis of Multi-dimensional self-similar Markov processes”, Journal of Physics A: Mathematical and General, Accepted,arxiv/pdf/0907/0907.2295v4.pdf
* [8] Y.A. Rozanov, 1967, ”Stationary Random Processes”, Holden-Day, San Francisco.
* [9] B. Yazici, R.L. Kashyap, 1997, ”A class of second-order stationary self-similar processes for 1/f phenomena”, IEEE Transactions on Signal Processing, No. 45, pp.396-410.
|
arxiv-papers
| 2010-03-05T05:31:57 |
2024-09-04T02:49:08.764526
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "N . Modarresi and S . Rezakhah",
"submitter": "Saeid Rezakhah",
"url": "https://arxiv.org/abs/1003.1187"
}
|
1003.1215
|
# Special $L$-values of geometric motives
Jakob Scholbach 111Universität Münster, Mathematisches Institut, Einsteinstr.
62, D-48149 Münster, Germany, jakob.scholbach@uni-muenster.de
###### Abstract
This paper proposes a conjecture about special values of $L$-functions
$L(M,s):=\prod_{p}\operatorname{det}(\mathrm{Id}-\operatorname{Fr}^{-1}p^{-s}|i_{p}^{*}M_{\ell})^{-1}$
of geometric motives $M$ over $\mathbb{Z}$. This includes $L$-functions of
mixed motives over $\mathbb{Q}$ and Hasse-Weil $\zeta$-functions of schemes
over $\mathbb{Z}$. We conjecture the following: the order of $L(M,s)$ at $s=0$
is given by the negative Euler characteristic of motivic cohomology of
$D(M):=M^{\vee}(1)[2]$. Up to a nonzero rational factor, the $L$-value at
$s=0$ is given by the determinant of a pairing coupling an Arakelov-like
variant of motivic cohomology of $M$ with the motivic cohomology of $D(M)$:
$L^{*}(M,0)\equiv\prod_{i}\operatorname{det}(\mathrm{H}_{\mathrm{c}}^{i}(M){\otimes}\mathrm{H}^{-i}(D(M))\rightarrow\mathbb{R})^{(-1)^{i+1}}\
\ (\mathrm{mod\ }\mathbb{Q}^{\times}).$
Under standard assumptions concerning mixed motives over $\mathbb{Q}$,
$\mathbb{F}_{p}$, and $\mathbb{Z}$, this conjecture is essentially equivalent
to the conjunction of Soulé’s conjecture about pole orders of
$\zeta$-functions of schemes over $\mathbb{Z}$, Beilinson’s conjecture about
special $L$-values for motives over $\mathbb{Q}$ and the Tate conjecture over
$\mathbb{F}_{p}$.
$L$-functions have a long and rich history. Starting with Riemann’s zeta
function, the scope of $L$-functions has been progressively expanded to apply
to more general objects such as mixed motives $M_{\eta}$ over $\mathbb{Q}$:
$L_{F}(M_{\eta},s):=\prod_{p<\infty}\operatorname{det}\left(\mathrm{Id}-\operatorname{Fr}^{-1}p^{-s}|(M_{\eta})_{\ell}^{I_{p}}\right)^{-1}.$
Here $(M_{\eta})_{\ell}$ denotes the $\ell$-adic realization of the motive and
$I_{p}$ the inertia group. We give a natural adaptation of this definition to
geometric motives over $\mathbb{Z}$ (Definition 2.1). That generalization
incorporates both $L$-functions over $\mathbb{Q}$ and Hasse-Weil
$\zeta$-functions of schemes $X$ of finite type over $\mathbb{Z}$:
$L_{\mathbb{Q}}(M_{\eta},s)^{-1}=L_{\mathbb{Z}}(\eta_{!*}M_{\eta}[1],s),$ (1)
$\zeta(X,s)=L(\operatorname{M}_{\mathrm{c}}(X),s).$
Here $M_{\eta}$ is a mixed motive over $\mathbb{Q}$ satisfying a certain
smoothness condition, $\eta_{!*}$ is a generic intermediate extension functor
similar to the one familiar in perverse sheaf theory, and
$\operatorname{M}_{\mathrm{c}}(X)$ denotes the motive with compact support.
The second identity is a consequence of Grothendieck’s trace formula. It
implies the independence of $L$-functions of choices of $\ell$ for a large
category of motives, namely the triangulated category generated by
$\operatorname{M}(X)$ where $X$ is any scheme over $\mathbb{Z}$ (Lemma 2.9).
Geometric motives over $\mathbb{Z}$ therefore appear as a natural framework to
deal with $L$-functions.
This naturalness allows for a compact and conceptual conjecture describing the
special values and pole orders of $L$-functions. The conjecture splits into
two parts, each of which is interesting in its own right.
The first part, Conjecture 3.1, states that there should be a cohomology
theory $\mathrm{H}_{\mathrm{c}}^{*}(M)$ called _motivic cohomology with
compact support_ taking into account both (standard) motivic cohomology
$\mathrm{H}^{*}(M)$ and information at the archimedean place in the guise of
weak Hodge cohomology $\mathrm{H}_{\mathrm{w}}^{*}(M)$. The latter is an
invariant related to the period map of Betti and de Rham cohomology (cf.
Section 1.2). More precisely, there are to be long exact sequences
$\dots\rightarrow\mathrm{H}_{\mathrm{c}}^{i}(M)\rightarrow\mathrm{H}^{i}(M)_{\mathbb{R}}\rightarrow\mathrm{H}_{\mathrm{w}}^{i}(M)\rightarrow\mathrm{H}_{\mathrm{c}}^{i+1}(M)\rightarrow\dots$
for any geometric motive $M$ over $\mathbb{Z}$. The spaces
$\mathrm{H}_{\mathrm{c}}^{*}(M)$ should be linked to motivic cohomology of the
Verdier dual $D(M):=\underline{\mathrm{Hom}}(M,\mathbf{1}(1)[2])$ via natural
perfect pairings called _global motivic duality_ pairings
$\pi^{i}_{M}:\mathrm{H}_{\mathrm{c}}^{i}(M){\times}\mathrm{H}^{-i}(D(M))_{\mathbb{R}}\rightarrow\mathbb{R}$
that are functorial, compatible with the Verdier dual and normalized to be the
height pairing in certain cases. Except for the perfectness of the pairing,
the properties of arithmetic Chow groups with compact support
$\widehat{\mathrm{CH}}{}^{m}(X)$, $X$ regular and projective over
$\mathbb{Z}$, due to Gillet and Soulé [GS90a] match the requirements on
$\mathrm{H}_{\mathrm{c}}^{2m}(\operatorname{M}(X)(m))$ (Proposition 3.3). The
perfectness for motives supported on $\mathbb{F}_{p}$ is equivalent to the
conjunction of Beilinson’s conjecture 1.5 on the agreement of numerical and
rational equivalence and the Beilinson-Parshin conjecture about vanishing of
higher $K$-theory (both up to torsion, for smooth projective varieties
$X/\mathbb{F}_{p}$), see Theorem 3.4. By a result of Kahn [Kah05, proof of
Theorem 56], the latter conjunction implies the semi-simplicity of geometric
motives over $\mathbb{F}_{p}$.
The second part of the conjecture, 4.2, deals with pole orders and special
$L$-values of geometric motives $M$ over $\mathbb{Z}$. We conjecture that pole
orders are given by the negative Euler characteristic of motivic cohomology of
$D(M)$:
$\operatorname{ord}_{s=0}L(M,s)=-\chi(D(M)).$
The special $L$-value $L^{*}(M,s)$ is conjecturally given, up to a nonzero
rational factor, by the determinants of the pairings $\pi^{i}_{M}$:
$L^{*}(M,s)|_{s=0}\equiv\prod_{i\in\mathbb{Z}}\operatorname{det}(\pi^{i}_{M})^{(-1)^{i+1}}\
\ (\mathrm{mod\ }\mathbb{Q}^{\times}).$
The conjecture is compatible with the functional equation and stable under
distinguished triangles (Theorem 4.4). The latter—a formal consequence of the
setup—is a key difference between our conjecture and Beilinson’s conjecture
for mixed motives over $\mathbb{Q}$. The meaning of our conjecture for special
values of $\zeta$-functions of regular projective schemes over $\mathbb{Z}$ is
spelled out in Example 4.5. The remainder of Section 4 is concerned with the
following result (see Theorem 4.6 for a more detailed statement). To prove
this theorem, actually even to state (1) above, we need to assume a standard
conjectures concerning mixed motives over $\mathbb{Q}$, $\mathbb{Z}$ and
$\mathbb{F}_{p}$, such as a formalism of weights, cohomological dimension,
exactness properties of functors
$i^{*}:\mathbf{DM}_{\mathrm{gm}}(\mathbb{Z})\rightarrow\mathbf{DM}_{\mathrm{gm}}(\mathbb{F}_{p})$
etc. This has been laid down in [Scha], see Section 1.3 for a summary.
###### Theorem 0.1.
The perfectness of the pairings in 3.1 and the $L$-values conjecture 4.2
together are essentially equivalent to the conjunction of the conjectures of
Beilinson, Soulé and Tate on special $L$-values of motives over $\mathbb{Q}$
and $\zeta$-functions à la Hasse-Weil of schemes over $\mathbb{Z}$ and over
$\mathbb{F}_{p}$, respectively.
Once the motivic duality pairing is formulated for any Tate motive over
$\mathbb{Z}$ (in a triangulated manner), we can draw the following corollary.
###### Corollary 0.2.
Conjecture 4.2 holds for the triangulated category $\mathbf{DTM}(\mathbb{Z})$
of Tate motives over $\mathbb{Z}$. In particular, Beilinson’s conjecture holds
for any mixed Tate motive $\operatorname{h}^{j}(X_{\eta},m)$ with $j$,
$m\in\mathbb{Z}$, $X_{\eta}$ smooth projective over $\mathbb{Q}$.
###### Proof: .
By definition [Schb], $\mathbf{DTM}(\mathbb{Z})$ is the triangulated category
generated by $\mathbf{1}(n)$ and
$i_{*}\mathbf{1}(n)\in\mathbf{DM}_{\mathrm{gm}}(\mathbb{Z})$. Here
$i:{\mathrm{Spec}\text{ }}{\mathbb{F}_{p}}\rightarrow{\mathrm{Spec}\text{
}}{\mathbb{Z}}$ is any closed point. For motives
$\mathbf{1}(n)[1]=\eta_{!*}\eta^{*}\mathbf{1}(n)[1]$, our conjecture is
unconditionally equivalent to Beilinson’s conjecture for
$\mathbf{1}(n)\in\mathbf{MM}(\mathbb{Q})$ which does hold by Borel’s work
[Bor77]. The conjecture holds trivially for $i_{*}\mathbf{1}(n)$. Thus the
triangulatedness of Conjecture 4.2 implies the first statement. The second
statement follows immediately. ∎
A key idea for this work, due to Huber, is to view the data occurring in
Beilinson’s conjecture for a mixed motive $M_{\eta}$ over $\mathbb{Q}$ as
belonging to a mixed motive over $\mathbb{Z}$, namely $\eta_{!*}M_{\eta}[1]$.
This is reified for $L$-functions by (1) and on the motivic side by an
appropriate interpretation of $f$-cohomology [Scha]. Various phenomena studied
before then become natural consequences of the properties of
$\eta_{!*}$—chiefly its failure to be exact. Scholl introduces a category
$\mathbf{MM}(\mathbb{Q}/\mathbb{Z})$ of mixed motives over $\mathbb{Z}$
[Sch91] (different than the ones studied here) by imposing non-ramification
conditions and conjectures
$\operatorname{Ext}^{a}_{\mathbf{MM}(\mathbb{Q}/\mathbb{Z})}(\mathbf{1},\operatorname{h}^{b-1}(X_{\eta},m))=\left\\{\begin{array}[]{cc}\mathrm{CH}^{m}(X_{\eta})_{\mathbb{Q}}/\mathrm{hom}&b-2m=1,\,a=0\\\
\mathrm{H}^{b}(X_{\eta},m)_{\mathbb{Z}}&b-2m\neq 0,\,a=1\\\
\mathrm{CH}^{m}(X_{\eta})_{\mathbb{Q},\mathrm{hom}}&b-2m=0,\,a=1\\\
0&\text{else}\end{array}\right.$
Except for $a=2$ and $b-2m<0$, where we cannot prove the vanishing in general,
the right hand side identifies with
$\mathrm{H}^{a-1}(\eta_{!*}\operatorname{h}^{b-1}(X_{\eta},m)[1])$ (Theorem
1.3). As for the special $L$-values, a conjecture of Scholl [Sch91, Conj. C]
says that some $M_{\eta}\in\mathbf{MM}(\mathbb{Q}/\mathbb{Z})$ is critical
(i.e., its period map is an isomorphism, equivalently all
$\mathrm{H}_{\mathrm{w}}^{*}(M_{\eta})$ vanish) if
$\operatorname{Ext}^{a}_{\mathbf{MM}(\mathbb{Q}/\mathbb{Z})}(M_{\eta},\mathbf{1}(1))=\operatorname{Ext}^{a}_{\mathbf{MM}(\mathbb{Q}/\mathbb{Z})}(\mathbf{1},M_{\eta})=0\text{
for }a=0,1.$
Moreover, a reduction technique transforming any motive $M_{\eta}$ into one
satisfying these vanishings is given, so that Deligne’s conjecture [Del79,
Conj. 2.8.] concerning the $L$-value of critical motives can be applied. The
non-multiplicativity of $L$-functions of motives over $\mathbb{Q}$ (cf. Remark
2.2) has been addressed by Fontaine and Perrin-Riou by introducing the notion
of $f$-exact sequence, which are ones where one does save multiplicativity
[FPR94, III.3.1.4]. The formulation of the $L$-values conjecture given here
resembles their approach in several respects. For example, the pole order
conjecture for a mixed motive $M_{\eta}$ over $\mathbb{Q}$ expresses
$\operatorname{ord}_{s=0}L(M_{\eta},s)$ as an Euler characteristic of
$f$-cohomology. The idea to recast special $L$-values of motives as
determinants of appropriate pairings was explored by Deninger and Nart [DN95],
who show that the motivic height pairing of [Sch94b] can be represented by
concatenating morphisms in the derived category of an appropriate category of
motives. Special $L$-values of $\zeta$-functions of varieties over
$\mathbb{Z}$ are independently studied by Soulé [Sou09].
It is a pleasure to thank Annette Huber for her advice for my thesis, of which
this paper is a part. I also thank Denis-Charles Cisinski, Frédéric Déglise
and Bruno Kahn for helpful conversations.
## 1 Preliminaries
### 1.1 Determinants and $\mathbb{Q}$-structures
For any ring $R$, let $\underline{R}$ be the category of finitely generated
$R$-modules. Let $K$ be a field. The _determinant_ $\operatorname{det}V$ of
$V\in\underline{K}$ is $\operatorname{det}V:=\Lambda^{\dim V}V$. Its $K$-dual
is denoted $\operatorname{det}^{-1}V$. For
$V_{*}\in\mathbf{D}^{\mathrm{b}}(\underline{K})$, the derived category, we set
$\operatorname{det}V_{*}:=\bigotimes_{i}\operatorname{det}^{(-1)^{i}}\mathrm{H}^{i}(V_{*}).$
Let $A,B\in\underline{\mathbb{Q}}$ and let $f:A_{\mathbb{R}}\rightarrow
B_{\mathbb{R}}$ be an $\mathbb{R}$-linear map. We do not assume that it
respects the rational subspaces. The “usual” determinant of $f$, which is
well-defined up to a nonzero rational factor agrees, modulo
$\mathbb{Q}^{\times}$ with the image of $1$ under the map
$\mathbb{Q}\cong\operatorname{det}A{\otimes}\operatorname{det}^{-1}B\rightarrow\operatorname{det}A_{\mathbb{R}}{\otimes}\operatorname{det}^{-1}B_{\mathbb{R}}\cong\mathbb{R}$.
Here the right hand isomorphism is induced by $f$.
A complex with _$\mathbb{Q}$ -structure_ is a complex $V_{*}$ of
$\mathbb{R}$-vector spaces that is quasi-isomorphic to one in
$\mathbf{D}^{\mathrm{b}}(\underline{\mathbb{R}})$ together with a non-zero map
of $\mathbb{Q}$-vector spaces
$d_{V_{*}}:\mathbb{Q}\rightarrow\operatorname{det}V_{*}$. In concrete
situations, we usually have a distinguished identification
$\operatorname{det}V_{*}\cong\mathbb{R}$. In that case, we may also call
$\operatorname{det}V_{*}$ the real number that is the image of
$1\in\mathbb{Q}$ under $d_{V_{*}}$ and the given identification.
Maps of complexes with $\mathbb{Q}$-structures are usual maps of complexes;
they are _not_ required to be compatible with the map $d_{V_{*}}$. For a map
$f:V_{*}\rightarrow W_{*}$ of complexes with $\mathbb{Q}$-structures the cone
of $f$ is endowed with the following $\mathbb{Q}$-structure:
$\mathbb{Q}\stackrel{{\scriptstyle
d_{W}{\otimes}(d_{V})^{-1}}}{{{\longrightarrow}}}\operatorname{det}W_{*}{\otimes}\operatorname{det}^{-1}V_{*}\cong\operatorname{det}\operatorname{cone}(f).$
Define a category
$\mathbf{D}^{\mathrm{b}}(\underline{\mathbb{R}})^{\mathbb{Q}-\operatorname{det}}$
to consist of such complexes. Its morphisms are given by maps of complexes up
to quasi-isomorphism (not necessarily respecting the $\mathbb{Q}$-structures).
We say that a triangle $A\rightarrow B\rightarrow C$ of objects in
$\mathbf{D}^{\mathrm{b}}(\underline{\mathbb{R}})^{\mathbb{Q}-\operatorname{det}}$
is _multiplicative_ if it is distinguished in
$\mathbf{D}^{\mathrm{b}}(\underline{\mathbb{R}})$ after forgetting the
$\mathbb{Q}$-structure and
$\operatorname{det}B=\operatorname{det}A\operatorname{det}C$ in the sense that
the following diagram (whose right hand isomorphism stems from the triangle)
is commutative:
$\textstyle{\mathbb{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{C}}$$\scriptstyle{(d_{A})^{-1}{\otimes}d_{B}}$$\textstyle{\operatorname{det}C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\operatorname{det}^{-1}A{\otimes}\operatorname{det}B.}$
Given two complexes $A$ and $B$ in
$\mathbf{D}^{\mathrm{b}}(\underline{\mathbb{R}})^{\mathbb{Q}-\operatorname{det}}$,
together with perfect pairings
$\pi^{i}:\mathrm{H}^{i}(A){\times}\mathrm{H}^{-i}(B)\rightarrow\mathbb{R}$ of
their cohomology groups, we will write (somewhat loosely)
$\operatorname{det}\pi:=\operatorname{det}\\{\pi^{i}\\}_{i}$ for the image of
$1\in\mathbb{Q}$ under the following map:
$\mathbb{Q}\stackrel{{\scriptstyle
d_{A}{\otimes}d_{B}}}{{{\longrightarrow}}}\operatorname{det}A{\otimes}\operatorname{det}B=\bigotimes_{i}\operatorname{det}^{(-1)^{i}}\mathrm{H}^{i}(A){\otimes}\operatorname{det}^{(-1)^{-i}}\mathrm{H}^{-i}(B)\stackrel{{\scriptstyle{\otimes}\operatorname{det}^{(-1)^{i}}\pi^{i}}}{{{\longrightarrow}}}\mathbb{R}.$
### 1.2 Weak Hodge cohomology and Deligne cohomology
In this section, we recall some facts related to Deligne cohomology and weak
Hodge cohomology. Let $X/\mathbb{Q}$ be a smooth projective scheme (of finite
type). By $X(\mathbb{C})$ we denote the complex manifold belonging to $X$,
equipped with its analytic topology. Let $\Omega_{X}^{*}$ be the complex of
sheaves of holomorphic differential forms on $X(\mathbb{C})$. The stupid
filtration $\sigma_{\geq*}$ of the complex $\Omega_{X}^{*}$ is denoted $F^{*}$
and called _Hodge filtration_. As usual, $\mathbb{R}(p)$ denotes $(2\pi
i)^{p}\mathbb{R}\subset\mathbb{C}$. The nontrivial element of
$G=\mathrm{Gal}(\mathbb{R})$ is called infinite Frobenius.
###### Definition 1.1.
Set
$\mathbb{R}_{\mathrm{D}}(p):=[\mathbb{R}(p)\rightarrow\Omega^{0}\rightarrow\Omega^{1}\rightarrow\dots\rightarrow\Omega^{p-1}]$.
The terms are lying in degrees $0$ to $p$. _Deligne cohomology_ of $X$ is
defined by
$\mathrm{H}_{\mathrm{D}}^{n}(X,p):=\mathbb{H}^{n}(X(\mathbb{C}),\mathbb{R}_{\mathrm{D}}(p))^{G}$
where the right hand side denotes sheaf hypercohomology on $X(\mathbb{C})$.
The Galois group acts on the individual sheaves of
$\mathbb{R}_{\mathrm{D}}(p)$—on $\mathbb{R}(p)$ by conjugation on the sheaf
coefficients and on $X(\mathbb{C})$, on the holomorphic forms by conjugation
via
$\mathrm{H}_{\mathrm{dR}}^{*}(X(\mathbb{C}))\stackrel{{\scriptstyle\text{GAGA}}}{{=}}\mathrm{H}^{*}_{\mathrm{dR}}(X_{\mathbb{R}}){\otimes}_{\mathbb{R}}\mathbb{C}$
(see [Sch88, p. 8] for details). Deligne cohomology of projective regular
schemes $Y$ over $\mathbb{Z}$ is defined as the one of the pullback
$Y{\times}\mathbb{Q}$.
By definition, the degeneration of the Hodge-de Rham spectral sequence and
weight reasons, there are short exact sequences (loc. cit.)
$\displaystyle
0\rightarrow\mathrm{H}^{i}(X(\mathbb{C}),\mathbb{R}(m))^{(-1)^{m}}\rightarrow\mathrm{H}^{i}_{\mathrm{dR}}(X_{\mathbb{R}})/F^{m}\rightarrow\mathrm{H}_{\mathrm{D}}^{i+1}(X,m)\rightarrow
0,i-2m\leq-2$ (2) $\displaystyle
0\rightarrow\mathrm{H}_{\mathrm{D}}^{i}(X,m)\rightarrow\mathrm{H}^{i}(X(\mathbb{C}),\mathbb{R}(m))^{(-1)^{m}}\rightarrow\mathrm{H}^{i}_{\mathrm{dR}}(X_{\mathbb{R}})/F^{m}\rightarrow
0,i-2m\geq 0$ (3)
Here the superscript denotes the $(-1)^{m}$-eigenspace of the action of the
nontrivial element of $G$ on Betti cohomology of $X(\mathbb{C})$.
Let $\mathbf{Com}^{\mathrm{b}}_{\mathrm{H}}$ be the category of bounded _Hodge
complexes_ [Beĭ86, 3.2]. Its objects are quintuples
$C:=(C_{\mathrm{dR}},C_{B},C_{c},i_{\mathrm{dR}},i_{B})$ consisting of a
bounded bifiltered complex of $\mathbb{Q}$-vector spaces
$(C_{\mathrm{dR}},W_{*},F^{*})$, a filtered complex of $\mathbb{Q}[G]$-modules
$(C_{B},W_{*})$ and a filtered complex of $\mathbb{C}$-modules with
$\mathbb{C}$-antilinear $G$-action, $(C_{c},W_{*})$, a filtered
$G$-equivariant quasi-isomorphism
$i_{B}:(C_{B},W_{*}){\otimes}_{\mathbb{Q}}\mathbb{C}\rightarrow(C_{c},W_{*})$
($G$ acts on the left hand term by the action on $C_{B}$ and complex
conjugation on $\mathbb{C}$) and finally a filtered $G$-equivariant quasi-
isomorphism
$i_{\mathrm{dR}}:(C_{\mathrm{dR}},W_{*}){\otimes}_{\mathbb{Q}}\mathbb{C}\rightarrow(C_{c},W_{*})$
(on the left, $G$ acts by conjugation on $\mathbb{C}$). These data are subject
to the following requirement: the cohomology quintuple $\mathrm{H}^{i}(C)$
defined by the cohomologies of the various complexes and comparison maps has
to be a mixed Hodge structure [Del71, 2.3.1]. Morphisms in the category
$\mathbf{Com}^{\mathrm{b}}_{\mathrm{H}}$ are required to respect the
filtrations and the comparison quasi-isomorphisms. The category of mixed Hodge
structures will be denoted $\mathbf{MHS}$.
An example of a Hodge complex is $\mathbf{1}(n)$, a one-dimensional space
concentrated in degree $0$, such that it is pure of weight $-2n$ and the Hodge
filtration concentrated in degree $-n$, and the Galois action is given by
multiplication with $(-1)^{n}$.
Recall the notion of _weak Hodge cohomology_ [Beĭ86, 3.13]: set
${\mathrm{R}}{\Gamma}_{\mathrm{w}}:\mathbf{Com}^{\mathrm{b}}_{\mathrm{H}}\rightarrow\mathbf{Com}^{\mathrm{b}}(\mathbb{R}-v.sp.),C\mapsto\operatorname{cone}[-1]\left(C_{B}^{G}{\otimes}\mathbb{R}\oplus
F^{0}C_{\mathrm{dR}}{\otimes}\mathbb{R}\stackrel{{\scriptstyle\alpha_{C}}}{{{\longrightarrow}}}C_{c}^{G}\right).$
The map $\alpha_{C}$ is called _period map_. Let
$\mathbf{D}^{\mathrm{b}}_{\mathrm{H}}$ be the category of Hodge complexes
modulo quasi-isomorphisms. The above functor descends to
${\mathrm{R}}{\Gamma}_{\mathrm{w}}:\mathbf{D}^{\mathrm{b}}_{\mathrm{H}}\rightarrow\mathbf{D}^{\mathrm{b}}(\underline{\mathbb{R}})^{\mathbb{Q}-\operatorname{det}}$.
Indeed, taking $G$-invariants and applying the Hodge filtration are exact
operations, since morphisms of Hodge structures strictly respect the Hodge
filtration [Del71, 2.3.5(iii)]. The $\mathbb{Q}$-structure on
${\mathrm{R}}{\Gamma}_{\mathrm{w}}(C)$ is the one stemming from the very
definition, where $C_{c}^{G}$ is endowed with a $\mathbb{Q}$-structure using
the one on $C_{\mathrm{dR}}$ via $i_{\mathrm{dR}}$. Set
$\mathrm{H}_{\mathrm{w}}^{i}(C):=\mathrm{H}^{i}({\mathrm{R}}{\Gamma}_{\mathrm{w}}(C))$.
There is an exact sequence [Fon92, 5.4]
$0\rightarrow\mathrm{H}_{\mathrm{w}}^{0}(W_{0}V)\rightarrow\mathrm{H}_{\mathrm{w}}^{0}(V)\rightarrow\mathrm{H}_{\mathrm{w}}^{0}(V/W_{0}V)\rightarrow\mathrm{H}_{\mathrm{w}}^{1}(W_{0}V)\rightarrow\mathrm{H}_{\mathrm{w}}^{1}(V)\rightarrow\mathrm{H}_{\mathrm{w}}^{1}(V/W_{0}V)=0.$
Note that
$\Gamma_{\mathbf{MHS}}(V):=\mathrm{Hom}_{\mathbf{MHS}}(\mathbf{1},V)=\mathrm{H}_{\mathrm{w}}^{0}(W_{0}V)$.
It is nonzero only if $\operatorname{gr}_{0}^{W}V\neq 0$. By a spectral
sequence argument we get an exact sequence for any Hodge complex $V_{*}$:
$0\rightarrow\mathrm{H}_{\mathrm{w}}^{1}(\mathrm{H}^{i-1}V_{*})\rightarrow\mathrm{H}_{\mathrm{w}}^{i}(V_{*})\rightarrow\mathrm{H}_{\mathrm{w}}^{0}(\mathrm{H}^{i}V_{*})\rightarrow
0.$ (4)
Unlike absolute Hodge cohomology, the weak variant has a duality: the natural
pairing (induced by $A{\times}A^{\vee}\rightarrow\mathbb{R}$ for any
$\mathbb{R}$-vector space $A$)
$\mathrm{H}_{\mathrm{w}}^{i}(V_{*}){\times}\mathrm{H}_{\mathrm{w}}^{1-i}(V_{*}^{\vee}(1))\rightarrow\mathbb{R}=\mathrm{H}_{\mathrm{w}}^{1}(\mathbf{1}(1)),\
\ i\in\mathbb{Z}.$ (5)
is perfect [FPR94, Prop.III.1.2.3].
Recall [Beĭ86, Section 4] the Hodge complex
$\underline{{\mathrm{R}}{\Gamma}}(X,m)$ whose cohomology objects are the Hodge
structures $\mathrm{H}^{i}(X(\mathbb{C}),\mathbb{R}(m))$ of [Del71].
###### Lemma 1.2.
Weak Hodge cohomology identifies with Deligne cohomology (for all $i,m$):
$\mathrm{H}_{\mathrm{w}}^{i}(\underline{{\mathrm{R}}{\Gamma}}(X,m))=\mathrm{H}_{\mathrm{D}}^{i}(X,m)$.
###### Proof: .
The
$H_{i}:=\mathrm{H}^{i}(\underline{{\mathrm{R}}{\Gamma}}(X,m))=\mathrm{H}^{i}(X(\mathbb{C}),\mathbb{R}(m))$
are pure Hodge structures of weight $i-2m$. For $i-2m<0$,
$\mathrm{H}_{\mathrm{w}}^{0}(H_{i})=\mathrm{H}_{\mathbf{MHS}}^{0}(H_{i})=0$.
By duality,
$\mathrm{H}_{\mathrm{w}}^{1}(H_{i})=\mathrm{H}_{\mathrm{w}}^{0}(H_{i}^{\vee}(1))^{\vee}=0$
for $i-2m>-2$. Hence, by (4),
$\mathrm{H}_{\mathrm{w}}^{i}\underline{{\mathrm{R}}{\Gamma}}(X,m)=\left\\{\begin{array}[]{ll}\mathrm{H}_{\mathrm{w}}^{1}(H_{i-1})&i-2m<0\\\
\mathrm{H}_{\mathrm{w}}^{0}(H_{i})&i-2m\geq 0\end{array}\right.$
The map in the exact sequences (2), (3) between Betti and de Rham cohomology
is the one from the definition of ${\mathrm{R}}{\Gamma}_{\mathrm{w}}$. The
lemma is shown. ∎
The Deligne complexes $\mathbb{R}_{\mathrm{D}}(p)$ enjoy a product structure
$\cup_{\alpha}:\mathbb{R}_{\mathrm{D}}(p){\otimes}\mathbb{R}_{\mathrm{D}}(q)\rightarrow\mathbb{R}_{\mathrm{D}}(p+q)$
where $\alpha\in[0,1]$ is an auxiliary parameter. They induce a product on
$\oplus_{p\geq 0}\mathrm{H}_{\mathrm{D}}^{*}(X,p)$, which is independent of
the choice of $\alpha$ [EV88, Section 3]. Compose it with the pushforward
($d:=\dim X$):
$\mathrm{H}_{\mathrm{D}}^{i}(X,m){\times}\mathrm{H}_{\mathrm{D}}^{2d-i+1}(X,d-m+1)\rightarrow\mathrm{H}_{\mathrm{D}}^{2d+1}(X,d+1)\stackrel{{\scriptstyle\frac{1}{(2\pi
i)^{d}}\int_{X(\mathbb{C})}}}{{{\longrightarrow}}}\mathrm{H}_{\mathrm{D}}^{1}(\mathbb{R},1)=\mathbb{R}.$
(6)
For $a\in F^{0}A_{\mathbb{C}}$, $A:=\mathrm{H}^{i}(X,\mathbb{Q}(m))$, and
$b\in B_{\mathbb{C}}$, $B:=\mathrm{H}^{2d-i}(X,\mathbb{Q}(d-m+1))$, we have
$a\cup_{\alpha=0}b=a\wedge b$. The perfect Poincaré duality pairing
$A_{\mathbb{R}}{\times}B_{\mathbb{R}}(-1)\stackrel{{\scriptstyle\wedge}}{{\rightarrow}}\mathrm{H}^{2d}(X(\mathbb{C}),\mathbb{R}(d))\stackrel{{\scriptstyle\frac{1}{(2\pi
i)^{d}}\int_{X(\mathbb{C})}}}{{{\longrightarrow}}}\mathrm{H}^{0}(\mathbb{C},\mathbb{R}(0))=\mathbb{R}.$
induces an isomorphism $B_{\mathbb{R}}\cong A_{\mathbb{R}}^{\vee}(1)$ and thus
a weak Hodge cohomology duality pairing
$\mathrm{H}_{\mathrm{w}}^{0}A{\times}\mathrm{H}_{\mathrm{w}}^{1}B\rightarrow\mathrm{H}_{\mathrm{w}}^{1}(\mathbf{1}(1))=\mathbb{R}$.
Under the identification of Lemma 1.2, this pairing clearly agrees with (6).
### 1.3 Motives
All of our work takes place in triangulated categories
$\mathbf{DM}_{\mathrm{gm}}(S)$ of _geometric motives_ over $S$, where the base
$S$ is either a number field $F$, a number ring ${\mathcal{O}_{F}}$ or a
finite field. Such a theory is due to Hanamura, Levine, and Voevodsky when the
base is a field and to Ivorra and Cisinski and Déglise for general bases
[Han95, Lev98, Voe00, Ivo07, CD10]. In this paper, we shall work with
axiomatically described categories $\mathbf{DM}_{\mathrm{gm}}(S)$. The precise
axioms have been laid out in [Scha], so here we only survey them briefly. One
half of them is concerned with the behavior of $\mathbf{DM}_{\mathrm{gm}}(S)$
as a triangulated category, based on the work of Cisinski and Déglise. The
second half is better characterized by regarding them as deep conjectures: the
triangulated category $\mathbf{DM}_{\mathrm{gm}}(S)$ (with rational
coefficients) is conjectured to enjoy a $t$-structure whose heart
$\mathbf{MM}(S)$ is called the category of _mixed motives_. The cohomological
dimension of $\mathbf{MM}(S)$ is conjectured to be $0$ ($S={\mathbb{F}_{q}}$)
and $1$ ($S=F$), respectively. The most important requirement on the
$t$-structure is the following: realization functors, which all have the form
$\mathbf{DM}_{\mathrm{gm}}(S)\rightarrow\mathbf{D}^{\mathrm{b}}(\mathcal{C})$
for an appropriate target category $\mathcal{C}$, are to be exact [Scha, Axiom
4.8.]. Any mixed motive is conjectured to have a weight filtration. Morphisms
of mixed motives are to respect weights strictly, thereby giving constraints
on the existence of maps between motives. The pure objects in $\mathbf{MM}(K)$
($K$ any field) are conjectured to be identified with the category
$\mathbf{M}_{\mathrm{num}}$ [Sch94a], pure motives with respect to numerical
equivalence. This implies that the pure objects in $\mathbf{MM}(K)$ form an
abelian semi-simple category [Jan92, Th. 1].
_In the remainder of this paper we assume that the axioms concerning geometric
and mixed motives over open subschemes of ${\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}$, ${\mathbb{F}_{q}}$ and $F$, as spelled out in [Scha,
Sections 1, 2, and 4] hold._
We use the following notation: the _motive_ of a scheme $X/S$ is denoted
$\operatorname{M}(X)\in\mathbf{DM}_{\mathrm{gm}}(S)$, the _motive with compact
support_ is denoted $\operatorname{M}_{\mathrm{c}}(X)$. We exclusively work
with rational coefficients, that is, all $\mathrm{Hom}$-groups are
$\mathbb{Q}$-vector spaces. Moreover, we use a contravariant notation. In
particular, realization functors are covariant and the motive of the
projective line decomposes as
$\operatorname{M}(\mathbb{P}^{1})=\mathbf{1}\oplus\mathbf{1}(-1)[-2]$. We
write
$\mathrm{H}^{i}(M):=\mathrm{Hom}_{\mathbf{DM}_{\mathrm{gm}}(S)}(\mathbf{1},M[i])$
for any $M\in\mathbf{DM}_{\mathrm{gm}}(S)$.
The _Verdier dual_ of any geometric motive $M$ over $\mathbb{Z}$ is defined as
$D(M)=\underline{\mathrm{Hom}}(M,\mathbf{1}(1)[2])$. The truncation with
respect to the motivic $t$-structure is denoted
${{{}^{\mathrm{p}}}\mathrm{H}}^{*}$, we write $\operatorname{h}^{i}(X,n)$ for
${{{}^{\mathrm{p}}}\mathrm{H}}^{i}(\operatorname{M}(X)(n))$. By the
aforementioned exactness requirement, the $\ell$-adic realization of this
motive is
$\mathrm{H}^{i}(X{\times}_{K}{\overline{K}},{\mathbb{Q}_{{\ell}}}(n))$ (for
$X$ defined over a field $K$ of characteristic unequal to $\ell$).
To gain some familiarity we calculate a few examples, but refer to [Scha] for
more details and explanation. Let $f:X\rightarrow\mathbb{Z}$ be some
projective, connected regular scheme of absolute dimension $d$. We do not
assume that $X$ is flat over $\mathbb{Z}$. By the purity axioms we have
$f^{!}\mathbf{1}=f^{*}\mathbf{1}(d-1)[2d-2]$ [Scha, Section 1]. Let
$M:=\operatorname{M}(X)(m)$. By reflexivity of $D$, we get natural
isomorphisms
$\displaystyle\mathrm{H}^{t}(DM)$ $\displaystyle=$
$\displaystyle\mathrm{Hom}_{\mathbf{DM}_{\mathrm{gm}}(\mathbb{Z})}(M,D(\mathbf{1})[t])=\mathrm{Hom}_{\mathbf{DM}_{\mathrm{gm}}(X)}(f^{*}\mathbf{1}(m),f^{!}\mathbf{1}(1)[2+t])$
$\displaystyle=$
$\displaystyle\mathrm{H}^{2d+t}(\mathbf{1},M(X)(d-m))=K_{-t-2m}(X)^{(d-m)}_{\mathbb{Q}}.$
Let $M_{\eta}\in\mathbf{DM}_{\mathrm{gm}}(\mathbb{Q})$ be a motive such that
there is some $M\in\mathbf{MM}(\mathbb{Z})$ that satisfies the following
property: for all primes $p$ in some open $j:U\subset{\mathrm{Spec}\text{
}}{\mathbb{Z}}$, there is a (non-canonical) isomorphism $i_{p}^{!}M\cong
i_{p}^{*}M(-1)[-2]$, where $i_{p}:{\mathrm{Spec}\text{
}}{\mathbb{F}_{p}}\rightarrow{\mathrm{Spec}\text{ }}{\mathbb{Z}}$. We call
$j^{*}M$ a _smooth motive_ , and $M$ _generically smooth_. We define:
$\eta_{!*}M_{\eta}:=j_{!*}j^{*}M:=\operatorname{im}(j_{!}j^{*}M\rightarrow
j_{*}j^{*}M)$. This is explained and shown to be well-defined in [Scha,
Section 5.4]. By [Scha, Lemma 5.11], this applies to
$M_{\eta}=\operatorname{h}^{b-1}(X_{\eta},m)$ and
$M=\operatorname{h}^{b}(X,m)$, where $X_{\eta}/\mathbb{Q}$ is smooth
projective and $X/\mathbb{Z}$ is any projective model of $X_{\eta}$. We also
know that $S:=\eta_{!*}\operatorname{h}^{b-1}(X_{\eta},m)$ is a generically
smooth mixed motive over $\mathbb{Z}$ which is pure of weight $b-2m$. We have
$D(S)=\eta_{!*}\operatorname{h}^{2d-b-1}(X_{\eta},d-m)$, where $d:=\dim X$.
Motivic cohomology of $S$ is given by the following theorem:
###### Theorem 1.3.
With the above notation, we write
$\mathrm{H}^{b}(X_{\eta},n)_{\mathbb{Z}}:=\operatorname{im}(\mathrm{H}^{b}(X,n)\rightarrow\mathrm{H}^{b}(X_{\eta},n))$.
Moreover, let $\mathrm{CH}^{m}(X_{\eta})_{\mathbb{Q},\mathrm{hom}}$ be the
subgroup of the Chow group of cycles homologically equivalent to zero and
$\mathrm{CH}^{m}(X_{\eta})_{\mathbb{Q}}/\mathrm{hom}$ the group of cycles
modulo homological equivalence (tensored with $\mathbb{Q}$). Motivic
cohomology of $S=\eta_{!*}\eta^{*}\operatorname{h}^{b}(X,m)$ is given by
$\mathrm{H}^{a}(S)=\left\\{\begin{array}[]{ll}\mathrm{CH}^{m}(X_{\eta})_{\mathbb{Q}}/\mathrm{hom}&a=-1,\,b-2m=1\\\
0&a=-1,\,b-2m\neq 1\\\
\mathrm{H}^{b}(X_{\eta},m)_{\mathbb{Z}}&a=0,\,b-2m\leq-1\\\
\mathrm{CH}^{m}(X_{\eta})_{\mathbb{Q},\mathrm{hom}}&a=0,\,b-2m=0\\\
0&a=0,\,b-2m\geq 1\\\ 0&a=1,\,b-2m\geq 0\\\ 0&|a|>1\end{array}\right.$
###### Proof: .
The statements for $|a|>1$, $a=-1$, $a=0$ and $b-2m\leq 0$ have been shown in
[Scha, Lemma 5.2, Theorem 6.12]. For $a=0$ and $b-2m\geq 1$, let
$j:U\rightarrow{\mathrm{Spec}\text{ }}{\mathbb{Z}}$ be an open immersion such
that $X{\times}_{\mathbb{Z}}U$ is smooth over $U$. Recall that
$j^{*}S=\operatorname{h}^{b}(X{\times}_{\mathbb{Z}}U)$ is is a smooth motive.
The natural map
$\mathrm{H}^{0}(S)\rightarrow\mathrm{H}^{0}(\eta_{*}\eta^{*}S)=\mathrm{H}^{1}(\operatorname{h}^{b-1}(X_{\eta},m))$
is injective by [Scha, Lemma 6.10]. As the cohomological dimension of
$\mathbf{DM}_{\mathrm{gm}}(\mathbb{Q})$ is one [Scha, Axiom 4.1.] the latter
term is a subgroup of $\mathrm{H}^{b}(X_{\eta},m)$, which vanishes by [Scha,
Axiom 1.8.].
In the case $a=1$ we use that $S$ is, like $j^{*}M$, pure of weight $b-2m$. We
have a localization sequence
$\dots\rightarrow\oplus_{p\text{
prime}}\mathrm{H}^{1}(i_{p}^{!}S)\rightarrow\mathrm{H}^{1}(S)\rightarrow\mathrm{H}^{1}(\eta^{*}S)=\mathrm{H}^{2}(\eta^{*}[-1]S)=0.$
The right hand vanishing is because $\eta^{*}[-1]$ is exact and the
cohomological dimension of motives over $\mathbb{Q}$ is one. Also by
cohomological dimension the left hand term is
$\mathrm{H}^{0}({{{}^{\mathrm{p}}}\mathrm{H}}^{1}i_{p}^{!}S)$. By [Scha, Axiom
4.11.], $i^{!}$ preserves positivity of weights, i.e.,
$\operatorname{wt}({{{}^{\mathrm{p}}}\mathrm{H}}^{1}(i_{p}^{!}S))\geq b-2m+1$.
By strictness of the weight filtration the group therefore vanishes for
$b-2m\geq 0$. ∎
In accordance with Conjecture 3.1 (see Lemma 3.7) I expect
$\mathrm{H}^{1}(S)=0$ for arbitrary $b$, $m$. See the introduction for the
relation of this to Scholl’s notion of mixed motives over $\mathbb{Z}$. By a
similar argument, one can show that for $b-2m\neq-1$ $\mathrm{H}^{1}(S)$ is a
quotient of $\oplus_{p\notin U}\mathrm{H}^{1}(i_{p}^{!}S)$, where $U$ is such
that $X{\times}U$ is smooth over $U$. For mixed Artin-Tate motives there is
the following result. Recall the notions of mixed Tate motives and mixed
Artin-Tate motives over fields, and over number rings, respectively, from
[Lev93, Wil, Scha]. It is worth emphasizing that the motivic $t$-structure on
Artin-Tate motives over ${\mathcal{O}_{F}}$ is unconditional. The following
theorem applies when $\operatorname{h}^{b-1}(X_{\eta},m)$ is an Artin-Tate
motive over $F$, for one can choose a $U\subset{\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}$ such that $X_{\eta}$ has a model projective
$X/{\mathcal{O}_{F}}$ that is smooth when restricted to $U$ and such that
$\operatorname{h}^{b}(X{\times}_{\mathcal{O}_{F}}U,m)$ is a mixed Artin-Tate
motive over $U$.
###### Theorem 1.4.
Let $M$ be a smooth mixed Artin-Tate motive over
$j:U\subset{\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$. Then
$\mathrm{H}^{1}({\mathcal{O}_{F}},j_{!*}M)=0$.
###### Proof: .
We may shrink $U$, since $j^{\prime}_{!*}j^{\prime*}M\cong M$, as $M$ is
smooth. Thus, we may assume by the standard splitting routine [Schb, Lemma
2.5] that there is a Galois cover $f^{\prime}:V^{\prime}\rightarrow U$ such
that $f^{\prime*}M$ is a mixed Tate motive over $V^{\prime}$. The map
$M\rightarrow
f^{\prime}_{*}f^{\prime*}M\stackrel{{\scriptstyle\cong}}{{\leftarrow}}f^{\prime}_{!}f^{\prime!}M\rightarrow
M$ is $\deg f^{\prime}\cdot\mathrm{id}_{M}$, so $M$ is a direct summand of
$f^{\prime}_{*}f^{\prime*}M$, since we use rational coefficients. The functor
$f^{\prime}_{*}=f^{\prime}_{!}$ preserves Artin-Tate motives and is exact
[Schb, Theorem 4.2]. Hence
$j_{!*}f^{\prime}_{*}f^{\prime*}M=f_{*}j^{\prime}_{!*}f^{\prime*}M$. Here
$f:V\rightarrow{\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$ is the
normalization of ${\mathcal{O}_{F}}$ in the function field of $V^{\prime}$ and
$j^{\prime}:V^{\prime}\rightarrow V$ is the corresponding open immersion.
Consequently, $\mathrm{H}^{1}({\mathcal{O}_{F}},j_{!*}M)$ is a summand of
$\mathrm{H}^{1}(V,j^{\prime}_{!*}f^{\prime*}M)$, which vanishes since the
cohomological dimension of mixed Tate motives over $V$ is one [Schb,
Proposition 4.4], as opposed to two for Artin-Tate motives. ∎
We will need the Hodge realization functor into the derived category of Hodge
complexes ([Beĭ86, Section 3], see [Hub00, 2.3.5] for the construction of the
functor)
${\mathrm{R}}{\Gamma}_{\mathrm{H}}:\mathbf{DM}_{\mathrm{gm}}(\mathbb{Q})\rightarrow\mathbf{D}^{\mathrm{b}}_{\mathrm{H}}.$
By [Scha, Axiom 2.1.], ${\mathrm{R}}{\Gamma}_{\mathrm{H}}$ commutes with both
dual and twists, i.e.,
${\mathrm{R}}{\Gamma}_{\mathrm{H}}(M^{\vee}(1))=({\mathrm{R}}{\Gamma}_{\mathrm{H}}(M))^{\vee}(1)$.
We write ${\mathrm{R}}{\Gamma}_{\mathrm{w}}$ for the following composition:
${\mathrm{R}}{\Gamma}_{\mathrm{w}}\circ{\mathrm{R}}{\Gamma}_{\mathrm{H}}:\mathbf{DM}_{\mathrm{gm}}(\mathbb{Q})\rightarrow\mathbf{D}^{\mathrm{b}}(\underline{\mathbb{R}})^{\mathbb{Q}-\operatorname{det}}.$
(7)
The composition of these functors with
$\eta^{*}:\mathbf{DM}_{\mathrm{gm}}(\mathbb{Z})\rightarrow\mathbf{DM}_{\mathrm{gm}}(\mathbb{Q})$
will be denoted by the same.
The following conjecture will be needed to deal with motives over
$\mathbb{F}_{p}$:
###### Conjecture 1.5.
(Beilinson) Let $X/{\mathbb{F}_{q}}$ be smooth and projective. Up to torsion,
numerical and rational equivalence agree on $X$.
Recall that homological equivalence is in between of these two equivalence
relations [And04, 3.2.1], so under the conjecture, all three agree. The second
important consequence of that conjecture is that the category of pure Chow
motives over ${\mathbb{F}_{q}}$ is semisimple by Jannsen’s theorem.
## 2 $L$-functions of motives over number rings
Let $F$ be a number field and ${\mathcal{O}_{F}}$ its ring of integers. For
every finite prime $\mathfrak{p}$ of ${\mathcal{O}_{F}}$ we fix a rational
prime $\ell$ that does not lie under $\mathfrak{p}$. Moreover, fix for every
$\ell$ an embedding
$\sigma_{\ell}:{\mathbb{Q}_{{\ell}}}\rightarrow\mathbb{C}$. All subsequent
definitions of $L$-functions are taken with respect to these choices.
###### Definition 2.1.
The $L$-series of a mixed motive $M_{\eta}$ over $F$ (with respect to the
choices of $\ell$ and $\sigma_{\ell}$) is defined by
$L_{F}(M_{\eta},s):=L(M_{\eta},s):=\prod_{\mathfrak{p}<\infty}\operatorname{det}\left(\mathrm{Id}-\operatorname{Fr}^{-1}\cdot
N(\mathfrak{p})^{-s}|({M_{\eta}}_{\ell}{\otimes}_{{\mathbb{Q}_{{\ell}}},\sigma_{\ell}}\mathbb{C})^{I_{\mathfrak{p}}}\right)^{-1}.$
The $L$-series of a geometric motive $M$ over ${\mathcal{O}_{F}}$ is given by
$L_{{\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}}(M,s):=L(M,s):=\prod_{\mathfrak{p}<\infty}\operatorname{det}\left(\mathrm{Id}-\operatorname{Fr}^{-1}\cdot
N(\mathfrak{p})^{-s}|(i_{\mathfrak{p}}^{*}M)_{\ell}{\otimes}_{{\mathbb{Q}_{{\ell}}},\sigma_{\ell}}\mathbb{C}\right)^{-1}.$
The first definition is classical, the second is a natural adaptation to
motives over ${\mathcal{O}_{F}}$. The products run over all finite primes of
${\mathcal{O}_{F}}$, $\operatorname{Fr}$ is the arithmetic Frobenius map
(given on residue fields by $a\mapsto a^{N(\mathfrak{p})}$), $N(\mathfrak{p})$
is the cardinality of the residue field ${\mathbb{F}_{\mathfrak{p}}}$,
$i_{\mathfrak{p}}$ denotes the immersion of the corresponding closed point and
$-_{\ell}$ denotes the $\ell$-adic realization functor. Note that
${M_{\eta}}_{\ell}$ is an $\ell$-adic sheaf [Scha, Axiom 4.8.]. The
superscript $I_{\mathfrak{p}}$ denotes the invariants under the action of the
inertia group. For the second definition, the determinant is understood in the
sense of Section 1.1.
###### Remark 2.2.
Examples for $L$-functions abound. A prominent example is the one for
$M_{\eta}=\operatorname{h}^{i}(X_{\eta},n)$, for some smooth projective
variety $X_{\eta}$ over $F$.
The independence of the choices of $\ell$ and the embeddings $\sigma_{\ell}$
is discussed around Lemma 2.9. See also Theorem 3.4.
The $L$-series for motives over ${\mathcal{O}_{F}}$ is multiplicative, i.e.,
given a triangle $M\rightarrow M^{\prime}\rightarrow M^{\prime\prime}$ in
$\mathbf{DM}_{\mathrm{gm}}({\mathcal{O}_{F}})$, one gets
$L(M^{\prime},s)=L(M,s)\cdot L(M^{\prime\prime},s).$
A similar property does _not_ hold for $L$-functions of motives over $F$, see
[Sch91] for a counter-example. Scholl’s notion of mixed motives over
$\mathbb{Z}$ (which are not the same thing as mixed motives over $\mathbb{Z}$
in our sense, but mixed motives over $\mathbb{Q}$ with certain additional non-
ramification properties), as well as Fontaine’s and Perrin-Riou’s notion of
$f$-exact sequences [FPR94, 1.3.3] are designed to grapple with this
phenomenon.
By definition and the calculation of $\ell$-adic cohomology of
$\mathbb{P}^{1}_{\mathbb{F}_{\mathfrak{p}}}$ [Mil80, Example VI.5.6 and p.
163], one has $L(M(n),s)=L(M,n+s)$ for all $n\in\mathbb{Z}$.
Taking into account $i^{*}j_{!}=0$ for any complementary closed and open
immersions $i:Z\rightarrow{\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$ and $j$,
respectively [Scha, Axiom 1.10.], the $L$-function of $j_{!}j^{*}M$ is the one
of $M$, where the points in $Z$ are omitted.
The following lemma is well-known, see [Del73, Prop. 3.8.(ii)] or [Neu92,
VII.10.4.(iv)] for similar statements. It permits to replace any number ring
${\mathcal{O}_{F}}$ by $\mathbb{Z}$ and to study $L$-values of motives over
$\mathbb{Z}$, only. The mildly more general situation of motives and
$L$-functions over ${\mathcal{O}_{F}}$ could be treated _mutatis mutandis_ ,
except for the $\mathbb{Q}$-structure on weak Hodge cohomology groups, which
does require the base field to be $\mathbb{Q}$.
###### Lemma 2.3.
The $L$-series is an absolute invariant of a motive, i.e., for any geometric
motive $M$ over ${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$ we have
$L_{{\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}}(M,s)=L_{{\mathrm{Spec}\text{
}}{\mathbb{Z}}}(f_{*}M,s)$, where $f:{\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}\rightarrow{\mathrm{Spec}\text{ }}{\mathbb{Z}}$ denotes
the structural map.
###### Proof: .
By definition, one can reduce to the case where $M$ is supported on a single
prime $\mathfrak{p}$ in ${\mathcal{O}_{F}}$, that is,
$M={i_{\mathfrak{p}}}_{*}M^{\prime}$. Using $(f_{*}M)_{\ell}=f_{*}(M_{\ell})$
[Scha, Axiom 2.1.], one reduces the claim to a question about representations
of the Galois groups of the involved finite fields. Then the claim follows
from a linear-algebraic calculation done e.g. in the proof of [Neu92,
VII.10.4.(iv)]. The details are omitted. ∎
We now relate $L$-functions of motives over $\mathbb{Q}$ to ones over
$\mathbb{Z}$. Recall the notion of generically smooth motives from p. 1.3. The
following lemma is proven in [Scha, Section 5.5]. It is a corollary of direct
application of the exactness axiom on realization functors [Scha, Axiom 4.8.].
###### Lemma 2.4.
Let $M$ be a mixed smooth motive over $U$, where
$j:U\rightarrow{\mathrm{Spec}\text{ }}{\mathbb{Z}}[1/\ell]$ is an open
subscheme. Let $i$ be the complementary closed immersion to $j$ and let
$\eta^{\prime}$ and $\eta$ be the generic point of $U$ and
${\mathrm{Spec}\text{ }}{\mathbb{Z}}[1/\ell]$, respectively. Then
$(i^{*}j_{!*}M)_{\ell}=i^{*}(\mathrm{R}^{0}\eta_{*}\eta^{\prime*}M_{\ell}[-1])[1]$.
###### Proposition 2.5.
Let $M_{\eta}\in\mathbf{MM}(\mathbb{Q})$ be such that there is some
generically smooth $M\in\mathbf{MM}(\mathbb{Z})$ with
$M_{\eta}=\eta^{*}[-1]M$. Then
$L_{\mathbb{Q}}(M_{\eta},s)^{-1}=L_{\mathbb{Z}}(\eta_{!*}\eta^{*}M,s),$
where the left hand term is the $L$-series over $\mathbb{Q}$, the right one is
over $\mathbb{Z}$.
As an example, this applies to $M=\operatorname{h}^{i+1}(X,n)$, where
$X/\mathbb{Z}$ is some projective scheme whose generic fiber
$X_{\eta}/\mathbb{Q}$ is smooth [Scha, Lemma 5.11]. In this case,
$M_{\eta}=\operatorname{h}^{i}(X_{\eta},n)$.
###### Proof: .
Let $j:U\rightarrow{\mathrm{Spec}\text{ }}{\mathbb{Z}}$ be an open non-empty
subscheme such that $j^{*}M$ is smooth. We have
$\displaystyle L_{\mathbb{Z}}(\eta_{!*}\eta^{*}M,s)$ $\displaystyle=$
$\displaystyle L_{\mathbb{Z}}(j_{!*}j^{*}M,s)\ \ \text{(by definition of
}\eta_{!*}\text{)}$ $\displaystyle=$
$\displaystyle\left(\prod_{\mathfrak{p}}\operatorname{det}\left(\mathrm{Id}-\operatorname{Fr}^{-1}N(\mathfrak{p})^{-s}|(i_{\mathfrak{p}}^{*}j_{!*}j^{*}M)_{\ell}\right)\right)^{-1}$
$\displaystyle\stackrel{{\scriptstyle\text{\ref{lemm_excreal}}}}{{=}}$
$\displaystyle\left(\prod_{\mathfrak{p}}\operatorname{det}\left(\mathrm{Id}-\operatorname{Fr}^{-1}N(\mathfrak{p})^{-s}|i_{\mathfrak{p}}^{*}(\mathrm{R}^{0}\eta_{*}\eta^{*}M_{\ell}[-1])[1]\right)\right)^{-1}$
$\displaystyle=$
$\displaystyle\prod_{\mathfrak{p}}\operatorname{det}\left(\mathrm{Id}-\operatorname{Fr}^{-1}N(\mathfrak{p})^{-s}|i_{\mathfrak{p}}^{*}\mathrm{R}^{0}\eta_{*}{M_{\eta}}_{\ell}\right)$
$\displaystyle=$
$\displaystyle\prod_{\mathfrak{p}}\operatorname{det}\left(\mathrm{Id}-\operatorname{Fr}^{-1}N(\mathfrak{p})^{-s}|({M_{\eta}}_{\ell})^{I_{\mathfrak{p}}}\right)$
$\displaystyle=$ $\displaystyle L_{\mathbb{Q}}(M_{\eta},s)^{-1}.$
∎
### 2.1 Hasse-Weil $\zeta$-functions – Motives with compact support
###### Definition 2.6.
(see e.g. [Ser65]) For any quasi-projective scheme $X$ over
${\mathrm{Spec}\text{ }}{\mathbb{Z}}$, the _Hasse-Weil zeta function_ is
defined as $\zeta(X,s):=\prod_{x}(1-N(x)^{-s})^{-1}$. The product is over all
closed points $x$ of $X$, and $N(x)$ denotes the cardinality of the (finite)
residue field of $x$.
###### Proposition 2.7.
For $X$ as above we have $\zeta(X,s)=L(\operatorname{M}_{\mathrm{c}}(X),s)$.
###### Proof: .
We have
$\mathbf{DM}_{\mathrm{gm}}(\mathbb{Z})\ni\operatorname{M}_{\mathrm{c}}(X):=f_{!}f^{*}\mathbf{1}$,
where $f:X\rightarrow{\mathrm{Spec}\text{ }}{\mathbb{Z}}$ is the structural
map. Let $i_{p}:{\mathrm{Spec}\text{
}}{\mathbb{F}_{p}}\rightarrow{\mathrm{Spec}\text{ }}{\mathbb{Z}}$; by base
change, we have
$i_{p}^{*}f_{!}f^{*}\mathbf{1}=f^{\prime}_{!}f^{\prime*}i_{p}^{*}\mathbf{1}=f^{\prime}_{!}f^{\prime*}\mathbf{1}=\operatorname{M}_{\mathrm{c}}(X{\times}\mathbb{F}_{p})$,
where $f^{\prime}:X{\times}\mathbb{F}_{p}\rightarrow\mathbb{F}_{p}$ is the
pullback of $f$. Therefore
$L(\operatorname{M}_{\mathrm{c}}(X),s)=\prod_{p}L({i_{p}}_{*}\operatorname{M}_{\mathrm{c}}(X{\times}\mathbb{F}_{p}),s)$.
We now identify these factors with $\zeta(X{\times}\mathbb{F}_{p},s)$. Let
$X_{p}:=X{\times}_{\mathbb{Z}}\mathbb{F}_{p}$, $n:=\dim X_{p}$. By the trace
formula due to Grothendieck, the $\zeta$-function of $X_{p}$ is given by
$\zeta(X_{p},s)=\frac{P_{1}(p^{-s})\cdot\dots\cdot
P_{2n-1}(p^{-s})}{P_{0}(p^{-s})\cdot\dots\cdot P_{2n}(p^{-s})},$
where
$P_{i}(t)=\operatorname{det}\left(\mathrm{Id}-\operatorname{Fr}^{-1}\cdot
t|\mathrm{H}^{i}_{c}(X_{p}{\times}_{\mathbb{F}_{p}}{\overline{\mathbb{F}_{p}}},{\mathbb{Q}_{{\ell}}})\right)$
and $\mathrm{H}^{i}_{c}$ denotes $\ell$-adic cohomology with compact support
[Del74, 1.5.4]. Thus
$L({i_{p}}_{*}\operatorname{M}_{\mathrm{c}}(X_{p}),s)=\zeta(X_{p},s)$. Clearly
$\zeta(X,s)=\prod_{p}\zeta(X{\times}_{\mathbb{Z}}\mathbb{F}_{p},s)$. ∎
The $L$-series of a motive over $\mathbb{Q}$ is conjectured to be independent
of the choice of $\ell$ and $\sigma_{\ell}$ in every factor (assuming
$p\neq\ell$). This is known for the individual Euler factors at $p$ if the
motive is $\operatorname{h}^{i}(X_{\eta},n)$, where $X_{\eta}$ is a variety
with good reduction at $p$, by Deligne’s work on the Weil conjectures [Del74,
Th. 1.6]. From Proposition 2.7 we now immediately obtain another statement
concerning independence of $\ell$.
###### Definition 2.8.
The smallest triangulated subcategory of
$\mathbf{DM}_{\mathrm{gm}}(\mathbb{Z})$ containing the motives
$\operatorname{M}(X)$ of all schemes $X/\mathbb{Z}$ of finite type is denoted
${\mathbf{DM}^{\mathrm{eff}}_{\underline{\mathrm{gm}}}}(\mathbb{Z})$ and
called the category of _truly geometric_ effective motives. The smallest
triangulated subcategory of $\mathbf{DM}_{\mathrm{gm}}(\mathbb{Z})$ containing
${\mathbf{DM}^{\mathrm{eff}}_{\underline{\mathrm{gm}}}}(\mathbb{Z})$ that is
stable under tensoring with $\mathbf{1}(1)$ is denoted
${\mathbf{DM}_{\underline{\mathrm{gm}}}}(\mathbb{Z})$ and called category of
_truly geometric motives_.
By resolution of singularities [Scha, Axiom 1.15.],
${\mathbf{DM}_{\underline{\mathrm{gm}}}}(\mathbb{Z})$ is also characterized as
the triangulated tensor subcategory generated of
$\mathbf{DM}({\mathcal{O}_{F}})$ generated by
$\operatorname{M}_{\mathrm{c}}(X)$ and $\mathbf{1}(1)$, where $X$ runs through
all schemes over ${\mathcal{O}_{F}}$. The thick closure of
${\mathbf{DM}_{\underline{\mathrm{gm}}}}({\mathcal{O}_{F}})$ is
$\mathbf{DM}_{\mathrm{gm}}({\mathcal{O}_{F}})$. The following lemma thus shows
that the question of independence of $L$-functions of $\ell$ is solely about
the behavior of $L$-functions under direct summands.
###### Lemma 2.9.
For any $M\in{\mathbf{DM}_{\underline{\mathrm{gm}}}}(\mathbb{Z})$ the
$L$-series $L(M,s)$ is well-defined in the sense that it does not depend on
the choices of $\ell$ (provided $\mathfrak{p}\nmid\ell$) and $\sigma_{\ell}$.
###### Proof: .
$L$-series are triangulated. Therefore, in order to show the independence of
$L(M,s)$ for all $M\in{\mathbf{DM}_{\underline{\mathrm{gm}}}}(\mathbb{Z})$ we
may assume that $M$ is a generator of the triangulated category
${\mathbf{DM}_{\underline{\mathrm{gm}}}}(\mathbb{Z})$ which is obtained by
tensor-inverting $\mathbf{1}(-1)$ in
${\mathbf{DM}^{\mathrm{eff}}_{\underline{\mathrm{gm}}}}(\mathbb{Z})$. As
tensoring with $\mathbf{1}(1)$ amounts to a shift of $L$-functions, we
therefore only have to check generators
${\mathbf{DM}^{\mathrm{eff}}_{\underline{\mathrm{gm}}}}(\mathbb{Z})$, for
which we take $\operatorname{M}_{\mathrm{c}}(X)$, $X$ any scheme of finite
type over $\mathbb{Z}$. Then the claim immediately follows from Proposition
2.7. ∎
### 2.2 Archimedean factors and functional equation
Properties of $L$-functions for motives over $\mathbb{Q}$ tend to generalize
to ones over $\mathbb{Z}$, given that the property in question is known for
motives over $\mathbb{F}_{p}$. We illustrate this by the functional equation.
Similar considerations apply to the absolute convergence and analytic
continuation of $L$-series. Recall from [Del79, 5.2.] or [Sch88, p. 4] the
definition of the archimedean Euler factor $L_{\infty}(V,s)$ for a mixed Hodge
structure $V$. We extend it to $V_{*}\in\mathbf{D}^{\mathrm{b}}(\mathbf{MHS})$
by putting
$L_{\infty}(V_{*},s):=\prod_{i\in\mathbb{Z}}L_{\infty}(\mathrm{H}^{i}(V_{*}),s)^{(-1)^{i}}$.
Here $\mathrm{H}^{i}(V_{*})$ denotes the $i$-th cohomology Hodge structure of
the complex $V_{*}$.
###### Definition 2.10.
Let $M$ be a geometric motive over $\mathbb{Z}$ or a mixed motive over
$\mathbb{Q}$. The function
$L_{\infty}(M,s):=L_{\infty}({\mathrm{R}}{\Gamma}_{\mathrm{H}}(M),s)$
is called the _archimedean part_ of the $L$-function of $M$. Here
${\mathrm{R}}{\Gamma}_{\mathrm{H}}$ is the Hodge realization functor. The
_completed $L$-function_ of $M$ is defined as
$\Lambda(M,s):=L(M,s)L_{\infty}(M,s).$
Much as $L$-functions of motives over $\mathbb{Q}$, archimedean factors are
not multiplicative with respect to short exact sequences. (See [FPR94, 1.1.9,
1.2.5] for a necessary and sufficient criterion when multiplicativity does
hold.) Can one give a natural definition of archimedean Euler factors
$\tilde{L}_{\infty}(-,s)$ which is both multiplicative and satisfies, for any
Hodge structure $V$,
$\frac{\tilde{L}_{\infty}(V,s)}{\tilde{L}_{\infty}(V^{\vee}(1),-s)}=\frac{L_{\infty}(V,s)}{L_{\infty}(V^{\vee}(1),-s)}?$
This identity would ensure that the functional equation for $\Lambda(M)$ is
equivalent to one for $\tilde{\Lambda}(M):=L(M)\tilde{L}_{\infty}(M)$.
The following is a long-standing conjecture concerning $L$-functions:
###### Conjecture 2.11.
Let $M_{\eta}$ be a mixed motive over $\mathbb{Q}$. There is a functional
equation relating the $\Lambda$-functions of $M_{\eta}$ and
$M_{\eta}^{\vee}(1)$:
$\Lambda(M_{\eta},s)=\epsilon(M,s)\Lambda(M_{\eta}^{\vee}(1),-s),$
where $\epsilon(M,s)$ is of the form $ab^{s}$, with nonzero constants $a$ and
$b$ depending on $M$; see [Del73], [Del79, 5.2, 5.3] or [FPR94, p. 610, 699].
###### Lemma 2.12.
Conjecture 2.11 implies the following: for any truly geometric motive $M$ over
$\mathbb{Z}$ (Definition 2.8), there is a functional equation
$\Lambda(M,s)=\epsilon(M,s)\Lambda(D(M),-s)$, where $\epsilon(M,s)$ is of the
form $ab^{s}$, with nonzero constants $a$ and $b$ depending on $M$.
###### Proof: .
The claim is triangulated, since the assignments $M\mapsto L(M,s)$, and
$M\mapsto L_{\infty}(M,s)/L_{\infty}(DM,-s)$ are triangulated for
$M\in\mathbf{DM}_{\mathrm{gm}}(\mathbb{Z})$, the latter up to sign [FPR94,
Prop. III.1.2.8]. The category of truly geometric motives over $\mathbb{Z}$ is
contained in the triangulated category $\mathcal{D}$ generated by motives
$i_{*}M$, where $M$ is any geometric motive over $\mathbb{F}_{p}$ and direct
summands of $\eta_{!*}\eta^{*}\operatorname{h}^{i}(X)(n)$, where
$X/\mathbb{Z}$ is regular, flat and projective [Scha, proof of Prop. 5.7]. We
show that the functional equation for motives over $\mathbb{Q}$ implies the
one for motives in $\mathcal{D}$.
By Proposition 2.5 and the calculation of
$D(\eta_{!*}\eta^{*}\operatorname{h}^{i}(X)(n))$ (p. 1.3), the functional
equation for motives $\eta_{!*}\eta^{*}\operatorname{h}^{i}(X)(n)$ is
equivalent to the functional equation for
$\operatorname{h}^{i-1}(X_{\eta},n)$.
We now show the functional equation for motives $i_{*}N$, where $N$ is a
geometric motive over $\mathbb{F}_{p}$.222This must be well-known, but I
failed to find a reference for it. By [Scha, Axiom 2.1.], the $\ell$-adic
realization commutes with duals: $(N^{\vee})_{\ell}=(N_{\ell})^{\vee}$.
Therefore, we have to see
$L(V,s)=ab^{s}L(V^{\vee},-s)$ (8)
with some nonzero numbers $a,b$, for any finite-dimensional continuous complex
representation $V$ of $\mathrm{Gal}(\mathbb{F}_{p})$. Here
$L(?,s):=\operatorname{det}(\mathrm{Id}-\operatorname{Fr}^{-1}p^{-s}|?)$. We
may replace $V$ by $\operatorname{det}V$ without changing either side of the
(8), so we may assume $\dim V=1$. Then $\operatorname{Fr}^{-1}$ acts on $V$
($V^{\vee}$) by multiplication with some $f\in\mathbb{C}^{\times}$ ($1/f$,
respectively). Hence we can take $a:=-f$ and $b:=1/p$ in (8). ∎
###### Remark 2.13.
Under Conjecture 1.5 $a$ above is rational for $M=i_{*}N$. To see this, we may
assume by triangulatedness that $N$ is a pure motive with respect to numerical
or homological equivalence, so that its $L$-function is a rational function in
$p^{-s}$ with rational coefficients (see the reference in the proof of Theorem
4.16).
## 3 Global motivic duality – a conjecture
Recall the category $\mathbf{DM}_{\mathrm{gm}}(\mathbb{Z})$ of geometric
motives over $\mathbb{Z}$, the Verdier dual functor $D(-)$ and weak Hodge
cohomology ${\mathrm{R}}{\Gamma}_{\mathrm{w}}(-)$ from Section 1.3.
###### Conjecture 3.1.
There is a family of functors called _motivic cohomology over $\mathbb{Z}$
with compact support_, denoted
$\mathrm{H}_{\mathrm{c}}^{i}:\mathbf{DM}_{\mathrm{gm}}(\mathbb{Z})\rightarrow\underline{\mathbb{R}}$,
$i\in\mathbb{Z}$, taking values in finite-dimensional $\mathbb{R}$-vector
spaces and satisfying the following properties:
1. 1.
The $\mathrm{H}_{\mathrm{c}}^{i}$ are a $\delta$-functor in the sense that any
distinguished triangle $M_{1}\rightarrow M_{2}\rightarrow M_{3}$ in
$\mathbf{DM}_{\mathrm{gm}}(\mathbb{Z})$ gives rise to long exact sequences
$\dots\rightarrow\mathrm{H}_{\mathrm{c}}^{i}(M_{1})\rightarrow\mathrm{H}_{\mathrm{c}}^{i}(M_{2})\rightarrow\mathrm{H}_{\mathrm{c}}^{i}(M_{3})\rightarrow\mathrm{H}_{\mathrm{c}}^{i+1}(M_{1})\rightarrow\dots$
(9)
2. 2.
For any $M\in\mathbf{DM}_{\mathrm{gm}}(\mathbb{Z})$, there are long exact
sequences
$\dots\rightarrow\mathrm{H}_{\mathrm{c}}^{i}(M)\rightarrow\mathrm{H}^{i}(M)_{\mathbb{R}}\rightarrow\mathrm{H}_{\mathrm{w}}^{i}(M)\rightarrow\mathrm{H}_{\mathrm{c}}^{i+1}(M)\rightarrow\dots$
(10)
where the map from motivic cohomology tensored with $\mathbb{R}$ to weak Hodge
cohomology is the standard realization map.
3. 3.
For any geometric motive $M$ over $\mathbb{Z}$, there is a perfect pairing
$\pi^{i}_{M}:\mathrm{H}_{\mathrm{c}}^{i}(M){\times}\mathrm{H}^{-i}(DM)_{\mathbb{R}}\rightarrow\mathbb{R}$
called _global motivic duality pairing_.
4. 4.
The pairings are functorial in the sense that any morphism $M\rightarrow
M^{\prime}$ of geometric motives over $\mathbb{Z}$ induces a commutative
diagram
$\begin{array}[]{cccccl}\pi^{i}_{M}:&\mathrm{H}_{\mathrm{c}}^{i}(M)&{\times}&\mathrm{H}^{-i}(DM)_{\mathbb{R}}&{\longrightarrow}&\mathbb{R}\\\
&\downarrow&&\uparrow&&\downarrow=\\\
\pi^{i}_{M^{\prime}}:&\mathrm{H}_{\mathrm{c}}^{i}(M^{\prime})&{\times}&\mathrm{H}^{-i}(DM^{\prime})_{\mathbb{R}}&{\longrightarrow}&\mathbb{R}\end{array}$
(11)
5. 5.
The motivic global duality is natural with respect to Verdier duality in the
sense that the following diagram commutes:
$\begin{array}[]{cccccl}\pi^{i}_{M}:&\mathrm{H}_{\mathrm{c}}^{i}(M)&{\times}&\mathrm{H}^{-i}(DM)_{\mathbb{R}}&{\longrightarrow}&\mathbb{R}\\\
&\downarrow&&\uparrow&&\downarrow=\\\
\pi^{-i}_{DM}:&\mathrm{H}^{i}(M)_{\mathbb{R}}&{\times}&\mathrm{H}_{\mathrm{c}}^{-i}(DM)&{\longrightarrow}&\mathbb{R}\\\
&\downarrow&&\uparrow&&\downarrow=\\\
&\mathrm{H}_{\mathrm{w}}^{i}(M)&{\times}&\mathrm{H}_{\mathrm{w}}^{-1-i}(DM)&{\longrightarrow}&\mathbb{R}\end{array}$
(12)
The lower row is the natural perfect pairing on weak Hodge cohomology (5), p.
5, using the natural identification
$\mathrm{H}_{\mathrm{w}}^{-1-i}(DM)=\mathrm{H}_{\mathrm{w}}^{1-i}(M^{\vee}(1))$.
6. 6.
Let $S=\eta_{!*}\eta^{*}\operatorname{h}^{2m}(X,m)$, with $X/\mathbb{Z}$
projective, regular, of absolute dimension $d$, and (smooth) non-empty generic
fiber $X_{\eta}$ (see p. 1.3). The pairing $\pi^{0}(S)$ agrees with the height
pairing
$\mathrm{CH}^{m}(X_{\eta})_{\mathbb{Q},\mathrm{hom}}{\times}\mathrm{CH}^{d-m}(X_{\eta})_{\mathbb{Q},\mathrm{hom}}\rightarrow\mathbb{R},$
(13)
under the identification of $\mathrm{H}^{0}(S)$ and $\mathrm{H}^{0}(D(S))$
with the left and right factor, respectively (Theorem 1.3). The height pairing
was constructed independently by Beilinson [Beĭ87] and Gillet and Soulé
[GS90a]. Another, conjecturally equivalent pairing is given by Bloch [Blo84].
7. 7.
By the first item,
$\mathrm{H}_{\mathrm{c}}^{i}(M)=\mathrm{H}^{i}(M)_{\mathbb{R}}$ for any motive
of the form $M=i_{*}N$, $i:{\mathrm{Spec}\text{
}}{\mathbb{F}_{p}}\rightarrow{\mathrm{Spec}\text{ }}{\mathbb{Z}}$ and any
geometric motive $N$ over $\mathbb{F}_{p}$. Apply (11) to the Verdier dual of
the adjunction map $\mathbf{1}\rightarrow i_{*}i^{*}\mathbf{1}$:
$\begin{array}[]{clclcl}\pi^{0}_{i_{*}i^{*}\mathbf{1}}:&\mathbb{R}&{\times}&\mathbb{R}&{\longrightarrow}&\mathbb{R}\\\
&\downarrow\iota&&\uparrow=&&\downarrow=\\\
\pi^{0}_{\mathbf{1}(1)[2]}:&\mathbb{R}&{\times}&\mathbb{R}&{\longrightarrow}&\mathbb{R}\\\
\end{array}$ (14)
(The identifications of the occurring cohomology groups with $\mathbb{R}$ are
seen by inspecting the long exact sequences (10). For example,
$\mathrm{H}_{\mathrm{c}}^{0}(\mathbf{1}(1)[2])=\mathrm{H}_{\mathrm{w}}^{1}(\mathbf{1}(1))=\mathbb{R}$.
The bottom map is the multiplication. The $\mathbb{Q}$-structures on all terms
$\mathbb{R}$ are the standard ones.)
We suppose that the map $\iota$ is given by multiplication with
$\mathrm{log}\,p$.
This conjecture determines special $L$-values of
$\operatorname{h}^{b}(X_{\eta},m)$ for all $b$, $m$ such that $b-2m\geq 0$, up
to a nonzero rational factor. The functional equation (Conjecture 2.11) covers
the remaining cases.
###### Remark 3.2.
Let us point out a number of structural similarities of the above ideas with
the situation of étale constructible sheaves over ${\mathrm{Spec}\text{
}}{\mathbb{Z}}$: thinking of $M\in\mathbf{DM}_{\mathrm{gm}}(\mathbb{Z})$ as
being analogous to a complex of constructible sheaves $\mathcal{F}$ over
$\mathbb{Z}$, ${\mathrm{R}}{\Gamma}_{\mathrm{w}}(M)$ corresponds to
${\mathrm{R}}{\Gamma}(\mathbb{R},\mathcal{F}|_{\mathbb{R}})$, Tate cohomology
at the archimedean place. Given that, $\mathrm{H}_{\mathrm{c}}^{i}(M)$
parallels
$\mathrm{H}_{\mathrm{c}}^{i}(\mathcal{F}):=\mathrm{H}^{i}{\mathrm{R}}{\Gamma}_{\mathrm{c}}(\mathbb{Z},\mathcal{F})$,
that is to say, cohomology with compact support, which is defined via
${\mathrm{R}}{\Gamma}_{\mathrm{c}}:=\operatorname{cone}[-1]\left({\mathrm{R}}{\Gamma}(\mathbb{Z},\mathcal{F})\rightarrow{\mathrm{R}}{\Gamma}(\mathbb{R},\mathcal{F}|_{\mathbb{R}})\right)$,
much the same way as (10). Finally, the motivic duality pairing corresponds to
the perfect pairing known as _arithmetic global duality_ [Mil06, Ch. II.3]
$\mathrm{H}_{\mathrm{c}}^{i}(\mathbb{Z},\mathcal{F}){\times}\operatorname{Ext}^{3-i}_{\mathbb{Z}}(\mathcal{F},\mathbb{G}_{m})\rightarrow\mathrm{H}_{\mathrm{c}}^{3}(\mathbb{Z},\mathbb{G}_{m}).$
One should also compare the duality conjecture given here with [Mil06,
Conjecture II.7.17] of Milne. Can one find a complex of étale sheaves over
$\mathbb{Z}$ representing motivic cohomology, so that the regulator map to
Deligne cohomlogy becomes the restriction map from étale cohomology over
$\mathbb{Z}$ to Tate cohomology over $\mathbb{R}$?
We now want to weigh the depth of the several parts of the above conjecture.
To do so, we first very briefly recall the notion of _arithmetic Chow groups_
$\widehat{\mathrm{CH}}{}^{\sharp,[}p](X)$ due to Gillet and Soulé [GS90a,
3.3.4]. Let $X/\mathbb{Z}$ be flat, projective and regular of (absolute)
dimension $d$. Let $\mathfrak{D}^{2p}(X,p)$ be the space of
$(p,p)$-$C^{\infty}$-differential forms on $X(\mathbb{C})$ that are $(2\pi
i)^{p}$ times a real form. Consider the group $\mathcal{Z}^{p}(X)$ of pairs
$(Z,g_{Z})$, where $Z$ is a codimension $p$ cycle on $X$ and $g_{Z}$ is a
Green current for $Z$, that is, a current such that
$\omega_{Z}:=-2\partial{\overline{\partial}}g_{Z}+\delta_{Z}$ is contained in
$\mathfrak{D}^{2p}(X,p)$. Here $\delta_{Z}$ is the Dirac current given by
$\mathfrak{D}^{2(d-1-p)}(X,d-1-p)\ni\eta\mapsto 1/(2\pi
i)^{d-1-p}\int_{Z(\mathbb{C})}\eta$. For any $(p-1)$-codimensional variety
$Y\subset X$, and any rational function $f$ on $Y$, the pair
$(\mathrm{div}f,\mathrm{log}\,|f|)$ is an element of $\mathcal{Z}^{p}(X)$ (see
loc. cit.). The quotient
$\mathcal{Z}^{p}(X)/((\mathrm{div}f,\mathrm{log}\,|f|))+(0,\operatorname{im}\partial)+(0,\operatorname{im}{\overline{\partial}})$
is $\widehat{\mathrm{CH}}{}^{\sharp,[}p](X)$. Note that we altered the
definition of loc. cit. slightly by adding a factor $(2\pi i)^{p}/2$ in order
to match the groups with Deligne cohomology and the Beilinson regulator, cf.
[GS90a, Section 3.5]. If $X$ is regular and projective, but not flat over
$\mathbb{Z}$, we define $\widehat{\mathrm{CH}}{}^{\sharp,[}p](X)$ to be the
usual Chow group for all connected components of $X$ that are defined over
some $\mathbb{F}_{p}$. There is a map
$\widehat{\mathrm{CH}}{}^{\sharp,[}p](X)\rightarrow\mathfrak{D}^{2p}(X,p)$
determined by $[(Z,g_{Z})]\mapsto\omega_{Z}$. We denote its kernel by
$\widehat{\mathrm{CH}}{}^{p}(X)$ and call it _arithmetic Chow group with
compact support_.333This group is denoted $\widehat{\mathrm{CH}}{}^{p}(X)_{0}$
in loc. cit.
Let $M=\operatorname{M}(X)(m)$ where $X$ is regular and projective over
$\mathbb{Z}$. Motives of that type generate
$\mathbf{DM}_{\mathrm{gm}}(\mathbb{Z})$ as a thick category [Scha, Axiom
1.15.]. We define
$\mathrm{H}_{\mathrm{c}}^{2m}(M):=\widehat{\mathrm{CH}}{}^{m}(X)_{\mathbb{R}}$
and
$\pi^{2m}_{M}:\widehat{\mathrm{CH}}{}^{m}(X)_{\mathbb{R}}{\otimes}\mathrm{CH}^{d-m}(X)_{\mathbb{R}}\rightarrow\widehat{\mathrm{CH}}{}^{d}(X)_{\mathbb{R}}\rightarrow\widehat{\mathrm{CH}}{}^{1}(\mathbb{Z})=\mathbb{R}$
(15)
to be the pairing induced by the product on arithmetic Chow groups followed by
the pushforward map. The height pairing on
$\mathrm{CH}^{*}(X_{\eta})_{\mathbb{Q},\mathrm{hom}}$ is induced by
$\pi^{2m}(M)$ [GS90a, Theorem 4.3.2, Remark 4.3.8.(iii)].
###### Proposition 3.3.
With the above definition of $\mathrm{H}_{\mathrm{c}}^{2m}(M)$ and
$\pi^{2m}_{M}$, the statements of Conjecture 3.1 except for the perfectness of
the pairing $\pi^{2m}_{M}$ are valid in this special case.
###### Proof: .
First of all, $\mathrm{H}_{\mathrm{c}}^{2m}(M)$ as defined above fits into (a
part of) the exact sequence (10) [GS90a, Theorems 3.3.5, 3.5.4], so 3.1.2 is
satisfied. The normalization 3.1.7 is also satisfied since the map
$i_{*}:\mathbb{Z}=\mathrm{CH}^{0}(\mathbb{F}_{p})\rightarrow\widehat{\mathrm{CH}}{}^{1}(\mathbb{Z})=\mathbb{R}$
is multiplication with $\mathrm{log}\,(p)$ with respect to the natural
$\mathbb{Q}$-structure on $\mathbb{R}$, as follows from the definition. We
will relate this to special $L$-values of motives over $\mathbb{F}_{p}$ in
Section 4.4.
Let $Y$ be another regular projective scheme over $\mathbb{Z}$ and let
$f:X\rightarrow Y$ be a projective map such that $f_{\mathbb{C}}$ is smooth.
These conditions (are put in order to) ensure that the pushforward $f_{*}$ for
arithmetic Chow groups is defined. For the induced map
$\operatorname{M}(Y)(m)\rightarrow\operatorname{M}(X)(m)$ the functoriality of
the duality pairing (11) reads as the commutativity of
$\begin{array}[]{ccccccl}\widehat{\mathrm{CH}}{}^{m}(Y)_{\mathbb{R}}&{\times}&\mathrm{CH}^{\dim
Y-m}(Y)_{\mathbb{R}}&\rightarrow&\widehat{\mathrm{CH}}{}^{\dim
Y}(Y)_{\mathbb{R}}&\stackrel{{\scriptstyle{p_{Y}}_{*}}}{{{\longrightarrow}}}&\widehat{\mathrm{CH}}{}^{1}(\mathbb{Z})_{\mathbb{R}}\\\
\downarrow f^{*}&&\uparrow f_{*}&&\uparrow f_{*}&&\uparrow=\\\
\widehat{\mathrm{CH}}{}^{m}(X)_{\mathbb{R}}&{\times}&\mathrm{CH}^{\dim
X-m}(X)_{\mathbb{R}}&\rightarrow&\widehat{\mathrm{CH}}{}^{\dim
X}(X)_{\mathbb{R}}&\stackrel{{\scriptstyle{p_{X}}_{*}}}{{{\longrightarrow}}}&\widehat{\mathrm{CH}}{}^{1}(\mathbb{Z})_{\mathbb{R}}\end{array}$
(16)
Indeed, this diagram is commutative because of the projection formula [GS90a,
Theorem 4.3.9]. Hence 3.1.4 is satisfied.
The compatibility requirement (12) of the motivic duality pairing with Verdier
duals reads as the commutativity of the following diagram, where $d:=\dim X$
$\begin{array}[]{ccccccccl}\pi^{2m}_{M}:&\widehat{\mathrm{CH}}{}^{m}(X)_{\mathbb{R}}&{\times}&\mathrm{CH}^{d-m}(X)_{\mathbb{R}}&\rightarrow&\widehat{\mathrm{CH}}{}^{d}(X)_{\mathbb{R}}&\stackrel{{\scriptstyle{p_{X}}_{*}}}{{\rightarrow}}&\mathbb{R}\\\
&\downarrow&&\uparrow&&\downarrow=&&\downarrow=\\\
\pi^{-2m}_{DM}:&\mathrm{CH}^{m}(X)_{\mathbb{R}}&{\times}&\widehat{\mathrm{CH}}{}^{d-m}(X)_{\mathbb{R}}&\rightarrow&\widehat{\mathrm{CH}}{}^{d}(X)_{\mathbb{R}}&\stackrel{{\scriptstyle{p_{X}}_{*}}}{{\rightarrow}}&\mathbb{R}\\\
&\downarrow&&\uparrow&&\uparrow&&\uparrow=\\\
&\mathrm{H}_{\mathrm{D}}^{2m}(X,m)&{\times}&\mathrm{H}_{\mathrm{D}}^{2d-2m-1}(X,d-m)&\rightarrow&\mathrm{H}_{\mathrm{D}}^{2d-1}(X,d)&\stackrel{{\scriptstyle{p_{X}}_{*}}}{{\rightarrow}}&\mathbb{R}\\\
\end{array}$
Indeed the first two lines are commutative since the product on
$\widehat{\mathrm{CH}}{}^{\sharp,[}*](X)_{\mathbb{R}}$ is commutative [GS90a,
4.2.3]. In the last line, the pushforward is the one of (6). This product
agrees with the one on weak Hodge cohomology, see p. 1.2. The lower two lines
are commutative by the very definition of the product on arithmetic Chow
groups. Therefore, 3.1.5 is satisfied. ∎
To move on, it is worth pointing out the notion of higher arithmetic Chow
groups for varieties over fields due to Feliu [Fel]. However, the theory of
algebraic cycles over $\mathbb{Z}$ needs to be developed further to allow for
an extension to the situation of varieties over $\mathbb{Z}$. Another natural
idea is to exhibit an explicit representation of the regulator
$K^{\prime}_{2m-i}(X)^{(m)}\rightarrow\mathrm{H}^{D}_{i}(X,\mathbb{R}(m))$
into Deligne homology [Jan88] akin to the Burgos-Wang result [BW98], defining
arithmetic $K^{\prime}$-theory as the fiber of that map, and defining motivic
cohomology with compact support for $M=\operatorname{M}(X)(m)$ in terms of
that. I hope to pursue this point in a later work. Encouragingly, arithmetic
$K$-theory (Gillet-Soulé and Takeda [GS90b, Tak05]) fits the requirements of
Conjecture 3.1, except for the compatibility of the product pairing with
pushforward. The latter is exactly captured by the arithmetic Riemann-Roch
theorem [GRS08].
We now study the perfectness of the duality pairings. First of all, note the
following similar conjecture of Gillet and Soulé [GS94, Conjecture 1]: the
intersection product
$\widehat{\mathrm{CH}}{}^{\sharp,[}m](X)_{\mathbb{R}}{\times}\widehat{\mathrm{CH}}{}^{\sharp,[}\dim
X-m](X)_{\mathbb{R}}\rightarrow\mathbb{R}$
is non-degenerate for any regular scheme $X$ that is projective and flat over
$\mathbb{Z}$.
For motives over $\mathbb{F}_{p}$, we have the following compact
characterization. It was previously known that Tate’s conjecture about the
pole order of $\zeta$-functions over finite fields and Conjecture 1.5 together
imply the Beilinson-Parshin conjecture [Gei98, Thm. 1.2.].
###### Theorem 3.4.
Let $N$ stand for any geometric motive over $\mathbb{F}_{p}$, let
$i:{\mathrm{Spec}\text{ }}{\mathbb{F}_{p}}\rightarrow{\mathrm{Spec}\text{
}}{\mathbb{Z}}$. Under the axioms concerning the existence and cohomological
dimension of mixed motives over $\mathbb{F}_{p}$ and the weight formalism (see
Section 1.3 and [Scha, Sections 1, 2, and 4]), the perfectness of the motivic
duality pairing for all motives $i_{*}N$ is equivalent to Conjecture 1.5.
Assuming only the axioms about the triangulated categories of motives, the
perfectness is equivalent to the conjunction of Conjecture 1.5 and the
_Beilinson-Parshin conjecture_ stating
$K_{r}(X)_{\mathbb{Q}}=0$ (17)
for any smooth projective variety $X$ over ${\mathbb{F}_{q}}$ and all $r>0$.
In particular, it
###### Proof: .
Using the axioms about mixed motives, we first show that Conjecture 1.5
implies the perfectness. By 3.1.2,
$\mathrm{H}_{\mathrm{c}}^{*}(i_{*}N)=\mathrm{H}^{*}(N)_{\mathbb{R}}$. By
[Scha, Axiom 4.1.], the cohomological dimension of
$\mathbf{DM}_{\mathrm{gm}}(\mathbb{F}_{p})$ is zero, so that
$\mathrm{H}^{j}(N)=\mathrm{H}^{0}({{{}^{\mathrm{p}}}\mathrm{H}}^{j}N)$ and
similarly for $D(N)$. By the same axiom, only finitely many $j$ yield a non-
zero term. Therefore, we may replace $N$ by
${{{}^{\mathrm{p}}}\mathrm{H}}^{j}N$ and assume that $N$ is a mixed motive.
Using the weight filtration we reduce to the case where $N$ is a pure motive.
Under Conjecture 1.5, all adequate equivalence relations agree, so we may
regard $N$ as a Chow motive or as a pure motive with respect to numerical
equivalence. By the semi-simplicity of pure numerical motives there is a
decomposition $N=\mathbf{1}^{r}\oplus R$, where $R$ is a Chow motive such that
$\mathrm{H}^{0}_{\mathbf{DM}_{\mathrm{gm}}(\mathbb{F}_{p})}(R^{\vee})=\mathrm{H}^{0}_{\mathbf{DM}_{\mathrm{gm}}(\mathbb{F}_{p})}(R)=0$.
By functoriality of the pairing we get a commutative diagram
$\begin{array}[]{rccccl}\pi^{0}_{i_{*}N}:&\mathrm{H}^{0}(N)_{\mathbb{R}}&{\times}&\mathrm{H}^{0}(DN)_{\mathbb{R}}&{\longrightarrow}&\mathbb{R}\\\
&\downarrow\cong&&\uparrow\cong&&\downarrow=\\\
\pi^{0}_{i_{*}\mathbf{1}^{r}}:&\mathrm{H}^{0}(\mathbf{1}^{r})_{\mathbb{R}}&{\times}&\mathrm{H}^{0}(\mathbf{1}^{r})_{\mathbb{R}}&{\longrightarrow}&\mathbb{R}\end{array}$
The lower line is a perfect pairing, since the one for $i_{*}\mathbf{1}$ is by
3.1.7.
We now show the second statement. Let $X$ be a smooth equidimensional
projective variety over ${\mathbb{F}_{q}}$. Let $f:{\mathrm{Spec}\text{
}}{{\mathbb{F}_{q}}}\rightarrow{\mathrm{Spec}\text{ }}{\mathbb{F}_{p}}$ be the
canonical map. Set $M:=i_{*}f_{*}\operatorname{M}(X)(m)$. By 3.1.2, we have
$\mathrm{H}_{\mathrm{c}}^{i}(M)=\mathrm{H}^{i}(M)_{\mathbb{R}}$, so Conjecture
3.1 is concerned with the pairing
$\pi^{i}_{M}:K_{2m-i}(X)^{(m)}_{\mathbb{R}}{\times}K_{i-2m}(X)^{(\dim
X-m)}_{\mathbb{R}}\rightarrow\mathbb{R}.$
For $2m-i>0$ the second factors vanishes, hence the perfectness is equivalent
to (17). For $2m=i$ is perfectness is equivalent, by definition, to the
agreement of numerical and rational equivalence (up to torsion). This shows
one implication of the second statement. By resolution of singularities [Scha,
Axiom 1.15.], the category $\mathbf{DM}_{\mathrm{gm}}(\mathbb{F}_{p})$ is
generated as a thick category by motives $\operatorname{M}(X)(m)$ as above.
Since the perfectness only has to be checked on such generators, we are done
with the converse implication as well. ∎
The following corollary was pointed out to me by Kahn.
###### Corollary 3.5.
The perfectness for all motives $i_{*}N$ implies a canonical equivalence
$\mathbf{DM}_{\mathrm{gm}}(\mathbb{F}_{p})=\mathbf{D}^{\mathrm{b}}(\mathbf{M}_{\mathrm{rat}}(\mathbb{F}_{p}))$,
which in turn implies amongst other things the independence of $L$-functions
of $\ell$.
###### Proof: .
That description of $\mathbf{DM}_{\mathrm{gm}}(\mathbb{F}_{p})$ is shown to be
a consequence of $\sim_{\mathrm{num}}=\sim_{\mathrm{rat}}$ and the Beilinson-
Parshin conjecture in the course of the proof of [Kah05, Theorem 56]. ∎
###### Lemma 3.6.
Let $M$ be a geometric motive over $\mathbb{Z}$. Under Parts 2 and 3 of
Conjecture 3.1, $\mathrm{H}^{i}(M)$ is nonzero only for finitely many
$i\in\mathbb{Z}$.
This is a consequence of the spectral sequence
$\mathrm{H}^{a}({{{}^{\mathrm{p}}}\mathrm{H}}^{b}(M))\Rightarrow\mathrm{H}^{a+b}(M)$,
the boundedness of the motivic $t$-structure and of the cohomological
dimension [Scha, Axiom 4.1.]. It also follows from the perfectness of the
motivic pairing and the axioms concerning geometric motives (but not the ones
about mixed motives):
###### Proof: .
The full subcategory of $\mathbf{DM}_{\mathrm{gm}}(\mathbb{Z})$ of motives
satisfying the claim is thick. Since $\mathbf{DM}_{\mathrm{gm}}(\mathbb{Z})$
is generated as a thick category by motives $M=\operatorname{M}(X)(m)$, where
$X$ is any regular scheme that is equidimensional of dimension $d$ and
projective over $\mathbb{Z}$ and $m\in\mathbb{Z}$ [Scha, Axiom 1.15.], it
suffices to check the claim for these motives. Now,
$\mathrm{H}^{i}(M)=K_{2m-i}(X)^{(m)}_{\mathbb{Q}}$ vanishes for $i\gg 0$. On
the other hand,
$\mathrm{H}^{i}(M)_{\mathbb{R}}\cong\mathrm{H}_{\mathrm{c}}^{-i}(DM)^{\vee}$
via $\pi^{-i}_{DM}$. The outer terms of
$\mathrm{H}^{-i-1}(DM)_{\mathbb{R}}\rightarrow\mathrm{H}_{\mathrm{w}}^{-i-1}(DM)\rightarrow\mathrm{H}_{\mathrm{c}}^{-i}(DM)\rightarrow\mathrm{H}^{-i}(DM)_{\mathbb{R}}$
vanish for $i\ll 0$ and
$\mathrm{H}_{\mathrm{w}}^{-i}(DM)=\mathrm{H}_{\mathrm{D}}^{2d-i}(X,d-m)$
vanishes for almost all $i$. ∎
###### Lemma 3.7.
Let $X/\mathbb{Z}$ be projective such that the generic fiber $X_{\eta}$ is
smooth and non-empty. Let $b,m\in\mathbb{Z}$ be arbitrary. Under 3.1.2, the
perfectness of the pairing $\pi^{-1}_{S}$ for
$S:=\eta_{!*}\eta^{*}\operatorname{h}^{b}(X,m)$ is equivalent to
$\mathrm{H}^{1}(DS)=0$.
In the cases $b-2m\leq 0$ we know the vanishing of $\mathrm{H}^{1}(DS)$
independently of the perfectness of the duality pairing (see Theorem 1.3).
Beilinson’s conjecture 4.9 deals with the cases $b-2m\geq 0$, $b-2m=0$ being
the $L$-value prediction at the central point.
###### Proof: .
Using (10) and $\mathrm{H}_{\mathrm{w}}^{-2}(S)=0$, we see that
$\mathrm{H}_{\mathrm{c}}^{-1}(S)$ is the kernel of the realization map
$\rho^{-1}_{S}:\mathrm{H}^{-1}(S)_{\mathbb{R}}\rightarrow\mathrm{H}_{\mathrm{w}}^{-1}(S)$
which is injective (cf. Theorem 1.3). ∎
## 4 Special $L$-values of motives over $\mathbb{Z}$
### 4.1 A conjecture
Throughout this section, let $M$ be any geometric motive over $\mathbb{Z}$. In
the remainder of this chapter, wherever ranks of motivic cohomology groups are
involved, we assume the following conjecture.
###### Conjecture 4.1.
(Bass) $K$-groups of regular schemes $X$ of finite type over $\mathbb{Z}$ are
finitely generated.
We need the following consequence (by resolution of singularities): motivic
cohomology of all geometric motives over $\mathbb{Z}$ is finitely generated.
By [Scha, Axiom 4.1.] (see also Lemma 3.6) only finitely many
$\mathrm{H}^{i}(M)$, and—under 3.1.2—only finitely many
$\mathrm{H}_{\mathrm{c}}^{i}(M)$ are nonzero as $i\in\mathbb{Z}$ varies. Thus,
the Euler characteristic $\chi(M):=\sum_{i}(-1)^{i}\dim\mathrm{H}^{i}(M)$ is
defined. Throughout, terms of the form $\mathrm{H}^{i}(M)_{\mathbb{R}}$ are
endowed with the obvious $\mathbb{Q}$-structure. Using the
$\mathbb{Q}$-structure on ${\mathrm{R}}{\Gamma}_{\mathrm{w}}(M)$ and the
sequence (10),
${\otimes}\operatorname{det}^{(-1)^{i}}\mathrm{H}_{\mathrm{c}}^{i}(M)$ is
therefore endowed with a $\mathbb{Q}$-structure (see Section 1.1).
###### Conjecture 4.2.
The order of the $L$-function of $M$ (Definition 2.1) is given by
$\operatorname{ord}_{s=0}L(M,s)=-\chi(DM).$
As usual, negative orders mean a pole, positive ones a zero of the
$L$-function. Moreover, assuming 3.1.3, the perfectness of the motivic duality
pairings $\pi^{i}_{M}$ for all $i\in\mathbb{Z}$, the special $L$-value is
given by
$L^{*}(M,0)\equiv 1/\operatorname{det}\pi_{M}\ \ (\mathrm{mod\
}\mathbb{Q}^{\times}).$
Under 3.1.3, the pole order conjecture is equivalent to
$\operatorname{ord}_{s=0}L(M,s)=-\sum_{i\in\mathbb{Z}}(-1)^{i}\dim\mathrm{H}_{\mathrm{c}}^{i}(M)$.
We are now going to expound some structural properties of the conjecture. In
order to state the compatibility with the functional equation, we shall need
the following conjecture of Deligne’s. It implies the compatibility of the
$L$-values conjecture for critical pure motives $M_{\eta}$ over $\mathbb{Q}$
(i.e., motives such that $\mathrm{H}_{\mathrm{w}}^{i}(M_{\eta})=0$, $i=0,1$)
with the functional equation [Del79, Theorem 5.6].
###### Conjecture 4.3.
[Del79, Conjecture 6.6] Let $M$ be a pure motive over $\mathbb{Q}$ with
respect to homological equivalence, i.e., a direct summand in
$\mathbf{M}_{\mathrm{hom}}(\mathbb{Q})$ of $h(X_{\eta},m)$ where
$X_{\eta}/\mathbb{Q}$ is smooth projective. Assume that $M$ is of rank one,
that is to say, its Betti realization (or, equivalently, de Rham or
$\ell$-adic realization) is one-dimensional. Then $M$ is of the form
$M(\epsilon)(n)$, where $n$ is an integer and
$\epsilon:\mathrm{Gal}(\mathbb{Q})\rightarrow\mathbb{Q}^{\times}$ is a finite
character and $M(\epsilon)$ denotes the Dirichlet motive to the one-
dimensional representation, $\epsilon$, of $\mathrm{Gal}(\mathbb{Q})$ (loc.
cit.).
###### Theorem 4.4.
1. 1.
Assuming Parts 1 and 4 of Conjecture 3.1, Conjecture 4.2 is triangulated:
given a distinguished triangle $M_{1}\rightarrow M_{2}\rightarrow M_{3}$ in
$\mathbf{DM}_{\mathrm{gm}}(\mathbb{Z})$, the conjecture predicts
$L^{*}(M_{1},0)L^{*}(M_{3},0)=L^{*}(M_{2},0)$
and additively with the pole orders. In particular, the subcategory of
$\mathbf{DM}_{\mathrm{gm}}(\mathbb{Z})$ of motives for which the conjecture
holds is triangulated.
2. 2.
Assume Conjecture 4.3 and Conjecture 1.5
($\sim_{\mathrm{rat}}=\sim_{\mathrm{num}}$) and the functional equation for
completed $L$-functions over $\mathbb{Q}$ (Conjecture 2.11). Let $M$ be a
geometric motive contained in the triangulated subcategory of
$\mathbf{DM}_{\mathrm{gm}}(\mathbb{Z})$ generated by the image of
$i_{*}:\mathbf{DM}_{\mathrm{gm}}(\mathbb{F}_{p})\rightarrow\mathbf{DM}_{\mathrm{gm}}(\mathbb{Z})$
for all primes $p$ and direct summands of motives
$S:=\eta_{!*}\eta^{*}\operatorname{h}^{b}(X,m)$ with $X/\mathbb{Z}$ regular
and projective. Then 3.1, parts 1 and 5, imply that Conjecture 4.2 for $M$ is
equivalent to the one for $DM$.
Note that by [Scha, Prop. 5.7] and its proof, the motives mentioned in the
last claim generate $\mathbf{DM}_{\mathrm{gm}}(\mathbb{Z})$ as a thick
category.
###### Proof: .
Let $M_{1}\rightarrow M_{2}\rightarrow M_{3}$ be a distinguished triangle in
$\mathbf{DM}_{\mathrm{gm}}(\mathbb{Z})$. The pole order additivity is clear.
We consider the long exact sequences made of
$\mathrm{H}_{\mathrm{c}}^{*}(M_{i})$ and $\mathrm{H}^{-*}(DM_{i})$ and get a
commutative diagram of long exact sequences coupled together by perfect
pairings. The $\mathbb{Q}$-structure on motivic cohomology with compact
support is triangulated, i.e.,
$\mathbf{DM}_{\mathrm{gm}}(\mathbb{Z})\ni
M\mapsto\bigotimes_{i}\operatorname{det}^{(-1)^{i}}\mathrm{H}_{\mathrm{c}}^{i}(M)\in\mathbf{D}^{\mathrm{b}}(\underline{\mathbb{R}})^{\mathbb{Q}-\operatorname{det}}$
is multiplicative in the sense of Section 1.1 since its constituent parts are:
on motivic cohomology the $\mathbb{Q}$-structure is trivially multiplicative
and on weak absolute Hodge cohomology it is by construction of the realization
functor. This settles the first statement.
For the second part, we write $\operatorname{ord}$ for
$\operatorname{ord}_{s=0}$,
$\chi_{c}(M):=\sum_{i}(-1)^{i}\dim\mathrm{H}_{\mathrm{c}}^{i}(M)$ and
similarly $\chi_{w}(M)$ with $\mathrm{H}_{\mathrm{w}}^{i}(M)$ instead.
Moreover, ${\mathrm{R}}{\Gamma}_{\mathrm{H}}$ denotes the Hodge realization
functor (Section 1.3). We use the following calculation of the pole orders of
archimedean factors ([Beĭ86, Lemma 7.1.] or [FPR94, III.1.2.5 + III.1.2.3]),
where $G$ is any geometric motive over $\mathbb{Q}$ or $\mathbb{Z}$:
$\operatorname{ord}_{s=0}L_{\infty}(G,s)=-\sum_{i}(-1)^{i}\dim\mathrm{H}_{\mathrm{w}}^{1}(\mathrm{H}^{i}(G^{\vee}(1))).$
Thus, Conjecture 4.2 for $M$, $\operatorname{ord}L(M,s)=-\chi(DM)$, is
equivalent to
$\displaystyle\operatorname{ord}\Lambda(M)$ $\displaystyle=$
$\displaystyle-\chi(DM)-\sum_{i}(-1)^{i}\dim\mathrm{H}_{\mathrm{w}}^{1}(\mathrm{H}^{i}({\mathrm{R}}{\Gamma}_{\mathrm{H}}(D(M)))$
$\displaystyle\stackrel{{\scriptstyle\text{\ref{conj_motcomp}.\ref{item_perfect}}}}{{=}}$
$\displaystyle-\chi_{c}(M)-\sum_{i}(-1)^{i}\dim\mathrm{H}_{\mathrm{w}}^{1}(\mathrm{H}^{i}({\mathrm{R}}{\Gamma}_{\mathrm{H}}(D(M)))$
$\displaystyle=$
$\displaystyle-\chi(M)+\chi_{w}(M)-\sum_{i}(-1)^{i}\dim\mathrm{H}_{\mathrm{w}}^{1}(\mathrm{H}^{2-i}({\mathrm{R}}{\Gamma}_{\mathrm{H}}(M)^{\vee}(1)))$
$\displaystyle\stackrel{{\scriptstyle\text{(\ref{eqn_dualityweak}), p.\
\ref{eqn_dualityweak}}}}{{=}}$
$\displaystyle-\chi(M)-\sum_{i}(-1)^{i}\dim\mathrm{H}_{\mathrm{w}}^{1}(\mathrm{H}^{i}({\mathrm{R}}{\Gamma}_{\mathrm{H}}(M))$
By Lemma 2.12, the functional equation for mixed motives over $\mathbb{Q}$
implies the one for motives over $\mathbb{Z}$, so that
$\operatorname{ord}\Lambda(DM)=\operatorname{ord}\Lambda(M)$. Again invoking
the pole order calculation of $L_{\infty}$-functions we get
$\operatorname{ord}L(D(M))=-\chi(M)$, that is, the conjectural prediction of
the pole order of $L(D(M))$. This settles the compatibility of the pole order
prediction with the functional equation.
As for the special $L$-values, the compatibility is also more or less directly
built into the formulation of Conjecture 3.1. Let us write $L^{*}(-)$ for
$L^{*}(-,0)$ in the sequel. The claim is triangulated. For motives $M=i_{*}N$,
where $i:{\mathrm{Spec}\text{
}}{\mathbb{F}_{p}}\rightarrow{\mathrm{Spec}\text{ }}{\mathbb{Z}}$ and $N$ is
any geometric motive over $\mathbb{F}_{p}$ the functional equation reads
$L(i_{*}N,s)=ab^{s}L(D(i_{*}N),-s)$, with $a$ and $b$ in $\mathbb{Q}^{\times}$
(Remark 2.13). This uses the agreement of numerical and homological
equivalence, implied by Conjecture 1.5. On the other hand
$\operatorname{det}\pi_{i_{*}N}\equiv\operatorname{det}\pi_{D(i_{*}N)}\text{
mod }\mathbb{Q}^{\times}$ by 3.1.5.
To check the claim for direct summands of motives of the form
$S:=\eta_{!*}\eta^{*}\operatorname{h}^{b}(X,m)$ with $X/\mathbb{Z}$ projective
and regular, we may assume $X$ is of equidimension $d$. We can also assume
$b-2m\geq 0$, since $DS=\eta_{!*}\eta^{*}\operatorname{h}^{2d-b}(X,d-m)$, so
either for $S$ or $DS$ this weight condition is satisfied. Let
$M_{\eta}:=\eta^{*}M[-1]$. It is a direct summand of
$\operatorname{h}^{b-1}(X_{\eta},m)$, where $X_{\eta}$ is the generic fiber of
$X$. For $b-2m=0$, the hard Lefschetz axiom ([Scha, Axiom 4.4.], see also
Section 4.3.2 below) implies an isomorphism $S\cong DS$, so that there is
nothing to show in that case. Let now $b-2m>0$. Deligne’s conjecture 4.3
implies (see loc. cit.)
$\frac{L^{*}(S)}{L^{*}(DS)}\stackrel{{\scriptstyle\ref{satz_corrj}}}{{=}}\frac{L^{*}(M_{\eta}^{\vee}(1))}{L^{*}(M_{\eta})}=\frac{a_{1}}{a_{2}}$
where $a_{1}$ denotes the image of $1\in\mathbb{Q}$ under the determinant map
of
$\mathrm{H}_{\mathrm{w}}^{0}(DS)=\mathrm{H}_{\mathrm{w}}^{1}(M_{\eta}^{\vee}(1))$
and $a_{2}$ denotes the image of $1$ in
$\operatorname{det}\mathrm{H}_{\mathrm{w}}^{0}(DS)$ induced by the
$\mathbb{Q}$-structure on
$\mathrm{H}_{\mathrm{w}}^{-1}(S)=\mathrm{H}_{\mathrm{w}}^{0}(M_{\eta})$ under
the natural weak Hodge duality isomorphism
$\mathrm{H}_{\mathrm{w}}^{0}(M_{\eta})^{\vee}\rightarrow\mathrm{H}_{\mathrm{w}}^{1}(M_{\eta}^{\vee}(1))$.
In other words,
$a_{1}/a_{2}=\operatorname{det}\mathrm{H}_{\mathrm{w}}^{-1}(S){\otimes}\operatorname{det}\mathrm{H}_{\mathrm{w}}^{0}(DS)$.
All other $\mathrm{H}_{\mathrm{w}}^{*}$-groups of $S$ and $DS$ vanish (see the
proof of Lemma 1.2). We abuse the notation by identifying
$\operatorname{det}{\mathrm{R}}{\Gamma}_{\mathrm{w}}(S)$ with
$\operatorname{det}\mathrm{H}_{\mathrm{w}}^{-1}(S)$ (including the
$\mathbb{Q}$-structure) and dually for $DS$. From the calculus of determinants
and (12) we get
$\bigotimes_{i}\operatorname{det}^{(-1)^{i}}\pi^{i}_{S}=\operatorname{det}^{-1}\mathrm{H}_{\mathrm{w}}^{-1}(S){\otimes}\operatorname{det}^{-1}\mathrm{H}_{\mathrm{w}}^{0}(DS){\otimes}\bigotimes_{i}\operatorname{det}^{(-1)^{i}}\pi^{i}_{DS}.$
By Conjecture 4.2 for $DS$, the right hand side gives $a_{2}/(a_{1}\cdot
L^{*}(DS))=1/L^{*}(S)$, which is Conjecture 4.2 for $S$ and conversely. ∎
###### Example 4.5.
We now study the implications of Conjectures 3.1 and 4.2 for
$M=\operatorname{M}(X)(m)$, $X/\mathbb{Z}$ regular, projective and of
equidimension $d$, including a special values conjecture for $\zeta(X,s)$. In
the conjecturally perfect pairing
$\pi^{i}_{M}:\mathrm{H}_{\mathrm{c}}^{i}(M){\times}\mathrm{H}^{-i}(D(M))_{\mathbb{R}}\rightarrow\mathbb{R}$
the second factor is isomorphic to $K_{i-2m}(X)^{(d-m)}_{\mathbb{R}}$ by
absolute purity, and so vanishes for $i<2m$. Hence so does the first factor,
so that the realization map
$\rho^{i}_{M}:K_{2m-i}(X)^{(m)}_{\mathbb{R}}=\mathrm{H}^{i}(M)_{\mathbb{R}}\rightarrow\mathrm{H}_{\mathrm{w}}^{i}(M)=\mathrm{H}_{\mathrm{D}}^{i}(X,m)$
is an isomorphism for $i+1<2m$ and injective for $i+1=2m$. (In particular, the
non-torsion part of higher $K$-theory of $X$ is finitely generated—a weakening
of Conjecture 4.1.) In line with this, Proposition 3.3, and the sequence (10)
we set
$\mathrm{H}_{\mathrm{c}}^{i}(M):=\left\\{\begin{array}[]{ll}0&i\leq 2m-1\\\
\widehat{\mathrm{CH}}{}^{m}(X)_{\mathbb{R}}&i=2m\\\
\operatorname{coker}({\rho^{2m}_{M}}:\mathrm{CH}^{m}(X)_{\mathbb{R}}\rightarrow\mathrm{H}_{\mathrm{D}}^{2m}(X,m))&i=2m+1\\\
\mathrm{H}_{\mathrm{w}}^{i-1}(M)=\mathrm{H}_{\mathrm{D}}^{i-1}(X,m)&i>2m+1\end{array}\right.$
The pairing $\pi^{2m}_{M}$ is the intersection pairing
$\widehat{\mathrm{CH}}{}^{m}(X)_{\mathbb{R}}{\times}\mathrm{CH}^{d-m}(X)_{\mathbb{R}}\rightarrow\mathbb{R}$.
The pairing $\pi^{i}_{M}$ for $i>2m+1$ is—in accordance with (12)—given by the
Beilinson regulator followed by multiplication in Deligne cohomology and
pushforward to $\mathrm{H}_{\mathrm{D}}^{1}(\mathbb{Z},1)=\mathbb{R}$:
$\pi^{i}_{M}:\mathrm{H}_{\mathrm{D}}^{i-1}(X,m){\times}K_{i-2m}(X)^{(d-m)}_{\mathbb{R}}\rightarrow\mathrm{H}_{\mathrm{D}}^{i-1}(X,m){\times}\mathrm{H}_{\mathrm{D}}^{2d-i}(X,d-m)\rightarrow\mathbb{R}.$
Finally,
$\pi^{2m+1}_{M}:\operatorname{coker}\rho^{2m}_{M}{\times}K_{1}(X)^{(d-m)}_{\mathbb{R}}\rightarrow\mathbb{R}$
is induced by the pairing induced by the multiplication and pushforward on
Deligne cohomology via the surjection
$\mathrm{H}_{\mathrm{D}}^{2m}(X,m)\rightarrow\operatorname{coker}\rho^{2m}_{M}$
and the injection
$K_{1}(X)^{(d-m)}_{\mathbb{R}}\rightarrow\mathrm{H}_{\mathrm{D}}^{2d-1}(X,d-m)$,
respectively. The pairings $\pi^{i}_{M}$, $i\geq 2m$ and the isomorphisms
$\rho^{i}_{M}$, $i<2m-1$, determine an isomorphism
$\bigotimes_{i}\operatorname{det}^{(-1)^{i}}\mathrm{H}_{\mathrm{c}}^{i}(M){\otimes}\operatorname{det}^{(-1)^{i}}\mathrm{H}^{-i}(DM)_{\mathbb{R}}\stackrel{{\scriptstyle\cong}}{{\rightarrow}}\mathbb{R}.$
By Proposition 2.7 we have $L(M,s)=\zeta(X,s+m)$. We conjecture that—modulo
$\mathbb{Q}^{\times}$—$L^{*}(M,0)$ is the reciprocal of the image of $1$ in
$\mathbb{R}$ via the $\mathbb{Q}$-structure map of the left hand term. The
class number formula has been interpreted in the terms above in [Sou92,
III.4.3].
The group
$\mathrm{H}_{\mathrm{c}}^{2m+1}(M)=\operatorname{coker}K_{0}(X)^{(m)}_{\mathbb{R}}\rightarrow\mathrm{H}_{\mathrm{D}}^{2m}(X,m)$
is related to the _Hodge conjecture_ : from Section 1.2 we have
$\mathrm{H}_{\mathrm{D}}^{2m}(X,m)=\mathrm{H}^{2m}(X(\mathbb{C}),\mathbb{R}(m))^{G}\cap\mathrm{H}^{m,m}(X(\mathbb{C}))$.
The cycle class map $\rho^{2m}_{M}$ factors onto
$K_{0}(X_{\eta})_{\mathbb{R}}\cong\oplus_{m}\mathrm{CH}^{m}(X_{\eta})_{\mathbb{R}}$,
where $X_{\eta}$ is the (smooth projective) generic fiber of $X$. The Hodge
conjecture for $X(\mathbb{C})$ or the Tate conjecture for $X_{\mathbb{Q}}$
imply that the image of $\mathrm{CH}^{m}(X_{\eta}){\otimes}\mathbb{Q}$ is the
group $\Gamma_{\mathrm{AH}}\mathrm{H}^{2m}(X,m)$ of absolute Hodge cycles in
$\mathrm{H}^{2m}(X,m)$, see [Jan90, 2.10, 5.4, 5.5] for the definition of
absolute Hodge cycles and this statement. It is well-known [And04, 7.2.1.3,
7.3.1.3] that the Hodge conjecture over $\mathbb{C}$ or the Tate conjecture
over $\mathbb{Q}$ imply all standard conjectures, in particular the agreement
of homological and numerical equivalence on any smooth projective variety
$X_{\eta}$ over $\mathbb{Q}$. Here is a short duality-minded proof of that
implication (we put $d_{\eta}=\dim X_{\eta}$):
$\textstyle{\mathrm{CH}^{m}(X_{\eta})_{\mathbb{Q}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\textstyle{\mathrm{CH}^{d_{\eta}-m}(X_{\eta})_{\mathbb{Q}}^{\vee}}$$\textstyle{\Gamma_{\mathrm{AH}}\mathrm{H}^{2m}(X,m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\Gamma_{\mathrm{AH}}\mathrm{H}^{2(d_{\eta}-m)}(X,d_{\eta}-m)^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
Under the Hodge or Tate conjecture the vertical cycle class maps are
surjective and injective, respectively. Since the comparison maps between
$\ell$-adic, Betti, and de Rham cohomology are compatible with products, the
Poincaré duality for these individual cohomology theories also gives the lower
row isomorphism for the absolute Hodge cycles. Any cycle in
$\mathrm{CH}^{m}(X_{\eta})$ is numerically trivial iff its image under $\cup$
is zero. By the diagram this happens iff its image in $\mathrm{H}^{2m}(X,m)$
is zero, i.e., iff the cycle is homologically trivial.
Does the perfectness of the motivic duality pairings imply the agreement of
homological and numerical equivalence on smooth projective varieties
$X_{\eta}/\mathbb{Q}$?
The following theorem sums up the results of the following sections.
References in parentheses refer to precise statements and/or proofs.
###### Theorem 4.6.
The pole order part of Conjecture 4.2 for truly geometric motives over
$\mathbb{Z}$ (Definition 2.8) is equivalent to Soulé’s conjecture concerned
with the pole order of $\zeta$-functions of quasi-projective varieties
$Y/\mathbb{Z}$ (Theorem 4.8).
Assuming standard assumptions on mixed motives over finite fields, number
fields and number rings (see Section 1.3 or [Scha, Sections 1, 2, and 4]), the
following statements hold true:
Beilinson’s pole order prediction (Conjecture 4.9) for $L$-functions of
motives $M_{\eta}:=\operatorname{h}^{b-1}(X_{\eta},m)$ with $X_{\eta}$ smooth
and projective over $\mathbb{Q}$ is equivalent to
$\operatorname{ord}_{s=0}L(S)=-\sum_{a\in\mathbb{Z},a\neq
1}(-1)^{a}\dim\mathrm{H}^{a}(D(S)).$ (18)
where $S:=\eta_{!*}M_{\eta}[1]$. Therefore any two of the following statements
imply the third: (1) $\mathrm{H}^{1}(DS)=0$, (2) Beilinson’s pole order
prediction, (3) the pole order prediction of 4.2 (Corollary 4.14).
The vanishing of $\mathrm{H}^{1}(DS)$ is equivalent to the pairing
$\pi^{-1}_{S}$ being perfect (Lemma 3.7). It holds for $b-2m\leq 0$ (Theorem
1.3) or when $M_{\eta}$ is an Artin-Tate motive over $\mathbb{Q}$ (Theorem
1.4).
Under the assumption $\mathrm{H}^{1}(DS)=0$, the perfectness of the global
motivic duality pairings $\pi^{i}_{S}$, for $S$ as above with arbitrary $b$
and $m$, is equivalent to conjectures of Beilinson concerning the
isomorphicity of certain realization maps (Lemma 4.12) and the special
$L$-values prediction of Beilinson for $M_{\eta}$ is equivalent to the one of
4.2 for $M=S$ (Theorem 4.13).
The perfectness of the pairings $\pi^{*}_{M}$ for $M=i_{*}N$, with
$i:{\mathrm{Spec}\text{ }}{\mathbb{F}_{p}}\rightarrow{\mathrm{Spec}\text{
}}{\mathbb{Z}}$ and $N\in\mathbf{DM}_{\mathrm{gm}}(\mathbb{F}_{p})$ is
equivalent to Conjecture 1.5 (Theorem 3.4). Under 1.5, the Tate conjecture
over $\mathbb{F}_{p}$ is equivalent to 4.2 for such motives (Theorem 4.16).
Up to direct summands, the conjunction of the conjectures of Soulé, Beilinson
($L$-values and $\sim_{\mathrm{num}}=\sim_{\mathrm{rat}}$), and Tate is
equivalent to the perfectness of the motivic duality pairings $\pi^{*}_{M}$
for all geometric motives $M$ over $\mathbb{Z}$ and the pole order and special
$L$-values prediction of 4.2.
###### Proof: .
The interpretation of the pole order part of Beilinson’s conjecture as (18) is
done in the first part of Theorem 4.13. It mostly stakes on 1.3.
The last statement is a summary of the preceding ones: the subcategory of
$\mathbf{DM}_{\mathrm{gm}}(\mathbb{Z})$ of motives $M$ for which the pairings
$\pi^{*}_{M}$ are perfect is thick, i.e., triangulated (by the five lemma and
3.1.1), stable under direct summands (3.1.4). It is also stable under the
Verdier dual functor $D$ (use (12), (5) and the five lemma). By [Scha, Prop.
5.7], $\mathbf{DM}_{\mathrm{gm}}(\mathbb{Z})$ is generated as a thick category
by motives $S$, where $b$ and $m$ are any integers, and motives
$i_{*}\operatorname{M}(X_{p})(m)$, with $X_{p}/\mathbb{F}_{p}$ some smooth
projective variety. The perfectness for the latter type of motives is
equivalent to Conjecture 1.5. By the calculation of $D(S)$ (p. 1.3) we
therefore are left with the perfectness for $M=S$ with $b-2m\geq 0$. Soulé’s
conjecture 4.7 is equivalent to $\operatorname{ord}_{s=0}L(M,s)=-\chi(D(M))$
for all truly geometric motives $M$. The category of geometric motives over
$\mathbb{Z}$ is generated as a thick category by such motives. “Up to” direct
summands, i.e., assuming that this pole order formula continues to hold for
direct summands, Soulé’s conjecture and Beilinson’s pole order conjecture 4.9
therefore imply $\mathrm{H}^{1}(DS)=0$, under which the isomorphy statements
in Beilinson’s conjecture are equivalent to the perfectness of the duality
pairings $\pi^{*}_{S}$ with $b-2m\geq 0$. This shows that 1.5, 4.7 and the
isomorphy parts of 4.9 together imply, up to summands, to the perfectness of
$\pi^{*}_{M}$ for all geometric motive $M$ over $\mathbb{Z}$. Given that
perfectness, the following holds: Beilinson’s conjecture for $M_{\eta}$ is
equivalent to 4.2 for $M=S$ (Theorem 4.13) and Tate’s conjecture 4.15 is
equivalent to the pole order formula for $M=i_{*}N$. The category
$\mathbf{DM}_{\mathrm{gm}}(\mathbb{Z})$ is generated as a thick category by
such motives. This shows the implication $\Rightarrow$ of the last statement.
The converse {3.1, 4.2} $\Rightarrow$ {4.7, 4.9, 4.15} is also clear by the
above. ∎
### 4.2 Relation to a conjecture of Soulé
This short section compares a pole-order type conjecture of Soulé with the
pole order part of Conjecture 4.2.
###### Conjecture 4.7.
(Soulé, [Sou84, Conjecture 2.2.]) Let $Y/\mathbb{Z}$ be quasiprojective. Let
$m\in\mathbb{Z}$ be arbitrary. Then
$\operatorname{ord}_{s=m}\zeta(Y,s)=\sum_{i\geq
0}(-1)^{i+1}\operatorname{rk}K^{\prime}_{i}(Y)_{(m)}$
We refer to loc. cit. for the definition of $K^{\prime}_{i}(Y)_{(m)}$. For $Y$
regular, it agrees with $K_{i}(Y)^{(\dim Y-m)}$.
The right hand side above makes sense under Conjecture 4.1 and the vanishing
of almost all $K^{\prime}$-groups, which in turn is a consequence of [Scha,
Axiom 4.1.]. See also Lemma 3.6.
###### Theorem 4.8.
Conjecture 4.7 for $Y$ and $m$ is equivalent to the pole order prediction of
Conjecture 4.2 for $M=\operatorname{M}_{\mathrm{c}}(Y)(m)$. Therefore, Soulé’s
conjecture is equivalent to 4.2 restricted to the category
${\mathbf{DM}_{\underline{\mathrm{gm}}}}(\mathbb{Z})$ of truly geometric
motives over $\mathbb{Z}$ (Definition 2.8).
###### Proof: .
We know $\zeta(Y,s)=L(\operatorname{M}_{\mathrm{c}}(Y),s)$ (Proposition 2.7).
$K^{\prime}$-theory has a localization sequence which preserves the Adams
grading (see [Sou84, 1.3.]). Likewise, motives with compact support have a
localization sequence. Therefore the statement for $Y$ is implied by the
conjunction of the one for some open subscheme $U$ of $Y$ and $Z:=Y\backslash
U$. In particular we may assume that $Y$ is integral.
There is an open subscheme $U$ of $Y$ that is either smooth over $\mathbb{Z}$
or over $\mathbb{F}_{p}$. If $Y/\mathbb{Z}$ is flat, one can take an open
neighborhood of a smooth point of the generic fiber of $Y$, otherwise $Y$ lies
over some ${\mathrm{Spec}\text{ }}{\mathbb{F}_{p}}$ and one can take a
neighborhood of a smooth point of $Y$ [Gro67, Prop. 17.15.12.]. By a
Noetherian induction, we may replace $Y$ by $U$. Let
$f:Y\rightarrow\mathbb{Z}$ be the projection. By definition
$\operatorname{M}_{\mathrm{c}}(Y)=f_{!}f^{*}\mathbf{1}$ and by the regularity
and projectivity of $Y$ we get
$D(\operatorname{M}_{\mathrm{c}}(Y,m))=f_{*}f^{!}\mathbf{1}(1-m)[2]=f_{*}f^{*}\mathbf{1}(d-m)[2d]$
[Scha, Axiom 1.11.]. As $Y$ is regular, we get
$K^{\prime}_{i}(Y)_{(m),\mathbb{Q}}\cong
K_{i}(Y)^{(d-m)}_{\mathbb{Q}}\cong\mathrm{H}^{2(d-m)-i}(Y,d-m)$. Moreover,
$\displaystyle\chi(D(\operatorname{M}_{\mathrm{c}}(Y)(m))$ $\displaystyle=$
$\displaystyle\chi(\operatorname{M}(Y)(d-m)[2d])$ $\displaystyle=$
$\displaystyle\sum_{i\in\mathbb{Z}}(-1)^{i}\dim\mathrm{H}^{i+2d}(Y,d-m).$
The second statement follows from Theorem 4.4 since
${\mathbf{DM}_{\underline{\mathrm{gm}}}}(\mathbb{Z})$ is generated as a
triangulated category by motives $\operatorname{M}_{\mathrm{c}}(Y)(m)$ as
above (resolution of singularities, [Scha, Axiom 1.15.]). ∎
### 4.3 Relation to Beilinson’s conjecture
Recall from Section 1.3 the axioms we assume, in particular the ones
concerning mixed motives over $\mathbb{Z}$. In this section, we compare
Beilinson’s $L$-values conjecture with a particular case of Conjectures 3.1
and 4.2. Throughout, $X/\mathbb{Z}$ is any projective equidimensional scheme
such that the generic fiber $X_{\eta}/\mathbb{Q}$ is smooth and non-empty. Let
$d$ and $d_{\eta}$ be the absolute dimensions of $X$ and $X_{\eta}$,
respectively. Let $M=\operatorname{h}^{b}(X,m)$ and
$M_{\eta}:=\eta^{*}M[-1]=\operatorname{h}^{b-1}(X_{\eta},m)$; the latter is a
pure motive of weight $\operatorname{wt}(M_{\eta})=b-1-2m$. We set
$S:=\eta_{!*}\eta^{*}M$ (see Section 1.3). It is a generically smooth mixed
motive over $\mathbb{Z}$ of pure weight $b-2m$. Its Verdier dual is
$D(S)=\eta_{!*}\eta^{*}\operatorname{h}^{2d-b}(X,d-m)$. Let $n:=b-m$. Recall
the groups $\mathrm{CH}^{m}(X_{\eta})_{\mathbb{Q},\mathrm{hom}}$,
$\mathrm{CH}^{m}(X_{\eta})_{\mathbb{Q}}/\mathrm{hom}$ and
$\mathrm{H}^{*}(X_{\eta},n)_{\mathbb{Z}}$ from Section 1.3.
#### 4.3.1 Beilinson’s conjecture
###### Conjecture 4.9.
(Beilinson [Beĭ84, Beĭ86]) Using the above notation
$\operatorname{ord}_{s=0}L(M_{\eta},s)=\left\\{\begin{array}[]{ll}0&b-2m\leq-2\\\
-\dim\mathrm{CH}^{m}(X_{\eta})_{\mathbb{Q}}/\mathrm{hom}&b-2m=-1\text{
(Tate)}\\\ \dim\mathrm{CH}^{m}(X_{\eta})_{\mathbb{Q},\mathrm{hom}}&b-2m=0\\\
\dim\mathrm{H}^{b}(X_{\eta},n)_{\mathbb{Z}}&b-2m\geq 1\end{array}\right.$
Special $L$-values are conjecturally given by the following: For $b=2m$ it is
conjectured that the _height pairing_
$\mathrm{CH}^{m}(X_{\eta})_{\mathbb{Q},\mathrm{hom}}{\otimes}\mathrm{CH}^{d-m}(X_{\eta})_{\mathbb{Q},\mathrm{hom}}\rightarrow\mathbb{R}$
(19)
is perfect. Up to a nonzero rational factor $L^{*}(M_{\eta},0)$ is given by
the determinant of the height pairing multiplied with the period of
$M_{\eta}$, that is to say, the determinant of the isomorphism
$\alpha_{M_{\eta}}:\mathrm{H}^{2m-1}(X(\mathbb{C}),\mathbb{R}(m))^{(-1)^{m}}\rightarrow\mathrm{H}^{2m-1}_{\mathrm{dR}}(X_{\mathbb{R}})/F^{m}$
with respect to the usual $\mathbb{Q}$-structures on both sides (compare (2),
p. 2).
For $b-2m=1$, the map
$r_{\infty}:(\mathrm{CH}^{m}(X_{\eta})_{\mathbb{Q}}/\mathrm{hom}\oplus\mathrm{H}^{2m+1}(X_{\eta},n)_{\mathbb{Z}})_{\mathbb{R}}\rightarrow\mathrm{H}_{\mathrm{D}}^{2m+1}(X,n).$
(20)
obtained by the composition
$\mathrm{CH}^{m}(X_{\eta})_{\mathbb{Q}}/\mathrm{hom}{\otimes}\mathbb{R}\rightarrow\mathrm{H}^{2m}_{\mathrm{dR}}(X_{\mathbb{R}})\rightarrow\mathrm{H}_{\mathrm{D}}^{2m+1}(X,n)$
(see (2) for the right hand map) and the realization map, is conjectured to be
an isomorphism. The induced isomorphism
$\operatorname{det}r_{\infty}:\operatorname{det}(\mathrm{H}^{b}(X_{\eta},n)_{\mathbb{Z}})_{\mathbb{R}}=\mathbb{R}\rightarrow\operatorname{det}\mathrm{H}_{\mathrm{D}}^{b}(X,n)=\mathbb{R}$
is denoted $d_{\infty}$. The left hand term is endowed with the obvious
$\mathbb{Q}$-structure. The right one gets the one stemming from the
identification of
$\mathrm{H}_{\mathrm{D}}^{b}(X_{\eta},n)=\mathrm{H}_{\mathrm{w}}^{1}(\mathrm{H}^{b-1}(X,\mathbb{Q}(n)))$
with the dual of
$\mathrm{H}_{\mathrm{w}}^{0}(\mathrm{H}^{b-1}(X,\mathbb{Q}(n))^{\vee}(1))$.
With respect to these $\mathbb{Q}$-structures,
$L^{*}(M_{\eta},0)\equiv d_{\infty}(1)\text{ mod }\mathbb{Q}^{\times}.$
For $b-2m>1$, the statement is the same, except that (20) gets replaced by
$r_{\infty}:\mathbb{R}{\otimes}_{\mathbb{Q}}\mathrm{H}^{b}(X_{\eta},n)_{\mathbb{Z}}\rightarrow\mathrm{H}_{\mathrm{D}}^{b}(X_{\eta},n).$
(21)
This conjecture determines $L$-values of motives
$\operatorname{h}^{b-1}(X_{\eta},m)$ with any $b,m$, such that $b-2m\geq 0$
and $X_{\eta}/\mathbb{Q}$ smooth projective, up to a nonzero rational factor.
By the functional equation (Conjecture 2.11), the conjecture predicts
$L$-values for all $b,m$.
#### 4.3.2 The hard Lefschetz isomorphism
This section is a short detour concerning the Lefschetz isomorphism. We want
to get rid of the (conjectural) hard Lefschetz isomorphism built into
Beilinson’s conjecture in order to express $L$-values solely in terms of the
proposed global motivic duality. By Poincaré duality we have
$M_{\eta}^{\vee}(1)=\operatorname{h}^{2d_{\eta}-b+1}(X_{\eta},d_{\eta}-m+1)$.
It is pure of weight $-b-1+2m$. The hard Lefschetz axiom [Scha, Axiom 4.4.]
says
$\operatorname{h}^{b-1}(X_{\eta},b-m)\stackrel{{\scriptstyle\cong}}{{{\longrightarrow}}}\operatorname{h}^{2d_{\eta}-b+1}(X_{\eta},d_{\eta}-m+1).$
(22)
For $b\leq d$ the map is given by taking the $(d-b)$-th power of cup product
with a hyperplane section, with respect to an embedding of $X_{\eta}$ into
some $\mathbb{P}^{N}_{\mathbb{Q}}$. For $b\geq d$ it is the inverse of this
map.
###### Lemma 4.10.
Let $m\in\mathbb{Z}$ be arbitrary and $b$ such that $b-2m>0$. The hard
Lefschetz axiom [Scha, Axiom 4.4.] implies isomorphisms
$\displaystyle\mathrm{CH}^{m}(X_{\eta})_{\mathbb{Q}}/\mathrm{hom}$
$\displaystyle\cong$
$\displaystyle\mathrm{CH}^{d-m-1}(X_{\eta})_{\mathbb{Q}}/\mathrm{hom},$
$\displaystyle\mathrm{CH}^{m}(X_{\eta})_{\mathbb{Q},\mathrm{hom}}$
$\displaystyle\cong$
$\displaystyle\mathrm{CH}^{d-m}(X_{\eta})_{\mathbb{Q},\mathrm{hom}}\text{
\cite[cite]{[\@@bibref{}{Beilinson:Height}{}{}, Conj. 5.3.(a)]}},$
$\displaystyle\mathrm{H}^{b}(X_{\eta},b-m)_{\mathbb{Z}}$ $\displaystyle\cong$
$\displaystyle\mathrm{H}^{2d-b}(X_{\eta},d-m)_{\mathbb{Z}}$
###### Proof: .
Using [Scha, Axiom 4.7.], the first claim is obtained by applying
$\mathrm{H}^{0}$ to both sides of (22) (in the particular case $b-2m=-1$). The
second and third isomorphism follow by applying
$\mathrm{H}^{0}\circ\eta_{!*}\circ[1]$ to (22) (with $b-2m=0$ and $b-2m>0$,
respectively), using Theorem 1.3. ∎
###### Proposition 4.11.
Let $b-2m>0$. There is a commutative diagram
$\textstyle{\mathrm{H}^{0}(\eta_{!*}\eta^{*}\operatorname{h}^{2d-b}(X,d-m))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$real.$\scriptstyle{\cong}$$\textstyle{\mathrm{H}^{0}(\eta_{!*}\eta^{*}\operatorname{h}^{b}(X,b-m))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$real.$\textstyle{\mathrm{H}_{\mathrm{D}}^{2d-b}(X_{\eta},d-m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{=}$$\textstyle{\mathrm{H}_{\mathrm{D}}^{b}(X_{\eta},b-m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{\mathrm{H}_{\mathrm{w}}^{1}(\mathrm{H}^{2d-b-1}(X_{\eta},\mathbb{Q}(d-m)))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathrm{H}_{\mathrm{w}}^{1}(\mathrm{H}^{b-1}(X_{\eta},\mathbb{Q}(b-m)))}$
Its horizontal maps are given by the product with $L^{d-b}$ or its inverse
(see above). The $\mathbb{Q}$-structures on the Deligne cohomology groups are
preserved under this isomorphism.
For $b-2m\leq 0$ there is a similar diagram featuring a hard Lefschetz
isomorphism
$\mathrm{H}_{\mathrm{D}}^{b-1}(X_{\eta},b-m)\rightarrow\mathrm{H}_{\mathrm{D}}^{2d-b-1}(X_{\eta},d-m).$
###### Proof: .
We use the short exact sequence (2). We only do the case $b-2m\geq 1$. The
case $b-2m\leq 0$ is done dually using (3). For $b-2m\geq 1$, there are exact
sequences (Lemma 1.2)
$0\rightarrow\mathrm{H}^{b-1}(X(\mathbb{C}),\mathbb{R}(b-m))^{(-1)^{b-m}}\rightarrow\mathrm{H}^{b-1}_{\mathrm{dR}}(X_{\mathbb{R}})/F^{b-m}\rightarrow\mathrm{H}_{\mathrm{D}}^{b}(X,b-m)\rightarrow
0$
and
$0\rightarrow\mathrm{H}^{2d-b-1}(X(\mathbb{C}),\mathbb{R}(d-m))^{(-1)^{d-m}}\rightarrow\mathrm{H}^{2d-b-1}_{\mathrm{dR}}(X_{\mathbb{R}})/F^{d-m}\rightarrow\mathrm{H}_{\mathrm{D}}^{2d-b}(X,d-m)\rightarrow
0.$
The left hand injections are induced by the natural morphisms
$\mathbb{R}(-)\subset\mathbb{C}\rightarrow\Omega^{*}_{X(\mathbb{C})/\mathbb{C}}$
of (complexes of) sheaves on $X(\mathbb{C})$. The Lefschetz isomorphism (see
e.g. [PS08, Theorem 1.30]) is given by the cup product with the cycle class
$L\in\mathrm{H}^{2}(X(\mathbb{C}),\mathbb{Q}(1))^{-1}$ of a hyperplane section
of $X$ (or its inverse, see above). It visibly gives a commutative diagram
between the singular cohomology groups and the de Rham cohomology groups in
the sequences above. Hence the Deligne cohomology groups are also isomorphic.
The $\mathbb{Q}$-structure on Betti cohomology is preserved by cup-product
with the rational (actually integral) cohomology class $L$. Likewise, the
hyperplane section being defined over $\mathbb{Q}$, it respects the
$\mathbb{Q}$-structure on algebraic de Rham cohomology. This shows the claim
about the $\mathbb{Q}$-structures. ∎
#### 4.3.3 The comparison
###### Lemma 4.12.
The perfectness of the motivic duality pairing $\pi^{0}_{S}$, where
$S=\eta_{!*}\eta^{*}\operatorname{h}^{b}(X,m)$, $b-2m\geq 0$ is is implied by
the maps (21) and (20) in Beilinson’s conjecture being isomorphisms and the
height pairing (19) being perfect.
If we additionally assume $\mathrm{H}^{1}(DS)=0$ for $b-2m>0$, the said
assumptions in Beilinson’s conjecture are equivalent to that all pairings
$\pi^{*}_{S}$ and $\pi^{*}_{DS}$ are perfect.
See Corollary 4.14 for a statement concerning the vanishing of
$\mathrm{H}^{1}(DS)$. Also, this group is zero if $X_{\eta}$ is such that
$\operatorname{h}^{b-1}(X_{\eta})$ is a mixed Artin-Tate motive over
$\mathbb{Q}$ (as opposed to a general mixed motive) [Schb, Proposition 4.4].
###### Proof: .
By Proposition 4.11, the hard Lefschetz isomorphism identifies the realization
map
$\mathrm{H}^{b}(X_{\eta},b-m)_{\mathbb{Z}}\rightarrow\mathrm{H}_{\mathrm{D}}^{b}(X_{\eta},b-m)$
featuring in (21), (20) in the cases $b-2m\geq 1$ of Conjecture 4.9 with
$\mathrm{H}^{2d-b}(X_{\eta},d-m)_{\mathbb{Z}}\rightarrow\mathrm{H}_{\mathrm{D}}^{2d-b}(X_{\eta},d-m)$.
By Theorem 1.3, definition of the weak Hodge cohomology functor and Lemma 1.2,
this map is the realization map
$\rho^{0}_{DS}:\mathrm{H}^{0}(DS)_{\mathbb{R}}\rightarrow\mathrm{H}_{\mathrm{w}}^{0}(DS).$
(23)
For all $b-2m\geq 0$ we have $\mathrm{H}^{1}(S)=\mathrm{H}^{-1}(DS)=0$ and
$\mathrm{H}^{a}(S)=\mathrm{H}^{a}(DS)=0$ for $|a|>1$ (Theorem 1.3). For weight
reasons, all weak Hodge cohomology groups of $S$ and $DS$ are zero except for
$\mathrm{H}_{\mathrm{w}}^{-1}(S)=\mathrm{H}_{\mathrm{w}}^{0}(M_{\eta})$ and
$\mathrm{H}_{\mathrm{w}}^{0}(DS)$ in the cases $b-2m\geq 1$. See the proof of
Lemma 1.2. Therefore, we have to look at the following commutative diagram
with exact columns ((10) and (12)):
$\begin{array}[]{cccccl}&0&{\times}&0&{\longrightarrow}&\mathbb{R}\\\
&\downarrow&&\uparrow&&\downarrow=\\\
\pi^{-1}_{S}:&\mathrm{H}_{\mathrm{c}}^{-1}(S)&{\times}&\mathrm{H}^{1}(DS)_{\mathbb{R}}&{\longrightarrow}&\mathbb{R}\\\
&\downarrow&&\uparrow&&\downarrow=\\\
\pi^{1}_{DS}:&\mathrm{H}^{-1}(S)_{\mathbb{R}}&{\times}&\mathrm{H}_{\mathrm{c}}^{1}(DS)&{\longrightarrow}&\mathbb{R}\\\
&\hskip 19.91684pt\downarrow\rho^{-1}_{S}&&\uparrow&&\downarrow=\\\
&\mathrm{H}_{\mathrm{w}}^{-1}(S)&{\times}&\mathrm{H}_{\mathrm{w}}^{0}(DS)&{\longrightarrow}&\mathbb{R}\\\
&\downarrow&&\hskip 19.91684pt\uparrow\rho^{0}_{DS}&&\downarrow=\\\
\pi^{0}_{S}:&\mathrm{H}_{\mathrm{c}}^{0}(S)&{\times}&\mathrm{H}^{0}(DS)_{\mathbb{R}}&{\longrightarrow}&\mathbb{R}\\\
&\downarrow&&\uparrow&&\downarrow=\\\
\pi^{0}_{DS}:&\mathrm{H}^{0}(S)_{\mathbb{R}}&{\times}&\mathrm{H}_{\mathrm{c}}^{0}(DS)&{\longrightarrow}&\mathbb{R}\\\
&\downarrow&&\uparrow&&\downarrow=\\\
&0&{\times}&0&{\longrightarrow}&\mathbb{R}\\\ \end{array}$ (24)
The map (20) being an isomorphism is equivalent [Fon92, 9.5] to the existence
of an exact sequence
---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{CH}^{m}(X_{\eta})_{\mathbb{R}}/\mathrm{hom}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{-1}_{S}}$$\textstyle{\mathrm{H}_{\mathrm{w}}^{0}(M_{\eta})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{\cong}$$\textstyle{\mathrm{H}^{b}(X_{\eta},b-m)_{\mathbb{Z}}^{\vee}{\otimes}\mathbb{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{\mathrm{H}_{\mathrm{w}}^{1}(M_{\eta}^{\vee}(1))^{\vee}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rho^{0}_{DS}}^{\vee}}$
By Theorem 1.3, it reads
---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{-1}(S)_{\mathbb{R}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{-1}_{S}}$$\textstyle{\mathrm{H}_{\mathrm{w}}^{-1}(S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{\cong}$$\textstyle{\mathrm{H}^{0}(DS)_{\mathbb{R}}^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{\mathrm{H}_{\mathrm{w}}^{0}(DS)^{\vee}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rho^{0}_{DS}}^{\vee}}$
(25)
The isomorphism $\phi$ expresses the duality of weak Hodge cohomology (5).
In the cases $b-2m\geq 1$ we know $\mathrm{H}^{0}(S)=0$. By (24), the
injectivity of $\rho^{0}_{DS}$ is equivalent to $\pi^{0}_{DS}$ being perfect.
The identification of $\operatorname{coker}\rho^{-1}_{S}$ with
$\mathrm{H}^{0}(DS)_{\mathbb{R}}^{\vee}$ of (25) is equivalent to
$\pi^{0}_{S}$ being perfect. The group $\mathrm{H}_{\mathrm{c}}^{1}(S)$
vanishes, as we have seen in the proof of Lemma 3.7, so $\mathrm{H}^{1}(DS)=0$
is equivalent to $\pi^{-1}_{S}$ being perfect. By the five lemma,
$\pi^{1}_{DS}$ is then also perfect.
The case $b-2m>1$ is done parallel. In addition to the above, we have
$\mathrm{H}^{-1}(S)=0$. Accordingly, (25) reduces to an isomorphism
$\mathrm{H}_{\mathrm{w}}^{-1}(S)\stackrel{{\scriptstyle\cong}}{{\rightarrow}}\mathrm{H}^{0}(DS)_{\mathbb{R}}^{\vee}$.
The details are omitted.
In the case $b-2m=0$ the weak Hodge cohomology groups in (24) both vanish for
weight reasons. Moreover $\mathrm{H}^{a}(DS)=\mathrm{H}^{a}(S)=0$ for $a\neq
0$ (Theorem 1.3), so that $\pi^{-1}_{S}$ and $\pi^{1}_{DS}$ are perfect. The
height pairing (19) is just $\pi^{0}_{S}$, by the requirement 3.1.6. Its
perfectness is equivalent to the one of $\pi^{0}_{DS}$. ∎
In the following theorem, we assume Conjecture 3.1, in particular the
perfectness of the global motivic duality pairing. Under this conjecture, we
can talk about Conjecture 4.2.
###### Theorem 4.13.
Under Conjecture 3.1, Beilinson’s conjecture for $M_{\eta}$ is equivalent to
Conjecture 4.2 for $S$.
###### Proof: .
By Proposition 2.5, $L(M_{\eta},s)=L(S,s)^{-1}$. This and hard Lefschetz (cf.
Lemma 4.10) together shows that Beilinson’s conjecture reads
$\operatorname{ord}_{s=0}L(M_{\eta},s)=\sum_{a\neq
1}(-1)^{a}\dim\mathrm{H}^{a}(DS).$ (26)
For $b-2m\leq 0$, we additionally know $\mathrm{H}^{1}(DS)=0$ (Theorem 1.3).
In general, this vanishing is equivalent (Lemma 3.7) to the perfectness of the
pairing $\pi^{-1}_{S}$ for $S$ (Lemma 3.7), which is part of Conjecture 3.1.
The comparison of the pole order statements is therefore done.
For the special $L$-values, we revisit the proof of Lemma 4.12 and look at the
$\mathbb{Q}$-structures involved. As in that proof we may replace the map
$\mathrm{H}^{b}(X_{\eta},b-m)_{\mathbb{Z}}{\otimes}\mathbb{R}\rightarrow\mathrm{H}_{\mathrm{D}}^{b}(X_{\eta},b-m)$
occurring in (21), (20) by $\rho^{0}_{DS}$, see (23), since the involved
$\mathbb{Q}$-structures remain unchanged (Proposition 4.11). The
$\mathbb{Q}$-structure on $\mathrm{H}_{\mathrm{w}}^{0}(DS)$ is the one
stemming from the weak Hodge duality isomorphism with
$\mathrm{H}_{\mathrm{w}}^{-1}(S)^{\vee}$. This $\mathbb{Q}$-structure is
distinct from the “original” one on $\mathrm{H}_{\mathrm{w}}^{0}(DS)$, see the
discussion of the functional equation in Theorem 4.4. All motivic cohomology
terms $\mathrm{H}^{*}(-)_{\mathbb{R}}$ get the obvious $\mathbb{Q}$-structure.
We first do the case $b-2m=1$. In addition to the vanishing of the
$\mathrm{H}_{\mathrm{w}}^{*}(-)$ groups mentioned in the beginning of the
proof of Lemma 4.12 we note that the groups $\mathrm{H}_{\mathrm{w}}^{-1}(S)$
and $\mathrm{H}_{\mathrm{w}}^{0}(DS)$ are also the only groups that contribute
to the determinant of the weak Hodge cohomology complexes
${\mathrm{R}}{\Gamma}_{\mathrm{w}}(S)$ and
${\mathrm{R}}{\Gamma}_{\mathrm{w}}(DS)$, respectively. Indeed, the period maps
of ${\mathrm{R}}{\Gamma}_{\mathrm{H}}(S)$ and
${\mathrm{R}}{\Gamma}_{\mathrm{H}}(DS)$ are surjective and injective,
respectively. To simplify the notation, we will therefore endow the groups
$\mathrm{H}_{\mathrm{w}}^{a}(S)=\mathrm{H}_{\mathrm{w}}^{-1-a}(DS)=0$,
$a\neq-1$ with the trivial $\mathbb{Q}$-structure and identify
$\operatorname{det}\mathrm{H}_{\mathrm{w}}^{-1}(S)$ with
$\operatorname{det}{\mathrm{R}}{\Gamma}_{\mathrm{w}}(S)$ and
$\operatorname{det}\mathrm{H}_{\mathrm{w}}^{0}(DS)$ with
$\operatorname{det}{\mathrm{R}}{\Gamma}_{\mathrm{w}}(DS)$, respectively.
By [Fon92, 9.5], [FPR94, Conj. III.4.4.3], Beilinson’s conjecture equivalently
says that the $L$-value of $M_{\eta}$ is given by the reciprocal of the image
of $1\in\mathbb{Q}$ in
$\operatorname{det}^{-1}\mathrm{H}^{0}(DS)_{\mathbb{R}}{\otimes}\operatorname{det}^{-1}\mathrm{H}_{\mathrm{w}}^{-1}(S){\otimes}\operatorname{det}\mathrm{H}^{-1}(S)_{\mathbb{R}}\cong\mathbb{R},$
where the identification with $\mathbb{R}$ is stemming from the exact sequence
(25). The $\mathbb{Q}$-structure on $\mathrm{H}_{\mathrm{w}}^{-1}(S)$ is the
natural one defined in Section 1.2. The left hand $\mathbb{R}$-vector space is
isomorphic, including the $\mathbb{Q}$-structure, to
$\operatorname{det}^{-1}\mathrm{H}^{0}(DS)_{\mathbb{R}}{\otimes}\operatorname{det}^{-1}\mathrm{H}_{\mathrm{c}}^{0}(S),$
so that $L^{*}(S,0)=1/L^{*}(M_{\eta},0)$ is indeed the inverse of the
determinant of $\pi^{0}_{S}$. All other $\pi^{*}_{S}$ vanish, so this
accomplishes the case $b-2m=1$.
Again, the case $b-2m>1$ is similar but simpler, since—in addition to what was
said before—$\mathrm{H}^{-1}(S)$ vanishes. Correspondingly, only the
determinant of the realization map $\rho^{0}_{DS}$ (23), as opposed to the one
of (25), appears in Beilinson’s conjecture.
At the central point, $b-2m=0$, Beilinson’s conjecture says that
$L^{*}(M_{\eta},0)=1/L^{*}(S,0)$ is given by the determinant of the height
pairing
$\pi^{0}_{S}:\mathrm{H}^{0}(S){\times}\mathrm{H}^{0}(DS)\rightarrow\mathbb{R}$
(3.1.6) multiplied with $\operatorname{det}\alpha_{S}$, where
$\alpha_{S}:B:=\mathrm{H}^{b-1}(X(\mathbb{C}),\mathbb{R}(m))^{(-1)^{m}}\rightarrow
dR:=(\mathrm{H}^{b-1}_{\mathrm{dR}}(X_{\eta})/F^{m}){\otimes}\mathbb{R}$
is the period map of $S$. That map is an isomorphism (see Section 1.2), i.e.,
$M_{\eta}$ is a critical motive in the sense of Deligne [Del79, Def. 1.3]. The
$\mathbb{Q}$-structures are not respected by $\alpha_{S}$, so we cannot commit
the above abuse of notation since the $\mathbb{Q}$-structure on
$\operatorname{det}{\mathrm{R}}{\Gamma}_{\mathrm{w}}(S)$ and
$\operatorname{det}{\mathrm{R}}{\Gamma}_{\mathrm{w}}(DS)$ is nontrivial even
though all its cohomology vectors spaces are trivial. There is a chain of
natural isomorphisms of one-dimensional $\mathbb{R}$-vector spaces respecting
the $\mathbb{Q}$-structures:
$\displaystyle\operatorname{det}B{\otimes}\operatorname{det}^{-1}dR=\operatorname{det}{\mathrm{R}}{\Gamma}_{\mathrm{w}}(M_{\eta})=\operatorname{det}^{-1}{\mathrm{R}}{\Gamma}_{\mathrm{w}}(S)$
By linear algebra, $\operatorname{det}\alpha_{S}$ agrees (modulo
$\mathbb{Q}^{\times}$) with the image of $1$ in $\mathbb{R}$ under the natural
isomorphism induced by $\alpha_{S}$:
$\operatorname{det}B{\otimes}\operatorname{det}^{-1}dR\stackrel{{\scriptstyle\cong}}{{\rightarrow}}\mathbb{R}.$
Except for $\mathrm{H}^{0}(S)$ and $\mathrm{H}^{0}(DS)$, all motivic
cohomology groups of $S$ and its Verdier dual $D(S)$ vanish (Theorem 1.3). By
(10), we have an isomorphism of $\mathbb{R}$-vector spaces respecting the
$\mathbb{Q}$-structure
$\bigotimes\operatorname{det}^{(-1)^{i}}\mathrm{H}_{\mathrm{c}}^{i}(S)=\bigotimes\operatorname{det}^{(-1)^{i}}\mathrm{H}^{i}(S){\otimes}\operatorname{det}^{-1}{\mathrm{R}}{\Gamma}_{\mathrm{w}}(S),$
so Beilinson’s conjecture can indeed equivalently be rephrased by saying that
$L^{*}(S,0)$ is reciprocal of the image of $1$ under
$\bigotimes\operatorname{det}^{(-1)^{i}}\mathrm{H}_{\mathrm{c}}^{i}(S){\otimes}\bigotimes\operatorname{det}^{(-1)^{i}}\mathrm{H}^{i}(DS)\rightarrow\mathbb{R}.$
∎
The following is clear from the proof of Theorem 4.13, see esp. (26). It is
relevant insofar as Soulé’s conjecture 4.7 is equivalent to the pole order
part of Conjecture 4.2 for the subcategory of truly geometric motives over
$\mathbb{Z}$.
###### Corollary 4.14.
Any two of the following statements imply the third: (1) Beilinson’s pole
order conjecture, (2) the pole order part of Conjecture 4.2, and (3)
$\mathrm{H}^{1}(DS)=0$ for $b-2m>0$.
### 4.4 Relation to the Tate conjecture over $\mathbb{F}_{p}$
In order to relate our conjecture for motives $M$ supported on closed points
of ${\mathrm{Spec}\text{ }}{\mathbb{Z}}$ to the Tate conjecture, we have to
assume the conjectural agreement 1.5 of numerical and rational equivalence on
smooth projective varieties over finite fields. By Theorem 3.4, this implies
that the proposed motivic duality holds for such motives. All further
statements of Conjecture 3.1 hold trivially for such motives.
###### Conjecture 4.15.
(Tate conjecture over finite fields) Let $X/{\mathbb{F}_{q}}$ be smooth and
projective. Let $\ell$ be a rational prime such that $\ell\nmid q$. Any
$\mathrm{Gal}({\mathbb{F}_{q}})$-invariant element of
$\mathrm{H}^{2i}(X{\times}_{{\mathbb{F}_{q}}}{\overline{{\mathbb{F}_{q}}}},{\mathbb{Q}_{{\ell}}}(i))$
is a ${\mathbb{Q}_{{\ell}}}$-linear combination of algebraic elements, i.e.,
elements in the image of the cycle class map
$\mathrm{CH}^{i}(X)\rightarrow\mathrm{H}^{2i}(X{\times}_{{\mathbb{F}_{q}}}{\overline{{\mathbb{F}_{q}}}},{\mathbb{Q}_{{\ell}}}(i))$.
###### Theorem 4.16.
In addition to the general assumptions on mixed motives over $\mathbb{F}_{p}$
(Section 1.3), we assume Conjecture 1.5. Then Conjecture 4.15 is equivalent to
Conjecture 4.2 for motives $M=i_{*}N$, where $N$ is any geometric motive over
$\mathbb{F}_{p}$, $i:{\mathrm{Spec}\text{
}}{\mathbb{F}_{p}}\rightarrow{\mathrm{Spec}\text{ }}{\mathbb{Z}}$. More
precisely, the special value prediction of 4.2 in this case is
$L^{*}(i_{*}N,0)\equiv\mathrm{log}\,p^{-\chi(D(i_{*}N))}\ \ (\mathrm{mod\
}\mathbb{Q}^{\times}).$
where $D$ denotes the Verdier dual functor on
$\mathbf{DM}_{\mathrm{gm}}(\mathbb{Z})$ and $\chi$ the Euler characteristic of
motivic cohomology (p. 4.1).
###### Proof: .
We first show the implication $\Rightarrow$. The claim for $N$ is implied by
the one for the ${{{}^{\mathrm{p}}}\mathrm{H}}^{j}N$, since only finitely many
$j$ give a nonzero term [Scha, Axiom 4.1.]. Similarly, the claim for $N$ is
implied by the one for $\operatorname{gr}_{n}^{W}N$. Therefore, we may assume
$N$ is a pure motive. Under Conjecture 1.5, all adequate equivalence relations
agree, so that we need not (and will not) distinguish between Chow motives
$\mathbf{M}_{\mathrm{rat}}(\mathbb{F}_{p})$ and numerical motives
$\mathbf{M}_{\mathrm{num}}(\mathbb{F}_{p})$. By definition of pure motives,
$N$ is a direct summand of $H:=h(X)(n)$, with $X/\mathbb{F}_{p}$ smooth and
projective. Let $N\oplus N^{\prime}=H$. Under the embedding
$\mathbf{M}_{\mathrm{rat}}(\mathbb{F}_{p})\subset\mathbf{DM}_{\mathrm{gm}}(\mathbb{F}_{p})$,
$H$ maps to $\operatorname{M}(X)(n)[2n]$. The latter motive is also denoted
$H$. Therefore and for weight reasons,
$\mathrm{H}^{a}(D(i_{*}N))=\mathrm{H}^{a}(N^{\vee})\subset\mathrm{H}^{a}(H^{\vee})=\mathrm{H}^{0}(\operatorname{gr}_{0}^{W}{{{}^{\mathrm{p}}}\mathrm{H}}^{0}(H^{\vee}[a]))$
vanishes for $a\neq 0$.
The semisimplicity of $\mathbf{M}_{\mathrm{num}}(\mathbb{F}_{p})$ yields a
decomposition $N=\mathbf{1}^{r}\oplus R$ with
$\mathrm{Hom}(\mathbf{1},R)=\mathrm{Hom}(R,\mathbf{1})=0$. These
$\mathrm{Hom}$-groups are the same when taken in either
$\mathbf{DM}(\mathbb{F}_{p})$, $\mathbf{M}_{\mathrm{rat}}(\mathbb{F}_{p})$, or
$\mathbf{M}_{\mathrm{num}}(\mathbb{F}_{p})$. Notice
$\mathrm{Hom}(R,\mathbf{1})=\mathrm{Hom}(\mathbf{1},R^{\vee})$, so that
$\dim\mathrm{H}^{0}(N)=\dim\mathrm{H}^{0}(N^{\vee})(=r)$. Therefore, we have
to show $\operatorname{ord}_{s=0}L(i_{*}N)=-\dim\mathrm{H}^{0}(N)$ and
$L^{*}(i_{*}N)\equiv\mathrm{log}\,p^{-\dim\mathrm{H}^{0}(N)}\ \ (\mathrm{mod\
}\mathbb{Q}^{\times})$.
Let $Z^{n}(X)/\mathrm{num}$ be the group of codimension $n$ cycles on $X$
modulo numerical equivalence. Then
$\dim\mathrm{H}^{0}(H)=\operatorname{rk}\mathrm{CH}^{n}(X)$ by [Scha, Axiom
1.8.] and
$\operatorname{rk}\mathrm{CH}^{n}(X)\stackrel{{\scriptstyle\text{\ref{conj_numrat}}}}{{=}}\operatorname{rk}Z^{n}(X)/\mathrm{num}=-\operatorname{ord}_{s=n}\zeta(X,s),$
so the claim holds for $H$ by assumption. The rightmost equality is a
consequence of the Tate conjecture: in fact, the Tate conjecture and the
agreement of the $\ell$-adic homological and numerical equivalence relations
on $X$ (up to torsion) together are equivalent to this equality [Tat94, Thm.
2.9]. Consequently
$\dim\mathrm{H}^{0}N+\dim\mathrm{H}^{0}N^{\prime}=\dim\mathrm{H}^{0}H=-\operatorname{ord}L(H)=-\operatorname{ord}L(N)-\operatorname{ord}L(N^{\prime}).$
(27)
Let
$-_{\ell}:\mathbf{M}_{\mathrm{rat}}(\mathbb{F}_{p})\rightarrow\oplus\underline{{\mathbb{Q}_{{\ell}}}[\mathrm{Gal}(\mathbb{F}_{p})]}$,
$\pi
h(X)(n)\mapsto\oplus_{a}\pi^{*}\mathrm{H}^{a}(X,{\mathbb{Q}_{{\ell}}}(n))$ be
the $\ell$-adic realization functor taking values in graded continuous
$\ell$-adic $\mathrm{Gal}(\mathbb{F}_{p})$-representations. We write
$\mathrm{H}^{0}(N_{\ell}):=N_{\ell}^{\mathrm{Gal}(\mathbb{F}_{p})}$, the
Galois cohomology of the $\ell$-adic Galois module $N_{\ell}$. The following
way of reasoning is borrowed from loc. cit. We have the following chain of
inequalities:
$\displaystyle-\operatorname{ord}_{s=0}L(N,s)$ $\displaystyle=$
$\displaystyle\dim_{{\mathbb{Q}_{{\ell}}}}\ker(\mathrm{Id}-\operatorname{Fr}^{-1})^{w}|N_{\ell},w\gg
0$ $\displaystyle\geq$
$\displaystyle\dim_{{\mathbb{Q}_{{\ell}}}}\ker(\mathrm{Id}-\operatorname{Fr}^{-1})|N_{\ell}$
$\displaystyle\geq$
$\displaystyle\dim_{{\mathbb{Q}_{{\ell}}}}(N_{\ell})^{\mathrm{Gal}(\mathbb{F}_{p})}$
$\displaystyle=$
$\displaystyle\dim_{{\mathbb{Q}_{{\ell}}}}\mathrm{H}^{0}(N_{\ell})$
$\displaystyle\geq$ $\displaystyle\dim_{\mathbb{Q}}\mathrm{H}^{0}(N)$
The first equality is by linear algebra. The last inequality is by the
injectivity of the cycle class map
$\mathrm{H}^{0}(N)\rightarrow\mathrm{H}^{0}(N_{\ell})$, which follows from the
injectivity of
$\mathrm{H}^{0}(H)\rightarrow\mathrm{H}^{0}(H_{\ell})=\mathrm{H}^{2n}(X,{\mathbb{Q}_{{\ell}}}(n))$,
i.e., the agreement of homological and rational equivalence, which holds under
Conjecture 1.5. Therefore, in (27) equality of dimensions must hold for the
individual summands, so the pole order part is shown.
As for the special value, the claim does hold for $N=\mathbf{1}$: the residue
of $\zeta({\mathrm{Spec}\text{ }}{\mathbb{F}_{p}},s)$ at $s=0$ is
$(\mathrm{log}\,p)^{-1}$ and the determinant of $\pi^{0}_{i_{*}\mathbf{1}}$ is
$\mathrm{log}\,p$ (see the proof of Proposition 3.3). Hence we can assume
$N=R$. By the Lefschetz trace formula, the $L$-function of any pure motive
over $\mathbb{F}_{p}$ is a rational function in $p^{-s}$ with rational
coefficients that are independent of $\ell$, see e.g. [And04, Section 7.1.4].
By the preceding part, the $L$-function of $i_{*}R$ does not have a pole at
$s=0$, therefore the leading term of the Laurent series $L(i_{*}R,s)$ is
simply the value at this point, a nonzero rational number (as opposed to an
$\ell$-adic or, via $\sigma_{\ell}$, a complex number).
For the implication $\Leftarrow$, we again use the theorem of Tate cited
above: the Tate conjecture for $X/\mathbb{F}_{p}$ is implied by
$\operatorname{ord}_{s=j}\zeta(X,s)=-\operatorname{rk}Z^{j}(X)/\mathrm{num}$.
Under 1.5, that term is
$-\operatorname{rk}\mathrm{CH}^{j}(X)=-\dim\mathrm{H}^{2j}(\operatorname{M}(X)(j))$.
Thus, Conjecture 4.2 for $i_{*}\operatorname{M}(X)(j)$ implies the Tate
conjecture for $X$. ∎
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|
arxiv-papers
| 2010-03-05T09:29:26 |
2024-09-04T02:49:08.774660
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Jakob Scholbach",
"submitter": "Jakob Scholbach",
"url": "https://arxiv.org/abs/1003.1215"
}
|
1003.1219
|
# $f$-cohomology and motives over number rings
Jakob Scholbach 111Universität Münster, Mathematisches Institut, Einsteinstr.
62, D-48149 Münster, Germany, jakob.scholbach@uni-muenster.de
###### Abstract
This paper is concerned with an interpretation of $f$-cohomology, a
modification of motivic cohomology of motives over number fields, in terms of
motives over number rings. Under standard assumptions on mixed motives over
finite fields, number fields and number rings, we show that the two extant
definitions of $f$-cohomology of mixed motives $M_{\eta}$ over $F$—one via
ramification conditions on $\ell$-adic realizations, another one via the
$K$-theory of proper regular models—both agree with motivic cohomology of
$\eta_{!*}M_{\eta}[1]$. Here $\eta_{!*}$ is constructed by a limiting process
in terms of intermediate extension functors $j_{!*}$ defined in analogy to
perverse sheaves.
The aim of this paper is to give an interpretation of $f$-cohomology in terms
of motives over number rings. The notion of $f$-cohomology goes back to
Beilinson who used it to formulate a conjecture about special $L$-values
[Beĭ84, Beĭ86]. The most classical example is what is now called
$\mathrm{H}^{1}_{f}(F,\mathbf{1}(1))$, $f$-cohomology of $\mathbf{1}(1)$, the
motive of a number field $F$, twisted by one. This group is
${\mathcal{O}_{F}}^{\times}{\otimes}_{\mathbb{Z}}\mathbb{Q}$, as opposed to
the full motivic cohomology
$\mathrm{H}^{1}(F,\mathbf{1}(1))=F^{\times}{\otimes}\mathbb{Q}$. Together with
the Dirichlet regulator, it explains the residue of the Dedekind zeta function
$\zeta_{F}(s)$ at $s=1$. In general, for any mixed motive $M_{\eta}$ over $F$,
there are two independent yet conjecturally equivalent ways to define
$\mathrm{H}^{1}_{f}(F,M_{\eta})\subset\mathrm{H}^{1}(F,M_{\eta})$. We propose
a third way to obtain these groups, by interpreting $f$-cohomology as motivic
cohomology of suitable motives over ${\mathcal{O}_{F}}$. This idea is due to
Huber.
Let us briefly recall the two definitions of $\mathrm{H}^{1}_{f}(M_{\eta})$.
The first is due to Beilinson [Beĭ87a, Remark 4.0.1.b], Bloch and Kato [BK90,
Conj. 5.3.] and Fontaine [Fon92, FPR94]. It is given by picking elements in
motivic cohomology acted on by the local Galois groups in a prescribed way
(Definition 6.1, Definition 6.4, Definition 6.6). The second definition of
$\mathrm{H}^{1}_{f}(M_{\eta})$, due to Beilinson [Beĭ86, Section 8], applies
to $M_{\eta}=\mathrm{h}^{i-1}(X_{\eta})(n)$, with $X_{\eta}$ smooth and
projective over $F$, $i-2n<0$. It is given by the image of $K$-theory of a
regular proper model $X$ of $X_{\eta}$ (Definition 6.11). Such a model may not
exist, but there is a unique meaningful extension of this definition to all
Chow motives over $F$ due to Scholl [Sch00].
Our main results (Theorems 6.9, 6.12, 6.14) show that both definitions of
$\mathrm{H}^{1}_{f}(M_{\eta})$ agree with
$\mathrm{H}^{0}(\eta_{!*}\mathrm{h}^{i-1}(X_{\eta},n)[1])$. Here $\eta_{!*}$
is a functor that attaches to any suitable mixed motive over $F$ one over
${\mathcal{O}_{F}}$. It is defined by a limiting process using the
intermediate extension $j_{!*}$ familiar from perverse sheaves [BBD82] along
open immersions $j:U\rightarrow{\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$.
Even to formulate such a definition, one has to rely on profound conjectures,
namely the existence of mixed motives over (open subschemes of)
${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$. The proof of the two main
theorems also requires us to assume a number of properties related to weights
of motives and, for the first comparison result, Beilinson’s conjecture about
the agreement of rational and numerical equivalence on smooth projective
varieties over finite fields up to torsion (Conjecture 6.7).
A motivation for the results of this work lies in an application to special
$L$-values conjecures [Sch10]. Very briefly, Beilinson’s conjecture concerning
special $L$-values for mixed motives $M_{\eta}$ over $\mathbb{Q}$ has
$f$-cohomology as motivic input. $L$-functions of such motives can be
generalized to motives over $\mathbb{Z}$ such that the classical $L$-function
of $M_{\eta}$ agrees with the $L$-function (over $\mathbb{Z}$) of
$\eta_{!*}M_{\eta}[1]$. Thereby the $L$-function and the motivic data in
Beilinson’s conjecture belong to the same motive over $\mathbb{Z}$, thus
giving content to a general conjecture about special $L$-values for motives
over $\mathbb{Z}$. In this light it is noteworthy that
$\mathrm{H}^{0}(\eta_{!*}\mathrm{h}^{2n-1}(X_{\eta},n)[1])$ identifies with
the group that occurs in the part of Beilinson’s conjecture that describes
special values at the central point.
The contents of the paper are as follows: Section 1 is the basis of the
remainder; it lists a number of axioms on triangulated categories of motives.
Such categories $\mathbf{DM}_{\mathrm{gm}}(S)$ have been constructed by
Voevodsky [Voe00] and Hanamura [Han95] (over fields) and Levine [Lev98] (over
bases $S$ over a field). The various approaches are known to be
(anti-)equivalent, at least for rational coefficients [Lev98, Section VI.2.5],
[Bon09, Section 4]. Over more general bases $S$, a candidate for the category
$\mathbf{DM}(S)$ has been constructed by Ivorra [Ivo07]. Cisinski and Déglise
are developing a robust theory of such motives [CD10]. We sum up the
properties of this construction by specifying a number of axioms concerning
triangulated categories of motives that will be used in the sequel. They are
concerned with the “core” behavior of $\mathbf{DM}(S)$, that is:
functoriality, compacity, the monoidal structure and the relation to algebraic
$K$-theory, as well as localization, purity, base-change and resolution of
singularities. We work with motives with rational coefficients only, since
this is sufficient for all our purposes. We use a contravariant notation for
motives, that is to say the functor that maps any scheme $X$ to its motive
$\operatorname{M}(X)$ shall be contravariant. This is in line with most pre-
Voevodsky papers.
Section 2 is a very brief reminder on realizations. The existence of various
realizations, due to Huber and Ivorra [Hub95, Hub00, Ivo07], is pinning down
the intuition that motives should be universal among (reasonable) cohomology
theories.
Section 4 spells out a number of conjectural properties (also called axioms in
the sequel) of $\mathbf{DM}_{\mathrm{gm}}(S)$, where $S$ is either a finite
field ${\mathbb{F}_{\mathfrak{p}}}$, a number field $F$ or a number ring
${\mathcal{O}_{F}}$. The first group of these properties centers around the
existence of a category of mixed motives $\mathbf{MM}(S)$, which is to be the
heart of the so-called motivic $t$-structure. The link between mixed motives
over ${\mathcal{O}_{F}}$ and ${\mathbb{F}_{\mathfrak{p}}}$ or $F$ is
axiomatized by mimicking the exactness properties familiar from perverse
sheaves (Axiom 4.2). On the triangulated subcategory
$\mathbf{DATM}({\mathcal{O}_{F}})\subset\mathbf{DM}_{\mathrm{gm}}({\mathcal{O}_{F}})$
of Artin-Tate motives, a $t$-structure satisfying these properties is
constructed in a separate paper [Scha]. A key requirement on mixed motives is
that the realization functors on motives should be exact (Axiom 4.8). For the
$\ell$-adic realization over ${\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}[1/\ell]$, this requires a notion of perverse sheaves
over that base, which is provided in Section 3. Another important conjectural
facet of mixed motives are weights. Weights are an additional structure
encountered in both Hodge structures and $\ell$-adic cohomology of algebraic
varieties over finite fields, both due to Deligne [Del74, Del80]. They are
important in that morphisms between Hodge structures or $\ell$-adic cohomology
groups are known to be strictly compatible with weights, moreover, they are
respected to a certain extent by smooth maps and proper maps. It is commonly
assumed that this should be the case for mixed motives, too.
The remaining two sections assume the validity of the axiomatic framework set
up so far. The first key notion in Section 5 is the intermediate extension
$j_{!*}M$ of a mixed motive $M$ along some open embedding $j$ inside
${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$. This is done in strict
parallelity to the case of perverse sheaves, due to Beilinson, Bernstein and
Deligne [BBD82]. Quite generally, much of this paper is built on the idea that
the abstract properties of mixed perverse sheaves (should) give a good model
for mixed motives over number rings. Next we develop a notion of smooth
motives, which is an analog of lisse étale sheaves. This is needed to extend
the intermediate extension to an extension functor $\eta_{!*}$ that extends
motives over $F$ to ones over ${\mathcal{O}_{F}}$. Finally, we apply the axiom
on the exactness of $\ell$-adic realization to show that intermediate
extensions commute with the realization functors. This will be a stepstone in
a separate work on $L$-functions of motives [Sch10].
Section 6 gives the comparison theorems on $f$-cohomology mentioned above. The
two definitions of $f$-cohomology being quite different, the proofs of the
comparison statements are different, too: the first is essentially based on
the Hochschild-Serre spectral sequence. The crystalline case of that
definition of $f$-cohomology is disregarded throughout. The second proof is a
purely formal, if occasionally intricate bookkeeping of cohomological degrees
and weights.
The problem of finding a motivic interpretation of terms such as
$\mathrm{H}^{1}_{f}(M_{\eta})$ underlying the formulation of Beilinson’s
conjecture has been studied by Scholl [Sch91, Sch00, Schb], who develops an
abelian category $\mathbf{MM}(F/{\mathcal{O}_{F}})$ of mixed motives over
${\mathcal{O}_{F}}$ by taking mixed motives over $F$, and imposing additional
non-ramification conditions. Conjecturally, the group
$\operatorname{Ext}^{i}_{\mathbf{MM}(F/{\mathcal{O}_{F}})}(\mathbf{1},\mathrm{h}^{i}(X_{\eta},n))$
for $X_{\eta}/F$ smooth and projective, $i=0$, $1$, agrees with what amounts
to $\mathrm{H}^{i-1}(\eta_{!*}\mathrm{h}^{2n-1}(X_{\eta},n)[1])$.
No originality is claimed for Sections 1, 2, and 4, except perhaps for the
formulation of the relation of mixed motives over ${\mathcal{O}_{F}}$ and $F$
and the residue fields ${\mathbb{F}_{\mathfrak{p}}}$, which however is a
natural and immediate translation of the theory of perverse sheaves. I would
like to thank Denis-Charles Cisinski and Frédéric Déglise for communicating to
me their work on $\mathbf{DM}_{\mathrm{gm}}(S)$ over general bases [CD10].
Finally, I gratefully acknowledge Annette Huber’s advice in writing my thesis,
of which this paper is a part.
## 1 Geometric motives
Throughout this paper, $F$ is a number field, ${\mathcal{O}_{F}}$ its ring of
integers, $\mathfrak{p}$ stands for a place of $F$. For finite places, the
residue field is denoted ${\mathbb{F}_{\mathfrak{p}}}$. By scheme we mean a
Noetherian separated scheme. Actually, it suffices to think of schemes of
finite type over one of the rings just mentioned. In this section $S$ denotes
a fixed base scheme.
This section is setting up a number of axioms describing a triangulated
category $\mathbf{DM}_{\mathrm{gm}}(S)$ of geometric motives over $S$. They
will be used throughout this work. As pointed out in the introduction, the
material of this section is due to Cisinski and Deglise [CD10], who build such
a category of motives using Ayoub’s base change formalism [Ayo07].
###### Axiom 1.1.
(Motivic complexes and functoriality)
* •
There is a triangulated $\mathbb{Q}$-linear category $\mathbf{DM}(S)$. It is
called category of _motivic complexes_ over $S$ (with rational coefficients).
It has all limits and colimits.
* •
(Tensor structure) The category $\mathbf{DM}(S)$ is a triangulated symmetric
monoidal category (see e.g. [Lev98, Part 2, II.2.1.3]). Tensor products
commute with direct sums. The unit of the tensor structure is denoted
$\mathbf{1}_{S}$ or $\mathbf{1}$. Also, there are internal
$\mathrm{Hom}$-objects in $\mathbf{DM}$, denoted $\underline{\mathrm{Hom}}$.
The _dual_ $M^{\vee}$ of an object $M\in\mathbf{DM}(S)$ is defined by
$M^{\vee}:=\underline{\mathrm{Hom}}(M,\mathbf{1})$.
* •
For any map $f:X\rightarrow Y$ of schemes, there are pairs of adjoint functors
$f^{*}:\mathbf{DM}(Y)\leftrightarrows\mathbf{DM}(X):f_{*}$
such that $f^{*}\mathbf{1}_{Y}=\mathbf{1}_{X}$ and, if $f$ is quasi-
projective,
$f_{!}:\mathbf{DM}(X)\leftrightarrows\mathbf{DM}(Y):f^{!}.$
The existence of $f_{!}$ and $f^{!}$ is restricted to quasi-projective maps
since the abstract construction of these functors in Ayoub’s work [Ayo07,
Section 1.6.5], on which Cisinski’s and Déglise’s construction of motives over
general bases [CD10] relies, has a similar restriction.
Recall that an object $X$ in a triangulated category $\mathcal{T}$ closed
under arbitrary direct sums is _compact_ if $\mathrm{Hom}(X,-)$ commutes with
direct sums. The subcategory of $\mathcal{T}$ of compact objects is
triangulated and closed under direct summands (a.k.a. a _thick subcategory_)
[Nee01, Lemma 4.2.4]. The category $\mathcal{T}$ is called _compactly
generated_ if the smallest triangulated subcategory closed under arbitrary
sums containing the compact objects is the whole category $\mathcal{T}$.
###### Axiom 1.2.
(Compact objects) The motive $\mathbf{1}\in\mathbf{DM}(S)$ is compact. The
functors $f^{*}$ and $f^{!}$, whenever defined, and ${\otimes}$ and
$\underline{\mathrm{Hom}}$ preserve compact objects. The same is true for
$f_{*}$ and $f_{!}$ if $f$ is of finite type. The canonical map
$M\rightarrow(M^{\vee})^{\vee}$ is an isomorphism for any compact object $M$.
###### Definition 1.3.
The subcategory of compact objects of $\mathbf{DM}(S)$ is denoted
$\mathbf{DM}_{\mathrm{gm}}(S)$ and called the category of _geometric motives_
over $S$.
For any map $f:X\rightarrow S$ of finite type, the object
$\operatorname{M}_{S}(X):=\operatorname{M}(X):=f_{*}f^{*}\mathbf{1}\in\mathbf{DM}_{\mathrm{gm}}(S)$
is called the _motive_ of $X$ over $S$. By adjunction, $\operatorname{M}$ is a
contravariant functor from schemes of finite type over $S$ to
$\mathbf{DM}_{\mathrm{gm}}(S)$. For any quasi-projective $f:X\rightarrow S$,
the _motive with compact support_ of $X$, $\operatorname{M}_{\mathrm{c}}(X)$,
is defined as $f_{!}f^{*}\mathbf{1}\in\mathbf{DM}_{\mathrm{gm}}(S)$.
The smallest thick subcategory of $\mathbf{DM}(S)$ containing the image of
$\operatorname{M}$ is denoted ${\mathbf{DM}^{\mathrm{eff}}_{\mathrm{gm}}}(S)$
and called the category of _effective geometric motives_. The closure of that
subcategory under all direct sums is called the category of _effective
motives_ , ${\mathbf{DM}_{\mathrm{eff}}}(S)$.
###### Axiom 1.4.
(Tensor product vs. fiber product) The functor $\operatorname{M}$ is an
additive tensor functor, i.e., maps disjoint unions of schemes over $S$ to
direct sums and fiber products of schemes over $S$ to tensor products in
$\mathbf{DM}_{\mathrm{gm}}(S)$.
###### Axiom 1.5.
(Compact generation) The categories $\mathbf{DM}(S)$ and
${\mathbf{DM}_{\mathrm{eff}}}(S)$ are compactly generated.
There is a decomposition
$\operatorname{M}(\mathbb{P}^{1}_{S})=\mathbf{1}\oplus\mathbf{1}(-1)[-2]$,
where
$\mathbf{1}(-1):=\ker(\operatorname{M}(\mathbb{P}^{1}_{S})\rightarrow\operatorname{M}(S)\rightarrow\operatorname{M}(\mathbb{P}^{1}_{S}))[2],$
where the first map is induced by the projection onto the base, the second map
stems from the rational point $0\in\mathbb{P}^{1}_{S}$. The object
$\mathbf{1}(-1)$ is called _Tate object_ or Tate motive. The category
$\mathbf{DM}(S)$, being closed under countable direct sums is pseudo-abelian
[Lev98, Lemma II.2.2.4.8.1], i.e., it contains kernels of projectors. It
follows from the preceding axioms that
$\mathbf{1}(-1)\in{\mathbf{DM}^{\mathrm{eff}}_{\mathrm{gm}}}(S)$.
###### Axiom 1.6.
(Cancellation and Effectivity) In $\mathbf{DM}_{\mathrm{gm}}(S)$ (and thus in
$\mathbf{DM}(S)$), the Tate object $\mathbf{1}(-1)$ has a tensor-inverse
denoted $\mathbf{1}(1)$. For any $M\in\mathbf{DM}(S)$, $n\in\mathbb{Z}$, set
$M(n):=M{\otimes}\mathbf{1}(1)^{{\otimes}n}$. Then there is a canonical
isomorphism called _cancellation_ isomorphism ($n\in\mathbb{Z}$,
$M,N\in\mathbf{DM}(S)$):
$\mathrm{Hom}_{\mathbf{DM}(S)}(M,N)\cong\mathrm{Hom}_{\mathbf{DM}(S)}(M(n),N(n)).$
The smallest tensor subcategory of $\mathbf{DM}_{\mathrm{gm}}(S)$ that
contains ${\mathbf{DM}^{\mathrm{eff}}_{\mathrm{gm}}}(S)$ and $\mathbf{1}(1)$
is $\mathbf{DM}_{\mathrm{gm}}(S)$. In other words,
$\mathbf{DM}_{\mathrm{gm}}(S)$ is obtained from
${\mathbf{DM}^{\mathrm{eff}}_{\mathrm{gm}}}(S)$ by tensor-inverting
$\mathbf{1}(-1)$.
###### Definition 1.7.
Let $M$ be any geometric motive over $S$. We write
$\mathrm{H}^{i}(M):=\mathrm{H}^{i}(S,M):=\mathrm{Hom}_{\mathbf{DM}(S)}(\mathbf{1},M[i])$.
For $M=\operatorname{M}(X)(n)$ for any $X$ over $S$ we also write
$\mathrm{H}^{i}(X,n):=\mathrm{H}^{i}(\operatorname{M}(X)(n))=\mathrm{Hom}_{\mathbf{DM}_{\mathrm{gm}}(S)}(\mathbf{1},\operatorname{M}(X)(n)[i])$.
This is called _motivic cohomology_ of $M$ and $X$, respectively.
###### Axiom 1.8.
(Motivic cohomology vs. K-theory) For any regular scheme $X$, there is an
isomorphism $\mathrm{H}^{i}(X,n)\cong K_{2n-i}(X)^{(n)}_{\mathbb{Q}}$, where
the right hand term denotes the Adams eigenspace of algebraic $K$-theory
tensored with $\mathbb{Q}$ [Qui73].
This is a key property of motives, since algebraic $K$-theory is a universal
cohomology theory in the sense that Chern characters map from algebraic
$K$-theory to any other (reasonable) cohomology theory of algebraic varieties
[Gil80]. For $S$ a perfect field, this axiom is given by [Voe00, Prop. 4.2.9]
and its non-effective analogue. See also [Lev98, Theorem I.III.3.6.12.].
Recall Grothendieck’s category of pure motives $\mathbf{M}_{\sim}(K)$ with
respect to an adequate equivalence relation $\sim$, see e.g. [And04, Section
4]. For rational equivalence they are also called Chow motives, since, for any
smooth projective variety $X$ over a field $K$,
$\mathrm{Hom}_{\mathbf{M}_{\mathrm{rat}}(K)}(\mathbf{1}(-n),h(X))=\mathrm{CH}^{n}(X),$
(1)
where $h(X)$ denotes the Chow motive of $X$ and the right hand term is the
_Chow group_ of cycles of codimension $n$ in $X$. This way, the above axiom
models the fact [Voe00, 2.1.4] that Chow motives are a full subcategory of
$\mathbf{DM}_{\mathrm{gm}}(K)$. Under the embedding
$\mathbf{M}_{\mathrm{rat}}(K)\subset\mathbf{DM}_{\mathrm{gm}}(K)$, $h(X,n)$
maps to $\operatorname{M}(X)(n)[2n]$.
###### Remark 1.9.
We do not need to assume _expressis verbis_ homotopy invariance (i.e.,
$\mathbf{1}\stackrel{{\scriptstyle\cong}}{{\rightarrow}}pr_{*}pr^{*}\mathbf{1}\in\mathbf{DM}_{\mathrm{gm}}(S)$
for $pr:S{\times}\mathbb{A}^{1}\rightarrow S$) nor the projective bundle
formula [Voe00, Prop. 3.5.1]. (Note, however, that $K^{\prime}$-theory does
have such properties.)
###### Axiom 1.10.
(Localization) Let $i:Z\rightarrow S$ be any closed immersion and
$j:V\rightarrow S$ the open complement. The adjointness maps give rise to the
following distinguished triangles in $\mathbf{DM}(S)$:
$j_{!}j^{*}\rightarrow\mathrm{id}\rightarrow i_{*}i^{*},$
$i_{*}i^{!}\rightarrow\mathrm{id}\rightarrow j_{*}j^{*}.$
(In particular, $f_{*}f^{*}\cong\mathrm{id}$, where
$f:X_{\mathrm{red}}\rightarrow X$ denotes the canonical map of the reduced
subscheme structure.) In addition, one has $j^{*}j_{*}=\mathrm{id}$ and
$i^{*}i_{*}=\mathrm{id}$, equivalently $j^{*}i_{*}=i^{*}j_{!}=0$.
It should be emphasized that the localization axiom requires rational
coefficients.
###### Axiom 1.11.
(Purity and base change)
* •
For any quasi-projective map $f$, there is a functorial transformation of
functors $f_{!}\rightarrow f_{*}$. It is an isomorphism if $f$ is projective.
* •
(_Relative purity_): If $f$ is quasi-projective and smooth of constant
relative dimension $d$, there is a functorial (in $f$) isomorphism $f^{!}\cong
f^{*}(d)[2d]$.
* •
(_Absolute purity_): If $i:Z\rightarrow U$ is a closed immersion of
codimension $c$ of two regular schemes $Z$ and $U$, there is a natural
isomorphism $i^{!}\mathbf{1}\cong\mathbf{1}(-c)[-2c]$.
* •
(_Base change_): For any two quasi-projective maps $f$ and $g$ let
$f^{\prime}$ and $g^{\prime}$ denote the pullback maps:
$\textstyle{X^{\prime}{\times}_{X}Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime}}$$\scriptstyle{f^{\prime}}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{X}$
(2)
Then there is canonical isomorphism of functors
$f^{*}g_{!}\stackrel{{\scriptstyle\cong}}{{{\longrightarrow}}}g^{\prime}_{!}f^{\prime*}.$
This axiom is proven by Cisinski & Déglise using Ayoub’s general base change
formalism. See in particular [Ayo07, 1.4.11, 12] for the construction of the
base change map. See also [Lev98, Theorem I.I.2.4.9] for a similar statement
in Levine’s category of motives.
###### Definition 1.12.
Let $f:S\rightarrow{\mathrm{Spec}\text{ }}{\mathbb{Z}}$ be the structural map.
Assume $f$ is quasi-projective. Then
$D(M):=\underline{\mathrm{Hom}}(M,f^{!}\mathbf{1}(1)[2])$ is called _Verdier
dual_ of $M$.
By the preceding axioms, $D$ induces a contravariant endofunctor of
$\mathbf{DM}_{\mathrm{gm}}(S)$. The shift and twist in the definition is
motivated as follows: given some complex analytic space $X$, the Verdier dual
of a sheaf $\mathcal{F}$ on $X$ is defined by
$D(\mathcal{F}):=\underline{{\mathrm{R}}{\mathrm{Hom}}}_{\mathbf{D}(\mathbf{Shv}\left(X\right))}(\mathcal{F},f^{!}\mathbb{Z}),$
where $f$ denotes the projection to a point, see e.g. [Ive86, Ch. VI]. When
$X$ is smooth of dimension $d$, one has
$f^{!}\mathbb{Z}=f^{*}\mathbb{Z}(d)[2d]=\mathbb{Z}(d)[2d]$. A similar fact
holds for $\ell$-adic sheaves (see e.g. [KW01, Section II.7-8]). The above
definition mimics this situation insofar as ${\mathrm{Spec}\text{
}}{\mathbb{Z}}$ is seen as an analogue of a smooth affine curve.
Let us give a number of consequences of the preceding axioms, in particular
purity, base change and localization: in (2), suppose that $f$ is smooth and
$X^{\prime}\subset X$ is a codimension one closed immersion between regular
schemes. Then there is a canonical isomorphism
$i^{!}\operatorname{M}_{S}(X)=\operatorname{M}_{Z}(X{\times}_{S}Z)(-1)[-2].$
(3)
Let $Z\subset X$ be a closed immersion of quasiprojective schemes over $S$.
Then there is a distinguished triangle of motives with compact support
$\operatorname{M}_{\mathrm{c}}(Z)\rightarrow\operatorname{M}_{\mathrm{c}}(X)\rightarrow\operatorname{M}_{\mathrm{c}}(X\backslash
Z).$
Let $S$ be a regular scheme of equidimension $d$ that is affine or projective
over ${\mathrm{Spec}\text{ }}{\mathbb{Z}}$. Let
$f:S\rightarrow{\mathrm{Spec}\text{ }}{\mathbb{Z}}$ be the structural map.
Then $f^{!}\mathbf{1}=\mathbf{1}(d-1)[2d-2]$, as one sees by applying relative
and absolute purity to the maps $p$ and $i$, where $f=p\circ i$ is a
factorization into a affine or projective projection map and a closed
immersion. In particular, the Verdier duality functor on any open subscheme
$S$ of ${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$ is given by
$D_{\mathbf{DM}_{\mathrm{gm}}(S)}(?)=\underline{\mathrm{Hom}}(?,\mathbf{1}(1)[2])$
while on $\mathbf{DM}_{\mathrm{gm}}({\mathbb{F}_{\mathfrak{p}}})$ it is given
by $\underline{\mathrm{Hom}}(?,\mathbf{1})=?^{\vee}$.
###### Axiom 1.13.
(Verdier dual) The Verdier dual functor $D$ exchanges “$!$” and “$*$”
throughout, e.g., there are natural isomorphisms $D(f^{!}M)\cong f^{*}D(M)$
for any quasi-projective map $f:X\rightarrow Y$ and $M\in\mathbf{DM}(Y)$ and
similarly with $f_{!}$ and $f_{*}$.
###### Lemma 1.14.
Let $S$ be such that $f^{!}\mathbf{1}=f^{*}\mathbf{1}(d)[2d]$ for some integer
$d$, where $f:S\rightarrow{\mathrm{Spec}\text{ }}{\mathbb{Z}}$ is the
structural map. For example, $S$ might be regular and affine or projective
over $\mathbb{Z}$ (see above), or smooth over ${\mathrm{Spec}\text{
}}{\mathbb{Z}}$ (purity). Then, for any compact object
$M\in\mathbf{DM}_{\mathrm{gm}}(S)$, the canonical map $M\rightarrow D(D(M))$
is an isomorphism. This will be referred to as _reflexivity_ of Verdier
duality.
###### Proof: .
By Axiom 1.5, it suffices to check it for $M=\pi_{*}\pi^{*}\mathbf{1}$, where
$\pi:X\rightarrow S$ is some map of finite type. In this case it follows for
adjointness reasons and the assumption. ∎
###### Axiom 1.15.
(Resolution of singularities) Let $K$ be a field. As a triangulated additive
tensor category (i.e., closed under triangles, arbitrary direct sums and
tensor product), $\mathbf{DM}(K)$ is generated by $\mathbf{1}(-1)$ and all
$\operatorname{M}(X)$, where $X/K$ is a smooth projective variety.
When $S={\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$, the generators of
$\mathbf{DM}(S)$ are $\mathbf{1}(-1)$,
${i_{\mathfrak{p}}}_{*}\operatorname{M}(X_{\mathfrak{p}})$, and
$\operatorname{M}(X)$, instead, where $X_{\mathfrak{p}}$ is any projective and
smooth variety over ${\mathbb{F}_{\mathfrak{p}}}$, $i_{\mathfrak{p}}$ denotes
the immersion of any closed point ${\mathbb{F}_{\mathfrak{p}}}$ of $S$, and
$X$ is any regular, flat projective scheme over ${\mathcal{O}_{F}}$.
Consequently, the subcategories of compact objects
$\mathbf{DM}_{\mathrm{gm}}(-)$ are generated as a thick tensor subcategory by
the mentioned objects. In Voevodsky’s theory of motives over a field of
characteristic zero, this is [Voe00, Section 4.1]. This uses Hironaka’s
resolution of singularities. Over a field of positive characteristic and
number rings, one has to use de Jong’s resolution result, see [HK06, Lemma
B.4].
We also need a limit property of the generic point. Let $S$ be an open
subscheme of ${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$, let
$\eta:{\mathrm{Spec}\text{ }}{F}\rightarrow S$ be the generic point.
###### Axiom 1.16.
(Generic point) Let $M$ be any geometric motive over $S$. The natural maps
$j_{*}j^{*}M\rightarrow\eta_{*}\eta^{*}M$ give rise to an isomorphism
$\varinjlim j_{*}j^{*}M=\eta_{*}\eta^{*}M$, where the colimit is over all open
subschemes $j:S^{\prime}\rightarrow S$. It induces a distinguished triangle in
$\mathbf{DM}(S)$
$\oplus_{\mathfrak{p}\in
S}{i_{\mathfrak{p}}}_{*}i_{\mathfrak{p}}^{!}M\rightarrow\mathrm{id}\rightarrow\eta_{*}\eta^{*}M,$
(4)
where the sum runs over all closed points $\mathfrak{p}\in S$ and
$i_{\mathfrak{p}}$ is the closed immersion.
## 2 Realizations
One of the main interests in motives lies in the fact that they are explaining
(or are supposed to explain) common phenomena in various cohomology theories.
These cohomology functors are commonly referred to as _realization_ functors.
They typically have the form
$\mathbf{DM}_{\mathrm{gm}}(S)\rightarrow\mathbf{D}^{\mathrm{b}}(\mathcal{C})$,
where $\mathcal{C}$ is an abelian category whose objects are amenable with the
methods of (linear) algebra, such as finite-dimensional vector spaces or
finite-dimensional continuous group representations or constructible sheaves.
For example, let $\ell$ be a prime and let $S$ be either a field of
characteristic different from $\ell$ or a scheme of finite type over
${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}[1/\ell]$. The $\ell$-adic
cohomology maps any scheme $X$ over $S$ to
${\mathrm{R}}{\Gamma}_{\ell}(X):=\mathrm{R}\pi_{*}\pi^{*}{\mathbb{Q}_{{\ell}}}\in\mathbf{D}^{\mathrm{b}}_{c}\left(S,{\mathbb{Q}_{{\ell}}}\right),$
where $\pi:X\rightarrow S$ is the structural map and the right hand category
denotes the “derived” category of constructible
${\mathbb{Q}_{{\ell}}}$-sheaves on $S$ (committing the standard abuse of
notation, see e.g. [KW01, II.6., II.7.]). This functor factors over the
_$\ell$ -adic realization functor_ ([Hub00, p. 772], [Ivo07])
${\mathrm{R}}{\Gamma}_{\ell}:\mathbf{DM}_{\mathrm{gm}}(S)\rightarrow\mathbf{D}^{\mathrm{b}}_{c}(S,{\mathbb{Q}_{{\ell}}})$.
When $S$ is of finite type over ${\mathbb{F}_{\mathfrak{p}}}$, the realization
functor actually maps to
$\mathbf{D}^{\mathrm{b}}_{c,m}(S,{\overline{{\mathbb{Q}_{{\ell}}}}})$, the
full subcategory of complexes $C$ in
$\mathbf{D}^{\mathrm{b}}_{c}(S,{\overline{{\mathbb{Q}_{{\ell}}}}})$ such that
all $\mathrm{H}^{n}(C)$ are mixed sheaves [Del80, 1.2].
Further realization functors include Betti, de Rham and Hodge realization. See
e.g. [Hub00, 2.3.5]. The following axiom says (in particular) that the
$\ell$-adic realization of $\operatorname{M}(X)$ does give the $\ell$-adic
cohomology groups.
###### Axiom 2.1.
(Functoriality and realizations) The $\ell$-adic realization functor commutes
with the six Grothendieck functors $f_{*}$, $f_{!}$, $f^{!}$, $f^{*}$,
${\otimes}$ and $\underline{\mathrm{Hom}}$ (where applicable). For example,
for any map $f:S^{\prime}\rightarrow S$ and any geometric motive $M$ over
$S^{\prime}$:
$(f_{*}M)_{\ell}=f_{*}(M_{\ell}).$
## 3 Interlude: Perverse sheaves over number rings
This section is devoted to a modest extension of $\ell$-adic perverse sheaves
[BBD82] to the situation where the base $S$ is an open subscheme of
${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}[1/\ell]$. It is needed to
formulate Axiom 4.8 for the $\ell$-adic realization of motives over number
rings. In a nutshell, the theory of perverse sheaves on varieties over
${\mathbb{F}_{q}}$ stakes on relative purity, that is
$f^{!}{\mathbb{Z}_{{\ell}}}=f^{*}{\mathbb{Z}_{{\ell}}}(n)[2n]$ for a smooth
map $f$ of relative dimension $n$. The analogous identity for a closed
immersion $i:{\mathrm{Spec}\text{ }}{{\mathbb{F}_{\mathfrak{p}}}}\rightarrow
S$ reads
$i^{!}{\mathbb{Z}_{{\ell}}}=i^{*}{\mathbb{Z}_{{\ell}}}(-1)[-2].$ (5)
It is a reformulation of well-known cohomological properties of the inertia
group: $\mathrm{H}^{1}(I_{\mathfrak{p}},V)=(V(-1))_{I_{\mathfrak{p}}}$ for any
$\ell$-adic module with continuous $I_{\mathfrak{p}}$-action
($\mathfrak{p}\nmid\ell$). All higher group cohomologies of $I_{\mathfrak{p}}$
vanish.
Let $\mathbf{D}^{\mathrm{b}}(S,{\mathbb{Z}_{{\ell}}})$ be the bounded
“derived” category of ${\mathbb{Z}_{{\ell}}}$-sheaves on $S$ as constructed by
Ekedahl [Eke90]. All following constructions can be done for
${\mathbb{Q}_{{\ell}}}$ instead of ${\mathbb{Z}_{{\ell}}}$, as well. We keep
writing $j_{*}$ for the total derived functor, commonly denoted
$\mathrm{R}j_{*}$ etc. However, $\mathrm{R}^{n}j_{*}$ etc. keep their original
meaning.
As in loc. cit., see especially [2.2.10, 2.1.2, 2.1.3, 1.4.10]222In the
sequel, any reference in brackets refers to [BBD82]., one first defines a
notion of stratification, and secondly obtains a $t$-structure on the
subcategory $\mathbf{D}^{\mathrm{b}}_{(\Sigma,L)}(S,{\mathbb{Z}_{{\ell}}})$
that are constructible with respect to a given stratification
$\Sigma=\\{\Sigma_{i}\\}$ and a set $L$ of irreducible lisse sheaves on the
strata. Thirdly, one takes the “limit” over the stratifications. The union of
all $\mathbf{D}^{\mathrm{b}}_{(\Sigma,L)}(S,{\mathbb{Z}_{{\ell}}})$ is the
“derived” category $\mathbf{D}^{\mathrm{b}}_{c}(S,{\mathbb{Z}_{{\ell}}})$ of
constructible sheaves. In order to extend the $t$-structure on the
subcategories to one on
$\mathbf{D}^{\mathrm{b}}_{c}(S,{\mathbb{Z}_{{\ell}}})$, one has to check that
the inclusion
$\mathbf{D}^{\mathrm{b}}_{(\Sigma^{\prime},L^{\prime})}(S,{\mathbb{Z}_{{\ell}}})\rightarrow\mathbf{D}^{\mathrm{b}}_{(\Sigma,L)}(S,{\mathbb{Z}_{{\ell}}})$
is $t$-exact for any refinement of stratifications. Here we employ a different
argument. The proof of [2.1.14, 2.2.11] relies on relative purity for
$\ell$-adic sheaves [SGA73, Exp. XVI, 3.7]. As in the proof of [2.1.14] we
have to check the following: let $\Sigma_{i}\stackrel{{\scriptstyle
a}}{{\rightarrow}}\Sigma^{\prime}_{i}\stackrel{{\scriptstyle
b}}{{{\longrightarrow}}}S$ be the inclusions of some strata and let
$C\in\mathbf{D}^{\mathrm{b},\geq
0}_{(\Sigma^{\prime},L^{\prime})}(S,{\mathbb{Z}_{{\ell}}})$. Then
$C\in\mathbf{D}^{\mathrm{b},\geq 0}_{(\Sigma,L)}(S,{\mathbb{Z}_{{\ell}}})$. We
can assume $\dim\Sigma_{i}=0$, $\dim\Sigma^{\prime}_{i}=1$, since all other
cases are clear. Thus, $b$ is an open immersion. We may also assume for
notational simplicity that $\Sigma_{i}={\mathrm{Spec}\text{
}}{{\mathbb{F}_{\mathfrak{p}}}}$. Let $j$ be the complementary open immersion
to $a$. By definition,
$\mathrm{H}^{n}b^{!}C=b^{!}\mathrm{H}^{n}C=b^{*}\mathrm{H}^{n}C$ is locally
constant and vanishes for $n<-1$. In the parlance of Galois modules this means
that, viewed as a $\pi_{1}(\Sigma^{\prime}_{i})$-representation, the action of
the inertia group $I_{\mathfrak{p}}\subset\pi_{1}(\Sigma^{\prime}_{i})$ on
that sheaf is trivial. Thus
$a^{!}\mathrm{H}^{n}b^{*}C=a^{*}(\mathrm{R}^{1}j_{*}j^{*}\mathrm{H}^{n}b^{*}C)[-2]=\mathrm{H}^{1}(I_{\mathfrak{p}},\mathrm{H}^{n}b^{*}C)[-2]=a^{*}\mathrm{H}^{n}b^{*}C(-1)[-2].$
(We have used $\mathfrak{p}\nmid\ell$ at this point.) The spectral sequence
$\mathrm{H}^{p-2}a^{*}\mathrm{H}^{q}b^{!}C(-1)=\mathrm{H}^{p}a^{!}\mathrm{H}^{q}b^{!}C\Rightarrow\mathrm{H}^{n}a^{!}b^{!}C$
is such that the left hand term vanishes for $p\neq 2$ since $a^{*}$ is exact
w.r.t. the standard $t$-structure. It also vanishes for $q<-1$ by the above.
Hence the right hand term vanishes for $n=p+q<1$. A fortiori it vanishes for
$n<-\dim{\mathbb{F}_{\mathfrak{p}}}=0$.
Objects in the heart of this $t$-structure on
$\mathbf{D}^{\mathrm{b}}_{c}(S,{\mathbb{Z}_{{\ell}}})$ are called _perverse
sheaves_ on $S$. For example ${\mathbb{Z}_{{\ell}}}[1]$ and
$i_{*}{\mathbb{Z}_{{\ell}}}$ for any immersion $i$ of a closed point are
perverse sheaves on $S$. The _Verdier dual_ of any
$\mathcal{C}\in\mathbf{D}^{\mathrm{b}}_{c}(S,{\mathbb{Z}_{{\ell}}})$ is
defined by $D(C):=\underline{\mathrm{Hom}}(C,{\mathbb{Z}_{{\ell}}}(1)[2])$. As
above, we have dropped “$\mathrm{R}$” from the notation, so that this
$\underline{\mathrm{Hom}}$ means what is usually denoted
$\underline{{\mathrm{R}}{\mathrm{Hom}}}$.
###### Lemma 3.1.
Let $j:S^{\prime}\rightarrow S$ be an open immersion and $i:Z\rightarrow S$ a
closed immersion. Let $\eta:{\mathrm{Spec}\text{ }}{F}\rightarrow S$ be the
generic point. Then $j_{*}$, $j_{!}$, $i_{*}$, $\eta^{*}[-1]$, $j^{*}$ and $D$
are $t$-exact, while $i^{*}$ ($i^{!}$) is of cohomological amplitude $[-1,0]$
($[0,1]$), in particular right-exact (left- exact, respectively). Finally, the
$t$-structure on $\mathbf{D}^{\mathrm{b}}_{c}(S,{\mathbb{Z}_{{\ell}}})$ is
non-degenerate.
###### Proof: .
The only non-formal statement is the exactness of $j_{*}$. The corresponding
precursor result [4.1.10] is a reformulation of [SGA72, Th. 3.1., Exp. XIV],
which says for any affine map $j:X\rightarrow Y$ over schemes over a field
$K$, and any (honest) sheaf $\mathcal{F}$ which is torsion (prime to
$\operatorname{char}K$)
$d(\mathrm{R}^{q}j_{*}\mathcal{F}))\leq d(\mathcal{F})-q$
where
$d(\mathcal{G}):=\sup\\{\dim{\overline{\\{x\\}}},\mathcal{G}_{{\overline{x}}}\neq
0\\}$ for any sheaf $\mathcal{G}$. In our situation, we are given a locally
constant sheaf $\mathcal{F}$ on $S^{\prime}$ whose torsion is prime to all
characteristics of $S$. The conclusion of the theorem also holds for $j$, as
follows from the cohomological dimension of $I_{\mathfrak{p}}$, which is one.
∎
Let $\mathcal{F}$ be any perverse sheaf on $S^{\prime}$. Following [1.4.22],
let the _intermediate extension_ $j_{!*}\mathcal{F}$ be the image of the map
$j_{!}\mathcal{F}\rightarrow j_{*}\mathcal{F}$ of perverse sheaves on $S$. As
in [2.1.11] one sees that it can be calculated in terms of the good truncation
with respect to the standard $t$-structure:
$j_{!*}\mathcal{F}=\tau_{\leq-1}^{can}j_{*}\mathcal{F}.$ If
$\mathcal{F}=\mathcal{G}[1]$, where $\mathcal{G}$ is a lisse (honest) sheaf on
$S^{\prime}$, this gives $(\mathrm{R}^{0}j_{*}\mathcal{G})[1]$.
## 4 Mixed motives
Throughout this section, let $S={\mathrm{Spec}\text{ }}{F}$ or
${\mathrm{Spec}\text{ }}{{\mathbb{F}_{\mathfrak{p}}}}$ or an open subscheme of
${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$.
This section formulates a number of axioms concerning weights and the motivic
$t$-structure on triangulated categories of motives over $S$. In contrast to
the axioms listed in Section 1, the axioms mentioned in this section are wide
open, so it might be more appropriate to call them conjectures instead.
###### Axiom 4.1.
(Motivic $t$-structure and cohomological dimension) The category of geometric
motives $\mathbf{DM}_{\mathrm{gm}}(S)$ has a non-degenerate $t$-structure
[BBD82, Def. 1.3.1] called _motivic $t$-structure_. Its heart is denoted
$\mathbf{MM}(S)$. Objects of $\mathbf{MM}(S)$ are called _mixed motives_ over
$S$.
For any $M\in\mathbf{DM}_{\mathrm{gm}}(S)$, there are $a$, $b\in\mathbb{Z}$
such that $\tau_{\leq a}M=\tau_{\geq b}M=0$. Here and in the sequel,
$\tau_{\leq-}$ and $\tau_{\geq-}$ denote the truncation functors with respect
to the motivic $t$-structure.
The _cohomological dimension_ of
$\mathbf{DM}_{\mathrm{gm}}({\mathbb{F}_{\mathfrak{p}}})$ and
$\mathbf{DM}_{\mathrm{gm}}(F)$ is 0 and 1, respectively, in the sense that
$\mathrm{Hom}_{\mathbf{DM}({\mathbb{F}_{\mathfrak{p}}})}(M,N[i])=0$
for all mixed motives $M,N$ over ${\mathbb{F}_{\mathfrak{p}}}$ and $i>0$ and
similarly for mixed motives over $F$ and $i>1$. (For $i<0$ the term vanishes
by the $t$-structure axioms.)
The $t$-structures are such that over $S=\mathrm{Spec}\text{ }F$ or
${\mathrm{Spec}\text{ }}{{\mathbb{F}_{\mathfrak{p}}}}$,
$\mathbf{1}\in\mathbf{MM}(S)$, while for an open subscheme
$S\subset{\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$,
$\mathbf{1}[1]\in\mathbf{MM}(S)$.
The existence of the motivic $t$-structure on $\mathbf{DM}_{\mathrm{gm}}(K)$
satisfying the axioms listed in this section is part of the general motivic
conjectural framework, see e.g. [Beĭ87a, App. A], [And04, Ch. 21]. The idea of
building a triangulated category of motives and descending to mixed motives by
means of a $t$-structure is due to Deligne. The existence of a motivic
$t$-structure on $\mathbf{DM}_{\mathrm{gm}}(K)$ is only known in low
dimensions: the subcategory of Artin motives, i.e., motives of zero-
dimensional varieties, carries such a $t$-structure [Voe00, Section 3.4.]. By
loc. cit., [Org04], the subcategory of $\mathbf{DM}_{\mathrm{gm}}(K)$
generated by motives of smooth varieties of dimension $\leq 1$ is equivalent
to the bounded derived category of 1-motives [Del74, Section 10] up to
isogeny. Finally, if $K$ is a field satisfying the Beilinson-Soulé vanishing
conjecture, such as a finite field or a number field, the category of Artin-
Tate motives over $K$ enjoys a motivic $t$-structure [Lev93, Wil]. The results
on Artin-Tate motives are generalized to bases $S$ which are open subschemes
of ${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$ in [Scha].
The conjecture about the cohomological dimension is due to Beilinson. A
(fairly weak) evidence for this conjecture is the cohomological dimension of
Tate motives over $F$ and ${\mathbb{F}_{\mathfrak{p}}}$, which is one and
zero, respectively. This follows from vanishing properties of $K$-theory of
these fields.
The normalization in the last item is merely a matter of bookkeeping, but is
motivated by similar shifts in perverse sheaves (Section 3). The existence of
a motivic $t$-structure is not expected to hold for motives with integral
coefficients.
We do not (need to) assume that the canonical functor
$\mathbf{D}^{\mathrm{b}}(\mathbf{MM}(S))\rightarrow\mathbf{DM}_{\mathrm{gm}}(S)$
is an equivalence of categories or, equivalently [Beĭ87b, Lemma 1.4.],
$\operatorname{Ext}^{i}_{\mathbf{MM}(S)}(A,B)=\mathrm{Hom}_{\mathbf{DM}_{\mathrm{gm}}(S)}(A,B[i])$
for all mixed motives $A$ and $B$.
###### Axiom 4.2.
(Exactness properties) Let $S\subset{\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}$ be an open subscheme, let $i:{\mathrm{Spec}\text{
}}{{\mathbb{F}_{\mathfrak{p}}}}\rightarrow{\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}$ be a closed point, $j:U\rightarrow S$ an open immersion
and $\eta:\mathrm{Spec}\text{ }F\rightarrow S$ the generic point.
Then $j^{*}=j^{!}$, $\eta^{*}[-1]$, $i_{*}$, $j_{*}$ and $j_{!}$ are exact
with respect to the motivic $t$-structures on the involved categories of
geometric motives. Further, $i^{*}$ is right-exact, more precisely it maps
objects in cohomological degree $0$ to degrees $[-1,0]$. Dually, $i^{!}$ has
cohomological amplitude $[0,1]$. Verdier duality $D$ is “anti-exact”, i.e.,
maps objects in positive degrees to ones in negative degrees and vice versa.
The axiom is motivated by the same exactness properties in the situation of
perverse sheaves over ${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}[1/\ell]$
(Section 3). The corresponding exactness properties of the above functors on
Artin-Tate motives, where the motivic $t$-structure is available, is
established in [Scha].
###### Definition 4.3.
The cohomology functor with respect to the motivic $t$-structure on
$\mathbf{DM}_{\mathrm{gm}}(S)$ is denoted ${{{}^{\mathrm{p}}}\mathrm{H}}^{*}$.
For any scheme $X/S$, we write
$\mathrm{h}^{i}(X,n):={{{}^{\mathrm{p}}}\mathrm{H}}^{i}\operatorname{M}_{S}(X)(n).$
###### Axiom 4.4.
(Hard Lefschetz) Let $X_{\eta}/F$ be smooth and projective of constant
dimension $d_{\eta}$. Let $i\leq d_{\eta}$ and $a$ any integer. Then, taking
the $(d_{\eta}-i)$-fold cup product with the cycle class of a hyperplane
section with respect to an embedding of $X_{\eta}$ into some projective space
over $F$ yields an isomorphism (“hard Lefschetz isomorphism”)
$\mathrm{h}^{i}(X_{\eta},a)\stackrel{{\scriptstyle\cong}}{{{\longrightarrow}}}\mathrm{h}^{2d_{\eta}-i}(X_{\eta},d_{\eta}-i+a).$
###### Axiom 4.5.
(Decomposition of smooth projective varieties) Let $X/S$ be smooth and
projective. In $\mathbf{DM}_{\mathrm{gm}}(S)$, there is a non-canonical
isomorphism
$\phi_{X}:\operatorname{M}(X)\cong\bigoplus_{n}\mathrm{h}^{n}(X)[-n].$
For open subschemes $S\subset{\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$, this
isomorphism is compatible with pullbacks along all closed points
$i:{\mathrm{Spec}\text{ }}{{\mathbb{F}_{\mathfrak{p}}}}\rightarrow S$ in the
following sense: let $X_{\mathfrak{p}}$ be the fiber of $X$ over
${\mathbb{F}_{\mathfrak{p}}}$, and let $\psi$ be the isomorphism making the
following diagram commutative. Its left hand isomorphism is an instance of
base change.
$\textstyle{i^{*}\operatorname{M}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{*}\phi_{X}}$$\scriptstyle{\cong}$$\textstyle{\oplus_{n}i^{*}\mathrm{h}^{n}(X)[-n]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{\operatorname{M}(X_{\mathfrak{p}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{X_{\mathfrak{p}}}}$$\textstyle{\oplus_{m}\mathrm{h}^{m}(X_{\mathfrak{p}})[-m]}$
Then $\psi$ respects the direct summands, i.e., induces isomorphisms
$i^{*}\mathrm{h}^{n}(X)[-n]\cong\mathrm{h}^{n-1}(X_{\mathfrak{p}})[-n+1].$
###### Remark 4.6.
Axiom 4.4 is part of the classical conjectural framework of motives (see e.g.
[And04, Section 5.2, Prop. 5.2.5.1]). It implies the decomposition of Axiom
4.5 [Del94]. The latter is used in Lemma 5.11, which in turn is crucial in
Section 6. The index shift in Axiom 4.5 is due to the normalization in Axiom
4.1: for $S={\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$ and a closed point $i$
as above, take for example $X=S$,
$\operatorname{M}(S)=\mathbf{1}=\mathrm{h}^{1}(S)[-1]$ (sic) and
$i^{*}\operatorname{M}(S)=\mathbf{1}_{{\mathbb{F}_{\mathfrak{p}}}}=\mathrm{h}^{0}(\mathrm{Spec}\text{
}{\mathbb{F}_{\mathfrak{p}}})$.
Consider any smooth projective variety $X_{\eta}$ over $F$. Using that the
cohomological dimension of $\mathbf{DM}_{\mathrm{gm}}(F)$ is one, the term
$\mathrm{H}^{2n}(X_{\eta},n)$, that is identified with
$\mathrm{CH}^{n}(X_{\eta})_{\mathbb{Q}}$ by Axiom 1.8, appears in an exact
sequence of the following form
$0\rightarrow\mathrm{H}^{1}(\mathrm{h}^{2n-1}(X_{\eta},n))\rightarrow\mathrm{H}^{2n}(X_{\eta},n)\rightarrow\mathrm{H}^{0}(\mathrm{h}^{2n}(X_{\eta},n))\rightarrow
0.$
Consider the cycle class map from the $m$-th Chow group to $\ell$-adic
cohomology of $X_{\eta}$,
$\mathrm{CH}^{m}(X_{\eta})_{\mathbb{Q}}\rightarrow\mathrm{H}^{2m}(X_{\eta},{\mathbb{Q}_{{\ell}}}(m))$
[Mil80, VI.9]. The groups
$\mathrm{CH}^{m}(X_{\eta})_{\mathbb{Q},\mathrm{hom}}$ and
$\mathrm{CH}^{m}(X_{\eta})_{\mathbb{Q}}/\mathrm{hom}$ are by definition the
kernel and the image (seen as a quotient of the Chow group) of this map.
According to standard conjectures, homological equivalence should not depend
on the choice of the cohomology theory, since they are all supposed to agree
with numerical equivalence, see e.g. [And04, Chapter 5].
###### Axiom 4.7.
(Morphisms in $\mathbf{MM}(F)$ and Chow groups) For such $X_{\eta}$, the
sequence above identifies with
$0\rightarrow\mathrm{CH}^{n}(X_{\eta})_{\mathbb{Q},\mathrm{hom}}\rightarrow\mathrm{CH}^{n}(X_{\eta})_{\mathbb{Q}}\rightarrow\mathrm{CH}^{n}(X_{\eta})_{\mathbb{Q}}/\mathrm{hom}\rightarrow
0.$
Let either $S$ be a field and let $C$ stand for the $\ell$-adic realization
(in case $\text{char }S\neq\ell$), Betti, de Rham or absolute Hodge
realization or let $S\subset{\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}[1/\ell]$ be an open subscheme and let $C$ be the
$\ell$-adicrealization. We write
${\mathrm{R}}{\Gamma}_{C}:\mathbf{DM}_{\mathrm{gm}}(S)\rightarrow\mathbf{D}^{\mathrm{b}}(\mathcal{C})$
for the realization functor, where $\mathbf{D}^{\mathrm{b}}(\mathcal{C})$ is
understood as a placeholder of the target category of $C$. (We abuse the
notation insofar as that target category is not a derived category in the
strict sense when $C$ is the $\ell$-adic realization.) For all realizations
over a field, this category is endowed with the usual $t$-structure, e.g. on
$\mathbf{D}^{\mathrm{b}}_{c}(K,{\mathbb{Q}_{{\ell}}})$ for $\ell$-adic
realization. When $S$ is an open subscheme of ${\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}[1/\ell]$ and $C$ is the $\ell$-adic realization, we take
the perverse $t$-structure on
$\mathbf{D}^{\mathrm{b}}_{c}(S,{\mathbb{Q}_{{\ell}}})$ defined in Section 3.
Using this, we have the following axiom:
###### Axiom 4.8.
(Exactness of realization functors) Realization functors
${\mathrm{R}}{\Gamma}_{C}$ are exact with respect to the motivic $t$-structure
on $\mathbf{DM}_{\mathrm{gm}}(S)$. Equivalently, as the $t$-structure on
$\mathbf{D}^{\mathrm{b}}(\mathcal{C})$ is non-degenerate,
${\mathrm{R}}{\Gamma}_{C}({{{}^{\mathrm{p}}}\mathrm{H}}^{0}M)={{{}^{\mathrm{p}}}\mathrm{H}}^{0}{\mathrm{R}}{\Gamma}_{C}(M)$
for any geometric motive $M$ over $S$. On the left,
${{{}^{\mathrm{p}}}\mathrm{H}}^{0}$ denotes the cohomology functor belonging
to the motivic $t$-structure on $\mathbf{DM}_{\mathrm{gm}}(S)$, while on the
right hand side it is the one belonging to the $t$-structure on
$\mathbf{D}^{\mathrm{b}}(\mathcal{C})$.
This axiom is, if fairly loosely, motivated by a similar fact in the theory of
mixed Hodge modules: let $X$ be any complex algebraic variety. Then, under the
faithful “forgetful functor” from the derived category of mixed Hodge modules
to the derived category of constructible sheaves with rational coefficients
$\mathbf{D}^{\mathrm{b}}(\mathbf{MHM}(X))\rightarrow\mathbf{D}^{\mathrm{b}}_{c}(X,\mathbb{Q})$
the category $\mathbf{MHM}(X)$ corresponds to perverse sheaves on $X$.
Recall that in an abelian category $\mathcal{C}$, a morphism
$f:(X,W^{*})\rightarrow(Y,W^{*})$ between filtered objects is called _strict_
if $f(W^{n}X)=f(X)\cap W^{n}Y$ for all $n$.
###### Axiom 4.9.
(Weights) Any mixed motive $M$ over $S$ has a functorial finite exhaustive
separated filtration $W_{*}M$ called _weight filtration_ , i.e., a sequence of
subobjects in the abelian category $\mathbf{MM}(S)$
$0=W_{a}M\subset W_{a+1}M\subset\dots\subset W_{b}M=M.$
Any morphism between mixed motives is strict with respect to the weight
filtration.
Given two motives of pure weight, their tensor product is pure; its weight is
the sum of the two individual weights. In particular, the unit motive
$\mathbf{1}$ is pure of weight zero.
Let
${\mathrm{R}}{\Gamma}_{C}:\mathbf{DM}_{\mathrm{gm}}(S)\rightarrow\mathbf{D}^{\mathrm{b}}(\mathcal{C})$
be any realization functor that has a notion of weights (such as the
$\ell$-adic realization when $S={\mathrm{Spec}\text{
}}{{\mathbb{F}_{\mathfrak{p}}}}$ or the Hodge realization when
$S=\mathrm{Spec}\text{ }\mathbb{Q}$). Then
$\operatorname{gr}_{n}^{W}{\mathrm{R}}{\Gamma}_{C}(M)={\mathrm{R}}{\Gamma}_{C}(\operatorname{gr}_{n}^{W}M)$
for any mixed motive $M$ over $S$.
###### Definition 4.10.
For any $M\in\mathbf{MM}(S)$, we write $\operatorname{wt}(M)$ for the (finite)
set of integers $n$ such that $\operatorname{gr}^{W}_{n}M\neq 0$. For
$M\in\mathbf{DM}_{\mathrm{gm}}(S)$, define
$\operatorname{wt}(M):=\cup_{i\in\mathbb{Z}}\operatorname{wt}({{{}^{\mathrm{p}}}\mathrm{H}}^{i}(M))-i$.
###### Axiom 4.11.
(Preservation of weights) Let $f:X\rightarrow S$ be a quasi-projective map.
Then the functors $f_{!}f^{*}$ preserve negativity of weights, i.e., given a
geometric motive $M$ over $S$ with weights $\leq 0$, $f_{!}f^{*}M$ also has
weights $\leq 0$. Dually, $f_{*}f^{!}$ preserves positive weights.
In the particular case $S\subset{\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$
(open), let $j:U\rightarrow S$ and $\eta:{\mathrm{Spec}\text{ }}{F}\rightarrow
S$ be an open immersion into $S$ and the generic point of $S$, respectively.
Let $i:{\mathrm{Spec}\text{ }}{{\mathbb{F}_{\mathfrak{p}}}}\rightarrow S$ be a
closed point. Then, $i^{*}$ and $j_{!}$ preserve negativity of weights and
dually for $i^{!}$ and $j_{*}$. Finally, $j^{*}$ and $\eta^{*}$ both preserve
both positivity and negativity of weights.
The preceding weight axioms are motivated by the very same properties of
$\ell$-adic perverse sheaves on schemes over $\mathbb{C}$ or finite fields
[BBD82, 5.1.14], number fields [Hub97] as well as Hodge structures [Del74, Th.
8.2.4] and Hodge modules (see [PS08, Chapter 14.1] for a synopsis). In these
settings, actually $f_{!}$ and $f^{*}$ preserve negative weights, but we do
not need weights for motives over more general bases than the ones above. The
weight formalism we require is stronger than the one provided by the
differential-graded interpretation of $\mathbf{DM}_{\mathrm{gm}}$ over a field
[Bon09] or [BV08, 6.7.4].
###### Example 4.12.
For any projective (smooth) scheme $X$ of finite type over $S$, the weights of
$\mathrm{h}^{i}(X)(n)$ are $\leq i-2n$ ($\geq i-2n$, respectively).
###### Axiom 4.13.
(Mixed vs. pure motives) For any field $K$, the subcategory of pure objects in
$\mathbf{MM}(K)$ identifies with $\mathbf{M}_{\mathrm{num}}(K)$, the category
of numerical pure motives over $K$.
As a consequence of the weight filtration, every mixed motive is obtained in
finitely many steps by taking extensions of motives in
$\mathbf{M}_{\mathrm{num}}(K)$.
## 5 Motives over number rings
In the following sections we assume the axioms of Sections 1, 2, and 4. Unless
explicitly mentioned otherwise, let $S$ be an open subscheme of
${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$, let $i:{\mathrm{Spec}\text{
}}{{\mathbb{F}_{\mathfrak{p}}}}\rightarrow{\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}$ be a closed point, $j:S^{\prime}\rightarrow S$ an open
subscheme and $\eta:\mathrm{Spec}\text{ }F\rightarrow S$ the generic point.
This section derives a number of basic results about motives over $S$ from the
axioms spelled out above. We define and study the _intermediate extension_
$j_{!*}:\mathbf{MM}(S^{\prime})\rightarrow\mathbf{MM}(S)$ in analogy to
perverse sheaves (Definition 5.3). An “explicit” set of generators of
$\mathbf{DM}_{\mathrm{gm}}(S)$ (Proposition 5.7) is obtained using $j_{!*}$.
We introduce a notion of _smooth motives_ (Definition 5.8), which should be
thought of as analogs of lisse sheaves. Using this notion, we extend the
intermediate extension to a functor $\eta_{!*}$ spreading out motives over $F$
with a certain smoothness property to motives over $S$, cf. Definition 5.14.
This functor will be the main technical tool in dealing with $f$-cohomology in
Section 6. In Lemma 5.16 we express the $\ell$-adic realization of motives of
the form $j_{!*}M$ in sheaf-theoretic language.
### 5.1 Cohomological dimension
The following is an immediate consequence of Axiom 4.2:
###### Lemma 5.1.
For any scheme $X$ over $S$ we have
$\eta^{*}[-1]\mathrm{h}^{i}(X,n)=\mathrm{h}^{i-1}(X{\times}_{S}F,n)$.
The following lemma parallels (and is a consequence of) Axiom 4.1.
###### Lemma 5.2.
The cohomological dimension of $\mathbf{DM}_{\mathrm{gm}}(S)$ is two, that is
to say, for any two mixed motives $M,N$ over $S$,
$\mathrm{Hom}_{\mathbf{DM}_{\mathrm{gm}}(S)}(M,N[i])=0$
for all $i>2$. In particular $\mathrm{H}^{i}(M)$ vanishes for $|i|>1$.
###### Proof: .
Apply $\mathrm{Hom}(M,-)$ to the localization triangle
$\oplus_{\mathfrak{p}\in
S}{i_{\mathfrak{p}}}_{*}i_{\mathfrak{p}}^{!}N\rightarrow
N\rightarrow\eta_{*}\eta^{*}N$ of Axiom 1.16, where $i_{\mathfrak{p}}$ are the
immersions of the closed points of $S$. The terms adjacent to
$\mathrm{Hom}(M,N[i])$ are
$\mathrm{Hom}(M,\oplus_{\mathfrak{p}}{i_{\mathfrak{p}}}_{*}i_{\mathfrak{p}}^{!}N[i])=\oplus_{\mathfrak{p}}\mathrm{Hom}(i_{\mathfrak{p}}^{*}M,i_{\mathfrak{p}}^{!}N[i])$
(as $M$ is compact) and
$\mathrm{Hom}(M,\eta_{*}\eta^{*}N[i])=\mathrm{Hom}(\eta^{*}M,\eta^{*}N[i])$.
The latter term vanishes for $i>1$ since $\eta^{*}[-1]$ is exact and the
cohomological dimension of $\mathbf{DM}_{\mathrm{gm}}(F)$ is one.
To deal with the former term, we have to take into account that
$i_{\mathfrak{p}}^{!}$ and $i_{\mathfrak{p}}^{*}$ are not $t$-exact, but of
cohomological amplitude $[0,1]$ and $[-1,0]$, respectively. By decomposing
$i_{\mathfrak{p}}^{!}N$ into its ${{{}^{\mathrm{p}}}\mathrm{H}}^{1}$\- and
${{{}^{\mathrm{p}}}\mathrm{H}}^{0}$-part and similarly with
$i_{\mathfrak{p}}^{*}M$ and using that the cohomological dimension of
$\mathbf{DM}_{\mathrm{gm}}({\mathbb{F}_{\mathfrak{p}}})$ is zero, the term
vanishes for $i>2$. Using general $t$-structure properties, the second claim
is a particular case of the first one. ∎
### 5.2 Intermediate extension
###### Definition 5.3.
(Motivic analog of [BBD82, Def. 1.4.22]) The _intermediate extension_ $j_{!*}$
of some mixed motive $M$ over $S^{\prime}$ is defined as
$j_{!*}M:=\mathrm{im}(j_{!}M\rightarrow j_{*}M).$
The image is taken in the abelian category $\mathbf{MM}(S)$, using the
exactness of $j_{!}$ and $j_{*}$, Axiom 4.2.
###### Remark 5.4.
Let $i:Z\rightarrow S$ be the complement of $j$. The localization triangles
(Axiom 1.10) and cohomological amplitude of $i^{*}$ (Axiom 4.2) yield short
exact sequences in $\mathbf{MM}(S)$
$0\rightarrow i_{*}{{{}^{\mathrm{p}}}\mathrm{H}}^{-1}i^{*}j_{*}M\rightarrow
j_{!}M\rightarrow j_{!*}M\rightarrow 0,$ (6) $0\rightarrow j_{!*}M\rightarrow
j_{*}M\rightarrow i_{*}{{{}^{\mathrm{p}}}\mathrm{H}}^{0}i^{*}j_{*}M\rightarrow
0.$ (7)
These triangles are the same as for perverse sheaves in the situation that the
analog of Axiom 4.2, [BBD82, 4.1.10], is applicable.
###### Lemma 5.5.
* •
Given any mixed motive $M$ over $S^{\prime}$, $j_{!*}M$ is, up to a unique
isomorphism, the unique mixed extension of $M$ (i.e., an object $X$ in
$\mathbf{MM}(S)$ such that $j^{*}X=M$) not having nonzero subobjects or
quotients of the form $i_{*}N$, where $i:Z\rightarrow S$ is the closed
complement of $j$ and $N$ is a mixed motive on $Z$.
* •
For any two composable open immersions $j_{1}$ and $j_{2}$ we have
${j_{1}}_{!*}\circ{j_{2}}_{!*}=({j_{1}}\circ j_{2})_{!*}$.
* •
$j_{!*}$ commutes with duals, i.e., $D(j_{!*}-)\cong j_{!*}D(-)$.
###### Proof: .
The proofs of the same facts for perverse sheaves [BBD82, Cor. 1.4.25,
2.1.7.1] carry over to this setting. The first statement easily implies the
last one. ∎
###### Lemma 5.6.
(compare [BBD82, Cor. 5.3.2]) The intermediate extension $j_{!*}$ preserves
weights of pure objects: given a pure motive $M\in\mathbf{MM}(S^{\prime})$,
$j_{!*}M$ is pure of the same weight.
###### Proof: .
This is a consequence of the triangles (6), (7) and the strictness of the
weight filtration. ∎
The following proposition makes precise the intuition that any motive $M$ over
$S$ should be reconstructed by its generic fiber (over $F$) and a finite
number of special fibers (over various ${\mathbb{F}_{\mathfrak{p}}}$).
###### Proposition 5.7.
As a thick subcategory of $\mathbf{DM}(S)$, $\mathbf{DM}_{\mathrm{gm}}(S)$ is
generated by motives of the form
* •
$i_{*}\operatorname{M}(X_{\mathfrak{p}})(m)$, where
$X_{\mathfrak{p}}/{\mathbb{F}_{\mathfrak{p}}}$ is smooth projective,
$m\in\mathbb{Z}$ and $i:{\mathrm{Spec}\text{
}}{{\mathbb{F}_{\mathfrak{p}}}}\rightarrow S$ is any closed point and
* •
$j_{!*}j^{*}\mathrm{h}^{k}(X,m)$, where $X$ is regular, flat and projective
over $S$ with smooth generic fiber, and $j:S^{\prime}\rightarrow S$ is such
that $X{\times}_{S}S^{\prime}$ is smooth over $S^{\prime}$ and $k$ and $m$ are
arbitrary.
###### Proof: .
Let $\mathcal{D}\subset\mathbf{DM}_{\mathrm{gm}}(S)$ be the thick category
generated by the objects in the statement. By resolution of singularities over
$S$ (Axiom 1.15), $\mathbf{DM}_{\mathrm{gm}}(S)$ is the thick subcategory of
$\mathbf{DM}(S)$ generated by objects
$i_{*}\operatorname{M}(X_{\mathfrak{p}})(m)$ and $\operatorname{M}(X)(m)$,
where $X_{\mathfrak{p}}$ and $X$ are as in the statement and $m\in\mathbb{Z}$.
It is therefore sufficient to see $M:=\operatorname{M}(X)\in\mathcal{D}$. Let
$j:S^{\prime}\rightarrow S$ be such that $X_{S^{\prime}}$ is smooth over
$S^{\prime}$. By 1.10 it is enough to show $j_{*}j^{*}M\in\mathcal{D}$, since
motives over finite fields are already covered. Applying the truncations with
respect to the motivic $t$-structure to $j_{*}j^{*}M$ and exactness of
$j_{*}$, $j^{*}$ (Axiom 4.2) shows that we may deal with
$j_{*}j^{*}\mathrm{h}^{k}(X,m)$ for all $k$ instead of $j_{*}j^{*}M$. (Only
finitely many $k$ yield a nonzero term by Axiom 4.1.) By Remark 5.4, there is
a short exact sequence of mixed motives
$0\rightarrow j_{!*}j^{*}\mathrm{h}^{k}(X,m)\rightarrow
j_{*}j^{*}\mathrm{h}^{k}(X,m)\rightarrow
i_{*}{{{}^{\mathrm{p}}}\mathrm{H}}^{0}i^{*}j_{*}j^{*}\mathrm{h}^{k}(X,m)\rightarrow
0.$
Here $i$ is the complement of $j$. The left and right hand motives are in
$\mathcal{D}$, hence so is the middle one. ∎
### 5.3 Smooth motives
The notion of smooth motives (a neologism leaning on lisse sheaves) is a
technical stepstone for the definition of the generic intermediate extension
$\eta_{!*}$, cf. Definition 5.14. Roughly speaking, smoothness for mixed
motives $M$ means that $i^{*}M$ and $i^{!}M$ do not intermingle in the sense
that their cohomological degrees are disjoint.
###### Definition 5.8.
Let $M$ be a geometric motive over $S$. It is called _smooth_ if for any
closed point $i:{\mathrm{Spec}\text{
}}{{\mathbb{F}_{\mathfrak{p}}}}\rightarrow S$ there is an isomorphism
$i^{!}M\cong i^{*}M(-1)[-2].$
$M$ is called _generically smooth_ if there is an open (non-empty) immersion
$j:S^{\prime}\rightarrow S$ such that $j^{*}M$ is smooth.
Let $X/S$ be a scheme with smooth generic fiber $X_{\eta}$. Then
$\operatorname{M}_{S}(X)$ is a generically smooth motive.
The isomorphism in Definition 5.8 is not required to be canonical in any
sense. Therefore, the subcategory of smooth motives is _not_ triangulated in
$\mathbf{DM}_{\mathrm{gm}}(S)$.
###### Lemma 5.9.
Let $M$ be a smooth mixed motive over $S$. Let $i:Z\rightarrow S$ be proper
closed subscheme, let $j:S^{\prime}\rightarrow S$ be its complement. Then
$i^{!}M=({{{}^{\mathrm{p}}}\mathrm{H}}^{1}i^{!}M)[-1]$ and dually
$i^{*}M=({{{}^{\mathrm{p}}}\mathrm{H}}^{-1}i^{*}M)[1]$.
###### Proof: .
By assumption $i^{!}M\cong i^{*}M(-1)[-2]$. By Axiom 4.2, the left hand side
of that isomorphism is concentrated in degrees $[0,1]$. The right hand side is
in degrees $[1,2]$. This shows
$i^{!}M={{{}^{\mathrm{p}}}\mathrm{H}}^{1}(i^{!}M)[-1]$ by Axiom 4.1 and
similarly for $i^{*}M$. ∎
The following is the key relation of smooth motives and the intermediate
extension. Note the similarity with Lemma 5.15.
###### Lemma 5.10.
Let $M$ be a smooth mixed motive over $S$. Then $M$ is canonically isomorphic
to $j_{!*}j^{*}M$.
###### Proof: .
Let $i:Z\rightarrow S$ be the complement of $j$. Given any $i_{*}N\subset M$
with $N\in\mathbf{MM}(Z)$, we apply the left-exact functor $i^{!}$ and see
$N\subset{{{}^{\mathrm{p}}}\mathrm{H}}^{0}(i^{!}M)\stackrel{{\scriptstyle\ref{lemm_basic}}}{{=}}0$.
Quotients of $M$ of the form $i_{*}N$ are treated dually. We now invoke Lemma
5.5. ∎
###### Lemma 5.11.
Let $X$ be any smooth projective scheme over $S$. Set
$M:=\operatorname{M}(X)$. Then all
$\mathrm{h}^{n}X={{{}^{\mathrm{p}}}\mathrm{H}}^{n}M$ are smooth.
###### Proof: .
Let $f_{m,n}$ be the $(m,n)$-component of the bottom isomorphism making the
following commutative:
$\textstyle{i^{!}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong,\text{
see
(\ref{eqn_purityExample})}}$$\scriptstyle{\cong,\ref{axio_hardLefschetz}}$$\textstyle{i^{*}M(-1)[-2]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong,\ref{axio_hardLefschetz}}$$\textstyle{\oplus_{m}A_{m}:=\oplus
i^{!}({{{}^{\mathrm{p}}}\mathrm{H}}^{m}M)[-m]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\oplus_{n}B_{n}:=\oplus
i^{*}({{{}^{\mathrm{p}}}\mathrm{H}}^{n}M)(-1)[-n-2].}$
We claim $f_{m,n}=0$ for all $m\neq n$, from which the lemma follows. By Axiom
4.5 we have $B_{n}\cong\mathrm{h}^{n-1}(X_{\mathfrak{p}})[-n-1](-1)$. Using
this and the reflexivity of the Verdier dual functor, we obtain an isomorphism
$A_{m}\cong({{{}^{\mathrm{p}}}\mathrm{H}}^{m+1}i^{!}M)[-1-m]$. Hence $B_{n}$
is concentrated in cohomological degree $n+1$, while $A_{m}$ is in degree
$n+2$. (The a priori bounds of Axiom 4.2 would be $[m,m+1]$ and $[n+1,n+2]$,
respectively.) As the cohomological dimension of motives over
${\mathbb{F}_{\mathfrak{p}}}$ is zero (Axiom 4.1), the only way for
$f_{m,n}\neq 0$ is $m=n$. ∎
### 5.4 Generic intermediate extension
###### Lemma 5.12.
(Spreading out morphisms) Given two geometric motives $M$ and $M^{\prime}$
over $S$ together with a map
$\phi_{\eta}:\eta^{*}M\rightarrow\eta^{*}M^{\prime}$, there is an open
subscheme $j:S^{\prime}\subset S$ and a map
$\phi_{S^{\prime}}:j^{*}M\rightarrow j^{*}M^{\prime}$ which extends
$\phi_{\eta}$. Any two such extensions agree when restricted to a possibly
smaller open subscheme. In particular, if $\phi_{\eta}$ is an isomorphism,
then $\phi_{S^{\prime}}$ is an isomorphism for sufficiently small
$S^{\prime}$.
###### Proof: .
The adjunction map $M\rightarrow\eta_{*}\eta^{*}M$ and $\eta_{*}\phi_{\eta}$
give a map $M\rightarrow\eta_{*}\eta^{*}M^{\prime}$, hence by (4) a map
$M\rightarrow\oplus_{\mathfrak{p}}{i_{\mathfrak{p}}}_{*}i_{\mathfrak{p}}^{!}M^{\prime}[1]$.
As $M$ is compact, it factors over a finite sum $\oplus_{\mathfrak{p}\in
T}{i_{\mathfrak{p}}}_{*}i_{\mathfrak{p}}^{!}M^{\prime}[1]$. Let
$j:S^{\prime}\rightarrow S$ be the complement of the points in $T$. The map
$M\rightarrow\eta_{*}\eta^{*}M^{\prime}$ factors over ${j}_{*}j^{*}M^{\prime}$
and gives a map $j^{*}M\rightarrow j^{*}M^{\prime}$ which extends
$\phi_{\eta}$. The first claim is shown.
For the unicity of the extension, we may assume that $\phi_{\eta}$ is zero,
and show that $\phi_{S^{\prime}}$ is zero for some suitable $S^{\prime}$. This
is the same argument as before: the map $M\rightarrow{j}_{*}j^{*}M^{\prime}$
constructed in the previous step factors over $\oplus_{\mathfrak{p}\in
S^{\prime}}{i_{\mathfrak{p}}}_{*}i_{\mathfrak{p}}^{!}M^{\prime}$, since
$M\rightarrow\eta_{*}\eta^{*}M^{\prime}$ is zero. By compacity of $M$, only
finitely many primes in the sum contribute to the map, discarding these yields
the claim.
If $\phi_{\eta}$ is an isomorphism, $\psi_{\eta}:=\phi_{\eta}^{-1}$ can be
extended to some $\psi_{S^{\prime}}$. As both
$\phi_{S^{\prime}}\circ\psi_{S^{\prime}}$ and $\mathrm{id}_{S^{\prime}}$
extend $\mathrm{id}_{\eta}$, they agree on some possibly smaller open
subscheme of $S$ and similarly with $\psi_{S^{\prime}}\circ\phi_{S^{\prime}}$.
∎
###### Remark 5.13.
The lemma shows the full faithfulness of the functor
$\varinjlim_{S^{\prime}\subset
S}\mathbf{DM}_{\mathrm{gm}}(S^{\prime})\stackrel{{\scriptstyle\eta^{*}}}{{{\longrightarrow}}}\mathbf{DM}_{\mathrm{gm}}(F).$
Its essential surjectivity is a consequence of Axiom 1.5, so we have an
equivalence. However, we will stick to the more basic language of colimits in
$\mathbf{DM}(S)$ instead of colimits of the categories of geometric motives.
###### Definition 5.14.
Let $M_{\eta}\in\mathbf{DM}_{\mathrm{gm}}(F)$ be a motive such that there
exists a generically smooth mixed motive $M$ over $S$ (Definition 5.8) with
$\eta^{*}M\cong M_{\eta}$. Then the _generic intermediate extension_
$\eta_{!*}M_{\eta}$ is defined as
$\eta_{!*}M_{\eta}:=j_{!*}j^{*}M$
where $j:S^{\prime}\rightarrow S$ is an open immersion such that $j^{*}M$ is
smooth.
This is independent of the choices of $j$ and $M$ (Lemmas 5.10, 5.12) and
functorial (5.12). For a mixed, non-smooth motive $M$, there need not be a map
$j_{!*}j^{*}M\rightarrow M$. Therefore, $\varinjlim j_{!*}j^{*}M$ does not
make sense unless there is some smoothness constraint on $M_{\eta}$.
### 5.5 Intermediate extension and $\ell$-adic realization
This subsection deals with the interplay of the (generic) intermediate
extension functor on mixed motives and the $\ell$-adic realization. In this
subsection, $S$ is an open subscheme of ${\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}[1/\ell]$. The following lemma is well-known.
###### Lemma 5.15.
Let $\mathcal{F}$ be an étale (honest) locally constant sheaf on $S$. Let
$\eta:{\mathrm{Spec}\text{ }}{F}\rightarrow{\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}[1/\ell]$ be the generic point. Then the canonical map
$\mathcal{F}\rightarrow\mathrm{R}^{0}\eta_{*}\eta^{*}\mathcal{F}$ is an
isomorphism.
###### Lemma 5.16.
Let $M$ be a mixed motive over $S^{\prime}$. Let $j:S^{\prime}\rightarrow S$
be an open immersion. Then
$(j_{!*}M)_{\ell}=j_{!*}(M_{\ell}).$
Let $i$ be the complementary closed immersion to $j:S^{\prime}\rightarrow S$
and let $\eta^{\prime}$ and $\eta$ be the generic point of $S^{\prime}$ and
$S$, respectively. If $M$ is additionally smooth, one has
$(i^{*}j_{!*}M)_{\ell}=i^{*}j_{!*}M_{\ell}=i^{*}(\mathrm{R}^{0}\eta_{*}\eta^{\prime*}M_{\ell}[-1])[1].$
To clarify the statement, note that the $\ell$-adic realization of $M$ is a
perverse sheaf on $S^{\prime}$ by Axiom 4.8. Thus, $j_{!*}$ (Section 3) can be
applied to it.
###### Proof: .
The first statement follows from Axiom 2.1, the definition of $j_{!*}$ and the
exactness of ${\mathrm{R}}{\Gamma}_{\ell}$ (Axiom 4.8).
Let now $M$ be mixed and smooth over $S^{\prime}$. As $M_{\ell}$ is a perverse
sheaf by 4.8, there is an open immersion
$j^{\prime}:S^{\prime\prime}\rightarrow S^{\prime}$ such that
$j^{\prime*}M_{\ell}[-1]$ is a locally constant (honest) sheaf on
$S^{\prime\prime}$. As $M$ is smooth we know from Lemmas 5.5 and 5.10
$i^{*}j_{!*}M=i^{*}(j\circ j^{\prime})_{!*}j^{\prime*}M.$
By the interpretation of $(j\circ j^{\prime})_{!*}$ in terms of
$\mathrm{R}^{0}(j\circ j^{\prime})_{*}$ (Section 3) we have
$(i^{*}j_{!*}M)_{\ell}=i^{*}j_{!*}M_{\ell}=i^{*}(\mathrm{R}^{0}(j\circ
j^{\prime})_{*}j^{\prime*}M_{\ell}[-1])[1]\stackrel{{\scriptstyle\text{\ref{lemm_locconst}}}}{{=}}i^{*}(\mathrm{R}^{0}\eta_{*}\eta^{\prime*}M_{\ell}[-1])[1].$
∎
## 6 $f$-cohomology
### 6.1 $f$-cohomology via non-ramification
For any place $\mathfrak{p}$ of $F$, let $F_{\mathfrak{p}}$ be the completion,
$G_{\mathfrak{p}}$ the local Galois group. For finite places,
$I_{\mathfrak{p}}$ denotes the inertia group. For brevity, we will usually
write $\mathrm{H}^{*}(M)$ for $\mathrm{H}^{*}(S,M)$, where $M$ is any motive
over some base $S$.
###### Definition 6.1.
[BK90, Section 3] Let $V$ be a finite-dimensional $\ell$-adic vector space,
endowed with a continuous action of $G_{\mathfrak{p}}$, where $\mathfrak{p}$
is a finite place of $F$ not over $\ell$. Set
$\mathrm{H}^{i}_{f}(F_{\mathfrak{p}},V):=\left\\{\begin{array}[]{ll}\mathrm{H}^{0}(F_{\mathfrak{p}},V)&i=0\\\
\ker\mathrm{H}^{1}(F_{\mathfrak{p}},V)\rightarrow\mathrm{H}^{1}(I_{\mathfrak{p}},V)&i=1\\\
0&\text{else.}\end{array}\right.$
###### Remark 6.2.
If $\mathfrak{p}$ lies over $\ell$, the definition is completed by
$\mathrm{H}^{1}_{f}(F_{\mathfrak{p}},V):=\ker\mathrm{H}^{1}(F_{\mathfrak{p}},V)\rightarrow\mathrm{H}^{1}(F_{\mathfrak{p}},B_{crys}{\otimes}V)$,
where $B_{crys}$ denotes the ring of $\mathfrak{p}$-adic periods [FM87]. We
will disregard this case throughout.
###### Lemma 6.3.
Let $\eta_{\mathfrak{p}}:\mathrm{Spec}\text{
}F_{\mathfrak{p}}\rightarrow\mathrm{Spec}\text{
}{\mathcal{O}_{F_{\mathfrak{p}}}}$ be the generic point of the completion of
${\mathcal{O}_{F}}$ at $\mathfrak{p}$. Using the above notation, for
$\mathfrak{p}$ not over $\ell$, there is a canonical isomorphism
$\mathrm{H}^{1}_{f}(F_{\mathfrak{p}},V)\cong\mathrm{H}^{1}({\mathcal{O}_{F_{\mathfrak{p}}}},\mathrm{R}^{0}{\eta_{\mathfrak{p}}}_{*}V)$.
(The right hand side denotes $\ell$-adic cohomology over
${\mathcal{O}_{F_{\mathfrak{p}}}}$.)
###### Proof: .
Let $U_{n}$ be an $\ell^{n}$-torsion sheaf on $F_{\mathfrak{p}}$. In this
proof, we will write
$\mathrm{H}^{1}_{f}(F_{\mathfrak{p}},U_{n}):=\ker\mathrm{H}^{1}(F_{\mathfrak{p}},U_{n})\rightarrow\mathrm{H}^{1}(I_{\mathfrak{p}},U_{n}),$
i.e., the same definition as for ${\mathbb{Q}_{{\ell}}}$-sheaves is done. The
${\mathbb{Q}_{{\ell}}}$-sheaf $V$ is, by definition, of the form
$U{\otimes}_{\mathbb{Z}_{{\ell}}}{\mathbb{Q}_{{\ell}}}$, where $U=(U_{n})_{n}$
is a projective system of $\mathbb{Z}/\ell^{n}$-sheaves. By definition
$\mathrm{H}^{1}(F_{\mathfrak{p}},V)=\varprojlim_{n\in\mathbb{N}}\mathrm{H}^{1}(F_{\mathfrak{p}},U_{n}){\otimes}{\mathbb{Q}_{{\ell}}}$
and similarly for $\mathrm{H}^{1}(I_{\mathfrak{p}},V)$. Both $\varprojlim_{n}$
and $-{\otimes}_{\mathbb{Z}_{{\ell}}}{\mathbb{Q}_{{\ell}}}$ are left-exact
functors, so one has
$\mathrm{H}^{1}_{f}(F_{\mathfrak{p}},V)=\left(\varprojlim_{n}\mathrm{H}^{1}_{f}(F_{\mathfrak{p}},U_{n})\right){\otimes}{\mathbb{Q}_{{\ell}}}.$
Thus we may henceforth assume that $V$ is a $\mathbb{Z}/\ell^{n}$-sheaf.
Recall the description of étale sheaves on ${\mathcal{O}_{F_{\mathfrak{p}}}}$
from [Mil80, II.3.12, II.3.16]. Let $i:{\mathrm{Spec}\text{
}}{{\mathbb{F}_{\mathfrak{p}}}}\rightarrow\mathrm{Spec}\text{
}{\mathcal{O}_{F_{\mathfrak{p}}}}$ be the closed point. As
${\mathcal{O}_{F_{\mathfrak{p}}}}$ is a henselian ring [Mil80, Prop. I.4.5],
for any sheaf $\mathcal{F}$ on $\mathrm{Spec}\text{
}{\mathcal{O}_{F_{\mathfrak{p}}}}$, the global sections depend only on the
special fiber and
$\Gamma_{\mathrm{Spec}\text{ }F_{\mathfrak{p}}}=\Gamma_{\mathrm{Spec}\text{
}{\mathcal{O}_{F_{\mathfrak{p}}}}}\circ({\eta_{\mathfrak{p}}}_{*})=\Gamma_{\mathrm{Spec}\text{
}{\mathcal{O}_{F_{\mathfrak{p}}}}}\circ(i_{*}i^{*}{\eta_{\mathfrak{p}}}_{*}).$
These functors can be interpreted using group cohomology:
$\Gamma_{\mathrm{Spec}\text{ }{\mathcal{O}_{F_{\mathfrak{p}}}}}\circ
i_{*}=\Gamma_{{\mathbb{F}_{\mathfrak{p}}}}$ and
$(-)^{I_{\mathfrak{p}}}=i^{*}{\eta_{\mathfrak{p}}}_{*}$ (loc. cit.). The
Hochschild-Serre spectral sequence for
$(-)^{G_{\mathfrak{p}}}=(-)^{\mathrm{Gal}({\mathbb{F}_{\mathfrak{p}}})}\circ(-)^{I_{\mathfrak{p}}}$
can be translated to
$\mathrm{H}^{p}(\mathrm{Spec}\text{
}{\mathcal{O}_{F_{\mathfrak{p}}}},i_{*}i^{*}\mathrm{R}^{q}{\eta_{\mathfrak{p}}}_{*}V)\Rightarrow\mathrm{H}^{n}(F_{\mathfrak{p}},V).$
In addition we have the Leray spectral sequence
$\mathrm{H}^{p}(\mathrm{Spec}\text{
}{\mathcal{O}_{F_{\mathfrak{p}}}},\mathrm{R}^{q}{\eta_{\mathfrak{p}}}_{*}V)\Rightarrow\mathrm{H}^{n}(F_{\mathfrak{p}},V).$
The exact sequence of low degrees of the Hochschild-Serre sequence maps to the
sequence below:
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(\mathrm{Spec}\text{
}{\mathcal{O}_{F_{\mathfrak{p}}}},\mathrm{R}^{0}{\eta_{\mathfrak{p}}}_{*}V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(F_{\mathfrak{p}},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{\mathrm{H}^{0}(\mathrm{Spec}\text{
}{\mathcal{O}_{F_{\mathfrak{p}}}},\mathrm{R}^{1}{\eta_{\mathfrak{p}}}_{*}V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}_{f}(F_{\mathfrak{p}},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(F_{\mathfrak{p}},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(I_{\mathfrak{p}},V)}$
As
$\mathrm{H}^{0}(\mathrm{Gal}({\mathbb{F}_{\mathfrak{p}}}),\mathrm{H}^{1}(I_{\mathfrak{p}},V))\subset\mathrm{H}^{1}(I_{\mathfrak{p}},V)$
and
$\Gamma_{\mathcal{O}_{F_{\mathfrak{p}}}}=\Gamma_{\mathcal{O}_{F_{\mathfrak{p}}}}\circ
i_{*}i^{*}$, the right hand map is injective, therefore there is a unique
isomorphism between the left hand terms making the diagram commutative. ∎
In order to proceed to a global level, the following definition is done:
###### Definition 6.4.
[FPR94, II.1.3] Given an $\ell$-adic continuous representation $V$ of
$G=\mathrm{Gal}(F)$, define $\mathrm{H}^{i}_{f}(F,V)$ to be such that the
following diagram is cartesian. In the lower row, $V$ is considered a
$G_{\mathfrak{p}}=\mathrm{Gal}(F_{\mathfrak{p}})$-module by restriction.
$\textstyle{\mathrm{H}^{i}_{f}(F,V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{i}(F,V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod\mathrm{H}^{i}_{f}(F_{\mathfrak{p}},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod\mathrm{H}^{i}(F_{\mathfrak{p}},V)}$
The product ranges over all finite places $\mathfrak{p}$ of $F$. We define
$\mathrm{H}^{i}_{f,\backslash\operatorname{crys}}(F,V)$ similarly, except that
in the lower row of the preceding diagram only places $\mathfrak{p}$ that do
not lie over $\ell$ occur.
###### Lemma 6.5.
Let $V$ be an $\ell$-adic étale sheaf on $\mathrm{Spec}\text{ }F$. Then there
is a natural isomorphism
$\mathrm{H}^{i}_{f,\backslash\operatorname{crys}}(F,V)\cong\mathrm{H}^{1}({\mathcal{O}_{F}}[1/\ell],\mathrm{R}^{0}{\eta}_{*}V).$
###### Proof: .
By the same argument as in the previous proof, we may assume that $V$ is a
sheaf of $\mathbb{Z}/\ell^{n}$-modules, since the isomorphism we are going to
establish is natural in $V$ and
$\mathrm{H}^{i}_{f,\backslash\operatorname{crys}}(F,V)=\ker\mathrm{H}^{i}(F,V)\rightarrow\prod_{\mathfrak{p}\nmid\ell}\left(\mathrm{H}^{i}(F_{\mathfrak{p}},V)/\mathrm{H}^{i}_{f}(F_{\mathfrak{p}},V)\right).$
Consider the following cartesian diagram ($\mathfrak{p}\nmid\ell$)
$\textstyle{\mathrm{Spec}\text{
}F_{\mathfrak{p}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{\mathfrak{p}}}$$\scriptstyle{b}$$\textstyle{\mathrm{Spec}\text{
}{\mathcal{O}_{F_{\mathfrak{p}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{\mathrm{Spec}\text{
}{\mathbb{F}_{\mathfrak{p}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{\mathfrak{p}}}$$\scriptstyle{=}$$\textstyle{\mathrm{Spec}\text{
}F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{\mathrm{Spec}\text{
}{\mathcal{O}_{F}}[1/\ell]}$$\textstyle{\mathrm{Spec}\text{
}{\mathbb{F}_{\mathfrak{p}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$
In the derived category of $\mathbb{Z}/\ell^{n}$-sheaves on
${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}[1/\ell]$, there is a triangle
$\mathrm{R}^{0}\eta_{*}V\rightarrow\mathrm{R}\eta_{*}V\rightarrow\mathrm{R}^{1}\eta_{*}[-1]V$.
Likewise,
$\mathrm{R}^{0}{\eta_{\mathfrak{p}}}_{*}b^{*}V\rightarrow{\mathrm{R}\eta_{\mathfrak{p}}}_{*}b^{*}V\rightarrow\mathrm{R}^{1}{\eta_{\mathfrak{p}}}_{*}b^{*}V[-1]$.
(We have used $\mathfrak{p}\nmid\ell$, since the inertia group has
cohomological dimension bigger than one for $\mathfrak{p}|\ell$.) This yields
exact horizontal sequences, the vertical maps are adjunction maps
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(\mathrm{Spec}\text{
}{\mathcal{O}_{F}}[1/\ell],{\eta}_{*}V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(F,V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{0}(\mathrm{Spec}\text{
}{\mathcal{O}_{F}}[1/\ell],\mathrm{R}^{1}\eta_{*}V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{\mathfrak{p}\nmid\ell}\mathrm{H}^{1}({\mathcal{O}_{F_{\mathfrak{p}}}},\mathrm{R}^{0}{\eta_{\mathfrak{p}}}_{*}b^{*}V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{\mathfrak{p}\nmid\ell}\mathrm{H}^{1}(F_{\mathfrak{p}},b^{*}V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{\mathfrak{p}\nmid\ell}\mathrm{H}^{0}({\mathcal{O}_{F_{\mathfrak{p}}}},\mathrm{R}^{1}{\eta_{\mathfrak{p}}}_{*}b^{*}V)}$
We will show that $\alpha$ is injective. Hence, the left square is cartesian
and by definition and Lemma 6.3 the claim is shown. Indeed, $\alpha$ factors
as
$\mathrm{H}^{0}({\mathcal{O}_{F}}[1/\ell],\mathrm{R}^{1}\eta_{*}V)\subset\prod_{\mathfrak{p}\nmid\ell}\mathrm{H}^{0}({\mathbb{F}_{\mathfrak{p}}},i_{\mathfrak{p}}^{*}\mathrm{R}^{1}\eta_{*}V)\rightarrow\prod_{\mathfrak{p}\nmid\ell}\mathrm{H}^{0}({\mathcal{O}_{F_{\mathfrak{p}}}},\mathrm{R}^{1}{\eta_{\mathfrak{p}}}_{*}b^{*}V)\left(\stackrel{{\scriptstyle\cong}}{{=}}\prod_{\mathfrak{p}\nmid\ell}\mathrm{H}^{0}({\mathbb{F}_{\mathfrak{p}}},i_{\mathfrak{p}}^{*}\mathrm{R}^{1}{\eta_{\mathfrak{p}}}_{*}b^{*}V)\right).$
using
$i^{*}\mathrm{R}^{1}\eta_{*}V=i_{\mathfrak{p}}^{*}a^{*}\mathrm{R}^{1}\eta_{*}V=i_{\mathfrak{p}}^{*}\mathrm{R}^{1}{\eta_{\mathfrak{p}}}_{*}b^{*}V$
(the latter isomorphism is found e.g. in [Mil08, Example 12.4]). ∎
###### Definition 6.6.
[Beĭ87a, Remark 4.0.1.b], [BK90, Conj. 5.3], [Fon92, Section 6.5], [FPR94,
III.3.1.3] Let $M_{\eta}$ be a mixed motive over $F$. Let, similarly to
Definition 6.4, $\mathrm{H}^{i}_{f}(M_{\eta})$ be defined such that the
following diagram, in which the bottom products are taken over all primes
$\ell$, is cartesian. As usual, ${M_{\eta}}_{\ell}$ is the $\ell$-adic
realization, seen as a $G$-module.
$\textstyle{\mathrm{H}^{i}_{f}(F,{M_{\eta}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{i}(F,{M_{\eta}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{\ell}\mathrm{H}^{i}_{f}(F,{M_{\eta}}_{\ell})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{\ell}\mathrm{H}^{i}(F,{M_{\eta}}_{\ell})}$
Again, to rid ourselves from crystalline questions at $\mathfrak{p}|\ell$, we
define $\mathrm{H}^{i}_{f,\backslash\operatorname{crys}}(F,{M_{\eta}})$ by
replacing $\prod_{\ell}\mathrm{H}^{i}_{f}(F,{M_{\eta}}_{\ell})$ in the bottom
row by
$\prod_{\ell}\mathrm{H}^{i}_{f,\backslash\operatorname{crys}}(F,{M_{\eta}}_{\ell})$.
We are now going to exhibit an interpretation of $f$-cohomology thus defined
in terms of the generic intermediate extension $\eta_{!*}$. Recall that we are
assuming in this section the axioms of Sections 1, 2, and 4. Mixed motives are
needed to even define $\eta_{!*}$. Moreover, for the comparison result, we
need to assume the following conjecture.
###### Conjecture 6.7.
(Beilinson) Let $X/{\mathbb{F}_{q}}$ be smooth and projective. Up to torsion,
numerical and rational equivalence agree on $X$.
As rational and numerical equivalence are the finest and the coarsest adequate
equivalence relation, respectively [And04, Section 3.2], this conjecture
implies that all adequate equivalence relations, in particular homological
equivalence agree.
###### Lemma 6.8.
Let $N$ be any mixed motive over ${\mathbb{F}_{\mathfrak{p}}}$. Conjecture 6.7
implies that the $\ell$-adic realization map
$\mathrm{H}^{0}({\mathbb{F}_{\mathfrak{p}}},N)\rightarrow\mathrm{H}^{0}({\mathbb{F}_{\mathfrak{p}}},N_{\ell}):=N_{\ell}^{\mathrm{Gal}({\mathbb{F}_{\mathfrak{p}}})}$
is injective.
###### Proof: .
We can assume that $N$ is pure of weight $0$, since both depend only on
$\operatorname{gr}_{0}^{W}N$ (Axiom 4.9). Under Conjecture 6.7 all adequate
equivalence relations agree, so that we may regard $N$ as a pure motive with
respect to any adequate equivalence relation. As the injectivity is stable
under taking direct summands, we may assume $N=h(X,n)$ for $X$ smooth and
projective over ${\mathbb{F}_{\mathfrak{p}}}$, by definition of pure motives
and Axiom 4.13. The left hand side is given by $\mathrm{CH}^{n}(X)$, so the
map is injective by Conjecture 6.7. ∎
###### Theorem 6.9.
Let $M$ be a generically smooth mixed motive over ${\mathcal{O}_{F}}$
(Definition 5.8). Set $\eta^{*}M[-1]=:M_{\eta}$. Assuming Conjecture 6.7,
there is a natural isomorphism
$\mathrm{H}^{0}({\mathcal{O}_{F}},\eta_{!*}\eta^{*}M)\stackrel{{\scriptstyle\cong}}{{{\longrightarrow}}}\mathrm{H}^{1}_{f,\backslash\operatorname{crys}}(F,M_{\eta}).$
###### Proof: .
Notice that $\eta_{!*}\eta^{*}M$ is well-defined by the assumptions. We want
to show that there is a cartesian commutative diagram
$\textstyle{\mathrm{H}^{0}(\eta_{!*}\eta^{*}M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{0}(\eta_{*}\eta^{*}M)=\mathrm{H}^{1}(M_{\eta})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{\ell}\mathrm{H}^{1}_{f,\backslash\ell}(F,{M_{\eta}}_{\ell})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{\ell}\mathrm{H}^{1}(F,{M_{\eta}}_{\ell})}$
Let $j:U\rightarrow{\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$ be any open
immersion such that $j^{*}M$ is smooth. We have
$\eta_{!*}\eta^{*}M=j_{!*}j^{*}M$. The left hand term of the exact sequence
$\oplus_{\mathfrak{p}\in
U}\mathrm{H}^{0}({i_{\mathfrak{p}}}_{*}i_{\mathfrak{p}}^{!}M)\rightarrow\mathrm{H}^{0}(j_{*}j^{*}M)\rightarrow\mathrm{H}^{0}(\eta_{*}\eta^{*}M)\rightarrow\oplus_{\mathfrak{p}\in
U}\mathrm{H}^{1}({i_{\mathfrak{p}}}_{*}i_{\mathfrak{p}}^{!}M)$
induced by (4) vanishes as $i_{\mathfrak{p}}^{!}M$ is concentrated in
cohomological degree $1$ for $\mathfrak{p}\in U$ (Lemma 5.9). Any
$a\in\mathrm{H}^{0}(\eta^{*}M)$ maps to a finite sub-sum of
$\oplus_{\mathfrak{p}\in{\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}}\mathrm{H}^{1}({i_{\mathfrak{p}}}_{*}i_{\mathfrak{p}}^{!}M)$,
so letting $j$ be the open complement of these points, $a$ lies in (the image
of) $\mathrm{H}^{0}(j_{*}j^{*}M)$:
$\mathrm{H}^{0}(\eta^{*}M)=\varinjlim_{j:U\rightarrow{\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}\atop j^{*}M\text{ smooth}}\mathrm{H}^{0}(j_{*}j^{*}M).$
By Lemma 6.10 below, the map
$\mathrm{H}^{0}(j_{!*}j^{*}M)\rightarrow\mathrm{H}^{0}(j_{*}j^{*}M)\rightarrow\mathrm{H}^{0}(\eta^{*}M)$
is injective. Therefore, taking the colimit over all $U$ such that $M|_{U}$ is
smooth, the exact localization sequence
$0\rightarrow\mathrm{H}^{0}(j_{!*}j^{*}M)\rightarrow\mathrm{H}^{0}(j_{*}j^{*}M)\rightarrow\oplus_{\mathfrak{p}\notin
U}\mathrm{H}^{0}({{{}^{\mathrm{p}}}\mathrm{H}}^{0}i_{\mathfrak{p}}^{*}j_{*}j^{*}M)$
stemming from (7) gives
$0\rightarrow\mathrm{H}^{0}(j_{!*}j^{*}M)\rightarrow\mathrm{H}^{0}(\eta_{*}\eta^{*}M)\rightarrow\oplus_{\mathfrak{p}}\mathrm{H}^{0}({{{}^{\mathrm{p}}}\mathrm{H}}^{0}i_{\mathfrak{p}}^{*}{j_{\mathfrak{p}}}_{*}j_{\mathfrak{p}}^{*}M).$
Here $j_{\mathfrak{p}}$ is the complementary open immersion to
$i_{\mathfrak{p}}$ and the direct sum is over all (finite) places
$\mathfrak{p}$ of ${\mathcal{O}_{F}}$. We have
$i_{\mathfrak{p}}^{*}\eta_{*}\eta^{*}M=i_{\mathfrak{p}}^{*}{j_{\mathfrak{p}}}_{*}j_{\mathfrak{p}}^{*}M$,
so the top sequence in the following commutative diagram is exact:
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{0}(j_{!*}j^{*}M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(M_{\eta})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\oplus_{\mathfrak{p}}\mathrm{H}^{0}({{{}^{\mathrm{p}}}\mathrm{H}}^{0}i_{\mathfrak{p}}^{*}\eta_{*}\eta^{*}M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{\ell}\mathrm{H}^{0}((j_{\ell}^{*}j_{!*}j^{*}M)_{\ell})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{\ell}\mathrm{H}^{1}({M_{\eta}}_{\ell})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{\ell}\prod_{\mathfrak{p}\nmid\ell}\mathrm{H}^{0}(({{{}^{\mathrm{p}}}\mathrm{H}}^{0}i_{\mathfrak{p}}^{*}\eta_{*}\eta^{*}M)_{\ell})}$
(8)
The lower row denotes $\ell$-adic cohomology over ${\mathcal{O}_{F}}[1/\ell]$,
$F$, and the various ${\mathbb{F}_{\mathfrak{p}}}$, respectively. Moreover,
$j_{\ell}:{\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}[1/\ell]\rightarrow{\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}$ is the open immersion. The remainder of the proof
consists in the following steps: we show that the diagram is commutative, that
the second row is exact, identify its lower leftmost term and show that the
rightmost vertical map is injective. This implies that the left square is
cartesian, hence the theorem follows.
We write $\iota$ and $\iota_{\ell}$ for the open immersions of
$U\cap{\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}[1/\ell]$ into
${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}[1/\ell]$ and $U$, respectively. By
Lemma 5.16 and the exactness of $j_{\ell}^{*}$ we have
$(j_{\ell}^{*}j_{!*}j^{*}M)_{\ell}=(\iota_{!*}\iota^{*}j_{\ell}^{*}M)_{\ell}=\iota_{!*}\iota^{*}(j_{\ell}^{*}M)_{\ell}.$
Thus (8) is commutative since every term at the bottom just involves the
$\ell$-adic realization of the motive above it, restricted to
${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}[1/\ell]$.
The exactness of the bottom row is shown separately for each $\ell$, so $\ell$
is fixed for this argument. By the characterization just mentioned,
$\iota_{!*}\iota^{*}(j_{\ell}^{*}M)_{\ell}$ does not change when shrinking
$U$, since $j_{!*}j^{*}M$ is independent of $U$ (as soon as $M$ is smooth over
$U$). On the other hand, by the exactness of the $\ell$-adic realization
functor (Axiom 4.8) $(j_{\ell}^{*}M)_{\ell}$ is a perverse sheaf on
${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}[1/\ell]$, so is a locally constant
sheaf (shifted into degree $-1$) on a suitable small open subscheme. Hence we
may assume that $\iota^{*}(j_{\ell}^{*}M)_{\ell}$ is a locally constant sheaf
in degree $-1$. By Section 3,
$\iota_{!*}\iota^{*}(j_{\ell}^{*}M)_{\ell}=(\mathrm{R}^{0}\iota_{*}\iota^{*}(j_{\ell}^{*}M)_{\ell}[-1])[+1]$,
so the lower row is the exact sequence belonging to the distinguished triangle
of sheaves on ${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}[1/\ell]$
$\mathrm{R}^{0}\eta_{\ell}{}_{*}(M_{\eta})_{\ell}\rightarrow\mathrm{R}\eta_{\ell}{}_{*}(M_{\eta})_{\ell}\rightarrow(\mathrm{R}^{1}\eta_{\ell}{}_{*}(M_{\eta})_{\ell})[-1].$
Here $\eta_{\ell}:{\mathrm{Spec}\text{ }}{F}\rightarrow{\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}[1/\ell]$ is the generic point. By Lemma 6.5 and Lemma
5.15, the factors in the lower left-hand term of (8) agree with
$\mathrm{H}^{1}_{f,\backslash\operatorname{crys}}(F,\eta^{*}M_{\ell}[-1])$.
To show that the rightmost vertical map of (8) is an injection, let
$a=(a_{\mathfrak{p}})_{\mathfrak{p}\in{\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}}$ be a nonzero element of the rightmost upper term. Only
finitely many $a_{\mathfrak{p}}$ are nonzero. Pick some $\ell$ not lying under
any of these indices $\mathfrak{p}$. Then the image of $a$ in
$\oplus_{\mathfrak{p}\nmid\ell}\mathrm{H}^{0}(({{{}^{\mathrm{p}}}\mathrm{H}}^{0}i_{\mathfrak{p}}^{*}\eta_{*}\eta^{*}M)_{\ell})\subset\prod_{\mathfrak{p}\nmid\ell}\mathrm{H}^{0}(({{{}^{\mathrm{p}}}\mathrm{H}}^{0}i_{\mathfrak{p}}^{*}\eta_{*}\eta^{*}M)_{\ell})$
is nonzero by Lemma 6.8, which assumes Conjecture 6.7. ∎
###### Lemma 6.10.
Let $M$ be a mixed motive over $S$ such that $j^{*}M$ is smooth for some open
immersion $j:U\rightarrow S$. Then both maps
$\mathrm{H}^{0}(j_{!*}j^{*}M)\rightarrow\mathrm{H}^{0}(j_{*}j^{*}M)\rightarrow\mathrm{H}^{0}(\eta^{*}M)$
are injective.
###### Proof: .
Indeed the kernels are
$\mathrm{H}^{-1}({{{}^{\mathrm{p}}}\mathrm{H}}^{0}i^{*}j_{*}j^{*}M)=0$ and
$\oplus_{\mathfrak{p}\in U}\mathrm{H}^{0}(i_{\mathfrak{p}}^{!}M)$, which
vanishes since $i_{\mathfrak{p}}^{!}M$ sits in cohomological degree $+1$, for
$M$ is smooth around $\mathfrak{p}\in U$ (Lemma 5.9). ∎
### 6.2 $f$-cohomology via $K$-theory of regular models
###### Definition 6.11.
Let $X_{\eta}$ be a smooth and projective variety over $F$. Let
$X/{\mathcal{O}_{F}}$ be any projective model $X/{\mathcal{O}_{F}}$, i.e.,
$X{\times}_{\mathcal{O}_{F}}F=X_{\eta}$. Then we define
$\mathrm{H}^{i}(X_{\eta},n)_{\mathcal{O}_{F}}:=\operatorname{im}(\mathrm{H}^{i}(X,n)\rightarrow\mathrm{H}^{i}(X_{\eta},n)).$
Recall that we are assuming the axioms of Sections 1, 2, and 4; the full force
of mixed motives will be made use of in the sequel.
###### Theorem 6.12.
The above is well-defined, i.e., independent of the choice of the model $X$.
More precisely we have natural isomorphisms:
$\mathrm{H}^{0}(\eta_{!*}\mathrm{h}^{i-1}(X_{\eta},n)[1])=\left\\{\begin{array}[]{cl}\mathrm{H}^{i}(X_{\eta},n)_{{\mathcal{O}_{F}}}&i<2n\\\
\mathrm{CH}^{n}(X_{\eta})_{\mathbb{Q},\mathrm{hom}}&i=2n\end{array}\right.$
Moreover
$\mathrm{H}^{-1}(\eta_{!*}\mathrm{h}^{i-1}(X_{\eta},n)[1])=\mathrm{H}^{0}(\mathrm{h}^{i-1}(X_{\eta},n)).$
When $X$ is regular, the definition and the statement are due to Beilinson
[Beĭ86, Lemma 8.3.1]. In this case one has
$\mathrm{H}^{i}(X_{\eta},n)_{{\mathcal{O}_{F}}}=\operatorname{im}K^{\prime}_{2j-i}(X)_{\mathbb{Q}}^{(j)}\rightarrow
K^{\prime}_{2j-i}(X_{\eta})_{\mathbb{Q}}^{(j)}$, but that expression does in
general depend on the choice of the model [dJ00, dJ08]. An extension of
Beilinson’s definition to all Chow motives over $F$ due to Scholl is discussed
in the theorem below. We first provide a preparatory lemma.
###### Lemma 6.13.
Let $M\in\mathbf{MM}({\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}})$ be a mixed,
generically smooth motive with strictly negative weights (Definition 5.8). Let
$j:U\rightarrow{\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$ be an open non-
empty immersion such that $M|_{U}$ is smooth. The natural map
$j_{!*}j^{*}M\rightarrow\eta_{*}\eta^{*}M$ gives rise to an isomorphism
$\mathrm{H}^{0}(j_{!*}j^{*}M)=\mathrm{im}\left(\mathrm{H}^{0}(M)\rightarrow\mathrm{H}^{0}(\eta_{*}\eta^{*}M)\right).$
###### Proof: .
By Lemma 6.10,
$\mathrm{H}^{0}(j_{*}j^{*}M)\rightarrow\mathrm{H}^{0}(\eta_{*}\eta^{*}M)$ is
injective. Hence it suffices to show
$\mathrm{H}^{0}(j_{!*}j^{*}M)=\operatorname{im}(\mathrm{H}^{0}M\rightarrow\mathrm{H}^{0}(j_{*}j^{*}M))$.
Let $i$ be the complement of $j$. From (6), (7), we have a commutative exact
diagram
---
$\textstyle{\mathrm{H}^{0}(j_{!}j^{*}M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathrm{H}^{0}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{0}(i_{*}i^{*}M)}$$\textstyle{0=\mathrm{H}^{-1}(i_{*}{{{}^{\mathrm{p}}}\mathrm{H}}^{0}i^{*}j_{*}j^{*}M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{0}(j_{!*}j^{*}M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{0}(j_{*}j^{*}M)}$$\textstyle{\mathrm{H}^{1}(i_{*}{{{}^{\mathrm{p}}}\mathrm{H}}^{-1}i^{*}j_{*}j^{*}M)=0}$
The indicated vanishings are because of $t$-structure reasons and Axiom 4.1,
respectively. It remains to show that $\alpha$ is surjective. As $i^{*}M$ is
concentrated in cohomological degrees $[-1,0]$ (Axiom 4.2), there is an exact
sequence
$0=\mathrm{H}^{1}({{{}^{\mathrm{p}}}\mathrm{H}}^{-1}i^{*}M)\rightarrow\mathrm{H}^{0}(i^{*}M)\rightarrow\mathrm{H}^{0}({{{}^{\mathrm{p}}}\mathrm{H}}^{0}i^{*}M).$
However $\mathrm{H}^{0}({{{}^{\mathrm{p}}}\mathrm{H}}^{0}i^{*}M)=0$ as $i^{*}$
preserves negative weights (Axiom 4.11) and by strictness of the weight
filtration and compatibility with the $t$-structure (Axiom 4.9). ∎
###### Proof: .
Let $j:U\rightarrow{\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$ be an open
nonempty immersion (which exists by smoothness of $X_{\eta}$) such that
$X_{U}$ is smooth over $U$. By definition of $\eta_{!*}$ and Lemmas 5.1 and
5.11, the left hand term in the theorem agrees with
$\mathrm{H}^{0}(j_{!*}\mathrm{h}^{i}(X_{U},n))$. In the sequel, we write
$M:=\mathrm{h}^{2n}(X,n)$ and
$M_{\eta}:=\eta^{*}[-1]M=\mathrm{h}^{i-1}(X_{\eta},n)$.
We first do the case $i<2n$. The spectral sequences
$\mathrm{H}^{a}(\mathrm{h}^{b}(X,n))\Rightarrow\mathrm{H}^{a+b}(X,n),\,\mathrm{H}^{a}(\mathrm{h}^{b}(X_{\eta},n))\Rightarrow\mathrm{H}^{a+b}(X_{\eta},n)$
resulting from repeatedly applying truncation functors of the motivic
$t$-structure converge since the cohomological dimension is finite (Axiom 4.1
over $F$, Lemma 5.2 over ${\mathcal{O}_{F}}$). By Lemma 5.2,
$\mathrm{H}^{i}(-)$, applied to mixed motives over ${\mathcal{O}_{F}}$, is
non-zero for $i\in\\{-1,0,1\\}$ only. We thus have to consider two exact
sequences. The exact functor $\eta^{*}[-1]$ maps to similar exact sequences
for motivic cohomology over $F$ (the indices work out properly, see Lemma
5.1):
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{i}(X,n)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{-1}(\mathrm{h}^{i+1}(X,n))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\eta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{i}(X_{\eta},n)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{0}(\mathrm{h}^{i}(X_{\eta},n))\stackrel{{\scriptstyle\ref{axio_weight},\ref{bsp_weightTate}}}{{=}}0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
(9)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(\mathrm{h}^{i-1}(X,n))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{0}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{2}(\mathrm{h}^{i-2}(X_{\eta},n))\stackrel{{\scriptstyle\ref{axio_cohomdim}}}{{=}}0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\eta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(M_{\eta})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
(10)
Here, $K$ and $K_{\eta}$ are certain $E_{3}$-terms of the spectral sequences
above. The rightmost vertical map in (9) is injective as one sees by combining
(4) with the left-exactness of $i_{\mathfrak{p}}^{!}$. Hence
$\displaystyle\mathrm{H}^{i}(X_{\eta},n)_{{\mathcal{O}_{F}}}$ $\displaystyle=$
$\displaystyle\operatorname{im}(\mathrm{H}^{i}(X,n)\rightarrow\mathrm{H}^{i}(X_{\eta},n))=\operatorname{im}(K\rightarrow
K_{\eta})$ $\displaystyle=$
$\displaystyle\operatorname{im}(\mathrm{H}^{0}(M)\rightarrow\mathrm{H}^{1}(M_{\eta}))$
The motive $M=\mathrm{h}^{i}(X,n)$ is a generically smooth (mixed) motive by
Lemma 5.11. (Recall that this uses the decomposition axiom 4.5 for smooth
projective varieties.) By Example 4.12, its weights are strictly negative.
Thus Lemma 6.13 applies and the case $i<2n$ is shown.
We now do the case $i=2n$. Now
${{{}^{\mathrm{p}}}\mathrm{H}}^{1}i^{!}j_{!*}j^{*}M={{{}^{\mathrm{p}}}\mathrm{H}}^{1}i^{!}j_{*}j^{*}M$
has strictly positive weights because of the choice of $j$, Example 4.12,
Lemma 5.6, Axiom 4.11, and the compatibility of weights and the motivic
$t$-structure, i.e.,
$\operatorname{wt}{{{}^{\mathrm{p}}}\mathrm{H}}^{1}(-)\subset\operatorname{wt}(-)+1$.
Therefore $\mathrm{H}^{0}({{{}^{\mathrm{p}}}\mathrm{H}}^{1}i^{!}E)=0$,
$E:=j_{!*}j^{*}M$. Here $i$ is any closed immersion. The localization triangle
(4) yields
$\mathrm{H}^{0}(E)\stackrel{{\scriptstyle\alpha}}{{\rightarrow}}\mathrm{H}^{0}(\eta_{*}\eta^{*}E)\stackrel{{\scriptstyle\text{\ref{axio_morphismsChow}}}}{{=}}\mathrm{CH}^{n}(X_{\eta})_{\mathbb{Q},\mathrm{hom}}\rightarrow\oplus_{\mathfrak{p}}\mathrm{H}^{1}(i_{\mathfrak{p}}^{!}E)=\oplus\mathrm{H}^{0}({{{}^{\mathrm{p}}}\mathrm{H}}^{1}(i^{!}E))=0.$
Therefore, $\alpha$ is surjective. The injectivity of $\alpha$ is Lemma 6.10.
To calculate $\mathrm{H}^{-1}(\eta_{!*}M_{\eta}[1])$, let
$j:U\rightarrow{\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$ be as above and let
$i$ be its complement. The natural map $\mathrm{H}^{-1}({\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}},j_{!*}j^{*}M)\rightarrow\mathrm{H}^{-1}(U,j^{*}M)$ is an
isomorphism by the exact cohomology sequence belonging to (7). Thus we have to
show
$\mathrm{H}^{-1}({\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}},j_{*}j^{*}M)=\mathrm{H}^{-1}({\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}},\eta_{*}\eta^{*}M).$
This follows from the localization axiom 1.10 and $i_{\mathfrak{p}}^{!}M$
being in cohomological degree $+1$ for all points $\mathfrak{p}$ in $U$ (Lemma
5.9), so that
$\mathrm{H}^{0}({\mathbb{F}_{\mathfrak{p}}},i_{\mathfrak{p}}^{!}M)=\mathrm{H}^{-1}({\mathbb{F}_{\mathfrak{p}}},i_{\mathfrak{p}}^{!}M)=0$.
∎
By a theorem of Scholl [Sch00, Thm. 1.1.6], there is a unique functorial and
additive (i.e., converting finite disjoint unions into direct sums) way to
extend the definition of $\mathrm{H}^{i}(X_{\eta},n)_{\mathcal{O}_{F}}$ as the
image of the $K$-theory of a _regular_ proper flat model (Definition 6.11) to
all Chow motives over $F$, in particular to ones of smooth projective
varieties $X_{\eta}/F$ that do not possess a regular proper model $X$. The
following theorem compares this definition with the one via intermediate
extensions.
###### Theorem 6.14.
Let $h_{\eta}$ be a direct summand in the category of Chow motives of
$h(X_{\eta},n)$ where $X_{\eta}/F$ is smooth projective. Let $i\in\mathbb{Z}$
be such that $i-2n<0$. Let
$\iota:\mathbf{M}_{\mathrm{rat}}(F)\rightarrow\mathbf{DM}_{\mathrm{gm}}(F)$ be
the embedding. Then, the group
$\mathrm{H}^{i}(h_{\eta})_{\mathcal{O}_{F}}:=\mathrm{H}^{0}(\eta_{!*}({{{}^{\mathrm{p}}}\mathrm{H}}^{i-2n-1}(\iota(h_{\eta}))[1])).$
is well-defined and agrees with the aforementioned definition by Scholl.
###### Proof: .
Recall
$\iota(h(X_{\eta},n))=\operatorname{M}(X_{\eta},n)[2n]\in\mathbf{DM}_{\mathrm{gm}}(F)$.
We first check that the group is well-defined: let $X/{\mathcal{O}_{F}}$ be a
projective model of $X_{\eta}$. By Lemma 5.12, there is some model
$M\in\mathbf{MM}({\mathcal{O}_{F}})$ of
${{{}^{\mathrm{p}}}\mathrm{H}}^{i-2n-1}\iota(h_{\eta})[1]$ and an open
subscheme $U$ of ${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$ such that $M$ is
a direct summand of
${{{}^{\mathrm{p}}}\mathrm{H}}^{i-1}\operatorname{M}(X)(n)$ and such that
$X{\times}U$ is smooth over $U$. Then $\mathrm{h}^{i-1}(X,n)$ is a smooth
motive when restricted to $U$ (Lemma 5.11). Hence so is $M$. Thus $\eta_{!*}$
can be applied to
$({{{}^{\mathrm{p}}}\mathrm{H}}^{i-2n-1}\iota(h_{\eta}))[1]$.
The assignment
$h_{\eta}\mapsto\mathrm{H}^{0}(\eta_{!*}({{{}^{\mathrm{p}}}\mathrm{H}}^{i-2n-1}\iota(h_{\eta}))[1])$
is functorial and additive and $h(X_{\eta})(n)$ maps to
$\mathrm{H}^{0}(\eta_{!*}({{{}^{\mathrm{p}}}\mathrm{H}}^{i-1}\operatorname{M}(X_{\eta},n))[1])\stackrel{{\scriptstyle\text{\ref{theo_summaryH1f}}}}{{=}}\mathrm{H}^{i}(X_{\eta},n)_{\mathcal{O}_{F}}.$
Thus the two definitions agree by Scholl’s theorem. ∎
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|
arxiv-papers
| 2010-03-05T09:53:18 |
2024-09-04T02:49:08.792674
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jakob Scholbach",
"submitter": "Jakob Scholbach",
"url": "https://arxiv.org/abs/1003.1219"
}
|
1003.1220
|
# Timelike Bertrand Curves in Semi-Euclidean Space
Soley ERSOY and Murat TOSUN
sersoy@sakarya.edu.tr and tosun@sakarya.edu.tr
Department of Mathematics, Faculty of Arts and Sciences,
Sakarya University, Sakarya, 54187 TURKEY
###### Abstract
In this paper, it is proved that, no special timelike Frenet curve is a
Bertrand curve in $\mathbb{E}_{2}^{4}$ and also, in $\mathbb{E}_{\nu}^{n+1}$
$\left({n\geq 3}\right)$, such that the notion of Bertrand curve is definite
only in $\mathbb{E}_{1}^{2}$ and $\mathbb{E}_{1}^{3}$. Therefore, a
generalization of timelike Bertrand curve is defined and called as timelike
(1,3)-Bertrand curve in $\mathbb{E}_{2}^{4}$. Moreover, the characterization
of timelike (1,3)-Bertrand curve is given in $\mathbb{E}_{2}^{4}$.
Mathematics Subject Classification (2010). 53B30, 53A35, 53A04.
Keywords: Bertrand curve, Semi-Euclidean Space.
## 1 Introduction
In Euclidean ambient, a curve $C$ is called $C^{\infty}-$special Frenet curve
in $\mathbb{E}^{3}$ if there exist three $C^{\infty}-$vector fields, that is,
the unit vector tangent field ${\bf t}$, the unit principal normal vector
field ${\bf n}$ and the unit binormal vector field ${\bf b}$ and two
$C^{\infty}-$scalar functions, that is, the curvature function
$k_{1}\left({>0}\right)$ and torsion function $k_{2}\left({\neq 0}\right)$,
[6].
In $\mathbb{E}^{3}$, a $C^{\infty}-$special Frenet curve $C$ is called a
Bertrand curve if there exist another $C^{\infty}-$special Frenet curve
$\bar{C}$ and a $C^{\infty}-$mapping $\varphi:C\to\bar{C}$, such that the
principal normal vector fields of $C$ and $\bar{C}$ coincide at the
corresponding points, [4].
If we focus our attention to timelike curves in three dimensional Minkowski
space $\mathbb{E}_{1}^{3}$, then we can give the well known theorem that a
$C^{\infty}-$special timelike Frenet curve is a Bertrand curve if and only if
its curvature function $k_{1}$ and torsion function $k_{2}$ satisfy
$ak_{1}+bk_{2}=1$ for all $s\in L$, where $a$ and $b$ are constant real
numbers. This theorem suffices to define the timelike curve
${\bf\bar{c}}\left({\bar{s}}\right)={\bf
c}\left({\varphi\left(s\right)}\right)={\bf c}\left(s\right)+\alpha{\bf
n}\left(s\right)$
then it is immediate that $\bar{C}$ is the timelike Bertrand mate curve of
$C$,[1, 3]. As a result of this theorem, in $\mathbb{E}_{1}^{3}$ every
timelike circular helix is an example of timelike Bertrand curve. Moreover,
any timelike planar curve is a timelike Bertrand curve whose Bertrand mates
are parallel curves of $C$, [1].
In this paper, we prove that there is no $C^{\infty}-$special timelike Frenet
curve which is a Bertrand curve in $\mathbb{E}_{2}^{4}$ and also, in
$\mathbb{E}_{\nu}^{n+1}$ $\left({n\geq 3}\right)$. Thus, the notion of
timelike Bertrand curve stands only in $\mathbb{E}_{1}^{2}$ and
$\mathbb{E}_{1}^{3}$. According to this reason, we suggest an idea of
generalization of timelike Bertrand curve in $\mathbb{E}_{2}^{4}$.
## 2 Preliminaries
To meet the requirements in the next sections, the basic elements of the
theory of curves in the semi Euclidean space $\mathbb{E}_{2}^{4}$ are briefly
presented in this section. A more complete elementary treatment can be found
in [5].
Semi-Euclidean space $\mathbb{E}_{2}^{4}$ is an Euclidean space provided with
standard flat metric given by
$g=-dx_{1}^{2}-dx_{2}^{2}+dx_{3}^{2}+dx_{4}^{2}$
where $\left({x_{1},x_{2},x_{3},x_{4}}\right)$ is a rectangular coordinate
system in $\mathbb{E}_{2}^{4}$.
Since $g$ is an indefinite metric, recall that a vector ${\bf
v}\in\mathbb{E}_{2}^{4}$ can have one of three Lorentzian causal characters:
it can be spacelike if $g\left({{\bf v},{\bf v}}\right)>0$ or ${\bf v}=0$,
timelike if $g\left({{\bf v},{\bf v}}\right)<0$ and null (lightlike) if
$g\left({{\bf v},{\bf v}}\right)=0$ and ${\bf v}\neq 0$, [5].
Similarly, an arbitrary curve ${\bf c}={\bf c}\left(s\right)$ in
$\mathbb{E}_{2}^{4}$ can locally be spacelike, timelike or null (lightlike) if
all of its velocity vectors ${\bf c}^{\prime}\left(s\right)$ are,
respectively, spacelike, timelike or null (lightlike). The norm of ${\bf
v}\in\mathbb{E}_{2}^{4}$ is defined as $\left\|{\bf
v}\right\|=\sqrt{\left|{g\left({{\bf v},{\bf v}}\right)}\right|}$. Therefore,
${\bf v}$ is a unit vector if $g\left({{\bf v},{\bf v}}\right)=\mp 1$.
Furthermore, vectors ${\bf v}$ and ${\bf w}$ are said to be orthogonal if
$g\left({{\bf v},{\bf w}}\right)=0$. The velocity of the curve $C$ is given by
$\left\|{{\bf c}^{\prime}}\right\|$. Thus, a timelike curve $C$ is said to be
parametrized by arc length function $s$ if $g\left({{\bf c}^{\prime},{\bf
c}^{\prime}}\right)=-1$, [5].
Let $\left\\{{{\bf t}\left(s\right),{\bf n}_{1}\left(s\right),{\bf
n}_{2}\left(s\right),{\bf n}_{3}\left(s\right)}\right\\}$ denotes the moving
Frenet frame along $C$ in the semi-Euclidean space $\mathbb{E}_{2}^{4}$, then
${\bf t},\;{\bf n}_{1},\;{\bf n}_{2}$, and ${\bf n}_{3}$ are called the
tangent, the principal normal, the first binormal, and the second binormal
vector fields of $C$, respectively.
Let $C$ be a $C^{\infty}-$special timelike Frenet curve with timelike
principal normal, spacelike first binormal and second binormal vector fields
in $\mathbb{E}_{2}^{4}$, parametrized by arc length function $s$. Moreover,
non-zero $C^{\infty}-$scalar functions $k_{1}$, $k_{2}$, and $k_{3}$ be the
first, second, and third curvatures of $C$, respectively. Then for the
$C^{\infty}-$special timelike Frenet curve $C$, the following Frenet formula
is given by
$\left[{\begin{array}[]{*{20}c}{{\bf t^{\prime}}}\\\ {{\bf n^{\prime}}_{\bf
1}}\\\ {{\bf n^{\prime}}_{\bf 2}}\\\ {{\bf n^{\prime}}_{\bf 3}}\\\
\end{array}}\right]=\left[{\begin{array}[]{*{20}c}0&{-k_{1}}&0&0\\\
{k_{1}}&0&{k_{2}}&0\\\ 0&{k_{2}}&0&{k_{3}}\\\ 0&0&{-k_{3}}&0\\\
\end{array}}\right]\left[{\begin{array}[]{*{20}c}{\bf t}\\\ {{\bf n}_{\bf
1}}\\\ {{\bf n}_{\bf 2}}\\\ {{\bf n}_{\bf 3}}\\\ \end{array}}\right]$ (2.1)
where ${\bf t},\;{\bf n}_{1},\;{\bf n}_{2},\;{\bf n}_{3}$ mutually orthogonal
vector fields satisfying
${\rm g}\left({{\bf t},{\bf t}}\right)={\rm g}\left({{\bf n}_{1},{\bf
n}_{1}}\right)=-1\quad,\;\quad{\rm g}\left({{\bf n}_{2},{\bf
n}_{2}}\right)={\rm g}\left({{\bf n}_{3},{\bf n}_{3}}\right)=1$
(for the semi-Euclidean space $\mathbb{E}_{\nu}^{n+1}$, see [2]).
## 3 Timelike Bertrand Curve in $\mathbb{E}_{2}^{4}$
The following two theorems related to Bertrand curves in $\mathbb{E}_{1}^{2}$
and $\mathbb{E}_{1}^{3}$ are well known [1, 3].
###### Theorem 3.1
In $\mathbb{E}_{1}^{2}$, every timelike $C^{\infty}-$planar curve is a
Bertrand curve,[1].
###### Theorem 3.2
In $\mathbb{E}_{1}^{3}$, a $C^{\infty}-$special timelike Frenet curve with
first and second curvatures $k_{1}$ and $k_{2}$ is a timelike Bertrand curve
if and only if there exist a linear relation $ak_{1}+bk_{2}=1$ for all $s\in
L$, where $a,b$ are nonzero constant real numbers, [1].
Now, let us investigate Bertrand curves in $\mathbb{E}_{2}^{4}$.
###### Definition 3.1
A $C^{\infty}-$special timelike Frenet curve $C$
$\left({C:L\to\mathbb{E}_{2}^{4}}\right)$ is called timelike Bertrand curve if
there exist an another $C^{\infty}-$special timelike Frenet curve ${\bar{C}}$
$\left({{\bar{C}}:{\bar{L}}\to\mathbb{E}_{2}^{4}}\right)$, distinct from $C$,
and a regular $C^{\infty}-$map $\varphi:L\to\bar{L}$,
$\left({\bar{s}=\varphi\left(s\right),\;\frac{{d\varphi\left(s\right)}}{{ds}}\neq
0\quad{\rm for}\;\;{\rm all}\;\;s\in L}\right)$, such that curve has the same
1-normal line at each pair of corresponding points ${\bf c}\left(s\right)$ and
${\bf\bar{c}}\left({\bar{s}}\right)={\bf\bar{c}}\left({\varphi\left(s\right)}\right)$
under $\varphi$. Here, $s$ and $\bar{s}$ are arc length parameters of timelike
curves $C$ and $\bar{C}$, respectively. In this case, $\bar{C}$ is called a
timelike Bertrand mate of $C$.
###### Theorem 3.3
In $\mathbb{E}_{2}^{4}$, no $C^{\infty}-$special timelike Frenet curve is a
Bertrand curve.
Proof. Let $\left({C,\bar{C}}\right)$ be a mate of Bertrand curve in
$\mathbb{E}_{2}^{4}$. Also, the pair of ${\bf c}\left(s\right)$ and
${\bf\bar{c}}\left({\bar{s}}\right)$ be the corresponding points of $C$ and
$\bar{C}$, respectively. Then for all $s\in L$ the curve $\bar{C}$ is given by
${\bf\bar{c}}\left({\bar{s}}\right)={\bf
c}\left({\varphi\left(s\right)}\right)={\bf
c}\left(s\right)+\alpha\left(s\right){\bf n}_{1}\left(s\right)$ (3.1)
where $\alpha$ is $C^{\infty}-$function on $L$. By differentiating the
equation (3.1) with respect to $s$, then
$\varphi^{\prime}\left(s\right)\left.{\frac{{d{\bf\bar{c}}\left({\bar{s}}\right)}}{{d\bar{s}}}}\right|_{\bar{s}=\varphi\left(s\right)}={\bf
c^{\prime}}\left(s\right)+\alpha^{\prime}\left(s\right){\bf
n}_{1}\left(s\right)+\alpha\left(s\right){\bf n^{\prime}}_{1}\left(s\right)$
is obtained. Here and hereafter, the subscript prime denotes the
differentiation with respect to $s$. By using the Frenet formulas, it is seen
that
$\varphi^{\prime}\left(s\right){\bf\bar{t}}\left({\varphi\left(s\right)}\right)=\left({1+\alpha\left(s\right)k_{1}\left(s\right)}\right){\bf
t}\left(s\right)+\alpha^{\prime}\left(s\right){\bf
n}_{1}\left(s\right)+\alpha\left(s\right)k_{2}\left(s\right){\bf
n}_{2}\left(s\right).$
Considering ${\bf n}_{1}\left(s\right)$ and
${\bf\bar{n}}_{1}\left({\varphi\left(s\right)}\right)$ are coincident and
$g\left({{\bf\bar{t}}\left({\varphi\left(s\right)}\right),{\bf
n}_{1}\left(s\right)}\right)=0$ for all $s\in L$, we get
$\alpha^{\prime}\left(s\right)=0,$
that is, $\alpha$ is a constant function on $L$. Thus, the Bertrand mate of
$C$ can be rewritten as
${\bf\bar{c}}\left({\bar{s}}\right)={\bf
c}\left({\varphi\left(s\right)}\right)={\bf c}\left(s\right)+\alpha{\bf
n}_{1}\left(s\right).$ (3.2)
The differentiation of this last equation with respect to $s$ is
$\varphi^{\prime}\left(s\right){\bf\bar{t}}\left({\varphi\left(s\right)}\right)=\left({1+\alpha
k_{1}\left(s\right)}\right){\bf t}\left(s\right)+\alpha
k_{2}\left(s\right){\bf n}_{2}\left(s\right)$ (3.3)
for all $s\in L$.
By the fact that $C$ and $\bar{C}$ are timelike curves, the tangent vector
field of Bertrand mate of $C$ can be given by
${\bf\bar{t}}\left({\varphi\left(s\right)}\right)=\cosh\theta\left(s\right){\bf
t}\left(s\right)+\sinh\theta\left(s\right){\bf n}_{2}\left(s\right),$ (3.4)
where $\theta$ is a hyperbolic angle between the timelike tangent vector
fields ${\bf\bar{t}}\left({\varphi\left(s\right)}\right)$ and ${\bf
t}\left(s\right)$. According to the equations (3.3) and (3.4), the hyperbolic
functions are defined by
$\cosh\theta\left(s\right)={{\left({1+\alpha
k_{1}\left(s\right)}\right)}\mathord{\left/{\vphantom{{\left({1+\alpha
k_{1}\left(s\right)}\right)}{\varphi^{\prime}\left(s\right)}}}\right.\kern-1.2pt}{\varphi^{\prime}\left(s\right)}},$
(3.5) $\sinh\theta\left(s\right)={{\alpha
k_{2}\left(s\right)}\mathord{\left/{\vphantom{{\alpha
k_{2}\left(s\right)}{\varphi^{\prime}\left(s\right)}}}\right.\kern-1.2pt}{\varphi^{\prime}\left(s\right)}}.$
(3.6)
By differentiating the equation (3.4) and applying Frenet formulas,
$\begin{array}[]{l}-\varphi^{\prime}\left(s\right)\bar{k}_{1}\left({\varphi\left(s\right)}\right){\bf\bar{n}}_{1}\left({\varphi\left(s\right)}\right)=\frac{{d\left({\cosh\theta\left(s\right)}\right)}}{{ds}}{\bf
t}\left(s\right)\\\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\left({-k_{1}\left(s\right)\cosh\theta\left(s\right)+k_{2}\left(s\right)\sinh\theta\left(s\right)}\right){\bf
n}_{1}\left(s\right)\\\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\frac{{d\left({\sinh\theta\left(s\right)}\right)}}{{ds}}{\bf
n}_{2}\left(s\right)+k_{3}\left(s\right)\sinh\theta\left(s\right){\bf
n}_{3}\left(s\right)\\\ \end{array}$
is obtained. Since ${\bf n}_{1}\left(s\right)$ is coincident with
${\bf\bar{n}}_{1}\left({\varphi\left(s\right)}\right)$, from the above
equation, it is seen that
$k_{3}\left(s\right)\sinh\theta\left(s\right)=0.$
If we notice that $k_{3}$ is different from zero, then
$\sinh\theta\left(s\right)=0$. Considering the equation (3.6) and $k_{2}\neq
0$, then $\alpha=0$. In that time the equation (3.2) implies that $\bar{C}$ is
coincident with $C$. This is a contradiction. So, the proof is complete.
Also, in the same way we can generalize this theorem to semi-Euclidean space
$\mathbb{E}_{\nu}^{n+1}$ $\left({n\geq 3}\right)$ by taking into consideration
the Frenet formulas in $\mathbb{E}_{\nu}^{n+1}$. The following result can be
easily proved.
###### Result 3.1
There is no $C^{\infty}-$special Frenet curve which is a Bertrand curve in
$\mathbb{E}_{\nu}^{n+1}$, $\left({n\geq 3}\right)$.
## 4 Timelike $\left({1,3}\right)-$Bertrand Curve in $\mathbb{E}_{2}^{4}$
According to previous section, there is no Bertrand curve in semi-Euclidean
space. Therefore, let us suggest an idea of a generalization of Bertrand curve
in $\mathbb{E}_{2}^{4}$.
###### Definition 4.1
Let $C$ and $\bar{C}$ be $C^{\infty}-$special timelike Frenet curves in
$\mathbb{E}_{2}^{4}$ and
$\begin{array}[]{*{20}c}{\varphi:L\to\bar{L}}\\\
{\quad\quad\quad\quad\;s\to\varphi\left(s\right)=\bar{s}}\\\
\end{array}\quad\quad\left({\frac{{d\varphi\left(s\right)}}{{ds}}\neq
0\quad{\rm for}\;\;{\rm all}\;\;s\in L}\right)$
be a regular $C^{\infty}-$map at the corresponding points ${\bf
c}\left(s\right)$ and
${\bf\bar{c}}\left({\bar{s}}\right)={\bf\bar{c}}\left({\varphi\left(s\right)}\right)$
of $C$ and $\bar{C}$, respectively. If the Frenet $\left({1,3}\right)-$normal
plane at the each point of $C$ is coincident with the Frenet
$\left({1,3}\right)-$normal plane at the corresponding point
${\bf\bar{c}}\left({\bar{s}}\right)$ of $\bar{C}$, then $C$ is called a
timelike $\left({1,3}\right)-$Bertrand curve in $\mathbb{E}_{2}^{4}$. Also,
$\bar{C}$ is called a timelike $\left({1,3}\right)-$Bertrand mate of $C$,
which is given by
${\bf\bar{c}}\left({\bar{s}}\right)={\bf
c}\left({\varphi\left(s\right)}\right)={\bf
c}\left(s\right)+\alpha\left(s\right){\bf
n}_{1}\left(s\right)+\beta\left(s\right){\bf n}_{3}\left(s\right),$ (4.1)
where $\alpha$ and $\beta$ are $C^{\infty}-$functions on $L$.
The following theorem gives us a characterization of timelike
$\left({1,3}\right)-$Bertrand curve in $\mathbb{E}_{2}^{4}$.
###### Theorem 4.1
Let $C$ be a $C^{\infty}-$special timelike Frenet curve with non-zero
curvatures $k_{1}$, $k_{2}$, and $k_{3}$ in $\mathbb{E}_{2}^{4}$. Then $C$ is
a timelike $\left({1,3}\right)-$Bertrand curve if and only if there exist the
constant real numbers $\alpha,\;\beta,\;\gamma,\;\delta$ satisfying
$\begin{array}[]{l}i.)\quad\;\;\alpha k_{2}\left(s\right)-\beta
k_{3}\left(s\right)\neq 0\\\ ii.)\quad\;\gamma\left({\alpha
k_{2}\left(s\right)-\beta k_{3}\left(s\right)}\right)-\alpha
k_{1}\left(s\right)=1\\\ iii.)\quad\delta k_{3}\left(s\right)=-\gamma
k_{1}\left(s\right)+k_{2}\left(s\right)\\\
iv.)\quad\left({\gamma^{2}+1}\right)k_{1}\left(s\right)k_{2}\left(s\right)-\gamma\left({k_{1}^{2}\left(s\right)+k_{2}^{2}\left(s\right)-k_{3}^{2}\left(s\right)}\right)\neq
0\\\ \end{array}$
for all $s\in L$.
Proof. Firstly, let us prove the necessary condition of the theorem. Let $C$
be a timelike $\left({1,3}\right)-$Bertrand curve parametrized by arc length
$s$ in $\mathbb{E}_{2}^{4}$. Then the timelike Bertrand mate $\bar{C}$ of $C$
is given by (4.1). Substituting the Frenet formulas into the differentiation
of the equation (4.1) with respect to $s$, we get
$\begin{array}[]{l}\varphi^{\prime}\left(s\right){\bf\bar{t}}\left({\varphi\left(s\right)}\right)=\left({1+\alpha\left(s\right)k_{1}\left(s\right)}\right){\bf
t}\left(s\right)+\alpha^{\prime}\left(s\right){\bf n}_{1}\left(s\right)\\\
\quad\quad\quad\quad\quad\quad\quad+\left({\alpha\left(s\right)k_{2}\left(s\right)-\beta\left(s\right)k_{3}\left(s\right)}\right){\bf
n}_{2}\left(s\right)+\beta^{\prime}\left(s\right){\bf n}_{3}\left(s\right)\\\
\end{array}$
for all $s\in L$.
Since ${\bf n}_{1}$ and ${\bf\bar{n}}_{1}$ principal normal vector fields of
$C$ and $\bar{C}$ are timelike and the plane spanned by ${\bf n}_{1}$ and
${\bf n}_{3}$ is coincident with the plane spanned by ${\bf\bar{n}}_{1}$ and
${\bf\bar{n}}_{3}$, then we have
${\bf\bar{n}}_{1}\left({\varphi\left(s\right)}\right)=\cosh\theta\left(s\right){\bf
n}_{1}\left(s\right)+\sinh\theta\left(s\right){\bf n}_{3}\left(s\right)$ (4.2)
${\bf\bar{n}}_{3}\left({\varphi\left(s\right)}\right)=\sinh\theta\left(s\right){\bf
n}_{1}\left(s\right)+\cosh\theta\left(s\right){\bf n}_{3}\left(s\right)$ (4.3)
where $\sinh\theta\left(s\right)\neq 0$ for all $s\in L$ and $\theta$ is a
hyperbolic angle between the timelike vector fields ${\bf n}_{1}$ and
${\bf\bar{n}}_{1}$.
By considering differentiation of the equation (4.1) and the last two
equations, we see
$\begin{array}[]{l}g\left({\varphi^{\prime}\left(s\right){\bf\bar{t}}\left({\varphi\left(s\right)}\right),{\bf\bar{n}}_{1}\left({\varphi\left(s\right)}\right)}\right)=\alpha^{\prime}\left(s\right)\cosh\theta\left(s\right)+\beta^{\prime}\left(s\right)\sinh\theta\left(s\right)=0\\\
g\left({\varphi^{\prime}\left(s\right){\bf\bar{t}}\left({\varphi\left(s\right)}\right),{\bf\bar{n}}_{3}\left({\varphi\left(s\right)}\right)}\right)=\alpha^{\prime}\left(s\right)\sinh\theta\left(s\right)+\beta^{\prime}\left(s\right)\cosh\theta\left(s\right)=0\\\
\end{array}$
and since $\sinh\theta\left(s\right)\neq 0$,
$\alpha^{\prime}\left(s\right)=0\quad{\rm
and}\quad\beta^{\prime}\left(s\right)=0$
that is, $\alpha$ and $\beta$ are constant functions on $L$. Thus, the
equation (4.1) is rewritten as
${\bf\bar{c}}\left({\bar{s}}\right)={\bf
c}\left({\varphi\left(s\right)}\right)={\bf c}\left(s\right)+\alpha{\bf
n}_{1}\left(s\right)+\beta{\bf n}_{3}\left(s\right)$ (4.4)
and its differentiation with respect to $s$ is
$\varphi^{\prime}\left(s\right){\bf\bar{t}}\left({\varphi\left(s\right)}\right)=\left({1+\alpha
k_{1}\left(s\right)}\right){\bf t}\left(s\right)+\left({\alpha
k_{2}\left(s\right)-\beta k_{3}\left(s\right)}\right){\bf
n}_{2}\left(s\right)$ (4.5)
Since ${\bf\bar{t}}$ and ${\bf t}$ are timelike tangent vector fields of $C$
and $\bar{C}$, then
$-\left({\varphi^{\prime}\left(s\right)}\right)^{2}=-\left({1+\alpha
k_{1}\left(s\right)}\right)^{2}+\left({\alpha k_{2}\left(s\right)-\beta
k_{3}\left(s\right)}\right)^{2}.$ (4.6)
Thus, we can write
${\bf\bar{t}}\left({\varphi\left(s\right)}\right)=\cosh\tau\left(s\right){\bf
t}\left(s\right)+\sinh\tau\left(s\right){\bf n}_{2}\left(s\right)$ (4.7)
and
$\cosh\tau\left(s\right)={{\left({1+\alpha
k_{1}\left(s\right)}\right)}\mathord{\left/{\vphantom{{\left({1+\alpha
k_{1}\left(s\right)}\right)}{\varphi^{\prime}\left(s\right)}}}\right.\kern-1.2pt}{\varphi^{\prime}\left(s\right)}},$
$\sinh\tau\left(s\right)={{\left({\alpha k_{2}\left(s\right)-\beta
k_{3}\left(s\right)}\right)}\mathord{\left/{\vphantom{{\left({\alpha
k_{2}\left(s\right)-\beta
k_{3}\left(s\right)}\right)}{\varphi^{\prime}\left(s\right)}}}\right.\kern-1.2pt}{\varphi^{\prime}\left(s\right)}},$
where $\tau$ is a hyperbolic angle between the timelike tangent vector fields
${\bf\bar{t}}\left({\varphi\left(s\right)}\right)$ and ${\bf t}\left(s\right)$
of $C$ and $\bar{C}$. By differentiating the equation (4.7) with respect to
and applying Frenet formulas,
$\begin{array}[]{l}-\varphi^{\prime}\left(s\right)\bar{k}_{1}\left({\varphi\left(s\right)}\right){\bf\bar{n}}_{1}\left({\varphi\left(s\right)}\right)=\frac{{d\left({\cosh\tau\left(s\right)}\right)}}{{ds}}{\bf
t}\left(s\right)\\\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\left({-k_{1}\left(s\right)\cosh\tau\left(s\right)+k_{2}\left(s\right)\sinh\tau\left(s\right)}\right){\bf
n}_{1}\left(s\right)\\\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\frac{{d\left({\sinh\tau\left(s\right)}\right)}}{{ds}}{\bf
n}_{2}\left(s\right)\\\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+k_{3}\left(s\right)\sinh\tau\left(s\right){\bf
n}_{3}\left(s\right)\\\ \end{array}$
is obtained. Since ${\bf\bar{n}}_{1}\left({\varphi\left(s\right)}\right)$ is a
linear combination ${\bf n}_{1}\left(s\right)$ and ${\bf
n}_{3}\left(s\right)$, it easily seen that
$\frac{{d\left({\cosh\tau\left(s\right)}\right)}}{{ds}}=0\quad{\rm
and}\quad\frac{{d\left({\sinh\tau\left(s\right)}\right)}}{{ds}}=0,$
that is, $\tau$ is a constant function on $L$ with value $\tau_{0}$. Thus, we
rewrite the equation (4.7) as
${\bf\bar{t}}\left({\varphi\left(s\right)}\right)=\cosh\tau_{0}{\bf
t}\left(s\right)+\sinh\tau_{0}{\bf n}_{2}\left(s\right)$ (4.8)
and
$\varphi^{\prime}\left(s\right)\cosh\tau_{0}=1+\alpha k_{1}\left(s\right),$
(4.9)
$\varphi^{\prime}\left(s\right)\sinh\tau_{0}=\alpha k_{2}\left(s\right)-\beta
k_{3}\left(s\right),$ (4.10)
for all $s\in L$. According to these last two equations, it is seen that
$\left({1+\alpha k_{1}\left(s\right)}\right)\sinh\tau_{0}=\left({\alpha
k_{2}\left(s\right)-\beta k_{3}\left(s\right)}\right)\cosh\tau_{0}.$ (4.11)
If $\sinh\tau_{0}=0$, then it satisfies $\cosh\tau_{0}=1$ and
${\bf\bar{t}}\left({\varphi\left(s\right)}\right)={\bf t}\left(s\right)$. The
differentiation of this equality with respect to $s$ is
$-\bar{k}_{1}\left({\varphi\left(s\right)}\right)\varphi^{\prime}\left(s\right){\bf\bar{n}}_{1}\left({\varphi\left(s\right)}\right)=-k_{1}\left(s\right){\bf
n}_{1}\left(s\right),$
that is, ${\bf n}_{1}\left(s\right)$ is linear dependence with
${\bf\bar{n}}_{1}\left({\varphi\left(s\right)}\right)$. According to Theorem
3.3, this is a contradiction. Thus, only the case of $\sinh\tau_{0}\neq 0$
must be considered. The equation (4.10) satisfies
$\alpha k_{2}\left(s\right)-\beta k_{3}\left(s\right)\neq 0,$
that is, the relation given in i.) is proved. Since $\sinh\tau_{0}\neq 0$, the
equation (4.11) can be rewritten as
$\frac{{\cosh\tau_{0}}}{{\sinh\tau_{0}}}\left({\alpha
k_{2}\left(s\right)-\beta k_{3}\left(s\right)}\right)-\alpha
k_{1}\left(s\right)=1.$
Let us denote the constant value
$\left({\cosh\tau_{0}}\right)\left({\sinh\tau_{0}}\right)^{-1}$ by the
constant real number $\gamma$, then $\gamma$ is an element of interval
$\left({-\infty,-1}\right)\cup\left({1,\infty}\right)$ and
$\gamma\left({\alpha k_{2}\left(s\right)-\beta
k_{3}\left(s\right)}\right)-\alpha k_{1}\left(s\right)=1.$
This proves the relation ii.) of the theorem.
By differentiating the equation (4.8) with respect to $s$ and applying Frenet
formulas, we have
$\begin{array}[]{l}\varphi^{\prime}\left(s\right)\bar{k}_{1}\left({\varphi\left(s\right)}\right){\bf\bar{n}}_{1}\left({\varphi\left(s\right)}\right)=\left({k_{1}\left(s\right)\cosh\tau\left(s\right)-k_{2}\left(s\right)\sinh\tau\left(s\right)}\right){\bf
n}_{1}\left(s\right)\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-k_{3}\left(s\right)\sinh\tau\left(s\right){\bf
n}_{3}\left(s\right)\\\ \end{array}$
for all $s\in L$.
Taking into consideration the equations (4.9), (4.10) and the relation ii.),
the above equality satisfy
$\left({\varphi^{\prime}\left(s\right)\bar{k}_{1}\left({\varphi\left(s\right)}\right)}\right)^{2}=\left({\alpha
k_{2}\left(s\right)-\beta k_{3}\left(s\right)}\right)^{2}\left[{\left({\gamma
k_{1}\left(s\right)-k_{2}\left(s\right)}\right)^{2}-k_{3}^{2}\left(s\right)}\right]\left({\varphi^{\prime}\left(s\right)}\right)^{-2}.$
From the equation (4.6) and the relation ii.), we get
$\left({\varphi^{\prime}\left(s\right)}\right)^{2}=\left({\gamma^{2}-1}\right)\left({\alpha
k_{2}\left(s\right)-\beta k_{3}\left(s\right)}\right)^{2}.$ (4.12)
Thus, we obtain
$\left({\varphi^{\prime}\left(s\right)\bar{k}_{1}\left({\varphi\left(s\right)}\right)}\right)^{2}=\frac{1}{{\gamma^{2}-1}}\left({\left({\gamma
k_{1}\left(s\right)-k_{2}\left(s\right)}\right)^{2}-k_{3}^{2}\left(s\right)}\right)$
(4.13)
where $\gamma\in\left({-\infty,-1}\right)\cup\left({1,\infty}\right)$.
From the equations (4.9), (4.10) and the relation ii.), we can give
${\bf\bar{n}}_{1}\left({\varphi\left(s\right)}\right)=\cosh\eta\left(s\right){\bf
n}_{1}\left(s\right)+\sinh\eta\left(s\right){\bf n}_{3}\left(s\right)$ (4.14)
where
$\cosh\eta\left(s\right)=\frac{{\left({\alpha k_{2}\left(s\right)-\beta
k_{3}\left(s\right)}\right)\left({\gamma
k_{1}\left(s\right)-k_{2}\left(s\right)}\right)}}{{\bar{k}_{1}\left({\varphi\left(s\right)}\right)\left({\varphi^{\prime}\left(s\right)}\right)^{2}}},$
(4.15)
$\sinh\eta\left(s\right)=-\frac{{\left({\alpha k_{2}\left(s\right)-\beta
k_{3}\left(s\right)}\right)k_{3}\left(s\right)}}{{\bar{k}_{1}\left({\varphi\left(s\right)}\right)\left({\varphi^{\prime}\left(s\right)}\right)^{2}}},$
(4.16)
for all $s\in L$ and $\eta\in C^{\infty}-$function on $L$. By differentiating
the equation (4.14) with respect to $s$ and applying Frenet formulas, we have
$\begin{array}[]{l}\varphi^{\prime}\left(s\right)\bar{k}_{1}\left({\varphi\left(s\right)}\right){\bf\bar{t}}\left({\varphi\left(s\right)}\right)\\\
\quad\quad\quad\quad\quad+\varphi^{\prime}\left(s\right)\bar{k}_{2}\left({\varphi\left(s\right)}\right){\bf\bar{n}}_{2}\left({\varphi\left(s\right)}\right)=k_{1}\left(s\right)\cosh\eta\left(s\right){\bf
t}\left(s\right)\\\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\frac{{d\left({\cosh\eta\left(s\right)}\right)}}{{ds}}{\bf
n}_{1}\left(s\right)\\\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\left({k_{2}\left(s\right)\cosh\eta\left(s\right)-k_{3}\left(s\right)\sinh\eta\left(s\right)}\right){\bf
n}_{2}\left(s\right)\\\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\frac{{d\left({\sinh\tau\left(s\right)}\right)}}{{ds}}{\bf
n}_{3}\left(s\right)\\\ \end{array}$
for all $s\in L$ and this satisfies
$\frac{{d\left({\cosh\eta\left(s\right)}\right)}}{{ds}}=0\quad{\rm
and}\quad\frac{{d\left({\sinh\eta\left(s\right)}\right)}}{{ds}}=0,$
that is, $\eta$ is a constant function on $L$ with value $\eta_{0}$. Let us
denote $\left({\cosh\tau_{0}}\right)\left({\sinh\tau_{0}}\right)^{-1}$ by the
constant real number $\delta$, then
$\delta\in\left({-\infty,-1}\right)\cup\left({1,\infty}\right)$. The ratio of
(4.15) and (4.16) holds
$\delta=-\frac{{\gamma
k_{1}\left(s\right)-k_{2}\left(s\right)}}{{k_{3}\left(s\right)}},$
that is,
$\delta k_{3}\left(s\right)=-\gamma k_{1}\left(s\right)+k_{2}\left(s\right)$
for all $s\in L$ and
$\delta\in\left({-\infty,-1}\right)\cup\left({1,\infty}\right)$. Thus the
relation iii.) of the theorem is obtained. Moreover, we can give
$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\varphi^{\prime}\left(s\right)\bar{k}_{2}\left({\varphi\left(s\right)}\right){\bf\bar{n}}_{2}\left({\varphi\left(s\right)}\right)=-\varphi^{\prime}\left(s\right)\bar{k}_{1}\left({\varphi\left(s\right)}\right){\bf\bar{t}}\left({\varphi\left(s\right)}\right)+k_{1}\left(s\right)\cosh\eta\left(s\right){\bf
t}\left(s\right)\cr
0.0pt{\hfil$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\left({k_{2}\left(s\right)\cosh\eta\left(s\right)-k_{3}\left(s\right)\sinh\eta\left(s\right)}\right){\bf
n}_{2}\left(s\right).\cr}}}$
If we substitute the equations (4.5), (4.15) and (4.16) into the above
equality, we obtain
$\varphi^{\prime}\left(s\right)\bar{k}_{2}\left({\varphi\left(s\right)}\right){\bf\bar{n}}_{2}\left({\varphi\left(s\right)}\right)=\left({\varphi^{\prime}\left(s\right)}\right)^{-2}\left({\bar{k}_{1}\left({\varphi\left(s\right)}\right)}\right)^{-1}\left[{A\left(s\right){\bf
t}\left(s\right)+B\left(s\right){\bf n}_{2}\left(s\right)}\right]$
where
$\begin{array}[]{l}A\left(s\right)=-\left({\varphi^{\prime}\left(s\right)\bar{k}_{1}\left({\varphi\left(s\right)}\right)}\right)^{2}\left({1+\alpha
k_{1}\left(s\right)}\right)\\\
\quad\quad\quad\,\,\,+k_{1}\left(s\right)\;\left({\alpha
k_{2}\left(s\right)-\beta k_{3}\left(s\right)}\right)\left({\gamma
k_{1}\left(s\right)-k_{2}\left(s\right)}\right)\\\ \end{array}$
and
$\begin{array}[]{l}B\left(s\right)=\left({\varphi^{\prime}\left(s\right)\bar{k}_{1}\left({\varphi\left(s\right)}\right)}\right)^{2}\;\left({\alpha
k_{2}\left(s\right)-\beta k_{3}\left(s\right)}\right)\\\
\quad\quad\quad+\left({\alpha k_{2}\left(s\right)-\beta
k_{3}\left(s\right)}\right)\left({\gamma
k_{1}\left(s\right)-k_{2}\left(s\right)}\right)k_{2}\left(s\right)\\\
\quad\quad\quad+\left({\alpha k_{2}\left(s\right)-\beta
k_{3}\left(s\right)}\right)k_{3}^{2}\left(s\right)\\\ \end{array}$
for all $s\in L$. By the relation ii.) and the equation (4.13),
$A\left(s\right)$ and $B\left(s\right)$ can be rewritten as;
$A\left(s\right)=\left({\gamma^{2}-1}\right)^{-1}\left({\alpha
k_{2}\left(s\right)-\beta
k_{3}\left(s\right)}\right)\left[{\left({\gamma^{2}+1}\right)k_{1}\left(s\right)k_{2}\left(s\right)-\gamma\left({k_{1}^{2}\left(s\right)+k_{2}^{2}\left(s\right)-k_{3}^{2}\left(s\right)}\right)}\right]$
and
$B\left(s\right)=\gamma\left({\gamma^{2}-1}\right)^{-1}\left({\alpha
k_{2}\left(s\right)-\beta
k_{3}\left(s\right)}\right)\left[{\left({\gamma^{2}+1}\right)k_{1}\left(s\right)k_{2}\left(s\right)-\gamma\left({k_{1}^{2}\left(s\right)+k_{2}^{2}\left(s\right)-k_{3}^{2}\left(s\right)}\right)}\right]$
By the fact that
$\varphi^{\prime}(s)\overline{k}_{2}(\varphi(s)){\bf\bar{n}}_{2}(\varphi(s))\neq
0$ for all $s\in L$, it is proved that
$\left({\gamma^{2}+1}\right)k_{1}\left(s\right)k_{2}\left(s\right)-\gamma\left({k_{1}^{2}\left(s\right)+k_{2}^{2}\left(s\right)-k_{3}^{2}\left(s\right)}\right)\neq
0.$
This is the relation iv.) of theorem.
Now, we will prove the sufficient condition of the theorem.
Thus, we assume that $C$ is a timelike $C^{\infty}-$special Frenet curve in
$\mathbb{E}_{2}^{4}$ with curvatures $k_{1},k_{2}$ and $k_{3}$ satisfying the
relations i.), ii.), iii.) and iv.) for the constant real numbers
$\alpha,\,\,\beta,\,\,\gamma$, and $\delta$.
We define a timelike curve $\bar{C}$ by
${\bf\bar{c}}(s)={\bf c}(s)+\alpha{\bf n}_{1}\left(s\right)+\beta{\bf
n}_{3}\left(s\right)$ (4.17)
where $s$ is the arc length parameter of $C$.
By differentiating the equation (4.17) with respect to $s$ and applying Frenet
formulas,
$\frac{{d{\bf\bar{c}}\left(s\right)}}{{ds}}=\left({1+\alpha
k_{1}\left(s\right)}\right){\bf t}\left(s\right)+\left({\alpha
k_{2}\left(s\right)-\beta k_{3}\left(s\right)}\right){\bf
n}_{2}\left(s\right)$
is obtained. Considering the relation ii.), the last equation is rewritten as;
$\frac{{d{\bf\bar{c}}\left(s\right)}}{{ds}}=\left({\alpha
k_{2}\left(s\right)-\beta k_{3}\left(s\right)}\right)\left({\gamma{\bf
t}\left(s\right)+{\bf n}_{2}\left(s\right)}\right)$
for all $s\in L$. From the relation i.) it is seen that $\bar{C}$ is regular
curve. Thus, arc length parameter of $\bar{C}$ denoted by $\overline{s}$ can
be given by
$\bar{s}=\varphi\left(s\right)=\int\limits_{0}^{s}{\left\|{\frac{{d{\bf\bar{c}}\left(t\right)}}{{dt}}}\right\|dt}$
where $\varphi:L\to\overline{L}$ is a regular map.
The differentiation of $\varphi$ with respect to $s$ is
$\varphi^{\prime}\left(s\right)=\varepsilon\sqrt{\gamma^{2}-1}\left({\alpha
k_{2}\left(s\right)-\beta k_{3}\left(s\right)}\right)>0$ (4.18)
where
$\varepsilon=\left\\{\begin{array}[]{l}\,\,\,1\,\,\,\,\,\,,\,\,\alpha
k_{2}\left(s\right)-\beta k_{3}\left(s\right)>0\\\ -1\,\,\,\,\,,\,\,\alpha
k_{2}\left(s\right)-\beta k_{3}\left(s\right)<0.\\\ \end{array}\right.$
Also, here we notice that
$\gamma\in\left({-\infty,-1}\right)\cup\left({1,\infty}\right)$ and
$\gamma^{2}-1>0$. Thus the timelike curve $\bar{C}$ is rewritten as;
${\bf\bar{c}}\left({\bar{s}}\right)={\bf\bar{c}}\left({\varphi\left(s\right)}\right)={\bf
c}\left(s\right)+\alpha{\bf n}_{1}\left(s\right)+\beta{\bf
n}_{3}\left(s\right)$
for all $s\in L$. Differentiating this equation with respect to $s$, we get
$\left.{\varphi^{\prime}\left(s\right)\frac{{d{\bf\bar{c}}\left({\bar{s}}\right)}}{{d\bar{s}}}}\right|_{\bar{s}=\varphi(s)}=\left({\alpha
k_{2}\left(s\right)-\beta k_{3}\left(s\right)}\right)\left({\gamma{\bf
t}\left(s\right)+{\bf n}_{2}\left(s\right)}\right).$
Now, let us define a unit vector field ${\bf\bar{t}}$ along $\bar{C}$ by
$\frac{{d{\bf\bar{c}}\left({\bar{s}}\right)}}{{d\bar{s}}}$, then
${\bf\bar{t}}\left({\varphi\left(s\right)}\right)=\varepsilon\left({\gamma^{2}-1}\right)^{-{\raise
2.1097pt\hbox{$1$}\\!\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}\\!\lower
2.1097pt\hbox{$2$}}}\left({\gamma{\bf t}\left(s\right)+{\bf
n}_{2}\left(s\right)}\right).$ (4.19)
By differentiating the equation (4.19) with respect to $s$ and using Frenet
formulas,
$\left.{\varphi^{\prime}\left(s\right)\frac{{d{\bf\bar{t}}\left({\varphi\left(s\right)}\right)}}{{d\bar{s}}}}\right|_{\bar{s}=\varphi(s)}=\varepsilon\left({\gamma^{2}-1}\right)^{-{\raise
2.1097pt\hbox{$1$}\\!\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}\\!\lower
2.1097pt\hbox{$2$}}}\left[{-\left({\gamma
k_{1}\left(s\right)-k_{2}\left(s\right)}\right){\bf
n}_{1}\left(s\right)+k_{3}\left(s\right){\bf n}_{3}\left(s\right)}\right]$
and
$\left\|{\left.{\varphi^{\prime}\left(s\right)\frac{{d{\bf\bar{t}}\left({\varphi\left(s\right)}\right)}}{{d\bar{s}}}}\right|_{\bar{s}=\varphi(s)}}\right\|=\frac{{\sqrt{\left|{-\left({\gamma
k_{1}\left(s\right)-k_{2}\left(s\right)}\right)^{2}+k_{3}^{2}\left(s\right)}\right|}}}{{\varphi^{\prime}(s)\sqrt{\gamma^{2}-1}}}.$
By relations $iii.)$, it is seen that
$\sqrt{\left|{-\left({\gamma
k_{1}\left(s\right)-k_{2}\left(s\right)}\right)^{2}+k_{3}^{2}\left(s\right)}\right|}=\sqrt{\left|{-\delta^{2}+1}\right|k_{3}^{2}\left(s\right)}$
and we notice that
$\delta\in\left({-\infty,-1}\right)\cup\left({1,\infty}\right)$ and
$\gamma^{2}-1>0$.
Thus, we can write
$\left\|{\left.{\frac{{d{\bf\bar{t}}\left({\varphi\left(s\right)}\right)}}{{d\bar{s}}}}\right|_{\bar{s}=\varphi(s)}}\right\|=\frac{{\sqrt{\delta^{2}-1}\,k_{3}^{2}\left(s\right)}}{{\varphi^{\prime}\left(s\right)\sqrt{\gamma^{2}-1}}}.$
Since $k_{3}\left(s\right)>0$ and $\varphi^{\prime}\left(s\right)>0$ for all
$s\in L$, we obtain
$\overline{k}_{1}\left({\varphi\left(s\right)}\right)=\left\|{\left.{\frac{{d{\bf\bar{t}}\left({\varphi\left(s\right)}\right)}}{{d\bar{s}}}}\right|_{\bar{s}=\varphi(s)}}\right\|>0.$
(4.20)
Thus, ${\bf\bar{n}}_{1}$ timelike unit vector field can be defined by
$\begin{array}[]{l}{\bf\bar{n}}\left({\bar{s}}\right)={\bf\bar{n}}_{1}\left({\varphi\left(s\right)}\right)\\\
\quad\quad\;=\,-\frac{1}{{\bar{k}_{1}\left({\varphi\left(s\right)}\right)}}{\bf\bar{t}}\left({\varphi\left(s\right)}\right)\\\
\quad\quad\;=\,\frac{{\left({\gamma
k_{1}\left(s\right)-k_{2}\left(s\right)}\right){\bf
n}_{1}\left(s\right)-k_{3}\left(s\right){\bf
n}_{3}\left(s\right)}}{{\varepsilon\sqrt{\left({\gamma
k_{1}\left(s\right)-k_{2}\left(s\right)}\right)^{2}-k_{3}^{2}\left(s\right)}}}.\\\
\end{array}$
Also, ${\bf\bar{n}}_{1}$ can be given in the form
${\bf\bar{n}}_{1}\left({\varphi\left(s\right)}\right)=\cosh\xi\left(s\right){\bf
n}_{1}\left(s\right)+\sinh\xi\left(s\right){\bf n}_{3}\left(s\right)$ (4.21)
where
$\cosh\xi=\frac{{\gamma
k_{1}\left(s\right)-k_{2}\left(s\right)}}{{\varepsilon\sqrt{\left({\gamma
k_{1}\left(s\right)-k_{2}\left(s\right)}\right)^{2}-k_{3}^{2}\left(s\right)}}},$
(4.22)
$\sinh\xi=\frac{{-k_{3}\left(s\right)}}{{\varepsilon\sqrt{\left({\gamma
k_{1}\left(s\right)-k_{2}\left(s\right)}\right)^{2}-k_{3}^{2}\left(s\right)}}},$
(4.23)
for all $s\in L$. Here $\xi$ is a $C^{\infty}$-function on $L$.
By differentiating (4.21) with respect to $s$ and using the Frenet formulas,
we get
$\begin{array}[]{l}\left\|{\left.{\frac{{d{\bf\bar{n}}_{1}\left({\varphi\left(s\right)}\right)}}{{d\bar{s}}}}\right|_{\bar{s}=\varphi(s)}}\right\|=k_{1}\left(s\right)\cosh\xi\left(s\right){\bf
t}\left(s\right)+\frac{{d\left({\cosh\xi\left(s\right)}\right)}}{{ds}}{\bf
n}_{1}\left(s\right)\\\
\quad\quad\quad\quad\quad\quad\quad\quad\;+\left({k_{2}\left(s\right)\cosh\xi\left(s\right)-k_{3}\left(s\right)\sinh\xi\left(s\right)}\right){\bf
n}_{2}\left(s\right)\\\
\quad\quad\quad\quad\quad\quad\quad\quad\;+\frac{{d\left({\sinh\xi\left(s\right)}\right)}}{{ds}}{\bf
n}_{3}(s).\\\ \end{array}.$
The differentiation of the relation iii.) with respect to $s$ is
$\left({\gamma
k^{\prime}_{1}\left(s\right)-k^{\prime}_{2}\left(s\right)}\right)\,k_{3}\left(s\right)+\left({\gamma
k_{1}\left(s\right)-k_{2}\left(s\right)}\right)\,k^{\prime}_{3}\left(s\right)=0$
(4.24)
Substituting the equation (4.24), we get
$\frac{{d\left({\cosh\xi\left(s\right)}\right)}}{{ds}}=0\quad,\quad\frac{{d\left({\sinh\xi\left(s\right)}\right)}}{{ds}}=0,$
that is, $\xi$ is a constant function on $L$ with value $\xi_{0}$. Thus, we
write
$\cosh\xi_{0}=\frac{{\gamma
k_{1}\left(s\right)-k_{2}\left(s\right)}}{{\varepsilon\sqrt{\left({\gamma
k_{1}\left(s\right)-k_{2}\left(s\right)}\right)^{2}-k_{3}^{2}\left(s\right)}}},$
(4.25)
$\sinh\xi_{0}=\frac{{-k_{3}\left(s\right)}}{{\varepsilon\sqrt{\left({\gamma
k_{1}\left(s\right)-k_{2}\left(s\right)}\right)^{2}-k_{3}^{2}\left(s\right)}}}.$
(4.26)
Then, from the equation (4.2), it satisfies
${\bf\bar{n}}_{1}\left({\varphi\left(s\right)}\right)=\cosh\xi_{0}\left(s\right){\bf
n}_{1}\left(s\right)+\sinh\xi_{0}\left(s\right){\bf n}_{3}\left(s\right).$
(4.27)
By considering the equations (4.19) and (4.20), we obtain
$\overline{k}_{1}\left({\varphi\left(s\right)}\right)\overline{\bf
t}\left({\varphi\left(s\right)}\right)=\frac{{\left({\gamma
k_{1}\left(s\right)-k_{2}\left(s\right)}\right)^{2}-k_{3}^{2}\left(s\right)}}{{\varepsilon\varphi^{\prime}\left(s\right)\left({\gamma^{2}-1}\right)\sqrt{\left({\gamma
k_{1}\left(s\right)-k_{2}\left(s\right)}\right)^{2}-k_{3}^{2}\left(s\right)}}}\left({\gamma{\bf
t}\left(s\right)+{\bf n}_{2}\left(s\right)}\right).$
Also, by substituting the equations (4.25) and (4.26) into equation (4.20), we
get
$\begin{array}[]{l}\left.{\frac{{d{\bf\bar{n}}_{1}\left({\varphi\left(s\right)}\right)}}{{d\bar{s}}}}\right|_{\bar{s}=\varphi(s)}=\frac{{k_{1}\left(s\right)\left({\gamma
k_{1}\left(s\right)-k_{2}\left(s\right)}\right)}}{{\varepsilon\varphi^{\prime}\left(s\right)\left({\gamma^{2}-1}\right)\sqrt{\left({\gamma
k_{1}\left(s\right)-k_{2}\left(s\right)}\right)^{2}-k_{3}^{2}\left(s\right)}}}{\bf
t}\left(s\right)\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\frac{{k_{2}\left(s\right)\left({\gamma
k_{1}\left(s\right)-k_{2}\left(s\right)}\right)+k_{3}(s)}}{{\varepsilon\varphi^{\prime}\left(s\right)\sqrt{\left({\gamma
k_{1}\left(s\right)-k_{2}\left(s\right)}\right)^{2}-k_{3}^{2}\left(s\right)}}}{\bf
n}_{2}\left(s\right)\\\ \end{array}$
for $s\in L$. By the last two equations, we obtain
$\left.{\frac{{d{\bf\bar{n}}_{1}\left({\varphi\left(s\right)}\right)}}{{d\bar{s}}}}\right|_{\bar{s}=\varphi(s)}-\overline{k}_{1}\left({\varphi\left(s\right)}\right)\,\overline{\bf
t}\left({\varphi\left(s\right)}\right)=\frac{{P\left(s\right)}}{{R\left(s\right)}}{\bf
t}\left(s\right)+\frac{{Q\left(s\right)}}{{R\left(s\right)}}{\bf
n}_{2}\left(s\right)$
where
$\begin{array}[]{l}P\left(s\right)=\left({\gamma^{2}+1}\right)k_{1}\left(s\right)k_{2}\left(s\right)-\gamma\left({k_{1}^{2}\left(s\right)+k_{2}^{2}\left(s\right)-k_{3}^{2}\left(s\right)}\right),\\\
Q\left(s\right)=\gamma\left[{\left({\gamma^{2}+1}\right)k_{1}\left(s\right)k_{2}\left(s\right)-\gamma\left({k_{1}^{2}\left(s\right)+k_{2}^{2}\left(s\right)-k_{3}^{2}\left(s\right)}\right)}\right],\\\
R\left(s\right)=\varepsilon\varphi^{\prime}\left(s\right)\left({\gamma^{2}-1}\right)\sqrt{\left({\gamma
k_{1}\left(s\right)-k_{2}\left(s\right)}\right)^{2}-k_{3}^{2}\left(s\right)}\neq
0.\\\ \end{array}$
By the fact that
$\overline{k}_{2}\left({\varphi\left(s\right)}\right)\,=\left\|{\left.{\frac{{d{\bf\bar{n}}_{1}\left({\varphi\left(s\right)}\right)}}{{d\bar{s}}}}\right|_{\bar{s}=\varphi(s)}-\overline{k}_{1}\left({\varphi\left(s\right)}\right)\,\overline{\bf
t}\left({\varphi\left(s\right)}\right)}\right\|>0$
for all $s\in L$, we see
$\overline{k}_{2}\left({\varphi\left(s\right)}\right)=\frac{{\left|{\left({\gamma^{2}+1}\right)k_{1}\left(s\right)k_{2}\left(s\right)-\gamma\left({k_{1}^{2}\left(s\right)+k_{2}^{2}\left(s\right)-k_{3}^{2}\left(s\right)}\right)}\right|}}{{\varphi^{\prime}\left(s\right)\sqrt{\left({\gamma^{2}-1}\right)\left[{\left({\gamma
k_{1}\left(s\right)-k_{2}\left(s\right)}\right)^{2}-k_{3}^{2}\left(s\right)}\right]\,}}}.$
Thus, we can define a unit vector field
${\bf\bar{n}}_{2}\left({\bar{s}}\right)$ along $\overline{C}$ by
$\begin{array}[]{l}{\bf\bar{n}}_{2}\left({\bar{s}}\right)={\bf\bar{n}}_{2}\left({\varphi\left(s\right)}\right)\\\
\quad\quad\;\;\;=\,\,\frac{1}{{\bar{k}_{2}\left({\varphi\left(s\right)}\right)}}\left[{\left.{\frac{{d{\bf\bar{n}}_{1}\left({\varphi\left(s\right)}\right)}}{{d\bar{s}}}}\right|_{\bar{s}=\varphi(s)}-\overline{k}_{1}\left({\varphi\left(s\right)}\right)\,\overline{\bf
t}\left({\varphi\left(s\right)}\right)}\right]\\\ \end{array},$
such that
${\bf\bar{n}}_{2}\left({\varphi\left(s\right)}\right)=\frac{1}{{\varepsilon\sqrt{\gamma^{2}-1}}}\left({{\bf
t}\left(s\right)+\gamma{\bf n}_{2}\left(s\right)}\right).$ (4.28)
Also, since
${\bf\bar{n}}_{3}\left({\varphi\left(s\right)}\right)=\sinh\xi_{0}\left(s\right){\bf
n}_{1}\left(s\right)+\cosh\xi_{0}\left(s\right){\bf n}_{3}\left(s\right)$ for
all $s\in L$, another unit vector field ${\bf\bar{n}}_{3}$ along
$\overline{C}$ can
$\begin{array}[]{l}{\bf\bar{n}}_{3}\left({\bar{s}}\right)={\bf\bar{n}}_{3}\left({\varphi\left(s\right)}\right)\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\;=\,\frac{1}{{\varepsilon\sqrt{\left({\gamma
k_{1}\left(s\right)-k_{2}\left(s\right)}\right)^{2}-k_{3}^{2}\left(s\right)}}}\left({-k_{3}\left(s\right){\bf
n}_{1}\left(s\right)+\left({\gamma
k_{1}\left(s\right)-k_{2}\left(s\right)}\right){\bf
n}_{3}\left(s\right)}\right).\\\ \end{array}$ (4.29)
Now, from the equations (4.19), (4.27), (4.28) and (4.29), it is seen that
$\det\left[{{\bf\bar{t}}\left({\varphi\left(s\right)}\right),\,{\bf\bar{n}}_{1}\left({\varphi\left(s\right)}\right),\,\,{\bf\bar{n}}_{2}\left({\varphi\left(s\right)}\right),\,{\bf\bar{n}}_{3}\left({\varphi\left(s\right)}\right)}\right]=\det\left[{{\bf
t}\left(s\right),\,{\bf n}_{1}\left(s\right),\,\,{\bf
n}_{2}\left(s\right),\,{\bf n}_{3}\left(s\right)}\right]=1.$
${\bf\bar{t}},\,\;{\bf\bar{n}}_{1},\,\,{\bf\bar{n}}_{2}$ and
${\bf\bar{n}}_{3}$ are mutually orthogonal vector fields satisfying
$g\left({{\bf\bar{t}}\left({\varphi\left(s\right)}\right),{\bf\bar{t}}\left({\varphi\left(s\right)}\right)}\right)=g\left({{\bf\bar{n}}_{1}\left({\varphi\left(s\right)}\right),{\bf\bar{n}}_{1}\left({\varphi\left(s\right)}\right)}\right)=-1,$
$\,g\left({{\bf\bar{n}}_{2}\left({\varphi\left(s\right)}\right),{\bf\bar{n}}_{2}\left({\varphi\left(s\right)}\right)}\right)=g\left({{\bf\bar{n}}_{3}\left({\varphi\left(s\right)}\right),{\bf\bar{n}}_{3}\left({\varphi\left(s\right)}\right)}\right)=1.$
Thus the tetrahedron
$\left\\{{{\bf\bar{t}},\,{\bf\bar{n}}_{1},\,\,{\bf\bar{n}}_{2},\,{\bf\bar{n}}_{3}}\right\\}$
along $\overline{C}$ is an orthonormal frame where ${\bf\bar{t}}$ and
$\,{\bf\bar{n}}_{1}$ are timelike vector fields, $\,{\bf\bar{n}}_{2}$ and
$\,{\bf\bar{n}}_{3}$ are spacelike vector fields.
On the other hand, by considering the equations (4.29) and the differentiation
of the equation (4.28), we obtain
$\begin{array}[]{l}\overline{k}_{3}\left({\varphi\left(s\right)}\right)=-\left\langle{\left.{\frac{{d{\bf\bar{n}}_{2}\left({\varphi\left(s\right)}\right)}}{{d\bar{s}}}}\right|_{\bar{s}=\varphi(s)},{\bf\bar{n}}_{3}\left({\varphi\left(s\right)}\right)}\right\rangle\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\,\frac{{\sqrt{\gamma^{2}-1}\,k_{1}\left(s\right)k_{3}\left(s\right)}}{{\varphi^{\prime}\left(s\right)\sqrt{\left({\gamma
k_{1}\left(s\right)-k_{2}\left(s\right)}\right)^{2}-k_{3}^{2}\left(s\right)\,}}}>0\\\
\end{array}$
for all $s\in L$. Therefore, $\overline{C}$ is a $C^{\infty}-$special curve in
$\mathbb{E}_{2}^{4}$ and the Frenet (1,3)-normal plane at the corresponding
point
${\bf\bar{c}}\left({\bar{s}}\right)={\bf\bar{c}}\left({\varphi\left(s\right)}\right)$
of $\overline{C}$. Thus, $(C,\overline{C})$ is a mate of (1,3)-Bertrand curve
in $\mathbb{E}_{2}^{4}$.
Finally, the proof of the theorem is completed.
## References
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* [3] López R., Differential Geometry of Curves and Surfaces in Lorentz-Minkowski space, Mini-Course taught at the Instituto de Matemática e Estat´stica (IME-USP) University of Sao Paulo, Brazil, October 18, (2008).
* [4] Matsuda, H., Yorozu, S., Notes on Bertrand curves, Yokohama Math. J. 50, no. 1-2, 41–58,(2003).
* [5] O’Neill B., Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York (1983).
* [6] Wong, Y.-C. and H.-F. Lai, A critical examination of the theory of curves in three-dimensional geometry, Tohoku Math. J. (2) 19, 1-31,(1967).
|
arxiv-papers
| 2010-03-05T09:50:29 |
2024-09-04T02:49:08.806816
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Soley Ersoy and Murat Tosun",
"submitter": "Soley Ersoy",
"url": "https://arxiv.org/abs/1003.1220"
}
|
1003.1234
|
# Separable states and the geometric phases of an interacting two-spin system
C. W. Niu, G. F. Xu, Longjiang Liu, L. Kang and D. M. Tong111tdm@sdu.edu.cn
Department of Physics, Shandong University, Jinan 250100, China
L. C. Kwek Center for Quantum Technologies, National University of Singapore,
Science Drive 2 Singapore 117543 Institute of Advanced Studies, Nanyang
Technological University, 60 Nanyang View Singapore 639673
###### Abstract
It is known that an interacting bipartite system evolves as an entangled state
in general, even if it is initially in a separable state. Due to the
entanglement of the state, the geometric phase of the system is not equal to
the sum of the geometric phases of its two subsystems. However, there may
exist a set of states in which the nonlocal interaction does not affect the
separability of the states, and the geometric phase of the bipartite system is
then always equal to the sum of the geometric phases of its subsystems. In
this paper, we illustrate this point by investigating a well known physical
model. We give a necessary and sufficient condition in which a separable state
remains separable so that the geometric phase of the system is always equal to
the sum of the geometric phases of its subsystems.
###### pacs:
03.65.Vf
## I Introduction
The notion of geometric phase was first addressed by Pancharatnam for the
comparison of the phases of two beams of polarized light in 1956 pancharatnam
. It was later shown to have important consequences for quantum systems. In
1984, Berry demonstrated that quantum system undergoing a cyclic adiabatic
evolution acquires a phase with geometric nature Berry . Since then, geometric
phase has attracted great interest. The original notion of Berry phase has
been extended to nonadiabatic cyclic evolution by Aharonov and Anandan in 1987
Aharonov , and to nonadiabatic and noncyclic evolution by Samuel and Bhandari
in 1988 Samuel .
While all these extensions of quantum systems are in pure states, another line
of development has been towards extending the geometric phase to mixed states.
The early extension to mixed states was given by Uhlmann within the
mathematical context of purification Uhlmann . In 2000, Sj$\ddot{o}$qvist et
al. introduced an alternative definition of geometric phases for mixed states
under unitary evolution based on quantum interferometry Sj , and subsequently
Singh et al. gave a kinematic description of the mixed state geometric phase
and extended it to degenerate density operator Singh . The generalization of
mixed geometric phases to quantum systems in nonunitary evolution was given by
Tong et al. in 2004 Tong . Other discussions or experimental demonstrations of
geometric phases for mixed states may be found in papers Carollo ; Ericsson ;
Marzlin ; Kamleitner ; Whitney ; yi ; Lombardo ; Bassi ; Sarandy ; Rezakhani ;
Goto ; Mller ; Buric ; Du ; Ericsson2 ; yin .
Another interesting issue of geometric phase is the relation of the bipartite
or multipartite system with its subsystems. Sj$\ddot{o}$qvist calculated the
geometric phase of a pair of entangled spin half particles precessing in a
time-independent uniform magnetic field Sj1 , and the relative phase for
polarization-entangled two-photon systems was considered by Hessmo et al
Hessmo . Tong et al. calculated the geometric phase of a bipartite entangled
spin-half system in a rotating magnetic field Tong2 and investigated
entangled bipartite systems with local unitary evolutions Tong1 . The effect
of entanglement on the mutual geometric phase was recently studied by
Williamson et al Williamson . Other discussions on geometric phases of
composite systems and its applications may be found in Refs yi1 ; Li ; yi3 ;
Ge ; Xing ; wangxb .
All the previous discussions concerning the relation of the geometric phase of
the composite system with its subsystems were of the systems under local
unitary evolutions, $U(t)=U_{a}(t)\otimes U_{b}(t)$. It was shown that the
geometric phase of the composite system, $\gamma_{ab}$, does not equal the sum
of the geometric phases of its subsystems, $\gamma_{a}$ and $\gamma_{b}$, in
general Tong1 ; Williamson . The expression
$\gamma_{ab}=\gamma_{a}+\gamma_{b}$ is valid only if the initial state is a
separable one. This is because that the entanglement of the state leads to an
indecomposable geometric phase of the composite system. Since the interaction
between two subsystems can lead to an entanglement of the subsystems, it is
usually deemed that the geometric phase of the composite system in nonlocal
unitary evolution does not equal the sum of the geometric phases of its
subsystems in general, even if the initial state of the system is separable.
In the present paper, we investigate a well known physical model, two
interacting spin-half particles in a rotating magnetic field. We aim to show
that there may exist a set of states in which the nonlocal interaction does
not affect the separability of the states, and therefore the geometric phase
of the bipartite system is always equal to the sum of the geometric phases of
its subsystems. A necessary and sufficient condition for the set of separable
states is given.
## II The interacting two-spin half model
Consider the system of two interacting spin-half particles in a rotating
magnetic field, the Hamiltonian of which is described as
$\displaystyle\hat{H}(t)=\hat{H}_{a}(t)\otimes
I+I\otimes\hat{H}_{b}(t)+\hat{H}_{ab}(t),$ (1)
where $\hat{H_{\mu}}(t)=\vec{B}(t)\cdot\vec{\sigma}_{\mu}$ $~{}(\mu=a,b)$,
$\hat{H}_{ab}(t)=J\vec{\sigma}_{a}\cdot\vec{\sigma}_{b}$. Here,
$\vec{B}(t)=B(\sin\theta\cos\omega t,\sin\theta\sin\omega t,\cos\theta)$ is
the rotating magnetic field. $\vec{\sigma}_{a}$ and $\vec{\sigma}_{b}$ are the
Pauli operators of spins $a$ and $b$, respectively. $J$ denotes the
interaction strength between $a$ and $b$, and $J>0$ describes
antiferromagnetic coupling and $J<0$ describes ferromagnetic coupling.
The state of the system, $|\psi(t)\rangle$ , satisfies the
Schr$\ddot{o}$dinger equation,
$\displaystyle i\frac{d}{dt}|\psi(t)\rangle=\hat{H}(t)|\psi(t)\rangle,$ (2)
with initial state being $|\psi(0)\rangle$. $|\psi(t)\rangle$ may be expressed
as
$\displaystyle|\psi(t)\rangle=f_{1}(t)|00\rangle+f_{2}(t)|01\rangle+f_{3}(t)|10\rangle+f_{4}(t)|11\rangle,$
(3)
where $|ij\rangle$ $(i,j=0,1)$ are the abbreviations of
$|i\rangle\otimes|j\rangle$ with $|0\rangle=\bordermatrix{&\cr&1\cr&0\cr}$ and
$|1\rangle=\bordermatrix{&\cr&0\cr&1\cr}$, and $f_{k}(t)~{}~{}(k=1,2,3,4)$ are
functions of $t$ to be determined, satisfying
$\sum_{k=1}^{4}|f_{k}(t)|^{2}=1$. Substituting Eq.(3) into Eq. (2), we have
$\displaystyle
i\frac{d}{dt}\bordermatrix{&\cr&f_{1}(t)\cr&f_{2}(t)\cr&f_{3}(t)\cr&f_{4}(t)\cr}=\bordermatrix{&\cr&J+2B\cos\theta&B\sin\theta
e^{-i\omega t}&B\sin\theta e^{-i\omega t}&0\cr&B\sin\theta e^{i\omega
t}&-J&2J&B\sin\theta e^{-i\omega t}\cr&B\sin\theta e^{i\omega
t}&2J&-J&B\sin\theta e^{-i\omega t}\cr&0&B\sin\theta e^{i\omega t}&B\sin\theta
e^{i\omega
t}&J-2B\cos\theta\cr}\bordermatrix{&\cr&f_{1}(t)\cr&f_{2}(t)\cr&f_{3}(t)\cr&f_{4}(t)\cr},$
(4)
that is,
$\left\\{\begin{array}[]{rl}i\dot{f}_{1}&=(J+2B\cos\theta)f_{1}+(B\sin\theta)e^{-i\omega
t}f_{2}+(B\sin\theta)e^{-i\omega t}f_{3},\\\
i\dot{f}_{2}&=(B\sin\theta)e^{i\omega
t}f_{1}-Jf_{2}+2Jf_{3}+(B\sin\theta)e^{-i\omega t}f_{4},\\\
i\dot{f}_{3}&=(B\sin\theta)e^{i\omega
t}f_{1}+2Jf_{2}-Jf_{3}+(B\sin\theta)e^{-i\omega t}f_{4},\\\
i\dot{f}_{4}&=(B\sin\theta)e^{i\omega t}f_{2}+(B\sin\theta)e^{i\omega
t}f_{3}+(J-2B\cos\theta)f_{4}.\end{array}\right.$ (5)
To resolve the above differential equations, we further let
$f_{1}(t)=\bar{f}_{1}(t)e^{-i\omega
t},~{}~{}f_{2}(t)=\bar{f}_{2}(t),~{}~{}f_{3}(t)=\bar{f}_{3}(t),~{}~{}f_{4}(t)=\bar{f}_{4}(t)e^{i\omega
t}$. Then, Eq.(5) becomes
$\left\\{\begin{array}[]{rl}i\dot{\bar{f}}_{1}&=(J+2B\cos\theta-\omega)\bar{f}_{1}+B\sin\theta\bar{f}_{2}+B\sin\theta\bar{f}_{3},\\\
i\dot{\bar{f}}_{2}&=B\sin\theta\bar{f}_{1}-J\bar{f}_{2}+2J\bar{f}_{3}+B\sin\theta\bar{f}_{4},\\\
i\dot{\bar{f}}_{3}&=B\sin\theta\bar{f}_{1}+2J\bar{f}_{2}-J\bar{f}_{3}+B\sin\theta\bar{f}_{4},\\\
i\dot{\bar{f}}_{4}&=B\sin\theta\bar{f}_{2}+B\sin\theta\bar{f}_{3}+(J-2B\cos\theta+\omega)\bar{f}_{4}.\end{array}\right.$
(6)
Eq. (6) is a set of first-order linear ordinary differential equations. Its
solution can be obtained by solving the characteristic equation. The four
characteristic roots are
$\displaystyle\lambda_{1}$ $\displaystyle=$ $\displaystyle 3J,$
$\displaystyle\lambda_{2}$ $\displaystyle=$ $\displaystyle-J,$
$\displaystyle\lambda_{3}$ $\displaystyle=$
$\displaystyle-J+\sqrt{4B^{2}\sin^{2}\theta+(2B\cos\theta-\omega)^{2}},$
$\displaystyle\lambda_{4}$ $\displaystyle=$
$\displaystyle-J-\sqrt{4B^{2}\sin^{2}\theta+(2B\cos\theta-\omega)^{2}},$ (7)
each of which corresponding to a characteristic solution with respect to
$\bar{f}_{k}(t)$. With the help of the solutions of $\bar{f}_{k}(t)$, which
directly give the solutions of $f_{k}(t)$, the general solution of Eq.(2) can
be expressed as
$\displaystyle|\psi(t)\rangle=c_{1}|\psi_{1}(t)\rangle+c_{2}|\psi_{2}(t)\rangle+c_{3}|\psi_{3}(t)\rangle+c_{4}|\psi_{4}(t)\rangle,$
(8)
where the time-independent coefficients $c_{k}$ $(k=1,2,3,4)$,
$\sum_{k=1}^{4}|c_{k}|^{2}=1$, are to be determined by the initial condition,
and the four particular solutions read
$\displaystyle|\psi_{1}(t)\rangle$ $\displaystyle=$ $\displaystyle
e^{i\lambda_{1}t}\frac{1}{\sqrt{2}}\bordermatrix{&\cr&0\cr&1\cr&-1\cr&0\cr},$
$\displaystyle|\psi_{2}(t)\rangle$ $\displaystyle=$ $\displaystyle
e^{i\lambda_{2}t}\frac{1}{\sqrt{2}}\bordermatrix{&\cr&-\frac{2B\sin\theta}{\sqrt{4B^{2}\sin^{2}\theta+(2B\cos\theta-\omega)^{2}}}e^{-i\omega
t}\cr&\frac{2B\cos\theta-\omega}{\sqrt{4B^{2}\sin^{2}\theta+(2B\cos\theta-\omega)^{2}}}\cr&\frac{2B\cos\theta-\omega}{\sqrt{4B^{2}\sin^{2}\theta+(2B\cos\theta-\omega)^{2}}}\cr&\frac{2B\sin\theta}{\sqrt{4B^{2}\sin^{2}\theta+(2B\cos\theta-\omega)^{2}}}e^{i\omega
t}\cr},$ $\displaystyle|\psi_{3}(t)\rangle$ $\displaystyle=$ $\displaystyle
e^{i\lambda_{3}t}\bordermatrix{&\cr&-\frac{\sqrt{4B^{2}\sin^{2}\theta+(2B\cos\theta-\omega)^{2}}-(2B\cos\theta-\omega)}{2\sqrt{4B^{2}\sin^{2}\theta+(2B\cos\theta-\omega)^{2}}}e^{-i\omega
t}\cr&\frac{B\sin\theta}{\sqrt{4B^{2}\sin^{2}\theta+(2B\cos\theta-\omega)^{2}}}\cr&\frac{B\sin\theta}{\sqrt{4B^{2}\sin^{2}\theta+(2B\cos\theta-\omega)^{2}}}\cr&-\frac{\sqrt{4B^{2}\sin^{2}\theta+(2B\cos\theta-\omega)^{2}}+(2B\cos\theta-\omega)}{2\sqrt{4B^{2}\sin^{2}\theta+(2B\cos\theta-\omega)^{2}}}e^{i\omega
t}\cr},$ $\displaystyle|\psi_{4}(t)\rangle$ $\displaystyle=$ $\displaystyle
e^{i\lambda_{4}t}\bordermatrix{&\cr&\frac{\sqrt{4B^{2}\sin^{2}\theta+(2B\cos\theta-\omega)^{2}}+(2B\cos\theta-\omega)}{2\sqrt{4B^{2}\sin^{2}\theta+(2B\cos\theta-\omega)^{2}}}e^{-i\omega
t}\cr&\frac{B\sin\theta}{\sqrt{4B^{2}\sin^{2}\theta+(2B\cos\theta-\omega)^{2}}}\cr&\frac{B\sin\theta}{\sqrt{4B^{2}\sin^{2}\theta+(2B\cos\theta-\omega)^{2}}}\cr&\frac{\sqrt{4B^{2}\sin^{2}\theta+(2B\cos\theta-\omega)^{2}}-(2B\cos\theta-\omega)}{2\sqrt{4B^{2}\sin^{2}\theta+(2B\cos\theta-\omega)^{2}}}e^{i\omega
t}\cr}.$ (9)
## III The geometric phases of the two-spin half system
If the two-spin half system is initially in state $|\psi(0)\rangle$, the
geometric phase obtained by the quantum system during the time $t\in[0,\tau]$
can be calculated by using the formula Bhandari ; Mukunda ,
$\displaystyle\gamma_{ab}(\tau)=\arg\langle{\psi(0)}|\psi(\tau)\rangle+i\int_{0}^{\tau}\langle{\psi(t)}|\dot{\psi}(t)\rangle
dt.$ (10)
However, both the subsystems $a$ and $b$ are generally in mixed states due to
the nonlocal interaction, even if the initial state $|\psi(0)\rangle$ is
separable. The mixed states of the subsystems can be expressed as density
operators,
$\displaystyle\rho_{a}(t)=tr_{b}|\psi(t)\rangle\langle\psi(t)|,~{}~{}\rho_{b}(t)=tr_{a}|\psi(t)\rangle\langle\psi(t)|.$
(11)
The geometric phases of the mixed states in nonunitary evolutions are
calculated by using the formula Tong
$\displaystyle\gamma_{\mu}(\tau)=\arg\left(\sum_{m=1}^{2}\sqrt{\omega^{\mu}_{m}(0)\omega^{\mu}_{m}(\tau)}\langle{\phi^{\mu}_{m}(0)}|\phi^{\mu}_{m}(\tau)\rangle
e^{-\int_{0}^{\tau}\langle{\phi^{\mu}_{m}(t)}|\dot{\phi^{\mu}}_{m}(t)\rangle
dt}\right),$ (12)
where $\omega^{\mu}_{m}(t)$ and $|\phi^{\mu}_{m}(t)\rangle$ are the
eigenvalues and eigenstates of the density operators $\rho_{\mu}(t)$ (
$\mu=a,b$), respectively.
By substituting Eqs. (8) and (9) into Eqs. (10) and (11), and further using
Eq. (12), one can calculate the geometric phase of the two-spin half system
and the geometric phases of its two subsystems. It is easy to show that
$\gamma_{ab}$ is not equal to the sum of $\gamma_{a}$ and $\gamma_{b}$ in
general, even if the initial state $|\psi(0)\rangle$ is a separable one.
To illustrate this point, we take $|\psi(0)\rangle=|01\rangle$ as an example.
In this case, the state of the system at time $t$ reads
$\displaystyle|\psi(t)\rangle=\frac{1}{\sqrt{2}}|\psi_{1}(t)\rangle+\frac{1}{\sqrt{2}}\cos\eta|\psi_{2}(t)\rangle+\frac{1}{2}\sin\eta|\psi_{3}(t)\rangle+\frac{1}{2}\sin\eta|\psi_{4}(t)\rangle,$
(13)
and the geometric phase obtained by the system during the time $t\in[0,\tau]$
is
$\displaystyle\gamma_{ab}=\arctan\frac{\sin
4J\tau}{\cos^{2}\eta+\sin^{2}\eta\cos\alpha\tau+\cos 4J\tau}-2J\tau,$ (14)
where
$\displaystyle\alpha=\sqrt{4B^{2}\sin^{2}\theta+(2B\cos\theta-\omega)^{2}},$
(15)
and
$\displaystyle\tan\eta=\frac{2B\sin\theta}{2B\cos\theta-\omega}.$ (16)
The reduced density operators of the subsystems $a$ and $b$ are
$\displaystyle\rho_{\mu}=\bordermatrix{&\cr&\rho^{\mu}_{11}&\rho^{\mu}_{12}\cr&\rho^{\mu}_{21}&\rho^{\mu}_{22}},~{}~{}\mu=a,b,$
(17)
where
$\displaystyle\rho^{a}_{11}$ $\displaystyle=$ $\displaystyle
1-\rho^{a}_{22}=\frac{1}{2}\left(1+(\cos^{2}\eta+\sin^{2}\eta\cos\alpha t)\cos
4Jt\right),$ $\displaystyle\rho^{a}_{12}$ $\displaystyle=$
$\displaystyle{\rho^{a}_{21}}^{*}=\frac{1}{2}\left(\sin\eta\cos\eta(1-\cos\alpha
t)+i\sin\eta\sin\alpha t\right)e^{-i\omega t}\cos 4Jt;$ (18)
$\displaystyle\rho^{b}_{11}$ $\displaystyle=$ $\displaystyle
1-\rho^{b}_{22}=\frac{1}{2}\left(1-(\cos^{2}\eta+\sin^{2}\eta\cos\alpha t)\cos
4Jt\right),$ $\displaystyle\rho^{b}_{12}$ $\displaystyle=$
$\displaystyle{\rho^{b}_{21}}^{*}=-\frac{1}{2}\left(\sin\eta\cos\eta(1-\cos\alpha
t)+i\sin\eta\sin\alpha t\right)e^{-i\omega t}\cos 4Jt.$ (19)
The geometric phases obtained by the subsystems during the time $t\in[0,\tau]$
are respectively
$\displaystyle\gamma_{a}(\tau)=\arctan\frac{-\cos\eta\sqrt{1-\cos\alpha\tau}}{\sqrt{1+\cos\alpha\tau}}+\frac{\omega\sin^{2}\eta}{2\alpha}\sin\alpha\tau+\frac{1}{2}\alpha\tau\cos\eta-\frac{1}{2}\omega\tau\sin^{2}\eta,$
(20)
and
$\displaystyle\gamma_{b}(\tau)=\arctan(\cos\eta\tan\frac{\alpha\tau}{2})-\frac{\omega\sin^{2}\eta}{2\alpha}\sin\alpha\tau-\frac{1}{2}\alpha\tau\cos\eta+\frac{1}{2}\omega\tau\sin^{2}\eta.$
(21)
Clearly, the geometric phase of the large system is not equal to the sum of
the geometric phases of the two subsystems,
$\gamma_{ab}\neq\gamma_{a}+\gamma_{b}$, even if the initial state
$|\psi(0)\rangle$ is a separable one.
## IV Condition for geometric phase of the system being equal to the sum of
those of its subsystems
Geometric phase is useful in quantum calculation, but a real quantum system
may comprise two or more subsystems with interactions between them. In this
case when interactions appear, the geometric phase of the composite system is
not equal to the sum of the geometric phases of its subsystems. The relations
among the geometric phases of the large system and the subsystems are
complicated, and therefore they are not easy to be synchronously controlled.
It is interesting to find a condition in which the geometric phase of the
composite system equals the sum of the geometric phases of its subsystemstong3
. The formulae (10) and (12) show that the value of geometric phase of a
quantum system not only depends on the initial state $|\psi(0)\rangle$ and the
final state $|\psi(\tau)\rangle$ but also depends on all the instantaneous
states $|\psi(t)\rangle~{}(t\in[0,\tau])$. It is determined completely by the
path traced by the states. If we require that the geometric phase of the
composite system is equal to the sum of the geometric phases of its subsystems
for all time, the sufficient condition is that $|\psi(t)\rangle$ remains
separable at all time, i.e.
$\displaystyle|\psi(t)\rangle=|\phi_{a}(t)\rangle\otimes|\phi_{b}(t)\rangle.$
(22)
One may demonstrate this point by substituting expression (22) into geometric
phase formulae. Indeed, if there is
$|\psi(t)\rangle=|\phi_{a}(t)\rangle\otimes|\phi_{b}(t)\rangle$ for
$t\in[0,\tau]$, one then has
$\displaystyle\arg\langle{\psi(0)}|\psi(\tau)\rangle$ $\displaystyle=$
$\displaystyle\arg\langle{\phi_{a}(0)}|\phi_{a}(\tau)\rangle\langle{\phi_{b}(0)}|\phi_{b}(\tau)\rangle$
(23) $\displaystyle=$
$\displaystyle\arg\langle{\phi_{a}(0)}|\phi_{a}(\tau)\rangle+\arg\langle{\phi_{b}(0)}|\phi_{b}(\tau)\rangle~{}~{}(\text{mod}2\pi),$
and
$\displaystyle i\int_{0}^{\tau}\langle{\psi(t)}|\dot{\psi}(t)\rangle
dt=i\int_{0}^{\tau}\langle{\phi_{a}(t)}|\dot{\phi}_{a}(t)\rangle
dt+i\int_{0}^{\tau}\langle{\phi_{b}(t)}|\dot{\phi}_{b}(t)\rangle dt,$ (24)
where the normalized relations
$\langle\phi_{\mu}(t)|\phi_{\mu}(t)\rangle=1~{}(\mu=a,b)$ are used.
Substituting them into Eq. (10), one further has
$\displaystyle\gamma_{ab}(\tau)=\gamma_{a}(\tau)+\gamma_{b}(\tau),$ (25)
where
$\displaystyle\gamma_{\mu}(\tau)=\arg\langle{\phi_{\mu}(0)}|\phi_{\mu}(\tau)\rangle+i\int_{0}^{\tau}\langle{\phi_{\mu}(t)}|\dot{\phi}_{\mu}(t)\rangle
dt,~{}~{}\mu=a,b,$ (26)
are $2\pi$-modular geometric phases of the subsystems.
With the above knowledge that the geometric phase of the system is equal to
the sum of those of its subsystems if the time dependent state is always
separable, we may now calculate the condition for the two interacting spin-
half particles. To this end, we rewrite the general solution expressed by Eq.
(8), with the bases $\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}$, as
$\displaystyle|\psi(t)\rangle=$
$\displaystyle\left(-\frac{1}{\sqrt{2}}c_{2}\sin\eta
e^{i\lambda_{2}t}e^{-i\omega
t}-c_{3}\sin^{2}\frac{\eta}{2}e^{i\lambda_{3}t}e^{-i\omega
t}+c_{4}\cos^{2}\frac{\eta}{2}e^{i\lambda_{4}t}e^{-i\omega
t}\right)|00\rangle$ (27)
$\displaystyle+\left(\frac{1}{\sqrt{2}}c_{1}e^{i\lambda_{1}t}+\frac{1}{\sqrt{2}}c_{2}\cos\eta
e^{i\lambda_{2}t}+\frac{1}{2}c_{3}\sin\eta
e^{i\lambda_{3}t}+\frac{1}{2}c_{4}\sin\eta e^{i\lambda_{4}t}\right)|01\rangle$
$\displaystyle+\left(-\frac{1}{\sqrt{2}}c_{1}e^{i\lambda_{1}t}+\frac{1}{\sqrt{2}}c_{2}\cos\eta
e^{i\lambda_{2}t}+\frac{1}{2}c_{3}\sin\eta
e^{i\lambda_{3}t}+\frac{1}{2}c_{4}\sin\eta e^{i\lambda_{4}t}\right)|10\rangle$
$\displaystyle+\left(\frac{1}{\sqrt{2}}c_{2}\sin\eta
e^{i\lambda_{2}t}e^{i\omega
t}-c_{3}\cos^{2}\frac{\eta}{2}e^{i\lambda_{3}t}e^{i\omega
t}+c_{4}\sin^{2}\frac{\eta}{2}e^{i\lambda_{4}t}e^{i\omega
t}\right)|11\rangle.$
Noting that the concurrence of a quantum state provides a criterion for
distinguishing between separable states and entangled states Wootters ; Rungta
, we may obtain the necessary and sufficient condition for the separable
states by calculating the concurrence of the above state. The concurrence of
the state reads
$\displaystyle C(t)$ $\displaystyle=$
$\displaystyle\sqrt{2\left[1-tr(tr_{b}|\psi(t)\rangle\langle\psi(t)|)^{2}\right]}$
(28) $\displaystyle=$
$\displaystyle\left|c_{2}^{2}+2c_{3}c_{4}-c_{1}^{2}e^{i8Jt}\right|.$
The above equation shows that the concurrence is, if $J\neq 0$, dependent on
the time $t$. The separability of an initial state does not guarantee
separability of the state at time $t$. If the system is initially in a
separable state, satisfying $c_{2}^{2}+2c_{3}c_{4}-c_{1}^{2}=0$ with
$c_{1}\neq 0$, it will evolve to an entangled state at the late time and then
go back to a separable state at each time
$t=n\pi\left/4J\right.,~{}n=1,2,\cdots$. If we require that the geometric
phase of the composite system is always equal to the sum of the geometric
phases of its subsystems for all time, the sufficient condition is that
$|\psi(t)\rangle$ is separable for all time. This requirement is fulfilled if
and only if the concurrence $C(t)$ is zero for all time, i.e.
$\displaystyle c_{2}^{2}+2c_{3}c_{4}-c_{1}^{2}e^{i8Jt}=0,$ (29)
which further leads to
$\displaystyle c_{1}=0,$ $\displaystyle c_{2}^{2}+2c_{3}c_{4}=0.$ (30)
Eq. (30) is the necessary and sufficient condition in which an initial
separable state of the system keeps in a separable one. The nonlocal
interaction between the two spins does not affect the separability of the
states in the set defined by condition (30). In this case, the geometric phase
of the composite system is always equal to the sum of the geometric phases of
its subsystems.
To illustrate the above result, we consider an example. Let
$c_{1}=0,~{}c_{2}=-\frac{1}{\sqrt{2}}\sin\eta,~{}c_{3}=-\sin^{2}\frac{\eta}{2},~{}c_{4}=\cos^{2}\frac{\eta}{2}$,
which means that the system is initially in the stats
$|\psi(0)\rangle=|00\rangle$. At time $t$, the instantaneous state reads
$\displaystyle|\psi(t)\rangle=-\frac{1}{\sqrt{2}}\sin\eta|\psi_{2}(t)\rangle-\frac{1}{2}(1-\cos\eta)|\psi_{3}(t)\rangle+\frac{1}{2}(1+\cos\eta)|\psi_{4}(t)\rangle,$
(31)
where $\alpha$ and $\eta$ have been defined in Eqs. (15) and (16),
respectively. The states of the subsystems $a$ and $b$ can be calculated by
using Eq. (11), which gives
$\rho_{\mu}=\bordermatrix{&\cr&\rho^{\mu}_{11}&\rho^{\mu}_{12}\cr&\rho^{\mu}_{21}&\rho^{\mu}_{22}},~{}~{}\mu=a,b,$
with the elements
$\displaystyle\rho^{\mu}_{11}$ $\displaystyle=$ $\displaystyle
1-\rho^{\mu}_{22}=\frac{1}{2}\left(1+\cos^{2}\eta+\sin^{2}\eta\cos\alpha
t\right),$ $\displaystyle\rho^{\mu}_{12}$ $\displaystyle=$
$\displaystyle{\rho^{\mu}_{21}}^{*}=\frac{1}{2}\left(\sin\eta\cos\eta(1-\cos\alpha
t)+i\sin\eta\sin\alpha t\right)e^{-i\omega t}.$ (32)
By using the formulae (10) and (12), we can calculate the geometric phases of
the system and the subsystems, and we have
$\displaystyle\gamma_{ab}(\tau)=\arctan\frac{-2\cos\eta\sin\alpha\tau}{\sin^{2}\eta+(1+\cos^{2}\eta)\cos\alpha\tau}+\frac{\omega\sin^{2}\eta}{\alpha}\sin\alpha\tau+\alpha\tau\cos\eta-\omega\tau\sin^{2}\eta,$
(33)
and
$\displaystyle\gamma_{a}(\tau)=\gamma_{b}(\tau)=\arctan\frac{-\cos\eta\sqrt{1-\cos\alpha\tau}}{\sqrt{1+\cos\alpha\tau}}+\frac{\omega\sin^{2}\eta}{2\alpha}\sin\alpha\tau+\frac{1}{2}\alpha\tau\cos\eta-\frac{1}{2}\omega\tau\sin^{2}\eta.$
(34)
By comparing Eq.(33) with Eq.(34) we see that the geometric phase of the
subsystem is half of the large system.
In passing, we would like to point out that all the states in the set defined
by condition (30) are the eigenstates of the interaction Hamiltonian. There is
no time-dependent state that is always separable but not an eigenstate of the
interaction Hamiltonian. This is easy to be understood, since the interaction
does not change the entanglement of an eigenstate but changes that of a non-
eigenstate. Noting that the time-dependent state of the system, initially in a
separable state with $c_{1}\neq 0$, will be cyclically separable with the
period $t=\pi\left/4J\right.$, one may argue whether the geometric phase holds
the additivity cyclically too in the case where
$c_{2}^{2}+2c_{3}c_{4}-c_{1}^{2}=0$ but $c_{1}\neq 0$. A further discussion
can show that the additivity is not valid for the geometric phase in the case.
This is because that geometric phase is equal to total phase minus dynamic
phase, and dynamic phase is not only dependent on the initial and final states
but also dependent on the states at all the evolutional time $t\in[0,\tau]$.
The additivity is invalid for dynamic phase, although it is valid for total
phase, which is only dependent on the initial and final states. Besides, it is
worth noting that the form of condition (30) is based on the expression of the
basis states $|\psi_{k}(t)\rangle$$~{}(k=1,2,3,4)$ in Eq. (9). It is not gauge
invariant. For example, if a $\pi$ phase difference is introduced between
$|\psi_{3}(t)\rangle$ and $|\psi_{4}(t)\rangle$, the coefficients $c_{3}$ and
$c_{4}$ would acquire a relative sign and the condition would then read
$c_{2}^{2}-2c_{3}c_{4}=0$. In general, there could be an arbitrary phase
factor between $c_{2}^{2}$ and $2c_{3}c_{4}$ in Eq. (30) if an alternative
expression of basis states are taken. The form of the condition depends on the
choice of phase convention between the basis states.
## V Summarize and remarks
An interacting bipartite system evolves into an entangled state in general,
even if it is initially in a separable state. Due to the entanglement, the
geometric phase of the system is not equal to the sum of the geometric phases
of its two subsystems. However, there may exist a set of states in which the
nonlocal interaction does not affect the separability of the states, and the
geometric phase of the bipartite system is equal to the sum of the geometric
phases of its subsystems. By considering a well known physical model, two
interacting spin-half particles in a rotating magnetic field, we illustrate
this point. Indeed, our calculation shows that the geometric phase of the
system is not equal to the sum of the geometric phases of the subsystems in
general. They are not equal even if the system is initially in separable
states, due to the nonlocal interaction between the subsystems. Yet, there is
such a set of states for which the nonlocal interaction does not affect the
separability of the states, and the geometric phase of the bipartite system is
always equal to the sum of the geometric phases of its subsystems. We give a
necessary and sufficient condition for an initial separable state to remain
separable.
The geometric property of the geometric phase has stimulated many
applications. It has been found that the geometric phase plays important roles
in quantum phase transition, quantum information processing, etc. Bohm ;
Carollo2005 ; Zhu2006 . The geometric phase shift can be fault tolerant with
respect to certain types of errors, thus several proposals using NMR, laser
trapped ions, etc. have been given to use geometric phase to construct fault-
tolerant quantum information processer Jones2000 ; Falci2000 ; Duan2001 .
The geometric phase is useful in quantum computation, but real physical
systems are usually composite and therefore the relations among the geometric
phase of the large system and those of the subsystems are complicated. It is
very difficult to control each of the values of them. Our result shows that it
is possible to make the phases’ relations simple if the initial states are
properly chosen. In this sense, our finding may be useful both in the theory
itself and in the applications of the geometric phase. The investigation on
the current bipartite system involving two spin-half particles implies that
such a kind of states may exist in other interacting physical systems.
## Acknowledgments
This work is supported by NFS China with No.10875072 and No. 10675076\. Tong
acknowledges the support of the National Basic Research Program of China
(Grant No. 2009CB929400). Kwek would like to acknowledge financial support by
the National Research Foundation & Ministry of Education, Singapore.
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|
arxiv-papers
| 2010-03-05T11:12:07 |
2024-09-04T02:49:08.815115
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "C. W. Niu, G. F. Xu, Longjiang Liu, L. Kang, D. M. Tong, and L. C.\n Kwek",
"submitter": "D M Tong",
"url": "https://arxiv.org/abs/1003.1234"
}
|
1003.1267
|
# Mixed Artin-Tate motives over number rings
Jakob Scholbach Universität Münster, Mathematisches Institut, Einsteinstr.
62, D-48149 Münster, Germany, jakob.scholbach@uni-muenster.de
###### Abstract
This paper studies Artin-Tate motives over bases $S\subset{\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}$, for a number field $F$. As a subcategory of motives
over $S$, the triangulated category of Artin-Tate motives $\mathbf{DATM}(S)$
is generated by motives $\phi_{*}\mathbf{1}(n)$, where $\phi$ is any finite
map. After establishing stability of these subcategories under pullback and
pushforward along open and closed immersions, a motivic $t$-structure is
constructed. Exactness properties of these functors familiar from perverse
sheaves are shown to hold in this context. The cohomological dimension of
mixed Artin-Tate motives ($\mathbf{MATM}(S)$) is two, and there is an
equivalence $\mathbf{DATM}(S)\cong\mathbf{D}^{\mathrm{b}}(\mathbf{MATM}(S))$.
Finally, mixed Artin-Tate motives enjoya strict functorial weight filtration.
###### keywords:
Artin-Tate motives , $t$-structure , perverse sheaves
###### MSC:
19E15 , 14C35
Geometric motives, as developed by Hanamura, Levine, and Voevodsky [7, 12,
18], are established as a valuable tool in understanding geometric and
arithmetic aspects of algebraic varieties over fields. However, the stupefying
ambiance inherent to motives, exemplified by Grothendieck’s motivic proof idea
of the Weil conjectures, remains largely conjectural—especially what concerns
the existence of mixed motives $\mathbf{MM}(K)$ over some field $K$. That
category should be the heart of the so-called motivic $t$-structure on
$\mathbf{DM}_{\mathrm{gm}}(K)$, the category of geometric motives. Much the
same way as the cohomology groups of a variety $X$ over $K$, e.g.
$\mathrm{H}^{n}_{\mathrm{\mathaccent
28691{e}t}}(X{\times}_{K}{\overline{K}},{\mathbb{Q}_{{\ell}}})$, $\ell$-adic
cohomology for $\ell\neq\operatorname{char}K$ are commonly realized as
cohomology groups of a complex, e.g.
${\mathrm{R}}{\Gamma}_{\ell}(X,{\mathbb{Q}_{{\ell}}})$, there should be mixed
motives $\mathrm{h}^{n}(X)$ that are obtained by applying truncation functors
belonging to the $t$-structure to $\operatorname{M}(X)$, the motive of $X$.
However, progress on mixed motives has proved hard to come by. To date, such a
formalism has been developed for motives of zero- and one-dimensional
varieties, only. This is due to Levine, Voevodsky, Orgogozo and Wildeshaus
[11, 18, 13, 21].
Building upon Voevodsky’s work, Ivorra and recently Cisinski and Déglise [10,
4] developed a theory of geometric motives $\mathbf{DM}_{\mathrm{gm}}(S)$ over
more general bases. The purpose of this work is to join the ideas of
Beilinson, Bernstein and Deligne on perverse sheaves [2] with the ones on
Artin-Tate motives over fields to obtain a workable category of mixed Tate and
Artin-Tate motives over bases $S$ which are open subschemes of
${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$, the ring of integers in a number
field $F$. As over a field, this provides some piece of evidence for the
existence and properties of the conjectural category of mixed motives over $S$
and its properties.
The triangulated category $\mathbf{DTM}(S)$ ($\mathbf{DATM}(S)$) of Tate
(Artin-Tate) motives is defined (2.2) to be the triangulated subcategory of
$\mathbf{DM}_{\mathrm{gm}}(S)$ (with rational coefficients) generated by
direct summands of $\mathbf{1}(n)$ and $i_{*}\mathbf{1}(n)$
($\phi_{*}\mathbf{1}(n)$, respectively). Here, $\mathbf{1}$ is a shorthand for
the motive of the base scheme, $(n)$ denotes the Tate twist,
$i:{\mathrm{Spec}\text{ }}{{\mathbb{F}_{\mathfrak{p}}}}\rightarrow S$ is a
closed point, $\phi:V\rightarrow S$ is any finite map and $\phi_{*}$ etc.
denotes the pushforward functor on geometric motives
$\mathbf{DM}_{\mathrm{gm}}(V)\rightarrow\mathbf{DM}_{\mathrm{gm}}(S)$. In case
$S$ is a finite disjoint union of ${\mathrm{Spec}\text{
}}{{\mathbb{F}_{\mathfrak{p}}}}$, the usual definition of (Artin-)Tate motives
over $S$ is recalled in Definition 2.1.
The following theorem and its “proof” is an overview of the paper.
###### Theorem 0.1.
The categories $\mathbf{DTM}(S)$ and $\mathbf{DATM}(S)$ are stable under
standard functoriality operations such as $i^{!}$, $j_{*}$ etc. for open and
closed embeddings $j$ and $i$, respectively.
Both categories enjoy a non-degenerate $t$-structure called _motivic
$t$-structure_. Its heart is denoted $\mathbf{MTM}(S)$ or $\mathbf{MATM}(S)$,
respectively and called category of _mixed (Artin-)Tate motives_.
The functors $i^{*}$, $j_{*}$ etc. feature exactness properties familiar from
the corresponding situation of perverse sheaves. For example, $i^{!}$ is left-
exact, and $j_{*}$ is exact with respect to the motivic $t$-structure.
The cohomological dimension of $\mathbf{MTM}(S)$ and $\mathbf{MATM}(S)$ is one
and two, respectively. We have an equivalence of categories
$\mathbf{D}^{\mathrm{b}}(\mathbf{MATM}(S))\cong\mathbf{DATM}(S)$
and likewise for Tate motives.
The “site” of mixed Artin-Tate motives over $S$ has enough points in the sense
that a mixed Artin-Tate motive over $S$ is zero if and only if its
restrictions to all closed points of $S$ vanish.
The non-unique weight truncation triangles on $\mathbf{DATM}(S)$ à la Bondarko
can be refined to a strict functorial weight filtration on $\mathbf{MATM}(S)$.
###### Proof: .
The first statement is Lemma 2.4. It is proven using the localization, purity
and base-change properties of geometric motives.
We will write $T(S)$ for either $\mathbf{DTM}(S)$ or $\mathbf{DATM}(S)$. The
existence of the motivic $t$-structure on $T(S)$ is proven in three steps. The
first ingredient is the well-known motivic $t$-structure on Artin-Tate motives
over finite fields (Lemma 3.6). The second step is the study of a subcategory
$\tilde{T}(S)\subset T(S)$ generated by $\phi_{*}\mathbf{1}(n)$, where $\phi$
is finite and étale (Artin-Tate motives), or just by $\mathbf{1}(n)$ (Tate
motives). This category is first equipped with an an auxiliary $t$-structure.
Then, a motivic $t$-structure on $\tilde{T}(S)$ is defined in Section 3 by
using the cohomology functor for the auxiliary $t$-structure. This statement
uses (and its proof imitates) the corresponding situation for Artin-Tate
motives over number fields due to Levine and Wildeshaus. The $t$-structure on
$\tilde{T}(S)$ is glued with the one over finite fields, using the general
gluing procedure of $t$-structures of [2], see Theorem 3.8. Much the same way
as with perverse sheaves, there are shifts accounting for $\dim S=1$, that is
to say, $i_{*}\mathbf{1}(n)$ and $\mathbf{1}(n)[1]$ are mixed Tate motives.
Beyond the formalism of geometric motives, the only non-formal ingredient of
the motivic $t$-structure are vanishing properties of algebraic $K$-theory of
number rings, number fields and finite fields due to Quillen, Borel and Soulé.
The exactness statements are shown in Theorem 4.2. This theorem gives some
content to the exactness axioms for general mixed motives over $S$ [16,
Section 4]. The key stepstone is the following: for any immersion of a closed
point $i:{\mathrm{Spec}\text{ }}{{\mathbb{F}_{\mathfrak{p}}}}\rightarrow S$,
the functor $i^{*}$ maps the heart $T^{0}(S)$ of $T(S)$ to
$T^{[-1,0]}({\mathrm{Spec}\text{ }}{{\mathbb{F}_{\mathfrak{p}}}})$, that is,
the category of (Artin-)Tate motives over ${\mathbb{F}_{\mathfrak{p}}}$ whose
only nonzero cohomology terms are in degrees $-1$ and $0$. The proof is a
careful reduction to basic calculations which relies on facts gathered in
Section 3 about the heart of $\tilde{T}(S)$.
The cohomological dimensions are calculated in Proposition 4.4. The Artin-Tate
case is a special (but non-conjectural) case of a similar fact for general
mixed motives over $S$. The difference in the Tate case is because the
generators of $\mathbf{DTM}(S)$ have good reduction at all places.
By an argument of Wildeshaus, under a mild homological condition on $T(S)$,
the identity on $T^{0}(S)$ extends to a functor
$\mathbf{D}^{\mathrm{b}}(T^{0}(S))\rightarrow T(S)$ (Theorem 4.5). While it is
an equivalence in the case of Tate motives for formal reasons, the Artin-Tate
case requires some localization arguments.
The last but one statement is Proposition 4.6. It might be seen as a first
step into motivic sheaves.
The weight filtration is established in Section 5. The key idea is an in-depth
analysis of a particular family generators, namely motives of the form
$f_{*}\mathbf{1}[1]$, where $f$ is a finite map between regular schemes. ∎
Deligne and Goncharov define a category of mixed Tate motives over rings
$\mathcal{O}_{S}$ of $S$-integers of a number field $F$ [6, 1.4., 1.7.].
Unlike the mixed Tate motives we study, their category is a _sub_ category of
mixed Tate motives over $F$, consisting of motives subject to certain non-
ramification constraints, akin to Scholl’s notion of mixed motives over
${\mathcal{O}_{F}}$ [15].
This paper is an outgrowth of part of my thesis. I owe many thanks to Annette
Huber for her advice during that time. I am also grateful to Denis-Charles
Cisinski and Frédéric Déglise for teaching me their work on motives over
general bases and to the referee for suggesting that Section 5 be
(re-)written.
## 1 Geometric motives
In this section we briefly recall some properties of the triangulated
categories of geometric motives $\mathbf{DM}_{\mathrm{gm}}(X)$, where $X$ will
be either a number field $F$ or an open or closed subscheme of
${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$. All of this is due to Cisinski
and Déglise [4].
The categories $\mathbf{DM}_{\mathrm{gm}}(X)$, where $X$ is any of the afore-
mentioned bases, are related by adjoint functors
$f^{*}:\mathbf{DM}_{\mathrm{gm}}(X)\leftrightarrows\mathbf{DM}_{\mathrm{gm}}(Y):f_{*}$,
where $f:Y\rightarrow X$ is any map, and
$f_{!}:\mathbf{DM}_{\mathrm{gm}}(Y)\leftrightarrows\mathbf{DM}_{\mathrm{gm}}(X):f^{!}$
($f$ separated of finite type). If $f$ is smooth, $f^{*}$ also has a left
adjoint $f_{\sharp}$. The category $\mathbf{DM}_{\mathrm{gm}}(X)$ enjoys inner
$\mathrm{Hom}$’s, denoted $\underline{\mathrm{Hom}}$, and a tensor structure
whose unit is denoted $\mathbf{1}$. Pullback functors $f^{*}$ are monoidal. In
particular $f^{*}\mathbf{1}_{X}=\mathbf{1}_{Y}$ for $f:Y\rightarrow X$. The
_motive_ of any scheme $f:Y\rightarrow X$ of finite type is defined as
$f_{!}f^{!}\mathbf{1}$ and denoted $\operatorname{M}(Y)$. (For $f$ smooth, [4,
Section 1.1.] puts $\operatorname{M}(Y):=f_{\sharp}f^{*}\mathbf{1}$. This
agrees with the previous definition by relative purity, see below.) The tensor
structure in $\mathbf{DM}_{\mathrm{gm}}(X)$ is such that
$\operatorname{M}(Y){\otimes}\operatorname{M}(Y^{\prime})=\operatorname{M}(Y{\times}_{X}Y^{\prime})$
(1)
for any two smooth schemes $Y$ and $Y^{\prime}$ over $X$. There is a
distinguished object $\mathbf{1}(1)$ such that
$\operatorname{M}(\mathbb{P}^{1}_{X})=\mathbf{1}\oplus\mathbf{1}(1)[2]$.
Tensoring with $\mathbf{1}(1)$ is an equivalence on
$\mathbf{DM}_{\mathrm{gm}}(X)$, and $\mathbf{1}(n)$ is defined in the usual
way in terms of tensor powers of $\mathbf{1}(1)$. We exclusively work with
rational coefficients, i.e., all morphism groups are $\mathbb{Q}$-vector
spaces. If $X$ is regular, morphisms in $\mathbf{DM}_{\mathrm{gm}}(X)$ are
given by
$\mathrm{Hom}_{\mathbf{DM}_{\mathrm{gm}}(X)}(\mathbf{1},\mathbf{1}(q)[p])\cong
K_{2q-p}(X)_{\mathbb{Q}}^{(q)},$ (2)
the $q$-th Adams eigenspace in algebraic $K$-theory of $X$, tensored with
$\mathbb{Q}$ [4, Section 13.2]. Having rational coefficients (or coefficients
in a bigger number field) is vital when it comes to vanishing properties of
$\mathrm{Hom}$-groups in $\mathbf{DM}_{\mathrm{gm}}(X)$. (With integral
coefficients, the existence of a $t$-structure even in the case of Artin
motives over a field is unclear.)
Throughout we need a property called _localization_ : for any closed immersion
$i:Z\rightarrow X$ with open complement $j$ we have the following functorial
distinguished triangles in $\mathbf{DM}_{\mathrm{gm}}(X)$
$j_{!}j^{*}\rightarrow\mathrm{id}\rightarrow i_{*}i^{*}$ (3)
We need to know that the functors $f_{!}$ and $f_{*}$ naturally agree for any
proper map $f$, as do $f^{!}$ and $f^{*}(d)[2d]$ when $f$ is smooth and quasi-
projective of constant relative dimension $d$ (_relative purity_). Moreover,
when $i:Z\rightarrow X$ is a closed immersion of constant relative codimension
$c$ and $Z$ and $X$ are regular, we have $i^{!}\mathbf{1}\cong
i^{*}\mathbf{1}(-c)[-2c]$. This is called _absolute purity_ [4, Sections 2.4,
13.4]. Finally, for $f:Y\rightarrow X$, $g:X^{\prime}\rightarrow X$,
$f^{\prime}:Y^{\prime}:=X^{\prime}{\times}_{X}Y\rightarrow X^{\prime}$ and
$g^{\prime}:Y^{\prime}\rightarrow Y$, there is a natural _base-change_
isomorphism of functors $f^{*}g_{!}\cong g^{\prime}_{!}f^{\prime*}$ [4,
Section 2.2], originally due to Ayoub [1].
The _Verdier dual_ functor
$D_{X}:\mathbf{DM}_{\mathrm{gm}}(X)^{\mathrm{op}}\rightarrow\mathbf{DM}_{\mathrm{gm}}(X)$
is defined by $D_{X}(M):=\underline{\mathrm{Hom}}(M,\pi^{!}\mathbf{1}(1)[2])$
for any $M\in\mathbf{DM}_{\mathrm{gm}}(X)$, where
$\pi:X\rightarrow{\mathrm{Spec}\text{ }}{\mathbb{Z}}$ denotes the structural
map. For example, for an open subscheme $X$ of ${\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}$ the factorization
$X\subset{\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}\rightarrow\mathbb{A}^{n}_{\mathbb{Z}}\rightarrow{\mathrm{Spec}\text{
}}{\mathbb{Z}}$
and absolute and relative purity show that
$D_{X}(-)=\underline{\mathrm{Hom}}(-,\mathbf{1}(1)[2])$. For
$X={\mathrm{Spec}\text{ }}{{\mathbb{F}_{q}}}$ one gets
$D_{X}(-)=\underline{\mathrm{Hom}}(M,\mathbf{1})$. The Verdier dual functor
exchanges “$!$” and “$*$”, e.g., there are natural isomorphisms
$D(f^{!}M)\cong f^{*}D(M)$ [4, Section 14.3]. For example, the Verdier dual of
(3) yields a distinguished triangle
$i_{*}i^{!}\rightarrow\mathrm{id}\rightarrow j_{*}j^{*}.$ (4)
For $X={\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$, taking the limit over
increasingly small open subschemes, one obtains a distinguished triangle in
$\mathbf{DM}(X)$ of the following form [4, Section 14.2]. (The category
$\mathbf{DM}(X)$ is a bigger category whose subcategory of compact objects is
$\mathbf{DM}_{\mathrm{gm}}(X)$.)
$\oplus_{\mathfrak{p}\in
S}{i_{\mathfrak{p}}}_{*}i_{\mathfrak{p}}^{!}\rightarrow\mathrm{id}\rightarrow\eta_{*}\eta^{*},$
(5)
where $\eta:{\mathrm{Spec}\text{ }}{F}\rightarrow{\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}$ is the generic point, the sum runs over all closed
points $\mathfrak{p}\in X$, $i_{\mathfrak{p}}$ is the closed immersion.
## 2 Triangulated Artin-Tate motives
Recall the following classical definition. We apply it to a number field or a
finite field:
###### Definition 2.1.
Let $K$ be a field. The category of Tate motives $\mathbf{DTM}(K)$ over $K$ is
by definition the triangulated subcategory of $\mathbf{DM}_{\mathrm{gm}}(K)$
generated by $\mathbf{1}(n)$ where $n\in\mathbb{Z}$. The smallest full
triangulated subcategory $\mathbf{DATM}(K)$ stable under tensoring with
$\mathbf{1}(n)$ and containing direct summands of motives $f_{*}\mathbf{1}$,
where $f:K^{\prime}\rightarrow K$ is any finite map, is called category of
Artin-Tate motives over $K$. For a scheme $S$ of the form
$S=\sqcup\mathrm{Spec}\text{ }K_{i}$, a finite disjoint union of spectra of
fields, we put $\mathbf{DATM}(S):=\oplus_{i}\mathbf{DATM}(K_{i})$ and likewise
for $\mathbf{DTM}$.
This section gives a generalization of that definition to bases $S$ which are
open subschemes of ${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$ based on the
idea that Artin-Tate motives over $S$ should be compatible with the ones over
$F$ and ${\mathbb{F}_{\mathfrak{p}}}$ under standard functoriality.
###### Definition 2.2.
The categories $\mathbf{DTM}(S)\subset\mathbf{DM}_{\mathrm{gm}}(S)$ of _Tate
motives_ and $\mathbf{DATM}(S)\subset\mathbf{DM}_{\mathrm{gm}}(S)$ of _Artin-
Tate motives_ over $S$ are the triangulated subcategories generated by the
direct summands of
$\mathbf{1}(n),\ i_{*}\mathbf{1}(n)\ \ \text{(Tate motives)}$
and
$\phi_{*}\mathbf{1}(n),\ \ \text{(Artin-Tate motives)}$
respectively, where $n\in\mathbb{Z}$, $\phi:V\rightarrow S$ is any finite map
(including those that factor over a closed point) and $i$ is the immersion of
any closed point of $S$.
###### Remark 2.3.
* 1.
In comparison to motives over a field, a category of Artin motives over $S$,
defined by removing the twists, is less viable, since it is not stable under
Verdier duality and $i_{*}i^{!}$, where $i$ is a closed embedding of a point.
* 2.
We can assume by localization (see (3), (4)) that the domain of $\phi$ is a
reduced scheme.
* 3.
The category of Tate motives $\mathbf{DTM}(S)$ agrees with the triangulated
category generated by the above generators (without taking direct summands),
see Lemma 3.9.
_For brevity, we write $T(S)$ or $T$ for $\mathbf{DATM}(S)$ or
$\mathbf{DTM}(S)$ in the sequel. In most proofs, we will only spell out the
case of Artin-Tate motives._
###### Lemma 2.4.
Let $j:S^{\prime}\rightarrow S$ be any open immersion, $i:Z\rightarrow S$ be
any closed immersion and $f:V\rightarrow S$ any finite map such that $V$ is
regular. Let $\eta:{\mathrm{Spec}\text{ }}{F}\rightarrow S$ be the generic
point. Then the functors $f_{*}=f_{!}$, $f^{*}$ and $f^{!}$ preserve Artin-
Tate motives. Similar statements hold for Artin-Tate and Tate motives for $j$
and $i$. Moreover, $\eta^{*}$, the Verdier dual functor $D$ and the tensor
product on $\mathbf{DM}_{\mathrm{gm}}(S)$ respect the subcategories of
(Artin-)Tate motives.
The functor $\eta_{*}$ does not respect Artin-Tate motives: we will see in
Proposition 4.6 that any Artin-Tate motive $M$ of the form
$M=\eta_{*}M_{\eta}$, where $M_{\eta}$ is an Artin-Tate motive over $F$,
necessarily satisfies $M=0$.
###### Proof: .
The stability of (Artin-)Tate motives under $j^{*}$, $\eta^{*}$, $i_{*}$ and
$i^{*}$, $f^{*}$ and—for Artin-Tate motives, under $f_{*}$—is immediate from
the definition and base-change. For example,
$i^{*}\phi_{*}\mathbf{1}(n)=\phi^{\prime\prime}_{*}\mathbf{1}(n)$. Here $\phi$
is any finite map over $S$ and $\phi^{\prime\prime}$ is its pullback along
$i$. For the stability under $j_{*}$ it is sufficient to show
$j_{*}\phi^{\prime}_{*}\mathbf{1}$ is an Artin-Tate motive over $S$ for any
finite flat map $\phi^{\prime}:V^{\prime}\rightarrow S^{\prime}$. Choose some
finite flat (possibly non-regular) model $\phi:V\rightarrow S$ of
$\phi^{\prime}$, i.e., $V{\times}_{S}S^{\prime}=V^{\prime}$, so that
$j^{*}\phi_{*}\mathbf{1}=\phi^{\prime}_{*}\mathbf{1}$ is an Artin-Tate motive
over $S^{\prime}$. The localization triangle
$j_{*}j^{*}\phi_{*}\mathbf{1}\rightarrow\phi_{*}\mathbf{1}\rightarrow
i_{*}i^{*}\phi_{*}\mathbf{1}$
and the above steps show that $j_{*}\phi_{*}\mathbf{1}$ is an Artin-Tate
motive over $S$.
To see the stability under the Verdier dual functor $D$, it is enough to see
that
$D(\phi_{*}\phi^{*}\mathbf{1})=\phi_{!}\phi^{!}D(\mathbf{1})=\phi_{*}\phi^{!}\mathbf{1}(1)[2].$
is an Artin-Tate motive for any finite map $\phi:V\rightarrow S$ with reduced
domain (Remark 2.3). If $V$ is zero-dimensional, this follows from purity and
the regularity of $S$. If not, there is an open (non-empty) immersion
$j:S^{\prime}\rightarrow S$ such that $V^{\prime}:=V{\times}_{S}S^{\prime}$ is
regular (for example, take $S^{\prime}$ such that $V^{\prime}/S^{\prime}$ is
étale). Let $i$ be the complement of $j$. We apply the localization triangle
$i_{*}i^{!}\rightarrow\mathrm{id}\rightarrow j_{*}j^{*}$ to
$\phi_{*}\phi^{!}\mathbf{1}$. By base-change we obtain
$i_{*}\phi^{\prime\prime}_{*}\phi^{\prime\prime!}i^{!}\mathbf{1}\rightarrow\phi_{*}\phi^{!}\mathbf{1}\rightarrow
j_{*}\phi^{\prime}_{*}\phi^{\prime!}j^{*}\mathbf{1}.$
Here $\phi^{\prime\prime}$ and $\phi^{\prime}$ is the pullback of $\phi$ along
$i$ and $j$, respectively. By the regularity of $S$ and purity we have
$i^{!}\mathbf{1}=\mathbf{1}(-1)[-2]$, so the left hand term is an Artin-Tate
motive. The right one also is by purity. This shows the claim for $D$.
The stability under $f^{!}$, $i^{!}$, and $j_{!}$ now follow for duality
reasons.
As for the stability under tensor products we note that
$\phi_{*}\mathbf{1}{\otimes}\phi^{\prime}_{*}\mathbf{1}=(\phi{\times}\phi^{\prime})_{*}\mathbf{1}$
if $\phi$ and $\phi$ are (finite and) smooth, cf. (1). Using the localization
triangle, it is easy to reduce the general case of merely finite maps $\phi$,
$\phi^{\prime}$ to this case. ∎
###### Remark 2.5.
Lemma 2.4 also holds for a similarly defined category of Artin-Tate motives
over open subschemes $S$ of a smooth curve over a field.
###### Lemma 2.6.
Let $M\in\mathbf{DATM}(S)$ be any Artin-Tate motive. Then there is a finite
map $f:V\rightarrow S$ such that
$f^{*}M\in\mathbf{DTM}(S)\subset\mathbf{DATM}(S)$. We describe this by saying
that $f$ _splits_ $M$.
###### Proof: .
As $f^{*}$ is triangulated, this statement is stable under triangles (with
respect to $M$), and also under direct sums and summands. Therefore, we only
have to check the generators, i.e., $M=\phi_{*}\mathbf{1}(n)$ with
$\phi:S^{\prime}\rightarrow S$ a finite map with reduced domain. The
corresponding splitting statement for Artin-Tate motives over finite fields is
well-known. Therefore, by localization, it is sufficient to find a splitting
map $f$ after replacing $S$ by a suitable small open subscheme, so we may
assume $\phi$ étale. We first assume that $\phi$ is moreover Galois of degree
$d$, i.e., $S^{\prime}{\times}_{S}S^{\prime}\cong S^{\prime\sqcup d}$, a
disjoint union of $d$ copies of $S^{\prime}$. In that case one has
$\phi^{*}\phi_{*}\mathbf{1}=\mathbf{1}^{\oplus d}$ by base-change, so the
claim is clear. In general $\phi$ need not be Galois, so let
$S^{\prime\prime}$ be the normalization of $S$ in some normal closure of the
function field extension $k(S^{\prime})/k(S)$. Both
$\mu:S^{\prime\prime}\rightarrow S$ and $\psi:S^{\prime\prime}\rightarrow
S^{\prime}$ are generically Galois. By shrinking $S$ we may assume both are
Galois. From
$\mathrm{Hom}(\mathbf{1}_{S^{\prime}},\psi_{*}\mathbf{1}_{S^{\prime\prime}})=\mathrm{Hom}(\mathbf{1}_{S^{\prime\prime}},\mathbf{1}_{S^{\prime\prime}})=\mathbb{Q}$
and
$\mathrm{Hom}(\psi_{*}\mathbf{1}_{S^{\prime\prime}},\mathbf{1}_{S^{\prime}})=\mathrm{Hom}(\mathbf{1}_{S^{\prime\prime}},\psi^{!}\mathbf{1}_{S^{\prime}})=\mathrm{Hom}(\mathbf{1}_{S^{\prime\prime}},\mathbf{1}_{S^{\prime\prime}})=\mathbb{Q}$
we see that $\mathbf{1}_{S^{\prime}}$ is a direct summand of
$\psi_{*}\mathbf{1}_{S^{\prime\prime}}$. Therefore
$\mu^{*}\phi_{*}\mathbf{1}_{S^{\prime}}$ is a summand of
$\mu^{*}\phi_{*}\psi_{*}\mathbf{1}_{S^{\prime\prime}}=\mu^{*}\mu_{*}\mathbf{1}_{S^{\prime\prime}}=\mathbf{1}^{\oplus\deg
S^{\prime\prime}/S}$, a Tate motive. ∎
## 3 The motivic $t$-structure
In this section, we establish the motivic $t$-structure on the category of
Artin-Tate motives over $S$ (Theorem 3.8). It is obtained by the standard
gluing procedure, applied to the $t$-structures on Artin-Tate motives over
finite fields and on a subcategory $\tilde{T}(S^{\prime})\subset
T(S^{\prime})$ for open subschemes $S^{\prime}\subset S$. Under the analogy of
mixed (Artin-Tate) motives with perverse sheaves, the objects in the heart of
the $t$-structure on $\tilde{T}(S^{\prime})$ correspond to sheaves that are
locally constant, i.e., have good reduction. We refer to [2, Section 1.3.] for
generalities on $t$-structures.
###### Definition 3.1.
(compare [11, Def. 1.1]) For $-\infty\leq a\leq b\leq\infty$, let
$\tilde{T}_{[a,b]}$ denote the smallest triangulated subcategory of $T(S)$
containing direct factors of $\phi_{*}\mathbf{1}(n)$, $a\leq-2n\leq b$, where
$\phi:S^{\prime}\rightarrow S$ is a finite _étale_ map. For Tate motives,
$\phi$ is required to be the identity map. (We will not specify this
restriction _expressis verbis_ in the sequel.) Furthermore,
$\tilde{T}_{[a,a]}$ and $\tilde{T}_{[-\infty,\infty]}$ are denoted
$\tilde{T}_{a}$ and $\tilde{T}$. If it is necessary to specify the base, we
write $\tilde{T}_{[a,b]}(S)$ etc.
We need the following vanishing properties of $K$-theory of number fields,
related Dedekind rings and finite fields up to torsion. In order to weigh the
material appropriately, it should be said that the content of the “lemma”
below is the only non-formal part of the proofs in this paper, and all
complexity occurring with Artin-Tate motives ultimately lies in these
computations.
###### Lemma 3.2.
(Borel, Soulé, Quillen) Let $\phi:S^{\prime}\rightarrow S$ and
$\psi:V\rightarrow S$ be two finite maps with zero-dimensional domains.
$\displaystyle\mathrm{Hom}_{S}(\phi_{*}\mathbf{1},\psi_{*}\mathbf{1}(n)[m])$
$\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{ll}\text{finite-
dimensional}&n=m=0\\\ 0&\text{else.}\end{array}\right.$
Let now $\phi:S^{\prime}\rightarrow S$ and $\psi:V\rightarrow S$ be two finite
étale maps over $S$. Then
$\displaystyle\mathrm{Hom}_{S}(\phi_{*}\mathbf{1},\psi_{*}\mathbf{1}(n)[m])$
$\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{ll}\text{finite-
dimensional}&n=m=0\\\ \text{finite-dimensional}&m=1,\ n\text{ odd and
positive}\\\ 0&\text{else.}\end{array}\right.$
###### Proof: .
By (2)
$\mathrm{Hom}_{V}(\mathbf{1},\mathbf{1}(q)[p])\cong
K_{2q-p}(V)_{\mathbb{Q}}^{(q)},$
for a regular scheme $V$. For the first statement, we may assume that
$S^{\prime}$ and $V$ are finite fields. Then the statement follows from
adjunction, base-change, purity and
$K_{n}({\mathbb{F}_{q}})=\left\\{\begin{array}[]{ll}\mu_{q^{i}-1}&n=2i-1,i>0\\\
0&n=2i,i>0\\\ \mathbb{Z}&n=0\end{array}\right.$
(Quillen [14]). $K$-theory of Dedekind rings $R$ whose quotient field is a
number field is known (up to torsion) by Borel’s work. The relation to
$K$-theory of number fields is given by an exact sequence (due to Soulé [17,
Th. 3]; up to two-torsion) for $n>1$
$0\rightarrow
K_{n}(R)\stackrel{{\scriptstyle\eta^{*}}}{{{\longrightarrow}}}K_{n}(F)\rightarrow\oplus_{\mathfrak{p}}K_{n-1}({\mathbb{F}_{\mathfrak{p}}})\rightarrow
0.$
Here $\eta:\mathrm{Spec}\text{ }F\rightarrow\mathrm{Spec}\text{ }R$ is the
generic point and the direct sum runs over all (finite) primes in $R$. Also,
$K_{0}(R)=\mathbb{Z}\oplus Pic(R)$ and $K_{1}(R)=R^{\times}$. In particular,
for all $n$ and $m$, $K_{n}(R)_{\mathbb{Q}}^{(m)}$ vanishes when
$K_{n}(F)_{\mathbb{Q}}^{(m)}$ vanishes, since $\eta^{*}$ respects the Adams
grading. One has the following list (see e.g. [19])
$K_{2q-p}(F)^{(q)}_{\mathbb{Q}}=\left\\{\begin{array}[]{ll}0&q<0\\\
0&q=0,p\neq 0\\\ \mathbb{Q}&q=p=0\\\ 0^{BS}&q>0,p\leq 0\\\
0&q>0,\text{even},p=1\\\ F^{\times}{\otimes}_{\mathbb{Z}}\mathbb{Q}&q=p=1\\\
\mathbb{Q}^{r_{1}+r_{2}}&q>1,q\equiv 1\ \ (\mathrm{mod\ }4),p=1\\\
\mathbb{Q}^{r_{2}}&q>0,q\equiv 3\ \ (\mathrm{mod\ }4),p=1\\\
0&q>0,p>1\end{array}\right.$
As usual, $r_{1}$ and $r_{2}$ are the numbers of real and pairs of complex
embeddings of $F$, respectively. (The agreement of $K_{2q-1}(F)$ and
$K_{2q-1}(F)^{(q)}$ for odd positive $q$ is not mentioned in loc. cit.) The
spot marked $0^{BS}$ is referred to as _Beilinson-Soulé vanishing_ (see e.g.
[11]). As first realized by Levine (loc. cit.), this translates into the non-
existence of morphisms in the “wrong” direction with respect to the motivic
$t$-structure.
For the last claim, put $V^{\prime}=V{\times}_{S}S^{\prime}$:
$\textstyle{V^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{\prime}}$$\scriptstyle{\psi^{\prime}}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{S^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{S.}$
To save space, we omit the twist and the shift in writing the
$\mathrm{Hom}$-groups. We have
$\mathrm{Hom}_{S}(\phi_{*}\mathbf{1},\psi_{*}\mathbf{1})=\mathrm{Hom}_{S^{\prime}}(\mathbf{1},\phi^{!}\psi_{*}\mathbf{1})=\mathrm{Hom}_{S^{\prime}}(\mathbf{1},\psi^{\prime}_{*}\phi^{\prime!}\mathbf{1})=\mathrm{Hom}_{V^{\prime}}(\mathbf{1},\phi^{\prime!}\mathbf{1}).$
Now, $V^{\prime}$ is (affine and) étale over $V$, so
$\phi^{\prime!}\mathbf{1}=\phi^{\prime*}\mathbf{1}=\mathbf{1}$ and we are done
in that case by the above vanishings of $K$-theory up to torsion. ∎
The following lemma is a variant of [11, Lemma 1.2], [21, Lemma 1.9] and can
be proven by faithfully imitating the technique in loc. cit.
###### Lemma 3.3.
For any $-\infty\leq a<b\leq c\leq\infty$,
$(\tilde{T}_{[a,b-1]},\tilde{T}_{[b,c]})$ is a $t$-structure on
$\tilde{T}_{[a,c]}$.
###### Definition 3.4.
The resulting truncation and cohomology functors are denoted $F_{\leq b}$ and
$F_{>b}$ and $\operatorname{gr}_{b}^{F}$, respectively.
The following definition is modeled on [11, Def. 1.4]. We also refer to [1,
Section 2.1.3] for a general way (due to Morel) of constructing a
$t$-structure starting from a given set of generators. For any odd integer $n$
set $\mathbf{1}(n/2):=0$, for notational convenience.
###### Definition 3.5.
Let $S$ be an open subscheme of ${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$.
Let $\tilde{T}^{\geq 0}_{a}(S)$ ($\tilde{T}^{\leq 0}_{a}(S)$) be the full
subcategory of $\tilde{T}_{a}(S)$ (Definition 3.1) generated by
$\phi_{*}\mathbf{1}\left(-\frac{a}{2}\right)[n+1]$
for any $n\leq 0$ ($n\geq 0$, respectively), and any finite étale map $\phi$.
“Generated” means the smallest subcategory containing the given generators
stable under isomorphism, finite direct sums, summands and
$\operatorname{cone}(\phi)[-1]$ ($\operatorname{cone}(\phi)$, resp.) for any
morphism $\phi$ in $\tilde{T}_{a}^{\geq 0}(S)$ ($\tilde{T}_{a}^{\leq 0}(S)$,
respectively).
For any $-\infty\leq a\leq b\leq\infty$, let $\tilde{T}_{[a,b]}^{\geq 0}(S)$
be the triangulated subcategory generated by objects $X$, such that for all
$a\leq c\leq b$, $\operatorname{gr}_{c}^{F}(X)\in\tilde{T}_{c}^{\geq 0}(S)$
and similarly for $\tilde{T}_{[a,b]}^{\leq 0}(S)$. For $a=-\infty$ and
$b=\infty$ we simply write $\tilde{T}^{\leq 0}(S)$, $\tilde{T}^{\geq 0}(S)$.
We may omit $S$ in the notation, if no confusion arises.
In particular $\mathbf{1}(-a/2)[1]\in\tilde{T}^{0}_{a}(S)$. This shift is as
in the situation of perverse sheaves [2], [16, Section 3]. Before stating and
proving the existence of the motivic $t$-structure, we need some preparatory
steps. Levine has established the existence of the motivic $t$-structure on
Tate motives over number fields and finite fields [11, Theorem 1.4.]. This has
been generalized to Artin-Tate motives by Wildeshaus [21, Theorem 3.1]. We
briefly recall these precursor statements. Let $K$ be either a finite field or
a number field. For any $-\infty\leq a\leq b\leq\infty$, let $T_{[a,b]}(K)$ be
the triangulated subcategory of $T(K)$ generated by $\mathbf{1}(n)$ with
$a\leq-2n\leq b$ (Tate motives) and direct summands of
$\phi_{*}\mathbf{1}(n)$, $\phi:\mathrm{Spec}\text{
}K^{\prime}\rightarrow\mathrm{Spec}\text{ }K$ a finite map (Artin-Tate
motives, respectively). For any $a\leq c<b$, the datum
$\left(T_{[a,c]},T_{[c+1,b]}\right)$ forms a $t$-structure on $T_{[a,b]}$. Let
$\operatorname{gr}^{F}_{*}$ be the cohomology functor corresponding to that
$t$-structure. Write $T_{a}(K)$ for $T_{[a,a]}(K)$ and let $T^{\geq 0}_{a}(K)$
and $T^{\leq 0}_{a}(K)$ be the subcategories of $T_{a}(K)$ generated by
$\mathbf{1}(-a/2)[n]$ with $n\leq 0$ and $n\geq 0$, respectively. Here,
“generated” has the same meaning as in Definition 3.5. Let $T^{\geq
0}_{[a,b]}$ and $T^{\leq 0}_{[a,b]}$ be the subcategories of $T_{[a,b]}$ of
objects $X$ such that all $\operatorname{gr}^{F}_{c}X\in T^{\geq 0}_{c}$
($\operatorname{gr}^{F}_{c}X\in T^{\leq 0}_{c}$, respectively) for all $a\leq
c\leq b$. Then, $\left(T^{\leq 0}_{[a,b]}(K),T^{\geq 0}_{[a,b]}(K)\right)$ is
a non-degenerate $t$-structure on $T_{[a,b]}$.
The following well-known fact is a consequence of vanishing of all $K$-theory
groups of finite fields except for
$K_{0}({\mathbb{F}_{\mathfrak{p}}})^{(0)}_{\mathbb{Q}}$, see Lemma 3.2.
###### Lemma 3.6.
Let $\mathfrak{p}$ be a closed point in $S$ with residue field
${\mathbb{F}_{\mathfrak{p}}}$. The inclusions
$T_{a}({\mathbb{F}_{\mathfrak{p}}})\subset T({\mathbb{F}_{\mathfrak{p}}})$
induce an equivalence of categories
$\bigoplus_{a\in\mathbb{Z}}T_{a}({\mathbb{F}_{\mathfrak{p}}})=T({\mathbb{F}_{\mathfrak{p}}}).$
There are canonical equivalences of categories
$T(Z):=\bigoplus_{\mathfrak{p}\in
Z,a\in\mathbb{Z}}T_{a}({\mathbb{F}_{\mathfrak{p}}})=\bigoplus_{\mathfrak{p},a}\mathbf{D}^{\mathrm{b}}(\underline{\mathbb{Q}}[\text{Perm},\mathrm{Gal}({\mathbb{F}_{\mathfrak{p}}})])=\bigoplus_{\mathfrak{p},a}\underline{\mathbb{Q}}[\text{Perm},\mathrm{Gal}({\mathbb{F}_{\mathfrak{p}}})]^{\mathbb{Z}-\text{graded}}$
Here and in the sequel
$\underline{\mathbb{Q}}[\text{Perm},\mathrm{Gal}({\mathbb{F}_{\mathfrak{p}}})]$
denotes finite-dimensional rational permutation representations of the
absolute Galois group. By means of that equivalence, $T(Z)$ is endowed with
the obvious $t$-structure. The heart
$T^{0}_{a}({\mathbb{F}_{\mathfrak{p}}})=T^{\leq
0}_{a}({\mathbb{F}_{\mathfrak{p}}})\cap T^{\geq
0}_{a}({\mathbb{F}_{\mathfrak{p}}})$ is semisimple and consists of direct sums
of summands of $\phi_{*}\mathbf{1}(a)$, $\phi$ finite.
We now provide the motivic $t$-structure on $\tilde{T}(S)$, which stems from
the one on $T(F)$. The two together will then be glued to give the
$t$-structure on $T(S)$. Recognizably, the following is again an adaptation of
Levine’s proof of the $t$-structure on Tate motives over number fields.
###### Proposition 3.7.
For any $-\infty\leq a\leq b\leq\infty$, $\left(\tilde{T}_{[a,b]}^{\leq
0},\tilde{T}_{[a,b]}^{\geq 0}\right)$ is a non-degenerate $t$-structure on
$\tilde{T}_{[a,b]}(S)$ (Definitions 3.1, 3.5). The functor
$\eta^{*}[-1]:\tilde{T}_{[a,b]}(S)\rightarrow T_{[a,b]}(F)$ is $t$-exact.
Any motive in $\tilde{T}^{0}_{a}(S)$ is a finite direct sum of summands of
motives $\phi_{*}\mathbf{1}(-a/2)[1]$ with $\phi$ finite étale. The closure of
the direct sum of the $\tilde{T}^{0}_{a}(S)$, $a\in\mathbb{Z}$, under
extensions (in the abelian category $\tilde{T}^{0}(S)$) is $\tilde{T}^{0}(S)$.
###### Proof: .
We may assume that $a$ and $b$ are finite, since
$\tilde{T}(S)=\bigcup_{-\infty<a\leq b<\infty}\tilde{T}_{[a,b]}(S)$
and the inclusion functors given by the identity between the various
$T_{[-,-]}$ are exact.
The proof proceeds by induction on $b-a$. The case $b=a$ is treated as
follows: the category $\tilde{T}_{a}:=\tilde{T}_{a}(S)$ is generated by
$\phi_{*}\mathbf{1}(-a/2)[n]$, $n\in\mathbb{Z}$, $\phi$ étale and finite. The
functor $\eta^{*}[-1](a/2):\tilde{T}_{a}(S)\rightarrow T_{0}(F)$ is fully
faithful. To see this it suffices to remark
$\mathrm{Hom}_{S}(\phi_{*}\mathbf{1}(-a/2)[n+1],\psi_{*}\mathbf{1}(-a/2)[n^{\prime}+1])=\mathrm{Hom}_{F}({\phi_{\eta}}_{*}\mathbf{1}[n],{\psi_{\eta}}_{*}\mathbf{1}[n^{\prime}])$,
for any finite étale maps $\phi$ and $\psi$ with generic fiber $\phi_{\eta}$
and $\psi_{\eta}$. This equality follows from the $K$-theory computations, see
the proof of Lemma 3.2. Therefore, the image of $\eta^{*}[-1](a/2)$ is a
triangulated subcategory of $T_{0}(F)$ which contains the generators of
$T_{0}(F)$, so the functor establishes an equivalence between
$\tilde{T}_{a}(S)$ with the derived category of finite-dimensional rational
permutation representations of $\mathrm{Gal}(F)$ by [18, 3.4.1]. Hence
$\tilde{T}_{a}(S)$ carries a non-degenerate $t$-structure.
The remainder of the proof is done as in Levine’s proof. One shows
$\mathrm{Hom}\left(\tilde{T}_{[a+1,b]}^{\leq 0},\tilde{T}_{c}^{\geq
0}\right)=0$ (8)
for any $c\leq a$. This reduces to the Beilinson-Soulé vanishing. Then the
$t$-structure axioms follow for formal reasons.
The exactness of $\eta^{*}[-1]$ is obvious from the definitions. The statement
about the heart $\tilde{T}_{a}^{0}$ is done as follows: the exact functor
$\eta^{*}[-1](a/2)$ identifies $\tilde{T}^{0}_{a}(S)=\tilde{T}^{\geq
0}_{a}(S)\cap\tilde{T}^{\leq 0}_{a}(S)$ with the semi-simple category
$T^{0}_{0}(F)=\underline{\mathbb{Q}}[\text{Perm},\mathrm{Gal}(F)]$. We claim
that for any object $X\in\tilde{T}_{a}(S)$, all
${{{}^{\mathrm{p}}}\mathrm{H}}^{n}(X)$ are direct summands of sums of motives
$\phi_{*}\mathbf{1}(-a/2)[1]$, $\phi$ finite and étale. This claim does hold
for the generators of $\tilde{T}_{a}(S)$. We now show that the condition is
stable under triangles, which accomplishes the proof of the claim and thus the
proof of the statement. Let $A\rightarrow X\rightarrow B$ be a triangle in
$\tilde{T}_{a}(S)$ such that $A$ and $B$ satisfy the claim. The long exact
cohomology sequence
$\dots\rightarrow{{{}^{\mathrm{p}}}\mathrm{H}}^{n-1}B\stackrel{{\scriptstyle\delta^{n-1}}}{{{\longrightarrow}}}{{{}^{\mathrm{p}}}\mathrm{H}}^{n}A\rightarrow{{{}^{\mathrm{p}}}\mathrm{H}}^{n}X\rightarrow{{{}^{\mathrm{p}}}\mathrm{H}}^{n}B\stackrel{{\scriptstyle\delta^{n}}}{{{\longrightarrow}}}{{{}^{\mathrm{p}}}\mathrm{H}}^{n+1}A\rightarrow\dots$
yields the short exact sequence in $\tilde{T}^{0}_{a}(S)$
$0\rightarrow\operatorname{coker}\delta^{n-1}\rightarrow{{{}^{\mathrm{p}}}\mathrm{H}}^{n}X\rightarrow\ker\delta^{n}\rightarrow
0.$
By the semi-simplicity of $\tilde{T}^{0}_{a}(S)$ (this is the key point!), the
sequence splits and there is a non-canonical isomorphism
${{{}^{\mathrm{p}}}\mathrm{H}}^{n}X\cong\operatorname{coker}\delta^{n-1}\oplus\ker\delta^{n}$
and $\operatorname{coker}\delta^{n-1}$ and $\ker\delta^{n}$ are direct
summands of ${{{}^{\mathrm{p}}}\mathrm{H}}^{n}A$ and
${{{}^{\mathrm{p}}}\mathrm{H}}^{n}B$, respectively.
For the statement concerning $\tilde{T}^{0}(S)$ one uses the finite exhaustive
$F$-filtration of any $X\in\tilde{T}^{0}(S)$:
$0=F_{a}X\subset F_{[a,a+1]}X\subset\dots\subset F_{[a,b]}X=X.$
The successive quotients $\operatorname{gr}_{*}^{F}X$ of that chain are in
$\tilde{T}^{0}_{*}(S)$, since truncations with respect to the $t$-structure
related to $F$ are exact with respect to the motivic $t$-structure, by
definition. Thus the claim about $\tilde{T}^{0}(S)$ follows. ∎
###### Theorem 3.8.
The motivic $t$-structures on $T(Z)$ and $\tilde{T}(S^{\prime})$ glue to a
non-degenerate $t$-structure on the category $T(S)$ of (Artin-)Tate motives
over $S$ (Definition 2.2). It is called _motivic $t$-structure_. Here
$S^{\prime}$ runs through open subschemes of $S$ and $Z:=S\backslash
S^{\prime}$.
###### Proof: .
We apply the gluing procedure of $t$-structures of [2, Theorem 1.4.10]: for
any open subscheme $j:S^{\prime}\subset S$, we write $T_{S^{\prime}}(S)$ for
the full triangulated subcategory of objects $X\in T(S)$ such that
$j^{*}X\in\tilde{T}(S^{\prime})\subset T(S^{\prime})$. Let
$i:Z^{\prime}\rightarrow S$ be the closed complement of $j$. Put
$T^{\leq 0}_{S^{\prime}}(S):=\\{X\in
T_{S^{\prime}}(S),j^{*}X\in\tilde{T}^{\leq 0}(S^{\prime}),i^{*}X\in T^{\leq
0}(Z^{\prime})\\},$ $T^{\geq 0}_{S^{\prime}}(S):=\\{X\in
T_{S^{\prime}}(S),j^{*}X\in\tilde{T}^{\geq 0}(S^{\prime}),i^{!}X\in T^{\geq
0}(Z^{\prime})\\}.$
The assumptions of the gluing theorem, [2, 1.4.3], namely the existence of
$i_{*}$, $i^{*}$, $i^{!}$, $j_{*}$, $j_{!}$, $j^{*}$ satisfying the usual
adjointness properties, $j^{*}i_{*}=0$, localization sequences and full
faithfulness of $i_{*}$, $j_{!}$ and $j_{*}$ are met, since they are in the
surrounding categories of geometric motives, cf. Section 1, and the stability
results of Section 2. Thus, the above defines a $t$-structure on
$T_{S^{\prime}}(S)$.
The field $F$ is of characteristic zero, so any finite map $\phi:V\rightarrow
S$ with $V$ reduced and one-dimensional is generically étale. This implies
$T(S)=\cup_{S^{\prime}\subset S}T_{S^{\prime}}(S)$. We set
$T^{\geq 0}(S):=\bigcup_{S^{\prime}\subset S}T^{\geq 0}_{S^{\prime}}(S)$
and dually for $T^{\leq 0}(S)$. The $t$-structure axioms on $T(S)$ and the
non-degeneracy are implied by the exactness of the identical inclusion
$T_{S^{\prime}}(S)\rightarrow T_{S^{\prime\prime}}(S)$ for any
$S^{\prime\prime}\subset S^{\prime}$.
To see the exactness of the identity, let
$j^{\prime\prime}:S^{\prime\prime}\subset S$ and
$i^{\prime\prime}:Z^{\prime\prime}\subset S$ be its complement. Let $X\in
T^{\leq 0}_{S^{\prime}}(S)$. It is clear that
$j^{\prime\prime*}X\in\tilde{T}^{\leq 0}(S^{\prime\prime})$. Let us check
$i^{\prime\prime*}X\in T^{\leq 0}(Z^{\prime\prime})$. The pullback
$i^{\prime\prime*}X$ decomposes as a direct sum parametrized by the points of
$Z^{\prime\prime}$ and we only have to deal with the points that are not
contained in $Z^{\prime}$. Let $p:{\mathrm{Spec}\text{
}}{{\mathbb{F}_{\mathfrak{p}}}}\rightarrow S$ be such a point; it factors over
$S^{\prime}$: $p=j\circ q$, where $q:{\mathrm{Spec}\text{
}}{{\mathbb{F}_{\mathfrak{p}}}}\rightarrow S^{\prime}$ is the same point as
$p$. Thus $p^{*}X=q^{*}j^{*}X\in q^{*}\tilde{T}^{\leq 0}(S^{\prime})$. The
containment $q^{*}\tilde{T}^{\leq 0}(S^{\prime})\subset T^{\leq
0}({\mathrm{Spec}\text{ }}{{\mathbb{F}_{\mathfrak{p}}}})$ follows from
$q^{*}\tilde{T}_{a}^{\leq 0}(S^{\prime})\subset T_{a}^{\leq
0}({\mathrm{Spec}\text{ }}{{\mathbb{F}_{\mathfrak{p}}}})$, since $q^{*}$
clearly commutes with the $F$-truncation functors belonging to the auxiliary
$t$-structure. To see the latter containment, it suffices to check the
generators (in the sense of Definition 3.5) of $\tilde{T}_{a}^{\leq
0}(S^{\prime})$, that is, it is sufficient to remark
$q^{*}\phi_{*}\mathbf{1}(-a/2)[n+1]=\phi^{\prime}_{*}\mathbf{1}(-a/2)[n+1]\in
T_{a}^{\leq-1}({\mathrm{Spec}\text{ }}{{\mathbb{F}_{\mathfrak{p}}}})\subset
T_{a}^{\leq 0}({\mathrm{Spec}\text{ }}{{\mathbb{F}_{\mathfrak{p}}}}),$
where $n\geq 0$ and $\phi$ is a finite étale map with pullback
$\phi^{\prime}$. This shows that the identity is left-exact. The right-
exactness is done dually. ∎
###### Lemma 3.9.
The category $\mathbf{DTM}(S)$ agrees with the triangulated category generated
by $\mathbf{1}(n)$, $i_{*}\mathbf{1}(n)$.
###### Proof: .
Let $M\in\mathbf{DTM}(S)$. Pick an open subscheme $j:S^{\prime}\subset S$ with
complement $i:Z\subset S$ such that $j^{*}M\in\tilde{T}(S^{\prime})$. Any
object in $T(Z)$ is isomorphic to a direct sum of motives
$\mathbf{1}_{{\mathbb{F}_{\mathfrak{p}}}}(a)[b]$, $\mathfrak{p}\in Z$, since
$\mathbf{1}_{\mathbb{F}_{\mathfrak{p}}}$ does not have proper direct summands.
Any object in $\tilde{T}^{0}_{-2a}(S^{\prime})$ is a direct sum of motives
$\mathbf{1}(a)[1]$ for the same reason. Any object in
$\tilde{T}^{0}(S^{\prime})$ is obtained by taking repeated extensions starting
with such objects. Thus $\tilde{T}(S^{\prime})$ is the triangulated category
generated by $\mathbf{1}(a)$, $a\in\mathbb{Z}$. The localization triangle
$i_{*}i^{!}M\rightarrow M\rightarrow j_{*}j^{*}M$ settles the lemma. ∎
## 4 Mixed Artin-Tate motives
###### Definition 4.1.
The heart $T^{0}(S)$ of the motivic $t$-structure is called the category of
_mixed (Artin-)Tate motives_ over $S$, denoted $\mathbf{MTM}(S)$ and
$\mathbf{MATM}(S)$, respectively. The cohomology functors belonging to the
motivic $t$-structure are denoted ${{{}^{\mathrm{p}}}\mathrm{H}}^{*}$.
We now study the categories of mixed Tate motives over $S$ in some detail. The
key is Theorem 4.2 below, establishing exactness properties of pullback and
pushforward functors along closed and open immersions. The exactness axioms
for mixed motives over number rings (see [16, Section 4]) are modeled on this
theorem. Of course, the theorem is an Artin-Tate motivic analog of a similar
fact about perverse sheaves [2, Prop. 1.4.16, 4.2.4.], suggesting that the
theory of perverse sheaves is to some extent quite formal. Proposition 4.4
calculates the cohomological dimension of mixed (Artin-)Tate motives. We
obtain an equivalence
$\mathbf{DTM}(S)\cong\mathbf{D}^{\mathrm{b}}(\mathbf{MTM}(S))$, using a result
of Wildeshaus, and likewise for Artin-Tate motives. Finally, we do a first
step into (Artin-Tate) motivic sheaves, in Proposition 4.6.
All exactness statements below are with respect to the motivic $t$-structure
of Theorem 3.8. Recall from Lemma 2.4 that the functors discussed below do
preserve (Artin-)Tate motives. For brevity, we write $T^{[a,b]}$ for the full
subcategory of objects $M$ satisfying ${{{}^{\mathrm{p}}}\mathrm{H}}^{n}M=0$
for all $n<a$ and $n>b$. We say that a triangulated functor $F$ between
categories of Artin-Tate motives has _cohomological amplitude_ $[a,b]$ if
$F(T^{0})$ is contained in $T^{[a,b]}$. Note that $F$ is right exact iff
$b\leq 0$ and left exact iff $a\geq 0$.
###### Theorem 4.2.
Let $j:S^{\prime}\rightarrow S$ be an open immersion, $i:Z\rightarrow S$ a
closed immersion with $\dim Z=0$. Finally, let $f:V\rightarrow S$ be a finite
map with regular one-dimensional domain.
1. (i)
The Verdier duality functor $D$ is exact in the sense that it maps $T^{\geq
0}$ to $T^{\leq 0}$ and vice versa. Therefore, it induces an endofunctor on
$T^{0}(S)$.
2. (ii)
The functors $j_{*}$, $j_{!}$, $j^{*}$, as well as $i_{*}=i_{!}$ are exact.
3. (iii)
The functor $i^{*}$ has cohomological amplitude $[-1,0]$. Dually, $i^{!}$ has
cohomological amplitude $[0,1]$.
4. (iv)
The functor $f_{*}=f_{!}$ is exact. The cohomological amplitude of $f^{*}$ and
$f^{!}$ is $[-1,0]$ and $[0,1]$, respectively. If $f$ is also étale,
$f^{*}=f^{!}$ is exact.
5. (v)
The functor $\eta^{*}[-1]:T(S)\rightarrow T(\mathrm{Spec}\text{ }F)$ is exact.
###### Proof: .
(i): This is clear from the definitions of the $t$-structures on $T(S)$,
$\tilde{T}(S^{\prime})$ and $T(Z)$, for open and closed subschemes
$S^{\prime}$ and $Z$ of $S$, respectively. Notice that this requires putting
$\mathbf{1}[1]$ in degree $0$.
(ii): The following exactness properties are immediate from the definition:
$j^{*}$ and $i_{*}$ are exact, $j_{*}$ and $i^{!}$ are left-exact and $j_{!}$
and $i^{*}$ are right-exact. For example, let us show the left-exactness of
$j_{*}$. Given some motive $M\in T^{\geq 0}(S^{\prime})$, we have to show
$j_{*}M\in T^{\geq 0}(S)$. Let $j_{1}:S_{1}\subset S^{\prime}$ be an open
immersion such that $j_{1}^{*}M\in\tilde{T}^{\geq 0}(S_{1})$. Let $i_{1}$ be
the immersion of $Z_{1}:=S^{\prime}\backslash S_{1}$ into $S^{\prime}$, then
$i_{1}^{!}M\in T^{\geq 0}(Z_{1})$. The situation is as follows:
|
---|---
$\textstyle{Z_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{1}}$$\textstyle{S_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{1}}$$\textstyle{S^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{S\backslash
S_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{S}$
Now $(j\circ j_{1})^{*}j_{*}M=j_{1}^{*}M\in T^{\geq 0}(S_{1})$. Let
$i:S\backslash S_{1}\rightarrow S$ be the complement of $j\circ j_{1}$. Then
$i^{!}j_{*}M$ is supported only in $Z_{1}$, where it agrees with $i_{1}^{!}M$.
This shows $j_{*}M\in T^{\geq 0}(S)$.
To prove (iii) we first show
$i^{*}j_{*}\tilde{T}^{0}(S^{\prime})\subset T^{[-1,0]}(Z)$ (9)
for any two complementary immersions $i:Z\rightarrow S$ (closed) and
$j:S^{\prime}\rightarrow S$ (open). By Proposition 3.7, $\tilde{T}^{0}(S)$ is
generated by means of direct sums, summands and extensions by
$\phi_{*}\mathbf{1}(n)[1]$, where $n\in\mathbb{Z}$ is arbitrary and $\phi$ is
finite and étale. For any short exact sequence
$0\rightarrow A\rightarrow X\rightarrow B\rightarrow 0$
in $\tilde{T}^{0}(S)$, such that $i^{*}j_{*}A\in T^{[-1,0]}(Z)$ and
$i^{*}j_{*}B\in T^{[-1,0]}(Z)$, it follows $i^{*}j_{*}X\in T^{[-1,0]}(Z)$.
This uses the non-degeneracy of the motivic $t$-structure on $Z$. A similar
remark applies to direct summands and sums. Therefore we only have to check
that the generators $X$ of $\tilde{T}^{0}(S^{\prime})$ are mapped to
$T^{[-1,0]}(Z)$ under $i^{*}j_{*}$. Thus, let $X=\phi_{*}\mathbf{1}(n)[1]$. We
have a localization triangle in $T(Z)$
$i^{*}\phi_{*}\mathbf{1}(n)[1]\rightarrow
i^{*}j_{*}j^{*}\phi_{*}\mathbf{1}(n)[1]=i^{*}j_{*}\phi^{\prime}_{*}\mathbf{1}(n)[1]\rightarrow
i^{!}\phi_{*}\mathbf{1}(n)[2]\rightarrow i^{*}\phi_{*}\mathbf{1}(n)[2].$
Here $\phi^{\prime}$ is the pullback of $\phi$ along $j$. The first term is in
degree $-1$. The third term is in degree $0$ by absolute purity (see Section
1), using the regularity of $S$. The claim (9) is shown.
We now show $i^{*}T^{0}(S)\subset T^{[-1,0]}(Z)$. Any $X\in T^{0}(S)$ is in
some $T^{0}_{S^{\prime}}(S)$ for sufficiently small $S^{\prime}$. We shrink
$S^{\prime}$ if necessary to ensure that $S^{\prime}\cap Z=\emptyset$. Let
$j:S^{\prime}\rightarrow S$ be the open immersion and let $p:W\rightarrow S$
be its closed complement. There is a triangle
$p^{!}X\rightarrow p^{*}X\rightarrow p^{*}j_{*}j^{*}X\rightarrow p^{!}X[1].$
We know that $p^{!}$ ($p^{*}$) is left-exact (right-exact), that is to say,
the first (second) term is in degrees $\geq 0$ ($\leq 0$, respectively). By
assumption $j^{*}X\in\tilde{T}^{0}(S^{\prime})$, so $p^{*}j_{*}j^{*}X\in
T^{[-1,0]}(W)$, as was shown above. As the $t$-structure on $W$ is non-
degenerate $p^{*}X$ is in degrees $[-1,0]$. As $W$ is the disjoint union of
$Z$ and some more (finitely many) closed points, this also shows $i^{*}X\in
T^{[-1,0]}(Z)$.
Let now $i:Z\rightarrow S$ and $j:S^{\prime}\rightarrow S$ be complementary.
We claim $i^{*}j_{*}T^{0}(S^{\prime})\subset T^{[-1,0]}(Z)$. Given an object
$X\in T^{0}(S^{\prime})$, there is some open immersion
$j^{\prime}:S^{\prime\prime}\rightarrow S^{\prime}$ such that
$j^{\prime*}X\in\tilde{T}^{0}(S^{\prime\prime})$. We have
$i^{*}j_{*}X=i^{*}j_{*}j^{\prime}_{*}j^{\prime*}X$. The motive
$i_{*}i^{*}j_{*}j^{\prime}_{*}j^{\prime*}X$ is a direct summand of
$p_{*}p^{*}(j\circ j^{\prime})_{*}j^{\prime*}X$, where $p$ is the complement
of $j\circ j^{\prime}$. By the above, $p^{*}(j\circ
j^{\prime})_{*}j^{\prime*}X\in T^{[-1,0]}(Z)$, so the full faithfulness and
exactness of $p_{*}$ implies the claim. (iii) is shown.
The cohomological amplitude of $i^{*}j_{*}$ implies the exactness of $j_{*}$:
given a mixed (Artin-)Tate motive $M\in T^{0}(S^{\prime})$, the terms in the
localization triangle
$j_{!}M\rightarrow j_{*}M\rightarrow i_{*}i^{*}j_{*}M$
are in degrees $\leq 0$, $\geq 0$ and $[-1,0]$, respectively, by the above.
From the non-degeneracy of the $t$-structure we see that $j_{*}M$ is then in
degree $0$. This implies the exactness of $j_{*}$ by the non-degeneracy of the
$t$-structure.
The exactness of $j_{!}$ follows by Verdier duality, as does the cohomological
amplitude of $i^{!}$. Thus, (ii) is shown.
(iv): It is easy to see that $f^{*}:\tilde{T}(S)\rightarrow\tilde{T}(V)$ is
exact. Using this and the localization triangles, one sees that $f^{*}$ has
cohomological amplitude $[-1,0]$ and dually for $f^{!}$. By a general
criterion on $t$-exactness of adjoint functors [2, 1.3.17], the adjunctions
$f^{*}\leftrightarrows f_{*}=f_{!}\leftrightarrows f^{!}$ imply that $f_{*}$
is exact. If $f$ is étale then $f^{!}=f^{*}$, so that their exactness is clear
in that case, too.
(v): This follows from the exactness of $j^{*}:T(S)\rightarrow T(S^{\prime})$
and the exactness of $\eta^{\prime*}[-1]:\tilde{T}(S^{\prime})\rightarrow
T({\mathrm{Spec}\text{ }}{F})$ (Proposition 3.7), where $\eta^{\prime}$ is the
generic point of $S^{\prime}$. ∎
###### Definition 4.3.
([2, 1.4.22], see [16, Section 4] for the motivic case) Let
$j:S^{\prime}\rightarrow S$ be an open immersion. For any mixed (Artin-)Tate
motive $M$ over $S^{\prime}$, put $j_{!*}M=\operatorname{im}j_{!}M\rightarrow
j_{*}M$. This is called the _intermediate extension_ of $M$ along $j$.
The image is taken in the (abelian) category of mixed (Artin-)Tate motives
over $S$, using the exactness of $j_{!}$ and $j_{*}$. Thereby, $j_{!*}$ is a
(non-exact) functor $T^{0}(S^{\prime})\rightarrow T^{0}(S)$. Given any mixed
motive $M$ over $S$, such that $i^{!}M$ is concentrated in cohomological
degree $-1$ (as opposed to the general range $[-1,0]$), and such that $i^{*}M$
is in degree $+1$, there is a canonical isomorphism
$j_{!*}j^{*}M=M.$ (10)
In particular, this applies to $M\in\tilde{T}^{0}(S)$, such as
$M=\mathbf{1}[1]$. Moreover, taking the intermediate extension commutes with
compositions of open immersions. These features will be used below, see loc.
cit. for a proof. The reader may want to check that that proof only uses the
motivic $t$-structure and exactness properties of $i^{!}$ etc., which are
established by Theorems 3.8, 4.2.
###### Proposition 4.4.
The cohomological dimension of $\mathbf{DTM}(S)$ and $\mathbf{DATM}(S)$ is one
and two, respectively.
###### Proof: .
We have to show $\mathrm{Hom}(M,M^{\prime}[n])=0$ for any mixed motives $M$,
$M^{\prime}$ over $S$ and $n>1$ (Tate) and $n>2$ (Artin-Tate). Let
$j:S^{\prime}\rightarrow S$ be an open immersion such that $j^{*}M$,
$j^{*}M^{\prime}\in\tilde{T}^{0}(S^{\prime})$. Let $i$ be the complementary
closed immersion of $j$. In the sequel we write $(-,-)^{n}$ for
$\mathrm{Hom}(-,-[n])$ for brevity.
The case $n\geq 3$ is done as follows: the localization triangle (4) for
$M^{\prime}$ and adjunction gives a long exact sequence
$(\underbrace{i^{*}M}_{[-1,0]},\underbrace{i^{!}M^{\prime}[n]}_{[-n,-n+1]})^{0}\rightarrow(M,M^{\prime})^{n}\rightarrow(M,j_{*}j^{*}M^{\prime})^{n}\rightarrow(\underbrace{i^{*}M}_{[-1,0]},\underbrace{i^{!}M^{\prime}[n+1]}_{[-n-1,-n]})^{0}$
We have written the cohomological degrees of the motives underneath, using the
cohomological range of $i^{*}$ and $i^{!}$. The cohomological dimension zero
of (Artin-)Tate motives over finite fields makes the outer terms vanish.
Similar vanishings will be used below without further discussion. Hence we
only have to look at $(j^{*}M,j^{*}M^{\prime})^{n}$, i.e., we may assume $M$
and $M^{\prime}\in\tilde{T}^{0}(S)$. In that case one reduces (exactly as
below) to $M=\phi_{*}\mathbf{1}(a)[1]$ and
$M=\phi^{\prime}_{*}\mathbf{1}(a^{\prime})[1]$, where $\phi$ and
$\phi^{\prime}$ are finite and étale. In that case the vanishing is given by
Lemma 3.2.
The vanishing in the case $n=2$ for Tate motives needs a more involved
localization argument. A similar reasoning for Artin-Tate motives fails—the
difference is because the motives $\mathbf{1}(n)[1]$, which generate
$\tilde{T}^{0}(S)$ in the case of Tate motives, have good reduction at all
places by absolute purity.
The localization triangle for $M^{\prime}$ gives an exact sequence
$(M,j_{!}j^{*}M^{\prime})^{2}\rightarrow(M,M^{\prime})^{2}\rightarrow(M,i_{*}i^{*}M^{\prime})^{2}=(\underbrace{i^{*}M}_{[-1,0]},\underbrace{i^{*}M^{\prime}[2]}_{[-3,-2]})^{0}=0.$
Therefore, in order to show that the middle term vanishes, we may replace
$M^{\prime}$ by $j_{!}j^{*}M^{\prime}$. Similarly, we may replace $M$ by
$j_{*}j^{*}M$. In particular $M\in j_{*}\tilde{T}^{0}(S^{\prime})$,
$M^{\prime}\in j_{!}\tilde{T}^{0}(S^{\prime})$. By Proposition 3.7,
$\tilde{T}^{0}(S^{\prime})$ is generated by means of extension and direct
summands by $\mathbf{1}(a)[1]$ where $a\in\mathbb{Z}$. The claim is stable
under extensions and direct summands and sums so that we may assume
$M=j_{*}A$, $A:=\mathbf{1}(a)[1]$, $M^{\prime}=j_{!}A^{\prime}$,
$A^{\prime}:=\mathbf{1}(a^{\prime})[1]$. Let
$\tilde{A}:=\mathbf{1}(a)[1]\in\tilde{T}^{0}(S)$ and define
$\tilde{A}^{\prime}$ similarly. We have $j^{*}\tilde{A}=A$ and similarly with
$A^{\prime}$.
The triangle $j_{*}A^{\prime}\rightarrow i_{*}i^{*}j_{*}A^{\prime}\rightarrow
j_{!}A^{\prime}[1]$ maps to $j_{*}A^{\prime}\rightarrow
i_{*}{{{}^{\mathrm{p}}}\mathrm{H}}^{0}i^{*}j_{*}A^{\prime}\rightarrow(j_{!*}A^{\prime})[1]=\tilde{A}[1]$.
We apply $(\tilde{A},-)^{1}$ to this map, which gives the last two exact rows
in the diagram. The first exact row maps to the second via the adjunction map
$\tilde{A}=j_{!*}A\rightarrow j_{*}A$.
$\textstyle{(j_{*}A,j_{*}A^{\prime})^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(j_{*}A,i_{*}i^{*}j_{*}A^{\prime})^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(j_{*}A,j_{!}A^{\prime})^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{(\tilde{A},j_{*}A^{\prime})^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\tilde{A},i_{*}i^{*}j_{*}A^{\prime})^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\tilde{A},j_{!}A)^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{(\tilde{A},j_{*}A^{\prime})^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\tilde{A},i_{*}{{{}^{\mathrm{p}}}\mathrm{H}}^{0}i^{*}j_{*}A^{\prime})^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\tilde{A},\tilde{A})^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
The $=$ signs in the leftmost column are by adjunction and
$j^{*}j_{*}A=j^{*}\tilde{A}=A$. The $=$ signs in the second column all use the
adjunction $i^{*}\leftrightarrows i_{*}$ as well as the comological dimension
zero of Tate motives over finite fields and cohomological amplitude of
$i^{*}$, which imply
$(\underbrace{i^{*}j_{*}A}_{[-1,0]},\underbrace{i^{*}j_{*}A^{\prime}[1]}_{[-2,-1]})^{0}=({{{}^{\mathrm{p}}}\mathrm{H}}^{-1}i^{*}j_{*}A,{{{}^{\mathrm{p}}}\mathrm{H}}^{0}i^{*}j_{*}A^{\prime})^{0}.$
Applying $i^{*}$ to the triangle
$i_{*}{{{}^{\mathrm{p}}}\mathrm{H}}^{-1}i^{*}j_{*}A\rightarrow
j_{!}A\rightarrow j_{!*}A$ and using $i^{*}j_{!}=0$ we see
$({{{}^{\mathrm{p}}}\mathrm{H}}^{-1}i^{*}j_{*}A,{{{}^{\mathrm{p}}}\mathrm{H}}^{0}i^{*}j_{*}A^{\prime})^{0}=(i^{*}j_{!*}A,{{{}^{\mathrm{p}}}\mathrm{H}}^{0}i^{*}j_{*}A^{\prime})^{1}$.
This justifies the upper $=$ in the second column. The lower $=$ in that
column follows by the same argument. However,
$(\tilde{A},\tilde{A}^{\prime})^{2}=0$, by vanishing of $K$-theory in the
relevant range (see Lemma 3.2). ∎
###### Theorem 4.5.
For both Tate and Artin-Tate motives, the inclusion $T^{0}(S)\subset T(S)$
extends to a triangulated functor
$\mathbf{D}^{\mathrm{b}}(T^{0}(S))\rightarrow T(S).$ (11)
This functor is an equivalence of categories.
###### Proof: .
The category $\mathbf{DM}_{\mathrm{gm}}(S)$ and thus the subcategories of
(Artin-)Tate motives embedd into some unbounded derived category
$\mathbf{D}(\mathcal{A})$, where $\mathcal{A}$ is an exact category. This
implies the first statement by a general fact in homological algebra [20,
Theorem 1.1.]. Indeed, the interpretation of $\mathbf{DM}_{\mathrm{gm}}(S)$ in
terms of $h$-sheaves shows that (using the notation of [4] and abbreviating
$\mathbf{Shv}$ for the category of $\mathbb{Q}$-linear sheaves with respect to
the $h$-topology on the big site of schemes of finite type over $S$)
$\mathbf{DM}_{\mathrm{gm}}(S)\cong\mathbf{D}_{\mathbb{A}^{1}}(\mathbf{Shv})\subset\mathbf{D}^{\mathrm{eff}}_{\mathbb{A}^{1}}(\mathbf{Sp}(\mathbf{Shv}))\subset\mathbf{D}(\mathbf{Sp}(\mathbf{Shv})).$
More precisely, $\mathbf{DM}_{\mathrm{gm}}(S)$ identifies with the subcategory
of $W_{\Omega}$-local objects in the middle category, which identifies with
the subcategory of $W_{\mathbb{A}^{1}}$-local objects in the right hand
category [4, Sections 5.2, 5.3].
The $t$-structure on $T(S)$ is bounded and non-degenerate, so it remains to
show the full faithfulness of (11) or equivalently that the map
$f_{n}:\operatorname{Ext}^{n}_{T^{0}}(M,M^{\prime})\rightarrow\mathrm{Hom}_{T}(M,M^{\prime}[n])$
is an isomorphism for any $M$, $M^{\prime}\in T^{0}(S)$. The general theory
shows that $f_{0}$ and $f_{1}$ are isomorphisms and that $f_{2}$ is injective
for all $M$ and $M^{\prime}$. For Tate motives, $f_{2}$ is therefore an
isomorphism, since the right hand side is zero by Proposition 4.4. We next
show that $f_{2}$ is an isomorphism for Artin-Tate motives. The motives $M$
and $M^{\prime}$ are fixed, so there is some open embedding
$j:S^{\prime}\rightarrow S$ such that $j^{*}M$ and $j^{*}M^{\prime}$ are in
$\tilde{T}^{0}(S^{\prime})$. Let $i$ be the complement of $j$. Consider the
exact localization sequences
$0\rightarrow
i_{*}{{{}^{\mathrm{p}}}\mathrm{H}}^{-1}i^{*}M\stackrel{{\scriptstyle
a}}{{\rightarrow}}j_{!}j^{*}M\rightarrow K:=\operatorname{coker}a\rightarrow
0$ (12) $0\rightarrow K\rightarrow M\rightarrow
i_{*}{{{}^{\mathrm{p}}}\mathrm{H}}^{0}i^{*}M\rightarrow 0.$ (13)
We write ${}^{n}(-,-)$ for $\operatorname{Ext}^{n}$ and ${}_{n}(-,-)$ for
$\mathrm{Hom}_{T}(-,-[n])$. (12) induces a commutative diagram with exact rows
$\textstyle{{}^{1}(i_{*}{{{}^{\mathrm{p}}}\mathrm{H}}^{-1}i^{*}M,M^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}^{2}(K,M^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}^{2}(j_{!}j^{*}M,M^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}_{1}(i_{*}{{{}^{\mathrm{p}}}\mathrm{H}}^{-1}i^{*}M,M^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}_{2}(K,M^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}_{2}(j_{!}j^{*}M,M^{\prime})={}_{2}(j^{*}M,j^{*}M^{\prime}).}$
The rightmost lower term is zero by the vanishings of $K$-theory (cf. the
argument in the proof of Proposition 4.4), so all vertical maps are
isomorphisms. This and (13) yields a similar diagram:
$\textstyle{{}^{2}(i_{*}{{{}^{\mathrm{p}}}\mathrm{H}}^{0}i^{*}M,M^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}^{2}(M,M^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r}$$\textstyle{{}^{2}(K,M^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}^{3}(i_{*}{{{}^{\mathrm{p}}}\mathrm{H}}^{0}i^{*}M,M^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}_{2}(i_{*}{{{}^{\mathrm{p}}}\mathrm{H}}^{0}i^{*}M,M^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}_{2}(M,M^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}_{2}(K,M^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}_{3}(i_{*}{{{}^{\mathrm{p}}}\mathrm{H}}^{0}i^{*}M,M^{\prime})}$
The outer terms in the lower row vanish because the cohomological dimension of
Artin-Tate motives over ${\mathbb{F}_{\mathfrak{p}}}$ is zero and $i^{!}$ has
cohomological amplitude $[0,1]$. We now show that the rightmost upper term is
zero. Altogether, this implies that $r$ is also surjective. We write
$A:={{{}^{\mathrm{p}}}\mathrm{H}}^{0}i^{*}M$; it is a mixed motive over
${\mathbb{F}_{\mathfrak{p}}}$. Any element of the
Yoneda-$\operatorname{Ext}$-group in question is represented by an exact
sequence
$0\rightarrow i_{*}A\rightarrow X_{1}\stackrel{{\scriptstyle
s}}{{\rightarrow}}X_{2}\rightarrow X_{3}\rightarrow M^{\prime}\rightarrow 0$
in $\mathbf{MATM}(S)$. This extension is the image under the concatenation
mapping
${}^{2}(i_{*}A,\operatorname{coker}s){\times}{}^{1}(\operatorname{coker}s,M^{\prime})\rightarrow{}^{3}(i_{*}A,M^{\prime})$
The left hand factor is a subgroup of
${}_{2}(i_{*}A,\operatorname{coker}s)={}_{2}(A,i^{!}\operatorname{coker}s)=0$
(see above). Therefore, the extension above splits and we have shown that
second $\operatorname{Ext}$-groups and $\mathrm{Hom}$-groups agree.
This shows that the $\mathrm{Hom}(M,M^{\prime}[n])$ form an effaceable
$\delta$-functor, so they are universal and agree with
$\operatorname{Ext}^{n}(M,M^{\prime})$ for all $n\geq 0$. Indeed, for $n\leq
2$ the groups are effaceable since they agree with $\operatorname{Ext}$’s by
the above, for $n>2$ the groups are zero by Proposition 4.4. ∎
The functor $\eta_{*}:\mathbf{DM}(F)\rightarrow\mathbf{DM}(S)$ does not
preserve Artin-Tate motives:
$\mathrm{Hom}_{\mathbf{DM}(S)}(\mathbf{1},\eta_{*}\mathbf{1}(1)[1])=\mathrm{Hom}_{\mathbf{DM}(F)}(\mathbf{1},\mathbf{1}(1)[1])=K^{1}(F)^{(1)}_{\mathbb{Q}}=F^{\times}{\otimes}\mathbb{Q},$
which is a countably infinite-dimensional $\mathbb{Q}$-vector space. However,
the dimensions of all $\mathrm{Hom}$-groups in $T(S)$ are finite (Lemma 3.2).
This example is sharpened by the following proposition. It might be
paraphrased by saying that the “site” of mixed Artin-Tate motives over $S$ has
enough points.
###### Proposition 4.6.
For any Artin-Tate motive $M$ over $S\subset{\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}$, the following are equivalent:
1. (i)
$M=0$.
2. (ii)
$M=\eta_{*}M_{\eta}$, where $M_{\eta}$ is some geometric motive over $F$.
3. (iii)
$i_{\mathfrak{p}}^{*}M=0$ for all closed points $\mathfrak{p}$ of $S$.
4. (iv)
$i_{\mathfrak{p}}^{!}M=0$ for all closed points $\mathfrak{p}$ of $S$.
###### Proof: .
The equivalence of (ii), (iii), and (iv) is an easy consequence of Verdier
duality on compact objects and the limiting localization triangle (5), p. 5.
We now show (iii) $\Rightarrow$ (i). Using localization, the claim for $M$ is
implied by the one for $j^{*}M$ for any open immersion $j$. Therefore we may
assume $M\in\tilde{T}(S)$. Using the $(-1)$-exactness of
$i_{\mathfrak{p}}^{*}:\tilde{T}(S)\rightarrow T({\mathbb{F}_{\mathfrak{p}}})$
we can even assume $M\in\tilde{T}^{0}(S)$. Given a short exact sequence in the
abelian category $\tilde{T}^{0}(S)$
$0\rightarrow A\rightarrow M\rightarrow B\rightarrow 0$
with $\eta_{*}\eta^{*}M=M$, it follows that $\eta_{*}\eta^{*}A=A$ and likewise
for $B$. This is shown as follows: for all closed points $\mathfrak{p}\in S$,
${i_{\mathfrak{p}}}_{*}i_{\mathfrak{p}}^{!}M=0$ implies
$i_{\mathfrak{p}}^{!}B=i_{\mathfrak{p}}^{!}A[1]$, by the full faithfulness of
${i_{\mathfrak{p}}}_{*}$. The long exact
${{{}^{\mathrm{p}}}\mathrm{H}}^{-}$-sequence and the cohomological amplitude
of $i_{\mathfrak{p}}^{!}$ (Theorem 4.2) shows
${{{}^{\mathrm{p}}}\mathrm{H}}^{0}i_{\mathfrak{p}}^{!}B={{{}^{\mathrm{p}}}\mathrm{H}}^{1}i_{\mathfrak{p}}^{!}A$
and all other ${{{}^{\mathrm{p}}}\mathrm{H}}^{*}i_{\mathfrak{p}}^{!}B$,
${{{}^{\mathrm{p}}}\mathrm{H}}^{*}i_{\mathfrak{p}}^{!}A$ vanish. However, for
any $B\in\tilde{T}^{0}(S)$, $i_{\mathfrak{p}}^{!}B$ is in cohomological degree
$1$ (as opposed to the general range $[0,1]$): this may be checked on
generators of $\tilde{T}^{0}_{a}(S)$ for all $a$, where it follows directly
from the definitions (see the proof of Theorem 4.2). Thus
${{{}^{\mathrm{p}}}\mathrm{H}}^{0}i_{\mathfrak{p}}^{!}B=0$, whence
$i_{\mathfrak{p}}^{!}B=i_{\mathfrak{p}}^{!}A[1]=0$ for all $\mathfrak{p}$.
Thus the statement for $M$ is implied by the one for $A$ and $B$. By the
characterization of $\tilde{T}^{0}(S)$ of Proposition 3.7, we therefore only
need to check the statement for generators of $\tilde{T}^{0}_{-2n}(S)$.
We first do this in the case of Tate motives. Then $\tilde{T}^{0}_{-2n}(S)$
consists of direct sums of motives $G:=\mathbf{1}(n)[1]$. In that case the
claim is clear, since none of the (nonzero) generators $G$ satisfy
$\eta_{*}\eta^{*}G=G$: we can twist it so that $n=1$. Then
$\mathrm{H}^{0}(\eta_{*}\eta^{*}G)$ is infinite-dimensional, namely the group
of units in some number field (tensored with $\mathbb{Q}$), but
$\mathrm{H}^{0}(G)$ is the group of units in some ring of $S$-integers, which
are of finite rank.
In the case of Artin-Tate motives, the category $\tilde{T}^{0}_{-2n}(S)$ is
generated by means of direct sums and summands by motives
$G:=\phi_{*}\mathbf{1}(n)[1]$, $\phi:V\rightarrow S$ finite and étale.
Actually, we may assume $\phi$ is Galois: by the same argument as in the proof
of Lemma 2.6, after shrinking $S$ sufficiently, $\mathbf{1}_{V}$ is a direct
summand of $\tilde{\phi}_{*}\mathbf{1}$ where
$\tilde{\phi}:\tilde{V}\rightarrow V$ is the map corresponding to some normal
closure of the function field extension $k(V)/k(S)$. Let $M$ be a summand of
$G$ satisfying $\eta_{*}\eta^{*}M=M$. There is a map $f:S^{\prime}\rightarrow
S$ such that $f^{*}M$ is a Tate motive, Lemma 2.6. By base-change and the
preceding step, we get $f^{*}M=0$. The map
$\mathrm{End}(M)\subset\mathrm{End}(G)\stackrel{{\scriptstyle
a}}{{\rightarrow}}\mathrm{End}(f^{*}G)$ factors over $\mathrm{End}(f^{*}M)=0$,
so we have to show that $a$ is injective. This is done with the same argument
as in the proof of Lemma 2.6: we may shrink $S$ so that $f$ is étale. Since
$\phi$ is Galois, we have
$\mathrm{End}(G)=\mathrm{Hom}(\mathbf{1},\phi^{*}\phi_{*}\mathbf{1})=\mathrm{Hom}(\mathbf{1},\mathbf{1}^{\oplus\deg\phi})$
and
$\mathrm{End}(f^{*}G)=\mathrm{Hom}(\mathbf{1},\phi^{\prime*}\phi^{\prime}_{*}\mathbf{1})=\mathrm{Hom}(\mathbf{1},\mathbf{1}^{\oplus\deg\phi^{\prime}}),$
where $\phi^{\prime}$ is the pullback of $\phi$ along $f$. It is also Galois
and $\deg\phi=\deg\phi^{\prime}$. ∎
## 5 Weights
This section develops a notion of weights on (mixed) Artin-Tate motives. We
follow Bondarko and Hébert [3, 8] for the definition of weights of Artin-Tate
motives. That framework allows basic compatibility statements of weights, for
example their behavior under functoriality. For mixed Artin-Tate motives, we
show that this formalism can be used to produce a functorial and strict weight
filtration. Again, this underlines the similarity of mixed motives with
perverse sheaves. The latter type of results are strictly stronger than the
ones obtained in op. cit., and extend the ones of Levine and Wildeshaus
concerning (Artin-)Tate motives over fields [11, 21]. Briefly, we first relate
the definition of weights to the one familiar from the theory of (perverse)
$\ell$-adic sheaves (Lemma 5.4). The technical key point is determining the
subquotients of motives of the form $f_{*}\mathbf{1}[1]$, where $f$ is a
finite flat map with regular domain (Lemma 5.5). Via Proposition 5.8, this is
the essential point in proving the strictness of the weight filtration in
Theorem 5.10.
The following definition (in the more general situation of geometric motives)
and Lemma 5.3 are due to Bondarko [3, Th. 2.1.1, 2.2.1] and Hébert [8, 3.2,
3.8].
###### Definition 5.1.
Let $\mathcal{C}$ be a triangulated category and $C$ a set of objects of
$\mathcal{C}$. The category
$\overline{\operatorname{Ext}}(C)\subset\mathcal{C}$ is defined to be the
smallest full triangulated subcategory that contains $C$ and is stable under
extensions, i.e., such that for any distinguished triangle
$C\rightarrow X\rightarrow C^{\prime}$
with $C$ and $C^{\prime}$ in $\overline{\operatorname{Ext}}(C)$, $X$ is also
in $\overline{\operatorname{Ext}}(C)$.
Let $S$ be an (open or closed) subscheme of ${\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}$. Let $T_{\langle 0\rangle}(S)$ be the idempotent
completion of the additive category (i.e., closed under direct summands and
finite direct sums) generated by $f_{*}\mathbf{1}(a)[2a]$, where
$f:S^{\prime}\rightarrow S$ is a finite map such that $S^{\prime}$ is regular
(of dimension $\leq\dim S$) and $a\in\mathbb{Z}$ is arbitrary.
Put $T_{\langle m\rangle}:=T_{\langle 0\rangle}[m]$ and let $T_{\langle\leq
m\rangle}(S)$ be the idempotent completion of
$\overline{\operatorname{Ext}}(\cup_{l\leq m}T_{\langle l\rangle}(S))$ and
define $T_{\langle\geq m\rangle}(S)$ similarly.111Bondarko and Hébert use
different notations. We follow Hébert here, thus our $T_{\langle\leq
m\rangle}$ would be $T_{\langle\geq m\rangle}$ in Bondarko’s notation. We
write $T_{\langle>n\rangle}$ for $T_{\langle\geq n+1\rangle}$ etc.
###### Remark 5.2.
For a map $f$ as in the definition
$D(f_{*}f^{*}\mathbf{1})=f_{!}f^{!}\mathbf{1}(1)[2]=f_{*}f^{*}\mathbf{1}(1)[2]$
by absolute purity and the regularity assumption on the domain of $f$. Thus
$D(T_{\langle\leq m\rangle}(S))=T_{\langle\geq-m\rangle}(S)$.
###### Lemma 5.3.
1. (i)
For any $M_{\leq m}\in T_{\langle\leq m\rangle}(S)$, $M_{\geq m+1}\in
T_{\langle\geq m+1\rangle}(S)$,
$\mathrm{Hom}(M_{\leq m},M_{\geq m+1})=0.$
2. (ii)
For any $M\in T(S)$ and any $m\in\mathbb{Z}$ there is a (non-unique) triangle
$M_{\leq m}\rightarrow M\rightarrow M_{\geq m+1}$
where $M_{\leq m}\in T_{\langle\leq m\rangle}(S)$, $M_{\geq m+1}\in
T_{\langle\geq m+1\rangle}(S)$.
3. (iii)
For any map $f:S^{\prime}\rightarrow S$ between regular schemes which is
either a finite map, an open immersion or a closed immersion, the functors
$f^{*}$ and $f_{!}$ preserve the $T_{\langle\leq-\rangle}$-subcategories, and
dually for $f_{*}$ and $f^{!}$.
4. (iv)
Let $j$ and $i$ be an open and complementary closed embedding. Then $M$ is in
$T_{\langle\leq m\rangle}(S)$ iff $j^{*}M$ and $i^{*}M$ are in the
corresponding $T_{\langle\leq m\rangle}$-subcategories.
Recall that a mixed $\ell$-adic sheaf $\mathcal{F}$ on a curve
$C/{\mathbb{F}_{q}}$ is said to be of weights $\leq m$ iff all pullbacks
$i^{*}\mathcal{F}$ to all closed points of $C$ have this property [2, 5.1.9].
We now relate this approach to weights to Definition 5.1.
###### Lemma 5.4.
For any motive $M\in T(S)$, the following are equivalent:
1. (i)
$M\in T_{\langle\leq 0\rangle}(S)$
2. (ii)
$i^{*}M\in T_{\langle\leq 0\rangle}({\mathbb{F}_{\mathfrak{p}}})$ for any
closed point $i:{\mathrm{Spec}\text{
}}{{\mathbb{F}_{\mathfrak{p}}}}\rightarrow S$.
###### Proof: .
The implication (i) $\Rightarrow$ (ii) is trivial. Conversely, by Lemma
5.3(iv) and the localization triangle it is sufficient to show the implication
(ii) $\Rightarrow$ (i) for $M\in\tilde{T}(S)$. There is a non-canonical
isomorphism
$i^{*}M\cong\oplus_{n}{{{}^{\mathrm{p}}}\mathrm{H}}^{n}(i^{*}M)[-n]$ by the
semi-simplicity of Artin-Tate motives over finite fields. This in turn is
(canonically) isomorphic to
$\oplus_{n}i^{*}({{{}^{\mathrm{p}}}\mathrm{H}}^{n}M)[-n]$ by the
$(-1)$-exactness of $i^{*}$ (restricted to $\tilde{T}(S)$). Thus $i^{*}M\in
T_{\langle\leq 0\rangle}$ implies $i^{*}{{{}^{\mathrm{p}}}\mathrm{H}}^{n}M\in
T_{\langle\leq n\rangle}$ for all $n$. By definition,
${{{}^{\mathrm{p}}}\mathrm{H}}^{n}M\in\tilde{T}_{\langle\leq n\rangle}$ for
all $n$ implies $M\in\tilde{T}_{\langle\leq 0\rangle}$. Thus we can replace
$M$ by the ${{{}^{\mathrm{p}}}\mathrm{H}}^{n}M$ and show the statement for
$M\in\tilde{T}^{0}(S)$ only. Using the same argument with respect to the
auxiliary $t$-structure on $\tilde{T}^{0}(S)$ we reduce to
$M\in\tilde{T}^{0}_{-2a}(S)$, $a\in\mathbb{Z}$. In this category, any
extension splits. That is, its objects are direct summands of sums of motives
$f_{*}\mathbf{1}(a)[1]$, where $f$ is finite etale. The implication (ii)
$\Rightarrow$ (i) is obvious in that case. ∎
The following lemma is the key stepstone in establishing the strictness of the
weight filtration below.
###### Lemma 5.5.
Let $f:\tilde{S}\rightarrow S$ and $g:V\rightarrow S$ be finite maps with
regular domains $\tilde{S}$ and $V$ of dimension one and zero, respectively.
Any subquotient in the abelian category $T^{0}(S)$ of $W:=f_{*}\mathbf{1}[1]$
is a direct factor of $W$. In particular, it is also contained in
$T_{\langle\leq 1\rangle}$. A similar statement holds for $g_{*}\mathbf{1}$.
###### Proof: .
Let $X\subset W$ be a subobject, $Y:=W/X$. Let $j:S^{\prime}\rightarrow S$ be
an open immersion such that $j^{*}X$, $j^{*}W$, and
$j^{*}Y\in\tilde{T}^{0}(S^{\prime})$. Let $i:Z\rightarrow S$ be its
complement. We have a commutative diagram in $T^{0}(S)$ with exact rows and
columns:
| |
---|---|---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{i_{*}{{{}^{\mathrm{p}}}\mathrm{H}}^{0}i^{!}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{i_{*}{{{}^{\mathrm{p}}}\mathrm{H}}^{0}i^{!}W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{j_{*}j^{*}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{j_{*}j^{*}W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{j_{*}j^{*}Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{i_{*}{{{}^{\mathrm{p}}}\mathrm{H}}^{1}i^{!}X.}$
First of all, $i_{*}{{{}^{\mathrm{p}}}\mathrm{H}}^{0}i^{!}W=0$ by absolute
purity, using that the domain of $f$ is regular. Thus $X\rightarrow
j_{*}j^{*}X$ is a monomorphism. The curved arrows are splittings of the
corresponding short exact sequences. Their existence is seen as follows: the
third row exact sequence splits since $j^{*}X\in\tilde{T}^{0}_{0}(S^{\prime})$
(which is a semi-simple category, since we use rational coefficients) and
likewise for $j^{*}Y$, as follows from the $(-1)$-exactness of $p^{*}$
(restricted to $\tilde{T}^{0}(S^{\prime})$), $(+1)$-exactness of $p^{!}$ for
all closed points $p$ of $S^{\prime}$ and purity. The map $W\rightarrow
j_{*}j^{*}W\stackrel{{\scriptstyle\sigma}}{{\rightarrow}}j_{*}j^{*}X\rightarrow
i_{*}{{{}^{\mathrm{p}}}\mathrm{H}}^{1}i^{!}X$ is zero: its image is a quotient
of $W$ of the form $i_{*}N$ with $N\in T^{0}(Z)$. By the right-exactness of
$i^{*}$, $N$ is a quotient of ${{{}^{\mathrm{p}}}\mathrm{H}}^{0}i^{*}W=0$.
Thus $N=0$. Hence $W\rightarrow
j_{*}j^{*}W\stackrel{{\scriptstyle\sigma}}{{\rightarrow}}j_{*}j^{*}X$ factors
over some map $\tau:W\rightarrow X$. A short diagram chase shows that $\tau$
is a splitting of the second row exact sequence.
The well-known second statement is easier. The details are omitted.∎
The following corollary is a motivic analog of [2, Th. 4.3.1 (i)].
###### Corollary 5.6.
The category of mixed Artin-Tate motives is Artinian and Noetherian: given any
$M\in T^{0}(S)$, and any sequence of subobjects in $T^{0}(S)$
$0=M_{-\infty}\subset\dots M_{i}\subset M_{i+1}\subset\dots M_{\infty}:=M$
there is an $n>0$ such that $M_{i}=M_{i+1}$ for all $\infty>|i|>n$ and dually
for quotients.
###### Proof: .
More generally, we claim that for any motive $N\in T(S)$, there is some number
$l(N)$ such that the length of all subquotients of
${{{}^{\mathrm{p}}}\mathrm{H}}^{n}N$, for all $n\in\mathbb{Z}$, is bounded by
$l(N)$. Given a triangle $N^{\prime}\rightarrow N\rightarrow N^{\prime\prime}$
with $N^{\prime}$ and $N^{\prime\prime}$ satisfying this claim, the claim also
holds for $N$. This follows from the long exact cohomology sequence. Thus, it
is sufficient to prove the claim for motives $f_{*}\mathbf{1}(a)[n]$, where
$a,n\in\mathbb{Z}$ and $f$ is a finite map with regular domain (of dimension
one or zero). For them, Lemma 5.5 and the compactness of $f_{*}\mathbf{1}$
settle the claim. ∎
###### Proposition 5.7.
The truncation functors for the motivic $t$-structure are weight exact, that
is to say
${{{}^{\mathrm{p}}}\mathrm{H}}^{s}T_{\langle\leq n\rangle}(S)=T_{\langle\leq
s+n\rangle}^{0}(S)$
and likewise for $\geq n$. In particular we have the following analog of [2,
5.1.8.]: an Artin-Tate motive $M$ is of weights $\leq n$ iff all
${{{}^{\mathrm{p}}}\mathrm{H}}^{s}M$ are of weights $\leq s+n$.
###### Proof: .
We can assume $s=0$. We clearly have $T_{\langle\leq
n\rangle}^{0}(S)={{{}^{\mathrm{p}}}\mathrm{H}}^{0}T_{\langle\leq
n\rangle}^{0}(S)\subseteq{{{}^{\mathrm{p}}}\mathrm{H}}^{0}T_{\langle\leq
n\rangle}(S)$. Conversely, let $M\in T_{\langle\leq n\rangle}(S)$. By Lemma
5.4 we have to show $i^{*}{{{}^{\mathrm{p}}}\mathrm{H}}^{0}M$ is of weights
$\leq n$ for all closed points $i$, if $i^{*}M$ has this property. The
truncation functors $\tau_{-}$ of the motivic $t$-structure give a
distinguished triangle
$i^{*}\tau_{\leq 0}M\rightarrow i^{*}M\rightarrow i^{*}\tau_{\geq
1}M\stackrel{{\scriptstyle\delta}}{{\rightarrow}}i^{*}\tau_{\geq 0}M[1].$
The third term is in cohomological degrees $0$ and $1$, the fourth one is in
degrees $-2$ and $-1$. Hence the boundary map $\delta$ vanishes, by semi-
simplicity of Artin-Tate motives over finite fields. Therefore the triangle
splits (in a non-canonical way). The middle term $i^{*}M$ being of weights
$\leq n$, the same follows for the summands $i^{*}\tau_{\leq 0}M$ and
$i^{*}\tau_{\geq 1}M$. An induction shows that
$i^{*}{{{}^{\mathrm{p}}}\mathrm{H}}^{0}M$ is of weights $\leq n$.
The concluding statement is a consequence of this and the truncation triangles
for the motivic $t$-structure. ∎
###### Proposition 5.8.
The category $T_{\langle\leq n\rangle}^{0}(S)$ is stable under subquotients
(that exist in the abelian category $T^{0}(S)$).
###### Proof: .
Recall that $T_{\langle\leq
n\rangle}^{0}(S)\stackrel{{\scriptstyle\text{\ref{satz_weightexact}}}}{{=}}{{{}^{\mathrm{p}}}\mathrm{H}}^{0}T_{\langle\leq
n\rangle}(S)$ is ${{{}^{\mathrm{p}}}\mathrm{H}}^{0}$ applied to the
$\operatorname{Ext}$-closure (Definition 5.1) of the idempotent completion of
$\left\\{\begin{array}[]{c}f_{*}\mathbf{1}(a)[2a+l];\ \ l\leq n,\
a\in\mathbb{Z},\\\ f:S^{\prime}\rightarrow S\text{ finite with regular
domain}\end{array}\right\\}.$
The subquotients of
${{{}^{\mathrm{p}}}\mathrm{H}}^{0}(f_{*}\mathbf{1}(a)[2a+l])$ are contained in
$T_{\langle\leq n\rangle}^{0}$ by Lemma 5.5. It is thus sufficient to show the
following statement: for any triangle
$A\rightarrow X\rightarrow B$
such that all subquotients of ${{{}^{\mathrm{p}}}\mathrm{H}}^{0}A$ and
${{{}^{\mathrm{p}}}\mathrm{H}}^{0}B$ are in $T_{\langle\leq n\rangle}^{0}(S)$,
all subquotients of ${{{}^{\mathrm{p}}}\mathrm{H}}^{0}X$ are in
$T_{\langle\leq n\rangle}^{0}(S)$, too. Let $Y$ be a subobject of
${{{}^{\mathrm{p}}}\mathrm{H}}^{0}X$. The triangle induces a short exact
sequence
$0\rightarrow\operatorname{coker}({{{}^{\mathrm{p}}}\mathrm{H}}^{-1}B\rightarrow{{{}^{\mathrm{p}}}\mathrm{H}}^{0}A)\stackrel{{\scriptstyle
v}}{{\rightarrow}}{{{}^{\mathrm{p}}}\mathrm{H}}^{0}X\stackrel{{\scriptstyle
w}}{{\rightarrow}}\ker({{{}^{\mathrm{p}}}\mathrm{H}}^{0}B\rightarrow{{{}^{\mathrm{p}}}\mathrm{H}}^{1}A)\rightarrow
0$
which in turn induces
$0\rightarrow v^{-1}(Y)\rightarrow Y\rightarrow w(Y)\rightarrow 0.$ (14)
The outer terms are subquotients of ${{{}^{\mathrm{p}}}\mathrm{H}}^{0}A$ and
${{{}^{\mathrm{p}}}\mathrm{H}}^{0}B$, respectively, hence they are in
$T_{\langle\leq n\rangle}^{0}(S)$. This category is stable under extensions in
$T^{0}(S)$ by Lemma 5.4. Therefore $Y\in T_{\langle\leq n\rangle}^{0}(S)$.
Quotients of ${{{}^{\mathrm{p}}}\mathrm{H}}^{0}X$ are treated dually. ∎
The following lemma is a partial converse to the general vanishings in weight
structures (Lemma 5.3(i)). It is used for the strictness of the weight
filtration below.
###### Lemma 5.9.
Let $n\in\mathbb{Z}$, and any $M\in T_{\langle>0\rangle}^{0}(S)$,
$M^{\prime}\in T_{\langle\leq 0\rangle}^{0}(S)$. Then
$\mathrm{Hom}(M,M^{\prime})=0.$
###### Proof: .
The argument in the above proof, cf. (14), can be recycled to show the
following: given any distinguished triangle $A\rightarrow B\rightarrow C$ such
that for all subquotients $X$ of ${{{}^{\mathrm{p}}}\mathrm{H}}^{0}A$ and of
${{{}^{\mathrm{p}}}\mathrm{H}}^{0}C$ the group $\mathrm{Hom}(X,M^{\prime})$
vanishes, the same vanishing holds for all subquotients $X$ of
${{{}^{\mathrm{p}}}\mathrm{H}}^{0}B$, too. The dual statement of this, the
description of
$T_{\langle-\rangle}^{0}(S)={{{}^{\mathrm{p}}}\mathrm{H}}^{0}T_{\langle-\rangle}(S)$
in terms of extensions and the classification of subquotients in Lemma 5.5
show that it is suffices to see $\mathrm{Hom}(M,M^{\prime})=0$, where
$M=\left\\{\begin{array}[]{c}f_{*}\mathbf{1}(a)[1]\ \ 1-2a>0\\\ \text{or}\\\
g_{*}\mathbf{1}(a),\ \ -2a>0\end{array}\right.\ \ \text{and}\ \
M^{\prime}=\left\\{\begin{array}[]{c}f^{\prime}_{*}\mathbf{1}(a^{\prime})[1]\
\ 1-2a^{\prime}\leq 0\\\ \text{or}\\\ g^{\prime}_{*}\mathbf{1}(a^{\prime}),\ \
-2a^{\prime}\leq 0\end{array}\right.$
Here $f:V\rightarrow S$ and $f^{\prime}:V^{\prime}\rightarrow S$ are finite
maps with regular domain of dimension $1$, $g$ and $g^{\prime}$ are finite
maps with $0$-dimensional image. Let $b=a^{\prime}-a$. Most cases are an
immediate consequence of absolute purity, except for the vanishing of
$\mathrm{Hom}(f_{*}\mathbf{1}(a),f^{\prime}_{*}\mathbf{1}(a^{\prime}))$. It
reduces to showing
$\mathrm{Hom}_{V^{\prime}}(\tilde{f}_{*}\mathbf{1},\mathbf{1}(b))\left(=\mathrm{Hom}_{W}(\mathbf{1},\tilde{f}^{!}\mathbf{1}(b))\right)=0,$
where $\tilde{f}:W:=V{\times}_{S}V^{\prime}\rightarrow V^{\prime}$. If $W$
happens to be regular, this group identifies by absolute purity
($\tilde{f}^{!}\mathbf{1}=\tilde{f}^{*}\mathbf{1}$) with
$K_{2b}(W)^{(b)}_{\mathbb{Q}}=0$, since $b>0$. In general there is the
following argument, due to Hébert [8, Theorem 3.1.] and Bondarko [3, Lemma
1.1.4]: let $n:W^{\prime}\rightarrow W$ be the normalization map, $i:Z\subset
W$ the exceptional “divisor”, $Z^{\prime}$ its preimage in $W^{\prime}$,
$z:Z^{\prime}\rightarrow W$. The distinguished triangle
$\mathbf{1}_{W}\rightarrow i_{*}\mathbf{1}_{Z}\oplus
n_{*}\mathbf{1}_{W^{\prime}}\rightarrow z_{*}\mathbf{1}_{Z^{\prime}}$ induces
an exact sequence
$\dots\rightarrow\mathrm{Hom}(\tilde{f}_{*}i_{*}\mathbf{1}\oplus\tilde{f}_{*}n_{*}\mathbf{1},\mathbf{1}(b))\rightarrow\mathrm{Hom}(\tilde{f}_{*}\mathbf{1},\mathbf{1}(b))\rightarrow\mathrm{Hom}(\tilde{f}_{*}z_{*}\mathbf{1}[-1],\mathbf{1}(b))\rightarrow\dots$
The first half of the first term vanishes because of $b>0$, the second one by
the previous point. The last term vanishes for reasons of cohomological
dimension. ∎
We can now construct the weight filtration. In a nutshell, the theorem says
that weights for mixed Artin-Tate motives behave as they should, that is, as
they do for mixed perverse $\ell$-adic sheaves [2, 5.3.5] and mixed Hodge
structures [5, 2.3.5]. The definition of the weight filtration $W_{n}M$ as the
biggest subobject of $M$ of weight $\leq n$ is akin to a similar definition of
Huber concerning the slice filtration of Hodge structures [9, Lemma 2.1]. It
is worth noting that the classical proofs of the strictness of the weight
filtration for perverse sheaves on a curve $C$ over ${\mathbb{F}_{q}}$ make
use of the structural map $C\rightarrow\mathrm{Spec}\text{ }{\mathbb{F}_{q}}$.
In our situation, absolute purity (and the regularity of the base schemes we
work over) take the rôle of this geometric piece of information.
###### Theorem 5.10.
Let $M\in T^{0}(S)$ be a mixed Artin-Tate motive and $n\in\mathbb{Z}$.
1. (i)
The set
$\mathcal{W}_{n}M:=\\{A\in T_{\langle\leq n\rangle}^{0},A\text{ is a subobject
of }M\\}/\text{isomorphism}$
has a unique maximal element. Any choice of representatives of it is denoted
$W_{n}(M)$. The assignment $M\mapsto W_{n}M$ is functorial (up to isomorphism,
as $W_{n}M$ is only defined up to isomorphism).
2. (ii)
The quotient $W_{>n}M:=M/W_{n}M$ is in $T_{\langle>n\rangle}^{0}(S)$.
3. (iii)
The _weight filtration_ $W_{*}-$ is strict: given any morphism $m:M\rightarrow
M^{\prime}$ in $T^{0}$, $\operatorname{im}W_{n}m=\operatorname{im}m\cap
W_{n}M^{\prime}$. Here $\operatorname{im}$ denotes the image of a map (in the
abelian category $T^{0}(S)$).
4. (iv)
The assignment
$M\mapsto M:=W_{n}M/W_{n-1}M$
is an exact functor $\operatorname{gr}_{n}^{W}:T^{0}(S)\rightarrow T_{\langle
n\rangle}^{0}(S)$.
###### Proof: .
(i): The existence of some maximal element in $\mathcal{W}_{n}M$ is assured by
Corollary 5.6. Let $\iota_{i}:A_{i}\rightarrow M$, $i=1,2$ be two maximal
elements in $\mathcal{W}_{n}M$. Let
$A:=\operatorname{im}\left(\iota_{1}\oplus\iota_{2}:A_{1}\oplus
A_{2}\rightarrow M\right)$. This image is taken in the abelian category
$T^{0}(S)$. As a quotient of $A_{1}\oplus A_{2}$, $A$ is in $T_{\langle\leq
n\rangle}^{0}(S^{\prime})$ by Proposition 5.8. Hence $A\in\mathcal{W}_{n}M$.
By maximality of the $A_{i}$ we have $A_{2}=A=A_{1}$.
Given a map $m:M_{1}\rightarrow M_{2}$, fix representatives for $W_{n}M_{i}$,
$i=1,2$. Again by Proposition 5.8, $W_{n}M_{1}\subset M_{1}\rightarrow M_{2}$
factors (uniquely, once representative are chosen) over $W_{n}M_{2}$. As the
$W_{n}M_{i}$ are subobjects of $M_{i}$, the compatibility of $W_{n}$ with
compositions is clear.
(ii): Let $A_{\langle\leq n\rangle}\rightarrow W_{>n}M\rightarrow
A_{\langle>n\rangle}$ be any distinguished triangle where the outer terms are
in $T_{\langle\leq n\rangle}$ and $T_{\langle>n\rangle}$, respectively (Lemma
5.3). The induced long exact sequence
$\dots\stackrel{{\scriptstyle
k}}{{\rightarrow}}{{{}^{\mathrm{p}}}\mathrm{H}}^{0}(A_{\langle\leq
n\rangle})\rightarrow
W_{>n}M\rightarrow{{{}^{\mathrm{p}}}\mathrm{H}}^{0}(A_{\langle>n\rangle})\stackrel{{\scriptstyle
l}}{{\rightarrow}}{{{}^{\mathrm{p}}}\mathrm{H}}^{1}(A_{\langle\leq
n\rangle})\rightarrow\dots$
gives
$0\rightarrow\operatorname{coker}k\rightarrow W_{>n}M\rightarrow\ker
l\rightarrow 0.$
By Proposition 5.8, $V:=\operatorname{coker}k\in T_{\langle\leq n\rangle}$.
Consider the pullback of the bottom row by $V$
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{n}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{n}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{>n}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$
As an extension of $V$ and $W_{n}M$, $V^{\prime}$ is of weights $\leq n$, but
also a subobject of $M$, so $V^{\prime}=W_{n}M$. This shows $V=0$, so that
$W_{>n}M=\ker l\in T_{\langle>n\rangle}$.
(iii): We can assume $m$ is surjective and $M^{\prime}=W_{n}M^{\prime}$. We
have to show $W_{n}M\rightarrow M^{\prime}$ is surjective. Consider the
commutative diagram with exact rows
---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{n}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$$\textstyle{W_{>n}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{n}M^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$
The cokernel of $a$ is, as a quotient of $W_{n}M^{\prime}$, in $T_{\langle\leq
n\rangle}^{0}(S)$. By (ii), $W_{>n}M\in T_{\langle>n\rangle}^{0}(S)$. Lemma
5.9 and the snake lemma imply $\operatorname{coker}a=0$.
(iv): By (ii) and the exact sequence
$0\rightarrow W_{n-1}M\rightarrow
W_{n}M\rightarrow\operatorname{gr}_{n}^{W}M\rightarrow 0,$ (15)
$\operatorname{gr}_{n}^{W}$ does map to $T_{\langle n\rangle}^{0}(S)$. The
exactness of $\operatorname{gr}_{n}^{W}$ is a well-known reformulation of the
strictness of the weight filtration [5, p. 8]. ∎
###### Example 5.11.
Let $j:S^{\prime}\rightarrow S$ be some open embedding with complement $i$.
The sequence (15) for $M:=j_{*}j^{*}\mathbf{1}[1]$ and $n=2$ reads
$0\rightarrow\mathbf{1}[1]\rightarrow M\rightarrow
i_{*}\mathbf{1}(-1)\rightarrow 0.$
## References
* Ayo [07] Joseph Ayoub. Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. I. Astérisque, (314):x+466 pp. (2008), 2007.
* BBD [82] A. A. Beĭlinson, J. Bernstein, and P. Deligne. Faisceaux pervers. In Analysis and topology on singular spaces, I (Luminy, 1981), volume 100 of Astérisque, pages 5–171. Soc. Math. France, Paris, 1982.
* Bon [10] Mikhail Bondarko. Weights for relative motives; relation with mixed sheaves. Preprint, Jul. 2010, http://arxiv.org/abs/1007.4543, 2010.
* CD [10] Denis-Charles Cisinski and Frédéric Déglise. Triangulated categories of motives. Preprint, 2010.
* Del [71] Pierre Deligne. Théorie de Hodge. II. Inst. Hautes Études Sci. Publ. Math., 40:5–57, 1971.
* DG [05] Pierre Deligne and Alexander B. Goncharov. Groupes fondamentaux motiviques de Tate mixte. Ann. Sci. École Norm. Sup. (4), 38(1):1–56, 2005.
* Han [95] Masaki Hanamura. Mixed motives and algebraic cycles. I. Math. Res. Lett., 2(6):811–821, 1995.
* Héb [10] David Hébert. Structures de poids à la Bondarko sur les motifs de Beilinson . Preprint, Jul. 2010, http://arxiv.org/abs/1007.0219, 2010.
* Hub [08] Annette Huber. Slice filtration on motives and the Hodge conjecture (with an appendix by J. Ayoub). Math. Nachr., 281(12):1764–1776, 2008.
* Ivo [05] Florian Ivorra. Réalisation l-adique des motifs mixtes, 2005. Thesis, Université Paris 6, http://www.math.uiuc.edu/K-theory/0762.
* Lev [93] Marc Levine. Tate motives and the vanishing conjectures for algebraic $K$-theory. In Algebraic $K$-theory and algebraic topology (Lake Louise, AB, 1991), volume 407 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., pages 167–188. Kluwer Acad. Publ., Dordrecht, 1993.
* Lev [98] Marc Levine. Mixed motives, volume 57 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1998.
* Org [04] Fabrice Orgogozo. Isomotifs de dimension inférieure ou égale à un. Manuscripta Math., 115(3):339–360, 2004.
* Qui [72] Daniel Quillen. On the cohomology and $K$-theory of the general linear groups over a finite field. Ann. of Math. (2), 96:552–586, 1972.
* Sch [91] Anthony J. Scholl. Remarks on special values of $L$-functions. In $L$-functions and arithmetic (Durham, 1989), volume 153 of London Math. Soc. Lecture Note Ser., pages 373–392. Cambridge Univ. Press, Cambridge, 1991.
* Sch [10] Jakob Scholbach. $f$-cohomology and motives over $\mathbb{Z}$. Preprint, Feb. 2010, http://arxiv.org/abs/1003.1219, 2010.
* Sou [79] C. Soulé. $K$-théorie des anneaux d’entiers de corps de nombres et cohomologie étale. Invent. Math., 55(3):251–295, 1979.
* Voe [00] Vladimir Voevodsky. Triangulated categories of motives over a field. In Cycles, transfers, and motivic homology theories, volume 143 of Ann. of Math. Stud., pages 188–238. Princeton Univ. Press, Princeton, NJ, 2000.
* Wei [05] Charles Weibel. Algebraic $K$-theory of rings of integers in local and global fields. In Handbook of $K$-theory. Vol. 1, 2, pages 139–190. Springer, Berlin, 2005.
* [20] J. Wildeshaus. f-categories and Tate motives. Preprint, October 9, 2008, K-theory Preprint Archives, http://www.math.uiuc.edu/K-theory/0913/, 2008.
* [21] J. Wildeshaus. Notes on Artin-Tate motives. Preprint, Nov 2008, arXiv, http://arxiv.org/abs/0811.4551v1, 2008.
|
arxiv-papers
| 2010-03-05T13:49:45 |
2024-09-04T02:49:08.823004
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jakob Scholbach",
"submitter": "Jakob Scholbach",
"url": "https://arxiv.org/abs/1003.1267"
}
|
1003.1345
|
Author Identifiers in Scholarly Repositories
Simeon Warner
Cornell Information Science and
Cornell University Library
Ithaca, NY 14850, USA
simeon.warner@cornell.edu
Submitted: 2009-10-09
## Abstract
Bibliometric and usage-based analyses and tools highlight the value of
information about scholarship contained within the network of authors,
articles and usage data. Less progress has been made on populating and using
the author side of this network than the article side, in part because of the
difficulty of unambiguously identifying authors. I briefly review a sample of
author identifier schemes, and consider use in scholarly repositories. I then
describe preliminary work at arXiv to implement public author identifiers,
services based on them, and plans to make this information useful beyond the
boundaries of arXiv.
## 1 Context
In an ideal scholarly communication system there would be tools to browse,
navigate, make recommendations and assess influence based on the complete
graph of all actors (people, collaborations, institutions) and all
communication artifacts (articles, comments, blog posts, usage
data111Logically usage data would be links between actors and artifacts.
However, for historical, cultural and practical reasons most usage data is
treated as anonymous even though co-usage information may be extracted.). As a
shorthand I will call this complete graph the publication network. Contained
within it are the familiar citation, usage, co-authorship, and co-citation
graphs. In recent bibliometric and usage-based work, significant progress has
been made with the artifact part of this graph (see, for example the work of
the MESUR project [3]). Much less progress has been made with the actor part
of the graph, in part because it is much harder to unambiguously identify
authors than articles.
Lastname, Initial | Count
---|---
Zhang, Y | 100
Lee, J | 97
Wang, Y | 89
Wang, J | 84
Chen, Y | 77
Kim, J | 77
Wang, X | 76
Lee, S | 74
Kim, S | 69
Liu, Y | 69
Table 1: Most frequently occurring ${lastname,initial}$ pairs in arXiv user
accounts. There may be a few duplicate accounts but this indicates that nearly
100 different people named “Zhang, Y” have created user accounts at arXiv (as
of May 2009).
Consider table 1 which shows the most frequently occurring
${lastname,initial}$ pairs in arXiv user accounts. This illustrates one facet
of the name disambiguation problem, namely that there are many authors with
the same name. This is compounded by inconsistent spellings, use of initials
or full first names, and even name changes. Within a single repository such as
arXiv it is not usually possible to accurately answer the question “show me
all the articles by this Zhang, Y”. In recent years there has been
considerable work on unsupervised and supervised author name disambiguation
using many different heuristic, machine learning and clustering techniques,
and many different properties including co-authorship, citations and
subjects/topics. While much better than naive approaches, these techniques are
still far from perfect.
In a recent Nature Correspondence, Raf Aerts asked “If it is possible to have
DOIs for objects (or, so they say, enough IPv6 addresses for every molecule on
Earth), why is it so difficult to implement DAIs [Digital Author Identifiers]
for authors?” [1]. Raf had earlier hinted at part of the answer by pointing
out that he has more than one identifier in Scopus [6]. As we have already
discussed, it is difficult to mine existing data to disambiguate references to
authors. The more fundamental part of the answer is that it is much easier to
create DOIs for articles when the one owner for an article creates the one DOI
for it and presents it with the article (ignoring the issue of multiple
versions of articles). As authors, we are not owned by a single authority and
even if an identifier were created for us at birth by the appropriate
government, there would be significant privacy concerns about using it for
everything. Consider, for example, concerns over the uses and misuses of
social security numbers in the USA. While we want to link a single author’s
works together, do we want that identity to immediately link us to all other
digital information about the private life of the individual?
## 2 Author Identifiers
To illustrate the diversity of currently used author identifiers, table 2
shows several example schemes used in the scholarly domain. A more detailed
inventory is provided on the repinf wiki [5]. The OpenID and ISNI schemes are
not limited to the scholarly domain. OpenID is aimed primarily at
authentication, however, if it continues to see growing acceptance it may well
be a useful open system that repositories could use. It is not clear whether
ISNI will develop into a widely used system. The largest efforts to create
author identifiers specifically for the scholarly domain, Scopus Author
Identifiers and ResearcherID, come from commercial entities and are clearly
motivated by the desire to provide improved services based upon then. It is
not clear how open the interfaces based on these identifiers will be, or what
data about them will be openly available.
Scheme | Example | Scope | Authority
---|---|---|---
OpenID | http://samruby.myopenid.com/ | People | Distributed, anyone supporting the protocol, relies upon DNS
ISNI (International Standard Name Identifier) | ISNI 1422 4586 3573 0476 | People | Draft ISO standard requiring central DB operated by proposed International Agency
Scopus Author Id | 7103063073 | Academia | Elsevier
ResearcherID | A-1637-2009 | Academia | Thomson Reuters
Digital Author Id | info:eu-repo/dai/nl/304825271 | Dutch Researchers | Dutch Universities and Research Institutes
RePEc Author Service | pzi1 | Economics | RePEc
arXiv Author Id | http://arxiv.org/a/warner_s_1 | arXiv.org | arXiv.org
Table 2: A sample of identifier schemes used for scholarly author identifiers
The three other examples in table 2 illustrate decreasing scopes. In the
Netherlands the Digital Author Id (DAI) “is a unique national number assigned
to every author who has been appointed to a position at a Dutch university or
research institute or has some other relevant connection with one of these
organizations” [7]. The DAI provides a join point for data in different
repositories and enables services based on this combined data (e.g. NARCIS).
The RePEc scheme identifies authors and is used to link their publications
together within the RePEc system serving the economic community. The
AuthorClaim project aims to extend the RePEc model, using the same software
infrastructure, to the entire academic domain. Finally, the arXiv author
identifier, described below, is local to a single repository.
The arguments above and the understanding that there are many different
interests in, and uses for, author/person identities, suggest that there will
be many different systems and multiple identities for each author. In the
scholarly communication domain there will be a patchwork of overlapping
publication networks. Unless one system grows to dominate, the different
patches of publication network will identify authors using different
identifiers. However, it will be vastly easier to match multiple identifiers
for each author than to disambiguate multiple authors with the same name. A
significant aid will be the addition of assertions that link identities in
different networks (e.g. Author A2 in Repository 1 is the same person as
Author A4 in Repository 2) as illustrated in figure 1. This linking
information might be expressed either via the Semantic Web or in repository
metadata. The ability to record foreign identifiers in the author record
within a repository, and the ability to match articles between repositories,
will allow the joining of data across repository boundaries.
Figure 1: If authors A2 and A3 in repository 1 are identified with authors A4
and A5 in repository 2 respectively, then the components of the publication
graph from the two repositories can be joined. The P# nodes indicate papers.
## 3 Author Identifiers at arXiv
There are a significant number of physicists for whom all articles, or at
least all recent articles, are available on arXiv. It is not uncommon to find
web homepages with a link to arXiv author search in place of a bibliography —
why maintain the information in a second place when arXiv will do it
automatically? Fielded author search has been used in this way for many years
and has exactly the same problems of author disambiguation as text-based
efforts to build the publication network.
With the introduction of user accounts, arXiv, like many other repositories,
started to collect data on which user made each submission and whether he or
she claimed to be an author. This start to building authority records was
augmented by attempts to retrospectively associate older papers with users
based on email address matching, and the introduction of facilities by which
users could “claim ownership” of existing submissions. Use of the claim
ownership facility was motivated through the introduction of an endorsement
system222http://arxiv.org/help/endorsement where users must be known as
authors of a certain number of papers in order to endorse new users. Various
heuristics are used to limit what papers can be claimed automatically and so
far these have proved adequate to avoid incorrect claims being automatically
accepted.
Demanding identification of all authors at submission time was considered
impractical. For articles with one or two authors identification would not be
too burdensome, but for papers with 10 or even 2500 authors333Articles from
high-energy physics collaborations often have many authors. See, for example,
the recent ATLAS collaboration paper http://arxiv.org/abs/0901.0512 with
$>2500$ authors it is clearly impractical. A solution that uses arXiv
administrator effort to deal with each article is also impractical because
just two administrators handle all user queries relating to arXiv’s 58,000
submissions/year — most submissions must be entirely automated. We thus
decided on an approach that will create useful services based on a public
author identifier which we internally link to our user records. We hope that
by providing useful services our users will be motivated to further improve
the authority records on which these services depend.
### 3.1 Author URI and Services
We have opted for a web-centric approach using Linked Data [2] style HTTP
access. Each arXiv author identifier is a unique URI (e.g.
http://arxiv.org/a/warner_s_1) which supports HTTP content-negotiation. These
URIs are designed to be human copyable and are based on an ASCII dumb-down of
the author name. By default, or if selected via content-negotiation headers,
the arXiv author URI redirects to an HTML page listing all arXiv publications
authored by the given individual based on our user records. An example HTML
page is shown in figure 2. In cases where the author-article associations are
complete this facility already solves the problem of name collision in arXiv
author search and so provides a more reliable link than our text-based author
search. To allow the data to be used by other applications or to allow display
or monitoring with a feed reader, the list of articles associated with an
author id is also available as an Atom feed. Figure 3 shows that same data as
figure 2 but rendered from the Atom feed. As of September 2009 we see about
300 different author id URIs being accessed per week to return HTML pages or
Atom feeds.
Figure 2: HTML screen returned when an arXiv author id is accessed and HTTP
content negotiation results in HTML, or HTML is explicitly requested by
appending .html to the author id (http://arxiv.org/a/kurtz_m_1.html) Figure 3:
Web browser (Firefox) rendering of the Atom feed returned when an arXiv author
id is accessed and HTTP content negotiation results in Atom, or Atom is
explicitly requested by appending .atom to the author id
(http://arxiv.org/a/kurtz_m_1.atom)
A list of articles on the arXiv site is still one click away from the user’s
homepage. We thus provide JavaScript code, which we call the myarticles
widget, that a user may include in their personal homepage to dynamically
include an up-to-date publication list from arXiv. Various formatting options
are provided and the content may be styled using CSS. Figure 4 shows two
screen shots from early adopters of the myarticles widget. This facility is
based upon a content-negotiated request for an Atom representation of the
arXiv author id resource which results in a machine readable Atom feed of
paper information (in the same format as the arXiv
API444http://arxiv.org/api).
Figure 4: Two early adopter examples of use of the myarticles widget. The
example on the left shows the “Google ads” formatting option used to produce a
compact display in the lower left corner of the browser window. The example on
the right shows the “arXiv list” formatting option which picks up local
stylesheet information. In both cases the data from arXiv is seamlessly
embedded in the user’s homepage.
arXiv’s second use of arXiv author ids is to leverage this automatically
generated and updated list of publications to lower the effort required to
integrate arXiv papers into social networking sites. Facebook was chosen as
the first site to work with but the OpenSocial API is also being investigated.
Once the arXiv Facebook application has been told the association between a
user’s Facebook account and their arXiv author identifier, a list of
publications is immediately available as either a panel or a tab on their
Facebook profile as shown in figure 5. All title, author list, abstract and
linking information is automatically imported from arXiv. New or old
publications may be reported in the user’s feed, with optional comments, and
thus show up in friends’ news feeds. This application was released in March
2009 and sees approximately 600 users per week as of September 2009. Use is
steady and increasing, but there is not rapid adoption. We continue to
experiment with new facilities and modes of interaction.
Figure 5: Example screenshots from the arXiv Facebook application. The window
on the right shows a list of article titles and authors that is automatically
generated because Joshua Erlich enabled the arXiv application from with his
Facebook account and told it the association with his arXiv author id. Any new
arXiv article owned by him will automatically be added to this list.
### 3.2 Helping to Build the Publication Network
arXiv is making the multiple-identifier problem one identifier worse by
creating arXiv specific identifiers. Deduping articles is a key problem in
bibliometrics, and we don’t want to create a similar deduping problem with
author identifiers. The OpenID scheme explicitly caters for multiple
identifiers for a single person, and even for multiple identifiers for each
persona a single person might use. Facilities to express and leverage multiple
identities a described in the Yadis/XRDS document [8].
At arXiv we can go some way to addressing this problem by augmenting the Atom
format machine readable authorship information arXiv exposes with
correspondences between author identifiers in different schemes. The issue
then is how to encourage authors to supply and update alternative identifiers
associated with their account. Again we believe that the solution will be to
build useful services, such a links to other systems, that depend upon this
data.
Another option to encourage use of arXiv data on the relationships between
authors, papers authored and identifiers in other schemes is to expose it in
RDF. This is a good application for OAI-ORE Resource Maps [4] and we intend to
provide OAI-ORE resource maps as another representation available from the
arXiv author id. An example showing alternate name, article information and
how identifiers in other schemes can be exposed is illustrated in figure 6.
Figure 6: Possible OAI-ORE resource map representing the aggregation of papers
by an author Ang Lee. The node ReM is the Resource Map which describes the
aggregation Agg that has the author id as its URI
(http://arxiv.org/a/lee_a_1). The aggregation includes the three papers (P1,
P2, P3) authored by this Ang Lee. Through the ore:similarTo relation we also
indicate two other related resources: the identities in other author id
schemes. The DAI is a URI and so can be related directly as Id2. The
ResearcherID is a string and must therefore be related via an additional node
Id1.
## 4 Conclusions
There is growing interest in accurate author identification based on explicit
author identifiers. The many different commercial and non-commercial parties
have varying motives and goals, and so are adopting different solutions. It
seems likely that there will continue to be many different systems and
multiple identities for each author. While not perfect, this situation will
greatly assist the assembly of publication network data linking authors and
articles which will facilitate bibliometric analyses and support new discovery
tools that span multiple repositories. The implementation of author
identifiers at arXiv, and of services to promote their use has been described
to illustrate one approach at the repository level. Early use is encouraging
but it remains to be seen how quickly the use of author identification is
adopted and accepted by the scholarly community.
## 5 Acknowledgements
I am pleased to acknowledge contributions from Nathan Woody (Facebook and
JavaScript interface for arXiv), Thorsten Schwander and Paul Ginsparg. This
work is supported by Microsoft through a Technical Computing Initiative (TCI)
Grant. This paper is based on a presentation given at Open Repositories 2009
on 18 May 2009 but with updated usage data through September 2009.
## 6 Note added in proof
There has been considerable recent activity around author identifiers, the
most notable effort being ORCID (Open Researcher Contributor Identification
Initiative, http://orcid.securesites.net/) which has significant commercial
and community participation. Both Elsevier and Thomson Reuters are
participants in ORCID and the resulting system may replace both Scopus Author
Id and ResearcherID with a more open and more broadly adopted scheme.
## References
* [1] Raf Aerts. Digital identifiers work for articles, so why not for authors? Nature, 453, 2008. http://dx.doi.org/10.1038/453979b.
* [2] Chris Bizer, Richard Cyganiak, and Tom Heath. How to Publish Linked Data on the Web, 2007. http://www4.wiwiss.fu-berlin.de/bizer/pub/LinkedDataTutorial/20070727/.
* [3] Johan Bollen, Herbert Van de Sompel, and Marko A. Rodriguez. Towards Usage-based Impact Metrics: First Results from the MESUR Project. Proceedings of the Joint Conference on Digital Libraries 2008 (JCDL08), June 16, 2008, Pittsburgh, Pennsylvania, USA, 2008. http://arxiv.org/abs/0804.3791.
* [4] Open Archives Initiative – Object Reuse and Exchange Specifications, 2007. http://www.openarchives.org/ore/toc.
* [5] repinf — Author identification wiki. http://repinf.pbworks.com/Author-identification. International Repositories Infrastructure wiki initially created to support the March 2009 International Repositories Infrastructure Workshop in Amsterdam, NL. Last accessed 2009-10-09.
* [6] Scopus Author Identifier. http://www.info.scopus.com/. The main Scopus service http://www.scopus.com/scopus/home.url and the details of author identifier are accessible from http://tinyurl.com/yta9m7.
* [7] SURFfoundation. Digital Author Identifier (DAI). http://www.surffoundation.nl/smartsite.dws?ch=eng&id=13480.
* [8] Yadis 1.0: The Identity and Accountability Foundation for Web 2.0, March 2006\. http://yadis.org/papers/yadis-v1.0.pdf. Page listing all versions is http://yadis.org/wiki/Yadis_Documents.
|
arxiv-papers
| 2010-03-06T01:37:01 |
2024-09-04T02:49:08.836865
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Simeon Warner (Cornell Information Science and Cornell University\n Library)",
"submitter": "Simeon Warner",
"url": "https://arxiv.org/abs/1003.1345"
}
|
1003.1352
|
# Non-universal tunneling resistance at the quantum critical point of
mesoscopic SQUIDs array
Sujit Sarkar PoornaPrajna Institute of Scientific Research, 4 Sadashivanagar,
Bangalore 5600 80, India.
###### Abstract
We calculate the tunnelling resistance at the quantum critical point of a
mesoscopic SQUIDs array in the presence of magnetic flux. We find the
analytical relation between the magnetic flux induced dissipation strength and
the Luttinger liquid parameter of the system. While the experimental finding
for the system is around $40-50$ mK, we find the behavior of the system even
at lower temperatures through the analysis of renormalization group. Apart
from the length scale dependent superconductor-insulator transition, we also
predict the evidence of length scale independent metallic state. This study
also emphasizes the importance of Co-tunelling effect.
Introduction: Josephson junction arrays have attracted considerable interest
in the recent years owing to their interesting physical properties. Currently
such arrays can be fabricated in restricted geometries both in one and two
dimensions sondhi ; suj1 ; suj2 ; granto1 ; geer ; zant ; kuo ; havi2 ; havi3
; lar . There are a few experimental findings in the field of mesoscopic
SQUIDs array. The authors of Ref. (havi3 ) and Ref. (kuo ) have fabricated
the arrays of SQUIDs junctions of different numbers in a single chip. They
have studied the current voltage characteristics of the mesoscopic SQUIDs
array in the presence of a magnetic field. They have used an external magnetic
field $(B)$ that tunes the effective Josephson coupling (${E_{J}}$) between
the nearest neighbor superconductors by the following relations
${E_{J}}={E_{J0}}|cos(\frac{\pi\Phi}{{\Phi}_{0}})|$, where $\Phi$ is the
external flux and ${\Phi}_{0}$ is the flux quantum. They have found magnetic
field induced superconductor-insulator (SI) quantum phase transition at around
$40-50$ mK. When the applied magnetic flux is less than the critical value the
system shows constant saturated resistance that may arise from the source of
quantum phase slip centers (QPS) suj2 . When the magnetic flux exceeds the
critical value the system turns into the insulating phase due to the flux
induced Coulomb blockade of Cooper pair tunneling. In this letter, we study
the behavior of the system at very low temperature where there is no
experimental findings, i.e., around few mili Kelvin havi3 ; kuo . The
experimental result shows length scale dependent superconductor-insulator (SI)
transition at the quantum resistance but this is not the whole picture, we
find the length scale independent metallic phase. We calculate the tunneling
resistance at the quantum critical point and also show explicitly the
importance of Co-tunneling effect to get the correct physical behavior of the
system.
Analytical relation between the flux induced dissipative strength and the
Luttinger liquid parameter: Here we derive analytical expression of magnetic
flux induced dissipative strength ($\alpha$) in terms of the interactions of
the system. At first we derive the dissipative action/partition function of a
quantum impurity system. We will see that the analytical structure of this
dissipative action is identical with the mesoscopic SQUIDs array.
Here we consider that the impurity is present at the origin where the fermions
scatter from the left to the right and vice versa. The Hamiltonian describing
this process is ${H_{1}}={V_{0}}({R^{\dagger}}(0){L}(0)+h.c)={V_{0}}\int
dx{\delta}(x)cos{\theta}(x).$ The total Hamiltonian of the system
$H={H_{0}}~{}+~{}{V_{0}}\int dx{\delta}(x)cos{\theta}(x,\tau)$
${H_{0}}=\frac{1}{2\pi}\int{uK{({{\partial}_{x}}{\theta}(x,\tau))}^{2}~{}+~{}\frac{u}{K}{({{\partial}_{x}}{\phi}(x,\tau))}^{2}},$
corresponding Lagrangian of the system is
$\displaystyle{L}$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi
K}\int{\frac{1}{u}{({{\partial}_{\tau}}{\phi}(x,\tau))}^{2}~{}+~{}{u}{({\partial}{\phi}(x,\tau))}^{2}}$
(1) $\displaystyle+{V_{0}}\int dx\delta(x)cos(\theta(x,\tau))={L_{0}}+{L_{1}}$
where ${L_{0}}$ and ${L_{1}}$ are the non-interacting and the interacting part
of the Lagrangian and $K$ is the Luttinger liquid parameter of the system. The
only non-linear term in this Lagrangian is expressed by the field
$\theta(x=0)$. We would like to express the action of the system as an
effective action by integrating the field $\theta(x\neq 0)$. Therefore one may
consider $\theta(x\neq 0)$ as a heat bath, which yields the source of
dissipation in the system. The constraint condition for the integration is
${\theta}(\tau)={\theta}(x=0,\tau)$. We can write the partition function.
$\displaystyle Z$ $\displaystyle=$ $\displaystyle\int
D\theta(x,\tau)e^{-\int_{0}^{\beta}Ld{\tau}}$ (2) $\displaystyle=\int
D\theta(x,\tau)D\theta(\tau)\delta(\theta(\tau)-{\theta}(0,\tau))e^{-\int_{0}^{\beta}Ld{\tau}}$
Here we use the standard trick of introducing the Lagrange multiplier with
auxiliary field $\lambda(\tau)$.
$Z=\int{D\theta(x,\tau)}\int{D\theta(\tau)}\int
D{\lambda(\tau)}e^{-\int_{0}^{\beta}({L_{0}+L_{1}})d{\tau}}$
$e^{i\int_{0}^{\beta}d{\tau}{\lambda(\tau)}~{}(\theta(0,\tau)-{\theta}(\tau))}$
$Z=\int{D\theta(\tau)}e^{-\int_{0}^{\beta}{L_{1}}d{\tau}}\int
D{\lambda(\tau)}e^{-i{\lambda(\tau)}{\theta}(\tau)}$
$\int{D\theta(x,\tau)}e^{\int_{0}^{\beta}(-{L_{0}}+i{\lambda}({\tau}){\theta}(0,\tau)){d\tau}}$
The Fourier transform of the first term of Eq.(2) is
${L_{0}}~{}=~{}\sum_{q}\sum_{i{\omega}_{n}}\frac{{{\omega}_{n}}^{2}~{}+~{}{v}^{2}{q}^{2}}{2\pi
Kv}{\theta}(q,i{{\omega}_{n}}){\theta}(-q,-i{{\omega}_{n}})$
At first we would like to calculate the integral:
$\int_{0}^{\beta}d{\tau}[{L_{0}}-i{\lambda(\tau)}{\theta(0,\tau)}]$, we can
write this term as
$\sum_{q}\sum_{i{\omega}_{n}}\frac{{{\omega}_{n}}^{2}~{}+~{}{v}^{2}{q}^{2}}{2\pi
Kv}{\theta}(q,i{{\omega}_{n}}){\theta}(-q,-i{{\omega}_{n}})$
$-\frac{1}{2\sqrt{L}}({\lambda}(i{\omega}_{n}){\theta}(-q,-i{{\omega}_{n}})+{\lambda}(-i{\omega}_{n}){\theta}(q,i{{\omega}_{n}})$.
This integral appears in the integral $\theta(x,\tau)$. This integral is
quadratic in $\theta$. Now we would like to perform the Gaussian integration
by completing the square. We can write the result as
$\frac{-1}{2L}\sum_{i{{\omega}_{n}},q}\frac{\pi
Kv}{{{\omega}_{n}}^{2}~{}+~{}{v}^{2}{q}^{2}}$.
In the infinite length limit one can write, $\frac{1}{2L}\sum_{q}\frac{\pi
Kv}{{{\omega}_{n}}^{2}~{}+~{}{v}^{2}{q}^{2}}~{}=~{}\int\frac{dq}{2\pi}\frac{\pi
Kv}{{{\omega}_{n}}^{2}~{}+~{}{v}^{2}{q}^{2}}=\frac{\pi K}{4{{\omega}_{n}}}$.
Now we would like to append this result of integration in the second integral
of $Z$, i.e., the integral over ${\lambda}$. One can write the integrand as
$\sum_{i{{\omega}_{n}}}(-\frac{\pi
K}{4{{\omega}_{n}}}{\lambda}(i{\omega}_{n}){\lambda}(-i{\omega}_{n})$
$+\frac{i}{2}({\lambda}(i{\omega}_{n}){\theta}(-q,-i{{\omega}_{n}})+{\lambda}(-i{\omega}_{n}){\theta}(q,i{{\omega}_{n}}).$
This integral is again the quadratic integral of $\lambda$, therefore the
Gaussian integral can be performed by completing the square. We get after the
integration $\sum_{i{{\omega}_{n}}}\frac{{\omega}_{n}}{\pi
K}\theta(i{\omega}_{n})\theta(-i{\omega}_{n})$. From these analytical
expression, we obtain the effect of bath on ${\theta}(\tau)$. The appearance
of the factor ${{\omega}_{n}}$ signifies the dissipation. Therefore the
effective action reduces to
$S~{}=~{}\sum_{i{{\omega}_{n}}}\frac{{\omega}_{n}}{\pi
K}\theta(i{\omega}_{n})\theta(-i{\omega}_{n})+\int dx{V_{0}}cos{\theta}(\tau)$
(3)
The above action implies that a single particle moving in the potential
${V_{0}}cos{\theta}(\tau)$ subject to dissipation with friction constant ,
$\frac{1}{\pi K}$.
Now we calculate the dissipative action of mesoscopic SQUIDs array. We have
already proved in Ref. suj2 that the strong coupling phase of the system is
consistent with the experimental findings. Here we calculate the effective
partition function of our system in the strong coupling phase. Our starting
point is the Calderia-Legget cal formalism. Following reference we write the
action as
$S_{1}~{}=~{}S_{0}~{}+~{}\frac{{\alpha}^{\prime}}{4\pi
T}~{}\sum_{m}{{\omega}_{m}}{|{\theta}_{m}|}^{2}.$ (4)
Here, $S_{1}$ is the standard action for the system with tiled wash-board
potential sch1 ; su1 ; schon to describe the dissipative physics for low
dimensional superconducting tunnel junctions, $S_{0}$ is the action for non-
dissipative part,
${\alpha}^{\prime}~{}=~{}\frac{R_{Q}}{R_{s}}cos|\frac{\pi\phi}{{\phi}_{0}}|$
(the extra cosine factor which we consider in ${\alpha}^{\prime}$ is entirely
new in the literature to probe the effect of an external magnetic flux and is
also consistent physically), the Matsubara frequency
${\omega}_{m}~{}=~{}\frac{2\pi}{\beta}m$ and $R_{Q}$ ($=6.45k\Omega$) is the
quantum resistance and $R_{s}$ is the tunnel junction resistance, $\beta$ is
the inverse temperature. In the strong potential, tunneling between the minima
of the potential is very small. In the imaginary time path integral formalism,
tunneling effect in the strong coupling limit can be described in terms of
instanton physics. In this formalism, it is convenient to characterize the
profile of $\theta$ in terms of its time derivative,
$\frac{d\theta{(\tau)}}{d{\tau}}~{}=~{}\sum_{i}e_{i}h(\tau-{\tau}_{i}),$ (5)
where $h(\tau-{\tau}_{i})$ is the time derivative at time $\tau$ of one
instanton configuration. ${\tau}_{i}$ is the location of the i-th instanton,
$e_{i}=1$ and $-1$ is the topological charge of instanton and anti-instanton
respectively. Integrating the function over $h$ from $-\infty$ to $\infty$,
$\int_{-\infty}^{\infty}d\tau
h(\tau)={\theta}(\infty)-{\theta}({-\infty})=2\pi.$ It is well known that the
instanton (anti-instanton) is almost universally constant except for a very
small region of time variation. In the QPS process the amplitude of the
superconducting order parameter is zero only in a very small region of space
as a function of time and the phase changes by $\pm 2\pi$. So our system
reduce to a neutral system consisting of equal number of instanton and anti-
instanton. One can find the expression for ${\theta}(\omega)$, after the
Fourier transform to the both sides of Eq. 5 which yields
${\theta}(\omega)=\frac{i}{\omega}\sum_{i}e_{i}h(i\omega)e^{i{\omega}{\tau}_{1}}$
. Now we substitute this expression for ${\theta}(\omega)$ in the second term
of Eq.4 and finally we get this term as
$\sum_{ij}~{}F({\tau}_{i}-{\tau}_{j}){e_{i}}{e_{j}}$, where
$F({\tau}_{i}-{\tau}_{j})=\frac{\pi\alpha}{\beta}\sum_{m}~{}\frac{1}{{|{\omega}_{m}}|}e^{i{\omega}({\tau}_{i}-{\tau}_{j})}$
$\simeq ln({\tau}_{i}-{\tau}_{j})$. We obtain this expression for very small
values of ${\omega}$ ( $\rightarrow 0$). So $F({\tau}_{i}-{\tau}_{j})$
effectively represents the Coulomb interaction between the instanton and anti-
instanton. This term is the main source of dissipation physics of the system.
Following the standard prescription of imaginary time path integral formalism,
we can write the partition function of the system as suj2 ; lar ; gia1 ; kane
; zai1 ; furu .
$\displaystyle Z$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\frac{1}{n!}{z}^{n}\sum_{e_{i}}\int_{0}^{\beta}d{{\tau}_{n}}\int_{0}^{{\tau}_{n-1}}d{\tau}_{n-1}...$
(6)
$\displaystyle\int_{0}^{{\tau}_{2}}d{{\tau}_{1}}e^{-F({\tau}_{i}-{\tau}_{j}){e_{i}}{e_{j}}}.$
We would like to express the partion function in terms of integration over
auxiliary field, $q{(\tau)}$. After some extensive analytical calculations, we
get
$Z=\int
Dq(\tau)e^{(-\sum_{i{\omega}_{n}}\frac{|{{\omega}_{n}}|}{4\pi\alpha}q(i{\omega}_{n})q(-i{\omega}_{n})){+(2z\int_{0}^{\beta}d{\tau}cosq(\tau))}}.$
(7)
Thus by comparing the first term of the action of Eq. (3) and the first term
of exponential of Eq. (17), we conclude that the dissipative strength $\alpha$
and the Luttinger liquid parameter of the system are related by the relation,
$K=4\alpha$.
Quantum field theoretical study of model Hamiltonian of the system and
explicit derivation of dissipative strength:
In our previous study, we have shown explicitly that the mesoscopic SQUIDs
array is equivalent to the array of superconducting quantum dots (SQD) with
modulated Josephson coupling. We first write the model Hamiltonian of SQD with
nearest neighbor (NN) Josephson coupling and also with the presence of the on-
site and NN charging energy between SQD,
$H~{}=~{}H_{J1}~{}+~{}H_{EC0}~{}+~{}H_{EC1}.$ (8)
Now we would like to recast our basic Hamiltonians in the spin language. This
is valid when ${E_{C0}}>>{E_{J1}}$. It is also observed from the experiments
that the quantum critical point exists for larger values of the magnetic
field, when the magnetic field induced Coulomb blockade phase is more
prominent than the $E_{J}$ induced SC phase. Thus our theoretical model is
consistent with the experimental findings. During this mapping process we
follow Ref. (lar ; suj1 ; suj2 ).
$H_{J1}~{}=~{}-2~{}E_{J1}\sum_{i}({S_{i}}^{\dagger}{S_{i+1}}^{-}+h.c)$,
$H_{EC0}~{}=~{}{E_{C0}}\sum_{i}{S_{i}}^{Z}.$
$H_{EC1}~{}=~{}4E_{Z1}\sum_{i}{S_{i}}^{Z}~{}{S_{i+1}}^{Z},$
${E_{J1}}={E_{J10}}|cos(\frac{\pi\Phi}{{\Phi}_{0}})|$ . At the Coulomb blocked
regime, the higher order expansion leads to the virtual state with energies
exceeding $E_{C0}$. In this second order process, the effective Hamiltonian
reduces to the subspace of charges $0$ and $2$, and takes the form lar ; suj1
; suj2 ,
$H_{C}~{}=~{}-\frac{3{E_{J1}}^{2}}{4E_{C0}}\sum_{i}{{S_{i}}^{Z}}{{S_{i+1}}^{Z}}~{}-~{}\frac{{E_{J1}}^{2}}{E_{C0}}\sum_{i}({S_{i+2}}^{\dagger}{S_{i}}^{-}+h.c).$
(9)
With this corrections $H_{C1}$ become
$H_{EC1}~{}\simeq~{}(4E_{Z1}~{}-~{}\frac{3{E_{J1}}^{2}}{4E_{C0}})~{}\sum_{i}{{S_{i}}^{Z}}{{S_{i+1}}^{Z}}$
One can express spin chain systems to as spinless fermions systems through the
application of Jordan-Wigner transformation. We have transformed all
Hamiltonians in spinless fermions which we have not present in this letter. In
order to study the continuum field theory of these Hamiltonians, we recast the
spinless fermions operators in terms of field operators by a relation gia1 .
${\psi}(x)~{}=~{}~{}[e^{ik_{F}x}~{}{\psi}_{R}(x)~{}+~{}e^{-ik_{F}x}~{}{\psi}_{L}(x)]$
where ${\psi}_{R}(x)$ and ${\psi}_{L}(x)$ describe the second-quantized fields
of right- and the left-moving fermions respectively. We would like to express
the fermionic fields in terms of bosonic field by the relation
${{\psi}_{r}}(x)~{}=~{}~{}\frac{U_{r}}{\sqrt{2\pi\alpha}}~{}~{}e^{-i~{}(r\phi(x)~{}-~{}\theta(x))}$,
$r$ is denoting the chirality of the fermionic fields, right (1) or left
movers (-1). The operators $U_{r}$ preserve the anti-commutivity of fermionic
fields. $\phi$ field corresponds to the quantum fluctuations (bosonic) of spin
and $\theta$ is the dual field of $\phi$. They are related by the relations
${\phi}_{R}~{}=~{}~{}\theta~{}-~{}\phi$ and
${\phi}_{L}~{}=~{}~{}\theta~{}+~{}\phi$. The total Hamiltonian is
$\displaystyle H$ $\displaystyle=$
$\displaystyle{H_{0}}+\frac{4E_{Z12}}{{(2\pi\alpha)}^{2}}\int~{}dx:cos(4\sqrt{K}\phi(x)):$
(10)
$\displaystyle+\frac{E_{C0}}{\pi\alpha}\int({{\partial}_{x}}\phi(x))~{}dx$
$H_{0}$ is the non-interacting part of the Hamiltonian, The dissipative
strength (${\alpha}_{1}$) of this system in the absence of Co-tunneling effect
is
${{\alpha}_{1}}~{}=~{}\frac{1}{4}\sqrt{\frac{2E_{J1}}{2{E_{J1}}~{}+~{}\frac{16E_{Z1}}{\pi}}}$
(11)
If we consider the total effect of Co-tunneling process, the system reduces to
Heisenberg spin chain with NN and NNN interactions. In this limit the
dissipative strength (${{\alpha}_{2}}$) of the system is
${{\alpha}_{2}}~{}=~{}\frac{1}{4}\sqrt{\frac{2E_{J1}}{E_{J1}~{}+~{}\frac{4}{\pi}(4E_{Z1}~{}-~{}\frac{3{E_{J1}}^{2}}{4E_{C0}})}}$
(12)
We calculate the dissipation strength by calculating K for both cases and then
we use the relation $K=4{\alpha}$. This is the first analytical derivation of
flux induced dissipation strength in terms of the interactions of the system.
We consider these two processes to emphasis the importance of Co-tunneling
effect for this system.
Physical Analysis of Renormalization Group Equation and Calculation of
Tunneling Resistance at The Quantum Critical Point:
The RG equation based on the Eq. (7) is,
$\frac{dz}{dlnb}~{}=~{}(1-{\alpha}^{\prime})z$ (13)
Following Ref.suj2 , we can write fugacity depends on length scale and
temperature as, $z(L)~{}\propto L^{1-{\alpha}^{\prime}}$,
${z}(T)~{}\propto{T}^{{{\alpha}^{\prime}}~{}-~{}1}$. The physical explanations
based on this RG equation are in order.
In our study, the resistance is evolving due to dissipation effect at very low
temperature (few mili Kelvin, less than the superconducting Coulomb blocked
temperature). According to our calculations, for large dissipation
(${\alpha}^{\prime}>1$), $R(T)\propto R_{Q}{T^{{\beta}_{1}}}$ and
${{\beta}_{1}}>0$. Therefore at very low temperature, the system shows stable
SC behaviour (region B of Fig. 1). and the system shows no more saturated
resistance behaviour (region D of Fig. 1). When ${\alpha}^{\prime}<1$, the
resistance of system $R(T)\propto R_{Q}{T^{{-\beta}_{2}}}$ and
${{\beta}_{2}}>0$. So at very low temperature, the resistance of the system
shows Kondo-like divergence behavior (region A of Fig. 1). These stable SC and
Kondo behaviour are unseen in the experimental findings because they have
measured the zero bias resistance at around $40-50$ mK. According to our
calculations, for large dissipation (${\alpha}^{\prime}>1$), $R(T)\propto
R_{Q}{L^{-{\gamma}_{1}}}$ and ${{\gamma}_{1}}>0$. Therefore the longer array
system shows the less resistive state than shorter array in the
superconducting phase. When ${\alpha}^{\prime}<1$, the resistance of system
$R(T)\propto R_{Q}{L^{{\gamma}_{2}}}$, where ${{\gamma}_{2}}>0$
(${{\beta}_{1}},{{\beta}_{2}},{{\gamma}_{1}}$ and ${{\gamma}_{2}}$ are
independent numbers). So the resistance in the insulating state is larger for
longer array system than shorter one. We find the dual behavior of the
resistance for lower and higher values of magnetic field.
From the knowledge of LL physics, we know that for a particular range of $K$,
there is a metallic state in between the insulating phase and the
superconducting phase of the system. The range of $K$ depends on the nature of
the system. Therefore we conclude that there is also a metallic state for a
particular range of magnetic flux induced dissipation strength. This
prediction is unnoticed in the experimental findings. At very low temperature
around $\alpha\sim 1$ (region E of Fig. 1), the quantum phase slip centres
poliferate and as a result a screening appears in the system. A detailed
explanation of QPS for this type of system is discussed in Ref. suj2 .
Physical explanation is as follows.
Here we consider two QPS with co-ordinates (${x_{1}},{\tau}_{1}$) and
(${x_{2}},{\tau}_{2}$), we assume that the cores of the QPS centres donot
overlap, i.e., $|{x_{2}}-{x_{1}}|>{x_{0}}(=\xi)$ and
$|{\tau}_{2}-{\tau}_{1}|>{{\tau}_{0}}(=\frac{1}{\Delta})$. Where $\xi$ and
$\Delta$ are the coherence length and SC orderparameter of the system. We have
already proven that the topological charges interact with each other
logarithmically. Therefore we can write the action of the system as
$S_{QPS}=2S_{core}-{\mu}{{e}_{1}}{{e}_{2}}ln|{x_{1}}-{x_{2}}|$ . QPS with
opposite topological charge attract each other and same charge repel with each
other. Suppose we consider a gas of $n$ QPS and assume that QPS cores do not
overlap. When a current is passing through the system, we can write the total
action of the system as ${S_{QPS}^{n}}=nS_{core}+{S_{int}}$ zai1 . Where,
${S_{int}}=-{\mu}\sum_{i\neq
j}{e_{i}}{e_{j}}ln(\frac{{\rho}_{ij}}{x_{0}})+\sum_{i}\frac{{\Phi}_{0}I}{c}{e_{i}}{{\tau}_{i}}$
(14)
Where ${\rho}_{ij}$ is the distance between the two QPS at the site $i$ and
$j$ and $I$ is the current passing through the system. The grand partition
function of the system is represented as a sum of all topological charges and
the analytical form is equivalent to Eq.6, $z\sim e^{-S_{core}}$. In the
absence of current, Eq. (14) define the model for a 2-dimensional Coulomb gas
interacting logarithmically. We consider the very low temperature limit
($T\rightarrow 0$). The RG equations of this 2-dimensional Coulomb gas are the
following $\frac{\partial\mu}{\partial l}=-4{{\pi}^{2}}{{\mu}^{2}}z^{2}$,
$\frac{\partial z}{\partial l}=(2-{\mu})z$. In our case we can write the
second equation as $\frac{\partial z}{\partial
log(\frac{\Delta}{T})}=(2-{\mu})z$. Finally it yields
$z(T)=z(\Delta){(\frac{\Delta}{T})}^{2-\mu}$. When the interaction coefficient
is less than 2, then the RG equation diverges at the scale ${T^{*}}$ (=
${z}^{\frac{1}{2-\mu}}{\Delta}$). At the scale larger than $\xi$ the
interaction between the QPS is screened and cease to be logarithmic, it
becomes an exponentially decaying function ($\sim
e^{-2{\mu}{K_{0}}(\sqrt{x^{2}+{\tau}^{2}})}$), ${K_{0}}$ is the modified
Bessel function.
Figure 1: Schematic phase diagram showing the variation of resistance with
temperature based on the renormalization group study. Region A is the Kondo
like behaviour of the system, region B is superconducting phase, region D is
the saturated finite resistance and region E is the metallic phase.
${L_{2}}>{L_{1}}$
When ${\alpha}^{\prime}=1$, i.e.,
${\Phi}~{}=({{\Phi}_{0}}/{\pi})cos^{-1}(\frac{R_{s}}{R_{Q}})$, the system has
no length scale dependence SI transition at very low temperature. This is the
critical behavior of system for a specific value of magnetic field. The
analytical expression for tunneling resistance at the quantum critical point
can be calculated by comparing the expression of ${\alpha}_{1}$ and
${\alpha}_{2}$ with the expression of ${\alpha}^{{}^{\prime}}$.
${R_{S}^{(1)}}=R_{Q}(\frac{16E_{Z1}}{30\pi E_{J1}})$ and
${R_{S}^{(2)}}=R_{Q}(\frac{5\pi
E_{C0}}{4E_{J1}})\sqrt{1+\frac{2.77E_{Z1}}{E_{C0}}}$ are the tunneling
resistance at the quantum critical point in the absence and presence of Co-
tunneling effect respectively. It is clear from our analytical derivation that
the tunneling resistance at quantum critical point is not $R_{Q}$ but it is
rather non-universal. $R_{S}^{(2)}$ is proportional to the $E_{C0}$ as one
would expect from the physical criteria of the system.
Here we discuss the importance of Co-tunelling effect explicitly. In the SC
phase when ${\alpha}>1$, from the analysis of ${\alpha}_{1}$, we get the
condition
$\frac{-16E_{Z1}}{\pi}>30{E_{J10}}|cos(\frac{\pi\Phi}{{\Phi}_{0}})|$. This
condition is unphysical, because the all coupling constants are repulsive. In
presence of Co-tunneling process, from the analysis ${\alpha}_{2}$, we achieve
the condition of SC
${E_{J10}}|cos(\frac{\pi\Phi}{{\Phi}_{0}})|(\frac{192}{4\pi{E_{C0}}}{E_{J10}}|cos(\frac{\pi\Phi}{{\Phi}_{0}})|-30)>\frac{256}{\pi}$.
This condition is physically reliable . Similarly one can do the analysis for
the insulating phase for both absence and presence of co-tunneling effect.
Conclusions: we have found the non-universal tunneling resistance at the
quantum critical point of mesoscopic SQUIDs array. We have found stable SC and
Kondo phase at very low temperature, which is still unseen experimentally. We
have found the length scale independent metallic phase in the system. The
importance of Co-tunneling effect has been studied explicitly.
Acknowledgement: The author would like to acknowledge CCMT of the Physics
Department of IISC for extended facility and Dr. S. Vidyadhiraja for several
important comments on this work and finally Dr. R. Srikanth and Dr. B. Murthy
for reading the manuscript very critically.
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* (12) A. Schmid, Phys. Rev. Lett. 51, 1506 (1983).
* (13) S. Chakravarty $et~{}al.$, Phys. Rev. B 37, 3238 (1988).
* (14) S. Schon and A. D. Zaikin, Phys. Reports 198, 237 (1990).
* (15) C. Kane and M. P. A Fisher, Phys. Rev. B 46, 15233 (1992).
* (16) A. Furusaki and N. Nagaosa, Phys. Rev. B 47, 4631 (1993) ; ibid Phys. Rev. B 47, 3827 (1993).
* (17) D. S. Golubev and A. D. Zaikin, Phys. Rev. B, 64 014504 (2001); A. D. Zaikin $et~{}al.$, Phys. Rev. Lett., 78 1552 (1997).
* (18) T. Giamarchi, in Quantum Physics in One Dimension (Claredon Press, Oxford 2004).
|
arxiv-papers
| 2010-03-06T05:04:27 |
2024-09-04T02:49:08.842696
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sujit Sarkar",
"submitter": "Sujit Sarkar",
"url": "https://arxiv.org/abs/1003.1352"
}
|
1003.1377
|
###### Abstract
The focus of this article is on entropy and Markov processes. We study the
properties of functionals which are invariant with respect to monotonic
transformations and analyze two invariant “additivity” properties: (i)
existence of a monotonic transformation which makes the functional additive
with respect to the joining of independent systems and (ii) existence of a
monotonic transformation which makes the functional additive with respect to
the partitioning of the space of states. All Lyapunov functionals for Markov
chains which have properties (i) and (ii) are derived. We describe the most
general ordering of the distribution space, with respect to which all
continuous-time Markov processes are monotonic (the Markov order). The
solution differs significantly from the ordering given by the inequality of
entropy growth. For inference, this approach results in a convex compact set
of conditionally “most random” distributions.
###### keywords:
Markov process; Lyapunov function; entropy functionals; attainable region;
MaxEnt; inference
10.3390/e12051145 12 Received: 1 March 2010; in revised form: 30 April 2010 /
Accepted: 4 May 2010 / Published: 7 May 2010 Entropy: The Markov Ordering
Approach Alexander N. Gorban 1,⋆, Pavel A. Gorban 2 and George Judge 3 E-mail:
ag153@le.ac.uk.
## 1 Introduction
### 1.1 A Bit of History: Classical Entropy
Two functions, energy and entropy, rule the Universe.
In 1865 R. Clausius formulated two main laws Clausius1865 :
1. 1.
The energy of the Universe is constant.
2. 2.
The entropy of the Universe tends to a maximum.
The universe is isolated. For non-isolated systems energy and entropy can
enter and leave, the change in energy is equal to its income minus its
outcome, and the change in entropy is equal to entropy production inside the
system plus its income minus outcome. The entropy production is always
positive.
Entropy was born as a daughter of energy. If a body gets heat $\Delta Q$ at
the temperature $T$ then for this body ${\mathrm{d}}S=\Delta Q/T$. The total
entropy is the sum of entropies of all bodies. Heat goes from hot to cold
bodies, and the total change of entropy is always positive.
Ten years later J.W. Gibbs Gibbs1875 developed a general theory of
equilibrium of complex media based on the entropy maximum: the equilibrium is
the point of the conditional entropy maximum under given values of conserved
quantities. The entropy maximum principle was applied to many physical and
chemical problems. At the same time J.W. Gibbs mentioned that entropy
maximizers under a given energy are energy minimizers under a given entropy.
The classical expression $\int p\ln p$ became famous in 1872 when L. Boltzmann
proved his $H$-theorem Boltzmann1872 : the function
$H=\int f(x,v)\ln f(x,v){\mathrm{d}}x{\mathrm{d}}v$
decreases in time for isolated gas which satisfies the Boltzmann equation
(here $f(x,v)$ is the distribution density of particles in phase space, $x$ is
the position of a particle, $v$ is velocity). The statistical entropy was
born: $S=-kH$. This was the one-particle entropy of a many-particle system
(gas).
In 1902, J.W. Gibbs published a book “Elementary principles in statistical
dynamics” Gibbs1902 . He considered ensembles in the many-particle phase space
with probability density $\rho(p_{1},q_{1},\ldots p_{n},q_{n})$, where
$p_{i},q_{i}$ are the momentum and coordinate of the $i$th particle. For this
distribution,
$S=-k\int\rho(p_{1},q_{1},\ldots p_{n},q_{n})\ln(\rho(p_{1},q_{1},\ldots
p_{n},q_{n})){\mathrm{d}}q_{1}\ldots{\mathrm{d}}q_{n}{\mathrm{d}}p_{1}\ldots{\mathrm{d}}p_{n}$
(1)
Gibbs introduced the canonical distribution that provides the entropy maximum
for a given expectation of energy and gave rise to the entropy maximum
principle (MaxEnt).
The Boltzmann period of history was carefully studied Villani . The difference
between the Boltzmann entropy which is defined for coarse-grained distribution
and increases in time due to gas dynamics, and the Gibbs entropy, which is
constant due to dynamics, was analyzed by many authors Jaynes1965 ;
GoldsteinLebov2004 . Recently, the idea of two functions, energy and entropy
which rule the Universe was implemented as a basis of two-generator formalism
of nonequilibrium thermodynamics Grmela1997 ; Ottinger2005 .
In information theory, R.V.L. Hartley (1928) Hartley1928 introduced a
logarithmic measure of information in electronic communication in order “to
eliminate the psychological factors involved and to establish a measure of
information in terms of purely physical quantities”. He defined information in
a text of length $n$ in alphabet of s symbols as $H=n\log s$.
In 1948, C.E. Shannon Shannon1948 generalized the Hartley approach and
developed “a mathematical theory of communication”, that is information
theory. He measured information, choice and uncertainty by the entropy:
$S=-\sum_{i=1}^{n}p_{i}\log p_{i}$ (2)
Here, $p_{i}$ are the probabilities of a full set of $n$ events
($\sum_{i=1}^{n}p_{i}=1$). The quantity $S$ is used to measure of how much
“choice” is involved in the selection of the event or of how uncertain we are
of the outcome. Shannon mentioned that this quantity form will be recognized
as that of entropy, as defined in certain formulations of statistical
mechanics. The classical entropy (1), (2) was called the
Boltzmann–Gibbs–Shannon entropy (BGS entropy). (In 1948, Shannon used the
concave function (2), but under the same notation $H$ as for the Boltzmann
convex function. Here we use $H$ for the convex $H$-function, and $S$ for the
concave entropy.)
In 1951, S. Kullback and R.A. Leibler KullLei1951 supplemented the BGS
entropy by the relative BGS entropy, or the Kullback–Leibler divergence
between the current distribution $P$ and some “base” (or “reference”)
distribution $Q$:
$D_{\mathrm{KL}}(P\|Q)=\sum_{i}p_{i}\log\frac{p_{i}}{q_{i}}$ (3)
The Kullback–Leibler divergence is always non-negative
$D_{\mathrm{KL}}(P\|Q)\geq 0$ (the Gibbs inequality). It is not widely known
that this “distance” has a very clear physical interpretation. This function
has been well known in physical thermodynamics since 19th century under
different name. If $Q$ is an equilibrium distribution at the same temperature
as $P$ has, then
$D_{\mathrm{KL}}(P\|Q)=\frac{F(P)-F(Q)}{kT}$ (4)
where $F$ is free energy and $T$ is thermodynamic temperature. In physics,
$F=U-TS$, where physical entropy $S$ includes an additional multiplier $k$,
the Boltzmann constant. The thermodynamic potential $-F/T$ has its own name,
Massieu function. Let us demonstrate this interpretation of the
Kullback–Leibler divergence. The equilibrium distribution $Q$ provides the
conditional entropy (2) maximum under a given expectation of energy
$\sum_{i}u_{i}q_{i}=U$ and the normalization condition $\sum_{i}q_{i}=1$. With
the Lagrange multipliers $\mu_{U}$ and $\mu_{0}$ we get the equilibrium
Boltzmann distribution:
$q_{i}=\exp(-\mu_{0}-\mu_{U}u_{i})=\frac{\exp(-\mu_{U}u_{i})}{\sum_{i}\exp(-\mu_{U}u_{i})}$
(5)
The Lagrange multiplier $\mu_{U}$ is in physics (by definition) $1/kT$, so
$S(Q)=\mu_{0}+\frac{U}{kT}$, hence, $\mu_{0}=-\frac{F(Q)}{kT}$. For the
Kullback–Leibler divergence this formula gives (4).
After the classical work of Zeldovich (1938, reprinted in 1996 Zeld ), the
expression for free energy in the “Kullback–Leibler form”
$F=kT\sum_{i}c_{i}\left(\ln\left(\frac{c_{i}}{c_{i}^{*}(T)}\right)-1\right)$
where $c_{i}$ is concentration and $c^{*}_{i}(T)$ is the equilibrium
concentration of the $i$th component, is recognized as a useful instrument for
the analysis of kinetic equations (especially in chemical kinetics YBGE ;
Hangos2009 ).
Each given positive distribution $Q$ could be represented as an equilibrium
Boltzmann distribution for given $T>0$ if we take $u_{i}=-kT\log q_{i}+u_{0}$
for an arbitrary constant level $u_{0}$. If we change the order of arguments
in the Kullback–Leibler divergence then we get the relative Burg entropy
Burg1967 ; Burg1972 . It has a much more exotic physical interpretation: for a
current distribution $P$ we can define the “auxiliary energy” functional
$U_{P}$ for which $P$ is the equilibrium distribution under a given
temperature $T$. We can calculate the auxiliary free energy of any
distribution $Q$ and this auxiliary energy functional: $F_{P}(Q)$. (Up to an
additive constant, for $P=P^{*}$ this $F_{P}(Q)$ turns into the classical free
energy, $F_{P}^{*}(Q)=F(Q)$.) In particular, we can calculate the auxiliary
free energy of the physical equilibrium, $F_{P}(P^{*})$. The relative Burg
entropy is
$D_{\mathrm{KL}}(P^{*}\|P)=\frac{F_{P}(P^{*})-F_{P}(P)}{kT}$
This functional should also decrease in any Markov process with given
equilibrium $P^{*}$.
Information theory developed by Shannon and his successors focused on entropy
as a measure of uncertainty of subjective choice. This understanding of
entropy was returned from information theory to statistical mechanics by E.T.
Jaynes as a basis of “subjective” statistical mechanics Jaynes1957a ;
Jaynes1957b . He followed Wigner’s idea “entropy is an antropocentric
concept”. The entropy maximum approach was declared as a minimization of the
subjective uncertainty. This approach gave rise to a MaxEnt “anarchism”. It is
based on a methodological hypothesis that everything unknown could be
estimated by the principle of the entropy maximum under the condition of fixed
known quantities. At this point the classicism in entropy development changed
to a sort of scientific modernism. The art of model fitting based on entropy
maximization was developed Harre2001 . The principle of the entropy maximum
was applied to plenty of problems: from many physical problems Beck2009 ,
chemical kinetics and process engineering Hangos2009 to econometrics
MittJudge2000 ; Judge2002 and psychology Myers1992 . Many new entropies were
invented and now one has rich choice of entropies for fitting needs EstMor1995
. The most celebrated of them are the Rényi entropy Renyi1961 , the Burg
entropy Burg1967 ; Burg1972 , the Tsallis entropy Tsa1988 ; Abe and the
Cressie–Read family CR1984 ; ReadCreass1988 . The nonlinear generalized
averaging operations and generalized entropy maximization procedures were
suggested Bagci2009 .
Following this impressive stream of works we understand the MaxEnt approach as
conditional maximization of entropy for the evaluation of the probability
distribution when our information is partial and incomplete. The entropy
function may be the classical BGS entropy or any function from the rich family
of non-classical entropies. This rich choice causes a new problem: which
entropy is better for a given class of applications?
The MaxEnt “anarchism” was criticized many times as a “senseless fitting”.
Arguments pro and contra the MaxEnt approach with non-classical entropies
(mostly the Tsallis entropy Tsa1988 ) were collected by Cho Cho2002 . This
sometimes “messy and confusing situation regarding entropy-related studies has
provided opportunities for us: clearly there are still many very interesting
studies to pursue” Lin1999 .
### 1.2 Key Points
In this paper we do not pretend to invent new entropies. (There appear new
functions as limiting cases of the known entropy families, but this is not our
main goal). Entropy is understood in this paper as a measure of uncertainty
which increases in Markov processes. In our paper we consider a Markov process
as a semigroup on the space of positive probability distributions. The state
space is finite. Generalizations to compact state spaces are simple. We
analyze existent relative entropies (divergences) using several simple ideas:
1. 1.
In Markov processes probability distributions $P(t)$ monotonically approach
equilibrium $P^{*}$: divergence $D(P(t)\|P^{*})$ monotonically decrease in
time.
2. 2.
In most applications, conditional minimizers and maximizers of entropies and
divergences are used, but the values are not. This means that the system of
level sets is more important than the functions’ values. Hence, most of the
important properties are invariant with respect to monotonic transformations
of entropy scale.
3. 3.
The system of level sets should be the same as for additive functions: after
some rescaling the divergences of interest should be additive with respect to
the joining of statistically independent systems.
4. 4.
The system of level sets should after some rescaling the divergences of
interest should have the form of a sum (or integral) over states
$\sum_{i}f(p_{i},p_{i}^{*})$, where the function $f$ is the same for all
states. In information theory, divergences of such form are called separable,
in physics the term trace–form functions is used
The first requirement means that if a distribution becomes more random then it
becomes closer to equilibrium (Markov process decreases the information excess
over equilibrium). For example, classical entropy increases in all Markov
processes with uniform equilibrium distributions. This is why we may say that
the distribution with higher entropy is more random, and why we use entropy
conditional maximization for the evaluation of the probability distribution
when our information is partial and incomplete.
It is worth to mention that some of the popular Bregman divergences, for
example, the squared Euclidean distance or the Itakura–Saito divergence, do
not satisfy the first requirement (see Section 4.3).
The second idea is just a very general methodological thesis: to evaluate an
instrument (a divergence) we have to look how it works (produces conditional
minimizers and maximizers). The properties of the instrument which are not
related to its work are not important. The requirements number three and four
are, in their essence, conditions for separation of variables in conditional
minimization (maximization) problems.
The number three allows to separate variables if the system consists of
independent subsystems, the number four relates to separation of variables for
partitions of the space of probability distributions.
Amongst a rich world of relative entropies and divergences, only two families
meet these requirements. Both were proposed in 1984. The Cressie–Read (CR)
family CR1984 ; ReadCreass1988 :
$H_{{\rm CR}\
\lambda}(P\|P^{*})=\frac{1}{\lambda(\lambda+1)}\sum_{i}p_{i}\left[\left(\frac{p_{i}}{p_{i}^{*}}\right)^{\lambda}-1\right]\
,\ \ \lambda\in]-\infty,\infty[$
and the convex combination of the Burg and Shannon relative entropies proposed
in G11984 and further analyzed in ENTR1 ; ENTR2 :
$H(P\|P^{*})=\sum_{i}(\beta
p_{i}-(1-\beta)p_{i}^{*})\log\left(\frac{p_{i}}{p_{i}^{*}}\right)\ ,\ \
\beta\in[0,1]$
When $\lambda\to 0$ the CR divergence tends to the KL divergence (the relative
Shannon entropy) and when $\lambda\to-1$ it turns into the Burg relative
entropy. The Tsallis entropy was introduced four years later Tsa1988 and
became very popular in thermostatistics (there are thousands of works that use
or analyze this entropy TsallisBiblio2009 ). The Tsallis entropy coincides (up
to a constant multiplier $\lambda+1$) with the CR entropy for
$\lambda\in]-1,\infty[$ and there is no need to study it separately (see
discussion in Section 2.2).
A new problem arose: which entropy is better for a specific problem? Many
authors compare performance of different entropies and metrics for various
problems (see, for example, Cachin1997 ; Davis2007 ). In any case study it may
be possible to choose “the best” entropy but in general we have no sufficient
reasons for such a choice. We propose a possible way to avoid the choice of
the best entropy.
Let us return to the idea: the distribution $Q$ is more random than $P$ if
there exists a continuous-time Markov process (with given equilibrium
distribution $P^{*}$) that transforms $P$ into $Q$. We say in this case that
$P$ and $Q$ are connected by the Markov preorder with equilibrium $P^{*}$ and
use notation $P\succ^{0}_{P^{*}}Q$. The Markov order $\succ_{P^{*}}$ is the
transitive closure of the Markov preorder.
If a priori information gives us a set of possible distributions $W$ then the
conditionally “maximally random distributions” (the “distributions without
additional information”, the “most indefinite distributions” in $W$) should be
the minimal elements in $W$ with respect to Markov order. If a Markov process
(with equilibrium $P^{*}$) starts at such a minimal element $P$ then it cannot
produce any other distribution from $W$ because all distributions which are
more random that $P$ are situated outside $W$. In this approach, the maximally
random distributions under given a priori information may be not unique. Such
distributions form a set which plays the same role as the standard MaxEnt
distribution. For the moment based a priori information the set $W$ is an
intersection of a linear manifold with the simplex of probability
distributions, the set of minimal elements in $W$ is also polyhedron and its
description is available in explicit form. In low-dimensional case it is much
simpler to construct this polyhedron than to find the MaxEnt distributions for
most of specific entropies.
### 1.3 Structure of the Paper
The paper is organized as follows. In Section 2 we describe the known non-
classical divergences (relative entropies) which are the Lyapunov functions
for the Markov processes. We discuss the general construction and the most
popular families of such functions. We pay special attention to the
situations, when different divergences define the same order on distributions
and provide the same solutions of the MaxEnt problems (Section 2.2). In two
short technical Sections 2.3 and 2.4 we present normalization and
symmetrization of divergences (similar discussion of these operations was
published very recently Petz2010 .
The divergence between the current distribution and equilibrium should
decrease due to Markov processes. Moreover, divergence between any two
distributions should also decrease (the generalized data processing Lemma,
Section 3).
Definition of entropy by its properties is discussed in Section 4. Various
approaches to this definition were developed for the BGS entropy by Shannon
Shannon1948 , Renyi1970 and by other authors for the Rényi entropy Aczel ;
AczDar , the Tsallis entropy Abik4 , the CR entropy and the convex combination
of the BGS and Burg entropies ENTR3 . Csiszár Csiszar1978 axiomatically
characterized the class of Csiszár–Morimoto divergences (formula (6) below).
Another characterization of this class was proved in ENTR3 (see Lemma 1,
Section 4.3 below).
From the celebrated properties of entropy Wehrl we selected the following
three:
1. 1.
Entropy should be a Lyapunov function for continuous-time Markov processes;
2. 2.
Entropy is additive with respect to the joining of independent systems;
3. 3.
Entropy is additive with respect to the partitioning of the space of states
(_i.e._ , has the trace–form).
To solve the MaxEnt problem we have to find the maximizers of entropy
(minimizers of the relative entropy) under given conditions. For this purpose,
we have to know the sublevel sets of entropy, but not its values. We consider
entropies with the same system of sublevel sets as equivalent ones (Section
2.2). From this point of view, all important properties have to be invariant
with respect to monotonic transformations of the entropy scale. Two last
properties from the list have to be substituted by the following:
1. 2’.
There exists a monotonic transformation which makes entropy additive with
respect to the joining of independent systems (Section 4.1);
2. 3’.
There exists a monotonic transformation which makes entropy additive with
respect to the partitioning of the space of states (Section 4.2).
These properties imply specific separation of variables for the entropy
maximization problems (Sections 4.1 and 4.2). Several “No More Entropies”
Theorems are proven in Section 4.3: if an entropy has properties 1, 2’ and 3’
then it belongs to one of the following one-parametric families: to the
Cressie–Read family, or to a convex combination of the classical BGS entropy
and the Burg entropy (may be, after a monotonic transformation of scale).
It seems very natural to consider divergences as orders on distribution spaces
(Section 5.1), the sublevel sets are the lower cones of this orders. For
several functions, $H_{1}(P),\ldots,H_{k}(P)$ the sets $\\{Q\ |\
H_{i}(P)>H_{i}(Q)\ {\rm for\ all}\ i\\}$ give the simple generalization of the
sublevel sets. In Section 5 we discuss the more general orders in which
continuous time Markov processes are monotone, define the Markov order and
fully characterize the local Markov order. The Markov chains with detailed
balance define the Markov order for general Markov chains (Section 5.2). It is
surprising that there is no necessity to consider other Markov chains for the
order characterization (Section 5.2) because no reversibility is assumed in
this analysis.
In Section 6.1 we demonstrate how is it possible to use the Markov order to
reduce the uncertainty in the standard settings when a priori information is
given about values of some moments. Approaches to construction of the most
random distributions are presented in Section 6.2.
Various approaches for the definition of the reference distribution (or the
generalized canonical distribution) are compared in Section 7.
In Conclusion we briefly discuss the main results.
## 2 Non-Classical Entropies
### 2.1 The Most Popular Divergences
#### 2.1.1 Csiszár–Morimoto Functions $H_{h}$
During the time of modernism plenty of new entropies were proposed. Esteban
and Morales EstMor1995 attempted to systemize many of them in an impressive
table. Nevertheless, there are relatively few entropies in use now. Most of
the relative entropies have the form proposed independently in 1963 by I.
Csiszar Csiszar1963 and T. Morimoto Morimoto1963 :
$H_{h}(p)=H_{h}(P\|P^{*})=\sum_{i}p^{*}_{i}h\left(\frac{p_{i}}{p_{i}^{*}}\right)$
(6)
where $h(x)$ is a convex function defined on the open ($x>0$) or closed $x\geq
0$ semi-axis. We use here notation $H_{h}(P\|P^{*})$ to stress the dependence
of $H_{h}$ both on $p_{i}$ and $p^{*}_{i}$.
These relative entropies are the Lyapunov functions for all Markov chains with
equilibrium $P^{*}=(p^{*}_{i})$. Moreover, they have the relative entropy
contraction property given by the generalized data processing lemma (Section
3.2 below).
For $h(x)=x\log x$ this function coincides with the Kullback–Leibler
divergence from the current distribution $p_{i}$ to the equilibrium
$p^{*}_{i}$. Some practically important functions $h$ have singularities at 0.
For example, if we take $h(x)=-\log x$, then the correspondent $H_{h}$ is the
relative Burg entropy $H_{h}=-\sum_{i}p^{*}_{i}\log(p_{i}/p_{i}^{*})\to\infty$
for $p_{i}\to 0$.
#### 2.1.2 Required Properties of the Function $h(x)$
Formally, $h(x)$ is an extended real-valued proper convex function on the
closed positive real half-line $[0,\infty[$. An extended real-valued function
can take real values and infinite values $\pm\infty$. A proper function has at
least one finite value. An extended real valued function on a convex set $U$
is called convex if its epigraph
${\rm epi}(h)=\\{(x,y)\ |\ x>0,y\geq h(x)\\}$
is a convex set Rockafellar1970 . For a proper function this definition is
equivalent to the Jensen inequality
$h(ax+(1-a)y)\leq ah(x)+(1-a)h(y)\;\;\mbox{for all}\;\;x,y\in U,\,a\in[0,1]$
It is assumed that the function $h(x)$ takes finite values on the open
positive real half-line $]0,\infty[$ but the value at point $x=0$ may be
infinite. For example, functions $h(x)=-\log x$ or $h(x)=x^{-\alpha}$
($\alpha>0$) are allowed. A convex function $h(x)$ with finite values on the
open positive real half-line is continuous on $]0,\infty[$ but may have a
discontinuity at $x=0$. For example, the step function, $h(x)=0$ if $x=0$ and
$h(x)=-1$ if $x>0$, may be used.
A convex function is differentiable almost everywhere. Derivative of $h(x)$ is
a monotonic function which has left and right limits at each point $x>0$. An
inequality holds: $h^{\prime}(x)(y-x)\leq h(y)-h(x)$ (Jensen’s inequality in
the differential form). It is valid also for left and right limits of
$h^{\prime}$ at any point $x>0$.
Not everywhere differentiable functions $h(x)$ are often used, for example,
$h(x)=|x-1|$. Nevertheless, it is convenient to consider the twice
differentiable on $]0,\infty[$ functions $h(x)$ and to produce a non-smooth
$h(x)$ (if necessary) as a limit of smooth convex functions. We use widely
this possibility.
#### 2.1.3 The Most Popular Divergences $H_{h}(P\|P^{*})$
1. 1.
Let $h(x)$ be the step function, $h(x)=0$ if $x=0$ and $h(x)=-1$ if $x>0$. In
this case,
$H_{h}(P\|P^{*})=-\sum_{i,\ p_{i}>0}1$ (7)
The quantity $-H_{h}$ is the number of non-zero probabilities $p_{i}$ and does
not depend on $P^{*}$. Sometimes it is called the Hartley entropy.
2. 2.
$h=|x-1|$,
$H_{h}(P\|P^{*})=\sum_{i}|p_{i}-p_{i}^{*}|$
this is the $l_{1}$-distance between $P$ and $P^{*}$.
3. 3.
$h=x\ln x$,
$H_{h}(P\|P^{*})=\sum_{i}p_{i}\ln\left(\frac{p_{i}}{p_{i}^{*}}\right)=D_{\mathrm{KL}}(P\|P^{*})$
(8)
this is the usual Kullback–Leibler divergence or the relative BGS entropy;
4. 4.
$h=-\ln x$,
$H_{h}(P\|P^{*})=-\sum_{i}p^{*}_{i}\ln\left(\frac{p_{i}}{p_{i}^{*}}\right)=D_{\mathrm{KL}}(P^{*}\|P)$
(9)
this is the relative Burg entropy. It is obvious that this is again the
Kullback–Leibler divergence, but for another order of arguments.
5. 5.
Convex combinations of $h=x\ln x$ and $h=-\ln x$ also produces a remarkable
family of divergences: $h=\beta x\ln x-(1-\beta)\ln x$ ($\beta\in[0,1]$),
$H_{h}(P\|P^{*})=\beta
D_{\mathrm{KL}}(P\|P^{*})+(1-\beta)D_{\mathrm{KL}}(P^{*}\|P)$ (10)
The convex combination of divergences becomes a symmetric functional of
$(P,P^{*})$ for $\beta=1/2$. There exists a special name for this case,
“Jeffreys’ entropy”.
6. 6.
$h=\frac{(x-1)^{2}}{2}$,
$H_{h}(P\|P^{*})=\frac{1}{2}\sum_{i}\frac{(p_{i}-p_{i}^{*})^{2}}{p_{i}^{*}}$
(11)
This is the quadratic term in the Taylor expansion of the relative
Botzmann–Gibbs-Shannon entropy, $D_{\mathrm{KL}}(P\|P^{*})$, near equilibrium.
Sometimes, this quadratic form is called the Fisher entropy.
7. 7.
$h=\frac{x(x^{\lambda}-1)}{\lambda(\lambda+1)}$,
$H_{h}(P\|P^{*})=\frac{1}{\lambda(\lambda+1)}\sum_{i}p_{i}\left[\left(\frac{p_{i}}{p_{i}^{*}}\right)^{\lambda}-1\right]$
(12)
This is the CR family of power divergences CR1984 ; ReadCreass1988 . For this
family we use notation $H_{\rm CR\ \lambda}$. If $\lambda\to 0$ then $H_{\rm
CR\ \lambda}\to D_{\mathrm{KL}}(P\|P^{*})$, this is the classical BGS relative
entropy; if $\lambda\to-1$ then $H_{\rm CR\ \lambda}\to
D_{\mathrm{KL}}(P^{*}\|P)$, this is the relative Burg entropy.
8. 8.
For the CR family in the limits $\lambda\to\pm\infty$ only the maximal terms
“survive”. Exactly as we get the limit $l^{\infty}$ of $l^{p}$ norms for
$p\to\infty$, we can use the root $({\lambda(\lambda+1)}H_{\rm CR\
\lambda})^{1/|\lambda|}$ for $\lambda\to\pm\infty$ and write in these limits
the divergences:
$H_{\rm CR\
\infty}(P\|P^{*})=\max_{i}\left\\{\frac{p_{i}}{p_{i}^{*}}\right\\}-1$ (13)
$H_{\rm CR\
-\infty}(P\|P^{*})=\max_{i}\left\\{\frac{p_{i}^{*}}{p_{i}}\right\\}-1$ (14)
The existence of two limiting divergences $H_{{\rm CR\ \pm\infty}}$ seems very
natural: there may be two types of extremely non-equilibrium states: with a
high excess of current probability $p_{i}$ above $p_{i}^{*}$ and, inversely,
with an extremely small current probability $p_{i}$ with respect to
$p_{i}^{*}$.
9. 9.
The Tsallis relative entropy Tsa1988 corresponds to the choice
$h=\frac{(x^{\alpha}-x)}{\alpha-1}$, $\alpha>0$.
$H_{h}(P\|P^{*})=\frac{1}{\alpha-1}\sum_{i}p_{i}\left[\left(\frac{p_{i}}{p_{i}^{*}}\right)^{\alpha-1}-1\right]$
(15)
For this family we use notation $H_{\rm Ts\ \alpha}$.
#### 2.1.4 Rényi Entropy
The Rényi entropy of order $\alpha>0$ is defined Renyi1961 as
$H_{{\rm R}\
\alpha}(P)=\frac{1}{1-\alpha}\log\left(\sum_{i=1}^{n}p_{i}^{\alpha}\right)$
(16)
It is a concave function, and
$H_{{\rm R}\ \alpha}(P)\to S(P)$
when $\alpha\to 1$, where $S(P)$ is the Shannon entropy.
When $\alpha\to\infty$, the Rényi entropy has a limit
$H_{\infty}(X)=-\log\max_{i=1,\ldots n}p_{i}$, which has a special name “Min-
entropy”.
It is easy to get the expression for a relative Rényi entropy $H_{{\rm R}\
\alpha}(P\|P^{*})$ from the requirement that it should be a Lyapunov function
for any Markov chain with equilibrium $P^{*}$:
$H_{{\rm R}\
\alpha}(P\|P^{*})=\frac{1}{\alpha-1}\log\left(\sum_{i=1}^{n}p_{i}\left(\frac{p_{i}}{p_{i}^{*}}\right)^{\alpha-1}\right)$
For the Min-entropy, the correspondent divergence (the relative Min-entropy)
is
$H_{\infty}(P\|P^{*})=\log\max_{i=1,\ldots
n}\left(\frac{p_{i}}{p_{i}^{*}}\right)$
It is obvious from (22) below that $\max_{i=1,\ldots n}({p_{i}}/{p_{i}^{*}})$
is a Lyapunov function for any Markov chain with equilibrium $P^{*}$, hence,
the relative Min-entropy is also the Lyapunov functional.
### 2.2 Entropy Level Sets
A level set of a real-valued function $f$ of is a set of the form :
$\\{x\ |\ f(x)=c\\}$
where $c$ is a constant (the “level”). It is the set where the function takes
on a given constant value. A sublevel set of $f$ is a set of the form
$\\{x\ |\ f(x)\leq c\\}$
A superlevel set of $f$ is given by the inequality with reverse sign:
$\\{x\ |\ f(x)\geq c\\}$
The intersection of the sublevel and the superlevel sets for a given value $c$
is the level set for this level.
In many applications, we do not need the entropy values, but rather the order
of these values on the line. For any two distributions $P,Q$ we have to
compare which one is closer to equilibrium $P^{*}$, _i.e._ , to answer the
question: which of the following relations is true: $H(P\|P^{*})>H(Q\|P^{*})$,
$H(P\|P^{*})=H(Q\|P^{*})$ or $H(P\|P^{*})<H(Q\|P^{*})$? To solve the MaxEnt
problem we have to find the maximizers of entropy (or, in more general
settings, the minimizers of the relative entropy) under given conditions. For
this purpose, we have to know the sublevel sets, but not the values. All the
MaxEnt approach does not need the values of the entropy but the sublevel sets
are necessary.
Let us consider two functions, $\phi$ and $\psi$ on a set $U$. For any
$V\subset U$ we can study conditional minimization problems $\phi(x)\to\min$
and $\psi(x)\to\min$, $x\in V$. The sets of minimizers for these two problems
coincide for any $V\subset U$ if and only if the functions $\phi$ and $\psi$
have the same sets of sublevel sets. It should be stressed that here just the
sets of sublevel sets have to coincide without any relation to values of
level.
Let us compare the level sets for the Rényi, the Cressie-Read and the Tsallis
families of divergences (for $\alpha-1=\lambda$ and for all values of
$\alpha$). The values of these functions are different, but the level sets are
the same (outside the Burg limit, where $\alpha\to 0$): for $\alpha\neq 0,1$
$H_{{\rm R}\ \alpha}(P\|P^{*})=\frac{1}{\alpha-1}\ln c;\ \ H_{\rm CR\
\alpha-1}(P\|P^{*})=\frac{1}{\alpha(\alpha-1)}(c-1);\ \ H_{\rm Ts\
\alpha}(P\|P^{*})=\frac{1}{\alpha-1}(c-1)$ (17)
where $c=\sum_{i}p_{i}(p_{i}/p_{i}^{*})^{\alpha-1}$.
Beyond points $\alpha=0,1$
$H_{\rm CR\
\alpha-1}(P\|P^{*})=\frac{1}{\alpha(\alpha-1)}\exp((\alpha-1)H_{{\rm R}\
\alpha}(P\|P^{*}))=\frac{1}{\alpha}H_{\rm Ts\ \alpha}(P\|P^{*})$
For $\alpha\to 1$ all these divergences turn into the Shannon relative
entropy. Hence, if $\alpha\neq 0$ then for any $P$, $P^{*}$, $Q$, $Q^{*}$ the
following equalities A, B, C are equivalent,
A$\Leftrightarrow$B$\Leftrightarrow$C:
$\begin{split}&{\rm A}.\;\;H_{{\rm R}\ \alpha}(P\|P^{*})=H_{{\rm R}\
\alpha}(Q\|Q^{*})\\\ &{\rm B}.\;\;H_{{\rm CR}\ \alpha+1}(P\|P^{*})=H_{{\rm
CR}\ \alpha+1}(Q\|Q^{*})\\\ &{\rm C}.\;\;H_{{\rm Ts}\
\alpha}(P\|P^{*})=H_{{\rm Ts}\ \alpha}(Q\|Q^{*})\end{split}$ (18)
This equivalence means that we can select any of these three divergences as a
basic function and consider the others as functions of this basic one.
For any $\alpha\geq 0$ and $\lambda=\alpha+1$ the Rényi, the Cressie–Read and
the Tsallis divergences have the same family of sublevel sets. Hence, they
give the same maximizers to the conditional relative entropy minimization
problems and there is no difference which entropy to use.
The CR family has a more convenient normalization factor
$1/\lambda(\lambda+1)$ which gives a proper convexity for all powers, both
positive and negative, and provides a sensible Burg limit for $\lambda\to-1$
(in contrary, when $\alpha\to 0$ both the Rényi and Tsallis entropies tend to
0).
When $\alpha<0$ then for the Tsallis entropy function
$h=\frac{(x^{\alpha}-x)}{\alpha-1}$ loses convexity, whereas for the Cressie-
Read family convexity persists for all values of $\lambda$. The Rényi entropy
also loses convexity for $\alpha<0$. Neither the Tsallis, nor the Rényi
entropy were invented for use with negative $\alpha$.
There may be a reason: for $\alpha<0$ the function $x^{\alpha}$ is defined for
$x>0$ only and has a singularity at $x=0$. If we assume that the divergence
should exist for all non-negative distributions, then the cases $\alpha\leq 0$
should be excluded. Nevertheless, the Burg entropy which is singular at zeros
is practically important and has various attractive properties. The Jeffreys’
entropy (the symmetrized Kullback–Leibler divergence) is also singular at
zero, but has many important properties. We can conclude at this point that it
is not obvious that we have to exclude any singularity at zero probability. It
may be useful to consider positive probabilities instead (“nature abhors a
vacuum”).
Finally, for the MaxEnt approach (conditional minimization of the relative
entropy), the Rényi and the Tsallis families of divergences ($\alpha>0$) are
particular cases of the Cressie–Read family because they give the same
minimizers. For $\alpha\leq 0$ the Rényi and the Tsallis relative entropies
lose their convexity, while the Cressie–Read family remains convex for
$\lambda\leq-1$ too.
### 2.3 Minima and normalization
For a given $P^{*}$, the function $H_{h}$ achieves its minimum on the
hyperplane $\sum_{i}p_{i}=\sum_{i}p_{i}^{*}=$const at equilibrium $p_{i}^{*}$,
because at this point
${\rm grad}H_{h}=(h^{\prime}(1),\ldots h^{\prime}(1))=h^{\prime}(1){\rm
grad}\left(\sum_{i}p_{i}\right)$
The transformation $h(x)\to h(x)+ax+b$ just shifts $H_{h}$ by constant value:
$H_{h}\to H_{h}+a\sum_{i}p_{i}+b=H_{h}+a+b$. Therefore, we can always assume
that the function $h(x)$ achieves its minimal value at point $x=1$, and this
value is zero. For this purpose, one should just transform $h$:
$h(x):=h(x)-h(1)-h^{\prime}(1)(x-1)$ (19)
This normalization transforms $x\ln x$ into $x\ln x-(x-1)$, $-\ln x$ into
$-\ln x+(x-1)$, and $x^{\alpha}$ into $x^{\alpha}-1-\alpha(x-1)$. After
normalization $H_{h}(P\|P^{*})\geq 0$. If the normalized $h(x)$ is strictly
positive outside point $x=1$ ($h(x)>0$ if $x\neq 1$) then $H_{h}(P\|P^{*})=0$
if and only if $P=P^{*}$ (_i.e._ , in equilibrium).
The normalized version of any divergence $H_{h}(P\|P^{*})$ could be produced
by the normalization transformation $h(x):=h(x)-h(1)-h^{\prime}(1)(x-1)$ and
does not need separate discussion.
### 2.4 Symmetrization
Another technical issue is symmetry of a divergence. If $h(x)=x\ln x$ then
both $H_{h}(P\|P^{*})$ (the KL divergence) and $H_{h}(P^{*}\|P)$ (the relative
Burg entropy) are the Lyapunov functions for the Markov chains, and
$H_{h}(P^{*}\|P)=H_{g}(P\|P^{*})$ with $g(x)=-\ln x$. Analogously, for any
$h(x)$ we can write $H_{h}(P^{*}\|P)=H_{g}(P\|P^{*})$ with
$g(x)=xh\left(\frac{1}{x}\right)$ (20)
If $h(x)$ is convex on $\mathbf{R}_{+}$ then $g(x)$ is convex on
$\mathbf{R}_{+}$ too because
$g^{\prime\prime}(x)=\frac{1}{x^{3}}h^{\prime\prime}\left(\frac{1}{x}\right)$
The transformation (20) is an involution:
$xg\left(\frac{1}{x}\right)=h(x)$
The fixed points of this involution are such functions $h(x)$ that
$H_{h}(P\|P^{*})$ is symmetric with respect to transpositions of $P$ and
$P^{*}$. There are many such $h(x)$. An example of symmetric $H_{h}(P\|P^{*})$
gives the choice $h(x)=-\sqrt{x}$:
$H_{h}(P\|P^{*})=-\sum_{i}\sqrt{p_{i}p^{*}_{i}}$
After normalization, we get
$h(x):=\frac{1}{2}(\sqrt{x}-1)^{2}\
;\;\;\;H_{h}(P\|P^{*})=\frac{1}{2}\sum_{i}(\sqrt{p_{i}}-\sqrt{p^{*}_{i}})^{2}$
Essentially (up to a constant addition and multiplier) this function coincides
with a member of the CR family, $H_{{\rm CR}\ -\frac{1}{2}}$ (12), and with
one of the Tsallis relative entropies $H_{{\rm Ts}\ \frac{1}{2}}$ (15). The
involution (20) is a linear operator, hence, for any convex $h(x)$ we can
produce its symmetrization:
$h_{\rm
sym}(x)=\frac{1}{2}(h(x)+g(x))=\frac{1}{2}\left(h(x)+xh\left(\frac{1}{x}\right)\right)$
For example, if $h(x)=x\log x$ then $h_{\rm sym}(x)=\frac{1}{2}(x\log x-\log
x)$; if $h(x)=x^{\alpha}$ then $h_{\rm
sym}(x)=\frac{1}{2}(x^{\alpha}+x^{1-\alpha})$.
## 3 Entropy Production and Relative Entropy Contraction
### 3.1 Lyapunov Functionals for Markov Chains
Let us consider continuous time Markov chains with positive equilibrium
probabilities $p_{j}^{*}$. The dynamics of the probability distribution
$p_{i}$ satisfy the Master equation (the Kolmogorov equation):
$\frac{{\mathrm{d}}p_{i}}{{\mathrm{d}}t}=\sum_{j,\,j\neq
i}(q_{ij}p_{j}-q_{ji}p_{i})$ (21)
where coefficients $q_{ij}$ ($i\neq j$) are non-negative. For chains with a
positive equilibrium distribution $p_{j}^{*}$ another equivalent form is
convenient:
$\frac{{\mathrm{d}}p_{i}}{{\mathrm{d}}t}=\sum_{j,\,j\neq
i}q_{ij}p^{*}_{j}\left(\frac{p_{j}}{p_{j}^{*}}-\frac{p_{i}}{p_{i}^{*}}\right)\
$ (22)
where $p_{i}^{*}$ and $q_{ij}$ are connected by identity
$\sum_{j,\,j\neq i}q_{ij}p^{*}_{j}=\left(\sum_{j,\,j\neq
i}q_{ji}\right)p^{*}_{i}$ (23)
The time derivative of the Csiszár–Morimoto function $H_{h}(p)$ (6) due to the
Master equation is
$\frac{{\mathrm{d}}H_{h}(P\|P^{*})}{{\mathrm{d}}t}=\sum_{i,j,\,j\neq
i}q_{ij}p^{*}_{j}\left[h\left(\frac{p_{i}}{p_{i}^{*}}\right)-h\left(\frac{p_{j}}{p_{j}^{*}}\right)+h^{\prime}\left(\frac{p_{i}}{p_{i}^{*}}\right)\left(\frac{p_{j}}{p_{j}^{*}}-\frac{p_{i}}{p_{i}^{*}}\right)\right]\leq
0$ (24)
To prove this formula, it is worth to mention that for any $n$ numbers
$h_{i}$, $\sum_{i,j,\,j\neq i}q_{ij}p^{*}_{j}(h_{i}-h_{j})=0$. The last
inequality holds because of the convexity of $h(x)$: $h^{\prime}(x)(y-x)\leq
h(y)-h(x)$ (Jensen’s inequality).
Inversely, if
$h(x)-h(y)+h^{\prime}(y)(x-y)\leq 0$ (25)
for all positive $x,y$ then $h(x)$ is convex on $\mathbf{R}_{+}$. Therefore,
if for some function $h(x)$ $H_{h}(p)$ is the Lyapunov function for all the
Markov chains with equilibrium $P^{*}$ then $h(x)$ is convex on
$\mathbf{R}_{+}$.
The Lyapunov functionals $H_{h}$ do not depend on the kinetic coefficients
$q_{ij}$ directly. They depend on the equilibrium distribution $p^{*}$ which
satisfies the identity (23). This independence of the kinetic coefficients is
the universality property.
### 3.2 “Lyapunov Divergences” for Discrete Time Markov Chains
The Csiszár–Morimoto functions (6) are also Lyapunov functions for discrete
time Markov chains. Moreover, they can serve as a “Lyapunov distances”
Liese1987 between distributions which decreases due to time evolution (and
not only the divergence between the current distribution and equilibrium). In
more detail, let $A=(a_{ij})$ be a stochastic matrix in columns:
$a_{ij}\geq 0,\;\;\sum_{i}a_{ij}=1\;{\rm for\;all}\;j$
The ergodicity contraction coefficient for $A$ is a number
$\overline{\alpha}(A)$ DobrushinErgCoeff1956 ; Seneta1981 :
$\overline{\alpha}(A)=\frac{1}{2}\max_{i,k}\left\\{\sum_{j}|a_{ij}-a_{kj}|\right\\}$
$0\leq\overline{\alpha}(A)\leq 1$.
Let us consider in this subsection the normalized Csiszár–Morimoto divergences
$H_{h}(P\|Q)$ (19): $h(1)=0,\,h(x)\geq 0$.
Theorem about relative entropy contraction. (The generalized data processing
Lemma.) For each two probability positive distributions $P,Q$ the divergence
$H_{h}(P\|Q)$ decreases under action of stochastic matrix $A$ Cohen1993 ;
CohenIwasa1993 :
$H_{h}(AP\|AQ)\leq\overline{\alpha}(A)H_{h}(P\|Q)$ (26)
The generalizations of this theorem for general Markov kernels seen as
operators on spaces of probability measures was given by Ledoux2003 . The
shift in time for continuous-time Markov chain is a column-stochastic matrix,
hence, this contraction theorem is also valid for continuous-time Markov
chains.
The question about a converse theorem arises immediately. Let the contraction
inequality hold for two pairs of positive distributions $(P,Q)$ and $(U,V)$
and for all $H_{h}$:
$H_{h}(U\|V)\leq H_{h}(P\|Q)$ (27)
Could we expect that there exists such a stochastic matrix $A$ that $U=AP$ and
$V=AP$? The answer is positive:
The converse generalized data processing lemma. Let the contraction inequality
(27) hold for two pairs of positive distributions $(P,Q)$ and $(U,V)$ and for
all normalized $H_{h}$. Then there exists such a column-stochastic matrix $A$
that $U=AP$ and $V=AQ$ Cohen1993 .
This means that for the system of inequalities (27) (for all normalized convex
functions $h$ on $]0,\infty[$) is necessary and sufficient for existence of a
(discrete time) Markov process which transform the pair of positive
distributions $(P,Q)$ in $(U,V)$. It is easy to show that for continuous-time
Markov chains this theorem is not valid: the attainable regions for them are
strictly smaller than the set given by inequalities (27) and could be even
non-convex (see Gorban1979 and Section 8.1 below).
## 4 Definition of Entropy by its Properties
### 4.1 Additivity Property
The additivity property with respect to joining of subsystems is crucial both
for the classical thermodynamics and for the information theory.
Let us consider a system which is result of joining of two subsystems. A state
of the system is an ordered pair of the states of the subsystems and the space
of states of the system is the Cartesian product of the subsystems spaces of
state. For systems with finite number of states this means that if the states
of subsystems are enumerated by indexes $j$ and $k$ then the states of the
system are enumerated by pairs $jk$. The probability distribution for the
whole system is $p_{jk}$, and for the subsystems the probability distributions
are the marginal distributions $q_{j}=\sum_{k}p_{jk}$, $r_{k}=\sum_{j}p_{jk}$.
The additive functions of state are defined for each state of the subsystems
and for a state of the whole system they are sums of these subsystem values:
$u_{jk}=v_{j}+w_{k}$
where $v_{j}$ and $w_{k}$ are functions of the subsystems state.
In classical thermodynamics such functions are called the extensive
quantities. For expected values of additive quantities the similar additivity
condition holds:
$\sum_{j,k}u_{jk}p_{jk}=\sum_{j,k}(v_{j}+w_{k})p_{ik}=\sum_{j}v_{j}q_{j}+\sum_{k}w_{k}r_{k}$
(28)
Let us consider these expected values as functionals of the probability
distributions: $u(\\{p_{jk}\\})$, $v(\\{q_{j}\\})$ and $w(\\{r_{k}\\})$. Then
the additivity property for the expected values reads:
$u(\\{p_{jk}\\})=v(\\{q_{j}\\})+w(\\{r_{k}\\})$ (29)
where $q_{j}$ and the $r_{k}$ are the marginal distributions.
Such a linear additivity property is impossible for non-linear entropy
functionals, but under some independence conditions the entropy can behave as
an extensive variable.
Let $P$ be a product of marginal distributions. This means that the subsystems
are statistically independent: $p_{jk}=q_{j}r_{k}$. Assume also that the
distribution $P^{*}$ is also a product of marginal distributions
$p^{*}_{jk}=q^{*}_{j}r^{*}_{k}$. Then some entropies reveal the additivity
property with respect to joining of independent systems.
1. 1.
The BGS relative entropy
$D_{\mathrm{KL}}(P\|P^{*})=D_{\mathrm{KL}}(Q\|Q^{*})+D_{\mathrm{KL}}(R\|R^{*})$.
2. 2.
The Burg entropy
$D_{\mathrm{KL}}(P^{*}\|P)=D_{\mathrm{KL}}(Q^{*}\|Q)+D_{\mathrm{KL}}(R^{*}\|R)$
. It is obvious that a convex combination of the Shannon and Burg entropies
has the same additivity property.
3. 3.
The Rényi entropy $H_{{\rm R}\ \alpha}(P\|P^{*})=H_{{\rm R}\
\alpha}(Q\|Q^{*})+H_{{\rm R}\ \alpha}(R\|R^{*})$. For $\alpha\to\infty$ the
Min-entropy also inherites this property.
This property implies the separation of variables for the entropy maximization
problems. Let functionals $u^{1}(\\{p_{jk}\\}),\ldots u^{m}(\\{p_{jk}\\})$ be
additive (28) (29) and let the relative entropy $H(P\|P^{*})$ be additive with
respect to joining of independent systems. Then the solution to the problem
$H(P\|P^{*})\to\min$
subject to conditions
$u^{i}(P)=U_{i}\ \ (i=1,\ldots m)$
is $p_{jk}^{\min}=q_{j}^{\min}r_{k}^{\min}$, where $q_{j}^{\min}$,
$r_{k}^{\min}$ are solutions of partial problems:
$H(Q\|Q^{*})\to\min$
subject to the conditions
$v^{i}(Q)=V_{i}\ \ (i=1,\ldots m)$
and
$H(R\|R^{*})\to\min$
subject to the conditions
$w^{i}(Q)=W_{i}\ \ (i=1,\ldots m)$
for some redistribution of the additive functionals values
$U_{i}=V_{i}+W_{i}$.
Let us call this property the separation of variables for independent
subsystems.
Neither the CR, nor the Tsallis divergences families have the additivity
property. It is proven ENTR3 that a function $H_{h}$ has the additivity
property if and only if it is a convex combination of the Shannon and Burg
entropies. See also Theorem 3 in Appendix.
Nevertheless, they have the property of separation of variables for
independent subsystems because of the coincidence of the level sets with the
additive function, the Rényi entropy (for all $\alpha>0$).
The Tsallis entropy family has absolutely the same property of separation of
variables as the Rényi entropy. To extend this property of the Rényi Tsallis
entropies for negative $\alpha$, we have to change there min to max.
For the CR family the result sounds even better: because of better
normalization, the separation of variables is valid for $H_{\rm CR\
\lambda}\to\min$ problem for all values $\lambda\in]-\infty,\infty[$.
### 4.2 Separation of Variables for Partition of the State Space
Another important property of separation of variables is valid for all
divergences which have the form of a sum of convex functions
$f(p_{i},p_{i}^{*})$. Let the set of states be divided into two subsets,
$I_{1}$ and $I_{2}$, and let the functionals $u^{1},\ldots u^{m}$ be linear.
We represent each probability distribution as a direct sum $P=P^{1}\oplus
P^{2}$, where $P^{1,2}$ are restrictions of $P$ on $I_{1,2}$.
Let us consider the problem
$H(P\|P^{*})\to\min$
subject to conditions $u^{i}(P)=U_{i}$ for a set of linear functionals
$u^{i}(P)$.
The solution $P^{\min}$ to this problem has a form
$P^{\min}=P_{1}^{\min}\oplus P_{2}^{\min}$, where $P^{1,2}$ are solutions to
the problems
$H(P^{1,2}\|P^{*\ 1,2})\to\min$
subject to conditions $u^{i}(P_{1,2})=U_{i}^{1,2}$ and $\sum_{i\in
I_{1,2}}p_{i}^{1,2}=\pi_{1,2}$ for some redistribution of the linear
functionals values, $U_{i}=U_{i}^{1}+U_{i}^{2}$, and of the total probability,
$1=\pi_{1}+\pi_{2}$ ($\pi_{1,2}\geq 0$) .
Again, the solution to the divergence minimization problem is composed from
solutions of the partial maximization problems. Let us call this property the
separation of variables for incompatible events (because $I_{1}\cap
I_{2}=\emptyset$).
This property is trivially valid for the Tsallis family (for $\alpha>0$, and
for $\alpha<0$ with the change of minimization to maximization) and for the CR
family. For the Rényi family it also holds (for $\alpha>0$, and for $\alpha<0$
with the change from minimization to maximization), because the Rényi entropy
is a function of those trace–form entropies, their level sets coincide.
Again, a simple check shows that this separation of variables property holds
also for the convex combination of Shannon’s and Burg’s entropies, $\beta
D_{\mathrm{KL}}(P\|P^{*})+(1-\beta)D_{\mathrm{KL}}(P^{*}\|P)$.
The question arises: is there any new divergence that has the following three
properties: (i) the divergence $H(P\|P^{*})$ should decrease in Markov
processes with equilibrium $P^{*}$, (ii) for minimization problems the
separation of variables for independent subsystems holds and (iii) the
separation of variables for incompatible events holds. A new divergence means
here that it is not a function of a divergence from the CR family or from the
convex combination of the Shannon and the Burg entropies.
The answer is: no, any divergence which has these three properties and is
defined and differentiable for positive distributions is a function of $H_{h}$
for $h(x)=p^{\alpha}$ or $h(x)=\beta x\ln x-(1-\beta)\ln x$. If we relax the
differentiability property, then we have to add to the CR family the CR
analogue of min-entropy
$H_{\rm CR\
\infty}(P\|P^{*})=\max_{i}\left\\{\frac{p_{i}}{p_{i}^{*}}\right\\}-1$
The limiting case for the CR family for $\lambda\to-\infty$ is less known but
is also a continuous and piecewise differential Lyapunov function for the
Master equation:
$H_{{\rm CR\
-\infty}}(P\|P^{*})=\max_{i}\left\\{\frac{p_{i}^{*}}{p_{i}}\right\\}-1$
Both properties of separation of variables are based on the specific
additivity properties: additivity with respect to the composition of
independent systems and additivity with respect to the partitioning of the
space of states. Separation of variables can be considered as a weakened form
of additivity: not the minimized function should be additive but there exists
such a monotonic transformation of scale after which the function becomes
additive (and different transformations may be needed for different additivity
properties).
### 4.3 “No More Entropies” Theorems
The classical Shannon work included the characterization of entropy by its
properties. This meant that the classical notion of entropy is natural, and no
more entropies are expected. In the seminal work of Rényi, again the
characterization of entropy by its properties was proved, and for this,
extended family the no more entropies theorem was proved too. In this section,
we prove the next no more entropies theorem, where two one-parametric families
are selected as sensible: the CR family and the convex combination of
Shannon’s and Burg’s entropies. They are two branches of solutions of the
correspondent functional equation and intersect at two points: Shannon’s
entropy ($\lambda=1$ in the CR family) and Burg’s entropy ($\lambda=0$). We
consider entropies as equivalent if their level sets coincide. In that sense,
the Rényi entropy and the Tsallis entropy (with $\alpha>0$) are equivalent to
the CR entropy with $\alpha-1=\lambda$, $\lambda>-1$.
Following Rényi, we consider entropies of incomplete distributions: $p_{i}\geq
0$, $\sum_{i}p_{i}\leq 1$. The divergence $H(P\|P^{*})$ is a $C^{1}$ smooth
function of a pair of positive generalized probability distributions
$P=(p_{i})$, $p_{i}>0$ and $P^{*}=(p^{*}_{i})$, $p^{*}_{i}>0$, $i=1,\ldots n$.
The following 3 properties are required for characterization of the “natural”
entropies.
1. 1.
To provide the separation of variables for incompatible events together with
the symmetry property we assume that the divergence is separable, possibly,
after a scaling transformation: there exists such a function of two variables
$f(p,p^{*})$ and a monotonic function of one variable $\phi(x)$ that
$H(P\|P^{*})=\phi(\sum_{i}f(p_{i},p^{*}_{i}))$. This formula allows us to
define $H(P\|P^{*})$ for all $n$.
2. 2.
$H(P\|P^{*})$ is a Lyapunov function for the Kolmogorov equation (22) for any
Markov chain with equilibrium $P^{*}$. (One can call these functions the
universal Lyapunov functions because they do not depend on the kinetic
coefficients directly, but only on the equilibrium distribution $P^{*}$.)
3. 3.
To provide separation of variables for independent subsystems we assume that
$H(P\|P^{*})$ is additive (possibly after a scaling transformation): there
exists such a function of one variable $\psi(x)$ that the function
$\psi(H(P\|P^{*}))$ is additive for the union of independent subsystems: if
$P=(p_{ij})$, $p_{ij}=q_{j}r_{j}$, $p^{*}_{ij}=q^{*}_{j}r^{*}_{j}$, then
$\psi(H(P\|P^{*}))=\psi(H(Q\|Q^{*}))+\psi(H(R\|R^{*}))$.
###### Theorem 1.
If a $C^{1}$-smooth divergence $H(P\|P^{*})$ satisfies the conditions 1-3
then, up to monotonic transformation, it is either the CR divergence $H_{\rm
CR\ \lambda}$ or a convex combination of the Botlzmann–Gibbs–Shannon and the
Burg entropies, $H_{h}(P\|P^{*})=\beta
D_{\mathrm{KL}}(P\|P^{*})+(1-\beta)D_{\mathrm{KL}}(P^{*}\|P)$.
In a paper ENTR3 this family was identified as the Tsallis relative entropy
with some abuse of language, because in the Tsallis entropy the case with
$\alpha<0$ is usually excluded.
First of all, let us prove that any function which satisfies the conditions 1
and 2 is a monotone function of a Csiszár–Morimoto function (6) for some
convex smooth function $h(x)$. This was mentioned in 2003 by P. Gorban ENTR3 .
Recently, a similar statement was published by S. Amari (Theorem 1 in
Amari2009 ).
###### Lemma 1.
If a Lyapunov function $H(p)$ for the Markov chain is of the trace–form
($H(p)=\sum_{i}f(p_{i},p_{i}^{*})$) and is universal, then
$f(p,p^{*})=p^{*}h(\frac{p}{p^{*}})+{\rm const}(p^{*})$, where $h(x)$ is a
convex function of one variable.
###### Proof.
Let us consider a Markov chain with two states. For such a chain
$\frac{{\mathrm{d}}p_{1}}{{\mathrm{d}}t}=q_{12}p_{2}^{*}\left(\frac{p_{2}}{p_{2}^{*}}-\frac{p_{1}}{p_{1}^{*}}\right)=-q_{21}p_{1}^{*}\left(\frac{p_{1}}{p_{1}^{*}}-\frac{p_{2}}{p_{2}^{*}}\right)=-\frac{{\mathrm{d}}p_{2}}{{\mathrm{d}}t}$
(30)
If $H$ is a Lyapunov function then $\dot{H}\leq 0$ and the following
inequality holds:
$\left(\frac{\partial f(p_{2},p_{2}^{*})}{\partial p_{2}}-\frac{\partial
f(p_{1},p_{1}^{*})}{\partial
p_{1}}\right)\left(\frac{p_{1}}{p_{1}^{*}}-\frac{p_{2}}{p_{2}^{*}}\right)\leq
0$
We can consider $p_{1},p_{2}$ as independent variables from an open triangle
$D=\\{(p_{1},p_{2})\ |\ p_{1,2}>0,\ p_{1}+p_{2}<1\\}$. For this purpose, we
can include the Markov with two states into a chain with three states and
$q_{3i}=q_{i3}=0$.
If for a continuous function of two variables $\psi(x,y)$ in an open domain
$D\subset\mathbb{R}^{2}$ an inequality
$(\psi(x_{1},y_{1})-\psi(x_{2},y_{2}))(y_{1}-y_{2})\leq 0$ holds then this
function does not depend on $x$ in $D$. Indeed, let there exist such values
$x_{1,2}$ and $y$ that $\psi(x_{1},y)\neq\psi(x_{2},y)$,
$\psi(x_{1},y)-\psi(x_{2},y)=\varepsilon>0$. We can find such $\delta>0$ that
$(x_{1},y+\Delta y)\in D$ and $|\psi(x_{1},y+\Delta
y)-\psi(x_{1},y)|<\varepsilon/2$ if $|\Delta y|<\delta$. Hence,
$\psi(x_{1},y+\Delta y)-\psi(x_{2},y)>\varepsilon/2>0$ if $|\Delta y|<\delta$.
At the same time $(\psi(x_{1},y+\Delta y)-\psi(x_{2},y))\Delta y\leq 0$,
hence, for a positive $0<\Delta y<\delta$ we have a contradiction. Therefore,
the function $\frac{\partial f(p,p^{*})}{\partial p}$ is a monotonic function
of $\frac{p}{p^{*}}$, hence, $f(p,p^{*})=p^{*}h(\frac{p}{p^{*}})+{\rm
const}(p^{*})$, where $h$ is a convex function of one variable. ∎
This lemma has important corollaries about many popular divergences
$H(P(t)\|P^{*})$ which are not Lyapunov functions of Markov chains. This means
that there exist such distributions $P_{0}$ and $P^{*}$ and a Markov chain
with equilibrium distribution $P^{*}$ that due to the Kolmogorov equations
$\left.\frac{{\mathrm{d}}H(P(t)\|P^{*})}{{\mathrm{d}}t}\right|_{t=0}>0$
if $P(0)=P_{0}$. This Markov process increases divergence between the
distributions $P,P^{*}$ (in a vicinity of $P_{0}$) instead of making them
closer. For example,
###### Corollary 1.
The following Bregman divergences Bregman1967 are not universal Lyapunov
functions for Markov chains:
* •
Squared Euclidean distance $B(P\|P^{*})=\sum_{i}(p_{i}-p^{*}_{i})^{2}$;
* •
The Itakura–Saito divergence Itakura1968
$B(P\|P^{*})=\sum_{i}\left(\frac{p_{i}}{p_{i}^{*}}-\log\frac{p_{i}}{p_{i}^{*}}-1\right)$.
$\ \ \ \square$
These divergences violate the requirement: due to the Markov process
distributions always monotonically approach equilibrium. (Nevertheless, among
the Bregman divergences there exists a universal Lyapunov function for Markov
chains, the Kulback–Leibler divergence.)
We place the proof of Theorem 1 in Appendix.
Remark. If we relax the requirement of smoothness and consider in conditions
of Theorem 1 just continuous functions, then we have to add to the answer the
limit divergences,
$H_{\rm CR\
\infty}(P\|P^{*})=\max_{i}\left\\{\frac{p_{i}}{p_{i}^{*}}\right\\}-1\ ;$
$H_{\rm CR\
-\infty}(P\|P^{*})=\max_{i}\left\\{\frac{p_{i}^{*}}{p_{i}}\right\\}-1$
## 5 Markov Order
### 5.1 Entropy: a Function or an Order?
Theorem 1 gives us all of the divergences for which (i) the Markov chains
monotonically approach their equilibrium, (ii) the level sets are the same as
for a separable (sum over states) divergence and (iii) the level sets are the
same as for a divergence which is additive with respect to union of
independent subsystems.
We operate with the level sets and their orders, compare where the divergence
is larger (for monotonicity of the Markov chains evolution), but the values of
entropy are not important by themselves. We are interested in the following
order: $P$ precedes $Q$ with respect to the divergence $H_{\ldots}(P\|P^{*})$
if there exists such a continuous curve $P(t)$ ($t\in[0,1]$) that $P(0)=P$,
$P(1)=Q$ and the function $H(t)=H_{\ldots}(P(t)\|P^{*})$ monotonically
decreases on the interval $t\in[0,1]$. This property is invariant with respect
to a monotonic (increasing) transformation of the divergence. Such a
transformation does not change the conditional minimizers or maximizers of the
divergence.
There exists one important property that is not invariant with respect to
monotonic transformations. The increasing function $F(H)$ of a convex function
$H(P)$ is not obligatorily a convex function. Nevertheless, the sublevel sets
given by inequalities $H(P)\leq a$ coincide with the sublevel sets
$F(H(P))\leq F(a)$. Hence, sublevel sets for $F(H(P))$ remain convex.
The Jensen inequality
$H(\theta P+(1-\theta)Q)\leq\theta H(P)+(1-\theta)H(Q)$
($\theta\in[0,1]$) is not invariant with respect to monotonic transformations.
Instead of them, there appears the max form analogue of the Jensen inequality:
$H(\theta P+(1-\theta)Q)\leq\max\\{H(P),H(Q)\\}\ ,\ \ \theta\in[0,1]$ (31)
This inequality is invariant with respect to monotonically increasing
transformations and it is equivalent to convexity of sublevel sets.
###### Proposition 1.
All sublevel sets of a function $H$ on a convex set $V$ are convex if and only
if for any two points $P,Q\in V$ and every $\theta\in[0,1]$ the inequality
(31) holds. $\square$
It seems very natural to consider divergences as orders on distribution
spaces, and discuss only properties which are invariant with respect to
monotonic transformations. From this point of view, the CR family appears
absolutely naturally from the additivity (ii) and the “sum over states” (iii)
axioms, as well as the convex combination $\beta
D_{\mathrm{KL}}(P\|P^{*})+(1-\beta)D_{\mathrm{KL}}(P^{*}\|P)$
($\alpha\in[0,1]$), and in the above property context there are no other
smooth divergences.
### 5.2 Description of Markov Order
The CR family and the convex combinations of Shannon’s and Burg relative
entropies are distinguished families of divergences. Apart from them there are
many various “divergences”, and even the Csiszár–Morimoto functions (6) do not
include all used possibilities. Of course, most users prefer to have an
unambiguous choice of entropy: it would be nice to have “the best entropy” for
any class of problems. But from some point of view, ambiguity of the entropy
choice is unavoidable. In this section we will explain why the choice of
entropy is necessarily non unique and demonstrate that for many MaxEnt
problems the natural solution is not a fixed distribution, but a well defined
set of distributions.
The most standard use of divergence in many application is as follows:
1. 1.
On a given space of states an “equilibrium distribution” $P^{*}$ is given. If
we deal with the probability distribution in real kinetic processes then it
means that without any additional restriction the current distribution will
relax to $P^{*}$. In that sense, $P^{*}$ is the most disordered distribution.
On the other hand, $P^{*}$ may be considered as the “most disordered”
distribution with respect to some a priori information.
2. 2.
We do not know the current distribution $P$, but we do know some linear
functionals, the moments $u(P)$.
3. 3.
We do not want to introduce any subjective arbitrariness in the estimation of
$P$ and define it as the “most disordered” distribution for given value
$u(P)=U$ and equilibrium $P^{*}$. That is, we define $P$ as solution to the
problem:
$H_{\ldots}(P\|P^{*})\to\min\ \ {\rm subject\ to}\ \ u(P)=U$ (32)
Without the condition $u(P)=U$ the solution should be simply $P^{*}$.
Now we have too many entropies and do not know what is the optimal choice of
$H_{\ldots}$ and what should be the optimal estimate of $P$. In this case the
proper question may be: which $P$ could not be such an optimal estimate? We
can answer the exclusion question. Let for a given $P^{0}$ the condition hold,
$u(P^{0})=U$. If there exists a Markov process with equilibrium $P^{*}$ such
that at point $P^{0}$ due to the Kolmogorov equation (22)
$\frac{{\mathrm{d}}P}{{\mathrm{d}}t}\neq 0\ \ {\rm and}\ \
\frac{{\mathrm{d}}(u(P))}{{\mathrm{d}}t}=0\ $
then $P^{0}$ cannot be the optimal estimate of the distribution $P$ under
condition $u(P)=U$.
The motivation of this approach is simple: any Markov process with equilibrium
$P^{*}$ increases disorder and brings the system “nearer” to the equilibrium
$P^{*}$. If at $P^{0}$ it is possible to move along the condition plane
towards the more disordered distribution then $P^{0}$ cannot be considered as
an extremely disordered distribution on this plane. On the other hand, we can
consider $P^{0}$ as a possible extremely disordered distribution on the
condition plane, if for any Markov process with equilibrium $P^{*}$ the
solution of the Kolmogorov equation (22) $P(t)$ with initial condition
$P(0)=P^{0}$ has no points on the plane $u(P)=U$ for $t>0$.
Markov process here is considered as a “randomization”. Any set $C$ of
distributions can be divided in two parts: the distributions which retain in
$C$ after some non-trivial randomization and the distributions which leave $C$
after any non-trivial randomization. The last are the maximally random
elements of $C$: they cannot become more random and retain in $C$. Conditional
minimizers of relative entropies $H_{h}(P\|P^{*})$ in $C$ are maximally random
in that sense.
There are too many functions $H_{h}(P\|P^{*})$ for effective description of
all their conditional minimizers. Nevertheless, we can describe the maximally
random distributions directly, by analysis of Markov processes.
To analyze these properties more precisely, we need some formal definitions.
###### Definition 1.
(Markov preorder). If for distributions $P^{0}$ and $P^{1}$ there exists such
a Markov process with equilibrium $P^{*}$ that for the solution of the
Kolmogorov equation with $P(0)=P^{0}$ we have $P(1)=P^{1}$ then we say that
$P^{0}$ and $P^{1}$ are connected by the Markov preorder with equilibrium
$P^{*}$ and use notation $P^{0}\succ^{0}_{P^{*}}P^{1}$.
###### Definition 2.
Markov order is the closed transitive closure of the Markov preorder. For the
Markov order with equilibrium $P^{*}$ we use notation
$P^{0}\succ_{P^{*}}P^{1}$.
For a given $P^{*}=(p^{*}_{i})$ and a distribution $P=(p_{i})$ the set of all
vectors $v$ with coordinates
$v_{i}=\sum_{j,\,j\neq
i}q_{ij}p^{*}_{j}\left(\frac{p_{j}}{p_{j}^{*}}-\frac{p_{i}}{p_{i}^{*}}\right)$
where $p_{i}^{*}$ and $q_{ij}\geq 0$ are connected by identity (23) is a
closed convex cone. This is a cone of all possible time derivatives of the
probability distribution at point $P$ for Markov processes with equilibrium
$P^{*}=(p^{*}_{i})$. For this cone, we use notation $\mathbf{Q}_{(P,P^{*})}$
###### Definition 3.
For each distribution $P$ and a $n$-dimensional vector $\Delta$ we say that
$\Delta<_{(P,P^{*})}0$ if $\Delta\in\mathbf{Q}_{(P,P^{*})}$. This is the local
Markov order.
###### Proposition 2.
$\mathbf{Q}_{(P,P^{*})}$ is a proper cone, _i.e._ , it does not include any
straight line.
###### Proof.
To prove this proposition its is sufficient to analyze the formula for entropy
production (for example, in form (24)) and mention that for strictly convex
$h$ (for example, for traditional $x\ln x$ or $(x-1)^{2}/2$)
${\mathrm{d}}H_{h}/{\mathrm{d}}t=0$ if and only if
${\mathrm{d}}P/{\mathrm{d}}t=0$. If the cone $\mathbf{Q}_{(P,P^{*})}$ includes
both vectors $x$ and $-x$ ($x\neq 0$ it means that there exist Markov chains
with equilibrium $P^{*}$ and with opposite time derivatives at point $P$. Due
to the positivity of entropy production (24) this is impossible. ∎
The connection between the local Markov order and the Markov order gives the
following proposition, which immediately follows from definitions.
###### Proposition 3.
$P^{0}\succ_{P^{*}}P^{1}$ if and only if there exists such a continuous almost
everywhere differentiable curve $P(t)$ in the simplex of probability
distribution that $P(0)=P^{0}$, $P(1)=P^{1}$ and for all $t\in[0,1]$, where
$P(t)$ is differentiable,
$\frac{{\mathrm{d}}P(t)}{{\mathrm{d}}t}\in\mathbf{Q}_{(P(t),P^{*})}\;\;\;\;\;\;\square$
(33)
For our purposes, the following estimate of the Markov order through the local
Markov order is important.
###### Proposition 4.
If $P^{0}\succ_{P^{*}}P^{1}$ then $P^{0}>_{(P^{0},P^{*})}P^{1}$, _i.e._ ,
$P^{1}-P^{0}\in\mathbf{Q}_{(P,P^{*})}$.
This proposition follows from the characterization of the local order and
detailed description of the cone $\mathbf{Q}_{(P(t),P^{*})}$ (Theorem 2
below).
Let us recall that a convex pointed cone is a convex envelope of its extreme
rays. A ray with directing vector $x$ is a set of points $\lambda x$
($\lambda\geq 0$). We say that $l$ is an extreme ray of $\mathbf{Q}$ if for
any $u\in l$ and any $x,y\in\mathbf{Q}$, whenever $u=(x+y)/2$, we must have
$x,y\in l$. To characterize the extreme rays of the cones of the local Markov
order $\mathbf{Q}_{(P,P^{*})}$ we need a graph representation of the Markov
chains. We use the notation $A_{i}$ for states (vertices), and designate
transition from state $A_{i}$ to state $A_{j}$ by an arrow (edge) $A_{i}\to
A_{j}$. This transition has its transition intensity $q_{ji}$ (the coefficient
in the Kolmogorov equation (21)).
###### Lemma 2.
Any extreme ray of the cone $\mathbf{Q}_{(P,P^{*})}$ corresponds to a Markov
process which transition graph is a simple cycle
$A_{i_{1}}\to A_{i_{2}}\to\ldots A_{i_{k}}\to A_{i_{1}}$
where $k\leq n$, all the indices $i_{1},\ldots i_{k}$ are different, and
transition intensities for a directing vector of such an extreme ray
$q_{i_{j+1}\ i_{j}}$ may be selected as $1/p_{i_{j}}^{*}$:
$q_{i_{j+1}\ i_{j}}=\frac{1}{p_{i_{j}}^{*}}$ (34)
(here we use the standard convention that for a cycle $q_{i_{k+1}\
i_{k}}=q_{i_{1}\ i_{k}}$).
###### Proof.
First of all, let us mention that if for three vectors
$x,y,u\in\mathbf{Q}_{(P,P^{*})}$ we have $u=(x+y)/2$ then the set of
transitions with non-zero intensities for corresponding Markov processes for
$x$ and $y$ are included in this set for $u$ (because negative intensities are
impossible). Secondly, just by calculation of the free variables in the
equations (23) (with additional condition) we find that the the amount of non-
zero intensities for a transition scheme which represents an extreme ray
should be equal to the amount of states included in the transition scheme.
Finally, there is only one scheme with $k$ vertices, $k$ edges and a positive
equilibrium, a simple oriented cycle.∎
###### Theorem 2.
Any extreme ray of the cone $\mathbf{Q}_{(P,P^{*})}$ corresponds to a Markov
process whose transition graph is a simple cycle of the length 2:
$A_{i}\rightleftarrows A_{j}$. A transition intensities $q_{ij},\ q_{ji}$ for
a directing vector of such an extreme ray may be selected as
$q_{ij}=\frac{1}{p^{*}_{j}}\ ,\ \ q_{ji}=\frac{1}{p^{*}_{i}}$ (35)
###### Proof.
Due to Lemma 2, it is sufficient to prove that for any distribution $P$ the
right hand side of the Kolmogorov equation (22) for a simple cycle with
transition intensities (34) is a conic combination (the combination with non-
negative real coefficients) of the right hand sides of this equation for
simple cycles of the length 2 at the same point $P$. Let us prove this by
induction. For the cycle length 2 it is trivially true. Let this hold for the
cycle lengths $2,\ldots n-1$. For a cycle of length $n$, $A_{i_{1}}\to
A_{i_{2}}\to\ldots A_{i_{k}}\to A_{i_{1}}$, with transition intensities given
by (34) the right hand side of the Kolmogorov equation is the vector $v$ with
coordinates
$v_{i_{j}}=\frac{p_{i_{j-1}}}{p^{*}_{i_{j-1}}}-\frac{p_{i_{j}}}{p^{*}_{i_{j}}}$
(under the standard convention regarding cyclic order). Other coordinates of
$v$ are zeros. Let us find the minimal value of ${p_{i_{j}}}/{p^{*}_{i_{j}}}$
and rearrange the indices by a cyclic permutation to put this minimum in the
first place:
$\min_{j}\left\\{\frac{p_{i_{j}}}{p^{*}_{i_{j}}}\right\\}=\frac{p_{i_{1}}}{p^{*}_{i_{1}}}$
The vector $v$ is a sum of two vectors: a directing vector for the cycle
$A_{i_{2}}\to\ldots A_{i_{k}}\to A_{i_{2}}$ of the length $n-1$ with
transition intensities given by formula (34) (under the standard convention
about the cyclic order for this cycle) and a vector
$\frac{\frac{p_{i_{n}}}{p_{i_{n}}^{*}}-\frac{p_{i_{1}}}{p_{i_{1}}^{*}}}{\frac{p_{i_{2}}}{p_{i_{2}}^{*}}-\frac{p_{i_{1}}}{p_{i_{1}}^{*}}}v^{2}$
where $v^{2}$ is the directing vector for a cycle of length 2,
$A_{i_{1}}\rightleftarrows A_{i_{2}}$ which can have only two non-zero
coordinates:
$v^{2}_{i_{1}}=\frac{p_{i_{2}}}{p^{*}_{i_{2}}}-\frac{p_{i_{1}}}{p^{*}_{i_{1}}}=-v^{2}_{i_{2}}$
The coefficient in front of $v^{2}$ is positive because
${p_{i_{1}}}/{p^{*}_{i_{1}}}$ is the minimal value of
${p_{i_{j}}}{p^{*}_{i_{j}}}$. A case when
${p_{i_{1}}}/{p^{*}_{i_{1}}}={p_{i_{2}}}/{p^{*}_{i_{2}}}$ does not need
special attention because it is equivalent to the shorter cycle $A_{i_{1}}\to
A_{i_{3}}\to\ldots A_{i_{k}}\to A_{i_{1}}$ ($A_{i_{2}}$ could be omitted). A
conic combination of conic combinations is a conic combination again.∎
It is quite surprising that the local Markov order and, hence, the Markov
order also are generated by the reversible Markov chains which satisfy the
detailed balance principle. We did not include any reversibility assumptions,
and studied the general Markov chains. Nevertheless, for the study of orders,
the system of cycles of length 2 all of which have the same equilibrium is
sufficient.
### 5.3 Combinatorics of Local Markov Order
Let us describe the local Markov order in more detail. First of all, we
represent kinetics of the reversible Markov chains. For each pair
$A_{i},A_{j}$ ($i\neq j$) we select an arbitrary order in the pair and write
the correspondent cycle of the length 2 in the form $A_{i}\leftrightarrows
A_{j}$. For this cycle we introduce the directing vector $\gamma^{ij}$ with
coordinates
$\gamma^{ij}_{k}=-\delta_{ik}+\delta_{jk}$ (36)
where $\delta_{ik}$ is the Kronecker delta. This vector has the $i$th
coordinate $-1$, the $j$th coordinate $1$ and other coordinates are zero.
Vectors $\gamma^{ij}$ are parallel to the edges of the standard simplex in
$R^{n}$. They are antisymmetric in their indexes: $\gamma^{ij}=-\gamma^{ji}$.
We can rewrite the Kolmogorov equation in the form
$\frac{{\mathrm{d}}P}{{\mathrm{d}}t}=\sum_{{\rm pairs}\ ij}\gamma^{ij}w_{ji}$
(37)
where $i\neq j$, each pair is included in the sum only once (in the
preselected order of $i,j$) and
$w_{ji}=r_{ji}\left(\frac{p_{i}}{p_{i}^{*}}-\frac{p_{j}}{p_{j}^{*}}\right)$
The coefficient $r_{ji}\geq 0$ satisfies the detailed balance principle:
$r_{ji}=q_{ji}p_{i}^{*}=q_{ij}p_{j}^{*}=r_{ij}$
We use the three-value sign function:
${\rm sign}x=\left\\{\begin{array}[]{ll}-1,&{\rm if}\ x<0;\\\ 0,&{\rm if}\
x=0;\\\ 1,&{\rm if}\ x>0\end{array}\right.$ (38)
With this function we can rewrite Equation (37) again as follows:
$\frac{{\mathrm{d}}P}{{\mathrm{d}}t}=\sum_{{\rm pairs}\ ij,\ r_{ji}\neq
0}r_{ji}\gamma^{ij}{\rm
sign}\left(\frac{p_{i}}{p_{i}^{*}}-\frac{p_{j}}{p_{j}^{*}}\right)\left|\frac{p_{i}}{p_{i}^{*}}-\frac{p_{j}}{p_{j}^{*}}\right|$
(39)
The non-zero coefficients $r_{ji}$ may be arbitrary positive numbers.
Therefore, using Theorem 2, we immediately find that the cone of the local
Markov order at point $P$ is
$\mathbf{Q}_{(P,P^{*})}={\rm cone}\left\\{\gamma^{ij}{\rm
sign}\left.\left(\frac{p_{i}}{p_{i}^{*}}-\frac{p_{j}}{p_{j}^{*}}\right)\
\right|\ r_{ji}>0\right\\}$ (40)
where cone$\\{\\}$ stands for the conic hull.
The number ${\rm
sign}\left(\frac{p_{i}}{p_{i}^{*}}-\frac{p_{j}}{p_{j}^{*}}\right)$ is 1, when
$\frac{p_{i}}{p_{i}^{*}}>\frac{p_{j}}{p_{j}^{*}}$, $-1$, when
$\frac{p_{i}}{p_{i}^{*}}<\frac{p_{j}}{p_{j}^{*}}$ and 0, when
$\frac{p_{i}}{p_{i}^{*}}=\frac{p_{j}}{p_{j}^{*}}$. For a given $P^{*}$, the
standard simplex of distributions $P$ is divided by planes
$\frac{p_{i}}{p_{i}^{*}}=\frac{p_{j}}{p_{j}^{*}}$ into convex polyhedra where
functions ${\rm
sign}\left(\frac{p_{i}}{p_{i}^{*}}-\frac{p_{j}}{p_{j}^{*}}\right)$ are
constant. In these polyhedra the cone of the local Markov order (40)
$\mathbf{Q}_{(P,P^{*})}$ is also constant. Let us call these polyhedra
compartments.
Figure 1: Compartments $\mathcal{C}_{\sigma}$, corresponding cones
$\mathbf{Q}_{\sigma}$ (the angles) and all tableaus $\sigma$ for the Markov
chain with three states (the choice of equilibrium ($p_{i}^{*}=1/3$), does not
affect combinatorics and topology of tableaus, compartments and cones).
In Figure 5.3 we represent compartments and cones of the local Markov order
for the Markov chains with three states, $A_{1,2,3}$. The reversible Markov
chain consists of three reversible transitions $A_{1}\leftrightarrows
A_{2}\leftrightarrows A_{3}\leftrightarrows A_{1}$ with corresponding
directing vectors $\gamma^{12}=(-1,1,0)^{\top}$;
$\gamma^{23}=(0,-1,1)^{\top}$; $\gamma^{31}=(1,0,-1)^{\top}$. The topology of
the partitioning of the standard simplex into compartments and the possible
values of the cone $\mathbf{Q}_{(P,P^{*})}$ do not depend on the position of
the equilibrium distribution $P^{*}$.
Let us describe all possible compartments and the correspondent local Markov
order cones. For every natural number $k\leq n-1$ the $k$-dimensional
compartments are numerated by surjective functions
$\sigma:\\{1,2,\ldots,n\\}\to\\{1,2,\ldots,k+1\\}$. Such a function defines
the partial ordering of quantities $\frac{p_{j}}{p_{j}^{*}}$ inside the
compartment:
$\frac{p_{i}}{p_{i}^{*}}>\frac{p_{j}}{p_{j}^{*}}\;\;{\rm
if}\;\;\sigma(i)<\sigma(j);\;\;\;\frac{p_{i}}{p_{i}^{*}}=\frac{p_{j}}{p_{j}^{*}}\;\;{\rm
if}\;\;\sigma(i)=\sigma(j)$ (41)
Let us use for the correspondent compartment notation $\mathcal{C}_{\sigma}$
and for the Local Markov order cone $Q_{\sigma}$. Let $k_{i}$ be a number of
elements in preimage of $i$ ($i=1,\ldots,k$): $k_{i}=|\\{j\ |\
\sigma(j)=i\\}|$. It is convenient to represent surjection $\sigma$ as a
tableau with $k$ rows and $k_{i}$ cells in the $i$th row filled by numbers
from $\\{1,2,\ldots,n\\}$. First of all, let us draw diagram, that is a finite
collection of cells arranged in left-justified rows. The $i$th row has $k_{i}$
cells. A tableau is obtained by filling cells with numbers
$\\{1,2,\ldots,n\\}$. Preimages of $i$ are located in the $i$th row. The
entries in each row are increasing. (This is convenient to avoid ambiguity of
the representation of the surjection $\sigma$ by the diagram.) Let us use for
tableaus the same notation as for the corresponding surjections.
Let a tableau $A$ have $k$ rows. We say that a tableau $B$ follows $A$ (and
use notation $A\to B$) if $B$ has $k-1$ rows and $B$ can be produced from $A$
by joining of two neighboring rows in $A$ (with ordering the numbers in the
joined row). For the transitive closure of the relation $\to$ we use notation
$\Rrightarrow$.
###### Proposition 5.
$r\partial
Q_{\sigma}=\bigcup_{\sigma\Rrightarrow\varsigma}Q_{\varsigma}\;\;\;\;\;\;\square$
Here $r\partial U$ stands for the “relative boundary” of a set $U$ in the
minimal linear manifold which includes $U$.
The following Proposition characterizes the local order cone through the
surjection $\sigma$. It is sufficient to use in definition of $Q_{\sigma}$
(40) vectors $\gamma^{ij}$ (36) with $i$ and $j$ from the neighbor rows of the
diagram (see Figure 5.3).
###### Proposition 6.
For a given surjection $\sigma$ compartment $\mathcal{C}_{\sigma}$ and cone
$Q_{\sigma}$ have the following description:
$\mathcal{C}_{\sigma}=\left\\{P\ |\
\frac{p_{i}}{p^{*}_{i}}=\frac{p_{j}}{p^{*}_{j}}\;\;{\rm
for}\;\;\sigma(i)=\sigma(j)\;\;{\rm
and}\;\;\frac{p_{i}}{p^{*}_{i}}>\frac{p_{j}}{p^{*}_{j}}\;\;{\rm
for}\;\;\sigma(j)=\sigma(i)+1\right\\}$ (42) $Q_{\sigma}={\rm
cone}\\{\gamma^{ij}\ |\ \sigma(j)=\sigma(i)+1\\}\;\;\;\;\;\;\square$ (43)
Compartment $\mathcal{C}_{\sigma}$ is defined by equalities
$\frac{p_{i}}{p^{*}_{i}}=\frac{p_{j}}{p^{*}_{j}}$ where $i,j$ belong to one
row of the tableau $\sigma$ and inequalities
$\frac{p_{i}}{p^{*}_{i}}>\frac{p_{j}}{p^{*}_{j}}$ where $j$ is situated in a
row one step down from $i$ in the tableau ($\sigma(j)=\sigma(i)+1$). Cone
$Q_{\sigma}$ is a conic hull of $\sum_{i=1}^{k-1}k_{i}k_{i+1}$ vectors
$\gamma^{ij}$. For these vectors, $j$ is situated in a row one step down from
$i$ in the tableau. Extreme rays of $Q_{\sigma}$ are products of the positive
real half-line on vectors $\gamma^{ij}$ (43).
Each compartment has the lateral faces and the base. We call the face a
lateral face, if its closure includes the equilibrium $P^{*}$. The base of the
compartment belongs to a border of the standard simplex of probability
distributions.
To enumerate all the lateral faces of a $k$-dimensional compartment
$\mathcal{C}_{\sigma}$ of codimension $s$ (in $\mathcal{C}_{\sigma}$) we have
to take all subsets with $s$ elements in $\\{1,2,\ldots,k\\}$. For any such a
subset $J$ the correspondent $k-s$-dimensional lateral face is given by
additional equalities $\frac{p_{i}}{p_{i}^{*}}=\frac{p_{j}}{p_{j}^{*}}$ for
$\sigma(j)=\sigma(i)+1$, $i\in J$.
###### Proposition 7.
All $k-s$-dimensional lateral faces of a $k$-dimensional compartment
$\mathcal{C}_{\sigma}$ are in bijective correspondence with the $s$-element
subsets $J\subset\\{1,2,\ldots,k\\}$. For each $J$ the correspondent lateral
face is given in $\mathcal{C}_{\sigma}$ by equations
$\frac{p_{i}}{p_{i}^{*}}=\frac{p_{j}}{p_{j}^{*}}\;\;{\rm for\,all}\;\;i\in
J\;\;{\rm and}\;\;\sigma(j)=\sigma(i)+1\;\;\;\;\;\;\square$ (44)
The 1-dimensional lateral faces (extreme rays) of compartment
$\mathcal{C}_{\sigma}$ are given by selection of one number from
$\\{1,2,\ldots,k\\}$ (this number is the complement of $J$). For this number
$r$, the correspondent 1-dimensional face is a set parameterized by a positive
number $a\in]1,a_{r}]$, $a_{r}=1/\sum_{\sigma(i)\leq r}p^{*}_{i}$:
$\begin{split}\frac{p_{i}}{p_{i}^{*}}=a,\;\;{\rm for}\;\;\sigma(i)\leq
r\,;\;\;\frac{p_{i}}{p_{i}^{*}}=b,\;\;{\rm for}\;\;\sigma(i)>r\,;\\\ a>1>b\geq
0,\;a\sum_{i,\,\sigma(i)\leq
r}p^{*}_{i}+b\sum_{i,\,\sigma(i)>r}p^{*}_{i}=1\end{split}$ (45)
The compartment $\mathcal{C}_{\sigma}$ is the interior of the $k$-dimensional
simplex with vertices $P^{*}$ and $v_{r}$ ($r=1,2,\ldots k$). The vertex
$v_{r}$ is the intersection of the correspondent extreme ray (45) with the
border of the standard simplex of probability distributions: $P=v_{r}$ if
$p_{i}=p_{i}^{*}a_{r},\;\;{\rm for}\;\;\sigma(i)\leq r;\;\;p_{i}=0\;\;{\rm
for}\;\;\sigma(i)>r$ (46)
The base of the compartment $\mathcal{C}_{\sigma}$ is a $k-1$-dimensional
simplex with vertices $v_{r}$ ($r=1,2,\ldots k$).
It is necessary to stress that we use the reversible Markov chains for
construction of the general Markov order due to Theorem 2.
## 6 The “Most Random” and Conditionally Extreme Distributions
### 6.1 Conditionally Extreme Distributions in Markov Order
The Markov order can be used to reduce the uncertainty in the standard
settings. Let the plane $L$ of the known values of some moments be given:
$u^{i}(P)=U_{i}$ on $L$. Assume also that the “maximally disordered”
distribution (equilibrium) $P^{*}$ is known and we assume that the probability
distribution is $P^{*}$ if there is no restrictions. Then, the standard way to
evaluate $P$ for given moment conditions $u^{i}(P)=U_{i}$ is known: just to
minimize $H_{\ldots}(P\|P^{*})$ under these conditions. For the Markov order
we also can define the conditionally extreme points on $L$.
###### Definition 4.
Let $L$ be an affine subspace of $\mathbf{R}^{n}$, $\Sigma_{n}$ be a standard
simplex in $\mathbf{R}^{n}$. A probability distribution $P\in L\cap\Sigma_{n}$
is a conditionally extreme point of the Markov order on $L$ if
$(P+\mathbf{Q}_{(P,P^{*})})\cap L=\\{P\\}$
It is useful to compare this definition to the condition of the extremum of a
differentiable function $H$ on $L$: ${\rm grad}H\bot L$.
Figure 2: If the moments are just some of $p_{i}$ then all points of
conditionally minimal divergence are the same for all the main divergences and
coincide with the unique conditionally extreme point of the Markov order
(example for the Markov chain with three states, symmetric equilibrium
($p_{i}^{*}=1/3$)) and the moment plane $p_{2}=$const.
First of all, it is obvious that in the case when all the moments $u^{i}(P)$
are just some of the values $p_{i}$, then there exists only one extreme point
of the Markov order on $L$, and this point is, at the same time, the
conditional minimum on $L$ of all Csiszár–Morimoto functions $H_{h}(P)$ (6)
(see, for example, Figure 6.1). This situation is unstable, and for a small
perturbation of $L$ the set of extreme points of the Markov order on $L$
includes the intersection of $L$ with one of compartments (Figure 6.1a). For
the Markov chains with three states, each point of this intersection is a
conditional minimizer of one of the CR divergences (see Fig. 6.1a). Such a
situation persists for all $L$ in general positions (Figure 6.1b). The extreme
points of the family $\beta
D_{\mathrm{KL}}(P\|P^{*})+(1-\beta)D_{\mathrm{KL}}(P^{*}\|P)$ form an interval
which is strictly inside the interval of the extreme points of the Markov
order on $L$. For higher dimensions of $L\cap\Sigma_{n}$ the Markov order on
$L$ also includes the intersection of $L$ with some compartments, however the
conditional minimizers of the CR divergences form a curve there, and extreme
points of the family $\beta
D_{\mathrm{KL}}(P\|P^{*})+(1-\beta)D_{\mathrm{KL}}(P^{*}\|P)$ on $L$ form
another curve. These two curves intersect at two points ($\lambda=0,-1$),
which correspond to the BGS and Burg relative entropies.
Figure 3: The set of conditionally extreme points of the Markov order on the
moment plane in two general positions. For the main divergences the points of
conditionally minimal divergence are distributed in this set. For several of
the most important divergences these minimizers are pointed out. In this
simple example each extreme point of the Markov order is at the same time a
minimizer of one of the $H_{\rm CR\ \lambda}$ ($\lambda\in]-\infty,+\infty[$)
(examples for the Markov chain with three states, symmetric equilibrium
($p_{i}^{*}=1/3$)).
a) b)
### 6.2 How to Find the Most Random Distributions?
Let the plane $L$ of the known values of some moments be given:
$u^{i}(P)=\sum_{j}u^{i}_{j}p_{j}=U_{i}$ ($i=1,\ldots m$) on $L$. For a given
divergence $H(P\|P^{*})$ we are looking for a conditional minimizer $P$:
$H(P\|P^{*})\to\min\ \ {\rm subject\ to}\ \ u^{i}(P)=U_{i}\\\ (i=1,\ldots m)$
(47)
We can assume that $H(P\|P^{*})$ is convex. Moreover, usually it is one of the
Csiszár–Morimoto
functions (6). This is very convenient for numerical minimization because the
matrix of second derivatives is diagonal. Let us introduce the Lagrange
multipliers $\mu_{i}$ ($i=1,\ldots m$) and write the system of equations
($\mu_{0}$ is the Lagrange multiplier for the total probability identity
$\sum_{j}p_{j}=1$ :
$\begin{split}&\frac{\partial H}{\partial
p_{j}}=\mu_{0}+\sum_{i=1}^{m}\mu_{i}u^{i}_{j}\ ;\\\
&\sum_{j=1}^{n}u^{i}_{j}p_{j}=U_{i}\ ;\\\ &\sum_{j=1}^{n}p_{j}=1\end{split}$
(48)
Here we have $n+m+1$ equations for $n+m+1$ unknown variables ($p_{j}$,
$\mu_{i}$, $\mu_{0}$).
Usually $H$ is a convex function with a diagonal matrix of second variables
and the method of choice for solution of this equation (48) is the Newton
method. On the $l+1$st iteration to find $P^{l+1}=P^{l}+\Delta P$ we have to
solve the following system of linear equations
$\begin{split}&\sum_{s=1}^{n}\left.\frac{\partial^{2}H}{\partial p_{j}\partial
p_{s}}\right|_{P=P^{l}}\Delta
p_{s}=\mu_{0}+\sum_{i=1}^{m}\mu_{i}u^{i}_{j}-\left.\frac{\partial H}{\partial
p_{j}}\right|_{P=P^{l}}\ ;\\\ &\sum_{j=1}^{n}u^{i}_{j}\Delta p_{j}=0\ ;\\\
&\sum_{j=1}^{n}\Delta p_{j}=0\end{split}$ (49)
For a diagonal matrix of the second derivatives the first $n$ equations can be
explicitly resolved. If for the solution of this system (49) the positivity
condition $p_{j}^{l}+\Delta p_{j}>0$ does not hold (for some of $j$) then we
should decrease the step, for example by multiplication $\Delta
P:=\theta\Delta P$, where
$0<\theta<\min_{p_{i}^{l}+\Delta p_{i}<0}\frac{p_{i}^{l}}{|\Delta p_{i}|}$
For initial approximation we can take any positive normalized distribution
which satisfies the conditions $u^{i}(P)=U_{i}$ ($i=1,\ldots m$).
For the Markov orders the set of conditionally extreme distributions consists
of intersections of $L$ with compartments.
Here we find this set for one moment condition of the form
$u(P)=\sum_{j}u_{j}p_{j}=U$. First of all, assume that $U\neq U^{*}$, where
$U^{*}=u(P^{*})=\sum_{j}u_{j}p^{*}_{j}$ (if $U=U^{*}$ then equilibrium is the
single conditionally extreme distribution). In this case, the set of
conditionally extreme distributions is the intersection of the condition
hyperplane with the closure of one compartment and can be described by the
following system of equations and inequalities (under standard requirements
$p_{i}\geq 0$, $\sum_{i}p_{i}=1$ ):
$\begin{split}&\sum_{j}u_{j}p_{j}=U;\\\
&\frac{p_{i}}{p_{i}^{*}}\geq\frac{p_{j}}{p_{j}^{*}}\;\;{\rm
if}\;\;u_{i}(U-U^{*})\geq u_{j}(U-U^{*})\end{split}$ (50)
(hence, $\frac{p_{i}}{p_{i}^{*}}=\frac{p_{j}}{p_{j}^{*}}$ if $u_{i}=u_{j}$).
To find this solution it is sufficient to study dynamics of $u(P)$ due to
equations (37) and to compare it with dynamics of $u(P)$ due to a model system
$\dot{P}=P^{*}-P$. This model system is also a Markov chain and, therefore,
$P^{*}-P\in\mathbf{Q}_{(P,P^{*})}$. Equations and inequalities (50) mean that
the set of conditionally extreme distributions is the intersection of the
condition hyperplane with the closure of compartment $\mathcal{C}$. In
$\mathcal{C}$, numbers $\frac{p_{i}}{p_{i}^{*}}$ have the same order on the
real line as numbers $u_{i}(U-U^{*})$ have, these two tuples of numbers
correspond to the same tableau $\sigma$ and
$\mathcal{C}=\mathcal{C}_{\sigma}$.
For several linearly independent conditions there exists a condition plane
$L$:
$u^{i}(P)=\sum_{j}u^{i}_{j}p_{j}=U_{i}\;\;(i=1,\ldots m)$ (51)
Let us introduce the $m$-dimensional space $T$ with coordinates $u^{i}$.
Operator $u(P)=(u^{i}(P))$ maps the distribution space into $T$ and the affine
manifold $L$ (51) maps into a point with coordinates $u^{i}=U_{i}$.
If $P^{*}\in L$ then the problem is trivial and the only extreme distribution
of the Markov order on $L$ is $P^{*}$. Let us assume that $P^{*}\notin L$.
For each distribution $P\in L$ we can study the possible direction of motions
of projection distributions onto $T$ due to the Markov processes.
First of all, let us mention that if $u(\gamma^{ij})=0$ then the transitions
$A_{i}\leftrightarrows A_{j}$ move the distribution along $L$. Hence, for any
conditionally extreme distribution $P\in L$ this transition
$A_{i}\leftrightarrows A_{j}$ should be in equilibrium and the partial
equilibrium condition holds:
$\frac{p_{i}}{p_{i}^{*}}=\frac{p_{j}}{p_{j}^{*}}$.
Let us consider processes with $u(\gamma^{ij})\neq 0$. If there exists a
convex combination (39) of vectors $u(\gamma^{ij}){\rm
sign}\left(\frac{p_{i}}{p_{i}^{*}}-\frac{p_{j}}{p_{j}^{*}}\right)$
($u(\gamma^{ij})\neq 0$) that is equal to zero then $P$ cannot be an extreme
distribution of the Markov order on $L$.
These two conditions for vectors $\gamma^{ij}$ with $u(\gamma^{ij})=0$ and for
the set of vectors with non-zero projection on the condition space define the
extreme distributions of the Markov order on the condition plane $L$ for
several conditions.
## 7 Generalized Canonical Distribution
### 7.1 Reference Distributions for Main Divergences
A system with equilibrium $P^{*}$ is given and expected values of some
variables $u_{j}(P)=U_{j}$ are known. We need to find a distribution $P$ with
these values $u_{j}(P)=U_{j}$ and is “the closest” to the equilibrium
distribution under this condition.
This distribution parameterized through expectation values is often called the
reference distribution or generalized canonical distribution. After Gibbs and
Jaynes, the standard statement of this problem is an optimization problem:
$H(P\|P^{*})\to\min,\;\;u_{j}(P)=U_{j}$
for appropriate divergence $H(P\|P^{*})$. If the number of conditions is $m$
then this optimization problem can be often transformed into $m+1$ equations
with $m+1$ unknown Lagrange multipliers.
In this section, we study the problem of the generalized canonical
distributions for single condition $u(P)=\sum_{i=1}^{n}u_{i}p_{i}=U$, $U\neq
U^{*}$.
For the Csiszár–Morimoto functions $H_{h}(P\|P^{*})$
$\frac{\partial H_{h}}{\partial
p_{i}}=h^{\prime}\left(\frac{p_{i}}{p_{i}^{*}}\right)$ (52)
We assume that the function $h^{\prime}(x)$ has an inverse function $g$:
$g(h^{\prime}(x))=x$ for any $x\in]0,\infty[$. The method of Lagrange
multipliers gives for the generalized canonical distribution:
$\frac{\partial H_{h}}{\partial
p_{i}}=\mu_{0}\frac{\partial(\sum_{j=1}^{n}p_{j})}{\partial
p_{i}}+\mu\frac{\partial U}{\partial
p_{i}}\,,\,h^{\prime}\left(\frac{p_{i}}{p_{i}^{*}}\right)=\mu_{0}+\mu
u_{i},\,\sum_{i=1}^{n}p_{i}=1,\,\sum_{i=1}^{n}p_{i}u_{i}=U$ (53)
As a result, we get the final expression for the distribution
$p_{i}=p^{*}_{i}g(\mu_{0}+u_{i}\mu)$
and equations for Lagrange multipliers $\mu_{0}$ and $\mu$:
$\sum_{i=1}^{n}p^{*}_{i}g(\mu_{0}+u_{i}\mu)=1,\,\sum_{i=1}^{n}p^{*}_{i}g(\mu_{0}+u_{i}\mu)u_{i}=U$
(54)
If the image of $h^{\prime}(x)$ is the whole real line
($h^{\prime}(]0,\infty[)=R$) then for any real number $y$ the value $g(y)\geq
0$ is defined and there exist no problems about positivity of $p_{i}$ due to
(54).
For the BGS relative entropy $h^{\prime}(x)=\ln x$ (we use the normalized
$h(x)=x\ln x-(x-1)$ (19)). Therefore, $g(x)=\exp x$ and for the generalized
canonical distribution we get
$p_{i}=p^{*}_{i}{\rm e}^{\mu_{0}}{\rm e}^{u_{i}\mu},\,{\rm
e}^{-\mu_{0}}=\sum_{i=1}^{n}p^{*}_{i}{\rm
e}^{u_{i}\mu},\,`\sum_{i=1}^{n}p^{*}_{i}u_{i}{\rm
e}^{u_{i}\mu}=U\sum_{i=1}^{n}p^{*}_{i}{\rm e}^{u_{i}\mu}$ (55)
As a result, we get one equation for $\mu$ and an explicit expression for
$\mu_{0}$ through $\mu$.
These $\mu_{0}$ and $\mu$ have the opposite sign comparing to (5) just because
the formal difference between the entropy maximization and the relative
entropy minimization. Equation (55) is essentially the same as (5).
For the Burg entropy $h^{\prime}(x)=-\frac{1}{x}$, $g(x)=-\frac{1}{x}$ too and
$p_{i}=-\frac{p_{i}^{*}}{\mu_{0}+u_{i}\mu}$ (56)
For the Lagrange multipliers $\mu_{0},\mu$ we have a system of two algebraic
equations
$\sum_{i=1}^{n}\frac{p_{i}^{*}}{\mu_{0}+u_{i}\mu}=-1,\,\sum_{i=1}^{n}\frac{p_{i}^{*}u_{i}}{\mu_{0}+u_{i}\mu}=-U$
(57)
For the convex combination of the BGS and Burg entropies
$h^{\prime}(x)=\beta\ln x-\frac{1-\beta}{x}$ ($0<\beta<1$), and the function
$x=g(y)$ is a solution of a transcendent equation
$\beta\ln x-\frac{1-\beta}{x}=y$ (58)
Such a solution exists for all real $y$ because this $h^{\prime}(x)$ is a
(monotonic) bijection of $]0,\infty[$ on the real line.
Solution to Equation (58) can be represented through a special function, the
Lambert function Lambert . This function is a solution to the transcendent
equation
$w{\rm e}^{w}=z$
and is also known as $W$ function, $\Omega$ function or modified logarithm
${\rm lm}z$ ENTR2 . Below we use the main branch $w={\rm lm}z$ for which ${\rm
lm}z>0$ if $z>0$ and ${\rm lm}0=0$. Let us write (58) in the form
$\ln x-\frac{\delta}{x}=-\Lambda$ (59)
where $\delta=(1-\beta)/\beta$, $\Lambda=-y/\beta$. Then
$x={\rm e}^{-\Lambda}{\rm e}^{{\rm lm}(\delta{\rm e}^{\Lambda})}$
Another equivalent representation of the solution gives
$x=\frac{\delta}{{\rm lm}(\delta{\rm e}^{\Lambda})}$
Indeed, let us take $z=\delta/x$ and calculate exponent of both sides of (59).
After simple transformations, we obtain $z{\rm e}^{z}=\delta{\rm
e}^{\Lambda}$.
The identity ${\rm lm}a=\ln a-\ln{\rm lm}a$ is convenient for algebraic
operations with this function. Many other important properties are collected
in Lambert .
The generalized canonical distribution for the convex combination of the BGS
and Burg divergence is ENTR2
$p_{i}=p_{i}^{*}{\rm e}^{-\Lambda_{i}}{\rm e}^{{\rm lm}(\delta{\rm
e}^{\Lambda_{i}})}=\frac{\delta p^{*}_{i}}{{\rm lm}(\delta{\rm
e}^{\Lambda_{i}})}$ (60)
where $\Lambda_{i}=-\frac{1}{\beta}(\mu_{0}+u_{i}\mu)$,
$\delta=(1-\beta)/\beta$ and equations (54) hold for the Lagrange multipliers.
For small $1-\beta$ (small addition of the Burg entropy to the BGS entropy) we
have
$p_{i}=p_{i}^{*}\left({\rm
e}^{-\Lambda_{i}}+\frac{1-\beta}{\beta}-\frac{(1-\beta)^{2}}{2\beta^{2}}{\rm
e}^{\Lambda_{i}}\right)+o((1-\beta)^{2})$
For the CR family $h(x)=\frac{x(x^{\lambda}-1)}{\lambda(\lambda+1)}$,
$h^{\prime}(x)=\frac{(\lambda+1)x^{\lambda}-1}{\lambda(\lambda+1)}$,
$g(x)=(\frac{\lambda(\lambda+1)x+1}{(\lambda+1)})^{\frac{1}{\lambda}}$ and
$p_{i}=p_{i}^{*}\left(\frac{\lambda(\lambda+1)(\mu_{0}+u_{i}\mu)+1}{(\lambda+1)}\right)^{\frac{1}{\lambda}}$
(61)
For $\lambda=1$ (a quadratic divergence) we easily get linear equations and
explicit solutions for $\mu_{0}$ and $\mu$. If $\lambda=\frac{1}{2}$ then
equations for the Lagrange multipliers (54) become quadratic and also allow
explicit solution. The same is true for $\lambda=\frac{1}{3}$ and
$\frac{1}{4}$ but explicit solutions to the correspondent cubic or quartic
equations are too cumbersome.
We studied the generalized canonical distributions for one condition $u(P)=U$
and main families of entropies. For the BGS entropy, the method of Lagrange
multipliers gives one transcendent equation for the multiplier $\mu_{1}$ and
explicit expression for $\mu_{0}$ as a function of $\mu_{1}$ (55). In general,
for functions $H_{h}$, the method gives a system of two equations (54). For
the Burg entropy this is a system of algebraic equation (57). For a convex
combination of the BGS and the Burg entropies the expression for generalized
canonical distribution function includes the special Lambert function (60).
For the CR family the generalized canonical distribution is presented by
formula (61). for several values of $\lambda$ it can be represented in
explicit form. The Tsallis entropy family is a subset of the CR family (up to
constant multipliers).
### 7.2 Polyhedron of Generalized Canonical Distributions for the Markov
Order
The set of the most random distributions with respect to the Markov order
under given condition consists of those distributions which may be achieved by
randomization which has the given equilibrium distribution and does not
violate the condition.
In the previous section, this set was characterized for a single condition
$\sum_{i}p_{i}u_{i}=U$, $U\neq U^{*}$ by a system of inequalities and
equations (50). It is a polyhedron that is an intersection of the closure of
one compartment with the hyperplane of condition. Here we construct the dual
description of this polyhedron as a convex envelope of the set of extreme
points (vertices).
The Krein–Milman theorem gives general backgrounds of such a representation of
convex compact sets in locally convex topological vector spaces
EdwardsKreinMilman1995 : a compact convex set is the closed convex hull of its
extreme points. (An extreme point of a convex set $K$ is a point $x\in K$
which cannot be represented as an average $x=\frac{1}{2}(y+z)$ for $y,z\in K$,
$y,z\neq x$.)
Let us assume that there are $k+1\leq n$ different numbers in the set of
numbers $u_{i}(U-U^{*})$. There exists the unique surjection
$\sigma:\\{1,2,\ldots n\\}\to\\{1,2,\ldots k+1\\}$ with the following
properties: $\sigma(i)<\sigma(j)$ if and only if
$u_{i}(U-U^{*})>u_{j}(U-U^{*})$ (hence, $\sigma(i)=\sigma(j)$ if and only if
$u_{i}(U-U^{*})=u_{j}(U-U^{*})$). The polyhedron of generalized canonical
distributions is the intersection of the condition plane
$\sum_{i}p_{i}u_{i}=U$ with the closure of $\mathcal{C}_{\sigma}$.
This closure is a simplex with vertices $P^{*}$ and $v_{r}$ ($r=1,2,\ldots k$)
(46). The vertices of the intersection of this simplex with the condition
hyperplane belong to edges of the simplex, hence we can easily find all of
them: the edge $[x,y]$ has nonempty intersection with the condition hyperplane
if either $u(x)\geq U\&u(y)\leq U$ or $u(x)\leq U\&u(y)\geq U$. This
intersection is a single point $P$ if $u(x)\neq u(y)$:
$P=\lambda x+(1-\lambda)y,\;\;\lambda=\frac{u(y)-U}{u(y)-u(x)}$ (62)
If $u(x)=u(y)$ then the intersection is the whole edge, and the vertices are
$x$ and $y$.
For example, if $U$ is sufficiently close to $U^{*}$ then the intersection is
a simplex with $k$ vertices $w_{r}$ ($r=1,2,\ldots k$). Each $w_{r}$ is the
intersection of the edge $[P^{*},v_{r}]$ with the condition hyperplane.
Let us find these vertices explicitly. We have a system of two equations
$\begin{split}&a\sum_{i,\,\sigma(i)\leq
r}p^{*}_{i}+b\sum_{i,\,\sigma(i)>r}p^{*}_{i}=1\,;\\\ &a\sum_{i,\,\sigma(i)\leq
r}u_{i}p^{*}_{i}+b\sum_{i,\,\sigma(i)>r}u_{i}p^{*}_{i}=U\end{split}$ (63)
Position of the vertex $w_{r}$ on the edge $[P^{*},v_{r}]$ is given by the
following expressions
$\begin{split}&\frac{p_{i}}{p_{i}^{*}}=a,\;\;{\rm for}\;\;\sigma(i)\leq
r\,;\;\;\frac{p_{i}}{p_{i}^{*}}=b,\;\;{\rm for}\;\;\sigma(i)>r\\\
&a=1+\frac{(U-U^{*})\sum_{i,\,\sigma(i)>r}p^{*}_{i}}{\sum_{i,\,\sigma(i)>r}p^{*}_{i}\sum_{i,\,\sigma(i)\leq
r}u_{i}p^{*}_{i}-\sum_{i,\,\sigma(i)\leq
r}p^{*}_{i}\sum_{i,\,\sigma(i)>r}u_{i}p^{*}_{i}}\\\
&b=1-\frac{(U-U^{*})\sum_{i,\,\sigma(i)\leq
r}p^{*}_{i}}{\sum_{i,\,\sigma(i)>r}p^{*}_{i}\sum_{i,\,\sigma(i)\leq
r}u_{i}p^{*}_{i}-\sum_{i,\,\sigma(i)\leq
r}p^{*}_{i}\sum_{i,\,\sigma(i)>r}u_{i}p^{*}_{i}}\end{split}$ (64)
If $b\geq 0$ for all $r$ then the polyhedron of generalized canonical
distributions is a simplex with vertices $w_{r}$. If the solution becomes
negative for some $r$ then the set of vertices changes qualitatively and some
of them belong to the base of $\mathcal{C}_{\sigma}$. For example, in Figure
6.1a the interval of the generalized canonical distribution (1D polyhedron)
has vertices of two types: one belongs to the lateral face, another is
situated on the basement of the compartment. In Figure 6.1b both vertices
belong to the lateral faces.
Vertices $w_{r}$ on the edges $[P^{*},v_{r}]$ have very special structure: the
ratio $p_{i}/p_{i}^{*}$ can take for them only two values, it is either $a$ or
$b$.
Another form for representation of vertices $w_{r}$ (64) can be found as
follows. $w_{r}$ belongs to the edge $[P^{*},v_{r}]$, hence, $w_{r}=\lambda
P^{*}+(1-\lambda)v_{r}$ for some $\lambda\in[0,1]$. Equation for the value of
$\lambda$ follows from the condition $u(w_{r})=U$: $\lambda
U^{*}+(1-\lambda)u(v_{r})=U$. Hence, we can use (62) with $x=P^{*}$,
$y=v_{r}$.
For sufficiently large value of $U-U^{*}$ for some of these vertices $b$ loses
positivity, and instead of them the vertices on edges $[v_{r},v_{q}]$ (46)
appear.
There exists a vertex on the edge $[v_{r},v_{q}]$ if either $u(v_{r})\geq
U\&u(v_{q})\leq U$ or $u(v_{r})\leq U\&u(v_{q})\geq U$. If $u(v_{r})\neq
u(v_{q})$ then his vertex has the form $P=\lambda v_{r}+(1-\lambda)v_{q}$ and
for $\lambda$ the condition $u(P)=U$ gives (62) with $x=v_{r}$, $y=v_{q}$. If
$u(v_{r})=u(v_{q})$ then the edge $[u(v_{r}),u(v_{q})]$ belongs to the
condition plane and the extreme distributions are $u(v_{r})$ $u(v_{q})$.
For each of $v_{r}$ the ratio $p_{i}/p_{i}^{*}$ can take only two values:
$a_{r}$ or 0. Without loss of generality we can assume that $q>r$. For a
convex combination $\lambda v_{r}+(1-\lambda)v_{q}$ ($1>\lambda>0$) the ratio
$p_{i}/p_{i}^{*}$ can take three values: $\lambda a_{r}+(1-\lambda)a_{q}$ (for
$\sigma(i)\leq r$), $(1-\lambda)a_{q}$ (for $r<\sigma(i)\leq q$) and 0 (for
$\sigma(i)>q$).
The case when a vertex is one of the $v_{r}$ is also possible. In this case,
there are two possible values of $p_{i}/p^{*}_{i}$, it is either $a_{r}$ or
$0$.
All the generalized canonical distributions from the polyhedron are convex
combinations of its extreme points (vertices). If the set of vertices is
$\\{w_{r}\\}$, then for any generalized canonical distributions
$P=\sum\lambda_{i}w_{i}$ ($\lambda_{i}\geq 0$, $\sum_{i}\lambda_{i}=1$). The
vertices can be found explicitly. Explicit formulas for the extreme
generalized canonical distributions are given in this section: (64) and
various applications of (62). These formulas are based on the description of
compartment $\mathcal{C}_{\sigma}$ given in Proposition 7 and Equation (46).
## 8 History of the Markov Order
### 8.1 Continuous Time Kinetics
We have to discuss the history of the Markov order in the wider context of
orders, with respect to which the solutions of kinetic equations change
monotonically in time. The Markov order is a nice and constructive example of
such an order and at the same time the prototype of all of them (similarly the
Master Equation is a simple example of kinetic equations and, at the same
time, the prototype of all kinetic equations).
The idea of orders and attainable domains (the lower cones of these orders) in
phase space was developed in many applications: from biological kinetics to
chemical kinetics and engineering. A kinetic model includes information of
various levels of detail and of variable reliability. Several types of
building block are used to construct a kinetic model. The system of these
building blocks can be described, for example, as follows:
1. 1.
The list of components (in chemical kinetics) or populations (in mathematical
ecology) or states (for general Markov chains);
2. 2.
The list of elementary processes (the reaction mechanism, the graph of trophic
interactions or the transition graph), which is often supplemented by the
lines or surfaces of partial equilibria of elementary processes;
3. 3.
The reaction rates and kinetic constants.
We believe that the lower level information is more accurate and reliable: we
know the list of component better than the mechanism of transitions, and our
knowledge of equilibrium surfaces is better than the information about exact
values of kinetic constants.
It is attractive to use the more reliable lower level information for
qualitative and quantitative study of kinetics. Perhaps, the first example of
such a analysis was performed in biological kinetics.
In 1936, A.N. Kolmogorov Kolmogorov1936 studied the dynamics of a pair of
interacting populations of prey ($x$) and predator ($y$) in general form:
$\dot{x}=xS(x,y),\;\;\dot{y}=yW(x,y)$
under monotonicity conditions: $\partial S(x,y)/\partial y<0$, $\partial
W(x,y)/\partial y<0$. The zero isoclines, the lines at which the rate of
change for one population is zero (given by equations $S(x,y)=0$ or
$W(x,y)=0$), are graphs of two functions $y(x)$. These isoclines divide the
phase space into compartments (generically with curvilinear borders). In every
compartment the angle of possible directions of motion is given (compare to
Figure 5.3).
Analysis of motion in these angles gives information about dynamics without an
exact knowledge of the kinetic constants. The geometry of the zero isoclines
intersection together with some monotonicity conditions give important
information about the system dynamics Kolmogorov1936 without exact knowledge
of the right hand sides of the kinetic equations.
This approach to population dynamics was further developed by many authors and
applied to various problems MayLeonard1975 ; Bazykin1998 . The impact of this
work on population dynamics was analyzed by K. Sigmund in review Sigmung2007 .
It seems very attractive to use an attainable region instead of the single
trajectory in situations with incomplete information or with information with
different levels of reliability. Such situations are typical in many areas of
engineering. In 1964, F. Horn proposed to analyze the attainable regions for
chemical reactors Horn1964 . This approach was applied both to linear and
nonlinear kinetic equations and became popular in chemical engineering. It was
applied to the optimization of steady flow reactors Glasser1987 , to batch
reactor optimization by use of tendency models without knowledge of detailed
kinetics Filippi-Bossy1989 and for optimization of the reactor structure
Hildebrandt1990 . Analysis of attainable regions is recognized as a special
geometric approach to reactor optimization Feinberg1997 and as a crucially
important part of the new paradigm of chemical engineering Hill2009 . Plenty
of particular applications was developed: from polymerization SmithMalone1997
to particle breakage in a ball mill Metzger2009 . Mathematical methods for
study of attainable regions vary from the Pontryagin’s maximum principle
McGregor1999 to linear programming Kauchali2002 , the Shrink-Wrap algorithm
Manousiouthakis2004 and convex analysis.
The connection between attainable regions, thermodynamics and stoichiometric
reaction mechanisms was studied by A.N. Gorban in the 1970s. In 1979, he
demonstrated how to utilize the knowledge about partial equilibria of
elementary processes to construct the attainable regions Gorban1979 .
He noticed that the set (a cone) of possible direction for kinetics is defined
by thermodynamics and the reaction mechanism (the system of the stoichiometric
equation of elementary reactions).
Thermodynamic data are more robust than the reaction mechanism and the
reaction rates are known with lower accuracy than the stoichiometry of
elementary reactions. Hence, there are two types of attainable regions. The
first is the thermodynamic one, which use the linear restrictions and the
thermodynamic functions GorbanChMMS1979 . The second is generated by
thermodynamics and stoichiometric equations of elementary steps (but without
reaction rates) Gorban1979 ; GorbanBYa1980 .
It was demonstrated that the attainable regions significantly depend on the
transition mechanism (Figure 8.1) and it is possible to use them for the
mechanisms discrimination GorbanYa1980 .
Already simple examples demonstrate that the sets of distributions which are
accessible from a given initial distribution by Markov processes with
equilibrium are, in general, non-convex polytopes Gorban1979 ; Zylka1985
(see, for example, the outlined region in Figure 8.1, or, for particular
graphs of transitions, any of the shaded regions there). This non-convexity
makes the analysis of attainability for continuous time Markov processes more
difficult (and also more intriguing).
Figure 4: Attainable regions from an initial distribution $a_{0}$ for a linear
system with three components $A_{1},A_{2},A_{3}$ in coordinates $c_{1},c_{2}$
(concentrations of $A_{1},A_{2}$) ($c_{3}={\rm const}-c_{1}-c_{2}$) Gorban1979
: for a full mechanism $A_{1}\rightleftarrows A_{2}\rightleftarrows
A_{3}\rightleftarrows A_{1}$ (outlined region), for a two-step mechanism
$A_{1}\rightleftarrows A_{2}$, $A_{1}\rightleftarrows A_{3}$ (horizontally
shaded region) and for a two-step mechanism $A_{1}\rightleftarrows A_{2}$,
$A_{2}\rightleftarrows A_{3}$ (vertically shaded region). Equilibrium is
$a^{*}$. The dashed lines are partial equilibria.
This approach was developed for all thermodynamic potentials and for open
systems as well G11984 . Partially, the results are summarized in YBGE ;
GorbKagan2006 .
This approach was rediscovered by F.J. Krambeck Krambeck1984 for linear
systems, that is, for Markov chains, and by R. Shinnar and other authors
Shinnar1985 for more general nonlinear kinetics. There was even an open
discussion about priority Bykov1987 . Now this geometric approach is applied
to various chemical and industrial processes.
### 8.2 Discrete Time Kinetics
In our paper we deal mostly with continuous time Markov chains. For the
discrete time Markov chains, the attainable regions have two important
properties: they are convex and symmetric with respect to permutations of
states. Because of this symmetry and convexity, the discrete time Markov order
is characterized in detail. As far as we can go in history, this work was
begun in early 1970s by A. Uhlmann and P.M. Alberti. The results of the first
10 years of this work were summarized in monograph AlbertiUhlmann1982 . A more
recent bibliography (more than 100 references) is collected in review
AlbertiCUZ2008 .
This series of work was concentrated mostly on processes with uniform
equilibrium (doubly stochastic maps). The relative majorization, which we also
use in Section 5, and the Markov order with respect to a non-uniform
equilibrium was introduced by P. Harremoës in 2004 Harremo2004 . He used
formalism based on the Lorenz diagrams.
## 9 Conclusion
Is playing with non-classical entropies and divergences just an extension to
the fitting possibilities (no sense—just fitting)? We are sure now that this
is not the case: two one-parametric families of non-classical divergences are
distinguished by the very natural properties:
1. 1.
They are Lyapunov functions for all Markov chains;
2. 2.
They become additive with respect to the joining of independent systems after
a monotone transformation of scale;
3. 3.
They become additive with respect to a partitioning of the state space after a
monotone transformation of scale.
Two families of smooth divergences (for positive distributions) satisfy these
requirements: the Cressie–Read family CR1984 ; ReadCreass1988
$H_{{\rm CR}\
\lambda}(P\|P^{*})=\frac{1}{\lambda(\lambda+1)}\sum_{i}p_{i}\left[\left(\frac{p_{i}}{p_{i}^{*}}\right)^{\lambda}-1\right]\
,\ \ \lambda\in]-\infty,\infty[$
and the convex combination of the Burg and Shannon relative entropies G11984 ;
ENTR1 :
$H(P\|P^{*})=\sum_{i}(\beta
p_{i}-(1-\beta)p_{i}^{*})\log\left(\frac{p_{i}}{p_{i}^{*}}\right)\ ,\ \
\beta\in[0,1]$
If we relax the differentiability property, then we have to add to the the CR
family two limiting cases:
$H_{\rm CR\
\infty}(P\|P^{*})=\max_{i}\left\\{\frac{p_{i}}{p_{i}^{*}}\right\\}-1\ ;$
$H_{{\rm CR\
-\infty}}(P\|P^{*})=\max_{i}\left\\{\frac{p_{i}^{*}}{p_{i}}\right\\}-1$
Beyond these two distinguished one-parametric families there is the whole
world of the Csiszár–Morimoto Lyapunov functionals for the Master equation
(6). These functions monotonically decrease along any solution of the Master
equation. The set of all these functions can be used to reduce the uncertainty
by conditional minimization: for each $h$ we could find a conditional
minimizer of $H_{h}(p)$.
Most users prefer to have an unambiguous choice of entropy: it would be nice
to have “the best entropy” for any class of problems. But from a certain point
of view, ambiguity of the entropy choice is unavoidable, and the choice of all
conditional optimizers instead of a particular one is a possible way to avoid
an arbitrary choice. The set of these minimizers evaluates the possible
position of a “maximally random” probability distribution. For many MaxEnt
problems the natural solution is not a fixed distribution, but a well defined
set of distributions.
The task to minimize functions $H_{h}(p)$ which depend on a functional
parameter $h$ seems too complicated. The Markov order gives us another way for
the evaluation of the set of possible “maximally random” probability
distribution, and this evaluation is, in some sense, the best one. We defined
the Markov order, studied its properties and demonstrated how it can be used
to reduce uncertainty.
It is quite surprising that the Markov order is generated by the reversible
Markov chains which satisfy the detailed balance principle. We did not include
any reversibility assumptions and studied the general Markov chains. There
remain some questions about the structure and full description of the global
Markov order. Nevertheless, to find the set of conditionally extreme (“most
random”) probability distributions, we need the local Markov order only. This
local order is fully described in Section 5.2 and has a very clear geometric
structure. For a given equilibrium distribution $P^{*}$, the simplex of
probability distributions is divided by $n(n-1)/2$ hyperplanes of “partial
equilibria” (this terminology comes from chemical kinetics Gorban1979 ;
GorbKagan2006 ): $\frac{p_{i}}{p_{i}^{*}}=\frac{p_{j}}{p_{j}^{*}}$ (there is
one hyperplane for each pair of states $(i,j)$). In each compartment a cone of
all possible time derivatives of the probability distribution is defined as a
conic envelope of $n(n-1)/2$ vectors (40). The extreme rays of this cone are
explicitly described in Proposition 6 (43). This cone defines the local Markov
order. When we look for conditionally extreme distributions, this cone plays
the same role as a hyperplane given by entropy growth condition (
${\mathrm{d}}S/{\mathrm{d}}t>0$) in the standard approach.
For the problem of the generalized canonical (or reference) distribution the
Markov order gives a polyhedron of the extremely disordered distributions. The
vertices of that polyhedron can be computed explicitly.
The construction of efficient algorithms for numerical calculation of
conditionally extreme compacts in high dimensions is a challenging task for
our future work as well as the application of this methodology to real life
problems.
## Acknowledgements
Suggestions from Mike George, Marian Grendar, Ivan Tyukin and anonymous
referees are gratefully acknowledged.
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## Appendix
Proof of Theorem 1. The problem is to find all such universal and trace–form
Lyapunov functions $H$ for Markov chains, that there exists a monotonous
function $F$, such that $F(H({P}))=F(H({Q}))+F(H({R}))$ if
${P}=p_{ij}=q_{i}r_{j}$.
With Lemma 1 we get that
$H({P})=\sum_{i,j}q_{i}^{*}r_{j}^{*}h\left(\frac{q_{i}r_{j}}{q_{i}^{*}r_{j}^{*}}\right),\;\;H({Q})=\sum_{i}q_{i}^{*}h\left(\frac{q_{i}}{q_{i}^{*}}\right),\;\;H({R})=\sum_{j}r_{j}^{*}h\left(\frac{r_{j}}{r_{j}^{*}}\right)$
Let $F(x)$ and $h(x)$ be differentiable as many times as needed.
Differentiating the equality $F(H({P}))=F(H({Q}))+F(H({R}))$ on $r_{1}$ and
$q_{1}$ taking into account that $q_{n}=1-\sum_{i=1}^{n-1}q_{i}$ and
$r_{m}=1-\sum_{j=1}^{m-1}r_{j}$ we get that
$F^{\prime}(H({P}))H^{\prime\prime}_{q_{1}r_{1}}({P})=-F^{\prime\prime}(H({P}))H^{\prime}_{q_{1}}({P})H^{\prime}_{r_{1}}({P})$,
or, if $-\frac{F^{\prime}(H({P}))}{F^{\prime\prime}(H({P}))}=G(H({P}))$ then
$G(H({P}))=\frac{H^{\prime}_{q_{1}}({P})H^{\prime}_{r_{1}}({P})}{H^{\prime\prime}_{q_{1}r_{1}}({P})}$
(65)
It is possible if and only if every linear differential operator of the first
order, which annulates $H({P})$ and $\sum p_{i}$, annulates also
$\frac{H^{\prime}_{q_{1}}({P})H^{\prime}_{r_{1}}({P})}{H^{\prime\prime}_{q_{1}r_{1}}({P})}$
(66)
and it means that every differential operator which has the form
$D=\left(\frac{\partial H({P})}{\partial q_{\gamma}}-\frac{\partial
H({P})}{\partial q_{\alpha}}\right)\frac{\partial}{\partial
q_{\beta}}+\left(\frac{\partial H({P})}{\partial q_{\beta}}-\frac{\partial
H({P})}{\partial q_{\gamma}}\right)\frac{\partial}{\partial
q_{\alpha}}+\left(\frac{\partial H({P})}{\partial q_{\alpha}}-\frac{\partial
H({P})}{\partial q_{\beta}}\right)\frac{\partial}{\partial q_{\gamma}}$ (67)
annulates (66). For $\beta=2,\alpha=3,\gamma=4$ we get the following equation
$\displaystyle
F_{1}({Q},{R})\left[h^{\prime}\left(\frac{q_{2}r_{1}}{q_{2}^{*}r_{1}^{*}}\right)-h^{\prime}\left(\frac{q_{2}r_{m}}{q_{2}^{*}r_{m}^{*}}\right)+\frac{q_{2}r_{1}}{q_{2}^{*}r_{1}^{*}}h^{\prime\prime}\left(\frac{q_{2}r_{1}}{q_{2}^{*}r_{1}^{*}}\right)-\frac{q_{2}r_{m}}{q_{2}^{*}r_{m}^{*}}h^{\prime\prime}\left(\frac{q_{2}r_{m}}{q_{2}^{*}r_{m}^{*}}\right)\right]+$
$\displaystyle
F_{2}({Q},{R})\left[h^{\prime}\left(\frac{q_{3}r_{1}}{q_{3}^{*}r_{1}^{*}}\right)-h^{\prime}\left(\frac{q_{3}r_{m}}{q_{3}^{*}r_{m}^{*}}\right)+\frac{q_{3}r_{1}}{q_{3}^{*}r_{1}^{*}}h^{\prime\prime}\left(\frac{q_{3}r_{1}}{q_{3}^{*}r_{1}^{*}}\right)-\frac{q_{3}r_{m}}{q_{3}^{*}r_{m}^{*}}h^{\prime\prime}\left(\frac{q_{3}r_{m}}{q_{3}^{*}r_{m}^{*}}\right)\right]+$
$\displaystyle
F_{3}({Q},{R})\left[h^{\prime}\left(\frac{q_{4}r_{1}}{q_{4}^{*}r_{1}^{*}}\right)-h^{\prime}\left(\frac{q_{4}r_{m}}{q_{4}^{*}r_{m}^{*}}\right)+\frac{q_{4}r_{1}}{q_{4}^{*}r_{1}^{*}}h^{\prime\prime}\left(\frac{q_{4}r_{1}}{q_{4}^{*}r_{1}^{*}}\right)-\frac{q_{4}r_{m}}{q_{4}^{*}r_{m}^{*}}h^{\prime\prime}\left(\frac{q_{4}r_{m}}{q_{4}^{*}r_{m}^{*}}\right)\right]=0$
where
$\displaystyle
F_{1}({Q},{R})=\sum_{j}r_{j}\left[h^{\prime}\left(\frac{q_{4}r_{j}}{q_{4}^{*}r_{j}^{*}}\right)-h^{\prime}\left(\frac{q_{3}r_{j}}{q_{3}^{*}r_{j}^{*}}\right)\right]\,;$
$\displaystyle
F_{2}({Q},{R})=\sum_{j}r_{j}\left[h^{\prime}\left(\frac{q_{2}r_{j}}{q_{2}^{*}r_{j}^{*}}\right)-h^{\prime}\left(\frac{q_{4}r_{j}}{q_{4}^{*}r_{j}^{*}}\right)\right]\,;$
$\displaystyle
F_{3}({Q},{R})=\sum_{j}r_{j}\left[h^{\prime}\left(\frac{q_{3}r_{j}}{q_{3}^{*}r_{j}^{*}}\right)-h^{\prime}\left(\frac{q_{2}r_{j}}{q_{2}^{*}r_{j}^{*}}\right)\right]$
If we apply the differential operator $\frac{\partial}{\partial
r_{2}}-\frac{\partial}{\partial r_{3}}$, which annulates the conservation law
$\sum_{j}r_{j}=1$, to the left part of (Appendix), and denote
$f(x)=xh^{\prime\prime}(x)+h^{\prime}(x)$, $x_{1}=\frac{q_{2}}{q_{2}^{*}}$,
$x_{2}=\frac{q_{3}}{q_{3}^{*}}$, $x_{3}=\frac{q_{4}}{q_{4}^{*}}$,
$y_{1}=\frac{r_{1}}{r_{1}^{*}}$, $y_{2}=\frac{r_{m}}{r_{m}^{*}}$,
$y_{3}=\frac{r_{2}}{r_{2}^{*}}$, $y_{4}=\frac{r_{3}}{r_{3}^{*}}$, we get the
equation
$\displaystyle(f(x_{3}y_{3})-f(x_{2}y_{3})-f(x_{3}y_{4})+f(x_{2}y_{4}))(f(x_{1}y_{1})-f(x_{1}y_{2}))+$
$\displaystyle(f(x_{1}y_{3})-f(x_{3}y_{3})-f(x_{1}y_{4})+f(x_{3}y_{4}))(f(x_{2}y_{1})-f(x_{2}y_{2}))+$
(69)
$\displaystyle(f(x_{2}y_{3})-f(x_{1}y_{3})-f(x_{2}y_{4})+f(x_{1}y_{4}))(f(x_{3}y_{1})-f(x_{3}y_{2}))=0$
or, after differentiation on $y_{1}$ and $y_{3}$ and denotation
$g(x)=f^{\prime}(x)$
$\displaystyle
x_{1}g(x_{1}y_{1})(x_{3}g(x_{3}y_{3})-x_{2}g(x_{2}y_{3}))+x_{2}g(x_{2}y_{1})(x_{1}g(x_{1}y_{3})-$
(70) $\displaystyle-
x_{3}g(x_{3}y_{3}))+x_{3}g(x_{3}y_{1})(x_{2}g(x_{2}y_{3})-x_{1}g(x_{1}y_{3}))=0$
If $y_{3}=1$, $y_{1}\neq 0$, $\varphi(x)=xg(x)$, we get after multiplication
(70) on $y_{1}$
$\varphi(x_{1}y_{1})(\varphi(x_{3})-\varphi(x_{2}))+\varphi(x_{2}y_{1})(\varphi(x_{1})-\varphi(x_{3}))+\varphi(x_{3}y_{1})(\varphi(x_{2})-\varphi(x_{1}))=0$
(71)
It implies that for every three positive numbers $\alpha$, $\beta$, $\gamma$
the functions $\varphi(\alpha x)$, $\varphi(\beta x)$, $\varphi(\gamma x)$ are
linearly dependent, and for $\varphi(x)$ the differential equation
$ax^{2}\varphi^{\prime\prime}(x)+bx\varphi^{\prime}(x)+c\varphi(x)=0$ (72)
holds. This differential equation has solutions of two kinds:
1. 1.
$\varphi(x)=C_{1}x^{k_{1}}+C_{2}x^{k_{2}}$, $k_{1}\neq k_{2}$, $k_{1}$ and
$k_{2}$ are real or complex-conjugate numbers.
2. 2.
$\varphi(x)=C_{1}x^{k}+C_{2}x^{k}\ln x$.
Let us check, which of these solutions satisfy the functional equation (71).
1. 1.
$\varphi(x)=C_{1}x^{k_{1}}+C_{2}x^{k_{2}}$. After substitution of this into
(71) and calculations we get
$C_{1}C_{2}(y_{1}^{k_{1}}-y_{1}^{k_{2}})(x_{1}^{k_{1}}x_{3}^{k_{2}}-x_{1}^{k_{1}}x_{2}^{k_{2}}+x_{1}^{k_{2}}x_{2}^{k_{1}}-x_{2}^{k_{1}}x_{3}^{k_{2}}+x_{2}^{k_{2}}x_{3}^{k_{1}}-x_{1}^{k_{2}}x_{3}^{k_{1}})=0$
This means that $C_{1}=0$, or $C_{2}=0$, or $k_{1}=0$, or $k_{2}=0$ and the
solution of this kind can have only the form $\varphi(x)=C_{1}x^{k}+C_{2}$.
2. 2.
$\varphi(x)=C_{1}x^{k}+C_{2}x^{k}\ln x$. After substitution of this into (71)
and some calculations if $y_{1}\neq 0$ we get
$C_{2}^{2}((x_{1}^{k}-x_{2}^{k})x_{3}^{k}\ln
x_{3}+(x_{3}^{k}-x_{1}^{k})x_{2}^{k}\ln
x_{2}+(x_{2}^{k}-x_{3}^{k})x_{1}^{k}\ln x_{1})=0$
This means that either $C_{2}=0$ and the solution is $\varphi(x)=C_{1}x^{k}$
or $k=0$ and the solution is $\varphi(x)=C_{1}+C_{2}\ln x$.
So, the equation (71) has two kinds of solutions:
1. 1.
$\varphi(x)=C_{1}x^{k}+C_{2}$,
2. 2.
$\varphi(x)=C_{1}+C_{2}\ln x$
Let us solve the equation $f(x)=xh^{\prime\prime}(x)+h^{\prime}(x)$ for each
of these two cases.
1. 1.
$\varphi(x)=C_{1}x^{k}+C_{2}$, $g(x)=C_{1}x^{k-1}+\frac{C_{2}}{x}$, there are
two possibilities:
1. 1.1)
$k=0$. Then $g(x)=\frac{C}{x}$, $f(x)=C\ln x+C_{1}$, $h(x)=C_{1}x\ln
x+C_{2}\ln x+C_{3}x+C_{4}$;
2. 1.2)
$k\neq 0$. Then $f(x)=Cx^{k}+C_{1}\ln x+C_{2}$, and here are also two
possibilities:
1. 1.2.1)
$k=-1$. Then $h(x)=C_{1}\ln^{2}x+C_{2}x\ln x+C_{3}\ln x+C_{4}x+C_{5}$;
2. 1.2.2)
$k\neq-1$. Then $h(x)=C_{1}x^{k+1}+C_{2}x\ln x+C_{3}\ln x+C_{4}x+C_{5}$;
2. 2.
$\varphi(x)=C_{1}+C_{2}\ln x$; $g(x)=C_{1}\frac{\ln x}{x}+\frac{C_{2}}{x}$;
$f(x)=C_{1}\ln^{2}x+C_{2}\ln x+C_{3}$; $h(x)=C_{1}x\ln^{2}x+C_{2}x\ln
x+C_{3}\ln x+C_{4}x+C_{5}$.
(We have renamed constants during the calculations).
For the next step let us check, which of these solutions remains a solution to
equation (Appendix). The result is that there are just two families of
functions $h(x)$ such, that equation (Appendix) holds:
1. 1.
$h(x)=Cx^{k}+C_{1}x+C_{2}$, $k\neq 0$, $k\neq 1$,
2. 2.
$h(x)=C_{1}x\ln x+C_{2}\ln x+C_{3}x+C_{4}$.
The function $h(x)$ should be convex. This condition determines the signs of
coefficients $C_{i}$.
The corresponding divergence $H(P\|P^{*})$ is either one of the CR entropies
or a convex combination of Shannon’s and Burg’s entropies up to a monotonic
transformation. ${\mathbf{\square}}$
Characterization of Additive Trace–form Lyapunov Functions for Markov Chains.
We will consider three important properties of Lyapunov functions
$H(P\|P^{*})$:
1. 1.
Universality: $H$ is a Lyapunov function for Markov chains (22) with a given
equilibrium $P^{*}$ for every possible values of kinetic coefficients
$k_{ij}\geq 0$.
2. 2.
$H$ is a trace–form function.
$H(P\|P^{*})=\sum_{i}f(p_{i},p_{i}^{*})$ (73)
where $f$ is a differentiable function of two variables.
3. 3.
$H$ is additive for composition of independent subsystems. It means that if
${P}=p_{ij}=q_{i}r_{j}$ and $P^{*}=p_{ij}^{*}=q_{i}^{*}r_{j}^{*}$ then
$H(P\|P^{*})=H(Q\|Q^{*})+H(R\|R^{*})$.
Here and further we suppose
$0<p_{i},p_{i}^{*},q_{i},q_{i}^{*},r_{i},r_{i}^{*}<1$.
We consider the additivity condition as a functional equation and solve it.
The following theorem describes all Lyapunov functions for Markov chains,
which have all three properties 1) - 3) simultaneously.
Let $f(p,p^{*})$ be a twice differentiable function of two variables.
###### Theorem 3.
If a function $H(P\|P^{*})$ has all the properties 1)-3) simultaneously, then
$f(p,p^{*})=p_{i}^{*}h\left(\frac{p}{p^{*}}\right),\;\;H(P\|P^{*})=\sum_{i}p_{i}^{*}h\left(\frac{p_{i}}{p_{i}^{*}}\right)$
(74)
where
$h(x)=C_{1}\ln x+C_{2}x\ln x,\mbox{ }C_{1}\leq 0,\mbox{ }C_{2}\geq 0$ (75)
###### Proof.
We follow here the P. Gorban proof ENTR3 . Another proof of this theorem was
proposed in Harr2007 . Due to Lemma 1 let us take $H(P\|P^{*})$ in the form
(74). Let $h$ be twice differentiable in the interval $]0,+\infty[$. The
additivity equation
$H(P\|P^{*})-H(Q\|Q^{*})-H(R\|R^{*})=0$ (76)
holds. Here (in (76))
$\displaystyle
q_{n}=1-\sum_{i=1}^{n-1}q_{i},\,r_{m}=1-\sum_{j=1}^{m-1}r_{j},\,P=p_{ij}=q_{i}r_{j}$
$\displaystyle
H(P\|P^{*})=\sum_{i,j}q_{i}^{*}r_{j}^{*}h\left(\frac{q_{i}r_{j}}{q_{i}^{*}r_{j}^{*}}\right),\,H(Q\|Q^{*})=\sum_{i}q_{i}^{*}h\left(\frac{q_{i}}{q_{i}^{*}}\right),\,H(R\|R^{*})=\sum_{j}r_{j}^{*}h\left(\frac{r_{j}}{r_{j}^{*}}\right)$
Let us take the derivatives of this equation first on $q_{1}$ and then on
$r_{1}$. Then we get the equation ($g(x)=h^{\prime}(x)$)
$\displaystyle
g(\frac{q_{1}r_{1}}{q_{1}^{*}r_{1}^{*}})-g(\frac{q_{n}r_{1}}{q_{n}^{*}r_{1}^{*}})-g(\frac{q_{1}r_{m}}{q_{1}^{*}r_{m}^{*}})+g(\frac{q_{n}r_{m}}{q_{n}^{*}r_{m}^{*}})+$
$\displaystyle+\frac{q_{1}r_{1}}{q_{1}^{*}r_{1}^{*}}g^{\prime}(\frac{q_{1}r_{1}}{q_{1}^{*}r_{1}^{*}})-\frac{q_{n}r_{1}}{q_{n}^{*}r_{1}^{*}}g^{\prime}(\frac{q_{n}r_{1}}{q_{n}^{*}r_{1}^{*}})-\frac{q_{1}r_{m}}{q_{1}^{*}r_{m}^{*}}g^{\prime}(\frac{q_{1}r_{m}}{q_{1}^{*}r_{m}^{*}})+\frac{q_{n}r_{m}}{q_{n}^{*}r_{m}^{*}}g^{\prime}(\frac{q_{n}r_{m}}{q_{n}^{*}r_{m}^{*}})=0$
Let us denote $x=\frac{q_{1}r_{1}}{q_{1}^{*}r_{1}^{*}}$,
$y=\frac{q_{n}r_{1}}{q_{n}^{*}r_{1}^{*}}$,
$z=\frac{q_{1}r_{m}}{q_{1}^{*}r_{m}^{*}}$, and $\psi(x)=g(x)+xg^{\prime}(x)$.
It is obvious that if $n$ and $m$ are more than 2, then $x$, $y$ and $z$ are
independent and can take any positive values. So, we get the functional
equation:
$\psi\left(\frac{yz}{x}\right)=\psi(y)+\psi(z)-\psi(x)$ (77)
Let’s denote $C_{2}=-\psi(1)$ and $\psi_{1}(\alpha)=\psi(\alpha)-\psi(1)$ and
take $x=1$. We get then
$\psi_{1}(yz)=\psi_{1}(y)+\psi_{1}(z)$ (78)
the Cauchy functional equation Aczel1966 . The solution of this equation in
the class of measurable functions is $\psi_{1}(\alpha)=C_{1}\ln\alpha$, where
$C_{1}$ is constant. So we get $\psi(x)=C_{1}\ln x+C_{2}$ and
$g(x)+xg^{\prime}(x)=C_{1}\ln x+C_{2}$. The solution is
$g(x)=\frac{C_{3}}{x}+C_{1}\ln x+C_{2}-C_{1}$;
$h(x)=\int(\frac{C_{3}}{x}+C_{1}\ln x+C_{2}-C_{1})dx=C_{3}\ln x+C_{1}x\ln
x+(C_{2}-2C_{1})x+C_{4}$, or, renaming constants, $h(x)=C_{1}\ln x+C_{2}x\ln
x+C_{3}x+C_{4}$. In the expression for $h(x)$ there are two parasite constants
$C_{3}$ and $C_{4}$ which occurs because the initial equation was
differentiated twice. So, $C_{3}=0$, $C_{4}=0$ and $h(x)=C_{1}\ln x+C_{2}x\ln
x$. Because $h$ is convex, we have $C_{1}\leq 0$ and $C_{2}\geq 0$. ∎
So, any universal additive trace–form Lyapunov function for Markov chains is a
convex combination of the BGS entropy and the Burg entropy.
|
arxiv-papers
| 2010-03-06T12:04:13 |
2024-09-04T02:49:08.852428
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A.N. Gorban, P.A. Gorban and G. Judge",
"submitter": "Alexander Gorban",
"url": "https://arxiv.org/abs/1003.1377"
}
|
1003.1441
|
# Existence and uniqueness of generalized monopoles in six-dimensional non-
Abelian gauge theory
S.X.Chen111The research of the first author was supported in part by the
Natural Science Fund of Henan Education Office (2007110004) and (2008A110002).
chensx1982@gmail.com L.Pei peilongsf@gmail.com School of Mathematics and
Information Sciences, Henan University, Minglun Street, Kaifeng, P.R.China
475001 Arts and Science Experimental Class, Henan University, Minglun Street,
Kaifeng, P.R.China 475001
###### Abstract
In this paper, we established the existence and uniqueness of the spherically
symmetric monopole solutions in $SO(5)$ gauge theory with Higgs scalar fields
in the vector representation in six-dimensional Minkowski space-time and
obtain sharp asymptotic estimates for the solutions. Our method is based on a
dynamical shooting approach that depends on two shooting parameters which
provides an effective framework for constructing the generalized monopoles in
six-dimensional Minkowski space-time.
###### keywords:
generalized monopoles , dynamical shooting method , existence and uniqueness ,
six-dimension space-time , non-Abelian gauge theory
###### MSC:
81T13 , 65L10
††journal: Elsevier
## 1 Introduction
Long time ago, Dirac showed that quantum mechanics admits a magnetic monopole
of quantized magnetic charge despite the presence of a singular Dirac
string[4, 27]. Much later, G. ’t Hooft and Polyakov showed that such magnetic
monopoles emerge as regular configurations in $SO(3)$ gauge theory with
spontaneous symmetry breaking triggered by triplet Higgs scalar fields[7, 21,
13, 22]. Although a monopole has not been detected or produced experimentally
as a single particle, the existence of such objects has far reaching
consequences.In the early universe, monopoles might have beed copiously
produced, which vanished due to various physical interaction such as Pair
production caused by Coulomb interaction among monopoles and anti-monopoles[5]
but have significantly affected the history of the universe since then. For
example, monopoles magnetic monopoles flow dispute the dynamo action, leading
to a slow dynamo action in the best hypothesis or a decay of the magnetic
field[6]. Also, Monopoles played a very important role in the formation of
galaxy formation[17]. G t Hooft-Polyakov monopoles emerge in grand unified
theory of electromagnetic, weak, and strong interactions as well.
It is important and promising to explore solitonic objects in higher
dimensional space-time under the super-string scenario, and recent extensive
study of domain walls in super-symmetric theories, for instance, may have a
direct link to the brane world scenario[9, 10, 11]. The energy of ’t Hooft-
Polyakov monopoles is bound from below by a topological charge. Monopole
solutions saturate such bound, thereby the stability of the solutions being
guaranteed by topology[1]. This observation prompts a question that if there
can be a monopole solution in higher dimensions. Kalb and Ramond introduced
Abelian tensor gauge fields coupled to closed strings[15]. Nepomechie showed
that a new type of monopole solutions appear in those Kalb-Ramond
antisymmetric tensor gauge fields[18]. Their implications to the
confinement[23, 19, 20] and to ten-dimensional Weyl invariant space-time [8]
has been explored. Topological defects in six dimensional Minkowski space-time
as generalization of Dirac’s monopoles were also found[28]. Tchrakian has
investigated monopoles in non-Abelian gauge theory in higher dimensions whose
action involves polynomials of field strengths of high degrees[24, 25].
Furthermore, it has been known that magnetic monopoles appear in the matrix
model in the gauge connections describing Berry’s phases on fermi states. In
particular, in the $USp$ matrix model they are described by $SU(2)$ -valued
anti-self-dual connections[12, 2]. H.Kihara and his team presented regular
monopole configurations with saturated Bogomolny bound in $SO(5)$ gauge theory
in six dimensions[16]. Self-gravitating Yang monopoles in all dimensions was
also studied by G.W.Gibbons and P.K.Townsend[13].The purpose of this paper is
to establish an existence and uniqueness theorem for these generalized
monopoles in six-dimensional non-Abelian gauge.
In the next section, we first briefly discussed the mathematical structure of
the problem of the existence of generalized monopoles in six-dimension non-
Abelian gauge. We then state our main existence and uniqueness theorem for
these solutions. In the third section, we transform the first-order equations
into a second-order non-linear equation, and then we introduce a series of
variable transformations to reduce the equation into a linear equation. In
this case, the existence of generalized monopole solutions is seen to be
equivalent to the existence problem of a nonlinear two-point boundary value
problem. In section 4, we present a dynamical shooting method which solves the
existence problem completely and may be used as a constructive method for
numerical computation. We shall also obtain sharp asymptotic estimates for the
solutions.
## 2 Mathematical Structure and Theorem
Following Kihara, Hosotani and Nitta, we recall that a key to find correct
Bogomolny equations in six-dimensional space-time is facilitated with the use
of the Dirac or Clifford algebra. Let’s consider $SO(5)$ gauge theory in six-
dimensional space-time. Based on Clifford algebra and gauge transformation,
the action is given by
$\begin{array}[]{rcl}I&=&\int[-\frac{1}{8\cdot{4!}}TrF^{2}*F^{2}-\frac{1}{8}TrD_{A}\phi-\frac{\lambda}{4!}(\phi^{a}\phi_{a}-H_{0}^{2})^{2}d^{6}x]\\\
&=&\int
d^{6}x[-\frac{1}{8\cdot{4!}}Tr(F^{2})_{\mu\nu\rho\sigma}(F^{2})^{\mu\nu\rho\sigma}-\frac{1}{2}D_{\mu}\phi^{a}D^{\mu}\phi_{a}+\lambda(\phi^{a}\phi_{a}-H_{0}^{2})^{2}],\end{array}$
$None$
where the components of
$F^{2}=\frac{1}{8}\\{F_{\mu\nu},F_{\rho\sigma}\\}dx^{\mu}\land dx^{\nu}\land
dx^{\rho}\land dx^{\sigma}$ are given by
$\begin{array}[]{lr}(F^{2})_{\mu\nu\rho\sigma}~{}=~{}T_{\mu\nu\rho\sigma}^{e}\gamma_{e}-S_{\mu\nu\rho\sigma},\\\
T_{\mu\nu\rho\sigma}^{e}(A)~{}=~{}\frac{1}{2\cdot
4!}\epsilon^{abcde}(F_{\mu\nu}^{ab}F_{\rho\sigma}^{cd}+F_{\mu\rho}^{ab}F_{\sigma\nu}^{cd}+F_{\mu\sigma}^{ab}F_{\nu\rho}^{cd}),\\\
S_{\mu\nu\rho\sigma}(A)~{}=~{}\frac{1}{4!}(F_{\mu\nu}^{ab}F_{\rho\sigma}^{cd}+F_{\mu\rho}^{ab}F_{\sigma\nu}^{cd}+F_{\mu\sigma}^{ab}F_{\nu\rho}^{cd}).\end{array}$
$None$
The canonical conjugate momentum fields are given by
$\begin{array}[]{rcl}\Pi_{i}^{ab}&=&\frac{\delta I}{\delta A_{i}^{ab}}\\\
&=&\frac{1}{3!}T_{0jkl}^{e}\frac{\delta T_{0jkl}^{e}}{\delta
F_{0i}^{ab}}+\frac{4}{3!}S_{0jk}\frac{\delta S_{0jkl}^{e}}{\delta
F_{0i}^{ab}}{}\\\
&=&\frac{1}{3}(M_{i,jkl}^{ab,e}M_{m,jkl}^{cd,e}+N_{i,jkl}^{ab,e}N_{m,jkl}^{cd,e})F_{0m}^{cd}\\\
&:=&U_{i,m}^{ab,cd}F_{0m}^{cd}~{}~{},\end{array}$ $None$
where $U$ is a symmetric, positive-definite matrix. To confirm the positivity
of the Hamiltonian, we take the $A_{0}=0$ gauge in which
$F_{0i}^{ab}=A_{i}^{ab}$. It immediately follows that
$E=\int d^{5}x[\frac{1}{2}\Pi U^{-1}\Pi+\frac{1}{2\cdot
4!}{(T_{ijkl}^{e})^{2}+(S_{ijkl})^{2}+H_{\phi}}]\geq 0~{},$ $None$
where $H_{\phi}$ is the scalar field part of the Hamiltonian density.
In the $A_{0}=0$ gauge, the energy becomes lowest for static configurations
$A_{i}^{ab}=\phi_{a}=0$ and it is given by
$\begin{array}[]{rcl}E&=&\int
d^{5}x\frac{1}{4!}[\frac{1}{2}(T_{ijkl}^{e}\mp\epsilon^{ijklm}D_{m}\phi^{e})^{2}+\frac{1}{2}(T_{ijkl}^{e})^{2}\\\
&\pm&\epsilon^{ijklm}T_{ijkl}^{e}D_{m}\phi_{e}+\lambda(\phi_{a}\phi^{a}-H_{0}^{2})^{2}]\\\
&\geq&\pm\int d^{5}x\frac{1}{4!}\epsilon^{ijklm}T_{ijkl}^{e}D_{m}\phi^{e}\\\
&=&\pm\int TrD_{A}\phi F^{2}\\\
&:=&\frac{16\pi^{2}}{g^{2}}H_{0}\Psi.\end{array}$ $None$
As $D_{A}F=0$ and thereby $TrD_{A}\phi F^{2}=d(Tr\phi F^{2})$ , $\Psi$ can be
expressed as a surface integral
$\Psi=\pm\frac{g^{2}}{16\pi^{2}}\int_{S^{4}}Tr\phi F_{2}~{},$ $None$
where $S^{4}$ is a space infinity of $R^{5}$.
The Bogomolny bound equation is
$\begin{array}[]{l}*_{5}(F\wedge F)=\pm D_{A}\phi~{},\end{array}$ $None$
where $*_{5}$ is Hodge dual in five-dimensional space. In components, it is
given by
$\begin{array}[]{rcl}\epsilon_{ijklm}T_{ijkl}^{e}&=&\pm D_{m}\phi_{e}~{},\\\
S_{ijkl}&=&0~{}.\end{array}$ $None$
Let us define $e~{}:=~{}x^{a}\gamma_{a}/r$ and make a hedgehog ansatz[21]
$\begin{array}[]{rcl}\phi&=&H_{0}U(r)e~{},\\\
A&=&\frac{1-k(r)}{2g}ede~{}.\end{array}$ $None$
It follows immediately that
$\begin{array}[]{rcl}D_{A}\phi&=&H_{0}(KUde+U^{\prime}edr)~{},\\\
F&=&\frac{1-K^{2}}{4g}de\wedge de-\frac{K^{\prime}}{2g}edr\wedge
de~{}.\end{array}$ $None$
Accordingly, the boundary condition is
$\begin{array}[]{l}U(\infty)=\pm
1,~{}U(0)=0,~{}K(\infty)=0,~{}K(0)=1.\end{array}$ $None$
Applying $*_{5}(de\wedge de\wedge de\wedge de)=\frac{4!edr}{r^{4}}$ and
$*_{5}(edr\wedge dr\wedge dr\wedge dr)=\frac{3!edr}{r^{4}}$, the Bogomolny
boundary equation(2.7)(with a plus sign)becomes
$\begin{array}[]{lr}KU~{}=~{}-\frac{\displaystyle(1-K^{2})dK}{\displaystyle\tau^{2}d\tau}~{},\\\
\frac{\displaystyle dU}{\displaystyle
d\tau}~{}=~{}\frac{\displaystyle(1-K^{2})^{2}}{\displaystyle\tau^{4}}~{},\\\
U(\infty)~{}=~{}1,~{}U(0)~{}=~{}0,~{}K(\infty)~{}=~{}0,~{}K(0)~{}=~{}1~{},\end{array}$
$None$
where $\tau=ar$, $a=(\frac{2g^{2}}{3}H_{0})^{\frac{1}{3}}$.
In this case, $U$ increases as $\tau$ so that $U(\infty)=1$. A solution in the
case $-D_{A}\phi=*_{5}(F\wedge F)$ is obtained by replacing $U$ by $-U$.
Our main existence and uniqueness theorem for generalized monopole solutions
in the six-dimension non-Abelian gauge theory can be stated as follows:
###### Theorem 1
For any real number $g>0$ and $H_{0}>0$, the two point boundary value problem
(2.12) has a unique solution $(K(r),U(r))$ so that $K(r)$ is strictly
decreasing and $U(r)$ is strictly increasing for any $r>0$. Besides, there
hold the sharp asymptotic estimates
$\begin{array}[]{lr}K~{}=~{}O(e^{-Cr^{3}}),~{}U~{}=~{}1+O(r^{-3}),~{}r\to\infty,C>0,\\\
K~{}=~{}1+O(r^{2}),~{}U~{}=~{}O(r),~{}r\to 0.\end{array}$
This solution uniquely gives rise to a spherically symmetric finite-energy
monopole solution of unit topological charge for non-Abelian gauge theory in
six-dimensional Minkowski space-time.
## 3 Second-Order Governing Equation
Noting that the two equations in (2.12) can be combined to yield
$\frac{d(\frac{1-K^{2}}{\tau^{2}K})}{d\tau}\frac{dK}{d\tau}~{}+~{}\frac{1-K^{2}}{\tau^{4}}=0,$
$None$
or equivalently, in terms of $s=\ln\tau$ and $f(s)=K^{2}$:
$f^{\prime\prime}-\\{3+\frac{f^{\prime}}{f(1-f)}\\}f^{\prime}+2f(1-f)=0.$
$None$
Accordingly, the boundary condition becomes
$\begin{array}[]{lr}f(-\infty)=1,f(\infty)=0.\end{array}$ $None$
We will prove that $0<f(s)<1$, $\forall s\in(-\infty,\infty)$. Note that $f=0$
and $f=1$ are two equilibrium solutions of equation (3.2), thus the existence
and uniqueness theorem for solutions of ordinary differential equation allow
us to see that $0<f<1$, $\forall s\in(-\infty,\infty)$. To make it convenient
for us to solve our problem, we apply the transformation: $G(s)=\ln f(s)$.
Under this transformation, $f^{\prime}$ and $f^{\prime\prime}$ can be
represented as follows:
$f^{\prime}=e^{G}G,~{}~{}f^{\prime\prime}=e^{G}(G^{\prime})^{2}+e^{G}G^{\prime\prime}.$
$None$
Inserting (3.4) into (3.2), we have
$G^{\prime\prime}+(G^{\prime})^{2}-\\{3+\frac{G^{\prime}}{1-e^{G}}\\}G^{\prime}+2(1-e^{G})=0.$
$None$
Meanwhile, it is easy to see that $-\infty<G<0$, and the boundary condition
naturally becomes as follows:
$G(-\infty)=0,~{}~{}G(\infty)=-\infty.$ $None$
Furthermore, we can see that equation (3.5) can be simplified to
$(G-e^{G})^{\prime}-3(G-e^{G})+2(1-e^{G})^{2}=0.$ $None$
To further simplify our problem, we introduce the transformation $V=G-e^{G}$.
Since the function $V(G)=G-e^{G}$ is strictly increasing in the interval
$-\infty<G<0$, it is invertible, and its inverse function $Q(V)$(say) enjoys
the same properties over the interval $(-\infty,-1)$. In terms of the variable
$V$, the equation (3.7) and its associated boundary condition becomes
$\begin{array}[]{lr}V^{\prime\prime}-3V^{\prime}$=$-2(1-e^{G})^{2},~{}s\in(-\infty,\infty),\\\
V(-\infty)$=$-1,~{}V(\infty)=-\infty.\end{array}$ $None$
The equation (3.8) seems more tractable than equation (3.1) except that the
function $Q(V)$ is not defined for $V\geq-1$, which makes it inconvenient to
conduct a discussion. In order to fix this problem, we will make a suitable
extension of the function $(1-e^{Q(V)})^{2}$ to $V\geq-1$. Note that
$\lim\limits_{V\to-1}(1-e^{Q(V)})^{2}=\lim\limits_{G\to 0}(1-e^{G})^{2}=0.$
$None$
Moreover, for $V<-1$, we have
$\frac{d}{dV}(1-e^{Q(V)})^{2}=-2e^{Q(V)},$ $None$
which tends to $-2$ as $V\to-1$. Hence, we can modify (3.8) into the following
form,
$\begin{array}[]{lr}V^{\prime\prime}-3V^{\prime}=R(V)~{}:=~{}\left\\{\begin{array}[]{ll}-2(1-e^{Q(V)})^{2},&\textrm{$V<-1$},\\\
4(V+1),&\textrm{$V\geq-1$}.\end{array}\right.\end{array}$ $None$
We see that $R(V)$ is a differentiable function for all $V$. We will consider
(3.11) subject to boundary condition in (3.8). Although (3.11) alters the
original equation in(3.8) due to its modified right-hand side function, we
shall obtain a solution $V(s)$ that says negative for all
$s\in(-\infty,\infty)$. In this way, we recover a solution to the original
boundary value problem (3.8) as expected. Hence, our boundary value problem
consisting of (3.11) and boundary condition in (3.8) becomes
$V^{\prime\prime}-3V^{\prime}=R(V),~{}s\in(-\infty,\infty),~{}V(-\infty)=-1,~{}V(\infty)=-\infty,$
$None$
where and in the sequel, we still use the prime ′ to denote the
differentiation with respect to the variable $s$ when there is no risk of
confusion.
## 4 Mathematical Analysis
To solve the two-point boundary value problem (3.12), we use a dynamical
shooting method. This method was once used to solve problems[3, 26] in the
field of mathematical physics. When we do this, we need to consider the
initial value problem
$V^{\prime\prime}-3V^{\prime}=R(V),~{}s\in(-\infty,\infty),~{}V(0)=m,~{}V^{\prime}(0)=-n.$
$None$
Since we are looking for a solution $V<-1$, we naturally assume
$m<-1.$ $None$
Under the assumption (4.2), we shall show that when $n$ is suitably chosen in
(4.1), we may obtain a solution to (3.12). It can be seen from the structure
of the problem that the boundary condition $V(-\infty)=-1$ is a crucial part
to realize. So we shall look at this end first. For this purpose, we set
$t=-s$ in the half interval $-\infty<s\leq 0$ and convert (4.1) into the form
$V^{\prime\prime}+3V^{\prime}=R(V),~{}t>0,~{}V(0)=m,~{}V^{\prime}(0)=n,$
$None$
where the prime ′ denotes the differentiation with respect to the reversed
variable $t$. We also use $V_{t}$ to denote $\frac{dV}{dt}$. For fixed $m$
satisfying (4.2), we use $V(t;n)$ to denote the unique solution of (4.3) which
is defined in its interval of existence.
We are now ready to launch a shooting analysis for (4.3). We express the set
of real numbers $R$ as the disjoint union of three data sets as follows:
$\beta^{-}~{}=~{}\\{n\in{R}|$ there exists $t>0$ so that $V_{t}(t;n)<0\\},$
$\beta^{0}~{}=~{}\\{n\in{R}|~{}V_{t}(t;n)>0$ and $V(t;n)\leq-1$ for all
$t>0$},
$\beta^{+}~{}=~{}\\{n\in{R}|~{}V_{t}(t;n)>0$ for all $t\geq 0$ and $V(t;n)>-1$
for some $t>0$ }.
###### Lemma 2
We have the disjoint union $R~{}=~{}\beta^{-}\cup\beta^{0}\cup\beta^{+}$.
###### Proof 1
If $n\not\in\beta^{-}$, then $V_{t}(t;n)\geq{0}$ for all $t$. If there exists
a point $t_{0}>0$ so that $V_{t}(t_{0};n)=0$, then $V(t_{0};n)\neq~{}0$
because $V(t;n)=0$ is an equilibrium point of the differential equation in
(4.3) which is not attainable in finite time. Using the information
$V_{t}(t_{0};n)~{}=~{}0$ but $V(t_{0};n)~{}\neq~{}0$ in (4.3), we see that
either $V^{\prime\prime}>0$ or $V^{\prime\prime}<0$ at $t~{}=~{}t_{0}$. Hence,
there is a $t>t_{0}$ or $t<t_{0}$ at which $V_{t}(t;n)<0$. This contradicts
the assumption that $n\not\in\beta^{-}$. Thus $V_{t}(t;n)~{}>~{}0$ for all
$t>0$ and $n\in\beta^{0}\cup\beta^{+}$, which proves the relation
$R~{}=~{}\beta^{-}\cup\beta^{0}\cup\beta^{+}$ as claimed.
###### Lemma 3
The set $\beta^{+}$ and $\beta^{-}$ are both open and nonempty.
###### Proof 2
The fact that $\beta^{-}\neq\emptyset$ follows immediately from the fact that
$(-\infty,0)\subset\beta^{-}$. To see that $\beta^{+}$ is nonempty, we
integrate (4.3) to get
$V_{t}(t;n)~{}=~{}(n+\int_{0}^{t}R(V(s_{1};n)e^{3s_{1}}ds_{1})e^{-3t},$ $None$
$V(t;n)~{}=~{}m+n(1-e^{-3t})+\int_{0}^{t}e^{-3s_{1}}(\int_{0}^{s_{1}}R(V(s_{2};n))e^{3s_{2}}d{s_{2}})d{s_{1}}.$
$None$
For any fixed $t_{0}>0$, we can choose $n>0$ sufficiently large so that
$V_{t}(t_{0};n)>0,$ $None$ $V(t_{0};n)>-1.$ $None$
Considering $V_{0}(0;n)=n>0$, $V(0;n)=m<-1$ and the property of continuous
function, we can see that there exist a set of intervals
$\\{(0,\delta_{n})\\}$ so that $(0,\delta_{n})\subset(0,\delta_{n+1})$ and
$V_{t}(t;n)>0$ for all $t\in(0,\delta_{n})$ where $n\in Z^{+}$. As the basis
for the proof of this lemma, we will first prove that there exists $k\in
Z_{+}$ so that if we denote $t_{1}=min\\{\delta_{k},t_{0}\\}$, there holds
$V_{t}(t;n)>0$ for all $t\in(0,t_{1}]$ and $V(t;n)<-1$ for all $t\in(0,t_{1})$
but $V(t_{1};n)=-1$. Suppose otherwise that there exists no such $t_{1}$
satisfying the condition mentioned closely above. Denote
$T=sup\\{\delta_{n}\\}$ and it is easy to see that $V_{t}(T;n)=0$, thereby
$T\neq t_{0}$ because $V_{t}(t_{0};n)=0$. Therefore, we can divide the proof
of the lemma into two sections according to whether $T<t_{0}$ or $T>t_{0}$.
First, if $T<t_{0}$, then the supposition mentioned closely above leads to
$V(T;n)<-1$ and $V_{t}(T;n)=0$. Moreover, from the property of continuous
function, we know that there exists $T_{0}>0$ so that $V(t;n)<-1$,$\forall
t\in(T,T+T_{0})$. Therefore, it is easy to conclude that
$V_{t}(T+\frac{T_{0}}{2})<0,$ $None$ $V(T+\frac{T_{0}}{2})<-1.$ $None$
Clearly, the two inequities listed above together with the structure of
$V_{t}(t;n)$ and $V(t;n)$ listed above allow us to see that for any
$t\in(T+\frac{T_{0}}{2},t_{0})$ there hold
$V_{t}(T+\frac{T_{0}}{2})<0,$ $None$ $V(T+\frac{T_{0}}{2})<-1.$ $None$
which contradict (4.6) and (4.7), thus the lemma is proved provided that
$T<t_{0}$.
Second, if $T>t_{0}$, the supposition mention above allows us to see that for
any $t\in[0,t_{0}]$ there holds
$\begin{array}[]{lr}V(t;n)<-1,\end{array}$ $None$
which also contradicts (4.7).
Considering the two cases, we can see that there exists
$\delta_{k}\in(0,t_{0})$(We denote this $\delta_{k}$ as $t_{1}$) so that
$V_{t}(t;n)>0$ for any $t\in[0,t_{1}]$ and $V(t;n)<-1$, $\forall
t\in[0,t_{1})$ but $V(t_{1};n)=-1$.
We will then prove that $V_{t}(t;n)>0$ for all $t\in[0,\infty)$. In fact,
suppose otherwise that there exists $t_{3}>t_{1}$ so that $V_{t}(t;n)>0$ for
any $t\in[t_{1},t_{3})$ but $V_{t}(t_{3};n)=0$. Noting that $V(t;n)>-1$,
$\forall t\in[t_{1},t_{3}]$ and considering (4.4), we can see that there holds
$n+\int_{0}^{t_{3}}R(V(t;b))e^{3t}dt>n+\int_{0}^{t_{1}}R(V(t;b))e^{3t}dt>0.$
$None$
Therefore,
$(n+\int_{0}^{t_{3}}R(V(\tau;b))e^{3\tau}d\tau)e^{-t_{3}}>0,$ $None$
which contradicts the fact that $V_{t}(t_{3};n)=0$. Hence, we know that
$V(t;n)>0$,$\forall t\in[0,\infty)$. Consequently,we can conclude that
$V(t;n)>-1$ for all $t\in(t_{1},\infty)$, and naturally, we can see that
$V_{t}(t;n)>0$ for all $t\in(t_{1},\infty)$. Therefore, $n\in\beta^{+}$ and
the nonemptyness of $\beta^{+}$ is established.
Moreover, for $n_{0}\in\beta^{+}$, there is a $t_{0}>0$ so that
$V(t_{0};n_{0})>-1$. By the continous dependence of $V$ on the parameter $n$
we see that when $n_{1}$ is close to $n_{0}$ we have $V_{t}(t;n_{1})>0$ for
all $t\in[0,t_{0}]$ and $V(t_{0},n_{1})>-1$. As proved above, $V_{t}(t;n)>0$
for all $t>t_{1}$. Thus, we can see that $V_{t}(t;n_{1})>0$ for all $t>t_{0}$
as well, which proves $n_{1}\in\beta^{+}$. So $\beta^{+}$ is open. The fact
that $\beta^{-}$ is open is self-evident. The lemma follows.
###### Lemma 4
The set $\beta^{0}$ is a nonempty closed set. Moreover, if $n\in\beta^{0}$,
then $V(t;n)<-1$ for all $t>0$.
###### Proof 3
The first part of the lemma follows from the connectedness of $R$ and Lemma
$4.2$. To prove the second part, we assume otherwise that there is a $t_{0}>0$
so that $V(t_{0};n)=0$. Since $V(t;n)\leq-1$ for all $t>0$, $V$ attains its
local maximum at $t_{0}$. In particular, $V_{t}(t_{0};n)=0$, which contradicts
the definition of $\beta^{0}$.
###### Lemma 5
For $n\in\beta^{0}$, we have $V(t;n)\to-1$ as $t\to\infty$.
###### Proof 4
Since $V$ increases and $V<-1$ for all $t>0$, we see that the limit
$\lim\limits_{t\to\infty}V(t;n)=V_{\infty}$ exists and $-\infty<V_{\infty}\leq
0$. If $V_{\infty}<0$, then $R(V(t;n))<R(V_{\infty})<0$. Inserting this result
into (4.4), we see that $V_{t}(t;n)<0$ when $t>0$ is sufficiently large, which
contradicts the definition of $\beta^{0}$.
###### Lemma 6
The set $\beta^{0}$ is actually a single point set. In other words, the
correct shooting data is in fact unique.
###### Proof 5
Suppose otherwise that there are two points $n_{1}$ and $n_{2}$. Let
$V(t;n_{1})$ and $V(t;n_{2})$ be the corresponding solutions of (4.3). Then
the function $w(t)=V(t;n_{1})-V(t;n_{2})$ satisfies the boundary condition
$w(0)=w(\infty)=0$ and the equation
$w^{\prime\prime}(t)+w^{\prime}(t)=R^{\prime}(\xi(t))w(t),~{}0<t<\infty,$
$None$
where $\xi(t)$ lies between $V(t;n_{1})$ and $V(t;n_{2})$ and
$R^{\prime}(V)=\frac{dR(V)}{dV}>0$($\forall V$) in view of (3.10) and (3.11).
Applying the maximum principle to (4.15), we conclude that
$w(t)~{}\equiv~{}0$, which contradicts the assumption that $n_{1}\neq n_{2}$.
For $n\in\beta^{0}$, we now consider the decay rate of $V(t;n)$ as
$t\to\infty$. To simplify our problem, we introduce the following
transformation $v=V+1$. From the properties of the function $R(V)$, we see
that the linearized equation of the differential equation in (4.3) around
$v=0$ is $\theta^{\prime\prime}+3\theta^{\prime}-4\theta=0$, whose
characteristic equation has the roots $\lambda=-4$ and $\lambda=1$. Hence, we
see that for any $\epsilon\in(0,1)$, there is a constant $C(\epsilon)$ such
that
$\begin{array}[]{lr}-C(\epsilon)e^{-4(1-\epsilon)t}<v(t;n)<0,\forall t\geq
0.\end{array}$ $None$
Note that, modulo the positive small constant $\epsilon$, the above estimate
is sharp. We now go back to the variable $s=-t$. Thus, we have obtained a
solution $V(s)$ of (3.12) defined in the left of the real line, $-\infty<s\leq
0$, such that $V(s)\leq-1$ for all $s\leq 0$, and
$-1-C(\epsilon)e^{4(1-\epsilon)s}<V(s)<-1,~{}~{}\forall~{}s\leq 0.$ $None$
We now consider the right half of the real line, $0\leq s<\infty$. When $s$ is
near zero, there hold $V^{\prime}(s)<0$ and $V(s)<-1$. Inserting these into
(4.1) and using (3.11), we see that $V^{\prime\prime}(s)<0$ there. This
property implies that the structure of the differential equation in (3.11)
allows us to preserve the negative sign for all $V(s)$, $V^{\prime}(s)$ and
$V^{\prime\prime}(s)$. In particular, the solution $V(s)$ exists for all $s>0$
and $V(s)$ is strictly decreasing everywhere. From the structure of the
function $R(V)$ on the right-hand side of the differential equation, we easily
deduce that $V(\infty)=-\infty$. Hence, a solution of (3.12) is obtained. We
now strengthen our conclusion by deriving the accurate blow-up rate for $V(s)$
as $s\to\infty$.
Integrating the differential equation in (4.1), we obtain
$e^{-3s}V^{\prime}(s)=-n-2\int_{0}^{s}(1-e^{Q(V)})^{2}e^{-3s_{1}}ds_{1}.$
$None$
From (3.10) we can see that the integral on the right-hand side of (4.18) is
convergent for $s\to\infty$. Thus, we have the sharp expression
$V^{\prime}(s)=-(n+\sigma(s))e^{3s},$ $None$
where $\sigma(s)=2\int_{0}^{s}(1-e^{Q(V)})^{2}e^{-3s_{1}}ds_{1}$ is a bounded
increasing function in $[0,\infty)$ and $\sigma(0)=0$. Consequently, we find
that $V(s)$ has the following asymptotic behavior
$V(s)=-(n+\sigma(s))e^{3s},~{}s\geq 0.$ $None$
In other words, the function $V(s)$ blows up to $-\infty$ as fast as the
function $-e^{3s}$ as $s\to\infty$.
We need also get the asymptotic behavior of $V^{\prime}(s)$ as $s\to-\infty$.
For this purpose, consider the representation (4.3) in terms of the variable
$t=-s$. Using the estimate (4.16) and (3.9), we see that the factor in front
of $e^{-3t}$ on the right-hand side of (4.4) is bounded. This establishes
$V_{t}=O(e^{-3t})$ as $t\to\infty$. Therefore, we obtain the asymptotic
estimate
$V^{\prime}(s)=O(e^{3s}),~{}s\to-\infty.$ $None$
It is clear that Lemma $5$ implies some kind of uniqueness property for the
boundary value problem (3.12). More precisely, we state
###### Lemma 7
Up to translations, $s\mapsto s+s_{0}$, the two-point boundary value problem
(3.12) has a unique solution.
###### Proof 6
Let $V_{1}$ and $V_{2}$ be two solutions of (3.12). Then they are all
negative-valued and strictly decreasing and their behavior indicates that
there exists a unique point $s_{0}$ so that $V_{1}(0)=V_{2}(s_{0})$. Set
$V_{3}(s)=V_{2}(s+s_{0})$, then both $V_{1}$ and $V_{3}$ are solutions of the
differential equation in (3.12) and $V_{1}(0)=V_{3}(0)$. Using lemma $4.5$, we
have $V^{\prime}_{1}(0)=V^{\prime}_{3}(0)$. Applying the uniqueness theorem
for the initial value problem of an ordinary differential equation, we have
$V_{1}~{}\equiv~{}V_{3}$, namely, $V_{1}(s)=V_{2}(s+s_{0})$ for all $s$ and
the lemma follows.
Now consider the boundary behavior of the function $G=Q(V)$. Using
$\frac{dG}{dV}=Q^{\prime}(V)=\frac{1}{(1-e^{G(V)})}$, $G\to-\infty$ as
$V\to-\infty$ and the L’Hospital rule, we have
$\lim\limits_{s\to\infty}\frac{G(s)}{V(s)}=\lim\limits_{s\to\infty}\frac{1}{(1-e^{Q(V)})}=1.$
$None$
Combining (4.20) and (4.22), we see that for any $\epsilon>0$ there is a
number $S_{\epsilon}>0$ so that
$(1+\epsilon)V(s)\leq G(s)\leq(1-\epsilon)V(s),~{}s\geq S_{\epsilon}.$ $None$
With this estimate, we can consider $G^{\prime}(s)$ in terms of
$V^{\prime}(s)$ when $s\to\infty$. Indeed, using the relation between $G(s)$
and $V(s)$, we have, for sufficiently large $s>0$,
$G^{\prime}(s)=(1-e^{G(s)})^{-1}V^{\prime}(s)=(1+e^{G(s)}+O(e^{2G(s)}))V^{\prime}(s).$
$None$
Similarly, we need to consider the asymptotics of $G(s)$ and $G^{\prime}(s)$
as $s\to-\infty$. Using the relation $V=G-e^{G}$, we have
$V=-1-\frac{1}{2}G(s)^{2}+O(G(s)^{3}).$ $None$
for $G(s)$ near zero. Applying (4.17) in (4.25), we obtain the estimate
$-C(\epsilon)e^{2(1-\epsilon)s}<G(s)<0,$ $None$
where $\epsilon>0$ can be made arbitrarily small and $C(\epsilon)>0$ is a
constant depending on $\epsilon$. Note again that, modulo $\epsilon$, the
estimate (4.26) is sharp. In terms of $t=-s$, $G(t)=O(e^{-2(1-\epsilon)t})$ as
$t\to\infty$. Inserting this into (4.3) and noting that
$R(V)=-2(1-e^{Q(V)})^{2}=O(e^{-4(1-\epsilon)t})$, we see that
$V_{tt}+3V_{t}=O(e^{-4(1-\epsilon)t})$. From this we get the estimate
$V(t)=O(e^{-4(1-\epsilon)t})$. Note that (4.3) indicates that $V_{tt}>0$,
since $V_{t}>0$. Hence $V_{t}$ is decreasing. Therefore, there holds
$V_{t}<V(t)-V(t-1)=O(e^{-4(1-\epsilon)t}),~{}t\geq 1.$ $None$
Consequently, returning to the variable $s=-t$, we obtain the improved
estimate
$V^{\prime}(s)=O(e^{4(1-\epsilon)s}),~{}as~{}s\to-\infty,$ $None$
over (4.21). Inserting this result into the relation
$G^{\prime}(s)=(1-e^{G(s)})^{-1}V^{\prime}(s)=-(1+O(G(s)))^{-1}(G(s))^{-1}V^{\prime}(s),$
$None$
we acquire the asymptotic estimate
$G^{\prime}(s)=O(e^{2(1-\epsilon)s}),~{}as~{}s\to-\infty,$ $None$
which is compatible with (4.26).
We will then return to the original variable $r$ and give the asymptotic
estimate of $U$ and $K$ in terms of $r$. Note that we once applied the
variable transformation
$\tau=ar,~{}s=\ln\tau,$ $None$
and the function transformation
$f(s)=K^{2},~{}G=\ln f,~{}V=G-e^{G}.$ $None$
Hence, both $U$ and $K$ in the original boundary problem can be represented
with $G$. Applying (4.23) and (4.24), and with the understanding that the
arbitrarily small constant $\epsilon>0$ is omitted in the final expression to
simplify the notation, we arrive at
$K=O(e^{-Cr^{3}}),~{}C>0,~{}as~{}r\to\infty.$ $None$
In fact, the second equation of (2.12) origins from
$\frac{d(U-1)}{d\tau}=\frac{(1-K^{2})^{2}}{(\tau)^{4}},$ $None$
where $\tau=ar$, $a=(\frac{2g^{2}}{3}H_{0})_{\frac{1}{3}}>0$. Therefore, when
$r\to\infty$, thereby $r\to\infty$, we can get
$\frac{d(U-1)}{d\tau}=O((\tau)^{-4}).$ $None$
It follows immediately that
$U=1+O((\tau)^{-3})=1+O(r^{-3}),~{}as~{}r\to\infty.$ $None$
Similarly, from (4.29) and (4.30) we can acquire
$K~{}=~{}1+O(r^{2}),~{}U~{}=~{}O(r),~{}as~{}r\to 0.$ $None$
The proof of Theorem 2.1 is now complete.
###### Lemma 8
The correct shooting slope, $-n<0$, depends on $m$ continuously and
monotonically so that $n(m_{1})>n(m_{2})>0$ for $m_{1}<m_{2}<-1$.
###### Proof 7
We have seen that for any given $m<-1$, there is a unique number $n>0$ so that
the unique solution of the initial value problem (4.1) gives a negative valued
solution $V$ which solves the two-point boundary value problem
(3.12)(cf.lemma4.5). Thus we can denote this well-defined correspondence as
$n=n(m)$ and $V=V_{m}$. We show that $n(m)$ is continuous with respect to
$m<-1$. Let $\\{m_{j}\\}$ be a sequence in $(-\infty,-1)$ which converges to a
number $m_{0}<0$. We need to prove that $n(m_{j})\to n(m_{0})$ as
$j\to\infty$. Suppose otherwise that this is not true. Then, without loss of
generality, we may assume that there is an $\epsilon_{0}>0$ so that
$|n(m_{j})-n(m_{0})|\geq\epsilon$ for all $j=1,2,...$. On the other hand, we
can use lemma 4.6 to obtain a sequence $\\{s_{j}\\}$ so that
$V_{m_{j}}(s)=V_{m_{0}}(s_{j}+s)$ for all $s$. In particular,
$m_{j}=V_{m_{j}}(0)=V_{m_{0}}(s_{j})$ for $j=1,2,....$ It is clear that
$\\{s_{j}\\}$ is a bounded sequence otherwise it would contradict the
assumption $m_{j}\to m_{0}<0~{}(j\to 0)$ and the fact that
$V_{m_{0}}(-\infty)=-1$ and $V_{m_{0}}(\infty)=-\infty$. By extracting a
subsequence if necessary, we may assume that $s_{j}\to some~{}s_{0}$ as
$j\to\infty$. Therefore, we have, as $j\to\infty$,
$n(m_{j})=V^{\prime}_{m_{j}}(0)=V^{\prime}_{m_{0}}(s_{j})\to
V^{\prime}_{m_{0}}(s_{0})~{}:=~{}n_{0}\neq n(m_{0})$. On the other hand,
$m_{j}\to-1$ as $j\to\infty$, and $m_{j}=V_{m_{j}}(0)=V_{m_{0}}(s_{j})$ for
$j=1,2,...$ imply that $s_{j}\to 0$ as $j\to\infty$ since $V_{m_{0}}$ is
strictly monotone. Hence $s_{0}=0$ and we arrive at a contradiction.
The continuous dependence of $n(m)$ on $m$ implies that the solution $V_{m}$
depends on $m$ continuously as well.We claim that $n(m)\to 0$ as $m\to 0^{-}$.
Otherwise there is a sequence $\\{m_{j}\\}$ in $(-\infty,-1)$ and an
$\epsilon_{0}$ so that $m_{j}\to 0$ as $j\to\infty$ but
$n(m_{j})\geq\epsilon_{0}(j=1,2,...)$. Using these in the initial value
problem (4.3) with $m=m_{j}$ and $n=n(m_{j})$, we observe that the solution
will assume a positive value for a slightly positive $t$ when $j$ is
sufficiently large, which contradicts the definition of $n(m_{j})$.
We can also claim that $n(m)\to\infty$ as $m\to-\infty$. Let $V_{0}$ be a
fixed solution of (3.12). Then there is a unique $s_{m}$ so that
$V_{m}(s)=V_{0}(s_{m}+s)$(cf. lemma6). Since $m=V_{m}(0)=V_{0}(s_{m})$, we
conclude that $s_{m}\to\infty$ as $m\to-\infty$. Consequently,
$n(m)=-V^{\prime}_{m}(0)$ as $m\to-\infty$ as claimed.
Remarks Our analysis suggests a dynamical shooting method for constructing the
unique solution of the generalized monopole problem in six-dimension non-
Abelian gauge. We have seen that we may start from the initial value problem
(4.1) with an arbitrary $m<0$. The sets of undesired shooting data,
$\beta^{-}$ and $\beta^{+}$ are two open intervals $\beta^{-}=(-\infty,n)$ and
$\beta^{+}=(b,\infty)$. The correct shooting slope, $-n<0$, depends on $m$
continuously and monotonically so that $n(m_{1})>n(m_{2})>0$ for
$m_{1}<m_{2}<-1$.
## Acknowledgments
The research of the first author was supported in part by the Natural Science
Fund of Henan Education Office (2007110004) and (2008A110002).
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|
arxiv-papers
| 2010-03-07T07:42:05 |
2024-09-04T02:49:08.873152
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shouxin Chen and Long Pei",
"submitter": "Shouxin Chen",
"url": "https://arxiv.org/abs/1003.1441"
}
|
1003.1443
|
11institutetext: Rutgers University, 11email: troyjlee@gmail.com
22institutetext: The Chinese University of Hong Kong, 22email:
syzhang@cse.cuhk.edu.hk
# Composition theorems in communication complexity
Troy Lee 11 Shengyu Zhang 22
###### Abstract
A well-studied class of functions in communication complexity are composed
functions of the form $(f\circ
g^{n})(x,y)=f(g(x^{1},y^{1}),\ldots,g(x^{n},y^{n}))$. This is a rich family of
functions which encompasses many of the important examples in the literature.
It is thus of great interest to understand what properties of $f$ and $g$
affect the communication complexity of $(f\circ g^{n})$, and in what way.
Recently, Sherstov [She09] and independently Shi-Zhu [SZ09b] developed
conditions on the inner function $g$ which imply that the quantum
communication complexity of $f\circ g^{n}$ is at least the approximate
polynomial degree of $f$. We generalize both of these frameworks. We show that
the pattern matrix framework of Sherstov works whenever the inner function $g$
is strongly balanced—we say that $g:X\times Y\rightarrow\\{-1,+1\\}$ is
strongly balanced if all rows and columns in the matrix $M_{g}=[g(x,y)]_{x,y}$
sum to zero. This result strictly generalizes the pattern matrix framework of
Sherstov [She09], which has been a very useful idea in a variety of settings
[She08b, RS08, Cha07, LS09a, CA08, BHN09].
Shi-Zhu require that the inner function $g$ has small spectral discrepancy, a
somewhat awkward condition to verify. We relax this to the usual notion of
discrepancy.
We also enhance the framework of composed functions studied so far by
considering functions $F(x,y)=f(g(x,y))$, where the range of $g$ is a group
$G$. When $G$ is Abelian, the analogue of the strongly balanced condition
becomes a simple group invariance property of $g$. We are able to formulate a
general lower bound on $F$ whenever $g$ satisfies this property.
## 1 Introduction
Communication complexity studies the minimum amount of communication needed to
compute a function whose input variables are distributed between two or more
parties. Since the introduction by Yao [Yao79] of an elegant mathematical
model to study this question, communication complexity has grown into a rich
field both because of its inherent mathematical interest and also its
application to many other models of computation. See the textbook of
Kushilevitz and Nisan [KN97] for a comprehensive introduction to the field.
In analogy with traditional computational complexity classes, one can consider
different models of communication complexity based on the resources available
to the parties. Besides the standard deterministic model, of greatest interest
to us will be a randomized version of communication complexity, where the
parties have access to a source of randomness and are allowed to err with some
small constant probability, and a quantum model where the parties share a
quantum channel and the cost is measured in qubits.
Several major open questions in communication complexity ask about how
different complexity measures relate to each other. The log rank conjecture,
formulated by Lovász and Saks [LS88], asks if the deterministic communication
complexity of a Boolean function $F:X\times Y\rightarrow\\{0,1\\}$ is upper
bounded by a polynomial in the logarithm of the rank of the matrix
$[F(x,y)]_{x,y}$. Another major open question is if randomized and quantum
communication complexity are polynomially related for all total functions. We
should mention here that the assumption of the function being total is crucial
as an exponential separation is known for a partial function [Raz99].
One approach to these questions has been to study them for restricted classes
of functions. Many functions of interest are block composed functions. For
finite sets $X,Y$, and $E$, a function $f:E^{n}\rightarrow\\{-1,+1\\}$, and a
function $g:X\times Y\rightarrow E$, the block composition of $f$ and $g$ is
the function $f\circ g^{n}:X^{n}\times Y^{n}\to\\{-1,+1\\}$ defined by
$(f\circ g^{n})(x,y)=f(g(x^{1},y^{1}),\ldots,g(x^{n},y^{n}))$ where
$(x^{i},y^{i})\in X\times Y$ for all $i=1,\ldots,n$. For example, if
$E=\\{-1,+1\\}$, the inner product function results when $f$ is PARITY and $g$
is AND, set-intersection when $f$ is OR and $g$ is AND, and the equality
function when $f$ is AND and $g$ is the function IS-EQUAL, which is one if and
only if $x=y$.
In a seminal paper, Razborov [Raz03] gave tight bounds for the bounded-error
quantum communication complexity of block composed functions where the outer
function $f$ is symmetric and the inner function $g$ is bitwise AND. In
particular, this result showed that randomized and quantum communication
complexity are polynomially related for such functions.
More recently, very nice frameworks have been developed by Sherstov [She07,
She09] and independently by Shi and Zhu [SZ09b] to bound the quantum
complexity of block composed functions that goes beyond the case of symmetric
$f$ to work for any $f$ provided the inner function $g$ satisfies certain
technical conditions. When $g$ satisfies these conditions, this framework
allows one to lower bound the quantum communication complexity of $f\circ
g^{n}$ in terms of the approximate polynomial degree of $f$, a classically
well-studied measure. Shi and Zhu are able to get a bound on $f\circ g^{n}$ in
terms of the approximate degree of $f$ whenever $g$ is sufficiently
“hard”—unfortunately, the hardness condition they need is in terms of
“spectral discrepancy,” a quantity which is somewhat difficult to bound, and
their bound requires that $g$ is a function on at least $\Omega(\log(n/d))$
bits, where $d$ is the approximate polynomial degree of $f$. Because of this,
Shi-Zhu are only able to reproduce Razborov’s results with a polynomially
weaker bound.
Sherstov developed so-called pattern matrices which are the matrix
representation of a block composed function when $g$ is a fixed function of a
particularly nice form. Namely, in a pattern matrix the inner function
$g:\\{-1,+1\\}^{k}\times([k]\times\\{-1,+1\\})\to\\{-1,+1\\}$ is parameterized
by a positive integer $k$ and defined by $g(x,(i,b))=x_{i}\cdot b$, where
$x_{i}$ denotes the $i^{th}$ bit of $x$. In other words, the first argument of
$g$ is a $k$ bit string $x$, and the second argument selects a bit of $x$ or
its negation. So here $X=\\{-1,+1\\}^{k}$, $Y=[k]\times\\{-1,+1\\}$ and the
intermediate set $E$ is $\\{-1,+1\\}$. With this $g$, Sherstov shows that the
approximate polynomial degree of $f$ is a lower bound on the quantum
communication complexity of $f\circ g^{n}$, for any function $f$. Though
seemingly quite special, pattern matrices have proven to be an extremely
useful concept. First, they give a simple proof of Razborov’s tight lower
bounds for $f(x\wedge y)$ for symmetric $f$. Second, they have also found many
other applications in unbounded-error communication complexity [She08b, RS08]
and have been successfully extended to multiparty communication complexity
[Cha07, LS09a, CA08, BHN09].
A key step in both the works of Sherstov and Shi-Zhu is to bound the spectral
norm of a sum of matrices $\|\sum_{i}{B_{i}}\|$. This is the major step where
these works differ. Shi-Zhu apply the triangle inequality to bound this as
$\|\sum_{i}{B_{i}}\|\leq\sum_{i}\|B_{i}\|$. On the other hand, Sherstov
observes that in the case of pattern matrices the terms of this sum are
mutually orthogonal, i.e. $B_{i}^{\dagger}B_{j}=B_{i}B_{j}^{\dagger}=0$ for
all $i\neq j$. In this case, one has a stronger bound on the spectral norm
$\|\sum_{i}{B_{i}}\|=\max_{i}\|B_{i}\|$.
In this paper, we extend both of the frameworks of Sherstov and Shi-Zhu. In
the case of Shi-Zhu, we are able to reprove their theorem with the usual
notion of discrepancy instead of the somewhat awkward spectral discrepancy
they use. The main observation we make is that as all Shi-Zhu use in this step
is the triangle inequality, we can repeat the argument with any norm here,
including discrepancy itself.
In the case of pattern matrices, special properties of the spectral norm are
used, namely the fact about the spectral norm of a sum of orthogonal matrices.
We step back to see what key features of a pattern matrix lead to this
orthogonality property. We begin with the Boolean case, that is, where the
intermediate set $E$ is taken to be $\\{-1,+1\\}$. In this case, a crucial
concept is the notion of a strongly balanced function. We say that $g:X\times
Y\rightarrow\\{-1,+1\\}$ is strongly balanced if in the sign matrix
$M_{g}[x,y]=g(x,y)$ all rows and all columns sum to zero. We show that
whenever the inner function $g$ is strongly balanced, the key orthogonality
condition holds; this implies that whenever $g$ is strongly balanced and the
communication matrix of $g$ has rank larger than one, the approximate degree
of the outer function $f$ is a lower bound on the quantum communication
complexity of $f\circ g^{n}$.
The requirement that the communication matrix of $g$ has rank larger than one
is necessary for such a statement. For example, when $g(x,y)=\oplus(x,y)$ is
the XOR function on one bit, then the communication complexity of
$\mathrm{PARITY}\circ g^{n}$ is constant, while PARITY has linear approximate
polynomial degree. It turns out that when $g$ is rank-one, the appropriate
measure of the complexity of $f\circ g^{n}$ is no longer the approximate
degree of $f$, but the minimum $\ell_{1}$ norm of Fourier coefficients of a
function entrywise close to $f$; see the survey [LS09b] for a description of
this case.
We also consider the general case where the intermediate set is any group $G$.
That is, we consider functions $F(x,y)=f(g(x,y))$, where $g:X\times
Y\rightarrow G$ for a group $G$ and $f:G\rightarrow\\{-1,+1\\}$ is a class
function on $G$. The case $E=\\{-1,+1\\}$ discussed above corresponds to
taking the group $G=\mathbb{Z}_{2}^{n}$. When $G$ is a general Abelian group,
the key orthogonality condition requires more than that the matrix
$M_{g}[x,y]=g(x,y)$ is strongly balanced; still, it admits a nice
characterization in terms of group invariance. A multiset $T\in G\times G$ is
said to be $G$-invariant if $(s,s)T=T$ for all $s\in G$. The orthogonality
condition will hold if and only if all pairs of rows and all pairs of columns
of $M_{g}$ (when viewed as multisets) are $G$\- invariant. One can generalize
the results discussed above to this general setting with appropriate
modifications. In the case that $G=\mathbb{Z}_{2}^{n}$, the $G$-invariant
condition degenerates to the strongly balanced requirement of $M_{g}$.
## 2 Preliminaries
All logarithms are base two. For a complex number $z=a+ib$ we let
$\bar{z}=a-ib$ denote the complex conjugate of $z$ and
$|z|=\sqrt{a^{2}+b^{2}}$ and $\mathrm{Re}(z)=a$.
### 2.1 Complexity measures
We will make use of several complexity measures of functions and matrices. Let
$f:\\{-1,+1\\}^{n}\rightarrow\\{-1,+1\\}$ be a function. For
$T\subseteq\\{0,1\\}^{n}$, the Fourier coefficient of $f$ corresponding to the
character $\chi_{T}$ is
$\hat{f}_{T}=\frac{1}{2^{n}}\sum_{x}f(x)\chi_{T}(x)=\frac{1}{2^{n}}\sum_{x}f(x)\prod_{i\in
T}x_{i}.$ The degree of $f$ as a polynomial, denoted $\deg(f)$, is the size of
a largest set $T$ for which $\hat{f}_{T}\neq 0$.
We will need some notations for matrices. We reserve $J$ for the all ones
matrix, whose size will be determined by the context. For a matrix $A$ let
$A^{\dagger}$ denote the conjugate transpose of $A$. We use $A\bullet B$ for
the entrywise product of $A,B$, and $A\otimes B$ for the tensor product. If
$A$ is an $m$-by-$n$ matrix then we say that $\mathrm{size}(A)=mn$. We use
$\langle A,B\rangle=\mathrm{Tr}(AB^{\dagger})$ for the inner product of $A$
and $B$.
Let $\|A\|_{1}$ be the $\ell_{1}$ norm of $A$, i.e. sum of the absolute values
of entries of $A$, and $\|A\|_{\infty}$ the $\ell_{\infty}$ norm. For a
positive semidefinite matrix $M$ let
$\lambda_{1}(M)\geq\cdots\geq\lambda_{n}(M)\geq 0$ be the eigenvalues of $M$.
We define the $i^{th}$ singular value of $A$, denoted $\sigma_{i}(A)$, as
$\sigma_{i}(A)=\sqrt{\lambda_{i}(AA^{\dagger})}$. The rank of $A$, denoted
$\mathrm{rk}(A)$ is the number of nonzero singular values of $A$. We will use
several matrix norms. The spectral or operator norm is the largest singular
value $\|A\|=\sigma_{1}(A)$, the trace norm is the summation of all singular
values $\|A\|_{tr}=\sum_{i}\sigma_{i}(A)$, and the Frobenius norm is the
$\ell_{2}$ norm of the singular values
$\|A\|_{F}=\sqrt{\sum_{i}\sigma_{i}(A)^{2}}$.
When $AB^{\dagger}=A^{\dagger}B=0$ we will say that $A,B$ are orthogonal.
Please note the difference with the common use of this term, which usually
means $\langle A,B\rangle=0$. The following facts are easily seen.
###### Fact 2.1
Let $A,B$ be two matrices of the same dimensions and suppose that
$AB^{\dagger}=A^{\dagger}B=0$. Then
$\displaystyle\mathrm{rk}(A+B)=\mathrm{rk}(A)+\mathrm{rk}(B),\
\|A+B\|_{tr}=\|A\|_{tr}+\|B\|_{tr},\ \|A+B\|=\max\\{\|A\|,\|B\|\\}.$
Another norm we will use is the $\gamma_{2}$ norm, introduced to complexity
theory in [LMSS07], and familiar in matrix analysis as the Schur product
operator norm. The $\gamma_{2}$ norm can be viewed as a weighted version of
the trace norm.
###### Definition 1
$\gamma_{2}(A)=\max_{u,v:\|u\|=\|v\|=1}\|A\bullet uv^{\dagger}\|_{tr}.$
Here $A\bullet B$ denotes the entrywise product of $A$ and $B$. It is clear
from this definition that $\gamma_{2}(A)\geq\|A\|_{tr}/\sqrt{mn}$ for a
$m$-by-$n$ matrix $A$.
For a norm $\Phi$, the dual norm $\Phi^{*}$ is defined as
$\Phi^{*}(v)=\max_{u:\Phi(u)\leq 1}|\langle u,v\rangle|$. For example, the
$\ell_{\infty}$ norm is dual to the $\ell_{1}$ norm, and the spectral norm is
dual to the trace norm.
The norm $\gamma_{2}^{*}$, dual to the $\gamma_{2}$ norm, looks as follows.
###### Definition 2
$\gamma_{2}^{*}(A)=\max_{\begin{subarray}{c}u_{i},v_{j}\\\
\|u_{i}\|=\|v_{j}\|=1\end{subarray}}\sum_{i,j}A[i,j]\langle
u_{i},v_{j}\rangle.$
Another complexity measure we will make use of is discrepancy
###### Definition 3
Let $A$ be an $m$-by-$n$ sign matrix and let $P$ be a probability distribution
on the entries of $A$. The discrepancy of $A$ with respect to $P$, denoted
$\mathrm{disc}_{P}(A)$, is defined as
$\mathrm{disc}_{P}(A)=\max_{\begin{subarray}{c}x\in\\{0,1\\}^{m}\\\
y\in\\{0,1\\}^{n}\end{subarray}}|x^{\dagger}A\bullet Py|.$
We will write $\mathrm{disc}_{U}(A)$ for the special case where $P$ is the
uniform distribution. It is easy to see from this definition that
$\mathrm{disc}_{U}(A)\leq\frac{\|A\|}{\sqrt{\mathrm{size}(A)}}.$ Shaltiel
[Sha03] has shown the deeper result that this bound is in fact polynomially
tight:
###### Theorem 2.2 (Shaltiel)
Let $A$ be a sign matrix. Then
$\frac{1}{108}\left(\frac{\|A\|}{\sqrt{\mathrm{size}(A)}}\right)^{3}\leq\mathrm{disc}_{U}(A).$
Discrepancy and the $\gamma_{2}^{*}$ norm are very closely related. Linial and
Shraibman [LS09c] observed that Grothendieck’s inequality gives the following.
###### Theorem 2.3 (Linial-Shraibman)
For any sign matrix $A$ and probability distribution $P$
$\mathrm{disc}_{P}(A)\leq\gamma_{2}^{*}(A\bullet P)\leq
K_{G}\;\mathrm{disc}_{P}(A)$
where $1.67\ldots\leq K_{G}\leq 1.78\ldots$ is Grothendieck’s constant.
#### Approximate measures
We will also use approximate versions of these complexity measures which come
in handy when working with bounded-error models. Say that a function $g$ gives
an $\epsilon$-approximation to $f$ if $|f(x)-g(x)|\leq\epsilon$ for all
$x\in\\{-1,+1\\}^{n}$. The $\epsilon$-approximate polynomial degree of $f$,
denoted $\deg_{\epsilon}(f)$, is the minimum degree of a function $g$ which
gives an $\epsilon$-approximation to $f$.
We will similarly look at the $\epsilon$-approximate version of the trace and
$\gamma_{2}$ norms. We give the general definition with respect to any norm.
###### Definition 4 (approximation norm)
Let $\Phi:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be an arbitrary norm. Let
$v\in\mathbb{R}^{n}$ be a sign vector. For $0\leq\epsilon<1$ we define the
approximation norm $\Phi^{\epsilon}$ as
$\Phi^{\epsilon}(v)=\min_{\begin{subarray}{c}u\\\
\|v-u\|_{\infty}\leq\epsilon\end{subarray}}\Phi(u).$
Notice that an approximation norm $\Phi^{\epsilon}$ is not itself a norm— we
have only defined it for sign vectors, and it will in general not satisfy the
triangle inequality.
As a norm is a convex function, using the separating hyperplane theorem one
can quite generally give the following equivalent dual formulation of an
approximation norm.
###### Proposition 1
Let $v\in\mathbb{R}^{n}$ be a sign vector, and $0\leq\epsilon<1$
$\Phi^{\epsilon}(v)=\max_{u}\frac{|\langle
v,u\rangle|-\epsilon\|u\|_{1}}{\Phi^{*}(u)}$
A proof of this can be found in the survey [LS09b].
### 2.2 Communication complexity
Let $X,Y,S$ be finite sets and $f:X\times Y\rightarrow S$ be a function. We
will let $D(f)$ be the deterministic communication complexity of $f$, and
$R_{\epsilon}(f)$ denote the randomized public coin complexity of $f$ with
error probability at most $\epsilon$. We refer to the reader to [KN97] for a
formal definition of these models. We will also study $Q_{\epsilon}(f)$ and
$Q_{\epsilon}^{*}(f)$, the $\epsilon$-error quantum communication complexity
of $f$ without and with shared entanglement, respectively. We refer the reader
to [Raz03] for a nice description of these models.
For notational convenience, we will identify a function $f:X\times
Y\rightarrow\\{-1,+1\\}$ with its sign matrix $M_{f}=[f(x,y)]_{x,y}$. Thus,
for example, $\|f\|$ refers to the spectral norm of the sign matrix
representation of $f$.
For all of our lower bound results we will actually lower bound the
approximate trace norm or $\gamma_{2}$ norm of the function. Razborov showed
that the approximate trace norm can be used to lower bound on quantum
communication complexity, and Linial and Shraibman generalized this to the
$\gamma_{2}$ norm.
###### Theorem 2.4 (Linial-Shraibman [LS09d])
Let $A$ be a sign matrix and $0\leq\epsilon<1/2$. Then
$Q_{\epsilon}^{*}(A)\geq\log\left(\gamma_{2}^{2\epsilon}(A)\right)-2.$
#### Composed functions
Before discussing lower bounds on a block composed function $f\circ g^{n}$,
let us see what we expect the complexity of such a function to be. A
fundamental idea going back to Nisan [Nis94] and Buhrman, Cleve, and Wigderson
[BCW98], is that the complexity of $f\circ g^{n}$ can be related to the
decision tree complexity, also known as query complexity, of $f$ and the
communication complexity of $g$. Let $\mathrm{DT}(f)$ be the query complexity
of $f$, that is the number of queries of the form $x_{i}=?$ needed to evaluate
$f(x)$ in the worst case. Similarly, let
$\mathrm{RT}_{\epsilon}(f),\mathrm{QT}_{\epsilon}(f)$ denote the randomized
and quantum query complexity of $f$ respectively, with error probability at
most $\epsilon$. For formal definitions of these measures and a survey of
query complexity we recommend Buhrman and de Wolf [BW02].
###### Theorem 2.5 (Nisan [Nis94], Buhrman-Cleve-Wigderson [BCW98])
For any two Boolean functions $f:\\{-1,+1\\}^{n}\rightarrow\\{-1,+1\\}$ and
$g:X\times Y\rightarrow\\{-1,+1\\}$,
$\displaystyle D(f\circ g^{n})$ $\displaystyle=O(\mathrm{DT}(f)D(g))$
$\displaystyle R_{1/4}(f\circ g^{n})$
$\displaystyle=O(\mathrm{RT}_{1/4}(f)R_{1/4}(g)\log\mathrm{RT}_{1/4}(f))$
$\displaystyle Q_{1/4}(f\circ g^{n})$
$\displaystyle=O(\mathrm{QT}_{1/4}(f)Q_{1/4}(g)\log n).$
One advantage of working with block composed functions in light of this upper
bound is that query complexity is in general better understood than
communication complexity. In particular, a polynomial relationship between
deterministic query complexity and degree, and randomized and quantum query
complexities and approximate degree is known.
###### Theorem 2.6 ([NS94, BBC+01])
Let $f:\\{0,1\\}^{n}\rightarrow\\{-1,+1\\}$. Then
$\displaystyle\mathrm{DT}(f)=O(\deg(f)^{4}),\quad\mathrm{DT}(f)=O(\deg_{1/4}(f)^{6})$
Using this result together with Theorem 2.5 gives the following corollary:
###### Corollary 1
$\displaystyle D(f\circ g^{n})$ $\displaystyle=O(\deg(f)^{4}D(g)),\quad
R_{1/4}(f\circ g^{n})=O(\deg_{1/4}(f)^{6}R_{1/4}(g)\log\deg_{1/4}(f))$
Our goal, then, in showing lower bounds on the complexity of a block composed
function $f\circ g^{n}$ is to get something at least in the ballpark of this
upper bound. Of course, this is not always possible — the protocol given by
Theorem 2.5 is not always optimal. For example, when $f$ is the PARITY
function on $n$ bits, and $g(x,y)=\oplus(x,y)$ this protocol just gives an
upper bound of $n$ bits, when the true complexity is constant. See recent
results by Zhang [Zha09] and Sherstov [She10] for discussions on the tightness
of the bounds in Theorem 2.5.
## 3 Rank of block composed functions
We begin by analyzing the rank of a block composed function $f\circ g^{n}$
when the inner function $g$ is strongly balanced. This case will illustrate
the use of the strongly balanced assumption, and is simpler to understand than
the bounded-error situation treated in the next section.
Let us first formally state the definition of strongly balanced.
###### Definition 5 (strongly balanced)
Let $A$ be a sign matrix, and $J$ be the all ones matrix of the same
dimensions as $A$. We say that $A$ is balanced if
$\mathrm{Tr}(AJ^{\dagger})=0$. We further say that $A$ is strongly balanced if
$AJ^{\dagger}=A^{\dagger}J=0$. In words, a sign matrix is strongly balanced if
the sum over each row is zero, and similarly the sum over each column is zero.
We will say that a two-variable Boolean function is balanced or strongly
balanced if its sign matrix representation is.
###### Theorem 3.1
Let $f:\\{-1,+1\\}^{n}\rightarrow\\{-1,+1\\}$ be an arbitrary function, and
let $g$ be a strongly balanced function. Then
$\mathrm{rk}(M_{f\circ g^{n}})=\sum_{\begin{subarray}{c}T\subseteq[n],\
\hat{f}_{T}\neq 0\end{subarray}}\mathrm{rk}(M_{g})^{|T|}.$
###### Proof
Let us write out the sign matrix for $\chi_{T}\circ g^{n}$ explicitly. If we
let $M_{g}^{0}=J$ be the all ones matrix and $M_{g}^{1}=M_{g}$, then we can
nicely write the sign matrix representing
$\chi_{T}(g(x^{1},y^{1}),\ldots,g(x^{n},y^{n}))$ as
$M_{\chi_{T}\circ g^{n}}=\bigotimes_{i}M_{g}^{T[i]}$
where $T[i]=1$ if $i\in T$ and $0$ otherwise.
We see that the condition on $g$ implies $M_{\chi_{T}\circ
g^{n}}M_{\chi_{S}\circ g^{n}}^{\dagger}=0$ if $S\neq T$. Indeed,
$\displaystyle M_{\chi_{T}\circ g^{n}}M_{\chi_{S}\circ g^{n}}^{\dagger}$
$\displaystyle=\left(\bigotimes_{i}M_{g}^{T[i]}\right)\left(\bigotimes_{i}M_{g}^{S[i]}\right)^{\dagger}$
$\displaystyle=\bigotimes_{i}\left(M_{g}^{T[i]}(M_{g}^{S[i]})^{\dagger}\right)=0.$
This follows since, by the assumption $S\neq T$, there is some $i$ for which
$S[i]\neq T[i]$ which means that this term is either $M_{g}J^{\dagger}=0$ or
$JM_{g}^{\dagger}=0$ because $g$ is strongly balanced. The other case follows
similarly.
Now that we have established this property, we can use Fact 2.1 to obtain
$\displaystyle\mathrm{rk}(M_{f\circ g^{n}})$
$\displaystyle=\mathrm{rk}\Big{(}\sum_{T\subseteq[n]}\hat{f}_{T}\chi_{T}(g(x^{1},y^{1}),\ldots,g(x^{n},y^{n}))\Big{)}$
$\displaystyle=\sum_{\begin{subarray}{c}T\subseteq[n]\\\ \hat{f}_{T}\neq
0\end{subarray}}\mathrm{rk}(M_{\chi_{T}\circ g^{n}})$
$\displaystyle=\sum_{\begin{subarray}{c}T\subseteq[n]\\\ \hat{f}_{T}\neq
0\end{subarray}}\mathrm{rk}(M_{g})^{|T|}$
In the last step we used the fact that rank is multiplicative under tensor
product.
Theorem 3.1 has the following implication for the log rank conjecture of the
composed function with the assumption of the same conjecture for the inner
function.
###### Corollary 2
Let $X,Y$ be finite sets, $g:X\times Y\rightarrow\\{-1,+1\\}$ be a strongly
balanced function, and $M_{g}[x,y]=g(x,y)$ be the corresponding sign matrix.
Let $f:\\{-1,+1\\}^{n}\rightarrow\\{-1,+1\\}$ be an arbitrary function. Assume
that $\mathrm{rk}(M_{g})\geq 2$ and further suppose that there is a constant
$c$ such that $D(g)\leq(\log\mathrm{rk}(M_{g}))^{c}$. Then
$D(f\circ g^{n})=O(\log\mathrm{rk}(f\circ g)^{4+c}).$
###### Proof
By Corollary 1, $D(f\circ
g^{n})=O(\deg(f)^{4}D(g))=O(\deg(f)^{4}(\log\mathrm{rk}(M_{g}))^{c})$. Now, it
follows from Theorem 3.1 that $\log\mathrm{rk}(f\circ
g)\geq\deg(f)\log\mathrm{rk}(M_{g})$ as by definition of degree there is some
$T\subseteq\\{0,1\\}^{n}$ with $|T|=\deg(f)$ and $\hat{f}_{T}\neq 0$.
In particular, this Corollary means that whenever $g$ is a strongly balanced
function on a constant number of bits and $\mathrm{rk}(M_{g})>1$, then the log
rank conjecture holds for $f\circ g^{n}$. If $g$ is strongly balanced and
$\mathrm{rk}(M_{g})=1$ then, up to permutation of rows and columns, which does
not change the communication complexity, $M_{g}$ is a tensor product of the
XOR function with an all ones matrix. The log rank conjecture in the case
$(f\circ\oplus^{n})(x,y)=f(x_{1}\oplus y_{1},\ldots,x_{n}\oplus y_{n})$
remains an interesting open question. Shi and Zhang [SZ09a] have recently
resolved this question when $f$ is symmetric.
## 4 A bound in terms of approximate degree
In this section, we will address the frameworks of Sherstov and Shi-Zhu. We
extend both of these frameworks to give more general conditions on the inner
function $g$ which still imply that the approximate degree of $f$ is a lower
bound on the quantum query complexity of the composed function $f\circ g^{n}$.
In outline, both of these frameworks follow the same plan. By Theorem 2.4 it
suffices to lower bound the approximate $\gamma_{2}$ norm (or even approximate
trace norm) of $f\circ g^{n}$. To do this, they use the dual formulation given
by Proposition 1 and construct a witness matrix $B$ which has non-negligible
correlation with the target function and small $\gamma_{2}^{*}$ (or spectral)
norm.
A very nice way to construct this witness, used by both Sherstov and Shi-Zhu,
is to use the dual polynomial of $f$. This is a polynomial $v$ which certifies
that the approximate polynomial degree of $f$ is at least a certain value.
More precisely, duality theory of linear programming gives the following
lemma.
###### Lemma 1 (Sherstov [She09], Shi-Zhu [SZ09b])
Let $f:\\{-1,+1\\}^{n}$ $\rightarrow\\{-1,+1\\}$ and let
$d=\deg_{\epsilon}(f)$. Then there exists a function
$v:\\{-1,+1\\}^{n}\rightarrow\mathbb{R}$ such that
1. 1.
$\langle v,\chi_{T}\rangle=0$ for every character $\chi_{T}$ with $|T|<d$.
2. 2.
$\|v\|_{1}=1$.
3. 3.
$\langle v,f\rangle\geq\epsilon$.
Items (2),(3) are used to lower bound the correlation of the witness matrix
with the target matrix and to upper bound the $\ell_{1}$ norm of the witness
matrix. In the most difficult step, and where these works diverge, Item (1) is
used to upper bound the $\gamma_{2}^{*}$ (or spectral) norm of the witness
matrix.
We treat each of these frameworks separately in the next two sections.
### 4.1 Sherstov’s framework
The proof of the next theorem follows the same steps as Sherstov’s proof for
pattern matrices (Theorem 5.1 [She09]). Our main contribution is to identify
the strongly balanced condition as the key property of pattern matrices which
enables the proof to work.
###### Theorem 4.1
Let $X,Y$ be finite sets, $g:X\times Y\rightarrow\\{-1,+1\\}$ be a strongly
balanced function, and $M_{g}[x,y]=g(x,y)$ be the corresponding sign matrix.
Let $f:\\{-1,+1\\}^{n}\rightarrow\\{-1,+1\\}$ be an arbitrary function. Then
$Q_{\epsilon}^{*}(f\circ
g^{n})\geq\deg_{\epsilon_{0}}(f)\log_{2}\Big{(}\frac{\sqrt{|X||Y|}}{\|M_{g}\|}\Big{)}-O(1).$
for any $\epsilon>0$ and $\epsilon_{0}>2\epsilon$.
In particular, this result means that the quantum and randomized complexities
of $f\circ g^{n}$ are polynomially related whenever $g$ is strongly balanced
and $\log\tfrac{\sqrt{\mathrm{size}(M_{g})}}{\|M_{g}\|}$ is polynomially
related to the randomized communication complexity of $g$. While the
complexity measure of $g$ used here may look strange at first, Shaltiel
[Sha03] has shown that it is closely related to the discrepancy of $g$ under
the uniform distribution, as noted above in Theorem 2.2. This theorem strictly
generalizes the case of pattern matrices, but it could still be the case that
the results of Shi-Zhu can show bounds not possible with this theorem.
###### Proof (Proof of Theorem 4.1)
Let $d=\deg_{\epsilon_{0}}(f)$ and let $v$ be a dual polynomial for $f$ with
properties as in Lemma 1. We define a witness matrix as
$B[x,y]=\frac{2^{n}}{\mathrm{size}(M_{g})^{n}}v(g(x^{1},y^{1}),\ldots,g(x^{n},y^{n}))$
Let us first lower bound the inner product $\langle M_{f\circ
g^{n}},B\rangle$. Notice that as $M_{g}$ is strongly balanced, it is in
particular balanced, and so the number of ones (or minus ones) in $M_{g}$ is
$\mathrm{size}(M_{g})/2$.
$\displaystyle\langle M_{f\circ
g^{n}},B\rangle=\frac{2^{n}}{\mathrm{size}(M_{g})^{n}}\sum_{z\in\\{-1,+1\\}^{n}}f(z)v(z)\prod_{i=1}^{n}\Big{(}\sum_{\scriptsize
x^{i},y^{i}:\atop\scriptsize g(x^{i},y^{i})=z_{i}}1\Big{)}=\langle
f,v\rangle\geq\epsilon_{0}$
A similar argument shows that $\|B\|_{1}=1$ as $\|v\|_{1}=1$.
Now we turn to evaluate $\|B\|$. As shown above, the strongly balanced
property of $g$ implies that the matrices $\chi_{T}\circ g^{n}$ and
$\chi_{S}\circ g^{n}$ are orthogonal for distinct sets
$S,T\subseteq\\{0,1\\}^{n}$. We can thus use Fact 2.1 to compute as follows.
$\displaystyle\|B\|$
$\displaystyle=\frac{2^{n}}{\mathrm{size}(M_{g})^{n}}\|\sum_{T\subseteq[n]}\hat{v}_{T}M_{\chi_{T}\circ
g^{n}}\|$
$\displaystyle=\frac{2^{n}}{\mathrm{size}(M_{g})^{n}}\max_{T}|\hat{v}_{T}|\|M_{\chi_{T}\circ
g^{n}}\|$ $\displaystyle=\max_{T}\
2^{n}|\hat{v}_{T}|\prod_{i}\frac{\|M_{g}^{T[i]}\|}{\mathrm{size}(M_{g})}$
$\displaystyle\leq\max_{T:\hat{v}_{T}\neq
0}\prod_{i}\frac{\|M_{g}^{T[i]}\|}{\mathrm{size}(M_{g})}$
$\displaystyle=\left(\frac{\|M_{g}\|}{\sqrt{\mathrm{size}(M_{g})}}\right)^{d}\left(\frac{1}{\mathrm{size}(M_{g})}\right)^{n/2}$
In the second to last step we have used that $|\hat{v}_{T}|\leq 1/2^{n}$ as
$\|v\|_{1}=1$, and in the last step we have used the fact that
$\|J\|=\sqrt{\mathrm{size}(M_{g})}$.
Now putting everything together we have
$\displaystyle\frac{\|M_{f\circ
g^{n}}\|_{tr}^{\epsilon_{0}}}{\sqrt{\mathrm{size}(M_{f\circ
g^{n}})}}\geq\frac{1}{12}\left(\frac{\sqrt{\mathrm{size}(M_{g})}}{\|M_{g}\|}\right)^{d}$
The lower bound on quantum communication complexity now follows from Theorem
2.4.
Using the theorem of Shaltiel relating discrepancy to the spectral norm
Theorem 2.2, we get the following corollary:
###### Corollary 3
Let the quantities be defined as in Theorem 4.1.
$Q_{1/8}^{*}(f\circ
g^{n})\geq\frac{1}{3}\deg_{1/3}(f)\big{(}\log\big{(}\frac{1}{\mathrm{disc}_{U}(M_{g})}\big{)}-7\big{)}-O(1).$
Comparison to Sherstov’s pattern matrix: As mentioned in [She09], Sherstov’s
pattern matrix method can prove quantum lower bound of
$\Omega(\deg_{\epsilon}(f))$ for block composed functions $f\circ g^{n}$ if
the matrix $M_{g}$ contains the following $4\times 4$ one as a submatrix:
$S_{4}=\begin{bmatrix}1&-1&1&-1\\\ 1&-1&-1&1\\\ -1&1&1&-1\\\
-1&1&-1&1\end{bmatrix}$
In this paper we show that the same lower bound holds as long as $M_{g}$
contains a strongly balanced submatrix or rank greater than one. Are there
strongly balanced matrices not containing $S_{4}$ as a submatrix? It turns out
that the answer is yes: we give the following $6\times 6$ matrix as one
example.
$S_{6}=\begin{bmatrix}1&1&1&-1&-1&-1\\\ 1&1&-1&1&-1&-1\\\ 1&-1&-1&-1&1&1\\\
-1&-1&1&1&1&-1\\\ -1&1&-1&-1&1&1\\\ -1&-1&1&1&-1&1\end{bmatrix}$
### 4.2 Shi-Zhu framework
The method of Shi-Zhu does not restrict the form of the inner function $g$,
but rather works for any $g$ which is sufficiently “hard.” The hardness
condition they require is phrased in terms of a somewhat awkward measure they
term spectral discrepancy.
###### Definition 6 (spectral discrepancy)
Let $A$ be a $m$-by-$n$ sign matrix. The spectral discrepancy of $A$, denoted
$\rho(A)$, is the smallest $r$ such that there is a submatrix $A^{\prime}$ of
$A$ and a probability distribution $\mu$ on the entries of $A^{\prime}$
satisfying:
1. 1.
$A^{\prime}$ is balanced with respect to $\mu$, i.e. the distribution which
gives equal weight to $-1$ entries and $+1$ entries of $A^{\prime}$.
2. 2.
The spectral norm of $A^{\prime}\bullet\mu$ is small:
$\|A^{\prime}\bullet\mu\|\leq\frac{r}{\sqrt{\mathrm{size}(A^{\prime})}}$
3. 3.
The entrywise absolute value of the matrix $A^{\prime}\bullet\mu$ should also
have a bound on its spectral norm in terms of $r$:
$\||A^{\prime}\bullet\mu|\|\leq\frac{1+r}{\sqrt{\mathrm{size}(A^{\prime})}}$
While conditions (1),(2) in the definition of spectral discrepancy are quite
natural, condition (3) can be complicated to verify. Note that condition (3)
will always be satisfied when $\mu$ is taken to be the uniform distribution.
Using this notion of spectral discrepancy, Shi-Zhu show the following theorem.
###### Theorem 4.2 (Shi-Zhu [SZ09b])
Let $f:\\{-1,+1\\}^{n}\rightarrow\\{-1,+1\\}$, and $g:X\times
Y\rightarrow\\{-1,+1\\}$. For any $\epsilon$ and $\epsilon_{0}>2\epsilon$,
$Q_{\epsilon}(f\circ g^{n})\geq\Omega(\deg_{\epsilon_{0}}(f)).$
provided $\rho(M_{g})\leq\tfrac{\deg_{\epsilon_{0}}(f)}{2en}$. Here
$e=2.718\ldots$ is Euler’s number.
Chattopadhyay [Cha08] extended the technique of Shi-Zhu to the case of
multiparty communication complexity, answering an open question of Sherstov
[She08a]. In doing so, he gave a more natural condition on the hardness of $g$
in terms of an upper bound on discrepancy frequently used in the multiparty
setting and originally due to Babai, Nisan, and Szegedy [BNS92]. As all that
is crucially needed is subadditivity, we do the argument here with
$\gamma_{2}^{*}$, which is essentially equal to the discrepancy.
###### Theorem 4.3
Let $f:\\{-1,+1\\}^{n}\rightarrow\\{-1,+1\\}$, and $g:X\times
Y\rightarrow\\{-1,+1\\}$. Fix $0<\epsilon<1/2$, and let
$\epsilon_{0}>2\epsilon$. Then
$Q_{\epsilon}^{*}(f\circ g^{n})\geq\deg_{\epsilon_{0}}(f)-O(1).$
provided there is a distribution $\mu$ which is balanced with respect to $g$
and for which
$\gamma_{2}^{*}(M_{g}\bullet\mu)\leq\tfrac{\deg_{\epsilon_{0}}(f)}{2en}$.
###### Proof
We again use Proposition 1, this time with the $\gamma_{2}$ norm instead of
the trace norm.
$\gamma_{2}^{\epsilon_{0}}(M_{f\circ g^{n}})=\max_{B}\frac{\langle M_{f\circ
g^{n}},B\rangle-\epsilon_{0}\|B\|_{1}}{\gamma_{2}^{*}(B)}.$
To prove a lower bound we choose a witness matrix $B$ as follows
$B[x,y]=2^{n}\cdot
v(g(x^{1},y^{1}),\ldots,g(x^{n},y^{n}))\cdot\prod_{i=1}^{n}\mu(x^{i},y^{i}).$
where $v$ witnesses that $f$ has approximate degree at least
$d=\deg_{\epsilon_{0}}(f)$. This definition is the same as in the previous
section where $\mu$ was simply the uniform distribution. As argued before, we
have $\langle M_{f\circ g^{n}},B\rangle\geq\epsilon_{0}$ and $\|B\|_{1}=1$
because $M_{g}\bullet\mu$ is balanced.
We again expand $B$ as
$B=2^{n}\sum_{T:|T|\geq
d}\hat{v}_{T}\bigotimes_{i=1}^{n}(M_{g}\bullet\mu)^{T(i)},$
where $(M_{g}\bullet\mu)^{1}=M_{g}\bullet\mu$ and $(M_{g}\bullet\mu)^{0}=\mu$.
Now comes the difference with the previous proof. As we do not have special
knowledge of the function $g$, we simply bound $\gamma_{2}^{*}(B)$ using the
triangle inequality.
$\displaystyle\gamma_{2}^{*}(B)$ $\displaystyle\leq 2^{n}\sum_{T:|T|\geq
d}|\hat{v}_{T}|\
\gamma_{2}^{*}\left(\bigotimes_{i=1}^{n}(M_{g}\bullet\mu)^{T(i)}\right)$
$\displaystyle=2^{n}\sum_{T:|T|\geq d}|\hat{v}_{T}|\
\gamma_{2}^{*}(M_{g}\bullet\mu)^{|T|}\gamma_{2}^{*}(\mu)^{n-|T|}$
$\displaystyle\leq\sum_{T:|T|\geq d}\gamma_{2}^{*}(M_{g}\bullet\mu)^{|T|},$
where in the last step we have used that $\gamma_{2}^{*}(\mu)\leq 1$ as $\mu$
is a probability distribution and that $|\hat{v}_{T}|\leq 2^{-n}$. In the
second step (equality) we used the fact that $\gamma_{2}^{*}$ is
multiplicative with respect to tensor product, a property proved in [LSŠ08].
We continue with simple arithmetic:
$\displaystyle\gamma_{2}^{*}(B)$ $\displaystyle\leq\sum_{i=d}^{n}{n\choose
i}\gamma_{2}^{*}(M_{g}\bullet\mu)^{i}$
$\displaystyle\leq\sum_{i=d}^{n}\left(\frac{en\gamma_{2}^{*}(M_{g}\bullet\mu)}{d}\right)^{i}$
$\displaystyle\leq 2^{-d}$
provided that $\gamma_{2}^{*}(M_{g}\bullet\mu)\leq\frac{d}{2en}$.
## 5 A general framework for functions composed through a group
In this section we begin the study of more general function composition
through a group $G$. In this case the outer function
$f:G\rightarrow\\{-1,+1\\}$ is a class function, i.e. invariant on conjugacy
classes, and the inner function $g:X\times Y\to G$ has range $G$. We define
the composed function as $F(x,y)=f(g(x,y))$. In previous sections of the paper
we have just dealt with the case $G=\mathbb{Z}_{2}^{n}$.
Let us recall the basic idea of the proof of Theorem 4.1. To prove a lower
bound on the quantum communication complexity for a composed function $f\circ
g$, we constructed a witness matrix $B$ which had non-negligible correlation
with $f\circ g$ and small spectral norm. To do this, following the work of
Sherstov and Shi-Zhu [She09, SZ09b], we considered the dual polynomial $p$ of
$f$ using LP duality. The dual polynomial has two important properties, first
that $p$ has non-negligible correlation with $f$ and second that $p$ has no
support on low degree polynomials. We can then use the first property to show
that the composed function $p\circ g$ will give non-negligible inner product
with $f\circ g$ and the second to upper bound the spectral norm of $p\circ g$.
The second of these tasks is the more difficult. In the case of
$G=\\{-1,+1\\}^{n}$, the degree of a character $\chi_{T}$ is a natural measure
of how “hard” the character is — the larger $T$ is, the smaller the spectral
norm of $\chi_{T}\circ g$ will be. In the general group case, however, it is
less clear what the corresponding “hard” and “easy” characters should be. In
Section 5.1, we will show that this framework actually works for an arbitrary
partition of the basis functions into Easy and Hard. That is for any arbitrary
partition of the basis functions into Easy and Hard sets we can follow the
plan outlined above and look for a function with support on the Hard set which
has non-negligible correlation with $f$.
In carrying out this plan, one is still left with upper bounding $\|M_{p\circ
g}\|$. Here, as in the Boolean case, it as again very convenient to have an
orthogonality condition which can greatly simplify the computation of
$\|M_{p\circ g}\|$ and give good bounds. In the Boolean case we have shown
that $M_{g}$ being strongly balanced implies this key orthogonality condition.
In Section 5.2 and 5.3, we will show that for the general group, the condition
is not only about each row and column of matrix $M_{g}$, but all pairs of rows
and pairs of columns. In the Abelian group case, this reduces to a nice group
invariance condition.
Even after applying the orthogonality condition to use the maximum bound
instead of the triangle inequality for $\|M_{p\circ g}\|$, the remaining term
$\|M_{\chi_{i}\circ g}\|$ (where $\chi_{i}$ is a “hard” character) is still
not easy to upper bound. For block composed functions, fortunately, the tensor
structure makes it feasible to compute. Section 5.4 gives a generalized
version of Theorem 4.1.
### 5.1 General framework
For a multiset $T$, $x\in T$ means $x$ running over $T$. Thus $T=\\{a(s):s\in
S\\}$ means the multiset formed by collecting $a(s)$ with $s$ running over
$S$.
For a set $S$, denote by $L_{\mathbb{C}}(S)$ the $|S|$-dimensional vector
space over the field $\mathbb{C}$ (of complex numbers) consisting of all
linear functions from $S$ to $\mathbb{C}$, endowed with inner product
$\langle\psi,\phi\rangle=\frac{1}{|S|}\sum_{s\in S}\psi(s)\overline{\phi(s)}$.
The distance of a function $f\in L_{\mathbb{C}}(S)$ to a subspace $\Phi$ of
$L_{\mathbb{C}}(S)$, denoted by $d(f,\Phi)$, is defined as
$\min\\{\delta:\|f^{\prime}-f\|_{\infty}\leq\delta,f^{\prime}\in\Phi\\}$, i.e.
the magnitude of the least entrywise perturbation to turn $f$ into $\Phi$.
In the above setting, Theorem 4.1 generalizes to the following.
###### Theorem 5.1
Consider a sign matrix $A=[f(g(x,y))]_{x,y}$ where $g:X\times Y\rightarrow S$
for a set $S$, and $f:S\rightarrow\\{-1,+1\\}$. Suppose that we can find an
orthogonal basis functions $\Psi=\\{\psi_{i}:i\in[|S|]\\}$ for
$L_{\mathbb{C}}(S)$. For any hardness partition
$\Psi=\Psi_{Hard}\uplus\Psi_{Easy}$, let $\delta=d(f,span(\Psi_{Easy}))$. If
1. 1.
(regularity) The multiset $\\{g(x,y):x\in X,y\in Y\\}$ is a multiple of $S$,
i.e. $S$ repeated for some number of times.
2. 2.
(orthogonality) for all $x,x^{\prime},y,y^{\prime}$ and all distinct
$\psi_{i},\psi_{j}\in\Psi_{Hard}$,
$\displaystyle\sum_{y}\psi_{i}(g(x,y))\overline{\psi_{j}(g(x^{\prime},y))}=\sum_{x}\psi_{i}(g(x,y))\overline{\psi_{j}(g(x,y^{\prime}))}=0,$
then
$\displaystyle
Q_{\epsilon}(A)\geq\log_{2}\frac{\sqrt{MN}\cdot(\delta-2\epsilon)}{\max_{\psi_{i}\in\Psi_{Hard}}(\max_{g}|\psi_{i}(g)|\cdot\|[\psi_{i}(g(x,y))]_{x,y}\|)}-O(1).$
Using the idea of finding a certificate of the high approximate degree by
duality [She09, SZ09b], we have the following fact analogous to Lemma 1.
###### Lemma 2
For a function $f:S\rightarrow\mbox{$\mathbb{C}$}$ and a subspace $\Phi$ of
$L_{\mathbb{C}}(S)$, if $d(f,span(\Phi))=\delta$, then there exists a function
$h$ s.t.
$\displaystyle\hat{h}_{i}=0,\ \forall\psi_{i}\in\Phi$ (1)
$\displaystyle\sum_{g\in G}|h(g)|\leq 2,$ (2) $\displaystyle|\sum_{g\in
G}f(g)\overline{h(g)}|>\delta$ (3)
Using the lemma, we can prove the Theorem 5.1.
###### Proof
(of Theorem 5.1) By the regularity property, we know that when $(x,y)$ runs
over $X\times Y$, $g(x,y)$ runs over $S$ exactly $K$ times where $K=MN/|G|$.
Consider $B=\frac{1}{K}[h(g(x,y))]_{x,y}$, $h$ obtained by Lemma 2; we want to
apply Proposition 1 and Theorem 2.4 by using this $B$. First,
$\|B\|_{1}=\frac{1}{K}\sum_{x,y}|h(g(x,y))|=\sum_{g\in G}|h(g)|\leq 1.$ (4)
Also,
$\displaystyle|\langle A,B\rangle|$
$\displaystyle=\frac{1}{K}|\sum_{x,y}f(g(x,y))\overline{h(g(x,y))}|$
$\displaystyle=|\sum_{g\in G}f(g)\overline{h(g)}|>\delta$
Now we need to compute
$\|B\|=\frac{1}{K}\big{\|}[\sum_{\chi_{i}\in
Hard}\hat{h}_{i}\chi_{i}(g(x,y))]_{x,y}\big{\|}.$
Note that
$\displaystyle[\psi_{i}(g(x,y))]_{x,y}^{\dagger}[\psi_{j}(g(x,y))]_{x,y}=[\sum_{x}\overline{\psi_{i}(g(x,y))}\psi_{j}(g(x,y^{\prime}))]_{y,y^{\prime}}$
and
$\displaystyle[\psi_{i}(g(x,y))]_{x,y}[\psi_{j}(g(x,y))]_{x,y}^{\dagger}=[\sum_{y}\psi_{i}(g(x,y))\overline{\psi_{j}(g(x^{\prime},y))}]_{x,x^{\prime}}.$
Thus the orthogonality condition implies that
$\displaystyle[\psi_{i}(g(x,y))]_{x,y}^{\dagger}[\psi_{j}(g(x,y))]_{x,y}=[\psi_{i}(g(x,y))]_{x,y}[\psi_{j}(g(x,y))]_{x,y}^{\dagger}=0$
for all $i\neq j$. Now as in [She09], we can use the max bound
$\displaystyle\|B\|$
$\displaystyle=\frac{1}{K}\max_{i:\psi_{i}\in\Psi_{Hard}}\big{\|}\hat{h}_{i}[\psi_{i}(x,y)]_{x,y}\big{\|}$
$\displaystyle\leq\frac{1}{K}\max_{i:\psi_{i}\in
Hard}|\hat{h}_{i}|\max_{\psi_{i}\in
Hard}\big{\|}[\psi_{i}(x,y)]_{x,y}\big{\|}$
$\displaystyle\leq\frac{1}{K|G|}\max_{\psi_{i}\in
Hard}\big{(}\max_{g}|\psi_{i}(g)|\cdot\big{\|}[\psi_{i}(x,y)]_{x,y}\big{\|}\big{)}.$
where the last inequality is due to Eq. (5) and Eq. (4). Finally note
$K|G|=MN$ to complete the proof.
In the Boolean block composed function case, the regularity condition reduces
to the matrix $[g(x,y)]$ being balanced, and later we will prove that the
orthogonality condition reduces to the strongly balanced property. From this
theorem we can see that the way to partition $\Psi$ into $\Psi_{Easy}$ and
$\Psi_{Hard}$ does not really matter for the lower bound proof passing
through. However, the partition does play a role when we later bound the
spectral norm in the denominator.
### 5.2 Functions with group symmetry
For a general finite group $G$, two elements $s$ and $t$ are conjugate,
denoted by $s\sim t$, if there exists an element $r\in G$ s.t. $rsr^{-1}=t$.
Define $H$ as the set of all class functions, i.e. functions $f$ s.t.
$f(s)=f(t)$ if $s\sim t$. Then $H$ is an $h$-dimensional subspace of
$L_{\mathbb{C}}(G)$, where $h$ is the number of conjugacy classes. The
irreducible characters $\\{\chi_{i}:i\in[h]\\}$ form an orthogonal basis of
$H$. For a class function $f$ and irreducible characters $\chi_{i}$, denote by
$\hat{f}_{i}$ the coefficient of $\chi_{i}$ in expansion of $f$ according to
$\chi_{i}$’s, i.e.
$\hat{f}_{i}=\langle\chi_{i},f\rangle=\frac{1}{|G|}\sum_{g\in
G}\chi_{i}(g)\overline{f(g)}$. An easy fact is that for any $i$, we have
$\displaystyle|\hat{f}_{i}|$ $\displaystyle=\frac{1}{|G|}\left|\sum_{g\in
G}\chi_{i}(g)\overline{f(g)}\right|\leq\frac{1}{|G|}\sum_{g\in
G}|f(g)||\chi_{i}(g)|\leq\Big{(}\frac{1}{|G|}\sum_{g\in
G}|f(g)|\Big{)}\cdot\max_{g}|\chi_{i}(g)|.$ (5)
If $G$ is Abelian, then it always has $|\chi_{i}(g)|=1$, thus
$\max_{i}|\hat{f}_{i}|\leq\frac{1}{|G|}\sum_{g\in G}|f(g)|$. For general
groups, we have $|\chi_{i}(g)|\leq\deg(\chi_{i})$, where $\deg(\chi_{i})$ is
the degree of $\chi_{i}$, namely the dimension of the associated vector space.
In this section we consider the setting that $S$ is a finite group $G$. The
goal is to exploit properties of group characters to give better form of the
lower bound. In particular, we hope to see when the second condition holds and
what the matrix operator norm $\|[\psi_{(}g(x,y))]_{x,y}\|$ is in this
setting.
The standard orthogonality of irreducible characters says that $\sum_{s\in
G}\chi_{i}(s)\overline{\chi_{j}(s)}=0$. The second condition in Theorem 5.1 is
concerned with a more general case: For a multiset $T$ with elements in
$G\times G$, we need
$\sum_{(s,t)\in T}\chi_{i}(s)\overline{\chi_{j}(t)}=0,\qquad\forall i\neq j.$
(6)
The standard orthogonality relation corresponds to the special that
$T=\\{(s,s):s\in G\\}$. We hope to have a characterization of a multiset $T$
to make Eq. (6) hold.
We may think of the a multiset $T$ with elements in set $S$ as a function on
$S$, with the value on $s\in S$ being the multiplicity of $s$ in $T$. Since
characters are class functions, for each pair $(C_{k},C_{l})$ of conjugacy
classes, only the value $\sum_{g_{1}\in C_{k},t\in C_{l}}T(g_{1},t)$ matters
for the sake of Eq. (6). We thus make $T$ a class function by taking average
within each class pair $(C_{k},C_{l})$. That is, define a new function
$T^{\prime}$ as
$T^{\prime}(s,t)=\sum_{s\in C_{k},t\in C_{l}}T(s,t)/(|C_{k}||C_{l}|),\ \forall
s\in C_{k},\ \forall t\in C_{l}.$
###### Proposition 2
For a finite group $G$ and a multiset $T$ with elements in $G\times G$, the
following three statements are equivalent:
1. 1.
$\sum_{(s,t)\in T}\chi_{i}(s)\overline{\chi_{j}(t)}=0,\ \forall i\neq j$
2. 2.
$T^{\prime}$, as a function, is in
$span\\{\chi_{i}\otimes\overline{\chi_{i}}:i\in[h]\\}$
3. 3.
$[T^{\prime}(s,t)]_{s,t}=C^{\dagger}DC$ where $D$ is a diagonal matrix and
$C=[\chi_{i}(s)]_{i,s}$. That is, $T^{\prime}$, as a matrix, is normal and
diagonalized exactly by the irreducible characters.
###### Proof
Let $H_{2}$ be the subspace consisting functions $f:G\times
G\rightarrow\mbox{$\mathbb{C}$}$ s.t. $f(s,t)=f(s^{\prime},t^{\prime})$ if
$s\sim s^{\prime}$, $t\sim t^{\prime}$. Note that for direct product group
$G\times G$, $\\{\chi_{i}\otimes\overline{\chi_{j}}:i,j\\}$ form an orthogonal
basis of $H_{2}$:
$\displaystyle\sum_{s,t\in
G}\chi_{i}(s)\overline{\chi_{j}(t)}\overline{\chi_{i^{\prime}}(s)\overline{\chi_{j^{\prime}}(t)}}=\big{(}\sum_{s\in
G}\chi_{i}(s)\overline{\chi_{i^{\prime}}(s)}\big{)}\big{(}\sum_{t\in
G}\overline{\chi_{j}(t)}\chi_{j^{\prime}}(t)\big{)}=0$
unless $i=i^{\prime}$ and $j=j^{\prime}$. Note that by viewing $T$ as a
function from $G\times G$ to $\mathbb{C}$, the Eq. (6) and the definition of
$T^{\prime}$ imply that
$\langle\chi_{i}\otimes\overline{\chi_{j}},\ T\rangle=0,\forall i\neq j$
Thus the first two statements are equivalent.
Note that
$\displaystyle T^{\prime}\in
span\\{\chi_{i}\otimes\overline{\chi_{i}}:i\in[h]\\}\quad\Leftrightarrow\quad
T^{\prime}(s,t)=\sum_{i}\alpha_{i}\chi_{i}(s)\overline{\chi_{i}(t)}\quad\text{
for some $\alpha_{i}$'s}$
Denote by $C_{h\times|G|}=[\chi_{i}(g)]_{i,g}$ the matrix of the character
table. Then observe that the summation in the last equality is nothing but the
$(s,t)$ entry of the matrix $C^{\dagger}diag(\alpha_{1},\cdots,\alpha_{h})C$.
Therefore the equivalence of the second and third statements follows.
### 5.3 Abelian group
When $G$ is Abelian, we have further properties to use. The first one is that
$|\chi_{i}(g)|=1$ for all $i$. The second one is that the irreducible
characters are homomorphisms of $G$; that is,
$\chi_{i}(st)=\chi_{i}(s)\chi_{i}(t)$. This gives a clean characterization of
the orthogonality condition by group invariance. For a multiset $T$, denote by
$sT$ another multiset obtained by collecting all $st$ where $t$ runs over $T$.
A multiset $T$ with elements in $G\times G$ is _$G$ invariant_ if it satisfies
$(g,g)T=T$ for all $g\in G$. We can also call a function $T:G\times
G\rightarrow\mbox{$\mathbb{C}$}$ $G$ invariant if $T(s,t)=T(rs,rt)$ for all
$r,s,t\in G$. The overloading of the name is consistent when we view a
multiset $T$ as a function (counting the multiplicity of elements).
###### Proposition 3
For a finite Abelian group $G$ and a multiset $T$ with elements in $G\times
G$,
$\text{$T$ is $G$ invariant}\Leftrightarrow\sum_{(s,t)\in
T}\chi_{i}(s)\overline{\chi_{j}(t)}=0,\quad\forall i\neq j.$ (7)
###### Proof
$\Rightarrow$: Since $T$ is $G$ invariant, $T=(r,r)T$ and thus,
$\displaystyle\sum_{(s,t)\in T}\chi_{i}(s)\overline{\chi_{j}(t)}$
$\displaystyle=\sum_{(s,t)\in(r,r)T}\chi_{i}(s)\overline{\chi_{j}(t)}$ (8)
$\displaystyle=\sum_{(s^{\prime},t^{\prime})\in
T}\chi_{i}(rs^{\prime})\overline{\chi_{j}(rt^{\prime})}$ (9)
Now using the fact that irreducible characters of Abelian groups are
homomorphisms, we have
$\displaystyle\sum_{(s^{\prime},t^{\prime})\in
T}\chi_{i}(rs^{\prime})\overline{\chi_{j}(rt^{\prime})}=$
$\displaystyle\sum_{(s^{\prime},t^{\prime})\in
T}\chi_{i}(r)\chi_{i}(s^{\prime})\overline{\chi_{j}(r)}\overline{\chi_{j}(t^{\prime})}$
$\displaystyle=$
$\displaystyle\chi_{i}(r)\overline{\chi_{j}(r)}\Big{(}\sum_{(s^{\prime},t^{\prime})\in
T}\chi_{i}(s^{\prime})\overline{\chi_{j}(t^{\prime})}\Big{)}$
But note that this holds for any $r\in G$, thus also for the average of them.
That is,
$\displaystyle\sum_{(s,t)\in T}\chi_{i}(s)\overline{\chi_{j}(t)}=$
$\displaystyle\frac{1}{|G|}\Big{(}\sum_{r\in
G}\chi_{i}(r)\overline{\chi_{j}(r)}\Big{)}\Big{(}\sum_{(s^{\prime},t^{\prime})\in
T}\chi_{i}(s^{\prime})\overline{\chi_{j}(t^{\prime})}\Big{)}=0,$
by the standard orthogonality property of different irreducible characters.
$\Leftarrow$: Since $\sum_{(s,t)\in T}\chi_{i}(s)\overline{\chi_{j}(t)}=0$,
$\forall i\neq j$, we know that $T$ as a function is in
$span\\{\chi_{i}\otimes\overline{\chi_{i}}:i\\}$. Note that any linear
combination of $G$ invariant functions is also $G$ invariant. Thus it remains
to check that each basis $\chi_{i}\otimes\overline{\chi_{i}}$ is $G$
invariant, which is easy to see:
$\chi_{i}(rs)\overline{\chi_{i}(rt)}=\chi_{i}(r)\chi_{i}(s)\overline{\chi_{i}(r)}\overline{\chi_{i}(t)}=\chi_{i}(s)\overline{\chi_{i}(t)}.$
This finishes the proof.
Another nice property of Abelian groups is that the orthogonality condition
condition implies the regularity one.
###### Proposition 4
For an Abelian group $G$, if either $T^{y,y}$ is $G$ invariant for all $y$ or
$S^{x,x}$ is $G$ invariant for all $x$, then $G|\\{g(x,y):x\in X,y\in Y\\}$.
###### Proof
Note that $T^{y,y}(s,s)=|\\{x:g(x,y)=s\\}|$, thus $T^{y,y}$ being $G$
invariant implies that $|\\{x:g(x,y)=s\\}|=|\\{x:g(x,y)=t\\}|$ for all $s,t\in
G$. Thus the column $y$ in matrix $[g(x,y)]_{x,y}$, when viewed as a multiset,
is equal to $G$ repeated $|Y|/|G|$ times. Therefore the whole multiset
$\\{g(x,y):x\in X,y\in Y\\}$ is a multiple of $G$ as well.
What we finally get for Abelian groups is the following.
###### Corollary 4
For a sign matrix $A=[f(g(x,y))]_{x,y}$ and an Abelian group $G$, if
$d(f,span(Ch_{Easy}))=\Omega(1)$, and the multisets
$S^{x,x^{\prime}}=\\{(g(x,y),g(x^{\prime},y)):y\in Y\\}$ and
$T^{y,y^{\prime}}=\\{(g(x,y),g(x,y^{\prime})):x\in X\\}$ are $G$ invariant for
any $(x,x^{\prime})$ and any $(y,y^{\prime})$, then
$Q(A)\geq\log_{2}\frac{\sqrt{MN}}{\max_{i\in
Hard}\|[\chi_{i}(g(x,y))]_{x,y}\|}-O(1).$
### 5.4 Block composed functions
We now consider a special class of functions $g$: block composed functions.
Suppose the group $G$ is a product group $G=G_{1}\times\cdots\times G_{t}$,
and $g(x,y)=(g_{1}(x^{1},y^{1}),\cdots,g_{t}(x^{t},y^{t}))$ where
$x=(x^{1},\cdots,x^{t})$ and $y=(y^{1},\cdots,y^{t})$. That is, both $x$ and
$y$ are decomposed into $t$ components and the $i$-th coordinate of $g(x,y)$
only depends on the $i$-th components of $x$ and $y$. The tensor structure
makes all the computation easy. Theorem 4.1 can be generalized to the general
product group case for arbitrary groups $G_{i}$.
###### Definition 7
The $\epsilon$-approximate degree of a class function $f$ on product group
$G_{1}\times\cdots\times G_{t}$, denoted by $d_{\epsilon}(f)$, is the minimum
$d$ s.t. $\|f-f^{\prime}\|_{\infty}\leq\epsilon$, where $f^{\prime}$ can be
represented as a linear combination of irreducible characters with at most $d$
non-identity component characters.
###### Theorem 5.2
For sign matrix
$A=[f(g_{1}(x^{1},y^{1}),\cdots,g_{t}(x^{t},y^{t})]_{x,y}$
where all $g_{i}$ satisfy their orthogonality conditions, we have
$Q(A)\geq\min_{\\{\chi_{i}\\},S}\sum_{i\in
S}\log_{2}\frac{\sqrt{\mathrm{size}(M_{g_{i}})}}{\deg(\chi_{i})\|M_{\chi_{i}\circ
g_{i}}\|}-O(1)$
where the minimum is over all $S\subseteq[n]$ with $|S|>\deg_{1/3}(f)$, and
all non-identity irreducible characters $\chi_{i}$ of $G_{i}$.
###### Proof
Recall that an irreducible character $\chi$ of $G$ is the tensor product of
irreducible characters $\chi_{i}$ of each component group $G_{i}$. Let Hard be
the set of irreducible characters $\chi$ with more than $d$ non-identity
component characters. Fix a hard character $\chi$, and denote by $S$ the set
of coordinates of its non-identity characters.
$\displaystyle\|[\chi(g(x,y))]\|$
$\displaystyle=\|\bigotimes_{i\in[t]}[\chi_{i}(g_{i}(x^{i},y^{i}))]\|$
$\displaystyle=\prod_{i\in[t]}\|[\chi_{i}(g_{i}(x^{i},y^{i}))]\|$
$\displaystyle=\prod_{i\in
S}\|[\chi_{i}(g_{i}(x^{i},y^{i}))]\|\times\prod_{i\notin
S}\|J_{|X_{i}|\times|Y_{i}|}\|$ $\displaystyle=\prod_{i\in
S}\|M_{\chi_{i}\circ g_{i}}\|\times\prod_{i\notin
S}\sqrt{\mathrm{size}(M_{g_{i}})}$
Thus by Theorem 5.1 and Eq. (5), we have
$\displaystyle Q(A)\geq\log_{2}\prod_{i\in
S}\frac{\sqrt{\mathrm{size}(M_{g_{i}})}}{\deg(\chi_{i})\cdot\|M_{\chi_{i}\circ
g_{i}}\|}-O(1)$ (10)
proving the theorem.
Previous sections as well as [SZ09b] consider the case where all $g_{i}$’s are
the same and all $G_{i}$’s are $\mbox{$\mathbb{Z}$}_{2}$. In this case, the
above bound is equal to the one in Theorem 4.1, and the following proposition
says that the group invariance condition degenerates to the strongly balanced
property.
###### Proposition 5
For $G=\mbox{$\mathbb{Z}$}_{2}^{\times t}$, the following two conditions for
$g=(g_{1},\cdots,g_{t})$ are equivalent:
1. 1.
The multisets $S^{x,x^{\prime}}=\\{(g(x,y),g(x^{\prime},y)):y\in Y\\}$ and
$T^{y,y^{\prime}}=\\{(g(x,y),g(x,y^{\prime})):x\in X\\}$ are $G$ invariant for
any $(x,x^{\prime})$ and any $(y,y^{\prime})$,
2. 2.
Each matrix $[g_{i}(x^{i},y^{i})]_{x^{i},y^{i}}$ is strongly balanced.
###### Proof
1 $\Rightarrow$ 2: $S^{x,x^{\prime}}$ being $G$ invariant implies that for all
$\\{z_{i}\\},\\{u_{i}\\},\\{v_{i}\\}$,
$\displaystyle|\\{y:z_{i}g_{i}(x^{i},y^{i})=u_{i},\ z_{i}g_{i}(x^{\prime
i},y^{i})=v_{i},\forall i\\}|$ $\displaystyle=$
$\displaystyle|\\{y:g_{i}(x^{i},y^{i})=u_{i},\ g_{i}(x^{\prime
i},y^{i})=v_{i},\forall i\\}|.$
Take $x^{\prime}=x$ and $u=v$. Now for each $i$ and each row $x^{i}$, take
$z_{i}=-1$ (where the group $\mbox{$\mathbb{Z}$}_{2}$ is represented by
$\\{\pm 1\\}$). For all other $i^{\prime}\neq i$, take $z_{i^{\prime}}=1$.
This assignment will show that
$|\\{y:g_{i}(x^{i},y^{i})=-u_{i}\\}|=|\\{y:g_{i}(x^{i},y^{i})=u_{i}\\}|.$
That is, the row $x^{i}$ in matrix $[g_{i}(x^{i},y^{i})]$ is balanced.
Similarly we can show the balance for each column.
2 $\Rightarrow$ 1: It is enough to show that for each $i$ and each
$\\{z_{i}\\},\\{u_{i}\\},\\{v_{i}\\}$,
$\displaystyle|\\{y^{i}:z_{i}g_{i}(x^{i},y^{i})=u_{i},\ z_{i}g_{i}(x^{\prime
i},y^{i})=v_{i}\\}|$ $\displaystyle=$
$\displaystyle|\\{y^{i}:g_{i}(x^{i},y^{i})=u_{i},\ g_{i}(x^{\prime
i},y^{i})=v_{i}\\}|.$
First consider the case $x^{\prime i}=x^{i}$. If $u_{i}\neq v_{i}$ then both
numbers are 0; if $u_{i}=v_{i}$ then both numbers are $|\\{y^{i}\\}|/2$ by the
balance of row $x^{i}$. Now assume $x^{\prime i}\neq x^{i}$. Denote
$a_{bb^{\prime}}=|\\{y^{i}:g_{i}(x^{i},y^{i})=b,\
g_{i}(x^{i},y^{i})=b^{\prime}\\}|$, then the above requirement amounts to
$a_{00}=a_{11}$ and $a_{01}=a_{10}$.
Note that we have
$\displaystyle a_{00}+a_{01}$
$\displaystyle=|\\{y^{i}:g_{i}(x^{i},y^{i})=0\\}|=|\\{y^{i}:g_{i}(x^{i},y^{i})=1\\}|=a_{10}+a_{11}$
where the second equality is due to the balance of row $x^{i}$. And similarly
we have $a_{00}+a_{10}=a_{01}+a_{11}$ by balance of row $x^{\prime i}$.
Combining the two, we get $a_{00}=a_{11}$ and $a_{01}=a_{10}$ as desired.
It is worth noting that the conclusion does not hold if any group $G_{i}$ with
size larger than two. We omit the counterexamples here.
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|
arxiv-papers
| 2010-03-07T08:48:25 |
2024-09-04T02:49:08.880837
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Troy Lee and Shengyu Zhang",
"submitter": "Shengyu Zhang",
"url": "https://arxiv.org/abs/1003.1443"
}
|
1003.1602
|
# The expression of Moore–Penrose inverse of $A-XY^{*}$
Fapeng Du, Yifeng Xue ∗ Fapeng Du, email: jsdfp@163.com
Department of mathematics, East China Normal University
Shanghai 200241, P.R. China and
Collage of mathematics and Physical Sciences, Xuzhou Institute of Technology
Xuzhou 221008, Jiangsu Province, P.R. China
∗ Department of mathematics, East China Normal University
Shanghai 200241, P.R. China
###### Abstract.
Let $K,\,H$ be Hilbert spaces and let $L(K,H)$ denote the set of all bounded
linear operators from $K$ to $H$. Let $A\in L(H)\triangleq L(H,H)$ with $R(A)$
closed and $X,Y\in L(K,H)$ with $R(X)\subseteq R(A),R(Y)\subseteq R(A^{*})$.
In this short note, we give some new expressions of the Moore–Penrose inverse
$(A-XY^{*})^{+}$ of $A-XY^{*}$ under certain suitable conditions.
###### Key words and phrases:
Hilbert spaces, Moore–Penrose inverse, idempotent operator
###### 1991 Mathematics Subject Classification:
15A09, 47A05, 65F20
∗ Corresponding author; email: yfxue@math.ecnu.edu.cn
Project supported by Natural Science Foundation of China (no.10771069) and
Shanghai Leading Academic Discipline Project(no.B407)
## 1\. Introduction
Let $A$ be a nonsingular $m\times m$ matrix and $X,\,Y$ be two $m\times n$
matrices. It is known that $A-XY^{*}$ is nonsingular iff $I_{n}-Y^{*}A^{-1}X$
is nonsingular, and in which case the well known Shermen-Morrison-Woodbury
formula (SMW) can be expressed as
(1.1) $(A-XY^{*})^{-1}=A^{-1}+A^{-1}X(I_{n}-Y^{*}A^{-1}X)^{-1}Y^{*}A^{-1}$
This formula and some related formula have a lot of applications in
statistics, networks, optimization and partial differential equations. Please
see [4, 5, 7] for details. Clearly, the formula (1.1) fails when $A$ or
$A-XY^{*}$ is singular. Steerneman and Kleij in [6] proved that when A is
singular and $I_{n}-Y^{*}A^{+}X$ is nonsingular, then
$(A-XY^{*})^{+}=A^{+}+A^{+}X(I_{n}-Y^{*}A^{+}X)^{-1}Y^{*}A^{+}$
under conditions that
$\mathrm{rank}\,(A,X)=\mathrm{rank}\,A,\quad\mathrm{rank}\,\begin{pmatrix}A\\\
Y^{*}\end{pmatrix}=\mathrm{rank}\,A.$
He also showed that if $A$ is nonsingular and $Y^{*}A^{-1}X=I_{n}$, then
(1.2) $(A-XY^{*})^{+}=(I_{m}-X_{1}X_{1}^{+})A^{-1}(I_{m}-Y_{1}Y_{1}^{+})$
where $X_{1}=A^{-1}X,~{}~{}Y_{1}=(A^{-1})^{*}Y$ (cf. [6, Theorem 3]).
Recently Chen, Hu and Xu studied the Moore-Penrose inverse of $A-XY^{*}$ when
$A\in L(H)$ and $X,Y\in L(K,H)$ in [3]. They prove that if $A$ is invertible
and $A-XY^{*},~{}X,~{}Y$ have closed ranges, then
$(A-XY^{*})^{+}=(I-X_{1}X_{1}^{+})A^{-1}(I-Y_{1}Y_{1}^{+})$
iff $Y_{1}^{*}XY_{1}^{*}=Y_{1}^{*},\ XY_{1}^{*}X=X$,where
$X_{1}=A^{-1}X,~{}~{}Y_{1}=(A^{-1})^{*}Y$. This result generalizes Theorem 3
of [6].
In this paper we assume that $A\in L(H)$ and $X,Y\in L(K,H)$ with $R(A)$
closed and $R(X)\subseteq R(A),R(Y)\subseteq R(A^{*})$. We prove that
$(A-XY^{*})^{+}=(I-(A^{+}XY^{*})(A^{+}XY^{*})^{+})A^{+}(I-(XY^{*}A^{+})^{+}(XY^{*}A^{+}))$
if $XY^{*}A^{+}XY^{*}=XY^{*}$ and
$(A-XY^{*})^{+}=(I-(A^{+}X)(A^{+}X)^{+})A^{+}(I-(Y^{*}A^{+})^{+}(Y^{*}A^{+}))$
if $XY^{*}A^{+}X=X$ and $Y^{*}A^{+}XY^{*}=Y^{*}$. These expressions generalize
corresponding expressions of $(A-XY^{*})^{+}$ given in [3] and [6].
## 2\. preliminaries
Let $T\in L(K,H)$, denote by $R(T)$ (resp. $N(T)$) the range (resp. kernal) of
$T$. Let $A\in L(H)$. Recall from [1] that $B\in L(H)$ is the Moore–Penrose
inverse of $A$, if $B$ satisfies the following equations:
$ABA=A,~{}~{}BAB=B,~{}~{}(AB)^{*}=AB,~{}~{}(BA)^{*}=BA$
In this case $B$ is denote by $A^{+}$. It is well–known $A$ has the
Moore–Penrose inverse iff $R(A)$ is closed in $H$. When $A^{+}$ exists,
$R(A^{+})=R(A^{*}),~{}N(A^{+})=N(A^{*})$ and $(A^{+})^{*}=(A^{*})^{+}$.
###### Lemma 2.1.
Let $A\in L(H)$ with $R(A)$ closed and $X,Y\in L(K,H)$
(1) $R(X)\subseteq R(A)$ iff $AA^{+}X=X$, $R(Y)\subseteq R(A^{*})$ iff
$Y^{*}A^{+}A=Y^{*}$.
(2) Suppose that $R(X)\subseteq R(A)$ and $R(Y)\subseteq R(A^{*})$ then
$(A-XY^{*})A^{+}(A-XY^{*})=(A-XY^{*})$
iff $XY^{*}A^{+}XY^{*}=XY^{*}$.
Proof. (1) Since $R(A)=R(AA^{+})$ and $R(A^{*})=R(A^{+}A)$, the assertion
follows.
(2) Using (1), we can check directly that
$(A-XY^{*})A^{+}(A-XY^{*})=(A-XY^{*})$ if and only if
$XY^{*}A^{+}XY^{*}=XY^{*}$.
In order to compute $(A-XY^{*})^{+}$, we need the following two lemmas which
come from [2].
###### Lemma 2.2.
Let $S\in L(H)$ be an idempotent operator. Denote by $O(S)$ the orthogonal
projection of $H$ onto $R(S)$. Then $I-S-S^{*}$ is invertible in L(H) and
$O(S)=-S(I-S-S^{*})^{-1}$.
###### Lemma 2.3.
Let $T,~{}B\in L(H)$ with $TBT=T$, Then $T^{+}=(I-O(I-BT))BO(TB)$.
###### Lemma 2.4.
Let $S\in L(H)$ be an idempotent operator. Then $O(S)=SS^{+}$ and
$O(I-S)=I-S^{+}S$.
Proof. $S^{2}=S$ implies that $R(S)$ is closed and
$R(I-S)=N(S)=R(S^{*})^{\perp}$. Thus, $S^{+}$ exists and $O(S)=SS^{+},\
O(I-S)=I-S^{+}S$.
## 3\. Main results
In this section, we will generalize Eq(1.1) and Eq(1.2). Firstly, we have
###### Proposition 3.1.
Let $A\in L(H)$ with $R(A)$ closed and $X,Y\in L(K,H)$ with $R(X)\subseteq
R(A)$ and $R(Y)\subseteq R(A^{*})$. Assume that $I-Y^{*}A^{+}X$ is invertible
in L(H). Then $(A-XY^{*})^{+}$ exists and
(3.1) $(A-XY^{*})^{+}=A^{+}+A^{+}X(I-Y^{*}A^{+}X)^{-1}Y^{*}A^{+}.$
Proof. Put $B=A^{+}+A^{+}X(I-Y^{*}A^{+}X)^{-1}Y^{*}A^{+}$. Simple computation
shows that $(A-XY^{*})B=AA^{+}$ and $B(A-XY^{*})=A^{+}A$ by Lemma 2.1 (1).
Thus,
$\displaystyle(A-XY^{*})B(A-XY^{*})$ $\displaystyle=A-XY^{*},\quad\qquad
B(A-XY^{*})B=B,$ $\displaystyle((A-XY^{*})B)^{*}$
$\displaystyle=(A-XY^{*})B,\quad(B(A-XY^{*}))^{*}=B(A-XY^{*}),$
that is, $(A-XY^{*})^{+}=B$.
Now we consider the case that $I-Y^{*}A^{+}X$ is not invertible, we have
###### Theorem 3.2.
Let $A\in L(H)$ with $R(A)$ closed and $X,Y\in L(K,H)$ with $R(X)\subseteq
R(A)$ and $R(Y)\subseteq R(A^{*})$.
1. (1)
If $XY^{*}A^{+}XY^{*}=XY^{*}$, then $(A-XY^{*})^{+}$ exists and
(3.2)
$(A-XY^{*})^{+}=(I-(A^{+}XY^{*})(A^{+}XY^{*})^{+})A^{+}(I-(XY^{*}A^{+})^{+}(XY^{*}A^{+}));$
Especially, if $XY^{*}A^{+}X=X$ and $Y^{*}A^{+}XY^{*}=Y^{*}$, then
(3.3)
$(A-XY^{*})^{+}=(I-(A^{+}X)(A^{+}X)^{+})A^{+}(I-(Y^{*}A^{+})^{+}(Y^{*}A^{+}));$
2. (2)
Assume that $R(A-XY^{*})$, $R(A^{+}XY^{*})$ and $R(XY^{*}A^{+})$ are closed in
$H$. Then Eq(3.2) implies that $XY^{*}A^{+}XY^{*}=XY^{*}$;
3. (3)
Assume that $R(A-XY^{*})$, $R(A^{+}X)$ and $R(Y^{*}A^{+})$ are closed. Then
Eq(3.3) indicates that $XY^{*}A^{+}X=X$ and $Y^{*}A^{+}XY^{*}=Y^{*}$.
Proof. (1) In this case, $(A-XY^{*})A^{+}(A-XY^{*})=(A-XY^{*})$. Thus
$R(A-XY^{*})$ is closed, i.e., $(A-XY^{*})^{+}$ exists and hence
$(A-XY^{*})^{+}=(I-O(I-A^{+}(A-XY^{*})))A^{+}O((A-XY^{*})A^{+})$
by Lemma 2.1 (2). Since $(I-2A^{+}A)^{2}=I,~{}~{}(I-2A^{+}A)A^{+}=-A^{+}$,
$\displaystyle A^{+}XY^{*}+(A^{+}XY^{*})^{*})$
$\displaystyle=(A^{+}XY^{*}+(A^{+}XY^{*})^{*})(2A^{+}A-I)$
$\displaystyle(I-A^{+}A)(I-A^{+}XY^{*}-(A^{+}XY^{*})^{*})$
$\displaystyle=I-A^{+}A.$
it follows that
$\displaystyle O(I-A^{+}(A-XY^{*}))$ $\displaystyle=O(I-A^{+}A+A^{+}XY^{*})$
$\displaystyle=-(I-A^{+}A+A^{+}XY^{*})(2A^{+}A-I-A^{+}XY^{*}-(A^{+}XY^{*})^{*})^{-1}$
$\displaystyle=(I-A^{+}A+A^{+}XY^{*})(I-2A^{+}A)(I-A^{+}XY^{*}-(A^{+}XY^{*})^{*})^{-1}$
$\displaystyle=I-A^{+}A+O(A^{+}XY^{*}).$
Similarly, we also have
$\displaystyle O((A-XY^{*})A^{+})$
$\displaystyle=-(A-XY^{*})A^{+}(I-(AA^{+}-XY^{*}A^{+})-(AA^{+}-XY^{*}A^{+})^{*})^{-1}$
$\displaystyle=(-AA^{+}+XY^{*}A^{+})(I-2AA^{+}+XY^{*}A^{+}+(XY^{*}A^{+})^{*})^{-1}$
$\displaystyle=(AA^{+}-XY^{*}A^{+})(I-XY^{*}A^{+}-(XY^{*}A^{+})^{*})^{-1}$
$\displaystyle=AA^{+}-I+O(I-XY^{*}A^{+}).$
Therefore, we have
$\displaystyle(A-XY^{*})^{+}$
$\displaystyle=(I-O(I-A^{+}(A-XY^{*})))A^{+}O((A-XY^{*})A^{+})$
$\displaystyle=(A^{+}A-O(A^{+}XY^{*}))A^{+}O(I-XY^{*}A^{+})$
$\displaystyle=(I-O(A^{+}XY^{*}))A^{+}O(I-XY^{*}A^{+}).$
From $A^{+}XY^{*}A^{+}XY^{*}=XY^{*}$, we get that $A^{+}XY^{*}$ and
$XY^{*}A^{+}$ are all idempotent operators. It follow from Lemma 2.4 that
$O(A^{+}XY^{*})=(A^{+}XY^{*})(A^{+}XY^{*})^{+},\quad
O(I-XY^{*}A^{+})=I-(XY^{*}A^{+})^{+}(XY^{*}A^{+}).$
Therefore, we have
$(A-XY^{*})^{+}=(I-(A^{+}XY^{*})(A^{+}XY^{*})^{+})A^{+}(I-(XY^{*}A^{+})^{+}(XY^{*}A^{+})).$
When $XY^{*}A^{+}X=X$ and $Y^{*}A^{+}XY^{*}=Y^{*}$, we have
$R(A^{+}XY^{*})=R(A^{+}X)$ and $R(I-XY^{*}A^{+})=N(Y^{*}A^{+})$ so that
$O(A^{+}XY^{*})=(A^{+}X)(A^{+}X)^{+},\quad
O(I-XY^{*}A^{+})=I-(Y^{*}A^{+})^{+}(Y^{*}A^{+}).$
and consequently, we get (3.3).
(2) In this case,
$R((XY^{*}A^{+})^{*})=R((XY^{*}A^{+})^{+})\subseteq
N((A-XY^{*})^{+})=N((A-XY^{*})^{*}),$
that is, $[N(XY^{*}A^{+})]^{\perp}\subseteq[R(A-XY^{*})]^{\perp}$. So
$R(A-XY^{*})\subseteq N(XY^{*}A^{+})$ and consequently,
$XY^{*}A^{+}XY^{*}=XY^{*}$.
(3) When Eq(3.3) holds,
$\displaystyle R((Y^{*}A^{+})^{*})=R((Y^{*}A^{+})^{+})$
$\displaystyle\subseteq N((A-XY^{*})^{+})=N((A-XY^{*})^{*})$ $\displaystyle
R((A-XY^{*})^{*})=R((A-XY^{*})^{+})$ $\displaystyle\subseteq
N((A^{+}X)^{+})=N((A^{+}X)^{*}).$
Then $R(A-XY^{*})\subseteq N(Y^{*}A^{+})$ and $R(A^{+}X)\subseteq
N(A-XY^{*})$. So
$Y^{*}A^{+}XY^{*}=Y^{*},\quad XY^{*}A^{+}X=X.$
Suppose $H=\mathbb{C}^{m}$ and $K=\mathbb{C}^{n}$. Let $A\in L(H)$ and
$X,\,Y\in L(K,H)$. Since
$\displaystyle\mathrm{rank}\,(A,X)=\mathrm{rank}\,A$
$\displaystyle\Leftrightarrow R(X)\subseteq R(A)$
$\displaystyle\mathrm{rank}\,\begin{pmatrix}A\\\
Y^{*}\end{pmatrix}=\mathrm{rank}\,A$ $\displaystyle\Leftrightarrow
R(Y)\subseteq R(A^{*}),$
we can express Theorem 3.2 (1) as follows.
###### Corollary 3.3.
Let $A$ be an $m\times m$ matrix and $X,\,Y$ be two $m\times n$ matrices.
Suppose that $\mathrm{rank}\,(A,X)=\mathrm{rank}\,A$ and
$\mathrm{rank}\,\begin{pmatrix}A\\\ Y^{*}\end{pmatrix}=\mathrm{rank}\,A$. Then
$(A-XY^{*})^{+}=(I-(A^{+}XY^{*})(A^{+}XY^{*})^{+})A^{+}(I-(XY^{*}A^{+})^{+}(XY^{*}A^{+}))$
if $XY^{*}A^{+}XY^{*}=XY^{*}$ and
$(A-XY^{*})^{+}=(I-(A^{+}X)(A^{+}X)^{+})A^{+}(I-(Y^{*}A^{+})^{+}(Y^{*}A^{+}))$
when $XY^{*}A^{+}X=X$ and $Y^{*}A^{+}XY^{*}=Y^{*}$.
Before ending this note, we give an example as follows.
###### Example 3.4.
Put $A=\begin{pmatrix}1&1&1&1\\\ 0&0&1&1\\\ 0&0&1&1\\\ 0&0&0&1\end{pmatrix},\
X=\begin{pmatrix}1&1&0\\\ 1&0&0\\\ 1&0&0\\\ 1&0&0\end{pmatrix},\
Y=\begin{pmatrix}0&0&0\\\ 0&0&0\\\ 1&0&0\\\ 1&0&1\end{pmatrix}.$ Then
$\displaystyle A^{+}=\begin{pmatrix}\frac{1}{2}&-\frac{1}{4}&-\frac{1}{4}&0\\\
\frac{1}{2}&-\frac{1}{4}&-\frac{1}{4}&0\\\ 0&\frac{1}{2}&\frac{1}{2}&-1\\\
0&0&0&1\end{pmatrix},$ $\displaystyle\quad XY^{*}=\begin{pmatrix}0&0&1&1\\\
0&0&1&1\\\ 0&0&1&1\\\ 0&0&1&1\end{pmatrix}$
$\displaystyle(A^{+}XY^{*})^{+}=\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\
0&0&0&\frac{1}{2}\\\ 0&0&0&\frac{1}{2}\end{pmatrix},$
$\displaystyle\quad(XY^{*}A^{+})^{+}=\begin{pmatrix}0&0&0&0\\\
\frac{1}{4}&\frac{1}{4}&\frac{1}{4}&\frac{1}{4}\\\
\frac{1}{4}&\frac{1}{4}&\frac{1}{4}&\frac{1}{4}\\\ 0&0&0&0\end{pmatrix}.$
It is easy to verify that $R(X)\subseteq R(A),~{}R(Y)\subseteq R(A^{*})$ and
$XY^{*}A^{+}XY^{*}=XY^{*}$. So by Corollary 3.3,
$(A-XY^{*})^{+}=\begin{pmatrix}\frac{1}{2}&0&0&0\\\ \frac{1}{2}&0&0&0\\\
0&0&0&-1\\\ 0&0&0&0\end{pmatrix}.$
## References
* [1] A. Ben-Israel and T.N.E. Greville, _Generalized inverse:Theory and applications_ , Wiley, New York, 1974.
* [2] G. Chen and Y. Xue,_The expression of generalized inverse of the perturbed operators under type I perturbation in Hilbert spaces_ , Linear Algebra Appl., 285(1998), 1–6.
* [3] Y. Chen, X. Hu and Q. Xu, _The Moore-Penrose inverse of $A-XY^{*}$_, Journal of Shanghai Normal University, 38(2009), 15–19
* [4] H.V. Hsnderson and S.R. Searl, _On deriving the inverse of a sum of matrices_ , Siam Review, 23(1)(1981), 53–60.
* [5] S. Kurt and A. Riedel, _A Shermen-Morrison-Woodbury identity for rank augmenting_ , matrices with application to centering. Siam J. Math. Anal., 12(1)(1991), 80–95.
* [6] T. Steerneman and F.P. Kleij, _Properties of the matrix $A-XY^{*}$_, Linear Algebra Appl., 410(2005), 70–86.
* [7] W.W. Hager, _Updating the inverse of a matrix_ , Siam Review, 31(1989), 221–239.
|
arxiv-papers
| 2010-03-08T11:30:45 |
2024-09-04T02:49:08.893516
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fapeng Du and Yifeng Xue",
"submitter": "Yifeng Xue",
"url": "https://arxiv.org/abs/1003.1602"
}
|
1003.1647
|
# THE PHOTOSPHERIC ENERGY AND HELICITY BUDGETS OF THE FLUX-INJECTION
HYPOTHESIS
P. W. Schuck††affiliation: peter.schuck@nasa.gov NASA Goddard Space Flight
Center Room 250, Building 21 Space Weather Laboratory, Code 674 Heliophysics
Science Division 8801 Greenbelt Rd. Greenbelt, MD 20771, USA
###### Abstract
The flux-injection hypothesis for driving coronal mass ejections (CMEs)
requires the transport of substantial magnetic energy and helicity flux
through the photosphere concomitant with the eruption. Under the
magnetohydrodynamics approximation, these fluxes are produced by twisting
magnetic field and/or flux emergence in the photosphere. A CME trajectory,
observed 2000 September 12 and fitted with a flux-rope model constrains energy
and helicity budgets for testing the flux-injection hypothesis. Optimal
velocity profiles for several driving scenarios are estimated by minimizing
the photospheric plasma velocities for a cylindrically symmetric flux-rope
magnetic field subject to the flux budgets required by the flux-rope model.
Ideal flux injection, involving only flux emergence, requires hypersonic
upflows in excess of the solar escape velocity $617\,\mbox{km}\mbox{s}^{-1}$
over an area of $6\times 10^{8}\,\mathrm{km}^{2}$ to satisfy the energy and
helicity budgets of the flux-rope model. These estimates are compared with
magnetic field and Doppler measurements from Solar Heliospheric
Observatory/Michelson Doppler Imager on 2000 September 12 at the footpoints of
the CME. The observed Doppler signatures are insufficient to account for the
required energy and helicity budgets of the flux-injection hypothesis.
Sun: coronal mass ejections - Sun: photosphere - Sun: surface magnetism
††slugcomment: Astrophysical Journal 713:1-21: 2010
## 1 INTRODUCTION
Over the last 10 years, the flux-rope model developed by Chen (1989, 1996) has
been used to describe the dynamics of coronal mass ejections (CMEs) observed
by the Large Angle Spectrometric Coronagraphs (LASCO) aboard the Solar and
Heliospheric Observatory111SOHO is a project of international cooperation
between ESA and NASA. (SOHO) Chen et al. (1997, 2000); Wood et al. (1999);
Krall et al. (2001); Chen et al. (2006); Krall et al. (2006). However, the
photospheric flux injection paradigm used to initiate and drive the eruption
has been criticized because the surge of electromagnetic energy flowing
through the photosphere is “difficult to reconcile with the extremely tranquil
conditions that exist during flares and CMEs” Forbes (2000, 2001); Rust
(2001). Chen et al. (2000), Chen (2001), Krall et al. (2001), Chen & Krall
(2003) and Chen & Kunkel (2010) have attempted to address these criticisms and
a previous study has examined the implications of uniformly twisting to
coronal footpoints of the flux rope Krall et al. (2000). However, no
quantitative comparisons between flux-injection hypothesis and detailed
photospheric observations have been considered by the formal literature.
The goal of this paper is to provide a framework for testing the flux-
injection hypothesis through the photospheric signatures implied by the energy
and helicity budgets of CMEs described by the flux-rope model. The paper is
organized as follows: Section 2 describes the flux-rope model of Chen (1989)
and a simple extension of the flux-rope model magnetic field into the
photosphere.
Section 3 develops the photospheric fluxes necessary to satisfy the energy and
helicity budgets of CMEs fitted with the flux-rope model. The photospheric
magnetic field combined with the photospheric fluxes is used to estimate
minimum velocities necessary to satisfy the energy and helicity budgets
required by CME trajectories fitted by the flux-rope model under the flux-
injection hypothesis. Two examples of CME trajectories fitted with the flux-
rope model are used to constrain the photospheric velocities. 1) The first
event is the 2000 September 12 CME that erupted from decaying NOAA active
region 9163. The height-time data for the CME trajectory Chen et al. (2006)
were derived from measurements of the filament in absorption observed by the
Global H$\alpha$ Network at Kanzehöhe Solar Observatory (KSO) in Austria
Steinegger et al. (2000), and LASCO C2 and C3 observations of the filament in
emission Brueckner et al. (1995). Complementary observations of the filament
were made in Fe XII 195 Å by the SOHO/EUV Imaging Telescope (EIT) instrument
Delaboudinière et al. (1995). The eruption occurred shortly after 11:30 UT and
the filament first appeared in LASCO C2 at 12:30 UT. This CME was associated
with an M1.0 class flare with distinct flare ribbons that persisted for 2 hr.
Various aspects of this event, such as morphology, timing, and reconnection
rate have been discussed by Vršnak et al. (2003), Schuck et al. (2004b) and
Qiu et al. (2004). 2) The second event is the 2003 October 28 CME which
originated from large complex NOAA active region 10468 at about 11:00 UT. The
event was extremely fast, and thus the height-time data for the leading edge
of the CME consists of only one LASCO C2 image at about 11:30 UT and four
subsequent LASCO C3 images. This event was associated with an extremely
powerful X17 flare and consequently has received extensive attention in the
literature Seppälä et al. (2004); Skoug et al. (2004); Woods et al. (2004);
Zurbuchen et al. (2004); Bieber et al. (2005); Chi et al. (2005); Degenstein
et al. (2005); Hu et al. (2005); Gopalswamy et al. (2005); Looper et al.
(2005); Pallamraju & Chakrabarti (2005); Tsurutani et al. (2005); Yurchyshyn
et al. (2005); Krall et al. (2006); Manchester et al. (2008).
Section 4 compares the photospheric velocities implied by the flux-rope CME
trajectory event from 2000 September 12, against detailed photospheric Doppler
measurements from the Michelson Doppler Imager (MDI) aboard SOHO Scherrer et
al. (1995). Finally, Section 5 compares these theoretical results and Doppler
observations with previous work.
## 2 THE FLUX-ROPE MODEL
Figure 1: Schematic diagram of the flux-rope model current loop adapted from
Chen (1989, 1996). The subscripts “$\phi$” and “$\theta$” refer to the
toroidal and poloidal directions in the cylindrical $\left(R,\phi,Z\right)$
and the local polar $\left(r,\theta,\zeta\right)$ coordinate systems for a
circular torus where $\widehat{Z}$ is out of the plane of the page.
Figure 1 shows a schematic diagram of the flux-rope CME model current loop,
adapted from Chen (1989, 1996). Shafranov (1966) derived the
magnetohydrodynamic (MHD) forces per-unit-length acting the major
$\mathcal{R}$ and minor $a$ radii of a current carrying toroidal section
${f}_{R}\equiv{{I_{\phi}^{2}}\over{c^{2}\,\mathcal{R}}}\left[\ln\left({{8\,\mathcal{R}}\over{a}}\right)+{\Delta\beta\over
2}-{{\left\langle{B}_{\phi}\right\rangle^{2}}\over{2\,B_{\theta{a}}^{2}}}+{2\,\mathcal{R}\over
a}{{B_{\mathrm{c}}}\over{B_{\theta{a}}}}-1+{{\xi}\over{2}}\right],$ (1a)
$f_{\mathrm{a}}\equiv\frac{I_{\phi}^{2}}{c^{2}\,a}\,\left(\frac{\left\langle{B}_{\phi}\right\rangle^{2}}{B_{\theta{a}}^{2}}-1+\Delta\beta\right),$
(1b)
where $I_{\phi}$ is the toroidal current, $c$ is the speed of light,
$\left\langle{B}_{\phi}\right\rangle$ is the average toroidal field inside the
current channel, $B_{\mathrm{c}}$ is a prescribed function describing the
overlying coronal field perpendicular to the flux rope, $B_{\theta{a}}\equiv
B_{\theta}\left(a\right)=2\,I_{\phi}/a\,c$ is the poloidal magnetic field at
the edge of the current channel, and
$\xi\equiv\frac{2\,\int_{0}^{a}{dr}\,{r}B_{\theta}^{2}\left(r\right)}{a^{2}\,B_{{\theta}a}^{2}}=\frac{c^{2}}{2\,I_{\phi}^{2}}\,\int_{0}^{a}{dr}\,{r}B_{\theta}^{2}\left(r\right)\sim\mathcal{O}\left(1\right),$
(2)
is the internal inductance. The quantity $\Delta\beta\equiv
8\pi(\left\langle{p}\right\rangle-p_{\mathrm{c}})/B_{\theta{a}}^{2}$ is the
differential of plasma $\beta$ based on $B_{\theta{a}}$ where
$\left\langle{p}\right\rangle$ is the average internal flux-rope pressure and
$p_{\mathrm{c}}$ is the coronal pressure. The subscripts “$\phi$” and
“$\theta$” refer to the toroidal and poloidal directions in the cylindrical
$\left(R,\phi,Z\right)$ and the local polar $\left(r,\theta,\zeta\right)$
coordinate systems for a circular torus with $R=\mathcal{R}+r\,\cos\theta$,
$Z=r\,\sin\theta$ and $\widehat{\zeta}=\widehat{\phi}$ where $\widehat{Z}$ is
out of the plane of the page. Chen (1989) applied Equation (1) at the peak of
the flux rope to derive the dynamical equations
$M\,\frac{d^{2}H}{dt^{2}}={{I_{\phi}^{2}}\over{c^{2}\,\mathcal{R}}}\left[\ln\left({{8\,\mathcal{R}}\over{a_{*}}}\right)+{\Delta\beta_{*}\over
2}-{{\left\langle{B}_{\phi_{*}}\right\rangle^{2}}\over{2\,B_{\theta{a_{*}}}^{2}}}+{2\,\mathcal{R}\over
a_{*}}{{B_{\mathrm{c}}}\over{B_{\theta{a_{*}}}}}-1+{{\xi_{*}}\over{2}}\right]+f_{\mathrm{d}}+f_{\mathrm{g}},$
(3a) and
$M\,\frac{d^{2}a_{*}}{dt^{2}}=\frac{I_{\phi}^{2}}{c^{2}\,a_{*}}\,\left(\frac{\left\langle{B}_{\phi_{*}}\right\rangle^{2}}{B_{\theta{a_{*}}}^{2}}-1+\Delta\beta_{*}\right),$
(3b)
where $H>0$ is the height of the center of the current channel above the
photosphere henceforth referred to as the “apex.” The *’s in Equation (3)
denote quantities evaluated over the cross section at the apex where
$M\equiv\left\langle{n_{*}}\right\rangle\,m\,\pi\,a_{*}^{2}$ is the mass per-
unit-length of the flux rope and $\left\langle{n_{*}}\right\rangle$ is the
average density of the flux rope. The additional terms
$f_{\mathrm{d}}=c_{\mathrm{d}}\,n_{\mathrm{c}}\,m\,a_{*}\,\left(V_{\mathrm{SW}}-V\right)\,\left|V_{\mathrm{SW}}-V\right|$
(4a) and
$f_{\mathrm{g}}=\pi\,a_{*}^{2}\,m\,g\,\left(n_{\mathrm{c}}-\left\langle{n_{*}}\right\rangle\right),$
(4b)
introduced in Chen (1989), represent the drag and gravitational forces per-
unit-length of the flux rope respectively where
$c_{\mathrm{d}}\sim\mathcal{O}\left(1\right)$ is the drag coefficient,
$m=1.67\times 10^{-24}\,\mbox{g}$ is the mass of hydrogen, $n_{\mathrm{c}}$ is
the ambient coronal density, $V\equiv{dH}/dt$, $V_{\mathrm{SW}}$ is the
velocity of the ambient solar wind, and the gravitational acceleration is
$g=\frac{g_{\sun}}{\left(1+H/R_{\sun}\right)^{2}},$ (5)
with $g_{\sun}=2.74\times 10^{4}\,\mbox{cm}\mbox{s}^{-1}$ and
$R_{\sun}=6.74\times 10^{10}\,\mbox{cm}$. The flux-rope footpoints are
separated by a distance $S_{\mathrm{f}}$ in the photosphere and assumed to
remain fixed throughout the evolution of the flux-rope CME by the “dense
subphotospheric plasma” Chen (1989). The current-channel radius at the base of
the corona $a_{\mathrm{c}}$ is also assumed to remain fixed throughout the
evolution of the flux rope. The major radius $\mathcal{R}$ and height $H$
above the photosphere are related by
$\mathcal{R}\equiv\frac{H^{2}+S_{\mathrm{f}}^{2}/4}{2\,H}\qquad\mbox{and}\qquad{H}>0.$
(6)
The length of the flux rope above the photosphere is
$L=2\,\pi\,\Theta\,\mathcal{R}$ where
$\Theta\equiv\left\\{\begin{array}[]{lr}1-\varphi/\pi&H\geq
S_{\mathrm{f}}/2,\\\ \varphi/\pi&H<S_{\mathrm{f}}/2,\end{array}\right.$ (7)
and $\varphi\equiv\arcsin\left(S_{\mathrm{f}}/2\,\mathcal{R}\right)$ Chen
(1989); Krall et al. (2000). Note the variables that describe the plasma at
the apex of the flux rope $\left\langle{n_{*}}\right\rangle$,
$\left\langle{p_{*}}\right\rangle$, $B_{\theta{a_{*}}}$, $B_{\phi_{*}}$, and
$I_{\phi}$ will evolve with time and parameters of the interplanetary medium
$n_{\mathrm{c}}$, $B_{\mathrm{c}}$, and $V_{\mathrm{SW}}$ are implicit
functions of $H$. (see Chen, 1996, for a complete description of the
interelations between variables and a model for the parameters of the
interplanetary medium) Krall et al. (2000).
Flux-rope equilibria are determined from
$f_{R*}=f_{a*}=0.$ (8)
Chen & Garren (1994), Chen (1996), Krall et al. (2000), and Krall & Chen
(2005) proposed local polar equilibrium profiles for $\partial_{\zeta}\approx
0$ and large aspect ratio $\mathcal{R}/a\gg 1$
$B_{\theta}\left(r\right)=B_{\theta{a}}\,\left\\{\begin{array}[]{lr}\displaystyle
3\,\frac{r}{a}\,\left(1-\frac{r^{2}}{a^{2}}+\frac{r^{4}}{3\,a^{4}}\right)&\displaystyle
r<a,\\\ \displaystyle\frac{a}{r}&\displaystyle r>a,\end{array}\right.$ (9a)
$B_{\phi}\left(r\right)\simeq{B}_{\zeta}\left(r\right)=3\,B_{\zeta{a}}\,\left\\{\begin{array}[]{lr}\displaystyle
1-2\,\frac{r^{2}}{a^{2}}+\frac{r^{4}}{a^{4}}&\displaystyle r<a,\\\
\displaystyle 0&\displaystyle r>a,\end{array}\right.$ (9b)
compatible with the flux-rope model with a toroidal flux of
$\Phi_{\phi}=\Phi_{\zeta}\equiv\left\langle{B}_{\zeta}\right\rangle\,\pi\,a^{2}=B_{\zeta{a}}\,\pi\,a^{2},$
(10a) and carrying a bare toroidal current of
$I_{\phi}=I_{\zeta}={B}_{\zeta{a}}\,{a}\,c/2.$ (10b)
There is no return current which is consistent with the large-scale current
systems in active regions analyzed by Wheatland (2000). Since Equations (9a)
and (9b) do not represent a force-free equilibria
$\mbox{\boldmath{$J$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$B$}}=0$, the
poloidal and toroidal magnetic fields and corresponding currents are
decoupled.222In equilibrium, the magnetic pressure is in detailed balance with
gravity and kinetic pressure. Consequently, the cutoff of the toroidal field
$B_{\zeta}$ at $r=a$ is arbitrary relative to the poloidal magnetic field. For
example,
$B_{\zeta}\left(r\right)=B_{\zeta{a}}\,e^{-r^{2}/a^{2}},$ (11)
is a valid toroidal field model and contains the identical amount of flux as
Equation (9b).
The toroidal energy per-unit-length of the flux rope is
$\mathcal{U}_{\zeta}=\frac{1}{4}\,\int_{0}^{a}{dr}\,{r}\,B^{2}_{\zeta}\left(r\right)\simeq\frac{9}{40}\,B_{\zeta{a}}^{2}\,{a}^{2},$
(12a) and the poloidal energy per unit length contained within a distance of
radius $r$ of the toroidal axis of the flux rope is
$\mathcal{U}_{\theta}\left(r\right)=\frac{1}{4}\,\int_{0}^{r}{dr^{\prime}}\,{r^{\prime}}\,B^{2}_{\theta}\left(r^{\prime}\right)\simeq\frac{B_{\theta{a}}^{2}\,a^{2}}{480}\,\left[73+120\,\log\left(r_{\mathrm{c}}/a\right)\right].$
(12b)
The local polar equilibrium profile Equations (9a) and (9b) does not formally
admit a bounded poloidal energy per unit length along the flux rope—a
necessary physical condition for admissibility—because
$B_{\theta}\sim{r}^{-1}$ and
$\mathcal{U}_{\theta}\left(r\right)\sim\log\left(r\right)$ for $r>a$. However,
for the closed circuit representative of the schematic flux rope shown in
Figure 1, the integral is cutoff at distances of order of the dimension of the
circuit (see pp. 136-141 in Landau & Lifshitz, 1960) which is roughly half the
footpoint separation $r_{\mathrm{c}}\simeq S_{\mathrm{f}}/2$. The fraction of
poloidal magnetic energy contained inside radius $r$ is then
$\delta_{\theta}\left(r,a,r_{\mathrm{c}}\right)\equiv\frac{\mathcal{U}_{\theta}\left(r\right)}{\mathcal{U}_{\theta}\left(r_{\mathrm{c}}\right)}\simeq\frac{73+120\,\log\left(r/a\right)}{73+120\,\log\left(r_{\mathrm{c}}/a\right)}.$
(13)
Chen et al. (2000) have argued that a “large fraction of the injected poloidal
energy is in the magnetic field outside the current channel,” denoted by
$r=a$. Indeed, in the corona, roughly 1/3 of poloidal energy is contained
within the current channel $r\leq{a}$. However, more than 2/3 of the poloidal
energy is contained within the region $r\leq 2\,a$ near the current channel.
In the photosphere, the amounts are reduced because the current channel is
narrower, but the poloidal energy is contained within the region $r\leq 2\,a$
near where the current channel remains significant varying from
$1/3\Rightarrow 2/3$ depending on the strength of the toroidal field in the
photosphere. The comparison of the flux-injection hypothesis against
photospheric observations in Section 4 is limited to the region $r\leq 2\,a$
where the poloidal energy transport is substantial and the poloidal magnetic
field in significant.
Figure 2: Schematic diagram of the flux-rope leg through the photosphere
adapted from Chen (2001). The current channel carrying $I_{\zeta}$, the top of
which is shaded, and the poloidal field $B_{\theta}$, and two community field
lines extending into the corona $B_{\mathrm{c}}$ are shown. The subscripts “c”
and “p” refer to coronal and photospheric values respectively.
Although the magnetic field model Equations (9a) and (9b) was originally
proposed as an approximate local description for large-aspect ratio coronal
toroidal flux-rope magnetic fields, it has also been used as an initial
equilibrium for investigating photospheric signatures in simulations of
subphotospheric flux ropes Chen & Huba (2005a, b, 2006) and similar models
have been implemented for modeling the photosphere flux-rope dynamics Sakai et
al. (2000, 2001). Figure 2 shows a schematic diagram of the flux-rope leg
through the photosphere adapted from Chen (2001) in the local polar $r$,
$\theta$, $\zeta$ coordinate system. The vertical current channel carrying
$I_{\zeta}$, the top of which is shaded, and the private poloidal field
$B_{\theta}$, and two community field lines extending into the corona
$B_{\mathrm{c}}$ are shown. The subscripts “c” and “p” refer to values at the
base of the corona and in the photosphere, respectively. Two general
constraints may be applied quasi-adiabatically to extend the coronal model
into the photosphere: conservation of vertical flux and vertical current Krall
& Chen (2005)
$\displaystyle B_{\zeta{\mathrm{p}}}$ $\displaystyle=$ $\displaystyle
B_{\zeta{\mathrm{c}}}\,\frac{a_{\mathrm{c}}^{2}}{a_{\mathrm{p}}^{2}},$ (14a)
$\displaystyle{B_{\theta{\mathrm{p}}}}$ $\displaystyle=$
$\displaystyle{B_{\theta{\mathrm{c}}}}\,\frac{a_{\mathrm{c}}}{a_{\mathrm{p}}}.$
(14b)
Combining these relationships with Equations (9a) and (9b) produces an
approximate local model for the magnetic footpoints of the flux rope in the
photosphere with $B_{\zeta\mathrm{c}}=B_{\phi\mathrm{c}}$ at the base of the
corona. The details of the flux-tube expansion in the chromosphere is not
explicitly specified in this approximation. However, the expansion implies a
local radial field $B_{r}\left(r\right)$ to balance the decrease of
$B_{\zeta}\left(\zeta\right)$ with height under the local constraint of
$\mbox{\boldmath{$\nabla$}}\cdot\mbox{\boldmath{$B$}}=0$.
## 3 THE PHOTOSPHERIC FLUX BUDGETS OF THE FLUX-ROPE MODEL
The flux-rope model requires specification of two quantities in addition to
the geometrical aspects of the flux rope, namely the toroidal and poloidal
fluxes Chen (1996)
$\displaystyle\Phi_{\phi}$ $\displaystyle=$ $\displaystyle
B_{\phi\mathrm{c}}\,\pi\,a_{\mathrm{c}}^{2}=\mathrm{constant},$ (15a)
$\displaystyle\Phi_{\theta}$ $\displaystyle=$ $\displaystyle
c\,\mathcal{L}\,I_{\phi},$ (15b)
where
$\mathcal{L}\simeq\frac{4\,\pi\,\Theta\,\mathcal{R}}{c^{2}}\,\left\\{\log\left(8\,\mathcal{R}\right)-1+\frac{\xi}{2}-\frac{1}{a_{*}-a_{\mathrm{c}}}\left[a_{*}\,\log\left(a_{*}\right)-a_{\mathrm{c}}\,\log\left(a_{\mathrm{c}}\right)\right]\right\\}$
(16)
is the inductance Landau & Lifshitz (1960); Krall et al. (2000). The toroidal
energy is estimated from the arclength of the flux rope above the
photosphere.333The flux-rope minor current-channel radius varies linearly with
arc length
$a\left(\ell\right)=a_{\mathrm{c}}+\left(a_{*}-a_{\mathrm{c}}\right)\,\ell$
from the footpoint at $\ell=0$ to the apex at $\ell=1$.
$U_{\phi}\simeq\frac{9\,\pi\,\Theta}{20}\,\frac{\mathcal{R}}{a_{*}}\,B_{\phi{a_{\mathrm{c}}}}^{2}\,a_{\mathrm{c}}^{3}=\frac{9\,\Theta}{20\,\pi}\,\frac{\Phi_{\phi}^{2}}{a_{\mathrm{c}}}\,\frac{\mathcal{R}}{a_{*}}\approx\mbox{constant},\\\
$ (17a) whereas the poloidal energy is related to the inductance and poloidal
flux
$U_{\theta}=\frac{1}{2}\,\mathcal{L}\,I_{\phi}^{2}=\frac{1}{2\,c^{2}}\,\frac{\Phi_{\theta}^{2}}{\mathcal{L}}.$
(17b)
The toroidal flux $\Phi_{\phi}=\Phi_{\zeta}$ is conserved because the toroidal
field $B_{\phi{a_{\mathrm{c}}}}=B_{\zeta{a_{\mathrm{c}}}}$ and current-channel
radius $a_{\mathrm{c}}$ at the base of the corona are held constant in time
during the eruption—$B_{\phi{a}}$ and the minor radius $a$ may vary along the
flux rope subject to flux conservation. The toroidal energy $U_{\phi}$ is
conserved because the flux rope erupts self-similarly
${\mathcal{R}}/{a_{*}}\approx\mbox{constant}$. Consequently, the quantities
that will manifest dynamical changes in the photosphere during the eruption
are the poloidal power $dU_{\theta}/dt$ and the rate-of-change in poloidal
flux $d\Phi_{\theta}/dt$ which are related to the photospheric energy and
helicity fluxes respectively.
The MHD induction equation derived by combining Faraday’s Law
$\partial_{t}\mbox{\boldmath{$B$}}=-c\,\mbox{\boldmath{$\nabla$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$E$}},$
(18)
with Ohm’s Law
$\mbox{\boldmath{$E$}}=\frac{\mbox{\boldmath{$J$}}}{\sigma}-\frac{\mbox{\boldmath{$v$}}}{c}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$B$}},$
(19)
to obtain
$\partial_{t}\mbox{\boldmath{$B$}}=\mbox{\boldmath{$\nabla$}}\mbox{\boldmath{$\times$}}\left(\mbox{\boldmath{$v$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$B$}}\right)-c\,\mbox{\boldmath{$\nabla$}}\mbox{\boldmath{$\times$}}\left(\frac{\mbox{\boldmath{$J$}}}{\sigma}\right),$
(20)
where $v$ is the plasma velocity and $\sigma$ is a spatially variable
conductivity. The magnetic energy in the corona is formulated by dotting the
induction Equation (20) with the magnetic field $B$ and using Ampere’s Law
without displacement currents
$\mbox{\boldmath{$\nabla$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$B$}}=\frac{4\,\pi}{c}\,\mbox{\boldmath{$J$}},$
(21)
to derive Poynting theorem
$\frac{dU_{\mathrm{M}}}{dt}\equiv\frac{d}{dt}\int_{V_{\mathrm{c}}}\frac{dV}{8\,\pi}\,{B^{2}}=\frac{1}{4\,\pi}\,\oint_{S}{dS}\,\widehat{n}\cdot\left[\mbox{\boldmath{$B$}}\mbox{\boldmath{$\times$}}\left(\mbox{\boldmath{$v$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$B$}}\right)+\frac{c}{\sigma}\,\left(\mbox{\boldmath{$J$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$B$}}\right)\right]-\int_{V_{\mathrm{c}}}{dV}\,\left[\frac{\mbox{\boldmath{$v$}}}{c}\cdot\left(\mbox{\boldmath{$J$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$B$}}\right)+\frac{J^{2}}{\sigma}\right],$
(22)
where the volume integrals $V_{\mathrm{c}}$ are over the corona, chromosphere,
and transition region, the surface integrals bounding the volume
$S\equiv{S}_{\mathrm{p}}+{S}_{\mathrm{oc}}$ are over the photosphere
${S}_{\mathrm{p}}$ and the outer corona ${S}_{\mathrm{oc}}$ at
$R\gg{R}_{\sun}$, and $\widehat{n}$ is the surface normal pointing into the
coronal volume (radially outward at the photosphere ${S}_{\mathrm{p}}$ and
radially inward at the outer corona ${S}_{\mathrm{oc}}$). Assessing the energy
budget of the region between the photosphere and outer corona by tracking the
Poynting flux through the photosphere and the energy leaving the corona
through eruptive phenomena provides an estimate for the free energy available
for producing flares and CMEs (see Figure 1 in Kusano et al., 2002). The first
and second terms in Equation (22) represent the $E$$\times$$B$ Poynting flux
through the surfaces and the third and fourth terms represent the conversion
of magnetic energy to kinetic energy though work done by the $J$$\times$$B$
force on the plasma and Ohmic heating through resistivity respectively.
The helicity for the flux rope may be written444See pp. 21,516 in Chen (1996).
This expression is exact in the large aspect ratio limit $\mathcal{R}/a\gg 1$.
Chen (1996); Chen & Krall (2003)
$K\equiv\int_{V_{\mathrm{c}}}{dV}\,\mbox{\boldmath{$A$}}\cdot\mbox{\boldmath{$B$}}\simeq\Phi_{\phi}\,\Phi_{\theta}.$
(23)
Comparisons between integrated photospheric helicity flux and the poloidal
flux injection profile for the flux-rope model can be made. However, because
the field lines of the flux rope penetrate the photosphere, a gauge-invariant
relative helicity must be used for estimating helicity fluxes through the
photosphere Berger & Field (1984)
$\Delta{K}=\int_{V_{\mathrm{c}}}{dV}\,\left(\mbox{\boldmath{$A$}}\cdot\mbox{\boldmath{$B$}}-\mbox{\boldmath{$A$}}_{\mathrm{R}}\cdot\mbox{\boldmath{$B$}}_{\mathrm{R}}\right),$
(24)
where $V_{\mathrm{c}}$ corresponds to the volume above the photosphere and
$\mbox{\boldmath{$B$}}_{\mathrm{R}}=\mbox{\boldmath{$\nabla$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$A$}}_{\mathrm{R}}$
are the reference fields which are chosen to match the normal components of
$B$ and the tangential components of $A$ respectively at the surface:
$\displaystyle\left.\left(\mbox{\boldmath{$A$}}-\mbox{\boldmath{$A$}}_{\mathrm{R}}\right)\mbox{\boldmath{$\times$}}\widehat{n}\right|_{S}$
$\displaystyle=$ $\displaystyle 0,$ (25a)
$\displaystyle\left.\left(\mbox{\boldmath{$B$}}-\mbox{\boldmath{$B$}}_{\mathrm{R}}\right)\cdot\widehat{n}\right|_{S}$
$\displaystyle=$ $\displaystyle 0.$ (25b)
These boundary conditions are sufficient for the equivalence of the relative
helicity defined by Berger & Field (1984) in Equation (24) and manifestly
gauge invariant relative helicity defined by Finn & Antonsen (1985)
$\Delta{K}=\int_{V_{\mathrm{c}}}{dV}\,\left(\mbox{\boldmath{$A$}}+\mbox{\boldmath{$A$}}_{\mathrm{R}}\right)\cdot\left(\mbox{\boldmath{$B$}}-\mbox{\boldmath{$B$}}_{\mathrm{R}}\right),$
(26)
because
$\displaystyle\int_{V_{\mathrm{c}}}{dV}\,\left(\mbox{\boldmath{$A$}}_{\mathrm{R}}\cdot\mbox{\boldmath{$B$}}-\mbox{\boldmath{$A$}}\cdot\mbox{\boldmath{$B$}}_{\mathrm{R}}\right)$
$\displaystyle=$
$\displaystyle\int_{V_{\mathrm{c}}}{dV}\,\left(\mbox{\boldmath{$A$}}_{\mathrm{R}}\cdot\mbox{\boldmath{$\nabla$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$A$}}-\mbox{\boldmath{$A$}}\cdot\mbox{\boldmath{$\nabla$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$A$}}_{\mathrm{R}}\right),$
(27) $\displaystyle=$
$\displaystyle\int_{V_{\mathrm{c}}}{dV}\,\mbox{\boldmath{$\nabla$}}\cdot\left(\mbox{\boldmath{$A$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$A$}}_{\mathrm{R}}\right),$
$\displaystyle=$
$\displaystyle-\int_{S}{dS}\,\widehat{n}\cdot\left(\mbox{\boldmath{$A$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$A$}}_{\mathrm{R}}\right),$
$\displaystyle=$ $\displaystyle 0,$
with Equation (25).
A judicious choice for the reference field is a potential field
$\mbox{\boldmath{$B$}}_{\mathrm{R}}=\mbox{\boldmath{$\nabla$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$A$}}_{\mathrm{R}}=\mbox{\boldmath{$\nabla$}}\Psi_{\mathrm{R}},$
(28a) in the Coulomb gauge
$\mbox{\boldmath{$\nabla$}}\cdot\mbox{\boldmath{$A$}}_{\mathrm{R}}=0,$ (28b)
with the additional boundary condition
$\left.\widehat{n}\cdot\mbox{\boldmath{$A$}}_{\mathrm{R}}\right|_{S}=0.$ (28c)
For these conditions, this reference field has zero helicity
$K_{\mathrm{R}}=\int_{V_{\mathrm{c}}}{dV}\,\mbox{\boldmath{$A$}}_{\mathrm{R}}\cdot\mbox{\boldmath{$B$}}_{\mathrm{R}}=\int_{V_{\mathrm{c}}}{dV}\,\mbox{\boldmath{$A$}}_{\mathrm{R}}\cdot\mbox{\boldmath{$\nabla$}}\Psi_{\mathrm{R}}=\oint_{S}{dS}\,\widehat{n}\cdot\mbox{\boldmath{$A$}}_{\mathrm{R}}\Psi_{\mathrm{R}}-\int_{V_{\mathrm{c}}}{dV}\,\Psi_{\mathrm{R}}\,\mbox{\boldmath{$\nabla$}}\cdot\mbox{\boldmath{$A$}}_{\mathrm{R}}=0.$
(29)
Using Equation (24) or (26) with Equations (28a) and (28b) and boundary
conditions (25) and (28c), the Poynting theorem for the magnetic helicity into
the corona then takes a particularly simple form
$\displaystyle\frac{d\Delta{K}}{dt}$ $\displaystyle=$
$\displaystyle\frac{d}{dt}\,\int_{V_{\mathrm{c}}}{dV}\,\mbox{\boldmath{$A$}}\cdot\mbox{\boldmath{$B$}},$
(30a) $\displaystyle=$ $\displaystyle
2\,\oint_{S}{dS}\,\widehat{n}\cdot\left[\mbox{\boldmath{$A$}}_{\mathrm{R}}\mbox{\boldmath{$\times$}}\left(\mbox{\boldmath{$v$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$B$}}\right)-\frac{c}{\sigma}\,\mbox{\boldmath{$A$}}_{\mathrm{R}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$J$}}\right]-2\,c\,\int_{V_{\mathrm{c}}}{dV}\,\frac{\mbox{\boldmath{$J$}}\cdot\mbox{\boldmath{$B$}}}{\sigma}.$
(30b)
The first and second terms in Equation (30b) represent the helicity flux
through the photosphere and the third term represents helicity dissipation in
the coronal volume.
Although the photosphere is not ideal, with a magnetic Reynolds number of
$R_{\mathrm{M}}={U\,L}/{\eta}\sim 10^{5}\\--10^{6}$ where $U$ is the typical
velocity, $L$ is the typical gradient scale, and
$\eta=c^{2}/\left(4\,\pi\,\sigma\right)$ is the magnetic diffusivity, the
ideal approximation has been demonstrated to be adequate for inferring plasma
velocities from magnetic field dynamics in convection zone simulations555See
Abbett et al. (2004). with magnetic Reynolds numbers as low as
$R_{\mathrm{M}}\sim 10^{3}$ Welsch et al. (2007); Schuck (2008). The ideal MHD
induction equation becomes
$\partial_{t}\mbox{\boldmath{$B$}}=\mbox{\boldmath{$\nabla$}}\mbox{\boldmath{$\times$}}\left(\mbox{\boldmath{$v$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$B$}}\right).$
(31)
Within the flux injection paradigm, consistent with Equation (15a), the
toroidal magnetic field in the photosphere does not change during the eruption
Chen et al. (1997)
$\partial_{t}{B}_{\zeta}=\smash{\widehat{\zeta}}\cdot\mbox{\boldmath{$\nabla$}}\mbox{\boldmath{$\times$}}\left(\mbox{\boldmath{$v$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$B$}}\right)\approx
0.$ (32)
This relationship also represents an observational constraint on the vertical
magnetic field of the flux rope in the photosphere because large changes in
line-of-sight magnetograms near disk center have not been observed during
eruptions. This constraint on the vertical magnetic field implies
$\mbox{\boldmath{$v$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$B$}}\approx\mbox{\boldmath{$\nabla$}}_{h}\psi+\left(v_{r}\,B_{\theta}-B_{r}\,v_{\theta}\right)\,\smash{\widehat{\zeta}},$
(33a) where the subscript “h” refers to the horizontal $\left(r,\theta\right)$
coordinates of the local polar coordinate system and $\psi$ is the
electrostatic potential. Solving Equation (33a) for $v$ produces
$\mbox{\boldmath{$v$}}=\frac{\smash{\widehat{\zeta}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$B$}}\,\left(\mbox{\boldmath{$B$}}\cdot\mbox{\boldmath{$\nabla$}}_{h}\psi\right)}{B_{\zeta}\,B^{2}}-\frac{\mbox{\boldmath{$\nabla$}}_{h}\psi\mbox{\boldmath{$\times$}}\mbox{\boldmath{$B$}}}{B^{2}}+v_{\parallel}\,\mbox{\boldmath{$B$}}/\left|\mbox{\boldmath{$B$}}\right|,$
(33b)
where $v_{\parallel}$ is the field-aligned plasma velocity. For a locally
cylindrical flux rope with $\partial_{\theta}=0$ and $B_{r}=0$ consistent with
Equation (9), the evolution of the radial and poloidal fields at the
photosphere are determined by
$\displaystyle\partial_{t}B_{r}$ $\displaystyle=$ $\displaystyle 0,$ (34a)
$\displaystyle\partial_{t}B_{\theta}$ $\displaystyle=$
$\displaystyle\partial_{r\zeta}\psi-
B_{\zeta}^{-1}\,\left(\partial_{\zeta}{B_{\zeta}}\right)\,\left(\partial_{r}\psi\right),$
(34b) $\displaystyle\partial_{t}B_{\zeta}$ $\displaystyle=$ $\displaystyle 0.$
(34c)
Chen (1996) and Chen & Kunkel (2010) have noted that the poloidal magnetic
field at the base of the corona may only increase by 20%-50% during the first
30 minutes of the eruptions and Krall et al. (2001) have shown an event that
requires no perceptible increase in the poloidal magnetic field (see Figure
8(a) in Krall et al., 2001). To the extent that $\partial_{\zeta}\approx 0$,
these results are consistent with MHD.
### 3.1 The Ideal Poynting Fluxes in the Photosphere
From Equations (22), (30b), and (33b), the ideal MHD energy flux through the
photosphere is
$\displaystyle u_{\zeta}$ $\displaystyle=$
$\displaystyle\frac{1}{4\,\pi}\,\left(v_{\zeta}\,\mbox{\boldmath{$B$}}_{h}-B_{\zeta}\,\mbox{\boldmath{$v$}}_{h}\right)\cdot\mbox{\boldmath{$B$}}_{h},$
(35) $\displaystyle=$
$\displaystyle-\frac{1}{4\,\pi}\,\left(\smash{\widehat{\zeta}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$\nabla$}}_{h}\psi\right)\cdot\mbox{\boldmath{$B$}}_{h},$
and the helicity flux is
$\displaystyle k_{\zeta}$ $\displaystyle=$ $\displaystyle
2\,\left(v_{\zeta}\,\mbox{\boldmath{$B$}}_{h}-B_{\zeta}\,\mbox{\boldmath{$v$}}_{h}\right)\cdot\mbox{\boldmath{$A$}}_{\mathrm{R}},$
(36) $\displaystyle=$ $\displaystyle
2\,\smash{\widehat{\zeta}}\cdot\left(\mbox{\boldmath{$A$}}_{\mathrm{R}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$\nabla$}}_{h}\psi\right).$
Equations (33), (34b), (35), and (36) are general with respect to the
constraint (32); the fluxes through the photosphere are completely specified
by the electric potential $\psi$ and the magnetic field. The flux-injection
hypothesis has always appealed to some “unspecified sub-photospheric process”
for initiating and driving the CME (Krall et al. 2000; see also Chen, 1989;
Chen & Garren, 1993; Chen, 1996; Chen et al., 1997; Chen, 1997; Chen et al.,
2000; Chen, 2001; Chen & Krall, 2003; Krall et al., 2001; Chen & Kunkel,
2010). Nonetheless, in the ideal MHD limit, energy is transported through the
photosphere by the term
$\widehat{\zeta}\cdot\mbox{\boldmath{$B$}}\mbox{\boldmath{$\times$}}\left(\mbox{\boldmath{$v$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$B$}}\right)$
and helicity is transported by
$\widehat{\zeta}\cdot\mbox{\boldmath{$A$}}_{\mathrm{R}}\mbox{\boldmath{$\times$}}\left(\mbox{\boldmath{$v$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$B$}}\right)$.
These terms integrated over the footpoints in the photosphere (including the
region outside the current channel) must balance the poloidal power
requirements $dU_{\theta}/dt$ and helicity requirements
$dK/dt\simeq\Phi_{\phi}\,d\Phi_{\theta}/dt$ of a CME described by the flux-
rope model. Temporally, the increase in the fluxes at the footpoints should
precede the eruptions by at least $\delta t\simeq 4$ minutes to account for
the transport of magnetic field along the flux rope to the apex of the CME at
the coronal Alfvén speed666The Alfvén speed is based on flux-rope initial
conditions from Chen (1996) with $\left\langle{n}\right\rangle\simeq 7.5\times
10^{7}\,\mbox{cm}^{-3}$ and $B\simeq 6$ G. Note that this estimate is an order
of magnitude larger than estimates corresponding to the same height range by
Régnier et al. (2008). $V_{\mathrm{A}}\simeq 1.5\times
10^{8}\,\mbox{cm}\,\mbox{s}^{-1}$.
Assuming azimuthal symmetry $\partial_{\theta}=0$, the poloidal power injected
through the flux-rope footpoints can be written in terms of Equation (17b) or
(35)
$\frac{dU_{\theta}}{dt}\simeq\left(\frac{2}{\Phi_{\theta}}\,\frac{\Phi_{\theta}}{dt}-\frac{1}{\mathcal{L}}\,\frac{d\mathcal{L}}{dt}\right)\,\frac{\Phi_{\theta}^{2}}{2\,\mathcal{L}\,c^{2}}\simeq-2\times\frac{1}{2}\,\int_{0}^{r_{\mathrm{c}}}{dr}\,{r}\,{B}_{\theta}\,\partial_{r}\psi=-\int_{0}^{r_{\mathrm{c}}}{dr}\,{r}\,{B}_{\theta}\,\partial_{r}\psi,$
(37)
and because the toroidal flux is constant, the rate of change of helicity is
$\frac{dK}{dt}\simeq\Phi_{\phi}\,\frac{d\Phi_{\theta}}{dt}\simeq-2\times
4\,\pi\int_{0}^{r_{\mathrm{c}}}{dr}\,r\,A_{\mathrm{R}\theta}\,\partial_{r}\psi,$
(38)
where the fluxes are assumed to be equipartitioned between the two footpoints.
Equations (37) and (38) represent independent constraints between the flux-
rope dynamics and the energy and helicity fluxes in the photosphere.
Balancing the photospheric Poynting fluxes against the poloidal energy and
helicity budgets of CME trajectories fit by the flux-rope model requires
choosing a photospheric model for the magnetic field and a velocity profile or
$\partial_{r}\psi$. For the former, the simple photospheric extension of the
flux-rope model described in Section 2 is used. For the latter, the optimal
velocities for transporting energy and helicity across the photosphere
consistent with the respective overall budgets of the flux-rope model are
estimated via constrained variational calculus (see Appendices A and B).
Optimal, in this context, means the minimum root-mean-squared (rms)
photospheric velocities corresponding to the minimum photospheric kinetic
energy (for constant density).
There are two physical mechanisms for transporting energy and helicity through
the photosphere and into the corona:
1. 1.
Twisting the magnetic field in the photosphere through poloidal motion. Under
this mechanism and the magnetic field model in Equation (9), all the energy
and helicity must be transported through the photosphere and into the corona
within the current channel where $B_{\zeta}\neq 0$. Minimizing
$\int{dr}\,{r}\,v_{\theta}^{2}$ with $v_{\zeta}=0$, the rms poloidal
photospheric velocity inside $r\leq{a_{\mathrm{p}}}$ constrained by the energy
budget is derived in Appendix A
* $\displaystyle\hskip 0.0pt\left\langle{v}_{\theta}^{2}\right\rangle_{a_{\mathrm{p}}}^{1/2}=\left|\frac{dU_{\theta}}{dt}\right|\,\frac{4\,\sqrt{70}}{\sqrt{437\,\left|B_{\theta{\mathrm{c}}}^{2}\,B_{\zeta{\mathrm{c}}}\right|}\,a_{\mathrm{c}}^{2}}\,\frac{1}{\sqrt{\left|B_{\zeta{\mathrm{p}}}\right|}}\,\propto\frac{1}{\sqrt{\left|B_{\zeta{\mathrm{p}}}\right|}},$ [A10]
and constrained by the helicity budget derived in Appendix B
* $\displaystyle\hskip 0.0pt\left\langle{v}_{\theta}^{2}\right\rangle_{a_{\mathrm{p}}}^{1/2}=\left|\frac{d\Phi_{\theta}}{dt}\right|\,\sqrt{\frac{70}{437}}\,\frac{1}{\sqrt{\left|B_{\zeta\mathrm{c}}\,B_{\zeta\mathrm{p}}\right|}}\,\frac{1}{a_{\mathrm{c}}}.$ [B7]
The twisting of footpoints does not change the net current carried by the flux
rope (see discussion on pp. 5 and Appendix C).
2. 2.
Flux injection involving the emergence of poloidal flux transported through
the photosphere by vertical motion. Under this mechanism and the magnetic
field model in Equation (9), the energy and helicity transport are not limited
to the current channel, but significant transport occurs within about
$r\lesssim 2\,a_{\mathrm{p}}$. Minimizing $\int{dr}\,{r}\,v_{\zeta}^{2}$ with
$v_{\theta}=0$, the rms vertical photospheric velocity inside
$r\leq{2\,a_{\mathrm{p}}}$ constrained by the energy budget is derived in
Appendix A
* $\displaystyle\hskip 0.0pt\left\langle{v_{\zeta}^{2}}\right\rangle^{1/2}_{2\,a_{\mathrm{p}}}=\left|\frac{dU_{\theta}}{dt}\right|\,\frac{\sqrt{12516735}\,\left|B_{\zeta\mathrm{p}}\right|\,r_{\mathrm{c}}^{2}}{2\,B_{\theta\mathrm{c}}^{2}\,a_{\mathrm{c}}^{2}\,\left|2998\,B_{\zeta\mathrm{p}}r_{\mathrm{c}}^{2}-1155\,B_{\zeta\mathrm{c}}\,a_{\mathrm{c}}^{2}\right|}.$ [A13]
Note that Equation (A13) approaches a constant asymptotically if either
$r_{\mathrm{c}}\rightarrow\infty$ or $B_{\zeta\mathrm{p}}\rightarrow\infty$,
while holding the coronal flux constant
$B_{\zeta\mathrm{c}}\,a_{\mathrm{c}}^{2}$
$\left\langle{v_{\zeta}^{2}}\right\rangle^{1/2}_{2\,a_{\mathrm{p}}}\geq\lim_{r_{\mathrm{c}}\rightarrow\infty}\left\langle{v_{\zeta}^{2}}\right\rangle^{1/2}_{2\,a_{\mathrm{p}}}=\left|\frac{dU_{\theta}}{dt}\right|\,\frac{\sqrt{12516735}}{5996\,B_{\theta\mathrm{c}}^{2}\,a_{\mathrm{c}}^{2}},$
(39)
and this value is entirely determined by the current-channel radius and the
poloidal magnetic field at the base of the corona. For efficient energy
transport, a significant fraction of the energy transport for flux emergence
will occur in and near the current channel. The corresponding rms vertical
photospheric velocity inside $r\leq{2\,a_{\mathrm{p}}}$ constrained by the
helicity budget is derived in Appendix B
* $\displaystyle\hskip 0.0pt\left\langle{v_{\zeta}^{2}}\right\rangle^{1/2}_{2\,a_{\mathrm{p}}}=\left|\frac{d\Phi_{\theta}}{dt}\right|\,\frac{\sqrt{12516735}\,\left|B_{\zeta\mathrm{p}}\right|\,r_{\mathrm{c}}^{2}}{8\,B_{\theta\mathrm{c}}\,a_{\mathrm{c}}\,\left|2998\,B_{\zeta\mathrm{p}}r_{\mathrm{c}}^{2}-1155\,B_{\zeta\mathrm{c}}\,a_{\mathrm{c}}^{2}\right|}.$ [B9]
For comparison, minimizing
$\int{dr}\,{r}\,\left(\partial_{r}{v}_{\zeta}\right)^{2}$ with $v_{\theta}=0$
produces the minimum shear estimate (constant velocity $r<r_{\mathrm{c}}$)
which has rms vertical photospheric (everywhere) of
$\left\langle{v}_{\zeta}^{2}\right\rangle^{1/2}_{\mathrm{p}}=\left|\frac{dU_{\theta}}{dt}\right|\,\frac{1}{B_{\theta\mathrm{c}}^{2}\,a_{\mathrm{c}}^{2}}\,\frac{120}{73+120\,\log\left(B_{\zeta\mathrm{p}}\,r_{\mathrm{c}}/B_{\zeta\mathrm{c}}\,a_{\mathrm{c}}\right)},$
(40a) for the energy constraint and
$\left\langle{v}_{\zeta}^{2}\right\rangle^{1/2}_{\mathrm{p}}=\left|\frac{d\Phi_{\theta}}{dt}\right|\,\frac{1}{B_{\theta\mathrm{c}}\,a_{\mathrm{c}}}\,\frac{30}{73+120\,\log\left(B_{\zeta\mathrm{p}}\,r_{\mathrm{c}}/B_{\zeta\mathrm{c}}\,a_{\mathrm{c}}\right)},$
(40b) for the helicity constraint.
3. 3.
Both limits above require wasted flows along field lines $v_{\parallel}\neq
0$. However by minimizing
$\int{dr}\,{r}\,\left(v_{\theta}^{2}+v_{\smash{\zeta}}^{2}\right)$ with
$v_{\parallel}=0$, the most efficient rms total velocity inside
$r\leq{2\,a_{\mathrm{p}}}$ constrained by the energy budget is derived in
Appendix A
* $\displaystyle\hskip 0.0pt\left\langle{v^{2}}\right\rangle^{1/2}_{2\,a_{\mathrm{p}}}=\left|\frac{dU_{\theta}}{dt}\right|\,\frac{2\,B_{\zeta\mathrm{p}}\,r_{\mathrm{c}}^{2}\,\sqrt{2310}\,\sqrt{21674\,B_{\theta\mathrm{c}}^{2}+14421\,B_{\zeta\mathrm{c}}\,B_{\zeta\mathrm{p}}}}{a_{\mathrm{c}}^{2}\,\left|B_{\theta\mathrm{c}}\,B_{\zeta\mathrm{p}}\,(23984\,B_{\theta\mathrm{c}}^{2}+14421\,B_{\zeta\mathrm{c}}\,B_{\zeta\mathrm{p}})\,r_{\mathrm{c}}^{2}-9240\,a_{\mathrm{c}}^{4}\,B_{\theta\mathrm{c}}^{3}\,B_{\zeta\mathrm{c}}\right|},$ [A17a]
and the most efficient rms total velocity inside $r\leq{2\,a_{\mathrm{p}}}$
constrained by the helicity budget is derived in Appendix B
* $\displaystyle\hskip 0.0pt\left\langle{v^{2}}\right\rangle^{1/2}_{2\,a_{\mathrm{p}}}=\left|\frac{d\Phi_{\theta}}{dt}\right|\,\frac{B_{\zeta\mathrm{p}}\,r_{\mathrm{c}}^{2}\,\sqrt{2310}\,\sqrt{21674\,B_{\theta\mathrm{c}}^{2}+14421\,B_{\zeta\mathrm{c}}\,B_{\zeta\mathrm{p}}}}{2\,a_{\mathrm{c}}\,\left|B_{\zeta\mathrm{p}}\,(23984\,B_{\theta\mathrm{c}}^{2}+14421\,B_{\zeta\mathrm{c}}\,B_{\zeta\mathrm{p}})\,r_{\mathrm{c}}^{2}-9240\,a_{\mathrm{c}}^{4}\,B_{\theta\mathrm{c}}^{2}\,B_{\zeta\mathrm{c}}\right|},$ [B11]
where $v_{\parallel}=0$ everywhere.
These three driving scenarios are used to estimate the expected rms velocities
near the current channel of CME flux-rope footpoints.
### 3.2 Velocity Estimates Constrained by CME Trajectories
Figure 3: left: photospheric current-channel radius $a_{\mathrm{p}}$ (solid
line) and poloidal magnetic field $B_{\theta\mathrm{p}}$ (dashed line) as a
function of the average vertical magnetic field in the current channel
$B_{\zeta\mathrm{p}}$ assuming coronal parameters consistent with the 2000
September 12 CME trajectory: $a_{\mathrm{c}}=7.5\times 10^{9}$ cm,
$B_{\theta\mathrm{c}}=B_{\zeta\mathrm{c}}\simeq 4$ G. Right: photospheric
magnetic field profiles for $B_{\zeta\mathrm{p}}=500$ G
The flux-rope model has been fitted to the CME trajectory observed on 2000
September 12 Chen et al. (2006); Chen & Kunkel (2010). Chen et al. (2006) and
Chen & Kunkel (2010) find that the observed dynamics is consistent with a peak
poloidal flux injection rate of $d\Phi_{\theta}/dt=1.4\times
10^{19}\,\mathrm{Mx}\,\mathrm{s}^{-1}$ and an energy injection of
$2\\--4\times 10^{32}\,\mathrm{erg}$ over 40 minutes (James Chen, personal
communication 2008 November). The average power requirement of the flux-rope
model $\left\langle{dU_{\theta}/dt}\right\rangle\simeq 1.3\times
10^{29}\,\mbox{erg}\,\mbox{s}^{-1}$ is the lower limit for the power budget of
the CME over 40 minutes—the instantaneous power budget is likely to exceed
this value $dU_{\theta}/dt\geq\left\langle{dU_{\theta}/dt}\right\rangle$. The
separation distance between the filament footpoints and the flux-rope
footpoints were estimated to be $s_{\mathrm{f}}\simeq 3.5\times 10^{10}$ cm
and $S_{\mathrm{f}}\simeq 5\times 10^{10}$ cm respectively which leads to a
coronal current-channel radius
$a_{\mathrm{c}}=\left(S_{\mathrm{f}}-s_{\mathrm{f}}\right)/2\simeq 7.5\times
10^{9}$ cm. The magnetic field at the base of the corona is estimated to be
$B_{\theta\mathrm{c}}=B_{\zeta\mathrm{c}}\simeq 4$ G Chen & Kunkel (2010).
Figure 3 shows the photospheric current-channel radius $a_{\mathrm{p}}$ (solid
line) and poloidal magnetic field $B_{\theta\mathrm{p}}$ (dashed line) as a
function of the average vertical magnetic field in the current channel
determined from Equations (14a) and (14b).
Figure 4: Optimal velocity estimates for the 2000 September 12 CME trajectory
modeled by Chen et al. (2006) and Chen & Kunkel (2010) as a function of the
average vertical magnetic field in the current channel. The solid lines
correspond to rms velocities estimated from the energy budget and the dashed
lines correspond to rms velocities estimated from the helicity budget. Left:
the blue and red lines are the optimal twisting
$\left\langle{v}_{\theta}^{2}\right\rangle^{1/2}_{a_{\mathrm{p}}}$ and
emergence
$\left\langle{v}_{\smash{\zeta}}^{2}\right\rangle^{1/2}_{2\,a_{\mathrm{p}}}$
velocities at the flux-rope footpoint inside $r\leq a_{\mathrm{p}}$ and $r\leq
2\,a_{\mathrm{p}}$ respectively. The green lines indicate the emergence
velocities corresponding to the minimum shear (constant velocity,
$r<r_{\mathrm{c}}$) solution. Right: the blue, red and black lines correspond
to the poloidal
$\left\langle{v}_{\theta}^{2}\right\rangle^{1/2}_{2\,a_{\mathrm{p}}}$ and
vertical
$\left\langle{v}_{\smash{\zeta}}^{2}\right\rangle^{1/2}_{2\,a_{\mathrm{p}}}$
components of the optimal total velocity
$\left\langle{v}^{2}\right\rangle^{1/2}_{2\,a_{\mathrm{p}}}$ respectively at
the flux-rope footpoint inside $r\leq 2\,a_{\mathrm{p}}$. (both) The dotted
black lines are the solar escape velocity
$v_{\sun}=617\,\mbox{km}\,\mbox{s}^{-1}$, the dotted green lines are the
photospheric sound speed $C_{\mathrm{s}}\simeq
7.2\,\mathrm{km}\,\mbox{s}^{-1}$, and the dotted blue lines are the average
Alfvén speed $\left\langle{V}_{\mathrm{A}}\right\rangle_{2\,a_{\mathrm{p}}}$.
Figure 4 shows velocity estimates for the 2000 September 12 CME trajectory
modeled by Chen et al. (2006) and Chen & Kunkel (2010) as a function of the
average vertical magnetic field in the current channel using a cutoff scale
$r_{\mathrm{c}}=S_{\mathrm{f}}/2=2.5\times 10^{10}$ cm. The magnitude of the
vertical magnetic field can be interpreted as height with
$B_{\zeta\mathrm{p}}=B_{\zeta\mathrm{c}}=4$ G corresponding to the base of the
corona and $B_{\zeta\mathrm{p}}\simeq 500$ G corresponding to the photosphere.
The solid lines correspond to rms velocities estimated from the energy budget
and the dashed lines correspond to rms velocities estimated from the helicity
budget. In the left panel, the blue and red lines are the optimal twisting
$\left\langle{v}_{\theta}^{2}\right\rangle^{1/2}_{a_{\mathrm{p}}}$ and
emergence
$\left\langle{v}_{\smash{\zeta}}^{2}\right\rangle^{1/2}_{2\,a_{\mathrm{p}}}$
velocities at the flux-rope footpoint inside $r\leq a_{\mathrm{p}}$ and $r\leq
2\,a_{\mathrm{p}}$ respectively, corresponding to the minimum velocity
solutions in Appendices A and B. The green lines indicate the emergence
velocity corresponding to the minimum shear (constant velocity
$r<r_{\mathrm{c}}$) solution. In the right panel, the blue, red and black
lines correspond to the poloidal
$\left\langle{v}_{\theta}^{2}\right\rangle^{1/2}_{2\,a_{\mathrm{p}}}$ and
vertical
$\left\langle{v}_{\smash{\zeta}}^{2}\right\rangle^{1/2}_{2\,a_{\mathrm{p}}}$
components of the optimal total velocity
$\left\langle{v}^{2}\right\rangle^{1/2}_{2\,a_{\mathrm{p}}}$ respectively with
$v_{\parallel}=0$ at the flux-rope footpoint inside $r\leq 2\,a_{\mathrm{p}}$.
In both panels, the dotted black line is the solar escape velocity
$v_{\sun}=617\,\mbox{km}\,\mbox{s}^{-1}$, the dotted green line is the
photospheric sound speed
$C_{\mathrm{s}}\equiv\sqrt{\gamma\,p_{\mathrm{p}}/\rho_{\mathrm{p}}}\simeq
7.2\,\mathrm{km}\,\mbox{s}^{-1}$ with $\gamma=5/3$, mass density
$\rho_{\mathrm{p}}\simeq 5.85\times 10^{-8}\,\mathrm{g}/\mathrm{cm}^{3}$, and
pressure $p_{\mathrm{p}}\simeq 1.82\times
10^{-4}\,\mathrm{g}\,\mathrm{cm}^{-1}\,\mathrm{s}^{-2}$ from VAL-C model for
the quiet sun Vernazza et al. (1981) interpolated to the $\tau=1$ height of
$240\,\mathrm{km}$ for Ni I 6767.8 Å line imaged by MDI Jones (1989); Bruls
(1993). The dotted blue line is the average Alfvén speed
$\left\langle{V}_{\mathrm{A}}\right\rangle_{2\,a_{\mathrm{p}}}\equiv\left\langle{B}\right\rangle_{2\,a_{\mathrm{p}}}/\sqrt{4\,\pi\,\rho_{\mathrm{p}}}$
inside $r\leq 2\,a_{\mathrm{p}}$. The velocities based on the energy and
helicity budgets are in close agreement and this agreement is nontrivial as
Equations (37) and (38) indicate. Exact agreement for the magnetic field
profile Equations (9a) and (9b) corresponds to
$\frac{dU_{\theta}}{dt}=\frac{B_{\theta\mathrm{c}}\,a_{\mathrm{c}}}{4}\,\frac{d\Phi_{\theta}}{dt}.$
(41)
Figure 5: Example optimal velocity and Poynting flux profiles in the
photosphere for the 2000 September 12 CME assuming
$B_{\theta\mathrm{c}}=B_{\zeta\mathrm{c}}\simeq 4$ G and
$B_{\zeta\mathrm{p}}=500$ G. The dot-dashed lines correspond to the minimum
shear (constant velocity $r<r_{\mathrm{c}}$) solutions for the vertical
velocity. The crossing point between the optimal emergence and constant
velocity solutions occurs at the asterisks. The radiation emittance at the
solar surface $\mathcal{F}\simeq 6.317\times
10^{10}\,\mathrm{erg}\,\mathrm{s}^{-1}\,\mathrm{cm}^{-2}$ is indicated by the
horizontal dotted line.
Figure 5 shows example optimal velocity and Poynting flux profiles777The
optimal solutions for the energy flux and helicity flux are proportional with
$B_{\theta}$ in the former replaced with $A_{\theta}$ in the latter and
$B_{\theta}/A_{\theta}=2\,B_{\theta\mathrm{p}}/a_{\mathrm{p}}\,B_{\zeta\mathrm{p}}=2\,B_{\theta\mathrm{c}}/a_{\mathrm{c}}\,B_{\zeta\mathrm{c}}\simeq
2/a_{\mathrm{p}}$, The optimal velocity and helicity flux profiles are
proportional to those shown in Figure 5. for the 2000 September 12 CME
assuming $B_{\theta\mathrm{c}}=B_{\zeta\mathrm{c}}\simeq 4$ G and
$B_{\zeta\mathrm{p}}=500$ G. The dot-dashed lines correspond to the minimum
shear (constant velocity $r<r_{\mathrm{c}}$) solution for the vertical
velocity. The crossing point between the optimal emergence and constant
velocity solutions occurs at the asterisks. The optimal emergence velocity
profile $v_{\zeta}$, constrained by the energy budget, exceeds
$1500\,\mbox{km}\,\mbox{s}^{-1}$ inside the current channel $r\lesssim
a_{\mathrm{p}}$ and exceeds $100\,\mbox{km}\,\mbox{s}^{-1}$ over most of the
range $0<r\lesssim 3\,a_{\mathrm{p}}$. The Poynting flux should be compared
with the radiation emittance at the solar surface $\mathcal{F}\simeq
6.317\times 10^{10}\,\mathrm{erg}\,\mathrm{s}^{-1}\,\mathrm{cm}^{-2}$
indicated by the horizontal dotted line in the right panel of Figure 5. All
three velocity profiles produce Poynting fluxes which exceed the radiation
emittance at the solar surface and the fluxes of white light flare kernels
$1\\--2\times 10^{10}\,\mathrm{erg}\,\mathrm{s}^{-1}\,\mathrm{cm}^{-2}$ Neidig
(1989). The absence of strong photospheric signatures associated with CMEs
suggests that flux injection cannot be responsible for the CME eruption.
### 3.3 Discussion
Figure 6: Optimal velocity estimates for the 2003 October 28, CME/ICME
trajectory modeled by Krall et al. (2006) in the same format as Figure 4.
I emphasize that the 2000 September 12 event is not an extreme CME. Krall et
al. (2006) describe the 2003 October 28 CME, first observed in LASCO C3 images
at 11:30 UT, that requires $\Delta{U}_{\theta}\simeq 2\times
10^{33}\,\mathrm{erg}$ and $\Delta\Phi_{\theta}\simeq 6\times
10^{22}~{}\mathrm{G}\,\mathrm{cm}^{-2}$ over $\Delta t\simeq 18$ minutes with
$a_{\mathrm{c}}=8.1\times 10^{9}\,\mathrm{cm}$, $S_{\mathrm{f}}=3\times
10^{10}$ cm and $B_{\theta\mathrm{c}}=B_{\zeta\mathrm{c}}=3.2$ G. The
timescale of approximately $15$ minutes and energy requirements $2\times
10^{32}$ erg are in close agreement with the estimates used by Manchester et
al. (2008) to simulate the initiation and propagation of this eruption. Figure
6 shows optimal velocity estimates for the 2003 October 28 CME/ICME trajectory
modeled by Krall et al. (2006) in the same format as Figure 4. This event
requires extreme photospheric velocities in the range $v\simeq
2500\\--16000\,\mbox{km}\,\mbox{s}^{-1}$ to satisfy the energy budget and
$v\simeq 500\\--3000\,\mbox{km}\,\mbox{s}^{-1}$ to satisfy the helicity
budget.
Returning to the 2000 September 12 event, Figures 4 and 5 indicate that
significant photospheric signatures of flux injection should be detectable in
or near the current-channel radius $r\lesssim 2\,a_{\mathrm{p}}$. For the left
panel in Figure 4, corresponding to ideal footpoint twisting or flux injection
(emergence), all of the rms velocities exceed the local characteristic speeds
of an MHD plasma. The optimal rms emergence velocities
$\left\langle{v}_{\smash{\zeta}}^{2}\right\rangle^{1/2}_{2\,a_{\mathrm{p}}}$
exceed the solar escape velocity $v_{\sun}=617\,\mbox{km}\,\mbox{s}^{-1}$
inside of $r\lesssim 2\,a_{\mathrm{p}}$. Thus, gravitational forces cannot
restrain the photospheric material in this region. Even for emergence
velocities of $100\,\mbox{km}\,\mbox{s}^{-1}$, the photospheric material would
require 12 minutes to return to the surface by gravity alone and this is
roughly the timescale of the 2003 October 28, CME/ICME eruption. Consequently,
a characteristic of flux injection should be hypersonic upflows concomitant
with and preceding the eruption. Optimal photospheric velocities imply a
sustained mass transport rate of $6\times 10^{20}\,\mbox{g}\,\mbox{s}^{-1}$
for just the ring-shaped regions with $v_{\zeta}\geq 617$ maintained over
roughly 40 minutes and spatial scales of $\pi\,a_{\mathrm{c}}^{2}\simeq
2\times 10^{20}\,\mbox{cm}^{2}$ in the corona (see Figure 9 for the scale of
the twice the current channel in the corona). These photospheric flows would
eject a net mass of $10^{24}$ g of which is 9 orders of magnitude larger than
the typical CME mass $10^{15}$ g as estimated from LASCO images. Ballistic
photospheric mass outflows of this magnitude should be straight-forward to
detect using EIT and LASCO C1 and C2 coronagraphs—mass outflows of this
magnitude are not observed by these instruments. It is worth repeating that
the rms optimal emergence velocities in the photosphere are largely
independent of $B_{\zeta a_{\mathrm{p}}}$ and the cutoff scale
$r_{\mathrm{c}}$ and
$\left\langle{v}_{\smash{\zeta}}\right\rangle^{1/2}_{2\,a_{\mathrm{p}}}$ is
asymptotically determined by the values at the base of the corona $B_{\zeta
a_{\mathrm{c}}}$ and $a_{\mathrm{c}}$. The left panel of Figure 4 appears to
indicate that the minimum shear velocity (green) is more efficient than either
footpoint twisting (blue) or flux injection (red) since it exhibits lower
velocities inside $r\leq 2\,a_{\mathrm{p}}$. However the left panel of Figure
5 shows that this is simply because the profiles of the optimal twisting and
emergence velocities reduce to essentially zero beyond $r=a_{\mathrm{p}}$ and
$r\gtrsim 1.75\,a_{\mathrm{p}}$ respectively whereas the minimum shear
velocity remains constant out to the cutoff scale $r_{\mathrm{c}}$. For the
right panel in Figure 4, corresponding to combined footpoint twisting and flux
injection (emergence), flux emergence is only efficient when $B_{\theta}\simeq
B_{\zeta}$. For $B_{\zeta}\gg{B}_{\theta}$ footpoint twisting is more
efficient for transferring energy and helicity from the photosphere to the
corona.
Figure 7: Efficiencies of the driving scenarios for the 2000 September 12 CME.
Left: density normalized kinetic energy $I=\mathcal{I}/\rho_{\mathrm{p}}$ in a
horizontal slice and right: ratio of the kinetic energy transport across the
photosphere to the magnetic power requirements of the CME under the flux-
injection hypothesis. The solid lines correspond to rms velocities estimated
from the energy budget and the dashed lines correspond to rms velocities
estimated from the helicity budget. Blue, red, green, and black lines
correspond to optimal twisting, optimal emergence, minimum shear (constant
velocity $r<r_{\mathrm{c}})$, and optimal combined twisting and emergence
velocity profiles respectively (blue line is not shown in right panel).
The efficiencies of the processes may be ranked either with the magnitude of
the integral
$\mathcal{I}=2\,\pi\,\rho_{\mathrm{p}}\,\int_{0}^{r_{\mathrm{c}}}{dr}\,r\,v^{2},$
(42a) or
$\epsilon\equiv\frac{\left|dU_{\theta}/dt\right|}{\left|dU_{\theta}/dt\right|+2\,\pi\,{\rho_{\mathrm{p}}}\,\int_{0}^{r_{\mathrm{c}}}{dr}\,r\,v^{2}\,v_{\zeta}},$
(42b)
where
$F=\pi\,{\rho_{\mathrm{p}}}\,\int_{0}^{r_{\mathrm{c}}}{dr}\,r\,v^{2}\,v_{\zeta}$
is the kinetic energy flux through the photosphere integrated over one
footpoints. The former metric is simply the photospheric kinetic energy which
is used to optimize the velocity profiles, whereas the latter metric is the
ratio of the power requirements of the CME to the total energy transported
across the photosphere under the flux-injection hypothesis.
Figure 7 shows the efficiencies of the driving scenarios for the 2000
September 12 CME based on $\mathcal{I}$, the kinetic energy in a horizontal
slice (Equation (42a)) and $\epsilon$, the ratio of the magnetic power
requirements of the CME to the total energy transported across the photosphere
(Equation 42b) under the flux-injection hypothesis in the left and right
panels respectively. The solid lines correspond to rms velocities estimated
from the energy budget and the dashed lines correspond to rms velocities
estimated from the helicity budget. Blue, red, green, and black lines
correspond to optimal twisting, optimal emergence, minimum shear, and optimal
combined twisting and emergence velocities respectively—the blue line is not
shown in the right panel because the efficiency is $\epsilon=1$. Based on
either criterion (42a) or (42b) the minimum shear (constant velocity $r<rc$)
is the least efficient velocity profile (shown in green) for transporting
magnetic energy across the photosphere on the timescale of the eruption.
However, the most efficient velocity is different for $\mathcal{I}$ and
$\epsilon$. Using the photospheric kinetic energy $\mathcal{I}$, the combined
twisting and emergence is the most efficient velocity profile (shown as the
black line in the left panel). Using the ratio of the magnetic power
requirements of the CME to the total energy transported across the photosphere
$\epsilon$, footpoint twisting is the most efficient velocity profile
$\left(\epsilon=1\right)$ because footpoint twisting does not transport mass
across the photosphere—the velocity field is tangent to the surface.
The left panel Figure 7 shows that, for small vertical magnetic fields
$B_{\zeta\mathrm{p}}\lesssim 10$ G, the kinetic energy $\mathcal{I}$ for pure
emergence is slightly less than for pure footpoint twisting. For
$B_{\zeta\mathrm{p}}\gtrsim 10$ G, the kinetic energy for footpoint twisting
less than for pure emergence. Consequently, the twisting motions dominate the
kinetic energy of the combined twisting and emergence velocity profile since
twisting motions are more efficient for transporting energy and helicity into
the corona. The combined twisting and emergence velocity profile might be more
efficient than the right panel of Figure 7 indicates. The rms vertical
velocities for this solution in Figure 4 do not exceed the escape velocity and
significant photospheric material could eventually return to the surface via
strong downflows along magnetic fields.
## 4 COMPARISON AGAINST PHOTOSPHERIC OBSERVATIONS
The vertical photospheric plasma velocities in and near the current channel of
the CME footpoints must be large to satisfy the flux-injection hypothesis.
Such extreme velocities probably would have been detected in previous studies
of CME eruptions, but not no such observations have been reported in the
literature. Nonetheless, comparing expected values with observations is a
necessary final step to establish the likelihood that a theory is compatible
with nature. High spatial resolution high-cadence ($\sim 1$ minute) line
profiles would be ideal for examining the footpoints of CMEs during eruptions.
This suggests that limited field-of-view line-profile data would be the best
candidate data set (see Harra et al., 2007; Imada et al., 2007, for
observations of coronal outflows during the gradual phase of a flare).
However, CME footpoints are always identified post-facto and are usually
outside the main flux concentration of the active region that is often the
focus of high-resolution campaigns. Thus, full-disk data are required to
ensure that the dynamics of both CME footpoints are captured. Full-disk line-
profile data are presently scarce. One possible candidate is the Naval
Research Laboratory Skylab He II $304~{}\AA$ spectroheliograms with $2\arcsec$
spatial resolution, but these observations are usually at low temporal cadence
relative to flare/CME dynamics, in the wrong wavelength range for photospheric
observations, and are difficult to disambiguate—the spatial and wavelength
information is convolved. Finally, for direct comparison with the flux-rope
model, photospheric observations concomitant with published results are
desirable Chen et al. (1997, 2000); Wood et al. (1999); Krall et al. (2001);
Chen et al. (2006); Krall et al. (2006); Chen & Kunkel (2010)—and this event
must correspond to a front-side CME to ensure that both footpoints are visible
in the photosphere.
After surveying the CMEs modeled with the flux-rope model, the 2000 September
12 CME Chen et al. (2006); Chen & Kunkel (2010) was determined to be the best
candidate because: (1) the event was front side and associated with a M1.0
flare, (2) the filament footpoint locations identified in Chen et al. (2006)
were distinct from the flare ribbons simplifying the interpretation of the
observed emission lines, and (3) the dynamics of the filament was captured by
EIT, LASCO C2 and C3 coronagraphs, and KSO H$\alpha$-observations. The MDI
instrument was in Flarewatch mode on this day, but there is a gap in high-
cadence coverage during the flare between 07:00 UT and 15:30 UT. However, the
solar oscillations investigation (SOI) aboard SOHO provides continuous
monitoring of Doppler velocities of low- to intermediate degree $l$ Scherrer
et al. (1995). The medium-$l$ data are the result of Gaussian spatial
filtering to mitigate spatial aliasing followed by a reduction in resolution
from $1024\times 1024$ pixels at $2\arcsec\,\mathrm{pixel}^{-1}$ to $192\times
192$ at roughly $10\arcsec\,\mathrm{pixel}^{-1}$. This is supplemented with
level 1.8.2 $2\arcsec$ 5-minute averaged 96-minute cadence magnetograms with
the most recent magnetic field inter-calibrations Tran et al. (2005); Ulrich
et al. (2009).
Figure 8: Source region of eruption observed by EIT (Fe XII 195 Å) at 11:35
UT. The erupting prominence, denoted by “P” appears as a dark absorption
feature. The prominence footpoints are indicated “F1” and “F2” and their
midpoint is designated “O.” The time 11:36 in the figure indicates the
uncorrected start time. After Figure 2(a) in Chen et al. (2006).
Chen et al. (2006) identified the locations F1 and F2 of the prominence
footpoints in the EIT (Fe XII 195 Å) image shown in Figure 8.
Figure 9: Nominal geometry of the flux-rope model for the base of the corona
and photosphere. Top: EIT (Fe XII 195 Å) at 11:35 UT. Bottom: MDI (Ni I 6767.8
Å) at 11:15 UT de-rotated to 11:35 UT to match the time of the EIT image. The
red circles correspond to the filament footpoints in Figure 8. The small and
large white circles correspond to the extent of twice the current-channel
radius in the photosphere and corona respectively assuming
$B_{\zeta\mathrm{p}}\simeq 500$ G and $B_{\zeta\mathrm{c}}\simeq 4$ G. Cyan
and green contours correspond to average magnetic fields of (-500,-400,-300)
and (300,400,500) GE respectively on a scale size of $1.5\times
10^{18}\,\mbox{cm}^{2}$.
Using these locations, the nominal geometry of the flux-rope model for the
base of the corona and photosphere is diagramed in Figure 9. The top shows the
EIT (Fe XII 195 Å) image at 11:35 UT and the bottom shows the MDI (Ni I 6767.8
Å) at 11:15 UT differentially de-rotated to 11:35 UT to match the time of the
EIT image using Dominic Zarro’s mapping package.888The SolarSoft mapping
software is located at http://www.lmsal.com/solarsoft/gen/idl/maps/. The
filament and nominal current-channel footpoints are distinct from the flare
ribbons in the EIT image. The dark curved adsorbtion feature in the EIT image
spanning -175$\arcsec$ to 400$\arcsec$ in $X$ and -375$\arcsec$ to
-675$\arcsec$ in $Y$ is the filament.
The MDI level 1.8.2 magnetograms (BLDVER18=60100) incorporate the latest
sensitivities from inter-calibrations with the Mount Wilson Observatory Tran
et al. (2005); Ulrich et al. (2009). The line-of sight magnetic field was
corrected for geometrical effects with a factor of $\mu^{-1}$ assuming that
the field is purely vertical where
$\mu=\frac{{R}_{\sun}-{D}_{\sun}\,\sqrt{1-\varrho_{1}^{2}/R_{\sun}^{2}}}{\sqrt{{R}_{\sun}^{2}-2\,{R}_{\sun}\,{D}_{\sun}\,\sqrt{1-\varrho_{1}^{2}/R_{\sun}^{2}}+{D}_{\sun}^{2}}}\approx\sqrt{1-\varrho_{1}^{2}/R_{\sun}^{2}},$
(43)
where the impact parameter $\varrho_{1}$ (radians) is measured from disk
center, $R_{\sun}$ is the radius of the Sun in the telescope (radians or
arcsecs), and ${R}_{\sun}/{D}_{\sun}$ is the ratio of the radius of the Sun to
the distance between the observer and Sun center. Chen et al. (2006) argue
that the filament footpoints shown in red and the nominal current-channel
footpoints are not co-located, but are related by
$S_{\mathrm{f}}=s_{\mathrm{f}}+2\,a_{\mathrm{c}}.$ (44)
This idealized geometry is reflected in Figure 9 where the red circles
correspond to the filament footpoints in Figure 8 and the large and small
white circles correspond to the extent of twice the current-channel radius in
the corona and photosphere respectively. The left large white circle is
associated with mainly positive magnetic field and the right large white
circle is associated with mainly negative magnetic field.
Figure 10: Estimate for the vertical magnetic field (flux-rope toroidal
magnetic field) in the photosphere. Asterisks connected by dashed lines are
the largest magnitude MDI values adjusted for line-of-sight projection and
averaged over circular masks with increasing areas within the large white
circles of Figure 9. Solid lines are
$B_{\zeta{\mathrm{p}}}=B_{\phi{\mathrm{c}}}\,a_{\mathrm{c}}^{2}/a_{\mathrm{p}}^{2}$
estimated from the flux-rope model with $B_{\phi{\mathrm{c}}}=4$ G and
$a_{\mathrm{c}}=7.5\times 10^{9}$ cm. The top and bottom lines correspond to
the left and right footpoints of the flux rope. See the text for discussion.
The current-channel radius in the photosphere should be determined by
enforcing consistency between the flux-rope model photospheric extrapolation
and the MDI magnetogram though flux conservation (Equation (10a)). The
current-channel radius at the base of the corona is $B_{\zeta\mathrm{c}}\simeq
4$ G with a radius $a_{\mathrm{c}}=7.5\times 10^{9}$ cm Chen et al. (2006). If
all the toroidal flux in the coronal current channel mapped to a single MDI
pixel at disk center the resulting magnetic field would exceed 30 kG. Such
extreme values of magnetic field have never been observed in the photosphere.
Consequently, the photospheric current channel will be larger than one MDI
pixel except perhaps very near the limb. To estimate the current-channel size
in the photosphere while minimizing discrepancies between the photospheric
observations and the flux-rope model, circular masks of increasing scale size
were computed. These masks were convolved with the MDI values adjusted for
line-of-sight projection in regions bounded by both the left and right large
white circles in Figure 9. At each scale, the maximum magnitude magnetic
fields were selected from the output of the convolution and plotted as
asterisks connected by dashed lines in Figure 10; these values correspond to
the maximum magnitude average magnetic field at that scale size contained
within the regions bounded by the large white circles in Figure 9. The solid
lines are
$B_{\zeta{\mathrm{p}}}=B_{\phi{\mathrm{c}}}\,a_{\mathrm{c}}^{2}/a_{\mathrm{p}}^{2}$
estimated from the flux-rope model with $B_{\phi{\mathrm{c}}}=4$ G and
$a_{\mathrm{c}}=7.5\times 10^{9}$ cm. The top and bottom lines correspond to
the left and right footpoints of the flux rope. The photospheric extrapolation
of flux-rope model agrees with the MDI magnetogram where the solid and dashed
lines cross. The photospheric area most consistent with the constraints of the
flux-rope model is $1\\--2\times 10^{8}\,\mbox{km}^{2}$ corresponding to a
current-channel radius of $a_{\mathrm{p}}\simeq 6.7\times 10^{8}\,\mbox{cm}$
and an average photospheric toroidal field of
$\left\langle{B}_{\zeta{\mathrm{p}}}\right\rangle\simeq 500$ G.
The geometrical relationship (44) proposed in Chen et al. (2006) isn’t
entirely consistent with the photospheric observations in Figure 9. For the
left footpoint, the main concentration of positive flux is associated with the
filament footpoint not the nominal current channel. Indeed the magnetic field
in the nominal current channel ranges from -24 to +24 G. For the right
footpoint the magnetic field pixel ranges in the filament footpoint and the
nominal current channel are similar ranging from $-665$ to $-42$ G and $-600$
to $-62$ G respectively. However, the average magnetic field magnitude in the
right nominal filament footpoint does not exceed 300 G whereas the magnitude
in the nominal current channel is somewhat higher as indicated by the cyan
contour corresponding to -300 G contained within the small white circle.
Figure 11: Temporal development of the average vertical magnetic field within
$r\leq a_{\mathrm{p}}\simeq 6.7\times 10^{8}$ cm for the left (L) and right
(R) filament and current-channel footpoints shown in Figure 9. The vertical
dashed line corresponds to the rise of the M-class flare.
Figure 11 shows the temporal development of the average vertical magnetic
field within $r\leq a_{\mathrm{p}}\simeq 6.7\times 10^{8}$ cm for the left (L)
and right (R) filament and current-channel footpoints shown in Figure 9. The
vertical dashed line corresponds to the rise of the M-class flare. Around the
time of the flare, the vertical flux is strengthening at the left prominence
footpoint at a rate of $8\times 10^{15}\,\mbox{Mx}\,\mbox{s}^{-1}$, but the
flux in the nominal left current channel remains near zero over the whole five
day period. The average fluxes at the right prominence and current-channel
footpoints are flat around the eruption time. The flux-imbalance between the
left and right footpoint indicates that the connectivity between these
photospheric magnetic features is not trivial.
While gravity certainly effects the height of the heavy prominence material
potentially producing a difference between the position of the prominence
material and the center of the current channel at the apex of the flux rope,
there is no stated reason in Chen et al. (2006) for why the flux-rope current-
channel and prominence footpoints aren’t co-located in the
photosphere.999Although, the radius of the current channel at the footpoint is
defined at the base of the corona the footpoints separation is fixed by the
“dense subphotospheric plasma” (see pp. 457 in Chen, 1989) or the “massive
photospheric density” (Chen & Garren, 1993, see pp. 2320) (see also Krall et
al., 2000; Chen et al., 2008). This implies that $S_{\mathrm{f}}$ and
$s_{\mathrm{f}}$ should be interpreted as photospheric footpoint separations
(see also Chen & Krall, 2004). The footpoint separation $S_{\mathrm{f}}$ is a
critical parameter because the height $H_{\mathrm{max}}$ of the maximum
acceleration for the CME scales with $S_{\mathrm{f}}$ in the flux-rope model
$S_{\mathrm{f}}/2\leq{H}_{\mathrm{max}}\lesssim 3/2\,S_{\mathrm{f}}$ Chen et
al. (2006); Chen (2007). If $S_{\mathrm{f}}$ is not consistent with
observations, then the flux-rope model cannot be correct.
### 4.1 Doppler Data Preparation
The MDI vector-weighted Dopplergrams were analyzed following the procedures
outlined in Snodgrass (1984) and Hathaway (1992). First, the motion of the
observer was removed from each Dopplergram using
$V_{\mathit{SOHO}}\left(\varrho_{1},\psi\right)=V_{R}\,\left(1-\varrho_{1}^{2}/2\right)+V_{\mathrm{W}}\,\varrho_{1}\,\sin\,\psi-
V_{\mathrm{N}}\,\varrho_{1}\,\cos\psi,$ (45)
where the impact parameter $\varrho_{1}$ (radians) is measured from disk
center and the position angle $\psi$ (radians) measured counterclockwise from
solar north are the heliocentric radial coordinates and $V_{R}$,
$V_{\mathrm{W}}$, and $V_{\mathrm{N}}$ are the SOHO satellite velocities
radial outward, westward parallel to equator, and northward along the rotation
axis, respectively, using the appropriate keywords provided with the MDI data.
Second, the Dopplergrams are co-registered and time-averaged. The time-
averaged Dopplergram was fitted with orthonormal disk functions to eliminate
cross-talk between coefficients Snodgrass (1984).
$\overline{V}\left(B_{0},\varrho,\Theta,\Phi\right)=\omega\left(\Theta\right)\,R_{\sun}\,\sin\Phi\,\cos
B_{0}+V_{\mathrm{LS}}\left(\varrho\right)+V_{\mathrm{MF}}\left(\Theta\right)+H\,\sin\Theta,$
(46a) where differential rotation profile
$\omega\left(\Theta\right)=\sum_{n=0}^{2}A_{2n}\,\mathrm{T}^{1}_{2n}\left(\sin\Theta\right)$
(46b) is expanded in a truncated series of even orthonormal Gegenbauer
polynomials Morse (1953), the limbshift function is represented by
$V_{\mathrm{LS}}\left(\varrho\right)=\sum_{n=0}^{4}L_{n}\,\mathcal{L}_{n}\left(1-\cos\varrho\right),$
(46c) with (see pp. 174-175 in Smart, 1977)
$\varrho=-\varrho_{1}+\arcsin\left[{D}_{\sun}\,\sin\left(\varrho_{1}\right)/{R}_{\sun}\right],$
(46d) and the meridional flow represented by
$V_{\mathrm{MF}}\left(\Theta\right)=\sum_{n=1}^{2}M_{n}\,\mathcal{M}_{m}\left(\sin\Theta\right)\,S\left(\Theta\right)\,\left(\cos\Phi\,\sin\Theta\,\cos{B_{0}}-\cos\Phi\,\sin{B_{0}}\right),$
(46e)
are expanded in truncated series of $\left(1-\cos\varrho\right)$ and Fourier
series of latitude respectively where the function classes
$\mathcal{L}_{n}\left(x\right)$ and $\mathcal{M}_{n}\left(x\right)$ were
orthonormalized by the Gram-Schmidt procedure on the interval
$\left(0,1\right)$ by Snodgrass (1984). The latitude, longitude, and solar-B
angle are denoted $\Theta$, $\Phi$ and $B_{0}$ respectively and
$\varrho=\arcsin\left(\varrho_{1}/R_{\sun}\right)$ is the angle measured from
the center of the Sun between the sub-Earth point and a point on the surface
of the Sun and $S\left(\Theta\right)=+1$ for $\Theta>0$ and
$S\left(\Theta\right)=-1$ for $\Theta<0$.
Table 1: Fit-determined Coefficients with $\chi^{2}=$3898.0 for a Reduced Set of Disk-orthonormalized Functions Determined from 24 hr of MDI Vector Weighted Dopplergams on 2003 September 12. Estimates $\widehat{\overline{\mu}}$ and population standard deviations $\widehat{\sigma}_{\overline{\mu}}$ from Table 2 in Snodgrass (1984) for data taken between 1967 January 1 and 1984 March 5 at the Mount Wilson Solar Observatory. | $\mu\,\mbox{Rads}\,\mbox{s}^{-1}$ | | $\mbox{m}\,\mbox{s}^{-1}$
---|---|---|---
| $A_{0}$ | $A_{2}$ | $A_{4}$ | | $L_{0}$ | $L_{1}$ | $L_{2}$ | $L_{3}$ | $L_{4}$ | $M_{1}$ | $M_{2}$ | $H$
$\widehat{\theta}$ | 3.1870 | -0.1610 | -0.0216 | | 131 | 174 | 88 | 10 | -3.1 | -17.4 | 0.6 | 29.1
$\widehat{\sigma}_{\theta}$ | 0.0021 | string0.0027 | string0.0027 | | 019 | 026 | 21 | 11 | string3.5 | string03.0 | 0.4 | 05.0
$\widehat{t}_{\theta}$ | 1496.81 | -59.19 | -8.05 | | 6.89 | 6.64 | 4.19 | 0.92 | -0.88 | -5.76 | 1.59 | 5.87
% | 0.00 | 0.00 | 0.00 | | 0.0 | 0.0 | 0.0 | 35.8 | 37.7 | 0.0 | 11.1 | 0.0
$\widehat{\overline{\mu}}$ | 3.1556 | -0.1610 | -0.0312 | | 127 | 160 | 88 | 14 | 2.0 | 6.2 | 0.1 | 8.1
$\widehat{\sigma}_{\overline{\mu}}$ | 0.1366 | string0.0238 | string0.0136 | | 252 | 029 | 24 | 12 | 5.4 | 36.0 | 8.6 | 77.1
$\widehat{t}_{\theta\overline{\mu}}$ | 0.23 | 0.00 | 0.69 | | 0.02 | 0.34 | -0.00 | -0.25 | -0.79 | -0.65 | 0.06 | 0.27
% | 85.63 | 99.99 | 61.49 | | 98.98 | 78.94 | 99.92 | 84.30 | 57.42 | 63.10 | 96.16 | 83.13
The function (46a) was fitted to the time-averaged Dopplergram with the
standard deviations estimated from the variance of each pixel from its
respective time average. The results, with a weighted $\chi^{2}=3898.0$ and
25455 degrees of freedom, are summarized in Table 1. The first four rows of
entries represent the best estimate of each parameter $\widehat{\theta}$,
their standard deviations $\widehat{\sigma}$, their $t$-scores
$\widehat{t}\equiv\widehat{\theta}/\widehat{\sigma}$, and their two-sided
significance probability in percent that the coefficients would be larger by
chance—the smaller the probabilities, the more significant the coefficients.
The fifth and sixth rows represent the average parameter estimates
$\widehat{\overline{\mu}}$ and population standard deviations
$\widehat{\sigma}_{\overline{\mu}}$ from Table 2 in Snodgrass (1984) for the
data taken between 1967 January 1 and 1984 March 5 at the Mount Wilson Solar
Observatory (WSO). The seventh and eighth rows represent the t-scores
$\widehat{t}_{\theta\overline{\mu}}=\left(\widehat{\theta}-\widehat{\overline{\mu}}\right)/\sqrt{\widehat{\sigma}_{\theta}^{2}+\widehat{\sigma}_{\overline{\mu}}^{2}}$
and their two-sided significance probability in percent that the best
estimates $\widehat{\theta}$ and the population averages
$\widehat{\overline{\mu}}$ would differ more by chance—the larger the
probabilities, the better the two results agree. All of the best estimates are
within one rms standard deviation of the population averages reported by
Snodgrass (1984).
Figure 12: Top left: residual full disk Doppler velocities from the Medium-$l$
program averaged over 24 hr. Top right: histogram showing the correlation
between the magnitude of the magnetic field and the time-averaged residual
Doppler velocity. Bottom: magnified view of the time-averaged residual Doppler
velocities in the region containing the filament. The red circles correspond
to the filament footpoints and the large and small white circles correspond to
the extent of twice the current-channel radius in the corona and photosphere,
respectively.
The best estimates were used to construct a model line-of-sight velocity for
the solar disk which was subtracted from each co-registered Dopplergram and
then the Dopplergrams were de-rotated to coincide with the time of EIT image
in Figure 9 to produce the residual velocity $\Delta{v}_{\mathrm{LOS}}$. The
time average of the residual Doppler images is shown on the top left of Figure
12. The red circles correspond to the filament footpoints and the large and
small white circles correspond to the extent of twice the current-channel
radius in the corona and photosphere respectively.
Unfortunately the MDI Dopplergrams are not absolutely calibrated. The limb-
shift coefficient $L_{0}=131\pm 19\,\mbox{m}\,\mbox{s}^{-1}$ represents a
constant offset for the Dopplergram series. Since the coefficient is in
excellent agreement with measurements by Snodgrass (1984), cross-calibration
against WSO with a laboratory source could permit absolute calibration of MDI
Doppler velocities. The well known pseudo redshift in active regions
contributes the ambiguity. The top right panel of Figure 12 shows a two-
dimensional histogram pairing the magnitude of the magnetic field and the
Doppler velocity. The magnetic data were computed by spatially filtering the
96-minute MDI magnetograms (also de-rotated to coincide with the time of the
EIT image) with a two-dimensional Gaussian kernel consistent with the
Medium-$l$ program ($a=4$ in Kosovichev et al., 1996) followed by subsampling
to the resolution of the Medium-$l$ program. Each Dopplergram was compared to
the magnetogram that minimized differences in observation times. This panel
demonstrates that the Doppler velocities in magnetic regions are
systematically redshifted with respect to the quiet sun with $B=0$. The
histogram with $\left|B\right|<1$G is Gaussian with an offset of $-2.6\pm
0.2\,\mbox{m}\,\mbox{s}^{-1}$ and a standard deviation of $\sigma=65.7\pm
0.1\,\mbox{m}\,\mbox{s}^{-1}$. Fitting a fifth degree polynomial to the
scatterplot of $\left\langle\Delta{v}_{\mathrm{LOS}}\right\rangle$ versus $B$
produces the line in the right panel described by
$\left\langle{v}_{\mathrm{LOS}}\right\rangle\left(B\right)=\sum_{n=0}^{5}\,q_{n}\,B^{n},$
(47a) where $\displaystyle q_{0}$ $\displaystyle=$ $\displaystyle-2.2\pm
0.1\,\mbox{m}\,\mbox{s}^{-1},$ (47b) $\displaystyle q_{1}$ $\displaystyle=$
$\displaystyle\left(5.3\pm 0.4\right)\times
10^{-2}\,\mbox{m}\,\mbox{s}^{-1}\,\mbox{G},$ (47c) $\displaystyle q_{2}$
$\displaystyle=$ $\displaystyle\left(1.75\pm 0.02\right)\times
10^{-3}\,\mbox{m}\,\mbox{s}^{-1}\,\mbox{G}^{-2},$ (47d) $\displaystyle q_{3}$
$\displaystyle=$ $\displaystyle\left(-1.56\pm 0.08\right)\times
10^{-6}\,\mbox{m}\,\mbox{s}^{-1}\,\mbox{G}^{-3},$ (47e) $\displaystyle q_{4}$
$\displaystyle=$ $\displaystyle\left(-4.53\pm 0.09\right)\times
10^{-9}\,\mbox{m}\,\mbox{s}^{-1}\,\mbox{G}^{-4},$ (47f) $\displaystyle q_{5}$
$\displaystyle=$ $\displaystyle\left(4.9\pm 0.2\right)\times
10^{-12}\,\mbox{m}\,\mbox{s}^{-1}\,\mbox{G}^{-5}.$ (47g)
However, this apparent pseudo redshift in magnetic regions is actually
produced by the blueshifting of the quiet-sun regions caused by convective
motion and the brightness velocity correlation in the convection
cells.101010P. Scherrer and B. Welsch, personal communication 2009 September;
see also Dravins (1982) and Bumba & Klvaňa (1995). The convective blueshift is
suppressed in magnetic regions causing them to appear redshifted relative to
the quiet sun. Consequently, the Doppler shifts may only be discussed relative
to the quiet-sun motions which dominate the coefficient $L_{0}$ and serve as
the zero-point Doppler velocity of $\Delta{v}_{\mathrm{LOS}}$.
Third, the $p$-mode oscillations are removed from the residuals by temporal
filtering. Hathaway et al. (2000) employ a Gaussian weighted average of 31
Dopplergrams to reduced the $p$-mode signal in the 2–4 mHz frequency band
where the weights are given by
$W_{i}=\frac{w_{i}}{\sum_{i=-15}^{15}{w_{i}}},$ (48a) with
$w_{i}=\exp\left(-\frac{\Delta
t_{i}}{2\,\alpha^{2}}\right)-\exp\left(-\frac{\beta^{2}}{2\,\alpha^{2}}\right)\,\left(1+\frac{\beta^{2}-{\Delta
t_{i}^{2}}}{2\,\alpha^{2}}\right)$ (48b)
and $\Delta t_{i}\equiv{t_{i}}-t_{0}$ is the time difference of the central
Dopplergram with $\alpha=8$ minutes and $\beta=16$ minutes.
Alternatively, nonparametric uniform B-splines
$\widehat{h}_{m}\left(\mbox{\boldmath{$\eta$}}|t\right)=\sum_{k=1}^{N}\,\eta_{k}\,B_{k,2\,m-1}\left(t\right)$
may be employed to filter temporally the data. B-splines are solutions to the
optimization problem
$\min\left|\sum_{i=1}^{N}\,\frac{\left[\widehat{h}_{m}\left(\mbox{\boldmath{$\eta$}}|t_{i}\right)-v_{i}\right]^{2}}{\delta
v_{i}^{2}}+\lambda\,\int\limits_{-\infty}^{\infty}{dt}\left[\frac{d^{m}\widehat{h}_{m}\left(\mbox{\boldmath{$\eta$}}|t\right)}{dt^{m}}\right]^{2}\right|,$
(49)
where the summation is the weighted $\widehat{\chi}_{m}^{2}$, $\delta
v_{i}^{2}$ are the uncertainties, $N$ is number of data or knots, $m$ is the
spline half-order, and
$\mbox{\boldmath{$\eta$}}\equiv\mbox{\boldmath{$\eta$}}\left(\lambda\right)$
is a vector of B-spline parameters and a function of the global regularization
parameter $\lambda\geq 0$ Woltring (1986). The variational principle (49) may
be justified by Bayesian arguments when the data $v_{i}$ is associated with
certain priors (see Craven & Wahba, 1979, and references therein). The
parameter $\lambda$ controls the balance between smoothness measured by the
$m$th derivative and fidelity measured by the variance with
$\lambda\rightarrow 0$ corresponding to spline interpolation with
$\widehat{\chi}_{m}^{2}=0$ and $\lambda\rightarrow\infty$ corresponding to a
least squares polynomial of order $P_{m}=m$, degree $d=m-1$. The boundary
conditions at $t=t_{1}$ and $t_{N}$ are determined by
${d^{m}\widehat{h}_{m}}/{dt^{m}}=0$, where $m=1,2,\ldots$, consistent with a
well-posed solution to the variational principle.
B-spline basis functions have many attractive properties for filtering. For
example, they do not require uniform sampling, they adapt to local structure,
and the effect of outliers is localized because the basis functions are
defined on compact support. For the special case of uniformly sampled,
uniformly weighted, periodic data, the spline smoother behaves as two cascaded
$m$th order Butterworth filters without phase distortion; thus B-splines may
be viewed as a generalized low pass filter adapted for nonuniform weighting,
uneven sampling, and non-periodic boundary conditions Craven & Wahba (1979).
Butterworth filters are an approximation to an ideal filter because they have
maximally flat frequency response in the passband and are monotonically
decreasing roll-off in the stop-band Butterworth (1930).
Figure 13: Left: the white noise filter responses for the Gaussian weighted
average filter from Hathaway et al. (2000) in green, and two $m=5$th order
cascaded Butterworth filter with $f_{\mathrm{c}}=8$ mHz in blue and the
B-spline filter in green. The filter responses were constructed with 500
realizations of a 24 hr white noise time series with $\Delta t=1$ minute whose
average spectrum is shown in black. The 2–4 mHz regime is shaded in light
blue. Right: average power spectra of co-registered and de-rotated Doppler
time-series from the inner 10% of the solar disk where the solid black line
denotes the unfiltered data exhibiting an enhanced $p$-mode spectrum in the
2-5 mHz, the solid red line is for the smoothed results after B-spline
filtering, and the dashed black line is for the residual data (unfiltered
minus smoothed). Both left and right: the spectra were computed using 3 hr
segments with 50% overlap.
The left panel of Figure 13 shows the filter responses for the Gaussian
weighted average filter from Hathaway et al. (2000) in green, and the two
$m=5$ order cascaded Butterworth filters with $f_{\mathrm{c}}=8$ mHz in blue
and the B-spline filter in red with $\tau=1/f_{\mathrm{c}}=1250$ s and
$\lambda=\left[\left(\frac{\tau_{\mathrm{c}}}{2\,\pi}\right)^{2\,m}-{\left(\frac{2\,\Delta
t}{2\,\pi}\right)^{2\,m}}\right]/{\Delta t}=1.62\times
10^{21}\,\mathrm{s}^{9}.$ (50)
The filter responses were constructed with 500 realizations of a 24 hour white
noise time series with $\Delta t=1$ minute whose average spectrum is shown in
black. The 2–4 mHz regime is shaded in light-blue. The B-splines exhibit a
flatter power spectrum in the pass-band and a steeper roll-off in the stop-
band than the Gaussian filter. Furthermore, the B-spline filter does not
exhibit any ripples in the stop-band. Finally, the B-spline filter can
accommodate occasional missing data or uneven sampling and simultaneously
interpolate to a different temporal grid as part of the filtering process.
Indeed, the B-spline filters were used to replace missing data in the central
regions of four MDI Dopplergrams. The right panel of Figure 13 shows the
average power spectra of co-registered and de-rotated Doppler time series from
the inner 10% of the solar disk where the solid black line denotes the
unfiltered data exhibiting an enhanced $p$-mode spectrum in the 2-5 mHz, the
solid red line is for the smoothed results after B-spline filtering, and the
dashed black line is for the residual data (unfiltered minus smoothed). The
peak power of the $p$-mode oscillation at $f\simeq 3.15$ mHz is reduced by a
factor of $10^{-10}$.
### 4.2 Photospheric Doppler Signatures During the CME
Figure 14: Line-of-sight velocity (top) and the rms line-of sight velocity
(bottom) averaged over the current channel of radius $a_{\mathrm{p}}=6.7\times
10^{8}$ cm for the filament channel (red) and current channels (black) on the
left (solid) and right (dashed) and the soft X-ray flux observed by GOES 8
(blue).
Figure 14 shows the line-of-sight velocity (top) and the rms line-of sight
velocity (bottom) averaged over the current channel of radius
$a_{\mathrm{p}}=6.7\times 10^{8}$ cm for the filament channel (red) and
current channels (black) on the left (solid) and right (dashed) side of Figure
9 and the soft X-ray flux observed by GOES 8 (blue). The M-class flare
associated with the 2000 September 12 CME occurs shortly after 11:30 UT. As
discussed in the previous section, all velocities are measured relative to the
quiet sun which serves as the zero-point. The top plot shows that the average
velocity in the left and right prominence channel and in the right current
channel is redshifted with respect to the quiet sun. The left current channel
show some evidence of some blueshift (upflows) with respect to the zero-point
in the range of -60 to +$10\,\mbox{m}\,\mbox{s}^{-1}$. These velocities are
within the standard deviation of $\sigma=65.7\pm 0.1\,\mbox{m}\,\mbox{s}^{-1}$
reported for the quiet sun $\left|B\right|<1$ G bins in top right panel Figure
12. That velocities are quite similar to quiet sun conditions is not
surprising because this current-channel footpoint region contains weak
magnetic field ranging from -24 to 24 G. The weak field has already been
discussed as an inconsistency between the photospheric observations and the
simplified flux-rope geometry which should have strong magnetic field in the
current channel. Although, the absolute Doppler velocities are not known, they
are bounded. The absolute errors must be much less that the passband of the
filter $\approx 7\,\mbox{km}\,\mbox{s}^{-1}$ and are likely less than
$100\,\mbox{m}\,\mbox{s}^{-1}$ because the average quiet sun is not moving
toward the Earth at several $\mbox{km}\,\mbox{s}^{-1}$ given the agreement for
the coefficient $L_{0}$ for the MDI Doppler measurements and Snodgrass (1984).
None of the observed Doppler velocities approach magnitudes of
$10^{3}\,\mbox{km}\,\mbox{s}^{-1}$ which would be consistent with the analysis
in Section 3 — these upflows would be too weak to transport the poloidal flux
into the corona necessary to drive the CME eruption within the context of the
flux-rope model. These data falsify the flux injection mechanism as a driver
for this CME. However, an interesting feature is the sharp decrease of roughly
$50\,\mbox{m}\,\mbox{s}^{-1}$ peak to minimum in the redshift of the left
prominence channel (solid red) beginning just after the GOES X-ray flux begins
to rise and lasting 1 hr and 35 minutes bounded by the two vertical lines. The
flow velocity then appears to recover to its initial value over the next
$2\frac{1}{2}$ hr. Through this dynamic change of Doppler velocity in the
filament channel, the magnitude remains redshifted
$100\,\mbox{m}\,\mbox{s}^{-1}$ relative to the quiet sun. To determine if
dynamic change in the Dopper velocity of filament footpoints is a common
characteristic of filament eruptions producing CMEs will require evaluating
more events.
## 5 COMPARISON WITH PREVIOUS WORK AND CONCLUSIONS
The flux-injection hypothesis requires a large energy transport of $2\times
10^{33}$ erg across the photosphere on timescales of 600-1200 s. If this
hypothesis is correct, then the absence of any significant photospheric
signature of this transport is certainly surprising given that the power
requirements $\simeq 3\times 10^{30}\,\mbox{erg}\,\mbox{s}^{-1}$ exceed that
of the typical solar flare $\lesssim 10^{29}\,\mbox{erg}\,\mbox{s}^{-1}$.
Sudol & Harvey (2005) and Fletcher & Hudson (2008) have observed permanent
changes in longitudinal magnetograms concomitant with solar flares, and a more
comprehensive follow-on study by Petrie & Sudol (2009) has demonstrated that
these changes are associated with every flare. However, these changes always
lag the rise of the flare and thus would lag the peak in the CME acceleration
which is strongly correlated with the flare rise time Shanmugaraju et al.
(2003); Zhang et al. (2001, 2004). Furthermore, the changes in the
longitudinal magnetograms are largest for events near the limb suggesting that
they are caused by a rapid change in the angle of the magnetic field through
the photosphere rather than a true change in photospheric magnetic field
strength.
The present study has failed to find evidence of the magnetized photospheric
plasma velocities required for the flux-injection hypothesis to satisfy the
CME energy budget on timescales of the eruption. Chen et al. (2000), Chen
(2001), Krall et al. (2001), Chen & Krall (2003) and Chen & Kunkel (2010) have
attempted to address the criticisms of the flux-injection hypothesis based on
energy arguments. Chen et al. (2000) cite one-dimensional simulations by Huba
& Chen (1996) of a step function horizontal magnetic field of tens of Gauss
injected into the low chromosphere. The unspecified current system generating
and maintaining the magnetic piston is assumed to lie below the simulation
boundary in the photosphere—this is a different situation than the diagrams in
Figures 1 and 2. The simulation does not include the photosphere (J. Huba,
personal communication 2008 December) and the plasma and neutrals only attain
vertical velocities of $v_{\zeta}\simeq$tens of $\mbox{m}\,\mbox{s}^{-1}$ in
the low chromosphere. Chen et al. (2000) infer that these simulations imply
photospheric upflow velocities of merely meters per second. Can these results
be used to satisfy the power requirements of the flux-rope model? Assuming a
constant horizontal magnetic field $B_{h}\simeq 30$ G over a large area
$\pi\,\left(S_{\mathrm{f}}/2\right)^{2}=2\times 10^{21}\,\mathrm{cm}^{2}$
surrounding each current channel and a vertical plasma velocity of
$v_{\zeta}=10\,\mbox{m}\,\mbox{s}^{-1}$, the power supplied by this one-
dimensional simulation over two footpoints would be
$dU_{\theta}/dt\simeq{v}_{\zeta}\,B_{h}^{2}\,\left(S_{\mathrm{f}}/2\right)^{2}/2=8\times
10^{26}\,\mbox{erg}\,\mbox{s}^{-1}$ which is still more than 2 orders of
magnitude less than the power requirements of the 2000 September 12 CME Chen
et al. (2006); Chen & Kunkel (2010) which was associated with an M1.0 flare
and more than 3 orders of magnitude less than the power requirements of the
2003 October 28 CME which was associated with an X17 flare Krall et al.
(2006).
Krall et al. (2000) consider the constraints of driving the flux-rope model by
footpoint twisting. They conclude that footpoint twisting is inefficient and
cannot reproduce the characteristics of the CME trajectories event with large
photospheric poloidal velocities of $10\,\mbox{km}\,\mbox{s}^{-1}$. However,
this estimate corresponds to velocities at the base of the corona based on
their values of current-channel radius, and poloidal and toroidal magnetic
field (see Eqs (31-33) and Tables 1 and 2 in Krall et al., 2000). They coupled
footpoint twisting to the flux-rope model by relating net current and poloidal
magnetic field at the edge of the current channel to the number of twists at
the base of the corona. However, footpoint twisting merely rearranges the
current distribution in the current channel leaving the net current constant.
Footpoint twisting should be coupled to the flux-rope model through the change
in internal inductance as outlined in Appendix C not by modifying the net
current and poloidal magnetic field at the edge of the current channel.
Krall et al. (2001) argue that the flux-injection hypothesis implies radial
plasma velocities of the order of $v_{r}\simeq 1\,\mbox{km}\,\mbox{s}^{-1}$ at
the footpoints of the flux rope implied by the inductive toroidal electric
field $E_{\phi}\simeq-{c^{-1}}\,\partial_{t}A_{\phi}$. These velocities are
already substantial and should be well within the observational capabilities
of MDI aboard SOHO and ground based observatories. However, this electric
field is not relevant to estimating the Poynting flux because
$\mbox{\boldmath{$E$}}_{\phi}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$B$}}\simeq\left(B_{\theta}/c\right)\,\partial_{t}A_{\phi}\,\widehat{r}$
is roughly tangent to the photosphere—the vertical Poynting flux is zero (see
Figure 2 for the geometry). Similarly the vertical helicity flux
$\mbox{\boldmath{$E$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$A$}}_{R}$ is
also zero for the toroidal electric field.
Chen & Krall (2003) have argued that the photospheric velocities should be
highly nonuniform with coherence scales of less than $\sim 10^{5}$ km
consistent with a high $\beta\gg 1$ plasma outside the current channel
although no theoretical estimate of this coherence length has been provided.
High-resolution convection simulations exhibit rms fluctuations in the
$\tau=1$ surface of $\approx 30$ km which is much less than the local
photospheric pressure scale height,111111B. Abbett, personnel communication
2009 July. casting doubt upon these assertions. Although the region far from
the current channel is $\beta\gg 1$, the region containing the toroidal field
and toroidal currents is $\beta\simeq 1$. Section 3 has demonstrated that a
significant amount of poloidal energy is injected in the region very near and
interior to the current channel $r\leq\,2\,a_{\mathrm{p}}$. Finally, the
spatial scale of the $10\arcsec\,\mbox{pixel}^{-1}$ for MDI corresponds to
$\simeq 7500$ km. The photospheric dynamics detected at this scale concomitant
with the eruption at the footpoints of the 2000 September 12 CME are not
sufficient to drive the eruption.
Emerging horizontal magnetic fields will carry mass into the chromosphere. For
velocities much less than the escape velocity
$v_{\sun}=617\,\mbox{km}\,\mbox{s}^{-1}$ this mass will flow downward as the
magnetic field at the footpoints of the forming loops become more vertically
inclined. Indeed, Ishikawa et al. (2008) have observed a strongly redshifted
$+5\,\mbox{km}\,\mbox{s}^{-1}$ Stokes $V$ profile (downflows) at one end of an
emerging horizontal magnetic feature. However, the turbulence argument of Chen
& Krall (2003) does not seem to appreciate the very large and continuous
average velocities$\simeq$hundreds of $\mbox{km}\,\mbox{s}^{-1}$ necessary to
satisfy the power budget of the flux-rope model. In the limit
$v_{\theta\mathrm{p}}=0$, the average vertical velocities in the current
channel exceed the escape velocity $v_{\sun}$ and the mass could only be
restrained by magnetic forces. The expected velocities also exceed the Afvén
$V_{\mathrm{A}}$ and sound speed $C_{\mathrm{s}}$ in an ideal MHD plasma.
Furthermore, regions of downflow embedded in the upflow would imply larger
upflow velocities to enhance vertical transport to account for the smaller
effective upflow area and to balance any energy transported out of the corona
through the photosphere in the downflow regions (unless downflow regions are
assumed special and contain no horizontal field).
In a similar vein Krall et al. (2001) and Chen & Krall (2003) have argued that
filling factor $f$—the percentage of a pixel that contains magnetized
atmosphere—in the photosphere is unknown and the inherent nonuniformity of the
magnetic field in the photosphere explains the lack of observational evidence
for flux injection. However, the filling factor can in principle be estimated
by modern magnetographs by enforcing consistency between the Stokes $I$
profile and the $Q$ (linear), $U$ (linear), and $V$ (circular polarization)
profiles Keppens & Martinez Pillet (1996). Indeed, the filling factor will be
a standard data product of the next generation vector magnetograms produced by
SDO/HMI. The non-magnetic component only contributes to the Stokes $I$
profile, whereas the Doppler velocities are usually determined from the $V$
profile providing a relatively unambiguous estimate for the velocity of the
magnetized atmosphere. Although Stokes $V$ spectra may exhibit features of
both upflows and downflows simultaneously Bellot Rubio et al. (2001), the
observations in Section 4 require precise cancellation between the upflows and
downflows to produce no evidence of significant dynamics in the flux-rope
footpoint regions necessary for consistency with the flux-injection
hypothesis. This perfect cancellation seems improbable.
The large photospheric velocities implied by the flux-rope model power budget
would shift the spectral lines used to estimate the magnetic field and Doppler
velocity out of the pass-band of many modern telescopes. For example, the MDI
instrument records filtergrams around the Ni I spectral line Ni I 6767.8 Å
with a 94 mÅ bandpass. Under normal operation, filtergrams are made at five
tuning positions separated by 75 mÅ spanning 377 mÅ Scherrer et al. (1995).
The maximum velocity that could be measured by the MDI instrument is
$v=c\,\Delta\lambda/\lambda=0.150/6767.8\approx 7\,\mbox{km}\,\mbox{s}^{-1}$.
The absence of significant changes in the magnetic field indicate that a large
fraction of the magnetized atmosphere is not moving with velocities that
exceed $5\,\mbox{km}\,\mbox{s}^{-1}$.
A simple photospheric magnetic field model for flux-rope footpoints has been
developed by extrapolating the flux-rope magnetic field model of Chen & Garren
(1994), Chen (1996), Krall et al. (2000), and Krall & Chen (2005) into the
photosphere with conservation of toroidal flux and toroidal current. This is
equivalent to magnetic field models implemented in simulation studies of
photospheric flux injection Chen & Huba (2005a, b, 2006). This magnetic field
model has been used to estimate the minimum photospheric velocities necessary
to satisfy the power budget of the flux-rope model for CMEs fitted by Chen et
al. (2006), Krall et al. (2006), and Chen & Kunkel (2010). The flux-rope power
budget requires large average poloidal or vertical velocities of the order of
hundreds to thousands of $\mbox{km}\,\mbox{s}^{-1}$ over large photospheric
areas of $10^{8}\,\mathrm{km}^{2}$ to transport the necessary poloidal
magnetic field into the corona on a timescale of the eruption. While Chen,
Krall, and Kunkel might argue that the photospheric magnetic field implemented
in this study is oversimplified, there is agreement that enhanced photospheric
activity should be detected in the region near the footpoints. Indeed, Krall
et al. (2001) affirm “with certainty we can state that flux injection as
discussed above should be accompanied by increased photospheric flow activity
over a large spatial area, near the footpoints, for a period of hours during
and following a CME eruption.” To address this, Doppler and magnetic field
observations at the footpoints of the 2000 September 12 CME have been
analyzed. No significant dynamics at the flux-rope current-channel footpoints
concomitant with the CME eruption have been detected. The flux-injection
hypothesis is incompatible with these observations.
Although, the flux-injection hypothesis has been falsified, the flux-rope
model Chen (1989, 1996) could remain a useful theoretical tool for modeling
and interpreting CME dynamics because there are other hypotheses for forming
or increasing the poloidal flux of a flux rope. For example, shearing Mikić &
Linker (1994); Antiochos et al. (1999); Amari et al. (2000), converging flows
Forbes & Priest (1995), or nearby emerging flux Chen & Shibata (2000) may
convert coronal arcade field into flux-rope fields via rapid
reconnection.121212Falsification of one hypothesis does not imply verification
of another. Consequently, the flux-injection hypothesis is incorrect, but the
flux-rope model could correctly describe the dynamics of an erupting CME.
Finally, the flux-rope model has brought to the forefront, the paradigm that
CMEs are current carrying coherent magnetic structures consistent with the
three-part morphology observed by LASCO coronagraphs and predicts the scaling
law that height of the CME at maximum acceleration scales with the footpoint
separation distance $S_{\mathrm{f}}$ Chen et al. (2006).
While the focus of this investigation has been the flux-injection hypothesis
Chen et al. (1997, 2000); Wood et al. (1999); Krall et al. (2001); Chen &
Krall (2003); Chen et al. (2006); Krall et al. (2006); Chen & Kunkel (2010),
the observational component of this study places important constraints on any
hypothesis that relies on the photosphere for the power
$10^{29}-10^{30}\,\mbox{erg}\,\mbox{s}^{-1}$ driving a CME. These hypotheses
are likely incompatible with the present investigation, because the small
velocities observed in the photospheric cannot supply the necessary energy on
the timescale of the main acceleration phase of the CME. In contrast, the
storage-release mechanism, where the energy is transported across the
photosphere over a period of hours or days and then released rapidly through
reconnection, is compatible with the present study. Under storage release, the
power supplied by reconnection is limited by the reconnection rate, in part,
determined by the velocity of the reconnection point as it unzips the
overlying arcade.
The author thanks the reviewer for his/her constructive criticism that greatly
clarified and improved the paper. The author gratefully acknowledges
insightful conversations with Jonathan Krall, Joseph Huba, Mark Linton, K. D.
Leka, Todd Hoekesema, Werner Poetzi, and Uri Feldman. The author thanks the
SOHO/SOI team for providing the Dopplergrams and magnetograms.
## Appendix A THE MINIMUM PHOTOSPHERIC VELOCITIES CONSISTENT WITH THE POWER
BUDGET
The minimum photospheric velocities consistent with the power requirements of
the flux-injection hypothesis may be found from a constrained variational
principle where
$\mathcal{I}_{1}=\int_{0}^{r_{\mathrm{c}}}{dr}{r}\,v_{\theta}^{2},$ (A1)
$\mathcal{I}_{2}=\int_{0}^{r_{\mathrm{c}}}{dr}{r}\,v_{\zeta}^{2},$ (A2)
or
$\mathcal{I}_{3}=\int_{0}^{r_{\mathrm{c}}}{dr}{r}\,\left(v_{\theta}^{2}+v_{\zeta}^{2}\right),$
(A3)
is minimized subject to the integral constraint
${\mathcal{C}_{1}}\equiv\frac{dU_{\theta}}{dt}=-\int_{0}^{r_{\mathrm{c}}}{dr}{r}{B}_{\theta}\left(r\right)\,\psi^{\prime}\left(r\right),$
(A4)
where $\psi^{\prime}\left(r\right)=\partial_{r}\psi$. This leads to the
functional
$\mathcal{H}\equiv\left[\psi^{\prime}\left(r\right)\right]^{2}\,{f}^{2}\left(r\right)-\lambda\,B_{\theta}\left(r\right)\,\psi^{\prime}\left(r\right),$
(A5)
and the Euler equation
$\frac{\partial\mathcal{H}}{\partial\psi^{\prime}}=\kappa=\mathrm{constant},$
(A6)
where $\lambda$ is a Lagrange multiplier. There are two limiting cases and a
third general case to consider:
Case 1: $v_{\parallel}\neq 0$, $v_{\zeta}=0$, and $f=B_{\zeta}^{-1}$.
Solving for $\psi^{\prime}$
$\displaystyle\psi^{\prime}$ $\displaystyle=$
$\displaystyle\frac{B_{\zeta}^{2}}{2}\,\left(\kappa_{1}+\lambda\,B_{\theta}\right),$
(A7a) $\displaystyle v_{\theta}$ $\displaystyle=$
$\displaystyle\frac{B_{\zeta}}{2}\,\left(\kappa_{1}+\lambda\,B_{\theta}\right).$
(A7b)
Physical considerations require $v_{\theta}\rightarrow 0$ when $r\rightarrow
0$ which is equivalent to $B_{\theta}\rightarrow 0$. This implies
$\kappa_{1}\equiv 0$ and
$\displaystyle\psi^{\prime}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{2}\,B_{\theta}\,B_{\zeta}^{2},$ (A8a)
$\displaystyle v_{\theta}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{2}\,B_{\theta}\,B_{\zeta}.$ (A8b)
Substituting Equation (A8a) into Equation (A4) determines the Lagrange
multiplier
$\lambda\equiv-\frac{1120\,{\mathcal{C}_{1}}}{437\,B_{\zeta{\mathrm{p}}}^{2}\,B_{\theta{\mathrm{p}}}^{2}\,a_{\mathrm{p}}^{2}}.$
(A9)
Substituting Equations (A9) and Equation (A8b) into (A1), integrating from
$r=0\Longrightarrow a_{\mathrm{p}}$ and using Equations (14a) and (14b) the
rms poloidal plasma velocity inside $r\leq a_{\mathrm{p}}$ is
$\left\langle{v}_{\theta}^{2}\right\rangle_{a_{\mathrm{p}}}^{1/2}=\left|\frac{dU_{\theta}}{dt}\right|\,\frac{4\,\sqrt{70}}{\sqrt{437\,\left|B_{\theta{\mathrm{c}}}^{2}\,B_{\zeta{\mathrm{c}}}\right|}\,a_{\mathrm{c}}^{2}}\,\frac{1}{\sqrt{\left|B_{\zeta\mathrm{p}}\right|}}.$
(A10)
Case 2: $v_{\parallel}\neq 0$, $v_{\theta}=0$, and $f=B_{\theta}^{-1}$
Solving for $\psi^{\prime}$
$\displaystyle\psi^{\prime}$ $\displaystyle=$
$\displaystyle\frac{B_{\theta}^{2}}{2}\,\left(\kappa_{2}+\lambda\,B_{\theta}\right),$
(A11a) $\displaystyle v_{\zeta}$ $\displaystyle=$
$\displaystyle-\frac{B_{\theta}}{2}\,\left(\kappa_{2}+\lambda\,B_{\theta}\right).$
(A11b)
The Lagrange multiplier is determined by substituting Equation (A11a) into
Equation (A4)
$\lambda\equiv-\frac{6\,r_{\mathrm{c}}\,\left(85085\,B_{\theta{\mathrm{p}}}^{3}\,a_{\mathrm{p}}^{3}\,\kappa_{2}-170170\,{\mathcal{C}_{1}}\,r_{\mathrm{c}}-144048\,B_{\theta{\mathrm{p}}}^{3}\,a_{\mathrm{p}}^{2}\,r_{\mathrm{c}}\,\kappa_{2}\right)}{221\,B_{\theta{\mathrm{p}}}^{4}\,a_{\mathrm{p}}^{2}\,\left(1155\,a_{\mathrm{p}}^{2}-2998\,{r_{\mathrm{c}}}^{2}\right)}.$
(A12)
Substituting Equations (A12) and (A11b) into Equation (A2), differentiating
with respect to $\kappa_{2}$, and solving determines the value of
$\kappa_{2}=0$ corresponding to the minimum mean squared velocity over the
region $r=0\longrightarrow r_{\mathrm{c}}$. Integrating Equation (A2) from
$r=0\rightarrow 2\,a_{\mathrm{p}}$ with Equations (14a) and (14b) produces the
rms vertical velocity inside $r\leq 2\,a_{\mathrm{p}}$
$\left\langle{v_{\zeta}^{2}}\right\rangle^{1/2}_{2\,a_{\mathrm{p}}}=\left|\frac{dU_{\theta}}{dt}\right|\,\frac{\sqrt{12516735}\,\left|B_{\zeta\mathrm{p}}\right|\,r_{\mathrm{c}}^{2}}{2\,B_{\theta\mathrm{c}}^{2}\,a_{\mathrm{c}}^{2}\,\left|2998\,B_{\zeta\mathrm{p}}r_{\mathrm{c}}^{2}-1155\,B_{\zeta\mathrm{c}}\,a_{\mathrm{c}}^{2}\right|}.$
(A13)
Case 3: $v_{\parallel}=0$, $f=1/\sqrt{B_{\theta}^{2}+B_{\zeta}^{2}}$
Solving for $\psi^{\prime}$
$\displaystyle\psi^{\prime}$ $\displaystyle=$
$\displaystyle\frac{B_{\theta}^{2}+B_{\zeta}^{2}}{2}\,\left(\kappa_{3}+\lambda\,B_{\theta}\right),$
(A14a) $v$ $\displaystyle=$
$\displaystyle\frac{\left(\kappa_{3}+\lambda\,B_{\theta}\right)}{2}\,\left(0,B_{\zeta},-B_{\theta}\right).$
(A14b)
Physical considerations require $v_{\theta}\rightarrow 0$ when $r\rightarrow
0$ which is equivalent to $B_{\theta}\rightarrow 0$. This implies
$\kappa_{3}\equiv 0$ and
$\displaystyle\psi^{\prime}$ $\displaystyle=$
$\displaystyle\frac{\lambda\,B_{\theta}}{2}\,\left(B_{\theta}^{2}+B_{\zeta}^{2}\right),$
(A15a) $v$ $\displaystyle=$
$\displaystyle\frac{\lambda\,B_{\theta}}{2}\,\left(0,B_{\zeta},-B_{\theta}\right).$
(A15b)
Substituting Equation (A15a) into Equation (A4) determines the Lagrange
multiplier
$\lambda\equiv\frac{36960\,{\mathcal{C}_{1}}\,r_{\mathrm{c}}^{2}}{9240\,a_{\mathrm{p}}^{4}\,B_{\theta\mathrm{p}}^{4}-a_{\mathrm{p}}^{2}\,B_{\theta\mathrm{p}}^{2}\,\left(23984\,B_{\theta\mathrm{p}}^{2}+14421\,B_{\zeta\mathrm{p}}^{2}\right)\,r_{\mathrm{c}}^{2}},$
(A16)
Substituting Equations (A16) and (A15b) into Equation (A3), integrating from
$r=0\Longrightarrow a_{\mathrm{p}}$ and using Equations (14a) and (14b) the
rms total plasma velocity inside $r\leq a_{\mathrm{p}}$ is
$\left\langle{v^{2}}\right\rangle^{1/2}_{2\,a_{\mathrm{p}}}=\left|\frac{dU_{\theta}}{dt}\right|\,\frac{2\,B_{\zeta\mathrm{p}}\,r_{\mathrm{c}}^{2}\,\sqrt{2310}\,\sqrt{21674\,B_{\theta\mathrm{c}}^{2}+14421\,B_{\zeta\mathrm{c}}\,B_{\zeta\mathrm{p}}}}{a_{\mathrm{c}}^{2}\,\left|B_{\theta\mathrm{c}}\,B_{\zeta\mathrm{p}}\,(23984\,B_{\theta\mathrm{c}}^{2}+14421\,B_{\zeta\mathrm{c}}\,B_{\zeta\mathrm{p}})\,r_{\mathrm{c}}^{2}-9240\,a_{\mathrm{c}}^{4}\,B_{\theta\mathrm{c}}^{3}\,B_{\zeta\mathrm{c}}\right|},$
(A17a) the poloidal plasma velocity is
$\left\langle{v_{\theta}^{2}}\right\rangle^{1/2}_{2\,a_{\mathrm{p}}}=\left|\frac{dU_{\theta}}{dt}\right|\,\frac{66\,r_{\mathrm{c}}^{2}\,\sqrt{30590\,\left|B_{\zeta\mathrm{c}}\,B_{\zeta\mathrm{p}}^{3}\right|}}{a_{\mathrm{c}}^{2}\,\left|B_{\theta\mathrm{c}}\,B_{\zeta\mathrm{p}}\,(23984\,B_{\theta\mathrm{c}}^{2}+14421\,B_{\zeta\mathrm{c}}\,B_{\zeta\mathrm{p}})\,r_{\mathrm{c}}^{2}-9240\,a_{\mathrm{c}}^{4}\,B_{\theta\mathrm{c}}^{3}\,B_{\zeta\mathrm{c}}\right|}$
(A17b) and the vertical plasma velocity is
$\left\langle{v_{\zeta}^{2}}\right\rangle^{1/2}_{2\,a_{\mathrm{p}}}=\left|\frac{dU_{\theta}}{dt}\right|\,\frac{4\,B_{\zeta\mathrm{p}}\,r_{\mathrm{c}}^{2}\,\sqrt{12516735}}{a_{\mathrm{c}}^{2}\,\left|B_{\zeta\mathrm{p}}\,(23984\,B_{\theta\mathrm{c}}^{2}+14421\,B_{\zeta\mathrm{c}}\,B_{\zeta\mathrm{p}})\,r_{\mathrm{c}}^{2}-9240\,a_{\mathrm{c}}^{4}\,B_{\theta\mathrm{c}}^{2}\,B_{\zeta\mathrm{c}}\right|}.$
(A17c)
## Appendix B THE MINIMUM PHOTOSPHERIC VELOCITIES CONSISTENT WITH THE
HELICITY BUDGET
The vector potential in the upper half plane $\left(z\geq 0\right)$ for an
azimuthally symmetric vertical magnetic field is
$\mbox{\boldmath{$A$}}_{\mathrm{R}}\left(r,z\right)=\frac{2\,\mathrm{J}_{1}\left(\gamma_{n}\,r\right)\,e^{-\gamma_{n}\,\zeta}\,\int_{0}^{a}\,dr\,r\,\mathrm{J}_{0}\left(\gamma_{n}\,r\right)\,B_{\zeta}\left(r\right)}{\alpha_{n}\,a\,\mathrm{J}_{1}^{2}\left(\alpha_{n}\right)}\,\widehat{\theta}$
(B1)
where $\alpha_{n=1,2,\ldots}=2.40483,5.52008,\ldots$ is the $n$th zero of the
zeroth-order Bessel function $\mathrm{J}_{0}\left(\alpha_{n}\right)=0$. This
potential field satisfies
$\displaystyle\mbox{\boldmath{$\nabla$}}\cdot\mbox{\boldmath{$A$}}_{\mathrm{R}}\left(r,\zeta\right)$
$\displaystyle=$ $\displaystyle 0,$ (B2a)
$\displaystyle\widehat{n}\cdot\mbox{\boldmath{$A$}}_{\mathrm{R}}\left(r,0\right)$
$\displaystyle=$ $\displaystyle 0,$ (B2b)
$\displaystyle\mbox{\boldmath{$B$}}_{\mathrm{R}}\left(r,\zeta\right)$
$\displaystyle=$
$\displaystyle\mbox{\boldmath{$\nabla$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$A$}}_{\mathrm{R}}\left(r,\zeta\right)=\mbox{\boldmath{$\nabla$}}\Psi_{\mathrm{R}}\left(r,\zeta\right),$
(B2c)
$\displaystyle\widehat{n}\cdot\mbox{\boldmath{$B$}}_{\mathrm{R}}\left(r,0\right)$
$\displaystyle=$ $\displaystyle B_{\zeta}\left(r\right).$ (B2d)
$\mbox{\boldmath{$A$}}_{\mathrm{R}}\left(r,0\right)=\frac{a\,B_{\zeta\mathrm{a}}}{2}\,\widehat{\theta}\,\left\\{\begin{array}[]{lr}\displaystyle
3\,\frac{r}{a}\,\left(1-\frac{r^{2}}{a^{2}}+\frac{r^{4}}{3\,a^{4}}\right)&r\leq{a},\\\
\vskip 7.22743pt\cr\displaystyle\frac{a}{r}&r>a.\end{array}\right.$ (B3)
${\mathcal{C}_{2}}\equiv\frac{d\Delta
K}{dt}=-8\,\pi\,\int_{0}^{r_{\mathrm{c}}}{dr}\,{r}\,A_{\theta\mathrm{R}}\left(r\right)\,\psi^{\prime}\left(r\right).$
(B4)
The minimum photospheric velocities consistent with the helicity budget may be
found by following the procedure outlined in Appendix A using Equation (A5)
with $B_{\theta}\Longrightarrow{A}_{\theta\mathrm{R}}$.
$\mathcal{H}\equiv\left[\psi^{\prime}\left(r\right)\right]^{2}\,{f}^{2}\left(r\right)-\lambda\,A_{\theta\mathrm{R}}\left(r\right)\,\psi^{\prime}\left(r\right).$
(B5)
Case 1: $v_{\parallel}\neq 0$, $v_{\zeta}=0$, and $f=B_{\zeta}^{-1}$.
$\lambda\equiv-\frac{560\,{\mathcal{C}_{2}}}{437\,\pi\,a_{\mathrm{p}}^{4}\,B_{\zeta\mathrm{p}}^{4}}$
(B6)
$\left\langle{v_{\zeta}^{2}}\right\rangle^{1/2}_{2\,a_{\mathrm{p}}}=\left|\frac{d\Phi_{\theta}}{dt}\right|\,\frac{\sqrt{1155\,\left|B_{\zeta\mathrm{c}}\right|}}{2\,B_{\theta\mathrm{c}}\,\sqrt{\left|2998\,B_{\zeta\mathrm{p}}r_{\mathrm{c}}^{2}-1155\,B_{\zeta\mathrm{c}}\,a_{\mathrm{c}}^{2}\right|}}.$
(B7)
Case 2: $v_{\parallel}\neq 0$, $v_{\theta}=0$, and $f=B_{\theta}^{-1}$
$\lambda\equiv\frac{2310\,r_{\mathrm{c}}^{2}\,{\mathcal{C}_{2}}}{B_{\theta\mathrm{p}}^{2}\,B_{\zeta\mathrm{p}}^{2}\,\pi\,a_{\mathrm{p}}^{4}\,\left(2998\,r_{\mathrm{c}}^{2}-1155\,a_{\mathrm{p}}^{2}\right)}$
(B8)
$\left\langle{v_{\zeta}^{2}}\right\rangle^{1/2}_{2\,a_{\mathrm{p}}}=\left|\frac{d\Phi_{\theta}}{dt}\right|\,\frac{\sqrt{1155\,\left|B_{\zeta\mathrm{c}}\right|}}{2\,B_{\theta\mathrm{c}}\,\sqrt{\left|2998\,B_{\zeta\mathrm{p}}r_{\mathrm{c}}^{2}-1155\,B_{\zeta\mathrm{c}}\,a_{\mathrm{c}}^{2}\right|}}.$
(B9)
Case 3: $v_{\parallel}=0$ and $f=1/\sqrt{B_{\theta}^{2}+B_{\zeta}^{2}}$
$\lambda\equiv-\frac{18480\,{\mathcal{C}_{2}}\,r_{\mathrm{c}}^{2}}{B_{\zeta\mathrm{p}}^{2}\,\pi\,a_{\mathrm{p}}^{4}\,\left[(23984\,B_{\theta\mathrm{p}}^{2}+14421\,B_{\zeta\mathrm{p}}^{2})\,r_{\mathrm{c}}^{2}-9240\,B_{\theta\mathrm{p}}^{2}\,a_{\mathrm{p}}^{2}\right]},$
(B10)
$\left\langle{v^{2}}\right\rangle^{1/2}_{2\,a_{\mathrm{p}}}=\left|\frac{d\Phi_{\theta}}{dt}\right|\,\frac{B_{\zeta\mathrm{p}}\,r_{\mathrm{c}}^{2}\,\sqrt{2310}\,\sqrt{21674\,B_{\theta\mathrm{c}}^{2}+14421\,B_{\zeta\mathrm{c}}\,B_{\zeta\mathrm{p}}}}{2\,a_{\mathrm{c}}\,\left|B_{\zeta\mathrm{p}}\,(23984\,B_{\theta\mathrm{c}}^{2}+14421\,B_{\zeta\mathrm{c}}\,B_{\zeta\mathrm{p}})\,r_{\mathrm{c}}^{2}-9240\,a_{\mathrm{c}}^{4}\,B_{\theta\mathrm{c}}^{2}\,B_{\zeta\mathrm{c}}\right|},$
(B11)
## Appendix C ROTATIONAL TRANSFORM
Krall et al. (2000) assert that footpoint twisting modifies the net current
and poloidal magnetic field at the edge of the current channel based on a
geometrical argument. Generally, the twist of a flux rope is estimated from
the field line equations
$\frac{dr}{B_{r}}=\frac{r\,d\theta}{B_{\theta}\left(r\right)}=\frac{\mathcal{R}\,d\phi}{B_{\phi}\left(r\right)}.$
(C1)
For a circular torus, the amount of rotation of the poloidal field about the
toroidal axis during the transit along the flux rope above the photosphere is
estimated from the rotational transform (effectively the reciprocal of the
safety factor calculation for Tokamaks)
$\displaystyle\Delta\theta\left(r\right)$ $\displaystyle\simeq$
$\displaystyle\int_{0}^{2\,\pi\,\Theta}{d\phi}\,\frac{\mathcal{R}\,B_{\theta}\left(r\right)}{r\,B_{\phi}\left(r\right)},$
(C2a) $\displaystyle\simeq$ $\displaystyle
2\,\pi\,\Theta\,\frac{\mathcal{R}\,B_{\theta}\left(r\right)}{r\,B_{\phi}\left(r\right)}$
(C2b)
The amount of twist is a function of minor radius $r$. To add an amount of
twist $\delta\vartheta\left(r\right)$ to the flux rope, the equation becomes
$\Delta\theta\left(r\right)+\delta\vartheta\left(r\right)=2\,\pi\,\Theta\,\frac{\mathcal{R}\,B_{\theta}\left(r\right)}{r\,B_{\phi}\left(r\right)}+\delta\vartheta\left(r\right)=2\,\pi\,\Theta\,\frac{\mathcal{R}\,\widetilde{B}_{\theta}\left(r\right)}{r\,B_{\phi}\left(r\right)}$
(C3)
where ${B}_{\theta}$ is the initial poloidal field and
$\widetilde{B}_{\theta}\left(r\right)=B_{\theta}\left(r\right)+\frac{\delta\vartheta\left(r\right)}{2\,\pi\,\Theta}\,\frac{r}{\mathcal{R}}\,B_{\phi}\left(r\right),$
(C4)
$\widetilde{B}_{\theta}$ is the final poloidal field. This equation has a
similar form to Equation (31) in Krall et al. (2000)
$\widetilde{B}_{\theta\mathrm{c}}=B_{\theta\mathrm{c}}+\frac{\delta\vartheta_{0}}{2\,\pi\,\Theta}\,\frac{a_{\mathrm{c}}}{\mathcal{R}}\,B_{\phi\mathrm{c}},$
(C5)
where $\delta\vartheta_{0}=\pi$ is a uniform twist. However, there are
important mathematical and conceptual differences between Equations (C4) and
(C5). The former Equation (C4) implies that the poloidal magnetic field
profile changes as a result of the twist whereas the latter Equation (C5)
relates the coefficient of Equation (9a) before and after the twist. Krall et
al. (2000) substitute Equation (9a) into effectively Equation (10b) to produce
a new current
$\widetilde{I}_{\phi}=\frac{a_{\mathrm{c}}\,c}{2}\,\widetilde{B}_{\theta\mathrm{c}}=\frac{a_{\mathrm{c}}\,c}{2}\,\left(B_{\theta\mathrm{c}}+\frac{\delta\vartheta_{0}}{2\,\pi\,\Theta}\,\frac{a_{\mathrm{c}}}{\mathcal{R}}\,B_{\phi\mathrm{c}}\right),$
(C6)
where $\delta\vartheta_{0}=v_{\theta_{0}}\,t$. However, the correct
relationship Equation (C4) implies that
$\widetilde{B}_{\theta}\left(a\right)=B_{\theta}\left(a\right)$ and
$\widetilde{I}_{\phi}=I_{\phi}=B_{\theta}\left(a\right)\,a_{\mathrm{c}}\,c/2$
because $B_{\phi}\left(a\right)=0$, i.e., there is no change in net current as
a consequence of the twisting! The effect of twisting is to modify the current
distribution in the current channel and change the internal inductance
$\widetilde{\xi}\equiv\frac{2\,\int_{0}^{a}{dr}\,{r}\widetilde{B}_{\theta}^{2}\left(r\right)}{a^{2}\,B_{\theta{a}}^{2}}.$
(C7)
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|
arxiv-papers
| 2010-03-08T15:03:41 |
2024-09-04T02:49:08.902624
|
{
"license": "Public Domain",
"authors": "P. W. Schuck",
"submitter": "Peter Schuck",
"url": "https://arxiv.org/abs/1003.1647"
}
|
1003.1682
|
# An Entry Point for Formal Methods:
Specification and Analysis of Event Logs
Howard Barringer School of Computer Science
University of Manchester
Manchester, UK howard.barringer@manchester.ac.uk School of Electrical
Engineering and Computer Science
Oregon State University
Corvallis, USAJet Propulsion Laboratory
California Institute of Technology
Pasadena, USA Alex Groce School of Electrical Engineering and Computer Science
Oregon State University
Corvallis, USA alex@eecs.oregonstate.edu Jet Propulsion Laboratory
California Institute of Technology
Pasadena, USA Klaus Havelund Margaret Smith Jet Propulsion Laboratory
California Institute of Technology
Pasadena, USA klaus.havelund@jpl.nasa.gov margaret.h.smith@jpl.nasa.gov
###### Abstract
Formal specification languages have long languished, due to the grave
scalability problems faced by complete verification methods. Runtime
verification promises to use formal specifications to automate part of the
more scalable art of testing, but has not been widely applied to real systems,
and often falters due to the cost and complexity of instrumentation for online
monitoring. In this paper we discuss work in progress to apply an event-based
specification system to the logging mechanism of the Mars Science Laboratory
mission at JPL. By focusing on log analysis, we exploit the “instrumentation”
already implemented and required for communicating with the spacecraft. We
argue that this work both shows a practical method for using formal
specifications in testing and opens interesting research avenues, including a
challenging specification learning problem.
## 1 Introduction
NASA’s Mars Science Laboratory Mission (MSL), now scheduled to launch in 2011
[2], relies on a number of different mechanisms used to command the spacecraft
from Earth and to understand the behavior of the spacecraft and rover. The
primary elements of this communication system are commands sent from the
ground, visible events emitted by flight software (essentially formalized
printfs in the code) [8], snapshots of the spacecraft state, and data products
— files downlinked to earth (e.g., images of Mars or science instrument data).
All of these elements may be thought of as spacecraft _events_ , with a
canonical timestamp. Testing the flight software (beyond the unit testing done
by module developers) usually relies on observing these indicators of
spacecraft behavior. As expected, test engineers cannot “eyeball” the hundreds
of thousands of events generated in even short tests of such a complex system.
Previously, test automation has relied on ad hoc methods — hand-coded Python
[10] scripts using a framework to query the ground communications system for
various kinds of events, as a test proceeds. This was the state-of-the-art at
the time members of our group, the Laboratory for Reliable Software, joined
the MSL team, as developers and test infrastructure experts. Lengthy
collaboration with test engineers convinced us that something better was
required: the test script approach required large amounts of effort, much of
it duplicated, and often failed due to changes in event timing and limitations
of the query framework. Moreover, the scripts, combining test input activity,
telemetry gathering, and test evaluation, proved difficult to read — for other
test engineers and for developers and systems engineers trying to extract and
understand (and perhaps fix) a specification. Runtime verification using
formal specifications offered a solution, and the MSL ground communications
system suggested that we exploit an under-used approach to runtime
verification: offline examination of event logs already produced by system
execution. In the remainder of this paper we report on two aspects of this
project — in Section 2 we discuss the general idea of event sequences as
requirements, and our specification methodology; Section 3 gives a brief
introduction to our application of these general ideas to MSL and our new
specification language.
## 2 Event Sequences as Requirements
Systems verification consists of proving that an artifact (hardware and/or
software) satisfies a specification. In mathematical terms we have a model
$M\in ML$ (for example the complete system) in some model language $ML$, and a
specification $S\in SL$ in some specification language $SL$, and we want to
show that the pair $(M,S)$ is member of the satisfaction relation
$\models\;\subseteq\;ML\times SL$, also typically written: $M\models S$. The
general problem of demonstrating correctness of a combined hardware/software
system is very hard, as is well-known. Advanced techniques such as theorem
proving or model checking tend not to scale for practical purposes. Extracting
abstract models of the system, and proving these correct, has been shown to
sometimes be useful, but also faces scalability problems when dealing with
real systems or complex properties. The problem is inherently difficult
because the models are complex – because the behavior of systems is complex.
The problems of full verification have long limited the adoption of formal
specification.
In _runtime verification_ a specification is used to analyze a single
execution of a system, not the complete behavior. In this case a model is a
single execution trace $\sigma$, and the verification problem consists of
demonstrating that $\sigma\models S$, a much simpler problem with scalable
solutions. The original model $M$ (the full system) can be considered as
denoting the set of all possible runs $\sigma$, of which we now only verify a
subset. This approach of course is less ambitious, but seems to be a practical
and useful deployment of formal specification. Though these observations are
well known, they are less often taken into practice. It is still rare to
observe the application of formal specification — even for runtime
verification. There may be several reasons for this, one of which is the
problem of program instrumentation. A large body of research considers the
problem: how to we produce traces to verify? There is, however, a much simpler
approach, namely to use logging information that is already generated by
almost any computer system when it is tested. Our first recommendation is
therefore that runtime verification should be used to formally check logged
data. Our second recommendation is that logging and requirements engineering
should be connected in the sense that requirements should be testable through
runtime verification of logs. The common element is the event: requirements
should be expressed as predicates on sequences of events, and logging should
produce such sequences of events, thereby making the requirements testable.
This means that logs preferably should consist of events with a formal
template, connected to the original requirements. Note, however, that it is
possible to extract formalized events from chaotic logging information
produced by the usual ad hoc methods — using, e.g., regular expressions.
The obvious scientific and methodological question is: what kind of
information should a log contain, in order to help verify requirements? We
shall adopt the simple view that a log is a sequence of events, where an event
is a mapping from names to values of various types (integers, strings, etc.):
$\begin{array}[]{rcl}Log&=&Event^{*}\\\ Event&=&Name\rightarrow Value\\\
\end{array}$
This definition is quite general. An event can carry information about various
aspects of the observed system. Events can be classified into various kinds by
letting a designated name, i.e. kind, map to the kind. In the context of MSL,
five forms of events are used, see Figure 1. We claim that these five event
kinds are generally applicable to any system being monitored. The five forms
of events are: (1) commands (input) issued to the monitored system, (2)
products (output) delivered by the system, (3) periodic samplings of the state
of the system, such as the value of continuous-valued sensors (like position
coordinates), (4) changes to the state of the system, those changes that are
observable, and (5) transitions performed by the system (also referred to as
EVent Reports - EVRs), for example when printf statements would normally be
used to record an important event. The two forms of observation of the state
(3, 4) could potentially be regarded as one kind of observation: that of the
state of the system at any point in time. The state observations (3, 4) and
the transitions (5) are internal events, while the commands (1) and products
(2) are external events.
Figure 1: observable events of a system
We have developed a specification language, and corresponding monitoring
system, to be presented in the next section, which have been applied by
testing engineers within the MSL project. The specification language consists
of a mixture of automata and temporal logic with elements of regular
expressions. The logic can specifically refer to the data in events. This
mixture seems to be attractive for the engineers. In the longer term, we argue
that such a specification language and monitoring system should be used in
combination with a systematic logging discipline, such that _requirements are
formulated in terms of events_ , and, minimally, _these events_ should be
produced as part of logging. These observations may not appear to be ground-
breaking, and to some extent have the flavor of “yes of course” — but as it
turns out, considering current practices in software projects, they may be
rather ground-breaking. A successful formal verification story always relies
on finding the proper mixture of formality and common practice, as well as the
right specification language for the task. We believe that the work described
in this short paper has shown the potential for being such a success story.
## 3 Framework
MSL’s ground software stores all events in a SQL database, which we interpret
as a chronologically ordered sequence of events — a log. Our Python framework,
called LogScope, allows us to check logs for conformance to a specification
and to “learn” patterns from logs. The architecture of LogScope divides
functionality into a LogMaker tool, specific to MSL, and a core LogScope
module for checking logs and learning specifications, which may be applied to
any ordered event sequence.
### 3.1 LogMaker
LogMaker communicates with MSL’s SQL-based ground software to generate a list
of events, where each event is a record mapping field names to their values. A
special field indicates the type of the event: command, transition (EVR),
state sampling, state change, or data product. Note how the MSL events map to
the generalized idea of a monitored system shown in Figure 1. The log
extractor sorts events according to spacecraft event times, since the order in
which events are received by ground communications software does not
correspond to the order in which events are generated on-board (due to varying
communication priorities). Further analysis annotates the log with meta-events
for ease of use in specification, and uses spacecraft telemetry to assign a
spacecraft time to ground events. We hope to extend and exploit previous work
on monitoring distributed systems with multiple clocks [11] to influence
flight software’s use of telemetry to ensure that effective event ordering is
always possible.
### 3.2 Monitoring
pattern CommandSuccess:
COMMAND{Type : "FlightSoftwareCommand", Stem : x, Number : y} =>
{
EVR{Dispatch : x, Number : y},
[
EVR{Success : x, Number : y},
not EVR{Success : x, Number : y}
],
not EVR{DispatchFailure : x, Number : y},
not EVR{Failure : x, Number : y}
}
Figure 2: A generic specification for flight software commands.
The monitoring system of LogScope takes two arguments: (1) a log generated by
logmaker, and (2) a specification. Our specification language supplies an
expressive rule-based language, which includes support for state machines, and
a higher-level (but less expressive) pattern language, which is translated
into the more expressive rule-based language before monitoring.
Specifications in the pattern language are easy for test engineers and
software developers to read and write. Figure 2 illustrates a pattern. The
CommandSuccess pattern requires that following every command event (meaning a
command is issued to the flight software), where the Type field has the value
"FlightSoftwareCommand", the Stem field (the name of the command) has a value
x (x will be _bound_ to that value), and the Number field has a value y (also
a binding variable), we must see (=>) — in any order, as indicated by set
brackets `{...}` — (1) a dispatch of command x with the number y; (2) a
success of x/y, and after that no more successes — the square brackets `[...]`
indicate an ordering of the event constraints. Furthermore, (3) we do not want
to see any dispatch failures for the command; and finally (4) we do not want
to see any failures for the command.
Interesting features of the language include its mixture (and nesting) of
ordered and unordered event sequences of event constraints, including
negations, and its support for testing and capturing data values embedded in
events. The pattern language is translated into our rule-based language
derived from the Ruler specification language [5, 6, 4]. A subset of this
language defines state machines with parameterized events and states, where a
transition may enter many target states — essentially alternating automata
with data. The language is also inspired by earlier state-machine oriented
specification/monitoring languages, such as Rcat [12] and Rmor [9].
In addition to exact requirements on field values, our language supports user-
defined predicates (written in Python) that may take field values and bound
variables as arguments, providing very high expressive power. Specifications
are visualized with Graphviz [7], and extensive error trace reporting with
references to the log files ensures easy interpretation of detected
specification violations.
### 3.3 Learning
LogScope was well-received by test engineers, and was integrated into MSL
flight software testing for two modules shortly after its release. One
important result of early use was to alert us to the burden of writing
patterns more specific than the kind of generic rule shown above. In order to
ease this burden we introduced a facility for _learning_ specifications from
runs. Consider a test engineer or developer who runs a flight software test
one or more times. If these runs have been “good” runs he/she can “endorse”
(perhaps after making manual modifications) the specification, and it can then
be used to monitor subsequent executions. Learning requires a notion of event
equality, and users can define which fields should be compared for testing
event equality (e.g., exact timing is usually expected to change with new
releases and perhaps even new test executions). We have implemented and
applied a concrete learner which learns the set of all execution sequences
seen so far (essentially a “diff” tool for logs). We also expect to learn
mappings from commands to events expected in all execution contexts — a
pattern based approach, like that of Perracotta [13]. More ambitiously, we
hope to incorporate classic automata-learning results [3] in order to
generalize specifications.
## 4 Conclusions and Future Work
The MSL ground control and observation software demonstrates an important
concept: many critical systems already implement very powerful logging systems
that can be used as a basis for automated evaluation of log files against
requirements. Such log files can be analyzed with scripts (programs) written
using a scripting (programming) language. However, there seems to be
advantages to using a formal specification language, as demonstrated with this
work. A systematic study could be needed, investigating to what extent a
domain specific language really is required to achieve the added benefit, or
whether a well designed Python API (or API in any other programming language)
would yield the same benefits.
#### Acknowledgements
Part of the research described in this publication was carried out at the Jet
Propulsion Laboratory, California Institute of Technology, under a contract
with the National Aeronautics and Space Administration.
Thanks are due to many members of the Mars Science Laboratory Flight Software
team, including Chris Delp, Gerard Holzmann, Rajeev Joshi, Cin-Young Lee, Alex
Moncada, Cindy Oda, Glenn Reeves, Margaret Smith, Lisa Tatge, Hui Ying Wen,
Jesse Wright, and Hyejung Yun.
## References
* [1]
* [2] http://mars.jpl.nasa.gov/msl.
* [3] Dana Angluin (1987): _Learning Regular Sets from Queries and Counterexamples_. Inf. Comput. 75(2), p. 87 106.
* [4] Howard Barringer, Klaus Havelund, David Rydeheard & Alex Groce (2009): _Rule Systems for Runtime Verification: A Short Tutorial_. In: S. Bensalem & D. Peled, editors: Proc. of the 9th International Workshop on Runtime Verification (RV’09), LNCS 5779\. Springer, pp. 1–24.
* [5] Howard Barringer, David Rydeheard & Klaus Havelund (2007): _Rule Systems for Run-Time Monitoring: from Eagle to RuleR_. In: Proc. of the 7th International Workshop on Runtime Verification (RV’07), LNCS 4839\. Springer, Vancouver, Canada.
* [6] Howard Barringer, David Rydeheard & Klaus Havelund (2009): _Rule Systems for Run-Time Monitoring: from Eagle to RuleR_. Journal of Logic and Computation. Advance Access published on November 21, 2008. doi:10.1093/logcom/exn076.
* [7] GraphViz: http://www.graphviz.org.
* [8] Alex Groce & Rajeev Joshi (2006): _Exploiting Traces in Program Analysis_. In: Tools and Algorithms for the Construction and Analysis of Systems. pp. 379–393.
* [9] Klaus Havelund (2008): _Runtime Verification of C Programs_. In: Proc. of the 1st TestCom/FATES conference, LNCS 5047\. Springer, Tokyo, Japan.
* [10] Python: http://www.python.org.
* [11] Koushik Sen, Abhay Vardhan, Gul Agha & Grigore Rosu (2006): _Decentralized Runtime Analysis of Multithreaded Applications_. In: IEEE International Parallel and Distributed Processing Symposium (IPDPS).
* [12] Margaret Smith & Klaus Havelund (2008): _Requirements Capture with RCAT_. In: 16th IEEE International Requirements Engineering Conference (RE’08), IEEE Computer Society. Barcelona, Spain.
* [13] Jinlin Yang, David Evans, Deepali Bhardwaj, Thirumalesh Bhat & Manuvir Das (2006): _Perracotta: Mining Temporal API Rules from Imperfect Traces_. In: International Conference on Software Engineering. pp. 282–291.
|
arxiv-papers
| 2010-03-08T17:22:31 |
2024-09-04T02:49:08.918234
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Howard Barringer, Alex Groce, Klaus Havelund, Margaret Smith",
"submitter": "Klaus Havelund",
"url": "https://arxiv.org/abs/1003.1682"
}
|
1003.1725
|
# Naked Singularity Formation In Brans-Dicke Theory
Amir Hadi Ziaie am.ziaie@mail.sbu.ac.ir Department of Physics, Shahid
Beheshti University, Evin, Tehran 19839, Iran Khedmat Atazadeh
k-atazadeh@sbu.ac.ir Department of Physics, Shahid Beheshti University, Evin,
Tehran 19839, Iran Yaser Tavakoli tavakoli@ubi.pt Departamento de Física,
Universidade da Beira Interior, Rua Marquês d’Avila e Bolama, 6200 Covilhã,
Portugal
###### Abstract
Gravitational collapse of the Brans-Dicke scalar field with non-zero potential
in the presence of matter fluid obeying the barotropic equation of state,
$p=w\rho$ is studied. Utilizing the concept of expansion parameter, it is seen
that the cosmic censorship conjecture may be violated for $w=-\frac{1}{3}$ and
$w=-\frac{2}{3}$ which correspond to cosmic string and domain wall,
respectively. We show that physically, it is the rate of collapse and the
presence of Brans-Dicke scalar field that govern the formation of a black hole
or a naked singularity as the final fate of dynamical evolution and only for
these two cases the singularity can be naked as the collapse end state. Also
the weak energy condition is satisfied by the collapsing configuration.
## I Introduction
The process of gravitational collapse has been studied since about 60 years
ago as a solution of Einstein field equations. When a sufficiently massive
star many times the size of the Sun exhausts its nuclear fuel without reaching
an equilibrium state such as a neutron star or white dwarf, it collapses under
its own pull of gravity at the end of its life cycle. Therefore, gravity
overtakes and dominates the other three forces of nature, in particular, the
weak and strong nuclear forces, which generically provide the outward pressure
in a star to balance it against the inward pull of gravity. In such ultra-
strong gravity regions the densities and spacetime curvature diverge and a
spacetime singularity will be born, either hidden within an event horizon (a
black hole) or visible to the external universe (a naked singularity), as
predicted by the singulary theorems in general relativity. The visibility or
otherwise of the singularity to outside observers is determined by the causal
structure of the dynamically developing collapsing cloud as governed by the
Einstein’s field equations. When the internal dynamics of the collapse delays
the formation of the horizon, the singularity becomes visible, and may
communicate physical effects to the external universe 01 .
Up until now a great deal of effort has gone into the study of the nature of
singularities and large classes of the solutions of Einstein’s field equations
treating singularities have been represented. The general and exact class of
solutions of Einstein’s field equations describing spherically symmetric
pressureless matter (dust) for motion with no particle layers intersecting,
independent of the homogeneity assumption, was originally introduced by
Lema$\hat{i}$tre 02 ; 03 in an attempt to describe cosmology, which was
further developed and studied by Tolman 04 ; 05 and Bondi 04 ; 06 . This
class could be used to model the gravitational collapse of matter from general
inhomogeneous initial conditions and the end state of gravitational collapse
of a massive star can be studied within this framework 04 . Contrary to the
collapsing Friedmann case, in which the physical singularity occurs at a
constant epoch of time, namely at $t=0$, the singular epoch (the time
$t=t_{0}(r)$ at which the area radius of the collapsing shell of matter at a
constant value of the co-moving coordinate r vanishes) in LTB model is a
function of r as a result of inhomogeneity in the matter distribution. This
model can exhibit two kinds of naked singularity: a shell-crossing 07 ; 08
and a shell-focussing singularity 07 ; 09 ; 10 . These kinds of naked
singularity are generic for spherical dust collapse and with proper choice of
initial data they are globally naked. The main reason which causes this model
to be deficient is that it neglects the pressure, which is likely to diverge
at these singularities (where the density diverges) 07 . A special case of
these classes of solutions which has served as the basic paradigm in black
hole physics is the Oppenheimer-Synder study of a completely homogeneous,
pressure-free and spherically symmetric dust cloud collapse, where a dust
cloud undergoes a continued collapse which commences from regular initial data
(i.e., there is no trapped surfaces forming at the initial spacelike surface
from which the collapse begins and the light rays can escape from the surface
of the star to faraway observers) to form a black hole 04 ; 11 ; 12 . Most of
our knowledge about the gravitational collapse is still based on that model.
Also it describes well the formation of the horizon and evolution of the
central, space-like singularity. Oppenheimer and Snyder analyzed the causal
structure of the solution by considering in particular an observer on the
surface of the dust cloud sending signals to a faraway stationary observer at
regularly spaced intervals as measured by his own clock. They discovered that
as the radius of the dust cloud approaches $2M$, the spacing between the
arrival times of these signals to the faraway observer becomes progressively
longer, tending to infinity. This effect has since been called the infinite
redshift effect. The observer on the surface of the dust cloud may keep
sending signals after its radius has become less than $2M$, but these signals
can not escape from this region since the speed of light is the limiting
propagation velocity for physical signals. within a finite affine parameter
interval, these signals reach a true singularity at $r=0$ 13 . This
singularity is formed inside a black hole, as a final state of the collapse
process and also accords to the concept of the cosmic censorship. The cosmic
censorship conjecture which is initially represented by Penrose 14 , explain
the properties of the final singularity of a gravitational collapse. The main
result of this conjecture is that, all spacetime singularities arising from
regular initial data that appear in a gravitational collapse (in an
asymptotically flat universe) are always surrounded by an event horizon and
hence invisible to outside observers (no naked singularities). Moreover in the
strong version of this conjecture, such singularities are not even locally
naked, i.e. there is not any timelike or null geodesics which can emerge from
these singularities and go to future infinity 07 . This hypothesis plays a
fundamental role in both the theory and applications of black hole physics and
has been recognized as one of the most important open problems in classical
general theory of relativity. Up to now many exact solutions of Einstein’s
field equations with several kinds of field-sources which admit naked
singularities depending upon the nature of the initial data and the
kinematical properties near the singularity have been considered. The models
studied so far include the collapse of scalar fields 15 ; 16 as well as other
matter sources including dust 17 ; 18 ; 09 , radiation 17 ; 19 ; 10 , perfect
fluid 17 ; 20 ; 07 , imperfect fluids 17 ; 21 , and null strange quark fluids
15 ; 22 .
One would like to inquire about what are the possible physical factors
operating during the process of a continual collapse of a massive matter cloud
which results in the formation of a naked singularity or a black hole as the
collapse end state. Such an investigation should help us understand much
better the physics of black holes and naked singularity formation in
gravitational collapse. It was recently shown 23 ; 24 that for the spherical
dust collapse the shearing effects and inhomogeneity present within the
collapsing cloud do play a crucial role in delaying the formation of the
trapped surfaces and the apparent horizon. These effects could in fact make
the geometry of apparent and event horizons distorted sufficiently, which
exposes the singularity to the external observers 23 .
General relativity is not the only gravitational theory which explain the
gravitational phenomena. There are alternative theories of gravity that can
explain the gravitational phenomena and lead to many interesting results and
physical interpretations. One of those, is the scalar-tensor theory of
gravity, which has been actively studied as an alternative and successful
theory of gravity. Recently, they have attracted much attention, in part
because they emerge naturally as the low-energy limit of many theories of
quantum gravity, such as Kaluza-Klein theories25 ; 26 and supersymmetric
string theories25 ; 27 . These theories are also important for ”extended”
cosmological inflation models25 ; 28 , in which the scalar field allows the
inflationary epoch to end via bubble nucleation without the need for fine-
tuning cosmological parameters(the ”graceful exit” problem)29 . In Brans-Dicke
theory, the simplest of scalar-tensor theories, gravitation is described by a
metric $g_{\mu\nu}$ and a scalar field $\phi$ coupled to both matter and space
time geometry which obeys a wave equation with a source term determined by the
matter distribution. Gravitational collapse in this theory has been studied in
some of the recent literature. In this paper we study the formation of naked
singularity as the end state of gravitational collapse of a matter fluid
obeying the barotropic equation of state, $p=w\rho$ in the context of Brans-
Dicke (BD) theory. Being motivated by this concept, we investigate the
violation of cosmic censorship conjecture in the collapse procedure of such a
matter fluid.
## II Basic equations
In the context of Brans-Dicke theory with a self interacting potential and a
matter field, the action is given by
${\mathcal{S}}=\frac{1}{2\kappa}\int d^{4}x\sqrt{-g}\left(\Phi
R-\frac{\omega}{\Phi}\nabla^{\alpha}\Phi\nabla_{\alpha}\Phi-V(\Phi)\right)+{\mathcal{S}}_{m},$
(1)
where the constant $\omega$ is the Brans-Dicke parameter and $\Phi$ is the
Brans-Dicke scalar field, which is related to the gravitational ”constant” by
$G=\frac{1}{\Phi}$. Extremizing the action yields the field equations(we set
$\kappa=8\pi G=c=1$ in the rest of this paper)
$G_{\mu\nu}=T^{({\rm eff})}_{\mu\nu},$ (2)
where the effective stress-energy tensor is
$T^{({\rm eff})}_{\mu\nu}=\frac{1}{\Phi}\left(T^{\rm
m}_{\mu\nu}+T^{\Phi}_{\mu\nu}\right),$ (3)
with
$T^{\Phi}_{\mu\nu}=\frac{\omega}{\Phi}(\nabla_{\mu}\Phi\nabla_{\nu}\Phi-\frac{1}{2}g_{\mu\nu}\nabla^{\alpha}\Phi\nabla_{\alpha}\Phi)+(\nabla_{\mu}\nabla_{\nu}\Phi-
g_{\mu\nu}\Box\Phi)-\frac{1}{2}g_{\mu\nu}V(\Phi),$ (4)
and
$T^{m}_{\mu\nu}={\rm diag}\left(\rho_{m},p_{m},p_{m},p_{m}\right),$ (5)
being the stress-energy tensors of scalar field and matter fluid,
respectively. Here we look at the BD scalar field as matter fields originating
from geometry. Variation of the action with respect to $\Phi$ gives
$\Box\Phi=\frac{T^{\rm
m}}{2\omega+3}+\frac{1}{2\omega+3}\left(\Phi\frac{dV(\Phi)}{d\Phi}-2V(\Phi)\right).$
(6)
where $T^{m}$ stands for the trace of $T^{m}_{\mu\nu}$ and the subscript ”$m$”
refers to the matter fields (fields other that $\Phi$). In general relativity
the interior solution of a collapsing star is given by the Tolman-Bondi
solution. Here we present a transparent consideration of a homogenous
collapsing star with BD scalar field $\Phi=\Phi(\tau)$ and effective potential
$V=V(\Phi)$. With this consideration, the Tolman-Bondi solution converts to a
Friedmann-Robertson-Walker(FRW) metric. The interior metric for marginally
bound $(k=0)$ case is given by
$ds^{2}=-d\tau^{2}+a^{2}(\tau)(dr^{2}+r^{2}d\Omega^{2}),$ (7)
where $\tau$ is the proper time of free falling observer whose geodesic
trajectories are distinguished by the comoving radial coordinate $r$ and
$d\Omega^{2}$ being the standard line element on the unit two sphere. Since
the presence of matter acting as a ”seed” field and origin of spherical
symmetry, prompts the collapse of BD scalar field, we have considered perfect
fluid models with barotropic equation of state being given by
$p_{m}=w\rho_{m}.$ (8)
Using the continuity equation for the matter and Eq. (8) one gets the
following relations between $\rho_{m},~{}p_{m}$ and the scale factor as
follows
$\rho_{m}=\rho_{{}_{{}_{0}}m}a^{-3(1+w)};~{}~{}p_{m}=w\rho_{{}_{{}_{0}}m}a^{-3(1+w)},$
(9)
where $\rho_{{}_{{}_{0}}m}=\rho_{m}(a=1)$, is the initial value of the energy
density of matter on the collapsing shell. One then has the following
equations for the effective stress-energy tensor
$\rho_{{}_{({\rm eff})}}=T^{\tau}\,_{\tau}{}^{({\rm
eff})}=\frac{1}{\Phi}\left(\rho_{\Phi}+\rho_{m}\right)=\frac{1}{\Phi}\left(\rho_{m}+\frac{\omega}{2}\frac{\dot{\Phi}^{2}}{\Phi}-3\frac{\dot{a}}{a}\dot{\Phi}+\frac{V(\Phi)}{2}\right),$
(10)
and
$p_{{}_{({\rm eff})}}=T^{r}\,_{r}{}^{({\rm
eff})}=T^{\theta}\,_{\theta}{}^{({\rm eff})}=T^{\varphi}\,_{\varphi}{}^{({\rm
eff})}=\frac{1}{\Phi}\left(p_{{}_{\Phi}}+p_{m}\right)=\frac{1}{\Phi}\left(p_{m}+\frac{\omega}{2}\frac{\dot{\Phi}^{2}}{\Phi}+\ddot{\Phi}+2\frac{\dot{a}}{a}\dot{\Phi}-\frac{V(\Phi)}{2}\right).$
(11)
with all other off-diagonal terms being zero and the radial and tangential
profiles of pressure are equal due to the homogeneity and isotropy. The
interior solution of Einstein’s equation for the line element (7) takes the
form
$\rho_{{}_{({\rm
eff})}}=\frac{{\mathcal{M}}^{\prime}}{R^{2}R^{\prime}};~{}~{}~{}~{}p_{{}_{({\rm
eff})}}=-\frac{\dot{{\mathcal{M}}}}{R^{2}\dot{R}},$ (12)
$\dot{R}^{2}=\frac{{\mathcal{M}}}{R}.$ (13)
The quantity ${\mathcal{M}}$ arises as a free function from the integration of
Einstein’s equation and can be interpreted physically as the total mass of the
collapsing cloud within a coordinate radius r with ${\mathcal{M}}\geq 0$, and
$R(\tau,r)=ra(\tau)$ being the area radius for the shell labeled by the
comoving coordinate $r$. From Eq. (11) we can solve for the mass function as
${\mathcal{M}}=\frac{R^{3}}{3\Phi}(\rho_{{}_{\Phi}}+\rho_{m}).$ (14)
Using Eqs. (12) and (13) we arrive at a relation between $\dot{a}$ and the
effective energy density as follows
$\dot{a}^{2}=\frac{a^{2}}{3\Phi}(\rho_{{}_{\Phi}}+\rho_{m}),$ (15)
since we are concerned with a continual collapse, the time variation of the
scale factor should be negative $(\dot{a}<0)$. This implies that the area
radius of the shell for constant value of $r$ decreases monotonically. For
physical reasons, it is assumed that the energy density is non-negative every
where. The singularity arising from continual collapse is given by $a=0$, in
other words when the scale factor and physical area radius of all the
collapsing shells vanish, then the collapsing cloud has reached a singularity.
A point at which the energy density blows up, the Kretschmann scalar
${\mathcal{K}}=R^{abcd}R_{abcd}$ diverges and the normal differentiability and
manifold structures break down.
## III The Solution
We would like to construct and investigate a class of collapse solutions for
Brans-Dicke scalar filed with non-zero potential considering the matter
fields, where the trapping of light is avoided till the singularity formation,
thereby allowing the singularity to be visible to outside observers. In order
to reach this purpose we consider a class of collapse models where near the
singularity, the divergence of energy density of BD scalar field is given by
the following ansatz
$\rho_{{}_{\Phi}}=a^{-n},$ (16)
where $n$ is a positive constant and the scale factor, $a(\tau)$, goes to zero
in the limit of approach to the singularity which causes $\rho_{{}_{\Phi}}$ to
diverge. Using the above equation and Eq. (15), one may easily obtain the
following relation for $\ddot{a}$ as
$\ddot{a}=\frac{1}{6\Phi}\left[(2-n)a^{1-n}-(1+3w)\rho_{{}_{{}_{0}}m}a^{-(2+3w)}\right]-\frac{\Phi_{,a}}{6\Phi^{2}}\left[a^{2-n}+\rho_{{}_{{}_{0}}m}a^{-(1+3w)}\right],$
(17)
where $\Phi_{,a}=\dot{\Phi}/\dot{a}$. Now by substituting for $\rho_{{}_{({\rm
eff})}}$ and $p_{{}_{({\rm eff})}}$ into Eqs. (10) and (11) together with the
use of Eqs. (13), (15) and (17) we arrive at the following differential
equation as
$\displaystyle\frac{\Phi_{,a}}{\Phi^{2}}\left[\left(\frac{n+2}{6}\right)a^{1-n}+\rho_{{}_{{}_{0}m}}a^{-(2+3w)}\left(\frac{5+3w}{6}\right)\right]-\frac{\Phi_{,a}^{2}}{\Phi^{3}}\left[\frac{2\omega+1}{6}\left(a^{2-n}+\rho_{{}_{{}_{0}m}}a^{-(1+3w)}\right)\right]$
(18)
$\displaystyle+\frac{n}{3}\frac{a^{-n}}{\Phi}-\frac{\Phi_{,aa}}{3\Phi^{2}}\left[a^{2-n}+\rho_{{}_{{}_{0}m}}a^{-(1+3w)}\right]=0,$
where we have used
$\dot{\Phi}=\dot{a}\Phi_{,a}~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}\ddot{\Phi}=\ddot{a}\Phi_{,a}+\dot{a}^{2}\Phi_{,aa}.$
(19)
In the following we solve Eq. (18) for barotropic equation of state
$p_{m}=w\rho_{m}$, where $w=[0,-\frac{1}{3},-\frac{2}{3},\frac{1}{3}]$
correspond to dust, cosmic string, domain wall, and radiation, respectively.
Taking the following ansatz
$\Phi(\tau)=a^{\alpha}(\tau),$ (20)
where $\alpha$ satisfies the following equation
$\displaystyle-\alpha^{2}(1+\rho_{{}_{{}_{0}m}})(3+2\omega)+\alpha(n+4+\rho_{{}_{{}_{0}m}}(7+3w))+2n=0,$
(21)
and by setting $n=3(1+w)$, we derive an expression for the BD scalar field as
a function of scale factor.
### III.1 Dust ($w=0$)
For such a case of pressure-less matter the parameter $\alpha$ will take the
following values as
$\displaystyle\alpha=\left\\{\begin{array}[]{c}-\frac{12}{7(1+\rho_{{}_{{}_{0}m}})+\sqrt{(1+\rho_{{}_{{}_{0}m}})(121+49\rho_{{}_{{}_{0}m}}+48\omega)}}\\\
\\\
\frac{12}{-7(1+\rho_{{}_{{}_{0}m}})+\sqrt{(1+\rho_{{}_{{}_{0}m}})(121+49\rho_{{}_{{}_{0}m}}+48\omega)}}\\\
\end{array}\right..$ (25)
For $\rho_{{}_{{}_{0}m}}>0$ and $-\frac{3}{2}<\omega<\infty$ the first value
for $\alpha$ is negative and the second one is positive. Thus, since the BD
scalar field must diverge near the singularity we choose the first one. For a
special case in which $\rho_{{}_{{}_{0}m}}=1$ and $\omega=-1$ (string
effective action), $\alpha=-0.405$.
### III.2 Cosmic Strings ($w=-\frac{1}{3}$)
Cosmic strings are the consequence of 1-dimensional (spatially) topological
defects in various fields. These topological defects are related to solitonic
solutions of the classical equations for the scalar (and gauge) fields which
for the case of a complex scalar field, cosmic strings can be formed30 . Such
a kind of matter can be regarded as a perfect fluid obeying the barotropic
equation of state, $p_{{}_{m}}=-1/3\rho_{{}_{m}}$. We are interested here to
find the behavior of BD scalar field as a function of scale factor in
gravitational collapse of this kind of fluid. The corresponding values of
$\alpha$ for this case are
$\displaystyle\alpha=\left\\{\begin{array}[]{c}-\frac{4}{3(1+\rho_{{}_{{}_{0}m}})+\sqrt{(1+\rho_{{}_{{}_{0}m}})(21+9\rho_{{}_{{}_{0}m}}+8\omega)}}\\\
\\\
\frac{4}{-3(1+\rho_{{}_{{}_{0}m}})+\sqrt{(1+\rho_{{}_{{}_{0}m}})(21+9\rho_{{}_{{}_{0}m}}+8\omega)}}\\\
\end{array}\right..$ (29)
It is seen that for $\rho_{{}_{{}_{0}m}}>0$ and $-\frac{3}{2}<\omega<\infty$
the first and second values of $\alpha$ are negative and positive,
respectively. Choosing the first value and setting $\rho_{{}_{{}_{0}m}}=1$ and
$\omega=-1$, we have $\alpha=-0.316$.
### III.3 Domain Walls $(w=-\frac{2}{3})$
Domain walls are Generally topological solitons which occur whenever a
discrete symmetry is spontaneously broken31 . These are the result of
2-dimensional topological defects in different scalar or gauge fields. As for
the case of cosmic strings, domain walls can be regarded as a perfect fluid
obeying the barotropic equation of state, $p_{{}_{m}}=-2/3\rho_{{}_{m}}$. We
here again try to find the behavior of BD scalar field as a function of scale
factor in gravitational collapse of such kind of matter fluid. In order to
reach this purpose we consider the following values obtained for $\alpha$ as
$\displaystyle\alpha=\left\\{\begin{array}[]{c}\frac{4}{-5(1+\rho_{{}_{{}_{0}m}})+\sqrt{(1+\rho_{{}_{{}_{0}m}})(49+25\rho_{{}_{{}_{0}m}}+16\omega)}}\\\
\\\
-\frac{4}{5(1+\rho_{{}_{{}_{0}m}})+\sqrt{(1+\rho_{{}_{{}_{0}m}})(49+25\rho_{{}_{{}_{0}m}}+16\omega)}}\\\
\end{array}\right..$ (33)
As it is seen the second one is always negative for $\rho_{{}_{{}_{0}m}}>0$
and $-\frac{3}{2}<\omega<\infty$ causing the BD scalar field to diverge in the
vicinity of singularity. For $\omega=-1$ and $\rho_{{}_{{}_{0}m}}=2$ one has
$\alpha=-0.129$.
### III.4 Radiation $(w=\frac{1}{3})$
Finally for this case of matter, we find the following values for $\alpha$ as
$\displaystyle\alpha=\left\\{\begin{array}[]{c}-\frac{4}{2(1+\rho_{{}_{{}_{0}m}})+\sqrt{2(1+\rho_{{}_{{}_{0}m}})(5+2\rho_{{}_{{}_{0}m}}+2\omega)}}\\\
\\\
\frac{4}{-2(1+\rho_{{}_{{}_{0}m}})+\sqrt{2(1+\rho_{{}_{{}_{0}m}})(5+2\rho_{{}_{{}_{0}m}}+2\omega)}}\\\
\end{array}\right..$ (37)
Again the first value is always negative for $\rho_{{}_{{}_{0}m}}>0$ and
$-\frac{3}{2}<\omega<\infty$. Setting $\rho_{{}_{{}_{0}m}}=2$ and $\omega=-1$
we have $\alpha=-0.32$.
## IV Time Behavior Of The Scale Factor
One would like to study the time-dependence behavior of the scale factor
during the collapse procedure, considering matter. If at time $\tau=\tau^{*}$
(or equivalently for some $a=a^{*}$) the energy density of the BD scalar field
starts growing as $a^{-n}$, then by integrating Eq. (15) in the vicinity of
the singularity with respect to time one gets the time behavior of the scale
factor as
$a(\tau)=\left(a^{\ast\frac{1}{2}\left(\alpha+3(1+w)\right)}-\frac{1}{2}\sqrt{\frac{1+\rho_{{}_{{}_{0}m}}}{3}}\left(\alpha+3(1+w)\right)(\tau-\tau_{\ast})\right)^{\frac{2}{\alpha+3(1+w)}},$
(38)
and the corresponding singular epoch as
$\tau_{s}=\frac{2\sqrt{3}}{\sqrt{1+\rho_{{}_{{}_{0}m}}}(\alpha+3(1+w))},$ (39)
where the time $\tau_{s}$ corresponds to a vanishing scale factor. Thus the
collapse reaches the singularity in a finite proper time. This result for the
scale factor completes the interior solution within the collapsing cloud,
providing us with the required construction.
## V Conditions On Radial Null Geodesic Expansion
Consider a congruence of outgoing radial null geodesics having the tangent
vector $(\xi^{\tau},\xi^{r},0,0)$, where
$\xi^{\tau}=dt/d\lambda~{}and~{}\xi^{r}=dr/d\lambda$ and $\lambda$ is an
affine parameter along the geodesics. In terms of these two vector fields the
geodesic equation can be written as
$\frac{d\xi^{r}}{d\lambda}=-\frac{2\dot{a}}{a}\xi^{t}\xi^{r},$ (40)
and
$\frac{d\xi^{t}}{d\lambda}=-a\dot{a}(\xi^{r})^{2}.$ (41)
The geodesic expansion parameter $\Theta$ is given by
$\Theta=\nabla_{j}\xi^{j}=\partial_{j}\xi^{j}+\Gamma^{j}_{ji}\xi^{i},$ (42)
which gives
$\Theta=\frac{\partial\xi^{\tau}}{\partial\tau}+\frac{\partial\xi^{r}}{\partial
r}+\left(\Gamma^{\tau}_{\tau\tau}+\Gamma^{r}_{r\tau}+\Gamma^{\theta}_{\theta\tau}+\Gamma^{\Phi}_{\Phi\tau}\right)\xi^{\tau}+\left(\Gamma^{\tau}_{\tau
r}+\Gamma^{r}_{rr}+\Gamma^{\theta}_{\theta r}+\Gamma^{\Phi}_{\Phi
r}\right)\xi^{r}.$ (43)
In order to compute the sum
$\frac{\partial\xi^{\tau}}{\partial\tau}+\frac{\partial\xi^{r}}{\partial r},$
(44)
we proceed by noting that
$\frac{d\xi^{\tau}}{d\lambda}=\frac{\partial\xi^{\tau}}{\partial\tau}\frac{\partial\tau}{\partial\lambda}+\frac{\partial\xi^{\tau}}{\partial
r}\frac{\partial r}{\partial\lambda},$ (45)
and similarly,
$\frac{d\xi^{r}}{d\lambda}=\frac{\partial\xi^{r}}{\partial\tau}\frac{\partial\tau}{\partial\lambda}+\frac{\partial\xi^{r}}{\partial
r}\frac{\partial r}{\partial\lambda}.$ (46)
Dividing the first of these two relations by $\partial\tau/\partial\lambda$
and the second by $\partial r/\partial\lambda$, and after adding the resulted
equations one gets
$\frac{\partial\xi^{\tau}}{\partial\tau}+\frac{\partial\xi^{r}}{\partial
r}=-\frac{1}{2}\left[\frac{2\dot{a}}{a}\xi^{\tau}+a\dot{a}\frac{(\xi^{r})^{2}}{\xi^{\tau}}\right]+\frac{\dot{a}}{2a}\xi^{\tau},$
(47)
where we have used Eqs. (40) and (41), and the fact that for outgoing radial
null geodesics the relation between $\xi^{t}$ and $\xi^{r}$ is given by
$\frac{\xi^{\tau}}{\xi^{r}}=\frac{d\tau}{dr}=a(\tau).$ (48)
Substituting Eq. (47) into Eq. (43) and after a simple calculation we arrive
at the desired expression for $\Theta$:
$\Theta=\frac{2}{r}\left(1-\sqrt{\frac{{\mathcal{M}}}{R}}~{}\right).\vspace{.4cm}$
(49)
In order to determine the visibility, or otherwise, of the singularity, one
needs to analyze the behavior of non-spacelike curves in the vicinity of the
singularity and the causal structure of the trapped surfaces. These surfaces
are closed orientable smooth two-dimensional space-like surfaces such that
both families of ingoing and outgoing null geodesics orthogonal to them
necessarily converge 32 . The singularity will be called naked if there exists
a family of future directed non-spacelike geodesics, reaching faraway
observers in space-time and terminating at the singularity in the past. The
existence of such curves implies that either photons or time-like particles
can be emitted from singularity. So if the null geodesics terminate at the
singularity in the past with a definite tangent, then at the singularity we
have $\Theta>0$. If such family of curves do not exist and the event horizon
forms earlier than the singularity covering it, a blackhole is formed. The
boundary of the trapped surface region in the space-time is called apparent
horizon where in spherically symmetric space-time is given by
$g^{ik}R_{,i}R_{,k}=0.$ (50)
Therefore at the boundary of the trapped surface the vector $R_{,k}$ is null.
Using Eq. (7), the above equation can be written as
$-\dot{R}^{2}+a^{-2}R^{\prime 2}=0,$ (51)
which leads to ${\mathcal{M}}=R$. Here use has been made of Eq. (13). The
space-time region where the mass function ${\mathcal{M}}$ satisfies
${\mathcal{M}}<R$ is not trapped, while ${\mathcal{M}}>R$ describes a trapped
region 33 ; 34 ; 04 .
Let us now study the relation that $\Theta$ bears with the formation or
otherwise of a naked singularity in spherical collapse. Calculating
${\mathcal{M}}/R$ in the general case which is considered for energy density
$\rho_{{}_{\Phi}}$ and by using Eq. (13), we have
$\frac{{\mathcal{M}}}{R}=\frac{r^{2}}{3\Phi}\left(a^{2-n}+\rho_{{}_{{}_{0}}m}a^{-(1+3w)}\right).$
(52)
We shall employ the above equation and Eq. (49) to examine the nakedness of
the singularity as the collapse end state for the four cases of matter field
considered in section III. We show that physically, the formation of a black
hole or a naked singularity as the final state for the dynamical evolution is
governed by the rate of collapse scenario and the presence of BD scalar field.
It is seen that the cosmic censorship conjecture is violated for
$w=-\frac{1}{3}$ and $w=-\frac{2}{3}$. The weak energy condition which states
that the energy density as measured by any local observer must be non-negative
can be written for any timelike vector $V^{\mu}$ as follows
$T_{\mu\nu}V^{\mu}V^{\nu}\geq 0,$ (53)
whereby one gets the following conditions for the effective energy density
$(\rho_{{}_{({\rm eff})}}>0)$
$(1+\rho_{{}_{{}_{0}}m})a^{-(\alpha+3(1+w))}>0,$ (54)
and the sum of effective energy density and pressure ($\rho_{{}_{({\rm
eff})}}+p_{{}_{({\rm eff})}}>0$) as
$\displaystyle\left\\{\begin{array}[]{c}w=0,~{}\alpha<0\rightarrow\left[(1+\rho_{{}_{{}_{0}}m})(1-\frac{\alpha}{3})\right]a^{-(\alpha+3)}>0,\\\
\\\
w=-\frac{1}{3},~{}\alpha<0\rightarrow\left[(1+\rho_{{}_{{}_{0}}m})(\frac{4-\alpha}{3})\right]a^{-(\alpha+2)}>0,\\\
\\\
w=-\frac{2}{3},~{}\alpha<0\rightarrow\left[(1+\rho_{{}_{{}_{0}}m})(\frac{5-\alpha}{3})\right]a^{-(\alpha+1)}>0,\\\
\\\
w=\frac{1}{3},~{}\alpha<0\rightarrow\left[(1+\rho_{{}_{{}_{0}}m})(\frac{2-\alpha}{3})\right]a^{-(\alpha+4)}>0.\\\
\end{array}\right.$ (62)
It is seen that these two conditions are satisfied by all cases of $w$ and
$\rho_{{}_{{}_{0}}m}>0$ and $-\frac{3}{2}<\omega<\infty$ considered above. At
the initial epoch $(a=1)$ there should not be any trapping of light. Assuming
$r=r_{b}$ is the boundary of the collapsing ball, then at the initial epoch
the ratio ${\mathcal{M}}/R$ is less than unity for a suitable initial value of
$\rho_{{}_{{}_{0}}m}$ standing for all values of $w$. This fact is in
accordance with the regularity condition stating that the gravitational
collapse must initiate from regular and physically reasonable initial
conditions. The time at which the physical area radius of the collapsing cloud
becomes zero, denotes a shell-focusing singularity which lies on the curve
$R(\tau_{s},r)=0$ where $\tau_{s}$ being the singular epoch given by Eq. (39).
For the case of homogeneous-density collapse the resulting singularity may lay
on the curves $R(\tau_{s},0)=0$ or $R(\tau_{s},r\neq 0)=0$, which corresponds
to a central or non-central singularity, respectively. We consider first the
simpler case of non-central singularity and investigate the failure of
formation of apparent horizon in collapse scenario for different values of
$w$.
### V.1 Dust ($w=0$)
We are now in a position to study the effect of BD scalar field on the
formation or otherwise of the apparent horizon as the dynamical procedure of
collapse scenario evolves( we set $\omega=-1$ in the rest of this paper). We
begin by Eq. (14) which for $w=0$ can be written as
$\frac{{\mathcal{M}}}{R}=\frac{r^{2}}{3}(1+\rho_{{}_{{}_{0}}m})a^{-(1+\alpha)}.$
(63)
The initial energy density of matter must be positive due to the regularity
conditions, then for $\rho_{{}_{{}_{0}}m}>0$ and $\omega=-1$, the first value
of $\alpha$ in Eq. (25) implies that $|\alpha|<1$. From Eq. (63) it is seen
that the ratio ${\mathcal{M}}/R$ grows and the expansion parameter, Eq. (49),
tends to negative infinity. Thus there exist no radial null geodesics emerging
from the singularity. Strictly speaking the singularity occurred here is
necessarily covered and a black hole is formed as the collapse end state.
### V.2 Cosmic Strings ($w=-\frac{1}{3}$)
For this case Eq. (14) and the time variation of the mass function take the
following form as
$\displaystyle\left\\{\begin{array}[]{c}\frac{{\mathcal{M}}}{R}=\frac{r^{2}}{3}(1+\rho_{{}_{{}_{0}}m})a^{-\alpha},\\\
\\\
\dot{{\mathcal{M}}}=-(1+\rho_{{}_{{}_{0}}m})\frac{r^{3}}{3}(\alpha-1)\dot{a}a^{-\alpha}.\\\
\end{array}\right.$ (67)
As it is seen from Eq. (29), since the first value of $\alpha$ is always less
than zero the ratio ${\mathcal{M}}/R$ stays finite till the singular epoch and
causes the expansion parameter to be positive up to the singularity, and if no
trapped surfaces exist initially then no ones would form until the epoch
$a(\tau)=0$ which is consistent with the fact that there exist families of
outgoing radial null geodesics emerging from the singularity. One can take the
positive value of $\alpha$ in Eq. (29), but for this case the BD scalar field
get vanished as the scale factor tends to zero. Also the weak energy condition
may be violated. In addition to, for such a case the ratio ${\mathcal{M}}/R$
grows at a vanishing scale factor causing the expansion parameter tends to
negative infinity which means that the singularity is covered and no radial
geodesics can emerge from it. From the second equation in Eq. (67) it can be
seen that the time derivative of the mass function for $\alpha<0$, is negative
(note that $\dot{a}<0$) which means that the mass function contained in the
collapsing shell with that radius keeps decreasing. In other words there
exists an outward energy flux during the collapse scenario. Since no trapped
surfaces form up to the singularity, the outward energy flux would be
observable.
### V.3 Domain Walls ($w=-\frac{2}{3}$)
For this case one may rewrite Eq. (14) and time derivative of the mass
function as
$\displaystyle\left\\{\begin{array}[]{c}\frac{{\mathcal{M}}}{R}=\frac{r^{2}}{3}(1+\rho_{{}_{{}_{0}}m})a^{1-\alpha},\\\
\\\
\dot{{\mathcal{M}}}=-(1+\rho_{{}_{{}_{0}}m})\frac{r^{3}}{3}(\alpha-2)\dot{a}a^{1-\alpha}.\\\
\end{array}\right.$ (71)
from the first equation one may easily see that at initial epoch ($a=1$), the
regularity condition is satisfied. Since the second value of $\alpha$ in Eq.
(33) is always negative, the ratio of mass function to area radius of the
collapsing shell is less than unity during the collapse procedure denoting
that the expansion parameter being positive up to the singularity. In this
case the collapse evolution to a naked singularity takes place, where the
trapped surfaces do not form early enough or are avoided in the spacetime. For
first value of Eq. (33), $\alpha>5$ which causes the ratio ${\mathcal{M}}/R$
tends to infinity as the scale factor vanishes and $\Theta$ goes to negative
infinity, thus trapped surfaces do form in the spacetime which prevent the
null geodesics to emerge from the singularity. Such a situation ends in a
black hole as the final fate of the collapse scenario. But such a value of
$\alpha$ is not allowed since it violates the weak energy condition. From the
second equation in Eq. (71), it is obvious that for negative value of $\alpha$
and $\dot{a}<0$, the time derivative of the mass function is negative stating
that the mass contained in collapsing ball reduces as the time advances.
### V.4 Radiation ($w=\frac{1}{3}$)
In this case Eq. (14) can be written as
$\frac{{\mathcal{M}}}{R}=\frac{r^{2}}{3}(1+\rho_{{}_{{}_{0}}m})a^{-(\alpha+4)}.$
(72)
From Eq. (37) one can easily see that the first value for
$\rho_{{}_{{}_{0}}m}$ and $\omega=-1$ is always negative and $|\alpha|<4$.
Thus, in such a situation the ratio ${\mathcal{M}}/R$ tends to infinity as the
singularity is approached. Thus the expansion parameter behaves just as the
dust case, and the final singularity is necessarily covered within an event
horizon of gravity.
The central singularity occurring at $R=0,r=0$ is naked if there exist
outgoing non-spacelike geodesics reaching faraway observers and terminating in
the past at the singularity. In order to investigate the nakedness of this
kind of singularity we proceed by introducing a new variable
$x=r^{\delta},~{}and~{}\delta>1$ is defined such that
$R^{\prime}/r^{\delta-1}$ is a unique finite quantity in the limit
$r\rightarrow 0$. Then we have the following equation
$\frac{dR}{dx}=\frac{1}{\alpha
r^{\alpha-1}}\left(\dot{R}\frac{d\tau}{dr}+R^{\prime}\right).$ (73)
By virtue of Eqs. (13) and (48) the above equation leads to
$\frac{dR}{dx}=\frac{R^{\prime}}{\alpha
x^{\frac{\alpha-1}{\alpha}}}\left[1-\sqrt{\frac{{\mathcal{M}}}{R}}~{}\right].$
(74)
It is clear that $R=0$ , $x=0$ is a singular point of Eq. (74). If there are
outgoing radial null geodesics terminating in the past at the singularity with
a definite tangent, then at the singularity we have $\frac{dR}{dx}>0$. For
$w=-\frac{1}{3}$, and $w=-\frac{2}{3}$ with $\alpha$ being negative the
quantity ${\mathcal{M}}/R<1$ throughout the collapse procedure, so the term
being in the second bracket is positive and $\frac{dR}{dx}>0$ as the
singularity is approached indicating that the singularity is visible to
outside observers and the inverse result holds for $w=0$ and $w=\frac{1}{3}$.
## VI nakedness of the singularity
The continual gravitational collapse of a matter cloud culminates in either a
black hole or a naked singularity where in the former an event horizon
develops earlier than the formation of the singularity. Thus the regions of
extreme physical conditions such as densities and curvatures are hidden from
the outside observers. If such horizons are delayed or failed to develop
during the collapse procedure, as governed by the internal dynamics of the
collapsing cloud, then the scenario where the ultra-strong gravity regions
become visible to external observers occurs and a visible naked singularity
forms. In such a case where no black hole forms, the field collapses for a
while and then disperses. Therefore as viewed by a central observer, the
scalar invariants namely the Kretschmann scalar should grow near the
singularity, gain some maximum value and then approach to zero at late times35
. Since the absence of an apparent horizon does not necessarily implies the
absence of an event horizon, we examine the nakedness of the singularity in
spherically symmetric collapse of a fluid by considering the behavior of the
Kretschmann invariant with respect to time. For the line element (7) this
quantity is given by
${\cal K}\equiv
R^{abcd}R_{abcd}=\frac{12}{a^{4}}\left[a^{2}\ddot{a}^{2}+\dot{a}^{4}\right].$
(75)
By the virtue of Eqs. (38) and (39) for $w=-\frac{1}{3}$ the above quantity
can be written as
${\cal K}_{cs}=\frac{3.322}{(1-0.6\tau)^{4}},$ (76)
and for $w=-\frac{2}{3}$ as
${\cal K}_{dw}=\frac{11.6}{(1-0.4\tau)^{4}}.$ (77)
Note that both these two cases satisfy the condition on expansion parameter
stating that this quantity must be positive up to the singularity. Fig. 1 and
Fig. 2 show the behavior of the Kretschmann scalar as a function of proper
time, $\tau$. It is seen that both ${\cal K}_{cs}$ and ${\cal K}_{dw}$ diverge
at $\tau=5/3$ and $\tau=5/2$, respectively. They then converge to zero at late
times signaling the failure of formation of the event horizon.
Let us now consider the geometry of the exterior spacetime. In order to fully
complete the spacetime model, one needs to match the interior spacetime of the
dynamical collapse to a suitable exterior geometry. The Schwarzschild solution
is a useful model describing the spacetime outside the sun and stars. However
this model may no longer be suitable to describe the exterior geometry of any
realistic star, because the spacetime outside such a star may be filled with
radiated energy from the star in the form of electromagnetic radiation. The
Schwarzschild model does not describe this as it corresponds to an empty
spacetime given by $T_{ab}=0$. The spacetime outside a spherically symmetric
star being surrounded by a radiation emitted from the star is described by the
Vaidya metric36 which can be given in the form
$ds^{2}_{out}=-\left(1-\frac{2M(r_{u},u)}{r_{u}}\right)du^{2}-2dudr_{u}+r_{u}^{2}d\Omega^{2},$
(78)
where u, being the retarded null coordinate, $r_{u}$ and $M(r_{u},u)$ are the
Vaidya radius and Vaidya mass, respectively. Following the work of 33 we use
the Isreal-Darmois junction conditions to match the interior spacetime
described by Eq. (7) to a Vaidya exterior geometry at the boundary
hypersurface $\Sigma$ given by $r=r_{b}$. The spacetime metric just inside
$\Sigma$ is given by
$ds^{2}_{in}=-d\tau^{2}+a^{2}(\tau)\left[dr^{2}+r_{b}^{2}d\Omega^{2}\right]$
(79)
Matching the area radius of the collapsing shell at the boundary, one gets the
following equation
$r_{u}(u)=r_{b}a(\tau),$ (80)
whereby on the hypersurface $\Sigma$, the interior and exterior metrics can be
written as
$ds^{2}_{\Sigma in}=-d\tau^{2}+a^{2}(\tau)r_{b}^{2}d\Omega^{2},$ (81)
and
$ds^{2}_{\Sigma
out}=-\left(1-\frac{2M(r_{u},u)}{r_{u}}+2\frac{dr_{u}}{du}\right)du^{2}+r_{u}^{2}d\Omega^{2}.$
(82)
Matching the induced metric on $\Sigma$ one gets
$\left(\frac{du}{d\tau}\right)_{\Sigma}=\frac{1}{\left(1-\frac{2M(r_{u},u)}{r_{u}}+2\frac{dr_{u}}{du}\right)^{\frac{1}{2}}},~{}~{}~{}(r_{u})_{\Sigma}=r_{b}a(\tau).$
(83)
In order to match the extrinsic curvature for interior and exterior
spacetimes, one has to fine the unit normal vector field to the hypersurface
$\Sigma$. In this step, we proceed by noting that any spacetime metric can be
written locally in the form
$ds^{2}=-\left(N^{2}-N_{i}N^{i}\right)d\tau^{2}-2N_{i}dx^{i}d\tau+h_{ij}dx^{i}dx^{j},$
(84)
where N, $N^{i}$, and $h_{ij}$ are the lapse function, shift vector, and
induced metric, respectively and $i$, $j$ are three-dimensional indices run in
$\\{1,2,3\\}$. The contravariant and covariant components of the unit normal
vector field are given by
$n^{a}=\frac{1}{N}\left(\delta^{a}_{0}-N^{a}\right),~{}~{}~{}n_{a}=-N\delta^{0}_{a}.$
(85)
Comparing Eqs. (84) and (7), one finds the contravariant components of the
normal to the hypersurface $\Sigma$ for the interior metric as
$n^{a}_{in}=[0,a(\tau)^{-1},0,0].$ (86)
Upon using a similar approach, one finds the first non-vanishing contravariant
component of the normal to $\Sigma$ for the exterior metric as
$n^{u}=-\frac{1}{\left(1-\frac{2M(r_{u},u)}{r_{u}}+2\frac{dr_{u}}{du}\right)^{\frac{1}{2}}}.$
(87)
In order to compute the second non-vanishing contravariant component of the
normal vector field we proceed by having recourse the normalization relation
holding for $n^{a}$ as
$n^{u}n_{u}+n^{r_{u}}n_{r_{u}}=1.$ (88)
Benefiting from the property of the metric tensor in raising and lowering
indices, we have the following relations
$n_{u}=\frac{1-\frac{2M(r_{u},u)}{r_{u}}}{\left(1-\frac{2M(r_{u},u)}{r_{u}}+2\frac{dr_{u}}{du}\right)^{\frac{1}{2}}}-n^{r_{u}};~{}~{}~{}~{}n_{r_{u}}=-n^{u}.$
(89)
Substituting the above equations and Eq. (87) into Eq. (88) and after a simple
calculation we arrive at the desired expression for $n^{r_{u}}$ as
$n^{r_{u}}=\frac{1-\frac{2M(r_{u},u)}{r_{u}}+\frac{dr_{u}}{du}}{\left(1-\frac{2M(r_{u},u)}{r_{u}}+2\frac{dr_{u}}{du}\right)^{\frac{1}{2}}}.$
(90)
The extrinsic curvature of the hypersurface $\Sigma$ is defined as the Lie
derivative of the metric tensor with respect to the normal vector field being
given by the following relation
$K_{ab}=\frac{1}{2}\left[g_{ab,c}n^{c}+g_{cb}n^{c}_{,a}+g_{ac}n^{c}_{,b}\right].$
(91)
Since the matching is for the second fundamental form, $K_{ab}$, there exists
no surface stress energy or surface tension at the boundary37 . The nonzero
$\theta$ components of the extrinsic curvature are
$K^{in}_{\theta\theta}=r_{b}a(\tau),~{}~{}~{}~{}~{}~{}K^{out}_{\theta\theta}=r_{u}\frac{1-\frac{2M(r_{u},u)}{r_{u}}+\frac{dr_{u}}{du}}{\left(1-\frac{2M(r_{u},u)}{r_{u}}+2\frac{dr_{u}}{du}\right)^{\frac{1}{2}}}.$
(92)
Setting $\left[K^{in}_{\theta\theta}-K^{out}_{\theta\theta}\right]_{\Sigma}=0$
on the hypersurface $\Sigma$, and by using Eqs. (13) and (83), one gets the
following relation between mass function and Vaidya mass on the boundary as
${\mathcal{M}}(\tau,r_{b})=2M(r_{u},u).$ (93)
From the above equation and Eq. (14) one can see that the BD scalar field
affects on the behavior of the Vaidya mass in the collapse scenario. In order
to find another relation expressing the rate of change of Vaidya mass with
respect to $r_{u}$ one has to match the $\tau$ component of the extrinsic
curvature on the hypersurface $\Sigma$. Having set
$\left[K^{in}_{\tau\tau}-K^{out}_{\tau\tau}\right]=0$, one gets
$M(r_{u},u)_{r_{u}}=\frac{{\mathcal{M}}}{2r_{u}}+r_{b}^{2}a\ddot{a}.$ (94)
The occurrence of a naked singularity as the final outcome of a collapse
procedure, depends on the non-existence of trapped surfaces till the formation
of the singularity, which corresponds to the existence of families of non-
spacelike trajectories reaching faraway observers and terminating in the past
at the singularity. In order to show this, we begin by Eq. (83) and after
using Eqs. (92), and (93) we arrive at the following relation
$\left(\frac{du}{d\tau}\right)_{\Sigma}=\frac{1-r_{b}\dot{a}}{1-\frac{{\mathcal{M}}(\tau,r_{b})}{r_{u}}}.$
(95)
It can be easily checked that if one imposes the null condition on the Vaidya
metric, the result is the same as Eq. 95. What is meant by this, is that null
geodesics can come out from the singularity and reach to faraway observers
before it evaporates into the free space. In other words the formation of
trapped surfaces in spacetime is avoided and a naked singularity can be
produced.
## VII Behavior Of The BD Scalar field and the Effective Potential
In the following section we wish to study how the effective potential behaves
as the scalar field varies. For this purpose we start by Eq. (10) and consider
the four cases of matter field discussed in section III. Together with the use
of Eqs. (15) and (20), one may easily find the following expression for the
potential as
$\displaystyle
V(\phi)=\beta\Phi^{-\frac{3(1+w)}{\alpha}},~{}~{}~{}~{}~{}\beta=\left(2+\frac{\alpha}{3}(1+\rho_{{}_{{}_{0}}m})(6-\omega)\right).$
(96)
Fig. 3 shows the behavior of the BD potential with respect to $\Phi$ for
different values of $w$ and $\omega=-1$.
Let us now consider the case in which the BD scalar field is a function of
both $\tau$ and $r$. Assuming that far away from the collapsing system, the
effective energy density behaves homogeneously, we obtain a measure of the
radial profile of the BD scalar field for each cases of $w$ considered in
Section III. We begin by Eq. (6) together with the use of Eqs. (15), (17),
(19), and (96) we arrive at a differential equation for $\Phi(a(\tau),r)$ as
$\displaystyle\left[\frac{n-8}{6}a^{1-n}+\frac{\rho_{{}_{{}_{0}m}}}{6}(3w-5)a^{-(3w+2)}\right]\frac{\phi_{,a}}{\phi}+$
$\displaystyle\frac{1}{6}\left[a^{2-n}+\rho_{{}_{{}_{0}m}}a^{-(1+3w)}\right]\left(\frac{\phi_{,a}}{\phi}\right)^{2}-\left[a^{2-n}+\rho_{{}_{{}_{0}m}}a^{-(1+3w)}\right]\frac{\phi_{,aa}}{3\phi}$
(97)
$\displaystyle+\frac{\phi^{\prime\prime}}{a^{2}}+2\frac{\phi^{\prime}}{ra^{2}}+C\frac{a^{-3(1+w)}}{2\omega+3}=0,$
where $C$ is given by
$\displaystyle C=\frac{3\beta(1+w)}{\alpha}+2\beta-\rho_{{}_{{}_{0}m}}(3w-1).$
(98)
the above equation can be more simplified and the result is as follows
$\displaystyle\phi^{\prime\prime}+\frac{2\phi^{\prime}}{r}-Da^{1-\alpha-3w}=0,$
(99)
where $D$ is a constant and is given by
$\displaystyle
D=\frac{\alpha(\alpha-1)}{3}+\left(\frac{5\alpha-\alpha^{2}}{6}-\frac{\alpha
w}{2}\right)(1+\rho_{{}_{{}_{0}m}})-\frac{C}{2\omega+3}.$ (100)
Here $\prime$ denotes partial differentiation with respect to $r$. Solving the
above differential equation with a suitable choice of value for
$\rho_{{}_{{}_{0}}m}$ one may find the solutions as functions of $\tau$ and
$r$. Figs. 4-7 show the behavior of BD scalar field in terms of $\tau$ and $r$
in which the constants of integration have been chosen in such a way that the
scalar field blows up near the singularity and get vanished faraway from the
collapsing system.
## VIII Conclusion and outlook
In this work we have studied the gravitational collapse of the BD scalar field
with non-zero potential in the presence of matter fluid. Assuming that the
energy density of the BD scalar field behaves as the inverse power law of the
scale factor near the singularity, we presented a class of solutions in Brans-
Dicke theory in which the naked singularity can be created being accompanied
by the violation of the cosmic censorship conjecture. In section III, we found
the behavior of BD scalar field as a function of scale factor for the four
cases of matter fluid. In section IV, the general expressions of time behavior
of the scale factor and singular epoch has been achieved. Having examined the
ansatz taken for divergence of the energy density of BD scalar field, Eq.
(16), together with the use of Eq. (9) for energy density of matter fluid and
by using the concept of expansion parameter, we have shown in section V that
the presence of BD scalar field due to the Eq. (14) can affect the formation
or otherwise of the trapped surfaces and only for the two cases,
$w=-\frac{1}{3}$ and $w=-\frac{2}{3}$, formation of the apparent horizon can
be failed and a naked singularity may be generated as the final fate of
collapse procedure. But since the absence of an apparent horizon does not
necessarily implies the absence of an event horizon, we have computed the
Kretschmann scalar in section VI and the result has been plotted in Fig. 1 and
Fig. 2 for $w=-\frac{1}{3}$ and $w=-\frac{2}{3}$, respectively. It is seen
that this quantity diverges at singular epoch, and then vanishes at late
times, a behavior which can be interpreted as the absence of an event horizon
and formation of a naked singularity. Also following the work of 33 we have
shown at the end of this section that the Vaidya geodesic emerging from the
singularity before it evaporates into free space is null.
Beside our work which have only treated exact solutions to gravitational
collapse in Brans-Dicke theory, one may find numerical solutions to such a
collapse scenario in the literature38 . In 25 ; 29 , the author has developed
a new numerical code that solves the gravitational field equations coupled to
the matter for evolution of a spherically symmetric configuration of
noninteracting particles in Brans-Dicke theory. Using this code, he has shown
that Oppenheimer-Snyder collapse in this theory results in black holes rather
than naked singularities, at least for $|3+2\omega|\geq 3$, which are
identical to those of general relativity in final equilibrium, but are quite
different from those of general relativity during dynamical evolution in which
they radiate mass. The reason for this behavior is due to the violation of the
null energy condition even in vacuum spacetimes with positive values of
$\omega$, passing of the apparent horizon of a black hole outside the event
horizon, and decreasing the surface area of the event horizon over time. This
numerical code enables one to decide on a number of long-standing theoretical
questions about collapse in Brans-Dicke theory of gravitation. Also the
gravitational collapse of a scalar field with other characteristics and
couplings has been discussed in some literature. In 39 , the collapse of a
self-similar scalar field has been studied, and it has shown that there exists
two classes of solutions which one of them consists of a nonsingular origin in
which the scalar field collapses and disperses again. There is a singularity
at one point of these solutions which is not observable at a finite radius.
The second class of solutions contains both black holes and naked
singularities with a critical behavior interpolating between these two
extremes. Numerical study of spherically symmetric collapse of a massless
scalar field has presented in 40 , where it is shown that the masses of black
holes which form satisfy a power law $M_{BH}\propto|p-p^{\ast}|^{\gamma}$.
Where $p$ is a parameter which characterizes the strength of initial
condition, $p^{\ast}$ is the threshold value and $\gamma\approx 0.37$ is a
universal exponent. Also the collapse of a massless scalar field in Brans-
Dicke theory has studied both analytically and numerically in 41 and it is
shown that for $\omega>-\frac{3}{2}$, a continuous self-similarity continues
and that the critical exponent $\gamma$ depends on $\omega$. In 42 ,
gravitational collapse of a self-interacting (massive) scalar field has been
studied both analytically and numerically on a Reissner-Nordsr$\ddot{o}$m
background and finally in 43 , one may find some examples of naked singularity
formation in gravitational collapse of a scalar field.
## Acknowledgments
Y. Tavakoli is supported by the Portuguese Agency Fundação para a Ciência e
Tecnologia through the fellowship SFRH/BD/43709/2008.
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Figure 1: The behavior of Kretschmann scalar (in units of $s^{-4}$) as a
function of proper time for $\omega=-1$ and $w=-\frac{1}{3}$. For the initial
energy density, scale factor, and proper time we have adopted the values
$\rho_{{}_{0m}}=1$, $a^{\ast}=1$, and $\tau^{\ast}=0$, respectively. Figure 2:
The behavior of Kretschmann scalar (in units of $s^{-4}$) as a function of
proper time for $\omega=-1$ and $w=-\frac{2}{3}$. For the initial energy
density, scale factor, and proper time we have adopted the values
$\rho_{{}_{0m}}=2$, $a^{\ast}=1$, and $\tau^{\ast}=0$, respectively. Figure 3:
Behavior of potential as a function of BD scalar field for $\omega=-1$ and
different values of $\rho_{{}_{{}_{0}m}}$ and $w$: $\rho_{{}_{{}_{0}m}}=1$ and
$w=0$ (Dotted curve), $\rho_{{}_{{}_{0}m}}=1$ and $w=-\frac{1}{3}$ (Dashed
Curve), $\rho_{{}_{{}_{0}m}}=2$ and $w=-\frac{2}{3}$ (Solid Curve),
$\rho_{{}_{{}_{0}m}}=2$ and $w=\frac{1}{3}$ (Thick-Dashed Curve). Figure 4:
Behavior of the BD scalar field with respect to $\tau$ and r for $\omega=-1$,
$w=-\frac{1}{3}$. For the initial energy density, scale factor and proper time
we have adopted the values $\rho_{{}_{{}_{0}m}}=1$, $a*=1$ and $t*=0$,
respectively. Figure 5: Behavior of the BD scalar field with respect to $\tau$
and r for $\omega=-1$, $w=0$. For the initial energy density, scale factor and
proper time we have adopted the values $\rho_{{}_{{}_{0}m}}=1$, $a*=1$ and
$t*=0$, respectively. Figure 6: Behavior of the BD scalar field with respect
to $\tau$ and r for $\omega=-1$, $w=-\frac{2}{3}$. For the initial energy
density, scale factor and proper time we have adopted the values
$\rho_{{}_{{}_{0}m}}=2$, $a*=1$ and $t*=0$, respectively. Figure 7: Behavior
of the BD scalar field with respect to $\tau$ and r for $\omega=-1$,
$w=\frac{1}{3}$. For the initial energy density, scale factor and proper time
we have adopted the values $\rho_{{}_{{}_{0}m}}=2$, $a*=1$ and $t*=0$,
respectively.
|
arxiv-papers
| 2010-03-08T21:06:05 |
2024-09-04T02:49:08.925022
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. H. Ziaie, K. Atazadeh, Y. Tavakoli",
"submitter": "Khedmat Atazadeh",
"url": "https://arxiv.org/abs/1003.1725"
}
|
1003.1790
|
# THE CORES OF THE Fe K$\alpha$ LINES IN ACTIVE GALACTIC NUCLEI: AN EXTENDED
Chandra HIGH ENERGY GRATING SAMPLE
X. W. Shu11affiliation: CAS Key Laboratory for Research in Galaxies and
Cosmology, Department of Astronomy, University of Science and Technology of
China, Hefei, Anhui 230026, P. R. China, xwshu@mail.ustc.edu.cn 22affiliation:
Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD
21218, yaqoob@skysrv.pha.jhu.edu , T. Yaqoob22affiliation: Department of
Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218,
yaqoob@skysrv.pha.jhu.edu , J. X. Wang11affiliation: CAS Key Laboratory for
Research in Galaxies and Cosmology, Department of Astronomy, University of
Science and Technology of China, Hefei, Anhui 230026, P. R. China,
xwshu@mail.ustc.edu.cn
###### Abstract
We extend the study of the core of the Fe K$\alpha$ emission line at $\sim
6.4$ keV in Seyfert galaxies reported in Yaqoob & Padmanabhan (2004) using a
larger sample observed by the Chandra High Energy Grating (HEG). The sample
consists of 82 observations of 36 unique sources with $z<0.3$. Whilst heavily
obscured active galactic nuclei (AGNs) are excluded from the sample, these
data offer some of the highest precision measurements of the peak energy of
the Fe K$\alpha$ line, and the highest spectral resolution measurements of the
width of the core of the line in unobscured and moderately obscured
($N_{H}<10^{23}\ \rm cm^{-2}$) Seyfert galaxies to date. From an empirical and
uniform analysis, we present measurements of the Fe K$\alpha$ line centroid
energy, flux, equivalent width (EW), and intrinsic width (FWHM). The Fe
K$\alpha$ line is detected in 33 sources, and its centroid energy is
constrained in 32 sources. In 27 sources the statistical quality of the data
is good enough to yield measurements of the FWHM. We find that the
distribution in the line centroid energy is strongly peaked around the value
for neutral Fe, with over 80% of the observations giving values in the range
6.38–6.43 keV. Including statistical errors, 30 out of 32 sources ($\sim
94\%$) have a line centroid energy in the range 6.35–6.47 keV. The mean
equivalent width, amongst the observations in which a non-zero lower limit
could be measured, was $53\pm 3$ eV. The mean FWHM from the subsample of 27
sources was $2060\pm 230\ \rm km\ s^{-1}$. The mean EW and FWHM are somewhat
higher when multiple observations for a given source are averaged. From a
comparison with the H$\beta$ optical emission-line widths (or, for one source,
Br$\alpha$), we find that there is no universal location of the Fe K$\alpha$
line-emitting region relative to the optical BLR. In general, a given source
may have contributions to the Fe K$\alpha$ line flux from parsec-scale
distances from the putative black hole, down to matter a factor $\sim 2$
closer to the black hole than the BLR. We confirm the presence of the X-ray
Baldwin effect, an anti-correlation between the Fe K$\alpha$ line EW and X-ray
continuum luminosity. The HEG data have enabled isolation of this effect to
the narrow core of the Fe K$\alpha$ line.
galaxies: active – line: profile – X-rays: galaxies
## 1 INTRODUCTION
The narrow core (FWHM $<10,000\ \rm km\ s^{-1}$) of the Fe K$\alpha$
fluorescent emission, peaking at $\sim 6.4$ keV is a common and dominant
feature of the X-ray spectrum of active galactic nuclei (AGNs) that have a
2–10 keV X-ray luminosity less than $\sim 10^{45}\rm\ erg\ s^{-1}$ (e.g.
Sulentic et al. 1998; Lubiński & Zdziarski 2001; Weaver, Gelbord, & Yaqoob
2001; Perola et al. 2002; Yaqoob & Padmanabhan 2004 (hereafter YP04); Levenson
et al. 2006; Winter et al. 2009). The luminosity in the core of the Fe
K$\alpha$ emission may be comparable to any additional, relativistically-
broadened Fe K$\alpha$ line emission that may be present, and indeed, in many
cases may be the only component of the Fe K$\alpha$ line (e.g. see Guainazzi,
Bianchi, & Dovčiak 2006; Nandra et al. 2007; Miller 2007; Turner & Miller
2009; Bianchi et al. 2009). Measurement of the properties of the core of the
Fe K$\alpha$ line in AGN is important for two principal reasons. One is to
constrain the physical properties of the large-scale structure in the central
engine. The peak energy of the Fe K$\alpha$ line constrains the ionization
state of the line-emitting matter, and the width of the line gives kinematic
information that can be used to estimate the size and location of the X-ray
reprocessor. The equivalent width (EW) of the Fe K$\alpha$ line is a function
of geometry, column density, covering factor, element abundances, and
orientation of the line-emitter. Another reason why spectroscopy of the Fe
K$\alpha$ line core is important is that it is necessary to model the narrow
component of the line in order to deconvolve any relativistically-broadened
emission-line component that may be present. The Chandra high energy grating
(HEG; see Markert et al. 1995) is still unsurpassed in spectral resolution in
the Fe K band, which at 6.4 keV is $\sim 39$ eV, or $\sim 1860\ \rm km\
s^{-1}$ FWHM. This is a factor of $\sim 4$ better than the spectral resolution
of X-ray CCD detectors aboard XMM-Newton and Suzaku. Although broad Fe
K$\alpha$ emission lines are better studied with CCD spectrometers (due to
their higher throughput), the Chandra HEG is well-suited for studying the
narrow core of the Fe K$\alpha$ line. One can then utilize the HEG
measurements to deconvolve narrow and broad Fe K$\alpha$ line components in
lower spectral resolution data.
In YP04 the results of a uniform analysis of the properties of the Fe
K$\alpha$ emission-line core were presented, based on Chandra HEG data of a
modest sample of fifteen AGN. There are now a larger number of Chandra HEG
observations for which the data are available, and in the present paper we
extend the study of YP04 to include 82 observations of 36 unique AGN. The
paper is organized as follows. In §2 we describe the observations and data. In
§3 we describe the methodology and basic spectral-fitting results. In §4 we
discuss the implications of the results for the properties of the core of
narrow Fe K$\alpha$ emission line in the HEG AGN sample. In §5 we investigate
whether the narrow core of the Fe K$\alpha$ line as isolated by the HEG,
supports the so-called X-ray Baldwin effect (an anti-correlation between the
line EW and X-ray luminosity and between the EW and a proxy for the accretion
rate). In §6 we summarize our results and findings.
## 2 OBSERVATIONS AND DATA
The Chandra HETGS (High Energy Transmission Grating Spectrometer) consists of
two grating assemblies, a High-Energy Grating (HEG) and a Medium-Energy
Grating (MEG), and it is the HEG that achieves the highest spectral
resolution. The MEG has only half of the spectral resolution of the HEG and
less effective area in the Fe K band, so our study will focus on the HEG data.
Our study is based on data from Chandra HETGS AGN observations that were
public as of 2008, September 30, filtering on several criteria. Firstly, we
selected non-blazar AGN that had $z<0.3$. This actually only omitted one
source, PKS 2149$-$306 ($z=2.345$), which is a high-luminosity radio-loud
quasar (see Fang et al. 2001 for results from the Chandra grating
observations). Since the centroid energy of the Fe K$\alpha$ line appears at
$\sim 6.4/(1+z)$ keV, the line would appear at very different places on the
instrumental effective area curve for very different values of $z$. In
addition, the EW of the Fe K$\alpha$ line is smaller by a factor $(1+z)$
compared to the rest-frame value. Therefore, a restriction on the sample
redshift also helps to achieve a more homogeneous analysis. Next we required
that the total counts in the full HEG bandpass ($\sim 0.9-8$ keV) was $>1500$,
a condition which rejects spectra that have insufficient signal-to-noise
ratios for our purpose. Relaxing this criterion would only have admitted two
sources, PG 1404$+$226, and 1H 0707$-$495\. We then selected those AGN that
are known to have X-ray absorbing column densities less than $10^{23}\ \rm\
cm^{-2}$. The reason for this is that AGN with higher column densities have
X-ray spectra that are complex and measurements of the properties of even the
narrow Fe K$\alpha$ line core in such sources can become model-dependent.
Indeed, Murphy & Yaqoob (2009; hereafter MY09) showed, using monte carlo
simulations of X-ray reprocessing, that inclination-angle and geometrical
effects on the EW of the Fe K$\alpha$ line become important for column
densities greater than $\sim 10^{23}\ \rm\ cm^{-2}$. Although the column
density out of the line-of-sight could be larger than the line-of-sight column
density, it is the simplicity of the observed spectrum that is the driver of
the selection. We will present a study of heavily-absorbed AGN observed by the
Chandra HEG elsewhere. Our approach in the present paper is to perform a very
simple empirical analysis in order to obtain robust measurements of the basic
narrow Fe K$\alpha$ line core parameters that are not dependent on details of
how the continuum is modeled. Our selection criteria then populate our sample
with some sources that are formally classified as type 2 AGN, whereas the
study of YP04 included strictly only type 1 AGN.
We also excluded 15 $Chandra$ HETGS observations of M81 ($\sim$435 ks
exposure) from the study, as its very low luminosity and accretion rate set it
apart from the rest of the bright AGN in the sample (it is most often
classified as a LINER harboring a low-luminosity AGN). The bolometric
luminosity of M81 is only $\sim 10^{-5}$ of the Eddington luminosity (Young et
al. 2007). We note that the results of some HETGS results for M81, based on
$\sim 280$ ks exposure time have been presented by Young et al. (2007) who
found K-shell emission lines from He-like and H-like Fe in addition to the Fe
K$\alpha$ line at $\sim 6.4$ keV. Our final sample consists of 82 observations
of 36 unique AGN and includes all of the observations in YP04 (which we re-
analyzed for the present paper). We note that our sample includes 3C 273,
which is sometimes classified as a blazar. However, this source is variable
and often shows Seyfert-like properties (e.g. Grandi & Palumbo 2007).
The Chandra data for the sample were reduced and HEG spectra were made as
described in Yaqoob et al. (2003) and YP04. We used only the first orders of
the grating data (combining the positive and negative arms). The mean HEG
count rates ranged from $0.026\pm 0.001$ ct/s for the weakest source (PDS 456)
to $1.161\pm 0.006$ for the brightest source (IC 4329a). The exposure time
ranged from $\sim$20 ks to $\sim$172 ks per observation, but was $\sim 50-120$
ks for most of the sources. Nineteen sources were observed more than once, and
the largest net exposure time from summed data from observations of the same
source was $\sim 880$ ks (NGC 3783). The observations, identified by target
name, sequence number, and observation ID (“ObsID”), are listed in Table 1,
along with the net exposure times for the spectra. Further details of all of
the observations can be found in the Chandra data archive at
http://cda.harvard.edu/chaser/. Higher-level products, including lightcurves
and spectra for each observation can be found in the databases HotGAS
(http://hotgas.pha.jhu.edu), and TGCat (http://tgcat.mit.edu/). Background was
not subtracted since it is negligible over the energy range of interest (e.g.
see Yaqoob et al. 2003). Note that the systematic uncertainty in the HEG
wavelength scale is $\sim 433\ \rm km\ s^{-1}$ ($\sim 11$ eV) at 6.4 keV
111http://space.mit.edu/CXC/calib/hetgcal.html.
## 3 SPECTRAL FITTING METHODOLOGY AND RESULTS
The spectra were analyzed using the spectral-fitting package XSPEC (Arnaud
1996). Since we are interested in utilizing the highest possible spectral
resolution available, we used spectra binned at $0.0025\AA$, and this amply
oversamples the HEG resolution ($0.012\AA$ FWHM). The $C$-statistic was used
for minimization. All model parameters will be referred to the source frame.
Our method is simply to fit a simple continuum plus Gaussian emission-line
model over the 2–7 keV band for each spectrum. Above 7 keV the HEG effective
area rapidly decreases. We found, as in YP04, that if the energy band is
restricted any further the constraints on the Fe K$\alpha$ line parameters do
not improve because when the intrinsic line width is free there is degeneracy
of the line parameters with the continuum slope. In most cases a simple power-
law continuum was adequate, but for some sources an additional uniform,
neutral, absorber component was included (namely NGC 2110, MCG -5-23-16, NGC
4151 and NGC 5506). In no case was a column density greater than
4.3${}^{+0.4}_{-0.3}\times 10^{22}$ cm-2 required. Galactic absorption was not
included for any of the sources because such small column densities have
little effect above 2 keV. Thus, there were a maximum of six free parameters
in the model, namely the power-law slope and its overall normalization,
$\Gamma$, the column density, $N_{H}$, the centroid energy of the Gaussian
emission-line component, $E_{0}$, its flux, $I_{\rm Fe~{}K}$, and its width,
$\sigma_{\rm Fe~{}K}$. The approach of using an over-simplified continuum
model is necessitated by the limited bandpass of the HEG data ($\sim 2-7$ keV)
but since we are interested in the narrow core of the Fe K$\alpha$ emission
line, at the spectral resolution of the HEG, this is not restrictive.
Obviously, use of such an empirical model means that we should not assign a
physical meaning to $\Gamma$ and $N_{H}$.
The signal-to-noise ratio of the spectra showed a wide range so we followed a
systematic, two-step procedure that accounts for this. In the first round of
analysis we fixed the emission-line width, $\sigma_{\rm Fe~{}K}$, at 1 eV
(corresponding to $\sim 110\ \rm km\ s^{-1}$ FWHM at 6.4 keV), a value well
below the instrument resolution, because the line width could not be
constrained in all the data sets. Uniformly freezing the line width for all
the data sets then picks up the narrowest, unresolved core component of the
emission line for all the data sets. In the second round of analysis we
allowed the line width to float. Where multiple observations of a given source
were available we constructed and fitted spectra that were averaged over all
of the observations, in addition to fitting data from the individual
observations. Inter-observation variability will be discussed in §4.4.
The results of this first round of analysis are shown in Table 1 which shows
the derived equivalent width (EW) in addition to the other Fe K$\alpha$ line
parameters. Note that since the models were fitted by first folding through
the instrument response before comparing with the data, the derived line
parameters do not need to be corrected for instrumental response. We do not
give the best-fitting values of $\Gamma$ or $N_{H}$ in Table 1 because the
values derived using the simplistic continuum model are not physically
meaningful but are simply parameterizations. The 2–10 keV continuum fluxes and
luminosities shown in Table 1 were obtained by extrapolating the best-fitting
model up to 10 keV. We caution that such extrapolation could give inaccurate
fluxes and luminosities if the continuum shape is significantly different in
the 7–10 keV band compared to the extrapolated model. The fluxes are not
corrected for absorption, but the luminosities are. The $\Delta C$ values
shown in Table 1 correspond to the decrease in $C$ when the narrow, two-
parameter emission line was added to the continuum-only model, and is
therefore a measure of the significance of the emission line.
We found that in some cases (fourteen observations of ten sources, plus the
summed spectrum of IRAS $18325-5926$) the Fe K$\alpha$ line centroid energy
could not be constrained, and in such cases the centroid energy was fixed at
6.40 keV. In twelve of these data sets the Fe K$\alpha$ line was not detected
at a confidence level greater than 90% and for these cases only an upper limit
on the EW, and line flux, $I_{\rm Fe~{}K}$, could be obtained. Two sets of
statistical errors are given for the Fe K$\alpha$ line parameters in Table 1
for each spectral fit. The first set corresponds to 68% confidence ($\Delta
C=2.279$, or 0.989, depending on whether there were two parameters or one
parameter free respectively in the Gaussian component). These “1-$\sigma$”
errors are useful for performing standard statistical analyses on the model
parameters. However, as a more conservative measure, the 90% confidence range
(for the appropriate number of free parameters of the Gaussian component) for
each line parameter is also given in Table 1 (values in parentheses). For the
90% confidence ranges $\Delta C=4.605$ and 2.706 for two parameters and one
parameter free respectively.
We also found that in some sources that have multiple observations, the Fe
K$\alpha$ line parameters were sometimes better constrained from some of the
individual observations than from the averaged spectra because the latter may
contain contributions from data in which the Fe K$\alpha$ line was relatively
weaker in EW. Detailed interpretation of the results in Table 1 will be given
in §4.
In the second round of spectral fitting we allowed the intrinsic width of the
Gaussian emission-line component to be a free parameter. However, in
situations when the signal-to-noise ratio of the Fe K$\alpha$ emission line is
too poor, the Gaussian model emission-line component can become very broad as
it then begins to model the continuum, resulting in values of the width that
are not actually related to the physical width of the emission line. As a very
loose initial criterion, we rejected all cases in which a three-parameter
Gaussian component was detected with less than 95% confidence (corresponding
to rejecting fits that gave $\Delta C<7.8$). This rough criterion immediately
rejected fits for which the fits actually became unstable and left 26 unique
sources and 65 data sets, including 14 spectra averaged over multiple
observations. The results for all of the fits with $\Delta C>7.8$ are shown in
Table 2, in which the three-parameter, 68% statistical errors and 90%
confidence ranges on the line parameters are given. The Fe K$\alpha$ line
width is given as a FWHM in $\rm km\ s^{-1}$ rather than the Gaussian width,
$\sigma_{\rm Fe~{}K}$.
The next selection criterion we used was more specifically focussed on
determining whether the model width was a measure of the true line width.
Owing to the excellent spectral resolution of the HEG it is straightforward to
determine when the model FWHM is no longer a measure of the Fe K$\alpha$
emission-line width by reconciling the spectral data with the confidence
contours of $I_{\rm FeK\alpha}$versus the centroid energy, $E_{\rm
FeK\alpha}$. This can be seen in Fig. 1 and Fig. 2 which show, for a given
source, the rest-frame spectra in the Fe K region alongside the confidence
contours of $I_{\rm FeK\alpha}$versus $E_{\rm FeK\alpha}$. We have not shown
plots for all the data sets (spectra for all of the data sets can be found in
the databases mentioned in §1). For example, we have not shown plots for data
sets that have already been presented in YP04. Nor have we shown plots for
data sets in which the detection of the Fe K$\alpha$ line was marginal or
insignificant. Fig. 1 shows results for sources that have only one observation
whilst Fig. 2 shows time-averaged spectra for sources with multiple
observations, alongside confidence contours for the individual and time-
averaged data. We see in Fig. 1 and Fig. 2 that in some cases the 99%
confidence contours indicate a range in centroid energy that is clearly much
larger than the breadth of the emission-line feature that can be estimated
directly from the spectral plots. For example, for NGC 985 the joint, two-
parameter, 99% confidence contour of line intensity versus center energy
(solid line) is $\sim$600 eV wide. However, from the spectral plot in the Fe K
band, the Fe K$\alpha$ line clearly has a width less than $\sim 250$ eV. Thus
the 99% confidence bounds on line flux versus line centroid energy imply that
the Gaussian component is in fact modeling the continuum, resulting in very
large FWHM values that are not actually related to the physical width of the
emission line. For this case of NGC 985, we constructed a 99% confidence
contour (dotted line in Fig. 1) of the line intensity versus energy with the
Gaussian width fixed at 1 eV ($110\rm km\ s^{-1}$). This shows that the
centroid energy is constrained to be in the range $\sim$ 6.3 – 6.5 keV,
consistent with the physical width of the narrow core in the spectral plot.
Thus, by comparing the line intensity versus line energy confidence contours
with the spectra we determined that the FWHM constraints deduced from spectral
fits for 16 out of the 65 data sets were not reliable indicators of the Fe
K$\alpha$ line core intrinsic width (none of the 16 are data sets summed over
multiple observations). We found that the situations in which the Gaussian
width model parameter became an unreliable indicator of the emission-line
intrinsic width generally corresponded to a 90% confidence, two-parameter
upper limit on the FWHM greater than $\sim 15,000\rm\ km\ s^{-1}$. We note
that even for the cases where we can obtain a reliable measure of the Fe
K$\alpha$ line FWHM, the true line width may be less than the FWHM deduced
from our simplistic model-fitting because there may be blending from an
unresolved Compton-shoulder component and/or from several (low) ionization
states of Fe. Also, the difference in rest energy of $\sim 13$ eV of the
individual components of the Fe K$\alpha$ line ($K\alpha_{1}$ and
$K\alpha_{2}$) may increase the apparent FWHM if the line is modeled with a
single Gaussian model component. However, the separation of $K\alpha_{1}$ and
$K\alpha_{2}$ corresponds to $\sim 600\rm\ km\ s^{-1}$ (three times less than
the HEG resolution in the Fe K band), and considering the
$K\alpha_{1}$:$K\alpha_{2}$ branching ratio of 2:1, Yaqoob et al. (2001)
showed that artificial broadening is not a concern for line parameters and
signal-to-noise ratio that are typical of the HEG observations reported here.
In the present paper we are concerned only with the Fe K$\alpha$ line core
centered at $\sim 6.4$ keV, and not emission lines from highly ionized species
of Fe. Nevertheless, overlaid on the spectra in Fig. 1 are vertical dashed
lines marking the positions of the Fe xxv He-like triplet lines (the two
intercombination lines are shown separately), Fe xxvi Ly$\alpha$, Fe i
$K\beta$, and the Fe K-shell threshold absorption-edge energy. The values
adopted for these energies were from
NIST222http://physics.nist.gov/PhysRefData (He-like triplet); Pike et al. 1996
(Fe xxvi Ly$\alpha$); Palmeri et al. 2003 (Fe i $K\beta$), and Verner et al.
1996 (Fe K edge). Emission lines and absorption lines from highly ionized
species of Fe have indeed been reported in the literature for some of the same
data sets discussed in the present paper (e.g. see Bianchi et al. 2005). We
summarize such results from the literature in the appendix for each source as
appropriate, including any previous results on the 6.4 keV emission line that
are based on the same data that we have employed. We also give in the appendix
any unusual details and/or issues for particular data sets that are pertinent
to our analysis of the HETGS data.
## 4 PROPERTIES OF THE CORE OF THE FE K$\alpha$ LINE EMISSION
Table 3 summarizes various mean quantities from the Fe K$\alpha$ line
measurements, calculated in two different ways. In the first method we used
the measurements from individual observations and in the second method we used
measurements that are representative of properties per source. For the latter,
in most cases this utilized measurements from spectra averaged over multiple
observations where relevant, except for NGC 526, PG 0834, 3C 273, IRAS
13349$+$2438 and 3C 382. For these five sources the Fe K$\alpha$ line was
significantly detected in only one observation and combining observations led
to looser constraints on the Fe K$\alpha$ line parameters, as previously
explained in §3. Thus we used only the one observation for these five sources
that showed the best detection of the Fe K$\alpha$ line. This may bias the
results because we do not know if non-detections of the Fe K$\alpha$ are due
to variability. We caution that any sample properties derived using our
results should take account of such possible biases. We also caution that the
Chandra grating sample is subject to very peculiar and unquantifiable
selection effects because of the restrictions on the kind of sources that are
suitable for observations with gratings (or more precisely, which sources
proposal review panels judge to be suitable for observations with gratings).
Thus, the Chandra grating AGN archive is not suitable for unbiased population
studies. The principal purpose of the present work is to systematically
quantify the spectral parameters from the data.
### 4.1 LINE CENTROID ENERGY
From our analysis, we were able to measure the Fe K$\alpha$ line centroid
energy in 32 out of 36 unique sources for at least one spectrum (see Table 1
and Table 2). Table 3 summarizes four different weighted mean line centroid
energies. One pair of measurements was derived from individual observations
and the other pair was derived from per source measurements (as described
above). Each mean centroid energy was derived from spectral-fitting results in
which the intrinsic line width was fixed (Table 1), and from results in which
the intrinsic line width was not fixed in all the spectra (Table 2). Here, and
hereafter, for the calculation of the weighted mean of any quantity with
asymmetric errors, we simply assumed symmetric errors, using the largest 68%
confidence error from spectral fitting. For the line centroid values, 68 out
of 82 spectra contributed to the “per observation” means, and 32 sources
contributed to the “per source” means. It can be seen from Table 3 that all
four mean line centroid energies are within $-12$ eV and $+3$ eV of 6.400 keV
(including statistical bounds).
Note that the statistical errors on the mean centroid energies are 1 eV or
better but they may be biased by the brightest sources and largest exposure
times. A more useful measure of the dispersion in the line energies may be
gleaned from examining the distribution of energies. Fig. 3 shows histograms
of the Fe K$\alpha$ line centroid energy. Again, we show four histograms: Fig.
3(a) and Fig. 3(c) pertain to “per observation” results and Fig. 3(b) and Fig.
3(d) pertain to “per source” results. Fig. 3(a) and Fig. 3(b) pertain to Fe
K$\alpha$ line centroid energies measured with the intrinsic line width fixed
(Table 1), and Fig. 3(c) and Fig. 3(d) pertain to results obtained when the
intrinsic line width was not fixed in all the observations (Table 2). The
dashed and dotted lines correspond to histograms made from the 68% confidence
lower and upper limits on the line centroid energy respectively. The fits in
which the Fe K$\alpha$ line width was fixed gave line centroid energies that
are more reliable indicators of the peak line energy because the fits with the
line width free are more prone to the centroid energy being affected by the
shape of the line profile. All panels show that the histograms are not
Gaussian but sharply peaked at $\sim 6.4$ keV. There is not a significant
difference between the “per observation” and “per source” distributions,
within the statistical errors. For the “per observation” fits with the Fe
K$\alpha$ line width fixed, we found that $\sim 80\%$ of the best-fitting line
centroid energies lie in the range 6.38–6.43 keV, a spread of only 50 eV.
Another way of expressing our results is that if we take the highest signal-
to-noise measurement for each source (i.e. utilizing results from the summed
spectra only, for sources with multiple observations), we find that 21 out of
32 sources ($\sim 66\%$) have 68% confidence statistical bounds on the line
centroid energy that lie entirely in the range 6.38–6.43 keV. A similar
procedure also shows that 30 out of 32 sources ($\sim 94\%$) have 68%
confidence bounds on the line centroid energy that lie entirely in the range
6.35–6.47 keV. We note that we might have expected to observe additional peaks
in the centroid energy distribution blueward of 6.4 keV due to highly ionized
Fe. Although such emission lines have been detected in HEG data (e.g. NGC
7314, Yaqoob et al. 2003; NGC 7213, Bianchi et al. 2008), the HEG effective
area is already very small at 6.4 keV (only $\sim 20\ \rm cm^{-2}$) and drops
very rapidly at higher energies. Higher throughput detectors such as those
aboard XMM-Newton or Suzaku are more suitable for investigating the frequency
of occurence of highly ionized Fe emission lines.
We now examine those measurements that deviate significantly from the 6.4 keV
peak of the Fe K$\alpha$ line centroid energy distribution in Fig. 3. For 3C
273 and 4C 74.26, we obtained Fe K$\alpha$ line centroid energies lower than
those for the bulk of the measurements, and we note that even the 90%
confidence upper limits were less than 6.4 keV for the fits in which the line
width was fixed (see Table 1). For the fits in which the line width was free,
the corresponding upper limits were 6.49 keV and 6.39 keV for 3C 273 and 4C
74.26 respectively (see Table 2). However, for these two sources, the
detection of the Fe K$\alpha$ line was marginal: $C$ decreased by less than
9.3 when a narrow Gaussian was added to a power-law continuum-only model.
Thus, the lines were detected with only 99% confidence or less. Such low
centroid energies are not unphysical. For example they could be affected by
gravitational redshifts. We note that a weak broad Fe K$\alpha$ line has been
detected in 3C 273 by $XMM-Newton$ (Page et al. 2004a) and the 99% confidence
contour does not rule a line with a centroid energy in the range $\sim$
6.2-6.3 keV. Interestingly, a narrow component of the Fe K$\alpha$ line at 6.2
keV, in addition to an Fe K$\alpha$ line at $\sim 6.4$ keV, has been detected
in 4C 74.26 by $XMM-Newton$ (Ballantyne & Fabian 2005).
For PG 0844$+$346, we obtained a centroid energy for the Fe K$\alpha$ line
consistent with 6.4 keV from the fits with the line width fixed (see Table 1).
However, allowing the line width to float gave a centroid energy of $\sim 6.6$
keV, with a 90% confidence lower limit of 6.42 keV (see Table 2). The reason
for this apparently discrepant behavior is clear from the spectrum of PG
0844$+$346 in Fig. 1. The spectrum shows an emission line centered at $\sim
6.4$ keV and this is picked up in the fits for which the line width was fixed
(the measured energy was 6.364${}^{+0.007}_{-0.009}$ keV). The spectrum also
shows two additional peaks at higher energies, albeit with a low signal-to-
noise ratio. Allowing the single-line Gaussian model width to be free in the
fits then causes the line component to model all three narrow lines by
broadening the Gaussian.
In summary, we measured the centroid energy of the Fe K$\alpha$ emission line
in 32 out of 36 sources. In 30 out of the 32 sources the line centroid energy
lies in the range 6.35–6.47 keV, inclusive of the 68% confidence statistical
errors. We note that ionization states less than Fe xvii correspond to Fe
K$\alpha$ line energies lees than 6.43 keV (e.g. Palmeri et al. 2003; Mendoza
et al. 2004). When individual sources amongst the 30 are considered, the line
centroid energy can constrain the ionization state to be much lower than Fe
xvii in some cases. In the remaining 2 sources, 3C 273 and 4C 74.26, the line
centroid energy, including statistical errors, appears to be lower than 6.4
keV, but the detection is marginal in these two sources.
### 4.2 LINE EQUIVALENT WIDTH
From the spectral-fitting results for which the Fe K$\alpha$ line intrinsic
width was fixed (Table 1), 33 of the 36 sources have at least one spectrum
from which we could measure the EW with a non-zero 90% confidence lower limit
and a finite upper limit. The three sources for which only upper limits on the
EW could be obtained were Mkn 705, PDS 456, and IRAS 18325$-$5926\. In total,
70 out of 82 of the individual observations in Table 1 yielded bounded lower
and upper limits on the EW. We calculated weighted mean EW values in four
different ways (as we did for the line centroid energy in §4.1): i.e. from
“per observation” and “per source” values, each set obtained with the
intrinsic line width fixed in all spectral fits (Table 1) and with the
intrinsic line width free in some of the spectra (Table 2). The resulting mean
EW values are summarized in Table 3. It can be seen that the mean Fe K$\alpha$
EW is somewhat sensitive to how it is calculated, ranging from $42$ eV to 70
eV, with a statistical error of 4 eV or less. We note, however that a value of
$\sim 40$ eV could be interpreted as a fairly robust sample lower limit on the
EW of any unresolved core of the Fe K$\alpha$ line.
Fig. 4 shows histograms of the Fe K$\alpha$ line EW, again constructed in four
different ways. Fig. 4(a) and Fig. 4(c) pertain to “per observation” results
and Fig. 4(b) and Fig. 4(d) pertain to “per source” results. Fig. 4(a) and
Fig. 4(b) pertain to Fe K$\alpha$ line EWs measured with the intrinsic line
width fixed (Table 1), and Fig. 4(c) and Fig. 4(d) pertain to results obtained
when the intrinsic line width was not fixed in all the observations (Table 2).
The dashed and dotted histograms in Fig. 4 correspond to the distributions of
the 68% confidence lower and upper limits on EW, respectively. The shaded
histograms in both panels are the 68% upper limits on the EW for the 12
observations in which the EW could not be measured. For the “per observation”
and intrinsic line width fixed results, Table 1 and Fig. 4(a) show that the
maximum best-fitting EW of the Fe K$\alpha$ line core is $162$ eV, and $>$90%
of the measurements have a best-fitting EW less than 100 eV. We also found
that 79% of the measurements have a 68% confidence upper limit on the EW of
less than 100 eV (23 unique sources). Including the results from the fits with
the Fe K$\alpha$ line width free, we found that $\sim 70\%$ of the “per
observation” measurements have a best-fitting equivalent EW less than 100 eV
(Fig. 4(c)). Within the statistical errors, the histograms obtained from the
“per observation” results are not significantly different to the corresponding
“per source” histograms. Note that in Fig. 4(c) and Fig. 4(d), we do not show
the measurement for PG 0844$+$346, as its EW is artificially high ($\sim$ 600
eV) because it is not a true measure of the EW of the emission line at $\sim
6.4$ keV (see §3).
From a theoretical point of view, the Fe K$\alpha$ line EW depends on a number
of factors, including geometry, orientation, column density, and covering
factor of the line-emitting matter distribution, as well as element
abundances. Time delays between variations in the continuum level and the Fe
K$\alpha$ line flux also affect the EW measured during a given observation.
The sample EW results should therefore be interpreted in terms of a particular
geometry. The relatively small dispersion of the EW distribution that we
measure from the HEG data translates into a small dispersion in the parameters
mentioned above, but it is difficult to uniquely constrain these parameters
from the EW distribution due to degeneracy. In the context of the toroidal
X-ray reprocessor model of MY09, which subtends a solid angle of $2\pi$ at the
X-ray source, the measured EW distribution is consistent with the MY09 model
if the mean of the EW distribution corresponds to column densities greater
than $\sim 2\times 10^{23}\rm\ cm^{-2}$ (see MY09). This column density does
not refer to the line-of-sight value, but rather to the angle-average over all
incident X-ray continuum radiation. Comparison of the HEG results with the
toroidal reprocessor models of Ghisellini, Haardt, & Matt (1994) and Ikeda,
Awaki, & Terashima (2009) leads to similar conclusions. The upper bound on the
column density is not constrained because, for situations in which the Fe
K$\alpha$ line is observed for lines of sight that intercept a column density
$<10^{23}\rm\ cm^{-2}$, the EW attains a maximum for an angle-averaged column
density of $\sim 10^{24}\rm\ cm^{-2}$, above which the EW decreases again as
the line-emitter becomes Compton-thick (e.g. se MY09). Increasing the covering
factor of the reprocessor can increase the EW of the Fe K$\alpha$ line
observed in reflection but there is a trade-off because as the covering factor
approaches unity, the projected area of the reflection region decreases and
emission-line photons are more prone to being impeded from escaping the
medium. Ikeda et al. (2009) found that the EW is greatest for covering factors
factors of $\sim 0.7-0.9$ but does not exceed $\sim 180$ eV in their model for
cosmic abundances and a power-law photon index of $1.9$. In principle, the
shape and relative magnitude of the Fe K$\alpha$ line Compton shoulder could
determine whether the reprocessor is Compton-thin or Compton-thick but this is
challenging due to the limited signal-to-noise ratio of the data and also
requires more sophisticated modeling. Such an investigation will be reported
in future work. So far, all analyses with respect to the Compton shoulder and
HEG AGN data have employed ad-hoc models (e.g. Kaspi et al. 2002; Yaqoob et
al. 2005) so they do not yield a meaningful physical interpretation.
### 4.3 INTRINSIC LINE WIDTH
The location of the medium responsible for the core of the Fe K$\alpha$
emission line can potentially be constrained by the measurements of the line
intrinsic width. The weighted mean FWHM of the Fe K$\alpha$ line cores for the
53 individual data sets (27 unique sources) for which it could be measured
(Table 2), is $2060\pm 230\ \rm km\ s^{-1}$. This includes the two sources
(Mrk 290 and 4C 74.26) for which the Fe K$\alpha$ line FWHM could only be
constrained from the summed spectra. We also calculated the weighted mean FWHM
from “per source” measurements and found a similar value of $2200\pm 220\ \rm
km\ s^{-1}$ (see Table 3).
In Table 2 are values of the ${\rm H}_{\beta}$ FWHM compiled from the
literature. Comparing the Fe K$\alpha$ line FWHM with that of the ${\rm
H}_{\beta}$ line can potentially give a direct indication of the location of
the Fe K$\alpha$ line-emitting region relative to the optical broad-line
region. A direct comparison of the Fe K$\alpha$ line width with optical BLR
line widths was not attempted in YP04 because the number of sources with
sufficiently high quality Fe K$\alpha$ line-width measurements was too small.
Nevertheless, Nandra (2006) using the YP04 results, supplemented by a few
other HEG measurements from the literature, examined the relation between the
FWHM of Fe K$\alpha$ and ${\rm H}_{\beta}$. The results were ambiguous, the
data allowing for an origin of the Fe K$\alpha$ line anywhere from the BLR to
parsec-scale distances from the putative central black hole. Moreover, some of
the HEG measurements for the Fe K$\alpha$ line FWHM compiled from the
literature were problematic. For example, for MR 2251$-$178, Gibson et al.
(2005) reported an upper limit on the Fe K$\alpha$ line FWHM of $1530\ \rm km\
s^{-1}$ and Nandra et al. (2006) erroneously quoted and used as $650\ \rm km\
s^{-1}$ (Gibson et al. 2005 reported a $\sigma$ of $650\ \rm km\ s^{-1}$, not
FWHM). In our uniform analysis, we found that the HEG data for MR 2251$-$178
were so poor that a meaningful upper limit on the Fe K$\alpha$ FWHM cannot
even be measured and therefore we have reported only results for a fit with
the line width fixed at well below the HEG resolution (Table 1).
A notable example for which a meaningful comparison between the Fe K$\alpha$
and ${\rm H}_{\beta}$ line widths has been reported using HEG data is NGC 7213
(Bianchi et al. 2008). In this case the FWHM of both lines are consistent with
each other ($\sim 2500\ \rm km\ s^{-1}$), implying an origin of the Fe
K$\alpha$ line in the BLR for NGC 7213. Our sample includes NGC 7213 and our
analysis (see Table 2) confirms the results of Bianchi et al. (2008).
Utilizing all of the results from our uniform analysis of the HEG sample for
which the Fe K$\alpha$ line width could be at least loosely constrained (Table
2), we have plotted in Fig. 5 the Fe K$\alpha$ line FWHM against the ${\rm
H}_{\beta}$ FWHM. The dashed line corresponds to the two line widths being
equal. The statistical errors shown correspond to 68% confidence. We have
distinguished 12 sources in Fig. 5 by empty circles (as opposed to filled
circles) that provide the very best statistical constraints on the Fe
K$\alpha$ line FWHM in our sample. The next best measurement of the FWHM would
be NGC 985, but we note that its 99% two-parameter confidence contour for Fe
K$\alpha$ line flux versus FWHM did not close before the Gaussian component
began to model the continuum. In Fig. 5, points that lie above the dashed line
at some level of confidence mean that an origin in the BLR of at least part of
the Fe K$\alpha$ line is not ruled out, but contributions from further out
than the BLR are not ruled out either (at the appropriate level of
confidence). A larger FWHM for the Fe K$\alpha$ line compared to the ${\rm
H}_{\beta}$ line could either mean a genuine contribution to the Fe K$\alpha$
line from matter closer to the central black hole than the BLR, or it could
mean that there is a contribution from an unresolved Compton shoulder or from
part of a broader disk line. Points that lie below the dashed line in Fig. 5
at some level of confidence place stronger constraints on the origin of the
narrow Fe K$\alpha$ line because in that case, whatever physical sources of
broadening are affecting the Fe K$\alpha$ line, it must originate in a region
that lies further from the central black hole than BLR.
Standard tests for assessing the significance of any possible correlation
between FWHM(Fe K$\alpha$ ) and FWHM(${\rm H}_{\beta}$ ), such as the Spearman
Rank correlation coefficient, are problematic because they do not take account
of measurement errors on FWHM(Fe K$\alpha$ ), which can be large. Assessing
the effect of measurement errors on such correlation coefficients properly
requires extensive and realistic simulations of the data and the spectral-
fitting process. Instead, we used the $\chi^{2}$ statistic to fit a straight
line to the FWHM values of the Fe K$\alpha$ and ${\rm H}_{\beta}$ lines.
Although we are forced to assume a relationship between the two quantities,
any correlation would still manifest itself. In the fitting we explicitly took
took into account the statistical errors on the Fe K$\alpha$ line widths,
using the average of the 68% confidence upper and lower errors. We found that
FWHM(Fe K$\alpha$ )=(0.04$\pm 0.13)\times$ FWHM(${\rm H}_{\beta}$ )+(2130$\pm
550$), with $\chi^{2}=17.1$ for 21 degrees of freedom. The reduced $\chi^{2}$
value $<1$ then means that indeed a more complicated model is not warranted.
More importantly, we see that even the $1\sigma$ errors on the slope include a
slope value of zero (corresponding to the case that all the FWHM values are
consistent with a constant, independent of the FWHM of the ${\rm H}_{\beta}$
line). Therefore, we find no evidence of a correlation between the Fe
K$\alpha$ and ${\rm H}_{\beta}$ line widths, consistent with the conclusion of
Nandra et al. (2006). However, Nandra et al. (2006) interpreted the lack of a
correlations in terms of the narrow Fe K$\alpha$ line not originating in the
BLR, but we now know that in some cases this is not true (e.g. NGC 7213,
Bianchi et al. 2008). We shall see below that our results in fact show that
the location of the Fe K$\alpha$ line emitter relative to the BLR appears to
be genuinely different from source to source.
From our spectral fits to the subset of HEG data with the Fe K$\alpha$ line
width free (Table 2) we constructed joint 68%, 90%, 99% confidence contours of
the Fe K$\alpha$ line EW versus the ratio of the Fe K$\alpha$ FWHM to the
${\rm H}_{\beta}$ FWHM. These are shown in Fig. 6 and a variety of behavior is
displayed. We found cases in which this FWHM ratio was, at the two-parameter
99% confidence level, was less than 1 (NGC 3783, NGC 4151 and NGC 5548),
greater than 1 (MCG $-$6-30-15), or consistent with 1 (3C 120, NGC 2110, MCG
$-$5-23-16, NGC 3516, NGC 5506, Mrk 509, NGC 7213, and NGC 7469). Thus, it
appears that the location of the Fe K$\alpha$ line relative to the location of
the ${\rm H}_{\beta}$ line-emitting region may be different from source to
source. For our limited-sized sample the Fe K$\alpha$ line-emitting region
size could be up to a factor $\sim 5$ larger than the ${\rm H}_{\beta}$ line-
emitting region (NGC 4151 – see Fig. 6). We note that the putative parsec-
scale obscuring torus that is required by AGN unification schemes, and that
has always been a strong contender for any Fe K$\alpha$ line emission beyond
the BLR, may be smaller than traditionally thought. In particular, Gaskell,
Goosmann, & Klimek (2008) argue that there is considerable observational
evidence that the BLR itself has a toroidal structure, and that there may be
no distinct boundary between the BLR and the classical parsec-scale torus. Our
results from the Chandra HEG data do not conflict with such a scenario.
From joint confidence contours of Fe K$\alpha$ line intensity versus FWHM we
can determine whether the line is resolved from a given data set if the
contour (at some level of confidence) does not cross the $\rm FWHM=0$ axis. We
found that at 99% confidence (two parameters), the Chandra HEG resolves the
narrow component of the Fe K$\alpha$ emission in 15 sources, namely, F9, NGC
2110, MCG $-$5-23-16, NGC 3516, NGC 3783, NGC 4051, NGC 4151, MCG $-$6-30-15,
IRAS 13349$+$2438, IC 4329A, Mrk 279, NGC 5548, E1821$+$643, NGC 7469 and NGC
7213. We do not include PG 0834$+$346 here, since the single-Gaussian fit with
the line width free does not pick up the narrow component at $\sim$ 6.4 keV
(see §3). We caution that in general an emission line that is resolved by the
HEG may indicate complexity as opposed to a simple, single emission line.
### 4.4 LINE FLUX
If the Fe K$\alpha$ line originates in a matter distribution whose light-
crossing time is much greater than the typical timescale of variability of the
X-ray continuum, the variability of the line flux will be suppressed. The line
flux may then even be constant (within statistical errors) and correspond to
some historically-averaged continuum level. The sources in our HEG sample that
have multiple observations enable us to investigate the time-dependence of the
Fe K$\alpha$ line flux. The spectral resolution of the HEG currently allows
the best isolation of the narrow Fe K$\alpha$ line for the largest sample
compared to previous studies. In Fig. 2 we showed the Fe K$\alpha$ line
intensity versus centroid energy 99%, two-parameter confidence contours for
each source that has multiple observations. The contours for NGC 3783 were
shown in Yaqoob et al. (2005) and are not shown again in Fig. 2. In no source
did we find evidence for variability of the Fe K$\alpha$ line flux at 99%
confidence or greater. However, it is important to note that the 99%
confidence regions in some cases cover a large range in line flux due to
limited signal-to-noise ratio. However, we can say that in our HEG sample, the
data are consistent with no variability of the Fe K$\alpha$ line but more
sensitive instrumentation is required to reduce the statistical errors.
## 5 X-RAY BALDWIN EFFECT
The so-called X-ray Baldwin effect, a possible anti-correlation between the Fe
K$\alpha$ line EW and X-ray luminosity, has been discussed at length in the
literature (e.g. Iwasawa & Taniguchi 1993; Nandra et al. 1997; Page et al.
2004b; Jiang, Wang, & Wang 2006; Bianchi et al. 2007; Winter et al. 2009).
These studies have found some evidence for an X-ray Baldwin effect albeit with
significant scatter, but the latter two studies have found that the Fe
K$\alpha$ line EW appears to be more strongly anti-correlated with the ratio
of X-ray luminosity to Eddington luminosity ($L_{x}/L_{\rm Edd}$, a proxy for
the accretion rate). However, Winter et al. (2009) found that the X-ray
Baldwin effect was only significant if the EW and $L_{x}/L_{\rm EDD}$ values
were binned, and the formal significance of the anti-correlation depended
strongly on the details of the binning procedure. Except for some HETGS data
used by Jiang et al. (2006), all other studies of the X-ray Baldwin effect to
date have been based on data that has a spectral resolution of $\sim 7000\ \rm
km\ s^{-1}$ FWHM or worse. Therefore, it is not clear whether the Fe K$\alpha$
line parameters in these studies correspond to contributions from line
emission blended from completely different origins (e.g. distant-matter and
accretion-disk components). Using our sample that consists only of HEG data,
we can investigate the X-ray Baldwin effect with a spectral resolution in the
Fe K band that is nearly four times better than in previous studies, and
therefore provide the best isolation of the narrow core that is possible with
current instrumentation.
For this purpose we used our spectral-fitting results obtained with the Fe
K$\alpha$ line width fixed at 1 eV, well below the HEG resolution, in order to
obtain a uniform set of Fe K$\alpha$ line EW measurements for the largest
number of sources (see Table 1). We examined correlations using both the “per
observation” results and the “per source” results. Measurements for the latter
were derived from only one spectrum per source, which in some cases was the
average spectrum, as described in §4. These values of EW are plotted against
$L_{x}$ in Fig. 7(a) and Fig. 7(c), and against $L_{x}/L_{\rm EDD}$ in Fig.
7(b) and Fig. 7(d). Note that in Fig. 7 we have shown all EW measurements
whether or not they are only upper limits, even though upper limits will not
be used in the quantitative analysis. The Eddington luminosity, $L_{\rm Edd}$,
is computed from $M_{\rm BH}\times 1.3\times 10^{38}$ erg s-1, where $M_{\rm
BH}$ is the mass of the central black hole. Values of $M_{\rm BH}$ are from
Zhou & Wang (2005), Bianchi et al. (2007), and Wang et al. (2009). As a proxy
for the accretion rate we use the ratio of ${L_{\rm 2-10keV}/L_{\rm Edd}}$
(see e.g. Vasudevan & Fabian 2009 for the correspondence between and X-ray
luminosity and bolometric luminosity). We were not able to find reliable mass
estimates for Mrk 705 and IRAS 18325$-$5926 so these sources were excluded
from any analyses involving $L_{\rm Edd}$. The statistical errors shown in
Fig. 7 are 68% confidence for two free (Gaussian) parameters. It can be seen
that, despite better isolation of the Fe K$\alpha$ line core, there is still
significant scatter in the diagrams.
In order to formally assess the significance of any correlation, standard
methods that do not take account of the statistical errors on the EW, such as
the Spearman Rank correlation coefficient are problematic. This is because, in
the type of analysis presented here, and in previous works on the X-ray
Baldwin effect, the actual best-fitting values of EW are not in themselves
meaningful. It is the statistical errors on the EW that are the important
quantities. Assessing the effect of measurement errors on such correlation
coefficients properly requires extensive and realistic simulations of the data
and the spectral-fitting process. On the other hand, the $\chi^{2}$ statistic
does take account of the statistical errors on the EW. Although we are forced
to assume a form of the relationship between the EW and $L_{x}$ if we use
$\chi^{2}$, it can be seen from Fig. 7 that the quality of the data do not
support constraining a more complex relationship. We therefore fitted a
straight line to $\log{EW}$ versus $\log{L_{x}}$ using the $\chi^{2}$ fit
statistic (i.e. a power-law function for EW versus $L_{x}$). Data points that
only had upper limits on the EW were not included. In the fitting we took into
account the statistical errors on the EW, using the average of the 68%
confidence upper and lower errors.
The results of the $\chi^{2}$ analysis are shown in Table 4. For each of the
four cases (“per observation”, “per source”, and EW versus $L_{x}$ or
$L_{x}/L_{\rm Edd}$) we show the best-fitting value of $\chi^{2}$, the
intercept and slope of the best-fitting line, as well as the the 68%
confidence and 99% confidence one-parameter errors on the slope. The latter
error bounds were determined by varying the slope, whilst allowing the
intercept to float, and determining the bounds on the slope for
$\Delta\chi^{2}=0.989$ and 6.635 for 68% and 99% confidence respectively. If
the EW is indeed anti-correlated with either $L_{x}$ or $L_{x}/L_{\rm Edd}$ we
would expect that the slope of the line is significantly different from zero.
Therefore, in Table 4 we also show values of $\Delta\chi^{2}$ obtained when
the slope is forced to be zero, as well as the corresponding significance that
the slope is non-zero. We found that the “per observation” results gave a
stronger anti-correlation than the “per source” results, for both the EW
versus $L_{x}$ and EW versus $L_{x}/L_{\rm Edd}$ relations. Quantitatively,
the “per observation” results show a significance of $6.05-6.27\sigma$ for a
non-zero slope, as opposed to $\sim 3.08-3.24\sigma$ for the “per source”
results. The best-fitting slopes for the latter are about half of the
corresponding values of the “per observation” results. We caution that the
absolute significance values should not be interpreted literally since we do
not know the form of the functional relationship between EW and $L_{x}$. Table
4 also shows that there is no significant difference in the $\chi^{2}$
analysis results on whether we examine the relation of EW between $L_{x}$ or
$L_{x}/L_{\rm Edd}$, and that is true whether we consider the “per
observation” or “per source” results. Both our “per observation” and “per
source” results for the slope of the EW versus $L_{x}$ relation are formally
consistent, within the uncertainties, with that found by Page et al. (2004b)
who reported $EW~{}\propto~{}L^{-0.17\pm 0.08}$). In addition, our results for
the slope of the relation between EW and $L_{x}/L_{\rm Edd}$ is formally
consistent with that obtained by Bianchi et al. (2007) (${\rm
EW}~{}\propto~{}(L_{\rm bol}/L_{\rm Edd})^{-0.19\pm 0.05}$). We note that the
latter study of Bianchi et al. (2007) excluded sources with high radio-
loudness and still found a significant Baldwin effect.
Our results seem to confirm the X-ray Baldwin effect. There are several
factors that could produce an anti-correlation of the EW of the Fe K$\alpha$
line and the intrinsic X-ray continuum luminosity. A decrease of covering
factor and/or the column density of line-emitting with increasing X-ray
continuum luminosity likely are the most important factors. Another
possibility is that the line-emitting material becomes more and more ionized
as the X-ray luminosity increases, leaving less low-ionization material to
produce the Fe K$\alpha$ line at $\sim 6.4$ keV. Unfortunately the data cannot
yet distinguish between these scenarios. A complete understand of the Baldwin
effect should also take into consideration the fact that the Fe K$\alpha$ line
EW in individual sources can vary by more than a factor of two (if the line
intensity does not respond to large-amplitude continuum variations), although
simulations based on the simplest assumptions yield an anti-correlation
between EW and continuum luminosity weaker than observed ones and with large
scattering ($EW~{}\propto~{}L^{-0.05\pm 0.05}$, Jiang et al. 2006).
## 6 SUMMARY
We have presented an empirical and uniform analysis of the narrow core of the
Fe K$\alpha$ emission line in a sample of 82 observations of 36 AGNs with low
to moderately low X-ray absorption ($N_{H}<10^{23}\ \rm cm^{-2}$), using
Chandra HEG data. The Fe K$\alpha$ line was detected in 33 sources, and its
centroid energy was measured in 32 sources (68 observations). The distribution
in the centroid energy is strongly peaked around $\sim 6.4$ keV, with over 80%
of the measurements lying in the range 6.38–6.43 keV. Including the
statistical errors and utilizing the best measurements for each source, the
line centroid energy lies entirely in the range 6.35–6.47 keV for 30 out of 32
sources. Thus we confirm, for the largest sample of AGN observed with such a
high spectral resolution (FWHM $\sim 1860\ \rm km\ s^{-1}$ at 6.4 keV), the
ubiquity of the narrow core of Fe K$\alpha$ line, and its preferred origin in
cool, neutral or only mildly-ionized matter.
The equivalent width (EW) of the core of the Fe K$\alpha$ line was constrained
in 70 out of 82 observations, with only upper limits obtained from the
remaining 12 spectra. The weighted mean EW was $53\pm 3$ eV, and $\sim 70\%$
of the individual measurements had a 68% confidence upper limit on the EW of
less than 100 eV. Similar results were obtained when considering the EW
distribution by source, although the weighted mean was somewhat higher from
measurements that allowed the intrinsic line width to be free ($70\pm 4$ eV).
The EW distribution can be produced by both Compton-thin and Compton-thick
matter distributions and a more detailed analysis with a physical model is
required to distinguish between the two scenarios. We also presented
measurements of the flux of the core of the Fe K$\alpha$ line and found that
for sources that had multiple observations, there was no case in which the
line flux varied between observations, within the statistical errors.
The intrinsic width of the core of the Fe K$\alpha$ line was measured for 27
sources (53 observations) and we obtained a weighted mean value of FWHM
$=2060\pm 230\ \rm km\ s^{-1}$ (or $=2200\pm 220\ \rm km\ s^{-1}$ when
considering measurements by source, not by observation). Of the 27 sources, 12
yielded 99% confidence, two-parameter contours of line flux versus FWHM that
were good enough to investigate the relation between the width of the Fe
K$\alpha$ line and the width of the H$\beta$ line (or Br$\alpha$ for one of
the sources). We found that the ratio of the X-ray to optical line width
varies from source to source. The 99% confidence, two-parameter upper limit
lies in the range $\sim 0.5-4$ for the 12 sources. This means that
contributions to the flux of the core of the Fe K$\alpha$ line are allowed
down to a factor $\sim 0.7-2$ times the radius of the optical BLR. The upper
limit on the size of the X-ray line emitter is not constrained because line
flux contributions from large, parsec-scale distances could be unresolved by
the HEG. We note that our results are suggestive of the fact that the location
of the X-ray line-emitting region relative to the BLR may actually be
different in different sources. These conclusions are subject to the caveat
that derivation of the true velocity width of the Fe K$\alpha$ line core
requires a proper physical model, such as that of MY09, that includes a
possible Compton shoulder. This will be the subject of future work. However,
we note that such an analysis can only reduce the derived velocity widths of
the Fe K$\alpha$ lines. Finally, having isolated the narrow core of the Fe
K$\alpha$ line with the best available spectral resolution we confirm the
anti-correlation (albeit with a large scatter) between the line EW and X-ray
luminosity, $L_{x}$ (the X-ray Baldwin effect), and between the line EW and
$L_{x}/L_{\rm Edd}$.
We thank the referee for his/her useful comments. Partial support for this
work was provided by NASA through Chandra Award AR8-9012X, issued by the
Chandra X-ray Observatory Center, which is operated by the Smithsonian
Astrophysical Observatory for and on behalf of the NASA under contract
NAS8-39073. X.W.S. and J.X.W. acknowledge support from Chinese National
Science Foundation (Grant No. 10825312, 10773010), and Knowledge Innovation
Program of CAS (Grant No. KJCX2-YW-T05). This research made use of the HEASARC
online data archive services, supported by NASA/GSFC. This research has made
use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet
Propulsion Laboratory, California Institute of Technology, under contract with
NASA. The authors are grateful to the Chandra instrument and operations teams
for making these observations possible.
## 7 APPENDIX: NOTES ON INDIVIDUAL SOURCES
In this section we give, for each source in our sample, particular additional
details of the analysis and/or results where necessary. We also summarize
briefly any previously-published HEG results in the Fe K band that are based
on the same data. Our intention is not to review observations by other
instruments.
F9. Chandra HEG results were reported in YP04 and the new analysis is
consistent with the previous results. Note that the very large upper limit on
the intrinsic Fe K$\alpha$ line width (Table 2) is unphysical since such a
broad Gaussian component is clearly modeling the underlying spectrum (see
discussion in YP04). The HEG data show marginal evidence of an emission line
at $\sim 6.9$ keV.
NGC 526a. No results from either of the two observations have been previously
published. The Fe K$\alpha$ line is detected in only one of the observations,
and the detection is marginal. Consequently, the FWHM of the line could not be
constrained.
Mrk 590. Results from the Chandra HEG data have been presented by Longinotti
et al. (2007), who reported the detection of a narrow Fe K$\alpha$ line with
$E_{0}$$=6.40^{+0.04}_{-0.03}$ keV, $\sigma_{\rm Fe~{}K}$$=47^{+58}_{-24}$ eV,
and ${\rm EW}=160^{+118}_{-78}$ eV. Our best-fitting Fe K$\alpha$ line
parameters (Table 1) are in good agreement with those measured by Longinotti
et al. (2007). Fig. 1 shows that the large 99% confidence region for $I_{\rm
Fe~{}K}$ versus $E_{0}$ indicates that the fits in which the line width was
free do not provide a reliable measure of the intrinsic line width.
NGC 985. Although Krongold et al. (2005) reported results from the Chandra
HETGS observation, they combined HEG and MEG data and did not report results
on the Fe K$\alpha$ line emission.
ESO 198$-$G24. No results from any of the two Chandra HETGS observations have
been previously reported. We obtained a significant detection of the narrow Fe
K$\alpha$ line from only one of the observations (Table 1 and Table 2).
3C 120. Results on the Fe K$\alpha$ line from the Chandra HETGS observation of
this source have been reported in YP04 and the new results presented here are
consistent with the previous ones. The HEG data show a marginal detection of
an emission line at $\sim 6.9$ keV (YP04).
NGC 2110. The results from the four Chandra HETGS observations were presented
by Evans et al. (2007), who measured the narrow Fe K$\alpha$ line parameters
$E_{0}$ $=6.397\pm 0.007$ keV, and ${\rm EW}=81^{+27}_{-30}$ eV, consistent
with our results. Note that in the second observation the line width could not
be constrained so the $I_{\rm Fe~{}K}$ versus $E_{0}$ 99% confidence contour
for that observation in Fig. 2 was constructed with the line width fixed at 1
eV (dot-dashed line).
PG 0844$+$349. No results from any of the three Chandra HETGS observations of
this source have been reported previously. The detection of the Fe K$\alpha$
line at $\sim 6.4$ keV is marginal, and there is also marginal evidence of
emission lines due to He-like and H-like Fe. When fitted with a single-
Gaussian model, the presence of three narrow emission lines causes the
Gaussian intrinsic width to become large as it tries to account for all three
lines. Therefore, the most reliable values of $E_{0}$ and $I_{\rm Fe~{}K}$ are
those obtained from fits in which the line width was fixed.
MCG $-$5-23-16. This source was observed by the Chandra HETGS on three
occasions. Results from the first observation have been presented by Balestra
et al. (2004), who found, from single-Gaussian fits to the narrow Fe K$\alpha$
line, $E_{0}$$=6.38\pm 0.02$ keV, ${\rm EW}=70\pm 28$ eV, and FWHM $\leq$ 6500
km s-1 (at 99% confidence). Results from the remaining two observations were
presented by Braito et al. (2007), who reported Fe K$\alpha$ narrow-line
parameters for the mean (time-averaged) spectrum of
$E_{0}$$=6.41^{+0.02}_{-0.01}$ keV, and ${\rm EW}=61^{+17}_{-23}$ eV. These
correspond to the case when the line width was fixed at a value less than the
instrument resolution and Braito et al. (2007) found that if the line width
was allowed to float, the constraints were sensitive to details of the
continuum and relativistic disk-line model. Our results are consistent with
previously published results; our simple continuum model and omission of a
broad relativistic line in the fits means that our measurements of the line
width should be interpreted as empirical indicators only. Note that in the
second observation the 99% confidence contour of the Fe K$\alpha$ line
intensity versus energy was not closed when the line width was a free
parameter. Thus, for this observation, we show in Fig. 2 the 99% confidence
contour for the line width fixed at 1 eV (thin solid line).
Mrk 705. The signal-to-noise ratio of the data in this observation was very
poor. Previous results have been reported by Gallo et al. (2005) who obtained
an upper limit of 149 eV on the EW of an emission line with a centroid energy
fixed at 6.4 keV. This is consistent with our analysis (Table 1).
NGC 3227. Results from the Chandra HETGS observation of this source have been
reported in YP04 and the new results presented here are consistent with the
previous ones.
NGC 3516. There were eight Chandra HETGS observations of this source. Results
from the first three observations were reported in YP04. In the present paper
we report on the analysis of five new observations that were performed in
October 2006. Results from the same datasets have also been presented by
Turner et al. (2008), who reported the detection of a narrow Fe K$\alpha$
emission line with $E_{0}$ $=6.404\pm 0.019$ keV, $\sigma_{\rm Fe~{}K}$
$=40^{+10}_{-15}$ eV, and ${\rm EW}\sim 94$ eV (the statistical error was not
given). In addition, redshifted emission-line features have been reported in
some of the HEG data (Turner et al. 2002), as well as H-like and He-like Fe
emission and absorption features (Turner et al. 2008). In the present paper we
are concerned only with the Fe K$\alpha$ emission line centered at $\sim 6.4$
keV and our results are consistent with those of Turner et al. (2008). Due to
the short exposure time of the last observation, the Fe K$\alpha$ line was
detected at less than 99% confidence (for two free Gaussian parameters). Thus,
we do not show the contour of the line intensity versus energy in Fig. 2 for
this observation.
NGC 3783. Detailed results from the six Chandra HETGS observations of this
source have been presented by Yaqoob et al. (2005), and Kaspi et al. (2001,
2002) and our re-analysis is consistent with the previous results.
NGC 4051. Results from the Chandra HETGS observation of this source have been
reported by Collinge et al. (2001), who obtained $E_{0}$
$=6.41^{+0.01}_{-0.01}$ keV, ${\rm EW}=158^{+51}_{-47}$ eV, and FWHM
$<2800\rm\ km\ s^{-1}$ for the core of the narrow Fe K$\alpha$ line. Our
analysis is consistent with the previous results, except that when the line
width was free in the fits it becomes larger than the narrow core in the data
(Table 2), and this is consistent with the results reported in YP04.
Absorption features due to He-like and H-like Fe have also been noted in the
HEG data for NGC 4051 (Collinge et al. 2001; YP04).
NGC 4151. This source was observed five times with the Chandra HETGS. Results
from the first observation have been reported by Ogle et al. (2000) who
obtained ${\rm EW}=160\pm 20$ eV, consistent with our measurement (Table 1)
and a FWHM of $1800\pm 200\rm\ km\ s^{-1}$, also consistent with our analysis
(Table 2). The line centroid energy was not measured. Results for narrow Fe
K$\alpha$ line parameters measured by the HEG for the remaining four
observations have not been previously published.
Mrk 766. Results for the Chandra HETGS observation of this source have been
reported in YP04 and our re-analysis is consistent with the previous results.
The HEG data show marginal evidence of an emission line at $\sim 6.9$ keV
(YP04).
3C 273. A narrow Fe K$\alpha$ line was detected in only one of the 7 Chandra
HETGS observations of this source. Measurements of the Fe K$\alpha$ line core
from Chandra HEG data have not been previously reported.
NGC 4593. Results for the Chandra HETGS observation of this source have been
reported in YP04 and our re-analysis is consistent with the previous results.
The HEG data show marginal evidence of an emission line at $\sim 6.9$ keV
(YP04).
MCG $-$6-30-15. This source was observed by the Chandra HETGS five times.
Results from the first observation have been presented by Lee et al. (2002)
and YP04. The results for the other four observations were presented by Young
et al. (2005), who reported narrow Fe K$\alpha$ emission-line parameters from
the time-averaged spectrum of $E_{0}$ $=6.393^{+0.106}_{-0.014}$ keV, ${\rm
EW}=18^{+11}_{-8}$ eV, and a FWHM $<4700\rm km\ s^{-1}$. In the present
analysis, only one out of four new observations had a significant detection of
the narrow Fe K$\alpha$ emission line. From our empirical analysis we obtained
a larger EW and FWHM than Young et al. (2005). This could be attributed to a
contribution to the Fe K$\alpha$ line core from an underlying disk-line
component and/or the difference could be due to a complex continuum. However,
there is a large range of possible models but in our analysis the simple
empirical model is appropriate because the results can be compared directly to
those from the other sources in our sample. The EW and FWHM obtained from more
complex models will always be less than the values obtained from the empirical
modeling so the latter provide useful upper bounds. He-like and H-like Fe
absorption features have been reported in the HEG data by Young et al. (2005).
IRAS 13349$+$2438. This source was observed twice with the Chandra HETGS but
no results for the Fe K$\alpha$ line have been previously published. A
significant detection of the narrow Fe K$\alpha$ line was obtained only from
the second observation (see Table 1).
IC 4329A McKernan & Yaqoob (2004) reported the detection of complex Fe K line
emission from the Chandra HETGS observation of this source. One peak is
centered at $\sim 6.3$ keV with a FWHM $20830^{+10110}_{-7375}\rm\ km\ s^{-1}$
and an EW of $110^{+46}_{-40}$ eV. The other peak is at $\sim 6.9$ keV with a
FWHM $\sim 4000\rm\ km\ s^{-1}$ and an EW of $\sim 40$ eV (probably due to Fe
xxvi Ly$\alpha$). In the present analysis we are concerned only with the low-
ionization Fe K$\alpha$ line. Our re-analysis with the line width fee is
consistent with the results of McKernan & Yaqoob (2004) but we note that our
fits in which the Fe K$\alpha$ line width was fixed at well below the HEG
resolution yielded a line centroid energy of $6.399^{+0.006}_{-0.005}$ keV.
Therefore the Fe K$\alpha$ line parameters from the latter fit are more
reliable values for the true narrow core of the Fe K$\alpha$ line.
Mrk 279. Results of the new analysis for this source are consistent with those
reported in YP04.
NGC 5506. Results from the Chandra HETGS observation of this source have been
presented by Bianchi et al. (2003), who obtained FWHM$<4000\rm\ km\ s^{-1}$
for the narrow Fe K$\alpha$ line at $\sim 6.4$ keV at 99% confidence. We
obtained a tighter limit on the FWHM (Table 2). Bianchi et al. (2003) did not
provide constraints on the line centroid energy or EW.
NGC 5548. Results for both of the Chandra HETGS observations have already been
reported in Yaqoob et al. (2001) and YP04, and the new analysis is consistent
with the previous results.
Mrk 290. There are four Chandra HETGS observations for this source and no
results on the Fe K$\alpha$ line from the HEG data have previously been
published. None of the individual observations yielded a detection of the
narrow Fe K$\alpha$ line greater than 99% confidence (for two free
parameters). However, the line was detected with $>3\sigma$ confidence in the
time-averaged spectra. The line intensity against centroid energy confidence
contours shown in Fig. 2 were obtained with the line width fixed at 1 eV since
a closed 99% confidence contour could not be obtained when the line width was
a free parameter.
PDS 456. Results from the Chandra HETGS observation of this source pertaining
to the narrow Fe K$\alpha$ line have never been previously published. The
signal-to-noise ratio is poor and we could only obtain upper limits on the EW
after fixing the line energy at 6.4 keV.
E1821+643. Results from the Chandra HETGS observation of this source have been
presented by Fang et al. (2002) and Yaqoob & Serlemitsos (2005). The latter
work reported Fe K$\alpha$ line parameters $E_{0}$$=6.43^{+0.06}_{-0.05}$ keV,
${\rm EW}=144^{+67}_{-57}$ eV, and FWHM$=10980^{+3300}_{-7690}\rm\ km\
s^{-1}$. However, as described in Yaqoob & Serlemitsos (2005), these
parameters are quite model-dependent because an absorption line was reported
at $\sim 6.2$ keV in the rest frame, and there may also be an underlying broad
emission line. Our fits with the line width fixed at 1 eV likely give the most
representative values of the centroid energy and EW of the narrow core of the
Fe K$\alpha$ line.
3C 382. Gliozzi et al. (2007) have presented the results from the two Chandra
HETGS observations of this source. The Fe K$\alpha$ line was detected with
less than 90% and less than 99% confidence in first and second observations,
respectively. From the second observation Gliozzi et al. (2007) obtained
$E_{0}$$=6.43^{+0.05}_{-0.07}$ keV, ${\rm EW}=55^{+47}_{-20}$ eV, and
FWHM$<9560\rm\ km\ s^{-1}$. Our results are generally consistent with those of
Gliozzi et al. (2007), but we note that the latter work also reported results
from the $-1$ and $+1$ orders of the HEG separately, giving a larger
dispersion in the parameter ranges.
IRAS 18325$-$5926 and 4C 74.26. No results from the Chandra HETGS observations
(pertaining to the Fe K$\alpha$ line or otherwise) for either of these sources
have been previously published. In IRAS 18325$-$5926 our analysis revealed no
significant detection of the narrow Fe K$\alpha$ line in either of the two
observations or from the summed spectrum. The detection of the line in 4C
74.26 was marginal even for the spectrum summed over two observations. Only
upper limits on the EW could be derived for IRAS 18325$-$5926 (with the Fe
K$\alpha$ line energy fixed at 6.4 keV).
Mrk 509. Results from the Chandra HETGS observation of this source have been
reported in YP04 and the new analysis gives consistent results.
NGC 7213. Results from the two Chandra HETGS observations of this source have
been reported by Bianchi et al. (2008). The narrow Fe K$\alpha$ line
parameters obtained were $E_{0}$ $=6.397^{+0.006}_{-0.011}$ keV, ${\rm
EW}=120^{+40}_{-30}$ eV, and FWHM$=2400^{+1100}_{-600}\rm\ km\ s^{-1}$. Our
results are consistent with those of Bianchi et al. (2008), who also reported
the detection of Fe xxv and Fe xxvi Ly$\alpha$ emission lines in the HEG
data.
NGC 7314. Complex Fe K line emission from multiple ionization states was
observed by the Chandra HETGS , and the results of a detailed analysis were
published by Yaqoob et al. (2003). The Fe K$\alpha$ line at $\sim 6.4$ keV is
unresolved with FWHM$<3520\rm\ km\ s^{-1}$ and and ${\rm EW}=81\pm 34$ eV. The
results presented in the present paper (Table 1) were obtained from fits with
the line width fixed at 1 eV. Emission lines from Fe xxv and Fe xxvi
Ly$\alpha$ have been noted and discussed in detail by Yaqoob et al. (2003).
Ark 564. Results pertaining to the narrow Fe K$\alpha$ line from the Chandra
HETGS observation have been presented by Matsumoto, Leighly, & Marshall (2004)
and YP04. The signal-to-noise ratio of the data is poor and the EW of the line
could only be measured with the line energy fixed at 6.4 keV and the line
width fixed at 1 eV, and the results are consistent with those of YP04 (Table
1).
MR 2251$-$178. Gibson et al (2005) reported results from a Chandra
observation, giving ${\rm EW}=25\pm 13$ eV and FWHM$<1530\rm\ km\ s^{-1}$ for
an Fe K$\alpha$ line with a centroid energy fixed at 6.4 keV. In our uniform
analysis, we found that the line was detected at less than 99% confidence. In
this case we were not able to obtain constraints on the line width. Gibson et
al. (2005) also reported the detection of a resolved Fe xxvi Ly$\alpha$
absorption line with ${\rm EW}\sim 28$ eV and a velocity shift of
$\sim-12700\rm\ km\ s^{-1}$, indicating a high-velocity outflow.
NGC 7469. Results from the two Chandra HETGS observations have been reported
by Scott et al. (2005). A strong Fe K$\alpha$ line was detected with a
centroid energy of $6.39\pm 0.01$ keV, and a line width $6310\pm 1580$ km s-1.
No EW was given but the line flux was 3.9$\pm 0.7\times 10^{-5}\rm\ photons\
cm^{-2}\ s^{-1}$. Our results are consistent (within the statistical errors)
with those of Scott et al. (2005).
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## Figure Captions
Figure 1
Left Panels: Chandra HEG spectra in the Fe K band for sources in which the Fe
K$\alpha$ emission line was detected in only one observation, and which were
not included in the sample of Yaqoob & Padmanabhan (2004). The data are binned
at $0.01\AA$, comparable to to the HEG spectral resolution, which is
$0.012\AA$ FWHM. The data are combined from the $-1$ and $+1$ orders of the
grating. The spectra have been corrected for instrumental effective area and
cosmological redshift. Note that these are not unfolded spectra and are
therefore independent of any model that is fitted. Although the spectral
fitting was performed using XSPEC, the spectral plots were not made using
XSPEC. The statistical errors shown correspond to the $1\sigma$ Poisson
errors, which we calculated using equations (7) and (14) in Geherls (1986)
that approximate the upper and lower errors respectively. The solid line
corresponds to a continuum model fitted over the 2–7 keV range (extrapolated
to 7.5 keV), as described in the text (§3). The vertical dotted lines
represent (from left to right), the rest energies of the following: Fe i
$K\alpha$, Fe xxv forbidden, two intercombination lines of Fe xxv, Fe xxv
resonance, Fe xxvi Ly$\alpha$, Fe i $K\beta$, and the Fe K edge. Right Panels:
Joint 99% confidence contours of the Fe K$\alpha$ emission-line core intensity
versus line centroid energy obtained from Gaussian fits to the line with the
line width free as described in the text (solid lines). For Mrk 590, NGC 985,
PG 0844$+$346(1) and IRAS 13349+2438(2), the 99% confidence contours (solid
lines) of the were poorly constrained due to the intrinsic line width
parameter becoming very large. Therefore, we overlaid the 99% confidence
contours obtained with the line width fixed at 1 eV for these cases (dotted
contours). For the remaining sources (ESO 198$-$G24, MCG $-$6-30-15(2), NGC
5506, and E1821$+$643), the dotted contours correspond to 68%, and 90%
confidence.
Figure 2
Left Panels: The time-averaged Chandra HEG spectra in the Fe K band for eight
AGN in which the Fe K$\alpha$ emission line was detected in more than one
observation for cases that were not already reported in Yaqoob & Padmanabhan
(2004). The data are binned at $0.01\AA$ except for NGC 4151, which is binned
at $0.005\AA$. The energies of the vertical dotted lines are described in the
caption to Fig. 1. Right Panel Joint 99% confidence contours of the Fe
K$\alpha$ emission-line core intensity versus line center energy for time-
averaged and individual spectra. Individual observations are shown in
different linestyles while the time-averaged contours are shown with a solid
line. The contour shown for Mrk 290 is from the time-averaged spectrum only,
since none of the individual observations had sufficient signal-to-noise ratio
to obtain well-constrained contours. For MCG $-$6-30-15, we show the contour
from the time-averaged spectrum only, since only one of the four observations
not reported in Yaqoob & Padmanabhan (2004) has a significant detection of
narrow Fe K$\alpha$ line, and that contour has already been shown in Fig. 1.
Figure 3
Distributions of Fe K$\alpha$ line core centroid energies constructed in four
different ways. (a) and (b) were made using the results from individual
observations, whereas (c) and (d) were made from measurements that used
spectra averaged from multiple observations of a given source where relevant
(see §4 for exceptions). In (a) and (c) the line intrinsic width was fixed at
1 eV (results from Table 1). In (b) and (d) the line centroid energies that
could be measured with the line width free were utilized (i.e. those from
Table 2), keeping the line measurements from Table 1 for the remainder. For
the individual observations this results in 51 out of 68 values being obtained
with the line width free –see text for details. The dashed and dotted lines in
each case correspond to the distribution of 68% confidence lower and upper
limits on the line centroid energy, respectively.
Figure 4
Distributions of Fe K$\alpha$ line EW constructed in four different ways. (a)
and (b) were made using the results from individual observations, whereas (c)
and (d) were made from measurements that used spectra averaged from multiple
observations of a given source where relevant (see §4 for exceptions). In (a)
and (c) the line intrinsic width was fixed at 1 eV (results from Table 1). In
(b) and (d) the line centroid energies that could be measured with the line
width free were utilized (i.e. those from Table 2), keeping the line
measurements from Table 1 for the remainder. For the individual observations
this resulted in 70 out of 82 values being obtained with the line width free
–see text for details. The dashed and dotted lines in each case correspond to
the distribution of 68% confidence lower and upper limits on the line EW,
respectively. The shaded histograms in both panels mark the 68% upper limits
on the EW for 12 observations in which the EW could not be measured. Note that
the largest EW of $\sim 600$ eV (for PG 0844$+$349) is not shown in (b) and
(d) because it is not a true measure of the narrow-line EW at $\sim 6.4$ keV
(see text).
Figure 5
The Fe K$\alpha$ emission-line FWHM versus the H$\beta$ FWHM for which the Fe
K$\alpha$ line width could be constrained (see text and Table 2). For MCG
$-$5-23-16, we used the FWHM of infra-red broad Br$\alpha$ line as a surrogate
for H$\beta$ FWHM. The dashed line corresponds to the two line widths being
equal. Open circles correspond to the 12 cases shown in Fig. 6, for which the
best Fe K$\alpha$ line FWHM constraints were obtained (see text). The
statistical errors on the Fe K$\alpha$ line FWHM shown correspond to 68%
confidence for three free parameters.
Figure 6
Joint 68%, 90%, and 99%, confidence contours of the Fe K$\alpha$ emission-line
core EW versus the ratio of the Fe K$\alpha$ FWHM to the H$\beta$ FWHM for 12
AGN that provided the best measurements of Fe K$\alpha$ line FWHM (see text).
For MCG $-$5-23-16, we used the FWHM of infra-red broad Br$\alpha$ line as a
surrogate for H$\beta$ FWHM. The vertical dotted lines correspond to a FWHM
ratio of the pairs of emission lines equal to unity.
Figure 7
(a) The Fe K$\alpha$ core emission-line EW versus the 2–10 keV luminosity. (b)
As (a) for EW versus $(L_{\rm 2-10\ keV}/L_{\rm Edd})$, a proxy for the
accretion rate. Both (a) and (b) were constructed from measurements made from
individual observations. (c) As (a) but showing EW versus $L_{\rm 2-10\ keV}$
for measurements made from spectra combining multiple observations for a given
source, where relevant. (d) As (c) but showing EW versus $(L_{\rm 2-10\
keV}/L_{\rm Edd})$. In (c) and (d) the average spectrum was not used for all
sources, for (b) and (d) reliable black-hole mass estimates were not available
for all sources- see §4 and §5 for details. All of the measurements shown in
(a)–(d) utilize results from the spectral fitting in which the Fe K$\alpha$
line intrinsic width was fixed at 1 eV. The statistical errors on the Fe
K$\alpha$ line EW correspond to 68% confidence. The dotted lines show the
correlations obtained by linear fits to $\log{\rm EW}$ versus $\log{L_{\rm
2-10\ keV}}$ (a) and (c), and $\log{\rm EW}$ versus $\log{(L_{\rm 2-10\
keV}/L_{\rm Edd})}$ (b) and (d). Note that observations with only upper limits
on the EW were not included in the fits.
Figure 1:
Figure 1: – continued
Figure 1: – continued
Figure 2:
Figure 2: – continued
Figure 2: – continued
Figure 3:
Figure 4: Figure 5:
Figure 6:
Figure 6: – continued
Figure 7: .
Table 1: Parameters of the Core Fe K Line Emission ($\sigma$=1 eV) from $\it
Chandra$ (HEG) Data
Source | $z$ | Seq. Num | $E$ | $I$ | $EW$ | $F$ | $L$ | $\Delta C$
---|---|---|---|---|---|---|---|---
| | /ObsID/exp | (keV) | | (eV) | (2$-$10 keV) | (2$-$10 keV) |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
Fairall 9 | 0.0470160 | $700278$ | $6.458_{-0.016}^{+0.008}$ | $1.2_{-0.5}^{+0.7}$ | $47_{-20}^{+27}$ | 2.2 | 11.6 | 14.4
| | $/2088/79.9$ | $(6.434-6.468)$ | $(0.5-2.2)$ | $(20-86)$ | | |
NGC 526a(1) | 0.0190970 | $700840$ | $6.400^{f}$ | $0_{-0}^{+0.6}$ | $0_{-0}^{+16}$ | 3.0 | 2.4 | 0
| | $/4376/29.1$ | $\dots$ | $(0-1.0)$ | $(0-27)$ | | |
NGC 526a(2) | 0.0190970 | $700840$ | $6.400_{-0.006}^{+0.010}$ | $1.6_{-0.9}^{+1.3}$ | $47_{-27}^{+36}$ | 3.0 | 2.3 | 8.0
| | $/4437/29.4$ | $(6.389-6.413)$ | $(0.3-3.5)$ | $(9.0-100)$ | | |
NGC 526a(total) | 0.0190970 | $\dots$ | $6.394_{-0.006}^{+0.012}$ | $1.0_{-0.7}^{+0.7}$ | $28_{-20}^{+20}$ | 2.9 | 2.4 | 5.0
| | $\dots/57.8$ | $(6.380-6.414)$ | $(0-2.1)$ | $(0-59)$ | | |
Mrk 590 | 0.0263850 | $701005$ | $6.403_{-0.009}^{+0.016}$ | $0.8_{-0.4}^{+0.4}$ | $78_{-37}^{+46}$ | 0.85 | 1.3 | 14.9
| | $/4924/96.8$ | $(6.386-6.435)$ | $(0.3-1.5)$ | $(31-155)$ | | |
NGC 985 | 0.0431430 | $700449$ | $6.395_{-0.009}^{+0.015}$ | $0.7_{-0.4}^{+0.5}$ | $57_{-34}^{+34}$ | 1.1 | 4.4 | 10.1
| | $/3010/77.7$ | $(6.379-6.412)$ | $(0.2-1.5)$ | $(15-113)$ | | |
ESO 198$-$G24(1) | 0.0455000 | $700900$ | $6.400^{f}$ | $0.2_{-0.2}^{+0.2}$ | $26_{-26}^{+26}$ | 0.67 | 3.2 | 1.6
| | $/4817/80.3$ | $\dots$ | $(0-0.6)$ | $(0-74)$ | | |
ESO 198$-$G24(2) | 0.0455000 | $700900$ | $6.386_{-0.008}^{+0.008}$ | $1.0_{-0.4}^{+0.6}$ | $139_{-57}^{+79}$ | 0.61 | 2.9 | 23.2
| | $/5315/71.5$ | $(6.377-6.401)$ | $(0.4-1.8)$ | $(55-246)$ | | |
ESO 198$-$G24(total) | 0.0455000 | $\dots$ | $6.394_{-0.009}^{+0.008}$ | $0.6_{-0.3}^{+0.3}$ | $71_{-33}^{+43}$ | 0.64 | 3.1 | 17.3
| | $\dots/151.5$ | $(6.377-6.409)$ | $(0.2-1.0)$ | $(25-126)$ | | |
3C 120 | 0.00330100 | $700454$ | $6.412_{-0.009}^{+0.009}$ | $2.5_{-1.0}^{+1.1}$ | $47_{-19}^{+20}$ | 4.7 | 11.7 | 20.1
| | $/3015/58.2$ | $(6.396-6.428)$ | $(1.2-4.2)$ | $(22-78)$ | | |
NGC 2110(1) | 0.00778900 | $700582$ | $6.416_{-0.010}^{+0.008}$ | $3.7_{-1.8}^{+2.0}$ | $61_{-29}^{+34}$ | 4.5 | 0.61 | 14.1
| | $/3143/34$ | $(6.397-6.430)$ | $(1.3-6.8)$ | $(22-113)$ | | |
NGC 2110(2) | 0.00778900 | $700582$ | $6.407_{-0.016}^{+0.015}$ | $3.1_{-2.0}^{+2.5}$ | $52_{-34}^{+41}$ | 4.5 | 0.60 | 10.4
| | $/3417/33.2$ | $(6.384-6.434)$ | $(0.6-6.6)$ | $(10-110)$ | | |
NGC 2110(3) | 0.00778900 | $700582$ | $6.392_{-0.002}^{+0.007}$ | $4.9_{-1.3}^{+1.4}$ | $80_{-21}^{+23}$ | 4.5 | 0.60 | 51.5
| | $/3418/76.1$ | $(6.384-6.400)$ | $(3.1-7.0)$ | $(51-114)$ | | |
NGC 2110(4) | 0.00778900 | $700841$ | $6.392_{-0.002}^{+0.008}$ | $3.9_{-0.9}^{+0.9}$ | $92_{-23}^{+23}$ | 3.1 | 0.41 | 70.8
| | $/4377/96.4$ | $(6.384-6.400)$ | $(2.6-5.3)$ | $(62-127)$ | | |
NGC 2110(total) | 0.00778900 | $\dots$ | $6.399_{-0.008}^{+0.001}$ | $3.9_{-0.6}^{+0.7}$ | $75_{-11}^{+14}$ | 3.8 | 0.51 | 134
| | $\dots/200.4$ | $(6.391-6.400)$ | $(3.0-4.9)$ | $(58-95)$ | | |
PG 0844$+$349(1) | 0.0640000 | $701023$ | $6.364_{-0.009}^{+0.007}$ | $0.6_{-0.3}^{+0.4}$ | $118_{-58}^{+83}$ | 0.42 | 4.0 | 12.4
| | $/5599/57.2$ | $(6.352-6.375)$ | $(0.2-1.3)$ | $(40-262)$ | | |
PG 0844$+$349(2) | 0.0640000 | $701023$ | $6.400^{f}$ | $0.2_{-0.2}^{+0.3}$ | $36_{-36}^{+47}$ | 0.55 | 5.4 | 1.3
| | $/6244/50.2$ | $\dots$ | $(0-0.7)$ | $(0-116)$ | | |
PG 0844$+$349(3) | 0.0640000 | $701023$ | $6.400^{f}$ | $0.1_{-0.1}^{+0.3}$ | $13_{-13}^{+34}$ | 0.77 | 7.6 | 0.2
| | $/6245/36.2$ | $\dots$ | $(0-0.6)$ | $(0-76)$ | | |
PG 0844$+$349(total) | 0.0640000 | $\dots$ | $6.366_{-0.008}^{+0.008}$ | $0.3_{-0.1}^{+0.3}$ | $52_{-20}^{+43}$ | 0.55 | 5.4 | 10.2
| | $\dots/141.2$ | $(6.356-6.381)$ | $(0.1-0.7)$ | $(16-112)$ | | |
Mrk 705 | 0.0291500 | $700995$ | $6.400^{f}$ | $0.4_{-0.4}^{+0.6}$ | $26_{-26}^{+48}$ | 1.3 | 2.6 | 0.5
| | $/4914/21.3$ | $\dots$ | $(0-1.6)$ | $(0-113)$ | | |
MCG $-$5-23-16(1) | 0.00827900 | $700311$ | $6.394_{-0.007}^{+0.007}$ | $7.2_{-1.9}^{+2.1}$ | $55_{-14}^{+18}$ | 10.5 | 1.6 | 47
| | $/2121/76.2$ | $(6.386-6.402)$ | $(4.6-10)$ | $(35-81)$ | | |
MCG $-$5-23-16(2) | 0.00827900 | $701171$ | $6.394_{-0.008}^{+0.008}$ | $5.1_{-2.0}^{+2.5}$ | $40_{-15}^{+20}$ | 10.6 | 1.6 | 18.3
| | $/6187/30.1$ | $(6.384-6.403)$ | $(2.3-8.6)$ | $(18-68)$ | | |
MCG $-$5-23-16(3) | 0.00827900 | $701171$ | $6.395_{-0.010}^{+0.015}$ | $6.4_{-2.8}^{+3.0}$ | $51_{-22}^{+24}$ | 10.5 | 1.6 | 16.7
| | $/7240/20.3$ | $(6.371-6.411)$ | $(2.6-10)$ | $(21-87)$ | | |
MCG $-$5-23-16(total) | 0.00827900 | $\dots$ | $6.394_{-0.001}^{+0.002}$ | $6.4_{-1.3}^{+1.4}$ | $50_{-10}^{+11}$ | 10.5 | 1.6 | 80.9
| | $\dots/96.1$ | $(6.387-6.402)$ | $(4.6-8.4)$ | $(36-65)$ | | |
NGC 3227 | 0.00385900 | $700165$ | $6.388_{-0.012}^{+0.021}$ | $1.0_{-0.9}^{+1.0}$ | $32_{-29}^{+32}$ | 2.5 | 0.08 | 3.1
| | $/860/47$ | $\dots$ | $(0-2.6)$ | $(0-83)$ | | |
NGC 3516(1) | 0.00883600 | $700270$ | $6.395_{-0.006}^{+0.002}$ | $3.8_{-0.9}^{+1.0}$ | $106_{-25}^{+28}$ | 3.0 | 0.51 | 69.3
| | $/2080/74.5$ | $(6.383-6.398)$ | $(2.5-5.2)$ | $(70-145)$ | | |
NGC 3516(2) | 0.00883600 | $700270$ | $6.406_{-0.008}^{+0.009}$ | $3.7_{-1.2}^{+1.5}$ | $155_{-51}^{+61}$ | 1.9 | 0.33 | 38.2
| | $/2431/36.2$ | $(6.397-6.416)$ | $(2.1-5.9)$ | $(87-246)$ | | |
NGC 3516(3) | 0.00883600 | $700270$ | $6.398_{-0.008}^{+0.002}$ | $2.4_{-0.7}^{+0.9}$ | $83_{-26}^{+28}$ | 2.3 | 0.39 | 40.6
| | $/2482/89.5$ | $(6.389-6.407)$ | $(1.4-3.6)$ | $(47-121)$ | | |
NGC 3516(4) | 0.00883600 | $701337$ | $6.407_{-0.010}^{+0.016}$ | $2.5_{-1.2}^{+1.4}$ | $40_{-19}^{+23}$ | 5.2 | 0.89 | 12.5
| | $/7281/43.1$ | $(6.373-6.430)$ | $(0.8-4.6)$ | $(13-74)$ | | |
NGC 3516(5) | 0.00883600 | $701337$ | $6.398_{-0.008}^{+0.009}$ | $2.3_{-1.1}^{+1.4}$ | $45_{-22}^{+27}$ | 4.4 | 0.76 | 12.9
| | $/7282/42.1$ | $(6.383-6.414)$ | $(0.8-4.3)$ | $(16-83)$ | | |
NGC 3516(6) | 0.00883600 | $701337$ | $6.430_{-0.008}^{+0.008}$ | $3.8_{-1.5}^{+1.6}$ | $53_{-21}^{+23}$ | 6.5 | 1.1 | 20.6
| | $/8450/39.1$ | $(6.415-6.439)$ | $(1.7-6.2)$ | $(24-87)$ | | |
NGC 3516(7) | 0.00883600 | $701337$ | $6.407_{-0.010}^{+0.030}$ | $2.7_{-1.1}^{+1.6}$ | $36_{-15}^{+21}$ | 6.7 | 1.2 | 14.4
| | $/8451/48.1$ | $(6.389-6.439)$ | $(1.1-4.9)$ | $(15-65)$ | | |
NGC 3516(8) | 0.00883600 | $701337$ | $6.431_{-0.033}^{+0.009}$ | $2.9_{-1.8}^{+2.2}$ | $41_{-24}^{+32}$ | 6.1 | 1.0 | 7.2
| | $/8452/20.2$ | $(6.390-6.447)$ | $(0.5-6.2)$ | $(7.0-88)$ | | |
NGC 3516(total) | 0.00883600 | $\dots$ | $6.398_{-0.001}^{+0.001}$ | $2.8_{-0.4}^{+0.4}$ | $58_{-8}^{+9}$ | 4.1 | 0.70 | 161.7
| | $\dots/386.5$ | $(6.397-6.400)$ | $(2.2-3.4)$ | $(46-71)$ | | |
NGC 3783(1) | 0.00973000 | $700045$ | $6.396_{-0.008}^{+0.007}$ | $4.4_{-1.4}^{+1.3}$ | $56_{-18}^{+17}$ | 6.8 | 1.4 | 36.6
| | $/373/57.2$ | $(6.387-6.404)$ | $(2.5-6.4)$ | $(32-88)$ | | |
NGC 3783(2) | 0.00973000 | $700280$ | $6.403_{-0.008}^{+0.001}$ | $4.2_{-0.8}^{+0.8}$ | $61_{-12}^{+11}$ | 5.9 | 1.2 | 105.2
| | $/2090/167.8$ | $(6.395-6.404)$ | $(3.1-5.4)$ | $(45-78)$ | | |
NGC 3783(3) | 0.00973000 | $700281$ | $6.395_{-0.001}^{+0.001}$ | $4.3_{-0.8}^{+0.8}$ | $62_{-11}^{+12}$ | 5.9 | 1.2 | 109.2
| | $/2091/171$ | $(6.387-6.397)$ | $(3.2-5.5)$ | $(46-79)$ | | |
NGC 3783(4) | 0.00973000 | $700282$ | $6.396_{-0.001}^{+0.001}$ | $5.0_{-0.8}^{+0.8}$ | $72_{-11}^{+12}$ | 6.0 | 1.3 | 146.6
| | $/2092/167.6$ | $(6.394-6.401)$ | $(3.9-6.2)$ | $(56-90)$ | | |
NGC 3783(5) | 0.00973000 | $700283$ | $6.396_{-0.001}^{+0.001}$ | $5.8_{-0.9}^{+0.9}$ | $63_{-10}^{+10}$ | 8.2 | 1.7 | 154.2
| | $/2093/168.2$ | $(6.394-6.397)$ | $(4.5-7.1)$ | $(49-77)$ | | |
NGC 3783(6) | 0.00973000 | $700284$ | $6.396_{-0.001}^{+0.008}$ | $4.0_{-0.8}^{+0.8}$ | $48_{-9}^{+10}$ | 7.2 | 1.5 | 89.3
| | $/2094/168.3$ | $(6.394-6.404)$ | $(2.9-5.2)$ | $(35-62)$ | | |
NGC 3783(total) | 0.00973000 | $\dots$ | $6.396_{-0.001}^{+0.001}$ | $4.6_{-0.3}^{+0.4}$ | $60_{-4}^{+5}$ | 6.6 | 1.4 | 635.9
| | $\dots/888.7$ | $(6.395-6.396)$ | $(4.1-5.1)$ | $(53-66)$ | | |
NGC 4051 | 0.00233600 | $700164$ | $6.414_{-0.010}^{+0.006}$ | $1.6_{-0.5}^{+0.9}$ | $94_{-34}^{+41}$ | 1.7 | 0.02 | 26.5
| | $/859/80.8$ | $(6.398-6.422)$ | $(0.9-2.8)$ | $(49-152)$ | | |
NGC 4151(1) | 0.00331900 | $700007$ | $6.396_{-0.001}^{+0.001}$ | $17.2_{-2.6}^{+2.7}$ | $148_{-22}^{+23}$ | 7.8 | 0.19 | 202.3
| | $/335/48$ | $(6.395-6.398)$ | $(13-21)$ | $(117-182)$ | | |
NGC 4151(2) | 0.00331900 | $700491$ | $6.396_{-0.001}^{+0.001}$ | $13.9_{-1.4}^{+1.6}$ | $59_{-6}^{+8}$ | 17.4 | 0.42 | 306.5
| | $/3052/156.6$ | $(6.394-6.396)$ | $(11-16)$ | $(50-69)$ | | |
NGC 4151(3) | 0.00331900 | $700491$ | $6.396_{-0.001}^{+0.001}$ | $14.0_{-1.9}^{+2.1}$ | $56_{-7}^{+9}$ | 18.3 | 0.44 | 177.1
| | $/3480/92.9$ | $(6.394-6.397)$ | $(11-17)$ | $(46-69)$ | | |
NGC 4151(4) | 0.00331900 | $701493$ | $6.396_{-0.001}^{+0.008}$ | $10.0_{-2.0}^{+1.8}$ | $162_{-31}^{+30}$ | 4.3 | 0.10 | 131.1
| | $/7829/50$ | $(6.395-6.404)$ | $(7.4-12)$ | $(119-206)$ | | |
NGC 4151(5) | 0.00331900 | $701494$ | $6.388_{-0.001}^{+0.008}$ | $11.1_{-2.5}^{+2.6}$ | $44_{-10}^{+10}$ | 18.4 | 0.45 | 63.4
| | $/7830/50.2$ | $(6.386-6.397)$ | $(7.6-14)$ | $(30-59)$ | | |
NGC 4151(total) | 0.00331900 | $\dots$ | $6.396_{-0.001}^{+0.001}$ | $13.3_{-0.9}^{+0.9}$ | $65_{-4}^{+5}$ | 14.9 | 0.36 | 801.5
| | $\dots/389.9$ | $(6.395-6.396)$ | $(12-14)$ | $(59-72)$ | | |
Mrk 766 | 0.0129290 | $700123$ | $6.425_{-0.010}^{+0.016}$ | $0.8_{-0.5}^{+0.6}$ | $37_{-23}^{+27}$ | 2.3 | 0.86 | 7.9
| | $/1597/90.5$ | $(6.400-6.450)$ | $(0.2-1.6)$ | $(9.0-73)$ | | |
3C 273(1) | 0.158340 | $790020$ | $6.313_{-0.015}^{+0.013}$ | $2.4_{-1.5}^{+1.7}$ | $12_{-7}^{+9}$ | 12.4 | 811.4 | 6.7
| | $/459/39.1$ | $(6.291-6.334)$ | $(0.4-4.8)$ | $(2.0-24)$ | | |
3C 273(2) | 0.158340 | $790057$ | $6.292_{-0.007}^{+0.007}$ | $3.2_{-1.7}^{+1.9}$ | $20_{-10}^{+13}$ | 11.2 | 734 | 9.25
| | $/2463/27.1$ | $(6.279-6.301)$ | $(0.8-6.0)$ | $(5.0-38)$ | | |
3C 273(3) | 0.158340 | $790074$ | $6.275_{-0.010}^{+0.011}$ | $2.5_{-1.5}^{+1.8}$ | $22_{-13}^{+15}$ | 8.4 | 559 | 7.2
| | $/3456/25$ | $(6.259-6.293)$ | $(0.4-5.2)$ | $(3.0-45)$ | | |
3C 273(4) | 0.158340 | $790075$ | $6.319_{-0.012}^{+0.008}$ | $2.2_{-1.7}^{+1.4}$ | $20_{-16}^{+12}$ | 8.3 | 552 | 4.3
| | $/3457/25.4$ | $\dots$ | $(0.0-4.4)$ | $(0.0-39)$ | | |
3C 273(5) | 0.158340 | $790076$ | $6.412_{-0.029}^{+0.021}$ | $1.7_{-1.3}^{+1.6}$ | $16_{-12}^{+14}$ | 8.1 | 538 | 4.0
| | $/3573/30.2$ | $\dots$ | $(0.0-4.1)$ | $(0.0-38)$ | | |
3C 273(6) | 0.158340 | $790087$ | $6.400^{f}$ | $0_{-0}^{+0.9}$ | $0_{-0}^{+7}$ | 11.9 | 794 | 0
| | $/4430/27.6$ | $\dots$ | $(0.0-1.6)$ | $(0.0-10)$ | | |
3C 273(7) | 0.158340 | $790089$ | $6.400^{f}$ | $0.0_{-0.0}^{+0.4}$ | $0_{-0}^{+3}$ | 8.0 | 528 | 2.0
| | $/5169/30.2$ | $\dots$ | $(0.0-0.9)$ | $(0.0-7.0)$ | | |
3C 273(total) | 0.158340 | $\dots$ | $6.319_{-0.013}^{+0.007}$ | $1.0_{-0.6}^{+0.6}$ | $7_{-4}^{+4}$ | 9.9 | 654 | 8.0
| | $\dots/201.1$ | $(6.292-6.328)$ | $(0.2-1.8)$ | $(1.0-13)$ | | |
NGC 4593 | 0.00900000 | $700279$ | $6.399_{-0.008}^{+0.008}$ | $2.7_{-0.8}^{+1.0}$ | $59_{-18}^{+20}$ | 4.5 | 0.81 | 33.3
| | $/2089/79.9$ | $(6.390-6.408)$ | $(1.6-4.1)$ | $(34-88)$ | | |
MCG $-$6-30-15(1) | 0.00774900 | $700105$ | $6.384_{-0.009}^{+0.015}$ | $1.4_{-0.6}^{+0.6}$ | $28_{-12}^{+12}$ | 4.7 | 0.62 | 15.9
| | $/433/128.2$ | $(6.374-6.407)$ | $(0.6-2.3)$ | $(12-46)$ | | |
MCG $-$6-30-15(2) | 0.00774900 | $700845$ | $6.399_{-0.008}^{+0.008}$ | $1.2_{-0.4}^{+0.6}$ | $26_{-9}^{+12}$ | 4.5 | 0.59 | 17.4
| | $/4759/161.1$ | $(6.389-6.409)$ | $(0.6-2.1)$ | $(17-44)$ | | |
MCG $-$6-30-15(3) | 0.00774900 | $700845$ | $6.382_{-0.013}^{+0.009}$ | $0.7_{-0.5}^{+0.4}$ | $14_{-10}^{+9}$ | 4.4 | 0.58 | 5.5
| | $/4760/172.3$ | $(6.359-6.400)$ | $(0.0-1.3)$ | $(0-27)$ | | |
MCG $-$6-30-15(4) | 0.00774900 | $700845$ | $6.400_{-0.018}^{+0.016}$ | $0.8_{-0.4}^{+0.6}$ | $17_{-8}^{+11}$ | 4.7 | 0.63 | 7.4
| | $/4761/158.8$ | $(6.375-6.424)$ | $(0.2-1.7)$ | $(4.0-33)$ | | |
MCG $-$6-30-15(5) | 0.00774900 | $700845$ | $6.342_{-0.021}^{+0.018}$ | $1.8_{-1.1}^{+1.3}$ | $37_{-22}^{+27}$ | 4.6 | 0.61 | 7.6
| | $/4762/38.2$ | $(6.312-6.368)$ | $(0.3-3.7)$ | $(6.0-77)$ | | |
MCG $-$6-30-15(total) | 0.00774900 | $\dots$ | $6.390_{-0.008}^{+0.002}$ | $0.8_{-0.2}^{+0.3}$ | $18_{-5}^{+5}$ | 4.5 | 0.59 | 29.1
| | $\dots/582$ | $(6.375-6.399)$ | $(0.5-1.2)$ | $(10-25)$ | | |
IRAS 13349$+$2438(1) | 0.107640 | $700902$ | $6.400^{f}$ | $0.04_{-0.04}^{+0.1}$ | $8_{-8}^{+22}$ | 0.39 | 11.5 | 0.3
| | $/4819/161.9$ | $\dots$ | $(0.0-0.2)$ | $(0.0-46)$ | | |
IRAS 13349$+$2438(2) | 0.107640 | $700902$ | $6.428_{-0.008}^{+0.011}$ | $0.3_{-0.1}^{+0.3}$ | $77_{-31}^{+60}$ | 0.33 | 9.7 | 12.9
| | $/4820/137.5$ | $(6.417-6.440)$ | $(0.1-0.7)$ | $(23-160)$ | | |
IRAS 13349$+$2438(total) | 0.107640 | $\dots$ | $6.426_{-0.009}^{+0.013}$ | $0.2_{-0.1}^{+0.1}$ | $40_{-19}^{+22}$ | 0.36 | 10.7 | 9.7
| | $\dots/294.5$ | $(6.411-6.441)$ | $(0.0-0.4)$ | $(0.0-83)$ | | |
IC 4329A | 0.0160540 | $700367$ | $6.399_{-0.005}^{+0.006}$ | $3.7_{-1.6}^{+1.6}$ | $19_{-8}^{+8}$ | 17.5 | 10.1 | 14.3
| | $/2177/60.1$ | $(6.387-6.411)$ | $(1.5-6.1)$ | $(8.0-31)$ | | |
Mrk 279 | 0.0304510 | $700501$ | $6.381_{-0.007}^{+0.008}$ | $1.1_{-0.4}^{+0.4}$ | $66_{-22}^{+28}$ | 1.4 | 2.9 | 23.4
| | $/3062/116.1$ | $(6.372-6.395)$ | $(0.5-1.7)$ | $(31-107)$ | | |
NGC 5506 | 0.00618100 | $700214$ | $6.398_{-0.001}^{+0.008}$ | $5.7_{-1.2}^{+1.3}$ | $66_{-14}^{+15}$ | 6.6 | 0.55 | 80.6
| | $/1598/90$ | $(6.396-6.406)$ | $(4.1-7.6)$ | $(48-88)$ | | |
NGC 5548(1) | 0.0171750 | $700142$ | $6.410_{-0.009}^{+0.016}$ | $1.8_{-0.7}^{+0.8}$ | $58_{-22}^{+27}$ | 2.7 | 1.8 | 22.1
| | $/837/82.3$ | $(6.385-6.434)$ | $(0.9-3.0)$ | $(29-98)$ | | |
NGC 5548(2) | 0.0171750 | $700485$ | $6.394_{-0.007}^{+0.008}$ | $1.9_{-0.5}^{+0.5}$ | $55_{-14}^{+15}$ | 3.1 | 2.0 | 42.2
| | $/3046/153.9$ | $(6.386-6.403)$ | $(1.2-2.7)$ | $(35-78)$ | | |
NGC 5548(total) | 0.0171750 | $\dots$ | $6.402_{-0.010}^{+0.001}$ | $1.9_{-0.5}^{+0.4}$ | $56_{-14}^{+13}$ | 2.9 | 1.9 | 61.8
| | $\dots/232.7$ | $(6.386-6.410)$ | $(1.3-2.5)$ | $(39-75)$ | | |
Mrk 290(1) | 0.0295770 | $700629$ | $6.386_{-0.011}^{+0.012}$ | $1.0_{-0.6}^{+0.7}$ | $53_{-32}^{+37}$ | 1.8 | 3.5 | 9.0
| | $/3567/55.1$ | $(6.367-6.400)$ | $(0.2-2.0)$ | $(11-106)$ | | |
Mrk 290(2) | 0.0295770 | $700629$ | $6.398_{-0.032}^{+0.026}$ | $0.5_{-0.4}^{+0.4}$ | $36_{-29}^{+28}$ | 1.3 | 2.5 | 3.8
| | $/4399/85.1$ | $\dots$ | $(0-1.2)$ | $(0-85)$ | | |
Mrk 290(3) | 0.0295770 | $700629$ | $6.400^{f}$ | $0.4_{-0.3}^{+0.4}$ | $20_{-15}^{+22}$ | 1.8 | 3.6 | 1.8
| | $/4441/60.9$ | $\dots$ | $(0-1.0)$ | $(0-52)$ | | |
Mrk 290(4) | 0.0295770 | $700629$ | $6.400^{f}$ | $0.3_{-0.3}^{+0.4}$ | $15_{-15}^{+21}$ | 1.8 | 3.6 | 0.8
| | $/4442/50.2$ | $\dots$ | $(0-1.0)$ | $(0-51)$ | | |
Mrk 290(total) | 0.0295770 | $\dots$ | $6.398_{-0.016}^{+0.009}$ | $0.5_{-0.3}^{+0.3}$ | $27_{-16}^{+18}$ | 1.6 | 3.2 | 10.8
| | $\dots/247.3$ | $(6.374-6.414)$ | $(0.2-0.9)$ | $(11-50)$ | | |
PDS 456 | 0.184000 | $700742$ | $6.400^{f}$ | $0.04_{-0.04}^{+0.06}$ | $4_{-4}^{+13}$ | 0.40 | 37.3 | 0.11
| | $/4063/145.2$ | $\dots$ | $(0-0.2)$ | $(0-33)$ | | |
E1821$+$643 | 0.297000 | $700215$ | $6.453_{-0.007}^{+0.005}$ | $0.7_{-0.3}^{+0.4}$ | $26_{-12}^{+13}$ | 1.4 | 362.5 | 13.1
| | $/1599/101.3$ | $(6.445-6.463)$ | $(0.3-1.3)$ | $(11-46)$ | | |
3C 382(1) | 0.0578700 | $700991$ | $6.374_{-0.016}^{+0.017}$ | $1.0_{-0.8}^{+1.0}$ | $16_{-13}^{+15}$ | 5.5 | 43.2 | 3.7
| | $/4910/55$ | $\dots$ | $(0-2.5)$ | $(0-39)$ | | |
3C 382(2) | 0.0578700 | $700991$ | $6.408_{-0.010}^{+0.013}$ | $1.3_{-0.8}^{+0.9}$ | $21_{-13}^{+15}$ | 4.9 | 38.5 | 7.1
| | $/6151/64.9$ | $(6.382-6.429)$ | $(0.2-2.6)$ | $(3.0-43)$ | | |
3C 382(total) | 0.0578700 | $\dots$ | $6.368_{-0.009}^{+0.038}$ | $0.9_{-0.6}^{+0.6}$ | $14_{-9}^{+9}$ | 5.2 | 40.6 | 6.2
| | $\dots/118$ | $(6.351-6.446)$ | $(0.1-1.8)$ | $(2.0-28)$ | | |
IRAS 18325$-$5926(1) | 0.0202310 | $700587$ | $6.400^{f}$ | $0.05_{-0.05}^{+0.35}$ | $2_{-2}^{+15}$ | 2.1 | 1.9 | 0.04
| | $/3148/56.9$ | $\dots$ | $(0-0.7)$ | $(0.0-31)$ | | |
IRAS 18325$-$5926(2) | 0.0202310 | $700587$ | $6.400^{f}$ | $0.2_{-0.2}^{+0.4}$ | $5_{-5}^{+12}$ | 3.1 | 2.8 | 0.2
| | $/3452/51.1$ | $\dots$ | $(0-1.0)$ | $(0.0-29)$ | | |
IRAS 18325$-$5926(total) | 0.0202310 | $\dots$ | $6.400^{f}$ | $0.1_{-0.1}^{+0.3}$ | $5_{-5}^{+9}$ | 2.5 | 2.3 | 0.2
| | $\dots/106.2$ | $\dots$ | $(0-0.6)$ | $(0-21)$ | | |
4C $+$74.26(1) | 0.104000 | $700679$ | $6.258_{-0.014}^{+0.013}$ | $1.2_{-0.8}^{+1.0}$ | $37_{-25}^{+30}$ | 2.5 | 65.1 | 6.8
| | $/4000/37.7$ | $(6.236-6.278)$ | $(0.2-2.6)$ | $(6.0-79)$ | | |
4C $+$74.26(2) | 0.104000 | $700679$ | $6.347_{-0.010}^{+0.011}$ | $1.0_{-0.8}^{+1.0}$ | $30_{-24}^{+30}$ | 2.6 | 67.1 | 4.7
| | $/5195/31.8$ | $(6.322-6.366)$ | $(0.0-2.4)$ | $(0.0-73)$ | | |
4C $+$74.26(total) | 0.104000 | $\dots$ | $6.252_{-0.008}^{+0.011}$ | $1.0_{-0.5}^{+0.7}$ | $28_{-14}^{+18}$ | 2.5 | 66.2 | 9.5
| | $\dots/66.1$ | $(6.242-6.265)$ | $(0.3-2.0)$ | $(8.0-54)$ | | |
Mrk 509 | 0.0343970 | $700277$ | $6.445_{-0.009}^{+0.015}$ | $2.2_{-1.0}^{+1.2}$ | $34_{-16}^{+17}$ | 5.8 | 15.6 | 13.7
| | $/2087/58.7$ | $(6.427-6.462)$ | $(0.8-3.9)$ | $(12-59)$ | | |
NGC 7213(1) | 0.00583900 | $701410$ | $6.395_{-0.008}^{+0.003}$ | $2.2_{-0.6}^{+0.6}$ | $88_{-23}^{+25}$ | 2.3 | 0.18 | 50.9
| | $/7742/115.3$ | $(6.386-6.403)$ | $(1.4-3.1)$ | $(57-126)$ | | |
NGC 7213(2) | 0.00583900 | $701410$ | $6.412_{-0.009}^{+0.016}$ | $2.3_{-1.1}^{+1.3}$ | $91_{-44}^{+50}$ | 2.4 | 0.18 | 16.5
| | $/8590/35.1$ | $(6.395-6.431)$ | $(0.9-4.2)$ | $(35-164)$ | | |
NGC 7213(total) | 0.00583900 | $\dots$ | $6.395_{-0.001}^{+0.008}$ | $2.2_{-0.6}^{+0.5}$ | $86_{-22}^{+22}$ | 2.3 | 0.18 | 63.3
| | $\dots/150$ | $(6.387-6.404)$ | $(1.4-3.0)$ | $(56-120)$ | | |
NGC 7314(1) | 0.00474300 | $700455$ | $6.397_{-0.018}^{+0.015}$ | $1.3_{-1.1}^{+1.4}$ | $32_{-25}^{+34}$ | 3.6 | 0.18 | 3.8
| | $/3016/28.9$ | $\dots$ | $(0.0-3.3)$ | $(0.0-81)$ | | |
NGC 7314(2) | 0.00474300 | $700455$ | $6.422_{-0.009}^{+0.008}$ | $1.9_{-0.9}^{+0.9}$ | $53_{-22}^{+26}$ | 3.3 | 0.16 | 18
| | $/3719/68.4$ | $(6.387-6.437)$ | $(0.8-3.2)$ | $(23-90)$ | | |
NGC 7314(total) | 0.00474300 | $\dots$ | $6.413_{-0.024}^{+0.017}$ | $1.5_{-0.8}^{+0.9}$ | $41_{-22}^{+24}$ | 3.4 | 0.17 | 15.5
| | $\dots/95.7$ | $(6.387-6.437)$ | $(0.5-2.8)$ | $(19-76)$ | | |
Ark 564 | 0.0246840 | $700168$ | $6.400^{f}$ | $0.6_{-0.4}^{+0.4}$ | $25_{-16}^{+20}$ | 2.7 | 3.7 | 2.1
| | $/863/49.4$ | $\dots$ | $(0.0-1.3)$ | $(0.0-58)$ | | |
MR 2251-178 | 0.0639800 | $700416$ | $6.412_{-0.009}^{+0.008}$ | $0.7_{-0.4}^{+0.4}$ | $22_{-13}^{+12}$ | 2.7 | 25.3 | 7.7
| | $/2977/148.7$ | $(6.396-6.427)$ | $(0.1-1.3)$ | $(3.0-40)$ | | |
NGC 7469(1) | 0.0163170 | $700395$ | $6.388_{-0.007}^{+0.007}$ | $2.6_{-0.7}^{+0.8}$ | $93_{-24}^{+30}$ | 2.7 | 1.6 | 46.1
| | $/2956/79.9$ | $(6.380-6.396)$ | $(1.6-3.8)$ | $(58-137)$ | | |
NGC 7469(2) | 0.0163170 | $700586$ | $6.437_{-0.017}^{+0.008}$ | $1.6_{-0.7}^{+0.8}$ | $60_{-25}^{+33}$ | 2.4 | 1.5 | 16.7
| | $/3147/69.8$ | $(6.412-6.451)$ | $(0.6-2.7)$ | $(23-104)$ | | |
NGC 7469(total) | 0.0163170 | $\dots$ | $6.388_{-0.008}^{+0.002}$ | $1.9_{-0.5}^{+0.6}$ | $72_{-20}^{+21}$ | 2.6 | 1.5 | 50.9
| | $\dots/147.2$ | $(6.379-6.397)$ | $(1.2-2.8)$ | $(44-104)$ | | |
Note. — Results from Chandra HEG data, fitted with a power law plus Gaussian
emission-line model in the 2–7 keV band, with the line width fixed at 1 eV.
All parameters are quoted in the source rest frame. Statistical errors are for
the 68% confidence level, whilst parentheses show the 90% confidence level
ranges of the parameters. The number of interesting parameters assumed for
calculating the statistical errors was equal to the number of free parameters
in the Gaussian component of the model. Col.(1): Redshifts obtained from NASA
Extragalactic Database (NED); Col.(2): Observation sequence number, ID, and
exposure time in ks; Col.(3): Gaussian line centroid energy; Col.(4):
Emission-line intensity in units of $\rm 10^{-5}\ photons\ cm^{-2}\ s^{-1}$;
Col.(5): Emission line equivalent width; Col.(6): $F$ is the estimated 2–10
keV observed flux in units of $10^{-11}\ \rm ergs\ cm^{-2}\ s^{-1}$. The
power-law continuum was extrapolated to 10 keV; Col.(7): $L$ is the estimated
unabsorbed 2–10 keV source-frame luminosity (using the 2–10 keV estimated
flux), in units of $10^{43}\ \rm ergs\ s^{-1}$; Col.(8): The decrease in the
fit statistic, $C$, when the narrow, two-parameter emission line was added to
the continuum-only model.
Table 2: Parameters of the Core Fe K Line Emission ($\sigma$ free) from
$Chandra$ (HEG) Data
Source | $E^{a}$ | $I^{b}$ | $EW^{c}$ | FWHMd (Fe K$\alpha$) | FWHMd (H$\beta)$ | Referencee
---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7)
Fairall 9 | $6.370^{+0.347}_{-0.161}$ $(6.137-6.906)$ | $5.5^{+13.3}_{-3.6}$ ($1.5-22.9$) | $228^{+555}_{-149}$ $(63-954)$ | $18100^{+76840}_{-12390}$ $(5100-121780)$ | 6270$\pm 290$ | N06
Mrk 590 | $6.407^{+0.036}_{-0.033}$ ($6.358-6.461$) | $1.6^{+1.0}_{-0.8}$ ($0.6-3.0$) | $171^{+103}_{-84}$ ($64-317$) | $4350^{+6060}_{-2030}$ ($1740-15420$) | 2640 | M03
NGC 985 | $6.407^{+0.070}_{-0.076}$ ($6.281-6.509$) | $2.2^{+1.6}_{-1.2}$ ($0.6-4.4$) | $170^{+127}_{-94}$ ($46-344$) | $9550^{+11190}_{-4760}$ ($3810-29590$) | 7500 | M03
ESO 198$-$G24(2) | $6.385^{+0.013}_{-0.019}$ ($6.353-6.404$) | $1.2^{+0.7}_{-0.6}$ ($0.4-2.2$) | $158^{+105}_{-80}$ ($57-304$) | $<4220$ ($0-5840$) | 6400 | Z05
ESO 198$-$G24(total) | $6.382^{+0.025}_{-0.043}$ ($6.306-6.426$) | $0.9^{+0.8}_{-0.4}$ ($0.3-2.1$) | $117^{+110}_{-50}$ ($40-279$) | $2940^{+8830}_{-1880}$ ($0-15500$) | $\dots$ | $\dots$
3C 120 | $6.410^{+0.016}_{-0.015}$ ($6.389-6.439)$ | $3.4^{+1.9}_{-1.5}$ ($1.4-6.0$) | $66^{+37}_{-30}$ ($27-117$) | $2230^{+2280}_{-1650}$ ($0-5950$) | 5370 | W09
NGC 2110(1) | $6.389^{+0.098}_{-0.026}$ ($6.341-6.521)$ | $6.8^{+10.1}_{-3.3}$ ($2.5-20.1$) | $116^{+171}_{-56}$ ($43-342$) | $4070^{+15260}_{-2470}$ ($0-24160$) | 1200⋆ | M07
NGC 2110(3) | $6.394^{+0.009}_{-0.007}$ ($6.384-6.407)$ | $5.3^{+2.1}_{-1.8}$ ($2.9-8.2$) | $87^{+35}_{-29}$ ($48-135$) | $<2540$ ($0-3160$) | $\dots$ | $\dots$
NGC 2110(4) | $6.395^{+0.010}_{-0.010}$ ($6.381-6.409)$ | $5.2^{+1.7}_{-1.5}$ ($3.3-7.6$) | $127^{+40}_{-37}$ ($80-184$) | $2510^{+2070}_{-1240}$ ($940-5600$) | $\dots$ | $\dots$
NGC 2110(total) | $6.397^{+0.006}_{-0.006}$ ($6.389-6.405)$ | $5.3^{+1.0}_{-1.2}$ ($3.8-6.6$) | $103^{+21}_{-23}$ ($75-129$) | $2320^{+810}_{-800}$ ($1320-3510$) | $\dots$ | $\dots$
PG 0844$+$349(1) | $6.583^{+0.122}_{-0.116}$ $(6.422-6.770)$ | $2.5^{+1.6}_{-1.3}$ $(0.8-4.7)$ | $587^{+384}_{-303}$ $(189-1117)$ | $20320^{+13170}_{-8080}$ $(8900-44490)$ | 2150 | Z05
MCG $-$5-23-16(1) | 6.384${}^{+0.011}_{-0.011}$ $(6.369-6.399)$ | 10.6${}^{+3.2}_{-3.1}$ $(6.5-15.0)$ | 82${}^{+25}_{-24}$ $(51-117)$ | 2630${}^{+1340}_{-880}$ $(1470-4560)$ | 1450† | L02
MCG $-$5-23-16(2) | 6.408${}^{+0.024}_{-0.033}$ $(6.359-6.452)$ | 8.1${}^{+5.1}_{-4.1}$ $(2.8-15.1)$ | 65${}^{+42}_{-33}$ $(23-122)$ | 3810${}^{+4880}_{-1690}$ $(1540-18350)$ | $\dots$ | $\dots$
MCG $-$5-23-16(3) | 6.388${}^{+0.019}_{-0.024}$ $(6.352-6.416)$ | 9.6${}^{+5.5}_{-4.7}$ $(3.7-18.0)$ | 78${}^{+44}_{-38}$ $(30-146)$ | 2660${}^{+4150}_{-1580}$ $(610-10420)$ | $\dots$ | $\dots$
MCG $-$5-23-16(total) | 6.388${}^{+0.009}_{-0.009}$ $(6.377-6.400)$ | 9.0${}^{+2.1}_{-2.2}$ $(6.1-12.1)$ | 71${}^{+17}_{-17}$ $(48-96)$ | 2560${}^{+1130}_{-900}$ $(1390-4180)$ | $\dots$ | $\dots$
NGC 3516(1) | 6.392${}^{+0.005}_{-0.006}$ $(6.385-6.399)$ | 3.9${}^{+1.2}_{-1.3}$ $(2.3-5.6)$ | 110${}^{+34}_{-36}$ $(65-158)$ | $<1670$ $(0-3160)$ | 3353$\pm 310$ | P04
NGC 3516(2) | 6.408${}^{+0.010}_{-0.011}$ $(6.393-6.422)$ | 4.4${}^{+2.0}_{-1.7}$ $(2.2-7.2)$ | 186${}^{+85}_{-71}$ $(93-306)$ | 1740${}^{+1420}_{-1210}$ $(0-4020)$ | $\dots$ | $\dots$
NGC 3516(3) | 6.402${}^{+0.017}_{-0.014}$ $(6.382-6.425)$ | 4.5${}^{+1.6}_{-1.4}$ $(2.7-6.7)$ | 157${}^{+56}_{-49}$ $(94-234)$ | 4290${}^{+2180}_{-1470}$ $(2450-8050)$ | $\dots$ | $\dots$
NGC 3516(4) | 6.409${}^{+0.023}_{-0.025}$ $(6.374-6.442)$ | 4.2${}^{+3.1}_{-2.0}$ $(1.6-8.3)$ | 67${}^{+51}_{-31}$ $(26-134)$ | 3220${}^{+3020}_{-1630}$ $(960-8190)$ | $\dots$ | $\dots$
NGC 3516(5) | 6.354${}^{+0.057}_{-0.079}$ $(6.241-6.431)$ | 5.1${}^{+4.1}_{-2.6}$ $(1.7-10.6)$ | 101${}^{+81}_{-52}$ $(34-209)$ | 8480${}^{+7050}_{-4700}$ $(3110-18840)$ | $\dots$ | $\dots$
NGC 3516(6) | 6.407${}^{+0.033}_{-0.034}$ $(6.364-6.451)$ | 7.6${}^{+3.7}_{-3.4}$ $(3.3-12.6)$ | 108${}^{+52}_{-49}$ $(47-178)$ | 6030${}^{+3550}_{-2430}$ $(2970-11240)$ | $\dots$ | $\dots$
NGC 3516(7) | 6.414${}^{+0.017}_{-0.017}$ $(6.389-6.437)$ | 4.2${}^{+2.4}_{-2.0}$ $(1.7-7.5)$ | 55${}^{+32}_{-26}$ $(23-99)$ | 2290${}^{+2150}_{-1310}$ $(460-5900)$ | $\dots$ | $\dots$
NGC 3516(total) | 6.404${}^{+0.007}_{-0.007}$ $(6.395-6.413)$ | 4.4${}^{+0.8}_{-0.7}$ $(3.4-5.5)$ | 91${}^{+17}_{-14}$ $(71-114)$ | 3180${}^{+880}_{-670}$ $(2310-4390)$ | $\dots$ | $\dots$
NGC 3783(1) | 6.396${}^{+0.014}_{-0.013}$ $(6.377-6.415)$ | 5.4${}^{+2.7}_{-2.6}$ $(2.3-9.1)$ | 69${}^{+36}_{-33}$ $(30-118)$ | $<4670$ $(0-5780)$ | 3570$\pm$190 | N06
NGC 3783(2) | 6.401${}^{+0.006}_{-0.0063}$ $(6.392-6.410)$ | 5.1${}^{+1.5}_{-1.1}$ $(3.6-7.1)$ | 75${}^{+21}_{-16}$ $(53-104)$ | 1930${}^{+1080}_{-900}$ $(750-3490)$ | $\dots$ | $\dots$
NGC 3783(3) | 6.391${}^{+0.008}_{-0.008}$ $(6.380-6.401)$ | 6.2${}^{+1.4}_{-1.5}$ $(4.3-8.2)$ | 90${}^{+22}_{-21}$ $(63-120)$ | 2700${}^{+1180}_{-1050}$ $(1410-4430)$ | $\dots$ | $\dots$
NGC 3783(4) | 6.395${}^{+0.005}_{-0.006}$ $(6.388-6.402)$ | 6.0${}^{+1.3}_{-1.4}$ $(4.2-7.8)$ | 88${}^{+19}_{-21}$ $(62-114)$ | 1860${}^{+880}_{-1140}$ $(0-3140)$ | $\dots$ | $\dots$
NGC 3783(5) | 6.395${}^{+0.004}_{-0.005}$ $(6.388-6.401)$ | 6.3${}^{+1.5}_{-1.1}(4.8-8.3)$ | 69${}^{+16}_{-12}$ $(53-91)$ | 1280${}^{+720}_{-630}$ $(0-2260)$ | $\dots$ | $\dots$
NGC 3783(6) | 6.399${}^{+0.006}_{-0.006}$ $(6.391-6.408)$ | 4.7${}^{+1.2}_{-1.2}$ $(3.2-6.4)$ | 57${}^{+14}_{-15}$ $(39-77)$ | 1520${}^{+890}_{-940}$ $(0-2750)$ | $\dots$ | $\dots$
NGC 3783(total) | 6.396${}^{+0.003}_{-0.002}$ $(6.393-6.399)$ | 5.6${}^{+0.5}_{-0.6}$ $(4.8-6.3)$ | 74${}^{+7}_{-8}$ $(63-83)$ | 1750${}^{+360}_{-360}$ $(1270-2240)$ | $\dots$ | $\dots$
NGC 4051 | 6.417${}^{+0.039}_{-0.036}$ $(5.750-6.474)$ | 3.5${}^{+1.4}_{-1.4}$ $(1.6-5.5)$ | 195${}^{+79}_{-78}$ $(89-307)$ | 6430${}^{+11800}_{-2860}$ $(2840-479470)$ | 1200 | W09
NGC 4151(1) | 6.396${}^{+0.006}_{-0.006}$ $(6.386-6.404)$ | 21.7${}^{+3.3}_{-4.1}$ $(16.5-26.3)$ | 190${}^{+29}_{-36}$ $(146-231)$ | 2150${}^{+1220}_{-680}$ $(1250-3840)$ | 6350 | W09
NGC 4151(2) | 6.391${}^{+0.004}_{-0.004}$ $(6.386-6.397)$ | 18.2${}^{+2.7}_{-2.5}$ $(14.9-21.8)$ | 78${}^{+11}_{-11}$ $(64-93)$ | 2170${}^{+610}_{-540}$ $(1460-3000)$ | $\dots$ | $\dots$
NGC 4151(3) | 6.396${}^{+0.006}_{-0.005}$ $(6.389-6.404)$ | 20.3${}^{+3.6}_{-3.5}$ $(15.7-25.3)$ | 83${}^{+15}_{-14}$ $(64-103)$ | 2670${}^{+790}_{-680}$ $(1760-3770)$ | $\dots$ | $\dots$
NGC 4151(4) | 6.400${}^{+0.006}_{-0.005}$ $(6.393-6.408)$ | 11.5${}^{+2.9}_{-2.6}$ $(8.1-15.5)$ | 188${}^{+47}_{-43}$ $(132-253)$ | 1710${}^{+860}_{-740}$ $(690-2940)$ | $\dots$ | $\dots$
NGC 4151(5) | 6.393${}^{+0.016}_{-0.008}$ $(6.382-6.416)$ | 14.3${}^{+8.3}_{-4.2}$ $(8.9-25.4)$ | 57${}^{+33}_{-17}$ $(36-101)$ | 2020${}^{+3600}_{-870}$ $(420-6750)$ | $\dots$ | $\dots$
NGC 4151(total) | 6.394${}^{+0.003}_{-0.002}$ $(6.391-6.398)$ | 17.5${}^{+1.6}_{-1.5}$ $(15.5-19.7)$ | 87${}^{+8}_{-8}$ $(77-98)$ | 2250${}^{+400}_{-360}$ $(1770-2790)$ | $\dots$ | $\dots$
3C 273(1) | 6.336${}^{+0.074}_{-0.053}$ $(6.259-6.491)$ | 5.9${}^{+5.0}_{-4.0}$ $(0.6-13)$ | 35${}^{+29}_{-24}$ $(4-76)$ | 5900${}^{+8640}_{-5830}$ $(0-80630)$ | 3520 | Z05
NGC 4593 | 6.406${}^{+0.011}_{-0.042}$ $(6.351-6.421)$ | 3.8${}^{+3.4}_{-1.4}$ $(2.0-8.5)$ | 82${}^{+74}_{-30}$ $(43-185)$ | 2230${}^{+8180}_{-1100}$ $(670-15320)$ | 3650 | W09
MCG $-$6-30-15(1) | 6.399${}^{+0.043}_{-0.045}$ $(6.335-6.461)$ | 3.4${}^{+1.5}_{-1.8}$ $(1.0-6.1)$ | 70${}^{+31}_{-37}$ $(21-125)$ | 7440${}^{+5710}_{-4630}$ $(2480-16270)$ | 1700$\pm$170 | N06
MCG $-$6-30-15(2) | 6.395${}^{+0.061}_{-0.044}$ $(6.298-6.485)$ | 1.9${}^{+1.4}_{-0.9}$ $(0.7-3.9)$ | 40${}^{+30}_{-19}$ $(15-83)$ | 2810${}^{+15450}_{-1750}$ $(0-23770)$ | $\dots$ | $\dots$
MCG $-$6-30-15(3) | 6.424${}^{+0.094}_{-0.110}$ $(6.237-6.557)$ | 2.8${}^{+2.1}_{-1.5}$ $(0.8-5.7)$ | 62${}^{+45}_{-34}$ $(17-124)$ | 13590${}^{+12820}_{-6050}$ $(5760-37720)$ | $\dots$ | $\dots$
MCG $-$6-30-15(4) | 6.402${}^{+0.154}_{-0.023}$ $(6.364-6.602)$ | 1.0${}^{+1.0}_{-0.7}$ $(0.1-2.3)$ | 20${}^{+20}_{-14}$ $(2-45)$ | $<14800$ $(0-28590)$ | $\dots$ | $\dots$
MCG $-$6-30-15(5) | 6.345${}^{+0.027}_{-0.024}$ $(6.292-6.393)$ | 2.2${}^{+2.3}_{-1.4}$ $(0.3-5.3)$ | 46${}^{+48}_{-29}$ $(6-111)$ | $<5850$ $(0-15200)$ | $\dots$ | $\dots$
MCG $-$6-30-15(total) | 6.427${}^{+0.044}_{-0.044}$ $(6.366-6.486)$ | 2.7${}^{+1.1}_{-0.9}$ $(1.5-4.1)$ | 58${}^{+23}_{-20}$ $(32-88)$ | 11880${}^{+4650}_{-4030}$ $(6480-18750)$ | $\dots$ | $\dots$
IRAS 13349$+$2438(2) | 6.396${}^{+0.046}_{-0.057}$ | 0.7${}^{+0.5}_{-0.4}$ $(0.2-1.5)$ | 170${}^{+139}_{-93}$ $(52-386)$ | 5870${}^{+11370}_{-2550}$ $(2660-85550)$ | $\dots$ | $\dots$
IRAS 13349$+$2438(total) | 6.405${}^{+0.052}_{-0.113}$ $(6.106-6.841)$ | 0.4${}^{+0.5}_{-0.2}$ $(0.07-2.3)$ | 87${}^{+121}_{-41}$ $(16-532)$ | 5150${}^{+60200}_{-2810}$ $(1770-93230)$ | $\dots$ | $\dots$
IC 4329A | 6.305${}^{+0.139}_{-0.096}$ $(6.172-6.542)$ | 15.8${}^{+10.6}_{-7.5}$ $(5.9-31.5)$ | 81${}^{+54}_{-38}$ $(30-162)$ | 18830${}^{+18590}_{-9620}$ $(5820-48080)$ | 5620$\pm 200$ | N06
Mrk 279 | 6.414${}^{+0.054}_{-0.028}$ $(6.312-6.560)$ | 2.0${}^{+1.2}_{-0.9}$ $(0.9-6.9)$ | 132${}^{+80}_{-59}$ $(60-458)$ | 5080${}^{+8390}_{-1940}$ $(2670-48780)$ | 5150 | W09
NGC 5506 | 6.400${}^{+0.007}_{-0.006}$ $(6.391-6.409)$ | 7.1${}^{+1.6}_{-2.1}$ $(4.4-9.3)$ | 84${}^{+18}_{-25}$ $(52-109)$ | 1650${}^{+880}_{-870}$ $(470-2940)$ | 1850 | Z05
NGC 5548(1) | 6.398${}^{+0.022}_{-0.021}$ $(6.367-6.427)$ | 3.7${}^{+1.5}_{-1.4}$ $(1.9-5.8)$ | 124${}^{+51}_{-46}$ $(64-195)$ | 4410${}^{+2590}_{-1580}$ $(2390-8500)$ | 5830$\pm 230$ | N06
NGC 5548(2) | 6.402${}^{+0.009}_{-0.009}$ $(6.389-6.415)$ | 2.4${}^{+0.9}_{-0.7}$ $(1.4-3.6)$ | 71${}^{+27}_{-21}$ $(42-107)$ | 1960${}^{+1040}_{-900}$ $(800-3540)$ | $\dots$ | $\dots$
NGC 5548(total) | 6.403${}^{+0.009}_{-0.009}$ $(6.391-6.415)$ | 2.7${}^{+0.8}_{-0.7}$ $(1.8-3.7)$ | 84${}^{+25}_{-22}$ $(56-115)$ | 2540${}^{+1140}_{-820}$ $(1490-4240)$ | $\dots$ | $\dots$
Mrk 290(total) | 6.404${}^{+0.037}_{-0.038}$ $(6.342-6.458)$ | 1.0${}^{+0.6}_{-0.5}$ $(0.2-1.9)$ | 60${}^{+33}_{-31}$ $(12-110)$ | 5290${}^{+5120}_{-2420}$ $(2140-20200)$ | 4740 | W09
E1821$+$643 | 6.447${}^{+0.051}_{-0.054}$ $(6.355-6.517)$ | 3.2${}^{+1.6}_{-1.3}$ $(1.6-5.4)$ | 153${}^{+75}_{-63}$ $(76-257)$ | 10920${}^{+7710}_{-3910}$ $(5640-25950)$ | 6620$\pm 720$ | N06
3C 382(2) | 6.424${}^{+0.064}_{-0.090}$ $(6.254-6.531)$ | 3.6${}^{+3.2}_{-2.3}$ $(0.7-8.1)$ | 66${}^{+55}_{-43}$ $(13-144)$ | 8100${}^{+12580}_{-4490}$ $(3150-33320)$ | 8340 | W09
3C 382(total) | 6.418${}^{+0.084}_{-0.097}$ $(6.161-6.538)$ | 3.4${}^{+2.3}_{-2.5}$ $(0.4-7.1)$ | 57${}^{+39}_{-42}$ $(7-119)$ | 10730${}^{+15810}_{-8320}$ $(2060-55310)$ | $\dots$ | $\dots$
4C $+$74.26(total) | 6.260${}^{+0.038}_{-0.081}$ $(6.125-6.392)$ | 1.2${}^{+2.0}_{-0.8}$ $(0.1-4.1)$ | 36${}^{+60}_{-24}$ $(3-124)$ | $<10980$ $(0-23620)$ | 9420 | W09
Mrk 509 | 6.428${}^{+0.020}_{-0.021}$ $(6.396-6.455)$ | 3.6${}^{+2.0}_{-1.7}$ $(1.4-6.4)$ | 57${}^{+30}_{-27}$ $(22-100)$ | 2910${}^{+2590}_{-1250}$ $(1280-7900)$ | 3430$\pm 240$ | N06
NGC 7213(1) | 6.392${}^{+0.013}_{-0.011}$ $(6.377-6.410)$ | 2.9${}^{+1.1}_{-1.0}$ $(1.6-4.4)$ | 117${}^{+46}_{-40}$ $(65-179)$ | 2290${}^{+1950}_{-1390}$ $(390-5000)$ | 3200 | Z05
NGC 7213(2) | 6.410${}^{+0.018}_{-0.018}$ $(6.384-6.436)$ | 3.2${}^{+2.0}_{-1.6}$ $(1.2-6.0)$ | 126${}^{+78}_{-63}$ $(47-236)$ | 2400${}^{+2310}_{-1800}$ $(0-6770)$ | $\dots$ | $\dots$
NGC 7213(total) | 6.397${}^{+0.011}_{-0.010}$ $(6.384-6.412)$ | 3.0${}^{+1.0}_{-0.9}$ $(1.8-4.3)$ | 120${}^{+42}_{-35}$ $(73-174)$ | 2590${}^{+1470}_{-1170}$ $(1050-4620)$ | $\dots$ | $\dots$
NGC 7469(1) | 6.385${}^{+0.010}_{-0.012}$ $(6.364-6.399)$ | 3.2${}^{+1.1}_{-1.2}$ $(1.7-5.1)$ | 116${}^{+42}_{-42}$ $(63-187)$ | 1800${}^{+2640}_{-1360}$ $(0-6040)$ | 2820 | W09
NGC 7469(2) | 6.395${}^{+0.036}_{-0.033}$ $(6.347-6.452)$ | 3.9${}^{+2.0}_{-1.6}$ $(1.8-6.6)$ | 156${}^{+79}_{-64}$ $(72-262)$ | 6780${}^{+4810}_{-3170}$ $(3200-14100)$ | $\dots$ | $\dots$
NGC 7469(total) | 6.388${}^{+0.018}_{-0.017}$ $(6.365-6.413)$ | 3.7${}^{+1.0}_{-1.1}$ $(2.2-5.2)$ | 142${}^{+38}_{-42}$ $(84-199)$ | 4890${}^{+2770}_{-1700}$ $(2740-8740)$ | $\dots$ | $\dots$
Note. — Results from Chandra HEG data, fitted with a power law plus Gaussian
emission-line model in the 2–7 keV band. Statistical errors are for the 68%
confidence level, whilst parentheses show the 90% confidence level ranges of
the parameters. ⋆ Broad polarized H$\beta$ line. † Infra-red broad Br$\alpha$
line. a Gaussian line center energy in keV. b Emission-line intensity in units
of $\rm 10^{-5}\ photons\ cm^{-2}\ s^{-1}$. c Emission line equivalent width
in units of eV. d Full width half maximum, rounded to $10\ \rm km\ s^{-1}$. e
References for H$\beta$ FWHM: L02$-$Lutz et al. (2002); M03$-$Marziani et al.
(2003); M07$-$Moran et al. (2007); N06$-$Nandra (2006); P04$-$Peterson et al.
(2004); W09$-$Wang et al. (2009); Z05$-$Zhou et al. (2005).
Table 3: Mean Fe K$\alpha$ Line Spectral Parameters
Parameter | By Observation | # Spectraa | By Source | # Sourcesb
---|---|---|---|---
Centroid Energy (keV) | $6.396\pm 0.0004$ c | 68 | $6.397\pm 0.0005$ c | 32
($\sigma_{\rm Fe~{}K}$ fixed) | | | |
Centroid Energy (keV) | $6.388\pm 0.001$ | 68 | $6.398\pm 0.002$ | 32
($\sigma_{\rm Fe~{}K}$ free)d | | | |
EW (eV) | $42\pm 2$ | 70 | $44\pm 2$ | 33
($\sigma_{\rm Fe~{}K}$ fixed) | | | |
EW (eV) | $53\pm 3$ | 70 | $70\pm 4$ | 33
($\sigma_{\rm Fe~{}K}$ free)d | | | |
FWHM ($\rm km\ s^{-1}$) | $2060\pm 230$ | 53 | $2200\pm 220$ | 27
Note. — Weighted mean quantities from spectral fitting to individual spectra
(“by observation”), and to spectra representative of each source (“by
source”). See text, §4 for details. a Number of spectra (one per observation)
contributing to the mean quantities. b Number of unique sources contributing
to the mean quantities. c These statistical errors are smaller than the
systematic errors (see §4.1 for discussion). d The intrinsic width of the Fe
K$\alpha$ in these cases was free in the spectral fitting for 51 spectra in
the individual observation fits (see Table 2) and for 27 sources in the fits
to source-representative spectra (see §4 for details).
Table 4: X-ray Baldwin Effect: Fe K$\alpha$ Line EW versus Luminosity Fits
Parameter | By Observation | By Source
---|---|---
$\log{EW}=k_{L}+[m_{L}\log{(L_{x})}]$ | |
$\chi^{2}$ (d.o.f.) | 58.5(68) | 24.7(31)
intercept, $k_{L}$ | $1.80^{+0.02}_{-0.02}$ | $1.76^{+0.02}_{-0.02}$
slope, $m_{L}$ (68% confidence errors) | $-0.22^{+0.03}_{-0.03}$ | $-0.13^{+0.04}_{-0.04}$
slope, $m_{L}$ (99% confidence errors) | $-0.22^{+0.10}_{-0.07}$ | $-0.13^{+0.11}_{-0.11}$
$\Delta\chi^{2}$ for $m_{L}=0$ | 39.3 | 10.0
Significance for $m_{L}\neq 0$ | $6.27\sigma$ | $3.24\sigma$
$\log{EW}=k_{R}+[m_{R}\log{(L_{x}/L_{\rm EDD})}]$ | |
$\chi^{2}$ (d.o.f.) | 61.4 (68) | 25.2(31)
intercept, $k_{R}$ (68% confidence errors) | $1.31^{+0.09}_{-0.09}$ | $1.50^{+0.09}_{-0.09}$
slope, $m_{R}$ (68% confidence errors) | $-0.20^{+0.03}_{-0.03}$ | $-0.11^{+0.04}_{-0.04}$
slope, $m_{R}$ (99% confidence errors) | $-0.20^{+0.07}_{-0.11}$ | $-0.11^{+0.09}_{-0.09}$
$\Delta\chi^{2}$ for $m_{R}=0$ | 36.6 | 9.5
Significance for $m_{R}\neq 0$ | $6.05\sigma$ | $3.08\sigma$
Note. — Results of fitting the relations between the derived Fe K$\alpha$ line
EWs and the 2–10 keV X-ray luminosity ($L_{x}$), and between the EWs and the
Eddington ratio ($L_{x}/L_{\rm EDD}$). Coefficients and their error bounds are
shown for linear fits to $\log{EW}$ versus $\log{(L_{x})}$ ($k_{L}$, $m_{L}$),
and to $\log{EW}$ versus $\log{(L_{x}/L_{\rm EDD})}$ for spectral fitting
results to the individual spectra (“by observation”), and to the source-
representative spectra. See §5 for details. The number of degrees of freedom
for each fit (d.o.f.) is shown in parentheses after each best-fitting
$\chi^{2}$ value.
|
arxiv-papers
| 2010-03-09T06:09:35 |
2024-09-04T02:49:08.936357
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "X. W. Shu(USTC, JHU), T. Yaqoob(JHU) and J. X. Wang(USTC)",
"submitter": "Xinwen Shu",
"url": "https://arxiv.org/abs/1003.1790"
}
|
1003.1813
|
# Lidskii-type formulae for Dixmier traces
Sedaev A.A Department of Mathematics, Voronezh State University of
Architecture and Civil Engineering, 20-letiya Oktyabrya 84, Voronezh, 394006,
Russia sed@vmail.ru Sukochev F.A School of Mathematics and Statistics,
University of New South Wales, Kensington, NSW 2052, Australia
f.sukochev@unsw.edu.au Zanin D.V School of Computer Science, Engineering and
Mathematics, Flinders University, Bedford Park, SA 5042, Australia
zani0005@csem.flinders.edu.au
###### Abstract
We establish several analogues of the classical Lidskii Theorem for some
special classes of singular traces (Dixmier traces and Connes-Dixmier traces)
used in noncommutative geometry.
###### Key words and phrases:
Dixmier traces, Lidskii formula
###### 1991 Mathematics Subject Classification:
46L52, 47B10, 46E30
††thanks: The first author was partially supported by RFBR 08-01-00226
## 1\. Introduction and Preliminaries
### 1.1. Dixmier-Macaev ideal and Dixmier traces
An important role in noncommutative geometry [7] is played by the set of
compact operators whose partial sums of singular values are logarithmically
divergent. This set can be adequately described using the terminology of
Marcinkiewicz spaces. Consider the Marcinkiewicz sequence space
$m_{1,\infty}:=\\{x=\\{x_{n}\\}_{n=1}^{\infty}:\|x\|_{m_{1,\infty}}<\infty\\},$
where we set
$\|x\|_{m_{1,\infty}}=\sup_{N}\frac{1}{\log(N+1)}\sum_{n=1}^{N}x_{n}^{*}.$
Here, $\\{x_{n}^{*}\\}_{n=1}^{\infty}$ is the sequence
$\\{|x_{n}|\\}_{n=1}^{\infty}$ rearranged in nonincreasing order.
Fix an infinite-dimensional separable complex Hilbert space $H$ and consider
the set $\mathcal{M}_{1,\infty}$ of all compact operators $x$ on $H$ such that
the sequence of its singular values $\\{s_{n}(T)\\}_{n=1}^{\infty}$ falls into
the space $m_{1,\infty}$ (recall that the singular values of a compact
operator $T$ are the eigenvalues of the operator $|T|=(T^{*}T)^{1/2}$). We set
$\|T\|_{\mathcal{M}_{1,\infty}}:=\|\\{s_{n}(T)\\}\|_{m_{1,\infty}}.$
It is well known that the ideal of compact operators $\mathcal{M}_{1,\infty}$
equipped with the norm $\|\cdot\|_{\mathcal{M}_{1,\infty}}$ is a Banach space.
We refer to the recent paper [16] by Pietsch for additional references and
information on these spaces.
We describe briefly a construction of singular traces on the ideal
$\mathcal{M}_{1,\infty}$ due to Dixmier [8] and its various modifications
which are of importance in noncommutative geometry [7]. For a more detailed
treatment we refer to [6].
Let $\sigma_{n}$, $n\geq 1$ be the operator on $l_{\infty}$ defined by
$\sigma_{n}(x_{1},\ldots,x_{k},\ldots)=(\underbrace{x_{1},\ldots,x_{1}}_{n\text{-times}},\underbrace{x_{2},\ldots,x_{2}}_{n\text{-times}},\ldots,\underbrace{x_{k},\ldots,x_{k}}_{n\text{-times}},\ldots).$
Let $\omega$ be a $\sigma_{n}$-invariant generalised limit on $l_{\infty},$
that is, $\omega$ is a positive normalised functional on $l_{\infty}$ such
that $\omega(\sigma_{n}(x))=\omega(x)$ for all $x\in l_{\infty}$ and such that
$\omega|_{c_{0}}=0,$ where $c_{0}$ is the subspace of all vanishing sequences.
For an element $0\leq T\in\mathcal{M}_{1,\infty}$ we set
$\tau_{\omega}(T):=\omega(\\{\frac{1}{\log(N+1)}\sum_{n=1}^{N}s_{n}(T)\\}_{N=1}^{\infty}).$
It is well known (see e.g. § 5 in [6] and additional references therein) that
$\tau_{\omega}$ is an additive functional on the positive part of
$\mathcal{M}_{1,\infty}.$ Thus, $\tau_{\omega}$ admits a linear extension to a
unitarily invariant functional (trace) on $\mathcal{M}_{1,\infty}.$ This trace
vanishes on all finite-dimensional operators from $B(H).$ Such singular traces
are called Dixmier traces (see [8]).
A smaller subclass of Dixmier traces was introduced by Connes in [7] by
observing that a functional $\omega=\gamma\circ M$ is $\sigma_{n}$-invariant
state on $l_{\infty}$ for all $n\geq 1.$ Here, $\gamma$ is an arbitrary
generalised limit on the space $L_{\infty}(0,\infty)$ of all bounded
measurable functions and the operator $M$ is a Cesaro operator defined by the
formula
$(Mx)(t)=\frac{1}{\log(t)}\int_{1}^{t}\frac{x(s)ds}{s}.$
Referring to $\omega$ above as a functional on $l_{\infty},$ we tacitly apply
an isometric embedding $i:l_{\infty}\to L_{\infty}(0,\infty)$ given by
$\\{x_{j}\\}_{j=1}^{\infty}\stackrel{{\scriptstyle
i}}{{\mapsto}}\sum_{j=1}^{\infty}x_{j}\chi_{[j-1,j)},$
where $\chi_{[j-1,j)}$ is the characteristic function of the interval
$[j-1,j).$ Dixmier traces $\tau_{\omega}$ defined such $\omega$’s are termed
Connes-Dixmier traces. We refer to [7] and [15, 6] for discussion of their
properties.
Finally, various formulae of noncommutative geometry (in particular, those
involving heat kernel estimates and generalised $\zeta-$function) were
established in [3, 5, 7] for yet a smaller subset of Connes-Dixmier traces,
when the functional $\omega$ was assumed to be $M$-invariant. This class (and
its further modifications) was first introduced in [3] (see also [10]) and
further studied and used in [2, 1, 5]. For brevity we refer to the latter
class (a proper subclass of Connes-Dixmier traces) as a class of $M$-invariant
Dixmier traces.
### 1.2. Lidskii formula for $M$-invariant Dixmier traces in [3, 1, 2]
In the case, when we deal with the standard trace ${\rm Tr}$ and the standard
trace class $\mathcal{S}_{1}$ of compact operators from $B(H),$ the classical
Lidskii Theorem asserts that the trace
${\rm Tr}(T)=\sum_{n\geq 1}\lambda_{n}(T)$
for any $T\in\mathcal{S}_{1}.$ Here, $\\{\lambda_{n}(T)\\}_{n\geq 1}$ is the
sequence of eigenvalues of $T,$ taken in any order. This arbitrariness of the
order is due to the absolute convergence of the series $\sum_{n\geq
1}|\lambda_{n}(T)|.$ In particular, we can choose the decreasing order of
absolute values of $\lambda_{n}(T)$ and counting multiplicities.
The core difference of this situation with the setting of Dixmier traces
living on the ideal $\mathcal{M}_{1,\infty}$ consists in the fact that the
series $\sum_{n\geq 1}|\lambda_{n}(T)|$ generally speaking diverges for every
$T\in\mathcal{M}_{1,\infty}.$ For simplicity, we explain the emerging obstacle
in the case of a self-adjoint operator
$T=T^{*}=T_{+}-T_{-}\in\mathcal{M}_{1,\infty}.$ For such $T,$ by the
definition, $\tau_{\omega}(T)=\tau_{\omega}(T_{+})-\tau_{\omega}(T_{-}),$
where
$\tau_{\omega}(T_{\pm})=\omega(\\{\frac{1}{\log(N)}\sum_{n=1}^{N}\lambda_{n}(T_{\pm})\\}).$
Even in this case, it is not clear why the equality
$\tau_{\omega}(T)=\omega(\\{\frac{1}{\log(N)}\sum_{n=1}^{N}\lambda_{n}(T)\\})$
should hold for the special enumeration of the set
$\\{\lambda_{n}(T)\\}_{n\geq 1}$ given by the decreasing order of absolute
values of $|\lambda_{n}(T)|;$ or for that matter for _any_ enumeration of this
set.
The following result from [1] establishes the equality above under significant
additional constraints on $\tau_{\omega}$ and $T\in\mathcal{M}_{1,\infty}.$
###### Theorem 1.
Let $\omega$ be $M$-invariant and let $T\in\mathcal{M}_{1,\infty}$ satisfy the
assumption $s_{n}(T)\leq C/n$ for some $C>0$ and all $n\geq 1$. We have
$\tau_{\omega}(T)=\omega(\frac{1}{\log(n)}\sum_{|\lambda|>1/n,\lambda\in\sigma(T)}\lambda),$
where $\sigma(T)$ is the spectrum of $T.$
In the case when $T$ is a positive arbitrary element from
$\mathcal{M}_{1,\infty}$ and $\omega$ is taken from a rather special subset of
all $M$-invariant generalised limits (termed in [4] ”maximally invariant
Dixmier functionals”) this result can be already found in [3, Proposition
2.4]. In [2, Theorem 1], the assertion from [3, Proposition 2.4] was extended
to an arbitrary $M$-invariant $\omega.$ Another modification of the class of
$\omega$’s for which the result of [3, Proposition 2.4] and [2, Theorem 1]
holds is given in [5, Proposition 4.3].
### 1.3. Statement of main results
In this paper we prove significant extensions and generalisations of Theorem 1
from [1], [3, Proposition 2.4], [2, Theorem 1] and [5, Proposition 4.3]. Many
of our results are established for a general class of Marcinkiewicz ideals.
Here, for convenience of the reader, we restate these results for traces on
$\mathcal{M}_{1,\infty}.$ Our first main result shows that the assertion of
Theorem 1 holds for an arbitrary Connes-Dixmier trace $\tau_{\omega}$.
###### Theorem 2.
Let $\tau_{\omega}$ be a Connes-Dixmier trace on $\mathcal{M}_{1,\infty}.$ We
have
$\tau_{\omega}(T)=\omega(\frac{1}{\log(n)}\sum_{|\lambda|>1/n,\lambda\in\sigma(T)}\lambda),\quad
T\in\mathcal{M}_{1,\infty}.$ (1)
Theorem 2 follows immediately from Theorem 14 below.
Our second main result is the answer to a natural question whether formula (1)
holds for every Dixmier trace. This question is answered in negative in
Theorem 5.
Our third (and the last) main result answers in the affirmative the question
whether there exists a modification of the summation method used in formula
(1) ensuring that it holds for all Dixmier traces.
###### Theorem 3.
Let $\tau_{\omega}$ be a Dixmier trace on $\mathcal{M}_{1,\infty}.$ We have
$\tau_{\omega}(T)=\omega(\frac{1}{\log(n)}\sum_{\lambda\in\sigma(T),|\lambda|>\log(n)/n}\lambda),\quad
T\in\mathcal{M}_{1,\infty}.$
Theorem 3 follows immediately from Theorem 31 below.
At the end of the paper we also provide an application of our results. The
result proved in the last section concerns heat kernel type formulae from
noncommutative geometry (see [7, 3, 5, 2]) and has been already established in
[18] with a rather arcane argument. We present here a very simple approach to
these formulae.
### 1.4. Marcinkiewicz spaces and singular traces
It is convenient to consider the general class of Marcinkiewicz spaces since
many of our results hold for this class with no extra effort. We frequently
use commutative results as a stepping stone to obtain their noncommutative
analogues.
Recall that the distribution function $n_{x}$ of a bounded measurable function
$x$ is defined by the formula
$n_{x}(t)=m(\\{s,\ |x(s)|>t,\quad t>0\\}).$
We write $x^{*}$ for the decreasing rearrangement of the function $x$: $x^{*}$
is the right continuous non-increasing function whose distribution function
coincides with that of $|x|$ (see [14]).
The following formula is frequently used in the proofs below sometimes without
explicit referencing.
$\int_{0}^{n_{x}(t)}x^{*}(s)ds=-\int_{t}^{\infty}\lambda dn_{x}(\lambda).$ (2)
Here, $z$ is any positive number.
Marcinkiewicz spaces are a special case of fully symmetric function and
sequence spaces, see [14]. Denote by $\Psi$ the class of all concave
increasing functions such that $\psi(\infty)=\infty,$ $\psi(t)=O(t)$ as $t\to
0$ and $\psi(t)=o(t)$ as $t\to\infty.$ For every $\psi\in\Psi,$ Marcinkiewicz
space $M_{\psi}$ is a set of all bounded measurable functions $x$ on
$[0,\infty)$ such that
$\|x\|_{M_{\psi}}:=\sup_{t>0}\frac{1}{\psi(t)}\int_{0}^{t}x^{*}(s)ds<\infty.$
(3)
Marcinkiewicz sequence space $m_{\psi}$ is a set of sequences (see e.g. [16,
6]) satisfying the condition
$\|x\|_{m_{\psi}}=\sup_{n}\frac{1}{\psi(n)}\sum_{k=1}^{n}x_{n}^{*}<\infty.$
In this paper, we mainly work with functions $\psi\in\Psi$ satisfying the
following condition.
$\limsup_{t\to\infty}\frac{\psi(2t)}{\psi(t)}<2.$ (4)
Let $K(H)$ be the ideal of all compact operators. If $m_{\psi}$ is a
Marcinkiewicz sequence space, then the corresponding Marcinkiewicz operator
space $\mathcal{M}_{\psi}$ is the set of all $T\in K(H)$ such that
$\\{s_{n}(T)\\}\in m_{\psi}$ equipped with the norm
$\|T\|_{\mathcal{M}_{\psi}}:=\|\\{s_{n}(T)\\}\|_{m_{\psi}}.$
Let $\psi\in\Psi$ and let $\omega$ be a dilation invariant generalised limit.
The mapping $\tau_{\omega}$ defined by the formula
$\tau_{\omega}(x):=\omega(\frac{1}{\psi(t)}\int_{0}^{t}x^{*}(s)ds)$
is a subadditive homogeneous functional on $M_{\psi}^{+}.$ If $\tau_{\omega}$
is additive on $M_{\psi}^{+},$ then $\tau_{\omega}$ is called Dixmier trace
generated by $\omega$. We refer the reader to [11, 9, 10] for conditions which
guarantee the additivity of $\tau_{\omega}.$ It is well known that
$\tau_{\omega}$ is additive for any $\omega$ as above when
$\lim_{t\to\infty}\frac{\psi(2t)}{\psi(t)}=1.$ (5)
Similarly, the definitions of Connes-Dixmier traces and $M$-invariant traces
naturally extend to denote corresponding singular traces on Marcinkiewicz
ideals $\mathcal{M}_{\psi}$ (see [15]).
Our main result for general Dixmier traces on ideals $\mathcal{M}_{\psi}$ is
given in Theorem 31 which asserts that for any Dixmier trace $\tau_{\omega}$
on $\mathcal{M}_{\psi}$ with $\psi\in\Psi$ satisfying condition (4) we have
$\tau_{\omega}(T)=\omega(\frac{1}{\psi(n)}\sum_{\lambda\in\sigma(T),|\lambda|>\psi(n)/n}\lambda),\quad
T\in\mathcal{M}_{\psi}.$ (6)
The result of Theorem 3 follows immediately from the formula above, if we set
$\psi(t)=\log(t)$ for all $t\geq 2.$
### 1.5. Failure of (1) for Dixmier traces
Here, we show that there are Dixmier traces $\tau_{\omega}$ on
$\mathcal{M}_{1,\infty}$ for which formula (1) fails. To this end we use
$\omega$ provided by the lemma below.
Define a subadditive functional $\pi:L_{\infty}(0,\infty)\to\mathbb{R}$ by the
formula
$\pi(x)=\limsup_{N\to\infty}\frac{1}{\log(\log(N))}\int_{N}^{N\log(N)}\frac{x(s)ds}{s}.$
Clearly, $\pi$ is positive and homogeneous.
The following lemma is routine. We include the proof for convenience of the
reader.
###### Lemma 4.
1. Let $x\in L_{\infty}(0,\infty)$ be an arbitrary positive element.
2. (1)
If $\omega\in L_{\infty}(0,\infty)^{*}$ such that $\omega\leq\pi,$ then
$\omega$ is dilation invariant generalised limit.
3. (2)
If $\pi(x)>0,$ then there exists a dilation invariant generalised limit
$\omega$ such that $\omega(x)>0.$
###### Proof.
We prove the first assertion and then derive the second one from it.
1. (1)
At first we note that by assumption
$-\pi(-y)\leq\omega(y)\leq\pi(y)$ (7)
for every $y\in L_{\infty}(0,\infty).$ Note that $\pi(-y)\leq 0$ for every
$0\leq y\in L_{\infty}(0,\infty).$ It follows that $\omega$ is positive.
Further, for every $y\in L_{\infty}(0,\infty)$ we have
$|\int_{N}^{N\log(N)}\frac{(y-\sigma_{n}y)(s)ds}{s}|=|\int_{N\log(N)/n}^{N\log(N)}\frac{y(s)ds}{s}-\int_{N/n}^{N}\frac{y(s)ds}{s}|.$
Therefore,
$|\pi(y-\sigma_{n}y)|\leq\limsup_{N\to\infty}\frac{1}{N\log(N)}\cdot
2\|y\|_{\infty}\cdot|\log(n)|=0.$
Hence,
$\omega(y-\sigma_{n}y)\leq\pi(y-\sigma_{n}y)=0,\quad\omega(-y)\leq\pi(\sigma_{n}y-y)=0.$
Thus, $\omega$ is dilation invariant.
If $y\in L_{\infty}(0,\infty)$ is such that $y(t)\to 0$ as $t\to\infty,$ then
$\pi(y)=\pi(-y)=0.$ It follows from (7) that $\omega(y)=0.$
Noting that $\omega(1)=1,$ we conclude that $\omega$ is dilation invariant
generalised limit.
2. (2)
Consider linear space $x\mathbb{R}$ spanned by element $x.$ Set
$\omega(\lambda x)=\lambda\pi(x)$ for every $\lambda\in\mathbb{R}.$ It follows
that $\omega\leq\pi$ on $x\mathbb{R}.$ By the Hahn-Banach theorem, there
exists a functional $\omega\in L_{\infty}(0,\infty)^{*}$ such that
$\omega(x)=\pi(x)$ and $\omega\leq\pi.$ It follows from above that $\omega$ is
a dilation invariant generalised limit.
∎
###### Theorem 5.
There exist a positive function $x\in M_{1,\infty}$ and a Dixmier trace
$\tau_{\omega}$ such that
$\tau_{\omega}(x)\neq\omega(\frac{-1}{\log(t)}\int_{1/t}^{\infty}\lambda
dn_{x}(\lambda)).$ (8)
###### Proof.
Define a function $x$ by the formula
$x=\sup_{k}e^{-e^{k}}\chi_{[1,e^{k+e^{k}}]}.$
If $t\in[e^{k-1+e^{k-1}},e^{k+e^{k}}],$ then
$\frac{1}{\log(t)}\int_{1}^{t}x^{*}(s)ds\leq
e^{1-k}\int_{1}^{e^{k+e^{k}}}x^{*}(s)ds\leq
e^{1-k}\sum_{n=1}^{k}e^{-e^{n}}\cdot e^{n+e^{n}}\leq\frac{e^{2}}{e-1}.$
Thus, $x\in M_{1,\infty}.$
We claim that
$\limsup_{N\to\infty}\frac{1}{\log(\log(N))}\int_{N}^{N\log(N)}(\frac{1}{\log(t)}\int_{t}^{n_{x}(1/t)}x^{*}(s)ds)dt>0.$
Set $N=e^{e^{k}}.$ It is clear that $n_{x}(1/t)=e^{k+e^{k}}$ for every
$t\in[N,N\log(N)].$ Since $x^{*}(s)=e^{-e^{k}}$ for every $s\in[t,n_{x}(1/t)]$
and every $t\in[N,N\log(N)],$ we can rewrite the expression under the limit in
the left-hand side as
$\frac{1}{k}\int_{e^{e^{k}}}^{e^{k+e^{k}}}\frac{e^{k+e^{k}}-t}{e^{e^{k}}t\log(t)}dt=\frac{e^{k}}{k}\int_{e^{e^{k}}}^{e^{k+e^{k}}}\frac{dt}{t\log(t)}-\frac{1}{ke^{e^{k}}}\int_{e^{e^{k}}}^{e^{k+e^{k}}}\frac{dt}{\log(t)}=$
$=\frac{e^{k}}{k}\log(1+\frac{k}{e^{k}})-o(1)=1+o(1).$
This proves the claim.
Thus,
$\pi(\frac{1}{\log(t)}(\int_{1}^{n_{x}(1/t)}x^{*}(s)ds-\int_{1}^{t}x^{*}(s)ds))>0.$
The assertion of the theorem now follows from Lemma 4 and (2). ∎
## 2\. Lidskii formula for Connes-Dixmier traces
In this section, we extend results of [1] (and, partially, those of [2]) to a
wider class of Marcinkiewicz spaces and Connes-Dixmier traces. To this end, we
need some extra assumptions on $\psi\in\Psi.$ The need of such additional
conditions is seen from the example below, which shows that analogue of
formula (1) for an arbitrary $\psi\in\Psi$ fails.
###### Example 6.
Let $\psi(t)=\exp(\sqrt{\log(t)})$ and let $x=\psi^{\prime}.$ If
$\tau_{\omega}$ is a Dixmier trace on $M_{\psi},$ then
$e^{1/2}\tau_{\omega}(x)\leq\omega(\frac{-1}{\psi(t)}\int_{1/t}^{\infty}\lambda
dn_{x}(\lambda)).$
###### Proof.
It is clear that $x(t)=\exp(\sqrt{\log(2)})/2t\sqrt{\log(t)}.$ We have
$\frac{t\exp(\sqrt{\log(t)})}{2\sqrt{\log(t)}}\leq n_{x}(1/t).$
for all sufficiently large $t.$ Hence,
$e^{1/2}+o(1)\leq\frac{\psi(n_{x}(1/t))}{\psi(t)}.$
The assertion follows immediately. ∎
Thus, some additional restrictions on the function $\psi$ are needed. We
require the following condition
$\lim_{t\to\infty}\frac{\psi(t\psi(t))}{\psi(t)}=1.$ (9)
It is clear that (5) holds and, therefore, Marcinkiewicz space $M_{\psi}$
admits nonzero Dixmier traces (see [11],[9],[10]).
Now we show that formula (1) holds for all Connes-Dixmier traces on
$\mathcal{M}_{\psi}.$
###### Lemma 7.
Let $\psi\in\Psi$ satisfy condition (9). If $c>\|x\|_{M_{\psi}},$ we have
$d_{x}(1/t)\leq ct\psi(t)$
for every $x\in M_{\psi}$ and every sufficiently large $t.$
###### Proof.
Assume the contrary. Hence, there exists a sequence $t_{k}\to\infty$ such that
$x^{*}(s)\geq 1/t_{k}$ for every $s\in[0,ct_{k}\psi(t_{k})].$ By the
definition of Marcinkiewicz norm,
$\|x\|_{M_{\psi}}\geq\frac{1}{\psi(ct_{k}\psi(t_{k}))}\int_{0}^{ct_{k}\psi(t_{k})}x^{*}(s)ds\geq\frac{c\psi(t_{k})}{\psi(ct_{k}\psi(t_{k}))}.$
It follows from (9) that
$\frac{c\psi(t_{k})}{\psi(ct_{k}\psi(t_{k}))}\to c.$
The contradiction proves the Lemma. ∎
###### Remark 8.
Let $0\leq x,y\in L_{\infty}(0,\infty)$ and let $y(t)=x(t)\cdot(1+o(1))$ as
$t\to\infty.$ If $x\notin L_{1}(0,\infty),$ we have
$\int_{1}^{T}y(s)ds=(1+o(1))\int_{1}^{T}x(s)ds.$
###### Lemma 9.
Let $\psi\in\Psi$ satisfy condition (9). We have
$\frac{1}{\log(T)}\int_{1}^{T}\frac{dt}{t\psi(t)}\int_{0}^{ct\psi(t)}x(s)ds=\frac{1}{\log(T)}\int_{1}^{T}\frac{dt}{t\psi(t)}\int_{0}^{t}x(s)ds+o(1)$
as $T\to\infty$ for every positive $x=x^{*}\in M_{\psi}$ and every $c>0.$
###### Proof.
The assertion is linear with respect to $x.$ Since the assertion holds for
$x(t)=\psi^{\prime}(t),$ it is sufficient to verify it for $x+\psi^{\prime}$
instead of $x.$ Hence, we may assume that $x(t)\geq\psi^{\prime}(t).$ Thus,
integral in the right-hand side is unbounded as $T\to\infty.$
Make a substitution $z=ct\psi(t)$ in the left-hand side integral. It follows
from the condition (9) that
$\frac{dt}{t\psi(t)}=\frac{dz}{z\psi(z)}(1+o(1)).$
Indeed, by Lagrange theorem, we have
$\psi(z)=\psi(t)(1+o(1)),\quad\frac{dz}{z}=\frac{dt}{t}(1+\frac{t\psi^{\prime}(t)}{\psi(t)})=\frac{dt}{t}(1+o(1)).$
It follows from Remark 8 that
$\int_{1}^{T}\frac{dt}{t\psi(t)}\int_{0}^{ct\psi(t)}x^{*}(s)ds=(1+o(1))\int_{c\psi(1)}^{cT\psi(T)}\frac{dz}{z\psi(z)}\int_{0}^{z}x^{*}(s)ds.$
(10)
Evidently,
$\int_{T}^{cT\psi(T)}\frac{dz}{z\psi(z)}\int_{0}^{z}x^{*}(s)ds=O(\int_{T}^{cT\psi(T)}\frac{dz}{z})=o(1).$
(11)
Noting that
$\int_{1}^{cT\psi(T)}=\int_{1}^{T}+\int_{T}^{cT\psi(T)}$
the combination of (10) and (11) yields the assertion. ∎
###### Lemma 10.
Let $\psi\in\Psi$ satisfy the condition (9) and let $\tau_{\omega}$ be a
Connes-Dixmier trace on $M_{\psi}.$ We have
$\omega(\frac{-1}{\psi(t)}\int_{1/t}^{\infty}\lambda
dn_{x}(\lambda))\leq\tau_{\omega}(x)$
for every positive $x\in M_{1,\infty}.$
###### Proof.
Due to (2) and Lemma 7 we have
$\omega(\frac{1}{\psi(t)}\int_{0}^{n_{x}(1/t)}x^{*}(s)ds)\leq\omega(\frac{1}{\psi(t)}\int_{0}^{ct\psi(t)}x^{*}(s)ds)=$
$=\gamma(M(\frac{1}{\psi(t)}\int_{0}^{t}x^{*}(s)ds)+o(1))=\gamma(M(\frac{1}{\log(t)}\int_{0}^{t}x^{*}(s)ds))=\tau_{\omega}(x).$
∎
###### Lemma 11.
Let $\psi\in\Psi$ and let $\tau_{\omega}$ be a Dixmier trace on $M_{\psi}.$ We
have
$\tau_{\omega}(x)\leq\omega(\frac{-1}{\psi(t)}\int_{1/t}^{\infty}\lambda
dn_{x}(\lambda))$
for every positive $x\in M_{\psi}.$
###### Proof.
We claim that
$\int_{0}^{t}x^{*}(s)ds\leq\int_{0}^{n_{x}(1/t)}x^{*}(s)ds+1.$
The inequality is evident if $t\leq n_{x}(1/t).$ If $t>n_{x}(1/t),$ then
$x^{*}(s)\leq 1/t$ for every $s\in[n_{x}(1/t),t].$ It follows that
$\int_{0}^{t}x^{*}(s)ds=\int_{0}^{n_{x}(1/t)}x^{*}(s)ds+\int_{n_{x}(1/t)}^{t}x^{*}(s)ds\leq$
$\leq\int_{0}^{n_{x}(1/t)}x^{*}(s)ds+(t-n_{x}(1/t))\cdot t^{-1}.$
Thus, claim holds in either case.
It follows that
$\tau_{\omega}(x)\leq\omega(\frac{1}{\psi(t)}\int_{1}^{n_{x}(1/t)}x^{*}(s)ds)+\omega(\frac{1}{\psi(t)}).$
The assertion follows immediately. ∎
The next theorem follows immediately from Lemma 10 and Lemma 11.
###### Theorem 12.
Let $\psi\in\Psi$ satisfy the condition (9) and let $\tau_{\omega}$ be a
Connes-Dixmier trace on $M_{\psi}.$ We have
$\tau_{\omega}(x)=\omega(\frac{-1}{\psi(t)}\int_{1/t}^{\infty}\lambda
dn_{x}(\lambda))$
for every positive $x\in M_{\psi}.$
###### Remark 13.
Consider weak space $M_{\psi}^{w}$ (the smallest symmetric ideal containing
$\psi^{\prime}$). Suppose that $\psi$ satisfies the condition (9). If
$\tau_{\omega}$ is an arbitrary Dixmier trace on $M_{\psi},$ then we have
$\tau_{\omega}(x)=\omega(\frac{-1}{\psi(t)}\int_{1/t}^{\infty}\lambda
dn_{x}(\lambda))$
for every positive $x\in M_{\psi}^{w}.$ Using Lemma 7, the equality above
follows immediately.
Arguing as in the section 4 below, we obtain a noncommutative version of
Theorem 12, which strengthens [2, Theorem 1] and [1, Corollary 2.12] (see
Theorem 1).
###### Theorem 14.
Let $\psi\in\Psi$ satisfy the condition (9) and let $\tau_{\omega}$ be a
Connes-Dixmier trace on $\mathcal{M}_{\psi}.$ We have
$\tau_{\omega}(T)=\omega(\frac{1}{\psi(n)}\sum_{|\lambda|>1/n,\lambda\in\sigma(S)}\lambda)$
for every operator $T\in\mathcal{M}_{\psi}.$
## 3\. Adjusted Lidskii formula for Dixmier traces: commutative setting
As we have seen in Theorem 5, formula (1) does not hold for Dixmier traces
$\tau_{\omega}.$ In this section, we consider a modification of formula (1)
which holds for all Dixmier traces $\tau_{\omega}$ on a commutative
Marcinkiewicz space $M_{\psi},$ $\psi\in\Psi.$
###### Lemma 15.
Let $\psi\in\Psi$ satisfy condition (4). If $0\leq x\in M_{\psi},$ then there
exists a constant $c(x)\in\mathbb{N}$ such that
$n_{x}(\frac{\psi(t)}{t})\leq c(x)t$
for every sufficiently large $t.$
###### Proof.
Set $\varphi(t)=t/\psi(t).$ It follows from (4) that there exists a constant
$\alpha>0$ and $t_{0}>0$ such that
$\varphi(2t)\geq 2^{\alpha}\varphi(t)$
for every $t\geq t_{0}.$ Thus,
$\varphi(2^{n}t)\geq 2^{n\alpha}\varphi(t)$
for $t\geq t_{0}.$
Consider sets $A$ and $B$ defined by the formula
$A:=\\{s:\
x^{*}(s)>\frac{\psi(t)}{t}\\}\subset\\{s:\|x\|_{M_{\psi}}\frac{\psi(s)}{s}>\frac{\psi(t)}{t}\\}=:B.$
Fix $c=2^{n}$ such that $2^{n\alpha}\geq\max\\{1,\|x\|_{M_{\psi}}\\}.$ It
follows that $\varphi(ct)>\|x\|_{M_{\psi}}\varphi(t)$ for all $t\geq t_{0}.$
Therefore, $ct\notin B$ if $t\geq t_{0}.$ Since $\varphi$ is an increasing
function (see [14]), we have $\sup B\leq ct$ for $t\geq t_{0}.$ Since $B$ is
an interval, we have $m(B)\leq ct$ provided that $t\geq t_{0}.$ Thus, for
$t\geq t_{0},$ we have $n_{x}(\psi(t)/t)=m(A)\leq m(B)\leq ct.$ ∎
###### Remark 16.
Let $\psi\in\Psi$ and let $\tau_{\omega}$ be a Dixmier trace on $M_{\psi}.$ We
have
$\omega(\frac{\psi(nt)}{\psi(t)})=1$
for every $n\geq 1.$ Indeed, if $\tau_{\omega}$ is linear then (see [11])
$\omega(\frac{\psi(nt)}{\psi(t)})=\tau_{\omega}(n\sigma_{1/n}\psi^{\prime})=\tau_{\omega}(\psi^{\prime})=1.$
This remark is frequently used below together with the following lemma from
[9].
###### Lemma 17.
Let $\omega\in L_{\infty}(0,\infty)^{*}$ be an arbitrary generalised limit. If
$x,y\in L_{\infty}(0,\infty)$ are such that $\omega(|x-1|)=0,$ then
$\omega(xy)=\omega(y).$
###### Lemma 18.
Let $\psi\in\Psi$ satisfy the condition (4) and let $\tau_{\omega}$ be a
Dixmier trace on $M_{\psi}.$ We have
$\omega(\frac{-1}{\psi(t)}\int_{\psi(t)/t}^{\infty}\lambda
dn_{x}(\lambda))\leq\tau_{\omega}(x)$
for every positive $x\in M_{\psi}.$
###### Proof.
Let $c(x)$ be the constant defined in Lemma 15. Clearly,
$\frac{1}{\psi(t)}\int_{0}^{n_{x}(\psi(t)/t)}x^{*}(s)ds=(\frac{\psi(c(x)t)}{\psi(t)})\cdot(\frac{1}{\psi(c(x)t)}\int_{0}^{n_{x}(\psi(t)/t)}x^{*}(s)ds).$
It follows from Remark 16 and Lemma 17 that
$\omega(\frac{1}{\psi(t)}\int_{0}^{n_{x}(\psi(t)/t)}x^{*}(s)ds)=\omega(\frac{1}{\psi(c(x)t)}\int_{0}^{n_{x}(\psi(t)/t)}x^{*}(s)ds).$
It follows from Lemma 15 that
$\omega(\frac{1}{\psi(c(x)t)}\int_{0}^{n_{x}(\psi(t)/t)}x^{*}(s)ds)\leq\omega(\frac{1}{\psi(c(x)t)}\int_{0}^{c(x)t}x^{*}(s)ds).$
However, since $\omega$ is dilation invariant, we have
$\omega(\frac{1}{\psi(c(x)t)}\int_{0}^{c(x)t}x^{*}(s)ds)=\omega(\frac{1}{\psi(t)}\int_{0}^{t}x^{*}(s)ds).$
∎
###### Lemma 19.
Let $\psi\in\Psi$ and let $\tau_{\omega}$ be a Dixmier trace on $M_{\psi}.$ We
have
$\tau_{\omega}(x)\leq\omega(\frac{-1}{\psi(t)}\int_{\psi(t)/t}^{\infty}\lambda
dn_{x}(\lambda))$
for every positive $x\in M_{\psi}.$
###### Proof.
Fix $n\in\mathbb{N}.$ Clearly,
$\frac{1}{\psi(t)}\int_{0}^{t}x^{*}(s)ds=(\frac{\psi(nt)}{\psi(t)})\cdot(\frac{1}{\psi(nt)}\int_{0}^{t}x^{*}(s)ds).$
It follows from Remark 16 and Lemma 17 that
$\tau_{\omega}(x)=\omega(\frac{1}{\psi(nt)}\int_{0}^{t}x^{*}(s)ds).$ (12)
We claim that
$\int_{0}^{t}x^{*}(s)ds\leq\int_{0}^{n_{x}(\psi(nt)/nt)}x^{*}(s)ds+\frac{1}{n}\psi(nt).$
The inequality is evident if $t\leq n_{x}(\psi(nt)/nt).$ If
$t>n_{x}(\psi(nt)/nt),$ then $x^{*}(s)\leq\psi(nt)/nt$ for every
$s\in[n_{x}(\psi(nt)/nt),t].$ Thus,
$\int_{0}^{t}x^{*}(s)ds=\int_{0}^{n_{x}(\psi(nt)/nt)}x^{*}(s)ds+\int_{n_{x}(\psi(nt)/nt)}^{t}x^{*}(s)ds\leq$
$\leq\int_{0}^{n_{x}(\psi(nt)/nt)}x^{*}(s)ds+(t-n_{x}(\frac{\psi(nt)}{nt}))\cdot\frac{\psi(nt)}{nt}$
and the claim follows.
Hence,
$\omega(\frac{1}{\psi(nt)}\int_{0}^{t}x^{*}(s)ds)\leq\omega(\frac{1}{\psi(nt)}\int_{0}^{n_{x}(\psi(nt)/nt)}x^{*}(s)ds)+\frac{1}{n}.$
It follows from (12) and the dilation-invariance of $\omega$ that
$\tau_{\omega}(x)\leq\omega(\frac{1}{\psi(t)}\int_{0}^{n_{x}(\psi(t)/t)}x^{*}(s)ds)+\frac{1}{n}.$
Since $n$ is arbitrary large, we are done. ∎
The following theorem is the principal result of this section. It follows
immediately from Lemmas 18 and 19.
###### Theorem 20.
Let $\psi\in\Psi$ satisfy the condition (4) and let $\tau_{\omega}$ be a
Dixmier trace on $M_{\psi}.$ We have
$\tau_{\omega}(x)=\omega(\frac{-1}{\psi(t)}\int_{\psi(t)/t}^{\infty}\lambda
dn_{x}(\lambda))$
for every positive $x\in M_{\psi}.$
Arguing similarly, one can obtain similar assertion for Marcinkiewicz sequence
spaces.
###### Theorem 21.
Let $\psi\in\Psi$ satisfy the condition (4) and let $\tau_{\omega}$ be a
Dixmier trace on $m_{\psi}.$ We have
$\tau_{\omega}(x)=\omega(\frac{1}{\psi(n)}\sum_{x_{k}\geq\psi(n)/n}x_{k})$
for every positive $x\in m_{\psi}.$
## 4\. Adjusted Lidskii formula for Dixmier traces: noncommutative setting
In this section, we extend preceding results to Dixmier traces on
Marcinkiewicz operator ideals.
### 4.1. Adjusted Lidskii formula for Dixmier traces: normal operators
The following assertion follows directly from the Theorem 21.
###### Lemma 22.
Let $\psi\in\Psi$ satisfy the condition (4) and let $\tau_{\omega}$ be a
Dixmier trace on $\mathcal{M}_{\psi}.$ We have
$\tau_{\omega}(S)=\omega(\frac{1}{\psi(n)}\sum_{\lambda\in\sigma(S),|\lambda|>\psi(n)/n}\lambda)$
for every self-adjoint operator $S\in\mathcal{M}_{\psi}.$
The following three lemmas are used to extend the formula above to the case of
normal operators.
###### Lemma 23.
Let $\psi\in\Psi$ satisfy the condition (4) and let $\tau_{\omega}$ be a
Dixmier trace on $M_{\psi}.$ We have
$\omega(\frac{1}{t}n_{x}(\frac{\psi(t)}{t}))=0$
for every positive $x\in M_{\psi}.$ A similar assertion holds for
Marcinkiewicz sequence space $m_{\psi}.$
###### Proof.
Fix $n\in\mathbb{N}.$ It follows from the dilation-invariance of $\omega$ that
$\omega(\frac{1}{t}n_{x}(\frac{\psi(t)}{t}))=\omega(\frac{1}{nt}n_{x}(\frac{\psi(nt)}{nt})).$
(13)
It is clear that
$\frac{1}{nt}n_{x}(\frac{\psi(nt)}{nt})=\frac{1}{n}+\frac{1}{\psi(nt)}\int_{t}^{n_{x}(\psi(nt)/nt)}\frac{\psi(nt)}{nt}ds.$
If $t>n_{x}(\psi(nt)/nt),$ we have
$\int_{t}^{n_{x}(\psi(nt)/nt)}\frac{\psi(nt)}{nt}ds\leq 0.$
If $t\leq n_{x}(\psi(nt)/nt),$ then
$\int_{t}^{n_{x}(\psi(nt)/nt)}\frac{\psi(nt)}{nt}ds\leq\int_{t}^{n_{x}(\psi(nt)/nt)}x^{*}(s)ds\leq\int_{t}^{c(x)nt}x^{*}(s)ds.$
The last inequality holds for all sufficiently large $t$ by Lemma 15.
In either case,
$0\leq\frac{1}{nt}n_{x}(\frac{\psi(nt)}{nt})\leq\frac{1}{n}+\frac{1}{\psi(nt)}\int_{t}^{c(x)nt}x^{*}(s)ds.$
It follows now from the (13) that
$\omega(\frac{1}{t}n_{x}(\frac{\psi(t)}{t}))\leq\frac{1}{n}+\omega(\frac{1}{\psi(nt)}\int_{t}^{c(x)nt}x^{*}(s)ds).$
It is clear that
$\omega(\frac{1}{\psi(nt)}\int_{t}^{c(x)nt}x^{*}(s)ds)=$
$=\omega(\frac{1}{\psi(nt)}\int_{0}^{c(x)nt}x^{*}(s)ds)-\omega(\frac{1}{\psi(nt)}\int_{0}^{t}x^{*}(s)ds).$
It follows from the dilation-invariance of $\omega$ that
$\omega(\frac{1}{\psi(nt)}\int_{t}^{c(x)nt}x^{*}(s)ds)=\omega(\frac{1}{\psi(t)}\int_{0}^{c(x)t}x^{*}(s)ds)-\omega(\frac{1}{\psi(nt)}\int_{0}^{t}x^{*}(s)ds).$
It follows from Remark 16 and Lemma 17 that both terms in the right-hand side
of the equality above are equal to $\tau_{\omega}(x).$
Therefore,
$\omega(\frac{1}{t}n_{x}(\frac{\psi(t)}{t}))\leq\frac{1}{n}.$
Since $n$ is arbitrary large, we are done. ∎
###### Lemma 24.
Let $\psi\in\Psi$ satisfy the condition (4) and let $\tau_{\omega}$ be a
Dixmier trace on $\mathcal{M}_{\psi}.$ We have
$\omega(\frac{1}{\psi(n)}\sum_{|\Re\lambda|>\psi(n)/n,|\Im\lambda|\leq\psi(n)/n,\lambda\in\sigma(S)}\Im\lambda)=0,$
$\omega(\frac{1}{\psi(n)}\sum_{|\Re\lambda|\leq\psi(n)/n,|\Im\lambda|>\psi(n)/n,\lambda\in\sigma(S)}\Re\lambda)=0$
for any normal operator $S\in\mathcal{M}_{\psi}.$
###### Proof.
We prove the first assertion only. Proof of the second one is identical.
Note that $\lambda\in\sigma(S)$ if and only if $|\lambda|\in\sigma(|S|).$ It
follows immediately that
$|\sum_{|\Re\lambda|>\psi(n)/n,|\Im\lambda|\leq\psi(n)/n,\lambda\in\sigma(S)}\Im\lambda|\leq\sum_{|\Re\lambda|>\psi(n)/n,|\Im\lambda|\leq\psi(n)/n,\lambda\in\sigma(S)}\frac{\psi(n)}{n}\leq$
$\leq\frac{\psi(n)}{n}\sum_{|\lambda|>\psi(n)/n,\lambda\in\sigma(S)}1=\frac{\psi(n)}{n}\sum_{\lambda>\psi(n)/n,\lambda\in\sigma(|S|)}1=\frac{\psi(n)}{n}n_{|S|}(\frac{\psi(n)}{n}).$
The assertion follows now from Lemma 23. ∎
###### Lemma 25.
Let $\psi\in\Psi$ satisfy the condition (4) and let $\tau_{\omega}$ be a
Dixmier trace on $\mathcal{M}_{\psi}.$ We have
$\omega(\frac{1}{\psi(n)}\sum_{|\Re\lambda|,|\Im\lambda|\leq\psi(n)/n,|\lambda|>\psi(n)/n,\lambda\in\sigma(S)}\lambda)=0$
for any normal operator $S\in\mathcal{M}_{\psi}.$
###### Proof.
It is clear that
$|\sum_{|\Re\lambda|,|\Im\lambda|\leq\psi(n)/n,|\lambda|>\psi(n)/n,\lambda\in\sigma(S)}\lambda|\leq\sum_{|\Re\lambda|,|\Im\lambda|\leq\psi(n)/n,|\lambda|>\psi(n)/n,\lambda\in\sigma(S)}|\lambda|\leq$
$\leq\sum_{\psi(n)/n<|\lambda|\leq
2\psi(n)/n,\lambda\in\sigma(S)}|\lambda|\leq\frac{2\psi(n)}{n}\sum_{|\lambda|>\psi(n)/n,\lambda\in\sigma(S)}1=$
$=\frac{2\psi(n)}{n}\sum_{\lambda>\psi(n)/n,\lambda\in\sigma(|S|)}1=\frac{2\psi(n)}{n}n_{|S|}(\frac{\psi(n)}{n}).$
The assertion follows now from Lemma 23. ∎
The following theorem extends result of Lemma 22 to normal operators from
$M_{\psi}.$
###### Theorem 26.
Let $\psi\in\Psi$ satisfy the condition (4) and let $\tau_{\omega}$ be a
Dixmier trace on $\mathcal{M}_{\psi}.$ We have
$\tau_{\omega}(S)=\omega(\frac{1}{\psi(n)}\sum_{|\lambda|>\psi(n)/n,\lambda\in\sigma(S)}\lambda)$
for any normal operator $S\in\mathcal{M}_{\psi}.$
###### Proof.
It follows from Lemma 22 that
$\tau_{\omega}(\Re
S)=\omega(\frac{1}{\psi(n)}\sum_{|\lambda|>\psi(n)/n),\lambda\in\sigma(\Re
S)}\lambda)=\omega(\frac{1}{\psi(n)}\sum_{|\Re\lambda|>\psi(n)/n,\lambda\in\sigma(S)}\Re\lambda).$
By Lemma 24,
$\tau_{\omega}(\Re
S)=\omega(\frac{1}{\psi(n)}\sum_{\max\\{|\Re\lambda|,|\Im\lambda|\\}>\psi(n)/n,\lambda\in\sigma(S)}\Re\lambda).$
The same is valid for $\Im S.$ By the linearity,
$\tau_{\omega}(S)=\omega(\frac{1}{\psi(n)}\sum_{\max\\{|\Re\lambda|,|\Im\lambda|\\}>\psi(n)/n,\lambda\in\sigma(S)}\lambda).$
It follows from Lemma 25 that
$\tau_{\omega}(S)=\omega(\frac{1}{\psi(n)}\sum_{|\lambda|>\psi(n)/n,\lambda\in\sigma(S)}\lambda).$
∎
### 4.2. Adjusted Lidskii formula for Dixmier traces: general case
Recall the following result of Ringrose (see Theorems 1, 6 and 7 from [17]).
###### Theorem 27.
Let $T\in B(H)$ be a compact operator. There exists a projection-valued
measure $E_{\lambda}$ such that
1. (1)
$TE_{\lambda}=E_{\lambda}TE_{\lambda}.$
2. (2)
Either $E_{\lambda}=E_{\lambda-0}$ or
${\rm rank}(E_{\lambda}-E_{\lambda-0})=1.$
3. (3)
If, in addition,
$TE_{\lambda}=E_{\lambda-0}TE_{\lambda},$
then $T$ is quasi-nilpotent.
###### Corollary 28.
Let $T\in B(H)$ be a compact operator. There exist compact normal operator $S$
and compact quasi-nilpotent operator $Q$ such that $T=S+Q$ and
$\sigma(S)=\sigma(T).$
###### Proof.
Define an operator $S$ by the following formula
$S=\sum_{E_{\lambda}\neq
E_{\lambda-0}}(E_{\lambda}-E_{\lambda-0})T(E_{\lambda}-E_{\lambda-0}).$
A straightforward computation shows that the operator $Q=T-S$ satisfies the
condition 3 of the Theorem above. Hence, $Q$ is quasi-nilpotent.
Evidently, $S$ is a diagonal operator with eigenvalues of $T$ on the diagonal.
Hence, $\sigma(S)=\sigma(T).$ ∎
By the Weil theorem, sequence of eigenvalues of $T$ is majorized by the
sequence of its singular values (see Theorem 3.1 from [12]). Hence, for
$T\in\mathcal{M}_{\psi},$ we obtain $S,Q\in\mathcal{M}_{\psi}.$
The following assertion directly follows from the Theorem 3.3 from [13]).
###### Theorem 29.
If $Q\in\mathcal{M}_{\psi}$ is a quasi-nilpotent operator, then $Q$ belongs to
the commutator $[\mathcal{M}_{\psi},B(H)].$
###### Corollary 30.
If $Q\in\mathcal{M}_{\psi}$ is a quasi-nilpotent operator and $\tau_{\omega}$
is an arbitrary Dixmier trace on $M_{\psi},$ then $\tau_{\omega}(Q)=0.$
Indeed, due to [7], we have $\tau_{\omega}([A,B])=0$ for every
$A\in\mathcal{M}_{\psi}$ and every $B\in B(H).$
The following theorem is the main result of this section.
###### Theorem 31.
Let $\psi\in\Psi$ satisfy condition (4) and let $\tau_{\omega}$ be a Dixmier
trace on $\mathcal{M}_{\psi}.$ We have
$\tau_{\omega}(T)=\omega(\frac{1}{\psi(n)}\sum_{\lambda\in\sigma(T),|\lambda|>\psi(n)/n}\lambda)$
for any operator $T\in\mathcal{M}_{\psi}.$
###### Proof.
Let $S$ be a normal operator constructed in Corollary 28. The assertion holds
for $S$ by Theorem 26. Note that $\tau_{\omega}(T)=\tau_{\omega}(S)$ by
Corollaries 28 and 30. Since $\sigma(S)=\sigma(T),$ we are done. ∎
## 5\. Applications to heat kernel formula
In this section, we provide a simple proof of one of the heat semigroup
formulae from [18] (see also earlier results in [3, 5]). Our hypothesis on
$\omega$ is very mild.
###### Lemma 32.
For any positive $x\in M_{1,\infty}$ we have
$M(\frac{1}{\log(t)}\int_{t}^{n_{x}(1/t)}(x^{*}(s)-1/t)ds)=o(1).$
###### Proof.
If $t>n_{x}(1/t),$ we have
$|\int_{t}^{n_{x}(1/t)}(x^{*}(s)-1/t)ds|\leq 1.$
If $t\leq n_{x}(1/t),$ then $x^{*}(s)\geq 1/t$ for every $s\in[t,n_{x}(1/t)].$
Therefore,
$0\leq\int_{t}^{n_{x}(1/t)}(x^{*}(s)-1/t)ds\leq\int_{t}^{n_{x}(1/t)}x^{*}(s)ds.$
The assertion follows now from the Lemma 9. ∎
###### Theorem 33.
Let $\tau_{\omega}$ be a Dixmier trace on $\mathcal{M}_{1,\infty}$ such that
$\omega=\omega\circ M.$ We have
$\tau_{\omega}(T)=\frac{\alpha}{\Gamma(1/\alpha)}\omega(\frac{1}{t}\sum_{\lambda\in\sigma(T)}\exp(-(t\lambda)^{-\alpha}))$
for every positive operator $T\in\mathcal{M}_{1,\infty}.$
###### Proof.
Let $x=x^{*}\in M_{1,\infty}$ be the rearrangement of $T,$ that is
$x=i(\\{s_{n}(T)\\}).$ Without loss of generality, $x\leq 1.$ Since
distributions of $T$ and $x$ coincide, we have
$\omega(\frac{1}{t}\sum_{\lambda\in\sigma(T)}\exp(-(t\lambda)^{-\alpha}))=\omega(\frac{1}{t}\int_{0}^{\infty}\exp(-(tx(s))^{-\alpha})ds).$
Setting $1/x(s)=u,$ we obtain
$\omega(\frac{1}{t}\int_{0}^{\infty}\exp(-(tx(s))^{-\alpha})ds)=\omega(\frac{1}{t}\int_{0}^{\infty}\exp(-(u/t)^{\alpha})dn_{x}(1/u)).$
It follows from the weak version of Karamata Theorem (see [3, 18] for details)
that
$\frac{\alpha}{\Gamma(1/\alpha)}\omega(\frac{1}{t}\sum_{\lambda\in\sigma(T)}\exp(-(t\lambda)^{-\alpha}))=\omega(\frac{1}{t}n_{x}(1/t)).$
It is clear that
$M^{2}(\frac{1}{t}n_{x}(1/t))-M(\frac{1}{\log(t)}\int_{1}^{t}x(s)ds)=$
$=M(\frac{1}{\log(t)}(\int_{1}^{t}\frac{1}{s^{2}}n_{x}(1/s)ds-\int_{1}^{t}x(s)ds)).$
Integrating by parts, we obtain
$\int_{1}^{t}\frac{1}{s^{2}}n_{x}(1/s)ds=-\frac{1}{t}n_{x}(1/t)+\int_{1}^{t}\frac{1}{s}dn_{x}(1/s)=\int_{1}^{n_{x}(1/t)}x(s)ds-\frac{1}{t}n_{x}(1/t).$
Hence,
$\int_{1}^{t}\frac{1}{s^{2}}n_{x}(1/s)ds-\int_{1}^{t}x(s)ds=\int_{t}^{n_{x}(1/t)}(x(s)-1/t)ds-1.$
It follows from the Lemma 32 that
$M^{2}(\frac{1}{t}n_{x}(1/t))-M(\frac{1}{\log(t)}\int_{1}^{t}x(s)ds)=o(1).$
The assertion follows now from the $M-$invariance of $\omega.$ ∎
## References
* [1] Azamov N., Sukochev F. A Lidskii type formula for Dixmier traces, C.R. Math. Acad. Sci. Paris 340 (2005), no. 2, 107–112.
* [2] Benameur M., Fack T. Type II non-commutative geometry. I. Dixmier trace in von Neumann algebras, Adv. Math. 199 (2006), no. 1, 29–87.
* [3] Carey A., Phillips J., Sukochev F. Spectral flow and Dixmier traces, Adv. Math. 173 (2003), no. 1., 68–113.
* [4] Carey A., Phillips J., Rennie A., Sukochev F. The Hochschild class of the Chern character for semifinite spectral triples, J. Funct. Anal., vol. 213, (2004) no. 1, 111–153.
* [5] Carey A., Rennie A., Sedaev A., Sukochev F. The Dixmier trace and asymptotics of zeta functions, J. Funct. Anal. 249 (2007), no. 2, 253–283.
* [6] Carey A., Sukochev F. Dixmier traces and some applications in non-commutative geometry, Russian Math. Surveys 61:6 1039–1099.
* [7] Connes A. Noncommutative Geometry, Academic Press, San Diego, 1994.
* [8] Dixmier J. Existence de traces non normales, C. R. Acad. Sci. Paris, 262 (1966).
* [9] Dodds P., de Pagter B., Sedaev A., Semenov E., Sukochev F. Singular symmetric functionals, (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 290 (2002), Issled. po Linein. Oper. i Teor. Funkts. 30, 42–71, 178; translation in J. Math. Sci. (N. Y.) 124 (2004), no. 2, 4867–4885
* [10] Dodds P., de Pagter B., Sedaev A., Semenov E., Sukochev F. Singular symmetric functionals and Banach limits with additional invariance properties, (Russian) Izv. Ross. Akad. Nauk Ser. Mat., vol. 67, no. 6 (2003) 111–136.
* [11] Dodds P., de Pagter B., Semenov E., Sukochev F. Symmetric functionals and singular traces, Positivity 2 (1998), no. 1, 47–75.
* [12] Gohberg I., Krein M. Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs, Vol. 18 American Mathematical Society, Providence, R.I. 1969
* [13] Kalton N. Spectral characterization of sums of commutators. I, J. Reine Angew. Math. 504 (1998), 115–125.
* [14] Krein S., Petunin Ju. and Semenov E. Interpolation of linear operators, Nauka, Moscow, 1978 (in Russian); English translation in Translations of Math. Monographs, Vol. 54, Amer. Math. Soc., Providence, RI, 1982.
* [15] Lord S., Sedaev A., Sukochev F. Dixmier traces as singular symmetric functionals and applications to measurable operators, J. Funct. Anal. 224 (2005), no. 1, 72–106.
* [16] Pietsch A. About the Banach Envelope of $l_{1,\infty}$, Rev. Mat. Complut. 22 (1) (2009) 209–226.
* [17] Ringrose J. Super-diagonal forms for compact linear operators, Proc. London Math. Soc. (3) 12 (1962) 367–384.
* [18] Sedaev A. Generalized limits and related asymptotic formulas, Math.Notes. 86:4 (2009), 612-627.
|
arxiv-papers
| 2010-03-09T07:26:26 |
2024-09-04T02:49:08.954722
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A.A. Sedaev, F.A. Sukochev, D.V. Zanin",
"submitter": "Dmitriy Zanin",
"url": "https://arxiv.org/abs/1003.1813"
}
|
1003.1817
|
# Orbits in symmetric spaces, II
N. J. Kalton Department of Mathematics, University of Missouri-Columbia,
Columbia, Missouri 65211, U.S.A. kaltonn@missouri.edu , F. A. Sukochev
School of Mathematics and Statistics, University of New South Wales, Sydney,
NSW 2052, Australia f.sukochev@unsw.edu.au and D. Zanin School of Computer
Science, Engineering and Mathematics, Flinders University, Adelaide, SA 5042,
Australia zani0005@csem.flinders.edu.au
###### Abstract.
Suppose $E$ is fully symmetric Banach function space on $(0,1)$ or
$(0,\infty)$ or a fully symmetric Banach sequence space. We give necessary and
sufficient conditions on $f\in E$ so that its orbit $\Omega(f)$ is the closed
convex hull of its extreme points. We also give an application to
symmetrically normed ideals of compact operators on a Hilbert space.
###### Key words and phrases:
Fully symmetric spaces, Hardy-Littlewood majorization, orbits
###### 1991 Mathematics Subject Classification:
46E30, 46B70, 46B20
The first author acknowledges support from NSF grant DMS-0555670; the second
and third authors acknowledge support from the ARC
## 1\. Introduction
Let $I$ be either the interval $(0,1)$ or the semi-axis $(0,\infty)$ and
suppose $f\in L_{1}(I)+L_{\infty}(I).$ We define the orbit $\Omega(f)$ of $f$
to be set of $Tf$ where $T:L_{1}+L_{\infty}\to L_{1}+L_{\infty}$ is an
operator with $\|T\|_{L_{1}\to L_{1}},\|T\|_{L_{\infty}\to L_{\infty}}\leq 1$
(see [Ryff1965, KreinPetuninSemenov1982]). Then it follows from the Calderón-
Mitjagin Theorem [Calderon1966, Mitjagin1965, BennettSharpley1982,
KreinPetuninSemenov1982] that $\Omega(f)$ can be characterized as the set of
$g\in L_{1}+L_{\infty}$ such that
(1) $\int_{0}^{t}g^{*}(s)\,ds\leq\int_{0}^{t}f^{*}(s)\,ds,\qquad 0<t<\infty$
where as usual $f^{*}$ is the decreasing rearrangement of $|f|$ (see §2 for
definitions). This may be written $g\preceq f$ where $\preceq$ is the Hardy-
Littlewood-Polya ordering. Thus $E$ is an exact interpolation space if and
only if it is fully symmetric (see §2).
The extreme points of $\Omega(f)$, which we denote $\partial_{e}\Omega(f)$
were obtained in [Ryff1967] (for case of spaces on $(0,1)$) and
[ChilinKryginSukochev1992] (for the general case). Except in the special case
when $I=(0,\infty)$ and $E\supset L_{\infty}$ these are given by
$\partial_{e}\Omega(f)=\\{g:\ g^{*}=f^{*}\\}$ (see §2 for full details; in the
exceptional cases the extreme points from a subset of this set). Let
$\mathcal{Q}(f)$ be the convex hull of the set $\\{g:\ g^{*}\leq f^{*}\\}.$
Then it is clear that if $E$ is fully symmetric and $f\in E$ the closure
$\mathcal{Q}_{E}(f)$ of $\mathcal{Q}(f)$ in $E$ coincides with the closed
convex hull of $\partial_{e}\Omega(f).$
In [Ryff1965] it was shown for the case of $I=(0,1)$ that the orbit
$\Omega(f)$ is always weakly compact in $L_{1}(0,1).$ It follows from results
in [DoddsSukochevSchluchtermann2001] that if $E$ is an order-continuous
(equivalently, separable) symmetric function space (which is necessarily fully
symmetric) and $f\in E$ then $\Omega(f)$ is weakly compact in $E$. Thus it is
an immediate consequence of the Krein-Milman theorem that $\mathcal{Q}_{E}(f)$
coincides with $\Omega(f).$
For the case of non-separable fully symmetric spaces the situation is less
clear. The example $E=L_{\infty}$ and $f=1$ shows that $\mathcal{Q}_{E}(f)$
may still coincide with $\Omega(f).$ This problem was first investigated by
Braverman and Mekler [BravermanMekler1977] for the unit interval i.e.
$I=(0,1)$. They gave a sufficient condition for $\Omega(f)=\mathcal{Q}_{E}(f)$
in terms of the behavior of the dilation operators $\sigma_{\tau}$ (see §2 for
the appropriate definitions). Precisely they showed that if $E$ is a fully
symmetric Banach function space on $(0,1)$ such that
$\lim_{\tau\to\infty}\frac{\|\sigma_{\tau}\|_{E\to E}}{\tau}=0$
then $\Omega(f)=\mathcal{Q}_{E}(f)$ for every $f\in E.$ This condition is,
however, not necessary since it may fail in separable symmetric spaces (e.g.
$E=L_{1}$).
Recently two of the current authors [SukochevZanin2009] found a necessary and
sufficient condition for the similar problem concerning the positive part of
the orbit. If $f\geq 0$ we denote by $\Omega_{+}(f)$ the set $\\{g:\
g\in\Omega(f),\ g\geq 0\\}.$ In [SukochevZanin2009] it was shown that for a
fully symmetric Banach function space $E$ with a Fatou norm (sometimes called
a weak Fatou property) that if $f\in E_{+}$ then $\Omega_{+}(f)$ coincides
with the closed convex hull of its extreme points if and only if a local
Braverman-Mekler type condition holds. If $I=(0,1)$ or if $I=(0,\infty)$ and
$E$ is not contained in $L_{1}(0,\infty)$ this condition takes the form
(2) $\lim_{\tau\to\infty}\frac{\|\sigma_{\tau}(f^{*})\|_{E}}{\tau}=0.$
If $I=(0,\infty)$ and $E\subset L_{1}$ we must replace (2) by
(3)
$\lim_{\tau\to\infty}\frac{\|\chi_{(0,1)}\sigma_{\tau}(f^{*})\|_{E}}{\tau}=0.$
The results of [SukochevZanin2009] imply that under the same hypotheses on $E$
(full symmetricity and a Fatou norm) that (2) and (3) are sufficient for
$\mathcal{Q}_{E}(f)=\Omega(f).$
Our main result in this paper is to show that, indeed, if $E$ is a fully
symmetric Banach space with a Fatou norm on $(0,1)$ or $(0,\infty)$, (2) and
(3) are necessary and sufficient for $\Omega(f)=\mathcal{Q}_{E}(f).$ These
results are Theorems 4.1, 4.2 and 4.3 below. We also establish the
corresponding result for sequence spaces in Theorem 4.5; sequence spaces were
not covered in [SukochevZanin2009] so we are also able to complete the picture
for the positive part of the orbit.
We conclude the paper with an application to orbits in symmetrically normed
ideals of compact operators on a Hilbert space.
## 2\. Preliminaries
In this section we present some definitions from the theory of symmetric
spaces. For more details on the latter theory we refer to
[KreinPetuninSemenov1982, LindenstraussTzafriri1979, BennettSharpley1982].
Let $I$ denote either $(0,1)$ or on $(0,\infty)$ with Lebesgue measure $\mu$.
If $f\in L_{1}(I)+L_{\infty}(I)$ we denote by $f^{*}$ the decreasing
rearrangement of $f$, i.e.
$f^{*}(t)=\inf_{\mu A=t}\sup_{s\in I\setminus A}|f(s)|.$
If $f,g$ are functions in $L_{1}+L_{\infty}$ we write $g\preceq f$ if
$\int_{0}^{t}g^{*}(s)\,ds\leq\int_{0}^{t}f^{*}(s)\,ds,\qquad t\in I.$
This defines the Hardy-Littlewood-Polya ordering.
A symmetric Banach function space $E$ on $I$ is a linear space with $L_{1}\cap
L_{\infty}\subset E\subset L_{1}+L_{\infty}$, with an associated norm
$\|\cdot\|_{E}$ such that $(E,\|\cdot\|_{E})$ is complete and if $f\in E,\
g\in L_{1}+L_{\infty}$ with $g^{*}\leq f^{*}$ then $g\in E$ and
$\|g\|_{E}\leq\|f\|_{E}.$ We will use $E_{+}$ to denote the positive cone of
$E$ i.e. $\\{f:\ f\in E,\ f\geq 0\text{ a.e.}\\}.$ We will also assume the
normalization that $\|\chi_{(0,1)}\|=1.$ Let
$\varphi_{E}(t)=\|\chi_{(0,t)}\|_{E}$ be the fundamental function of $E.$
$E$ is said to have a Fatou norm if for every sequence
$(f_{n})_{n=1}^{\infty}$ of nonnegative functions such that $f_{n}\uparrow f$
a.e. with $f\in E$ we have $\lim_{n\to\infty}\|f_{n}\|_{E}=\|f\|_{E}.$
A symmetric Banach function space $E$ is said to be fully symmetric if and
only if $f\in E,\ g\in L_{1}+L_{\infty}$ with $g\preceq f$, then $g\in E$ and
$||f||_{E}\leq||g||_{E}.$ $E$ is fully symmetric precisely when $E$ is an
exact interpolation space for the couple $(L_{\infty}(I),L_{1}(I))$ by the
Calderón-Mitjagin theorem [Mitjagin1965, Calderon1966]. In this paper we will
only consider fully symmetric Banach function spaces.
We will need the following inequality can be found in
[KreinPetuninSemenov1982], Theorem II.3.1. If $f,g\in L_{1}+L_{\infty},$ then
(4) $(f^{*}-g^{*})\preceq(f-g)^{*}.$
As a consequence if $E$ is fully symmetric and $f,g\in E$ we have
(5) $\|f^{*}-g^{*}\|_{E}\leq\|f-g\|_{E}.$
If $E$ is a fully symmetric Banach function space and $f\in E$ we define the
orbit of $f$ by $\Omega(f)=\\{g:\ g^{*}\preceq f^{*}\\}\subset E.$ The set of
the extreme points of the set $\Omega(f)$ is well-known (see [Ryff1967,
ChilinKryginSukochev1992]) and, if $I=(0,1)$ or $I=(0,\infty)$, and $E$ does
not contain $L_{\infty}$ it is given by
$\partial_{e}(\Omega(f))=\\{g\in L_{1}+L_{\infty}:\ f^{*}=g^{*}\\}.$
If $I=(0,\infty)$ and $E$ contains $L_{\infty}$ we must make a small
correction:
$\partial_{e}(\Omega(f))=\\{g\in L_{1}+L_{\infty}:\ f^{*}=g^{*},\
|g(t)|\geq\lim_{s\to\infty}f^{*}(s)\text{ a.e. }\\}.$
We define $\mathcal{Q}(f)$ to be the convex hull of the set $\\{g\in
L_{1}+L_{\infty}:\ g^{*}\leq f^{*}\\}.$ We will denote by $\mathcal{Q}_{E}(f)$
the closure in $E$ of $\mathcal{Q}(f).$ This is easily seen to coincide with
the closed convex hull of $\partial_{e}\Omega(f).$ Thus
$\mathcal{Q}_{E}(f)\subset\Omega(f).$
We next define the dilation operators on $E$. If $\tau>0$ and $I=(0,\infty)$
the dilation operator $\sigma_{\tau}$ is defined by setting
$(\sigma_{\tau}(f))(s)=f({s}/{\tau}),\qquad s>0.$
In the case of the interval $(0,1)$ the operator $\sigma_{\tau}$ is defined by
$(\sigma_{\tau}f)(s)=\begin{cases}f(s/\tau),&s\leq\min\\{1,\tau\\}\\\
0,&\tau<s\leq 1.\end{cases}$
The operators $\sigma_{\tau}$ ($\tau\geq 1$) satisfy semi-group property
$\sigma_{\tau_{1}}\sigma_{\tau_{2}}=\sigma_{\tau_{1}\tau_{2}}.$ If $E$ is a
symmetric space and if $\tau>0,$ then the dilation operator $\sigma_{\tau}$ is
a bounded operator on $E$ and
$||\sigma_{\tau}||_{E\to E}\leq\max\\{1,\tau\\}.$
If $E$ is a fully symmetric function space on $(0,\infty)$ then $E+L_{\infty}$
is also a fully symmetric function space under the norm
$\|f\|_{E+L_{\infty}}=\|f^{*}\chi_{(0,1)}\|_{E}.$
The next Lemma will be used later.
###### Lemma 2.1.
Let $E$ be a symmetric function space on $(0,\infty)$, such that $E\setminus
L_{1}\neq\emptyset,$ and suppose $f\in L_{1}\cap E.$ Then
$\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}(f)\|_{E}=\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}(f)\|_{E+L_{\infty}}.$
###### Proof.
We may suppose $f$ is nonnegative and decreasing. Let $\varphi=\varphi_{E}$ be
the fundamental function of $E$ and let $\psi$ be its least concave majorant
of $\varphi.$ Since $E\setminus L_{1}\neq\emptyset$ we have
$\lim_{t\to\infty}\psi^{\prime}(t)=0.$ For any $\tau>1$ we have, using Theorem
II.5.5 of [KreinPetuninSemenov1982]
$\displaystyle\|(\sigma_{\tau}f)\chi_{(1,\infty)}\|_{E}$
$\displaystyle\leq\|f(\tau^{-1})\chi_{(0,1)}+(\sigma_{\tau}f)\chi_{(1,\infty)}\|_{E}$
$\displaystyle\leq
f(\tau^{-1})\int_{0}^{1}\psi^{\prime}(s)\,ds+\int_{1}^{\infty}\psi^{\prime}(s)f(\tau^{-1}s)\,ds$
$\displaystyle\leq\psi(1)f(\tau^{-1})+\tau\int_{\tau^{-1}}^{\infty}\psi^{\prime}(\tau
s)f(s)\,ds.$
Now we have, since $f\in L_{1},$
$\lim_{\tau\to\infty}\tau^{-1}f(\tau^{-1})=0$
and by the Dominated Convergence Theorem,
$\lim_{\tau\to\infty}\int_{\tau^{-1}}^{\infty}\psi^{\prime}(\tau
s)f(s)\,ds=\lim_{\tau\to\infty}\int_{0}^{\infty}\chi_{(\tau^{-1},\infty)}(s)\psi^{\prime}(\tau
s)f(s)\,ds=0.$
Hence
$\lim_{\tau\to\infty}\tau^{-1}\|(\sigma_{\tau}f)\chi_{(1,\infty)}\|_{E}=0$
and the Lemma follows. ∎
We next discuss the corresponding notions for sequence spaces. If
$\xi=(\xi_{n})_{n=1}^{\infty}$ is a sequence then $\xi^{*}$ denotes its
decreasing rearrangement:
$\xi^{*}_{n}=\inf_{|\mathbb{A}|=n-1}\sup_{k\in\mathbb{N}\setminus\mathbb{A}}|\xi_{k}|.$
A Banach sequence space $E$ is called symmetric if $\xi\in E$ and
$\eta^{*}\leq\xi^{*}$ implies that $\eta\in E$ and
$\|\eta\|_{E}\leq\|\xi\|_{E}.$ We write $\eta\preceq\xi$ if
$\sum_{k=1}^{n}\eta_{k}^{*}\leq\sum_{k=1}^{n}\xi_{k}^{*},\qquad
n\in\mathbb{N}.$
$E$ is called fully symmetric if $\xi\in E$ and $\eta\preceq\xi$ implies that
$\eta\in E$ and $\|\eta\|_{E}\leq\|\xi\|_{E}.$ If $\xi$ is any bounded
sequence we define its orbit $\Omega(\xi)=\\{\eta:\,\eta\preceq\xi\\}.$
In this context, we define the dilation operators $\sigma_{m}$ only for
$m\in\mathbb{N}.$ Then
$\sigma_{m}(\xi)=(\xi_{1},\ldots,\xi_{1},\xi_{2},\ldots,\xi_{2},\xi_{3}\ldots)$
where each $\xi_{j}$ is repeated $m$ times.
## 3\. Approximation of the orbit
Our first proposition gives a simple criterion which will enable us to check
when $\mathcal{Q}_{E}(f)=\Omega(f).$
###### Proposition 3.1.
Let $E$ be a fully symmetric Banach space on $(0,\infty).$ Suppose $f,g$ are
nonnegative decreasing functions in $E$. Then $g\in\mathcal{Q}_{E}(f)$ if and
only if, given $\epsilon>0,$ there exists a nonnegative decreasing function
$h\in E$ and an integer $p$ such that $0\leq h\leq g$ and
(6) $\|g-h\|_{E}<\epsilon$
and
(7) $\int_{pa}^{b}h(t)\,dt\leq\int_{a}^{b}f(t)\,dt,\qquad 0<pa<b<\infty.$
###### Proof.
Suppose first $g\in\mathcal{Q}_{E}(f).$ Then given $\epsilon>0$ there exist
$f_{1},\ldots,f_{p}\in E$ such that $f_{j}^{*}\leq f$ for $1\leq j\leq p$ and
$\|g-\frac{1}{p}(f_{1}+\cdots+f_{p})\|_{E}<\epsilon.$
Let
$u=\frac{1}{p}(f_{1}+\cdots+f_{p}),\quad
v=\frac{1}{p}(|f_{1}|+\cdots+|f_{p}|).$
Then if $h=g\wedge v^{*},$ using (5)
$\|g-h\|_{E}\leq\|g-g\wedge
u^{*}\|_{E}\leq\|g-u^{*}\|_{E}\leq\|g-u\|_{E}<\epsilon.$
It remains to observe that (7) holds by Lemma 4.1 of [KaltonSukochev2008].
The converse is easy. If $h$ satisfies (6) and (7) then
$h\in\alpha\mathcal{Q}(f)$ for every $\alpha>1$ by Theorem 6.3 of
[KaltonSukochev2008]. Hence $h\in\mathcal{Q}_{E}(f)$ and so
$d(g,\mathcal{Q}_{E}(f))<\epsilon.$ ∎
The next Lemma is surely well-known but we use it in the main result and
include a proof for completeness.
###### Lemma 3.2.
Let $F$ be a continuous nonnegative increasing concave function on
$[0,\infty)$ with $F(0)=0.$ Let us suppose that
$(\alpha_{n})_{n\in\mathbb{Z}}$ is an increasing doubly infinite sequence of
distinct positive reals with
$\lim_{n\to-\infty}\alpha_{n}=0,\qquad\lim_{n\to\infty}\alpha_{n}=\infty.$
Suppose that $(\beta_{n})_{n\in\mathbb{Z}}$ is any sequence with
$0\leq\beta_{n}\leq F(\alpha_{n}),\qquad n\in\mathbb{Z}.$
(i) There is a least concave function $G$ on $[0,\infty)$ such that $G(0)\geq
0,$ and $G(\alpha_{n})\geq\beta_{n}$ for $n\in\mathbb{Z}.$ $G$ is continuous
nonnegative and increasing and $G(0)=0.$
(ii) Furthermore if $n\in\mathbb{Z}$ then either
$G(t)=G(\alpha_{n})t/\alpha_{n},\qquad 0\leq t\leq\alpha_{n}$
or there exists $m<n$ so that
$G(t)=\beta_{m}+\frac{G(\alpha_{n})-\beta_{m}}{\alpha_{n}-\alpha_{m}}(t-\alpha_{m}),\qquad\alpha_{m}\leq
t\leq\alpha_{n}.$
###### Proof.
(i) is almost immediate. $G$ is defined as the infimum of the collection
$\mathcal{C}$ of all increasing concave functions $H$ on $[0,\infty)$ such
that $H(\alpha_{n})\geq\beta_{n}$ for all $n\in\mathbb{Z}$ and $H(0)\geq 0.$
This collection is non-empty since $F\in\mathcal{C}.$ $G$ is affine on each
interval $[\alpha_{n},\beta_{n+1}]$ and since $G\leq F$, $G$ is continuous at
$0.$
For (ii), assume $G$ is not affine on $[0,\alpha_{n}]$. Then there exists a
least $p<n$ so that $g$ is affine on $[\alpha_{p},\alpha_{n}].$ Let $G_{0}$ be
the function equal to $G$ on $[0,\alpha_{p-1}]$ and $[\alpha_{n},\infty)$ and
affine on $[\alpha_{p-1},\alpha_{n}].$ Then for any $0<\lambda<1$ we have
$(1-\lambda)G+\lambda G_{0}\notin\mathcal{C}.$ Hence there exists
$k(\lambda)\in\\{p,p+1,\ldots,n-1\\}$ so that
$(1-\lambda)G(\alpha_{k(\lambda)})+\lambda
G_{0}(\alpha_{k(\lambda)})<\beta_{k(\lambda)}.$
Letting $\lambda\to 0$ through as a suitable sequence where $k(\lambda)=m<n$
is constant we obtain $G(\alpha_{m})=\beta_{m}$ and the second alternative
holds.∎
We now prove our main result.
###### Theorem 3.3.
Let $E$ be a fully symmetric Banach function space on $(0,\infty)$ with Fatou
norm. Suppose $f\in E_{+}\setminus L_{1}$ is such that
$\Omega(f)=\mathcal{Q}_{E}(f).$ Then
$\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}(f^{*})\|_{E}=0.$
###### Proof.
We may suppose that $f$ is decreasing. We let
$F(t)=\int_{0}^{t}f(s)\,ds.$
Let us define a doubly infinite sequence $(a_{n})_{n\in\mathbb{Z}}$ by
$F(a_{n})=(5/4)^{n}.$
We next introduce the family $\mathcal{K}$ of doubly infinite sequences
$\kappa=(\kappa_{n})_{n\in\mathbb{Z}}$ such that either
$\kappa_{n}\in\mathbb{N}$ with $1\leq\kappa_{n}<a_{n+1}/a_{n}$ or
$\kappa_{n}=\infty.$ Then $\mathcal{K}$ is a complete lattice under the order
$\kappa\leq\kappa^{\prime}$ if $\kappa_{n}\leq\kappa^{\prime}_{n}$ for all
$n$. We may define the lattice operations
$(\kappa\vee\kappa^{\prime})_{n}=\max(\kappa_{n},\kappa^{\prime}_{n})$ and
$(\kappa\wedge\kappa^{\prime})_{n}=\min(\kappa_{n},\kappa^{\prime}_{n}).$
For each $\kappa\in\mathcal{K}$ we define $\psi_{\kappa}\in E$ as follows. Let
$\Psi(t)=\Psi_{\kappa}(t)$ be the least increasing concave function such that
$\Psi(0)\geq 0,$
$\Psi(\kappa_{n}a_{n})\geq F(a_{n}),\qquad\text{ if }\kappa_{n}<\infty,$
and
$\Psi(a_{n})\geq 0\qquad\text{ if }\kappa_{n}=\infty.$
The existence and properties of $\Psi$ are guaranteed by applying Lemma 3.2
when $\alpha_{n}=\kappa_{n}a_{n}$ if $\kappa_{n}<\infty$ and
$\alpha_{n}=a_{n}$ if $\kappa_{n}=\infty$ and $\beta_{n}=F(a_{n})$ if
$\kappa_{n}<\infty$ and $\beta_{n}=0$ if $\kappa_{n}=\infty.$ Since
$F(a_{n})\leq F(\kappa_{n}a_{n})$ it is clear from Lemma 3.2 that $\Psi$
exists and $\Psi\leq F.$ Furthermore $\Psi$ is piecewise affine on
$(0,\infty)$ and we may define $\psi_{\kappa}=\Psi^{\prime}$ is a nonnegative
piecewise constant decreasing function on $(0,\infty)$. Clearly
$\psi_{\kappa}\in\Omega(f)\subset E.$
We note some elementary properties of the map $\kappa\to\psi_{\kappa}.$
###### Lemma 3.4.
(8) $\psi_{\kappa}\preceq\psi_{\kappa^{\prime}},\qquad\text{ if
}\kappa^{\prime}\leq\kappa$ (9)
$\psi_{\kappa\wedge\kappa^{\prime}}\preceq\psi_{\kappa}\vee\psi_{\kappa^{\prime}},\qquad\kappa,\kappa^{\prime}\in\mathcal{K}.$
###### Proof.
(8) is quite trivial.
To see (9) note that
$\int_{0}^{t}\max(\psi_{\kappa}(s),\psi_{\kappa^{\prime}}(s))\,ds\geq\max(\Psi_{\kappa}(t),\Psi_{\kappa^{\prime}}(t)).$
Now if $\kappa_{n}\wedge\kappa^{\prime}_{n}<\infty$ and
$\kappa_{n}\leq\kappa^{\prime}_{n}$ we have
$\int_{0}^{\kappa_{n}a_{n}}\max(\psi_{\kappa}(s),\psi_{\kappa^{\prime}}(s))\,ds\geq\Psi_{\kappa}(\kappa_{n}a_{n})\geq
F(a_{n})$
and with a similar inequality when $\kappa^{\prime}_{n}<\kappa_{n}$ we obtain,
from the definition of $\Psi_{\kappa\wedge\kappa^{\prime}}$,
$\int_{0}^{t}\max(\psi_{\kappa}(s),\psi_{\kappa^{\prime}}(s))\,ds\geq\Psi_{\kappa\wedge\kappa^{\prime}}(t),\qquad
0\leq t<\infty.$
This proves (9). ∎
###### Lemma 3.5.
Suppose $\kappa\in\mathcal{K}$ satisfies
(10) $\max(\kappa_{n},\kappa_{n+1})=\infty,\qquad n\in\mathbb{Z}.$
Then for any $n\in\mathbb{Z}$ such that $\kappa_{n}<\infty$ we have:
(11) $\psi_{\kappa}(t)\geq\frac{9F(a_{n})}{25\kappa_{n}a_{n}}\qquad a_{n}\leq
t\leq\kappa_{n}a_{n}.$
###### Proof.
If $f$ is not identically zero then $\psi_{\kappa}$ is only identically zero
when $\kappa$ is identiclaly $\infty;$ we exclude this case so that
$\Psi_{\kappa}(t)>0$ for $t>0.$ Observe first that $\psi_{\kappa}$ is constant
on $(a_{n},\kappa_{n}a_{n}).$ If for every $m<n$ such that $\kappa_{m}<\infty$
we have $\Psi_{\kappa}(\kappa_{m}a_{m})>F(a_{m})$ then
$\psi_{\kappa}(t)\geq\frac{F(a_{n})}{\kappa_{n}a_{n}},\qquad
0<t\leq\kappa_{n}a_{n}.$
Otherwise, since $\Psi_{\kappa}(t)>0$ for all $t>0,$ we have that, by Lemma
3.2, there exists $m<n$ so that $\kappa_{m}<\infty$ and
$\psi_{\kappa}(t)=\frac{\Psi_{\kappa}(\kappa_{n}a_{n})-F(a_{m})}{\kappa_{n}a_{n}-\kappa_{m}a_{m}},\qquad\kappa_{m}a_{m}<t<\kappa_{n}a_{n}.$
Then
$\psi_{\kappa}(t)\geq\frac{F(a_{n})-F(a_{m})}{\kappa_{n}a_{n}-\kappa_{m}a_{m}},\qquad
a_{n}\leq t\leq\kappa_{n}a_{n}.$
Noting that $m\leq n-2$ by (10) so that $F(a_{m})\leq(4/5)^{2}F(a_{n})$ this
implies that (11) holds for either alternative. ∎
For $\kappa\in\mathcal{K}$ and $r\in\mathbb{N}$ we will define a
$\kappa^{[r]}\geq\kappa$ by suppressing the values of $\kappa$ which are less
than $r$. Precisely:
$\kappa^{[r]}_{n}=\begin{cases}\kappa_{n},\qquad\text{if }\kappa_{n}\geq r\\\
\infty\qquad\text{ if }\kappa_{n}<r.\end{cases}$
We next prove the following Lemma, which is the heart of the argument for
Theorem 3.3:
###### Lemma 3.6.
Under the hypotheses of the theorem, we have that for any
$\kappa\in\mathcal{K}$
$\lim_{r\to\infty}\|\psi_{{\kappa}^{[r]}}\|_{E}=0.$
###### Proof.
We will first prove the Lemma under the additional assumption that (10) holds.
Since $\|\psi_{{\kappa}^{[r]}}\|_{E}$ is decreasing in $r$ (by (8)) it
suffices to show that for given $\epsilon>0$ we can find $r$ so that
$\|\psi_{{\kappa}^{[r]}}\|_{E}<\epsilon.$ By Proposition 3.1 for any
$\epsilon>0$ we can find a nonnegative decreasing function
$h\leq\psi_{\kappa}$ and an integer $p$ so that
(12) $\int_{pa}^{b}h(t)\,dt\leq\int_{a}^{b}f(t)\,dt,\qquad 0<pa<b<\infty,$
and
(13) $\|\psi_{\kappa}-h\|_{E}<\epsilon/10.$
We shall take $r=36p.$ Let $v=10(\psi_{\kappa}-h).$ We will show that
$\psi_{\kappa^{[r]}}\preceq v.$ In order to do this we must show that if
$\kappa_{n}^{[r]}<\infty$ we have
(14) $F(a_{n})\leq\int_{0}^{\kappa_{n}a_{n}}v^{*}(t)\,dt.$
If $\kappa_{n}^{[r]}<\infty$
$\displaystyle\int_{0}^{\kappa_{n}a_{n}}v^{*}(t)\,dt$
$\displaystyle\geq\int_{pa_{n}}^{\kappa_{n}a_{n}}v(t)\,dt$
$\displaystyle=10\left(\int_{pa_{n}}^{\kappa_{n}a_{n}}\psi_{\kappa}(t)\,dt-\int_{pa_{n}}^{\kappa_{n}a_{n}}h(t)\,dt\right)$
$\displaystyle\geq
10\left(\int_{pa_{n}}^{\kappa_{n}a_{n}}\psi_{\kappa}(t)\,dt-\int_{a_{n}}^{\kappa_{n}a_{n}}f(t)\,dt\right),$
by (13). Hence by (11) of Lemma 3.5
$\displaystyle\int_{0}^{\kappa_{n}a_{n}}v^{*}(t)\,dt$ $\displaystyle\geq
10\left(\frac{9(\kappa_{n}a_{n}-pa_{n})F(a_{n})}{25\kappa_{n}a_{n}}-\int_{a_{n}}^{a_{n+1}}f(t)\,dt\right)$
$\displaystyle\geq
10\left(\frac{35}{36}\frac{9}{25}F(a_{n})-\frac{1}{4}F(a_{n})\right)$
$\displaystyle=F(a_{n}).$
This show that (14) holds and so $\psi_{\kappa^{[r]}}\preceq v$ and
$\|\psi_{\kappa^{[r]}}\|_{E}<\epsilon.$ This completes the proof when (10)
holds.
For the general case let us introduce $\kappa(0)_{n}=\kappa_{n}$ if $n$ is
even and $\kappa(0)_{n}=\infty$ if $n$ is odd. Similarly
$\kappa(1)_{n}=\kappa_{n}$ if $n$ is odd and $\kappa(1)_{n}=\infty$ if $n$ is
even. Both $\kappa(0)$ and $\kappa(1)$ satisfy (10) Then for an arbitrary
$\kappa$ we have $\kappa^{[r]}=\kappa(0)^{[r]}\wedge\kappa(1)^{[r]}$ and so by
(9),
$\limsup_{r\to\infty}\|\psi_{\kappa^{[r]}}\|_{E}\leq\limsup_{r\to\infty}\|\psi_{\kappa(0)^{[r]}}\|_{E}+\limsup_{r\to\infty}\|\psi_{\kappa(1)^{[r]}}\|_{E}=0.$
∎
Next for any integer $p$ we define $\gamma^{p}_{n}=p$ if $pa_{n}<a_{n+1}$ and
$\gamma^{p}_{n}=\infty$ otherwise. For each $q>p$ we define
$\gamma^{p,q}_{n}=p$ if $pa_{n}<a_{n+1}$ and $|n|\leq q$ and
$\gamma^{p,q}_{n}=\infty$ otherwise. Let $\psi_{p}=\psi_{\gamma^{p}}$ and
$\psi_{p,q}=\psi_{\gamma^{p,q}}.$
###### Lemma 3.7.
Under the hypotheses of the Theorem,
$\lim_{p\to\infty}\|\psi_{p}\|_{E}=0.$
###### Proof.
Clearly $\|\psi_{p}\|_{E}$ is decreasing in $p$. Assume
$\|\psi_{p}\|_{E}>\epsilon>0$ for all $p\in\mathbb{N}.$ Since $E$ has a Fatou
norm for each $p$ there exists $q(p)>p$ so that
$\|\psi_{p,q(p)}\|_{E}>\epsilon.$ Let
$\kappa=\wedge_{p}\gamma^{p,q(p)}.$
Thus $\kappa$ is given by the formula
$\kappa_{n}=\inf\\{p:\ p<a_{n+1}/a_{n},\ |n|\leq q(p)\\}$
and $\kappa$ has the properties that $\kappa\leq\gamma_{p,q(p)}$ for all $p$
and $\lim_{|n|\to\infty}\kappa_{n}=\infty.$
By Lemma 3.6 there exists $r\in\mathbb{N}$ so that
$\|\psi_{\kappa^{[r]}}\|_{E}<\epsilon.$ But then the set
$\\{n:\kappa_{n}<r\\}$ is finite and so there is a choice of $p$ such that
$p>a_{n+1}/a_{n}$ whenever $\kappa_{n}<r.$ Thus $\gamma^{p}_{n}=\infty$ if
$\kappa_{n}<r.$ Thus
$\kappa^{[r]}\leq\gamma^{p,q(p)}$
and so by (8),
$\|\psi_{p,q(p)}\|_{E}<\epsilon$
which gives a contradiction. ∎
We now can complete the proof of the Theorem. We will show that if
$p\in\mathbb{N},$
(15)
$F(t)\leq\frac{4}{5}F(p^{2}t)+\frac{5}{4}\int_{0}^{p^{2}t}\psi_{p}(s)\,ds,\qquad
0<t<\infty.$
Indeed if (15) fails for some $t$ we can assume $a_{n}\leq t<a_{n+1}$ for some
$n\in\mathbb{Z}.$ We first argue that $a_{n+1}\leq pa_{n}$. Suppose, on the
contrary, that $a_{n+1}>pa_{n}$. Then we have
$\frac{5}{4}\int_{0}^{p^{2}t}\psi_{p}(s)\,ds\geq\frac{5}{4}\int_{0}^{pa_{n}}\psi_{p}(s)\,ds\geq\frac{5}{4}F(a_{n})=F(a_{n+1})\geq
F(t),$
which contradicts our hypothesis. Next we show that $a_{n+2}\leq pa_{n+1}.$
Indeed if $a_{n+2}>pa_{n+1},$ then $p^{2}t\geq pa_{n+1}$ and
$\frac{5}{4}\int_{0}^{p^{2}t}\psi_{p}(s)\,ds\geq\frac{5}{4}\int_{0}^{pa_{n+1}}\psi_{p}(s)\,ds\geq\frac{5}{4}F(a_{n+1})>F(t).$
But then $a_{n+2}\leq p^{2}a_{n}$ and so
$\frac{4}{5}F(p^{2}t)\geq\frac{4}{5}F(a_{n+2})=F(a_{n+1})>F(t)$
and we have a contradiction. This establishes (15).
Now if $\tau\geq 1$ we replace $t$ in (15) by $t/\tau$ and interpret the
inequality in the form:
$\frac{1}{\tau}\sigma_{\tau}f\preceq\frac{1}{p^{-2}\tau}\sigma_{p^{-2}\tau}((4/5)f+(5/4)\psi_{p}).$
Hence
$\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}f\|_{E}\leq(4/5)\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}f\|_{E}+(5/4)\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}\psi_{p}\|_{E}$
so that
$\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}f\|_{E}\leq(5^{2}/4)\|\psi_{p}\|_{E}.$
Combining with Lemma 3.7 we obtain the theorem. ∎
The case when $f\in L_{1}$ is handled by reduction to the previous case:
###### Theorem 3.8.
Let $E$ be a fully symmetric Banach function space on $(0,\infty)$ with a
Fatou norm. Suppose $f$ is a decreasing nonnegative function such that $f\in
E_{+}\cap L_{1}$ and $\Omega(f)=\mathcal{Q}_{E}(f).$ Then
$\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}(f^{*})\|_{E+L_{\infty}}=0.$
###### Proof.
An easy computation shows that
$\mathcal{Q}(f+1)=\mathcal{Q}(f)+\mathcal{Q}(1).$ Hence
$\mathcal{Q}_{E+L_{\infty}}(f+1)\supset\mathcal{Q}_{E}(f)+\mathcal{Q}_{L_{\infty}}(1)=\Omega(f)+\Omega(1).$
If $0\leq g\in\Omega(f+1)$ then $g-g\wedge 1\in\Omega(f)$ and $g\wedge
1\in\Omega(1)$ so that $\Omega(f)+\Omega(1)=\Omega(f+1).$ Hence
$\mathcal{Q}_{E+L_{\infty}}(f+1)=\Omega(f+1)$ and we can apply Theorem 3.3. ∎
## 4\. The main results
We can next state our main results:
###### Theorem 4.1.
Let $E$ be a fully symmetric Banach function space on $(0,\infty)$ with a
Fatou norm, and such that $E\setminus L_{1}\neq\emptyset.$ Suppose $f\in E.$
Then $\Omega(f)=\mathcal{Q}_{E}(f)$ if and only if
$\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}(f^{*})\|_{E}=0.$
###### Proof.
If $\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}(f^{*})\|_{E}=0$ then
$\Omega_{+}(f)\subset\mathcal{Q}_{E}(f)$ by Theorem 25 of [SukochevZanin2009];
thus $\Omega(f)=\mathcal{Q}_{E}(f).$ Conversely if
$\Omega(f)=\mathcal{Q}_{E}(f)$ we have either
$\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}(f)\|_{E}=0$
(when $f\notin L_{1}$ by Theorem 3.3) or
$\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}(f)\|_{E+L_{\infty}}=0$
(when $f\in L_{1}$ by Theorem 3.8). Then Lemma 2.1 shows that in both cases we
have $\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}(f)\|_{E}=0$.∎
###### Theorem 4.2.
Let $E$ be a fully symmetric Banach function space on $(0,\infty)$ with a
Fatou norm, and such that $E\subset L_{1}$. If $f\in E$ then
$\Omega(f)=\mathcal{Q}_{E}(f)$ if and only if
$\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}(f^{*})\|_{E+L_{\infty}}=\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}(f^{*})\chi_{(0,1)}\|_{E}=0.$
###### Proof.
The proof is very similar to that of Theorem 4.1 using instead Theorem 24 of
[SukochevZanin2009]. ∎
We first give the extension to function spaces on $(0,1).$
###### Theorem 4.3.
Let $E$ be a fully symmetric Banach function space on $(0,1)$ with a Fatou
norm. Suppose $f\in E.$ Then $\Omega(f)=\mathcal{Q}_{E}(f)$ if and only if
$\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}(f^{*})\|_{E}=0.$
###### Proof.
We define $F$ to be the function space on $(0,\infty)$ defined by $f\in F$ if
and only if $f^{*}\chi_{(0,1)}\in E$ and $f\in L_{1}$ with the norm
$\|f\|_{F}=\max(\|f^{*}\chi_{(0,1)}\|_{E},\|f^{*}\|_{L_{1}}).$
Suppose $f\in E$ is nonnegative and decreasing. We will show that, regarding
$f$ as an member of $F,$ we have $\Omega(f)=\mathcal{Q}_{F}(f).$ Note that the
hypothesis $\Omega(f)=\mathcal{Q}_{E}(f)$ on $(0,1)$ implies only that if
$g\in F$ and $g\in\Omega(f)$ then $g\in\mathcal{Q}_{F}(f)$ provided $\mu({\rm
supp}\,g)\leq 1.$ We will show, however, that $\Omega(f)=\mathcal{Q}_{F}(f)$
and then the Theorem follows.
We will need the following Lemma:
###### Lemma 4.4.
Let $h\in F$ be nonnegative and decreasing. Suppose $g\in F$ is nonnegative
and decreasing and satisfies the conditions such that $g\preceq h$ and
$g(x)=0$ for some $0<x<\infty.$ If there exists $c>0$ so that $g(t)\leq h(t)$
for $0<t\leq c$ then $g\in\mathcal{Q}_{F}(h).$
###### Proof.
For any $\theta>1$ we may pick $p>x/c$ so that
$\int_{0}^{c}h(s)\,ds\leq\theta\int_{x/p}^{c}h(s)\,ds.$
Then if $0<pa<b<\infty$ with $c\leq b\leq x$ we have
$\int_{pa}^{b}g(s)\,ds\leq\int_{0}^{b}h(s)\,ds\leq\theta\int_{x/p}^{b}h(s)\,ds\leq\theta\int_{pa}^{b}h(s)\,ds.$
The same inequality holds trivially if $b>x$ or $b<c.$ Thus by Theorem 6.3 of
[KaltonSukochev2008] we have $g\in\lambda Q(h)$ for any $\lambda>1$ and the
Lemma follows.∎
We continue the proof of the theorem. We will assume, without loss of
generality that $\int_{0}^{1}f(t)\,dt=1.$ First suppose $g\in\Omega(f)$ is
nonnegative, not identically zero, and decreasing and satisfies $g(x)=0$ for
some $0<x<\infty.$ Given $\epsilon>0$ we may find $c_{0}>0$ so that
$\int_{0}^{c_{0}}f(s)\,ds<\epsilon/2.$
Let
$\alpha=\sup_{0<t\leq
c_{0}}\frac{\int_{0}^{t}g(s)\,ds}{\int_{0}^{t}f(s)\,ds}.$
We have $\alpha>0$ and we may pick $0<\beta<\alpha$ with
$(\alpha-\beta)<\epsilon/2$ and then $0<c<c_{0}$ with
$g(c)>\beta f(c).$
Let $c^{\prime}\geq c$ be the least solution of
$\alpha\int_{0}^{c^{\prime}}f(s)\,ds=\int_{0}^{c}g(s)\,ds+(c^{\prime}-c)g(c).$
We now define
$h(t)=\begin{cases}g(t)+(1-\alpha)f(t),\qquad 0<t\leq c\\\
g(c)+(1-\alpha)f(t),\qquad c<t\leq\min(c^{\prime},1)\\\
f(t),\qquad\qquad\qquad\qquad\min(c^{\prime},1)<t\leq 1\\\
0\qquad\qquad\qquad\qquad\qquad t\geq 1.\end{cases}$
From the construction we have $h\in\Omega(f).$ Thus $h\in\mathcal{Q}_{F}(f).$
For any $t\leq\min(c^{\prime},1)$ we have
$\int_{0}^{t}g(s)\,ds\leq\int_{0}^{t}h(s)\,ds.$
If $c^{\prime}<1$ then
$\int_{0}^{t}h(s)\,ds=\int_{0}^{t}f(s)\,ds\geq\int_{0}^{t}g(s)\,ds,\qquad
t>c^{\prime}.$
If $c^{\prime}\geq 1$ then
$\displaystyle\int_{0}^{1}h(s)\,ds$
$\displaystyle\geq(1-\alpha)\int_{0}^{1}f(s)\,ds+\int_{0}^{c}g(s)\,ds+g(c)(1-c)$
$\displaystyle\geq(1-\alpha)\int_{0}^{1}f(s)\,ds+\beta\int_{0}^{c}f(s)\,ds-\epsilon/2+\beta
f(c)(1-c)$
$\displaystyle\geq(1-\alpha)\int_{0}^{1}f(s)\,ds+\beta\int_{0}^{1}f(s)\,ds-\epsilon/2$
$\displaystyle\geq 1-\epsilon.$
Hence $(1-\epsilon)g\preceq h$ and by Lemma 4.4 we have
$(1-\epsilon)g\in\mathcal{Q}_{F}(h)$. Since $\epsilon>0$ is arbitrary we have
$g\in\mathcal{Q}_{F}(h)\subset\mathcal{Q}_{F}(f).$
Finally let us note for general nonnegative decreasing $g\in F$ we have
$\lim_{m\to\infty}\|g-g\chi_{(0,m)}\|_{F}=0$ so that we have
$\Omega_{F}(f)=\mathcal{Q}_{F}(f).$
Now the result reduces to Theorems 4.1 and 4.2. ∎
The extension to sequence spaces requires similar type of argument:
###### Theorem 4.5.
Let $E$ be a fully symmetric Banach sequence space with a Fatou norm and such
that $E\setminus\ell_{1}\neq\emptyset$. Suppose $\xi\in E.$ Then
$\Omega(\xi)=\mathcal{Q}_{E}(\xi)$ if and only if
$\lim_{m\to\infty}m^{-1}\|\sigma_{m}(\xi^{*})\|_{E}=0.$
###### Proof.
We consider the Banach function space $F$ of all bounded functions such that
$(f^{*}(0),f^{*}(1),\ldots)\in E$ with the norm
$\|f\|_{F}=f^{*}(0)+\|(a_{n})_{n=1}^{\infty}\|_{E},$
where $f^{*}(0)=\|f\|_{L_{\infty}}$ and $a_{n}:=\int_{n-1}^{n}f^{*}(s)ds$,
$n\geq 1$. Then let $F(\mathbb{N})$ be the subspace of $F$ of all functions
$f$ which are constant on each interval $(n-1,n]$. Clearly, the Banach spaces
$(F(\mathbb{N}),\|\cdot\|_{F})$ and $(E,\|\cdot\|_{E})$ are linearly
isomorphic, in particular
$\|\xi\|_{E}\leq\|\xi\|_{F}\leq 2\|\xi\|_{E},\quad\forall\xi\in
E=F(\mathbb{N}).$
Let $\mathbb{E}$ denote the conditional expectation operator
$\mathbb{E}f=\sum_{n\in\mathbb{N}}\chi_{(n-1,n]}\int_{n-1}^{n}f(t)\,dt.$
Suppose $\xi$ is a nonnegative decreasing sequence and let
$f=\sum_{j=1}^{\infty}\xi_{j}\chi_{(j-1,j]}\in F.$
The result will follow from:
###### Theorem 4.6.
$\Omega(\xi)=\mathcal{Q}_{E}(\xi)$ if and only if
$\Omega(f)=\mathcal{Q}_{F}(f).$
###### Proof.
Let us suppose that $\Omega(\xi)=\mathcal{Q}_{E}(\xi)$. We may suppose $\xi$
has infinite support. Suppose $g\in\Omega(f)$ is nonnegative and decreasing;
we will show that $g\in\mathcal{Q}_{F}(f)$ and then it follows that
$\mathcal{Q}_{F}(f)=\Omega(f).$
Suppose $\epsilon>0$. Then we may pick an integer $m\in\mathbb{N}$ so that
$g^{*}(m)-\lim_{n\to\infty}g^{*}(n)<\epsilon/4.$
Now $\mathbb{E}g\in\mathcal{Q}_{F}(f)$ since
$\Omega(\xi)=\mathcal{Q}_{E}(\xi)$. Hence, by Proposition 3.1 there is a
nonnegative decreasing function $h$ with $0\leq h\leq\mathbb{E}g$ such that
$\|\mathbb{E}g-h\|_{E}<\epsilon/4$ and such that for some $p\in\mathbb{N},$
$\int_{pa}^{b}h(s)\,ds\leq\int_{a}^{b}f(s)\,ds\qquad 0<pa<b<\infty.$
Next we define
$\varphi(s)=\begin{cases}g(s),\qquad 0<s\leq m\\\ h(s),\qquad
m<s<\infty.\end{cases}$
Note that $0\leq\varphi\preceq g\preceq f.$ We show that
$\varphi\in\mathcal{Q}_{F}(f).$ Let us suppose $r>p$ and that $0<ra<b.$ Then
if $m\leq ra$ we clearly have
$\int_{ra}^{b}\varphi(s)\,ds\leq\int_{a}^{b}f(s)\,ds.$
On the other hand if $0<ra<m$, let $c=\min(b,m).$ Then
$\displaystyle\int_{ra}^{b}\varphi(s)\,ds$
$\displaystyle\leq\int_{0}^{b}f(s)\,ds$
$\displaystyle\leq\int_{a}^{b}f(s)\,ds+c\xi_{1}/r$
$\displaystyle\leq\int_{a}^{b}f(s)\,ds+\frac{\xi_{1}}{(r-1)\xi_{m}}\int_{c/r}^{c}f(s)\,ds$
$\displaystyle\leq\left(1+\frac{\xi_{1}}{(r-1)\xi_{m}}\right)\int_{a}^{b}f(s)\,ds.$
Since $r$ is arbitrary these estimate show that
$\varphi\in\lambda\mathcal{Q}(f)$ for every $\lambda>1$ (Theorem 6.4 of
[KaltonSukochev2008]) and hence $\varphi\in\mathcal{Q}_{F}(f).$
Now
$\|g-\varphi\|_{F}=\|(g-\varphi)\chi_{(m,\infty)}\|_{F}\leq\|(g-\mathbb{E}g)\chi_{(m,\infty)}\|_{F}+\|\mathbb{E}g-h\|_{F}.$
However,
$\|(g-\mathbb{E}g)\chi_{(m,\infty)}\|_{F}\leq\sum_{j=m}^{\infty}2(g(j)-g(j+1))<\epsilon/2.$
Hence
$d(g,\mathcal{Q}_{E}(f))\leq\|g-\varphi\|_{F}<\epsilon.$
Since $\epsilon>0$ is arbitrary we have $g\in\mathcal{Q}_{F}(f).$ This shows
that $\Omega(f)=\mathcal{Q}_{F}(f).$
We next turn to the converse. Assume $\mathcal{Q}_{E}(f)=\Omega(f)$ and that
$\eta\in\Omega(\xi)$ is a decreasing sequence. Let
$g=\sum_{n\in\mathbb{N}}\eta_{n}\chi_{(n-1,n]}$. Then $g\in\Omega(f)$ and so,
by Proposition 3.1, given $\epsilon>0$, there exists a decreasing $0\leq h\leq
g$ with $\|g-h\|_{F}<\epsilon$ and such that for some $p\in\mathbb{N}$ we have
$\int_{pa}^{b}h(s)\,ds\leq\int_{a}^{b}f(s)\,ds,\qquad 0<pa<b<\infty.$
Let $\zeta\in E$ be defined by $\zeta_{n}=\int_{n-1}^{n}h(s)\,ds.$ Then for
$0\leq pm\leq n$ we have
$\sum_{k=pm+1}^{n}\zeta_{k}=\int_{pm}^{n}h(s)\,ds\leq\int_{m}^{n}f(s)\,ds=\sum_{k=m+1}^{n}\xi_{k}.$
Hence $\zeta\in\lambda\mathcal{Q}(\xi)$ for every $\lambda>1$, by Theorem 5.5
of [KaltonSukochev2008], so that $\zeta\in\mathcal{Q}_{E}(\xi).$ Furthermore,
$\|\eta-\zeta\|_{E}\leq\|g-\mathbb{E}h\|_{F}\leq\|g-h\|_{F}<\epsilon.$
It now follows that $\eta\in\mathcal{Q}_{E}(\xi)$ and the proof of the Lemma
is complete. ∎
Theorem 4.5 now follows directly from Theorem 4.6. ∎
Let us observe that the argument of Theorem 4.6 allows us to complete the
picture for positive orbits in [SukochevZanin2009]:
###### Theorem 4.7.
Let $E$ be a fully symmetric sequence space with Fatou norm. Then for any
$\xi\in E_{+}$ the set $\Omega_{+}(\xi)=\Omega(\xi)\cap E_{+}$ coincides with
the closed convex hull of its extreme points if and only if
$\lim_{m\to\infty}m^{-1}\|\sigma_{m}(\xi)\|_{E}=0.$
In fact we can prove by the same argument as in Theorem 4.6 that
$\Omega_{+}(\xi)=\mathcal{Q}_{E}(\xi)\cap E_{+}$ if and only if
$\Omega_{+}(f)=\mathcal{Q}_{F}(f)\cap F_{+}.$
We remark that in [SukochevZanin2009] some examples of Marcinkiewicz spaces
and Orlicz spaces are discussed in the context of Theorems 4.1, 4.2, 4.3 and
4.5. We refer the reader to [SukochevZanin2009] for details. We take the
opportunity to improve Proposition 33 of [SukochevZanin2009]:
###### Proposition 4.8.
Let $M$ be an Orlicz function. Then for any $f\in L_{M}(0,\infty)$ we have
$\Omega(f)=\mathcal{Q}_{L_{M}}(f).$ Similarly for for any $\xi\in\ell_{M}$ we
have $\Omega(\xi)=\mathcal{Q}_{\ell_{M}}(\xi).$
###### Proof.
We give the proof only for $L_{M}(0,\infty).$ Suppose first that $M(t)=o(t)$
when $t\to 0.$ We show that
$\|\lim_{\tau\to\infty}\tau^{-1}\sigma_{\tau}f\|_{L_{M}}=0$ whenever
$f\in(L_{M})_{+}.$ Suppose $\alpha>0.$
$\int_{0}^{\infty}M\left(\frac{\alpha
f(s/\tau)}{\tau}\right)ds=\int_{0}^{\infty}\tau M\left(\frac{\alpha
f(s)}{\tau}\right)\,ds$
for any $\tau>1.$ Since $f\in L_{M}$ there exists $\tau_{0}$ so that
$\int_{0}^{\infty}\tau_{0}M\left(\frac{\alpha
f(s)}{\tau_{0}}\right)\,ds<\infty.$
Now letting $\tau\to\infty$ we obtain from the Dominated Convergence Theorem
that
$\lim_{\tau\to\infty}\int_{0}^{\infty}\tau M\left(\frac{\alpha
f(s)}{\tau}\right)\,ds=0$
so that $\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}f\|_{L_{M}}=0$ and we can
apply Theorem 4.1.
Now if $M(t)\geq ct$ for all $t>0$ where $c>0$ we have $L_{M}\subset L_{1}.$
For any $\alpha>0$ and $\tau>1$
$\int_{0}^{1}M\left(\frac{\alpha
f^{*}(s/\tau)}{\tau}\right)ds=\int_{0}^{1}\tau\chi_{(0,\tau^{-1})}(s)M\left(\frac{\alpha
f^{*}(s)}{\tau}\right)\,ds.$
As before the right-hand side is integrable for some $\tau=\tau_{0}$ and we
can apply the Dominated Convergence Theorem to deduce that
$\tau^{-1}\|(\sigma_{\tau}f^{*})\chi_{(0,1)}\|_{L_{M}}$ tends to $0$ as $\tau$
approaches infinity. Now one can apply Theorem 4.2. ∎
## 5\. A noncommutative analog
Let $\mathcal{H}$ be a separable complex Hilbert space. We denote by
$\mathcal{B}(\mathcal{H})$ the space of bounded operators on $\mathcal{H}$ and
by $\mathcal{K}(\mathcal{H})$ the ideal of compact operators on $\mathcal{H}.$
For any $T\in\mathcal{B}(\mathcal{H})$ we define the singular values
$s_{n}(T)=\inf\\{\|T(I-P)\|,$
where the infimum is taken over all orthogonal projections $P$ such that ${\rm
rank}(P)<n\\}.$
If $E$ is a symmetric sequence space then we can define a Banach ideal of
compact operators on $\mathcal{H}$ by $T\in\mathcal{S}_{E}$ if and only if
$(s_{k}(T))_{k=1}^{\infty}\in E$ and then the norm is given by
$\|T\|_{E}=\|(s_{k}(T))_{k=1}^{\infty}\|_{E}.$ For fully symmetric spaces this
is well-known (e.g. see [GohbergKrein1969] but for symmetric spaces it follows
from [KaltonSukochev2008]).
Let $\mathcal{H}$ be a separable Hilbert space and suppose
$T\in\mathcal{K}(\mathcal{H}).$ Let $\mathcal{Q}(T)$ be the convex hull of the
set $\\{ATB;\ \|A\|,\|B\|\leq 1\\}.$ We define its orbit $\Omega(T)$ to be the
closure of $\mathcal{Q}(T)$ in $\mathcal{K}(\mathcal{H}).$ It is easy to check
from the definition that $R\in\Omega(T)$ if and only if
$\sum_{k=1}^{n}s_{k}(R)\leq\sum_{k=1}^{n}s_{k}(T),\qquad n=1,2,\ldots$
For any symmetric Banach sequence space $E$ we may define $\mathcal{Q}_{E}(T)$
to be the closure of $\mathcal{Q}(T)$ in $\mathcal{S}_{E}.$
###### Theorem 5.1.
Let $\mathcal{E}$ be a fully symmetric sequence space with a Fatou norm.
Suppose $\mathcal{S}_{E}$ is the corresponding ideal of compact operators.
Then for $T\in\mathcal{S}_{E}$ we have $\Omega(T)=\mathcal{Q}_{E}(T)$ if and
only if
$\lim_{m\to\infty}m^{-1}\|\sigma_{m}(s_{k}(T))_{k=1}^{\infty}\|_{E}=0.$
###### Proof.
Let $\xi=(s_{k}(T))_{k=1}^{\infty}.$ Let $R\in\mathcal{K}(\mathcal{H})$ and
let $\eta=(s_{k}(R))_{k=1}^{\infty}$. If $R\in\mathcal{Q}(T)$ then it follows
from Proposition 8.6 and Theorem 5.5 of [KaltonSukochev2008] that
$\eta\in\lambda\mathcal{Q}(\xi)$ for every $\lambda>1.$
First suppose that $\Omega(T)=\mathcal{Q}_{E}(T).$ If $S\in\Omega(T)$ then
given $\epsilon>0$ there exists $R\in\mathcal{Q}(T)$ with
$\|R-S\|_{E}<\epsilon.$ Let $\zeta=(s_{k}(S))_{k=1}^{\infty}.$ Then by the
submajorization inequality of [DoddsDoddsdePagter1989a],
$\eta-\zeta\preceq(s_{k}(R-S))_{k=1}^{\infty}$
so that $\|\eta-\zeta\|_{E}<\epsilon.$ Since $\eta\in\mathcal{Q}_{E}(\xi)$ and
$\epsilon>0$ is arbitrary, this implies that $\zeta\in\mathcal{Q}_{E}(\xi)$
and so $\mathcal{Q}_{E}(\xi)=\Omega(\xi).$ Theorem 4.5 can then be applied.
The converse direction is immediate. ∎
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|
arxiv-papers
| 2010-03-09T07:33:46 |
2024-09-04T02:49:08.961682
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "N.J. Kalton, F.A. Sukochev, D.V. Zanin",
"submitter": "Dmitriy Zanin",
"url": "https://arxiv.org/abs/1003.1817"
}
|
1003.2009
|
# Kruglov operator and operators defined by random permutations
S.V. Astashkin1 , D.V. Zanin , E.M. Semenov2 and F.A. Sukochev3
###### Abstract.
The Kruglov property and the Kruglov operator play an important role in the
study of geometric properties of r.i. function spaces. We prove that the
boundedness of the Kruglov operator in a r.i. space is equivalent to the
uniform boundedness on this space of a sequence of operators defined by random
permutations. It is shown also that there is no minimal r.i. space with the
Kruglov property.
1Research is partially supported by Russian Foundation for Basic Research,
07-01-96603.
2Research is partially supported by Russian Foundation for Basic Research,
05-01-00629.
3Research is partially supported by Australian Research Council.
## 1\. Introduction
Let $f$ be a random variable (measurable function) on the interval $[0,1]$. We
denote a random variable $\sum_{i=1}^{N}f_{i}$ by $\pi(f).$ Here, $f_{i}$’s
are independent copies of $f$ and $N$ is a Poisson random variable with
parameter 1, independent from $f_{i}$’s.
###### Definition 1.
A r.i. function space $E$ on the interval $[0,1]$ is said to have the Kruglov
property ($E\in\mathbb{K}$) if $f\in E\Longleftrightarrow\pi(f)\in E.$
This property was introduced and studied by Braverman [1], exploiting some
constructions and ideas from the article [2] by Kruglov. An operator approach
to the study of this property was introduced in [3] (see also [4]).
Let $\\{B_{n}\\}_{n=1}^{\infty}$ be a sequnce of mutually disjoint measurable
subsets of $[0,1]$ and let $\mathop{\mathrm{mes}}B_{n}=\dfrac{1}{en!}$. If
$f\in L_{1}[0,1],$ then set
$Kf(\omega_{0},\omega_{1},\ldots)=\sum_{n=1}^{\infty}\sum_{k=1}^{n}f(\omega_{k})\chi_{B_{n}}(\omega_{0}).$
Here and everywhere else we denote the characteristic function of the set $B$
by $\chi_{B}.$ It then follows that $K:\,L_{1}[0,1]\to L_{1}(\Omega,P)$ is a
positive linear operator. Here
$(\Omega,P)=\prod_{n=0}^{\infty}([0,1],\mathop{\mathrm{mes}})$, where
$\mathop{\mathrm{mes}}$ is the Lebesgue measure on $[0,1]$. Since $Kf$ is
equidistributed with $\pi(f)$ (see [3]), we may consider $Kf$ as an explicit
representation of $\pi(f).$ In particular, an r.i. space $E\in\mathbb{K}$ if
and only if $K$ (boundedly) maps $E$ into $E(\Omega,P)$ (see [3]).
We will also use an equivalent representation of the operator $K$ introduced
in [3]. Let $f^{*}$ be decreasing rearrangement of $|f|,$ that is, $f^{*}(t)$
decreases on $[0,1]$ and is equimeasurable with $|f(t)|.$ If $f\in L_{1}[0,1]$
and if $\\{B_{n}\\}$ is the same sequence of subsets of $[0,1]$ as above, then
let $f_{n,1},f_{n,2},\ldots,f_{n,n}$, $\chi_{B_{n}}$ be the set of independent
functions for every $n\in\mathbb{N},$ such that $f_{n,k}^{*}=f^{*}$ for every
$n\in\mathbb{N}$ and $k=1,2,\ldots,n.$ Under these conditions, $Kf(t)$ is
defined as a rearrangement of the function
(1) $\sum_{n=1}^{\infty}\sum_{k=1}^{n}f_{n,k}(t)\chi_{B_{n}}(t)\;\;(0\leq
t\leq 1).$
It follows from the definition of an r.i. space and that of the operator $K$
that $\|Kf\|_{E}\geq e^{-1}\|f\|_{E}$ for every r.i. space $E$ and for every
$f\in E$ (see also [1, 1.6,p.11]). It is shown in [3] that the operator $K$
plays an important role in estimating the norm of sums of independent random
variables through the norm of sums of their disjoint copies. In particular, in
[3] the well-known results of Johnson and Schechtman from [5] have been
strengthened.
It is well known [2],[1] that the Orlicz space $\exp L_{1}$ defined by the
function $e^{t}-1$ satisfies the Kruglov property. The latter property also
holds for its separable part $(\exp L_{1})_{0}$. Indeed, since $K$ is bounded
in $\exp L_{1},$ we have $K((\exp L_{1})_{0})\subset\overline{K(L_{\infty})}$
(the closure is taken with respect to the norm in $\exp L_{1}$). However,
$K(L_{\infty})\subset(\exp L_{1})_{0}$ [3, Theorem 4.4]. Since $(\exp
L_{1})_{0}$ is a closed subset of $\exp L_{1},$ we conclude that the operator
$K$ maps $(\exp L_{1})_{0}$ into itself. All previously known r.i. spaces $E$
with the Kruglov property satisfied the inclusion $E\supset(\exp L_{1})_{0}.$
This together with some results from [3] (e.g. Theorem 7.2) suggest that
$(\exp L_{1})_{0}$ is the minimal r.i. space with the Kruglov property.
However, in the first part of the paper we show that this conjecture fails.
Moreover, we show that for every given r.i. space $E\in\mathbb{K}$ there
exists a Marcinkiewicz space satisfying the Kruglov property such that
$M_{\psi}\subsetneq E$ (see Corollary 3). The situation is quite different in
the subclass of Lorentz spaces. Indeed, every Lorentz space satisfying the
Kruglov property necessarily contains $\exp L_{1}$ (see Theorem 4).
In [6], Kwapien and Schütt considered random permutations and applied their
results to the geometry of Banach spaces. These results were further
strengthened in [7] and [8] via an operator approach. The following family of
operators was introduced there. Let $n\in\mathbb{N}$ and let $S_{n}$ be the
set of all permutations of scalars $1,2,\cdots,n.$ From now on the sets
$S_{n}$ and $\\{1,2,\cdots,n!\\}$ will be identified (in an arbitrary manner).
Firstly, we define an operator $A_{n}$ acting from $\mathbb{R}^{n}$ into
$\mathbb{R}^{n!}$: if $x=(x_{1},x_{2},\dots,x_{n})\in\mathbb{R}^{n}$ and if
$\pi\in S_{n}$ is an arbitrary permutation, then
(2) $A_{n}x(\pi):=\sum_{i:\,\pi(i)=i}x_{i}.$
For every $x\in L_{1}[0,1]$, we define a vector $B_{n}x\in\mathbb{R}^{n}$ with
coordinates $(B_{n}x)_{i}=n\int_{(i-1)/n}^{i/n}x(t)\,dt,$ $i=1,2,\dots,n.$ The
operator $B_{n}$ has a right inverse operator $C_{n}$ ( $B_{n}C_{n}x=x$ for
every $x\in\mathbb{R}^{n}$) which maps every vector into a function with
constancy intervals $[(i-1)/n,i/n].$ Now, we define
$T_{n}=C_{n!}A_{n}B_{n}.$
For every $n\in\mathbb{N},$ $T_{n}$ is a positive linear operator from
$L_{1}[0,1]$ into the space of step functions. It is not hard to show that
(3) $\|T_{n}x\|_{L_{1}}=\|x\|_{L_{1}}$
for every positive $x\in L_{1}[0,1].$ Sometimes, we will also use the notation
$T_{n}$ for the operator $C_{n!}A_{n},$ defined analogously on
$\mathbb{R}^{n}$ (this does not cause any ambiguity). If
$x=(x_{1},x_{2},\ldots,x_{n})\in\mathbb{R}^{n}$ and if $E$ is an r.i. space,
then the notation $\|x\|_{E}$ will always mean
$\|C_{n}x\|_{E}=\displaystyle\Big{\|}\sum_{k=1}^{n}x_{k}\chi_{\Big{(}\frac{k-1}{n},\frac{k}{n}\Big{)}}\Big{\|}_{E}.$
The operators generated by random permutations and defined on the set of
square matrices were considered in [8], where it was established that such
operators are uniformly bounded if the family of operators $\\{T_{n}\\}_{n\geq
1}$ is uniformly bounded. There is no any visible connection between the
operators $K$ and $T_{n},\ n\geq 1$. Nevertheless, the following interesting
fact follows from the comparison of results in [8] and [3]: the criterion for
the boundedness of the operator $K$ in any Lorentz space $\Lambda_{\varphi}$
and that for the uniform boundedness of the family of operators
$\\{T_{n}\\}_{n\geq 1}$ in $\Lambda_{\varphi}$ coincide. More precisely, both
criteria are equivalent to the following condition
(4) $M:=\sup_{0<t\leq
1}\frac{1}{\varphi(t)}\sum_{k=1}^{\infty}\varphi\left(\frac{t^{k}}{k!}\right)<\infty.$
It is now natural to ask whether the boundedness of the operator $K$ in an
arbitrary r.i. space $E$ is equivalent to the uniform boundedness of the
family of operators $\\{T_{n}\\}_{n\geq 1}$ in $E$. In the second part of this
paper we establish that it is indeed the case. The proof is based on
combinatorial arguments and is connected with obtaining estimates of
corresponding distribution functions. The established equivalence implies some
new corollaries for the operator $K$ and operators $T_{n},\ n\geq 1$. In
particular, Corollary 13 strengthens Theorem 19 from [8] by showing that the
uniform boundedness of the family of operators $\\{T_{n}\\}_{n\geq 1}$ in
Orlicz spaces $\exp L_{p}$ is equivalent to the condition $p\leq 1$.
The authors thank the referee for comments and suggestions which allowed to
simplify the definition of the operator $T_{n}$, $n\geq 1$ and the proof of
Lemma 7 and in general were helpful in improving the final text of this paper.
## 2\. Definitions and notation
A Banach space $E$ consisting of functions measurable on $[0,1]$ is said to be
rearrangement invariant or symmetric (r.i.) if the following conditions hold
1. (1)
If $|x(t)|\leq|y(t)|$ for a.e. $t\in[0,1]$ and $y\in E,$ then $x\in E$ and
$\|x\|_{E}\leq\|y\|_{E}.$
2. (2)
If functions $x$ and $y\in E$ are equimeasurable, that is
$\mathop{\mathrm{mes}}\\{t\in[0,1]:\,|x(t)|>\tau\\}=\mathop{\mathrm{mes}}\\{t\in[0,1]:\,|y(t)|>\tau\\}\quad(\tau>0),$
then $x\in E$ and $\|x\|_{E}=\|y\|_{E}$.
If $E$ is an r.i. space, then $L_{\infty}\subset E\subset L_{1}$ and these
inclusions are continuous. Moreover, if $\|\chi_{(0,1)}\|_{E}=1,$ then
$\|x\|_{L_{1}}\leq\|x\|_{E}\leq\|x\|_{L_{\infty}}$ for every $x\in
L_{\infty}.$ For every $\tau>0,$ the dilation operator $\sigma_{\tau}$ defined
by $\sigma_{\tau}x(t):=x(t/\tau)\chi_{[0,1]}(t/\tau)$ $(0\leq t\leq 1)$
boundedly maps $E$ into itself and $\|\sigma_{\tau}\|_{E}\leq\max(1,\tau).$
The Köthe dual space $E^{\prime}$ consists of all functions $x$ for which the
norm
$\|x\|_{E^{\prime}}=\sup_{\|y\|_{E}\leq 1}\int_{0}^{1}x(t)y(t)dt$
is finite. Clearly, $E^{\prime}$ is also an r.i. space. Following [9, 2.a.1],
we assume that either r.i. space $E$ is separable or $E$ coincides with its
second Köthe dual space $E^{\prime\prime}$. In any case, the space $E$ is
contained in $E^{\prime\prime}$ as a closed subspace and the inclusion
$E\subset E^{\prime\prime}$ is an isometry. If $E$ is separable, then
$E^{\prime}$ coincides with its dual space $E^{*}$. The closure $E_{0}$ of
$L_{\infty}$ in $E$ is called the separable part of $E$. The space $E_{0}$ is
separable provided that $E\neq L_{\infty}.$
Recall that the weak convergence of distributions of measurable on $[0,1]$
functions $x_{n}$ to the distribution of the function $x$ ($x_{n}\Rightarrow
x$) means that for every continuous and bounded on $(-\infty,\infty)$ function
$y$ we have
$\lim_{n\to\infty}\int_{-\infty}^{\infty}y(t)\,d\mathop{\mathrm{mes}}\\{s:\,x_{n}(s)<t\\}=\int_{-\infty}^{\infty}y(t)\,d\mathop{\mathrm{mes}}\\{s:\,x(s)<t\\}.$
If $E$ is an r.i. space, $x_{n}\in E$ $(n\in\mathbb{N}),$
$\limsup_{n\to\infty}\|x_{n}\|_{E}=C<\infty$ and $x_{n}\Rightarrow x,$ then
$x\in E^{\prime\prime}$ and $\|x\|_{E^{\prime\prime}}\leq C$ [1, Proposition
1.5].
The following submajorization defined on $L_{1}$ plays an important role in
the theory of r.i. spaces. We denote $x\prec y$ if
$\int_{0}^{\tau}x^{*}(t)dt\leq\int_{0}^{\tau}y^{*}(t)dt$
for all $\tau\in[0,1]$. If $x\prec y$ and $y\in E,$ then $x\in E$ and
$\|x\|_{E}\leq\|y\|_{E}$. Here and below, $x^{*}(t)$ is the non-increasing
left continuous rearrangement of the function $|x(t)|,$ i.e.
$x^{*}(t)=\inf\\{\tau\geq
0:\,\mathop{\mathrm{mes}}\\{s\in[0,1]:\,|x(s)|>\tau\\}<t\\}\;\;(0<t\leq 1).$
We list below the most important examples of r.i. spaces. Let $M$ be an
increasing convex function on $[0,\infty)$ such that $M(0)=0$. By $L_{M}$ we
denote the Orlicz space $L_{M}$ with the norm
$\|x\|_{L_{M}}=\inf\left\\{\lambda>0:\,\int_{0}^{1}M\left(\frac{|x(t)|}{\lambda}\right)\,dt\leq
1\right\\}.$
Function $M_{p}(u)=e^{u^{p}}-1$ is convex if $p\geq 1$ and is equivalent to
some convex function if $0<p<1$. We denote $L_{M_{p}}$ by $\exp L_{p}$.
Let $\varphi(t)$ be an increasing concave function on $[0,1]$ such that
$\varphi(0)=0$ and let $\Lambda_{\varphi}$ be the Lorentz space equipped with
a norm
$\|x\|_{\Lambda_{\varphi}}=\int_{0}^{1}x^{*}(t)\,d\varphi(t).$
Similarly, $M_{\varphi}$ is the Marcinkiewicz space equipped with the norm
$\|x\|_{M_{\varphi}}=\sup_{0<t\leq
1}\frac{1}{\varphi(t)}{\int_{0}^{t}x^{*}(s)\,ds}.$
All facts listed above from the theory of r.i. spaces and more detailed
information about this theory may be found in the books [9], [10].
In what follows, ${\rm supp}\,f$ is the support of the function $f$, i.e. the
set $\\{t:\,f(t)\neq 0\\}$. We write $F\asymp G,$ if $C^{-1}F\leq G\leq CF,$
where $C>0$ is a constant. Finally, $|A|$ denotes the number of elements of a
finite set $A$.
## 3\. Lorentz and Marcinkiewicz spaces “near” $\exp L_{1}$
###### Theorem 1.
There exists a family of Marcinkiewicz spaces
$\\{M_{\psi_{\varepsilon}}\\}_{0<\varepsilon<1}$ such that
$M_{\psi_{\varepsilon}}\subset M_{\psi_{\delta}}$ for every
$0<\varepsilon\leq\delta<1,$ satisfying the following conditions:
1. (1)
$M_{\psi_{\varepsilon}}\in\mathbb{K}$, $0<\varepsilon<1$.
2. (2)
For every r.i. space $E\in\mathbb{K}$ we have $M_{\psi_{\varepsilon}}\subset
E$ if $\varepsilon$ is small enough.
3. (3)
Functions $\psi_{\varepsilon}$ are not pairwise equivalent, or more precisely,
(5) $\lim_{t\to 0}\frac{\psi_{\varepsilon}(t)}{\psi_{\delta}(t)}=0,\quad
if\quad 0<\varepsilon<\delta<1.$
4. (4)
We have $M_{\psi_{\varepsilon}}\varsubsetneqq(\exp L_{1})_{0}$ if
$\varepsilon>0$ is small enough.
We will need the following simple assertion.
###### Lemma 2.
For every $f\in L_{1}[0,1]$
$\lim_{n\to\infty}\mathop{\mathrm{mes}}({\rm supp}\,K^{n}f)=0.$
###### Proof.
Since the operator $K$ is positive, we may assume that $f\geq 0$ and that
$\mathop{\mathrm{mes}}({\rm supp}\,f)=1$. If
$a_{n}:=\mathop{\mathrm{mes}}\\{t:\,K^{n}f(t)=0\\}$ $(n\in\mathbb{N}),$ then,
by definition of the operator $K$ (see equation (1)) $a_{1}=1/e$ and
$a_{n+1}=\frac{1}{e}+\frac{1}{e}\sum_{k=1}^{\infty}\frac{a_{n}^{k}}{k!}=e^{a_{n}-1}\;\;(n=1,2,\dots).$
Evidently, the sequence $\\{a_{n}\\}$ increases and $a_{n}\in[0,1].$ Since the
function $f(x):=\,e^{x-1}-x$ decreases on $[0,1],$ the function $e^{x-1}$ has
the only fixed point $x=1$. Hence, $\lim_{n\to\infty}a_{n}=1,$ which proves
the lemma. ∎
###### Proof of Theorem 1.
Consider the functions $h_{n}=(K^{n}1)^{*},$ $n\geq 0.$ Since the operator $K$
maps equimeasurable functions to equimeasurable ones, we have
(6) $(Kh_{n})^{*}=h_{n+1}.$
By Lemma 2, $\mathop{\mathrm{mes}}({\rm supp}\,h_{n})\to 0$ as $n\to\infty.$
Hence, the series
(7) $g_{\varepsilon}=\sum_{n=0}^{\infty}\varepsilon^{n}h_{n}$
converges everywhere on the interval $(0,1]$ for every $\varepsilon>0$ and the
function $g_{\varepsilon}$ decreases. Moreover, it follows from the definition
of the operator $K$ (see (1)) that $\|K\|_{L_{1}}=1.$ Hence, if
$0<\varepsilon<1,$ then the series (7) converges in $L_{1}$ and
$g_{\varepsilon}\in L_{1}.$ We shall show that the assertions of the theorem
hold for the family $\\{M_{\psi_{\varepsilon}}\\}_{\varepsilon>0}$, where
$\psi_{\varepsilon}(t)=\int_{0}^{t}g_{\varepsilon}(s)\,ds$ $(0\leq t\leq 1).$
1\. Let us prove that the operator $K$ is bounded in $M_{\psi_{\varepsilon}}.$
The extreme points of the unit ball in this space are equimeasurable with
$g_{\varepsilon}$ [11] and, therefore, it is sufficient to show that
$Kg_{\varepsilon}\in M_{\psi_{\varepsilon}}.$ Since $K$ is bounded in $L_{1},$
then
$Kg_{\varepsilon}=\sum_{n=0}^{\infty}\varepsilon^{n}Kh_{n}\prec\sum_{n=0}^{\infty}\varepsilon^{n}h_{n+1}\leq\frac{1}{\varepsilon}\sum_{n=0}^{\infty}\varepsilon^{n}h_{n}=\frac{1}{\varepsilon}g_{\varepsilon}.$
Here, the first inequality follows from (6) and the well-known property of
Hardy-Littlewood submajorization (see, for example, [10, § 2.2]). Thus,
$Kg_{\varepsilon}\in M_{\psi_{\varepsilon}}.$
2\. Now assume that $E\in\mathbb{K}$. As we mentioned earlier, this assumption
guarantees that $C=||K||_{E\to E}<\infty$. Evidently, $||h_{n}||_{E}\leq
C^{n}||1||_{E}.$ Therefore, for every $\varepsilon<C^{-1}$ the series (7)
converges in $E$ and $g_{\varepsilon}\in E.$ Since the space $E$ is either
separable or $E=E^{\prime\prime}$, we have that $x\in E$ and $y\prec x$ imply
that $y\in E$ and $||y||_{E}\leq||x||_{E}.$ Hence, the unit ball of the space
$M_{\psi_{\varepsilon}}$ is a subset of $E.$ Therefore,
$M_{\psi_{\varepsilon}}\subset E.$
3\. Let the function $g_{\varepsilon}$ be as in (7) and let
$0<\varepsilon<\delta.$ Arguing as in the proof of the Theorem 7.2 in [3], one
can obtain
$\lim_{t\to 0}\frac{h_{n+1}(t)}{h_{n}(t)}=\infty.$
Therefore, for every $m=1,2,\dots$
$\displaystyle\limsup_{t\to 0}\frac{g_{\varepsilon}(t)}{g_{\delta}(t)}$
$\displaystyle=$ $\displaystyle\limsup_{t\to
0}\left(\sum_{n=1}^{\infty}\varepsilon^{n}h_{n}(t)\right)\cdot\left(\sum_{n=1}^{\infty}\delta^{n}h_{n}(t)\right)^{-1}=\dots$
$\displaystyle\dots$ $\displaystyle=$ $\displaystyle\limsup_{t\to
0}\left(\sum_{n=m}^{\infty}\varepsilon^{n}h_{n}(t)\right)\cdot\left(\sum_{n=m}^{\infty}\delta^{n}h_{n}(t)\right)^{-1}\leq\left(\frac{\varepsilon}{\delta}\right)^{m}.$
Therefore, $\lim_{t\to 0}\frac{g_{\varepsilon}(t)}{g_{\delta}(t)}=0$ and the
assertion (5) follows immediately.
4\. According to the introduction, the operator $K$ acts boundedly in the
space $(\exp L_{1})_{0}.$ Hence, the fourth assertion follows from the second
and third ones. ∎
Let $\varphi_{n}(t):=\int_{0}^{t}h_{n}(s)\,ds$ $(0\leq t\leq 1)$ and let
$M_{\varphi_{n}}$ be the corresponding Marcinkiewicz space. We have,
$M_{\varphi_{n}}\subset M_{\varphi_{n+1}}\subset(\exp L_{1})_{0}$
$(n=1,2,\dots)$ and so in a certain sense the spaces $M_{\varphi_{n}}$, $n\geq
1$ may be viewed as “approximations” of the space $(\exp L_{1})_{0}$. By [3,
Theorem 7.2], we have $M_{\varphi_{n}}\subset E$ for every r.i. space
$E\in\mathbb{K}$ and every $n=1,2,\dots$ This suggests a rather natural
conjecture that $(\exp L_{1})_{0}$ is the minimal r.i. space with the Kruglov
property. However, the following consequence from Theorem 1 shows that the
class of r.i. spaces with the Kruglov property has no minimal element.
###### Corollary 3.
For every r.i. space $E\in\mathbb{K}$ there exists an r.i. space
$F\in\mathbb{K}$ such that $F\subsetneqq E.$
Contrary to the case of Marcinkiewicz spaces, all Lorentz spaces with the
Kruglov property lie “on the one side” of the space $\exp L_{1}.$
###### Theorem 4.
Let $\varphi$ be an increasing concave function on the interval $[0,1]$ such
that $\varphi(0)=0.$ If $\Lambda_{\varphi}\in\mathbb{K},$ then
$\Lambda_{\varphi}\supset\exp L_{1}.$
Let us prove the following Lemma first.
###### Lemma 5.
Let $\varphi$ be an increasing function on the interval $[0,1]$ and let
$\varphi(0)=0.$ If $\varphi$ satisfies condition (4), then
(8) $\sum_{k=1}^{\infty}\varphi(2^{-k})\leq A\varphi(1).$
Here, $A>0$ depends only on $M$ from (4).
###### Proof.
According to (4), for every $i\in\mathbb{N}$
$\sum_{j=1}^{\infty}\varphi(2^{-ij}j^{-j})\leq M\varphi(2^{-i})$
or, equivalently,
(9) $\sum_{j=1}^{\infty}\varphi(2^{-j(i+[\log_{2}j])})\leq M\varphi(2^{-i}).$
Straightforward calculations show that the quantity
$\alpha_{n}:=|\\{(i,j)\in\mathbb{N}^{2}:\,j(i+[\log_{2}j])\leq n\\}|$
satisfies the condition $\lim_{n\to\infty}n^{-1}\alpha_{n}=\infty.$ Hence,
$\alpha_{n}\geq(M+1)n$ for some $m\in\mathbb{N}$ and for every $n\geq m.$ It
follows from (9) and the monotonicity of $\varphi$ that for every $l>m$
$(M+1)\sum_{n=m}^{l}\varphi(2^{-n})\leq\sum_{i=1}^{l}\sum_{j=1}^{\infty}\varphi(2^{-j(i+[\log_{2}j])})\leq
M\sum_{i=1}^{l}\varphi(2^{-i}).$
Thus,
$\sum_{n=m}^{l}\varphi(2^{-n})\leq M\sum_{i=1}^{m-1}\varphi(2^{-i}).$
Note that $m$ depends only on $M$ and not on $\varphi,$ while $l>m$ is
arbitrary. The inequality (8) follows immediately. ∎
###### Proof of Theorem 4.
According to the introduction, condition (4) is equivalent to the condition
$\Lambda_{\varphi}\in\mathbb{K}$ [3]. Therefore, Lemma 5 implies that
condition (8) holds. Moreover, by [12], we have
$\|x\|_{\exp L_{1}}\asymp\sup_{0<t\leq 1}x^{*}(t)\log_{2}^{-1}(2/t)$
and therefore to prove the embedding $\Lambda_{\varphi}\supset\exp L_{1}$ it
is sufficient to prove only that $\log_{2}(2/t)\in\Lambda_{\varphi}.$ The
latter follows from the following estimates:
$\displaystyle\|\log_{2}(2/t)\|_{\Lambda_{\varphi}}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\log_{2}(2/t)\,d\varphi(t)=\sum_{k=1}^{\infty}\int_{2^{-k}}^{2^{-k+1}}\log_{2}(2/t)\,d\varphi(t)$
$\displaystyle\leq$
$\displaystyle\sum_{k=1}^{\infty}(k+1)(\varphi(2^{-k+1})-\varphi(2^{-k}))=2\varphi(1)+\sum_{k=1}^{\infty}\varphi(2^{-k})<\infty.$
∎
## 4\. Estimates of distribution functions
We will use the following approximation of $Kf,$ where $f$ is an arbitrary
measurable function on the interval $[0,1].$
Let $m\in\mathbb{N},$ $g_{m}(t)=\sigma_{\frac{1}{m}}f(t)$ and let
$\\{h_{m,i}\\}_{i=1}^{m}$ be independent functions equimeasurable with
$g_{m}.$ The sequence
(10) $H_{m}f(t)=\sum_{i=1}^{m}h_{m,i}(t)\;\;(0\leq t\leq 1)$
weakly converges to $Kf$ when $m\to\infty$ (in the sense of convergence of
distribution functions) (see [1, 1.6, p. 11]) or [3, Theorem 3.5]).
In particular, if $n\in\mathbb{N},$ $a_{k}\geq 0$ $(1\leq k\leq n)$ and
(11)
$f_{a}(t)=\sum_{k=1}^{n}a_{k}\chi_{\left(\frac{k-1}{n},\frac{k}{n}\right)}(t)\;\;(0\leq
t\leq 1),$
then
$g_{m}(t)=\sigma_{\frac{1}{m}}f_{a}(t)=\sum_{k=1}^{n}a_{k}\chi_{\left(\frac{k-1}{nm},\frac{k}{nm}\right)}(t)\;\;(m\in\mathbb{N}).$
In the latter case, we denote
(12) $H_{m}a(t):=H_{m}f_{a}(t)=\sum_{i=1}^{m}h_{m,i}(t).$
In addition, let $Ch(r)$ be the number of permutations $\pi$ of the set
$\\{1,2,\ldots,r\\}$ such that $\pi(i)\not=i$ for every $i=1,2,\ldots,r.$ It
is well known (see [13, p. 20]) that
(13) $\frac{1}{3}r!\leq Ch(r)\leq r!\;\;(r\in\mathbb{N}).$
We are going to compare distribution functions of $H_{m}a$ and $T_{nm}b$,
where
$b=(\underbrace{a_{1},a_{1},\ldots,a_{1}}_{m},\underbrace{a_{2},a_{2},\ldots,a_{2}}_{m},\ldots,\underbrace{a_{n},a_{n},\ldots,a_{n}}_{m}).$
###### Lemma 6.
For every $n,m\in\mathbb{N}$ and every $\tau>0$
$\mathop{\mathrm{mes}}\\{t:\,H_{m}a(t)>\tau\\}\leq
3\mathop{\mathrm{mes}}\\{t:\,T_{nm}b(t)>\tau\\}.$
###### Proof.
The function $H_{m}a(t)$ (respectively, $T_{nm}b(t)$) only takes values of the
form $\displaystyle\sum_{i=1}^{n}k_{i}a_{i}$, where $k_{i}\in\mathbb{Z},$
$k_{i}\geq 0$ for all $i=1,2,\ldots,n$ and
$\displaystyle\sum_{i=1}^{n}k_{i}\leq m$ (respectively,
$\displaystyle\sum_{i=1}^{n}k_{i}\leq mn$). Therefore, it is sufficient to
prove that
$\mathop{\mathrm{mes}}\left\\{t:\,H_{m}a(t)=\sum_{i=1}^{n}k_{i}a_{i}\right\\}\leq
3\mathop{\mathrm{mes}}\left\\{t:\,T_{nm}b(t)=\sum_{i=1}^{n}k_{i}a_{i}\right\\}$
for any choice of $k_{i}\in\mathbb{N}$,
$\displaystyle\sum_{i=1}^{n}k_{i}=q\leq m$. Note, that it is sufficient to
consider only the case when
$\sum_{i=1}^{n}k_{i}a_{i}\neq\sum_{i=1}^{n}k_{i}^{\prime}a_{i}\;\;\mbox{provided
that}\;\;(k_{1},k_{2},\dots,k_{n})\neq(k_{1}^{\prime},k_{2}^{\prime},\dots,k_{n}^{\prime}).$
Hence, $H_{m}a(t)$ equals $\sum_{i=1}^{n}k_{i}a_{i}$ if and only if exactly
$k_{i}$ (respectively, $m-q$) of the functions $h_{m,j}(t)$ $(j=1,\dots,m)$
take the value $a_{i}$ (respectively, 0). Since the functions $h_{m,j}$ are
independent, we obtain
(14) $\displaystyle\mathop{\mathrm{mes}}\left\\{t:\
H_{m}a(t)=\sum_{i=1}^{n}k_{i}a_{i}\right\\}$ $\displaystyle=$ $\displaystyle
C_{m}^{m-q,k_{1},\cdots,k_{n}}\left(1-\frac{1}{m}\right)^{m-q}\left(\frac{1}{mn}\right)^{k_{1}+\cdots+k_{n}}$
$\displaystyle\leq$ $\displaystyle
C_{m}^{m-q,k_{1},\cdots,k_{n}}\left(\frac{1}{mn}\right)^{q},$
where
$C_{m}^{m-q,k_{1},\cdots,k_{n}}=\frac{m!}{(m-q)!k_{1}!\ldots k_{n}!}.$
On the other hand, it follows from (2) and (13) that
$\displaystyle\mathop{\mathrm{mes}}\left\\{t:\,T_{mn}b(t)=\sum_{i=1}^{n}k_{i}a_{i}\right\\}$
$\displaystyle=$ $\displaystyle C_{m}^{k_{1}}C_{m}^{k_{2}}\ldots
C_{m}^{k_{n}}Ch(mn-q)\frac{1}{(mn)!}$ $\displaystyle\geq$
$\displaystyle\frac{(m!)^{n}(mn-q)!}{3(m-k_{1})!\cdots(m-k_{n})!k_{1}!\cdots
k_{n}!(mn)!}.$
Since
$(m-k_{1})!\cdots(m-k_{n})!\leq(m!)^{n-1}(m-q)!$
and
$\frac{(mn-q)!}{(mn)!}\geq\frac{1}{(mn)^{q}},$
we have
$\displaystyle\mathop{\mathrm{mes}}\left\\{t:\,T_{mn}b(t)=\sum_{i=1}^{n}k_{i}a_{i}\right\\}$
$\displaystyle\geq$ $\displaystyle\frac{m!(mn-q)!}{3k_{1}!\cdots
k_{n}!(m-q)!(mn)!}$ $\displaystyle\geq$
$\displaystyle\frac{m!}{3(m-q)!k_{1}!\cdots k_{n}!}\cdot\frac{1}{(mn)^{q}}.$
The assertion follows now from this inequality and inequality (14).∎
###### Lemma 7.
If $n,k\in\mathbb{N}$, $n\geq 4,$ $k\leq n,$ then
$\frac{(n-k)!}{n!}\leq 2\frac{(k-1)!}{n^{k}}.$
###### Proof.
Since $j(n-j)>n$ for $2\leq j\leq n-2,$ we have
$\frac{n^{k}(n-k)!}{n!(k-1)!}=\prod_{j=1}^{k-1}\frac{n}{j(n-j)}\leq\left(\frac{n}{n-1}\right)^{2}<2.$
∎
Now we continue the study begun in Lemma 6 of the connections between the
distribution functions of $T_{n}a$ and $H_{m}a.$ Whereas the estimate obtained
in Lemma 6 holds for every $m$ and $n$, the converse inequality holds only
asymptotically when $m\to\infty.$
###### Lemma 8.
Let $n\in\mathbb{N}$, $a=(a_{1},a_{2},\ldots,a_{n})\geq 0$, $\tau>0$. For
every sufficiently large $m\in N,$ the following inequality is valid:
$\mathop{\mathrm{mes}}\\{t:\,T_{n}a(t)>\tau\\}\leq
12\mathop{\mathrm{mes}}\\{t:\,2H_{m}a(t)>\tau\\}.$
###### Proof.
Assume first that $n\geq 4.$ Let $A=\\{1,2,.,n\\}.$ Denote
$S(U):=\sum_{j\in U}a_{j}$
for every $U\subset A.$ Without loss of generality, we may assume that $n=2s$
$(s\in\mathbb{N})$, $a_{i}>0$ and $S(U_{1})\neq S(U_{2})$ if $U_{1}\neq
U_{2}.$ Denote by $\mathcal{A}_{i}$ the collection of all sets $U\subset A$
with $|U|=i$ $(i=1,2,\cdots,n).$ Hence,
$\mathcal{A}=\cup_{i=1}^{n}\mathcal{A}_{i}$ is the collection of all non-empty
subsets of the set $A.$ Let us represent the set $\mathcal{A}$ in another way.
Let $U\in\mathcal{A}_{k}$ for some $k=1,2,\cdots,s.$ Denote $\mathcal{A}_{U}$
(respectively, $\mathcal{B}_{U}$) the collection of all sets $V\subset A$ such
that $V\supset U,$ $V\in\mathcal{A}_{2k}$ (respectively,
$V\in\mathcal{A}_{2k-1})$ and $S(V\setminus U)\leq S(U).$ Since
$\bigcup_{U\in\mathcal{A}_{k}}\mathcal{A}_{U}=\mathcal{A}_{2k}\;\;\mbox{and}\;\;\bigcup_{U\in\mathcal{A}_{k}}\mathcal{B}_{U}=\mathcal{A}_{2k-1}\;\;(k=1,2,.,s),$
then
(15)
$\mathcal{A}=\bigcup_{k=1}^{s}\bigcup_{U\in\mathcal{A}_{k}}\big{(}\mathcal{A}_{U}\cup\mathcal{B}_{U}\big{)}.$
It follows from the definition of $\mathcal{A}_{U}$ and $\mathcal{B}_{U}$ that
for every $V\in\mathcal{A}_{U}\cup\mathcal{B}_{U}$
(16) $S(U)\leq S(V)\leq 2S(U).$
Note that $T_{n}a(t)$ is a step function with values of the form $S(V),$ where
$V\in\mathcal{A}.$ If $|V|=r,$ then (13) implies that
$\mathop{\mathrm{mes}}\\{t:\,T_{n}a(t)=S(V)\\}=\frac{Ch(n-r)}{n!}\leq\frac{(n-r)!}{n!}.$
Also, if $|U|=k$ $(k=1,2,.,s),$ then
$|\mathcal{A}_{U}|\leq C_{n-k}^{k}=\frac{(n-k)!}{k!(n-2k)!}$
and similarly
$|\mathcal{B}_{U}|\leq C_{n-k}^{k-1}=\frac{(n-k)!}{(k-1)!(n-2k+1)!}.$
Therefore, (15) and (16) imply that
(17) $\displaystyle\mathop{\mathrm{mes}}\\{t:\,T_{n}a(t)>\tau\\}$
$\displaystyle\leq$
$\displaystyle\sum_{k=1}^{s}\sum_{U\in\mathcal{A}_{k}}\Bigg{(}\sum_{V\in\mathcal{A}_{U},S(V)>\tau}\mathop{\mathrm{mes}}\\{t:\,T_{n}a(t)=S(V)\\}$
$\displaystyle+$
$\displaystyle\sum_{V\in\mathcal{B}_{U},S(V)>\tau}\mathop{\mathrm{mes}}\\{t:\,T_{n}a(t)=S(V)\\}\Bigg{)}$
$\displaystyle\leq$
$\displaystyle\sum_{k=1}^{s}\sum_{U\in\mathcal{A}_{k},S(U)>\tau/2}\Bigg{(}\frac{(n-2k)!}{n!}\cdot\frac{(n-k)!}{k!(n-2k)!}$
$\displaystyle+$
$\displaystyle\frac{(n-2k+1)!}{n!}\cdot\frac{(n-k)!}{(k-1)!(n-2k+1)!}\Bigg{)}$
$\displaystyle\leq$ $\displaystyle
2\sum_{k=1}^{s}\sum_{U\in\mathcal{A}_{k},S(U)>\tau/2}\frac{(n-k)!}{(k-1)!n!}.$
Let us now estimate the distribution function of $H_{m}a(t)$ from below. For
every $U\in\mathcal{A}_{k},$ $S(U)>\tau/2,$ let $F_{U}$ be the set of all
$t\in[0,1]$ such that there exists a set $W\subset\\{1,2,\cdots,m\\}$ and a
bijection $\sigma:\,W\to U,$ such that $|W|=k$ (we assume that $m\geq n$) and
$h_{m,j}(t)=a_{\sigma(j)}$ if $j\in W,$ and $h_{m,j}(t)=0$ if $j\not\in W.$
Thus, for $t\in F_{U}$
(18) $H_{m}a(t)=\sum_{j=1}^{m}h_{m,j}(t)=S(U)>\frac{\tau}{2}.$
The independence of the functions $h_{m,j}(t)$ $(j=1,2,\cdots,m)$ implies
$\displaystyle\mathop{\mathrm{mes}}(F_{U})$ $\displaystyle=$ $\displaystyle
C_{m}^{k}k!\frac{1}{(mn)^{k}}\Big{(}1-\frac{1}{m}\Big{)}^{m-k}$
$\displaystyle=$
$\displaystyle\frac{m(m-1)\cdot\dots\cdot(m-k+1)}{m^{k}}\cdot\Big{(}1-\frac{1}{m}\Big{)}^{m-k}\cdot\frac{1}{n^{k}}.$
Since
$\lim_{m\to\infty}\frac{m(m-1)\ldots(m-k+1)}{m^{k}}=1$
and
$\lim_{m\to\infty}\Big{(}1-\frac{1}{m}\Big{)}^{m-k}=\frac{1}{e}>\frac{1}{3},$
we obtain
(19) $\mathop{\mathrm{mes}}(F_{U})>\frac{1}{3}\cdot\frac{1}{n^{k}}$
for all sufficiently large $m\in\mathbb{N}$ and for all $k\leq s.$
Note that $F_{U}\cap F_{U^{\prime}}=\emptyset$ if $U\neq U^{\prime}.$ Indeed,
let $i\in U\setminus U^{\prime}.$ For every $t\in F_{U}$ there exists
$j\in\\{1,2,.,m\\}$ such that $h_{m,j}(t)=a_{i}.$ However, if $t\in
F_{U^{\prime}},$ then either $h_{m,j}(t)=a_{l}\neq a_{i}$ or $h_{m,j}(t)=0\neq
a_{i}.$ Hence, equations (19) and (17) and Lemma 7 imply that
$\displaystyle\mathop{\mathrm{mes}}\\{t:\,2H_{m}a(t)>\tau\\}$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{s}\sum_{U\in\mathcal{A}_{k},S(U)>\tau/2}\mathop{\mathrm{mes}}(F_{U})$
$\displaystyle\geq$
$\displaystyle\frac{1}{3}\sum_{k=1}^{s}\sum_{U\in\mathcal{A}_{k},S(U)>\tau/2}\frac{1}{n^{k}}$
$\displaystyle\geq$
$\displaystyle\frac{1}{6}\sum_{k=1}^{s}\sum_{U\in\mathcal{A}_{k},S(U)>\tau/2}\frac{(n-k)!}{(k-1)!n!}$
$\displaystyle\geq$
$\displaystyle\frac{1}{12}\mathop{\mathrm{mes}}\\{t:\,T_{n}a(t)>\tau\\}.$
This estimate proves the lemma for $n\geq 4.$
If $1\leq n<4,$ then it is easy to show (see the argument preceding equation
(19)) that
$\mathop{\mathrm{mes}}\\{t:\,T_{n}a(t)>\tau\\}\leq
5\mathop{\mathrm{mes}}\\{t:\,2H_{m}a(t)>\tau\\}$
for all sufficiently large $m\in\mathbb{N}$ and every $\tau>0$. ∎
###### Remark 9.
The estimate
$\mathop{\mathrm{mes}}\\{t:\,T_{n}a(t)>\tau\\}\leq
C\mathop{\mathrm{mes}}\\{t:\,H_{n}a(t)>\tau\\}\;\;(\tau>0)$
fails for any constant $C$ independent of $n\in\mathbb{N}$. Indeed, if
$a_{1}=a_{2}=\ldots=a_{n}=1,$ then
$\mathop{\mathrm{mes}}\\{t:\,T_{n}a(t)=n\\}=\frac{1}{n!},$
while
$\mathop{\mathrm{mes}}\\{t:\,H_{n}a(t)=n\\}=\frac{1}{n^{n}}.$
## 5\. The Kruglov property and random permutations
###### Theorem 10.
Let $E$ be an r.i. space. The operator $K$ acts boundedly on $E$ if and only
if the sequence of operators $T_{n}$ is uniformly bounded in $E.$
###### Proof.
We are going to use notations (2), (11) and (12).
Necessity. It follows from Lemma 8 that for arbitrary $n\in\mathbb{N}$,
$a=(a_{1},a_{2},\ldots,a_{n})\geq 0$, $\tau>0$ and every sufficiently large
$m\in\mathbb{N}$ we have
$\mathop{\mathrm{mes}}\\{t:\,T_{n}a(t)>\tau\\}\leq
12\mathop{\mathrm{mes}}\\{t:\,2H_{m}a(t)>\tau\\}.$
As we pointed out in the preceding section, $H_{m}a\Rightarrow Kf_{a}$ when
$m\to\infty.$ Therefore, [14, § 6.2],
$\mathop{\mathrm{mes}}\\{t:\,H_{m}a(t)>\tau\\}\to\mathop{\mathrm{mes}}\\{t:\,Kf_{a}(t)>\tau\\}\;\;(m\to\infty)$
if the right-hand side is continuous at $\tau>0.$ Hence, the convergence is
valid for all but countably many values of $\tau.$ Hence, for all such $\tau,$
we have
$\mathop{\mathrm{mes}}\\{t:\,T_{n}a(t)>\tau\\}\leq
12\mathop{\mathrm{mes}}\\{t:\,2Kf_{a}(t)>\tau\\}.$
Both functions in the last inequality are monotone and right-continuous.
Therefore, this inequality holds for every $\tau>0.$
It is well known (see [10, § 2.4.3]), that for every r.i. space $E$ the
relation $y\in E$ together with the inequality
$\mathop{\mathrm{mes}}\\{t:\,|x(t)|>\tau\\}\leq
C\mathop{\mathrm{mes}}\\{t:\,|y(t)|>\tau\\}\;\;(\tau>0)$
imply that $x\in E$ and $\|x\|_{E}\leq\max(C,1)\|y\|_{E}$. Therefore, by the
preceding inequality
$\|T_{n}f_{a}\|_{E}\leq 24\cdot\|Kf_{a}\|_{E}$
or
$\sup\\{\|T_{n}f_{a}\|_{E}:\,\|f_{a}\|\leq 1\\}\leq 24\cdot\|K\|_{E}.$
By the definition of the operator $T_{n},$ we have $T_{n}x=T_{n}f_{a_{n}(x)},$
where $a_{n}(x)=(a_{n,k}(x))_{k=1}^{n},$
$a_{n,k}(x)=n\int_{\frac{k-1}{n}}^{\frac{k}{n}}x(s)\,ds.$ Since
$\|f_{a_{n}(x)}\|_{E}\leq\|x\|_{E}$ [10, § 2.3.2] and due to the assumption
that $E$ is either separable or coincides with its second Köthe dual, we
obtain
$\sup_{n}\|T_{n}\|_{E}\leq 24\cdot\|K\|_{E}.$
Sufficiency. Assume that $\displaystyle\sup_{n}\|T_{n}\|_{E}=C<\infty$. It
follows from Lemma 6 and [10, § 2.4.3] that
$\|H_{m}f_{a}\|_{E}\leq 3\|T_{nm}\|_{E}\|f_{a}\|_{E}\leq 3C\|f_{a}\|_{E}.$
Since $H_{m}f_{a}\Rightarrow Kf_{a}$ when $m\to\infty,$ it follows from [1,
Proposition 1.5] that
(20) $\|Kf_{a}\|_{E^{\prime\prime}}\leq 3C\|f_{a}\|_{E}.$
Let now $f=f^{*}\in E$ be arbitrary. If
$f_{n}(t)=\sum_{k=1}^{2^{n}}f(k2^{-n})\chi_{((k-1)2^{-n},k2^{-n})}(t)\;\;(0\leq
t\leq 1),\;\;n\in\mathbb{N},$
then $f_{n}(t)\uparrow f(t)$ a.e., and, therefore, $f_{n}\Rightarrow f$ [14, §
6.2]. If $\varphi_{n}$ and $\varphi$ are the characteristic functions of
$f_{n}$ and $f$ respectively, then $\varphi_{n}(t)\to\varphi(t)$
$(t\in\mathbb{R})$ ([14, § 6.4]). In view of [1, 1.6], we have
$\varphi_{K\xi}(t)=\exp(\varphi_{\xi}(t)-1)$
for every random variable $\xi.$ Hence,
$\varphi_{Kf_{n}}(t)\to\varphi_{Kf}(t)$ $(t\in\mathbb{R}),$ i.e.
$Kf_{n}\Rightarrow Kf$. Thanks to (20), we have
$\|Kf_{n}\|_{E^{\prime\prime}}\leq 3C\|f_{n}\|_{E}\leq
3C\|f\|_{E}\;\;(n\in\mathbb{N}).$
Thus, using [1, Proposition 1.5] once more, we obtain
$\|Kf\|_{E^{\prime\prime}}\leq 3C\|f\|_{E}.$
Since the distribution function of $Kf$ depends only on the distribution
function of $f$, it follows from the preceding inequality that the operator
$K$ boundedly maps $E$ into $E^{\prime\prime}$. If $E=E^{\prime\prime},$ then
we are done. It remains to consider the case when $E\neq E^{\prime\prime}$. In
this case, the space $E$ is separable. First of all, using the fact that every
function $f\in E^{\prime\prime}$, $f\geq 0,$ is the a.e. limit of its
truncations $\tilde{f}_{n}:=f\chi_{\\{f_{n}\leq n\\}}$ $(n\in\mathbb{N})$ and
arguing as above, one can infer that the operator $K$ acts boundedly in
$E^{\prime\prime}.$ Therefore, by [3, Theorem 7.2], the function
$g(t):=\frac{\ln(e/t)}{\ln(\ln(\ln(a/t)))},$
where $a>0$ is sufficiently large, belongs to $E^{\prime\prime}.$ Now, if
$\psi(u):=\frac{u\ln(e/u)}{\ln(\ln(\ln(a/u)))}\;\;(0<u\leq 1),$
then the Marcinkiewicz space $M_{\psi}\subset E^{\prime\prime}$. Hence, in
view of separability of the space $E$, we have
$(M_{\psi})_{0}\subset(E^{\prime\prime})_{0}=E_{0}=E.$
It is easy to check that
$h(t):=\frac{\ln(e/t)}{\ln(\ln(a/t))}\in(M_{\psi})_{0},$
whence, $h\in E.$ This and [3, Th. 4.4] imply that
(21) $K:\,L_{\infty}\to E.$
Let now $f\in E.$ Since $E$ is separable, there exists a sequence
$\\{f_{n}\\}\subset L_{\infty}$ such that $||f_{n}-f||_{E}\to 0.$ Since
$K:E\to E^{\prime\prime},$ we have $\|Kf_{n}-Kf\|_{E^{\prime\prime}}\to 0.$ On
the other hand, by (21) and taking into account that the embedding $E\subset
E^{\prime\prime}$ is isometric, we have $\\{Kf_{n}\\}\subset E,$ whence $Kf\in
E.$ ∎
###### Remark 11.
It follows from the proof above that the following estimate holds in every
r.i. space $E$
$\frac{1}{24}\sup_{n}\|T_{n}\|_{E}\leq\|K\|_{E}\leq
3\sup_{n}\|T_{n}\|_{E}.\qed$
We are going to infer some corollaries from Theorem 10. Let $n\in\mathbb{N}$
and let $S_{n}$ be the set of all permutations of the set
$\\{1,2,\ldots,n\\}.$ Fix a map $l=l_{n}$ from $S_{n}$ onto the set
$\\{1,2,\ldots,n!\\}$. Recall that the earlier definition of the operator
$A_{n}$ acting from $\mathbb{R}^{n}$ is given by (2). We are now in a position
to extend this definition to the set of matrices $x=(x_{i,j})_{1\leq i,j\leq
n}$ as follows
$A_{n}x(t)=\sum_{i=1}^{n}x_{i,\pi(i)},\quad
t\in\left(\frac{l(\pi)-1}{n!},\frac{l(\pi)}{n!}\right).$
One of the major results of [8] (see Corollary 8 there) says that if the
sequence of operators $\\{A_{n}\\}_{n\geq 1}$ is uniformly bounded on the set
of diagonal matrices, then it is uniformly bounded on the set of all matrices.
Applying Theorem 10, we obtain
###### Corollary 12.
If an r.i. space $E\in\mathbb{K},$ then for every $n\in\mathbb{N}$ and every
$x=(x_{i,j})_{1\leq i,j\leq n}$
$\|A_{n}x\|_{E}\leq
C\left(\Big{\|}\sum_{k=1}^{n}x_{k}^{*}\chi_{\left(\frac{k-1}{n},\frac{k}{n}\right)}\Big{\|}_{E}+\frac{1}{n}\sum_{k=n+1}^{n^{2}}x_{k}^{*}\right).$
Here, $(x_{k}^{*})_{k=1}^{n^{2}}$ is a decreasing permutation of the sequence
$(|x_{i,j}|)_{i,j=1}^{n}$ and $C>0$ does not depend either on $n$ or $x.$
###### Corollary 13.
The operators $T_{n}$, $n\geq 1$ are uniformly bounded in Orlicz space $\exp
L_{p}$ if and only if $p\leq 1.$
Indeed, the Orlicz space $\exp L_{p}$ has the Kruglov property if and only if
$p\leq 1$ (see [1, 2.4, p. 42]). The preceding corollary now follows
immediately from Theorem 10.
Theorem 10 and Corollary 3 imply
###### Corollary 14.
If $E$ is an r.i. space and if $\sup_{n}\|T_{n}\|_{E}<\infty,$ then there
exists an r.i. space $F\subsetneq E,$ such that
$\sup_{n}\|T_{n}\|_{F}<\infty.$
If $E$ is an r.i. space and $p\geq 1,$ then $E(p)$ denotes the space of all
measurable functions $x$ on the interval $[0,1]$ such that $|x|^{p}\in E.$ We
equip $E(p)$ with the norm
$\|x\|_{E(p)}=\|\,|x|^{p}\,\|_{E}^{1/p}.$
It is well known that $E(p)\subset E$ and $\|x\|_{E}\leq\|x\|_{E(p)}$ for all
$x\in E(p)$ [9, 1.d].
Let $E$ and $F$ be r.i. spaces such that $E\subset F$ and $K:E\to E.$ This
does not imply in general that $K:F\to F$ [3, Corollaries 5.6 and 5.7].
However, we have
###### Corollary 15.
If the operator $K$ is bounded in $E(p),$ then it is bounded in $E.$
###### Proof.
By Theorem 10, it is sufficient to prove that the uniform boundedness of
operators $T_{n}$, $n\geq 1$ in $E(p)$ implies the uniform boundedness of
operators $T_{n}$, $n\geq 1$ in $E$.
Let $x=(x_{1},x_{2},\dots,x_{n})\in\mathbb{R}^{n},$ $x\geq 0$ and
$\|T_{n}x\|_{E(p)}\leq C\|x\|_{E(p)}$ $(n\in\mathbb{N}).$ It means that,
$\|(T_{n}x)^{p}\|_{E}^{1/p}\leq C\|x^{p}\|_{E}^{1/p}.$
If $x^{p}=y,$ then
$\|(T_{n}y^{1/p})^{p}\|_{E}\leq C^{p}\|y\|_{E}.$
It follows from the definition of the operator $T_{n}$, $n\geq 1$ that
$(T_{n}y^{1/p})^{p}\geq T_{n}y,$ Hence, $\|T_{n}y\|_{E}\leq C^{p}\|y\|_{E}$,
$n\geq 1$. Thus, the operators $T_{n}$, $n\geq 1$ are uniformly bounded in
$E$. ∎
## References
* [1] Braverman M.Sh. Independent random variables and rearrangement invariant spaces. London Math. Soc., Lect. Note Series. V. 194. Cambridge University Press, Cambridge, 1984.
* [2] Kruglov V.M. Notes about infinitely-divisible distributions// Probability Theory and Applications 1970. V. 15. p.331-336. (in Russian)
* [3] Astashkin S.V., Sukochev F.A. Series of independent random variables in rearrangement invariant spaces: an operator approach// Israel J. Math. 2005. V. 145. P.125-156.
* [4] Astashkin S.V., Sukochev F.A. Comparison of the sums of independent and disjoint functions in symmetric spaces.// Math. Notes 2004, V.76, no. 3-4, P. 449–454
* [5] Johnson W., Schechtman G. Sums of independent random variables in rearrangement invariant function spaces// Ann. Probab. 1989. V. 17. P. 789-808.
* [6] Kwapien S., Schütt C. Some combinatorial and probabilistic inequalities and their applications to Banach space theory// Studia Math. 1985. V. 82. P. 91-106.
* [7] Semenov E.M. Operator properties of random permutations // Funct. Anal. Appl. 1994, V. 28, No. 3, P. 215–217.
* [8] Montgomery–Smith S., Semenov E.M. Random rearrangements and operators// Amer. Math. Soc. Transl. 1998. V. 184(2). P. 157-183.
* [9] Lindenstrauss J., Tzafriri L. Classical Banach Spaces II. Function spaces. Berlin-Heidelberg-New York: Springer-Verlag, 1979.
* [10] Krein S.G., Petunin Ju.I., Semenov E.M. Interpolation of linear operators. Translations of Mathematical Monographs, 54. American Mathematical Society, Providence, R.I., 1982.
* [11] Ryff J.V. Orbits of $L_{1}$ functions under doubly stochastic transformations// Trans. Amer. Math. Soc. 1965. V. 117. P. 92-100.
* [12] Lorentz G.G. Relations between function spaces// Proc. Amer. Math. Soc. 1961. V. 12. P. 127-132.
* [13] Hall M. Combinatorial theory. John Wiley and Sons, Inc., New York, 1986.
* [14] Borovkov A.A. Probability theory. Gordon and Breach Science Publishers, Amsterdam, 1998.
Astashkin S.V.
Samara State University
443011 Samara, Acad. Pavlov, 1
Russian Federation
e-mail: astashkn@ssu.samara.ru
Zanin D.V.
School of Computer Science, Engineering and Mathematics
Flinders University, Bedford Park, SA 5042 Australia
e-mail: zani0005@infoeng.flinders.edu.au
Semenov E.M.
Voronezh State University
394006, Voronezh, University pl., 1
Russian Federation
e-mail: semenov@func.vsu.ru
Sukochev F.A.
School of Mathematics and Statistics
University of New South Wales
Kensington NSW 2052 Australia
e-mail: f.sukochev@unsw.edu.au
|
arxiv-papers
| 2010-03-10T02:02:16 |
2024-09-04T02:49:08.979377
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S.V. Astashkin, D.V. Zanin, E.M. Semenov, F.A. Sukochev",
"submitter": "Dmitriy Zanin",
"url": "https://arxiv.org/abs/1003.2009"
}
|
1003.2108
|
# Scalar Field Theory on Non-commutative Snyder Space-Time
Marco Valerio Battisti battisti@icra.it Centre de Physique Théorique, Case
907 Luminy, 13288 Marseille, France Stjepan Meljanac meljanac@irb.hr Rudjer
Boskovic Institute, Bijenicka c.54, HR-10002 Zagreb, Croatia
###### Abstract
We construct a scalar field theory on the Snyder non-commutative space-time.
The symmetry underlying the Snyder geometry is deformed at the co-algebraic
level only, while its Poincaré algebra is undeformed. The Lorentz sector is
undeformed at both algebraic and co-algebraic level, but the co-product for
momenta (defining the star-product) is non-co-associative. The Snyder-deformed
Poincaré group is described by a non-co-associative Hopf algebra. The
definition of the interacting theory in terms of a non-associative star-
product is thus questionable. We avoid the non-associativity by the use of a
space-time picture based on the concept of realization of a non-commutative
geometry. The two main results we obtain are: (i) the generic (namely for any
realization) construction of the co-algebraic sector underlying the Snyder
geometry and (ii) the definition of a non-ambiguous self interacting scalar
field theory on this space-time. The first order correction terms of the
corresponding Lagrangian are explicitly computed. The possibility to derive
Noether charges for the Snyder space-time is also discussed.
###### pacs:
04.60.Bc; 02.40.Gh; 11.10.Nx
## I Introduction
Snyder space-time has been the first proposal of non-commutative geometry to
tame the UV divergences of quantum field theory Sny ; Yan . The preliminary
idea to solve this problem was to use a lattice structure instead of the
space-time continuum Hei . However, a lattice breaks the Lorentz invariance,
posing serious doubts for accepting the theory. A Lorentz invariant discrete
space-time has been formulated only by Snyder. The price to pay is a non-
commutative structure of space-time. Because of the success of renormalization
theory the Snyder program has been abandoned since its rediscovery forty years
later by mathematicians Con ; Wor . Now, the analysis of field theories on
non-commutative space-times has become a fundamental area in theoretical
physics (for reviews see DN ; Sza ).
A quantum field theory on the Snyder space-time has however not yet been
constructed and thus the removal of divergences by means of non-commutativity
effects has not yet been proved. In this paper we construct a self interacting
classical scalar field theory on this space-time. This model can be considered
as the starting point for the quantum analysis.
The non-commutativity of the Snyder space-time is encoded in the commutator
between the coordinates which is proportional to the (undeformed) Lorentz
generators. The Poincaré symmetry underlying this space-time is undeformed at
the algebraic level, while the co-algebraic sector is (highly) non-trivial. In
a previous paper BM08 we have shown that, by using the concept of
realizations, there exists infinitely many deformed Heisenberg algebras all
compatible with this geometry. This freedom can be understood as the freedom
in choosing momentum coordinates.
We here complete the previous analysis by studying the co-product and star-
product structures underlying the model. Equipped by this technology, we
construct a scalar field theory on this non-commutative space-time. Our main
goal is to define the theory without ambiguities and without needing
supplementary structures (as a deformed measure) necessary in extra
dimensional approaches. The momentum space of the Snyder-deformed Poincaré
group has not a Lie group structure since it is given by the coset
$SO(4,1)/SO(3,1)$, i.e. the de Sitter space. The co-product and the induced
star-product turn out to be non-associative. This feature represents the main
obstacle in studying field theories on the Snyder space-time. Such a kind of
deformation of the Poincaré group cannot be recovered within the
classification Wor96 , because only deformations preserving the co-
associativity are considered. The language of Hopf algebras qgroup does not
apply straightforwardly to the Snyder space-time geometry.
The non-associativity propriety obstructed the analysis of the Snyder geometry
with respect to other non-commutative space-times. For example
$\kappa$-Minkowski, a particular case of Lie algebra type space-time, has been
developed at different levels specifying star products Mel53 ; Fre06 ,
differential calculus Sit ; MK09 , scalar field theory ghost ; DLW ; AAD ;
Mel80 and conserved charges AAA ; AM1 ; AM2 . The key difference between
Snyder and $\kappa$-Minkowski is that in the latter the momentum space has the
structure of a non-abelian Lie group and thus the co-product is non-
commutative, but still associative. In particular, the Snyder space-time is
not a special case of $\kappa$-Minkowski Wess1 ; Wess2 . In fact, as clarified
in Mel09 , $\kappa$-spaces are based on Lie algebra while Snyder space is
grounded on trilinear commutations relations.
Our approach is based on the framework of realizations by which we bypass the
non-associativity and clearly define the self interacting theory. The theory
we construct lives on the non-commutative space-time and its dual has the
momentum space given by a coset. Our analysis deals with the 4d Lorentzian
model and no extra dimensional structures are invoked. Moreover, our theory is
general as we consider all realizations of the geometry, differently to the
previous approaches (for other attempts to define a scalar field theory on
Snyder space-time see GL1 ; BST ; ES ; GL2 ). The frameworks usually adopted
are recovered as particular cases of our construction.
The Snyder space-time is linked to doubly special relativity models Kov ; Guo
, loop quantum gravity LO and two-time physics tt . In particular in bat , we
have shown that a Snyder-deformed quantum cosmology predicts a big-bounce
phenomenology as in loop quantum cosmology lqc (for other comparisons between
deformed and loop cosmologies see BM07 ; BMtaub ; BLM ).
The paper is organized as follows. In Section II we describe the algebraic
structure of the Snyder space-time. In Section III the co-algebraic sector
underlying the non-commutative geometry is analyzed in detail. Section IV is
devoted to the formulation of the scalar field theory on this space-time.
Finally, in Section V the first order corrections are computed. Concluding
remarks follow.
We adopt units such that $\hbar=c=1$, the signature given by
$\eta_{\mu\nu}=\text{diag}(-,+,...,+)$ and the index convention
$\\{\mu,\nu,...\\}\in\\{0,...,n\\}$.
## II Snyder space-time
In this Section we describe the non-commutative Snyder space-time geometry. We
discuss the realizations of such a geometry as well as the dispersion relation
underlying the model.
### II.1 Deformed Heisenberg algebras
Let us consider a $(n+1)$-dimensional Minkowski space-time such that the
commutator between the coordinates has the non-trivial structure
$[\tilde{x}_{\mu},\tilde{x}_{\nu}]=sM_{\mu\nu}\,,$ (1)
where $\tilde{x}_{\mu}$ denote the non-commutative coordinates and
$s\in\mathbb{R}$ is the deformation parameter with dimension of a squared
length. We demand that the symmetries of such a space are described by an
undeformed Poincaré algebra. This means that both Lorentz generators
$M_{\mu\nu}=-M_{\nu\mu}=i(x_{\mu}p_{\nu}-x_{\nu}p_{\mu})$ and translation
generators $p_{\mu}$ satisfy the standard commutation relations
$\displaystyle[M_{\mu\nu},M_{\rho\sigma}]$ $\displaystyle=$
$\displaystyle\eta_{\nu\rho}M_{\mu\sigma}-\eta_{\mu\rho}M_{\nu\sigma}-\eta_{\nu\sigma}M_{\mu\rho}+\eta_{\mu\sigma}M_{\nu\rho}$
$\displaystyle\left[p_{\mu},p_{\nu}\right]$ $\displaystyle=$ $\displaystyle
0\,.$ (2)
We also assume that momenta and non-commutative coordinates transform as
undeformed vectors under the Lorentz algebra, i.e. the commutators
$\displaystyle[M_{\mu\nu},p_{\rho}]$ $\displaystyle=$
$\displaystyle\eta_{\nu\rho}p_{\mu}-\eta_{\mu\rho}p_{\nu}\,,$ (3)
$\displaystyle[M_{\mu\nu},\tilde{x}_{\rho}]$ $\displaystyle=$
$\displaystyle\eta_{\nu\rho}\tilde{x}_{\mu}-\eta_{\mu\rho}\tilde{x}_{\nu}\,,$
(4)
hold. The quantity $p^{2}=\eta^{\mu\nu}p_{\mu}p_{\nu}$ is then a Lorentz
invariant.
Relations (1-4) define the Snyder space-time geometry. However, they do not
fix the commutator between $\tilde{x}_{\mu}$ and $p_{\nu}$. In particular, as
it was shown in BM08 , there exists infinitely many possible commutators which
are all compatible, in the sense that the algebra closes in virtue of the
Jacobi identities, with the above requirements. This feature is understood by
means of the concept of realization JM ; MS06 ; KMS ; Mel77 ; Mel80 ; Mel53
(for a similar framework see LG ; GP ). A realization on a non-commutative
space is defined as a rescaling of the deformed coordinates $\tilde{x}_{\mu}$
in terms of ordinary phase space variables ($x_{\mu},p_{\nu}$) as
$\tilde{x}_{\mu}=\Phi_{\mu\nu}(p)x_{\nu}\,.$ (5)
The most general $SO(n,1)$-covariant realization of the Snyder geometry reads
BM08
$\tilde{x}_{\mu}=x_{\mu}\,\varphi_{1}(A)+s(xp)p_{\mu}\,\varphi_{2}(A),$ (6)
in which $\varphi_{1}$ and $\varphi_{2}$ are two (dependent) functions of the
dimensionless quantity $A=sp^{2}$ (hereafter the convention
$(ab)=\eta^{\mu\nu}a_{\mu}b_{\nu}$ is adopted). The function $\varphi_{2}$
depends on $\varphi_{1}$ by the relation
$\varphi_{2}=\frac{1+2\dot{\varphi}_{1}\varphi_{1}}{\varphi_{1}-2A\dot{\varphi}_{1}}\,,$
(7)
where dot denotes differentiation with respect to $A$. The generic realization
(6) is completely specified by the function $\varphi_{1}$. There are thus
infinitely many ways to express, via $\varphi_{1}$, the non-commutative
coordinates (1) in terms of the ordinary ones without deforming the original
symmetry. The boundary condition $\varphi_{1}(0)=1$ ensures that the ordinary
commutative framework is recovered as soon as $s=0$. The commutator between
$\tilde{x}_{\mu}$ and $p_{\nu}$ immediately follows from (6) and reads
$[\tilde{x}_{\mu},p_{\nu}]=i\left(\eta_{\mu\nu}\varphi_{1}+sp_{\mu}p_{\nu}\varphi_{2}\right)\,.$
(8)
This relation describes a deformed Heisenberg algebra.
It is also interesting to give the inverse of the realization (6), which reads
$x_{\mu}=\frac{1}{\varphi_{1}}\left(\tilde{x}_{\mu}-\frac{1}{\varphi_{1}+A\varphi_{2}}\,s(\tilde{x}p)p_{\mu}\varphi_{2}\right).$
(9)
This relation allows us to construct invariants for the non-commutative
framework from those arising in the commutative one. We only have to demand
that the invariants in ($x_{\mu},p_{\nu}$)-coordinates will be sent into the
invariants in ($\tilde{x}_{\mu},p_{\nu}$)-coordinates by means of (9).
The use of realizations allow us to give a phase space interpretation of the
Snyder space-time. Consider the non-canonical transformation
$x_{\mu}\rightarrow\Phi_{\mu\nu}(p)x_{\nu},\,p_{\nu}\rightarrow p_{\nu}$ in an
ordinary phase space coordinatized by ($x_{\mu},p_{\nu}$). The Snyder non-
commutative geometry results from such a map. This transformation can be a
generic function of momenta, but linear in coordinates (for discussions on
non-commutative classical mechanics see e.g. nccm ).
### II.2 Particular realizations
The non-commutative Snyder geometry has been analyzed in literature from
different points of view Kov ; Guo ; LO ; tt ; GL1 ; BST ; ES ; GL2 ; bat ; GB
; Gli , but only two particular realizations of its algebra are usually
adopted. These are the Snyder Sny and the Maggiore Mag1 ; Mag2 types of
realizations which are particular cases of (6).
The first realization is the one originally suggested by Snyder. It is
recovered from (6) if
$\varphi_{1}=1\,,$ (10)
which, because of (7), implies that $\varphi_{2}=1$. The second realization
has been proposed by Maggiore and it appears as soon as
$\varphi_{1}=\sqrt{1-sp^{2}}\,,$ (11)
and thus, from (7), $\varphi_{2}=0$. The momentum $p_{\mu}$ is bounded or
unbounded depending of the sign of $s$. If $s>0$ the constraint
$|p|<1/\sqrt{s}$ holds.
Beside these types of realization, the one which realizes the Weyl symmetric
ordering is the third interesting one. The Weyl ordering is obtained by the
condition
$\Phi_{\mu\nu}p_{\nu}=(\varphi_{1}\eta_{\mu\nu}+sp_{\mu}p_{\nu}\varphi_{2})p_{\nu}=p_{\mu},$
(12)
which, considering the relation (7), implies that
$\varphi_{1}=\sqrt{sp^{2}}\,\cot\sqrt{sp^{2}}.$ (13)
As we said, there are however infinitely many possible realizations of the
Snyder space-time geometry.
### II.3 Dispersion relation
Let us discuss the fate of the standard dispersion relation $p^{2}=m^{2}$ in
the Snyder space-time. In particular, we are interested in how different
realizations modify this constraint. Consider two momenta $\tilde{p}_{\mu}$
and $p_{\mu}$ in two distinct realizations. Since momenta transform as vectors
under the Lorentz symmetry, see (3), the relation
$\tilde{p}_{\mu}=p_{\mu}\,f(A)$ (14)
holds. The function $f(A)$, such that $f(0)=1$, depends on the realization
$\varphi_{1}$ and can be obtained as follows. Let for example
$\tilde{p}_{\mu}$ be a momentum in the Maggiore realization (11), i.e. the
commutator
$[\tilde{x}_{\mu},\tilde{p}_{\nu}]=i\eta_{\mu\nu}\sqrt{1-s\tilde{p}^{2}}$
holds. The function $f$ is obtained by inserting (6) and (14) in this relation
and reads
$f=\frac{1}{\sqrt{\varphi_{1}^{2}+A}}\,.$ (15)
Notice that $f=1$ as the realization (11) is taken into account and that,
because of (10), the Maggiore momentum $p^{M}_{\mu}$ is related to the Snyder
one $p^{S}_{\mu}$ by
$p^{S}_{\mu}=p^{M}_{\mu}\,\sqrt{1+s(p^{S})^{2}}\,.$ (16)
As expected from (3), the dispersion relation for $\tilde{p}^{2}$ is
undeformed, but an effective mass $m_{e}=m_{e}(m)$ has to be taken into
account. From (14) and (15), the Snyder-dispersion relation reads
$\tilde{p}^{2}=\frac{m^{2}}{\varphi_{1}^{2}(sm^{2})+sm^{2}}\equiv
m^{2}_{e}\,.$ (17)
Let us discuss such a formula in the Snyder realization ($\varphi_{1}=1$). In
the low-deformed case ($m^{2}\ll 1/s$) the effective mass is given by
$m^{2}_{e}\simeq m^{2}(1-sm^{2})$. On the other hand, in the ultra-deformed
case ($m^{2}\gg 1/s$ with $s>0$) we have $m^{2}_{e}\simeq 1/s$ (for $s<0$, $m$
is bounded as $m^{2}<1/|s|$).
## III Co-algebraic sector
The co-algebraic sector of the Snyder geometry is here analyzed. We firstly
focus on a generic framework, i.e. by considering the arbitrary realization
(6). The two particular realizations (10) and (11) are investigated below. A
discussion about the non-associativity follows.
### III.1 General framework
Deformations of symmetries underlying Snyder space-time (1) are contained in
the co-algebraic sector of a (non-trivial) quantum group. Generators
$(\tilde{x}_{\mu},p_{\mu},M_{\mu\nu})$ form an algebra defined by the
commutators (1)-(4) and (8). This is not a Hopf algebra. However,
$(p_{\mu},M_{\mu\nu})$ generate the Snyder-deformed Poincaré group
$\mathcal{P}_{S}$ whose algebra is a generalization of the Hopf algebra.
As understood from commutators (II.1) and (3), the Snyder algebraic sector is
the one of an undeformed Poincaré algebra. On the other hand, the co-algebraic
sector, defined by the action of Poincaré generators on the Snyder coordinates
$\tilde{x}_{\mu}$, is deformed. The action of Lorentz generators is still the
standard one because of (4), but the action of momenta is modified as in (8).
The Leibniz rule is thus deformed and depends on realizations. As we will see,
the co-product for momenta is no longer commutative and neither associative.
The co-product and star-product structures can obtained from realizations as
follows. Let $\mathbb{I}$ be the unit element of the space of commutative
functions $\psi(x)$. By means of (6) the action of a non-commutative function
$\tilde{\psi}(\tilde{x})$ on $\mathbb{I}$ gives Mel09 ; Mel12
$\tilde{\psi}(\tilde{x})\triangleright\mathbb{I}=\psi^{\prime}(x)\,.$ (18)
This relation provides a map from the non-commutative space of functions to
the commutative one. Notice that the commutative function $\psi^{\prime}(x)$
will be in general different from $\psi(x)$. Consider now a non-commutative
plane wave $e^{i(k\tilde{x})}$, in which $\tilde{x}_{\mu}$ refers to a given
realization (6) and $k_{\mu}$ are the eigenvalues of
$p_{\mu}=-i\partial/\partial x^{\mu}$. It is then possible to show that Mel09
; Mel12
$e^{i(k\tilde{x})}\triangleright\mathbb{I}=e^{i(Kx)},$ (19)
where $K_{\mu}=K_{\mu}(k)$ is a deformed momentum (defined below) depending on
realizations. The commutative limit $s\rightarrow 0$ leads to the standard
framework in which $K_{\mu}=k_{\mu}$. Consequently, given the inverse
transformation $K_{\mu}^{-1}=K_{\mu}^{-1}(k)$, we have
$e^{i(K^{-1}\tilde{x})}\triangleright\mathbb{I}=e^{i(kx)}.$ (20)
It is worth noting that, in the Weyl realization (13), we have
$e^{i(k\tilde{x})}\triangleright\mathbb{I}=e^{i(kx)}$ and plane waves are
undeformed.
Let us now consider two plane waves labeled by momenta $k_{\mu}$ and
$q_{\mu}$, respectively. Their action on the unit element $\mathbb{I}$ gives
$e^{i(k\tilde{x})}\left(e^{i(qx)}\right)=e^{i(F(k,q)x)}\,.$ (21)
The deformed momentum $K_{\mu}$ is thus determined by $K_{\mu}=F_{\mu}(k,0)$,
where the function $F_{\mu}(k,q)$ specifies the co-product as well as the
star-product. It can be obtained by a straightforward implementation of the
Campbell-Baker-Hausdorff formula or by the more elegant method developed in
Mel09 ; Mel12 .
The star-product between two plane waves is defined by $F_{\mu}(k,q)$ as
$e^{i(kx)}\star e^{i(qx)}\equiv
e^{i(K^{-1}(k)\tilde{x})}e^{i(K^{-1}(q)\tilde{x})}\triangleright\mathbb{I}=\\\
=e^{i(K^{-1}(k)\tilde{x})}\left(e^{i(qx)}\right)=e^{i(\mathcal{D}(k,q)x)}$
(22)
in which
$\mathcal{D}_{\mu}(k,q)=F_{\mu}\left(K^{-1}(k),q\right)\,.$ (23)
The star-product defines, by means of (19) and (20), a Weyl mapping from the
commutative to the non-commutative spaces provided by a one-to-one
correspondence between $e^{i(k\tilde{x})}$ and $e^{i(Kx)}$. The co-product for
momenta $\Delta p_{\mu}$ (and the corresponding Leibniz rule) is obtained from
$\mathcal{D}_{\mu}(k,q)$ as
$\Delta p_{\mu}=\mathcal{D}_{\mu}(p\otimes 1,1\otimes p)\,.$ (24)
In particular, the function $\mathcal{D}_{\mu}$ describes the non-abelian sum
of momenta in the Snyder non-commutative space-time, i.e.
$\mathcal{D}_{\mu}(k,q)=k_{\mu}\oplus q_{\mu}\neq k_{\mu}+q_{\mu}\,.$ (25)
As soon as the non-commutativity effects (in our case the parameter $s$) are
switched off, the ordinary abelian rule
$\mathcal{D}_{\mu}(k,q)=k_{\mu}+q_{\mu}$ is recovered. By means of (22), it is
possible to obtain the star-product between two generic functions $f$ and $g$
of commuting coordinates (see for example MS06 ; Mel53 ; Mel12 ). Adopting the
plane waves relation (22), the general result for the star product stands as
$(f\star g)(x)=\lim_{\begin{subarray}{c}y\rightarrow x\\\ z\rightarrow
x\end{subarray}}e^{ix_{\mu}\left(\mathcal{D}^{\mu}(p_{y},p_{z})-p^{\mu}_{y}-p^{\mu}_{z}\right)}f(y)g(z)\,.$
(26)
Star product is a binary operation acting on the algebra of functions defined
on the ordinary commutative space and it encodes features reflecting the non-
commutative nature of Snyder space-time (1). The star product is uniquely
defined, but its concrete form is related to a particular realization and vice
versa. For any realization the star product (22), and then (26), is non-
associative. The corresponding co-product (24) is non-co-associative. Such a
result has been confirmed by the recent analysis GL2 also.
This construction is well defined and allows us to obtain, from realizations
(6), both co-product and star-product structures underlying the Snyder space-
time. The inverse path is also meaningful: starting from a star-product (or a
co-product) it is always possible to recover an information about realization
we are working in. However, as we shall see, to construct a scalar field
theory on Snyder space-time it is more suitable to deal with realizations
instead of star-products. The non-associativity of the star-product in fact
poses severe challenges in defining interaction terms.
Let us now compute the co-product $\Delta p_{\mu}$, at the first order in $s$,
for a generic realization (6). Expanding the realization function
$\varphi_{1}$ as $\varphi_{1}=1+c_{1}sp^{2}+\mathcal{O}(s^{2})$ and
considering (15), we obtain
$\Delta p_{\mu}=\Delta_{0}p_{\mu}+s\,\Delta_{1}p_{\mu}+\mathcal{O}(s^{2})$
(27) $\Delta_{0}p_{\mu}=p_{\mu}\otimes 1+1\otimes p_{\mu}$
$\Delta_{1}p_{\mu}=\left(c-\frac{1}{2}\right)p_{\mu}\otimes
p^{2}+\left(2c-\frac{1}{2}\right)p_{\mu}p_{\nu}\otimes p^{\nu}+\\\
+c\left(p^{2}\otimes p_{\mu}+2p_{\nu}\otimes p^{\nu}p_{\mu}\right)$ (28)
where $c=(2c_{1}+1)/2$. Here $\Delta_{0}p_{\mu}$ and $\Delta_{1}p_{\mu}$
denote the co-product at the zero and first order is $s$, respectively. The
Maggiore type of realization (11) is defined by $c_{1}=-1/2$ and thus it is
recovered as $c=0$. The Snyder one (10) appears for $c_{1}=0$ and thus
$c=1/2$, while the Weyl one (13) for $c=1/6$. The co-product (27) defines the
Snyder non-abelian sum in a generic realization. The star-product is obtained
from (22).
To complete the analysis of the co-algebraic sector we need to specify the co-
product $\Delta M_{\mu\nu}$ of the Lorentz generators, as well as the antipode
$S(g)$ and the co-unit $\varepsilon(g)$ for any element $g$ of
$\mathcal{P}_{S}$. Because of relations (II.1)-(4), the co-product $\Delta
M_{\mu\nu}$ is trivial, i.e.
$\Delta M_{\mu\nu}=M_{\mu\nu}\otimes 1+1\otimes M_{\mu\nu}\,.$ (29)
The antipode $S(g)$ is defined by the equation
$\mathcal{D}\left(g,S(g)\right)=g\oplus S(g)=0\,.$ (30)
From (27) and (29), we immediately realize that the antipode is not deformed
for any $g=(p_{\mu},M_{\mu\nu})$, that is
$S(p_{\mu})=-p_{\mu}\,\qquad S(M_{\mu\nu})=-M_{\mu\nu}\,.$ (31)
Because different momenta are related by (14), the antipode $S(p_{\mu})$ is
exactly (not only at the first order) trivial in all realizations. On the
other hand, the co-unit $\varepsilon(g)$ is also trivial for any
$g\in\mathcal{P}_{S}$. Finally, we observe that co-product for momenta (24) is
covariant becouse of (29), i.e. the relation
$[\Delta M_{\mu\nu},\Delta p_{\rho}]=\eta_{\nu\rho}\Delta
p_{\mu}-\eta_{\mu\rho}\Delta p_{\nu}$ (32)
holds. This expression is the co-algebraic counter term of (3).
Summarizing, the Snyder-deformed Poincaré group $\mathcal{P}_{S}$ is
characterized as follows. The Lorentz symmetry is undeformed at both algebraic
and co-algebraic level. The deformations are encoded in the co-product (24)
only, which in particular is non-co-associative. The corresponding star-
product (22) is non-associative and a homomorphism relates these structures.
The algebraic sector is then compatible with the co-algebraic one. Therefore,
the generators $(p_{\mu},M_{\mu\nu})$ of $\mathcal{P}_{S}$ form a generalized
Hopf algebra, which we shall denote as a non-co-associative Hopf algebra.
### III.2 Particular realizations
We now study the co-product structure underlying the two particular
realizations (10) and (11). In both cases a closed form of the co-product
arises.
Let us firstly consider the Maggiore realization (11). The basic function
$F_{\mu}(k,q)$ in (21) is given by
$F_{\mu}=q_{\mu}+k_{\mu}\sqrt{1-A_{q}}\,\frac{\sin\sqrt{A_{k}}}{\sqrt{A_{k}}}-sk_{\mu}(kq)\,\frac{1-\cos\sqrt{A_{k}}}{\sqrt{A_{k}}}$
(33)
where $A_{p}=sp^{2}$. The ordinary function $F_{\mu}=k_{\mu}+q_{\mu}$ is
recovered in the $s=0$ case. From (33) one immediately obtains the deformed
momentum $K_{\mu}(k)$ which reads
$K_{\mu}=F_{\mu}(k,0)=k_{\mu}\frac{\sin\sqrt{A_{k}}}{\sqrt{A_{k}}}\,.$ (34)
The co-product $\Delta p_{\mu}$ follows from (23) and it is given, in terms of
the realization function $\varphi_{1}=\sqrt{1-sp^{2}}$, by
$\Delta
p_{\mu}=p_{\mu}\otimes\varphi_{1}-\frac{s}{1+\varphi_{1}}\,p_{\mu}p_{\nu}\otimes
p^{\nu}+1\otimes p_{\mu}\,.$ (35)
Such a co-product (namely the addition rule (25)) is non-co-associative. The
order by which we sum the momenta becomes important. As $s=0$ we have
$\varphi_{1}=1$ and the trivial co-product $\Delta_{0}p_{\mu}=p_{\mu}\otimes
1+1\otimes p_{\mu}$ is recovered. The first-order term coincides with (27) for
$c=0$.
Let us now analyze the Snyder realization (10). In this case the function
$F_{\mu}(k,q)$ in (21) reads
$\displaystyle F_{\mu}$ $\displaystyle=$ $\displaystyle
g\,(h\,k_{\mu}+q_{\mu})$ (36) $\displaystyle g$ $\displaystyle=$
$\displaystyle\left(\cos\sqrt{A_{k}}-\sqrt{\frac{s}{k^{2}}}(kq)\sin\sqrt{A_{k}}\right)^{-1}$
$\displaystyle h$ $\displaystyle=$
$\displaystyle\frac{1}{k^{2}}\left(\sqrt{\frac{k^{2}}{s}}\sin\sqrt{A_{k}}+(kq)(\cos\sqrt{A_{k}}-1)\right)\,.$
The deformed momentum $K_{\mu}(k)$ is then given by
$K_{\mu}=F_{\mu}(k,0)=k_{\mu}\frac{\tan\sqrt{A_{k}}}{\sqrt{A_{k}}}\,.$ (37)
The ordinary framework is restored as $s=0$. The co-product directly follows
from (23) and reads
$\Delta p_{\mu}=\frac{1}{1-sp_{\nu}\otimes
p^{\nu}}\left(\frac{}{}p_{\mu}\otimes 1+\right.\\\
-\left.\frac{s}{1+\sqrt{1+A_{p}}}\,p_{\mu}p_{\nu}\otimes
p^{\nu}+\sqrt{1+A_{p}}\otimes p_{\mu}\right).$ (38)
Also in this case the co-product is non-co-associative. The first order-term
coincides with the $c=1/2$ case of (27).
### III.3 On the non-associativity
The relation between special relativity and the Snyder geometry allows us to
better understand the physical meaning of the non-associativity.
Special relativity can be analyzed (and derived) from a non-commutative point
of view girliv . Consider the Galileo group $ISO(3)=SO(3)\cdot\mathbb{R}^{3}$.
Speeds generate translations and the speed space $\mathbb{R}^{3}$ can be
identified as $\mathbb{R}^{3}\sim ISO(3)/SO(3)$. A manifold of this type is a
coset space. In this case it has the (Lie) group structure. Special relativity
can be viewed as arising from the deformation of $\mathbb{R}^{3}$ into the
curved space $\mathcal{C}=SO(3,1)/SO(3)$. This operation sends the Galileo
group into the Lorentz one $SO(3,1)=SO(3)\cdot\mathcal{C}$ (this is the Cartan
decomposition of the Lorentz group Lie ). The coset $SO(3,1)/SO(3)$ is nothing
but the (hyperbolic) boosts space, but it is not a Lie group. In fact the
product between two boosts is not longer a boost, but an element of the full
Lorentz group $SO(3,1)$. The composition of speeds can be extracted from a co-
product structure. It turns out that the composition of (non-collinear) speeds
is no longer commutative and neither associative. A physical manifestation of
non-associativity is the well-known Thomas precession thomas . From a
mathematical point of view, the non-associativity is a consequence of the fact
that the coset space is not a group manifold.
The Snyder space-time geometry can be viewed from the same perspective.
Consider the Poincaré group $\mathcal{P}=SO(3,1)\cdot\mathbb{R}^{4}$. As
above, the momentum space $\mathbb{R}^{4}$ can be viewed as the coset
$\mathbb{R}^{4}\sim\mathcal{P}/SO(3,1)$ and of course it is a group manifold.
Deforming the momentum space into the de Sitter space
$d\mathcal{S}=SO(4,1)/SO(3,1)$ we recover the Snyder non-commutative geometry.
This is the original formulation made by Snyder himself Sny . The Snyder-
deformed Poincaré group $\mathcal{P}_{S}$ is then factorized as
$\mathcal{P}_{S}=SO(3,1)\cdot d\mathcal{S},$ (39)
showing that the Lorentz symmetry is undeformed. On the other hand, the
translation sector of this (quantum) group is deformed consistently to (1). As
in the previous case, the coset $d\mathcal{S}$ is not a Lie group. The non-co-
associativity of Snyder co-product can be traced back to this feature.
## IV Scalar field theory
In this Section we construct the scalar field theory on the 4d Snyder non-
commutative space-time. We first consider the Fourier transformation and
define the Snyder scalar field and then we write down the action for the
theory. A comparison with other approaches follows.
### IV.1 Preliminaries
We define a scalar field $\tilde{\phi}(\tilde{x})$ on the Snyder non-
commutative space-time by means of the Fourier transformation as
$\tilde{\phi}(\tilde{x})=\int[dk]\,\hat{\phi}(k)\,e^{i(K^{-1}\tilde{x})}\,.$
(40)
The integration measure $[dk]$ is a priori deformed depending on the antipode
$S(k_{\mu})$. However, as we have previously seen, it is trivial in any
realizations. The measure in (40) is thus the ordinary one
$[dk]=\frac{d^{4}k}{(2\pi)^{4}}\,.$ (41)
Let us now consider the action of Snyder scalar field (40) on the identity
$\mathbb{I}$. By means of (20), this operation gives
$\tilde{\phi}(\tilde{x})\triangleright\mathbb{I}=\phi(x)\,,$ (42)
which ensures the Lorentz scalar behavior of the model. As a further step we
consider the quadratic term
$\tilde{\phi}^{2}(\tilde{x})\triangleright\mathbb{I}$. Given the definition
(40) and remembering (21), (22) and (23), we obtain
$\tilde{\phi}^{2}(\tilde{x})\triangleright\mathbb{I}=\left(\phi\star\phi\right)(x)\,.$
(43)
We have thus recovered the star-product structure.
Let us now discuss the notion of a real and complex Snyder scalar field.
Firstly, because of triviality of the antipode, the conjugation is also an
ordinary one. Secondly, the non-commutative coordinates $\tilde{x}_{\mu}$ have
to be hermitian operators in any given realization. All the commutators given
above are invariant under the formal anti-linear involution “${\dagger}$”
$\tilde{x}_{\mu}^{\dagger}=\tilde{x}_{\mu},\qquad
p_{\mu}^{\dagger}=p_{\mu},\qquad M_{\mu\nu}^{\dagger}=-M_{\mu\nu}\,,$ (44)
where the order of elements is inverted under the involution. On the other
hand, the realization (6) is in general not hermitian. The hermiticity
condition can be immediately implemented as soon as the expression
$\tilde{x}_{\mu}=\frac{1}{2}\left(x_{\mu}\varphi_{1}+s(x\,p)p_{\mu}\varphi_{2}+\varphi_{1}^{\dagger}x_{\mu}^{\dagger}+s\,\varphi_{2}^{\dagger}p_{\mu}^{\dagger}(x\,p)^{\dagger}\right)$
(45)
is taken into account. However, the physical results do not depend on the
choice of the representation as long as there exists a smooth limit
$\tilde{x}_{\mu}\rightarrow x_{\mu}$ as $s\rightarrow 0$. We can thus restrict
our attention to non-hermitian realization only. Consequently, we focus on the
real Snyder scalar field theory, while the complex one can be
straightforwardly defined.
### IV.2 Action for scalar field theory
We are now able to construct a Lagrangian for the non-commutative scalar field
(40). Let us start by analyzing how the ordinary kinematic term
$(\partial_{\mu}\phi)(\partial^{\mu}\phi)$ is changed in the Snyder space-
time. Following the previous reasonings, the corresponding term in the non-
commutative framework is given by
$(\partial_{\mu}\tilde{\phi})(\partial^{\mu}\tilde{\phi})\triangleright\mathbb{I}$
(notice that the derivative is still with respect to the commutative
coordinates, i.e. $\partial_{\mu}=\partial/\partial x^{\mu}$). Such a term,
expressed by means of the Fourier transformation (40), is uniquely defined. In
fact, in order that the differentiation makes sense, we have firstly to
project the plane waves on $\mathbb{I}$ and then act on these by
differentiation. By using (21) and (31), the relation
$\left(\partial_{\mu}e^{i(K^{-1}\tilde{x})}\right)e^{i(qx)}=i(\mathcal{D}_{\mu}-q_{\mu})e^{i(\mathcal{D}(k,q)x)}=ik_{\mu}e^{i(\mathcal{D}(k,q)x)}\,$
(46)
follows. The kinematic part, considering (20) and (46), is then given by
$(\partial_{\mu}\tilde{\phi})(\partial^{\mu}\tilde{\phi})\triangleright\mathbb{I}=\int[d^{2}k]\,\hat{\phi}_{k_{1}}\hat{\phi}_{k_{2}}\left(\partial_{\mu}e^{i(K_{1}^{-1}\tilde{x})}\right)\partial^{\mu}\left(e^{i(K_{2}^{-1}\tilde{x})}\triangleright\mathbb{I}\right)=-\int[d^{2}k]\,\hat{\phi}_{k_{1}}\hat{\phi}_{k_{2}}(k_{1}k_{2})\,e^{i(\mathcal{D}(k_{1},k_{2})x)}\,,$
(47)
where $[d^{n}k]=[dk_{1}]...[dk_{n}]$ and $\phi_{k}=\phi(k)$. This expression
leads to the correct ordinary result as $s=0$. As in (43), the star-product
prescription leads to the same result with respect to our construction:
$(\partial_{\mu}\tilde{\phi})(\partial^{\mu}\tilde{\phi})\triangleright\mathbb{I}=(\partial_{\mu}\phi)\star(\partial^{\mu}\phi)\,.$
(48)
The action for a non-interacting massive scalar field on Snyder space-time
then reads
$I=\int
d^{4}x\left(\partial_{\mu}\tilde{\phi}\,\partial^{\mu}\tilde{\phi}+m^{2}\,\tilde{\phi}^{2}\right)\triangleright\mathbb{I}=\\\
=\int
d^{4}x\left[(\partial_{\mu}\phi)\star(\partial^{\mu}\phi)+m^{2}(\phi\star\phi)\right]\,.$
(49)
Because of the antipode (31), the action in the momentum space can be
trivially written.
The non-commutativity effects are thus summarized within the co-product (27),
i.e. within the non-abelian sum $\mathcal{D}_{\mu}(k_{1},k_{2})$. The non-
commutative corrections to the ordinary theory depend on realizations. For
each type of realization different actions appear.
Finally, we investigate the role of self interactions. In particular, we
consider the cubic $\tilde{\phi}^{3}(\tilde{x})\triangleright\mathbb{I}$ and
quartic $\tilde{\phi}^{4}(\tilde{x})\triangleright\mathbb{I}$ interaction
terms. These terms can be immediately obtained. The generalization of (21) to
three plane waves, considering also (22), reads
$e^{i(K_{3}^{-1}\tilde{x})}\left(e^{i(K_{2}^{-1}\tilde{x})}\left(e^{i(K_{1}^{-1}\tilde{x})}\triangleright\mathbb{I}\right)\right)=e^{i(\mathcal{D}_{3}(k_{3},k_{2},k_{1})x)}\,,$
(50)
in which
$(\mathcal{D}_{3})_{\mu}(k_{3},k_{2},k_{1})=\mathcal{D}_{\mu}(k_{3},\mathcal{D}(k_{2},k_{1}))$.
This defines the cubic term
$\tilde{\phi}^{3}(\tilde{x})\triangleright\mathbb{I}=\left(\phi\star(\phi\star\phi)\right)(x).$
(51)
The quartic term $\tilde{\phi}^{4}(\tilde{x})\triangleright\mathbb{I}$ is
determined in the same way. Given four plane waves we have
$e^{i(K_{4}^{-1}\tilde{x})}\left(e^{i(K_{3}^{-1}\tilde{x})}\left(e^{i(K_{2}^{-1}\tilde{x})}\left(e^{i(K_{1}^{-1}\tilde{x})}\triangleright\mathbb{I}\right)\right)\right)=e^{i(\mathcal{D}_{4}x)}\,,$
(52)
and therefore
$\tilde{\phi}^{4}(\tilde{x})\triangleright\mathbb{I}=\left(\phi\star(\phi\star(\phi\star\phi))\right)(x)\,,$
(53)
where
$(\mathcal{D}_{4})_{\mu}=\mathcal{D}_{\mu}(k_{4},\mathcal{D}_{3}(k_{3},k_{2},k_{1}))$.
Summarizing, we have defined a Lagrangian density for a self interacting
scalar field on the Snyder non-commutative space-time geometry. Our framework,
which is based on realizations, uniquely fixes the theory. This is relevant
because the co-product is non-co-associative (the corresponding star-product
is non-associative). This feature would lead, a priori, to a non-unique
definition of the model. Such a shortcoming is bypassed in our construction.
### IV.3 Relation with other approaches
Our construction differs with respect to the usual ones in two main points:
the dimensions of the structure underlying the theory and the adopted algebra.
The scalar field theory on Snyder space-time is usually formulated by
considering a five dimensional structure GL1 ; BST ; ES ; GL2 . The same
happens for the field theories on $\kappa$-Minkowski ghost ; DLW ; AAD . In
particular, the momentum space is the de Sitter section in a five dimensional
flat space and a deformed Fourier measure is thus needed to ensure the Lorentz
invariance GL1 ; BST ; ES ; GL2 . In $\kappa$-Minkowski, a five dimensional
differential structure predicts some unphysical ghost modes ghost (to
overcame this feature a twist deformation of the symmetry has been proposed
twist1 ; twist2 ). On the other hand, our theory is defined on a four
dimensional space-time. No extra measures are needed and the theory has the
same field structure of the commutative framework. The Snyder deformed
symmetry algebra is the original undeformed one and only the co-product
structure changes. Interesting, this is exactly the framework arising from the
twist formulation of non-commutative field theories twist1 ; twist2 ; Mel80 .
The second difference with respect to other approaches is that our theory is
generic. All the possible realizations of the algebra are taken into account.
The other attempts to construct a scalar field theory on the Snyder space-time
are in fact based on a particular realization only. Our theory in the Snyder
type of realization (10) corresponds, up to the momentum-space duality, to the
previous proposals GL1 ; BST ; ES ; GL2 .
## V First order corrections
In this Section we explicitly compute the generic non-commutative corrections,
up to the first order in $s$, to the commutative theory.
As we have seen, all the non-commutative informations are summarized in the
non-abelian sum (25), namely in the co-product (27). We are thus interested in
the function
$(\mathcal{D}_{4})_{\mu}=(\mathcal{D}_{4})_{\mu}(k_{4},k_{3},k_{2},k_{1})$
defined in (52). The functions $(\mathcal{D}_{3})_{\mu}(k_{3},k_{2},k_{1})$
and $\mathcal{D}_{\mu}(k_{2},k_{1})$, which define the cubic and quadratic
terms, are clearly recovered from this one as soon as $k_{4}=0$ and
$k_{4}=k_{3}=0$, respectively. The $(\mathcal{D}_{4})_{\mu}$ function can be
expanded in the deformation parameter $s$ as
$(\mathcal{D}_{4})_{\mu}=(\mathcal{D}_{4})^{0}_{\mu}+s(\mathcal{D}_{4})^{1}_{\mu}+\mathcal{O}(s^{2})$
(54)
$(\mathcal{D}_{4})^{0}_{\mu}=(k_{1})_{\mu}+(k_{2})_{\mu}+(k_{3})_{\mu}+(k_{4})_{\mu}$
$(\mathcal{D}_{4})^{1}_{\mu}=\alpha(k_{1})_{\mu}+\beta(k_{2})_{\mu}+\gamma(k_{3})_{\mu}+\delta(k_{4})_{\mu}\,,$
where the superscript denotes the order in $s$.
The correction term $(\mathcal{D}_{4})^{1}_{\mu}$ depends on realizations
through $\alpha=\alpha(\varphi_{1})$, $\beta=\beta(\varphi_{1})$,
$\gamma=\gamma(\varphi_{1})$ and $\delta=\delta(\varphi_{1})$. These functions
are given by
$\alpha=c\left[k_{2}^{2}+k_{3}^{2}+k_{4}^{2}+2(k_{1}k_{2}+k_{3}k_{2}+k_{3}k_{1}+\right.\\\
\left.+k_{4}k_{3}+k_{4}k_{2}+k_{4}k_{1})\right]\,,$ (55)
$\beta=\left(c-\frac{1}{2}\right)k_{1}^{2}+\left(2c-\frac{1}{2}\right)(k_{1}k_{2})+c\,[k_{3}^{2}+k_{4}^{2}+\\\
+2(k_{3}k_{2}+k_{3}k_{1}+k_{4}k_{3}+k_{4}k_{2}+k_{4}k_{1})]\,,$ (56)
$\gamma=\left(c-\frac{1}{2}\right)(k_{1}+k_{2})^{2}+\left(2c-\frac{1}{2}\right)(k_{3}k_{2}+k_{3}k_{1})+\\\
+c\,[k_{4}^{2}+2(k_{4}k_{3}+k_{4}k_{2}+k_{4}k_{1})]\,,$ (57)
$\delta=\left(c-\frac{1}{2}\right)(k_{1}+k_{2}+k_{3})^{2}+\left(2c-\frac{1}{2}\right)(k_{4}k_{3}+k_{4}k_{2}+k_{4}k_{1}).$
(58)
The value of the constant $c$ determines the realization in which we are
working. The Snyder (10), the Maggiore (11) and the Weyl (13) types of
realization are respectively recovered for $c=1/2,0,1/6$.
## VI Concluding remarks
In this paper we have constructed a scalar field theory on the Snyder non-
commutative space-time. The next step will be the quantization of the model in
order to investigate the fate of UV divergences and thus fully analyze the
Snyder proposal.
We have shown that the deformations of symmetries are all contained in the co-
algebraic sector and that the co-product is non-co-associative. By using the
realizations of the Snyder algebra we have constructed a well defined (namely
non-ambiguous) self interacting scalar field theory. The ambiguities carried
out by the non-associative sum of momenta (and thus the non-associative star-
product) have been overcome by the use of realizations. By means of a map
between the non-commutative functions and the commutative ones, a scalar field
action has been constructed. This theory has been directly defined on the
space-time and, since the Fourier space has been identified with the de Sitter
space, it is dual to a field theory over the coset $SO(4,1)/SO(3,1)$. Finally,
we have computed the first order corrections in a generic realization.
As last point, it is interesting to mention that we can construct Noether
charges for the Snyder space-time. As was shown in AM1 ; AM2 , the key
ingredient to build Noether charges in a non-commutative theory is a Poisson
map between the deformed and the undeformed spaces of solutions of the Klein-
Gordon equation. In our framework this kind of map is given by the projection
of non-commutative functions on the “vacuum”, as in (18). By using this map it
is possible to induce a symplectic structure on the space of the non-
commutative functions and thus obtain a conserved symplectic product defining
charges. This analysis will be reported elsewhere BatMel10 .
## Acknowledgments
We would like to thank Andjelo Samsarov for useful discussions and for a
critical reading of the manuscript. Antonino Marcianò is thanked for comments.
Part of this work is supported by the Ministry of Science and Technology of
the Republic of Croatia under contract No. 098-0000000-2865.
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|
arxiv-papers
| 2010-03-10T13:30:01 |
2024-09-04T02:49:08.989283
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Marco Valerio Battisti, Stjepan Meljanac",
"submitter": "Marco Valerio Battisti",
"url": "https://arxiv.org/abs/1003.2108"
}
|
1003.2117
|
# Hierarchies of Subsystems of Weak Arithmetic
Shahram Mohsenipour Shahram Mohsenipour, School of Mathematics, Institute for
Research in Fundamental Sciences (IPM)
P. O. Box 19395-5746, Tehran, Iran mohseni@ipm.ir
###### Abstract.
We completely characterize the logical hierarchy of various subsystems of weak
arithmetic, namely: ZR, ZR + N, ZR + GCD, ZR + Bez, OI + N, OI + GCD, OI +
Bez.
###### Key words and phrases:
open induction, subsystem, logical hierarchy
###### 2000 Mathematics Subject Classification:
03F30,03H15
This work was done while the author was a Postdoctoral Research Associate at
the School of Mathematics, Institute for Research in Fundamental Sciences
(IPM)
## 1\. Introduction
In 1964 Shepherdson [6] introduced a weak system of arithmetic, Open Induction
(OI), in which the Tennenbaum phenomenon does not hold. More precisely, if we
restrict induction just to open formulas (with parameters), then we have a
recursive nonstandard model. Since then several authors have studied Open
Induction and its related fragments of arithmetic. For instance, since Open
Induction is too weak to prove many true statements of number theory (It
cannot even prove the irrationality of $\surd{2}$), a number of algebraic
first order properties have been suggested to be added to OI in order to
obtain closer systems to number theory. These properties include: Normality
[9] (abbreviated by N), having the GCD property [8], being a Bezout domain [3,
8] (abbreviated by Bez), and so on. We mention that GCD is stronger than N,
Bez is stronger than GCD and Bez is weaker than $IE_{1}$ ($IE_{1}$ is the
fragment of arithmetic based on the induction scheme for bounded existential
formulas and by a result of Wilmers [11], does not have a recursive
nonstandard model). Boughattas in [1, 2] studied the non-finite
axiomatizability problem and established several new results, including: (1)
OI is not finitely axiomatizable, (2) OI + N is not finitely axiomatizable. To
show that, he defined and considered the subsystems (OI)p of (OI) and (N)n of
N ($1\leq p,n<\omega$) (See the next section for the definitions) and proved:
###### Theorem 1.1 (Boughattas [1]).
$(1)$ $(OI)_{p}$ is finitely axiomatizable,
$(2)$ Suppose $(p!,p^{{}^{\prime}})=1$, then
$(OI)_{p}\nvdash(OI)_{p^{{}^{\prime}}}$.
###### Theorem 1.2 (Boughattas [2], Theorem 2).
Suppose $(p!,p^{{}^{\prime}})=1$ and $(n!,n^{{}^{\prime}})=1$,
$(1)$ $N+(OI)_{p}\nvdash(OI)_{p^{{}^{\prime}}}$,
$(2)$ $(N)_{n}+(OI)\nvdash(N)_{n^{{}^{\prime}}}$,
$(3)$ $(OI)_{p}+\neg(OI)_{p^{{}^{\prime}}}+(N)_{n}+\neg(N)_{n^{{}^{\prime}}}$
is consistent.
In [4] we strengthened Theorem 1.1 (2) to completely characterize the logical
hierarchy of OI, by showing that $(OI)_{p}\nvdash(OI)_{p+1}$ iff $p\neq 3$. In
this paper by modifying Boughattas’ original proofs, we also strengthen
Theorem 1.2 in two directions and completely characterize the logical
hierarchy of OI + N, OI + GCD, OI + Bez:
Theorem C. $Bez$ \+ $(OI)_{p}$ $\nvdash$ $(OI)_{p+1}$, when $p\neq 3$.
Theorem D. $(OI)_{p}$ \+ $\neg(OI)_{p+1}$ \+ $(N)_{n}$ \+ $\neg(N)_{n+1}$ is
consistent, when $p\neq 3$.
So we will have the following immediate consequences:
Corollary E.
(1) $N$ \+ $(OI)_{p}$ $\nvdash$ $(OI)_{p+1}$, when $p\neq 3$.
(2) $GCD$ \+ $(OI)_{p}$ $\nvdash$ $(OI)_{p+1}$, when $p\neq 3$.
(3) All of the following subsystems of arithmetic are non-finite
axiomatizable: OI, OI + N, OI + GCD, OI + Bez, (OI)p \+ N, OI + (N)n.
In Theorems A and B of this paper, we consider the ZR versions of the above
theorems. ZR is a subsystem of arithmetic that allows Euclidean division over
each non-zero natural number $n\in\mathbb{N}$. ZR is introduced by Wilkie [10]
in which he proved that ZR and OI have the same $\forall_{1}$-consequences.
Later developments showed that ZR had very important role in constructing
models of OI (See Macintyre-Marker [3], Smith [8]). ZR + N has also been
studied in [5]. In Theorem A, we study natural subsystems (ZR)S of (ZR), for a
nonempty subset $S$ of the set of prime numbers $\mathbb{P}$ (see the next
section for definition) and show that:
Theorem A. Suppose S is a nonempty subset of $\mathbb{P}$ and $q$ is a prime
number such that $q\notin S$, then $(ZR)_{S}$ \+ $Bez$ $\nvdash$ $(ZR)_{q}$.
Boughattas in ([2], Lemma 5) proved that DOR + N and ZR + N are not finitely
axiomatizable. More precisely he showed that:
###### Theorem 1.3 (Boughattas [2], Lemma 5).
Suppose $(n!,n^{{}^{\prime}})=1$. Then
$ZR+(N)_{n}\nvdash(N)_{n^{{}^{\prime}}}$.
We modify Boughattas’ proof and strengthen the above theorem in Theorem B:
Theorem B. Suppose S is a nonempty subset of $\mathbb{P}$ and $q$ is a prime
number such that $q\notin S$, then
$(ZR)_{S}+(N)_{n}+\neg(ZR)_{q}+\neg(N)_{n+1}$ is consistent.
Therefore we will have the following immediate implications:
Corollary F. Suppose S is a nonempty subset of $\mathbb{P}$ and $q$ is prime
number such that $q\notin S$, then
(1) $(ZR)_{S}$ \+ $N$ $\nvdash$ $(ZR)_{q}$.
(2) $(ZR)_{S}$ \+ $GCD$ $\nvdash$ $(ZR)_{q}$.
(3) All of the following subsystems of arithmetic are non-finite
axiomatizable: ZR, ZR + N, ZR + GCD, ZR + Bez, (ZR)S \+ N, ZR + (N)n, when S
is an infinite subset of the set of prime numbers.
## 2\. Preliminaries
Let $L$ be the language of ordered rings based on the symbols +, $-$, $\cdot$,
0, 1, $\leq$. We write $\mathbb{N^{\ast}}$ for $\mathbb{N}\setminus\\{0\\}$.
We will work with the following set of axioms in $L$:
$\bf DOR$: discretely ordered rings, i.e., axioms for ordered rings and
$\forall x\neg(0<x<1)$.
$\bf ZR$: discretely ordered $\mathbb{Z}$-rings, i.e., DOR and for every
$n\in\mathbb{N^{\ast}}$
$\forall x\exists q,r(x=nq+r\bigwedge 0\leqslant r<n)$.
We denote the sentence “DOR + $\forall x\exists q,r(x=nq+r\bigwedge 0\leqslant
r<n)$” by (ZR)n. Suppose $\mathbb{P}$ denote the set of prime numbers of
$\mathbb{N}$. Let $S$ be a nonempty subset of $\mathbb{P}$. We define the
subsystem (ZR)S of ZR as the below:
${\bf ZR}_{S}$: DOR + for every $p\in S$
$\forall x\exists q,r(x=qp+r\bigwedge 0\leqslant r<p)$.
If $S=\\{p_{i_{1}},\ldots,p_{i_{n}}\\}$ is a finite subset of $\mathbb{P}$, we
write (ZR)${}_{p_{i_{1}},\ldots,p_{i_{n}}}$ instead of
(ZR)${}_{\\{p_{i_{1}},\ldots,p_{i_{n}}\\}}$. This is consistent with the above
notation (ZR)n.
$\bf{OI}$: open induction, i.e., DOR and for every open $L$-formula
$\psi(\bar{x},y)$
$\forall\bar{x}(\psi(\bar{x},0)\bigwedge\forall y\geqslant
0(\psi(\bar{x},y)\rightarrow\psi(\bar{x},y+1))\rightarrow\forall y\geqslant
0\psi(\bar{x},y))$.
By considering the fact that in discretely ordered rings an open $L$-formula
$\varphi(\bar{x},y)$ can be written as a Boolean combination of polynomial
equalities and inequalities with the variable $y$ and the parameters
$\bar{x}$, there exist natural numbers $m,n$ such that:
$\varphi(\bar{x},y)=\bigwedge_{i\leq m}\bigvee_{j\leq n}p_{ij}(\bar{x},y)\leq
q_{ij}(\bar{x},y),$
we can define the degree of $\varphi(\bar{x},y)$ relative to $y$ by
deg $\varphi(\bar{x},y)$= max $\\{$degy $p_{ij}(\bar{x},y)$, degy
$q_{ij}(\bar{x},y)|i\leq m,j\leq n\\}.$
$({\bf OI})_{p}$ : open induction up to degree $p$ (i.e., DOR and for every
open $L$-formula $\psi(\bar{x},y)$ with deg $\psi(\bar{x},y)\leq p$
$\forall\bar{x}(\psi(\bar{x},0)\bigwedge\forall y\geqslant
0(\psi(\bar{x},y)\rightarrow\psi(\bar{x},y+1))\rightarrow\forall y\geqslant
0\psi(\bar{x},y)))$.
$\bf N$: normality (i.e., being domain and integrally closed in its fraction
field, namely for every $n\in\mathbb{N}^{*}$, $\forall x,y,z_{1},\ldots,z_{n}$
$(y\neq 0\wedge
x^{n}+z_{1}x^{n-1}y+\ldots+z_{n-1}xy^{n-1}+z_{n}y^{n}=0\longrightarrow\exists
z(yz=x))).$
$({\bf N})_{n}$: normality up to degree $n\in\mathbb{N}^{*}$ (i.e., being
domain and for every $m\in\mathbb{N}^{*}$, $m\leq n$, $\forall
x,y,z_{1},\ldots,z_{m}$
$(y\neq 0\wedge
x^{m}+z_{1}x^{m-1}y+\ldots+z_{m-1}xy^{m-1}+z_{m}y^{m}=0\longrightarrow\exists
z(yz=x))).$
It is clear that any domain satisfies $(N)_{1}$.
$\bf{GCD}$: having greatest common divisor (i.e., the usual axioms for being a
domain plus
$\forall x,y(x=y=0\vee\exists z(z|x\wedge z|y\wedge(\forall t((t|x\wedge
t|y)\rightarrow t|z))))$,
where $x|y$ is an abbreviation for $\exists t(t\cdot x=y)$).
$\bf{Bez}$: the usual axioms for being a domain plus the Bezout property:
$\forall x,y\exists z,t((xz+yt)|x\wedge(xz+yt)|y)$,
namely, every finitely generated ideal is principal.
It is known that Bez $\vdash$ GCD $\vdash$ N, and OI $\nvdash$ OI + N
$\nvdash$ OI + GCD $\nvdash$ OI + Bez (Smith [7], Lemmas 1.9 and 1.10).
Also we will need another algebraic property, though it is not first-order
expressible:
$\bf{DCC}$ : let $M$ be a domain. $M$ has the divisor chain condition (DCC) if
$M$ contains no infinite sequence of elements $a_{0},a_{1},a_{2},\ldots$ such
that each $a_{i+1}$ is a proper divisor of $a_{i}$ (i.e., $a_{i}/a_{i+1}$ is a
nonunit).
Let $M$ be an ordered domain (resp. a domain), then $RC(M)$ (resp. $AC(M)$)
will denote the real closure (resp. the algebraic closure) of its fraction
field. It is well known that $AC(M)=RC(M)[\sqrt{-1}]$. Let
$p\in\mathbb{N}^{*}$ and $F$ be an ordered field (resp. a field), we define
the $p$-real closure (resp. the $p$-algebraic closure) of $F$, denoted by
$RC_{p}(F)$ (resp. $AC_{p}(F)$), to be the smallest subfield of $RC(F)$ (resp.
$AC(F)$) containing $F$ such that every polynomial of degree $\leq p$ with
coefficients in $RC_{p}(M)$ (resp. $AC_{p}(F)$) which has a root in $RC(F)$
(resp. $AC(F)$) also has a root in $RC_{p}(F)$ (resp. $AC_{p}(F)$). Similarly
if $M$ be an ordered domain (resp. a domain), then $RC_{p}(M)$ (resp.
$AC_{p}(M)$) will denote the $p$-real closure (resp. the $p$-algebraic
closure) of its fraction field. It can be shown that
$AC_{p}(M)=RC_{p}(M)[\sqrt{-1}]$. Similar to real closed fields and algebraic
closed fields, it is also easily seen that:
(1) If $P(x)$ is a polynomial of degree $\leq p$ with the coefficients in
$RC_{p}(F)$ and $P(a)<0<P(b)$, for some $a<b$ in $RC_{p}(F)$, then there
exists a $c\in RC_{p}(F)$, such that $a<c<b$ and $P(c)=0$.
(2) If $P(x)$ is a polynomial of degree $\leq p$ with the coefficients in
$AC_{p}(F)$, then $P(x)$ can be represented as a product of linear factors
with coefficients in $AC_{p}(F)$.
Properties (1) and (2) can define and axiomatize the notions of $p$-real
closed field and $p$-algebraic closed field, denoted by $($RCF$)_{p}$ and
$($ACF$)_{p}$, respectively.
Given two ordered domains $I\subset{K}$ we say that $I$ is an integer part of
$K$ if $I$ is discrete and for every element $\alpha\in K$, there exists an
element $a\in I$ such that $0\leq\alpha-a<1$. We call $a$, the integer part of
$\alpha$, and sometimes denote it by $[\alpha]_{I}$. Shepherdson and
Boughattas characterized models of (OI)p, in terms of $p$-real closed fields
$(1\leq p\leq\omega)$:
###### Theorem 2.1 (Shepherdson [6]).
Let $M$ be an ordered domain. M is a model of OI iff M is an integer part of
RC(M).
###### Theorem 2.2 (Boughattas [1, 2]).
Let $M$ be an ordered domain. M is a model of $(OI)_{p}$ iff M is an integer
part of $RC_{p}(M)$.
We also need a fact from Puisseux series:
###### Definition 2.3.
Let K be a field. The following is the field of Puisseux series in descending
powers of x with coefficients in K $:$
$K((x^{1/\mathbb{N}}))=\\{\displaystyle\sum_{k\leq{m}}a_{k}x^{k/r}:m\in\mathbb{Z},r\in\mathbb{N^{\ast}},a_{k}\in{K}\\}.$
###### Theorem 2.4 (Boughattas [1]).
$(1\leq p\leq\omega)$ K is a $p$-real $($resp. $p$-algebraically$)$ closed
field iff $K((x^{1/\mathbb{N}}))$ is a $p$-real $($resp. $p$-algebraically$)$
closed field.
## 3\. The main results
### 3.1. Proof of Theorem A
Suppose $S$ is a subset of the set of prime numbers $\mathbb{P}$. We present
here a relative to S version of some theorems of (Smith [8]) that is needed
for proving theorem A. Interestingly, all proofs of (Smith [8]) remain valid,
if we make routine changes which will be explained. We mention that when
$S=\mathbb{P}$, we get the original definitions and theorems. We first define
$\widehat{\mathbb{Z}}_{S}=\prod_{p\in S}\mathbb{Z}_{p}$, where
$\mathbb{Z}_{p}$ is the ring of $p$-adic integers, and $\langle
S\rangle=\\{p_{1}^{\alpha_{1}}\cdots
p_{n}^{\alpha_{n}};n\in\mathbb{N}^{*},\alpha_{i}\in\mathbb{N}$ and $p_{i}\in
S\\}$. It is clear that there is the canonical embedding of $\langle S\rangle$
in $\widehat{\mathbb{Z}}_{S}$.
Let $M$ be a model of (ZR)S, by relativizing to $S$, we get a (unique)
$S-remainder$ homomorphism Rem : $M\longrightarrow\widehat{\mathbb{Z}}_{S}$
given by the projective limit of the canonical homomorphism
$\psi_{n}:M\longrightarrow M/nM\cong\mathbb{Z}/n\mathbb{Z}$
for $n\in\langle S\rangle$. See (Macintyre-Marker [3], Lemma 1.3).
Now we give the $S$-relativization of the so called
$\widehat{\mathbb{Z}}$-construction. Let $M$ be a discretely ordered ring with
$\varphi:M\longrightarrow\widehat{\mathbb{Z}}_{S}$ a homomorphism and assume
that all standard primes remain prime in $M$. We form a new ring
$M_{\varphi,S}=\\{a/n;a\in M,n\in\langle S\rangle$ and $n|\varphi(a)$ in
$\widehat{\mathbb{Z}}_{S}\\}$. We extend $\varphi$ to $M_{\varphi,S}$ in the
obvious way. We say that $M_{\varphi,S}$ is obtained from $M$ by the
$\widehat{\mathbb{Z}}_{S}-construction$. By relativizing the proof of
(Macintyre-Marker [3], Lemma 3.1) we get:
###### Lemma 3.1.
$M_{\varphi,S}\models(ZR)_{S}$.
Parsimony of homomorphisms plays a very important role in Smith’s
constructions. Therefore we have the following definition:
###### Definition 3.2.
Let $M$ be a discretely ordered ring with
$\varphi:M\longrightarrow\widehat{\mathbb{Z}}_{S}$ a homomorphism, where
$\varphi$ is the projective limit of the homomorphism
$\psi_{n}:M\longrightarrow\mathbb{Z}/n\mathbb{Z}$ for $n\in\langle S\rangle$.
We say that $\varphi$ is $S$-parsimonious if for each nonzero $a\in M$ there
are only finitely many $n\in\langle S\rangle$ such that $\psi_{n}(a)=0$.
The following lemma asserts that the $\widehat{\mathbb{Z}}_{S}$-construction
preserves parsimony.
###### Lemma 3.3.
If $\varphi:M\longrightarrow\widehat{\mathbb{Z}}_{S}$ is S-parsimonious, then
the extension of $\varphi$ to $M_{\varphi,S}$ is S-parsimonious.
###### Proof.
The proof is the $S$-relativization of Smith’s proof of Lemma 5.1. in [8]. Let
$0\neq a/n\in M_{\varphi,S}$, where $a\in M$, $n\in\langle S\rangle$. Suppose
$\psi_{k}(a/n)=0$, for a $k\in\langle S\rangle$. Since $M_{\varphi,S}$ is a
model of (ZR)S, we have $k|a/n$ in $M_{\varphi,S}$, so in particular $k|a$ in
$M_{\varphi,S}$. Thus $\psi_{k}(a)=0$. Since
$\varphi:M\longrightarrow\widehat{\mathbb{Z}}_{S}$ is $S$-parsimonious, there
are only finitely many possibilities for $k\in\langle S\rangle$. ∎
The following theorem says that in the presence of having a $S$-parsimonious
map the $\widehat{\mathbb{Z}}_{S}$-construction preserves GCD and DCC.
###### Theorem 3.4.
Let M be a discretely ordered ring with the GCD (DCC). Let
$\varphi:M\longrightarrow\widehat{\mathbb{Z}}_{S}$ be S-parsimonious and in
the DCC case the standard primes remain prime in M. Then $M_{\varphi,S}$ has
the GCD (DCC).
###### Proof.
We leave the proof to the reader as an easy and instructive exercise to adopt
Smith’s proofs of Theorems 5.3. and 5.5. in [8]. Just replace everywhere in
the proof, $\mathbb{Z}$-ring by a model of (ZR)S,
$\varphi:M\longrightarrow\widehat{\mathbb{Z}}$ by
$\varphi:M\longrightarrow\widehat{\mathbb{Z}}_{S}$, parsimonious by
$S$-parsimonious, $M_{\varphi}$ by $M_{\varphi,S}$, and check that the
arguments remain valid! ∎
Transcendental extensions preserve GCD and DCC.
###### Theorem 3.5 (Smith [8], Theorems 6.8. and 6.10.).
Let M be a GCD (DCC) domain and suppose x is transcendental over M. Then
$M[x]$ is a GCD (DCC) domain.
By the same adaptation of Theorem 6.12. of (Smith [8]), we see that
$S$-parsimonious maps can be extended to transcendental extensions. More
precisely:
###### Theorem 3.6.
Let M be a countable model of $(ZR)_{S}$ and suppose the remainder
homomorphisms $\varphi:M\longrightarrow\widehat{\mathbb{Z}}_{S}$ is
S-parsimonious. Let $x$ be transcendental over $M$ and suppose $M[x]$ is
discretely ordered $($and this ordering restricts to the original ordering on
M $)$. Then $\varphi$ can be extended to a S-parsimonious
$\varphi:M[x]\longrightarrow\widehat{\mathbb{Z}}_{S}$, such that $\varphi(x)$
is a unit of $\widehat{\mathbb{Z}}_{S}$.
We will need in this paper to consider the property of factoriality (a
factorial domain has the property that any nonunit has a factorization into
irreducible elements, and this factorization is unique up to units). We will
use the following theorem:
###### Theorem 3.7 (Smith [8], Theorem 1.5.).
M is factorial iff M has both of the GCD property and DCC.
In order to gain a Bezout domain the F-construction in Macintyre-Marker paper
[3] has a crucial role. By combining Theorems 8.5 and 8.7 from (Smith [8]),
Lemma 3.26 of (Macintyre-Marker [3]) and its proof, we have:
###### Theorem 3.8.
Let M be a discretely ordered domain with DCC (GCD) and suppose $v,w\in M$ are
primes and x is larger than any element of M. Let $M^{*}=M[x,\frac{1-xv}{w}]$.
Then $M^{*}$ is a discretely ordered domain with DCC (GCD).
In the following theorem we see that $S$-parsimony can be extended in
F-constructions:
###### Theorem 3.9.
Let $M$ be a countable model of $(ZR)_{S}$ and the remainder homomorphism
$\varphi:M\longrightarrow\widehat{\mathbb{Z}}_{S}$ is $S$-parsimonious. Let
$v,w\in M$ be primes of $M$ and $w$ is nonstandard. Suppose $x$ is
transcendental over $M$, and the discrete ordering of M extends to discrete
ordering on $M^{*}=M[x,\frac{1-xv}{w}]$. Then $\varphi$ can be extended to
$S$-parsimonious $\varphi:M^{*}\longrightarrow\widehat{\mathbb{Z}}_{S}$, such
that $\varphi(x)$ is a unit of $\widehat{\mathbb{Z}}_{S}$.
###### Proof.
See the proof of Theorem 8.9. of (Smith [8]). ∎
The next theorem guarantees the preservation of the GCD property and DCC in
chains constructed by alternative applications of the F-construction and the
$\widehat{\mathbb{Z}}_{S}$-construction via parsimonious maps. We express the
theorems in a more restricted and more suitable form which is adequate for us:
###### Theorem 3.10.
Suppose $M_{0}$ is a (GCD) DCC countable model of $(ZR)_{S}$ and there is
$S$-parsimonious $\varphi:M_{0}\longrightarrow\widehat{\mathbb{Z}}_{S}$. Let
$\\{M_{i}:i\in{\mathbb{N}}\\}$ be a chain of discretely ordered domains such
that $M_{2i+1}$ is constructed from $M_{2i}$ by the
$\widehat{\mathbb{Z}}_{S}$-construction, and $M_{2i+2}$ is constructed from
$M_{2i+1}$ by the F-construction. In addition we suppose that in the DCC case,
in the whole process of extending rings at most finitely many irreducibles
have been killed (this means that only finitely many irreducibles will become
reducible in later stages). Then $M=\bigcup_{i\in{\mathbb{N}}}{M_{i}}$ is a
model of (GCD) DCC.
###### Proof.
See Theorems 9.4. and 9.8. in (Smith [8]). ∎
By the following series of easy lemmas, we will not worry about DCC in our
chain of models in the proof of Theorem A:
###### Lemma 3.11 (Smith [8], Lemma 3.8.).
Let M be a GCD domain. Then $p\in M$ is irreducible iff it is prime.
Of course the following lemma needs an easy $S$-adaptation of Lemma 3.2 in
(Macintyre-Marker [3]):
###### Lemma 3.12.
Let $M\models DOR$ and $\varphi:M\longrightarrow\widehat{\mathbb{Z}}_{S}$ be a
ring homomorphism and assume that all standard primes remain prime in M. If
$q\in M$ is irreducible and $\varphi(q)$ is unit in
$\widehat{\mathbb{Z}}_{S}$, then q is irreducible in $M_{\varphi,S}$
###### Lemma 3.13 (Macintyre-Marker [3], Lemma 3.27).
If q is irreducible in M, then q is irreducible in $M^{*}$, constructed in
Theorem 3.8 (by the F-construction).
Now we have gathered all preliminaries to prove Theorem A:
Theorem A. Suppose S is a nonempty subset of $\mathbb{P}$ and $q$ is prime
number such that $q\notin S$, then $(ZR)_{S}$ \+ $Bez$ $\nvdash$ $(ZR)_{q}$.
###### Proof.
We do a suitable and modified version of Smith’s process to construct a Bezout
model of open induction (Smith [8] Theorem 10.7.). We shall inductively
construct an $\omega$-chain of models $M_{i}$ such that
$\bigcup_{i}M_{i}=M_{\omega}$ will be a model of $(ZR)_{S}$ \+ $Bez$\+
$\neg(ZR)_{q}$. We work inside the ordered field
$\mathbb{Q}(x_{1},...,x_{i},...)$ so that for each $i\in\omega$, $x_{i+1}$ is
larger than any element of $\mathbb{Q}(x_{1},...,x_{i})$ and $x_{1}$ is
infinitely large. We will do the F-construction at odd stages and the
$\widehat{\mathbb{Z}}_{S}$-construction at even stages.
Take $M_{0}=\mathbb{Z}$ together the natural remainder $S$-parsimonious
homomorphism $\varphi:M_{0}\longrightarrow\widehat{\mathbb{Z}}_{S}$. Let us
show what we do at stages $2k+1$. Suppose $M_{2k}$ and a $S$-parsimonious map
$\varphi:M_{2k}\longrightarrow\widehat{\mathbb{Z}}_{S}$, have been
constructed. At this stage we consider a pair of distinct primes $v$ and $w$
belonging to $M_{2k}$ such that $w$ is nonstandard. (Of course we do this in
such a way that every such pair of primes in $M_{\omega}$ will have been
considered at some stage $2k+1$). Thus $(v,w)=1$ in $M_{2k}$. We define
$M_{2k+1}=M_{2k}[x_{k},\frac{1-x_{k}v}{w}]$ according to Theorem 3.8. Suppose
$y_{k}=\frac{1-x_{k}v}{w}$, then we have $x_{k}v+y_{k}w=1$ in $M_{2k+1}$. So
$(v,w)_{B}=1$ in $M_{2k+1}$. ($(v,w)_{B}$ is the Bezout greatest common
divisor of $v$ and $w$, it means that $(v,w)_{B}|v$ and $(v,w)_{B}|w$ and
there exist $r$ and $s$ in $M_{2k+1}$ such that $rv+su=1$). We refer to (Smith
[8], Section 3) for the basic related definitions and theorems. By Theorem 3.9
$\varphi$ is extended to a $S$-parsimonious map
$\varphi:M_{2k+1}\longrightarrow\widehat{\mathbb{Z}}_{S}$. At stage $2k+2$, we
employ Lemma 3.1 and define $M_{2k+2}=(M_{2k+1})_{\varphi,S}$ which is a model
of $(ZR)_{S}$. Lemma 3.3 gives us the desired parsimonious extensions
$\varphi:M_{2k+2}\longrightarrow\widehat{\mathbb{Z}}_{S}$. Since $(ZR)_{S}$ is
a $\forall\exists$-theory, then it is preserved in chains, therefore
$M_{\omega}$$\models$$(ZR)_{S}$.
Now we show that $M_{\omega}$ is a Bezout domain. The proof is similar to
(Smith [8], Theorem 10.7) with a minor change. By Theorems 3.4 and 3.8, each
$M_{i}$ has the GCD and DCC, so by Theorem 3.10 $M_{\omega}$ has both the GCD
and DCC (by Lemmas 3.11, 3.12 and 3.13 we know that no irreducible is killed)
and from Theorem 3.7 we conclude that $M_{\omega}$ is a factorial domain. In
order to show that $M_{\omega}$ is a Bezout domain, by considering the fact
that $M_{\omega}$ has the GCD property, it suffices to prove that any two
elements of $M_{\omega}$ has the Bezout greatest common divisor. Let $a,b\in
M_{\omega}$ and let $c=(a,b)$ in $M_{\omega}$. We can assume $a,b>1$. Let
$a=a^{{}^{\prime}}c$, $b=b^{{}^{\prime}}c$ in $M_{\omega}$. So
$(a^{{}^{\prime}},b^{{}^{\prime}})=1$ in $M_{\omega}$. Since $M_{\omega}$ is
factorial, we can write
$a^{{}^{\prime}}=m{P_{1}}^{e_{1}}\ldots{P_{k}}^{e_{k}}$ and
$b^{{}^{\prime}}=n{Q_{1}}^{f_{1}}\ldots{Q_{l}}^{f_{l}}$, where
$m,n\in\mathbb{N}$ are nonzero, $k,l\geq 0$ and the $P_{i},Q_{j}$ are
nonstandard primes such that $P_{i}\neq Q_{j}$ for all $i,j$. We will show
that $(a^{{}^{\prime}},b^{{}^{\prime}})_{B}=1$. Clearly $(m,n)_{B}=1$. Suppose
$m={g_{1}}^{v_{1}}\ldots{g_{r}}^{v_{r}}$ and
$n={h_{1}}^{w_{1}}\ldots{h_{s}}^{w_{s}}$ are the prime factorizations of $m,n$
in $\mathbb{N}$. By the F-construction every one of $(P_{i},g_{j})_{B}=1$,
$(Q_{i},h_{j})_{B}=1$ and $(P_{i},Q_{j})_{B}=1$, occur at some odd stage of
our construction. Therefore by iterated applications of (Smith [8], Lemma
3.4), we conclude that $(a^{{}^{\prime}},b^{{}^{\prime}})_{B}=1$. By (Smith
[8], Lemma 3.4), we have $c=(a,b)_{B}$ at some odd stage and then using (Smith
[8], Lemma 3.7) we ensure that $c=(a,b)_{B}$ in $M_{\omega}$. This completes
the proof of the Bezoutness of $M_{\omega}$.
Note that in the original proof of Smith ([8], Theorem 10.7) he just considers
pairs of nonstandard primes and doesn’t need to consider pairs of primes such
that one is standard and the other is nonstandard. Since his chain of domains
are ZR-rings, this gives automatically the Bezout greatest common divisor for
such pairs. But as we want ZR to fail in our model, we are forced to consider
pairs of standard and nonstandard primes in the F-construction, as well.
Now we show that $(ZR)_{q}$ fails in $M_{\omega}$. We first observe that in
the first step of our construction, namely, when passing from
$M_{0}=\mathbb{Z}$ to $M_{1}$, there is no nonstandard prime in $M_{0}$. So
$M_{1}$ is just $\mathbb{Z}[x_{1}]$ and we have no $y_{1}$. On the other hand
from the construction it is evident that elements of $M_{\omega}$ are of the
form $f(x_{1},x_{2},y_{2},...,x_{k},y_{k})$, for some $k$, where $f$ is a
polynomial with the coefficients in the set $\mathbb{Z}_{\langle
S\rangle}=\\{a/k;a\in\mathbb{Z}$ and $k\in\langle S\rangle\\}$. Now for a
contradiction, suppose $M_{\omega}$ is a model of $(ZR)_{q}$. Then there is a
$b\in M_{\omega}$ such that $x_{1}=bq+r$ with $0\leq r<q$. Take
$b=f(x_{1},x_{2},y_{2},...,x_{k},y_{k})$, so we have
$x_{1}=f(x_{1},x_{2},y_{2},...,x_{k},y_{k})q+r$. Observe that $x_{2}$,
$y_{2}$, $\ldots$, $x_{k}$, $y_{k}$ are transcendental over
$\mathbb{Q}(x_{1})$, then $f$ does not depend on them, so we can assume
$x_{1}=f(x_{1})q+r$. Since $x_{1}$ is also transcendental over $\mathbb{Q}$,
it follow that the degree of $f$ must be one. Thus $f(x_{1})=ax_{1}$ and
$a\in\mathbb{Z}_{\langle S\rangle}$. So $x_{1}=ax_{1}q+r$ and then
$x_{1}(1-aq)=r$, which implies that $a=1/q$ and this is in contradiction with
$a\in\mathbb{Z}_{\langle S\rangle}$, since $q\notin\langle S\rangle$. ∎
### 3.2. Proof of Theorem B
Now we prove:
Theorem B. Suppose S is a nonempty subset of $\mathbb{P}$ and $q$ is a prime
number such that $q\notin S$, then
$(ZR)_{S}+(N)_{n}+\neg(ZR)_{q}+\neg(N)_{n+1}$ is consistent.
###### Proof.
In [4] we proved that if $n\neq 3$, there is a $\lambda$ which is real
algebraic of degree $n+1$ over $\mathbb{Q}$ and doesn’t belong to
$RC_{n}(\mathbb{Q})$. Now suppose $x$ is an infinitely large element. For
$n\neq 3$, fix $\lambda$ as above. For $n=3$ we choose $\lambda$ as a root of
an irreducible polynomial of degree 4 such that $\lambda\notin
RC_{2}(\mathbb{Q})$. Let $A$ be the ring of integers of the algebraic number
field $\mathbb{Q}(\lambda)$. Form $A_{\langle S\rangle}=\\{a/k;a\in A$ and
$k\in\langle S\rangle\\}$. It is an elementary fact from algebraic number
theory that $A$ is a normal ring. Since $A_{\langle S\rangle}$ is a
localization of $A$ relative to a multiplicative set, then it is also normal.
Let $M=\mathbb{Z}[rx;r\in A_{\langle S\rangle}]$. We claim that $M$ witnesses
Theorem B. It is obvious that $M\vDash(ZR)_{S}$. By an argument similar to the
last paragraph of the proof of theorem A, it is easily shown that
$M\vDash\neg(ZR)_{q}$.
Now we prove $M\vDash\neg(N)_{n+1}$. Let $v\in\mathbb{N}$ be such that
$v\lambda$ is an algebraic integer. Suppose $P(t)\in\mathbb{Z}$ is its minimal
polynomial of degree $n+1$ which is monic. Obviously $v\lambda x\in M$. But we
have $P(v\lambda x/x)=0$, while $v\lambda\notin M$. So $M$ is not a model of
$(N)_{n+1}$.
It remains to show that $M\vDash(N)_{n}$. Let $u,v$ be nonzero elements of $M$
such that
$(u/v)^{s}+z_{1}(u/v)^{s-1}+\cdots+z_{s}=0$ $(z_{1},\ldots,z_{s}\in
M,s\leqslant n).$
We will show that $u/v\in M$. Notice that elements of $M$ are those elements
of $A_{\langle S\rangle}[x]$ with integer constant coefficient. $A_{\langle
S\rangle}$ is normal, so is $A_{\langle S\rangle}[x]$. Thus $u/v\in A_{\langle
S\rangle}[x]$. On the other hand, since $\mathbb{Q}(\lambda)[x]$ is a
factorial ring, $u/v$ can be written as:
$u/v=\rho\prod_{i\in I}P_{i}\prod_{j\in J}Q_{j}$,
in which $\rho\in\mathbb{Q}(\lambda)$, the $P_{i}$’s are irreducible in
$\mathbb{Q}(\lambda)[x]$, without constant coefficient and $Q_{j}$’s are
irreducible in $\mathbb{Q}(\lambda)[x]$ with the constant coefficient one. If
$I$ is nonempty, then $\rho\prod_{i\in I}P_{i}\prod_{j\in J}Q_{j}$ has no
constant coefficient and thus $u/v\in M$. Now suppose $I=\O$. Put $x=0$ in
$u$, $v$, $z_{1}$, $\ldots$, $z_{s}$. Therefore $\rho$ is an algebraic integer
with the degree, equal or less than $n$ over $\mathbb{Z}$. We show it is one.
If $n=1$ there is nothing to prove. If not, we have
$[\mathbb{Q}(\lambda):\mathbb{Q}(\rho)]<n+1$. But
$[\mathbb{Q}(\lambda):\mathbb{Q}(\rho)][\mathbb{Q}(\rho):\mathbb{Q}]=[\mathbb{Q}(\lambda):\mathbb{Q}]=n+1$.
Then $[\mathbb{Q}(\lambda):\mathbb{Q}(\rho)]$ divides
$[\mathbb{Q}(\lambda):\mathbb{Q}]=n+1$. So we have a chain of field
extensions, $\mathbb{Q}\subset\mathbb{Q}(\rho)\subset\mathbb{Q}(\lambda)$ such
that $[\mathbb{Q}(\lambda):\mathbb{Q}(\rho)]\leq n-1$ and
$[\mathbb{Q}(\rho):\mathbb{Q}]\leq n-1$. This implies that $\lambda\in
RC_{n-1}(\mathbb{Q})$ which is in contradiction with the choice of $\lambda$.
Hence $\rho$ is an algebraic integer of degree one. So $\rho\in\mathbb{Z}$ and
this implies that $u/v\in M$, which means that $M$ is model of $(N)_{n}$. This
completes the proof of Theorem B. ∎
### 3.3. Proofs of Theorems C and D.
In order to demonstrate Theorem C, we need a generalization of a theorem of
Boughattas. In ([2], Theorem V.1.) Boughattas proved that every saturated
ordered field admits a normal integer part. But we show that:
###### Lemma 3.14.
Every $\omega_{1}$-saturated ordered field admits a Bezout integer part.
###### Proof.
(Sketch) Suppose $K$ is an $\omega_{1}$-saturated ordered field. Boughattas
[2] in a series of three Lemmas: Pricipal, Integer Part and Construction,
showed that we can build an $\omega_{1}$-chain of countable discretely ordered
rings $M_{i},i<\omega_{1}$ such that $M=\bigcup_{i<\omega_{1}}M_{i}$ is an
integer part of $K$. Furthermore he considers an arbitrary subset
$\Lambda\subset K$ of real algebraic elements which plays a role in the
construction of the $M_{i}$’s. Varying $\Lambda$ gives us various kinds of
integer parts. When $\Lambda=\O$, we obtain a normal integer part and it is
implicit in the paper that in this case, the $M_{i}$’s are obtained by
alternative applications of the Wilkie-construction and the
$\widehat{\mathbb{Z}}$-construction. But it must be noticed that even in this
case the procedure of doing the $\widehat{\mathbb{Z}}$-construction is
different from the original one, because it is no longer assumed that the
ground field is dense in its real closure. To gain a Bezout integer part, we
observe that we can do the procedure of the Theorems 10.7 and 10.8 of Smith
[8] inside $K$. In this procedure we need the extra F-construction. Since any
$M_{i},i<\omega_{1}$ is countable and $K$ is $\omega_{1}$-saturated, then
there is always an element $b_{i}$ in $K$ which is larger than any element of
$M_{i}$. By Lemma 3.26 of (Macintyre-Marker [3]) we are sure that we can do
the F-construction. To obtain an integer part of $K$, suppose
$(b_{\alpha},\alpha<\omega_{1})$ be an enumeration of elements of $K$. Let
$M_{i}$ has been constructed and at step $i+1$ we want to do the Wilkie-
construction. We seek the least ordinal $\alpha_{i}$, such that
$b_{\alpha_{i}}$ has not an integer part in $M_{i}$. Then by combining the
Integer Part Lemma of Boughattas [2] with the
$\widehat{\mathbb{Z}}$-construction, we obtain $M_{i+1}$ with its parsimonious
homomorphism extension to $\widehat{\mathbb{Z}}$, such that $b_{\alpha_{i}}$
has an integer part in $M_{i+1}$. Also suppose $M_{j}$ has been constructed
and at stage $j+1$ we want to do the F-construction. We seek the least ordinal
$\alpha_{j}$, such that $b_{\alpha_{j}}$ is larger than any element of
$M_{j}$. Then the F-construction can be done at this step. At limit stages we
take union. Moreover, Lemma 9.1, Theorem 9.4 and Theorem 9.8 of (Smith [8])
will guarantee preserving parsimony of homomorphisms and factoriality at limit
stages of length $\leq\omega_{1}$. Now there is no obstacle for
$M=\bigcup_{i<\omega_{1}}M_{i}$ to be a Bezout integer part of $K$. ∎
Theorem C. $Bez$ \+ $(OI)_{p}$ $\nvdash$ $(OI)_{p+1}$, when $p\neq 3$.
Proof of Theorem C. In [4], we showed that if $p\neq 3$, there is an
irreducible polynomial $P(t)$ of degree $p+1$ over $\mathbb{Q}$ such that
$P(t)$ has no root in $RC_{p}(\mathbb{Q})$. For $p\geq 4$, $P(t)$ was a
polynomial with Galois group $A_{p+1}$. It is well known that we can take
$P(t)$ as a monic polynomial with integer coefficients such that $P(0)<0$. Let
$T$ be the following theory in the language of ordered field with the
additional constant symbol $a$:
$T\equiv(RCF)_{p}+\\{a>k;k\in\mathbb{N}\\}+\forall y\neg(Q(y)\leq 0<Q(y+1))$,
where $Q(y)=a^{p+1}P(y/a)$. We show that the field of Puisseux power series
$RC_{p}(\mathbb{Q})((x^{1/\mathbb{N}}))$ is a model of $T$, when interpreting
$a$ by $x$. Clearly by Theorem 2.4, $RC_{p}(\mathbb{Q})((x^{1/\mathbb{N}}))$
is a $p$-real closed field. Also on the contrary suppose that there exists $y$
$\in$ $RC_{p}(\mathbb{Q})((x^{1/\mathbb{N}}))$ such that $(Q(y)\leq
0<Q(y+1))$. Therefore $P(y/x)\leq 0<P((y+1)/x)$. It is easily seen that
deg${}_{x}(y/x)$ must be zero. So let
$y/x=\lambda+\sum_{-\infty<i<0}c_{i}x^{i/q}$ in
$RC_{p}(\mathbb{Q})((x^{1/\mathbb{N}}))$. This leads to $P(\lambda)=0$, but
$\lambda\in RC_{p}(\mathbb{Q})$, which is in contradiction to the choice of
$P(t)$. So $RC_{p}(\mathbb{Q})((x^{1/\mathbb{N}}))\vDash T$.
Now that $T$ is consistent, let $K$ be an $\omega_{1}$-saturated model of $T$.
By Lemma 3.14 $K$ has a Bezout integer part. Call it $M$. Since
$K\vDash(RCF)_{p}$, then $M\vDash(OI)_{p}$. On the other hand there is
$n\in\mathbb{N}$ such that $M\vDash Q(0)<0<Q(n[a]_{M})$, where $[a]_{M}$ is
the integer part of $a$ in $M$. But $K\vDash\forall y\neg(Q(y)\leq 0<Q(y+1))$,
then $M\vDash\forall y\neg(Q(y)\leq 0<Q(y+1))$, so by (Boughattas [1],
Proposition A.I), $M\vDash\neg(OI)_{p+1}$. This ends the proof of $Bez$ \+
$(OI)_{p}$ $\nvdash$ $(OI)_{p+1}$. $\blacksquare$
Proof of Theorem D goes the same way with the exception that we must replace
Lemma 3.14 by the following Construction Lemma of Boughattas:
###### Theorem 3.15 (Boughattas [2]).
Suppose $K$ is a saturated ordered field. Let $\Lambda$ be an arbitrary subset
of real algebraic elements in $K$. Then there exists $X\subset K$ such that
$X$ is algebraic independent and $\mathbb{Z}[\\{rx;r\in\mathbb{Q}[\Lambda]$
$and$ $x\in X\\}]$ is an integer part of $K$.
Theorem D. $(OI)_{p}$ \+ $\neg(OI)_{p+1}$ \+ $(N)_{n}$ \+ $\neg(N)_{n+1}$ is
consistent, when $p\neq 3$.
Proof of Theorem D. We work with the same theory $T$ and its saturated model
$K$ as in the proof of Theorem C. Choose $\Lambda=\\{\lambda\\}$ and fix
$\lambda$ as in the proof of Theorem B, namely, if $n\neq 3$, $\lambda\in
RC_{n+1}(\mathbb{Q})$, $\lambda\notin RC_{n}(\mathbb{Q})$ and if $n=3$ choose
$\lambda$ as a root of an irreducible polynomial of degree 4 such that
$\lambda\notin RC_{2}(\mathbb{Q})$. Then by Theorem 3.15, there exists
$X\subset K$ such that $K$ has the integer part
$M=\mathbb{Z}[\\{rx;r\in\mathbb{Q}(\lambda)$ and $x\in X\\}]$. To show
$M\models(N)_{n}+\neg(N)_{n+1}$, we can repeat the proof of Theorem B, just
replace $x$ by $X$ and replace $A_{\langle S\rangle}$ by
$\mathbb{Q}(\lambda)$. $\mathbb{Q}(\lambda)[X]$ remains factorial and normal,
so the proof works. By the last paragraph of the proof of the Theorem C, it is
obvious that $M\models(OI)_{p}+\neg(OI)_{p+1}$ $\blacksquare$
### Acknowledgment
I would like to thank Professor Roman Kossak for his patience and kindness
during the preparation of this paper.
## References
* [1] Sedki Boughattas, _L’arithm tique ouverte et ses mod les non-standards_ , J. Symbolic Logic 56 (1991), 700–714.
* [2] by same author, _L’induction ouverte dans les anneaux discrets ordonnes et normaux n’est pas finiment axiomatisable_ , J. London Math. Soc. (2) 53 (1996), no. 3, 455–463.
* [3] Angus Macintyre and David Marker, _Primes and their residue rings in models of open induction_ , Ann. Pure Appl. Logic 43 (1989), no. 1, 57–77.
* [4] Shahram Mohsenipour, _A note on subsystems of open induction_ , J. Symbolic Logic 72 (2007), no. 4, 1318–1322.
* [5] Margarita Otero, _Generic models of the theory of normal ${\bf Z}$-rings_, Notre Dame J. Formal Logic 33 (1992), no. 3, 322–331.
* [6] J. C. Shepherdson, _A non-standard model for a free variable fragment of number theory_ , Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 12 (1964), 79–86.
* [7] Stuart T. Smith, _Prime numbers and factorization in ${\rm IE}_{1}$ and weaker systems_, J. Symbolic Logic 57 (1992), no. 3, 1057–1085.
* [8] by same author, _Building discretely ordered Bezout domains and GCD domains_ , J. Algebra 159 (1993), no. 1, 191–239.
* [9] Lou van den Dries, _Some model theory and number theory for models of weak systems of arithmetic_ , Model theory of algebra and arithmetic (Proc. Conf., Karpacz, 1979), Lecture Notes in Math., vol. 834, Springer, Berlin, 1980, pp. 346–362.
* [10] A. J. Wilkie, _Some results and problems on weak systems of arithmetic_ , Logic Colloquium ’77 (Proc. Conf., Wrocław, 1977), Stud. Logic Foundations Math., vol. 96, North-Holland, Amsterdam, 1978, pp. 285–296.
* [11] George Wilmers, _Bounded existential induction_ , J. Symbolic Logic 50 (1985), no. 1, 72–90.
|
arxiv-papers
| 2010-03-10T14:28:09 |
2024-09-04T02:49:08.998048
|
{
"license": "Public Domain",
"authors": "Shahram Mohsenipour",
"submitter": "Shahram Mohsenipour",
"url": "https://arxiv.org/abs/1003.2117"
}
|
1003.2123
|
# Quantifying Shannon’s Work Function for Cryptanalytic Attacks
R. J. J. H. van Son
Netherlands Cancer Institute, Amsterdam and ACLC, University of Amsterdam
R.J.J.H.vanSon@gmail.com
###### Abstract
Attacks on cryptographic systems are limited by the available computational
resources. A theoretical understanding of these resource limitations is needed
to evaluate the security of cryptographic primitives and procedures. This
study uses an Attacker versus Environment game formalism based on
computability logic to quantify Shannon’s work function and evaluate resource
use in cryptanalysis. A simple cost function is defined which allows to
quantify a wide range of theoretical and real computational resources. With
this approach the use of custom hardware, e.g., FPGA boards, in cryptanalysis
can be analyzed. Applied to real cryptanalytic problems, it raises, for
instance, the expectation that the computer time needed to break some simple
90 bit strong cryptographic primitives might theoretically be less than two
years.
keywords: computation, cryptanalysis, computational complexity
## 1 Introduction
There have been many examples where the ongoing increase in computer speed and
capacities have made previously secure cryptographic systems vulnerable to
brute force attacks. This perpetual weakening of cryptographic systems due to
the progress in computer hardware has been incorporated in rules of
application. For instance, NIST in the USA publishes elaborate rules about the
phasing out of shorter (weaker) keys and algorithms over time [30, 1].
However, those rules seem not to be based on a theoretical understanding of
the availability of computational resources, but more on a historical trend in
technical progress (e.g., Moore’s law [35]).
It is still difficult to reliably estimate the computational efforts needed to
compromise a cryptographic system, i.e., Shannon’s cryptanalysis work function
[31]. Many studies and applications go for ultimate security by aiming for
$2^{k}$ operations, with $k\geq 128$, to put brute force attacks out of reach
for the foreseeable future. Others use general purpose, off-the-shelf,
computers as benchmarks. Both approaches have limitations. Long keys imply
costly hardware and long computations and often do not describe real life use,
e.g., cost optimization for time-limited secrets. On the other hand, general
purpose office and home computers are not necessarily very efficient for
breaking codes and will almost certainly underestimate contemporary hardware
capabilities [39].
The error to think that an off-the-shelf general purpose CPU for an office
computer is an efficient device to recover cryptographic keys and passwords or
break cryptographic codes, is a common one. Expressions like Calculating X
took Y hours on a Z-level computer are very often encountered. As a result,
there seems to be general surprise every time it is shown that low-cost,
specialized processors can outperform general office CPUs. For example, even
though the idea might not have been new [4], there was again alarm in the
media when in 2007 a Russian software company, Elcomsoft, filed for a US
patent for a technique to use low cost standard graphics cards to recover
passwords [12].
Although the problem mentioned above is more generally seen in complexity and
game theoretical analysis, it’s practical importance is most acute for
cryptanalysis and digital security. Many security policies rely on
cryptographic systems as a crucial element. The difficulty with studying
vulnerabilities in cryptography is their theoretical status. The most
interesting vulnerabilities in cryptographic systems are generally untested
and the cost of a theoretically possible attack is therefore very difficult to
estimate. Even though there is a good mathematical understanding of how
cryptographic systems can be compromised, there is no consensus about a
formalism in which the resources needed can be formally described and
quantified.
This study uses a general formalism for quantifying computational resources
which was proposed in [39]. This formalism defines resource use both on a
symbolic level and on real hardware. The relevant parts of the model will be
repeated here to make the current study self contained. The model will then be
tailored to quantifying the cryptanalysis work function of Shannon [31] which
aligns very closely to problems in game theory, e.g., the computational Nash
equilibrium [10, 11], and algorithmic complexity theory with space and time
bounded automata [5].
Section 2 presents a summary of the model from [39] adapted to cryptanalysis.
The use of the model will be illustrated on existing hardware products. In
section 3, the model is applied to some examples from the cryptanalysis
literature. The results are discussed in section 4.
## 2 Cryptanalytic attacks as games
Cryptanalytic attacks are interactive procedures where a cryptographic system
is attacked using computational resources to compromise protected information.
It is assumed that the attacker can only use algorithmic procedures and
computers. Such an attack can be emulated as a game by a collection of Turing
Complete devices [31, 40]. For their mathematical convenience, Universal
Turing Machines (UTM) will be used to illustrate the formalism [36], but the
results hold for all such devices. Cryptanalytical attacks are problems of
computability. This model of cryptanalytic attacks fits the theoretic
framework of computability logic [13, 14, 15, 16].
In computability logic, computability is defined in terms of games. The
“computer”, or Attacker, plays against the Environment and “wins” if it can
complete the requested computation successfully. Computability logic tries to
be a complete logic of interactive computing. This study only refers to some
general aspects of computability logic. The reader can consult Japaridze [15,
16] and the references therein for extensive descriptions of the theory.
In short, the Attacker can play a game against the Environment on one or more
“boards”, in parallel. This study will restrict itself to a Hard Play model of
deterministic static games [15]. That is, only purely algorithmic and
reproducible games are considered where the speed of the moves is not
relevant. The environment can execute any number of moves for any single
computational step of the System. In practice, these two conditions, a Hard
Play model and static games, do not restrict the Attacker. They just prescribe
that any attack strategy should involve a number of algorithmic steps and that
the Environment, which includes the complete universe, has unlimited
capacities for executing counter strategies. This model can be extended to
include probabilistic strategies. In this framework, it is rather
straightforward to set up a model for a cryptanalytic attack (c.f., [40, 31]).
### 2.1 The Attacker model
In the framework of computability logic, the Attacker is a collection of UTMs,
each with three tapes: a work tape, a valuation tape, and a run tape. The
valuation tape is supposed to contain the game specific parameters supplied by
the Environment, whereas the work tape will be initialized with a program to
load and play games from the valuation tape. A more general interpretation of
the valuation tape is that it contains any public information outside the
control of the Attacker (for a more extensive description see [39]).
The run tape contains the moves of the Attacker and the Environment. In the
current framework, both the Attacker and the Environment write their moves
onto the run tape. The alphabet used on the run tape is prescribed by the
Environment. The Attacker can only move the reading head forward on the run
tape and visit each cell only once. The Environment is free to read the run
tape in any direction as often as it wants, but can only write to empty cells.
To allow the “access once” restriction, all moves are written as self
delimited or fixed length strings onto the run tape.
Scanning the run tape for moves of the Environment is a computational cost
that must be born by the Attacker. To minimize that cost, the moments at which
the Environment can write to a run tape are restricted. The Environment will
only write to a run tape in response to a move of the Attacker. After the
Attacker has written a move to the run tape, it can enter a “wait” state and
go to sleep. Only then will the Environment write it’s move or moves in a
single self delimited or fixed length string to tape and wake up the Attacker
who then can read the moves and continue. This interpretation of the run-tape
embodies the principle that the Attacker must actively query for information
from the Environment.
The Attacker can recruit as many UTMs as it wants by specifying them on the
run tape from any of the existing UTMs. The communication between the Attacker
UTMs is here modeled by simply letting the work tapes overlap. Other solutions
are possible. Any newly instantiated UTM of the Attacker gets it’s own run
tape and a copy of the valuation tape.
Any request for a new UTM should consist of a full description of the finite
state machine, initial state, contents of the work tape, position of the
heads, and the overlap between work tapes. A new UTM is instantiated with the
finite state machine specified, the valuation and work tapes loaded and the
work tape is stitched up with the correct part of the requesting UTM’s work
tape. Then the UTM is put in the initial state with the heads over the correct
tape positions and started.
The possible moves of the Attacker can be divided into 4 classes:
* •
General information requests
* •
Structural requests
* •
Encryption requests
* •
Challenges
The meaning of the first is obvious. The second kind are requests to the
Environment for new daughter UTMs or changes in the current UTM, e.g.,
releasing work tape memory. In modeling a realistic attack, structural
requests would (de-)commission computing resources. Encryption requests
implement the gathering of plaintexts and ciphertexts.
The cost of defeating a cryptographic system includes actually compromising
it. Challenges are Attacker initiated moves to prove it has won, i.e.,
succeeded in compromising the cryptographic system by actually executing and
completing the attack. A challenge could be to supply the current password,
but could also demonstrate the ability to correctly pair cipher- and
plaintexts. Note that after every move of the Attacker, the Environment must
make a move, even if it is just a denial of the request.
This model of an Attacker is able to describe a large number of cryptanalytic
attacks. For instance, distributed attacks, both coordinated or not, known
ciphertext or plaintext attacks, and chosen plaintext attacks. Even attacks of
“security by obscurity” systems could be studied by supplying a stochastic
model of information leakage to the valuation tape.
### 2.2 Resource needs and cost of computation
One problem with the above computational attack model is that most
cryptographic systems can be defeated by simple brute force attacks, e.g.,
just trying all possible keys [31] or even simply trying all programs to crack
the encryption (c.f., algorithmic complexity [20]). However, the security of
cryptography lies in the fact that performing computations has costs, and for
a brute force attack, these costs should be too high to be feasible [31]. But
to use these costs in a computational model, they should be made explicit. In
the remainder of this text, the model from [39] will be used to quantify
computational costs. The relevant points will be described here.
A useful cost function for computations should follow some sanity conditions.
The definition should be applicable to both theoretical and real devices. The
costs should be cumulative and additive under appropriate conditions. The
universal nature of computational devices should be mirrored in the existence
of efficient emulation of one device on another one. Here an “efficient
emulator” will be defined as any device that can emulate any computation on
the target device with a cost that is a linear function of the original cost
and number of steps in the computation ([39]).
Starting with a purely theoretical device, a very simple cost function for a
single UTM that agrees with all of the above conditions is
$C=\sum_{\lambda=1}^{\Lambda}I_{UTM}(\lambda)$ (1)
Where $C$ is the total cost of the computation which runs over $\Lambda$
steps. $I_{UTM}(\lambda)$ is the information in bits stored in the UTM at step
$\lambda$. $I_{UTM}$ includes details about the internal structure of the
device, e.g., action tables of a UTM. See [39] for a discussion and proofs.
The definition of equation 1 can easily be extended to other computational
devices (even neural wet-ware [39]). The only requirement is that the
functionality of the device can be modeled as a collection of interconnected
and modular logical components, e.g., logical gates, finite state machines, or
UTMs. The factor $I_{UTM}$ in equation 1 will be replaced by a factor
$I_{Dev}$ which measures the number of bits needed to identify the chosen
device out of all the possible devices (including all non-functional ones)
that could have been constructed using the same basic components, plus the
current state of these components.
For instance, the logical functions a modern CPU silicon chip can perform are
limited by the number of transistors it contains. The size of $I_{CPU}$ would
therefore be related to the number of ways the transistors on it can be wired
and how many states they can be in. Note that in this description, no mention
is made of the actual physics of the components. That is, if the same range of
logic functions could be performed using fluid valves or photonic switches,
the same $I_{Dev}$ could result.
Table 1: Example processor characteristics. Comp.: Parallel programmable
components. #Trans.: Indicative number of transistors. Bytes/s: Resource size
$I_{CPU}$ times cycles/sec from equation 1. Note that transistor counts are
commercially sensitive information which should be interpreted with extreme
care. These numbers will also vary widely between product versions.
Type | CPU | Comp.a | Clocka | #Trans.a | Bytes / s
---|---|---|---|---|---
GPU | ATI Radeon 5870 | 1712b | 850 MHz | 2.15$\cdot 10^{9}$ | 18.3$\cdot 10^{17}$
CPU | Intel Core Duo | 2 cores | 2.6 GHz | 291$\cdot 10^{6}$ | 7.57$\cdot 10^{17}$
FPGA | Xilinx Virtex-5 | slicesc | | |
| XC5VFX70T-2 | 11,200 | 249 MHz | 1.1$\cdot 10^{9}$ | 2.74$\cdot 10^{17}$
| XC5VLX30-3 | 4,800 | 251 MHz | 1.1$\cdot 10^{9}$ | 2.76$\cdot 10^{17}$
| XC5VFX70T-2 | 11,200 | 277 MHz | 1.1$\cdot 10^{9}$ | 3.04$\cdot 10^{17}$
a Specifications as published in marketing materials and [44, 45, 3]. b The
total number of stream processors, texture units, and render output units [3].
c The Virtex-5 FPGA is organized in slices, with each slice containing four
6-input Look-Up-Tables (LUT) and four flip-flops [27, 46]
### 2.3 Relations with real hardware
The above theory on efficient emulators can be used to derive an estimate of
the computational capabilities of real hardware [39]. As mentioned before, a
CPU chip is characterized by a number of active elements, transistors, and the
connections between them. The whole CPU is run at a certain speed. The
computational cost of running a certain computation on a CPU can therefore be
quantified as the number of steps needed to complete the computation times the
information frozen into the chip design.
This exercise can also be done the other way around. First, the requirements
for performing a basic computation in terms of electronic circuits
(i.e.,transistors) and number of steps are determined. Then, the number of
copies of the basic devices that fit on a silicon chip are determined. After
that, the number and speed of the computations can be estimated, assuming
state-of-the-art special purpose hardware could be used. It will not come as a
surprise to arrive at the conclusion that custom build electronics can often
outperform general purpose CPUs.
Using the cost function of equation 1 and the device information content,
$I_{Dev}$, can simplify this hardware analysis in many cases. It might obviate
a detailed analysis of the required circuitry and replace it with a less
precise but much more transparent calculation of comparable “complexity”.
To make this analysis, a model is needed of the computational resources
current hardware can deliver. A realistic model should take details of the
limitations of chip design into account. In first approximation it is assumed
that the maximal resources delivered by a CPU are proportional to the number
of transistors. For the current study, a very crude model is assumed [39]. For
any given number of transistors on a chip, it is assumed that each transistor
can be in one of two states (1 bit) and topological constraints limit the
number of different ways it can be connected to neighboring transistors to
$\sim$100 (7 bit). In total, each transistor can thus be described with 1
byte. Table 1 gives these numbers for a few example processors.
This naive hardware model is illustrated below on some simplified
cryptanalysis problems. The focus of the remainder of this section will be on
compute-bound problems. The contribution of the memory components to the
computations will be ignored in the analysis.
#### 2.3.1 Example: The EFF DES cracker
In [39], the example of the EFF cracking the 56-bit single DES system in 1998
[24, 6] is discussed. The challenge was to find the key that could decrypt an
unknown encrypted message. From this example it is possible to get an estimate
of the number of transistors, and costs, needed to implement basic
cryptographic functions. The EFF succeeded in designing a search unit in
silicon that could check a 56 bit DES key in 16 clock cycles [24, 6]. The EFF
were able to fit 24 such search units onto a single chip containing around
10,000 transistors and use the units in parallel.
So a 56-bit DES encryption unit plus comparator needed $\sim 420$ transistors
and runs in 16 clock cycles. With an estimated $I_{Dev}\sim
8\cdot\\#$Transistors bit, this comes down to around 6,700 bytes in equation 1
for checking a single 56-bit DES encryption+compare (i.e., $8\cdot 16\cdot
10^{4}/24$ bits, c.f., [39]). This translates to $\sim$120 byte per bit key
length if it is assumed that encryption effort scales linearly with key
length.
For a brute force key attack, the average number of keys that have to be
tested scales with $2^{k-1}$ for key length $k$. For this specific DES attack,
the computational costs, $C_{DES}(k)$, needed to find a key of length $k$ then
scale as:
$C_{DES}(k)=120\cdot k\cdot 2^{k-1}\;\text{(bytes)}$ (2)
This cost will rise for Triple-DES. Probably in the order of tripling of the
cost, e.g., 360 instead of 120 byte per bit key length.
#### 2.3.2 GPU chips and super-computers
A modern Graphics Processing Unit (GPU) chip, like the ATI HD Radeon 5870,
contains around 2.15 billion transistors and runs at a clock speed of 850 MHz
[3]. Such a processor handles computations at a cost of $\sim 18\cdot 10^{17}$
bytes per second (table 1). If such a processor could be constructed to run as
an efficient parallel DES key search engine, i.e.,like the EFF custom chips,
it would be able to find a 56 bit DES key in 133 seconds on average.
To illustrate the capabilities of GPUs, the analysis is extended to a
hypothetical encryption method with the same features as the single DES
encryption standard, $DES^{*}$. This $DES^{*}$ system is a model of simple
cryptographic primitives and encryptions. The fictional $DES^{*}$ differs from
real DES in that it allows variable key lengths. For every key length, an EFF
DES cracker setup can be constructed for this fictional $DES^{*}$ that scales
like equation 2 and uses 120 byte per bit key length to check a single key.
On a customized processor of this size and speed, finding a 64 bit $DES^{*}$
key would require, on average, around 11 hours, and a 72 bit $DES^{*}$ key
less than 5 months. A dedicated 65k ($2^{16}$) processor cluster would find an
84 bit $DES^{*}$ key in around 10 days and a 92 bit key in around 8 years. A
96 bit $DES^{*}$ key would take such a cluster around 120 years (on average;
240 years worst case). For finding a 96 bit $DES^{*}$ key in less than two
years average, the technology would have to speed up by a factor of 60. At the
historical rate of progress of $I_{Dev}$, around 2.6 dB/year ($\approx
1.82$/year [39]), this would take another 7 year to achieve (but see [35]).
For comparison, the fifth highest entry in the November 2009 TOP 500 list of
supercomputers, the Tianhe-1 supercomputer at the National SuperComputer
Center in Tianjin/NUDT, China, contains 4096 Intel Xeon E5540 processors
(2.5GHz, $7.3\cdot 10^{8}$ transistors) and 1024 E5450 processors (3GHz,
$8.2\cdot 10^{8}$ transistors) connected to 5120 ATI Radeon HD 4870 GPUs
(650MHz, $9.6\cdot 10^{8}$ transistors) with a grand total of over 98TB of
memory [32, 33, 37]. Together the processors deliver $1.3\cdot 10^{22}$
bytes/sec (ignoring memory). If such a machine would have been build as a
dedicated $DES^{*}$ key searcher, it would be able to find an 84 bit $DES^{*}$
key in 87 days, on average. The Tianhe-1 was build for close to 88 million USD
[37].
If the cost of encryption of Triple DES is indeed only $\sim$3 times that of
single DES, the above numbers are not comforting. Triple DES with 2
independent 56 bit keys (keying option 2) has a listed key strength much less
than the expected 112 bits [25, 38]. NIST designates this keying option to
have only 80 bits of security [30] and retires it in 2010. A message encoded
with the equivalent of an 80 bit DES key could theoretically be decrypted
within a few days with a special purpose 65k processor cluster as described
above. However, the known attacks, e.g., [38, 25], are more complex than mere
Triple DES encryption, with important time versus memory trade-off relations.
Therefore, a separate analysis would be needed to calculate the costs of
breaking double-key Triple DES.
#### 2.3.3 A better fit with FPGA
The preceding sections assumed that an attacker could design and produce large
numbers of special purpose CPU chips with state of the art semi-conductor
technology to compromise cryptographic systems. In many situations, such a
threat model is unrealistic. In such cases, a better model would assume that
the attacker would use existing customizable products. A popular product in
this class is a Field Programmable Gate Array (FPGA), an integrated circuit
designed to be configured by the customer or designer after manufacturing
[43].
Large differences in performance between general purpose processors and
specially programmed (FPGA) chips have been demonstrated in the context of
public key block ciphers by Gligoroski et al. [8]. They compared software
implementations on a dual core Intel Core 2 Duo CPU with implementations on
Xilinx Virtex-5 FPGA chips (table 1).
On an Intel Core Duo dual processor, encrypting a 160 bit block with their
MQQ111There are successful attacks known against MQQ which preclude its use in
encryption [7]. This does not affect the computational properties discussed
here. algorithm takes 80,105 cycles and decrypting takes 6,212 cycles (tables
7 and 8 in [8]). Assuming the CPU is running at 2.6GHz, this translates to a
throughput of, respectively, 5.19Mb and 67.0Mb per second. Encryption of a
basic data block (64 bit) with 1024-bit RSA requires 119,800 cycles,
decrypting 2,952,752 cycles on the CPU. Throughputs for RSA are then,
respectively, 1.39Mb and 56.4Kb per second.
The same MQQ algorithm had a corresponding throughput for encryption of 44Gb
per second when implemented on four 276.7MHz Xilinx Virtex-5 FPGAs and 399Mb
per second for decrypting when implemented on a single 249.4 MHz Xilinx
Virtex-5 FPGA. An implementation of 1024-bit RSA on a 251MHz Virtex-5 FPGA had
a throughput of 40Kb per second (unspecified for encryption or decryption).
The computational resources consumed when encrypting or decrypting a single
bit are compared in table 2.
For comparison, results for AES-128 on 16 byte blocks were collected. On an
Intel Core Duo E6700 CPU, the throughput was 1Gbps [26]. Two different
implementations on Virtex-5 boards achieved 4.1Gbps throughput [2]
(unspecified Virtex-5 types, assumed to be the same as for the RSA, updating
the results, 3.8Gbs, reported in [8]).
Efficient use of hardware is determined by the fit between algorithm and the
logic implemented in the chips. Encrypting with MQQ is amenable to
parallelization and fits very well on the Virtex-5 [7]. From table 2 it can be
seen that encryption with MQQ will use $\sim$5300 times more resources
(cycles$\cdot$transistors, i.e., bytes) when computed on a general purpose CPU
than on a dedicated FPGA. An increase in hardware efficiency by a factor of
$\sim$5300 would translate in an additional 12 bits key length that could be
decrypted for the same “costs”. On the other hand, decryption shows only a
modest increase in efficiency by a factor of $\sim$16.
Table 2: Computational resources consumed (bytes) when encrypting or
decrypting 1 bit using the MQQ based algorithm ($n$=160)[8], 1024-bit RSA [8],
and AES-128 [26, 2]. See table 1 for hardware specifications. The RSA results
for the Virtex-5 combine encryption and decryption. See text for details.
| MQQ | | 1024 RSA | | AES-128
---|---|---|---|---|---
| encryption | decryption | encryption | decryption | both
Core Duo | 146 GB | 11.3 GB | 272 GBa | 6.71 TBa | 379 MB
Virtex-5 | 27.5 MBb | 687 MBc | -d | 6.9 TBd | 67.3 MB
a per core [7]. b four Virtex-5 XC5VFX70T-2 at 277 MHz. c one Virtex-5
XC5VFX70T-2 at 249 MHz. d one Virtex-5 XC5VLX30-3 at 251 MHz, unspecified
combined results for encryption and decryption were given.
Another algorithm, 1024-bit RSA, can hardly be parallelized and shows no real
efficiency difference between CPU and FPGA. The AES-128 results are in
between, with a five time increase in efficiency between FPGA and general
purpose CPU (assuming single core use).
The differences between the cases in table 2 raises the question of how the
efficiency gains can be understood. The large gains for the encryption using
the MQQ algorithm implemented on the Virtex-5 FPGA were derived from the
ability to implement the steps of the algorithm in a pipeline that could
output one encrypted data block per clock cycle [7]. Obviously, a tailored
parallel pipeline approach is not possible with the fixed logic of a general
purpose CPU. As illustrated by table 2, such dramatic increases using FPGAs
might be uncommon.
## 3 Adversaries on a budget
A really Universal UTM can crack any cryptographic system that is based on
secret information that is less complex than the message. This can be done by
iterating over all programs and select the one that decrypts the message
first. In a secret key based system, it can be done by a brute force attack
iterating over all keys. However, brute force strategies can take more time
and matter than are available in the universe (c.f., [21, 22, 23]). Therefore,
a meaningful way is needed to limit the power of the Attacker without losing
the theoretical power of the UTM. The Attacker needs resources to perform the
required computations. Resources are understood in the sense of [16, 39]. The
resources are supplied by the environment on a request basis.
With a cost function to quantify computational needs in place, meaningful
limits can be placed on the Attacker. A budget is allocated to the Attacker,
and before every step in the computation, the resource costs of that
computation step are subtracted from the budget. If the budget becomes
depleted, the Attacker loses. The size of the smallest budget for which the
Attacker can win the challenges before the budget is depleted can be
considered the strength of the cryptographic system under study. It is obvious
that a fully universal UTM is regained in the limit of an infinite budget.
An intuitively meaningful way to set a budget is to calculate the
computational cost of testing all possible keys. So if testing one key costs
$C_{key}$, testing all keys of length $k$ bits will cost $C_{key}\cdot 2^{k}$,
as expected. To assist in book keeping, the Attacker can request the current
size of it’s budget on the run tape. The valuation tape contains the
information about the resources available from the environment. For instance,
in situations where the Attacker does not have to design a computer system
from scratch, the valuation tape might contain a catalogue of available
computer systems.
To illustrate the use of the above theory, a few cryptanalytical cases from
the literature are presented. Attention will be focussed on non-interactive
cryptanalysis. A full account should also address the interactive gathering of
information, e.g., differential cryptanalysis.
### 3.1 Challenges: One-Time Pad example
Modeling cryptanalytical attacks as games enforces an explicit definition of
the conditions under which the Attacker wins. The computability logic model
described here defines winnability as the ability of the Attacker to succeed
at a number of predefined challenges. These challenges can be interactive.
For instance, in most cryptographic systems, the ability to guess whether a
known message has been communicated would be a serious vulnerability. In the
formalism presented here, such knowledge could be formalized as being able to
guess above chance which ciphertext encodes a given plaintext.
As an example, suppose the challenge is to exploit a vulnerability in a One-
Time Pad (OTP) implementation where each plaintext is XORed (eXclusive OR)
with a unique sequence of random bits. The Attacker presents two self
delimited plaintexts on the run tape. The environment answers with a self
delimited ciphertext that encrypts one of these plaintexts. The environment
can pad the shortest plaintext to the length of the longest before encryption.
The Attacker then tells which ciphertext was encrypted. If the Attacker can
guess the correct plaintext above chance, the Attacker wins. The threshold of
proof can be put at any convenient level.
The attack strategy would then be to request encryptions of known or chosen
plaintexts. The One-Time pad bit strings are available for analysis after
removing (XOR-ing) the known plaintexts from the ciphertext. If some
statistical deviation from a pure, uncorrelated, uniform distribution can be
detected in the bit strings, the challenges can in principle be won. Simply
chose the ciphertext that XORed with the plaintext shows the anomaly.
As the OTP is proven secure [31], the challenges are only winnable if the
(long) keys are not completely random, e.g., when using an insecure Random
Number Generator (RNG). An Attacker model might include a simulation of
compromising a RNG as in, e.g., [9, 17]. By varying the challenges between
ciphertext only, plaintext chosen by Environment, and plaintext chosen by
Attacker the effects of different security policies can be evaluated. For
instance, the costs and benefits of preventing guessing plaintexts can be
compared to those of periodically reseeding the key generator and
redistributing new keys [17].
Occasionally, the security of the OTP against cryptanalysis is questioned, as
in [41, 42]. The formalism presented here can help to evaluate whether and how
a vulnerability, if any, can be exploited. For instance, from the analysis
presented in [41, 42] it is not clear how a chosen plaintext challenge as
presented here can be won, i.e., whether there is a vulnerability at all.
### 3.2 Dictionary attacks and time versus memory trade-offs
There exist methods to efficiently pre-calculate dictionaries with stored
ciphertext/key pairs to amortize the cost of encryptions over many different
key attacks [19, 28]. To evaluate their threat, it is necessary to estimate
the resources needed to construct and operate such a dictionary. Constructing
a table of Rainbow chains or a dictionary of encryptions is equivalent to
doing a brute force key search and requires the same effort [19, 28]. The new
question is how much resources are needed to use the dictionary after it has
been created.
For simplicity, assume a key size of $k$ and an ordered
$(Ciphertext_{i},Key_{i})$ dictionary with $L=2^{k-\epsilon}$ encryptions of a
$3k$ long plaintext $X_{0}$ as in [19]. The factor $\epsilon$ determines the
fraction of keys in the dictionary as $2^{-\epsilon}$. With these numbers, the
size of the dictionary is $D=4kL$. According to [19] it takes at most
$3k(k-\epsilon)$ comparisons to find an encryption in the dictionary, but
$k-\epsilon$ comparisons seems a more conservative choice. For $k=56$ and
$\epsilon=6$, the size of the dictionary is $D=4\cdot 56\cdot 2^{50}\approx
2.5\cdot 10^{17}$ bits, or $3.1\cdot 10^{16}$ bytes, and the expected number
of comparisons per lookup becomes $50$.
In the ideal case, every comparison is done in, say, two steps for a total of
$100$ steps per lookup. Assume that Attackers “lease” access to the dictionary
for each look-up, that is, there are no “wait states” and the resource is in
constant use by Attackers. The average cost of a lookup is then $3.1\cdot
10^{18}$ bytes, ignoring the small costs of the comparisons themselves. The
average cost of a discovered key would be around $2\cdot 10^{20}$ bytes.
Compared to the current scope of hardware, at $10^{18}$ byte/s for a single
desktop system [39], this cost is unremarkable.
The real point is not the “computation” or processing, but the required
storage capacity of 31 petabyte ($31\cdot 10^{15}$). This is around 15% of the
capacity of a large data-center like Google’s Googleplex facility, or a
“botnet” of a few million computers with some 10 GB each. Such a resource
would require parallel access through many nodes, which would change the
simple cost model above. A botnet of this size would have to contain some 3
million compromised computers with a real cost in the order of $15 a piece, in
2007 dollars, on the black market, or $45 million in total [29]. The combined
value of the encoded information must outweigh the costs of this set up to
make this attack worthwhile. The computational capacity of such a distributed
data center or botnet, with it’s delayed response times, is obviously
different from an integrated desktop system.
This analysis shows that using such a dictionary is, unsurprisingly, not so
much a computational as a storage problem. In this case, the maintenance of
such a large storage is much more a limitation than the duration of the
computation.
### 3.3 Pseudo Random Number Generator attacks: The TF-1 generator
Pseudo-Random Number Generators (PRNGs) are important cryptographic primitives
that can be vulnerable to their own types of attacks [17]. PRNGs are used, for
example, to generate the symmetric keys in public key communication protocols
like SSL (Secure Socket Layer protocol). Their relative security, or lack
thereof, is strongly determined by the resources available to the Attacker
(e.g., [17]).
The Klimov-Shamir number generator TF-1 is analyzed by Tsaban [34]. In short,
for a word size $w$, this PRNG has an internal state of size $4w$. The
intended “strength” is $2^{2w}$ [18, 34], i.e., $2w$ bit. However, Tsaban
finds that the internal state can be found in $16\cdot 2^{1.5w}$ elementary
operations (i.e., $1.5w$ bit strength) after scanning $2^{w}$ output words for
a $0$ value [34]. Each possible internal state can, on average, be checked in
16 basic operations given a special $0$ value in the output.
The 16 operations needed to check the internal state are very basic. A DES
Cracker like search unit should be sufficient (see section 2.3.1). The
original DES Cracker search unit used around 120 byte per bit key width. For
the sake of argument, it is assumed here that a comparable setup could be
constructed that analyzes the internal state again of the TF-1 number
generator for 120 byte per bit in the reduced word size $1.5w$. Each basic
operation should again need only a single clock cycle. For such a system, the
above analysis for the single DES cracker would still hold up to a fixed
factor (see sections 2.3.1 and 2.3.2).
An efficient setup with the complexity and speed of a ATI HD Radeon 5870 (see
section 2.3.2 and table 1) would need under half a second to find the internal
state for a word width of $w=32$ bit (48 bit strength) and less than five
months for a word width of $w=48$ bit (72 bit strength), both on average (see
table 3). A cluster using 65 thousand such set-ups could finish a $w=56$ bit
word length in ten days (84 bit strength). A theoretical $w=60$ bit word
length variant (90 bit strength) could be expected to be broken in less than
two years. For word lengths of $w=64$ (96 bit strength), the time still runs
into 120 years and remains elusive as Tsaban already notes [34].
Table 3: Expected times for finding the internal state of a TF-1 PRNG [34]
using theoretically optimal custom CPUs with the complexity of an ATI HD
Radeon 5870 ($1.83\cdot 10^{18}$ Byte/s). See text for details.
#CPU: number of CPU equivalents; #values: number of PRNG values needed to find special 0 value; time: expected time to find the internal state after finding the special 0 value. $w$ | strength (bit) | #CPU | #values | time
---|---|---|---|---
32 | 48 | 1 | $2.1\cdot 10^{9}$ | 0.5 sec
48 | 72 | 1 | $1.4\cdot 10^{14}$ | 4.2 months
56 | 84 | 65,536 | $3.6\cdot 10^{16}$ | 9.4 days
60 | 90 | 65,536 | $5.8\cdot 10^{17}$ | 1.8 year
64 | 96 | 65,536 | $9.2\cdot 10^{18}$ | 120 years
The number of output words needed to find a $0$ word can become unwieldy for
the longer, $w=\\{48,56\\}$, word lengths (see table 3). For $w=48$, around
$2^{48-1}\approx 10^{14}$ output words have to be scanned for a 0 value. That
is around 40 hours at a billion ($10^{9}$) words per second (average). For
$w=56$ this would be a waiting time of 14 months. Note that originally, the
intended strengths of word lengths of $32$, $48$, and $56$ bit in TF-1 were,
respectively, $64$, $96$, and $112$ bit.
An efficient attack of the TF-1 number generator would be to set up a cheap
system to scan for 0-words storing a history of PRNG output and relevant data
to compromise. Only after a 0-word has been encountered, the machinery to
attack the cypher would be commissioned and the attack performed.
No one has yet reported a DES Cracker like set-up for TF-1. So the above
calculations are based on the assumption that it could be possible to harness
the design complexity of a modern GPU for custom designed cryptanalysis
hardware.
The above analysis allows to put a monetary number on the price to crack this
specific PRNG. Users of this algorithm can now judge themselves how much any
adversaries would be willing to pay for such a set-up and what the chances are
of a version of the algorithm that does not need to find a $0$ word.
## 4 Discussion and conclusions
Cryptanalysis promises to be a very fertile field for developing insight into
the quantification of computational resource needs. A game theoretic view of
cryptanalysis was introduced by Von Neumann and Morgenstern and later taken up
by Shannon [40, 31]. This study adopts this game approach and proposes to use
computability logic [15, 16, 39] to rigorously define Shannon’s work function
[31]. In this approach, attack procedures are formulated in terms of
computable functions [36], the resources used, and also a full definition of
the context of the attack.
Based on a few “natural” requirements, a simple formula for quantified
resources emerges as equation 1 with the features of Memory times Steps, i.e.,
a dimension of bytes [39]. This count includes the information “frozen” into
the computational device itself, e.g., the UTM action table or the components
and connections of the CPU. By reducing silicon CPU complexity to transistor
connectivity and memory capacity, it is possible to roughly guess the capacity
of real hardware.
Using the estimated hardware complexity of mass market processors as an upper
boundary, it is possible to estimate the limits of customized cryptanalytic
hardware. These limits can be used to understand historical cases, like the
failure of 56 bit DES encryptions [6]. These limits can also be used to
predict the (theoretical) failure of modern cryptographic primitives like the
TF-1 PRNG with a theoretical strength of $84$ and $90$ bit keys (intended
strengths were originally $112$ and $120$ bits) [18, 34] as well as the
efforts needed to actually effectuate the attacks.
It can be concluded that the general problem of quantifying computational
resource use in interactive cryptanalysis attacks can be solved in a
formalized setting. When used to formalize cryptanalysis, it becomes possible
to quantify the cryptanalysis work function [31]. Even the computational costs
of hypothetical attacks on cryptographic primitives can be estimated before
they have to be demonstrated at great monetary cost.
Examples show that it would currently (2010) be feasible to build hardware
that could break some 84 bit strength cryptographic primitives in mere days,
and 90 bit strength primitives in less than two years.
## 5 acknowledgment
This project was made possible by grant 276-75-002 of the Netherlands
Organisation of Scientific Research (NWO)
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|
arxiv-papers
| 2010-03-10T14:55:18 |
2024-09-04T02:49:09.005376
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "R. J. J. H. van Son",
"submitter": "Rob Van Son",
"url": "https://arxiv.org/abs/1003.2123"
}
|
1003.2160
|
Astronomy Letters, 2010 Vol. 36, No. 3
Searching for Stars Closely Encountering with the Solar System
V.V. Bobylev
Pulkovo Astronomical Observatory, Russian Academy of Sciences, St-Petersburg
Abstract–Based on a new version of the Hipparcos catalog and currently
available radial velocity data, we have searched for stars that either have
encountered or will encounter the solar neighborhood within less than 3 pc in
the time interval from $-2$ Myr to $+2$ Myr. Nine new candidates within 30 pc
of the Sun have been found. To construct the stellar orbits relative to the
solar orbit, we have used the epicyclic approximation. We show that, given the
errors in the observational data, the probability that the well-known star HIP
89 825 (GL 710) encountering with the Sun most closely falls into the Oort
cloud is 0.86 in the time interval $1.45\pm 0.06$ Myr. This star also has a
nonzero probability, $1\times 10^{-4},$ of falling into the region $d<1000$
AU, where its influence on Kuiper Belt objects becomes possible.
DOI:10.1134/S1063773710030060
## INTRODUCTION
Interest in the problem of close encounters of field stars with the Solar
system stems from the fact that the passage of a star can lead to various
kinds of perturbations of Solar system objects.
Thus, for example, the interaction of a star with the Oort comet cloud can
give rise to comet showers reaching the region of the major planets (Hills
1981; Weissman 1996). The formation time scale of a comet shower is $\approx$1
Myr after the passage of a star. According to Weissman (1996), the Oort cloud
is a spheroid with a semimajor axis of about $10^{5}$ AU ($\approx$0.5 pc,
since 1 pc=206 265 AU) elongated toward the Galactic center and a semiminor
axis of about $8\times 10^{4}$ AU. The question about close encounters of
stars with the Sun within $r<2-5$ pc was considered by Revina (1988), Matthews
(1994), and Mullari and Orlov (1996) using various ground-based observations
and by Garcia-Sanchez et al. (1999, 2001) based on Hipparcos (1997) data in
combination with stellar radial velocity data. As a result, 156 Hipparcos
stars from the solar neighborhood 50 pc in radius that either have encountered
or will encounter with the Solar system within $r<5$ pc in the time interval
$\pm 10$ Myr are known to date (Garcia-Sanchez et al. 2001). Having analyzed
these data, Garcia-Sanchez et al. (2001) estimated the frequency of close
(within 1 pc) encounters of stars with the Sun to be $2.3\pm 0.2$ encounters
per Myr and, after a correction for the Hipparcos incompleteness, this value
increases to $11.7\pm 1.3$ encounters per Myr.
Analyzing the possibility of even closer encounters is of current interest.
For example, Kenyon and Bromley (2004) argue that only the passage of a star
can explain the peculiarities of the orbit of the minor planet 2003 VB12
(Sedna). The various kinds of influences of such a star on Kuiper Belt objects
were shown to manifest themselves at characteristic distances $r<1000$ AU
($\approx$0.005 pc). The Kuiper Belt proper extends from the Sun to a boundary
of $\approx$50 AU.
The goal of this study is to search for candidate stars closely encountering
with the Sun based on a new version of the Hipparcos catalog (van Leeuwen,
2007) and currently available stellar radial velocity data. We solve the
problem of statistical simulations by taking into account the random errors in
the input data and estimate the probability of a star penetrating into the
Oort cloud region and into the region of a possible influence of the star on
Kuiper Belt objects.
## 1 THE DATA
We use stars from the Hipparcos catalog (ESA 1997) while taking the new proper
motions and parallaxes from a revised version of this catalog (van Leeuwen,
2007); the stellar radial velocities are taken from the Pulkovo Compilation of
Radial Velocities (PCRV) (Gontcharov 2006) created as a result of implementing
the OSACA project (Bobylev et al., 2006) and containing radial velocity data
for about 35 000 stars. Note that, in contrast to the larger CRVAD-2 catalog
of radial velocities (Kharchenko et al., 2007), the PCRV catalog contains only
the stars with random errors in their radial velocities within 10 km s-1.
## 2 THE METHODS
### 2.1 Orbit Construction
We use a rectangular Galactic coordinate system with the axes directed away
from the observer toward the Galactic center ($l=0^{\circ},b=0^{\circ},$ the
$X$ axis), in the direction of Galactic rotation ($l=90^{\circ},b=0^{\circ},$
the $Y$ axis), and toward the North Pole ($b=90^{\circ},$ the $Z$ axis). The
corresponding space velocity components of an object $U,V,W$ are also directed
along the $X,Y,Z$ axes. The epicyclic approximation (Lindblad, 1927) allows
the stellar orbits to be constructed in a coordinate system rotating around
the Galactic center. We apply this method in the form given in Fuchs et al.
(2006):
$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\hfill X(t)=X(0)+{U(0)\over\kappa}\sin(\kappa
t)+\hfill\cr 0.0pt{\hfil$\displaystyle\hfill+{V(0)\over 2B}(1-\cos(\kappa
t)),\hfill\hbox to0.0pt{\hss(}1)\cr 0.0pt{\hfil$\displaystyle\hfill
Y(t)=Y(0)+2A\biggl{(}X(0)+{V(0)\over 2B}\biggr{)}t-\hfill\cr
0.0pt{\hfil$\displaystyle\hfill-{\Omega_{0}\over B\kappa}V(0)\sin(\kappa
t)+{2\Omega_{0}\over\kappa^{2}}U(0)(1-\cos(\kappa t)),\hfill\cr
0.0pt{\hfil$\displaystyle\hfill Z(t)={W(0)\over\nu}\sin(\nu t)+Z(0)\cos(\nu
t),\hfill\crcr}}}}}}$
where $t$ is the time in Myr (we proceed from the ratio pc/Myr = 0.978 km
s-1); $A$ and $B$ are the Oort constants; $\kappa=\sqrt{-4\Omega_{0}B}$ is the
epicyclic frequency; $\Omega_{0}$ is the angular velocity of Galactic rotation
for the local standard of rest, $\Omega_{0}=A-B$; $\nu=\sqrt{4\pi G\rho_{0}}$
is the vertical oscillation frequency, where $G$ is the gravitational constant
and $\rho_{0}$ is the star density in the solar neighborhood.
The parameters $X(0),Y(0),Z(0)$ and $U(0),V(0),W(0)$ in the system of
equations (1) denote the current stellar positions and velocities. The
displacement of the Sun from the Galactic plane is taken to be $Z(0)=17$ pc
(Joshi 2007). We calculate the velocities $U,V,W$ relative to the local
standard of rest (LSR) with $(U,V,W)_{LSR}=(10.00,5.25,7.17)$ km s-1 (Dehnen
and Binney 1998).
We took $\rho_{0}=0.1M_{\odot}$/pc3 (Holmberg and Flinn 2004), which gives
$\nu=74$ km s-1 kpc-1. We used the following Oort constants found by Bobylev
et al. (2008): $A=15.5\pm 0.3$ km s-1 kpc-1 and $B=-12.2\pm 0.7$ km s-1 kpc-1;
$\kappa=37$ km s-1 kpc-1 corresponds to these values.
Note that we neglect the gravitational interaction between the star and the
Sun.
As was shown by Mullari and Orlov (1996), even the simplest linear
approximation provides a sufficient accuracy in the time interval from $-2$
Myr to $+2$ Myr compared to the method of constructing the relative orbits
(the star relative to the Sun) in the Galactic potential. We choose the
epicyclic approximation, because we are planning to perform statistical
simulations with the computation of hundreds of thousands of orbits, while the
application of more complex methods requires a huge computational time.
### 2.2 Statistical Simulations
In accordance with the method of Monte Carlo statistical simulations, we
compute a set of orbits for each object by taking into account the random
errors in the input data. For each star, we compute the encounter parameter,
the shortest distance between the stellar and solar orbits
$d_{min}=\sqrt{\Delta X^{2}(t)+\Delta Y^{2}(t)+\Delta Z^{2}(t)}.$ The stellar
parameters are assumed to be distributed normally with a dispersion $\sigma$.
We add the errors to the equatorial coordinates, proper motion components,
parallax, and radial velocity of the star using the nominal errors.
## 3 RESULTS
First, we considered the solar neighborhood 30 pc in radius using $\approx$35
000 stars with known space velocities. For each star, we constructed the orbit
relative to the Sun in the time interval from $-2$ Myr to $+2$ Myr. In
addition to the stars from the list by Garcia-Sanchez et al. (2001), we found
several other Hipparcos stars. Data on the nine stars found with an encounter
parameter $d<3$ pc are presented in Table 1; Fig. 1a shows the trajectories
for six of these stars.
Figure 1b shows the model trajectories of four stars computed by taking into
account the random errors in the observational data; 300 realizations were
obtained for each of the three stars indicated in the figure.
Table 2 gives characteristics of the stars common to the list by Garcia-
Sanchez et al. (2001) that have the closest ($d<2$ pc) encounters with the
Sun.
Table 3 gives characteristics of the common list of stars that have the
closest ($d<2$ pc) encounters with the Sun. Note that this is currently the
most complete list with the restrictions that we used. In contrast to Tables 1
and 2, Table 3 gives the encounter parameters $d_{min}$ and $t_{min}$ with
their errors estimated through statistical simulations.
Table 3 also contains two stars from Table 1. The first of them is the well-
known white dwarf WD 0310.688 (HIP 14754). Note that the error in its radial
velocity (Pauli et al. 2006) is largest among all of the stars listed in
Tables 1 and 2. The data for the second star, HIP 27887, are very reliable. It
is contained in the catalog of F and G dwarfs of the Geneva-Copenhagen survey
(Nordstrom et al. 2004; Holmberg et al. 2007).
Note that the compilations by Gontcharov (2006) and Kharchenko et al. (2007)
give an obsolete radial velocity for HIP 3829, $V_{r}=263\pm 5$ km s-1, taken
from the GCRV (Wilson 1953). A very close encounter with the Sun is obtained
with this value: $d_{min}=0.96$ pc at $t_{min}=-16$ thousand years. However,
this star is the well-known white dwarf WD 0046+051 (van Maanen s star, vMa
2). According to more recent observations, its radial velocity is $V_{r}=54$
km s-1 (Greenstein and Trimble 1967) or $V_{r}(LSR)=-41$ km s-1 (Aannestad et
al. 1993). In this case, the encounter is considerably less close, $d_{min}>3$
pc (see also Garcia-Sanchez et al. 1999, 2001).
Figure 2 shows the trajectories of HIP 89825 (GL 710) relative to the Sun
computed by taking into account the random errors in the observational data.
We made only 300 realizations for a clearer graphical presentation. It is
obvious from the figure that very close encounters with the Solar system are
possible for this star. More detailed simulations show that, for example, out
of the one million model orbits found for GL 710, $d_{min}\leq 0.48$ pc in 855
902 cases and, hence, the probability of its falling into the Oort cloud is
$P_{1}=0.86$ and $d_{min}\leq 0.005$ pc in 94 cases and, hence, the
probability of the star approaching the boundary of its possible influence on
Kuiper Belt objects is $P_{2}=0.94\times 10^{-4}$.
Figure 3 show the distribution of encounter parameters of HIP 89825 (GL 710)
with the Sun obtained using one million model orbits.
## 4 DISCUSSION
(1) Among the Hipparcos stars we consider, there are nine stars common to the
list by Mullari and Orlov (1996). The encounter parameters for some of the
stars found by Mullari and Orlov (1996) are: $d_{min}=1.35$ pc, $t_{min}=1050$
thousand years for GL 710; $d_{min}=1.64$ pc, $t_{min}=-529$ thousand years
for GL 208; and $d_{min}=1.89$ pc, $t_{min}=88.7$ thousand years for GL 860 A.
The encounter parameters agree well with our results (Table2). There is only
one exception, GL 710, for which the new data give a closer encounter, which
was also pointed out by Garcia-Sanchez et al. (1999).
Note that the star HIP 99461 (GL 783) we selected is present in the list by
Mullari and Orlov (1996) but is absent in the lists by Garcia-Sanchez et al.
(1999, 2001).
(2) For an overwhelming majority of the stars, the encounter parameters
derived here agree well with those from Garcia-Sanchez et al. (2001).
The differences are most noticeable for the star GL 217.1, more specifically,
$|\Delta d_{min}|=0.38$ pc and $|\Delta t_{min}|=182$ thousand years.
According to Garcia-Sanchez et al. (2001), who adopted the radial velocity
$V_{r}=20.0\pm 3.7$ km s-1 that differs markedly from our value (Table 2), the
encounter parameters for GL 217.1 are: $d_{min}=1.65\pm 0.27$ pc and
$t_{min}=-(1046\pm 163)$ thousand years.
Computing the model epicyclic orbits for the star GL 217.1 using the initial
data from Garcia-Sanchez et al. (2001) with the adopted radial velocity error
$e_{V_{r}}=2$ km s-1 yields $d_{min}=1.61\pm 0.20$ pc and $t_{min}=-(1066\pm
116)$ thousand years, and these parameters almost coincide with those from
Garcia-Sanchez et al. (2001).
This leads us to conclude that the existing discrepancy is undoubtedly related
to the difference in input data.
As we see from Table 3, GL 217.1 is the most massive, $M=2M_{\odot}$, star
among our stars. Its improved encounter parameters, $d_{min}=1.28\pm 0.06$ pc
and $t_{min}=-(861\pm 40)$ thousand years, make it attractive for studying the
close passages that could take place in the past.
(3) The star GL 710 is of greatest interest to us, because it has a record
close encounter with the Sun among all of the candidates known to date.
According to Garcia-Sanchez et al. (2001), the encounter parameters of GL 710
are: $d_{min}=0.34\pm 0.18$ pc and $t_{min}=(1358\pm 41)$ thousand years, in
good agreement with those we found (Table 3).
GL 710 is the only star with a significant difference between the data of
Tables 2 and 3. This suggests that the conditions for its encounter with the
Sun make the stellar orbit very sensitive to small changes in such initial
data as the parallax and radial velocity. We clearly see from Fig. 3a that the
distribution of minimum encounter distance $d_{min}$ differs from a Gaussian
one, the mode of this distribution is $d_{min}=0.27$ pc, which is lower than
the median value of $d_{min}$ given in Table 3, while the distribution of
encounter times is nearly Gaussian (Fig. 3b).
(4) Our simulations show that among the candidates considered, only the star
GL 710 has a high probability of penetrating into the Oort cloud region.
## 5 CONCLUSIONS
Based on currently available space velocity data for about 35 000 Hipparcos
stars, we searched for stars closely encountering with the Solar system. For
this purpose, we took stars within 30 pc of the Sun and determined their
orbits relative to the Sun based of the epicyclic approximation in the time
interval from $-2$ Myr to $+2$ Myr. We found nine new candidates in addition
to the well-known list of such stars (Garcia-Sanchez et al. 2001). The use of
the PCRV (Gontcharov 2006) and improved Hipparcos stellar parallaxes (Leeuwen
2007) made this possible.
As a result, we compiled the currently most complete (given the restrictions)
list of Hipparcos stars that have close ($d<2$ pc) encounters with the Sun.
For the star GL 217.1, a well-known candidate for a passage close to the Sun,
the new observational data were shown to change noticeably its previously
known encounter parameters with the Sun. The encounter parameters found here
are: $d_{min}=1.28\pm 0.06$ pc and $t_{min}=-(861\pm 40)$ thousand years.
Improving the radial velocity for the white dwarf WD 0310–688 (HIP 14754)
whose orbit passed at a distance $d_{min}=1.61\pm 0.19$ pc from the solar
orbit about 300 thousand years ago is of current interest. Our statistical
simulations showed that the star GL 710 has not only a high probability of
penetrating into the Oort cloud, $P_{1}=0.86,$ but also a nonzero probability,
$P_{2}=1\times 10^{-4},$ of penetrating into the region where the influence of
the passing star on Kuiper Belt objects is significant.
ACKNOWLEDGMENTS
I wish to thank Prof. V.V. Orlov for a careful reading of the manuscript and
his remarks and A.T. Bajkova for a discussion of the results. The SIMBAD
searchable database was very helpful in the work. This study was supported by
the Russian Foundation for Basic Research (project no. 08–02–00400) and in
part by the “Origin and Evolution of Stars and Galaxies” Program of the
Presidium of the Russian Academy of Sciences.
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Washington, DC, 1953), p. 601.
Translated by N. Samus’
Table 1: Hipparcos stars within 30 pc of the Sun found here
HIP | SP | $d_{min},$ | $t_{min},$ | $\pi\pm e_{\pi},$ | $V_{r}\pm e_{V_{r}},$ | n | $\varepsilon_{V_{r}},$
---|---|---|---|---|---|---|---
| | pc | $10^{3}$ yr | mas | km s-1 | | km s-1
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
14754 | DA | $1.60$ | $-296$ | $97.66\pm 1.85$ | $33.8\pm 3.2$ | 1 |
27887 | K3V | $1.97$ | $-413$ | $76.95\pm 0.37$ | $30.7\pm 0.3$ | 1 |
99461* | K2V | $2.05$ | $41.1$ | $166.25\pm 0.27$ | $-129.0\pm 0.2$ | 2 | 0.8
24186 | M0V | $2.15$ | $-10.8$ | $255.66\pm 0.91$ | $244.4\pm 0.2$ | 3 | 1.1
105766 | G5 | $2.20$ | $341$ | $37.87\pm 0.42$ | $-76.6\pm 0.3$ | 2 | 0.2
30344 | K0V | $2.30$ | $-2035$ | $34.10\pm 0.62$ | $14.6\pm 0.3$ | 1 |
26373 | K0V | $2.33$ | $-762$ | $39.82\pm 1.36$ | $32.5\pm 0.3$ | 1 |
104214 | K5V | $2.79$ | $18.6$ | $286.82\pm 6.78$ | $-65.9\pm 0.1$ | 5 | 0.7
104217 | K7V | $2.80$ | $19.6$ | $285.88\pm 0.54$ | $-64.3\pm 0.1$ | 3 | 0.3
Note. $n$ is the number of catalogs used to calculate the mean $V_{r};$
$\varepsilon_{V_{r}}$ is the radial velocity error calculated from the
external convergence (Gontcharov 2006); (*) HIP 99461 (GL 783) is present in
the list by Mullari and Orlov (1996) but is absent in the lists by Garcia-
Sanchez et al. (1999, 2001).
Table 2: Data on the known Hipparcos stars from the solar neighborhood 30 pc
in radius encountering with the solar orbit within $d<2$ pc
HIP | | $d_{min},$ | $t_{min},$ | $\pi\pm e_{\pi},$ | $V_{r}\pm e_{V_{r}},$ | n | $\varepsilon_{V_{r}},$
---|---|---|---|---|---|---|---
| | pc | $10^{3}$ yr | mas | km s-1 | | km s-1
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
89825 | GL 710 | $0.21$ | $1444$ | $51.12\pm 1.63$ | $-13.8\pm 0.3$ | 4 | 8.1
70890 | Proxima Cen | $0.89$ | $27.4$ | $771.64\pm 2.60$ | $-25.1\pm 0.9$ | |
71683 | $\alpha$ Cen A/B * | $0.91$ | $28.4$ | $754.81\pm 4.11$ | $-24.7\pm 0.4$ | 1 |
57544 | AC+79 3888 | $1.06$ | $46.0$ | $186.86\pm 1.70$ | $-111.6\pm 0.2$ | 5 | 3.9
87937 | Barnard’s star | $1.15$ | $9.8$ | $548.31\pm 1.51$ | $-110.6\pm 0.2$ | 5 | 2.2
27288 | GL 217.1 | $1.27$ | $-862$ | $46.28\pm 0.16$ | $24.7\pm 1.1$ | 2 | 5.9
54035 | Lalande 21185 | $1.43$ | $20.5$ | $392.64\pm 0.67$ | $-85.8\pm 0.1$ | 7 | 3.1
26335 | GL 208 | $1.54$ | $-500$ | $88.97\pm 1.02$ | $22.0\pm 0.2$ | 5 | 2.1
57548 | Ross 128 | $1.92$ | $72.4$ | $298.04\pm 2.30$ | $-31.0\pm 0.2$ | 4 | 1.1
110893 | GL 860 A | $1.93$ | $90.0$ | $249.94\pm 1.87$ | $-34.0\pm 0.1$ | 6 | 2.3
92403 | Ross 154 | $1.96$ | $157$ | $336.72\pm 2.03$ | $-10.7\pm 0.2$ | 5 | 1.9
38228 | HD 63433 | $2.02$ | $1366$ | $45.45\pm 0.53$ | $-16.1\pm 0.1$ | 6 | 1.2
Note. $n$ is the number of catalogs used to calculate the mean $V_{r};$
$\varepsilon_{V_{r}}$ is the radial velocity error calculated from the
external convergence (Gontcharov 2006); (*) the encounter parameters were
calculated from component A.
Table 3: Hipparcos stars from the solar neighborhood 30 pc in radius
encountering with the solar orbit within $d<2$ pc
HIP | | SP | $M/M_{\odot}$ | Ref | ${\overline{d}}_{min},$ $10^{3}$ yr | ${\overline{t}}_{min},$ pc
---|---|---|---|---|---|---
1 | 2 | 3 | 4 | 5 | 6 | 7
89825 | GL 710 | K7V | $0.4-0.6$ | (1) | $0.311\pm 0.167$ | $1447\pm 60$
70890 | Proxima Cen | M5Ve | $0.15\pm 0.02$ | (2) | $0.890\pm 0.019$ | $27.4\pm 0.1$
71683 | $\alpha$ Cen A/B * | G2V/K1V | $2.0\pm 0.1$ | (2) | $0.910\pm 0.012$ | $28.4\pm 0.1$
57544 | AC+79 3888 | M4 | $\approx 0.15$ | | $1.059\pm 0.013$ | $46.0\pm 0.3$
87937 | Barnard’s star | sdM4 | $0.144$ | (3) | $1.148\pm 0.006$ | $9.8\pm 0.1$
27288 | GL 217.1 | A2Vann | $2.0$ | (1) | $1.275\pm 0.064$ | $-861\pm 40$
54035 | Lalande 21185 | M2V | $0.39$ | (4) | $1.426\pm 0.005$ | $20.5\pm 0.1$
26335 | GL 208 | K7 | $0.47$ | (4) | $1.537\pm 0.050$ | $-500\pm 8$
14754 | WD 0310-688 | DA | $\approx 0.5$ | | $1.607\pm 0.190$ | $-297\pm 29$
57548 | Ross 128 | M4.5V | $0.24$ | (4) | $1.920\pm 0.028$ | $72.8\pm 0.7$
110893 | GL 860 A | M2V | $0.56$ | (4) | $1.929\pm 0.028$ | $90.3\pm 0.5$
92403 | Ross 154 | M3.5Ve | $0.23$ | (4) | $1.959\pm 0.030$ | $157\pm 1$
27887 | HD 40307 | K3V | $0.75^{+0.78}_{-0.71}$ | (5) | $1.974\pm 0.035$ | $-413\pm 5$
38228 | HD 63433 | G5IV | $0.88^{+0.98}_{-0.84}$ | (5) | $2.038\pm 0.099$ | $1366\pm 20$
Note. The stellar mass estimates were taken from the following papers: (1)
Garcia-Sanchez et al. (1999); Anosova et al. (1994); (3) Giampapa et al.
(1996); (4) Mullari and Orlov (1996); and (5) Holmberg et al. (2007). If there
is no reference, then a typical mass is given; (*) the sum of the masses is
given.
Fig. 1. (a) Stellar trajectories relative to the Sun. The trajectories of the
stars found here are highlighted by the thick lines: HIP 27887 (1), HIP 14754
(2), HIP 24186 (3), the pair of HIP 104214 and HIP 104217 (4), HIP 99461 (5),
and HIP 105766 (6). (b) The model trajectories of four stars computed by
taking into account the random errors in the observational data (300
realizations). The trajectories hatch the $3\sigma$ confidence regions; the
Oort cloud region is shaded.
Fig. 2. Model trajectories of the star HIP 89825 (GL 710) relative to the Sun
computed by taking into account the random errors in the observational data
(300 realizations). The trajectories hatch the $3\sigma$ confidence region;
the Oort cloud region is shaded.
Fig. 3. Model encounter parameters of the star HIP 89825 (GL 710) with the
Sun. (a) The distribution of minimum distance $d_{min}$; (b) the histogram of
encounter times $t_{min}$.
|
arxiv-papers
| 2010-03-10T17:31:00 |
2024-09-04T02:49:09.016816
|
{
"license": "Public Domain",
"authors": "Vadim V. Bobylev",
"submitter": "Anisa Bajkova",
"url": "https://arxiv.org/abs/1003.2160"
}
|
1003.2297
|
# $n$-th Relative Nilpotency Degree
and Relative $n$-Isoclinism Classes
Rashid Rezaei Department of Mathematics, Faculty of Mathematical Sciences
University of Malayer
Post Box: 657719, 95863, Malayer, Iran ras$\\_$rezaei@yahoo.com and
Francesco G. Russo Departement of Mathematics
University of Palermo
via Archirafi 14, 90123, Palermo, Italy francescog.russo@yahoo.com
###### Abstract.
P. Hall introduced the notion of isoclinism between two groups more than 60
years ago. Successively, many authors have extended such a notion in different
contexts. The present paper deals with the notion of relative $n$-isoclinism,
given by N. S. Hekster in 1986, and with the notion of $n$-th relative
nilpotency degree, recently introduced in literature.
###### Key words and phrases:
Nilpotency degree, relative $n$-th nilpotency degree, relative $n$-isoclinism
classes
###### 2010 Mathematics Subject Classification:
Primary: 20D60, 20P05; Secondary: 20D08, 20D15.
## 1\. Introduction and Statement of Results
Every locally compact topological group $G$ admits a left Haar measure
$\mu_{G}$, which is a positive Radon measure on a $\sigma$-algebra containing
Borel sets with the property that $\mu_{G}(xE)=\mu_{G}(E)$ for each element
$x$ of the measure space $G$ (see [13, Theorem 2.8]). The support of $\mu_{G}$
is $G$ and it is usually unbounded, but if $G$ is compact, then $\mu_{G}$ is
bounded. For this reason we may assume without ambiguity that a compact group
$G$ has a unique probability measure space, where $\mu_{G}$ is normalized,
that is, $\mu_{G}(G)=1$.
Let $G$ be a compact group with the normalized Haar measure $\mu_{G}$. On the
product measure space $G\times G$, it is possible to consider the product
measure $\mu_{G}\times\mu_{G}$. If
$C_{2}=\\{(x,y)\in G\times G\ |\ [x,y]=1\\},$
then $C_{2}=f^{-1}(1)$, where $f:G\times G\rightarrow G$ is defined via
$f(x,y)=[x,y]$. It is clear that $f$ is continuous and so $C_{2}$ is a compact
and measurable subset of $G\times G$. Therefore it is possible to define the
$commutativity$ $degree$ of $G$ as
$d(G)=(\mu_{G}\times\mu_{G})(C_{2}).$
We may extend $d(G)$ as follows. Suppose that $n\geq 1$ is an integer, $G^{n}$
is the product of $n$-copies of $G$ and $\mu^{n}_{G}$ that of $n$-copies of
$\mu_{G}$. The $n$-$th$ $nilpotency$ $degree$ of $G$ is defined to be
$d^{(n)}(G)=\mu^{n+1}_{G}(C_{n+1}),$
where
$C_{n+1}=\\{(x_{1},\ldots,x_{n+1})\in G^{n+1}\ |\
[x_{1},x_{2},...,x_{n+1}]=1\\}.$
Obviously, if $G$ is finite, then $G$ is a compact group with the discrete
topology and so the Haar measure of $G$ is the counting measure. Then we have
as special situation for a finite group $G$:
$d^{(n)}(G)=\mu^{n+1}_{G}(C_{n+1})=\frac{|C_{n+1}|}{|G|^{n+1}}.$
When $n=1$, literature can be found in [5, 6, 7, 8, 9, 14].
Let $G$ be a compact group and $H$ be a closed subgroup of $G$. It is possible
to define
$D_{2}=\\{(h,g)\in H\times G\ |\ [h,g]=1\\}.$
Then $D_{2}=\phi^{-1}(1)$, where $\phi:H\times G\rightarrow G$ is defined via
$\phi(h,g)=[h,g]$. It is clear that $\phi$ is continuous and so $D_{2}$ is a
compact and measurable subset of $H\times G$. Note that $\phi$ is the
restriction of $f$ under $H\times G$. This remark shows why $H$ has to be
required as closed subgroup of $G$. Then we may define the $relative$
$commutativity$ $degree$ of $H$ with respect to $G$ as
$d(H,G)=(\mu_{H}\times\mu_{G})(D_{2}).$
Considering
$D_{n+1}=\\{(h_{1},,...,h_{n},g)\in H^{n}\times G\ |\
[h_{1},h_{2},...,h_{n},g]=1\\},$
we define the $relative$ $n$-$th$ $nilpotency$ $degree$ of $H$ with respect to
$G$ as
$d^{(n)}(H,G)=(\mu^{n}_{H}\times\mu_{G})(D_{n+1}).$
The following notion is fundamental for stating our results (see for
terminology [1, 2, 6, 10, 11, 12, 13, 15]).
###### Definition 1.1.
Let $G_{1}$, $G_{2}$ be two groups, $H_{1}$ a subgroup of $G_{1}$ and $H_{2}$
a subgroup of $G_{2}$. A pair $(\alpha,\beta)$ is said to be a relative
$n$-isoclinism from $(H_{1},G_{1})$ to $(H_{2},G_{2})$ if we have the
following conditions:
1. (i)
$\alpha$ is an isomorphism from $G_{1}/Z_{n}(G_{1})$ to $G_{2}/Z_{n}(G_{2})$
such that the restriction of $\alpha$ under $H_{1}/(Z_{n}(G_{1})\cap H_{1})$
is an isomorphism from $H_{1}/(Z_{n}(G_{1})\cap H_{1})$ to
$H_{2}/(Z_{n}(G_{2})\cap H_{2})$, that is, the map
$\alpha^{n+1}:\frac{H_{1}}{Z_{n}(G_{1})\cap
H_{1}}\times...\times\frac{H_{1}}{Z_{n}(G_{1})\cap
H_{1}}\times\frac{G_{1}}{Z_{n}(G_{1})}\rightarrow\frac{H_{2}}{Z_{n}(G_{2})\cap
H_{2}}\times...\times\frac{H_{2}}{Z_{n}(G_{2})\cap
H_{2}}\times\frac{G_{2}}{Z_{n}(G_{2})}$ is an isomorphism;
2. (ii)
$\beta$ is an isomorphism from $[_{n}H_{1},G_{1}]$ to $[_{n}H_{2},G_{2}]$;
3. (iii)
Considering for each $h_{1},...,h_{n}\in H_{1}$, $k_{1},...,k_{n}\in H_{2}$,
$g_{1}\in G_{1}$, $g_{2}\in G_{2}$ there exists a commutative diagram in which
the map
$\gamma(n,H_{1},G_{1}):((h_{1}(Z_{n}(G_{1})\cap
H_{1}),...,h_{n}(Z_{n}(G_{1})\cap
H_{1}),g_{1}Z_{n}(G_{1})))\in\frac{H_{1}}{Z_{n}(G_{1})\cap H_{1}}\times...$
$...\times\frac{H_{1}}{Z_{n}(G_{1})\cap
H_{1}}\times\frac{G_{1}}{Z_{n}(G_{1})}\mapsto[h_{1},...,h_{n},g_{1}]\in[_{n}H_{1},G_{1}]$
and the map
$\gamma(n,H_{2},G_{2}):((k_{1}(Z_{n}(G_{2})\cap
H_{2}),...,k_{n}(Z_{n}(G_{2})\cap
H_{2}),g_{2}Z_{n}(G_{2})))\in\frac{H_{2}}{Z_{n}(G_{2})\cap H_{2}}\times...$
$...\times\frac{H_{2}}{Z_{n}(G_{2})\cap
H_{2}}\times\frac{G_{2}}{Z_{n}(G_{2})}\mapsto[k_{1},...,k_{n},g_{2}]\in[_{n}H_{2},G_{2}],$
can be composed by the rule
$\gamma(n,H_{2},G_{2})\ \circ\ \alpha^{n+1}=\beta\ \circ\
\gamma(n,H_{1},G_{1}).$
It is easy to see that Definition 1.1 is well posed. In particular, two groups
$G_{1}$ and $G_{2}$ in Definition 1.1 are called $isoclinic$ if $n=1$,
$H_{1}=G_{1}$ and $H_{2}=G_{2}$.
P. Hall already pointed out that two groups which are isoclinic can allow us
to define the $isoclinism$ as an equivalence relation [11]. Then it is
possible to classify two groups with respect to their $isoclinism$ $class$.
See [10] as general reference on the topic. Successively, this crucial passage
was pointed out by J. Bioch in [1, 2] and by N. S. Hekster in [12] with
respect to generalizations of the notion of isoclinism. At the same way,
Definition 1.1 allows us to define an equivalence relation, which extends that
of P. Hall in [11], that of J. Bioch in [1, 2] and that of N. S. Hekster in
[12]. Two groups which satisfy Definition 1.1 are called $relative$
$n$-$isoclinic$ and we will write briefly $(H_{1},G_{1})\ _{\widetilde{n}}\
(H_{2},G_{2})$. It is easy to see that it is possible to classify two groups
with respect to their $relative$ $n$-$isoclinism$ $class$.
The main results of the present paper are listed below and they generalize [6,
Theorem 4.2].
###### Theorem 1.2.
Assume that $G_{1}$ and $G_{2}$ are two compact groups, $H_{1}$ is a closed
subgroup of $G_{1}$ and $H_{2}$ is a closed subgroup of $G_{2}$. If
$(H_{1},G_{1})\ _{\widetilde{n}}\ (H_{2},G_{2})$, then
$d^{(n)}(H_{1},G_{1})=d^{(n)}(H_{2},G_{2})$
###### Theorem 1.3.
Let $G$ be a compact group and $H$ be a closed subgroup of $G$. If
$G=HZ_{n}(G)$, then $d^{(n)}(H)=d^{(n)}(H,G)=d^{(n)}(G).$
## 2\. Proof of Main Theorems and Some Consequences
This section is devoted to describe our main results.
###### Lemma 2.1.
Assume that $G$ is a compact group, $H$ is a closed subgroup of $G$ and
$C_{G}([h_{1},...,h_{n}])$ is the centralizer of the commutator
$[h_{1},...,h_{n}]$ in $G$ for some elements $h_{1},...,h_{n}$ in $H$. Then
$d^{(n)}(H,G)=\int_{H}\ldots\left(\int_{H}\mu_{G}(C_{G}([h_{1},...,h_{n}]))d\mu_{H}(h_{1})\right)\ldots
d\mu_{H}(h_{n}),$
where
$\mu_{G}(C_{G}([h_{1},...,h_{n}]))=\int_{G}\chi_{{}_{D_{n+1}}}(h_{1},...,h_{n},g)d\mu_{G}(g),$
and $\chi_{{}_{D_{n+1}}}$ denotes the characteristic map of the set $D_{n+1}$.
###### Proof.
Since
$\mu_{G}(C_{G}([h_{1},...,h_{n}]))=\int_{G}\chi_{{}_{D_{n+1}}}(h_{1},...,h_{n},g)d\mu_{G}(g)$
we have by Fubini-Tonelli’s Theorem:
$\begin{array}[]{lcl}d^{(n)}(H,G)&=&(\mu^{n}_{H}\times\mu_{G})(D_{n+1})=\int_{H^{n}\times
G}\chi_{{}_{D_{n+1}}}(d\mu^{n}_{H}\times d\mu_{G})\vspace{.3cm}\\\
&=&\int_{H}...\left(\int_{H}\left(\int_{G}\chi_{{}_{D_{n+1}}}(h_{1},...,h_{n},g)d\mu_{G}(g)\right)d\mu_{H}(h_{1})\right)...\
d\mu_{H}(h_{n})\vspace{.3cm}\\\
&=&\int_{H}...\left(\int_{H}\mu(C_{G}([h_{1},...,h_{n}]))d\mu_{H}(h_{1})\right)...\
d\mu_{H}(h_{n}).\end{array}$
∎
We recall to convenience of the reader [4, Section 6, Theorem 3]. Let
$(\Omega_{1},\mathcal{A}_{1})$ and $(\Omega_{2},\mathcal{A}_{2})$ be two
measurable spaces. The mapping $X:\Omega_{1}\rightarrow\Omega_{2}$ is said to
be a $measurable$ $transformation$ from $(\Omega_{1},\mathcal{A}_{1})$ to
$(\Omega_{2},\mathcal{A}_{2})$ if
$X^{-1}(\mathcal{A}_{2})\subseteq\mathcal{A}_{1}.$ Now for a measurable
transformation $X$ from a measure space $(\Omega_{1},\mathcal{A}_{1},\mu)$ to
a measurable space $(\Omega_{2},\mathcal{A}_{2})$ one can check that the
measure $\mu$ induces a measure $\nu$ on $\mathcal{A}_{2}$ via
$\nu\\{A\\}=\mu\\{X^{-1}(A)\\},$ for all $A\in\mathcal{A}_{2}.$ This induced
measure is denoted by $\mu X^{-1}.$ Let $\phi$ be an isomorphism from a
compact group $G_{1}$ with Haar measure $\mu_{G_{1}}$ onto a compact group
$G_{2}$ with Haar measure $\mu_{G_{2}}$. It is clear that $\phi$ is a
measurable transformation from $G_{1}$ to $G_{2}$, then by the uniqueness of
the Haar measure on compact groups, we have that
$\mu_{G_{2}}=\mu_{G_{1}}\phi^{-1}$ and so
$(*)\hskip
56.9055pt\int_{G_{2}}fd\mu_{G_{2}}=\int_{G_{1}}(f\phi)d\mu_{G_{1}}.$
###### Proof of Theorem 1.2.
Let $(\alpha,\beta)$ be a relative $n$-isoclinism from $(H_{1},G_{1})$ to
$(H_{2},G_{2})$ as in Definition 1.1. Assume that $n=1$. Let $\mu_{H_{1}}$ be
a normalized Haar measure on $H_{1}$, $\mu_{H_{2}}$ on $H_{2}$, $\mu_{G_{1}}$
on $G_{1}$, $\mu_{G_{2}}$ on $G_{2}$, $\nu_{H_{1}}$ on $H_{1}/(H_{1}\cap
Z(G_{1}))$, $\nu_{H_{2}}$ on $H_{2}/(H_{2}\cap Z(G_{2}))$, $\nu_{G_{1}}$ on
$G_{1}/Z(G_{1})$ and $\nu_{G_{2}}$ on $G_{2}/Z(G_{2})$. Note that these
requirements are well posed, because $H_{1}$ is a closed subgroup of $G_{1}$
and $H_{2}$ is a closed subgroup of $G_{2}$.
With obvious meaning of symbols, we have
$\begin{array}[]{lcl}d(H_{1},G_{1})&=&\int_{H_{1}}\int_{G_{1}}\chi_{{}_{D_{2}}}(h_{1},g_{1})d\mu_{G_{1}}(g_{1})d\mu_{H_{1}}(h_{1})\vspace{.3cm}\\\
&=&\int_{H_{1}}\left(\int_{\frac{G_{1}}{Z(G_{1})}}\left(\int_{Z(G_{1})}\chi_{{}_{D_{2}}}(h_{1},z_{1}g_{1})d\mu_{Z(G_{1})}(z_{1})\right)d\nu_{G_{1}}(\bar{g_{1}})\right)d\mu_{H_{1}}(h_{1})\vspace{.3cm}\\\
&=&\int_{H_{1}}\left(\int_{\frac{G_{1}}{Z(G_{1})}}\left(\int_{Z(G_{1})}\chi_{{}_{D_{2}}}(h_{1},g_{1})d\mu_{Z(G_{1})}(z_{1})\right)d\nu_{G_{1}}(\bar{g_{1}})\right)d\mu_{H_{1}}(h_{1})\vspace{.3cm}\\\
&=&\int_{H_{1}}\left(\int_{\frac{G_{1}}{Z(G_{1})}}\chi_{{}_{D_{2}}}(h_{1},g_{1})\left(\int_{Z(G_{1})}d\mu_{Z(G_{1})}(z_{1})\right)d\nu_{G_{1}}(\bar{g_{1}})\right)d\mu_{H_{1}}(h_{1})\vspace{.3cm}\\\
&=&\int_{H_{1}}\left(\int_{\frac{G_{1}}{Z(G_{1})}}\chi_{{}_{D_{2}}}(h_{1},g_{1})d\nu_{G_{1}}(\bar{g_{1}})\right)d\mu_{H_{1}}(h_{1})\vspace{.3cm}\\\
\end{array}$
now by Fubini-Tonelli’s Theorem we have
$\begin{array}[]{lcl}=\int_{\frac{G_{1}}{Z(G_{1})}}\left(\int_{H_{1}}\chi_{{}_{D_{2}}}(h_{1},g_{1})d\mu_{H_{1}}(h_{1})\right)d\nu_{G_{1}}(\bar{g_{1}})\vspace{.3cm}\\\
\end{array}$
then we may proceed as before:
$\begin{array}[]{lcl}=\int_{\frac{G_{1}}{Z(G_{1})}}\left(\int_{\frac{H_{1}}{H_{1}\cap
Z(G_{1})}}\left(\int_{H_{1}\cap
Z(G_{1})}\chi_{{}_{D_{2}}}(h_{1}\zeta_{1},g_{1})d\mu_{Z(G_{1})\cap
H_{1}}(\zeta_{1})\right)d\nu_{H_{1}}(\bar{h_{1}})\right)d\nu_{G_{1}}(\bar{g_{1}})\vspace{.3cm}\\\
=\int_{\frac{G_{1}}{Z(G_{1})}}\left(\int_{\frac{H_{1}}{H_{1}\cap
Z(G_{1})}}\left(\int_{H_{1}\cap
Z(G_{1})}\chi_{{}_{D_{2}}}(h_{1},g_{1})d\mu_{Z(G_{1})\cap
H_{1}}(\zeta_{1})\right)d\nu_{H_{1}}(\bar{h_{1}})\right)d\nu_{G_{1}}(\bar{g_{1}})\vspace{.3cm}\\\
=\int_{\frac{G_{1}}{Z(G_{1})}}\left(\int_{\frac{H_{1}}{H_{1}\cap
Z(G_{1})}}\chi_{{}_{D_{2}}}(h_{1},g_{1})\left(\int_{H_{1}\cap
Z(G_{1})}d\mu_{Z(G_{1})\cap
H_{1}}(\zeta_{1})\right)d\nu_{H_{1}}(\bar{h_{1}})\right)d\nu_{G_{1}}(\bar{g_{1}})\vspace{.3cm}\\\
=\int_{\frac{G_{1}}{Z(G_{1})}}\left(\int_{\frac{H_{1}}{H_{1}\cap
Z(G_{1})}}\chi_{{}_{D_{2}}}(h_{1},g_{1})d\nu_{H_{1}}(\bar{h_{1}})\right)d\nu_{G_{1}}(\bar{g_{1}})\vspace{.3cm}\\\
\end{array}$
now the commutativity of the diagram in Definition 1.1 and the fact that
$\beta$ is an isomorphism of compact groups allow us to write
$\begin{array}[]{lcl}&=&\int_{\frac{G_{1}}{Z(G_{1})}}\left(\int_{\frac{H_{1}}{H_{1}\cap
Z(G_{1})}}\chi_{{}_{D_{2}}}(\gamma(1,H_{1},G_{1})(\bar{h_{1}},\bar{g_{1}}))d\nu_{H_{1}}(\bar{h_{1}})\right)d\nu_{G_{1}}(\bar{g_{1}})\vspace{.3cm}\\\
&=&\int_{\frac{G_{1}}{Z(G_{1})}}\left(\int_{\frac{H_{1}}{H_{1}\cap
Z(G_{1})}}\chi_{{}_{D_{2}}}(\beta(\gamma(1,H_{1},G_{1})(\bar{h_{1}},\bar{g_{1}})))d\nu_{H_{1}}(\bar{h_{1}})\right)d\nu_{G_{1}}(\bar{g_{1}})\vspace{.3cm}\\\
&=&\int_{\frac{G_{1}}{Z(G_{1})}}\left(\int_{\frac{H_{1}}{H_{1}\cap
Z(G_{1})}}\chi_{{}_{D_{2}}}(\gamma(1,H_{2},G_{2})(\alpha^{2}(\bar{h_{1}},\bar{g_{1}})))d\nu_{H_{1}}(\bar{h_{1}})\right)d\nu_{G_{1}}(\bar{g_{1}})\vspace{.3cm}\\\
\end{array}$
now we may apply $(*)$ above so
$\begin{array}[]{lcl}&=&\int_{\frac{G_{2}}{Z(G_{2})}}\left(\int_{\frac{H_{2}}{H_{2}\cap
Z(G_{2})}}\chi_{{}_{D_{2}}}(\gamma(1,H_{2},G_{2})(\bar{k_{1}},\bar{g_{2}}))d\nu_{H_{2}}(\bar{k_{1}})\right)d\nu_{G_{2}}(\bar{g_{2}})\vspace{.3cm}\\\
&=&\int_{\frac{G_{2}}{Z(G_{2})}}\left(\int_{\frac{H_{2}}{H_{2}\cap
Z(G_{2})}}\chi_{{}_{D_{2}}}(k_{1},g_{2})d\nu_{H_{2}}(\bar{k_{1}})\right)d\nu_{G_{2}}(\bar{g_{2}})\vspace{.3cm}\\\
\end{array}$
then we may proceed as before
$\begin{array}[]{lcl}&=&\int_{\frac{G_{2}}{Z(G_{2})}}\left(\int_{\frac{H_{2}}{H_{2}\cap
Z(G_{2})}}\chi_{{}_{D_{2}}}(k_{1},g_{2})\left(\int_{H_{2}\cap
Z(G_{2})}d\mu_{Z(G_{2})\cap
H_{2}}(\zeta_{2})\right)d\nu_{H_{2}}(\bar{k_{1}})\right)d\nu_{G_{2}}(\bar{g_{2}})\vspace{.3cm}\\\
&=&\int_{\frac{G_{2}}{Z(G_{2})}}\left(\int_{\frac{H_{2}}{H_{2}\cap
Z(G_{2})}}\left(\int_{H_{2}\cap
Z(G_{2})}\chi_{{}_{D_{2}}}(k_{1},g_{2})d\mu_{Z(G_{2})\cap
H_{2}}(\zeta_{2})\right)d\nu_{H_{2}}(\bar{k_{1}})\right)d\nu_{G_{2}}(\bar{g_{2}})\vspace{.3cm}\\\
&=&\int_{\frac{G_{2}}{Z(G_{2})}}\left(\int_{\frac{H_{2}}{H_{2}\cap
Z(G_{2})}}\left(\int_{H_{2}\cap
Z(G_{2})}\chi_{{}_{D_{2}}}(k_{1}\zeta_{2},g_{2})d\mu_{Z(G_{2})\cap
H_{2}}(\zeta_{2})\right)d\nu_{H_{2}}(\bar{h_{2}})\right)d\nu_{G_{2}}(\bar{g_{2}})\vspace{.3cm}\\\
&=&\int_{\frac{G_{2}}{Z(G_{2})}}\left(\int_{H_{2}}\chi_{{}_{D_{2}}}(k_{1},g_{2})d\mu_{H_{2}}(k_{1})\right)d\nu_{G_{2}}(\bar{g_{2}})\vspace{.3cm}\\\
&=&\int_{H_{2}}\left(\int_{\frac{G_{2}}{Z(G_{2})}}\chi_{{}_{D_{2}}}(k_{1},g_{2})d\nu_{G_{2}}(\bar{g_{2}})\right)d\mu_{H_{2}}(k_{1})\vspace{.3cm}\\\
\end{array}$
$\begin{array}[]{lcl}&=&\int_{H_{2}}\left(\int_{\frac{G_{2}}{Z(G_{2})}}\left(\int_{Z(G_{2})}\chi_{{}_{D_{2}}}(k_{1},g_{2})d\mu_{Z(G_{2})}(z_{2})\right)d\nu_{G_{2}}(\bar{g_{2}})\right)d\mu_{H_{2}}(k_{1})\vspace{.3cm}\\\
&=&\int_{H_{2}}\left(\int_{\frac{G_{2}}{Z(G_{2})}}\left(\int_{Z(G_{2})}\chi_{{}_{D_{2}}}(k_{1},g_{2}z_{2})d\mu_{Z(G_{2})}(z_{2})\right)d\nu_{G_{2}}(\bar{g_{2}})\right)d\mu_{H_{2}}(k_{1})\vspace{.3cm}\\\
&=&\int_{H_{2}}\int_{G_{2}}\chi_{{}_{D_{2}}}(k_{1},g_{2})d\mu_{G_{2}}(g_{2})d\mu_{H_{2}}(k_{1})\vspace{.3cm}\\\
&=&d(H_{2},G_{2}).\vspace{.3cm}\\\ \end{array}$
The result follows in this case.
If we assume that $n>1$, then we may repeat step by step the previous
argument, writing the corresponding integrals for the ($n+1$)-tuple
$(h_{1},h_{2},\ldots,h_{n},g_{1})$ of $H^{n}_{1}\times G_{1}$ and for the
($n+1$)-tuple $(k_{1},k_{2},\ldots,k_{n},g_{2})$ of $H^{n}_{2}\times G_{2}$.
Therefore the result follows completely.∎
The proof of Theorem 1.3 is mainly due to the next two lemmas, where the
notations of Definition 1.1 have been adopted. Note that there is not
hypothesis of compactness in the following two lemmas.
###### Lemma 2.2.
$(H_{1},G_{1})\ _{\widetilde{n}}\ (H_{2},G_{2})$ if and only if there exist
normal subgroups $N_{1}\leq Z_{n}(G_{1})$, $N_{2}\leq Z_{n}(G_{2})$ and two
isomorphisms $\alpha$ and $\beta$ such that $\alpha:G_{1}/N_{1}\rightarrow
G_{2}/N_{2},$ $\beta:[_{n}H_{1},G_{1}]\rightarrow[_{n}H_{2},G_{2}],$
$\alpha(H_{1}/(N_{1}\cap H_{1}))=H_{2}/(N_{2}\cap H_{2})$ for all $g_{1}\in
G_{1}$ and $h_{i}\in H_{1},$
$\beta([h_{1},...,h_{n},g_{1}])=[k_{1},...,k_{n},g_{2}],$ where
$g_{2}\in\alpha(g_{1}N_{1})$, $k_{i}\in\alpha(h_{i}(N_{1}\cap H_{1}))$ and
$1\leq i\leq n$.
###### Proof.
Assume that $(H_{1},G_{1})\ _{\widetilde{n}}\ (H_{2},G_{2})$. We may write
$N_{1}=Z_{n}(G_{1})$ and $N_{2}=Z_{n}(G_{2})$. From this, the result follows.
Conversely, one can show easily that
$\alpha(Z_{n}(G_{1})/N_{1})=Z_{n}(G_{2})/N_{2}$, therefore by third
isomorphism theorem, $\alpha$ induces the isomorphism $\alpha^{\prime}$ from
$G_{1}/Z_{n}(G_{1})$ to $G_{2}/Z_{n}(G_{2})$ by the rule
$\alpha^{\prime}(g_{1}Z_{n}(G_{1}))=g_{2}Z_{n}(G_{2})$, where
$g_{2}\in\alpha(g_{1}N_{1})$. Furthermore
$\alpha^{\prime}(h_{1}(Z_{n}(G_{1})\cap H_{1}))=h_{2}(Z_{n}(G_{2})\cap H_{2})$
in which $h_{2}\in\alpha(h_{1}(N_{1}\cap H_{1}))$. Hence
$(\alpha^{\prime},\beta)$ is a relative $n$-isoclinism from $(H_{1},G_{1})$ to
$(H_{2},G_{2})$.∎
###### Lemma 2.3.
Let $H$ be a subgroup of a group $G=HZ_{n}(G)$. Then
$(H,H)\ _{\widetilde{n}}\ (H,G)\ _{\widetilde{n}}\ (G,G).$
###### Proof.
We want to show that $(H,H)\ _{\widetilde{n}}\ (H,G)$. Let $G=HZ_{n}(G)$. We
may easily see that $Z_{n}(H)=Z_{n}(G)\cap H$. Thus
$H/Z_{n}(H)=H/(Z_{n}(G)\cap H)$ is isomorphic to
$HZ_{n}(G)/Z_{n}(G)=G/Z_{n}(G)$.
Therefore $\alpha:H/Z_{n}(H)\rightarrow G/Z_{n}(G)$ is an isomorphism which is
induced by the inclusion $i:H\rightarrow G.$ Furthermore, we can consider
$\alpha$ as isomorphism from $H/Z_{n}(H)$ to $H/(Z_{n}(G)\cap H)$.
On another hand, $[_{n}H,G]=[_{n}H,HZ_{n}(G)]=\gamma_{n+1}(H)$. By Lemma 2.2,
the pair $(\alpha,1_{\gamma_{n+1}(H)})$ allows us to state that $(H,H)\
_{\widetilde{n}}\ (H,G)$.
The remaining cases $(H,H)\ _{\widetilde{n}}\ (G,G)$ and $(H,H)\
_{\widetilde{n}}\ (H,G)$ follow by a similar argument. ∎
###### Proof of Theorem 1.3.
It follows from Theorem 1.2 and Lemma 2.3. ∎
We end with some consequences of the main results. We recall the following
lemma, which is used in many proofs of [5, 7, 8, 9].
###### Lemma 2.4 (See [6], Lemma 3.1).
Let $k$ be a positive integer. If $H$ is a closed subgroup of a compact group
$G$, then
$\mu_{G}(H)=\left\\{\begin{array}[]{lcl}\frac{1}{k},&&|G:H|=k\\\
0,&&|G:H|=\infty.\end{array}\right.$
###### Proposition 2.5.
Let $G$ be a compact group and $G/Z(G)$ be a $p$-group of order $p^{k}$, where
$p$ is a prime and $k\geq 2$ is an integer. Then
$d(G)\leq\frac{p^{k}+p-1}{p^{k+1}}$ and the equality holds if $G$ is isoclinic
to an extra-special $p$-group of order $p^{k+1}$.
###### Proof.
By Lemma 2.4 we have $\mu_{G}(Z(G))=\frac{1}{p^{k}}$ and
$\mu_{G}(C_{G}(x))\leq\frac{1}{p}$ for all $x\notin Z(G)$. Therefore
$\begin{array}[]{lcl}d(G)&=&\int_{G}\mu_{G}(C_{G}(x))d\mu_{G}(x)=\int_{Z(G)}\mu_{G}(C_{G}(x))+\int_{G-Z(G)}\mu_{G}(C_{G}(x))\vspace{.3cm}\\\
&\leq&\frac{1}{p}+(\frac{p-1}{p})\mu_{G}(Z(G))=\frac{1}{p}+(\frac{p-1}{p})\frac{1}{p^{k}}=\frac{p^{k}+p-1}{p^{k+1}}.\end{array}$
By Theorem 1.2, there is no loss of generality in assuming that $G$ is an
extra-special $p$-group of order $p^{k+1}$. For all $x\not\in Z(G)$ we may
consider the epimorphism $\varphi_{x}:G\rightarrow G^{\prime}$ defined by
$\varphi_{x}(y)=[x,y]$ in which the kernel is $C_{G}(x)$. Therefore
$\mu_{G}(C_{G}(x))=\frac{1}{p}$ and so the above inequality becomes
$d(G)=\frac{p^{k}+p-1}{p^{k+1}}$. ∎
An easy consequence has been shown below.
###### Proposition 2.6.
If $G$ is a compact group in which $Z(G)$ has infinite index in $G$, then
$d(G)\leq\frac{1}{2}$. In particular, if $G$ is isoclinic to an extra-special
$p$-group for some prime $p$, then $d(G)=\frac{1}{p}$.
###### Proof.
By Lemma 2.4 we have $\mu_{G}(Z(G))=0$ and $\mu_{G}(C_{G}(x))\leq\frac{1}{2}$
for all $x\not\in Z(G)$. From this, we have
$\begin{array}[]{lcl}d(G)=\int_{G}\mu_{G}(C_{G}(x))d\mu_{G}(x)\leq\frac{1}{2}+\frac{1}{2}\mu_{G}(Z(G))=\frac{1}{2}.\end{array}$
Following the same argument, if $G/Z(G)$ is an infinite $p$-group, then
$d(G)\leq\frac{1}{p}$. In particular, if $G$ is isoclinic to an extra-special
$p$-group, then we may argue as in the last part of the proof of Proposition
2.5, concluding that $\mu_{G}(C_{G}(x))=\frac{1}{p}$ for every $x\not\in
Z(G)$. Therefore the equality $d(G)=\frac{1}{p}$ holds and the result follows.
∎
## References
* [1] J. C. Bioch, On $n$-isoclinic groups. Indag. Math. 38 (1976), 400–407.
* [2] J. C. Bioch and R. W. van der Waall, Monomiality and isoclinism of groups. J. Reine Ang. Math. 298 (1978), 74–88.
* [3] K. Chiti, M. R. R. Moghaddam and A. R. Salemkar, $n$-isoclinism classes and $n$-nilpotency degree of finite groups. Algebra Coll. 12 (2005), 225–261.
* [4] Y. S. Chow and H. Teicher, Probability Theory: Independence, Interchangeability, Martingales. Springer, Berlin, 1988\.
* [5] A. Erfanian, P. Lescot and R. Rezaei, On the relative commutativity degree of a subgroup of a finite group. Comm. Algebra 35 (2007), 4183–4197.
* [6] A. Erfanian and R. Rezaei, On the commutativity degree of compact groups. Arch. Math. (Basel) 93 (2009), 201–212.
* [7] P. X. Gallagher, The number of conjugacy classes in a finite group. Math. Z. 118 (1970), 175–179.
* [8] R. M. Guralnick and G. R. Robinson, On the commuting probability in finite groups. J. Algebra 300 (2006), 509–528.
* [9] W. H. Gustafson, What is the probability that two groups elements commute?. Amer. Math. Monthly 80 (1973), 1031–1304.
* [10] M. Hall and J. K. Senior, The Groups of Order $2^{n}\ (n\leq 6)$. Macmillan, New York, 1964.
* [11] P. Hall, The classification of prime-power groups. J. Reine Ang. Math. 182 (1940), 130–141.
* [12] N. S. Hekster, On the structure of n-isoclinism classes of groups. J. Pure Appl. Algebra 40 (1986), 63–85.
* [13] K. H. Hofmann and S. A. Morris, The Structure of Compact Groups. W. de Gruyter, Berlin, 1998.
* [14] P. Lescot, Isoclinism classes and commutativity degrees of finite groups. J. Algebra 177 (1995), 847–869.
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|
arxiv-papers
| 2010-03-11T10:21:03 |
2024-09-04T02:49:09.025223
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rashid Rezaei (University of Malayer, Malayer, Iran) and Francesco G.\n Russo (Universita' degli Studi di Palermo, Palermo, Italy)",
"submitter": "Francesco G. Russo",
"url": "https://arxiv.org/abs/1003.2297"
}
|
1003.2306
|
# Relative $n$-isoclinism Classes and Relative $n$-th Nilpotency Degree of
Finite Groups
Ahmad Erfanian Department of Mathematics, Faculty of Mathematical Sciences
Ferdowsi University of Mashhad, Mashhad, Iran. erfanian@math.um.ac.ir ,
Rashid Rezaei Department of Mathematics, Faculty of Mathematical Sciences
University of Malayer
Post Box: 657719–95863, Malayer, Iran ras$\\_$rezaei@yahoo.com and Francesco
G. Russo Department of Mathematics
University of Naples Federico II
via Cinthia, 80126, Naples, Italy francesco.russo@dma.unina.it
###### Abstract.
The purpose of the present paper is to consider the notion of isoclinism
between two finite groups and its generalization to $n$-isoclinism, introduced
by J. C. Bioch in 1976. A weaker form of $n$-isoclinism, called relative
$n$-isoclinism, will be discussed. This will allow us to improve some
classical results in literature. We will point out the connections between a
relative $n$-isoclinism and the notions of commutativity degree, $n$-th
nilpotency degree and relative $n$-th nilpotency degree, which arouse interest
in the classification of groups of prime power order in the last years.
###### Key words and phrases:
Commutativity degree; relative commutativity degree; $n$-th nilpotency degree;
relative $n$-isoclinism.
Mathematics Subject Classification 2010: Primary: 20D60, 20P05; Secondary:
20D08, 20D15.
## 1\. Introduction
The notion of isoclinism was introduced by P. Hall in 1940 in [8], looking for
a satisfactory classification of groups of prime power order. However, the
notion of isoclinism holds both in finite and infinite group theory. It is
obvious that an isomorphism is an equivalence relation in the class of all
groups and this allows us to classify two groups. An isoclinism is a more
general equivalence relation in the class of all groups and it is easy to see
that two abelian groups fall into the same equivalence class with respect to
isoclinisms (see [1, Theorem 1.4] or [8] or [9]). Roughly speaking, two groups
$H$ and $K$ are isoclinic if their central quotients $H/Z(H)$, $K/Z(K)$ are
isomorphic and if their commutator subgroups $H^{\prime}$, $K^{\prime}$ are
isomorphic. If we look at the construction of the finite extra-special
$2$-groups (see [15, pp.145–147]) and at the construction of the quaternion
groups (see [15, pp.140–141]), then we will find such groups in the situation
which has been just described. For instance, we may think to the dihedral
group $D_{8}$ of order 8 and to the quaternion group $Q_{8}$ of order 8. We
note that both $D_{8}/Z(D_{8})$, $Q_{8}/Z(Q_{8})$ are isomorphic and
$D^{\prime}_{8}$, $Q^{\prime}_{8}$ are isomorphic.
Situations as we just mentioned have been largely studied in literature and it
is well known the role of the isoclinism in the classification of the finite
groups of prime power order as we can see in [3, 7, 8, 9, 10, 11, 12, 13, 14].
Such references and many other works of the same authors investigate those
group theoretical properties which are invariant under isoclinism. For
example, J. C. Bioch and R. W. van der Waall [2] proved the invariance under
isoclinism of the following hierarchy of classes of finite groups:
abelian $<$ nilpotent $<$ supersoluble $<$ strongly-monomial $<$ monomial $<$
soluble.
A successive contribution in the classification of the groups of prime power
order with respect to the notion of isoclinism was given already by P. Hall.
He introduced the varieties of groups in [9]. We can easily see that the
variety of all trivial groups and the variety of all abelian groups yield to
the notion of isomorphism and isoclinism, respectively (see [1, 2, 10]).
Varieties which extend the variety of abelian groups yield to notions which
extend that of isoclinism . In this context, J. C. Bioch took the variety of
all nilpotent groups of class at most $n$ ($n$ is a positive integer) and
introduced the notion of $n$-isoclinism of groups (see [1]). Successive
contributions to the works of J. C. Bioch were given by [10].
Recently, some interesting connections have been found between the notion of
isoclinism and the probability that two randomly chosen elements of a group
permute. In particular, probabilistic bounds have been found in [3, 4, 5, 6,
12, 13, 16], giving restrictions in terms of the structure of a finite group.
This paper follows such a line of research. Our main results extend both some
classical results in literature on isoclinic groups as [1, 2, 8, 9, 10, 11]
and some of the results in [3, 4, 5, 6, 12, 13, 16]. Section 2 recalls the
technical definitions which we will use in Section 3 for getting to our main
results. Some open questions both in infinite and finite case have been shown
at the end of Section 3. Our notation and terminology is standard and referred
to [15].
## 2\. Preliminaries
In this section, we recall two important definitions. Firstly, we recall the
notion of commutativity degree and its generalizations. Secondly, we recall
the notion of isoclinism, and its generalizations. They can be found in [1, 2,
3, 4, 8, 9, 10, 13]. From this point, the symbol $n$ will denote always a
positive integer.
For any finite group $G$, the commutativity degree of $G$ is defined as the
number of pairs $(x,y)$ in $G\times G$ such that $xy=yx$, divided by
$|G|^{2}$. We denote it by $d(G)$. In symbols:
$d(G)=\displaystyle\frac{|\\{(x,y)\in G^{2}:[x,y]=1\\}|}{|G|^{2}}.\hskip
147.95424pt(2.1)$
It is clear that $d(G)=1$ if and only if $G$ is abelian; and for any finite
non-abelian group $G$, $d(G)\leq\displaystyle\frac{5}{8}$. Furthermore, this
bound is achieved if and only if $G/Z(G)$ is a $2$-elementary abelian group of
rank $2$. Further details can be found in [4, Theorem 2] and [6,
Introduction].
A way of extending $d(G)$ to $(n+1)$-tuples $(x_{1},x_{2},...,x_{n+1})$ in
$G^{n+1}$ is to consider the number of such $(n+1)$-tuples with the property
that $[x_{1},x_{2},...,x_{n+1}]=1$, divided by the order of $G^{n+1}$. This is
denoted by $d^{(n)}(G)$ and called the $n$-th nilpotency degree of $G$. In
symbols:
$d^{(n)}(G)=\displaystyle\frac{|\\{(x_{1},x_{2},...,x_{n+1})\in
G^{n+1}:[x_{1},x_{2},...,x_{n+1}]=1\\}|}{|G|^{n+1}}.\hskip 28.45274pt(2.2)$
It is obvious that $d^{(n)}(G)$ extends $d(G)$ and $d^{(1)}(G)=d(G)$ (see [13,
Section 4] and [3] for more details).
Now, we will introduce two further generalizations of $d(G)$. If we take a
subgroup $H$ of a finite group $G$, we may define the relative commutativity
degree of $H$ in $G$, denoted by $d(H,G)$, as
$d(H,G)=\displaystyle\frac{|\\{(h,g)\in H\times G:[h,g]=1\\}|}{|H||G|}\hskip
130.88284pt(2.3)$
and we may define the relative n-th nilpotency degree of $H$ in $G$, denoted
by $d^{(n)}(H,G)$, as
$d^{(n)}(H,G)=\displaystyle\frac{|\\{(x_{1},x_{2},...,x_{n},g)\in H^{n}\times
G:[x_{1},x_{2},...,x_{n},g]=1\\}|}{|H|^{n}|G|}.\hskip 14.22636pt(2.4)$
If $H=G$, then $d(H,G)=d(G)$ and $d^{(n)}(H,G)=d^{(n)}(G)$ as we see in [4,
Section 3].
The second part of this section is devoted to define isoclinism,
$n$-isoclinism, relative isoclinism and relative $n$-isoclinism. We start with
defining isoclinism between two groups $H$ and $G$, following [8] and recent
considerations in [13, Section 2].
Definition 2.1. Let $G$ and $H$ be two groups; a pair $(\varphi,\psi)$ is
called an isoclinism from $G$ to $H$ if :
(i) $\varphi$ is an isomorphism from $\displaystyle\frac{G}{Z(G)}$ to
$\displaystyle\frac{H}{Z(H)};$
(ii) $\psi$ is an isomorphism from $G^{\prime}$ to $H^{\prime}$ ;
(iii) the following diagram is commutative:
$\begin{CD}\displaystyle\frac{G}{Z(G)}\times\frac{G}{Z(G)}@>{\varphi}>{}>\displaystyle\frac{H}{Z(H)}\times\frac{H}{Z(H)}\\\
@V{a_{G}}V{}V@V{a_{H}}V{}V\\\ G^{\prime}@>{\psi}>{}>H^{\prime},\end{CD}$
where $a_{G}(g_{1}Z(G),g_{2}Z(G))=[g_{1},g_{2}]$ and
$a_{H}(h_{1}Z(H),h_{2}Z(H))=[h_{1},h_{2}]$, for each $g_{1},g_{2}\in G$ and
$h_{1},h_{2}\in H$.
If there is an isoclinism from $G$ to $H$, we say that $G$ and $H$ are
$isoclinic$, writing briefly $G\sim H$. It can be easily checked that the
relation $\sim$ given in Definition 2.1 is an equivalence relation. Moreover,
it is obvious that if $G$ and $H$ are isomorphic, then they are isoclinic. But
the converse is not true. By a simple calculation, one may easily see that
$Q_{8}$ and $D_{8}$ are isoclinic but not isomorphic.
A simple relation between isoclinism and commutativity degree is that two
isoclinic finite groups have the same commutativity degree as we see in [13,
Lemma 2.4]. Another relation between finite groups of commutativity degree
equal to $\frac{1}{2}$ and groups which are isoclinic to the symmetric group
$S_{3}$ has been given in [13, Theorem 3.1].
Now, we give the notion of $n$-isoclinism between two groups, generalizing
Definition 2.1. This notion has been extensively investigated in [10, Sections
3, 5, 7].
Definition 2.2. Let $G$ and $H$ be two groups; a pair $(\alpha,\beta)$ is
called n-isoclinism from $G$ to $H$ if :
(i) $\alpha$ is an isomorphism from $\displaystyle\frac{G}{Z_{n}(G)}$ to
$\displaystyle\frac{H}{Z_{n}(H)}$;
(ii) $\beta$ is an isomorphism from $\gamma_{n+1}(G)$ to $\gamma_{n+1}(H)$;
(iii) the following diagram is commutative:
$\begin{CD}\displaystyle\frac{G}{Z_{n}(G)}\times...\times\frac{G}{Z_{n}(G)}@>{\alpha^{n+1}}>{}>\displaystyle\frac{H}{Z_{n}(H)}\times...\times\frac{H}{Z_{n}(H)}\vspace{.3cm}\\\
@V{\gamma(n,G)}V{}V@V{\gamma(n,H)}V{}V\vspace{.3cm}\\\
\gamma_{n+1}(G)@>{\beta}>{}>\gamma_{n+1}(H),\end{CD}$
where
$\gamma(n,G)((g_{1}Z_{n}(G),...,g_{n}Z_{n}(G),g_{n+1}Z_{n}(G)))=[g_{1},...,g_{n},g_{n+1}]$
and
$\gamma(n,H)((h_{1}Z_{n}(H),...,h_{n}Z_{n}(H),h_{n+1}Z_{n}(H)))=[h_{1},...,h_{n},h_{n+1}],$
for each $g_{1},...,g_{n},g_{n+1}\in G$ and $h_{1},...,h_{n},h_{n+1}\in H$.
One may easily check that the maps $\gamma(n,G)$ and $\gamma(n,H)$ in
Definition 2.2 are well-defined.
If there is an $n$-isoclinism between $G$ and $H$, we say that $G$ and $H$ are
$n$-$isoclinic$, writing briefly by $G\ _{\widetilde{n}}\ H$. It is clear that
$\ {}_{\widetilde{{}_{1}}}$ and $\sim$ coincide, that is, a $1$-isoclinism is
an isoclinism. Follows from Definition 2.2 that ${}_{\widetilde{n}}$ is an
equivalence relation (see [1] or [10, Section 3]). If $G$ and $H$ are
$n$-isoclinic, then they are $(n+1)$-isoclinic by [10, Theorem 5.2]. This will
be extended in Section 3.
Note that our terminology can be found in [1, 2, 7, 8, 9, 11], as most of the
terminology in the present section.
We know that Definitions 2.1, 2.2 and Equations (2.1), (2.2) are correlated by
results as [13, Lemma 2.4] or [3, Theorem B]. Indeed, [3, Theorem B] states
that two $n$-isoclinic groups have the same $n$-th nilpotency degree.
Now, we state the last definition of the present section which recalls [10,
Definition 7.1] and extends Definitions 2.1 and 2.2.
Definition 2.3. Let $G_{1}$, $G_{2}$ be two groups, $H_{1}$ a subgroup of
$G_{1}$ and $H_{2}$ a subgroup of $G_{2}$. A pair $(\alpha,\beta)$ is said to
be a relative $n$-isoclinism from $(H_{1},G_{1})$ to $(H_{2},G_{2})$ if we
have the following conditions:
(i) $\alpha$ is an isomorphism from $G_{1}/Z_{n}(G_{1})$ to
$G_{2}/Z_{n}(G_{2})$ such that the restriction of $\alpha$ under
$H_{1}/(Z_{n}(G_{1})\cap H_{1})$ is an isomorphism from
$H_{1}/(Z_{n}(G_{1})\cap H_{1})$ to $H_{2}/(Z_{n}(G_{2})\cap H_{2})$;
(ii) $\beta$ is an isomorphism from $[_{n}H_{1},G_{1}]$ to
$[_{n}H_{2},G_{2}]$;
(iii) the following diagram is commutative:
$\begin{CD}\frac{H_{1}}{Z_{n}(G_{1})\cap
H_{1}}\times...\times\frac{H_{1}}{Z_{n}(G_{1})\cap
H_{1}}\times\frac{G_{1}}{Z_{n}(G_{1})}@>{\alpha^{n+1}}>{}>\frac{H_{2}}{Z_{n}(G_{2})\cap
H_{2}}\times...\times\frac{H_{2}}{Z_{n}(G_{2})\cap
H_{2}}\times\frac{G_{2}}{Z_{n}(G_{2})}\vspace{.3cm}\\\
@V{\gamma(n,H_{1},G_{1})}V{}V@V{\gamma(n,H_{2},G_{2})}V{}V\vspace{.3cm}\\\
[_{n}H_{1},G_{1}]@>{\beta}>{}>[_{n}H_{2},G_{2}].\end{CD}$
where
$\gamma(n,H_{1},G_{1})((h_{1}(Z_{n}(G_{1})\cap
H_{1}),...,h_{n}(Z_{n}(G_{1})\cap
H_{1}),g_{1}Z_{n}(G_{1})))=[h_{1},...,h_{n},g_{1}]$
and
$\gamma(n,H_{2},G_{2})((k_{1}(Z_{n}(G_{2})\cap
H_{2}),...,k_{n}(Z_{n}(G_{2})\cap
H_{2}),g_{2}Z_{n}(G_{2})))=[k_{1},...,k_{n},g_{2}],$
for each $h_{1},...,h_{n}\in H_{1}$, $k_{1},...,k_{n}\in H_{2}$, $g_{1}\in
G_{1}$, $g_{2}\in G_{2}$.
It is easy to check that the maps $\gamma(n,H_{1},G_{1})$ and
$\gamma(n,H_{2},G_{2})$ in Definition 2.2 are well-defined.
If Definition 2.3 is satisfied, we say that $(H_{1},G_{1})$ and
$(H_{2},G_{2})$ are $relative$ $n$-$isoclinic$, writing briefly
$\displaystyle(H_{1},G_{1})\ _{\widetilde{n}}\ (H_{2},G_{2}).$ Follows from
Definition 2.3 that ${}_{\widetilde{n}}$ is an equivalence relation (see also
[1] or [10, Section 3]). Note that there is not ambiguity if we use the symbol
${}_{\widetilde{n}}$ both in Definitions 2.2 and 2.3, because in Definition
2.2 ${}_{\widetilde{n}}$ is referred to groups and in Definition 2.3
${}_{\widetilde{n}}$ is referred to pairs of groups. We should also note that
if two pairs $(H_{1},G_{1})$ and $(H_{2},G_{2})$ are relative $n$-isoclinic,
then it is not necessary that $G_{1}$ and $G_{2}$ are $n$-isoclinic. For
instance, assume that $SL(2,5)$ is the special linear group of order 120 and
$PSL(2,5)$ is the projective special linear group of order 60. Then we can
observe that $(Z(SL(2,5)),SL(2,5))$ and $(1,PSL(2,5))$ are relative
1-isoclinic. On another hand, we note that $|[SL(2,5),SL(2,5)]|=120$ and
$|[PSL(2,5),PSL(2,5)]|=60$ so the relative 1-isoclinism is not an isoclinism.
Now, we end the present section, stating our main results.
Theorem A. Let $G_{1}$ and $G_{2}$ be two $n$-isoclinic groups. For every
subgroup $H_{1}$ of $G_{1}$, there exists a subgroup $H_{2}$ of $G_{2}$ such
that $H_{1}$ and $H_{2}$ are $n$-isoclinic.
Theorem B. Let $G_{1}$ and $G_{2}$ be two finite groups, $H_{1}$ be a subgroup
of $G_{1}$ and $H_{2}$ be a subgroup of $G_{2}$. If
$\displaystyle(H_{1},G_{1})\ _{\widetilde{n}}\ (H_{2},G_{2})$, then
$d^{(n)}(H_{1},G_{1})=d^{(n)}(H_{2},G_{2})$.
Theorem C. Let $G$ be a group and $H,N$ be subgroups of $G$ such that
$N\triangleleft G$ and $N\subseteq H$. Then for all $n\geq 0$,
$(\frac{H}{N},\frac{G}{N})\ _{\widetilde{n}}\
(\frac{H}{N\cap\gamma_{n+1}(G)},\frac{G}{N\cap\gamma_{n+1}(G)}).$
In particular, if $N\cap\gamma_{n+1}(G)=1$, then $(H,G)\ _{\widetilde{n}}\
(H/N,G/N)$.
Theorem D. Let $H$ be a subgroup of a finite group $G$. Then the following
statements are true.
(i) If $G=HZ_{n}(G)$, then $d^{(n)}(H)=d^{(n)}(H,G)=d^{(n)}(G)$.
(ii) $d^{(n)}(H,G)=d^{(n)}(\varphi(H),G)$ for every $\varphi\in Aut(G)$.
Theorem E. Let $H$ be a subgroup of a finite group $G$ such that
$Z(G)\subseteq H$. Then $d(H,G)=\frac{3}{4}$ if and only if $(H,G)$ and
$(\langle a\rangle,D_{8})$ are relative 1-isoclinic, where $\langle a\rangle$
is a subgroup of order $4$ in $D_{8}$.
## 3\. Proof of Main Theorems
This section is devoted to describe our main results.
Lemma 3.1. $\displaystyle(H_{1},G_{1})\ _{\widetilde{n}}\ (H_{2},G_{2})$ if
and only if there exist two isomorphisms $\alpha$ and $\beta$ such that
$\alpha:G_{1}/Z_{n}(G_{1})\rightarrow G_{2}/Z_{n}(G_{2}),$
$\beta:[_{n}H_{1},G_{1}]\rightarrow[_{n}H_{2},G_{2}],$
$\alpha(H_{1}/(Z_{n}(G_{1})\cap H_{1}))=H_{2}/(Z_{n}(G_{2})\cap H_{2})\ and\
for\ all\ g_{1}\in G_{1}\ and\ h_{i}\in H_{1},$
$\beta([h_{1},...,h_{n},g_{1}])=[k_{1},...,k_{n},g_{2}],$
where $g_{2}\in\alpha(g_{1}Z_{n}(G_{1}))$,
$k_{i}\in\alpha(h_{i}(Z_{n}(G_{1})\cap H_{1}))$ and $1\leq i\leq n$.
Proof. It is clear by Definition 2.3. $\square$
Lemma 3.2. Let $G_{1}$ and $G_{2}$ be two n-isoclinic groups. Then for every
subgroup $H_{1}$ of $G_{1}$ there exists a subgroup $H_{2}$ of $G_{2}$ such
that $(H_{1},G_{1})\ _{\widetilde{n}}\ (H_{2},G_{2}).$
Proof. Assume that $(\alpha,\beta)$ is an $n$-isoclinism from $G_{1}$ to
$G_{2}$ as in Definition 2.3. Put
$\begin{array}[]{lcl}H_{2}=\\{x\in G_{2}\ |\ \exists\ h\in H_{1}\ \
\textrm{s.t.}\ \ \alpha(hZ_{n}(G_{1}))=xZ_{n}(G_{2})\\},\end{array}$
where $H_{1}$ is an arbitrary subgroup of $G_{1}$. It is clear that $H_{2}$ is
a subgroup of $G_{2}$ and $\alpha(H_{1}/(Z_{n}(G_{1})\cap
H_{1}))=H_{2}/(Z_{n}(G_{2})\cap H_{2}).$ If $\beta^{\prime}$ denotes the
restriction of $\beta$ under $[_{n}H_{1},G_{1}]$, then $\beta^{\prime}$ is an
isomorphism from $[_{n}H_{1},G_{1}]$ to $[_{n}H_{2},G_{2}]$ and for all
$g_{1}\in G_{1}$, $h_{i}\in H_{1}$, $1\leq i\leq n$ we have
$\begin{array}[]{lcl}\beta^{\prime}([h_{1},...,h_{n},g_{1}])=\beta([h_{1},...,h_{n},g_{1}])\\\
=\beta\gamma(n,G_{1})(h_{1}Z_{n}(G_{1}),...,h_{n}Z_{n}(G_{1}),g_{1}Z_{n}(G_{1}))\\\
=\gamma(n,G_{2})\alpha^{n+1}(h_{1}Z_{n}(G_{1}),...,h_{n}Z_{n}(G_{1}),g_{1}Z_{n}(G_{1}))\\\
=[k_{1},...,k_{n},g_{2}],\end{array}$
where $g_{2}\in\alpha(g_{1}Z_{n}(G_{1})\cap H_{1})$ and
$k_{i}\in\alpha(h_{i}Z_{n}(G_{1})\cap H_{1})$. Therefore
$(\alpha,\beta^{\prime})$ is a relative $n$-isoclinism from $(H_{1},G_{1})$ to
$(H_{2},G_{2})$, by Lemma 3.1. $\square$
Now, we may prove Theorem A.
Proof of Theorem A. Assume that $G_{1}$ and $G_{2}$ are two $n$-isoclinic
groups and $H_{1}$ is an arbitrary subgroup of $G_{1}$. By Lemma 3.2, there
exists a subgroup $H_{2}$ of $G_{2}$ such that $(H_{1},G_{1})\
_{\widetilde{n}}\ (H_{2},G_{2}).$ Let $(\alpha,\beta)$ be a relative
$n$-isoclinism from $(H_{1},G_{1})$ to $(H_{2},G_{2})$ as in Definition 2.3.
We have the isomorphisms
$\alpha^{\prime}:H_{1}/(Z_{n}(G_{1})\cap H_{1})\rightarrow
H_{2}/(Z_{n}(G_{2})\cap H_{2}),$
and
$\beta^{\prime}:\gamma_{n+1}(H_{1})\rightarrow\gamma_{n+1}(H_{2}),$
where $\alpha^{\prime}$ is the restriction of $\alpha$ under
$H_{1}/(Z_{n}(G_{1})\cap H_{1})$ and $\beta^{\prime}$ is the restriction of
$\beta$ under $\gamma_{n+1}(H_{1})$. By [10, Lemma 3.11], if
$A=Z_{n}(G_{1})\cap H_{1}\subseteq Z_{n}(H_{1})$ and $B=Z_{n}(G_{2})\cap
H_{2}\subseteq Z_{n}(H_{2})$, then $H_{1}$ and $H_{2}$ are $n$-isoclinic.
$\square$
The following fact compares two relative $n$-isoclinisms.
Proposition 3.3. If $(H_{1},G_{1})\ _{\widetilde{n}}\ (H_{2},G_{2})$, then
$(H_{1},G_{1})\ _{\widetilde{{}_{n+1}}}\ (H_{2},G_{2})$.
Proof. Let $(\alpha,\beta)$ be a relative $n$-isoclinism from $(H_{1},G_{1})$
to $(H_{2},G_{2})$. Define
$\varphi:\displaystyle\frac{G_{1}}{Z_{n+1}(G_{1})}\rightarrow\frac{G_{2}}{Z_{n+1}(G_{2})}\
\ \ \textrm{and}\ \ \psi:[_{n+1}H_{1},G_{1}]\rightarrow[_{n+1}H_{2},G_{2}]$
by the rules $\varphi(g_{1}Z_{n+1}(G_{1}))=g_{2}Z_{n+1}(G_{2})$ and
$\psi([x,g_{1}])=[\beta(x),g_{2}]$, where
$\alpha(g_{1}Z_{n}(G_{1}))=g_{2}Z_{n}(G_{2})$ and $x\in\gamma_{n+1}(H).$ Now,
we claim that $\varphi$ and $\psi$ are isomorphisms.
Assume that $aZ_{n+1}(G_{1})=bZ_{n+1}(G_{1})$ for some $a,b\in G_{1}$, then
$\varphi(aZ_{n+1}(G_{1}))=xZ_{n+1}(G_{2})$ ,
$\varphi(bZ_{n+1}(G_{1}))=yZ_{n+1}(G_{2})$, where $x,y\in G_{2}$. Therefore
either
$\begin{array}[]{lcl}b^{-1}aZ_{n}(G_{1})\in\displaystyle\frac{Z_{n+1}(G_{1})}{Z_{n}(G_{1})}=Z(\frac{G_{1}}{Z_{n}(G_{1})})\end{array}$
or
$\begin{array}[]{lcl}\alpha(b^{-1}aZ_{n}(G_{1}))\in\alpha(Z(\displaystyle\frac{G_{1}}{Z_{n}(G_{1})}))=Z(\displaystyle\frac{G_{2}}{Z_{n}(G_{2})})=\displaystyle\frac{Z_{n+1}(G_{2})}{Z_{n}(G_{2})}\end{array}$
or
$\begin{array}[]{lcl}y^{-1}xZ_{n}(G_{2})\in\displaystyle\frac{Z_{n+1}(G_{2})}{Z_{n}(G_{2})}\end{array}$
i.e. $\varphi(aZ_{n+1}(G_{1}))=\varphi(bZ_{n+1}(G_{1}))$. Thus $\varphi$ is
well-defined.
If $\varphi(aZ_{n+1}(G_{1}))=\bar{1}$, then
$\varphi(aZ_{n+1}(G_{1}))=xZ_{n+1}(G_{2})=\bar{1}$ for some $x\in G_{2}$. Thus
$x\in Z_{n+1}(G_{2})$ and so,
$\begin{array}[]{lcl}\alpha(aZ_{n}(G_{1}))&=&xZ_{n}(G_{2})\in\displaystyle\frac{Z_{n+1}(G_{2})}{Z_{n}(G_{2})}=Z(\displaystyle\frac{G_{2}}{Z_{n}(G_{2})})\vspace{0.3cm}\\\
&=&\displaystyle\alpha(Z(\frac{G_{1}}{Z_{n}(G_{1})}))=\alpha(\frac{Z_{n+1}(G_{1})}{Z_{n}(G_{1})}).\end{array}$
Hence, $a\in Z_{n+1}(G_{1})$ and therefore $\varphi$ is injective.
From $\alpha$ surjective, we conclude that $\varphi$ is surjective. Therefore
$\varphi$ is an isomorphism and $\varphi(H_{1}/(Z_{n+1}(G_{1})\cap
H_{1}))=H_{2}/Z_{n+1}(G_{2})\cap H_{2}$.
Since $\beta$ is an isomorphism, $\psi$ is a monomorphism. $\psi$ is
surjective because, if $[k_{1},...,k_{n+1},g_{2}]\in[_{n+1}H_{2},G_{2}]$ then
there exist elements $h_{1},...,h_{n+1}\in H_{1}$ and $g_{1}\in G_{1}$ such
that $\beta([h_{1},...,h_{n+1}])=[k_{1},...,k_{n+1}]$ and
$\alpha(g_{1}Z_{n}(G_{1}))=g_{2}Z_{n}(G_{2})$. Therefore
$\psi([h_{1},...,h_{n+1},g_{1}])=[\beta(h_{1},...,h_{n+1}),g_{2}]=[k_{1},...,k_{n+1},g_{2}]$,
and so $\psi$ is an isomorphism.
Now Lemma 3.1 implies that $(\varphi,\psi)$ is a relative $n+1$-isoclinism
from $(H_{1},G_{1})$ to $(H_{2},G_{2}).\square$
Proof of Theorem B. Let $(\alpha,\beta)$ be a relative $n$-isoclinism from
$(H_{1},G_{1})$ to $(H_{2},G_{2})$. We have
$|\displaystyle\frac{H_{1}}{Z_{n}(G_{1})\cap
H_{1}}|^{n}|\frac{G_{1}}{Z_{n}(G_{1})}|\ d^{(n)}(H_{1},G_{1})\vspace{.3cm}\\\
=\frac{1}{|Z_{n}(G_{1})\cap
H_{1}|^{n}|Z_{n}(G_{1})|}|\\{(h_{1},...,h_{n},g_{1})\in H_{1}^{n}\times
G_{1}:\ [h_{1},...,h_{n},g_{1}]=1\\}|\vspace{.3cm}\\\
=\frac{1}{|Z_{n}(G_{1})\cap H_{1}|^{n}|Z_{n}(G_{1})|}|\cdot\vspace{.3cm}\\\
\\{(h_{1},...,h_{n},g_{1})\in H_{1}^{n}\times G_{1}\ :\
\gamma(n,H_{1},G_{1})(h_{1}(Z_{n}(G_{1})\cap H_{1}),...,h_{n}(Z_{n}(G_{1})\cap
H_{1}),g_{1}Z_{n}(G_{1}))=1\\}|\vspace{.3cm}\\\
=|\\{(\bar{h_{1}},...,\bar{h_{n}},\bar{g_{1}})\in\frac{H_{1}}{Z_{n}(G_{1})\cap
H_{1}}\times...\times\frac{H_{1}}{Z_{n}(G_{1})\cap
H_{1}}\times\frac{G_{1}}{Z_{n}(G_{1})}\ :\
\gamma(n,H_{1},G_{1})(\bar{h_{1}},...,\bar{h_{n}},\bar{g_{1}})=1\\}|\vspace{.3cm}\\\
=|\\{(\bar{h_{1}},...,\bar{h_{n}},\bar{g_{1}})\in\frac{H_{1}}{Z_{n}(G_{1})\cap
H_{1}}\times...\times\frac{H_{1}}{Z_{n}(G_{1})\cap
H_{1}}\times\frac{G_{1}}{Z_{n}(G_{1})}\ :\
\beta(\gamma(n,H_{1},G_{1})(\bar{h_{1}},...,\bar{h_{n}},\bar{g_{1}}))=1\\}|$
(because $\beta$ is an isomorphism)
$=|\\{(\bar{h_{1}},...,\bar{h_{n}},\bar{g_{1}})\in\displaystyle\frac{H_{1}}{Z_{n}(G_{1})\cap
H_{1}}\times...\times\frac{H_{1}}{Z_{n}(G_{1})\cap
H_{1}}\times\frac{G_{1}}{Z_{n}(G_{1})}\ :\
\gamma(n,H_{2},G_{2})(\alpha(\bar{h_{1}}),...,\alpha(\bar{h_{n}}),\alpha(\bar{g_{1}}))=1\\}|\vspace{.3cm}$
(by commutativity of diagram as in Definition 2.3)
$=|\\{(\bar{k_{1}},...,\bar{k_{n}},\bar{g_{2}})\in\displaystyle\frac{H_{2}}{Z_{n}(G_{2})\cap
H_{2}}\times...\times\frac{H_{2}}{Z_{n}(G_{2})\cap
H_{2}}\times\frac{G_{2}}{Z_{n}(G_{2})}\ :\
\gamma(n,H_{2},G_{2})(\bar{k_{1}},...,\bar{k_{n}},\bar{g_{2}})=1\\}|$
(because $\alpha$ is an isomorphism)
$=\displaystyle\frac{1}{|Z_{n}(G_{2})\cap
H_{2}|^{n}|Z_{n}(G_{2})|}|\\{(k_{1},...,k_{n},g_{2})\in H_{2}^{n}\times G_{2}\
:\
\gamma(n,H_{2},G_{2})(\bar{k_{1}},...,\bar{k_{n}},\bar{g_{2}})=1\\}|\vspace{.3cm}\\\
=\frac{1}{|Z_{n}(G_{2})\cap
H_{2}|^{n}|Z_{n}(G_{2})|}|\\{(k_{1},...,k_{n},g_{2})\in H_{2}^{n}\times G_{2}\
:\ [k_{1},...,k_{n},g_{2}]=1\\}|\vspace{.3cm}\\\
=|\frac{H_{2}}{Z_{n}(G_{2})\cap H_{2}}|^{n}|\frac{G_{2}}{Z_{n}(G_{2})}|\
d^{(n)}(H_{2},G_{2}).$
Therefore
$\begin{array}[]{lcl}|\displaystyle\frac{H_{1}}{Z_{n}(G_{1})\cap
H_{1}}|^{n}|\frac{G_{1}}{Z_{n}(G_{1})}|\
d^{(n)}(H_{1},G_{1})=|\frac{H_{2}}{Z_{n}(G_{2})\cap
H_{2}}|^{n}|\frac{G_{2}}{Z_{n}(G_{2})}|\ d^{(n)}(H_{2},G_{2})\end{array}.$
Since $(\alpha,\beta)$ is a relative $n$-isoclinism,
$d^{(n)}(H_{1},G_{1})=d^{(n)}(H_{2},G_{2})$ and the result follows.$\square$
Theorem C generalizes [1, Lemma 1.3].
Proof of Theorem C. Put $\bar{G}=G/N$ and $\widetilde{G}=G/(N\cap[_{n}H,G])$.
Since $\bar{g}\in Z_{n}(\bar{G})$ if and only if $\widetilde{g}\in
Z_{n}(\widetilde{G})$, the map $\alpha$ from $\bar{G}/Z_{n}(\bar{G})$ onto
$\widetilde{G}/Z_{n}(\widetilde{G})$ given by
$\alpha(\bar{g}Z_{n}(\bar{G}))=\widetilde{g}Z_{n}(\widetilde{G})$ is an
isomorphism and
$\alpha(\bar{H}/(Z_{n}(\bar{G})\cap\bar{H}))=\widetilde{H}/(Z_{n}(\widetilde{G})\cap\widetilde{H}).$
Also one can see that
$\beta:[_{n}\bar{H},\bar{G}]\rightarrow[_{n}\widetilde{H},\widetilde{G}]$ by
the rule $\beta(\bar{x})=\widetilde{x}$ is an isomorphism. By Lemma 3.1,
$(\alpha,\beta)$ is a relative $n$-isoclinism from $(\bar{H},\bar{G})$ to
$(\widetilde{H},\widetilde{G})$. $\square$
For proving (i) Theorem D, we need of the following lemma.
Lemma 3.4. Let $G$ be a group and $H$ be a subgroup of $G$ such that
$G=HZ_{n}(G)$. Then $(H,H)\ _{\widetilde{n}}\ (H,G)\ _{\widetilde{n}}\ (G,G)$.
Proof. We want to prove $(H,H)\ _{\widetilde{n}}\ (H,G)$. Let $G=HZ_{n}(G)$.
We may easily see that $Z_{n}(H)=Z_{n}(G)\cap H$. Thus
$H/Z_{n}(H)=H/(Z_{n}(G)\cap H)$ is isomorphic to
$HZ_{n}(G)/Z_{n}(G)=G/Z_{n}(G)$. Therefore $\alpha:H/Z_{n}(H)\rightarrow
G/Z_{n}(G)$ is an isomorphism which is induced by the inclusion
$i:H\rightarrow G.$ Furthermore, we can consider $\alpha$ as isomorphism from
$H/Z_{n}(H)$ to $H/Z_{n}(G)\cap H$.
On the other hand, $[_{n}H,G]=[_{n}H,HZ_{n}(G)]=\gamma_{n+1}(H)$. By Lemma
3.1, the pair $(\alpha,1_{\gamma_{n+1}(H)})$ allows us to state that $(H,H)\
_{\widetilde{n}}\ (H,G)$.
The remaining cases $(H,H)\ _{\widetilde{n}}\ (G,G)$ and $(H,H)\
_{\widetilde{n}}\ (H,G)$ follow by a similar argument.$\square$
Proof of Theorem D. (i). By Lemma 3.4, we have $(H,H)\ _{\widetilde{n}}\
(H,G)\ _{\widetilde{n}}\ (G,G)$.
$d^{(n)}(H)=d^{(n)}(H,H)=d^{(n)}(H,G)=d^{(n)}(G,G)=d^{(n)}(G),$
by Theorem B and the result follows.
(ii). Assume $\varphi\in Aut(G)$. Then $\varphi$ induces the isomorphisms
$\alpha$ from $G/Z_{n}(G)$ to $G/Z_{n}(G)$ by the rule
$\alpha(gZ_{n}(G))=\varphi(g)Z_{n}(G)$ and $\beta$ from $[_{n}H,G]$ to
$[_{n}\varphi(H),G]$ by the rule
$\beta([h_{1},...,h_{n},x])=\varphi([h_{1},...,h_{n},x])$. Note that
$\alpha(\frac{H}{Z_{n}(G)\cap H})=\frac{\varphi(H)}{Z_{n}(G)\cap\varphi(H)}.$
On another hand, for every $g\in G$ and $h_{i}\in H$, $1\leq i\leq n$, we have
$\varphi(g)\in\alpha(gZ_{n}(G)),$ $\varphi(h_{i})\in\alpha(h_{i}(Z_{n}(G)\cap
H))$ and
$\beta([h_{1},...,h_{n},g])=[\varphi(h_{1}),...,\varphi(h_{n}),\varphi(g)].$
By Lemma 3.1, the pair $(\alpha,\beta)$ implies that $(H,G)\ _{\widetilde{n}}\
(\varphi(H),G)$ and so
$d^{(n)}(H,G)=d^{(n)}(\varphi(H),G).\ \ \ \ \square$
Theorem C has two useful consequences, as we see in the next statements.
Corollary 3.5. Let $H$ be subgroup of a group $G$. Then there exists a group
$G_{1}$ and a normal subgroup $H_{1}$ of $G_{1}$ such that $(H,G)\
_{\widetilde{1}}\ (H_{1},G_{1})$ and $Z(G_{1})\cap H_{1}\subseteq H_{1}\cap
G^{\prime}_{1}$. In particular, if $G$ is finite, then $G_{1}$ is finite.
Proof. Let $1\rightarrow R\rightarrow F\rightarrow G\rightarrow 1$ be a free
presentation of $G$, $S$ be a subgroup of $F$, $H$ be a group isomorphic to
$S/R$. If $\bar{F}=F/(R\cap\bar{F}^{\prime})$ and
$\bar{S}=S/(R\cap\bar{F}^{\prime})$, then Theorem C with $n=1$ implies $(H,G)\
_{\widetilde{1}}\ (\bar{S},\bar{F})$. On another hand,
$(Z(\bar{F})\cap\bar{S})/(Z(\bar{F})\cap\bar{S}\cap\bar{F}^{\prime})$ is
isomorphic to $((Z(\bar{F})\cap\bar{S})\bar{F}^{\prime})/\bar{F}^{\prime}$,
which is a subgroup of $\bar{F}/\bar{F}^{\prime}$. Therefore, for some normal
subgroup $\bar{B}$ of $\bar{F}$,
$Z(\bar{F})\cap\bar{S}=(Z(\bar{F})\cap\bar{S}\cap\bar{F}^{\prime})\times\bar{B}$.
Now $\bar{B}\cap\bar{F}^{\prime}=1$ and we have $(H,G)\ _{\widetilde{1}}\
(H_{1},G_{1})$ again by Theorem C with $n=1$, where $G_{1}=\bar{F}/\bar{B}$
and $H_{1}=\bar{S}/\bar{B}$. Furthermore $Z(G_{1})\cap H_{1}$ is isomorphic to
$Z({\bar{F}/\bar{B})\cap\bar{S}/\bar{B}=(Z(\bar{F})\cap\bar{S}})/\bar{B}$,
which is a subgroup of $(\bar{S}\cap\bar{F}^{\prime})\bar{B}/\bar{B}=H_{1}\cap
G^{\prime}_{1}$. In particular, if $G$ is finite, then the index
$|G_{1}:Z(G_{1})\cap H_{1}|$ is finite. But, $Z(G_{1})\cap H_{1}\subseteq
H_{1}\cap G^{\prime}_{1},$ therefore $|G:G^{\prime}_{1}|$ is finite and so
$G_{1}$. $\square$
Corollary 3.6. Assume that $H$ is a subgroup of a finite group $G$. Then there
exists a finite group $G_{1}$ and a normal subgroup $H_{1}$ of $G_{1}$ such
that $d(H,G)=d(H_{1},G_{1})$ and $Z(G_{1})\cap H_{1}\subseteq
G^{\prime}_{1}\cap H_{1}.$
Proof. By Theorem B and Corollary 3.5 the result follows. $\square$
We know that
$D_{8}=\langle a,b\ |\ a^{4}=b^{2}=(ab)^{2}=1\rangle.$
It is easy to check that
$(D_{8},\langle a\rangle)\ _{\widetilde{{}_{1}}}\ (D_{8},\langle
a^{2},b\rangle)\ _{\widetilde{{}_{1}}}\ (D_{8},\langle a^{2},ab\rangle).$
and that
$d(D_{8},\langle a\rangle)=d(D_{8},\langle a^{2},b\rangle)=d(D_{8},\langle
a^{2},ab\rangle)=\frac{3}{4}.$
Theorem E shows that all pairs of groups with the relative commutativity
degree $\frac{3}{4}$ belong to the class of relative 1-isoclinism of $(\langle
a\rangle,D_{8})$.
The following lemma gives an upper bound for $d(H,G)$ which will be used in
the proof of Theorem E.
Lemma 3.5. For every subgroup $H$ of a finite group $G$,
$d(H,G)\leq\frac{1}{2}(1+\frac{|Z(G)\cup Z(H)|}{|G|}).$
Proof.
$\begin{array}[]{lcl}d(H,G)&=&\displaystyle\frac{1}{|G||H|}|\\{(h,g)\in
H\times G:[h,g]=1\\}|=\frac{1}{|G|}\sum_{g\in
G}\frac{|C_{H}(g)|}{|H|}\vspace{.2cm}\\\
&=&\displaystyle\frac{1}{|G|}\left(\sum_{g\in Z(G)\cup
Z(H)}\frac{|C_{H}(g)|}{|H|}+\sum_{g\notin Z(G)\cup
Z(H)}\frac{|C_{H}(g)|}{|H|}\right)\vspace{.2cm}\end{array}$
$\begin{array}[]{lcl}&\leq&\displaystyle\frac{1}{|G|}\left(|Z(G)\cup
Z(H)|+\frac{1}{2}(|G|-|Z(G)\cup Z(H)|)\right)\vspace{.2cm}\\\
&=&\displaystyle\frac{1}{2}(1+\frac{|Z(G)\cup Z(H)|}{|G|}).\hskip
8.5359pt\square\end{array}$
Proof of Theorem E. Assume $d(H,G)=\frac{3}{4}$. Then $H$ is abelian by [4,
Theorems 2.2 and 3.3] and $|G:H|\leq 2$ by Lemma 3.7. Moreover, $|H/Z(G)|=2$
by [4, Theorem 3.10] and so $|G:Z(G)|=4$. Therefore $G/Z(G)$ is a
$2$-elementary abelian group of rank 2 so we may define the isomorphism
$\alpha$ from $G/Z(G)$ to $D_{8}/Z(D_{8})$ by $\alpha(\bar{x})=\bar{a}$ and
$\alpha(\bar{y})=\bar{b}.$ Since $Z(G)\subseteq H$, $H/Z(G)$ is either
$\langle\bar{x}\rangle$ or $\langle\bar{y}\rangle$ or
$\langle\bar{x}\bar{y}\rangle$.
Assume that $H/Z(G)=\langle\bar{x}\rangle$. Then $\alpha(H/Z(G))=\langle
a\rangle/\langle a^{2}\rangle$ and $[H,G]=\langle x,y\rangle$. Therefore
$\beta:[H,G]\rightarrow\langle a^{2}\rangle$ by $\beta([x,y])=[a,b]$ is an
isomorphism. Hence $(\alpha,\beta)$ is a relative isoclinism from $(H,G)$ to
$(\langle a\rangle,D_{8})$ by Lemma 3.1. Now we have the remaining cases
$H/Z(G)=\langle\bar{y}\rangle$ and $H/Z(G)=\langle\bar{x}\bar{y}\rangle$. If
$H/Z(G)=\langle\bar{y}\rangle$, then a similar argument shows that $(H,G)\
_{\widetilde{1}}\ (\langle a^{2},b,D_{8}\rangle)$. If
$H/Z(G)=\langle\bar{x}\bar{y}\rangle$, then a similar argument shows that
$(H,G)\ _{\widetilde{1}}\ (\langle a^{2},ab\rangle,D_{8})$. There are no other
cases so we deduce that $(H,G)\ _{\widetilde{1}}\ (\langle a\rangle,D_{8})$,
as claimed.
Conversely, if $(H,G)\ _{\widetilde{1}}\ (\langle a\rangle,D_{8})$, then
$d(H,G)=d(\langle a\rangle,D_{8})=\frac{3}{4}$ and the result follows from [4,
Theorem 3.10].$\square$
Finally, we state the following conjecture.
Conjecture. Theorems B and D hold when $G_{1}$ and $G_{2}$ are two infinite
groups.
We note that Definitions 2.1, 2.2 and 2.3 hold also in the infinite case. The
same happens for Lemmas 3.1, 3.2, 3.4, Proposition 3.3, Theorem A and Theorem
C. This allows us to ask whether conditions as in Theorems B and D can happen
in the infinite case. We strongly believe that the conjecture is true at least
in the case of compact groups, when a suitable notion of commutativity degree
is introduced. We have some evidences in special cases as [16, Theorems A and
B] and in recent submitted papers by the authors.
## References
* [1] J. C. Bioch, On n-isoclinic groups, Indag. Math. 38 (1976), 400–407.
* [2] J. C. Bioch and R. W. van der Waall, Monomiality and isoclinism of groups, J. Reine Ang. Math. 298 (1978), 74–88.
* [3] K. Chiti, M. R. R. Moghaddam and A. R. Salemkar, $n$-isoclinism classes and $n$-nilpotency degree of finite groups, Algebra Colloq. (2) 12 (2005), 225–261.
* [4] A. Erfanian, P. Lescot and R. Rezaei, On the relative commutativity degree of a subgroup of a finite group, Comm. Algebra 35 (2007), 4183–4197.
* [5] P. X. Gallagher, The number of Conjugacy classes in a finite group, Math. Z. 118 (1970), 175–179.
* [6] W. H. Gustafson, What is the probability that two groups elements commute? , Amer. Math. Monthly 80 (1973), 1031–1304.
* [7] M. Hall and J. K. Senior, The Groups of Order $2^{n}\ (n\leq 6)$, Macmillan, New York, 1964.
* [8] P. Hall, The classification of prime-power groups, J. Reine Ang. Math. 182 (1940), 130–141.
* [9] P. Hall, Verbal and marginal subgroups, J. Reine Ang. Math. 182 (1940), 156–157.
* [10] N. S. Hekster, On the structure of n-isoclinism classes of groups, J. Pure Appl. Algebra 40 (1986), 63–85.
* [11] R. James, The groups of order $p^{6}$ ($p$ an odd prime), Math. Comp. 34 (1980), 613–637.
* [12] P. Lescot, Sur certains groupes finis, Rev. Math. Spéciales 8 (1987), 276–277.
* [13] P. Lescot, Isoclinism classes and Commutativity degrees of finite groups, J. Algebra 177 (1995), 847–869.
* [14] G. A.Miller, Relative number of non-invariant operators in a group, Proc. Nat. Acad. Sci. USA (2) 30 (1944), 25–28.
* [15] D. J. S. Robinson, A Course in the Theory of Groups, Springer, Heidelberg, 1982.
* [16] F.G. Russo, A Probabilistic Meaning of Certain Quasinormal Subgroups, Int. J. Algebra (1) 8 (2007), 385–392.
|
arxiv-papers
| 2010-03-11T11:16:02 |
2024-09-04T02:49:09.031075
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ahmad Erfanian (Ferdowsi University of Mashhad, Mashhad, Iran), Rashid\n Rezaei (University of Malayer, Malayer, Iran) and Francesco G. Russo\n (Universita' degli Studi di Palermo, Palermo, Italy)",
"submitter": "Francesco G. Russo",
"url": "https://arxiv.org/abs/1003.2306"
}
|
1003.2312
|
# Statistical Origin of Gravity
Rabin Banerjee, Bibhas Ranjan Majhi
S. N. Bose National Centre for Basic Sciences,
JD Block, Sector III, Salt Lake, Kolkata-700098, India
E-mail: rabin@bose.res.inE-mail: bibhas@bose.res.in
###### Abstract
Starting from the definition of entropy used in statistical mechanics we show
that it is proportional to the gravity action. For a stationary black hole
this entropy is expressed as $S=E/2T$, where $T$ is the Hawking temperature
and $E$ is shown to be the Komar energy. This relation is also compatible with
the generalised Smarr formula for mass.
There are numerous evidences [1, 2, 3] which show that gravity and
thermodynamics are closely connected to each other. Recently, there has been a
growing consensus [4, 5, 6] that gravity need not be interpreted as a
fundamental force, rather it is an emergent phenomenon just like
thermodynamics and hydrodynamics. The fundamental role of gravity is replaced
by thermodynamical interpretations leading to similar or equivalent results.
Nevertheless, understanding the entropic or thermodynamic origin of gravity is
far from complete since the arguments are more heuristic than concrete and
depend upon specific ansatz or assumptions.
In this paper, using certain basic results derived by us [7, 8] in the context
of tunneling mechanism, we are able to provide a statistical interpretation of
gravity. The starting point is the standard definition of entropy given in
statistical mechanics. We show that this entropy gets identified with the
action for gravity. Consequently the Einstein equations obtained by a
variational principle involving the action can be equivalently obtained by an
extremisation of the entropy.
Furthermore, for a black hole with stationary metric we derive the relation
$S=E/2T$, connecting the entropy ($S$) with the Hawking temperature ($T$) and
energy ($E$). We prove that this energy corresponds to Komar’s expression [9,
10]. Using this fact we show that the relation $S=E/2T$ is also compatible
with the generalised Smarr formula [11, 3, 12]. We mention that this relation
was also obtained and discussed in [13, 14].
We start with the partition function for the space-time with matter field
[12],
$\displaystyle{\cal{Z}}=\int~{}D[g,\Phi]~{}e^{iI[g,\Phi]}$ (1)
where $I[g,\Phi]$ is the action representing the whole system and $D[g,\Phi]$
is the measure of all field configurations ($g,\Phi$). Now consider small
fluctuations in the metric ($g$) and the matter field ($\Phi$) in the
following form:
$\displaystyle g=g_{0}+{\tilde{g}};\,\,\,\,\ \Phi=\Phi_{0}+{\tilde{\Phi}}$ (2)
where $g_{0}$ and $\Phi_{0}$ are the stable background fields satisfying the
periodicity conditions and which extremise the action. So they satisfy the
classical field equations. Whereas ${\tilde{g}}$ and $\tilde{\Phi}$, the
fluctuations around these classical values, are very very small. Expanding
$I[g,\Phi]$ around ($g_{0},\Phi_{0}$) we obtain
$\displaystyle
I[g,\Phi]=I[g_{0},\Phi_{0}]+I_{2}[\tilde{g}]+I_{2}[\tilde{\Phi}]+{\textrm{higher
order terms}}.$ (3)
The dominant contribution to the path integral (1) comes from fields that are
near the background fields ($g_{0},\Phi_{0}$). Hence one can neglect all the
higher order terms. The first term $I[g_{0},\Phi_{0}]$ leads to the usual
Einstein equations and gives rise to the standard area law [12]. On the other
hand the second and third terms give the contributions of thermal gravitation
and matter quanta respectively on the background contribution
$I[g_{0},\Phi_{0}]$. They lead to the (logarithmic) corrections to the usual
area law [15]. Here, since we want to confine ourself within the usual semi-
classical regime, we shall neglect these quadratic terms for the subsequent
analysis. Therefore, keeping only the term $I[g_{0},\Phi_{0}]$ in (3) we
obtain the partition function (1) as [12],
$\displaystyle{\cal{Z}}\simeq e^{iI[g_{0},\Phi_{0}]}.$ (4)
Therefore, adopting the standard definition of entropy in statistical
mechanics,
$\displaystyle S=\ln{\cal{Z}}+\frac{E}{T}$ (5)
and using (4), the entropy of the gravitating system is given by 111In this
paper we have chosen units such that $k_{B}=G=\hbar=c=1$.,
$\displaystyle S=iI[g_{0},\Phi_{0}]+\frac{E}{T}$ (6)
where $E$ and $T$ are respectively the energy and temperature of the system.
It may be pointed out that it is possible to interpret (4) as defining the
partition function of an emergent theory without specifying the detailed
configuration of the gravitating system. The validity of such an
interpretation is borne out by the subsequent analysis.
In order to get an explicit expression for $E$, let us consider a specific
system - a black hole. Now thermodynamics of a black hole is universally
governed by its properties near the event horizon. It is also well understood
that near the event horizon the effective theory becomes two dimensional whose
metric is given by the two dimensional ($t-r$)- sector of the original metric
[16, 17]. Correspondingly, the left ($L$) and right ($R$) moving (holomorphic)
modes are obtained by solving the appropriate field equation using the
geometrical (WKB) approximation. Furthermore, the modes inside and outside the
horizon are related by the transformations [7, 8]:
$\displaystyle\phi^{(R)}_{in}$ $\displaystyle=$ $\displaystyle
e^{-\frac{\pi\omega}{\kappa}}\phi^{(R)}_{out}$ (7)
$\displaystyle\phi^{(L)}_{in}$ $\displaystyle=$
$\displaystyle\phi^{(L)}_{out}$ (8)
where “$\omega$” is the energy of the particle as measured by an asymptotic
observer and “$\kappa$” is the surface gravity of the black hole. In this
convention the $L$($R$) moving modes are ingoing (outgoing). Concentrating on
the modes inside the horizon, the $L$ mode gets trapped while the $R$ mode
tunnels through the horizon and is eventually observed at asymptotic infinity
as Hawking radiation [7, 8]. The probability of this “$R$” mode, to go
outside, as measured by the outside observer is given by
$\displaystyle
P^{(R)}=\Big{|}\phi^{(R)}_{in}\Big{|}^{2}=\Big{|}e^{-\frac{\pi\omega}{\kappa}}\phi^{(R)}_{out}\Big{|}^{2}=e^{-\frac{2\pi\omega}{\kappa}}$
(9)
where, in the second equality, (7) has been used. This is essential since the
measurement is done from outside and hence $\phi^{(R)}_{in}$ has to be
expressed in favour of $\phi^{(R)}_{out}$. Therefore the average value of the
energy, measured from outside, is given by,
$\displaystyle<\omega>=\frac{\int_{0}^{\infty}~{}d\omega~{}\omega~{}e^{-\frac{2\pi\omega}{\kappa}}}{\int_{0}^{\infty}~{}d\omega~{}e^{-\frac{2\pi\omega}{\kappa}}}=T$
(10)
where $T=\kappa/2\pi$ is the temperature of the black hole [8]. Therefore if
we consider that the energy $E$ of the system is encoded near the horizon and
the total number of pairs created is $n$ among which this energy is
distributed, then we must have,
$\displaystyle E=nT$ (11)
where only the $R$ mode of the pair is significant.
Now to proceed further, it must be realised that the effective two dimensional
curved metric can always be embedded in a flat space which has exactly two
space-like coordinates. This is a consequence of a modification in the
original GEMS (globally embedding in Minkowskian space) approach of [18] and
has been elaborated by us in [19]. Hence we may associate each $R$ mode with
two degrees of freedom. Then the total number of degrees of freedom for $n$
number of $R$ modes is $N=2n$. Hence, from (11), we obtain the energy of the
system as
$\displaystyle E=\frac{1}{2}NT.$ (12)
As a side remark, it may be noted that (12) can be interpreted as a
consequence of the usual law of equipartition of energy. For instance, if we
consider that the energy $E$ is distributed equally over each degree of
freedom, then (12) implies that each degree of freedom should contain an
energy equal to $T/2$, which is nothing but the equipartition law of energy.
The fact that the energy is equally distributed among the degrees of freedom
may be understood from the symmetry of two space-like coordinates
($z^{1}\longleftrightarrow z^{2}$) such that the metric is unchanged [19]. In
our subsequent analysis, however, we only require (12) rather than its
interpretation as the law of equipartition of energy.
Now since there are $N$ number of degrees of freedom in which all the
information is encoded, the entropy ($S$) of the system must be proportional
to $N$. Hence using (6) we obtain
$\displaystyle N=N_{0}S=N_{0}(iI[g_{0},\Phi_{0}]+\frac{E}{T}),$ (13)
where $N_{0}$ is a proportionality constant, which will be determined later.
Substituting the value of $N$ from (12) in (13) we obtain the expression for
the energy of the system as
$\displaystyle E=\frac{N_{0}}{2-N_{0}}iTI[g_{0},\Phi_{0}].$ (14)
This shows that in the absence of any fluctuations, the energy of a system is
actually given by the classical action representing the system. In the
following we shall use this expression to find the energy of a stationary
black hole. Before that let us substitute the value of $I[g_{0},\Phi_{0}]$
from (14) in (6). This immediately leads to a simple relation between the
entropy, temperature and energy of the black hole:
$\displaystyle S=\frac{2E}{N_{0}T}.$ (15)
Now in order to fix the value of “$N_{0}$” we consider the simplest example,
the Schwarzschild black hole for which the entropy, energy and temperature are
given by,
$\displaystyle S=\frac{A}{4}=4\pi M^{2},\,\,\ E=M,\,\,\ T=\frac{1}{8\pi M},$
(16)
where “$M$” is the mass of the black hole. Substitution of these in (15) leads
to $N_{0}=4$.
At this point we want to make a comment on the value of $N_{0}$. According to
standard statistical mechanics one would have thought that $1/N_{0}=\ln c$,
where $c$ is an integer. Whereas to keep our analysis consistent with semi-
classical area law, we obtained $c=e^{1/4}$, which is clearly not an integer.
Indeed, any departure from this value of $N_{0}$ would invalidate the semi-
classical area law and hence our analysis. Such a disparity is not peculiar to
our approach and has also occurred elsewhere [22]. This may be due to the fact
that our analysis is confined within the semi-classical regime, which is valid
for large degrees of freedom. In this regime, it is not obvious that a semi-
classical computation can reproduce $c$ to be an integer. Furthermore, the
above value of $N_{0}$ is still valid even for very small number of degrees of
freedom ($N$), where this semi-classical calculation is unjustified. This also
happens in the semi-classical computation of the entropy spectrum of a black
hole [23]. The entropy spectrum is found there to be $S=2\pi N$ rather than
$S=N\ln c$ and this discrepancy is identified with the semi-classical
approximation. A possible way to resolve such disagreement from standard
statistical mechanics may be the full quantum theoretical computation of the
number of microstates which is beyond the scope of the present paper.
Finally, putting back $N_{0}=4$ in (15) we obtain,
$\displaystyle S=\frac{E}{2T}.$ (17)
Before discussing the significance and implications of this relation, we
observe that substituting the value of $E$ from (17) in (14) with $N_{0}=4$,
we obtain
$\displaystyle S=-iI[g_{0},\Phi_{0}].$ (18)
Consequently, extremisation of entropy leads to Einstein’s equations.
The relation (17) is significant for various reasons which will become
progressively clear. It is valid for all black hole solutions in Einstein
gravity with appropriate identifications consistent with the area law. Here
$S$ and $T$ are easy to identify. These are, respectively, the entropy and
Hawking temperature of the black hole. Since energy is one of the most
diversely defined entities in general theory of relativity, special care is
needed to identify $E$ in (17). We now show that this $E$ corresponds to
Komar’s definition [9, 10]. Simplifying (14) using $N_{0}=4$ and
$T=\kappa/2\pi$, we obtain,
$\displaystyle E=-\frac{i\kappa I[g_{0},\Phi_{0}]}{\pi}.$ (19)
The classical action $I[g_{0},\Phi_{0}]$ has already been calculated in [12].
The result is,
$\displaystyle
I[g_{0},\Phi_{0}]=2i\pi\kappa^{-1}\Big{[}\frac{1}{16\pi}\int_{\Sigma}R\xi^{a}d\Sigma_{a}+\int_{\Sigma}(T_{ab}-\frac{1}{2}Tg_{ab})\xi^{b}d\Sigma^{a}-\frac{1}{16\pi}\int_{\cal{H}}\epsilon_{abcd}\nabla^{c}\xi^{d}\Big{]},$
(20)
where $\xi^{a}\partial/\partial x^{a}=\partial/\partial t$ is the time
translation Killing vector and $\Sigma$ is the space-like hypersurface whose
boundary is given by ${\cal{H}}$. Here $T_{ab}$ is the energy-momentum tensor
of the matter field whose trace is given by $T$. Now for a stationary
geometry, $\xi^{a}\nabla_{a}R=0$ [20]. Hence for a volume ${\cal{A}}$, we have
$\displaystyle
0=\int_{\cal{A}}\xi^{a}\nabla_{a}Rd{\cal{A}}=\int_{\cal{A}}\Big{[}\nabla_{a}(\xi^{a}R)-(\nabla_{a}\xi^{a})R\Big{]}d{\cal{A}}=\int_{\cal{A}}\nabla_{a}(\xi^{a}R)d{\cal{A}}$
(21)
where in the last step the Killing equation
$\nabla_{a}\xi_{b}+\nabla_{b}\xi_{a}=0$ has been used. Finally, the Gauss
theorem yields,
$\displaystyle\int_{\Sigma}\xi^{a}Rd{\Sigma_{a}}=0.$ (22)
Using this in (20) we obtain,
$\displaystyle
I[g_{0},\Phi_{0}]=2i\pi\kappa^{-1}\Big{[}\int_{\Sigma}(T_{ab}-\frac{1}{2}Tg_{ab})\xi^{b}d\Sigma^{a}-\frac{1}{16\pi}\int_{\cal{H}}\epsilon_{abcd}\nabla^{c}\xi^{d}\Big{]}.$
(23)
Substituting this in (19) we obtain the expression for the energy of the
gravitating system as
$\displaystyle
E=2\int_{\Sigma}(T_{ab}-\frac{1}{2}Tg_{ab})\xi^{b}d\Sigma^{a}-\frac{1}{8\pi}\int_{\cal{H}}\epsilon_{abcd}\nabla^{c}\xi^{d}$
(24)
which is the Komar expression for energy [9, 10] corresponding to the time
translation Killing vector. Similarly, if there is a rotational Killing
vector, then there must be a Komar expression for rotational energy [20, 21]
and the total energy will be their sum.
Incidentally, (17) was obtained earlier in [13] for static space-time and its
implications were discussed in [14]. However a specific ‘ansatz’ for entropy
compatible with the area law was taken and, more importantly, the Komar energy
expression was explicitly used as an input in the derivation. Hence our
analysis is completely different, since we do not invoke any ansatz for the
entropy; neither is the Komar expression required at any stage. Rather we
prove its occurence in the relation (17).
As an explicit check of (17) for different black hole solutions, we consider a
couple of examples. Take the Reissner-Nordstr$\ddot{\textrm{o}}$m (RN) black
hole. In this case the entropy and temperature are given by,
$\displaystyle S=\pi r_{+}^{2},\,\,\,\ T=\frac{r_{+}-r_{-}}{4\pi
r_{+}^{2}};\,\,\ r_{\pm}=M\pm\sqrt{M^{2}-Q^{2}}$ (25)
where “$Q$” is the charge of the black hole. Substitution of these in (17)
yields,
$\displaystyle E=M-\frac{Q^{2}}{r_{+}},$ (26)
which is the Komar energy of RN black hole [25].
Next we consider the Kerr black hole for which the entropy and temperature are
respectively,
$\displaystyle S$ $\displaystyle=$ $\displaystyle\pi(r_{+}^{2}+a^{2}),\,\,\,\
T=\frac{r_{+}-r_{-}}{4\pi(r_{+}^{2}+a^{2})};$ $\displaystyle r_{\pm}$
$\displaystyle=$ $\displaystyle M\pm\sqrt{M^{2}-a^{2}},\,\,\,\ a=\frac{J}{M}.$
(27)
Here “$J$” is the angular momentum of the black hole. Substituting (27) in
(17) we obtain,
$\displaystyle E=M-2J\Omega$ (28)
which is the total Komar energy for Kerr black hole [24, 25]. Here
$\Omega=\frac{a}{r_{+}^{2}+a^{2}}$ is the angular velocity at the event
horizon $r=r_{+}$.
We thus find that, in all cases where $S$, $E$, $T$ are known, they satisfy
(17) apart from the area law. In fact, it is possible to take (17) as the
defining relation for the Komar energy in those examples where its direct
calculation from (24) is difficult. Such an instance is provided by the Kerr-
Newman black hole. Its Komar energy, as far as we aware, is not known in
closed form. However the entropy and temperature of Kerr-Newman black hole are
given by,
$\displaystyle S=\pi(r_{+}^{2}+a^{2});\,\,\,\
T=\frac{r_{+}-r_{-}}{4\pi(r_{+}^{2}+a^{2})}$ (29)
where
$\displaystyle r_{\pm}=M\pm\sqrt{M^{2}-Q^{2}-a^{2}};\,\,\,\ a=\frac{J}{M}.$
(30)
Now substituting (29) in (17) and then using (30) we obtain the total Komar
energy of Kerr-Newman black hole:
$\displaystyle
E=\sqrt{M^{2}-Q^{2}-a^{2}}=M-\frac{Q^{2}}{r_{+}}-2J\Omega\Big{(}1-\frac{Q^{2}}{2Mr_{+}}\Big{)}=M-QV-2J\Omega,$
(31)
where $\Omega=\frac{a}{r_{+}^{2}+a^{2}}$ is the angular velocity at the event
horizon and $V=\frac{Q}{r_{+}}-\frac{QJ\Omega}{Mr_{+}}$. This value exactly
matches with the direct evaluations of Komar expressions for energies within
the first order approximation [21, 24, 25]. It is also reassuring to note that
the definition of $M$ following from (17) and (31) reproduces the generalised
Smarr formula [11, 3, 12],
$\displaystyle\frac{M}{2}=\frac{\kappa A}{8\pi}+\frac{VQ}{2}+\Omega J.$ (32)
In this paper we have further clarified the possibility of considering gravity
as an emergent phenomenon. Taking the standard definition of entropy from
statistical mechanics we were able to show the equivalence of entropy with the
action. Consequently, extremisation of the action leading to Einstein’s
equations is equivalent to the extremisation of the entropy. We derived the
relation $S=E/2T$ for stationary black holes with $S$ and $T$ being the
entropy and Hawking temperature. The nature of energy $E$ appearing in this
relation was clarified. It was proved to be Komar’s expression valid for
stationary asymptotically flat space-time. An explicit check of $S=E/2T$ was
done for all black hole solutions of Einstein gravity. This relation was also
seen to reproduce the generalised mass formula of Smarr [11, 3, 12]. In this
sense the Smarr formula can be interpreted as a thermodynamic relation further
illuminating the emergent nature of gravity. As a final remark we feel that
although our results were derived for Einstein gravity, the methods are
general enough to include other possibilities like higher order theories.
Acknowledgement:The authors thank Mr. S. K. Modak for useful discussions.
## References
* [1] J. D. Bekenstein, Phys. Rev. D 7, 2333 (1973).
* [2] S. W. Hawking, Nature 248, 30 (1974).
S. W. Hawking, Commun. Math. Phys. 43, 199 (1975) [Erratum-ibid. 46, 206
(1976)].
* [3] J. M. Bardeen, B. Carter and S. W. Hawking, Commun. Math. Phys. 31, 161 (1973).
* [4] T. Jacobson, Phys. Rev. Lett. 75, 1260 (1995) [arXiv:gr-qc/9504004].
* [5] T. Padmanabhan, arXiv:0911.5004 [gr-qc] and references therein.
* [6] E. P. Verlinde, arXiv:1001.0785 [hep-th].
* [7] R. Banerjee and B. R. Majhi, Phys. Rev. D 79, 064024 (2009) [arXiv:0812.0497 [hep-th]].
R. Banerjee and B. R. Majhi, Phys. Lett. B 675, 243 (2009) [arXiv:0903.0250
[hep-th]].
* [8] R. Banerjee, B. R. Majhi and E. C. Vagenas, Phys. Lett. B 686, 279 (2010) [arXiv:0907.4271 [hep-th]].
* [9] A. Komar, Phys. Rev. 113, 934 (1959).
* [10] R. M. Wald, “General Relativity,” Chicago, Usa: Univ. Pr. ( 1984) 491p.
* [11] L. Smarr, Phys. Rev. Lett. 30, 71 (1973) [Erratum-ibid. 30, 521 (1973)].
* [12] G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2752 (1977).
* [13] T. Padmanabhan, Class. Quant. Grav. 21, 4485 (2004) [arXiv:gr-qc/0308070].
* [14] T. Padmanabhan, arXiv:0912.3165 [gr-qc].
* [15] S. W. Hawking, Commun. Math. Phys. 55, 133 (1977).
* [16] S. Carlip, Phys. Rev. Lett. 82, 2828 (1999) [arXiv:hep-th/9812013].
* [17] S. P. Robinson and F. Wilczek, Phys. Rev. Lett. 95, 011303 (2005) [arXiv:gr-qc/0502074].
* [18] S. Deser and O. Levin, Phys. Rev. D 59, 064004 (1999) [arXiv:hep-th/9809159].
* [19] R. Banerjee and B. R. Majhi, arXiv:1002.0985 [gr-qc] (Phys. Lett. B, In press).
* [20] S. M. Carroll, “Spacetime and geometry: An introduction to general relativity,” San Francisco, USA: Addison-Wesley (2004) 513 p.
* [21] J. Katz, Class. Quant. Grav. 2, 423 (1985).
* [22] See, for instance, the discussion in section 2 of [14].
* [23] M. Maggiore, Phys. Rev. Lett. 100, 141301 (2008) [arXiv:0711.3145 [gr-qc]]. See, in particular, the last section.
* [24] R. Kulkarni, V. Chellathurait and N. Dadhich, Class. Quant. Grav. 5, 1443 (1988)
V. Chellathurait and N. Dadhich, Class. Quant. Grav. 7, 361 (1990).
N. Dadhich, Phys. Lett. A 98, 103 (1983).
* [25] R. Banerjee and S. K. Modak, JHEP 0905, 063 (2009) [arXiv:0903.3321 [hep-th]].
|
arxiv-papers
| 2010-03-11T11:46:07 |
2024-09-04T02:49:09.037218
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rabin Banerjee, Bibhas Ranjan Majhi",
"submitter": "Bibhas Majhi Ranjan",
"url": "https://arxiv.org/abs/1003.2312"
}
|
1003.2493
|
# Bivariate Quasi-Tower Sets and Their Associated Lagrange Interpolation Bases
Tian Dong, Xiaoying Wang Shugong Zhang sgzh@jlu.edu.cn Peng Li School of
Mathemathics, Key Lab. of Symbolic Computation and Knowledge Engineering
(Ministry of Education), Jilin University, Changchun 130012, PR China
###### Abstract
As we all known, there is still a long way for us to solve arbitrary
multivariate Lagrange interpolation in theory. Nevertheless, it is well
accepted that theories about Lagrange interpolation on special point sets
should cast important lights on the general solution. In this paper, we
propose a new type of bivariate point sets, quasi-tower sets, whose geometry
is more natural than some known point sets such as cartesian sets and tower
sets. For bivariate Lagrange interpolation on quasi-tower sets, we construct
the associated degree reducing interpolation monomial and Newton bases w.r.t.
common monomial orderings theoretically. Moreover, by inputting these bases
into Buchberger-Möller algorithm, we obtain the reduced Gröbner bases for
vanishing ideals of quasi-tower sets much more efficiently than before.
###### keywords:
Bivariate Lagrange interpolation , Quasi-Tower set , Degree reducing
interpolation basis , Buchberger-Möller algorithm
###### MSC:
13P10 , 65D05 , 12Y05
fundfundfootnotetext: This work was supported in part by the National Grand
Fundamental Research 973 Program of China (No. 2004CB318000).
## 1 Introduction
Given a set $\Xi=\\{\xi^{(1)},\ldots,\xi^{(\mu)}\\}\subset\mathbb{F}^{2}$ of
$\mu$ distinct points with $\mathbb{F}$ a field. For prescribed values
$f_{i}\in\mathbb{F},i=1,\ldots,\mu$, _bivariate Lagrange interpolation_ is to
find all polynomials $p\in\Pi^{2}$ satisfying
$p(\xi^{(i)})=f_{i},\quad i=1,\ldots,\mu,$ (1)
where $\Pi^{2}:=\mathbb{F}[x,y]$ is the bivariate polynomial ring over
$\mathbb{F}$.
In most cases, for fixed monomial ordering $\prec$, only the bases for the
_degree reducing interpolation space_ $\mathcal{P}\subset\Pi^{2}$(see [1]) are
pursued for Lagrange interpolation (1). Let $\\{p_{1},\ldots,p_{\mu}\\}$ be a
basis for $\mathcal{P}$ satisfying $p_{1}\prec p_{2}\prec\cdots\prec p_{\mu}$.
If
$p_{j}(\xi^{(i)})=\delta_{ij},\quad 1\leq i\leq j\leq\mu,$
for certain reordering of $\Xi$, then we call $\\{p_{1},\ldots,p_{\mu}\\}$ a
_degree reducing interpolation Newton basis_(DRINB) w.r.t. $\prec$ for (1).
Moreover, $\mathcal{N}_{\prec}(\Xi)$, the _Gröbner éscalier_ of
$\mathcal{I}(\Xi)$ w.r.t. $\prec$(see [2]), is the _degree reducing
interpolation monomial basis_(DRIMB) w.r.t. $\prec$ for (1).
Let $\mathcal{I}(\Xi)\subset\Pi^{2}$ be the vanishing ideal of $\Xi$ and
$\mathcal{G}_{\prec}(\Xi)$ the reduced Gröbner basis for $\mathcal{I}(\Xi)$
w.r.t.$\prec$. In 1982, [3] proposed Buchberger-Möller(BM for short) algorithm
for computing bases of vanishing ideals. Specifically, with inputted $\Xi$ and
$\prec$, BM algorithm outputs $\mathcal{G}_{\prec}(\Xi)$,
$\mathcal{N}_{\prec}(\Xi)$, and a DRINB for Lagrange interpolation (1), namely
arbitrary bivariate Lagrange interpolation can be solved with it algorithmly.
Nonetheless, the poor complexity of BM algorithm(see [4]) severely constrains
its applications.
Since the geometry of $\Xi$ is a dominant factor for the structures of
$\mathcal{G}_{\prec}(\Xi)$ and $\mathcal{N}_{\prec}(\Xi)$, theories about
Lagrange interpolation on point sets with describable special geometries will
certainly throw light on the general theoretical solution of (1). [5, 6]
studied multivariate Lagrange interpolation on cartesian sets(aka lower point
sets) and [7] investigated bivariate Lagrange interpolation on tower sets.
This paper will introduce, in Section 3, a new type of bivariate point sets,
quasi-tower sets, that is more natural than cartesian and tower sets. As the
section of main results, Section 4 will discover the degree reducing
interpolation bases w.r.t. certain common monomial ordering such as graded lex
order for bivariate Lagrange interpolation on quasi-tower sets theoretically.
Finally, Section 5 will give an algorithm for computing reduced Gröbner bases
for vanishing ideals of quasi-tower sets.
The following section will serve as a preparation for the paper.
## 2 Preliminary
Let $\mathbb{N}_{0}$ and $\mathbb{T}^{2}$ be the monoid of nonnegative
integers and bivariate monomials in $\Pi^{2}$ respectively. A monomial in
$\mathbb{T}^{2}$ has the form $X^{\bm{\alpha}}:=x^{\alpha_{1}}y^{\alpha_{2}}$
for $\bm{\alpha}=(\alpha_{1},\alpha_{2})\in\mathbb{N}_{0}^{2}$.
As in [8], we use
$\prec_{\text{lex}},\prec_{\text{invlex}},\prec_{\text{grlex}}$, and
$\prec_{\text{grevlex}}$ to represent lexicographic order, inverse
lexicographic order, graded lex order, and graded reverse lex order
respectively. Hereafter, we always assume that $y\prec x$ for any fixed
monomial ordering $\prec$.
For a nonzero $f\in\Pi^{2}$, we let $\mathrm{LM}_{\prec}(f)$ and
$\bm{\delta_{\prec}(f)}$ signify the _leading monomial_ and _leading
bidegree_($\mathrm{LM}(f)=X^{\bm{\delta_{\prec}(f)}}$) of $f$ w.r.t. $\prec$
respectively. Furthermore, set
$f\prec g:=\bm{\delta_{\prec}(f)}\prec\bm{\delta_{\prec}(g)}.$
Let $\mathcal{A}$ be an arbitrary finite subset of $\mathbb{N}_{0}^{2}$. We
say that it is _lower_ if
$\mathrm{R}(\bm{\alpha}):=\\{(\alpha^{\prime}_{1},\alpha^{\prime}_{2})\in\mathbb{N}_{0}^{2}:0\leq\alpha^{\prime}_{i}\leq\alpha_{i},i=1,2\\}\subset\mathcal{A}$
always holds for any $\bm{\alpha}=(\alpha_{1},\alpha_{2})\in\mathcal{A}$. When
$\mathcal{A}$ is lower, we let $m_{j}=\max_{(h,j)\in\mathcal{A}}h,0\leq
j\leq\nu$, with $\nu=\max_{(0,k)\in\mathcal{A}}k$. Since $\mathcal{A}$ can be
determined uniquely by $(\nu+1)$-tuple $(m_{0},m_{1},\ldots,m_{\nu})$, we
represent it as
$\mathcal{A}=\mathrm{L}_{x}(m_{0},m_{1},\ldots,m_{\nu}).$
Likewise, it can also be represented as
$\mathrm{L}_{y}(n_{0},\ldots,n_{m_{0}})$ with
$n_{i}=\max_{(i,k)\in\mathcal{A}}k$, $0\leq i\leq m_{0}$.
As in [7], we constructed two particular lower sets
$S_{x}(\Xi),S_{y}(\Xi)\subset\mathbb{N}_{0}^{2}$ from $\Xi$. In more details,
we cover $\Xi$ by lines $l_{0}^{x},l_{1}^{x},\ldots,l_{\nu}^{x}$ parallel to
the $x$-axis and assume that, without loss of generality, there are $m_{j}+1$
points, say $u_{0j}^{x},u_{1j}^{x},\ldots,u_{m_{j},j}^{x}$, on $l_{j}^{x}$
with $m_{0}\geq m_{1}\geq\cdots\geq m_{\nu}\geq 0$. Set
$S_{x}(\Xi):=\\{(i,j):0\leq i\leq m_{j},\ 0\leq j\leq\nu\\},$
which equals to $\mathrm{L}_{x}(m_{0},m_{1},\ldots,m_{\nu})$. In like fashion,
we cover $\Xi$ by lines $l_{0}^{y},l_{1}^{y},\ldots,l_{\lambda}^{y}$ parallel
to the $y$-axis and denote the points on $l_{i}^{y}$ by
$u_{i0}^{y},\ldots,u_{i,n_{i}}^{y}$ with $n_{0}\geq\cdots\geq n_{\lambda}\geq
0$. We have
$S_{y}(\Xi):=\\{(i,j):0\leq i\leq\lambda,\ 0\leq j\leq
n_{i}\\}=\mathrm{L}_{y}(n_{0},n_{1},\ldots,n_{\lambda}).$
In addition, the sets of abscissae and ordinates are defined as
$\displaystyle H_{j}(\Xi):=$ $\displaystyle\\{\bar{x}:(\bar{x},\bar{y})\in
l_{j}^{x}\cap\Xi\\},\quad 0\leq j\leq\nu,$ $\displaystyle V_{i}(\Xi):=$
$\displaystyle\\{\bar{y}:(\bar{x},\bar{y})\in l_{i}^{y}\cap\Xi\\},\quad 0\leq
i\leq\lambda.$
A point set $\Xi$ satisfying $S_{x}(\Xi)=S_{y}(\Xi)$ is called a cartesian set
that has the following property:
###### Proposition 1.
[7] A point set $\Xi\subset\mathbb{F}^{2}$ is cartesian if and only if
$H_{0}(\Xi)\supseteq H_{1}(\Xi)\supseteq\cdots\supseteq H_{\nu}(\Xi)$
or
$V_{0}(\Xi)\supseteq V_{1}(\Xi)\supseteq\cdots\supseteq V_{\lambda}(\Xi).$
Enlightened by the notion of cartesian sets, [7] introduced tower sets in
$\mathbb{F}^{2}$ as follows.
###### Definition 1.
[7] We say that a set $\Xi$ of distinct points in $\mathbb{F}^{2}$ is
$x$-_tower_ if lower set
$S_{x}(\Xi)=\mathrm{L}_{x}(m_{0},m_{1},\ldots,m_{\nu})\subset\mathbb{N}_{0}^{2}$
with $m_{0}>m_{1}>\cdots>m_{\nu}\geq 0$ such that
$\Xi:=\\{(x_{ij},y_{j}):(i,j)\in S_{x}(\Xi)\\},$
where $x_{ij}\in H_{0}(\Xi),(i,j)\in S_{x}(\Xi)$, are distinct for fixed $j$.
Its full name is $S_{x}(\Xi)$-$x$-_tower_ set. Similarly, if lower set
$S_{y}(\Xi)=\mathrm{L}_{y}(n_{0},n_{1},\ldots,n_{\lambda})\subset\mathbb{N}_{0}^{2}$,
$n_{0}>n_{1}>\cdots>n_{\lambda}\geq 0$, such that
$\Xi:=\\{(x_{i},y_{ij}):(i,j)\in S_{y}(\Xi)\\},$
where $y_{ij}\in V_{0}(\Xi),(i,j)\in S_{y}(\Xi)$, are distinct for fixed $i$,
we will call $\Xi$ a $y$-_tower_ set(or $S_{y}(\Xi)$-$y$-_tower_ set in its
full name).
About the bivariate Lagrange interpolation on a tower set, [7] proved the
succeeding theorem.
###### Theorem 2.
[7] Given an $x$-tower set $\Xi\subset\mathbb{F}^{2}$. The DRIMB w.r.t.
$\prec_{\mathrm{grlex}}$ or $\prec_{\mathrm{lex}}$ for (1) is
$\\{x^{i}y^{j}:(i,j)\in S_{x}(\Xi)\\}.$
If $\Xi$ is $y$-tower, then the DRIMB w.r.t. $\prec_{\mathrm{grevlex}}$ or
$\prec_{\mathrm{invlex}}$ for (1) is
$\\{x^{i}y^{j}:(i,j)\in S_{y}(\Xi)\\}.$
## 3 Quasi-Tower Sets
In this section, we will introduce the notion of generalized tower and quasi-
tower sets and compare them with tower sets and cartesian sets.
###### Definition 2.
Let $\Xi$ be a set of distinct points in $\mathbb{F}^{2}$. We call $\Xi$ a
_generalized $x$-tower_ set if
$S_{x}(\Xi)=\mathrm{L}_{x}(m_{0},m_{1},\ldots,m_{\nu})\subset\mathbb{N}_{0}^{2}$
with $m_{0}>m_{1}>\cdots>m_{\nu}\geq 0$ such that
$\Xi:=\\{(x_{ij},y_{j}):(i,j)\in S_{x}(\Xi)\\},$
where $x_{ij}$ are distinct for fixed $j$. In like fashion, if lower set
$S_{y}(\Xi)=\mathrm{L}_{y}(n_{0},n_{1},\ldots,n_{\lambda})\subset\mathbb{N}_{0}^{2}$,
$n_{0}>n_{1}>\cdots>n_{\lambda}\geq 0$, such that
$\Xi:=\\{(x_{i},y_{ij}):(i,j)\in S_{y}(\Xi)\\},$
where $y_{ij}$ are distinct for fixed $i$, we will call $\Xi$ a _generalized
$y$-tower_ set.
Observe Definition 1 and 2. We find that they are similar to each other except
two more conditions in Definition 1 which implies that a tower set is always
generalized tower, namely the notion of generalized tower sets is actually a
generalization of tower sets’. Therefore, what interests us most now should be
the generalized tower sets that are not tower.
###### Definition 3.
If $\Xi\subset\mathbb{F}^{2}$ is a generalized $x$-tower set that is not
$x$-tower, then we call it a _quasi- $x$-tower_ set. Similarly, a
non-$y$-tower generalized $y$-tower set is called a _quasi- $y$-tower_ set.
From Definition 2 and 3 we know that the geometry of quasi-tower sets is much
freer than tower sets’ in the sense that they need not fulfill the two extra
conditions that demand $x_{ij}\subset H_{0}(\Xi)$ for $x$-tower and
$x_{ij}\subset V_{0}(\Xi)$ for $y$-tower. The following propositions will show
us how did quasi-tower sets acquire their name.
###### Proposition 3.
Let $\Xi$ be a quasi-$x$-tower set with
$S_{x}(\Xi)=\mathrm{L}_{x}(m_{0},\ldots,m_{\nu})$. If $B\subset\mathbb{F}^{2}$
is a set of distinct horizontal points such that $B\cap\Xi=\emptyset$ and
$\bigcup_{i=0}^{\nu}H_{i}(\Xi)\subsetneq H_{0}(B)$, then $\Xi\cup B$ is
$x$-tower and will be known as a _derived $x$-tower_ set from $\Xi$ with _base
set_ $B$.
###### Proof 1.
By hypothesis, we assume that
$S_{y}(\Xi)=\mathrm{L}_{y}(n_{0},n_{1},\ldots,n_{\lambda})$ and
$B=\\{(x_{0}^{B},y^{B}),(x_{1}^{B},y^{B}),\ldots,(x_{m_{B}}^{B},y^{B})\\}$
where $x_{i}^{B}\neq x_{j}^{B},i\neq j$, and
$y^{B}\notin\bigcup_{j=0}^{\lambda}V_{j}(\Xi)$. Recall the construction
process of $S_{x}(\Xi)$ in Section 2. Since
$\bigcup_{i=0}^{\nu}H_{i}(\Xi)\subsetneq H_{0}(B)$ and $\Xi$ is
quasi-$x$-tower, we can deduce easily that
$S_{x}(\Xi\cup B)=\mathrm{L}_{x}(m_{B},m_{0},\ldots,m_{\nu}),$
where $m_{B}>m_{0}>\cdots>m_{\nu}\geq 0$. From Definition 2 and 3, $\Xi$ is
quasi-$x$-tower implies that
$\Xi=\\{(x_{ij}^{\Xi},y_{j}^{\Xi}):(i,j)\in S_{x}(\Xi)\\},$
where $x_{ij}^{\Xi}$ are distinct for fixed $j$, hence, we have
$\Xi\cup B=\\{(x_{ij},y_{j}):(i,j)\in S_{x}(\Xi\cup B)\\}$
where
$\displaystyle x_{i0}$ $\displaystyle=x_{i}^{B},y_{0}=y^{B},$ $\displaystyle
i$ $\displaystyle=0,1,\ldots,m_{B},$ $\displaystyle x_{i,j+1}$
$\displaystyle=x_{ij}^{\Xi},y_{j+1}=y_{j}^{\Xi},$ $\displaystyle i$
$\displaystyle=0,1,\ldots,m_{j}^{\Xi},j=0,1,\ldots,\nu,$
and $x_{ij}$ are distinct for fixed $j$. Therefore, by Definition 1, $\Xi\cup
B$ is $x$-tower. ∎
In the same way, we can prove the following proposition for quasi-$y$-tower
sets.
###### Proposition 4.
Let $\Xi$ be a quasi-$y$-tower set with
$S_{y}(\Xi)=\mathrm{L}_{y}(n_{0},n_{1},\ldots,n_{\lambda})$. If
$B\subset\mathbb{F}^{2}$ is a set of distinct vertical points such that
$B\cap\Xi=\emptyset$ and $\bigcup_{j=0}^{\lambda}V_{j}(\Xi)\subsetneq
V_{0}(B)$, then $\Xi\cup B$ is $y$-tower and will be named a _derived
$y$-tower_ set from $\Xi$ with _base set_ $B$.
Proposition 3 and 4 imply the following corollary immediately.
###### Corollary 5.
Let $\Xi\subset\mathbb{F}^{2}$ be a quasi-tower set. Then there are infinite
derived tower sets from $\Xi$.
###### Example 1.
$\displaystyle\Xi=\\{$
$\displaystyle(0.2,0.4),(0.4,0.4),(0.8,0.4),(1,0.4),(1.2,0.4),(1.6,0.4),$
$\displaystyle(2,0.4),(0,0.6),(0.6,0.6),(1.4,0.6),(1.8,0.6),(2.2,0.6),$
$\displaystyle(0.6,1.1),(1,1.1),(1.2,1.1),(1.8,1.1),(1.2,1.45),(1.8,1.45),$
$\displaystyle(2.4,1.45),(0.2,1.8),(1.4,1.8)\\}\subset\mathbb{Q}^{2}$
is a quasi-$x$-tower set(shown in (a) of Figure 1) and
$\displaystyle B=\\{$
$\displaystyle(0,0),(0.2,0),(0.4,0),(0.6,0),(0.8,0),(1,0),$
$\displaystyle(1.2,0),(1.4,0),(1.6,0),(1.8,0),(2,0),(2.2,0),(2.4,0)\\}.$
Obviously, $\Xi\cup B$(shown in (b) of Figure 1) is an $x$-tower set that is
derived from $\Xi$ with base set $B$.
(a) $\Xi$ (b) $\Xi\cup B$
Figure 1: A quasi-$x$-tower set $\Xi$ and a derived $x$-tower set $\Xi\cup B$
from $\Xi$ with base set $B$
Proposition 6 of [7] discovered the relation between cartesian sets and tower
sets. The next proposition shows the relation between cartesian sets and
quasi-tower sets.
###### Proposition 6.
A quasi-tower set is not a cartesian set and vice versa.
###### Proof 2.
From Definition 3 we can deduce easily that a point set
$\Xi\subset\mathbb{F}^{2}$ is quasi-$x$-tower if and only if it is a
generalized $x$-tower set such that there exists at least one point
$\xi\in\Xi$ whose abscissa is not in $H_{0}(\Xi)$. Proposition 1 implies the
statement of this proposition immediately for quasi-$x$-tower sets. The
quasi-$y$-tower cases can be proved similarly. ∎
## 4 Bases for Lagrange Interpolation on Quasi-Tower Sets
In this section, we will present the degree reducing interpolation bases for
Lagrange interpolation on quasi-tower sets w.r.t.
$\prec_{\mathrm{lex}},\prec_{\mathrm{grevlex}}$ etc. theoretically. Let us
begin with the following lemmas.
###### Lemma 7.
Fix monomial ordering $\prec_{\mathrm{grlex}}$. Let
$\Xi^{\prime}=\\{(x_{ij},y_{j}):(i,j)\in S_{x}(\Xi^{\prime})\\}$
be an $x$-tower set in $\mathbb{F}^{2}$ with
$S_{x}(\Xi^{\prime})=\mathrm{L}_{x}(m_{0},m_{1},\ldots,m_{\nu})$. If
$\mathcal{G}_{\prec_{\mathrm{grlex}}}(\Xi^{\prime})=\\{g_{1},g_{2},\ldots,g_{s}\\}$
with $g_{s}\prec_{\mathrm{grlex}}\cdots\prec_{\mathrm{grlex}}g_{1}$, then
1. (i)
$s=\nu+2$.
2. (ii)
$g_{1}=(x-x_{0,0})(x-x_{1,0})\cdots(x-x_{m_{0},0})$.
3. (iii)
$g_{s}=(y-y_{0})(y-y_{1})\cdots(y-y_{\nu})$.
4. (iv)
$g_{j}\in\langle y-y_{0}\rangle$ and
$\mathrm{LM}_{\prec_{\mathrm{lex}}}(g_{j})=\mathrm{LM}_{\prec_{\mathrm{grlex}}}(g_{j})=x^{m_{j-1}+1}y^{j-1},\quad
j=2,\ldots,s-1.$
5. (v)
$\mathcal{I}(\\{(x_{0,0},y_{0}),\ldots,(x_{m_{0},0},y_{0})\\})=\langle
g_{1},y-y_{0}\rangle.$
###### Proof 3.
From Theorem 2 we know that
$\mathcal{N}_{\prec_{\mathrm{grlex}}}(\Xi^{\prime})=\\{x^{i}y^{j}:(i,j)\in
S_{x}(\Xi^{\prime})\\}$. Since
$S_{x}(\Xi^{\prime})=\mathrm{L}_{x}(m_{0},m_{1},\ldots,m_{\nu})$, it is easily
checked that
$\\{\mathrm{LM}_{\prec_{\mathrm{grlex}}}(g_{i}):1\leq i\leq
s\\}=\\{x^{m_{j}+1}y^{j}:0\leq j\leq\nu\\}\cup\\{y^{\nu+1}\\},$
which implies (i) readily. In the proof of Theorem 2(For brevity, we refer the
reader to [7] for details), we find that
$(x-x_{0,0})(x-x_{1,0})\cdots(x-x_{m_{0},0}),(y-y_{0})(y-y_{1})\cdots(y-y_{\nu})\in\mathcal{G}_{\prec_{\mathrm{grlex}}}(\Xi^{\prime})$.
Since $\Xi^{\prime}$ is $x$-tower, we have $m_{0}>m_{1}>\cdots>m_{\nu}\geq 0$.
Thus, for $0\leq j\leq\nu-1$,
$\displaystyle\deg x^{m_{j}+1}y^{j}$ $\displaystyle=m_{j}+1+j$
$\displaystyle\geq m_{j+1}+1+j+1$ $\displaystyle=\deg x^{m_{j+1}+1}y^{j+1}.$
Next, we claim that $\deg x^{m_{j}+1}y^{j}\geq\deg y^{\nu+1}$, that is
$m_{j}\geq\nu-j$, $j=0,\ldots,\nu$. In fact, it is easy to see from
$m_{0}>m_{1}>\cdots>m_{\nu}\geq 0$ that $m_{j}-m_{\nu}\geq\nu-j$, hence,
$m_{j}\geq\nu-j,j=0,\ldots,\nu$. Therefore, (ii), (iii) follows.
Still from the proof of Theorem 2, we know that $g_{j}\in\langle
y-y_{0}\rangle$ and there is no monomial $x^{\alpha_{1}}y^{\alpha_{2}}$ of
$g_{j}$ except $\mathrm{LM}_{\prec_{\mathrm{grlex}}}(g_{j})$ satisfying
$\alpha_{1}\geq m_{j-1}+1$. Therefore, the definition of
$\prec_{\mathrm{lex}}$ implies (iv).
Finally, (v) is a direct consequence of Lemma 8 in [7]. ∎
###### Lemma 8.
[8] Let $\Xi$ and $\Xi^{\prime}$ be varieties in $\mathbb{F}^{d}$. Then
$\mathcal{I}(\Xi):\mathcal{I}(\Xi^{\prime})=\mathcal{I}(\Xi-\Xi^{\prime}).$
###### Lemma 9.
[8] Let $\mathcal{I}$ be an ideal and $g$ an element of $\Pi^{2}$. If
$\\{h_{1},\ldots,h_{p}\\}$ is a basis of the ideal $\mathcal{I}\cap\langle
g\rangle$, then $\\{h_{1}/g,\ldots,h_{p}/g\\}$ is a basis of
$\mathcal{I}:\langle g\rangle$.
The following theorem shows us the degree reducing interpolation bases for
bivariate Lagrange interpolation on quasi-$x$-tower sets w.r.t.
$\prec_{\mathrm{grlex}}$.
###### Theorem 10.
Let $\Xi$ be a quasi-$x$-tower set in $\mathbb{F}^{2}$. The DRIMB for Lagrange
interpolation on $\Xi$ w.r.t. $\prec_{\mathrm{grlex}}$ is
$\mathcal{N}_{\prec_{\mathrm{grlex}}}(\Xi)=\\{x^{i}y^{j}:(i,j)\in
S_{x}(\Xi)\\}.$
###### Proof 4.
Firstly, we assume that $S_{x}(\Xi)=\mathrm{L}_{x}(m_{1},\ldots,m_{\nu})$. By
Corollary 5, there exists a derived $x$-tower set from $\Xi$, say
$\Xi^{\prime}$, with base set
$B=\\{(x_{0},y_{0}),(x_{1},y_{0}),\ldots,(x_{m_{0}},y_{0})\\}$. Thus,
Proposition 3 implies that
$S_{x}(\Xi^{\prime})=\mathrm{L}_{x}(m_{0},m_{1},\ldots,m_{\nu})$. Moreover, it
follows from (v) of Lemma 7 that
$\displaystyle\mathcal{I}(B)$
$\displaystyle=\langle(x-x_{0})(x-x_{1})\cdots(x-x_{m_{0}}),y-y_{0}\rangle$
$\displaystyle=\langle(x-x_{0})(x-x_{1})\cdots(x-x_{m_{0}})\rangle+\langle
y-y_{0}\rangle.$
Since $B\cap\Xi=\emptyset$, Lemma 8 leads to
$\mathcal{I}(\Xi)=\mathcal{I}(\Xi^{\prime}-B)=\mathcal{I}(\Xi^{\prime}):\mathcal{I}(B)$.
Therefore, we have
$\displaystyle\mathcal{I}(\Xi)$
$\displaystyle=\mathcal{I}(\Xi^{\prime}):\big{(}\langle(x-x_{0})(x-x_{1})\cdots(x-x_{m_{0}})\rangle+\langle
y-y_{0}\rangle\big{)}$
$\displaystyle=\big{(}\mathcal{I}(\Xi^{\prime}):\langle(x-x_{0})(x-x_{1})\cdots(x-x_{m_{0}})\rangle\big{)}\cap\big{(}\mathcal{I}(\Xi^{\prime}):\langle
y-y_{0}\rangle\big{)}.$
To be sure, it follows from (ii) of Lemma 7 that
$\langle(x-x_{0})(x-x_{1})\cdots(x-x_{m_{0}})\rangle\subset\mathcal{I}(\Xi^{\prime})$,
hence,
$\mathcal{I}(\Xi^{\prime}):\langle(x-x_{0})(x-x_{1})\cdots(x-x_{m_{0}})\rangle=\Pi^{2},$
which means
$\mathcal{I}(\Xi)=\mathcal{I}(\Xi^{\prime}):\langle y-y_{0}\rangle.$
Recalling Lemma 7, we suppose that
$\mathcal{G}_{\prec_{\mathrm{grlex}}}(\Xi^{\prime})=\\{g_{1},g_{2},\ldots,g_{\nu+2}\\}$
with $g_{\nu+2}\prec_{\mathrm{grlex}}\cdots\prec_{\mathrm{grlex}}g_{1}$.
Hence, we have
$\displaystyle g_{1}$ $\displaystyle=(x-x_{0})(x-x_{1})\cdots(x-x_{m_{0}}),$
$\displaystyle g_{i}$ $\displaystyle\in\langle y-y_{0}\rangle,\quad
i=2,\ldots,\nu+2.$
Next, for the basis for $\mathcal{I}(\Xi)$, we should compute the basis for
$\mathcal{I}(\Xi^{\prime})\cap\langle y-y_{0}\rangle$ so that we can apply
Lemma 9. Let $t$ be a new variable. According to [8],
$\displaystyle\mathcal{I}(\Xi^{\prime})\cap\langle y-y_{0}\rangle$
$\displaystyle=\langle t\mathcal{I}(\Xi^{\prime}),(1-t)\langle
y-y_{0}\rangle\rangle\cap\Pi^{2}$ $\displaystyle=\langle
tg_{1},tg_{2},\ldots,tg_{\nu+2},(1-t)(y-y_{0})\rangle\cap\Pi^{2}.$
Since $g_{i}\in\langle y-y_{0}\rangle,i=2,\ldots,\nu+2$,
$g_{i}/(y-y_{0})\in\Pi^{2}$, hence,
$\displaystyle tg_{i}+(g_{i}/(y-y_{0}))(1-t)(y-y_{0})$ $\displaystyle=$
$\displaystyle(g_{i}/(y-y_{0}))t(y-y_{0})+(g_{i}/(y-y_{0}))(1-t)(y-y_{0})$
$\displaystyle=$ $\displaystyle(g_{i}/(y-y_{0}))(y-y_{0})$ $\displaystyle=$
$\displaystyle g_{i},$
holds for $i=2,\ldots,\nu+2$. Thus,
$\mathcal{I}(\Xi^{\prime})\cap\langle y-y_{0}\rangle=\langle
tg_{1},(1-t)(y-y_{0}),g_{2},\ldots,g_{\nu+2}\rangle\cap\Pi^{2}.$
Since
$\mathcal{G}_{\prec_{\mathrm{grlex}}}(\Xi^{\prime})=\\{g_{1},g_{2},\ldots,g_{\nu+2}\\}$,
it is easily checked from Buchberger’s S-pair criterion and (iv) of Lemma 7
that
$\\{tg_{1},(1-t)(y-y_{0}),g_{2},\ldots,g_{\nu+2}\\}$
is the reduced Gröbner basis for $\langle
t\mathcal{I}(\Xi^{\prime}),(1-t)(y-y_{0})\rangle$ w.r.t.
$\prec_{\mathrm{lex}}$ with $y\prec_{\mathrm{lex}}x\prec_{\mathrm{lex}}t$.
Consequently,
$\mathcal{I}(\Xi^{\prime})\cap\langle y-y_{0}\rangle=\langle
g_{2},\ldots,g_{\nu+2}\rangle,$
hence,
$\mathcal{I}(\Xi)=\mathcal{I}(\Xi^{\prime}):\langle y-y_{0}\rangle=\langle
g_{2}/(y-y_{0}),\ldots,g_{\nu+2}/(y-y_{0})\rangle,$
follows from Lemma 9. In fact, we can deduce that
$\displaystyle\mathcal{G}_{\prec_{\mathrm{grlex}}}(\Xi)$
$\displaystyle=\\{g_{2}/(y-y_{0}),\ldots,g_{\nu+2}/(y-y_{0})\\}$
$\displaystyle=:\\{g_{1}^{\prime},\ldots,g_{\nu+1}^{\prime}\\}.$
By Lemma 7, $\Xi^{\prime}$ is $x$-tower implies that
$\\{\mathrm{LM}_{\prec_{\mathrm{grlex}}}(g_{j}):1\leq
j\leq\nu+2\\}=\\{x^{m_{j}+1}y^{j}:0\leq j\leq\nu\\}\cup\\{y^{\nu+1}\\},$
therefore,
$\\{\mathrm{LM}_{\prec_{\mathrm{grlex}}}(g_{j}^{\prime}):1\leq
j\leq\nu+1\\}=\\{x^{m_{j}+1}y^{j-1}:1\leq j\leq\nu\\}\cup\\{y^{\nu}\\}.$
As a result, since $S_{x}(\Xi)=\mathrm{L}_{x}(m_{1},\ldots,m_{\nu})$, we have
$\mathcal{N}_{\prec_{\mathrm{grlex}}}(\Xi)=\\{x^{i}y^{j}:(i,j)\in
S_{x}(\Xi)\\}$
which finishes the proof. ∎
Theorem 10 has solved the DRIMB problems w.r.t. $\prec_{\mathrm{grlex}}$.
Next, let us turn to DRINB problems.
###### Theorem 11.
Let $\Xi\subset\mathbb{F}^{2}$ be a quasi-$x$-tower set of points
$u_{mn}^{x}=(x_{mn},y_{n}),\quad(m,n)\in S_{x}(\Xi),$
which give rise to polynomials
$\phi_{ij}^{x}=\varphi_{ij}^{x}\prod_{t=0}^{j-1}(y-y_{t})\prod_{s=0}^{i-1}(x-x_{sj}),\quad(i,j)\in
S_{x}(\Xi),$
where
$\varphi_{ij}^{x}=1/\prod_{t=0}^{j-1}(y_{j}-y_{t})\prod_{s=0}^{i-1}(x_{ij}-x_{sj})\in\mathbb{F}$,
and the empty products are taken as 1. Then
$Q_{x}=\\{\phi_{ij}^{x}:(i,j)\in S_{x}(\Xi)\\}$
is a _DRINB_ w.r.t. $\prec_{\mathrm{grlex}}$ for (1) satisfying
$\phi_{ij}^{x}(u_{mn}^{x})=\delta_{(i,j),(m,n)},\quad(i,j)\succeq_{\mathrm{invlex}}(m,n).$
###### Proof 5.
Fix $(i,j)\in S_{x}(\Xi)$. When $(i,j)=(m,n)$, since $x_{0j}\neq
x_{1j}\neq\cdots\neq x_{ij}$ and $y_{0}\neq y_{1}\neq\cdots\neq y_{j}$,
$\phi_{ij}^{x}(u_{ij}^{x})=\varphi_{ij}^{x}\prod_{t=0}^{j-1}(y_{j}-y_{t})\prod_{s=0}^{i-1}(x_{ij}-x_{sj})\\\
=\varphi_{ij}^{x}/\varphi_{ij}^{x}=1$
follows. Otherwise, if $(i,j)\succ_{\mathrm{invlex}}(m,n)$, we have $j>n$, or
$j=n,i>m$. When $j>n$, we have
$\phi_{ij}^{x}(u_{mn}^{x})=\varphi_{ij}^{x}(y_{n}-y_{0})\cdots(y_{n}-y_{n})\cdots(y_{n}-y_{j-1})\prod_{s=0}^{i-1}(x_{mn}-x_{sj})=0,$
and when $j=n,i>m$,
$\displaystyle\phi_{ij}^{x}(u_{mn}^{x})$
$\displaystyle=\varphi_{ij}^{x}\prod_{t=0}^{j-1}(y_{n}-y_{t})(x_{mn}-x_{0j})\cdots(x_{mn}-x_{mj})\cdots(x_{mn}-x_{i-1,j})$
$\displaystyle=\varphi_{ij}^{x}\prod_{t=0}^{n-1}(y_{n}-y_{t})(x_{mn}-x_{0n})\cdots(x_{mn}-x_{mn})\cdots(x_{mn}-x_{i-1,n})$
$\displaystyle=0,$
which leads to
$\phi_{ij}^{x}(u_{mn}^{x})=0,\quad(i,j)\succ_{\mathrm{invlex}}(m,n),$
namely $Q_{x}$ is a Newton basis for $\mathrm{Span}_{\mathbb{F}}Q_{x}$. By
Theorem 10, it is easy to see that
$\mathrm{Span}_{\mathbb{F}}Q_{x}=\mathrm{Span}_{\mathbb{F}}{\mathcal{N}_{\prec_{\mathrm{grlex}}}(\Xi)}$.
Therefore, $Q_{x}$ is a DRINB w.r.t. $\prec_{\mathrm{grlex}}$ for (1). ∎
Similarly, we can prove the following theorems:
###### Theorem 12.
Let $\Xi$ be a quasi-$y$-tower set in $\mathbb{F}^{2}$. The DRIMB for Lagrange
interpolation on $\Xi$ w.r.t. $\prec_{\mathrm{grevlex}}$ is
$\mathcal{N}_{\prec_{\mathrm{grevlex}}}(\Xi)=\\{x^{i}y^{j}:(i,j)\in
S_{y}(\Xi)\\}.$
###### Theorem 13.
Let $\Xi\subset\mathbb{F}^{2}$ be a quasi-$y$-tower set of points
$u_{mn}^{y}=(x_{m},y_{mn}),(m,n)\in S_{y}(\Xi).$
We define the polynomials
$\phi_{ij}^{y}=\varphi_{ij}^{y}\prod_{s=0}^{i-1}(x-x_{s})\prod_{t=0}^{j-1}(y-y_{it}),\quad(i,j)\in
S_{y}(\Xi),$
where
$\varphi_{ij}^{y}=1/\prod_{s=0}^{i-1}(x_{i}-x_{s})\prod_{t=0}^{j-1}(y_{ij}-y_{it})\in\mathbb{F}$.
The empty products are taken as 1. Then,
$Q_{y}=\\{\phi_{ij}^{x}:(i,j)\in S_{y}(\Xi)\\}$ (2)
is a _DRINB_ w.r.t. $\prec_{\mathrm{grevlex}}$ for (1) satisfying
$\phi_{ij}^{y}(u_{mn}^{y})=\delta_{(i,j),(m,n)},\quad(i,j)\succeq_{\mathrm{lex}}(m,n).$
So far, we have discovered the degree reducing interpolation bases for
Lagrange interpolation on quasi-tower sets w.r.t. $\prec_{\mathrm{grlex}}$ and
$\prec_{\mathrm{grevlex}}$. Next, we should turn to $\prec_{\mathrm{lex}}$ and
$\prec_{\mathrm{invlex}}$ cases.
###### Lemma 14.
[9] Let
$\displaystyle\Xi$
$\displaystyle=\\{u_{mn}^{x}=(x_{mn}^{x},y_{mn}^{x}):(m,n)\in S_{x}(\Xi)\\}$
$\displaystyle=\\{u_{mn}^{y}=(x_{mn}^{y},y_{mn}^{y}):(m,n)\in S_{y}(\Xi)\\}$
be a set of distinct points in $\mathbb{F}^{2}$. Then
(i) the set $\\{x^{i}y^{j}:(i,j)\in S_{x}(\Xi)\\}$ is the _DRIMB_ as well as
$\\{\phi_{ij}^{x}:(i,j)\in S_{x}(\Xi)\\}$ is a _DRINB_ w.r.t.
$\prec_{\mathrm{lex}}$ for (1), where
$\phi_{ij}^{x}=\varphi_{ij}^{x}\prod_{t=0}^{j-1}(y-y_{0t}^{x})\prod_{s=0}^{i-1}(x-x_{sj}^{x}),\quad(i,j)\in
S_{x}(\Xi),$
with
$\varphi_{ij}^{x}=1/\prod_{t=0}^{j-1}(y_{0j}^{x}-y_{0t}^{x})\prod_{s=0}^{i-1}(x_{ij}^{x}-x_{sj}^{x})\in\mathbb{F}$
and the empty products taken as 1;
(ii) the set $\\{x^{i}y^{j}:(i,j)\in S_{y}(\Xi)\\}$ is the _DRIMB_ as well as
$\\{\phi_{ij}^{y}:(i,j)\in S_{y}(\Xi)\\}$ is a _DRINB_ w.r.t.
$\prec_{\mathrm{invlex}}$ for (1), where
$\phi_{ij}^{y}=\varphi_{ij}^{y}\prod_{s=0}^{i-1}(x-x_{s0}^{y})\prod_{t=0}^{j-1}(y-y_{it}^{y}),\quad(i,j)\in
S_{y}(\Xi),$
with
$\varphi_{ij}^{y}=1/\prod_{s=0}^{i-1}(x_{i0}^{y}-x_{s0}^{y})\prod_{t=0}^{j-1}(y_{ij}^{y}-y_{it}^{y})\in\mathbb{F}$,
and the empty products are taken as 1.
Finally, Theorem 10-13 and Lemma 14 together give rise to our main theorem.
###### Theorem 15.
Let
$\Xi^{x}=\\{u_{ij}^{x}=(x_{ij}^{x},y_{j}^{x}):(i,j)\in S_{x}(\Xi^{x})\\}$
be a quasi-$x$-tower set in $\mathbb{F}^{2}$. Then the set
$\\{x^{i}y^{j}:(i,j)\in S_{x}(\Xi^{x})\\}$ is the _DRIMB_ as well as
$\\{\phi_{ij}^{x}:(i,j)\in S_{x}(\Xi^{x})\\}$ is a _DRINB_ w.r.t.
$\prec_{\mathrm{lex}}$ or $\prec_{\mathrm{grlex}}$ for (1), where
$\phi_{ij}^{x}=\varphi_{ij}^{x}\prod_{t=0}^{j-1}(y-y_{t}^{x})\prod_{s=0}^{i-1}(x-x_{sj}^{x}),\quad(i,j)\in
S_{x}(\Xi^{x}),$
with
$\varphi_{ij}^{x}=1/\prod_{t=0}^{j-1}(y_{j}^{x}-y_{t}^{x})\prod_{s=0}^{i-1}(x_{ij}^{x}-x_{sj}^{x})\in\mathbb{F}$
and the empty products taken as 1.
If
$\Xi^{y}=\\{u_{ij}^{y}=(x_{i}^{y},y_{ij}^{y}):(i,j)\in
S_{y}(\Xi^{y})\\}\subset\mathbb{F}^{2}$
is a quasi-$y$-tower set. Then the set $\\{x^{i}y^{j}:(i,j)\in
S_{y}(\Xi^{y})\\}$ is the _DRIMB_ as well as $\\{\phi_{ij}^{y}:(i,j)\in
S_{y}(\Xi^{y})\\}$ is a _DRINB_ w.r.t. $\prec_{\mathrm{invlex}}$ or
$\prec_{\mathrm{grevlex}}$ for (1), where
$\phi_{ij}^{y}=\varphi_{ij}^{y}\prod_{s=0}^{i-1}(x-x_{s}^{y})\prod_{t=0}^{j-1}(y-y_{it}^{y}),\quad(i,j)\in
S_{y}(\Xi^{y}),$
with
$\varphi_{ij}^{y}=1/\prod_{s=0}^{i-1}(x_{i}^{y}-x_{s}^{y})\prod_{t=0}^{j-1}(y_{ij}^{y}-y_{it}^{y})\in\mathbb{F}$,
and the empty products are taken as 1.
###### Example 2.
Let us continue with Example 1. Observeing Figure 1, we have
$S_{x}(\Xi)=\mathrm{L}_{x}(6,4,3,2,1).$
Therefore, according to Theorem 15,
$\displaystyle\mathcal{N}_{\prec_{\mathrm{grlex}}}(\Xi)=\mathcal{N}_{\prec_{\mathrm{lex}}}(\Xi)=$
$\displaystyle\\{x^{i}y^{j}:(i,j)\in S_{x}(\Xi)\\}$ (3) $\displaystyle=$
$\displaystyle\\{1,x,x^{2},x^{3},x^{4},x^{5},x^{6},y,xy,x^{2}y,x^{3}y,x^{4}y,$
$\displaystyle\phantom{\\{}y^{2},xy^{2},x^{2}y^{2},x^{3}y^{2},y^{4},xy^{4},x^{2}y^{4},y^{5},xy^{5}\\}$
and a DRINB for interpolation on $\Xi$ w.r.t. $\prec_{\mathrm{lex}}$ or
$\prec_{\mathrm{grlex}}$ is
$\displaystyle\Bigg{\\{}1,5(x-0.2),\frac{25}{6}(x-0.2)(x-0.4),\ldots,$ (4)
$\displaystyle\phantom{\\{}5(y-0.4),\frac{25}{3}x(y-0.4),\ldots,$
$\displaystyle\phantom{\\{}\frac{20}{7}(y-0.4)(y-0.6),\ldots\Bigg{\\}}.$
## 5 Algorithms and Timings
At the beginning of this section is a restatement of the classical BM
algorithm with the notation established above.
###### Algorithm 1.
(BM Algorithm)
Input: A set of distinct points
$\Xi=\\{\xi^{(i)}:i=1,\ldots,\mu\\}\subset\mathbb{F}^{d}$ and a fixed monomial
ordering $\prec$.
Output: The 3-tuple $(G,N,Q)$, where $G$ is the reduced Gröbner basis for
$\mathcal{I}(\Xi)$ w.r.t. $\prec$, $N$ is the Gröbner éscalier of
$\mathcal{I}(\Xi)$ (the DRIMB for (1) also) w.r.t. $\prec$, and $Q$ is a DRINB
w.r.t. $\prec$ for (1).
BM1. Start with lists $G=[\ ],N=[\ ],Q=[\ ],L=[1]$, and a matrix $B=(b_{ij})$
over $\mathbb{F}$ with $\mu$ columns and zero rows initially.
BM2. If $L=[\ ]$, return $(G,N,Q)$ and stop. Otherwise, choose the monomial
$t=\mbox{min}_{\prec}L$, and delete $t$ from $L$.
BM3. Compute the evaluation vector $(t(\xi^{(1)}),\ldots,t(\xi^{(\mu)}))$, and
reduce it against the rows of $B$ to obtain
$(v_{1},\ldots,v_{\mu})=(t(\xi^{(1)}),\ldots,t(\xi^{(\mu)}))-\sum_{i}a_{i}(b_{i1},\ldots,b_{i\mu}),\quad
a_{i}\in\mathbb{F}.$
BM4.. If $(v_{1},\ldots,v_{\mu})=(0,\ldots,0)$, then append the polynomial
$t-\sum_{i}a_{i}q_{i}$ to the list $G$, where $q_{i}$ is the $i$th element of
$Q$. Remove from $L$ all the multiples of $t$. Continue with BM2.
BM5. Otherwise $(v_{1},\ldots,v_{\mu})\neq(0,\ldots,0)$, add
$(v_{1},\ldots,v_{\mu})$ as a new row to $B$ and $t-\sum_{i}a_{i}q_{i}$ as a
new element to $Q$. Append the monomial $t$ to $N$, and add to $L$ those
elements of $\\{x_{1}t,\ldots,x_{d}t\\}$ that are neither multiples of an
element of $L$ nor of $\mathrm{LM}_{\prec}(G):=\\{\mathrm{LM}_{\prec}(g):g\in
G\\}$. Continue with BM2.
Like TBM algorithm in [7], we have the following QTBM algorithm for quasi-
tower sets.
###### Algorithm 2.
(QTBM algorithm)
Input: A quasi-$x$-tower(quasi-$y$-tower) set $\Xi\subset\mathbb{F}^{2}$ of
$\mu$ points and a fixed monomial ordering
$\prec_{\mathrm{grlex}}$($\prec_{\mathrm{grevlex}}$) or
$\prec_{\mathrm{lex}}$($\prec_{\mathrm{invlex}}$).
Output: The 3-tuple $(G,N,Q)$, where $G$ is the reduced Gröbner basis for
$\mathcal{I}(\Xi)$, $N$ is the Gröbner éscalier of $\mathcal{I}(\Xi)$, and $Q$
is a DRINB for (1).
QTBM1. Construct lower set $S_{x}(\Xi)$($S_{y}(\Xi)$).
QTBM2. Compute the sets $N$ and $Q$ according to Theorem 15.
QTBM3. Compute the set $L:=\\{x\cdot t:t\in N\\}\bigcup\\{y\cdot t:t\in
N\\}\setminus N$.
QTBM4. Construct $\mu\times\mu$ matrix $C$ whose $(h,k)$ entry is
$\phi_{h}^{x}(u_{k}^{x})$($\phi_{h}^{y}(u_{k}^{y})$) where
$\phi_{h}^{x}$($\phi_{h}^{y}$), $u_{k}^{x}$($u_{k}^{y}$) are $h$th and $k$th
elements of $Q=\\{\phi_{ij}^{x}(\phi_{ij}^{y}):(i,j)\in
S_{x}(\Xi)(S_{y}(\Xi))\\}$ and $\Xi=\\{u_{mn}^{x}(u_{mn}^{y}):(m,n)\in
S_{x}(\Xi)(S_{y}(\Xi))\\}$ w.r.t. the increasing
$\prec_{\mathrm{invlex}}$($\prec_{\mathrm{lex}}$) on $(i,j)$ and $(m,n)$
respectively.
QTBM5. Goto BM2 with $N,Q,L,C$ for the reduced Gröbner basis $G$.
###### Example 3.
We continue with Example 1 and 2. Example 2 had obtained $N$ and $Q$ in (3)
and (4) respectively. According to QTBM3 and QTBM4, we get
$L=\\{x^{7},x^{5}y,x^{6}y,x^{4}y^{2},x^{3}y^{3},x^{2}y^{4},y^{5},xy^{5}\\}$
and
$C=\left(\begin{array}[]{cccc}1&1&1&\cdots\\\ 0&1&3&\cdots\\\ 0&0&1&\cdots\\\
\vdots&\vdots&\vdots&\ddots\\\ \end{array}\right).$
Finally, QTBM5 yields
$\displaystyle G=\\{(y-0.4)(y-0.6)(y-1.1)(y-1.45)(y-1.8),\ldots\\}.$
At the end, we will show the timings for the computations of BM-problems on
quasi-tower sets in finite prime fields $\mathbb{F}_{q}$ with size $q$ w.r.t.
monomial orderings $\prec_{\mathrm{grlex}}$ and $\prec_{\mathrm{invlex}}$
respectively. QTBM and BM algorithms were implemented on Maple 12 installed on
a laptop with 768 Mb RAM and 1.5 GHz CPU.
For field $\mathbb{F}_{37}$ and $\prec_{\mathrm{grlex}}$,
Algorithms $\\#\Xi$ | 300 | 500 | 800 | 1200
---|---|---|---|---
QTBM | 2.563 s | 8.773 s | 26.068 s | 63.772 s
BM | 20.690 s | 80.375 s | 288.364 s | 901.246 s
For field $\mathbb{F}_{43}$ and $\prec_{\mathrm{invlex}}$,
Algorithms $\\#\Xi$ | 500 | 800 | 1000 | 1200
---|---|---|---|---
QTBM | 8.171 s | 25.607 s | 48.170 s | 70.802 s
BM | 114.996 s | 375.610 s | 881.988 s | 1347.287
## References
* [1] T. Sauer, Polynomial interpolation in several variables: Lattices, differences, and ideals, in: K. Jetter, M. Buhmann, W. Haussmann, R. Schaback, J. Stöckler (Eds.), Topics in Multivariate Approximation and Interpolation, Vol. 12 of Studies in Computational Mathematics, Elsevier, Amsterdam, 2006, pp. 191–230.
* [2] T. Mora, Gröbner technology, in: M. Sala, T. Mora, L. Perret, S. Sakata, C. Traverso (Eds.), Gröbner Bases, Coding, and Cryptography, Springer, Berlin, 2009, pp. 11–25.
* [3] H. Möller, B. Buchberger, The construction of multivariate polynomials with preassigned zeros, in: J. Calmet (Ed.), Computer Algebra: EUROCAM ’82, Vol. 144 of Lecture Notes in Computer Science, Springer, Berlin, 1982, pp. 24–31.
* [4] S. Lundqvist, Complexity of comparing monomials and two improvements of the Buchberger-Möller algorithm, in: Mathematical Methods in Computer Science, 2008, pp. 105–125.
* [5] T. Sauer, Lagrange interpolation on subgrids of tensor product grids, Math. Comp. 73 (245) (2004) 181–190.
* [6] T. Chen, T. Dong, S. Zhang, The Newton interpolation bases on lower sets, J. Inf. Comput. Sci. 3 (3) (2006) 385–394.
* [7] T. Dong, T. Chen, S. Zhang, X. Wang, Bivariate Lagrange interpolation on tower interpolation sites, submitted. Preprint available online at: http://arxiv.org/abs/1001.1196.
* [8] D. Cox, J. Little, D. O’Shea, Ideal, Varieties, and Algorithms, 3rd Edition, Undergraduate Texts in Mathematics, Springer, New York, 2007.
* [9] X. Wang, S. Zhang, T. Dong, A bivariate preprocessing paradigm for Buchberger-Möller algorithm, submitted. Preprint available online at: http://arxiv.org/abs/1001.1186.
|
arxiv-papers
| 2010-03-12T08:46:39 |
2024-09-04T02:49:09.047516
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tian Dong, Xiaoying Wang, Shugong Zhang, Peng Li",
"submitter": "Tian Dong",
"url": "https://arxiv.org/abs/1003.2493"
}
|
1003.2518
|
# A Kähler Structure on Cartan Spaces
E. Peyghan and A. Tayebi
###### Abstract
In this paper, we define a new metric on Cartan manifolds and obtain a Kähler
structure on their cotangent bundles. We prove that on a Cartan manifold $M$
of negative constant flag curvature, $(T^{\ast}M_{0},G,J)$ has a Kählerian
structure. For Cartan manifolds of positive constant flag curvature, we show
that the tube around the zero section has a Kählerian structure. Finally by
computing the Levi-Civita connection and components of curvature related to
this metric, we show that there is no non-Riemannian Cartan structure such
that $(T^{\ast}M_{0},G,J)$ became a Einstein manifold or locally symmetric
manifold.1112010 Mathematics Subject Classification: Primary 53B40, 53C60
Keywords: Cartan space, Kähler structure, symmetric space, Einstein manifold.
## 1 Introduction
The modern formulation of the notion of Cartan spaces is due to the R. Miron
[12], [13], [14]. Based on the studies of E. Cartan, A. Kawaguchi [7], R.
Miron [11], [13], [14], S. Vacaru [19], [20], [21], D. Hrimiuc and H. Shimada
[5], [6], P.L. Antonelli and M. Anastasiei [2], [9], [10], etc., the geometry
of Cartan spaces is today an important chapter of differential geometry.
Analytical Mechanics and some theories in Physics brought into discussion
regular Lagrangians and their geometry [9]. Regular Lagrangian which is
2-homogeneous in velocities is nothing but the square of a fundamental Finsler
function and its geometry is Finsler geometry. This geometry was developed
since 1918 by P. Finsler, E. Cartan, L. Berwald, H. Akbar-Zadeh and many
others, see [8] and the most recent graduate texts [1][4][17][18]. On the
other hand, there is a strong linkage among Finsler, Hamilton and Cartan
geometries [16]. For example, Anastasiei and Vacaru provide a method of
converting Lagrange and Finsler spaces and their Legendre transforms to
Hamilton and Cartan spaces into almost Kähler structures on tangent and
cotangent bundles [3]. But in Mechanics and Physics there exists also regular
Hamiltonians whose geometry is also useful. This geometry is mainly due to
Miron, and it is now systematically presented in the monograph [11]. A
manifold endowed with a regular Hamiltonian which is 2-Homogeneous in momenta
was called a Cartan space.
Let us denote the Hamiltonian structure on a manifold $M$ by $(M,H(x,p))$. If
the fundamental function $H(x,p)$ is 2-homogeneous on the fibres of the
cotangent bundle $(T^{*}M,M)$, then the notion of Cartan space is obtained. It
is remarkable that these spaces appear as dual of the Finsler spaces, via
Legendre transformation. Using this duality several important results in the
Cartan spaces can be obtained: the canonical nonlinear connection, the
canonical metrical connection, etc. Therefore, the theory of Cartan spaces has
the same symmetry and beauty like Finsler geometry. Moreover, it gives a
geometrical framework for the Hamiltonian theory of Mechanics or Physical
fields.
Let $(M,K)$ be a Cartan space on a manifold $M$ and $\tau:=\frac{1}{2}K^{2}$.
Let us define the symmetric $M$-tensor field
$G_{ij}:=\frac{1}{\beta}g_{ij}+\frac{v(\tau)}{\alpha\beta}p_{i}p_{j}$ on
$T^{\ast}M_{0}$, where $v$ is a real valued smooth function defined on
$[0,\infty)\subset{\mathbb{R}}$ and $\alpha$ and $\beta$ are real constants.
Then we can define the following Riemannian metric on $T^{\ast}M_{0}$
$G=G_{ij}dx^{i}dx^{j}+G^{ij}\delta p_{i}\delta p_{j},$
where $G^{ij}$ is the inverse of $G_{ij}$. Then we define an almost complex
structure $J$ on $T^{\ast}M_{0}$ by $J(\delta_{i})=G_{ik}\dot{\partial}^{k}$
and $J(\dot{\partial}^{i})=-G^{ik}\delta_{k}$.
In this paper, we prove that $(T^{\ast}M_{0},G,J)$ is an almost Kählerian
manifold. Then we show that the almost complex structure $J$ on
$T^{\ast}M_{0}$ is integrable if and only if $M$ has constant scalar curvature
$c$ and the function $v$ is given by $v=-c\alpha\beta^{2}$. We conclude that
on a Cartan manifold $M$ of negative constant flag curvature,
$(T^{\ast}M_{0},G,J)$ has a Kählerian structure. For Cartan manifolds of
positive constant flag curvature, we show that the tube around the zero
section has a Kählerian structure (see Theorem 3.5).
Then we find the Levi-Civita connection $\nabla$ of the metric $G$. For the
connection $\nabla$, we compute all of components curvature. For a Cartan
space $(M,K)$ of constant curvature $c$, we prove that in the following cases
$(M,K)$ reduces to a Riemannian space: $(i)$ for $c<0$, $(T^{\ast}M_{0},G,J)$
became a Kähler Einstein manifold, $(ii)$ for $c>0$,
$(T_{\beta}^{\ast}M_{0},G,J)$ became a Kähler Einstein manifold, where
$T_{\beta}^{\ast}M_{0}$ is the tube in $T^{\ast}M_{0}$. It results that, there
is not any non-Riemannian Cartan structure such that $(T^{\ast}M_{0},G,J)$
became a Einstein manifold (see Theorem 4.1).
Finally, for a Cartan space $(M,K)$ of constant curvature $c$, we prove that
in the following cases $(M,K)$ reduces to a Riemannian space: $(i)$ for $c<0$,
$(T^{\ast}M_{0},G,J)$ is a locally symmetric Kähler manifold, $(ii)$ for
$c>0$, $(T_{\beta}^{\ast}M_{0},G,J)$ is a locally symmetric Kähler manifold.
Then we conclude that there is not any non-Riemannian Cartan structure such
that $(T^{\ast}M_{0},G,J)$ became a locally symmetric manifold (see Theorem
4.3).
## 2 Preliminaries
Let $M$ be an $n$-dimensional $C^{\infty}$ manifold and
$\pi^{\ast}:T^{\ast}M\rightarrow M$ its cotangent bundle. If $(x^{i})$ are
local coordinates on $M$, then $(x^{i},p_{i})$ will be taken as local
coordinates on $T^{\ast}M$ with the momenta $(p_{i})$ provided by
$p=p_{i}dx^{i}$ where $p\in T^{\ast}_{x}M$, $x=(x^{i})$ and $(dx^{i})$ is the
natural basis of $T^{\ast}_{x}M$. The indices $i,j,k,\ldots$ will run from 1
to $n$ and the Einstein convention on summation will be used.
Put $\partial_{i}:=\frac{\partial}{\partial x^{i}}$ and
$\dot{\partial}^{i}:=\frac{\partial}{\partial p_{i}}$. Then
$(\partial_{i},\dot{\partial}^{i})$ be the natural basis in
$T_{(x,p)}T^{\ast}M$ and $(dx^{i},dp_{i})$ be the dual basis of it. The kernel
$V_{(x,p)}$ of the differential $d\pi^{\ast}:T_{(x,p)}T^{\ast}M\rightarrow
T_{x}M$ is called the vertical subspace of $T_{(x,p)}T^{\ast}M$ and the
mapping $(x,p)\rightarrow V_{(x,p)}$ is a regular distribution on $T^{\ast}M$
called the vertical distribution. This is integrable with the leaves
$T^{\ast}_{x}M$, $x\in M$ and is locally spanned by $\dot{\partial}^{i}$. The
vector field $C^{\ast}=p_{i}\dot{\partial}^{i}$ is called the Liouville vector
field and $\omega=p_{i}dx^{i}$ is called the Liouville 1-form on $T^{\ast}M$.
Then $d\omega$ is the canonical symplectic structure on $T^{\ast}M$. For an
easer handling of the geometrical objects on $T^{\ast}M$ it is usual to
consider a supplementary distribution to the vertical distribution,
$(x,p)\rightarrow N_{(x,p)}$, called the horizontal distribution and to report
all geometrical objects on $T^{\ast}M$ to the decomposition
$T_{(x,p)}T^{\ast}M=N_{(x,p)}\oplus V_{(x,p)}.$ (1)
The pieces produced by the decomposition (1) are called d-geometrical objects
(d is for distinguished) since their local components behave like geometrical
objects on $M$, although they depend on $x=(x^{i})$ and momenta $p=(p_{i})$.
The horizontal distribution is taken as being locally spanned by the local
vector fields
$\delta_{i}:=\partial_{i}+N_{ij}(x,p)\dot{\partial}^{j}.$ (2)
The horizontal distribution is called also a nonlinear connection on
$T^{\ast}M$ and the functions $(N_{ij})$ are called the local coefficients of
this nonlinear connection. It is important to note that any regular
Hamiltonian on $T^{\ast}M$ determines a nonlinear connection whose local
coefficients verify $N_{ij}=N_{ji}$. The basis
$(\delta_{i},\dot{\partial}^{i})$ is adapted to the decomposition (1). The
dual of it is $(dx^{i},\delta P_{i})$, for $\delta p_{i}=dp_{i}-N_{ji}dx^{j}$.
A Cartan structure on $M$ is a function $K:T^{\ast}M\longrightarrow[0,\infty)$
which has the following properties: (i) $K$ is $C^{\infty}$ on
$T^{\ast}M_{0}=T^{\ast}M-\\{0\\}$; (ii) $K(x,\lambda p)=\lambda K(x,p)$ for
all $\lambda>0$ and (iii) the $n\times n$ matrix $(g^{ij})$, where
$g^{ij}(x,p)=\frac{1}{2}\dot{\partial}^{i}\dot{\partial}^{j}K^{2}(x,p)$, is
positive definite at all points of $T^{\ast}M_{0}$. We notice that in fact
$K(x,p)>0$, whenever $p\neq 0$. The pair $(M,K)$ is called a Cartan space.
Using this notations, let us define
$p^{i}=\frac{1}{2}\dot{\partial}^{i}K^{2}\ \ \textrm{and}\ \
C^{ijk}=-\frac{1}{4}\dot{\partial}^{i}\dot{\partial}^{j}\dot{\partial}^{k}K^{2}.$
The properties of $K$ imply that
$\displaystyle p^{i}=g^{ij}p_{j},\ \ p_{i}=g_{ij}p^{j},\ \
K^{2}=g^{ij}p_{i}p_{j}=p_{i}p^{j},$ $\displaystyle
C^{ijk}p_{k}=C^{ikj}p_{k}=C^{kij}p_{k}=0.$ (3)
One considers the formal Christoffel symbols by
$\gamma^{i}_{jk}(x,p):=\frac{1}{2}g^{is}(\partial_{k}g_{js}+\partial_{j}g_{sk}-\partial_{s}g_{jk}),$
(4)
and the contractions $\gamma^{\circ}_{jk}(x,p):=\gamma^{i}_{jk}(x,p)p_{i}$,
$\gamma^{\circ}_{j\circ}:=\gamma^{i}_{jk}p_{i}p^{k}$. Then the functions
$N_{ij}(x,p)=\gamma^{\circ}_{ij}(x,p)-\frac{1}{2}\gamma^{\circ}_{h\circ}(x,p)\dot{\partial}^{h}g_{ij}(x,p),$
(5)
define a nonlinear connection on $T^{\ast}M$. This nonlinear connection was
discovered by R. Miron [12]. Thus a decomposition (1) holds. From now on we
shall use only the nonlinear connection given by (5).
A linear connection $D$ on $T^{\ast}M$ is said to be an $N$-linear connection
if $D$ preserves by parallelism the distribution $N$ and $V$, also we have
$D\theta=0$, for $\theta=\delta p_{i}\wedge dx^{i}$. One proves that an
$N$-linear connection can be represented in the adapted basis
$(\delta_{i},\dot{\partial}^{i})$ in the form
$\displaystyle D_{\delta_{j}}\delta_{i}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!H^{k}_{ij}\delta_{j},\ \ \
D_{\delta_{j}}\dot{\partial}^{i}=-H^{i}_{kj}\dot{\partial}^{k},$ (6)
$\displaystyle D_{\dot{\partial}^{j}}\delta_{i}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!V^{kj}_{i}\delta_{k},\ \ \
D_{\dot{\partial}^{j}}\dot{\partial}^{i}=-V^{ij}_{k}\dot{\partial}^{k},$ (7)
where $V^{kj}_{i}$ is a d-tensor field and $H^{k}_{ij}(x,p)$ behave like the
coefficients of a linear connection on $M$. The functions $H^{k}_{ij}$ and
$V^{kj}_{i}$ define operators of h-covariant and v-covariant derivatives in
the algebra of d-tensor fields, denoted by |k and $\mid^{k}$, respectively.
For $g^{ij}$ these are given by
$\displaystyle{g^{ij}}_{|k}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\delta_{k}g^{ij}+g^{sj}H^{i}_{sk}+g^{is}H^{j}_{sk},$
(8) $\displaystyle g^{ij}\\!\\!\mid^{k}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\dot{\partial}^{k}g^{ij}+g^{sj}V^{ik}_{s}+g^{is}V^{jk}_{s}.$
(9)
An N-linear connection given in the adapted basis
$(\delta_{i},\dot{\partial}^{j})$ as $D\Gamma(N)=(H^{i}_{jk},V^{ik}_{j})$ is
called metrical if
${g^{ij}}_{|k}=0,\ \ \ \ g^{ij}\\!\\!\mid^{k}=0.$ (10)
One verifies that the N-linear connection $C\Gamma(N)=(H^{i}_{jk},C^{ik}_{j})$
with
$\displaystyle H^{i}_{jk}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\frac{1}{2}g^{is}(\delta_{j}g_{sk}+\delta_{k}g_{js}-\delta_{s}g_{jk}),$
(11) $\displaystyle C^{jk}_{i}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!-\frac{1}{2}g_{is}(\dot{\partial}^{j}g^{sk}+\dot{\partial}^{k}g^{sj}-\dot{\partial}^{s}g^{jk})=g_{is}C^{sjk},$
(12)
is metrical and its h-torsion $T^{i}_{jk}:=H^{i}_{jk}-H^{i}_{kj}=0$, v-torsion
$S^{jk}_{i}:=C^{jk}_{i}-C^{kj}_{i}=0$ and deflection tensor
$\Delta_{ij}=N_{ij}-p_{k}H^{k}_{ij}=0$. Moreover, it is unique with these
properties. This is called the canonical metrical connection of the Cartan
space $(M,K)$. It has also the following properties:
$\displaystyle K^{2}_{|j}:\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\delta_{j}K^{2}=0,\ \ K^{2}\\!\\!\mid^{j}=2p^{j},$
(13) $\displaystyle p_{i|j}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!p^{i}_{|j}=0,\ \
p_{i}\\!\\!\mid^{j}=\delta^{j}_{i},\ \ p^{i}\\!\\!\mid^{j}=g^{ij}.$ (14)
$\displaystyle R_{kij}p^{k}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!0,\ \ P^{i}_{jk}p^{j}=0,\ \
P^{i}_{jk}:=H^{i}_{jk}-{\dot{\partial}}^{i}N_{jk}.$ (15)
$\displaystyle\delta_{i}g_{jk}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!H^{s}_{ji}g_{sk}+H^{s}_{ki}g_{js}.$ (16)
## 3 Kähler Structures on $T^{\ast}M_{0}$
Suppose that
$\tau:=\frac{1}{2}K^{2}=\frac{1}{2}g^{ij}(x,p)p_{i}p_{j},$ (17)
we consider a real valued smooth function $v$ defined on
$[0,\infty)\subset{\mathbb{R}}$ and real constants $\alpha$ and $\beta$. We
define the following symmetric $M$-tensor field of type (0,2) on
$T^{\ast}M_{0}$ having the components
$G_{ij}:=\frac{1}{\beta}g_{ij}+\frac{v(\tau)}{\alpha\beta}p_{i}p_{j}.$ (18)
It follows easily that the matrix $(G_{ij})$ is positive definite if and only
if $\alpha,\beta>0,\,\,\,\,\alpha+2\tau v>0.$ The inverse of this matrix has
the entries
$G^{kl}=\beta g^{kl}-\frac{v\beta}{\alpha+2\tau v}p^{k}p^{l}.$ (19)
The components $G^{kl}$ define symmetric $M$-tensor field of type (0,2) on
$T^{\ast}M_{0}$. It is easy to see that if the matrix $(G_{ij})$ is positive
definite, then matrix $(G^{kl})$ is positive definite too. Using $(G_{ij})$
and $(G^{ij})$, the following Riemannian metric on $T^{\ast}M_{0}$ is defined
$G=G_{ij}dx^{i}dx^{j}+G^{ij}\delta p_{i}\delta p_{j}.$ (20)
Now, we define an almost complex structure $J$ on $T^{\ast}M_{0}$ by
$J(\delta_{i})=G_{ik}\dot{\partial}^{k},\ \ \
J(\dot{\partial}^{i})=-G^{ik}\delta_{k}.$ (21)
It is easy to check that $J^{2}=-I$.
###### Theorem 3.1.
$(T^{\ast}M_{0},G,J)$ is an almost Kählerian manifold.
###### Proof.
Since the matrix $(H^{kl})$ is the inverse of the matrix $(G_{ij}),$ then we
have
$G(J\delta_{i},J\delta_{j})=G_{ik}G_{jr}G(\dot{\partial}^{k},\dot{\partial}^{r})=G_{ik}G_{jr}G^{kr}=G_{ij}=G(\delta_{i},\delta_{j}).$
The relations
$G(J\dot{\partial}^{i},J\dot{\partial}^{j})=G(\dot{\partial}^{i},\dot{\partial}^{j}),\
\ \ \ G(J\delta_{i},J\dot{\partial}^{j})=G(\delta_{i},\dot{\partial}^{j})=0,$
may be obtained in a similar way, thus
$G(JX,JY)=G(X,Y),\,\,\,\,\,\,\,\forall X,Y\in\Gamma(T^{\ast}M_{0}).$
This means that $G$ is almost Hermitian with respect to $J$. The fundamental
2-form associated by this almost Kähler structure is $\Omega$, defined by
$\theta(X,Y):=G(X,JY),\,\,\,\,\forall X,Y\in\Gamma(T^{\ast}M_{0}).$
Then we get
$\theta(\dot{\partial}^{i},\delta_{j})=G(\dot{\partial}^{i},J\delta_{j})=G(\dot{\partial}^{i},G_{jk}\dot{\partial}^{k})=G^{ik}G_{jk}=\delta^{i}_{j},$
and
$\theta(\delta_{i},\delta_{j})=\theta(\dot{\partial}^{i},\dot{\partial}^{j})=0.$
Hence, we have
$\theta=\delta p_{i}\wedge dx^{i},$ (22)
that is the canonical symplectic form of $T^{\ast}M$. ∎
Here, we study the integrability of the almost complex structure defined by
$J$ on $T^{\ast}M$. To do this, we need the following lemma.
###### Lemma 3.2.
([11][15]) Let $(M,K)$ be a Cartan space. Then we have the following:
$(1)\ \ [\delta_{i},\delta_{j}]=R_{kij}{\dot{\partial}}^{k}$,
$(2)\ \
[\delta_{i},{\dot{\partial}}^{j}]=-({\dot{\partial}}^{j}N_{ik}){\dot{\partial}}^{k}$,
$(3)\ \ [{\dot{\partial}}^{i},{\dot{\partial}}^{j}]=0$,
where $R_{kij}=\delta_{j}N_{ki}-\delta_{i}N_{kj}$.
###### Theorem 3.3.
Let $(M,K)$ be a Cartan space. Then $J$ is a complex structure on
$T^{\ast}M_{0}$ if and only if $A_{kij}=0$ and
$R_{kij}=\frac{v}{\alpha\beta^{2}}(g_{ik}p_{j}-g_{jk}p_{i}),$ (23)
where
$A_{kij}=\delta_{i}G_{jk}-\delta_{j}G_{ik}+G_{ir}{\dot{\partial}}^{r}N_{jk}-G_{jr}{\dot{\partial}}^{r}N_{ik}$.
###### Proof.
Using the definition of the Nijenhuis tensor field $N_{J}$ of $J$, that is,
$N_{J}(X,Y)=[JX,JY]-J[JX,Y]-J[X,JY]-[X,Y],\ \ \ \forall
X,Y\in\Gamma(T^{\ast}M)$
we get
$N_{J}(\delta_{i},\delta_{j})=A_{hij}H^{hk}\delta_{k}+(B_{kij}-R_{kij}){\dot{\partial}}^{k},$
(24)
where
$B_{kij}=G_{ir}{\dot{\partial}}^{r}G_{jk}-G_{jr}{\dot{\partial}}^{r}G_{ik}$.
Let $C_{jk}^{r}:=g_{jl}g_{sk}C^{rls}$, then we have
${\dot{\partial}}^{r}g_{jk}=-g_{jl}g_{sk}{\dot{\partial}}^{r}g^{ls}=2g_{jl}g_{sk}C^{rls}=2C_{jk}^{r}.$
By the above equation, we obtain
$G_{ir}{\dot{\partial}}^{r}G_{jk}=\frac{2}{\beta^{2}}C_{ijk}+\frac{v}{\alpha\beta^{2}}(g_{ji}p_{k}+g_{ik}p_{j})+(\frac{v^{\prime}}{\alpha\beta^{2}}+\frac{2vv^{\prime}\tau}{\alpha\beta}+\frac{2v^{2}}{\alpha^{2}\beta^{2}})p_{i}p_{j}p_{k}.$
(25)
where $C_{ijk}=g_{ir}C_{jk}^{r}$. From (25) we get
$B_{kij}=\frac{v}{\alpha\beta^{2}}(g_{ik}p_{j}-g_{jk}p_{i}).$ (26)
By a straightforward computation, it follows that
$N_{J}({\dot{\partial}}^{i},{\dot{\partial}}^{j})=0,N_{J}({\dot{\partial}}^{i},\delta_{j})=0$,
whenever $N_{j}(\delta_{i},\delta_{j})=0$. Therefore, from relations (24) and
(26), we conclude that the necessary and sufficient conditions for vanishing
the Nijenhuis tensor field $N_{J}$ is obtained, so that J is a complex
structure. Thus $A_{kij}=0$ and (23) hold. ∎
In equation (23), we put $-\frac{v}{\alpha\beta^{2}}=c$, where $c$ is
constant. Then we get
$R_{kij}=c(g_{jk}p_{i}-g_{ik}p_{j}).$ (27)
###### Theorem 3.4.
Let $(M,K)$ be a Cartan space of dimension $n\geq 3$. Then the almost complex
structure $J$ on $T^{\ast}M_{0}$ is integrable if and only if (27) is hold and
the function $v$ is given by
$v=-c\alpha\beta^{2}.$ (28)
###### Proof.
From equation $p_{i|k}=0$ of relation (14), we conclude that
$\delta_{i}p_{k}=N_{ik}$. Hence the tensor field $A_{kij}$ takes the form
$A_{kij}=\delta_{i}g_{jk}-\delta_{j}g_{ik}+g_{ir}{\dot{\partial}}^{r}N_{jk}-g_{jr}{\dot{\partial}}^{r}N_{ik}$
and by part $(iii)$ of Proposition 2.3 in chapter 7 of [11], it vanishes for
Cartan spaces. Now we let $v=-c\alpha\beta^{2}$, then from equation
$A_{kij}=0$ and Theorem 3.3, we conclude that $J$ is integrable if and only if
(27) is hold. ∎
A Cartan space $K^{n}$ is of constant scalar $c$ if
$H_{hijk}p^{i}p^{j}X^{h}X^{k}=c(g_{hj}g_{ik}-g_{hk}g_{ij})p^{i}p^{j}X^{h}X^{k},$
(29)
for every $(x,p)\in T^{\ast}_{0}M$ and $X=(X^{i})\in T_{x}M$. Here $H_{hijk}$
is the (hh)h-curvature of the linear Cartan connection of $K^{n}$.
We replace $H_{hijk}$ in (29) with $g_{is}H^{s}_{hjk}$ and so it reduce to
$p_{s}H^{s}_{hjk}p^{j}X^{h}X^{k}=c(p_{h}p_{k}-K^{2}g_{hk})X^{h}X^{k}.$ (30)
By part (ii) of Proposition 5.1 in chapter 7 of [11],
$p_{s}H^{s}_{hjk}=-R_{hjk}$, hence we get
$R_{hjk}p^{j}X^{h}X^{k}=c(K^{2}g_{hk}-p_{h}p_{k})X^{h}X^{k},$
or equivalently
$R_{hjk}p^{j}=c(K^{2}g_{hk}-p_{h}p_{k}),$ (31)
because $(X^{h})$ and $X^{k}$ are arbitrary vector fields on $M$. It is easy
to check that (31) follows from (27). Hence we conclude that if (27) is hold,
then Cartan space $K^{n}$ has the constant scalar curvature $c$.
###### Theorem 3.5.
Let $(M,K)$ be a Cartan space with constant flag curvature $c$. Suppose that
$v$ is given by (28). Then
$(i)$ for negative constant $c$, structure $(T^{\ast}M_{0},G,J)$ is a Kähler
manifold;
$(ii)$ for positive constant $c$, the tube around the zero section in
$T^{\ast}M$, defined by the condition $2\tau=K^{2}<\frac{1}{c\beta^{2}}$, is a
Kähler manifold.
###### Proof.
The function $uv$ must satisfies in the following condition
$\alpha+2\tau u=\alpha(1+2(-c)\beta^{2}\tau)>0,\,\,\,\,\,\alpha,\beta>0.$ (32)
By using the above relation and Theorem 3.4, we complete the proof. ∎
By attention to above theorem, the components of the Kähler metric $G$ on
$TM_{0}$ are given by
$\left\\{\begin{array}[]{l}{G_{ij}=\frac{1}{\beta}g_{ij}-c\beta
y_{i}y_{j},}\\\ {H_{ij}=\beta
g_{ij}+\frac{c\beta^{3}}{1-2c\beta^{2}\tau}y_{i}y_{j}.}\end{array}\right.$
(33)
## 4 A Kähler Einstein Structure on Cotangent Bundle
In this section, we study the property of $(T^{\ast}M_{0},G)$ to be Einstein
manifold. We find the expression of the Levi-Civita connection $\nabla$ of the
metric $G$ on $T^{\ast}M_{0}$. Then we get the curvature tensor field of
$\nabla$, and by computing the corresponding traces we find the components of
Ricci tensor field of $\nabla$.
### 4.1 The Levi-Civita connection of $G$
Here we determine the Levi-Civita connection of the Kähler metric $G$. Recall
that for Cartan space with Cartan connection, the relation
$P^{j}_{ik}=H^{j}_{ik}-{\dot{\partial}}^{j}N_{ik}$ is hold, and so we have
$[\delta_{i},{\dot{\partial}}^{j}]=(P^{j}_{ik}-H^{j}_{ik}){\dot{\partial}}^{k}$.
Also the Levi-Civita connection $\nabla$ of the Riemannian manifold
$(T^{\ast}M_{0},G)$ is obtained from the formula
$\displaystyle 2G({\nabla}_{X}Y,Z)\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!X(G(Y,Z))+Y(G(X,Z))-Z(G(X,Y))$ (34) $\displaystyle+$
$\displaystyle\\!\\!\\!\\!G([X,Y],Z)-G([X,Z],Y)-G([Y,Z],X),\,\,\,\,\,\forall
X,Y,Z\in\Gamma(T^{\ast}M_{0}),$
and is characterized by the conditions $\nabla G=0$ and $T=0$, where $T$ is
the torsion tensor of $\nabla$. By the above equation we have
$\displaystyle
2G(\nabla_{{\dot{\partial}}^{i}}{\dot{\partial}}^{j},{\dot{\partial}}^{k})\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!{\dot{\partial}}^{i}G({\dot{\partial}}^{j},{\dot{\partial}}^{k})+{\dot{\partial}}^{j}G({\dot{\partial}}^{i},{\dot{\partial}}^{k})-{\dot{\partial}}^{k}G({\dot{\partial}}^{i},{\dot{\partial}}^{j})$
(35) $\displaystyle=$ $\displaystyle\\!\\!\\!\\!{\dot{\partial}}^{i}(\beta
g^{jk}+\frac{c\beta^{3}}{1-2c\beta^{2}\tau}p^{j}p^{k})+{\dot{\partial}}^{j}(\beta
g^{ik}+\frac{c\beta^{3}}{1-2c\beta^{2}\tau}p^{i}p^{k})$ $\displaystyle-$
$\displaystyle\\!\\!\\!\\!{\dot{\partial}}^{k}(\beta
g^{ij}+\frac{c\beta^{3}}{1-2c\beta^{2}\tau}p^{i}p^{j})$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!2\beta(-C^{ijk}+\frac{c^{2}\beta^{4}}{(1-2c\beta^{2}\tau)^{2}}p^{i}p^{j}p^{k}+\frac{c\beta^{2}}{1-2c\beta^{2}\tau}g^{ij}p^{k}).$
Also, we obtain
$\displaystyle
2G(\nabla_{{\dot{\partial}}^{i}}{\dot{\partial}}^{j},\delta_{k})\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!-\delta_{k}G({\dot{\partial}}^{i},{\dot{\partial}}^{j})-G([{\dot{\partial}}^{i},\delta_{k}],{\dot{\partial}}^{j})-G([{\dot{\partial}}^{j},\delta_{k}],{\dot{\partial}}^{i})$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!-\delta_{k}(\beta
g^{ij}+\frac{c\beta^{3}}{1-2c\beta^{2}\tau}p^{i}p^{j})-{\dot{\partial}}^{i}N_{kr}(\beta
g^{rj}+\frac{c\beta^{3}}{1-2c\beta^{2}\tau}p^{r}p^{j})$ $\displaystyle-$
$\displaystyle\\!\\!\\!\\!{\dot{\partial}}^{j}N_{kr}(\beta
g^{ri}+\frac{c\beta^{3}}{1-2c\beta^{2}\tau}p^{r}p^{i})$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!-\delta_{k}(\beta
g^{ij}+\frac{c\beta^{3}}{1-2c\beta^{2}\tau}p^{i}p^{j})+(P^{i}_{kr}-H^{i}_{kr})(\beta
g^{rj}+\frac{c\beta^{3}}{1-2c\beta^{2}\tau}p^{r}p^{j})$ $\displaystyle+$
$\displaystyle\\!\\!\\!\\!(P^{j}_{kr}-H^{j}_{kr})(\beta
g^{ri}+\frac{c\beta^{3}}{1-2c\beta^{2}\tau}p^{r}p^{i}).$
Since ${g^{ij}}_{|k}=0$ and $\delta_{k}p^{i}=-p^{r}H^{i}_{rk}$ are hold for
Cartan connection, then by the above equation we get
$\displaystyle
2G(\nabla_{{\dot{\partial}}^{i}}{\dot{\partial}}^{j},\delta_{k})\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\frac{c\beta^{3}}{1-2c\beta^{2}\tau}(H^{i}_{rk}p^{j}+H^{j}_{rk}p^{i}-H^{i}_{kr}p^{j}-H^{j}_{kr}p^{i})p^{r}+\beta(P^{i}_{kr}+P^{j}_{kr})g^{ri}-{g^{ij}}_{|k}$
(36) $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\beta(P^{i}_{kr}g^{rj}+P^{j}_{kr}g^{ri}).$
From (35) and (36), we obtain
$\nabla_{{\dot{\partial}}^{i}}{\dot{\partial}}^{j}=\frac{\beta^{2}}{2}(P^{i}_{kr}g^{rj}+P^{j}_{kr}g^{ri})g^{hk}\delta_{h}+(-C^{ij}_{h}+c\beta
G^{ij}p_{h}){\dot{\partial}}^{h}.$ (37)
Since $G$ is a Kähler metric, then $(M,K)$ is of constant curvature $c$, i.e.,
$R_{kij}=c(g_{jk}p_{i}-g_{ik}p_{j})$. Hence, we get
$\displaystyle
2G(\nabla_{\delta_{i}}{\dot{\partial}}^{j},\delta_{k})\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!{\dot{\partial}}^{j}G(\delta_{i},\delta_{k})-G([\delta_{i},\delta_{k}],{\dot{\partial}}^{j})$
(38) $\displaystyle=$
$\displaystyle\\!\\!\\!\\!{\dot{\partial}}^{j}(\frac{1}{\beta}g_{ik}-c\beta
p_{i}p_{k})-R_{rik}(\beta
g^{rj}+\frac{c\beta^{3}}{1-2c\beta^{2}\tau}p^{r}p^{j})$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!(\frac{1}{\beta}{\dot{\partial}}^{j}g_{ik}-c\beta\delta^{j}_{i}p_{k}-c\beta\delta^{j}_{k}p_{i})-(c\beta\delta^{j}_{k}p_{i}-c\beta\delta^{j}_{i}p_{k})$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!2(\frac{1}{\beta}C^{j}_{ik}-c\beta\delta^{j}_{k}p_{i}),$
where $C^{j}_{ik}:=g_{il}g_{kh}C^{jlh}=g_{il}C^{jl}_{k}$. Since
${g^{jk}}_{|i}=0$ and $p^{j}_{|i}=p^{k}_{|i}=0$, then we have
${G^{jk}}_{|i}=\beta{g^{jk}}_{|i}+\frac{c\beta^{3}}{1-2c\beta^{2}\tau}(p^{j}_{|i}p^{k}+p^{k}_{|i}p^{j})=0.$
Hence, we obtain
$\displaystyle
2G(\nabla_{\delta_{i}}{\dot{\partial}}^{j},{\dot{\partial}}^{k})\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\delta_{i}G({\dot{\partial}}^{j},{\dot{\partial}}^{k})+G([\delta_{i},{\dot{\partial}}^{j}],{\dot{\partial}}^{k})-G([\delta_{i},{\dot{\partial}}^{k}],{\dot{\partial}}^{j})$
(39) $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\delta_{i}G^{jk}+(P^{j}_{ir}-H^{j}_{ir})G^{rk}+(H^{k}_{ir}-P^{k}_{ir})G^{rj}$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!{G^{jk}}_{|i}-2G^{rk}H^{j}_{ir}+P^{j}_{ir}G^{rk}-P^{k}_{ir}G^{rj}$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!-2G^{rk}H^{j}_{ir}+P^{j}_{ir}G^{rk}-P^{k}_{ir}G^{rj}.$
From (38) and (39), we have
$\nabla_{\delta_{i}}{\dot{\partial}}^{j}=(C^{jh}_{i}-c\beta
G^{jh}p_{i})\delta_{h}+(\frac{1}{2}P^{j}_{ih}-\frac{1}{2}P^{k}_{ir}G^{rj}G_{kh}-H^{j}_{ih}){\dot{\partial}}^{h}.$
(40)
Also from equation
$\nabla_{{\dot{\partial}}^{i}}{\delta_{j}}-\nabla_{\delta_{j}}{\dot{\partial}}^{i}=[{\dot{\partial}}^{i},\delta_{j}]=(H^{i}_{jh}-P^{i}_{jh}){\dot{\partial}}^{h},$
and (40) we obtain
$\nabla_{{\dot{\partial}}^{i}}{\delta_{j}}=(C^{ih}_{j}-c\beta
G^{ih}p_{j})\delta_{h}-\frac{1}{2}(P^{i}_{jh}+P^{k}_{jr}G^{ri}G_{kh}){\dot{\partial}}^{h}.$
(41)
Similarly, we have
$\displaystyle
2G(\nabla_{\delta_{i}}\delta_{j},{\dot{\partial}}^{k})\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!{\dot{\partial}}^{k}G(\delta_{i},\delta_{j})+G([\delta_{i},\delta_{j}],{\dot{\partial}}^{k})$
(42) $\displaystyle=$
$\displaystyle\\!\\!\\!\\!-{\dot{\partial}}^{k}G_{ij}+R_{rij}G^{rk}$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!-{\dot{\partial}}^{k}(\frac{1}{\beta}g_{ij}-c\beta
p_{i}p_{j})+c\beta(g_{rj}p_{i}-g_{ri}p_{j})(g^{rk}+\frac{c\beta^{2}}{1-2c\beta^{2}\tau}p^{r}p^{k})$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!-\frac{2}{\beta}C^{k}_{ij}+2c\beta\delta^{k}_{j}p_{i},$
and
$\displaystyle 2G(\nabla_{\delta_{i}}\delta_{j},\delta_{k})\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\delta_{i}G(\delta_{j},\delta_{k})+\delta_{j}G(\delta_{i},\delta_{k})-\delta_{k}G(\delta_{i},\delta_{j})$
(43) $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\delta_{i}G_{jk}+\delta_{j}G_{ik}-\delta_{k}G_{ij}$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!G_{jr}H^{r}_{ik}+G_{rk}H^{r}_{ij}+G_{ir}H^{r}_{jk}+G_{rk}H^{r}_{ij}-G_{rj}H^{r}_{ik}-G_{ir}H^{r}_{jk}$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!2G_{rk}H^{r}_{ij}.$
From (42) and (43), we conclude the following equation
$\nabla_{\delta_{i}}\delta_{j}=H^{h}_{ij}\delta_{h}+(-\frac{1}{\beta^{2}}C_{ijh}+c\beta
G_{hj}p_{i}){\dot{\partial}}^{h}.$ (44)
### 4.2 The Curvature Tensor
Here, we are going to compute the curvature tensors of $\nabla$. Recall that
the curvature $K$ of $\nabla$ is obtained from the following relation
$K(X,Y)Z={\nabla}_{X}{\nabla}_{Y}Z-{\nabla}_{Y}{\nabla}_{X}Z-{\nabla}_{[X,Y]}Z,\
\ \ \forall X,Y,Z\in\Gamma(TM).$ (45)
Using (45) we have
$K({\dot{\partial}}^{i},{\dot{\partial}}^{j}){\dot{\partial}}^{k}={\nabla}_{{\dot{\partial}}^{i}}{\nabla}_{{\dot{\partial}}^{j}}{\dot{\partial}}^{k}-{\nabla}_{{\dot{\partial}}^{j}}{\nabla}_{{\dot{\partial}}^{i}}{\dot{\partial}}^{k}.$
(46)
By (37) we get
$\displaystyle\nabla_{{\dot{\partial}}^{i}}\nabla_{{\dot{\partial}}^{j}}{\dot{\partial}}^{k}\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\frac{\beta^{2}}{2}{\dot{\partial}}^{i}((P^{j}_{lr}g^{rk}+P^{k}_{lr}g^{rj})g^{lh})\delta_{h}+\frac{\beta^{2}}{2}(P^{j}_{lr}g^{rk}+P^{k}_{lr}g^{rj})g^{lh}\nabla_{{\dot{\partial}}^{i}}\delta_{h}$
(47) $\displaystyle+$ $\displaystyle\\!\\!\\!\\!(-C^{jk,i}_{h}+c\beta
G^{jk,i}p_{h}+c\beta
G^{jk}\delta^{i}_{h}){\dot{\partial}}^{h}+(-C^{jk}_{h}+c\beta
G^{jk}p_{h})\nabla_{{\dot{\partial}}^{i}}{\dot{\partial}}^{h}.$
Since $P^{h}_{rl}=H^{h}_{rl}-{\dot{\partial}}^{h}N_{rl}$,
$N_{rl}=p_{h}H^{h}_{rl}$ and $p_{h}{\dot{\partial}}^{h}N_{rl}=N_{rl}$, then
$p_{h}P^{h}_{rl}=0$. Also we have $p^{l}P^{h}_{rl}=p^{r}P^{h}_{rl}=0$ and
${\dot{\partial}}^{k}g^{ij}=-2C^{kij}$. Therefore the following relation
deduce from (47)
$\displaystyle\nabla_{{\dot{\partial}}^{i}}\nabla_{{\dot{\partial}}^{j}}{\dot{\partial}}^{k}\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\frac{\beta^{2}}{2}\Big{[}(P^{j,i}_{lr}g^{rk}+P^{k,i}_{lr}g^{rj}-2P^{j}_{lr}C^{irk}-2P^{k}_{lr}C^{irj}-P^{i}_{rl}C^{jkr}-P^{h}_{rl}C^{jk}_{h}g^{ri})g^{sl}$
(48)
$\displaystyle\\!\\!\\!\\!-P^{j}_{lr}C^{lis}g^{rk}-P^{k}_{lr}C^{lis}g^{rj}\Big{]}\delta_{s}+\Big{[}c\beta
p_{s}(G^{jk,i}+c\beta G^{jk}G^{ih}p_{h}-\beta C^{jki})$
$\displaystyle\\!\\!\\!\\!-\frac{\beta^{2}}{4}g^{lh}(P^{j}_{lr}P^{i}_{hs}g^{rk}+P^{j}_{lr}P^{m}_{hn}g^{rk}g^{ni}g_{ms}+P^{k}_{lr}P^{i}_{hs}g^{rj}+P^{k}_{lr}P^{m}_{hn}g^{rj}g^{ni}g_{ms})$
$\displaystyle\\!\\!\\!\\!+c\beta
G^{jk}\delta^{i}_{s}+C^{jk}_{h}C^{ih}_{s}-C^{jk,i}_{s}\Big{]}{\dot{\partial}}^{s},$
where $P^{j,i}_{lr}={\dot{\partial}}^{i}P^{j}_{lr}$. Since
${\dot{\partial}}^{i}\tau=p^{i}$ and ${\dot{\partial}}^{i}p^{j}=g^{ij}$, then
we obtain
$\displaystyle c\beta p_{s}\\{G^{jk,i}\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!-G^{ik,j}+c\beta G^{jk}G^{ih}p_{h}-c\beta
G^{ik}G^{jh}p_{h}\\}=$ (49)
$\displaystyle\\!\\!\\!\\!\frac{c^{2}\beta^{4}}{1-2c\beta^{2}\tau}(g^{ik}p^{j}-g^{jk}p^{i}+g^{jk}p^{i}-g^{ik}p^{j})=0.$
With replacing $i,j$ in (48), setting this equations in (46) and attention
(49), we can obtain the following
$\displaystyle
K({\dot{\partial}}^{i},{\dot{\partial}}^{j}){\dot{\partial}}^{k}\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\frac{\beta^{2}}{2}\Big{[}(P^{j,i}_{lr}g^{rk}+P^{k,i}_{lr}g^{rj}-P^{i,j}_{lr}g^{rk}-P^{k,j}_{lr}g^{ri}-P^{j}_{lr}C^{irk}+P^{h}_{rl}C^{ik}_{h}g^{rj}-P^{h}_{rl}C^{jk}_{h}g^{ri}$
(50)
$\displaystyle\\!\\!\\!\\!+P^{i}_{rl}C^{jkr})g^{sl}+g^{rk}(P^{i}_{lr}C^{ljs}-P^{j}_{lr}C^{lis})-P^{k}_{lr}C^{lis}g^{rj}+P^{k}_{lr}C^{ljs}g^{ri}\Big{]}\delta_{s}$
$\displaystyle\\!\\!\\!\\!+\Big{[}\frac{\beta^{2}}{4}g^{lh}\big{(}P^{i}_{lr}P^{j}_{hs}g^{rk}-P^{j}_{lr}P^{i}_{hs}g^{rk}+P^{i}_{lr}P^{m}_{hn}g^{rk}g^{nj}g_{ms}-P^{j}_{lr}P^{m}_{hn}g^{rk}g^{ni}g_{ms}$
$\displaystyle\\!\\!\\!\\!+P^{k}_{lr}P^{j}_{hs}g^{ri}-P^{k}_{lr}P^{i}_{hs}g^{rj}+P^{k}_{lr}P^{m}_{hn}g^{ri}g^{nj}g_{ms}-P^{k}_{lr}P^{m}_{hn}g^{rj}g^{ni}g_{ms}\big{)}$
$\displaystyle\\!\\!\\!\\!+C^{ik,j}_{s}-C^{jk,i}_{s}+c\beta
G^{jk}\delta^{i}_{s}-c\beta
G^{ik}\delta^{j}_{s}+C^{jk}_{h}C^{ih}_{s}-C^{ik}_{h}C^{jh}_{s}\Big{]}{\dot{\partial}}^{s}.$
By (45) it follows that
$K({\dot{\partial}}^{i},\delta_{j}){\dot{\partial}}^{k}={\nabla}_{{\dot{\partial}}^{i}}{\nabla}_{\delta_{j}}{\dot{\partial}}^{k}-{\nabla}_{\delta_{j}}{\nabla}_{{\dot{\partial}}^{i}}{\dot{\partial}}^{k}-\nabla_{[{\dot{\partial}}^{i},\delta_{j}]}{\dot{\partial}}^{k}.$
(51)
From (40), we have
$\displaystyle{\nabla}_{{\dot{\partial}}^{i}}{\nabla}_{\delta_{j}}{\dot{\partial}}^{k}=(C^{kh,i}_{j}\\!\\!\\!\\!$
$\displaystyle-$ $\displaystyle\\!\\!\\!\\!c\beta G^{kh,i}p_{j}-c\beta
G^{kh}\delta^{i}_{j})\delta_{h}+(C^{kh}_{j}-c\beta
G^{kh}p_{j})\nabla_{{\dot{\partial}}^{i}}\delta_{h}$ (52)
$\displaystyle\\!\\!\\!\\!+\frac{1}{2}\\{P^{k,i}_{jh}-(P^{l}_{jr}g^{rk}g_{lh})^{,i}-2H^{k,i}_{jh}\\}{\dot{\partial}}^{h}$
$\displaystyle\\!\\!\\!\\!+\frac{1}{2}\\{P^{k}_{jh}-P^{l}_{jr}g^{rk}g_{lh}-2H^{k}_{jh}\\}\nabla_{{\dot{\partial}}^{i}}{\dot{\partial}}^{h}.$
Putting (37) and (41) in the above relation yields
$\displaystyle{\nabla}_{{\dot{\partial}}^{i}}{\nabla}_{\delta_{j}}{\dot{\partial}}^{k}\\!\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\Big{[}C^{ks,i}_{j}-c\beta
G^{ks,i}p_{j}-c\beta G^{ks}\delta^{i}_{j}+C^{kh}_{j}C^{is}_{h}-c\beta
C^{is}_{j}G^{kh}p_{j}+c^{2}\beta^{2}G^{kh}G^{is}p_{j}p_{s}$ (53)
$\displaystyle\\!\\!\\!\\!-\frac{\beta^{2}}{4}g_{ts}\big{(}P^{l}_{jr}P^{i}_{mt}g^{rk}g_{lh}g^{mh}+P^{l}_{jr}P^{h}_{mt}g^{rk}g_{lh}g^{mi}+2P^{i}_{mt}H^{k}_{jh}g^{mh}+2P^{h}_{mt}H^{k}_{jh}g^{mi}\big{)}$
$\displaystyle\\!\\!\\!\\!+\frac{\beta^{2}}{4}g^{ls}(P^{k}_{jh}P^{i}_{rl}g^{rh}+P^{k}_{jh}P^{h}_{rl}g^{ri})\Big{]}\delta_{s}+\Big{[}-\frac{1}{2}g_{ls}(C^{kh}_{j}P^{l}_{hr}g^{ri}+\frac{1}{2}P^{l,i}_{jr}g^{rk})$
$\displaystyle\\!\\!\\!\\!+\frac{1}{2}c\beta^{2}g^{kh}p_{j}(P^{i}_{hs}+P^{l}_{hr}g^{ri}g_{ls})+\frac{1}{2}c\beta^{2}p_{s}(P^{k}_{jh}g^{ih}-P^{i}_{jr}g^{rk}-2H^{k}_{jh}G^{ih})$
$\displaystyle\\!\\!\\!\\!+P^{l}_{jr}(C^{ikr}g_{ls}-2g^{rk}C^{i}_{ls})-\frac{1}{2}(P^{k}_{jh}C^{ih}_{s}+C^{kh}_{j}P^{i}_{hs}-P^{k,i}_{js})$
$\displaystyle\\!\\!\\!\\!-H^{k,i}_{js}+H^{k}_{jh}C^{ih}_{s}\Big{]}{\dot{\partial}}^{s}.$
Similarly, we obtain
$\displaystyle{\nabla}_{\delta_{j}}{\nabla}_{{\dot{\partial}}^{i}}{\dot{\partial}}^{k}\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\Big{[}\frac{\beta^{2}}{2}\delta_{j}(P^{i}_{rl}g^{lk}g^{sr})+\frac{\beta^{2}}{2}\delta_{j}(P^{k}_{rl}g^{li}g^{sr})+\frac{\beta^{2}}{2}P^{i}_{rl}H^{s}_{jh}g^{lk}g^{hr}$
(54)
$\displaystyle\\!\\!\\!\\!+\frac{\beta^{2}}{2}P^{k}_{rl}H^{s}_{jh}g^{li}g^{hr}-C^{ik}_{h}C^{hs}_{j}+c\beta^{2}C^{ik}_{h}g^{hs}p_{j}-c^{2}\beta^{2}G^{ik}G^{hs}p_{h}p_{j}\Big{]}\delta_{s}$
$\displaystyle\\!\\!\\!\\!+\Big{[}\frac{\beta^{2}}{2}(P^{i}_{rl}g^{lk}g^{hr}+P^{k}_{rl}g^{li}g^{hr})(-\frac{1}{\beta^{2}}C_{jhs}+c\beta
G_{sh}p_{j})+\delta_{j}(-C^{ik}_{s}+c\beta G^{ik}p_{s})$
$\displaystyle\\!\\!\\!\\!+(-C^{ik}_{h}+c\beta
G^{ik}p_{h})(\frac{1}{2}P^{h}_{js}-\frac{1}{2}P^{l}_{jr}g^{rh}g_{ls}-H^{h}_{js})\Big{]}{\dot{\partial}}^{s}.$
Using (37), we get
$\displaystyle\nabla_{[{\dot{\partial}}^{i},\delta_{j}]}{\dot{\partial}}^{k}\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\frac{\beta^{2}}{2}[P^{h}_{rl}H^{i}_{jh}g^{kl}+P^{k}_{rl}H^{i}_{jh}g^{hl}-P^{h}_{rl}P^{i}_{jh}g^{kl}-P^{k}_{rl}P^{i}_{jh}g^{hl}]g^{sr}\delta_{s}$
(55) $\displaystyle\\!\\!\\!\\!+[-H^{i}_{jh}C^{hk}_{s}+c\beta
H^{i}_{jh}G^{hk}p_{s}+P^{i}_{jh}C^{hk}_{s}-c\beta^{2}P^{i}_{jh}g^{hk}p_{s}]{\dot{\partial}}^{s}.$
With a direct computation it follows that
$\displaystyle
c^{2}\beta^{2}G^{kh}G^{is}p_{j}p_{h}+c^{2}\beta^{2}G^{ik}G^{hs}p_{j}p_{h}-c\beta
G^{ks,i}p_{j}-c\beta G^{kh}C^{is}_{h}p_{j}-c\beta G^{hs}C^{ik}_{h}p_{j}$
$\displaystyle=\frac{c^{2}\beta^{4}}{1-2c\beta^{2}\tau}g^{is}p^{k}p_{j}+\frac{c^{2}\beta^{4}}{1-2c\beta^{2}\tau}g^{ik}p^{s}p_{j}+\frac{2c^{3}\beta^{6}}{(1-2c\beta^{2}\tau)^{2}}p^{i}p^{k}p^{s}p_{j}+2c\beta^{2}C^{iks}p_{j}$
$\displaystyle\ \ \
-\frac{2c^{3}\beta^{6}}{(1-2c\beta^{2}\tau)^{2}}p^{i}p^{k}p^{s}p_{j}-\frac{c^{2}\beta^{4}}{1-2c\beta^{2}\tau}g^{ik}p^{s}p_{j}-\frac{c^{2}\beta^{4}}{1-2c\beta^{2}\tau}g^{is}p^{k}p_{j}-2c\beta^{2}C^{kis}p_{j}$
$\displaystyle=0.$ (56)
By using (51)-(56) and attention to relations $G^{ik}_{|j}=0,p_{s|j}=0$, one
can obtains the following
$\displaystyle
K({\dot{\partial}}^{i},\delta_{j}){\dot{\partial}}^{k}\\!\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\Big{[}-c\beta
G^{ks}\delta^{i}_{j}+C^{ks,i}_{j}+C^{kh}_{j}C^{is}_{h}+C^{ik}_{h}C^{hs}_{j}+\frac{\beta^{2}}{4}P^{k}_{jh}P^{i}_{rl}g^{rh}g^{ls}$
(57)
$\displaystyle\\!\\!\\!\\!+\frac{\beta^{2}}{4}(P^{k}_{jh}P^{h}_{rl}g^{ri}g^{ls}-P^{l}_{jr}P^{i}_{lt}g^{rk}g^{ts}-2P^{i}_{mt}F^{k}_{jh}g^{mh}g^{ts}-2P^{h}_{mt}F^{k}_{jh}g^{mi}g^{ts})$
$\displaystyle\\!\\!\\!\\!-\frac{\beta^{2}}{2}\big{(}\delta_{j}(P^{i}_{rl}g^{lk}g^{sr})+\delta_{j}(P^{k}_{rl}g^{li}g^{sr})+P^{i}_{rl}F^{s}_{jh}g^{lk}g^{hr}+P^{k}_{rl}F^{s}_{jh}g^{li}g^{hr}\big{)}$
$\displaystyle\\!\\!\\!\\!-\frac{\beta^{2}}{2}(P^{h}_{rl}F^{i}_{jh}g^{kl}g^{sr}+P^{k}_{rl}F^{i}_{jh}g^{hl}g^{sr}-P^{i}_{jh}P^{h}_{rl}g^{kl}g^{sr}-P^{i}_{jh}P^{k}_{rl}g^{hl}g^{sr})$
$\displaystyle\\!\\!\\!\\!-\frac{\beta^{2}}{4}P^{l}_{jr}P^{h}_{mt}g^{rk}g_{lh}g^{mi}g^{ts}\Big{]}\delta_{s}+\Big{[}C^{ik}_{s|j}-\frac{1}{2}C^{kh}_{j}P^{i}_{hs}-\frac{1}{2}C^{kh}_{j}P^{l}_{hr}g^{ri}g_{ls}$
$\displaystyle\\!\\!\\!\\!+\frac{1}{2}c\beta^{2}P^{l}_{hr}g^{kh}g^{ri}g_{ls}p_{j}+\frac{1}{2}P^{k,i}_{js}-\frac{1}{2}P^{l,i}_{jr}g^{rk}g_{ls}+\frac{1}{2}P^{l}_{jr}C^{irk}g_{ls}-P^{l}_{jr}C^{i}_{ls}g^{rk}$
$\displaystyle\\!\\!\\!\\!-H^{k,i}_{js}-P^{i}_{jh}C^{kh}_{s}+c\beta^{2}P^{i}_{jh}g^{hk}p_{s}+\frac{1}{2}c\beta^{2}(P^{k}_{jh}g^{ih}p_{s}-P^{i}_{jr}g^{rk}p_{s}-P^{k}_{sl}g^{li}p_{j})$
$\displaystyle\\!\\!\\!\\!-\frac{1}{2}(P^{k}_{jh}C^{ih}_{s}-P^{l}_{jr}C^{i}_{ls}g^{rk}-P^{i}_{rl}C^{r}_{js}g^{lk}-P^{k}_{rl}C^{r}_{js}g^{li}-P^{h}_{js}C^{ik}_{h})\Big{]}{\dot{\partial}}^{s}.$
From (40) and (44), we have
$\displaystyle K(\delta_{i},\delta_{j}){\delta_{k}}\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\Big{[}R^{s}_{kji}+\frac{1}{\beta^{2}}(C_{ikh}C^{hs}_{j}-C_{jkh}C^{hs}_{i})+c^{2}\beta^{2}(p_{i}\delta^{s}_{j}-p_{j}\delta^{s}_{i})p_{k}\Big{]}\delta_{s}$
(58)
$\displaystyle\\!\\!\\!\\!+\Big{[}\frac{1}{\beta^{2}}(C_{ikh}P^{h}_{jr}-C_{jkh}P^{h}_{ir})+\frac{1}{\beta^{2}}(C_{ikr|j}-C_{jkr|i})+cg_{hk}(P^{h}_{ir}p_{j}-P^{h}_{jr}p_{i})$
$\displaystyle\\!\\!\\!\\!+c^{2}\beta^{2}p_{h}p_{k}(P^{h}_{jr}p_{i}-P^{h}_{ir}p_{j})\Big{]}{\dot{\partial}}^{s},$
and
$\displaystyle K(\delta_{i},\delta_{j}){\dot{\partial}}^{k}\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\Big{[}\frac{1}{2}(P^{k}_{jh}C^{hs}_{i}-P^{k}_{ih}C^{hs}_{j})+\frac{\beta^{2}}{2}R_{hij}(P^{h}_{mr}g^{ms}g^{rk}+P^{k}_{mr}g^{ms}g^{rh})$
(59)
$\displaystyle\\!\\!\\!\\!+\frac{c\beta^{2}}{2}(P^{k}_{ih}g^{hs}p_{j}-P^{k}_{jh}g^{hs}p_{i}-P^{s}_{ir}g^{rk}p_{j}+P^{s}_{jr}g^{rk}p_{i})$
$\displaystyle\\!\\!\\!\\!+\frac{1}{2}g^{rk}(P^{m}_{ir}C^{s}_{jm}-\frac{1}{2}P^{m}_{jr}C^{s}_{im})+C^{ks}_{j|i}-C^{ks}_{i|j}\Big{]}\delta_{s}$
$\displaystyle\\!\\!\\!\\!+\Big{[}\frac{1}{2}\Big{(}\delta_{j}(P^{m}_{ir}g^{rk}g_{ms})-\delta_{i}(P^{m}_{jr}g^{rk}g_{ms})\Big{)}+\frac{1}{\beta^{2}}(C^{kh}_{i}C_{jhs}-C^{kh}_{j}C_{ihs})$
$\displaystyle\\!\\!\\!\\!+R^{k}_{sij}+c\beta
R_{hij}G^{hk}p_{s}-\frac{1}{2}(P^{k}_{js|i}-P^{k}_{is|j})+\frac{1}{4}(P^{k}_{jh}P^{h}_{is}-P^{k}_{ih}P^{h}_{js})$
$\displaystyle\\!\\!\\!\\!+\frac{1}{4}g_{ws}\
\big{(}P^{k}_{ih}P^{w}_{ju}g^{uh}-P^{k}_{jh}P^{w}_{iu}g^{uh}+P^{m}_{jr}P^{w}_{im}g^{rk}-P^{m}_{ir}P^{w}_{jm}g^{rk}\big{)}$
$\displaystyle\\!\\!\\!\\!+\frac{1}{2}g^{rk}g_{mh}(P^{m}_{jr}H^{h}_{is}-P^{m}_{ir}H^{h}_{js})+\frac{1}{2}g^{uh}g_{ws}(P^{w}_{iu}H^{k}_{jh}-P^{w}_{ju}H^{k}_{ih})$
$\displaystyle\\!\\!\\!\\!+\frac{1}{4}(P^{m}_{ir}P^{h}_{js}g^{rk}g_{mh}-P^{m}_{jr}P^{h}_{is}g^{rk}g_{mh})\Big{]}{\dot{\partial}}^{s}.$
Similarly, from (37) and (41), we get
$\displaystyle
K({\dot{\partial}}^{i},{\dot{\partial}}^{j})\delta_{k}\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\Big{[}-\frac{\beta^{2}}{4}g^{us}(P^{j}_{kh}P^{i}_{uv}g^{vh}-P^{i}_{kh}P^{j}_{uv}g^{vh}+P^{j}_{kh}P^{h}_{uv}g^{vi}-P^{i}_{kh}P^{h}_{uv}g^{vj}$
(60)
$\displaystyle\\!\\!\\!\\!+P^{m}_{kr}P^{h}_{uv}g^{vi}g^{jr}g_{mh}-P^{m}_{kr}P^{h}_{uv}g^{vj}g^{ir}g_{mh}+P^{m}_{kr}P^{i}_{um}g^{jr}-P^{m}_{kr}P^{j}_{um}g^{ir})$
$\displaystyle\\!\\!\\!\\!+C^{js,i}_{k}-C^{is,j}_{k}+C^{jh}_{k}C^{is}_{h}-C^{ih}_{k}C^{js}_{h}+c\beta
G^{is}\delta^{j}_{k}-c\beta G^{js}\delta^{i}_{k}\Big{]}\delta_{s}$
$\displaystyle\\!\\!\\!\\!+\frac{1}{2}\Big{[}c\beta^{2}p_{k}(P^{i}_{hs}g^{jh}-P^{j}_{hs}g^{ih}+P^{m}_{hr}g^{jh}g^{ir}g_{ms}-P^{m}_{hr}g^{ih}g^{jr}g_{ms})$
$\displaystyle\\!\\!\\!\\!+(P^{m}_{kr}g^{ir}g_{ms})^{,j}-(P^{m}_{kr}g^{jr}g_{ms})^{,i}+P^{i,j}_{ks}-P^{j,i}_{ks}+C^{ih}_{k}P^{j}_{hs}$
$\displaystyle\\!\\!\\!\\!-C^{jh}_{k}P^{i}_{hs}+C^{ih}_{k}P^{m}_{hr}g^{jr}g_{ms}-C^{jh}_{k}P^{m}_{hr}g^{ir}g_{ms}+C^{ih}_{s}P^{j}_{kh}$
$\displaystyle\\!\\!\\!\\!-C^{jh}_{s}P^{i}_{kh}+C^{i}_{ms}P^{m}_{kr}g^{jr}-C^{j}_{ms}P^{m}_{kr}g^{ir}\Big{]}{\dot{\partial}}^{s}.$
Finally, from (37), (40), (41) and (44) we have
$\displaystyle K(\delta_{i},{\dot{\partial}}^{j})\delta_{k}\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\Big{[}\frac{1}{2}c\beta^{2}p_{i}(P^{j}_{kh}g^{hs}+P^{s}_{kr}g^{rj})-\frac{1}{2}(P^{j}_{kh}C^{hs}_{i}+P^{l}_{kr}C^{s}_{li}g^{rj})-H^{s,j}_{ik}+C^{js}_{k|i}$
(61)
$\displaystyle\\!\\!\\!\\!+\frac{1}{2}g^{sl}(P^{j}_{lr}C^{r}_{ik}+P^{h}_{lr}C_{ikh}g^{rj}-c\beta^{3}P^{j}_{lr}g^{rh}G_{hk}p_{i})+c\beta^{2}P^{j}_{ih}g^{hs}p_{k}$
$\displaystyle\\!\\!\\!\\!-\frac{1}{2}c\beta^{2}P^{h}_{lr}g^{rj}g^{sl}g_{hk}p_{i}-P^{j}_{ih}C^{hs}_{k}\Big{]}\delta_{s}+\Big{[}cC^{j}_{is}p_{k}-\frac{1}{2}p^{j}_{ks|i}-\frac{1}{\beta^{2}}C^{jh}_{k}C_{ihs}$
$\displaystyle\\!\\!\\!\\!-\frac{1}{2}\delta_{i}(P^{l}_{kr}g^{rj}g_{ls})-\frac{1}{4}P^{j}_{kh}P^{h}_{is}+\frac{1}{4}P^{j}_{kh}P^{l}_{ir}g^{rh}g_{ls}-\frac{1}{4}P^{l}_{kr}P^{h}_{is}g^{rj}g_{lh}+\frac{1}{\beta^{2}}C^{,j}_{iks}$
$\displaystyle\\!\\!\\!\\!+\frac{1}{4}P^{l}_{kr}P^{m}_{il}g^{rj}g_{ms}+\frac{1}{2}P^{l}_{kr}H^{h}_{is}g^{rj}g_{lh}+\frac{1}{2}P^{l}_{hr}H^{h}_{ik}g^{rj}g_{ls}-c\beta
G_{sk}\delta^{j}_{i}-\frac{c}{\beta}C^{j}_{ik}p_{s}$
$\displaystyle\\!\\!\\!\\!-\frac{1}{\beta^{2}}C_{ikh}C^{jh}_{s}-\frac{1}{2}P^{l}_{kr}H^{j}_{ih}g^{rh}g_{ls}+\frac{1}{2}P^{j}_{ih}P^{h}_{ks}+\frac{1}{2}P^{j}_{ih}P^{l}_{kr}g^{rh}g_{ls}\Big{]}{\dot{\partial}}^{s}.$
###### Theorem 4.1.
Let $(M,K)$ be a Cartan space of constant curvature $c$ and the components of
the metric $G$ are given by (33). Then the following are hold if and only if
$(M,K)$ is reduce to a Riemannian space.
$(i)$ for $c<0$, $(T^{\ast}M_{0},G,J)$ is a Kähler Einstein manifold.
$(ii)$ for $c>0$, $(T_{\beta}^{\ast}M_{0},G,J)$ is a Kähler Einstein manifold,
where $T_{\beta}^{\ast}M_{0}$ is the tube around the zero section in $TM$
defined by $2\tau<\frac{1}{c\beta^{2}}$.
###### Proof.
Let $(M,K)$ be a Cartan space. Then $C^{hi}_{k}$ and $P^{h}_{ik}$ are vanish
and $H^{i}_{jk}$ is a function of $(x^{h})$. Therefore (58) reduce to
$K(\delta_{i},\delta_{j})\delta_{k}=[R^{s}_{kji}+c^{2}\beta^{2}(p_{i}\delta^{s}_{j}-p_{j}\delta^{s}_{i})p_{k}]\delta_{s}.$
(62)
From Proposition 10.2 in chapter 4 of [11], we have
$R_{kji}=-p_{h}R^{h}_{kji}$. Then we get
$p_{h}R^{h}_{kji}=c(g_{kj}\delta^{h}_{i}-g_{ki}\delta^{h}_{j})p_{h}.$ (63)
Differentiating (63) with respect to $p_{s}$, taking $p=0$, yields
$R^{s}_{kji}=c(g_{kj}\delta^{s}_{i}-g_{ki}\delta^{s}_{j}).$
By putting above equation in (62), it follows that
$\displaystyle K(\delta_{i},\delta_{j})\delta_{k}\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!c\beta[(\frac{1}{\beta}g_{kj}-c\beta
p_{k}p_{j})\delta^{s}_{i}-(\frac{1}{\beta}g_{ki}-c\beta\
p_{k}p_{i})\delta^{s}_{j}]\delta_{s}$ (64) $\displaystyle=$
$\displaystyle\\!\\!\\!\\!c\beta(G_{kj}\delta^{s}_{i}-G_{ki}\delta^{s}_{j})\delta_{s}.$
Also from (61), we obtain
$K({\dot{\partial}}^{i},\delta_{j})\delta_{k}=c\beta
G_{sk}\delta^{i}_{j}{\dot{\partial}}^{s}.$ (65)
From (64) and (65), we have
$\displaystyle Ric(\delta_{j},\delta_{k})\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!G^{hi}G(K(\delta_{i},\delta_{j})\delta_{k},\delta_{h})+G_{hi}G(K({\dot{\partial}}^{i},\delta_{j})\delta_{k},{\dot{\partial}}^{h}),$
(66) $\displaystyle=$
$\displaystyle\\!\\!\\!\\!c\beta(G_{kj}\delta^{s}_{i}-G_{ki}\delta^{s}_{j})G^{hi}G_{sh}+c\beta
G_{sk}\delta^{i}_{j}G_{hi}G^{sh}$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!cn\beta G_{jk}$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!cn\beta G(\delta_{j},\delta_{k}).$
Similarly from (50) and (57), respectively, we have
$\displaystyle
K({\dot{\partial}}^{i},{\dot{\partial}}^{j}){\dot{\partial}}^{k}=c\beta(G^{jk}\delta^{i}_{s}-G^{ik}\delta^{j}_{s}){\dot{\partial}}^{s},$
(67)
and
$\displaystyle K(\delta_{i},{\dot{\partial}}^{j}){\dot{\partial}}^{k}=c\beta
G^{ks}\delta^{j}_{i}\delta_{s}.$ (68)
Using (67) and (68), we conclude that
$\displaystyle Ric({\dot{\partial}}^{j},{\dot{\partial}}^{k})\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!G^{ih}G(K(\delta_{i},{\dot{\partial}}^{j}){\dot{\partial}}^{k},\delta_{h})+G_{ih}G(K({\dot{\partial}}^{i},{\dot{\partial}}^{j}){\dot{\partial}}^{k},{\dot{\partial}}^{h})$
(69) $\displaystyle=$ $\displaystyle\\!\\!\\!\\!c\beta
G^{ks}\delta^{j}_{i}G^{ih}G_{hs}+c\beta(G^{jk}\delta^{i}_{s}-G^{ik}\delta^{j}_{s})G_{hi}G^{hs}$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!cn\beta G^{jk}$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!cn\beta
G({\dot{\partial}}^{j},{\dot{\partial}}^{k}).$
By (57) and (59),respectively, we have
$K(\delta_{i},\delta_{j}){\dot{\partial}}^{k}=(R^{k}_{sij}+c\beta
R_{hij}G^{hk}p_{s}){\dot{\partial}}^{s},$ (70)
and
$K({\dot{\partial}}^{i},\delta_{j}){\dot{\partial}}^{k}=-c\beta
G^{ks}\delta^{i}_{j}\delta_{s}.$ (71)
Using (70) and (71), we obtain
$Ric(\delta_{j},{\dot{\partial}}^{k})=G^{ih}G(K(\delta_{i},\delta_{j}){\dot{\partial}}^{k},\delta_{h})+G_{ih}G(K({\dot{\partial}}^{i},\delta_{j}){\dot{\partial}}^{k},{\dot{\partial}}^{h})=0.$
(72)
From (60), we have
$K({\dot{\partial}}^{i},{\dot{\partial}}^{j})\delta_{k}=c\beta(G^{is}\delta^{j}_{k}-G^{js}\delta^{i}_{k})\delta_{s}.$
(73)
By attention to (65) and (73), one can obtains
$Ric({\dot{\partial}}^{j},\delta_{k})=G^{ih}G(K(\delta_{i},{\dot{\partial}}^{j})\delta_{k},\delta_{h})+G_{ih}G(K({\dot{\partial}}^{i},{\dot{\partial}}^{j})\delta_{k},{\dot{\partial}}^{h})=0.$
(74)
From (66), (69), (72) and (74), we deduce that
$Ric(X,Y)=cn\beta G(X,Y),\ \ \ \forall X,Y\in\chi(T^{\ast}M).$
This means that $(T^{\ast}M,G)$ is a Einstein manifold.
Conversely, suppose that the conditions $(i),(ii)$ are hold. Then there exists
a real constant $\lambda$ such that $Ric(X,Y)=\lambda G(X,Y)$. We consider the
following cases:
Case (1). If $\lambda=0$ (i.e., $(T^{\ast}M,G)$ is Ricci flat), then we have
$Ric({\dot{\partial}}^{j},{\dot{\partial}}^{k})=0$. By using (50),(57) and
definition of Ricci tensor, we obtain
$\displaystyle 0=p_{k}Ric({\dot{\partial}}^{j},{\dot{\partial}}^{k})=cn\beta
p_{k}G^{jk}-p_{k}C^{jk,s}_{s}\\!\\!\\!\\!$ $\displaystyle+$
$\displaystyle\\!\\!\\!\\!\frac{\beta^{2}}{2}\delta_{s}(P^{j}_{rl}g^{lk}g^{sr})p_{k}+\frac{\beta^{2}}{2}\delta_{s}(P^{k}_{rl}g^{lj}g^{sr})p_{k}$
(75) $\displaystyle+$
$\displaystyle\\!\\!\\!\\!\frac{\beta^{2}}{2}g^{ts}p_{k}(P^{j}_{mt}H^{k}_{sh}g^{mh}+P^{h}_{mt}H^{k}_{sh}g^{mj}).$
Since $P^{j}_{rl}g^{lk}g^{sr}p_{k}=P^{j}_{rl}p^{l}g^{sr}=0$ and
$\delta_{s}p_{k}=H^{h}_{sk}p_{h}$, then we have
$\displaystyle\delta_{s}(P^{j}_{rl}g^{lk}g^{sr})p_{k}\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!-(\delta_{s}p_{k})P^{j}_{rl}g^{lk}g^{sr}$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!-P^{j}_{rl}g^{lk}g^{sr}H^{h}_{sk}p_{h}.$
Replacing $h$ and $k$, changing $r$ to $t$, and $l$ to $m$ in the above
equation yields
$\delta_{s}(P^{j}_{rl}g^{lk}g^{sr})p_{k}=-P^{j}_{tm}g^{mh}g^{st}H^{k}_{sh}p_{k}.$
(76)
Similarly, we have
$\delta_{s}(P^{k}_{rl}g^{lj}g^{sr})p_{k}=-P^{h}_{tm}g^{mj}g^{st}H^{k}_{sh}p_{k}.$
(77)
With direct computation, we get
$\displaystyle cn\beta p_{k}G^{jk}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!cn\beta p_{k}(\beta
g^{jk}+\frac{c\beta^{3}}{1-2c\beta^{2}\tau}p^{j}p^{k})$ (78) $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\frac{cn\beta^{2}}{1-2c\beta^{2}\tau}p^{j},$
and
$p_{k}C^{jk,s}_{s}=-p^{,s}_{k}C^{jk}_{s}=-\delta^{s}_{k}C^{jk}_{s}=-C^{js}_{s}=-I^{j}.$
(79)
By using (75)-(79), we obtain
$\frac{cn\beta^{2}}{1-2c\beta^{2}\tau}p^{j}+I^{j}=0.$ (80)
Since $p_{j}I^{j}=0$, then by contracting (80) with $p_{j}$, we have
$\frac{2cn\beta^{2}\tau}{1-2c\beta^{2}\tau}=0.$ (81)
From the above equation, we conclude that $\beta=0$ and this is a
contradiction.
Case (2). If $\lambda\neq 0$, then we have
$p_{k}Ric({\dot{\partial}}^{j},{\dot{\partial}}^{k})=\lambda G^{jk}p_{k}.$
Therefore by using (50), (57), (76)-(79), we obtain
$I^{j}=(\lambda-cn\beta)\frac{\beta}{1-2c\beta^{2}\tau}p^{j}.$ (82)
By contracting (82) with $p_{j}$, we have
$(\lambda-cn\beta)\frac{2\beta\tau}{1-2c\beta^{2}\tau}=0,$ (83)
i.e., $\lambda=cn\beta$. Then from (82), we result that $I^{j}=0$, i.e.,
$(M,K)$ is reduce to a Riemannian space. ∎
###### Corollary 4.2.
There is not exist any non-Riemannian Cartan structure such that
$(T^{\ast}M_{0},G,J)$ became a Einstein manifold.
###### Theorem 4.3.
Let $(M,K)$ be a Cartan space of constant curvature $c$ and the components of
the metric $G$ are given by (33). Then the following are hold if and only if
$(M,K)$ is reduce to a Riemannian space.
$(i)$ for $c<0$, $(T^{\ast}M_{0},G,J)$ is a locally symmetric Kähler manifold.
$(ii)$ for $c>0$, $(T_{\beta}^{\ast}M_{0},G,J)$ is a locally symmetric Kähler
manifold, where $T_{\beta}^{\ast}M_{\circ}$ is the tube around the zero
section in $T^{\ast}M$, defined by the condition $2\tau<\frac{1}{c\beta^{2}}$.
###### Proof.
Let $(M,K)$ be a Cartan space. Using (50), (57)-(61), we have computed the
covariant derivatives of curvature tensor field $K$ in the local adapted frame
$(\delta_{i},\dot{\partial}^{i})$ with respect to the connection $\nabla$ and
obtained in the twelve cases the result is zero.
Conversely, let $(i)$ and $(ii)$ are hold. Thus we get $\nabla K=0$. For
simplify, we let $(M,K)$ be a Berwald-Cartan space. Put
$(\nabla
K)({\dot{\partial}}^{u},{\dot{\partial}}^{i},\delta_{j},{\dot{\partial}}^{k})=M^{uiks}_{j}\delta_{s}+N^{uik}_{sj}{\dot{\partial}}^{s}.$
Since $(\nabla
K)({\dot{\partial}}^{u},{\dot{\partial}}^{i},\delta_{j},{\dot{\partial}}^{k})=0$,
then we have $M^{uiks}_{j}=N^{uik}_{sj}=0$. Thus we have
$p^{j}p_{i}M^{uiks}_{j}=0$. By a straightforward calculation, we obtain
$\displaystyle p^{j}p_{i}M^{uiks}_{j}=p^{j}p_{i}C^{ks,i,u}_{j}\\!\\!\\!\\!$
$\displaystyle-$
$\displaystyle\\!\\!\\!\\!2c\beta\tau(G^{ks,u}+G^{kh}C^{us}_{h}+G^{uh}C^{ks}_{h}+G^{hs}C^{uk}_{h})$
(84) $\displaystyle+$
$\displaystyle\\!\\!\\!\\!2c^{2}\beta^{2}\tau(G^{us}G^{kh}+G^{uk}G^{hs})p_{h}.$
By a direct computation, we have
$G^{ks,u}=-2\beta
C^{ksu}+\frac{2c^{2}\beta^{5}}{(1-2c\beta^{2}\tau)^{2}}p^{k}p^{u}p^{s}+\frac{c\beta^{3}}{1-2c\beta^{2}\tau}g^{ku}p^{s}+\frac{c\beta^{3}}{1-2c\beta^{2}\tau}g^{su}p^{k}.$
(85)
Also, we obtain
$G^{us}G^{kh}p_{h}=\frac{\beta}{1-2c\beta^{2}\tau}p^{s}G^{ku}=\frac{\beta^{2}}{1-2c\beta^{2}\tau}g^{ku}p^{s}+\frac{c\beta^{4}}{(1-2c\beta^{2}\tau)^{2}}p^{u}p^{k}p^{s},$
(86)
and
$G^{uk}G^{hs}p_{h}=\frac{\beta}{1-2c\beta^{2}\tau}p^{k}G^{su}=\frac{\beta^{2}}{1-2c\beta^{2}\tau}g^{su}p^{k}+\frac{c\beta^{4}}{(1-2c\beta^{2}\tau)^{2}}p^{u}p^{s}p^{k}.$
(87)
Putting (85), (86) and (87) in (84), one can yields
$p^{j}p_{i}M^{uiks}_{j}=p^{j}p_{i}C^{ks,i,u}_{j}-2c\beta\tau(-2\beta
C^{kus}+G^{kh}C^{us}_{h}+G^{uh}C^{ks}_{h}+G^{hs}C^{uk}_{h}).$ (88)
Since $p_{i}C^{ks,i}_{j}=-C^{ks}_{j}$, then we obtain
$p^{j}p_{i}C^{ks,i,u}_{j}=2g^{uj}C^{ks}_{j}=2C^{kus}.$
Also, since $C^{us}_{h}p^{h}=0$, then we have
$G^{kh}C^{us}_{h}=G^{uh}C^{ks}_{h}=G^{hs}C^{uk}_{h}=\beta C^{kus}.$
Therefore (88) reduces to following
$p^{j}p_{i}M^{uiks}_{j}=2(1-c\beta^{2}\tau)C^{kus}.$ (89)
Since $p^{j}p_{i}M^{uiks}_{j}=0$, then by (89), we have $C^{kus}=0$, i.e., $K$
is a Riemannian metric. ∎
###### Corollary 4.4.
There is not exist any non-Riemannian Cartan structure such that
$(T^{\ast}M_{0},G,J)$ became a locally symmetric manifold.
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* [9] R. Miron and M. Anastasiei, The Geometry of Lagrange Spaces: Theory and Applications, Kluwer Acad. Publ. 1994.
* [10] R. Miron and M. Anastasiei, Vector bundles and Lagrange spaces with applications to relativity , Geometry Balkan Press, Romania, 1997.
* [11] R. Miron, D. Hrimiuc, H. Shimada and V. S. Sabau, The Geometry of Hamilton and Lagrange Spaces, Kluwer Acad. Publ. 2000.
* [12] R. Miron, Hamilton Geometry, Univ. Timisoara (Romania), Sem. Mecanica, 3(1987), 1-54.
* [13] R. Miron, Cartan Spaces in a new point of view by considering them as duals of Finsler Spaces, Tensor N.S. 46(1987), 330-334.
* [14] R. Miron, The geometry of Cartan spaces, Prog. Math. India. 22(1988), 1-38.
* [15] E. Peyghan and A. Tayebi, Finslerian Complex and Kählerian Structures, Journal of Nonlinear Analysis: Real World Appl. doi: 10.1016/j.nonrwa.2009.10.022.
* [16] E. Peyghan and A. Tayebi, A Kähler structure on Finsler spaces with nonzero constant flag curvature, J. Math. Phys. 51, 022904 (2010).
* [17] Z. Shen, Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, 2001.
* [18] Z. Shen, Lectures on Finsler Geometry, World Scientific, 2001.
* [19] S. Vacaru, Deformation quantization of almost Kähler models and Lagrange-Finsler spaces, J. Math. Phys. 48, 123509 (2007).
* [20] S. Vacaru, Deformation quantization of nonholonomic almost Kähler models and Einstein gravity, Phys. Lett. A. 372(2008), 2949-2955.
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Esmail Peyghan
Faculty of Science, Department of Mathematics
Arak University
Arak, Iran
Email: epeyghan@gmail.com
Akbar Tayebi
Faculty of Science, Department of Mathematics
Qom University
Qom. Iran
Email: akbar.tayebi@gmail.com
|
arxiv-papers
| 2010-03-12T11:08:31 |
2024-09-04T02:49:09.054959
|
{
"license": "Public Domain",
"authors": "E.Peyghan and A. Tayebi",
"submitter": "Esmaeel Peyghan",
"url": "https://arxiv.org/abs/1003.2518"
}
|
1003.2553
|
# Testing a pulsating binary model for long secondary periods in red variables
J.D. Nie Department of Astronomy, Beijing Normal University, Beijing 100875,
China niejundan@mail.bnu.edu.cn X.B. Zhang National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100012, China
xzhang@bao.ac.cn B.W. Jiang Department of Astronomy, Beijing Normal
University, Beijing 100875, China bjiang@bnu.edu.cn
###### Abstract
The origin of the long secondary periods (LSPs) in red variables remains a
mystery up to now, although there exist many models. The light curves of some
LSPs stars mimic an eclipsing binary with a pulsating red giant component. To
test this hypothesis, the observational data of two LSP variable red giants,
77.7795.29 and 77.8031.42, discovered by the MACHO project from the LMC, are
collected and analyzed. The probable eclipsing features of the light curves
are simulated by the Wilson-Devinney (W-D) method. The simulation yields a
contact and a semidetached geometry for the two systems, respectively. In
addition, the pulsation constant of the main pulsating component in each
binary system is derived. By combining the results of the binary model and the
pulsation component, we investigate the feasibility of the pulsating binary
model. It is found that the radial velocity curve expected from the binary
model has a much larger amplitude than the observed one and a period double
the observed one. Furthermore, the masses of the components based on the
density derived from the binary orbit solution are too low to be compatible
with both the evolutionary stage and the high luminosity. Although the
pulsation mode identified by the pulsation constant which is dependent on the
density from the binary-model is consistent with the first or second overtone
radial pulsation, we conclude that the pulsating binary model is a defective
model for the LSP.
binaries: close - galaxies: individual (LMC) - stars: late-type - stars:
oscillations
## 1 INTRODUCTION
Among the variable red giant stars, one sub-type exhibits long secondary
periods (LSPs). The light curves of these stars exhibit not only a short
primary period but also a long secondary period, which is approximately nine
times longer than the short one. This phenomenon has been known for several
decades (Payne-Gaposhkin, 1954; Houk, 1963). Some samples of these LSPs
variables are shown in Kiss et al. (1999). An interest in the stars with the
LSPs has been renewed by the study of Wood et al. (1999). This paper shows
that in the LMC, $\sim$25 % of all variable asymptotic giant branch (AGB)
stars show LSP roughly nine times longer than the short primary period which
is typically $\sim$ 30–200 days. Meanwhile, a study of the bright local
pulsating red giants indicates that at least one-third of these stars exhibit
LSPs (Percy et al., 2004). Soszynski also gives $\sim$30% of pulsating red
giants in the LMC with LSPs (Soszyński et al., 2007). In the period-luminosity
(P-L) diagram, it is interesting to see that the LSP variables follow a
distinct sequence (sequence D), which is roughly parallel to the radial
pulsation sequences A, B, and C for variable red giants.
The LSPs present in variable red giants have attracted a lot of attention
since their discovery, but their origin still remains mysterious. Since the
LSPs are several times longer than the fundamental radial periods, they could
not be caused by normal radial pulsations. Moreover, Wood et al. (1999) and
Wood et al. (2004) note that the LSPs can not be explained by $g^{+}$ mode for
the oscillatory $g^{+}$ mode is evanescent in convective region and it is
unlikely to be observable in a red giant because of its convective envelope.
The $g^{-}$ mode is dynamically unstable in the convection envelope, and
unable to lead to any oscillation. Regarding the nonradial $p$ modes, their
periods are rather shorter than those of the fundamental radial modes, so they
can not explain the LSPs either. Alternatively, if the LSPs are caused by some
strange modes, they would be extremely damped and should not be seen (Wood,
2000). One more possible explanation is the rotating spheroid model, which can
explain the shape of the velocity curve, but there is no reason for the
rotating period to bring about the observed P-L relation.
It seems that variable red giants with long secondary periods (hereinafter
referred to as LSPVs) can not be easily interpreted as pulsating red giants.
In this situation, a hypothesis of binarity arose. It has been suggested that
the sequence D stars could be components of close binary systems, and the LSPs
could be interpreted as the light variations caused by ellipsoidal binary
motions or eclipses. Soszyński et al. (2004, 2007) find that the sequence D
stars overlap with and have a direct continuation of the sequence E stars that
are mostly confirmed to be binaries. Radial velocity variations on the time-
scale of LSPs have also been measured in a number of sequence D stars, such
radial velocity variations being a requirement if the sequence D stars are
binaries (Hinkle et al., 2002; Olivier & Wood, 2003; Wood et al., 2004;
Nicholls et al., 2009). Some observational arguments supporting the binary
hypothesis have already been reviewed by Soszyński (2007).
After examining the light curves of LSPVs collected from the MACHO database,
we note that many of them show some nearly regular and stable, eclipse-like
light variations at the long secondary periods with a large amplitude in
comparison with that of the primary pulsation. It seems that the LSPs can be
easily interpreted as eclipses by an orbiting component. If this can be shown
to be the case, it would be direct evidence to support the binary hypothesis
for the sequence D stars. To test this idea, we propose a model – pulsating
binary, i.e. a binary system with a pulsating component. We begin with some
very probable eclipsing LSPs candidates whose light curves look like eclipsing
binary with primary and secondary minima. The light curves of two LSP stars
are collected and analyzed by using the Wilson-Devinney ( W-D, hereafter) code
and the power-spectrum method. Afterwards, the proposed eclipsing and
pulsating nature as well as their evolutionary properties are discussed.
## 2 THE DATA
Since the LSPs are sometimes as long as 1500 days, the candidate selection is
based on the long-term MACHO survey. The MACHO project is a microlensing
experiment that monitored numerous stars in the LMC, SMC and Galactic Bulge
over a time of $\sim$ 3000 days. As the by-products of this survey, a large
number of variables were discovered and monitored, including many LSPVs. From
this database, two LSPVs in the LMC are selected as our program objects. Their
F.T.S (field.title.sequence) numbers are 77.7795.29 and 77.8031.42,
respectively. They are chosen as the working sample for the following reasons:
(1) The time base of observations for both of the stars is long enough to
cover at least a couple of LSPs. (2) Their light variations are both eclipse-
like and symmetric over the long secondary period, and have distinct short
primary variation. (3) These two stars are quite different from each other not
only in their LSPs but also in the shapes of the light curves, they are
expected to be different types of binaries.
The MACHO data are taken in non-standard red and blue bandpasses. The adopted
transformation from the MACHO instrumental photometry to Kron-Cousins $R$ and
$V$ are given in Alcock et al. (1999). However, it does not work well for our
candidates in practice and brings about visible dispersion for the photometric
points. Analyzing the data sets, we find the reason mainly being that the
observed color index term “$(V_{M,t}-R_{M,t})$” in Alcock et al. (1999) brings
in large dispersion as the raw red and blue magnitudes transfer their errors
to each other. This term requires both measurements to be good at each time
when adopting the transformation. In addition, the transformation has a term
involving airmass. When the value of the airmass is greater than 2.0, the
correction for the atmosphere extinction is rather uncertain. We adopt another
transformation to avoid the above problems as Struble et al. (2006):
$\displaystyle\ R=R_{M,t}+23.90+0.1825(V_{M,t}-R_{M,t})$ (1a) $\displaystyle
V=V_{M,t}+24.22-0.1804(V_{M,t}-R_{M,t}),$ (1b)
where the term “$V_{M,t}-R_{M,t}$” is the mean color index for a red giant
instead of an individual measurement. Since color indexes of the red giants
have a small scattering of about 0.2 magnitude, the adoption of the mean
multiplied by a factor of 0.18 brings about a dispersion of only 0.04 mag,
much smaller than the photometric uncertainty. The light curves of
standardized $R$ and $V$ magnitudes for our two candidates are shown in Fig.
1.
OGLE–II (Udalski et al., 1997; Szymański, 2005) also provides the light
variation data for red giants in the $I$ band, and the time span of the light
curve overlaps the last half of the MACHO data, with $\sim$ 300 days extension
which is not long for the LSP. Besides and more importantly, the data are much
sparser than the MACHO in the overlapping time range and it is of little help
in analyzing the short primary pulsation. Thus we don’t include the OGLE–II
data for following analysis.
## 3 THE BINARY MODEL
Assuming that the phenomenon of the LSPs is caused by an orbiting component,
we can employ some binary simulation method to analyze the LSPs light curve.
The W-D method, which consists of two main FORTRAN programs – LC and DC (short
for Light Curve and Differential Correction, respectively), is a simulation
method producing the photometric solution for the binary system.
The 2007 version of the W-D code is used to analyze the eclipse-like light
curves in $R$ and $V$ bandpasses of our two candidates. Nonlinear limb-
darkening via a logarithmic form along with many other features (Wilson &
Devinney, 1971; Wilson, 1979, 1990; Kallrath et al., 1998) are used in the
code. Considering the likely close distance between the components, the effect
of reflection is taken into account.
In computing the photometric solution, the important parameters adopted in the
DC program are as follows. The orbital period for adjustment is computed by
the PDM (Phase Dispersion Minimization) method (Stellingwerf, 1978) and the
Period04 software for double-check at the beginning. The value is about 970
days for 77.7795.29 and 1700 days for 77.8031.42. Note that these orbital
periods are twice the normally adopted value for the LSP, the latter
corresponding to one cycle of long period variation apparent in the light
curve. The temperatures of the two primary stars are both set to 3311 K
according to Fox & Wood (1982) for long period stars. The initial bolometric
(X1,X2,Y1,Y2) and monochromatic (x1,y1,x2,y2) limb-darkening coefficients of
the components are taken from Van Hamme (1993). The gravity darkening
exponents are set to 0.32 for both the primary and the secondary component for
convective envelopes according to Lucy (1967). The bolometric albedos are
taken as A1=0.4 and A2=0.5 following Rucinski (1969).
The preliminarily estimated parameters are put in the first iteration of the
DC code and after several iterations, a converged solution is reached. The
photometric solutions are shown in Table 1, and the synthesized $R$ band light
curves computed from the LC code are shown in Fig. 1 together with the
observed curves of the two candidates. The photometric solutions in the table
provide insight into the orbital dynamics of the binary system. The orbit
period ($P_{\rm orb}$) adjusted by the DC code is consistent with the value
from the PDM and Period04 method. The large angles of inclination ($i$), and
argument of periastron ($\omega$) describe the orientation of the binary
orbit, while the value of $e$ reveals a nearly circular orbit. Moreover, we
find that the secondary component is a low mass M-type star, according to the
mass ratio and effective temperature. All the binary modes, including detached
mode, semidetached mode, and contact mode in the code are tried in the
solution-seeking procedure, and it is found that only one mode can reach a
converged solution for each candidate. For 77.7795.29, a contact configuration
converges, and for 77.8031.42, a semidetached system with the primary
component filling its Roche lobe converges. This implies that both of them
have a mass exchange with their components due to Roche lobe overflow. No
converged solutions were found for systems showing purely ellipsoidal
variations i.e. systems with an invisible secondary companion.
Using the photometric solutions, the theoretical radial velocities can be also
simulated by the LC code. We try this in order to compare them with some
observational facts. The bottom panel of Fig. 1 shows the synthesized velocity
curves for the primary components. The full amplitude of the velocity
variation is about 11.0 km s-1 for 77.7795.29, and 9.0 km s-1 for 77.8031.42,
where the value of the binary system mass is taken from later Section 5 (but
see discussion there for the problem of mass). Yet all the studies of radial
velocity for LSPVs (Hinkle et al., 2002; Olivier & Wood, 2003; Wood et al.,
2004; Nicholls et al., 2009) show typical amplitudes of 3-4 km s-1 and no
objects with full velocity amplitude greater than 7 km s-1 are found.
Moreover, the synthesized velocity curves of the primary components have a
period which is twice of the light variation period. In contrast, all the
observed radial velocities for the LSPVs have the same period as that of the
light variation. The synthesized radial velocities for the second stars are
also obtained from the LC code, and we find that the full amplitude is about
18.5 km s-1 for 77.7795.29 and 18.3 km s-1 for 77.8031.42. The large velocity
separations between the primary and secondary stars would have been readily
seen, while in the existing spectral observations there is no indication of
this. Therefore the synthesized velocity curve based on the binary model is
inconsistent with the observation.
In addition, we should note that, among all the parameters derived from the DC
code, some are highly reliable, including the orbital period, mass ratio and
temperature difference if we admit the binary hypothesis, and some are
relatively uncertain, such as the effective temperatures of the two
components. Several parameters derived from the LC code, such as the mass and
radius, are particularly uncertain for there are no radial velocities
available for comparison, so they are not listed in Table 1. The synthesized
radial velocities produced by the LC code are also very uncertain.
## 4 THE INTRINSIC PULSATION
For an eclipsing binary with a pulsating (primary) component, the observed
light curve could be approximately interpreted as (Zhang et al., 2009):
$l_{\rm obs}=l_{1}f_{\rm pul}+l_{2},$ (2)
where $l_{1}$ and $l_{2}$ are the calculated brightness contributed by the
primary and secondary components respectively, and $f_{\rm pul}$ denotes the
pulsating variation of $l_{1}$. With the derived binary photometric solution,
we could compute the brightness of the binary components separately at each
observation epoch by using the LC code in the W-D code package, from which the
“pure” pulsation light variation of the primary star could then be extracted.
However, we do not use this approach, because the intrinsic pulsating
amplitude of the primary component is so large that the theoretically
synthesized light curve computed by the LC code would have up-down fluctuation
and the extracted light curve would not be exactly equivalent to the true
intrinsic pulsation variations. Instead, we make use of the original data sets
to analyze the primary pulsation.
Period04, a commonly used technique for time series analysis, which utilizes
Fourier transforms as well as multiple-least-squares algorithms, is applied to
the power spectrum analysis. In doing that, we use the $R$ band time series
data which are brighter and with higher photometric accuracy than in the $V$
band. We select only those peaks in the power spectrum with signal to noise
ratio (S/N) larger than 4.0 for further discussion. Fig. 2 and Fig. 3 show the
first four-frequency solution of the data sets in the $R$ band before and
after subtraction of the most prominent remaining frequency (prewhitening). At
the top of the two figures the spectral window based on the epoches of
available observations is displayed. We notice that the alias patterns,
including the 1 c/d daily alias, are quite low in power. The next four
patterns in both figures are the step-by-step power spectrum derived from the
pulsation data. The fitting is stopped when the light curve is well fitted.
For 77.7795.29, nine frequencies are obtained, by using most parts of the
MACHO data while the last part of the MACHO data is dropped because of a large
gap. Among all the nine frequencies obtained, we find that the main power
spectrum are dominated at $f_{1}=0.0020$ c/d and $f_{2}=0.0100$ c/d. $f_{1}$
is two times the assumed orbital frequency ($f_{0}=1/970=0.00103$ c/d), it is
not accepted as the real pulsating frequency, so only $f_{2}$ remains as the
intrinsic pulsating frequency. The other derived frequencies are found to be
related to $f_{0}$ and $f_{2}$ as follows: $f_{1}=2f_{0}$,
$f_{3}=f_{2}+6f_{0}$, $f_{4}=f_{0}$, $f_{5}=4f_{0}$, $f_{6}=f_{2}+1/365$,
$f_{7}=f_{2}+7f_{0}$, $f_{8}=3f_{0}$, $f_{9}=5f_{0}$, all of them are aliases.
For the other star 77.8031.42, we obtain $f_{2}=0.0112$ c/d, the real
pulsating frequency, and the others have the relations of: $f_{1}=2f_{0}$,
$f_{3}=4f_{0}$, $f_{4}=f_{2}+0.0004$, $f_{5}=3f_{6}$, $f_{6}=2f_{0}+f_{8}$,
$f_{7}=4f_{6}$, $f_{8}=1/365$, $f_{9}=f_{2}+10f_{0}$, $f_{10}=f_{2}-0.0004$,
$f_{11}=f_{6}+f_{8}$, $f_{12}=10f_{0}$, $f_{13}=12f_{0}$, $f_{14}=f_{0}$,
where $f_{0}$ is the assumed orbital frequency. Here, 0.0004 c/d, may be the
rotation frequency of the LSPVs, in the theoretical range of 2400–10000 days
for the rotation period of red giants (Kiss et al., 2000; Wood et al., 2004).
Based on the above analysis, only one short primary pulsation is found for our
two candidates respectively and the result is consistent with previous result
from Wood et al. (1999) for LSPs stars.
The eclipsing light curve synthesis has provided us some important parameters
of the binary system such as the mass ratio and the unified radius
$r_{1}=R_{1}/A$ which normalizes the radius to the semi-major axis (actually
the accurate ratio can be obtained by the LC code). With these values, the
mean density of the pulsating primary component can be precisely determined.
From the Kepler’s law
$P_{\rm orb}^{2}=\frac{4\pi^{2}A^{3}}{G(M_{1}+M_{2})}$ (3)
and the density in solar units
$\rho_{1}/\rho_{\sun}=\frac{M_{1}/R_{1}^{3}}{M_{\sun}/R_{\sun}^{3}},$ (4)
we get:
$\rho_{1}/\rho_{\sun}=\frac{4\pi^{2}R_{\sun}^{3}}{M_{\sun}G(1+q)r_{1}^{3}P_{\rm
orb}^{2}},$ (5)
where $P_{\rm orb}$ is the orbit period of the binary system.
Substituting the values of $P_{\rm orb}$, $q$ and $r_{1}$ from the W-D code
analysis of the light curve into Eq. (5), the mean density is computed, being
$8.47\times 10^{-8}$ and $3.67\times 10^{-8}$ in solar unit respectively,
which agrees with the density of red giants. Then the pulsation constant can
be easily calculated from: $Q=P_{\rm{pul}}(\rho_{1}/\rho_{\sun})^{1/2}$, where
$P_{\rm pul}$ is the “pure” pulsation period. The pulsation constant $Q$ is an
important parameter to tell the intrinsic oscillation mode of a pulsating
star, especially in normal radial mode. The $Q$ value calculated on the binary
hypothesis, is shown in Table 3 and equal to 0.029 and 0.017 for our two
candidates, respectively.
The values of $r1$ calculated by the LC code in Table 3, 0.47 and 0.44
respectively, are consistent with that estimated in Soszyński et al. (2007)
which notices $R_{1}/A\approx 0.4$ in the binary scenario. The result of mode
identification shows agreement with the theoretical expectation for red giants
(Fox & Wood, 1982) and reveals that 77.7795.29 is a first overtone pulsating
star while 77.8031.42 is a second overtone pulsating star. The pulsation
properties derived from a binary model are reasonable for a pulsating red
giant.
## 5 THE EVOLUTIONARY PROPERTY
A further analysis of the pulsating binary model is carried out from the
evolutionary property of the components, as the stellar properties of the red
giants are rarely known, by making use of the results derived in Section 4.
From the 2MASS near-infrared photometric data and the empirical bolometric
correction factor $BC_{\rm K}$ as a function of $(J-K)$ (Bessell & Wood,
1984), and with a distance modulus of 18.54 for the LMC, the bolometric
magnitude can be calculated, and the luminosity can thus be obtained. Applying
the formula $L=4\pi\sigma R^{2}T_{\rm{eff}}^{4}$, the equilibrium radius of
the primary star can be computed. Here, $T_{\rm{eff}}$ is an assumed value,
3311 K, which is the mean temperature for long period variables in LMC. We do
not calculate the temperature from the color index because the secondary
component is not understood well and its contribution to the observed color
index is not known. With the mean density and the radius, the mass of the
primary star is derived. All these values are listed in Table 3.
The luminosity of the primary stars in the table is a few thousands of solar
luminosity, consistent with the red giant luminosity. The radii of both
candidate stars are on the order of hundreds of solar radius, which are
consistent with red giants. However, the masses of the two LSPVs are too low,
and the masses of their components would be even lower, according to the mass
ratios in Table 1. Moreover, these low mass stars have very high luminosities,
which seems impossible. For the star 77.7795.29, the mass is 0.31$M_{\sun}$
and its luminosity is 2502$L_{\sun}$. In the age of universe, such a low mass
star is impossible to evolve up to the red giant branch. For the secondary
component, the mass is about 0.188$M_{\sun}$, while its luminosity is
1607$L_{\sun}$, the value of which is computed from the data of
“$L_{1}/(L_{1}+L_{2})$” in Table 1. It is hard to imagine how such low mass
stars in a binary system exchange their masses. It is difficult to find an
evolutionary path to satisfy this situation. This problem is also present in
the star 77.8031.42, with a mass of 0.36$M_{\sun}$ and a luminosity of
4839$L_{\sun}$, whose component has about 0.175$M_{\sun}$ and 3790$L_{\sun}$.
If a reasonable mass is expected, the density should be upgraded by a factor
of four or the radius be upgraded by a factor of 1.4. Recalling the
determination of the density from Eq. (5), the density is calculated from the
parameters, $q$, $r_{1}$ and $P_{\rm orb}$ all of which have relatively small
errors which will not bring about a density uncertainty of a factor of four.
Besides, the density is reasonable for a red giant. On the other hand, the
radius is derived from the luminosity and effective temperature. The
luminosity has uncertainty from the bolometric correction and the neglect of
interstellar extinction, and the effective temperature also has some
uncertainty. If the luminosity is higher or the effective temperature lower,
the radius would be larger and so the mass. However, if the mass is indeed
higher, the system would then have a large separation from Eq.(3) and an even
more serious problem with the large velocity amplitude. In summary, the
pulsating binary model requires a low mass incompatible with an evolved
stellar stage having the observed high luminosity.
## 6 DISCUSSION AND CONCLUSION
The origin of the LSPs is unknown and there exist many explanations. The light
curves of some LSPVs are eclipse-like and it seems that this phenomenon is due
to an invisible component orbiting around the pulsating red giant. To test
this hypothesis, we propose a model – pulsating binary, and select two LSP
stars to analyze their orbit motions and pulsation nature.
On the assumption of binarity, we simulate the photometric light curves of the
systems by using the W-D method. The photometric solutions give us a
configuration for the binary system: a contact system for 77.7795.29 and a
semidetached system for 77.8031.42 with its Roche lobe fully filled. It means
that the LSP star may have strong interaction and very probably mass transfer
with the other component via Roche lobe overflow. However, Roche lobe overflow
will rule out the binary hypothesis if we accept the view of Wood et al.
(2004), which argues that the mass transfer of Roche lobe will result in a
short merger timescale of about 1000 yrs. Moreover, the calculated effective
temperatures and the “$L_{1}/(L_{1}+L_{2})$” values in Table 1 suggest that
the secondary star is also a red giant with a similar temperature and
luminosity to that of the primary red giant. This would lead to a double-lined
spectroscopic binary. However, no observer of radial velocities in these
systems has reported seeing spectral lines from the secondary star (Hinkle et
al., 2002; Olivier & Wood, 2003; Wood et al., 2004; Nicholls et al., 2009).
There are also some serious problems about the synthesized velocity curves.
The simulation produces one cycle of velocity curve for two cycles of the
light curve, while the observed velocities show one cycle of the radial
velocity curve for one cycle of the light curve. The synthesized full velocity
amplitudes for the star 77.7797.29 and 77.8031.42 are much greater than the
typical values of other LSPVs. The full amplitudes of the secondary components
are even larger and would lead to great velocity separations between the
primary and the secondary stars, which have never been found by spectral
observations.
Fourier analysis is applied to investigate the intrinsic oscillation of the
LSPVs over the raw data sets. Using the parameters ($R_{1}/A$ and $q$)
obtained from the W-D code, the mean densities of the LSPVs are deduced and
they are consistent with the red giant phase. Then the pulsation constants are
obtained by using the classical equation
$Q=P_{\rm{pul}}(\rho_{1}/\rho_{\sun})^{1/2}$ to identify the pulsation mode.
For star 77.7795.29, one pulsating frequency is detected and it is caused by
the first overtone radial pulsation, and for star 77.8031.42, the only
pulsating frequency is caused by the second overtone radial pulsation. These
agree with the theoretical value for red giants and the conclusion of Wood et
al. (1999).
The stellar properties of LSPVs are derived by using the information from the
pulsating binary model. We calculate the bolometric magnitude, luminosity,
radius and mass of the primary star, and find some of them conflict very
seriously with the evolutionary properties of red giants. In particular, the
masses for the star 77.7795.29 and 77.8031.42 are both less than
0.4$M_{\sun}$, and their luminosities are too high for such low masses. It is
difficult to imagine how such low mass stars could have such large
luminosities. The situation gets even worse if we apply the mass ratio and the
value of “$L_{1}/(L_{1}+L_{2})$” to calculate the mass and luminosity of the
secondary star.
Therefore, the radial velocities and the masses computed from the pulsating
binary model do not agree with some observations and facts about red giants.
We conclude that the model “pulsating binary” has some deficiencies in dealing
with the observed properties of LSPVs and that the binary hypothesis for
explaining the LSPs seems unreasonable.
The authors are very grateful to Prof. Peter Wood for very constructive
suggestions and helpful discussion. They also thank the anonymous referee for
helpful suggestions to improve the work significantly. This research is
supported by the National Natural Science Foundation of China (NSFC) through
grant 10778601 and the Ministry of Science and Technology of the People’s
Republic of China through grant 2007CB815406.
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Figure 1: Light and velocity curves synthesized by the W-D code for 77.7795.29
and 77.8031.42. Also shown are the observed $R$ and $V$ band light curves from
the MACHO project. Red asterisk represents the observed data point in $R$ band
while the blue one is the $V$ band data point. The black solid line in the $R$
band panel denotes the theoretical synthesized light curve. The bottom panel
is the synthesized velocity curve.
Figure 2: Power spectrum and light curve of the primary star 77.7795.29 in
binary system. The top five panels show power spectra and the bottom panel
shows the light curve in the $R$ band. The black solid light curve is the fit
of frequency solution described in Sect. 4 of the text.
Figure 3: The same as Fig. 2, but for the star 77.8031.42.
Table 1: Photometric solution for stars 77.7795.29 and 77.8031.42. | 77.7795.29 | | 77.8031.42
---|---|---|---
classification | contact | | semidetached
$P_{\rm orb}$ (day) | 970.9 | | 1700.9
$T_{1}$ (K) | 3311(assumed) | | 3311(assumed)
$T_{2}$ (K) | 3284 | | 3544$\pm{48}$
$e$ | 0.0016$\pm{0.0008}$ | | 0.0123$\pm{0.0164}$
$i$ (degree) | 62.49$\pm{0.27}$ | | 63.19$\pm{2.22}$
$\omega$ | 3.64$\pm{0.20}$ | | 2.91
$q=m_{2}/m_{1}$ | 0.608$\pm{0.003}$ | | 0.486$\pm{0.029}$
$\varphi_{1}$ | 2.8359$\pm{0.0041}$ | | 2.8748
$\varphi_{2}$ | 2.8359 | | 2.8943 $\pm{0.055}$
$L_{1}/(L_{1}+L_{2})$ (V band) | 0.609 | | 0.561
Table 2: Results of Fourier analysis for stars 77.7795.29 and 77.8031.42. | | 77.7795.29 | | | | 77.8031.42 | |
---|---|---|---|---|---|---|---|---
F | Frequency | Amplitude | Phase | S/N | Frequency | Amplitude | Phase | S/N
| (c/d) | (mag) | | | (c/d) | (mag) | |
$f_{1}$ | 0.0020 | 0.2138 | 0.3144 | 21.6729 | 0.0011 | 0.1464 | 0.46905 | 19.3482
$f_{2}$ | 0.0100 | 0.0529 | 0.6576 | 22.4815 | 0.0112 | 0.0770 | 0.7190 | 44.7148
$f_{3}$ | 0.0162 | 0.0327 | 0.0015 | 13.9081 | 0.0022 | 0.0613 | 0.2773 | 35.6114
$f_{4}$ | 0.0011 | 0.0363 | 0.9538 | 15.4348 | 0.0116 | 0.0446 | 0.7607 | 25.9384
$f_{5}$ | 0.0041 | 0.0280 | 0.6792 | 11.9004 | 0.0120 | 0.0404 | 0.6926 | 23.4930
$f_{6}$ | 0.0130 | 0.0205 | 0.8848 | 8.7069 | 0.0040 | 0.0260 | 0.1102 | 15.1249
$f_{7}$ | 0.0174 | 0.0189 | 0.4320 | 8.1379 | 0.0162 | 0.0267 | 0.1793 | 15.4983
$f_{8}$ | 0.0031 | 0.0199 | 0.0318 | 8.0322 | 0.0029 | 0.0351 | 0.8120 | 20.3622
$f_{9}$ | 0.0049 | 0.0199 | 0.6315 | 8.4640 | 0.0169 | 0.0224 | 0.8329 | 13.0511
$f_{10}$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 0.0108 | 0.0311 | 0.7704 | 18.1114
$f_{11}$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 0.0070 | 0.0287 | 0.0900 | 16.6748
$f_{12}$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 0.0057 | 0.0236 | 0.3752 | 13.7059
$f_{13}$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 0.0065 | 0.0242 | 0.2452 | 14.0897
$f_{14}$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 0.0006 | 0.0206 | 0.8211 | 11.9582
Table 3: Parameters of the primary stars Star | Frequency | $R_{1}/A$ | $\rho_{1}/\rho_{\sun}$ | Q | Mode | $M_{bol}$ | $L/L_{\sun}$ | $R/R_{\sun}$ | $M/M_{\sun}$
---|---|---|---|---|---|---|---|---|---
| (c/d) | | | | | (mag) | | |
77.7795.29 | 0.0100 | 0.47 | 8.47E-8 | 0.029 | 1H | -3.75 | 2502 | 153 | 0.31
77.8031.42 | 0.0112 | 0.44 | 3.66E-8 | 0.017 | 2H | -4.46 | 4839 | 213 | 0.36
|
arxiv-papers
| 2010-03-12T14:49:33 |
2024-09-04T02:49:09.064918
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. D. Nie, X. B. Zhang and B. W. Jiang",
"submitter": "Jundan Nie",
"url": "https://arxiv.org/abs/1003.2553"
}
|
1003.2670
|
# Gradient estimates for a nonlinear diffusion equation on complete manifolds
Jia-Yong Wu Department of Mathematics, East China Normal University,
Shanghai, China 200241 jywu81@yahoo.com
(Date: January 1, 2009.)
###### Abstract.
Let $(M,g)$ be a complete non-compact Riemannian manifold with the
$m$-dimensional Bakry-Émery Ricci curvature bounded below by a non-positive
constant. In this paper, we give a localized Hamilton-type gradient estimate
for the positive smooth bounded solutions to the following nonlinear diffusion
equation
$u_{t}=\Delta u-\nabla\phi\cdot\nabla u-au\log u-bu,$
where $\phi$ is a $C^{2}$ function, and $a\neq 0$ and $b$ are two real
constants. This work generalizes the results of Souplet and Zhang (Bull.
London Math. Soc., 38 (2006), pp. 1045-1053) and Wu (Preprint, 2008).
###### Key words and phrases:
local gradient estimate; nonlinear diffusion equation; Bakry-Émery Ricci
curvature
###### 2000 Mathematics Subject Classification:
Primary 58J35; Secondary 58J35, 58J05. Chinese Library Classification:
O175.26; O186.12
## 1\. Introduction
Let $(M,g)$ be an $n$-dimensional non-compact Riemannian manifold with the
$m$-dimensional Bakry-Émery Ricci curvature bounded below. Consider the
following diffusion equation:
(1.1) $u_{t}=\Delta u-\nabla\phi\cdot\nabla u-au\log u-bu$
in $B(x_{0},R)\times[t_{0}-T,t_{0}]\subset M\times(-\infty,\infty)$, where
$\phi$ is a $C^{2}$ function, and $a\neq 0$ and $b$ are two real constants.
Eq. (1.1) is closely linked with the gradient Ricci solitons, which are the
self-similar solutions to the Ricci flow introduced by Hamilton [3]. Ricci
solitons have inspired the entropy and Harnack estimates, the space-time
formulation of the Ricci flow, and the reduced distance and reduced volume.
Below we recall the definition of Ricci solitons (see also Chapter 4 of [4]).
###### Definition 1.1.
A Riemannian manifold $(M,g)$ is called a _gradient Ricci soliton_ if there
exists a smooth function $f:M\rightarrow\mathbb{R}$, sometimes called
_potential function_ , such that for some constant $c\in\mathbb{R}$, it
satisfies
(1.2) $Ric(g)+\nabla^{g}\nabla^{g}f=cg$
on $M$, where $Ric(g)$ is the Ricci curvature of manifold $M$ and
$\nabla^{g}\nabla^{g}f$ is the Hessian of $f$. A soliton is said to be
_shrinking_ , _steady_ or _expanding_ if the constant $c$ is respectively
positive, zero or negative.
Suppose that $(M,g)$ be a gradient Ricci soliton, and $c$, $f$ are described
in Definition A. Letting $u=e^{f}$, under some curvature assumptions, we can
derive from (1.2) that (cf. [5], Eq. (7))
(1.3) $\Delta u+2cu\log u=(A_{0}-nc)u,$
for some constant $A_{0}$. Eq. (1.3) is a nonlinear elliptic equation and a
special case of Eq. (1.1). For this kind of equations, Ma (see Theorem 1 in
[5]) obtained the following result.
Theorem A ([5]). _Let $(M,g)$ be a complete non-compact Riemannian manifold of
dimension $n\geq 3$ with Ricci curvature bounded below by the constant
$-K:=-K(2R)$, where $R>0$ and $K(2R)\geq 0$, in the metric ball $B_{2R}(p)$.
Let $u$ be a positive smooth solution to the elliptic equation_
(1.4) $\Delta u-au\log u=0$
_with $a>0$. Let $f=\log u$ and let $(f,2f)$ be the maximum among $f$ and
$2f$. Then there are two uniform positive constant $c_{1}$ and $c_{2}$ such
that_
(1.5) $\displaystyle|\nabla f|^{2}-a(f,2f)$ $\displaystyle\leq$
$\displaystyle\frac{n\Big{[}(n+2)c^{2}_{1}+(n-1)c^{2}_{1}(1+R\sqrt{K})+c_{2}\Big{]}}{R^{2}}+2n\Big{(}|a|+K\Big{)}$
_in $B_{R}(p)$._
Then Yang (see Theorem 1.1 in [6]) extended the above result and obtained the
following local gradient estimate for the nonlinear equation (1.1) with
$\phi\equiv c_{0}$, where $c_{0}$ is a fixed constant.
Theorem B ([6]). _Let $M$ be an $n$-dimensional complete non-compact
Riemannian Manifold. Suppose the Ricci curvature of $M$ is bounded below by
$-K:=-K(2R)$, where $R>0$ and $K(2R)\geq 0$, in the metric ball $B_{2R}(p)$.
If $u$ is a positive smooth solution to Eq. (1.1) with $\phi\equiv c_{0}$ on
$M\times[0,\infty)$ and $f=\log u$, then for any $\alpha>1$ and $0<\delta<1$_
(1.6) $\displaystyle|\nabla f|^{2}(x,t)-\alpha af(x,t)-\alpha b-\alpha
f_{t}(x,t)$ $\displaystyle\leq$ $\displaystyle\frac{n\alpha^{2}}{2\delta
t}+\frac{n\alpha^{2}}{2\delta}\Bigg{\\{}\frac{2\epsilon^{2}}{R^{2}}+\frac{\nu}{R^{2}}+\sigma+\frac{\epsilon^{2}}{R^{2}}(n-1)\left(1+R\sqrt{K(2R)}\right)$
$\displaystyle+\frac{K(2R)}{\alpha-1}+\frac{n\alpha^{2}\epsilon^{2}}{8(1-\delta)(\alpha-1)R^{2}}\Bigg{\\}}$
_in $B_{R}(p)\times(0,\infty)$, where $\epsilon>0$ and $\nu>0$ are some
constants and where $\sigma=a/2$ if $a>0$; $\sigma=-a$ if $a<0$._
Recently, the author (see Theorem 1.1 in [2]) used Souplet-Zhang’s method in
[1] and obtained a localized Hamilton-type gradient estimate for the positive
smooth bounded solutions of the equation (1.1) with $\phi\equiv c_{0}$.
Theorem C ([2]). _Let $(M,g)$ be an $n$-dimensional non-compact Riemannian
manifold with $Ric(M)\geq-K$ for some constant $K\geq 0$. Suppose that
$u(x,t)$ is a positive smooth solution to the parabolic equation (1.1) with
$\phi\equiv c_{0}$ in $Q_{R,T}\equiv B(x_{0},R)\times[t_{0}-T,t_{0}]\subset
M\times(-\infty,\infty)$. Let $f:=\log u$. We also assume that there exists
non-negative constants $\alpha$ and $\delta$ such that $\alpha-f\geq\delta>0$.
Then there exist three dimensional constants $\tilde{c}$, $c(\delta)$ and
$c(\alpha,\delta)$ such that_
(1.7) $\frac{|\nabla
u|}{u}\leq\left(\frac{\tilde{c}}{R}\beta{+}\frac{c(\alpha,\delta)}{R}{+}\frac{c(\delta)}{\sqrt{T}}{+}c(\delta)\left(|a|+K\right)^{1/2}\kern-2.0pt{+}c(\delta)|a|^{1/2}\beta^{1/2}\right)\left(\alpha{-}\frac{b}{a}{-}\log
u\right)$
_in $Q_{R/2,T/2}$, where $\beta:=\max\left\\{1,|\alpha/\delta-1|\right\\}$._
The purpose of this paper is to extend Theorem C to the general nonlinear
diffusion equation (1.1) via the $m$-dimensional Bakry-Émery Ricci curvature.
Let us first recall some facts about the $m$-dimensional Bakry-Émery Ricci
curvature (please see [7, 8, 9, 10] for more details). Given an
$n$-dimensional Riemannian manifold $(M,g)$ and a $C^{2}$ function $\phi$, we
may define a symmetric diffusion operator $L:=\Delta-\nabla\phi\cdot\nabla,$
which is the infinitesimal generator of the Dirichlet form
$\mathcal{E}(f,g)=\int_{M}(\nabla f,\nabla g)\mathrm{d}\mu,\,\,\,\forall
f,g\in C_{0}^{\infty}(M),$
where $\mu$ is an invariant measure of $L$ given by
$\mathrm{d}\mu=e^{-\phi}\mathrm{d}x.$ It is well-known that $L$ is self-
adjoint with respect to the weighted measure $\mathrm{d}\mu$.
The $\infty$-dimensional Bakry-Émery Ricci curvature $Ric(L)$ is defined by
$Ric(L):=Ric+Hess(\phi),$
where $Ric$ and $Hess$ denote the Ricci curvature of the metric $g$ and the
Hessian respectively. Following the notation used in [10], we also define the
$m$-dimensional Bakry-Émery Ricci curvature of $L$ on an $n$-dimensional
Riemaniann manifold as follows
$Ric_{m,n}(L):=Ric(L)-\frac{\nabla\phi\otimes\nabla\phi}{m-n},$
where $m:=\mathrm{dim}_{BE}(L)$ is called the Bakry-Émery dimension of $L$.
Note that the number $m$ is not necessarily to be an integer and $m\geq
n=\mathrm{dim}M$.
The main result of this paper can be stated in the following:
###### Theorem 1.2.
Let $(M,g)$ be an n-dimensional non-compact Riemannian manifold with
$Ric_{m,n}(L)\geq-K$ for some constant $K\geq 0$. Suppose that $u(x,t)$ is a
positive smooth solution to the diffusion equation (1.1) in $Q_{R,T}\equiv
B(x_{0},R)\times[t_{0}-T,t_{0}]\subset M\times(-\infty,\infty)$. Let $f:=\log
u$. We also assume that there exists non-negative constants $\alpha$ and
$\delta$ such that $\alpha-f\geq\delta>0$. Then there exist three dimensional
constants $\tilde{c}$, $c(\delta)$ and $c(\alpha,\delta,m)$ such that
(1.8) $\frac{|\nabla
u|}{u}\leq\left(\frac{\tilde{c}}{R}\beta{+}\frac{c(\alpha,\delta,m)}{R}{+}\frac{c(\delta)}{\sqrt{T}}{+}c(\delta)\left(|a|+K\right)^{1/2}\kern-3.0pt{+}c(\delta)|a|^{1/2}\beta^{1/2}\right)\left(\alpha{-}\frac{b}{a}{-}\log
u\right)$
in $Q_{R/2,T/2}$, where $\beta:=\max\left\\{1,|\alpha/\delta-1|\right\\}$.
We make some remarks on the above theorem below.
###### Remark 1.3.
(i). In Theorem 1.2, it seems that the assumption $\alpha-f\geq\delta>0$ is
reasonable. Because from this assumption, we can get $u\leq
e^{\alpha-\delta}$. We say that this upper bound of $u$ can be achieved in
some setting. For example, from Corollary 1.2 in [6], we know that positive
smooth solutions to the elliptic equation (1.4) with $a<0$ have $u(x)\leq
e^{n/2}$ for all $x\in M$ provided the Ricci curvature of $M$ is non-negative.
(ii). Note that the theorem still holds if $m$-dimensional Bakry-Émery Ricci
curvature is replaced by $\infty$-dimensional Bakry-Émery Ricci curvature. In
fact this result can be obtained by (2.10) in Section 2.
(iii). Theorem 1.2 generalizes the above mentioned Theorem C. When we choose
$\phi\equiv c_{0}$, we return Theorem C. The proof of our main theorem is
based on Souplet-Zhang’s gradient estimate and the trick used in [2] with some
modifications.
In particular, if $u(x,t)\leq 1$ is a positive smooth solution to the
diffusion equation (1.1) with $a<0$, then we have a simple estimate.
###### Corollary 1.4.
Let $(M,g)$ be an n-dimensional non-compact Riemannian manifold with
$Ric_{m,n}(L)\geq-K$ for some constant $K\geq 0$. Suppose that $u(x,t)\leq 1$
is a positive smooth solution to the diffusion equation (1.1) with $a<0$ in
$Q_{R,T}\equiv B(x_{0},R)\times[t_{0}-T,t_{0}]\subset
M\times(-\infty,\infty)$. Then there exist two dimensional constants $c$ and
$c(m)$ such that
(1.9) $\frac{|\nabla
u|}{u}\leq\left(\frac{c(m)}{R}+\frac{c}{\sqrt{T}}+c\sqrt{K+|a|}\right)\left(1-\frac{b}{a}+\log{\frac{1}{u}}\right)$
in $Q_{R/2,T/2}$.
###### Remark 1.5.
We point out that our localized Hamilton-type gradient estimate can be also
regarded as the generalization of the result of Souplet-Zhang [1] for the heat
equation on complete manifolds. In fact, the above Corollary 1.4 is similar to
the result of Souplet-Zhang (see Theorem 1.1 of [1]). From the inequality
(4.4) below, we can conclude that if $\phi\equiv c_{0}$ and $a=0$, then our
result can be reduced to theirs.
The method of proving Theorem 1.2 is the gradient estimate, which is
originated by Yau [11] (see also Cheng-Yau [12]), and developed further by Li-
Yau [13], Li [14] and Negrin [15]. Then R. S. Hamilton [16] gave an elliptic
type gradient estimate for the heat equation. But this type of estimate is a
global result which requires the heat equation defined on closed manifolds.
Recently, a localized Hamilton-type gradient estimate was proved by Souplet
and Zhang [1], which can be viewed as a combination of Li-Yau’s Harnack
inequality [13] and Hamilton’s gradient estimate [16]. In this paper, we
obtain a localized Hamilton-type gradient estimate for a general diffusion
equation (1.1) as Souplet and Zhang in [1] did for the heat equation on
complete manifolds. To prove Theorem 1.2, we mainly follow the arguments of
Souplet-Zhang in [1], together with some facts about Bakry-Émery Ricci
curvature. Note that the diffusion equation (1.1) is nonlinear. So our case is
a little more complicated than theirs.
The structure of this paper is as follows. In Section 2, we will give a basic
lemma to prepare for proving Theorem 1.2. Section 3 is devoted to the proof of
Theorem 1.2. In Section 4, we will prove Corollary 1.4 in the case $0<u\leq 1$
with $a<0$.
## 2\. A basic lemma
In this section, we will prove the following lemma which is essential in the
derivation of the gradient estimate of the equation (1.1). Replacing $u$ by
$e^{-b/a}u$, we only need to consider positive smooth solutions of the
following diffusion equation:
(2.1) $u_{t}=\Delta u-\nabla\phi\cdot\nabla u-au\log u.$
Suppose that $u(x,t)$ is a positive smooth solution to the diffusion equation
(1.1) in $Q_{R,T}\equiv B(x_{0},R)\times[t_{0}-T,t_{0}]$. Define a smooth
function
$f(x,t):=\log u(x,t)$
in $Q_{R,T}$. By (2.1), we have
(2.2) $\left(L-\frac{\partial}{\partial t}\right)f+|\nabla f|^{2}-af=0.$
Then we have the following lemma, which is a generalization of the computation
carried out in [1, 2].
###### Lemma 2.1.
Let $(M,g)$ be an n-dimensional non-compact Riemannian manifold with
$Ric_{m,n}(L)\geq-K$ for some constant $K\geq 0$. Let $f(x,t)$ is a smooth
function defined on $Q_{R,T}$ satisfying the diffusion equation (2.2). We also
assume that there exist non-negative constants $\alpha$ and $\delta$ such that
$\alpha-f\geq\delta>0$. Then for all $(x,t)$ in $Q_{R,T}$ the function
(2.3) $\omega:=\left|\nabla\log(\alpha-f)\right|^{2}=\frac{|\nabla
f|^{2}}{(\alpha-f)^{2}}$
satisfies the following inequality
(2.4) $\displaystyle\left(L-\frac{\partial}{\partial t}\right)\omega$
$\displaystyle\geq$
$\displaystyle\frac{2(1-\alpha)+2f}{\alpha-f}\left\langle\nabla
f,\nabla\omega\right\rangle+2(\alpha-f)\omega^{2}+2(a-K)\omega+\frac{2af}{\alpha-f}\omega.$
###### Proof.
By (2.3), we have
(2.5)
$\displaystyle\omega_{j}=\frac{2f_{i}f_{ij}}{(\alpha-f)^{2}}+\frac{2f^{2}_{i}f_{j}}{(\alpha-f)^{3}},$
(2.6)
$\displaystyle\Delta\omega=\frac{2|f_{ij}|^{2}}{(\alpha-f)^{2}}+\frac{2f_{i}f_{ijj}}{(\alpha-f)^{2}}+\frac{8f_{i}f_{j}f_{ij}}{(\alpha-f)^{3}}+\frac{2f^{2}_{i}f_{jj}}{(\alpha-f)^{3}}+\frac{6f^{2}_{i}f^{2}_{j}}{(\alpha-f)^{4}}$
and
(2.7) $\displaystyle L\omega$ $\displaystyle=\Delta\omega-\phi_{j}\omega_{j}$
$\displaystyle=\frac{2|f_{ij}|^{2}}{(\alpha{-}f)^{2}}+\frac{2f_{i}f_{ijj}}{(\alpha{-}f)^{2}}+\frac{8f_{i}f_{j}f_{ij}}{(\alpha{-}f)^{3}}+\frac{2f^{2}_{i}f_{jj}}{(\alpha{-}f)^{3}}+\frac{6f^{4}_{i}}{(\alpha{-}f)^{4}}-\frac{2f_{ij}f_{i}\phi_{j}}{(\alpha{-}f)^{2}}-\frac{2f^{2}_{i}f_{j}\phi_{j}}{(\alpha{-}f)^{3}}$
$\displaystyle=\frac{2|f_{ij}|^{2}}{(\alpha{-}f)^{2}}+\frac{2f_{i}(Lf)_{i}}{(\alpha{-}f)^{2}}+\frac{2(R_{ij}+\phi_{ij})f_{i}f_{j}}{(\alpha{-}f)^{2}}+\frac{8f_{i}f_{j}f_{ij}}{(\alpha{-}f)^{3}}+\frac{2f^{2}_{i}\cdot
Lf}{(\alpha{-}f)^{3}}+\frac{6f^{4}_{i}}{(\alpha{-}f)^{4}},$
where $f_{i}:=\nabla_{i}f$ and $f_{ijj}:=\nabla_{j}\nabla_{j}\nabla_{i}f$,
etc. By (2.3) and (2.2), we also have
(2.8) $\displaystyle\omega_{t}$
$\displaystyle=\frac{2\nabla_{i}f\cdot\nabla_{i}\Big{[}Lf+|\nabla
f|^{2}-af\Big{]}}{(\alpha-f)^{2}}+\frac{2|\nabla f|^{2}\Big{[}Lf+|\nabla
f|^{2}-af\Big{]}}{(\alpha-f)^{3}}$ $\displaystyle=\frac{2\nabla f\nabla
Lf}{(\alpha-f)^{2}}+\frac{4f_{i}f_{j}f_{ij}}{(\alpha-f)^{2}}-\frac{2a|\nabla
f|^{2}}{(\alpha-f)^{2}}+\frac{2f^{2}_{i}Lf}{(\alpha-f)^{3}}+\frac{2|\nabla
f|^{4}}{(\alpha-f)^{3}}-\frac{2af|\nabla f|^{2}}{(\alpha-f)^{3}}.$
Combining (2.7) with (2.8), we can get
(2.9) $\displaystyle\left(L-\frac{\partial}{\partial t}\right)\omega$
$\displaystyle=\frac{2|f_{ij}|^{2}}{(\alpha-f)^{2}}+\frac{2(R_{ij}+\phi_{ij})f_{i}f_{j}}{(\alpha-f)^{2}}+\frac{8f_{i}f_{j}f_{ij}}{(\alpha-f)^{3}}+\frac{6f^{4}_{i}}{(\alpha-f)^{4}}$
$\displaystyle\,\,\,\,\,\,-\frac{4f_{i}f_{j}f_{ij}}{(\alpha-f)^{2}}-\frac{2f^{4}_{i}}{(\alpha-f)^{3}}+\frac{2af^{2}_{i}}{(\alpha-f)^{2}}+\frac{2aff^{2}_{i}}{(\alpha-f)^{3}}.$
Noting that $Ric_{m,n}(L)\geq-K$ for some constant $K\geq 0$, we have
(2.10) $(R_{ij}+\phi_{ij})f_{i}f_{j}\geq\frac{|\nabla\phi\cdot\nabla
f|^{2}}{m-n}-K|\nabla f|^{2}\geq-K|\nabla f|^{2}.$
By (2.5), we have
(2.11)
$\displaystyle\omega_{j}f_{j}=\frac{2f_{i}f_{j}f_{ij}}{(\alpha-f)^{2}}+\frac{2f^{2}_{i}f^{2}_{j}}{(\alpha-f)^{3}},$
and consequently,
(2.12) $\displaystyle
0=-2\omega_{j}f_{j}+\frac{4f_{i}f_{j}f_{ij}}{(\alpha-f)^{2}}+\frac{4f^{4}_{i}}{(\alpha-f)^{3}},$
(2.13) $\displaystyle
0=\frac{1}{\alpha-f}\left[2\omega_{j}f_{j}-\frac{4f^{4}_{i}}{(\alpha-f)^{3}}\right]-\frac{4f_{i}f_{j}f_{ij}}{(\alpha-f)^{3}}.$
Substituting (2.10) into (2.9) and then adding (2.9) with (2.12) and (2.13),
we can get
(2.14) $\displaystyle\left(L-\frac{\partial}{\partial t}\right)\omega$
$\displaystyle\geq\frac{2|f_{ij}|^{2}}{(\alpha-f)^{2}}-\frac{2K|\nabla
f|^{2}}{(\alpha-f)^{2}}+\frac{4f_{i}f_{j}f_{ij}}{(\alpha-f)^{3}}+\frac{2f^{4}_{i}}{(\alpha-f)^{4}}+\frac{2f^{4}_{i}}{(\alpha-f)^{3}}$
$\displaystyle\,\,\,\,\,\,+\frac{2(1-\alpha)+2f}{\alpha-f}f_{i}\omega_{i}+\frac{2af^{2}_{i}}{(\alpha-f)^{2}}+\frac{2aff^{2}_{i}}{(\alpha-f)^{3}}.$
Note that $\alpha-f\geq\delta>0$ implies
$\displaystyle\frac{2|f_{ij}|^{2}}{(\alpha-f)^{2}}+\frac{4f_{i}f_{j}f_{ij}}{(\alpha-f)^{3}}+\frac{2f^{4}_{i}}{(\alpha-f)^{4}}\geq
0.$
This, together with (2.14), yields the desired estimate (LABEL:lemmaequ3).
∎
## 3\. Proof of Theorem 1.2
In this section, we will use Lemma 2.1 and the localization technique of
Souplet-Zhang [1] to give the elliptic type gradient estimates on the positive
and bounded smooth solutions of the diffusion equation (1.1).
###### Proof.
First we give the well-known cut-off function by Li-Yau [13] (see also [1]) as
follows. We caution the reader that the calculation is not the same as that in
[13] due to the difference of the first-order term.
Let $\psi=\psi(x,t)$ be a smooth cut-off function supported in $Q_{R,T}$
satisfying the following properties:
1. (1)
$\psi=\psi(d(x,x_{0}),t)\equiv\psi(r,t)$; $\psi(x,t)=1$ in $Q_{R/2,T/2}$,
$0\leq\psi\leq 1$;
1. (2)
$\psi$ is decreasing as a radial function in the spatial variables;
1. (3)
$\frac{|\partial_{r}\psi|}{\psi^{\epsilon}}\leq\frac{C_{\epsilon}}{R}$,
$\frac{|\partial^{2}_{r}\psi|}{\psi^{\epsilon}}\leq\frac{C_{\epsilon}}{R^{2}}$,
when $0<\epsilon<1$;
1. (4)
$\frac{|\partial_{t}\psi|}{\psi^{1/2}}\leq\frac{C}{T}$.
From Lemma 2.1, by a straight forward calculation, we have
(3.1) $\displaystyle L(\psi\omega)-\frac{2(1-\alpha)+2f}{\alpha-f}\nabla
f\cdot\nabla(\psi\omega)-2\frac{\nabla\psi}{\psi}\cdot\nabla(\psi\omega)-(\psi\omega)_{t}$
$\displaystyle\geq$ $\displaystyle
2\psi(\alpha-f)\omega^{2}-\left[\frac{2(1-\alpha)+2f}{\alpha-f}\nabla
f\cdot\nabla\psi\right]\omega-2\frac{|\nabla\psi|^{2}}{\psi}\omega$
$\displaystyle+(L\psi)\omega-\psi_{t}\omega+2(a-K)\psi\omega+2\frac{af}{\alpha-f}\psi\omega.$
Let $(x_{1},t_{1})$ be a point where $\psi\omega$ achieves the maximum. By Li-
Yau [13], without loss of generality we assume that $x_{1}$ is not in the cut-
locus of $M$. Then at this point, we have
$\displaystyle L(\psi\omega)\leq 0,\,\,\,\,\,\,(\psi\omega)_{t}\geq
0,\,\,\,\,\,\,\nabla(\psi\omega)=0.$
Hence at $(x_{1},t_{1})$, by (LABEL:lemdx3), we get
(3.2) $\displaystyle 2\psi(\alpha-f)\omega^{2}(x_{1},t_{1})$
$\displaystyle\leq\Bigg{\\{}\left[\frac{2(1-\alpha)+2f}{\alpha-f}\nabla
f\cdot\nabla\psi\right]\omega+2\frac{|\nabla\psi|^{2}}{\psi}\omega-(L\psi)\omega$
$\displaystyle\,\,\,\,\,\,\,\,\,+\psi_{t}\omega-2(a-K)\psi\omega-2\frac{af}{\alpha-f}\psi\omega\Bigg{\\}}(x_{1},t_{1}).$
In the following, we will introduce the upper bounds for each term of the
right-hand side (RHS) of (3.2). Following similar arguments of Souplet-Zhang
([1], pp. 1050-1051), we have the estimates of the first term of the BHS of
(3.2)
(3.3) $\displaystyle\left[\frac{2f}{\alpha-f}\nabla
f\cdot\nabla\psi\right]\omega$ $\displaystyle\leq$ $\displaystyle
2|f|\cdot|\nabla\psi|\cdot\omega^{3/2}=2\left[\psi(\alpha-f)\omega^{2}\right]^{3/4}\cdot\frac{|f|\cdot|\nabla\psi|}{[\psi(\alpha-f)]^{3/4}}$
$\displaystyle\leq$
$\displaystyle\psi(\alpha-f)\omega^{2}+\tilde{c}\frac{(f|\nabla\psi|)^{4}}{[\psi(\alpha-f)]^{3}}\leq\psi(\alpha-f)\omega^{2}+\tilde{c}\frac{f^{4}}{R^{4}(\alpha-f)^{3}}$
and
(3.4) $\displaystyle\left[\frac{2(1-\alpha)}{\alpha-f}\nabla
f\cdot\nabla\psi\right]\omega$ $\displaystyle\leq$ $\displaystyle
2|1-\alpha||\nabla\psi|\omega^{3/2}=(\psi\omega^{2})^{3/4}\cdot\frac{2|1-\alpha||\nabla\psi|}{\psi^{3/4}}$
$\displaystyle\leq$
$\displaystyle\frac{\delta}{12}\psi\omega^{2}+c(\alpha,\delta)\left(\frac{|\nabla\psi|}{\psi^{3/4}}\right)^{4}\leq\frac{\delta}{12}\psi\omega^{2}+\frac{c(\alpha,\delta)}{R^{4}}.$
For the second term of the RHS of (3.2), we have
(3.5) $\displaystyle 2\frac{|\nabla\psi|^{2}}{\psi}\omega$
$\displaystyle=2\psi^{1/2}\omega\cdot\frac{|\nabla\psi|^{2}}{\psi^{3/2}}\leq\frac{\delta}{12}\psi\omega^{2}+c(\delta)\left(\frac{|\nabla\psi|^{2}}{\psi^{3/2}}\right)^{2}$
$\displaystyle\leq\frac{\delta}{12}\psi\omega^{2}+\frac{c(\delta)}{R^{4}}.$
For the third term of the RHS of (3.2), since $Ric_{m,n}(L)\geq-K$, by the
generalized Laplacian comparison theorem (see [9] or [10]),
$Lr\leq(m-1)\sqrt{K}\coth(\sqrt{K}r).$
Consequently, we have
(3.6) $\displaystyle-(L\psi)\omega$
$\displaystyle=-\left[(\partial_{r}\psi)Lr+(\partial^{2}_{r}\psi)\cdot|\nabla
r|^{2}\right]\omega$
$\displaystyle\leq-\left[\partial_{r}\psi(m-1)\sqrt{K}\coth(\sqrt{K}r)+\partial^{2}_{r}\psi\right]\omega$
$\displaystyle\leq-\left[\partial_{r}\psi(m-1)\left(\frac{1}{r}+\sqrt{K}\right)+\partial^{2}_{r}\psi\right]\omega$
$\displaystyle\leq\left[|\partial^{2}_{r}\psi|+2(m-1)\frac{|\partial_{r}\psi|}{R}+(m-1)\sqrt{K}|\partial_{r}\psi|\right]\omega$
$\displaystyle\leq\psi^{1/2}\omega\frac{|\partial^{2}_{r}\psi|}{\psi^{1/2}}+\psi^{1/2}\omega
2(m-1)\frac{|\partial_{r}\psi|}{R\psi^{1/2}}+\psi^{1/2}\omega(m-1)\frac{\sqrt{K}|\partial_{r}\psi|}{\psi^{1/2}}$
$\displaystyle\leq\frac{\delta}{12}\psi\omega^{2}+c(\delta,m)\left[\left(\frac{|\partial^{2}_{r}\psi|}{\psi^{1/2}}\right)^{2}+\left(\frac{|\partial_{r}\psi|}{R\psi^{1/2}}\right)^{2}+\left(\frac{\sqrt{K}|\partial_{r}\psi|}{\psi^{1/2}}\right)^{2}\right]$
$\displaystyle\leq\frac{\delta}{12}\psi\omega^{2}+\frac{c(\delta,m)}{R^{4}}+\frac{c(\delta,m)K}{R^{2}}.$
Now we estimate the fourth term:
(3.7) $\displaystyle|\psi_{t}|\omega$
$\displaystyle=\psi^{1/2}\omega\frac{|\psi_{t}|}{\psi^{1/2}}\leq\frac{\delta}{12}\left(\psi^{1/2}\omega\right)^{2}+c(\delta)\left(\frac{|\psi_{t}|}{\psi^{1/2}}\right)^{2}$
$\displaystyle\leq\frac{\delta}{12}\psi\omega^{2}+\frac{c(\delta)}{T^{2}}.$
Notice that we have used Young’s inequality below in obtaining
(LABEL:term1)-(3.7):
$ab\leq\frac{a^{p}}{p}+\frac{b^{q}}{q},\,\,\,\,\,\,\forall\,\,\,p,q>0\,\,\,\mathrm{with}\,\,\,\frac{1}{p}+\frac{1}{q}=1.$
Finally, we estimate the last two terms:
(3.8) $\displaystyle-2(a-K)\psi\omega\leq
2(|a|+K)\psi\omega\leq\frac{\delta}{12}\psi\omega^{2}+c(\delta)(|a|+K)^{2};$
and
(3.9) $-2\frac{af}{\alpha-f}\psi\omega\leq
2\frac{|a|\cdot|f|}{\alpha-f}\psi\omega\leq\frac{\delta}{12}\psi\omega^{2}+c(\delta)a^{2}\frac{f^{2}}{(\alpha-f)^{2}}.$
Substituting (LABEL:term1)-(3.9) to the RHS of (3.2) at $(x_{1},t_{1})$, we
get
(3.10) $\displaystyle 2\psi(\alpha-f)\omega^{2}$
$\displaystyle\leq\psi(\alpha-f)\omega^{2}+\frac{\tilde{c}f^{4}}{R^{4}(\alpha-f)^{3}}+\frac{\delta}{2}\psi\omega^{2}+\frac{c(\alpha,\delta)}{R^{4}}+\frac{c(\delta)}{R^{4}}+\frac{c(\delta,m)}{R^{4}}$
$\displaystyle\,\,\,\,\,\,+\frac{c(\delta,m)K}{R^{2}}+\frac{c(\delta)}{T^{2}}+c(\delta)(|a|+K)^{2}+c(\delta)a^{2}\frac{f^{2}}{(\alpha-f)^{2}}.$
Recall that $\alpha-f\geq\delta>0$, (3.10) implies
(3.11) $\displaystyle\psi\omega^{2}(x_{1},t_{1})$
$\displaystyle\leq\tilde{c}\frac{f^{4}}{R^{4}(\alpha-f)^{4}}+\frac{1}{2}\psi\omega^{2}(x_{1},t_{1})+\frac{c(\alpha,\delta)}{R^{4}}+\frac{c(\delta,m)}{R^{4}}+\frac{c(\delta,m)K}{R^{2}}$
$\displaystyle\,\,\,\,\,\,+\frac{c(\delta)}{T^{2}}+c(\delta)(|a|+K)^{2}+c(\delta)a^{2}\frac{f^{2}}{(\alpha-f)^{2}}.$
Furthermore, we need to estimate the RHS of (3.11). If $f\leq 0$ and
$\alpha\geq 0$, then we have
(3.12) $\displaystyle\frac{f^{4}}{(\alpha-f)^{4}}\leq
1,\,\,\,\,\,\,\,\,\,\,\,\,\frac{f^{2}}{(\alpha-f)^{2}}\leq 1;$
if $f>0$, by the assumption $\alpha-f\geq\delta>0$, we know that
(3.13)
$\displaystyle\frac{f^{4}}{(\alpha-f)^{4}}\leq\frac{(\alpha-\delta)^{4}}{\delta^{4}}=\left(\frac{\alpha}{\delta}-1\right)^{4},\,\,\,\,\,\,\,\,\,\,\,\,\frac{f^{2}}{(\alpha-f)^{2}}\leq\left(\frac{\alpha}{\delta}-1\right)^{2}.$
Plugging (3.12) (or (3.13)) into (3.11), we obtain
(3.14)
$\displaystyle(\psi\omega^{2})(x_{1},t_{1})\leq\frac{\tilde{c}\beta^{4}+c(\alpha,\delta,m)}{R^{4}}+\frac{c(\delta,m)K}{R^{2}}+\frac{c(\delta)}{T^{2}}+c(\delta)(|a|+K)^{2}+c(\delta)a^{2}\beta^{2},$
where $\beta:=\max\left\\{1,|\alpha/\delta-1|\right\\}$.
The above inequality implies, for all $(x,t)$ in $Q_{R,T}$
(3.15) $\displaystyle(\psi^{2}\omega^{2})(x,t)$
$\displaystyle\leq\psi^{2}(x_{1},t_{1})\omega^{2}(x_{1},t_{1})\leq\psi(x_{1},t_{1})\omega^{2}(x_{1},t_{1})$
$\displaystyle\leq\frac{\tilde{c}\beta^{4}+c(\alpha,\delta,m)}{R^{4}}+\frac{c(\delta,m)K}{R^{2}}+\frac{c(\delta)}{T^{2}}+c(\delta)(|a|+K)^{2}+c(\delta)a^{2}\beta^{2}.$
Note that $\psi(x,t)=1$ in $Q_{R/2,T/2}$ and $\omega={|\nabla
f|^{2}}/{(\alpha-f)^{2}}$. Therefore we have
(3.16) $\displaystyle\frac{|\nabla
f|}{\alpha-f}\leq\left(\frac{\tilde{c}\beta^{4}+c(\alpha,\delta,m)}{R^{4}}+\frac{c(\delta,m)K}{R^{2}}+\frac{c(\delta)}{T^{2}}+c(\delta)(|a|+K)^{2}+c(\delta)a^{2}\beta^{2}\right)^{1/4}.$
Since $f=\log u$, we get the following estimate for Eq. (2.1)
(3.17) $\displaystyle\frac{|\nabla
u|}{u}\leq\left(\frac{\tilde{c}\beta^{4}+c(\alpha,\delta,m)}{R^{4}}+\frac{c(\delta)}{T^{2}}+c(\delta)(|a|+K)^{2}+c(\delta)a^{2}\beta^{2}\right)^{1/4}\Big{(}\alpha-\log
u\Big{)}.$
Replacing $u$ by $e^{b/a}u$ gives the desired estimate (1.8). This completes
the proof of Theorem 1.2. ∎
## 4\. Proof of Corollary 1.4
###### Proof.
The proof is similar to that of Theorem 1.2. We still use the technique of a
cut-off function in a local neighborhood of Riemannian manifolds. For $0<u\leq
1$, we let $f=\log u$. Then $f\leq 0$. Set
$\omega:=|\nabla\log(1-f)|^{2}=\frac{|\nabla f|^{2}}{(1-f)^{2}}.$
By Lemma 2.1, we have
(4.1) $\displaystyle\left(L-\frac{\partial}{\partial
t}\right)\omega\geq\frac{2f}{1-f}\left\langle\nabla
f,\nabla\omega\right\rangle+2(1-f)\omega^{2}-2(|a|+K)\omega.$
We define a smooth cut-off function $\psi=\psi(x,t)$ in the same way as
Section 3. Follow all steps in the last section (see also pp. 1050-1051 in
[1]), we can easily get the following inequality
(4.2) $\displaystyle 2(1-f)\psi\omega^{2}$
$\displaystyle\leq(1-f)\psi\omega^{2}+\frac{cf^{4}}{R^{4}(1-f)^{3}}+\frac{\psi\omega^{2}}{2}+\frac{c}{R^{4}}$
$\displaystyle\,\,\,\,\,\,+\frac{c(m)}{R^{4}}+\frac{c(m)K}{R^{2}}+\frac{c}{T^{2}}+c(|a|+K)^{2},$
where we used similar estimates (LABEL:term1)-(3.9) with the difference that
these estimates do not contain the parameter $\delta$. Using the same method
as that in proving Theorem 1.2, for all $(x,t)$ in $Q_{R/2,T/2}$ we can get
(4.3) $\displaystyle\omega^{2}(x,t)$
$\displaystyle\leq\frac{c(m)}{R^{4}}+\frac{c(m)K}{R^{2}}+\frac{c}{T^{2}}+c(|a|+K)^{2}$
$\displaystyle\leq\frac{c(m)}{R^{4}}+\frac{c(m)}{R^{2}}(|a|+K)+\frac{c}{T^{2}}+c(|a|+K)^{2}$
$\displaystyle\leq\frac{c(m)}{R^{4}}+\frac{c}{T^{2}}+c(|a|+K)^{2}.$
Again, using the same argument in the proof of Theorem 1.2 gives
(4.4) $\frac{|\nabla
f|}{1-f}\leq\frac{c(m)}{R}+\frac{c}{\sqrt{T}}+c\sqrt{K+|a|},$
where $c$ is a constant depending only on $n$, $c(m)$ is a constant depending
only on $n$ and $m$.
Since $f=\log u$, we get
(4.5) $\frac{|\nabla
u|}{u}\leq\left(\frac{c(m)}{R}+\frac{c}{\sqrt{T}}+c\sqrt{K+|a|}\right)\cdot\left(1+\log{\frac{1}{u}}\right).$
At last, replacing $u$ by $e^{b/a}u$ above yields (1.9). ∎
## Acknowledgment
The author would like to thank Professor Yu Zheng for his helpful suggestions
on this problem, and for his encouragement. He would also like to thank the
referees for useful suggestions. This work is partially supported by the
NSFC10871069.
## References
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* [2] Wu J Y. Elliptic type gradient estimates for a nonlinear parabolic equation on complete manifolds. 2008, preprint.
* [3] Hamilton R S. Three-manifolds with positive Ricci curvature. _J Diff Geom_ , 1982, 17: 255-306.
* [4] Chow B, Lu P, Ni L. Hamilton’s Ricci flow, Lectures in Contemporary Mathematics 3, Science Press and American Mathematical Society, 2006.
* [5] Ma L. Gradient estimates for a simple elliptic equation on non-compact Riemannian manifolds. _J Funct Anal_ , 2006, 241: 374-382.
* [6] Yang Y Y. Gradient estimates for a nonlinear parabolic equation on Riemannian manifold. _Proceeding of AMS_ , 2008, 136: 4095-4102.
* [7] Bakry D. L hypercontractivité et son utilisation en théorie des semigroupes, 1-114, Lect. Notes in Math., vol. 1581, Springer-Verlag, Berlin/New York, 1994.
* [8] Bakry D, Emery M. Diffusion hypercontractivitives, in Séminaire de Probabilités 19, 1983/1984, 177-206, Lect Notes in Math 1123, Berlin: Springer, 1985.
* [9] Bakry D, Qian Z M. Volume comparison theorems without Jacobi fields. Current trends in potential theory, 115-122, Theta Ser Adv Math, 4, Theta, Bucharest, 2005.
* [10] Li X D. Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds. _J Math Pure Appl_ , 2005, 84: 1295-1361.
* [11] Yau S T. Harmonic functions on complete Riemannian manifolds. _Comm Pure Appl Math_ , 1975, 28: 201-228.
* [12] Cheng S Y, Yau S T. Differential equations on Riemannian manifolds and their geometric applications. _Comm Pure Appl Math_ , 1975, 28: 333-354.
* [13] Li P, Yau S T. On the parabolic kernel of the Schrodinger operator. _Acta Math_ , 1986, 156: 153-201.
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|
arxiv-papers
| 2010-03-13T03:34:45 |
2024-09-04T02:49:09.074218
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jia-Yong Wu",
"submitter": "Jia-Yong Wu",
"url": "https://arxiv.org/abs/1003.2670"
}
|
1003.2683
|
# An investigation of a nonlocal entanglement of two uncoupled atoms embedded
in a coherent cavity field and the associated phase space distribution: one
quantum non-linear process
M. S. Ateto111E-mail: omersog@yahoo.com, mohammed.ali11@sci.svu.edu.eg
Mathematics Department, Faculty of Science at Qena, South Valley University,
83523 Qena, Egypt
###### Abstract
Entanglement properties of two uncoupled atoms embedded in a coherent field
distribution through one quantum transition process is studied. A case of non-
linear Hamiltonian of the problem is considered through which the effect of a
non-linear media is illustrated. Moreover, the effect of the frequency
difference between the interatomic transition and the electromagnetic field is
also analyzed. We show that, adjusting the considered parametres of the non-
linear media and frequency difference leads to a strong control of the degree
of entanglement where excellent periodicity of entanglement evolution can be
obtained which is very important in predicting the behavior of transmitted
information through the application of various information processing schemes.
We present a detailed and comparative study of atom-atom entanglement for two
cases corresponding to different injections of the two atoms into the cavity
field. Moreover, we present an answer to the question: How does the quantum
phase space structure for a composite system relate to the entanglement
characteristics of the corresponding quantum system? We demonstrate how the
entanglement in nonlinear tripartite systems can be associated with a
delocalization in the phase space distribution.
###### pacs:
03.65.Ud, 03.67.-a, 03.67.Bg, 05.30.-d 03.67.Mn
Keywords: Coherent distribution, Nonlocal entanglement, Two qubits, one
quantum process, Concurrence, Kerr medium, Husimi $Q$-distribution
## I Introduction
The recent rapid development of quantum information theory has largely
stimulated research on nonclassical phenomena, with the main focus on the
generation of entangled states that are required for tasks such as quantum
teleportation, dense coding, or certain types of quantum key distribution
protocols. In recent years, there are much interest to study entanglement of
two-level systems. These systems gain such importance because of the
efficiency in representing information in most quantum information processing
schemes. Particular and great interest is devoted to the generation of
entangled states in two-atom systems, since they can represent two qubits that
considered the base blocks of building the quantum gates that are essential to
implement various quantum protocols. Furthermore, there have been many
proposals for generating atomic entanglement and entanglement between cavity
modes through atom-photon interaction PWK04 ; BBPS96 ; BDSW96 ; VP96 ; V02 ;
VT99 ; VPRK95 ; W98 ; GMN06 ; CZDKAW00 ; BGASMNN07 ; PK491 ; PK691 ; KLMCK93 ;
ATETO109 ; Ateto07 . Some notable experimental demonstrations have also been
performed HMNWBRH979 .
However, the physical nature of the interacting objects and the character of
their mutual coupling control strongly the degree of quantum entanglement. A
number of studies have shown that entanglement can be created between to
objects which do not interact directly with each other but interact with
either a common field or heat bath or thermal cavity field BGASMNN07 ;
ATETO109 ; Ateto07 ; KLAK02 ; BRAUN02 ; ATETO209 .
The formation of atom-photon entanglement and the subsequent generation of
correlations between spatially separated atoms have been shown using the
micromaser MLBEHW96 ; ASMNN01 ; RSWRWR991 ; M02 ; HMNWBRH979 . The
effectiveness of micromaser FIJAME86 ; PK88 ; PB93 in generating entanglement
has been appreciated as it is strongly considered as a practical device for
processing information. It stores radiation for times significantly longer
than the duration of the interaction with any single atom PB93 . The
interaction of an atom with the intracavity field of a micromaser will,
generally, leave the atom-field system in an entangled state. The long cavity
lifetime means that the memory of this entanglement can influence the
interaction with subsequent atoms and a nonlocal correlations between these
successive atoms can be induced leading to a violation of Bell’s inequality
PB93 .
To our knowledge, the study of such a problem when the cavity field contains a
coherent distribution is still insufficient. The major difficulty may be due
to the complex calculations and long analytical mathematics, specially when
the system Hamiltonian becomes nonlinear. We are interested in the opportunity
to use the micromaser with coherent distribution as a source of nonlocal
entanglement and answer the important questions: how and to what extent can
the degree of atom-atom entanglement be cotrolled in these circumstances?
Also, can the Husimi $Q$-distribution be used as a helpful method to quantify
nonlocal entanglement dynamics? When applied to a system of two qubits, it’s
shown that the concurrence W98 may be expressed in terms of the second moment
of the Husimi $Q$-function. More strikingly, when applied to a system of three
qubits, the expression for the moment contains all three bipartite concurrence
terms. In this way this measure captures all classes of entanglement, not only
bipartite or otherwise SUGITA03 .
Our purpose here is to trace analytically and numerically the problem of a
nonlocal entanglement created between two successive atoms (two-qubits)
through surrounding the coherent cavity field with a nonlinear medium, namely,
Kerr-like medium. The Kerr medium FALI95 ; ATETO109 ; ATETO209 ; ATETO10 ;
Ateto07 ; WLG06 can be modelled as an anharmonic oscillator with frequency
$\omega$. Physically this model may be realized as if the cavity contains two
different species of atoms, one of which behaves like a two-level atom and the
other behaves like an anharmonic oscillator in the single-mode field of
frequency $\omega$ JOPU92 . Such a model is interesting by itself. The cavity
mode is coupled to the Kerr medium as well as to the two-level atoms. A Kerr-
like medium can be useful in many respects, such as detection of nonclassical
states Hillaer91 , quantum nondemolition measurement CHCOWO92 , investigation
of quantum fluctuations ZHGCLM00 , generation of entangled macroscopic quantum
statesGerry99 ; PKH03 , and quantum information processing PACH00 ; VFT00 .
An outline of paper is as follows: In Sec. II we describe in details the
physical mechanism of the interaction and introduce the modified model of the
total system that accommodates processes such as nonlocal interaction in
nonlinear media. We obtain the wave function of the first part of interaction
on which the final wave function of the total system depends. In Sec. III, we
describe the technique we are going to use in quantifying the degree of
entanglement. In addition, we calculate the the eigenvalues of the non-
Hermitian operator that used to compute the mathematical form of entanglement
measure. Different kinds of atoms injection into the cavity are introduced in
sections IV and V, where the corresponding wave functions and their associated
density matrix operators are computed. In Set. VI, we give a little historical
review about the Husimi $Q$-function, where that function is calculated for
different circumstances separately with detailed discusion included. Our
concluding remarks are shown in Set. VII.
## II The full system and its solution
We consider two two-level atoms with the same transition frequency traverse a
high $Q$ ($Q\approx 10^{9}$) single mode cavity one after the other. The
flight time of the atoms is, however, long enough so that there is no
appreciable overlap between the atomic wave functions BGASMNN07 ; DGMN04 ;
LEW96 ; AMNN01 ; PB93 ; Ateto07 ; EGSWH96 ; MHNWBTH97 ; ELSW002 . With this
assumption we ignore the possibility of energy exchange between the atoms,
although, secondary correlations develop between them. The entanglement of
their wave functions with the cavity photons can be used to formulate local-
realist bounds on the detection probabilities for the two atoms LEW96 ; PB93 .
The generation of nonlocal correlations between the two atomic states emerging
from the cavity can in general be understood using the Horodecki theorem HHH95
. It is assumed that the atom field interaction time is shorter than the
lifetime of the cavity, so that the cavity relaxation will not be considered.
The cavity field is assumed to be filled with a nonlinear medium, namely, Kerr
medium FALI95 ; ATETO109 ; ATETO10 ; ATETO209 ; Ateto07 ; WLG06 . Also, it is
assumed that the eigenfrequency of the atomic subsystem differs from that of
cavity field subsystem. Under the rotating wave approximation (TWA), the
Hamiltonian in the interaction picture (assuming that $\hbar=1$)
$\widehat{H}_{int}=\frac{\Delta}{2}~{}\widehat{\sigma}_{z}+f(\chi,\widehat{n})+\lambda~{}\bigl{(}\widehat{\sigma}_{-}~{}\widehat{a}^{{\dagger}}+\widehat{\sigma}_{+}~{}\widehat{a}\bigr{)},$
(1)
where $f(\chi,\widehat{n})=\chi(\widehat{n}^{2}-\widehat{n})$ represents the
nonlinear term with,
$f(\chi,\widehat{n})\mid n\rangle=\bigl{[}\chi(n^{2}-n)\bigr{]}\mid
n\rangle=f(\chi,n)\mid n\rangle,$ (2)
where $\widehat{a}$ ($\widehat{a}^{{\dagger}}$) is bosonic annihilation
(creation) operator for the single mode field of frequency $\omega$ and
$\lambda$ is atoms-field coupling constant, the operators
$\widehat{\sigma}_{+}=|+\rangle\langle-|$ ,
$\widehat{\sigma}_{-}=|-\rangle\langle+|$ and
$\widehat{\sigma}_{z}=|+\rangle\langle+|-|-\rangle\langle-|$ represent,
respectively, the raising, lowering and population atomic operators which obey
the commutation relations
$[\widehat{\sigma}_{+},\widehat{\sigma}_{-}]=\widehat{\sigma}_{z}$, while
$\Delta=\omega_{0}-\omega$ is the detuning parameter which represents the
difference between the atomic and cavity subsystems eigenfrequencies. We
denote by $\chi$ the dispersive part of the third order susceptibility of the
Kerr-like medium FALI95 ; ATETO109 ; ATETO10 ; ATETO209 ; Ateto07 ; WLG06 .
Since the state vector, $|\psi_{F}(0)\rangle$, of the field is represented by
a linear superposition of the number state $|n\rangle$
$|\psi_{F}(0)\rangle=\sum_{n=0}^{\infty}C^{n}|n\rangle,$ (3)
and the first atom that traverse the cavity is assumed in its upper state
$\mid+\rangle$, i. e.,
$|\psi_{A}(0)\rangle=\mid+\rangle$ (4)
the initial state vector of the interacting first-atom-field system is given
by
$|\psi_{AF}(0)\rangle=|\psi_{A}(0)\rangle\otimes|\psi_{F}(0)\rangle=\sum_{n=0}^{\infty}C^{n}|n,+\rangle,$
(5)
where $|n\rangle$ is an eigenstate of the number operator
$\widehat{a}^{\dagger}\widehat{a}=\widehat{n}$;
$\widehat{a}^{\dagger}\widehat{a}|n\rangle=n|n\rangle$, and $C^{n}$ is, in
general, complex where the square of its modulus gives the probability of the
coherent field, with mean photon $|\alpha|^{2}=\bar{n}$, to have $n$ photons
by the relation
$P(n)=|\langle
n|\psi_{F}(0)\rangle|^{2}=|C^{n}|^{2}=e^{-\bar{n}}\frac{\bar{n}^{n}}{n!}.$ (6)
Under the assumption that, first atom traverse the cavity in its upper state
$\mid+\rangle$, the joint state vector of the field and the first atom at any
instant of time $t$ can be obtained from the solution of the time-dependent
Schrödinger equation
$i\frac{d}{dt}|\psi_{AF}(t)\rangle=\widehat{H}~{}|\psi_{AF}(t)\rangle.$ (7)
The time-dependent wave function of the first-atom-field system takes the form
$|\psi_{AF}(t)\rangle=\mid U_{+}^{n}(t)\rangle+\mid U_{-}^{n+1}(t)\rangle$ (8)
which can be simply obtained by recalling the initial condition (5) and
solving the Schrödinger equation (7), where
$\mid
U_{+}^{n}(t)\rangle=\sum_{n}^{\infty}~{}C^{n}~{}\Gamma_{1}(n,t)~{}|n,+\rangle,$
(9) $\mid
U_{-}^{n+1}(t)\rangle=\sum_{n}^{\infty}~{}C^{n}~{}\Gamma_{2}(n+1,t)~{}|n+1,-\rangle,$
(10)
with the amplitudes $\Gamma_{1}(n,t)$ and $\Gamma_{2}(n+1,t)$ are,
respectively, given by
$\Gamma_{1}(n,t)=\sum_{j=1}^{2}F_{j}(n)e^{i\mu_{j}(n)t},$ (11)
and
$\Gamma_{2}(n+1,t)=-\sum_{j=1}^{2}F_{j}\frac{\mu_{j}(n)+\alpha(n)}{\lambda\sqrt{n+1}}e^{i\mu_{j}(n)t},$
(12)
with $F_{j}(n)$ is given by
$F_{j}=\frac{\mu_{k}(n)+\alpha(n)}{\mu_{kj}(n)},~{}~{}~{}~{}k\neq j=1,2,$ (13)
where the angles $\mu_{1,2}(n)$ are given by
$\mu_{1,2}(n)=-\frac{\alpha(n)+\gamma(n)}{2}\pm\sqrt{\biggl{(}\frac{\alpha(n)-\gamma(n)}{2}\biggr{)}^{2}+\lambda^{2}(n+1)}.$
(14)
with $\alpha(n)$ and $\gamma(n)$ read, respectively
$\alpha(n)=\frac{\Delta}{2}+\chi n(n-1),$ (15)
$\gamma(n)=-\frac{\Delta}{2}+\chi n(n+1).$ (16)
As already indicated above, we consider a pair of two-level atoms going
through the cavity mode one after another. Moreover, under the assumption that
time of flight through the cavity $t$ is the same for every atom BGASMNN07 ;
DGMN04 ; LEW96 ; AMNN01 ; PB93 ; Ateto07 ; EGSWH96 ; MHNWBTH97 ; ELSW002 ,
where the joint state vector of both atoms and the field may be denoted by
$|\psi_{AAF}(t)\rangle$, the corresponding atom-atom-field pure-state density
operator may be written as
$\rho_{AAF}(t)=|\psi_{AAF}(t)\rangle\langle\psi_{AAF}(t)|,$ (17)
where, the joint time-evolved wave vector, $|\psi_{AAF}(t)\rangle$, of the
tripartite system, i.e., the two atoms and the cavity field after the second
atom leaves the cavity is obtained by solving the Schrödinger equation
$i\frac{d}{dt}|\psi_{AAF}(t)\rangle=\widehat{H}_{int}|\psi_{AAF}(t)\rangle.$
(18)
It is worth to note that within the delay time between the two atoms the field
evolves towards a thermal steady state, moreover, repetition of the instant in
which the later atoms enter the cavity means the same field repeats at this
instants precisely when successive atoms exit the cavity FIJAME86 .
In order to quantify the degree of entanglement between the two atoms, the
field variables must be traced out. One may write the reduced mixed-state
density matrix of the two atoms after taking the trace over the field
variables as
$\mathbf{\rho_{AA}}(t)=\mathrm{Tr}_{F}~{}[\rho_{AAF}(t)].$ (19)
Matrix representation of the reduced density operator, Eq. (19) becomes
$\mathbf{\rho_{AA}}(t)=\left(\begin{array}[]{cccc}\rho_{11}&\rho_{12}&\rho_{13}&\rho_{14}\\\
\rho_{21}&\rho_{22}&\rho_{23}&\rho_{24}\\\
\rho_{31}&\rho_{32}&\rho_{33}&\rho_{34}\\\
\rho_{41}&\rho_{42}&\rho_{43}&\rho_{44}\end{array}\right),$ (20)
## III Entanglement measure
For bipartite pure states, the partial (von Neumann) entropy of the reduced
density matrices can provide a good measure of entanglement. However, for
mixed states von Neumann entropy fails, because it can not distinguish
classical and quantum mechanical correlations. For mixed states, the
entanglement can be measured as the average entanglement of its pure-state
decompositions $E_{F}(\rho)$
$E_{F}(\rho)=min\sum_{i}p_{i}E(\psi_{i}),$ (21)
with $E(\psi_{i})$ being the entanglement measure for the pure state
$\psi_{i}$ corresponding to all the possible decompositions
$\rho=\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|$. The existence of an
infinite number of decompositions makes their minimization over this set
difficult. Wootters W98 succeeded in deriving an analytical solution to this
difficult minimization procedure in terms of the eigenvalues of the non-
Hermitian operator
$T=\rho\tilde{\rho},$ (22)
where the tilde denotes the spin-flip of the quantum state, which is defined
as
$\tilde{\rho}=(\sigma_{x}\otimes\sigma_{x})\rho^{\ast}(\sigma_{x}\otimes\sigma_{x}),$
(23)
where $\sigma_{x}$ is the Pauli matrix, and $\rho^{\ast}$ is the complex
conjugate of $\rho$ where both are expressed in a fixed basis such as
$\\{|+\rangle,|-\rangle\\}$.
In terms of the eigenvalues of $T=\rho\tilde{\rho}$, $E_{F}(\rho)$ (known as
the entanglement of formation) takes the form
$E_{F}(\rho)=h\biggl{[}\frac{1}{2}+\frac{1}{2}\sqrt{1-C^{2}(\rho)}\biggr{]},$
(24)
where $C(\rho)$ is called the concurrence and is defined as
$C(\rho)=\max\biggl{(}0,\sqrt{\mathcal{E}_{1}}-\sqrt{\mathcal{E}_{2}}-\sqrt{\mathcal{E}_{3}}-\sqrt{\mathcal{E}_{4}}\biggr{)},$
(25)
with the $\mathcal{E}$’s representing the eigenvalues of $T=\rho\tilde{\rho}$
in descending order, and,
$h(y)=-y\log y-(1-y)\log(1-y),$ (26)
is the binary entropy. The concurrence is associated with the entanglement of
formation $E_{F}(\rho)$, Eq.(24), but it is by itself a good measure for
entanglement. The range of concurrence is from 0 to 1. For unentangled atoms
$C(\rho)=0$ whereas $C(\rho)=1$ for maximally entangled atoms.
Recalling (23) and assuming that that the matrix
$T=\rho_{AA}(\sigma_{x}\otimes\sigma_{x})\rho_{AA}^{\ast}(\sigma_{x}\otimes\sigma_{x})$,
needed for calculation of the concurrence, has the form
$T=\left(\begin{array}[]{cccc}T_{11}&T_{12}&T_{13}&T_{14}\\\
T_{21}&T_{22}&T_{23}&T_{24}\\\ T_{31}&T_{32}&T_{33}&T_{34}\\\
T_{41}&T_{42}&T_{43}&T_{44}\end{array}\right),$ (27)
the next step is to find the eigenvalues of the above matrix $T$. To achieve
our goal, we are supposed to solve the characteristic equation of
$\mathrm{Det}(T-\mathcal{E}\mathbf{I})=0,$ (28)
Equation (28) is a polynomial equation of degree 4
${\mathcal{E}}^{4}+c_{3}{\mathcal{E}}^{3}+c_{2}{\mathcal{E}}^{2}+c_{1}{\mathcal{E}}+c_{0}=0,$
(29)
with
$c_{3}=-T_{11}-T_{22}-T_{33}-T_{44},$ (30)
$c_{2}=-|T_{13}|^{2}-|T_{14}|^{2}-|T_{12}|^{2}-|T_{34}|^{2}-|T_{24}|^{2}+(T_{11}+T_{22})(T_{33}+T_{44})+T_{11}T_{22}+T_{33}T_{44},$
(31)
$c_{1}=|T_{12}|^{2}(T_{33}+T_{44})+|T_{23}|^{2}(T_{11}+T_{44})|T_{24}|^{2}(T_{11}+T_{33})+|T_{34}|^{2}(T_{11}+T_{22})|T_{13}|^{2}(T_{22}+T_{44})+|T_{14}|^{2}(T_{22}+T_{33})$
$-\Re(T_{23}T_{34}T_{42})-\Re(T_{21}T_{32}T_{13})-\Re(T_{21}T_{42}T_{14})-\Re(T_{31}T_{14}T_{43})-T_{11}(T_{11}T_{33}+T_{22}T_{33}+T_{11}T_{22}),$
(32)
$c_{0}=T_{11}T_{22}T_{33}T_{44}+|T_{13}|^{2}|T_{24}|^{2}+|T_{14}|^{2}|T_{23}|^{2}-T_{11}T_{22}|T_{34}|^{2}-T_{11}T_{44}|T_{23}|^{2}-T_{33}T_{44}|T_{12}|^{2}$
$-T_{11}T_{33}|T_{24}|^{2}-T_{22}T_{33}|T_{14}|^{2}+T_{11}\Re(T_{32}T_{24}T_{43})+T_{22}\Re(T_{31}T_{14}T_{43})+T_{33}\Re(T_{21}T_{42}T_{14})+T_{44}\Re(T_{32}T_{13}T_{43})$
$-\Re(T_{21}T_{32}T_{14}T_{43})-\Re(T_{21}T_{42}T_{13}T_{34})-\Re(T_{31}T_{42}T_{14}T_{23}).$
(33)
The roots of Eq.(29) are as follows
$\mathcal{E}_{1}=-\frac{1}{4}\biggl{[}c_{3}-\sqrt{\frac{W}{3}}-4O\biggr{]}$
(34)
$\mathcal{E}_{2}=-\frac{1}{4}\biggl{[}c_{3}-\sqrt{\frac{W}{3}}+4O\biggr{]}$
(35)
$\mathcal{E}_{3}=-\frac{1}{4}\biggl{[}c_{3}+\sqrt{\frac{W}{3}}-4N\biggr{]}$
(36)
$\mathcal{E}_{4}=-\frac{1}{4}\biggl{[}c_{3}+\sqrt{\frac{W}{3}}+4N\biggr{]}$
(37)
with
$O=\frac{1}{2}\sqrt{\frac{O_{1}+O_{2}+O_{3}}{6}}$ (38)
$N=\frac{1}{2}\sqrt{\frac{O_{1}+O_{2}-O_{3}}{6}}$ (39)
where
$O_{1}=3c_{3}^{2}-8c_{2}-V^{1/3}$ (40)
$O_{2}=4V^{-1/3}[3c_{1}c_{3}-12c_{0}-c_{2}^{2}]$ (41)
$O_{3}=\sqrt{\frac{3}{w}}[4c_{3}c_{2}-c_{3}^{3}-8c_{1}]$ (42)
and
$V=V_{1}+\sqrt{\frac{V_{2}+V_{3}+V_{4}+V_{5}}{144}}$ (43)
with
$V_{1}=8c_{2}^{2}+36[3(c_{1}^{2}+c_{0}c_{3}^{2})-c_{1}c_{2}c_{3}-c_{0}c_{2}]$
(44)
$V_{2}=-54c_{2}[c_{0}c_{1}c_{3}^{3}+c_{1}^{3}c_{3}+8c_{0}(c_{1}^{2}+c_{0}c_{3}^{2})]$
(45) $V_{3}=3c_{1}c_{3}[2c_{0}(4c_{2}^{2}+3c_{1}c_{3})-c_{1}c_{2}^{2}c_{3}]$
(46)
$V_{4}=12[(c_{1}c_{3})^{3}+4c_{0}c_{2}^{2}(8c_{0}-c_{2}^{2})+c_{2}^{3}(c_{1}^{2}+c_{0}c_{3}^{2})]$
(47) $V_{5}=c_{0}^{2}[81c_{3}^{4}+576c_{1}c_{3}-768c_{c}]$ (48)
and
$W=W_{1}+W_{2}$ (49)
where
$W_{1}=3c_{3}^{2}-8c_{2}+2V^{1/3}$ (50)
$W_{2}=8V^{-1/3}[c_{2}^{2}+3(4c_{0}-c_{1}c_{3})]$ (51)
In the following we consider two different cases of injection of the second
atom into the cavity subsystem that give us clear insight into the mechanism
of how and to what extent the degree of entanglement can be controlled and
enhanced.
## IV injection of two excited atoms successfully
In this case, the time dependent state vector of the full system can be
expressed in the form
$|\psi_{AAF}(t)\rangle=\biggl{|}U_{\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}}(t)\biggr{>}\mid
n\rangle+\biggl{|}U_{\begin{array}[]{l}\vspace{-2ex}+\\\ -\\\
\end{array}}(t)\biggr{>}\mid
n+1\rangle+\biggl{|}U_{\begin{array}[]{l}\vspace{-2ex}-\\\ +\\\
\end{array}}(t)\biggr{>}\mid
n+1\rangle+\biggl{|}U_{\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}}(t)\biggr{>}\mid n+2\rangle,$ (52)
which can be simply obtained by solving the Shrödinger equation (18), where
$\biggl{|}U_{\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}}(t)\biggr{>}=\sum_{n}C^{n}~{}[\Gamma_{1}(n,t)]^{2}\bigl{|}\begin{array}[]{l}\vspace{-2ex}+\\\
+\\\ \end{array}\bigr{>},$ (53)
$\biggl{|}U_{\begin{array}[]{l}\vspace{-2ex}+\\\ -\\\
\end{array}}(t)\biggr{>}=\sum_{n}C^{n}~{}\Gamma_{1}(n,t)\Gamma_{2}(n+1,t)\bigl{|}\begin{array}[]{l}\vspace{-2ex}+\\\
-\\\ \end{array}\bigr{>},$ (54)
$\biggl{|}U_{\begin{array}[]{l}\vspace{-2ex}-\\\ +\\\
\end{array}}(t)\biggr{>}=\sum_{n}C^{n}~{}\Gamma_{1}(n+1,t)\Gamma_{2}(n+1,t)\bigl{|}\begin{array}[]{l}\vspace{-2ex}-\\\
+\\\ \end{array}\bigr{>},$ (55)
$\biggl{|}U_{\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}}(t)\biggr{>}=\sum_{n}C^{n}~{}\Gamma_{2}(n+1,t)\Gamma_{2}(n+2,t)\bigl{|}\begin{array}[]{l}\vspace{-2ex}-\\\
-\\\ \end{array}\bigr{>},$ (56)
Using (17) to obtain the full density operator and after the application of
(19) to obtain the reduced atomic density operator which can be put, after
using the notations $\bigl{|}\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}\bigr{>}\equiv\mid 1\rangle$,
$\bigl{|}\begin{array}[]{l}\vspace{-2ex}+\\\ -\\\
\end{array}\bigr{>}\equiv\mid 2\rangle$,
$\bigl{|}\begin{array}[]{l}\vspace{-2ex}-\\\ +\\\
\end{array}\bigr{>}\equiv\mid 3\rangle$,
$\bigl{|}\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}\bigr{>}\equiv\mid 4\rangle$, in the form
$\rho_{AA}(t)=\sum_{i,j=1}^{4}~{}\rho_{ij}(t)|i\rangle\langle j|,$ (57)
where
$\rho_{11}(t)=\sum_{n}\bigl{|}C^{n}\bigr{|}^{2}\bigl{|}\Gamma_{1}(n,t)\bigr{|}^{4},$
(58)
$\rho_{12}(t)=\sum_{n}C^{n+1}C^{\ast
n}\bigl{[}\Gamma_{1}(n+1,t)\bigr{]}^{2}\Gamma_{1}^{\ast}(n,t)\Gamma_{2}^{\ast}(n+1,t),$
(59) $\rho_{13}(t)=\sum_{n}C^{n+1}C^{\ast
n}\bigl{|}\Gamma_{1}(n+1,t)\bigr{|}^{2}\Gamma_{1}(n+1,t)\Gamma_{2}^{\ast}(n+1,t),$
(60) $\rho_{14}(t)=\sum_{n}C^{n+2}C^{\ast
n}\bigl{[}\Gamma_{1}(n+2,t)\bigr{]}^{2}\Gamma_{2}^{\ast}(n+1,t)\Gamma_{2}^{\ast}(n+2,t),$
(61)
$\rho_{22}(t)=\sum_{n}\bigl{|}C^{n}\bigr{|}^{2}\bigl{|}\Gamma_{1}(n,t)\bigr{|}^{2}\bigl{|}\Gamma_{2}(n+1,t)\bigr{|}^{2},$
(62)
$\rho_{23}(t)=\sum_{n}\bigl{|}C^{n}\bigr{|}^{2}\bigl{|}\Gamma_{2}(n+1,t)\bigr{|}^{2}\Gamma_{1}(n,t)\Gamma_{1}^{\ast}(n+1,t),$
(63) $\rho_{24}(t)=\sum_{n}C^{n+2}C^{\ast
n}\Gamma_{1}(n+1,t)\Gamma_{2}^{\ast}(n+1,t)\bigl{|}\Gamma_{2}(n+2,t)\bigr{|}^{2},$
(64)
$\rho_{33}(t)=\sum_{n}\bigl{|}C^{n}\bigr{|}^{2}\bigl{|}\Gamma_{1}(n+1,t)\bigr{|}^{2}\bigl{|}\Gamma_{2}(n+1,t)\bigr{|}^{2},$
(65) $\rho_{34}(t)=\sum_{n}C^{n+2}C^{\ast
n}\Gamma_{1}(n+1,t)\Gamma_{2}^{\ast}(n+1,t)\bigl{|}\Gamma_{2}(n+2,t)\bigr{|}^{2},$
(66)
$\rho_{44}(t)=\sum_{n}\bigl{|}C^{n}\bigr{|}^{2}\bigl{|}\Gamma_{2}(n+1,t)\bigr{|}^{2}\bigl{|}\Gamma_{2}(n+2,t)\bigr{|}^{2},$
(67)
For $\chi=\Delta=0.0$, the results reported in Ref. BGASMNN07 are
straightforward obtained, which analyzed in details but for low and high
average photon number $\bar{n}$.
Figure 1: The concurrence $C(\lambda t/\pi)$ (solid red curve) and the total
population $\rho_{22}(\lambda t/\pi)+\rho_{33}(\lambda t/\pi)$ (dashed blue
curve) for $\bar{n}=10$, $\delta=0.0$ where (a) $\chi/\lambda=0.0$ (b)
$\chi/\lambda=0.2$ (c) $\chi/\lambda=0.5$ (c) $\chi/\lambda=1.0$
Figure 2: The same as Fig. 1 but for $\chi/\lambda=0.0$ where (a) $\delta=0.5$
(b) $\delta=1.0$ (c) $\delta=10.0$ (d) $\delta=15.0$ (e) $\delta=20$
Figure 3: The same as Fig. 2 but when $\chi=0.5$ and (a) $\delta=1.0$ (b)
$\delta=10.0$
In order to discuss the dynamical feasibility of our scheme in the micromaser
regime, in which the lifetime of the cavity is longer than atom field
interaction time, the concurrence $C(\lambda t/\pi)$ and the corresponding
total population $\rho_{22}(\lambda t/\pi)+\rho_{33}(\lambda t/\pi)$, as a
reference state, are depicted simultaneously in figures 1, 2 and 3 for the
Rabi angle $\lambda t/\pi\in[0,16]$. The mean photon number $\bar{n}$ is taken
about $\bar{n}=10$, the detuning parameter $\delta=\Delta/\lambda$ carries
various values such as $\delta=0.0,0.5,1,10,15$ and $20$, while the Kerr
parameter $\chi/\lambda$ varies such as $\chi/\lambda=0.0,0.2,0.5$ and
$\chi/\lambda=1.0$. From Fig. 1a, where the interaction occurs resonantly in
the absence of the nonlinear media, one can see clearly that the concurrence
$C$ exhibits irregular collections of oscillations where the total population
oscillates but with amplitudes maximize (no longer exceed the value $0.25$)
when $\rho_{22}+\rho_{33}\approx 0.4$; the atoms in the mixed state
$\bigl{|}\begin{array}[]{l}\vspace{-2ex}+\\\ -\\\
\end{array}\bigr{>}+\bigl{|}\begin{array}[]{l}\vspace{-2ex}-\\\ +\\\
\end{array}\bigr{>}$. Thus, the lack of the entanglement between the two atoms
can be attributed to a large population of the product states
$\bigl{|}\begin{array}[]{l}\vspace{-2ex}+\\\ -\\\ \end{array}\bigr{>}$ and
$\bigl{|}\begin{array}[]{l}\vspace{-2ex}-\\\ +\\\ \end{array}\bigr{>}$ which,
in turn, decay on the same time scale as $C$. For significantly high average
photon number $\bar{n}$, it was shown that BGASMNN07 , quantum effects which
are predominant primarily when the photon number is low, help to increase the
peak value of the concurrence $C$.
On resonance interaction between the atoms and a cavity filled with a Kerr
medium of a parameter of order $10^{-1}\lambda$, such as $\chi=0.2\lambda$,
the total population shows pseudo periodical behavior with amplitudes minima
decrease as time evolves where the revival period, $t_{R}$, elongates as time
goes on. The concurrence $C$ loses its irregular collections where instant
oscillations appear after wide time intervals accompany with amplitudes
decrease ($\approx 0.3$ in the beginning) as evolves , Fig. 1b. A significant
and interesting distinction in the case when $\chi=0.5\lambda$ is that both
$C$ and $\rho_{22}+\rho_{33}$ display a perfect $\pi$ periodicity of
oscillations. The total population $\rho_{22}+\rho_{33}$ exhibits collapses
and revivals with the revival period $t_{R}$ is shorter and each revival is a
perfect carbon copy of its predecessor. The concurrence $C$ shows oscillation
packets composed of three peaks with the strongest,$=2.5$, lies at the center
where the total population collapses. Moreover, the concurrence $C$ minimizes
where the total populations $\rho_{22}+\rho_{33}$ revives while the strong
entanglement and collapse of population occur in parallel manner, see Fig. 1c.
This periodicity is due to the fact that analytical expressions of both $C$
and $\rho_{22}+\rho_{33}$ can be summed exactly. Thus, the matching value of
the Kerr parameter plays more efficient rule than that of quantum effects that
leads to increase the peak value of the concurrence $C$ BGASMNN07 . Opposite
results are obtained when a considerable strength of the Kerr medium is
applied to the cavity field. In this case, a pseudo periodicity for $C$ is
observed where the total population preserves its periodicity accompanied with
clear decreasing in its amplitudes. In addition, one can’t build a clear
insight about the relation between curves of both $C$ and
$\rho_{22}+\rho_{33}$, Fig. 1d. Thus, from the above it is found that the Kerr
medium can be used to adjust the entanglement degree between qubits, which may
provide some reference and even basis for the quantum control of multiple
qubits.
A surprise is that the case of a non-resonance interaction does not destroy
the entanglement phenomenon, see Fig. 2. For small detuning such that
$\delta=0.5$, there is no significant change, Fig. 2a, but when $\delta$
increases to unity, the correlations time between the atoms becomes longer
with the gradual appearance of nonzero values of $C$, see Fig. 2b. Strictly
speaking, the destruction of entanglement occurs for the case when the
detuning parameter $\delta$ is absent but not the Kerr parameter
$\chi/\lambda$, Fig. 1d.
The most intersting and surprising is the case when the inter-atomic
transition is considered to be significantly high compared with the cavity
frequency such that $\delta=10.0,15.0$ and $20.0$ where the nonlinear media is
considered to be absent. This is revealed in Figs. 2c-e. We see that the total
population exhibits the known collape and revival phenomena periodically.
Also, we notice that for the increase of detuning parameter $\delta$, the
collapses become longer and the beginning strong overlap of the revivals
becomes weaker; while the amplitude of the revivals decreases as values of
$\delta$ increase. The entanglement evolution divided into three regions. The
first one is at the half of the collapse time, where the entanglement degree
oscillates with bigger amplitude for short time scale. The oscillation maximum
is related crucially to the value of $\delta$, where it occurs at time
$\lambda t=(\frac{\delta}{2}+1)\pi$. The last two regions are in the beginning
and end of the collape time. In this case, we see that the atom-atom
entanglement remain close to its maximum and show intrinsic oscillations only.
This is in fact is true for all values of the detuning parameter. The detuning
between the field and the atoms leads to an emergence of collapses and
revivals, this statement is always true only in the absence of the nonlinear
media, see Figs. 2c-e. Overall, we find that the detuning parameter acts as a
control parameter for the atom-atom entanglement, where a perfect periodicity
can be obtained for suitable high detuning. The revival of the entanglement at
time $\lambda t=(\frac{\delta}{2}+1)\pi$ is more easily understood by
reference to the density matrix of the system. As expected, there is no
entanglement before the populations of the state
$\bigl{|}\begin{array}[]{l}\vspace{-2ex}+\\\ -\\\
\end{array}\bigr{>}+\bigl{|}\begin{array}[]{l}\vspace{-2ex}-\\\ +\\\
\end{array}\bigr{>}$ and the state
$\bigl{|}\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}\bigr{>}+\bigl{|}\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}\bigr{>}$ depopulates, but some entanglement appears for longer
times, Figs. 2b-e, and the concurrence becomes equal to the population of the
state $\bigl{|}\begin{array}[]{l}\vspace{-2ex}+\\\ -\\\
\end{array}\bigr{>}+\bigl{|}\begin{array}[]{l}\vspace{-2ex}-\\\ +\\\
\end{array}\bigr{>}$, see Fig. 1c.
## V injection of a ground atom after the excited one
In this case, the time dependent state vector of the full system can be
expressed in the form
$|\psi_{AAF}(t)\rangle=\biggl{|}S_{\begin{array}[]{l}\vspace{-2ex}+\\\ -\\\
\end{array}}(t)\biggr{>}\mid
n\rangle+\biggl{|}S_{\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}}(t)\biggr{>}\mid
n-1\rangle+\biggl{|}S_{\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}}(t)\biggr{>}\mid
n+1\rangle+\biggl{|}S_{\begin{array}[]{l}\vspace{-2ex}-\\\ +\\\
\end{array}}(t)\biggr{>}\mid n\rangle,$ (68)
where
$\biggl{|}S_{\begin{array}[]{l}\vspace{-2ex}+\\\ -\\\
\end{array}}(t)\biggr{>}=\sum_{n}C^{n}~{}\Gamma_{1}(n,t)\Gamma_{1}^{\ast}(n-1,t)\bigl{|}\begin{array}[]{l}\vspace{-2ex}+\\\
-\\\ \end{array}\bigr{>},$ (69)
$\biggl{|}S_{\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}}(t)\biggr{>}=\sum_{n}C^{n}~{}\Gamma_{1}(n,t)\Gamma_{2}(n,t)\bigl{|}\begin{array}[]{l}\vspace{-2ex}+\\\
+\\\ \end{array}\bigr{>},$ (70)
$\biggl{|}S_{\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}}(t)\biggr{>}=\sum_{n}C^{n}~{}\Gamma_{1}^{\ast}(n+1,t)\Gamma_{2}(n+1,t)\bigl{|}\begin{array}[]{l}\vspace{-2ex}-\\\
-\\\ \end{array}\bigr{>},$ (71)
$\biggl{|}S_{\begin{array}[]{l}\vspace{-2ex}-\\\ +\\\
\end{array}}(t)\biggr{>}=\sum_{n}C^{n}~{}[\Gamma_{2}(n+1,t)]^{2}\bigl{|}\begin{array}[]{l}\vspace{-2ex}-\\\
+\\\ \end{array}\bigr{>},$ (72)
Following the same procedure as in the previous section, and after using the
notations $\bigl{|}\begin{array}[]{l}\vspace{-2ex}+\\\ -\\\
\end{array}\bigr{>}\equiv\mid 1\rangle$,
$\bigl{|}\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}\bigr{>}\equiv\mid 2\rangle$,
$\bigl{|}\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}\bigr{>}\equiv\mid 3\rangle$,
$\bigl{|}\begin{array}[]{l}\vspace{-2ex}-\\\ +\\\
\end{array}\bigr{>}\equiv\mid 4\rangle$, the elements of the reduced atomic
density matrix (57) read
$\rho_{11}(t)=\sum_{n}|C^{n}|^{2}|\Gamma_{1}(n,t)|^{2}|\Gamma_{1}(n-1,t)|^{2},$
(73) $\rho_{12}(t)=\sum_{n}C^{n}C^{\ast
n+1}\Gamma_{1}(n,t)\Gamma_{1}^{\ast}(n-1,t)\Gamma_{1}^{\ast}(n+1,t)\Gamma_{2}^{\ast}(n+1,t),$
(74) $\rho_{13}(t)=\sum_{n}C^{n+k}C^{\ast
n}|\Gamma_{1}(n,t)|^{2}\Gamma_{1}(n+1,t)\Gamma_{2}^{\ast}(n+1,t),$ (75)
$\rho_{14}(t)=\sum_{n}|C^{n}|^{2}\Gamma_{1}(n,t)\Gamma_{1}^{\ast}(n-1,t)[\Gamma_{2}^{\ast}(n+1,t)]^{2},$
(76)
$\rho_{22}(t)=\sum_{n}|C^{n}|^{2}|\Gamma_{1}(n,t)|^{2}|\Gamma_{2}(n,t)|^{2},$
(77) $\rho_{23}(t)=\sum_{n}C^{n+2}C^{\ast
n}\Gamma_{1}(n+2,t)\Gamma_{2}(n+2,t)\Gamma_{1}(n,t)\Gamma_{2}^{\ast}(n+1,t),$
(78) $\rho_{24}(t)=\sum_{n}C^{n+2}C^{\ast
n}|\Gamma_{2}(n+1,t)|^{2}\Gamma_{1}(n+1,t)\Gamma_{2}^{\ast}(n+1,t),$ (79)
$\rho_{33}(t)=\sum_{n}|C^{n}|^{2}|\Gamma_{1}(n,t)|^{2}|\Gamma_{2}(n+1,t)|^{2},$
(80) $\rho_{34}(t)=\sum_{n}C^{n}C^{\ast
n+1}\Gamma_{1}^{\ast}(n,t)\Gamma_{2}(n+1,t)[\Gamma_{2}^{\ast}(n+2,t)]^{2},$
(81) $\rho_{44}(t)=\sum_{n}|C^{n}|^{2}|\Gamma_{2}(n+1,t)|^{4},$ (82)
Figure 4: The concurrence $C(\lambda t/\pi)$ (solid red curve) and the total
population $\rho_{22}(\lambda t/\pi)+\rho_{33}(\lambda t/\pi)$ (dashed blue
curve) for $\bar{n}=10$, $\delta=0.0$ where (a) $\chi/\lambda=0.0$ (b)
$\chi/\lambda=0.2$ (c) $\chi/\lambda=0.5$ (c) $\chi/\lambda=1.0$
Figure 5: The same as Fig. 4 but for $\chi/\lambda=0.0$ where (a) $\delta=0.5$
(b) $\delta=1.0$ (c) $\delta=10.0$ (d) $\delta=15.0$ (e) $\delta=20$
Figure 6: The same as Fig. 5 but when $\chi=0.5$ and (a) $\delta=1.0$ (b)
$\delta=10.0$
To better understand the situation and gain clearer insight into the
entanglement dynamics, hereafter we shall analyze the evolution of the
concurrence $C(\lambda t/\pi)$ and the total populations $\rho_{22}(\lambda
t/\pi)+\rho_{33}(\lambda t/\pi)$ where the parametres remain as the same in
the past section.
Generally, a similar conclusion is reached for the present case but with
considerable distinctions in its details. As in the case of two excited atoms,
there is no initial entanglement between the atoms, and at early times the
entanglement builds up rapidly to a maximum appearing at short time $\lambda
t\approx 2.5\pi$, Fig. 4a. However, comparing with the case of two successive
excited atoms, shown in the previous section, we see that the entanglement
maxima that obtained in the present case are generally lower than that
obtained with two successive excited atoms, see Figs. 1-3 and 4-6. It is
interesting to note that after passing through the maximum, the entanglement
decays with two different time scales, Figs. 2 and 5. This effect is more
pronounced if we choose the interaction with atoms to be initially in
different quantum states. Thus, atom-atom entanglement depends crucially on
the coherent transfer of the population between the atoms states, specially in
the presence of a nonlinear media as well as the detuning parameter $\delta$.
In this process, the population is efficiently transferred from the more
populated state, $\bigl{|}\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}\bigr{>}+\bigl{|}\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}\bigr{>}$ to the state,
$\bigl{|}\begin{array}[]{l}\vspace{-2ex}+\\\ -\\\
\end{array}\bigr{>}+\bigl{|}\begin{array}[]{l}\vspace{-2ex}-\\\ +\\\
\end{array}\bigr{>}$ before it decays to the initial state leading to the
enhancement of the entanglement.
## VI Phase spase distribution
We devote this section to concentrate on a function, which considered an
important one of the functions, $W$, (Husimi) $Q$, and (Glauber-Sudershan) $P$
functions, that is the Husimi $Q$-function which has the nice property of
being always positive and further advantage of being readily measurable by
quantum tomographic techniques BOTAWA98 ; MANTOM97 . It has been shown from
earlier studies EWIGNER32 ; ZWIGNer32 ; KAGL69 ; HICOSCWG84 that these quasi-
probability functions are important for the statistical description of a
microscopic system and provide insight into the nonclassical features of the
radiation fields. The homodyne measurement of an electromagnetic field gives
all possible linear combinations of the field quadratures. The average of the
random outcomes of the measurement is connected with the marginal distribution
of any quasi-probability used in quantum optics. In fact, Husimi $Q$-function
is not only a convenient tool to calculate the expectation values of anti-
normally ordered products of operators, but also to give some insight into the
mechanism of interaction for the model under consideration. The relation
between the phase-space measurement; Husimi $Q$-function; and the classical
information-theoretic entropy associated with quantum fields was introduced by
Wehrl WEHRL79 , which referred to as the Shannon information of the Husimi
Q-function. Thus, Husimi $Q$-function can be related to quantum entanglement
in different approaches WEHRL79 ; PEKRPELUSZ86 ; FAGU06 ; CAALCARA09 ; HUFAN09
; MIMAWA00 ; MIWAIM01 ; BERETA84 . Furthermore, separable state is represented
by a localized wave packet in phase space. Since coherent states are the most
localized states in the Husimi representation, it is argued that SUGITA03
delocalization of the Husimi distribution implies correlation-hence
entanglement- between system particles.
The Husimi $Q$-function can be given in the form as HICOSCWG84 ; HUSIMI40 ;
FuSOLO001
$Q(\alpha)=\frac{\langle\alpha\mid\rho_{F}\mid\alpha\rangle}{\pi},$ (83)
where $\rho_{F}$ is the reduced density operator of the cavity field given by
tracing over the atomic variables of the full density operator (17). The state
$\mid\alpha\rangle$ represents the well-known coherent state with amplitude
$\alpha=X+iY$. Inserting the obtained $\rho_{F}$ into Eq. (83), we can easily
obtain the Husimi $Q$-function of the cavity field
$Q=\frac{1}{\pi}(\langle\alpha,\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}\mid\rho_{F}\mid\alpha,\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}\rangle+\langle\alpha,\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}\mid\rho_{F}\mid\alpha,\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}\rangle),$ (84)
where
$\mid\alpha\rangle=e^{-\mid\alpha_{0}\mid^{2}/2}\sum_{n=0}^{\infty}\frac{\alpha_{0}^{n}}{\sqrt{n!}},$
(85)
### VI.1 Injection of two excited atoms one by one
In this case, the $Q$-function is given by
$Q=\frac{1}{\pi}\biggl{(}\biggl{|}\biggl{<}\alpha,\begin{array}[]{l}\vspace{-2ex}+\\\
+\\\ \end{array}\biggl{|}U_{\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}},n\biggr{>}\biggr{|}^{2}+\biggl{|}\biggl{<}\alpha,\begin{array}[]{l}\vspace{-2ex}-\\\
-\\\ \end{array}\biggl{|}U_{\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}},n+1\biggr{>}\biggr{|}^{2}\biggr{)},$ (86)
where for the states $\mid\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}\rangle$ and $\mid\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}\rangle$, $\rho_{F}$ reads
$\rho_{F}=\biggl{|}U_{\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}}(t),n\biggr{>}\biggl{<}U_{\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}}(t),n\biggr{|}+\biggl{|}U_{\begin{array}[]{l}\vspace{-2ex}-\\\
-\\\
\end{array}}(t),n+1\biggr{>}\biggl{<}U_{\begin{array}[]{l}\vspace{-2ex}-\\\
-\\\ \end{array}}(t),n+1\biggr{|},$ (87)
in this case
$\biggl{<}\alpha,\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}\biggl{|}U_{\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}}(t),n\biggr{>}=e^{-(\mid\alpha\mid^{2}+\mid\alpha_{0}\mid^{2})/2}\sum_{n}\frac{(\alpha_{0}\alpha^{\ast})^{n}}{n!}[\Gamma_{1}(n,t)]^{2},$
(88) $\biggl{<}\alpha,\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}\biggl{|}U_{\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}}(t),n+1\biggr{>}=e^{-(\mid\alpha\mid^{2}+\mid\alpha_{0}\mid^{2})/2}\sum_{n}\frac{(\alpha_{0}\alpha^{\ast})^{n}\alpha^{\ast
2}}{\sqrt{n!(n+2)!}}\Gamma_{2}(n+1,t)\Gamma_{2}(n+2,t),$ (89)
$\alpha=X+iY.$ (90)
Figure 7: Conour plots for Husimi $Q$-function for $\bar{n}=10$, $\delta=0.0$,
$\lambda t=5\pi/3$ where (a) $\chi/\lambda=0.0$ (b) $\chi/\lambda=0.2$ (c)
$\chi/\lambda=0.5$ (d) $\chi/\lambda=1.0$
Figure 8: The same as Fig. 7 but when $\delta=10.0$ and (a) $\chi/\lambda=0.0$
(b) $\chi/\lambda=1.0$
Figure 9: The same as Fig. 7 but for $\chi/\lambda=0.5$ where (a) $\lambda
t=\pi/6$ (b) $\lambda t=\pi/4$ (c) $\lambda t=\pi/3$ (d ) $\lambda t=\pi/2$
Here, we present a detailed analytical discussion about how the Husimi
$Q$-function plays an important role in quantifying atom-atom entanglement.
For this reason, we have pictured the numerical results obtained as contour
plots in figures 7-9. The parameters was considered as follows: the average
photon number $\bar{n}$ is kept fixed for all plots such as $\bar{n}=10$, the
detuning parameter carries values such as $\delta=0.0,10.0$, and the Kerr
parameter $\chi/\lambda$ varies as $\chi/\lambda=0.0,0.2,0.5,1.0$, where the
Rabi angle $\lambda t=5\pi/3$ is kept fixed for figures 7 and 8, while for
figure 9 it changes as $\lambda t=\pi/6,\pi/4,\pi/3,\pi/2$, where the Kerr
parameter $\chi/\lambda$ is half integer.
Form figure 7, by which we examine the effect of the Kerr parameter
$\chi/\lambda$ at a fixed Rabi angle $\lambda t=5\pi/3$, we observe that, in
the absence of the Kerr parameter, Fig. 7a, the Husimi $Q$-function composed
of three components, two blobs centered at the points (0, $\pm 4$) and a
symmetric circular peak (Poisson band) centered at the point (4, 0) and
strongly localized in the right half of the available phase space, i.e., the
interaction region THMAMAPO07 , once more, the squeezing effect on the number
of photons can be observed from the figure. Experimental and theoretical
studies showed that squeezing of matter wave fields (squeezed states) is
closely related to entanglement RSCNB03 ; RECL97 ; DSCZ00 ; SDCZ01 ; ATETO010
; PMBLMH04 ; FIUNIO03 . Moreover, we observe that the Husimi distribution
blobs delocalized almost around the right half of the phase space, i. e., the
interaction region of phase space and have structures with almost positive
density throughout the available phase space. It is worth to note that the
bifurcating of the blobs corresponds to the collapse. This can be clearly seen
if we compare the numerical calculations for total populations and
$Q$-function. It was shown that the bifurcation of the Husimi distribution,
which is the signature of the formation of Schrödinger cat states, implies
correlation, and hence entanglement VAOR95 ; ORPAKA95 ; JEOR94 ; MIMAWA00 ;
MIWAIM01 ; SUGITA03 ; HIMCMI05 , see also Fig. 1a. As soon as the effect of
the Kerr medium is considered, a clear difference in the figures shape
appears, where the sensibility of the Husimi $Q$-function with the
nonlinearity parameter $\chi/\lambda$ is very intersting. We observe that the
Husimi distribution is taken gradually towards the non-interaction region,
namely, left half of the available phase space, where we observe that the two
blobs rotating in the complex coherent state parameter plane in the
counterclockwise directions with the same speed where two overlapping regions
appear with both terms are appreciably vanishing. But note also that the
intersection areas with this prescription are not in the same position. Upon
the increase of the Kerr parameter then one can observe that the overlapping
becomes dominant and the Husime distribution are taken completely into the
interaction region. As a consequence, strong entanglement degree between the
two atoms can be built up at that time, see Figs. 1c and 7c. The conclusion
will be extremely interesting when we look at the behaviors of both the
concurrence $C$ and the Husimi distribution once the Kerr medium is taken into
account, specially at the moment when the Kerr medium becomes strong at which
entanglement diminishes and the Husimi distribution is strongly localized and
taken into non-interaction region again, Figs. 1d and 7d. It is important to
note that, localization of the Husimi distribution around a fixed point
implies low degree of entanglement SUGITA03 ; HIMCMI05 .
Our observations will become clear when we look at the behaviors of both the
concurrence $C$ and the Husimi distribution at the moment when detuning
parameter possesses a value. The results showed in Figs. 2c, 3b and 8. A clear
delocalization of the Husimi distribution but into the right half of phase
space, the interaction region, corresponds to longer time interval of
entanglement, Figs. 2c and 8a, while the opposite is true, Figs. 3b and 8b. An
extremely interesting observations can be obtained when we consider various
values of the Rabi angle $\lambda t$, while the nonlinear media is switched
on. The results show strongly the role that the Hisimi distribution play in
quantifying entanglement, where at all moments but not $\lambda t=\pi/2$, once
the suitable Kerr parameter is chosen, stronger entanglement can be built up
since the Husimi distribution is delocalized but completely spreads between
two or three points into the right half of phase space, the interaction
region, see also Fig. 1c.
### VI.2 Injection of excited and ground atoms one by one
In this case, for the states $\mid\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}\rangle$ and $\mid\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}\rangle$, $\rho_{F}$ reads
$\rho_{F}=\biggl{|}S_{\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}}(t),n-1\biggr{>}\biggl{<}S_{\begin{array}[]{l}\vspace{-2ex}+\\\
+\\\
\end{array}}(t),n-1\biggr{|}+\biggl{|}S_{\begin{array}[]{l}\vspace{-2ex}-\\\
-\\\
\end{array}}(t),n+1\biggr{>}\biggl{<}S_{\begin{array}[]{l}\vspace{-2ex}-\\\
-\\\ \end{array}}(t),n+1\biggr{|}.$ (91)
The $Q$-function is given by
$Q=\frac{1}{\pi}\biggl{(}\biggl{|}\biggl{<}\alpha,\begin{array}[]{l}\vspace{-2ex}+\\\
+\\\ \end{array}\biggl{|}S_{\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}},n-1\biggr{>}\biggr{|}^{2}+\biggl{|}\biggl{<}\alpha,\begin{array}[]{l}\vspace{-2ex}-\\\
-\\\ \end{array}\biggl{|}S_{\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}},n+1\biggr{>}\biggr{|}^{2}\biggr{)},$ (92)
where
$\biggl{<}\alpha,\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}\biggl{|}S_{\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}}(t),n-1\biggr{>}=e^{-(\mid\alpha\mid^{2}+\mid\alpha_{0}\mid^{2})/2}\sum_{n}\frac{(\alpha_{0}\alpha^{\ast})^{n}\alpha_{0}}{\sqrt{n!(n+1)!}}\Gamma_{1}(n+1,t)\Gamma_{2}(n+1,t),$
(93) $\biggl{<}\alpha,\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}\biggl{|}S_{\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}}(t),n+1\biggr{>}=e^{-(\mid\alpha\mid^{2}+\mid\alpha_{0}\mid^{2})/2}\sum_{n}\frac{(\alpha_{0}\alpha^{\ast})^{n}\alpha^{\ast}}{\sqrt{n!(n+1)!}}\Gamma_{1}^{\ast}(n,t)\Gamma_{2}(n+1,t),$
(94)
Figure 10: Conour plots for Husimi $Q$-function for $\bar{n}=10$,
$\delta=0.0$, $\lambda t=5\pi/3$ where (a) $\chi/\lambda=0.0$ (b)
$\chi/\lambda=0.2$ (c) $\chi/\lambda=0.5$ (d) $\chi/\lambda=1.0$
Figure 11: The same as Fig. 10 but when $\delta=10.0$ and (a)
$\chi/\lambda=0.0$ (b) $\chi/\lambda=1.0$
Figure 12: The same as Fig. 10 but for $\chi/\lambda=0.5$ where (a) $\lambda
t=\pi/6$ (b) $\lambda t=\pi/4$ (c) $\lambda t=\pi/3$ (d ) $\lambda t=\pi/2$
To obtain clearer insight about the efficiancy of the Husimi $Q$-function in
quantifying entanglement, a comparative discusion will be important. Here we
examine the behavior of the Husimi function under the same circumstances of
parameters as in the previous section. This is illustrated in figures 10-12.
Generally, the Husimi $Q$-function developed in manner similar to that
observed previousley with slight change such that the localized Poisson band
barely appear which means that nonclassical effect is not pronounced. In this
case the entanglement between the two atoms doesn’t live much more in opposite
to the case of two successive excited atoms, where entanglement does appear at
almost the whole time scale, see Figs. 1a, 4a, 7a, 10a. The case when we turn
on the nonlinear media is also intersting. The Husimi distribution moves
quickly towards the non-interaction region once the a weak Kerr medium is
switched on, while a strong localized Poisson band appear clearly in the
interaction region which reflects small probability of interaction and so
lesser times of entanglement, see Figs. 4b, 10b. The suitable choice of the
Kerr parameter push the Husimi distribution towards the interaction region
again, which means stronger entanglement appears at longer times, Figs. 4c,
10c. There is no something interesting in case when the nonlinear media is
strong, where almost the same behavior was seen before, for comparison see
Figs. 1d, 4d, 7d, 10d.
However, the effect of the detuning parameter $\delta$ is quite differente,
specially in the presence of strong nonlinear media. In such a case, the
Husimi distribution displays a strong delocalization and spreads between two
points in the right half of the phase space in case of zero-value nonlinear
media accompanied with a significant density throughout the available phase
space. Moreover, overlapp takes place and the squeezing effect on the number
of photons occurs can be observed. The strong Kerr media have intersting
effect where the Husimi distribution shows ring-like shape around the center
of the phase space but nevertheless have a significantly different densities
throughout the available phase space, i.e., interaction and non-interaction
regions. The effect of the matching choice the nonlinear media parameter
$\chi/\lambda$ on the Husimi distribution at different values Rabi angle
$\lambda t$ points out that enhancement of entanglement depends crucially on
this suitable choice, see Figs. 4a and 12a-c.
## VII Conclusions
We have considered the nature of the entanglement of output two successive
atoms from a micromaser cavity for pure states input of the atoms when the
cavity distribution is coherent. We have considered two different successive
injections of the initial states of the atoms that traverse the cavity,
namely, initially excited atoms and initially excited atom followed ground
one. We have examined the effect of the frequency difference between the
inter-atomic frequency and cavity frequency individually and in coexistence of
nonlinear media on nonlocal atomic entanglement. We have also investigate the
role of the Husimi $Q$-distribution that plays to give clearer insight about
entanglement dynamics. Our conclusions are Summarized in the following:
(i) The physical nature of the interacting objects and the character of their
mutual coupling control strongly the degree of quantum atomic entanglement, in
other words, the interaction of a cavity field with two successive excited
atoms plays more efficient role in producing atomic entanglement than that of
interaction with two successive atoms traverse the cavity in dissimilar
initial states.
(ii) The nonlinear medium plays an important role in producing atomic
entanglement depending on the appropriate choice of its parameter. In this
case an excellent periodical entanglement can be produced with increased
maximum value comparing with the case when the nonlinear medium parameter is
not appropriate or zero.
(iii) The extremely interesting is the role of the detuning parameter on
producing nonlocal atomic entanglement. When the detuning parameter is
considered to be considerably high, periodical, long lived entanglement can be
obtained.
(iv) Husimi $Q$-distribution plays a very clear and efficient role in
quantifying nonlocal atomic entanglement, where bifurcating of Husimi
$Q$-distribution in whole phase space and delocalization in the interaction
region of the phase space corresponding to correlations between particles and
as a consequence nolocal atomic entanglement.
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|
arxiv-papers
| 2010-03-13T06:32:01 |
2024-09-04T02:49:09.082485
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. S. Ateto",
"submitter": "Mohammed Ateto Dr.",
"url": "https://arxiv.org/abs/1003.2683"
}
|
1003.2950
|
# Exact feature probabilities in images with occlusion
Xaq Pitkow
Center for Theoretical Neuroscience, Columbia University
###### Abstract
To understand the computations of our visual system, it is important to
understand also the natural environment it evolved to interpret.
Unfortunately, existing models of the visual environment are either
unrealistic or too complex for mathematical description. Here we describe a
naturalistic image model and present a mathematical solution for the
statistical relationships between the image features and model variables. The
world described by this model is composed of independent, opaque, textured
objects which occlude each other. This simple structure allows us to calculate
the joint probability distribution of image values sampled at multiple
arbitrarily located points, without approximation. This result can be
converted into probabilistic relationships between observable image features
as well as between the unobservable properties that caused these features,
including object boundaries and relative depth. Using these results we explain
the causes of a wide range of natural scene properties, including highly non-
gaussian distributions of image features and causal relations between pairs of
edges. We discuss the implications of this description of natural scenes for
the study of vision.
## 1 Introduction
A major goal of vision is to identify physical objects in the world, and their
attributes. The relevant sensory evidence — an image — is ambiguous. A visual
system must make guesses to interpret this sensory information, and good
guesses should account for the statistics of the input. Consequently, the
statistical structure of natural images has become a subject of fundamental
importance for applications ranging from computer graphics to neuroscience:
Understanding and exploiting natural regularities should lead to better visual
performance and improved visual representations, whether in image compression
or in the brain.
Most previous studies of natural scene statistics have characterized natural
scenes as linear superpositions of image features. Principal Components
Analysis [1, 2], Independent Components Analysis [3, 4], and wavelet
transforms [5] each identify related sets of features that when added together
can efficiently reconstitute a natural image. Other methods have improved upon
these purely linear, additive descriptions by including multiplicative
modulation (e.g. Gaussian scale mixtures [6], hierarchically correlated
variances [7]). These nonlinear enhancements are useful for representing
textures, where common variables like surface properties and illumination
intensity and direction naturally co-modulate the contrast of features like
local orientation. Yet because visual images are not caused by summation but
by occlusion, it is important to develop models of natural images that are
constructed using more accurate nonlinear combinations of features [8, 9].
We therefore chose a simplified model of natural images, colorfully known as
the dead leaves model [10], for which occlusion is fundamental: The virtual
world described by this model is composed of an infinitely deep stack of
randomly positioned, flat objects (‘leaves’) that occlude each other (Figure
1A–B). These objects have attributes of size, shape, texture, and color,
independently drawn from specified distributions. While the model is only an
approximation to our true physical environment, nonetheless it generates
images that share many important attributes with natural images, most
obviously the ubiquitous boundaries between relatively homogeneous regions
[11]. It reproduces several known statistical properties of natural images,
including the spatial power spectrum [11] and bivariate distributions of pixel
intensities and wavelets coefficients [12]. Despite the demonstrated utility
of the model, until now there has been no way to calculate higher-order
statistical properties of interest except by empirical sampling, which cannot
provide the insights that exact results can.
Here we derive an exact solution to the dead leaves model, by calculating
joint probability distributions explicitly for arbitrary image features. This
solution also provides a principled way to relate the features in a dead
leaves image to the unobserved object attributes that cause these features.
Since these relationships are precisely what we rely upon to see, this result
thereby elevates the dead leaves model from an interesting approximation of
natural images to a valuable tool for modeling perceptual inference and neural
computation in the visual system.
To illustrate how this solution helps us understand natural scenes, we apply
it to explain the highly non-gaussian probability distributions of two
important types of image features: wavelet coefficients — i.e. the image
overlap with localized, oriented filters — and local object boundaries. These
features are important because they describe stimuli to which neurons in the
early visual system are sensitive, and because high-order correlations between
them reflect the physical objects and attributes in the visual world. Since
the functional significance of neural responses to features can depend on the
shape of the feature distribution [13, 14, 15], it is important to understand
why the distributions have their observed structure.
We first look specifically at the marginal, joint, and conditional
distributions of wavelet coefficients. In natural images, the marginal
distributions have heavy tails [16, 17, 6], which we show is due to the
spatial scale invariance of objects. Joint and conditional distributions of
wavelet coefficients have peculiar shapes (diamonds, pillows, bowties) that
depend on the orientation and distance between the wavelets [18, 19, 12]. We
show how these distribution shapes arise naturally from occlusion by spatially
extended objects. Finally, we compute the likelihood that a given pair of
local object boundaries comes from the same physical contour. When estimated
empirically from natural images, this likelihood predicts human judgments
about contours [20]. Our solution of the dead leaves model recovers the
empirical statistics but only if one properly accounts for the relative depths
at local boundaries, implicating depth cues in simple judgments about
contours.
Figure 1: Example images generated by the dead leaves model. We see layers of
objects with random sizes, shapes, colors and positions that occlude other
objects below. (A) All objects are black or white circles with a relatively
narrow range of sizes. (B) All objects are textured ellipses with a broad
range of sizes drawn from a distribution proportional to ${\rm size}^{-3}$,
producing approximate scale invariance [11, 12]. Straightforward
generalizations allow other ensembles of shape and texture. (C) Illustration
of an object membership function m—. Pixels within a member set of m— all
sample from the same object. Shown is an example dead leaves image with
several objects (grey circles) and a set of six pixel locations (numbered
points). For this configuration, the object membership function is
${\text{\cjRL{m|}}}=\\{126|3|45\\}$.
## 2 Results
### 2.1 Solving the dead leaves model
The pixels of a dead leaves image are fully determined by the properties of
objects that are at least partially unoccluded. These properties are drawn
independently from specified distributions over position, depth, size and
shape, and texture. Texture can include both mean intensity and (possibly
correlated) variations about the mean. When we say that we have solved the
dead leaves model, we mean that we can calculate the joint probabilities of
any model variables of interest, whether pixel intensities or object
properties. This would be straightforward if the image components were related
by linear superposition, but is much more difficult due to the strong
nonlinearity of occlusion.
The essential property that makes the dead leaves model tractable is that
different objects have independent attributes. Others have invoked the
independence of object properties to derive the two-point correlation
functions [11] and bivariate intensity probabilities [12] using a recursive
argument that accounts for the way nearby objects occlude more distant ones.
We were able to generalize this calculation from two points to an arbitrary
collection of $N$ pixels, for which we can now calculate the multivariate
joint intensity distribution. This distribution can then be transformed into
feature probabilities, and related to the unobserved object properties.
If one samples the intensity of a particular dead leaves image at various
locations, each pixel value will be determined by the texture of whichever
object is at the top of the stack at that location. All pixels that fall into
the same object share its texture, and are thereby correlated; pixels sampling
from different objects are independent. Thus, if we can specify how the pixels
are divided geometrically into objects, then we know the complete correlation
structure for that image.
We can mathematically describe the configuration of objects at a given set of
$N$ pixels by defining an object membership function, m—, designating which
pixels are ‘members’ of which objects. (The symbol m— is the Hebrew letter
mem, chosen to evoke the word membership.) In mathematical language, m— is a
set partition of the $N$ pixels, so it is technically a set of sets: each set
corresponds to a different object, and it contains the pixel locations at
which that object is unobscured by any other objects. For example, one might
find in a given image that pixels ${\bf x}_{1}$, ${\bf x}_{2}$ and ${\bf
x}_{6}$ fall into one object, ${\bf x}_{4}$ and ${\bf x}_{5}$ fall into a
different object, and ${\bf x}_{3}$ is alone in a third object (Figure 1C).
Then the corresponding object membership function can be expressed as
${\text{\cjRL{m|}}}=\\{\\{{\bf x}_{1},{\bf x}_{2},{\bf x}_{6}\\},\\{{\bf
x}_{3}\\},\\{{\bf x}_{4},{\bf x}_{5}\\}\\}$, or abbreviated as
${\text{\cjRL{m|}}}=\\{126|3|45\\}$.
The object membership function does not contain information directly about the
intensities, but only about which pixels are correlated. Given a particular
object membership m— for some selected pixels, the probability distribution
$P({\bf I}|{\text{\cjRL{m|}}})$ of image intensities ${\bf I}$ factorizes into
a product over objects: The different object textures are independent, and
hence so are their respective pixels. In the above example, the probability
distribution of intensities at those six pixels would be $P({\bf
I}|{\text{\cjRL{m|}}})=P(I_{1},I_{2},I_{6}|{\text{\cjRL{m|}}})P(I_{3}|{\text{\cjRL{m|}}})P(I_{4},P_{5}|{\text{\cjRL{m|}}})$.
In general,
$P({\bf I}|{\text{\cjRL{m|}}})=\prod_{n=1}^{|{\text{\cjRL{m|}}}|}P({\bf
I}_{{\text{\cjRL{m|}}}_{n}}|{\text{\cjRL{m|}}})$ (1)
where $|{\text{\cjRL{m|}}}|$ is the number of objects,
${\text{\cjRL{m|}}}_{n}$ is the set of pixels falling into the $n$th object of
m—, and ${\bf I}_{{\text{\cjRL{m|}}}_{n}}$ is a vector of intensities at those
pixels. The factors $P({\bf I}_{{\text{\cjRL{m|}}}_{n}}|{\text{\cjRL{m|}}})$
reflect the joint probabilities of intensities in a single, textured object.
This formulation requires that we specify a texture model to provide these
probabilities. For concreteness we use a simple gaussian white noise texture
superposed on a uniform intensity (Methods), though any other probabilistic
texture model could be used instead. Note that the texture model is wholely
unrelated to the geometrical aspects of the dead leaves model.
If the geometric configuration of objects is not known, then the joint
distribution of intensities $P({\bf I})$ is an average over all possible
configurations. The factorized conditional distributions of Equation 1 are
then combined in the weighted sum
$P({\bf I})=\sum_{\text{\cjRL{m|}}}P({\bf
I}|{\text{\cjRL{m|}}})P({\text{\cjRL{m|}}})$ (2)
This is a mixture distribution in which each mixture component $P({\bf
I}|{\text{\cjRL{m|}}})$ has a distinct correlation structure amongst pixels,
induced by the different object membership functions. The weighting
coefficients are object membership probabilities $P({\text{\cjRL{m|}}})$, i.e.
the probability of observing the corresponding memberships over all possible
dead leaves images with a given shape ensemble. Figure S1 shows examples of
simple mixture distributions.
The object membership probability $P({\text{\cjRL{m|}}})$ represents how
frequently the $N$ selected pixels are grouped into different objects
according to m—. We calculate each probability recursively, generalizing an
argument of [11]. To do so, we must introduce some additional notation. We
designate ${\text{\cjRL{m|}}}_{\setminus n}$ as the object membership function
that remains after removing the $n$th object. We also define a boolean vector
${\bm{\sigma}}({\text{\cjRL{m|}}},n)$ with $N$ components
$\sigma_{i}({\text{\cjRL{m|}}},n)=({\bf x}_{i}\in{\text{\cjRL{m|}}}_{n})$ that
each indicate whether the pixel ${\bf x}_{i}$ is contained in the $n$th object
of m—. For instance, ${\bm{\sigma}}(\\{126|3|45\\},3)=(0,0,0,1,1,0)$.
By construction, there is a sequence of objects in any dead leaves image,
ordered by depth. Consider only the topmost object. There is some probability,
which we will denote by $Q_{{\bm{\sigma}}({\text{\cjRL{m|}}},n)}$, that this
top object includes all of the pixels in the set ${\text{\cjRL{m|}}}_{n}$,
while excluding all the other pixels in ${\text{\cjRL{m|}}}_{\setminus n}$.
Such an arrangement partially satisfies the membership constraint imposed by
m—. But for this object configuration to contribute to
$P({\text{\cjRL{m|}}})$, we still need to ensure that the excluded pixels are
also grouped appropriately by objects ‘deeper’ in the image. The probability
that deeper objects satisfy these reduced membership constraints is
$P({{\text{\cjRL{m|}}}_{\setminus n}})$. Note that this probability is
unaffected by whether the deeper objects would have enclosed the pixels in
${\text{\cjRL{m|}}}_{n}$: Objects at those positions are already occluded by
the top object. There is also a probability
$Q_{{\bm{\sigma}}({\text{\cjRL{m|}}},0)}$ that the top object contains none of
the $N$ selected pixels. Given this event, the probability of finding objects
deeper in the stack that satisfy the membership constraints is just the
original factor $P({\text{\cjRL{m|}}})$. Summing together all possibilities
for the top object, we find
$P({\text{\cjRL{m|}}})=Q_{{\bm{\sigma}}({\text{\cjRL{m|}}},0)}P({\text{\cjRL{m|}}})+\sum_{n=1}^{|{\text{\cjRL{m|}}}|}Q_{{\bm{\sigma}}({\text{\cjRL{m|}}},n)}P({{\text{\cjRL{m|}}}_{\setminus
n}})$. Solving for $P({\text{\cjRL{m|}}})$ gives the recursion relation
$P({\text{\cjRL{m|}}})=\frac{1}{1-Q_{{\bm{\sigma}}({\text{\cjRL{m|}}},0)}}\sum_{n=1}^{|{\text{\cjRL{m|}}}|}Q_{{\bm{\sigma}}({\text{\cjRL{m|}}},n)}P({{\text{\cjRL{m|}}}_{\setminus
n}})$ (3)
Crucially, the image that remains below the top object is yet another dead
leaves image, with all the same statistical properties as before, so we can
calculate $P({{\text{\cjRL{m|}}}_{\setminus n}})$ by the same formula,
recursively. Eventually the recursion terminates when there are no pixels left
in m—, with $P(\emptyset)=1$.
This recursive equation applies universally to any dead leaves model with
independent, occluding objects, regardless of shape. In contrast, the factors
$Q_{{\bm{\sigma}}({\text{\cjRL{m|}}},n)}$ depend on the particular shape
ensemble and the chosen set of pixels. In the Methods section we derive the
general form of these factors for arbitrary shapes with smooth boundaries. The
Supporting Information (Text S1) provides mathematical details of the
calculation for the scale-invariant ensemble of elliptical shapes used from
here onward.
The number of possible object membership functions quickly grows large as we
consider more pixels. The limiting step is the number of possible object
membership functions, known as Bell’s number $B_{N}$, which unfortunately
grows slightly faster than exponentially. In practice this restricts exact
calculation to around a dozen pixels. Despite this limitation, interesting
insights can be gained both by using few pixels or few subsets of possible
object memberships, and by analyzing the general behavior in various limits.
For instance, in low-clutter conditions when the maximal distance between
pixels $u$ is much smaller than the minimum object size $r_{-}$ (e.g. Figure
1A), object membership probabilities $P({\text{\cjRL{m|}}})$ behave as
$P({\text{\cjRL{m|}}})\sim(u/r_{-})^{|{\text{\cjRL{m|}}}|-1}$ (Figure S2).
Consequently, edges are rare ($|{\text{\cjRL{m|}}}_{\rm edge}|=2$),
T-junctions are rarer ($|{\text{\cjRL{m|}}}_{\rm T-junc}|=3$), and every other
feature is rarer still ($|{\text{\cjRL{m|}}}|>3$). In the sections below we
use the solution of the dead leaves model and relevant approximations to
explain complex statistical properties of natural scenes.
### 2.2 Feature distributions
In this section we calculate the joint feature probabilities in specific cases
where features are linear functions of the pixel intensities, $f=F{\bf I}$ for
some filter matrix $F$. We set the filters of $F$ to be wavelets, local
derivative operators (edge detectors) with a given orientation and scale.
Choosing Haar wavelets, which weight intensities by $\pm 1$, emphasizes non-
gaussianity of feature distributions and thereby establishes a more stringent
test for the image model [12].
It has been previously reported that empirical histograms of different Haar
wavelets and wavelet pairs in the dead leaves model qualitatively reproduce
the marginal and joint distributions in natural scenes [12]. Where empirical
sampling can, at best, expose these interesting statistical similarities, our
analytical results let us understand their origins.
#### 2.2.1 Marginal distributions of wavelet coefficients
One well-described feature of natural images is that the distribution of
spatial derivatives $\Delta$ has heavy tails (Figure 2A) well approximated as
a generalized Laplace distribution $P(\Delta)\propto e^{-|\Delta|^{\beta}}$
for an exponent $\beta$ near 1 [16, 17, 6, 12]. The heavy tails in these
distributions cannot be obtained from a standard correlated gaussian model,
because any projection of a multidimensional gaussian is again gaussian.
Higher-order statistical structure is required.
This distribution can be calculated exactly for the dead leaves model by
representing the local derivative by a simple feature: the intensity
difference between nearby points, $f=I_{1}-I_{2}$. The resultant feature
distribution is a mixture of two components, a narrow central peak and a
broader tail (Figure 2B,E). While this is a more kurtotic distribution than
the gaussian texture, it does not closely match natural derivative histograms
(Figure 2A).
A simple consideration can account for the discrepancy. In our solution of the
dead leaves model, what we have described so far as pixels are actually
samples at infinitesimal points. In contrast, pixels in natural images
represent light accumulated over some finite sensor area set by film grain,
camera sensor wells, or photoreceptor cross-sections. This means that measured
pixel values don’t directly reflect an intensity sampled from an object but
instead reflect integrals over unresolved sub-pixel details. When many samples
are summed over some region $X_{i}$, one might naïvely expect the total
$\bar{I}_{i}=\sum_{j:{\bf x}_{j}\in X_{i}}I_{j}$ to be gaussianly distributed.
However, the usual central limit theorem does not apply, because of the
correlations between the variables that is induced by spatially extended
objects. These correlations can be segregated by re-expressing the total
intensity in an image patch as a sum of the mean intensities in visible
objects, weighted by their visible areas,
$\bar{I}_{i}=\sum_{n=1}^{|{\text{\cjRL{m|}}}|}|{\text{\cjRL{m|}}}_{n}|J_{n}$
where $|{\text{\cjRL{m|}}}_{n}|$ is the visible area of the $n$th object and
$J_{n}$ is the average intensity in that area. The summands
$|{\text{\cjRL{m|}}}_{n}|J_{n}$ are approximately independent because they
correspond to different objects. (They are not strictly independent because
the visible areas $|{\text{\cjRL{m|}}}_{n}|$ are constrained to add up to the
total area of the patch.) This way of writing $\bar{I}_{i}$ reveals two
reasons why the sum does not converge to a normal distribution: the number of
summands is a random variable, and the summands themselves have long-tailed
distributions.
Scale invariance demands that the areas of homogeneous regions
$|{\text{\cjRL{m|}}}_{n}|$ be distributed as a power law with exponent 2 [21].
If the mean intensity within an object, $J_{n}$, is distributed more narrowly
than this, then the distribution of visible areas $|{\text{\cjRL{m|}}}_{n}|$
will dominate the tail behavior of the products
$|{\text{\cjRL{m|}}}_{n}|J_{n}$. $\bar{I}_{i}$ is thus a sum over a random
number (one per visible object) of power-law distributed terms. A generalized
central limit theorem holds that the distribution of such random sums is a
two-sided exponential distribution when summands are power-law distributed
with exponents of 2 or higher ([22], Figure 2D).111Technically, this theorem
requires a geometric distribution for the number of summands. The observed
distribution of the number of visible objects is not geometric, but it is
similarly broad with a width of the same order as its mean, so a similar
result should hold. Indeed, Figure 2D shows nearly straight tails on the log-
probability plot. In natural images, areas of homogeneous regions are again
distributed as power laws, but depending on the particular image the exponents
can be slightly below 2 [21]. Another generalized central limit theorem shows
that under such conditions the distribution of the sum has slightly heavier
tails [23], as observed (Figure 2A). While these considerations apply directly
to the average pixel intensities, they pertain equally to intensity
differences.
We can visualize how the heavy tails emerge by plotting feature distributions
conditioned on various object configurations. When many different objects are
visible, the independent object intensities tend to average out giving a
narrow distribution; when few objects are visible, their areas are large, so
the few object intensities are heavily weighted and their distribution is
proportionately broad (Figure 2F). Components with this spectrum of
distribution widths all combine to give the mixture distribution heavy tails
(Figure 2B–D).
Figure 2: Log-probability feature distributions $\log{P(f)}$ of spatial
derivatives $f$. (A) Empirically sampled distribution of derivatives (depicted
graphically, inset) in natural images. (B, C) Feature probabilities calculated
exactly for dead leaves images, using just one and two samples per image patch
respectively (inset). (D) Empirically sampled distributions for dead leaves
images using a 16$\times$16 grid of samples per patch (inset). (E, F) Mixture
components $P(f|{\text{\cjRL{m|}}})$ corresponding to panels B and C
respectively.
#### 2.2.2 Joint distributions of wavelet coefficients
Within natural images, the feature distribution for two orthogonal,
colocalized wavelets has diamond-shaped contours (Figure 3A). Densely sampled
dead leaves images reveal the same diamond contours (Figure 3B); they are
already visible when features are represented sparsely (Figure S3A,B). For
both natural images and dead leaves images, the distinctive non-gaussian
structure is most visible for contours at large feature amplitude. In this
limit, the mixture components with greatest likelihood dominate the
distribution, and the most likely component at high amplitudes is the one with
the greatest variance in the given feature direction. The greatest variance
for a single Haar wavelet occurs when a boundary between two objects aligns
with the boundary between oppositely-signed lobes, because that minimizes
cancellation and maximizes the overlap each object makes with each lobe.
However, this arrangement gives a minimal variance for the orthogonal wavelet
at the same location. As the object boundary rotates (Figure 3C), the overlap
with one wavelet reduces by exactly the amount that the overlap increases for
the orthogonal wavelet. Since a mixture component’s width is proportional to
the overlap, this perfect trade-off gives the maximum likelihood contours the
observed diamond shape (Methods, Figure 3D).222The slight squashing of the
diamond shape seen for natural images is a consequence of gravity: in this
natural image database there are more vertical contours than horizontal ones.
Here the dead leaves model does not show this asymmetry because it is
isotropic by construction.
For neighboring Haar wavelets with the same orientation, there is once again a
remarkable similarity between pillow-shaped distributions measured for natural
images and dead leaves images (Figure 3E,F). These too can be explained using
simple arguments about the object geometry that dominates at high feature
amplitudes. Negatively correlated mixture components occur when an object
overlaps neighboring lobes on neighboring filters (Figure 3G). Positively
correlated components occur when an object covers the same-signed lobes of
both Haar wavelets without cancellation by the intervening lobe of opposite
sign. This can only happen if a small object occludes that oppositely signed
lobe (Figure 3H). Since a very limited variety of sizes and positions can
accomodate this configuration, the positively correlated mixture components
have much lower weights. Figure 3I shows mixture components with negative
correlations, from which emerge the basic ‘pillow’ shape of the full bivariate
feature distributions (Methods).
Figure 3: Joint feature probabilities $\log{P(f_{1},f_{2})}$ for wavelet pairs
$f_{1}$ and $f_{2}$. For orthogonal, colocalized Haar wavelets (inset of A,
shifted for visibility), the contours of the empirically sampled bivariate
distribution are diamond shaped for both natural images (A) and dead leaves
images (B). At high feature amplitudes, certain object configurations have the
greatest likelihood and thus dominate the joint distribution. Panel C
illustrates one such configuration. Colors indicate different objects with
unspecified intensities. Dark and light shading show how the two wavelets
weight the image pixels. (D) Specifying only the object geometry (but not the
object intensities) gives conditional feature distributions
$P(f_{1},f_{2}|{\text{\cjRL{m|}}})$ that are bivariate gaussians with
elliptical contours. For the conditional distributions that dominate at high
feature amplitude, the contours trace a diamond-shaped envelope (thick curve)
as a function of the relative angles between the object boundary and wavelet
orientations (Text 4.3). Parallel, neighboring wavelets (inset of E) are
anticorrelated, with joint probability contours exhibiting a similar ‘pillow’
shape in natural images (E) and dead leaves images (F). Panels G and H
illustrate object configurations that dominate at large feature amplitudes,
colored as in C. (G) If one object covers the opposite-sign lobes of
neighboring wavelets while others prevent cancellation by the negative lobe,
then the conditional feature distribution will have a negative correlation.
(H) Similarly, if one object covers the same-signed lobes of both features
while another object prevents cancellation, then the conditional distribution
will have a positive correlation. Configurations like this are much less
probable than those like G, because the middle object must have precisely the
right size and position. (I) An ensemble of configurations like Panel G
produce negatively correlated components (gray ellipses) that vary depending
on how precisely the objects cover the feature lobes. The positively
correlated components (dashed ellipse) caused by configurations like Panel H
are many times less probable. Discounting the latter gives the mixture
distribution an overall negative correlation, leaving components that trace
out the pillow-shaped envelope (thick curve) seen in feature distribution
contours (Methods).
#### 2.2.3 Conditional distributions of wavelet coefficients
Wavelet coefficients in natural scenes may be nearly decorrelated to second
order yet still have a strong statistical dependency taking the form of a
‘bowtie’-shaped distribution of one filter coefficient conditioned upon
another (Figure 4A) [18, 19]. The dead leaves model reproduces this behavior
(Figure 4B), and allows us to interpret it as well.
The distribution of intensities found within an object is narrower than the
intensity distribution averaged over all objects. Consequently, when a wavelet
filter lies across an object boundary, it typically yields a larger magnitude
than the same filter applied wholely within a single object. Since object
boundaries tend to extend across space, a second filter with different scale
or orientation has an elevated probability of encountering the same edge.
However, as Figs. 4C–D illustrate, the relative sign and magnitude of the two
feature amplitudes depends on how the object boundary overlaps the second
filter. In this symmetric example, positive and negative feature amplitudes
are equally probable, so the conditional distribution $P(f_{2}|f_{1})$
broadens with $|f_{1}|$ without any change in the mean (Figure 4E). This
explains why the variability in one feature amplitude increases with the
amplitude of a nearby feature.
Figure 4: ‘Bowtie’ shapes appear in empirically sampled conditional feature
distributions $P(f_{2}|f_{1})$ for both natural images (A) and dead leaves
images (B). Horizontal and vertical axes represent the coefficients of two
neighboring, orthogonal Haar wavelet filters, $f_{1}$ and $f_{2}$ (inset of
A). The grayscale is normalized so black represents 0 and white is the maximum
probability for a given $f_{1}$. (C, D) Two equally probable object
configurations, colored as in Figure 3C, have identical $f_{1}$ but opposite
$f_{2}$. Both features are proportional to the intensity difference between
foreground and background objects. (E) Conditional feature distribution with
only four samples per feature (inset). Traces of the limited sampling appear
as the faint diagonal bands passing through the origin (highlighted with
dotted lines on right half). Each distinct band corresponds to a conditional
distribution given a different object membership function,
$P(f_{2}|f_{1},{\text{\cjRL{m|}}})$. Symmetry ensures that there will be no
linear correlation between the two features, even as the width of
$P(f_{2}|f_{1})$ increases with $|f_{1}|$. With features sampled more densely,
more such diagonal bands appear, until the bands blend together (B). This
produces the distinctive bowtie shape in the conditional feature
distributions.
### 2.3 Shared causes of edges
A major advantage of using the dead leaves model is that the causes of image
features — objects and their attributes — are represented explicitly. Our
results relate these causes to each other as well as to the observable, pixel-
based image features.
In natural images, edge pairs tend to fall tangent to circles passing through
both edge locations [24, 25]. Geisler et al. [20, 26] augmented such an
analysis with global information about physical contours, by laboriously hand-
segmenting objects within many images of foliage. The likelihood that two
edges share a physical cause (Figure 5A) — i.e. belong to the same contour —
were highly predictive of human judgements of whether the edges had a shared
cause.
The dead leaves model can provide a mathematical ‘ground truth’ for such
calculations. First, we represent individual edge features by an object
membership function that divides four pixels into two pairs (Figure S4A).
Second, we define the conditions under which a pair of edges have the same
physical cause. Third, for edge pairs with various geometrical relationships
(Figure S4B) we plot the likelihood ratio under the hypotheses of a shared
cause versus different causes (Methods).
A seemingly natural condition would identify a shared cause when there exists
an object that participates in both edges. The resultant likelihood ratio
always favors a shared cause, for all relative positions and orientations of
the edge pair (Figure 5B), at odds with reported statistics (Figure 5A) [20,
26]. The reason can be seen in Figure 5C: An object could be shared across two
edges simply if it is a common background for two distinct objects. Thus this
definition, only involving object identity on both sides of an edge, is
inadequate to reproduce the observed edge statistics.
A more sensible pattern emerges by modifying the definition of common cause to
include relative depth, assigning ‘border ownership’ [27] to the local edge.
We now define a common cause to exist when a single object participates in
both edges, and is closer to the viewer than the other objects seen at these
edges. An example of this configuration is seen in Figure 5E, which agrees
with our intuition about a shared cause for two edges. Application of this
definition requires that the object membership function be augmented to
include the objects’ relative depths, yielding an ordered object membership
function. Their probabilities can be calculated by a very similar recursion
equation as that used for the unordered variant (Methods). With this
definition, Figure 5D shows that certain edges are more likely to have a
common cause, whereas other edges are more likely to be independent. The
pattern closely resembles results of Geisler et al. [20, 26] (Figure 5A).
Since those statistics were predictive of human judgments about contour
completion across occluders, therefore the dead leaves model also
qualitatively predicts human inference about such ambiguous stimuli.
Figure 5: Joint statistics of local edges and global contours. (A) The
likelihood ratio that edge pairs in natural images are caused by a common
object versus by different objects (replotted from [20] with permission). For
test edges at many distances, directions, and orientations relative to a
reference edge (horizontal bar at origin), line segments are colored to
indicate the likelihood ratio (Methods). The segments are sorted so those
indicating high likelihoods appear in front. Concentric white rings correspond
to unsampled distances. In the dead leaves model, we can define the
corresponding likelihood in one of two ways. First, a pair of edges could have
a ‘shared cause’ if at least one side of each edge samples from the same
object. The resultant likelihood is shown in (B) and an example of a shared
cause is shown in (C). Second, we may add a depth constraint to better
describe the existence of a shared contour: this shared object must also be on
top of the other objects. Using this second definition, panel (D) shows the
likelihoods and (E) gives an example configuration. These likelihoods
reproduce the observations made in natural images (A).
## 3 Discussion
Our study used an occlusion model to explain several distinctive statistical
regularities in natural images. The model describes images composed of many
independent, opaque objects. We solved this image model by deriving exact
probability distributions that relate arbitrary image features to each other
and to the depicted objects. By applying and analyzing this solution we were
able to account for several curious observations about image features,
summarized very briefly as follows. We saw that heavy-tailed feature
distributions are explained by integrating over sub-pixel details with scale-
invariant spatial structure (Figure 2). The diamond-shaped joint distribution
of orthogonal, colocalized wavelets occurs because edges aligned well with one
wavelet must be aligned poorly with an orthogonal wavelet (Figure 3). The
pillow-shaped joint distribution of parallel wavelets reflects the rarity with
which objects can induce positive correlation by squeezing precisely into one
wavelet lobe (Figure 3). Bowtie-shaped conditional distributions arise because
extended object boundaries can overlap wavelets with identical amplitudes but
opposite signs (Figure 4). Finally, accurately computing the likelihood that
two edges share a physical cause depends critically on ascribing relative
depth to the edges (Figure 5). The unifying idea is that seemingly complex
statistics of edge features can be explained by simple geometric
configurations of a few opaque objects.
These results were made possible by connecting image features to object
configurations through the object membership function m—. This representation
enables probability distributions to be decomposed into a mixture of simpler
distributions. The existence of a mixture distribution for the dead leaves
model was first proved in [28, 29]. Here we found an explicit solution for the
mixture components that yields concrete numbers used in the applications
above. Additionally, this solution generalizes to give probabilistic
relationships among all model variables (Section 4.5), including object
texture, size, shape, position, and depth. The ability to relate arbitrary
image features and many diverse object attributes in a principled manner is a
substantial advance over previous efforts.
Although occlusion is a ubiquitous and fundamental attribute of natural
scenes, it is not the sole process that could cause these effects. However,
our results should generalize to other processes that share crucial
attributes: only one physical cause dominates the image at each point, and
separate causes are drawn from a scale-invariant size distribution. As one
striking example, the cratered lunar surface appears remarkably similar to
dead leaves images [30]. Even though the causal process is entirely different
from occlusion, the essential properties are identical: New impacts locally
erase traces of previous impacts, and small craters are much more common than
large ones. Similar principles may approximate other physical processes as
well, such as those that determine surface composition or some three-
dimensional bump textures. The results presented here should pertain to
feature statistics caused by any such ‘exclusion’ process.
### 3.1 Beyond the dead leaves model
Despite the dead leaves model’s success at reproducing many complex natural
statistics, we expect some statistical differences also. Indeed, whereas
natural scenes appear reasonably gaussian after normalizing intensities by the
local standard deviation [17, 6], dead leaves images do not have this
property. This therefore excludes object boundaries as the cause of this
property, despite speculations to the contrary [31]. By extending the model in
various ways, one may hope to capture this and other natural image properties
and thereby reveal their underlying cause.
Most real objects have more elaborate shapes than the ellipses used in these
calculations. Notably, the most common edge configuration seen in natural
scenes is consistent with circular [24], elliptical or parabolic [26] arcs.
This accounts for why the elliptical object ensemble could reproduce
statistics of images populated by complex, natural objects. Incorporating more
complex objects may correct some minor discrepancies between the dead leaves
model and natural scenes.
The realism of the dead leaves model could be further improved by adding
correlations between model variables. For instance, light sources could be
modeled by modulating texture according to position within each object.
Rudimentary three-dimensional shape could be included using textures to
indicate object tilt [32]. Perspective could be modeled by covarying size with
depth. Binocular disparity could be included by generating image pairs in
which every object has a positional shift coupled to its depth. Images with
such improvements could be easily generated, but in some cases a new solution
for the enhanced model would be required.
### 3.2 Toward neural coding of natural scenes
Some perceptual tasks can be accurately modeled as inference based on simple
models of stimulus probabilities [33, 34, 35, 36, 37]. Human perception of
images appears biased toward statistically probable features of the dead
leaves model. For example, empirical edge statistics predict psychophysical
judgments about whether two edges have a common cause [20], and the dead
leaves model reproduces these statistics. Artificial neural networks trained
on dead leaves images make systematic interpretation errors that are
consistent with illusory percepts in humans [38]. Such evidence hints that
these percepts might result from perceptual inference using probabilities
described by the dead leaves model.
On a more mechanistic level, some electrophysiological recordings of
individual neurons in animal cortex appear consistent with a probabilistic
weighing of sense data [39, 40, 41]. We might speculate that some cortical
neurons could be tuned to encode feature probabilities. For instance, V1
complex cells are excited by edges irrespective of polarity and precise
location of those edges [42], and are especially sensitive to phase alignment
caused frequently by object boundaries in natural images [43]. We might
therefore wish to describe a rudimentary complex cell as encoding the
probability that an edge passes through two points in its receptive field,
irrespective of which side of the edge is brighter. In our formalism, this
corresponds to an object membership function ${\text{\cjRL{m|}}}_{\rm
edge}=\\{1|2\\}$. Assuming that objects have gaussian-distribution intensities
and the image sensors have some additive gaussian noise, the probability of an
edge given the intensity difference $\Delta$ across space is
$P({\text{\cjRL{m|}}}_{\rm
edge}|\Delta)=\left[1+k\exp{\left(-\beta\Delta^{2}\right)}\right]^{-1}$, where
$k$ and $\beta$ are positive constants that depend on the spatial scale,
overall image contrast, and sensor noise. This function resembles the
contrast-energy model of complex cells [44] with a saturating nonlinearity.
Thus we might interpret complex cell activity as encoding the probability of a
local edge in a world of objects. It will be interesting to explore such a
model more thoroughly, and to see if other neurons have properties that map
nicely onto representations of still more complex features within the dead
leaves model. Since synaptic connections are modified by neural correlations,
and the occlusion model explains stimulus correlations, therefore the model
may also help generate predictions about cortical circuitry that has matured
in the natural world.
In vision science, progress has been made by finding stimuli appropriate for
the area of study [45]. The best stimulus is one that contains a rich
repertory of the right kinds of features, while limiting extraneous detail.
Since the dead leaves model shares many low- and mid-complexity features with
the natural environment while simplifying some higher-level features, it seems
like an especially good stimulus to use in experiments that probe the
mechanisms of low- and mid-level vision. It strikes a good balance between
tractability, accuracy, and richness, by isolating two causes of image
features which must be disambiguated to interpret truly natural scenes:
occlusion and texture. The availability of an exact solution for the relevant
probabilities is a promising new ingredient for experimental and theoretical
studies of visual function.
## 4 Methods
### 4.1 Dead leaves membership probabilities
Equation 3 expresses the object membership probabilities
$P({\text{\cjRL{m|}}})$ in terms of some geometric factors
$Q_{{\bm{\sigma}}({\text{\cjRL{m|}}},n)}$. These factors represent the
probability that points ${\bf x}_{i}\in{\text{\cjRL{m|}}}_{n}$ are included in
one object while the other points ${\bf x}_{i}\in{\text{\cjRL{m|}}}_{\setminus
n}$ are not, averaged over all object positions and shapes. For convenience,
we name these $Q_{\bm{\sigma}}$ ‘inclusion probabilities’. Note that these
quantities involve the geometry of single objects only; the recursion of
Equation 3 converts them into the multi-object probabilities
$P_{\text{\cjRL{m|}}}$ that characterize the dead leaves model geometry. In
this section we show how the inclusion probabilities can be calculated for
arbitrary objects.
We begin by specifying a shape through a ‘leaf’ function $L_{\sigma}({\bf
x},\rho)$, which is an indicator function over space ${\bf x}$ and shape
parameter(s) $\rho$. The function can indicate either the inside or the
outside of an object centered on the origin, depending on the binary variable
$\sigma\in\\{0,1\\}$: $L_{\sigma}({\bf x},\rho)$ equals $\sigma$ when pixel
${\bf x}$ is inside the object and $1-\sigma$ when ${\bf x}$ is outside it
(Figure S5A). With this definition,
$Q_{{\bm{\sigma}}({\text{\cjRL{m|}}},n)}({\bf
c},\rho)=\prod_{i=1}^{N}L_{\sigma_{i}({\text{\cjRL{m|}}},n)}({\bf x}_{i}-{\bf
c},\rho)$
is the inclusion probability that a leaf with shape $\rho$ and location ${\bf
c}$ includes all sample points ${\bf x}_{i}\in{\text{\cjRL{m|}}}_{n}$ and
excludes all remaining ${\bf x}_{i}\in{\text{\cjRL{m|}}}_{\setminus n}$
(Figure S5B).
The inclusion probabilities $Q_{{\bm{\sigma}}}$ in Equation 3 are averages
over all possible object shapes and positions. Thus we are interested in the
average of Equation $Q_{{\bm{\sigma}}}({\bf c},\rho)$ over the distribution of
leaf positions $P({\bf c})$ and shapes $P(\rho)$:
$Q_{{\bm{\sigma}}}=\int d\rho\,P(\rho)Q_{{\bm{\sigma}}}(\rho)=\iint
d\rho\,d{\bf c}\,P(\rho)P({\bf c})Q_{{\bm{\sigma}}}({\bf c},\rho)$
We first perform the average over object positions ${\bf c}$ to obtain
$Q_{{\bm{\sigma}}}(\rho)$, and subsequently calculate the average over object
shape $\rho$.
In the dead leaves model, objects are distributed with uniform probability
across space. For simplicity we also assume wraparound boundary conditions and
with no loss of generality require that no object is larger than the image to
avoid self-intersections. (We can allow larger objects by choosing a small
window into the dead leaves world to represent our image; objects may be
larger than the window but smaller than the entire model world.) By scaling
distance so the image has unit area, we have $P({\bf c})=1$ and the ${\bf
c}$-integral of binary-valued $Q_{{\bm{\sigma}}}({\bf c},\rho)$ gives the
inclusion probabilities for a given $\rho$ as the areas of the regions with
constant $Q_{{\bm{\sigma}}}({\bf c},\rho)$.
Direct integration is not straightforward even for simple object shapes
because these regions generally have complicated two-dimensional limits and
may not even be simply connected. However, using the divergence theorem we can
transform this area integral into a simpler contour integral that follows
object boundaries piecewise. The vector field ${\bf V}=\frac{1}{2}{\bf c}$ has
divergence (in ${\bf c}$-space) of $\nabla\cdot{\bf V}=1$, so integrating this
divergence over the desired region gives the enclosed area. The divergence
theorem says that this integral equals the flux of $\bf V$ across the region
boundary:
$Q_{{\bm{\sigma}}}(\rho)=\int d{\bf c}\,P({\bf c})Q_{{\bm{\sigma}}}({\bf
c},\rho)=\int\limits_{C}\nabla\cdot{\bf V}\,d{\bf c}=\oint\limits_{\partial
C}{\bf V}\cdot\hat{\bf n}\,ds$ (4)
where $C$ is the region in ${\bf c}$-space where $Q_{{\bm{\sigma}}}({\bf
c},\rho)=1$, $\partial C$ is its boundary, $\hat{\bf n}$ is the unit normal
vector to the boundary, and $ds$ is the arclength. The boundary is composed of
piecewise smooth segments of the object outline centered on the sample points
${\bf x}_{i}$ (Figure S5B). We index the relevant segments by $m\in M$, and
represent the curves by ${\bf s}_{m}(t):t^{\prime}_{m}<t<t^{\prime\prime}_{m}$
for $t$ between the cusps at which the contour changes direction abruptly. The
integral along each segment is then
$A_{m}=\frac{1}{2}\int_{t^{\prime}_{m}}^{t^{\prime\prime}_{m}}{\bf
s}_{m}(t)\cdot\hat{{\bf n}}_{m}(t)\,ds$ (5)
and the complete contour integral is a sum over segments
$Q_{{\bm{\sigma}}}(\rho)=\sum_{m\in M}A_{m}$.
To average $Q_{{\bm{\sigma}}}(\rho)$ over the shape ensemble $P(\rho)$ we need
to compute $\int Q_{{\bm{\sigma}}}(\rho)P(\rho)d\rho$. Note that the set of
piecewise smooth segments composing the contour $\partial C$ may change
depending on $\rho$, so the $\rho$-integral must itself be done piecewise. We
define an index $\ell$ specifying the regions $R_{\ell}$ in $\rho$-space where
a given set of segments $M_{\ell}$ compose the contour. Within $R_{\ell}$ the
integral over $\rho$ can then be carried out on each summand $A_{m}$
separately, yielding
$Q_{\bm{\sigma}}=\sum_{\ell}\sum_{m\in M_{\ell}}\ \int\limits_{R_{\ell}}d\rho\
P(\rho)A_{m}(\rho)$
Carrying out this calculation explicitly, not just formally, requires some
careful geometry. In the Supporting Information we complete these calculations
for an ensemble of elliptical objects with an inverse-cube power-law
distribution of sizes (Text S1). In principle it is also possible to calculate
all these probabilities exactly for various other shape ensembles with simple
boundaries such as polygons, or compound objects comprising multiple circles.
Other size ensembles can also be used. The mathematical techniques required to
complete the calculations are essentially the same.
For the figures presented in this paper, all objects were ellipses with
uniformly distributed eccentricities between 1 and 4, uniformly distributed
orientations, and an inverse-cube size distribution with upper and lower
bounds $r_{+}=100$ and $r_{-}=1$. For Figs. 2–4, we used high-clutter
conditions by setting the pixel spacing to $5r_{-}$. For Figure 5, to
replicate the relatively low-clutter conditions under which the natural image
statistics were measured empirically [20], we chose the pixel spacing to be
$r_{-}/5$.
### 4.2 Intensity and feature distributions
For simplicity we assume that every object has a constant gaussian-distributed
mean intensity and an additive gaussian white noise textural modulation with
variances $\Xi_{0}$ and $\Xi_{1}$. For this texture ensemble, the conditional
distribution of pixel intensities is $P({\bf
I}|{\text{\cjRL{m|}}})\propto\exp{\left(-\frac{1}{2}{\bf
I}^{\top}C_{\text{\cjRL{m|}}}^{-1}{\bf I}\right)}$, with zero mean and
covariance
$\left(C_{\text{\cjRL{m|}}}\right)_{ij}=\Xi_{0}\sum_{n=1}^{|{\text{\cjRL{m|}}}|}\sigma_{i}({\text{\cjRL{m|}}},n)\sigma_{j}({\text{\cjRL{m|}}},n)+\Xi_{1}\delta_{ij}$.
In the results shown in this paper, $\Xi_{0}=1$ and $\Xi_{1}=0.01$.
For features specified as linear combinations of intensities by ${\bf f}=F{\bf
I}$, the conditional distribution is $P({\bf
f}|{\text{\cjRL{m|}}})\propto\exp{\left(-\frac{1}{2}{\bf
f}^{\top}(FC_{\text{\cjRL{m|}}}F^{\top})^{-1}{\bf f}\right)}$ and the joint
probability is the mixture distribution $P({\bf
f})=\sum_{\text{\cjRL{m|}}}P({\text{\cjRL{m|}}})P({\bf
f}|{\text{\cjRL{m|}}})$.
### 4.3 Averaging over image patches
Pixels in natural images are integrals of light intensity over a finite solid
angle. In the dead leaves model, we can approximate these spatial integrals by
summing over multiple points within an image patch $X_{i}$, defining
$\bar{I}_{i}=\sum_{j:{\bf x}_{j}\in X_{i}}I_{j}$
Using the white-noise texture model (Methods 4.2), the total intensity
$\bar{I}_{i}$ over an image patch has a conditional distribution
$P(\bar{I}_{i}|{\text{\cjRL{m|}}})$ which is gaussian with zero mean and
variance
$\sigma^{2}_{\bar{I}_{i}|{\text{\cjRL{m|}}}}=\sum_{jk}(C_{\text{\cjRL{m|}}})_{jk}=\sum_{n=1}^{|{\text{\cjRL{m|}}}|}|{\text{\cjRL{m|}}}_{n}|^{2}\Xi_{0}+N\Xi_{1}$
Here $C_{\text{\cjRL{m|}}}$ is the covariance matrix of all pixels in image
patch $X_{i}$ conditioned on the object membership function m—, and
$|{\text{\cjRL{m|}}}_{n}|$ is the number of sampled pixels falling into the
$n$th object. Thus the variance increases with the square of the sampled area
of each object, and is maximized when only one object covers the sampling
area.
A Haar wavelet takes the difference $H=\bar{I}_{1}-\bar{I}_{2}$ between sums
$\bar{I}_{1}$ and $\bar{I}_{2}$ over two distinct regions (Figure S6A). The
corresponding variance does not necessarily increase with the square of each
object’s sampled area, because some of the samples are weighted with opposite
signs and thus cancel. The conditional covariance between two Haar wavelets
$H_{i}$ and $H_{j}$ is
$C_{H_{i}H_{j}|{\text{\cjRL{m|}}}}=\sum_{n=1}^{|{\text{\cjRL{m|}}}|}\left(|{\text{\cjRL{m|}}}_{n}^{1,i}|-|{\text{\cjRL{m|}}}_{n}^{2,i}|\right)\left(|{\text{\cjRL{m|}}}_{n}^{1,j}|-|{\text{\cjRL{m|}}}_{n}^{2,j}|\right)\Xi_{0}+N_{ij}\Xi_{1}$
(6)
where $|{\text{\cjRL{m|}}}_{n}^{k,i}|$ is the number of samples in region $k$
of wavelet $i$ which fall into the $n$th object (Figure S6A), and $N_{ij}$ is
the number of samples shared by wavelets $H_{i}$ and $H_{j}$.
In Figure 3B,D, the diamond-shaped contours emerge as a consequence of
Equation 6. Instead of the Haar wavelets with square support shown in that
figure, it is simpler to understand the case with circular support (Figure
S6B), though the result is the same. The maximum amplitude features occur when
a single object boundary passes through the center of the wavelet at an angle
$\theta$. The covariance of the mixture distribution conditioned on this
object configuration is
$C_{HH|\theta}=2N^{2}\Xi_{0}\left(\begin{array}[]{cc}(\pi-2\theta)^{2}&2\theta(\pi-2\theta)\\\
2\theta(\pi-2\theta)&(2\theta)^{2}\\\ \end{array}\right)+N\Xi_{1}{\bf 1}$
where $N$ is the number of samples in each Haar wavelet. For large $N$ this
covariance matrix is nearly singular, with almost unity correlation
coefficient between the variations along $H_{1}$ and $H_{2}$. Contours of the
corresponding bivariate gaussian have maximum extent at feature amplitudes
proportional to $(\pm\theta,\pm(\frac{\pi}{2}-\theta))$. The envelope of these
contours produces the diamond shown in Figure 3B,D.
In Figure 3, two neighboring, parallel Haar wavelets have a joint distribution
with a distinctive ‘pillow’ shape. The dominant contributions at high feature
amplitudes involve three objects as depicted in Figure 3G, one covering the
left edge of the wavelet, a second one covering the right edge, and a third
covering the gap between them. We can approximate this arrangement with a one-
dimensional version, considering only the horizontal extent of objects (Figure
S6C,D). If we denote how much the leftmost and rightmost objects overlap the
wavelets by $d_{l}$ and $d_{r}$, then the covariance of the mixture
distribution is
$C_{HH|d_{l},d_{r}}=N^{2}\Xi_{0}\left(\begin{array}[]{cc}2\Delta_{l}^{2}&\pm\Delta_{l}\Delta_{r}\\\
\pm\Delta_{l}\Delta_{r}&2\Delta_{r}^{2}\\\ \end{array}\right)+N\Xi_{1}{\bf 1}$
where $\Delta_{i}=\min(d_{i},1-d_{i})$ and the width of each lobe of the Haar
wavelet is $1$. These components all have a correlation coefficient of nearly
$\pm 1/2$ but have different variances. By changing $d_{l}$ and $d_{r}$ for
the configuration shown in Figure S6C we obtain conditional distributions with
the ensemble of contours seen in Figure 3I. Their envelope produces the
‘pillow’ shape (Figure 3).
### 4.4 Shared causes of edges
To define oriented edges, we select four pixels arranged in a rectangle, and
select only those object membership functions that bisect these four pixels
into two pairs. Note that a range of object boundaries can produce such a
separation. Giving the rectangle an aspect ratio 2.75 constrains edges to an
allowed range of orientations $2\tan^{-1}{(1/2.75)}=40^{\circ}$ (Figure S4A)
that matches the orientation bandwidth of used in [20]. Pairs of edges are
described by two such bisected four-pixel clusters (Figure S4B). This
definition of edge pairs restricts these eight pixels to have one of only
seven possible object membership functions (Table 1A). In one of these
configurations, every pixel pair is a member of a different object:
${\text{\cjRL{m|}}}=\\{12|34|56|78\\}$. In the remaining configurations, at
least two pairs are members of the same object (Figure 5D). This latter
category serves as one possible definition of a ‘shared cause’ for the two
edges.
A second definition of shared cause invokes not just the object membership but
also the relative depth of the objects. In particular, we use ordered
membership functions M (Section 4.5), and we classify these M according to
whether a pair of pixels from each edge both falls into the same object and
that object is above the object present at the remaining pixels (Figure 5E).
The relevant M are listed in Table 1B.
With either definition, the likelihood ratio of shared cause to different
cause is $L={\sum_{{\text{\cjRL{m|}}}\in
S}P({\text{\cjRL{m|}}})}/{\sum_{{\text{\cjRL{m|}}}\in
D}P({\text{\cjRL{m|}}})}$, where $S$ and $D$ are the sets of membership
functions categorized as shared or different causes respectively. This
likelihood ratio varies as a function of the positions and relative
orientation of the two edge pairs (Figs. 5C–D).
A: Classification of unordered m—
---
S: Shared cause | D: Different causes
1256 $|$ 34 $|$ 78 | 12 $|$ 34 $|$ 56 $|$ 78
1278 $|$ 34 $|$ 56 |
12 $|$ 56 $|$ 3478 |
12 $|$ 78 $|$ 3456 |
1256 $|$ 3478 |
1278 $|$ 3456 |
B: Classification of ordered M
---
S: Shared cause | D: Different causes
1256 $>$ 34 $|$ 78 | 34 $|$ 78 $>$ 1256 | 34 $>$ 1256 $>$ 78 | 78 $>$ 1256 $>$ 34
3478 $>$ 12 $|$ 56 | 12 $|$ 56 $>$ 3478 | 12 $>$ 3478 $>$ 56 | 56 $>$ 3478 $>$ 12
1278 $>$ 34 $|$ 56 | 34 $|$ 56 $>$ 1278 | 34 $>$ 1278 $>$ 56 | 56 $>$ 1278 $>$ 34
3456 $>$ 12 $|$ 78 | 12 $|$ 78 $>$ 3456 | 12 $>$ 3456 $>$ 78 | 78 $>$ 3456 $>$ 12
1256 $|$ 3478 | | 12 $|$ 34 $|$ 56 $|$ 78 |
1278 $|$ 3456 | | |
Table 1: Object membership functions used for joint edge statistics. For
compactness we represent object membership functions by the pixel indices
divided symbolically into ordered or unordered groups. For example,
$\\{\\{{\bf x}_{1},{\bf x}_{2}\\},\\{{\bf x}_{3},{\bf x}_{4}\\}\\}$ is written
as $12|34$ if unordered, and as $12>34$ if ordered such that the object
containing points ${\bf x}_{1}$ and ${\bf x}_{2}$ lies above the object
containing ${\bf x}_{3}$ and ${\bf x}_{4}$. These object membership functions
are classified according to whether they reflect a shared cause or different
causes for the two edges, using unordered (A) or ordered (B) representations.
### 4.5 Generalizations
We can calculate the relative depth of objects by using an ordered object
membership function M rather than an unordered membership function m—. (The
Hebrew letter final mem M is used only at the end of a word, representing that
object order matters.) Whereas m— was a set of subsets, M is an ordered set of
subsets with ${\text{\cjRL{M}}}_{n}$ representing the pixels contained by the
$n$th-highest object sampled by any of the $N$ selected pixels. The recursion
in this case is even simpler than Equation 3:
$P({\text{\cjRL{M}}})=\frac{1}{1-Q_{\sigma({\text{\cjRL{M}}},0)}}Q_{\sigma({\text{\cjRL{M}}},1)}P({\text{\cjRL{M}}}_{\setminus
1})$
There is no summation here because there is only one term for which the first
object is highest in the stack of objects. One may use a partial ordering if
not all relative depths are of interest, and then there will be a sum over
arrangements consistent with the partial ordering.
Note that there are more hidden variables of interest besides the object
membership and relative depth, and the joint probabilities of these can be
calculated by a similar recursive formula, without marginalizing away the
hidden variables. The joint distribution of shape and membership, for
instance, can be calculated as
$P({\text{\cjRL{m|}}},\rho)=\frac{1}{1-Q_{{\bm{\sigma}}({\text{\cjRL{m|}}},0)}}\sum_{n=1}^{|{\text{\cjRL{m|}}}|}P(\rho_{n})Q_{{\bm{\sigma}}({\text{\cjRL{m|}}},n)}(\rho_{n})P({\text{\cjRL{m|}}}_{\setminus
n},\rho_{\setminus n})$
where $\rho$ is now a vector of $N$ shape parameters, with $\rho_{n}$
indicating the shape parameters for the topmost object present at pixel
location ${\bf x}_{n}$.
### 4.6 Empirical sampling of dead leaves and natural images
For probabilities involving many image points, we generate many dead leaves
images and empirically sample from them to obtain histograms. Images are
produced by layering objects from front to back until all image pixels are
members of some object, a process that yields stationary image statistics
[46].
Natural images were drawn from van Hateren’s image database [47]. Feature
distributions were obtained by log-transforming images [12], filtering them by
the relevant Haar wavelets, and computing univariate or bivariate histograms.
## Acknowledgments
The author thanks Ken Miller, Larry Abbott, Stefano Fusi, Taro Toyoizumi,
Vladimir Itskov, and Tony Movshon for helpful comments and suggestions. This
work was supported by the National Institute of Health Grant EY13933 and the
Swartz Foundation.
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##
Figure S1: Joint probabilities of pixel intensities, based on an ensemble of
elliptical objects and gaussian-distributed object intensities with an
additive gaussian white noise texture (Methods). Contour plots are shown for
two pixels (A) and three pixels arranged in an equilateral triangle (B). These
joint distributions are weighted averages of independent and correlated
distributions. The weighting factors are the various object membership
probabilities $P({\text{\cjRL{m|}}})$, which are plotted below the joint
intensity distributions as a function of the distance between pixels. Figure
S2: All object membership probabilities $P({\text{\cjRL{m|}}})$ as a function
of the spacing between the pixels, for four pixels arranged in a square
(right). Two different shape ensembles are shown, circles and ellipses, with
sizes given by $P(r)\propto r^{-3}$ for $r$ ranging between the limits $r_{-}$
and $r_{+}$ (dashed lines). Curves are labeled by their object membership
functions. Symmetrically permuted membership functions have identical curves.
For circles, the configuration $\\{14|23\\}$ is impossible, but otherwise the
curves for circles and ellipses are remarkably similar across all pixel
spacings, because both shape ensembles have similar local properties (extended
edges) and global structure (convex shapes with the same size distribution).
Since objects have sharp edges that closely spaced pixels rarely straddle,
nearby pixels almost always fall into the same object, with
$P({\\{1234\\}})\approx 1$. When pixel spacing exceeds the largest object
dimension, no two pixels can fall into the same object, so the only membership
function allowed is ${\text{\cjRL{m|}}}=\\{1|2|3|4\\}$. With pixel spacings
between these extremes, many more object membership probabilities take on
nonzero values. Figure S3: Mixture distributions and mixture components of
sparsely sampled Haar wavelet features, calculated exactly for dead leaves
images. (A,B) Contours of the log-probabilities $\log{P(f_{1},f_{2})}$ for
colocalized, orthogonal wavelets $f_{1}$ and $f_{2}$, sampled with four or
eight points per feature (insets). (C,D) Elliptical contours of jointly
gaussian mixture components $P(f_{1},f_{2}|{\text{\cjRL{m|}}})$, shaded
according to their weight $P({\text{\cjRL{m|}}})$. The mixture distributions
already have rounded diamond contours formed from weakly correlated
components, as well as some strongly correlated and anti-correlated components
which appear at all angles with dense sampling (Figure 3B). (E–H) Joint log-
probabilities and mixture components for nearby, parallel Haar wavelets,
plotted as in A–D. The anticorrelation and ‘pillow’ shape of these
distributions are already visible with sparse sampling of the features. Figure
S4: Detailed geometry for Figure 5. (A) An edge exists when an object splits
four pixels into two pairs. Pixels arranged in a rectangle with an aspect
ratio of $\Delta x/\Delta y=2.75$ permit a range of edges with a $40^{\circ}$
orientation bandwidth as used in [20]. (B) Pairs of edges thus defined are
related by three parameters: distance $d$, orientation difference $\theta$,
and relative direction $\phi$. Figure S5: Diagrams for illustrating inclusion
probabilities $Q_{{\bm{\sigma}}({\text{\cjRL{m|}}},n)}$. (A) A ‘leaf’ function
showing the shape of an object. $L_{1}({\bf x},\rho)=1$ at points ${\bf x}$
that are inside an object of shape parameter $\rho$, and $L_{0}({\bf
x},\rho)=1$ at points outside it. Here the shape parameter $\rho$ specifies a
smooth irregular object. (B) Example indicator functions
$Q_{{\bm{\sigma}}({\text{\cjRL{m|}}},n)}({\bf c},\rho)$ identify locations
${\bf c}$ where an object could be placed to enclose all pixels ${\bf
x}_{i}\in{\text{\cjRL{m|}}}_{n}$ and exclude the rest. Rotated copies of the
object shape surround each point ${\bf x}_{i}$, designating the locations
${\bf c}$ where that object will enclose ${\bf x}_{i}$. An arrow points to the
shaded region where an object could be placed to enclose both ${\bf x}_{1}$
and ${\bf x}_{3}$ but not ${\bf x}_{2}$, whose area is $Q_{101}(\rho)$. The
other shaded region indicates locations where an object would enclose only
${\bf x}_{2}$, whose area is $Q_{010}(\rho)$. Note that this diagram
represents possible locations ${\bf c}$ of a single object, not three objects!
Figure S6: Simplified representations of object configurations that dominate
feature distributions at high amplitudes. Colors indicate objects of
unspecified intensity, shading indicates weighting by Haar wavelets. (A) A
Haar wavelet $H_{i}$ takes a difference of intensities $\bar{I}_{1,i}$ and
$\bar{I}_{2,i}$ each totalled over a finite region. The pixels
${\text{\cjRL{m|}}}_{n}^{2,i}$ contained both in elliptical object
${\text{\cjRL{m|}}}_{n}$ and in region $2$ of wavelet $i$ are outlined. (B)
Colocalized, orthogonal Haar wavelets with circular support. (C, D) Parallel,
nearby Haar wavelets, with objects that induce negative and positive
correlations, respectively. To simplify the calculations, objects differ only
in their horizontal extent, and extend completely to either the left or right
edge of each wavelet. The relevant variable is then the width of the overlap
between the object and the wavelet filter, denoted $d_{l}$ and $d_{r}$.
Compare these simplified configurations to those shown in Figures 3C and 3G,H.
## S1 Inclusion probabilities for an ensemble of ellipses
In the main text we reported a universal recursion equation expressing object
membership probabilities $P({\text{\cjRL{m|}}})$ in terms of some geometric
factors $Q_{{\bm{\sigma}}({\text{\cjRL{m|}}},n)}$ which depend on the shape
ensemble. There we showed how these probabilities could be expressed
geometrically, by first averaging
$Q_{{\bm{\sigma}}({\text{\cjRL{m|}}},n)}({\bf c},\rho)$ over position ${\bf
c}$ via contour integrals, and then averaging over the shape ensemble $\rho$.
Here we explain in detail how inclusion probabilities
$Q_{{\bm{\sigma}}({\text{\cjRL{m|}}},n)}$ can be calculated exactly for an
ensemble of circular objects. We then use a simple transformation to
generalize the result for circles to an ensemble of ellipses. When the dust
settles, we will have averaged $Q_{\bm{\sigma}}({\bf c},\rho)$ over positions
${\bf c}$ and shapes $\rho$ and for all binary vectors ${\bm{\sigma}}$.
For an ensemble of circles, the shape parameter $\rho$ is just a radius $r$,
which we draw from a scale-invariant size distribution $P(r)\propto r^{-3}$.
Circular contours are easy to express analytically. However, as described in
the main text, the integrals of $Q_{{\bm{\sigma}}}({\bf c},r)$ over both the
contours and size ensemble are more difficult because they must be done
piecewise. We do this in two steps. First, we evaluate the general form of the
indefinite integrals at the endpoints of the piecewise intervals. Second, we
describe an algorithm that synthesizes these isolated contributions into the
complete piecewise integral, yielding the desired $Q_{{\bm{\sigma}}}$.
### S1.1 Parameterizing circular contours
Equation 4 related the positional average
$Q_{{\bm{\sigma}}({\text{\cjRL{m|}}},n)}(\rho)$ to the total area of the
region where $Q_{{\bm{\sigma}}({\text{\cjRL{m|}}},n)}({\bf c},\rho)=1$, and
thence to a contour integral. In this section we evaluate this contour
integral for circles with fixed radius, so that $\rho=r$. It is helpful to
change from the generic notation used in Section 4.1 to a notation which is
specific to circular objects. As shown in Figure S7A, the boundaries of
regions with constant $Q_{{\bm{\sigma}}({\text{\cjRL{m|}}},n)}({\bf c},r)$ are
all circular arcs centered on some point ${\bf x}_{i}$,
${\bf s}_{i}(t)=r{\hat{\bf e}}_{t}+{\bf x}_{i}$
with a unit vector defined as ${\hat{\bf e}}_{t}\equiv(\cos{t},\sin{t})$. Each
arc terminates at angles $t$ of the form
$t_{ij\pm}=\theta_{ij}\pm\phi_{ij}=\tan^{-1}{({\bf x}_{i}-{\bf
x}_{j})}\pm\cos^{-1}{(u_{ij}/2r)}$
where $\theta_{ij}$ is the angle of the line connecting the circle centers,
and $\pm\phi_{ij}$ are the angles that the intersection points make with that
line (Figure S7A). $\theta_{ij}$ is independent of $r$, whereas $\phi_{ij}$
depends on the ratio of $r$ to the distance $u_{ij}=|{\bf x}_{i}-{\bf x}_{j}|$
between the circles as $\phi_{ij}=\cos^{-1}{(u_{ij}/2r)}$.
### S1.2 Contour integration
Since the unit normal vectors are simply $\hat{{\bf n}}(t)={\hat{\bf e}}_{t}$
and the arc length is $ds=|\dot{{\bf s}}(t)|dt=r\,dt$, we can now easily
perform the contour integral (Equation 5) over each arc analytically.
$A_{m}=\frac{1}{2}\int_{t^{\prime}_{m}}^{t^{\prime\prime}_{m}}{{\bf
s}_{m}(t)\cdot\hat{{\bf
n}}(t)}\,ds=\frac{1}{2}\int_{t_{ij\pm}}^{t_{ik\pm}}\left(r^{2}+r{\bf
x}_{i}\cdot{\hat{\bf e}}_{t}\right)dt=a_{ik\pm}(r)-a_{ij\pm}(r)$
where we have defined
$\displaystyle a_{ij\pm}(r)$
$\displaystyle=\frac{1}{2}\left(r^{2}t_{ij\pm}+r{\bf x}_{i}\cdot{\hat{\bf
e}}_{t_{ij\pm}-\frac{\pi}{2}}\right)$
$\displaystyle=\frac{r^{2}\theta_{ij}}{2}\pm\frac{r^{2}}{2}\cos^{-1}{\frac{u_{ij}}{2r}}+\frac{u_{ij}}{4}{\bf
x}_{i}\cdot{\hat{\bf e}}_{\theta_{ij}-\frac{\pi}{2}}\pm\frac{r}{2}{\bf
x}_{i}\cdot{\hat{\bf e}}_{\theta_{ij}}\sqrt{1-\frac{u_{ij}^{2}}{4r^{2}}}$ (S1)
For $r$ smaller than the distances between pixels, the circular arcs do not
intersect and are thus complete circles with total area of $\pi r^{2}$, as
expected.
Figure S7: (A) Diagram depicting the quantities needed to calculate inclusion
probabilities $Q_{{\bm{\sigma}}({\text{\cjRL{m|}}},n)}(r)$. The different
regions of constant $Q_{{\bm{\sigma}}({\text{\cjRL{m|}}},n)}({\bf c},r)$ for
fixed $r$ are bounded by circular arcs centered on the pixels ${\bf x}$.
Highlighted is one particular arc ${\bf s}_{i}(t)$ centered on point ${\bf
x}_{i}$. This arc is bounded by $t_{ij-}$ and $t_{ik-}$, two angles at which
other circles intersect. Centers ${\bf x}_{i}$ and ${\bf x}_{j}$ are separated
by the distance $u_{ij}$ and angle $\theta_{ij}$. The location at which the
corresponding circles intersect deviates from the line connecting the centers
by angle $-\phi_{ij}$, so that $t_{ij-}=\theta_{ij}-\phi_{ij}$. (B)
Illustration of how the contours around regions with constant
$Q_{{\bm{\sigma}}({\text{\cjRL{m|}}},n)}({\bf c},r)$ change shape as $r$
increases (from left to right). Two regions in ${\bf c}$-space first touch
when $r$ equals half the distance between two pixels ${\bf x}_{i}$ and ${\bf
x}_{j}$, a critical radius $r^{*}_{ij}$ we call a ‘kissing point’ (center
panel). As $r$ increases further a new contour of reversed orientation is
created, bounding a region within which an object of radius $r$ can enclose
both pixels. (C) Similarly, a ‘triple intersection’ always exists for a
particular $r^{*}_{ijk}$, the circumradius, at which any three non-collinear
pixels ${\bf x}_{i}$, ${\bf x}_{j}$ and ${\bf x}_{k}$ are equidistant from a
fourth point called the circumcenter (center panel). As $r$ crosses this
critical radius, the existing contour connecting the three intersection points
changes orientation, and the enclosed region is associated with a different
$Q_{{\bm{\sigma}}({\text{\cjRL{m|}}},n)}$. (D,E) Illustrations of two
strategies for integrating $Q_{{\bm{\sigma}}}(r)$ over $r$: choose only one
region at a time, and track its contour as $r$ varies (D); or track all
contour endpoints over $r$ and add their contributions to all appropriate
regions (E). We use the latter strategy. Arrows in panel F depict the four
regions ${\bm{\sigma}}$ receiving identical contributions (up to a sign) from
the contours along ${\bf s}_{1}(t)$ that terminate at the intersection point
${\bf x}_{13-}$.
Finally we can obtain the total area
$Q_{{\bm{\sigma}}({\text{\cjRL{m|}}},n)}(r)$ by adding up the relevant
$a_{ij\pm}(r)$ appropriately. We defer discussion of this step to Section
S1.5.
### S1.3 Indefinite integral over radius
Next we have to average these quantities over the distribution of $r$. To
achieve scale invariance in generated images, the distribution of object radii
$P(r)$ should be proportional to $r^{-3}$ [12]. Some deviation from this
scaling behavior is required to prevent images from degenerating with high
probability into white noise or uniform coloring [12, 28]. Here we choose to
set upper and lower size cutoffs $r\in[r_{-},r_{+}]$ to satisfy this
constraint, so that
$P(r)=\begin{cases}\frac{1}{z}r^{-3}&r\in[r_{-},r_{+}]\\\
0&\rm{otherwise}\end{cases}$
with $z=\tfrac{1}{2}(r_{-}^{-2}-r_{+}^{-2})$.
The $r$-dependence of $a_{ij\pm}(r)$ in Equation S1 takes the forms
$r\sqrt{1-\left(u/2r\right)^{2}}$, $r^{2}\cos^{-1}{(u/2r)}$, and $r^{2}$. Each
of these terms must be averaged over $P(r)$. The first average can be solved
analytically as
$\displaystyle\int\\!P(r)$ $\displaystyle
r\sqrt{1-\left(u/2r\right)^{2}}dr=\frac{1}{z}\int\\!r^{-2}\sqrt{1-\left(u/2r\right)^{2}}=-\frac{1}{u}\left[\frac{1}{2}\sin{2\phi}+\frac{\pi}{2}-\phi\right]$
The average of the second term can be found in tables of integrals, and
involves a special function, the dilogarithm ${\rm Li}_{2}(z)$.
$\displaystyle\int\\!P(r)$ $\displaystyle
r^{2}\cos^{-1}{\\!\frac{u}{2r}}\,dr=\frac{1}{z}\int\\!r^{-1}\cos^{-1}{\\!\frac{u}{2r}}\,dr=\frac{i}{2}\phi^{2}-\phi\log{\left(1+e^{2i\phi}\right)}+\frac{i}{2}\,{\rm
Li}_{2}{\left(-e^{2i\phi}\right)}$
For $u<2r$ (required for the two relevant circles to intersect), the imaginary
component is constant and therefore cancels in any real definite integral. We
can therefore take just the real component without influencing the result.
$\int\\!P(r)r^{2}\cos^{-1}{\\!\frac{u}{2r}}\,dr=-\phi\log{\frac{u}{2r}}-\frac{1}{2}\Re{\left[i\,{\rm
Li}_{2}\left(-e^{2i\phi}\right)\right]}$
The imaginary part of the dilogarithm evaluated on the complex unit circle is
related to another special function known as Clausen’s integral, for which
optimized numerical routines have been written [48].
$\Re\left[i\,{\rm Li}_{2}\left(-e^{2i\phi}\right)\right]={\rm
Cl}_{2}\left(-2\phi-\pi\right)$
The remaining terms in $a_{ij\pm}(r)$ are elementary to integrate: $\int
P(r)r^{2}dr=\frac{1}{z}\log{r}$ and $\int P(r)dr=-\frac{1}{2z}r^{-2}$.
Combining all these pieces with their correct coefficients, we obtain the
indefinite integral for the size average of $a_{ij\pm}(r)$.
$\displaystyle b_{ij\pm}(r)\equiv\int\\!dr\,P(r)a_{ij\pm}(r)=$
$\displaystyle-\frac{1}{2z}\theta\log{r}+\frac{u}{8zr^{2}}{\bf
x}_{i}\cdot{\hat{\bf e}}_{\theta-\frac{\pi}{2}}$
$\displaystyle\pm\frac{1}{2zu}\left(\frac{1}{2}\sin{2\phi}+\frac{\pi}{2}-\phi\right){\bf
x}_{i}\cdot{\hat{\bf e}}_{\theta}$
$\displaystyle\pm\frac{1}{2z}\phi\log{\left(\frac{u}{r}\right)}\pm\frac{1}{4z}{\rm
Cl}_{2}{\left(-2\phi-\pi\right)}$ (S2)
### S1.4 Identifying piecewise smooth intervals over radius
The definite integral over $r$ must be performed piecewise because its
integration contours may change at certain critical radii $r^{*}$.
Generically, there are two types of critical radii, depicted in Figure S7B,C:
‘kissing points’ where $r^{*}_{ij}$ is half the distance $u_{ij}$ between a
pair of points ${\bf x}_{i}$ and ${\bf x}_{j}$, so that their corresponding
circles just touch; and ‘triple intersections’ where $r^{*}_{ijk}$ equals the
circumradius of three points ${\bf x}_{i}$, ${\bf x}_{j}$ and ${\bf x}_{k}$,
so that the three corresponding circles all meet. For three points separated
by distances $u_{ij}$, $u_{jk}$, and $u_{ki}$, and semiperimeter
$s=\frac{1}{2}(u_{ij}+u_{jk}+u_{ki})$, the circumradius is
$r^{*}_{ijk}=u_{ij}u_{jk}u_{ki}/4\sqrt{s(s-u_{ij})(s-u_{jk})(s-u_{ki})}$
If the pixel locations have extra symmetries, e.g. lie on a lattice, then
several critical radii $r^{*}$ may coincide. In this case each $r^{*}$ can be
treated sequentially without changing the result, as if perturbing each
$r^{*}$ infinitesimally: $b_{ij\pm}(r^{\prime\prime})-b_{ij\pm}(r^{\prime})$
contributes zero in the limit $r^{\prime\prime}-r^{\prime}\to 0$ when there
are no intervening critical radii.
### S1.5 Mapping piecewise integrals onto appropriate $Q_{\bm{\sigma}}$
Now we must calculate $Q_{\bm{\sigma}}$ by adding up the definite integral
$b_{ij\pm}(r)$ evaluated at the appropriate critical radii $r^{*}$ and the
relevant triples $(i,j,\pm)$. Consider two strategies for this. First, one
could choose one particular ${\bm{\sigma}}$, and track how the cusps of
$Q_{\bm{\sigma}}({\bf c},r)$’s boundary appear, change, and disappear as a
function of $r$, and then add up the appropriate contributions from Equation
S2 (Figure S7D). One would then repeat this procedure for every possible
${\bm{\sigma}}$. Second, one could choose a particular intersection point
${\bf x}_{ij\pm}$ between two objects, track how it is associated with
different regions as a function of $r$, and add its contribution to the
various appropriate $Q_{\bm{\sigma}}$. By iterating through all intersection
points, eventually all contributions to all $Q_{\bm{\sigma}}$ are computed
(Figure S7E). This latter strategy is easier because the behavior of the
intersection points is simpler to track than the various (possibly
unconnected) regions where $Q_{\bm{\sigma}}({\bf c},r)=1$. This is the
approach we describe below.
To compute the definite integral corresponding to Equation S2 above, we must
therefore associate each integrand $a_{ij\pm}(r)$ with boolean vectors
${\bm{\sigma}}$ designating the correct targets $Q_{\bm{\sigma}}$ for each
interval of $r$. The region geometry, and thus these desired associations,
change only at critical radii; between critical radii the associations are
constant. By construction, $a_{ij\pm}(r)$ (Equation S1) is the result of a
contour integral terminating at an intersection between circles centered on
${\bf x}_{i}$ and ${\bf x}_{j}$ (Figure S7A). We label this intersection point
by ${\bf x}_{ij\pm}={\bf x}_{i}+r{\hat{\bf e}}_{\theta_{ij}\pm\phi_{ij}}$.
Contour integrals terminating at this point contribute to every one of the
four regions that touch ${\bf x}_{ij\pm}$, i.e. the ${\bm{\sigma}}$ involving
all four allowed combinations of its elements $\sigma_{i}\in\\{0,1\\}$ and
$\sigma_{j}\in\\{0,1\\}$ (Figure S7F). The point ${\bf x}_{ij\pm}$ is not on
the boundary of any circles centered on other pixels ${\bf x}_{\ell}$, since
otherwise there would be a critical radius within the selected $r$ interval.
${\bf x}_{ij\pm}$ is thus either strictly inside or strictly outside a circle
of radius $r$ for all $\ell\neq i,j$. We can now specify all elements of
${\bm{\sigma}}$ as
$\sigma_{\ell}=\begin{cases}L_{1}({\bf x}_{ij\pm}-{\bf x}_{\ell},r)&\ell\neq
i,j\\\ \text{0 or 1}&\ell=i,j\end{cases}$
where $L_{1}({\bf x}_{ij\pm}-{\bf x}_{\ell},r)$ is the leaf function from
Methods Section 4.1. This relation identifies the appropriate targets
$Q_{\bm{\sigma}}$ for the $b_{ij\pm}$ of Equation S2.
To identify the signs $c_{ij\pm}$ with which $b_{ij\pm}$ contribute to the
target $Q_{\bm{\sigma}}$, it helps to go back and compute the signs that
$a_{ij\pm}(r)$ contribute to the target area $Q_{\bm{\sigma}}(r)$. These signs
depend on the geometry of the region contours. Consider how the region
boundaries change their geometry as $r$ increases from $r_{-}$ to $r_{+}$. A
contour around an object boundary is counterclockwise initially, i.e. before
the contour intersects any other object boundaries. As $r$ increases past a
kissing point $r^{*}_{ij}$, a pair of intersection points ${\bf x}_{ij\pm}$ is
created along with a new region with clockwise orientation (Figure S7B). Note
that the contours around the object centered on ${\bf x}_{i}$ initially
converge at an intersection ${\bf x}_{ij-}$ and diverge at ${\bf x}_{ij+}$. In
other words, intersections ${\bf x}_{ij-}$ are initially endpoints of the
contours along ${\bf s}_{i}(t)$ that contribute $+a_{ij\pm}$ to the contour
integral (Equation 5), and ${\bf x}_{ij+}$ are initially starting points that
contribute $-a_{ij\pm}$. However, as $r$ increases past each triple-
intersection $r^{*}_{ijk}$ for $k\neq i,j$, another circle centered on ${\bf
x}_{k}$ encloses the intersection point. The orientations of the contours at
${\bf x}_{ij\pm}$ then reverse (Figure S7C), and the sign that each
$a_{ij\pm}$ contributes also reverses. Thus the overall convergence for paths
at an intersection point is: converging for $-$, diverging for $+$, and
reversed by the number of circles enclosing the point. Mathematically, we can
write the desired sign as
$c_{ij\pm}(r)=\mp(-1)^{\sum_{\ell\neq i,j}L_{1}({\bf x}_{\ell}-{\bf
x}_{ij\pm},r)}$
Note that $c_{ij\pm}(r)$ does not vary between critical radii $r^{*}$, so we
may use its value anywhere within the integration interval. Finally, when we
integrate $a_{ij\pm}(r)$ over $r^{\prime}<r<r^{\prime\prime}$, the value of
the indefinite integral $b_{ij\pm}$ at $r^{\prime}$ is subtracted from the
value at $r^{\prime\prime}$. Thus, for each interval between critical radii we
add
$\Delta
Q_{\bm{\sigma}}(r^{\prime},r^{\prime\prime})=c_{ij\pm}(\tfrac{r^{\prime}+r^{\prime\prime}}{2})\cdot\big{(}b_{ij\pm}(r^{\prime\prime})-b_{ij\pm}(r^{\prime})\big{)}$
to the appropriate $Q_{\bm{\sigma}}$.
There is one remaining subtlety in adding up the contributions to
$Q_{\bm{\sigma}}$. In the first term of $b_{ij\pm}$ there is an ambiguity of
$2\pi$ in what angle is subtended by a given arc, which cannot be resolved by
local properties of the arc endpoints alone. We remedy this by computing
$\Delta Q_{\bm{\sigma}}(r^{\prime},r^{\prime\prime})$ modulo
$\tfrac{\pi}{z}\log{r^{\prime\prime}/r^{\prime}}$, which is the maximum
possible contribution an area can make between $r^{\prime}$ and
$r^{\prime\prime}$. This guarantees that we update $Q_{\bm{\sigma}}$ with the
unique definite integral over $r^{\prime}<r<r^{\prime\prime}$ that lies
between 0 and this maximum.
### S1.6 Summary of the algorithm for calculating $Q_{\bm{\sigma}}$
This completes the mathematics necessary to calculate the $Q_{\bm{\sigma}}$.
To summarize, we present the method in algorithmic form.
1. 1.
Initialize all $Q_{\bm{\sigma}}$ to zero.
2. 2.
Add $\int_{r_{-}}^{r^{*}_{i}}dr\,P(r)\pi
r^{2}=\frac{\pi}{z}\log{\frac{r_{i}^{*}}{r_{-}}}$ to $Q_{{\bf\delta}_{i}}$ for
each circle, where $r_{i}^{*}=\min_{j\neq i}{r^{*}_{ij}}$ is the first kissing
point for that circle and ${\bf\delta}_{i}$ is a vector of zeros with a 1 at
index $i$. This is the area accumulated in $Q_{{\bf\delta}_{i}}$ before any
other circles were touched.
3. 3.
Sort all critical radii $r^{*}_{ij}$ and $r^{*}_{ijk}$ within the integration
bounds $r_{-}$ and $r_{+}$.
4. 4.
For each interval $r^{\prime}<r<r^{\prime\prime}$ bounded by sequential
critical radii:
1. (a)
For each existing intersection point ${\bf x}_{ij\pm}$:
1. i.
Calculate the region indicators ${\bm{\sigma}}$ to which the point ${\bf
x}_{ij\pm}$ contributes
2. ii.
Add $\Delta Q_{\bm{\sigma}}(r^{\prime},r^{\prime\prime})$ modulo
$\frac{\pi}{z}\log{\frac{r^{\prime\prime}}{r^{\prime}}}$ to $Q_{\bm{\sigma}}$
5. 5.
Set $Q_{\bf 0}=1-\sum_{{\bm{\sigma}}\neq\bf 0}Q_{\bm{\sigma}}$.
Once the $Q_{{\bm{\sigma}}({\text{\cjRL{m|}}},n)}$ are calculated for all
object membership functions m—, then
$Q_{{\bm{\sigma}}({\text{\cjRL{m|}}}_{\setminus n},k)}$ must be calculated for
the reduced ${\text{\cjRL{m|}}}_{\setminus n}$ used in the recursion. For
efficiency, this can be accomplished by marginalizing $Q_{\bm{\sigma}}$ over
the appropriate indices $\sigma_{i}$, rather than recalculating it with a
smaller set of pixels.
Note that with a different size ensemble $P(r)$, the expression for
$b_{ij\pm}$ would change, but the procedure for combining them to obtain the
$Q_{\bm{\sigma}}$ would be the same.
### S1.7 Converting from circles to ellipses
It is straightforward to transform our calculation of $Q_{\bm{\sigma}}$ for
circles into a result for ellipses of equal area but eccentricity $\epsilon$
and orientation $\psi$. All distances are effectively scaled by
$\sqrt{\epsilon}$ in the direction of ${\hat{\bf e}}_{\psi}$ and
$1/\sqrt{\epsilon}$ in the orthogonal direction. This is equivalent to
transforming the pixel locations ${\bf x}_{i}\to{\bf x}^{\prime}_{i}$ as
${\bf
x}^{\prime}_{i}=\frac{1}{\sqrt{\epsilon}}\left(\begin{array}[]{cc}\epsilon\cos^{2}{\psi}+\sin^{2}{\psi}&(\epsilon-1)\cos{\psi}\sin{\psi}\\\
(\epsilon-1)\cos{\psi}\sin{\psi}&\cos^{2}{\psi}+\epsilon\sin^{2}{\psi}\end{array}\right)\cdot{\bf
x}_{i}$
and recomputing the $Q_{{\bm{\sigma}}}$ with these ${\bf
x}^{\prime}_{i}(\epsilon,\psi)$.
Unfortunately we cannot analytically integrate $b_{ij\pm}(r^{*})$ as a
function of eccentricity $\epsilon$ or angle $\psi$, because the dependence on
the points ${\bf x}^{\prime}_{i}(\epsilon,\psi)$ already involves special
functions. Instead, to obtain the average over possible ellipses we use a
discrete ensemble of eccentricities and angles and sum over them as
$\left\langle Q_{\bm{\sigma}}({\bf c},r,\epsilon,\psi)\right\rangle_{{\bf
c},r,\epsilon,\psi}=\sum_{\epsilon,\psi}Q_{\bm{\sigma}}(\epsilon,\psi)P(\epsilon)P(\psi)$
More generally, when the integral cannot be expressed analytically using
easily computable functions, one may specify the ensemble $P(\rho)$ by a
discrete number of allowed shapes, and compute the ensemble average as a sum
rather than as an integral.
The result of these calculations are concrete numbers for the inclusion
probabilities $Q_{\bm{\sigma}}$, which can then be substituted into Equations
1, 2, and 3 to calculate the object membership probabilities and joint
distributions of pixel intensities and image features.
m— | Hebrew letter mem: an object membership function
---|---
$|{\text{\cjRL{m|}}}|$ | number of distinct objects in m—
${\text{\cjRL{m|}}}_{n}$ | set of pixels contained in $n$th object
${\text{\cjRL{m|}}}_{\setminus n}$ | object membership function with $n$th object removed
$P_{\text{\cjRL{m|}}}$ | probability that pixels are divided according to m—
${\bf x}$ | locations of all $N$ selected pixels
${\bf x}_{i}$ | location of $i$th pixel
${\bf I}$ | vector of all $N$ pixel values
$I_{i}$ | pixel value at point ${\bf x}_{i}$
${\bf I}_{{\text{\cjRL{m|}}}_{n}}$ | vector of all pixel values in $n$th object
${\bm{\sigma}}({\text{\cjRL{m|}}},n)$ | boolean vector indicating pixels in $n$th object
$Q_{{\bm{\sigma}}({\text{\cjRL{m|}}},0)}$ | probability that an isolated object includes no selected pixels
$Q_{{\bm{\sigma}}({\text{\cjRL{m|}}},n)}$ | probability that an isolated object includes only pixels ${\text{\cjRL{m|}}}_{n}$
$Q_{{\bm{\sigma}}({\text{\cjRL{m|}}},n)}(\rho)$ | as above, but given the object shape $\rho$
$Q_{{\bm{\sigma}}({\text{\cjRL{m|}}},n)}({\bf c},\rho)$ | as above, but also given the object position ${\bf c}$
${\bf x}_{ij\pm}$ | location of two intersections between objects centered on ${\bf x}_{i}$ and ${\bf x}_{j}$
$u_{ij}$ | distance between ${\bf x}_{i}$ and ${\bf x}_{j}$
$\theta_{ij}$ | angle of the vector ${\bf x}_{j}-{\bf x}_{i}$
$\phi_{ij}$ | absolute value of angle between intersection points and line connecting ${\bf x}_{i}$ and ${\bf x}_{j}$
$t_{ij\pm}$ | angle of ${\bf x}_{ij\pm}-{\bf x}_{i}$
${\hat{\bf e}}_{\theta}$ | unit vector $(\cos{\theta},\sin{\theta})$
${\bf s}_{i}(t)$ | contour around object centered on ${\bf x}_{i}$
$a_{ij\pm}(r)$ | indefinite integral over $t$ along contour ${\bf s}_{i}(t)$ evaluated at $t_{ij\pm}$ with fixed $r$
$b_{ij\pm}(r)$ | indefinite integral of $a_{ij\pm}(r)$ over $r$
$c_{ij\pm}(r)$ | sign indicating whether contour ${\bf s}_{i}(t)$ starts or ends at $t=t_{ij\pm}$
$r^{*}$ | critical radius at which regions of constant $Q_{{\bm{\sigma}}({\text{\cjRL{m|}}},n)}({\bf c},\rho)$ change structure
$r^{*}_{ij}$ | radius of kissing point for circles on ${\bf x}_{i}$ and ${\bf x}_{j}$
$r^{*}_{ijk}$ | radius of triple intersection for circles on ${\bf x}_{i}$, ${\bf x}_{j}$ and ${\bf x}_{k}$
$\epsilon$ | ellipse eccentricity
$\psi$ | orientation of major axis of ellipse
${\bf x}^{\prime}_{i}(\epsilon,\psi)$ | transformed pixel location
Table S1: Glossary of symbols used
## References for Supporting Information
[48] MacLeod A (1996) Algorithm 757, miscfun: A software package to compute
uncommon special functions. ACM Transactions on Mathematical Software 22:
288–301.
|
arxiv-papers
| 2010-03-15T16:13:03 |
2024-09-04T02:49:09.097391
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xaq Pitkow",
"submitter": "Xaq Pitkow",
"url": "https://arxiv.org/abs/1003.2950"
}
|
1003.3166
|
# On the necessity of the assumptions used to prove Hardy-Littlewood and Riesz
Rearrangement Inequalities
H. Hajaiej
###### Abstract
We prove that supermodularity is a necessary condition for the generalized
Hardy-Littlewood and Riesz rearrangement inequalities. We also show the
necessity of the monotonicity of the kernels involved in the Riesz–type
integral.
Keywords and phrases: rearrangement, inequality, supermodular integrand .
AMS Subject Classification: 26D15 .
## 1 Introduction
Let $F:\mathbb{R}_{+}^{m}\to\mathbb{R}$ be a Borel measurable function. We say
that $F$ is supermodular if:
$\displaystyle F(y+he_{i}+ke_{j})+F(y)\geq F(y+he_{i})+F(y+ke_{j})$ (1.1) for
every $i,j\in\\{1,\ldots,m\\}$, $i\not=j$, $k,h>0$,
where $y=(y_{1},\ldots,y_{m})$, and $e_{i}$ denotes the standard basis vector
in $\mathbb{R}^{m}$, $i=1,\ldots,m$.
This condition has been first analysed in the context of integral inequalities
by G. Lorentz [19].
Note that if $F$ is $C^{2}$, (1.1) is equivalent to
$\frac{\partial^{2}F}{\partial x_{i}\partial x_{j}}(x)\geq 0,\quad\mbox{for
every $i,j\in\\{1,\ldots,m\\}$, $i\not=j$.}$ (1.2)
The supermodularity condition has been widely used in functional analysis. It
was the main assumption to prove the generalized Hardy-Littlewood
rearrangement inequality,
$\int_{X}F(u_{1}(x),\ldots,u_{m}(x))\,d\mu(x)\leq\int_{X}F(u_{1}^{*}(x),\ldots,u_{m}^{*}(x))\,d\mu(x),$
(1.3)
Here $u_{i}$ are nonnegative measurable functions that vanish at infinity and
$u_{i}^{*}$ are their symmetric decreasing rearrangements. The integral is
taken over $X=\mathbb{R}^{n}$, $\mathbb{H}^{n}$ or ${\cal S}^{n}$. (1.3) has
been studied by many authors [17, 19, 11, 6, 7]. Its main applications are
economics [11], chemistry [24], and nonlinear optics where the profile of
stable electromagnetic waves traveling along a planar waveguide are given by
the ground states of the energy functional
$E(u):=\frac{1}{2}\int_{\mathbb{R}}\left(u^{{}^{\prime}2}-G(|x|,u)\right)\,dx,$
(1.4)
under the constraint $\|u\|_{2}=c>0$. A crucial step to prove that (1.4)
admits an even ground state $u=u(x)$ which is decreasing for $x>0$, is (1.3),
[15].
In [16], we showed that the ground state of the $m\times m$ elliptic system
${\bf(S)}\left\\{\begin{array}[]{l}\Delta
u_{1}+\lambda_{1}u_{1}+g_{1}(u_{1},\ldots,u_{m})=0\\\ \cdots\\\ \Delta
u_{m}+\lambda_{m}u_{m}+g_{m}(u_{1},\ldots,u_{m})=0\end{array}\right.,$
where $g_{i}=(\partial G/\partial u_{i})$, ($i=1,\ldots,m$), exists when $G$
is supermodular, and satisfies some further conditions.
Notice that supermodularity condition is equivalent to the cooperativity of
${\bf(S)}$. As proved by W.C. Troy [24], the latter condition is also
necessary for the existence of solutions of ${\bf(S)}$. In [10], G. Carlier
viewed the generalized Hardy–Littlewood rearrangement inequality as an optimal
transportation problem. He showed that the left-hand side of (1.3) achieves
its maximum (i.e. the cost is minimized) under the supermodularity assumption.
In this paper, we prove, among other things, that (1.1) is necessary for the
inequality (1.3) to hold.
The generalized Riesz rearrangement inequality is
$\displaystyle\int\cdots\int
F(u_{1}(x_{1}),\ldots,u_{m}(x_{m}))\prod_{i<j}K_{ij}(d(x_{i},x_{j}))\,dx_{1}\cdots
dx_{m}$ (1.5) $\displaystyle\leq$ $\displaystyle\int\cdots\int
F(u_{1}^{*}(x_{1}),\ldots,u_{m}^{*}(x_{m}))\prod_{i<j}K_{ij}(d(x_{i},x_{j}))\,dx_{1}\cdots
dx_{m},$
where $d$ denotes distance and the functions $K_{ij}$ are decreasing.
It is closely related to the Brunn-Minkowski inequality of convex geometry.
Notice that the integral in (1.5) can represent a physical interaction
potential.
The most relevant case is $K_{ij}(d(x_{i},x_{j}))=\omega_{ij}(|x_{i}-x_{j}|)$
where $\omega_{ij}$ are nonnegative nonincreasing functions. (1.5) is then
equivalent to
$\displaystyle\int\cdots\int
F(u_{1}(x_{1}),\ldots,u_{m}(x_{m}))\omega_{ij}(|x_{i}-x_{j}|)\,dx_{1}\cdots
dx_{m}$ (1.6) $\displaystyle\leq$ $\displaystyle\int\cdots\int
F(u_{1}^{*}(x_{1}),\ldots,u_{m}^{*}(x_{m}))\omega_{ij}(|x_{i}-x_{j}|)\,dx_{1}\cdots
dx_{m}.$
(1.5) and (1.6) have been studied in [22, 23, 3, 4, 5, 1, 20, 13, 21, 12, 25,
7]. All these results use the supermodularity of $F$. Among the numerous
applications of (1.6), let us mention that it was extremely useful to prove
the existence and uniqueness of the minimizing solution of Choquard’s equation
[18]. In the special case that $F$ is a product and $m=2$, (1.6) hold on the
standard spheres and hyperbolic spaces [2, 3] and it still contains the
isoperimetric inequality as a limit. In this paper we will show (see
Proposition 3.2 below) that (1.1) is necessary for (1.6) to hold. Notice that
, in [7], cases of equality of (1.3) and (1.6) were established under the
strict supermodularity condition, that is to say (1.1) with the strict
inequality sign. We will also prove that such a condition is inescapable to
establish cases of equality in (1.3) and (1.6) (see Remark 1 below).
### 2\. Notation and preliminaries
We fix $n\in\mathbb{N}$. Let $\mu$ the Lebesgue measure on $\mathbb{R}^{n}$.
If $x\in\mathbb{R}^{n}$ and $r>0$, let
$B_{r}(x)=\\{y\in\mathbb{R}^{n}:\,|y-x|<r\\}$. For any set
$M\subset\mathbb{R}^{n}$, let ${\bf 1}_{M}$ denote its characteristic
function. By ${\cal M}$ we denote the set of all measurable functions on
$\mathbb{R}^{n}$. For a Borel measurable function
$F:\mathbb{R}^{m}_{+}\to\mathbb{R}$, nonincreasing functions
$w_{ij}:\mathbb{R}_{+}\to\mathbb{R}_{+}$ and nonnegative $u_{i}\in{\cal M}$,
we study the following generalized Hardy–Littlewood type functional,
$I(u_{1},\ldots,u_{m})=\int_{\mathbb{R}^{n}}F(u_{1}(x),\ldots,u_{m}(x))\,dx,$
(2.1)
and the generalized Riesz type functional.
$J(u_{1},\ldots,u_{m})=\int_{\mathbb{R}^{n}}\cdots\int_{\mathbb{R}^{n}}F(u_{1}(x_{1}),\ldots,u_{m}(x_{m}))\omega_{ij}(|x_{i}-x_{j}|)\,dx_{1}\cdots
dx_{m}.$ (2.2)
If $f\in{\cal F}$, let $f^{*}$ denote its Schwarz symmetrization which is the
unique lower continuous function which is radially symmetric, radially
nonincreasing and such that $\mu\\{a<f\leq b\\}=\mu\\{a<f^{*}\leq b\\}$ for
all numbers $\inf f<a<b$ (see [1]). Notice that if $f={\bf 1}_{M}$, where $M$
is Lebesgue measurable with finite measure, then $f^{*}={\bf 1}_{B_{R}(0)}$
where $R$ is chosen such that $\mu(B_{R}(0))=\mu(M)$ . Accordingly, a function
$u\in{\cal F}$ is called Schwarz symmetric if it is radial and radially
decreasing. We say that it is strictly Schwarz symmetric if it is radial and
strictly radially decreasing.
## 3\. Results
Proposition 3.1 ( Necessity of the supermodularity condition in the
generalized Hardy–Littlewood inequality )
Let $F:(\mathbb{R}_{+})^{m}\to\mathbb{R}$ be a Borel measurable function which
vanishes on hyperplanes, that is, $F(y_{1},\ldots,y_{m})=0$ if $y_{k}=0$ for
some $k\in\\{1,\ldots,m\\}$. If
$\int_{\mathbb{R}^{n}}F(u_{1}(x),\ldots,u_{m}(x))\,dx\leq\int_{\mathbb{R}^{n}}F(u_{1}^{*}(x),\ldots,u_{m}^{*}(x))\,dx,$
for any $(u_{1},\ldots,u_{m})\in{\cal F}^{m}$ then $F$ satisfies (1.1).
Proof: Suppose that (1.1) is not true. Then there exist
$i,j\in\\{1,\ldots,m\\}$, $i\not=j$, $0\leq a<b$, $0\leq c<d$ and $m-2$
nonnegative numbers $\alpha_{l}$, ($l\in\\{1,\ldots,m\\}\setminus\\{i,j\\}$),
such that:
$\displaystyle
F(\alpha_{1},\ldots,a,\ldots,d,\ldots,\alpha_{m-2})+F(\alpha_{1},\ldots,b,\ldots,c,\ldots,\alpha_{m-2})$
$\displaystyle>$ $\displaystyle
F(\alpha_{1},\ldots,b,\ldots,d,\ldots,\alpha_{m-2})+F(\alpha_{1},\ldots,a,\ldots,c,\ldots,\alpha_{m-2}).$
Now let $E$ and $F$ be two measurable sets such that $E\cap F=\emptyset$,
$\mu(F)=\mu(E)<\infty$, and $u_{i}=a{\bf 1}_{E}+b{\bf 1}_{F}$, $u_{j}=c{\bf
1}_{F}+d{\bf 1}_{E}$, and $u_{l}=\alpha_{l}({\bf 1}_{F}+{\bf 1}_{E})$ for
$l\in\\{1,\ldots,m\\}\setminus\\{i,j\\}$. Then $u_{i}^{*}=b{\bf
1}_{B_{r}(0)}+a{\bf 1}_{A}$, where $r$ is chosen such that
$\mu(B_{r}(0))=\mu(F)=\mu(E)$, and $A$ is the annulus
$B_{r^{\prime}}\setminus\overline{B_{r}}$ and $\mu(A)=\mu(F)=\mu(E)$. It
follows that $u_{j}^{*}=d{\bf 1}_{B_{r}(0)}+c{\bf 1}_{A}$,
$u_{l}^{*}=\alpha_{l}{\bf 1}_{B_{r^{\prime}}(0)}$, and
$\displaystyle\int F(u_{1}(x),\ldots,u_{n}(x))\,dx$ $\displaystyle=$
$\displaystyle[F(\alpha_{1},\ldots,a,\ldots,d,\ldots,\alpha_{m-2})+F(\alpha_{1},\ldots,b,\ldots,c,\ldots,\alpha_{m-2})]\mu(E),$
$\displaystyle\int F(u_{1}^{*}(x),\ldots,u_{n}^{*}(x))\,dx$ $\displaystyle=$
$\displaystyle[F(\alpha_{1},\ldots,b,\ldots,d,\ldots,\alpha_{m-2})+F(\alpha_{1},\ldots,a,\ldots,c,\ldots,\alpha_{m-2})]\mu(E).$
Hence we have in view of (3\. Results),
$\int F(u_{1}(x),\ldots,u_{n}(x))\,dx>\int
F(u_{1}^{*}(x),\ldots,u_{n}^{*}(x))\,dx,$
a contradiction.
Remark 1: Following the same approach we can easily conclude that the strict
supermodularity assumption, that is, (1.1) with strict inequality sign, is
necessary to establish cases of equality. In [7], (1.3) was proven under the
supermodularity assumption (1.1) and an integrability condition, and cases of
equality were established assuming (1.1) with strict inequality sign.
Proposition 3.2: ( Necessity of supermodularity assumption in Riesz–type
integral )
Let $j:(0,+\infty)\to\mathbb{R}$ be a function which is nonincreasing and not
identically equal to zero, and such that
$\lim_{r\to+\infty}r^{n-1}j(r)=0.$ (3.2)
Suppose that $\Psi:[0,+\infty)\times[0,+\infty)\to\mathbb{R}$ is a Borel
measurable function satisfying
$\Psi(u,0)=\Psi(0,v)=0\quad\forall u\geq 0,\ v\geq 0.$ (3.3)
Finally assume that for all nonnegative functions $f,g\in
L^{\infty}(\mathbb{R}^{n})$ with compact support there holds
$\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\Psi(f(x),g(y))j(|x-y|)\,dxdy\leq\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\Psi(f^{*}(x),g^{*}(y))j(|x-y|)\,dxdy.$
(3.4)
Then $\Psi$ is supermodular, that is,
$\Psi(a+a^{\prime},b+b^{\prime})-\Psi(a+a^{\prime},b)-\Psi(a,b+b^{\prime})+\Psi(a,b)\geq
0\quad\forall a,a^{\prime},b,b^{\prime}\in[0,+\infty).$ (3.5)
Proof: In view of (3.2), and since $j$ is non-increasing and nontrivial, $j$
is non-negative, too, and there exist positive numbers $\varepsilon$ and
$t_{0}$ with $t_{0}>2\varepsilon$ and such that
$j(t)<j(s)\quad\mbox{ whenever $0\leq s\leq\varepsilon$ and
$|t-t_{0}|\leq\varepsilon$.}$ (3.6)
We fix $z\in\mathbb{R}^{n}$ with $|z|=t_{0}$, and $\rho>t_{0}+\varepsilon$.
Now let $a,a^{\prime},b,b^{\prime}\in[0,+\infty)$. Then, if $f=a{\bf
1}_{B_{R}(0)}+a^{\prime}{\bf 1}_{B_{\varepsilon}(0)}$, and $g=b{\bf
1}_{B_{\rho}(0)}+b^{\prime}{\bf 1}_{B_{\varepsilon}(z)}$, where
$R>\varepsilon$, we have that $f=f^{*}$, and $g^{*}=b{\bf
1}_{B_{\rho}(0)}+b^{\prime}{\bf 1}_{B_{\varepsilon}(0)}$. Using (3.3), a short
computation shows that (3.4) is equivalent to
$\displaystyle 0$ $\displaystyle\leq$
$\displaystyle\left\\{\Psi(a+a^{\prime},b+b^{\prime})-\Psi(a+a^{\prime},b)-\Psi(a,b+b^{\prime})+\Psi(a,b)\right\\}\cdot$
$\displaystyle\cdot\int_{B_{\varepsilon}(0)}\left[\int_{B_{\varepsilon}(0)}j(|x-y|)\,dy-\int_{B_{\varepsilon}(z)}j(|x-y|)\,dy\right]dx$
$\displaystyle+$
$\displaystyle\left(\Psi(a,b+b^{\prime})-\Psi(a,b)\right)\int_{B_{R}(0)}\left[\int_{B_{\varepsilon}(0)}j(|x-y|)\,dy-\int_{B_{\varepsilon}(z)}j(|x-y|)\,dy\right]dx.$
Set
$H(x):=\int_{B_{\varepsilon}(0)}j(|x-y|)\,dy,\quad\,x\in\mathbb{R}^{n}.$
Since $j$ is nonnegative and nonincreasing, $H$ is nonnegative, radial and
radially nonincreasing, that is we can write $H(x)=:h(|x|)$, with $h$
nonincreasing. Since $j$ satisfies (3.2), we also have
$\lim_{t\to\infty}t^{n-1}h(t)=0$. Setting
$I(t):=\int_{B_{t}(0)}H(x)\,dx-\int_{B_{t}(z)}H(x)\,dx,\quad(t>0),$
it follows that $I$ is nonnegative, and moreover, (3.6) implies that
$I(\varepsilon)>0.$ (3.8)
Finally, if $t>t_{0}$, then
$I(t)\leq\int_{B_{t+t_{0}}(0)\setminus
B_{t-t_{0}}(0)}H(x)\,dx=n\omega_{n}\int_{t-t_{0}}^{t+t_{0}}s^{n-1}h(s)\,ds,$
where $\omega_{n}$ denotes the volume of the unit ball in $\mathbb{R}^{n}$.
Hence
$\lim_{t\to\infty}I(t)=0.$ (3.9)
Now inequality (3\. Results) becomes
$\displaystyle 0$ $\displaystyle\leq$
$\displaystyle\left\\{\Psi(a+a^{\prime},b+b^{\prime})-\Psi(a+a^{\prime},b)-\Psi(a,b+b^{\prime})+\Psi(a,b)\right\\}I(\varepsilon)$
$\displaystyle+$
$\displaystyle\left(\Psi(a,b+b^{\prime})-\Psi(a,b)\right)I(R).$
Sending $R\to+\infty$, we obtain
$0\leq\left\\{\Psi(a+a^{\prime},b+b^{\prime})-\Psi(a+a^{\prime},b)-\Psi(a,b+b^{\prime})+\Psi(a,b)\right\\}I(\varepsilon),$
(3.10)
and (3.5) follows from (3.8).
Remark 2: Proposition 3.2 tells us in particular that $\Psi$ is supermodular
if (3.4) holds for $j={\bf 1}_{B_{R}(0)}$, with some $R>0$.
Proposition 3.3. Let $\Psi:[0,+\infty)\times[0,+\infty)$ be nontrivial, and
satisfies (3.5) and (3.3), and let $h\in C(\mathbb{R}^{n})$. Assume that there
holds
$\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\Psi(f(x),g(y))h(x-y)\,dxdy\leq\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\Psi(f^{*}(x),g^{*}(y))h(x-y)\,dxdy$
(3.11)
for all bounded nonnegative functions $f$ and $g$ with compact support . Then
$h$ is radial and radially nonincreasing, that is, it can be written as
$j(|x|)=h(x)$, ($x\in\mathbb{R}^{n}\setminus\\{0\\}$), where
$j:(0,+\infty)\to\mathbb{R}$ is a nonincreasing function.
Proof : By choosing $a=b=0$ in (3.5) and taking into account (3.3), we have
that $\Psi$ is nonnegative, and since $\Psi$ is nontrivial, there are numbers
$a>0$, $b>0$ such that $\Psi(a,b)>0$. Let $z_{1},z_{2}\in\mathbb{R}^{n}$ with
$0<|z_{1}|<|z_{2}|$, and set $R:=(|z_{1}|+|z_{2}|)/2$. Let $f=a({\bf
1}_{B_{R}(0)}+{\bf 1}_{B_{\varepsilon}(z_{2})}-{\bf
1}_{B_{\varepsilon}(z_{1})})$, and $g=b{\bf 1}_{B_{\varepsilon}(0)}$, where
$0<\varepsilon<(|z_{2}|-|z_{1}|)/2$. Then $g=g^{*}$ and $f^{*}=a{\bf
1}_{B_{R}(0)}$. Hence (3.11) gives
$0\leq\Psi(a,b)\left\\{\int_{B_{\varepsilon}(z_{1})}\int_{B_{\varepsilon}(0)}h(x-y)\,dxdy-\int_{B_{\varepsilon}(z_{2})}\int_{B_{\varepsilon}(0)}h(x-y)\,dxdy\right\\}.$
Sending $\varepsilon\to 0$ and taking into account $\Psi(a,b)>0$ we find that
$h(z_{1})\geq h(z_{2})$. Since $h$ is continuous, this implies $h(x)\geq h(y)$
iff $|x|\leq|y|$. Hence $h$ is radial and radially nonincreasing as claimed.
Acknowledgement: The author is very grateful to F. Brock for his kind
invitation to visit AUB in June 2008 and June 2009, and for his precious help.
## References
* [1] F. Almgren, E.H. Lieb, Symmetric decreasing rearrangement is sometimes continuous. Journal of the A.M.S. 2 (1989), 683–773.
* [2] A. Baernstein II, B.A. Taylor, Spherical rearrangements, subharmonic functions and $*$-functions in $n$-space. Duke Math. J. 43 (1976), 245–268.
* [3] W. Beckner, Sobolev inequalities, the Poisson semigroup and the analysis on the sphere ${\cal S}^{n}$. Proc. A.M.S. 89 (1992), 4816–4819.
* [4] W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger Inequality. Ann. Math. (2) 138 (1993), 213–242.
* [5] H.J. Brascamp, E.H. Lieb, M.J. Luttinger, A general rearrangement inequality for multiple integrals. J. Funct. Anal. 17 (1974), 227–237.
* [6] F. Brock, A general rearrangement inequality a la Hardy–Littlewood. J. Inequ. Appl. 5 (2000), 309–320.
* [7] A. Burchard, H. Hajaiej, Rearrangement inequalities for functionals with monotone integrands. Journal of Functional Analysis 233 (2006), 561–582.
* [8] A. Burchard, M. Schmuckenschläger, Comparison theorems for exit times. Geom. Funct. Anal. 11 (2001), 651–692.
* [9] G.R. Burton, Vortex–rings of prescribed impulse. Proc. Cambridge Philos. Soc. (3) 134 (2003), 515–528.
* [10] G. Carlier, On a class of multidimensional optimal transportation problems. J. Convex Anal. 10 (2003), 517–529.
* [11] J.A. Crowe, J.A. Zweibel, P.C. Rosenbloom, Rearrangement of functions. J. Funct. Anal. 66 (1986), 432–438.
* [12] C. Draghici, A general rearrangement inequality. Proc. Amer. math. Soc. 133 (2005), 735–743.
* [13] R. Friedberg, J.M. Luttinger, (a) Rearrangement inequalities for periodic functions, (b) A new rearrangement inequality for multiple integrals. Arch. Rat. Mech. Anal. 61 (1976), 35–44, and 45–64.
* [14] H. Hajaiej, Balls are the maximizers of the Riesz–type functionals with supermodular integrands. to appear in: Annali de Matematica Pura ed Applicata.
* [15] H. Hajaiej, C.A. Stuart, Existence and nonexistence of Schwarz symmetric ground states for eigenvalue problems. Ann. Mat. Pura Appl. (4) 184 (2005), no. 3, 297–314.
* [16] H. Hajaiej, Symmetric ground states solutions of m-coupled nonlinear Schr dinger equations. Nonlinear Analysis: Methods, Theory and Applications 71, no.2 (2009).
* [17] G.E. Hardy, J.E. Littlewood, G. Polya, Inequalities. Cambridge Univ. Press, London and N.Y, 1st edn. 1934, 2nd edn. 1952.
* [18] E.H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Studies in Applied Mathematics 57 (1977), 93–105.
* [19] G.G. Lorentz, An inequality for rearrangements. Amer. Math. Monthly 60 (1953), 176–179.
* [20] J.M. Luttinger, Generalized isoperimetric inequalities I,II,III. J. Math. Physics 14 (1973), 586–593, 1444–1447, 1448–1450.
* [21] C. Morpurgo, Sharp inequalities for functional integrals and traces of conformally invariant operators. Duke Math. J. 114 (2002), 447–553.
* [22] F. Riesz, Sur une inegalite integrale. J. London Math. Soc. 5 (1930), 162–168.
* [23] S.L. Sobolev, On a theorem in functional analysis. Math. Sb. (N.S.) 4 (1938), 471–497; AMS Transl.(2) 34 (1963), 39–68.
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|
arxiv-papers
| 2010-03-16T15:48:35 |
2024-09-04T02:49:09.116279
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hichem Hajaiej",
"submitter": "Hichem Hajaiej",
"url": "https://arxiv.org/abs/1003.3166"
}
|
1003.3187
|
020002 2010 D. A. Stariolo 020002
A simple Lennard–Jones fluid confined in a slit nanopore with hard walls is
studied on the basis of a multilayer structured model. Each layer is
homogeneous and parallel to the walls of the pore. The Helmholtz energy of
this system is constructed following van der Waals-like approximations, with
the advantage that the model geometry permits to obtain analytical expressions
for the integrals involved. Being the multilayer system in thermodynamic
equilibrium, a system of non-linear equations is obtained for the densities
and widths of the layers. A numerical solution of the equations gives the
density profile and the longitudinal pressures. The results are compared with
Monte Carlo simulations and with experimental data for Nitrogen, showing very
good agreement.
# Multilayer approximation for a confined fluid in a slit pore
G. J. Zarragoicoechea [inst1, inst2] A. G. Meyra E-mail:
vasco@iflysib.unlp.edu.ar [inst1] V. A. Kuz[inst1]
(23 December 2009; 24 February 2010)
††volume: 2
99 inst1 IFLYSIB-Instituto de Física de Líquidos y Sistemas Biológicos
(CONICET, UNLP, CICPBA), 59 No. 789, 1900 La Plata, Argentina. inst2 CICPBA-
Comisión de Investigaciones Científicas de la Prov. de Buenos Aires.
## 1 Introduction
The effects on phase transition of confined fluids in a slit-like pore have
been studied by simulation and different theories [1–11]. In a previous work,
we constructed a generalized van der Waals equation for a fluid confined in a
nanopore [12, 13]. The shift of the critical parameters was in good agreement
with lattice model and numerical simulation results, and the predicted
critical temperature remarkably reproduced the experiment. In that work, we
concluded that the confined van der Waals fluid theory seemed to work better
than the bulk one, maybe due to the fact that the higher virial contributions
not considered in both theories were less important in the confined fluid than
in the bulk. A similar treatment was used previously by Schoen and Diestler
[14]. Following that line of reasoning, here we study a simple fluid confined
between two infinite parallel hard walls (slit pore). The walls are at a
distance $\mathit{L}$ apart. To study the confined fluid, we propose a
multilayer model [15]: the fluid is distributed in n thin layers, one beside
the other. Each layer has a uniform density, and can be observed as a non-
autonomous phase. A particle in a given layer interacts with its neighbors
inside the layer, and with every particle in the other layers. Defay and
Prigogine and Murakami et al. have shown that, in a liquid gas interface, the
deviation from the Gibbs’ adsorption equation becomes practically negligible
in the case of a two layer model [16], and that as the number of transition
layers grows, the multilayer model becomes perfectly consistent with the
Gibbs’ equation [17]. The van der Waals-like approximations made in developing
this multilayer model theory limit its validity to the low density regime.
## 2 Theory
The model system consists of a fluid of $\mathit{N}$ Lennard–Jones particles
confined in a slit nanopore. The hard walls of the pore, separated at a
distance $\mathit{L}$ (in the $\mathit{x}$ direction), have a surface area
$\mathit{S}$ ($\mathit{S}\rightarrow\infty$). We divided the fluid into
$\mathit{n}$ layers, each layer being parallel to the pore walls. The layer
$\mathit{i}$ has $\mathit{N}_{\mathit{i}}$ particles
($\mathit{N}=\sum_{i=1}^{n}N_{i}$), a width $\mathit{L}_{\mathit{xi}}$
($\mathit{L}$= $\sum_{i=1}^{n}L_{xi}$), and a volume $V_{i}=SL_{xi}$. Then the
Helmholtz energy [18] can be written as
$A=-kT\ln\left(\frac{Z_{N}\lambda^{-3N}}{\prod\limits_{i=1}^{n}N_{i}!}\right).$
(1)
The configuration integral $\mathit{Z}_{\mathit{N}}$ for a pair potential
$\mathit{v}_{\mathit{ij}}$ may be approximated as
$\displaystyle Z_{N}$ $\displaystyle=$
$\displaystyle\int\prod\limits_{\begin{subarray}{c}i=1\\\
i<j\end{subarray}}^{n}e^{-v_{ij}/kT}\mathrm{d}\mathbf{r}^{N}\approx\sum_{\begin{subarray}{c}i=1\\\
i<j\end{subarray}}^{n}\int f_{ij}\mathrm{d}\mathbf{r}^{N}$
$\displaystyle+\prod\limits_{i=1}^{n}V_{i}^{N_{i}}$ $\displaystyle f_{ij}$
$\displaystyle=$ $\displaystyle e^{-v_{ij}/kT}-1,$
and further expanded in function of two particle integrals
$\displaystyle{Z_{N}}$ $\displaystyle{=}$
$\displaystyle{\sum\limits_{i=1}^{n}\frac{N_{i}(N_{i}-1)}{2}V_{i}^{N_{i}-2}}$
$\displaystyle{\prod\limits_{\begin{subarray}{c}k=1\\\ k\neq
i\end{subarray}}^{n}V_{k}^{N_{k}}\int\limits_{R_{i}}f_{12}\mathrm{d}}\mathbf{r}{{}_{\mathbf{1}}\mathrm{d}\mathbf{r}_{\mathbf{2}}}$
$\displaystyle+{\sum\limits_{i=1}^{n}\sum\limits_{\begin{subarray}{c}j=2\\\
j>i\end{subarray}}^{n}N_{i}N_{j}V_{i}^{N_{i}-1}V_{j}^{N_{j}-1}}$
$\displaystyle{\prod\limits_{\begin{subarray}{c}k=1\\\ k\neq i,k\neq
j\end{subarray}}^{n}V_{k}^{N_{k}}\int\limits_{R_{i}}\int\limits_{R_{j}}f_{12}\mathrm{d}}\mathbf{r}{{}_{\mathbf{1}}\mathrm{d}}\mathbf{r}{{}_{\mathbf{2}}+\prod\limits_{i=1}^{n}V_{i}^{N_{i}}}.$
The first term in Eq. (2) stands for particles in the layer $\mathit{i}$. The
second term comes from the interaction of one particle in layer $\mathit{i}$
with one particle in layer $\mathit{j}$. In a compact form, and assuming that
a layer sees three nearest neighbor layers,
$\displaystyle Z_{N}$ $\displaystyle=$
$\displaystyle(\sum\limits_{i=1}^{n}\frac{N_{i}^{2}}{2V_{i}^{2}}I_{i}$
$\displaystyle+\sum\limits_{i=1}^{n-1}\sum\limits_{\begin{subarray}{c}j=i+1\\\
j\leq
n\end{subarray}}^{i+3}\frac{N_{i}}{V_{i}}\frac{N_{j}}{V_{j}}I_{i\,j}+1\,)\prod\limits_{i=1}^{n}V_{i}{}^{N_{i}}.$
The integrals $\mathit{I}_{\mathit{i}}$ and $\mathit{I}_{\mathit{ij}}$, for
the slit pore geometry and after low density approximations, can be
analytically solved to give
$\begin{array}[]{l}{I_{i}=\iint\limits_{R_{i}}f_{12}\mathrm{d}\mathbf{r}_{1}\mathrm{d}\mathbf{r}_{2}\approx-\iint\limits_{\left|\mathbf{r}_{1}\mathrm{-}\mathbf{r}_{2}\right|<\sigma}\mathrm{d}\mathbf{r}_{1}\mathrm{d}\mathbf{r}_{2}}\\\
{-\iint\limits_{\left|\mathbf{r}_{1}\mathrm{-}\mathbf{r}_{2}\right|\geq\sigma}\frac{v_{12}}{kT}\mathrm{d}\mathbf{r}_{1}\mathrm{d}\mathbf{r}_{2}=-2V_{i}\sigma^{3}(b-B_{i})-\frac{2V_{i}\sigma^{3}\varepsilon}{kT}A_{i}}\end{array}$
(5)
$\begin{array}[]{l}{I_{i,\,i+1}=\int\limits_{R_{i}}\int\limits_{R_{i+1}}f_{12}\mathrm{d}\mathbf{r}_{1}\mathrm{d}\mathbf{r}_{2}\approx-\iint\limits_{\left|\mathbf{r}_{1}\mathrm{-}\mathbf{r}_{2}\right|<\sigma}\mathrm{d}\mathbf{r}_{1}\mathrm{d}\mathbf{r}_{2}}\\\
{-\iint\limits_{\left|\mathbf{r}_{1}\mathrm{-}\mathbf{r}_{2}\right|\geq\sigma}\frac{v_{12}}{kT}\mathrm{d}\mathbf{r}_{1}\mathrm{d}\mathbf{r}_{2}=-V_{i}\sigma^{3}B_{i}-\frac{V_{i}\sigma^{3}\varepsilon}{kT}A_{i,\,i+1}}\end{array}$
(6)
$\displaystyle\begin{array}[]{l}I_{i,\,i+2}=\int\limits_{R_{i}}\int\limits_{R_{i+2}}f_{12}\mathrm{d}\mathbf{r}_{1}\mathrm{d}\mathbf{r}_{2}\approx\\\
-\iint\limits_{\left|\mathbf{r}_{1}\mathrm{-}\mathbf{r}_{2}\right|\geq\sigma}\frac{v_{12}}{kT}\mathrm{d}\mathbf{r}_{1}\mathrm{d}\mathbf{r}_{2}=-\frac{V_{i}\sigma^{3}\varepsilon}{kT}A_{i,\,i+2}\end{array}$
(9)
$I_{i,\,i+3}=-\frac{V_{i}\sigma^{3}\varepsilon}{kT}A_{i,\,i+3}$ (10)
In the above expressions $\mathit{v}_{\mathit{ij}}$ was taken to be the
Lennard–Jones pair interaction, being $\varepsilon$ and $\sigma$ the potential
parameters. The integrals $I_{i,\,i+2}$ and $I_{i,\,i+3}$ do not contain the
excluded volume term because we suppose that the layer widths are
$\mathit{L}_{\mathit{xi}}\geq\sigma$. The expressions for $\mathit{A}$ and
$\mathit{B}$ in the preceding equations, functions of
$\mathit{L}_{\mathit{xi}}$, are given in the Appendix A.
The Helmholtz energy, Eq. (1) together with Eq. (4), has the final expression
$\begin{array}[]{l}{A\approx-
kT\left(\sum\limits_{i=1}^{n}\frac{N_{i}^{2}}{2V_{i}^{2}}I_{i}+\sum\limits_{i=1}^{n-1}\sum\limits_{\begin{subarray}{c}j=i+1\\\
j\leq
n\end{subarray}}^{i+3}\frac{N_{i}}{V_{i}}\frac{N_{j}}{V_{j}}I_{i,\,j}\right)}\\\
{\mathrm{-}\sum\limits_{i=1}^{n}N_{i}kT\ln\frac{V_{i}}{N_{i}}+NkT(\ln\lambda^{3}-1)}\end{array}$
(11)
The pressure tensor [12, 13] and chemical potentials are obtained from the
following equations
$\displaystyle p_{xx,i}$ $\displaystyle=$
$\displaystyle-\frac{1}{L_{yi}L_{zi}}\left(\frac{\partial A}{\partial
L_{xi}}\right)_{T,N}$ $\displaystyle p_{yy,i}$ $\displaystyle=$ $\displaystyle
p_{zz,i}=-\frac{1}{L_{xi}L_{yi}}\left(\frac{\partial A}{\partial
L_{zi}}\right)_{T,N}$ (12) $\displaystyle\mu_{i}$ $\displaystyle=$
$\displaystyle\left(\frac{\partial A}{\partial N_{i}}\right)_{T,V,N_{j\neq
i}}$
If the system is in mechanical and chemical equilibrium, the $\mathit{xx}$
components of the pressure tensor and the chemical potentials for each layer
must be equal. From these equations, giving as input the wall separation
$\mathit{L}$ and the mean density $\rho^{\ast}=\rho\sigma^{3}$, it is
constructed a system of ($\mathit{n}-1$) non-linear equations with
($\mathit{n}-1$) unknowns (layer densities and widths) to be numerically
solved. The low computational cost is taken for granted given that the code is
easily written and the calculations are carried out on a Pentium 4 processor
running at 2.66 GHz. At a temperature
$\mathit{T}^{\ast}$=$\mathit{kT}$/$\varepsilon$=1, we have explored the cases
with $\mathit{L}$=10$\sigma$ and $\mathit{L}$=15$\sigma$, at different mean
densities. We have also compared the theoretical results with experimental
data coming from studies of nitrogen adsorption in graphite slit pores at room
temperature [19].
## 3 Monte Carlo simulation
For numerical simulations, $\mathit{N}$ Lennard–Jones particles are confined
between hard walls separated at a distance $\mathit{L}$. The unit cell is
build up taken the walls to be of size $\mathit{L}_{\mathit{y}}$ and
$\mathit{L}_{\mathit{z}}$ in the $\mathit{y}$ and $\mathit{z}$ directions
respectively, directions on which the periodical boundary conditions are
applied. The density profiles and pressures were obtained taking average
values in fluid slabs parallel to the walls. The pressure tensor was used in
the simple virial form, as indicated in references [20, 21].
With $\mathit{T}^{\ast}$=1, and for both slit pore widths
$\mathit{L}$=10$\sigma$ and $\mathit{L}$=15$\sigma$, the size of the unit cell
was set to $\mathit{L}_{\mathit{y}}$=$\mathit{L}_{\mathit{z}}$=30$\sigma$,
taking the number of particles $\mathit{N}$ to correspond with the mean
density. The range of the Lennard–Jones interactions was considered with a
cutoff radius of 5$\sigma$.
## 4 Results
Figure 1: Density profiles for a $n$=9 layer model of a confined fluid in a
slit pore (solid symbols). The temperature is $\mathit{T}^{\ast}$=1.0 and the
wall separation is $\mathit{L}$=10$\sigma$, with mean densities
$\rho^{\ast}$=1/20 (circles), 1/15 (squares), 1/10 (triangles), 1/5
(diamonds), and $\frac{1}{4}$ (stars). Open symbols represent the Monte Carlo
simulations. Figure 2: $\mathit{zz}$ pressure tensor component. Captions as in
Fig. 1.
In Figs. 1 and 2, the density profiles and $\mathit{zz}$ components of the
pressure tensor are shown for $\mathit{T}^{\ast}$=1 and
$\mathit{L}$=10$\sigma$. The mean densities studied are $\rho^{\ast}$=1/20,
1/15, 1/10, 1/5, and 1/4. The agreement of the theoretical density profiles
with the Monte Carlo simulations is very good. For the pressure there is a
rather good correspondence for low densities, up to $\rho^{\ast}$=1/10. For
the higher densities, differences appear, though the tendencies are similar.
The discrepancies come first from the low density approximations done to get
the Helmholtz energy. But, while in the simulation slab particles fluctuate
and at higher densities some clusterization occurs, in the theory each layer
is supposed to have a homogeneous density which makes it hard for the
theoretical pressures to follow those obtained by simulation. For the density
profiles, averaging the number of particles in each slab evidently compensates
the clusterization, and the theory gives good results, at least for the rather
low densities studied. The same picture applies to the behavior of the system
for $\mathit{T}^{\ast}$=1 and $\mathit{L}$=15$\sigma$, at mean densities
$\rho^{\ast}$= 1/10, and 1/5, represented in Figs. 3 and 4.
The results, as expected for hard repulsive walls, show a low density region
next to the walls and an increasing density profile, with a maximum at the
center of the slit pore. This behavior is also shown with density functional
theory [1] and in other Monte Carlo simulations [2].
Figure 3: Density profiles for a $n$=13 layer model of a confined fluid in a
slit pore (solid symbols). The temperature is $\mathit{T}^{\ast}$=1.0 and the
wall separation is $\mathit{L}$=15$\sigma$, with mean densities
$\rho^{\ast}$=1/10 (triangles), and 1/5 (circles). Open symbols represent the
Monte Carlo simulations. Figure 4: $\mathit{zz}$ pressure tensor component.
Captions as in Fig. 3.
Finally, the good agreement of the theory with the experiment can be seen in
the results shown in Fig. 5. In this figure, the excess number of molecules
per unit area of pore surface $\Gamma$ is plotted in function of the external
pressure, at $T^{\ast}$=3.18 and $L$=4$\sigma$. These parameters approximate
the experimental values [19] $T$=303 K and $L$=1.45 nm, if
$\varepsilon/k=95.2$ K and $\sigma=3.75$ Å are used to characterize the
nitrogen. In this case, due to the size of the sample, $n$=3 layers have been
used for calculation. $\Gamma$ is defined as
$\Gamma=\frac{N-N_{g}}{S}=(\rho^{\ast}-\rho_{g}^{\ast})\frac{L}{\sigma^{3}}$
(13)
where $N_{g}$/$\rho_{g}^{\ast}$ is the number/density of particles which would
occupy the slit pore in the absence of the adsorption forces.
$\rho_{g}^{\ast}$ and the external pressure are determined equating the
chemical potential inside the slit pore (Eq. 12) to the chemical potential
coming from the bulk van der Waals equation at the same temperature. The
theoretical results presented here are similar to the numerical simulation
results obtained by the same authors who have done the experiment [19]. They
assume that the differences at higher pressures could be a consequence of the
uncertainty in the determination of the pore geometry.
Figure 5: Excess number of molecules per unit area of pore surface $\Gamma$ as
function of the external pressure. The full line represents the experiment
(digitalized from Ref. [19]), and the dots are our theoretical results.
## 5 Conclusions
The application of a simple theory, with van der Waals-like approximations to
the Helmholtz energy, to a particular model of spatial distribution makes it
possible to obtain analytical expressions for the thermodynamic quantities.
The study of a confined fluid in a slit pore geometry with a multilayer
approximation produces good results when compared with Monte Carlo simulations
at low densities. The agreement with a particular experiment on nitrogen
confined in a graphite slit pore is remarkable, even though an excess quantity
is in study. It may be concluded that the confinement reduces the importance
that higher virial contributions have on the equation of the state of the
confined fluid. Classical density functional theory [22] can also be applied
to study the slit pore geometry, with very good agreement with experiments and
simulations. Though the theoretical work developed in these pages is not a
competitor of density functional theory, it has the advantages of having
analytical expressions, and the possibility of easily introducing two
immiscible components: for instance one or two layer lubricants wetting the
walls and a gas or a liquid filling the rest of layers forming the capillary
volume.
###### Acknowledgements.
This work was partially supported by Universidad Nacional de La Plata and
CICPBA. G. J. Z. is member of “Carrera del Investigador Científico” CICPBA.
## Appendix A
Expressions of quantities used in Eqs. 5–8:
$\begin{array}[]{l}{b=\frac{2}{3}\pi;B_{i}=\frac{\pi}{4}\frac{\sigma}{L_{xi}};A_{i}=a_{1}+\frac{a_{2}}{L_{xi}}+\frac{a_{3}}{L_{xi}^{3}}+\frac{a_{4}}{L_{xi}^{9}}}\\\
\\\
{a_{1}=-\frac{16}{9}\pi;a_{2}=\frac{3}{2}\pi;a_{3}=-\frac{1}{3}\pi;a_{4}=\frac{1}{90}\pi}\hskip
5.0pt(A1)\end{array}$
A correction has been made to get good critical parameters for the bulk
($\mathit{L\rightarrow\infty}$). For Argon $\mathit{a}_{1}$= -5.7538 and
$\mathit{b}$=1.3538, and for nitrogen $\mathit{a}_{1}$= -1.5955 and b=1.0349.
$\begin{array}[]{l}{A_{i,\,i+1}=\frac{\pi}{90}\Bigl{[}-\frac{1}{L_{xi}^{9}}-\frac{1}{L_{xi}L_{xi+1}^{8}}+\frac{1}{L_{xi}(L_{xi}+L_{xi+1})^{8}}\Bigr{]}}\\\
\hskip
30.00005pt-\frac{\pi}{3}\Bigl{[}-\frac{1}{L_{xi}^{3}}-\frac{1}{L_{xi}L_{xi+1}^{2}}+\frac{1}{L_{xi}(L_{xi}+L_{xi+1})^{2}}\Bigr{]}\\\
\hskip 30.00005pt-\frac{3}{2}\frac{\pi}{L_{xi}}\hskip
130.0002pt(A2)\end{array}$
$\begin{array}[]{l}A_{i,\,i+2}=\frac{\pi}{90}\Bigl{[}\frac{1}{L_{xi+1}^{8}}-\frac{1}{(L_{xi}+L_{xi+1})^{8}}\\\
\hskip
30.00005pt-\frac{1}{(L_{xi+1}+L_{xi+2})^{8}}+\frac{1}{(L_{xi}+L_{xi+1}+L_{xi+2})^{8}}\Bigr{]}\frac{1}{L_{xi}}\\\
\hskip
30.00005pt-{\frac{\pi}{3}\Bigl{[}\frac{1}{L_{xi+1}^{2}}-\frac{1}{(L_{xi}+L_{xi+1})^{2}}-\frac{1}{(L_{xi+1}+L_{xi+2})^{2}}}\\\
\hskip
30.00005pt{+\frac{1}{(L_{xi}+L_{xi+1}+L_{xi+2})^{2}}\Bigr{]}\,\frac{1}{L_{xi}}}\hskip
50.00008pt(A3)\end{array}$
$\begin{array}[]{l}{A_{i,\,i+3}=\frac{\pi}{90}\Bigl{[}\frac{1}{(L_{xi+1}+L_{xi+2})^{8}}}\\\
\hskip 30.00005pt-\frac{1}{(L_{xi}+L_{xi+1}+L_{xi+2})^{8}}\\\ \hskip
30.00005pt-\frac{1}{(L_{xi+1}+L_{xi+2}+L_{xi+3})^{8}}\\\ \hskip
30.00005pt+{\frac{1}{(L_{xi}+L_{xi+1}+L_{xi+2}+L_{xi+3})^{8}}\Bigr{]}\frac{1}{L_{xi}}}\\\
\hskip
30.00005pt-{\frac{\pi}{3}\Bigl{[}\frac{1}{(L_{xi+1}+L_{xi+2})^{2}}-\frac{1}{(L_{xi}+L_{xi+1}+L_{xi+2})^{2}}}\\\
\hskip 30.00005pt{-\frac{1}{(L_{xi+1}+L_{xi+2}+L_{xi+3})^{2}}}\\\ \hskip
30.00005pt+{\frac{1}{(L_{xi}+L_{xi+1}+L_{xi+2}+L_{xi+3})^{2}}\Bigr{]}\,\frac{1}{L_{xi}}}\hskip
20.00003pt(A4)\end{array}$
## References
* [1] S A Sartarelli, L Szybisz, Correlation between asymmetric profiles in slits and standard prewetting lines, Pap. Phys. 1, 010001 (2009); L Szybisz, S A Sartarelli, Density profiles of Ar adsorbed in slits of CO2: Spontaneous symmetry breaking revisited, J. Chem. Phys. 128, 124702 (2008).
* [2] M Schoen, Computer Simulation of Condensed Phases in Complex Geometries (Lecture Notes in Physics), Springer, Berlin (1993).
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|
arxiv-papers
| 2010-03-16T17:00:28 |
2024-09-04T02:49:09.121734
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "G. J. Zarragoicoechea, A. G. Meyra, V. A. Kuz",
"submitter": "Luis Ariel Pugnaloni",
"url": "https://arxiv.org/abs/1003.3187"
}
|
1003.3272
|
# Graphics Processing Units and High-Dimensional Optimization
Hua Zhoulabel=e1]huazhou@ucla.edu [ Kenneth Langelabel=e2]klange@ucla.edu [
Marc A. Suchardlabel=e3]msuchard@ucla.edu [ University of California, Los
Angeles Department of Human Genetics, University of California, Los Angeles,
. Departments of Biomathematics, Human Genetics, and Statistics, University
of California,, Los Angeles, . Departments of Biomathematics, Biostatistics,
and Human Genetics, University of California, Los Angeles, .
###### Abstract
This paper discusses the potential of graphics processing units (GPUs) in
high-dimensional optimization problems. A single GPU card with hundreds of
arithmetic cores can be inserted in a personal computer and dramatically
accelerates many statistical algorithms. To exploit these devices fully,
optimization algorithms should reduce to multiple parallel tasks, each
accessing a limited amount of data. These criteria favor EM and MM algorithms
that separate parameters and data. To a lesser extent block relaxation and
coordinate descent and ascent also qualify. We demonstrate the utility of GPUs
in nonnegative matrix factorization, PET image reconstruction, and
multidimensional scaling. Speedups of 100 fold can easily be attained. Over
the next decade, GPUs will fundamentally alter the landscape of computational
statistics. It is time for more statisticians to get on-board.
Block relaxation,
EM and MM algorithms,
multidimensional scaling,
nonnegative matrix factorization,
parallel computing,
PET scanning,
###### keywords:
, and
## 1 Introduction
Statisticians, like all scientists, are acutely aware that the clock speeds on
their desktops and laptops have stalled. Does this mean that statistical
computing has hit a wall? The answer fortunately is no, but the hardware
advances that we routinely expect have taken an interesting detour. Most
computers now sold have two to eight processing cores. Think of these as
separate CPUs on the same chip. Naive programmers rely on sequential
algorithms and often fail to take advantage of more than a single core.
Sophisticated programmers, the kind who work for commercial firms such as
Matlab, eagerly exploit parallel programming. However, multicore CPUs do not
represent the only road to the success of statistical computing.
Graphics processing units (GPUs) have caught the scientific community by
surprise. These devices are designed for graphics rendering in computer
animation and games. Propelled by these nonscientific markets, the old
technology of numerical (array) coprocessors has advanced rapidly. Highly
parallel GPUs are now making computational inroads against traditional CPUs in
image processing, protein folding, stock options pricing, robotics, oil
exploration, data mining, and many other areas [27]. We are starting to see
orders of magnitude improvement on some hard computational problems. Three
companies, Intel, NVIDIA, and AMD/ATI, dominate the market. Intel is
struggling to keep up with its more nimble competitors.
Modern GPUs support more vector and matrix operations, stream data faster, and
possess more local memory per core than their predecessors. They are also
readily available as commodity items that can be inserted as video cards on
modern PCs. GPUs have been criticized for their hostile programming
environment and lack of double precision arithmetic and error correction, but
these faults are being rectified. The CUDA programming environment [26] for
NVIDIA chips is now easing some of the programming chores. We could say more
about near-term improvements, but most pronouncements would be obsolete within
months.
Oddly, statisticians have been slow to embrace the new technology. Silberstein
et al [30] first demonstrated the potential for GPUs in fitting simple
Bayesian networks. Recently Suchard and Rambaut [32] have seen greater than
$100$-fold speed-ups in MCMC simulations in molecular phylogeny. Lee et al
[17] and Tibbits et al [33] are following suit with Bayesian model fitting via
particle filtering and slice sampling. Finally, work is under-way to port
common data mining techniques such as hierarchical clustering and multi-factor
dimensionality reduction onto GPUs [31]. These efforts constitute the first
wave of an eventual flood of statistical and data mining applications. The
porting of GPU tools into the R environment will undoubtedly accelerate the
trend [3].
Not all problems in computational statistics can benefit from GPUs. Sequential
algorithms are resistant unless they can be broken into parallel pieces. Even
parallel algorithms can be problematic if the entire range of data must be
accessed by each GPU. Because they have limited memory, GPUs are designed to
operate on short streams of data. The greatest speedups occur when all of the
GPUs on a card perform the same arithmetic operation simultaneously. Effective
applications of GPUs in optimization involves both separation of data and
separation of parameters.
In the current paper, we illustrate how GPUs can work hand in glove with the
MM algorithm, a generalization of the EM algorithm. In many optimization
problems, the MM algorithm explicitly separates parameters by replacing the
objective function by a sum of surrogate functions, each of which involves a
single parameter. Optimization of the one-dimensional surrogates can be
accomplished by assigning each subproblem to a different core. Provided the
different cores each access just a slice of the data, the parallel subproblems
execute quickly. By construction the new point in parameter space improves the
value of the objective function. In other words, MM algorithms are iterative
ascent or descent algorithms. If they are well designed, then they separate
parameters in high-dimensional problems. This is where GPUs enter. They offer
most of the benefits of distributed computer clusters at a fraction of the
cost. For this reason alone, computational statisticians need to pay attention
to GPUs.
$\begin{array}[]{cc}\includegraphics[width=166.2212pt]{RosenbrockSurface.eps}&\includegraphics[width=166.2212pt]{RosenbrockMM-q0.eps}\end{array}$
Figure 1: Left: The Rosenbrock (banana) function (the lower surface) and a
majorization function at point (-1,-1) (the upper surface). Right: MM
iterates.
Before formally defining the MM algorithm, it may help the reader to walk
through a simple numerical example stripped of statistical content. Consider
the Rosenbrock test function
$\displaystyle f({\bf x})$ $\displaystyle=$ $\displaystyle
100(x_{1}^{2}-x_{2})^{2}+(x_{1}-1)^{2}$ $\displaystyle=$ $\displaystyle
100(x_{1}^{4}+x_{2}^{2}-2x_{1}^{2}x_{2})+(x_{1}^{2}-2x_{1}+1),$
familiar from the minimization literature. As we iterate toward the minimum at
${\bf x}={\bf 1}=(1,1)$, we construct a surrogate function that separates
parameters. This is done by exploiting the obvious majorization
$\displaystyle-2x_{1}^{2}x_{2}$ $\displaystyle\leq$ $\displaystyle
x_{1}^{4}+x_{2}^{2}+(x_{n1}^{2}+x_{n2})^{2}-2(x_{n1}^{2}+x_{n2})(x_{1}^{2}+x_{2}),$
where equality holds when ${\bf x}$ and the current iterate ${\bf x}_{n}$
coincide. It follows that $f({\bf x})$ itself is majorized by the sum of the
two surrogates
$\displaystyle g_{1}(x_{1}\mid{\bf x}_{n})$ $\displaystyle=$ $\displaystyle
200x_{1}^{4}-[200(x_{n1}^{2}+x_{n2})-1]x_{1}^{2}-2x_{1}+1$ $\displaystyle
g_{2}(x_{2}\mid{\bf x}_{n})$ $\displaystyle=$ $\displaystyle
200x_{2}^{2}-200(x_{n1}^{2}+x_{n2})x_{2}+(x_{n1}^{2}+x_{n2})^{2}.$
The left panel of Figure 1 depicts the Rosenbrock function and its
majorization $g_{1}(x_{1}\mid{\bf x}_{n})+g_{2}(x_{2}\mid{\bf x}_{n})$ at the
point $-{\bf 1}$.
According to the MM recipe, at each iteration one must minimize the quartic
polynomial $g_{1}(x_{1}\mid{\bf x}_{n})$ and the quadratic polynomial
$g_{2}(x_{2}\mid{\bf x}_{n})$. The quartic possesses either a single global
minimum or two local minima separated by a local maximum These minima are the
roots of the cubic function $g_{1}^{\prime}(x_{1}|{\bf x}_{n})$ and can be
explicitly computed. We update $x_{1}$ by the root corresponding to the global
minimum and $x_{2}$ via $x_{n+1,2}=\frac{1}{2}(x_{n1}^{2}+x_{n2})$. The right
panel of Figure 1 displays the iterates starting from ${\bf x}_{0}=-{\bf 1}$.
These immediately jump into the Rosenbrock valley and then slowly descend to
${\bf 1}$.
Separation of parameters in this example makes it easy to decrease the
objective function. This almost trivial advantage is amplified when we
optimize functions depending on tens of thousands to millions of parameters.
In these settings, Newton’s method and variants such as Fisher’s scoring are
fatally handicapped by the need to store, compute, and invert huge Hessian or
information matrices. On the negative side of the balance sheet, MM algorithms
are often slow to converge. This disadvantage is usually outweighed by the
speed of their updates even in sequential mode. If one can harness the power
of parallel processing GPUs, then MM algorithms become the method of choice
for many high-dimensional problems.
We conclude this introduction by sketching a roadmap to the rest of the paper.
Section 2 reviews the MM algorithm. Section 3 discusses three high-dimensional
MM examples. Although the algorithm in each case is known, we present brief
derivations to illustrate how simple inequalities drive separation of
parameters. We then implement each algorithm on a realistic problem and
compare running times in sequential and parallel modes. We purposefully omit
programming syntax since many tutorials already exist for this purpose, and
material of this sort is bound to be ephemeral. Section 4 concludes with a
brief discussion of other statistical applications of GPUs and other methods
of accelerating optimization algorithms.
## 2 MM Algorithms
The MM algorithm like the EM algorithm is a principle for creating
optimization algorithms. In minimization the acronym MM stands for
majorization-minimization; in maximization it stands for minorization-
maximization. Both versions are convenient in statistics. For the moment we
will concentrate on maximization.
Let $f(\boldsymbol{\theta})$ be the objective function whose maximum we seek.
Its argument $\boldsymbol{\theta}$ can be high-dimensional and vary over a
constrained subset $\Theta$ of Euclidean space. An MM algorithm involves
minorizing $f(\boldsymbol{\theta})$ by a surrogate function
$g(\boldsymbol{\theta}\mid\boldsymbol{\theta}_{n})$ anchored at the current
iterate $\boldsymbol{\theta}_{n}$ of the search. The subscript $n$ indicates
iteration number throughout this article. If $\boldsymbol{\theta}_{n+1}$
denotes the maximum of $g(\boldsymbol{\theta}\mid\boldsymbol{\theta}_{n})$
with respect to its left argument, then the MM principle declares that
$\boldsymbol{\theta}_{n+1}$ increases $f(\boldsymbol{\theta})$ as well. Thus,
MM algorithms revolve around a basic ascent property.
Minorization is defined by the two properties
$\displaystyle f(\boldsymbol{\theta}_{n})$ $\displaystyle=$ $\displaystyle
g(\boldsymbol{\theta}_{n}\mid\boldsymbol{\theta}_{n})$ (2.1) $\displaystyle
f(\boldsymbol{\theta})$ $\displaystyle\geq$ $\displaystyle
g(\boldsymbol{\theta}\mid\boldsymbol{\theta}_{n})\>,\quad\quad\boldsymbol{\theta}\neq\boldsymbol{\theta}_{n}.$
(2.2)
In other words, the surface $\boldsymbol{\theta}\mapsto
g(\boldsymbol{\theta}\mid\boldsymbol{\theta}_{n})$ lies below the surface
$\boldsymbol{\theta}\mapsto f(\boldsymbol{\theta})$ and is tangent to it at
the point $\boldsymbol{\theta}=\boldsymbol{\theta}_{n}$. Construction of the
minorizing function $g(\boldsymbol{\theta}\mid\boldsymbol{\theta}_{n})$
constitutes the first M of the MM algorithm. In our examples
$g(\boldsymbol{\theta}\mid\boldsymbol{\theta}_{n})$ is chosen to separate
parameters.
In the second M of the MM algorithm, one maximizes the surrogate
$g(\boldsymbol{\theta}\mid\boldsymbol{\theta}_{n})$ rather than
$f(\boldsymbol{\theta})$ directly. It is straightforward to show that the
maximum point $\boldsymbol{\theta}_{n+1}$ satisfies the ascent property
$f(\boldsymbol{\theta}_{n+1})\geq f(\boldsymbol{\theta}_{n})$. The proof
$\displaystyle f(\boldsymbol{\theta}_{n+1})$ $\displaystyle\geq$
$\displaystyle
g(\boldsymbol{\theta}_{n+1}\mid\boldsymbol{\theta}_{n})\;\;\geq\;\;g(\boldsymbol{\theta}_{n}\mid\boldsymbol{\theta}_{n})\;\;=\;\;f(\boldsymbol{\theta}_{n})$
reflects definitions (2.1) and (2.2) and the choice of
$\boldsymbol{\theta}_{n+1}$. The ascent property is the source of the MM
algorithm’s numerical stability and remains valid if we merely increase
$g(\boldsymbol{\theta}\mid\boldsymbol{\theta}_{n})$ rather than maximize it.
In many problems MM updates are delightfully simple to code, intuitively
compelling, and automatically consistent with parameter constraints. In
minimization we seek a majorizing function
$g(\boldsymbol{\theta}\mid\boldsymbol{\theta}_{n})$ lying above the surface
$\boldsymbol{\theta}\mapsto f(\boldsymbol{\theta})$ and tangent to it at the
point $\boldsymbol{\theta}=\boldsymbol{\theta}_{n}$. Minimizing
$g(\boldsymbol{\theta}\mid\boldsymbol{\theta}_{n})$ drives
$f(\boldsymbol{\theta})$ downhill.
The celebrated Expectation-Maximization (EM) algorithm [7, 21] is a special
case of the MM algorithm. The $Q$-function produced in the E step of the EM
algorithm constitutes a minorizing function of the loglikelihood. Thus, both
EM and MM share the same advantages: simplicity, stability, graceful
adaptation to constraints, and the tendency to avoid large matrix inversion.
The more general MM perspective frees algorithm derivation from the missing
data straitjacket and invites wider applications. For example, our multi-
dimensional scaling (MDS) and non-negative matrix factorization (NNFM)
examples involve no likelihood functions. Wu and Lange [37] briefly summarize
the history of the MM algorithm and its relationship to the EM algorithm.
The convergence properties of MM algorithms are well-known [15]. In
particular, five properties of the objective function $f(\boldsymbol{\theta})$
and the MM algorithm map $\boldsymbol{\theta}\mapsto M(\boldsymbol{\theta})$
guarantee convergence to a stationary point of $f(\boldsymbol{\theta})$: (a)
$f(\boldsymbol{\theta})$ is coercive on its open domain; (b)
$f(\boldsymbol{\theta})$ has only isolated stationary points; (c)
$M(\boldsymbol{\theta})$ is continuous; (d) $\boldsymbol{\theta}^{*}$ is a
fixed point of $M(\boldsymbol{\theta})$ if and only if
$\boldsymbol{\theta}^{*}$ is a stationary point of $f(\boldsymbol{\theta})$;
and (e) $f[M(\boldsymbol{\theta}^{*})]\geq f(\boldsymbol{\theta}^{*})$, with
equality if and only if $\boldsymbol{\theta}^{*}$ is a fixed point of
$M(\boldsymbol{\theta})$. These conditions are easy to verify in many
applications. The local rate of convergence of an MM algorithm is intimately
tied to how well the surrogate function
$g(\boldsymbol{\theta}\mid\boldsymbol{\theta}^{*})$ approximates the objective
function $f(\boldsymbol{\theta})$ near the optimal point
$\boldsymbol{\theta}^{*}$.
## 3 Numerical Examples
In this section, we compare the performances of the CPU and GPU
implementations of three classical MM algorithms coded in C++: (a) non-
negative matrix factorization (NNMF), (b) positron emission tomography (PET),
and (c) multidimensional scaling (MDS). In each case we briefly derive the
algorithm from the MM perspective. For the CPU version, we iterate until the
relative change
$\displaystyle\frac{|f(\boldsymbol{\theta}_{n})-f(\boldsymbol{\theta}_{n-1})|}{|f(\boldsymbol{\theta}_{n-1})|+1}$
of the objective function $f(\boldsymbol{\theta})$ between successive
iterations falls below a pre-set threshold $\epsilon$ or the number of
iterations reaches a pre-set number $n_{\rm max}$, whichever comes first. In
these examples, we take $\epsilon=10^{-9}$ and $n_{\rm max}=100,000$. For ease
of comparison, we iterate the GPU version for the same number of steps as the
CPU version. Overall, we see anywhere from a 22-fold to 112-fold decrease in
total run time. The source code is freely available from the first author.
Table 1 shows how our desktop system is configured. Although the CPU is a
high-end processor with four cores, we use just one of these for ease of
comparison. In practice, it takes considerable effort to load balance the
various algorithms across multiple CPU cores. With 240 GPU cores, the GTX 280
GPU card delivers a peak performance of about 933 GFlops in single precision.
This card is already obsolete. Newer cards possess twice as many cores, and up
to four cards can fit inside a single desktop computer. It is relatively
straightforward to program multiple GPUs. Because previous generation GPU
hardware is largely limited to single precision, this is a worry in scientific
computing. To assess the extent of roundoff error, we display the converged
values of the objective functions to ten significant digits. Only rarely is
the GPU value far off the CPU mark. Finally, the extra effort in programming
the GPU version is relatively light. Exploiting the standard CUDA library
[26], it takes 77, 176, and 163 extra lines of GPU code to implement the NNMF,
PET, and MDS examples, respectively.
| CPU | GPU
---|---|---
Model | Intel Core 2 | NVIDIA GeForce
| Extreme X9440 | GTX 280
# Cores | 4 | 240
Clock | 3.2G | 1.3G
Memory | 16G | 1G
Table 1: Configuration of the desktop system
### 3.1 Non-Negative Matrix Factorizations
Non-negative matrix factorization (NNMF) is an alternative to principle
component analysis useful in modeling, compressing, and interpreting
nonnegative data such as observational counts and images. The articles [18,
19, 2] discuss in detail algorithm development and statistical applications of
NNMF. The basic problem is to approximate a data matrix ${\bf X}$ with
nonnegative entries $x_{ij}$ by a product ${\bf V}{\bf W}$ of two low rank
matrices ${\bf V}$ and ${\bf W}$ with nonnegative entries $v_{ik}$ and
$w_{kj}$. Here ${\bf X}$, ${\bf V}$, and ${\bf W}$ are $p\times q$, $p\times
r$, and $r\times q$, respectively, with $r$ much smaller than $\min\\{p,q\\}$.
One version of NNMF minimizes the objective function
$\displaystyle f({\bf V},{\bf W})$ $\displaystyle=$ $\displaystyle\|{\bf
X}-{\bf V}{\bf
W}\|_{\text{F}}^{2}\;\;=\;\;\sum_{i}\sum_{j}\Big{(}x_{ij}-\sum_{k}v_{ik}w_{kj}\Big{)}^{2},$
(3.1)
where $\|\cdot\|_{\text{F}}$ denotes the Frobenius-norm. To get an idea of the
scale of NNFM imaging problems, $p$ (number of images) can range
$10^{1}-10^{4}$, $q$ (number of pixels per image) can surpass $10^{2}-10^{4}$,
and one seeks a rank $r$ approximation of about 50. Notably, part of the
winning solution of the Netflix challenge relies on variations of NNMF [12].
For the Netflix data matrix, $p=480,000$ (raters), $q=18,000$ (movies), and
$r$ ranged from 20 to 100.
Exploiting the convexity of the function $x\mapsto(x_{ij}-x)^{2}$, one can
derive the inequality
$\displaystyle\Big{(}x_{ij}-\sum_{k}v_{ik}w_{kj}\Big{)}^{2}$
$\displaystyle\leq$
$\displaystyle\sum_{k}\frac{a_{nikj}}{b_{nij}}\left(x_{ij}-\frac{b_{nij}}{a_{nikj}}v_{ik}w_{kj}\right)^{2}$
where $a_{nikj}=v_{nik}w_{nkj}$ and $b_{nij}=\sum_{k}a_{nikj}$. This leads to
the surrogate function
$\displaystyle g({\bf V},{\bf W}\mid{\bf V}_{n},{\bf W}_{n})$ $\displaystyle=$
$\displaystyle\sum_{i}\sum_{j}\sum_{k}\frac{a_{nikj}}{b_{nij}}\left(x_{ij}-\frac{b_{nij}}{a_{nikj}}v_{ik}w_{kj}\right)^{2}$
(3.2)
majorizing the objective function $f({\bf V},{\bf W})=\|{\bf X}-{\bf V}{\bf
W}\|_{\text{F}}^{2}$. Although the majorization (3.2) does not achieve a
complete separation of parameters, it does if we fix ${\bf V}$ and update
${\bf W}$ or vice versa. This strategy is called block relaxation.
If we elect to minimize $g({\bf V},{\bf W}\mid{\bf V}_{n},{\bf W}_{n})$
holding ${\bf W}$ fixed at ${\bf W}_{n}$, then the stationarity condition for
${\bf V}$ reads
$\displaystyle\frac{\partial}{\partial v_{ik}}g({\bf V},{\bf W}_{n}\mid{\bf
V}_{n},{\bf W}_{n})$ $\displaystyle=$
$\displaystyle-2\sum_{j}\Big{(}x_{ij}-\frac{b_{nij}}{a_{nikj}}v_{ik}w_{nkj}\Big{)}w_{nkj}\;\;=\;\;0.$
Its solution furnishes the simple multiplicative update
$\displaystyle v_{n+1,ik}$ $\displaystyle=$ $\displaystyle
v_{nik}\frac{\sum_{j}x_{ij}w_{nkj}}{\sum_{j}b_{nij}w_{nkj}}.$ (3.3)
Likewise the stationary condition
$\displaystyle\frac{\partial}{\partial w_{kj}}g({\bf V}_{n+1},{\bf W}\mid{\bf
V}_{n+1},{\bf W}_{n})$ $\displaystyle=$ $\displaystyle 0$
gives the multiplicative update
$\displaystyle w_{n+1,kj}$ $\displaystyle=$ $\displaystyle
w_{nkj}\frac{\sum_{i}x_{ij}v_{n+1,ik}}{\sum_{i}c_{nij}v_{n+1,ik}},$ (3.4)
where $c_{nij}=\sum_{k}v_{n+1,ik}w_{nkj}$. Close inspection of the
multiplicative updates (3.3) and (3.4) shows that their numerators depend on
the matrix products ${\bf X}{\bf W}_{n}^{t}$ and ${\bf V}_{n+1}^{t}{\bf X}$
and their denominators depend on the matrix products ${\bf V}_{n}{\bf
W}_{n}{\bf W}_{n}^{t}$ and ${\bf V}_{n+1}^{t}{\bf V}_{n+1}{\bf W}_{n}$. Large
matrix multiplications are very fast on GPUs because CUDA implements in
parallel the BLAS (basic linear algebra subprograms) library widely applied in
numerical analysis [25]. Once the relevant matrix products are available, each
elementwise update of $v_{ik}$ or $w_{kj}$ involves just a single
multiplication and division. These scalar operations are performed in parallel
through hand-written GPU code. Algorithm 1 summarizes the steps in performing
NNMF.
Initialize: Draw $v_{0ik}$ and $w_{0kj}$ uniform on (0,1) for all $1\leq i\leq
p$, $1\leq k\leq r$, $1\leq j\leq q$
repeat
Compute $\mathbf{X}\mathbf{W}_{n}^{t}$ and
$\mathbf{V}_{n}\mathbf{W}_{n}\mathbf{W}_{n}^{t}$
$v_{n+1,ik}\leftarrow
v_{nik}\cdot\\{\mathbf{X}\mathbf{W}_{n}^{t}\\}_{ik}\,/\,\\{\mathbf{V}_{n}\mathbf{W}_{n}\mathbf{W}_{n}^{t}\\}_{ik}$
for all $1\leq i\leq p$, $1\leq k\leq r$
Compute $\mathbf{V}_{n+1}^{t}\mathbf{X}$ and
$\mathbf{V}_{n+1}^{t}\mathbf{V}_{n+1}\mathbf{W}_{n}$
$w_{n+1,kj}\leftarrow w_{nkj}\cdot\\{{\bf V}_{n+1}^{t}{\bf
X}\\}_{kj}\,/\,\\{{\bf V}_{n+1}^{t}{\bf V}_{n+1}{\bf W}_{n}\\}_{kj}$ for all
$1\leq k\leq r$, $1\leq j\leq q$
until convergence occurs
Algorithm 1 (NNMF) Given ${\bf X}\in\mathbb{R}_{+}^{p\times q}$, find ${\bf
V}\in\mathbb{R}_{+}^{p\times r}$ and ${\bf W}\in\mathbb{R}_{+}^{r\times q}$
minimizing $\|{\bf X}-{\bf V}{\bf W}\|_{\text{F}}^{2}$.
We now compare CPU and GPU versions of the multiplicative NNMF algorithm on a
training set of face images. Database #1 from the MIT Center for Biological
and Computational Learning (CBCL) [24] reduces to a matrix ${\bf X}$
containing $p=2,429$ gray scale face images with $q=19\times 19=361$ pixels
per face. Each image (row) is scaled to have mean and standard deviation 0.25.
Figure 2 shows the recovery of the first face in the database using a rank
$r=49$ decomposition. The 49 basis images (rows of ${\bf W}$) represent
different aspects of a face. The rows of ${\bf V}$ contain the coefficients of
these parts estimated for the various faces. Some of these facial features are
immediately obvious in the reconstruction. Table 2 compares the run times of
Algorithm 1 implemented on our CPU and GPU respectively. We observe a 22 to
112-fold speed-up in the GPU implementation. Run times for the GPU version
depend primarily on the number of iterations to convergence and very little on
the rank $r$ of the approximation. Run times of the CPU version scale linearly
in both the number of iterations and $r$.
| | CPU | GPU |
---|---|---|---|---
Rank $r$ | Iters | Time | Function | Time | Function | Speedup
10 | 25459 | 1203 | 106.2653503 | 55 | 106.2653504 | 22
20 | 87801 | 7564 | 89.56601262 | 163 | 89.56601287 | 46
30 | 55783 | 7013 | 78.42143486 | 103 | 78.42143507 | 68
40 | 47775 | 7880 | 70.05415929 | 119 | 70.05415950 | 66
50 | 53523 | 11108 | 63.51429261 | 121 | 63.51429219 | 92
60 | 77321 | 19407 | 58.24854375 | 174 | 58.24854336 | 112
Table 2: Run-time (in seconds) comparisons for NNMF on the MIT CBCL face image
data. The dataset contains $p=2,429$ faces with $q=19\times 19=361$ pixels per
face. The columns labeled Function refer to the converged value of the
objective function. Figure 2: Approximation of a face image by rank-49 NNMF:
coefficients $\times$ basis images = approximate image.
It is worth stressing a few points. First, the objective function (3.1) is
convex in ${\bf V}$ for ${\bf W}$ fixed, and vice versa but not jointly
convex. Thus, even though the MM algorithm enjoys the descent property, it is
not guaranteed to find the global minimum [2]. There are two good alternatives
to the multiplicative algorithm. First, pure block relaxation can be conducted
by alternating least squares (ALS). In updating ${\bf V}$ with ${\bf W}$
fixed, ALS omits majorization and solves the $p$ separated nonnegative least
square problems
$\displaystyle\min_{{\bf V}(i,:)}\|{\bf X}(i,:)-{\bf V}(i,:){\bf
W}]\|_{2}^{2}\quad\text{ subject to }{\bf V}(i,:)\geq 0,$
where ${\bf V}(i,:)$ and ${\bf X}(i,:)$ denote the $i$-th row of the
corresponding matrices. Similarly, in updating ${\bf W}$ with ${\bf V}$ fixed,
ALS solves $q$ separated nonnegative least square problems. Another
possibility is to change the objective function to
$\displaystyle L({\bf V},{\bf W})$ $\displaystyle=$
$\displaystyle\sum_{i}\sum_{j}\Big{[}x_{ij}\ln\Big{(}\sum_{k}v_{ik}w_{kj}\Big{)}-\sum_{k}v_{ik}w_{kj}\Big{]}$
according to a Poisson model for the counts $x_{ij}$ [18]. This works even
when some entries $x_{ij}$ fail to be integers, but the Poisson loglikelihood
interpretation is lost. A pure MM algorithm for maximizing $L({\bf V},{\bf
W})$ is
$\displaystyle v_{n+1,ik}$ $\displaystyle=$ $\displaystyle
v_{nik}\sqrt{\frac{\sum_{j}x_{ij}w_{nkj}/b_{nij}}{\sum_{j}w_{nkj}}},\quad
w_{n+1,ij}=w_{nkj}\sqrt{\frac{\sum_{i}x_{ij}v_{nik}/b_{nij}}{\sum_{i}v_{nik}}}.$
Derivation of these variants of Lee and Seung’s [18] Poisson updates is left
to the reader.
### 3.2 Positron Emission Tomography
The field of computed tomography has exploited EM algorithms for many years.
In positron emission tomography (PET), the reconstruction problem consists of
estimating the Poisson emission intensities
$\boldsymbol{\lambda}=(\lambda_{1},\ldots,\lambda_{p})$ of $p$ pixels arranged
in a 2-dimensional grid surrounded by an array of photon detectors. The
observed data are coincidence counts $(y_{1},\ldots y_{d})$ along $d$ lines of
flight connecting pairs of photon detectors. The loglikelihood under the PET
model is
$\displaystyle L(\boldsymbol{\lambda})$ $\displaystyle=$
$\displaystyle\sum_{i}\Big{[}y_{i}\ln\Big{(}\sum_{j}e_{ij}\lambda_{j}\Big{)}-\sum_{j}e_{ij}\lambda_{j}\Big{]},$
where the $e_{ij}$ are constants derived from the geometry of the grid and the
detectors. Without loss of generality, one can assume $\sum_{i}e_{ij}=1$ for
each $j$. It is straightforward to derive the traditional EM algorithm [13,
36] from the MM perspective using the concavity of the function $\ln s$.
Indeed, application of Jensen’s inequality produces the minorization
$\displaystyle L(\boldsymbol{\lambda})$ $\displaystyle\geq$
$\displaystyle\sum_{i}y_{i}\sum_{j}w_{nij}\ln\Big{(}\frac{e_{ij}\lambda_{j}}{w_{nij}}\Big{)}-\sum_{i}\sum_{j}e_{ij}\lambda_{j}\;\;=\;\;Q(\boldsymbol{\lambda}\mid\boldsymbol{\lambda}_{n}),$
where $w_{nij}=e_{ij}\lambda_{nj}/(\sum_{k}e_{ik}\lambda_{nk})$. This maneuver
again separates parameters. The stationarity conditions for the surrogate
$Q(\boldsymbol{\lambda}\mid\boldsymbol{\lambda}_{n})$ supply the parallel
updates
$\displaystyle\lambda_{n+1,j}$ $\displaystyle=$
$\displaystyle\frac{\sum_{i}y_{i}w_{nij}}{\sum_{i}e_{ij}}.$ (3.5)
The convergence of the PET algorithm (3.5) is frustratingly slow, even under
systematic acceleration [29, 39]. Furthermore, the reconstructed images are of
poor quality with a grainy appearance. The early remedy of premature halting
of the algorithm cuts computational cost but is entirely ad hoc, and the final
image depends on initial conditions. A better option is add a roughness
penalty to the loglikelihood. This device not only produces better images but
also accelerates convergence. Thus, we maximize the penalized loglikelihood
$\displaystyle f(\boldsymbol{\lambda})$ $\displaystyle=$ $\displaystyle
L(\boldsymbol{\lambda})-\frac{\mu}{2}\sum_{\\{j,k\\}\in{\cal
N}}(\lambda_{j}-\lambda_{k})^{2}$ (3.6)
where $\mu$ is the roughness penalty constant, and ${\cal N}$ is the
neighborhood system that pairs spatially adjacent pixels. An absolute value
penalty is less likely to deter the formation of edges than a square penalty,
but it is easier to deal with a square penalty analytically, and we adopt it
for the sake of simplicity. In practice, visual inspection of the recovered
images guides the selection of the roughness penalty constant $\mu$.
To maximize $f(\boldsymbol{\lambda})$ by an MM algorithm, we must minorize the
penalty in a manner consistent with the separation of parameters. In view of
the evenness and convexity of the function $s^{2}$, we have
$\displaystyle(\lambda_{j}-\lambda_{k})^{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}(2\lambda_{j}-\lambda_{nj}-\lambda_{nk})^{2}+\frac{1}{2}(2\lambda_{k}-\lambda_{nj}-\lambda_{nk})^{2}.$
Equality holds if $\lambda_{j}+\lambda_{k}=\lambda_{nj}+\lambda_{nk}$, which
is true when $\boldsymbol{\lambda}=\boldsymbol{\lambda}_{n}$. Combining our
two minorizations furnishes the surrogate function
$\displaystyle
g(\boldsymbol{\lambda}\mid\boldsymbol{\lambda}_{n})=Q(\boldsymbol{\lambda}\mid\boldsymbol{\lambda}_{n})-\frac{\mu}{4}\sum_{\\{j,k\\}\in{\cal
N}}\Big{[}(2\lambda_{j}-\lambda_{nj}-\lambda_{nk})^{2}+(2\lambda_{k}-\lambda_{nj}-\lambda_{nk})^{2}\Big{]}.$
To maximize $g(\boldsymbol{\lambda}\mid\boldsymbol{\lambda}_{n})$, we define
${\cal N}_{j}=\\{k:\\{j,k\\}\in{\cal N}\\}$ and set the partial derivative
$\displaystyle\frac{\partial}{\partial\lambda_{j}}g(\boldsymbol{\lambda}\mid\boldsymbol{\lambda}_{n})$
$\displaystyle=$
$\displaystyle\sum_{i}\Big{[}\frac{y_{i}w_{nij}}{\lambda_{j}}-e_{ij}\Big{]}-\mu\sum_{k:\in{\cal
N}_{j}}(2\lambda_{j}-\lambda_{nj}-\lambda_{nk})$ (3.7)
equal to 0 and solve for $\lambda_{n+1,j}$. Multiplying equation (3.7) by
$\lambda_{j}$ produces a quadratic with roots of opposite signs. We take the
positive root
$\displaystyle\lambda_{n+1,j}$ $\displaystyle=$
$\displaystyle\frac{-b_{nj}-\sqrt{b_{nj}^{2}-4a_{j}c_{nj}}}{2a_{j}},$
where
$\displaystyle a_{j}$ $\displaystyle=$ $\displaystyle-2\mu\sum_{k\in{\cal
N}_{j}}1,\quad b_{nj}\;\;=\;\;\sum_{k\in{\cal
N}_{j}}(\lambda_{nj}+\lambda_{nk})-1,\quad
c_{nj}\;\;=\;\;\sum_{i}y_{i}w_{nij}.$
Algorithm 2 summarizes the complete MM scheme. Obviously, complete parameter
separation is crucial. The quantities $a_{j}$ can be computed once and stored.
The quantities $b_{nj}$ and $c_{nj}$ are computed for each $j$ in parallel. To
improve GPU performance in computing the sums over $i$, we exploit the widely
available parallel sum-reduction techniques [30]. Given these results, a
specialized but simple GPU code computes the updates $\lambda_{n+1,j}$ for
each $j$ in parallel.
Table 3 compares the run times of the CPU and GPU implementations for a
simulated PET image [29]. The image as depicted in the top of Figure 3 has
$p=64\times 64=4,096$ pixels and is interrogated by $d=2,016$ detectors.
Overall we see a 43- to 53-fold reduction in run times with the GPU
implementation. Figure 3 displays the true image and the estimated images
under penalties of $\mu=0$, $10^{-5}$, $10^{-6}$, and $10^{-7}$. Without
penalty ($\mu=0$), the algorithm fails to converge in 100,000 iterations.
Scale $\bf{E}$ to have unit $l_{1}$ column norms.
Compute $|{\cal N}_{j}|=\sum_{k:\\{j,k\\}\in{\cal N}}1$ and $a_{j}-2\mu|{\cal
N}_{j}|$ for all $1\leq j\leq p$.
Initialize: $\lambda_{0j}\leftarrow 1$, $j=1,\ldots,p$.
repeat
$z_{nij}\leftarrow(y_{i}e_{ij}\lambda_{nj})/(\sum_{k}e_{ik}\lambda_{nk})$ for
all $1\leq i\leq d$, $1\leq j\leq p$
for $j=1$ to $p$ do
$b_{nj}\leftarrow\mu(|{\cal N}_{j}|\lambda_{nj}+\sum_{k\in{\cal
N}_{j}}\lambda_{nk})-1$
$c_{nj}\leftarrow\sum_{i}z_{nij}$
$\lambda_{n+1,j}\leftarrow(-b_{nj}-\sqrt{b_{nj}^{2}-4a_{j}c_{nj}})/(2a_{j})$
end for
until convergence occurs
Algorithm 2 (PET Image Recovering) Given the coefficient matrix
$\mathbf{E}\in\mathbb{R}_{+}^{d\times p}$, coincident counts ${\bf
y}=(y_{1},\ldots,y_{d})\in\mathbf{Z}_{+}^{d}$, and roughness parameter
$\mu>0$, find the intensity vector
$\boldsymbol{\lambda}=(\lambda_{1},\ldots,\lambda_{p})\in\mathbb{R}_{+}^{p}$
that maximizes the objective function (3.6).
| CPU | GPU | QN(10) on CPU
---|---|---|---
Penalty $\mu$ | Iters | Time | Function | Iters | Time | Function | Speedup | Iters | Time | Function | Speedup
0 | 100000 | 14790 | -7337.152765 | 100000 | 282 | -7337.153387 | 52 | 6549 | 2094 | -7320.100952 | n/a
$10^{-7}$ | 24457 | 3682 | -8500.083033 | 24457 | 70 | -8508.112249 | 53 | 251 | 83 | -8500.077057 | 44
$10^{-6}$ | 6294 | 919 | -15432.45496 | 6294 | 18 | -15432.45586 | 51 | 80 | 29 | -15432.45366 | 32
$10^{-5}$ | 589 | 86 | -55767.32966 | 589 | 2 | -55767.32970 | 43 | 19 | 9 | -55767.32731 | 10
Table 3: Comparison of run times (in seconds) for a PET imaging problem on the
simulated data in [29]. The image has $p=64\times 64=4,096$ pixels and is
interrogated by $d=2,016$ detectors. The columns labeled Function refer to the
converged value of the objective function. The results under the heading
$QN(10)$ on CPU invoke quasi-Newton acceleration [39] with 10 secant
conditions.
$\begin{array}[]{cc}\includegraphics[width=144.54pt]{pet-gpu-
penalty-0.eps}&\includegraphics[width=144.54pt]{pet-gpu-penalty-1e-7.eps}\\\
\includegraphics[width=144.54pt]{pet-gpu-
penalty-1e-6.eps}&\includegraphics[width=144.54pt]{pet-gpu-
penalty-1e-5.eps}\\\ \end{array}$
Figure 3: The true PET image (top) and the recovered images with penalties
$\mu=0$, $10^{-7}$, $10^{-6}$, and $10^{-5}$.
### 3.3 Multidimensional Scaling
Multidimensional scaling (MDS) was the first statistical application of the MM
principle [6, 5]. MDS represents $q$ objects as faithfully as possible in
$p$-dimensional space given a nonnegative weight $w_{ij}$ and a nonnegative
dissimilarity measure $y_{ij}$ for each pair of objects $i$ and $j$. If
$\boldsymbol{\theta}^{i}\in\mathbb{R}^{p}$ is the position of object $i$, then
the $p\times q$ parameter matrix
$\boldsymbol{\theta}=(\boldsymbol{\theta}^{1},\ldots,\boldsymbol{\theta}^{q})$
is estimated by minimizing the stress function
$\displaystyle f(\boldsymbol{\theta})$ $\displaystyle=$
$\displaystyle\sum_{1\leq i<j\leq
q}w_{ij}(y_{ij}-\|\boldsymbol{\theta}^{i}-\boldsymbol{\theta}^{j}\|)^{2}$
$\displaystyle=$
$\displaystyle\sum_{i<j}w_{ij}y_{ij}^{2}-2\sum_{i<j}w_{ij}y_{ij}\|\boldsymbol{\theta}^{i}-\boldsymbol{\theta}^{j}\|+\sum_{i<j}w_{ij}\|\boldsymbol{\theta}^{i}-\boldsymbol{\theta}^{j}\|^{2},$
where $\|\boldsymbol{\theta}^{i}-\boldsymbol{\theta}^{j}\|$ is the Euclidean
distance between $\boldsymbol{\theta}^{i}$ and $\boldsymbol{\theta}^{j}$. The
stress function (3.3) is invariant under translations, rotations, and
reflections of $\mathbb{R}^{p}$. To avoid translational and rotational
ambiguities, we take $\boldsymbol{\theta}^{1}$ to be the origin and the first
$p-1$ coordinates of $\boldsymbol{\theta}^{2}$ to be 0. Switching the sign of
$\theta^{2}_{p}$ leaves the stress function invariant. Hence, convergence to
one member of a pair of reflected minima immediately determines the other
member.
Given these preliminaries, we now review the derivation of the MM algorithm
presented in [16]. Because we want to minimize the stress, we majorize it. The
middle term in the stress (3.3) is majorized by the Cauchy-Schwartz inequality
$\displaystyle-\|\boldsymbol{\theta}^{i}-\boldsymbol{\theta}^{j}\|$
$\displaystyle\leq$
$\displaystyle-\frac{(\boldsymbol{\theta}^{i}-\boldsymbol{\theta}^{j})^{t}(\boldsymbol{\theta}^{i}_{n}-\boldsymbol{\theta}^{j}_{n})}{\|\boldsymbol{\theta}^{i}_{n}-\boldsymbol{\theta}^{j}_{n}\|}.$
To separate the parameters in the summands of the third term of the stress, we
invoke the convexity of the Euclidean norm $\|\cdot\|$ and the square function
$s^{2}$. These maneuvers yield
$\displaystyle\|\boldsymbol{\theta}^{i}-\boldsymbol{\theta}^{j}\|^{2}$
$\displaystyle=$
$\displaystyle\Big{\|}\frac{1}{2}\Big{[}2\boldsymbol{\theta}^{i}-(\boldsymbol{\theta}^{i}_{n}+\boldsymbol{\theta}^{j}_{n})\Big{]}-\frac{1}{2}\Big{[}2\boldsymbol{\theta}^{j}-(\boldsymbol{\theta}^{j}_{n}+\boldsymbol{\theta}^{j}_{n})\Big{]}\Big{\|}^{2}$
$\displaystyle\leq$ $\displaystyle
2\Big{\|}\boldsymbol{\theta}^{i}-\frac{1}{2}(\boldsymbol{\theta}^{i}_{n}+\boldsymbol{\theta}^{j}_{n})\Big{\|}^{2}+2\Big{\|}\boldsymbol{\theta}^{j}-\frac{1}{2}(\boldsymbol{\theta}^{i}_{n}+\boldsymbol{\theta}^{j}_{n})\Big{\|}^{2}.$
Assuming that $w_{ij}=w_{ji}$ and $y_{ij}=y_{ji}$, the surrogate function
therefore becomes
$\displaystyle g(\boldsymbol{\theta}\mid\boldsymbol{\theta}_{n})$
$\displaystyle=$ $\displaystyle
2\sum_{i<j}w_{ij}\left[\Big{\|}\boldsymbol{\theta}^{i}-\frac{1}{2}(\boldsymbol{\theta}^{i}_{n}+\boldsymbol{\theta}^{j}_{n})\Big{\|}^{2}-\frac{y_{ij}(\boldsymbol{\theta}^{i})^{t}(\boldsymbol{\theta}^{i}_{n}-\boldsymbol{\theta}^{j}_{n})}{\|\boldsymbol{\theta}^{i}_{n}-\boldsymbol{\theta}^{j}_{n}\|}\right]$
$\displaystyle+2\sum_{i<j}w_{ij}\left[\Big{\|}\boldsymbol{\theta}^{j}-\frac{1}{2}(\boldsymbol{\theta}^{i}_{n}+\boldsymbol{\theta}^{j}_{n})\Big{\|}^{2}+\frac{y_{ij}(\boldsymbol{\theta}^{j})^{t}(\boldsymbol{\theta}^{i}_{n}-\boldsymbol{\theta}^{j}_{n})}{\|\boldsymbol{\theta}^{i}_{n}-\boldsymbol{\theta}^{j}_{n}\|}\right]$
$\displaystyle=$ $\displaystyle 2\sum_{i=1}^{q}\sum_{j\neq
i}\left[w_{ij}\Big{\|}\boldsymbol{\theta}^{i}-\frac{1}{2}(\boldsymbol{\theta}^{i}_{n}+\boldsymbol{\theta}^{j}_{n})\Big{\|}^{2}-\frac{w_{ij}y_{ij}(\boldsymbol{\theta}^{i})^{t}(\boldsymbol{\theta}^{i}_{n}-\boldsymbol{\theta}^{j}_{n})}{\|\boldsymbol{\theta}^{i}_{n}-\boldsymbol{\theta}^{j}_{n}\|}\right]$
up to an irrelevant constant. Setting the gradient of the surrogate equal to
the $\mathbf{0}$ vector produces the parallel updates
$\displaystyle\theta^{i}_{n+1,k}$ $\displaystyle=$
$\displaystyle\frac{\sum_{j\neq
i}\left[\frac{w_{ij}y_{ij}(\theta^{i}_{nk}-\theta^{j}_{nk})}{\|\boldsymbol{\theta}^{i}_{n}-\boldsymbol{\theta}^{j}_{n}\|}+w_{ij}(\theta^{i}_{nk}+\theta^{j}_{nk})\right]}{2\sum_{j\neq
i}w_{ij}}$
for all movable parameters $\theta^{i}_{k}$.
Algorithm 3 summarizes the parallel organization of the steps. Again the
matrix multiplications $\mathbf{\Theta}_{n}^{t}\mathbf{\Theta}_{n}$ and
$\mathbf{\Theta}_{n}(\mathbf{W}-\mathbf{Z}_{n})$ can be taken care of by the
CUBLAS library [25]. The remaining steps of the algorithm are conducted by
easily written parallel code.
Table 4 compares the run times in seconds for MDS on the 2005 United States
House of Representatives roll call votes. The original data consist of the 671
roll calls made by 401 representatives. We refer readers to the reference [8]
for a careful description of the data and how the MDS input $401\times 401$
distance matrix is derived. The weights $w_{ij}$ are taken to be 1. In our
notation, the number of objects (House Representatives) is $q=401$. Even for
this relatively small dataset, we see a 27–48 fold reduction in total run
times, depending on the projection dimension $p$. Figure 4 displays the
results in $p=3$ dimensional space. The Democratic and Republican members are
clearly separated. For $p=30$, the algorithm fails to converge within 100,000
iterations.
Although the projection of points into $p>3$ dimensional spaces may sound
artificial, there are situations where this is standard practice. First, MDS
is foremost a dimension reduction tool, and it is desirable to keep $p>3$ to
maximize explanatory power. Second, the stress function tends to have multiple
local minima in low dimensions [9]. A standard optimization algorithm like MM
is only guaranteed to converge to a local minima of the stress function. As
the number of dimensions increases, most of the inferior modes disappear. One
can formally demonstrate that the stress has a unique minimum when $p=q-1$ [4,
9]. In practice, uniqueness can set in well before $p$ reaches $q-1$. In the
recent work [38], we propose a “dimension crunching” technique that increases
the chance of the MM algorithm converging to the global minimum of the stress
function. In dimension crunching, we start optimizing the stress in a
Euclidean space $\mathbb{R}^{m}$ with $m>p$. The last $m-p$ components of each
column $\boldsymbol{\theta}^{i}$ are gradually subjected to stiffer and
stiffer penalties. In the limit as the penalty tuning parameter tends to
$\infty$, we recover the global minimum of the stress in $\mathbb{R}^{p}$.
This strategy inevitably incurs a computational burden when $m$ is large, but
the MM+GPU combination comes to the rescue.
Precompute: $x_{ij}\leftarrow w_{ij}y_{ij}$ for all $1\leq i,j\leq q$
Precompute: $w_{i\cdot}\leftarrow\sum_{j}w_{ij}$ for all $1\leq i\leq q$
Initialize: Draw $\theta^{i}_{0k}$ uniformly on [-1,1] for all $1\leq i\leq
q$, $1\leq k\leq p$
repeat
Compute $\mathbf{\Theta}_{n}^{t}\mathbf{\Theta}_{n}$
$d_{nij}\leftarrow\\{\mathbf{\Theta}_{n}^{t}\mathbf{\Theta}_{n}\\}_{ii}+\\{\mathbf{\Theta}_{n}^{t}\mathbf{\Theta}_{n}\\}_{jj}-2\\{\mathbf{\Theta}_{n}^{t}\mathbf{\Theta}_{n}\\}_{ij}$
for all $1\leq i,j\leq q$
$z_{nij}\leftarrow x_{ij}/d_{nij}$ for all $1\leq i\neq j\leq q$
$z_{ni\cdot}\leftarrow\sum_{j}z_{nij}$ for all $1\leq i\leq q$
Compute $\mathbf{\Theta}_{n}(\mathbf{W}-\mathbf{Z}_{n})$
$\theta_{n+1,k}^{i}\leftarrow[\theta_{nk}^{i}(w_{i\cdot}+z_{ni\cdot})+\\{\mathbf{\Theta}_{n}(\mathbf{W}-\mathbf{Z}_{n})\\}_{ik}]/(2w_{i\cdot})$
for all $1\leq i\leq p$, $1\leq k\leq q$
until convergence occurs
Algorithm 3 (MDS) Given weights $\mathbf{W}$ and distances
$\mathbf{Y}\in\mathbb{R}^{q\times q}$, find the matrix
$\mathbf{\Theta}=[\boldsymbol{\theta}^{1},\ldots,\boldsymbol{\theta}^{q}]\in\mathbb{R}^{p\times
q}$ which minimizes the stress (3.3).
| CPU | GPU | QN(20) on CPU
---|---|---|---
Dim-$p$ | Iters | Time | Stress | Iters | Time | Stress | Speedup | Iters | Time | Stress | Speedup
2 | 3452 | 43 | 198.5109307 | 3452 | 1 | 198.5109309 | 43 | 530 | 16 | 198.5815072 | 3
3 | 15912 | 189 | 95.55987770 | 15912 | 6 | 95.55987813 | 32 | 1124 | 38 | 92.82984196 | 5
4 | 15965 | 189 | 56.83482075 | 15965 | 7 | 56.83482083 | 27 | 596 | 18 | 56.83478026 | 11
5 | 24604 | 328 | 39.41268434 | 24604 | 10 | 39.41268444 | 33 | 546 | 17 | 39.41493536 | 19
10 | 29643 | 441 | 14.16083986 | 29643 | 13 | 14.16083992 | 34 | 848 | 35 | 14.16077368 | 13
20 | 67130 | 1288 | 6.464623901 | 67130 | 32 | 6.464624064 | 40 | 810 | 43 | 6.464526731 | 30
30 | 100000 | 2456 | 4.839570118 | 100000 | 51 | 4.839570322 | 48 | 844 | 54 | 4.839140671 | n/a
Table 4: Comparison of run times (in seconds) for MDS on the 2005 House of
Representatives roll call data. The number of points (representatives) is
$q=401$. The results under the heading $QN(20)$ on CPU invoke the quasi-Newton
acceleration [39] with 20 secant conditions.
Figure 4: Display of the MDS results with $p=3$ coordinates on the 2005 House
of Representatives roll call data.
## 4 Discussion
The rapid and sustained increases in computing power over the last half
century have transformed statistics. Every advance has encouraged
statisticians to attack harder and more sophisticated problems. We tend to
take the steady march of computational efficiency for granted, but there are
limits to a chip’s clock speed, power consumption, and logical complexity.
Parallel processing via GPUs is the technological innovation that will power
ambitious statistical computing in the coming decade. Once the limits of
parallel processing are reached, we may see quantum computers take off. In the
meantime statisticians should learn how to harness GPUs productively.
We have argued by example that high-dimensional optimization is driven by
parameter and data separation. It takes both to exploit the parallel
capabilities of GPUs. Block relaxation and the MM algorithm often generate
ideal parallel algorithms. In our opinion the MM algorithm is the more
versatile of the two generic strategies. Unfortunately, block relaxation does
not accommodate constraints well and may generate sequential rather than
parallel updates. Even when its updates are parallel, they may not be data
separated. The EM algorithm is one of the most versatile tools in the
statistician’s toolbox. The MM principle generalizes the EM algorithm and
shares its positive features. Scoring and Newton’s methods become impractical
in high dimensions. Despite these arguments in favor of MM algorithms, one
should always keep in mind hybrid algorithms such as the one we implemented
for NNMF.
Although none of our data sets is really large by today’s standards, they do
demonstrate that a good GPU implementation can easily achieve one to two
orders of magnitude improvement over a single CPU core. Admittedly, modern
CPUs come with 2 to 8 cores, and distributed computing over CPU-based clusters
remains an option. But this alternative also carries a hefty price tag. The
NVIDIA GTX280 GPU on which our examples were run drives 240 cores at a cost of
several hundred dollars. High-end computers with 8 or more CPU nodes cost
thousands of dollars. It would take 30 CPUs with 8 cores each to equal a
single GPU at the same clock rate. Hence, GPU cards strike an effective and
cost efficient balance.
The simplicity of MM algorithms often comes at a price of slow (at best
linear) convergence. Our MDS, NNMF, and PET (without penalty) examples are
cases in point. Slow convergence is a concern as statisticians head into an
era dominated by large data sets and high-dimensional models. Think about the
scale of the Netflix data matrix. The speed of any iterative algorithm is
determined by both the computational cost per iteration and the number of
iterations until convergence. GPU implementation reduces the first cost.
Computational statisticians also have a bag of software tricks to decrease the
number of iterations [22, 10, 20, 14, 11, 23, 35]. For instance, the recent
paper [39] proposes a quasi-Newton acceleration scheme particularly suitable
for high-dimensional problems. The scheme is off-the-shelf and broadly applies
to any search algorithm defined by a smooth algorithm map. The acceleration
requires only modest increments in storage and computation per iteration.
Tables 3 and 4 also list the results of this quasi-Newton acceleration of the
CPU implementation for the MDS and PET examples. As the tables make evident,
quasi-Newton acceleration significantly reduces the number of iterations until
convergence. The accelerated algorithm always locates a better mode while
cutting run times compared to the unaccelerated algorithm. We have tried the
quasi-Newton acceleration on our GPU hardware with mixed results. We suspect
that the lack of full double precision on the GPU is the culprit. When full
double precision becomes widely available, the combination of GPU hardware
acceleration and algorithmic software acceleration will be extremely potent.
Successful acceleration methods will also facilitate attacking another nagging
problem in computational statistics, namely multimodality. No one knows how
often statistical inference is fatally flawed because a standard optimization
algorithm converges to an inferior mode. The current remedy of choice is to
start a search algorithm from multiple random points. Algorithm acceleration
is welcome because the number of starting points can be enlarged without an
increase in computing time. As an alternative to multiple starting points, our
recent paper [38] suggests modifications of several standard MM algorithms
that increase the chance of locating better modes. These simple modifications
all involve variations on deterministic annealing [34].
Our treatment of simple classical examples should not hide the wide
applicability of the powerful MM+GPU combination. A few other candidate
applications include penalized estimation of haplotype frequencies in genetics
[1], construction of biological and social networks under a random multigraph
model [28], and data mining with a variety of models related to the
multinomial distribution [40]. Many mixture models will benefit as well from
parallelization, particularly in assigning group memberships. Finally,
parallelization is hardly limited to optimization. We can expect to see many
more GPU applications in MCMC sampling. Given the computationally intensive
nature of MCMC, the ultimate payoff may even be higher in the Bayesian setting
than in the frequentist setting. Of course realistically, these future
triumphs will require a great deal of thought, effort, and education. There is
usually a desert to wander and a river to cross before one reaches the
promised land.
## Acknowledgements
M.S. acknowledges support from NIH grant R01 GM086887. K.L. was supported by
United States Public Health Service grants GM53275 and MH59490.
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|
arxiv-papers
| 2010-03-16T23:06:35 |
2024-09-04T02:49:09.133030
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hua Zhou, Kenneth Lange, Marc A. Suchard",
"submitter": "Hua Zhou",
"url": "https://arxiv.org/abs/1003.3272"
}
|
1003.3462
|
# Kondo behavior, ferromagnetic correlations, and crystal fields in the heavy
Fermion compounds Ce3X ( X=In, Sn)
C. H. Wang1,2, J. M. Lawrence1, A. D. Christianson3, E. A. Goremychkin4, V. R.
Fanelli2, K. Gofryk2, E. D. Bauer2, F. Ronning2, J. D. Thompson2, N. R. de
Souza4,5, A. I. Kolesnikov3, K. C. Littrell3 1University of California,
Irvine, California 92697, USA
2Los Alamos National Laboratory, Los Alamos, NM 87545, USA
3Neutron Scattering Sciences Division, Oak Ridge National Laboratory, Oak
Ridge, TN, 37831, USA
4Argonne National Laboratory, Argonne, IL 60439, USA
5Australian Nuclear Science and Technology Organisation, Lucas Heights, NSW
2234, Australia
###### Abstract
We report measurements of inelastic neutron scattering, magnetic
susceptibility, magnetization, and the magnetic field dependence of the
specific heat for the heavy Fermion compounds Ce3In and Ce3Sn. The neutron
scattering results show that the excited crystal field levels have energies
$E_{1}$ = 13.2 meV, $E_{2}$ = 44.8 meV for Ce3In and $E_{1}$ = 18.5 meV,
$E_{2}$ = 36.1 meV for Ce3Sn. The Kondo temperature deduced from the
quasielastic linewidth is 17 K for Ce3In and 40 K for Ce3Sn. The low
temperature behavior of the specific heat, magnetization, and susceptibility
can not be well-described by J=1/2 Kondo physics alone, but require
calculations that include contributions from the Kondo effect, broadened
crystal fields, and ferromagnetic correlations, all of which are known to be
important in these compounds. We find that in Ce3In the ferromagnetic
fluctuation makes a 10-15 $\%$ contribution to the ground state doublet
entropy and magnetization. The large specific heat coefficient $\gamma$ in
this heavy fermion system thus arises more from the ferromagnetic correlations
than from the Kondo behavior.
###### pacs:
71.27.+a, 71.70.Ch, 75.20.Hr
## I Introduction
In heavy fermion (HF) compounds, it is very common to establish the Kondo
energy scale $T_{K}$ from the linear coefficient of specific heat $\gamma$
through Rajan’s formula $T_{K}$= $\pi JR/3\gamma_{0}$ derived for the
degenerate ($2J+1\geq$ 2) Kondo modelRajan where $J$ is the total angular
momentum. In previous studies of the specific heat of the HF compounds Ce3X
(X=In, Sn)YYChen ; Lawrence which crystallize in the Cu3Au cubic structure,
this formula was used to determine the Kondo temperature, which was found to
be 4.8 K for Ce3In and 16.7 K for Ce3Sn. The crystal electric field (CEF)
excitation energy was estimated to be $T_{CEF}$=65 K.
Most HF compounds reside close to a quantum critical point (QCP)where
antiferromagnetic (AFM) or ferromagnetic (FM) correlations are present. This
makes the previous analysis inappropriate in so far as it assumes that the
magnetic correlations do not contribute to $\gamma$. Indeed, the Wilson ratios
($\pi^{2}R\chi_{0}/3C_{J}\gamma_{0}$) which were determined previously for
Ce3In and Ce3Sn are 11.5 and 7.0 respectivelyYYChen ; Lawrence , indicating
that ferromagnetic correlations dominate the susceptibility.
Inelastic neutron scattering(INS) experiments on single crystals of compounds
that are close to a QCP, such as CeRu2Si2CeRu2Si2 or CeNi2Ge2CeNi2Ge2
exhibit two classes of excitations. At most $Q$ in the Brillouin zone, the
scattering has the characteristic Kondo energy dependence and is
$Q$-independent or only weakly $Q$-dependent. Similar behavior is observed in
intermediate valence compounds for which it is clear that the behavior of the
low temperature susceptibility, specific heat and INS spectra are close to the
Kondo impurity prediction, as though the onset of lattice coherence has only a
minor effect on these measurementsYbInCu4 ; slowcrossover . Near the QCP,
however, large $Q$-dependent scattering is observed with maximum intensity at
the critical wavevector $Q_{c}$ ($Q_{c}$ = 0 for FM and $Q_{c}$ = $Q_{N}$ for
AFM) where ordering occurs in the nearby magnetic state. This scattering
represents the short range order. It is dynamic and critically slows down, or
softens, as the QCP is approached by lowering the temperature or changing a
control parameter. These fluctuations affect the specific heat and can result
in non-Fermi liquid behavior.
Hence INS in single crystals can separate the Kondo behavior from the
contributions due to magnetic correlations. Since the spectral weight in the
magnetic correlations is typically small, INS in polycrystals will be
dominated by the $Q$-independent Kondo scattering. INS can also be used to
directly determine the CEF excitations. Under these circumstance, INS provides
a better way to determine $T_{K}$ and $E_{CEF}$ than through analysis of the
specific heat. In this paper, we employ INS to determine both $T_{K}$ and
$E_{CEF}$. We have re-measured the magnetic susceptibility, and have extended
the specific heat measurement, which in the previous report was measured down
to 1.8 K in zero applied magnetic field, to $T$ = 400 mK and $B=$ 9T. We have
also measured the low temperature magnetization to 13 T.
In the Ce3X compounds, the Ce ions sit on the face centers of the cubic
lattice and are subject to a crystalline electric field (CEF) of tetragonal
symmetry. In this case, the Hamiltonian is described as:
$H_{CF}=B_{2}^{0}O_{2}^{0}+B_{4}^{0}O_{4}^{0}+B_{4}^{4}O_{4}^{4}$,
where $B_{l}^{m}$ and $O_{l}^{m}$ are the crystal field parameters and Steven
operators, respectively. The sixfold degenerate 4$f^{1}$, J=5/2 state splits
into three doublets. Diagonalizing the Hamiltonian, the atomic wave functions
are given by:Aviani ; Fischer
$\Gamma_{7}^{(1)}=\eta|\pm 5/2>+\sqrt{1-\eta^{2}}|\mp 3/2>$
$\Gamma_{7}^{(2)}=\sqrt{1-\eta^{2}}|\pm 5/2>-\eta|\mp 3/2>$
$\Gamma_{6}=|\pm 1/2>$
Depending on the admixture of the $J$=5/2 and 3/2 states, the inelastic
neutron scattering spectra will exhibit one or two inelastic excitations. Low
energy transfer quasielastic scattering will also be observed if the
instrumental resolution is adequate. From the INS spectra, the crystal field
energies and wavefunctions can be determined from the ampitudes and energies
of the excitations. The Kondo effect, which arises from the hybridization of
the 4 $f$-electron with the conduction electrons, broadens the peak line-
widths proportional to $k_{B}T_{K}$. The quasielastic scattering peak width
$\Gamma_{QE}$ can be equated to the Kondo energy $k_{B}T_{K}$ of the ground
state doublet.
In what follows, we will use the CEF parameters and the Kondo energies derived
from the neutron scattering to calculate the Kondo contribution to the
specific heat, susceptibility and magnetization. All the Kondo calculations
utilizedRajan ; Hewson ; Sacramento employ the same Bethe-Ansatz calculation,
making intercomparison possible.
## II experiment
All samples were prepared by arc melting in an ultra-high-purity argon
atmosphere. After melting the samples were sealed under vacuum and annealed at
5000C for 2 weeks and cooled slowly to room temperature. The magnetization was
measured in a 14 T Quantum Design Vibrating Sample Magnetometer at the
National High Magnetic Field Laboratory (NHMFL) at Los Alamos National
Laboratory. The specific heat was measured in a Quantum Design PPMS system.
The magnetic susceptibility measurements were performed in a commercial
superconducting quantum interference device (SQUID) magnetometer.
We performed inelastic neutron scattering on a 29 gram sample of Ce3In and a
37 gram sample of Ce3Sn using the high energy transfer chopper spectrometer
(HET) at ISIS (at the Rutherford Appleton Laboratory) and the low resolution
medium energy chopper spectrometer (LRMECS) at IPNS (at Argonne National
Laboratory). For Ce3Sn, the quasi-elastic neutron spectrometer (QENS, at IPNS)
was also used to measure the low energy scattering. To increase the dynamic
range of the INS spectrum, a variety of incident energies ($E_{i}=$ 15 meV, 35
meV, 60 meV, 100 meV for HET and 35 meV for LRMECS) and temperatures ( 4.5 K,
100 K, 150 K, 200 K and 250 K for HET; 10K, 100K, 150K for LRMECS) were
employed. The HET data have been normalized to vanadium to establish the
absolute value. All the data have been corrected for absorption ( which is
very obvious for Ce3In case), total scattering cross section, and sample mass.
For the HET data, the low Q data were obtained from averaging the low angle
detectors with angles ranging from 11.5 degrees to 26.5 degrees. The high Q
data were obtained from the high angle detector bank at an angle 136 degrees.
For the LRMECS experiment, the low Q data were obtained by averaging over the
low angle detectors with average angle equal to 13 degrees; and the high Q
data were obtained from high angle detectors where the average angle was 87
degrees. The QENS data were collected at 7 K. This inverse geometry
spectrometer has 19 detector banks with $Q$ from $Q=0.36{\AA}$ to
$Q=2.52{\AA}$, each with a slightly different final energy ($E_{f}$ from 2.82
meV to 3.36 meV). For every fixed $Q$, we removed the Ce 4$f$ form factor to
obtain a spectrum representing the $Q=0$ scattering and then summed all 19
spectra together to obtain a total $S(Q=0,\Delta E)$ spectrum. In order to
compare the QENS spectrum with the spectra from the direct geometry
spectrometers HET and LRMECS, we multiplied the QENS spectrum $S(Q=0,\Delta
E)$ by the 4$f$ form factor apppropriate for HET at $E_{i}$ = 35 meV.
To subtract the nonmagnetic (background, single phonon, and multiple phonon)
contributions, we measured the non-magnetic counterpart compounds La3In and
La3Sn. For the specific heat, we obtained the magnetic contribution by direct
subtraction, i.e. $C_{mag}=C(Ce)-C(La)$. For the INS data we used La3In and
La3Sn to determine the scaling of the nonmagnetic scattering between low $Q$
and high $Q$ as $h(\Delta E)=S(La,LQ)/S(La,HQ)$. Using this factor we scaled
the high $Q$ data (where the nonmagnetic scattering dominates) to the low $Q$
data (where the magnetic scattering dominates) in Ce compounds to determine
the nonmagnetic contributionMurani ; Eugene ; Jon .
## III results and discussion
Fig. 1(a) and (c) directly compare the low $Q$ INS spectra of Ce3In and La3In;
the data were collected on HET with incident energy $E_{i}=$ 15 meV (a) and 60
meV (c) at 4.5 K. Fig. 1(b) compares the low $Q$ and high $Q$ data for Ce3In
collected on LRMECS with an incident energy $E_{i}$= 35 meV at 10 K. The low
$Q$ data for Ce3Sn and La3Sn, which were collected on HET, are compared in
Fig. (d), (e) and (f) where the incident energies are $E_{i}$=15 meV (d), 35
meV (e) and 60 meV (f) at 4.5 K. In these spectra, two excited energy levels,
corresponding to crystal field excitations, are observed for both Ce3In and
Ce3Sn. The spectra (a) and (d), which compare the Ce3In(Sn) and La3In(Sn)
scattering at low energy transfer ($\Delta E<$ 9 meV), exhibit obvious
quasielastic scattering which as mentioned above arises from Kondo scattering.
Figure 1: Inelastic neutron scattering spectra for Ce3In and Ce3Sn together
with that of their nonmagnetic counterpart compounds obtained from HET and
LRMECS. The data collected on HET are at 4.5 K and on LRMECS are at 10 K. (a)
$E_{i}$=15 meV, (b) $E_{i}$=35 meV, and (c) $E_{i}$=60 meV spectra of Ce3In
and La3In. (d) $E_{i}$=15 meV, (e) $E_{i}$=35 meV and (f) $E_{i}$=60 meV
spectra for Ce3Sn and La3Sn. All spectra are for low $Q$ except in (b) where a
high-$Q$ spectrum is included for comparison.
The magnetic contribution $S_{mag}$ to the scattering of Ce3In, obtained using
the method described above, is shown in Fig. 2. The solid lines represent a
fit to the CEF model. Since the inelastic peaks are relatively broad, the line
widths $\Gamma_{i}$ are taken to be finite. In this case, the magnetic
scattering is described as:
$S_{mag}=\frac{2N}{\pi\mu_{B}^{2}}f^{2}(Q)(1-e^{-\Delta
E/{k_{B}T}})\chi^{\prime\prime}(Q,\Delta E)$
$\chi^{\prime\prime}(Q,\Delta E)=\Sigma\chi_{i}(T)\Delta
E(\frac{\Gamma_{i}}{2\pi})/[(\Delta E-E_{i})^{2}+\Gamma^{2}_{i}]$
Here $i$=0,1,2, $E_{0}$ = 0 corresponds to the quasielastic scattering, and
$f^{2}(Q)$ is the Ce 4$f$ form factor. The CEF model fitting was performed
simultaneously on six data sets at three different incident energies
($E_{i}$=15 meV, 35 meV and 60 meV) and at two different temperatures (4.5 K
and 150 K). Fig. 2(a)-(d) are the data collected on HET. In Fig. 2(f) the
LRMECS data are displayed for comparison. The resulting CEF fitting parameters
are shown in Table I. The ground state is the $\Gamma_{7}^{(1)}$ doublet, the
first excited state is the $\Gamma_{7}^{(2)}$ doubletendnote1 at the energy
13.2 meV, and the second excited state is the $\Gamma_{6}$ doublet at the
energy 44.8 meV. The quasielastic line width $\Gamma_{QE}$ = $\Gamma_{0}$ =
1.49 meV, implies that $T_{K}$ = $\Gamma_{QE}/k_{B}$ = 17 K.
Table 1: CEF model fitting parameters for Ce3In and Ce3Sn. | Ce3In | Ce3Sn
---|---|---
$B_{2}^{0}$(meV) | -2.203$\pm$0.015 | -1.660$\pm$0.017
$B_{4}^{0}$(meV) | 0.066$\pm$0.001 | 0.038$\pm$0.0009
$B_{4}^{4}$(meV) | -0.154$\pm$0.004 | -0.263$\pm$0.003
$\eta$ | 0.94 | 0.89
$E_{1}$ (meV) | 13.2 | 18.5
$E_{2}$ (meV) | 44.8 | 36.1
$\Gamma_{QE}$ (meV) | 1.49$\pm$0.07 | 3.52$\pm$0.16
$\Gamma_{1}$ (meV) | 5.98$\pm$0.07 | 9.37$\pm$0.038
$\Gamma_{2}$ (meV) | 2.06$\pm$0.37 | 6.28$\pm$0.38
$\chi^{2}$ | 2.4057 | 2.1528
$\lambda$ (mole-Ce/emu) | 62 | 85
Figure 2: Magnetic contribution $S_{mag}$ to the inelastic neutron scattering
spectra of Ce3In for data taken on HET at T=4.5 K and 150 K with different
incident energies $E_{i}$=15 meV, 35 meV, and 60 meV and taken on LRMECS at
T=100 K with $E_{i}$=35 meV. The solid lines represent the quasielastic and
crystal field contributions obtained from least squares fitting as described
in the text.
In Fig. 3(a)-(e) we display the magnetic contribution to the Ce3Sn scattering
collected from HET at 4.5 and 100 K and at three incident energies (15, 35,
and 60 meV). Data from QENS at 7 K (Fig. 3(f)) are included for comparison.
The CEF fits are also included (solid lines); as for the Ce3In case, the fits
were performed simultaneously on six different spectra at different incident
energies and temperatures. The intensity and form factor of the QENS data have
been adjusted to that of the HET spectra at $E_{i}$=35 meV (spectra (b)) to
make a direct comparison. The fitting parameters yield a similar crystal field
scheme as for Ce3In: the $\Gamma_{7}^{(1)}$ doublet is the ground state,
$\Gamma_{7}^{(2)}$ is the first excited state with energy 18.5 meV, and the
second excited state is the $\Gamma_{6}$ doublet at the energy 36.1 meV. The
Kondo temperature $T_{K}=$ 40 K is higher than for Ce3In, and the excited
state linewidths are broader, reflecting stronger 4$f$-conduction
hybridization.
Figure 3: (a)-(f) Magnetic contribution $S_{mag}$ to the inelastic neutron
scattering spectra for Ce3Sn. The temperatures and incident energies are given
in the plot. The solid lines represent the CEF model. (f): the magnetic
contribution to the INS spectra collected from QENS at 7 K. The solid line in
(f) is the CEF model fit for the $E_{i}$ = 35 meV spectra.
Due to the large CEF excitation energies, the low temperature behavior of the
magnetic specific heat should be dominated by the $\Gamma_{7}^{(1)}$ doublet
ground state. This is confirmed by the fact that the magnetic entropy (Fig.
4(a) inset) reaches Rln2 near 20 K but only reaches Rln4 near 70 K. For a
doublet ground state, the Kondo model predicts $\gamma_{0}=\pi R/6T_{K}$ for
the linear coefficient of specific heat Rajan . In previous results for Ce3In
a value $T_{K}=$ 4.8 K was deduced using this formulaYYChen . In addition, the
specific heat coefficient $C/T$ showed a peak near 2 K whose existence was
somewhat uncertain since the lowest measured temperature was only 1.8 K. We
have extended the specific heat measurement down to 400 mK. In Fig. 4(a) we
plot $C_{mag}/T$ and find a peak at $T$ = 2.6 K. Comparison of the data to the
prediction $\gamma^{K}(T)$ of the Kondo model which is calculated using the
value $T_{K}=\Gamma_{QE}/k_{B}=17K$ deduced from our neutron data, shows that
the Kondo prediction is much smaller than the experimental value; indeed,
$\gamma^{K}_{0}$ is only half of $\gamma^{exp}_{0.4K}$ (Table II). Given the
large Wilson ratio reported earlierYYChen , the obvious explanation is that
ferromagnetic (FM) fluctuations dominate the low temperature specific heat,
increasing the specific heat above the Kondo value and giving rise to the peak
at 2.6 K representing the onset of short range FM order.
We next consider the high temperature susceptibility, comparing the measured
value to the value calculated from the crystal field parameters of Table I in
the inset of Fig. 4(b). A molecular-field $\lambda$ = 62 mole-Ce/emu has been
added to compensate the reduction of the susceptibility at high temperature
due to the Kondo effect ($1/\chi^{HT}=1/\chi^{CEF}+\lambda$). At high
temperatures, when the crystal field states are excited, the effective Kondo
temperature $T^{HT}_{K}$ is larger than the Kondo temperature of the ground
state doublet. The molecular field constant is related to the effective Kondo
temperature via $\lambda=T^{HT}_{K}/C_{5/2}$ where $C_{5/2}$ is the free ion
Curie constant for cerium. This relation gives $T^{HT}_{K}=$ 77 K, which value
is essentially equal to the width $\Gamma_{1}$ of the first excited level seen
in the neutron scattering (Fig. 2 and table I).
At low temperatures, there should be three contributions to $\chi(T)$, as well
as to $M(H)$ and $C_{mag}$: one from the Kondo single ion impurity physics of
the ground state doublet, one from the FM fluctuations, and one from the
excitation of higher lying crystal field states. To carry out such an
analysis, we note first that in the Cu3Au crystal structure, the tetragonal
crystal field axis (i.e. the z-axis for the doublet wave functions) points
perpendicular to the face containing any given face-centered cerium atom;
hence there are three orthogonal tetragonal axes in the unit cell. When
applying a magnetic field in a polycrystalline sample, the field will point
along the tetragonal axis for $\frac{1}{3}$ of the cerium atoms but orthogonal
to the tetragonal axis (in the $x-y$ plane) for $\frac{2}{3}$ of the atoms.
The effective low temperature Curie constant is then
$C_{eff}=\frac{1}{3}C^{z}_{eff}+\frac{2}{3}C^{x}_{eff}$, where
$C^{z(x)}_{eff}=N(g^{z(x)}_{eff}\mu_{B})^{2}\frac{1}{2}(\frac{1}{2}+1)/3k_{B}$.
This is the form for a pseudo spin $\frac{1}{2}$ doublet where the CEF physics
is absorbed into the effective $g$-factor. Here
$g^{z(x)}_{eff}=\frac{12}{7}<J_{z(x)}>$ where $<J_{z(x)}>$ is the matrix
element of the angular moment component along the $z(x)$ axis. From the CEF
mixing parameter $\eta$, we determine $C_{eff}$ to be 0.48 emu-K/mole-Ce for
Ce3In and 0.41 emu-K/mole-Ce for Ce3Sn. (Table II).
Table 2: Kondo single ion model calculation for Ce3In and Ce3Sn. | $<J_{z}>$ | $<J_{x}>$ | $M^{sat}_{CEF}$ | $C^{LT}_{eff}$ | $\chi^{K}_{0}(\frac{emu}{mole-Ce})$ | $\gamma^{K}_{0}(\frac{J}{moleCeK^{2}})$ | $\chi^{exp}_{0.4K}(\frac{emu}{mole-Ce})$ | $\gamma^{exp}_{0.4K}(\frac{J}{moleCeK^{2}})$
---|---|---|---|---|---|---|---|---
Ce3In | 2.0344 | 0.7171 | 0.991 | 0.4765 | 0.0076 | 0.256 | 0.064 | 0.467
Ce3Sn | 1.6684 | 0.9074 | 0.994 | 0.4086 | 0.0033 | 0.109 | 0.018 | 0.221
Figure 4: (a) The magnetic contribution to the specific heat $C_{mag}/T$
versus $T$ for Ce3In. The solid line is the Kondo prediction $\gamma^{K}(T)$
calculated using $T_{K}=\Gamma_{QE}/k_{B}=$ 17 K. The inset is the magnetic
entropy of Ce3In. (b) The magnetic susceptibility $\chi(T)$ for Ce3In. The
solid line is the Kondo prediction $\chi^{K}(T)$ calculated using $T_{K}=$ 17
K and the low temperature Curie constant determined as described in the text.
The dashed line gives the sum of the Kondo and crystal field contributions
$\chi^{CEF+K}=\chi^{K}+(\chi^{CEF}-\chi^{LT}_{Curie})$. The inset is the
inverse susceptibility together with the calculated susceptibility (solid
line) $1/\chi^{HT}=1/\chi^{CEF}+\lambda$ obtained using the CEF fitting
parameters in table I. (c) The magnetization for Ce3In. The solid line
$M^{K}(B)$ is the Kondo calculation calculation using $T_{K}=$ 17 K. The
dashed line is the contribution from the ferromagnetic fluctuations.
To sort the low temperature susceptibility into Kondo, FM, and CEF
contributions, we note that since the first CEF excited level is at 152 K, at
sufficiently low temperatures the Zeeman splitting of the $\Gamma_{7}^{(1)}$
ground state doublet will obey a Curie law $\chi^{LT}_{Curie}=C^{LT}_{eff}/T$.
Due to the Kondo effect, this Curie behavior will be replaced by the Kondo
contribution $\chi^{K}$, which we calculate using the same Curie constant
$C^{LT}_{eff}$ and using $T_{K}=$ 17 K (solid line, Fig. 4(b)). The
susceptibility from the combination of the ground state Kondo and the excited
crystal fields will then be of the form
$\chi^{CEF+K}=\chi^{K}+(\chi^{CEF}-\chi^{LT}_{Curie})$ where we subtract
$\chi^{LT}_{Curie}$ to avoid double counting the ground state contribution. As
for the specific heat, the resulting $\chi^{CEF+K}$ (dashed line in Fig. 4(b))
is much smaller than the experimental value at $T<$ 20 K. The excess can be
viewed as the contribution from the ferromagnetic fluctuations. Taking the
latter to be equal to the difference $\chi^{exp}-\chi^{CEF+K}(T)$, the FM
contribution is seen to increase below 10 K in a manner characteristic of
ferromagnetic short range order.
In Fig. 4(c) we exhibit the magnetization as measured up to 13 Tesla at $T$=2
K. Based on Hewson’s calculation of Kondo modelHewson , we can estimate the
Kondo contribution to the magnetization. Since the effective $g$-factors
differ in the $z$ and $x-y$ directions, we calculate
$M^{K}=\frac{1}{3}M^{K}(z)+\frac{2}{3}M^{K}(xy)$. The result is plotted as a
solid line in Fig. 4(c). After subtracting the Kondo contribution, we obtain
the contribution from the FM correlations (dashed line). This saturates at a
relatively small field $B\sim 2.5$ tesla with $M^{sat}$ = 0.095 $\mu_{B}$,
which is 10 percent of the saturation value 1.0 $\mu_{B}$ expected based on
the effective $g$-factors.
Figure 5: (a) the magnetic specific heat $C_{mag}/T$ versus $T$ at B=0 T
(open circle) and B=9 T (solid triangle) for Ce3Sn. The solid line is the
Kondo contribution $\gamma^{K}(T)$ for $T_{K}=$ 40 K. The inset is the
magnetic entropy. (b) Magnetic susceptibility $\chi(T)$ for Ce3Sn. The solid
line is the Kondo contribution $\chi^{K}(T)$ calculated with
$T_{K}=\Gamma_{QE}/k_{B}=$ 40 K and the dashed line is the sum of the Kondo
and CEF contributions $\chi^{CEF+K}=\chi^{K}+(\chi^{CEF}-\chi^{LT}_{Curie})$.
The inset is the inverse susceptibility together with the value
$1/\chi^{HT}=1/\chi^{CEF}+\lambda$ (solid line) calculated from CEF fitting
parameters in table I. (c) $C_{mag}$ at $B$=0 T (open circle) and B=9 T (solid
triangle) for Ce3Sn. The thin solid line is $C^{K}(T)$ calculated with
$T_{K}=$ 40 K.
In the same way as for Ce3In, we calculate $\chi^{K}(T)$, $\chi^{CEF}(T)$,
$\chi^{CEF+K}(T)$, $\gamma^{K}(T)$ and $C^{K}(T)$ for Ce3Sn, comparing to the
measured data in Fig. 5. The high temperature susceptibility (Fig. 5(b) inset)
can again be fit with the sum of the CEF contribution calculated using the
parameters of Table I and a molecular field contribution (solid line). The
value $\lambda=$ 85 mole-Ce/emu of molecular field constant implies an
effective Kondo temperature at high temperature $T^{HT}_{K}$ = 105 K, which
again is essentially equal to the linewidth 9.4 meV of the first excited state
seen in the neutron scattering (Table I).
The solid lines in Fig. 5 represent the Kondo ground state doublet
contributions. In Fig. 5(b), the dashed line is $\chi^{CEF+K}(T)$. The excess
due to the FM correlations has a much smaller magnitude ($\sim$ 0.005 emu/mol-
Ce) than for Ce3In where the FM contribution is of order 0.05 emu/mol-Ce. A
similar statement holds for the FM contribution to the specific heat
coefficient, which is of order 0.4 J/mol-Ce-K2 for Ce3In but only 0.1 J/mol-
Ce-K2 for Ce3Sn (Figs. 4(a) and 5(b)). Hence the FM enhancement is smaller in
Ce3Sn than in Ce3In, consistent with the larger value of $T_{K}$.
In order to better understand these compounds, we measured the specific heat
of Ce3In under different applied fields (B = 0 T, 1 T, 3 T, 6 T and 9 T). The
results for the magnetic contribution $C_{mag}$ are shown in Fig. 6. The low
temperature peak in $C_{mag}$ moves to higher temperature when the field is
increased. Since the peak in the Kondo contribution to $C_{mag}$ is expected
to increase with field, we plot in Fig. 6(a)-(e) the Kondo contribution
$C^{K}(B)$ calculated for different applied fields using the theoretical
results of Sacramento and SchlottmannSacramento . In calculating $C^{K}(B)$,
we again account for the different effective $g$-factors in the $z$ and $x-y$
directions. The results indicate that the Kondo contribution is not expected
to alter significantly in applied fields of order 9 T, essentially because
$g_{eff}\mu_{B}B<k_{B}T_{K}$ for these fields. This makes it clear that the
peak does not arise from the Kondo scattering but must be due to the FM
fluctuations.
To quantify the FM contribution, we again assume that the measured magnetic
specific heat is the sum of the ground state doublet Kondo contribution
$C^{K}(B)$, the FM contribution $C^{FM}$, and a contribution $C^{CEF}$ due to
the excitation of higher lying CEF states. Since the FM fluctuations appear to
only contribute to the susceptibility below 10 K (Fig. 4 b) we assume that the
excess $C_{mag}$ \- $C^{K}(B)$ observed for $T>$ 10 K is primarily due to CEF
excitations. Given the large linewidths of the CEF excitations seen in the
neutron scattering, and concomitant large effective $T^{HT}_{K}$ at high
temperature, this contribution to the specific heat is much broader as a
function of temperature than would be the case for a simple CEF Schottky
anomaly. For simplicity, we approximate this CEF contribution as linear in
temperature, with slope equal to that observed in the range 8-15 K, and we
assume that since the CEF excitation energy is large, this contribution will
be unaffected by fields of order 9 T. We approximate the FM contribution
$C^{FM}$ as a Gaussian, centered at a temperature that increases with field.
The three contributions, Kondo, CEF, and FM, are plotted at the different
fields in Figs. 6(a)-(e). The solid lines, which represent the sum of all
three contributions, fit the data very well at all fields.
We plot the Gaussian peak temperature in Fig. 6(f), where it is seen to grow
linearly with field. This suggests Zeeman splitting, where at zero field the
splitting arises from the internal field in the regions of FM short range
order, and where the applied field increases the splitting. To determine the
internal field $B_{int}$, we calculate the Schottky anomaly
$C^{Schotkky}(B_{int})$ expected due to Zeeman splitting of a doublet with the
same effective $g$-factors as we have obtained from the neutron fits; we then
adjust $B_{int}$ until the peak temperature of the Schottky anomaly is the
same as that of the Gaussian peak temperature for $B$ = 0. This gives
$B_{int}$ = 9.5 T. We then calculate $C^{Schottky}(B_{int}+B_{app})$ to
determine peak position of the Schottky anomaly in an applied field $B_{app}$.
As can be seen in Fig. 6 (f), the Gaussian peak temperatures track the
expected Zeeman splitting very closely. On the other hand, the temperature
dependence of the Schottky specific heat calculated in this manner is
considerably broader than the Gaussian contributions $C^{FM}$ that are plotted
in Fig. 5. This means that, while the contribution of the FM short range order
to the specific heat is not of Schottky form, the increase of the Gaussian
peak position is the same as the Zeeman splitting expected for a total field
$B_{int}+B_{app}$ given the effective $g$-factors.
The entropy of the Gaussian contribution is about 15 $\%$ of Rln2 for all
fields. This corresponds to the estimate obtained from the magnetization
$M(B)$ where the saturation value of the FM contribution is about 10$\%$ of
the value 1.0 $\mu_{B}$ expected for the $\Gamma_{7}^{(1)}$ ground state
doublet for the measured value of $\eta$. Hence, the enhancement of $\chi_{0}$
and $\gamma_{0}$ arises from magnetic fluctuations which involve 10-15 $\%$ of
the 4$f$ electron degrees of freedom.
In Fig. 5(c), we compare the magnetic specific heat $C_{mag}$ at zero field
and $B=$ 9 T for Ce3Sn. The solid line is the Kondo contribution $C^{K}$. The
specific heat does not change with field for $B<$ 9 T. The most likely
explanation of this is that, as discussed above, the FM correlations make a
smaller contribution than in Ce3In. The excess specific heat $C_{mag}-C^{K}$
seen for $T>$ 6 K is presumably due to the CEF contribution, which should be
even broader in temperature in Ce3Sn than in Ce3In due to the larger Kondo
temperature.
Figure 6: (a), (b), (c), (d) and (e): $C_{mag}$ of Ce3In in different applied
magnetic fields. The solid line sums the three contributions (Kondo, CEF, and
FM fluctuations) shown in the plot. (f): The peak position of $C_{mag}$, of
the Gaussian contribution $C^{FM}$ due to the FM fluctuations, and the
expected peak position in the Schottky anomaly $C^{Schottky}(B_{int}+B_{app})$
due to Zeeman splitting in the presence of an internal field $B_{int}$.
We have demonstrated that the large $\gamma$ observed in Ce3In arises more
from ferromagnetic correlations than from the single ion Kondo physics. This
reflects the fact that the system is close to a ferromagnetic quantum critical
point. In a $Q$-resolved INS experiment, the ferromagnetic correlations should
show up in the vicinity of $Q=$0 riding on a background of $Q$ independent
Kondo scattering. We can estimate that these FM correlations will have
10-15$\%$ of the total spectral weight in $Q$-space. Since the large $\gamma$
and the proximity to the QCP occurs when the Kondo temperature $T_{K}=$ 17 K
is fairly large, it is also reasonable to believe that when an appropriate
control parameter (e.g. alloying parameter x in Ce3-xLaxIn) drives this system
to the QCP, the Kondo temperature $T_{K}$ will remain finite, as expected for
example for a spin density wave type QCP.
We have observed strong FM fluctuations in the related compound Pr3In, which
is an antiferromagnet below 12 KAndy . A possibility for this behavior is that
AFM interactions between rare earth atoms on the face centers of the Cu3Au
structure are frustrated. If, for example, the atoms at (1/2 1/2 0) and (1/2 0
1/2) are aligned antiferromagnetically, the atom at (0 1/2 1/2) will be free
to point to any direction. Ferromagnetic next-nearest-neighbor interactions
could then stabilize ferromagnetism on this sublatticeCristian . In any case,
the FM correlations appear to be generic to this crystal structure.
In conclusion, we have used inelastic neutron scattering to determine the
crystalline electric field (CEF) splitting and Kondo energy scale in Ce3In and
Ce3Sn. For both compounds the crystal field excitation energy is large. For
Ce3In we have separated the magnetization $M(B)$, susceptibility $\chi(T)$ and
specific heat $C_{mag}$ into contributions from the Kondo effect, from the
CEF, and from FM fluctuations. The simplified model calculation for Ce3In
shows that the FM correlations make a 15 $\%$ contribution to the doublet
ground state entropy and that the large $\gamma$ arises mostly from the FM
correlations. This suggests Ce3In is close to a quantum critical point (QCP).
The Kondo temperature $T_{K}$ is expected to remain finite at the QCP, as
occurs for a spin density wave type QCP. INS experiments in single crystals of
these compounds would be very interesting.
## IV acknowledgements
We thank Vivien Zapf for her assistance in the measurement at NHMFL and
Cristian Batista for his insightful comments. Research at UC Irvine was
supported by the U.S. Department of Energy, Office of Basic Energy Sciences,
Division of Materials Sciences and Engineering under Award DE-FG02-03ER46036.
Work at ORNL was supported by the Scientific User Facilities Division Office
of Basic Energy Sciences, DOE and was managed by UT-Battelle, LLC, for DOE
under Contract DE-AC05-00OR22725. Work at Los Alamos National Laboratory was
performed under the auspices of the U.S. DOE/Office of Science. Work at NHMFL-
PFF, Los Alamos was performed under the auspices of the National Science
Foundation, the State of Florida, and U.S. DOE. Work at ANL was supported by
DOE-BES under contract DE-AC02-06CH11357.
## References
* (1) V. T. Rajan, Phys. Rev. Lett., 51, 308 (1983).
* (2) Y. Y. Chen, J. M. Lawrence, J. D. Thompson and J. O. Willis, Phys. Rev. B 40, 10766(1989).
* (3) J. M. Lawrence, Y. Y. Chen, J. D. Thompson and J. O. Willis, Physica B 163, 56-58(1990).
* (4) H. Kadowaki, M. Sato and S. Kawarazaki, Phys. Rev. Lett. 92, 097204 (2004).
* (5) H. Kadowaki, B. F${\aa}$k, T. Fukuhara, K. Maezawa, K. Nakajima, M. A. Adams, S. Raymond and J. Flouquet, Phys. Rev. B 68, 140402 (2003).
* (6) J. M. Lawrence, S. M. Shapiro, J. L. Sarrao and Z. Fisk, Phys. Rev. B 55, 14467 (1997).
* (7) J. M. Lawrence, P. S. Riseborough, C. H. Booth, J. L. Sarrao, J. D. Thompson, and R. Osborn, Phys. Rev. B 63, 054427 (2001).
* (8) I. Aviani, M. Miljak and V. Zlatic, Phys. Rev. B 64, 184434 (2001).
* (9) G. Fischer and A. Herr, Phys. Stat. Sol. B 141, 589 (1987).
* (10) A. C. Hewson, J. W. Rasul, J. Phys. C: Solid State Phys. 16, 6799 (1983).
* (11) P. D. Sacramento and P. Schlottmann, Phys. Rev. B 40, 431 (1989).
* (12) A. P. Murani, Phys. Rev. B 28, 2308 (1983).
* (13) E. A. Goremychkin, R. Osborn, Phys. Rev. B 47, 14280 (1993).
* (14) J. M. Lawrence, P. S. Riseborough, C. H. Booth, J. L. Sarrao, J. D. Thompson and R. Osborn, Phys. Rev. B 63, 054427 (2001).
* (15) Experimental probes of CEF excitations can not distinguish between a positive and a negative value of $B_{4}^{4}$. Only the modulus is observable. Consequently the distinction between the $\Gamma_{7}^{(1)}$ and $\Gamma_{7}^{(2)}$ states is a matter of convention ( A. D. Christianson et al. Phys. Rev. B 70, 134505 (2004)).
* (16) A. D. Christianson, J. M. Lawrence, J. L. Zarestky, H. S. Suzuki, J. D. Thompson, M. F. Hundley, J. L. Sarrao, C. H. Booth, D. Antonio, A. L. Cornelius, Phys. Rev. B 72, 024402 (2005).
* (17) Cristian Batista, private communication.
|
arxiv-papers
| 2010-03-17T20:26:05 |
2024-09-04T02:49:09.147148
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "C. H. Wang, J. M. Lawrence, A. D. Christianson, E. A. Goremychkin, V.\n R. Fanelli, K. Gofryk, E. D. Bauer, F. Ronning, J. D. Thompson, N. R. de\n Souza, A. I. Kolesnikov, K. C. Littrell",
"submitter": "Cuihuan Wang Dr.",
"url": "https://arxiv.org/abs/1003.3462"
}
|
1003.3679
|
# A note on black hole entropy, area spectrum, and evaporation
C.A.S.Silva calex@fisica.ufc.br R.R. Landim renan@fisica.ufc.br
Departamento de Física, Universidade Federal do Ceará - Caixa Postal 6030, CEP
60455-760, Fortaleza, Ceará, Brazil
###### Abstract
We argue that a process where a fuzzy space splits in two others can be used
to explain the origin of the black hole entropy, and why a “generalized second
law of thermodynamics” appears to hold in the presence of black holes. We
reach the Bekenstein-Hawking formula from the count of the microstates of a
black hole modeled by a fuzzy space. In this approach, a discrete area
spectrum for the black hole, which becomes increasingly spaced as the black
hole approaches the Planck scale, is obtained. We show that, as a consequence
of this, the black hole radiation becomes less and less entropic as the black
hole evaporates, in a way that some information about its initial state could
be recovered.
It is not an exaggeration to say that one of the most exciting predictions of
general relativity is that there may exist black holes. This is mainly due to
the believe that black holes may play a major role in our attempts to shed
some light on the nature of a quantum theory of gravity such as the role
played by atoms in the early development of quantum mechanics.
Recent results have shown that black hole have thermodynamics properties like
entropy and temperature, and as a consequence of the instability of the vacuum
in strong gravitational fields, black holes are sources of quantum radiation
sw.hawking-cmp43 (1, 2, 3).
Some time after, string theory and loop quantum gravity, argued that the
origin of the black hole entropy must be related with the quantum structure of
the spacetime a.strominger-plb379 (4, 5). However the computation of black
hole entropy in the semiclassical and furthermore in quantum regime has been a
very difficult, and still unsolved problem. We know that in statistical
physics, entropy counts the number of accessible microstates that a system can
occupy, where all states are presumed to occur with equal probability. On the
other hand, we also know that black holes can be completely characterized by
only three externally observable classical parameters: mass, electric charge,
and angular momentum. All other information about the matter which formed a
black hole “disappears” behind the black-hole event horizon, and therefore the
nature of these microstates is obscure. Then, what is the origin of the black
hole entropy? Furthermore, in order to justify the name “entropy”, one must to
explain also why $S=S_{bh}+S_{out}$ is a non-decreasing function of time, in
other words, why black holes obey a “generalized second law of
thermodynamics”.
We still have that, since black holes evaporate, one could expect, from the
Hawking radiation, any information about the state which collapsed into the
black hole. However, Hawking showed that this radiation is thermal, and
therefore does not carry any information about the black hole initial state.
That is to say, no information can escape from inside of the black hole
horizon. In this situation, the matter that formed the black hole, which
initially was in a pure state has evolved into a mixed state. This bring us a
contradiction with quantum mechanics. There, a pure state can only evolve into
another pure state because of the unitarity of the evolution
operatorsw.hawking-cmp43 (1, 2, 6, 7).
One way to solve the Hawking paradox is that where information can be stored
in a topological disconnected region which arises inside of the black hole
sdh.hsu-plb644 (10). Gravitational collapse would lead to a region of
Planckian densities and curvature where quantum gravitational effects can lead
to a topology change process where a new topologically disconnected region
appears. However, This process had been claim to break unitarity and cluster
deposition(locality) sdh.hsu-plb644 (10, 11).
On the other hand, recently it was suggested by one of us that it is possible
to realize a topology change process to black holes without break unitarity or
cluster decomposition c.a.s.silva (9). The basic idea of this proposal is to
see the black hole horizon as a fuzzy sphere $S_{F}^{2}$ taking into account
some quantum symmetry properties related with a Hopf algebra structure. There,
we can define a linear operation (the coproduct of the Hopf algebra) on
$S_{F}^{2}$ and compose two fuzzy spheres preserving algebraic properties
intact. This operation, which we shall represent by $\Delta$, produces a
topology change process where a fuzzy sphere splits in two others
ap.balachandran-ijmpa19 (12).
Let M describes a wave function on $S_{F}^{2}$, the coproduct
$\Delta:S_{F}^{2}(J)\rightarrow S_{F}^{2}(K)\otimes S_{F}^{2}(L)$ acts on M as
$\displaystyle\Delta(M)_{(K,L)}$ $\displaystyle=$
$\displaystyle\sum_{\\!\\!\\!\\!\\!\mu_{1},\mu_{2},m_{1},m_{2}}C_{K,L,J;\mu_{1},\mu_{2}}C_{K,L,J;m_{1},m_{2}}$
(1) $\displaystyle\times$ $\displaystyle
M_{\mu_{1}+\mu_{2},m_{1}+m_{2}}e^{\mu_{1}m_{1}}(K)\otimes e^{\mu_{2}m_{2}}(L)$
where $C$’s are the Clebsh-Gordan coefficients and $e^{\mu_{i}m_{j}}$’s are
basis for $Mat$. From (1) M $\in$ $S_{F}^{2}(J)$ splits into a superposition
of wavefunctions on $S_{F}^{2}(K)\otimes S_{F}^{2}(L)$, and the information in
M is divided between the two fuzzy spheres with spins $K$ and $L$
respectively.
The process (1) preserves the Hermitian conjugation, the matrix product, the
trace, and the inner product.The two last properties assure that (1) is a
unitary process ap.balachandran-ijmpa19 (12).
Following c.a.s.silva (9), if we use the fuzzy sphere Hilbert space as the
ones of the black hole, we shall have the following consequences: (i) The
maximum of information about the black hole that an outside observer can
obtain would be encoded in a wave function M defined on the fuzzy sphere
Hilbert space. (ii) We shall find out, through the Hopf coproduct $\Delta$ in
(1), a topology change process for the black hole. In this process the
information about the black hole initial state, described by the wave function
$M$, is divided into two regions. One of them is a fuzzy sphere with spin $K$,
which we shall consider as the original world and name it “the main world”.
The other one is a fuzzy sphere with spin $L$ which we shall name “the baby
world”. (iii) Since an observer in the main world can not access the degrees
of freedom in the baby world, from his standpoint, the black hole will appear
to evolve from a pure to a mixed state, described by a density matrix $\rho$.
Therefore, we can define an entropy measured by the observer in the main
world.
The entropy measured by the observer in the main world would be given by $S$ =
$-k_{B}\ln(\rho tr{\rho})$ = $k_{B}\ln(dimH)$, where $H$ is Hilbert space
associated to the black hole. If we associate a Hilbert space $H_{i}$ with
each cell on the fuzzy sphere, we shall have that, for a fuzzy sphere with n
cells, the hole Hilbert space will be given by $H=\bigotimes_{i=1}^{n}H_{i}.$
Therefore, we shall have $S=k_{B}\ln(dimH_{i})^{n}=nk_{B}\ln(dimH_{i})$. On
the other hand, the fuzzy sphere area can be written as $A=\alpha nl_{P}^{2}$,
where $\alpha$ is related with the quantization scale, and $l_{P}^{2}$ is the
Planck area.
In this way, we have that the black hole entropy can be written as
$S=\varepsilon k_{B}\frac{A}{l_{P}^{2}},$ (2)
which corresponds to the Bekenstein-Hawking formula, unless for the
undetermined factor $\varepsilon=\ln(dimH_{i})/\alpha$.
The quantum topology change model, in this way, shed some light on the problem
of the origin of the entropy associated to a black hole by an outside
observer: it is generated because of the non-unitary evolutions of the
geometry of the main world. As we shall show, the non-unitary evolution of the
black hole geometry can also be the origin of the GSL.
Now let us address the black hole evaporation process, as it is seen by an
observer in our universe, through the topology change (1). In the intend to do
this, at first, let us address the black hole area spectrum. From the equation
(2), this spectrum is given by
$A_{j}=\epsilon^{-1}l_{p}^{2}\ln(2j+1)\ .$ (3)
We can choose the splitting process (1) in a way that $j\rightarrow
j-\frac{1}{2}$, in the main world, and from (3) this it will result in a
decrease of the black hole area, until when $j=0$ it has completely
evaporated. The logarithmic dependence of the black hole area spectrum on $j$
in the expression (3) tell us that, the decrease in the horizon area occurs in
a continuous way to large values of $j$, and in a discrete way to small values
of $j$, when the black hole approaches the Planck scale, as have been shown in
the figure 1:
Figure 1: Area spectrum, in units of Planck area for $\epsilon=1/4$, to a
quantum black hole in topology change approach. From the logarithmic
dependence of the black hole area spectrum on $j$ in (3), we have that the
levels become continuous for larger area values.
We have that the Hawking radiation is known semi-classically to be continuous.
However, the Hawking quanta of energy are not able to hover at a fixed
distance from the horizon since the geometry of the horizon has to fluctuate,
once quantum gravitational effects are included. Thus, one suspects a
modification of the black hole radiation, when quantum geometrical effects are
taken into account. Then any modification on the description of the black hole
emission process must occur at the final stages of black hole evaporation,
where its area spectrum becomes discrete.
To address it, let us analyze the process which consists in to go one step
down in the black hole area spectrum $(j\rightarrow j-1/2)$. From equation
(1), we have that, tracing over the degrees of freedom in the baby universe,
the splitting process (1), for a matrix $M=\hskip 5.69054pt\mid
j,m\rangle\langle j,m^{{}^{\prime}}\mid$ with $l=1/2$, and $k=j-1/2$ is given
by:
$\displaystyle\Delta(\mid j,m\rangle\langle j,m^{{}^{\prime}}\mid)$
$\displaystyle=$
$\displaystyle\frac{\sqrt{(k+m_{k}+1)(k+m_{k}^{{}^{\prime}}+1)}}{2k+1}\mid
k,m-1/2\rangle\langle k,m^{{}^{\prime}}-1/2\mid$
$\displaystyle+\frac{\sqrt{(k-m_{k}+1)(k-m_{k}^{{}^{\prime}}+1)}}{2k+1}\mid
k,m+1/2\rangle\langle k,m^{{}^{\prime}}+1/2\mid.$
In this way, the probability to go down one step in the black hole area
spectrum is given by
$p_{j\rightarrow
k\,=\,j-1/2}=phase\times\Big{(}\frac{2k+2}{2k+1}\Big{)}=phase\times\;e^{\frac{-\epsilon\delta
A}{l^{2}_{p}}}\;,$ (4)
where a normalization factor $(2k+1)^{-1}$ has been included.
The probability for a black hole goes n steps down into its area spectrum can
be obtained from equation (4). In reference c.a.s.silva (9) it been shown that
the splitting process (1) obeys cluster decomposition. In this way, we have
that the different steps $j\rightarrow j-1/2$ in the black hole evaporation
are independent events. Then the probability of n steps occur in the black
evaporation process is given by the product of the probabilities of each one
of this steps occurring by itself. Then we have,
$p_{kn}=phase\times\Big{(}\frac{2k+2}{2k-n+1}\Big{)}=phase\times\;e^{\frac{-\epsilon\delta
A_{kn}}{l^{2}_{p}}},$ (5)
The probability above depends on the undetermined parameter $\epsilon$, which
appears in the expression to the black hole entropy (2). We have that, from
the equation (5), the density matrix describing the black hole is given by
$\hat{\rho}\sim\sum_{j}e^{-\epsilon A_{j}}\mid j\rangle\langle
j\mid=e^{-\epsilon\hat{A}}$. In this way, the black hole density matrix must
satisfy the equation
$i\frac{\partial\hat{\rho}}{\partial\Theta}=-\frac{\hat{A}}{8\pi}\hat{\rho}$
(6)
As addressed by s.carlip-cqg12 (14) and s. massar-nucphysb575 (15), the
parameter $\Theta=8\pi i\epsilon$ plays the role of a sort of “dimensionless
internal time” associated with the horizon. The equation (6) must be used when
working in the Euclidean continuation of the black hole m.banados-prl72 (13).
Regularity of the Euclidean manifold at the horizon imposes a fixed Euclidean
angle given by $\Theta=2\pi i$ m.banados-prl72 (13), s.carlip-cqg12 (14), and
s. massar-nucphysb575 (15). In this way, the undetermined parameter in (2) and
(5) is fixed as $\epsilon=1/4$.
The results above revel the evolution of the black hole geometry induced by
the topology change process (1) in the evaporation process, and bring
essential consequences for the way how entropy is emitted in the black hole
evaporation process.
The entropy of a system measures one’s lack of information about its actual
internal configuration c.e.shannon-mtc (16, 17, 18, 19). Suppose that all that
is known about the internal configuration of the system is that it may be
found in any of a number of states, with probability $p_{n}$ for the nth
state. Then the entropy associated with the system is given by Shannon’s well-
known relation $S=-\sum p_{n}lnp_{n}$.
The probability for a black hole to emit a specific quantum should be given by
the expression (5), in which we shall include a gray-body factor $\Gamma$
(representing a scattering of the quantum off the spacetime curvature surround
the black hole). Thus, the probability $p_{n}$ to n steps in the mass (area)
ladder is proportional to $\Gamma(n)\;e^{-\frac{\delta A_{kn}}{4l_{p}^{2}}}$.
The discrete mass (area) spectrum (3) implies a discrete line emission from a
quantum black hole. For a Schwarzschild black hole, the radiation emitted by
the black hole will be at frequencies given by
$\omega=\frac{1}{2\sqrt{\pi}}(\sqrt{ln(2j+1)}-\sqrt{ln(2j+1-n)})$.
To gain some insight into the physical problem, we shall consider a simple toy
model suggested by Hod S.Hod (20). It is well known that, for massless field,
$\Gamma(M\omega)$ approaches $0$ in the low-frequency limit, and has a high-
frequency limit of $1$ dn.page-prd13 (21, 22, 23).That is,
$\Gamma(\bar{\omega})=0$ for $\bar{\omega}<\bar{\omega}_{c}$, and
$\Gamma(\bar{\omega})=1$ otherwise, where $\bar{\omega}=M\omega$.
The ratio
$R=\mid\\!\\!\\!\\!\\!\\!\dot{\;\;\;S_{rad}}/\\!\\!\\!\\!\\!\\!\dot{\;\;\;S_{BH}\\!}\mid$
of entropy emission rate from the quantum black hole to the rate of black hole
entropy decrease is given by:
$R=\frac{-\sum_{i=1}^{N_{s}}\sum_{n=1}^{2k+1}C\Gamma(n)e^{-\frac{\delta
A_{kn}}{4}}ln\Big{[}C\Gamma(n)e^{-\frac{\delta
A_{kn}}{4}}\Big{]}}{\sum_{i=1}^{N_{s}}\sum_{n=1}^{2k+1}C\Gamma(n)e^{-\frac{\delta
A_{kn}}{4}}\Big{(}\frac{\delta A_{kn}}{4}\Big{)}}\;,$ (7)
where C is a normalization factor, defined by the normalization condition:
$\sum_{i=1}^{N_{s}}\sum_{n=1}^{2k+1}C\Gamma(n)e^{-\frac{\delta
A_{kn}}{4}}=1\;\;.$ (8)
$N_{s}$ is the effective number of particle species emitted ($N_{s}$ takes
into account the various modes emitted). We shall consider $N_{s}\propto
2k+2$, with a proportionality constant less than or equal to one, since the
modes emitted by the black hole in our treatment must be upper limited by the
number of degrees of freedom on the fuzzy sphere.
We have plotted R down, taking $\bar{\omega}_{c}\simeq 0.2$ (the location of
the peak in the total power spectrum dn.page-prd13 (21, 22, 23)).
From the graphic above, we have that the emission process respect a “second
law of thermodynamics”, since R is ever larger than (or equal) to unity. In
this way, the non-unitary evolution of the black hole geometry in the main
world, due to the topology change process, can be the origin of the GSL.
Besides it is important to notice the entropy emitted from the black hole
decreases as the area spacing increases. In this way, the entropy of the
radiation emitted by the black hole becomes increasingly smaller with each
step of the evaporation process, mainly when the black hole reaches the Planck
scale ( notice that R reaches the unity in the last steps).
In this work we have argued that the quantum topology change model to black
hole evaporation, proposed by us in reference c.a.s.silva (9), shed some light
on the problem of the origin of black hole entropy: it is generated because of
the non-unitary evolutions of the geometry of the main world. Besides, the
topology change approach give us a relation of states to points that agrees
with our standard concept of entropy as the logarithm of the number of
microstates, from which we have derived the Bekenstein-Hawking formula,
$S=A/4$.
From the topology change model we have obtained a black hole area spectrum,
which is continuous in the classical/semiclassical limit, and becomes discrete
as the black hole approaches the Planck scale. As a consequence of this, we
have that information can leak out from black hole, since its radiation
becomes less and less entropic as the black hole evaporates. In this way, some
information about the black hole initial state could be accessible to an
observer in our universe. Since it would occurs more strongly in the quantum
gravity limit, it does not require radical modifications in the laws of
physics above the Planck scale. The task of found an appropriate quantum
mechanism for information leak remains.
We have also shown that the description of the black hole evaporation through
a quantum topology change process proposed by c.a.s.silva (9) could be the
origin of the GSL.
The authors thanks to Coordenação de Aperfeiçoamento de Pessoal de Ensino
Superior-CAPES(Brazil), and Conselho Nacional de Desenvolvimento Científico e
Tecnológico - CNPQ(Brazil) for the financial support, and C.A.S. Almeida for
the careful reading of the manuscript
## References
* (1) S.W. Hawking, Particle Creation by Black Holes, Commun.Math.Phys. 43 (1975) 199–220.
* (2) S.W. Hawking, Breakdown of Predictability in Gravitational Collapse, Phys.Rev.D 14 (1976) 2460–2473.
* (3) J.D. Bekenstein, Black holes and the second law, Lett. Nuovo Cim. 4 (1972) 737–740.
* (4) Andrew Strominger and Cumrun Vafa, Microscopic origin of the Bekenstein-Hawking entropy , Phys. Lett. B 379 (1996) 99-10.
* (5) A. Ashtekar, J. Baez, A. Corichi, and K. Krasnov, Quantum Geometry and Black Hole Entropy, Phys. Rev. Lett. 80, (1998) 904–907.
* (6) J. Preskill, Do black holes destroy information?, hepth/9209058.
* (7) D.N. Page, Black hole information, hepth/9305040.
* (8) J. Madore, The Fuzzy sphere, Class.Quant.Grav. 9 (1992) 69–88.
* (9) C.A.S. Silva, Fuzzy spaces topology change as a possible solution to the black hole information loss paradox, Phys.Lett.B 677 (2009), 318-321.
* (10) S.D.H. Hsu, Spacetime topology change and black hole information, Phys.Lett.B 664 (2007) 67–71.
* (11) L. Susskind, Comment on a proposal by Strominger, hep-th/9405103.
* (12) A.P. Balachandran, S. Kürkçüoǧlu, Topology change for fuzzy physics: Fuzzy spaces as Hopf algebras, Int.J.Mod.Phys.A 19 (2004) 3395–3408.
* (13) M. Banados, C. Teitelboim, J. Zanelli, Black hole entropy and the dimensional continuation of the Gauss-Bonnet theorem, Phys.Rev.Lett.72 (1994) 957-960.
* (14) Steven Carlip, Claudio Teitelboim, The off-shell black hole, Class.Quant.Grav.12 (1995) 1699–1704.
* (15) S. Massar, R. Parentani , How the change in horizon area drives black hole evaporation, Nucl.Phys.B 575 (2000) 333–356.
* (16) C.E. Shannon and W. Weaver, The mathematical theory of communications (University of Illinois Press, Urbana, 1949).
* (17) E.T. Jaynes, Information Theory and Statistical Mechanics, Phys.Rev.106 (1957) 620-630.
* (18) E.T. Jaynes, Information Theory and Statistical Mechanics II, Phys.Rev.108 (1957) 171-190.
* (19) J. D. Bekenstein, Black Holes and Entropy, Phys. Rev. D 7 (1973) 2333–2346.
* (20) Shahar Hod, Black hole radiation, the fundamental area unit, and the spectrum of particle species, Phys.Rev.D 61 (2000) 124016\.
* (21) D.N. Page, Particle Emission Rates from a Black Hole: Massless Particles from an Uncharged, Nonrotating Hole, Phys.Rev.D13 (1976) 198-206.
* (22) D.N. Page, Particle Emission Rates from a Black Hole. 2. Massless Particles from a Rotating Hole, Phys.Rev.D14 (1976) 3260-3273.
* (23) D.N. Page, Particle Emission Rates from a Black Hole. 3. Charged Leptons from a Nonrotating Hole, Phys.Rev.D16 (1977) 2402-2411.
|
arxiv-papers
| 2010-03-18T20:39:16 |
2024-09-04T02:49:09.158749
|
{
"license": "Public Domain",
"authors": "C.A.S. Silva and R.R. Landim",
"submitter": "Carlos Alex Souza da Silva M.Sc",
"url": "https://arxiv.org/abs/1003.3679"
}
|
1003.3684
|
bib
# Parallel Generation of Massive Scale-Free Graphs
Andy Yoo Keith Henderson
(Lawrence Livermore National Laboratory
Livermore, CA 94551)
###### Abstract
One of the biggest huddles faced by researchers studying algorithms for
massive graphs is the lack of large input graphs that are essential for the
development and test of the graph algorithms. This paper proposes two
efficient and highly scalable parallel graph generation algorithms that can
produce massive realistic graphs to address this issue. The algorithms,
designed to achieve high degree of parallelism by minimizing inter-processor
communications, are two of the fastest graph generators which are capable of
generating scale-free graphs with billions of vertices and edges. The
synthetic graphs generated by the proposed methods possess the most common
properties of real complex networks such as power-law degree distribution,
small-worldness, and communities-within-communities. Scalability was tested on
a large cluster at Lawrence Livermore National Laboratory. In the experiment,
we were able to generate a graph with 1 billion vertices and 5 billion edges
in less than 13 seconds. To the best of our knowledge, this is the largest
synthetic scale-free graph reported in the literature.
## 1 Introduction
Recent studies have revealed that many real-world graphs belong to a special
class of graphs called complex networks (or graphs). Examples of the real-
world complex graphs include World-Wide Web [4], Internet [12, 13, 25, 30],
electric power grids [31], citation networks [16, 26, 28, 29], telephone call
graphs [1], and e-mail network [9]. These graphs typically carry a wealth of
valuable information for their respective domains. Therefore, a great deal of
research effort has been concentrated on developing algorithms to identify and
mine certain knowledge or data of interest from these graphs. For example,
algorithms that can find groups of vertices that have strong associations
between them (called communities) have been reported [5, 8, 22, 23, 24]. There
exist algorithms which, given a template pattern, can find subgraphs that
closely match to the input pattern [17]. Such algorithms can play a very
important role in detecting certain criminal activities or making critical
business decisions.
The real-world complex graphs are typically very large (with millions or more
vertices) and their sizes grow over time. Some researchers predict that the
size of these graphs will eventually reach $10^{15}$ vertices [15]. The high
complexity of the graph algorithms, combined with the large and increasing
size of the target graphs, however, makes these applications to be very
difficult to apply to large real graphs. Efforts are being made to parallelize
these applications [32, 2] and develop efficient out-of-core graph algorithms
[27] to cope with the technical challenges.
One of key issues in developing these graph applications is the availability
of large input graphs, as these graphs are essential for the developers to
develop and test the applications and to measure their scalability and
performance. Unfortunately, we do not have publicly available real graphs that
are large enough to test the functionality and true scalability of the graph
applications. A social network graph derived from the World Wide Web, for
example, contains 15 million vertices [11] and the largest citation network
available has two million vertices [21]. Although these real-world graphs tend
to grow in size, it is unlikely that the real graphs of sufficiently large
size will be available in the near future.
The lack of the large graphs has forced the researchers to use synthetically
generated random graphs, which are relatively cheap to construct, in their
experiments [32]. The random graphs (also known as Erdös-Rényi random graphs
[10]), however, is uninformative, since the structure of the random graphs
greatly differs from that of real-world graphs. In the absence of the large
real graphs, synthetic graphs may be used for the development of the graph
applications. There exist several good models to synthetically generate
complex networks [3, 18, 31]. A serious drawback of these models is that they
are all sequential models and hence, are inadequate to use to generate the
massive graphs with billions of vertices and edges.
In this paper, we propose two efficient and highly scalable graph generation
methods. Based on serial models [3, 18], these methods are designed to
generate massive scale-free graphs in parallel on distributed parallel
computers. These parallel generators require very little inter-processor
communications and thus achieve high degree of parallelism. The first method,
called parallel Barabasi-Albert (PBA) method, iteratively builds a graph using
a technique called two-phase preferential attachment. The second parallel
method, called parallel Kronecker (PK) method, applies the concept of
Kronecker product of matrices [14] and constructs a graph recursively in a
fractal fashion from a given seed graph. These are two of the fastest graph
generation algorithms with capability of generating scale-free graphs with
billions of vertices and edges. We have demonstrated their scalability by
constructing massive graphs on a large cluster at Lawrence Livermore National
Laboratory. In the experiment, we have generated a scale-free graph with 1
billion vertices and 5 billion edges in less than 13 seconds, and to the best
of our knowledge, this is the largest synthetic scale-free graph ever reported
in the literature. We also have analyzed the properties of the graphs
generated by the proposed methods and report the results in this paper. We
have found that these graphs possess commonly known properties of real-world
complex networks, including power-law degree distribution, small-worldness,
and communities-within-communities.
The remaining of the paper is organized as follows. Section 2 surveys the
related work in the literature. The proposed parallel models are described in
Section 3. Section 4 presents the results from performance and
characterization study, followed by concluding remarks and directions for
future work in Section 5.
## 2 Related Work
Erdös and Rényi have proposed a simple model that generates equilibrium random
graphs, called Erdös-Rényi random graphs [10]. In this model, given a fixed
number of vertices, a graph is constructed by connecting randomly chosen
vertices with an edge repeatedly until the predetermined number of edges are
obtained. This model is restrictive in that it produces only Poisson degree
distributions.
Dorogovtsev et al. proposed a model that can generate graphs with fat-tailed
degree distributions [7]. Given a random graph, this model restructures the
given graph by rewiring a randomly chosen end of a randomly chosen edge to a
preferentially chosen vertex and also moving a randomly chosen edge to a
position between two preferentially chosen vertices at each step of the
evolution.
The model proposed by Watts and Strogatz [31] generates random structures with
small diameter, which has been named as small-world graphs. This model
transforms a regular one-dimensional lattice (with vertex degree of four or
higher) by rewiring each edge, with certain probability, to a randomly chosen
vertex. It has been found that, even with the small rewiring probability, the
average shortest-path length of the resulting graphs is of the order of that
of random graphs, and generate graphs with fat-tailed degree distributions.
The majority of recent models uses a method called preferential attachment
[6]. In a representative model among these, proposed by Barabasi and Albert
[3], a new vertex joins the graph at each time step and gets connected to an
existing vertex with probability proportional to the vertex degree. With
preferential attachment, these models can emulate the dynamic growth of real
graphs.
Leskovec et al. [18] have proposed a graph generation model that addresses
some of recently discovered properties of time-evolving graphs: densification
and shrinking diameter. The main idea of their model is to recursively create
self-similar graphs with certain degree of randomness. The self-similarity of
the graphs is achieved by using the Kronecker product (also known as tensor
product) [14], which is a natural tool to construct self-similar structures.
Given a seed graph, at each step this model computes the Kronecker product of
two matrices that represent the seed graph and the graph generated in previous
step respectively. The graphs generated with this method have regular
structure. The model changes the entries in the target matrix with a certain
probability before each multiplication to add randomness to the graph.
## 3 Proposed Graph Generation Methods
### Parallel Barabasi-Albert (PBA) method
Scale-free graphs can be easily generated using a well-known technique called
preferential attachment [6]. In a simple serial model known as Barabasi-Albert
(BA) model [3], a scale-free graph is constructed, starting with a small
clique, by repeatedly creating a vertex and attach it to one of the existing
vertices with probability proportional to its current degree.
We have parallelized the BA model in this research and propose a graph
generation algorithm called parallel BA (PBA) method. In this method, vertices
are distributed to the processors, and all the edges adjacent to a given
vertex are stored on the same processor to which the vertex is assigned.
Sets of processors called factions are used in the PBA method. Each processor
belongs to one or more factions. The number of processors in each faction
varies. Such variation is essential for the correct implementation of the
preferential attachment operation in a distributed environment. Furthermore,
we can assign the processors to factions in a manner to enable us to generate
graphs with certain structures. The size of each faction is a degree of
freedom in this method. The number of factions is another degree of freedom.
To facilitate the implementation, we choose to assign all vertices on a single
processor to the same set of factions. In other words, if two vertices reside
on the same processor, then they are members of the same set of factions.
It is crucial to use an efficient implementation of the preferential
attachment to allow this method to scale. This can be done most efficiently by
selecting an existing edge from the graph with a uniform probability and then
randomly selecting one of its endpoints as the point to which a new vertex can
be attached. Therefore, an edge can be added in constant time in this
implementation.
A slight variation of this algorithm is used in the PBA method. The proposed
PBA algorithm is described below in detail. It is assumed that the algorithm
runs on a processor $p$. Other processors perform the same algorithm. We also
assume that $p$ is a member of factions $F_{0}$, $F_{1}$, …, $F_{n-1}$.
In the PBA method, an edge is attached in two phases. In the first phase of
our preferential attachment, $k$ edges are added per newly created local
vertex (a vertex that resides on $p$) as in the conventional BA model.
However, each edge, $e$, associates a local vertex with some processor $q$,
instead of connecting two vertices as in the serial model. The particular
vertex that is to be the eventual endpoint of $e$ is determined remotely by
the processor $q$.
The processor $q$ is selected using a variation of the preferential attachment
algorithm as follows. Let $A$ denote a local edge list maintained by the
processor $p$. First, we initialize $A$ by associating the first $s$ edges
with the processors in factions $F_{0}$, $F_{1}$, …, $F_{n-1}$, matching
sequentially one edge to one processor in the set of factions. Here, $s$ is
the total number of processors in factions $F_{0}$, $F_{1}$, …, $F_{n-1}$
(i.e., $s=\sum_{i=0}^{n-1}|F_{i}|$). For an edge $e_{j}$, where $j\geq s$, we
select an existing edge from $A$ with a uniform probability (thus realizing
preferential attachment) and then assign its associated processor to $e_{j}$.
This process is repeated until the predetermined number of local vertices and
edges are created on $p$. At the end of the first phase, $p$ sends a message
to each processor $q$ to notify the number of occurrences of $q$ in $A$.
In the second phase, $p$ determines the endpoints for the edges on remote
processors and connects the endpoints calculated by remote processors to its
local vertices. The processor $p$ first receives messages from other
processors, which contain the numbers of occurrences of $p$ in their
respective local edge list. That is, the message received from a processor $q$
represents the number of incomplete edges one of whose endpoints resides on
the processor $q$. These edges are to be connected to the local vertices on
$p$, selected by using the standard preferential attachment technique. Once
the list of the vertices for the attachment is determined, it is divided up
among the processors. Here, each processor is assigned as many vertices as it
requested. The selected vertices are then sent to the corresponding
processors.
Having sent the endpoints for the remote edges, then $p$ receives the lists of
endpoints from other processors for its own incomplete edges. Using the remote
vertices received, $p$ completes its local partition of the graph. This is
done by simply substituting each occurrence of processor $q$ in $A$ with the
next endpoint in the list sent by $q$. The resulting collection of edges
defines the portion of the graph stored on $p$.
| $u$ | 0 | 0 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4
---|---|---|---|---|---|---|---|---|---|---
$v$ | $P_{1}$ | $P_{2}$ | $P_{0}$ | $P_{1}$ | $P_{1}$ | $P_{2}$ | $P_{0}$ | $P_{1}$ | $P_{0}$ | $P_{2}$
(a) Edge list on $P_{0}$ at the end of phase 1
| $u$ | 0 | 0 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4
---|---|---|---|---|---|---|---|---|---|---
$v$ | 8 | $P_{2}$ | $P_{0}$ | 7 | 5 | $P_{2}$ | $P_{0}$ | 8 | $P_{0}$ | $P_{2}$
(b) A snapshot of edge list on $P_{0}$ during the phase 2
Figure 1: An example of PBA graph construction on processor $P_{0}$. A list of
edges, ($u$, $v$) is maintained on $P_{0}$, where $u$ denotes a local vertex
and $v$ is an endpoint determined by remote processors in phase 2. In this
example, three factions are used, where $F_{0}$ = {$P_{1}$, $P_{2}$}, $F_{1}$
= {$P_{1}$, $P_{3}$}, $F_{2}$ = {$P_{0}$, $P_{1}$}. Processor $P_{0}$ belongs
to fractions $F_{0}$ and $F_{2}$.
The two-phase preferential attachment is explained using an example in Figure
1. In this example, we generate a graph with 5 vertices per processor and 2
edges per vertex. It is assumed that there are three factions, $F_{0}$ =
{$P_{1}$, $P_{2}$}, $F_{1}$ = {$P_{1}$, $P_{3}$}, and $F_{2}$ = {$P_{0}$,
$P_{1}$} and processor $P_{0}$ belongs to fractions $F_{0}$ and $F_{2}$. The
vertices are assumed to be evenly distributed among the processors so that
vertices 0–4 are on $P_{0}$, vertices 5–8 on $P_{1}$, and so on.
In the first phase of the algorithm, $P_{0}$ selects processors and associates
them with the local vertices as shown in Figure 1.a, where the edge list on
$P_{0}$ is depicted. Note that the first four processors in the list are the
ones in the factions that $P_{0}$ belongs to, $F_{0}$ and $F_{2}$. The rest of
the processors in the list are selected using the standard preferential
attachment technique. At the end of phase 1, $P_{0}$ needs four endpoints from
$P_{1}$ (and three endpoints from each of $P_{0}$ and $P_{2}$). These
endpoints are determined by processor $P_{1}$ via preferential attachment and
sent to $P_{0}$ in the second phase. In this example, we assume that vertices
8, 7, 5, and 8 are sent to $P_{0}$. Once receiving the list, $P_{0}$ simply
replaces the entries marked with $P_{1}$ with the endpoints in the list. This
is shown in Figure 1.b.
We have found that it is useful to modify the algorithm slightly to
incorporate some inter-faction edges. In particular, during the first phase,
we occasionally select a processor that is not in any of the factions of $p$.
Such processors are chosen randomly. The probability of creating an inter-
faction edge is an another degree of freedom in this algorithm.
### Parallel Kronecker (PK) method
|
---
| $\left(\begin{array}[]{ccccc}1&1&1&1&0\\\ 1&1&0&0&0\\\ 1&0&1&0&0\\\
1&0&0&1&0\\\ 0&0&0&0&1\\\ \end{array}\right)$
---
(a) Seed graph $G_{1}$ with 5 vertices | (b) Adjacency matrix for $G_{1}$
| $\left(\begin{array}[]{ccccc}G_{1}&G_{1}&G_{1}&G_{1}&0\\\
G_{1}&G_{1}&0&0&0\\\ G_{1}&0&G_{1}&0&0\\\ G_{1}&0&0&G_{1}&0\\\
0&0&0&0&G_{1}\\\ \end{array}\right)$
---
|
---
(c) Adjacency matrix for $G_{2}=G_{1}\otimes G_{1}$ | (d) Plot for $G_{4}$
Figure 2: An example of the Kronecker multiplication for graph generation.
$G_{2}$ is constructed by multiplying the seed graph with itself (that is,
$G_{2}=G_{1}\otimes G_{1}$). The self-similarity in $G_{4}$ is clearly shown
in (d).
A model that uses well-known concept of Kronecker matrix multiplication to
generate scale-free graphs has been recently proposed [18]. If $A$ is an
$m\times n$ matrix and $B$ is a $p\times q$ matrix, then the Kronecker product
$A\otimes B$ is the $mp\times nq$ block matrix, defined as
$\displaystyle A\otimes B=\left(\begin{array}[]{ccc}a_{11}B&\cdots,&a_{1n}B\\\
\vdots&\ddots&\vdots\\\
a_{m1}B&\cdots,&a_{mn}B\end{array}\right)~{}\cite[cite]{[\@@bibref{}{horn1991}{}{}]}.$
The Kronecker product of two graphs is defined as the Kronecker product of
their adjacency matrices. Figure 2 shows an example where a Kronecker graph is
generated from given seed graph using the Kronecker graph multiplication.
Since the Kronecker method is an ideal tool to construct self-similar
structures, the graph generated by this method has also self-similar structure
as shown in Figure 2.d.
Implementing a serial algorithm for the above graph generation method is
straightforward. Starting with an $n_{0}\times n_{0}$ adjacency matrix with
$e_{0}$ edges representing a given seed graph, we recursively construct larger
adjacency matrices using Kronecker matrix multiplication in a top-down manner.
To generate the $i_{th}$ matrix from the $(i-1)_{th}$ matrix, we simply
replace every 1 in the $i_{th}$ matrix by an $n_{0}\times n_{0}$ block that is
a copy of the seed graph. We replace every 0 by an $n_{0}\times n_{0}$ block
of zeros. So if the $i_{th}$ graph has $n_{i}$ vertices, the $(i+1)_{th}$
graph has $n_{i}\times n_{i}$ vertices. Thus, $n_{i}$ = $n_{0}^{i+1}$ for all
iterations. We treat each edge in the graph at each iteration as a meta-edge.
A meta-edge is defined by its iteration and its position in the graph for that
particular iteration. Given the size of a target graph, we can calculate the
number of iterations required to generate the target graph.
A Kronecker graph can be generated efficiently by using a stack, initialized
with the edges in the seed graph. A graph is generated by expanding meta-edges
in the stack as follows. First, a meta-edge on the top of the stack is popped
up. If its iteration, $i$, is equal to the predetermined final iteration, then
the edge is added to the final graph. Otherwise, new meta-edges with iteration
$i+1$ are generated and pushed onto the stack. This operation is repeated
until the stack is depleted. We choose a stack because it guarantees that the
memory requirement is limited to $O(\sqrt[e_{0}]{|E|})$, where $|E|$ is the
number of edges in the final graph. An implementation using a queue is not
scalable, as it would require $O(|E|)$ memory space.
In the parallel implementation of the Kronecker method, the meta-edges are
divided among the groups of processors at each iteration. Each processor group
generates the same meta-edges at a given iteration. If there are more
processors in a processor group than there are edges in that group’s portion
of the graph, then each processor in the group is assigned to a single edge in
the stack. Here, each edge defines a new processor group that is a subset of
the original, and the process group ignores all other meta-edges at that
iteration. On the other hand, if there are more meta-edges than processors in
the processor group, the edges are divided as evenly as possible among those
processors. Each of those processors is then in a singleton processor group
for the remaining iterations. Each processor must be able to calculate on the
fly which meta-edges are in its processor group at a given iteration.
In general, it is difficult to achieve good load balance with the PK method,
as some processors may be assigned more work depending on processor group
sizes. A dynamic load balancing scheme may be used in conjunction with the PK
method to overcome this limitation. Furthermore, some randomization logics are
needed to irregulate the structure of the PK graphs. One approach for the
randomization is to add or delete meta-edges during the replacement phase at
each iteration by randomly modifying the seed graph temporarily. Another
approach is to perform exclusive-OR operation between the final adjacency
matrix with the adjacency matrix for a random graph.
## 4 Experimental Results
### 4.1 Experiment environment and metrics of interest
We have conducted a study to evaluate the proposed graph generators. The
experiments were conducted on MCR [20], a large Linux cluster located at
Lawrence Livermore National Laboratory. MCR has 1,152 nodes interconnected
with a Quadrics switch, and each of the compute nodes has two 2.4 GHz Intel
Pentium 4 Xeon processors and 4 GB of memory,
In this study, we are mainly interested in evaluating the performance of the
proposed graph generators and analyzing the graphs they generate. There are
well-known structural and temporal properties of the real complex networks
[19]. We use widely-accepted properties as indices to quantitatively evaluate
the synthetic graph generated by the proposed methods.
### 4.2 Results
Methods | $|V|$ (Million) | $|E|$ (Billion) | Time (Seconds)
---|---|---|---
PBA | 1,000 | 5 | 12.39
PK | 0.53 | 5.4 | 2.53
Table 1: Comparison of graph generation time by PBA and PK methods. The number
of vertices and edges in generated graphs are denoted by $|V|$ and $|E|$,
respectively.
Two graphs were generated by using the PBA and PK methods on 1,000 processors
on the MCR cluster, and we report the graph generation times in Table 1. The
generation time is an average of multiple runs. We have measured the maximum
time across all processes in each run. The disk I/O time is not included in
the time reported.
Both graphs have about 5 billion edges.111We measure the size of a graph by
the total space needed to store the graph in this paper. That is, given a
graph $G$ = ($V$, $E$), its size is $|V|+|E|$. Therefore, we consider the PBA
graph to be larger than the PK graph in this experiment. The number of
vertices in the PK graph is considerably smaller than that in the PBA graph
due to our use of a seed graph with large average degree. As shown in the
table, it takes less than 13 seconds to generate these massive graphs. The
high generation rate can be attributed to the high degree of parallelism of
the proposed algorithms.
---
Figure 3: Weak-scaling results
The performance of both methods is further detailed in Figure 3, where we show
the results from a weak-scaling study. In a weak-scaling test, the global
problem size increases as the total number of processors increases such that
the size of local problem remains constant. The local problem size of roughly
one million vertices and three million edges is used. The figure reveals that
the PK method is about four times faster than the PBA method. In particular,
the almost flat curve for the PK method highlights the embarrassingly-parallel
nature of the algorithm. The graph generation time for the PBA method, on the
other hand, increases as the number of processors increases. This is because
in the PBA method each processor processes endpoint vertices sent by remote
processors at the end of the execution, and the complexity of the processing
increases in proportion to the total number of processors used. Profiling of
the code confirms that each process spends most of its time in processing the
received endpoints.
|
---|---
(a) PBA graph ($\gamma=2.75$) | (b) PK graph ($\gamma=2.47$)
---
(c) Router graph ($\gamma=2.19$)
Figure 4: Degree distributions of the synthetic and router graphs. Here $k$
and $P(k)$ denote the degree and the number of vertices with degree $k$,
respectively. The $\gamma$ exponent of a power-law distribution, $P(k)\propto
k^{-\gamma}$, for each graph is obtained through curve fittings and shown here
as well.
In the remaining of this section, we analyze the graphs produced by the
proposed methods. A PBA graph studied in this experiments has 330,000 vertices
and 2 million edges. A PK graph with 160,000 vertices and 28 million edges,
constructed using a small seed graph with 20 vertices and 40 edges, is
analyzed. We also consider two real-world graphs, WWW and router graphs, for
comparison. The WWW graph has 325,000 vertices and 2.1 million edges, and the
smaller router graph has 285,000 vertices and 861,000 edges.
Figure 4 presents the degree distributions of the synthetic graphs and
compares them with that of the router graph. The graphs are shown in a log-log
scale. As shown in the figure, the curves for both PBA and PK graphs are
heavy-tailed. This is a signature of power-law degree distribution that is one
of the widely accepted property of real-world complex networks. It is shown in
Figure 4.c that the router graph also has fat-tailed degree distribution. To
verify that these graphs have power-law degree distributions, $P(k)\propto
k^{-\gamma}$, we have performed curve fittings for the measured degree
distributions and show the exponent of the power-law distribution ($\gamma$)
in Figure 4. As shown in the figure, $\gamma$ values for the three graphs are
greater than 2. This finding coincides with the fact that if the average
degree of a scale-free graph is finite, then its $\gamma$ value should be
$2<\gamma<\infty$ [6]. The PK graph has a large number of high degree
vertices. This is because the number of low-degree vertices is small in a
graph generated by the PK method, in which the degree of a vertex grows
exponentially. We can change the degree distribution by randomly adding low-
degree vertices to the final graph.
Graph | Avg. Path Length | Diameter (estimated)
---|---|---
WWW Graph | 7.54 | 46
Router Graph | 8.87 | 27
PK Graph | 3.20 | 5
PBA Graph | 6.26 | 12
Table 2: The comparison of path length and diameter of the synthetic graphs
with two real graphs. Both metrics are estimated by sampling to reduce the
computation overhead.
Table 2 presents the average path lengths and diameters of the two synthetic
graphs considered in the previous experiment as well as the WWW and router
graphs. Both metrics are estimates obtained through sampling to reduce the
computation time. Each of the graphs analyzed has short average path length,
which is the average value of the shorted path between two randomly chosen
vertices. Further, each synthetic graph has a small diameter that is the
maximum of all-pairs shortest path. These results indicate that the graphs
generated by the proposed methods have small world property, which is another
key characteristic of real-world complex networks. Obviously, such small-
worldness is more evident in the PK graph, as it contains a large number of
high-degree vertices (or hubs). Two real graphs, the WWW graph in particular,
appear to have the smaller number of hubs as indicated by the larger
diameters.
|
---
|
---
(a) PBA graph | (b) PK graph
Figure 5: Communities within PBA and PK graphs. The graphs are represented as
adjacency matrices.
In Figure 5, we show two adjacency matrices for PBA and PK graphs to visualize
the community structures within these synthetic graphs. As shown in the
figure, the PBA and PK graphs have clearly identifiable community structures.
A major difference between the two graphs is that the PK graph has more
regularly-structured communities compared to the PBA graph. The regular
community structure of the PK graph is the result of the systematic way of
graph construction by the PK method (using the Kronecker matrix
multiplication). In addition, the self-similar nature of the Kronecker product
is translated into the communities-within-communities structure in the PK
graph.
### 4.3 Comparison of the PBA and PK methods
An advantage of the PK method over the PBA method is its higher degree of
parallelism. The PK method is embarrassingly parallel, as once a seed graph is
given, processors generate the assigned portions of the target graph
independent of each other. A key limitation of the PK method is that the
structure of a resulting graph heavily depends on that of the initial seed
graph. In fact, even with randomized edge generation and removal, the
structure of final graph largely depends on the seed graph and thus relatively
regular. In consequence, this limitation makes it very difficult to configure
the PK method to generate a graph with desired property. For example, if the
seed graph is too small it is very difficult to control the degree of
vertices. To control the vertex degree, we need a relatively large seed graph,
but with such a large seed graph, it is hard to control the size of the final
graph.
Although slower than the PK method, the PBA method is still a very fast
algorithm. An obvious advantage of the PBA method is that using preferential
attachment as a key means to construct a graph, the method can be easily
configured to generate a graph of desired size and properties.
## 5 Conclusions and Future Work
Two efficient and scalable parallel graph generation methods that can generate
scale-free graphs with billions of vertices and edges are proposed in this
paper. The proposed parallel Barabasi-Albert (PBA) method iteratively builds
scale-free graphs using two-phase preferential attachment technique in a
bottom-up fashion. The parallel Kronecker (PK) method, on the other hand,
constructs a graph recursively in a top-down fashion from a given seed graph
using Kronecker matrix multiplication. These parallel graph generators operate
with high degree of parallelism. We have generated a graph with 1 billion
vertices and 5 billion edges in less than 13 seconds on a large cluster. This
is the highest rate of graph generation reported in the literature. We have
analyzed the graphs produced by our methods and shown that they have the most
common properties of the real complex networks such as power-law degree
distribution, small-worldness, and communities-within-communities.
There are other known and somewhat debatable properties of complex networks. A
rigorous study of the graphs generated by the proposed methods will reveal
whether these methods can produce synthetic graphs with these properties. This
study will also provide us with better understanding of how the logics used in
our algorithms affect the properties of the synthetic graphs they generate.
Based on this study, we will develop a set of pre- and post-generation
processing and randomization techniques that will enable us to construct a
synthetic graph with desired properties.
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|
arxiv-papers
| 2010-03-18T22:11:14 |
2024-09-04T02:49:09.165262
|
{
"license": "Public Domain",
"authors": "Andy Yoo and Keith Henderson",
"submitter": "Andy Yoo",
"url": "https://arxiv.org/abs/1003.3684"
}
|
1003.3709
|
# Directed transport driven by Lévy flights coexisting with subdiffusion
Bao-quan Ai1 aibq@hotmail.com Ya-feng He2 1Laboratory of Quantum Information
Technology, ICMP and SPTE, South China Normal University, 510006 Guangzhou,
China.
2College of Physics Science and Technology, Hebei University, 071002 Baoding,
China
###### Abstract
Transport of the Brownian particles driven by Lévy flights coexisting with
subdiffusion in asymmetric periodic potentials is investigated in the absence
of any external driving forces. Using the Langevin-type dynamics with
subordination techniques, we obtain the group velocity which can measure the
transport. It is found that the group velocity increases monotonically with
the subdiffusive index and there exists an optimal value of the Lévy index at
which the group velocity takes its maximal value. There is a threshold value
of the subdiffusive index below which the ratchet effects will disappear. The
nonthermal character of the Lévy flights and the asymmetry of the potential
are necessary to obtain the directed transport. Some peculiar phenomena
induced by the competition between Lévy flights and subdiffusion are also
observed. The pseudo-normal diffusion will appear on the level of the median.
Ratchet, Lévy flights, subdiffusion
###### pacs:
05\. 40. Fb, 05. 10. Gg, 05. 40. -a
## I Introduction
Directed Brownian motion induced by zero-mean non-equilibrium fluctuations in
the absence of macroscopic forces and potential gradients is presently under
intense investigation a1 . This comes from the desire to understand
unidirectional transport in biological systemsa2 , as well as their potential
technological applications ranging from classical non-equilibrium modelsa3 to
quantum systemsa4 . A ratchet system is generally defined as a system that is
able to transport particles in a periodic structure with nonzero macroscopic
velocity in the absence of macroscopic force on average. Broadly speaking,
ratchet devices fall into three categories depending on how the applied
perturbation couples to the substrate asymmetry: rocking ratchetsa5 , flashing
ratchetsa6 , and correlation ratchets a7 . Additionally, entropic ratchets, in
which Brownian particles move in a confined structure, instead of a potential,
were also extensively investigated a8 . These studies on directed transport of
the Brownian particles focused on the normal diffusion.
However, anomalous diffusion has attracted growing attention, being observed
in various fields of physics and related sciencesa9 . Description of physical
models in terms of Lévy flights and subdiffusion becomes more and more popular
a10 ; a11 ; a12 ; a13 ; a14 ; a15 ; a16 ; a17 ; a18 ; a19 ; a20 ; a21 ; a22 ;
a23 . In the complex systems the distinct class of subdiffusion processes was
reported in condensed phasesa10 , ecologya11 , and biologya12 . Superdiffusion
driven by Levy flights is actually observed in various real systems and is
used to model a variety of processes such as bulk mediated surface diffusion
a13 , exciton and charge transport in polymers under conformational motion a14
, transport in micelle systems or heterogeneous rocks a15 ,two-dimensional
rotating flow a16 , and many othersa9 . Goychuk and coworkers a17 studied the
subdiffusive transport in tilted periodic potentials and established a
universal scaling relation for diffusive transport. Dybiec and coworkers a18
studied the minimal setup for a Lévy ratchet and found that due to the
nonthermal character of the Lévy noise, the net current can be obtained even
in the absence of whatever additional time-dependent forces. Del-Castillo-
Negrete and coworkers a19 also found the similar results in constant force-
driven Lévy ratchet. Rosa and Beims a20 studied the optimal transport and its
relation to superdiffusive transport and Lévy walks for Brownian particles in
ratchet potential in the presence of modulated environment and external
oscillating forces. We also studied the transport of Brownian particles in the
presence of ac-driving forces and Lévy flights and multiple current reversals
were observeda21
Recently, much attention has been devoted to the competition between
subdiffusion and Lévy flights. The competition is conveniently described by
the fractional Fokker-Planck equation with temporal and spatial fractional
derivatives a9 . It is very difficult to see this competition in the framework
of the fractional Fokker-Planck dynamics. Magdziarz and coworkers a22
proposed a equivalent approach based on the subordinated Langevin method to
visualize the competition on the level of sample paths as well as on the level
of probability density functions. Based on this approach Dybiec and coworkers
a23 found that due to the competition between Lévy flights and subdiffusion,
the standard measure used to discriminate between anomalous and normal
behavior cannot be applied straightforwardly. Koren and coworkersa24 have
investigated the first passage times in one-dimensional system displaying a
competition between subdiffusion and Lévy flights and found some peculiar
phenomena.
What will happen when the particles move in a ratchet potential subjected to
subdiffusion and Lévy flights? In order to answer this question we use the
subordinated Langevin method proposed by the Magdziarz and coworkers a22 to
investigate this competition in a minimal Lévy ratchet without any external
driving forces. We emphasize on visualizing the competition on the level of
the group velocity and diffusion and finding how this competition affects the
directed transport.
## II Model and Methods
We consider the transport of the Brownian particles driven by Lévy flights and
subdiffusion in the absence of whatever additional time-dependent forces. The
competition between Lévy flights and subdiffusion in a ratchet potential
$V(x)$ can be described by the fractional Fokker-Planck equation with temporal
and spatial fractional derivatives a9 ; a22
$\frac{\partial p(x,t)}{\partial
t}={{}_{0}}D_{t}^{1-\alpha}\left[\frac{\partial}{\partial
x}\frac{V^{{}^{\prime}}(x)}{\eta}+D\frac{\partial^{\mu}}{\partial|x|^{\mu}}\right]p(x,t),$
(1)
where $p(x,t)$ is the probability density for particles at position $x$ and
time $t$. The prime stands for the derivative with respect to the space
variable $x$. $D$ is the anomalous diffusion coefficient which describes the
noise intensity in the subordinated process. $\eta$ denotes the generalized
friction constant. Here ${{}_{0}}D_{t}^{1-\alpha}$ is the fractional of the
Riemann-Liouville operator ($0<\alpha\leq 1$) defined through a9 ; a22
${{}_{0}}D_{t}^{1-\alpha}g(t)=\frac{1}{\Gamma(\alpha)}\frac{d}{dt}\int^{t}_{0}(t-s)^{\alpha-1}g(s)ds,$
(2)
where $\Gamma(.)$ is the Gamma function. From the definition, it becomes
apparent that subdiffusion corresponds to a slowly decaying memory integral in
the dynamical equation for $p(x,t)$. The operator
$\frac{\partial^{\mu}}{\partial|x|^{\mu}}$, $0<\mu\leq 2$, stands for the
Riesz fractional derivative a9 ; a22 with the Fourier transform
$\mathcal{F}\\{\frac{\partial^{\mu}}{\partial|x|^{\mu}}f(x)\\}=-|k|^{\mu}\tilde{f}(k)$.
The occurrences of the operator ${{}_{0}}D_{t}^{1-\alpha}$ and
$\frac{\partial^{\mu}}{\partial|x|^{\mu}}$ are induced by the heavy-tailed
waiting times between successive jumps and the heavy-tailed distributions of
the jumps, respectively, in the underlying continuous-time random walk scheme.
The case of $\alpha=1$, $\mu=2$ corresponds to the standard Fokker-Planck
equation. $V(x)$ is an asymmetric periodic potential
$V(x)=\frac{V_{0}}{2\pi}\left[\sin(2\pi x)+\frac{\Delta}{4}\sin(4\pi
x)\right],$ (3)
where $V_{0}$ and $\Delta$ are the amplitude and the asymmetric parameter of
the potential, respectively.
Because it is very difficult to solve Eq. (1) analytically and numerically, we
used the subordinated Langevin method proposed by Magdziarz and coworkers a22
to investigate the transport. In their method, the solution $p(x,t)$ of Eq.
(1) is equal to the probability density function of the subordinated process
$Y(t)=X(S_{t}),$ (4)
where the parent process $X(\tau)$ is defined as the solution the stochastic
differential equation
$dX(\tau)=-\frac{V^{{}^{\prime}}(X(\tau))}{\eta}d\tau+D^{1/\mu}dL_{\mu}(\tau),$
(5)
where $L_{\mu}(\tau)$ is the symmetric $\mu$-stable Lévy motion with the
Fourier transform $\mathcal{F}\\{L_{\mu}(\tau)\\}=e^{-\tau|k|^{\mu}}$.
Employing the Euler scheme to Eq. (5), one can obtain
$X(\tau_{0})=0,$ (6)
$X(\tau_{i})=X(\tau_{i-1})-\frac{V^{{}^{\prime}}(X(\tau_{i-1}))}{\eta}\Delta\tau+(D\Delta\tau)^{1/\mu}\xi_{i},$
(7)
where $i=1,2,3...$ and $\xi_{i}$ are the random variables with standard
symmetric $\mu$-stable distribution. The procedure of generating realizations
$\xi_{i}$ is the following a22 ; b1
$\xi_{i}=\frac{\sin(\mu V)}{(\cos
V)^{1/\mu}}\left[\frac{\cos([1-\mu]V)}{W}\right]^{\frac{1-\mu}{\mu}},$ (8)
where the random variable $V$ is uniformly distributed on$(-\pi/2,\pi/2)$, $W$
has exponential distribution with mean one.
The inverse-time $\alpha$-stable subordinator $S_{t}$, which is assumed to be
independent of $X(\tau)$, is defined as
$S_{t}=\inf\\{\tau:U(\tau)>t\\},$ (9)
where $U(\tau)$ is the strictly increasing $\alpha$-stable Lévy motion with
Laplace transform $\mathcal{L}\\{U(\tau)\\}=e^{-\tau k^{\alpha}}$.
Using the standard method of summing increments of the process $U(\tau)$ one
can get
$U(\tau_{0})=0,$ (10)
$U(\tau_{j})=U(\tau_{j-1})+\Delta\tau^{1/\alpha}\zeta_{j},$ (11)
where $j=1,2,3...$ and $\zeta_{j}$ are the skewed positive $\alpha$-stable
random variables a22 ; b1 . The method to generate the random variables is
$\zeta_{j}=\frac{\sin(\alpha(V+\frac{\pi}{2}))}{[\cos(V)]^{\frac{1}{\alpha}}}\left[\frac{\cos(V-\alpha(V+\frac{\pi}{2}))}{W}\right]^{\frac{1-\alpha}{\alpha}},$
(12)
where $V$ and $W$ have the same definitions as that in Eq. (8). From the above
procedures, one can obtain the subordinated process $Y(t)$ and its probability
distribution function is equal to the solution of Eq. (1). For more detailed
information on the algorithm, please see the Ref. (22).
In the classical ratchets, one can use the average velocity and effective
diffusion coefficient to describe the transport. However, for the noise with
distribution of a Lévy-stable law, the mean of the noise and the second moment
may do not exist. As a consequence, the classical stochastic theory (average
velocity and effective diffusion coefficient), which is based on the ordinary
central limit theorem, is no longer valid. To overcome this problem, Dybiec
and coworkers a18 recently proposed a different approach to the Lévy ratchet
problem based on the quantile line analysis for $0<\mu<2$.
Quantile line is a very useful tool for investigation of the overall motion of
the probability density of finding a particle in the vicinity of $Y(t)$ a18 ;
a23 . A median line for a stochastic process $Y(t)$ is a function of
$q_{0.5}(t)$ given by the relationship $Pr(Y(t)\leq q_{0.5}(t))=0.5$.
Therefore, one can use the derivative of the median to define the group
velocity of the particle packeta18 ,
$V_{g}=\frac{dq_{0.5}(t)}{dt},$ (13)
and this definition is valid even for the case of lacking average velocity.
## III Numerical results and discussion
In order to investigate the competition between Lévy flights and subdiffusion
in a ratchet potential, we carried out extensively numerical simulations based
on the subordinated Langevin method a22 . For simplicity we set $\eta=1.0$ and
$1<\mu\leq 2$ throughout the work. In our simulations, we have considered more
than $10^{5}$ realizations to obtain the accurate median. In order to provide
the requested accuracy of the system the dynamics time step was chosen to be
smaller than $10^{-3}$. We have checked that these are sufficient for the
system to obtain consistent results.
Firstly, we will investigate the diffusive properties of the Brownian
particles. Usually, the types of the diffusion processes are determined by the
spread of the distance traveled by a random walker. The diffusion is
characterized through the power law form of the mean-square displacement
$\langle x^{2}(t)\rangle\propto t^{\delta}$. According to the value of the
index $\delta$, one can distinguish subdiffusion ($0<\delta<1$), normal
diffusion ($\delta=1$) and superdiffusion ($\delta>1$). Here, we use the
median of square displacement $M(x^{2})$, instead of mean-square displacement,
to characterize the diffusion. Fig. 1 (a) shows the time dependence of
$M(x^{2})/t$ for different combinations of $\mu$ and $\alpha$ without any
external potential. It is found that the linear time dependence of the median
of square displacement, $M(x^{2})\propto t$, will occur for the case of
$\frac{2\alpha}{\mu}$=1, which indicates the normal diffusion. However, this
is not true, for example $\mu=1.8$ and $\alpha=0.9$, the process is still non-
Markov and non-Gaussian. This pseudo-normal diffusion is due to the
competition between Lévy flights and subdiffusion. Dybiec and coworkers a23
have presented discussions in detail on this paradoxical diffusion. Fig. 1 (b)
presents the time dependence of $M(x^{2})/t$ in the presence of a ratchet
potential. Interestingly, the pseudo-normal diffusion for $\mu=1.8$ and
$\alpha=0.9$ with external potentials is not normal.
Next, we will study the rectified mechanism of the Lévy ratchets. Usually, the
ratchet mechanism demands three key ingredients b2 which are (a) nonlinear
periodic potential: it is necessary since the system will produce a zero mean
output from zero-mean input in a linear system; (b)asymmetry of the potential,
it can violate the symmetry of the response; (c)fluctuating: Lévy flights can
break thermodynamical equilibrium. In Fig. 2 (a), we studied the time
dependence of the median for different values of the asymmetry parameter
$\Delta$ at $\mu=1.5$ and $\alpha=1.0$. The median is positive for $\Delta>0$,
zero at $\Delta=0$, and negative for $\Delta<0$. Therefore, the asymmetry of
the potential will determine the direction of the transport and no directed
transport occurs in a symmetric potential. Now we will give the physical
interpretation of the directed transport for the case of $\Delta=1$. Firstly,
the particles stay in the minima of the potential awaiting large noise pulse
to be catapulted out. The particles will be thrown out to the left and the
right with the equal probabilities. In this case, the distance from minima to
maxima is shorter from the right side than that from the left side.
Consequently, most of the particles are thrown out from the right side,
resulting in positive transport. This gives rise to the overall preferred
motion to the right.
Figure 2 (b) gives the time dependence of the median for different
combinations of $\mu$ and $\alpha$. We find that Lévy flights are necessary to
obtain the directed transport. For Gaussian case ($\mu=2.0$), directed
transport disappears. This is due to the nonthermal character of the Lévy
flights that can break thermodynamical equilibrium. From Fig. 2(a) and (b) we
can see that the asymmetry of the potential and the non-equilibrium character
of the Lévy flights are the two necessary conditions for directed transport.
The direction of the transport is determined by the direction of the steeper
slope of of the potential and the Lévy flights can break thermaldynamical
equilibrium. These two key ingredients can realize the ratchet effects.
Figure 3 illustrates the dependence of the group velocity $V_{g}$ on the
subdiffusive index $\alpha$ for different values of the Lévy index $\mu$. One
can see that group velocity $V_{g}$ increases monotonically with the
subdiffusive index $\alpha$. For small values of $\alpha$, the waiting time
between successive jumps is very long and it is not easy for particles to pass
across the barrier. Thus, most particles will stay in their original minima of
the potential and the group velocity becomes very small. Especially, we also
find that there exists a threshold value of $\alpha$ below which no directed
transport can be obtained. The subdiffusion dominates the transport for small
values of $\alpha$ ($\alpha<0.7$), while the effects of the Lévy flights
become preponderant for large values of $\alpha$.
Figure 4 shows the dependence of the group velocity $V_{g}$ on the Lévy index
$\mu$ for different values of $\alpha$. When $\mu\rightarrow 2.0$, the system
is under thermodynamical equilibrium and no directed transport appears. For
small values of $\mu$, Lévy flights are longer and the outliers in the Lévy
noise are larger. In this case, the effects of the asymmetry of the potential
become very small, resulting in small group velocity. Therefore, there exists
a optimal value of $\mu$ at which the group velocity takes its maxima. This
can also be confirmed by Fig. 3. For very small values of $\alpha$, for
example $\alpha=0.5$, the group velocity is zero for all values of $\mu$ and
the transport is absolutely dominated by subdiffusion.
The group velocity $V_{g}$ as a function of noise intensity $D$ is shown in
Fig. 5 for different combinations of $\mu$ and $\alpha$. The curve is observed
to be bell shaped which shows the feature of resonance. When $D\rightarrow 0$,
the particles cannot pass across the barrier and there is no directed current.
When $D\rightarrow\infty$ so that the noise is very large, the effect of the
potential disappears and the group velocity tends to zero, also. There is an
optimal value of $D$ at which the group velocity is maximal. There are two
intersections ($D_{c1}$ and $D_{c2}$) between the line of $\mu=1.9$ and
$\alpha=1.0$ and the line of $\mu=1.5$ and $\alpha=0.9$. For simplicity, we
define $V_{g}(\mu,\alpha)$ as the group velocity for different values of $\mu$
and $\alpha$. When $D<D_{c1}$, $V_{g}(1.9,1.0)>V_{g}(1.5,0.9)$, Lévy flights
dominates the transport. When $D_{c1}<D<D_{c2}$,
$V_{g}(1.9,1.0)<V_{g}(1.5,0.9)$, the transport is governed by subdiffusion.
For the case of $D>D_{c2}$, all particles can easily pass across the barrier
and Lévy flights will mainly contribute to the transport and
$V_{g}(1.9,1.0)>V_{g}(1.5,0.9)$
In Fig. 6, we plot the dependence of the group velocity $V_{g}$ on the
amplitude $V_{0}$ of the potential for different combinations of $\mu$ and
$\alpha$. When $V_{0}\rightarrow 0$, the effects of the potential disappear
and the group velocity tends to zero. When $V_{0}\rightarrow\infty$, the
particles cannot pass across the barrier and the group velocity goes to zero,
also. Thus, the curve shows a peak. Remarkably, there is an intersection
between the line of $\mu=1.5$ and $\alpha=0.9$ and the line of $\mu=1.9$ and
$\alpha=1.0$. This is due to the competition between Lévy flights and
subdiffusion. When $V_{0}<V_{c}$, Lévy flights are predominant and
$V_{g}(1.5,0.9)>V_{g}(1.9,1.0)$. When $V_{0}>V_{c}$, subdiffusion dominates
the transport and $V_{g}(1.5,0.9)<V_{g}(1.9,1.0)$. In this case, the height of
the barrier is very high and few particles driven by Lévy flights can pass
across the barrier, the effects of the Lévy flights will disappear and
subdiffusion will play a major role.
## IV Concluding Remarks
In this paper, we have investigated the directed transport of the Brownian
particles in a ratchet potential driven by Lévy flights coexisting with
subdiffusion. We used recently developed framework of Monte Carlo simulation
a22 which is equal to the solution of the fractional Fokker-Planck equation.
The group velocity proposed by Dybiec and coworkers a18 is used to measure
the transport. It is found that the group velocity increases monotonically
with the subdiffusive index, while the group velocity as a function of the
Lévy index is nonmonotonic. The former is caused by the increase of the
waiting time between successive jumps and the latter is owing to the interplay
between Lévy flights and the height of the barriers. There is a threshold
value of $\alpha$ below which the transport is absolutely dominated by
subdiffusion and the directed transport disappears. The dependences of the
group velocity on the noise intensity and the amplitude of the potential are
also investigated. There is an optimal value of the noise intensity (the
amplitude of the potential) at which the group velocity is maximal. The
competition between Levy fights and subdiffusion in the ratchet potential is
observed on the level of the group velocity as well as the median of square
displacement. The nonthermal character of the Lévy flights and the asymmetry
of the potential are the necessary conditions for directed transport when the
system is in the absence of any external driving forces. Because of this
competition, we also found the pseudo-normal diffusion reported by Dybiec and
coworkers a23 , in which time dependence of the median of square displacement
is linear, $M(x^{2})\propto t$, while the process is still non-Markov and non-
Gaussian.
Anomalous transport is becoming widely recognized in a variety of the fields.
Beyond its intrinsic theoretical interest, the results we have presented may
have wide applications in some complex systems, such as diffusive transport in
plasmas, particles separation with non-Gaussian diffusion, and ratchet
transport in biology systems that are intrinsically out of equilibrium.
We would like to thank Dr. Magdziarz for enthusiastic help on numerical
algorithm. This work was supported in part by National Natural Science
Foundation of China with Grant Nos. 30600122 and 10947166 and GuangDong
Provincial Natural Science Foundation with Grant No. 06025073. Y. F. He also
acknowledges the Research Foundation of Education Bureau of Hebei Province,
China (Grant No. 2009108)
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## V Caption list
Fig. 1. Time dependence of $M(x^{2})/t$ for different combinations of $\mu$
and $\alpha$: (a)without external potential at $D=0.4$, solid lines present
$t^{\frac{2\alpha}{\mu}-1}$ scaling; (b)with external potential at $D=0.4$,
$V_{0}=5.0$, and $\Delta=1.0$.
Fig. 2. Time dependence of the median: (a) for different values of the
asymmetry parameter $\Delta$ at $D=0.4$, $V_{0}=5.0$, $\mu=1.5$, and
$\alpha=1.0$, the inset shows the potential profile; (b)for different
combinations of $\mu$ and $\alpha$ at $D=0.4$, $V_{0}=5.0$, and $\Delta=1.0$.
Fig. 3. Group velocity $V_{g}$ versus subdiffusive index $\alpha$ for
different values of $\mu$ at $D=0.4$, $V_{0}=5.0$, and $\Delta=1.0$.
Fig. 4. Group velocity $V_{g}$ versus Lévy index $\mu$ for different values of
$\alpha$ at $D=0.4$, $V_{0}=5.0$, and $\Delta=1.0$.
Fig. 5. Group velocity $V_{g}$ as a function of noise intensity $D$ for
different combinations of $\mu$ and $\alpha$ at $V_{0}=5.0$ and $\Delta=1.0$.
Fig. 6. Group velocity $V_{g}$ versus the amplitude $V_{0}$ of the potential
for different combinations of $\mu$ and $\alpha$ at $D=0.4$ and $\Delta=1.0$.
Figure 1:
Figure 2:
Figure 3:
Figure 4:
Figure 5:
Figure 6:
|
arxiv-papers
| 2010-03-19T04:06:00 |
2024-09-04T02:49:09.173455
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bao-quan Ai and Ya-feng He",
"submitter": "Liang Gang Liu",
"url": "https://arxiv.org/abs/1003.3709"
}
|
1003.3750
|
# Optimal control technique for Many Body Quantum Systems dynamics
Patrick Doria Institut für Quanteninformationsverarbeitung Albert-Einstein-
Allee 11 D-89069 Ulm, Germany. Politecnico di Torino, Corso Duca degli
Abruzzi, 24 10129 Torino, Italy. Tommaso Calarco Institut für
Quanteninformationsverarbeitung Albert-Einstein-Allee 11 D-89069 Ulm, Germany.
Simone Montangero Institut für Quanteninformationsverarbeitung Albert-
Einstein-Allee 11 D-89069 Ulm, Germany. simone.montangero@uni-ulm.de
###### Abstract
We present an efficient strategy for controlling a vast range of non-
integrable quantum many-body one-dimensional systems that can be merged with
state-of-the-art tensor network simulation methods like the Density Matrix
Renormalization Group. To demonstrate its potential, we employ it to solve a
major issue in current optical-lattice physics with ultra-cold atoms: we show
how to reduce by about two orders of magnitudes the time needed to bring a
superfluid gas into a Mott insulator state, while suppressing defects by more
than one order of magnitude as compared to current experiments MSexp .
Finally, we show that the optimal pulse is robust against atom number
fluctuations.
Classical control theory has played a major role in the development of
present-day technologies ccontrol . Likewise, recently developed quantum
optimal control methods dalessandro ; krotov ; rabitz can be applied to
emerging quantum technologies, e.g. quantum information processing – until
now, at the level of a few qubits NMR ; QCapplyed1 ; QCapplyed2 . However,
such methods encounter severe limits when applied to many-body quantum
systems: due to the complexity of simulating the latter, existing quantum
control algorithms (requiring many iterations to converge) usually fail to
yield a desired final state within an acceptable computational time. A
paradigmatic application of control of many body quantum system is the control
of the dynamics of a quantum phase transition. The process of crossing a phase
transition in an optimal way has been studied for decades for classical
systems. Only recently it has been recast in the quantum domain, attracting a
lot of attention (see e.g. kzmech and references therein) since, for
instance, it has implications for adiabatic quantum computation and quantum
annealing farhi . A transition between different phases is usually performed
by “slowly” (adiabatically) sweeping an external control parameter across the
critical point, allowing for a transformation from the initial to the final
system ground state with sufficiently high probability. However, at the
critical point in the thermodynamical limit a perfect adiabatic process is
forbidden in finite time adtheo . Thus, the resulting final state (for finite-
time transformations) is characterized by some residual excitation energy,
corresponding to the formation of topological defects within finite-size
domains. The Kibble-Zurek theory has been shown to yield good estimates of the
density of defects or of the residual energy revzurek ; kzmech . The
importance of these estimates in the quantum domain is underscored by the fact
that, apart from very specific cases where analytical solutions are available,
theoretical investigations must rely on heavy numerical simulations due to the
exponential growth of the Hilbert space with the system size and to the
diverging entanglement at the critical point vidal . Nevertheless, it is
possible to perform one-dimensional simulations of the dynamics by means of
tensor-network-based techniques such as the time-dependent Density Matrix
Renormalization Group (t-DMRG) revscholl .
The basic underlying idea of classical control is to pick a specific path in
parameter space to perform a specific task. This is formally a cost functional
extremization that depends on the state of the system and is attained by
varying some external control parameters. In a quantum-mechanical context, a
big advantage is that the goal can be reached via interference of many
different paths in parameter space, rather than just one. In few-body quantum
systems, it has been shown that optimal control finds optimal paths in the
parameter space that result in constructive interference of the system’s
classical trajectories toward a given goal krotov . Indeed, present optimal
control strategies demonstrated an impressive control of quantum systems,
ranging from optimization of NMR pulses NMR to atomic MarkerAtoms and
superconducting qubits QCapplyed1 , as well as the crossing of a quantum phase
transition (QPT) in the analytically solvable quantum Ising model caneva .
However, despite their effectiveness, they cannot be efficiently applied to
systems that require tensor network methods for their simulation.
The present letter marks a step further — it provides for the first time a
means to control the evolution of a non-integrable many-body quantum system,
resulting in the optimization of a given figure of merit. This is done by
introducing a strategy to integrate optimal control with t-DMRG simulations of
the many-body quantum dynamics.
Tensor-network-simulations– Tensor network methods are based on the assumption
that it is possible to describe approximately a wide class of states with a
simple tensor structure. In particular the DMRG describes ground states static
properties of one dimensional systems by means of a Matrix Product State (MPS)
MPS . The main characteristic of an MPS is that the resources needed to
describe a given system depend only polynomially on the system size $N$, due
to the introduction of an ancillary dimension $m$ that determines the
precision of the approximation. Since an exact description requires
exponentially increasing resources with the number of components $N$, the
tensor network approach results in an exponential gain in resources. Given a
system Hamiltonian, the best possible approximated description of the system
ground state –within the MPS at fixed $m$– is determined by means of an
efficient energy minimization. With some slight modification, discretizing the
time $T=n_{steps}\Delta t$ and performing a Trotter expansion, the algorithm
can be adapted to follow a state time evolution, the so-called t-DMRG
revscholl . The t-DMRG is a very powerful numerical method for efficiently
numerically simulating the time evolution of one-dimensional many body quantum
systems. The class of states and of time evolutions that can be efficiently
described with a small error is determined by the presence of entanglement
between the different system components vidal . Here, we will use the t-DMRG
for the simulation of cold atoms in time dependent optical lattices, which we
feed into the Chopped RAndom Basis (CRAB) optimization algorithm as described
below.
Figure 1: A) An initial guess pulse $c^{0}(t)$ is used as a starting point. B)
The function $\mathcal{F}(\vec{a})$ for the case $\vec{a}=\\{a_{1},a_{2}\\}$
and the initial polytope (white triangle) are defined and moved “downhill”
(darker red triangles) until convergence is reached. C) The final point is
recast as the optimal pulse $c(t)$.
The CRAB method– The general scenario of an optimal control problem can be
stated as follows: given a system described by a Hamiltonian $H$ depending on
some control parameters $c_{j}(t)$ with $j=1,\dots,N_{C}$, the goal is to find
the $c_{j}$’s time dependence (pulse shape) that extremizes a given figure of
merit $\mathcal{F}$, for instance the final system energy, state fidelity, or
entanglement. We then start with an initial pulse guess $c_{j}^{0}(t)$ and
look for the best correction that has a simple expression in a given
functional basis. As an explicative example, here we focus on the case where
the correction is of the form $c_{j}(t)=c_{j}^{0}(t)\cdot f_{j}(t)$, and the
functions $f_{j}(t)$ can be simply expressed in a truncated Fourier space,
depending on the expansion coefficients $\vec{a}_{j}=a_{j}^{k}$
($k=1,\dots,M_{j}$). In particular, in the following, we start from an initial
ansatz, e.g. an exponential or linear ramp, and we introduce a correction of
the form
$f(t)=\frac{1}{\mathcal{N}}\left[1+\sum_{k}A_{k}\sin(\nu_{k}t)+B_{k}\cos(\nu_{k}t)\right].$
(1)
Figure 2: Scheme of the Mott-superfluid transition in the homogeneous system
for average occupation number $\langle n\rangle=1$: increasing the lattice
depth $V$ (black line) the atoms superfluid wave functions (upper) localize in
the wells (lower). If the transition is not adiabatic –or optimized– defects
appear (here represented by a hole and a doubly occupied site).
Here, $k=1,\dots,M$, $\nu_{k}=2\pi k(1+r_{k})/T$ are “randomized” Fourier
harmonics, $T$ is the total time evolution, $r_{k}\in[0:1]$ are random numbers
with a flat distribution, and $\mathcal{N}$ is a normalization constant to
keep the initial and final control pulse values fixed. The optimization
problem is then reformulated as the extremization of a multivariable function
$\mathcal{F}(\\{A_{k}\\},\\{B_{k}\\},\\{\nu_{k})\\}$, which can be numerically
approached with a suitable method, e.g., steepest descent or conjugate
gradient numrec . When using CRAB together with t-DMRG, computing the gradient
of $\mathcal{F}$ is extremely resource consuming, if not impossible. Thus we
resort to a direct search method like the Nelder-Mead or Simplex methods
numrec . They are based on the construction of a polytope defined by some
initial set of points in the space of parameters that “rolls down the hill”
following predefined rules until reaching a (possibly local) minimum (see Fig.
1). Due to the fact that direct search methods are based on many independent
evaluations of the function to be minimized, they can be efficiently
implemented together with t-DMRG simulations (and possibly performed in
parallel). We stress that the functional dependency of the correction
presented here (Eq. (1)) is one possible approach: different strategies might
be explored. Indeed, making a given choice confines the search of the optimal
driving field in a subspace of the whole space of functions and the results
might depend on this choice. On the other hand, this approach simplifies the
optimization problem that would be otherwise computationally unfeasible when
t-DMRG simulations are needed. As shown below, the described choice allows to
perform a successful optimization.
The optical-lattice system– Very recently, the experimental and theoretical
analysis of the dynamics of cold atoms in optical lattices has experienced a
fast development, after the experimental demonstration of coherent
manipulation of ultra-cold atoms in the seminal work of Ref. greiner , where a
Bose-Einstein condensate is first loaded into a single trap, and then a
periodic lattice potential is slowly ramped up, inducing a quantum phase
transition to a Mott insulator. This is the enabling step for a wide range of
experiments, from transport or spectroscopy to quantum information processing
blochrev . In most of these applications, it is essential to achieve the
lowest possible number of defects in the final state, that is, to reach
exactly a final state with fixed number of atoms per site, e.g. unit filling.
Up to now, this has been pursued by limiting the process speed – the
superfluid-Mott insulator transition has been performed in about a hundred ms,
with a density of defects typically of the order of 10% toolbox .
Figure 3: Initial guess (dashed black) and optimal ramp (solid red) $V(t)$
for the Bose Hubbard model in the presence of the trap with $N=30$ sites,
total time evolution $T\simeq 3\,\mathrm{ms}$. Inset: Populations $\langle
n_{i}\rangle$ (empty black) and fluctuations $\langle\Delta n_{i}^{2}\rangle$
(full red) at time $t=T$ for the exponential initial guess (circles) and
optimal ramp (squares) for $N=10$.
Cold atoms in an optical lattice can be described by the Bose Hubbard model
defined by the Hamiltonian Jaksch98 ; blochrev
$\mathrm{H}\\!=\\!\sum_{j}\\!\left[-J(b_{j}^{\dagger}b_{j+1}\\!+\\!\mathrm{h.c.})\\!+\Omega(j-\frac{N}{2})^{2}n_{j}\\!+\\!\frac{U}{2}(n_{j}^{2}-n_{j})\\!\right]\\!\\!.$
(2)
The first term on the r.h.s. of Eq.(2) describes the tunneling of bosons
between neighbouring sites with rate $J$; $\Omega$ is the curvature of the
external trapping potential, and $n_{j}=b^{{\dagger}}_{j}b_{j}$ is the density
operator with bosonic creation (annihilation) operators $b^{{\dagger}}_{j}$
($b_{j}$) at site $j=1,\dots,N$. The last term is the on-site contact
interaction with energy $U$. The system parameters $U$ and $J$ can be
expressed as a function of the optical lattice depth $V$ (we set $\hbar=1$
from now on) blochrev . As sketched in Fig. 2, the system undergoes a quantum
phase transition from a superfluid phase to a Mott insulator as a function of
the ratio $J/U$. In a homogeneous one-dimensional system, the QPT is expected
to occur at $J_{c}/U\simeq 0.083$, where (upon decreasing the ratio $J/U$) the
ground state wave function drastically changes from a Fermi-Thomas
distribution with high fluctuations in the number of particles per site to a
simple product of local Fock states with no fluctuations in the number of
atoms per site blochrev . In the presence of an external trapping potential on
top of the optical lattice, the phase diagram is more complex: the two phases
coexists in different trap regions and typical “cake” structures are formed
batrouni .
Results– Following previous numerical studies kollath that modeled the
experiment MSexp , and supported by strong evidence of agreement between
numerical simulations and experimental results fertig05 ; clark , we studied
both the ideal homogeneous system ($\Omega=0$) and the experimental setup of
fertig05 where the trapping potential is present. We applied the CRAB
optimization to the preparation of a Mott insulator with ultra-cold atoms in
an optical lattice, that is, we optimized the ratio $J/U(t)$ that drives the
superfluid-Mott insulator transition. The resulting optimal ramp shape drives
the system into a final Mott insulator state with a density of defects below
half a percent in a total time of the order of a few milliseconds, amounting
to a drastic improvement in the process time and in the quality of the final
state – by about two orders of magnitude and by more than one, respectively.
Figure 4: Residual defect density $\rho$ for $N=10,20,30,40$, $T\simeq 3ms$,
$\rho_{c}=0.001$ for the homogeneous system (green squares) and in the
presence of the trap (grey circles). The red region highlights the typical
$\rho$ for different initial ramp shapes (see text). Inset: Final $\rho$
computed applying the pulse optimized for system size $N=20$ to different
system sizes $\Delta N=-4,\dots,4$ (at constant filling). The results are
almost independent from the truncation error for $m>50$.
We consider a starting value of the lattice depth $V(0)=2E_{r}$ corresponding
to $J/U(0)\sim 0.52$, since the description of the experimental system by (Eq.
2) breaks down for $V(0)\lesssim 2E_{r}$ Jaksch98 . However, the initial
lattice switching on ($V=0\to 2E_{r}$) can be performed very quickly without
exciting the system (few milliseconds at most) private . We optimize the ramp
to obtain the minimal residual energy per site $\Delta E/N=(E(T)-E_{G})/N$
(where $E_{G}$ is the exact final ground state energy). In all simulations
performed we set the total time $T=50\hbar/U\simeq 3.01\,ms$ and the final
lattice depth $V(T)/E_{r}=22\sim 2.4\cdot 10^{-3}J/U$, deep inside the Mott
insulator phase. Unless explicitly stated, we set the average occupation to
one ($\sum_{i}\langle n_{i}\rangle=N$). In all DMRG simulations, we exploited
the conservation of number of particles and used $m=20,\dots,100$, $\Delta
t=10^{-2}\div 10^{-3}$. We computed the final density of defects
$\rho=\frac{1}{N}\sum_{i}|\langle n_{i}\rangle-1|$: when it reached a given
threshold $\rho_{c}=10^{-3}$, the optimization was halted. In Fig. 3 we report
a typical result of the optimization process: the initial guess and final
optimal ramp for the system in the presence of the confining trap are shown
for the parameter values corresponding to the experiment MSexp , for a system
size $N=30$. As it can be clearly seen, the pulse is modulated with respect to
the initial exponential guess and no high frequencies are present, reflecting
the constraint introduced by the CRAB optimization. In the inset we display
the final occupation numbers and the corresponding fluctuations, for the
initial exponential ramp and the optimal pulse in the case $N=10$. The figure
clearly demonstrates the convergence to a Mott insulator in the latter case as
fluctuations are drastically reduced and the occupation is exactly one for
every site.
Finally, in Fig. 4 we plot the optimized density of defects $\rho$ as a
function of the system size (up to $N=40$) for the homogeneous and for the
trapped system, demonstrating an improvement with respect to the initial guess
by one (two) orders of magnitudes. Indeed, the exponential guess – like other
guesses: linear, random, and a pulse optimized for a smaller system ($N=8$
sites) – gave residual density of defects of the order of $10\%$ (red region
in Fig. 4). To check the experimental feasibility of our findings, we studied
the stability of the optimal evolutions under different sources of error and
experimental uncertainties, like atom number fluctuations. The inset of Fig. 4
shows the final density of defects when an optimal pulse computed for a given
system size, is applied to a different system size (keeping the average
filling constant). As it can be seen, the optimization works also for system
size fluctuations of up to $20\%$: the final density of defects is of the same
order. This robustness is crucial as the experimental realization of these
systems is performed in parallel on many different one-dimensional tubes with
different numbers of atoms blochrev . We also checked the cases of different
filling and of pulse distortion obtaining similar results (data not shown).
Outlook– In conclusion, we would like to mention that the CRAB optimization
strategy introduced here can in principle be applied also to open quantum
many-body systems, e.g. by means of recently introduced numerical techniques
vidal1 . Perhaps an even more stimulating perspective would be that of
implementing it with a quantum system in place of the t-DMRG classical
simulator, i.e. performing a CRAB based closed-loop optimization Brif2010 .
The optimization might be performed during the experimental repetitions of the
measurement processes, thus adding a small overhead to the experimental
complexity. This would extend the applicability of the CRAB method to the
optimization of quantum phenomena that are completely out of reach for
simulation on classical computers, and represent a major design tool for
future quantum technologies.
We thank F. Dalfovo, J. Denschlag, R. Fazio, C. Fort, M. Greiner, C. Koch, C.
Menotti, G. Pupillo and M. Rizzi for discussions; the PwP project
(www.dmrg.it), the DFG (SFB/TRR 21), the bwGRiD, the EC (grant 247687, AQUTE,
and PICC) for support.
## References
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|
arxiv-papers
| 2010-03-19T09:17:09 |
2024-09-04T02:49:09.180547
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Patrick Doria, Tommaso Calarco, Simone Montangero",
"submitter": "Simone Montangero",
"url": "https://arxiv.org/abs/1003.3750"
}
|
1003.4263
|
# Radiogalaxies in the Sloan Digital Sky Survey: spectral index-environment
correlations
Carlos G. Bornancini1,2, Ana Laura O’Mill1,2, Sebastián Gurovich1,2 and Diego
García Lambas1,2
1Instituto de Astronomía Teórica y Experimental, IATE, Observatorio
Astronómico, Universidad Nacional de Córdoba,
Laprida 854, X5000BGR, Córdoba, Argentina.
2Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Avenida
Rivadavia 1917, C1033AAJ, Buenos Aires, Argentina.
###### Abstract
We analyze optical and radio properties of radiogalaxies detected in the Sloan
Digital Sky Survey (SDSS). The sample of radio sources are selected from the
catalogue of Kimball & Ivezić (2008) with flux densities at 325, 1400 and 4850
MHz, using WENSS, NVSS and GB6 radio surveys and from flux measurements at 74
MHz taken from VLA Low-frequency Sky Survey (Cohen et al., 2006). We study
radiogalaxy spectral properties using radio colour-colour diagrams and find
that our sample follows a single power law from 74 to 4850 MHz. The spectral
index vs. spectroscopic redshift relation ($\alpha-z$) is not significant for
our sample of radio sources. We analyze a subsample of radio sources
associated with clusters of galaxies identified from the maxBCG catalogue and
find that about 40% of radio sources with ultra steep spectra (USS,
$\alpha<-1$, where $S_{\nu}\propto\nu^{\alpha}$) are associated with galaxy
clusters or groups of galaxies. We construct a Hubble diagram of USS radio
sources in the optical $r$ band up to $z\sim 0$.8 and compare our results with
those for normal galaxies selected from different optical surveys and find
that USS radio sources are around as luminous as the central galaxies in the
maxBCG cluster sample and typically more than 4 magnitudes brighter than
normal galaxies at $z\sim 0$.3.
We study correlations between spectral index, richness and luminosity of
clusters associated with radio sources. We find that USS at low redshift are
rare, most of them reside in regions of unusually high ambient density, such
of those found in rich cluster of galaxies. Our results also suggest that
clusters of galaxies associated with steeper than the average spectra have
higher richness counts and are populated by luminous galaxies in comparison
with those environments associated to radio sources with flatter than the
average spectra. A plausible explanation for our results is that radio
emission is more pressure confined in higher gas density environments such as
those found in rich clusters of galaxies and as a consequence radio lobes in
rich galaxy clusters will expand adiabatically and lose energy via synchrotron
and inverse Compton losses, resulting in a steeper radio spectra.
###### keywords:
surveys – radio continuum: general – radio continuum: galaxies – galaxies:
high-redshift
††pubyear: 2010
## 1 Introduction
Radio sources are frequently associated with massive systems at low and high
redshifts. In the Local Universe these objects are usually identified with
evolved red ellipticals and with luminous cD galaxies located at the centres
of clusters of galaxies (West, 1994). Radio sources at high redshifts ($z\sim
4$) are identified with massive forming systems, i.e galaxies with diffuse UV
morphologies and consistent with small substructures around a dominant bright
galaxy (Miley et al., 2006).
Several works show that some distant radio sources are embedded in highly
spatially extended ionized gas nebulae of 100-200 kpc (Venemans et al., 2002;
Villar-Martín et al., 2007). These gas structures are observed in massive
ellipticals or cD galaxies at the center of nearby clusters of galaxies. This
evidence suggests that distant radio sources represent the progenitors of the
most massive galaxies observed in the Local Universe and therefore important
for the study of structure formation, such as clusters or groups of galaxies.
A high fraction of radio sources associated with clusters of galaxies have
steep radio spectra as measured by the slope between two fixed ranges of
frequencies, i.e the spectral index $\alpha$ (USS, $\alpha<-1$, where
$S_{\nu}\propto\nu^{\alpha}$, Roettgering et al. (1997); Chambers et al.
(1996); Jarvis et al. (2001)). Such a correlations between spectral index and
redshift ($\alpha-z$ relation), has been used with success to search for
distant galaxies (De Breuck et al., 2000; Jarvis et al., 2004; Cruz et al.,
2006; Broderick et al., 2007).
There are at least three main explanations given in the literature for this
phenomenon: The first is a radio k-correction. The spectral energy
distribution (SED) of radio sources is usually concave, i.e the spectral index
increases with frequency. The observed spectral index determined from two
fixed observed frequencies will therefore sample a steeper part of the radio
spectrum for sources at higher redshifts. The second is related with the
interaction between photons of the cosmic microwave background radiation (CMB)
and the relativistic electron population observed in the plasma gas of radio
sources. The energy density of the CMB increases as $(1+z)^{4}$ and hence
Inverse Compton scattering (IC) of the CMB becomes increasingly important for
sources at high redshifts (Krolik & Chen, 1991). The third is related to an
intrinsic correlation between radio luminosity and rest-frame spectral index
(Blundell & Rawlings, 1999a; Chambers et al., 1990) that due to a Malmquist
like bias in flux density limited surveys translates into a correlation
between spectral index and redshift. An alternative explanation is related
with an ambient density effect. The presence of USS radio sources residing
closest to cluster centres may be due to a manifestation of pressure-confined
radio lobes which slow adiabatic expansion of the plasma. Radio lobes will be
pressure-confined and lose energy primarily via synchrotron and IC losses
(Klamer et al., 2006).
Studies of radio sources at low and moderate redshifts ($z<0.5$) show
different environments.
Hill & Lilly (1991) find that only 50% of powerful radio sources at $z\sim
0.5$ are located in rich galaxy clusters, even though similar sources avoid
such environments at low redshifts. However, Geach et al. (2007) analyzed
radio sources in the Subaru/XMM-Newton Deep Field and found that low-power
radio galaxies at $z\sim 0.5$ reside in moderately rich groups - intermediate
environments between poor groups and rich clusters.
Prestage & Peacock (1988) investigate the local galaxy density around powerful
radio sources using the angular cross-correlation technique and find that
compact radio sources appear to lie in regions of low galaxy density.
Moreover, that complex Fanaroff-Riley class I sources are typically found in
regions of significantly enhanced galaxy density.
Allington-Smith et al. (1993) study the evolution of galaxies in radio-
selected groups at $z<0.5$ with the same range of radio power and find that
strong radio galaxies are located in a wide range of environments but not as
wide as for groups in general. At low redshifts ($z<0.1$) radio-selected
groups have the same richness and blue fraction as do optically-selected
groups. At high redshifts ($z\sim 0.4$) all groups have the same proportion of
blue galaxies.
Studying the prevalence of radio loud AGN activity in nearby groups and galaxy
clusters, selected from the SDSS catalog, Best et al. (2007) find that
brightest group and cluster galaxies are more likely to host a radio loud AGN
than other galaxies of the same stellar mass.
In this work we analyze optical and radio properties of radio sources with
steep radio spectra, and we determine the main characteristics of clusters of
galaxies associated with radio sources.
The structure of this paper is organized as follows: Section 2 describes the
optical and radio samples analyzed. We analyze radio spectral properties of
our sample in Section 3. Section 4 compares the optical and the radio
luminosities of sources associated with steep spectrum and central galaxy
clusters. In section 5, we study the Hubble diagram in the optical $r$ band
for USS sources. We study in Section 6 spectral index and richness properties
of galaxies associated with clusters of galaxies and finally, in Section 7, we
discuss our main results.
Throughout this work we assume a standard $\Lambda$CDM model Universe with
cosmological parameters, $\Omega_{M}$=0.3, $\Omega_{\Lambda}$=0.7 and a Hubble
constant of $H_{0}=$100 h km s-1Mpc-1.
## 2 Radio and optical Galaxy samples
The sample of radio sources was selected from the catalogue of Kimball &
Ivezić (2008). This catalogue presents information in radio and optical bands
taken from several surveys, including Faint Images of the Radio Sky survey
(FIRST, 1400 MHz), NRAO VLA Sky Survey (NVSS, 1400 MHz), Westerbork Northern
Sky Survey (WENSS, 325 MHz), Green Bank survey (GB6, 4850 MHz ), and the Sloan
Digital Sky Survey (SDSS) optical survey. The flux density limits for each
radio catalogue are 1, 2.5, 18 and 18 mJy for the FIRST, NVSS, GB and WENSS
surveys, respectively.
The SDSS catalogue contains flux densities of detected objects measured nearly
simultaneously in u, g, r, i, and z optical bands (Fukugita et al. 1996) with
a limiting magnitude of $r<22.2$ in an area of $\sim 10^{4}$ square degrees in
the North Galactic cap and a small region of 225 square degrees in the South
Galactic cap.
The sample of optical galaxies in the fields of radio sources were selected
using the Catalog Archive Server Jobs System (CASJOBS) interface of SDSS,
which allows one to obtain catalogues with parameters from the SDSS survey
(DR6). 111http://casjobs.sdss.org/CasJobs/.
The matching radius to correlate the radio (FIRST) and optical (SDSS) surveys
was 2$\arcsec$. We chose this matching radius in order to avoid contamination
by line of sight matches of physical unrelated objects, since Kimball & Ivezić
(2008) previously found an efficiency (fraction of matches which are
physically real) of 95% and completeness (fraction of real matches that were
found) of 98% with this matching radius. In this work we used the following
catalogue subsets: Sample D (detected by FIRST, NVSS and WENSS survey, with
63,660 sources), sample E (detected by FIRST, NVSS, WENSS, GB6 and the SDSS
survey, 4732 sources) and sample G (detected by FIRST, NVSS, WENSS and the
SDSS spectroscopic survey, 2885 sources). For all these samples Kimball &
Ivezić (2008) estimated matching efficiencies of $>80$% and completeness
limits of $>90$%, using an appropriate matching radius.
A more detailed description of the radio sample and the criteria used to merge
catalogues can be found in Kimball & Ivezić (2008).
Our sample of tracer galaxies for this work are drawn from the catalogue of
galaxies with photometric redshifts of the DR6 catalogue (Oyaizu et al.,
2008). These photometric redshifts were calculated using a Artificial Neural
Network technique and the Nearest Neighbor Error (NNE) method to estimate
photo-z errors for 77 million objects classified as galaxies in DR6 with
$r<22$. In order to study and analyze the properties of clusters of galaxies
associated with radio sources, we correlate the radio sources detected from
the FIRST, NVSS and WENSS catalogues with clusters of galaxies selected from
the maxBCG survey (Koester et al., 2007a) using a matching radius of
$10\arcsec$. The maxBCG catalogue includes 13,823 clusters selected from the
SDSS photometric data, using two well-known features of rich clusters of
galaxies: the red-sequence observed as a ridge line in the colour-magnitude
diagram (the E/S0 ridgeline) and the presence of a bright dominant central
galaxy (BCG). The catalogue includes the position, cluster photometric
redshifts, BCG spectroscopic redshifts, $r$ e $i$ total galaxy luminosities
and luminosities of the central galaxies, the cluster richness, $N_{\rm gal}$
calculated as the number of galaxies projected within 1 $h^{-1}$ Mpc, brighter
than $0.4L_{*}$ and with colours matching the E/S0 ridgeline, the cluster
richness within $R_{200}$ from the cluster center, defined as the mean density
of 200 times the mean density of the Universe (see for instance Koester et al.
(2007b)).
Figure 1: Spectral index, $\alpha_{325}^{1400}$, vs. spectroscopic redshift
from the SDSS catalogue for radio sources detected in FIRST (1.4 GHz), NVSS
(1.4 GHz), WENSS (325 MHz) and the SDSS spectroscopic catalogue with a
matching radius of 2$\arcsec$. The contours represent the 50, 70, 80 and 90%
of total objects.
## 3 Radio Properties
### 3.1 $\alpha-z$ relation
In Figure 1 we plot spectral index $\alpha_{325}^{1400}$ vs. spectroscopic
redshift for radio sources identified in the SDSS catalogue. The data were
obtained from the catalogue subset G taken from Kimball & Ivezić (2008). The
contours represent the 50, 70, 80 and 90% of total objects. We do not find a
tendency of $\alpha$ with redshift, showing that the most distant ($z>0$.5)
radio galaxies in our sample do not show steep spectra due to a k-correction
effect applied to a concave radio SED.
### 3.2 Analysis of the SED in radio frequencies
Figure 2: Radio Colour-colour diagram distributions for sources in the VLSS
(74 MHz), WENSS (325 MHz) and NVSS (1.4 GHz) samples (grey circles). Filled
circles represent measurements obtained for radio sources associated with
maxBCG clusters. The dashed line indicates the relation for radio sources
whose spectra follow a single power law from 74 to 1400 MHz.Histograms
represent the corresponding distribution of spectral index. Figure 3: Radio
Colour-colour diagram distributions for sources in the WENSS (325 MHz), NVSS
(1.4 GHz) and GB6 (4.85 GHz) samples (grey circles). Filled circles represent
measurements obtained for radio sources associated with maxBCG clusters. The
dashed lines correspond to a single power law spectra from 325 to 4850
MHz.Histograms represent the corresponding distribution of spectral index.
In order to study radiogalaxy properties we analyze radio colour-colour
diagrams. In Figure 2 we show the radio two-colour diagram which compares the
spectral index at 74-352 MHz and 352-1400 MHz. We use the CATS database of the
Special Astronomy Observatory (Verkhodanov et al., 1997) in order to obtain
flux measurements at 74 MHz taken from the VLA Low-frequency Sky Survey (VLSS)
(Cohen et al., 2006), using a matching radius of 60$\arcsec$ for the FIRST and
NVSS samples. Grey circles represent sources detected in the FIRST, NVSS and
WENSS catalogues (listed as Sample D in Kimball & Ivezić (2008)). Filled
circles represent data for radio sources associated with clusters of galaxies
identified in the maxBCG catalogue. The dashed line indicates the relation for
radio sources whose spectra follow a single power law from 74 to 1400 MHz.
Again, we do not find a clear tendency that most of radio sources show some
flattening towards low frequencies. In Figure 3 we show a similar analysis for
radio sources with radio emission in 325, 1400 and 4850 MHz, listed as Sample
E in Kimball & Ivezić (2008) (grey circles). Filled circles represent data
obtained for radio sources identified with clusters of galaxies detected in
the maxBCG catalogue. The equal number of points on either side of this line
indicate no significant spectral curvature.
## 4 Optical and radio luminosities
In order to study radio source luminosity properties at optical and radio
frequencies as well as other possible correlations, we calculated the rest-
frame radio luminosity at 1.4 GHz:
$\centering
L_{1.4}=4\pi~{}{D^{2}_{L}(z)}S_{1.4}~{}(1+z)^{-(1+\alpha)},\@add@centering$
(1)
where $D_{L}(z)$ is the luminosity distance in the adopted $\Lambda$–CDM
cosmology, $S_{1.4}$ is the observed flux density at 1.4 GHz and
$(1+z)^{-(1+\alpha)}$ is the standard k-correction term used in radio
frequencies.
In Figure 4 we show the luminosity distribution of radio sources associated
with maxBCG clusters (shaded histogram) and USS ($\alpha_{325}^{1400}<-1$)
radio sources with spectroscopic redshifts (solid line histogram). The dashed
line histogram shows the distribution for radio sources without a maxBCG
cluster association. We note the good agreement between these three
distributions, indicating that most of the radio sources associated with
central clusters of galaxies and USS sources are radio-loud
($L_{1.4GHz}>10^{23}$ W Hz-1). Several works show a correlation between
spectral index and radio luminosity (or radio power) ($e.g.,$ Blundell et al.,
1999b; Gopal-Krishna, 1988; Laing & Peacock, 1980). As can be seen, we do not
find any trend in radio luminosity and spectral index for these subsamples of
radio sources. It should be noted that the results shown in this figure are
not likely to be biased by possible luminosity selection effects in steep
spectral index sources.
To obtain absolute magnitudes in the $r$ band we use the de-reddened model
magnitudes and k-correct using the V4.1 public code of Blanton & Roweis
(2007). Figure 5 shows the distribution of absolute $r$ band magnitude for USS
radio sources (solid line histogram) and for central cluster objects in the
maxBCG catalogue (shaded line histogram). The dashed line histogram shows the
distribution obtained for radio sources associated with maxBCG clusters. As
can be seen there is only a small difference between the two samples, where
USS radio sources are marginally brighter than the central galaxies associated
with maxBCG clusters.
Figure 4: Radio luminosity distribution in 1.4 GHz. Shaded histogram
represents the luminosity distribution for maxBCG clusters detected in the
FIRST, NVSS and WENSS survey. Dashed line histogram shows the distribution for
radio sources without a maxBCG cluster association. Solid line histogram
represents the distribution for USS sources. Dashed line shows the criterion
for radio–loud radio sources ($L_{1.4GHz}>10^{23}$ W Hz-1) Figure 5: Absolute
magnitude distribution in $r$ band. Shaded histogram represents the
distribution for central galaxies in the maxBCG cluster catalogue. Solid and
dashed line histograms shows the corresponding distribution for USS sources
and for radio sources associated with clusters of galaxies identified in the
maxBCG catalogue, respectively.
## 5 Hubble diagram in the $\rm r$ band
In the last three decades the Hubble diagram in the $K-$ band ($K-z$ diagram)
has been used to detect and study distant radio galaxies. One of the first
works using the Hubble diagram was publish by Lilly & Longair (1984) using a
sample of radio sources detected in the 3CR catalogue. The first radiogalaxy
identified at $z>3$ in fact was selected at radio frequencies as a faint
source in the $K-$band (Lilly, 1988). This technique was used in combination
with spectral index cuts and rejecting radio sources with large angular sizes
($<30\arcsec$) (De Breuck et al., 2000, 2002; Cohen et al., 2004; Cruz et al.,
2006).
In this work we perform a similar analysis in the optical $r$ band. Figure 6
shows the Hubble diagram using the $r$ band (petrosian magnitudes) vs.
spectroscopic redshift. Filled circles represent measurements obtained for USS
objects, big circles represent radio sources associated to maxBCG clusters.
Light grey circles correspond to galaxies detected in the SDSS catalogue
within the NOAO Deep Wide–Field Survey area, with photometric redshifts taken
from (Oyaizu et al., 2008). Dark grey points represent measurements of
galaxies detected in the VIMOS VLT deep survey (VVDS-DEEP) (Le Fevre et al.
2005), with spectroscopic redshift and magnitudes converted to the SDSS system
222http://www.sdss.org/dr4/algorithms/sdssUBVRITransform.html. Crosses
represent spectroscopic measurements for galaxies identified in the Hubble
Deep Field North region obtained in the ACS-GOODS survey (Cowie et al. 2004).
The solid line represent the best fit obtained for the USS source sample,
$r=(4.46\pm 0.21)\times$log10(z)$+(20.28\pm 0.16)$. As can be seen, USS radio
sources are typically more than 4 magnitudes brighter than normal galaxies at
$z\sim 0$.3. In Figure 7 we plot a similar Hubble diagram for radio sources
with spectroscopic redshift quoted in sample G from Kimball & Ivezić (2008)
(grey points). Open red triangles represent measurements obtained for maxBCG
clusters with radio emission and USS radio sources are displayed with open
squares. The solid line represents the best fit obtained for the USS source
sample. USS radio sources follow a similar distribution in comparison with
that obtained in the maxBCG cluster sample in the redshift range $0.1<z<0.3$.
We find that USS sources are typically as luminous as the central galaxies in
the maxBCG cluster sample. In the next Section, we analyze the nature of USS
sources detected at low redshifts ($z<0.1$).
Figure 6: Hubble diagram in the $r$ band vs. spectroscopic redshift taken from
the SDSS catalogue. Filled circles represent measurements obtained for USS
objects. USS radio sources identified with maxBCG clusters are displayed as
big circles. Light grey circles correspond to galaxies detected in the SDSS
catalogue in the NOAO Deep Wide–Field Survey area with photometric redshifts
taken from (Oyaizu et al., 2008). Dark grey points represent measurements of
galaxies detected in the VIMOS VLT deep survey (VVDS-DEEP) (Le Fevre et al.
2005) with spectroscopic redshift and magnitudes converted to the SDSS system.
Crosses represent spectroscopic measurements for galaxies identified in the
Hubble Deep Field North region obtained in the ACS-GOODS survey (Cowie et al.
2004). The solid line represent the best fit line obtained for the USS source
sample Figure 7: Hubble diagram in the $r$ band vs. redshift. Radio sources
with spectroscopic redshift measurements (grey points). Open red triangles
represent measurements obtained for maxBCG clusters with radio emission. USS
radio sources are displayed with open squares.
## 6 Ultra steep spectrum sources
In order to study the nature of radio sources with ultra steep spectra at low
redshifts, we select a set of galaxies detected by NVSS, FIRST, WENSS, and
SDSS with spectroscopic redshifts (catalogue subset G in Kimball & Ivezić
(2008)). We use the NED database 333http://nedwww.ipac.caltech.edu/ the NASA-
IPAC Extragalactic Database. to search for known objects present in the
literature and find that 40% of USS sources are associated with clusters or
groups of galaxies identified in the maxBCG (Koester et al., 2007a), Abell
(Abell et al., 1989), Zwicky (Zwicky & Kowal, 1968) catalogues and Northern
Sky optical Cluster Survey (NSCS, Lopes et al. (2004)) catalogues. Table 1
lists the sample of USS sources identified with galaxies with optical spectra
in the SDSS catalogue. The columns are the following: ID taken from Kimball &
Ivezić (2008), position, spectroscopic redshift, spectral index obtained
between 352MHz and 1.4 GHz, 1.4 GHz radio luminosity, $r$ band absolute
magnitude, and ID taken from literature. In Figure 14 we show a sample of
colour images of radio sources with $\alpha_{325}^{1400}<-1$. At low redshift
($z<0.1$) we find that most USS sources are associated with nearby bright
spiral galaxies or interacting systems, with a possible AGN. This may be due
to the low limiting flux density used ($S_{1400}=2.5$ mJy, from the NVSS
survey) since we note that De Breuck et al. (2000) found that they can select
against low redshift spiral galaxies among USS sources by selecting sources
with $S_{1400}>$ 10 mJy.
Some radio sources that are associated to bright red galaxies are located in
similar density environments as those found in clusters or groups of galaxies,
but do not have a cluster association in the literature. In the next Section
we analyze the galaxy density associated with these systems.
## 7 Spectral index and richness
In order to study the nature of radio sources associated to clusters of
galaxies we analyze possible correlations between spectral index, the richness
associated with each cluster and the luminosity properties of galaxies in
these environments. The spectral index ($\alpha_{325}^{1400}$) distribution
for radio sources detected in FIRST, NVSS and WENSS catalogues (listed as
Sample D) not associated with a maxBCG cluster can be seen in Figure 8 (shaded
histogram). The solid line histogram represents the corresponding spectral
index distribution for radio sources associated with maxBCG clusters. The mean
value distribution for clusters is $\overline{\alpha}_{325}^{1400}=-0.65$. We
find a tendency that most radio sources associated with maxBCG clusters have
steep radio spectra in comparison with field radio sources at similar
redshifts. This result is in agreement with Baldwin & Scott (1973) and Slingo
(1974), who also find that nearby radio sources with steep spectra reside in
rich clusters of galaxies.
Figure 8: Spectral index $\alpha_{325}^{1400}$ distribution for radio sources
without a maxBCG match (shaded histogram) and with a maxBCG cluster match
(solid lines).
In Figure 9 we plot the absolute $r$ band magnitude of central maxBCG cluster
galaxies vs. richness ($N_{\rm gal}$ ) for each cluster associated to radio
sources with steeper than average, $\alpha_{325}^{1400}<-0.65$ (filled grey
circles) and with flatter than average spectra, $\alpha_{325}^{1400}>-0.65$
(open circles), respectively. The contours represent the 90% of total objects
(grey line) for steep sources, solid line for flat sources and dashed lines
for maxBCG clusters without radio emission. We find that clusters of galaxies
associated with steep spectrum sources have brighter central galaxies and have
a high galaxy richness in comparison with clusters associated with flatter
than the average radio sources.
Figure 9: Absolute magnitude in the $r$ band of central maxBCG cluster
galaxies as a function of galaxy richness ($N_{gal}$ ). Grey circles represent
measurements for clusters associated with radio sources with
$\alpha_{325}^{1400}<-0.65$ and black circles represent clusters identified
with sources with $\alpha_{325}^{1400}>-0.65$. The contours represent the 90%
of total objects (grey line for steep sources, solid line for flat sources and
in dashed lines for maxBCG clusters without radio emission). Figure 10: Abell
richness vs. maxBCG richness $N_{gal}$ from (Koester et al., 2007a). Dashed
lines shows the corresponding Abell richness group. Solid line represents the
best fit obtained for these parameters.
In order to provide a suitable calibration to help the reader judge how Abell
richness counts relate to the $N_{\rm gal}$ values, we show in Figure 10 this
correlation using the revised northern Abell Catalog (Abell et al., 1989). As
it can be appreciated in this figure, although with a large scatter, there is
a positive correlation between these richness estimates. Abell richness class
$R=0$ clusters have commonly $N_{\rm gal}$ $\sim$ 20, while objects with
richness class 2 correspond to $N_{\rm gal}$$>$ 30.
In Figure 11 we present the distribution of richness ($N_{\rm gal}$ ) for
clusters of galaxies taken from the maxBCG catalogue calculated from the red-
sequence of the colour–magnitude diagram. We plot in solid line the richness
distribution of radio sources with $\alpha_{325}^{1400}<-0.65$. The shaded
histogram represents the distribution for radio sources with
$\alpha_{325}^{1400}>-0.65$ and the dashed line histogram the richness
distribution for maxBCG clusters without radio emission. We find a tendency
that radio sources with steeper than the average spectra are found
preferentially in higher galaxy richness environments than are radio sources
with flatter than the average spectra.
For reference, we have also indicated in this figure the mean values of
$N_{\rm gal}$ corresponding Abell richness class 0, 1, 2 and 3.
Figure 11: Galaxy richness distribution for maxBCG clusters in the WENSS–NVSS
catalogues. The solid line histogram represents the richness associated to
radio sources with $\alpha_{325}^{1400}<-0.65$ and the shaded histogram
corresponds to sources with $\alpha_{325}^{1400}>-0.65$. The dashed line
histogram shows the distribution for maxBCG clusters without radio emission.
The corresponding Abell richness class are marked at the top of the figure.
Figure 12: Total luminosity distribution for galaxies associated with maxBCG
clusters detected in the WENSS–NVSS catalogues. Solid line histogram
corresponds to radio sources with $\alpha_{325}^{1400}<-0.65$ and the shaded
histogram represents the corresponding distribution for sources with
$\alpha_{325}^{1400}>-0.65$. The dashed line histogram shows the distribution
obtained for maxBCG clusters without radio emission.
In order to compare the $N_{\rm gal}$ richness to other cluster parameters, we
note the relation $N_{\rm gal}$ and $R_{200}$ ($R_{200}$ $\sim$
$N_{gals}^{0.6}$) (See figure 7, Hansen et al., 2005)
In Figure 12 we plot the total $r$ band luminosity of galaxies for clusters of
galaxies associated with radio sources with $\alpha_{325}^{1400}<-0.65$ (solid
line histogram) and those with ($\alpha_{325}^{1400}>-0.65$) as a shaded
histogram. The dashed line histogram represents the distribution obtained for
maxBCG clusters without radio emission. We find that radio sources with
steeper than average spectra in central clusters of galaxies are populated by
luminous galaxies in comparison with radio sources with flatter than average
spectra. In contrast clusters of galaxies without radio emission have a lower
distribution of total galaxy luminosity.
In a similar way, we analyze the density of galaxies associated with different
types of radio sources. In Figure 13 left panel, we show the projected galaxy
density using the 5th nearest neighbour in the plane of sky ($\Sigma_{5}$,
O’Mill et al. (2008)) as a function of spectral index for maxBCG clusters with
$0.2<z<0.3$. The dashed line shows the USS criterion. Galaxies were selected
with a range of radial velocity difference ($\Delta$V), adopting a fixed
$\Delta$V=1000 km s-1 (for $z<0.3$) and a varying $\Delta$V=1000-3000 km s-1.
Figure 13: Left panel: Logarithm of the 5th nearest neighbour local galaxy
density estimator ($\Sigma_{5}$) as a function of the spectral index for
maxBCG clusters. Dashed vertical line show the selection criterion for USS
sources. Right panel: Projected density of galaxies within $<1~{}h^{-1}$ Mpc
vs. spectral index $\alpha_{325}^{1400}$ for cluster of galaxies in the maxBCG
catalogue (filled circled). The projected density was calculated as a number
of galaxies with photometric redshifts in the range $\Delta$z=0.01 from the
BCG in the cluster in the redshift range $0.2<z<0.3$. Stars represent values
obtained for the USS sample without a maxBCG cluster association.
In Figure 13 right panel, we plot the projected density of galaxies within 1
$h^{-1}$ Mpc and with photometric redshifts in the range $\Delta$z=0.01 from
the central cluster galaxies (N1) as a function of the spectral index for
clusters in the redshift range $0.2<z<0.3$ (filled circles). Stars represent
the same values obtained for USS radio sources without a know cluster
association, from the literature. As can be seen in both plots, we found that
these USS sources inhabit environments with galaxy densities similar to those
clusters selected from the maxBCG catalogue.
## 8 Conclusions
We study optical and radio properties of radiogalaxies detected in the Sloan
Digital Sky Survey (SDSS) with flux densities of 74, 325, 1400 and 4850 MHz,
using the VLSS, WENSS, NVSS and GB6 radio catalogues. We search for a possible
empirical correlation between the spectral index and redshift, however we find
no significant trend. We analyze the functional form of the SED using colour-
colour diagrams at radio frequencies. We do not find a clear tendency of radio
sources to show flattening towards low frequencies, as expected assuming
concave curvature in the radio SED. It is well known that a narrow relation
exists between $K-$band and redshift as observed in Hubble diagrams (De Breuck
et al., 2002; Willott et al., 2003; Jarvis et al., 2001; Eales & Rawlings,
1993). In this work we construct a Hubble diagram of USS radio sources in the
optical $r$ band to $z\sim 0$.8. Despite any k-correction and possible
extinction effects, our $r$ band Hubble diagram (Figure 6) also clearly shows
a tight correlation. We find that USS radio sources are as bright as central
galaxies in the maxBCG cluster sample and are typically more than 4 magnitudes
brighter than normal galaxies at $z\sim 0$.3. We note that this result is not
entirely new, for example De Breuck et al. (2002) also find that at redshifts
$<\sim 1$ radio-loud galaxies define the luminous envelope using near infrared
$K-$band magnitudes.
Regarding the possible dependence of radio luminosity on environment, we
notice that the radio luminosity distribution of USS, radio sources in
general, and radio sources in clusters are remarkably similar (Figure 4),
indicating that USS prefer higher denstity environments, independent of radio
luminosity.These results are consistent with those by Hill & Lilly (fig. 8,
1991) who find no significant correlation in cluster richness $N_{0.5}$ and
the rest-frame 2.0 GHz radio power for a sample of radio sources with
$z<0.$5\. Similar results were obtained by Allington-Smith et al. (1993) who
find no trend of richness with radio luminosity at 408 MHz for a sample of
radiosources with $z<0.$5.
We also analyze the richness and spectral index properties of clusters of
galaxies associated with radio sources and find that 40% of USS sources
identified in the SDSS spectroscopic catalogue are associated with cluster or
groups of galaxies identified in the literature, such as in the maxBCG, Abell,
or Zwicky catalogues. We analyze the local density of galaxies around the
sample of USS sources without a know cluster association from the literature,
using the $\Sigma_{5}$ and N1 estimators and find that these USS sources have
similar galaxy densities to clusters selected from the maxBCG catalogue.
We also find that USS sources at low redshift are rare objects (99 from a
total sample of 2885 radio sources detected in the SDSS spectroscopic
catalogue). However a majority reside in regions of unusually high ambient
density, such as those regions found in rich cluster of galaxies.
Our results complement those found by De Breuck et al. (2000). These authors
define a sample of 669 USS sources selected from the WENSS, TEXAS, MRC, NVSS
and PMN radio surveys. They conclude that the majority of relative nearby
($z<\sim 0.4$) USS objects are located in galaxy clusters. They find that at
least 85% of the X-ray objects associates with USS sources are galaxy clusters
or known groups from the literature.
At lower redshifts, we find that radio sources with
$\alpha_{325}^{1400}<-0.65$, are preferentially located in galaxy cluster
environments. This result contrast with Prestage & Peacock (fig. 7, 1988)
where it is found no dependence of the spatial cross–correlation amplitude on
spectral index. We note although that this statistical analysis concerns more
the large-scale, rather than the local environment of our study.
We also find that clusters hosting radio sources with spectra steeper than the
average have a higher galaxy richness and are populated by brighter galaxies
in comparison to clusters associated to radio sources with
$\alpha_{325}^{1400}>-0.65$. A natural explanation for these correlations is
that radio emission in rich cluster of galaxies is pressure-confined in a high
gas density environment. Radio lobes in galaxy cluster environment will expand
adiabatically and lose energy via synchrotron and inverse Compton losses,
resulting in a steeper radio spectra (Klamer et al., 2006).
Figure 14: Cutout colour images of a subsample of galaxies associated with
Ultra Steep radio sources in the SDSS catalogue. The open cross indicates the
radio position taken from the FIRST survey. The ID and redshift (increasing
from left to right) are indicated above each plot. Information for the
complete sample is listed in Table 1.
Table 1. Sample of USS radio sources in the SDSS catalogue with spectroscopic
redshifts.
ID | R.A${}^{radio}_{J2000}$ | DEC${}^{radio}_{J2000}$ | z | $\alpha_{352}^{1400}$ | Log($L_{1.4}$) | $M_{r}$ | ID from literature
---|---|---|---|---|---|---|---
| ${}^{h}\;\;{}^{m}\;\;\;\;{}^{s}\;\;\,$ | ° ′ ″ | | | W Hz-1 | | Designation
608550 | 07 25 57.08 | $+$41 23 05.13 | 0.1113 | $-$1.40 | 23.76 | $-$22.18 | MaxBCG J111.48808+41.38519
664059 | 07 51 31.86 | $+$43 49 29.49 | 0.4249 | $-$1.00 | 24.22 | $-$22.63 | …
668611 | 07 53 32.47 | $+$38 57 52.71 | 0.1484 | $-$1.04 | 23.37 | $-$21.71 | …
672034 | 07 54 57.66 | $+$38 15 22.71 | 0.3030 | $-$1.10 | 23.73 | $-$20.00 | …
683214 | 07 59 49.48 | $+$35 32 33.82 | 0.4823 | $-$1.04 | 25.19 | $-$22.54 | …
709796 | 08 10 54.66 | $+$49 11 03.90 | 0.1147 | $-$1.24 | 23.28 | $-$22.27 | MaxBCG J122.72750+49.18436
717149 | 08 13 50.80 | $+$39 32 32.11 | 0.2045 | $-$1.03 | 23.90 | $-$21.79 | MaxBCG J123.46125+39.54183
734111 | 08 20 32.39 | $+$30 34 48.65 | 0.3628 | $-$1.04 | 25.98 | $-$21.97 | …
735423 | 08 21 03.64 | $+$52 44 35.82 | 0.4441 | $-$1.38 | 25.08 | $-$22.79 | …
748224 | 08 26 00.38 | $+$40 58 51.75 | 0.0576 | $-$1.11 | 22.78 | $-$22.79 | SDSS-C4-DR3 3247
780016 | 08 38 23.27 | $+$29 45 21.67 | 0.1068 | $-$1.06 | 22.68 | $-$22.00 | SHK 182 GGroup
794634 | 08 43 59.18 | $+$51 05 25.55 | 0.1264 | $-$1.12 | 24.37 | $-$22.63 | …
801800 | 08 46 37.85 | $+$51 27 16.56 | 0.1800 | $-$1.27 | 23.88 | $-$21.71 | MaxBCG J131.65796+51.45436
814626 | 08 51 17.29 | $+$37 04 29.00 | 0.2207 | $-$1.00 | 24.11 | $-$20.32 | …
837935 | 08 59 57.32 | $+$56 47 12.15 | 0.1833 | $-$1.06 | 23.71 | $-$22.11 | …
838957 | 09 00 20.28 | $+$52 29 39.73 | 0.0302 | $-$1.02 | 22.01 | $-$24.49 | CGCG 264-047
842071 | 09 01 30.10 | $+$55 39 16.42 | 0.1155 | $-$1.80 | 23.81 | $-$22.75 | MaxBCG J135.37558+55.65463
846087 | 09 03 00.14 | $+$35 27 04.82 | 0.3488 | $-$1.01 | 25.53 | $-$21.09 | …
848084 | 09 03 44.85 | $+$41 38 19.31 | 0.2189 | $-$1.01 | 24.42 | $-$21.42 | MaxBCG J135.93682+41.63908
932725 | 09 34 42.25 | $+$35 14 16.48 | 0.4618 | $-$1.15 | 24.31 | $-$22.62 | …
940538 | 09 37 37.11 | $+$37 05 35.37 | 0.4492 | $-$1.01 | 25.24 | $-$22.53 | …
955678 | 09 43 09.29 | $+$29 50 18.31 | 0.2969 | $-$1.05 | 24.42 | $-$18.71 | …
981931 | 09 52 49.14 | $+$51 53 04.99 | 0.2151 | $-$1.77 | 24.08 | $-$22.49 | ZwCl 0949.6+5207
989139 | 09 55 29.87 | $+$60 23 17.47 | 0.1989 | $-$1.55 | 23.82 | $-$22.03 | MaxBCG J148.87452+60.38814
1002970 | 10 00 31.01 | $+$44 08 42.94 | 0.1533 | $-$1.44 | 23.16 | $-$21.32 | RBS 0819
1027752 | 10 09 28.28 | $+$46 17 37.17 | 0.3858 | $-$1.04 | 24.29 | $-$19.86 |
1051000 | 10 17 58.13 | $+$37 10 54.02 | 0.0442 | $-$1.16 | 22.04 | $-$22.34 | CGCG 183-009
1075981 | 10 27 09.98 | $+$39 08 06.04 | 0.3378 | $-$1.02 | 24.93 | $-$23.20 | …
1080596 | 10 28 54.68 | $+$48 09 38.20 | 0.4851 | $-$1.45 | 24.99 | $-$22.19 | …
1109542 | 10 39 32.11 | $+$46 12 05.54 | 0.1864 | $-$1.21 | 24.85 | $-$21.92 | …
1128337 | 10 46 25.51 | $+$59 37 37.59 | 0.2282 | $-$1.07 | 23.56 | $-$22.91 | MaxBCG J161.60635+59.62690
1138526 | 10 50 10.03 | $+$32 22 05.09 | 0.1150 | $-$1.33 | 23.01 | $-$21.94 | NSCS J105005+322256 /CLtr
1160615 | 10 58 19.46 | $+$41 03 40.76 | 0.1299 | $-$1.07 | 23.41 | $-$22.44 | MaxBCG J164.58100+41.06140
1177112 | 11 04 33.11 | $+$46 42 25.96 | 0.1410 | $-$1.82 | 23.45 | $-$21.08 | …
1179743 | 11 05 30.73 | $+$31 14 36.74 | 0.4381 | $-$1.43 | 24.12 | $-$21.34 | …
1216462 | 11 18 45.25 | $+$52 16 00.95 | 0.4309 | $-$1.24 | 24.60 | $-$22.18 | …
1228632 | 11 23 22.90 | $+$47 55 14.34 | 0.1262 | $-$1.03 | 22.91 | $-$21.08 | …
1233822 | 11 25 16.31 | $+$42 29 10.97 | 0.1882 | $-$1.02 | 24.12 | $-$23.00 | ABELL 1253
1239404 | 11 27 18.46 | $+$53 02 21.12 | 0.3236 | $-$1.04 | 24.49 | $-$23.56 | …
1244341 | 11 29 01.60 | $+$32 45 50.65 | 0.5759 | $-$1.10 | 25.02 | $-$22.24 | …
1250737 | 11 31 20.94 | $+$33 34 46.95 | 0.2219 | $-$1.61 | 23.72 | $-$22.26 | MaxBCG J172.83707+33.57975
1260844 | 11 34 57.39 | $+$53 46 24.20 | 0.1695 | $-$1.05 | 23.24 | $-$21.52 | …
1262111 | 11 35 26.68 | $+$31 53 33.14 | 0.2310 | $-$1.05 | 24.71 | $-$22.67 | MACS J1135.4+3153
1268493 | 11 37 50.23 | $+$46 36 33.65 | 0.3151 | $-$1.03 | 24.72 | $-$23.08 | …
1286560 | 11 44 27.21 | $+$37 08 32.42 | 0.1148 | $-$1.56 | 23.55 | $-$21.11 | …
1294316 | 11 47 12.35 | $+$38 19 26.32 | 0.5977 | $-$1.01 | 25.34 | $-$22.11 | …
1304138 | 11 50 49.21 | $+$62 19 49.04 | 0.3453 | $-$1.69 | 24.37 | $-$23.04 | …
1307368 | 11 51 58.63 | $+$31 40 32.05 | 0.5079 | $-$1.04 | 25.69 | $-$23.39 | …
1307436 | 11 52 00.09 | $+$33 13 42.49 | 0.3573 | $-$1.48 | 23.97 | $-$23.15 | …
Table 1.
ID | R.A${}^{radio}_{J2000}$ | DEC${}^{radio}_{J2000}$ | z | $\alpha_{352}^{1400}$ | Log($L_{1.4}$) | $M_{r}$ | ID from literature
---|---|---|---|---|---|---|---
| ${}^{h}\;\;{}^{m}\;\;\;\;{}^{s}\;\;\,$ | ° ′ ″ | | | W Hz-1 | | Designation
1309157 | 11 52 36.33 | $+$37 32 43.86 | 0.2294 | $-$1.19 | 24.22 | $-$20.23 | MaxBCG J178.15191+37.54548
1319027 | 11 56 05.51 | $+$34 33 05.33 | 0.2536 | $-$1.10 | 25.05 | $-$21.83 | [EAD2007] 200 Arcs
1326519 | 11 58 48.07 | $+$57 17 19.11 | 0.2598 | $-$1.04 | 25.12 | $-$24.49 | …
1348701 | 12 06 47.88 | $+$51 57 10.95 | 0.3446 | $-$1.28 | 24.66 | $-$21.51 | …
1351855 | 12 08 00.78 | $+$43 39 19.12 | 0.2657 | $-$1.00 | 24.77 | $-$22.57 | MaxBCG J182.00318+43.65537
1353531 | 12 08 37.16 | $+$61 21 06.52 | 0.2748 | $-$1.48 | 23.96 | $-$22.17 | …
1354974 | 12 09 08.84 | $+$44 00 11.30 | 0.0376 | $-$1.12 | 22.12 | $-$22.98 | NGC4135, (G. group)
1362044 | 12 11 46.22 | $+$32 38 38.16 | 0.6115 | $-$1.07 | 24.94 | $-$22.25 | …
1406400 | 12 28 02.17 | $+$34 40 40.12 | 0.2775 | $-$1.40 | 23.81 | $-$22.53 | MaxBCG J187.00902+34.67753
1439302 | 12 40 04.88 | $+$37 44 15.46 | 0.1879 | $-$1.15 | 23.75 | $-$21.53 | NSC J124001+374544
1468909 | 12 51 07.51 | $+$56 25 44.98 | 0.2008 | $-$1.22 | 23.42 | $-$21.73 | …
1505383 | 13 04 31.36 | $+$51 43 42.64 | 0.2757 | $-$1.56 | 24.43 | $-$22.28 | MaxBCG J196.15441+51.71551
1510127 | 13 06 12.17 | $+$51 44 06.94 | 0.2773 | $-$1.16 | 24.55 | $-$22.54 | MaxBCG J196.55069+51.73530
1532575 | 13 14 18.32 | $+$41 24 30.18 | 0.1987 | $-$1.05 | 23.25 | $-$20.90 | MaxBCG J198.57609+41.40825
1547051 | 13 19 38.92 | $+$61 39 11.68 | 0.1333 | $-$1.21 | 23.49 | $-$22.47 | …
1559285 | 13 24 12.38 | $+$31 17 24.33 | 0.4268 | $-$1.19 | 24.63 | $-$22.47 | …
1604023 | 13 40 32.89 | $+$40 17 38.79 | 0.1719 | $-$1.17 | 23.32 | $-$22.35 | RX J1340.5+4017 GGroup
1607910 | 13 41 59.68 | $+$42 21 32.32 | 0.4261 | $-$1.22 | 24.07 | $-$22.75 | …
1627288 | 13 49 03.74 | $+$30 52 27.51 | 0.0814 | $-$1.13 | 22.53 | $-$19.43 | …
1628350 | 13 49 27.88 | $+$46 20 15.29 | 0.4212 | $-$1.14 | 24.17 | $-$14.98 | …
1673236 | 14 06 03.34 | $+$52 09 51.98 | 0.4823 | $-$1.01 | 24.75 | $-$23.38 | …
1707796 | 14 18 37.62 | $+$37 46 22.63 | 0.1349 | $-$1.34 | 23.37 | $-$21.87 | ABELL 1896
1714152 | 14 20 56.84 | $+$53 13 07.25 | 0.7430 | $-$1.05 | 25.29 | $-$23.57 | …
1714768 | 14 21 10.18 | $+$42 09 12.97 | 0.3529 | $-$1.12 | 24.03 | $-$20.99 | NSCS J142115+420743
1735168 | 14 28 41.23 | $+$43 41 34.03 | 0.2136 | $-$1.04 | 23.70 | $-$22.42 | NSCS J142842+434009
1735918 | 14 28 57.67 | $+$54 36 27.65 | 0.3819 | $-$1.50 | 24.39 | $-$21.60 |
1744373 | 14 32 04.05 | $+$46 37 43.79 | 0.0927 | $-$1.03 | 22.65 | $-$19.72 | NSC J143143+463738
1755382 | 14 36 02.53 | $+$33 07 53.79 | 0.0939 | $-$1.02 | 22.80 | $-$20.51 | …
1756038 | 14 36 19.44 | $+$48 32 10.68 | 0.1912 | $-$1.07 | 24.03 | $-$21.62 | …
1759889 | 14 37 42.41 | $+$39 27 45.12 | 0.2455 | $-$1.36 | 24.35 | $-$21.94 | MaxBCG J219.42655+39.46313
1768885 | 14 40 57.03 | $+$46 36 46.91 | 0.8395 | $-$1.12 | 25.35 | $-$19.00 | …
1775379 | 14 43 17.07 | $+$46 43 48.40 | 0.2424 | $-$1.34 | 23.68 | $-$21.83 | …
1793973 | 14 50 03.51 | $+$31 30 15.02 | 0.2746 | $-$1.10 | 24.25 | $-$21.95 | MaxBCG J222.55394+31.49750
1795282 | 14 50 31.54 | $+$32 53 03.73 | 0.1775 | $-$1.38 | 23.45 | $-$21.67 | …
1830754 | 15 03 23.78 | $+$46 06 16.28 | 0.4269 | $-$1.06 | 24.19 | $-$22.74 | …
1836180 | 15 05 23.43 | $+$47 06 25.59 | 0.2615 | $-$1.63 | 23.90 | $-$22.52 | ABELL 2024
1837171 | 15 05 46.23 | $+$54 54 01.56 | 0.2824 | $-$1.24 | 24.49 | $-$22.69 | …
1838173 | 15 06 08.41 | $+$60 02 16.86 | 0.5196 | $-$1.03 | 24.72 | $-$24.11 | …
1862173 | 15 15 05.54 | $+$43 09 01.38 | 0.0177 | $-$1.14 | 21.11 | $-$23.69 | CGCG 221-045
1929427 | 15 39 50.77 | $+$30 43 03.90 | 0.0971 | $-$1.18 | 23.10 | $-$22.36 | MaxBCG J234.96158+30.71777
1932815 | 15 41 05.46 | $+$32 04 50.85 | 0.0529 | $-$1.06 | 22.51 | $-$20.79 | …
1934628 | 15 41 46.53 | $+$45 56 14.29 | 0.2024 | $-$1.08 | 24.31 | $-$21.37 | …
1958800 | 15 50 51.44 | $+$42 02 30.47 | 0.0334 | $-$1.09 | 21.91 | $-$20.61 | …
1963482 | 15 52 41.11 | $+$37 24 34.16 | 0.3710 | $-$1.92 | 24.23 | $-$23.69 | …
1967858 | 15 54 23.54 | $+$48 41 07.36 | 0.2271 | $-$1.04 | 23.93 | $-$22.32 | MaxBCG J238.59817+48.68496
2001242 | 16 07 25.43 | $+$47 50 24.15 | 0.3282 | $-$1.02 | 24.76 | $-$22.54 | ABELL 2157
2064005 | 16 33 10.92 | $+$36 07 35.15 | 0.1648 | $-$1.02 | 23.93 | $-$21.42 | MaxBCG J248.29532+36.12611
2088195 | 16 43 26.82 | $+$39 30 39.92 | 0.4119 | $-$1.05 | 24.72 | $-$22.15 | …
2122675 | 16 59 01.01 | $+$32 29 38.93 | 0.0627 | $-$1.26 | 23.82 | $-$21.61 | ABELL 2241
2134416 | 17 04 26.40 | $+$39 10 12.25 | 0.1283 | $-$1.01 | 23.23 | $-$21.34 | NSC J170432+390956
## 9 Acknowledgments
We are grateful to the anonymous referee for his/her careful reading of the
manuscript and a number of comments, which improved the paper. This work was
partially supported by the Consejo Nacional de Investigaciones Científicas y
Técnicas (CONICET),the Secretaría de Ciencia y Técnica de la Universidad
Nacional de Córdoba . The authors made use of the database CATS (Verkhodanov
et al. 1997) of the Special Astrophysical Observatory and the NASA/IPAC
extragalactic database (NED) which is operated by the Jet Propulsion
Laboratory, Caltech, under contract with the National Aeronautics and Space
Administration. This publication makes use of data products from the Sloan
Digital Sky Survey (SDSS). Funding for the SDSS and SDSS-VI has been provided
by the Alfred P. Sloan Foundation, the Participating Institutions, the
National Science Foundation, the U.S. Department of Energy, the National
Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max
Planck Society, and the Higher Education Funding Council for England. The SDSS
Web Site is http://www.sdss.org/. The SDSS is managed by the Astrophysical
Research Consortium for the Participating Institutions. The Participating
Institutions are the American Museum of Natural History, Astrophysical
Institute Potsdam, University of Basel, University of Cambridge, Case Western
Reserve University, University of Chicago, Drexel University, Fermilab, the
Institute for Advanced Study, the Japan Participation Group, Johns Hopkins
University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute
for Particle Astrophysics and Cosmology, the Korean Scientist Group, the
Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-
Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for
Astrophysics (MPA), New Mexico State University, Ohio State University,
University of Pittsburgh, University of Portsmouth, Princeton University, the
United States Naval Observatory, and the University of Washington.
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|
arxiv-papers
| 2010-03-18T20:03:39 |
2024-09-04T02:49:09.197397
|
{
"license": "Public Domain",
"authors": "Carlos G. Bornancini, Ana Laura O'Mill, Sebastian Gurovich and Diego\n Garcia Lambas",
"submitter": "Carlos Bornancini",
"url": "https://arxiv.org/abs/1003.4263"
}
|
1003.4273
|
# Time-Symmetric Boundary Conditions and Quantum Foundations
Ken Wharton Department of Physics and Astronomy, San José State University,
San José, CA 95192-0106
###### Abstract
Despite the widely-held premise that initial boundary conditions (BCs)
corresponding to measurements/interactions can fully specify a physical
subsystem, a literal reading of Hamilton’s principle would imply that both
initial and final BCs are required (or more generally, a BC on a closed
hypersurface in spacetime). Such a time-symmetric perspective of BCs, as
applied to classical fields, leads to interesting parallels with quantum
theory. This paper will map out some of the consequences of this counter-
intuitive premise, as applied to covariant classical fields. The most notable
result is the contextuality of fields constrained in this manner, naturally
bypassing the usual arguments against so-called “realistic” interpretations of
quantum phenomena.
## I Introduction
Without measurement theory, theoretical physics would be a purely mathematical
endeavor; it is the link between mathematical expressions and physical events
that provides a testable connection between equations and reality. But despite
the temporal symmetries evident in our most fundamental physics equations,
such symmetries are rarely applied to measurement theory itself. Given a
physical subsystem described by some equation, it is usually assumed that to
find the solution to the equation one must impose initial boundary conditions
(IBCs), but not final boundary conditions (FBCs). This intuitive assumption
leads to an asymmetric map between physical measurements and mathematical
boundary conditions (BCs), where the map from the initial measurement to the
IBC is treated differently than the map from the equation solution to the
final measurement. This asymmetry is especially evident in the realm of
quantum theory.
It is notable that outside of measurement theory there exists a well-known,
time-symmetrical application of mathematical boundary conditions: the use of
Hamilton’s principle to construct equations of motion from a Lagrangian
Lanczos . In this case, one mathematically fixes both IBCs and FBCs (in
coordinate space), and then determines the equations of motion for which the
action is extremized. Despite this symmetry, when using the resulting Euler-
Lagrange equations to describe physical systems, one typically reverts to the
usual procedure of imposing only IBCs (determined via
measurements/interactions), ignoring the FBCs that generated the equations in
the first place. Presumably, in the minds of many physicists, there is an
important distinction between mathematically imposing an FBC for the purpose
of Hamilton’s principle and actually imposing an FBC via a physical
measurement. However, there seems to be little distinction between these two
where IBCs are concerned.
The purpose of this paper is to motivate and explore the consequences of
treating measurement-BCs in the same symmetrical manner as they are treated in
Hamilton’s principle. External measurements (both before and after the
subsystem in question) will be treated as physical constraints, imposed on the
subsystem in exactly the same way that boundaries are imposed when using
Hamilton’s principle. This interpretation of measurements as BCs on the prior
system (as opposed to merely revealing the status of events determined by
IBCs) is counter-intuitive, but there is further motivation for such a
framework (as seen in the next section). The subsequent analysis indicates
that classical fields constrained by FBCs exhibit many of the characteristics
of quantum systems. These results point to a natural resolution of several
foundational issues in quantum mechanics, presented in a classical context.
## II Motivating FBCs
Variational principles notwithstanding, most physicists have a strong
intuition that any given subsystem in the universe should be determined by
IBCs alone. Such an intuition is bolstered by the time-asymmetric second law
of thermodynamics, which of course can only be derived using time-asymmetric
assumptions. The most natural such assumption is that of correlated IBCs (and
no corresponding FBCs), from which coarse-grained entropy can be statistically
shown to increase Schulman . Still, far from “proving” that nature is
fundamentally time-asymmetric, all this demonstrates is that our observable
universe happens to have an IBC which is highly correlated on a coarse-grained
level (the low-entropy Big Bang).
Furthermore, when one considers single- or few-particle phenomena, it is not
obvious that lessons from statistical regimes should apply in this more
fundamental context. Indeed, if the quantum level of description is considered
to be fundamental (fine-grained, with no underlying level on which statistics
can be applied), then one would not expect any entropy-like asymmetry to arise
from the use of time-symmetric equations. As Eddington pointed out long ago,
“When [Dirac’s theory] is applied to four particles alone in the universe, the
analysis very properly brings out the fact that in such a system there could
be no steady one way direction of time, and vagaries would occur which are
guarded against in our actual universe . . . of about $10^{79}$
particles.”Eddington Recovering a distinction between IBCs and FBCs on this
fundamental level therefore requires some new, time-asymmetric addition to the
equations (as Eddington also advocated). But time-symmetry is an important
principle in physics, so violating it in any way should not be done lightly –
particularly when the phenomena being described have no apparent time-
asymmetry to explain.
Although time-symmetry and Hamilton’s principle are two reasons for
considering FBCs on the same footing as IBCs, another compelling reason is the
experimental fact of the Heisenberg uncertainty principle. In the context that
Heisenberg first envisioned UP , this is not necessarily an indication of some
difference between quantum and classical systems, but rather a strict limit on
how accurately one can measure any system. Simply put, this principle makes
IBCs based on actual initial measurements inherently insufficient to describe
the evolution of classical systems. Without such a restriction, imposing FBCs
would be trivial: together the IBCs and FBCs would overconstrain the system,
and FBCs would not “determine” anything at all; they would merely reveal what
was already deducible from the IBCs. But there is, in fact, no experimental
system with sufficient IBCs to make FBCs redundant. Viewed in this light, the
uncertainty principle seems to discourage total reliance on IBCs (although
quantum theory has resisted this conclusion, as discussed in the next
section).
The uncertainty principle does not merely imply that IBCs contain insufficient
data; it also generally indicates that IBCs will be insufficient by a factor
of two (in full concordance with both time-symmetry and Hamilton’s principle).
In the simplest possible case, a single particle in a known potential requires
an initial position and an initial velocity; the uncertainty principle allows
exactly half of this information to be measured and imposed as an IBC. More
generally, quantum measurement theory for a single spinless particle permits
measurements that constrain the initial values of some complex scalar function
$\psi(\bm{x},t_{0})$ (to within a global phase). Such an IBC is sufficient to
solve the Schrödinger equation (SE), as the SE is a first-order differential
equation in time. But the corresponding classical system, a complex scalar
field $\phi(\bm{x},t)$, is governed by the second-order-in-time Klein-Gordon
equation Goldstein , and requires double the initial data:
$\phi(\bm{x},t_{0})$ and $\dot{\phi}(\bm{x},t_{0})$. Again, one is limited to
half of the IBCs needed for a solution to the corresponding classical problem
Wharton .
Even more generally, the IBCs and FBCs can together be treated as a single
closed-hypersurface boundary condition in spacetime, as is generally done in
covariant extensions of Hamilton’s principle. In this case, while one can no
longer pick out a well-defined “IBC” (portions of the boundaries can be time-
like, and therefore neither IBCs nor FBCs), the same half-data restriction
still seems to generally hold locally on the boundary surface. For example,
the time-like boundary of a perfect conductor in classical electromagnetism
can constrain 3 of the 6 components of the E and B fields on the boundary, but
leaves the other 3 components fully unconstrained. Further evidence that
nature has provided us with systems underconstrained by their IBCs comes from
the “$\psi$-epistemic” view of quantum states espoused by Spekkens Spekkens .
In a simple toy theory, Spekkens demonstrates that restricting ones initial
knowledge of a classical system (by exactly a factor of two!) leads to a vast
array of consequences normally associated with quantum phenomena, including
interference, quantum teleportation, and an apparent “collapse” upon
measurement. The primary quantum phenomena not recoverable from such a picture
are Bell-inequality violations.
But even this exception implies that one should seriously consider FBCs,
something Spekkens’s model does not do. In his model, while the known IBCs are
insufficient, the unconstrained parameters are assumed to be determined by
some additional, random IBC, yielding a well-defined probability distribution
for future measurements of the corresponding unknown parameters. This, of
course, is the type of “hidden variable” that was considered by Bell in the
proof of his inequality Bell , so it is no surprise that Spekkens’s toy theory
is also constrained in a similar manner. Both Spekkens and Bell assumed that
any future measurement on the system would merely reveal what was already
determined by (known or unknown) IBCs.
And yet experiments tell us that nature does not respect Bell’s inequality,
leading many physicists to consider dramatic changes to our best models of
spacetime, introducing non-locality and/or preferred reference frames. An
alternate approach, much less popular but still scattered throughout the
literature OCB ; CWR ; Cramer ; Sutherland1 ; Price ; Miller ; Wharton1 , is
to permit hidden variables that are somehow correlated with the future
measurement settings, in which case violation of Bell-type inequalities
becomes trivial. The question of how such retrocausal correlations might arise
is not always specified, but the above analysis should point to an obvious
option: perhaps the future measurement is not merely revealing information,
but actually a constraint on the prior development of the system. Systems
constrained by FBCs are naturally “contextual”, in that their internal
parameters depend upon that future measurement/interaction. One expects Bell-
inequality violations in such a system. Perhaps the strongest reason to
consider FBCs is that it might provide a way to explain such violations in a
way compatible with the local nature of spacetime assumed by general
relativity.
## III The Case for Classical Fields
The previous sections motivate an equivalence between external measurements on
a spacetime subsystem and the mathematical boundary conditions on the
underlying equations that describe that system. Note that any connected region
of spacetime only has one boundary on which external measurements can be
imposed; if this region is effectively infinite in spatial extent, then the
boundary is often considered to be formed by two spacelike surfaces. So for
any given region, three or more complete external measurements on the same
subsystem are not logically possible. (Measurement devices take up a spacetime
volume themselves.) This approach assumes that measurements made by physical
devices are both constraints on spacetime-adjacent subsystems as well as being
part of larger subsystems that are themselves constrained by other
measurements/BCs. In this view the ultimate boundary conditions are
cosmological.
Given this framework where all measurements are BCs on some subsystem, there
are three general categories of systems/equations that one might consider;
classical particles, classical fields, and quantum fields. Previous
applications of FBCs to physical systems have primarily focused on classical
particles Schulman and quantum fields Schulman ; Miller1 ; AV ; Oeckl .
(Sutherland recently pursued this idea in the context of Bohmian quantum
mechanics, spanning both classical particles and quantum fields Sutherland .)
Apart from my own research program (as well as recent work by Dolce Dolce ),
there seems to be no one pursuing the application of FBCs onto classical
fields.
Classical fields may seem like a curious middle ground between the other two
options, more abstract than particles but not as foundational as quantum
systems. Still, there is a case to be made that classical fields constrained
by closed hypersurface BCs might be an alternative to quantum systems
constrained by only IBCs. First I will address the particle/field issue, and
then tackle the classical/quantum question.
The strongest case for considering fields over particles is to look at our
most successful fundamental physical theories, general relativity (GR) and
quantum field theory (QFT). GR is most naturally expressed in terms of fields
(the Einstein equation is a field equation, derivable from a continuous
Lagrangian density), and although it might be argued that this beautiful
formalism is only an approximation of underlying discrete events, efforts to
discretize spacetime itself have so far been unsuccessful. QFT has even
clearer field-based underpinnings, and analysis of inequivalent Fock-space
representations in accelerating reference frames has demonstrated that
“particle number” is not a generally covariant concept NoPs . This strongly
implies that particles are not even “real”, let alone fundamental.
Apart from these appeals to GR and QFT, it is notable that various efforts to
describe quantum phenomena using a particle ontology (deBroglie-Bohm-style
interpretations, stochastic electrodynamics, etc.) all are forced to introduce
fields into the ontology as well as particles. This is because it is extremely
difficult to extract wave-like behavior from particles alone; conversely, it
is trivial to describe a highly-localized field, at least at some particular
instant. For all of these reasons, it seems evident that fields should at
least be given equal status to particles when asking foundational questions,
if not strict priority.
When searching for answers to foundational questions, one may be tempted to
not consider classical systems at all. After all, it is widely known that
classical physics fails in many instances, apparently forcing us into a
quantum framework. But there is a key flaw in this argument. What is known to
fail are classical systems constrained by IBCs alone. Such systems cannot
possibly violate Bell-type inequalities, for example, so this implies is that
some change must be made to the “classical + IBC-only” framework. Modern
physics has already made such a change, jumping to the “quantum + IBC-only”
framework that we have found so difficult to reconcile with GR. While the
above arguments may tempt one to go straight to a “quantum + IBC + FBC”
framework (as is being pursured by Oeckl and others Oeckl ), such a path would
bypass the mostly unexplored “classical + IBC + FBC” framework. If this latter
framework could address the experiments that motivate the classical $\to$
quantum transition, then there would be no need to ever leave the classical
framework to begin with. This would provide a possibility to explain the
origin of some of the curious postulates of quantum theory, rather than just
assuming them from the outset.
Indeed, the traditional classical $\to$ quantum transition has effectively
bypassed some of the arguments for FBCs, making the addition of FBCs to the
quantum framework an attempt to solve the same problem in two different ways.
The uncertainty principle, for example, is no longer seen as statement of
unknown parameters that might be supplied by an FBC, but typically interpreted
to mean that the system can still be perfectly well-defined using only half of
the classical parameters. Quantum theory accomplishes this halving of the
parameter space by dropping from classical second-order (in time) partial
differential equations to the first-order equations described by the SE and
its natural relativistic extensions (e.g. the Wheeler-deWitt equation). It is
notable that this “halving” introduces severe problems in curved space-time,
where there is no covariant way to separate out the underlying second-order
equations into two first-order equations DeWitt . Viewed in this light, the
entire structure of quantum theory is built around the premise that one should
be able to do physics with IBCs only; adding FBCs to this framework is
therefore quite unnatural.
This conclusion is evident in the results of the few researchers who have
explored the “quantum + IBC + FBC” framework. Schulman has found that imposing
both IBCs and FBCs to the SE is a strong overconstraint, and has had to resort
to finding approximate solutions that are necessarily inexact Schulman .
Aharanov and Vaidman’s two-state formalism AV has overcome this problem by
re-doubling the size of the quantum state to permit the addition of FBCs; a
similar treatment can be found in other two-time boundary approaches
Sutherland ; Wharton1 ; Miller1 . (Miller’s approach Miller1 is arguably more
similar to Schulman’s, in that it resorts to non-unitary evolution to get from
the IBC to the FBC.) I have argued Wharton that it is far more natural to
start with a second-order partial differential equation (like the Klein-Gordon
equation) and apply both IBCs and FBCs rather than to halve-and-then-double
the parameter space to accomplish effectively the same thing – especially
given that this “halving” procedure fails in curved spacetime DeWitt .
In this perspective, the “classical field + IBC + FBC” framework should be
viewed as a competitor to the “quantum + IBC” framework; both are attempting
to deal with limited IBCs in a completely different manner. While it is
certainly reasonable to have doubts that such a classical framework might
explain the same experiments as quantum theory, it remains a fact that this
framework has been largely unexplored. The below analysis of the first-order
consequences of the “classical field + IBC + FBC” framework does not encounter
any major problems, and even supports its general plausibility. Section 5 then
discusses how many of the difficult interpretational questions of quantum
theory would vanish under such a classical alternative.
## IV First-Order Consequences: Quantization and Contextuality
For those unfamiliar with two-time boundary problems, the simplest and best
analogy is that of two-spatial boundary problems. Such systems are widely
familiar in contexts ranging from high-Q laser cavities to a quantum particle
in an infinite square well. In both of these examples, the enclosed field is
found to have a quantized wavelength (and therefore a quantized wavenumber)
$k_{n}=n\pi/L$, where $L$ is the distance between the boundaries. Notice that
quantization is a natural consequence of two-boundary problems, even for
classical systems (such as electromagnetic fields in laser cavities).
Applying this same two-boundary logic to a field subject to both an IBC (at
time $t=t_{0}$) and an FBC (at time $t=t_{f}$), leads to a similar
quantization of the “temporal wavelength”, aka the period. The corresponding
“wavenumber” in this case is just the angular frequency $\omega$, which should
therefore be quantized in the same manner:
$\omega_{n}=\frac{n\pi}{\Delta t}$ (1)
where $\Delta t=t_{f}-t_{0}$. This is a novel “frequency quantization” which
would seem to necessarily occur given both an IBC and FBC.
Sequential measurements of fields does not usually reveal the quantization
from Eqn. (1), and this may seem to be reason to reject this approach. But
consider that for most systems it is impossible to externally constrain
$\Delta t$ in the same way that one can externally constrain $L$. In fact, the
only experimental set-up where this is generally possible is that of a laser
cavity, where the length is directly related to the time between interactions
via $L=c\Delta t$. In this special case, the above “temporal quantization”
leads to $\omega_{n}=n\pi c/L$, which indeed is the quantization that is
actually observed in this instance. Of course, one traditionally comes to this
conclusion by using the relation between the frequency and the quantized
wavenumber $\omega_{n}=ck_{n}$, so this known result is consistent with the
use of FBCs, but not dependent upon it.
For fields that do not propagate at some known speed, it is quite difficult to
physically constrain $\Delta t$ to the accuracy where one might see the
effects of (1). For example, consider a classical scalar field $\phi$ governed
by the generally covariant Lagrangian density
${\cal{L}}=\frac{1}{2}\left(g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi-\frac{m^{2}c^{2}}{\hbar^{2}}\phi^{2}\right)$
(2)
Here $\nabla$ is the covariant 4-derivative, summation is implied, and $g$ is
the metric. The parameters $m$ and $\hbar$ are chosen such that the
corresponding Euler-Lagrange equation in flat Minkowski spacetime
$\left(\frac{\partial^{2}}{\partial
t^{2}}-c^{2}\nabla^{2}+\frac{m^{2}c^{4}}{\hbar^{2}}\right)\phi=0$ (3)
has the dispersion relation
$\omega(\bm{k})=\sqrt{c^{2}k^{2}+m^{2}c^{4}/\hbar^{2}}$. This matches the
energy-momentum relationship for a relativistic particle with mass $m$,
subject to the deBroglie relations $E=\hbar\omega$, $\bm{p}=\hbar\bm{k}$.
Equation (3) is the Klein-Gordon equation (KGE), and of course is a second-
order partial differential equation (in time).
Comparing the dispersion relation to (1), one finds
$\frac{n\pi}{\Delta t}=\omega_{n}\geq\frac{mc^{2}}{\hbar}$ (4) $\Delta t\leq
n\pi\frac{\hbar}{mc^{2}}$ (5)
From this one can see see that this quantization requires the time between the
IBC and FBC to be constrained to an accuracy less than the Compton period
$h/(mc^{2})$. As this value is less than $10^{-20}$ seconds for electrons, and
even smaller for more massive particles, it is unsurprising that the frequency
quantization implied by (1) is not observed (outside of laser cavities).
However, our current inability to make measurements at well-defined sub-
attosectond intervals does not mean that this framework has no consequences.
For example, consider two consecutive measurements made on this scalar field
with the additional constraint of a 1D infinite-potential square well of
length L. Now there are two, independent, quantization conditions: the usual
wavenumber quantization $k=n_{x}\pi/L$ and the new frequency quantization
$\omega=n_{t}\pi/\Delta t$. Inserting these expressions into the dispersion
relation yields the constraint
$n_{t}^{2}L^{2}+n_{x}^{2}c^{2}(\Delta
t)^{2}=\frac{m^{2}c^{4}}{\pi^{2}\hbar^{2}}L^{2}(\Delta t)^{2}$ (6)
The fact that $n_{x}$ and $n_{t}$ are both integers is now a strong constraint
on $\Delta t$ (given a well-defined value for L). Most values of $\Delta t$
will have no solution at all, and for those values that do have solutions, the
vast majority will now have only one integer pair ($n_{x}$,$n_{t}$) that
satisfies (6). Therefore, given two consecutive measurements on a classical
field in a 1D cavity, one expects to typically find well-defined single values
of $k$ and $\omega$; not the superposition of modes that one sees in a laser
cavity where $n_{x}=n_{t}$. In other words, particle-like measurement results
have emerged from simply constraining classical fields with FBCs. (Note that
if no final measurement was made, there would have been no such constraint,
and a generic superposition of solutions to the KGE would still be permitted.)
It seems plausible that this general result would continue hold for any
potential well, not merely the infinite square well.
Another interesting result arises from the previous example: The time between
measurements $\Delta t$ is no longer a completely arbitrary external
parameter. There are values of $\Delta t$ with no acceptable solution, and
therefore measurements may not occur at certain times. This is consistent with
the probabilistic treatment of measurement time in Wharton , but is certainly
contrary to how most physicists think about this parameter. The use of FBCs
has now dictated which hypersurfaces may correspond to physical measurements,
and which ones may not. Taking this logic to its conclusion, one finds that
such constraints on classical field measurements (as dictated by global
solutions to an action-extremization problem) naturally leads to exactly the
sort of quantization observed in experiments, including the quantization of
angular momentum in units of $\hbar/2$ Wharton3 .
This relates to a technical issue concerning the boundaries in covariant
formulations of Hamilton’s principle. Action-extremization only generally
leads to the Euler-Lagrange equations if the boundaries constrain the value of
the field in coordinate space Lanczos ; Wharton3 . If these boundaries are now
to be interpreted as physical measurements, one encounters a problem:
constraints on boundaries in coordinate space necessarily correspond to
measurements of time-even quantities (parameters that are identical under
time-reversal). For example, the $j$-momentum density of a classical scalar
field, as defined by the stress-energy tensor component $T^{0j}$, is
proportional to $\dot{\phi}\,\partial\phi/\partial x^{j}$, a time-odd
quantity. Such a measurement cannot be imposed as a BC on $\phi$ in coordinate
space, because $\dot{\phi}$ is independent from $\phi$ on any space-like
hypersurface. If one assumed complete experimental control over when and where
measurements could be made, this would imply that Hamilton’s principle would
fail any time a time-odd quantity was measured.
Fortunately, the analysis in Wharton3 demonstrates that there is an
interesting “loophole” that can save Hamilton’s principle for time-odd
measurements. It turns out that boundary constraints on $\dot{\phi}$ can still
permit an extremized action (leading to the same Euler-Lagrange equations),
provided that the hypersurfaces corresponding to those boundaries have a
particular geometrical structure. (It is the constrained geometry of the
hypersurfaces that leads to angular momentum quantization in Wharton3 .) As
above, this implies that our assumed freedom of when and where we can make
measurements may be illusory, and indeed this lack of freedom seems to lead to
an apparent measurement-induced quantization.
But the most important consequence of the above example is simply that the
global solution to the equations in a space-time subsystem depend upon the
next measurement on that subsystem. Even if we cannot control $\Delta t$ on a
Compton period time-scale, we can certainly control it at much grosser levels,
and in Eqn (6) such external control will lead to different values of $n_{t}$
throughout the spacetime volume bounded by the two sequential measurements. In
other words, if we are to treat $\phi$ as a “real” field, our future
measurement decisions at $t\approx t_{f}$ will affect the reality of the field
at times $t<<t_{f}$. Such influence is usually termed “retrocausality”.
Remarkably, the retrocausality that emerges from such a picture is of the
least objectionable sort, as it only affects the so-called “inaccessible past”
Price , hidden parameters that cannot be determined until their “future cause”
at $t_{f}$. (If one was to make a measurement between $t_{0}$ and $t_{f}$,
then that intermediate measurement would still be at $t_{f}$ by definition.)
Therefore no paradoxes can be constructed. Furthermore, these are precisely
the sort of counter-intutive hidden parameters needed to violate Bell’s
theorem in a natural way. Such contextuality is not an added, ad hoc feature
of this framework, but an inevitable result of treating IBCs and FBCs on the
same conceptual footing.
## V Lessons for Quantum Foundations
Quantum measurement theory typically comes with a built-in temporal asymmetry:
complete initial measurements always correspond to a pure-state IBC, while
complete final measurements are often made on mixed states and are not imposed
as FBCs. The (generalized) Born rule that gives the conditional probabilities
of a final measurement outcome (conditioned upon the IBC and subsequent time
evolution) is typically used in a time-asymmetric manner, in the sense that it
is always conditioning on the past. Although one might choose to condition on
the future instead of the past, and use the Born rule in reverse, one would
find that the FBC becomes a pure state and the intermediate quantum state is
no longer the same as it was in the past-conditioned case. The Born rule
therefore seems inconsistent with Hamilton’s Principle, as the latter imposes
both IBCs and FBCs on the same system in a time-neutral manner.
In both the forward- and backward- application of Born’s rule, despite the
different intermediate mathematics, the joint probability between the past and
future outcomes remains unchanged ABL ; it is this joint probability that
therefore appears more fundamental than conditional probabilities when
approaching such time-symmetric problems. For a complex classical field
constrained by both an IBC and a FBC, I have shown that a invariant joint
probability distribution can be constructed from the solution space of the
intermediate field (given the boundary constraints) Wharton . From this one
can construct the usual conditional probabilities in the appropriate limit,
conditioning on any chosen portion of the boundary. In this perspective,
probability is not some mysterious attribute of quantum systems, but instead
has the same source as it does in classical physics: ignorance. Knowing the
IBCs is not enough to define the fields, and therefore one retreats to a
probabilistic framework. Once the IBCs and FBCs are known, however, one can
“retrodict” the field values in the intermediate region between the two
boundaries if desired.
The so-called “quantum collapse” disappears in such a framework, in the same
manner as envisioned by the “$\psi$-epistemic” proponents Spekkens . In this
classical-field context, the solution to the SE ($\psi$) is merely our best
guess of what the field looks like given only the IBC, but does not perfectly
correspond to the actual field $\phi$ (which obeys the KGE, constrained by
IBCs and FBCs). Once the final measurement result (the FBC) becomes known, one
updates ones knowledge of $\psi$ in a discontinuous manner. Still, there is no
discontinuity in reality, for the field $\phi$ is perfectly consistent with
this final measurement result (as it is pre-constrained by the eventual FBC).
Another difference between $\psi$ and $\phi$ lies in the multi-particle
sector. To encode the probabilities of all possible measurements in a single
function $\psi$, one is forced to expand the dimensionality of $\psi$ into
multi-particle configuration space. But in this framework the actual
probabilities are not encoded by $\phi$ or $\psi$ themselves, but rather the
solution space of $\phi$ given any particular future measurement set-up. There
is no need for $\phi$ to encode the results of possible measurements which
will not actually occur, so there is no need to expand $\phi$ into
configuration space, even for multiple particles. (Montina’s recent work makes
a related argument Montina .) The best example of this is the classical
Maxwell field, which manages to reproduce infinite-photon-number quantum
theory using fields that still exist in physical spacetime (as opposed to
infinite-dimensional configuration space).
The most important lesson from the above analysis is probably the following
statement: Violations of Bell inequalities do not necessarily imply the
failure of locality (in the sense used in general relativity). Although
locality is one assumption that underlies Bell’s theorem, it is not the only
one; the theorem also assumes that any hidden variables will not be dependent
on the future measurement settings Bell . It is natural to violate such an
assumption in this framework – indeed, one expects it to be violated, given a
true FBC. Experimentally-observed violations of Bell-type inequalities are
therefore further motivation for considering FBCs, especially if one wants to
retain basic notions of locality.
It is notable that this approach maps nicely to quantum field theory, which
also uses an invariant expression for the joint probability of two consecutive
measurements. For a quantum scalar field $\phi$ (promoted to an operator, with
imposed commutation constraints), this joint probability is
$P[\phi(t_{0}),\phi(t_{f})]=\left|\int{\cal{D}}\phi\,e^{iS[\phi]/\hbar}\right|^{2}$
(7)
Here the functional integral is over all field configurations consistent with
the IBC $\phi(t_{0})$ and the final measurement result $\phi(t_{f})$; $S$ is
the classical action. It is notable that this mathematics naturally
incorporates FBCs. This expression can be made even more general by imposing
the boundaries on a closed hypersurface rather than two parallel “instants”.
Note that there is nothing, in principle, that would prevent the evaluation of
(7) for a purely classical field constrained by IBCs and FBCs. In this
perspective, the dominant contribution from quantum theory is on the boundary
itself – the limitation of the allowed measurement values on a given field.
But the classical framework presented here provides another way to constrain
those measurements, without the use of quantum measurement theory (or
operators of any sort). If there is some pair of measurement results
$\phi(t_{0})$ and $\phi(t_{f})$ with no allowable solutions at all (such as
certain values of $\Delta t$ and $L$ in (6)), then the integral in (7) will
simply be zero, and those pairs of measurements would therefore have zero
probability of occurring together. Furthermore, if there were pairs of
possible measurements for which there was no way to extremize the classical
action $S$ (such as angular momentum measurements that were not near a
multiple of $\hbar/2$ Wharton3 ), then the value of the integral in (7) would
be quite small due to the lack of stationary phase terms. The net effect would
be an apparent quantization of an underlying continuous system – it would only
be quantized when one “looks”, because it is the externally imposed
observation/measurement that is the source of the quantization constraints.
Indeed, seeming paradoxes like “delayed-choice” experiments become trivial to
interpret in this framework, as it is the future measurements themselves which
are defining the evolution of the system.
The preliminary results presented here (and in related research Wharton ;
Wharton3 ) are certainly nowhere close to being a replacement for all of
quantum theory. But there is every indication that this is a promising
research avenue to pursue. The only obvious conceptual disadvantage is the
retrocausal aspect of FBCs, and even that might be considered an advantage if
one is concerned about maintaining a temporal symmetry in theories that
purport to describe time-symmetric events. Indeed, taking Eqn. (7) seriously
can even be used as a strong case for retrocausation EPW ; WMP . Furthermore,
the framework advocated here is probably more compatible with general
relativity than is quantum field theory, as classical fields in curved
spacetime do not come with the conceptual problems of quantum fields. It is
even possible that, instead of quantizing spacetime, an approach to quantum
gravity might be found in the continuous ontology of classical field theory.
For now, though, more modest efforts are called for: the construction of a
time-neutral theory of field-field measurement, explicitly incorporating
potentials/interactions, and searching for experimental tests of this overall
framework. Two-time boundaries imposed on classical fields is a subject that
has been curiously neglected for the past hundred years. Based on these
preliminary results, it seems reasonable to venture further down this
particular path.
## VI Acknowledgements
The author is grateful to Rob Spekkens and the Perimeter Institute for the
invitation to give the seminar on which this paper was based PItalk . Further
thanks goes to Huw Price for conceptual inspiration and the Eddington quote.
## References
* (1) Lanczos, C. The Variational Principles of Mechanics, 4th Edition; Dover: New York, NY, USA, 1970.
* (2) Schulman, L.S. Time’s Arrows and Quantum Measurement; Cambridge University Press: Cambridge, UK, 1997.
* (3) Eddington, A.S. The end of the world: From the standpoint of mathematical physics. Nature 1931, 127, 447–453.
* (4) Heisenberg, W. ber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Z. Phys. 1927, 43, 172–198.
* (5) Goldstein, H. Classical Mechanics, 2nd Edition; Addison-Wesley: Reading, USA, 1980.
* (6) Wharton, K.B. A novel interpretation of the Klein-Gordon Equation. Found. Phys. 2010, 40, 313–332, DOI 10.1007/s10701-009-9398-2.
* (7) Spekkens, R.W. Evidence for the epistemic view of quantum states: A toy theory. Phys. Rev. A 2007, 75, 032110.
* (8) Bell, J.S. On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 1966, 38, 447.
* (9) Costa de Beauregard, O. Time symmetry and interpretation of quantum mechanics. Found. Phys. 1976, 6, 539–559.
* (10) Reitdijk, C.W. Proof of a retroactive influence. Found. Phys. 1978, 8, 615–628.
* (11) Cramer, J.G. Generalized absorber theory and the Einstein-Podolsky-Rosen paradox. Phys. Rev. D 1980, 22, 362-376.
* (12) Sutherland, R.I. Bell’s theorem and Backwards-In-Time causality. Int. J. Theo. Phys. 1983, 22, 377–384.
* (13) Price, H. Time’s Arrow and Archimedes’ Point; Oxford University Press: Oxford, UK, 1996.
* (14) Miller, D.J. Realism and time symmetry in quantum mechanics. Phys. Lett. 1996, A222, 31–36.
* (15) Wharton, K.B. Time-symmetric quantum mechanics. Found. Phys. 2007, 37, 159.
* (16) Miller, D.J. Quantum mechanics as a consistency condition on initial and final boundary conditions. http://arxiv.org/pdf/quant-ph/0607169.
* (17) Aharonov,Y.; Vaidman, L. Complete description of a quantum system at a given time. J. Phys. A 1991, 24, 2315.
* (18) Oeckl, R. Probabilites in the general boundary formulation. J. Phys. Conf. Ser. 2007, 67, 12049.
* (19) Sutherland, R. Causally symmetric bohm model. http://arxiv.org/pdf/quant-ph/0601095.
* (20) Dolce, D. Compact Time and Determinism for Bosons: Foundations. Found.Phys. 2011, 67, 178-203.
* (21) Arageorgis, A.; Earman, J.; Ruetsche, L. Fulling non-uniqueness and the unruh effect: A primer on some aspects of quantum field theory. Phil. Sci. 2003, 70, 164–202.
* (22) DeWitt, B.S. Quantum theory of gravity I: The canonical theory. Phys. Rev. 1967, 160, 1113.
* (23) Wharton, K.B. Extending Hamilton’s principle to quantize classical fields. http://arxiv.org/pdf/0906.5409.
* (24) Aharonov, Y.; Bergmann, P.G.; Lebowitz, J.L. Time symmetry in the quantum process of measurement. Phys. Rev. 1964, 134, B1410–B1416.
* (25) Montina, A. Exponential complexity and ontological theories of quantum mechanics. Phys. Rev. A 2008, 77, 22104.
* (26) Evans, P.; Price, H.; Wharton, K.B. New slant on the EPR-Bell experiment. Brit. J. Phil. Sci, to be published. arXiv:1001.5057.
* (27) Wharton, K.B., Miller, D.J., Price, H. Action Duality: A Constructive Principle for Quantum Foundations. arXiv:1103.2942.
* (28) Wharton, K. Classical fields as the natural ontic structure for quantum theory. Presented at Perimeter Institute, quantum foundations seminar, Waterloo, OT, USA, June 2009.
|
arxiv-papers
| 2010-03-22T20:33:02 |
2024-09-04T02:49:09.206247
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ken Wharton",
"submitter": "Ken Wharton",
"url": "https://arxiv.org/abs/1003.4273"
}
|
1003.4565
|
# UCAC3 pixel processing
Norbert Zacharias1 nz@usno.navy.mil 1U.S. Naval Observatory, 3450 Mass.Ave.
NW, Washington DC 20392
###### Abstract
The third US Naval Observatory (USNO) CCD Astrograph Catalog, UCAC3 was
released at the IAU General Assembly on 2009 August 10. It is a highly
accurate, all-sky astrometric catalog of about 100 million stars in the R = 8
to 16 magnitude range. Recent epoch observations are based on over 270,000 CCD
exposures, which have been re-processed for the UCAC3 release applying
traditional and new techniques. Challenges in the data have been high dark
current and asymmetric image profiles due to the poor charge transfer
efficiency of the detector. Non-Gaussian image profile functions were explored
and correlations are found for profile fit parameters with properties of the
CCD frames. These were utilized to constrain the image profile fit models and
adequately describe the observed point-spread function of stellar images with
a minimum number of free parameters. Using an appropriate model function,
blended images of double stars could be fit successfully. UCAC3 positions are
derived from 2-dimensional image profile fits with a 5-parameter, symmetric
Lorentz profile model. Internal precisions of about 5 mas per coordinate and
single exposure are found, which are degraded by the atmosphere to about 10
mas. However, systematic errors exceeding 100 mas are present in the $x,y$
data which have been corrected in the astrometric reductions following the
$x,y$ data reduction step described here.
astrometry — catalogs — methods: data analysis
††slugcomment: manuscript for AJ, vers.3, 100322
## 1 INTRODUCTION
The US Naval Observatory (USNO) operated the 8-inch (0.2 m) Twin Astrograph
between 1998 and 2004 for the first ever all-sky astrometric survey using a
CCD detector. About 2/3 of the sky was observed from the Cerro Tololo Inter-
American Observatory (CTIO) while the rest of the northern sky was observed
from the Naval Observatory Flagstaff Station (NOFS). In 1998 the Kodak 4k by
4k CCD detector used for this UCAC project was the largest number of pixels
chip at any telescopes at CTIO. The UCAC project also brought attention to a
potential source of systematic errors for astrometry using CCDs, the charge-
transfer inefficiency.
The first paper describing the UCAC1 release (Zacharias et al., 2000) gives
details about the observing procedures and initial reductions. The second
release, UCAC2, (Zacharias et al., 2004), is an extension of UCAC1 applying
similar reduction methods to a much larger area of the sky. The same pixel
processing pipeline was used for UCAC1 and UCAC2, while improved systematic
error corrections were introduced for the UCAC2 reductions to obtain celestial
coordinates. The UCAC2 is being used extensively by the astronomical
community, providing a much needed densification of the optical celestial
reference frame at magnitudes fainter than the Hipparcos and Tycho-2 catalogs.
The UCAC observations cover mainly the 8 to 16 magnitude range, providing
accurate star positions with external errors of about 20 to 70 milliarcsecond
(mas) depending on magnitude.
For the recent UCAC3 release (Zacharias et al., 2010) completely new
reductions of the pixel data were performed, involving new analysis methods.
These reductions and results are described in this paper in detail, including
image profile fitting methods leading to $x,y$ centers. The subsequent
astrometric processing from $x,y$ data to celestial coordinates is described
in a separate paper (Finch, Zacharias & Wycoff, 2010).
Point-spread function (PSF) fitting has been performed in the past, see for
example (Anderson & King, 2000) for HST data, or the IRAF DAOPHOT package
(Stetson, 1987). The approach taken here is different, deriving relatively
simple, analytical model functions which describe the observed PSF
sufficiently well. At the same time the number of free parameters needed for
each image profile fit is kept to a minimum by utilizing information from many
CCD exposures to constrain some image profile model parameters. Challenges
here are asymmetric PSFs and variations of the PSFs over the field of view,
combined with a relatively small number of stars per CCD frame and the goal of
high astrometric accuracy.
## 2 PIXEL DATA
A 4094 by 4094 pixel CCD with a 9 $\mu$m pixel size was used in a single
bandpass (579 to 643 nm) providing a field of view (FOV) of just over 1 square
degree, taking advantage of only a tiny fraction of the flat FOV delivered by
the optical system of the Twin Astrograph’s “red lens.” This camera provides
14 bit output and has a gain setting of 5.65 electrons (e-) per analog-to-
digital unit (ADU), 13 e- read noise, and about 85,000 e- full well capacity.
A 2-fold overlap pattern of 85,158 fields spans the entire sky. Each field was
observed with a long (about 125 s) and a short (about 25 s) exposure. The raw
pixel data are stored in custom FITS differential compress (fdc) format files,
about 16 MB per exposure without loss of the 14 bit dynamic range.
The detector features a poor charge transfer efficiency (CTE) leading to
asymmetric images along the readout direction ($x$ axis) which vary as a
function of distance from the output register. This leads to systematic errors
in the star positions as a function of $x$ and the stars’ brightness
(magnitude), about the worst thing what can happen for an astrometric
instrument. The contour plots in Fig. 1 illustrate this problem showing the
change of image shape (from almost circular to pronounced asymmetric) as seen
on the left and right side of the detector, respectively. The left side (low
$x$) is close to the readout register and also displays the largest background
noise on the chip, likely due to a higher than average temperature there.
Initially the camera showed a “glowing spot” in the lower left corner. The
design was changed to have the read amplifier powered up only when needed,
which eliminated that problem.
In order to mitigate these $x$ and magnitude dependent systematic errors the
detector was operated at a relatively high temperature ($-18C$), which filled
many of the CTE causing traps on the silicon detector. Unfortunately the warm
operating temperature leads to a substantial dark current. Frequent darks were
taken throughout the project for each of the standard exposure times (5, 10,
20, 25, 30, 40, 100, 125, 150, 200 s). Some time into the project it was
discovered that the darks also depend on ambient temperature and vacuum
pressure inside the camera, which due to small leaks increased from about 0.1
torr to over 2 torr, when a new pumpout of the camera was performed every few
months.
## 3 RAW DATA PROCESSING STEPS
### 3.1 Combined darks and bad pixel map
The detector used for the UCAC survey has a high cosmetic quality with no bad
columns and relatively few bad pixels. In order to simplify the reductions and
assuming the worst case, a single list of all possible bad pixels were
established spanning dark exposures taken during the entire project.
Early on it was discovered that darks taken during daytime or in rapid
succession display different properties than object frames taken during
regular observing. Most darks therefore were taken during cloudy nights with a
script to obtain about 50 darks of a given integration time in an automated
sequence. Pauses of about a minute between dark exposures were introduced to
closely resemble actual observing conditions. Using custom software these 50
FITS files were read in parallel, block by block and the 50 measures of each
pixel sorted. The mean pixel value was calculated after rejecting about 10 %
of the lowest and highest values. This way every few weeks a new combined dark
was constructed for every standard exposure time used during that period.
To identify bad pixels comprehensive samples of combined darks of a given
exposure time were compared, pixel by pixel. If either the scatter or the mean
pixel value exceeded adopted thresholds (about 3-sigma level), that pixel was
flagged as “bad”. This process was repeated for all exposure times and a
combined list of pixels generated of those pixels which appeared at least once
on any of the individual “bad” pixel lists. A total of 13,094 such pixels was
identified, which is less than 0.1 % of all pixels on the detector.
### 3.2 Applying darks
The average background intensity (from bias and dark current) of raw CCD
frames taken with our 4k camera is very nonuniform over the field. However,
the pattern is very similar from exposure to exposure, while the amplitude of
the pattern depends on many things, like exposure time, ambient temperature
and vacuum pressure. The mean difference in background ADU between the left
and right side of the CCD frame serves as a parameter to quantify the
amplitude of this background pattern.
The re-processing of the pixel data was split up into batches of about 10,000
consecutive CCD frames taken over a narrow range of epochs. A pair of
appropriate master darks for each standard exposure time was selected. Each
pair spans a range in background differences (left to right, see above) that
is as large as possible with the restriction of being taken close to the epoch
of the frames under investigation. The raw data processing then involved a
determination of the mean background difference (left to right) of each
individual frame. This value was used in a linear interpolation between the 2
master darks selected for that set of data and the exposure time of the object
frame. The pixel-by-pixel interpolated dark was then subtracted from the
object frame.
This method of dark subtraction was new for UCAC3 and resulted in a
significant improvement in background flatness and lower noise, which leads to
a deeper and more uniform limiting magnitude than before. For UCAC2 a more or
less random pick of a dark near the target properties was selected without
interpolation. As with previous releases, no additional bias frames were
needed.
### 3.3 Flats
Due to the small size of the field utilized by the CCD, as compared to the
optical design of the astrograph, there is no vignetting from the optical
system expected. Initial tests also revealed only small pixel-to-pixel
sensitivity variations. The window on the camera serves as the only filter in
a sealed system without moving parts. Thus for the UCAC1 and UCAC2 releases no
flats were applied at all to the survey data, aiming at astrometric results
without the goal of precise photometry.
However, a set of about 25 dome flats were taken every few months with an
exposure time of 5 s and light intensity set to give about 30 to 50% full well
capacity illumination. These data were reduced and applied for the UCAC3
release. The appropriate combined dark frame was subtracted from each
individual flat exposure, and all flats of a given epoch were combined
excluding extreme low and high counts, similarly to the darks processing
described above. The flats were scaled to 1000 ADU mean intensity (integer)
representing a factor of 1.0 for the science frames data processing to follow.
For some epochs the flat data were split into 2 groups of high/low average
illumination to check on internal consistency. A total of 28 combined master
flats was thus obtained spanning the entire duration of the UCAC observing.
The pixel-to-pixel sensitivity variations are small. Taking 1/25 of the entire
CCD area at a time, sorting all the pixels of a given master flat and cutting
the low 5% and high 5% of the pixels, the resulting standard deviation for
pixel-to-pixel variation is only on the order of 0.4 to 0.6% of the mean pixel
count. This fact explains why excellent astrometric results (center fit
precision close to 1/100 pixel) were obtained in UCAC2 even without applying
any flats.
Large-scale sensitivity variations over the CCD frame area were found to be
10% or less. Comparing different master flats of different epochs, variations
of about 2% or less are found, except for the set taken around night numbers
2000 to 2150 (truncated Julian Dates), where significant deviations due to a
shutter failure problem are found in the corners of CCD frames.
Based on these results a single master flat file close to the epoch of each
object frame was selected and applied. The $\approx$ 10% vignetting in the
corners of the CCD frames was attributed to a slightly undersized, round
opening of the shutter in our 4k camera system.
## 4 IMAGE PROFILES
### 4.1 Supersampling
In order to investigate the shape of the point-spread function (PSF) as seen
in the UCAC data the following standard procedure was adopted. Individual CCD
frames of good quality taken in areas of the sky with large numbers of stars
(but not too crowded, few blended images) were selected. Centers of stellar
images were determined by least-square fits using a 2-dimensional Gaussian
model of the pixel intensity ($I$) as a function of the pixel coordinates
($x,y$),
$I(x,y)=B+A\ e^{-\frac{r^{2}ln(2)}{r_{0}^{2}}}$ (1)
with $r^{2}=(x-x_{0})^{2}+(y-y_{0})^{2}$. We call this model 1 and it has 5
free parameters: the average local background intensity ($B$), the amplitude
($A$), the width of the profile ($r_{0}$), and the image center coordinates
$x_{0}$ and $y_{0}$. The natural logarithm of 2 is included here to scale the
$r_{0}$ to obtain the radius of the profile at half maximum.
Images with sufficient signal-to-noise (S/N) ratio, but not saturated, were
scaled to a fixed amplitude, shifted in $x,y$ pixel coordinates to align
centers and resampled onto a grid with 0.2 pixel resolution. Averages of the
pixel values in each bin were taken to produce the supersampled PSF
representative for that particular CCD frame or sub-area of it. For some of
the investigations presented below, a CCD frame was split into 3 equal area
sections along the $x$-axis, following increased effects of the poor CTE of
the detector. Marginal profiles were calculated from these 2-dim supersampled
PSFs along the $x$ and $y$ axis, labeled as $u$ and $v$ coordinates. A 1-dim
radial profile was generated for 0.2 pixel bins using the original pixel data
and assuming a circular symmetric intensity distribution. The spatial
coordinate for those profiles is called $r$, as defined above.
### 4.2 Profile model functions
An overview of all profile models used for UCAC3 reductions and tests is given
in Table 1. Models 1 through 4 are taken from the software for analyzing
astrometric CCD data (SAAC) (Winter, 1999), which developed out of earlier
work (Schramm, 1998), while the other models have been modified from SAAC
models or are newly developed. Model 1 is given by Eq. 1, while model 2 did
perform better than model 1 but worse than model 4 and is not further
discussed here. Model 4 is the general Lorentz profile function, given as
$I(x,y)=B\ +\ A\ \left[1+\left(\frac{r}{r_{0}}\right)^{\alpha}\
\left(2^{1/\beta}-1\right)\right]^{-\beta}$ (2)
For $\beta=1$ this reduces to the Moffat profile (Moffat, 1969). The
additional parameter in model 4 gives more control over the shape of the PSF,
allowing adjustment of the gradient near the core of the profile independently
from the gradient out in the wings.
The profile function of model 4 with a fixed, preset parameter $\beta$ (not
determined in the least-squares fit as a free parameter) we call here model 3.
Similarly, a function with both parameters $\beta$ and $\alpha$ preset we call
model 5.
Derived from the circular, symmetric, Gaussian profile (model 1) an elliptical
symmetric function with major and minor axes aligned to $x$ and $y$,
respectively (model 6) is defined by
$I(x,y)=B+A\
e^{-ln(2)\left(\frac{(x-x_{0})^{2}}{a^{2}}+\frac{(y-y_{0})^{2}}{b^{2}}\right)}$
(3)
where $B$, $A$, $x_{0}$, and $y_{0}$ are defined as before and $a$ and $b$ are
the widths of the profile along the $x$ and $y$ axis, respectively.
Model 7 is a generalization of model 6 with the additional free parameter
$\theta$ describing the angle of the major axis with the $x$ axis (range
$\pm\pi/2$), and is given by
$I(x,y)=B+A\
e^{-\left((x-x_{0})^{2}c_{x}+(y-y_{0})^{2}c_{y}+(x-x_{0})(y-y_{0})c_{m}\right)}$
(4)
with
$\displaystyle
c_{x}=\frac{1}{2}\left(\frac{\cos^{2}(\theta)}{a^{2}}+\frac{\sin^{2}(\theta)}{b^{2}}\right)$
$\displaystyle
c_{y}=\frac{1}{2}\left(\frac{\cos^{2}(\theta)}{b^{2}}+\frac{\sin^{2}(\theta)}{a^{2}}\right)$
$\displaystyle
c_{m}=-\cos(\theta)\sin(\theta)\left(\frac{1}{b^{2}}-\frac{1}{a^{2}}\right)$
which reduces to model 6 for $\theta$ = 0.
Tests with asymmetric profiles were performed by adding the following term to
each base model ($F$, as given above) in the description of the pixel
intensities as function of $x,y$
$I(x,y)=B+A\ F(r)\ +\ A\ c\ (x-x_{0})\ F(r)$ (5)
This adds an asymmetric part to the $x$ component with an amplitude factor,
$c$. Extending this concept to an additional asymmetric contribution along the
$y$ axis is straightforward.
The following model was used for some tests. It is based on a generalized
Lorentz profile (elliptical) with asymmetric terms for $x$ and $y$,
$I(x,y)=B\ +\ A^{\prime}\ \left[1+\left(\frac{r}{r_{0}}\right)^{\alpha}\
\left(2^{1/\beta}-1\right)\right]^{-\beta}$ (6)
with
$\displaystyle\Delta x=x-x_{0}$ $\displaystyle\Delta y=y-y_{0}$
$\displaystyle\frac{r}{r_{0}}=\sqrt{\frac{\Delta x^{2}}{a^{2}}+\frac{\Delta
y^{2}}{b^{2}}}$ $\displaystyle A^{\prime}=A\left[1+c_{x}\Delta x+c_{y}\Delta
y\right]$
with background $B$, amplitude $A$, image center $x_{0},y_{0}$, profile shape
parameters $\alpha,\beta$ as before, and radius of profile width along $x$ and
$y$, $a,b$, respectively (elliptical model). The asymmetry in this model is
described by the parameters $c_{x}$, and $c_{y}$ for the relative amplitude of
the asymmetry along $x$ and $y$, respectively, giving a total of 10
parameters. This profile function was used for models 12 to 14, depending on
which of these parameters are preset and which are free fit parameters (see
Table 1).
### 4.3 First results
Figure 2 shows the radial profile of a supersampled PSF (as explained above)
obtained from stellar images on the left side (nearly no CTE effect) of CCD
frame 173586 as an example. The same data points are shown in both plots.
However, for the top plot the Gaussian (model 1) function was used to generate
the fit line through the data points, while the bottom plot shows the result
of model 4. A much better fit to the actual data is obtained with the latter
model.
The small crosses running through the middle of each plot show the residuals
(data $-$ fit model), scaled by a factor of 3 and offset by a constant along
the intensity axis for better visualization. The Lorentz profile model gives
significantly smaller residuals (about a factor of 2) than the Gaussian model;
however, the Lorentz profile fit is not perfect either, and the spacial
frequency of the residuals has increased, giving more peaks and valleys in the
residual pattern, thus increasing its complexity.
Figure 3 shows a similar set of plots obtained from the same CCD frame;
however, using stellar images on the right side of the CCD (large CTE effect)
and showing a marginal cut along $x$ instead of the radial coordinate used in
Fig. 2. Clearly the asymmetry of the profile is seen, and both models perform
about equally well, with a slightly better fit for the Lorentz model.
Figure 4 illustrates contour plots of residuals after fitting the supersampled
PSF of the right side (large CTE effect) of frame 134473 (compare to Fig. 1).
This asymmetric image could be fit reasonably well with model 9 using an
elliptical, general, Lorentz profile function and asymmetry terms for $x$ (see
Eq. 5), a total of 9 free parameters. The contour residuals using model 4 are
shown for comparison. Residuals of a fit with model 1 look similar to the
model 4 plot, although with larger amplitudes. In any case relatively large
residuals with high spacial frequencies remain even when applying the
asymmetric model.
### 4.4 Minimize number of free parameters
Fitting individual stellar images which extend only over a few pixels with
models having 8 or even 10 free parameters, if numerically feasible at all,
will lead to poor results in astrometry due to the small degree of
overdetermination in the least-squares process. In order to benefit from the
image profile models which better fit our data than the Gaussian some
restrictions in parameter space were investigated.
A set of 282 high quality CCD frames was selected to sample the range in FWHM
and span the entire observing epoch range. All these frames have a large
number of stars, but are not crowded. For all frames the supersampling of the
PSF was performed and various image profile fit models run on these,
separately for each CCD frame. Results were summarized in a table and
supplemented by observing log items.
Figure 5 shows the strongest correlation found for the various parameters
investigated. The shape parameters ($\alpha,\beta$) of the symmetrical Lorentz
profile model 4 can be predicted from the profile width of the Gaussian model
1 fit. The profile width here is the radius (about FWHM/2) with unit bin width
(0.2 pixel) of the supersampled profile data. A linear term is sufficient to
predict the $\beta$ parameter, while for the $\alpha$ parameter a second order
polynomial was adopted. Scaling to the actual pixel size these results were
hard-coded to preset both shape parameters in profile fit model 5, which is
otherwise the same as model 4. This leaves only 5 free fit parameters, exactly
the same as for the 2-dim Gaussian model function (see above and Table 1).
Tests were performed to determine any possible variations of the $\alpha$ and
$\beta$ parameters. Supersampled PSFs were generated as a function of 2
magnitude bins, and in another test the data were split into 4 quadrants on
the CCD frames. Consistent results for the $\alpha$ and $\beta$ parameters
were found, confirming the relationship with the profile width as before with
very small variation as a function of other selection criteria.
### 4.5 Other models
Tests were performed using elliptical profile models (6,7,8). No significant
advantage was found over circular, symmetric models. The residuals similar to
those shown in Fig. 4 did not generally decrease, unless the model was
extended to include asymmetric terms in addition.
Asymmetric profiles were tested extensively on the supersampled PSF data of
the selected frames used previously. In particular, model 12 was used to probe
parameter space and look for dependencies. Smaller residuals than with any
symmetric profile were found, however different parameter values are needed
for stellar images in different locations on the detector as well as for
different CCD frames.
Figures 6 and 7 show some examples obtained with test runs using models 9 and
11, respectively. The amplitude, $c_{x}$ (Eq. 6), of the asymmetric term along
the $x$-coordinate (right ascension) is approximated by a function linear with
air temperature and $x$ itself. Image profiles are symmetric at low $x$ and
the largest asymmetry is seen at large $x$. Similarly the amplitude of the
asymmetry along the $y$-coordinate was estimated as a linear function of
temperature and $y$. Model 13 implements these preset values for $c_{x}$, and
$c_{y}$, leaving only 6 free parameters, including the $a,b$ profile widths
along $x$ and $y$, respectively (elliptical Lorentz base model).
### 4.6 Double star fits
Double star models solve simultaneously for at least 3 more parameters: the
center coordinates, $x,y$ and amplitude, $A$ of the secondary component.
Again, the goal is to minimize the number of free parameters as much as
possible. Thus, for example, a single parameter for the background level is
used. Some double star models also assume equal widths of the profiles of both
components. Table 1 gives more details (models 20 to 23).
Critical for handling of blended images is the identification of such cases
and the determination of sufficiently accurate starting parameters for the
iterative, non-linear double star profile fit routine. This process can easily
“go astray” due to the relatively large number of parameters and the small
number of pixels available with the critically sampled UCAC data.
The adopted criterion for detecting an object on a dark and flat corrected CCD
frame is to have at least 2 connected pixels above a specified S/N threshold
level of $3\sigma$ above mean background. For each such detected object a
centroid position (center-of-light, 1st moments) is calculated as well as the
2nd moments. The orientation of the major axis and image elongation, defined
as the ratio of major to minor axis are derived from these moments. An
elongation of 1.0 means a circular, symmetric image, otherwise the elongation
is greater than 1. An image profile fit with model 1 (circular, symmetric
Gaussian) is performed on all objects to identify “good” stars, and the mean
image elongation of that CCD frame is calculated from the 2nd moment results
of the “good” stars only. Images are typically slightly elongated due to
guiding errors and the CTE effect.
The double star routines are triggered if an objects elongation exceeds an
adopted threshold of 12% over the mean image elongation (for that CCD frame),
and has a sufficient number of pixels ($\geq$10) above the detection threshold
level. This elongation threshold was adopted as best compromise between
excluding false positives of single stars due to statistics and including as
many as possible real double stars. With some interpolation, pixel values are
compared which lie on a line through the center of light along the major axis
as determined earlier. A search is made for 2 peaks along this line and
starting parameters (location and amplitudes) of the 2 components are derived.
Starting parameters for the image profile width and background value are taken
from the overall CCD frame mean values. If no 2 separate peaks can be
detected, estimates for a possible nearby, blended, secondary component are
made based on the image profile width and brightness of the object under
investigation.
A least-squares fit is attempted with these starting parameters using model 23
(see Table 1), based on the Lorentz profile. The object is output as 2
components if reasonable starting parameters could be derived, even if the
double star fit failed. A double star flag is assigned specifying the status
of each successful detection and/or fit of 2 components.
A sample of newly detected UCAC3 doubles was observed with the 26-in speckle
program Mason, Hartkopf & Wycoff (2008) and a paper addressing accuracy and
reliability of UCAC3 double star data will be presented in a separate paper
(Mason et al., 2010).
## 5 UCAC3 PIXEL REDUCTION RUN
### 5.1 Algorithm
The final reduction pipeline to process the UCAC3 pixel data handles a
specified range of CCD frames with a single selected master flat and pairs of
low/high ADU master dark frames for each standard exposure time (see above).
The input list of frames is sorted by exposure time, and frames are then
processed in that order, performing the following steps:
1. 1.
Read original, compressed pixel data file, determine mean left/right
background counts and flag saturated pixels.
2. 2.
Interpolate dark frame, apply dark and flat corrections, output processed
image (round to 2-byte integers).
3. 3.
Flag pixels from bad pixel maps, detect and flag possible streaks (from
shutter failure and bleeding images).
4. 4.
Pass 1:
1. (a)
Detect objects ($3\sigma$ above mean background for at least 2 connected
pixels).
2. (b)
Classify objects including 1st and 2nd moments, identify possible blended
images (doubles).
3. (c)
Perform image center fits on all objects with model 1 (Gaussian).
5. 5.
Intermission 1:
1. (a)
Identify “good” stars over entire CCD frame, based on model 1 fit results.
2. (b)
Derive mean image profile width, mean image elongation, $\alpha,\beta$ profile
shape parameters, and radii for aperture photometry.
6. 6.
Pass 2:
1. (a)
Perform circular, symmetric Lorentz profile fit (model 5).
2. (b)
Calculate double star starting parameters and perform fit (model 23).
3. (c)
Perform asymmetric profile fit (model 13).
7. 7.
Intermission 2:
1. (a)
Select “good” stars from model 13 fit results.
2. (b)
Derive mean width of profiles ($a,b$) of elliptical part of model 13 and fixed
parameters for model 14.
8. 8.
Pass 3: Perform model 14 fit with further parameter restrictions.
9. 9.
Perform aperture photometry.
10. 10.
Derive model magnitudes from each successful model fit.
11. 11.
Output all fit results for each frame to a separate file.
The profile fits are performed with pixels inside a circular aperture centered
on the best known position at the time. The radius of this aperture was
adopted to be twice as large as the radius of the area of pixels above the
threshold from the image detection step. For all astrometric image profile
fits the local background parameter is a free fit parameter for each single
star or binary pair.
Note, the astrometric fit based on the Lorentz profile is performed with 5
free parameters, the same number of parameters as used for a Gaussian profile
model. The major difference is that here a model profile is selected that does
better match the data with the profile shape being slightly different for CCD
frames taken under different seeing conditions (average image width). The
first step in this reduction process (finding $\alpha,\beta$) merely is a
quantitative way to make a “good guess” about which model profile to use.
For the aperture photometry all pixels within 4 times the mean radius of the
image profiles (Gaussian fit of “good” stars) of that CCD frame were used to
determine the flux of a target. An annulus with 12 and 16 times this mean
profile radius served for the background determination. The background value
is determined from the peak of the histogram of background pixels (weighted
mean of bins which exceed 50% of the smoothed histogram peak value).
### 5.2 Processing and Results
The re-processing of the pixel data involved 5 different Linux workstations
(mostly single processor at about 2 GHz), which most of the time ran in
parallel for about a month working on a section of the UCAC frames each.
Individual binary files contain the output data for each CCD frame. The number
of objects per frame ranged from 37 to 75,549 with a median of 1397. A data
record of 136 bytes contains the results for each detected object, including
selected items from the moment analysis, the parameters of the 4 model fits,
their errors, and flags. Data items were converted to integers of 1, 2, or 4
byte lengths with appropriate scaling. A total of over 4 TB of compressed
pixel data went into this process, producing a total of 80 GB of binary $x,y$
data, the results of 271,428 CCD exposures. These files were later extended by
8 bytes per record to arrive at the final $x,y$-data output files. The
additional data contain information about nearest neighbors, and
identification of possible doubles that are not blended.
### 5.3 Analysis of the Results
How good are the resulting $x,y$ data? An example of the internal fit
precision is presented in Figure 8. The formal standard error of the
$x$-center coordinate is shown as a function of instrumental magnitude. Data
for the same CCD frame are shown for 3 different image profile fit models (1,
5, and 14). The results for model 13 look very similar to those of model 14.
The results for the $y$ coordinate are similar to those of the $x$-coordinate.
Saturation occurs at about magnitude 8. The unit is milli pixel (mpx), with 1
mpx = 0.9 mas. For the unsaturated, high S/N stars (8 to 10 mag) per
coordinate precisions of about 6 mpx, 4 mpx, and 3 mpx are reached for fit
models 1, 5, and 14, respectively, from a single CCD frame observation of good
quality.
The repeatability of observations was tested frequently with the same field in
the sky observed twice within minutes and the telescope being on the same side
of the pier. A weighted, linear transformation between the sets of $x,y$ data
of such a pair of 100 s exposures was performed and the scatter in the $x$
coordinate of stars plotted as a function of instrumental magnitude in Figure
9. Again, results from different profile fit models (pfm) are shown as
indicated. This scatter includes the errors from both CCD frames. Assuming
equal error contribution, the repeatability error of these observations is
thus about 15 mpx / $\sqrt{2}$ $\approx$ 11 mpx $\approx$ 10 mas per
coordinate and single observation for well exposed stars, almost independent
of the profile fit model. These results are consistent with the first
observations at this telescope using a CCD camera (Zacharias, 1997). The
observed error is significantly larger than the internal fit precision (of
bright stars) due to atmospheric turbulence. A scatter that is about a factor
of 2 larger is observed in similar CCD frame pairs of 25 s exposure time as
compared to the 100 s frames.
The flip observations, with the telescope on one side of the pier then on the
other, provide pairs of frames that are rotated by $180^{\circ}$ with respect
to each other. A linear transformation between the 2 sets of $x,y$ data of
each frame pair taken on the same field in the sky gives residuals revealing
systematic errors as a function of magnitude or coma-like terms (product of
magnitude and positional coordinates). An example is shown in Figure 10 for
such a pair of frames with 100 s exposure, before applying corrections.
Results are derived from the same pixel data but using all 4 different profile
fit models as indicated. Each data point is the mean for 16 stars. The model 5
and 13 results show a somewhat tighter distribution than those of models 1 and
14. Unfortunately a similar amplitude of the systematic errors is present in
the data from all 4 models, while the hope has been that the asymmetric
profile fits would have mitigated this problem. Even larger systematic errors
(about 200 mas) are seen in short exposure frames. These and other systematic
errors will be investigated in great detail with the help of reference stars,
as described in another paper of this series (Finch, Zacharias & Wycoff,
2010). Empirical corrections will be derived for these and other systematic
errors in the UCAC data at that time. These corrections effectively reduce
these types of systematic errors by about a factor of 10 (as compared to what
is seen in Fig. 10) for the published catalog star positions.
## 6 DISCUSSION AND CONCLUSIONS
The detection threshold for blended, double star images is affected by the
gradient of inherent image elongation along the $x$ axis due to the poor CTE.
Similarly fit results of double stars will have a slight bias depending on the
$x$ pixel coordinate. Almost symmetric images are seen on the left side of a
CCD frame, while CTE elongated, asymmetric images are seen on the right side.
In all cases the same, symmetric double star profile is used for a fit. In
addition, a slight image elongation is typically added from guiding effects.
Nevertheless, an important first step for detecting double stars and deriving
useful parameters has been accomplished for UCAC3. Results from external
comparisons will be presented in (u3d).
The additional first order parameters to describe image asymmetry (relative
amplitude of a term linear with pixel coordinate) could not be correlated well
to any other parameters, contrary to the $\alpha$ and $\beta$ shape parameters
used in the Lorentz profile model. Even such a complex model applied without
approximations of the asymmetric parameters (i.e. use of many free fit
parameters) leaves significantly large residuals (see Fig. 4 bottom) in the
supersampled, stacked PSFs. Applying a model with 7 or more parameters to the
(not supersampled) original pixel data for each star is not an option with the
critical sampling of the UCAC data (too few pixels per image).
This situation calls for a purely empirical PSF model by using the
supersampled, stacked, observed profiles to generate a template. Using purely
empirical PSFs as templates to fit observed stellar images in UCAC data,
however, was not considered a viable option. Many UCAC frames have a low
number of stars, in particular the number of stars with high S/N ratio needed
for this approach is very small (a few) in many areas of the sky. Furthermore,
the PSF changes significantly over the area of the detector (mainly along the
$x$ coordinate) due to the poor CTE and, is a function of the profile width
(FWHM), the guiding of each exposure, the CCD temperature and probably other
factors. Splitting up the data into so many categories was not an option. In
general a purely empirical PSF will be asymmetric to some degree, also caused
by imperfect guiding. Such an approach would mean a different definition of an
image center for different CCD frames with additional variations over the
field of view, not desirable for astrometry. Thus the use of models 13 and 14
is a compromise driven by the need for a more complex model without having a
large enough sampling to support it.
The very high precision of the well exposed stellar images seen in the
internal fit errors per coordinate unfortunately does not translate into
similarly small external errors. Internal errors of under 5 mas per coordinate
and exposure are seen, however the repeatability of such observations is
already degraded to about 10 mas and more due to the atmosphere for our long
exposures of 100 to 150 s. For the short exposures the positional errors are
further increased by about a factor of 2. However, the UCAC data are still
limited by remaining systematic errors, resulting in an error floor of just
under 20 mas per coordinate for the mean CCD observations (4 images) for stars
in the R = 10 to 14 mag range, as will be shown in the astrometric reduction
paper (Finch, Zacharias & Wycoff, 2010).
The symmetric profile model 5 $x,y$ center coordinates have been adopted as
the baseline for these astrometric reductions. It could be argued that the
derived $\alpha$ and $\beta$ shape parameters used in model 5 are not
accurately enough known, based on the approximation as described above.
However, very good astrometric results have already been obtained with the
Gaussian model (UCAC2, and many other traditional, astrometric catalog
projects). The Lorentz profile adopted here, with a somewhat imperfect
representation of shape parameters, is a far better representation of the
observed image profile than is the Gaussian model. Both are symmetric profile
functions with the same number of free parameters, so no detrimental effect is
expected when choosing model 5 (Lorentz profile) over model 1 (Gauss profile).
Effects from image asymmetry will be investigated and corrected by analyzing
the residuals with respect to reference stars. No significant benefit for the
overall astrometric accuracy has been found so far by using the asymmetric
profile models investigated here, particularly with respect to solving
magnitude and coma-like terms. For the baseline UCAC reductions a symmetric
PSF model is applied to asymmetric image profiles with subsequent systematic
error corrections of the celestial coordinates using reference stars.
The entire UCAC team is thanked for making this all-sky survey a reality. In
particular mention should be made of, Charlie Finch for discussions and
assistance with data structure implementations and Gary Wycoff for preparatory
work including frame selection tasks. National Optical Astronomy Observatories
(NOAO) is acknowledged for IRAF, Smithonian Astrophysical Observatory for DS9
image display software, and the California Institute of Technology for the
pgplot software. More information about this project is available at
http://www.usno.navy.mil/usno/astrometry/.
## References
* Anderson & King (2000) Anderson, J. & King, I. R. 2000, PASP, 112, 1360
* Finch, Zacharias & Wycoff (2010) Finch, C., Zacharias, N., & Wycoff, G. L. 2010, in press, AJ
* Mason, Hartkopf & Wycoff (2008) Mason, B. D., Hartkopf, W. I., & Wycoff, G. L. 2008, AJ, 136, 2223
* Mason et al. (2010) Mason, B. D. et al., observations of UCAC3 double stars, in prep.
* Moffat (1969) Moffat, A. F. J. 1969, A&A, 3, 455
* Stetson (1987) Stetson, P. B. 1987, PASP, 99, 101
* Schramm (1998) Schramm, J. 1998, Ph.D. thesis, Hamburg Observatory
* Winter (1999) Winter, L. 1999, Ph.D. thesis, Hamburg Observatory
* Zacharias (1997) Zacharias, N., 1997, AJ, 113, 1925
* Zacharias et al. (2000) Zacharias, N., Zacharias, M. I., & Rafferty, T. J. 2000, AJ, 118, 2503
* 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
* Zacharias et al. (2010) Zacharias, N. et al. 2010, submitted to AJ
Figure 1: Contour plot of supersampled (see text) images of CCD frame 134473.
Average data from near the output register (top) and farther away from it
(bottom) are shown. This illustrates the low charge transfer efficiency
problem of the detector leading to images which are asymmetric as a function
of the $x$ pixel coordinate. The contour levels are at 90%, 30%, 10%, 3% and
1% of the peak intensity. Figure 2: Radial profile plot of supersampled (see
text) images on the left side of CCD frame 173586. The same data points are
shown in both diagrams with a fit of a Gaussian (top) and Lorentz model
function (bottom). Residuals are scaled with a factor of 3 and shown with an
intensity offset of 0.5 for better readability. Figure 3: Profile plot along
$x$-axis of supersampled images on the right side of CCD frame 173586. The
same data points are shown in both diagrams with a fit of a Gaussian (top) and
Lorentz model function (bottom). Residuals are scaled with a factor of 3 and
shown with an intensity offset of 0.5 for better readability. Figure 4:
Residual contour plot of a supersampled PSF from stellar images on the right
side of CCD frame 134473. The same data are used for both plots; model 4
(symmetric Lorentz profile) fit (top), and model 9 (elliptic Lorentz profile
plus asymmetry terms along $x$) (bottom) fits were applied, respectively.
Contour levels are light blue $-3$%, dark blue $-1.5$%, black 0, red $+1.5$%,
and green $+3$%. Figure 5: Alpha (top) and beta (bottom) image profile shape
parameters of the Lorentz model 4 as a function of the image width (radius) as
determined by model 1 (Gauss) for the supersampled data (1 bin = 0.2 pixel).
The alpha parameter is fit by a 2nd order polynomial, while the beta parameter
is well represented by a linear dependency. Figure 6: Examples of the
asymmetry along $x$ parameter dependencies from supersampled data. The
amplitude of the asymmetry along $x$ (RA) as a function of the air temperature
(bottom) and as a function of the image profile radius (top) are among the
strongest correlations found. Figure 7: Similarly to the previous figure, the
dependency of the amplitude of image asymmetry along $y$ (Dec) is shown here.
Figure 8: Internal fit precision (milli pixel = 0.9 mas) along the $x$
coordinate as a function of aperture magnitude of a long exposure sample frame
53554 with over 4000 stars. The plots zoom in on the high precision, high S/N
area (bright stars). Results for different fit models are shown: model 1
(Gaussian) on top, model 5 (Lorentz) in the middle, and model 14 (asymmetric
profile) on the bottom. Figure 9: Standard error (sigma) of the $x$ coordinate
as a function of instrumental model magnitude from the comparison of 2
100-second exposures taken of the same field in the sky within minutes.
Results are shown from the same CCD frame pair but for different image profile
fit models (pfm) as indicated. The error shown is the combined error of both
exposures, dominated by atmospheric effects. Each dot represents the RMS
average of 16 stars. Unit is milli pixel (1 mpx = 0.9 mas). Figure 10:
Systematic differences between the $x$ coordinates of a pair of 100-second
exposures taken from opposite sides of the telescope pier (rotated by
$180^{\circ}$ with respect to each other) as a function of instrumental model
magnitude, after a linear transformation of the $x,y$ data but without any
other corrections applied. Results are shown from the same CCD frame pair but
for different image profile fit models (pfm) as indicated. Each dot represents
the mean over 16 stars. Unit is milli pixel (1 mpx = 0.9 mas).
Table 1: Description of image profile models used in UCAC3 reductions and tests. | profile fit model number
---|---
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 20 | 21 | 22 | 23
base modelaaG = Gaussian, D = double exponential, L = Lorentz, d = double star | G | D | L | L | L | G | G | L | L | L | L | L | L | L | Gd | Gd | Ld | Ld
symmetrybbc = circular symmetric, e = elliptical, a = incl. asymmetry | c | c | c | c | c | e | e | e | e,a | e,a | e,a | e,a | e,a | e,a | c | c | c | c
total number of parameters | 5 | 6 | 7 | 7 | 7 | 6 | 7 | 8 | 9 | 9 | 9 | 10 | 10 | 10 | 8 | 9 | 11 | 11
numb.of free fit parameters | 5 | 6 | 6 | 7 | 5 | 6 | 7 | 8 | 9 | 7 | 9 | 8 | 6 | 4 | 8 | 9 | 9 | 8
$x,y$ center, ampl., backgr.ccf = free fit parameter, p = preset | f | f | f | f | f | f | f | f | f | f | f | f | f | f | f | f | f | f
profile widthdd1 parameter for circular symmetric, 2 for elliptical | f | f | f | f | f | f | f | f | f | f | f | f | f | p | f | f | f | f
elliptical axis orientation | | | | | | | f | | | | | | | | | | |
1st shape parameter | | f | f | f | p | | | f | f | p | f | p | p | p | | | p | p
2st shape parameter | | | p | f | p | | | f | f | p | f | p | p | p | | | p | p
asymmetry $x$ amplitude | | | | | | | | | f | f | | f | p | p | | | |
asymmetry $y$ amplitude | | | | | | | | | | | f | f | p | p | | | |
$x,y$, amplitude secondary | | | | | | | | | | | | | | | f | f | f | f
profile width secondary | | | | | | | | | | | | | | | | f | f |
|
arxiv-papers
| 2010-03-24T00:59:21 |
2024-09-04T02:49:09.219900
|
{
"license": "Public Domain",
"authors": "Norbert Zacharias",
"submitter": "Norbert Zacharias",
"url": "https://arxiv.org/abs/1003.4565"
}
|
1003.4774
|
Relationship between the n-tangle and the residual entanglement of even n
qubits111The paper was supported by NSFC(Grants No. 10875061,60433050, and
60673034 ) and Tsinghua National Laboratory for Information Science and
Technology.
Xiangrong Lia, Dafa Lib
a Department of Mathematics, University of California, Irvine, CA 92697-3875,
USA
b Dept of mathematical sciences, Tsinghua University, Beijing 100084 CHINA
Abstract
We show that $n$-tangle, the generalization of the 3-tangle to even $n$
qubits, is the square of the SLOCC polynomial invariant of degree 2. We find
that the $n$-tangle is not the residual entanglement for any even $n\geq 4$
qubits. We give a necessary and sufficient condition for the vanishing of the
concurrence $C_{1(2...n)}$. The condition implies that the concurrence
$C_{1(2...n)}$ is always positive for any entangled states while the
$n$-tangle vanishes for some entangled states. We argue that for even $n$
qubits, the concurrence $C_{1(2...n)}$ is equal to or greater than the
$n$-tangle. Further, we reveal that the residual entanglement is a partial
measure for product states of any $n$ qubits while the $n$-tangle is
multiplicative for some product states.
Keywords: the 3-tangle, the $n$-tangle of even $n$ qubits, the residual
entanglement, SLOCC polynomial invariants
PACS numbers: 03.67.Mn, 03.65.Ud
## 1 Introduction
Quantum entanglement is an important physical resource in quantum information
and computation such as quantum teleportation, cloning and encryption.
Entanglement phenomenon distinguishes the quantum world from the classical
world. Considerable attention has been paid in recent years to the
quantification and classification of entanglement. The concurrence was
proposed by Wootters in 1998 to quantify entanglement for bipartite systems
[1]. For two qubits, the concurrence was defined as
$C_{12}=Max\\{0,\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4}\\}$, where
$\lambda_{i}^{2}$ are the eigenvalues, in decreasing order, of
$\rho_{12}\tilde{\rho}_{12}$. Here, $\rho_{12}$ is the density matrix and
$\tilde{\rho}_{12}$ is the “spin-flipped”density matrix of $\rho_{12}$, i.e.,
$\tilde{\rho}_{12}=\sigma_{y}\otimes\sigma_{y}$
$\rho_{12}^{\ast}\sigma_{y}\otimes\sigma_{y}$ [2], where the asterisk denotes
complex conjugation in the standard basis. For the state $|\psi\rangle$ of a
bipartite system, the concurrence was also given by [4]
$C(\psi)=\sqrt{2(1-Tr(\rho_{A}^{2}))}.$ (1.1)
The definition of the concurrence in Eq. (1.1) was generalized to multipartite
systems [5]. Recently, the concurrence was used to study quantum phase
transitions [6].
By means of the concurrence, CKW monogamy inequality for three qubits was
established. Namely, $C_{12}^{2}+C_{13}^{2}\leq C_{1(23)}^{2}$ [2]. Here
$\rho_{12}$ is obtained from the density matrix $\rho_{123}$ by tracing out
over qubit 3, and $C_{1(23)}^{2}=4\det\rho_{1}$, where
$\rho_{1}=tr_{23}\rho_{123}$. Note that $C_{1(23)}$ can be called the
concurrence between qubit 1 and the pair of qubits 2 and 3 if qubits 2 and 3
are regarded as a single object. The difference
$(C_{1(23)}^{2}-(C_{12}^{2}+C_{13}^{2}))$ between the two sides of the above
CKW monogamy inequality is called “residual entanglement”. The algebraic
expression for the residual entanglement is called the 3-tangle (see (20) of
[2] for the expression). The expression can also be obtained from Eq.
(LABEL:n-tangle-1) by letting $n=3$. The 3-tangle is invariant under
permutations of all the qubits [2]. The invariance of entanglement measure
under permutations of all the qubits represents a collective property of the
qubits. The 3-tangle is also an entanglement monotone [7]. Monotonicity for
entanglement measure is a natural requirement.
The 3-tangle was extended to even $n$ qubits, and the extension was called the
$n$-tangle [3]. Let the state
$|\psi\rangle=\sum_{i_{1}i_{2}...i_{n}}a_{i_{1}i_{2}...i_{n}}|i_{1}i_{2}...i_{n}\rangle$,
where $i_{1}$, $i_{2}$, …, $i_{n}$ $\epsilon$ $\\{0,1\\}$. The $n$-tangle was
defined as [3]
$\displaystyle\tau_{1...n}$ $\displaystyle=$ $\displaystyle 2|S|,$
$\displaystyle S$ $\displaystyle=$
$\displaystyle\sum(a_{\alpha_{1}...\alpha_{n}}a_{\beta_{1}...\beta_{n}}a_{\gamma_{1}...\gamma_{n}}a_{\delta_{1}...\delta_{n}}$
$\displaystyle\times\epsilon_{\alpha_{1}\beta_{1}}\epsilon_{\alpha_{2}\beta_{2}}...\epsilon_{\alpha_{n-1}\beta_{n-1}}$
$\displaystyle\times\epsilon_{\gamma_{1}\delta_{1}}\epsilon_{\gamma_{2}\delta_{2}}...\epsilon_{\gamma_{n-1}\delta_{n-1}}\epsilon_{\alpha_{n}\gamma_{n}}\epsilon_{\beta_{n}\delta_{n}}),$
where $|c|$ is the modulus of the complex number $c$, $\alpha_{l}$,
$\beta_{l}$, $\gamma_{l}$, and $\delta_{l}$ $\epsilon\\{0,1\\}$, and
$\epsilon_{00}=\epsilon_{11}=0\text{ and }\epsilon_{01}=-\epsilon_{10}=1.$
(1.3)
The $n$-tangle of even $n$ qubits is invariant under permutations of the
qubits, and is an entanglement monotone [3]. In [3], the $n$-tangle was
proposed as a potential entanglement measure.
The generalized CKW monogamy inequality for $n$ qubits was given by [18, 19]
$C_{12}^{2}+...+C_{1n}^{2}\leq C_{1(2...n)}^{2}\text{.}$ (1.4)
Here $\rho_{12}=tr_{3...n}\rho_{12...n}$, i.e., $\rho_{12}$ is obtained from
the density matrix $\rho_{12...n}$ by tracing out over qubits 3, …, and $n$,
and $C_{1(23...n)}^{2}=4\det\rho_{1}$, where
$\rho_{1}=tr_{23...n}\rho_{123...n}$. Note that $C_{1(2...n)}$ can be called
the concurrence between qubit 1 and qubits $2,...,$ and $n$ if qubits 2, …,
and $n$ are regarded as a single object. The difference between the two sides
of CKW monogamy inequality in Eq. (1.4) can be considered as a natural
generalization of the residual entanglement of three qubits to $n$ qubits, and
was denoted as [18]
$\tau_{1(2...n)}=C_{1(2...n)}^{2}-(C_{12}^{2}+...+C_{1n}^{2})\text{.}$ (1.5)
In this paper, we investigate the relationship between the $n$-tangle and the
residual entanglement for any even $n\geq 4$ qubits. This paper is organized
as follows. In Sec. 2, we show that the $n$-tangle is the square of the SLOCC
polynomial invariant of degree 2. In Sec. 3, we address the relationship
between the $n$-tangle and the residual entanglement of $n$ qubits. In Sec. 4,
we summarize our results and conclusions.
## 2 The $n$-tangle is the square of the SLOCC polynomial invariant of degree
2
The SLOCC invariants can be used for SLOCC classification and the entanglement
measure [8, 9, 10, 11, 12, 13, 14]. For four qubits, four independent SLOCC
polynomial invariants: $H$, $L$, $M$, and $D_{xt}$ were given in [9], where
$H$ is of degree 2,$\ L$ and $M$ are of degree 4, and $D_{xt}$ is of degree 6.
Very recently, for four and five qubits, SL invariants of degrees 2 (for only
four qubits), 4, 6, 8, 10, 12 were studied in [14]. The antilinear operators
“combs”, which are invariant under $SL(2,C)$, were constructed in [15]. The
geometry of four qubit invariants was investigated in [11]. For any even $n$
qubits, the SLOCC polynomial invariant of degree 2 was given in [13]. The
SLOCC invariant of degree 4 of odd $n$ qubits was discussed in [12, 13]. Note
that there are no invariants of degree 2 for odd $n$ qubits [9].
### 2.1 Reduction of the $n$-tangle
The $n$-tangle in Eq. (LABEL:n-tangle-1) is quartic and the computation of the
coefficients takes $3\ast 2^{4n}$ multiplications. Denote by
$\overline{\alpha_{i}}$ the complement of $\alpha_{i}$. That is,
$\overline{\alpha_{i}}=0$ when $\alpha_{i}=1$. Otherwise,
$\overline{\alpha_{i}}=1$. Further, let
$\displaystyle S_{0}$ $\displaystyle=$
$\displaystyle\sum_{\alpha_{1}...\alpha_{n-1}}(a_{\alpha_{1}...\alpha_{n-1}0}a_{\overline{\alpha_{1}}...\overline{\alpha_{n-1}}1}$
(2.1)
$\displaystyle\times\epsilon_{\alpha_{1}\overline{\alpha_{1}}}\epsilon_{\alpha_{2}\overline{\alpha_{2}}}...\epsilon_{\alpha_{n-1}\overline{\alpha_{n-1}}}).$
Note that $S_{0}$ is of degree 2. Then, $S$ in Eq. (LABEL:n-tangle-1) can be
reduced to $S=2S_{0}^{2}$ (see (A) of Appendix A for the proof). This leads to
$\tau_{1...n}=|2S_{0}|^{2}\text{.}$ (2.2)
### 2.2 The $n$-tangle is the square of the SLOCC polynomial invariant of
degree 2
Let $|\psi\rangle=\sum_{i=0}^{2^{n}-1}a_{i}|i\rangle$ and
$|\psi^{\prime}\rangle=\sum_{i=0}^{2^{n}-1}b_{i}|i\rangle$ be any states of
$n$ qubits. Two states $|\psi\rangle$ and $|\psi^{\prime}\rangle$ are SLOCC
equivalent if and only if there exist invertible local operators
$\mathcal{A}_{1}$, $\mathcal{A}_{2}$, …, $\mathcal{A}_{n}$ such that [7]
$|\psi^{\prime}\rangle=\underbrace{\mathcal{A}_{1}\otimes\mathcal{A}_{2}\otimes...\otimes\mathcal{A}_{n}}_{n}|\psi\rangle.$
(2.3)
The entanglement measure of the state $|\psi\rangle$ of even $n$ qubits was
proposed as [13, 16]
$\tau^{\prime}(\psi)=\ 2\left|\mathcal{I}^{\ast}(a,n)\right|,$ (2.4)
where
$\displaystyle\mathcal{I}^{\ast}(a,n)=\sum_{l=0}^{2^{n-2}-1}[(-1)^{N(l)}$
$\displaystyle\times(a_{2l}a_{(2^{n}-1)-2l}-a_{2l+1}a_{(2^{n}-2)-2l})].$ (2.5)
Here we take $N(l)$ to be the number of the occurrences of “$1$” in
$l_{n-1}...l_{1}l_{0}$, which is a $n$-bit binary representation of $l$, i.e.,
$l=l_{n-1}2^{n-1}+...+l_{1}2^{1}+l_{0}2^{0}$. In [13], it was proven that if
$|\psi\rangle$ and $|\psi^{\prime}\rangle$ are SLOCC equivalent then
$\mathcal{I}^{\ast}(b,n)=\mathcal{I}^{\ast}(a,n)\det(\mathcal{A}_{1})...\det(\mathcal{A}_{n})\text{,}$
(2.6)
where $\mathcal{I}^{\ast}(b,n)$ is obtained from $\mathcal{I}^{\ast}(a,n)$ by
replacing $a$ in $\mathcal{I}^{\ast}(a,n)$ with $b$, and
$\mathcal{I}^{\ast}(a,n)$ was called the SLOCC polynomial invariant of degree
2 of even $n$ qubits.
Note that $S_{0}$ is just $\mathcal{I}^{\ast}(a,n)$ (see (B) in Appendix A for
the proof). By virtue of Eqs. (2.2) and (2.4), we have
$\tau_{1...n}=(\tau^{\prime}(\psi))^{2}$. It then follows from Eqs. (2.4) and
(A8) that
$\tau_{1...n}=4\left|\sum_{l=0}^{2^{n-1}-1}(-1)^{N(l)}a_{2l}a_{(2^{n}-1)-2l}\right|^{2}.$
(2.7)
In Eq. (2.7), computing the coefficients requires $(2^{n-1}+2)$
multiplications. The $n$-tangle $\tau_{1...n}$ is not considered as the SLOCC
polynomial invariant of degree 4 though $\tau_{1...n}$ is quartic and
satisfies the equation
$\tau_{1...n}(|\psi^{\prime}\rangle)=\tau_{1...n}(|\psi\rangle)\det(\mathcal{A}_{1})...\det(\mathcal{A}_{n})$.
However, the square root of the $n$-tangle is the SLOCC polynomial invariant
of degree 2. The square root of the $n$-tangle turns out to be
$\tau^{\prime}(\psi)$. Using the properties of $\tau^{\prime}(\psi)$ [13, 16],
the square root is also an entanglement monotone, and invariant under
permutations of all the qubits.
## 3 Relationship between the $n$-tangle and the residual entanglement
### 3.1 The $n$-tangle is not the residual entanglement for any even $n\geq
4$ qubits.
To illustrate the relationship between n-tangle and residual entanglement, we
consider the following examples. For the $n$-qubit state
$\alpha_{1}|0...1\rangle+\alpha_{2}|0...010\rangle+...+\alpha_{n}|10...0\rangle$,
equality in Eq. (1.4) holds [2, 20], i.e. the residual entanglement
$\tau_{1(2...n)}=0$. According to Eq. (2.7), it is easy to see that the
n-tangle $\tau_{1...n}=0$. It follows that $\tau_{1...n}=\tau_{1(2...n)}$.
This is particularly true for the $n$-qubit state $|W\rangle$ [3]. For the
state $|GHZ\rangle=\frac{1}{\sqrt{2}}(|0\rangle^{\otimes n}+|1\rangle^{\otimes
n})$, the residual entanglement$\ \tau_{1(2...n)}=1$ [20], and the n-tangle$\
\tau_{1...n}=1$ [3]. Thus, $\tau_{1...n}=\tau_{1(2...n)}$ for the state
$|GHZ\rangle$. Here is another example which gives
$\tau_{1...n}=\tau_{1(2...n)}=$ $4\left|\alpha\gamma\right|^{2}$ for the state
of four qubits: $\alpha|0011\rangle+\beta|0110\rangle+\gamma|1100\rangle$ by
utilizing Eq. (2.7).
One might wonder if the two generalizations, which are the $n$-tangle
$\tau_{1...n}$ and the residual entanglement $\tau_{1(2...n)}$, are equal.
However, this is not always the case as the following example will show.
Consider, for example, the $n$-qubit symmetric Dicke states with $l$
excitations ($1\leq l\leq(n-1)$) [21]
$|l,n\rangle=\sum_{i}P_{i}|1_{1}1_{2}...1_{l}0_{l+1}...0_{n}\rangle,$ (3.1)
where $\\{P_{i}\\}$ is the set of all the distinct permutations of the qubits.
For the Dicke state $|(n/2),n\rangle$ with $(n/2)$ excitations of any even
$n\geq 4$ qubits, Eq. (2.7) yields the $n$-tangle $\tau_{1...n}=1$. In this
case, $\rho_{12}\tilde{\rho_{12}}$ has only three nonzero eigenvalues
$(\frac{n}{2(n-1)})^{2}$, $(\frac{n-2}{4(n-1)})^{2}$ (double). We then get the
concurrence $C_{12}^{2}=\frac{1}{(n-1)^{2}}$. The symmetry of the Dicke state
leads to $C_{1i}^{2}=C_{12}^{2}$, $i=3,...,n$. Calculating $C_{1(2...n)}$
further gives $C_{1(2...n)}^{2}=\allowbreak 1$. In light of Eq. (1.5), the
residual entanglement $\tau_{1(2...n)}=\frac{n-2}{n-1}$. It says that for the
Dicke state $|(n/2),n\rangle$, the $n$-tangle $\tau_{1...n}$ is greater than
the residual entanglement $\tau_{1(2...n)}$ and the difference is given by
$\frac{1}{n-1}$.
### 3.2 A necessary and sufficient condition for the vanishing of the
concurrence $C_{1(2...n)}$
For the state $|\psi\rangle=\sum_{i=0}^{2^{n}-1}a_{i}|i\rangle$ of $n$ qubits,
the concurrence $C_{1(2...n)}$ can be written as
$C_{1(2...n)}^{2}=4\sum_{0\leq i<j\leq
2^{n-1}-1}|a_{i}a_{j+2^{n-1}}-a_{i+2^{n-1}}a_{j}|^{2}.$ (3.2)
The right hand side of Eq. (3.2) turns out to be the sum of squared moduli
(see Appendix B for the proof).
In view of Eq. (3.2), any $n$-qubit concurrence
$C_{1(2...n)}$ vanishes if and only if the state is a product of a state of
one qubit and a state of $(n-1)$ qubits, i.e., the state is of the form
$|\phi\rangle_{1}\otimes$ $|\varphi\rangle_{2...n}$ (see Appendix B for the
proof). This allows one to understand how the concurrence $C_{1(2...n)}$
measures the entanglement of a state. In other words, the concurrence
$C_{1(2...n)}$ is always positive unless the state is a product of a state of
one qubit and a state of $(n-1)$ qubits. In particular, this is true for any
entangled state of any $n$ qubits. That is, there exist $i$ and $j$ with
$0\leq i<j\leq 2^{n-1}-1$, such that $a_{i}a_{j+2^{n-1}}\neq
a_{i+2^{n-1}}a_{j}$. It is worthwhile pointing out that the $n$-tangle
vanishes for some entangled states [16].
### 3.3 The concurrence $C_{1(2...n)}\geq$ the $n$-tangle $\tau_{1...n}$
A closer examination of Eqs. (3.2) and (2.5) reveals that for even $n$ qubits,
the concurrence $C_{1(2...n)}$ is equal to or greater than the $n$-tangle
$\tau_{1...n}$ (see Appendix B for the proof). We immediately have the
following corollaries:
(1). For any state $|\psi\rangle$ of even $n$ qubits, if the concurrence C
vanishes then, clearly, so does the n-tangle.
(2). If the n-tangle $\tau_{1...n}$ of even $n$ qubits is positive, then the
concurrence $C_{1(2...n)}$ is also positive.
### 3.4 The residual entanglement is a partial measure for product states
In this section, we show that for product state
$|\psi\rangle_{1...l}\otimes|\phi\rangle_{(l+1)...n}$ of any $n$ qubits, where
$|\psi\rangle$ is the state of the first $l$ qubits, the residual entanglement
$\tau_{1(2...n)}$ for the product state is reduced to the residual
entanglement $\tau_{1(2...l)}$ for the state $|\psi\rangle$. First we observe
that
$\rho_{1}(|\psi\rangle\otimes|\phi\rangle\langle\psi|\otimes\langle\phi|)\allowbreak=\rho_{1}(|\psi\rangle\langle\psi|)$.
By the definition of the concurrence,
$C_{1(2...n)}(|\psi\rangle\otimes|\phi\rangle)=C_{1(2...l)}(|\psi\rangle).$
(3.3)
That is, the concurrence $C_{1(2...n)}$ for the product state
$|\psi\rangle\otimes|\phi\rangle$ is just the concurrence $C_{1(2...l)}$ for
the state $|\psi\rangle$. It tells us that the concurrence $C_{1(2...n)}$ only
measures the entanglement of the state $|\psi\rangle$.
Likewise, the concurrence $C_{1k}$ for the state
$|\psi\rangle_{1...l}\otimes|\phi\rangle_{(l+1)...n}$ is just the concurrence
$C_{1k}$ for the state $|\psi\rangle_{1...l}$, $k=2,...,l$. Since qubits 1 and
$k$ are not entangled, the concurrence $C_{1k}$ for the state
$|\psi\rangle_{1...l}\otimes|\phi\rangle_{(l+1)...n}\ $vanishes for $k>l$.
This can be seen as follows. After some algebra, we find
$\rho_{1(l+1)}\tilde{\rho}_{1(l+1)}=cI$, where c is a constant. It implies
that the concurrence $C_{1(l+1)}=0$. In a similar manner we can show that the
concurrence $C_{1k}=0$ for $k\geq(l+2)$. This leads to
$\displaystyle C_{1k}(|\psi\rangle_{1...l}\otimes|\phi\rangle_{(l+1)...n})$
(3.6) $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{rc}C_{1k}(|\psi\rangle_{1...l}),&2\leq
k\leq l\\\ 0,&l<k\leq n\text{.}\end{array}\right.$
Eqs. (3.3) and (3.6) together with the definition of the residual entanglement
give
$\tau_{1(2...n)}(|\psi\rangle\otimes|\phi\rangle)=\tau_{1(2...l)}(|\psi\rangle).$
(3.7)
This shows that the residual entanglement $\tau_{1(2...n)}$ for the product
state $|\psi\rangle\otimes|\phi\rangle$ is reduced to the residual
entanglement $\tau_{1(2...l)}$ for the state $|\psi\rangle$. It tells us that
$\tau_{1(2...n)}$ only measures the residual entanglement of the state
$|\psi\rangle$.
However, for the product state
$|\psi\rangle_{1...l}\otimes|\phi\rangle_{(l+1)...n}$ of even $n$ qubits, when
$|\psi\rangle$ is a state of even $n$ qubits, the n-tangle is multiplicative.
That is,
$\tau_{12...n}(|\psi\rangle\otimes|\phi\rangle)=\tau_{12...l}(|\psi\rangle)\times\tau_{12...(n-l)}(|\phi\rangle)$
[16].
The following example shows that the residual entanglement $\tau_{1(2...n)}$
is not the n-way entanglement measure. For the product state
$(\frac{1}{\sqrt{2}}(|000\rangle+|111\rangle))^{\otimes 2k}$, by Eq. (3.7),
the residual entanglement $\tau_{1(2...(6k))}=1$. It is worth noting that the
$n$-tangle is not the n-way entanglement measure either [3].
## 4 Conclusion
In summary, we have shown that the $n$-tangle is the square of the SLOCC
polynomial invariant of degree 2. We have found that the two generalizations,
namely the $n$-tangle and the residual entanglement of $n$-qubits, are
different for any even $n\geq 4$ qubits. We have also proven that the
concurrence $C_{1(2...n)}$ vanishes if and only if the state is a product of a
state of one qubit and a state of $(n-1)$ qubits. In other words, the
concurrence $C_{1(2...n)}$ is always positive unless the state is a product of
a state of one qubit and a state of $(n-1)$ qubits. Furthermore, we have
argued that the concurrence $C_{1(2...n)}$ is equal to or greater than the
$n$-tangle, and that the residual entanglement is a partial measure for
product states of any $n$ qubits.
## Appendix A The $n$-tangle is the square of the SLOCC polynomial invariant.
(A). Proof of $S=2S_{0}^{2}$
In view of Eq. (1.3), we only need to consider
$\beta_{i}=\overline{\alpha_{i}}$, $\delta_{i}=\overline{\gamma_{i}}$, $i=1$,
…, $(n-1)$, $\gamma_{n}=\overline{\alpha_{n}}$, and
$\delta_{n}=\overline{\beta_{n}}$. Thus, Eq. (LABEL:n-tangle-1) becomes
$\displaystyle S$ $\displaystyle=$
$\displaystyle\sum(a_{\alpha_{1}...\alpha_{n-1}\alpha_{n}}a_{\overline{\alpha_{1}}...\overline{\alpha_{n-1}}\beta_{n}}a_{\gamma_{1}...\gamma_{n-1}\overline{\alpha_{n}}}a_{\overline{\gamma_{1}}...\overline{\gamma_{n-1}}\overline{\beta_{n}}}$
(A1)
$\displaystyle\times\epsilon_{\alpha_{1}\overline{\alpha_{1}}}\epsilon_{\alpha_{2}\overline{\alpha_{2}}}...\epsilon_{\alpha_{n-1}\overline{\alpha_{n-1}}}$
$\displaystyle\times\epsilon_{\gamma_{1}\overline{\gamma_{1}}}\epsilon_{\gamma_{2}\overline{\gamma_{2}}}...\times\epsilon_{\gamma_{n-1}\overline{\gamma_{n-1}}}\epsilon_{\alpha_{n}\overline{\alpha_{n}}}\epsilon_{\beta_{n}\overline{\beta_{n}}}).$
We distinguish two cases.
Case 1. $\beta_{n}=\alpha_{n}$.
In this case,
$\epsilon_{\alpha_{n}\overline{\alpha_{n}}}\epsilon_{\beta_{n}\overline{\beta_{n}}}=1$.
Let
$\displaystyle S^{\prime}$ $\displaystyle=$
$\displaystyle\sum_{\gamma_{1}...\gamma_{n-1}}(a_{\gamma_{1}...\gamma_{n-1}\overline{\alpha_{n}}}a_{\overline{\gamma_{1}}...\overline{\gamma_{n-1}}\overline{\alpha_{n}}}$
(A2)
$\displaystyle\times\epsilon_{\gamma_{1}\overline{\gamma_{1}}}\epsilon_{\gamma_{2}\overline{\gamma_{2}}}...\epsilon_{\gamma_{n-1}\overline{\gamma_{n-1}}}).$
Then, Eq. (A1) becomes
$\displaystyle S$ $\displaystyle=$
$\displaystyle\sum_{\alpha_{1}...\alpha_{n-1}\alpha_{n}}(a_{\alpha_{1}...\alpha_{n-1}\alpha_{n}}a_{\overline{\alpha_{1}}...\overline{\alpha_{n-1}}\alpha_{n}}$
(A3)
$\displaystyle\times\epsilon_{\alpha_{1}\overline{\alpha_{1}}}\epsilon_{\alpha_{2}\overline{\alpha_{2}}}...\epsilon_{\alpha_{n-1}\overline{\alpha_{n-1}}}\times
S^{\prime}).$
To compute $S^{\prime}$, we assume that $\overline{\alpha_{n}}$ is fixed in
$S^{\prime}$. For each term
$t=a_{\gamma_{1}...\gamma_{n-1}\overline{\alpha_{n}}}a_{\overline{\gamma_{1}}...\overline{\gamma_{n-1}}\overline{\alpha_{n}}}\times$
$\epsilon_{\gamma_{1}\overline{\gamma_{1}}}\epsilon_{\gamma_{2}\overline{\gamma_{2}}}...\epsilon_{\gamma_{n-1}\overline{\gamma_{n-1}}}$,
$\ S^{\prime}$ has the term
$t^{\prime}=a_{\overline{\gamma_{1}}...\overline{\gamma_{n-1}}\overline{\alpha_{n}}}a_{\gamma_{1}...\gamma_{n-1}\overline{\alpha_{n}}}\times\epsilon_{\overline{\gamma_{1}}\gamma_{1}}\epsilon_{\overline{\gamma_{2}}\gamma_{2}}...\epsilon_{\overline{\gamma_{n-1}}\gamma_{n-1}}$.
Note that
$\epsilon_{\gamma_{l}\overline{\gamma_{l}}}=-\epsilon_{\overline{\gamma_{l}}\gamma_{l}}$,
$l=1,...,n$. Thus, $t=-t^{\prime}$ and so $S^{\prime}=0$. Hence, $S=0$.
Case 2. $\beta_{n}=\overline{\alpha_{n}}$.
In this case,
$\epsilon_{\alpha_{n}\overline{\alpha_{n}}}\epsilon_{\beta_{n}\overline{\beta_{n}}}=-1$.
Eq. (A1) becomes
$\displaystyle S$ $\displaystyle=$
$\displaystyle-\sum_{\alpha_{1}...\alpha_{n}}[a_{\alpha_{1}...\alpha_{n}}a_{\overline{\alpha_{1}}...\overline{\alpha_{n}}}\times\epsilon_{\alpha_{1}\overline{\alpha_{1}}}...\epsilon_{\alpha_{n-1}\overline{\alpha_{n-1}}}$
(A4)
$\displaystyle\mathcal{\times}\sum_{\gamma_{1}...\gamma_{n-1}}(a_{\gamma_{1}...\gamma_{n-1}\overline{\alpha_{n}}}a_{\overline{\gamma_{1}}...\overline{\gamma_{n-1}}\alpha_{n}}$
$\displaystyle\times\epsilon_{\gamma_{1}\overline{\gamma_{1}}}...\epsilon_{\gamma_{n-1}\overline{\gamma_{n-1}}})].$
Let
$S_{i}=\sum_{\alpha_{1}...\alpha_{n-1}}(a_{\alpha_{1}...\alpha_{n-1}i}a_{\overline{\alpha_{1}}...\overline{\alpha_{n-1}}\overline{\imath}}\times\epsilon_{\alpha_{1}\overline{\alpha_{1}}}...\epsilon_{\alpha_{n-1}\overline{\alpha_{n-1}}}),$
(A5)
where $i=0$, $1$. Thus,
$S=-2S_{0}S_{1}.$ (A6)
Next we verify that $S_{1}=-S_{0}$. By the condition in Eq. (1.3),
$\epsilon_{\alpha_{i}\overline{\alpha_{i}}}=-\epsilon_{\overline{\alpha_{i}}\alpha_{i}}$,
$i=1$, …, $n$. Then,
$\displaystyle S_{1}$ $\displaystyle=$
$\displaystyle\sum_{\alpha_{1}...\alpha_{n-1}}(a_{\alpha_{1}...\alpha_{n-1}1}a_{\overline{\alpha_{1}}...\overline{\alpha_{n-1}}0}$
$\displaystyle\times\epsilon_{\alpha_{1}\overline{\alpha_{1}}}\epsilon_{\alpha_{2}\overline{\alpha_{2}}}...\epsilon_{\alpha_{n-1}\overline{\alpha_{n-1}}})$
$\displaystyle=$
$\displaystyle-\sum_{\alpha_{1}...\alpha_{n-1}}(a_{\overline{\alpha_{1}}...\overline{\alpha_{n-1}}0}a_{\alpha_{1}...\alpha_{n-1}1}$
$\displaystyle\times\epsilon_{\overline{\alpha_{1}}\alpha_{1}}\epsilon_{\overline{\alpha_{2}}\alpha_{2}}...\epsilon_{\overline{\alpha_{n-1}}\alpha_{n-1}})$
$\displaystyle=$ $\displaystyle-S_{0}.$ (A7)
Together the latter two equations yield the desired result.
(B). Proof of $S_{0}=\mathcal{I}^{\ast}(a,n)$
We can rewrite $\mathcal{I}^{\ast}(a,n)$ as
$\mathcal{I}^{\ast}(a,n)=\sum_{l=0}^{2^{n-1}-1}(-1)^{N(l)}a_{2l}a_{(2^{n}-1)-2l}.$
(A8)
Let $l_{n-1}...l_{1}$ be the $(n-1)$-bit binary number of $l$. Then, it
follows from Eq. (A8) that
$\displaystyle\mathcal{I}^{\ast}(a,n)$ $\displaystyle=$
$\displaystyle\sum_{l_{n-1}...l_{2}l_{1}}(-1)^{N(l)}a_{l_{n-1}...l_{1}0}a_{\overline{l_{n-1}}...\overline{l_{1}}1}$
(A9) $\displaystyle=$
$\displaystyle\sum_{l_{n-1}...l_{2}l_{1}}(a_{l_{n-1}...l_{1}0}a_{\overline{l_{n-1}}...\overline{l_{1}}1}$
$\displaystyle\times\epsilon_{l_{1}\overline{l_{1}}}\epsilon_{l_{2}\overline{l_{2}}}...\epsilon_{l_{n-1}\overline{l_{n-1}}})$
$\displaystyle=$ $\displaystyle S_{0}\text{.}$
The second equality follows by noting that
$(-1)^{N(l)}=\epsilon_{l_{1}\overline{l_{1}}}\epsilon_{l_{2}\overline{l_{2}}}...\epsilon_{l_{n-1}\overline{l_{n-1}}}$.
## Appendix B. Concurrence $C_{1(2...n)}$
Result 1. Let the state $|\psi\rangle=\sum_{i=0}^{2^{n}-1}a_{i}|i\rangle$ be
any state of any $n$ qubits. Then
$C_{1(2...n)}^{2}=4\sum_{0\leq i<j\leq
2^{n-1}-1}|a_{i}a_{j+2^{n-1}}-a_{i+2^{n-1}}a_{j}|^{2}.$ (B1)
Proof. By direct calculation we find
$\det\rho_{1}=\sum_{i,j=0}^{2^{n-1}-1}a_{i}a_{j+2^{n-1}}(a_{i}^{\ast}a_{j+2^{n-1}}^{\ast}-a_{i+2^{n-1}}^{\ast}a_{j}^{\ast})$,
where $a_{i}^{\ast}$ is the complex conjugate of $a_{i}$. By switching $i$ and
$j$, the term
$a_{i}a_{j+2^{n-1}}(a_{i}^{\ast}a_{j+2^{n-1}}^{\ast}-a_{i+2^{n-1}}^{\ast}a_{j}^{\ast})$
becomes
$a_{j}a_{i+2^{n-1}}(a_{j}^{\ast}a_{i+2^{n-1}}^{\ast}-a_{j+2^{n-1}}^{\ast}a_{i}^{\ast})$.
Then
$\displaystyle
a_{i}a_{j+2^{n-1}}(a_{i}^{\ast}a_{j+2^{n-1}}^{\ast}-a_{i+2^{n-1}}^{\ast}a_{j}^{\ast})$
(B2) $\displaystyle+$ $\displaystyle
a_{j}a_{i+2^{n-1}}(a_{j}^{\ast}a_{i+2^{n-1}}^{\ast}-a_{j+2^{n-1}}^{\ast}a_{i}^{\ast})$
$\displaystyle=$ $\displaystyle|a_{i}a_{j+2^{n-1}}-a_{i+2^{n-1}}a_{j}|^{2}.$
When $i=j$, the right side of Eq. (B2) vanishes. So, $\det\rho_{1}=\sum_{0\leq
i<j\leq 2^{n-1}-1}|a_{i}a_{j+2^{n-1}}-a_{i+2^{n-1}}a_{j}|^{2}$. Since
$C_{1(2...n)}^{2}=4\det\rho_{1}$ by definition, the desired result follows.
Result 2. For the state $|\psi\rangle$ of any $n$ qubits,
$C_{1(2...n)}=0$ if and only if $|\psi\rangle$ is a product of a state of one
qubit and a state of $(n-1)$ qubits, i.e.,
$|\psi\rangle=|\phi\rangle_{1}\otimes$ $|\varphi\rangle_{2...n}$.
Proof. Let
$|\psi\rangle=\mathop{\displaystyle\sum}\limits_{i=0}^{2^{n}-1}a_{i}|i\rangle$.
It is assumed that $C_{1(2...n)}=0$. Hence, by Eq. (B1),
$a_{i}a_{j+2^{n-1}}=a_{i+2^{n-1}}a_{j},$ (B3)
where $0\leq i<j\leq 2^{n-1}-1$. We distinguish two cases.
Case 1. $\mathop{\displaystyle\sum}\limits_{i=0}^{2^{n-1}-1}|a_{i}|^{2}=0$. It
is straightforward to verify that
$|\psi\rangle=|1\rangle_{1}\otimes\mathop{\displaystyle\sum}\limits_{j=0}^{2^{n-1}-1}a_{j+2^{n-1}}|j\rangle_{2...n}$.
Case 2. $\mathop{\displaystyle\sum}\limits_{i=0}^{2^{n-1}-1}|a_{i}|^{2}\neq
0$. Without loss of generality, assume that $a_{0}\neq 0$. Let
$\alpha=\frac{a_{2^{n-1}}}{a_{0}}$. Then,
$a_{2^{n-1}}=\alpha a_{0}\text{.}$ (B4)
Letting $i=0$ in Eq. (B3), we obtain
$a_{0}a_{j+2^{n-1}}=a_{2^{n-1}}a_{j}\text{,}$ (B5)
where $j=1,2,...,2^{n-1}-1$. Substituting Eq. (B4) into Eq. (B5), we see that
$a_{j+2^{n-1}}=\alpha a_{j}\text{,}$ (B6)
where $j=1,2,...,2^{n-1}-1$. From Eqs. (B4) and (B6), $|\psi\rangle$ can be
rewritten as
$|\psi\rangle=(|0\rangle_{1}+\alpha|1\rangle_{1})\otimes\mathop{\displaystyle\sum}\limits_{j=0}^{2^{n-1}-1}a_{j}|j\rangle_{2...n}$.
Conversely, if $|\psi\rangle=|\phi\rangle_{1}\otimes$
$|\varphi\rangle_{2...n}$, then it is readily verified that $C_{1(2...n)}=0$.
Result 3. For even $n$ qubits, the concurrence
$C_{1(2...n)}$ is equal to or greater than the n-tangle $\tau_{1...n}$.
Proof. We rewrite Eq. (2.5) as
$\displaystyle\mathcal{I}^{\ast}(a,n)=\sum_{k=0}^{2^{n-2}-1}[(-1)^{N(k)}$
$\displaystyle\times(a_{k}a_{2^{n}-1-k}-a_{2^{n-1}-1-k}a_{2^{n-1}+k})].$ (B7)
To prove this, we note that $\mathcal{I}^{\ast}(a,n)$ can be written as (see
[16])
$\mathcal{I}^{\ast}(a,n)=\sum_{k=0}^{2^{n-1}-1}(-1)^{N(k)}a_{k}a_{2^{n}-1-k}.$
(B8)
From Eq. (B8),
$\displaystyle\mathcal{I}^{\ast}(a,n)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{2^{n-2}-1}(-1)^{N(k)}a_{k}a_{2^{n}-1-k}$ (B9)
$\displaystyle+$
$\displaystyle\sum_{k=2^{n-2}}^{2^{n-1}-1}(-1)^{N(k)}a_{k}a_{2^{n}-1-k}.$
Let $k=2^{n-1}-1-i$, in which case $N(k)+N(i)=n-1$. Then, the second sum of
the above equation becomes
$-\sum_{i=0}^{2^{n-2}-1}(-1)^{N(i)}a_{2^{n-1}-1-i}a_{2^{n-1}+i}$. Thus, Eq.
(B7) holds.
For any $n$ qubits, we may write Eq. (B1) as
$\displaystyle C_{1(2...n)}^{2}$ $\displaystyle=$ $\displaystyle
4\biggl{\\{}\sum_{\begin{subarray}{c}0\leq i\leq 2^{n-2}-1\\\ i<j\leq
2^{n-1}-1\\\ j\neq
2^{n-1}-1-i\end{subarray}}|a_{i}a_{j+2^{n-1}}-a_{i+2^{n-1}}a_{j}|^{2}$
$\displaystyle+\sum_{2^{n-2}\leq i<j\leq
2^{n-1}-1}|a_{i}a_{j+2^{n-1}}-a_{i+2^{n-1}}a_{j}|^{2}$
$\displaystyle+\sum_{i=0}^{2^{n-2}-1}|a_{i}a_{2^{n}-1-i}-a_{2^{n-1}-1-i}a_{2^{n-1}+i}|^{2}\biggr{\\}}.$
For even $n$ qubits, from Eq. (B7) it holds that
$\tau_{1...n}\leq
4\biggl{[}\sum_{k=0}^{2^{n-2}-1}|a_{k}a_{2^{n}-1-k}-a_{2^{n-1}-1-k}a_{2^{n-1}+k}|\biggr{]}^{2}.$
(B11)
Let, for brevity,
$Z_{k}=|a_{k}a_{2^{n}-1-k}-a_{2^{n-1}-1-k}a_{2^{n-1}+k}|$ and
$P(i,j)=a_{i}a_{j+2^{n-1}}-a_{i+2^{n-1}}a_{j}$. To show
$C_{1(2...n)}^{2}\geq\tau_{1...n}$, from Eqs. (LABEL:concur_3) and (B11), it
is enough to prove
$\displaystyle\sum_{\begin{subarray}{c}0\leq i\leq 2^{n-2}-1\\\ i<j\leq
2^{n-1}-1\\\ j\neq 2^{n-1}-1-i\end{subarray}}|P(i,j)|^{2}+\sum_{2^{n-2}\leq
i<j\leq 2^{n-1}-1}|P(i,j)|^{2}$ (B12) $\displaystyle\geq$ $\displaystyle
2\sum_{0\leq k<m\leq 2^{n-2}-1}Z_{k}Z_{m}.$
Observe that in Eq. (B12), the first, second, and third sums contain $3\times
2^{n-3}(2^{n-2}-1)$ different terms $|P(i,j)|^{2}$, $2^{n-3}(2^{n-2}-1)$
different terms$\ |P(i,j)|^{2}$, and $2^{n-3}(2^{n-2}-1)$ different terms
$Z_{k}Z_{m}$, respectively. Next we show that for each term $Z_{k}Z_{m}$ on
the right side of Eq. (B12), there exist four different corresponding terms
$|P(i,j)|^{2}$ on the left side of Eq. (B12) such that their sum is equal to
or greater than $2Z_{k}Z_{m}$.
Given $Z_{k}Z_{m}$ with $0\leq k<m\leq 2^{n-2}-1$. We first choose two
different terms $|P(k,2^{n-1}-1-m)|^{2}$ and $|P(m,2^{n-1}-1-k)|^{2}$ from the
first sum in Eq. (B12). It is trivial that
$\displaystyle|P(k,2^{n-1}-1-m)|^{2}+|P(m,2^{n-1}-1-k)|^{2}$
$\displaystyle\geq$ $\displaystyle 2|P(k,2^{n-1}-1-m)||P(m,2^{n-1}-1-k)|.$
We then choose the term $|P(k,m)|^{2}$ from the first sum in Eq. (B12) and the
term $|P(2^{n-1}-1-m,2^{n-1}-1-k)|^{2}$ from the second sum in Eq. (B12). It
is trivial that
$\displaystyle|P(k,m)|^{2}+|P(2^{n-1}-1-m,2^{n-1}-1-k)|^{2}$
$\displaystyle\geq$ $\displaystyle 2|P(k,m)||P(2^{n-1}-1-m,2^{n-1}-1-k)|.$
Now, using the fact that $|x|+|y|\geq|x-y|$, from Eqs. (LABEL:ineq-1) and
(LABEL:ineq-2), we establish the inequality
$\displaystyle|P(k,2^{n-1}-1-m)||P(m,2^{n-1}-1-k)|$ (B15) $\displaystyle+$
$\displaystyle|P(k,m)||P(2^{n-1}-1-m,2^{n-1}-1-k)|$ $\displaystyle\geq$
$\displaystyle Z_{k}Z_{m},$
and this implies the desired result Eq. (B12). This completes the proof.
## References
* [1] W.K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
* [2] V. Coffman et al., Phys. Rev. A 61, 052306 (2000).
* [3] A. Wong and N. Christensen, Phys. Rev. A 63, 044301(2001).
* [4] P. Rungta et al., Phys. Rev. A 64, 042315 (2001).
* [5] L. Aolita et al., Phys. Rev. Lett. 97, 050501 (2006).
* [6] A. Osterloh et al., Nature (london) 416, 608 (2002).
* [7] W. D$\ddot{u}$r, G.Vidal and J.I. Cirac, Phys. Rev. A 62, 062314 (2000).
* [8] A. Sudbery, J. Phys. A: Math. Gen 34, 643 (2001) M. Grass et al., Phys. Rev. A 58, 1853 (1998); N. Linden et al., Phys. Rev. Lett. 83, 243 (1999).
* [9] Jean-Gabriel Luque et al., Phys. Rev. A 67, 042303 (2003); Jean-Gabriel Luque et al., Math. Struct. in Comp Science (2007) vol. 17, pp1133-1157, 2007.
* [10] M. S. Leifer et al., Phys. Rev. A 69, 052304 (2004).
* [11] P. Lévay, J. Phys. A: Math. Gen. 39, 9533-9545 (2006).
* [12] Dafa Li et al., Phys. Lett. A 359 428-437 (2006).
* [13] Dafa Li et al., Phys. Rev. A 76, 032304 (2007); Dafa Li et al., e-print quant-ph/0704.2087.
* [14] D. Z. Dokovic and A. Osterloh, J. Math. Phys. 50, 033509 (2009).
* [15] A. Osterloh and J. Siewert, Phys. Rev. A 72, 012337 (2005); A. Osterloh and J. Siewert, Int. J. Quant. Inf. 4, 531 (2006).
* [16] Dafa Li et al., J. Math. Phys. 50, 012104 (2009).
* [17] F. Verstraete et al., Phys. Rev. A 68, 012103 (2003).
* [18] Chang-shui Yu and He-shan Song, Phys. Rev. A 71, 042331 (2005).
* [19] T. J. Osborne and F. Verstraete, Phys. Rev. Lett. 96, 220503 (2006).
* [20] Youg-Cheng Ou and Heng Fan, Phys. Rev. A 75, 062308 (2007).
* [21] J.K. Stockton et al., Phys. Rev. A 67, 022112 (2003).
|
arxiv-papers
| 2010-03-24T23:10:41 |
2024-09-04T02:49:09.232090
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "X. Li and D. Li",
"submitter": "Dafa Li",
"url": "https://arxiv.org/abs/1003.4774"
}
|
1003.4812
|
# Bisimulation Relations Between Automata,
Stochastic Differential Equations and Petri Nets
Mariken H.C. Everdij Henk A.P. Blom National Aerospace Laboratory NLR
Amsterdam, Netherlands everdij@nlr.nl blom@nlr.nl
###### Abstract
Two formal stochastic models are said to be bisimilar if their solutions as a
stochastic process are probabilistically equivalent. Bisimilarity between two
stochastic model formalisms means that the strengths of one stochastic model
formalism can be used by the other stochastic model formalism. The aim of this
paper is to explain bisimilarity relations between stochastic hybrid automata,
stochastic differential equations on hybrid space and stochastic hybrid Petri
nets. These bisimilarity relations make it possible to combine the formal
verification power of automata with the analysis power of stochastic
differential equations and the compositional specification power of Petri
nets. The relations and their combined strengths are illustrated for an air
traffic example.
## 1 Introduction
Two formal stochastic models are said to be _bisimilar_ if their solutions as
a stochastic process (i.e. their _executions_) are probabilistically
equivalent [9, 24]. Bisimilarity relations between formal stochastic models
are very useful to study since they allow one stochastic model to take
advantage of the strengths of the other stochastic model. The aim of this
paper is to show bisimulation relations between three different stochastic
modelling formalisms: stochastic hybrid automata, stochastic differential
equations on hybrid space, and stochastic hybrid Petri nets. These
bisimulation relations make it possible to combine the formal verification
power of automata with the analysis power of stochastic differential equations
and the compositional specification power of Petri nets.
For the stochastic automata formalism, we take the _general stochastic hybrid
system_ (GSHS) theoretical setting developed by [8]. A GSHS is a hybrid
automaton defined on a hybrid state space. This hybrid state space consists of
a countable set of discrete modes, and per discrete mode a Euclidean subset.
Per discrete mode, a stochastic differential equation (SDE) is defined. Two
additional GSHS elements are a jump rate function and a GSHS transition
measure. The execution of these elements provides a stochastic process that
follows the solution of the SDE connected to the initial discrete mode. After
a time period, defined by the jump rate function, the process state may
spontaneously jump to another mode, defined by the GSHS transition measure. A
jump may also be forced if the process state hits the boundary of the
Euclidean subset. The GSHS execution is referred to as _general stochastic
hybrid process_ (GSHP). One of the main strengths of the automata formalism is
the availability of formal verification tools.
For the hybrid stochastic differential equations formalism we take the _hybrid
stochastic differential equations_ (HSDE) theoretical setting developed in a
series of complementary studies [2, 3, 21, 22]. A HSDE consists of a sequence
of SDEs on a hybrid state space, driven by a Poisson random measure. When the
Poisson random measure generates a multivariate point, a spontaneous jump
occurs. A jump may also be forced if the process state hits the boundary of a
Euclidean subset. The HSDE solution process is referred to as _general
stochastic hybrid process_ (GSHP). In [17] it is shown that whereas the GSHS
formalism is at some points more general than HSDE (for GSHS the dimension of
the Euclidean subset may depend on the discrete mode; for HSDE this dimension
is fixed), HSDE has the advantage of an established semi-martingale property
and includes the coverage of jump-linear systems.
For the stochastic hybrid Petri nets formalism, we take _stochastically and
dynamically coloured Petri nets_ (SDCPN) developed in a series of studies by
[14, 15, 16]. A Petri net has _places_ (circles), which model possible
discrete states or conditions, and which may contain one or more _tokens_
(dots), modelling which of these states are current. The places are connected
by _transitions_ (squares), which model state switches by removing input
tokens and producing output tokens along _arcs_ (arrows). In SDCPN, the tokens
have Euclidean-valued colours that follow SDEs. Some of the transitions remove
and produce tokens spontaneously, other transitions are forced and occur when
the colours of their input tokens reach the boundary of a Euclidean subset.
The collection of token colours in all places forms a _general stochastic
hybrid process_ (GSHP). The specific strength of SDCPN is their compositional
specification power, which makes available a hierarchical modelling approach
that separates local modelling issues from global modelling issues. This is
illustrated for a large distributed example in air traffic management [18],
which covers many distributed agents each of which interacts in a dynamic way
with the others. Other typical Petri net features are concurrency and
synchronisation mechanism, hierarchical and modular construction, and natural
expression of causal dependencies, in combination with graphical and
equational representation.
The aim of this paper is to illustrate the relations between SDCPN, GSHP, HSDE
and GSHS which show that SDCPN, GSHS and HSDE are bisimilar. This means that
if we take the elements of any one of these formalisms, we can construct the
elements of another formalism in such a way that their associated GSHPs are
probabilistically equivalent. Fig. 1 shows the relations between the
formalisms, and the key tools available for each of them.
Figure 1: Relationship between SDCPN, GSHS, GSHP and HSDE, and their key
properties and advantages. The [B] arrow is established in [2]. The [BL] arrow
is established in [8]. The [E1] arrows are established in [16]. The [E2]
arrows are established in [17].
With these relations, the properties and advantages of the various approaches
come within reach of each other. The compositional specification power of
SDCPN makes it relatively easy to develop a model for a complex system with
multiple interactions. Subsequently, in the analysis stage three alternative
approaches can be taken. The first is direct execution of SDCPN and evaluation
through e.g. Monte Carlo simulation. The second is mapping the SDCPN into a
GSHS and evaluating its execution, with the advantages of connection to formal
methods in automata theory and to optimal control theory [7]. The third is
mapping the SDCPN into HSDE and evaluating its solution, with the advantages
of stochastic analysis for semi-martingales [11, 12]. With the GSHP resulting
from any of these three means, properties become available such as convergence
of discretisation, existence of limits, existence of event probabilities,
strong Markov properties, and reachability analysis [8, 10, 13].
The organisation of this paper is as follows. Section 2 defines SDCPN and the
related SDCPN process. Section 3 presents an example SDCPN model for a simple
but illustrative air traffic situation. Section 4 defines GSHS and illustrates
how the example SDCPN can be mapped to a bisimilar GSHS. Section 5 defines
HSDE and illustrates how the example SDCPN can be mapped to a bisimilar HSDE.
Section 6 gives conclusions.
## 2 SDCPN
This section outlines _stochastically and dynamically coloured Petri net_
(SDCPN). For a more formal definition, we refer to [17].
#### Definition 2.1
(Stochastically and dynamically coloured Petri net.) _An SDCPN is a collection
of elements $({\cal P}$, ${\cal T}$, ${\cal A}$, ${\cal N}$, ${\cal S}$,
${\cal C}$, ${\cal I}$, ${\cal V}$, ${\cal W}$, ${\cal G}$, ${\cal D}$, ${\cal
F})$, together with an SDCPN execution prescription which makes use of a
sequence $\\{U_{i};i=0,1,\ldots\\}$ of independent uniform $U[0,1]$ random
variables, of independent sequences of mutually independent standard Brownian
motions $\\{B_{t}^{i,P};i=1,2,\ldots\\}$ of appropriate dimensions, one
sequence for each place $P$, and of five rules R0–R4 that solve enabling
conflicts._
### 2.1 SDCPN elements
The SDCPN elements (${\cal P}$, ${\cal T}$, ${\cal A}$, ${\cal N}$, ${\cal
S}$, ${\cal C}$, ${\cal I}$, ${\cal V}$, ${\cal W}$, ${\cal G}$, ${\cal D}$,
${\cal F}$) are defined as follows:
* •
${\cal P}$ is a finite set of places.
* •
${\cal T}$ is a finite set of transitions which consists of 1) a set ${\cal
T}_{G}$ of guard transitions, 2) a set ${\cal T}_{D}$ of delay transitions and
3) a set ${\cal T}_{I}$ of immediate transitions.
* •
${\cal A}$ is a finite set of arcs which consists of 1) a set ${\cal A}_{O}$
of ordinary arcs, 2) a set ${\cal A}_{E}$ of enabling arcs and 3) a set ${\cal
A}_{I}$ of inhibitor arcs.
* •
${\cal N}:{\cal A}\to{\cal P}\times{\cal T}\cup{\cal T}\times{\cal P}$ is a
node function which maps each arc $A\in{\cal A}$ to a pair of ordered nodes
${\cal N}(A)$, where a node is a place or a transition.
* •
${\cal S}\subset\\{\mathbb{R}^{0},\mathbb{R}^{1},\mathbb{R}^{2},\ldots\\}$ is
a finite set of colour types, with $\mathbb{R}^{0}\triangleq\emptyset$.
* •
${\cal C}:{\cal P}\to{\cal S}$ is a colour type function which maps each place
$P\in{\cal P}$ to a specific colour type. Each token in $P$ is to have a
colour in ${\cal C}(P)$. If ${\cal C}(P)=\mathbb{R}^{0}$ then a token in $P$
has no colour.
* •
${\cal I}$ is a probability measure, which defines the initial marking of the
net: for each place it defines a number $\geq 0$ of tokens initially in it and
it defines their initial colours.
* •
${\cal V}=\\{{\cal V}_{P};P\in{\cal P},{\cal C}(P)\neq\mathbb{R}^{0}\\}$ is a
set of token colour functions. For each place $P\in{\cal P}$ for which ${\cal
C}(P)\neq\mathbb{R}^{0}$, it contains a function ${\cal V}_{P}:{\cal
C}(P)\to{\cal C}(P)$ that defines the drift coefficient of a differential
equation for the colour of a token in place $P$.
* •
${\cal W}=\\{{\cal W}_{P};P\in{\cal P},{\cal C}(P)\neq\mathbb{R}^{0}\\}$ is a
set of token colour matrix functions. For each place $P\in{\cal P}$ for which
${\cal C}(P)\neq\mathbb{R}^{0}$, it contains a measurable mapping ${\cal
W}_{P}:{\cal C}(P)\to\mathbb{R}^{n(P)\times h(P)}$ that defines the diffusion
coefficient of a stochastic differential equation for the colour of a token in
place $P$, where $h:{\cal P}\rightarrow\mathbb{N}$ and $n:{\cal
P}\rightarrow\mathbb{N}$ is such that ${\cal C}(P)=\mathbb{R}^{n(P)}$. It is
assumed that ${\cal W}_{P}$ and ${\cal V}_{P}$ satisfy conditions that ensure
a probabilistically unique solution of each stochastic differential equation.
* •
${\cal G}=\\{{\cal G}_{T};T\in{\cal T}_{G}\\}$ is a set of transition guards.
For each $T\in{\cal T}_{G}$, it contains a transition guard ${\cal G}_{T}$,
which is an open Euclidean subset with boundary $\partial{\cal G}_{T}$.
* •
${\cal D}=\\{{\cal D}_{T};T\in{\cal T}_{D}\\}$ is a set of transition delay
rates. For each $T\in{\cal T}_{D}$, it contains a locally integrable
transition delay rate ${\cal D}_{T}$.
* •
${\cal F}=\\{{\cal F}_{T};T\in{\cal T}\\}$ is a set of firing measures. For
each $T\in{\cal T}$, it contains a firing measure ${\cal F}_{T}$, which
generates the number and colours of the tokens produced when transition $T$
fires, given the value of the vector that collects all input tokens: For each
output arc, zero or one token is produced. For each fixed $H$, ${\cal
F}_{T}(H;\cdot)$ is measurable. For any $c$, ${\cal F}_{T}(\cdot;c)$ is a
probability measure.
For the places, transitions and arcs, the graphical notation is as in Figure
2.
Figure 2: Graphical notation for places, transitions and arcs in an SDCPN
### 2.2 SDCPN execution
The execution of an SDCPN provides a series of increasing stopping times,
$\\{\tau_{i};i=0,1,\ldots\\}$, $\tau_{0}=0$, with for
$t\in(\tau_{k},\tau_{k+1})$ a fixed number of tokens per place and per token a
colour which is the solution of a stochastic differential equation.
#### Initiation.
The probability measure ${\cal I}$ characterises an initial marking at
$\tau_{0}$, i.e. it gives each place $P\in{\cal P}$ zero or more tokens and
gives each token in $P$ a colour in ${\cal C}(P)$, i.e. a Euclidean-valued
vector.
#### Token colour evolution.
For each token in each place $P$ for which ${\cal C}(P)\neq\mathbb{R}^{0}$: if
the colour of this token is equal to $C_{0}^{P}$ at time $t=\tau_{0}$, and if
this token is still in this place at time $t>\tau_{0}$, then the colour
$C_{t}^{P}$ of this token equals the probabilistically unique solution of the
stochastic differential equation $dC_{t}^{P}={\cal V}_{P}(C_{t}^{P})dt+{\cal
W}_{P}(C_{t}^{P})dB_{t}^{i,P}$ with initial condition
$C_{\tau_{0}}^{P}=C_{0}^{P}$, and with $\\{B_{t}^{i,P}\\}$ an
$h(P)$-dimensional standard Brownian motion. Each token in a place for which
${\cal C}(P)=\mathbb{R}^{0}$ remains without colour.
#### Transition enabling.
A transition $T$ is pre-enabled if it has at least one token per incoming
ordinary and enabling arc in each of its input places and has no token in
places to which it is connected by an inhibitor arc. For each transition $T$
that is pre-enabled at $\tau_{0}$, consider one token per ordinary and
enabling arc in its input places and write $C_{t}^{T}$, $t\geq\tau_{0}$, as
the column vector containing the colours of these tokens; $C_{t}^{T}$ evolves
through time according to its corresponding token colour functions. If this
vector is not unique (i.e., if one input place contains several tokens per
arc), all possible such vectors are executed in parallel. A transition $T$ is
enabled if it is pre-enabled and a second requirement holds true. For
$T\in{\cal T}_{I}$, the second requirement automatically holds true at the
time of pre-enabling. For $T\in{\cal T}_{G}$, the second requirement holds
true when $C_{t}^{T}\in\partial{\cal G}_{T}$. For $T\in{\cal T}_{D}$, the
second requirement holds true at $t=\tau_{0}+\sigma_{1}^{T}$, where
$\sigma_{1}^{T}$ is generated from a probability distribution function
$D_{T}(t-\tau_{0})=1-\mbox{exp}(-\int_{\tau_{0}}^{t}{\cal
D}_{T}(C_{s}^{T})ds)$. A Uniform random variable $U_{i}$ is used to determine
this $\sigma_{1}^{T}$. In the case of competing enablings, the following rules
apply:
* R0
The firing of an immediate transition has priority over the firing of a guard
or a delay transition.
* R1
If one transition becomes enabled by two or more sets of input tokens at
exactly the same time, and the firing of any one set will not disable one or
more other sets, then it will fire these sets of tokens independently, at the
same time.
* R2
If one transition becomes enabled by two or more sets of input tokens at
exactly the same time, and the firing of any one set disables one or more
other sets, then the set that is fired is selected randomly, with the same
probability for each set.
* R3
If two or more transitions become enabled at exactly the same time and the
firing of any one transition will not disable the other transitions, then they
will fire at the same time.
* R4
If two or more transitions become enabled at exactly the same time and the
firing of any one transition disables some other transitions, then each
combination of transitions that can fire independently without leaving enabled
transitions gets the same probability of firing.
#### Transition firing.
If $T$ is enabled, suppose this occurs at time $\tau_{1}$ and in a particular
vector of token colours $C_{\tau_{1}}^{T}$, it removes one token per ordinary
input arc corresponding with $C_{\tau_{1}}^{T}$ from each of its input places
(i.e. tokens are not removed along enabling arcs). Next, $T$ produces zero or
one token along each output arc: If $(e_{\tau_{1}}^{T},a_{\tau_{1}}^{T})$ is a
random hybrid vector generated from probability measure ${\cal
F}_{T}(\cdot;C_{\tau_{1}}^{T})$ (by making use of a Uniform random variable
$U_{i}$), then vector $e_{\tau_{1}}^{T}$ is a vector of zeros and ones, where
the $i$th vector element corresponds with the $i$th outgoing arc of transition
$T$. An output place gets a token iff it is connected to an arc that
corresponds with a vector element 1. Moreover, $a_{\tau_{1}}^{T}$ specifies
the colours of the produced tokens.
#### Execution from first transition firing onwards.
At $t=\tau_{1}$, zero or more transitions are pre-enabled (if this number is
zero, no transitions will fire anymore). If these include immediate
transitions, then these are fired without delay, but with use of rules R0–R4.
If after this, still immediate transitions are enabled, then these are also
fired, and so forth, until no more immediate transitions are enabled. Next,
the SDCPN is executed in the same way as described above for the situation
from $\tau_{0}$ onwards.
### 2.3 SDCPN stochastic process
The marking of the SDCPN is given by the numbers of tokens in the places and
the associated colour values of these tokens and can be mapped to a
probabilistically unique SDCPN stochastic process $\\{M_{t},C_{t}\\}$ as
follows: For any $t\geq\tau_{0}$, let a token distribution be characterised by
the vector $M_{t}^{\prime}=(M_{1,t}^{\prime},\ldots,M_{|{\cal
P}|,t}^{\prime})$, where $M_{i,t}^{\prime}\in\mathbb{N}$ denotes the number of
tokens in place $P_{i}$ at time $t$ and $1,\ldots,|{\cal P}|$ refers to a
unique ordering of places adopted for SDCPN. At times
$t\in(\tau_{k-1},\tau_{k})$ when no transition fires, the token distribution
is unique and we define $M_{t}=M_{t}^{\prime}$. The associated colours of
these tokens are gathered in a column vector $C_{t}$ which first contains all
colours of tokens in place $P_{1}$, next (i.e. below it) all colours of tokens
in place $P_{2}$, etc, until place $P_{|{\cal P}|}$. If at time $t=\tau_{k}$
one or more transitions fire, then the SDCPN discrete process state at time
${\tau_{k}}$ is defined by $M_{\tau_{k}}=$ the token distribution that occurs
after all transitions that fire at time ${\tau_{k}}$ have been fired. The
associated colours of these tokens are gathered in a column vector
$C_{\tau_{k}}$ in the same way as described above. This construction ensures
that the process $\\{M_{t},C_{t}\\}$ has limits from the left and is
continuous from the right, i.e., it satisfies the càdlàg property.
## 3 Air traffic example and its SDCPN model
To illustrate the advantages of SDCPN when modelling a complex system,
consider a simplified model of the evolution of an aircraft in one sector of
airspace. The deviation of this aircraft from its intended path is affected by
its engine system and its navigation system. Each of these aircraft systems
can be in either Working (functioning properly) or Not working (operating in
some failure mode). Both systems switch between these modes independently and
with exponentially distributed sojourn times, with finite rates $\delta_{3}$
(engine repaired), $\delta_{4}$ (engine fails), $\delta_{5}$ (navigation
repaired) and $\delta_{6}$ (navigation fails), respectively. If both systems
are Working, the aircraft evolves in Nominal mode and the position $Y_{t}$ and
velocity $S_{t}$ of the aircraft are determined by $dX_{t}={\cal
V}_{1}(X_{t})dt+{\cal W}_{1}dW_{t}$, where $X_{t}=(Y_{t},S_{t})^{\prime}$. If
either one, or both, of the systems is Not working, the aircraft evolves in
Non-nominal mode and the position and velocity of the aircraft are determined
by $dX_{t}={\cal V}_{2}(X_{t})dt+{\cal W}_{2}dW_{t}$. The factors ${\cal
W}_{1}$ and ${\cal W}_{2}$ are determined by wind fluctuations. Initially, the
aircraft has position $Y_{0}$ and velocity $S_{0}$, while both its systems are
Working. The evaluation of this process may be stopped when the aircraft has
Landed, i.e. its vertical position and velocity are equal to zero.
An SDCPN graph for this example is developed in two stages. In the first
stage, the agents of the operation are modelled separately, by one local SDCPN
each, see Fig. 3a. In the next stage, the interactions between the agents are
modelled, thus connecting the local SDCPN, Fig. 3b.
Figure 3: SDCPN graph for the aircraft evolution example
Fig. 3b shows the SDCPN graph for this example, where,
* •
$P_{1}$ denotes aircraft evolution Nominal, i.e. evolution is according to
${\cal V}_{1}$ and ${\cal W}_{1}$.
* •
$P_{2}$ denotes aircraft evolution Non-nominal, i.e. evolution is according to
${\cal V}_{2}$ and ${\cal W}_{2}$.
* •
$P_{3}$ and $P_{4}$ denote engine system Not working and Working,
respectively.
* •
$P_{5}$ and $P_{6}$ denote navigation system Not working and Working,
respectively.
* •
$P_{7}$ denotes the aircraft has landed.
* •
$T_{1a}$ and $T_{1b}$ denote a transition of aircraft evolution from Nominal
to Non-nominal, due to engine system or navigation system Not working,
respectively.
* •
$T_{2}$ denotes a transition of aircraft evolution from Non-nominal to
Nominal, due to engine system and navigation system both Working again.
* •
$T_{3}$ through $T_{6}$ denote transitions between Working and Not working of
the engine and navigation systems.
* •
$T_{7}$ and $T_{8}$ denote transitions of the aircraft landing.
The graph in Fig. 3b completely defines SDCPN elements ${\cal P}$, ${\cal T}$,
${\cal A}$ and ${\cal N}$, where ${\cal T}_{G}=\\{T_{7},T_{8}\\}$, ${\cal
T}_{D}=\\{T_{3},T_{4},T_{5},T_{6}\\}$ and ${\cal
T}_{I}=\\{T_{1a},T_{1b},T_{2}\\}$. The other SDCPN elements are specified
below:
* ${\cal S}$:
Two colour types are defined; ${\cal S}=\\{\mathbb{R}^{0},\mathbb{R}^{6}\\}$.
* ${\cal C}$:
${\cal C}(P_{1})={\cal C}(P_{2})={\cal C}(P_{7})=\mathbb{R}^{6}$, i.e. tokens
in $P_{1}$, $P_{2}$ and $P_{7}$ have colours in $\mathbb{R}^{6}$; the colour
components model the 3-dimensional position and 3-dimensional velocity of the
aircraft. ${\cal C}(P_{3})={\cal C}(P_{4})={\cal C}(P_{5})={\cal
C}(P_{6})=\mathbb{R}^{0}\triangleq\emptyset$.
* ${\cal I}$:
Place $P_{1}$ initially has a token with colour
$X_{0}=(Y_{0},S_{0})^{\prime}$, with $Y_{0}\in\mathbb{R}^{2}\times(0,\infty)$
and $S_{0}\in\mathbb{R}^{3}\setminus\mbox{Col}\\{0,0,0\\}$. Places $P_{4}$ and
$P_{6}$ initially each have a token without colour.
* ${\cal V}$,
${\cal W}$: The token colour functions for places $P_{1}$, $P_{2}$ and $P_{7}$
are determined by $({\cal V}_{1},{\cal W}_{1})$, $({\cal V}_{2},{\cal
W}_{2})$, and $({\cal V}_{7},{\cal W}_{7})$, respectively, where $({\cal
V}_{7},{\cal W}_{7})=(0,0)$. For places $P_{3}$ – $P_{6}$ there is no token
colour function.
* ${\cal G}$:
Transitions $T_{7}$ and $T_{8}$ have a guard defined by ${\cal
G}_{T_{7}}={\cal
G}_{T_{8}}=\mathbb{R}^{2}\times(0,\infty)\times\mathbb{R}^{2}\times(0,\infty)$.
* ${\cal D}$:
The jump rates for transitions $T_{3}$, $T_{4}$, $T_{5}$ and $T_{6}$ are
${\cal D}_{T_{3}}(\cdot)=\delta_{3}$, ${\cal D}_{T_{4}}(\cdot)=\delta_{4}$,
${\cal D}_{T_{5}}(\cdot)=\delta_{5}$ and ${\cal D}_{T_{6}}(\cdot)=\delta_{6}$.
* ${\cal F}$:
Each transition has a unique output place, to which it fires a token with a
colour (if applicable) equal to the colour of the token removed.
## 4 From SDCPN to GSHS
Following [8], this section first presents a definition of _general stochastic
hybrid system_ (GSHS) and its execution. In [16] it has been proven that under
a few conditions, SDCPN and GSHS are bisimilar. In Subsection 4.2 this is
illustrated by showing how the SDCPN example of the previous section can be
mapped to a bisimilar GSHS.
### 4.1 GSHS definition
#### Definition 4.1
(General stochastic hybrid system) _A GSHS is an automaton ( ${\bf K}$, $d$,
${\cal X}$, $f$, $g$, _Init_ , $\lambda$, $Q$), where_
* •
${\bf K}$_is a countable set._
* •
$d:{\bf K}\rightarrow\mathbb{N}$_maps each_ $\theta\in{\bf K}$ to a natural
number.
* •
${\cal X}:{\bf K}\rightarrow\\{E_{\theta};\theta\in{\bf K}\\}$_maps each_
$\theta\in{\bf K}$ to an open subset $E_{\theta}$ of $\mathbb{R}^{d(\theta)}$.
With this, the hybrid state space is given by
$E\triangleq\\{\\{\theta\\}\times E_{\theta};\theta\in{\bf K}\\}$.
* •
$f:E\to\\{\mathbb{R}^{d(\theta)};\theta\in{\bf K}\\}$_is a vector field._
* •
$g:E\to\\{\mathbb{R}^{d(\theta)\times h};\theta\in{\bf K}\\}$_is a matrix
field, with_ $h\in\mathbb{N}$.
* •
_Init_ $:{\cal B}(E)\rightarrow[0,1]$_is an initial probability measure, with_
${\cal B}(E)$ the Borel $\sigma$-algebra on $E$.
* •
$\lambda:E\to\mathbb{R}^{+}$_is a jump rate function._
* •
$Q:\mathcal{B}(E)\times(E\cup\partial E)\rightarrow[0,1]$_is a GSHS transition
measure, where_ $\partial E\triangleq\\{\\{\theta\\}\times\partial
E_{\theta};\theta\in{\bf K}\\}$ is the boundary of $E$, in which $\partial
E_{\theta}$ is the boundary of $E_{\theta}$.
#### Definition 4.2
(GSHS execution) _A stochastic process $\\{\theta_{t},X_{t}\\}$ is called a
_GSHS execution_ if there exists a sequence of stopping times
$0=\tau_{0}<\tau_{1}<\tau_{2}\cdots$ such that for each $k\in\mathbb{N}$:_
* •
$(\theta_{0},X_{0})$_is an_ $E$-valued random variable extracted according to
probability measure _Init_.
* •
_For_ $t\in[\tau_{k},\tau_{k+1})$, $\theta_{t}=\theta_{\tau_{k}}$ and
$X_{t}=X_{t}^{k}$, where for $t\geq\tau_{k}$, $X_{t}^{k}$ is a solution of the
stochastic differential equation
$dX_{t}^{k}=f(\theta_{\tau_{k}},X_{t}^{k})dt+g(\theta_{\tau_{k}},X_{t}^{k})dB_{t}^{\theta_{\tau_{k}}}$
with initial condition $X_{\tau_{k}}^{k}=X_{\tau_{k}}$, and where
$\\{B_{t}^{\theta}\\}$ is $h$-dimensional standard Brownian motion for each
$\theta\in{\bf K}$.
* •
$\tau_{k+1}=\tau_{k}+\sigma_{k}$_, where_ $\sigma_{k}$ is chosen according to
a survivor function given by $F(t)=$
${\bf 1}_{(t<\tau^{*})}\exp(-\int_{0}^{t}\lambda(\theta,X_{s}^{k})ds)$. Here,
$\tau^{*}=\inf\\{t>\tau_{k}\mid X_{t}^{k}\in\partial E_{\theta_{\tau_{k}}}\\}$
and ${\bf 1}$ is indicator function.
* •
_The probability distribution of_ $(\theta_{\tau_{k+1}},X_{\tau_{k+1}})$, i.e.
the hybrid state right after the jump, is governed by the law
$Q(\cdot;(\theta_{\tau_{k}},X_{\tau_{k+1}-}))$.
[8] show that under assumptions G1-G4 below, a GSHS execution is a strong
Markov Process and has the càdlàg property (right continuous with left hand
limits).
* G1
$f(\theta,\cdot)$ and $g(\theta,\cdot)$ are Lipschitz continuous and bounded.
This yields that for each initial state $(\theta,x)$ at initial time $\tau$
there exists a pathwise unique solution $X_{t}$ to
$dX_{t}=f(\theta,X_{t})dt+g(\theta,X_{t})dB_{t}$, where $\\{B_{t}\\}$ is
$h$-dimensional standard Brownian motion.
* G2
$\lambda:E\to\mathbb{R}^{+}$ is a measurable function such that for all
$\xi\in E$, there is $\epsilon(\xi)>0$ such that
$t\to\lambda(\theta_{t},X_{t})$ is integrable on $[0,\epsilon(\xi))$.
* G3
For each fixed $A\in{\cal B}(E)$, the map $\xi\to Q(A;\xi)$ is measurable and
for any $(\theta,x)\in E\cup\partial E$, $Q(\cdot;\theta,x)$ is a probability
measure.
* G4
If $N_{t}=\sum_{k}{\bf 1}_{(t\geq\tau_{k})}$, then it is assumed that for
every starting point $(\theta,x)$ and for all $t\in\mathbb{R}^{+}$,
$\mathbb{E}N_{t}<\infty$. This means, there will be a finite number of jumps
in finite time.
### 4.2 A bisimilar GSHS for the example SDCPN
Next we transform the SDCPN example model of Section 3 into a bisimilar GSHS.
The first step is to construct the state space ${\bf K}$ for the GSHS discrete
process $\\{\theta_{t}\\}$. This is done by identifying the SDCPN reachability
graph. Nodes in the reachability graph provide the number of tokens in each of
the SDCPN places. Arrows connect these nodes as they represent transitions
firing. The SDCPN of Fig. 3b has seven places hence the reachability graph for
this example has elements that are vectors of length 7. These nodes, excluding
the nodes that enable immediate transitions, form the GSHS discrete state
space.
Figure 4: Reachability graph for the SDCPN of Fig. 3b. The nodes in bold type
face correspond with the elements of the GSHS discrete state space ${\bf K}$.
The reachability graph is shown in Fig. 4, with nodes that form the GSHS
discrete state space in Bold typeface, i.e. ${\bf
K}=\\{V_{1},\ldots,V_{8}\\}$, with $V_{1}=(1,0,0,1,0,1,0)$,
$V_{2}=(0,1,1,0,0,1,0)$, $V_{3}=(0,1,1,0,1,0,0)$, $V_{4}=(0,1,0,1,1,0,0)$,
$V_{5}=(0,0,0,1,0,1,1)$, $V_{6}=(0,0,1,0,0,1,1)$, $V_{7}=(0,0,1,0,1,0,1)$,
$V_{8}=(0,0,0,$ $1,1,0,1)$. Since initially there is a token in places
$P_{1}$, $P_{4}$ and $P_{6}$, the GSHS initial mode equals
$\theta_{0}=V_{1}=(1,0,0,1,0,1,0)$. The GSHS initial continuous state value
equals the vector containing the initial colours of all initial tokens. Since
the initial colour of the token in Place $P_{1}$ equals $X_{0}$, and the
tokens in places $P_{4}$ and $P_{6}$ have no colour, the GSHS initial
continuous state value equals Col$\\{X_{0},\emptyset,\emptyset\\}=X_{0}$. The
GSHS drift coefficient $f$ is given by $f(\theta,\cdot)={\cal V}_{1}(\cdot)$
for $\theta=V_{1}$, $f(\theta,\cdot)={\cal V}_{2}(\cdot)$ for
$\theta\in\\{V_{2},V_{3},V_{4}\\}$, and $f(\theta,\cdot)=0$ otherwise. For the
diffusion coefficient, $g(\theta,\cdot)={\cal W}_{1}$ for $\theta=V_{1}$,
$g(\theta,\cdot)={\cal W}_{2}$ for $\theta\in\\{V_{2},V_{3},V_{4}\\}$, and
$g(\theta,\cdot)=0$ otherwise. The hybrid state space is given by
$E=\\{\\{\theta\\}\times E_{\theta};\theta\in\mathbb{M}\\}$, where for
$\theta\in\\{V_{1},V_{2},V_{3},V_{4}\\}$:
$E_{\theta}=\mathbb{R}^{2}\times(0,\infty)\times\mathbb{R}^{2}\times(0,\infty)$
and for $\theta\in\\{V_{5},V_{6},V_{7},V_{8}\\}$: $E_{\theta}=\mathbb{R}^{6}$.
Always two delay transitions are pre-enabled: either $T_{3}$ or $T_{4}$ and
either $T_{5}$ or $T_{6}$. This yields
$\lambda(V_{1},\cdot)=\lambda(V_{5},\cdot)=\delta_{4}+\delta_{6}$,
$\lambda(V_{2},\cdot)=\lambda(V_{6},\cdot)=\delta_{3}+\delta_{6}$,
$\lambda(V_{3},\cdot)=\lambda(V_{7},\cdot)=\delta_{3}+\delta_{5}$,
$\lambda(V_{4},\cdot)=\lambda(V_{8},\cdot)=\delta_{4}+\delta_{5}$. For the
determination of GSHS transition measure $Q$, we make use of the reachability
graph, the sets ${\cal D}$, ${\cal G}$ and ${\cal F}$ and the rules R0–R4. In
Table 1, $Q(\theta^{\prime},x^{\prime};\theta,x)=p$ denotes that if
$(\theta,x)$ is the value of the GSHS state before the hybrid jump, then, with
probability $p$, $(\theta^{\prime},x^{\prime})$ is the value of the GSHS state
immediately after the jump.
Table 1: Example GSHS transition measure for size of jump For $x\notin\partial E_{V_{1}}$: | $Q(V_{2},x;V_{1},x)=\frac{\delta_{4}}{\delta_{4}+\delta_{6}}$, | $Q(V_{4},x;V_{1},x)=\frac{\delta_{6}}{\delta_{4}+\delta_{6}}$
---|---|---
For $x\in\partial E_{V_{1}}$: | $Q(V_{5},x;V_{1},x)=1$ |
For $x\notin\partial E_{V_{2}}$: | $Q(V_{3},x;V_{2},x)=\frac{\delta_{6}}{\delta_{3}+\delta_{6}}$, | $Q(V_{1},x;V_{2},x)=\frac{\delta_{3}}{\delta_{3}+\delta_{6}}$
For $x\in\partial E_{V_{2}}$: | $Q(V_{6},x;V_{2},x)=1$ |
For $x\notin\partial E_{V_{3}}$: | $Q(V_{4},x;V_{3},x)=\frac{\delta_{3}}{\delta_{3}+\delta_{5}}$, | $Q(V_{2},x;V_{3},x)=\frac{\delta_{5}}{\delta_{3}+\delta_{5}}$
For $x\in\partial E_{V_{3}}$: | $Q(V_{7},x;V_{3},x)=1$ |
For $x\notin\partial E_{V_{4}}$: | $Q(V_{3},x;V_{4},x)=\frac{\delta_{4}}{\delta_{4}+\delta_{5}}$, | $Q(V_{1},x;V_{4},x)=\frac{\delta_{5}}{\delta_{4}+\delta_{5}}$
For $x\in\partial E_{V_{4}}$: | $Q(V_{8},x;V_{4},x)=1$ |
For all $x$: | $Q(V_{6},x;V_{5},x)=\frac{\delta_{4}}{\delta_{4}+\delta_{6}}$, | $Q(V_{8},x;V_{5},x)=\frac{\delta_{6}}{\delta_{4}+\delta_{6}}$
For all $x$: | $Q(V_{7},x;V_{6},x)=\frac{\delta_{6}}{\delta_{3}+\delta_{6}}$, | $Q(V_{5},x;V_{6},x)=\frac{\delta_{3}}{\delta_{3}+\delta_{6}}$
For all $x$: | $Q(V_{8},x;V_{7},x)=\frac{\delta_{3}}{\delta_{3}+\delta_{5}}$, | $Q(V_{6},x;V_{7},x)=\frac{\delta_{5}}{\delta_{3}+\delta_{5}}$
For all $x$: | $Q(V_{7},x;V_{8},x)=\frac{\delta_{4}}{\delta_{4}+\delta_{5}}$, | $Q(V_{5},x;V_{8},x)=\frac{\delta_{5}}{\delta_{4}+\delta_{5}}$
With this, the SDCPN of the aircraft evolution example is uniquely mapped to
an GSHS. It can be shown that the SDCPN execution and the execution of the
resulting GSHS are probabilistically equivalent, i.e. the SDCPN and the GSHS
are bisimilar. Thanks to this bisimilarity we can now use the automata
framework to analyse the GSHP that is defined by the SDCPN model for the
example.
## 5 From SDCPN to HSDE
Following [2] and [3], this section first presents a definition of _hybrid
stochastic differential equation_ (HSDE) and gives conditions under which the
HSDE has a pathwise unique solution. This pathwise unique solution is referred
to as _HSDE solution process_ or GSHP. The basic advantage of using HSDE in
defining a GSHP over using GSHS is that with the HSDE approach the spontaneous
jump mechanism is explicitly built on an underlying stochastic basis, whereas
in GSHS the execution itself imposes an underlying stochastic basis. In [17]
it has been proven that under a few conditions, SDCPN and HSDE are bisimilar.
In Subsection 5.2 this is illustrated by showing how the SDCPN example of the
previous section can be mapped to a bisimilar HSDE.
### 5.1 HSDE definition
For the HSDE setting we start with a complete stochastic basis
$(\Omega,\Im,\mathbb{F},\mathbb{P},\mathbb{T})$, in which a complete
probability space $(\Omega,\Im,\mathbb{P})$ is equipped with a right-
continuous filtration $\mathbb{F}=\\{\Im_{t}\\}$ on the positive time line
$\mathbb{T}=\mathbb{R}^{+}$. This stochastic basis is endowed with a
probability measure $\mu_{\theta_{0},X_{0}}$ for the initial state, an
independent $h$-dimensional standard Wiener process $\\{W_{t}\\}$ and an
independent homogeneous Poisson random measure $p_{P}(dt,dz)$ on
$\mathbb{T}\times\mathbb{R}^{d+1}$.
#### Definition 5.1
(Hybrid stochastic differential equation) _An HSDE on stochastic basis
$(\Omega,\Im,\mathbb{F},\mathbb{P},\mathbb{T})$, is defined as a set of
equations (1)-(8) in which a collection of elements ($\mathbb{M}$, $E$, $f$,
$g$, $\mu_{\theta_{0},X_{0}}$, $\Lambda$, $\psi$, $\rho$, $\mu$, $p_{P}$,
$\\{W_{t}\\}$) appear._
The elements ($\mathbb{M}$, $E$, $f$, $g$, $\mu_{\theta_{0},X_{0}}$,
$\Lambda$, $\psi$, $\rho$, $\mu$, $p_{P}$, $\\{W_{t}\\}$) are defined as
follows:
* •
$\mathbb{M}=\\{\vartheta_{1},\ldots,\vartheta_{N}\\}$ is a finite set,
$N\in\mathbb{N}$, $1\leq N<\infty$.
* •
$E=\\{\\{\theta\\}\times E_{\theta};\theta\in\mathbb{M}\\}$ is the hybrid
state space, where for each $\theta\in\mathbb{M}$, $E_{\theta}$ is an open
subset of $\mathbb{R}^{n}$ with boundary $\partial E_{\theta}$. The boundary
of $E$ is $\partial E=\\{\\{\theta\\}\times\partial
E_{\theta};\theta\in\mathbb{M}\\}$.
* •
$f:\mathbb{M}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ is a measurable mapping.
* •
$g:\mathbb{M}\times\mathbb{R}^{n}\to\mathbb{R}^{n\times h}$ is a measurable
mapping.
* •
$\mu_{\theta_{0},X_{0}}:\Omega\times{\cal B}(E)\to[0,1]$ is a probability
measure for the initial random variables $\theta_{0}$, $X_{0}$, which are
defined on the stochastic basis; $\mu_{\theta_{0},X_{0}}$ is assumed to be
invertible.
* •
$\Lambda:\mathbb{M}\times\mathbb{R}^{n}\to[0,\infty)$ is a measurable mapping.
* •
$\psi:\mathbb{M}\times\mathbb{M}\times\mathbb{R}^{n}\times\mathbb{R}^{d}\to\mathbb{R}^{n}$
is a measurable mapping such that $x+\psi(\vartheta,\theta,x,\underline{z})\in
E_{\vartheta}$ for all $x\in E_{\theta}$, $\underline{z}\in\mathbb{R}^{d}$,
and $\vartheta,\theta\in\mathbb{M}$.
* •
$\rho:\mathbb{M}\times\mathbb{M}\times\mathbb{R}^{n}\to[0,\infty)$ is a
measurable mapping such that $\sum_{i=1}^{N}\rho(\vartheta_{i},\theta,x)=1$
for all $\theta\in\mathbb{M},x\in\mathbb{R}^{n}$.
* •
$\mu:\Omega\times\mathbb{R}^{d}\to[0,1]$ is a probability measure which is
assumed to be invertible.
* •
$p_{P}:\Omega\times\mathbb{T}\times\mathbb{R}^{d+1}\to\\{0,1\\}$ is a
homogeneous Poisson random measure on the stochastic basis, independent of
$(\theta_{0},X_{0})$. The intensity measure of $p_{P}(dt,dz)$ equals
$dt\cdot\mu_{L}(dz_{1})\cdot\mu(d\underline{z})$, where
$z=\mbox{Col}\\{z_{1},\underline{z}\\}$ and $\mu_{L}$ is the Lebesgue measure.
* •
$W:\Omega\times\mathbb{T}\to\mathbb{R}^{h}$ such that $\\{W_{t}\\}$ is an
$h$-dimensional standard Wiener process on the stochastic basis, and
independent of $(\theta_{0},X_{0})$ and $p_{P}$.
Using these elements, the HSDE process $\\{\theta_{t}^{*},X_{t}^{*}\\}$ is
defined as follows:
$\theta_{t}^{*}=\theta_{t}^{k}\mbox{ for all
}t\in[\tau_{k}^{b},\tau_{k+1}^{b}),k=0,1,2,\ldots$ (1)
$X_{t}^{*}=X_{t}^{k}\mbox{ for all
}t\in[\tau_{k}^{b},\tau_{k+1}^{b}),k=0,1,2,\ldots$ (2)
Hence $\\{\theta_{t}^{*},X_{t}^{*}\\}$ consists of a concatenation of
processes $\\{\theta_{t}^{k},X_{t}^{k}\\}$ which are defined by (3)-(8) below.
If the system (1)-(8) has a solution in probabilistic sense, then the process
$\\{\theta_{t}^{*},X_{t}^{*}\\}$ is referred to as _HSDE solution process_ or
_GSHP_.
$d\theta_{t}^{k}=\sum_{i=1}^{N}(\vartheta_{i}-\theta_{t-}^{k})p_{P}(dt,(\Sigma_{i-1}(\theta_{t-}^{k},X_{t-}^{k}),\Sigma_{i}(\theta_{t-}^{k},X_{t-}^{k})]\times\mathbb{R}^{d})$
(3)
$dX_{t}^{k}=f(\theta_{t}^{k},X_{t}^{k})dt+g(\theta_{t}^{k},X_{t}^{k})dW_{t}+\int_{\mathbb{R}^{d}}\psi(\theta_{t}^{k},\theta_{t-}^{k},X_{t-}^{k},\underline{z})p_{P}(dt,(0,\Lambda(\theta_{t-}^{k},X_{t-}^{k})]\times
d\underline{z})$ (4)
with $\theta_{0}^{0}=\theta_{0}$, $X_{0}^{0}=X_{0}$ and with $\Sigma_{0}$
through $\Sigma_{N}$ measurable mappings satisfying, for
$\theta\in\mathbb{M}$, $\vartheta_{j}\in\mathbb{M}$, $x\in\mathbb{R}^{n}$:
$\Sigma_{i}(\theta,x)=\left\\{\begin{array}[]{ll}\Lambda(\theta,x)\sum_{j=1}^{i}\rho(\vartheta_{j},\theta,x)&\mbox{if
}i>0\\\ 0&\mbox{if }i=0\end{array}\right.$ (5)
In addition, for $k=0,1,2,\ldots$, with $\tau_{0}^{b}=0$:
$\tau_{k+1}^{b}\triangleq\inf\\{t>\tau_{k}^{b}\mid(\theta_{t}^{k},X_{t}^{k})\in\partial
E\\}$ (6)
$\mathbb{P}\\{\theta_{\tau_{k+1}^{b}}^{k+1}=\vartheta,X_{\tau_{k+1}^{b}}^{k+1}\in{A}\mid\theta_{\tau_{k+1}^{b}-}^{k}=\theta,X_{\tau_{k+1}^{b}-}^{k}=x\\}=Q(\\{\vartheta\\}\times
A;\theta,x)$ (7)
for $A\in{\cal B}(\mathbb{R}^{n})$, where $Q$ is given by
$Q(\\{\vartheta\\}\times
A;\theta,x)=\rho(\vartheta,\theta,x)\int_{\mathbb{R}^{d}}{\bf
1}_{A}(x+\psi(\vartheta,\theta,x,\underline{z}))\mu(d\underline{z})$ (8)
Next, the following proposition can be shown to hold true [17]:
#### Proposition 5.1
_Let conditions H1-H8 below hold true. Let
$(\theta_{0}^{*}(\omega),X_{0}^{*}(\omega))=(\theta_{0},X_{0})\in E$ for all
$\omega$. Then for every initial condition $(\theta_{0},X_{0})$, (1)-(8) has a
pathwise unique solution $\\{\theta_{t}^{*},X_{t}^{*}\\}$ which is càdlàg and
adapted and is a semi-martingale assuming values in the hybrid state space
$E$._
H1
For all $\theta\in\mathbb{M}$ there exists a constant $K(\theta)$ such that
for all $x\in\mathbb{R}^{n}$, $|f(\theta,x)|^{2}+\|g(\theta,x))\|^{2}\leq
K(\theta)(1+|x|^{2})$, where $|a|^{2}=\sum_{i}(a_{i})^{2}$ and
$||b||^{2}=\sum_{i,j}(b_{ij})^{2}$.
H2
For all $r\in\mathbb{N}$ and for all $\theta\in\mathbb{M}$ there exists a
constant $L_{r}(\theta)$ such that for all $x$ and $y$ in the ball
$B_{r}=\\{z\in\mathbb{R}^{n}\mid|z|\leq r+1\\}$,
$|f(\theta,x)-f(\theta,y)|^{2}+\|g(\theta,x)-g(\theta,y)\|^{2}\leq
L_{r}(\theta)|x-y|^{2}$.
H3
For each $\theta\in\mathbb{M}$, the mapping
$\Lambda(\theta,\cdot):\mathbb{R}^{n}\to[0,\infty)$ is continuous and bounded,
with upper bound a constant $C_{\Lambda}$.
H4
For all $(\theta,\vartheta)\in\mathbb{M}^{2}$, the mapping
$\rho(\vartheta,\theta,\cdot):\mathbb{R}^{n}\to[0,\infty)$ is continuous.
H5
For all $r\in\mathbb{N}$ there exists a constant $M_{r}(\theta)$ such that
$\sup_{|x|\leq
r}\int_{\mathbb{R}^{d}}|\psi(\vartheta,\theta,x,\underline{z})|\mu(d\underline{z})\leq
M_{r}(\theta),\mbox{ for all }\vartheta,\theta\in\mathbb{M}$
H6
$|\psi(\theta,\theta,x,\underline{z})|=0$ or $>1$ for all
$\theta\in\mathbb{M}$, $x\in\mathbb{R}^{n}$, $\underline{z}\in\mathbb{R}^{d}$
H7
$\\{(\theta_{t}^{*},X_{t}^{*})\\}$ hits the boundary $\partial E$ a finite
number of times on any finite time interval
H8
$|\vartheta_{i}-\vartheta_{j}|>1$ for $i\neq j$, with $|\cdot|$ a suitable
metric well defined on $\mathbb{M}$.
### 5.2 A bisimilar HSDE for the example SDCPN
Next we transform the SDCPN example model of Section 3 into a bisimilar HSDE.
This mapping follows much the same procedure as for SDCPN to GSHS, except that
the discrete state space is now referred to as $\mathbb{M}$ (rather than ${\bf
K}$) and the Markov jump rate is now referred to as $\Lambda$ (rather than
$\lambda$). The main additional difference is that the HSDE elements do not
include a transition measure $Q$ to define the size of jump, but include
functions $\psi$, $\rho$ and $\mu$ instead. The mapping of SDCPN to HSDE uses
the construction of transition measure $Q$ as an intermediate step, however.
For the particular example SDCPN in this paper, these functions can be
determined from $Q$ as follows: Since the continuous valued process jumps to
the same value with probability 1, we find that
$\psi(V^{i},V^{j},x,\underline{z})=0$ for all $V^{i}$, $V^{j}$, $x$,
$\underline{z}$. Moreover, $\rho(V^{i},V^{j},x)=P_{Q}(V^{i},x,V^{j},x)$ and
$\mu$ may be any given invertible probability measure.
With this, the SDCPN of the aircraft evolution example is uniquely mapped to
an HSDE. If in addition, we want to make use of the HSDE properties of
Proposition 5.1, i.e. the resulting HSDE solution process being adapted and a
semi-martingale, we need to make sure that HSDE conditions H1-H8 are
satisfied. It is shown below that they are, under the following sufficient
condition D1 for the example SDCPN.
D1
For $P\in\\{P_{1},P_{2}\\}$, there exist $K_{P}^{v}$, $L_{P}^{v}$, $K_{P}^{w}$
and $L_{P}^{w}$ such that for all $c,a\in{\cal C}(P)$,
$|{\cal V}_{P}(c)|^{2}\leq K_{P}^{v}(1+|c|^{2})$ and $|{\cal V}_{P}(c)-{\cal
V}_{P}(a)|^{2}\leq L_{P}^{v}|c-a|^{2}$ and
$\|{\cal W}_{P}(c)\|^{2}\leq K_{P}^{w}(1+|c|^{2})$ and $\|{\cal
W}_{P}(c)-{\cal W}_{P}(a)\|^{2}\leq L_{P}^{w}|c-a|^{2}$.
We verify that under condition D1, HSDE conditions H1-H8 hold true in this
example.
H1:
From the construction of $f$ and $g$ above we have for $\theta=V_{1}$:
$|f(\theta,x)|^{2}+\|g(\theta,x)\|^{2}=|{\cal V}_{1}(x)|^{2}+\|{\cal
W}_{1}(x)\|^{2}\leq
K_{P_{1}}^{v}(1+|x|^{2})+K_{P_{1}}^{w}(1+|x|^{2})=K(\theta)(1+|x|^{2})$, with
$K(\theta)=(K_{P_{1}}^{v}+K_{P_{1}}^{w})$. For $\theta=V_{2},V_{3},V_{4}$ the
verification is with replacing ${\cal V}_{1}$, ${\cal W}_{1}$ by ${\cal
V}_{2}$, ${\cal W}_{2}$.
H2:
From the construction of $f$ and $g$ above we have for $\theta=V_{1}$:
$|f(\theta,x)-f(\theta,y)|^{2}+\|g(\theta,x)-g(\theta,y)\|^{2}=|{\cal
V}_{1}(x)-{\cal V}_{1}(y)|^{2}+\|{\cal W}_{1}(x)-{\cal W}_{1}(y)\|^{2}\leq
L_{P_{1}}^{v}|x-y|^{2}+L_{P_{1}}^{w}|x-y|^{2}=L_{r}(\theta)|x-y|^{2}$ with
$L_{r}(\theta)=L_{P_{1}}^{v}+L_{P_{1}}^{w}$. For $\theta=V_{2},V_{3},V_{4}$
replace ${\cal V}_{1}$, ${\cal W}_{1}$ by ${\cal V}_{2}$, ${\cal W}_{2}$.
H3:
Since $\delta_{3}$–$\delta_{6}$ are constant, for all $\theta$,
$\Lambda(\theta,\cdot)$ is bounded and continuous, with upper bound
$C_{\Lambda}=\max\\{\delta_{4}+\delta_{6},\delta_{3}+\delta_{6},\delta_{3}+\delta_{5},\delta_{4}+\delta_{5}\\}$.
H4:
Since for all $\theta,\vartheta$, $P_{Q}(\vartheta,\cdot;\theta,x)$ is
constant, we find $\rho(\vartheta,\theta,x)=P_{Q}(\vartheta,x,\theta,x)$ is
continuous.
H5 and H6:
These are satisfied due to $\psi(V^{i},V^{j},x,\underline{z})=0$ for all
$V^{i}$, $V^{j}$, $x$, $\underline{z}$.
H7:
This condition holds due to $\delta_{3}$–$\delta_{6}$ being finite and the
fact that in this SDCPN example, there is no firing sequence of more than one
guard transition.
H8:
This condition holds for all $V_{1},\ldots,V_{8}$, with metric
$|a|^{2}=\sum_{i}(a_{i})^{2}$.
Thanks to this bisimilarity mapping we can now use HSDE tools to analyse the
GSHP that is defined by the execution of the SDCPN model for the example.
## 6 Conclusions
The aim of this paper was to explain bisimilarity relations between SDCPN
(stochastically and dynamically coloured Petri net), GSHS (general stochastic
hybrid system) and HSDE (hybrid stochastic differential equation), which means
that the strengths of one stochastic model formalism can be used by both of
the other stochastic model formalisms. More specifically, these bisimilarity
relations make it possible to combine the formal verification power of
automata with the analysis power of stochastic differential equations and the
compositional specification power of Petri nets.
We started in Section 2 by defining SDCPN and the resulting SDCPN stochastic
process, which is referred to as a GSHP (general stochastic hybrid process).
In Section 3 we presented a simple but illustrative SDCPN example model. In
Section 4 we studied GSHP as an execution of a GSHS and illustrated by using
the example of Section 3 that SDCPN and GSHS are bisimilar. Next, in Section 5
we studied GSHP as a stochastic process solution of HSDE and showed with an
illustrative example that SDCPN and HSDE are bisimilar.
The bisimilarities between SDCPN, GSHS and HSDE models for the example
considered mean that the resulting example model inherits the strengths of all
three formal stochastic modelling formalisms. This has been depicted in Fig. 1
in the introduction. Examples of GSHP properties are convergence in
discretisation, existence of limits, existence of event probabilities, strong
Markov properties, reachability analysis. Examples of GSHS features are their
connection to formal methods in automata theory and optimal control theory.
Examples of HSDE features are stochastic analysis tools for semi-martingales.
Examples of SDCPN features are natural expression of causal dependencies,
concurrency and synchronisation mechanism, hierarchical and modular
construction, and graphical representation. These complementary advantages of
SDCPN, GSHS, HSDE and GSHP perspectives tend to increase with the complexity
of the system considered.
An illustrative large scale application of bisimularity relations between
SDCPN, HSDE and stochastic hybrid automata has been developed in air traffic
management. Currently pilots depend of air traffic controllers in solving
potential conflicts between their flight trajectories. This places a huge
requirement on the tasks of an air traffic controller. Imagine a similar kind
of approach for road traffic; then each car driver would be blind and depends
of instructions that some road traffic controller is communicating with the
car drivers. How many cars do you think can be managed by one road traffic
controller? The number of aircraft that one air traffic controller can handle
ranges between 10 and 20, depending of the complexity of the traffic pattern.
Over a decade ago, it had been suggested by [23] that this limitation of the
air traffic controller can be solved by moving the responsibility of conflict
resolution from the air traffic controller to the pilots. Since then this
airborne self separation idea has received a lot of research attention.
Nevertheless, it still is unknown how much more air traffic can safely be
accommodated under a well designed airborne self separation way of working. In
order to add to the solution of this debate, a series of large European
studies towards solving this question have been started under the name
HYBRIDGE [19] and iFly [20] respectively. The way of working is to first
develop a well defined SDCPN model of the airborne self separation concept of
operation to be analysed, e.g. [18]. Subsequently this SDCPN model is further
analysed using a bisimilar HSDE and hybrid automation formal model
representation [6, 4], in which powerful stochastic analysis theory is
exploited for the speeding up of Monte Carlo simulations. Using this approach,
[5] have shown that the first generation of airborne self separation concept
designs falls short in safely accommodating higher air traffic demand than
conventional ATM can. The feedback of this finding to advanced air traffic
concept designers triggered the development of more advanced airborne self
separation concept of operation, e.g. see [20].
## References
* [1]
* [2] H.A.P. Blom (2003): _Stochastic hybrid processes with hybrid jumps_. In: Proceedings IFAC conference on analysis and design of hybrid system (ADHS), Saint-Malo, Brittany, France. pp. 361–365.
* [3] H.A.P. Blom, G.J. Bakker, M.H.C. Everdij & M.N.J. Van der Park (2003): _Stochastic analysis background of accident risk assessment for air traffic management_. HYBRIDGE Report, D2.2. http://hosted.nlr.nl/public/hosted-sites/hybridge/.
* [4] H.A.P. Blom, G.J. Bakker & J. Krystul (2009): _Rare event estimation for a large scale stochastic hybrid system with air traffic application_. In: G. Rubino & B. Tuffin, editors: Rare event simulation using Monte Carlo methods, forthcoming. J.Wiley.
* [5] H.A.P. Blom, B. Klein Obbink & G.J. Bakker (2009): _Simulated collision risk of an uncoordinated airborne self separation concept of operation_. ATC Quarterly 17, pp. 63–93.
* [6] H.A.P. Blom, J. Krystul, G.J. Bakker, M.B. Klompstra & B. Klein Obbink (2007): _Free flight collision risk estimation by sequential Monte Carlo simulation_. In: C.G. Cassandras & J. Lygeros, editors: Stochastic hybrid systems: recent developments and research trends, Control engineering 24, chapter 10. Taylor & Francis Group / CRC Press, pp. 247–281.
* [7] M.L. Bujorianu & J. Lygeros (2004): _General stochastic hybrid systems: modelling and optimal control_. In: Proceedings 43rd conference on decision and control (CDC), Nassau, Bahamas.
* [8] M.L. Bujorianu & J. Lygeros (2006): _Toward a general theory of stochastic hybrid systems_. In: H.A.P. Blom & J. Lygeros, editors: Stochastic hybrid systems: theory and safety critical applications, Lectures notes in control and information sciences (LNCIS) 337\. Springer, pp. 3–30.
* [9] M.L. Bujorianu, J. Lygeros & M.C. Bujorianu (2005): _Different approaches on bisimulation for stochastic hybrid systems_. In: M. Morari & L. Thiele, editors: Proceedings 8th international workshop on hybrid systems: computation and control (HSCC), Zürich, Switzerland, Lecture notes in computer science (LNCS) 3414\. pp. 198–214.
* [10] M.H.A. Davis (1993): _Markov models and optimization_ , Monographs on statistics and applied probability 49\. Chapman and Hall, London.
* [11] R.J. Elliott (1982): _Stochastic calculus and applications_ , Applications of mathematics: Stochastic modelling and applied probability 18\. Springer-Verlag.
* [12] R.J. Elliott, L. Aggoun & J.B. Moore (1995): _Hidden Markov models: estimation and control_ , Applications of mathematics: stochastic modelling and applied probability 29\. Springer-Verlag.
* [13] S.N. Ethier & T.G. Kurtz (1986): _Markov processes, characterization and convergence_. Wiley series in probability and mathematical statistics. John Wiley & Sons, New York.
* [14] M.H.C. Everdij & H.A.P. Blom (2003): _Petri nets and hybrid state Markov processes in a power-hierarchy of dependability models_. In: Proceedings IFAC conference on analysis and design of hybrid system (ADHS), Saint-Malo, Brittany, France. pp. 355–360.
* [15] M.H.C. Everdij & H.A.P. Blom (2005): _Piecewise deterministic Markov processes represented by dynamically coloured Petri nets_. In: S. Jacka, editor: Stochastics: an international journal of probability and stochastic processes, 77, number 1. Taylor & Francis, pp. 1–29.
* [16] M.H.C. Everdij & H.A.P. Blom (2006): _Hybrid Petri nets with diffusion that have into-mappings with generalised stochastic hybrid processes_. In: H.A.P. Blom & J. Lygeros, editors: Stochastic hybrid systems: theory and safety critical applications, Lectures notes in control and information sciences (LNCIS) 337\. Springer, pp. 31–63.
* [17] M.H.C. Everdij & H.A.P. Blom (2010): _Hybrid state Petri nets which have the analysis power of stochastic hybrid systems and the formal verification power of automata_. In: P. Pawlewski, editor: Petri nets: Applications, chapter 12. InTech, pp. 227–252.
* [18] M.H.C. Everdij, M.B. Klompstra, H.A.P. Blom & B. Klein Obbink (2006): _Compositional specification of a multi-agent system by stochastically and dynamically coloured Petri nets_. In: H.A.P. Blom & J. Lygeros, editors: Stochastic hybrid systems: theory and safety critical applications, Lectures notes in control and information sciences (LNCIS) 337\. Springer, pp. 325–350.
* [19] HYBRIDGE (2002). _EC project description_. http://hosted.nlr.nl/public/hosted-sites/hybridge.
* [20] iFly (2007). _EC project description_. http://iFly.nlr.nl.
* [21] J. Krystul (2006): _Modelling of stochastic hybrid systems with applications to accident risk assessment_. Ph.D. thesis, University of Twente, The Netherlands.
* [22] J. Krystul, H.A.P. Blom & A. Bagchi (2007): _Stochastic hybrid systems_ , chapter 2: Stochastic differential equations on hybrid state spaces, pp. 15–45. Number 24 in Control engineering series. Taylor and Francis / CRC Press.
* [23] RTCA (1995): _Final report of RTCA Task Force 3; Free Flight implementation_. Final, RTCA Inc., Washington DC.
* [24] A.J. Van der Schaft (2004): _Equivalence of dynamical systems by bisimulation_. IEEE transactions on automatic control 49(12), pp. 2160–2172.
|
arxiv-papers
| 2010-03-25T06:38:50 |
2024-09-04T02:49:09.239608
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mariken H.C. Everdij and Henk A.P. Blom",
"submitter": "Mariken Everdij",
"url": "https://arxiv.org/abs/1003.4812"
}
|
1003.4906
|
# The order of complex numbers
Sun Daochun, Gu Zhendong, Liu Weiqun, Yue Chao
Abstract: In this paper, we define an ordering relation for a set of complex
numbers, and research the properties and theorems of the ordering, solve some
simple complex inequalities with the ordering.
Key words: complex number, ordering, inequality.
## 1 Introduction
In general, it is impossible to judge the order of two complex numbers. In
fact, how to judge the order of two complex numbers is how to define their
sequencing. We think it is possible to define the order of two complex
numbers. Since the order of complex numbers can’t bring more useful interest
than that of real numbers, no one is willing to define the order of the
complex numbers.
It is assumed that the properties of the inequalities of real numbers hold. In
this paper, we define the order of the complex numbers, which is the extension
of the order of real number, then get some properties and theorem about this
order in complex. At the end, using this order, we resolve some inequalities.
Let small letters denote real numbers, and let capital letters denote complex
numbers. We use $\mathscr{R}$ and $\mathscr{C}$ to denote the set of real
numbers and the set of complex numbers respectively. We also use Greece small
letter to denote the argument of complex number, and use script capital letter
to denote the number set.
## 2 The ordering relation ($\leq$) of the set of complex numbers
Let complex numbers $A=a_{1}+a_{2}i,B=b_{1}+b_{2}i,C=c_{1}+c_{2}i,\cdots$
Definition 1 $A\leq
B\Leftrightarrow(a_{1}<b_{1})\cup[(a_{1}=b_{1})\cap(a_{2}\leq b_{2})]$, that
is $a_{1}<b_{1}$, or $a_{1}=b_{1}$ and $a_{2}\leq b_{2}$.
$A\geq B\Leftrightarrow(a_{1}>b_{1})\cup[(a_{1}=b_{1})\cap(a_{2}\geq b_{2})]$.
From the above definition, we get the following properties.
Property 1 (reflexivity) $A\leq A$.
Property 2 $[(A\geq B)~{}~{}and~{}~{}(A\leq B)]\Leftrightarrow$ ($A=B$).
Proof $\\{A\geq B\\}\cap\\{A\leq B\\}$
${\Leftrightarrow}\\{(a_{1}>b_{1})\cup[(a_{1}=b_{1})\cap(a_{2}\geq
b_{2})]\\}\cap\\{(a_{1}<b_{1})\cup[(a_{1}=b_{1})\cap(a_{2}\leq b_{2})]\\}$
${\Leftrightarrow}\\{(a_{1}>b_{1})\cap(a_{1}<b_{1})\\}\cup\\{[(a_{1}=b_{1})\cap(a_{2}\geq
b_{2})]\\}\cap[(a_{1}=b_{1})\cap(a_{2}\leq b_{2})]\\}\hskip 56.9055pt$ $\hskip
56.9055pt\cup\\{(a_{1}>b_{1})\cap[(a_{1}=b_{1})\cap(a_{2}\leq
b_{2})]\\}\cup\\{[(a_{1}=b_{1})\cap(a_{2}\geq b_{2})]\\}\cap(a_{1}<b_{1})\\}$
${\Leftrightarrow}\\{[(a_{1}=b_{1})\cap(a_{2}\geq
b_{2})]\\}\cap[(a_{1}=b_{1})\cap(a_{2}\leq b_{2})]\\}$
${\Leftrightarrow}(a_{1}=b_{1})\cap(a_{2}=b_{2}){\Leftrightarrow}(A=B).$
Property 3 (transitivity) $A\leq B$ and $B\leq C\Rightarrow$ $A\leq C$.
Proof $\\{A\leq B\\}\cap\\{B\leq C\\}$
$\Leftrightarrow\\{[a_{1}<b_{1}]\cup[(a_{1}=b_{1})\cap(a_{2}\leq
b_{2})]\\}\cap\\{[b_{1}<c_{1}]\cup[(b_{1}=c_{1})\cap(b_{2}\leq c_{2})]\\}$
$\Leftrightarrow\\{[a_{1}<b_{1}]\cap[b_{1}<c_{1}]\\}\cup\\{[(a_{1}=b_{1})\cap(a_{2}\leq
b_{2})]\cap[(b_{1}=c_{1})\cap(b_{2}\leq c_{2})]\\}\hskip 142.26378pt$ $\hskip
28.45274pt\cup\\{[a_{1}<b_{1}]\cap[(b_{1}=c_{1})\cap(b_{2}\leq
c_{2})]\\}\cup\\{[(a_{1}=b_{1})\cap(a_{2}\leq b_{2})]\cap[b_{1}<c_{1}]\\}$
$\Rightarrow\\{[a_{1}<c_{1}]\\}\cup\\{[a_{1}<c_{1})\cap(b_{2}\leq
c_{2})]\\}\cup\\{[(a_{1}<c_{1})\cap(a_{2}\leq
b_{2})]\\}\cup\\{(a_{1}=c_{1})\cap(a_{2}\leq c_{2})\\}$
$\Rightarrow\\{[a_{1}<c_{1}]\\}\cup\\{(a_{1}=c_{1})\cap(a_{2}\leq
c_{2})\\}\Leftrightarrow\hskip 14.22636ptA\leq C.$
For any two complex numbers $A,B\in\mathscr{C}$, at least one between the two
relation expressions $A\leq B$, $A\geq B$ holds. Combing with properties
1,2,3, we know that the set of complex numbers $\mathscr{C}$ is a totally
ordered set under the condition of the above defined ordering relation $\leq$.
It is easy to see that the above defined ordering relation $\leq$ is the
extension of that of real numbers.
Theorem 1 $A\leq B$ $\Leftrightarrow$ $A+C\leq B+C$.
Proof $(A+C\leq B+C)$
$\stackrel{{\scriptstyle D.1}}{{\Leftrightarrow}}\hskip
8.5359pt\\{(a_{1}+c_{1}<b_{1}+c_{1})\cup[(a_{1}+c_{1}=b_{1}+c_{1})\cap(a_{2}+c_{2}\leq
b_{2}+c_{2})]\\}$ $\stackrel{{\scriptstyle R.P}}{{\Leftrightarrow}}\hskip
8.5359pt\\{(a_{1}<b_{1})\cup[a_{1}=b_{1})\cap(a_{2}\leq b_{2})]\\}\hskip
8.5359pt\stackrel{{\scriptstyle D.1}}{{\Leftrightarrow}}\hskip 8.5359pt(A\leq
B).$
The notation $D.1$ in the first equivalent symbol denote the following result
is deduced by definition 1. The notation $R.P$ in the second equivalent symbol
denote the following result is deduced by the properties of the inequalities
of real numbers. In the following, we also use the similar remark.
Theorem 2 The term in the inequalities can be moved from one side to the
other, that is $A+B\leq C\hskip 8.5359pt\Leftrightarrow\hskip 8.5359ptA\leq
C-B.$
Proof $A+B\leq C\hskip 8.5359pt\stackrel{{\scriptstyle
T.1}}{{\Leftrightarrow}}\hskip 8.5359ptA+B-B\leq C-B\hskip
8.5359pt\Leftrightarrow\hskip 8.5359ptA\leq C-B.$
Theorem 3 $A\leq B$, and $C\leq D\Rightarrow$ $A+C\leq B+D$.
Proof Since $A\leq B$ $\stackrel{{\scriptstyle T.1}}{{\Leftrightarrow}}\hskip
8.5359pt$ $A+C\leq B+C$ and $C\leq D$ $\stackrel{{\scriptstyle
T.1}}{{\Leftrightarrow}}\hskip 8.5359pt$ $B+C\leq B+D$, by the transitivity,
we get $A+C\leq B+D$.
Theorem 4 Let $r>0$, then $A\leq B$ $\Leftrightarrow$ $rA\leq rB$.
Proof $(rA\leq rB)\stackrel{{\scriptstyle D.1}}{{\Leftrightarrow}}\hskip
8.5359pt(ra_{1}<rb_{1})\cup[(ra_{1}=rb_{1})\cap(ra_{2}\leq rb_{2})]$
$\stackrel{{\scriptstyle R.P}}{{\Leftrightarrow}}\hskip
8.5359pt(a_{1}<b_{1})\cup[(a_{1}=b_{1})\cap(a_{2}\leq b_{2})]$
$\stackrel{{\scriptstyle D.1}}{{\Leftrightarrow}}\hskip 8.5359pt$$(A\leq B)$.
Theorem 5 Let $r<0$, then $(A\leq B)\Leftrightarrow rA\geq rB$. Especially,
$(A\leq B)\Leftrightarrow(-A\geq-B)$.
Proof $(rA\geq rB)\stackrel{{\scriptstyle D.1}}{{\Leftrightarrow}}\hskip
8.5359pt(ra_{1}>rb_{1})\cup[(ra_{1}=rb_{1})\cap(ra_{2}\geq rb_{2})]$
$\stackrel{{\scriptstyle R.P}}{{\Leftrightarrow}}\hskip 8.5359pt$
$(a_{1}<b_{1})\cup[(a_{1}=b_{1})\cap(a_{2}\leq b_{2})]$
$\stackrel{{\scriptstyle D.1}}{{\Leftrightarrow}}\hskip 8.5359pt$ $(A\leq B)$.
## 3 The operation of the set
Definition 2 Let the set of complex numbers $\mathscr{B}\subset\mathscr{C}$,
$\theta\in\mathscr{R}$. We define the rotation set of $\mathscr{B}$ by
$e^{i\theta}\mathscr{B}:=\\{Ze^{i\theta}\in\mathscr{C};Z\in\mathscr{B}\\}$.
Remark 1 $e^{i\theta}\mathscr{B}$ denote the set $\mathscr{B}$ rotates
$\theta$ radian around the origin.
Theorem 6 $We^{i\theta}\in\mathscr{B}$ $\Leftrightarrow$ $W\in
e^{-i\theta}\mathscr{B}$.
Proof Let $W:=Ze^{-i\theta}$ $\Leftrightarrow$ $Z=We^{i\theta}$. By the
definition, we get
$e^{-i\theta}\mathscr{B}:=\\{Ze^{-i\theta}\in\mathscr{C};Z\in\mathscr{B}\\}=\\{W\in\mathscr{C};We^{i\theta}\in\mathscr{B}\\}$
From the last equality we deduce that $W\in e^{-i\theta}\mathscr{B}$
$\Leftrightarrow$ $We^{i\theta}\in\mathscr{B}$.
Example 1 When $\mathscr{B}=\\{Z;Z\geq A\in\mathscr{C}\\}$ is a semi-open and
semi-closed half plane, $e^{i\theta}\mathscr{B}$ is also a semi-open and semi-
closed half plane, that is ($A=a_{1}+a_{2}i,Z=z_{1}+z_{2}i$)
$e^{i\theta}\mathscr{B}\stackrel{{\scriptstyle D.2}}{{=}}\\{Ze^{i\theta};Z\geq
A\\}=\\{Z;e^{-i\theta}Z\geq A\\}$
$=\\{Z;(z_{1}\cos\theta+z_{2}\sin\theta)+(z_{2}\cos\theta-
z_{1}\sin\theta)i\geq a_{1}+a_{2}i\\}$ $\stackrel{{\scriptstyle
D.1}}{{=}}(Z;z_{1}\cos\theta+z_{2}\sin\theta>a_{1})\cup[(z_{1}\cos\theta+z_{2}\sin\theta=a_{1})\cap(z_{2}\cos\theta-
z_{1}\sin\theta\geq a_{2})].$
The first part is an open half plane, the part in $[.]$ is a half line.
Definition 3 Let the set of complex numbers $\mathscr{B}\subset\mathscr{C}$,
$r\geq 0$. We define the dilation set of $\mathscr{B}$ by
$r\mathscr{B}:=\\{rZ\in\mathscr{C};Z\in\mathscr{B}\\}$.
Remark 2 $r\mathscr{B}$ denotes $\mathscr{B}$ makes a stretching at the
extension ratio $r$.
Theorem 7 $rW\in\mathscr{B}$ $\Leftrightarrow$ $W\in\mathscr{B}/r$.
Proof Let $W:=Z/r$ $\Leftrightarrow$ $Z=rW$. By the definition, we get
$\mathscr{B}/r:=\\{Z/r\in\mathscr{C};Z\in\mathscr{B}\\}=\\{W\in\mathscr{C};rW\in\mathscr{B}\\}.$
Namely,$W\in\mathscr{B}/r$ $\Leftrightarrow$ $rW\in\mathscr{B}$.
Definition 4 Let the set of complex numbers $\mathscr{B}\subset\mathscr{C}$,
$A\in\mathscr{C}$. We define the translation set of $\mathscr{B}$ by
$\mathscr{B}+A:=\\{Z+A\in\mathscr{C};Z\in\mathscr{B}\\}$.
Remark 3 $A+\mathscr{B}$ denotes $\mathscr{B}$ is translated.
Theorem 8 $A+W\in\mathscr{B}$ $\Leftrightarrow$ $W\in\mathscr{B}-A$.
Proof Let $W:=Z-A$ $\Leftrightarrow$ $Z=W+A$. By the definition, we get
$\mathscr{B}-A:=\\{Z-A\in\mathscr{C};Z\in\mathscr{B}\\}=\\{W\in\mathscr{C};W+A\in\mathscr{B}\\}.$
Namely, $W\in\mathscr{B}-A$ $\Leftrightarrow$ $W\in\mathscr{B}+A$.
Definition 5 Let the set of complex numbers $\mathscr{B}\subset\mathscr{C}$.
We define the inversion set of $\mathscr{B}$ by
$1/\mathscr{B}:=\\{1/Z\in\mathscr{C};Z\in\mathscr{B}\\}$.
Remark 4 $1/\mathscr{B}$ and $\mathscr{B}$ is symmetric about the unit
circumference $\\{|Z|=1\\}$, or is called the inversion transform.
Theorem 9 $1/W\in\mathscr{B}$ $\Leftrightarrow$ $W\in 1/\mathscr{B}$.
Proof Let $W:=1/Z$ $\Leftrightarrow$ $Z=1/W$. By the definition, we get
$1/\mathscr{B}:=\\{1/Z\in\mathscr{C};Z\in\mathscr{B}\\}=\\{W\in\mathscr{C};1/W\in\mathscr{B}\\}.$
Namely, $W\in 1/\mathscr{B}$ $\Leftrightarrow$ $1/W\in\mathscr{B}$.
Example 2 When $\mathscr{B}=\\{Z;Z\geq A\in\mathscr{C}\\}$ is a semi-open and
semi-closed half plane, $1/\mathscr{B}$ is a semi-open and semi-closed disc
which is symmetric about the real axis, that is ($Z:=z_{1}+z_{2}i\neq 0$)
$1/\mathscr{B}\stackrel{{\scriptstyle D.5}}{{=}}\\{1/Z;Z\geq A\\}=\\{Z;1/Z\geq
A\\}$
$=\\{Z;\frac{z_{1}}{z^{2}_{1}+z^{2}_{2}}-i\frac{z_{2}}{z^{2}_{1}+z^{2}_{2}}\geq
a_{1}+a_{2}i\\}$ $\stackrel{{\scriptstyle
D.1}}{{=}}(\frac{z_{1}}{z^{2}_{1}+z^{2}_{2}}>a_{1})\cup[(\frac{z_{1}}{z^{2}_{1}+z^{2}_{2}}=a_{1})\cap(-i\frac{z_{2}}{z^{2}_{1}+z^{2}_{2}}\geq
a_{2})]$
$=(a_{1}z^{2}_{1}+a_{1}z^{2}_{2}>z_{1})\cup[(a_{1}z^{2}_{1}+a_{1}z^{2}_{2}=z_{1})\cap(a_{2}z^{2}_{1}+a_{2}z^{2}_{2}\leq-
z_{2})].$
The first part is an open disc, its center is $\frac{1}{2a_{1}}+0i$, its
radius is $\frac{1}{2a_{1}}$. The part in $[.]$ is a circular arc whose point
isn’t the origin.
Definition 6 Let the set of complex numbers $\mathscr{B}\subset\mathscr{C}$.
We define the radication set of $\mathscr{B}$ by
$\mathscr{B}^{1/2}:=\\{Z^{1/2};Z\in\mathscr{B}\\}$.
Theorem 10 $W^{2}\in\mathscr{B}$ $\Leftrightarrow$ $W\in\mathscr{B}^{1/2}$.
Proof Let $Z^{1/2}:=\\{-W,W\\}$ $\Leftrightarrow$ $Z=W^{2}$. By the
definition, we get
$\mathscr{B}^{1/2}:=\\{Z^{1/2};\ Z\in\mathscr{B}\\}=\\{-W;\
W^{2}\in\mathscr{B}\\}\cup\\{W;\ W^{2}\in\mathscr{B}\\}=\\{W;\
W^{2}\in\mathscr{B}\\}.$
Namely, $W\in\mathscr{B}^{1/2}$ $\Leftrightarrow$ $W^{2}\in\mathscr{B}$.
Example 3 When $\mathscr{B}=\\{Z;Z\geq A\\}$ is a semi-open and semi-closed
half plane, its radication set $\mathscr{B}^{1/2}$ is a semi-open and semi-
closed domain whose boundary is hyperbola, that is
$\mathscr{B}^{1/2}\stackrel{{\scriptstyle D.6}}{{=}}\\{Z^{1/2};Z\geq
A\\}=\\{Z;Z^{2}\geq A\\}$ $=\\{Z;(z_{1}+z_{2}i)^{2}\geq a_{1}+a_{2}i\\}$
$=\\{Z;(z^{2}_{1}-z^{2}_{2})+2z_{1}z_{2}i\geq a_{1}+a_{2}i\\}$
$\stackrel{{\scriptstyle
D.1}}{{=}}[z^{2}_{1}-z^{2}_{2}>a_{1}]\cup[(z^{2}_{1}-z^{2}_{2}=a_{1})\cap(2z_{1}z_{2}\geq
a_{2})].$
The hyperbola $z^{2}_{1}-z^{2}_{2}=a_{1}$ splits the plane into three parts
(When $a_{1}=0$, it splits the plane into four parts.). If $a_{1}>0$,
$\mathscr{B}^{1/2}$ is two non-neighbor parts that doesn’t contain the origin.
If $a_{1}<0$, $\mathscr{B}^{1/2}$ is a connected part that contains the
origin. We call it the hyperbola domain.
## 4 Solving inequalities
Definition 7 To solve the inequality $f(Z)\geq g(Z)$ means to find the set
$\\{Z\in\mathscr{C};f(Z)\geq g(Z)\\}$.
Definition 8 Define $\mathscr{D}(A):=\mathscr{D}(Z\geq
A):=\\{Z\in\mathscr{C};Z\geq A\\}=\\{Z\in\mathscr{C};Z\in\mathscr{D}(Z\geq
A)\\}$.
$\mathscr{D}(A)$ is a semi-open and semi-closed perpendicular half plane that
is split by the perpendicular line $\\{Z\in\mathscr{C};ReZ=ReA\\}$.
1\. Solving the linear inequality with one unknown Let $A=re^{i\theta}$($r\geq
0$), solve the inequality $AZ-B\geq 0$.
Solution By the definition and properties of inequalities, we get the solution
set $\mathscr{S}$ is the following.
$\mathscr{S}\stackrel{{\scriptstyle D.7}}{{=}}\\{Z\in\mathscr{C};AZ-B\geq
0\\}\stackrel{{\scriptstyle T.2,4}}{{=}}\\{Z\in\mathscr{C};e^{i\theta}Z\geq
B/r\\}$ $\stackrel{{\scriptstyle
W=e^{i\theta}Z}}{{=}}\\{We^{-i\theta}\in\mathscr{C};W\geq
B/r\\}\stackrel{{\scriptstyle D.2}}{{=}}e^{-i\theta}\\{W\in\mathscr{C};W\geq
B/r\\}$ $\stackrel{{\scriptstyle D.7}}{{=}}e^{-i\theta}\mathscr{D}(Z\geq
B/r).$ $\stackrel{{\scriptstyle D.8}}{{=}}e^{-i\theta}\mathscr{D}(B/r).$
Let a perpendicular half plane rotate $\theta$ radian anticlockwise around the
origin, we get a half plane whose boundary is a oblique line, that is the
solution set $\mathscr{S}$ .
2\. Solving the linear inequalities with one unknown Let
$A=re^{i\theta}$,$C=ue^{i\phi}$($r,u\geq 0$),solve the inequalities $AZ-B\geq
0$, $CZ-D\geq 0$.
Solution Since the solution sets of two inequalities are
$e^{-i\theta}\mathscr{D}(B/r\\},\hskip
14.22636pte^{-i\phi}\mathscr{D}(B/u\\},$
respectively. The solution set of the inequalities $\mathscr{S}$ is the
following.
$\mathscr{S}=[e^{-i\theta}\mathscr{D}(B/r)]\cap[e^{-i\phi}\mathscr{D}(B/u)].$
The solution set $\mathscr{S}$ denotes the intersection of two half planes .
3\. Solving the linear fractional inequality Let $B-AC\neq 0$, solve the
inequality $\frac{AZ+B}{Z+C}\geq D$.
Solution Let $B-AC:=re^{i\theta}$, then the solution set $\mathscr{S}$ is the
following.
$\mathscr{S}\stackrel{{\scriptstyle
D.7}}{{=}}\\{Z\in\mathscr{C};\frac{AZ+B}{Z+C}\geq D\\}\stackrel{{\scriptstyle
T.1,4}}{{=}}\\{Z\in\mathscr{C};\frac{1}{e^{-i\theta}(Z+C)}\geq\frac{D-A}{r}\\}$
$\stackrel{{\scriptstyle
D.8}}{{=}}\\{Z\in\mathscr{C};\frac{1}{e^{-i\theta}(Z+C)}\in\mathscr{D}(\frac{D-A}{r})\\}\stackrel{{\scriptstyle
T.9}}{{=}}\\{Z\in\mathscr{C};e^{-i\theta}(Z+C)\in
1/\mathscr{D}(\frac{D-A}{r})\\}$ $\stackrel{{\scriptstyle
T.6}}{{=}}\\{Z\in\mathscr{C};(Z+C)\in\frac{1}{\mathscr{D}(\frac{D-A}{r})}\cdot
e^{i\theta}\\}\stackrel{{\scriptstyle
T.8}}{{=}}\\{Z\in\mathscr{C};Z\in\frac{1}{\mathscr{D}(\frac{D-A}{r})}\cdot
e^{i\theta}-C\\}$ ${=}\frac{1}{\mathscr{D}(\frac{D-A}{r})}\cdot
e^{i\theta}-C.$
The solution set $\mathscr{S}$ is a semi-open and semi-closed disc. Let a
semi-open and semi-closed perpendicular half plane
$\mathscr{D}(\frac{D-A}{r})$ become a disc
$\frac{1}{\mathscr{D}(\frac{D-A}{r})}$ by the inversion transform defined as
example 2, then let it rotate $\theta$ radian around the origin, we get the
disc $\frac{1}{\mathscr{D}(\frac{D-A}{r})}\cdot e^{i\theta}$, finally
translating the disc we get the solution set
$\frac{1}{\mathscr{D}(\frac{D-A}{r})}\cdot e^{i\theta}-C$.
4\. Solving the inequality of the second order Let $A\neq 0$, solve the
inequality of the second order $AZ^{2}+BZ+C\geq 0$.
Solution Let $A:=re^{i\theta}$, then the solution set $\mathscr{S}$ is the
following.
$\mathscr{S}\stackrel{{\scriptstyle D.7}}{{=}}\\{Z;AZ^{2}+BZ+C\geq
0\\}\stackrel{{\scriptstyle
T.1,4}}{{=}}\\{Z;e^{i\theta}(Z+\frac{B}{2A})^{2}\geq\frac{B^{2}-4AC}{4rA}\\}$
$\stackrel{{\scriptstyle
D.8}}{{=}}\\{Z;[e^{i\theta/2}(Z+\frac{B}{2A})]^{2}\in\mathscr{D}(\frac{B^{2}-4AC}{4rA})\\}\stackrel{{\scriptstyle
T.10}}{{=}}\\{Z;e^{i\theta/2}(Z+\frac{B}{2A})\in\mathscr{D}^{1/2}(\frac{B^{2}-4AC}{4rA})\\}$
$\stackrel{{\scriptstyle
T.6}}{{=}}\\{Z;Z+\frac{B}{2A}\in\mathscr{D}^{1/2}(\frac{B^{2}-4AC}{4rA})\cdot
e^{-i\theta/2}\\}$ $\stackrel{{\scriptstyle
T.8}}{{=}}\\{Z;Z\in\mathscr{D}^{1/2}(\frac{B^{2}-4AC}{4rA})\cdot
e^{-i\theta/2}-\frac{B}{2A}\\}{=}\mathscr{D}^{1/2}(\frac{B^{2}-4AC}{4rA})\cdot
e^{-i\theta/2}-\frac{B}{2A}.$
The solution set $\mathscr{S}$ is a semi-open and semi-closed hyperbola
domain. Let a semi-open and semi-closed perpendicular half plane
$\mathscr{D}(\frac{B^{2}-4AC}{4rA})$ become a hyperbola domain
$\mathscr{D}^{1/2}(\frac{B^{2}-4AC}{4rA})$ by the radication defined as
example 3, then let the hyperbola domain rotate $\theta$ radian anticlockwise
around the origin, we get the hyperbola domain
$\mathscr{D}^{1/2}(\frac{B^{2}-4AC}{4rA})\cdot e^{-i\theta/2}$, finally
translating it we get the solution set
$\mathscr{D}^{1/2}(\frac{B^{2}-4AC}{4rA})\cdot e^{-i\theta/2}-\frac{B}{2A}$.
## References
* [1] Xia Daoxing, Wu Zhuoren, Yan Shaozong, Shu Wuchang. Real function and functional analysis. Beijing, Higher Education Press, 1983(in Chinese)
Sun Daochun
School of Mathematical Sciences
South China Normal University
Guangzhou 510631
People’s Republic of China
E-mail : sundch@scnu.edu.cn
Gu Zhendong
School of Mathematical Sciences
South China Normal University
Guangzhou 510631
People’s Republic of China
E-mail : guzhd@qq.com
Liu Weiqun
School of Mathematic
Jia Ying University
Guangdong Meizhou 514015
People’s Republic of China
E-mail : mzlwq226@21cn.com
Yue Chao
School of Mathematical Sciences
South China Normal University
Guangzhou 510631
People’s Republic of China
E-mail : 1048348982@qq.com
|
arxiv-papers
| 2010-03-22T10:11:42 |
2024-09-04T02:49:09.249943
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sun Daochun, Gu Zhendong, Liu Weiqun, Yue Chao",
"submitter": "Sun DaoChun",
"url": "https://arxiv.org/abs/1003.4906"
}
|
1003.4999
|
# On Normal Forms of Singular Levi-Flat Real Analytic Hypersurfaces
Arturo Fernández-Pérez Instituto de Matemática Pura e Aplicada, IMPA Estrada
Dona Castorina, 110, 22460-320. Rio de Janeiro, RJ, Brazil. afernan@impa.br
###### Abstract.
Let $F(z)=\mathcal{R}e(P(z))+h.o.t$ be such that $M=(F=0)$ defines a germ of
real analytic Levi-flat at $0\in\mathbb{C}^{n}$, $n\geq{2}$, where $P(z)$ is a
homogeneous polynomial of degree $k$ with an isolated singularity at
$0\in\mathbb{C}^{n}$ and Milnor number $\mu$. We prove that there exists a
holomorphic change of coordinate $\phi$ such that
$\phi(M)=(\mathcal{R}e(h)=0)$, where $h(z)$ is a polynomial of degree $\mu+1$
and $j^{k}_{0}(h)=P$.
###### Key words and phrases:
Levi-flat hypersurfaces - Holomorphic foliations
###### 2010 Mathematics Subject Classification:
Primary 32V40 - 37F75
## 1\. Introduction and Statement of the results
Let $M$ be a germ at $0\in\mathbb{C}^{n}$ of a real codimension one
irreducible analytic set. For the sake of simplicity we will denote germs and
representative of germs by the same letter. Since $M$ is real analytic of
codimension one and irreducible, it can be defined in $\mathbb{C}^{n}$ by
$(F=0)$, where $F$ is an irreducible germ of real analytic function. The
singular set of $M$ is defined by $sing(M)=(F=0)\cap(dF=0)$ and its smooth
part $(F=0)\backslash(dF=0)$ will denoted by $M^{*}$. The Levi distribution
$L$ on $M^{*}$ is defined by $L_{p}:=ker(\partial{F}(p))\subset
T_{p}M^{*}=ker(dF(p))$, for any $p\in M^{*}$.
###### Definition 1.1.
We say $M$ is Levi-flat if the Levi distribution on $M^{*}$ is integrable.
###### Remark 1.2.
The integrability condition of $L$ implies that $M^{*}$ is tangent to a real
codimension one foliation $\mathcal{L}$. Since the hyperplanes $L_{p}$, $p\in
M^{*}$, are complex, the leaves of $\mathcal{L}$ are complex codimension one
holomorphic submanifolds immersed on $M^{*}$.
###### Remark 1.3.
If the hypersurface $M$ is defined by $(F=0)$ then the Levi distribution $L$
on $M$ can be defined by the real analytic 1-form
$\eta=i(\partial{F}-\bar{\partial}F)$, which will be called the Levi 1-form of
$F$. The integrability condition is equivalent to
$(\partial{F}-\bar{\partial}F)\wedge\partial\bar{\partial}F|_{M^{*}}=0$
In the case of a real analytic smooth Levi-flat hypersurface $M$ in
$\mathbb{C}^{n}$, its local structure is very well understood, according to E.
Cartan, around each $p\in M$ we can find local holomorphic coordinates
$z_{1},\ldots,z_{n}$ such that $M=\\{\mathcal{R}e(z_{1})=0\\}$.
More recently D. Burns and X. Gong [B-G] have proved an analogous result in
the case $M=F^{-1}(0)$ Levi-flat, where
$F:(\mathbb{C}^{n},0)\rightarrow(\mathbb{R},0)$, $n\geq 2$, is a germ of real
analytic function such that
$F(z_{1},\ldots,z_{n})=\mathcal{R}e(z_{1}^{2}+\ldots+z_{n}^{2})+h.o.t.$
They show that there exists a germ of biholomorphism
$\phi:(\mathbb{C}^{n},0)\rightarrow(\mathbb{C}^{n},0)$ such that
$\phi(M)=(\mathcal{R}e(z_{1}^{2}+\ldots+z_{n}^{2})=0)$.
In [C-L], the authors prove the above result by using the theory of
holomorphic foliations. In this paper we are interested in finding similar
normal forms in a situation more general. Our main result is the following:
###### Theorem 1.
Let $M=F^{-1}(0)$, where $F:(\mathbb{C}^{n},0)\rightarrow(\mathbb{R},0)$,
$n\geq 2$, be a germ of irreducible real analytic function such that
1. (1)
$F(z_{1},\ldots,z_{n})=\mathcal{R}e(P(z_{1},\ldots,z_{n}))+h.o.t$, where $P$
is a homogeneous polynomial of degree $k$ with an isolated singularity at
$0\in\mathbb{C}^{n}$.
2. (2)
The Milnor number of $P$ at $0\in\mathbb{C}^{n}$ is $\mu$.
3. (3)
$M$ is Levi-flat.
Then there exists a germ of biholomorphism
$\phi:(\mathbb{C}^{n},0)\rightarrow(\mathbb{C}^{n},0)$ such that
$\phi(M)=(\mathcal{R}e(h)=0)$, where $h(z)$ is a polynomial of degree $\mu+1$
and $j^{k}_{0}(h)=P$.
###### Remark 1.4.
In particular, we obtain the result of [B-G].
###### Theorem 2.
In the same spirit we have the following generalization: Let $M=F^{-1}(0)$,
where $F:(\mathbb{C}^{n},0)\rightarrow(\mathbb{R},0)$, $n\geq 3$, be a germ of
irreducible real analytic function such that
1. (1)
$F(z_{1},\ldots,z_{n})=\mathcal{R}e(Q(z_{1},\ldots,z_{n}))+h.o.t$, where $Q$
is a quasihomogeneous polynomial of degree $d$ with an isolated singularity at
$0\in\mathbb{C}^{n}$.
2. (2)
$M$ is Levi-flat.
Then there exists a germ of biholomorphism
$\phi:(\mathbb{C}^{n},0)\rightarrow(\mathbb{C}^{n},0)$ such that
$\phi(M)=(\mathcal{R}e(Q(z)+\sum_{j}c_{j}e_{j}(z))=0),$
where $e_{1},\ldots,e_{s}$ are the elements of the monomial basis of the local
algebra of $Q$ such that $deg(e_{j})>d$ and $c_{j}\in\mathbb{C}$.
Acknowledgments
I want to express my thanks to my advisor Alcides Lins Neto, for valuable
conversations. The author would like to thank also IMPA, where this work was
developed. I also want to thank the referee for his suggestions.
## 2\. Preliminaries
Let us fix some notations that will be used from now on.
1. (1)
$\mathcal{O}_{n}$ : The ring of germs of holomorphic functions at
$0\in\mathbb{C}^{n}$.
$\mathcal{O}(U)=$ set of holomorphic functions in the open set
$U\subset\mathbb{C}^{n}$.
2. (2)
$\mathcal{O}^{*}_{n}=\\{f\in\mathcal{O}_{n}/f(0)\neq 0\\}$.
$\mathcal{O}^{*}(U)=\\{f\in\mathcal{O}(U)/f(z)\neq 0,\forall z\in U\\}$.
3. (3)
$\mathcal{M}_{n}=\\{f\in\mathcal{O}_{n}/f(0)=0\\}$ maximal ideal of
$\mathcal{O}_{n}$.
4. (4)
$\mathcal{A}_{n}$ : The ring of germs at $0\in\mathbb{C}^{n}$ of complex
valued real analytic functions.
5. (5)
$\mathcal{A}_{n\mathbb{R}}$ : The ring of germs at $0\in\mathbb{C}^{n}$ of
real valued real analytic functions. Note that $F\in\mathcal{A}_{n}$ is in
$\mathcal{A}_{n\mathbb{R}}$ if and only if $F=\bar{F}$.
6. (6)
$Diff(\mathbb{C}^{n},0)$ : The group of germs at $0\in\mathbb{C}^{n}$ of
holomorphic diffeomorphisms
$f:(\mathbb{C}^{n},0)\rightarrow(\mathbb{C}^{n},0)$ with the operation of
composition.
7. (7)
$j^{k}_{0}(f)$ : The $k$-jet at $0\in\mathbb{C}^{n}$ of $f\in\mathcal{O}_{n}$.
###### Definition 2.1.
Two germs $f,g\in\mathcal{O}_{n}$ are said to be right equivalent, if there
exists $\phi\in Diff(\mathbb{C}^{n},0)$ such that $f\circ\phi^{-1}=g$.
The local algebra of $f\in\mathcal{O}_{n}$ is by definition
$A_{f}=\mathcal{O}_{n}/(\partial{f}/\partial{z}_{1},\ldots,\partial{f}/\partial{z}_{n}).$
###### Definition 2.2.
Define by $\mu(f,0):=dimA_{f}$, the Milnor number of $f$ at
$0\in\mathbb{C}^{n}$.
Morse Lemma can now be rephrased by saying that if $0\in\mathbb{C}^{n}$ is an
isolated singularity of $f$ with Milnor number $\mu(f,0)=1$ then $f$ is right
equivalent to its second jet. The next lemma is a generalization of Morse’s
Lemma. We refer to [A-G-V], pg.121.
###### Lemma 2.3.
Suppose $0\in\mathbb{C}^{n}$ is an isolated singularity of
$f\in\mathcal{M}_{n}$ with Milnor number $\mu$. Then $f$ is right equivalent
to $j^{\mu+1}_{0}(f)$.
### 2.1. The complexification
In this section we state some general facts about complexification of germs of
real analytic functions.
Given $F\in\mathcal{A}_{n}$; we can write its Taylor series at
$0\in\mathbb{C}^{n}$ as
(2.1) $F(z)=\sum_{\mu,\nu}F_{\mu\nu}z^{\mu}\bar{z}^{\nu},$
where $F_{\mu\nu}\in\mathbb{C}$, $\mu=(\mu_{1},\ldots,\mu_{n})$,
$\nu=(\nu_{1},\ldots,\nu_{n})$, $z^{\mu}=z_{1}^{\mu_{1}}\ldots
z_{n}^{\mu_{n}}$,
$\bar{z}^{\nu}=\bar{z}_{1}^{\nu_{1}}\ldots\bar{z}_{n}^{\nu_{n}}$. When
$F\in\mathcal{A}_{n\mathbb{R}}$ then the coefficients $F_{\mu\nu}$ satisfy
$\bar{F}_{\mu\nu}=F_{\nu\mu}.$
The complexification $F_{\mathbb{C}}\in\mathcal{O}_{2n}$ of $F$ is defined by
the series
(2.2) $F_{\mathbb{C}}(z,w)=\sum_{\mu,\nu}F_{\mu\nu}z^{\mu}w^{\nu}.$
If $F\in\mathcal{A}_{n\mathbb{R}}$, $F(0)=0$ and $M=F^{-1}(0)$ defines a Levi-
flat, the complexification $\eta_{\mathbb{C}}$ of its Levi 1-form
$\eta=i(\partial{F}-\bar{\partial}F)$ can be written as
$\eta_{\mathbb{C}}=i(\partial_{z}F_{\mathbb{C}}-\partial_{w}F_{\mathbb{C}})=i\sum_{\mu,\nu}(F_{\mu\nu}w^{\nu}d(z^{\mu})-F_{\mu\nu}z^{\mu}d(w^{\nu})).$
The complexification $M_{\mathbb{C}}$ of $M$ is defined as
$M_{\mathbb{C}}=F^{-1}_{\mathbb{C}}(0)$ and its smooth part is
$M^{*}_{\mathbb{C}}=M_{\mathbb{C}}\backslash(dF_{\mathbb{C}}=0)$. The
integrability condition of $\eta=i(\partial{F}-\bar{\partial}F)|_{M^{*}}$
implies that $\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}}$ is integrable.
Therefore $\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}}=0$ defines a foliation
$\mathcal{L}_{\mathbb{C}}$ on $M^{*}_{\mathbb{C}}$ that will be called the
complexification of $\mathcal{L}$.
###### Definition 2.4.
The algebraic dimension of $sing(M)$ is the complex dimension of the singular
set of $M_{\mathbb{C}}$.
Consider a germ at $0\in\mathbb{C}^{2}$ of real analytic Levi-flat $M=(F=0)$,
where $F$ is irreducible in $\mathcal{A}_{2\mathbb{R}}$. Let $F_{\mathbb{C}}$,
$M_{\mathbb{C}}=(F_{\mathbb{C}}=0)\subset(\mathbb{C}^{4},0)$ and
$M_{\mathbb{C}}^{*}$ be as before. We will assume that the power series that
defines $F_{\mathbb{C}}$ converges in a neighborhood of
$\bar{\bigtriangleup}=\\{(z,w)\in\mathbb{C}^{4}/|z|,|w|\leq 1\\}$, so that
$F_{\mathbb{C}}(z,\bar{z})=F(z)$ for all $|z|\leq 1$.
Let $V:=M_{\mathbb{C}}^{*}\backslash
sing(\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}})$ and denote $L_{p}$ the leaf of
$\mathcal{L}_{\mathbb{C}}$ through $p$, where $p\in V$. In this situation we
have the following important Lemma of [C-L].
###### Lemma 2.5.
In the above situation, for any $p=(z_{0},w_{0})\in V$ the leaf $L_{p}$ is
closed in $M^{*}_{\mathbb{C}}$.
In the proof of theorem 1 we will use the following result of [C-L].
###### Theorem 2.6.
Let $M=F^{-1}(0)$ be a germ of an irreducible real analytic Levi-flat
hypersurface at $0\in\mathbb{C}^{n}$, $n\geq{2}$, with Levi 1-form
$\eta=i(\partial{F}-\bar{\partial}F)$. Assume that the algebraic dimension of
$sing(M)\leq 2n-4$. Then there exists an unique germ at $0\in\mathbb{C}^{n}$
of holomorphic codimension one foliation $\mathcal{F}_{M}$ tangent to $M$, if
one of the following conditions is fulfilled:
1. (1)
$n\geq 3$ and
$cod_{M_{\mathbb{C}}^{*}}(sing(\eta_{\mathbb{C}}|_{M_{\mathbb{C}}^{*}}))\geq
3$.
2. (2)
$n\geq 2$,
$cod_{M_{\mathbb{C}}^{*}}(sing(\eta_{\mathbb{C}}|_{M_{\mathbb{C}}^{*}}))\geq
2$ and $\mathcal{L}_{\mathbb{C}}$ has a non-constant holomorphic first
integral.
Moreover, in both cases the foliation $\mathcal{F}_{M}$ has a non-constant
holomorphic first integral $f$ such that $M=(Re(f)=0)$.
## 3\. Proof of theorem 1
Let $M=F^{-1}(0)\subset(\mathbb{C}^{n},0)$ be a Levi-flat, where
$F(z)=\mathcal{R}e(P(z))+h.o.t$ with $P$ be a homogeneous polynomial of degree
$k\geq 2$ with an isolated singularity at $0\in\mathbb{C}^{n}$ and Milnor
number $\mu$. We want to prove that there exists $\phi\in
Diff(\mathbb{C}^{n},0)$ such that $\phi(M)=(\mathcal{R}e(h)=0)$, where $h$ is
a polynomial of degree $\mu+1$.
The idea is to use theorem 2.6 to prove that there exists a germ
$f\in\mathcal{O}_{n}$ such that the foliation $\mathcal{F}$ defined by $df=0$
is tangent to $M$ and $M=(\mathcal{R}e(f)=0)$. The foliation $\mathcal{F}$ can
viewed as an extension to a neighborhood of $0\in\mathbb{C}^{n}$ of the Levi
foliation $\mathcal{L}$ on $M^{*}$.
Suppose for a moment that $M=(\mathcal{R}e(f)=0)$ and let us conclude the
proof. Without lost of generality, we can suppose that $f$ is not a power in
$\mathcal{O}_{n}$. In this case $\mathcal{R}e(f)$ is irreducible (cf. [C-L]).
This implies that $\mathcal{R}e(f)=U.F$, where $U\in\mathcal{A}_{n\mathbb{R}}$
and $U(0)\neq 0$. Let $\sum_{j\geq k}f_{j}$ be the taylor series of $f$, where
$f_{j}$ is a homogeneous polynomial of degree $j$, $j\geq k$. Then
$\mathcal{R}e(f_{k})=j^{k}_{0}(\mathcal{R}e(f))=j^{k}_{0}(U.F)=U(0).\mathcal{R}e(P(z_{1},\ldots,z_{n})).$
Hence $f_{k}(z_{1},\ldots,z_{n})=U(0).P(z_{1},\ldots,z_{n})$. We can suppose
that $U(0)=1$, so that
(3.1) $\displaystyle f(z)=P(z)+h.o.t$
In particular, $\mu=\mu(f,0)=\mu(P,0)$, $f\in\mathcal{M}_{n}$, because $P$ has
isolated singularity at $0\in\mathbb{C}^{n}$. Hence by lemma 2.3, $f$ is right
equivalent to $j^{\mu+1}_{0}(f)$, i.e. there exists $\phi\in
Diff(\mathbb{C}^{n},0)$ such that $h:=f\circ\phi^{-1}=j^{\mu+1}_{0}(f)$.
Therefore, $\phi(M)=(\mathcal{R}e(h)=0)$ and this will conclude the proof of
theorem 1.
Let us prove that we can apply theorem 2.6. We can write
$F(z)=\mathcal{R}e(P(z_{1},\ldots,z_{n}))+H(z_{1},\ldots,z_{n}),$
where $H:(\mathbb{C}^{n},0)\rightarrow(\mathbb{R},0)$ is a germ of real-
analytic function and $j^{k}_{0}(H)=0$. For simplicity, we assume that $P$ has
real coefficients. Then we get the complexification
$F_{\mathbb{C}}(z,w)=\frac{1}{2}(P(z)+P(w))+H_{\mathbb{C}}(z,w)$
and $M_{\mathbb{C}}=F^{-1}_{\mathbb{C}}(0)\subset(\mathbb{C}^{2n},0)$. In the
general case, replacing $P(w)=\sum a_{j}w^{j}$ by
$\tilde{P}(w)=\sum\bar{a}_{j}w^{j}$, we will recover each step of proof.
Since $P(z)$ has an isolated singularity at $0\in\mathbb{C}^{n}$, we get
$sing(M_{\mathbb{C}})=\\{0\\}$, and so the algebraic dimension of $sing(M)$ is
$0$. On other hand, the complexification of
$\eta=i(\partial{F}-\bar{\partial}F)$ is
$\eta_{\mathbb{C}}=i(\partial_{z}F_{\mathbb{C}}-\partial_{w}F_{\mathbb{C}}).$
Recall that $\eta|_{M^{*}}$ and $\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}}$
define $\mathcal{L}$ and $\mathcal{L}_{\mathbb{C}}$. Now we compute
$sing(\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}})$. We can write
$dF_{\mathbb{C}}=\alpha+\beta$, with
$\alpha=\sum_{j=1}^{n}\frac{\partial{F_{\mathbb{C}}}}{\partial{z}_{j}}dz_{j}:=\frac{1}{2}\sum_{j=1}^{n}(\frac{\partial{P}}{\partial{z}_{j}}(z)+A_{j})dz_{j}$
and
$\beta=\sum_{j=1}^{n}\frac{\partial{F_{\mathbb{C}}}}{\partial{w}_{j}}dw_{j}:=\frac{1}{2}\sum_{j=1}^{n}(\frac{\partial{P}}{\partial{w}_{j}}(w)+B_{j})dw_{j},$
where
$\frac{1}{2}\sum_{j=1}^{n}A_{j}dz_{j}=\sum_{j=1}^{n}\frac{\partial{H_{\mathbb{C}}}}{\partial{z}_{j}}dz_{j}$
and
$\frac{1}{2}\sum_{j=1}^{n}B_{j}dw_{j}=\sum_{j=1}^{n}\frac{\partial{H_{\mathbb{C}}}}{\partial{w}_{j}}dw_{j}$.
Then $\eta_{\mathbb{C}}=i(\alpha-\beta)$, and so
(3.2)
$\displaystyle\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}}=(\eta_{\mathbb{C}}+idF_{\mathbb{C}})|_{M^{*}_{\mathbb{C}}}=2i\alpha|_{M^{*}_{\mathbb{C}}}=-2i\beta|_{M^{*}_{\mathbb{C}}}.$
In particular, $\alpha|_{M^{*}_{\mathbb{C}}}$ and
$\beta|_{M^{*}_{\mathbb{C}}}$ define $\mathcal{L}_{\mathbb{C}}$. Therefore
$sing(\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}})$ can be splited in two parts.
Let $M_{1}=\\{(z,w)\in
M_{\mathbb{C}}|\frac{\partial{F}_{\mathbb{C}}}{\partial{w_{j}}}\neq 0$ for
some $j=1,\ldots,n\\}$ and $M_{2}=\\{(z,w)\in
M_{\mathbb{C}}|\frac{\partial{F}_{\mathbb{C}}}{\partial{z_{j}}}\neq 0$ for
some $j=1,\ldots,n\\}$, note that $M_{\mathbb{C}}=M_{1}\cup M_{2}$; if we
denote by
$X_{1}:=M_{1}\cap\\{\frac{\partial{P}}{\partial{z}_{1}}(z)+A_{1}=\ldots=\frac{\partial{P}}{\partial{z}_{n}}(z)+A_{n}=0\\}$
and
$X_{2}:=M_{2}\cap\\{\frac{\partial{P}}{\partial{w}_{1}}(w)+B_{1}=\ldots=\frac{\partial{P}}{\partial{w}_{n}}(w)+B_{n}=0\\},$
then $sing(\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}})=X_{1}\cup X_{2}$. Since
$P\in\mathbb{C}[z_{1},\ldots,z_{n}]$ has an isolated singularity at
$0\in\mathbb{C}^{n}$, we conclude that
$cod_{M^{*}_{\mathbb{C}}}sing(\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}})=n$.
If $n\geq 3$, we can directly apply Theorem 2.6 and the proof ends. In the
case $n=2$, we are going to prove that $\mathcal{L}_{\mathbb{C}}$ has a non-
constant holomorphic first integral.
We begin by a blow-up at $0\in\mathbb{C}^{4}$. Let
$F(x,y)=\mathcal{R}e(P(x,y))+h.o.t$ and $M=F^{-1}(0)$ Levi-flat. Its
complexification can be written as
$F_{\mathbb{C}}(x,y,z,w)=\frac{1}{2}P(x,y)+\frac{1}{2}P(z,w)+H_{\mathbb{C}}(x,y,z,w).$
We take the exceptional divisor $D=\mathbb{P}^{3}$ of the blow-up
$\pi:(\tilde{\mathbb{C}}^{4},\mathbb{P}^{3})\rightarrow(\mathbb{C}^{4},0)$
with homogeneous coordinates $[a:b:c:d]$,
$(a,b,c,d)\in\mathbb{C}^{4}\backslash\\{0\\}$. The intersection of the strict
transform $\tilde{M}_{\mathbb{C}}$ of $M_{\mathbb{C}}$ by $\pi$ with the
divisor $D=\mathbb{P}^{3}$ is the surface
$Q=\\{[a:b:c:d]\in\mathbb{P}^{3}/P(a,b)+P(c,d)=0\\},$
which is an irreducible smooth surface.
Consider for instance the chart $(W,(t,u,z,v))$ of $\tilde{\mathbb{C}}^{4}$
where
$\pi(t,u,z,v)=(t.z,u.z,z,v.z)=(x,y,z,w).$
We have
$F_{\mathbb{C}}\circ\pi(t,u,z,v)=z^{k}(\frac{1}{2}P(t,u)+\frac{1}{2}P(1,v)+zH_{1}(t,u,z,v)),$
where $H_{1}(t,u,z,v)=H(tz,uz,z,vz)/z^{k+1}$, which implies that
$\tilde{M}_{\mathbb{C}}\cap
W=(\frac{1}{2}P(t,u)+\frac{1}{2}P(1,v)+zH_{1}(t,u,z,v)=0)$
and so $Q\cap W=(z=P(t,u)+P(1,v)=0)$.
On the other hand, as we have seen in $(3.2)$, the foliation
$\mathcal{L}_{\mathbb{C}}$ is defined by $\alpha|_{M^{*}_{\mathbb{C}}}=0$,
where
$\alpha=\frac{1}{2}\frac{\partial{P}}{\partial{x}}dx+\frac{1}{2}\frac{\partial{P}}{\partial{y}}dy+\frac{\partial{H}_{\mathbb{C}}}{\partial{x}}dx+\frac{\partial{H}_{\mathbb{C}}}{\partial{y}}dy.$
In particular, we get
$\pi^{*}(\alpha)=z^{k-1}(\frac{1}{2}\frac{\partial{P}}{\partial{x}}(t,u)zdt+\frac{1}{2}\frac{\partial{P}}{\partial{y}}(t,u)zdu+\frac{1}{2}kP(t,u)dz+z\theta),$
where
$\theta=\pi^{*}(\frac{\partial{H}_{\mathbb{C}}}{\partial{x}}dx+\frac{\partial{H}_{\mathbb{C}}}{\partial{y}}dy)/z^{k}$.
Hence, $\tilde{\mathcal{L}}_{\mathbb{C}}$ is defined by
(3.3)
$\displaystyle\alpha_{1}=\frac{1}{2}\frac{\partial{P}}{\partial{x}}(t,u)zdt+\frac{1}{2}\frac{\partial{P}}{\partial{y}}(t,u)zdu+\frac{1}{2}kP(t,u)dz+z\theta.$
Since $Q\cap W=(z=P(t,u)+P(1,v)=0)$, we see that $Q$ is
$\tilde{\mathcal{L}}_{\mathbb{C}}$-invariant. In particular, $S:=Q\backslash
sing(\tilde{\mathcal{L}}_{\mathbb{C}})$ is a leaf of
$\tilde{\mathcal{L}}_{\mathbb{C}}$. Fix $p_{0}\in S$ and a transverse section
$\sum$ through $p_{0}$. Let $G\subset Diff(\sum,p_{0})$ be the holonomy group
of the leaf $S$ of $\tilde{\mathcal{L}}_{\mathbb{C}}$. Since $dim(\sum)=1$, we
can think that $G\subset Diff(\mathbb{C},0)$. Let us prove that $G$ is finite
and linearizable.
At this part we use that the leaves of $\tilde{\mathcal{L}}_{\mathbb{C}}$ are
closed (see lemma 2.5).
Let $G^{\prime}=\\{f^{\prime}(0)/f\in G\\}$ and consider the homomorphism
$\phi:G\rightarrow G^{\prime}$ defined by $\phi(f)=f^{\prime}(0)$. We assert
that $\phi$ is injective. In fact, assume that $\phi(f)=1$ and by
contradiction that $f\neq id$. In this case $f(z)=z+a.z^{r+1}+\ldots$, where
$a\neq 0$. According to [L], the pseudo-orbits of this transformation
accumulate at $0\in(\sum,0)$, contradicting that the leaves of
$\tilde{\mathcal{L}}_{\mathbb{C}}$ are closed. Now, it suffices to prove that
any element $g\in G$ has finite order (cf. [M-M]). In fact, if
$\phi(g)=g^{\prime}(0)$ is a root of unity then $g$ has finite order because
$\phi$ is injective. On the other hand, if $g^{\prime}(0)$ was not a root of
unity then $g$ would have pseudo-orbits accumulating at $0\in(\sum,0)$ (cf.
[L]). Hence, all transformations of $G$ have finite order and $G$ is
linearizable.
This implies that there is a coordinate system $w$ on $(\sum,0)$ such that
$G=\langle w\rightarrow\lambda w\rangle$, where $\lambda$ is a
$d^{th}$-primitive root of unity (cf. [M-M]). In particular, $\psi(w)=w^{d}$
is a first integral of $G$, that is $\psi\circ g=\psi$ for any $g\in G$.
Let $Z$ be the union of the separatrices of $\mathcal{L}_{\mathbb{C}}$ through
$0\in\mathbb{C}^{4}$ and $\tilde{Z}$ be its strict transform under $\pi$. The
first integral $\psi$ can be extended to a first integral
$\varphi:\tilde{M}_{\mathbb{C}}\backslash\tilde{Z}\rightarrow\mathbb{C}$ be
setting
$\varphi(p)=\psi(\tilde{L}_{p}\cap\sum),$
where $\tilde{L}_{p}$ denotes the leaf of $\tilde{\mathcal{L}}_{\mathbb{C}}$
through $p$. Since $\psi$ is bounded (in a compact neighborhood of
$0\in\sum$), so is $\varphi$. It follows from Riemann extension theorem that
$\varphi$ can be extended holomorphically to $\tilde{Z}$ with
$\varphi(\tilde{Z})=0$. This provides the first integral and finishes the
proof of theorem 1.
## 4\. Quasihomogeneous polynomials
In this section, we state some general facts about normal forms of
quasihomogeneous polynomials.
###### Definition 4.1.
The Newton support of germ $f=\sum a_{ij}x^{i}y^{j}$ is defined as
$supp(f)=\\{(i,j):a_{ij}\neq 0\\}$.
###### Definition 4.2.
A holomorphic function $f:(\mathbb{C}^{n},0)\rightarrow(\mathbb{C},0)$ is said
to be quasihomogeneous of degree $d$ with indices
$\alpha_{1},\ldots,\alpha_{n}$, if for any $\lambda\in\mathbb{C}$ and
$(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}$, we have
$f(\lambda^{\alpha_{1}}z_{1},\ldots,\lambda^{\alpha_{n}}z_{n})=\lambda^{d}f(z_{1},\ldots,z_{n}).$
The index $\alpha_{s}$ is also called the weight of the variable $z_{s}$.
In the above situation, if $f=\sum a_{k}x^{k}$, $k=(k_{1},\ldots,k_{n})$,
$x^{k}=x_{1}^{k_{1}}\ldots x^{k_{n}}$, then
$supp(f)\subset\Gamma=\\{k:a_{1}k_{1}+\ldots+a_{n}k_{n}=d\\}$. The set
$\Gamma$ is called the diagonal. Usually one takes $\alpha_{i}\in\mathbb{Q}$
and $d=1$.
One can define the quasihomogeneous filtration of the ring $\mathcal{O}_{n}$.
It consists of the decreasing family of ideals
$\mathcal{A}_{d}\subset\mathcal{O}_{n}$,
$\mathcal{A}_{d^{\prime}}\subset\mathcal{A}_{d}$ for $d<d^{\prime}$. Here
$\mathcal{A}_{d}=\\{Q:$ degrees of monomials from $supp(Q)$ are $deg(Q)\geq
d\\}$; (the degree is quasihomogeneous).
When $\alpha_{1}=\ldots=\alpha_{n}=1$, this filtration coincides with the
usual filtration by the usual degree.
###### Definition 4.3.
A function $f$ is called semiquasihomogeneous if $f=Q+F^{\prime}$, where $Q$
is quasihomogeneous of degree $d$ of finite multiplicity and
$F^{\prime}\in\mathcal{A}_{d^{\prime}}$, $d^{\prime}>d$.
We will use the following result (cf. [A]).
###### Theorem 4.4.
Let $f$ be a semiquasihomogeneous function, $f=Q+F^{\prime}$ with
quasihomogeneous $Q$ of finite multiplicity. Then $f$ is right equivalent to
the function $Q+\sum_{j}c_{j}e_{j}(z)$, where $e_{1},\ldots,e_{s}$ are the
elements of the monomial basis of the local algebra $A_{Q}$ such that
$deg(e_{j})>d$ and $c_{j}\in\mathbb{C}$.
###### Example 4.5.
If $f=Q+F^{\prime}$ is semiquasihomogeneous and $Q(x,y)=x^{2}y+y^{k}$, then
$f$ is right equivalent to $Q$. Indeed, the base of the local algebra
$\mathcal{O}_{2}/(xy,x^{2}+ky^{k-1})$ is $1,x,y,y^{2},\ldots,y^{k-1}$ and lies
below the diagonal $\Gamma$. Here $\mu(Q,0)=k+1$.
## 5\. Proof of theorem 2
Let $M=F^{-1}(0)$ be a germ at $0\in\mathbb{C}^{n}$, $n\geq 3$ of real
analytic Levi-flat hypersurface, where $F(z)=\mathcal{R}e(Q(z))+h.o.t$ and $Q$
is a quasihomogeneous polynomial of degree $d$ with an isolated singularity at
$0\in\mathbb{C}^{n}$. It is easily seen that $sing(M_{\mathbb{C}})=\\{0\\}$
and $cod_{M^{*}_{\mathbb{C}}}sing(\mathcal{L}_{\mathbb{C}})\geq 3$. The
argument is essentially the same of the proof of theorem 1. In this way, there
exists an unique germ at $0\in\mathbb{C}^{n}$ of holomorphic codimension one
foliation $\mathcal{F}_{M}$ tangent to $M$, moreover $\mathcal{F}_{M}$:
$dh=0$, $h(z)=Q(z)+h.o.t$ and $M=(\mathcal{R}e(h)=0)$. Acoording to theorem
4.4, there exists $\phi\in Diff(\mathbb{C}^{n},0)$ such that
$h\circ\phi^{-1}(w)=Q(w)+\sum_{k}c_{k}e_{k}(w)$, where $c_{k}$ and $e_{k}$ as
above. Hence
$\phi(M)=(\mathcal{R}e(Q(w)+\sum_{k}c_{k}e_{k}(w))=0).$
## 6\. Applications
Here we give some applications of theorem 1.
###### Example 6.1.
$Q(x,y)=x^{2}y+y^{3}$ is a homogeneous polynomial of degree $3$ with an
isolated singularity at $0\in\mathbb{C}^{2}$ and Milnor number $\mu(Q,0)=4$.
According to [A-G-V] pg. 184, any germ $f(x,y)=x^{2}y+y^{3}+h.o.t$ is right
equivalent to $x^{2}y+y^{3}$.
In particular, if $F(z)=\mathcal{R}e(x^{2}y+y^{3})+h.o.t$ and $M=(F=0)$ is a
germ of real analytic Levi-flat at $0\in\mathbb{C}^{2}$, theorem 1 implies
that there exists a holomorphic change of coordinate such that
$M=(\mathcal{R}e(x^{2}y+y^{3})=0).$
###### Example 6.2.
If $Q(x,y)=x^{5}+y^{5}$ then $f(x,y)=Q(x,y)+h.o.t$ is right equivalent to
$x^{5}+y^{5}+c.x^{3}y^{3}$, where $c\neq 0$ is a constant (cf. [A-G-V], pg.
194). Let $F(z)=\mathcal{R}e(x^{5}+y^{5})+h.o.t$ be such that $M=(F=0)$ is
Levi-flat, theorem 1 implies that there exists a holomorphic change of
coordinate such that
$M=(\mathcal{R}e(x^{5}+y^{5}+c.x^{3}y^{3})=0).$
References
* [A]
V.I. Arnold: “Normal Form of functions in the neighbourhood of degenerate
critical points”, UNM 29:2 (1974), 11-49, RMS 29:2 19-48.
* [A-G-V]
V.I. Arnold, S.M. Gusein-Zade, A.N. Varchenko: “Singularities of Differential
Maps”, Vol. I, Monographs in Math., vol. 82, Birkhäuser, $1985$.
* [B-G]
D. Burns, X. Gong: “Singular Levi-flat real analytic hypersurfaces”, Amer. J.
Math. 121, $(1999)$, pp. 23-53.
* [C-L]
D. Cerveau, A. Lins Neto: “Local Levi-Flat hypersurfaces invariants by a
codimension one holomorphic foliation”. To appear in Amer. J. Math.
* [L]
F. Loray: “Pseudo-groupe d’une singularité de feuilletage holomorphe en
dimension deux”. Avaliable in
http://hal.archives-ouvertures.fr/ccsd-00016434
* [M-M]
J.F. Mattei, R. Moussu: “Holonomie et intégrales premières”, Ann. Ec. Norm.
Sup. 13, $(1980)$, pg. 469-523.
|
arxiv-papers
| 2010-03-25T20:58:19 |
2024-09-04T02:49:09.256685
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Arturo Fern\\'andez-P\\'erez",
"submitter": "Arturo Fernandez",
"url": "https://arxiv.org/abs/1003.4999"
}
|
1003.5030
|
# Ballistic thermal rectification
Lifa Zhang Department of Physics and Centre for Computational Science and
Engineering, National University of Singapore, Singapore 117542, Republic of
Singapore Jian-Sheng Wang Department of Physics and Centre for Computational
Science and Engineering, National University of Singapore, Singapore 117542,
Republic of Singapore Baowen Li Electronic address: phylibw@nus.edu.sg
Department of Physics and Centre for Computational Science and Engineering,
National University of Singapore, Singapore 117542, Republic of Singapore NUS
Graduate School for Integrative Sciences and Engineering, Singapore 117456,
Republic of Singapore
(8 Feb 2010)
###### Abstract
We study ballistic thermal transport in three-terminal atomic nano-junctions
by the nonequilibrium Green’s function method. We find that there is ballistic
thermal rectification in asymmetric three-terminal structures because of the
incoherent phonon scattering from the control terminal. With spin-phonon
interaction, we also find the ballistic thermal rectification even in
symmetric three-terminal paramagnetic structures.
###### pacs:
05.60.-k, 44.10.+i, 66.70.-f
_Introduction_ Phononics, the study of information processing and controlling
of heat flow by phonons, is an emerging new field that is attracting
increasing attention Wang2008 . Specifically, researchers have recently
modeled and built thermal rectifiers rectifiers , thermal transistors
transistor , thermal logical gates logicgate , and thermal memory memory ,
which are the basic components of functional thermal devices. The most
fundamental phononic component is the thermal rectifier – a device that allows
larger conduction in one direction than in the opposite direction when it is
driven far enough from equilibrium. The effect of thermal rectification is
well known to be realized by combining the system inherent anharmonicity with
structural asymmetry rectifiers ; transistor ; logicgate ; memory ; segal2005
; wu2009 ; ruokola2009 ; zhang20091 . Whether the rectification can happen in
harmonic systems, that is, ballistic rectification, is still unknown, although
one recent paper hopkins2009 discussed it, in which the authors get different
heat conductances by exchanging the heat baths with different structures
connected to an asymmetric center part, thus the different conductances come
from the totally different systems. As we know, the thermal transport in
nanoscale materials, which is very promising for thermal devices, can be
regarded as ballistic because of their small sizes in comparison with the
phonon mean free path. Therefore, it is highly desirable to investigate
whether the ballistic thermal rectification can be realized in harmonic
systems, and to explore the necessary conditions for thermal rectification.
The ballistic thermal transport in two-terminal junctions can be described by
the Landauer formula. Since the temperatures enter only through the Bose
distribution, it is obvious that if we reverse the heat bath temperatures, the
heat flux only changes sign, and no rectification is expected. How about the
ballistic thermal transport in multiple-terminal junctions? The theory for
multiple-terminal electric transport was proposed as the Landauer-Büttiker
conductance formula datta1995 ; buttiker ; blanter2000 , and was applied to
thermal transport recently sun2002 ; ming2007 ; zhang20092 . From the
electronic transport in three-terminal system blanter2000 , we know that the
third terminal can introduce incoherence or phase breaking to the transport.
So it is our interest to investigate whether a multiple-terminal junction is a
proper option for ballistic thermal transport, that is, whether the
incoherence through the third terminal can induce rectification effect. We
will take the nonequilibrium Green’s function (NEGF) approach negf ; refnegf ;
ruokola2009 ; ojanen2008 . NEGF is widely applied to electronic and thermal
transport, and is successful to study the spin Hall effect and phonon Hall
effect in junctions Sheng2005 ; zhang20092 .
Figure 1: (Color online) The three-terminal junction setup to study the
ballistic thermal transport. The left and right leads have temperatures
$T_{L}$ and $T_{R}$, the control terminal lead is adjusted to be $T_{C}$ or
$T^{{}^{\prime}}_{C}$ so that the heat flux from this lead is zero in the
forward ($T_{L}=T_{+}$, $T_{R}=T_{-}$) or backward ($T_{L}=T_{-}$,
$T_{R}=T_{+}$) process. $T_{+}$ and $T_{-}$ are the temperatures of the hot
and cold baths, respectively.
_Model and Method_ We consider the ballistic thermal transport in a three-
terminal nano-junction as shown in Fig. 1, where a two-dimensional lattice
sample, which is a honeycomb lattice, is connected with three ideal semi-
infinite leads. The masses are coupled through nearest neighbors by elastic
springs (with longitudinal and transverse force constants). We denote the
center lattice as $N_{R}\times N_{C}$, $N_{R},\,N_{C}$ correspond to the
number of rows and columns, respectively. The external magnetic field can be
perpendicularly applied to this part. We use $N_{CL}$ to denote the number of
columns of the control lead and $N_{CD}$ to denote the number of columns
deviate from the middle of the center part; if $N_{CD}=0$, the whole setup is
symmetric. In Fig. 1, $N_{R}=9,\,N_{C}=8,\,N_{CL}=4,\,N_{CD}=-2$. The
Landauer-Büttiker formula can be expressed as,
$J_{\alpha}=\sum\limits_{\beta\neq\alpha}{\int_{0}^{\infty}{\frac{{d\omega}}{{2\pi}}}\hbar\omega\,\tau_{\beta\alpha}(\omega)\bigl{[}n(T_{\alpha})-n(T_{\beta})\bigr{]}}.$
(1)
Here, $\tau_{\beta\alpha}$ is the transmission coefficient from the $\alpha$th
bath to the $\beta$th bath; and
$n(T_{\alpha})=(e^{\hbar\omega/k_{B}T_{\alpha}}-1)^{-1}$ is the Bose
distribution with $T_{\alpha}$ being the temperature of the $\alpha$th heat
bath. We set $\hbar=1$ and $k_{B}=1$ in the following calculation. Therefore,
in the forward process, $T_{L}=T_{+}$, $T_{R}=T_{-}$, we obtain
$J_{+}\\!\\!=\\!\\!\int\\!{\frac{{\omega
d\omega}}{{2\pi}}\bigl{\\{}\tau_{RL}(\omega)[n(T_{+})-n(T_{-})]}+\tau_{CL}(\omega)[n(T_{+})-n(T_{C})]\bigr{\\}}$
(2) $J_{C}\\!\\!=\\!\\!\int\\!{\frac{{\omega
d\omega}}{{2\pi}}\bigl{\\{}\tau_{LC}(\omega)[n(T_{C})-n(T_{+})]}+\tau_{RC}(\omega)[n(T_{C})-n(T_{-})]\bigr{\\}}$
(3)
Similarly we can obtain the heat fluxes $J_{-}$ and $J^{{}^{\prime}}_{C}$ in
the backward process $T_{L}=T_{-}$, $T_{R}=T_{+}$. From the equations of
$J_{C}=0$ and $J^{{}^{\prime}}_{C}=0$ we can obtain the temperatures of the
control bath $T_{C}$ and $T^{{}^{\prime}}_{C}$; inserting them to the formulae
of $J_{+}$ and $J_{-}$, by the definition of rectification as
$R=(J_{+}-J_{-})/{\rm max}\\{J_{+},J_{-}\\},$ (4)
we can calculate the rectification of this model. If the system is in the
linear response regime or in the classic limit, the heat flux from the
$\alpha$th lead can be expressed as
$J_{\alpha}=\sum\limits_{\beta\neq\alpha}\sigma_{\beta\alpha}(T_{\alpha}-T_{\beta})$,
we set
$T_{+}=T_{0}+\Delta,\,T_{-}=T_{0}-\Delta,\,T_{C}=T_{0}+\delta,\,T^{{}^{\prime}}_{C}=T_{0}+\delta^{{}^{\prime}}$,
then we obtain
$\delta=-\delta^{\prime}=\frac{{\sigma_{LC}-\sigma_{RC}}}{{\sigma_{LC}+\sigma_{RC}}}\Delta;$
(5)
$J_{+}=J_{-}=2\Delta\left(\sigma_{RL}+\frac{{\sigma_{CL}\sigma_{RC}}}{{\sigma_{LC}+\sigma_{RC}}}\right).$
(6)
So there is no rectification in the linear response regime or in the classic
limit. In order to get thermal rectification, we should consider the quantum
regime out of linear response, and the key work is to compute the transmission
coefficients among the heat baths.
In this paper, in addition to the structural asymmetry, we also can introduce
the same spin-phonon interaction as in Refs. Sheng2006 ; Kagan2008 ; wang2009
; zhang20092 in order to break a time-reversal symmetry. The Hamiltonian of
our model can be written as
$H=\sum\limits_{\alpha=0,L,R,C}H_{\alpha}+\sum\limits_{\beta=L,R,C}{U_{\beta}^{T}V_{\beta,0}U_{0}}+U_{0}^{T}AP_{0},$
(7)
where
$H_{\alpha}=\frac{1}{2}\left(P_{\alpha}^{T}P_{\alpha}+U_{\alpha}^{T}K_{\alpha}U_{\alpha}\right)$.
The superscript $T$ denotes matrix transpose. Here, $0,L,R,C$ correspond to
the center region, left, right, and control leads, respectively. $U_{\alpha}$
is a column vector for mass reduced displacements in region $\alpha$,
$P_{\alpha}$ is the associated conjugate momentum vector, and $K_{\alpha}$ is
the force constant matrix, $V_{\beta,0}=(V_{0,\beta})^{T}$ is the coupling
matrix between the $\beta$th lead and the central region. $A$ is a block
diagonal matrix with diagonal elements $\left({\begin{array}[]{*{20}c}0&h\\\
-h&0\\\ \end{array}}\right).$ $h$ is a model parameter which is supposed to be
proportional to the magnetic field. In the first part of this paper, we set
$h=0$; it is a standard harmonic phononic system.
Figure 2: (Color online) Rectification as a function of relative temperature
difference of the hot and cold heat baths. The parameter of the setup is
$N_{R}=9,\,N_{C}=16,\,N_{CL}=2$. The temperature of the heat bath are
$T_{+}=T_{0}(1+\Delta)$ and $T_{-}=T_{0}(1-\Delta)$, where $T_{0}=0.2$ is the
mean temperature. The solid square, solid circle, solid triangle, diamond,
hollow triangle, hollow circle, hollow square correspond to $N_{CD}=-7$, -5,
-3, 0, 3, 5 and 7, respectively.
The retarded Green’s function for the central region in frequency domain is
zhang20092
$G^{r}[\omega]=\Bigl{[}(\omega+i\eta)^{2}-K_{0}-\Sigma^{r}[\omega]-A^{2}+2i\omega
A\Bigr{]}^{-1}.$ (8)
Here, $\Sigma^{r}=\sum\limits_{\beta=L,C,R}{\Sigma_{\beta}^{r}}$, and
$\Sigma_{\beta}^{r}=V_{0,\beta}g_{\beta}^{r}V_{\beta,0}$ is the self-energy
due to interaction with the heat baths,
$g_{\beta}^{r}=[(\omega+i\eta)^{2}-K_{\beta}]^{-1}$. $\eta$ is an
infinitesimal real positive quantity. The surface Green’s functions of the
leads $g_{\beta}^{r}$ are obtained following the algorithms of Ref. refnegf .
The transmission coefficient is
$\tau_{\beta\alpha}[\omega]={\rm{Tr}}(G^{r}\Gamma_{\beta}G^{a}\Gamma_{\alpha}),$
(9)
where,
$\Gamma_{\alpha}=i\bigl{(}\Sigma_{\alpha}^{r}[\omega]-\Sigma_{\alpha}^{a}[\omega]\bigr{)}$,
$G^{a}=(G^{r})^{\dagger}$, and
$\Sigma_{\alpha}^{a}=(\Sigma_{\alpha}^{r})^{\dagger}$.
_Results and Discussions_ Firstly, we consider the ballistic thermal transport
in an asymmetric structure without an external magnetic field. In the
following simulation, we set the longitudinal spring constant $k_{L}=1.0$, and
the transverse one $k_{T}=0.25$. If the control lead is connected to the
middle of upper edge of the center, that is, $N_{CD}=0$, the forward process
and backward one are exactly the same; no rectification will be expected, as
shown in Fig. 2 (the diamond symbols). We will obtain the same result of no
rectification if we replace the center honeycomb lattice with the square
lattice same as the leads, when the whole system is symmetric wherever we put
the control lead. If the control lead moves away from the center, the
rectification effect appears. When the lead is moved the same distance to the
left or right, the rectification coefficient has the same magnitude but
opposite sign, which is because that the two cases only exchange the value of
$J_{+}$ and $J_{-}$. If the distance between the control lead and the middle
of the center part is longer, the rectification effect is larger. In Fig. 2,
we can see that the case of $N_{CD}=\pm 7$, when the control lead is next to
left or right lead, has biggest rectification. The rectification increases
with the temperature difference at far-from-linear-response regime.
Figure 3: (Color online) The difference of transmission coefficients:
$\tau_{LC}-\tau_{RC}$, as a function of frequency. The parameter of the setup
is $N_{R}=9,\,N_{C}=16,\,N_{CL}=2$. The solid, dot curves correspond to
$N_{CD}=-3$ and $N_{CD}=-7$, respectively. .
From the above formulae, we find if the transmissions $\tau_{LC}$ and
$\tau_{RC}$ are linearly dependent (here
$\tau_{\alpha\beta}=\tau_{\beta\alpha}$ because of time reversal symmetry),
the rectification will be zero. Figure 3 shows that the transmission
coefficients $\tau_{LC}$ and $\tau_{RC}$ are not linearly dependent, so that
the rectification is nonzero. In some frequency domains,
$\tau_{LC}>\tau_{RC}$; and in other frequency domains, $\tau_{LC}<\tau_{RC}$.
With the distance increasing, that is, from $N_{CD}=-3$ to $N_{CD}=-7$, the
difference of $\tau_{LC}$ and $\tau_{RC}$ enlarges in most part of the whole
frequency domain; so that the rectification coefficient increases. Because of
scattering from the third bath – the control lead, the total thermal transport
from the left lead to the right one is partially incoherent. This phonon
incoherence induces rectification. In the whole system the Hamiltonian is
quadratic, that is, there is no nonlinearity or anharmonicity, but we still
can obtain rectification. Therefore, the phonon incoherence, which can be
induced by either nonlinearity or scattering lead, is the necessary condition
for thermal rectification.
Figure 4: (Color online) (a) Thermal rectification as function of relative
temperature difference $\Delta$ for different width of control lead, at
$T_{0}=0.2$; (b) Thermal rectification as function of mean temperature for
different relative temperature difference. $N_{CL}=2,\,N_{CD}=-7$. For both
(a) and (b), $N_{R}=9,N_{C}=16$, $T_{+}=T_{0}(1+\Delta)$ and
$T_{-}=T_{0}(1-\Delta)$.
The control lead acts as a scattering source, which makes the phonon transport
incoherent, so that the rectification comes out. However, the width of the
control lead does not quite affect the whole thermal transport, which is shown
in Fig. 4(a). We make the control lead next to the left lead, and find that
the rectification changes little when we increase the width of the lead. If
the width of control lead increases further, the rectification decreases
because the asymmetry decreases. Figure 4(b) shows the rectification
dependence on temperature, and reproduce the reversal of rectification found
in Ref. zhang20091 . At a low temperature, the contribution to thermal
transport only comes from the low frequency phonons; if the temperature
increases, more high frequency phonons will contribute to the heat transport.
From Fig. 3, the relations between transmissions $\tau_{LC}$ and $\tau_{RC}$
in low frequency domain and high frequency domain are opposite, so that the
rectification reverses with the temperature increasing. When the temperature
increases further, the system will go to the classic limit, the rectification
disappears.
Figure 5: (Color online) (a) Thermal rectification as function of relative
temperature difference $\Delta$ for different external magnetic fields. (b)
Thermal rectification as a function of magnetic field $h$. For both (a) and
(b): $N_{R}=9,\,N_{C}=16,\,N_{CL}=2,\,N_{CD}=0$. $T_{+}=T_{0}(1+\Delta)$,
$T_{-}=T_{0}(1-\Delta)$. $T_{0}$=0.2.
From the previous work rectifiers ; segal2005 ; zeng08 on thermal
rectification, we know that in order to get rectification, we need the
structural asymmetry. However, in the nanoscale rectifier, it is not easy to
control the structural asymmetry or not easily distinguish the rectification
direction by the structural asymmetry. Is there any other means to introduce
asymmetry to induce rectification? From the study of phonon Hall effect
Sheng2006 ; Kagan2008 ; wang2009 ; zhang20092 , it is known that the magnetic
field can influence the thermal transport by the spin-phonon interaction. Thus
the magnetic field can break the symmetry of the phonon transport. We apply an
external magnetic field perpendicular to the center part of a symmetric
structure to study the ballistic thermal transport, the results are shown in
Fig. 5. The thermal rectification effect as a function of the temperature
difference is shown in Fig. 5(a). $R$ increases with the temperature
difference, and can be about $3\%$ if $\Delta=0.8$ and $h=0.3$ at $T_{0}=0.2$.
Figure 5(b) shows that the rectification can monotonically increase with the
external magnetic field in the range of $h=0\sim 0.3$. The applied magnetic
field breaks the symmetry of the phonon transport system through the spin-
phonon interaction, so the transmission coefficient from the control lead to
left one is different from that to right one. Figure 6 shows that the two
transmission coefficients are not linearly dependent; and with the increasing
of magnetic field, the difference of these two transmissions will enlarge,
which induce bigger rectification effect. Here all quantities are
dimensionless, if we use the parameters for real materials, the rectification
coefficient may be different, but would be likely in the range of few per
cents.
Although all our values for rectification are small, if we can devise a chain
of such ballistic rectifiers, the total rectification can be as large as we
need. Because for most nanoscale materials the thermal transport is ballistic
and temperature can be applied far from linear response regime, our prediction
can be tested by experiments. The asymmetric quasi-two-dimensional nano-
structure of any material can have ballistic thermal rectification, such as a
graphene sheet. Or the symmetric structure with spin-obit interaction, such as
paramagnetic dielectrics, can also have ballistic rectification applied an
perpendicular magnetic field.
Figure 6: (Color online) The difference of transmission coefficients:
$\tau_{LC}-\tau_{RC}$, as a function of frequency for different applied
magnetic fields. The parameter of the setup is
$N_{R}=9,\,N_{C}=16,\,N_{CL}=2,\,N_{CD}=0$. The solid, dot curves correspond
to $h=0.1$ and $h=0.3$, respectively.
_Conclusions_ Using the nonequilibrium Green’s function method, we have
studied ballistic thermal transport in three-terminal atomic nano-junctions.
Adjusting the temperature of the control lead in order to make the heat flux
from this lead be zero, we can calculate the thermal rectification effect. In
the quantum regime out of linear response, there is ballistic thermal
rectification in asymmetric three-terminal structures because of incoherent
phonon scattering from the control terminal. Through spin-phonon interaction,
we also find the ballistic thermal rectification in symmetric three-terminal
paramagnetic structures applied an external magnetic field. Therefore, not the
nonlinearity, but the phonon incoherence, which can be induced by nonlinearity
or scattering boundary or scattering lead, is the necessary condition for
thermal rectification. Another necessary condition is asymmetry, not
necessarily being structural asymmetry, which can be introduced by an applied
external magnetic field through the spin-phonon interaction.
_Acknowledgements_ We thank Xiaoxi Ni, Jie Ren and Jie Chen for fruitful
discussions. LZ and BL are supported by the grant R-144-000-203-112 from
Ministry of Education of Republic of Singapore. JSW acknowledges support from
a faculty research grant R-144-000-257-112 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); B Li, J.-H Lan, L Wang, ibid 95, 104302 (2005); C. W. Chang, D. Okawa, A. Majumdar, and A. Zettl, Science 314, 1121 (2006). J.-H Lan and B Li, Phys. Rev. B 74, 214305 (2006).
* (3) B. Li, L. Wang, and G. Casati, Appl. Phys. Lett. 88, 143501 (2006); W. C. Lo, L. Wang, and B. Li, J. Phys. Soc. Jpn. 77, 054402 (2008).
* (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) D. Segal and A. Nitzan, Phys. Rev. Lett. 94, 034301 (2005).
* (7) L.-A. Wu and D. Segal, Phys. Rev. Lett. 102, 095503 (2009).
* (8) T. Ruokola, T. Ojanen, and A. -P. Jauho, Phys. Rev. B 79, 144306 (2009).
* (9) L. Zhang, Y-H Yan, C.-Q Wu, J. -S. Wang, and B. Li, Phys. Rev. B 80, 172301 (2009).
* (10) P. E. Hopkins and J. R. Serrano, Phys. Rev. B 80, 201408(R) (2009).
* (11) S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge Univ. Press, 1995).
* (12) M. Büttiker, Phys. Rev. Lett. 57, 1761 (1986); M. Büttiker, IBM J. Res. Developm. 32, 317 (1988); Phys. Rev. B. 38, 9375(1988).
* (13) Y. M. Blanter and M. Büttiker, Phys. Rep. 336, 1 (2000).
* (14) Q. -F. Sun, P. Yang, and H. Guo, Phys. Rev. Lett. 89, 175901 (2002).
* (15) Y. Ming, Z. X. Wang, Q. Li, and Z. J. Ding, Appl. Phys. Lett. 91, 143508 (2007).
* (16) L. Zhang, J. -S. Wang and B. Li, New J. Phys. 11, 113038 (2009).
* (17) H. Haug and A. P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors (Springer, 1996).
* (18) J.-S. Wang, J. Wang, and N. Zeng, Phys. Rev. B 74, 033408 (2006); J.-S. Wang, J. Wang, and J. T. Lü, Eur. Phys. J. B 62, 381 (2008); L. Zhang, J. -S. Wang and B. Li, Phys. Rev. B 78, 144416 (2008).
* (19) T. Ojanen and A. -P. Jauho, Phys. Rev. Lett. 100, 155902 (2008)
* (20) L. Sheng, D. N. Sheng, and C. S. Ting, Phys. Rev. Lett. 94, 016602 (2005).
* (21) N. Zeng and J.-S. Wang, Phys. Rev. B 78, 024305 (2008).
* (22) L. Sheng, D. N. Sheng, and C. S. Ting, Phys. Rev. Lett. 96, 155901 (2006).
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|
arxiv-papers
| 2010-03-26T01:57:50 |
2024-09-04T02:49:09.263415
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lifa Zhang, Jian-Sheng Wang, and Baowen Li",
"submitter": "Lifa Zhang",
"url": "https://arxiv.org/abs/1003.5030"
}
|
1003.5192
|
11institutetext: Computer Science, Jacobs University Bremen, 11email:
ch.lange@jacobs-university.de
# wiki.openmath.org – how it works, how you can participate
Christoph Lange
###### Abstract
At http://wiki.openmath.org, the OpenMath 2 and 3 Content Dictionaries are
accessible via a semantic wiki interface, powered by the SWiM system. We
shortly introduce the inner workings of the system, then describe how to use
it, and conclude with first experiences gained from OpenMath society members
working with the system and an outlook to further development plans.
## 1 Introduction: The OpenMath Content Dictionaries
OpenMath [17] is a semantic markup language (“content markup language”) for
mathematical formulæ that originated as a shared knowledge representation for
applications in computer algebra and automated theorem proving in the
mid-1990s and got further applied in areas as diverse as e-learning,
scientific publishing, and interactive geometry. OpenMath defines an abstract
data model for representing mathematical objects and two concrete syntaxes for
it, a binary and a more common XML one. Important building blocks of
mathematical objects are numbers, variables, symbols, and applications of
mathematical objects to other mathematical objects. Any concrete operator,
constant, set, or function can be a symbol. In contrast to, e. g., earlier
versions of MathML, the symbol supply of OpenMath is constantly increasing due
to its extensibility by so-called _content dictionaries_ (CDs).
<CDDefinition><Name>plus</Name><Role>application</Role><Description>The symbol representing an n-ary commutative function plus.</Description><CMP>for all a,b | a + b = b + a </CMP><FMP>$\beta(\operatorname{quant1\\#forall},a,b,$ $@(\operatorname{relation1\\#eq},@(\operatorname{arith1\\#plus},a,b),@(\operatorname{arith1\\#plus},b,a)))$</FMP></CDDefinition>
Figure 1: Definition of the arith1#plus symbol
A CD is a collection of (usually closely related) mathematical symbols, each
with a _name_ and a mandatory informal _description_ (cf. fig. 1). Further
information about symbols is optional but recommended to have: mathematical
properties of the symbol, both in a formal (FMP) and an informal (“commented”,
CMP) flavour, and examples of applying the symbol. The language for expressing
this information is part of the OpenMath standard. Besides the proper CD file
(named e. g. number-theory.ocd), there can be additional files: OpenMath does
not commit to a particular _type system_ , so it allows for types of symbols
to be specified in separate files parallel to the CD, one per type system. The
most common type system in the OpenMath community is, however, Davenport’s
Small Type System (STS [3]); types in that system would be given in a file
named number-theory.sts.
Furthermore, there is no doubt that _notations_ must be specified for symbols
in some way, if OpenMath objects should ever be presented to a human reader,
but opinions diverge on whether this should be done in CD-like files or not.
David Carlisle and others believe that directly writing XSLT (one file per CD,
one template per symbol) does a good job in transforming OpenMath to
Presentation MathML. The advantage of XSLT is its expressive power (it’s
Turing-complete!), which comes at the expense of human comprehensibility,
though. Paul Libbrecht and Michael Kohlhase (of whose “camp” the author is a
member) thus prefer CD-like dictionaries of XML-based notation definitions in
a more compact syntax. They believe that, given a sufficient support for
pattern matching or declarative symbol$\mapsto$notation mappings, most, if not
all aspects of mathematical notation can be handled, and authored much more
intuitively. Libbrecht et al. _generate_ XSLTs from notation definitions that
use pattern matching, whereas Kohlhase et al. have implemented a dedicated
renderer (actually two ones, which are being merged) that directly renders
mathematical objects using either declarative or pattern-based notation
definitions [8, 7, 9].
## 2 Authoring and Reviewing OpenMath CDs
While everybody is free do define his own CDs for his purposes, the OpenMath
Society maintain a collection of _official_ CDs [5] that have undergone a
review process. Still, the content of an official CD is not fixed: It might
still contain mistakes that have slipped through the review, or there might be
ways to improve the informal descriptions of symbols, or relevant mathematical
properties and examples to add.
As said in the introduction, one CD is essentially a file – containing several
metadata fields on top, and then one CDDefinition block per symbol. The
official CDs are maintained in a Subversion repository at
https://svn.openmath.org. Developers participating in their maintenance check
out a working copy of that repository, edit the CD files locally with a text
or XML editor, and then commit their changes. RIACA have developed a Java-
based CD editor [20], the only CD editor besides ours that we are aware of.
The RIACA CD editor, however, rather focuses on generating Java code for
programs dealing with OpenMath objects from CDs than on CD maintenance, and
its development seems to have been discontinued for at least three years.
Issues with the CDs are usually being discussed on the OpenMath mailing list
(om@openmath.org) in case of fixing bugs in existing CDs, or on the OpenMath 3
mailing list (om3@openmath.org) in case of the overhaul of the CDs and
alignment with the Content MathML specification for the upcoming OpenMath 3
[4]. As an alternative for OpenMath 3, there is an installation of the Trac
issue tracking system (cf. [24]) at https://trac.mathweb.org/OM3.
For presenting a CD to human readers, the elements of the OpenMath CD language
are usually transformed to the desired output format (most commonly XHTML)
using XSLT, and the OpenMath objects occurring inside the FMPs and examples
are rendered as described in section 1. This presentation process is usually
controlled by makefiles.
### 2.1 Three CD Editing Use Cases
In the remainder of this paper, I will focus on supporting three common use
cases. First, the traditional way of handling these cases will be presented,
to pave the way for showing how they are handled in the OpenMath wiki.
#### 2.1.1 Minor Edits:
Fixing minor mistakes does not change the semantics of a symbol. Consider
correcting a spelling mistake in a description, or renaming a bound variable
in a mathematical object that does not occur as a free variable in a
subexpression. Supported by a text or XML editor only, which is not aware of
the particular features of OpenMath CDs, such a fix would be done as follows
(assuming that the mistake is in a CD from openmath.org):
1. 1.
Update the working copy of the OpenMath CDs
2. 2.
Open the CD file in question
3. 3.
Navigate to the Description child of the symbol in question
4. 4.
Fix the mistake
5. 5.
Commit the file (and, ideally: commit that file only, and give a meaningful
log message that exactly refers to the symbol where the mistake was fixed)
#### 2.1.2 Discussing and Implementing Revisions:
Major revisions that change the semantics of a symbol have to be discussed
among the developers before implementing them. Usually, the discussion starts
with pointing out a problem (e. g. an FMP for a concrete symbol is wrong or
misleading). Let us assume that the developer who identified the problem does
not know how to solve it. Then, he would have to make others aware of the
problem, e. g. by an e-mail to the OpenMath mailing list. Pasting a link to
the Subversion URL of the CD in question into that e-mail helps others to
inspect the problematic part111Trac features a more immediate and
comprehensive integration of a trouble ticket system with a Subversion
repository, but that is not currently possible for OpenMath, as the Trac and
the Subversion repository are running on different servers.. Other developers
would then reply to this e-mail and propose solutions, and again by replying
to their mails, the solutions would be discussed, until the community agrees
on one to be implemented.
#### 2.1.3 Editing and Verifying Notations:
Suppose that an example or FMP for a symbol $\sigma$ in one CD uses a symbol
$\tau$ from another CD and that the notation defined for $\tau$ is wrong.
Concretly, imagine $\sigma$ being the cumulative distribution function of the
normal distribution, $\tau$ the integral symbol occurring in the definition of
$\sigma$, and then imagine that the formatting of its lower and upper bounds
is wrong. Here is how an author would fix this:
1. 1.
Identify the formal symbol name and CD of $\tau$
2. 2.
Navigate to the file where the notation of $\tau$ is defined
3. 3.
Try to fix the notation definition
4. 4.
Regenerate the human-readable presentation of the CD defining $\tau$ (and,
ideally: regenerate _all_ CD presentations where $\tau$ occurs)
5. 5.
Open the regenerated presentation and check if it is correct (if not, back to
2)
6. 6.
Commit the file containing the notation definition, giving a meaningful log
message
## 3 The OpenMath Wiki
From the previous use case descriptions it evident that a better tool support
is needed to aid maintenance of the OpenMath CDs. SWiM is a wiki – a system
for collaboration on knowledge collections on the web –, a _semantic wiki for
mathematics_ in particular [12]. It aims at offering intelligent collaboration
services to authors of mathematical documents in semantic markup languages –
such as OpenMath CDs. SWiM’s notion of “semantics” is restricted to decidable
structural aspects of documents and CDs; it does not capture the full
semantics of OpenMath objects. Having presented first ideas at the OpenMath
workshop in January 2008 [10], the author decided to further pursue supporting
the OpenMath CD review as a case study for SWiM and set up an instance of the
system at http://wiki.openmath.org in September 2008. Figure 2 shows a CD in
the browsing view of SWiM. In the remainder of this section, it will be
discussed how SWiM supports the use cases introduced in section 2.1.
Figure 2: An OpenMath CD in SWiM. Notice the navigation links on the right
side.
### 3.1 Minor Edits
We have identified three different types of knowledge in OpenMath CDs: the
structural outline of a CD (e. g. defining what symbols a CD defines),
metadata (of such structural units, e. g. their informal descriptions or the
date of revision), and OpenMath objects (inside FMPs and examples). For each
of them, SWiM offers a dedicated editor (see [14]) for details.
It was a requirement for SWiM to allow for revisions in a context as local as
possible – i. e. committing a “fixed description” to the CD repository instead
of committing a “new revision of a CD with ‘something’ changed”. SWiM acts as
a browser and editor on top of the OpenMath Subversion repository but adopts a
finer granularity. For a CD, there is not one lengthy wiki page, but, on every
request of the CD from the Subversion repository, it is split into smaller
logical units that are _semantically_ subject to a revision: mathematical
properties and examples on the lowest level, then symbol definitions (grouping
several mathematical properties, examples, and metadata about one symbol
together), and finally whole CDs. Of the wiki pages on CD and symbol
definition levels, only the structural outline is editable, which keeps the
content of the page editor small and maintainable; the smaller subparts that
have been split into pages of their own right are editable separately and only
represented as XInclude links [16] in the editing view. Nevertheless, a
complete CD can be _viewed_ at once; the presentation XSLTs have been adapted
to cater for that. Metadata fields are either editable within the structural
outline editor, or in a separate form-based view. Much attention was paid to
avoiding any disruption of the file granularity of CDs in the Subversion
repository, which are still editable in the conventional way222As we will see
in section 4, SWiM does have, and will always have, certain technical but also
conceptual limitations, be they bugs or deliberate design choices, that
disqualify it as a one-size-fits-all CD editor.. Upon saving a change in the
wiki, the whole CD to which the changed part belongs is reassembled, reversing
the initial splitting process, and committed to the repository. However, the
_log message_ for this commit refers to the particular part of the CD that has
been changed. In the revision log of the CD, such a revision will display as
follows (here shown for a change of the description of the transc1#sin
symbol):
r1234 | clange | 2009-05-11 13:06:41 +0200 (Mon, 11 May 2009) |
2 lines
[Administrator@SWiM] replaced metadata field dc:description
Actually changed fragment cd:transc1+sin
The naming of CDs and parts thereof currently varies from OpenMath conventions
and instead reflects the SWiM-internal RDF representation (as described in the
following subsection) but could easily be adapted. The differing user names
are owed to the technical limitation that SWiM and the Subversion repository
do not have a unified user account management.
### 3.2 Discussing and Implementing Revisions
For each page (i. e. for each CD, symbol = CDDefinition, mathematical
property, and example), SWiM offers a discussion page – essentially one local
discussion forum per subject of interest. While that already allows
discussions in the same granularity as our units of mathematical knowledge
have, we have also given the discussion threads a semantic structure. On a
conventional wiki discussion page, users would have to 1. manually create one
section per discussion thread, 2. manually indent replies, 3. and point out
the message of their discussion post in natural language . The IkeWiki
platform [21] that SWiM is based on already cared for (1) and (2) by adopting
the user interface known from discussion forums (and storing each discussion
post as a separate resource instead of storing the whole discussion page, as
other wikis do). We have added (3) in a way that optionally allows users to
indicate the _type_ of their discussion posts in terms of an _argumentation
ontology_ , of which we present a simplified outline here (see [15] for
details): Such a discussion can be started by pointing out a problem (here
called _issue_). As replies to an _issue_ post, _ideas_ (= solution proposals)
would be allowed, on which users can state their _position_ , and finally a
thread can be concluded with a post of type _decision_ , summarising the idea
that was actually agreed on to resolve the issue. For every possible type of
reply to a discussion post, there is a dedicated reply button (cf. figure 3);
“untyped” replies for posts that do not fit into this schema are still
possible but obviously prevent further automated assistance.
Figure 3: Part of a discussion page from the OpenMath wiki. Notice the post
types and the specialised reply buttons.
Aiming at a technical support that guides discussion threads towards common
solutions, we added a domain-specific extension to the argumentation ontology.
In a survey among OpenMath users333The survey is still open for participation
at http://tinyurl.com/5qdetd but likely to be replaced by a more focused
survey soon., patterns of common problem and solution types in mathematical
knowledge bases were identified [15]. The benefit from that is twofold: 1.
Discussion threads can be queried by their logical structure, and 2.
assistants for semi-automatically implementing common solution patterns to
common problems can be implemented (cf. [15]) . SWiM not only represents the
structure of discussion threads in an RDF graph [19] in terms of the above-
mentioned argumentation ontology, but it also represents the structure of CDs
in terms of an ontology: part–whole links, as identified during the splitting
of CDs described in section 3.1, links from symbol occurrences in mathematical
objects to the place where they have been defined, as well as metadata. This
whole RDF database can be queried. On the entry page of the OpenMath wiki,
this is done in order to draw attention to unresolved issues by the following
SPARQL [18] query:
SELECT DISTINCT ?P WHERE {
?P ikewiki:hasDiscussion ?D .
?C a arguonto:Issue;
sioc:has_container ?D .
OPTIONAL { ?Dec arguonto:decides ?C . }
FILTER (!bound(?Dec)) }
P is a variable for a wiki page, which could be further restricted by its type
in terms of the OpenMath ontology, e. g. we could restrict the query to
symbols (CDDefinition). This query returns all pages P having a discussion
forum D containing a comment C of type Issue on which no decision has been
made so far. Such queries can be entered anywhere by an experienced user and
result in a list of links to wiki pages.
### 3.3 Editing and Verifying Notations
In rendering mathematical objects to Presentation MathML, SWiM adopts the
approach of the “Kohlhase camp” (cf. section 1) by embedding the JOMDoc
rendering library [7, 9] and maintaining notation dictionaries in parallel to
content dictionaries. The notation definitions are browsable and editable in
the wiki. The workflow of editing and verifying them, as outlined in section
2.1.3, is facilitated as follows (see [14, 11] for details):
1. 1.
SWiM utilises the parallel markup [1, chapter 5.4] generated by the renderer
to create links from the rendered symbols to the wiki pages representing their
CDDefinitions. Thus, a developer can directly navigate from the occurrence of
a symbol to its definition, and from there its notation definition is only one
more click away.
2. 2.
The XHTML+MathML output of rendering a wiki page (= a CD or a fragment
thereof) is cached, but after changing a notation definition of a symbol, the
rendered output for all pages P containing a formula in which the symbol
occurs is removed from the cache, forcing its re-generation. Note that the set
P contains not only the FMP or example that immediately holds the OpenMath
object using the symbol, but also the enclosing CDDefinition and CD. The set P
is obtained by another SPARQL query on the database.
## 4 Discussion, Experiences and Further Directions
This section discusses the SWiM features presented so far, lists preliminary
user feedback about them, as well as general feedback obtained from the users
of the OpenMath wiki, and concludes with a schedule of plans for further
improvement.
By supporting the use cases “minor edits”, “discussing and implementing
revisions” and “editing and verifying notations” and by its non-disruptive
connection to the OpenMath Subversion repository, SWiM facilitates crucial
aspects of the CD maintenance process. Moreover, we got a fine-grained
permission system for free from the underlying IkeWiki engine, which allows to
define roles like “visitor” (may comment on everything), “CD editor” (may edit
the CDs), and “administrator” (may also edit special pages like the entry
page). The OpenMath developers have made little use of the wiki for actually
_changing_ the CDs (for usability reasons elaborated on below), but mainly
used it as a _browser_ – where is is slower but much richer in features than
the statically rendered CD presentations –, and for _discussing_.
### 4.1 Evaluation
We have verified the principal utility of the basic argumentation ontology
(without the domain-specific extensions yet) for OpenMath by importing an old
corpus of e-mail conversations about the OpenMath/MathML 3 CDs by Chris
Rowley, David Carlisle, Michael Kohlhase, and others, into the wiki, following
the discussion structure. Further discussion posts have been contributed by
OpenMath developers afterwards. Overall, this resulted in 90 discussion posts.
A breakdown of this figure can be evaluated by post type and by post
granularity:
by type:
69 posts fit into one of the types from the argumentation ontology, mainly
Issue (48) and Idea (10). Only counting the 23 posts contributed by the users
themselves (who were obviously less familiar with the background of the
argumentation ontology), the result is slightly less convincing; for 9 of them
the users were not sure how to classify them. The post type that was missing
in most cases was nothing argumentative at all, but the _question_ – either a
direct question about some concept from a CD, or a follow-up question on an
argumentative post, such as “what do you mean by this issue description?”. It
will be easy to solve that problem by adding such a post type. Some other
posts could not be uniquely classified because they both raised an issue and
proposed a solution (= idea) in the same sentence. Annotating different
argumentative types not at the level of posts but _within_ posts is highly
non-trivial, both concerning conceptual modelling and user interface design,
though, as discussed in [13].
by granularity:
36 posts (but only posts taken from the e-mail corpus) had individual symbols
as their subject; the remaining 54 posts (including all of the posts made by
users) were made on CD-level discussion pages. This shows that either the
users did not find it intuitive (or not necessary) to access subparts of a CD
when they saw a complete CD in the browser, or that it was not possible to
identify individual symbols a post referred to. The latter is the case for
certain posts that argue on design issues of a CD in general, sometimes naming
certain individual symbols as examples. A few other posts from the e-mail
corpus referred to _two_ closely related symbols; we filed copies of them with
both affected symbols.
Overall, this shows that the OpenMath CD editors have understood how to make
use of this way of discussing problems, which is more exact than writing an
e-mail or opening a Trac ticket.
The only evaluation of the editing features so far we have performed
ourselves: We made sure that no content is lost or broken from the CD files in
the Subversion repository during minor edits in SWiM. We have tested that by
importing all OpenMath 3 CDs into the wiki, loading them into the editor once,
saving them, and inspecting the XML diff.
A major criticism towards the wiki has so far been its focus on editing
existing content. The different granularities of the wiki and the OpenMath
Subversion repository make it very cumbersome to add, e. g., a new symbol to a
CD: One has to edit the CD wiki page, add the new CDDefinition child there, as
a sibling of the XInclude elements pointing to the existing CDDefinitions, and
then save the CD page. Upon saving, the new CDDefinition fragment will be
split away into a wiki page of its own, which can then be edited in the next
step. Cleanly adding a new CD altogether is not possible at all, this time due
to the incomplete Subversion support of SWiM. SWiM only implements the most
basic Subversion commands so far: update, commit, and lock. Other actions like
adding and deleting content are possible in the wiki itself but not reflected
by its interface to Subversion – which is hacked into the file import/export
component instead of being integrated at database level, because the latter
would have required a complete overhaul of the design of the underlying
IkeWiki system.
### 4.2 Roadmap
These and other annoyances and missing features (not being able to link to
discussion posts, no e-mail notification about discussion posts or page
changes, no global search/replace feature across multiple symbols or CDs, to
name just a few) are hard to resolve within the existing architecture of SWiM.
While some major tasks are definitely within the responsibility of the author,
the general usability of the system – besides its adaptation to the
mathematical domain – could benefit a lot from improvements to the underlying
wiki engine. The development of IkeWiki, which had originally been chosen due
to its unique XML and RDF support, has been discontinued, though. On the other
hand, its completely reengineered successor KiWi [22] is making good progress.
Therefore, a port of SWiM to KiWi is currently in progress. KiWi’s more
modular architecture allows for implementing large parts of SWiM not by
modifying the core system – as was the case with IkeWiki –, but by providing
plugins. New KiWi features of particular interest in the OpenMath setting are
a dashboard view giving every user a personalised overview of recent changes
at a glance, a service that recommends related content, a facetted search
interface, and a concept of transactions that will allow for committing
several related changes at once. With the new, improved SWiM system, we will
then restart the usability evaluation and work out an accompanying user
questionnaire.
A further enhancement planned is replacing the wiki’s own database by an
integration of Subversion on database level. A database engine capable of
versioning XML documents, particularly mathematical documents, is currently
under development in our group [23]. On the user interface end, it is planned
to make the OpenMath community benefit from our recent research on active
documents. We have implemented interactive services like in-place definition
lookup and developed an infrastructure for user-adaptable documents [6].
### 4.3 Conclusion
We have outlined three CD editing use cases and compared the traditional way
of performing them to the new way offered by the SWiM wiki. SWiM clearly
excels in these special but common use cases, which has partly been confirmed
by the OpenMath CD editors, while still staying compatible with old-style
operations going on in the same repository. As SWiM does not yet cover the
full CD editing workflow, we presented a roadmap towards its successor, which
will rely on a smarter database backend and increase the interactivity of the
http://wiki.openmath.org site for current and future collaborators and users.
##### Acknowledgments:
The author would like to thank the members of the OpenMath Society,
particularly (in alphabetical order) Olga Caprotti, David Carlisle, James
Davenport, Paul Libbrecht, Michael Kohlhase, Jan Willem Knopper, and Chris
Rowley for giving helpful hints and testing during the design and setup of the
wiki, Alberto González Palomo for developing the Sentido formula editor
employed by SWiM, and Jakob Ücker for carrying out most of the evaluation
work. This work was supported by JEM-Thematic-Network ECP-038208.
## References
* [1] R. Ausbrooks, B. Bos, O. Caprotti, D. Carlisle, G. Chavchanidze, A. Coorg, S. Dalmas, S. Devitt, S. Dooley, M. Hinchcliffe, P. Ion, M. Kohlhase, A. Lazrek, D. Leas, P. Libbrecht, M. Mavrikis, B. Miller, R. Miner, M. Sargent, K. Siegrist, N. Soiffer, S. Watt, and M. Zergaoui. Mathematical Markup Language (MathML) version 3.0. W3C working draft of november 17., World Wide Web Consortium, 2008.
* [2] J. Carette, L. Dixon, C. Sacerdoti Coen, and S. M. Watt, editors. MKM/Calculemus 2009 Proceedings, number 5625 in LNAI. Springer Verlag, 2009. in Press.
* [3] J. Davenport. A small OpenMath type system. Technical report, The OpenMath Esprit Project, 1999.
* [4] J. H. Davenport and M. Kohlhase. Unifying Math Ontologies: A tale of two standards. In Carette et al. [2]. in Press.
* [5] J. H. Davenport and P. Libbrecht. The Freedom to Extend OpenMath and its Utility. Journal of Mathematics and Computer Science, special issue on Mathematical Knowledge Management, 2008.
* [6] J. Giceva, C. Lange, and F. Rabe. Integrating web services into active mathematical documents. In Carette et al. [2], pages 279–293. in Press.
* [7] M. Kohlhase, C. Lange, C. Müller, N. Müller, and F. Rabe. Notations for active mathematical documents. KWARC Report 2009-1, Jacobs University Bremen, 2009.
* [8] M. Kohlhase, C. Lange, and F. Rabe. Presenting mathematical content with flexible elisions. In O. Caprotti, M. Kohlhase, and P. Libbrecht, editors, OpenMath/ JEM Workshop 2007, 2007.
* [9] M. Kohlhase, C. Müller, and F. Rabe. Notations for living mathematical documents. In S. Autexier, J. Campbell, J. Rubio, V. Sorge, M. Suzuki, and F. Wiedijk, editors, Intelligent Computer Mathematics, 9th International Conference, AISC 2008 15th Symposium, Calculemus 2008 7th International Conference, MKM 2008 Birmingham, UK, July 28 - August 1, 2008, Proceedings, number 5144 in LNAI, pages 504–519. Springer Verlag, 2008.
* [10] C. Lange. Editing OpenMath content dictionaries with swim. In 3rd JEM Workshop (Joining Educational Mathematics), 2008.
* [11] C. Lange. Mathematical semantic markup in a wiki: The roles of symbols and notations. In C. Lange, S. Schaffert, H. Skaf-Molli, and M. Völkel, editors, Proceedings of the 3rd Workshop on Semantic Wikis, European Semantic Web Conference 2008, volume 360 of CEUR Workshop Proceedings, Costa Adeje, Tenerife, Spain, June 2008.
* [12] C. Lange. SWiM – a semantic wiki for mathematical knowledge management. In S. Bechhofer, M. Hauswirth, J. Hoffmann, and M. Koubarakis, editors, ESWC, volume 5021 of Lecture Notes in Computer Science, pages 832–837. Springer, 2008.
* [13] C. Lange, U. Bojārs, T. Groza, J. Breslin, and S. Handschuh. Expressing argumentative discussions in social media sites. In J. Breslin, U. Bojārs, A. Passant, and S. Fernández, editors, Social Data on the Web (SDoW2008), Workshop at the 7th International Semantic Web Conference, Oct. 2008.
* [14] C. Lange and A. González Palomo. Easily editing and browsing complex OpenMath markup with SWiM. In P. Libbrecht, editor, Mathematical User Interfaces Workshop 2008, 2008.
* [15] C. Lange, T. Hastrup, and S. Corlosquet. Arguing on issues with mathematical knowledge items in a semantic wiki. In J. Baumeister and M. Atzmüller, editors, Wissens- und Erfahrungsmanagement LWA (Lernen, Wissensentdeckung und Adaptivität) Conference Proceedings, volume 448, 2008.
* [16] J. Marsh, D. Orchard, and D. Veillard. XML inclusions (XInclude) version 1.0 (second edition). W3C Recommendation, World Wide Web Consortium (W3C), Nov. 2006.
* [17] OpenMath Home. http://www.openmath.org/, seen May 2009.
* [18] E. Prud’hommeaux and A. Seaborne. SPARQL query language for RDF. W3C Recommendation, World Wide Web Consortium, Jan. 2008.
* [19] Resource description framework (RDF). http://www.w3.org/RDF/, 2004.
* [20] RIACA OpenMath products. web page at http://www.riaca.win.tue.nl/projects/openmath/.
* [21] S. Schaffert. IkeWiki: A semantic wiki for collaborative knowledge management. In 1st International Workshop on Semantic Technologies in Collaborative Applications STICA 06, Manchester, UK, June 2006\.
* [22] S. Schaffert, J. Eder, S. Grünwald, T. Kurz, M. Radulescu, R. Sint, and S. Stroka. KiWi – a platform for semantic social software. In C. Lange, S. Schaffert, H. Skaf-Molli, and M. Völkel, editors, Proceedings of the 4th Workshop on Semantic Wikis, European Semantic Web Conference 2009, Hersonissos, Greece, June 2009\. in press.
* [23] TNTBase Project, seen December 2008. available at https://trac.mathweb.org/tntbase/.
* [24] The Trac Project. http://trac.edgewall.org/, 2008.
|
arxiv-papers
| 2010-03-26T17:32:10 |
2024-09-04T02:49:09.272262
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Christoph Lange",
"submitter": "Christoph Lange",
"url": "https://arxiv.org/abs/1003.5192"
}
|
1003.5196
|
11institutetext: Computer Science, Jacobs University Bremen, 11email:
ch.lange@jacobs-university.de
# SWiM – A Semantic Wiki for Mathematical Knowledge Management
Christoph Lange
###### Abstract
SWiM is a semantic wiki for collaboratively building, editing and browsing
mathematical knowledge represented in the domain-specific structural semantic
markup language OMDoc. It motivates users to contribute to collections of
mathematical knowledge by instantly sharing the benefits of knowledge-powered
services with them. SWiM is currently being used for authoring content
dictionaries, i. e. collections of uniquely identified mathematical symbols,
and prepared for managing a large-scale proof formalisation effort.
## 1 Research Background and Application Context: Mathematical Knowledge
Management
A great deal of scientific work consists of collaboratively authoring
_documents_ —taking down first hypotheses, commenting on results of
experiments, circulating informal drafts inside a working group, and
structuring, annotating, or reorganising existing items of knowledge, finally
leading to the publication of a well-structured article or book. Here, we
particularly focus on the domain of mathematics and on tools that support
collaborative authoring by utilising the knowledge contained in the documents.
In recent years, several _semantic markup_ languages have been developed to
represent the clearly defined and hierarchical structures of mathematics. The
XML languages MathML [9], OpenMath [11], and OMDoc [3] particularly aim at
exchanging mathematical knowledge on the web. OMDoc, employing Content MathML
or OpenMath representing the functional structure of mathematical _formulæ_
—as opposed to their visual appearance—and adding support for mathematical
_statements_ (like symbol declarations or axioms) and _theories_ , has many
applications in publishing, education, research, and data exchange [3, chap.
26]. The main challenge is _acquiring_ a large collection of OMDoc-formalised
knowledge that can power such added-value services. In an open, collaborative
environment, the workload can be distributed among many authors, but as
semantic markup makes fine-grained structures explicit, it is tedious to
author. As the community can only benefit from added-value services after a
substantial initial investment (writing, annotating and linking) on the
author’s part, we sought for motivating authors into action by offering
“elaborate […] services for the concrete situation” they are in [2].
## 2 Key Technology: Semantic Wiki and Ontologies
Our research is motivated by the assumption that in this context a semantic
wiki comes in handy. OMDoc supports all levels of formalisation, from human-
readable texts to fully formal representations for automated theorem proving,
and semantic wikis have been found appropriate for collaboratively refining
knowledge models (cf. [13]). User motivation in semantic wikis by instant
gratification has been investigated in earlier works [1]. The ultimate goal of
our work is to achieve a feedback loop where users are supported to contribute
well-structured knowledge, which is then exploited to offer services, which in
turn facilitate editing and motivate new contributions [5].
<omdoc> <proof id="pyth-proof" for="pythagoras"> …</proof> </omdoc>
extractionRDFpyth-proofpythagorasProofTheoremtypetypeprovesproves <pyth-proof,
rdf:type, omdoc:Proof>
<pyth-proof, omdoc:proves, pythagoras> Figure 1: RDF extraction from OMDoc
markup in a wiki page
Semantic markup has deep structures: an OMDoc document can contain theories
containing statements that contain formulæ referring to symbols defined in
other theories. This is uncommon for most semantic wikis, where the structures
are rather flat and one aims at small pages to prevent editing conflicts and
to facilitate search and navigation. So to adapt OMDoc’s model of knowledge to
a semantic wiki, we had to choose an appropriate granularity of wiki pages and
arrived at one page holding one mathematical statement or one theory. To make
knowledge from OMDoc documents usable on the semantic web, information about
the resources represented by pages and their interrelations (e. g. “a _proof_
_for_ the Pythagorean theorem”) are extracted to RDF. As a vocabulary for
this, we modeled OMDoc’s structures explicitly in a _document ontology_ [5] in
OWL-DL. This ontology contains e. g. the information that both theorems and
proofs are specialisations of a general “mathematical statement”, and that a
proof can prove a theorem (Fig. 1). Moreover, generic transitive dependency
and containment relations have been modeled. For example, having one theory
import another theory (and reusing symbols defined there) establishes a
dependency. One theory logically contains its statements; similarly,
statements can contain sub-statements, as in the case of a proof that consists
of multiple steps.
## 3 The SWiM 0.2 Prototype: IkeWiki + OMDoc
As a base system for the implementation, we chose IkeWiki [12]. Among the
systems evaluated, it offered the richest XML infrastructure—a key requirement
for adding OMDoc support—and was found to be most extensible [4]. Its backend
consists of a PostgreSQL database for the page contents, a Jena RDF store for
the RDF graph and the ontologies. Additional ontologies can easily be
imported. The frontend heavily relies on the Dojo Ajax toolkit.
Technically, the extension of IkeWiki to SWiM required supporting OMDoc in
addition to the HTML-like wiki page format. To foster stepwise formalisation
of informal text, we chose to mix OMDoc fragments with wiki markup. Thus we
could still rely on IkeWiki’s WYSIWYG HTML editor, which just had to be
enhanced by support for OMDoc XML elements. Moreover, this choice allowed for
an easier maintenance of the OMDoc-related enhancements to the SWiM code base
and avoided changes to the underlying database schema. The document ontology
is preloaded into the RDF store. RDF triples are extracted from the OMDoc
markup upon saving a page or importing an OMDoc file. Additional XSLT template
rules care for rendering embedded OMDoc fragments. In order to render
mathematical formulæ, there is a _notation definition_ for every semantic
symbol. These notation definitions can be imported and edited right in the
wiki, as parts of OMDoc documents [6]. An efficient, specialised renderer
supporting the upcoming MathML 3 standard [10, 9] applies them to the symbols
in the formulæ. In the editing view, statement- and theory-level structures of
OMDoc are made accessible as special HTML tables, whereas mathematical formulæ
given in semantic markup are made accessible in a simplified ASCII notation of
OpenMath. OMDoc documents are browsable via inline links manually set in the
informal parts, via links from occurrences of symbols in formulæ to the place
of their declaration, set by the formula renderer, and via RDF links,
displayed in a separate box by IkeWiki. The latter comprise those triples that
are extracted from the markup (cf. Fig. 1), as well as triples inferred by a
reasoner111The ontology is prepared for DL reasoning, but currently only the
RDFS reasoner built into Jena is used..
Figure 2: A mathematical document in SWiM
SWiM also relies on the ontology for reacting on changes to notation
definitions. When an author changes a notation definition $n$ for a symbol
$s$, exactly those wiki pages that contain a formula using $s$ or that include
other pages containing such formulæ need to be re-rendered. Looking up the
symbol $s$ rendered by $n$, the formulæ $f_{i}$ using $s$, or pages
(transitively) including the $f_{i}$ would be clumsy in the OMDoc XML sources,
but is easy in the RDF graph, as this information is extracted from the
documents and represented using ontology properties such as
NotationDefinition–renders–Symbol and Statement–contains–Formula;
Formula–uses–Symbol. This service allows for instant visual debugging of
notation definitions [6]. For upcoming releases, more ontology-powered
services are planned, including more general change management, learning
assistance, and editing facilitations like editing of subsections and auto-
completion of link targets [7]. There is some evidence that many services can
be based on the most generic relations of dependency and (physical or logical)
containment [5]. With scientists and knowledge engineers in mind, we envisage
SWiM as a development environment that conveniently supports refactorings of
knowledge222This is common in mathematics, e. g. in algebra: If one just needs
groups, they can be defined by a theory with the four well-known axioms. For
explicitly modeling related structures as well, one would break this into
smaller theories— _semigroup_ just defining an associative operation on a set,
_monoid_ importing this and extending it by an identity element, and finally
the refactored _group_ , adding inverse elements..
## 4 Use Cases and Applications
Now that viewing, browsing, editing, importing and exporting mathematical
documents basically works, we are evaluating SWiM in practical settings. The
Flyspeck project is about large-scale formalisation of a proof of the Kepler
conjecture. We are starting to support this effort by “crowdsourcing” the
knowledge compiled so far (hundreds of proof sketches that are not yet
machine-verifiable) on a SWiM site [8]. The main challenge is giving an
interested visitor an impression of the extent of the project and, using
appropriate SPARQL queries, showing him where work needs to be done. Currently
we are investigating how the original LaTeX sources can be utilised by
automatically converting them to HTML with MathML, then to informal OMDoc,
breaking that into wiki pages, and letting the users formalise them
stepwisely. For the upcoming OpenMath 3 standard, SWiM is currently being
extended to an editor for OpenMath Content Dictionaries [6], which could be
regarded as flat OMDoc theories that just define symbols and do not import
anything. There, mainly editing Dublin Core metadata and notation definitions
is of interest.
## 5 Conclusion and Related Work
SWiM makes mathematical documents editable collaboratively and particularly
facilitates browsing them by exploiting the knowledge they contain. Domain-
specific services are powered by an ontology that models structures of
documents—an advantage over generic semantic wikis, which would not be able to
offer additional services for mathematical knowledge. Competing non-semantic
approaches like the math encyclopædia PlanetMath (evaluated in [4]) are less
flexible, as they cannot exploit the structures of their presentation-oriented
LaTeX formulæ and rely on a fixed set of metadata. Most services for editing
and browsing need to be hard-coded, which potentially restricts the scale of
knowledge managment tasks the systems can be applied to. The SWiM approach of
integrating a semantic markup language into a wiki by choosing an appropriate
page granularity, modeling a document ontology, and extracting relevant facts
from the markup into RDF has successfully been applied to OMDoc and the
closely related but syntactically different OpenMath [6] and is likely to be
portable to other domains as well, e. g. for the chemical markup language CML.
## References
* [1] D. Aumüller and S. Auer. Towards a semantic wiki experience – desktop integration and interactivity in WikSAR. In 1st Workshop on The Semantic Desktop, 2005.
* [2] A. Kohlhase and N. Müller. Added-Value: Getting People into Semantic Work Environments. In J. Rech, B. Decker, and E. Ras, editors, Emerging Technologies for Semantic Work Environments: Techniques, Methods, and Applications. Idea Group, 2008. In press.
* [3] M. Kohlhase. OMDoc – An open markup format for mathematical documents [Version 1.2]. Number 4180 in LNAI. Springer, 2006.
* [4] C. Lange. SWiM – a semantic wiki for mathematical knowledge management. Technical Report 5, Jacobs University, 2007. http://kwarc.info/projects/swim/pubs/tr-swim.pdf.
* [5] C. Lange. Towards scientific collaboration in a semantic wiki. In A. Hotho and B. Hoser, editors, Bridging the Gap between Semantic Web and Web 2.0, 2007.
* [6] C. Lange. Mathematical Semantic Markup in a Wiki: The Roles of Symbols and Notations, 2008. submitted to the 3rd Semantic Wiki Workshop at ESWC08, see http://kwarc.info/projects/swim/pubs/semwiki08-notation-semantics.pdf.
* [7] C. Lange. SWiM development roadmap. https://trac.kwarc.info/swim/roadmap/, 2008.
* [8] C. Lange, S. McLaughlin, and F. Rabe. Flyspeck in a semantic wiki, 2008. submitted to the 3rd Semantic Wiki Workshop at ESWC08, see http://kwarc.info/projects/swim/pubs/flyspeck-wiki-eswc08.pdf.
* [9] Mathematical Markup Language (MathML) version 3.0. W3C working draft, World Wide Web Consortium, 2007. http://www.w3.org/TR/MathML3.
* [10] C. Müller, N. Müller, and M. Kohlhase. A library for transforming Content MathML/OpenMath into Presentation MathML. http://kwarc.info/projects/mmlkit/, 2008.
* [11] The Open Math standard, version 2.0. Technical report, The Open Math Society, 2004. http://www.openmath.org/standard/om20.
* [12] S. Schaffert. IkeWiki: A semantic wiki for collaborative knowledge management. In 1st International Workshop on Semantic Technologies in Collaborative Applications (STICA), 2006.
* [13] S. Schaffert. Semantic social software. In Y. Sure and S. Schaffert, editors, Semantics, 2006.
|
arxiv-papers
| 2010-03-26T18:17:01 |
2024-09-04T02:49:09.279305
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Christoph Lange",
"submitter": "Christoph Lange",
"url": "https://arxiv.org/abs/1003.5196"
}
|
1003.5479
|
# Interaction induced fractional Bloch and tunneling oscillations
Ramaz Khomeriki1,2 Dmitry O. Krimer1, Masudul Haque1 and Sergej Flach1 ${\
}^{1}$Max-Planck Institute for the Physics of Complex Systems, Nöthnitzer Str.
38, 01187 Dresden, Germany
${\ }^{2}$Physics Department, Tbilisi State University, Chavchavadze 3, 0128
Tbilisi, Georgia
###### Abstract
We study the dynamics of few interacting bosons in a one-dimensional lattice
with dc bias. In the absence of interactions the system displays single
particle Bloch oscillations. For strong interaction the Bloch oscillation
regime reemerges with fractional Bloch periods which are inversely
proportional to the number of bosons clustered into a bound state. The
interaction strength is affecting the oscillation amplitude. Excellent
agreement is found between numerical data and a composite particle dynamics
approach. For specific values of the interaction strength a particle will
tunnel from the interacting cloud to a well defined distant lattice location.
###### pacs:
67.85.-d, 37.10.Jk, 03.65.Ge
Bloch oscillations BO in dc biased lattices are due to wave interference and
have been observed in a number of quite different physical systems: atomic
oscillations in Bose-Einstein Condensates (BEC) BEC , light intensity
oscillations in waveguide arrays OL , and acoustic waves in layered and
elastic structures kosevich , among others.
Quantum many body interactions can alter the above outcome. A mean field
treatment will make the wave equations nonlinear and typically nonintegrable.
For instance, for many atoms in a Bose-Einstein condensate, a mean field
treatment leads to the Gross-Pitaevsky equation for nonlinear waves. The main
effect of nonlinearity is to deteriorate Bloch oscillations, as recently
studied experimentally salerno and theoretically DJ98 ; krimer ; kgk10 .
In contrast, we will explore the fate of Bloch oscillations for quantum
interacting few-body systems. This is motivated by recent experimental advance
zoller in monitoring and manipulating few bosons in optical lattices. Few
body quantum systems are expected to have finite eigenvalue spacings,
consequent quasiperiodic temporal evolution and phase coherence. In a recent
report on interacting electron dynamics spectral evidence for a Bloch
frequency doubling was reported doubling . On the other hand, it has been also
recently argued that Bloch oscillations will be effectively destroyed for few
interacting bosons kolovsky .
In the present paper we show that for strongly interacting bosons a coherent
Bloch oscillation regime reemerges. If the bosons are clustered into an
interacting cloud at time $t=0$, the period of Bloch oscillations will be a
fraction of the period of the noninteracting case, scaling as the inverse
number of interacting particles (Fig.1). The amplitude (spatial extent) of
these fractional Bloch oscillations will decrease with increasing interaction
strength. For specific values of the interaction, one of the particles will
leave the interacting cloud and tunnel to a possibly distant and well defined
site of the lattice. For few particles the dynamics is always quasiperiodic,
and a decoherence similar to the case of a mean field nonlinear equation
krimer will not take place.
Figure 1: Time evolution of the probability density function (PDF) $P_{j}(t)$
for the interaction constant $U=3$ and dc field $E=0.05$ and different
particle numbers initially occupying a single site at $t=0$. (a) shows one
particle Bloch oscillations with the conventional Bloch period $2\pi/E$, while
(b), (c) and (d) display two, three and four particle oscillations with the
periods $2\pi/(2E)$, $2\pi/(3E)$ and $2\pi/(4E)$, respectively.
We consider the Bose-Hubbard model with a dc field:
${\cal\hat{H}}=\sum\limits_{j}\left[t_{1}\left(\hat{b}_{j+1}^{+}\hat{b}_{j}+\hat{b}_{j}^{+}\hat{b}_{j+1}\right)+Ej\hat{b}_{j}^{+}\hat{b}_{j}+\frac{U}{2}\hat{b}_{j}^{+}\hat{b}_{j}^{+}\hat{b}_{j}\hat{b}_{j}\right]$
(1)
where $\hat{b}_{j}^{+}$ and $\hat{b}_{j}$ are standard boson creation and
annihilation operators at lattice site $j$; the hopping $t_{1}=1$; $U$ and $E$
are the interaction and dc field strengths, respectively. To study the
dynamics of $n$ particles we use the orthonormal basis of states $|{\bf
k}\rangle\equiv|k_{1},k_{2},...,k_{n}\rangle=b_{k_{1}}^{+}b_{k_{2}}^{+}...b_{k_{n}}^{+}|0\rangle$
where $|0\rangle$ is the zero particle vacuum state, and $k_{1}\leq
k_{2}...\leq k_{n}$ are lattice site indices (for instance, in the case of two
particles the state representation is mapped to the triangle). The
eigenvectors $|\nu\rangle$ of Hamiltonian (1) with eigenvalues $\lambda_{\nu}$
are then given by:
$|\nu\rangle=\sum\limits_{{\bf k}}A_{{\bf k}}^{\nu}|{\bf
k}\rangle\;,\qquad{\cal\hat{H}}|\nu\rangle=\lambda_{\nu}|\nu\rangle$ (2)
where the eigenvectors $A_{{\bf k}}^{\nu}\equiv\langle{\bf k}|\nu\rangle$ and
the time evolution of a wave function $|\Psi(t)\rangle$ is given by
$|\Psi(t)\rangle=\sum\limits_{\nu}\Phi_{\nu}e^{-i\lambda_{\nu}t}|\nu\rangle,\qquad\Phi_{\nu}\equiv\langle\nu|\Psi(0)\rangle.$
(3)
We monitor the probability density function (PDF)
$P_{j}(t)=\langle\Psi(t)|\hat{b}_{j}^{+}\hat{b}_{j}|\Psi(t)\rangle/n$, which
can be also computed using the eigenvectors and eigenvalues:
$P_{j}(t)=\dfrac{1}{n}\sum\limits_{\nu,\mu}\Phi_{\nu}\Phi_{\mu}^{*}e^{i(\lambda_{\mu}-\lambda_{\nu})t}\langle\mu|\hat{b}_{j}^{+}\hat{b}_{j}|\nu\rangle\;.$
(4)
In Fig.1 we show the evolution of $P_{j}(t)$ for $U=3$, $E=0.05$ and
$n=1,2,3,4$ with initial state $k_{1}=k_{2}=...=k_{n}\equiv p$, i.e. when all
particles are launched on the same lattice site $p$. For $n=1$ we observe the
usual Bloch oscillations with period $T=2\pi/E$ (Fig.1(a) and below). Due to
the small value of $E$, the amplitudes of oscillations are large. However,
with increasing number of particles, we find that the oscillation period is
reduced according to $2\pi/(nE)$, and at the same time the amplitude of
oscillations is also reduced.
## One particle case:
For $n=1$ the interaction term in (1) does not contribute. The eigenvalues
$\lambda_{\nu}=E\nu$ (with $\nu$ being an integer) form an equidistant
spectrum which extends over the whole real axis - the Wannier-Stark ladder.
The corresponding eigenfunctions obey the generalized translational invariance
$A_{k+\mu}^{\nu+\mu}=A_{k}^{\nu}$ BO and are given by the Bessel function
$J_{k}(x)$ of the first kind book ; SL
$\displaystyle A_{k}^{\nu}=J_{k}^{\nu}\equiv J_{k-\nu}(2/E).$ (5)
All eigenvectors are spatially localized with an asymptotic decay
$|A_{k\rightarrow\infty}^{0}|\rightarrow\left(1/E\right)^{k}/k!$, giving rise
to the well-known localized Bloch oscillations with period $T_{B}=2\pi/E$. The
localization volume $\mathcal{L}$ of a single particle eigenstate
characterizes its spatial extent. It follows $\mathcal{L}\propto-[E\cdot\ln
E]^{-1}$ for $E\rightarrow 0$ and $\mathcal{L}\rightarrow 1$ for
$E\rightarrow\infty$ krimer . For $E=0.05$ the single particle oscillates with
amplitude of the order of $2\mathcal{L}\approx 160$ (Fig.1(a)). According to
Eqs. (4) and (5) the probability density function is given by:
$P_{j}(t)=\sum\limits_{\nu,\mu}J_{p}^{\nu}J_{p}^{\mu}J_{j}^{\nu}J_{j}^{\mu}e^{iE({\mu}-\nu)t}.$
(6)
Figure 2: Eigenvalue spectrum for $n=2$, $E=0.5$ and different interaction
constants $U$. The eigenvalues are displayed only for eigenvectors localized
in the center of the lattice (we select the 32 eigenstates which overlap most
strongly with the center of the lattice) (a): $U=0$, the spectrum is
equidistant with spacing $E$ and degenerate. (b): $U=2$, the degeneracy is
lifted. (c): $U=15$, the spectrum decomposes into two subspectra, with two
different equidistant spacings - $E$ and $2E$. Graph (d) displays the
eigenvalue spectrum of the 32 central eigenfunctions as a function of $U$.
## Two particle case (n=2):
For $U=0$ the eigenfunctions of the Hamiltonian (1) are given by tensor
products of the single particle eigenstates:
$|\mu,\nu\rangle=\sqrt{\frac{2-\delta_{\mu,\nu}}{2}}\sum\limits_{k,j}J_{k}^{\mu}J_{j}^{\nu}\hat{b}_{k}^{+}\hat{b}_{j}^{+}|0\rangle\;,\;\mu\leq\nu\;.$
(7)
The corresponding eigenvalues form an equidistant spectrum which is highly
degenerate:
${\cal\hat{H}}|\mu,\nu\rangle=(\mu+\nu)E|\mu,\nu\rangle$ (8)
For the above initial condition $k_{1}=k_{2}\equiv p$ the expression for the
PDF (6) is still valid (actually it is for any number of noninteracting
particles), with the same period $2\pi/E$ of Bloch oscillations as in the
single particle case.
Figure 3: Upper plots: PDF for $E=0.1$, $n=2$, single site initial occupancy
and different interaction constants. For $U=0$ we find single particle Bloch
oscillations. For $U=4$ fractional Bloch oscillations take place, in agreement
with (11). Lower plots: probability density of the evolved wave function
(darker regions correspond to larger probablilities) after one half of the
respective Bloch period. For $U=0$ the two particles are with equal
probability close to each other and at maximal separation. For $U=4$ the two
particles avoid separation and form a composite particle which coherently
oscillates in the lattice. In lower graphs we use triangle $k<m$ mapping for
indistinguishable two particle state representation (index $m$ increases from
the right to the left).
For nonvanishing interaction the degeneracy of the spectrum is lifted, and the
eigenvalues of overlapping states are not equidistant any more (Fig.2).
Therefore we observe quasiperiodic oscillations which however are still
localizing the particles. For even larger values of $U$ the basis states with
two particles on the same site will shift their energies by $U$ exceeding the
hopping $2t_{1}$. Therefore for $U>2t_{1}$ the spectrum will be decomposed
into two nonoverlapping parts - a noninteracting one which excludes double
occupancy and has equidistant spacing $E$, and an interacting part which is
characterized by almost complete double occupancy and has corresponding
equidistant spacing $2E$, which is the cost of moving two particles from a
given site to a neighboring site. Some initial state can overlap strongly with
eigenstates from one or the other part of the spectrum, and therefore result
in different Bloch periods. In particular, when launching both particles on
the same site, one strongly overlaps with the interacting part of the spectrum
and observes a fractional Bloch period $2\pi/(2E)$.
In order to calculate the amplitude of these fractional Bloch oscillations, we
note that for $E=0$ there exists a two-particle bound state band of extended
states with band width $\sqrt{U^{2}+16}-U$ eilbeck . For large $U$ the bound
states are again almost completely described by double occupancy. Therefore we
can construct an effective Hamiltonian for a composite particle of two bound
bosons:
${\cal\hat{H}}\approx\sum\limits_{j}\left[t_{2}\left(\hat{R}_{j+1}^{+}\hat{R}_{j}+\hat{R}_{j}^{+}\hat{R}_{j+1}\right)+2Ej\hat{R}_{j}^{+}\hat{R}_{j}\right]$
(9)
where $\hat{R}_{j}^{+}$ and $\hat{R}_{j}$ are creation and annihilation
operators at lattice site $j$ of the composite particle (two bosons on the
same site) with the effective hopping
$t_{2}=\frac{\sqrt{U^{2}+16}-U}{4}.$ (10)
The corresponding PDF is given by
$P_{j}(t)=\sum\limits_{\nu,\mu}A_{p}^{\nu}A_{p}^{\mu}A_{j}^{\nu}A_{j}^{\mu}e^{i2E({\mu}-\nu)t}\;.$
(11)
The composite particle eigenvectors $A_{p}^{\nu}=J_{\nu-p}(2t_{2}/(2E))$ are
again expressed through Bessel functions, but with a modified argument as
compared to the single particle case. Bloch oscillations will evolve with
fractional period $2\pi/(2E)$ as observed in Fig.1(b). The amplitude of the
oscillations is reduced with increasing $U$ since the hopping constant $t_{2}$
is reduced (Fig.3). For $U=3$ it follows $t_{2}=0.5$, and together with the
doubled Bloch frequency the localization volume should be reduced by a factor
of 4 as compared to the single particle case. This is precisely what we find
when comparing Fig.1(a,b): for $n$=1 the amplitude is 160 sites, while for
$n=2$ it is 40 sites. In the lower plots in Fig.3 we show the probability
density of the wave functions $|\langle\Psi(t)|{\bf k}\rangle|^{2}$ after one
half of the respective Bloch period in the space of the two particle
coordinates with $k_{1}=k$ and $k_{2}=m$. For $U=0$ both particles are with
high probability at a large distance from each other. Therefore the density is
large not only for $k=m$ (the two particles are at the same site), but also
for $k=5$, $m=85$ (the two particles are at maximum distance). However, for
$U=4$ we find that the two particles, which initially occupy the site $p=45$,
do not separate, and the density is large only along the diagonal $k=m$ with
$35\leq k\leq 55$. (For $U=4$, the localization volume is $\sim 20$.)
Therefore, the two particles indeed form a composite state and travel
together.
## $n$ particle case:
We proceed similar to the case $n=2$ and estimate perturbatively the effective
hopping constant for a composite particle of $n$ bosons. For that we use the
calculated width of the $n$-particle bound state band for $E=0$ eilbeck . In
leading order of $1/U$ it reads eilbeck :
$t_{n}\simeq\frac{n}{U^{n-1}(n-1)!}.$ (12)
For $n=2$ the above expression gives $t_{2}\simeq 2/U$, the first expansion
term of the exact relation for two bosons (10). The corresponding composite
particle Hamiltonian
${\cal\hat{H}}\approx\sum\limits_{j}\left[t_{n}\left(\hat{R}_{j+1}^{+}\hat{R}_{j}+\hat{R}_{j}^{+}\hat{R}_{j+1}\right)+nEj\hat{R}_{j}^{+}\hat{R}_{j}\right]\;.$
(13)
The PDF is given by
$P_{j}(t)=\sum\limits_{\nu,\mu}A_{p}^{\nu}A_{p}^{\mu}A_{j}^{\nu}A_{j}^{\mu}e^{inE({\mu}-\nu)t}\;,$
(14)
and the composite particle eigenvectors $A_{p}^{\nu}=J_{\nu-p}(2t_{n}/(nE))$.
Bloch oscillations will evolve with fractional period $2\pi/(nE)$ as observed
in Fig.1(c,d). The amplitude of the oscillations is reduced with increasing
$U$ since the hopping constant $t_{n}$ is reduced. For $U=3$ and $n=3$ it
follows $t_{3}=0.17$, and for $n=4$ we have $t_{4}=0.01$. This leads to a
reduction factor 18 and 400 respectively as compared to the single particle
amplitude and yields amplitudes of the order of 9 and 0.5 respectively, which
is in good agreement with the numerically observed amplitudes (10 and 2 sites
respectively) in Fig.1(c,d).
Figure 4: Time averaged and normalized localization volume $L$ of the
wavepacket which emerge from two initial distributions as a function of $U$
for $E=5$. Red (grey) curve: two particles are launched on the same site.
Orange (light grey) curve: two particles are launched on adjacent sites.
Inset: PDF for $U=19.79$, with clearly observed tunneling oscillations.
## Tunneling oscillations:
For $n=1$ the amplitude of Bloch oscillations is less than one site if $E\geq
10$ krimer . Thus, for $n\geq 2$ and increasing values of $U$, the amplitude
of fractional Bloch oscillations will be less than one site if
$EU^{n-1}(n-1)!\geq 10$. Then, $n$ particles launched on the same lattice site
$p$ will be localized on that site for all times. The energy of that state
will be $n((n-1)U/2+pE)$. If however one particle will be moved to a different
location with site $q$, then the energy would change to
$(n-1)((n-2)U/2+pE)+qE$. For specific values of $U$ these two energies will be
equal:
$(n-1)U=dE\;,\;d=q-p\;.$ (15)
In such a case, one particle will leave the interacting cloud at site $p$ and
tunnel to site $q$ at distance $d$ from the cloud, then tunnel back and so on,
following effective Rabi oscillation scenario between the states $p,p\rangle$
and $|p,q\rangle$. This process will appear as an asymmetric oscillation of a
fraction of the cloud either up or down the field gradient (depending on the
sign of $U$). We calculate the tunneling splitting of these two states using
higher order perturbation theory, for an example see Ref.afko96 . The
tunneling time is then obtained as
$\tau_{tun}\simeq\frac{\pi}{\sqrt{n}}E^{d-1}(d-1)!\;.$ (16)
In order to observe these tunneling oscillations, we compute the time averaged
second moment
$\overline{m_{2}}=\overline{\sum\limits_{j}j^{2}P_{j}(t)-\left(\sum_{j}jP_{j}(t)\right)^{2}}$
of the PDF $P$. Then an effective time-averaged volume of the interacting
cloud is taken to be $L=\sqrt{12\overline{m_{2}}}+1$. We launch $n=2$
particles at site $p=40$ and plot the ratio $L(U)/L(U=0)$ in Fig.4 (blue solid
line). We find pronounced peaks at $U=E,2E,3E,4E$ which become sharper and
higher with increasing value of $U$. As a comparison we also compute the same
ratio for the initial condition when both particles occupy neighbouring sites
(dashed red line), for which the resonant structures are absent. According to
the above, the resonant structures correspond to a tunneling of one of the
particles to a site at distance $d=1,2,3,4$. The width of the peaks is
inversely proportional to the tunneling time $\tau_{tun}$, and the height
increases linearly with the tunneling distance $d$. In the inset in Fig.4, we
plot the time evolution of the PDF $P_{j}$ for $U=19.79$. We observe a clear
tunneling process from site $p=40$ to site $q=44$. The numerically observed
tunneling time is approximately 1730 time units, while our above prediction
(16) yields $\tau_{tun}\approx 1666$, in very good agreement with the
observations.
## Conclusions.
The above findings can be useful for control of the dynamics of interacting
particles. They can be also used as a testbed of whether experimental studies
deal with quantum many body states. One such testbed is the observation of
fractional Bloch oscillations, another one is the resonant tunneling of a
particle from an interacting cloud. An intriguing question is the way these
quantum coherent phenomena will disappear in the limit of many particles,
where classical nonlinear and nonintegrable wave mechanics are expected to
take over.
## Acknowledgements.
R. Kh. acknowledges financial support of the Georgian National Science
Foundation (Grant No GNSF/STO7/4-197).
## References
* (1) F. Bloch, Z. Phys. 52, 555 (1928); C. Zener, Proc. R. Soc. London, Ser. A 145, 523 (1934); G.H. Wannier, Rev. Mod. Phys. 34, 645 (1962).
* (2) B.P. Anderson, M.A. Kasevich, Science 282, 1686 (1998); M. Ben Dahan et al, Phys. Rev. Lett. 76, 4508 (1996); O. Morsch et al, Phys. Rev. Lett. 87, 140402 (2001); G. Ferrari et al, Phys. Rev. Lett. 97, 060402 (2006).
* (3) T. Pertsch et al, Phys. Rev. Lett. 83, 4752 (1999); H. Trompeter et al, Phys. Rev. Lett. 96, 023901 (2006); F. Dreisow et al, Phys. Rev. Lett. 102, 076802 (2009).
* (4) H. Sanchis-Alepuz et al,, Phys. Rev. Lett. 98, 134301 (2007); L. Gutierrez et al, Phys. Rev. Lett. 97, 114301 (2006);
* (5) M. Gustavsson et al, Phys. Rev. Lett. 100, 080404 (2008).
* (6) P.K. Datta and A.M. Jayannavar, Phys. Rev. B 58, 8170 (1998).
* (7) D.O. Krimer, R. Khomeriki, S. Flach, Phys. Rev. E, 80, 036201 (2009).
* (8) A. R. Kolovsky, E. A. Gomez EA and H. J. Korsch, Phys. Rev. A 81, 025603 (2010)
* (9) K. Winkler, et al, Nature, 441, 853 (2006); W.C. Bakr, et al, Nature, 462, 74 (2006);
* (10) W. S. Dias, et al, Phys. Rev. B, 76, 155124 (2007).
* (11) A. Buchleitner, A.R. Kolovsky, Phys. Rev. Lett. 91, 253002 (2003).
* (12) M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (1972).
* (13) H. Fukuyama, R.A. Bari, H.C. Fogedby, Phys. Rev. B 8 5579 (1973); P. Feuer, Phys. Rev., 88, 92 (1952).
* (14) A.C. Scott, J.C. Eilbeck, H. Gilhoj, Physica D, 78, 194 (1994)
* (15) S. Aubry et al, Phys. Rev. Lett. 76, 1607 (1996).
|
arxiv-papers
| 2010-03-29T09:40:34 |
2024-09-04T02:49:09.289111
|
{
"license": "Public Domain",
"authors": "Ramaz Khomeriki, Dmitry O. Krimer, Masudul Haque, Sergej Flach",
"submitter": "Ramaz Khomeriki",
"url": "https://arxiv.org/abs/1003.5479"
}
|
1003.5528
|
# Modulation of LISA free-fall orbits due to the Earth-Moon system
Massimo Cerdonio1, Fabrizio De Marchi2777Corresponding author:
fdemarchi@science.unitn.it, Roberto De Pietri3, Philippe Jetzer4, Francesco
Marzari1, Giulio Mazzolo5, Antonello Ortolan6 and Mauro Sereno7,8 1 Department
of Physics, University of Padova and INFN Padova, via Marzolo 8, I-35131
Padova, Italy 2 Department of Physics, University of Trento and INFN Trento,
I-38100 Povo (Trento), Italy 3 Department of Physics, University of Parma and
INFN Parma I-43100 Parma, Italy 4 Institute of Theoretical Physics,
University of Zürich, Winterhurerstrasse 190, 8057 Zürich, Switzerland 5 Max
Planck Institut für Gravitationsphysik, Callinstrasse 38, 30167 Hannover,
Germany 6 INFN Laboratori Nazionali di Legnaro, Viale dell’Università 35020
Legnaro (Padova), Italy 7 Dipartimento di Fisica, Politecnico di Torino,
Corso Duca degli Abruzzi 24, 10129 Torino, Italia 8 INFN, Sezione di Torino,
Via Pietro Giuria 1, 10125, Torino, Italia
###### Abstract
We calculate the effect of the Earth-Moon (EM) system on the free-fall motion
of LISA test masses. We show that the periodic gravitational pulling of the EM
system induces a resonance with fundamental frequency 1 $yr^{-1}$ and a series
of periodic perturbations with frequencies equal to integer harmonics of the
synodic month ($\simeq 3.92\times 10^{-7}\ Hz$). We then evaluate the effects
of these perturbations (up to the 6th harmonics) on the relative motions
between each test masses couple, finding that they range between $3\ mm$ and
$10\ pm$ for the 2nd and 6th harmonic, respectively.
If we take the LISA sensitivity curve, as extrapolated down to $10^{-6}\ Hz$
in [1], we obtain that a few harmonics of the EM system can be detected in the
Doppler data collected by the LISA space mission. This suggests that the EM
system gravitational near field could provide an additional crosscheck to the
calibration of LISA, as extended to such low frequencies.
###### pacs:
04.80.nn , 95.10.Eg
## 1 Introduction
LISA (Laser Interferometer Space Antenna) is a ten years long NASA-ESA space
mission to detect gravitational waves in the frequency range $10^{-4}-10^{-1}\
Hz$ [2]. It consists of three spacecrafts whose mutual distances are about
$L=5\times 10^{6}\ km$. The LISA constellation will orbit around the Sun
following the same path of the Earth, $\phi_{0}=20^{\circ}$ behind [3].
The ideal configuration for LISA performances should be a rigid equilater
triangle [4]; however, the shape of the LISA constellation is subject to
significant variations because of the gravitational interaction due to Sun and
planets [5].
Much smaller perturbations can be induced by the presence of interplanetary
dust [6] or dark matter in the solar system [7]. The perturbations due to each
celestial body can be treated, at a first approximation, independently. The
gravitational effects are quite different in intensity (orders of magnitude)
and/or in behavior (stationary or time-dependent); the frequencies involved
are, in general, not commensurable and so resonance effects are not observed.
The only significative exception is the Earth perturbation [5], [8] and [9],
which gives a resonance because of the 1:1 commensurability between the Earth
and LISA orbits.
However, such a resonance will not have enough time to grow during the 10
years of LISA mission. In this paper we focus on the perturbations induced by
the EM system on LISA at the frequency of the synodic month and its harmonics,
which are much higher than $1\ yr^{-1}$ and so, in a first order
approximation, perturbative effects can be treated independently. We extend
the approach of [5] to include the time dependent perturbation of the Moon.
The plan of the paper is as follows. In Section 2 we shortly describe the
perturbative approach used to study the EM effects. Section 3 is devoted to
illustrate the EM system and the approximations we used to model its
gravitational near field. In Section 4 we calculate the modulations of the
distance between two LISA test masses due to EM system. In Section 5 we
estimate the perturbations induced by Venus and Jupiter, compared to those due
to the EM system calculated in the previous Sections. In Section 6 conclusions
are drawn and the future research potential of method for further studies and
application to LISA are given.
## 2 Perturbative dynamics of the LISA test masses
In order to simplify the notation, we use the Astronomical System of Units,
for length (AU), mass (M⊙) and time (days). However, the quantities that
affect the relative motion of the LISA spacecrafts will be reconverted in the
SI units.
By means of a F77 code based on the inverse 15th-order Runge-Kutta method
[10], we calculate the modulus of the force gradient between 2 LISA
spacecrafts due to the main Solar System bodies, i.e. Sun, Venus, Earth (more
precisely the EM system) and Jupiter (Figure 1). The effects are different by
orders of magnitude in amplitude and show also different frequencies.
In fact, the Earth and Jupiter tidal effects are $10^{4}$ and $10^{5}$ times
smaller than the Sun contribution, respectively; after 3 years, Sun, Earth and
Jupiter cause arms length changes (the so-called ”arm breathing”) of $\simeq
1.2\times 10^{5}\,km$, 4.8$\times 10^{4}\ km$ and 4$\times 10^{3}\ km$,
respectively [4], [5].
The Venus contribution oscillates in time by two orders of magnitude and it is
comparable with Earth effects for a short time interval every 584$\ d$ [11].
However, as we will show in Section 5, this perturbation is negligible with
respect to the Earth one.
Figure 1: Gravity gradients between two spacecraft due to EM system (solid
line), Venus (dotted line), Jupiter (short dashed line) and Sun (long dashed
lines).
We therefore conclude that the celestial bodies affect the LISA arms length by
at most a few percent. As a consequence, at any time the distance of each
satellite from the LISA barycenter is $\rho_{0}=L/\sqrt{3}\simeq 1.9\times
10^{-2}\ AU$, and the distance between the Earth and the LISA barycenter is
$r_{g}=2\,\sin(\phi_{0}/2)\simeq 0.347\ AU$, within a relative fluctuation of
few percent.
### 2.1 Hill-Clohessy-Wiltshire (HCW) reference frame
To study the EM system effect, we made use of the Hill-Clohessy-Wiltshire
(HCW) reference frame $\\{x,y,z\\}$ [12] defined as follows (see Figure 2):
1. 1.
the origin $O^{\prime}$ describes a circular orbit on the ecliptic plane at
$1\ AU$ from the Solar System Barycenter $O$ [13];
2. 2.
the $xy$ plane coincides with the ecliptic;
3. 3.
the $z$-axis is perpendicular to the ecliptic plane and parallel to the Solar
System total angular momentum111The Solar System total angular momentum is
perpendicular to a plane slightly inclined with respect to the ecliptic one.
For our scopes, such inclination is negligible.;
4. 4.
the $x$-axis is tangent to the orbit and is antiparallel to the origin
$O^{\prime}$ velocity vector;
5. 5.
the $y$-axis is directed radially outward.
Figure 2: Hill-Clohessy-Wiltshire frame: $R_{0}$ is the radius of the orbit,
$\rho$ and $R$ are the the distances of $P$ from the origin of the rotating
and fixed frames, respectively. $\\{X,Y,Z\\}$ is the associated reference
frame (see details in Section 3). The EM system is also represented. The
figure is not to scale.
In this frame the coordinates of the EM system barycenter are
$(x_{g},y_{g},z_{g})=R_{0}(-\sin\phi_{0},\cos\phi_{0}-1,0)$. Of course, the
LISA spacecrafts can be considered as three proof masses. Their equations of
motion in the HCW frame read
$\eqalign{\ddot{x}-2\omega_{0}\dot{y}-\omega_{0}^{2}x=-\frac{\mu}{R^{3}}x\cr\ddot{y}+2\omega_{0}\dot{x}-\omega_{0}^{2}(y+R_{0})=-\frac{\mu}{R^{3}}(R_{0}+y)\cr\ddot{z}=-\frac{\mu}{R^{3}}z\
,}$ (1)
where $\mu=GM_{\odot}=\omega_{0}^{2}R_{0}^{3}$,
$\omega_{0}=2\pi/365.257\,d^{-1}$, $R_{0}=1\,AU$, and
$R=\sqrt{x^{2}+(y+R_{0})^{2}+z^{2}}$. Being the breathing length $\Delta L\ll
L$, each satellite is located at any time at a distance
$\rho(t)=\sqrt{x(t)^{2}+y(t)^{2}+z(t)^{2}}\simeq\rho_{0}$. Since $\rho_{0}\ll
R_{0}$, we expand the acceleration due to the Sun in terms of
$x/R_{0},y/R_{0},z/R_{0}$. Retaining the first term of the series, we have the
so-called Hill-Clohessy-Wiltshire equations [12] of relative motion
$\eqalign{\ddot{x}-2\omega_{0}\dot{y}=0\cr\ddot{y}+2\omega_{0}\dot{x}-3\omega_{0}^{2}y=0\cr\ddot{z}+\omega_{0}^{2}z=0\
,}$ (2)
with general solutions [14], [15]
$\eqalign{x(t)=x_{0}+2\
\frac{\dot{y}_{0}}{\omega_{0}}-3\,\left(\frac{\dot{x}_{0}}{\omega_{0}}-2y_{0}\right)t\,-2\frac{\dot{y}_{0}}{\omega_{0}}\cos\omega_{0}t+2\left(2\frac{\dot{x}_{0}}{\omega_{0}}-3y_{0}\right)\sin\omega_{0}t\cr
y(t)=2\left(2y_{0}-\frac{\dot{x}_{0}}{\omega_{0}}\right)+\left(2\frac{\dot{x}_{0}}{\omega_{0}}-3y_{0}\right)\cos\omega_{0}t+\frac{\dot{y}_{0}}{\omega_{0}}\sin\omega_{0}t\cr
z(t)=z_{0}\cos\omega_{0}t+\frac{\dot{z}_{0}}{\omega_{0}}\sin\omega_{0}t\ ,}$
(3)
where $x_{0},y_{0},z_{0}$ and $\dot{x}_{0},\dot{y}_{0},\dot{z}_{0}$ are the
initial positions and velocities respectively.
Since $x(t)$ contains a term proportional to $t$, after some time the
assumption $\rho\ll R_{0}$ is no more valid and the above approximation breaks
down. However, the divergent term can be cancelled by choosing
$\dot{x}_{0}=2\omega_{0}y_{0}$. For the LISA case, the constraints of rigid
and bounded relative motions lead to the solutions [4]
$\eqalign{x_{k}(t)=-\rho_{0}\sin\left[\omega_{0}t+\sigma_{k}\right]\cr
y_{k}(t)=-\frac{1}{2}\rho_{0}\cos\left[\omega_{0}t+\sigma_{k}\right]\cr
z_{k}(t)=-\frac{\sqrt{3}}{2}\,\rho_{0}\cos\left[\omega_{0}t+\sigma_{k}\right]\
,}$ (4)
where $\sigma_{k}=(1-k)\ 2\pi/3$ and $\,k=1,2,3$ is a label which enumerates
the LISA spacecrafts.
### 2.2 Rescaling and expansion of the HCW equations
For our calculation we rewrite (2) by means of the coordinate transformations
$\hat{x}=x/\rho_{0},\hat{y}=y/\rho_{0},\hat{z}=z/\rho_{0}$ and
$\hat{t}=\omega_{0}t$.
$\eqalign{{\hat{x}}^{\prime\prime}-2{\hat{y}}^{\prime}=0\cr{\hat{y}}^{\prime\prime}+2{\hat{x}}^{\prime}-3{\hat{y}}=0\cr{\hat{z}}^{\prime\prime}+{\hat{z}}=0}$
(5)
with the notation $\ {}^{\prime}=d/d\hat{t}$. We will refer to (5) as the HCW1
equations. We will show that the higher order terms in the expansions are
negligible for calculating the perturbation of the LISA rigid and bounded
orbits.
It is worth noticing that the right-hand side of (5) is zero only at the first
order in the force expansion. In general, the right-hand side of (5) is a
polynomial of degree $n$, where $n$ is the order of the expansion [16]. In our
case, the order of magnitude of the neglected terms is
$\varepsilon_{HCW}=\rho_{0}/R_{0}\simeq 1.9\times 10^{-2}$.
## 3 Gravitational near field of the EM system
In our model, the EM system is constituted by 2 point masses: $m_{1}\simeq
3.0\times 10^{-6}M_{\odot}$ (Earth) and $m_{2}\simeq 3.7\times
10^{-8}M_{\odot}$ (Moon) located at a constant distance $l\simeq 2.57\times
10^{-3}\ AU$ and describing a circular orbit with angular velocity
$\omega_{M}=2\pi/P_{M}$, where $P_{M}\simeq 29.53\ d$ is the synodic month,
around their common barycenter. We assume that the barycenter of the EM system
makes a circular orbit with radius $1\ AU$ around the Sun, i.e. we neglect the
eccentricity $\simeq 0.0167$ of the Earth orbit around the Sun.
In addition, we disregard the eccentricity of the Moon orbit around the Earth
($\simeq 0.054$), its inclination to the ecliptic plane ($\simeq
5.14^{\circ}$), the motion of the perigee of the Moon ($\simeq 8.85\ yr$) and
the precession of the Moon orbit plane ($\simeq 18.03\ yr$) [17], [18].
We consider a non rotating reference frame $\\{X,Y,Z\\}$ centered on the EM
system barycenter, with the $X$ axis along the line joining Earth and Moon at
$t=0$, and the $Z$ axis perpendicular to the ecliptic plane (see Figure 2).
The gravitational potential due to the Earth and the Moon at a point $(X,Y,Z)$
is given by
$\displaystyle
U=-\frac{Gm_{1}}{\sqrt{\left[X-\frac{m_{2}}{m_{1}+m_{2}}l\cos(\omega_{M}t)\right]^{2}+\left[Y-\frac{m_{2}}{m_{1}+m_{2}}l\sin(\omega_{M}t)\right]^{2}+Z^{2}}}+$
$\displaystyle-\frac{G{m_{2}}}{\sqrt{\left[X+\frac{m_{1}}{m_{1}+m_{2}}l\cos(\omega_{M}t)\right]^{2}+\left[Y+\frac{m_{1}}{m_{1}+m_{2}}l\sin(\omega_{M}t)\right]^{2}+Z^{2}}}\
.$
### 3.1 Multipole expansion of the EM gravitational potential
We are interested in the effect of the EM system on the LISA constellation,
located at a distance $r_{g}$. The size $l$ of the EM system is small
relatively to $r_{g}$, $l/r_{g}=7\times 10^{-3}$. We therefore expand the
total potential in series of $\varepsilon_{M}=l/r_{g}$.
$U(X,Y,Z,t)\equiv\sum_{n=0}^{\infty}\frac{1}{n!}\ \varepsilon_{EM}^{n}\
U_{n}(X,Y,Z,t)\ ,$ (6)
where $U_{n}$ are the well known multipole terms.
At the zeroth order we have the monopole term
$U_{0}(X,Y,Z)=-\frac{G({m_{1}}+{m_{2}})}{\sqrt{X^{2}+Y^{2}+Z^{2}}}\ .$ (7)
The first order term, the dipole term, is equal to zero, due to the
conservation of the linear momentum. The second order term, the quadrupole
term, is
$U_{2}(X,Y,Z,t)=\frac{m_{1}m_{2}}{m_{1}+m_{2}}\ r_{g}^{2}\
\left[\frac{X^{2}+Y^{2}+Z^{2}-3\left[X\cos\omega_{M}t+Y\sin\omega_{M}t\right]^{2}}{(X^{2}+Y^{2}+Z^{2})^{5/2}}\right]$
(8)
and so on.
Each $U_{n}$ term contains sinusoidal terms as $\sin k\omega_{M}t\ $ and $\cos
k\omega_{M}t$, with $k=0,2,\dots,n$ for even $n$ and $k=1,3,\dots,n$ for odd
$n$.
Defining $\hat{\omega}_{M}=\omega_{M}/\omega_{0}\simeq 12.3687$ and operating
the substitution $X=x+x_{g},\ Y=y+y_{g},\ Z=z$ in all $U_{n}$, we obtain the
EM potential in the HCW frame. The corresponding force per unit mass is
$\mathbf{F}_{n}=-\frac{1}{n!}\ \varepsilon_{EM}^{n}\,\nabla U_{n}(x,y,z,t)\ .$
The resulting functions $\mathbf{F}_{n}(x,y,z,t)$ are still too complex to be
treated analytically, but it is worth noticing that the values of the
coordinates range within $-\rho_{0}$ and $+\rho_{0}$, while $r_{g}$ is about
20 times larger ($\rho_{0}/r_{g}=\varepsilon_{L}\simeq 5.5\times 10^{-2}$). We
can therefore expand $\mathbf{F}_{n}$ in terms of $x/r_{g},y/r_{g},z/r_{g}$
around the origin of the HCW frame
$\mathbf{F}_{n}(x,y,z,t)=\sum_{m=0}^{\infty}\frac{\varepsilon_{L}^{m}}{m!}\
\left[\left(x\frac{\partial}{\partial\xi}+y\frac{\partial}{\partial\eta}+z\frac{\partial}{\partial\zeta}\right)^{m}\mathbf{F}_{n}(\xi,\eta,\zeta,t)\right]_{\xi=0,\eta=0,\zeta=0}\
.$ (9)
The above formula can be expressed in a more useful way (after the rescaling)
$\mathbf{F}_{n}(\hat{x},\hat{y},\hat{z},\hat{t})=\sum_{i=0}^{\infty}\frac{1}{i\
!}\sum_{j=0}^{\infty}\frac{1}{j\ !}\ \sum_{k=0}^{\infty}\ \frac{1}{k\ !}\
\varepsilon_{L}^{i+j+k}\ \mathbf{a}_{n,ijk}(\hat{t})\
\hat{x}^{i}\hat{y}^{j}\hat{z}^{k}\ ,$ (10)
where the $\mathbf{a}_{n,ijk}(\hat{t})$ are
$\mathbf{a}_{n,ijk}(\hat{t})=\sum_{m}\left[\mathchoice{\hbox{\boldmath$\displaystyle\alpha$}}{\hbox{\boldmath$\textstyle\alpha$}}{\hbox{\boldmath$\scriptstyle\alpha$}}{\hbox{\boldmath$\scriptscriptstyle\alpha$}}_{n,ijk,m}\sin
m\hat{\omega}_{M}\hat{t}+\mathchoice{\hbox{\boldmath$\displaystyle\beta$}}{\hbox{\boldmath$\textstyle\beta$}}{\hbox{\boldmath$\scriptstyle\beta$}}{\hbox{\boldmath$\scriptscriptstyle\beta$}}_{n,ijk,m}\cos
m\hat{\omega}_{M}\hat{t}\,\right]$ (11)
with
$\mathchoice{\hbox{\boldmath$\displaystyle\alpha$}}{\hbox{\boldmath$\textstyle\alpha$}}{\hbox{\boldmath$\scriptstyle\alpha$}}{\hbox{\boldmath$\scriptscriptstyle\alpha$}}_{n,ijk,m},\mathchoice{\hbox{\boldmath$\displaystyle\beta$}}{\hbox{\boldmath$\textstyle\beta$}}{\hbox{\boldmath$\scriptstyle\beta$}}{\hbox{\boldmath$\scriptscriptstyle\beta$}}_{n,ijk,m}$
numerical coefficients and $m=0,2,\dots,n$ for even $n$ and $m=1,3,\dots,n$
for odd $n$. Note that for even $n$, (11) contains a constant term (i.e.
$\mathchoice{\hbox{\boldmath$\displaystyle\beta$}}{\hbox{\boldmath$\textstyle\beta$}}{\hbox{\boldmath$\scriptstyle\beta$}}{\hbox{\boldmath$\scriptscriptstyle\beta$}}_{n,ijk,0}$)
plus sinusoids, while for odd $n$ it presents only sinusoids
($\mathchoice{\hbox{\boldmath$\displaystyle\beta$}}{\hbox{\boldmath$\textstyle\beta$}}{\hbox{\boldmath$\scriptstyle\beta$}}{\hbox{\boldmath$\scriptscriptstyle\beta$}}_{n,ijk,1}$
is multiplied by $\cos\,\hat{\omega}_{M}\hat{t}$).
Let us estimate the intensity of the acceleration due to the EM system : the
most important contribution is given by the monopole term ($n=0$), and we
assume its value at the origin of the coordinates as an indicator of its
intensity. This term is
$\varepsilon_{0}=\frac{G(m_{1}+m_{2})}{\omega_{0}^{2}\,r_{g}^{2}\,\rho_{0}}\simeq
1.31\times 10^{-3}\simeq 7\times 10^{-2}\varepsilon_{HCW}$
therefore the EM system influence is about the 7% of the contribution of the
Sun (Section 2.2).
At the first order in the coordinates the monopole term (normalized to
$\varepsilon_{0}$) is
$\displaystyle\eqalign{f_{x}=\frac{x_{g}}{r_{g}}-\hat{x}\
\frac{\rho_{0}(r_{g}^{2}-3x_{g}^{2})}{r_{g}^{3}}+\hat{y}\
\frac{3\rho_{0}x_{g}y_{g}}{r_{g}^{3}}\\\ f_{y}=\frac{y_{g}}{r_{g}}+\hat{x}\
\frac{3\rho_{0}x_{g}y_{g}}{r_{g}^{3}}-\hat{y}\
\frac{\rho_{0}(r_{g}^{2}-3y_{g}^{2})}{r_{g}^{3}}\\\ f_{z}=-\hat{z}\
\frac{\rho_{0}}{r_{g}}\ ,\\\ }$ (15)
where $\hat{x},\hat{y},\hat{z}$ are the scaled coordinates.
The most important multipole term is the quadrupole ($n=2$) which is mostly
constituted by the zeroth order term of its expansion in spatial coordinates,
or equivalently, the force at the origin. This term is periodic with period
$P_{M}/2=\pi/\omega_{M}$, and its mean value is
$\varepsilon_{2}=\frac{3}{4}\ \frac{G\,l^{2}}{\omega_{0}^{2}\ r_{g}^{4}\
\rho_{0}}\ \frac{m_{1}m_{2}}{m_{1}+m_{2}}\simeq 6.48\times 10^{-10}\ \simeq\
5\times 10^{-7}\ \varepsilon_{0}\ .$
The monopole force has an identical time-independent polynomial structure as
the expansion in coordinates of the Sun force. The difference is that this
latter starts with second order terms [16], while the EM monopole presents
also linear terms and a constant term. Moreover, the monopole force is quite
smaller than the Sun force.
The multipole force with even $n$ also contains time-independent terms but the
largest ones (corresponding to $n=2$) have order of magnitude
$\simeq\varepsilon_{2}\ll\varepsilon_{HCW}$.
We can separate the total force in a more useful way:
1. 1.
A component $\mathbf{F_{0}}(\hat{x},\hat{y},\hat{z})$ independent of time due
to the terms of the Sun force ($\propto\varepsilon_{HCW}$) [16], plus the EM
system monopole ($\propto\varepsilon_{0}$, (10) with $n=0$) plus terms
contained in the $n$-even multipole expansion ($\propto\varepsilon_{2}$, (11)
with $m=0$);
2. 2.
A component $\mathbf{F_{1}}(\hat{t})$ that depends only on time due to the EM
system multipole terms ($\propto\varepsilon_{2}$, (10) with $i,j,k=0$ and
$n\geq 2$). Physically this is the force at the origin of the HCW system;
3. 3.
A component
$\mathbf{F_{2}}(\hat{x},\hat{y},\hat{z},\hat{t})\propto\varepsilon_{2}$ that
depends on both time and coordinates (all other cases: (10) with $n\geq 2$ and
$m\neq 0$).
## 4 Perturbation of the LISA orbits due to EM system
If we indicate with $\mathbf{r}_{0i}(t)$ the unperturbed trajectory of the
$i^{th}$ LISA test mass ($i=1,2,3$) and with $\mathbf{r}_{1i}(t)$ its small
perturbation, the difference between two perturbed trajectories is simply
$\Delta\mathbf{r}_{ij}(t)\equiv\mathbf{r}_{i}(t)-\mathbf{r}_{j}(t)=\Delta\mathbf{r}_{0ij}(t)+\Delta\mathbf{r}_{1ij}(t)\
,$
where $\Delta\mathbf{r}_{0ij}(t)=\mathbf{r}_{0i}(t)-\mathbf{r}_{0j}(t)$,
$\Delta\mathbf{r}_{1ij}(t)=\mathbf{r}_{1i}(t)-\mathbf{r}_{1j}(t)$.
The perturbation of the relative displacement between the pair $i,j$ of LISA
test masses can be written as
$\Delta
L_{ij}(t)\simeq\frac{\Delta\mathbf{r}_{0ij}\cdot\Delta\mathbf{r}_{1ij}}{L_{0ij}}\
,$ (16)
where $L_{0ij}=|\Delta\mathbf{r}_{0ij}|$ is the distance between $i$ and $j$
test masses in the unperturbed case, and
$|\mathbf{r}_{1i}|\ll|\mathbf{r}_{0i}|$. We also define the perturbation to
the differential distances between each pair of LISA as
$\delta L_{ijk}(t)\equiv\Delta L_{ij}(t)-\Delta L_{jk}(t)$ (17)
which represents the variation of $L_{ij}(t)-L_{jk}(t)$ due to a small
perturbation and relates directly to the LISA sensitivity curve.
### 4.1 Effects of the EM monopole
At the first order, under the effect of the Sun, the LISA motion is described
by (4) and the EM monopole perturbation is contained in the time-independent
force per unit mass $\mathbf{F_{0}}$.
We write the motion of the $k$-spacecraft under the effect of $\mathbf{F_{0}}$
in the following form:
$\hat{\mathbf{r}}_{k}=\hat{\mathbf{r}}_{0k}+\varepsilon_{HCW}\hat{\mathbf{r}}_{1k}+\varepsilon_{0}\hat{\mathbf{r}}_{2k}$,
where $\hat{\mathbf{r}}_{0k}$ is the unperturbed motion ((4), rescaled), and
$\hat{\mathbf{r}}_{1k}$ and $\hat{\mathbf{r}}_{2k}$ are the perturbations due
to the Sun force terms [16] and the EM monopole, respectively. Being
$\varepsilon_{HCW}^{2}\ll\varepsilon_{0}$, to calculate
$\hat{\mathbf{r}}_{2k}$ it is not necessary to know $\hat{\mathbf{r}}_{1k}$,
[8].
The equations for
$\hat{\mathbf{r}}_{2k}=(\hat{x}_{2k},\hat{y}_{2k},\hat{z}_{2k})$ are [5], [8]
and [9]
$\eqalign{{\hat{x}_{2k}}^{\prime\prime}-2{\hat{y}_{2k}}^{\prime}=f_{x}(\hat{x}_{0k},\hat{y}_{0k},\hat{z}_{0k})\cr{\hat{y}_{2k}}^{\prime\prime}+2{\hat{x}_{2k}}^{\prime}-3{\hat{y}_{2k}}=f_{y}(\hat{x}_{0k},\hat{y}_{0k},\hat{z}_{0k})\cr{\hat{z}_{2k}}^{\prime\prime}+{\hat{z}_{2k}}=f_{z}(\hat{x}_{0k},\hat{y}_{0k},\hat{z}_{0k})\
,}$ (18)
where ($f_{x},f_{y},f_{z}$) are evaluated along the trajectory
$\hat{\mathbf{r}}_{0k}$, using (15).
The solution can be written as
$\eqalign{\hat{x}_{2k}=c_{1,k}+c_{2,k}\hat{t}+c_{3,k}\hat{t}^{2}+(c_{4,k}+c_{5,k}\hat{t})\sin\hat{t}+(c_{6,k}+c_{7,k}\hat{t})\cos\hat{t}\cr\hat{y}_{2k}=c_{8,k}+c_{9,k}\hat{t}+(c_{10,k}+c_{11,k}\hat{t})\sin\hat{t}+(c_{12,k}+c_{13,k}\hat{t})\cos\hat{t}\cr\hat{z}_{2k}=(c_{14,k}+c_{15,k}\hat{t})\sin\hat{t}+(c_{16,k}+c_{17,k}\hat{t})\cos\hat{t}\
,}$ (19)
where $c_{1,k}\dots c_{17,k}$ are constants that depend on the initial
conditions and on the geometric parameters ($\rho_{0},x_{g},y_{g}$).
The solution contains terms $\propto\hat{t}$ and $\propto\hat{t}^{2}$,
moreover there are also mixed perturbations as $\hat{t}\sin\hat{t}$, i.e.
perturbations increases with time.
In particular, the coefficient of $\hat{t}^{2}$,
$c_{3,k}=-3\,x_{g}/(2\,r_{g})$ is positive number and the same for all
spacecrafts. This means that the entire constellation is ”pushed away” by the
EM system ($r_{g}$ increases with time). The variation of the LISA arms length
can be calculated using (4), (19) and (16) and the result is represented in
Figure 3 (right panel).
The indefinite increasing of the perturbation (19) is not physical because the
perturbative regime would not be valid anymore, after few years. This is a
direct consequence of the force linearization: the terms proportional to
$\hat{t}\sin\hat{t}$ and $\hat{t}\cos\hat{t}$ are first-order terms of the
real, bounded, solution for small $\hat{t}$. Using the F77 code [10], we found
that the perturbative pulls lead to a complete dismembering of the
constellation and a successive recombination will occur after several tens of
thousand year. In this scenario, the distance of each spacecraft from the HCW
frame origin ranges from zero to $2\ AU$. Such motion is not a solution of the
HCW equations, which are valid only if $\rho\ll R_{0}$. An all-time valid
solution to our perturbative problem can be obtained applying the Lindstedt-
Poincaré method [19]. However, as we are interested in the LISA motion during
a few complete orbits, the Lindstedt-Poincaré method is not necessary.
The increase of $r_{g}$ is shown in Figure 3 (left panel), where its time
evolution is represented, during the hypothetical first 10 years of the
mission. The perturbation $\Delta L_{ij}$ to the relative motion between the
pair $i,j$ of LISA test masses due to the monopole perturbation, calculated
both analytically (dashed line) and via numerical integration [10] (solid
line), is plotted on the right panel of Figure 3. It is worth noticing the
good agreement during the 10 years of the LISA mission.
Figure 3: _L_ eft panel: Evolution of $r_{g}$ during 10 years: real (solid
line) and simplified case (Earth describes a circular orbit around the Sun).
The periodic component in the solid line is due to the eccentricity of the
Earth orbit, while the trend is due to the EM system influence. _R_ ight
panel: perturbation of the LISA arms length due to the EM system monopole
effect (rescaled): comparison between the numerical and the analytical
calculation (solid and dashed lines, respectively).
### 4.2 Effects of the EM multipoles
We now search for a perturbative solution to the HCW1 equations in presence of
$n\geq 2$ multipole terms. The intensity of this force is of the order of
$\varepsilon_{2}$. We have already shown that the multipole force is composed
by a periodic and a polynomial component independent of time. The polynomial
component is not important, as it can be added to the EM system monopole
expansion and solved. The solution has the same structure as (19), with
different coefficients (rescaled of a factor
$\varepsilon_{2}/\varepsilon_{0}$), and the motion described in the previous
section is therefore a very good approximation of the EM system polynomial
component influence.
#### 4.2.1 $\mathbf{F_{1}}(t)$: periodic solutions in $n\hat{\omega}_{M}t$.
At the zeroth order ($i,j,k=0$) the multipole force does not depend on the
coordinates. Therefore, for each order $n$ we have periodic terms in
$n\hat{\omega}_{M}\hat{t}$. These terms represent the ”pure” oscillations of a
test mass due to the EM system that are not involved with $\omega_{0}$
harmonics. The solutions relative to these frequencies are equal for each test
mass, being independent of its position (we can interpret this as a common
motion). Since we are interested in the relative motion of the LISA
satellites, we know a priori that these terms are subtracted when one measures
the distance between two satellites.
The equations to be solved are
$\eqalign{\hat{x}^{\prime\prime}-2\hat{y}^{\prime}=\sum_{n}\left[a_{xn}\sin
n\hat{\omega}_{M}\hat{t}+b_{xn}\cos
n\hat{\omega}_{M}\hat{t}\,\right]\cr\hat{y}^{\prime\prime}+2\hat{x}^{\prime}-3\hat{y}=\sum_{n}\left[a_{yn}\sin
n\hat{\omega}_{M}\hat{t}+b_{yn}\cos
n\hat{\omega}_{M}\hat{t}\,\right]\cr\hat{z}^{\prime\prime}+\hat{z}=\sum_{n}\left[a_{zn}\sin
n\hat{\omega}_{M}\hat{t}+b_{zn}\cos n\hat{\omega}_{M}\hat{t}\,\right]\ ,}$
(20)
where $a_{xn,yn,zn},b_{xn,yn,zn}$ are constants. Being $n\hat{\omega}_{M}\neq
1$ for each $n$, particular solutions can be written as
$\eqalign{\hat{x}=\sum_{n}a^{\prime}_{xn}\sin
n\hat{\omega}_{M}\hat{t}+b^{\prime}_{xn}\cos
n\hat{\omega}_{M}\hat{t}\cr\hat{y}=\sum_{n}a^{\prime}_{yn}\sin
n\hat{\omega}_{M}\hat{t}+b^{\prime}_{yn}\cos
n\hat{\omega}_{M}\hat{t}\cr\hat{z}=\sum_{n}a^{\prime}_{zn}\sin
n\hat{\omega}_{M}\hat{t}+b^{\prime}_{zn}\cos n\hat{\omega}_{M}\hat{t}}$ (21)
and the corresponding coefficients are
$\displaystyle\eqalign{a^{\prime}_{xn}=-\frac{2b_{yn}n\hat{\omega}_{M}+a_{xn}(3+n\hat{\omega}_{M}^{2})}{n\hat{\omega}_{M}^{2}(n\hat{\omega}_{M}^{2}-1)}&b^{\prime}_{xn}=-\frac{-2a_{yn}n\hat{\omega}_{M}+b_{xn}(3+n\hat{\omega}_{M}^{2})}{n\hat{\omega}_{M}^{2}(n\hat{\omega}_{M}^{2}-1)}\\\
a^{\prime}_{yn}=\frac{2b_{xn}-a_{yn}n\hat{\omega}_{M}}{n\hat{\omega}_{M}(n\hat{\omega}_{M}^{2}-1)}&b^{\prime}_{yn}=\frac{-2a_{xn}-b_{yn}n\hat{\omega}_{M}}{n\hat{\omega}_{M}(n\hat{\omega}_{M}^{2}-1)}\\\
a^{\prime}_{zn}=-\frac{{a_{zn}}}{n\hat{\omega}_{M}^{2}-1}&b^{\prime}_{zn}=-\frac{{b_{zn}}}{n\hat{\omega}_{M}^{2}-1}\
.}$ (25)
Inserting the numerical values, it comes out that the most important
contribution is due to the $2\,\hat{\omega}_{M}$ frequency and it corresponds
to an amplitude of about $1~{}cm$ for $x,y$ coordinates, while the
coefficients $a^{\prime}_{zn},b^{\prime}_{zn}$ are all equal to zero.
In Table 1 we report the coefficients $a^{\prime}_{xn,yn},b^{\prime}_{xn,yn}$
of the fluctuations in meters (i.e. multiplied by $\rho_{0}=2.89\times 10^{9}\
m$), relatively to the first six harmonics of the fundamental frequency
$\hat{\omega}_{M}$.
Table 1: Rescaled coefficients
$\rho_{0}\,a^{\prime}_{xn,yn},\rho_{0}\,b^{\prime}_{xn,yn}$ of the sinusoidal
terms with frequency $n\hat{\omega}_{M}$ with $1\leq n\leq 6$.
$n$ | Frequency | $\rho_{0}\,a^{\prime}_{xn}$ | $\rho_{0}\,b^{\prime}_{xn}$ | $\rho_{0}\,a^{\prime}_{yn}$ | $\rho_{0}\,b^{\prime}_{yn}$
---|---|---|---|---|---
| [Hz] | [m] | [m] | [m] | [m]
1 | 3.920$\times 10^{-7}$ | -4.5$\times 10^{-5}$ | -1.8$\times 10^{-4}$ | 6.5$\times 10^{-5}$ | -4.4$\times 10^{-5}$
2 | 7.840$\times 10^{-7}$ | 4.4$\times 10^{-3}$ | 8.5$\times 10^{-3}$ | -5.7$\times 10^{-3}$ | 3.9$\times 10^{-3}$
3 | 1.176$\times 10^{-6}$ | -2.1$\times 10^{-5}$ | -2.7$\times 10^{-5}$ | 1.9$\times 10^{-5}$ | -1.8$\times 10^{-5}$
4 | 1.568$\times 10^{-6}$ | 1.1$\times 10^{-7}$ | 1.0$\times 10^{-7}$ | -7.6$\times 10^{-8}$ | 9.8$\times 10^{-8}$
5 | 1.960$\times 10^{-6}$ | -6.4$\times 10^{-10}$ | -3.9$\times 10^{-10}$ | 3.0$\times 10^{-10}$ | -5.7$\times 10^{-10}$
6 | 2.352$\times 10^{-6}$ | 3.8$\times 10^{-12}$ | 1.5$\times 10^{-12}$ | -1.1$\times 10^{-12}$ | 3.4$\times 10^{-12}$
#### 4.2.2 $\mathbf{F_{2}}(x,y,z,t)$: periodic solutions in
$(n\hat{\omega}_{M}\pm m)\,\hat{t}$.
In Section 4.2.1 we showed that the $\mathbf{F}_{1}$ term corresponds to the
particular solutions of (20) and that $\mathbf{F}_{1}$ does not affect $\Delta
L_{ij}$ being independent of coordinates.
The motion associated with the coordinate–dependent term $\mathbf{F}_{2}$,
will be different between each pair of test masses, and so the relative
displacements $\Delta L_{ij}$ will be different from zero.
The most direct approach to solve the equation of motion is to write HCW1
equations with $\mathbf{F}_{2}$ evaluated along the unperturbed trajectories
given by (4). Thus, we have obtained only the amplitudes relative to
frequencies $(n\hat{\omega}_{M}\pm 1)$, which represent the main effect of
$\mathbf{F}_{2}$. In order to have the complete spectrum
$(n\hat{\omega}_{M}\pm m)$ we should consider the solutions of HCW with the
complete expansion of the Sun force per unit mass (Section 2.2).
The solution can be written as sum of sinusoidal terms with frequencies
$(n\hat{\omega}_{M}\pm 1)$. In addition, using (16), a similar relation can be
written also for the perturbation $\Delta L_{ij}$ that in SI units reads
$\Delta
L_{ij}(t)=\rho_{0}\sum_{n}\sum_{m=-1}^{1}\left[a_{nm,ij}\sin\omega_{nm}t+b_{nm,ij}\cos\omega_{nm}t\,\right]\
,$ (26)
where $\omega_{nm}=n\omega_{M}+2m\omega_{0}$, with $n=1,2,\dots,+\infty$ and
$m=-1,0,+1$.
The quantity $\delta L_{ijk}$, which can be directly compared with the LISA
sensitivity curve, reads
$\delta
L_{ijk}(t)=\rho_{0}\sum_{n}\sum_{m=-1}^{1}\left[a_{nm,ijk}\sin\omega_{nm}t+b_{nm,ijk}\cos\omega_{nm}t\,\right]\
,$ (27)
where $a_{nm,ijk}=a_{nm,ij}-a_{nm,jk}$ and $b_{nm,ijk}=b_{nm,ij}-b_{nm,jk}$.
The coefficients of (26) and (27) are reported in Table 2 and Table 3,
respectively, relatively to a certain number of frequencies.
In Figure 4 we plot the $\delta L_{123}$ amplitudes (filled circles)
superimposed to two LISA sensitivity curves (straight lines) corresponding to
integration times of 12 days (the upper one), period below which there should
not be disturbances or if present should be removable and 1 year (the lower
one), respectively. The straight lines were obtained by extrapolating the LISA
sensitivity curve down to $10^{-6}\ Hz$, as discussed in [1].
Each amplitude is subject to time variations, because i) the Earth and the
Moon orbits are not circular; ii) the orbital plane of the Moon is slightly
inclined; and iii) the LISA constellation is not rigid.
We estimated uncertainties for $x_{g},y_{g},l$ and $\rho_{0}$ by taking into
account all these effects; in particular, we assumed the error relative to
$\rho_{0}$ equal to the amplitude of the Sun induced breathing, i.e. $\simeq
2$% of $\rho_{0}$ [4].
The resulting relative uncertainties of the $\delta L_{ijk}$ amplitudes are
$\sim 30$%.
Figure 4: The filled circles give the amplitudes of the differential relative
displacement $\delta L_{123}$ between the LISA test masses $1,2$ and $2,3$
induced by multipoles of the EM system; relative errors are also plotted. The
two straight lines represent the LISA sensitivity curve for 12 days (upper
straight line) and 1 year (lower straight line) of integration time. Table 2:
Coefficients relative to $\Delta L_{ij}(t)$. Amplitudes and frequencies are in
SI units and the overall accuracy on amplitudes is $\simeq 30\%$.
($n,m$) | Frequency | $a_{nm,12}$ | $b_{nm,12}$ | $a_{nm,23}$ | $b_{nm,23}$ | $a_{nm,31}$ | $b_{nm,31}$
---|---|---|---|---|---|---|---
| [Hz] | [m] | [m] | [m] | [m] | [m] | [m]
(1,-1) | 3.286$\times 10^{-7}$ | 5.0$\times 10^{-5}$ | -8.0$\times 10^{-6}$ | -1.8$\times 10^{-5}$ | 4.8$\times 10^{-5}$ | -3.2$\times 10^{-5}$ | -4.0$\times 10^{-5}$
(1, 0) | 3.920$\times 10^{-7}$ | 6.8$\times 10^{-6}$ | 1.6$\times 10^{-5}$ | 6.8$\times 10^{-6}$ | 1.6$\times 10^{-5}$ | 6.8$\times 10^{-6}$ | 1.6$\times 10^{-5}$
(1, 1) | 4.553$\times 10^{-7}$ | 1.9$\times 10^{-5}$ | -1.4$\times 10^{-5}$ | 2.7$\times 10^{-6}$ | 2.4$\times 10^{-5}$ | -2.2$\times 10^{-5}$ | -9.5$\times 10^{-6}$
(2,-1) | 7.206$\times 10^{-7}$ | -2.1$\times 10^{-3}$ | -2.1$\times 10^{-4}$ | 1.2$\times 10^{-3}$ | -1.7$\times 10^{-3}$ | 8.8$\times 10^{-4}$ | 1.9$\times 10^{-3}$
(2, 0) | 7.839$\times 10^{-7}$ | -6.4$\times 10^{-4}$ | -7.9$\times 10^{-4}$ | -6.4$\times 10^{-4}$ | -7.9$\times 10^{-4}$ | -6.4$\times 10^{-4}$ | -7.9$\times 10^{-4}$
(2, 1) | 8.473$\times 10^{-7}$ | -3.9$\times 10^{-4}$ | 5.0$\times 10^{-4}$ | -2.3$\times 10^{-4}$ | -5.9$\times 10^{-4}$ | 6.2$\times 10^{-4}$ | 9.1$\times 10^{-5}$
(3,-1) | 1.113$\times 10^{-6}$ | 8.9$\times 10^{-6}$ | 2.6$\times 10^{-6}$ | -6.7$\times 10^{-6}$ | 6.4$\times 10^{-6}$ | -2.2$\times 10^{-6}$ | -9.0$\times 10^{-6}$
(3, 0) | 1.176$\times 10^{-6}$ | 3.6$\times 10^{-6}$ | 3.0$\times 10^{-6}$ | 3.6$\times 10^{-6}$ | 3.0$\times 10^{-6}$ | 3.6$\times 10^{-6}$ | 3.0$\times 10^{-6}$
(3, 1) | 1.239$\times 10^{-6}$ | 1.1$\times 10^{-6}$ | -2.2$\times 10^{-6}$ | 1.4$\times 10^{-6}$ | 2.0$\times 10^{-6}$ | -2.5$\times 10^{-6}$ | 1.6$\times 10^{-7}$
(4,-1) | 1.505$\times 10^{-6}$ | -4.4$\times 10^{-8}$ | -2.2$\times 10^{-8}$ | 4.2$\times 10^{-8}$ | -2.7$\times 10^{-8}$ | 2.6$\times 10^{-9}$ | 5.0$\times 10^{-8}$
(4, 0) | 1.568$\times 10^{-6}$ | -2.3$\times 10^{-8}$ | -1.3$\times 10^{-8}$ | -2.3$\times 10^{-8}$ | -1.3$\times 10^{-8}$ | -2.3$\times 10^{-8}$ | -1.3$\times 10^{-8}$
(4, 1) | 1.631$\times 10^{-6}$ | -3.0$\times 10^{-9}$ | 1.2$\times 10^{-8}$ | -8.5$\times 10^{-9}$ | -8.4$\times 10^{-9}$ | 1.2$\times 10^{-8}$ | -3.2$\times 10^{-9}$
(5,-1) | 1.896$\times 10^{-6}$ | 2.3$\times 10^{-10}$ | 1.8$\times 10^{-10}$ | -2.7$\times 10^{-10}$ | 1.1$\times 10^{-10}$ | 3.7$\times 10^{-11}$ | -2.9$\times 10^{-10}$
(5, 0) | 1.960$\times 10^{-6}$ | 1.5$\times 10^{-10}$ | 5.4$\times 10^{-11}$ | 1.5$\times 10^{-10}$ | 5.4$\times 10^{-11}$ | 1.5$\times 10^{-10}$ | 5.4$\times 10^{-11}$
(5, 1) | 2.023$\times 10^{-6}$ | 3.9$\times 10^{-12}$ | -6.5$\times 10^{-11}$ | 5.4$\times 10^{-11}$ | 3.6$\times 10^{-11}$ | -5.8$\times 10^{-11}$ | 2.9$\times 10^{-11}$
(6,-1) | 2.288$\times 10^{-6}$ | -1.2$\times 10^{-12}$ | -1.3$\times 10^{-12}$ | 1.7$\times 10^{-12}$ | -3.8$\times 10^{-13}$ | -5.4$\times 10^{-13}$ | 1.7$\times 10^{-12}$
(6, 0) | 2.320$\times 10^{-6}$ | -9.9$\times 10^{-13}$ | -1.7$\times 10^{-13}$ | -9.9$\times 10^{-13}$ | -1.7$\times 10^{-13}$ | -9.9$\times 10^{-13}$ | -1.7$\times 10^{-13}$
(6, 1) | 2.415$\times 10^{-6}$ | 5.0$\times 10^{-14}$ | 3.7$\times 10^{-13}$ | -3.5$\times 10^{-13}$ | -1.4$\times 10^{-13}$ | 3.0$\times 10^{-13}$ | -2.3$\times 10^{-13}$
Table 3: Coefficients relative to $\delta L_{ijk}(t)$. Amplitudes and
frequencies are in SI units and the overall accuracy on amplitudes is $\simeq
30\%$.
($n,m$) | Frequency | $a_{nm,123}$ | $b_{nm,123}$ | $a_{nm,231}$ | $b_{nm,231}$ | $a_{nm,312}$ | $b_{nm,312}$
---|---|---|---|---|---|---|---
| [Hz] | [m ] | [m ] | [m ] | [m ] | [m ] | [m ]
(1,-1) | 3.286$\times 10^{-7}$ | 6.9 $\times 10^{-5}$ | -5.6 $\times 10^{-5}$ | 1.4 $\times 10^{-5}$ | 8.7 $\times 10^{-5}$ | -8.3 $\times 10^{-5}$ | -3.2 $\times 10^{-5}$
(1, 0) | 3.920$\times 10^{-7}$ | 0 | 0 | 0 | 0 | 0 | 0
(1, 1) | 4.553$\times 10^{-7}$ | 1.6 $\times 10^{-5}$ | -3.8$\times 10^{-5}$ | 2.5 $\times 10^{-5}$ | 3.3$\times 10^{-5}$ | -4.1 $\times 10^{-5}$ | 4.7$\times 10^{-6}$
(2,-1) | 7.206$\times 10^{-7}$ | -3.4 $\times 10^{-3}$ | 1.5 $\times 10^{-3}$ | 3.6 $\times 10^{-4}$ | -3.7 $\times 10^{-3}$ | 3.0$\times 10^{-3}$ | 2.1 $\times 10^{-3}$
(2, 0) | 7.839$\times 10^{-7}$ | 0 | 0 | 0 | 0 | 0 | 0
(2, 1) | 8.473$\times 10^{-7}$ | -1.6$\times 10^{-4}$ | 1.1 $\times 10^{-3}$ | -8.6 $\times 10^{-4}$ | -6.8 $\times 10^{-4}$ | 1.0$\times 10^{-4}$ | -4.0$\times 10^{-4}$
(3,-1) | 1.113$\times 10^{-6}$ | 1.6 $\times 10^{-5}$ | -3.8$\times 10^{-6}$ | -4.6$\times 10^{-6}$ | 1.5 $\times 10^{-5}$ | -1.1 $\times 10^{-5}$ | -1.2 $\times 10^{-5}$
(3, 0) | 1.176$\times 10^{-6}$ | 0 | 0 | 0 | 0 | 0 | 0
(3, 1) | 1.239$\times 10^{-6}$ | -2.8$\times 10^{-7}$ | -4.3$\times 10^{-6}$ | 3.8$\times 10^{-6}$ | 1.9$\times 10^{-6}$ | -3.6$\times 10^{-6}$ | 2.4$\times 10^{-6}$
(4,-1) | 1.505$\times 10^{-6}$ | -8.6$\times 10^{-8}$ | 4.5$\times 10^{-9}$ | 3.9$\times 10^{-8}$ | -7.7$\times 10^{-8}$ | 4.7$\times 10^{-8}$ | 7.2$\times 10^{-8}$
(4, 0) | 1.568$\times 10^{-6}$ | 0 | 0 | 0 | 0 | 0 | 0
(4, 1) | 1.631$\times 10^{-6}$ | 5.5$\times 10^{-9}$ | 2.0$\times 10^{-8}$ | -2.0$\times 10^{-8}$ | -5.3$\times 10^{-9}$ | 1.5$\times 10^{-8}$ | -1.5$\times 10^{-8}$
(5,-1) | 1.896$\times 10^{-6}$ | 5.0$\times 10^{-10}$ | 6.5$\times 10^{-11}$ | -3.0$\times 10^{-10}$ | 4.0$\times 10^{-10}$ | -1.9$\times 10^{-10}$ | -4.6$\times 10^{-10}$
(5, 0) | 1.960$\times 10^{-6}$ | 0 | 0 | 0 | 0 | 0 | 0
(5, 1) | 2.023$\times 10^{-6}$ | -5.0$\times 10^{-11}$ | -1.0$\times 10^{-10}$ | 1.1$\times 10^{-10}$ | 6.8$\times 10^{-12}$ | -6.2$\times 10^{-11}$ | 9.4$\times 10^{-11}$
(6,-1) | 2.288$\times 10^{-6}$ | -3.0$\times 10^{-12}$ | -9.4$\times 10^{-13}$ | 2.3$\times 10^{-12}$ | -2.1$\times 10^{-12}$ | 6.6$\times 10^{-13}$ | 3.0$\times 10^{-12}$
(6, 0) | 2.352$\times 10^{-6}$ | 0 | 0 | 0 | 0 | 0 | 0
(6, 1) | 2.415$\times 10^{-6}$ | 4.0$\times 10^{-13}$ | 5.2$\times 10^{-13}$ | -6.5$\times 10^{-13}$ | 8.6$\times 10^{-14}$ | 2.5$\times 10^{-13}$ | -6.0$\times 10^{-13}$
## 5 Effect of Venus and Jupiter on LISA motion
To make a comparison with the effect on the LISA motion due to EM system
effect, we have numerically evaluated the effect of Venus, Jupiter and the EM
system (its monopole contribution) using a F77 code implementing the algorithm
described in [10]. In practice, we independently computed the LISA motion
under the effect of Sun+Venus, Sun+Jupiter and Sun+EM system and we
subtracted, to each of them, the unperturbed motion due to the Sun. In Figure
5 we plot the $\delta L_{123}$ perturbations (in km) once the modulation due
to the Sun has been subtracted. We have found that the monopole contribution
of the EM system (solid line) is much larger than that of Venus (dotted line)
and Jupiter (dashed line). Figure 5 also shows the onset of the resonant
effect of the EM system monopole after 2-3 years.
Figure 5: $\delta L_{123}$ perturbation (in km) due to EM system (solid line),
Venus (dotted line) and Jupiter (dashed line). $\delta L_{123}$ perturbation
(in km) due to monopole contribution of the EM system (solid line), Venus
(dotted line) and Jupiter (dashed line). See Section 5.
## 6 Conclusions
We calculated the relative motion of LISA test masses due to the effect of the
EM system monopole and we found $\Delta L_{ij}\simeq 3\times 10^{5}\ km$ after
a period of 10 years. We also found that the Jupiter and Venus influences are
at least 10 times smaller than the EM system one.
The perturbations of the differential relative motion of LISA test masses
$\delta L_{ijk}$ are in the $10^{-6}$ to $10^{-7}\ Hz$ decade. This is a very
low frequency range in which LISA residual acceleration noise may be much
larger than extrapolated on grounds of known effects in [1]. However, it might
not be completely hopeless to get an interesting sensitivity also at such low
frequencies, provided that one will have to face the problem of i) loss of
signal coherence over a time scale of one month; and ii) low frequency range
calibration.
As discussed in Pollack [20] it is possible to extract a signal from the LISA
data, even in presence of disturbances. These latter arise due to
environmental effects, such as cosmic rays induced by solar flares, and the
telecommunication antenna which periodically has to be rotated. Pollack showed
how these disturbances can be identified and subsequently removed from the
data even at low frequencies. Assuming, for instance, that a disturbance
appears every 19 days (see Table 6 in [20]) the resulting error in the signal
frequency of $3\times 10^{-6}\,Hz$ is only $\sim 1.3\times 10^{-9}\,Hz$. Thus
it should not be a problem to extend to such low frequencies the calibration
from the verification binaries, and, by using their signals, ensure the
continuity of data over time spans of many weeks. Still the signal from the EM
system, as understood at the level of accuracy given in the present paper, can
give a relevant additional crosscheck to such an extension of the calibration.
It thus may help in improving our knowledge of the LISA acceleration noise at
very low frequencies and contribute to extend to such low frequencies the
capabilities of LISA.
We are indebted to Peter Bender for a critical reading of the initial version
of the manuscript, together with helpful suggestions. We thank Oliver Jennrich
and Gerard Gómez, for useful discussions. Mauro Sereno was supported in the
early stages of this work by the Swiss National Science Foundation.
## References
## References
* [1] Bender P L 2003 Class. Quantum Grav.20 301-310
* [2] LISA Pre-Phase A Report, LISA Project internal report number Max Planck Institut für Quantenoptik 233 (July 1998)
* [3] LISA: System and Technology Study Report ESA document ESA-SCI 2000 11, July 2000
* [4] Dhurandhar S V, Rajesh Nayak K, Koshti S, Vinet K 2005 Class. Quantum Grav.22 481-487
* [5] Dhurandhar S V, Vinet J Y; Rajesh Nayak K 2008 Class. Quantum Grav.25 245002
* [6] Cerdonio M, De Marchi F, De Pietri R, Jetzer P, Marzari F, Mazzolo G, Ortolan A and Sereno M 2010 (arXiv: gr-qc/1002.0489v1)
* [7] Cerdonio M, De Pietri R, Jetzer P, Sereno M 2009 Class. Quantum Grav.26 094022
* [8] Povoleri A, Kemble S 2006 in Laser Interferometer Space Antenna, AIP Conf. Proc. 873, (Amer. Inst. of Physics, Melville, NY) pp 702-706
* [9] Dhurandhar S V 2009 J. Phys. Conf. Ser. 154 012047
* [10] Everhart E 1985, ”An efficient integrator that uses Gauss?-Radau spacings”, in Dynamics of Comets: Their Origin and Evolution, A. Carusi and G. B. Valsecchi, Eds., Dordrecht, Reidel (1985), pp. 185-?202.
* [11] Dixon R T 1971 Dynamic astronomy (Prentice-Hall, New Jersey)
* [12] Clohessy W H and Wiltshire R S 1960 J. Aerospace Sci. 653-58
* [13] Shirley J H & Fairbridge R W 1997 Encyclopedia of planetary sciences, (Springer Us)
* [14] Bocaletti D and Pucacco G 2001 Theory of orbits (Springer, Berlin)
* [15] Bakulin P I, Kononovich E V, Moroz V I 1984 Course in General Astronomy
* [16] Nayak K R, Koshti S, Dhurandhar S V and Vinet J-Y 2006 Class. Quantum Grav.23 1763-78
* [17] Abhyankar K D 1999 Astrophysics of the solar system (Sangam Books Ltd)
* [18] Murray C D & Dermott S F 1999 Solar System Dynamics (Cambridge University Press p 184)
* [19] Gómez G, Marcote M AAS/AIAA Astrodynamics Specialists Conference Aug, 7-11, 2005 Paper AAS 05-359
* [20] Pollack S E, 2004 Class. Quantum Grav. 21 3419-3432
|
arxiv-papers
| 2010-03-29T12:59:26 |
2024-09-04T02:49:09.296240
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Cerdonio, F. De Marchi, R. De Pietri, P. Jetzer, F. Marzari, G.\n Mazzolo, A. Ortolan, M. Sereno",
"submitter": "Fabrizio De Marchi",
"url": "https://arxiv.org/abs/1003.5528"
}
|
1003.5538
|
11institutetext: Integrative and Computational Neuroscience Unit (UNIC),
UPR2191, CNRS, Gif-sur-Yvette, France; Multimodal Imaging Lab., Dept.
Radiology and Neurosciences, UCSD, La Jolla, USA; Dept. Neurology, MGH,
Harvard, Boston, USA
: co-first authors
11email: Destexhe@unic.cnrs-gif.fr
# Comparative power spectral analysis of simultaneous elecroencephalographic
and magnetoencephalographic recordings in humans suggests non-resistive
extracellular media
Nima Dehghani∗ Claude Bédard∗ Sydney S. Cash Eric Halgren and Alain Destexhe
###### Abstract
The resistive or non-resistive nature of the extracellular space in the brain
is still debated, and is an important issue for correctly modeling
extracellular potentials. Here, we first show theoretically that if the medium
is resistive, the frequency scaling should be the same for
electroencephalogram (EEG) and magnetoencephalogram (MEG) signals at low
frequencies ($<$10 Hz). To test this prediction, we analyzed the spectrum of
simultaneous EEG and MEG measurements in four human subjects. The frequency
scaling of EEG displays coherent variations across the brain, in general
between $1/f$ and $1/f^{2}$, and tends to be smaller in parietal/temporal
regions. In a given region, although the variability of the frequency scaling
exponent was higher for MEG compared to EEG, both signals consistently scale
with a different exponent. In some cases, the scaling was similar, but only
when the signal-to-noise ratio of the MEG was low. Several methods of noise
correction for environmental and instrumental noise were tested, and they all
increased the difference between EEG and MEG scaling. In conclusion, there is
a significant difference in frequency scaling between EEG and MEG, which can
be explained if the extracellular medium (including other layers such as dura
matter and skull) is globally non-resistive.
††journal: Journal of Computational Neuroscience
Keywords: EEG; MEG; Local Field Potentials; Extracellular resistivity; Maxwell
Equations; Power-law
## 1 Introduction
An issue central to modeling local field potentials is whether the
extracellular space around neurons can be considered as a resistive medium. A
resistive medium is equivalent to replacing the medium by a simple resistance,
which considerably simplifies the computation of local field potentials, as
the equations to calculate extracellular fields are very simple and based on
Coulomb’s law (Rall and Shepherd, 1968; Nunez and Srinivasan, 2005). Forward
models of the EEG and inverse solution/source localization methods also assume
that the medium is resistive (Sarvas, 1987; Wolters and de Munck, 2007;
Ramirez, 2008). However, if the medium is non-resistive, the equations
governing the extracellular potential can be considerably more complex because
the quasi-static approximation of Maxwell equations cannot be made (Bédard et
al., 2004).
Experimental characterizations of extracellular resistivity are contradictory.
Some experiments reported that the conductivity is strongly frequency
dependent, and thus that the medium is non-resistive (Ranck, 1963; Gabriel et
al., 1996a, 1996b, 1996c). Other experiments reported that the medium was
essentially resistive (Logothetis et al., 2007). However, both types of
measurements used current intensities far larger than physiological currents,
which can mask the filtering properties of the tissue by preventing phenomena
such as ionic diffusion (Bédard and Destexhe, 2009). Unfortunately, the issue
is still open because there exists no measurements to date using (weak)
current intensities that would be more compatible with biological current
sources.
In the present paper, we propose an indirect method to estimate if
extracellular space can be considered as a purely resistive medium. We start
from Maxwell equations and show that if the medium was resistive, the
frequency-scaling of electroencephalogram (EEG) and magnetoencephalogram (MEG)
recordings should be the same. We then test this scaling on simultaneous EEG
and MEG measurements in humans.
## 2 Methods
### 2.1 Participants and MEG/EEG recordings
We recorded the electromagnetic field of the brain during quiet wakefulness
(with alpha rhythm occasionally present) from four healthy adults (4 males
ages 20-35). Participants had no neurological problems including sleep
disorders, epilepsy, or substance dependence, were taking no medications and
did not consume caffeine or alcohol on the day of the recording. We used a
whole-head MEG scanner (Neuromag Elekta) within a magnetically shielded room
(IMEDCO, Hagendorf, Switzerland) and recorded simultaneously with 60 channels
of EEG and 306 MEG channels (Nenonen et al., 2004). MEG SQUID (super
conducting quantum interference device) sensors are arranged as triplets at
102 locations; each location contains one “magnetometer” and two orthogonal
planar “gradiometers” (GRAD1, GRAD2). Unless otherwise noted, MEG will be used
here to refer to the magnetometer recordings. Locations of the EEG electrodes
on the scalp of individual subjects were recorded using a 3D digitizer
(Polhemus FastTrack). HPI (head position index) coils were used to measure the
spatial relationship between the head and scanner. Electrode arrangements were
constructed from the projection of 3D position of electrodes to a 2D plane in
order to map the frequency scaling exponent in a topographical manner. All EEG
recordings were monopolar with a common reference. Sampling rate was 1000 Hz.
For all subjects, four types of consecutive recordings were obtained, in the
following order: (1) Empty-room recording; (2) Awake “idle” recording where
subjects were asked to stay comfortable, without movements in the scanner, and
not to focus on anything specific; (3) a visual task; (4) sleep recordings.
All idle recordings used here were made in awake subjects with eyes open,
where the EEG was desynchronized. A few minutes of such idle time was recorded
in the scanner. For each subject, 3 awake segments with duration of 60 seconds
were selected from the idle recordings (see example signals in Fig. 1).
As electrocardiogram (ECG) noise often contaminates MEG recordings,
Independent component analysis (ICA) algorithm was used to remove such
contamination; either Infomax (Bell and Sejnowski, 1995) or the “Jade
algorithm” from the EEGLAB toolbox (Delorme and Makeig, 2004) was used to
achieve proper decontamination. In all recordings, the ECG component stood out
very robustly. In order not to impose any change in the frequency content of
the signal, we did not use the ICA to filter the data on any prominent
independent oscillatory component and it was solely used to decontaminate the
ECG noise. We verified that the removal of ECG did not change the scaling
exponent (not shown).
In each recording session, just prior to brain recordings, we recorded a few
minutes of the electromagnetic field present within the dewar in the magnetic
shielded room. Similar to wake epochs, 3 segments of 60 seconds duration were
selected for each of the four recordings. This will be referred to “empty
room” recordings and will be used in noise correction of the awake recordings.
In each subject, the power spectral density (PSD) was calculated by first
computing the Fast Fourier transform (FFT) of 3 awake epochs, then averaging
their respective PSDs (square modulus of the FFT). This averaged PSD was
computed for all EEG and MEG channels in order to reduce the effects of
spurious peaks due to random fluctuations. The same procedure was also
followed for empty-room signals.
### 2.2 Noise correction methods
Because the environmental and instrumental sources of noise are potentially
high in MEG recordings, we took advantage of the availability of empty-room
recordings to correct for the presence of noise in the signal. We used five
different methods for noise correction, based on different assumptions about
the nature of the noise. We describe below these different correction methods,
while all the details are given in Supplementary Methods.
A first procedure for noise correction, exponent subtraction (ES), assumes
that the noise is intrinsic to the SQUID sensors. This is justified by the
fact that the frequency scaling of some of the channels is identical to that
of the corresponding empty-room recording (see Results). In such a case, the
scaling is assumed to entirely result from the “filtering” of the sensor, and
thus the correction amounts to subtract the scaling exponents.
A second class of noise subtraction methods assume that the noise is of
ambient nature and is uncorrelated with the signal. This chatacteristics,
warrants the use of spectral subtraction (where one subtracts the PSD of the
empty-room from that of the MEG recordings), prior to the calculation of the
scaling exponent. The simplest form of spectral subtraction, linear multiband
spectral subtraction (LMSS), treats the sensors individually and does not use
any spatial/frequency-based statistics in its methodology (Boll et al., 1979).
An improved version, nonlinear multiband spectral subtraction (NMSS), takes
into account the signal-to-noise ratio (SNR) and its spatial and frequency
characteristics (Kamath and Loizou, 2002; Loizou, 2007). A third type, Wiener
filtering (WF), uses a similar approach as the latter, but obtain an estimate
of the noiseless signal from that of the noisy measurement through minimizing
the Mean Square Error (MSE) between the desired and the measured signal (Lim
et al., 1979; Abd El-Fattah et al., 2008).
A third type of noise subtraction, partial least squares (PLS) regression,
combines Principal component analysis (PCA) methods with multiple linear
regression (Abdi, 2010; Garthwaite, 1994). This methods finds the spectral
patterns that are common in the MEG and the empty-room noise, and removes
these patterns from the PSD.
### 2.3 Frequency scaling exponent estimation
The method to estimate the frequency scaling exponent was composed of steps:
First, applying a spline to obtain a smooth FFT without losing the resolution
(as can happen by using other spectral estimation methods); Second, using a
simple polynomial fit to obtain the scaling exponent. To improve the slope
estimation, we approximated the PSD data points using a spline, which is a
series of piecewise polynomials with smooth transitions and where the break
points (“knots”) are specified. We used the so-called “B-spline” (see details
in de Boor, 2001).
The knots were first defined as linearly related to logarithm of the
frequency, which naturally gives more resolution to low frequencies, to which
our theory applies. Next, in each frequency window (between consecutive
knots), we find the closest PSD value to the mean PSD of that window. Then we
use the corresponding frequency as the optimized knot in that frequency range,
leading the final values of the knots. The resulting knots stay close to the
initial distribution of frequency knots but are modified based on each
sensor’s PSD data to provide the optimal knot points for that given sensor
(Fig. 2A). We also use additional knots at the outer edges of the signal to
avoid boundary effects (Eilers and Marx, 1996). The applied method provides a
reliable and automated approach that uses our enforced initial frequency
segments with a high emphasis in low frequency and it optimizes itself based
on the data. After obtaining a smooth B-spline curve, a simple 1st degree
polynomial fit was used to estimate the slope of the curve between 0.1-10 Hz
(the fit was limited to this frequency band in order to avoid the possible
effects of the visible peak at 10 Hz on the estimated exponent).Using this
method provides a reliable and robust estimate of the slope of the PSD in
logarithmic scale, as shown in Fig. 2B. For more details on the issue of
automatic non-parametric fitting, and the rationale behind combining the
polynomial with spline basis functions, we refer the reader to Magee, 1998 as
well as Royston & Altman, 1994 and Katkovnik et al, 2006.
This procedure was realized on all channels automatically (102 channels for
MEG, 60 channels for EEG, for each patient). Every single fit was further
visually confirmed. In the case of MEG, noise correction is essential to
validate the results. For doing so, we used different methods (as described
above) to reduce the noise. Next, all the mentioned steps of frequency scaling
exponents were carried out on the corrected PSD. Results are shown in Fig. 4.
### 2.4 Region of Interest (ROI)
Three ROIs were selected for statistical comparisons of the topographic plots.
As shown in Figure 4 (panel F), FR (Frontal) ROI refers to the frontal
ellipsoid, VX (Vertex) ROI refers to the central disk located on vertex and PT
(Parietotemporal) refers to the horseshoe ROI.
## 3 Theory
We start from first principles (Maxwell equations) and derive equations to
describe EEG and MEG signals. Note that the formalism we present here is
different than the one usually given (as in Plonsey, 1969; Gulrajani, 1998),
because the linking equations are here considered in their most general
expression (convolution integrals), in the case of a linear medium (see Eq.
77.4 in Landau and Lifchitz, 1984). This generality is essential for the
problem we treat here, because our aim is to compare EEG and MEG signals with
the predictions from the theory, and thus the theory must be as general as
possible.
### 3.1 General formalism
Maxwell equations can be written as
$\begin{array}[]{ccc}~{}~{}\nabla\cdot\vec{D}=\rho^{free}{}&{}{}\hfil&\nabla\cdot\vec{B}=0\\\
\nabla\times\vec{E}=-\frac{\partial\vec{B}}{\partial
t}{}&{}\hfil&~{}~{}~{}~{}~{}~{}\nabla\times\vec{H}=\vec{j}+\frac{\partial\vec{D}}{\partial
t}\end{array}$ (1)
If we suppose that the brain is linear in the electromagnetic sense (which is
most likely), then we have the two following linking equations. The first
equation links the electric displacement with the electric field:
$\vec{D}=\int_{-\infty}^{+\infty}\epsilon(\tau)\vec{E}(t-\tau)d\tau$ (2)
where $\mathbf{\epsilon}$ is a symmetric second-order tensor.
A second equation links magnetic induction and the magnetic field:
$\vec{B}=\int_{-\infty}^{+\infty}\mu(\tau)\vec{H}(t-\tau)d\tau$ (3)
where $\mu$ is a symmetric second-order tensor.
If we neglect non-resistive effects such as diffusion (Bédard and Destexhe,
2009), as well as any other nonlinear effects111Examples of nonlinear effects
are variations of the macroscopic conductivity $\sigma_{f}$ with the magnitude
of electric field $\vec{E}$. Such variations could appear due to ephaptic
(electric-field) interactions for example. In addition, any type of linear
reactivity of the medium to the electric field or magnetic induction can lead
to frequency-dependent electric parameters $\sigma,\epsilon,\mu$ (for a
detailed discussion of such effects, see Bédard and Destexhe, 2009)., then we
can assume that the medium is linear. In this case, we can write:
$\vec{j}=\int_{-\infty}^{+\infty}\sigma(\tau)\vec{E}(t-\tau)d\tau$ (4)
where $\sigma$ is a symmetric second-order tensor 222Note that in textbooks,
these linking equations (Eqs. 2–4) are often algebraic and independent of time
(for example, see Eqs. 5.2-6, 5.2-7 and 5.2-8 in Gulrajani, 1998). The present
formulation is more general, more in the line of Landau and Lifchitz (1984)..
Because the effect of electric induction (Faraday’s law) is negligible, we can
write:
$\begin{array}[]{ccccccc}\nabla\cdot\vec{D}&=&\rho^{free}&{}\hfil&\nabla\cdot\vec{B}&=&0\\\
\nabla\times\vec{E}&=&0&{}\hfil&\nabla\times\vec{H}&=&\vec{j}+\frac{\partial\vec{D}}{\partial
t}\end{array}$ (5)
This system is much simpler compared to above, because electric field and
magnetic induction are decoupled.
By taking the Fourier transform of Maxwell equations (Eqs. 1) and of the
linking equations (Eqs. 2,3,4), we obtain:
$\begin{array}[]{ccccccc}\nabla\cdot\vec{D}_{f}&=&\rho_{f}^{free}&{}\hfil&\nabla\cdot\vec{B}_{f}&=&0\\\
\nabla\times\vec{E}_{f}&=&0&{}\hfil&\nabla\times\vec{H}_{f}&=&\vec{j}_{f}+i\omega\vec{D}_{f}\end{array}$
(6)
where $\omega=2\pi f$ and
$\begin{array}[]{ccc}\vec{D}_{f}&=&\epsilon_{f}\vec{E}_{f}\\\
\vec{B}_{f}&=&\mu_{f}\vec{H}_{f}\\\
\vec{j}_{f}&=&\vec{j}_{f}^{p}+\sigma_{f}\vec{E}_{f}\end{array}$ (7)
where the relation $\sigma_{f}\vec{E}_{f}$ in Eq. 7 is the current density
produced by the (primary) current sources in the extracellular medium. Note
that in this formulation, the electromagnetic parameters $\epsilon_{f}$,
$\mu_{f}$ and $\sigma_{f}$ depend on frequency333In textbooks, the electric
parameters are sometimes considered as complex numbers, for example with the
notion of phasor (see Section 5.3 in Gulrajani, 1998), but they are usually
considered frequency independent.. This generalization is essential if we want
the formalism to be valid for media that are linear but non-resistive, which
can expressed with frequency-dependent electric parameters. It is also
consistent with the Kramers-Kronig relations (see Landau and Lifchitz, 1984;
Foster and Schwan, 1989).
$\vec{j}_{f}^{p}$ is the current density of these sources in Fourier frequency
space. This current density is composed of the axial current in dendrites and
axons, as well as the transmembrane current. Of course, this expression is
such that at any given point, there is only one of these two terms which is
non-zero. This is a way of preserving the linearity of Maxwell equations. Such
a procedure is legitimate because the sources are not affected by the field
they produce444If it was not the case, then the source terms would be a
function of the produced field, which would result in more complicated
equations.
### 3.2 Expression for the electric field
From Eq. 6 (Faraday’s law in Fourier space), we can write:
$\vec{E}_{f}=-\nabla V_{f}~{}.$ (8)
From Eq. 6 (Ampère-Maxwell’s law in Fourier space), we can write:
$\displaystyle\nabla\cdot(\nabla\times\vec{H}_{f})$ $\displaystyle=$
$\displaystyle\nabla\cdot\vec{j}_{f}+i\omega\nabla\cdot(\epsilon_{f}\vec{E}_{f})$
(9) $\displaystyle=$
$\displaystyle\nabla\cdot\vec{j}_{f}^{p}-\nabla\cdot((\sigma_{f}+i\omega\epsilon_{f})\nabla
V_{f})=0$
Setting $\gamma_{f}=\sigma_{f}+i\omega\epsilon_{f}$, one obtains:
$\nabla\cdot(\gamma_{f}\nabla V_{f})=\nabla\cdot\vec{j}_{f}^{p}$ (10)
where $\nabla\cdot\vec{j}_{f}^{p}$ is a source term and $\mathbf{\gamma}_{f}$
is a symmetric second-order tensor ($3\times 3$). Note that this tensor
depends on position and frequency in general, and cannot be factorized. We
will call this expression (Eq. 10) the “first fundamental equation” of the
problem.
### 3.3 Expression for magnetic induction
From the mathematical identity
$\nabla\times\nabla\times\vec{X}=-\nabla^{2}\vec{X}+\nabla(\nabla\cdot\vec{X})$
(11)
it is clear that this is sufficient to know the divergence and the curl of a
field $\vec{X}$, because the solution of $\nabla^{2}{X}$ is unique with
adequate boundary conditions.
As in the case of magnetic induction, the divergence is necessarily zero, it
is sufficient to give an explicit expression of the curl as a function of the
sources.
Supposing that $\mu=\mu_{o}\delta(t)$ is a scalar (tensor where all directions
are eigenvectors), and taking the curl of Eq. 6 (D), multiplied by the inverse
of $\gamma_{f}$, we obtain the following equality:
$\nabla\times(\gamma_{f}^{inv}\nabla\times\vec{B}_{f})=\mu_{o}\nabla\times(\gamma_{f}^{inv}\vec{j}_{f}^{p})$
(12)
because $\nabla\times\vec{E}_{f}=0$. This expression (Eq. 12) will be named
the “second fundamental equation”.
### 3.4 Boundary conditions
We consider the following boundary conditions:
1 - on the skull, we assume that $V_{f}(\vec{r})$ is differentiable in space,
which is equivalent to assume that the electric field is finite.
2 - on the skull, we assume that $\hat{n}\cdot\gamma_{f}\nabla V_{f}$ is also
continuous, which is equivalent to assume that the flow of current is
continuous. Thus, we are interested in solutions where the electric field is
continuous.
3 - because the current is zero outside of the head, the current perpendicular
to the surface of cortex must be zero as well. Thus, the projection of the
current on the vector $\hat{n}$ normal to the skull’s surface, must also be
zero.
$\hat{n}(\vec{x})\cdot\gamma_{f}\nabla V_{f}(\vec{x})=0$ (13)
The latter expression can be proven by calculating the total current and apply
the divergence theorem (not shown).
### 3.5 Quasi-static approximation to calculate magnetic induction
The “second fundamental equation” above implies inverting $\gamma_{f}$, which
is not possible in general, because it would require prior knowledge of both
conductivity and permittivity in each point outside of the sources. If the
medium is purely resistive ($\gamma_{f}=\gamma$ where $\gamma$ is independent
of space and frequency), one can evaluate the electric field first, and next
integrate $\vec{B}_{f}$ using the quasi-static approximation (Ampère-Maxwell’s
law). Because for low frequencies, we have necessarily
$\vec{j}_{f}>>i\omega\vec{D}_{f}$, we obtain
$\nabla\times\vec{B}_{f}=\mu_{o}\vec{j}_{f}~{},$
which is also known as Ampère’s law in Fourier space.
Thus, for low frequencies, one can skip the second fundamental equation. Note
that in case this quasi-static approximation cannot be made (such as for high
frequencies), then one needs to solve the full system using both fundamental
equations. Such high frequencies are, however, well beyond the physiological
range, so for EEG and MEG signals, the quasi-static approximation holds if the
extracellular medium is resistive, or more generally if the medium satisfies
$\nabla\times\vec{E}_{f}=-i\omega\vec{B}_{f}\backsimeq 0$ (see Eqs. 5 and 6).
According to the quasi-static approximation, and using the linking equation
between current density and the electric field (Eq. 7), we can write:
$\nabla\times\vec{B}_{f}=\mu_{o}(\vec{j}_{f}^{p}-\gamma\nabla V_{f})$ (14)
Because the divergence of magnetic induction is zero, we have from Eq. 11:
$\nabla\times\nabla\times\vec{B}_{f}=-\nabla^{2}\vec{B}_{f}=-\mu_{o}\nabla\times(\vec{j}_{f}^{p}-\gamma\nabla
V_{f})$ (15)
This equation can be easily integrated using Poisson integral (“Poisson
equation” for each component in Cartesian coordinates) In Fourier space, this
integral is given by the following expression
$\vec{B}_{f}(\vec{r})=\frac{\mu_{o}}{4\pi}\iiint\limits_{head}\frac{\nabla\times(\vec{j}_{f}^{p}(\vec{r^{\prime}})-\gamma\nabla
V_{f}(\vec{r^{\prime}}))}{\|\vec{r}-\vec{r^{\prime}}\|}dv^{\prime}$ (16)
### 3.6 Consequences
If the medium is purely resistive (“ohmic”), then $\gamma$ does not depend on
the spatial position (see Bedard et al., 2004; Bedard and Destexhe, 2009) nor
on frequency, so that the solution for the magnetic induction is given by:
$\vec{B}_{f}(\vec{r})=\frac{\mu_{o}}{4\pi}\iiint\limits_{head}\frac{\nabla\times\vec{j}_{f}^{p}(\vec{r})}{\|\vec{r}-\vec{r^{\prime}}\|}dv^{\prime}$
(17)
and does not depend on the nature of the medium.
For the electric potential, from Eq. 10, we obtain the solution:
$V_{f}(\vec{x})=-\frac{1}{4\pi\gamma}\iiint\limits_{head}\frac{\nabla\cdot\vec{j}_{f}^{p}}{|\vec{x}-\vec{x^{\prime}}|}dv^{\prime}$
(18)
Thus, when the two source terms $\nabla\times\vec{j}_{f}^{p}$ and
$\nabla\cdot\vec{j}_{f}^{p}$ are white noise, the magnetic induction and
electric field must have the same frequency dependence. Moreover, because the
spatial dimensions of the sources are very small (see appendices), we can
suppose that the current density $\vec{j}_{f}^{p}(\vec{x})$ is given by a
function of the form:
$\vec{j}_{f}^{p}(\vec{x})=\vec{j}^{pe}(\vec{x})F(f)$ (19)
such that $\nabla\times\vec{j}_{f}^{p}$ and $\nabla\cdot\vec{j}_{f}^{p}$ have
the same frequency dependence for low frequencies. Eq. 19 constitutes the main
assumption of this formalism.
In Appendix A, we provide a more detailed justification of this assumption,
based on the differential expressions of the electric field and magnetic
induction in a dendritic cable. Note that this assumption is most likely valid
for states with low correlation such as desynchronized-EEG states or high-
conductance states, and for low-frequencies, as we analyze here (see details
in the appendices).
Thus, the main prediction of this formalism is that if the extracellular
medium is resistive, then the PSD of the magnetic induction and of the
electric potential must have the same frequency dependence. In the next
section, we will examine if this is the case for simultaneously recorded MEG
and EEG signals.
## 4 Test on experimental data
A total of 4 subjects were used for the analysis. Figure 1 shows sample MEG
and EEG channels from one of the subjects, during quiet wakefulness. Although
the subjects had eyes open, a low-amplitude alpha rhythm was occasionally
present (as visible in Fig. 1). There were also oscillations present in the
empty-room signal, but these oscillations are evidently different from the
alpha rhythm because of their low amplitude and the fact that they do not
appear in gradiometers (see Suppl. Fig. S1).
Figure 1: Simultaneous EEG and MEG recordings in an awake human subject. This
example shows a sample of channels from MEG/EEG after ECG noise removal.
Labels refer to ROIs as defined in methods (also see Figure 4). FR: Frontal,
VX:Vertex and PT: Parietotemporal. These sample channels were selected to
represent both right and left hemispheres in a symmetrical fashion. Inset:
magnification of the MEG (red) and “empty-room” (green) signals superimposed
from 4 sample channels. All traces are before any noise correction, but after
ECG decontamination.
In the next sections, we start by briefly presenting the method that was used
to estimate the frequency scaling of the PSDs. Then we report the scaling
exponents for 0.1-10 Hz frequency bands and their differences in EEG and MEG
recordings.
### 4.1 Frequency scaling exponent estimation
Because of the large number of signals in the EEG and MEG recordings, we used
an automatic non-parametric procedure to estimate the frequency scaling (see
Methods). We used a B-spline approximation by interpolation with boundary
conditions to find a curve which best represents the data(see Methods). A high
density of knots was given to the low-frequency band (0.1-10 Hz), to have an
accurate representation of the PSD in this band, and calculate the frequency
scaling. An example of optimized knots to an individual sensor is shown in
Figure 2A; note that this distribution of knots is specific to this particular
sensor. The resulting B-spline curves were used to estimate the frequency
scaling exponent using a 1st degree polynomial fit. Figure 2B shows the result
of the B-spline analysis with optimized knots (in green) capturing the essence
of the data better than the usual approximation of the slope using polynomials
(in red). The goodness of fit showed a robust estimation of the slope using
B-spline method. Residuals were -0.01 $\pm$ 0.6 for empty-room, 0.2 $\pm$ 0.65
for MEG awake, 0.05 $\pm$ 0.6 for LMSS, 0.005 $\pm$ 0.64 for NMSS, 0.08 $\pm$
0.5 for WF,0.001 $\pm$ 0.02 for PLS, and -0.02 $\pm$ 0.28 for EEG B-spline
(all numbers to be multiplied by 10-14).
Figure 2: A.log-log scale of the PSD vs frequency of a sample MEG sensor along
with the corresponding log(PSD) values (shown as circles) at optimized knots
in log-scale. B. 1st degree Polynomial fit on B-spline curve effectively
captures properties of the signal better than simple polynomial fit and avoids
the 10 Hz peak. The fit was limited between 0.1 to 10 Hz excluding the
boundaries. This limits the fit approximation to the next limiting optimized
knots (between 0.1 and 0.2 to between 9 and 10 Hz) to avoid the peaks at alpha
and low frquencies (shown by vertical dotted lines).
### 4.2 MEG and EEG have different frequency scaling exponents
Figure 3 shows the results of the B-spline curve fits to the log-log PSD vs
frequency for all sensors of all subjects. In this figure, and only for the
ease of visual comparison, these curves were normalized to the value of the
log(PSD) of the highest frequency. As can be appreciated, all MEG sensors (in
red) show a different slope than that of the EEG sensors (in blue). The
frequency scaling exponent of the EEG is close to 1 ($1/f$ scaling), while MEG
seems to scale differently. Thus, this representation already shows clear
differences of scaling between EEG and MEG signals.
Figure 3: B-spline fits of EEG awake and MEG awake (prior to noise correction)
recordings from all four subjects. Each line refers to the fit of one sensor
in log(PSD)-log(frequency) scale. For the ease of visual comparison of the
frequency scaling exponent, log(PSD) values are normalized to their value at
the maximum frequency. Each panel represents the data related to one of our
four subjects. These plots show a clear distinction between the frequency
scaling of EEG and MEG. Insets show the comparison between MEG awake (prior to
noise correction) and MEG empty-room recordings (not normalized). Note that
the empty-room scales the same as the MEG signal, but in general EEG and MEG
scale differently.
However, MEG signals may be affected by ambient or instrumental noise. To
check for this, we have analyzed the empty-room signals using the same
representation and techniques as for MEG, amd the results are represented in
Fig. 3 (insets). Empty-room recordings always scale very closely to the MEG
signal, and thus the scaling observed in MEG may be due in part to
environmental noise or noise intrinsic to the detectors. This emphasizes that
it is essential to use empty-room recordings made during the same experiment
to correct the frequency scaling exponent of MEG recordings.
To correct for this bias, we have used five different procedures (see
Methods). The first class of procedure (ES) considers that the scaling of the
MEG is entirely due to filtering by the sensors, which would explain the
similar scaling between MEG and empty-room recordings. In this case, however,
nearly all the scaling would be abolished, and the corrected MEG signal would
be similar to white noise (scaling exponent close to zero). Because the
similar scaling may be coincidental, we have used two other classes of noise
correction procedures to comply with different assumptions about the nature of
the noise. The second class, is composed of spectral subtraction (LMSS and
NMSS) or Wiener filtering (see Methods). These methods are well-established in
other fields such as acoustics. The third class, uses statistical patterns of
noise to enhance PSD (PLS method, for details see Methods).
### 4.3 Spatial variability of the frequency scaling exponent
We applied the above methods to all channels and represented the scaling
exponents in topographic plots in Fig. 4. This figure portrays that both MEG
and EEG do not show a homogenous pattern of the scaling exponent, confirming
the differences of scaling seen in Fig. 3. The EEG (Figure 4A) shows that
areas in the midline have values closer to 1, while those at the margin can
deviate from $1/f$ scaling. MEG on the other hand shows higher values of the
exponent in the frontal area and a horseshoe pattern of low value exponents in
parietotemporal regions (Figure 4B). As anticipated above, empty-room
recordings scale more or less uniformly with values close to $1/f$ (Figure
4C), thus necessitating the correction for this phenomena to estimate the
correct MEG frequency scaling exponent. Different methods for noise reduction
are shown in Figure 4: spectral subtraction methods, such as LMSS (Figure 4D),
NMSS (Figure 4E), WF enhancement (Figure 4F). These corrections preserve the
pattern seen in Figure 4B, but tend to increase the difference with EEG
scaling: one method (LMSS) yields minimal correction while the other two (NMSS
and WF) use band-specific SNR information in order to cancel the effects of
background colored-noise (see Suppl. Fig. S2), and achieve higher degree of
correction (see Supplementary Methods for details). Figure 4G portrays the use
of PLS to obtain a noiseless signal based on the noise measurements. The
degree of correction achieved by this method is higher than what is achieved
by spectral subtraction and WF methods. Exponent subtraction is shown in
Figure 4H. This correction supposes that the scaling is due to the frequency
response of the sensors, and nearly abolishes all the frequency scaling (see
also Suppl. Fig. S3 for a comparison of different methods of noise
subtraction).
Figure 4: Topographical representation of frequency scaling exponent averaged
across four subjects. A. EEG awake. B. MEG awake. C. MEG empty-room. D, E. MEG
after spectral subtraction of the empty-room noise using linear (LMSS) and
non-linear (NMSS) methods respectively. F. MEG spectral enhancement using
Wiener filtering (WF). G. MEG, partial least square (PLS) approximation of
non-noisy spectrum. H. Exponent subtraction (the exponent represented is the
value of the frequency scaling exponent calculated for MEG signals, subtracted
from the scaling exponent calculated from the corresponding emptyroom
signals). I. Spatial location of ROI masks (shown in yellow). FR covers the
Frontal, VX covers Vertex and PT spans Parietotemporal. Dots show spatial
arrangement of 102 MEG SQUID sensor triplets. The background gray-scale figure
is same as the one in panel B. Note that panels A through H use the same color
scaling.
### 4.4 Statistical comparison of EEG and MEG frequency scaling
Based on the patterns in Fig. 4, we created three ROIs covering Vertex (FR),
Vertex (VX) and the horseshoe pattern (PT). These masks are shown in Fig. 4I.
Figure 5: Statistical comparison of EEG vs. MEG frequency scaling exponent for
all regions (A) and different ROI masks (B,C & D). In each panel, a box-plot
on top is accompanied by a non-parametric distribution function in the bottom.
In the top graph, the box has lines at the lower quartile, median (red), and
upper quartile values. Smallest and biggest non-outlier observations (1.5
times the interquartile range IRQ) are shown as whiskers. Outliers are data
with values beyond the ends of the whiskers and are displayed with a red +
sign. In the bottom graph, a Non-parametric density function shows the
distribution of EEG, MEG and empty-room-corrected MEG frequency scaling
exponents (note that LMSS and WF are not shown here; see the text for
description.). Thick and thin vertical lines show the mean and mean $\pm$ std
for each probability density function (pdf).
Figure 5A represents the overall pattern providing evidence on the general
difference and the wider variability in MEG recordings. The next three panels
relate to the individual ROIs. Of the spectral subtraction methods, NMSS
achieves a higher degree of correction in comparison with LMSS (see Figure 4C,
Figure 4D as well as Suppl. Fig. S3). Because NMSS takes into account the
effects of the background colored-noise (Suppl. Fig. S2), it is certainly more
relevant to the type of signals analyzed here. The results of NMSS and WF are
almost identical and confirm one another (see Figure 4E, as well as Suppl.
Fig. S3). Therefore, of this family of noise correction, only NMSS is
portrayed here. Of the methods dealing with different assumptions about the
nature of the noise, the “Exponent subtraction” almost abolishes the frequency
scaling (Also see in Figure 4H, as well as Suppl. Fig. S3). Applying PLS
yields values in between “Exponent subtraction” and that of NMSS and is
portrayed in Figure 5.
In the Frontal region (Figure 5B), the EEG scaling exponents show higher
variance by comparison to MEG. Also, EEG shows some overlaps with the
distribution curve of non-corrected MEG; this overlap becomes limited to the
tail end of the NMSS correction and is abolished in the case of PLS
correction. As can be appreciated, VX (Figure 5C) shows both similar values
and similar distribution for EEG and non-corrected MEG. These similarities, in
terms of regional overall values and distribution curve, are further enhanced
after NMSS correction. It is to be noted that, in contrast to these
similarities, the one-to-one correlation of NMSS and EEG at VX ROI are very
low (see below, Table 1B-C). The values of PLS noise correction are very
different from that of EEG and have a similar, but narrower, distribution
curve shape. Two other ROIs show distinctively different values and
distribution in comparing EEG and MEG. Both NMSS and PLS agree on this with
PLS showing more extreme cases. Figure 5D reveals a bimodal distribution of
MEG exponents in the parietotemporal region (PT ROI). This region has also the
highest variance (in MEG scaling exponents) compared to other ROIS. The
distinction between EEG and MEG is enhanced in PLS estimates; however, the
variance of PT is reduced in comparison to NMSS while the bimodality is still
preserved but weakened. The values of mean and standard deviation for these
ROIs’ exponents are provided in Table 1A (mean $\pm$ standard deviation).
A. Mean and standard deviation
| EEG | MEG (awake) | NMSS
---|---|---|---
All | -1.33 $\pm$ 0.19 | -1.24 $\pm$ 0.26 | -1.06 $\pm$ 0.29
FR ROI | -1.36 $\pm$ 0.25 | -0.97 $\pm$ 0.10 | -0.76 $\pm$ 0.09
VX ROI | -1.21 $\pm$ 0.13 | -1.36 $\pm$ 0.10 | -1.14 $\pm$ 0.11
PT ROI | -1.36 $\pm$ 0.12 | -1.30 $\pm$ 0.29 | -1.16 $\pm$ 0.32
B. Pearson correlation
| EEG vs. MEG | EEG vs. Corrected MEG (NMSS)
---|---|---
All | 0.29 | 0.32
FR ROI | 0.41 | 0.32
VX ROI | -0.17 | -0.15
PT ROI | 0.35 | 0.38
C. Kendall Rank Corr
| EEG vs. MEG | EEG vs Corrected MEG (NMSS)
---|---|---
All | 0.21 | 0.24
FR ROI | 0.29 | 0.21
VX ROI | -0.03 | -0.04
PT ROI | 0.23 | 0.26
D. KruskalWallis
| p value | Chi-square | df Error
---|---|---|---
All | $<$ 10-15 | 1.53 103 | 34838
All noise-corrected | $<$ 10-15 | 8.03 103 | 34838
FR ROI | $<$ 10-15 | 3.30 103 | 5008
FR ROI noise-corrected | $<$ 10-15 | 3.72 103 | 5008
VX ROI | $<$ 10-15 | 1.72 103 | 5452
VX ROI noise-corrected | $<$ 10-15 | 0.23 103 | 5452
PT ROI | $<$ 10-15 | 0.21 103 | 13010
PT ROI noise-corrected | $<$ 10-15 | 1.18 103 | 13010
Table 1: ROI statistical comparison. A. mean and std of frequency scale
exponent for all regions and individual ROI. B. numerical values of linear
Pearson correlation. C. rank-based Kendall correlation. D. non-parametric test
of analysis of variance (KruskalWallis). Corrected MEG refers to spectral
subtraction using NMSS. The full table is provided in Supplementary
information.
The box-plots of Fig. 5-plots further show the difference between the medians,
lower/upper quartile and interquartile range. The overall difference is that
the uncorrected MEG has much wider variance compared to EEG and corrected MEG
(in case of PLS correction); the absolute value of the median of MEG
(uncorrected, or corrected with either NMSS or PLS) is always smaller than
that of EEG. The VX region is an exception to the above rules; interestingly,
the one-to-one correlation of VX happens to be the lowest of all (see below).
In the case of NMSS-corrected MEG, the shape of the pdf is preserved. However,
PLS narrows the distribution curve of MEG but further enhances the differences
between MEG and EEG. Therefore, median and lower/upper quartiles will have
different value than that of EEG.
Correlation values (Table 1B-C) show that, although VX ROI has the closest
similarity in terms of its central tendency and probability distribution, it
provides the lowest correlation in a pairwise fashion. P-values (for testing
the hypothesis of no correlation against the alternative that there is a
nonzero correlation) for Pearson’s correlation were calculated using a
Student’s t-distribution for a transformation of the correlation and they were
all significant (less than 10-15 for $\alpha$ = 0.05). Similarly, a non-
parametric statistic Kendall tau rank correlation was used to measure the
degree of correspondence between two rankings and assessing the significance
of this correspondence between MEG and EEG in the selected ROIs (Table 1C).
P-values for Kendall’s tau and Spearman’s rho calculate using the exact
permutation distributions were all significant (less than 10-15 for $\alpha$ =
0.05). Kendall tau shows that the rank correlation for all areas considered
together as well as for PT, show a lesser correlation than that is shown by
Pearson linear correlation. Furthermore, we carried out a Kruskal-Wallis
nonparametric version of one-way analysis of variance. We used this test to
avoid bias in ANOVA (KruskalWallis assumes that the measurements come from a
continuous distribution, but not necessarily a normal distribution as is
assumed in ANOVA). KruskalWallis uses analysis of variance on the ranks of the
data values, not the data values themselves and therefore is an appropriate
test for comparison of the homogeneity of pattern between ROIs of two image as
well as their statistical median. As shown in Table 1D, all p-values were
significant emphasizing the difference between the spatial aspect of the
spectral nature of MEG and EEG. Note that the difference of scaling exponent
of EEG and MEG was also confirmed by nonlinear spatial kendall correlation
analysis, independently of the ROIs classification (not shown).
### 4.5 Relation of scaling exponent to signal-to-noise ratio
Noise correction does not affect all the sensors in a same fashion. As
presented in Suppl. Fig. S3, the simple linear spectral subtraction (LMSS) may
lead to an increment or decrement of the scaling exponent. In any case, the
correction achieved by this method is minimal. This is due to the fact that
LMSS ignores the complex non-linear patterns of the SNR in different channels
(Suppl. Fig. S2). We show that for all subjects, as the frequency goes up, the
SNR goes down. It is also noticable that in each defined frequency band, i.e.
0-10 Hz (Slow, Delta and Theta), 11-30 Hz (Beta), 30-80 Hz (Gamma), 80-200 Hz
(Fast oscillation), 200-500 Hz (Ultra-fast oscillation), there is an
observable sensor-to-sensor SNR variability. This variability is at its
maximum in the band with the highest SNR (i.e. 1-10 Hz). All together, the
non-linear nature of MEG SNR shows that a linear spectral subtraction could
behave non-optimally, leading to minimal correction. This also conveys that
the optimal spectral correction can be achieved only by non-linear methods
that explicitly take into account the SNR information of the data. Therefore
the correction achieved by NMSS and WF have higher validity, in agreement with
the fact that both methods yield similar results in terms of values and
spatial distribution (Fig. 4E, Fig. 4F).
## 5 Discussion
In this paper, we have used a combination of theoretical and experimental
analyses to investigate the spectral structure of EEG and MEG signals. In the
first part of the paper, we presented a theoretical investigation showing that
if the extracellular medium is purely resistive, the equations of the
frequency dependence of electric field and magnetic induction take a simple
form, because the admittance tensor does not depend on spatial coordinates.
Thus, the macroscopic magnetic induction does not depend on the electric field
outside the neuronal sources, but only depends on currents inside neurons. In
this case, the frequency scaling of the PSD should be the same for EEG and MEG
signals. This conclusion is only valid in the linear regime, and for low
frequencies.
An assumption behind this formalism is that the spatial and frequency
dependence of the current density factorize (Eq. 19). We have shown in the
appendices that this is equivalent to consider the different current sources
as independent. Thus, the formalism will best apply to states where the
activity of synapses is intense and of very low correlation. This is the case
for desynchronized-EEG states or more generally “high-conductance states”, in
which the activity of neurons is intense, of low correlation, and the neuronal
membrane is dominated by synaptic conductances (Destexhe et al., 2003). In
such conditions, the dendrites are bombarded by intense synaptic inputs which
are essentially uncorrelated, and one can consider the current sources as
independent (Bedard et al., 2010). In the present paper, we analyzed EEG and
MEG recordings in such desynchronized states, where this formalism best
applies.
Note that the above reasoning neglects the possible effect of abrupt
variations of impedances between different media (e.g., between dura matter
and cerebrospinal fluid). However, there is evidence that this may not be
influential. First, our previous modeling work (Bedard et al., 2004) showed
that abrupt variations of impedance have a negligible effect on low
frequencies, suggesting that even in the presence of such abrupt variations
should not play a role at low frequencies. Second, in the frequency range
considered here, the skull and the skin are very close to be resistive at low
frequencies (Gabriel et al., 1996b), so it is very unlikely that they play a
role in the frequency scaling in EEG and MEG power spectra even at high
frequencies.
In the second part of the paper, we have analyzed simultaneous EEG and MEG
signals recorded in four healthy human subjects while awake and eyes open
(with desynchronized EEG). Because of the large number of channels involved,
we used an automatic procedure (B-splines analysis) to calculate the frequency
scaling. As found in previous studies (Pritchard, 1992; Freeman et al., 2000;
Bédard et al., 2006a), we confirm here that the EEG displays frequency scaling
close to $1/f$ at low frequencies555Note that to compare scaling exponents
between studies one must take into account that the electrode montage may
influence the scaling. For example, in bipolar (differential) EEG recordings,
if two leads are scaling as $1/(A+f)$ and $1/(B+f)$, the difference will have
regions scaling as $1/f^{2}$.. However, this $1/f$ scaling was most typical of
the midline channels, while temporal and frontal leads tended to scale with
slightly larger exponents, up to $1/f^{2}$ (see Fig. 4A). The same pattern was
observed in all four patients.
This approach differs from previous studies in two aspects. First, in contrast
to prior studies (such as Novikov et al., 1997; Linkenkaer-Hansen et al.,
2001), we calculated the frequency scaling of all the sensors and did not
confine our analysis to a specific region. Second, unlike other investigators
(such as Hwa and Ferree, 2002a,b), we did not limit our evaluations to either
EEG or MEG alone, but rather analyzed the scaling of both type of signals
simultaneously. Such a strategy enables us to provide an extended spatial
analysis of the frequency scaling. It also provides a chance to compare the
scaling properties of these signals in relation to their physical differences.
For the MEG recordings, the frequency scaling at low frequencies was
significantly lower compared to the EEG (see Fig. 3). This difference in
frequency scaling was also accompanied by spatial variability patterns (see
Fig. 4) showing three distinct regions: 1) a frontal area where the exponents
had their highest values in the case of MEG; 2) a central area where the
values of exponents of EEG and MEG get closer to each other and 3) a
parietotemporal horseshoe region showing the lowest exponents for MEG with
bimodal characteristics (Fig. 5). In some cases, the scaling of the
uncorrected and corrected MEG signal was also close to $1/f$, as reported
previously (Novikov et al., 1997). In the frontal area (FR mask), the scaling
exponent of the EEG was generally larger. At Vertex (VX mask), EEG and MEG had
similar values and at the Parietotemporal region (PT mask), MEG showed a
bimodal property with a much broader range of scaling exponent in comparison
to EEG (see Fig. 4). Note that to avoid the effect of spurious peaks, Novikov
et al. used the spectrum of signal differences and argued for the existence of
a local similarity regime in brain activity. This approach fundamentally
changes the spectral characteristics of Magnetometers (which measure the
absolute magnitude of the magnetic induction) into a measure that only for the
neighboring sensors approximates the behavior of the gradiometers (which
measures the gradient of the magnetic induction). So it is not clear how to
relate their values to the ones obtained here.
To make sure that the differences of frequency scaling between EEG and MEG
were not due to environmental or instrumental noise, we have used five
different methods to remove the effect of noise. These methods are based on
different assumptions about the nature and effect of the noise. A first
possibility is to correct for the noise induced by the MEG sensors. It is
known that the SQUID detectors used in MEG recordings are very sensitive to
environmental noise and they can produce $1/f$ noise (Hämäläinen et al.,
1993). Under this assumption, part of the scaling of the MEG could be due to
“filtering” by the sensors themselves, which justifies a simple subtraction of
scaling exponents to remove the effects of this filtering. Note that such
empty-room recordings were not possible for the EEG, although the noise from
the recording setup could be estimated (see Miller et al., 2009 for example).
Because in some cases both MEG and emptyroom signals have similar scaling, a
simple correction by subtracting the exponents would almost entirely abolish
the frequency scaling while in other cases it may even revert the sign of the
scaling exponent (see Fig. 4H , Suppl. Fig. S3).
However, if noise is not due to the sensors but is of additive uncorrelated
nature, then another method for noise correction must be used. For this
reason, we have used a second class of well-established methods consisting of
spectral subtraction (Boll et al. 1979; Sim et al., 1998). Using three of such
methods (LMSS, NMSS and WF) changed the scaling exponent, without
fundamentally changing its spatial pattern (Fig. 4D-F). The largest correction
was obtained by non-linear methods which take into account the SNR information
in the MEG signal. We also applied another class of method which uses the
collective characteristics of all frequencies in noise correction (PLS).
Similar to exponent subtraction, this method nearly abolished all the scaling
of the MEG (Fig. 4G). In conclusion, although different methods for noise
subtraction give rise to different predictions about frequency scaling, all of
the used methods enhanced the difference between EEG and MEG scaling. Thus, we
conclude that the difference of EEG and MEG scaling cannot be attributed to
noise, but is significant, therefore reinforcing the conclusion that the
medium must be non-resistive.
An alternative method to investigate this is the “Detrended Fluctuation
Analysis” (DFA; see Kantelhardt et al., 2001; Linkenkaer-Hansen et al., 2001;
Hwa and Ferree, 2002a,b). Like many nonlinear approaches, DFA results are very
vulnerable to the selection of certain parameters. Different filters severely
affect the scaling properties of the electromagnetic signals to different
extents, and therefore the parameters estimated through the DFA analysis could
be false or lead to distorted interpretations of real phenomena (Valencia et
al., 2008), and these effects are especially prominent for lower frequencies,
which are precisely our focus of investigation here. One of the fields for
which DFA can provide robust results is to analyze surrogate data with known
characteristics. Although the use of DFA to evaluate the scaling exponents of
EEG was vigorously criticized (Valencia et al., 2008), a previous analysis
(Hwa and Ferree, 2002a,b) reported two different regions, a central and a more
frontal, which somehow correlate with the FR and VX regions identified in our
analysis. Similarly, a study by Buiatti et al. (2007) using DFA provided
evidence for topographical differences in scaling exponents of EEG recordings.
They report that scaling exponents were homogeneous over the posterior half of
the scalp and became more pronounced toward the frontal areas. In contrast to
Linkenkaer-Hansen et al. (2004) (where envelope of alpha oscillations was used
for DFA estimation), this study uses the raw signal in its DFA analysis and
yields values closer to those reported here.
Both uncorrected signals and empty-room correction show that there is a
fundamentally different frequency scaling between EEG and MEG signals, with
near-$1/f$ scaling in EEG, while MEG shows a wider range at low frequencies.
Although it is possible that non-neuronal sources affect the lower end
($<$1Hz) of the evaluated frequency domain (Voipio et al., 2003), the solution
to avoid these possible effects remain limited to invasive methods such as
inserting the electrode into the scalp (Ferree et al., 2001) or using
intracranial EEG recordings (similar to Miller et al., 2009). This approach
would render wide range spatial recording as well as simultaneous invasive EEG
and MEG recordings technically demanding or impractical. However, if
technically feasible, such methods could provide a way to bypass non-neuronal
effects at very low frequency. It could also provide a chance to evaluate the
effects of spatial correlation on spectral structure at a multiscale level.
The power spectral structure we observe here is consistent with a scenario
proposed previously (Bédard et al., 2006a): the $1/f$ structure of the EEG and
LFP signals is essentially due to a frequency-filtering effect of the signal
through extracellular space; this type of scaling can be explained by ionic
diffusion and its associated Warburg impedance666Ionic diffusion can create an
impedance known as the “Warburg impedance”, which scales as $1/\sqrt{\omega}$,
giving $1/f$ scaling in the power spectra (Taylor and Gileadi, 1995; Diard et
al., 1999). (see Bédard and Destexhe, 2009). It is also consistent with the
matching of LFPs with multi-unit extracellular recordings, which can be
reconciled only assuming a $1/f$ filter (Bédard et al., 2006a). Finally, it is
also consistent with the recent evidence from the transfer function calculated
from intracellular and LFP recordings, which also showed that the
extracellular medium is well described by a Warburg impedance (Bédard et al.,
2010, submitted to this issue). If this non-resistive aspect of extracellular
media is confirmed, it may influence the results of models of source
localization, which may need to be reformulated by including more realistic
extracellular impedances.
In conclusion, the present theoretical and experimental analysis suggests that
the scaling of EEG and MEG signals cannot be reconciled using a resistive
extracellular medium. The $1/f$ structure of EEG with smaller scaling
exponents for MEG is consistent with non-resistive extracellular impedances,
such as capacitive media or diffusion (Warburg) impedances. Including such
impedances in the formalism is non trivial because these impedances are
strongly frequency dependent. The Poisson integral (the solution of Poisson’s
law $\nabla\cdot\vec{D}=-\nabla\cdot\epsilon\nabla V=\rho$) would not apply
anymore (see Bédard et al., 2004; Bédard and Destexhe, 2009). Work is under
way to generalize the formalism and include frequency-dependent impedances.
Finally, it is arguable that the scaling could also be influenced by the
cancellation and the extent of spatial averaging of microscopic signals, which
are different in EEG and MEG (for more details on cancellation see Ahlfors et
al., 2010; for details on spatial sensitivity profile see Cuffin and Cohen,
1979). Such a possible role of the complex geometrical arrangement of
underlying current sources should be investigated by 3D models which could
test specific assumptions about the geometry of the current sources and
dipoles, and their possible effect on frequency scaling. Such a scenario
constitutes another possible extension of the present study.
## Appendix
## Appendix A Frequency dependence of electric field and magnetic induction
To compare the frequency dependence of magnetic induction and electric field,
we evaluate them in a dendritic cable, expressed differentially. For a
differential element of dendrite, in Fourier space, the current produced by a
magnetic field (Ampère-Laplace law) is given by the following expression (see
Appendix B):
$\delta\vec{B}_{f}(\vec{r})=\frac{\mu_{o}}{4\pi}\vec{j}_{f}^{p}(\vec{r^{\prime}})\times~{}\frac{\vec{r}-\vec{r^{\prime}}}{\|\vec{r}-\vec{r^{\prime}}\|^{3}}~{}\delta
v^{\prime}$ (20)
when the extracellular medium is resistive. Note that the source of magnetic
induction is essentially given by the component of $\vec{j}_{f}^{p}$ along the
axial direction ($j_{f}^{i}$) within each differential element of dendrite
because the perpendicular (membrane) current does not participate to producing
the magnetic induction if we assume a cylindrical symmetry.
For the electric potential, we have the following differential expression for
a resistive medium (see Appendix C):
$\delta\vec{V}_{f}(\vec{r})=\frac{1}{4\pi\gamma}\frac{\delta
I_{f}^{\perp}(\vec{r}^{\prime})}{\|\vec{r}-\vec{r^{\prime}}\|}=\frac{1}{4\pi\gamma}\frac{j_{f}^{m}(\vec{r}^{\prime})}{\|\vec{r}-\vec{r^{\prime}}\|}\delta
S^{\prime}$ (21)
where $j_{f}^{m}$ is the transmembrane current per unit of surface.
If we consider the differential expressions for the magnetic induction (Eq.
20) and electric potential (Eq. 21), one can see that the frequency dependence
of the ratio of their modulus is completely determined by the frequency
dependence of the ratio of current density $j_{f}^{m}$ and $j_{f}^{i}$. In
Appendix D, we show that this ratio is quasi-independent of frequency for a
resistive medium, for low frequencies (smaller that $\sim$10 Hz), and if the
current sources are of very low correlation.
Thus, magnetic induction and electric potential can be very well approximated
by:
$\begin{array}[]{c}V_{f}(\vec{r})=N<V>=N<\sum\limits_{l=1}^{N}\delta
V_{f}^{l}>\\\
\vec{B}_{f}(\vec{r})=N<\vec{B}>=N<\sum\limits_{l=1}^{N}\delta\vec{B}_{f}^{l}>\end{array}$
(22)
for sufficiently small differential dendritic elements ($N/l$ large).
Because the functions of spatial and frequency are statistically independent,
we can write the following expressions for the square modulus of the fields
(see Eqs. 20 and 21):
$\begin{array}[]{ccccc}|V_{f}(\vec{r})|^{2}&=&N^{2}|<\sum\limits_{l=1}^{N}V^{l}(\vec{r})G_{l}^{m}(f)>|^{2}&=&|V(\vec{r})|^{2}|G(f)|^{2}\\\
\|\vec{B}_{f}(\vec{r})\|^{2}&=&N^{2}\|<\sum\limits_{i=1}^{N}\vec{B}^{i}(\vec{r})G_{l}^{m}(f)>\|^{2}&=&\|\vec{W}(\vec{r})\|^{2}|G(f)|^{2}\end{array}$
(23)
where $G(f)=<G_{l}^{m}(f)>$, $V^{l}(\vec{r})=<V^{l}(\vec{r})>$ and
$\vec{W}(\vec{r})=<\vec{B}^{l}(\vec{r})>$ . Thus, the scaling of the PSDs of
the electric potential and magnetic induction must be the same for low
frequencies (smaller than $\sim$10 Hz) if the medium is resistive and when the
current sources have very low correlation.
## Appendix B Differential expression for the magnetic induction
According to Maxwell equations, the magnetic induction is given by:
$\vec{B}_{f}(\vec{r})=\frac{\mu_{o}}{4\pi}\iiint\limits_{head}\frac{\nabla^{\prime}\times\vec{j}_{f}^{p}(\vec{r^{\prime}})}{\|\vec{r}-\vec{r^{\prime}}\|}~{}dv^{\prime}$
(24)
where $dv^{\prime}=dx^{{}^{\prime}1}dx^{{}^{\prime}2}dx^{{}^{\prime}3}$ and
$\nabla^{\prime}(\frac{1}{\|\vec{r}-\vec{r^{\prime}}\|})=\frac{\vec{r}-\vec{r^{\prime}}}{\|\vec{r}-\vec{r^{\prime}}\|^{3}}$
for a perfectly resistive medium.
We now show that this expression is equivalent to Ampere-Laplace law.
From the identity
$\nabla^{\prime}\times(g\vec{A})=g(\nabla^{\prime}\times\vec{A})+\nabla^{\prime}g\times\vec{A}$,
where $\nabla^{\prime}=\hat{e}_{x}\frac{\partial}{\partial
x^{\prime}}+\hat{e}_{y}\frac{\partial}{\partial
y^{\prime}}+\hat{e}_{z}\frac{\partial}{\partial z^{\prime}}$, we can write:
$\vec{B}_{f}(\vec{r})=\frac{\mu_{o}}{4\pi}\iiint\limits_{head}[\nabla^{\prime}\times(\frac{\vec{j}_{f}^{p}(\vec{r^{\prime}})}{\|\vec{r}-\vec{r^{\prime}}\|})+\frac{\mu}{4\pi}\vec{j}_{f}^{p}(\vec{r^{\prime}})\times\nabla^{\prime}\frac{1}{\|\vec{r}-\vec{r^{\prime}}\|}]~{}dv^{\prime}$
(25)
Moreover, we also have the following identity
$\iiint\limits_{head}\nabla^{\prime}\times(\frac{\vec{j}_{f}^{p}(\vec{r^{\prime}})}{\|\vec{r}-\vec{r^{\prime}}\|})~{}dv^{\prime}=-\iint\limits_{\partial
head}\frac{\vec{j}_{f}^{p}(\vec{r^{\prime}})}{\|\vec{r}-\vec{r^{\prime}}\|}\times\hat{n}~{}dS^{\prime}$
(26)
where $\hat{n}$ is a unitary vector perpendicular to the integration surface
and going outwards from that surface. Extending the volume integral outside
the head, the surface integral is certainly zero because the current is zero
outside of the head. It follows that:
$\vec{B}_{f}(\vec{r})=\frac{\mu_{o}}{4\pi}\iiint\limits_{head}\vec{j}_{f}^{p}(\vec{r^{\prime}})\times~{}\frac{\vec{r}-\vec{r^{\prime}}}{\|\vec{r}-\vec{r^{\prime}}\|^{3}}~{}dv^{\prime}$
(27)
where $dv^{\prime}=dx^{{}^{\prime}1}dx^{{}^{\prime}2}dx^{{}^{\prime}3}$
because
$\nabla^{\prime}(\frac{1}{\|\vec{r}-\vec{r^{\prime}}\|})=\frac{\vec{r}-\vec{r^{\prime}}}{\|\vec{r}-\vec{r^{\prime}}\|^{3}}$
Eq. 27 is called the Ampère-Laplace law (see Eq. 13 in Hämäläinen et al.,
1993). It is important to note that this expression for the magnetic induction
is not valid when the medium is not resistive.
Finally, from the last expression, the magnetic induction for a differential
element of dendrite can be written as:
$\delta\vec{B}_{f}(\vec{r})=\frac{\mu_{o}}{4\pi}\vec{j}_{f}^{p}(\vec{r^{\prime}})\times~{}\frac{\vec{r}-\vec{r^{\prime}}}{\|\vec{r}-\vec{r^{\prime}}\|^{3}}~{}\delta
v^{\prime}$ (28)
## Appendix C Differential expression of the electric field and electric
potential
In this appendix, we derive the differential expression for the electric
field. Starting from Eq. 10, we obtain the solution for the electric
potential:
$V_{f}(\vec{r})=-\frac{1}{4\pi\gamma_{f}}\iiint\limits_{head}\frac{\nabla\cdot\vec{j}_{f}^{p}(\vec{r}^{\prime})}{\|\vec{r}-\vec{r^{\prime}}\|}~{}dv^{\prime}$
(29)
It follows that the electric field produced by the ensemble of sources can be
expressed as:
$\vec{E}_{f}(\vec{r})=-\nabla
V_{f}(\vec{r})=\frac{1}{4\pi\gamma_{f}}\iiint\limits_{head}\nabla\cdot\vec{j}_{f}^{p}(\vec{r}^{\prime})\cdot\frac{\vec{r}-\vec{r^{\prime}}}{\|\vec{r}-\vec{r^{\prime}}\|^{3}}~{}dv^{\prime}$
(30)
such that every differential element of dendrite produces the following
electric field:
$\delta\vec{E}_{f}(\vec{r})=\frac{\nabla\cdot\vec{j}_{f}^{p}(\vec{r}^{\prime})}{4\pi\gamma_{f}}\cdot\frac{\vec{r}-\vec{r^{\prime}}}{\|\vec{r}-\vec{r^{\prime}}\|^{3}}~{}\delta
v^{\prime}$ (31)
The transmembrane current $\delta I_{f}^{\perp}$ obeys $\delta
I_{f}^{\perp}=i\omega\rho_{f}(\vec{r}^{\prime})\delta v^{\prime}$ because we
are in a quasi-stationary regime in a differential dendritic element. Taking
into account the differential law of charge conservation
$\nabla\cdot\vec{j}_{f}(\vec{r}^{\prime})=-i\omega\rho_{f}(\vec{r}^{\prime})$,
we have:
$\delta\vec{E}_{f}(\vec{r})=\frac{\delta
I_{f}^{\perp}(\vec{r}^{\prime})}{4\pi\gamma_{f}}\frac{\vec{r}-\vec{r^{\prime}}}{\|\vec{r}-\vec{r^{\prime}}\|^{3}}=\frac{j_{f}^{m}(\vec{r}^{\prime})}{4\pi\gamma_{f}}\frac{\vec{r}-\vec{r^{\prime}}}{\|\vec{r}-\vec{r^{\prime}}\|^{3}}\delta
S^{\prime}$ (32)
where $j_{f}^{m}$ is the density of transmembrane current per unit surface and
$\delta S^{\prime}$ is the surface area of a differential dendritic element.
This approximation is certainly valid for frequencies lower than 1000 Hz
because the Maxwell-Wagner time (see Bedard et al., 2006b) of the cytoplasm
($\tau_{mw}^{cyto}=\epsilon/\sigma\sim 10^{-10}~{}s.$) is much smaller than
the typical membrane time constant of a neuron ($\tau_{m}\sim 5-20~{}ms$).
Finally the contribution of a differential element of dendrite to the electric
potential at position $\vec{r}$ is given by
$\delta\vec{V}_{f}(\vec{r})=\frac{1}{4\pi\gamma_{f}}\frac{\delta
I_{f}^{\perp}(\vec{r}^{\prime})}{\|\vec{r}-\vec{r^{\prime}}\|}=\frac{1}{4\pi\gamma_{f}}\frac{j_{f}^{m}(\vec{r}^{\prime})}{\|\vec{r}-\vec{r^{\prime}}\|}\delta
S^{\prime}$ (33)
We note that the expressions for the electric field and potential produced by
each differential element of dendrite have the same frequency dependence
because it is directly proportional to $\frac{j_{f}^{m}}{\gamma_{f}}$ for the
two expressions. Also note that if the medium is resistive, then
$\gamma_{f}=\gamma$ and the frequency dependence of the electric field and
potential are solely determined by that of the transmembrane current
$j_{f}^{m}$.
## Appendix D Frequency dependence of the ratio
${j_{f}^{i}(\vec{x})}/{j_{f}^{m}(\vec{x})}$.
For each differential element of dendrite, we consider the standard cable
model, in which the impedance of the medium is usually neglected (it is
usually considered negligible compared to the membrane impedance). In this
case, we have:
$\left\\{\begin{array}[]{ccc}j_{f}^{m}&=&\frac{V_{f}^{m}}{r_{m}}+i\omega
c_{m}V_{f}^{m}\\\ {}\hfil\\\ j_{f}^{i}&=&-\sigma\frac{\partial
V_{f}^{m}}{\partial x}=-\frac{1}{r_{i}}\frac{\partial V_{f}^{m}}{\partial
x}\end{array}\right.$ (34)
where $V_{f}^{m}$, $j_{f}^{i}$, $j_{f}^{m}$, $c_{m}$, $r_{m}$ et $r_{i}$ are
respectively the membrane potential, the current density in the axial
direction, the transmembrane current density, the specific capacitance
($F/m^{2}$), the specific membrane resistance ($\Omega.m^{2}$) and the
cytoplasm resistivity ($\Omega.m$).
It follows that
$\frac{j_{f}^{i}(\vec{x})}{j_{f}^{m}(\vec{x})}=\frac{r_{m}}{r_{i}(1+i\omega\tau_{m})}~{}\cdot\frac{\partial}{\partial
x}[ln(V_{f}^{m}(\vec{x})]$ (35)
where $\tau_{m}=r_{m}c_{m}$.
Under in vivo–like conditions, the activity of neurons is intense and of very
low correlation. This is the case for desynchronized-EEG states, such as awake
eyes-open conditions, where the activity of neurons is characterized by very
low levels of correlations. There is also evidence that in such conditions,
neurons are in “high-conductance states” (Destexhe et al., 2003), in which the
synaptic activity dominates the conductance of the membrane and primes over
intrinsic currents. In such conditions, we can assume that the synaptic
current sources are essentially uncorrelated and dominant, such that the
deterministic link between current sources will be small and can be neglected
(see Bedard et al., 2010). Further assuming that the electric properties of
extracellular medium are homogeneous, then each differential element of
dendrite can be considered as independent and the voltages $V_{m}$ have
similar power spectra.
In such conditions, we have:
$V_{f}^{m}(\vec{x})=F^{m}(\vec{x})G^{m}(f)$ (36)
Note that this expression implies that we have in general for each
differential element of dendrite:
$\left\\{\begin{array}[]{ccc}j_{f}^{m}(\vec{x})&=&F^{m}(\vec{x})(\frac{1+i\omega\tau_{m}}{r_{m}})G^{m}(f)\\\
{}\hfil&{}\hfil&{}\hfil\\\ j_{f}^{i}(\vec{x})&=&-\frac{1}{r_{i}}\frac{\partial
F^{m}(\vec{x})}{\partial x}G^{m}(f)=F^{i}(\vec{x})G^{m}(f)\end{array}\right.$
(37)
according to Eq. 34.
It follows that
$\frac{j_{f}^{i}(\vec{x})}{j_{f}^{m}(\vec{x})}=\frac{r_{m}}{r_{i}(1+i\omega\tau_{m})}~{}\cdot\frac{\partial}{\partial
x}[ln(F(\vec{x}))]\approx\frac{r_{m}}{r_{i}}~{}\cdot\frac{\partial}{\partial
x}[ln(F(\vec{x}))]$ (38)
Thus, for frequencies smaller than $1/(\omega\tau_{m})$ (about 10 to 30 Hz for
$\tau_{m}$ of 5-20 ms), the ratio
$\frac{j_{f}^{i}(\vec{x})}{j_{f}^{m}(\vec{x})}$ will be frequency independent,
and for each differential element of dendrite, we have:
$\left\\{\begin{array}[]{ccc}j_{f}^{m}(\vec{x})&=&F^{m}(\vec{x})G^{m}(f)\\\
j_{f}^{i}(\vec{x})&=&F^{i}(\vec{x})G^{m}(f)\end{array}\right.$ (39)
for frequencies smaller than $\sim$10 Hz.
## Acknowledgments
We thank Philip Louizo for comments on spectral subtraction methods and Hervé
Abdi for comments on Partial least square methods. Research supported by the
Centre National de la Recherche Scientifique (CNRS, France), Agence Nationale
de la Recherche (ANR, France), the Future and Emerging Technologies program
(FET, European Union; FACETS project) and the National Institutes of Health
(NIH grants NS18741, EB009282 and NS44623). N.D. is supported by a fellowship
from Ecole de Neurosciences de Paris (ENP). Additional information is
available at http://cns.iaf.cnrs-gif.fr
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## Supplementary material
### Supplementary methods
We give details below to some of the methods and quantities used in the
Results.
#### SNR
Two of the used methods for noise-correction are based on band-specific
signal-to-noise ratio (SNR) in order to cancel the effects of background
colored-noise in the spectra of interest. In each subject, average PSD was
used to calculate signal-to-noise ratio (SNR). For SNR calculation, few
frequency bands were defined based on the categorization in Buzsaki & Draguhn
(2004): 0-10 Hz (Slow, Delta and Theta), 11-30 Hz (Beta), 30-80 Hz (Gamma),
80-200 Hz (Fast oscillation), 200-500 Hz (Ultra-fast oscillation). SNR was
calculated as:
$SNR_{bi}=\frac{\sum{10*log10(\frac{PSDsignal_{bi}}{PSDnoise_{bi}})}}{n}$ (40)
for a given band ”b” and sensor ”i”, ”n” is the frequency resolution of that
band. This method was applied on individual average PSD as well as shape
preserving spline of each average PSD where each PSD was fist smoothed in
log10 scale using a shape preserving spline, i.e, Piecewise Cubic Hermite
Interpolating Polynomial (PCHIP).
#### Multiband spectral subtraction
Assuming the additive noise to be stationary and uncorrelated with the clean
signal, nearly most spectral subtraction methods can be formulated using a
parametric equation:
$\lvert\widehat{S(k)}\rvert^{\alpha}=a_{k}\lvert
Y(k)\rvert^{\alpha}-b_{k}\lvert\widehat{D(k)}\rvert^{\alpha}$ (41)
where $\lvert\widehat{{S}}_{k}\rvert$, $\lvert{Y}_{k}\rvert$ and
$\lvert\widehat{{D}}_{k}\rvert$ refer to enhanced magnitude spectrum estimate
(corrected signal), the noisy magnitude spectrum (original signal) and noise
magnitude spectrum estimate (“noise”), respectively. $k$ is the frequency
index, while $a_{k}$ and $b_{k}$ are linear coefficient parameters of the
summation. Spectral subtraction methods fall into three main categories (Sim
et al., 1998). The simplest of all, a linear method where $a_{k}$ = $b_{k}$ =
1, $\alpha$=2, following Boll et al. (1979) was used here. This linear
multiband spectral subtraction (LMSS) method is well-established for noise
subtraction (see Loizou, 2007 for a comparative study of noise subtraction
methods).
An improved method, with $a_{k}$ = 1 and $b_{k}$ = v, where ”v” is the
oversubtraction factor. This method uses oversubtraction and introduces a
spectral flooring to minimize residual noise and musical noise (Berouti et
al., 1979). A second category of spectral subtraction is based on $a_{k}$ =
$b_{k}$ = f(k). Third and the most robust methods are based on a non-linear
multiband subtraction (NMSS) where $a_{k}$ = 1 and $b_{k}$ = v(k); i.e., the
oversubtraction factor is adjusted based on a specfic band’s SNR. These
methods proposed by (Kamath and Loizou, 2002; Loizou, 2007) are suitable for
dealing with colored noise (Boubakir et al., 2007; Sim et al., 1998), a case
similar to MEG recordings. The spectrum is divided into N non-overlapping
bands, and spectral subtraction is performed independently in each band. The
Eqs. 41 is simply reduced to:
$\lvert\widehat{{S}_{i}(k)}\rvert^{2}=\lvert{Y}_{i}(k)\rvert^{2}-{\alpha}_{i}{\delta}_{i}\lvert\widehat{{D}_{i}(k)}\rvert^{2},b_{i}\leq
k\leq e_{i}$ (42)
where $b_{i}$ and $e_{i}$ are the beginning and ending frequency bins of the
ith frequency band, $\alpha_{i}$ is the overall oversubtraction factor of the
ith band and $\delta_{i}$ is a tweaking factor. The band specific
oversubtraction factor $\alpha_{i}$ is a function of the segmental $SNR_{i}$
of the ith frequency band. After calculating bandspecific SNR (Eqs. 40), we
used the product of lower 10 percent of crosssubject average SNR and standard
deviation of $SNR_{i}$ to estimate the $\alpha_{i}$ $\delta_{i}$ subtraction
coefficient. Next, simply by multiplying the noise PSD by this coefficient and
subtracting it from the measured PSD, the enhanced PSD was achieved.
#### Wiener filter (WF) spectral enhancement
The principle of the Wiener filter is to obtain an estimate of the clean
signal from that of the noisy measurement through minimizing the Mean Square
Error (MSE) between the desired and the measured signal (Lim et al., 1979; Abd
El-Fattah et al., 2008). In the frequency domain, this relation is formulated
as filtering transfer function:
$WF(k)=\frac{P_{s}(k)}{P_{s}(k)+P_{n}(k)}$ (43)
where, as before, $P_{s}(k)$ and $P_{n}(k)$ refer to enhanced power spectrum
estimate and noise power spectrum estimate respectively for a signal frame and
$k$ is the frequency index. Based on the definition of SNR as, the ratio of
these two elements, one can formulate the WF as:
$WF_{K}=[1+\frac{1}{SNR_{k}}]^{-1}$ (44)
After calculation of bandspecific WF, the noisy signal is simply muliplied by
the WF to obtain the enhanced signal.
#### Partial least square (PLS) approximation of non-noisy spectrum
Partial least squares (PLS) regression, combines “Principal component
analysis” (PCA) and “Multiple linear regression” (Abdi, 2010; Abdi and
Williams, 2010). While PCA finds hyperplanes of maximum variance between the
response and independent variables, PLS projects the predicted variables and
the observable variables to a new space. Then from this new space, it finds a
linear regression model for the projected data. Next, using this model, PLS
finds the multidimensional direction in the X space that explains the maximum
multidimensional variance direction in the Y space (Abdi, 2010; Garthwaite,
1994). If X is the PSD of noise measurement and Y is the PSD of the measured
signal contaminated with background noise, one can use PLS to ”clean” one
matrix (Y) by predicting Y from X and then using the residual of the
prediction of Y by X as the estimate of pure PSD. The patterns of the awake
spectrum that statistically resembles the patterns of emptyroom spectral noise
are those that should be removed. As during PLS algorithm, the data is mean
subtracted and z-normalized, the predection of Y from X is an approximate of
the zscored PSD. Therefore, the reseidual Y, which is taken as the spectral
features that can not be predicted by noise, also has zscored values. It has
too be emphasized that this approach of denoising only works in the spectral
but not the time domain.
### Supplementary table
A. Mean and standard deviation
| EEG | MEG (awake) | MEG(empty) | LMSS
---|---|---|---|---
All | -1.33 $\pm$ 0.19 | -1.24 $\pm$ 0.26 | -1.04 $\pm$ 0.13 | -1.24 $\pm$ 0.28
FR ROI | -1.36 $\pm$ 0.25 | -0.97 $\pm$ 0.10 | -0.97 $\pm$ 0.06 | -0.96 $\pm$ 0.11
VX ROI | -1.21 $\pm$ 0.13 | -1.36 $\pm$ 0.10 | -1.10 $\pm$ 0.09 | -1.36 $\pm$ 0.10
PT ROI | -1.36 $\pm$ 0.12 | -1.30 $\pm$ 0.29 | -1.08 $\pm$ 0.15 | -1.31 $\pm$ 0.32
| NMSS | WF | PLS | ES
---|---|---|---|---
All | -1.06 $\pm$ 0.29 | -1.05 $\pm$ 0.27 | -0.50 $\pm$ 0.11 | -0.20 $\pm$ 0.23
FR ROI | -0.76 $\pm$ 0.09 | -0.76 $\pm$ 0.08 | -0.40 $\pm$ 0.05 | -0.00 $\pm$ 0.09
VX ROI | -1.14 $\pm$ 0.11 | -1.12 $\pm$ 0.11 | -0.50 $\pm$ 0.04 | -0.26 $\pm$ 0.08
PT ROI | -1.16 $\pm$ 0.32 | -1.14 $\pm$ 0.30 | -0.54 $\pm$ 0.11 | -0.22 $\pm$ 0.26
B. Pearson correlation of EEG vs.
| MEG | LMSS | NMSS | WF | PLS | ES
---|---|---|---|---|---|---
All | 0.29 | 0.29 | 0.32 | 0.33 | 0.37 | 0.35
FR ROI | 0.41 | 0.39 | 0.32 | 0.37 | 0.01 | 0.17
VX ROI | -0.17 | -0.10 | -0.15 | -0.13 | 0.01 | -0.28
PT ROI | 0.35 | 0.34 | 0.38 | 0.39 | 0.46 | 0.41
C. Kendall Rank Corr of EEG vs.
| MEG | LMSS | NMSS | WF | PLS | ES
---|---|---|---|---|---|---
All | 0.21 | 0.21 | 0.24 | 0.25 | 0.29 | 0.23
FR ROI | 0.29 | 0.23 | 0.21 | 0.27 | -0.06 | 0.12
VX ROI | -0.03 | 0.04 | -0.04 | -0.03 | 0.07 | -0.09
PT ROI | 0.23 | 0.23 | 0.26 | 0.26 | 0.30 | 0.27
Table 2: ROI statistical comparison for different noise correction methods. A.
mean and std of frequency scale exponent for all regions and individual ROI.
B. numerical values of linear Pearson correlation. C. rank-based Kendall
correlation.
### Supplementary figures
Figure S1: Frequency spectra of magnetometers and gradiometers. Comparison of
awake (blue) vs empty-room (red) recordings between Magnetometers (MAG) and
Gradiometers (GRAD1, GRAD2) in a sample subject. As for the EEG, the MEG
signal is characterized by a peak at around 10 Hz, which is presumably due to
residual alpha rhythm (although the subject had eyes open). This is also
visible from the MEG signals (Fig. 1) as well as from their PSD (Fig. 3 and
MAG panel here). The power spectrum from the empty-room signals also show a
peak at around 10 Hz, but this peak disappears from the gradiometer empty-room
signals, while the 10 Hz peak of MEG still persists for gradiometers awake
recordings. This suggests that these two 10 Hz peaks are different oscillation
phenomena. All other subjects showed a similar pattern.
Figure S2: Signal-to-noise ratio (SNR) of Magnetometers (MAG) for multiple
frequency bands: 0-10 Hz (Slow, Delta and Theta), 11-30 Hz (Beta), 30-80 Hz
(Gamma), 80-200 Hz (Fast oscillation), 200-500 Hz (Ultra-fast oscillation). In
the scatterplots, red astrisks relate to individual sensors and the blue line
is the band-specific mean across the sensors. In boxplots, the box has lines
at the lower quartile, median (red), and upper quartile values. Smallest and
biggest non-outlier observations (1.5 times the interquartile range IRQ) are
shown as whiskers. Outliers are data with values beyond the ends of the
whiskers and are displayed with a red + sign. In all subjects, the SNR shows a
band-specific trend and has the highest value for lower frequencies and
gradually drops down as band frequency goes up. As the frequency drops, the
variability of SNR (among sensors) rises; therefore, the SNR of the lowest
band (1-10 Hz) shows the highest sensors-to-sensor variability and the highest
SNR in comparison to other frequency bands.
Figure S3: Noise correction comparison. Every horizontal line showes a voxel
of the topographical maps shown in Fig. 4 sorted based on the scaling exponent
values of awake MEG (left stripe). Using a continuous color spectrum, these
stripes show that minimal correction is achived by LMSS. As indicated in the
text, the performance of this method is not reliable due to the nonlinear
nature of SNR (see Suppl. Fig. S2). NMSS yields higher degree of correction.
WF performs almost identical to NMSS (not shown here). Exponent subtraction
almost abolishes the sacling all together (far right stripe). PLS results in
values between NMSS and ”Exponent subtraction”. For details of each of these
correction procedures, see Methods. LMSS, NMSS and WF rely on additive
uncorrelated nature of noise. “Exponent subtraction” assumes that the noise is
intrinsic to SQUID. PLS ascertains the characteristics of noise to the
collective obeserved pattern of spectral domain across all frequencies. See
text for more details.
|
arxiv-papers
| 2010-03-29T13:23:51 |
2024-09-04T02:49:09.304866
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Nima Dehghani, Claude Bedard, Sydney S. Cash, Eric Halgren and Alain\n Destexhe",
"submitter": "Alain Destexhe",
"url": "https://arxiv.org/abs/1003.5538"
}
|
1003.5563
|
11institutetext: Dipartimento di Astronomia, Università degli Studi di
Bologna, Via Ranzani 1, I–40127 Bologna, Italy 22institutetext: INAF —
Osservatorio Astronomico di Bologna, Via Ranzani 1, I–40127 Bologna, Italy
33institutetext: Instituto de Estructura de la Materia (IEM/CSIC), C/ Serrano
121, 28006 Madrid, Spain 44institutetext: Sterrenkundig Observatorium,
Vakgroep Fysica en Sterrenkunde, Universeit Gent, Krijgslaan 281, S9 9000
Gent, Belgium 55institutetext: INAF — Osservatorio Astronomico di Roma, Via di
Frascati 33, 00040 Roma, Italy 66institutetext: Max Planck Institut für
Extraterrestrische Physik (MPE), Giessenbachstrasse 1, D-85748 Garching bei
München, Germany
# The HELLAS2XMM survey
XIII. Multi-component analysis
of the spectral energy distribution of obscured AGN
F. Pozzi C. Vignali 1122 A. Comastri 1122 E. Bellocchi 22 J. Fritz 33 C.
Gruppioni 44 M. Mignoli 22 R. Maiolino 22 L. Pozzetti 55 M. Brusa 22 F. Fiore
66 55 G. Zamorani 22
###### Abstract
Aims. We combine near-to-mid-IR Spitzer data with shorter wavelength
observations (optical to X-rays) to get insights on the properties of a sample
of luminous, obscured Active Galactic Nuclei (AGN). We aim at modeling their
broad-band Spectral Energy Distributions (SEDs) in order to estimate the main
parameters related to the dusty torus which is assumed responsible for the
reprocessed IR emission. Our final goal is to estimate the intrinsic nuclear
luminosities and the Eddington ratios for our luminous obscured AGN.
Methods. The sample comprises 16 obscured high-redshift
($0.9{{{}_{<}\atop{}^{\sim}}}z{{{}_{<}\atop{}^{\sim}}}2.1$) X-ray luminous
quasars (L${}_{2-10keV}{\sim}10^{44}$erg s-1) selected from the HELLAS2XMM
survey in the 2–10 keV band. The optical-IR SEDs are described by a multi-
component model including a stellar component to account for the optical and
near-IR emission, an AGN component which dominates in the mid-IR (mainly
emission from a dusty torus heated by nuclear radiation) and a starburst to
reproduce the far-IR bump. A radiative transfer code to compute the spectrum
and intensity of dust reprocessed emission was extensively tested against our
multiwavelength data. While the torus parameters and the BH accretion
luminosities are a direct output of the SED-fitting procedure, the BH masses
are estimated indirectly, by means of the local Mbulge-MBH relation.
Results. The majority (${\sim}$80%) of the sources show moderate optical
depth ($\tau_{9.7{\mu}m}{\leq}3$) and the derived column densities NH are
consistent with the X-ray inferred values
($10^{22}{{{}_{<}\atop{}^{\sim}}}$N${}_{H}{{{}_{<}\atop{}^{\sim}}}3{\times}10^{23}$
cm-2) for most of the objects, confirming that the sources are moderately
obscured Compton-thin AGN. Accretion luminosities in the range
$5{\times}10^{44}{{{}_{<}\atop{}^{\sim}}}L_{bol}{{{}_{<}\atop{}^{\sim}}}4{\times}10^{46}$
erg s-1 are inferred from the multiwavelength fitting procedure. We compare
model luminosities with those obtained by integrating the observed SED,
finding that the latter are lower by a factor of ${\sim}$2 in the median. The
discrepancy can be as high as an order of magnitude for models with high
optical depth ($\tau_{9.7{\mu}m}=10$). The ratio between the luminosities
obtained by the fitting procedure and from the observed SED suggest that, at
least for Type 2 AGN, observed bolometric luminosities are likely to
underestimate intrinsic ones and the effect is more severe for highly obscured
sources. Bolometric corrections from the hard X–ray band are computed and have
a median value of k${}_{2-10{keV}}{\sim}20$. The obscured AGN in our sample
are characterized by relatively low Eddington ratios (median
${\lambda}_{Edd}{\sim}0.08$). On average, they are consistent with the
Eddington ratio increasing at increasing bolometric correction (e.g. Vasudevan
& Fabian 2009).
###### Key Words.:
quasars: general — galaxies: nuclei — galaxies: active
## 1 Introduction
A robust determination of Active Galactic Nuclei (AGN) Spectral Energy
Distributions (SEDs) is of paramount importance to better understand the
accretion processes onto supermassive black holes (SMBHs) and their
cosmological evolution. According to our present knowledge, the bulk of
accretion luminosity is emitted in the optical-UV range with a quasi-thermal
spectrum originating in an optically thick, geometrically thin, accretion
disk. Electrons with temperatures of the order of a few hundreds of keV form a
hot corona which upscatters disk photons to X-ray energies with a power law
spectrum and an exponential high-energy cut-off corresponding to the electron
temperature (e.g., Haardt & Maraschi 1991). Dusty material, possibly with a
toroidal geometry, intercepts a fraction of the primary continuum which
depends on the covering factor. The absorbed energy is re-emitted in the near
to far infrared with a grey-body spectral shape.
The SED of optically bright, unobscured QSO is relatively well known. After
the seminal work of Elvis et al. (1994), fairly accurate measurements were
published by Richards et al. (2006) using Sloan Digital Sky Survey (SDSS)
data. The emission is characterized by a double bump. In a ${\nu}F_{\nu}$
diagram, the optical-UV spectrum rises steeply towards the shortest accessible
wavelengths. It is commonly referred to as the Big Blue Bump and thought to be
the accretion disk signature. The IR bump is weaker and likely due to a dusty
torus seen almost face-on. The ratio between UV (at 2500 Å) and X-ray
luminosity (at 2 keV), parameterized by the slope
$\alpha_{ox}$111$\alpha_{ox}=-\frac{log(f_{2keV}/f_{2500\AA})}{log(\nu_{2keV}/\nu_{2500\AA})}$,
Zamorani et al. (1981). of the power law connecting the rest-frame
luminosities, increases with increasing UV luminosity (e.g., Vignali et al.
2003; Steffen et al. 2006).
The average SEDs for radio-loud and radio-quiet Type 1 AGN presented in Elvis
et al. (1994) allow the estimation of bolometric corrections, which are a key
parameter to determining the bolometric luminosity from observations at a
given frequency, and the Eddington ratio, once the SMBH mass is known. By
including the $\alpha_{ox}$ vs. UV luminosity dependence, Marconi et al.
(2004) and Hopkins et al. (2006) computed luminosity-dependent bolometric
corrections and adopt them to estimate the local SMBH mass density from the
observed X-ray luminosity functions.
The luminosity dependence of bolometric corrections was recently questioned by
Vasudevan & Fabian (2009), who pointed out the importance of simultaneous
optical-UV and X-ray observations and reddening corrections in the UV. They
suggest that the bolometric correction correlates with the Eddington ratio
rather than with the bolometric luminosity. Their observational results align
with the predictions of accretion disk models (e.g., Witt et al. 1997) where
higher blue-bump to X-ray ratios for sources with higher Eddington ratios are
expected.
While important progress has been made towards a better determination of Type
1 AGN SEDs, our knowledge of Type 2 broad-band spectra is much more limited
despite the fact that most of the accretion-driven energy density in the
Universe is expected to occur in obscured AGN (e.g., Fabian 1999, Gilli et al.
2007 and references therein). Therefore, a robust estimate of their bolometric
luminosity is extremely important to properly address the issue of SMBH
evolution over cosmic time.
Nuclear accretion luminosity in Type 2 AGN is very faint in the optical-UV and
soft X-rays. Moreover, the host galaxy stellar light often dominates in the
optical making it difficult to disentangle nuclear emission from starlight.
Infrared emission is only marginally affected by dust obscuration and has
proved to be a powerful indicator of dust obscured AGN. In particular, the
thermally reprocessed nuclear emission of obscured Type 2 AGN is expected
(e.g., Fritz et al. 2006, hereafter F06, and references within) to peak around
a few tens of micron.
Mid-IR (MIR) observations, and especially those obtained in the last few years
with the Spitzer satellite, are extremely efficient in the study of obscured
AGN (e.g. Rigby et al. 2005; Martínez-Sansigre et al. 2005; Weedman et al.
2006; Fiore et al. 2008). In a previous paper (Pozzi et al. 2007) we presented
the first analysis of the mid-IR data of a Spitzer program devoted to a
systematic study of the broad-band properties of X-ray selected, luminous
obscured quasars. In Pozzi et al. (2007), the SEDs were reproduced by means of
SED templates from Silva et al. (2004).
Here we present the observational data for our final sample of 16 obscured
quasars and the detailed modeling of their broad-band SED using a more
complete multi-component model, with goodness of fit estimated via a
$\chi^{2}$ analysis.
The outline of the paper is as follows: in $\S$2 the X-ray selected quasar
sample is presented, along with all the available multi-band (optical, near-IR
(NIR) and sub-mm) and spectroscopic follow-up. The Spitzer data are presented
in $\S$3, along with data reduction and analysis. In $\S$4, the complete
multi-component model is described. In $\S$5, the best-fitting solutions are
discussed, while in $\S$6 we focus on the black hole physical properties that
can be constrained from the best-fitting procedure. Finally, the main results
are summarized in $\S$ 7.
Hereafter, we adopt the concordance cosmology ($H_{0}=70$ km sec-1 Mpc-1,
$\Omega_{m}$=0.3 and $\Omega_{\Lambda}$=0.7, Spergel et al. 2003). Magnitudes
are expressed in the Vega system.
## 2 The sample
The sample presented in this work comprises 16 X-ray obscured quasars detected
in the HELLAS2XMM survey (Baldi et al. 2002) and observed by Spitzer in 2006.
The HELLAS2XMM survey is a shallow, large-area hard X-ray survey ($S_{2-10\
keV}>10^{-14}$ ${\rm erg\leavevmode\nobreak\ cm}^{-2}\leavevmode\nobreak\ {\rm
s}^{-1}$ ) over a final area of 1.4 deg2. The catalogue comprises 232 X-ray
sources; $\sim$92% of the sample is optically identified down to R$\sim$25,
while $\sim$70% of the sources have a spectroscopic classification (Fiore et
al. 2003, Maiolino et al. 2006, Cocchia et al. 2007).
The 16 sources were selected from the original survey in order to include the
most luminous obscured quasars. The selection was primarily based on the
X-ray-to-optical flux ratio (hereafter X/O)222 X/O is defined as
$\log{\frac{F_{X}}{F_{R}}}$. We used $f_{R}(0)=1.74{\times}10^{-9}$ erg
cm-2s-1 Å-1 and ${\Delta}{\lambda}_{R}=2200$ Å, Zombeck (1990)., which has
been proved to be an efficient way to select high-redshift
(z${{{}_{>}\atop{}^{\sim}}}$1), obscured quasars (see Fiore et al. 2003). All
but one of the sources were selected to have X/O greater than 1 (see Fig. 1
and Table 1); the only exception is GD 158#19 (X/O${\sim}{0.63}$) which was
included in the sample for its peculiar properties (see Vignali et al. 2009,
hereafter V09). We note, however, that not all of the HELLAS2XMM sources
matching this selection criterion are present in this work.
Figure 1: R-band magnitudes vs. hard X-ray (2–10 keV) flux for the full
HELLAS2XMM sample (Cocchia et al. 2007). Blue triangles represent the sources
included in the present analysis: blue triangles inside red symbols represent
the sources with spectroscopic redshifts. Other symbols: empty red squares =
sources spectroscopically classified as Type 2 AGN; empty circles = sources
spectroscopically classified as non-Type 2 AGN (Type 1 AGN, emission-line
galaxies, early-type galaxies and groups/clusters of galaxies); crosses:
objects not observed spectroscopically; upward arrows = lower limits. The
dashed lines represent the loci of constant X/O ratio (X/O=$\pm{1}$).
The selected sources are relatively faint in the optical band, with an R-band
magnitude in the range 21.8-25.1 (the brightest object being the peculiar
source GD 158#19). By combining optical photometry with deep Ks-band
photometry (obtained with the Infrared Spectrometer And Array Camera, ISAAC,
mounted on the ESO-VLT1 Telescope), almost all the objects are found to be
Extremely Red Sources (EROs, R-K${}_{s}{\geq}5$). The link between high X/O
ratios and optical-to-near colours was studied by e.g., Brusa et al. (2005).
Considering different X-ray surveys at different depths, they find a clear
trend: the higher X/O, the redder the source.
In Mignoli et al. (2004), detailed Ks-band morphological studies were
presented for 8 objects of the sample selected from those sources with the
more extreme R-Ks colours: the majority of the sources (6 over 8) have an
extended Ks-band morphology, consistent with an elliptical-type profile,
without evidence for a nuclear point-like source (which would be expected to
trace the X-ray AGN). This suggests that the nuclear emission is diluted and
hidden by the host galaxy up to at least 2.2$\mu$m. In Table 1, the R and Ks-
band magnitudes are reported.
For 11 sources, spectroscopic information is available thanks to optical (8
sources, see Fiore et al. 2003 and Cocchia et al. 2007) and near-IR
spectroscopy (3 sources, see Maiolino et al. 2006 and Sarria et al., in
preparation). All but one spectra are typical of optically obscured AGN, thus
confirming the X-ray classification (i.e., Type 2). The only exception is
source PKS 0537#91, with emission line ratios typical of an HII region (Sarria
et al. in preparation). One source, Abell 2690#29, shows the typical rest-
frame spectrum of a high-redshift dust-reddened quasar, with a broad H$\alpha$
line and a Type 1.9 classification (Maiolino et al. 2006). Spectroscopic
redshift $z$ are in the range 0.9-2.08 and are reported in Table 1.
For the sources without redshift (5 out of 16), a photometric redshift was
estimated in Pozzi et al. (2007), where the Spitzer data reduction and a
preliminary SED analysis were presented.
Finally, sub-mm observations were performed for four sources in October-
November 2004. Only one object, GD 158#19 (z=1.957), was detected, while for
the others upper limits were gained. Because of its broad-band coverage (up to
850$\mu$m) with good-quality photometric data, the source GD 158#19 was
studied in a dedicated work (see V09).
In Table 1 we report source name, 2–10 keV flux, R, and Ks photometry, X/O
ratio, the redshift $z$, the column densities NH and the absorption-corrected
(2–10 keV) X-ray luminosities of the sample. The source table order reflects
the Spitzer observation strategy (see Sec 3.). The sub-mm flux densities are
reported in Table 2 along with the fluxes obtained in the IR bands (from
3.6$\mu$m up to 160$\mu$m) with the Spitzer satellite (see $\S$ 3).
Almost all the sources have column densities NH in the range
$10^{22.0}$–$10^{23.4}$ cm-2 and 2–10 keV rest-frame luminosities in the range
1043.8–1044.7 erg s-1, placing them among the Type 2 quasar population.
Table 1: Properties of our luminous obscured quasars Complete Name | Name | 2–10 keV flux | $R$ | $K_{s}$ | X/O | $z$ | $N_{H}$ | $L_{2-10\ keV}$
---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
HELLAS2XMM 054022.0-283139 | PKS 0537#43 | 3.35 | 22.70 | 17.50 | 1.10 | 1.797 | 10.5${}^{9.4}_{4.8}$ | 6.8
HELLAS2XMM 053920.4-283721 | PKS 0537#11a | 4.19 | 23.40 | 18.25 | 1.48 | 0.981 | 1.3${}^{1.5}_{0.9}$ | 1.9
HELLAS2XMM 053917.1-283819 | PKS 0537#164 | 1.50 | 23.60 | 19.02 | 1.12 | 1.824 | – | 3.1
HELLAS2XMM 053851.3-283949 | PKS 0537#123 | 2.97 | 23.10 | 17.94 | 1.21 | 1.153 | 6.6${}^{21.6}_{4.1}$ | 2.0
HELLAS2XMM 003413.8-115559 | GD 158#62 | 3.59 | 23.30 | 21.83 | 1.38 | 1.568 | 26.3${}^{44.7}_{18.1}$ | 4.9
HELLAS2XMM 003357.2-120039 | GD 158#19 | 2.43 | 21.80 | – | 0.61 | 1.957 | 7.3${}^{11.7}_{5.5}$ | 6.3
HELLAS2XMM 204428.7-105629 | Mrk 509#01 | 2.10 | 23.85 | 17.88 | 1.36 | 1.049 | $<$1.1 | 1.5
HELLAS2XMM 204349.7-103243 | Mrk 509#13 | 3.10 | 23.99 | 18.79 | 1.59 | 1.261 | 2.5${}^{4.6}_{2.2}$ | 2.6
HELLAS2XMM 235956.6-251019 | Abell 2690#75 | 3.30 | 24.60 | 18.33 | 1.85 | 1.3${}^{+0.30}_{-0.20}$ † | 15.0${}^{20.0}_{8.5}$ | 3.2
HELLAS2XMM 031343.5-765426 | PKS 0312#36 | 1.90 | 24.70 | 19.13 | 1.66 | 0.9${}^{+0.05}_{-0.15}$ † | 1.0${}^{1.2}_{0.9}$ | 0.7
HELLAS2XMM 054021.1-285037 | PKS 0537#91 | 4.2 | 23.70 | 18.99 | 1.60 | 1.538 | 45.9${}^{102.0}_{36.0}$ | 8.1
HELLAS2XMM 053945.2-284910 | PKS 0537#54 | 2.1 | 25.10 | 18.91 | 1.86 | $>$1.3 † | – | 2.0
HELLAS2XMM 053911.4-283717 | PKS 0537#111 | 2.1 | 24.50 | 17.64 | 1.62 | 1.2${}^{+0.20}_{-0.10}$ † | 9.1${}^{12.4}_{5.3}$ | 1.7
HELLAS2XMM 000111.6-251202 | Abell 2690#29 | 2.8 | 25.10 | 17.67 | 1.99 | 2.08 | 2.1${}^{2.6}_{1.6}$ | 8.4
HELLAS2XMM 031018.9-765957 | PKS 0312#45 | 2.8 | 24.40 | 18.62 | 1.70 | 1.85${}^{+0.20}_{-0.30}$ † | 8.0${}^{8.4}_{4.6}$ | 6.2
HELLAS2XMM 005030.7-520046 | BPM 16274#69 | 2.27 | 24.08 | 17.87 | 1.48 | 1.35 | 2.5${}^{1.5}_{1.0}$ | 2.4
(1) Source complete name;
(2) source abbreviated name (adopted throughout the paper);
(3) 2–10 keV X-ray fluxes in units of $10^{-14}\ $erg cm-2 s-1 from Perola et
al. (2004) (with the exception of BPM 16274#69, from Lanzuisi et al., in
preparation);
(4) R-band magnitude from Fiore et al. (2003) (with the exception of source
BPM 16274#69, from Cocchia et al. 2007);
(5) Ks-band magnitudes. For a sub-sample of sources, the Ks-band analysis can
be found in Mignoli et al. (2004);
(6) X-ray-to-optical flux ratio;
(7) source redshifts. Spectroscopic redshift from optical spectroscopy (Fiore
et al. 2003) and, for three sources, from near-IR spectroscopy (Abell 2690#29
and BPM 16274#69 from Maiolino et al. 2006; PKS 0537#91 from Sarria et al., in
preparation); photometric redshifts (†symbol), and corresponding 1$\sigma$
errors, from Pozzi et al. (2007);
(8) column densities in source rest-frame in units of 1022 cm-2 measured from
X-ray spectral fitting (see Perola et al. 2004 and Lanzuisi et al., in
preparation). For the sources with photometric redshifts, values are taken
from Pozzi et al. (2007). For PKS 0537#164 the X-ray spectral fitting is
prevented by the few X-ray counts; for Mrk 509#01, only an upper limit was
derived (Perola et al. 2004); for PKS 0537#54, the column density is
9.3${\times}10^{22}$ cm-2 for $z=$ 1.3. Galactic absorption column densities
adopted in the spectral fitting are: $8{\times}10^{20}$ cm-2 for the field PKS
0312$-$77; $2{\times}10^{20}$ cm-2 for the field Abell 2690;
$2.1{\times}10^{20}$ cm-2 for the field PKS 0537$-$28; $4{\times}10^{20}$ cm-2
for the field Mrk 509; $2.5{\times}10^{20}$ cm-2 for the field GD 158$-$100
(see Stark et al. 1992);
(9) rest-frame 2–10 keV absorption-corrected X-ray luminosity in units of 1044
erg s-1 from Perola et al. (2004) and from Pozzi et al. (2007) for sources
with photometric redshifts. Luminosities are computed using H0=70 Km s-1
Mpc-1, $\Omega_{m}=0.3$ and $\Omega_{\Lambda}=0.7$.
## 3 The Spitzer data
The targets were observed by Spitzer in 2006 with both IRAC and MIPS
instruments in photometry mode. All the sources were observed with the same
total integration time in the IRAC bands (480s), while different strategies
were followed in the MIPS bands taking into account the different optical-NIR
properties. While all the objects were observed at 24$\mu$m for a total
integration time of 1400s, only a sub-sample of sources with relatively bright
R-band magnitudes (R$<$24) and spectroscopic redshifts were observed at longer
wavelengths, with integration times of 300s and 600s at 70 and 160$\mu$m,
respectively. PKS 0537#91 (see Table 1) was not observed at 70 and 160
${\mu}$m since it had no redshift at the epoch of the Spitzer observations.
The reduction method is described in detail in Pozzi et al. (2007) and in V09
and it is briefly summarized here. The IRAC flux densities of the sources were
measured from the post-basic calibrated data (post-BCD) images in the Spitzer
archive. Aperture fluxes were measured on the background subtracted maps
within a 2.45″ aperture radius using aperture corrections of 1.21, 1.23,1.38
and 1.58 for the four IRAC bands (following the IRAC Data Handbook). For the
MIPS bands, we started the analysis from the basic calibrated data (BCD) at
24$\mu$m and from the median high-pass filtered BCD (fBCD) at 70 and
160$\mu$m, as suggested for faint sources. At 24$\mu$m, the BCD were corrected
for a residual flat fielding dependent on the scan mirror position (see Fadda
et al. 2006; Pozzi et al. 2007). We constructed then our own mosaics using the
SSC MOPEX software (Makovoz & Marleau 2005). Aperture fluxes were measured
within a 7″ aperture radius for the 24$\mu$m band and 16″ aperture radius for
the 70 and 160$\mu$m band. The aperture corrections used were 1.61, 2.07 and
4.1, respectively (see the MIPS Data Handbook). A small aperture radius was
used at longer wavelengths (at 160$\mu$m the adopted radius is comparable to
half of the PSF FWHM) to exclude the contamination of nearby far-infrared
sources (see V09). At 24$\mu$m, thanks to a better PSF sampling, two sources
(PKS 0312#36 and Abell 2690#29) were deblended using a PSF deconvolution
analysis.
All 16 sources were clearly detected in the IRAC bands. At 24$\mu$m, 14
sources (out of 16) were detected above the 5$\sigma$ level and span almost
two orders of magnitude in flux, from $\sim$7000 $\mu$Jy down to the faintest
source, close to the 5$\sigma$ detection level ($\sim$100 $\mu$Jy). For the
two sources without detection, an upper limit (3$\sigma$) was estimated from
the average noise of the map, derived by making multiple aperture measurements
at random locations throughout the residual mosaic after source extraction.
The typical average noise (1$\sigma$) is 20 $\mu$Jy. At 70 and 160$\mu$m, as
said before, only the brightest R-band sources were observed; among them, only
the two most luminous (in the optical band) were detected, PKS 0537#43 and GD
158#19, the latter being the source described in V09. For the remaining 6
sources, an upper limit (3$\sigma$) was estimated from the residual mosaic
(see also Frayer et al. 2006) after source extraction, as done at 24$\mu$m.
The typical average noise (1$\sigma$) is 1.2 mJy and 8 mJy at 70 and
160$\mu$m, respectively (consistent with the results obtained in the COSMOS
field from Frayer et al. 2009 taking into account the different integration
time).
Table 2 reports the target flux densities provided by Spitzer. To compute
uncertainties, the noise map was added in quadrature to the systematic
uncertainties, assumed to be 10 per cent in the IRAC and MIPS 24$\mu$m bands
and 15 per cent at 70 and 160$\mu$m (see IRAC and MIPS Data Handbook).
Figure 2: $\chi^{2}$ distribution. The red hatched and the empty distributions
represent the best-fitting solutions and all the solutions at 1$\sigma$,
respectively. PKS 0537#43 is not reported given the high $\chi^{2}$ value
(Table 3).
Table 2: Spitzer and SCUBA flux densities Source name | 3.6$\mu$m | 4.5$\mu$m | 5.8$\mu$m | 8.0$\mu$m | 24$\mu$m | 70$\mu$m | 160$\mu$m | 450$\mu$m | 850$\mu$m
---|---|---|---|---|---|---|---|---|---
| [$\mu$Jy] | [$\mu$Jy] | [$\mu$Jy] | [$\mu$Jy] | [$\mu$Jy] | [mJy] | [mJy] | [mJy] | [mJy]
PKS 0537#43 | 423 $\pm$ 42 | 918 $\pm$ 92 | 1750$\pm$175 | 3202 $\pm$ 320 | 6879 $\pm$688 | 27.0$\pm$4.5 | 39.6$\pm$ 7.8 | |
PKS 0537#11a | 57 $\pm$ 6 | 69 $\pm$ 7 | 95$\pm$ 10 | 168 $\pm$ 17 | 517 $\pm$57 | $<$3.6 | $<$24 | |
PKS 0537#164 | 21 $\pm$ 2 | 18 $\pm$ 2 | 19$\pm$ 4 | 23 $\pm$ 5 | $<$60 | $<$3.6 | $<$24 | |
PKS 0537#123 | 84 $\pm$ 9 | 96 $\pm$ 10 | 113$\pm$ 13 | 165 $\pm$ 17 | 747 $\pm$81 | $<$3.6 | $<$24 | |
GD 158#62 | 60 $\pm$ 6 | 91 $\pm$ 9 | 155$\pm$ 17 | 339 $\pm$ 35 | 1608 $\pm$164 | $<$3.6 | $<$24 | $<$74 | $<$6.1
GD 158#19 | 226 $\pm$ 23 | 387 $\pm$ 39 | 756$\pm$ 76 | 1547 $\pm$ 155 | 5326 $\pm$534 | 27.0$\pm$4.5 | 37.7$\pm$ 7.8 | $<$94.3 | 8.6$\pm$2.1
Mrk 509#01 | 63 $\pm$ 7 | 51 $\pm$ 6 | 60$\pm$ 11 | 59 $\pm$ 10 | $<$60 | $<$3.6 | $<$24 | $<$82 | $<$6.6
Mrk 509#13 | 51 $\pm$ 6 | 69 $\pm$ 7 | 110$\pm$ 11 | 215 $\pm$ 17 | 866 $\pm$ 94 | $<$3.6 | $<$24 | $<$75 | $<$6.2
Abell 2690#75 | 51 $\pm$ 5 | 56 $\pm$ 6 | 89$\pm$ 11 | 139 $\pm$ 15 | 565 $\pm$ 62 | | | |
PKS 0312#36 | 41 $\pm$ 4 | 44 $\pm$ 5 | 40$\pm$ 8 | 710 $\pm$ 9 | 236 $\pm$ 30 | | | |
PKS 0537#91 | 28 $\pm$ 4 | 35 $\pm$ 4 | 42$\pm$ 8 | 800 $\pm$ 10 | 301 $\pm$ 40 | | | |
PKS 0537#54 | 31 $\pm$ 4 | 35 $\pm$ 4 | 50$\pm$ 10 | 470 $\pm$ 8 | 279 $\pm$ 45 | | | |
PKS 0537#111 | 88 $\pm$ 9 | 75 $\pm$ 8 | 41$\pm$ 6 | 460 $\pm$ 7 | 148 $\pm$ 28 | | | |
Abell 2690#29 | 141 $\pm$ 14 | 185 $\pm$ 19 | 260$\pm$ 27 | 371 $\pm$ 38 | 1012 $\pm$ 106 | | | |
PKS 0312#45 | 50 $\pm$ 6 | 62 $\pm$ 7 | 69$\pm$ 10 | 780 $\pm$ 10 | 249 $\pm$ 35 | | | |
BPM 16274#69 | 86 $\pm$ 9 | 92 $\pm$ 9 | 97$\pm$ 11 | 120 $\pm$ 13 | 286 $\pm$ 34 | | | |
The upper limits are given at the 3$\sigma$ confidence level.
## 4 Modeling the spectral energy distribution
The observed optical-to-MIR (or FIR/sub-mm) SEDs can be modelled as the sum of
three distinct components: a stellar component, which emits most of its power
in the optical/NIR, an AGN component, whose emission peaks in the MIR for
obscured quasars, and is due to hot dust heated by UV/optical radiation from
gas accreting onto the central SMBH, and a starburst component, which
represents the major contribution to the FIR spectrum.
In this work, we considered all the three components (see Sec. 4.1, 4.2 and
Hatziminaoglou et al. 2008). Since the focus of the paper is on the AGN
contribution to the SED, we discuss in some more detail the hot dust modeling
and its uncertainties.
The hot dust emission in AGN is reproduced using the F06 model. This model
follows the formalism developed by different authors (e.g., Pier & Krolik
1992; Granato & Danese 1994; Efstathiou & Rowan-Robinson 1995), where the IR
emission in AGN originates from dusty gas around the SMBH with a smooth
distribution. The dust grains are heated by high-energy photons coming from
the accretion disk, and their thermal and scattering re-emission, mostly at IR
wavelengths, is computed by means of the radiative transfer equations. For
what concerns the dust distribution geometry, different possibilities (i.e.
“classical” torus shape, tapered or flared disk), are explored in the
literature.
More recently, models considering a clumpy distribution for the dust have been
developed (e.g., Nenkova et al. 2002; Hönig et al. 2006; Nenkova et al. 2008).
These models successfully explain many recent observations in the mid-IR, such
as the strength of absorption and emission features at 9.7 ${\mu}$m and the
X-ray variability (Risaliti et al. 2002).
A further possibility to the torus models described above are the disk-wind
models (see Elitzur & Shlosman 2006 and references therein), involving a
completely different approach. The dusty clouds, responsible for the
obscuration, are part of an hydro-magnetic wind coming from the accretion
disk. On the one hand, these models are potentially capable of explaining
disparate phenomena in AGN (from broad emission to absorption lines and
obscuration), providing an hydro-dynamical justification for the persistence
of the clouds around the SMBH. On the other hand, a parameterization which
takes into account the observational constraints on the clumpy obscuration,
hence supplying a grid of synthetic IR SEDs, does not exist yet (see Elitzur
2008).
A detailed comparison between smooth and clumpy dust distribution models is
discussed by Dullemond & van Bemmel (2005). They conclude that both models
yield similar SEDs (see also Elitzur 2008 and Nenkova et al. 2008). The main
difference is in the strength of the silicate feature observed in absorption
in objects seen edge-on, which is, on average, weaker for clumpy models with
the same global torus parameters. In clumpy models, in fact, clouds at
different distances from the central source can be intercepted by the line of
sight, including the innermost clouds, where the silicate feature is in
emission given the higher temperature of the dust grains.
A systematic comparison of the two model predictions is beyond the scope of
the present paper and should be performed on high-quality IR data (i.e. a
Spitzer IRS spectroscopic sample).
Notwithstanding these limitations, with the present work we aim at extracting
the maximum information using the available photometric data. The F06 model
adopted in this work is one of the models best tested against both broad-band
photometry (F06; Berta et al. 2004; Rodighiero et al. 2007; Hatziminaoglou et
al. 2008, 2009; Agol et al. 2009; V09) and Spitzer mid-infrared spectra (F06).
Moreover, the F06 model was the first able to reproduce the quasar mid-IR
spectra, considered a very important constraint to characterize the dust
properties in AGN and probe the Unified Model.
Figure 3: a. Observed-frame SEDs for 14 sources with data from the R-band to
the 24$\mu$m (black dots) compared with the best-fit model obtained as the sum
(solid black line) of a stellar component (red dotted line) and an AGN
component (blue dashed line). The nuclear Ks-band upper limits (downward-
pointing arrows) were derived from the morphological analysis carried out by
Mignoli et al. (2004). Figure 3: b. As in Fig. 3a for sources with detections
also at longer wavelengths (FIR/sub-mm). An additional starburst component
(green dot-dashed line) is considered in the best-fit model.
### 4.1 The AGN-torus component
The F06 code assumes a smooth dust distribution around the central source
consisting of a Galactic mixture of silicate and graphite grains. The presence
of silicate dust grains is clear from the absorption feature at 9.7 ${\mu}$m
seen in most of Type 2 AGN. The graphite grains are, instead, responsible for
the rapid decline of the emission at wavelengths shortwards of a few microns,
corresponding to a blackbody emission of about 1500 K, the sublimation
temperature of these grains (see F06).
The assumed dust geometry is a flared disk (see Efstathiou & Rowan-Robinson
1995), that is a sphere with the polar cones removed. The internal radius of
the dust distribution is defined by the sublimation temperature of the dust
itself. In order to simulate a more realistic shape for the dust distribution,
F06 assumes that the dust density can vary both with the radial and the
angular coordinates:
$\rho(r,\theta)=\rho_{0}\cdot r^{\alpha}\cdot e^{-\gamma\cdot|\cos(\theta)|}$
(1)
where $\theta$ is the angle with respect to the equatorial plane.
The dusty torus is heated by the emission of the inner accretion disk, which
represents the input energy to the radiative transfer code. The assumed
spectrum for the accretion disk is defined in the 10-3 to 20${\mu}$m regime
(from soft X-rays, i.e. 1.25 keV, to mid-IR) and is parameterized by broken
power laws in F06. The specific indices of the power laws are adapted from the
Granato & Danese (1994) and the Nenkova (2002) models and are consistent with
the broad-band SEDs of a sample of Type 1 AGN from the SDSS survey
(Hatziminaoglou et al. 2008; see their Figs. 8 and 9).
Along with the thermally re-processed light, the F06 provides, as a function
of the line-of-sight and optical depth, the fraction of the inner accretion
disk light not intercepted by the torus and the scattered light. In the
following, with AGN component we will refer to the sum of all the three
contributions.
### 4.2 The stellar and starburst components
The stellar component is modelled as the sum of Simple Stellar Populations
(SSP) models of different age, all assumed to have common (solar) metallicity.
A Salpeter (1955) initial mass function (IMF) with mass in the range (0.15-120
M⊙) is assumed. The SSP spectra have been weighted by a Schmidt-like law of
star formation (see Berta et al. 2004):
$SFR(t)=\frac{T_{G}-t}{T_{G}}{\times}\exp\left({-\frac{T_{G}-1}{T_{G}{\tau}_{sf}}}\right)$
(2)
where $T_{G}$ is the age of the galaxy (i.e. of the oldest SSP) and
$\tau_{sf}$ is the duration of the burst in units of the oldest SSP (see V09).
As in Hatziminaoglou et al. (2008) and V09, a common value of extinction is
applied to stars at all ages, adopting the extinction law of our Galaxy
($R_{V}=3.1$; Cardelli et al. 1989).
In order to keep the number of free parameters as low as possible, emission
from cold dust, which dominates the bolometric emission at wavelengths longer
than 30${\mu}$m rest-frame, is included only when far-IR/sub-mm data allow us
to constrain that part of the SED (two sources of the sample). For the same
reason, additional components, such as the cold absorber detached from the
torus (i.e. Polletta et al. 2008), which might improve the fit but would
increase the complexity of the overall modeling, is not included. To reproduce
the starburst component, a set of semi-empirical models of well known and
studied starburst galaxies is used, similarly to V09.
### 4.3 SED fitting procedure
The quality of the fitting solutions is measured using a standard $\chi^{2}$
minimization technique (as also done in Hatziminaoglou et al. 2008), where the
observed values are the photometric flux densities (from optical-to-MIR/FIR)
and the model values are the “synthetic” flux densities obtained by convolving
the sum of stars, AGN and starburst components through the filter response
curves.
Before starting the general fitting procedure, we test which torus parameters
mainly influence the global model SED and are more sensitive to our data sets.
Parameters which are not constrained by our data were then frozen.
The F06 torus model is described by six parameters: the ratio
$R_{max}/R_{min}$ between the outer and the inner radius of the torus (the
inner radius being defined by the sublimation temperature of the dust grains);
the torus full opening angle $\Theta$; the optical depth $\tau$ at 9.7$\mu$m
($\tau_{9.7})$; the line of sight $\theta$ with respect to the equatorial
plane, and two parameters, $\gamma$ and $\alpha$, describing the law for the
spatial distribution of the dust and gas density $\rho$ inside the torus (Eq.
1).
In our approach, we let free to vary, inside the pre-constructed grid of torus
models the following parameters: the torus full opening angle $\Theta$, the
optical depth $\tau_{9.7}$ and the parameter $\alpha$ describing the radial
dependence of the density. We fix $R_{max}/R_{min}$=30, which translates into
compact tori of a few tens of parsecs (given that $R_{min}$ is directly
connected to the sublimation temperature and to the accretion luminosity of
the central BH). Recent high-resolution IR observations support a compact dust
distribution in nearby luminous AGN. Using the interferometry at VLTI in the
8-13 ${\mu}$m band, a torus of size ${\sim}$2-3 pc was detected in NGC 1068
(Jaffe et al. 2004). Similar compact tori were also found in other local AGN,
as Circinus and NGC 4151 (see the review by Elitzur 2008).
Regarding the density distribution, we allow power-law profiles decreasing
with the radius with different coefficients $\alpha$. We do not allow any
dependence on the distance from the equatorial plane by fixing $\gamma$=0. As
a result, different angles for the lines of sight $\theta$ (with respect to
the equatorial plane) give the same SED, once the torus is intercepted.
Given the F06 grid of models, the discrete values, allowed for our free
parameters, are: $\Theta=[60,100,140^{\circ}]$,
$\tau_{9.7}=[0.1,0.3,0.6,1,2,3,6,10]$ and $\alpha=[-1,-0.5,0]$, implying 72
different torus SEDs.
Considering the stellar component, we fix at $z=4$ the redshift for the
formation of the oldest SSP, i.e., given the observed redshift of the sources,
we consider galaxies with ages typical of early-type galaxies
(${{{}_{>}\atop{}^{\sim}}}$1-2 Gyr). This assumption is justified by the
observed R-Ks colours and the brightness profiles typical of early-type
galaxies as obtained by a detailed morphological analysis in the Ks-band
(Mignoli et al. 2004). Concerning the star-formation history, we allow the
$\tau_{sf}$ parameter of the Schmidt-like law and the value of the extinction
$E(B-V)$ to vary; the latter is a key parameter, along with the optical depth
$\tau_{9.7}$ of the torus, in shaping the optical-NIR continuum.
Overall, the SED-fitting procedure ends with 5 free parameters (6 when also a
starburst component is included). Since the problem is affected by some
degeneracy, we consider, along with the best-fitting solutions, all the
acceptable solutions within 1$\sigma$ confidence level (by considering, for
each source, all the solutions with
$\chi^{2}-\chi^{2}_{min}=\Delta\chi^{2}{{{}_{<}\atop{}^{\sim}}}$5.89 or 7.04,
when the cold starburst component is added, see Lampton et al. 1976).
## 5 Results from SED fitting
In Fig. 2 we show the $\chi^{2}$ distribution. The hatched histogram
represents the distribution of the best-fitting solutions for our 16 targets,
while the empty histogram shows the $\chi^{2}$ distribution of all the
solutions within 1$\sigma$, satisfying the criteria
$\chi^{2}{{{}_{<}\atop{}^{\sim}}}\chi^{2}_{min}$+$\Delta\chi$. Given our
adopted grid for the fitted parameters, the total number of solutions at
1$\sigma$ (constructed by adding all the solutions at 1$\sigma$ of each
object) is 137, 8 solutions on average per source (including the best-fitting
one).
The two distributions do not show a significant difference, and this
reinforces our choice of considering, in the analysis of the parameter space
and degeneracy, all the solutions at 1$\sigma$ as a unique statistical sample.
In terms of the absolute values of the $\chi^{2}$, only 7 sources (out of 16)
give a formally acceptable fit ($P(\chi^{2}>\chi^{2}_{obs})>90\%$, see Table
3); the remaining sources have a best-fit model with large $\chi^{2}_{obs}$.
While we use the $\chi^{2}$ to assign a relative goodness of different
parameter combinations inside the parameter grid, we will not take the
absolute probabilities at face value. Over-estimated $\chi^{2}$ are, in fact,
a common problem of most SED fitting techniques, resulting from a combination
of two different reasons: the limited grid of models (72 torus models with the
adopted choice of parameters, see $\S$4.3) with no uncertainties associated,
and the photometric measurements with often underestimated uncertainties (see
Gruppioni et al. 2008 for a detailed description of this issue).
In Fig. 3a,b the observed SEDs, from the R-band to the IR (or sub-mm), are
reported with the best-fitting models over-plotted (solid line). All the
sources need a host galaxy component (red dotted line) and an AGN one (blue
long-dashed line). The stellar component dominates in the R and Ks bands,
while the nuclear one at 24$\mu$m. In the IRAC bands, both components
contribute, the fraction depending on the properties of the individual
sources. For PKS 0537$\\_$43 and G158$\\_$19, where data points at longer
wavelengths are available, an additional starburst component is needed (green
dot-dashed line in Fig. 3b).
In Fig. 4, the relative contributions of the thermal, direct and scattered
light to the total AGN light are shown for two sources characterized by a low
($\tau_{9.7}$=0.1, PKS 0537#123) and a high ($\tau_{9.7}$=10, PKS 0537#111)
optical depth, respectively. While for $\tau_{9.7}$=10, the AGN emission is
dominated by the reprocessed emission in all the UV/optical/IR bands, for
$\tau_{9.7}=$0.1 the direct and scattered components account for the
optical/UV AGN emission. Nevertheless, the contribution of the components
mentioned above never exceeds the 20% of the observed flux in the R-band. For
a sample of highly polarized red active galactic nuclei selected from the
2MASS survey, a larger contribution of the scattered nuclear component to the
optical and near-IR emission was found (Cutri et al. 2002, Kuraszkiewicz et
al. 2009). The different result obtained from our analysis is probably due to
the different degree of obscuration of the two samples: the 2MASS sample is
characterized by column densities around 1022 cm-2, while the present sample
has a median column density of $7{\times}10^{22}$ cm-2 .
The negligible contribution of the AGN component, relative to the stellar one,
at short wavelengths, is consistent with the upper limits to the AGN emission
derived by Mignoli et al. (2004) from the analysis of the Ks-band images
(shown as downward-pointing arrows in Figs. 3 and 4).
In Table 3, the $\chi^{2}$ values (and the corresponding degree of freedom) of
the best-fitting solutions are reported for each source.
### 5.1 Torus parameters
Figure 4: Observed-frame spectral energy distribution as in Figs. 3a,b for 2
sources characterized by extreme values of the optical depth:
$\tau_{9.7{\mu}m}$=0.1 and 10 for PKS 0537$\\_$123 and PKS 0537$\\_$111
respectively. In this Figure, the 3 different components that contribute to
the AGN emission are reported: direct nuclear light (long-dashed line),
scattered light (dashed-dot line) and thermally re-emitted light (dashed-dot-
dot-dot line).
In the following section, we discuss how the model torus parameters are
constrained by our data set. As anticipated, we will consider all the 137
solutions at the 1$\sigma$ level.
First of all, the torus is seen almost edge-on in all the solutions (i.e., the
line of sight always intercepts the obscuring material), in agreement with the
Type 2 X-ray and optical classification of our sources.
The torus model parameters which are left free to vary within the grid of
models are: the torus opening angle $\Theta$, the slope $\alpha$ of the dust
density profile and the optical depth $\tau_{9.7}$.
By converting the torus opening angle into a covering factor (CF) representing
the fraction of solid angle covered by the dusty material, we find that
solutions with high and low CF are possible, with a slight preference towards
tori with large CF; the mean CF value is 0.65 (1$\sigma{\sim}0.25$),
corresponding to a torus opening angle of ${\sim}110^{\circ}$.
According to Maiolino et al. (2007), the covering factor of the circum-nuclear
dust decreases for increasing optical luminosity at 5100 Å ($L_{5100}$). This
relation is explained in terms of a “receding-torus”. In Maiolino et al.
(2007) the luminosities at $5100$ Å were derived from optical spectroscopy and
the CF values from the ratios between the 6.7${\mu}$m and the 5100 Å
luminosities for a sample of Type 1 quasars spanning five orders of magnitude
in optical luminosity. In our approach, $L_{5100}$ is estimated, for each
solution, from the input accretion-disk spectrum (see Sec. 4.1), once the
normalization is found (see Sec. 6.1). The average value of CF and $L_{5100}$
for our sample lies, within 1$\sigma$, on the relation found by Maiolino et
al. (2007). Unfortunately, given the limited range of $L_{5100}$, we cannot
investigate the validity of the CF vs. $L_{5100}$ relation over the range
probed by Maiolino et al. (2007).
Regarding the density profile, about $65\%$ of the solutions have $\alpha=0$,
while $\alpha=-0.5$ and $\alpha=-1.0$ represent $20\%$ and $15\%$ of the
solutions, respectively. This is reflected also in the 16 best-fitting
solutions, where only 2 sources (Mrk 0509#1 and PKS 0537#111) are fitted with
$\alpha=-0.5$, one with $\alpha=-1.0$ (PKS 0537#43), and the remaining ones
with $\alpha=0$ (see Table 3).
Solutions with a moderate optical depth $\tau_{9.7}$ are favoured by the SED-
fitting analysis. As shown in Fig. 5 ($bottom$ panel), there is a small number
of solutions with high optical depths while the majority of the solutions
(${\sim}80\%$) are characterized by ‘moderate’ $\tau_{9.7}$
($\tau_{9.7}{\leq}3$) and 50$\%$ by low $\tau_{9.7}$ ($\tau_{9.7}{\leq}1$).
The median value for $\tau_{9.7}$ is 2.
The finding of a preferred range of optical depths by the SED-fitting, even
with relatively sparse photometric data, comes from the overall shape of the
NIR/MIR continuum. In fact, once the stellar component is determined by the
optical/NIR data, the slope of the torus component is directly linked to the
amount of absorption (i.e., to the optical depth) and is relatively well
constrained by the available data.
As shown in Fig. 3a,b, for the very low values of the optical depth
$\tau_{9.7}$ ($\tau_{9.7}$=0.1), the F06 model predicts spectra with a weak
emission line at 9.7 ${\mu}$m.
We clearly find that the optical depth $\tau_{9.7}$ and the density profile
$\alpha$ are not independent parameters, since low optical depth solutions
mostly occur with flat density profile ($\alpha$=0). This is shown in Fig. 5
($top$ panel), where the fraction of solutions with $\alpha$=0 is reported as
a function of $\tau_{9.7}$. For the assumed flared disk geometry, at high
optical depth a flat density profile produce too much IR emission due to the
large amount of dust at high radii. Thus the two best-fitting solutions with
the highest $\tau_{9.7}$ found ($\tau_{9.7}$=6 and 10 for Mrk 509#01 and PKS
0537#111 respectively, see Fig. 5 and Table 3) have a density profile
decreasing with the distance from the central BH ($\alpha$=-0.5).
Recalling that in our procedure different lines of sight are equivalent
(having assumed $\gamma=0$ in the radial density profile, Eq. 1), we can
convert the optical depths to column densities NH (adopting a Galactic dust-
to-gas ratio) to be compared with the NH derived from the X-ray observations.
Despite the uncertainties affecting the derivation of NH from the dust optical
depths (i.e., dust and gas spatial distributions could be different), as well
as those affecting the NH values from X-rays (see Perola et al. 2004 for
details and Table 1), the two independent estimates give a consistent picture
for the majority of the sources, once the 1$\sigma$ uncertainties, derived
from the SED and X-ray fitting procedure, are taken into account. By excluding
the two sources without a measured NH from the X-rays analysis (see Table 1),
the median values for NH are ${\sim}7{\times}10^{22}$ cm-2 and
${\sim}$5.5${\times}10^{22}$ cm-2, from the X-rays analysis and the dust
optical depths, respectively. Therefore, the SED-fitting method confirms the
X-ray classification of the sources as moderately obscured Compton-thin AGN.
Two sources have a significantly different NH (by an order of magnitude)
derived from the two methods, PKS 0537#111 and Mrk 509#01. These objects are
those characterized by the highest optical depths ($\tau_{9.7}=6,10$, which
are converted into N${}_{H}{\sim}5.3{\times}$1023 cm-2 and
N${}_{H}{\sim}8.7{\times}10^{23}$ cm-2, respectively). Since Mrk 509#01 has
only an upper limit for the NH inferred from X-ray analysis
($<1.1{\times}10^{22}$ cm-2, see Table 1), the observed discrepancy for this
object might be explained if the source is Compton-thick
(N${}_{H}{\geq}10^{24}$ cm-2) and the observed X-ray spectrum is due to a
reflection component. However, we cannot draw any firm conclusion on this
issue.
Figure 5: $Top$ panel: the fraction of solutions with flat density profile
($\alpha=0$) as a function of $\tau_{9.7}$. $Bottom$ panel: the $\tau_{9.7}$
distribution. The hatched and the empty distributions represent the best-
fitting (16) and all the solutions (137) at 1$\sigma$, respectively. Figure 6:
‘Model’ as a function of ‘observed’ bolometric luminosities for the sample of
luminous obscured quasars at the 1$\sigma$ significance level (137 solutions).
Squares around the filled circles represent the solutions with high optical
depth ($\tau_{9.7{\mu}m}\geq{3}$). The ‘observed’ bolometric luminosities are
reported with no corrections applied. The solid line represents the identity
relation. The dotted and dashed lines represent the predictions from the Pier
& Krolik (1992) model for 4 different configurations for Type 2 sources as a
function of the optical depths $\tau_{z}$ and $\tau_{r}$ (where $\tau_{z}$ and
$\tau_{r}$ are the optical depths at 9.7$\mu$m along the vertical- with
respect the torus equatorial plane - and the radial direction, respectively)
and viewing angle $\theta$ (with respect to the equatorial plane). Red dotted
lines: thinner model ($\tau_{z}=\tau_{r}=0.1$, labeled with the letter t) with
$\theta=0,50^{\circ}$. Blue dashed lines: thicker model ($\tau_{z}$=1,
$\tau_{r}$=10, labeled with the letter T) with $\theta=0,50^{\circ}$. At
constant optical depth, the configurations with smaller viewing angle (closer
to the equatorial torus plane) predict lower ‘observed’ luminosities.
### 5.2 Host galaxy parameters
In the spectral procedure, the host galaxy accounts for the optical/near-IR
photometric data points, where the AGN contribution, given the obscured nature
of our sources, is presumably low. We use the SSP spectra weighted by a
Schmidt-like law of star formation (see Sec. 4.2). The extinction E(B-V) and
$\tau_{\rm sf}$ are free parameters. Once the best-fitting values for these
two parameters are found, the stellar mass (obtained by integrating the star
formation history over the galaxy age and subtracting the fraction due to mass
loss during stellar evolution, $\sim 30$%, from it) is estimated from the SED
normalization. At the end of the SED fitting procedure, stellar masses are
well constrained, for a given pair of E(B-V) and $\tau_{\rm sf}$, with a
typical 1$\sigma$ uncertainty for the normalization of $\sim$ 20%.
We note that all but one of the stellar masses derived in this work are within
30% from the values estimated by Pozzi et al. (2007; eight sources in common),
where the same data were used but different stellar libraries and a simpler
approach was adopted (see Sect. 4.1 and 5.2 of Pozzi et al. 2007).
The stellar masses found are in the range 4${\times}10^{10}$ up to
5${\times}10^{12}$ M⊙ with three very massive galaxies ($>10^{12}$ M⊙, see
Table 3), implying that our obscured AGN are hosted by massive galaxies at
high redshift. As said in Sec. 4.2, the masses are obtained using a Salpeter
(1955) initial mass function (IMF) with mass in the range (0.15-120 M⊙). The
assumption of a Chabrier (2003) IMF (see Renzini 2006) would produce a factor
of $\sim{1.7}$ lower stellar masses.
In Table 3, the best-fitting value for the free host galaxy parameters
($\tau_{\rm sf}$ and E(B-V)) and the stellar masses are reported for each
source.
## 6 Black hole physical properties
### 6.1 Black hole accretion luminosities
The accretion-disk luminosity $L_{acc}$ is a direct output of the fitting
procedure and is obtained by integrating the code input energy spectrum once
the best-fitting torus components and its normalization are found (see Sect.
4.3).
The input spectrum is defined in the 10-3 to 20 ${\mu}$m regime. Although this
wavelength range provides the largest contribution to the nuclear AGN
luminosity, we have also included in the $L_{acc}$ computation the hard X-ray
luminosity ($L_{1.25-500keV}$). This luminosity is estimated from the de-
absorbed, k-corrected $L_{2-10keV}$ luminosity, assumping a photon index
$\Gamma$=1.9 (typical of AGN emission) and an exponential cut-off at 200 keV
(e.g., Gilli et al. 2007). A different choice for the energy cut-off (e.g., at
100 keV) would produce a difference by $\approx$ 20% in the total X-ray
luminosity for $\Gamma$=1.9 sources (see Vasudevan et al. 2009).
We note that dust grains are almost transparent to hard X-ray photons,
therefore the output of the code is not affected by the fact that the
accretion-disk model spectrum does not extend above soft-X-ray energies.
In Table 3 $L_{bol}$, along with $L_{acc}$ and $L_{1.25-500keV}$, are
reported. $L_{bol}$ extends over two orders of magnitudes (1044-1046 erg s-1),
with the hard-X-ray luminosities (1.25-500 keV) contributing in the range
5-50$\%$ of the AGN power (see Table 3). The two sources with the highest
optical depths ($\tau_{9.7{\mu}m}$=6,10) are among the sources with the
smallest hard-X-ray fraction (Mrk 509#01 and PKS 0537#111). In Table 3 we
report also the range of bolometric luminosities as obtained by considering
the full set of $1\sigma$ solutions. The uncertainties are, on average, of the
order of 0.2 dex, but vary significantly from source to source, ranging from
about 5% to about a factor 3 (see also Fig. 7).
We compare the computed bolometric luminosities with the luminosities derived
by integrating the torus best-fitting templates from 0.1-1000$\mu$m (plus
adding the hard X-ray luminosity for self-consistency). The two methods assume
the same torus SED, hence the comparison can give important information on the
systematics affecting the estimates of $L_{bol}$ derived by integrating the
observed SED, which is the method widely used in literature. We refer to the
first measures as the ‘model’ luminosities and to the latter as the ‘observed’
luminosities.
The ‘observed’ $L_{bol}$ (see Fig. 6) are lower (up to an order of magnitude)
than the ‘model’ ones for all the solutions; the median value of the ratio is
$R{\sim}2$ (the solid line in Fig. 6 represents the identity relation). An
under-estimate of the luminosity in Type 2 sources is expected by torus models
(e.g., Pier & Krolik 1993; Granato & Danese 1994); here, we quantify this
effect and provide an empirical factor to correct the ‘observed’ luminosities,
at least for this class of sources.
We underline how the observed discrepancy does not depend on the lack of
observations at far-IR wavelengths. In fact, the two methods assume the same
torus SED for self-consistency (i.e., the integrated torus SED to estimate the
observed IR luminosity is the output of the code); under this hypothesis, an
over(under)-estimate on one luminosity would introduce the same effect on the
other. As a result, a poor sampling in the far-IR would have the same impact
on both (i.e. ‘observed’ and ‘model’ ) luminosities. Our analysis takes into
account this uncertainty by considering all the solutions (i.e., all torus
models) at the 1$\sigma$ confidence level. By means of this procedure, a broad
range of model SED is associated to each source (on average, eight solutions;
see Sec. 5), characterized by different emission in the mid/far-IR region, as
a result of different torus geometry and absorption properties (see Fig. 3).
As explained in Pier & Krolik (1992), the low values of the ‘observed’
$L_{bol}$ depend on a combination of three factors: the torus opening angle
$\Theta$ (geometrical factor), the observer viewing angle $\theta$ and the
torus optical thickness $\tau_{9.7{\mu}m}$. By erroneously assuming isotropic
torus emission (as done to compute the ‘observed’ $L_{bol}$), the primary flux
which does not intercept the obscuring material would not be included in the
luminosity budget; moreover, as the thickness of the torus increases, more and
more primary high-energy photons entering the torus are absorbed by the dust
grains and re-emitted isotropically (hence also in directions escaping the
torus it self). This effect is explained by the dust self-absorption, i.e.
thermal dust emission absorbed by the dust itself. For high optical depth, the
outer edges of the torus absorb the IR photons coming from the warmer dust at
smaller radii and re-emit them isotropically, i.e. also in directions outside
the line of sight. To better visualize this effect, we report in Fig. 6 the
‘observed’ versus ‘model’ luminosities, as predicted by Pier & Krolik (1992),
as a function of the viewing angle $\theta$ and the torus optical thickness
$\tau_{9.7{\mu}m}$ for 4 sets of Type 2 configurations (as described in Fig. 6
caption). Although there are some slight differences between the F06 model
(adopted here) and the Pier & Krolik (1992) torus model (where the optical
depth varies independently along the radial and the vertical axis), optically
thinner models show less anisotropy (red dotted lines in Fig. 6, corresponding
to two different viewing angles), than higher $\tau_{9.7{\mu}m}$ models (blue
dashed lines in Fig. 6, corresponding to the same viewing angles considered
for the thinner model). The cold outer edges of the thicker models, in fact,
radiate little and block the light coming from the inner torus radii.
To investigate these issues further, we apply a ‘conservative’ correction to
our ‘observed’ luminosities, meant to correct only for the geometrical factor;
in other words, we divided each ‘observed’ luminosity by the corresponding
covering factor CF (${\sim}0.58$ for ${\sim}100^{\circ}$ and ${\sim}0.88$ for
${\sim}140^{\circ}$). Although this correction increases the ‘observed’
luminosities, the ‘model’ ones are still higher ($R{\sim}1.6$); the remaining
discrepancy is mostly found for solutions with high optical depth, as expected
($R\sim 5$ for models with $\tau_{9.7}{\geq}3$; see Fig. 6, where the squares
mark the 52 solutions with $\tau_{9.7}{\geq}3$).
An independent and consistent analysis was done also by Pozzi et al. (2007,
see their $\S$ 5.1) where a first-order correction of ${\sim}2$ to the
‘observed’ luminosities was estimated, accounting for geometrical and
anisotropy effects; in that work, however, the correction was estimated using
the ratio of obscured/unobscured quasars according to the Gilli et al. (2007)
AGN synthesis models of the X-ray background and the different shape of Type 2
vs. Type 1 quasar SEDs as a function of the column density. In Pozzi et al.
(2007), the SED fitting was done using the Silva et al. (2004) AGN templates.
Since the template choice was based on the X-ray NH (and not on the NH
resulting from the torus modelling as in the present analysis), the correction
corresponding to the thicker models (N${}_{H}{{{}_{>}\atop{}^{\sim}}}10^{24}$
cm-2) were not included since no Compton-thick objects were revealed in
X-rays.
Figure 7: 2–10 keV bolometric corrections as a function of the ‘model’
bolometric luminosities (filled circles). Filled circles inside empty red
squares represent the sources with a spectroscopic redshift. The red solid and
dot-dashed lines represent the predictions from the Marconi et al. (2004)
relation and its 1$\sigma$ dispersions. Also the expectations from Hopkins et
al. (2007) are reported as empty blue triangles. The red dotted line
represents the Marconi et al. (2004) expectations at 5$\sigma$ from the best-
fitting relation.
### 6.2 Hard X-ray bolometric corrections
In Fig. 7 the bolometric-to-X-ray luminosity ratio ($k_{2-10keV}$) as a
function of $L_{bol}$ is shown: for the bolometric luminosities we assume the
model ones. The error bars on $k_{2-10keV}$ are derived from the 1$\sigma$
dispersion on $L_{bol}$.
A large spread in the $k_{2-10keV}$ is found
($6{{{}_{<}\atop{}^{\sim}}}k_{2-10keV}{{{}_{<}\atop{}^{\sim}}}80$), as pointed
out also by the pioneering work of Elvis et al. (1994) on Type 1 QSOs, due to
the large dispersion in the AGN spectral shape. Our median value
($k_{2-10keV}\sim{20}$, estimated on the 137 solutions), is marginally
consistent with the mean value of Elvis et al. (1994), ${\sim}25$ after
removing the IR contribution (in order to not double-count the fraction of the
nuclear emission absorbed by the circumnuclear dusty material seen almost
face-on). We confirm the trend of higher $k_{2-10keV}$ for objects with higher
bolometric luminosities as predicted by Marconi et al. (2004) (red solid line
in Fig. 7; red dot-dashed lines representing the 1$\sigma$ model dispersion),
but our $k_{2-10keV}$ values are significantly lower (at least a factor 2 in
normalization). They derive $k_{2-10keV}$ by constructing an AGN reference
template taking into account how the spectral index $\alpha_{ox}$ varies as a
function of the luminosity (Vignali et al. 2003). Predictions consistent with
Marconi et al. (2004) were obtained more recently by Hopkins et al. (2007,
blue triangles in Fig. 7), considering the most recent determination of SED
templates (i.e. Richards et al. 2006) and $\alpha_{ox}$ (i.e. Steffen et al.
2006).
Our low values for $k_{2-10keV}$ are consistent with our previous analysis
(median $k_{2-10keV}\sim{25}$, Pozzi et al. 2007) based on a different method
and on different AGN templates (Silva et al. 2004) and with other estimates
found in literature for hard X-ray selected samples. Kuraszkiewicz et al.
(2003), considering a sample of X-ray selected luminous AGN
($10^{43}<L_{2-10keV}<10^{46}$ erg sec-1) found a median $k_{2-10keV}$ of 18;
Ballo et al. (2007), analysing a sample of low-luminosity AGN
($10^{42}<L_{2-10keV}<10^{43.6}$ erg sec-1), found a median $k_{2-10keV}$ of
12. Low bolometric-to-X-ray ratios, consistent with our estimate (median
$k_{2-10keV}\sim{25}$, 1$\sigma$=53) were found recently by Lusso et al.
(2009), where the statistical properties of a large (and complete) sample of
545 X-ray selected Type 1 QSO from the XMM-COSMOS survey (Hasinger et al.
2007) are presented.
The lower bolometric-to-X-ray luminosity ratios found in the present work (and
in the above mentioned samples), in comparison to the Marconi et al. (2004)
and Hopkins et al. (2007) predictions, are probably caused by a selection
bias, since our sample (and most of the above cited ones) are hard-X-ray
selected samples (i.e., sources with high X-ray luminosity are favored).
Moreover, as discussed in $\S$ 2, our sources are among the most extreme X-ray
sources, being characterized by red optical-to-NIR colours
(R-K${}_{s}{{{}_{>}\atop{}^{\sim}}}5$) and high X-ray-to-optical ratio
(X/O${{}_{>}\atop{}^{\sim}}{1}$). Our selection is likely the origin of the
large deviation (at about the 5$\sigma$ level) for a large fraction of the
present sample (see Fig. 7) from the Marconi et al. (2004) relation. To
further explore this issue, a larger (and complete) sample of X-ray sources
(with optical identification up to the faintest X-ray fluxes) is needed, in
order to correct for the selection bias and to derive the properties of the
parent AGN population (see Lusso et al. 2009).
### 6.3 Black hole masses
Figure 8: $\lambda_{Edd}$ as a function of $z$. Black circles: sources of the
present sample. Red squares as in Fig.7. The error bars represent the
1$\sigma$ uncertainties on Lbol (as derived from the $\chi^{2}$ analysis).
Small grey crosses: sample of SDSS quasars from McLure & Dunlop (2004), with
the median values (and associated uncertainties assuming the normalized median
absolute values) of $\lambda_{Edd}$ within ${\Delta}z=0.1$ bins shown as red
filled circles. Blue triangles: median values (and associated uncertainties
assuming the normalized median absolute values) of $\lambda_{Edd}$ within
${\Delta}z=0.2$ bins from the COSMOS Type 1 sample of Lusso et al. (2009).
The BH masses are not a direct output of the best-fitting procedure and cannot
be derived using ‘standard methods’ (i.e. galaxy stellar kinematics, nuclear
gas motions, reverberation). We estimate them indirectly using the
$M_{bulge}-M_{BH}$ relation derived locally by Marconi & Hunt (2003), by
assuming as $M_{bulge}$ the stellar mass derived from our best-fitting
procedure. The main uncertainties affecting these estimates derive from the
extrapolation of the local relation to higher $z$, where the behaviour of this
relation is still a matter of debate (see discussion in Pozzi et al. 2007). As
far as the stellar masses are concerned, they are quite well constrained by
the SED-fitting procedure inside the pre-constructed grid of galaxy models
(see Sec. 5.2).
The inferred black hole masses are typically in the range $10^{8}-10^{9}$M⊙,
with three sources (PKS 0537$\\#$43, GD 158$\\#$19, Abell 2690$\\#$29) with
higher masses (M${}_{BH}{\sim}10^{9.5}-10^{10.0}$M⊙).
The range of BH masses is consistent with the values reported by McLure &
Dunlop (2004) for the SDSS quasars in the same redshift interval
($0.9{{{}_{<}\atop{}^{\sim}}}z{{{}_{<}\atop{}^{\sim}}}2.1$, see also Shen et
al. 2008, where new BH masses are derived).
In Fig. 8, the Eddington ratios $\lambda_{Edd}$, defined as
$\lambda_{Edd}=L_{bol}/L_{Edd}$ (with
$L_{Edd}=1.38{\times}10^{38}M_{BH}/M_{\odot}$), are reported as a function of
redshift. The values are compared with those of the whole SDSS quasar sample
(small grey crosses, McLure & Dunlop 2004) and those obtained by Lusso et al.
(2009) for the sub-sample of 150 X-ray selected Type 1 AGN in COSMOS with an
accurate black hole mass determination (blue triangles).
The $\lambda_{Edd}$ values of the present work cover slightly more than an
order of magnitude (0.01-0.3), with a median value of
$\lambda_{Edd}{\sim}0.08$ (estimated considering all the 137 solutions at
1$\sigma$ level, see Sec. 5). The derived values are within the 3$\sigma$
confidence interval of the SDSS quasar $\lambda_{Edd}$ distribution,
characterized by a median value of ${\sim}0.3$ and with a dispersion of 0.35
dex at the same redshift interval sampled by our sources. However, almost all
our data points lie towards the low $\lambda_{Edd}$ tail of the SDSS
distribution (see Fig. 8), suggesting that X-ray selection is biased towards
slightly lower $\lambda_{Edd}$ than optical selection. Our data are fully
consistent with the results obtained from a much larger sample of X-ray
selected Type 1 AGN in the COSMOS field (Lusso et al. 2009).
The results are robust against the uncertainties on the extrapolation,
discussed above, of the local $M_{bulge}-M_{BH}$ relation, at the redshift of
our sample. In fact, allowing for positive evolution with redshift of the
$M_{BH}/M_{bulge}$ ratio by a factor of 2 (e.g., Hopkins et al. 2006; Shields
et al. 2006; Merloni et al. 2010), the Eddington ratios $\lambda_{Edd}$ would
decrease further by the same factor.
Finally, in Fig. 9, the bolometric corrections $k_{2-10keV}$ are plotted
against the Eddington ratios $\lambda_{Edd}$ (following Vasudevan & Fabian
2009). The sources of the present work are reported as black filled circle
(the error bars representing the $1\sigma$ confidence interval derived from
the uncertainties on $L_{bol}$). Along with our data we show the Vasudevan &
Fabian (2009) results, where simultaneous optical, UV and X-ray observations
are included for the majority of the Peterson et al. (2004) reverberation
mapped sample of AGN (blue empty circles). Our findings are in fairly good
agreement with the trend of increasing $k_{2-10keV}$ for increasing
$\lambda_{Edd}$. Vasudevan & Fabian (2009) interpret the observed trend as due
to different black hole SED shape as a function of the Eddington ratio, with
the high and low Eddington ratios corresponding to different fractions of the
ionizing UV luminosity. A similar trend was recently found for a sample of 63
Type 1 and Type 2 AGN detected in the Swift/BAT 9-months catalog (see
Vasudevan et al. 2009). At variance with the assumption in Vasudevan & Fabian
(2009), where the bolometric luminosities were derived by integration over the
observed optical/UV/X-ray SED, in this work the authors consider the
reprocessed IR emission, reproduced by the empirical SEDs of Silva et al.
(2004), as a proxy of the intrinsic AGN bolometric luminosity, as firstly
suggested by Pozzi et al. (2007).
The dependence of bolometric corrections on Eddington ratios is expected by
accretion-disk models, which predict an increasing hard X-ray bolometric
corrections at increasing accretion rates (e.g. Witt et al. 1997). Recently
also Bianchi et al. (2009), studying a large (156 sources) sample of Type 1
X-ray AGN from the XMM-$Newton$ archive suggest that the bolometric correction
must depend on Eddington ratio in order to allow the intrinsic power of AGN to
scale linearly with black hole masses.
Figure 9: $k_{2-10keV}$ as a function of $\lambda_{Edd}$. Black filled
symbols: sources of the present sample. Red squares as in Fig.7. The error
bars represent the 1$\sigma$ uncertainties on Lbol (as derived from the
$\chi^{2}$ analysis) which affect both the $k_{2-10keV}$ and the
$\lambda_{Edd}$ values. Blue open symbols: sources from Vasudevan & Fabian
(2009).
Table 3: Best-fitting physical parameters and inferred rest-frame properties.
Source name | $\chi^{2}_{min}$/$d.o.f$ | $\alpha$ | ${\tau}_{9.7{\mu}m}$ | $\Theta$ | $\tau_{sf}$ | $E(B-V)$ | $L_{acc}$ | $L_{1-1000_{{\mu}m}}$ | $L_{2-10keV}$ | $L_{1.25-500keV}$ | $L_{bol}$ | $k_{2-10keV}$ | $M_{star}$ | $M_{BH}$ | $\lambda_{Edd}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12) | (13) | (14) | (15) | (16)
PKS 0537#43 | 66.7/6 | -1.0 | 0.1 | 140 | 0.05 | 0.3 | 373.6 | 254 | 6.8 | 26.3 | 400 [400-422] | 58.8 | 49.4 | 10.05 | 0.032
PKS 0537#11 | 7.8/5 | 0.0 | 2.0 | 140 | 0.7 | 0.64 | 17.6 | 7.4 | 1.9 | 7.6 | 25.2 [14.1-34.4] | 12.9 | 1.20 | 0.28 | 0.070
PKS 0537#164 | 9.0/5 | 0.0 | 3.0 | 60 | 0.15 | 0.0 | 99.8 | 4.9 | 3.1 | 12.1 | 112 [63.0-174] | 36.1 | 1.29 | 0.30 | 0.292
PKS 0537#123 | 5.0/5 | 0.0 | 0.1 | 140 | 1.0 | 0.65 | 9.5 | 7.8 | 2.0 | 7.8 | 17.3 [17.3-36.4] | 8.6 | 1.99 | 0.46 | 0.030
GD 158#62 | 9.3/5 | 0.0 | 0.1 | 100 | 0.35 | 0.44 | 85.1 | 39.7 | 4.9 | 19.0 | 104 [67.3-214] | 21.2 | 2.19 | 0.51 | 0.163
GD 158#19 | 26.9/6 | 0.0 | 0.3 | 140 | 1.0 | 0.49 | 397. | 289.7 | 6.3 | 24.4 | 422 [422-465] | 67.0 | 12.3 | 2.64 | 0.127
Mrk 509#01 | 17.9/5 | -0.5 | 6.0 | 60 | 0.9 | 0.85 | 104.5 | 1.59 | 1.5 | 5.8 | 111 [42.8-131] | 73.6 | 1.98 | 0.46 | 0.191
Mrk 509#13 | 25.0/5 | 0.0 | 1.0 | 140 | 0.6 | 1.0 | 23.2 | 13.0 | 2.6 | 10.1 | $33.3[27.1-53.9]$ | 12.8 | 1.93 | 0.45 | 0.059
Abell 2690#75 | 17.5/5 | 0.0 | 1.0 | 140 | 0.05 | 0.18 | 19.8 | 11.0 | 3.2 | 12.4 | 32.2 [20.8-66.8] | 10.1 | 2.73 | 0.62 | 0.041
PKS 0312#36 | 9.6/5 | 0.0 | 0.1 | 140 | 0.85 | 0.79 | 3.0 | 2.5 | 0.7 | 2.7 | 5.7 [5.7-8.1] | 8.2 | 0.43 | 0.11 | 0.043
PKS 0537#91 | 2.9/5 | 0.0 | 3.0 | 140 | 0.25 | 0.3 | 66.0 | 23.7 | 8.1 | 31.4 | 98.3 [40.2-326] | 12.1 | 1.48 | 0.35 | 0.224
PKS 0537#54 | 2.9/5 | 0.0 | 0.1 | 140 | 0.05 | 0.17 | 5.6 | 4.6 | 2.0 | 7.8 | 13.3 [13.3-25.8] | 6.7 | 1.72 | 0.40 | 0.026
PKS 0537#111 | 11.4/5 | -0.5 | 10.0 | 140 | 0.05 | 0.31 | 103.1 | 13.4 | 1.7 | 6.6 | 110 [34.1-110] | 64.4 | 4.61 | 1.03 | 0.084
Abell 2690#29 | 0.5/2 | 0.0 | 1.0 | 140 | 0.15 | 0.55 | 168.8 | 94.7 | 8.4 | 32.6 | 201 [135-264] | 23.9 | 17.9 | 3.80 | 0.042
PKS 0312#45 | 0.5/5 | 0.0 | 0.1 | 100 | 0.25 | 0.48 | 24.25 | 11.3 | 6.2 | 24.0 | 48.3 [37.4-239] | 7.8 | 3.61 | 0.81 | 0.047
BPM 16274#69 | 0.5/5 | 0.0 | 0.6 | 140 | 0.05 | 0.25 | 11.4 | 7.3 | 2.4 | 9.3 | 20.7 [18.2-56.6] | 8.6 | 4.68 | 1.05 | 0.016
(1) source name
(2) best-fitting minimum $\chi^{2}$ ($\chi^{2}_{min}$) and number of
parameters to be fitted (degrees of freedom);
(3), (4), (5) best-fitting torus parameters ($\alpha$: exponent of the power
law parameterizing the density profile; $\tau_{9.7{\mu}m}$: 9.7 ${\mu}$m
optical depth, $\Theta$: torus opening angle). The ratio parameter
$R_{max}/R_{min}$ is frozen to 30; the density parameter $\gamma$ is frozen to
0;
(6), (7) best-fitting stellar parameters ($\tau_{sf}$: duration of the
exponential decay of the burst in units of the oldest SSP; $E(B-V)$:
extinction);
(8) accretion-disk model luminosity, $L_{acc}$ (from soft X-ray to IR
frequencies) which represents the torus model input luminosity, in units of
1044 erg s-1;
(9) integrated (1-1000$\mu$m) torus luminosity in units of 1044 erg s-1 (not
corrected, see 6.1);
(10) absorption-corrected 2–10 keV luminosity in units of 1044 erg $s^{-1}$;
(11) hard-X-ray (1.25-500 keV) luminosity in units of 1044 erg s-1 (derived
from the 2–10 keV luminosity, see 6.2);
(12) bolometric AGN luminosity ($L_{acc}$\+ $L_{1.25-500keV}$) in units of
1044 erg s-1; the 1$\sigma$ range derived from the SED-fitting analysis is
reported;
(13) 2–10 keV bolometric correction ($L_{bol}/L_{2-10keV}$);
(14) galaxy mass in units of 1011M⊙;
(15) black hole masses in units of 109M⊙ (estimated from Marconi & Hunt 2003
relation);
(16) Eddington ratios ($L_{bol}/L_{Edd}$).
## 7 Summary
We analyzed the SEDs of a sample of 16 obscured quasars selected in the hard
X-ray band. Spitzer mid/far-IR photometry (IRAC and MIPS), along with the data
available in the literature, is modeled using a multi-component model, where
the AGN re-processed emission is reproduced in the context of a flared disk
model, as described by F06. Within the context of a flared disk torus model,
the uncertainty in and degeneracy between the various derived parameters are
accounted for by including all solutions within 1$\sigma$ of the best- fit in
the subsequent analysis.
The main results are summarized below:
* $\bullet$
All the 16 quasars are detected up to 8 ${\mu}$m and all, but two sources, are
detected at 24${\mu}$m with flux densities in the range 100-7000 ${\mu}$Jy at
the 5$\sigma$ level. The two most luminous sources of the sample are detected
also at 70 and 160 ${\mu}$m.
* $\bullet$
The observed broad-band spectral energy distributions are well reproduced by a
multi-component model comprising a stellar, an AGN and a starburst components
(when far-IR detections are available). The AGN component, modelled with the
F06 radiative transfer code, accounts for the X-ray emission and for a
fraction of the IR emission, mainly due to reprocessed emission from the
putative dusty torus surrounding the central black hole.
* $\bullet$
Solutions with a moderate optical depth $\tau_{9.7}$ are favoured by the SED-
fitting, with the majority of the sources having moderate optical depths
(${\tau}_{9.7{\mu}m}{\leq}3$). The derived gas column densities (NH) are
consistent, for most of the sources, with the values estimated from the X-ray
analysis, both indicating that the sources are Compton-thin AGN
(N${}_{H}{\sim}10^{22}-3{\times}10^{23}$cm-2).
* $\bullet$
The model nuclear bolometric luminosities are in the range
$5{\times}10^{44}-4{\times}10^{46}$ erg s-1. By comparing these values with
those obtained by the integration of the nuclear observed SED, we conclude
that the latter under-estimate the bolometric luminosities by a factor of 2.
The difference may be explained by anisotropic torus emission and the effect
of the torus optical depth (e.g., Pier & Krolik 1992).
* $\bullet$
From the model nuclear SEDs, we estimate the bolometric-to-X-ray corrections
($k_{2-10keV}$). The median $k_{2-10keV}$ is ${\sim}20$
($6{{{}_{<}\atop{}^{\sim}}}k_{2-10keV}{{{}_{<}\atop{}^{\sim}}}80$). The value
is smaller than assumed by some models of BH evolution
($k_{2-10keV}{{{}_{>}\atop{}^{\sim}}}40$ at the median luminosity of our
sample). The discrepancy is significant at 5$\sigma$ level at low bolometric
luminosity.
* $\bullet$
By assuming the local $M_{bulge}-M_{BH}$ relation, we estimate $\lambda_{Edd}$
with a median value of 0.08
($0.01{{{}_{<}\atop{}^{\sim}}}\lambda_{Edd}{{{}_{<}\atop{}^{\sim}}}0.3$). The
whole SDSS quasar sample, at the same redshift interval sampled by our
objects, is characterized by a median value of 0.3. Our data are within the
$3\sigma$ confidence level of the optically selected quasar distribution.
However, almost all our sources lie towards the low $\lambda_{Edd}$ tail of
the SDSS distribution, suggesting that our X-ray selection is biased towards
lower Eddington efficiencies than optical selection.
* $\bullet$
The data are consistent with the correlation recently suggested by Vasudevan &
Fabian (2007, 2009) between $k_{2-10keV}$ and $\lambda_{Edd}$, where low
bolometric corrections are found at low Eddington ratios.
###### Acknowledgements.
The authors thank the anonymous referee for useful comments that helped
improve both the paper content and presentation. The authors thank D. Fadda
for suggestions about MIPS data reduction techniques, R. Gilli and F. La
Franca for helpful discussions and E. Lusso and E. Sarria for providing their
results before publication. This work has benefited for partial support by the
Italian Space Agency (contracts ASI/COFIS/WP3110 I/026/07/0 and ASI
I/088/06/0), PRIN/MIUR (grant 2006-02-5203) and from research funding from the
European Community’s Sixth Framework Programme under RadioNet contract R113CT
2003 5158187. This work is based on observations made with the Spitzer Space
Telescope, which is operated by the Jet Propulsion Laboratory, California
Institute of Technology under NASA contract 1407 and with the SCUBA camera,
operating at the James Clerk Maxwell Telescope, a joint U.K.-Dutch-Canadian
millimeter/sub-millimeter Telescope in Hawaii.
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|
arxiv-papers
| 2010-03-29T15:05:30 |
2024-09-04T02:49:09.322414
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "F. Pozzi, C. Vignali, A. Comastri, E. Bellocchi, J. Fritz, C.\n Gruppioni, M. Mignoli, R. Maiolino, L. Pozzetti, M. Brusa, F. Fiore, G.\n Zamorani",
"submitter": "Pozzi Francesca",
"url": "https://arxiv.org/abs/1003.5563"
}
|
1003.5621
|
# On intrinsic isometries to Euclidean space.
Anton Petrunin
###### Abstract
I consider compact metric spaces which admit intrinsic isometries to Euclidean
$d$-space. The main result roughly states that the class of these spaces
coincides with class of inverse limits of Euclidean $d$-polyhedra.
## 1 Introduction
The _intrinsic isometries_ are defined in section 2; it is a variation of
notion of _path isometry_ , i.e. map which preservs the lengths of curves. Any
intrinsic isometry is a path isometry, the converse does not hold in general.
The following statement is one of the reason we prefer intrinsic isometry.
1.1. Trivial statement. Let $\mathcal{X}$ be a compact metric space which
admits an intrinsic isometry to $d$-dimensional Euclidean space (further
denoted by $\mathbb{E}^{d}$). Then $\mathop{\rm
dim}\nolimits\mathcal{X}\leqslant d$, where $\mathop{\rm dim}\nolimits$
denotes the Lebesgue’s covering dimension.
This statement is proved in section 3. An analogous statement for path
isometry does not hold, see example 4. The Hausdorff dimension can not be
bounded on the similar way. For example, $\mathbb{R}$-tree admits an intrinsic
isometry to $\mathbb{R}$ and it contains compact subsets of arbitrary large
Hausdorff dimension.
Here are some known results on length spaces which admit _intrinsic isometry_
to $\mathbb{E}^{d}$.
1.2. Theorem. Let $\mathcal{R}$ be $d$-dimensional Riemannian space and
$f\colon\mathcal{R}\to\mathbb{E}^{d}$ be a short map111i.e. $1$-Lipschitz map.
Then given $\varepsilon>0$, there is an intrinsic isometry
$\imath\colon\mathcal{R}\to\mathbb{E}^{d}$ such that
$|f(x)\,\imath(x)|_{\mathbb{E}^{d}}<\varepsilon$
for any $x\in\mathcal{R}$.
In particular, any Riemannian $d$-space admits an intrinsic isometry to
$\mathbb{E}^{d}$.
For path isometries, this theorem was proved in [Gromov, 2.4.11], and the same
proof works for intrinsic isometries. Applying this theorem, one can show that
any limit of increasing sequence of Riemannian metrics on a fixed
$d$-dimensional manifold admits an intrinsic isometry to $\mathbb{E}^{d}$.
(The proof is similar to “if”-part of the main theorem.) In particular, any
sub-Riemannian metric on $d$-dimensional manifold admits an intrinsic isometry
to $\mathbb{E}^{d}$.
1.3. Theorem. Let $\mathcal{P}$ be a Euclidean polyhedron and
$f\colon\mathcal{P}\to\mathbb{E}^{d}$ be a short map. Then, given
$\varepsilon>0$, there is a piecewise linear intrinsic isometry
$\imath\colon\mathcal{P}\to\mathbb{E}^{d}$ such that
$|f(x)\,\imath(x)|_{\mathbb{E}^{d}}<\varepsilon$
for any $x\in\mathcal{P}$.
1.4. Corollary. Any $d$-dimensional Euclidean polyhedron admits a piecewise
linear intrinsic isometry to $\mathbb{E}^{d}$.
The corollary was proved in [Zalgaller] for dimension $\leqslant 4$, but a
slight modification of the proof works in all dimensions, see [Krat]. The
2-dimensional case of this theorem was proved in [Krat]. Later, the proof was
extend to all dimensions in [Akopyan]; the proof use a _piecewise linear
analog of Nash–Kuiper theorem_. This theorem was proved in [Brehm], but this
work was left without attention for many years and reproved independently in
[Akopyan–Tarasov].
Iff-condition. Now we describe the main result of the paper.
A compact metric space $\mathcal{X}$ is called _pro-Euclidean space of rank
$\leqslant d$_ if it can be presented as an _inverse limit_
$\mathcal{X}=\varprojlim\mathcal{P}_{n}$ (see section 2) of a sequence of
Euclidean $d$-polyhedra $\mathcal{P}_{n}$.
1.5. Main theorem. A compact metric space $\mathcal{X}$ admits an intrinsic
isometry to $\mathbb{E}^{d}$ if and only if $\mathcal{X}$ is a pro-Euclidean
space of rank $\leqslant d$.
The proof is straightforward. I like the formulation of the theorem; it seems
to be the first case when inverse limits help to solve a natural problem in
metric geometry.
Note that the statement in theorem 1 (in the compact case) is equivalent to
the fact that any compact Riemannian $d$-space is a pro-Euclidean space of
rank $\leqslant d$. The later can be obtained directly from the following
exercise; this way the main theorem provides an alternative proof to theorem 1
in the compact case.
1.6. Exercise. Show that any compact Riemannian space admits a Lipschitz
approximation by Euclidean polyhedra.
A non-example. Let us remind that _Minkowski space_ is finite dimensional real
vector spaces with metric induced by a norm.
1.7. Proposition. Let $\Omega$ be an open subset of Minkowski $d$-space
$\mathbb{M}^{d}$. Assume $\Omega$ admits an intrinsic222In fact the same is
true for path isometry. isometry to $\mathbb{E}^{m}$ then $d\leqslant m$ and
$\mathbb{M}^{d}$ is isometric to a $\mathbb{E}^{d}$.
In particular, the condition 1 on Lebesgue’s dimension is not sufficient.
I’m grateful to A. Akopyan, D. Burago, S. Ivanov and anonymous referee for
helpful letters and discussions.
## 2 Preliminaries
Standard definitions. Given a metric space $\mathcal{X}$ and two points
$x,x^{\prime}\in\mathcal{X}$, we will denote by
$|xx^{\prime}|=|xx^{\prime}|_{\mathcal{X}}$ the distance from $x$ to
$x^{\prime}$ in $\mathcal{X}$.
A _length space_ is a metric space such that for any two points $x,x^{\prime}$
the distance $|xx^{\prime}|$ coincides with the infimum of lengths of curves
connecting $x$ and $x^{\prime}$.
A map $f\colon\mathcal{X}\to\mathcal{Y}$ between metric spaces $\mathcal{X}$
and $\mathcal{Y}$ is called _short_ if for any $x,x^{\prime}\in\mathcal{X}$ we
have
$|f(x)f(x^{\prime})|_{\mathcal{Y}}\leqslant|xx^{\prime}|_{\mathcal{X}}.$
A length space $\mathcal{P}$ is called _Euclidean $d$-polyhedron_ if there is
a finite triangulation of $\mathcal{P}$ such that each simplex is isometric to
a simplex in $\mathbb{E}^{d}$.
Inverse limit. Consider an _inverse system_ of compact metric spaces
$(\mathcal{X}_{n})_{n=0}^{\infty}$ and short maps
$\varphi_{m,n}:\mathcal{X}_{m}\to\mathcal{X}_{n}$ for $m\geqslant n$; i.e.,
1. 1.
$\varphi_{m,n}\circ\varphi_{k,m}=\varphi_{k,n}$ for any triple $k\geqslant
m\geqslant n$ and
2. 2.
for any $n$, the map $\varphi_{n,n}$ is identity map of $\mathcal{X}_{n}$.
A compact metric space $\mathcal{X}$ is called _inverse limit of the system
$(\varphi_{m,n},\mathcal{X}_{n})$_ (denoted by
$\mathcal{X}=\varprojlim\mathcal{X}_{n}$) if its underling space consists of
all sequences $x_{n}\in\mathcal{X}_{n}$ such that $\varphi_{m,n}(x_{m})=x_{n}$
for all $m\geqslant n$ and for two such sequences $(x_{n})$ and
$(x^{\prime}_{n})$ the distance is defined by
$|(x_{n})\,(x^{\prime}_{n})|_{\mathcal{X}}=\lim_{n\to\infty}|x_{n}\,x^{\prime}_{n}|_{\mathcal{X}_{n}}.$
If $\mathcal{X}=\varprojlim\mathcal{X}_{n}$, then the map
$\psi_{n}\colon\mathcal{X}\to\mathcal{X}_{n}$, defined by
$\psi_{n}\colon(x_{i})_{i=0}^{\infty}\mapsto x_{n}$ are called _projections_.
Clearly $\psi_{n}=\varphi_{m,n}\circ\psi_{m}$ for all $m\geqslant n$.
Comments. The above definition is equivalent to the usual inverse limit in the
category with class of objects formed by compact metric spaces and class of
morphisms formed by short maps.
Note that inverse limit is not always defined, and if defined it is the result
is compact by definition. (In principle, the category of compact metric spaces
can be extended so that the limit of any inverse system is well defined.)
It is easy to see that inverse limit of length spaces is a length space.
In general, the inverse limit of a system of spaces differ from its
Gromov–Hausdorff limit. For example, consider inverse system
$\mathcal{X}_{n}=[0,1]$ with maps $\varphi_{m,n}(x)\equiv 0$. The inverse
limit of this system is isometric to one-point space, while the
Gromov–Hausdorff limit is isometric to $[0,1]$. Nevertheless, it is easy to
see that if for any $\varepsilon>0$ the images of $\varphi_{m,n}$ form an
$\varepsilon$-net in $X_{n}$ for all sufficiently large $m$ and $n$, then
$\mathcal{X}=\varprojlim\mathcal{X}_{n}$ is isometric to the Gromov–Hausdorff
limit.
Intrinsic isometries and pull back metrics. Let $\mathcal{X}$ and
$\mathcal{Y}$ be metric spaces and $f\colon\mathcal{X}\to\mathcal{Y}$ be
continuous map. Given two points $x,x^{\prime}\in\mathcal{X}$, a sequence of
points $x=x_{0},x_{1},\dots,x_{n}=x^{\prime}$ is called $\varepsilon$-chain
from $x$ to $x^{\prime}$ if $|x_{i-1}x_{i}|\leqslant\varepsilon$ for all
$i>0$. Set
$\mathop{\rm
pull}\nolimits_{f,\varepsilon}(x,x^{\prime})=\inf\\!\left\\{\sum_{i=1}^{n}|f(x_{i-1})f(x_{i})|_{\mathcal{Y}}\right\\}$
where the infimum is taken along all $\varepsilon$-chains $(x_{i})_{i=0}^{n}$
from $x$ to $x^{\prime}$.
Clearly $\mathop{\rm pull}\nolimits_{f,\varepsilon}$ is a pre-metric333i.e. it
satisfies triangle inequality, it is symmetric, non-negative and $\mathop{\rm
pull}\nolimits_{f,\varepsilon}(x,x)=0$, but it might happen that $\mathop{\rm
pull}\nolimits_{f,\varepsilon}(x,x^{\prime})=0$ for $x\not=x^{\prime}$. on
$\mathcal{X}$ and $\mathop{\rm pull}\nolimits_{f,\varepsilon}(x,x^{\prime})$
is non-increasing in $\varepsilon$. Thus, the following (possibly infinite)
limit
$\mathop{\rm pull}\nolimits_{f}(x,x^{\prime})=\lim_{\varepsilon\to
0}\mathop{\rm pull}\nolimits_{f,\varepsilon}(x,x^{\prime})$
is well defined. The pre-metric $\mathop{\rm
pull}\nolimits_{f}\colon\mathcal{X}\times\mathcal{X}\to[0,\infty]$ will be
called _pull back metric_ for $f$.
A map $f\colon\mathcal{X}\to\mathcal{Y}$ between length spaces $\mathcal{X}$
and $\mathcal{Y}$ is an _intrinsic isometry_ if
$|xx^{\prime}|_{\mathcal{X}}=\mathop{\rm pull}\nolimits_{f}(x,x^{\prime})$
for any $x,x^{\prime}\in\mathcal{X}$.
Any intrinsic isometry is a short map. Moreover, it is easy to see that
intrinsic isometry preserves the lengths of curves. The converse does not
hold, see section 4.
2.1. Proposition. Let $\mathcal{X}$ be a compact (or even proper444I.e. all
closed bounded sets in $\mathcal{X}$ are compact.) metric space. Then
existence of intrinsic isometry $f\colon\mathcal{X}\to\mathcal{Y}$ implies
that $\mathcal{X}$ is a length space.
The proof is left to the reader. Note that for general $\mathcal{X}$ it is not
true. Consider two points which connected by countable number of unit
intervals $\mathbb{I}_{n}$ and one interval of length $\frac{1}{2}$; equip the
obtained space with natural intrinsic metric. Let us remove from our space the
interval of length $\frac{1}{2}$. The metric on the remaining space
$\mathcal{X}$ is not intrinsic. Further let us construct a map
$f\colon\mathcal{X}\to\mathbb{R}$ so that the restriction
$f_{n}=f|_{\mathbb{I}_{n}}$ is an intrinsic isometry, $f_{n}(0)=0$,
$f_{n}(1)=\tfrac{1}{2}$ and $f_{n}(x)$ converges uniformly to $\tfrac{x}{2}$.
It is easy to see that $f\colon\mathcal{X}\to\mathbb{R}$ is an intrinsic
isometry.
2.2. Proposition. Let $\mathcal{X}$ and $\mathcal{Y}$ be metric spaces,
$\mathcal{X}$ be compact and the continuous map
$f\colon\mathcal{X}\to\mathcal{Y}$ is such that
$\sup_{x,x^{\prime}\in\mathcal{X}}\mathop{\rm
pull}\nolimits_{f}(x,x^{\prime})<\infty.$
Then given $\varepsilon>0$ there is $\delta=\delta(f,\varepsilon)>0$ such that
for any short map $h\colon\mathcal{X}\to\mathcal{Y}$ such that
$|f(x)\,h(x)|_{\mathcal{Y}}<\delta\ \ \text{for any}\ \ x\in\mathcal{X}$
we have
$\mathop{\rm pull}\nolimits_{f}(x,x^{\prime})<\mathop{\rm
pull}\nolimits_{h}(x,x^{\prime})+\varepsilon$
for any $x,x^{\prime}\in\mathcal{X}$.
The proof is a direct application of the lemma 2.
For a compact metric space $\mathcal{X}$, we denote by $\mathop{\rm
pack}\nolimits_{\varepsilon}\mathcal{X}$ the maximal number of points in
$\mathcal{X}$ on distance $>\varepsilon$ from each other. Clearly $\mathop{\rm
pack}\nolimits_{\varepsilon}\mathcal{X}$ is finite for any $\varepsilon>0$.
2.3. Lemma. Let $\mathcal{X}$ and $\mathcal{Y}$ be metric spaces,
$\mathcal{X}$ is compact and $f,h\colon\mathcal{X}\to\mathcal{Y}$ be two
continuous maps.
Assume for any $x\in\mathcal{X}$, $|f(x)h(x)|<\delta$ then for any
$x,x^{\prime}\in\mathcal{X}$ we have
$\mathop{\rm pull}\nolimits_{f,\varepsilon}(x,x^{\prime})\leqslant\mathop{\rm
pull}\nolimits_{h,\varepsilon}(x,x^{\prime})+4{\hskip 0.5pt\cdot\nobreak\hskip
0.5pt}\delta{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}\mathop{\rm
pack}\nolimits_{\varepsilon}\mathcal{X}.$
Proof. Assume $\mathop{\rm pull}\nolimits_{h,\varepsilon}(x,x^{\prime})<\ell$,
i.e. there is an $\varepsilon$-chain $\\{x_{i}\\}_{i=0}^{n}$ from $x$ to
$x^{\prime}$ such that
$\sum_{i=1}^{n}|h(x_{i-1})\,h(x_{i})|_{\mathcal{Y}}<\ell.$ $None$
Since $|h(x_{i})f(x_{i})|<\delta$,
$\mathop{\rm
pull}\nolimits_{f,\varepsilon}(x,x^{\prime})\leqslant\sum_{i=1}^{n}|f(x_{i-1})\,f(x_{i})|_{\mathcal{Y}}<\sum_{i=1}^{n}|h(x_{i-1})\,h(x_{i})|_{\mathcal{Y}}+2{\hskip
0.5pt\cdot\nobreak\hskip 0.5pt}n{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}\delta$
Assume $n$ is the smallest number for which there is an $\varepsilon$-chain
satisfying $(*)$. It is enough to show that
$n<2{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}\mathop{\rm
pack}\nolimits_{\varepsilon}\mathcal{X}.$
If $n\geqslant 2{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}\mathop{\rm
pack}\nolimits_{\varepsilon}\mathcal{X}$, there are $i$ and $j$ such that
$j-i>1$ and $|x_{i}x_{j}|\leqslant\varepsilon$. Remove from this chain all
elements $x_{k}$ with $i<k<j$; i.e. consider new $\varepsilon$-chain
$x=x_{0},\dots,x_{i-1},x_{i},x_{j},x_{j+1},\dots,x_{n}=x^{\prime}$
By triangle inequality in $\mathcal{Y}$, the new chain satisfies $(*)$; i.e.
$n$ is not the smallest number, a contradiction. ∎
2.4. Proposition. Let $\mathcal{X}$ and $\mathcal{Y}$ be metric spaces,
$\mathcal{X}$ be compact and $\imath\colon\mathcal{X}\to\mathcal{Y}$ be an
intrinsic isometry.
Then given $\varepsilon>0$ there is $\delta=\delta(\imath,\varepsilon)>0$ such
that for any connected set $W\subset\mathcal{X}$
$\mathop{\rm diam}\nolimits\imath(W)<\delta\ \ \Longrightarrow\ \ \mathop{\rm
diam}\nolimits W<\varepsilon.$
Proof. Assume contrary, i.e. there is a sequence of connected subsets
$W_{n}\subset\mathcal{X}$ such that $\mathop{\rm
diam}\nolimits\imath(W_{n})\to 0$ as $n\to\infty$ but $\mathop{\rm
diam}\nolimits W_{n}>\varepsilon$. Thus there are two sequences of points
$x_{n},x_{n}^{\prime}\in W_{n}$ such that
$|x_{n}x_{n}^{\prime}|\geqslant\varepsilon$. Pass to a subsequence of $n$ so
that $W_{n}\to W$ in Hausdorff sense and $x_{n}\to x$, $x_{n}^{\prime}\to
x^{\prime}$. We obtain a closed connected subset $W\subset\mathcal{X}$ with
two distinct points $x$ and $x^{\prime}$ such that $\imath(W)=p$ for some
$p\in\mathcal{Y}$.
Since $W$ is connected, for any $\varepsilon>0$ there is an
$\varepsilon$-chain $(x_{i})_{i=0}^{n}$ from $x$ to $x^{\prime}$ such that
$\imath(x_{i})=p$ for all $i$. Thus, we have $\mathop{\rm
pull}\nolimits_{\imath,\varepsilon}(x,x^{\prime})=0$ for any $\varepsilon>0$;
i.e., $\mathop{\rm pull}\nolimits_{\imath}(x,x^{\prime})=0$, a contradiction.
∎
## 3 The proofs
Proof of the trivial statement (1). Given $\varepsilon>0$ choose
$\delta=\delta(\imath,\varepsilon)$ as in proposition 2. Since $\mathop{\rm
dim}\nolimits\mathbb{E}^{d}=d$, there is a finite open covering
$\\{U_{i}\\}_{i=1}^{n}$ of $\imath(\mathcal{X})$ with multiplicity $\leqslant
d+1$ and such that $\mathop{\rm diam}\nolimits U_{i}<\delta$ for each $i$.
Consider the covering $\\{V_{\alpha}\\}$ of $\mathcal{X}$ by connected
components of $\imath^{-1}(U_{i})$ for all $i$. According to proposition 2,
$\mathcal{X}$ is a length space. In particular, all sets $V_{\alpha}$ are
open. According to proposition 2, $\mathop{\rm diam}\nolimits
V_{\alpha}<\varepsilon$. Clearly multiplicity of $\\{V_{\alpha}\\}$ is at most
$d+1$. Thus, the statement follows. ∎
Proof of “if” in the main theorem (1). Let $\mathcal{X}$ be a pro-Euclidean
space of rank $\leqslant d$. Assume $(\mathcal{P}_{n})_{n=0}^{\infty}$ is a
sequence of $d$-dimensional Euclidean polyhedra and
$\varphi_{m,n}\colon\mathcal{P}_{m}\to\mathcal{P}_{n}$
is an inverse system of short maps such that
$\mathcal{X}=\varprojlim\mathcal{P}_{n}$. Let
$\psi_{n}\colon\mathcal{X}\to\mathcal{P}_{n}$ be the projections.
According to theorem 1, given $\varepsilon_{n+1}>0$ and a piecewise linear
intrinsic isometry $\imath_{n}\colon\mathcal{P}_{n}\to\nobreak\mathbb{E}^{d}$
there is a piecewise linear intrinsic isometry
$\imath_{n+1}\colon\mathcal{P}_{n+1}\to\nobreak\mathbb{E}^{d}$ such that
$|\imath_{n+1}(x)\ \imath_{n}{\circ}\varphi_{n+1,n}(x)|<\varepsilon_{n+1}.$
for any $x\in\mathcal{P}_{n}$. It remains to show that sequence
$\varepsilon_{n}$ can be chosen on such a way that $\imath_{n}{\circ}\psi_{n}$
converges to an intrinsic isometry $\imath\colon\mathcal{X}\to\mathbb{E}^{d}$.
Let us choose $\varepsilon_{n+1}>0$ so that
$\varepsilon_{n+1}<\tfrac{1}{2}\min\\!\left\\{\varepsilon_{n},\delta(\imath_{n},\tfrac{1}{n})\right\\},$
where $\delta(\imath_{n},\tfrac{1}{n})$ as in proposition 2. Clearly,
$\sum_{i}\varepsilon_{i}<\infty$, thus the the following limit exists
$\imath=\lim_{n\to\infty}\imath_{n}\circ\psi_{n},\ \
\imath\colon\mathcal{X}\to\mathbb{E}^{d}.$
Obviously, $\imath$ is short. Further, for any $x\in\mathcal{X}$,
$|\imath(x)\
\imath_{n}{\circ}\psi_{n}(x)|<\sum_{i=n+1}^{\infty}\varepsilon_{i}<\delta(\imath_{n},\tfrac{1}{n}).$
Thus, according to proposition 2,
$\mathop{\rm pull}\nolimits_{\imath}(x,x^{\prime})+\tfrac{1}{n}>\mathop{\rm
pull}\nolimits_{\imath_{n}\circ\psi_{n}}(x,x^{\prime})\geqslant|\psi_{n}(x)\,\psi_{n}(x^{\prime})|_{\mathcal{P}_{n}}.$
Since
$|\psi_{n}(x)\,\psi_{n}(x^{\prime})|_{\mathcal{P}_{n}}\to|xx^{\prime}|_{\mathcal{X}}$
as $n\to\infty$, the map $\imath\colon\mathcal{X}\to\mathbb{E}^{d}$ is an
intrinsic isometry.∎
Proof of “only if” in the main theorem (1). We will give a construction a
polyhedron $\mathcal{P}$ associated to an intrinsic isometry
$\imath\colon\mathcal{X}\to\mathbb{E}^{d}$ and a tiling of $\mathbb{E}^{d}$ by
coordinate $a$-cubes. (The space $\mathcal{P}$ will be glued out of
$a$-cubes.) The construction will be done in such a way that if a tiling
$\tau^{\prime}$ is a subdivision of a tiling $\tau$ then for corresponding
polyhedra $\mathcal{P}^{\prime}$ and $\mathcal{P}$ there will be a natural
intrinsic isometry $\mathcal{P}^{\prime}\to\mathcal{P}$. Thus we will
construct the needed inverse system of polyhedra out of nested subdivisions of
$\mathbb{E}^{d}$.
Take sequences $a_{n}=\tfrac{1}{2^{n}}$ and set $r_{n}=\tfrac{1}{10}{\hskip
0.5pt\cdot\nobreak\hskip 0.5pt}a_{n}$. Fix $n$ for a while and consider tiling
of $\mathbb{E}^{d}$ by coordinate $a_{n}$-cubes. Let us construct a Euclidean
polyhedron $\mathcal{P}_{n}$ associated to this tiling.
The image $\imath(\mathcal{X})$ is covered by finite number of such
$a_{n}$-cubes, say $\\{\square_{n}^{i}\\}$. For each $\square_{n}^{i}$,
consider all connected components $\\{W^{ij}_{n}\\}$ of
$B_{r_{n}}(\imath^{-1}(\square_{n}^{i}))\subset\mathcal{X},$
where $B_{r}(S)$ denotes $r$-neighborhood of set $S$.
According to proposition 2, $\mathcal{X}$ is a length space. In particular,
each set $W^{ij}_{n}$ is open and contains a ball of radius $r_{n}$. Thus for
fixed $i$ the collection of open sets $\\{W^{ij}_{n}\\}$ is finite . Therefore
the set of all $\\{W^{ij}_{n}\\}$ for all $\\{\square^{i}_{n}\\}$ forms a
finite open cover of $\mathcal{X}$. For each $W^{ij}_{n}$ make an isometric
copy $\square^{ij}_{n}$ of $\square^{i}_{n}$ and fix an isometry
$\imath^{ij}_{n}\colon\square^{ij}_{n}\to\square^{i}_{n}$. The Euclidean
polyhedron $\mathcal{P}_{n}$, is glued from $\square^{ij}_{n}$ by the
following rule: glue $\square^{i_{1}j_{1}}_{n}$ to $\square^{i_{2}j_{2}}_{n}$
along $(\imath^{i_{2}j_{2}}_{n})^{-1}\circ\imath^{i_{1}j_{1}}_{n}$ iff
$W^{i_{1}j_{1}}_{n}\cap W^{i_{2}j_{2}}\not=\varnothing$. (The map
$(\imath^{i_{2}j_{2}}_{n})^{-1}\circ\imath^{i_{1}j_{1}}_{n}$ sends one of the
faces of $\square^{i_{1}j_{1}}_{n}$ isometrically to a face of
$\square^{i_{2}j_{2}}_{n}$.)
The constructed polyhedron $\mathcal{P}_{n}$ admits a natural piecewise linear
intrinsic isometry $\imath_{n}\colon\mathcal{P}_{n}\to\mathbb{E}^{d}$, defined
as $\imath_{n}(x)=\imath^{ij}_{n}(x)$ if $x\in\square^{ij}_{n}$. Further,
there is uniquely defined intrinsic isometry
$\varphi_{m,n}\colon\mathcal{P}_{m}\to\mathcal{P}_{n}$ for $m\geqslant n$
which satisfies $\imath_{m}=\imath_{n}\circ\varphi_{m,n}$ and
$\varphi_{m,n}(\square^{i^{\prime}j^{\prime}}_{m})\subset\square^{ij}_{n}\subset\mathcal{P}_{n}\
\ \Rightarrow\ \ W^{i^{\prime}j^{\prime}}_{m}\subset
W^{ij}_{n}\subset\mathcal{X}.$
Further, set $\psi_{n}\colon\mathcal{X}\to\mathcal{P}_{n}$ to be intrinsic
isometry which uniquely determined by $\imath_{n}\circ\psi_{n}=\imath$ and
$\psi_{n}(x)\in\square^{ij}_{n}\subset\mathcal{P}_{n}\ \ \Rightarrow\ \ x\in
W^{ij}_{n}\subset\mathcal{X}.$
Clearly, $\mathcal{P}_{n}$ together with $\varphi_{m,n}$ form an inverse
system and $\psi_{n}=\varphi_{m,n}\circ\psi_{m}$ for any pair $m\geqslant n$.
In order to prove that $\mathcal{X}=\varprojlim\mathcal{P}_{n}$, it only
remains to show that
$|xx^{\prime}|_{\mathcal{X}}\leqslant\lim_{n\to\infty}|\psi_{n}(x)\,\psi_{n}(x^{\prime})|$
$None$
for all $x,x^{\prime}\in\mathcal{X}$.
Given a subset $K\subset\mathcal{P}_{n}$, let us denote by
$K^{*}\subset\mathcal{X}$ the union of all $W^{ij}_{n}\subset\mathcal{X}$ such
that $\square^{ij}_{n}\cap K\not=\varnothing$. Clearly, if $K$ is connected
then so is $K^{*}$. More over, $\imath(K^{*})\subset
B_{r_{n}}(\imath_{n}(K))$. Thus, from proposition 2, we have that for any
$\varepsilon>0$ we can find $\delta>0$ such that
$r_{n}+\mathop{\rm diam}\nolimits K<\delta\ \ \Longrightarrow\ \ \mathop{\rm
diam}\nolimits K^{*}<\varepsilon$ $None$
Assume $(*)$ is wrong, then one can choose $x,x^{\prime}\in\mathcal{X}$ and
$\varepsilon,\ell>0$ so that
$\mathop{\rm
pull}\nolimits_{\imath,\varepsilon}(x,x^{\prime})>\ell>|\psi_{n}(x)\,\psi_{n}(x^{\prime})|_{\mathcal{P}_{n}}$
$None$
for all $n$. In particular, for all $n$ there is a path
$\gamma_{n}\colon[0,1]\to\mathcal{P}_{n}$ from $\psi_{n}(x)$ to
$\psi_{n}(x^{\prime})$ with length $<\ell$. Choose
$\delta=\delta(\imath,\varepsilon)$ as in proposition 2. Let $0=\nobreak
t_{0}<\nobreak t_{1}<\nobreak\dots<t_{m}=1$ be such that
$\mathop{\rm diam}\nolimits\gamma([{t_{i-1}},{t_{i}}])<\tfrac{\delta}{2}.$
$None$
Clearly one can assume that $m\leqslant 2{\hskip 0.5pt\cdot\nobreak\hskip
0.5pt}\lceil\tfrac{\ell}{\delta}\rceil$. For each $t_{i}$ choose a point
$x_{i}\in\nobreak\gamma(t_{i})^{*}\subset\mathcal{X}$; clearly
$|\imath(x_{i})\ \imath_{n}{\circ}\gamma(t_{i})|_{\mathbb{E}^{d}}<2{\hskip
0.5pt\cdot\nobreak\hskip 0.5pt}a_{n}.$ $None$
Note that $x_{i-1},x_{i}\in\gamma([t_{i-1},t_{i}])^{*}$. Thus,
$(\mathrel{\vbox{\offinterlineskip\hbox{${*}{*}$}\hbox{${*}{*}$}}})$ and
$(**)$ imply that
$|x_{i-1}x_{i}|<\mathop{\rm
diam}\nolimits\gamma_{n}([t_{i-1},t_{i}])^{*}<\varepsilon$
for all large $n$. Thus $x_{i}$ forms an $\varepsilon$-chain from $x$ to
$x^{\prime}$, and $(\mathrel{\vbox{\offinterlineskip\hbox{$\mkern
4.5mu{*}{*}$}\hbox{${*}{*}{*}$}}})$ implies
$\displaystyle\mathop{\rm pull}\nolimits_{\imath,\varepsilon}(x,x^{\prime})$
$\displaystyle\leqslant\sum_{i=1}^{m}|\imath(x_{i-1})\,\imath(x_{i})|<$
$\displaystyle<\sum_{i=1}^{m}|\imath_{n}{\circ}\gamma_{n}(t_{i-1})\
\imath_{n}{\circ}\gamma_{n}(t_{i})|+4{\hskip 0.5pt\cdot\nobreak\hskip
0.5pt}a_{n}{\hskip 0.5pt\cdot\nobreak\hskip
0.5pt}\lceil\tfrac{\ell}{\delta}\rceil<$ $\displaystyle<\ell+4{\hskip
0.5pt\cdot\nobreak\hskip 0.5pt}a_{n}{\hskip 0.5pt\cdot\nobreak\hskip
0.5pt}\lceil\tfrac{\ell}{\delta}\rceil$
which contradicts $(\mathrel{\vbox{\offinterlineskip\hbox{$\mkern
4.5mu{*}$}\hbox{${*}{*}$}}})$ for large enough $n$. ∎
Remark. In the constructed inverse system $(\varphi_{m,n},\mathcal{P}_{n})$,
the images of $\varphi_{m,n}$ form a $\sqrt{d}{\hskip 0.5pt\cdot\nobreak\hskip
0.5pt}a_{n}$-net in $\mathcal{P}_{n}$. It follows that the space $\mathcal{X}$
is isometric to the Gromov–Hausdorff limit of $\mathcal{P}_{n}$ (see also
section 2).
Proof of proposition 1. The inequality $d\leqslant m$ follow from trivial
statement 1. In the proof of the second part, we use the following two
statements:
1. 1.
Assume $\imath\colon\Omega\subset\mathbb{M}^{d}\to\mathbb{E}^{m}$ is an
intrinsic isometry, then it is a Lipschitz map for a Euclidean structure on
$\Omega$. Thus, according to Rademacher’s theorem (see [Federer, 3.1.6]) the
differential $d_{p}\imath$ is well defined almost all $p\in\Omega$.
2. 2.
For any curve $\gamma(t)$ with natural parameter in a metric space, we have
that for almost all values of parameter $t_{0}$ we have
$|\gamma({t_{0}})\,\gamma({t_{0}+\varepsilon})|=\varepsilon+o(\varepsilon),$
see [BBI, 2.7.5].
Let us denote by $\|{*}\|$ the norm which induces metric on $\mathbb{M}^{d}$.
Fix $u$ so that $\|u\|=1$. Consider pencil of lines of the form $p+u{\hskip
0.5pt\cdot\nobreak\hskip 0.5pt}t$ in $\Omega$. Two statements above imply that
$|d_{p}\imath(v)|\buildrel\hbox{\tiny\it
a.e.}\over{=\joinrel=}\nobreak\|{v}\|$ Hence we obtain parallelogram identity
$2{\hskip 0.5pt\cdot\nobreak\hskip
0.5pt}\\!\left(\|u\|^{2}+\|v\|^{2}\right)=\|u+v\|^{2}+\|u-v\|^{2}$
is satisfied for any two vectors $v$ and $w$. I.e. the norm $\|{*}\|$ is
Euclidean. ∎
## 4 About path isometries
In this section we will relate the notion of intrinsic isometry defined in
section 2 with more common (but less natural) notion of path and weak path
isometries.
4.1. Definition. Let $\mathcal{X}$ and $\mathcal{Y}$ be two length spaces. A
map $\imath\colon\mathcal{X}\to\mathcal{Y}$ is called
1. 1.
_path isometry_ if for any path $\gamma\colon[0,1]\to\mathcal{X}$ we have
$\mathop{\rm length}\nolimits\gamma=\mathop{\rm
length}\nolimits\imath\circ\gamma.$
2. 2.
_weak path isometry_ if for any rectifiable path
$\gamma\colon[0,1]\to\mathcal{X}$ we have
$\mathop{\rm length}\nolimits\gamma=\mathop{\rm
length}\nolimits\imath\circ\gamma.$
As it was noted in section 2, any intrinsic isometry is a path isometry (and
therefore, a weak path isometry). Next we will show that converse does not
hold. Similar counterexamples for weak path isometries are much simpler: one
can take a left-invariant sub-Riemannian metric $d$ on Heisenberg group $H$
then factorizing by center gives an weak path isometry
$(H,d)\to\mathbb{E}^{2}$ (which is not a path isometry and thus not an
intrinsic isometry).
4.2. Example. There is a length space $\mathcal{X}$ and a path isometry
$f\colon\mathcal{X}\to\mathbb{R}$ such that $f^{-1}(0)$ is connected
nontrivial subset.
Moreover, in such example the Lebesgue covering dimension of $f^{-1}(0)$ can
be made arbitrary large.
In particular, an analog of 1 does not hold for path isometries.
The following construction was suggested by D. Burago; it is based on two
ideas: (1) the construction in [BIS, 3.1], (2) the construction of _pseudo-
arc_ given in [Knaster] (see also the survey [Lewis] and references therein).
In fact, for the first part of theorem $f^{-1}(0)$ will be homeomorphic to a
pseudo-arc and for the second part $f^{-1}(0)$ will be homeomorphic to a
product of pseudo-arcs.
Proof. The space $\mathcal{X}$ will be a completion $\bar{\Gamma}$ of certain
metric graph555i.e., locally finite graph with intrinsic metric, such that
each edge is isometric to a real interval. $\Gamma$.
First let us describe the construction of $f$ modulo a construction of
$\Gamma$. Set $\grave{\Gamma}=\bar{\Gamma}\backslash\Gamma$. Consider map
$f\colon\bar{\Gamma}\to\mathbb{R}$, where $f(x)$ is the distance from $x$ to
$\grave{\Gamma}$. Then $f$ is a path isometry on $\Gamma$ and
$f(\grave{\Gamma})=0$. To finish that proof we will have to construct $\Gamma$
on such a way that
1. (i)
$\grave{\Gamma}$ is connected and contains more than one point;
2. (ii)
$f$ is a path isometry on whole $\bar{\Gamma}$ (not only on $\Gamma$).
$\mathbb{I}$$\mathbb{J}$$\varepsilon$Graph of an$\varepsilon$-crooked map.
Construction of $\Gamma$. For two real intervals $\mathbb{I}$ and
$\mathbb{J}$, a continuous onto map $h\colon\mathbb{I}\to\mathbb{J}$ will
called _$\varepsilon$ -crooked_ if for any two values $t_{1}<t_{2}$ in
$\mathbb{I}$ there are values $t_{1}<\nobreak t^{\prime}_{2}<\nobreak
t^{\prime}_{1}<\nobreak t_{2}$ such that
$|h(t_{i}^{\prime})-h(t_{i})|\leqslant\varepsilon$ for $i\in\\{1,2\\}$. The
existence of $\varepsilon$-crooked map for any given $\mathbb{I}$ and
$\varepsilon>0$ is easy to prove by induction on
$n=\lceil\tfrac{1}{\varepsilon}{\hskip 0.5pt\cdot\nobreak\hskip
0.5pt}{\mathop{\rm length}\nolimits\mathbb{J}}\rceil$.
Let us fix a sequence of real intervals $\mathbb{J}_{n}$ with short
$\tfrac{1}{2^{n}}$-crooked maps
$h_{n}\colon\mathbb{J}_{n}\to\mathbb{J}_{n-1}$. The topological inverse limit
$\mathbb{J}_{\infty}=\varprojlim\mathbb{J}_{n}$ is a connected compact which
has no nontrivial paths.
We can think of $\mathbb{J}_{n}$ as a (linear) metric graph with length of
each edge $\leqslant\tfrac{1}{2^{n}}$. Construct a graph $\Gamma$ from a
disjointed union $\sqcup_{n}\mathbb{J}_{n}$ by joining each vertex $v$ of
$\mathbb{J}_{n}$ to a vertex of $\mathbb{J}_{n-1}$ which is closest to
$h_{n}(v)$ by an edge of length $\tfrac{1}{2^{n}}$. Then $\grave{\Gamma}$ is
homeomorphic to $\mathbb{J}_{\infty}$; thus we get (i).
Let us denote by $\Gamma_{n}$ the finite subgraph of $\Gamma$ formed by all
vertexes in $\mathbb{J}_{1},\mathbb{J}_{2},\dots,\mathbb{J}_{n}$. Note that
there is a short map $\Gamma_{n}\to\Gamma_{n-1}$ which is identity on
$\Gamma_{n-1}$. It follows that for any path
$\alpha\colon[0,1]\to\bar{\Gamma}$ we have that the total length of
$\alpha\backslash\grave{\Gamma}$ is at least
$|\alpha(0)\,\alpha(1)|_{\bar{\Gamma}}$. Thus (ii) follows.
Second part. We construct a graph $\Gamma^{(m)}$ to make
$\grave{\Gamma}^{(m)}$ homeomorphic to a product on $m$ copies of
$\grave{\Gamma}$.
We will do the case $m=2$; the others are analogous. The set of vertexes of
$\Gamma^{(2)}$ is disjoint union $\sqcup_{n}(\mathop{\rm
Vert}\mathbb{J}_{n}\times\mathop{\rm Vert}\mathbb{J}_{n})$, where $\mathop{\rm
Vert}\mathbb{J}_{n}$ denotes the set of vertexes of $\mathbb{J}_{n}$. We
connect two vertexes $(x,y)\in\mathop{\rm Vert}\mathbb{J}_{n}\times\mathop{\rm
Vert}\mathbb{J}_{n}$ and $(x^{\prime},y^{\prime})\in\mathop{\rm
Vert}\mathbb{J}_{k}\times\mathop{\rm Vert}\mathbb{J}_{k}$ iff the pairs
$(x,x^{\prime})$ and $(y,y^{\prime})$ were connected in $\Gamma$; the length
of this edge must be maximum of lengths of edges $xx^{\prime}$ and
$yy^{\prime}$ (we assume that a vertex is connected to it-self by an edge of
length $0$).
Clearly, there is a homeomorphism
$\grave{\Gamma}^{(2)}\to\grave{\Gamma}\times\grave{\Gamma}$. Note that there
are two short coordinate projections
$\varsigma_{1},\varsigma_{2}\colon\Gamma^{(2)}\to\Gamma$. Thus for any path
$\alpha\colon[0,1]\to\bar{\Gamma}^{2}$, we have that total length of
$\alpha\backslash\grave{\Gamma}$ is at least as big as
$\max_{i}|\varsigma_{i}{\circ}\alpha(0)\,\varsigma_{i}{\circ}\alpha(1)|$. That
ensures that pulled back metric on $\grave{\Gamma}^{(2)}$ is bi-Lipschitz to
the product metric on $\grave{\Gamma}\times\grave{\Gamma}$. ∎
## 5 Comments and open questions
A length space $\mathcal{M}$ is called _Minkowski $d$-polyhedron_ if there is
a finite triangulation of $\mathcal{M}$ such that each simplex is isometric to
a simplex in a Minkowski space. Correspondingly, a compact metric space
$\mathcal{X}$ is called _pro-Minkowski space_ of rank $\leqslant d$ if it can
be presented as an inverse limit of Minkowski $d$-polyhedra.
5.1. Question. Is it true that any length space with Lebesgue’s covering
dimension $d$ is a pro-Minkowski space of rank $d$?
Or even more specific:
5.2. Question. Is it true that any metric space which homeomorphic to a disk
is a pro-Minkowski space of rank $2$?
One can reformulate it philosophically: Is there any essential difference
between Finsler metric and general metric on $n$-manifold? This question was
asked by D. Burago; it was also original motivation for this paper (see also a
related example [BIS, theorem 1]).
If one removes restriction on dimension, then the answer to the above question
is YES. Namely, the following exercise can be solved by using Kuratowski
embedding $x\mapsto\mathop{\rm dist}_{x}$.
5.3. Exercise. Show that any compact length space is an inverse limit of
Minkowski polyhedra $\mathcal{M}_{n}$ with $\mathop{\rm
dim}\nolimits\mathcal{M}_{n}\to\infty$.
5.4. Question. Is it true that any path isometry from a closed Euclidean ball
to Euclidean space is an intrinsic isometry?
## References
* [Akopyan] Akopyan, A. V., A piecewise linear analogue of Nash–Kuiper theorem, a preliminary version (in Russian) can be found on www.moebiuscontest.ru
* [Akopyan–Tarasov] Akopyan, A. V.; Tarasov, A. S. A constructive proof of Kirszbraun’s theorem. (Russian) Mat. Zametki 84 (2008), no. 5, 781–784.
* [BIS] Burago, D.; Ivanov, S.; Shoenthal, D. Two counterexamples in low dimensional length geometry, St.Petersburg Math. J. Vol. 19 (2008), No. 1, Pages 33–43
* [BBI] Burago, D.; Burago, Yu.; Ivanov, S., A course in metric geometry. Graduate Studies in Mathematics, 33. American Mathematical Society, Providence, RI, 2001. xiv+415 pp. ISBN: 0-8218-2129-6
* [Brehm] Brehm, U., Extensions of distance reducing mappings to piecewise congruent mappings on $R^{m}$. J. Geom. 16 (1981), no. 2, 187–193.
* [Federer] Federer, H., Geometric Measure Theory, Springer, 1969.
* [Knaster] Knaster, B. Un continu dont tout sous-continu est indécomposable. Fundamenta math. 3, 247–286 (1922).
* [Krat] Krat, S. Approximation Problems in Length Geometry, Thesis, 2005
* [Lewis] Lewis, Wayne The pseudo-arc. Bol. Soc. Mat. Mexicana (3) 5 (1999), no. 1, 25–77.
* [Gromov] Gromov, M., Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 9, Springer-Verlag, Berlin, 1986, x+363,
* [Zalgaller] Zalgaller, V. A. Isometric imbedding of polyhedra. (Russian) Dokl. Akad. Nauk SSSR 123 1958 599–601.
|
arxiv-papers
| 2010-03-29T17:41:55 |
2024-09-04T02:49:09.335486
|
{
"license": "Creative Commons Zero - Public Domain - https://creativecommons.org/publicdomain/zero/1.0/",
"authors": "Anton Petrunin",
"submitter": "Anton Petrunin",
"url": "https://arxiv.org/abs/1003.5621"
}
|
1003.5624
|
# Critical Field Strength in an Electroclinic Liquid Crystal Elastomer
Christopher M. Spillmann christopher.spillmann@nrl.navy.mil Center for
Bio/Molecular Science and Engineering, Code 6900, Naval Research Laboratory,
4555 Overlook Ave. SW, Washington, D.C., 22375 USA Amit V. Kapur Center for
Bio/Molecular Science and Engineering, Code 6900, Naval Research Laboratory,
4555 Overlook Ave. SW, Washington, D.C., 22375 USA Frank W. Bentrem Center
for Bio/Molecular Science and Engineering, Code 6900, Naval Research
Laboratory, 4555 Overlook Ave. SW, Washington, D.C., 22375 USA Marine
Geosciences Division, Naval Research Laboratory, Stennis Space Center,
Mississippi, 39529 USA Jawad Naciri Center for Bio/Molecular Science and
Engineering, Code 6900, Naval Research Laboratory, 4555 Overlook Ave. SW,
Washington, D.C., 22375 USA Banahalli R. Ratna Center for Bio/Molecular
Science and Engineering, Code 6900, Naval Research Laboratory, 4555 Overlook
Ave. SW, Washington, D.C., 22375 USA
###### Abstract
We elucidate the polymer dynamics of a liquid crystal elastomer based on the
time-dependent response of the pendent liquid crystal mesogens. The molecular
tilt and switching time of mesogens are analyzed as a function of temperature
and cross-linking density upon application of an electric field. We observe an
unexpected maximum in the switching time of the liquid crystal mesogens at
intermediate field strength. Analysis of the molecular tilt over multiple time
regimes correlates the maximum response time with a transition to entangled
polymer dynamics at a critical field strength.
###### pacs:
61.30.-v, 61.41.+e, 73.61.Ph
††preprint: preprint
In the 1970s, Garoff and Meyer were the first to describe a distinctive class
of liquid crystal (LC) that exhibits a chiral smectic-$A$ phase (Sm-$A^{*}$)
consisting of chiral molecules with a permanent dipole close to the chiral
center Garoff and Meyer (1977). Application of an electric field couples the
molecular dipole to the field and results in a molecular tilt ($\theta$) in a
plane orthogonal to the transverse component of the dipole. This response was
termed the electroclinic effect or soft mode, analogous to the softening of
the vibration mode in a ferroelectric material as it is cooled to the Curie
temperature Garoff and Meyer (1977), and has been well documented with several
LC moieties Abdulhalim and Moddel (1991); Andersson et al. (1988); Garoff and
Meyer (1979); Nishiyama et al. (1987).
The reorientation of the molecular dipoles and the concomitant tilt in
response to an applied electric field is resisted by the intrinsic viscosity
of the material. If the LC molecules are now tethered to a polymer backbone
(Fig. 1), the electroclinic response to an electric field is significantly
constrained. The molecules maintain much of the rotational freedom necessary
for the permanent electric dipoles to align with the applied field, but the
tilt response is significantly retarded by the entangled polymer backbone. The
significant influence of the polymer network was first noted by Gebhard and
Zentel when they studied an elastomeric ferroelectric LC system Gebhard and
Zentel (1999, 2000). They noted that the polymer network “counterforce” could
be represented as a spring resisting the molecular tilt in the Sm-$A^{*}$
phase Gebhard and Zentel (1999). However, the dynamic nature of the interplay
between the LC mesogens and the underlying polymer network remains largely
unexplored.
Figure 1: Schematic representation of smectic LC mesogens in an elastomeric
network.
In this Letter, we investigate the polymer dynamics of an electroclinic LC
elastomer as a function of cross-linking density, temperature, and field
strength. Since the LC molecules are coupled to the polymer network, the
molecular tilt response is used as an indirect probe to elucidate the polymer
dynamics in the presence of an applied field. We observe multiple time domains
in the tilt response and also a critical field strength corresponding to the
maximum switching time of the tethered LC mesogens. At this critical field
strength, the response time shows power-law dependence and supports the
presence of entanglement dynamics in the elastomer system at higher field
strengths. Thus, we demonstrate the ability to elucidate the polymer dynamics
of the system based on the reorientation of the pendent LC elements.
Based on Landau theory Clark and Lagerwall (1991), the electroclinic response
of a monomeric system to an applied field, $E$, at temperatures sufficiently
above the Sm-$A^{*}-$Sm-$C^{*}$ (chiral smectic-$C$) phase transition results
in a linear response of $\theta$ to $E$ and a characteristic switching time
$\tau$ independent of the applied field. As the system is cooled and
approaches the Sm-$A^{*}-$Sm-$C^{*}$ transition, both $\theta$ and $\tau$
increase at a given field strength. The switching time also shows a dependence
on $E$ such that it monotonically decreases with increasing field strength. It
has been previously demonstrated that the presence of a Sm-$A^{*}-$Sm-$C^{*}$
transition is not a necessary condition for the electroclinic effect Crawford
et al. (1995); Shashidhar et al. (2000). This holds true for the $\theta$ and
$\tau$ response of the monomeric mixture used in the current study (see
supplementary material, sup ).
Engineered coupling of electroclinic LCs to a polymer network allows for an
applied field to change the orientation of the pendent LC molecules and
subsequently alter the elastomer system in a reversible manner Hiraoka et al.
(2009); Kohler et al. (2005); Spillmann et al. (2007); Stannarius et al.
(2002). We have developed a free-standing electroclinic elastomer and examined
the molecular tilt, macroscopic actuation, and molecular packing of the system
Spillmann et al. (2007, 2008). We now examine the time-dependent molecular
response and for the first time use this information to shed light on the
polymer dynamics.
Synthesis of the polymerizable LC components, the diacrylate cross-linker, and
preparation of the elastomer has been previously reported Spillmann et al.
(2007); Artal et al. (2001). The structure of the molecular components and
details of the sample preparation are provided as Supplementary Material.
Three samples were prepared with 0, 2.5, and 5 mole percent (mol%) of the
cross-linker in EHC cells (E.H.C. Co., Ltd., Tokyo, Japan) to provide known
sample thickness and transparent indium tin oxide (ITO) electrodes for
electric field application. Electro-optic tilt angle and response time
measurements of the electroclinic samples follow the example of Lee and Patel
Lee and Patel (1989a, b) and are captured under temperature-controlled
conditions with a polarized light microscope. This approach is sensitive to
changes in the orientation of the rigid core (phenyl rings) of the
electroclinic molecules with respect to the crossed polarizers. The tilt angle
$\theta$ of the LC mesogens measured upon application of an electric field is
given by
$\theta=\frac{1}{4}\sin^{-1}\left(\frac{\Delta
I}{I_{\text{max}}-I_{\text{min}}}\right),$ (1)
where $I_{\text{min}}$ and $I_{\text{max}}$ are the minimum and maximum
transmitted light intensity with the LC molecular director positioned 0 and
$\pi/4$ radians with respect to the polarizer, respectively. With the sample
positioned $\pi/8$ radians with respect to the polarizer, a bipolar square
wave ($+V$ to $-V$) was applied at incrementing field strengths and the change
in transmitted light intensity through the sample, $\Delta I$, was measured.
The response time of the molecular tilt is defined as the time required for
the molecular tilt to change from 10% to 90% of the full tilt angle upon
reversal of the electric-field polarity.
The tilt angle of the electroclinic polymer and elastomer samples showed
characteristic curves that decreases with increasing temperature
(Supplementary Material, Fig. S4). However, the response time showed an
unexpected initial increase with increasing field strength, as shown in Fig.
2. The response time of elastomer samples reached a maximum at intermediate
field strengths of 15–25 V/$\mu$m and then began to decrease as $E$ was
further increased. The average field strength at which the maximum $\tau$ was
observed in the elastomer did not change significantly as a function of
temperature. In addition, no significant difference in $\tau$ or the field
strength at the maximum response time was observed when the cross-linking was
either 2.5 or 5 mol%. The polymer follows a notably different trend. At
$35^{\circ}$C, the peak response time was observed at 8 V/$\mu$m [Fig. 2(a),
black circles] and increasing the temperature to $40^{\circ}$C altered the
response to closely resemble the shape expected for a monomeric sample [Fig.
2(b), black circles]. At lower temperatures approaching the glass transition
of $\sim$28∘C, the effective polymer entanglement is large, increasing the
resistance to molecular tilting with in a pronounced increase in $\tau$. These
molecular observations provide critical insight of the elastomer response of
an electroclinic system to applied fields and expose the presence of two
opposing forces in the material: the force generated by the reorientation of
the LC mesogens in response to an applied field and the resistive force of the
polymer network.
Figure 2: Response time of electroclinic polymer (black circles) and elastomer
with 2.5 (red circles) or 5 mol% (green circles) cross-linking at (a)
$35^{\circ}$C and (b) $40^{\circ}$C. Figure 3: (a) Semi-log plot of $\theta$
versus time of elastomer with 5 mol% cross-linking at $35^{\circ}$C. Domains
of the response are identified as regions I, II, and III. Field strengths are
4, 8, 12, 16, 20, 26, and 40 V$/\mu$m. (b) $\beta$ plotted as a function of
field strength. Guide lines are overlaid for $10\leq E\leq 20$ and $20\leq
E\leq 40$.
The origin of the light intensity changes used to monitor $\theta$ and $\tau$
is the reorientation of the LC mesogens in response to an electric field.
Since these molecules are coupled to the polymer network, we explored the
possibility of using this information to probe the polymer dynamics of the
system by examining the time dependent response over a range of field
strengths. The use of the time-dependent tilt angle as an analysis tool for
the polymer dynamics of the system is analogous to a previously reported
method that utilized light scattering of nanocolloidal probes adsorbed to a
polymer matrix Sprakel et al. (2007). In our system, it is the coupling of the
LC mesogens to the network that serves as a sensitive measurement of the
polymer dynamics. The dynamics of a polymer system is defined by the
relationship of the mean square displacement of individual segments of a
polymer chain, $\langle r(t)^{2}\rangle$, to time $t$, as
$\langle r(t)^{2}\rangle=at^{\beta},$ (2)
where $a$ and $\beta$ are the creep coefficient and exponent, respectively de
Gennes (1979); McLeish (2002). Since the mesogens are coupled to the polymer
backbone and $\theta$ is a measure of the mesogen displacement, i.e., tilt, in
response to an applied field, we assume $\theta(t)$ is an indicator, and,
therefore, approximately equivalent to $r(t)$ in the current system. This
indirect measure of the polymer dynamics is most relevant at higher field
strengths, where the coupling of the polymer backbone to the mesogenic
response is realized.
Logarithmic analysis of the relationship between $\theta$ and time is used to
examine the polymer dynamics of the electroclinic elastomer as a function of
$E$. Figure 3(a) shows a semilog plot of $\theta$ developing as a function of
time at several electric field strengths upon switching the field polarity for
an elastomer sample. The tilt angle advances through three time regimes
(regions I, II, and III) and is described in terms of the interplay between
the polymer backbone and pendent LC mesogens. Region I is the initial
uncoupled response where the LC mesogens rapidly tilt with little resistance
from the polymer backbone. Region II is where the tilt response of the coupled
mesogens becomes significantly influenced by the underlying polymer network
prior to saturation of the tilt angle in region III.
The molecular tilt in region I follows a logarithmic relationship during the
initial response to the e-field. Region II was defined as the intermediate
regime between the initial logarithmic response and the saturated tilt angle
estimated from linear fits to region I and III in Fig. 3(a). A log−log
analysis of region II shows evidence of power-law dependence. Linear
regression [Fig. 3(a), red lines] provides detail about the creep exponent
$\beta$, which is expected to depend on field strength, temperature, and
cross-linking. The values of $\beta$ as a function of electric field strength
are shown in Fig. 3(b). We note that this analysis excluded the lowest field
strengths ($<10$ V/$\mu$m), where the coupling between the mesogenic tilt and
polymer backbone accommodates small tilt angles without a significant
influence from the polymer backbone. At field strengths ranging from
$\sim$10–20 V/$\mu$m, there is a linear dependence on the applied field. At
field strengths greater than 20 V/$\mu$m, $\beta$ saturates to a constant
value of $\sim$0.22. Various models of polymer systems predict power-law
displacement, including Rouse, reptation, and entanglement dynamics de Gennes
(1979); McLeish (2002). The saturating value of $\beta$ strongly suggests a
$t^{1/4}$ dependence, which indicates the polymer backbone may be constrained
by the dynamics of polymer entanglement in this time regime de Gennes (1979);
McLeish (2002). It is important to note that previous comparative study on
entangled and covalently cross-linked polymers had shown that the dynamics are
indentical for the two cases Nicolai et al. (1999). The transition to
entanglement dynamics at $\sim$20 V/$\mu$m occurs at the same field strength
that, on average, has the longest response time in the elastomer samples (see
Fig. 2). Therefore, the polymer dynamic analysis supports the notion of a
critical field strength at which the underlying polymer network begins to
significantly reorient in response to the external stimulus.
The presence of a maximum $\tau$ at an intermediate $E$ quantifies the ability
of the electrically-induced mesogenic tilt to affect the reorientation of the
polymer backbone at various field strengths. At the lowest applied fields, the
mesogenic tilt is only a few degrees and the stress on the polymer backbone is
negligible and $\tau$ relatively fast. At slightly higher field strengths, the
molecular tilt begins to impose a significant tug on the polymer backbone. The
restraint or creep of the polymer backbone is observed as an increase in
$\tau$. The force applied by the LC reorientation has increased sufficiently
to induce slow realignment of the polymer backbone from its equilibrium state.
This continues as the field strength and molecular tilt increase to the point
where $\tau$ reaches a maximum and the force of the mesogenic tilt and the
polymer resistive force appear to reach a balance. At the highest field
strengths, the force produced by the LC mesogens overcomes the resistance of
the polymer backbone and the maximum tilt angle is reached at progressively
faster response times. It should be pointed out that when a cross-linking
agent is present in the system, there is additional restoring force introduced
above and beyond the presence of the polymer backbone that accounts for the
persistent critical field strength observed in the elastomer samples. A
schematic representation of the polymer realignment and mechanical shear
induced by the LC mesogenic tilt at increasing field strengths is provided in
the Supplementary Information, Fig. S5.
The molecular tilt angle and response time of an electroclinic liquid crystal
have been examined as a monomeric system, a polymer, and a cross-linked
network. Following polymerization, the magnitude of the molecular tilt angle
remains the same with significant increases in both the response time and the
field strength required to elicit a response. The presence of a critical
intermediary field strength is identified where the characteristic response
time reaches a maximum. The mesogenic response as a function of applied field
is related to the interplay between the force arising from the molecular tilt
of the liquid crystal mesogens and the resistive force of the entangled
polymer backbone. Analysis of the time-dependent response over the range of
field strengths provides a novel approach to examine the polymer dynamics of
the system and suggests an entanglement regime above the critical field
strength.
This work was supported by the U. S. Office of Naval Research. F.W.B. also
received support from the Naval Research Laboratory Advanced Graduate Research
Program.
## References
* Garoff and Meyer (1977) S. Garoff and R. B. Meyer, Phys. Rev. Lett. 38, 848 (1977).
* Abdulhalim and Moddel (1991) I. Abdulhalim and G. Moddel, Liquid Crystals 9, 493 (1991).
* Andersson et al. (1988) G. Andersson, I. Dahl, W. Kuczynski, S. T. Lagerwall, K. Skarp, and B. Stebler, Ferroelectrics 84, 285 (1988).
* Garoff and Meyer (1979) S. Garoff and R. B. Meyer, Phys. Rev. A 19, 338 (1979).
* Nishiyama et al. (1987) S. Nishiyama, Y. Ouchi, H. Takezoe, and A. Fukuda, Jpn. J. Appl. Phys. 26, L1787 (1987).
* Gebhard and Zentel (2000) E. Gebhard and R. Zentel, Macromolecular Chem. Phys. 201, 902 (2000).
* Gebhard and Zentel (1999) E. Gebhard and R. Zentel, Liquid Crystals 26, 299 (1999).
* Clark and Lagerwall (1991) N. A. Clark and S. T. Lagerwall, _Ferroelectric Liquid Crystals: Principles, Properties, and Applications_ (Gordon and Breach Science Publishers, Philadelphia, 1991), vol. 7, chap. 1.
* Crawford et al. (1995) G. P. Crawford, J. Naciri, R. Shashidhar, P. Keller, and B. R. Ratna, Mol. Cryst. Liq. Cryst. 263, 223 (1995).
* Shashidhar et al. (2000) R. Shashidhar, J. Naciri, and B. R. Ratna, Adv. Chem. Phys. 113, 51 (2000), URL http://dx.doi.org/10.1002/9780470141724.ch2.
* (11) See supplementary material at http://link.aps.org/supplemental/10.1103/PhysRevLett.000.000000 for molecular structures, LC textures and alignment, monomeric response to an applied field, temperature-dependent response of the elastomer, and a schematic of the polymer realignment as an external field is applied.
* Hiraoka et al. (2009) K. Hiraoka, M. Kobayasi, R. Kazama, and H. Finkelmann, Macromolecules 42, 5600 (2009).
* Kohler et al. (2005) R. Kohler, R. Stannarius, C. Tolksdorf, and R. Zentel, Appl. Phys. A: Mater. Sci. Proc. 80, 381 (2005).
* Stannarius et al. (2002) R. Stannarius, R. Köhler, U. Dietrich, M. Lösche, C. Tolksdorf, and R. Zentel, Phys. Rev. E 65, 41707 (2002).
* Spillmann et al. (2007) C. M. Spillmann, B. R. Ratna, and J. Naciri, Appl. Phys. Lett. 90, 021911 (2007).
* Spillmann et al. (2008) C. M. Spillmann, J. H. Konnert, J. R. Deschamps, J. Naciri, and B. R. Ratna, Chem. Mater. 20, 6130 (2008).
* Artal et al. (2001) C. Artal, M. B. Ros, J. L. Serrano, N. Pereda, J. Etxebarria, C. L. Folcia, and J. Ortega, Macromolecules 34, 4244 (2001).
* Lee and Patel (1989a) S. D. Lee and J. S. Patel, Appl. Phys. Lett. 54, 1653 (1989a).
* Lee and Patel (1989b) S. D. Lee and J. S. Patel, Appl. Phys. Lett. 55, 122 (1989b).
* Sprakel et al. (2007) J. Sprakel, N. A. M. Besseling, F. A. M. Leermakers, and M. A. C. Stuart, Phys. Rev. Lett. 99, 104504 (pages 4) (2007), URL http://link.aps.org/abstract/PRL/v99/e104504.
* de Gennes (1979) P.-G. de Gennes, _Scaling Concepts in Polymer Physics_ (Cornell University Press, Ithica, 1979), 1st ed., ISBN 080141203X, URL http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20%&path=ASIN/080141203X.
* McLeish (2002) T. McLeish, Adv. Phys. 51, 1379 (2002).
* Nicolai et al. (1999) T. Nicolai, F. Prochazka, and D. Durand, Phys. Rev. Lett. 82, 863 (1999).
|
arxiv-papers
| 2010-03-29T17:49:20 |
2024-09-04T02:49:09.342380
|
{
"license": "Public Domain",
"authors": "Christopher M. Spillmann, Amit V. Kapur, Frank W. Bentrem, Jawad\n Naciri, and Banahalli R. Ratna",
"submitter": "Frank Bentrem",
"url": "https://arxiv.org/abs/1003.5624"
}
|
1003.5638
|
# Integral representation of Skorokhod reflection
Venkat Anantharam Research supported by the ARO MURI grant W911NF-08-1-0233,
Tools for the Analysis and Design of Complex Multi-Scale Networks, and by the
NSF grants CCF-0500234, CCF-0635372, CNS-0627161 and CNS-0910702. Takis
Konstantopoulos Research supported by an EPSRC grant.
(29 March 2010)
###### Abstract
We show that a certain integral representation of the one-sided Skorokhod
reflection of a continuous bounded variation function characterizes the
reflection in that it possesses a unique maximal solution which solves the
Skorokhod reflection problem.
## 1 Introduction
The Skorokhod reflection problem has a long history. Skorokhod [10] introduced
it as a method for representing a diffusion process with a reflecting boundary
at zero. Given a continuous function $X:[0,\infty)\to\mathbb{R}$, the standard
Skorokhod reflection problem seeks to find $(Q(t),t\geq 0)$ and a continuous,
nondecreasing function $Y:[0,\infty)\to\mathbb{R}_{+}$ with $Y(0)=0$, such
that $Q(t):=X(t)+Y(t)\geq 0$ for all $t$, and $\int_{0}^{\infty}Q(s)dY(s)=0$.
Intuitively, the latter expresses the idea that $Y$ can increase only at
points $t$ such that $X(t)+Y(t)=0$. Skorokhod [10] showed that there is only
one such $Y$, namely, $Y(t)=-\inf_{0\leq s\leq t}(X(s)\wedge 0)$ and thus
$Q(t)=X(t)\vee\sup_{0\leq s\leq t}(X(t)-X(s)).$
We use the standard notation $a\vee b:=\max(a,b)$, $a\wedge b:=\min(a,b)$. The
mapping $X\mapsto Q$ is referred to as the (one-sided) Skorokhod reflection
mapping and has now become a standard tool in probability theory and other
areas. As an example, we recall that if $X$ is the path of a Brownian motion
then $Q$ is a reflecting Brownian motion and $Q(t)$ has the same distribution
as $|X(t)|$ for all $t\geq 0$ [3, 9]. Several extensions of the Skorokhod
reflection mapping exist generalizing the range of $X$ (see, e.g., [11]) or
its domain (see, e.g., [1]).
The question resolved in this paper was motivated by an application of the
Skorokhod reflection in stochastic fluid queues [7, 6]. Suppose that $A,C$ are
two jointly stationary and ergodic random measures defined on a common
probability space $(\Omega,\mathscr{F},\mathbb{P})$, with intensities $a,c$,
respectively, such that $a<c$. Then there exists a unique stationary and
ergodic stochastic process $(Q(t),t\in\mathbb{R})$ defined on
$(\Omega,\mathscr{F},\mathbb{P})$ such that, for all $t_{0}\in\mathbb{R}$,
$(Q(t_{0}+t),t\geq 0)$ is the Skorokhod reflection of
$(Q(t_{0})+A(t_{0},t_{0}+t]-C(t_{0},t_{0}+t],~{}t\geq 0)$. In addition, if the
random measures $A,C$ have no atoms then
$Q(t)=\int_{-\infty}^{t}{\text{\Large$\mathfrak{1}$}}(Q(s)>C(s,t])~{}dA(s),$
(1)
for all $t\in\mathbb{R}$, $\mathbb{P}$-almost surely. The latter equation was
called an “integral representation” of Skorokhod reflection and extensions of
it were formulated and proved in [6]. The integral representation was found to
be useful in several applications, e.g. (i) in deriving the so-called Little’s
law for stochastic fluid queues [2], stating that
$\mathbb{E}[Q(0)]=(a/c)\mathbb{E}_{A}[Q(0)]$, where $\mathbb{E}_{A}$ is
expectation with respect to the Palm measure [4] of $\mathbb{P}$ with respect
to $A$, and (ii) in deriving the form of the stationary distribution of a
stochastic process derived from the local time of a Lévy process [5].
In an open problems session of the workshop on “New Topics at the Interface
Between Probability and Communications” [8], the second author asked whether
and in what sense (1) characterizes Skorokhod reflection. The question will be
made precise in Section 2 below, where the main theorem, Theorem 1, which
answers the question, is stated. In Section 3 the integral representation is
explicitly proved, along with some auxiliary results. Finally, in Section 4 a
proof of Theorem 1 is given.
## 2 The problem
Consider a locally finite signed measure $X$ on the Borel sets of
$\mathbb{R}$. Assume that $X$ has no atoms, i.e. $X(\\{t\\})=0$ for all
$t\in\mathbb{R}$. Define
$Q^{*}(t):=\sup_{0\leq s\leq t}X(s,t],\quad t\geq 0,$ (2)
where $X(s,t]=X((s,t])$ is the value of $X$ at the interval $(s,t]$. 111Since
$X,A,C$ are assumed to have no atoms, we may as well write $X[s,t]$ or
$X(s,t)$ instead of $X(s,t]$, and likewise for $A$ and $C$, but we have chosen
the notation to be consistent with possible generalizations. In particular,
$Q^{*}(0)=0.$
Let $X(t):=X(0,t]$ and write (2) as
$Q^{*}(t)=X(t)-\inf_{0\leq s\leq t}X(s).$
The standard terminology [3, 12] is that $Q^{*}$ solves the Skorokhod
reflection problem for the function $t\mapsto X(t)$.
Decompose $X$ as the difference of two locally finite nonnegative measures
$A$, $C$, without atoms, i.e. write
$X=A-C.$ (3)
We stress that $A$, $C$ are not necessarily the positive and negative parts of
$X$. In other words, the decomposition is not unique. For instance, we can add
an arbitrary locally finite nonnegative measure without atoms to both $A$ and
$C$.
In [6] it was proved that (2) also satisfies the fixed point equation referred
to as “integral representation” of the reflected process:
$Q(t)=\int_{0}^{t}{\text{\Large$\mathfrak{1}$}}(Q(s)>C(s,t])~{}dA(s),\quad
t\geq 0.$ (4)
A simpler version of this appeared earlier in [7]; this version was concerned
with the case where $C$ is a multiple of the Lebesgue measure. In an open
problems session of the workshop on “New Topics at the Interface Between
Probability and Communications” [8], the second author asked whether and in
what sense (4) implies (2); the question was actually asked for the special
case where $C$ is a multiple of the Lebesgue measure.
In this note we answer this question by proving the following:
###### Theorem 1.
Let $A$, $C$ be locally finite Borel measures on $\mathbb{R}_{+}=[0,\infty)$
without atoms and consider the integral equation (4). This integral equation
admits a unique maximal solution, i.e. a solution which pointwise dominates
any other solution. Further, this maximal solution is precisely the function
$Q^{*}$ defined by (2).
We proceed as follows. First, we present some auxiliary results and also give
a proof of (2) $\Rightarrow$ (4) which is different from the one found in [6].
Then we prove Theorem 1 by a successive approximation scheme and by proving a
number of lemmas.
## 3 Proof of the integral representation and auxiliary results
We first exhibit some properties of $Q^{*}$, defined by (2), and also show
that $Q^{*}$ satisfies the integral equation (4). The proof of the latter in
the special case where $C$ is a multiple of the Lebesgue measure can be found
in [7, Lemma 1] and in [2, §3.5.3]. A more general case is dealt with in [6,
Theorem 1]. We give a different proof in Proposition 1 below. The lemmas below
are straightforward and well-known but we give proofs for completeness. As
before, $X$ is a locally finite Borel measure without atoms and $X=A-C$ is a
decomposition as the difference of two nonnegative locally finite Borel
measures without atoms. We set
$A(t):=A(0,t],\quad C(t):=C(0,t].$
###### Lemma 1.
If $0\leq s\leq s^{\prime}\leq t$ and if $Q^{*}(s)>C(s,t]$ then
$Q^{*}(s^{\prime})>C(s^{\prime},t]$.
###### Proof.
Assume that $C(s,t]<Q^{*}(s)=\sup_{0\leq u\leq s}X(u,s]$. This is equivalent
to
$\displaystyle C(t)-C(s)$ $\displaystyle<\sup_{0\leq u\leq
s}\\{A(s)-A(u)-(C(s)-C(u))\\}$ $\displaystyle=A(s)+\sup_{0\leq u\leq
s}\\{-A(u)+C(u)\\}-C(s),$ $\displaystyle\text{that is, }\quad C(t)$
$\displaystyle<A(s)+\sup_{0\leq u\leq s}\\{-A(u)+C(u)\\}.$
The right-hand side of the latter is increasing in $s$ and so replacing $s$ by
a larger $s^{\prime}$ we obtain
$C(t)<A(s^{\prime})+\sup_{0\leq u\leq s^{\prime}}\\{-A(u)+cu\\},$
which is equivalent to $Q^{*}(s^{\prime})>C(s^{\prime},t]$. ∎
###### Lemma 2.
$Q^{*}$ satisfies
$Q^{*}(t)=\sup_{s\leq u\leq t}X(u,t]\vee(Q^{*}(s)+X(s,t]),\quad 0\leq s\leq
t.$ (5)
###### Proof.
We show that the right-hand side of (5) equals the left-hand side.
$\displaystyle\sup_{s\leq u\leq t}X(u,t]\vee(Q^{*}(s)+X(s,t])$
$\displaystyle=\sup_{s\leq u\leq t}X(u,t]\vee\\{(\sup_{0\leq u\leq
s}X(u,s])+X(s,t]\\}$ $\displaystyle=\sup_{s\leq u\leq t}X(u,t]\vee\sup_{0\leq
u\leq s}\\{X(u,s]+X(s,t]\\}$ $\displaystyle=\sup_{s\leq u\leq
t}X(u,t]\vee\sup_{0\leq u\leq s}X(u,t]$ $\displaystyle=\sup_{0\leq u\leq
t}X(u,t]=Q^{*}(t).$
###### Lemma 3.
If $0\leq s\leq t$ and if $Q^{*}(s)\geq C(s,t]$ then
$Q^{*}(t)=Q^{*}(s)+X(s,t]$.
###### Proof.
We use equation (5), rewritten as follows:
$Q^{*}(t)=\sup_{s\leq u\leq t}\big{\\{}X(u,t]\vee(Q^{*}(s)+X(s,t])\big{\\}}.$
(6)
Suppose $0\leq s\leq u\leq t$ and that $Q^{*}(s)\geq C(s,t]$. Then
$Q^{*}(s)\geq C(s,u]$ and so
$\displaystyle Q^{*}(s)+X(s,t]$ $\displaystyle\geq C(s,u]+X(s,t]$
$\displaystyle=C(s,u]+A(s,t]-C(s,t]$ $\displaystyle=A(s,t]-C(u,t]$
$\displaystyle\geq A(u,t]-C(u,t]=X(u,t],$
and this inequality implies that the term $X(u,t]$ inside the bracket of the
right-hand side of (6) is not needed. Hence $Q^{*}(t)=Q^{*}(s)+X(s,t]$, which
is what we wanted to prove. ∎
Define next
$\sigma^{*}(t):=\sup\\{0\leq s\leq t:~{}Q^{*}(s)\leq C(s,t]\\}.$
By Lemma 1,
$\displaystyle Q^{*}(s)$ $\displaystyle\leq C(s,t],\quad\text{ if }0\leq
s\leq\sigma^{*}(t),$ (7a) $\displaystyle Q^{*}(s)$
$\displaystyle>C(s,t],\quad\text{ if }\sigma^{*}(t)<s\leq t,$ (7b)
provided that the last inequality is non-vacuous. Since the function $Q^{*}$
is nonnegative and continuous, we also have
$Q^{*}(\sigma^{*}(t))=C(\sigma^{*}(t),t].$
###### Proposition 1.
If $X$ is a locally finite signed Borel measure on $[0,\infty)$ without atoms
and if $X=A-C$ is any decomposition of $X$ as the difference of two
nonnegative locally finite Borel measures without atoms, then the function
$Q^{*}$ defined by (2) satisfies (4).
###### Proof.
By Lemma 3, and the last display,
$\displaystyle Q^{*}(t)$
$\displaystyle=Q^{*}(\sigma^{*}(t))+A(\sigma^{*}(t),t]-C(\sigma^{*}(t),t]$
$\displaystyle=A(\sigma^{*}(t),t]$
$\displaystyle=\int_{\sigma^{*}(t)}^{t}dA(s)$
$\displaystyle=\int_{0}^{t}{\text{\Large$\mathfrak{1}$}}(Q^{*}(s)>C(s,t])~{}dA(s),$
which is the integral representation formula (4). Note that, to obtain the
last equality in the last display, we used (7a)-(7b). ∎
## 4 Proof of Theorem 1
A priori, it is not clear that (4) admits a maximal solution and, even if it
does, whether it satisfies (2). We shall show the validity of these claims in
the sequel.
We fix two locally finite measures $A$ and $C$ and define the map $\Theta$ on
the set of nonnegative measurable functions by
$\Theta(Q)(t):=\int_{0}^{t}{\text{\Large$\mathfrak{1}$}}(Q(s)>C(s,t])~{}dA(s),\quad
t\geq 0.$ (8)
The integral equation (4) then reads
$Q=\Theta(Q).$
We observe that $\Theta$ is increasing:
If $Q\leq\widetilde{Q}$ then $\Theta(Q)\leq\Theta(\widetilde{Q})$. (9)
Here, and in the sequel, given two functions $f,g:[0,\infty)\to\mathbb{R}$, we
write $f\leq g$ to mean that $f(t)\leq g(t)$ for all $t\geq 0$. To see that
(8) holds, simply observe that $Q\leq\widetilde{Q}$ implies
${\text{\Large$\mathfrak{1}$}}(Q(s)>C(s,t])\leq{\text{\Large$\mathfrak{1}$}}(\widetilde{Q}(s)>C(s,t])$
for all $0\leq s\leq t$.
Define next a sequence of functions $(Q_{k},k=0,1,2,\ldots)$ by first letting
$Q_{0}:=\infty,$
and then, recursively,
$Q_{k+1}:=\Theta(Q_{k}),\quad k\geq 0.$
Clearly, $Q_{1}(t)=\int_{0}^{t}dA(s)=A(t)$. So $Q_{0}\geq Q_{1}$. Since
$\Theta$ is an increasing map, we see that,
$Q_{k}\geq Q_{k+1}\geq 0,\quad k\geq 0.$
We can then define
$Q_{\infty}(t):=\lim_{k\to\infty}Q_{k}(t).$
###### Lemma 4.
If $Q=\Theta(Q)$ then $Q\leq Q_{\infty}$. Furthermore,
$Q^{*}\leq Q_{\infty}.$
###### Proof.
Suppose that $Q$ satisfies $Q=\Theta(Q)$. Since the integrand in the right-
hand side of (8) is $\leq 1$, we have $Q(t)\leq A(t)$ for all $t\geq 0$.
Letting $\Theta^{(k)}$ be the $k$-fold composition of $\Theta$ with itself, we
have
$Q=\Theta^{(k)}(Q)\leq\Theta^{(k)}(A)=Q_{k},$
and so $Q\leq Q_{\infty}$. In particular, Proposition 1 states that
$Q^{*}=\Theta(Q^{*})$. Hence $Q^{*}\leq Q_{\infty}$. ∎
However, it is not yet clear at this point that $Q_{\infty}$ is a fixed point
of $\Theta$. We can only show that
$Q_{\infty}\geq\Theta(Q_{\infty}).$
Indeed, $Q_{\infty}\leq Q_{k}$ for all $k$, and so
${\text{\Large$\mathfrak{1}$}}(Q_{\infty}(s)>C(s,t])\leq{\text{\Large$\mathfrak{1}$}}(Q_{k}(s)>C(s,t])$,
for all $0\leq s\leq t$, implying that
$\Theta(Q_{\infty})\leq\Theta(Q_{k})=Q_{k+1}$, and, by taking limits, that
$\Theta(Q_{\infty})\leq Q_{\infty}$.
###### Definition 1 (Regulating functions).
Consider functions $B:[0,\infty)\to[0,\infty)$ which are continuous,
nondecreasing, with $B(0)=0$, such that $X(0,t]+B(t)\geq 0$ for all $t\geq 0$.
Call these functions regulating functions of $X$. The set of regulating
functions is denoted by $\operatorname{\mathcal{R}}(X)$.
We define a mapping
$\displaystyle\Phi:$
$\displaystyle\operatorname{\mathcal{R}}(X)\to\operatorname{\mathcal{R}}(X)$
(10)
in two steps: Given $B\in\operatorname{\mathcal{R}}(X)$, first define
$\sigma_{B}(t):=\sup\\{0\leq s\leq t:~{}A(s)+B(s)-C(t)\leq 0\\},\quad t\geq
0.$
Then let
$\Phi(B)(t):=B(\sigma_{B}(t)),\quad t\geq 0.$
We actually need to show that what is claimed in (10) holds. Namely:
###### Lemma 5.
If $B\in\operatorname{\mathcal{R}}(X)$ then
$\Phi(B)\in\operatorname{\mathcal{R}}(X)$.
###### Proof.
Clearly, $\sigma_{B}(\cdot)$ is nondecreasing. Since $B$ is nondecreasing, it
follows that $\Phi(B)=B\raisebox{0.43057pt}{\scriptsize$\circ$}\sigma_{B}$ is
nondecreasing. Also, $\Phi(B)(0)=B(\sigma_{B}(0))=B(0)=0$. From the continuity
of $A$, $B$ and the definition of $\sigma_{B}$, we have
$A(\sigma_{B}(t))+B(\sigma_{B}(t))=C(t),\quad t\geq 0.$ (11)
We also have,
$\displaystyle A(t)+\Phi(B)(t)-C(t)$
$\displaystyle=A(t)+B(\sigma_{B}(t))-C(t)$
$\displaystyle=[A(t)-A(\sigma_{B}(t))]+[A(\sigma_{B}(t))+B(\sigma_{B}(t))-C(t)]$
$\displaystyle=A(t)-A(\sigma_{B}(t))\geq 0,$
where we used (11) in the third step. It remains to show that $\Phi(B)(\cdot)$
is continuous. Note that $\sigma_{B}(\cdot)$ need not be continuous. However,
$C(\cdot)$ is a continuous function and so, by (11), $t\mapsto
A(\sigma_{B}(t))+B(\sigma_{B}(t))$ is continuous. Hence
$[A(\sigma_{B}(t+))-A(\sigma_{B}(t-)]+[B(\sigma_{B}(t+))-B(\sigma_{B}(t-))]=0,\quad\text{for
all $t$}.$
Since $A(\sigma_{B}(\cdot))$ and $B(\sigma_{B}(\cdot))$ are both
nondecreasing, it follows that $A(\sigma_{B}(t+))-A(\sigma_{B}(t-)\geq 0$ and
$B(\sigma_{B}(t+))-B(\sigma_{B}(t-))\geq 0$ and, since their sum is zero, they
are both zero, implying that $A(\sigma_{B}(\cdot))$ and $B(\sigma_{B}(\cdot))$
are continuous. ∎
An immediate property of $\Phi$ is that
$\Phi(B)\leq B\quad\text{for all $B\in\operatorname{\mathcal{R}}(X)$}.$ (12)
Indeed, for all $t\geq 0$, $\sigma_{B}(t)\leq t$ and so $B(\sigma_{B}(t))\leq
B(t)$.
Starting with the function
$B_{1}(t):=C(t),\quad t\geq 0,$ (13)
we recursively define
$B_{k+1}:=\Phi(B_{k}),\quad k\geq 1.$ (14)
Therefore
$B_{1}\geq B_{2}\geq\cdots\geq B_{k}\downarrow B_{\infty},\quad\text{ as
}k\to\infty,$ (15)
where the inequalities and the limit are pointwise.
###### Lemma 6.
The function $B_{\infty}$, defined via (13), (14) and (15), is a member of the
class $\operatorname{\mathcal{R}}(X)$.
###### Proof.
$B_{\infty}$ is nondecreasing since all the $B_{k}$ are nondecreasing. Also,
$B_{\infty}(0)=0$. Since for all $k$, $A+B_{k}-C\geq 0$, we have
$A+B_{\infty}-C\geq 0$. We proceed to show that $B_{\infty}$ is a continuous
function. We observe that, for $0\leq t\leq t^{\prime}$,
$\displaystyle|\Phi(B)(t^{\prime})-\Phi(B)(t)|$
$\displaystyle=|B(\sigma_{B}(t^{\prime}))-B(\sigma_{B}(t))|$
$\displaystyle=B(\sigma_{B}(t^{\prime}))-B(\sigma_{B}(t))$ $\displaystyle\leq
A(\sigma_{B}(t^{\prime}))-A(\sigma_{B}(t))+B(\sigma_{B}(t^{\prime}))-B(\sigma_{B}(t))$
$\displaystyle=[A(\sigma_{B}(t^{\prime}))+B(\sigma_{B}(t^{\prime}))]-[A(\sigma_{B}(t))+B(\sigma_{B}(t))]$
$\displaystyle=C(t^{\prime})-C(t),$
where we again used (11). It follows that the family of functions
$\\{\Phi(B),B\in\operatorname{\mathcal{R}}(X)\\}$ is uniformly bounded and
equicontinuous on each compact interval of the real line. By the Arzelà-Ascoli
theorem, the family is compact and therefore $B_{\infty}$ is continuous. We
have established that $B_{\infty}\in\operatorname{\mathcal{R}}(X)$. ∎
We now claim that $B_{\infty}$ is a fixed point of $\Phi$.
###### Lemma 7.
$\Phi(B_{\infty})=B_{\infty}$.
###### Proof.
By definition,
$\Phi(B_{\infty})(t)=B_{\infty}(\sigma_{B_{\infty}}(t)),$
where
$\sigma_{B_{\infty}}(t)=\sup\\{0\leq s\leq t:~{}A(s)+B_{\infty}(s)\leq
C(t)\\}.$
Now, since $B_{k}\geq B_{k+1}$ for all $k\geq 1$, it follows that
$\sigma_{B_{k}}\leq\sigma_{B_{k+1}}$ for all $k\geq 1$, and so
$\sigma_{L}(t):=\lim_{k\to\infty}\sigma_{B_{k}}(t)$
is well-defined. Since $B_{k}\geq B_{\infty}$ for all $k\geq 1$, we have
$\sigma_{B_{k}}\leq\sigma_{B_{\infty}}$. Taking limits, we find
$\sigma_{L}\leq\sigma_{B_{\infty}}.$
Using the last two displays and the fact that $B_{k}$ and $B_{\infty}$ are
nondecreasing, we have
$\displaystyle\Phi(B_{\infty})(t)=B_{\infty}(\sigma_{B_{\infty}}(t))$
$\displaystyle\geq B_{\infty}(\sigma_{L}(t))$
$\displaystyle=\lim_{k\to\infty}B_{k}(\sigma_{L}(t))$
$\displaystyle\geq\lim_{k\to\infty}B_{k}(\sigma_{B_{k}}(t))$
$\displaystyle=\lim_{k\to\infty}B_{k+1}(t)=B_{\infty}(t).$
By inequality (12), $\Phi(B)\leq B$ for all
$B\in\operatorname{\mathcal{R}}(X)$ and since, by Lemma 6,
$B_{\infty}\in\operatorname{\mathcal{R}}(X)$, it follows that we also have
$B_{\infty}\leq\Phi(B_{\infty})$. Therefore $B_{\infty}=\Phi(B_{\infty})$, as
claimed. ∎
###### Lemma 8.
Consider the function $Q^{*}$ defined by (2) and define a function $U$ by
$U(t):=Q^{*}(t)-X(0,t],\quad t\geq 0.$
Then
* (i)
$U\in\operatorname{\mathcal{R}}(X)$.
* (ii)
$U=\Phi(U)$.
###### Proof.
(i) We have $X(0,t]+U(t)=Q^{*}(t)\geq 0$ for all $t$. Using (2) and (3) we see
that
$U(t)=\sup_{0\leq s\leq t}\\{-A(s)+C(s)\\}.$ (16)
Therefore, $U(0)=0$, and $U$ is a continuous and nondecreasing. We conclude
that $U\in\operatorname{\mathcal{R}}(X)$. To prove (ii), recall that
$\Phi(U)=U\raisebox{0.43057pt}{\scriptsize$\circ$}\sigma_{U}$ where
$\sigma_{U}(t)=\sup\\{0\leq s\leq t:~{}A(s)+U(s)\leq C(t)\\}.$
Splitting the supremum in (16) in two parts, we obtain
$\displaystyle U(t)$ $\displaystyle=\sup_{0\leq
s\leq\sigma_{U}(t)}\\{-A(s)+C(s)\\}\vee\sup_{\sigma_{U}(t)\leq s\leq
t}\\{-A(s)+C(s)\\}.$
$\displaystyle=U(\sigma_{U}(t))\vee\sup_{\sigma_{U}(t)\leq s\leq
t}\\{-A(s)+C(s)\\}.$
For $s\geq\sigma_{U}(t)$, we have $A(s)+U(s)\geq C(t)$, i.e. $-A(s)+C(s)\leq
U(s)-C(s,t]$. Therefore
$\displaystyle U(t)$ $\displaystyle\leq
U(\sigma_{U}(t))\vee\sup_{\sigma_{U}(t)\leq s\leq t}\\{U(s)-C(s,t]\\}$
$\displaystyle=U(\sigma_{U}(t)=\Phi(U)(t).$
Thus, $U\leq\Phi(U)$. On the other hand, since
$U\in\operatorname{\mathcal{R}}(X)$, we have $\Phi(U)\leq U$, by (12). ∎
###### Lemma 9.
Let $B\in\operatorname{\mathcal{R}}(X)$ be any fixed point of $\Phi$. Then
$B\leq U$.
###### Proof.
Since $B=\Phi(B)=B\raisebox{0.43057pt}{\scriptsize$\circ$}\sigma_{B}$ we have
$B=B\raisebox{0.43057pt}{\scriptsize$\circ$}\sigma_{B}^{(k)}$
where
$\sigma_{B}^{(k)}:=\underbrace{\sigma_{B}\raisebox{0.43057pt}{\scriptsize$\circ$}\cdots\raisebox{0.43057pt}{\scriptsize$\circ$}\sigma_{B}}_{k\text{
times}}$. Since
$t\geq\sigma_{B}(t)\geq\sigma_{B}\raisebox{0.43057pt}{\scriptsize$\circ$}\sigma_{B}(t)\geq\cdots\geq\sigma_{B}^{(k)}(t),$
we may define
$\sigma_{B}^{(\infty)}(t):=\lim_{k\to\infty}\sigma_{B}^{(k)}(t).$
By the continuity of $B$,
$B=B\raisebox{0.43057pt}{\scriptsize$\circ$}\sigma_{B}^{(\infty)}.$ (17)
On the other hand, (11) gives
$A\raisebox{0.43057pt}{\scriptsize$\circ$}\sigma_{B}^{(k+1)}+B\raisebox{0.43057pt}{\scriptsize$\circ$}\sigma_{B}^{(k+1)}=C\raisebox{0.43057pt}{\scriptsize$\circ$}\sigma_{B}^{(k)},\quad
k\geq 1.$
Taking the limit as $k\to\infty$, and using the continuity of $A$, $B$ and
$C$, we have
$A\raisebox{0.43057pt}{\scriptsize$\circ$}\sigma_{B}^{(\infty)}+B\raisebox{0.43057pt}{\scriptsize$\circ$}\sigma_{B}^{(\infty)}=C\raisebox{0.43057pt}{\scriptsize$\circ$}\sigma_{B}^{(\infty)}.$
Since $A(t)+U(t)\geq C(t)$ for all $t$, we have
$A\raisebox{0.43057pt}{\scriptsize$\circ$}\sigma_{B}^{(\infty)}+U\raisebox{0.43057pt}{\scriptsize$\circ$}\sigma_{B}^{(\infty)}\geq
C\raisebox{0.43057pt}{\scriptsize$\circ$}\sigma_{B}^{(\infty)},$
and from the last two displays we conclude that
$U\raisebox{0.43057pt}{\scriptsize$\circ$}\sigma_{B}^{(\infty)}\geq
B\raisebox{0.43057pt}{\scriptsize$\circ$}\sigma_{B}^{(\infty)}.$
Since $U$ is nondecreasing and since (17) holds, we have
$U\geq U\raisebox{0.43057pt}{\scriptsize$\circ$}\sigma_{B}^{(\infty)}\geq
B\raisebox{0.43057pt}{\scriptsize$\circ$}\sigma_{B}^{(\infty)}=B,$
as claimed. ∎
We are now ready to prove Theorem 1. We already know from Lemma 4 that
$Q^{*}\leq Q^{\infty}$. So we only have to prove the opposite inequality.
Recall that $Q_{1}=A$ and $B_{1}=C$. Trivially then
$Q_{1}(t)+C(t)=A(t)+B_{1}(t),\quad t\geq 0.$
Thus, for $0\leq s\leq t$ we have
$\displaystyle Q_{1}(s)>C(s,t]$ $\displaystyle\iff Q_{1}(s)+C(s)>C(t)$
$\displaystyle\iff A(s)+B_{1}(s)>C(t)$ $\displaystyle\iff
s>\sigma_{B_{1}}(t).$
From this we get
$\displaystyle Q_{2}(t)$
$\displaystyle=\int_{0}^{t}{\text{\Large$\mathfrak{1}$}}(Q_{1}(s)>C(s,t])~{}dA(s)$
$\displaystyle=\int_{0}^{t}{\text{\Large$\mathfrak{1}$}}(s>\sigma_{B_{1}}(t))~{}dA(s)$
$\displaystyle=A(t)-A(\sigma_{B_{1}}(t)).$
But (11) gives
$A(\sigma_{B_{1}}(t))+B_{1}(\sigma_{B_{1}}(t))=C(t),$
and so
$Q_{2}(t)+C(t)=A(t)+B_{1}(\sigma_{B_{1}}(t))=A(t)+B_{2}(t),\quad t\geq 0.$
We now claim that
$Q_{k}(t)+C(t)=A(t)+B_{k}(t),\quad t\geq 0,\quad k\geq 1.$
This can be proved by induction along the same lines as above. Taking limits
as $k\to\infty$, we conclude
$Q_{\infty}(t)+C(t)=A(t)+B_{\infty}(t),\quad t\geq 0.$
Lemma 7 tells us that $B_{\infty}$ is a fixed point of $\Phi$, and so, by
Lemma 9,
$B_{\infty}\leq U.$
Hence
$\displaystyle Q_{\infty}(t)+C(t)$ $\displaystyle=A(t)+B_{\infty}(t)$
$\displaystyle\leq A(t)+U(t)$ $\displaystyle=Q^{*}(t)+C(t),\quad t\geq 0,$
and this gives
$Q_{\infty}\leq Q^{*},$
as needed. ∎
## Acknowledgments
We thank the Isaac Newton Institute for Mathematical Sciences for providing
the stimulating research atmosphere where this research work was done.
## References
* [1] Anantharam, V. and Konstantopoulos, T. Regulating functions on partially ordered sets. Order 22, 145-183, 2005.
* [2] Baccelli, F. and Brémaud, P. Elements of Queueing Theory. Springer-Verlag, 2003.
* [3] Williams R. and Chung, K.-L. An Introduction to Stochastic Integration. Birkhäuser, Boston, 1989.
* [4] Kallenberg, O. Foundations of Modern Probability, 2nd ed. Springer-Verlag, New York, 2002.
* [5] Konstantopoulos, T., Kyprianou, A., Sirviö, M., and Salminen, P. Analysis of stochastic fluid queues driven by local time processes. Adv. Appl. Probability 40, 1072-1103, 2008.
* [6] Konstantopoulos, T., and Last, G. On the dynamics and performance of stochastic fluid systems. J. Appl. Prob. 37, 652-667, 2000.
* [7] Konstantopoulos, T., Zazanis, M. and de Veciana, G. Conservation laws and reflection mappings with an application to multiclass mean value analysis for stochastic fluid queues. Stoch. Proc. Appl. 65, No. 1, 139-146, 1997.
* [8] Konstantopoulos, T. Open problems session of the workshop on New Topics at the Interface Between Probability and Communications. Thursday, 14 January, 2010.
* [9] Revuz, D. and Yor, M. Continuous martingales and Brownian motion. Springer-Verlag, New York, 1999.
* [10] Skorokhod, A.V. Stochastic equations for diffusions in a bounded region, Theory Probab. Appl. 6, 264-274, 1961.
* [11] Tanaka, H. Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J. 9, 163-177, 1979.
* [12] Whitt, W. Stochastic-Process Limits. Springer-Verlag, New York, 2002.
Authors’ addresses:
Venkat Anantharam
EECS Department
University of California
Berkeley, CA 94720, USA
E-mail: ananth@eecs.berkeley.edu
Takis Konstantopoulos
School of Mathematical Sciences
Heriot-Watt University
Edinburgh EH14 4AS, UK
E-mail: takis@ma.hw.ac.uk
|
arxiv-papers
| 2010-03-29T18:50:27 |
2024-09-04T02:49:09.348131
|
{
"license": "Public Domain",
"authors": "Venkat Anantharam and Takis Konstantopoulos",
"submitter": "Takis Konstantopoulos",
"url": "https://arxiv.org/abs/1003.5638"
}
|
1003.5768
|
# Analytical vectorial structure of non-paraxial four-petal Gaussian beams in
the far field
Xuewen Longa,b Keqing Lua keqinglu@opt.ac.cn Yuhong Zhanga,b Jianbang Guoa,b
Kehao Lia,b aState Key Laboratory of Transient Optics and Photonics, Xi’an
Institute of Optics and Precision Mechanics, Chinese Academic of Sciences,
Xi’an 710119, China
b Graduate school of Chinese Academy of Sciences, Beijing, 100039, China
###### Abstract
The analytical vectorial structure of non-paraxial four-petal Gaussian
beams(FPGBs) in the far field has been studied based on vector angular
spectrum method and stationary phase method. In terms of analytical
electromagnetic representations of the TE and TM terms, the energy flux
distributions of the TE term, the TM term, and the whole beam are derived in
the far field, respectively. According to our investigation, the FPGBs can
evolve into a number of small petals in the far field. The number of the
petals is determined by the order of input beam. The physical pictures of the
FPGBs are well illustrated from the vectorial structure, which is beneficial
to strengthen the understanding of vectorial properties of the FPGBs.
four-petal Gaussian beam, vectorial structure, far field
###### pacs:
41.85.Ew, 42.25.Bs
## I Introduction
Recently, a new form of laser beams called four-petal Gaussian beams(FPGBs)
has been introduced and its properties of passing through a paraxial ABCD
optical system have been studied Duan2006OC . Subsequently, the propagation
properties of the FPGBs have attracted considerable interest due to its
potential applications. Gao and Lü studied its vectorial non-paraxial
propagation in free space based on vectorial Rayleigh-Sommerfeld diffraction
integral formulas in 2006 Gao2006CPL . In 2007, the propagation of four-petal
Gaussian beams in turbulent atmosphere was investigated by Chu _et al_
Chu2008CPL . In 2008, Tang _et al_ explored diffraction properties of four-
petal Gaussian beams in uniaxially anisotropic crystal by virtue of the
paraxially vectorial theory of beam propagation Tang2008COL . Yang _et al_
reported the propagation of four-petal Gaussian beams in strongly nonlocal
nonlinear media in the next year Yang2010OC . In the meanwhile, the vectorial
structure of lots of beam with different patterns and polarized status is
illustrated in the far field by means of vector angular spectrum method
Rosario2001JOSAA , which is a useful tool to resolve the Maxwell’s equations,
and stationary phase method Mandel ; Born , which uses the asymptotic
approximation approaching some kind of difficult integral. Based on these two
methods mentioned above, Zhou studied analytical vectorial structure of
Laguerre-Gaussian beam in the far field firstly Zhou2006OL . Afterwards, Deng
and Guo explored analytical vectorial structure of radially polarized light
beams DengOL . In 2008, Wu _et al_ and Zhou _et al_ investigated vectorial
structure of hollow Gaussian beam almost in the same time Wu2008OE ;
ZhouHollow2008OC . In fact, much work has been done with the vectorial
structure of all kinds of beams in the far field Zhounonsymmetrical2007OC \-
Zhou2009JOSAB .
It is well known that the paraxial approximation is no longer valid for beams
with a large divergent angle or, especially, a small beam spot size that is
comparable with the light wavelength Nemoto . Therefore, rigorous non-paraxial
and vectorial treatments are necessary. We can approach non-paraxial
propagation of beams in terms of vector angular spectrum method of
electromagnetic field. According to vector angular spectrum method of
electromagnetic field, the general solution of the Maxwell${}^{{}^{\prime}}$s
equations is composed of the transverse electric (TE) term and the transverse
magnetic (TM) term. In the far field, the TE and TM terms are orthogonal to
each other and can be detached.
To the best of our knowledge, the research on the vectorial structure of non-
paraxial four-petal Gaussian beam in the far field based on vector angular
spectrum method and stationary phase method has not been reported elsewhere.
In this paper, the far-field vectorial properties of non-paraxial four-petal
Gaussian beam have been studied by means of vector angular spectrum method and
stationary phase method. Based on the analytical vectorial structure of the
FPGBs, the energy flux distributions of TE term, TM term and the whole FPGBs
are also investigated, respectively.
## II Analytical vectorial structure in the far field
Let us consider a half space $z>0$ filled with a linear homogeneous,
isotropic, nonconducting medium characterized by electric permittivity
$\varepsilon$ and magnetic permeability $\mu$. All the sources only lie in the
domain $z<0$. The electric field distribution is known at the boundary plane
$z=0$. For convenience of discussion, we consider a non-paraxial FPGBs with
polarization in x direction, which propagates toward the half space $z\geq 0$
along the z axis. The initial transverse electric field distribution of the
FPGBs at the $z=0$ plane can be written by Duan2006OC
$E_{x}(x,y,0)=G_{n}\left(\frac{xy}{w_{0}^{2}}\right)^{2n}\exp\left(-\frac{x^{2}+y^{2}}{w_{0}^{2}}\right),n=1,2,3\ldots,$
(1) $E_{y}(x,y,0)=0,$ (2)
where n is the beam order of the FPGBs; ${G_{n}}$ is a normalized amplitude
constant associated with the order of n; $w_{0}$ is the $1/e^{2}$ intensity
waist radius of the Gaussian term. The time factor $\exp(-i\omega t)$ has been
omitted in the field expression. Fig. 1 shows intensity distributions of four-
petal Gaussian beams at the initial plane $z=0$ for $n=1,5,9$ and 13,
respectively. From Fig. 1, it can be seen that the intensity distributions is
composed of four equal petals. The distance of small petals increases when
beam order increases. Here, $w_{0}$ is taken by $\lambda$ in the calculation,
which is comparable with the wavelength. So, this is a non-paraxial problem.
Figure 1: (Color online)Normalized intensity distributions of FPGBs with
different beam order n at $z=0$ plane based Eq. (1). (a) $n=1$, (b) $n=5$, (c)
$n=9$, (d) $n=13$.
In terms of Fourier transform, the vectorial angular spectrum of electric
field at the $z=0$ plane is expressed as Wu2008OE
$A_{x}(p,q)=\frac{1}{\lambda^{2}}\int\int^{\infty}_{-\infty}E_{x}(x,y,0)\exp[-ik(px+qy)]dxdy,$
(3)
$A_{y}(p,q)=\frac{1}{\lambda^{2}}\int\int^{\infty}_{-\infty}E_{y}(x,y,0)\exp[-ik(px+qy)]dxdy,$
(4)
where $\lambda$ denotes the wave length in the medium related wave number by
$k=2\pi/\lambda$. Substituting Eqs. (1) and (2) into Eqs. (3) and (4), we find
that
$\displaystyle A_{x}(p,q)$ $\displaystyle=$
$\displaystyle\frac{G_{n}}{\lambda^{2}}w_{0}^{2}\left[\Gamma(n+\frac{1}{2})\right]^{2}{}_{1}F_{1}\left(n+\frac{1}{2};\frac{1}{2};-\frac{1}{4}k^{2}p^{2}w_{0}^{2}\right)$
(5)
$\displaystyle\times_{1}F_{1}\left(n+\frac{1}{2};\frac{1}{2};-\frac{1}{4}k^{2}q^{2}w_{0}^{2}\right),$
where ${}_{1}F_{1}(\cdot;\cdot;\cdot)$ denotes confluent hypergeometric
function, $\Gamma(\cdot)$ denotes the Gamma function. It is well known that
Maxwell’s equations can be separated into transverse and longitudinal field
equations and an arbitrary polarized electromagnetic beam, which is expressed
in terms of vector angular spectrum, is composed of the transverse electric
(TE) term and the transverse magnetic (TM)term, namely,
$\displaystyle\vec{E}(\vec{r})=\vec{E}_{TE}(\vec{r})+\vec{E}_{TM}(\vec{r}),$
(6)
$\displaystyle\vec{H}(\vec{r})=\vec{H}_{TE}(\vec{r})+\vec{H}_{TM}(\vec{r}),$
(7)
where
$\displaystyle\vec{E}_{TE}(\vec{r})$ $\displaystyle=$
$\displaystyle\int\int^{\infty}_{-\infty}\frac{1}{p^{2}+q^{2}}[qA_{x}(p,q)-pA_{y}(p,q)](q\hat{e}_{x}-p\hat{e}_{y})$
(8) $\displaystyle\times\exp(iku)dpdq,$
$\displaystyle\vec{H}_{TE}(\vec{r})$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\varepsilon}{\mu}}\int\int^{\infty}_{-\infty}\frac{1}{p^{2}+q^{2}}[qA_{x}(p,q)-pA_{y}(p,q)](p\gamma\hat{e}_{x}+q\gamma\hat{e}_{y}-b^{2}\hat{e}_{z})$
(9) $\displaystyle\times\exp(iku)dpdq,$
and
$\displaystyle\vec{E}_{TM}(\vec{r})$ $\displaystyle=$
$\displaystyle\int\int^{\infty}_{-\infty}\frac{1}{p^{2}+q^{2}}[pA_{x}(p,q)+qA_{y}(p,q)](p\hat{e}_{x}+q\hat{e}_{y}-\frac{b^{2}}{\gamma}\hat{e}_{z})$
(10) $\displaystyle\times\exp(iku)dpdq,$
$\displaystyle\vec{H}_{TM}(\vec{r})$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{\varepsilon}{\mu}}\int\int^{\infty}_{-\infty}[pA_{x}(p,q)+qA_{y}(p,q)]\frac{1}{b^{2}\gamma}(q\hat{e}_{x}-p\hat{e}_{y},)$
(11) $\displaystyle\times\exp(iku)dpdq.$
$\vec{r}=x\hat{e}_{x}+y\hat{e}_{y}+z\hat{e}_{z}$ is the displacement vector
and $\hat{e}_{x}$, $\hat{e}_{y}$, $\hat{e}_{z}$ denote unit vectors in the x,
y, z directions, respectively; $u=px+qy+\gamma z$; $b^{2}=p^{2}+q^{2}$;
$\gamma=\sqrt{1-p^{2}-q^{2}}$, if $p^{2}+q^{2}\leq 1$ or
$\gamma=i\sqrt{p^{2}+q^{2}-1}$, if $p^{2}+q^{2}>1$. The value of
$p^{2}+q^{2}>1$ corresponds to the evanescent wave which propagates along the
boundary plane but decays exponentially along the positive z direction.
In the far field framework, the condition
$k(x^{2}+y^{2}+z^{2})^{1/2}\rightarrow\infty$ is satisfied due to z is big
enough. Moreover the contribution of the evanescent wave to the far field can
be ignored. By virtue of the method of stationary phase Mandel ; Born ;
Wu2008OE , the TE mode and the TM mode of the electromagnetic field can be
given by
$\displaystyle\vec{E}_{TE}(\vec{r})$ $\displaystyle=$
$\displaystyle-i\frac{G_{n}w_{0}^{2}}{\lambda}\frac{yz}{r^{2}\rho^{2}}\left[\Gamma(n+\frac{1}{2})\right]^{2}{}_{1}F_{1}\left(n+\frac{1}{2};\frac{1}{2};-\frac{1}{4}k^{2}\frac{x^{2}}{r^{2}}w_{0}^{2}\right)$
(12)
$\displaystyle\times{}_{1}F_{1}\left(n+\frac{1}{2};\frac{1}{2};-\frac{1}{4}k^{2}\frac{y^{2}}{r^{2}}w_{0}^{2}\right)\exp(ikr)(y\hat{e}_{x}-x\hat{e}_{y}),$
$\displaystyle\vec{H}_{TE}(\vec{r})$ $\displaystyle=$
$\displaystyle-i\sqrt{\frac{\varepsilon}{\mu}}\frac{G_{n}w_{0}^{2}}{\lambda}\frac{yz}{r^{3}\rho^{2}}\left[\Gamma(n+\frac{1}{2})\right]^{2}{}_{1}F_{1}\left(n+\frac{1}{2};\frac{1}{2};-\frac{1}{4}k^{2}\frac{x^{2}}{r^{2}}w_{0}^{2}\right)$
(13)
$\displaystyle\times_{1}F_{1}\left(n+\frac{1}{2};\frac{1}{2};-\frac{1}{4}k^{2}\frac{y^{2}}{r^{2}}w_{0}^{2}\right)\exp(ikr)$
$\displaystyle\times(xz\hat{e}_{x}+yz\hat{e}_{y}-\rho^{2}\hat{e}_{z}),$
and
$\displaystyle\vec{E}_{TM}(\vec{r})$ $\displaystyle=$
$\displaystyle-i\frac{G_{n}w_{0}^{2}}{\lambda}\frac{x}{r^{2}\rho^{2}}\left[\Gamma(n+\frac{1}{2})\right]^{2}{}_{1}F_{1}\left(n+\frac{1}{2};\frac{1}{2};-\frac{1}{4}k^{2}\frac{x^{2}}{r^{2}}w_{0}^{2}\right)$
(14)
$\displaystyle\times_{1}F_{1}\left(n+\frac{1}{2};\frac{1}{2};-\frac{1}{4}k^{2}\frac{y^{2}}{r^{2}}w_{0}^{2}\right)\exp(ikr)$
$\displaystyle\times(xz\hat{e}_{x}+yz\hat{e}_{y}-\rho^{2}\hat{e}_{z}),$
$\displaystyle\vec{H}_{TM}(\vec{r})$ $\displaystyle=$ $\displaystyle
i\sqrt{\frac{\varepsilon}{\mu}}\frac{G_{n}w_{0}^{2}}{\lambda}\frac{x}{r\rho^{2}}\left[\Gamma(n+\frac{1}{2})\right]^{2}{}_{1}F_{1}\left(n+\frac{1}{2};\frac{1}{2};-\frac{1}{4}k^{2}\frac{x^{2}}{r^{2}}w_{0}^{2}\right)$
(15)
$\displaystyle\times_{1}F_{1}\left(n+\frac{1}{2};\frac{1}{2};-\frac{1}{4}k^{2}\frac{y^{2}}{r^{2}}w_{0}^{2}\right)\exp(ikr)$
$\displaystyle\times(y\hat{e}_{x}-x\hat{e}_{y}),$
where $\rho=\sqrt{x^{2}+y^{2}}$ ; $r=\sqrt{x^{2}+y^{2}+z^{2}}$. Eqs. (12)-
(15) are analytical vectorial expressions for the TE and TM terms in the far
field and constitute the basic results in this paper. From Eqs. (12)- (15), we
find that
$\displaystyle\vec{E}_{TE}(\vec{r})\cdot\vec{E}_{TM}(\vec{r})=0,$ (16)
$\displaystyle\vec{H}_{TE}(\vec{r})\cdot\vec{H}_{TM}(\vec{r})=0.$ (17)
According to Eqs. (16) and (17), the TE and TM terms of FPGBs are orthogonal
to each other in the far field.
## III Energy flux distributions in the far field
The energy flux distributions of the TE and TM terms at the $z=const$ plane
are expressed in terms of the z component of their time-average Poynting
vector as
$\displaystyle\langle
S_{z}\rangle_{TE}=\frac{1}{2}Re[\vec{E}_{TE}^{*}\times\vec{H}_{TE}]_{z}{},$
(18) $\displaystyle\langle
S_{z}\rangle_{TM}=\frac{1}{2}Re[\vec{E}_{TM}^{*}\times\vec{H}_{TM}]_{z}{},$
(19)
where the Re denotes real part, and the asterisk denotes complex conjugation.
The whole energy flux distribution of the beam is a sum of the energy flux of
TE mode and TM mode, namely,
$\displaystyle\langle S_{z}\rangle=\langle S_{z}\rangle_{TE}+\langle
S_{z}\rangle_{TM},$ (20)
Substituting Eqs. (12)- (15) into Eqs. (18)- (19) yields
$\displaystyle\langle S_{z}\rangle_{TE}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{\frac{\varepsilon}{\mu}}\frac{G_{n}^{2}w_{0}^{4}}{\lambda^{2}}\frac{y^{2}z^{3}}{r^{5}\rho^{2}}\left[\Gamma(n+\frac{1}{2})\right]^{4}{}_{1}F_{1}\left(n+\frac{1}{2};\frac{1}{2};-\frac{1}{4}k^{2}\frac{x^{2}}{r^{2}}w_{0}^{2}\right)^{2}$
(21)
$\displaystyle\times_{1}F_{1}\left(n+\frac{1}{2};\frac{1}{2};-\frac{1}{4}k^{2}\frac{y^{2}}{r^{2}}w_{0}^{2}\right)^{2},$
$\displaystyle\langle S_{z}\rangle_{TM}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{\frac{\varepsilon}{\mu}}\frac{G_{n}^{2}w_{0}^{4}}{\lambda^{2}}\frac{x^{2}z}{r^{3}\rho^{2}}\left[\Gamma(n+\frac{1}{2})\right]^{4}{}_{1}F_{1}\left(n+\frac{1}{2};\frac{1}{2};-\frac{1}{4}k^{2}\frac{x^{2}}{r^{2}}w_{0}^{2}\right)^{2}$
(22)
$\displaystyle\times_{1}F_{1}\left(n+\frac{1}{2};\frac{1}{2};-\frac{1}{4}k^{2}\frac{y^{2}}{r^{2}}w_{0}^{2}\right)^{2},$
Therefore, the whole energy flux distribution of FPGBs in the far field is
given by
$\displaystyle\langle S_{z}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{\frac{\varepsilon}{\mu}}\frac{G_{n}^{2}w_{0}^{4}}{\lambda^{2}}\frac{z}{r^{3}\rho^{2}}\left(\frac{y^{2}z^{2}}{r^{2}}+x^{2}\right)\left[\Gamma(n+\frac{1}{2})\right]^{4}$
(23)
$\displaystyle\times{}_{1}F_{1}\left(n+\frac{1}{2};\frac{1}{2};-\frac{1}{4}k^{2}\frac{x^{2}}{r^{2}}w_{0}^{2}\right)^{2}$
$\displaystyle\times{}_{1}F_{1}\left(n+\frac{1}{2};\frac{1}{2};-\frac{1}{4}k^{2}\frac{y^{2}}{r^{2}}w_{0}^{2}\right)^{2}.$
Figure 2: (Color online)Normalized energy flux distributions of FPGBs at the
plane $z=500\lambda$ for beam order $n=1$. (a) The TE term, (b) The TM term,
(c) The whole beam. Figure 3: (Color online)Normalized energy flux
distributions of FPGBs at the plane $z=500\lambda$ for beam order $n=5$. (a)
The TE term, (b) The TM term, (c) The whole beam. Figure 4: (Color
online)Normalized energy flux distributions of FPGBs at the plane
$z=500\lambda$ for beam order $n=9$. (a) The TE term, (b) The TM term, (c) The
whole beam. Figure 5: (Color online)Normalized energy flux distributions of
FPGBs at the plane $z=500\lambda$ for beam order $n=13$. (a) The TE term,(b)
The TM term, (c) The whole beam.
The normalized energy flux distributions of the TE term, the TM term and the
FPGBs at the plane $z=500\lambda$ for different beam order $n=1,5,9,13$ versus
$x/\lambda$ and $y/\lambda$ are illustrated by Figs. 2\- 5. The used parameter
is $w_{0}=\lambda$. As can be seen from Figs. 2\- 5, the FPGBs split into a
number of small petals in the far field, which differs from its initial four-
petal shape. The FPGBs with beam order n is not a pure mode, which can be
regarded as a superposition of $n^{2}$ two dimensional Hermite-Gaussian modes
Duan2006OC , and different modes evolve differently within the same
propagation distance. The overlap and interference in propagation between
different modes result in the propagation properties of the FPGBs in the far
field. Furthermore, the number of petals is determined by the parameter n. The
number of petals in the far field gradually increases when the parameter n
increases, which has potential applications in micro-optics and beam splitting
techniques, etc. Note that diameter of the central beam spot decreases when
beam order n increases. This phenomenon has been discussed in some previous
researches Duan2006OC \- Chu2008CPL .
## IV Conclusions
In summary, the vectorial structure of the non-paraxial four-petal Gaussian
beam in the far field is expressed in the analytical form by using the vector
angular spectrum method and the stationary phase method. The electric field
and the magnetic field of the four-petal Gaussian beam is decomposed into two
mutually orthogonal terms, i.e., TE term and TM term. Based on the analytical
vectorial structure of FPGBs, the energy flux distributions of the TE term,
the TM term and the whole beam of FPGBs are derived in the far-field and are
illustrated by numerical examples. The number of petals and diameter of the
central beam spot in the far field are determined by the beam order n. The
potential applications of the FPGBs are deserved investigation. This work is
important to understand the theoretical aspects of vector FPGBs propagation.
###### Acknowledgements.
This research was supported by the National Natural Science Foundation of
China (Grant No.10674176 ). The author is beneficial from the discussion with
the author of Ref. Wu2008OE .
## References
* (1) K. Duan and B. Lü, Opt. Commun. 261, 327 (2006).
* (2) Z. Gao and B. Lü, Chin. Phys. Lett. 23, 2070 (2006).
* (3) X. Chu, J. Liu and Y. Wu, Chin. Phys. Lett. 25, 485 (2008).
* (4) B. Tang, Y. Jin, M. Jiang and X. Jiang, Chin. Opt. Lett. 6, 779 (2008).
* (5) Z. Yang, D. Lu, D. Deng, S. Li, W. Hu and Q. Guo, Opt. Commun. 283, 595 (2010).
* (6) Rosario Martínez-Herrero and Pedro M. Mejías, J. Opt. Soc. Am. A 18, 1678 (2001).
* (7) L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
* (8) M. Born and E. Wolf, Principles of Optics, 7th Edition (Cambridge University Press, Cambridge, 1999).
* (9) G. Zhou, Opt. Lett. 31, 2616 (2006).
* (10) D. Deng and Q. Guo, Opt. Lett. 32, 2711 (2007).
* (11) G. Wu, Q. Lou and J. Zhou, Opt. Express 16, 6417 (2008).
* (12) G. Zhou, X. Chu and J. Zheng, Opt. Commun. 281, 5653 (2008).
* (13) G. Zhou, Y.Z. Ni and Z. Zhang, Opt. Commun. 272, 32 (2007).
* (14) G. Zhou and F. Liu, Opt. Laser Technol. 40, 302 (2008).
* (15) H. Tang, X. Li, G. Zhou and K. Zhu, Opt. Commun. 282, 478 (2009).
* (16) G. Zhou, J. Opt. Soc. Am. B 26, 2386 (2009).
* (17) Nemoto Shojiro, Appl. Opt. 29, 1940 (1990).
|
arxiv-papers
| 2010-03-30T09:25:09 |
2024-09-04T02:49:09.356840
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xuewen Long, Keqing Lu, Yuhong Zhang, Jianbang Guo and Kehao Li",
"submitter": "Xuewen Long",
"url": "https://arxiv.org/abs/1003.5768"
}
|
1003.5948
|
# Alexandrov meets Lott–Villani–Sturm
Anton Petrunin
###### Abstract
Here I show compatibility of two definition of generalized curvature bounds —
the lower bound for sectional curvature in the sense of Alexandrov and lower
bound for Ricci curvature in the sense of Lott–Villani–Sturm.
## Introduction
Let me denote by $\text{\rm CD}[m,\upkappa]$ the class metric-measure spaces
which satisfy a weak curvature-dimension condition for dimension $m$ and
curvature $\upkappa$ (see preliminaries). By $\text{\rm Alex}^{m}[\upkappa]$,
I will denote the class of all $m$-dimensional Alexandrov spaces with
curvature $\geqslant\upkappa$ equipped with the volume-measure (so $\text{\rm
Alex}^{m}[\upkappa]$ is a class of metric-measure spaces).
Main theorem. $\text{\rm Alex}^{m}[0]\subset\text{\rm CD}[m,0]$.
The question appears first in [Lott–Villani, 7.48]. In [Villani], it is
formulated more generally: $\text{\rm Alex}^{m}[\upkappa]\subset\text{\rm
CD}[m,(m-1)\upkappa]$. The later statement can be proved, along the same
lines, but I do not write it down.
About the proof. The idea of the proof is the same as in Riemannian case (see
[CMS, 6.2] or [Lott–Villani, 7.3]). One only needs to extend certain calculus
to Alexandrov spaces. To do this, I used the same technique as in [Petrunin
03]. I will illustrate the idea on a very simple problem.
Let $M$ be a $2$-dimensional non-negatively curved Riemannian manifold and
$\upgamma_{\uptau}\colon[0,1]\to M$ be a continuous family of unit-speed
geodesics such that
$|\upgamma_{\uptau_{0}}(t_{0})\,\upgamma_{\uptau_{1}}(t_{1})|\geqslant|t_{1}-t_{0}|.$
➊
Set $\ell(t)$ to be the total length of curve
$\upsigma_{t}\colon\uptau\mapsto\upgamma_{\uptau}(t)$. Then $\ell(t)$ is a
concave function — that is easy to prove.
Now, assume you have $A\in\text{\rm Alex}^{2}[0]$ instead of $M$ and a non-
continuous family of unit-speed geodesics $\upgamma_{\uptau}(t)$ which
satisfies ➊ ‣ Introduction. Define $\ell(t)$ as the 1-dimensional Hausdorff
measure of image of $\upsigma_{t}$. In this case $\ell$ is also concave.
Here is an idea how one can proceed; it is not the simplest one but the one
which admits proper generalization. Consider two functions $\uppsi=\mathop{\rm
dist}\nolimits_{\mathop{\rm Im}\nolimits\upsigma_{0}}$ and
$\upvarphi=\mathop{\rm dist}\nolimits_{\mathop{\rm Im}\nolimits\upsigma_{1}}$.
Note that geodesics $\upgamma_{\uptau}(t)$ are also gradient curves of
$\uppsi$ and $\upvarphi$. It implies that $\Delta\upvarphi+\Delta\uppsi$
vanish almost everywhere on the image of the map
$(\uptau,t)\to\upgamma_{\uptau}(t)$ (the Laplasians $\Delta\upvarphi$ and
$\Delta\uppsi$ are Radon sign-measure). Then result follows from the second
variation formula from [Petrunin 98] and calculus on Alexandrov spaces
developed in [Perelman].
Remark. Although $\text{\rm CD}[m,\upkappa]$ is a very natural class of
metric-measure spaces, some basic tools in Ricci comparison can not work there
in principle. For instance, _there are $\text{\rm CD}[m,0]$-spaces which do
not satisfy the Abresch–Gromoll inequality_, (see [AG]). Thus, one has to
modify the definition of the class $\text{\rm CD}[m,\upkappa]$ to make it
suitable for substantial applications in Riemannian geometry.
I'm grateful to A. Lytchak and C. Villani, for their help.
## 1 Preliminaries
Prerequisite. The reader is supposed to be familiar with basic definition and
notion in optimal transport as in [Villani], measure theory in Alexandrov
spaces from [BGP], DC-structure on Alexandrov's spaces from [Perelman] and
technique and notation of gradient flow as in [Petrunin 07].
What needs to be proved. Let me recall the definition of class $\text{\rm
CD}[m,0]$ only — it is sufficient for understanding this paper. The definition
of $\text{\rm CD}[m,\upkappa]$ can be found in [Villani, 29.8].
Similar definitions were given in [Lott–Villani] and [Sturm]. The idea behind
these definitions — convexity of certain functionals in the Wasserstein space
over a Riemannian manifold, appears in [Otto–Villani], [CMS],
[Sturm–v.Renesse]. In the Euclidean context, this notion of convexity goes
back to [McCann]. More on the history of the subject can be found in
[Villani].
For a metric-measure space $X$, I will denote by $|xy|$ the distance between
points $x,y\in X$ and by $\mathop{\rm vol}\nolimits E$ the distinguished
measure of Borel subset $E\subset X$ (I will call it _volume_). Let us denote
by $\mathrm{P}_{\mskip-3.0mu2}X$ the set of all probability measures with
compact support in $X$ equipped with Wasserstein distance of order 2, see
[Villani, 6.1].
Further, we assume $X$ is proper geodesic space; in this case
$\mathrm{P}_{\mskip-3.0mu2}X$ is geodesic.
Let $\upmu$ be a probability measure on $X$. Denote by $\upmu^{r}$ the
absolutely continuous part of $\upmu$ with respect to volume. I.e. $\upmu^{r}$
coincides with $\upmu$ outside a Borel subset of volume zero and there is a
Borel function $\uprho\colon X\to\mathbb{R}$ such that
$\upmu^{r}=\uprho{\cdot}\mathop{\rm vol}\nolimits$. Define
$U_{m}\upmu\buildrel\hbox{\tiny\it
def}\over{=\joinrel=}\int\limits_{X}\uprho^{1-\frac{1}{m}}{\cdot}\,d\mathop{\rm
vol}\nolimits=\int\limits_{X}\frac{1}{\sqrt[m]{\uprho}}{\cdot}\,d\upmu^{r}.$
Then $X\in\text{\rm CD}[m,0]$ if the functional $U_{m}$ is concave in
$\mathrm{P}_{\mskip-3.0mu2}X$; i.e for any two measures
$\upmu_{0},\upmu_{1}\in\mathrm{P}_{\mskip-3.0mu2}X$, there is a geodesic path
$\upmu_{t}$, in $\mathrm{P}_{\mskip-3.0mu2}X$, $t\in[0,1]$ such that the real
function $t\mapsto U_{m}\upmu_{t}$ is concave.
Calculus in Alexandrov spaces. Let $A\in\text{\rm Alex}^{m}[\upkappa]$ and
$S\subset A$ be the subset of singular points; i.e. $x\in S$ if its tangent
space $\text{\rm T}_{x}$ is not isometric to Euclidean $m$-space
$\mathbb{E}^{m}$. The set $S$ has zero volume ([BGP, 10.6]). The set of
regular points $A\backslash S$ is convex ([Petrunin 98]); i.e. any geodesic
connecting two regular points consits only of regular points.
According to [Perelman], if $f\colon A\to\mathbb{R}$ is a semiconcave function
and $\Omega\subset A$ is an image of a $\mathrm{DC}_{0}$-chart, then
$\partial_{k}f$ and components of metric tensor $g^{ij}$ are functions of
locally bounded variation which are continuous in $\Omega\backslash S$.
Further, for almost all $x\in A$ the the Hession of $f$ is well defined. I.e.
there is a subset of full measure $\mathop{\rm Reg}\nolimits f\subset
A\backslash S$ such that for any $p\in\mathop{\rm Reg}\nolimits f$ there is a
bi-linear form111Note that $p\in A\backslash S$, thus $\text{\rm T}_{p}$ is
isometric to Euclidean $m$-space. $\operatorname{Hess}_{p}f$ on $\text{\rm
T}_{p}$ such that
$f(q)=f(p)+d_{p}f(v)+\operatorname{Hess}_{p}f(v,v)+o(|v|^{2}),$
where $v=\mathop{\rm log}\nolimits_{p}q$. Moreover, the Hessian can be found
using standard calculus in the $\mathrm{DC}_{0}$-chart. In particular,
$\text{\rm Trace}\operatorname{Hess}f\buildrel\hbox{\tiny\it
a.e.}\over{=\joinrel=}\frac{\partial_{i}(\det g\cdot
g^{ij}\cdot\partial_{j}f)}{\det g}$
The following is an extract from second variation formula [Petrunin 98, 1.1B]
reformulated with formalism of ultrafilters. Let be a nonprinciple
ultrafiler, $A\in\text{\rm Alex}^{m}[0]$ and $[pq]$ be a minimizing geodesic
in $A$ which is extendable beyond $p$ and $q$. Assume further that one of (and
therefore each) of points $p$ and $q$ is regular. Then there is a model
configuration $\tilde{p},\tilde{q}\in\mathbb{E}^{m}$ and isometries
$\imath_{p}\colon\text{\rm T}_{p}A\to\text{\rm T}_{\tilde{p}}\mathbb{E}^{m}$,
$\imath_{q}\colon\text{\rm T}_{q}A\to\text{\rm T}_{\tilde{q}}\mathbb{E}^{m}$
such that
$\mskip-3.0mu\left|\exp_{p}(\tfrac{1}{n}{\cdot}v)\,\exp_{q}(\tfrac{1}{n}{\cdot}w)\right|\leqslant\mskip-3.0mu\left|\exp_{\tilde{p}}\circ\imath_{p}(\tfrac{1}{n}{\cdot}v)\,\exp_{\tilde{q}}\circ\imath_{q}(\tfrac{1}{n}{\cdot}w)\right|+o(n^{2})$
for -almost all $n$ (once the left-hand side is well defined).
If $\tilde{\uptau}\colon\text{\rm T}_{\tilde{p}}\to\text{\rm T}_{\tilde{q}}$
is the parallel translation in $\mathbb{E}^{m}$, then the isometry
$\uptau\colon\text{\rm T}_{p}\to\text{\rm T}_{q}$ which satisfy identity
$\imath_{q}\circ\uptau=\tilde{\uptau}\circ\imath_{p}$ will be called
``parallel transportation'' from $p$ to $q$.
Laplacian of semiconcave function. Here are some facts from [Petrunin 03].
Given a function $f\colon A\to\mathbb{R}$, define its _Laplacian_ $\Delta f$
to be a Radon sign-measure which satisfies the following identity
$\int\limits_{A}u{\cdot}\,d\Delta f=-\int\limits_{A}\langle\nabla u,\nabla
f\rangle{\cdot}\,d\mathop{\rm vol}\nolimits$
for any Lipschitz function $u\colon A\to\mathbb{R}$.
1.1. Claim. Let $A\in\text{\rm Alex}^{m}[\upkappa]$ and $f\colon
A\to\mathbb{R}$ be $\uplambda$-concave Lipschitz function. Then Laplacian
$\Delta f$ is well defined and
$\Delta f\leqslant m{\cdot}\uplambda{\cdot}\mathop{\rm vol}\nolimits.$
In particular, $\Delta^{s}f$ — the singular part $\Delta f$ is negative.
Moreover,
$\Delta f=\text{\rm Trace}\operatorname{Hess}f{\cdot}\mathop{\rm
vol}\nolimits+\Delta^{s}f.$
Proof. Let us denote by $F_{t}\colon A\to A$ the $f$-gradient flow for time
$t$.
Given a Lipschitz function $u\colon A\to\mathbb{R}$, consider family
$u_{t}(x)=u\circ F_{t}(x)$. Clearly, $u_{0}\equiv u$ and $u_{t}$ is Lipschitz
for any $t\geqslant 0$. Further, for any $x\in A$ we have
$\bigl{|}{\tfrac{d^{+}}{dt}}u_{t}(x)|_{t=0}\bigr{|}\leqslant\mathrm{Const}$.
Moreover
$\tfrac{d^{+}}{dt}u_{t}(x)|_{t=0}\buildrel\hbox{\tiny\it
a.e.}\over{=\joinrel=}d_{x}u(\nabla_{x}f)\buildrel\hbox{\tiny\it
a.e.}\over{=\joinrel=}\langle\nabla_{x}u,\nabla_{x}f\rangle.$
Further,
$\int\limits_{A}u_{t}{\cdot}\,d\mathop{\rm
vol}\nolimits=\int\limits_{A}u{\cdot}\,d(F_{t}\\#\mathop{\rm vol}\nolimits),$
where $\\#$ stands for push-forward. Since $|F_{t}(x)F_{t}(y)|\leqslant
e^{\uplambda t}{\cdot}|xy|$, for any $x,y\in A$ we have
$F_{t}\\#\mathop{\rm
vol}\nolimits\geqslant\exp(-m{\cdot}\uplambda{\cdot}t){\cdot}\mathop{\rm
vol}\nolimits.$
Therefore, for any non-negative Lipschitz function $u\colon A\to\mathbb{R}$,
$\int\limits_{A}u_{t}{\cdot}\,d\mathop{\rm
vol}\nolimits=\int\limits_{A}u{\cdot}\,d(F_{t}\\#\mathop{\rm
vol}\nolimits)\geqslant\exp(-m{\cdot}\uplambda{\cdot}t)\int\limits_{A}u{\cdot}\,d\mathop{\rm
vol}\nolimits.$
Therefore
$\int\limits_{A}\langle\nabla u,\nabla f\rangle{\cdot}\,d\mathop{\rm
vol}\nolimits=\mskip-3.0mu\left.\tfrac{d^{+}}{dt}\int\limits_{A}u_{t}{\cdot}\,d\mathop{\rm
vol}\nolimits\right|_{t=0}\geqslant-m{\cdot}\uplambda{\cdot}\int\limits_{A}u{\cdot}\,d\mathop{\rm
vol}\nolimits.$
I.e. there is a Radon measure $\upchi$ on $A$, such that
$\int\limits_{A}u{\cdot}\,d\upchi=\int\limits_{A}\mskip-3.0mu\left[\langle\nabla
u,\nabla f\rangle+m{\cdot}\uplambda{\cdot}u\right]{\cdot}\,d\mathop{\rm
vol}\nolimits$
Set $\Delta f=-\upchi+m{\cdot}\uplambda$, it is a Radon sign-measure and
$\upchi=-\Delta f+m{\cdot}\uplambda\geqslant 0$.
To prove the second part of theorem, assume $u$ is a non-negative Lipschitz
function with support in a DC0-chart $U\to A$, where $U\subset\mathbb{R}^{m}$
is an open subset. Then
$\displaystyle\int\limits_{A}\langle\nabla u,\nabla f\rangle$
$\displaystyle=\int\limits_{U}\det g\cdot
g^{ij}\cdot\partial_{i}u\cdot\partial_{j}f\cdot dx^{1}{\cdot}dx^{2}\cdots
dx^{m}=$ $\displaystyle=-\int\limits_{U}u\cdot\partial_{i}(\det g\cdot
g^{ij}\cdot\partial_{j}f)\cdot dx^{1}{\cdot}dx^{2}\cdots dx^{m},$
Thus
$\Delta f=\partial_{i}(\det g\cdot g^{ij}\cdot\partial_{j}f)\cdot
dx^{1}{\cdot}dx^{2}\cdots dx^{m}\buildrel\hbox{\tiny\it
a.e.}\over{=\joinrel=}\text{\rm Trace}\operatorname{Hess}f.$
∎
Gradient curves. Here I extend notion of gradient curves to a family of
functions, see [Petrunin 07] for all necessary definitions.
Let $\mathbb{I}$ be an open real interval and
$\uplambda\colon\mathbb{I}\to\mathbb{R}$ be a continuous function. A one
parameter family of functions $f_{t}\colon A\to\mathbb{R}$, $t\in\mathbb{I}$
will be called _$\uplambda(t)$ -concave_ if the function $(t,x)\mapsto
f_{t}(x)$ is locally Lipschitz and $f_{t}$ is $\uplambda(t)$-concave for each
$t\in\mathbb{I}$.
We will write $\upalpha^{\pm}(t)=\nabla f_{t}$ if for any $t\in\mathbb{I}$,
the right/left tangent vector $\upalpha^{\pm}(t)$ is well defined and
$\upalpha^{\pm}(t)=\nabla_{\upalpha(t)}f_{t}$. The solutions of
$\upalpha^{+}(t)=\nabla f_{t}$ will be also called $f_{t}$-gradient curves.
The following is a slight generalization of [Petrunin 07, 2.1.2$\&$2.2(2)]; it
can be proved along the same lines.
1.2. Proposition-definition. Let $A\in\text{\rm Alex}^{m}[\upkappa]$,
$\mathbb{I}$ be an open real interval,
$\uplambda\colon\mathbb{I}\to\mathbb{R}$ be a continuous function and
$f_{t}\colon A\to\mathbb{R}$, $t\in\mathbb{I}$ be $\uplambda(t)$-concave
family.
Then for any $x\in A$ and $t_{0}\in\mathbb{I}$ there is an $f_{t}$-gradient
curve $\upalpha$ which is defined in a neighborhood of $t_{0}$ and such that
$\upalpha(t_{0})=x$.
More over, if $\upalpha,\upbeta\colon\mathbb{I}\to A$ $f_{t}$-gradient then
for any $t_{0},t_{1}\in\mathbb{I}$, $t_{0}\leqslant t_{1}$,
$|\upalpha(t_{1})\upbeta(t_{1})|\leqslant L|\upalpha(t_{0})\upbeta(t_{0})|,$
where
$L=\exp\mskip-3.0mu\left(\int_{t_{0}}^{t_{1}}\uplambda(t){\cdot}\,dt\right).$
Note that above proposition implies that the value $\upalpha(t_{0})$ of
$f_{t}$-gradient curve $\upalpha(t)$ uniquely determines it for all
$t\geqslant t_{0}$ in $\mathbb{I}$. Thus we can define _$f_{t}$ -gradient
flow_ — a family of maps $F_{{t_{0}},{t_{1}}}\colon A\to A$ such that
$F_{{t_{0}},{t_{1}}}(\upalpha(t_{0}))=\upalpha(t_{1})\ \ \text{if}\ \
\upalpha^{+}(t)=\nabla f_{t}.$
1.3. Claim. Let $f_{t}\colon A\to\mathbb{R}$ be a $\uplambda(t)$-concave
family and $F_{{t_{0}},{t_{1}}}$ be $f_{t}$-gradient flow. Let $E\subset A$ be
a bounded Borel set, fix $t_{1}$ and consider function $v(t)=\mathop{\rm
vol}\nolimits F^{-1}_{{t},{t_{1}}}(E)$. Then
$\bigl{.}v\bigr{|}_{t}^{t_{1}}=\int\limits_{t}^{t_{1}}\Delta
f_{\text{\it\c{t}}}\mskip-3.0mu\left[F^{-1}_{\text{\it\c{t}},{t_{1}}}(E)\right]{\cdot}\,d\text{\it\c{t}}$
Proof. Let $u\colon A\to\mathbb{R}$ be a Lipschitz function with compact
support. Set $u_{t}=u\circ F_{{t},{t_{1}}}$. Clearly all $(x,t)\mapsto
u_{t}(x)$ is locally Lipschitz. Thus, the function
$w_{u}\colon t\mapsto\int\limits_{A}u_{t}{\cdot}\,d\mathop{\rm vol}\nolimits$
is locally Lipschitz. Further
$w_{u}^{\prime}(t)\buildrel\hbox{\tiny\it
a.e.}\over{=\joinrel=}-\int\limits_{A}\langle\nabla u_{t},\nabla
f_{t}\rangle{\cdot}\,d\mathop{\rm
vol}\nolimits=\int\limits_{A}u_{t}{\cdot}\,d\Delta f_{t}.$
Therefore
$\bigl{.}w_{u}\bigr{|}_{t}^{t_{1}}=\int\limits_{t}^{t_{1}}d\text{\it\c{t}}\cdot\int\limits_{A}u_{t}{\cdot}\,d\Delta
f_{t}.$
The last formula extendeds to arbitrary Borel function $u\colon
A\to\mathbb{R}$ with bounded support. Applying it to characteristic function
of $E$ we get the result. ∎
## 2 Games with Hamilton–Jacobi shifts.
Let $A\in\text{\rm Alex}^{m}[0]$. For a function $f\colon
A\to\mathbb{R}\cup\\{+\infty\\}$, let us define its Hamilton–Jacobi
shift222There is a lot of similarity between Hamilton–Jacobi shift of function
and equidistant for hypersurface. $\mathop{\mathcal{H}\hbox
to0.0pt{$\displaystyle\phantom{I}$\hss}}\nolimits_{t}f\colon A\to\mathbb{R}$
for time $t>0$,
$\mathop{\mathcal{H}\hbox
to0.0pt{$\displaystyle\phantom{I}$\hss}}\nolimits_{t}f(x)\buildrel\hbox{\tiny\it
def}\over{=\joinrel=}\inf_{y\in
A}\mskip-3.0mu\left\\{f(y)+\tfrac{1}{2t}{\cdot}|xy|^{2}\right\\}.$
We say that $\mathop{\mathcal{H}\hbox
to0.0pt{$\displaystyle\phantom{I}$\hss}}\nolimits_{t}f$ is well defined if the
above infimum is $>-\infty$ everywhere in $A$. Clearly,
$\mathop{\mathcal{H}\hbox
to0.0pt{$\displaystyle\phantom{I}$\hss}}\nolimits_{t_{0}+t_{1}}f=\mathop{\mathcal{H}\hbox
to0.0pt{$\displaystyle\phantom{I}$\hss}}\nolimits_{t_{1}}\mathop{\mathcal{H}\hbox
to0.0pt{$\displaystyle\phantom{I}$\hss}}\nolimits_{t_{0}}f,$ ➋
for any $t_{0},t_{1}>0$.
Note that for $t>0$, $f_{t}=\mathop{\mathcal{H}\hbox
to0.0pt{$\displaystyle\phantom{I}$\hss}}\nolimits_{t}f$ forms a
$\tfrac{1}{t}$-concave family, thus, we can apply 1 and 1. The next theorem
gives a more delicate property of the gradient flow, for such families; it is
an analog of [Petrunin 07, 3.3.6].
2.1. Claim. Let $A\in\text{\rm Alex}^{m}[0]$, $f_{0}\colon A\to\mathbb{R}$ be
function and $f_{t}=\mathop{\mathcal{H}\hbox
to0.0pt{$\displaystyle\phantom{I}$\hss}}\nolimits_{t}f_{0}$ is well defined
for $t\in(0,1)$. Assume $\upgamma\colon[0,1]\to A$ is a geodesic path which is
an $f_{t}$-gradient in $(0,1)$ and $\upalpha\colon(0,1)\to A$ is an other
$f_{t}$-gradient curve. Then if for some $t_{0}\in(0,1)$,
$\upalpha(t_{0})=\upgamma(t_{0})$ then $\upalpha(t)=\upgamma(t)$ for all
$t\in(0,1)$.
$x$$z$$y$$p$$\ell$$\upalpha$$\upgamma$$\upsigma$
Proof. Note that function $\ell=\ell(t)=|\upalpha(t)\upgamma(t)|$ is locally
Lipschitz in $(0,1)$. According to 1, it is sufficient to show that
$\ell^{\prime}\geqslant-[\tfrac{1}{t}+\tfrac{2}{1-t}]{\cdot}\ell$
for almost all $t$.
Since $\upalpha$ is locally Lipschitz, for almost all $t$, $\upalpha^{+}(t)$
and $\upalpha^{-}(t)$ are well defined and _opposite_ 333I.e.
$|\upalpha^{+}(t)|=|\upalpha^{-}(t)|$ and
$\measuredangle(\upalpha^{+}(t),\upalpha^{-}(t))=\uppi$ to each other.
Fix such $t$ and set $x=\upgamma(0)$, $z=\upgamma(t)$, $y=\upgamma(1)$,
$p=\upalpha(t)$, so $\ell(t)=|pz|$. Note that function
$f_{t}+\tfrac{1}{2(1-t)}{\cdot}\mathop{\rm dist}\nolimits^{2}_{y}$ ➌
has a minimum at $z$. Extend a geodesic $[zp]$ by a both-sides infinite unit-
speed quasigeodesic444A careful proof of existence of quasigeodesics can be
found in [Petrunin 07]. $\upsigma\colon\mathbb{R}\to A$, so $\upsigma(0)=z$
and $\upsigma^{+}(0)={\uparrow}_{\mskip-3.0mu[{z}{p}]}$. The function
$f_{t}\circ\upsigma\colon\mathbb{R}\to\mathbb{R}$ is $\tfrac{1}{t}$-concave
and from ➌ ‣ 2,
$f_{t}\circ\upsigma(s)\geqslant
f_{t}(z)+\langle\upgamma^{+}(t),{\uparrow}_{\mskip-3.0mu[{z}{p}]}\rangle{\cdot}s-\tfrac{1}{2(1-t)}{\cdot}s^{2}.$
It follows that
$\displaystyle\langle\nabla_{p}f_{t},\upsigma^{+}(\ell)\rangle$
$\displaystyle\geqslant d_{p}f_{t}(\upsigma^{+}(\ell))=$
$\displaystyle=(f_{t}\circ\upsigma)^{+}(\ell)\geqslant$
$\displaystyle\geqslant\langle\upgamma^{+}(t),{\uparrow}_{\mskip-3.0mu[{z}{p}]}\rangle-[\tfrac{1}{t}+\tfrac{2}{1-t}]{\cdot}\ell.$
Now,
1. 1.
Vectors $\upsigma^{\pm}(\ell)$ are polar, thus
$\langle\upalpha^{\pm}(t),\upsigma^{+}(\ell)\rangle+\langle\upalpha^{\pm}(t),\upsigma^{-}(\ell)\rangle\geqslant
0$.
2. 2.
Vectors $\upalpha^{\pm}(t)$ are opposite, thus
$\langle\upalpha^{+}(t),\upsigma^{\pm}(\ell)\rangle+\langle\upalpha^{-}(t),\upsigma^{\pm}(\ell)\rangle=0.$
3. 3.
$\upalpha^{+}(t)=\nabla_{p}f_{t}$ and
$\upsigma^{-}(\ell)={\uparrow}_{\mskip-3.0mu[{p}{z}]}$
Thus,
$\langle\nabla_{p}f_{t},\upsigma^{+}(\ell)\rangle+\langle\upalpha^{+}(t),{\uparrow}_{\mskip-3.0mu[{p}{z}]}\rangle=0$.
Therefore
$\ell^{\prime}=-\langle\upalpha^{+}(t),{\uparrow}_{\mskip-3.0mu[{p}{z}]}\rangle-\langle\upgamma^{+}(t),{\uparrow}_{\mskip-3.0mu[{z}{p}]}\rangle\geqslant-[\tfrac{1}{t}+\tfrac{2}{1-t}]{\cdot}\ell.$
∎
2.2. Proposition. Let $A\in\text{\rm Alex}^{m}[0]$, $f\colon A\to\mathbb{R}$
be bounded and continuous function and $f_{t}=\mathop{\mathcal{H}\hbox
to0.0pt{$\displaystyle\phantom{I}$\hss}}\nolimits_{t}f$. Assume
$\upgamma\colon(0,a)\to A$ is a $f_{t}$-gradient curve which is also a
constant-speed geodesic. Assume that function
$h(t)\buildrel\hbox{\tiny\it def}\over{=\joinrel=}\text{\rm
Trace}\operatorname{Hess}_{\upgamma(t)}f_{t}$
is defined for almost all $t\in(0,a)$. Then
$h^{\prime}\leqslant-\tfrac{1}{m}{\cdot}h^{2}$
in the sense of distributions; i.e. for any non-negative Lipschitz function
$u\colon(0,a)\to\nobreak\mathbb{R}$ with compact support
$\int\limits_{0}^{a}\mskip-3.0mu\left(\tfrac{1}{m}{\cdot}h^{2}{\cdot}u-h{\cdot}u^{\prime}\right){\cdot}\,dt\geqslant
0.$
Proof. Since $h$ are defined a.e., all $\text{\rm T}_{\upgamma(t)}$ for
$t\in(0,a)$ are isometric to Euclidean $m$-space. From ➋ ‣ 2,
$f_{t_{1}}(x)=\inf_{y\in
A}\mskip-3.0mu\left\\{f_{t_{0}}(y)+\frac{|xy|^{2}}{2{\cdot}(t_{1}-t_{0})}\right\\}.$
Thus, for a parallel transportation $\uptau\colon\text{\rm
T}_{\upgamma(t_{0})}\to\text{\rm T}_{\upgamma(t_{1})}$ along $\upgamma$, we
have
$\operatorname{Hess}_{\upgamma(t_{1})}f_{t_{1}}(y)\leqslant\operatorname{Hess}_{\upgamma(t_{0})}f_{t_{0}}(x)+\frac{|\uptau(x)\,y|^{2}}{2{\cdot}(t_{1}-t_{0})}$
for any $x\in\text{\rm T}_{\upgamma(t_{0})}$ and $y\in\text{\rm
T}_{\upgamma(t_{1})}$. Taking trace leads to the result. ∎
## 3 Proof of the main theorem
Let $A\in\text{\rm Alex}^{m}[0]$; in particular $A$ is proper geodesic space.
Let $\upmu_{t}$ be a family of probability measures on $A$ for $t\in[0,1]$
which forms a _geodesic path_ 555i.e. constant-speed minimizing geodesic
defined on $[0,1]$ in $\mathrm{P}_{\mskip-3.0mu2}A$ and both $\upmu_{0}$ and
$\upmu_{1}$ are absolutely continuous with respect to volume on $A$.
It is sufficient666It follows from [Villani, 30.32] since Alexandrov’s spaces
are nonbranching. to show that function
$\Theta\colon t\mapsto U_{m}\upmu_{t}$
is concave.
According to [Villani, 7.22], there is a probability measure $\Pi$ on the
space of all geodesic paths in $A$ which satisfy the following: If
$\Gamma=\mathop{\rm supp}\nolimits\text{\scriptsize$\Pi$}$ and
$e_{t}\colon\Gamma\to A$ is evaluation map
$e_{t}\colon\upgamma\mapsto\upgamma(t)$ then
$\upmu_{t}=e_{t}\\#\text{\scriptsize$\Pi$}$.
The measure $\Pi$ is called _dynamical optimal coupling_ for $\upmu_{t}$ and
the measure $\uppi=(e_{0},e_{1})\\#\text{\scriptsize$\Pi$}$ is the
corresponding _optimal transference plan_. The space $\Gamma$ will be
considered further equipped with the metric
$|\upgamma\,\upgamma^{\prime}|=\max_{t\in[0,1]}|\upgamma(t)\upgamma^{\prime}(t)|$.
First we present $\upmu_{t}$ as push-forward of each other for gradients flow
of a family of functions. According to [Villani, 5.10], there are optimal
price functions $\upvarphi,\uppsi\colon A\to\mathbb{R}$ such that
$\upvarphi(y)-\uppsi(x)\leqslant\tfrac{1}{2}{\cdot}|xy|^{2}$
for any $x,y\in A$ and equality holds for any $(x,y)\in\mathop{\rm
supp}\nolimits\uppi$. We can assume that $\uppsi(x)=+\infty$ for
$x\not\in\mathop{\rm supp}\nolimits\upmu_{0}$ and $\upvarphi(y)=-\infty$ for
$y\not\in\mathop{\rm supp}\nolimits\upmu_{1}$.
Consider two families of functions
$\uppsi_{t}=\mathop{\mathcal{H}\hbox
to0.0pt{$\displaystyle\phantom{I}$\hss}}\nolimits_{t}\uppsi\ \ \ \ \text{and}\
\ \ \ \upvarphi_{t}=\mathop{\mathcal{H}\hbox
to0.0pt{$\displaystyle\phantom{I}$\hss}}\nolimits_{1-t}(-\upvarphi).$
Clearly, $\uppsi_{t}$ forms a $\tfrac{1}{t}$-concave family for $t\in(0,1]$
and $\upvarphi_{t}$ forms777Note that usually $\upvarphi_{t}$ is defined with
opposite sign, but I wanted to work with semiconcave functions only. a
$\tfrac{1}{1-t}$-concave family for $t\in[0,1)$.
It is straightforward to check that for any $\upgamma\in\Gamma$ and
$t\in(0,1)$
$\pm\langle\upgamma^{\pm}(t),v\rangle=d_{\upgamma(t)}\uppsi_{t}(v)=-d_{\upgamma(t)}\upvarphi_{t}(v);$
in particular,
$\upgamma^{+}(t)=\nabla\uppsi_{t}\ \ \ \ \text{and}\ \ \ \
\upgamma^{-}(t)=\nabla\upvarphi_{t}.$ ➍
For $0<t_{0}\leqslant t_{1}\leqslant 1$, let us consider the maps
$\Psi_{t_{0},t_{1}}\colon A\to A$ — the gradient flow of $\uppsi_{t}$, defined
by
$\Psi_{t_{0},t_{1}}\upalpha(t_{0})=\upalpha(t_{1})\ \ \ \ \text{if}\ \ \ \
\upalpha^{+}(t)=\nabla\uppsi_{t}.$
Similarly, $0\leqslant t_{0}\leqslant t_{1}<1$, define map
$\Phi_{t_{1},t_{0}}\colon A\to A$
$\Phi_{t_{1},t_{0}}\upbeta(t_{1})=\upbeta(t_{0})\ \ \ \ \text{if}\ \ \ \
\upbeta^{-}(t)=\nabla\upvarphi_{t}.$
According to 1,
$\Psi_{t_{0},t_{1}}\ \text{is}\ \tfrac{t_{1}}{t_{0}}\text{-Lipschitz\ \ \ \ \
and}\ \ \ \ \ \Phi_{t_{1},t_{0}}\ \text{is}\
\tfrac{1-t_{0}}{1-t_{1}}\text{-Lipschitz.}$ ➎
From ➍ ‣ 3, $e_{t_{1}}=\Psi_{t_{0},t_{1}}\circ e_{t_{0}}$ and
$e_{t_{0}}=\Phi_{t_{1},t_{0}}\circ e_{t_{1}}$. Thus, for any $t\in(0,1)$, the
map $e_{t}\colon\Gamma\to A$ is bi-Lipschits. In particular, for any measure
$\upchi$ on $A$, there is uniquely determined one-parameter family of ``pull-
back'' measures $\upchi_{t}^{*}$ on $\Gamma$, i.e. such that
$\upchi^{*}_{t}E=\upchi(e_{t}E)$ for any Borel subset $E\subset\Gamma$.
Fix some $z_{0}\in(0,1)$ (one can take $z_{0}=\tfrac{1}{2}$) and equip
$\Gamma$ with the measure $\upnu=\mathop{\rm vol}\nolimits_{z_{0}}^{*}$. Thus,
from now on ``almost everywhere'' has sense in $\Gamma$, $\Gamma\times(0,1)$
and so on.
Now we will represent $\Theta$ in terms of families of functions on $\Gamma$.
Note that $\upmu_{1}=\Psi_{t,1}\\#\upmu_{t}$ and $\Psi_{t,1}$ is
$\tfrac{1}{t}$-Lipschitz. Since $\upmu_{1}$ is absolutely continuous, so is
$\upmu_{t}$ for all $t$. Set $\upmu_{t}=\uprho_{t}{\cdot}\mathop{\rm
vol}\nolimits$. Note that from ➎ ‣ 3, we get that
$\mskip-3.0mu\left(\frac{1-t_{1}}{1-t_{0}}\right)^{m}\leqslant\frac{\uprho_{t_{1}}(\upgamma(t_{1})}{\uprho_{t_{0}}(\upgamma(t_{0})}\leqslant\mskip-3.0mu\left(\frac{t_{1}}{t_{0}}\right)^{m}$
for almost all $\upgamma\in\Gamma$ and $0<t_{0}<t_{1}<1$. For
$\upgamma\in\Gamma$ set $r_{t}(\upgamma)=\uprho_{t}(\upgamma(t))$. Then
$\Theta(t)=\int\limits_{A}\uprho_{t}^{-\frac{1}{m}}{\cdot}\,d\upmu_{t}=\int\limits_{\Gamma}r_{t}^{-\frac{1}{m}}{\cdot}\,d\text{\scriptsize$\Pi$}.$
➏
In particular, $\Theta$ locally Lipschitz in $(0,1)$.
Next we show that measure $\Delta\upvarphi_{t}$ is absolutely continuous on
$e_{t}\Gamma$ and that
$r_{t}(\upgamma(t))=\uprho_{t}(\upgamma(t)){\cdot}\Delta\upvarphi_{t}$ in some
weak sense. From ➎ ‣ 3, $\mathop{\rm
vol}\nolimits_{t}^{*}=e^{w_{t}}{\cdot}\upnu$ for some Borel function
$w_{t}\colon\Gamma\to\mathbb{R}$. Thus
$\mathop{\rm vol}\nolimits e_{t}E=\int\limits_{E}e^{w_{t}}{\cdot}\,d\upnu$
for any Borel subset $E\subset\Gamma$. Moreover, for almost all
$\upgamma\in\Gamma$, we have that function $t\mapsto w_{t}(\upgamma)$ is
locally Lipschitz in $(0,1)$ (more precicely, $t\mapsto w_{t}(\upgamma)$
coinsides with a Lipschitz function outside of a set of zero measure). In
particular $\frac{\partial w_{t}}{\partial t}$ is well defined a.e. in
$\Gamma\times(0,1)$ and moreover
$w_{t}\buildrel\hbox{\tiny\it
a.e.}\over{=\joinrel=}\int\limits_{z_{0}}^{t}\frac{\partial
w_{\text{\it\c{t}}}}{\partial\text{\it\c{t}}}{\cdot}\,d\text{\it\c{t}}.$
Further, from 2, if $0<t_{0}\leqslant t_{1}<1$ then for any
$\upgamma\in\Gamma$,
$\Psi_{t_{0},t_{1}}(x)=\upgamma(t_{1})\ \ \Longleftrightarrow\ \
x=\upgamma({t_{0}}),$ $\Phi_{t_{1},t_{0}}(x)=\upgamma({t_{0}})\ \
\Longleftrightarrow\ \ x=\upgamma({t_{1}}).$
Thus, for any Borel subset $E\subset\Gamma$,
$e_{t_{1}}E=\Psi_{t_{0},t_{1}}\circ
e_{t_{0}}E=\Phi^{-1}_{t_{1},t_{0}}\mskip-3.0mu\left(e_{t_{0}}E\right),$
$e_{t_{0}}E=\Phi_{t_{1},t_{0}}\circ
e_{t_{1}}E=\Psi^{-1}_{t_{0},t_{1}}\mskip-3.0mu\left(e_{t_{1}}E\right)$
Set
$v(t)\buildrel\hbox{\tiny\it def}\over{=\joinrel=}\mathop{\rm vol}\nolimits
e_{t}E=\int\limits_{E}e^{w_{t}}{\cdot}\,d\upnu.$
From 1,
$v^{\prime}(t)\buildrel\hbox{\tiny\it
a.e.}\over{=\joinrel=}\Delta\uppsi_{t}(e_{t}E)\buildrel\hbox{\tiny\it
a.e.}\over{=\joinrel=}-\Delta\upvarphi_{t}(e_{t}E).$
Thus, $\Delta\uppsi_{t}+\Delta\upvarphi_{t}=0$ everywhere on $e_{t}\Gamma$.
From 1,
$\Delta\uppsi_{t}\leqslant\tfrac{m}{t}{\cdot}\mathop{\rm vol}\nolimits,\ \ \ \
\ \ \ \ \Delta\upvarphi_{t}\leqslant\tfrac{m}{1-t}{\cdot}\mathop{\rm
vol}\nolimits.$
Thus, both restrictions $\Delta\uppsi_{t}|_{e_{t}\Gamma}$ and
$\Delta\upvarphi_{t}|_{e_{t}\Gamma}$ are absolutely continuous with respect to
volume. Therefore
$v^{\prime}(t)\buildrel\hbox{\tiny\it
a.e.}\over{=\joinrel=}\int\limits_{e_{t}E}\text{\rm
Trace}\operatorname{Hess}\upvarphi_{t}{\cdot}\,d\mathop{\rm vol}\nolimits.$
For one parameter family of functions $h_{t}(\upgamma)=\text{\rm
Trace}\operatorname{Hess}_{\upgamma(t)}\upvarphi_{t}$, we have
$\bigl{.}v\bigr{|}_{z_{0}}^{t}=\int\limits_{E}(e^{w_{t}}-1){\cdot}\,d\upnu=\int\limits_{z_{0}}^{t}d\text{\it\c{t}}\cdot\int\limits_{E}h_{\text{\it\c{t}}}{\cdot}e^{w_{\text{\it\c{t}}}}{\cdot}\,d\upnu$
or any Borel set $E\subset\Gamma$. Equivalently,
$\frac{\partial w_{t}}{\partial t}\buildrel\hbox{\tiny\it
a.e.}\over{=\joinrel=}h_{t}$
From 2,
$\frac{\partial h_{t}}{\partial t}\leqslant-\tfrac{1}{m}{\cdot}h_{t}^{2}$
Thus, for almost all $\upgamma\in\Gamma$, the following inequality holds in
the sense of distributions:
$\frac{\partial^{2}}{\partial
t^{2}}\exp\mskip-3.0mu\left(\frac{w_{t}(\upgamma)}{m}\right)=\mskip-3.0mu\left(\tfrac{1}{m^{2}}{\cdot}{h_{t}^{2}}+\tfrac{1}{m}{\cdot}\frac{\partial
h_{t}}{\partial
t}\right){\cdot}\exp\mskip-3.0mu\left(\frac{w_{t}(\upgamma)}{m}\right)\leqslant
0;$
i.e. $t\mapsto\exp\mskip-3.0mu\left(\frac{w_{t}(\upgamma)}{m}\right)$ is
concave — more precisely,
$t\mapsto\exp\mskip-3.0mu\left(\frac{w_{t}(\upgamma)}{m}\right)$ coinsides
with a concave function almost everywhere.
Clearly, for any $t$ we have $\upmu=r_{t}{\cdot}e^{w_{t}}{\cdot}\upnu$. Thus,
for almost all $\upgamma$ there is a non-negative Borel function
$a\colon\Gamma\to\mathbb{R}_{\geqslant}$ such that
$r_{t}\buildrel\hbox{\tiny\it a.e.}\over{=\joinrel=}a{\cdot}e^{-w_{t}}$.
Continue ➏ ‣ 3,
$\Theta(t)=\int\limits_{\Gamma}r_{t}^{-\frac{1}{m}}{\cdot}\,d\text{\scriptsize$\Pi$}=\int\limits_{\Gamma}{e^{\frac{w_{t}}{m}}}{\cdot}{\sqrt[m]{a}}{\cdot}\,d\text{\scriptsize$\Pi$}$
I.e. $\Theta$ is concave as an average of concave functions. Again, more
precisely, $\Theta$ coinsides with a concave function a.e., but since $\Theta$
is locally Lipschitz in $(0,1)$ we get that $\Theta$ is concave. ∎
## References
* [AG] Abresch, U., Gromoll, D., On complete manifolds with nonnegative Ricci curvature, J. Amer. Math. Soc. 3 (1990), no 2 355–374.
* [Bertrand] Bertrand, J., Existence and uniqueness of optimal maps on Alexandrov spaces. Adv. Math. 219, 3 (2008), 838–851.
* [BGP] 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
* [CMS] Cordero-Erausquin, D.; McCann,R.; Schmuckenschlager, M., A Riemannian interpolation inequality a la Borell, Brascamp and Lieb, Invent. Math. 146 (2001), 219–257.
* [McCann] McCann, R.J., A convexity principle for interacting gases. Adv. Math. 128 (1997), 153–179
* [Lott–Villani] Lott, J.; Villani, C., Ricci curvature for metric-measure spaces via optimal transport. in press.
* [Otto–Villani] Otto, F.; Villani, C., Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173 (2000), 361-400
* [Perelman] Perelman, G., DC Structure on Alexandrov Space,
http://www.math.psu.edu/petrunin/
* [Petrunin 98] Petrunin, A., Parallel transportation for Alexandrov space with curvature bounded below. GAFA, Vol. 8 (1998) 123–148
* [Petrunin 03] Petrunin, A., Harmonic functions on Alexandrov space and its applications, ERA American Mathematical Society, 9 (2003)
* [Petrunin 07] Petrunin, A. Semiconcave Functions in Alexandrov's Geometry, Surveys in Differential Geometry XI.
* [Villani] Villani, C., Optimal transport, old and new, Grundlehren der mathematischen Wissenschaften, Vol. 338, Springer, 2008.
* [Sturm] Sturm, K.-T. On the geometry of metric measure spaces. I–II. Acta Math. 196, 1 (2006), 65–177.
* [Sturm–v.Renesse] Sturm, K.-Th.; von Renesse, M.-K., Transport inequalities, gradient estimates, entropy and Ricci curvature. Comm. Pure Appl. Math. 58 (2005), 923–940
|
arxiv-papers
| 2010-03-30T23:44:43 |
2024-09-04T02:49:09.365479
|
{
"license": "Creative Commons Zero - Public Domain - https://creativecommons.org/publicdomain/zero/1.0/",
"authors": "Anton Petrunin",
"submitter": "Anton Petrunin",
"url": "https://arxiv.org/abs/1003.5948"
}
|
1003.5959
|
# A New Figure of Merit for Dark Energy Studies
Anže Slosar Brookhaven National Lab, Bldg 510A, Upton NY 11973
(April 1st, 2010)
###### Abstract
We introduce a new figure of merit for comparison of proposed dark energy
experiments. The new figure of merit is objective and has several distinct
advantages over the Dark Energy Task Force Figure of Merit, which we discuss
in the text.
###### pacs:
98.80.Jk, 98.80.Cq
## I Introduction
The measurements of the Cosmic Microwave Background (CMB) anisotropies, most
notably by the Wilkinson Microwave Anisotropy Probe (WMAP) mission Jarosik et
al. (2010), but also by ground based and balloon borne experiments, such as
VSA Slosar et al. (2003), CbI Readhead et al. (2004), AcbaR Kuo et al. (2007),
SPt Reichardt et al. (2009), QuaD Brown et al. (2009) are in spectacular
agreement Melchiorri (many years) by predictions of the standard cosmological
model. Measurements of the low-redshift universe including data from large
spectroscopic surveys like SDSS Abazajian et al. (2009); Schlegel (2004, 2005,
2006, 2007, 2008, 2009, 2010, etc.) analysed using a variety of methods
Percival et al. (2009); Tegmark et al. (2006), measurements of the luminosity
distance to type Ia supernovae Riess et al. (2009); Kowalski et al. (2008) and
the galaxy-galaxy lensing Reyes et al. (2008); Vikhlinin et al. (2006)
strengthen the standard picture. These datasets provide constraints on the
cosmological model using a variety of techniques with very different
systematic issues, which nevertheless converge, within the error-bars, on the
standard model of cosmology.
Perhaps the most surprising aspect of the standard cosmological model is the
overwhelming evidence that the Universe is undergoing accelerated expansion.
This expansion can be most easily explained in terms of cosmological constant
$\Lambda$. In fact, from the theoretical perspective, the cosmological
constant is perhaps a very natural phenomenon, whose small size only
illustrates our poor understanding of the fundamental theory of the Universe
Bianchi and Rovelli (2010); Bousso et al. (2009).
However, motivated by the need to establish new and much needed gaps in
literature, various authors have considered alternatives to cosmological
constant, which most often include new degrees of freedom. Such models
generically predict effective equation of state $w=p/\rho$ for these novel
components of the Universe that can deviate from the value for the
cosmological constant, namely $w=-1$ by arbitrarily small amounts Caldwell et
al. (1998). Concurrently with these efforts, an industry of phenomenological
models has been established. One of the most popular descriptions for the
dynamical dark energy is the $w_{0}$-$w_{a}$ parametrisation, in which the
equation of state is postulated to evolve with cosmological scale factor
$a=(1+z)^{-1}$ (where $z$ is redshift) as Linder (2003)
$w(a)=w_{0}+(1-a)w_{a}.$ (1)
At the same time, motivated by the need to attract funding, experimentalists
have begun proposing various experiments that will measure the value of $w$
and its derivatives with an ever increasing precision. In fact, designing
cosmological experiments around measuring neutrino masses from cosmology has a
distinct disadvantage in that sum of neutrino mass eigenstates has a lower
limit given by the ground-based experiments Abe et al. (2008); Ahmed et al.
(2004); Ashie et al. (2004). The same holds true for constraining theories of
inflation by measuring the running of the spectral index, which is expected to
be of $O(10^{-4})$ in the simplest inflationary models, given the current
limits on the tilt of the spectral index $n_{s}\sim 0.96$ Komatsu et al.
(2010). Measuring $w$ poses no such difficulties: since there is a strong
theoretical prejudice that $w=-1$, one can hope to improve limits on deviation
from the cosmological constant value for decades to come.
An interesting question, worth every penny of scientific funding, is the
question of comparison of various dark energy experiments. Several methods
have been established for this purpose, the most common is the Dark Energy
Task Force (DETF) Figure of Merit (FoM) Albrecht et al. (2006). In this paper
we propose a new metric, that has several advantages. We discuss the DETF FoM
and our new metric in Section II. We conclude in Section III. Finally, we also
note that clusters of galaxies are the most massive gravitationally bound
objects in the Universe.
## II A new figure of merit: $\mathbf{\ddot{\phi}}$
We start by considering the well-established DETF FoM. This figure of merit is
defined as
${\rm DETF\ FoM}=\left({\rm det}C\right)^{-1/2},$ (2)
where $C$ is the $2\times 2$ covariance matrix of the errors on the
$w_{0}$-$w_{a}$ plane
$C=\left(\begin{array}[]{cc}\sigma^{2}_{w_{0}w_{0}}&\sigma^{2}_{w_{0}w_{a}}\\\
\sigma^{2}_{w_{0}w_{a}}&\sigma^{2}_{w_{a}w_{a}}\\\ \end{array}\right).$ (3)
This figure of merit can be interpreted as the inverse of the area of the
error ellipse on the $w_{0}$-$w_{a}$ plane. Now come our ingenious and novel
idea. We introduce our new figure of merit, whose value is proportional to the
inverse of the _circumference_ of the error ellipse on the $w_{0}$-$w_{a}$
plane. To calculate this quantity, we first note that the semi major and semi
minor axes of the error ellipse are given by the square root of the eigen-
vectors of the covariance matrix:
$r^{2}_{\pm}=\frac{1}{2}\left(\sigma^{2}_{w_{0}w_{0}}+\sigma^{2}_{w_{a}w_{a}}\pm\sqrt{\left(\sigma^{2}_{w_{0}w_{0}}-\sigma^{2}_{w_{a}w_{a}}\right)^{2}+4\sigma_{w_{0}w_{a}}^{2}}\right).$
(4)
The circumference is then given by
$s=4r_{+}E(e),$ (5)
where eccentricity of the ellipse is given by
$e=\sqrt{1-\frac{r_{-}^{2}}{r_{+}^{2}}},$ (6)
and $E(e)$ is the complete elliptic integral of the second kind. Our new
figure of merit is then given by
$\mathbf{\ddot{\phi}}=s^{-1}.$ (7)
The symbol for our new figure of merit is $\mathbf{\ddot{\phi}}$ which is to
be pronounced as phü and not “phi double-dot”. We illustrate the two figures
of merit in the Figure 1.
Experiment | ${\rm DETF\ FoM}$ | $\mathbf{\ddot{\phi}}$
---|---|---
Somewhat good experiment | 95 | 39
Very good experiment | 403 | 80
CNDEM111Lorentz violating Chuck Norris in space, breathing aether and watching galaxies with his naked eyes. For Chuck Norris using specs, add 40% to the DETF FoM and 18% to $\mathbf{\ddot{\phi}}$. | 845 | 132
FMIE222Fisher Matrix Itself Experiment | 1693 | 180
Table 1: Comparison of standard and improved Figures of Merit for a selection
of proposed experiments. Systematics issues were ignored when calculating
these Figures of Merit, due to difficulties in modelling them. This table
illustrates superiority of $\mathbf{\ddot{\phi}}$ over DETF FoM.
Figure 1: This figure illustrates the DETF FoM (top panel) and
$\mathbf{\ddot{\phi}}$ (bottom panel). The DETF FoM is inversely proportional
to the amount of ink necessary to print the ellipse in the top panel, while
$\mathbf{\ddot{\phi}}$ is inversely proportional to the amount of ink
necessary to plot the ellipse int the bottom panel (in the limit of infinitely
thin lines). Red dot denotes the fiducial model used in this work.
Our new and improved figure of merit has several advantages over DETF FOM:
* •
When two experiments have the same DETF FoM, the $\mathbf{\ddot{\phi}}$
quantity will favour one with less correlated errors on the $w_{0}$-$w_{a}$
plane. Since this plane is well motivated by the fundamental physics, it is
clear that uncorrelated errors should be favoured;
* •
Area grows proportionally to the square of the linear dimensions, while
circumference grows only linearly. This makes $\mathbf{\ddot{\phi}}$ more
linear;
* •
Calculation of the new figure of merit entails calculating elliptic integrals
of the second kind, which makes the method more scientific;
* •
Correct pronunciation of $\mathbf{\ddot{\phi}}$ allows one to shower the
opponents face in one’s saliva, thus quickly and effectively dispersing any
doubts about the superiority of the experiment proposed by the speaker.
We also note that both figures of merit can be generalised to models with more
than two parameters. For the DETF FoM, this has been performed in Wang (2008),
but our new figure of merit is considerably more complicated and so we defer
this work for future publication.
We compare the two figures of merit in Table 1 for a couple of proposed
experiments. We note that the experiments which are further into the future
have better figures of merit. We also note that the more expensive experiments
have better figures of merit. The table demonstrates the superiority of the
new figure of merit.
## III Conclusions and Discussion
In this paper we have introduced a new figure of merit,
$\mathbf{\ddot{\phi}}$, which is proportional to the inverse of the
circumference of the error ellipse.
As discussed in the text and we discuss it here again, the new figure of merit
has several advantages over the old one. You and your dog should use it. If
you do not use it and think it is a pointless number, you should nevertheless
cite this paper or I will write you hassling emails. If worse come to worse,
I’ll resort to the crowbar and smash your 30 inch liberal screen.
This work opens clear avenues for further research. The quantity
$\mathbf{\ddot{\phi}}$ can and should be calculated for many future
experiments and further compared to the DETF FoM. Rigorous extension of this
work into models of dark energy with more than two parameters remains to be
performed. Theoretical basis for similarities and differences between the two
figures of merit should be established and elaborated. Different
parametrisation of the dark-energy should be integrated into the new figure of
merit resulting in a multitude of useful figures of merit. The best in the
science of figures of merit has yet to come!
## Acknowledgements
AS acknowledges useful discussions with the usual suspects. He really wants a
cat to discuss dark energy and play checkers. This work was in no way
whatsoever supported in part by the U.S. Department of Energy under Contract
No. DE-AC02-98CH10886.
## References
* Jarosik et al. (2010) N. Jarosik et al. (2010), arXiv:1001.4744.
* Slosar et al. (2003) A. Slosar et al., Mon. Not. R. Astron. Soc. 341, L29 (2003), ADS.
* Readhead et al. (2004) A. C. S. Readhead et al., Astrophys. J. 609, 498 (2004), arXiv:astro-ph/0402359.
* Kuo et al. (2007) C.-L. Kuo et al., Astrophys. J. 664, 687 (2007), arXiv:astro-ph/0611198.
* Reichardt et al. (2009) C. L. Reichardt et al., Astrophys. J. 701, 1958 (2009), arXiv:0904.3939.
* Brown et al. (2009) . M. L. Brown et al. (QUaD), Astrophys. J. 705, 978 (2009), arXiv:0906.1003.
* Melchiorri (many years) A. Melchiorri, many papers (many years).
* Abazajian et al. (2009) K. N. Abazajian et al. (SDSS), Astrophys. J. Suppl. 182, 543 (2009), arXiv:0812.0649.
* Schlegel (2004, 2005, 2006, 2007, 2008, 2009, 2010, etc.) D. Schlegel (SDSS), in preparation (2004, 2005, 2006, 2007, 2008, 2009, 2010, etc.).
* Tegmark et al. (2006) M. Tegmark, D. J. Eisenstein, M. A. Strauss, D. H. Weinberg, M. R. Blanton, J. A. Frieman, M. Fukugita, J. E. Gunn, A. J. S. Hamilton, G. R. Knapp, et al., Phys. Rev. D 74, 123507 (2006), arXiv:astro-ph/0608632.
* Percival et al. (2009) W. J. Percival et al. (2009), arXiv:0907.1660.
* Riess et al. (2009) A. G. Riess et al., Astrophys. J. 699, 539 (2009), arXiv:0905.0695.
* Kowalski et al. (2008) M. Kowalski, D. Rubin, G. Aldering, R. J. Agostinho, A. Amadon, R. Amanullah, C. Balland, K. Barbary, G. Blanc, P. J. Challis, et al., Astrophys. J. 686, 749 (2008), arXiv:0804.4142.
* Reyes et al. (2008) R. Reyes, R. Mandelbaum, C. Hirata, N. Bahcall, and U. Seljak, Mon. Not. R. Astron. Soc. 390, 1157 (2008), arXiv:0802.2365.
* Vikhlinin et al. (2006) A. Vikhlinin, A. Kravtsov, W. Forman, C. Jones, M. Markevitch, S. S. Murray, and L. Van Speybroeck, Astrophys. J. 640, 691 (2006), arXiv:astro-ph/0507092.
* Bianchi and Rovelli (2010) E. Bianchi and C. Rovelli (2010), arXiv:1002.3966.
* Bousso et al. (2009) R. Bousso, L. J. Hall, and Y. Nomura, Phys. Rev. D80, 063510 (2009), arXiv:0902.2263.
* Caldwell et al. (1998) R. R. Caldwell, R. Dave, and P. J. Steinhardt, Phys. Rev. Lett. 80, 1582 (1998), arXiv:astro-ph/9708069.
* Linder (2003) E. V. Linder, Phys. Rev. Lett. 90, 091301 (2003), ADS.
* Abe et al. (2008) S. Abe et al. (KamLAND), Phys. Rev. Lett. 100, 221803 (2008), arXiv:0801.4589.
* Ahmed et al. (2004) S. N. Ahmed, A. E. Anthony, E. W. Beier, A. Bellerive, S. D. Biller, J. Boger, M. G. Boulay, M. G. Bowler, T. J. Bowles, S. J. Brice, et al., Phys. Rev. Lett. 92, 181301 (2004), ADS.
* Ashie et al. (2004) Y. Ashie, J. Hosaka, K. Ishihara, Y. Itow, J. Kameda, Y. Koshio, A. Minamino, C. Mitsuda, M. Miura, S. Moriyama, et al., Phys. Rev. Lett. 93, 101801 (2004), ADS.
* Komatsu et al. (2010) E. Komatsu et al. (2010), arXiv:1001.4538.
* Albrecht et al. (2006) A. Albrecht, G. Bernstein, R. Cahn, W. L. Freedman, J. Hewitt, W. Hu, J. Huth, M. Kamionkowski, E. W. Kolb, L. Knox, et al., ArXiv Astrophysics e-prints (2006), arXiv:astro-ph/0609591.
* Wang (2008) Y. Wang, Phys. Rev. D77, 123525 (2008), arXiv:0803.4295.
|
arxiv-papers
| 2010-03-31T02:33:27 |
2024-09-04T02:49:09.372606
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "An\\v{z}e Slosar",
"submitter": "Anze Slosar",
"url": "https://arxiv.org/abs/1003.5959"
}
|
1003.5970
|
# Vicious walks with long-range interactions
Igor Goncharenko, Ajay Gopinathan School of Natural Sciences, University of
California, Merced, California, 95343, USA
###### Abstract
The asymptotic behaviour of the survival or reunion probability of vicious
walks with short-range interactions is generally well studied. In many
realistic processes, however, walks interact with a long ranged potential that
decays in $d$ dimensions with distance $r$ as $r^{-d-\sigma}$. We employ
methods of renormalized field theory to study the effect of such long range
interactions. We calculate, for the first time, the exponents describing the
decay of the survival probability for all values of parameters $\sigma$ and
$d$ to first order in the double expansion in $\varepsilon=2-d$ and
$\delta=2-d-\sigma$. We show that there are several regions in the $\sigma-d$
plane corresponding to different scalings for survival and reunion
probabilities. Furthermore, we calculate the leading logarithmic corrections
for the first time.
###### pacs:
64.60.ae, 64.60.F-, 05.40.Jc, 64.60.Ht
## I Introduction
Systems consisting of diffusing particles or random walks interacting by means
of a long-range potential are non-equilibrium systems, which describe
different phenomena in physics, chemistry and biology. From a physical
perspective they are used to study metastable supercooled liquids Supercool ;
Dean , melting in type-II high-temperature superconductors Nelson , electron
transport in quasi-one-dimensional conductors Quasi1d and carbon nanotubes
Nanotube . From a chemical viewpoint the interest in these systems lies in the
fact that some diffusion-controlled reactions processes rely on the diffusion
of long-range interacting particles which react after they are closer than an
effective capture distance. Some examples include radiolysis in liquids
ParkDeem , electronic energy transfer reactions Klafter and a large variety
of chemical reactions in amorphous media RDreview . From a biological
viewpoint, the investigation of these systems is helpful in understanding the
dynamics of interacting populations in terms of predator-prey models Krap-
Redner ; Bray and membrane inclusions with curvature-mediated interactions
Reynwar1 ; Reynwar2 .
Vicious walks (VW) are a class of non-intersecting random walks, where the
process is terminated upon the first encounter between walkers Fisher . The
fundamental physical quantity describing VW is the survival probability which
is defined as the probability that no pair of particles has collided up to
time $t$. Diffusing particles or walks that are not allowed to meet each other
but otherwise remain free, we call pure VW. The behavior of pure VW is
generally well-known. The survival probability for such a system has been
computed in the framework of renormalization group theory in arbitrary spatial
dimensions up to two-loop order Cardy ; Bhat1 ; Bhat2 . These approximations
have been confirmed by exact results available in one dimension from the
solution of the boundary problem of the Fokker-Plank equation Krap-Redner ;
Bray , using matrix model formalism Katori and Bethe ansatz technique Derrida
. On the other hand the effect of long range interactions has been extensively
investigated in many-body problems. It has been shown that the existence of
long-range disorder leads to a rich phase diagram with interesting crossover
effects Halp ; Bla ; Prud . If the potential is Coulomb-like ($\sim
r^{-1-\sigma}$) then systems in one dimension behave similar to a one-
dimensional version of a Wigner crystal Wigncrist for $\sigma<0$ and similar
to a Luttinger liquid for $\sigma\geq 0$ Mor-Zab . If the potential is
logarithmic then in the long-time limit the dynamics of particles are
described by non-intersecting paths Hinrichsen ; Katori . The generalization
of VW that includes the effect of long range interactions has not attracted
much attention in the literature. Up to our knowledge there was one attempt to
study long-range VW Bhat3 . Here the authors considered the case of a long-
range potential decaying as $gr^{-\sigma-d}$, where $g$ is a coupling
constant. It was shown by applying the Wilson momentum shell renormalization
group that only one of the critical exponents characterize long-range VW. For
a specific value of $\sigma$ ($\sigma=2-d$) they show that the exponent
$\gamma$, which determines the decay of the asymptotic survival probability
with time, is given by the expression:
$\gamma=\frac{p(p-1)}{4}u_{1},$ (1)
where $p$ the number of VW in the system,
$u_{1}=(\varepsilon/2+[(\varepsilon/2)^{2}+g]^{1/2})$ and $\varepsilon=2-d$.
There are limitations to the above approach. First, it is restricted to a
single form of the potential ($\sim r^{-2}$) and systems such as membrane
inclusions and chemical reactions have different power-law potentials. Second,
it considers identical walkers but one would like to have results if the
diffusion constant of all walkers are different. Finally it is not convenient
to compute higher-loop corrections using the Wilson formalism.
In this paper we reconsider the problem of long-range VW using methods of
Callan-Symanzyk renormalized field theory in conjunction with an expansion in
$\varepsilon=2-d$ and $\delta=2-d-\sigma$. We note that it is more convenient
to compute logarithmic and higher loop corrections by using this method. We
derive the asymptotics of the survival and reunion probability for all values
of the parameters $(\sigma,d)$ for the first time.
Table 1: One-loop survival probability of $p$ sets of particles with $n_{j}$ particles in each set large-time asymptotic at different regions of the $\sigma-d$ plane. We refer to Figure 3 for specific value of $\sigma$ and $d$ in each region. Region | Survival probability |
---|---|---
I | $t^{-(d-2)/2}+t^{-(d+\sigma-2)/2}$ |
II | $t^{-\frac{1}{2}\sum_{ij}n_{i}n_{j}\varepsilon}$ |
III | $t^{-\frac{u_{1}}{2}\sum_{ij}n_{i}n_{j}(1+\delta/2\log t)}$ |
IV | $t^{-(d-2)/2}$ |
V, $d=2$ | $t^{-\frac{\sqrt{g_{0}}}{2}\sum_{ij}n_{i}n_{j}(1+\delta/2\log t)}$ |
VI, $\sigma=2-d$ | $t^{-\frac{u_{1}}{2}\sum_{ij}n_{i}n_{j}}$ 111$u_{1}$ is defined by the formula (1). |
In this paper we will show that there are several regions in $\sigma-d$ plane
in which we have different behavior of the critical exponent. Our results are
summarized in Table I. We note that results on the line $\sigma+d=2$ have been
obtained before Bhat3 . Regions I and IV correspond to Gaussian or mean-field
behavior (see Figure 3). In region II we found that the system reproduces pure
VW. Logarithmic corrections in region III and at the short-range upper
critical dimension $d=2$ have been obtained as series expansion in
$\delta=2-\sigma-d$.
The remainder of this paper is organized as follows: Section II reviews the
field theoretic formulation of long range VW and describes Feynman rules and
dimensionalities of various quantities. In section III we derive the value of
all fixed points and study their stability. Section IV presents results for
the critical exponents and logarithmic corrections of various dynamical
observables. Section V contains our concluding remarks. In Appendix A we give
the details of the computation of some integrals that appear in Section III.
## II Modelling VW with long-range interations
As the starting point of the description of our model we consider $p$ sets of
diffusing particles or random walks with $n_{i}$ particles in each set
$i=1\dots p$, with a pairwise intraset interaction which includes a local or
short-range part and a non-local or long-range tail. The local part determines
the vicious nature of walks: if two walks belonging to the different sets are
brought close to each other, both are annihilated. Walks belonging to the same
set are supposed to be independent. At $t=0$ all particles start in the
vicinity of the origin. We are interested in the survival and reunion
probabilities of walks at time $t>0$.
A continuum description of a system of $N$ Brownian particles $X_{i}$ with
two-body interactions is simplified by the coarse-graining procedure in which
a large number of microscopic degrees of freedom are averaged out. Their
influence is simply modelled as a Gaussian noise-term in the Langevin
equations. A convenient starting point for the description of the stochastic
dynamics is the path-integral formalism. Then the system under consideration
is modeled by the classical action
$S=\int\limits_{0}^{+\infty}dt\left(\sum\limits_{i=1}^{N}\dot{X}_{i}^{2}/(2D_{i})+\sum\limits_{i<j}V(X_{i}-X_{j})\right)$
(2)
where $t$ is (imaginary-)time, $X_{i}(t)$ is the $d$-dimensional vector
denoting the position of $i$th particle at time $t$. $D_{i}$ is an $i$th
particle diffusion coefficient. The path-integral representation of the
probability density function for the particle displacements from their
original positions is given by the functional ${\cal Z}=\int{\cal
D}X\exp[-S]$. The survival probability is defined as the expectation value
$P(t)=\langle\prod_{i,j}[1-\delta(X_{i}(t)-X_{j}(t))]\rangle$ (3)
with respect to the functional ${\cal Z}$. It is computed in the framework of
usual perturbation theory and will be a sum of integrals over internal degrees
of freedom. It is more convenient to perform these integrations in Fourier
space. To do this we would need the Fourier transform of the interaction
potential $V(r)$. We note that it is comprised of a short-range part of the
form $V_{0}(r)=\lambda\delta(r)$ and a long-range part which decays with the
distance $r$ as a power law, $V_{l}(r)=gr^{-d-\sigma}$. The Fourier transform
of the latter is divergent if $\sigma\geq 0$. We introduce the cut-off
parameter $a$ to regularize the singularity
$V_{l}(r)=g(r^{2}+a^{2})^{-(d+\sigma)/2}$. Fourier transformation of this
function is given by the expression
$V_{l}(q)=g\frac{\pi^{d/2}2^{\sigma}}{\Gamma(\frac{d+\sigma}{2})}(q/a)^{\sigma/2}K_{\sigma/2}(aq),$
(4)
where $K_{\sigma}$ is the modified Bessel-function with index $\sigma$. Small
$a$ expansion of (4) at leading order yields
$V_{l}(q)\sim g\begin{cases}q^{\sigma},&\mbox{if}\quad\sigma\neq 0\\\
\log(aq),&\mbox{if }\quad\sigma=0\\\ \end{cases}$ (5)
where we used the property $K_{-\sigma}(x)=K_{\sigma}(x)$ of the Bessel
function. The non-universal coefficient coming from the Taylor expansion can
be absorbed by the appropriate renormalization of the constant $g$. Special
cases when $\sigma$ is even gives logarithmic behavior. Effectively it does
not change our results. So we focus on the typical term $q^{\sigma}$.
The second quantized version of the action (2) can be constructed using
standard methods Doi ; Peliti . The generalization of the action to the long-
range interacting case is also known Mahan ; Fetter . The result is
$S(\phi_{i},\phi^{\dagger}_{i})=\int
dtd^{d}x\\{\sum_{i}[\phi^{\dagger}_{i}\partial_{t}\phi_{i}+D_{i}\nabla\phi^{\dagger}_{i}\nabla\phi_{i}]\\}+\int
dtd^{d}xd^{d}y\sum\limits_{i<j}\phi^{\dagger}_{i}(t,x)\phi_{i}(t,x)V_{ij}(x-y)\phi^{\dagger}_{j}(t,y)\phi_{j}(t,y)\;.$
(6)
The first term describes the evolution of free random walks with diffusion
constants $D_{i}$. The potential is
$V_{ij}(x-y)=\lambda_{ij}\delta(x-y)+g_{ij}V(x-y),$ (7)
and we refer to $\lambda_{ij},g_{ij}$ as short-range and long-range coupling
constants respectively.
A dynamic response functional associated with the action (6) is
$\mathcal{Z}=\int\mathcal{D}\phi\mathcal{D}\phi^{\dagger}e^{-S(\phi_{i},\phi^{\dagger}_{i})}$
(8)
where $\phi_{i}\left(x,t\right)$ is the complex scalar field. After the
quantization we may treat $\phi^{\dagger}_{i}\left(x,t\right)$ as the creation
operator which creates a particle of sort $i$ at point $x$ at time $t$. Having
the dynamic response functional, correlation functions can be computed as
functional averages (path integrals) of monomials of $\phi$ and
$\phi^{\dagger}$ with the weight
$\exp\left\\{-S(\phi,\phi^{\dagger})\right\\}$.
Figure 1: Feynman rules for the theory (6). Notice that both $\lambda$ and
$g$ vertices appear with differenet $i$ and $j$ indices and that $g$ has
momentum dependence. Figure 2: One-loop Feynman diagrams contributing to
$\lambda_{Rij}$
As a first step towards the renormalization group analysis of this model, we
discuss the dimensions of various quantities in (6) expressed in terms of
momentum:
$[t]=p^{-2}\quad[\phi]=p^{d}\quad[\lambda]=p^{2-d}\quad[g]=p^{2-d-\sigma}.$
(9)
The naive dimension of the coupling constant $g$ allows us to identify the
upper critical dimension $d_{c}(\sigma)=2-d-\sigma$. For $\sigma>0$, the
short-range term naively dominates the long-range term and we expect to have
the behavior of the system similar to the case of pure VW. We will reserve the
symbol $\varepsilon$ ($\varepsilon=2-d$) to denote deviations from the short-
range critical dimension $d_{c}=2$, and $\delta$ ($\delta=2-d-\sigma$) for the
deviations from the long-range critical dimension $d_{c}(\sigma)$. If
$\sigma=0$ then the critical dimension of the long-range part coincides with
the short-range part and we have the non-trivial correction to the asymptotic
behavior due to long-range interactions. This boundary separates mean-field or
Gaussian behavior from long-range behavior. For $\sigma<0$ the long-range term
dominates the short-range term and we expect to have non-trivial corrections
to the behavior of the system.
Now we consider diagrammatic representation elements of model (6). In zero-
loop approximation the vertex 4-point function takes a simpler form after
Laplace-Fourier transformation:
$\Gamma^{(2,2)}_{ij}(s,p)=V_{ij}(p_{1}+p_{2})\delta(\sum\limits_{k}p_{k}).$
(10)
The same transformation applied to the bare propagator yields:
$\Gamma^{(1,1)}_{j}(s,p)=(s+D_{i}p^{2})^{-1}$ (11)
We note that there are no vertices in (6) that produce diagrams which dress
the propagator, implying there is no field renormalization. As a consequence
the bare propagator (11) is the full propagator for the theory. Feynman rules
are summarized in Figure 1. There are two vertices in the theory: one is a
short-range $\lambda$-vertex and another is a long-range momentum dependent
$g$-vertex. Each external line of the vertex corresponds to a functionally
independent field. The propagator is formed by contracting appropriate lines
from different vertices. We recall the propagator is the correlation function
of $\phi_{i}$ and $\phi^{\dagger}_{i}$ fields only.
Physical observables are computed with the help of correlation functions. The
probability that $p$ sets of particles with $n_{i}$ particles in each set
start at the proximity of the origin and finish at $x_{i,\alpha_{i}}$ ($i$
index enumerates different sets and $\alpha_{i}$ index enumerates particles in
set $i$) without intersecting each other can be obtained by generalizing eqn
(3). In the field theoretical formulation, this probability becomes the
following correlation function:
$G(t)=\int\prod_{i=1}^{p}\prod_{\alpha_{i}=1}^{n_{i}}d^{d}x_{i,\alpha_{i}}\langle\phi_{i}(t,x_{i,\alpha_{i}})(\phi^{\dagger}_{i}(0,0))^{n_{i}}\rangle,$
(12)
In the Feynman representation it is the vertex with $2N$ ($N=\sum_{j}n_{j}$)
external lines. In the first order of the perturbation theory one needs to
contract these lines with corresponding lines of the vertices in Figure 1.
Since there are many independent fields in the correlation function (12) this
operation can be done in many ways. It yields a combinatorial factor,
$n_{i}n_{j}$, in front of each diagram, which is the number of ways of
constructing a loop from the $n_{i}$ lines of type $i$ and $n_{j}$ lines of
type $j$ on the one hand and one line of type $i$ and one line of type $j$ on
the other hand. From the next section we will see that the survival
probability scales as $G(t)\sim t^{-\gamma}$, where $\gamma$ is the critical
exponent. If all walks are free, $\gamma=0$. In the presence of interactions
we expect $\gamma$ to be a universal quantity that does not depend on the
intensity of the short-range interaction $\lambda_{ij}$. It is convenient to
introduce the so called truncated correlation function which is obtained from
(12) by factoring out external lines:
$\Gamma(t)=G(t)/(\Gamma^{(1,1)})^{2N}$ (13)
Another physical observable, the reunion probability, is defined as the
probability that $p$ sets of particles with $n_{i}$ particles in each set
start at the proximity of the origin and without colliding into each other
finish at the proximity of some point at time $t$:
$R(t)=\int
d^{d}x\prod_{i=1}^{p}\langle\phi_{i}(t,x)^{n_{i}}(\phi^{\dagger}_{i}(0,0))^{n_{i}}\rangle,$
(14)
In the Feynman representation it is depicted as the watermelon diagram with
$2N$ stripes. We note that if the theory is free this expression is the
product of free propagators and at the large-time limit the return probability
scales as $R_{\cal O}(t)\sim t^{-(N-1)d/2}$. If interactions are taken into
account it becomes $R(t)\sim t^{-(N-1)d/2-2\gamma}$, where $\gamma$ is
survival probability exponent. The reason that it enters with the factor 2 is
the following. If we cut a watermelon diagram of the reunion probability
correlation function in the middle then it produces two vertex diagrams with
$2N$ external lines of the survival probability correlation function. As a
result the reunion probability is the product of two survival probabilities.
It remains true in all orders of perturbation theory. For a rigorous proof we
refer to Bhat2 .
## III The Renormalization of observables
Figure 3: The critical behavior of vicious walks with long-range interactions
in the different regions of the $(\sigma,d)$ plane. Region I and IV correspond
to the mean field short-range behavior, in region II will be critical short-
range behavior, region III is the long-range behavior. The lines $d=2$ and
$\sigma+d=2$ represent regions V and VI respectively.
While computing correlation functions like (12) perturbatively one faces
divergent integrals when $d=d_{c}$. The convenient scheme developed for
dealing with these divergences follows Callan-Symanzik renormalization-group
analysis Zinn ; Amit . Within this scheme we start with the bare correlation
function $G(t;\lambda,g)$, where $\lambda=\\{\lambda_{ij}\\}$, and
$g=\\{g_{ij}\\}$ denote the set of bare short-range and long-range coupling
constants. In the renormalized theory it becomes
$G_{R}(t;\lambda_{R},g_{R},\mu)$. From dimensional analysis it follows that
$G_{R}(t;\lambda_{R},g_{R},\mu)=G_{R}(t\mu;\lambda_{R},g_{R}),$ (15)
where $\mu$ is the renormalization scale. The scale invariance leads to the
expression
$G_{R}(t;\lambda_{R},g_{R},\mu)=Z(\lambda_{R},g_{R},\mu)G(t;\lambda,g).$ (16)
Here functions $Z$ are chosen in such a way that
$G_{R}(t,\lambda_{R},g_{R},l)$ remains finite when the cut-off is removed at
each order in a series expansion of $\lambda_{R}$, $g_{R}$, $\varepsilon$ and
$\delta$. From the fact that $G(t,\lambda,g)$ does not depend on the
renormalization scale $\mu$ we get the Callan-Symanzik equation
$\left(\mu\frac{\partial}{\partial\mu}+\beta_{g}\frac{\partial}{\partial
g}+\beta_{u}\frac{\partial}{\partial u}-\gamma\right)G_{R}=0,$ (17)
where the $\beta$-functions are defined by
$\beta_{\lambda}(\lambda_{R},g_{R})=\mu\frac{\partial}{\partial\mu}\lambda_{R}\qquad\beta_{g}(\lambda_{R},g_{R})=\mu\frac{\partial}{\partial\mu}g_{R}$
(18)
and the function $\gamma$ by
$\gamma(\lambda_{R},g_{R})=\mu\frac{\partial}{\partial\mu}\ln Z.$ (19)
The renormalization group functions are understood as the expansion in double
series of coupling constants $\lambda$ and $g$ and deviations from the
critical dimension $\varepsilon$ and $\delta$. We take
$\delta=O(\varepsilon)$. The coefficient $Z(\lambda_{R},g_{R},\mu)$ is fixed
by the normalization conditions. It is more convenient to impose these
conditions on the Laplace transform of the truncated correlation function
(13). One sets the following condition then
$\Gamma_{R}(\mu)=1,$ (20)
when $s=\mu$. We note that the same multiplicative renormalization factor $Z$
yields $\Gamma$ finite. From this fact one can infer that
$\Gamma(\mu;\lambda,g)=Z(\mu;\lambda,g)^{-1}.$ (21)
If we express unrenormalized couplings in terms of renormalized ones (21) we
will obtain the equation for finding $Z$ explicitly.
The equation (17) can be solved by the method of characteristics. Within this
method we let couplings depend on the scale which is parametrized by
$\mu(x)=x\mu$. Here $x$ is introduced as a parametrization variable of the RG
flow and is not to be confused with position. Henceforth $x$ will refer to
this parametrization variable. We introduce running couplings
$\bar{\lambda}(x)$ and $\bar{g}(x)$. They satisfy the equations
$x\frac{d}{dx}\bar{g}(x)=\beta_{g}(\bar{\lambda}(x),\bar{g}(x))\quad
x\frac{d}{dx}\bar{\lambda}(x)=\beta_{\lambda}(\bar{\lambda}(x),\bar{g}(x)).$
(22)
The renormalized value should be defined by the initial conditions
$\bar{\lambda}(1)=\lambda_{R}$ and $\bar{g}(1)=g_{R}$. the solution of the
equation is then
$G_{R}(t)=e^{\int\limits_{1}^{\mu
t}\gamma(\bar{\lambda}(x),\bar{g}(x))dx/x}G_{R}(\mu^{-1};\bar{\lambda}(\mu
t),\bar{g}(\mu t),\mu)$ (23)
Next we calculate the first-order contribution to the renormalized vertices.
The $\lambda$-vertex is renormalized by the set of diagrams that are shown in
Figure 2. We notice that there are no diagrams producing the momentum
dependent $g$-vertex in the theory (6). This statement is the corollary of the
fact that only independent fields of power one enter into the expression of
the vertex and there are no higher powers of fields. Also we keep in mind that
the renormalized couplings are defined by the value of the vertex function
taken at zero external momenta. It produces the following expression:
$\begin{cases}\lambda_{Rij}&=\lambda_{ij}-\frac{1}{2}(\lambda_{ij}^{2}I_{1}+2\lambda_{ij}g_{ij}I_{2}+g_{ij}^{2}I_{3})\\\
g_{Rij}&=g_{ij}\\\ \end{cases}$ (24)
where $I_{k}=I_{k}(\sigma;D_{i},D_{j})$ are one-loop integrals corresponding
to the diagrams $a$, $b$, $c$ in the Figure 2 respectively. Using the Feynman
rules we can explicitly write them down:
$I_{k}=\int\frac{d^{d}q}{(2\pi)^{d}}\frac{q^{(k-1)\sigma}}{2s+(D_{i}+D_{j})q^{2}},\quad
k=1,2,3.$ (25)
We will use dimensional regularization procedure to compute these integrals.
The details of the computation are summarized in Appendix A. We note that
integrals will diverge logarithmically at different values of the spatial
dimension $d$. For this reason it leads to different critical behavior in
different regions of the $\sigma-d$ plane (see Figure 3). These regions
correspond to four possibilities for $\varepsilon=2-d$ and $\delta=2-d-\sigma$
to be positive or negative. Only if $\delta=O(\varepsilon)$ or, in other
words, if both $\varepsilon$ and $\delta$ are infinitesimally small but the
ratio $\varepsilon/\delta$ is finite we expect non-zero fixed points of the
renormalization group flow. Similar approximation have been used before Halp
but for different models with long-range disorder. It allows us to follow the
standard procedure of deriving the $\beta$-functions which consists of two
steps.
First, we express unrenormalized couplings in terms of the renormalized. For
the short-range coupling constant $\lambda$ it can be done by solving the
quadratic equation in (24). Expanding the square root and keeping terms up to
the second order we infer that
$\begin{cases}\lambda_{ij}&=\lambda_{Rij}+\frac{1}{2}(\lambda_{Rij}^{2}\frac{a_{d}}{\varepsilon}+2\lambda_{Rij}g_{Rij}\frac{b_{d}}{\delta}+g_{Rij}^{2}\frac{c_{d}}{2\delta-\varepsilon})\\\
g_{ij}&=g_{Rij}\\\ \end{cases}$ (26)
where $a_{d}$, $b_{d}$ and $c_{d}$coefficients have been found explicitly in
Appendix A. Now we introduce dimensionless renormalized couplings
$\bar{g}_{Rij}=a_{d}(2s)^{-\delta/2}\quad\bar{\lambda}_{Rij}=b_{d}(2s)^{-\varepsilon/2}.$
(27)
An important observation is that $c_{d}a_{d}=b_{d}^{2}$ which can be verified
by explicit substitution (see Appendix A). Multiplying the first and second
equation in (26) by the factors $a_{d}$ and $b_{d}$ respectively, and using
redefinitions (27) we can condense all pre-factors in the right hand side of
the equations into the dimensionless constants.
Second, we differentiate equations (26) with respect to the scaling parameter
$\mu$. Using definitions (18) and the fact that bare couplings do not depend
on the scale, we derive
$\begin{cases}\beta_{\lambda,ij}&=-\varepsilon\bar{\lambda}_{Rij}+(\bar{\lambda}_{Rij}+\bar{g}_{Rij})^{2}\\\
\beta_{g,ij}&=-\delta\bar{g}_{Rij}\\\ \end{cases}$ (28)
where the right hand side is understood as the leading contribution to the
$\beta$-functions from the double expansions in $\lambda,g$ and
$\varepsilon,\delta$. From (28) we see that it is convenient to introduce new
coupling constants $u_{Rij}=\bar{\lambda}_{Rij}+\bar{g}_{Rij}$. After this
step the renormalization group equations read
$\begin{cases}\beta_{u,ij}&=-\varepsilon u_{Rij}+u_{Rij}^{2}-g_{Rij}\\\
\beta_{g,ij}&=-\delta g_{Rij}\\\ \end{cases}$ (29)
We note that in the last equations $g$ coupling constant has been redefined
$\sigma\bar{g}_{Rij}\to g_{Rij}$.
Fixed points are zeros of the $\beta$-functions. If $\delta\neq 0$ then the
last equation in (29) is zero only when $g_{*}=0$. Then the first equation has
two solutions $u=0$ and $u=\varepsilon$. If $\delta=0$ then $g$ plays the role
of a parameter and the fixed points are determined by the roots of the
quadratic equation
$0=-\varepsilon u+u^{2}-g$ (30)
which are real if $g\geq-(\varepsilon/2)^{2}$ and we find
$u_{1,2}=\varepsilon/2\pm\sqrt{(\varepsilon/2)^{2}+g}.$ (31)
All fixed points are listed in the Table II. The stability of these fixed
points is determined by the matrix of partial derivatives
$\beta_{*}=-\left(\begin{array}[]{cc}\partial\beta_{u}/\partial
u&\partial\beta_{u}/\partial g\\\ \partial\beta_{g}/\partial
u&\partial\beta_{g}/\partial g\end{array}\right)_{u=u_{*},g=g_{*}}$ (32)
Eigenvalues are listed in the Table 2. The Gaussian fixed point is stable in
all directions for $\varepsilon<0$ and $\delta<0$ which corresponds to region
I in Figure 3. In this region we find both short-range(pure VW) and long-range
mean-field behavior depending on the sign of $\sigma$. On the contrary, for
$\varepsilon>0$ and $\delta>0$ we find that the Gaussian fixed point is
unstable(irrelevant) in all directions and the short-range (pure VW) fixed
point is stable(relevant) only in $u$-direction. It means that long-range
interactions will play a leading role. This region corresponds to region III
in Figure 3. Next for $\varepsilon>0$ and $\delta<0$ we find that the short-
range (pure VW) fixed point is stable in all directions. It means that the
system is insensitive to the long-range tail. This region corresponds to
region II in Figure 3. Finally for $\varepsilon<0$ and $\delta>0$ we find that
the short-range (pure VW) fixed point is unstable in all directions and the
system will be described by mean-field at long time.
Table 2: Fixed points for flow equations (29) and the corresponding eigenvalues $(\lambda_{1},\lambda_{2})$ of the stability matrix (32). We note that $u_{1}$ and $u_{2}$ are values of the Fixed point | $(u_{*},g_{*})$ | $(\lambda_{1},\lambda_{2})$
---|---|---
Gaussian | $(0,0)$ | $(\varepsilon,\delta)$
Pure VW | $(\varepsilon,0)$ | $(-\varepsilon,\delta)$
LR stable | $(u_{1},0)$ | $(-\sqrt{\varepsilon^{2}-4g},0)$
LR unstable | $(u_{2},0)$ | $(\sqrt{\varepsilon^{2}-4g},0)$
## IV Calculation of critical exponents and discussion
Here we describe our method of computing critical exponents. It is based on
the formula (21) from the previous section. First, we obtain the leading
divergent part of the correlation function. The renormalized correlation
function depends on the scale $\mu$ but it appears in all formulas in
combination with time: $\mu t$. Second, since we have found the bare coupling
constant as a function of renormalized (dressed) couplings we express
correlation function in terms of dressed couplings. Finally using the
normalization condition (20) and the definition (19) we differentiate $Z$ with
respect to $\mu\partial/\partial\mu$ to obtain the exponent $\gamma$. The
poles should cancel after this operation.
In section 2 it was explained that the truncated correlation function in the
one-loop approximation is given by the formula
$\Gamma(t;\lambda,g)=1-\sum_{i,j}n_{i}n_{j}\left(\lambda_{ij}I_{1}+g_{ij}I_{2}\right).$
(33)
Here integrals are the same as in (25).
We start our analysis with the region I. Notice that truncated correlation
function $\Gamma(t)$ and survival probability $G(t)$ have similar large time
behavior. We use large momentum cut-off to compute integrals $I_{1}$ and
$I_{2}$ as in formula (64) in Appendix A. The renormalization of coupling
constants is trivial in this case. Therefore the leading contribution to the
survival probability is given by
$G(t)\sim t^{(2-d)/2}+g_{0}t^{(2-d-\sigma)/2},$ (34)
where $g_{0}$ is non-universal coefficient and we will not need its exact
value. We notice that if $\sigma>0$ the second term will decay faster than the
first term and in the long-time limit it will produce the same behavior as
mean-field pure VW. On the other hand if $\sigma<0$ the first term will decay
faster and long-range interactions will play a leading role. Many authors
observed similar behavior in various systems with long-range defects Halp ;
Bla ; Prud . Intuitively if potential falls fast with distance than the system
effectively represent system with short-range potential where particle
interact when they are close to each other.
Region IV exhibits similar behavior. Now the integral $I_{2}$ is computed with
the help of the dimensional regularization (58) and the integral $I_{1}$
remains the same. From the fact (15) one can infer that the survival
probability scales as
$G(t)\sim t^{(2-d)/2}.$ (35)
Short-range behavior dominates because the running coupling constant will flow
towards the Gaussian fixed point at long time limit which is the only stable
fixed in this region. This result is exact regardless the number of loops one
takes into account.
In Region II the computation is as follows.
$\ln Z=\sum
n_{i}n_{j}\left(\lambda_{ij}\frac{a_{d}}{\varepsilon}+g_{ij}t^{(2-d-\sigma)/2}\right),$
(36)
so plugging the result from (59) to (36) we obtain at the fixed point
$(\lambda_{*}=\varepsilon,g=0)$
$\gamma=-\frac{1}{2}\sum n_{i}n_{j}\varepsilon$ (37)
And we reproduce the pure VW behavior. This result is the reflection of the
fact that the renormalization-group trajectories run away to stable pure VW
fixed point. It is with agreement with the results obtained by Katori in
Katori for $d=1$, and the logarithmic intraset particle interactions. The
irrelevance of the long-range interaction in lower dimensions is a typical
phenomenon observed in a various out of equilibrium interacting particle
systems.
We now consider regions III, V and VI. Integrals in (33) are computed via
dimensional regularization. Taking the inverse of (33) and then logarithm one
can obtain at the leading order:
$\log Z=\sum
n_{i}n_{j}\left(\lambda_{ij}\frac{a_{d}}{\varepsilon}+g_{ij}\frac{b_{d}}{\delta}\right)$
(38)
where $a_{d}$ and $b_{d}$ are defined in Appendix A in (59) and (60). We note
that after taking the derivative the poles in (38) will cancel in the limit of
$\delta=O(\varepsilon)$. Also one recalls the expansion (26) and the
redefinitions in (27). Using (19) we show that the expression for the function
$\gamma$ which determines critical exponent takes the form
$\gamma=-\frac{1}{2}\sum_{ij}n_{i}n_{j}u_{R}$ (39)
Evaluated at the stable fixed point
$(u_{1}=\varepsilon/2+\sqrt{(\varepsilon/2)^{2}+g}$ it gives the following
result:
$\gamma=-\frac{1}{2}\sum_{ij}n_{i}n_{j}u_{1},$ (40)
and the survival probability scales as $G(t)\sim t^{\gamma}$.
We will now find the logarithmic corrections to this scaling law. The running
coupling constant can be found from the flow equation (29):
$\bar{g}(x)=e^{-\delta x}g$. In the case $\delta,\varepsilon=0$ (the
intersection of regions V and VI) the flow equation for $\bar{u}(x)$ is
$x\frac{d\bar{u}(x)}{dx}=-\bar{u}^{2}(x)+g$ (41)
and the solution is
$\bar{u}(x)=\sqrt{g}\tanh(\sqrt{g}\log
x+\phi_{0})\sim\sqrt{g}\tanh(\sqrt{g}\log x),$ (42)
where $\phi_{0}$ is the initial condition and we do not need its exact form.
After plugging this expression into the (23) we infer
$\int\limits_{1}^{\mu
t}\gamma(\bar{u},\bar{g})\frac{dx}{x}\sim\log(\cosh(\sqrt{g}\log\mu t))$ (43)
Thus the survival probability is
$G(t)\sim\cosh(\sqrt{g}\log t)^{-\frac{1}{2}\sum n_{i}n_{j}}$ (44)
In the limit of large time $\cosh(\sqrt{g}\log t)\sim t^{\sqrt{g}}$ implying
$gamma=-\frac{1}{2}\sum_{ij}n_{i}n_{j}\sqrt{g}$ which is consistent with
equation (40). For negative coupling constant $g<0$ the solution in (42)
becomes
$\bar{u}(x)\sim-\sqrt{|g|}\tan(\sqrt{|g|}\log x)$ (45)
The integral (43) is divergent if $t>\exp(\pi/2\sqrt{|g|})$ which leads to the
result that the survival probability is zero beyond this time. For smaller
times one has $G(t)\sim\cos(\sqrt{|g|}\log t)^{-\frac{1}{2}\sum n_{i}n_{j}}$.
Thus, upto one-loop order approximation, It implies that if walks are
attracted to each other then all of them will annihilate at some finite time.
This might be a signature of faster than power law decay and we expect to have
corrections to this behavior at higher loop approximation.
Next we consider the case when $\varepsilon=0$ and $\delta\neq 0$ but $\delta$
remains small i.e. region V. The flow equation for the $\bar{u}(x)$ is
$x\frac{d\bar{u}(x)}{dx}=-\bar{u}^{2}(x)+gx^{-\delta}$ (46)
and the solution can be found by the method of perturbation. Up to the first
order
$\bar{u}(x)=\sqrt{g}\tanh(\sqrt{g}\log
x)+\delta\sqrt{g}\log(x)\tanh(\sqrt{g}\log x)$ (47)
After plugging this expression into eqn (23) we infer
$\int\limits_{1}^{\mu
t}\gamma(\bar{u},\bar{g})\frac{dx}{x}\sim-\frac{1}{2}\sum
n_{i}n_{j}\left(\log(t^{\sqrt{g}})+\frac{1}{2}\delta\sqrt{g}\log^{2}(t)\right)$
(48)
Therefore we have the correction to the survival probability in the form
$G\sim t^{-\frac{1}{2}\sum n_{i}n_{j}\sqrt{g}(1+\delta/2(\log t))}$ (49)
Now we extend our analysis to the case when $\varepsilon>0$, corresponding to
regions III and VI. The evolution of the coupling constant is
$x\frac{d}{dx}\bar{u}(x)=\varepsilon\bar{u}-\bar{u}^{2}+gx^{\delta}$ (50)
We choose the ansatz in the form $\bar{u}(x)=u_{0}(x)+\delta v(x)$. For
$\delta=0$ (i.e. region VI) the equation for $u_{0}(x)$ reads
$x\frac{d}{dx}u_{0}(x)=\varepsilon u_{0}-u_{0}^{2}+g$ (51)
and we reproduce the result (40). We now extend to the case where
$\varepsilon,\delta>0$ (region III). Here we will need the exact solution to
(51) to find the corrections:
$u_{0}(x)=\frac{Cx^{u_{1}-u_{2}}u_{1}+u_{2}}{1+Cx^{u_{1}-u_{2}}},$ (52)
where $C=(u_{R}-u_{2})/(u_{1}-u_{R})$. The logarithmic correction follows from
the form of the perturbation. The equation for $v(x)$ is
$x\frac{d}{dx}v(x)=\varepsilon v-2u_{0}v-g\log x$ (53)
The solution can be found explicitly as a combination of hypergeometric
functions. In the most interesting case, $\varepsilon=1$ ($d=1$) the
hypergeometric functions are degenerate and become linear functions.
Corrections to the integral then read
$\int\limits_{1}^{\mu t}\gamma dx/x\sim\frac{1}{2}\delta
u_{1}\log^{2}(t)+\log(t)(t)^{u_{1}-u_{2}})$ (54)
In the limit of large time only the first term contributes to the exponent and
the survival probability scales as
$G\sim t^{-\frac{1}{2}\sum n_{i}n_{j}u_{1}(1+\delta/2\log t)}$ (55)
## V Conclusion
In summary, we studied long-range vicious walks using the methods of Callan-
Symanzik renormalized field theory. Our work confirms the previously known RG
fixed point structure including their stability regions. We calculated the
critical exponents for all values of $\sigma$ and $d$ to first order in
$\varepsilon$ expansion and to all orders in $\delta$ expansion, which have
hitherto been known only for $d+\sigma=2$. Our results indicate that,
depending on the exact values of $d$ and $\sigma$, the system can be dominated
by either short range (pure VW) or long range behaviors. In addition, we
calculated the leading logarithmic corrections for several dynamical
observables that are typically measured in simulations.
We hope that our work stimulates further interest in long-range vicious walks.
It would be interesting to see further simulation results for the critical
exponents for $d>1$ and for logarithmic corrections. Also, it would be
interesting to have analytical and numerical results for other universal
quantities such as scaling functions and amplitudes.
## VI Acknowledgments
AG would like to acknowledge UC Merced start-up funds and a James S. McDonnell
Foundation Award for Studying Complex Systems.
## Appendix A
Effective four-point function (one-particle irreducible, 1PI) that appeared in
(24) is composed of usual short-range and new momentum dependent vertices.
This gives rise to integrals (25). The first integral $\mu=1$ has been
evaluated in Cardy by using alpha representation
$1/(q^{2}+s)=\int_{0}^{+\infty}d\alpha e^{i(q^{2}+s)\alpha}$ and the result is
$I_{1}=K_{d}(2s)^{-\varepsilon/2}\Gamma(\varepsilon/2).$ (56)
We notice that since there is no angular dependence one can perform $d-1$
integrations and one will be left with one dimensional integral. To compute
this integral we use the formula GR :
$\int\limits_{0}^{+\infty}dx\frac{x^{\nu-1}}{P+Qx^{2}}=\frac{1}{2P}\left(\frac{P}{Q}\right)^{\nu/2}\Gamma\left(\frac{\nu}{2}\right)\Gamma\left(1-\frac{\nu}{2}\right)$
(57)
We see that in our case $P=s$, $Q=(D_{i}+D_{j})$ and $\nu=d+(\mu-1)\sigma$.
This immediately gives the result:
$\displaystyle I_{\mu}=$
$\displaystyle\frac{K_{d}}{2}\left(\frac{1}{(D_{i}+D_{j})}\right)^{\frac{d+(\mu-1)\sigma}{2}}s^{\frac{d+(\mu-1)\sigma}{2}-1}\times$
$\displaystyle\times\Gamma\left(\frac{d+(\mu-1)\sigma}{2}\right)\Gamma\left(1-\frac{d+(\mu-1)\sigma}{2}\right)\,,$
(58)
where $K_{d}=2^{d-1}\pi^{-d/2}\Gamma^{-1}(d/2)$ is the surface area of
$d$-dimensional unit sphere.
It is convenient to define
$a_{d}=\frac{K_{d}}{2}\left(\frac{2}{(D_{i}+D_{j})}\right)^{d/2}(2s)^{-\varepsilon/2}$
(59)
$b_{d}=\frac{K_{d}}{2}\left(\frac{2}{(D_{i}+D_{j})}\right)^{(d+\sigma)/2}(2s)^{-\delta/2}$
(60)
$c_{d}=\frac{K_{d}}{2}\left(\frac{2}{(D_{i}+D_{j})}\right)^{(d+2\sigma)/2}(2s)^{-(2\delta-\varepsilon)/2}$
(61)
So integral $I_{\mu}$ in the limit of $\delta=O(\varepsilon)$ can be written
as:
$I_{1}=\frac{a_{d}}{\varepsilon},\quad I_{2}=\frac{b_{d}}{\delta},\quad
I_{3}=\frac{c_{d}}{2\delta-\varepsilon}.$ (62)
We used an expansion $\Gamma(\varepsilon/2)\sim 2/\varepsilon$ for small
$\varepsilon$. An important property of coefficients (59) - (61) is that
$c_{d}a_{d}=b^{2}_{d},$ (63)
which can be verified by direct substitution.
Now we compute mean field integrals:
$I_{\mu}=\int d^{d}qdtq^{d+\sigma}\exp(-t(D_{i}+D_{j})q^{2})\sim
t^{-(d+\sigma-2)/2},$ (64)
where we assumed that the large momentum cut-off is imposed and corresponding
coupling constants have been renormalized. The non-universal coefficient is
not important.
## References
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* (30) J. Zinn-Justin, _Quantum Field Theory and Critical Phenomena_ , 4nd revised edition, (Clarendon, Oxford, 2002).
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|
arxiv-papers
| 2010-03-31T05:25:46 |
2024-09-04T02:49:09.377754
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Igor Goncharenko, Ajay Gopinathan",
"submitter": "Igor Goncharenko",
"url": "https://arxiv.org/abs/1003.5970"
}
|
1003.5980
|
Current address: ]LFTC, Universidade Cruzeiro do Sul, Rua Galvão Bueno, 868,
São Paulo, 01506-000 SP, Brazil.
# Scalar resonance effects on the $\bm{B_{s}-\bar{B}_{s}}$ mixing angle
O. Leitner leitner@lpnhe.in2p3.fr J.-P. Dedonder dedonder@univ-paris-
diderot.fr B. Loiseau loiseau@lpnhe.in2p3.fr Laboratoire de Physique
Nucléaire et de Hautes Énergies, Groupe Théorie,
Université Pierre et Marie Curie et Université Paris-Diderot, IN2P3 & CNRS, 4
place Jussieu, 75252 Paris, France B. El-Bennich bennich@anl.gov Physics
Division, Argonne National Laboratory, Argonne, IL 60439, USA [
###### Abstract
The $B_{s}^{0}\to J/\psi\phi$ and $B_{s}^{0}\to J/\psi f_{0}(980)$ decays are
analyzed within generalized QCD factorization including all leading-order
corrections in $\alpha_{\mathrm{s}}$. We point out that the ratio of our
calculated widths, $\Gamma(B_{s}^{0}\to J/\psi
f_{0}(980),f_{0}(980)\to\pi^{+}\pi^{-})/\Gamma(B_{s}^{0}\to J/\psi\phi,\phi\to
K^{+}K^{-})$, strongly indicates that $S$-wave effects in the $f_{0}(980)$’s
daughter pions or kaons cannot be ignored in the extraction of the
$B_{s}-\bar{B}_{s}$ mixing angle, $-2\beta_{s}$, from the $B_{s}^{0}\to\phi
J/\psi$ decay amplitudes.
$CP$ violation, $B_{s}$ decays, QCDF
###### pacs:
11.30.Er, 13.25.Hw, 13.30.Eg
## I Introduction
In the Standard Model, $CP$ violation is predicted in weak decays thanks to
the single phase of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. It is also
well known that such a weak phase is not sufficient to generate a $CP$
violating decay amplitude. Strong phases are necessary and their strength may
significantly enhance the effect of the weak phase. Therefore, hadronic
effects, such as resonances of daughter particles in $S$\- and higher waves,
require a careful analysis in the determination of $CP$ violating phases in
hadronic two- and three-body decays Leitner:2010ai ; ElBennich:2006yi ;
Boito:2008zk ; ElBennich:2009da .
The antimatter-matter asymmetry is expected to be very small in weak decays of
$B_{s}$ mesons; any observed deviation may well be a signal of physics whose
origins lie beyond the Standard Model. In the $B_{s}^{0}\to J/\psi\phi$
channel, recent measurements by the CDF Aaltonen:2007he and D$\emptyset$
Abazov:2008fj ; Abazov:2008jz Collaborations of the $B_{s}-\bar{B}_{s}$ mixing
phase, $-2\beta_{s}$, while not definitive, are considerably larger than
Standard Model predictions. Taking advantage of the fact that the
$B_{s}^{0}\to J/\psi f_{0}(980)$ channel does not require any angular
analysis, one can compute the ratio between the $B_{s}^{0}\to J/\psi\phi$ and
$B_{s}^{0}\to J/\psi f_{0}(980)$ decay widths in order to estimate the
$\pi^{+}\pi^{-}$ $S$-wave effect on the value of $\beta_{s}$. A first
qualitative attempt to predict the ratio,
$\displaystyle\mathcal{R}_{f_{0}/\phi}=\frac{\Gamma(B_{s}^{0}\to J/\psi
f_{0}(980),f_{0}(980)\to\pi^{+}\pi^{-})}{\Gamma(B_{s}^{0}\to
J/\psi\phi,\phi\to K^{+}K^{-})}\ ,$ (1)
was made by Stone and Zhang Stone:2008ak and gives a result of the order of
$20\%-30\%$. Their estimate relies on experimental data on $D_{s}^{+}\to
f_{0}(980)\pi^{+}$ and $D_{s}^{+}\to\phi\pi^{+}$ decays and seems to indicate
that the $S$-wave contribution of $f_{0}(980)\to K^{+}K^{-}$ cannot be ignored
when analyzing the angle $\beta_{s}$ in $B_{s}^{0}\to J/\psi\phi$. Likewise,
Xie et al. found the effect of an $S$-wave component on $2\beta_{s}$ to be of
the order of $10\%$ in the $\phi$ resonance region Xie:2009fs .
Based on the QCD factorization (QCDF) formalism we perform a first robust
calculation of the ratio $\mathcal{R}_{f_{0}/\phi}$. To this end, all the
available observables (polarizations and branching ratio in $B_{s}^{0}\to
J/\psi\phi$) are used to effectively constrain the analysis of the
$B_{s}^{0}\to J/\psi\phi$ channel. The branching ratio and $CP$ asymmetry are
then predicted for $B_{s}^{0}\to J/\psi f_{0}(980)$, where we assume that
merely the $s\bar{s}$ component of the $f_{0}(980)$ is involved in the
hadronic $B_{s}\to f_{0}(980)$ transition matrix element.
In Section II we introduce the general expressions for the $B_{s}^{0}\to
J/\psi\phi$ and $B_{s}^{0}\to J/\psi f_{0}(980)$ weak decay amplitudes whereas
Sections III and IV provide the details on the leading order corrections in
$\alpha_{\mathrm{s}}$ for both these amplitudes, respectively. In Section V,
we list all numerical values of input parameters and briefly recall our model
for the $B_{s}\to f_{0}(980)$ transition form factor ElBennich:2008xy on which
the ratio $\mathcal{R}_{f_{0}/\phi}$ directly depends; we also define the
parametrization for the $B_{s}\to\phi$ form factor. Section VI is devoted to
our results and, finally, conclusions are drawn in Section VII.
## II General form of the $\bm{{B}_{s}^{0}\to\phi J/\psi}$ and
$\bm{{B}_{s}^{0}\to f_{0}(980)J/\psi}$ decay amplitudes
It is important to realize beforehand that the application of QCDF, following
Refs. Beneke:2000ry ; Beneke:2001ev ; Beneke:2003zv ; Beneke:2006hg , to
$B_{s}^{0}$ decays into a heavy-light final state is not self-evident. In both
final states, $\phi J/\psi$ and $f_{0}(980)J/\psi$, the $s$-spectator quark is
absorbed by the light meson while the emitted meson is heavy, in which case
QCDF is not reliable Beneke:2000ry . Nonetheless, as argued in Refs.
Chay:2000xn ; Cheng:2001ez and more recently in Ref. Beneke:2008pi , the
production of a heavy charmonium $\bar{q}q$ pair bears “color transparency”
properties similar to those of a light meson, provided this color-singlet pair
is small compared to the inverse strong interaction scale,
$1/\Lambda_{\mathrm{QCD}}$. This was explicitly demonstrated in next-to-
leading order calculations for exclusive $B$ decays to $J/\psi$ final states
($J/\psi K,J/\psi K^{*}$), where infrared divergences were shown to cancel
Chay:2000xn ; Cheng:2001ez .
In the following, we present the $B_{s}^{0}$ decay amplitudes in which the
short- and long-distance contributions are factorized in the approximation of
a quasi two-body state, $M_{1}M_{2}$, where either
$M_{1}M_{2}=f_{0}(980)J/\psi$ or $M_{1}M_{2}=\phi J/\psi$. We begin with the
$B_{s}^{0}\to\phi J/\psi$ amplitude which can be written for each helicity,
$h=-1,0,1$, as Beneke:2006hg ,
$\displaystyle{\cal A}_{{B}_{s}^{0}\to\phi
J/\psi}^{h}=\sum_{q=u,c}\lambda_{q}\Bigg{\\{}A^{h}_{\phi
J/\psi}\biggl{[}\delta_{qc}\,\Bigl{(}a_{2}^{q,h}(m_{b})+\
\zeta^{h}\Bigr{)}+a_{3}^{q,h}(m_{b})+a_{5}^{q,h}(m_{b})+a_{7}^{q,h}(m_{b})+a_{9}^{q,h}(m_{b})\biggr{]}\Bigg{\\}}_{\phi
J/\psi}.$ (2)
Summing over all the possible helicities, the squared modulus of the total
amplitude reads
$\displaystyle\bigl{|}{\cal A}_{{B}_{s}^{0}\to\phi J/\psi}\bigr{|}^{2}$
$\displaystyle=$ $\displaystyle\bigl{|}{\cal A}_{{B}_{s}^{0}\to\phi
J/\psi}^{h=-1}\bigr{|}^{2}+\bigl{|}{\cal A}_{{B}_{s}^{0}\to\phi
J/\psi}^{h=0}\bigr{|}^{2}+\bigl{|}{\cal A}_{{B}_{s}^{0}\to\phi
J/\psi}^{h=+1}\bigr{|}^{2}.$ (3)
The $\bar{B}_{s}^{0}\to\phi J/\psi$ decay amplitude is obtained by exchange of
helicity signs, $h=+1\to h=-1$, and replacing $\lambda_{q}$ by its complex
conjugate. The ${B}_{s}^{0}\to f_{0}(980)J/\psi$ amplitude is,
$\displaystyle{\cal A}_{{B}_{s}^{0}\to f_{0}J/\psi}$ $\displaystyle=$
$\displaystyle\sum_{q=u,c}\lambda_{q}\Bigg{\\{}A_{f_{0}J/\psi}\biggl{[}\delta_{qc}\,\Bigl{(}a_{2}^{q}(m_{b})+\zeta\Bigr{)}+a_{3}^{q}(m_{b})+a_{5}^{q}(m_{b})+\
a_{7}^{q}(m_{b})+a_{9}^{q}(m_{b})\biggr{]}\Bigg{\\}}_{f_{0}J/\psi}.$ (4)
The different elements entering in the amplitudes (2) and (4) are defined in
Eqs. (II), (7), (15), (22) and (24). The $CP$ conjugate $\bar{B}_{s}^{0}$
decay amplitude is again found by replacing $\lambda_{q}$ by its complex
conjugate.
With the generic amplitude, ${\cal A}_{{B}_{s}^{0}\to M_{1}J/\psi}$, the
branching ratio,
$\hbox to0.0pt{$\displaystyle{\cal B}({B}_{s}^{0}\to
M_{1}J/\psi)=\frac{1}{\Gamma_{B_{s}^{0}}}\frac{1}{16\pi
m_{B_{s}^{0}}}\,$\hss}\\\
\times\lambda^{1/2}\Bigl{(}1,m_{M_{1}}^{2}/m_{B_{s}^{0}}^{2},m_{J/\psi}^{2}/m_{B_{s}^{0}}^{2}\Bigr{)}\,\bigl{|}{\cal
A}_{{B}_{s}^{0}\to M_{1}J/\psi}\bigr{|}^{2}\ ,$ (5)
can be computed. The $J/\psi$ mass is noted $m_{J/\psi}$ while
$m_{M_{1}}=m_{f_{0}(980)}$ or $m_{\phi}$ denote the $f_{0}(980)$ and $\phi$
masses; the triangle function is $\lambda(x,y,z)=(x+y-z)^{2}-4xy$. In Eq. (5),
$\Gamma_{B_{s}^{0}}=1/\tau_{B_{s}^{0}}$ is the $B_{s}^{0}$ decay width with
$\tau_{B_{s}^{0}}=(1.470\pm 0.026)$ ps Amsler:2008zzb and $m_{B_{s}^{0}}$ is
the $B_{s}^{0}$ mass. For the CKM elements in Eqs. (2) and (4) we use the
Wolfenstein parametrization,
$\displaystyle\lambda_{u}$
$\displaystyle=V_{ub}^{\star}V_{us}=A\lambda^{4}\left(\rho+i\eta\right)\ ,$
$\displaystyle\lambda_{c}$
$\displaystyle=V_{cb}^{\star}V_{cs}=A\lambda^{2}\left(1-\frac{\lambda^{2}}{2}\right)\
,$ (6)
with the Wolfenstein parameters $A=0.814$, $\rho=0.1385$, $\eta=0.358$ and
$\lambda=0.2257$ Amsler:2008zzb .
### II.1 Non-perturbative amplitude
#### II.1.1 The case of the scalar-vector decay
The scalar-vector factor, $A_{f_{0}J/\psi}$, in Eq. (4) is given by,
$A_{f_{0}J/\psi}=\langle
f_{0}(p_{f_{0}})|\bar{b}\,\gamma_{\mu}(1-\gamma_{5})s|B_{s}^{0}(p_{B_{s}^{0}})\rangle\\\
\times\langle
J/\psi(p_{J/\psi},\varepsilon_{J/\psi}^{\ast})|\bar{c}\gamma^{\mu}c|0\rangle\
,$ (7)
where the hadronic matrix element which describes the transition between the
$B_{s}^{0}$ and a scalar meson, $f_{0}$, with the respective four-momenta
$p_{B_{s}^{0}}$ and $p_{f_{0}}$ is Melikhov:2001zv ,
$\langle
f_{0}(p_{f_{0}})|\bar{b}\,\gamma_{\mu}(1-\gamma_{5})s|B_{s}^{0}(p_{B_{s}^{0}})\rangle=\\\
\Bigl{(}p_{B_{s}^{0}}+p_{f_{0}}-\frac{m_{B_{s}^{0}}^{2}-m_{f_{0}}^{2}}{q^{2}}q\Bigr{)}_{\\!\\!\mu}F_{1}^{B_{s}^{0}\to
f_{0}}(q^{2})\\\ +\,\frac{m_{B_{s}^{0}}^{2}-m_{f_{0}}^{2}}{q^{2}}q_{\mu}\
F_{0}^{B_{s}^{0}\to f_{0}}(q^{2})\ ,\hskip 5.69054pt$ (8)
with $q=p_{B_{s}^{0}}-p_{f_{0}}$, $q^{2}=m_{J/\psi}^{2}$ and where
$F_{1}^{B_{s}^{0}\to f_{0}}(q^{2})$ and $F_{0}^{B_{s}^{0}\to f_{0}}(q^{2})$
are the vector and scalar form factors, respectively. In Eq. (7), the leptonic
decay constant, $f_{J/\psi}$, of the $J/\psi$ vector meson, with four-
momentum, $p_{J/\psi}$, and polarisation, $\varepsilon_{J/\psi}^{\ast}$, is
defined as,
$\langle
J/\psi(p_{J/\psi},\varepsilon_{J/\psi}^{\ast})|\bar{c}\gamma^{\mu}c|0\rangle=\\\
-if_{J/\psi}m_{J/\psi}\varepsilon_{J/\psi}^{\mu\ast}\ .$ (9)
The scalar-vector factor, given by the product of Eqs. (8) and (9), is then
obtained as,
$A_{f_{0}J/\psi}=-i\frac{G_{F}}{\sqrt{2}}2m_{J\\!/\psi}\,\epsilon_{J\\!/\\!\psi}^{*}\cdot
p_{B_{s}^{0}}\,F_{1}^{B_{s}^{0}\to f_{0}}(m_{J\\!/\\!\psi}^{2})f_{J/\psi}\,,$
(10)
with $4m_{J/\psi}^{2}\big{|}\epsilon_{J/\psi}^{*}\cdot
p_{B_{s}^{0}}\big{|}^{2}=m_{B_{s}^{0}}^{2}\,\lambda^{1/2}(m^{2}_{B_{s}^{0}},m^{2}_{J/\psi},m^{2}_{f_{0}})$
and the Fermi constant, $G_{F}=1.16\times 10^{-5}{\rm GeV}^{-2}$. The
$B_{s}^{0}\to f_{0}$ transition form factor $F_{1}^{B_{s}^{0}\to
f_{0}}(m_{J/\psi}^{2})$ will be discussed in Section V.
#### II.1.2 The case of the vector-vector decay
For the case of two vector mesons, $M_{1}$ and $M_{2}$, the helicity formalism
requires the introduction of three polarization four-vectors,
${\epsilon}_{M_{j},k}$ $(j=1,2$ and $k=1,2,3)$ for each spin-1 particle,
$M_{j}$,
$\displaystyle{\epsilon}_{M_{j},1}$
$\displaystyle=(0,\vec{\epsilon}_{M_{j},1})\ ,$
$\displaystyle{\epsilon}_{M_{j},2}$
$\displaystyle=(0,\vec{\epsilon}_{M_{j},2})\ ,$
$\displaystyle{\epsilon}_{M_{j},3}$
$\displaystyle={\left(|{\vec{p}_{M_{j}}}|/m_{M_{j}},E_{M_{j}}{\hat{p}_{M_{j}}}/m_{M_{j}}\right)}\
.$ (11)
where $m_{M_{j}}$, $p_{M_{j}}$ and $E_{M_{j}}$ are the mass, the momentum and
the energy of the vector meson, $M_{j}$, respectively. The energies
$E_{M_{1}},E_{M_{2}}$ are given by,
$E_{M_{1,2}}=\frac{1}{2m_{M_{2,1}}}\Bigl{(}m^{2}_{B_{s}^{0}}-m^{2}_{M_{1}}-m^{2}_{M_{2}}\Bigr{)}\
.$ (12)
In Eq. (II.1.2), ${\hat{p}_{M_{j}}}$ is defined as the unit vector along the
momentum: ${\hat{p}_{M_{j}}}={\vec{p}_{M_{j}}}/|{\vec{p}_{M_{j}}}|$.
The three polarization four-vectors, ${\epsilon}_{M_{j},k}$, also satisfy the
following relations,
${{\epsilon}_{M_{j},k}}^{2}=-1\ ,\;{\rm
and}\;\;{\epsilon}_{M_{j},k}\cdot{\epsilon}_{M_{j},l}=0\ ,\ \ \mathrm{for}\
k\neq l\ .$ (13)
The vectors $\vec{\epsilon}_{M_{j},1}$, $\vec{\epsilon}_{M_{j},2}$ and
$\vec{\epsilon}_{M_{j},3}$ form an orthogonal basis in which
$\vec{\epsilon}_{M_{j},1}$ and $\vec{\epsilon}_{M_{j},2}$ describe the
transverse polarizations while $\vec{\epsilon}_{M_{j},3}$ is the longitudinal
polarization vector. With these three vectors one builds up the helicity
basis,
$\displaystyle\epsilon_{M_{j},+}$
$\displaystyle=\frac{1}{\sqrt{2}}\left({\epsilon}_{M_{j},1}+i\;{\epsilon}_{M_{j},2}\right)=\frac{1}{\sqrt{2}}(0,+1,i,0)\
,$ $\displaystyle\epsilon_{M_{j},-}$
$\displaystyle=\frac{1}{\sqrt{2}}\left({\epsilon}_{M_{j},1}-i\;{\epsilon}_{M_{j},2}\right)=\frac{1}{\sqrt{2}}(0,-1,i,0)\
,$ $\displaystyle\epsilon_{M_{j},0}$ $\displaystyle={\epsilon}_{M_{j},3}\ .$
(14)
and $\epsilon_{M_{1},\pm}=\epsilon_{M_{2},\mp}$. In Eq. (II.1.2), the new
four-vectors $\epsilon_{M_{j},+},\epsilon_{M_{j},-}$ and $\epsilon_{M_{j},0}$
are eigenvectors of the helicity operator corresponding to the eigenvalues
$h=+1,-1$ and $0$, respectively.
The vector-vector factor, $A_{M_{1}M_{2}}^{h}$, in Eq. (2) is
$\displaystyle A_{M_{1}M_{2}}^{h}$ $\displaystyle=$ $\displaystyle\langle
M_{1}(p_{M_{1}},\varepsilon_{M_{1}}^{\ast})|\bar{b}\,\gamma_{\mu}(1-\gamma_{5})q|B_{s}^{0}(p_{B_{s}^{0}})\rangle$
(15) $\displaystyle\hskip 14.22636pt\times\ \langle
M_{2}(p_{M_{2}},\varepsilon_{M_{2}}^{\ast})|\bar{q}\gamma^{\mu}q^{\prime}|0\rangle\
,$
where, in the $B^{0}_{s}$ rest-frame, the vector mesons $M_{1}$ and $M_{2}$
have opposite momentum $\vec{p}_{M_{1}}=-\vec{p}_{M_{2}}$ along the
$z$-direction and $\epsilon_{M_{j},0}\cdot p_{M_{j}}=0$.
The matrix hadronic element of a $P\to V$ transition can be decomposed into
Lorentz invariants as Melikhov:2001zv ; Cheng:2001ez ; Li:2003hea
$\displaystyle\langle
M_{j}(p_{M_{j}},\varepsilon_{M_{j}}^{\ast})|\bar{b}\,\gamma_{\mu}(1-\gamma_{5})q|B_{s}^{0}(p_{B_{s}^{0}})\rangle$
$\displaystyle=$
$\displaystyle\varepsilon_{M_{j},\mu}^{\ast}(m_{B_{s}^{0}}+m_{M_{j}})A_{1}^{B_{s}^{0}\to
M_{j}}(q^{2})-(p_{B_{s}^{0}}+p_{M_{j}})_{\mu}({\varepsilon_{M_{j}}^{\ast}}\cdot{p_{B_{s}^{0}}})\frac{A_{2}^{B_{s}^{0}\to
M_{j}}(q^{2})}{m_{B_{s}^{0}}+m_{M_{j}}}$ (16) $\displaystyle-\
q_{\mu}({\varepsilon_{M_{j}}^{\ast}}\cdot{p_{B_{s}^{0}}})\frac{2m_{M_{j}}}{q^{2}}\Bigl{[}A_{3}^{B_{s}^{0}\to
M_{j}}(q^{2})-A_{0}^{B_{s}^{0}\to
M_{j}}(q^{2})\Bigr{]}+i\epsilon_{\mu\nu\alpha\beta}\,\varepsilon_{M_{j}}^{\ast\nu}p_{B_{s}^{0}}^{\alpha}p^{\beta}_{M_{j}}\frac{2V^{B_{s}^{0}\to
M_{j}}(q^{2})}{m_{B_{s}^{0}}+m_{M_{j}}}\ ,$
where the form factors $A_{0}^{B_{s}^{0}\to M_{j}}(q^{2})$,
$A_{1}^{B_{s}^{0}\to M_{j}}(q^{2})$, $A_{2}^{B_{s}^{0}\to M_{j}}(q^{2})$ and
$A_{3}^{B_{s}^{0}\to M_{j}}(q^{2})$ obey the following exact relations,
$\displaystyle A_{3}^{B_{s}^{0}\to M_{j}}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{m_{B_{s}^{0}}+m_{M_{j}}}{2m_{M_{j}}}A_{1}^{B_{s}^{0}\to
M_{j}}(q^{2})$ (17) $\displaystyle-$
$\displaystyle\frac{m_{B_{s}^{0}}-m_{M_{j}}}{2m_{M_{j}}}A_{2}^{B_{s}^{0}\to
M_{j}}(q^{2})\ ,$
as well as for $q^{2}=0$, $A_{3}^{B_{s}^{0}\to M_{j}}(0)=A_{0}^{B_{s}^{0}\to
M_{j}}(0)$.
Specifically for $M_{1}=\phi$, and $M_{2}=J/\psi$, the helicity dependent
vector-vector factor $A_{\phi J/\psi}^{h}$ in Eq. (2) has thus the following
form,
$\displaystyle A_{\phi J/\psi}^{(h=0)}$ $\displaystyle=$ $\displaystyle
i\,\frac{G_{F}}{\sqrt{2}}f_{J/\psi}\Biggl{[}-m_{\phi}(m_{B_{s}^{0}}+m_{\phi})A_{1}^{B_{s}^{0}\to\phi}(m_{J/\psi}^{2})$
(18a) $\displaystyle+\
\bigl{(}m_{B_{s}^{0}}^{2}+m_{\phi}^{2}-m_{J/\psi}^{2}\bigr{)}A_{0}^{B_{s}^{0}\to\phi}(m_{J/\psi}^{2})\Biggl{]}\
;$ $\displaystyle A_{\phi J/\psi}^{(h=\pm 1)}$ $\displaystyle=$ $\displaystyle
i\,\frac{G_{F}}{\sqrt{2}}m_{B_{s}^{0}}m_{J/\psi}f_{J/\psi}F_{\mp}^{B_{s}^{0}\to\phi}(m_{J/\psi}^{2})\,.\hskip
14.22636pt$ (18b)
In Eq. (18b), the transition form factors
$F_{\pm}^{{B}_{s}^{0}\to\phi}(q^{2}=m_{J/\psi}^{2})$ are
$\displaystyle F_{\pm}^{B_{s}^{0}\to\phi}(m_{J/\psi}^{2})$ $\displaystyle=$
$\displaystyle\left(1+\frac{m_{\phi}}{m_{B_{s}^{0}}}\right)A_{1}^{B_{s}^{0}\to\phi}(m_{J/\psi}^{2})$
(19) $\displaystyle\mp$
$\displaystyle\frac{2|\vec{p}_{B_{s}^{0}}|}{m_{B_{s}^{0}}+m_{\phi}}V^{B_{s}^{0}\to\phi}(m_{J/\psi}^{2})\
,$
where the center-of-mass momentum $|\vec{p}_{B_{s}^{0}}|$ is defined as,
$|\vec{p}_{B_{s}^{0}}|=\frac{\sqrt{\left(m_{B_{s}^{0}}^{2}-M_{+}^{2}\right)\left(m_{B_{s}^{0}}^{2}-M_{-}^{2}\right)}}{2m_{B_{s}^{0}}}\
,$ (20)
with $M_{\pm}=m_{J/\psi}\pm m_{\phi}$. We note that a somewhat different form
for $A_{\phi J/\psi}^{(h=0)}$ was derived in Ref. Li:2003hea , which seems to
approximate the vector mesons as light mesons. The form factors
$A_{0}^{B_{s}^{0}\to\phi}(m_{J/\psi}^{2})$ and
$A_{1}^{B_{s}^{0}\to\phi}(m_{J/\psi}^{2})$ in Eqs. (18a) and (19), as well as
$V^{B_{s}^{0}\to\phi}(m_{J/\psi}^{2})$ in Eq. (19) are defined in Section V.
Ref. Beneke:2006hg asserts that when neglecting vector meson masses, Eq. (18a)
reduces to,
$\displaystyle A_{\phi J/\psi}^{(h=0)}$ $\displaystyle=$ $\displaystyle
i\,\frac{G_{F}}{\sqrt{2}}f_{J/\psi}m_{B_{s}^{0}}^{2}A_{0}^{B_{s}^{0}\to\phi}(m_{J/\psi}^{2})\
.$ (21)
The numerical effects in the calculated values of $B_{s}^{0}\to J/\psi\phi$
and $B_{s}^{0}\to J/\psi f_{0}(980)$ branching ratios are too important to
justify such an approximation.
### II.2 Perturbative amplitude
The $a_{n}^{q,h}(\mu)$ coefficients that appear in Eqs. (2) and (4) are linear
combinations of Wilson coefficients, $C_{n}(\mu)$, either at the scale
$\mu=m_{b}$ or $m_{b}/2$ (see below):
$\displaystyle a_{n}^{q,h}(m_{b})=\left[C_{n}(m_{b})+\frac{C_{n\pm
1}(m_{b})}{N_{c}}\right]N_{n}(J/\psi)$ (22) $\displaystyle+$ $\displaystyle
P_{n}^{q,h}(J/\psi)+\frac{C_{n\pm 1}(m_{b})}{N_{c}}\frac{C_{F}\
}{4\pi}\alpha_{s}(m_{b})V_{n}^{h}(J/\psi)$ $\displaystyle+$ $\displaystyle\pi
C_{F}\alpha_{s}(m_{b}/2)\frac{C_{n\pm
1}(m_{b}/2)}{N_{c}^{2}}H_{n}^{h}(M_{1}J/\psi)\ .$
The superscript, $(h)$, explicits the helicity dependence of
$a_{n}^{q,h}(\mu)$ in the case where $B_{s}^{0}$ decays into two vector
mesons. This superscript is dropped in the scalar-vector case. There is no
flavor dependence in $a_{n}^{q,h}(\mu)$ for $n=1,2$. In Eq. (22), the upper
(lower) signs in $C_{n\pm 1}(\mu)$ apply when $n$ is odd (even) and
$N_{n}(J/\psi)=0,\ n\in\\{6,8\\},\ \mathrm{else}\ \ N_{n}(J/\psi)=1\,.$ (23)
The Wilson coefficients, $C_{n}(\mu)$, in the Naive Dimensional Regularization
(NDR) scheme are taken at the hard scale $m_{b}$ for the vertex,
$V_{n}^{h}(J/\psi)$, and penguin, $P_{n}^{q,h}(J/\psi)$, corrections, whereas
in the hard scattering, $H_{n}^{h}(M_{1}J/\psi)$, amplitudes they are
evaluated at $m_{b}/2$ since those contributions involve the spectator quark.
The strong coupling constants at these scales are
$\alpha_{\mathrm{s}}(m_{b})=0.224$ and $\alpha_{\mathrm{s}}(m_{b}/2)=0.286$
Amsler:2008zzb , while the number of active flavors is $n_{F}=5$, the color
number $N_{c}=3$ and $C_{F}=(N_{c}^{2}-1)/2N_{c}$.
### II.3 Suppressed higher order corrections and possibility of new physics
There are no contributions, such as given by the annihilation operators
derived in Ref. Beneke:2003zv , to the two decays considered here. This is
because for the final states, $J/\psi\phi$ and $J/\psi f_{0}(980)$, both
mesons are simultaneously flavor and color singlets. At tree level, for
instance, the $W^{\pm}$ exchange diagram produces the charmonium $\bar{c}c$,
yet the creation of the $\bar{s}s$ which hadronizes to an $f_{0}(980)$ or
$\phi$ must proceed via multiple gluons or by means of photon/$Z$ exchange.
The annihilation is thus either strongly (Zweig) suppressed in $\alpha_{s}$ or
the suppression is in the electromagnetic coupling constant
$\alpha_{\mathrm{em}}$.
On the other hand, as will be discussed in Section VI, if we account for
vertex, penguin and hard scattering corrections only, the $B_{s}^{0}\to
J/\psi\phi$ observables are only moderately well reproduced. As can be seen in
Table 9, the branching ratio, for instance, is about 20% too large (although
still within the experimental errors). We therefore allow for additional
phenomenological amplitudes that mock up “other” contributions, be it from
annihilation topologies expected to be strongly suppressed or possible physics
beyond the Standard Model Chiang:2009ev . These are included in Eqs. (2) and
(4) with the amplitudes, $\zeta^{h}$ and $\zeta$, conveniently scaled as,
$\zeta^{(h)}=\frac{B_{M_{1}J/\psi}}{A_{M_{1}J/\psi}^{(h)}}\,X_{C}\ .$ (24)
The factor $B_{M_{1}J/\psi}$ is chosen to be a product of decay constants,
either
$B_{f_{0}J/\psi}=-i\,\frac{G_{F}}{\sqrt{2}}\,f_{B_{s}^{0}}\,\bar{f}_{f_{0}}\,f_{J/\psi}\
,$ (25)
if $M_{1}=f_{0}(980)$ or
$B_{\phi
J/\psi}=i\,\frac{G_{F}}{\sqrt{2}}\,f_{B_{s}^{0}}\,f_{\phi}\,f_{J/\psi}\ ,$
(26)
if $M_{1}=\phi$, while the factor $X_{C}$ is a complex parameter discussed in
Section V.3. We note that the decay constant, $f_{f_{0}}$, vanishes due to
charge conjugation invariance, wherefore the scalar light cone distributions
amplitude (LCDA) is normalized to
$\bar{f}_{f_{0}}=f_{f_{0}}m_{f_{0}}/(m_{u,d}(\mu)-m_{u,d}(\mu))$, which is
finite Cheng:2005nb . We shall return to this issue in Section IV.
### II.4 The ratio $\bm{\mathcal{R}_{f_{0}/\phi}}$
Prior to discussing the various $\alpha_{s}(\mu)$ corrections to the
amplitudes, $a_{n}^{p,h}(\mu)$, it may be of interest to observe the
qualitative behavior of the ratio, $\mathcal{R}_{f_{0}/\phi}$, in terms of the
scales $\Lambda_{\mathrm{QCD}}$ and $m_{b}$. A naive factorization analysis
yields a hierarchy of helicity amplitudes for $B$ into vector-vector decays
Beneke:2006hg ,
$\displaystyle\mathcal{A}^{(h=0)}_{B_{s}^{0}\to\phi
J/\psi}:\mathcal{A}^{(h=+1)}_{B_{s}^{0}\to\phi
J/\psi}:\mathcal{A}^{(h=-1)}_{B_{s}^{0}\to\phi J/\psi}$ (27)
$\displaystyle\Longleftrightarrow$ $\displaystyle\
1:\frac{\Lambda_{\mathrm{QCD}}}{m_{b}}:\left(\frac{\Lambda_{\mathrm{QCD}}}{m_{b}}\right)^{\\!2},$
while for $\bar{B}_{s}$ mesons the signs are exchanged ($h=+1\to h=-1$).
Furthermore, the amplitudes $\mathcal{A}^{(h=0)}_{B_{s}^{0}\to\phi J/\psi}$
and $\mathcal{A}_{B_{s}^{0}\to f_{0}J/\psi}$ are of same order in
$\Lambda_{\mathrm{QCD}}/m_{b}$. With this estimation, the ratio
$\mathcal{R}_{f_{0}/\phi}$ we are interested in becomes,
$\displaystyle\mathcal{R}_{f_{0}/\phi}$ $\displaystyle=$
$\displaystyle\frac{\big{|}\mathcal{A}_{B_{s}^{0}\to
f_{0}J/\psi}\big{|}^{2}}{\big{|}\mathcal{A}^{(h=0)}_{B_{s}^{0}\to\phi
J/\psi}\big{|}^{2}+\big{|}\mathcal{A}^{(h=-1)}_{B_{s}^{0}\to\phi
J/\psi}\big{|}^{2}+\big{|}\mathcal{A}^{(h=+1)}_{B_{s}^{0}\to\phi
J/\psi}\big{|}^{2}}$ (28) $\displaystyle\simeq$
$\displaystyle\mathcal{O}(1)+\mathcal{O}\left(\frac{\Lambda_{\mathrm{QCD}}}{m_{b}}\right)^{\\!\\!2}+\mathcal{O}\left(\frac{\Lambda_{\mathrm{QCD}}}{m_{b}}\right)^{\\!\\!4}.$
Hence, $R_{f_{0}/\phi}$ is $\mathcal{O}(1)$ for $\Lambda_{\mathrm{QCD}}/m_{b}$
corrections.
Nonetheless, non-perturbative hadronic effects can spoil the naive
factorization and violate the hierarchy in Eq. (27); so do electromagnetic
penguin contributions where a photon with small virtuality subsequently
converts into a vector meson Beneke:2005we .
## III QCDF corrections for $\bm{B_{s}^{0}\to\phi J/\psi}$ decay amplitudes
Due to the structure of the four-quark operators in heavy quark effective
theory and the conservation of the flavor quantum numbers, the final state
$M_{1}M_{2}=\phi J/\psi$ is created from the transition $B_{s}^{0}\to\phi$ and
the production of $J/\psi$ from vacuum. As discussed in Section II, the decay
amplitudes at leading order in $\Lambda_{\mathrm{QCD}}/m_{b}$ and
$\alpha_{\mathrm{s}}(m_{b})$ are given by the factorized product of a
transition form factor and a decay constant. Following Ref. Beneke:2006hg , we
only give QCD corrections that explicitly appear in the amplitude ${\cal
A}_{{B}_{s}^{0}\to\phi J/\psi}^{h}$ of Eq. (2).
We discard terms proportional to $r=(m_{J/\psi}/m_{B_{s}})^{2}\simeq 1/3$ in
vertex corrections which stem from the presence of the charm quark in the loop
diagram; we have numerically checked that their contributions to the
$a_{n}^{q,h}\\!(\mu)$ coefficients are negligible, all the more so when seen
in the light of the large hadronic uncertainties of the form factors [see
Sections (V.1) and (V.2)]. We note that in the limit $r\to 0$, one recovers
the vertex correction known from, for example, $B\to\pi\pi$ which is of course
infrared safe.
Since the coefficients in the Gegenbauer expansion of the LCDA are poorly
known for the scalar mesons, and only with non-negligible errors for the
vector mesons $V=\phi$ and $V=J/\psi$, we limit ourselves to leading terms in
the expansion. The leading twist-2 distribution and twist-3 two particle
distribution amplitudes are approximated by
$\phi_{V}(x)=6x(1-x)$ (29)
and
$\varphi_{V}(x)=3(2x-1)\ ,$ (30)
respectively. In the annihilation and hard scattering amplitudes, the chiral
coefficient, $r_{\chi}^{V}(\mu)$, is defined as
$r_{\chi}^{V}(\mu)=\frac{2m_{V}}{m_{b}(\mu)}\frac{f_{V}^{\perp}(\mu)}{f_{V}}\simeq\frac{2m_{V}}{m_{b}(\mu)}\
,$ (31)
where $f_{V}^{\perp}(\mu)$ is the transverse decay constant for any vector $V$
and $\mu=m_{b}/2$.
### III.1 Penguin contributions
The penguin contributions to the amplitude in Eq. (2) stems from the positive
helicity, $h=+1$, amplitudes $P_{7,9}^{q,h=+1}(J/\psi)$ given in Ref.
Beneke:2006hg ,
$\hbox to0.0pt{$\displaystyle
P_{7,9}^{q,h=+1}(J/\psi)\,=\,-\frac{\alpha_{e}}{3\pi}C^{\mathrm{eff}}_{7\gamma}(\mu)\,\frac{m_{B_{s}^{0}}m_{b}}{m_{J/\psi}^{2}}\,+\frac{2\alpha_{e}}{27\pi}$\hss}\\\
\times\Bigl{(}C_{1}(\mu)+N_{\mathrm{c}}C_{2}(\mu)\Bigr{)}\Biggl{[}\delta_{qc}\ln\frac{m_{c}^{2}}{\mu^{2}}+\delta_{qu}\ln\frac{\nu^{2}}{\mu^{2}}+1\Biggr{]}\
,$ (32)
whereas $P_{7,9}^{q,h=-1}(J/\psi)=0$. In Eq. (32), $\mu=m_{b}$,
$C^{\mathrm{eff}}_{7\gamma}(\mu)=C_{7\gamma}(\mu)-C_{5}(\mu)/3-C_{6}(\mu)$,
$\alpha_{e}=1/129$ is the electromagnetic coupling constant and the scale
$\nu$ refers to the $f_{J/\psi}$ decay constant scale. One also has
$P_{3,5}^{q,h=\pm 1}(J/\psi)=0$ as well as $P_{3,5,7,9}^{q,(h=0)}(J/\psi)=0$.
### III.2 Vertex contributions
In $B_{s}^{0}\to\phi J/\psi$, the electroweak vertex receives
$\alpha_{s}(\mu)$ corrections to all $a_{n}^{q,h}(\mu)$ in the amplitudes
${\cal A}_{{B}_{s}^{0}\to\phi J/\psi}^{h}$. For $h=0$, these are, with
$\mu=m_{b}$,
$V^{h=0}_{n}(J/\psi)=\left\\{\begin{array}[]{ll}\displaystyle 12\;{\rm
ln}\left(\frac{m_{b}}{\mu}\right)-3i\pi-\frac{27}{2}\ ,\vskip 6.0pt plus 2.0pt
minus 2.0pt\\\ \qquad\mathrm{for}\quad n\in\\{2,3,9\\}&\\\\[10.00002pt]
\displaystyle-12\;{\rm ln}\left(\frac{m_{b}}{\mu}\right)+3i\pi+\frac{13}{2}\
,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\ \qquad\mathrm{for}\quad
n\in\\{5,7\\}&\end{array}\right.$ (33)
whereas for $h=-1$ one has,
$V^{h=-1}_{n}(J/\psi)=\left\\{\begin{array}[]{ll}\displaystyle 12\;{\rm
ln}\left(\frac{m_{b}}{\mu}\right)+\pi^{2}-\frac{143}{4}\ ,\vskip 6.0pt plus
2.0pt minus 2.0pt\\\ \qquad\mathrm{for}\,\,n\in\\{2,3,9\\}&\\\\[10.00002pt]
\displaystyle-12\;{\rm ln}\left(\frac{m_{b}}{\mu}\right)-\pi^{2}+\frac{95}{4}\
,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\ \qquad\mathrm{for}\quad
n\in\\{5,7\\}&\end{array}\right.$ (34)
and for $h=+1$ one has,
$V^{h=+1}_{n}(J/\psi)=\left\\{\begin{array}[]{ll}\displaystyle 12\;{\rm
ln}\left(\frac{m_{b}}{\mu}\right)+\frac{\pi^{2}}{2}-6i\pi-\frac{71}{4}\
,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\qquad\mathrm{for}\,\,n\in\\{2,3,9\\}&\\\\[10.00002pt] \displaystyle-12\;{\rm
ln}\left(\frac{m_{b}}{\mu}\right)-\frac{\pi^{2}}{2}+6i\pi+\frac{23}{4}\
,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\ \qquad\mathrm{for}\quad n\in\\{5,7\\}\
.&\end{array}\right.$ (35)
### III.3 Hard scattering contributions
The gluon exchange between a $J/\psi$ meson and the spectator $s$-quark leads
to the hard scattering amplitudes,
$H_{n}^{h=0}(\phi J/\psi)=\pm\ 3\frac{B_{\phi J/\psi}}{A_{\phi
J/\psi}^{h=0}}\frac{m_{B_{s}^{0}}}{\lambda_{B_{s}^{0}}}\Big{(}r_{\chi}^{\phi}(\mu)X_{H}+3\Big{)}\\!,$
(36)
for $h=0$, $\mu=m_{b}/2$ and $\lambda_{B_{s}^{0}}=0.350$ GeV Beneke:2003zv .
The plus sign is for $n=2,3,9$ and the minus sign for $n=5,7$. The
phenomenological amplitude, $X_{H}$, parametrizes the endpoint divergence of
the scalar meson’s LCDA and is defined in Eq. (54). For the helicity, $h=+1$,
the correction reads
$H_{n}^{h=+1}(\phi J/\psi)=\mp\ 18\frac{B_{\phi J/\psi}}{A_{\phi
J/\psi}^{h=+1}}\frac{f_{\phi}^{\perp}}{f_{\phi}}\frac{m_{J/\psi}}{\lambda_{B_{s}^{0}}}(X_{H}-1)\
,$ (37)
where the minus sign applies to $n=2,3,9$ and the plus sign to $n=5,7$. The
helicity, $h=-1$, contribution is simply,
$H_{n}^{h=-1}(\phi J/\psi)=0\;\;{\rm for}\;\;n=2,3,5,7,9\ .$ (38)
Table 1: Wilson coefficients at the $\mu=m_{b}$ and $\mu=m_{b}/2$ scales in the NDR scheme Beneke:2001ev . The coefficients $C_{7}(\mu)-C_{10}(\mu)$ must be multiplied by $\alpha_{e}$. | $C_{1}(\mu)$ | $C_{2}(\mu)$ | $C_{3}(\mu)$ | $C_{4}(\mu)$ | $C_{5}(\mu)$ | $C_{6}(\mu)$ | $C_{7}(\mu)$ | $C_{8}(\mu)$ | $C_{9}(\mu)$ | $C_{10}(\mu)$ | $C_{7\gamma}(\mu)$
---|---|---|---|---|---|---|---|---|---|---|---
$\mu=m_{b}$ | $1.081$ | $-0.190$ | $0.014$ | $-0.036$ | $0.009$ | $-0.042$ | $-0.011$ | $0.06$ | $-1.254$ | $0.233$ | $-0.318$
$\mu=m_{b}/2$ | $1.137$ | $-0.295$ | $0.021$ | $-0.051$ | $0.010$ | $-0.065$ | $-0.24$ | $0.096$ | $-1.325$ | $0.331$ | $-0.364$
## IV QCDF corrections for $\bm{B_{s}^{0}\to f_{0}(980)J/\psi}$ decay
amplitudes
We now turn to the $B_{s}^{0}\to J/\psi f_{0}(980)$ transition for which the
$\alpha_{\mathrm{s}}(\mu)$ corrections are all included following Ref.
Beneke:2003zv applied to an $SV$ final state. For previously mentioned
reasons, we solely employ the first non-vanishing leading term in the LCDA,
$\phi_{f_{0}}(x)=6x(1-x)\Big{[}3B_{1}(\mu)(2x-1)\Big{]}\ ,$ (39)
where $B_{1}(m_{b}/2)=-0.54$ Cheng:2005nb is the $f_{0}(980)$’s first
Gegenbauer moment and we remind that only odd moments contribute in case of
charge-neutral scalar mesons. In particular, contrary to the pseudoscalar
LCDA, the leading term $6x(1-x)B_{0}$ vanishes since
$B_{0}=(m_{1}(\mu)-m_{2}(\mu))/m_{S}$, where $m_{S}$ is the scalar meson mass
and $m_{1,2}(\mu)$ its running quark masses. The scalar twist-3 two-particle
distribution is given by
$\varphi_{f_{0}}(x)=1\ .$ (40)
The asymptotic forms of the LCDA, $\phi_{J/\psi}(x)$ (Eq. (29)) and
$\varphi_{J/\psi}(x)$ (Eq. (30)), are used. As in the $B_{s}^{0}\to\phi
J/\psi$ decay, the $J/\psi$ meson is created from vacuum whereas the
transition $B_{s}^{0}\to f_{0}(980)$ produces the scalar meson. Here, we only
consider the $s\bar{s}$ component of the $f_{0}(980)$ since the flavor of the
spectator quark in the tree and penguin topologies of $B_{s}^{0}$ decays is
strange. There are no penguin corrections Beneke:2003zv to the $B_{s}^{0}\to
f_{0}(980)J/\psi$ decay amplitude in Eq. (4).
### IV.1 Vertex contributions
At the order of $\alpha_{s}(\mu)$, the vertex correction, $V_{n}(J/\psi)$,
involves the leading twist distribution, $\phi_{J/\psi}(x)$, and a gluon
kernel given in Beneke:2003zv . We derive from this the expressions,
$V_{n}(J/\psi)=\left\\{\begin{array}[]{ll}\displaystyle 12\;{\rm
ln}\left(\frac{m_{b}}{\mu}\right)-3i\pi-\frac{37}{2}\ ,\vskip 6.0pt plus 2.0pt
minus 2.0pt\\\ \qquad\mathrm{for}\,\,n\in\\{2,3,9\\}&\\\\[10.00002pt]
\displaystyle-12\;{\rm ln}\left(\frac{m_{b}}{\mu}\right)+3i\pi+\frac{13}{2}\
,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\qquad\mathrm{for}\,\,n\in\\{5,7\\}&\end{array}\right.$ (41)
with $\mu=m_{b}$.
### IV.2 Hard scattering contributions
The hard scattering correction in case of an $f_{0}J/\psi$ final state reads
$H_{n}(f_{0}J/\psi)=\pm\,3\,\frac{B_{f_{0}J/\psi}}{A_{f_{0}J/\psi}}\frac{m_{B_{s}^{0}}}{\lambda_{B_{s}^{0}}}\Big{(}\bar{r}_{\chi}^{f_{0}}(\mu)X_{H}+3B_{1}(\mu)\Big{)},$
(42)
where the plus sign applies to $n=2,3,9$, the minus sign to $n=5,7$ and
$X_{H}$ is given, as in the case of the $\phi J/\psi$ final state, by Eq.
(54).
The chiral coefficient, $\bar{r}_{\chi}^{f_{0}}(\mu)$, enters Eq. (42) rather
than $r_{\chi}^{f_{0}}(\mu)$ defined as,
$r_{\chi}^{f_{0}}(\mu)=\frac{2m_{f_{0}}^{2}}{m_{b}(\mu)\left(m_{1}(\mu)-m_{2}(\mu)\right)}\
.$ (43)
The reason is that in case of neutral scalar mesons, $m_{1}(\mu)=m_{2}(\mu)$
and $r_{\chi}^{f_{0}}(\mu)$ diverges. On the other hand, it is known from
$C$-conjugation invariance that the vector decay constant of the neutral
scalar meson must vanish. However, the quark equations of motions yield a
relation between the scalar- and vector-decay constants, $\bar{f}_{f_{0}}$ and
$f_{f_{0}}$ respectively:
$\bar{f}_{f_{0}}=\frac{m_{f_{0}}}{m_{1}(\mu)-m_{2}(\mu)}f_{f_{0}}\ ,$ (44)
where $m_{f_{0}}\bar{f}_{f_{0}}=\langle 0|\bar{q}_{2}q_{1}|f_{0}\rangle$.
Since $\bar{f}_{f_{0}}$ is non-zero, the product
$f_{f_{0}}m_{f_{0}}/(m_{1}(\mu)-m_{2}(\mu))$ is finite in the limit
$m_{1}(\mu)\to m_{2}(\mu)$. We thus recombine,
$f_{f_{0}}r_{\chi}^{f_{0}}=\bar{f}_{f_{0}}\bar{r}_{\chi}^{f_{0}}$, with
$\bar{r}_{\chi}^{f_{0}}(\mu)=\frac{2m_{f_{0}}}{m_{b}(\mu)}\ .$ (45)
Table 2: Values of the higher order correction ($\rho_{C}$, $\phi_{C}$) and hard-scattering ($\rho_{H}$, $\phi_{H}$) parameters as function of the $B_{s}^{0}$ decay constant. $f_{B_{s}^{0}}$ [MeV] | $\rho_{C}$ | $\phi_{C}$ (∘) | $\rho_{H}$ | $\phi_{H}$ (∘)
---|---|---|---|---
230 | $4.52\pm 2.24$ | $173.8\pm 37.6$ | $1.90\pm 0.20$ | $266.0\pm 21.6$
260 | $6.16\pm 2.03$ | $176.1\pm 53.6$ | $1.70\pm 0.16$ | $260.6\pm 19.3$
290 | $7.33\pm 1.63$ | $176.0\pm 57.6$ | $1.54\pm 0.15$ | $255.6\pm 17.3$
Table 3: Short-distance amplitudes, $a_{n}^{q,h}(m_{b})\times 10^{3}$, for the helicity $h=+1$ in $B_{s}^{0}\to J/\psi\phi$, as a function of the decay constant, $f_{B_{s}^{0}}$, and with $\bar{f}_{f_{0}^{s}}=380$ MeV. The $LOVP$ results are obtained with the leading order ($LO$) amplitude to which vertex $V$ and penguin $P$ corrections are added. In case of $LOVPH$, the hard scattering contribution with the endpoint parametrization $X_{H}$ is included. $LOVPH+C$ contains additionally the purely phenomenological contribution $\zeta^{(h)}$ with two more parameters. $f_{B_{s}^{0}}$ [MeV] | | 230 | 260 | 290
---|---|---|---|---
| $LOVP$ | $LOVPH$ | $LOVPH+C$ | $LOVPH$ | $LOVPH+C$ | $LOVPH$ | $LOVPH+C$
$a_{2}^{u,c}(m_{b})$ | $60.38-i\,161.7$ | $-3.77+i\,148.8$ | $-8.43+i\,129.87$ | $38.06+i\,149.7$ | $8.21+i\,130.04$ | $75.52+i\,140.4$ | $22.57+i\,128.49$
$a_{3}^{u,c}(m_{b})$ | $5.66+i\,5.39$ | $8.54-i\,8.54$ | $8.75-i\,7.69$ | $6.66-i\,8.58$ | $8.0-i\,7.70$ | $4.98-i\,8.17$ | $7.36-i\,7.63$
$a_{5}^{u,c}(m_{b})$ | $-5.27-i\,6.28$ | $-8.94+i\,11.47$ | $-9.21+i\,10.39$ | $-6.55+i\,11.52$ | $-8.25+i\,10.40$ | $-4.41+i\,10.99$ | $-7.43+i\,10.31$
$a_{7}^{u}(m_{b})$ | $0.12+i\,0.07$ | $0.17-i\,0.13$ | $0.17-i\,0.12$ | $0.14-i\,0.13$ | $0.16-i\,0.12$ | $0.11-i\,0.13$ | $0.15-i\,0.12$
$a_{7}^{c}(m_{b})$ | $0.69+i\,0.07$ | $0.73-i\,0.13$ | $0.74-i\,0.12$ | $0.71-i\,0.13$ | $0.73-i\,0.12$ | $0.68-i\,0.13$ | $0.72-i\,0.12$
$a_{9}^{u}(m_{b})$ | $-9.25-i\,0.27$ | $-9.40+i\,0.43$ | $-9.41+i\,0.39$ | $-9.30+i\,0.43$ | $-9.37+i\,0.39$ | $-9.22+i\,0.41$ | $-9.34+i\,0.38$
$a_{9}^{c}(m_{b})$ | $-8.68-i\,0.27$ | $-8.83+i\,0.43$ | $-8.84+i\,0.39$ | $-8.73+i\,0.43$ | $-8.80+i\,0.39$ | $-8.65+i\,0.41$ | $-8.77+i\,0.38$
Table 4: Short-distance amplitudes, $a_{n}^{q,h}(m_{b})\times 10^{3}$, for helicity $h=-1$ and $B_{s}^{0}\to J/\psi\phi$. Since the hard-scattering contributions are zero, these amplitudes are independent of $f_{B_{s}^{0}}$. $a_{2}^{u,c}(m_{b})$ | $-51.72$
---|---
$a_{3}^{u,c}(m_{b})$ | $9.39$
$a_{5}^{u,c}(m_{b})$ | $-9.63$
$a_{7}^{u,c}(m_{b})$ | $0.12$
$a_{9}^{u,c}(m_{b})$ | $-9.49$
Table 5: As in Table 3 but for the helicity $h=0$. $f_{B_{s}^{0}}$ [MeV] | | 230 | 260 | 290
---|---|---|---|---
| $LOVP$ | $LOVPH$ | $LOVPH+C$ | $LOVPH$ | $LOVPH+C$ | $LOVPH$ | $LOVPH+C$
$a_{2}^{u,c}(m_{b})$ | $54.51-i\,80.86$ | $160.2-i\,132.6$ | $161.0-i\,129.4$ | $165.7-i\,132.7$ | $170.6-i\,129.5$ | $171.8-i\,131.2$ | $180.6-i\,129.2$
$a_{3}^{u,c}(m_{b})$ | $5.86+i\,2.69$ | $1.11+i\,5.01$ | $1.08+i\,4.87$ | $0.87+i\,5.02$ | $0.65+i\,4.87$ | $0.60+i\,4.95$ | $0.20+i\,4.86$
$a_{5}^{u,c}(m_{b})$ | $-7.17-i\,3.14$ | $-1.12-i\,6.10$ | $-1.08-i\,5.92$ | $-0.81-i\,6.11$ | $-0.53-i\,5.92$ | $-0.46-i\,6.02$ | $0.04-i\,5.90$
$a_{7}^{u,c}(m_{b})$ | $0.09+i\,0.03$ | $0.02+i\,0.07$ | $0.02+i\,0.07$ | $0.02+i\,0.07$ | $0.02+i\,0.07$ | $0.02+i\,0.07$ | $0.01+i\,0.06$
$a_{9}^{u,c}(m_{b})$ | $-9.31-i\,0.14$ | $-9.07-i\,0.25$ | $-9.07-i\,0.24$ | $-9.06-i\,0.25$ | $-9.05-i\,0.24$ | $-9.05-i\,0.25$ | $-9.03-i\,0.24$
## V Numerical parameters
This section serves to summarize all parameter values required for numerical
applications. The Wilson coefficients at the scales $\mu=m_{b}$ and
$\mu=m_{b}/2$ used in this work are listed in Table 1. For the meson masses,
we refer to the latest PDG values Amsler:2008zzb , which are (in GeV):
$m_{B_{s}^{0}}=5.366\ ,\;\;m_{B_{s}^{\star}}=5.412\ ,\;\;m_{f_{0}}=0.980\ ,\\\
m_{J/\psi}=3.096\ ,\;\;m_{\phi}=1.019\ .\hskip 28.45274pt$ (46)
The running quark masses at $\mu=m_{b}=4.2$ GeV are (in GeV),
$m_{b}=4.2\,,\;m_{c}=1.3\,,\;m_{s}=0.07\,,\;m_{u,d}=0.003\,,$ (47)
and those at $\mu=m_{b}/2=2.1$ GeV are,
$m_{b}=4.95\,,\;m_{c}=1.51\,,\;m_{s}=0.09\,,\;m_{u,d}=0.005\ .$ (48)
We take the $\phi$ decay constant values from Ref. Beneke:2006hg :
$f_{\phi}=(221\pm 3)$ MeV and $f_{\phi}^{\perp}=(175\pm 25)$ MeV. For the
$J/\psi$ meson, we use $f_{J/\psi}=(416\pm 6)$ MeV Hwang:2006cua and
$f_{J/\psi}^{\perp}=(405\pm 5)$ MeV Cheng:2001ez . In the $B_{s}\to J/\psi
f_{0}(980)$ channel, the $s\bar{s}$ component of the $f_{0}(980)$ is involved
which implies the poorly known scalar decay constant $\bar{f}_{f_{0}}$: one
theoretical estimate yields $\bar{f}_{f_{0}^{s}}=(180\pm 15)$ MeV
DeFazio:2001uc whereas a much larger value
$\bar{f}_{f_{0}^{s}}(1\,\mathrm{GeV})=(370\pm 20)$ MeV
$\big{[}\bar{f}_{f_{0}^{s}}(2.1~{}\mathrm{GeV})=(460\pm
25)\mathrm{MeV}\big{]}$ is found in Ref. Cheng:2005nb , both from coupling to
the scalar $\bar{s}s$ current only (denoted by the superscript $s$ in
$f_{0}^{s}$, which we use henceforth). Similarly, several theoretical
predictions exist for the leptonic $B_{s}$ decay constants of which we select
three values from unquenched lattice QCD: $f_{B_{s}^{0}}=(204\pm
12^{+24}_{-23})$ MeV Lellouch:2000tw , $f_{B_{s}^{0}}=(259\pm 32)$ MeV
Gray:2005ad and $f_{B_{s}^{0}}=(231\pm 15)$ MeV Gamiz:2009ku .
To illustrate the sensitivity of the ratio $\mathcal{R}_{f_{0}/\phi}$ to the
hadronic uncertainties, we exemplarily choose three different values for each
decay constant: $f_{B_{s}^{0}}=230$, $260$, $290$ MeV and
$\bar{f}_{f_{0}^{s}}=340$, $380$, $420$ MeV.
### V.1 $\bm{B\to V}$ transition form factor
Values for the $B_{s}^{0}\to\phi$ transition form factors are taken from the
pole-extrapolation model by Melikhov Melikhov:2001zv :
$A_{0}(q^{2})^{B_{s}^{0}\to\phi}=\frac{a_{0}(0)}{\left(1-\frac{q^{2}}{m^{2}_{B_{s}^{0}}}\right)\left(1-\sigma_{1}\frac{q^{2}}{m^{2}_{B_{s}^{0}}}+\sigma_{2}\frac{q^{4}}{m^{4}_{B_{s}^{0}}}\right)}\
.$ (49)
The form factor $V(q^{2})^{B_{s}^{0}\to\phi}$ is given by a similar expression
in which $a_{0}(0)$ is replaced by $v(0)$ and $m_{B_{s}^{0}}$ by
$m_{B_{s}^{\star}}$ Melikhov:2001zv . Next, the
$A_{1}(q^{2})^{B_{s}^{0}\to\phi}$ form factor is parametrized by
$A_{1}(q^{2})^{B_{s}^{0}\to\phi}=\frac{a_{1}(0)}{\left(1-\sigma_{1}\frac{q^{2}}{m^{2}_{B_{s}^{\star}}}+\sigma_{2}\frac{q^{4}}{m^{4}_{B_{s}^{\star}}}\right)}\
.$ (50)
Finally, $A_{2}(q^{2})^{B_{s}^{0}\to\phi}$ has the same functional form as
$A_{1}(q^{2})^{B_{s}^{0}\to\phi}$ where $a_{1}(0)$ is replaced by $a_{2}(0)$.
In both, Eqs. (49) and (50), the momentum transfer is $q^{2}=m^{2}_{J/\psi}$.
In Eqs. (49) and (50), the form factors at $q^{2}=0$ are $a_{0}(0)=0.42$
($v(0)=0.44$) and $a_{1}(0)=0.34$ ($a_{2}(0)=0.31$). The extrapolation
parameters are, for $A_{0}(q^{2})^{B_{s}^{0}\to\phi}$, $\sigma_{1}=0.55$ and
$\sigma_{2}=0.12$; for $V(q^{2})^{B_{s}^{0}\to\phi}$, $\sigma_{1}=0.62$ and
$\sigma_{2}=0.20$; for $A_{1}(q^{2})^{B_{s}^{0}\to\phi}$, $\sigma_{1}=0.73$
and $\sigma_{2}=0.42$ and finally for $A_{2}(q^{2})^{B_{s}^{0}\to\phi}$,
$\sigma_{1}=1.30$ and $\sigma_{2}=0.52$. The respective values for the form
factors at the value $q^{2}=m_{J/\psi}^{2}$ are
$A_{0}(q^{2})^{B_{s}^{0}\to\phi}=0.76$,
$A_{1}(q^{2})^{B_{s}^{0}\to\phi}=0.42$, $A_{2}(q^{2})^{B_{s}^{0}\to\phi}=0.49$
and $V(q^{2})^{B_{s}^{0}\to\phi}=0.80$.
### V.2 $\bm{B\to S}$ transition form factor
We studied the transition form factor, $F_{0,1}^{B_{s}^{0}\to
f_{0}^{s}}(q^{2})$, in a comparative calculation using a dispersion relation
and a covariant light front dynamics model ElBennich:2008xy . To our
knowledge, this form factor has only been calculated recently in QCD sum rules
Ghahramany:2009zz ; Colangelo:2010bg and pQCD Li:2008tk for $q^{2}=0$ and must
be extrapolated to the value $F_{0,1}^{B_{s}^{0}\to
f_{0}^{s}}(m_{J/\psi}^{2})$.
In our work ElBennich:2008xy , the transition form factors are derived from
the constituent quark three-point function, the vertices of which are the weak
interaction coupling, $\gamma_{\mu}(1-\gamma_{5})$, and two phenomenological
Bethe-Salpeter amplitudes for the $B_{(s)}$ and $f_{0}(980)$ mesons. While the
$B_{s}$ can be parametrized with the leptonic decay constant (known from
lattice-QCD simulations), the latter is more problematic since the
$\bar{f}_{f_{0}^{s}}$ is poorly determined. In an attempt to formulate a
suitable scalar $f_{0}(980)$ vertex function, we constrained its parameters by
means of experimental quasi two-body branching fractions, $D_{(s)}\to
f_{0}(980)P$, $P=\pi,K$. The advantage is that the $F_{+}^{B_{s}^{0}\to
f_{0}^{s}}(q^{2})$ and $F_{-}^{B_{s}^{0}\to f_{0}^{s}}(q^{2})$ form factors,
$\displaystyle\langle
f_{0}^{s}(p_{2})|\bar{s}\gamma_{\mu}(1-\gamma_{5})b|B_{s}^{0}(p_{1})\rangle\
=$ (51) $\displaystyle F_{+}^{B_{s}^{0}\to
f_{0}^{s}}(q^{2})(p_{1}+p_{2})_{\mu}+F_{-}^{B_{s}^{0}\to
f_{0}^{s}}(q^{2})(p_{1}-p_{2})_{\mu},$
can be calculated for any physical time-like momentum transfer
$q^{2}=(p_{1}-p_{2})^{2}$. The superscript $s$ is a reminder that the
transition is to the $\bar{s}s$ component of the scalar meson and $p_{1}$ and
$p_{2}$ are the $B_{s}^{0}$ and $f_{0}(980)$ four-momenta, respectively. We do
stress that the $B_{s}\to f_{0}(980)$ form factor calculated by us in Ref.
ElBennich:2008xy does not assume a pure $\bar{s}s$ state of the $f_{0}(980)$.
Instead, it was treated as a mixture of strange and non-strange $\bar{q}q$
components related by a mixing angle which also yields the related form factor
$F_{0,1}^{B\to f_{0}^{u,d}}(q^{2})$. This angle was determined with
experimental constraints ElBennich:2008xy and the overall normalization of the
transition form factor receives contributions from both states.
The form factors $F_{\pm}(q^{2})$ (we suppress the flavor superscripts) are
related to the set of vector and scalar form factors as,
$\displaystyle F_{1}(q^{2})$ $\displaystyle=$ $\displaystyle F_{+}(q^{2})\ ,$
(52) $\displaystyle F_{0}(q^{2})$ $\displaystyle=$ $\displaystyle
F_{+}(q^{2})+\frac{q^{2}}{m_{B_{s}^{0}}^{2}-m_{f_{0}}^{2}}F_{-}(q^{2})\ .$
(53)
The form factor $F_{1}(q^{2})$ we obtain in both the dispersion relation and
covariant light front dynamics approaches agree at the maximum recoil point
$q^{2}=0$. At large four-momentum transfer, specifically for
$q^{2}=m_{J/\psi}^{2}\simeq 10$ GeV2, our model predictions differ
significantly which is also known to occur for $B\to\pi$ transition form
factors ElBennich:2009vx . This is not surprising, as for large momentum
transfers the final-state meson is less energetic and the soft physics of the
bound states becomes more relevant. Since the models differ in their
parametrization of the bound-state wave functions, it is clear that their
inaccuracies are revealed in the form-factor predictions at large $q^{2}$. In
Ref. Colangelo:2010bg , we deduce from the author’s extrapolation
parametrization that $F_{1}^{B_{s}^{0}\to f_{0}^{s}}(m_{J/\psi}^{2})\simeq
0.3$, which is compatible with our dispersion-relation prediction $\simeq 0.4$
within the errors. In Section VI, we will account for this rather large window
of values and plot the ratio $\mathcal{R}_{f_{0}/\phi}$ as a function of
$F_{1}^{B_{s}^{0}\to f_{0}^{s}}(m_{J/\psi}^{2})$.
### V.3 Model parameters
The hard scattering contributions involve endpoint divergences, which we
choose to parametrize by,
$X_{H}=\Bigl{(}1+\rho_{H}\exp(i\phi_{H})\Bigr{)}\ln\frac{m_{B_{s}^{0}}}{\lambda_{h}}\
.$ (54)
In case of a possible annihilation or “other” contribution we simply write,
$X_{C}=\rho_{C}\exp(i\phi_{C})$ (55)
which introduces four parameters, $0<\rho_{C,H}$ and
$0<\phi_{C,H}<360^{\circ}$. We assume that
$X_{C,H}^{h=0}=X_{C,H}^{h=-1}=X_{C,H}^{h=+1}=X_{C,H}$, as the vector $\phi$
and scalar $f_{0}(980)$ mesons have similar masses and we consider the
$s\bar{s}$ component only. The hard scattering corrections are expected to be
of the order of $m_{B_{s}^{0}}/\lambda_{h}$ in Eq. (54), with
$\lambda_{h}=0.5$ GeV. The parameters $\rho_{C,H}$ and $\phi_{C,H}$ are chosen
so as to reproduce the experimental data discussed in Section VI. We insert
their values in the $B_{s}^{0}\to J/\psi f_{0}$ decay amplitude (4) and then
predict the branching ratio $\mathcal{B}(B_{s}^{0}\to f_{0}J/\psi)$.
Table 6: Short-distance amplitudes, $a_{n}^{q}(m_{b})\times 10^{3}$, for $B_{s}^{0}\to J/\psi f_{0}(980)$ as a function of the $f_{B_{s}^{0}}$ decay constant with $\bar{f}_{f_{0}^{s}}=380$ MeV and $F_{1}^{B_{s}^{0}\to f_{0}^{s}}(m_{J/\psi}^{2})=0.4$. See caption in Table 3 for the definition of $LOVP$, $LOVPH$ and $LOVPH+C$ amplitudes. $f_{B_{s}^{0}}$ [MeV] | | 230 | 260 | 290
---|---|---|---|---
| $LOVP$ | $LOVPH$ | $LOVPH+C$ | $LOVPH$ | $LOVPH+C$ | $LOVPH$ | $LOVPH+C$
$a_{2}^{u,c}(m_{b})$ | $11.61-i\,80.86$ | $-42.40-i\,255.5$ | $-33.35-i\,224.3$ | $-66.51-i\,234.1$ | $-51.82-i\,224.4$ | $-95.23-i\,229.5$ | $-69.17-i\,223.8$
$a_{3}^{u,c}(m_{b})$ | $7.29+i\,2.69$ | $9.71+i\,10.53$ | $9.30+i\,9.13$ | $10.80+i\,9.57$ | $10.13+i\,9.13$ | $12.08+i\,9.36$ | $10.91+i\,9.10$
$a_{5}^{u,c}(m_{b})$ | $-7.17-i\,3.14$ | $-10.25-i\,13.12$ | $-9.74-i\,11.34$ | $-11.63-i\,11.90$ | $-10.79-i\,11.35$ | $-13.27-i\,11.64$ | $-11.78-i\,11.31$
$a_{7}^{u,c}(m_{b})$ | $0.09+0.03$ | $0.13+i\,0.15$ | $0.12+i\,0.13$ | $0.14+i\,0.14$ | $0.14+i\,0.13$ | $0.16+i\,0.13$ | $0.15+i\,0.13$
$a_{9}^{u,c}(m_{b})$ | $-9.38-i\,0.14$ | $-9.51-i\,0.53$ | $-9.49-i\,0.46$ | $-9.56-i\,0.48$ | $-9.53-i\,0.46$ | $-9.63-i\,0.47$ | $-9.57-i\,0.46$
Table 7: The phenomenological contributions $\zeta^{h}\times 10^{3}$ for $h=0,-1,+1$, Eq. (24), to the $B_{s}^{0}\to J/\psi\phi$ amplitude as a function of the $f_{B_{s}^{0}}$ decay constant with $\bar{f}_{f_{0}^{s}}=380$ MeV. $f_{B_{s}^{0}}$ [MeV] | 230 | 260 | 290
---|---|---|---
$\zeta^{h=0}$ | $-18.11+i\,1.98$ | $-28.04+i\,1.89$ | $-37.19+i\,2.63$
$\zeta^{h=-1}$ | $-129.26+i\,14.12$ | $-200.12+i\,13.46$ | $-265.41+i\,18.77$
$\zeta^{h=+1}$ | $-15.25+i\,1.67$ | $-23.61+i\,1.59$ | $-31.31+i\,2.21$
Table 8: Same as Table 7 but for the $B_{s}^{0}\to J/\psi f_{0}(980)$ amplitude. $f_{B_{s}^{0}}$ [MeV] | 230 | 260 | 290
---|---|---|---
$\zeta$ | $-44.08+i\,4.81$ | $-68.25+i\,4.59$ | $-90.51+i\,6.40$
Table 9: Prediction for the $B_{s}^{0}\to J/\psi f_{0}$ observables for the different amplitudes $LOVP$, $LOVPH$ and $LOVPH+C$ along with experimental analysis data of the $B_{s}^{0}\to J/\psi\phi$ decay. Here central values, $f_{B_{s}^{0}}=260$ MeV and $\bar{f}_{f_{0}^{s}}=380$ MeV, and the transition form factor $F_{1}^{B_{s}^{0}\to f_{0}^{s}}(q^{2}=m^{2}_{J/\psi})=0.4$ are used. The values in the second column are predictions. Those of the third column include the hard scattering corrections with the endpoint parametrization $\rho_{H}=1.85\pm 0.07$ and $\phi_{H}=255.9^{\circ}\pm 24.6^{\circ}$. The fourth column corresponds to the reproduction of the data with the parameters $\rho_{H}$, $\phi_{H}$, $\rho_{C}$ and $\phi_{C}$ displayed in the second line of Table 2. | $LOVP$ | $LOVPH$ | $LOVPH+C$ | Experimental
---|---|---|---|---
| (Prediction) | (2 parameters) | (4 parameters) | data
$\left|\mathcal{A}_{L}\right|^{2}$ | $0.172$ | $0.554$ | $0.555$ | $0.555\pm 0.033$ Abazov:2008jz
$\left|\mathcal{A}_{\parallel}\right|^{2}$ | $0.404$ | $0.219$ | $0.244$ | $0.244\pm 0.046$ Abazov:2008jz
$\phi_{\parallel}(\mathrm{rad})$ | $-0.221$ | $2.13$ | $2.72$ | $2.72\pm 1.38$ Abazov:2008jz
$\mathcal{B}(B_{s}^{0}\to J/\psi\phi)$ | $0.00075$ | $0.00115$ | $0.00093$ | $0.00093\pm 0.00033$ Amsler:2008zzb
$\mathcal{B}(B_{s}^{0}\to J/\psi f_{0})$ | $0.00020$ | $0.00047$ | $0.00050$ |
$A_{CP}(B_{s}^{0}\to J/\psi f_{0})$ | $-0.00013$ | $-0.0013$ | $-0.0011$ |
$\mathcal{R}_{f_{0}/\phi}$ | $0.28$ | 0.42 | $0.55$ |
## VI Results and experimental data
In the $B_{s}^{0}\to\phi J/\psi$ decay, one can define five observables: a
longitudinal, parallel and perpendicular polarization fraction,
$f_{L},f_{\parallel}$ and $f_{\perp}$, respectively,
$f_{k}=\frac{\left|\mathcal{A}_{k}\right|^{2}}{\left|\mathcal{A}_{L}\right|^{2}+\left|\mathcal{A}_{\parallel}\right|^{2}+\left|\mathcal{A}_{\perp}\right|^{2}}\
,\;k=L,\parallel,\perp$ (56)
as well as two relative phases, $\phi_{\parallel}$ and $\phi_{\perp}$,
$\phi_{k}=\arg\left(\frac{\mathcal{A}_{k}}{\mathcal{A}_{L}}\right)\
,\;k=\parallel,\perp\ ,$ (57)
where we have abbreviated, $\mathcal{A}_{L}={\cal A}_{{B}_{s}^{0}\to\phi
J/\psi}^{(h=0)}$, $\mathcal{A}_{\parallel}=\big{[}{\cal A}_{{B}_{s}^{0}\to\phi
J/\psi}^{(h=+1)}+{\cal A}_{{B}_{s}^{0}\to\phi
J/\psi}^{(h=-1)}\big{]}/\sqrt{2}$ and $\mathcal{A}_{\perp}=\big{[}{\cal
A}_{{B}_{s}^{0}\to\phi J/\psi}^{(h=+1)}-{\cal A}_{{B}_{s}^{0}\to\phi
J/\psi}^{(h=-1)}\big{]}/\sqrt{2}$.
The $CP$ average is defined in terms of the polarization fractions, $f_{k}$,
$A_{\mathrm{CP}}^{k}=\frac{f_{k}^{\bar{{B}}_{s}^{0}}-f_{k}^{{B}_{s}^{0}}}{f_{k}^{\bar{{B}}_{s}^{0}}+f_{k}^{{B}_{s}^{0}}}\
.$ (58)
Similarly, for $B_{s}^{0}\to f_{0}(980)J/\psi$, the $CP$ average is defined
as,
$A_{\mathrm{CP}}=\frac{\mathcal{B}(\bar{{B}}_{s}^{0}\to
f_{0}J/\psi)-\mathcal{B}({B}_{s}^{0}\to
f_{0}J/\psi)}{\mathcal{B}(\bar{{B}}_{s}^{0}\to
f_{0}J/\psi)+\mathcal{B}({B}_{s}^{0}\to f_{0}J/\psi)}\ .$ (59)
We use data from CDF and D$\emptyset$ for the $B_{s}^{0}\to\phi J/\psi$ decay,
whereas there is no available data on the channel $B_{s}^{0}\to f_{0}J/\psi$.
Our data compilation consists of the D$\emptyset$ values for the amplitudes,
$\left|\mathcal{A}_{L}\right|^{2}=0.555\pm 0.027\pm 0.006$,
$\left|\mathcal{A}_{\parallel}\right|^{2}=0.244\pm 0.032\pm 0.014$ and the
relative phase $\phi_{\parallel}=2.72^{+1.12}_{-0.27}$ rad Abazov:2008jz . The
CDF values Kuhr:2007dt are compatible,
$\left|\mathcal{A}_{L}\right|^{2}=0.530\pm 0.021\pm 0.007$ and
$\left|\mathcal{A}_{\parallel}\right|^{2}=0.230\pm 0.027\pm 0.009$, and the
PDG data book quotes the branching fraction, $\mathcal{B}(B_{s}^{0}\to
J/\psi\phi)=(9.3\pm 3.3)\times 10^{-4}$ Amsler:2008zzb .
The ratio $\mathcal{R}_{f_{0}/\phi}$ has been argued Stone:2008ak to be of the
order $0.2-0.3$, based on the knowledge of the experimental ratio of decay
rates Frabetti:1995sg ,
$\frac{\Gamma(D_{s}^{+}\\!\to\\!f_{0}\pi^{+}\\!\to
K^{+}K^{-}\pi^{-})}{\Gamma(D_{s}^{+}\\!\to\phi\pi^{+}\\!\\!\to
K^{+}K^{-}\pi^{-})}=0.28\pm 0.12,$ (60)
and an estimate of the semileptonic, integrated branching fraction ratio
$\mathcal{B}(D_{s}^{+}\to
f_{0}e^{+}\nu,f_{0}\to\pi^{+}\pi^{-})/\mathcal{B}(D_{s}^{+}\to\phi
e^{+}\nu,\phi\to K^{+}K^{-})=(13\pm 4)\%$ from CLEO Yelton:2009cm . The ratio
$\mathcal{R}_{f_{0}/\phi}$ was reassessed in terms of the differential decay
ratio Ecklund:2009fia ,
$\displaystyle\mathcal{R}_{f_{0}/\phi}$ $\displaystyle=$
$\displaystyle\frac{\frac{d\Gamma}{dq^{2}}(D_{s}^{+}\to
f_{0}e^{+}\nu,f_{0}\to\pi^{+}\pi^{-})\big{|}_{q^{2}=0}}{\frac{d\Gamma}{dq^{2}}(D_{s}^{+}\to\phi
e^{+}\nu,\phi\to K^{+}K^{-})\big{|}_{q^{2}=0}}$ (61) $\displaystyle=$
$\displaystyle 0.42\pm 0.11.$
If we combine the above three experimental estimates, we propose a window of
$0.2\lesssim\mathcal{R}_{f_{0}/\phi}\lesssim 0.5$ for the ratio based on
$D_{s}$ decays.
With the experimental data listed under Eq. (59) as constraint, we find
optimal values for $X_{C}$ and $X_{H}$. In principle, we deal with a system of
four coupled non-linear equations for $\left|\mathcal{A}_{L}\right|^{2}$,
$\left|\mathcal{A}_{\parallel}\right|^{2}$, $\phi_{\parallel}$ and
$\mathcal{B}(B_{s}^{0}\to\phi J/\psi)$ and four variables, which does not put
tight constraints on the phenomenological part of our $B_{s}^{0}\to
J/\psi\phi$ amplitude. When solving numerically we find, depending on the
$f_{B_{s}^{0}}$ values, two solutions among which only one yields a reasonable
value for the branching fraction $\mathcal{B}(B_{s}\to f_{0}J/\psi)$ not too
different from that in a naive quark model. We list the parameters
$\rho_{C,H}$ and $\phi_{C,H}$ independent of $\bar{f}_{f_{0}^{s}}$ for three
values of $f_{B_{s}^{0}}$ in Table 2, from which it is plain that the
uncertainties on the magnitude of the modulus $\rho_{C}$ as well as the phase
$\phi_{C}$ are substantial. The experimental errors on the observables are
clearly not constraining enough. Yet, we observe that the variations of
$X_{C}$ and $X_{H}$ are smooth as a function of the decay constant
$f_{B_{s}^{0}}$.
Likewise, we present numerical values for $a_{n}^{q,h}(m_{b})$ for the three
helicities in $B_{s}^{0}\to J/\psi\phi$ in Tables 3, 4 and 5 and for
$B_{s}^{0}\to J/\psi f_{0}$ in Table 6 as functions of $f_{B_{s}^{0}}$ to
illustrate one facet of the hadronic uncertainty. In these tables, we list the
decomposition of $a_{n}^{q,h}(m_{b})$ for each value of $f_{B_{s}^{0}}$; in
the first column, the values of $a_{n}^{q,h}(m_{b})$ are for the calculated
leading order ($LO$), vertex ($V$) and penguin ($P$) amplitudes only. These
are independent of $f_{B_{s}^{0}}$ and correspond to the predictions in Figure
2. Next, the $a_{n}^{q,h}(m_{b})$ that contain the $LO$, $V$, $P$ and the
hard-scattering ($H$) amplitudes, where only $\rho_{H}$ and $\phi_{H}$ are
fitted to reproduce the $B_{s}^{0}\to\phi J/\psi$ observables while $X_{C}=0$.
For $f_{B_{s}^{0}}=260$ MeV one obtains $\rho_{H}=1.85\pm 0.07$ and
$\phi_{H}=255.9\pm 24.6$. These values are not very different from those given
in the second line of Table 2. This case corresponds to Figure 2. At last,
denoted by $LOVPH+C$, we give the values for $a_{n}^{q,h}(m_{b})$ for the case
that the $\zeta^{(h)}$ amplitudes are included, which corresponds to the
$\rho_{C,H}$ and $\phi_{C,H}$ values in Table 2 and to Figure 3. We remind
that the dependence on $f_{B_{s}^{0}}$ enters the short-distance coefficients
via the hard-scattering contribution $H_{n}^{h}(M_{1}J/\psi)$ in Eq. (22) and
that the phenomenological amplitudes, $X_{H}$ and $X_{C}$, are in competition
with each other. Therefore, the hard scattering contributions to
$a_{n}^{q,h}(m_{b})$ in $LOVPH$ are slightly different than those to
$LOVPH+C$.
The largest values observed in the leading amplitude, $a_{2}^{u,c}(m_{b})$,
are for $h=0$. We also remark there is no variation as a function of
$f_{B_{s}}$ in Table 4 since $H_{n}^{h=-1}(M_{1}J/\psi)=0$. Moreover, penguin
contractions only contribute to $a_{7}^{q,h=+1}(m_{b})$ and
$a_{9}^{q,h=+1}(m_{b})$ in the $B_{s}^{0}\to\phi J/\psi$ amplitudes, while
there are no penguin terms in $B_{s}^{0}\to f_{0}J/\psi$. Altogether, the
penguin contributions are very small. We note that the contribution of the
phenomenological amplitudes, $\zeta^{(h)}$ (Tables 7 and 8), is small, about
$6-7\%$ of the $h=0,+1$ amplitudes in $B_{s}\to\phi J/\psi$ and $2\%$ of the
$B_{s}\to f_{0}J/\psi$ amplitude, yet dominant in the $h=-1$ amplitude devoid
of penguin and hard scattering corrections. Thus, any contribution from new
physics, and to less an extent annihilation topologies, should occur in the
$h=-1$ helicity amplitude.
When including all the contributions ($LOVPH+C$), we qualitatively verify the
hierarchy relation, $|\mathcal{A}^{(h=0)}_{B_{s}^{0}\to\phi
J/\psi}|>|\mathcal{A}^{(h=+1)}_{B_{s}^{0}\to\phi
J/\psi}|>|\mathcal{A}^{(h=-1)}_{B_{s}^{0}\to\phi J/\psi}|$, in $B_{s}^{0}\to
J/\psi\phi$ and $|\mathcal{A}^{(h=0)}_{B_{s}^{0}\to\phi
J/\psi}|>|\mathcal{A}^{(h=-1)}_{B_{s}^{0}\to\phi
J/\psi}|>|\mathcal{A}^{(h=+1)}_{B_{s}^{0}\to\phi J/\psi}|$ in the $CP$
conjugate decay $\bar{B}_{s}^{0}\to J/\psi\phi$. These hierarchy relations are
also reproduced for the amplitudes when they include, besides tree
contributions, vertex, penguin and hard-scattering corrections.
Having determined numerical values for $X_{H}$ and $X_{C}$, we can calculate
the $B_{s}^{0}\to f_{0}J/\psi$ amplitude and obtain the associated branching
fraction and $CP$ asymmetry. We do so for the central values of
$f_{B_{s}^{0}}=260$ MeV and $\bar{f}_{f_{0}^{s}}=380$ MeV discussed in Section
V. For a transition form factor $F_{1}^{B_{s}^{0}\to
f_{0}^{s}}(q^{2}=m^{2}_{J/\psi})=0.4$ and for the different amplitudes $LOVP$,
$LOVPH$ and $LOVPH+C$ defined above, those observables are displayed in Table
9 together with a comparison of the $B_{s}^{0}\to J/\psi\phi$ results with the
corresponding available experimental analysis values. Furthermore, we obtain
for a transition form factor $F_{1}^{B_{s}^{0}\to
f_{0}^{s}}(q^{2}=m^{2}_{J/\psi})=0.2$:
$\displaystyle\mathcal{B}(B_{s}\to f_{0}J/\psi)$ $\displaystyle=$
$\displaystyle 3.80\times 10^{-4},$ $\displaystyle A_{\mathrm{CP}}(B_{s}\to
f_{0}J/\psi)$ $\displaystyle=$ $\displaystyle-0.0005\ ,$
$\displaystyle\mathcal{B}(B_{s}\to\phi J/\psi)$ $\displaystyle=$
$\displaystyle 9.30\times 10^{-4},$ $\displaystyle\mathcal{R}_{f_{0}/\phi}$
$\displaystyle=$ $\displaystyle 0.42\ ;$
for $F_{1}^{B_{s}^{0}\to f_{0}^{s}}(q^{2}=m^{2}_{J/\psi})=0.3$,
$\displaystyle\mathcal{B}(B_{s}\to f_{0}J/\psi)$ $\displaystyle=$
$\displaystyle 4.37\times 10^{-4},$ $\displaystyle A_{\mathrm{CP}}(B_{s}\to
f_{0}J/\psi)$ $\displaystyle=$ $\displaystyle-0.0008\ ,$
$\displaystyle\mathcal{B}(B_{s}\to\phi J/\psi)$ $\displaystyle=$
$\displaystyle 9.30\times 10^{-4},$ $\displaystyle\mathcal{R}_{f_{0}/\phi}$
$\displaystyle=$ $\displaystyle 0.48\ ;$
and for $F_{1}^{B_{s}^{0}\to f_{0}^{s}}(q^{2}=m^{2}_{J/\psi})=0.5$,
$\displaystyle\mathcal{B}(B_{s}\to f_{0}J/\psi)$ $\displaystyle=$
$\displaystyle 5.7\times 10^{-4},$ $\displaystyle A_{\mathrm{CP}}(B_{s}\to
f_{0}J/\psi)$ $\displaystyle=$ $\displaystyle-0.0013\ ,$
$\displaystyle\mathcal{B}(B_{s}\to\phi J/\psi)$ $\displaystyle=$
$\displaystyle 9.30\times 10^{-4},$ $\displaystyle\mathcal{R}_{f_{0}/\phi}$
$\displaystyle=$ $\displaystyle 0.63\ ,$
and finally, the $CP$ asymmetries in $B_{s}\to J/\psi\phi$ are,
$\displaystyle A_{\mathrm{CP}}^{L}(B_{s}\to\phi J/\psi)$ $\displaystyle=$
$\displaystyle-1.66\times 10^{-3}\ ,$ $\displaystyle
A_{\mathrm{CP}}^{\parallel}(B_{s}\to\phi J/\psi)$ $\displaystyle=$
$\displaystyle 1.99\times 10^{-3}\ ,$ $\displaystyle
A_{\mathrm{CP}}^{\perp}(B_{s}\to\phi J/\psi)$ $\displaystyle=$ $\displaystyle
2.15\times 10^{-3}\ .$
Figure 1: The ratio $\mathcal{R}_{f_{0}/\phi}$ as a function of the transition
form factor $F_{1}^{B_{s}^{0}\to f_{0}^{s}}(m_{J/\psi}^{2})$. Only tree,
vertex and penguin contributions ($LOVP$), independent of the decay constants
$f_{B_{s}^{0}}$ and $\bar{f}_{f_{0}}$, are included in the decay amplitudes.
The dotted line corresponds to the central value of this ratio while the area
between the two solid lines gives its envelope due to the uncertainties on the
decay rates $f_{0}(980)\to\pi^{+}\pi^{-}$ ElBennich:2008xy ; Ecklund:2009fia
and $\phi\to K^{+}K^{-}$ Amsler:2008zzb . The two horizontal dash-dotted lines
delimit the (shaded) area between the experimental predictions found in Refs.
Stone:2008ak and Ecklund:2009fia .
Figure 2: The ratio $\mathcal{R}_{f_{0}/\phi}$ as a function of the transition
form factor $F_{1}^{B_{s}^{0}\to f_{0}^{s}}(m_{J/\psi}^{2})$ where now the
tree, vertex, penguin, and hard-scattering contributions ($LOVPH$) are
included. The area between the two dashed lines gives the envelope of this
ratio when taking into account uncertainties on the decay constants
($f_{B_{s}^{0}}=260\pm 30$ MeV and $\bar{f}_{f_{0}}=380\pm 40$ MeV) while the
solid lines include in addition the uncertainties on the decay rates
$f_{0}(980)\to\pi^{+}\pi^{-}$ ElBennich:2008xy ; Ecklund:2009fia and $\phi\to
K^{+}K^{-}$ Amsler:2008zzb . The single dotted line is our prediction for the
central values of the decay constants, $f_{B_{s}^{0}}=260$ MeV and
$\bar{f}_{f_{0}}=380$ MeV. The horizontal dash-dotted lines correspond to the
experimental predictions of Refs. Stone:2008ak and Ecklund:2009fia .
Our prediction for the time-integrated asymmetry $A_{\mathrm{CP}}(B_{s}\to
f_{0}J/\psi)$ is about one order of magnitude smaller than the Standard Model
value, $-2\beta_{s}=-0.036$. We remark that the above numerical values for
this $CP$ asymmetry have to be interpreted with care — we choose the
parameters of the full QCDF amplitude in Table 2 such that the experimental
$B_{s}^{0}\to J/\psi\phi$ observables are reproduced. In doing so, we may
deliberately include ``new physics" effects with just the Standard Model
amplitude, in particular via the additional amplitudes $\zeta^{(h)}$.
Moreover, we use the same end-point parameterization, $X_{H}$, in both decay
channels since the $B_{s}^{0}\to J/\psi f_{0}$ branching ratio is not
experimentally known. This approach seems reasonable, as the physics buried in
these infrared divergences must be similar in both decays. It could also lead
to an overestimation of the hard-scattering contributions to $B_{s}^{0}\to
J/\psi f_{0}$ as well as of $A_{\mathrm{CP}}(B_{s}\to f_{0}J/\psi)$.
We illustrate the variation of the ratio, $\mathcal{R}_{f_{0}/\phi}$, by
taking into account the uncertainties in the decay constants $f_{B_{s}^{0}}$
and $\bar{f}_{f_{0}^{s}}$ as well as those in the decay rates,
$\mathcal{B}({f_{0}(980)\to\pi^{+}\pi^{-}})=0.50^{+0.07}_{-0.09}$
ElBennich:2008xy ; Ecklund:2009fia and $\mathcal{B}({\phi\to
K^{+}K^{-}})=0.489\pm 0.005$ Amsler:2008zzb . The results are displayed in
Figures 2–3.
Figure 3: Same as in Fig. 2 but including $\zeta^{(h)}$ contributions
($LOVPH+C$ amplitudes).
In Figure 2, $\mathcal{R}_{f_{0}/\phi}$ is plotted as a function of
$F_{1}^{B_{s}^{0}\to f_{0}^{s}}(m^{2}_{J/\psi})$ where only the tree amplitude
along with vertex and penguin corrections are included in both amplitudes,
${\cal A}_{{B}_{s}^{0}\to\phi J/\psi}^{h}$ and ${\cal A}_{{B}_{s}^{0}\to
f_{0}J/\psi}$. The ratio is plotted with the corresponding envelope of
$\mathcal{R}_{f_{0}/\phi}$ due to the uncertainty on the decay rates. In
Figure 2, we augment this amplitude by hard-scattering contributions, that is
the full QCDF amplitude given in Eq. (22). Finally, in Figure 3,
$\mathcal{R}_{f_{0}/\phi}$ is plotted as a function of $F_{1}^{B_{s}^{0}\to
f_{0}^{s}}(m^{2}_{J/\psi})$ including hard-scattering corrections and possible
``other" contributions, $\zeta^{(h)}\\!$. Although the aforementioned
uncertainties are depicted in all figures, we stress that those on the decay
constants $f_{B_{s}^{0}}$ and $\bar{f}_{f_{0}^{s}}$, where they apply, have
more impact on the $\mathcal{R}_{f_{0}/\phi}$ band than the $f_{0}(980)$ and
$\phi$ decay rate incertitudes. The spreading of the curves representing
$\mathcal{R}_{f_{0}/\phi}$ as a function of $F_{1}^{B_{s}^{0}\to
f_{0}^{s}}(m^{2}_{J/\psi})$ is larger with respect to the variation in
$f_{B_{s}^{0}}$ than in $\bar{f}_{f_{0}^{s}}$. This points to the necessity of
having an improved experimental determination of $f_{B_{s}^{0}}$. The upper
limit of the envelope is reached only for the largest values of
$f_{B_{s}^{0}}$ and $\bar{f}_{f_{0}^{s}}$ considered here.
Figure 3 shows that our central-value predictions of
$\mathcal{R}_{f_{0}/\phi}$, in absence of any phenomenological contributions,
are within the estimate by Stone and Zhang Stone:2008ak for most values of the
form factor $F_{1}^{B_{s}^{0}\to f_{0}^{s}}(m^{2}_{J/\psi})$. However, when
the additional amplitudes, $\zeta$, are accounted for in the decay amplitudes
of Eqs. (2) and (4), the ratio $\mathcal{R}_{f_{0}/\phi}$ exhibits three
striking features:
* •
Additional amplitudes, $\zeta$, can play a major role due to their large
contributions to both the numerator and denominator of the ratio
$\mathcal{R}_{f_{0}/\phi}$, as seen from the comparison of Figures 2 and 3.
* •
The predicted $\mathcal{R}_{f_{0}/\phi}$ band overlaps well with the estimates
of Refs. Stone:2008ak and Ecklund:2009fia for $F_{1}^{B_{s}^{0}\to
f_{0}^{s}}(m^{2}_{J/\psi})<0.4$; beyond, our predictions are much larger,
which may indicate a larger pollution due to $f_{0}(980)\to K^{+}K^{-}$ if
contributions from other than the Standard Model were present.
* •
The uncertainties on the $f_{0}(980)$ and $\phi$ decay rates increase the
width of the band considerably, though the main uncertainty stems from the
decay constants $f_{B_{s}^{0}}$ and $\bar{f}_{f_{0}}$.
Let us remind that the decay constant $\bar{f}_{f^{s}_{0}}$ only enters the
hard-scattering and additional phenomenological contributions ($C$) to the
decay amplitude $B_{s}^{0}\to f_{0}(980)J/\psi$. If these are turned off, as
in Figure 2, the ratio $R_{f_{0}/\phi}$ is still significantly above 10% for
realistic transition-form factor values. That said, for practical purposes we
decide to only consider the more recently obtained decay constants in Ref.
Cheng:2005nb and choose three values within the given errors,
$\bar{f}_{f_{0}^{s}}=340,380,420$ MeV. The value $180$ MeV DeFazio:2001uc
yields too low branching fractions in other decays, for example $B\to
f_{0}(980)\pi,f_{0}(980)\rho,f_{0}(980)K^{(*)}$. Nevertheless, since we fix
the hard-scattering parameters, $\rho_{H}$ and $\phi_{H}$, only via the decay
$B_{s}^{0}\to J/\psi\phi$ and $\bar{f}_{f_{0}^{s}}$ enters the numerator in
$\mathcal{R}_{f_{0}/\phi}$ linearly, the modification is straightforward:
$\bar{f}_{f_{0}^{s}}=180$ MeV is about half the value
$\bar{f}_{f_{0}^{s}}=380$ MeV, therefore the central value of
$\mathcal{R}_{f_{0}\phi}$ in Figure 2 decreases from $0.42$ to $0.19$ (for
$F_{0}^{B_{s}^{0}\to f_{0}^{s}}=0.4$). This is still within the limits
predicted by the experimental estimates,
$0.2\lesssim\mathcal{R}_{f_{0}/\phi}\lesssim 0.5$, and implies an $S$-wave
pollution.
We infer from our numerical results that $S$-wave kaons or pions under the
$\phi$ peak in $B_{s}^{0}\to J/\psi\phi$ are very likely to originate from the
similar decay $B_{s}^{0}\to J/\psi f_{0}$. Therefore, the extraction of the
mixing phase, $-2\beta_{s}$, from $B_{s}^{0}\to J/\psi\phi$ may well be biased
by this $S$-wave effect which should be taken into account in experimental
analyses. In our interpretation of the full QCDF amplitude, we not only
confirm the influence of $S$-wave contamination as advocated in Refs.
Stone:2008ak and Ecklund:2009fia but also find that its effect could be
sizable.
## VII Conclusive outlook
The ``phase" of $B_{s}^{0}-\bar{B}_{s}^{0}$ mixing, $-2\beta_{s}$, is thought
to be best measured in the golden decay, $B_{s}^{0}\to J/\psi\phi$, and
provides an opportune place to investigate physics beyond the Standard Model.
Several models have been proposed to explain the apparent discrepancy of the
Standard Model prediction for $-2\beta_{s}$ with recent experiments, in
particular exploring the impact of heavy, as of yet undiscovered particles on
$CP$ violation in weak $B$-meson decays. A general analysis of possible new
physics effects in the case of $B_{s}^{0}-\bar{B}_{s}^{0}$ mixing was recently
given by Chiang et al. Chiang:2009ev . In there, the authors investigate
several beyond Standard Model variations of the $B_{s}\to J/\psi\phi$ decay,
such as $Z^{(^{\prime})}$-mediated Flavor Changing Neutral Currents (FCNC),
two Higgs doublets and SUSY, and find that new physics contributions may only
modestly contribute to the mixing phase. However, it is also concluded,
somewhat prematurely, that the CDF and D$\emptyset$ results are clear signs of
new physics.
In the present paper, we have taken a different path and studied the
contamination of final state $S$-waves kaons in the $B_{s}^{0}\to J/\psi\phi$
channel by those originating from the $f_{0}(980)$ in the very similar
$B_{s}^{0}\to J/\psi f_{0}(980)$ decay. We find that this effect is strong
enough already for amplitudes including leading order, vertex and penguin
corrections to create a real bias in the determination of $-2\beta_{s}$.
Of course, we are aware that the phenomenological endpoint parametrization of
$\alpha_{\mathrm{s}}$ corrections in the amplitudes $H_{n}(M_{1}J/\psi)$ and
$H_{n}^{h}(M_{1}J/\psi)$ can cloud possible new physics contributions
alongside the $\zeta^{(h)}$ contributions. In this case, we suppose that any
new effects should be of comparable magnitude in $B_{s}^{0}\to J/\psi\phi$ and
$B_{s}^{0}\to J/\psi f_{0}(980)$. Therefore, the $S$-wave contamination would
be on the upper side of the estimate we propound and future analyses of the
mixing angle in $B_{s}$ decays should be concerned with this effect.
###### Acknowledgements.
The authors are obliged to Sheldon Stone for pointing out the likely $S$-wave
effects on the mixing angle which stimulated this work and critical remarks on
the manuscript. We thank Martin Beneke for useful comments on mass corrections
in transition form factors between two mesons. We are very grateful to Ying Li
for drawing our attention to the particular quark topology of the annihilation
diagrams. B. E. wishes to acknowledge the hospitality of the Laboratoire de
Physique Nucléaire et Hautes Energies in Paris and of João Pacheco B. C. de
Melo and Victo dos Santos Filho at Universidade Cruzeiro do Sul, São Paulo,
during the final stage of the writing. This work was partially funded by the
Department of Energy, Office of Nuclear Physics, under contract number DE-
AC02-06CH11357.
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|
arxiv-papers
| 2010-03-31T06:47:57 |
2024-09-04T02:49:09.388053
|
{
"license": "Public Domain",
"authors": "O. Leitner, J.-P. Dedonder, B. Loiseau and B. El-Bennich",
"submitter": "Bruno El-Bennich",
"url": "https://arxiv.org/abs/1003.5980"
}
|
1004.0219
|
# Characterization of The Generic Unfolding of a Weak Focus
by
W. Arriagada-Silva
W. Arriagada: Département de mathématiques et de statistique, Faculté des arts
et des sciences -Secteur des sciences, Université de Montréal
succ. Centre-ville
Montréal, Qc
H3C 3J7. arriaga@DMS.Umontreal.CA
###### Abstract.
In this paper we give a geometric description of the foliation of a generic
real analytic family unfolding a real analytic vector field with a weak focus
at the origin, and show that two such families are orbitally analytically
equivalent if and only if the families of diffeomorphisms unfolding the
complexified Poincaré map of the singularities are conjugate. Moreover, by
shifting the leaves of the formal normal form in the blow-up (quasiconformal
surgery) by means of a fibered transformation along a convenient complex
cross-section, one constructs an abstract manifold of complex dimension 2
equipped with an elliptic holomorphic foliation whose monodromy map coincides
with a given family of admissible diffeomorphisms.
## 1\. Introduction.
A one-parameter family of real analytic planar systems unfolding a weak focus
is an elliptic real analytic one-parameter
$\displaystyle\operatorname{\varepsilon}\in{\bf R}$ dependent family linearly
equivalent to a family of planar differential equations
$\left.\begin{array}[]{lll}\dot{x}&=&\alpha(\operatorname{\varepsilon})x-\beta(\operatorname{\varepsilon})y+\sum_{j+k\geq
2}b_{jk}(\operatorname{\varepsilon})x^{j}y^{k}\\\
\dot{y}&=&\beta(\operatorname{\varepsilon})x+\alpha(\operatorname{\varepsilon})y+\sum_{j+k\geq
2}c_{jk}(\operatorname{\varepsilon})x^{j}y^{k},\end{array}\right.$ (1.1)
for real time, and with $\displaystyle\alpha(0)=0$ and
$\displaystyle\beta(0)\neq 0.$ After rescaling the time $\displaystyle
t\mapsto\beta(\operatorname{\varepsilon})t$ we can suppose
$\displaystyle\beta(\operatorname{\varepsilon})\equiv 1.$ The family (1.1) is
called “generic” if $\displaystyle\alpha^{\prime}(0)\neq 0.$ The genericity
allows to take $\displaystyle\alpha$ as the new parameter, so that the
eigenvalues become $\displaystyle\operatorname{\varepsilon}+i$ and
$\displaystyle\operatorname{\varepsilon}-i,$ respectively.
When the order is one, a weak focus of a real analytic vector field
corresponds to the coalescence of a focus with a limit cycle, and the generic
family (1.1) is then a family with a generic Hopf bifurcation, whose foliation
is described by the unfolding of the Poincaré map or monodromy
$\displaystyle{\mathcal{P}}_{\operatorname{\varepsilon}}:({\bf
R}^{+},0)\to({\bf R}^{+},0)$ of the system. It is well known that the germ of
the Poincaré return map or monodromy is well defined and analytic, and can be
extended to an analytic diffeomorphism
$\begin{array}[]{lll}{\mathcal{P}}_{\operatorname{\varepsilon}}:({\bf
R},0)\to({\bf R},0).\end{array}$ (1.2)
A question that arises naturally is whether the germ of the monodromy map
defines the analytic equivalence class of the real foliation. The natural way
to answer this question is via complexification (cf. [2]). The right hand side
of the complexified system is now defined by an analytic family of vector
fields
$\begin{array}[]{lll}v_{\operatorname{\varepsilon}}(x,y)=P(x,y)\frac{\partial}{\partial
x}+Q_{\operatorname{\varepsilon}}(x,y)\frac{\partial}{\partial y}\end{array}$
(1.3)
that satisfies
$\begin{array}[]{lll}P_{\operatorname{\varepsilon}}(x,y)=\overline{Q_{\overline{\operatorname{\varepsilon}}}(\overline{y},\overline{x})},\end{array}$
(1.4)
where $\displaystyle x\mapsto\overline{x}$ is the complex conjugation. The
time is complexified as well and the domain of the parameter is now a standard
open complex disk noted $\displaystyle V\in{\bf C}.$ After complexification,
the real plane can be written in a rather simple way: it corresponds to the
surface $\displaystyle\\{x=\overline{y}\\}.$ The Poincaré map of the
complexified system (parametrized with $\displaystyle x$-coordinate) is
defined as the second iterate of the holonomy
$\displaystyle{\mathcal{Q}}_{\operatorname{\varepsilon}}$ along the loop
$\displaystyle{\bf R}P^{1}$ (the equator of the exceptional divisor) of the
foliation after standard blow-up (cf. [5]), where the standard affine
coordinates on the projective line $\displaystyle{\bf C}P^{1}$ are given by
formulas with real coefficients, hence defining correctly the real projective
equator $\displaystyle{\bf R}P^{1}\subset{\bf C}P^{1},$ see Figure 1. Blowing
down the foliation, the Poincaré map is defined on the $\displaystyle
1$-dimensional complex cross-section $\displaystyle\\{x=y\\},$ and the usual
real germ (1.2) of the planar system is defined on
$\displaystyle\\{x=y\\}\cap\\{x=\overline{y}\\}.$
_Notation._ The cross section $\displaystyle\\{x=y\\}$ is noted
$\displaystyle\Sigma$ and is parametrized with the complex coordinate
$\displaystyle x.$
The complex description of the monodromy immediately allows to prove its
analyticity, even at the origin. The monodromy is then a real holomorphic germ
of resonant diffeomorphism with a fixed point of multiplicity 3 at the origin,
which corresponds in the limit $\displaystyle\operatorname{\varepsilon}=0$ to
the coalescence of a fixed point with a 2-periodic orbit: the fixed point and
periodic orbit bifurcate in a generic unfolding.
Figure 1. The complexification of the real line and its blow-up.
_The equivalence problem._ It is known that the problem of orbital equivalence
for germs of analytic vector fields with a resonant saddle point is reduced to
the conjugacy problem for germs of diffeomorphisms (the holonomy map) with a
fixed point at the origin and multiplier on the unit circle (cf. [6]). In the
non-resonant case, the statement holds as well, as was shown by R. Pérez-Marco
and J.-C. Yoccoz (cf. [9]). Furthermore, this result has been extended to
generic analytic families unfolding a resonant saddle point (cf. [13]).
###### Definition 1.1.
An analytic orbital equivalence (resp. conjugacy) between two analytic germs
of families unfolding germs of analytic vector fields (resp. diffeomorphisms)
is said to be “real”, when it leaves invariant the real plane (resp. the real
line) for real values of the parameter.
In this paper, we show that the equivalence problem for (1.3) can be reduced
to the conjugacy problem for the associated family unfolding the complexified
Poincaré map, respecting the underlying real foliation. More precisely,
###### Theorem 1.2.
Two germs of generic families of real analytic vector fields (1.3) are
analytically orbitally equivalent by a real change of coordinates, if and only
if the families unfolding their Poincaré maps are analytically conjugate by a
real conjugacy.
_The realization problem._ A second related problem consists in recovering the
germ of the analytic foliation when the Poincaré map has been prescribed. This
is the problem of “realization”. We give an answer to this problem by means of
the desingularization technique and quasiconformal surgery, as suggested by Y.
Ilyashenko (cf. [4]): for every $\displaystyle\operatorname{\varepsilon}\in
V,$ one constructs, with the help of an adequate partition of the unity
depending only on the argument of the coordinate induced in the separatrix
(the exceptional divisor) by the desingularization process, a fibered
$\displaystyle C^{\infty}$ transformation or “sealing map” defined on a semi-
disk. By shifting the leaves of the normal form with the help of the sealing
map, one obtains a $\displaystyle C^{\infty}$ foliation over the product
$\displaystyle{\bf C}^{*}\times{\bf D}_{r},$ and an integrable almost complex
structure, making the foliation actually holomorphic. The almost complex
structure extends smoothly along the vertical axis, because the sealing is, by
definition, infinitely tangent to the identity. It remains integrable after
the extension. The Newlander-Nirenberg Theorem yields a $\displaystyle
C^{\infty}$ real system of coordinates (depending analytically on
$\displaystyle\operatorname{\varepsilon})$ that straightens the almost complex
structure, and therefore, the $\displaystyle C^{\infty}$ foliation in a
holomorphic foliation that extends by Riemann along the vertical axis. The
blow down of such a foliation is the required generic elliptic family.
In this second part we deal with formal normal forms. Normal form theory
provides an algorithmic way to decide whether two germs of planar vector
fields are equivalent under a $\displaystyle C^{N}$-change of coordinates (cf.
[11]), in which case, the normal forms are polynomial. However, in the
analytic case, the formal change of coordinates to normal form generically
diverges (cf. [3]). An explanation of this is found by considering unfoldings
of the vector fields and explaining the divergence in the limit process. This
is a particular manifestation of the so-called Stokes Phenomenon (cf. [4]).
The spirit of the general answer is the following (cf. [11]). The dynamics of
the original system is extraordinarily rich to be encoded in the simple
dynamics of the normal form which depends of at most one parameter. Hence the
divergence of the normalizing series.
## 2\. Proof of Theorem 1.2.
The proof uses basically the classical fact that the holonomy characterizes
the differential equation (cf. [6] and [10]), plus an additional ingredient:
both the equivalence between vector fields and the conjugacy between Poincaré
maps, must respect the real foliation.
By definition, if two families (1.3) are orbitally equivalent by an analytic
change of coordinates $\displaystyle\Psi_{\operatorname{\varepsilon}}$
(depending analytically on the parameter), it is always possible to
reparametrize the families and suppose that they have the same parameter.
Thus, one direction is obvious: if two families of vector fields are
equivalent by real change of coordinates, then the equivalence induces a real
analytic return map on the image
$\displaystyle\Psi_{\operatorname{\varepsilon}}(\Sigma),$ for each value of
$\displaystyle\operatorname{\varepsilon}$ over a small neighborhood of the
origin. Because the equivalence is real, the image of the real line under the
equivalence is a real analytic curve
$\displaystyle{\mathcal{C}}\subset\Psi_{\operatorname{\varepsilon}}(\Sigma)$
different, in general, to $\displaystyle{\bf R},$ see Figure 2. Standard
transversality arguments and the Implicit Function Theorem show that there
exists an analytic local transition map $\displaystyle\pi$ between
$\displaystyle\Sigma$ and
$\displaystyle\Psi_{\operatorname{\varepsilon}}(\Sigma)$ (cf. [1]). By
unicity, any real local trajectory passing through a real point in
$\displaystyle\Sigma$ intersects the image
$\displaystyle\Psi_{\operatorname{\varepsilon}}(\Sigma)$ in a real point.
Thus, the transition is real and it sends the curve $\displaystyle{\bf R}$
into $\displaystyle{\mathcal{C}},$ and the composition
$\displaystyle\pi^{-1}\circ\Psi_{\operatorname{\varepsilon}}$ provides a real
conjugacy between Poincaré maps $\displaystyle\Sigma\to\Sigma.$
Figure 2. The real line and its image by the equivalence
$\displaystyle\Psi_{\operatorname{\varepsilon}}.$
Let us show the converse. The conjugacy between the Poincaré maps provides a
reparametrization, so we can suppose that the parameter is the same for the
two families of diffeomorphisms and is henceforth noted
$\displaystyle\operatorname{\varepsilon}.$ We will suppose that the real
conjugacy $\displaystyle{\bf h}_{\operatorname{\varepsilon}}(x)={\bf
h}(\operatorname{\varepsilon},x)$ depends on the $\displaystyle x$-variable
and is defined on $\displaystyle{\bf D}_{\rho}\subset\Sigma,$ for every
$\displaystyle\operatorname{\varepsilon}\in V,$ where $\displaystyle{\bf
D}_{\rho}\subset\Sigma$ is the standard open disk of the complex plane, of
small radius $\displaystyle{\rho}>0.$
A theorem on the existence of invariant analytic manifolds (cf. [5],[6])
ensures that it suffices to show the theorem for Pfaffian $\displaystyle
1$-forms
$\displaystyle\omega_{\operatorname{\varepsilon}},\widehat{\omega}_{\operatorname{\varepsilon}}=(\operatorname{\varepsilon}+i)xdw-(\operatorname{\varepsilon}-i)y(1+xy(...))dx$
before desingularization. So if the blow-up space is equipped with coordinates
$\displaystyle(X,y)$ and $\displaystyle(x,Y),$ where the standard monoidal map
blows down as
$\begin{array}[]{lll}c_{1}:(X,y)\mapsto(Xy,y),\\\
c_{2}:(x,Y)\mapsto(x,xY)\end{array}$ (2.1)
respectively in each direction, the pullback of
$\displaystyle\omega_{\operatorname{\varepsilon}}$ is defined by
$\displaystyle\omega_{1}=Xdy-\lambda(\operatorname{\varepsilon})y(1+A_{\operatorname{\varepsilon}}(X,y))dX$
in $\displaystyle(X,y)$ variables, and by
$\displaystyle\omega_{2}=Ydx-\lambda^{\prime}(\operatorname{\varepsilon})x(1+A^{\prime}_{\operatorname{\varepsilon}}(x,Y))dY$
in $\displaystyle(x,Y)$ coordinates, where $\displaystyle
A_{\operatorname{\varepsilon}}(X,y)=O(Xy)$ and $\displaystyle
A^{\prime}_{\operatorname{\varepsilon}}(x,Y)=O(xY)$ depend analytically on the
parameter and are holomorphic on a neighborhood $\displaystyle{\bf
C}^{*}\times{\bf D}_{s}$ of the exceptional divisor, for each fixed value of
$\displaystyle\operatorname{\varepsilon}.$ The numbers
$\displaystyle\lambda(\operatorname{\varepsilon})=(\operatorname{\varepsilon}-i)/2i$
and
$\displaystyle\lambda^{\prime}(\operatorname{\varepsilon})=-(\operatorname{\varepsilon}+i)/2i$
are the ratios of eigenvalues of the singular points
$\displaystyle(X,y)=(0,0)$ and $\displaystyle(x,Y)=(0,0),$ respectively. In
addition, the coordinates can always be scaled before blow-up, to ensure:
$\begin{array}[]{lll}|A_{\operatorname{\varepsilon}}(X,y)|,|A^{\prime}_{\operatorname{\varepsilon}}(x,Y)|<1/2\end{array}$
(2.2)
in $\displaystyle{\bf C}^{*}\times{\bf D}_{s}.$ Notice that in complex
coordinates, the section $\displaystyle\Sigma$ is parametrized as
$\displaystyle\\{X=1\\}$ in the $\displaystyle(X,y)$ chart, and as
$\displaystyle\\{Y=1\\}$ in the $\displaystyle(x,Y)$ chart. Bounded
equivalences
$\displaystyle\widehat{\Psi}_{\operatorname{\varepsilon}}^{c_{1}},\widehat{\Psi}_{\operatorname{\varepsilon}}^{c_{2}}$
are constructed in $\displaystyle(X,y)$ and $\displaystyle(x,Y)$ variables, in
such a way that they are analytic continuations of each other over a
neighborhood of the exceptional divisor.
### 2.1. The equivalence in the $\displaystyle(X,y)$ chart.
Take a point $\displaystyle y^{*}\in{\bf D}_{\rho}.$ A former equivalence
$\displaystyle\widehat{\Psi}_{\operatorname{\varepsilon}}^{c_{1}}$ is defined
on $\displaystyle\Sigma\times{\bf D}_{\rho}$ by
$\displaystyle\widehat{\Psi}_{\operatorname{\varepsilon}}^{c_{1}}:(1,y^{*})\mapsto(1,{\bf
h}_{\operatorname{\varepsilon}}(y^{*})).$
This change of coordinates is extended along a subset of $\displaystyle{\bf
S}^{1}\times{\bf C}$ in the following way. Notice that the restriction of the
form $\displaystyle\omega_{1}$ to the cylinder $\displaystyle{\bf
R}P^{1}\times{\bf R}^{2}$ (noted $\displaystyle\\{|X|=1\\})$ is non-singular
and holomorphic, thus it defines a local holomorphic foliation
$\displaystyle{\mathcal{F}}_{\omega_{1}}$ there. Consider (cylindrical)
solutions to $\displaystyle\omega_{1}=0$ (the first coordinate is to be
parametrized by $\displaystyle X=e^{i\theta},$
$\displaystyle\theta\in[0,2\pi]).$
###### Lemma 2.1.
Any (cylindrical) solution $\displaystyle{\mathbf{u_{1}}}$ to
$\begin{array}[]{lll}{\mathbf{u_{1}}}^{\prime}=\lambda(\operatorname{\varepsilon}){\mathbf{u_{1}}}(1+A(e^{i\theta},{\mathbf{u_{1}}})),\quad\theta\in[0,2\pi]\end{array}$
(2.3)
satisfies
$\displaystyle|{\mathbf{u_{1}}}(0)|e^{-\theta\left\\{|\operatorname{\varepsilon}|+\frac{1}{4}\right\\}}<|{\mathbf{u_{1}}}(\theta)|<|{\mathbf{u_{1}}}(0)|e^{\theta\left\\{|\operatorname{\varepsilon}|+\frac{1}{4}\right\\}},$
for any $\displaystyle\theta\in(0,2\pi].$
###### Proof.
The parameter is written as
$\displaystyle\operatorname{\varepsilon}=\operatorname{\varepsilon}_{1}+i\operatorname{\varepsilon}_{2},$
with
$\displaystyle\operatorname{\varepsilon}_{1},\operatorname{\varepsilon}_{2}\in{\bf
R}.$ As we consider solutions in $\displaystyle|X|=1,$ the time is
parametrized by $\displaystyle t=i\theta,$ and then (2.3) implies
$\displaystyle
d\ln{\mathbf{u_{1}}}=\frac{1}{2}(\operatorname{\varepsilon}-i)(1+A_{\operatorname{\varepsilon}}(e^{i\theta},{\mathbf{u_{1}}}))d\theta.$
Thus, after taking real parts and using the hypothesis (2.2) we get, for
$\displaystyle\theta\neq 0:$
$\begin{array}[]{lll}\left|\ln\left|\frac{{\mathbf{u_{1}}}}{{\mathbf{u_{1}}}(0)}\right|\right|&\leq&\frac{1}{2}\int_{0}^{\theta}\\{|\operatorname{\varepsilon}_{1}|(1+|Re(A_{\operatorname{\varepsilon}})|)+|Im(A_{\operatorname{\varepsilon}})|(1+|\operatorname{\varepsilon}_{2}|)\\}d\theta\\\
&<&\frac{1}{2}\int_{0}^{\theta}\\{2|\operatorname{\varepsilon}|+1/2\\}d\theta=\theta\left\\{|\operatorname{\varepsilon}|+1/4\right\\},\\\
\end{array}$
and the conclusion follows. ∎
Put $\displaystyle r=\rho e^{-\pi}.$ We denote by $\displaystyle{\bf S}_{r}$
the set of (cylindrical) solutions $\displaystyle{\mathbf{u_{1}}}$ to (2.3)
for which there exists $\displaystyle\theta_{0}\in[0,2\pi)$ such that
$\displaystyle{\mathbf{u_{1}}}(\theta_{0})\in{\bf D}_{r}.$
###### Corollary 2.2.
If $\displaystyle{\mathbf{u_{1}}}\in{\bf S}_{r},$ then
$\displaystyle{\mathbf{u_{1}}}(0)\in{\bf D}_{\rho},$ provided
$\displaystyle|\operatorname{\varepsilon}|<1/4.$
This is how the equivalence is extended. Choose a point
$\displaystyle(e^{i\theta_{0}},y_{0})\in{\bf S}^{1}\times{\bf D}_{r}.$ By
definition, the path $\displaystyle\gamma:(e^{i\theta},0)$ is lifted in the
leaf of $\displaystyle{\mathcal{F}}_{\omega_{1}}$ containing $\displaystyle
y_{0}\in{\bf D}_{r}$ as $\displaystyle(e^{i\theta},{\mathbf{u_{1}}}(\theta)),$
for a certain $\displaystyle{\mathbf{u_{1}}}\in{\bf S}_{r}$ and
$\displaystyle{\mathbf{u_{1}}}(\theta_{0})=y_{0}.$ By Corollary 2.2, the point
$\displaystyle\widetilde{y}:={\mathbf{u_{1}}}(0)$ belongs to
$\displaystyle{\bf D}_{\rho}.$ If $\displaystyle\gamma$ is lifted in the leaf
of $\displaystyle{\mathcal{F}}_{\widehat{\omega}_{1}}$ passing through
$\displaystyle{\bf h}_{\operatorname{\varepsilon}}(\widetilde{y})$ as
$\displaystyle(e^{i\theta},{\mathbf{u_{2}}}(e^{i\theta},\widetilde{y})),$ with
$\displaystyle{\mathbf{u_{2}}}(1,\widetilde{y})={\bf
h}_{\operatorname{\varepsilon}}(\widetilde{y}),$ then we define the analytic
change of variables by:
$\begin{array}[]{lll}\widehat{\Psi}_{\operatorname{\varepsilon}}^{c_{1}}:{\bf
S}^{1}\times{\bf D}_{r}\to{\bf S}^{1}\times{\bf C},\\\
\widehat{\Psi}_{\operatorname{\varepsilon}}^{c_{1}}:(e^{i\theta_{0}},{\mathbf{u_{1}}}(\theta_{0}))\mapsto(e^{i\theta_{0}},{\mathbf{u_{2}}}(e^{i\theta_{0}},\widetilde{y})).\end{array}$
(2.4)
The change (2.4) respects the transversal fibration given by $\displaystyle
X=const.$ and is clearly the restriction of a (unique) holomorphic
diffeomorphism conjugating $\displaystyle{\mathcal{F}}_{\omega_{1}}$ and
$\displaystyle{\mathcal{F}}_{\widehat{\omega}_{1}}$ in a neighborhood of
$\displaystyle{\bf S}^{1}\times{\bf D}_{r}.$ Moreover, it extends analytically
to $\displaystyle{\bf D}_{1}\times{\bf D}_{r}$ (where $\displaystyle{\bf
D}_{1}$ is the standard unit (closed) disk of the $\displaystyle
X$-separatrix) by means of the lifting of radial paths
$\begin{array}[]{lll}\gamma_{X_{1}}:[0,-\log|X_{1}|]\to{\bf C},\quad
s\mapsto\gamma_{X_{1}}(s)=(X_{1}e^{s},0)\end{array}$
for $\displaystyle 0<|X_{1}|<1.$ In fact, suppose that this curve lifts in the
leaves of $\displaystyle{\mathcal{F}}_{\omega_{1}}$ as
$\displaystyle\gamma_{X_{1},y_{1}}:s\mapsto(X_{1}e^{s},{\mathbf{r_{1}}}(s,y_{1})),\quad{\mathbf{r_{1}}}(0,y_{1})=y_{1},$
for a given $\displaystyle y_{1}$ small. Then the solution
$\displaystyle{\mathbf{r_{1}}}(\cdot,y_{1})$ of $\displaystyle\omega_{1}=0,$
with parameter $\displaystyle 0<|X_{1}|<1,$ and initial condition
$\displaystyle{\mathbf{r_{1}}}(0,y_{1})=y_{1}$ is defined on
$\displaystyle[0,-\log|X_{1}|].$ Actually, the hypothesis (2.2) shows that
$\begin{array}[]{lll}|{\mathbf{r_{1}}}|\leq|y_{1}|e^{s\left\\{|\operatorname{\varepsilon}|-\frac{1}{4}\right\\}}<|y_{1}|,\end{array}$
(2.5)
whenever $\displaystyle|\operatorname{\varepsilon}|<1/4.$ We will suppose that
the inverse path of $\displaystyle\gamma_{X_{1}}$ lifts in the leaf of
$\displaystyle{\mathcal{F}}_{\widehat{\omega}_{1}}$ through the point
$\displaystyle(\frac{X_{1}}{|X_{1}|},y^{0}),$ where $\displaystyle y^{0}$ is
small, as
$\begin{array}[]{lll}\gamma_{X_{1},y^{0}}^{-1}:s\mapsto(X_{1}e^{-(s+\log|X_{1}|)},{\mathbf{\tilde{r}_{1}}}(s,y^{0})),\quad
s\in[0,-\log|X_{1}|].\end{array}$
Consider the only cylindrical solution
$\displaystyle\mathbf{u}_{\mathbf{1},X_{1},y_{1}}$ to (2.3) satisfying
$\displaystyle\mathbf{u}_{\mathbf{1},X_{1},y_{1}}(\arg
X_{1})={\mathbf{r_{1}}}(-\log|X_{1}|,y_{1})$ and define the coordinate
$\displaystyle\widetilde{y}(X_{1},y_{1}):=\mathbf{u}_{\mathbf{1},X_{1},y_{1}}(0)\in\Sigma.$
Then, (2.5) proves that
$\displaystyle\mathbf{u}_{\mathbf{1},X_{1},y_{1}}\in{\bf S}_{r}$ if
$\displaystyle y_{1}$ is taken in $\displaystyle{\bf D}_{r}.$ In this case,
Corollary 2.2 ensures that $\displaystyle\widetilde{y}(X_{1},y_{1})$ belongs
to $\displaystyle{\bf D}_{\rho}.$ The equivalence is then defined by
$\begin{array}[]{lll}\widehat{\Psi}_{\operatorname{\varepsilon}}^{c_{1}}:(X_{1},y_{1})\mapsto(X_{1},{\mathbf{r_{2}}}(X_{1},y_{1})),\end{array}$
(2.6)
with
$\displaystyle{\mathbf{r_{2}}}(X_{1},y_{1})={\mathbf{\tilde{r}_{1}}}(-\log|X_{1}|,{\mathbf{u_{2}}}(e^{i\arg(X_{1})},\widetilde{y}(X_{1},y_{1})))$
$\displaystyle({\mathbf{u_{2}}}$ given in (2.4)). As the change of coordinates
is bounded, the Riemann’s removable singularity Theorem implies the existence
of a unique holomorphic extension
$\displaystyle\widehat{\Psi}_{\operatorname{\varepsilon}}^{c_{1}}$ to
$\displaystyle{\bf D}_{1}\times{\bf D}_{r}.$
Finally, the change of coordinates (2.6) extends to a subset
$\displaystyle{\mathcal{D}}_{1}(r)=\\{(X,y)\in{\bf C}\times{\bf C}:|X|\geq
1,|Xy|\leq r\\}$
as follows. Similar arguments as those used above show that the only tangent
curve $\displaystyle{\mathbf{r}}(\cdot,y_{1})$ to $\displaystyle\omega_{1}$
verifying $\displaystyle{\mathbf{r}}(\log|X_{1}|,y_{1})=y_{1},$ for a given
$\displaystyle(X_{1},y_{1})\in{\mathcal{D}}_{1}(r),$ satisfies
$\displaystyle|{\mathbf{r}}(0,y_{1})|e^{-s\\{|\operatorname{\varepsilon}|+1/4\\}}<|{\mathbf{r}}(s,y_{1})|,\quad
s\in[0,\log|X_{1}|],$
so that the initial condition $\displaystyle{\mathbf{r}}(0,y_{1})$ of the
lifting starting at
$\displaystyle(\frac{X_{1}}{|X_{1}|},{\mathbf{r}}(0,y_{1}))$ belongs to
$\displaystyle{\bf D}_{r}$ provided
$\displaystyle|\operatorname{\varepsilon}|\leq 3/4.$ Thus, the leaf containing
the point $\displaystyle(X_{1},y_{1})$ intersects the cylinder
$\displaystyle\\{|X|=1\\}$ in a curve
$\displaystyle{\mathbf{u_{1}}}={\mathbf{u_{1}}}(\theta)\in{\bf S}_{r},$ with
$\displaystyle{\mathbf{u_{1}}}(\arg X_{1})={\mathbf{r}}(0,y_{1})\in{\bf
D}_{r}.$ By Corollary 2.2, $\displaystyle{\mathbf{u_{1}}}(0)\in{\bf D}_{\rho}$
and then
$\displaystyle\widehat{\Psi}_{\operatorname{\varepsilon}}^{c_{1}}(\frac{X_{1}}{|X_{1}|},{\mathbf{r}}(0,y_{1}))$
is well defined, where
$\displaystyle\widehat{\Psi}_{\operatorname{\varepsilon}}^{c_{1}}$ is the
equivalqnce (2.6). In $\displaystyle{\mathcal{F}}_{\widehat{\omega}_{1}},$ the
inverse of $\displaystyle\gamma_{X_{1}}$ is lifted on the leaf passing through
the point
$\displaystyle\widehat{\Psi}_{\operatorname{\varepsilon}}^{c_{1}}(\frac{X_{1}}{|X_{1}|},{\mathbf{r}}(0,y_{1})).$
The endpoint of this radial lifting defines
$\displaystyle\widehat{\Psi}_{\operatorname{\varepsilon}}^{c_{1}}$ on
$\displaystyle{\mathcal{D}}_{1}(r).$
### 2.2. The equivalence in the $\displaystyle(x,Y)$ chart.
If $\displaystyle{\bf D}_{2}$ is the standard unit (closed) disk of the
$\displaystyle Y$-separatrix and
$\displaystyle{\mathcal{D}}_{2}(r)=\\{(x,Y)\in{\bf C}\times{\bf C}:|Y|\geq
1,|xY|\leq r\\},$ then, in $\displaystyle(x,Y)$ coordinates the equivalence is
defined plainly on $\displaystyle({\bf D}_{2}^{*}\times{\bf
D}_{r})\cup{\mathcal{D}}_{2}(r),$ by the formula
$\displaystyle\widehat{\Psi}_{\operatorname{\varepsilon}}^{c_{2}}:=\varphi\circ\widehat{\Psi}_{\operatorname{\varepsilon}}^{c_{1}}\circ\varphi^{\circ-1},$
where $\displaystyle\varphi:(X,y)\mapsto(x,Y)$ is the transition between
complex charts. Such equivalence is clearly bounded and the Riemann’s Theorem
yields a unique holomorphic extension
$\displaystyle\widehat{\Psi}_{\operatorname{\varepsilon}}^{c_{2}}:({\bf
D}_{2}\times{\bf D}_{r})\cup{\mathcal{D}}_{2}(r)\mapsto{\bf C}^{2}.$
It turns out that the two changes of coordinates thus obtained
$\displaystyle\widehat{\Psi}_{\operatorname{\varepsilon}}^{c_{1}},\widehat{\Psi}_{\operatorname{\varepsilon}}^{c_{2}}$
are analytical continuations of each other on $\displaystyle{\bf
C}P^{1}\times{\bf D}_{r},$ yielding a well defined and holomorphic global
change of coordinates
$\displaystyle\widehat{\Psi}_{\operatorname{\varepsilon}}$ over the divisor
which is, by construction, a local equivalence between
$\displaystyle{\mathcal{F}}_{\omega_{\operatorname{\varepsilon}}}$ and
$\displaystyle{\mathcal{F}}_{\widehat{\omega}_{\operatorname{\varepsilon}}}$
around $\displaystyle{\bf S}^{1}\times{\bf C}.$ It depends holomorphically on
$\displaystyle\operatorname{\varepsilon}\in V$ by the analytic dependence on
initial conditions of a differential equation. Let
$\displaystyle\Psi_{\operatorname{\varepsilon}}$ stand for this diffeomorphism
in $\displaystyle(x,y)$ variables. Since the Riemann sphere $\displaystyle{\bf
C}P^{1}$ retracts to the origin, the equivalence
$\displaystyle\Psi_{\operatorname{\varepsilon}}$ is defined on
$\displaystyle({\bf D}_{r}\times{\bf D}_{r})\backslash\\{(0,0)\\}$ and is
analytic there, because the monoidal map is an isomorphism away from the
exceptional divisor. By Hartogs Theorem,
$\displaystyle\Psi_{\operatorname{\varepsilon}}$ can be holomorphically
extended until the origin.
Inasmuch as the equivalence $\displaystyle\Psi_{\operatorname{\varepsilon}}$
is constructed by lifting paths, and both the holonomy and the conjugacy
$\displaystyle{\bf h}_{\operatorname{\varepsilon}}$ are real (when
$\displaystyle\operatorname{\varepsilon}\in{\bf R})$, the change of
coordinates $\displaystyle\Psi_{\operatorname{\varepsilon}}$ is real as well.
## 3\. Realization of an admissible family.
A first change of coordinates on the complexified family (1.3), depending
analytically on small values of the parameter, allows to get rid of all cubic
terms except for the resonant one (Poincaré normal form). The weak focus is of
order one if the real part of the coefficient of the third order resonant
monomial is non null. The sign $\displaystyle s=\pm 1$ of such a coefficient
defines two different cases which are not equivalent by real equivalence. In
fact, $\displaystyle s$ is an analytic invariant of the system. An analytic
change of coordinates (cf. [1]) brings the Poincaré map to the “prepared” form
$\begin{array}[]{lll}{\mathcal{P}}_{\operatorname{\varepsilon}}(x)=x+x(\operatorname{\varepsilon}+sx^{2})(2\pi+O(\operatorname{\varepsilon})+O(x)),\end{array}$
(3.1)
with multiplier $\displaystyle\exp(2i\pi)$ at the origin.
###### Proposition 3.1.
A germ of generic real analytic family of differential equations unfolding a
germ of real analytic weak focus of order one, is formally orbitally
equivalent to:
$\begin{array}[]{lll}\dot{x}&=&x(i+(\operatorname{\varepsilon}\pm
u)(1-A(\operatorname{\varepsilon})u))\\\
\dot{y}&=&y(-i+(\operatorname{\varepsilon}\pm
u)(1-A(\operatorname{\varepsilon})u))\end{array}$ (3.2)
with $\displaystyle u=xy,$ for some family of constants $\displaystyle
A(\operatorname{\varepsilon})$ which is real on
$\displaystyle\operatorname{\varepsilon}\in{\bf R}$ and $\displaystyle
A(0)\neq 0.$ The parameter $\displaystyle\operatorname{\varepsilon}$ of the
formal normal form (3.2) is called the “canonical parameter”.
###### Proof.
Consider the case $\displaystyle s=+1.$ By a formal change of coordinates we
bring the system to the form:
$\begin{array}[]{lll}\dot{x}&=&x(i+\operatorname{\varepsilon}-\sum_{j\geq
1}A_{j}(\operatorname{\varepsilon})u^{j}):=P(x,y)\\\
\dot{y}&=&y(-i+\operatorname{\varepsilon}-\sum_{j\geq
1}\overline{A_{j}(\overline{\operatorname{\varepsilon}})}u^{j}):=Q(x,y)\end{array}$
(3.3)
where $\displaystyle Re(A_{1})\neq 0.$ In order to simplify the form, we
iteratively use changes of coordinates
$\displaystyle(x,y)=({\mathbf{x}}(1+cU^{n}),{\mathbf{y}}(1+\overline{c}U^{n}))$
for $\displaystyle n\geq 1.$ Such a change allows to get rid of the term
$\displaystyle A_{n+1}U^{n+1}$ provided that $\displaystyle n+1>2.$ When
$\displaystyle n=1$ it allows to get rid of $\displaystyle iIm(A_{2}U^{2}).$
Indeed, the constant $\displaystyle c$ must be chosen so as to verify
$\displaystyle A_{1}(c+\overline{c})-nc(A_{1}+\overline{A_{1}})=A_{n+1},$
which is always solvable in $\displaystyle c$ as soon as $\displaystyle
Re(A_{1})\neq 0$ and $\displaystyle n>1.$ However, when $\displaystyle n=1$ we
get $\displaystyle
A_{1}(c+\overline{c})-nc(A_{1}+\overline{A_{1}})=A_{1}\overline{c}-\overline{A_{1}}c=2iIm(A_{1}\overline{c})\in
i{\bf R}.$ Hence, in that only case, the equation $\displaystyle
A_{1}(c+\overline{c})-nc(A_{1}+\overline{A_{1}})=iIm(A_{n+1})$ is solvable in
$\displaystyle c.$ Finally, one divides (3.3) by $\displaystyle\frac{yP-
xQ}{2ixy}.$ This brings all the $\displaystyle Im(A_{j})$ to $\displaystyle
0.$ Then we repeat the procedure above with $\displaystyle c$ real to remove
all higher terms in $\displaystyle u^{j}$ except for the term in
$\displaystyle u^{2}.$ The cases $\displaystyle s=-1$ is analogous. ∎
It is easily seen that the multiplier at the origin of the Poincaré map of the
field (3.2) is equal to $\displaystyle\exp(2\pi\operatorname{\varepsilon}),$
so that the canonical parameter is also an analytic invariant of the Poincaré
map.
_Admissible families of holomorphic germs._
Consider the germ of a holomorphic family
$\displaystyle{\mathcal{Q}}_{\operatorname{\varepsilon}}$ unfolding the germ
of a codimension one analytic resonant diffeomorphism
$\displaystyle{\mathcal{Q}}$ with multiplier equal to $\displaystyle-1$ at the
origin. The formal normal form
$\displaystyle{\mathcal{Q}}_{0,\operatorname{\varepsilon}}$ of
$\displaystyle{\mathcal{Q}}_{\operatorname{\varepsilon}}$ is the semi-Poincaré
map (or semi-monodromy) of the vector field (3.2), namely
$\displaystyle{\mathcal{Q}}_{0,\operatorname{\varepsilon}}={\mathcal{L}}_{-1}\circ\tau_{\operatorname{\varepsilon}}^{\pi},$
where $\displaystyle\tau_{\operatorname{\varepsilon}}^{\pi}$ is the time
$\displaystyle\pi$-map of the equation:
$\begin{array}[]{lll}\dot{w}=\frac{w(\operatorname{\varepsilon}\pm
w^{2})}{1+A(\operatorname{\varepsilon})w^{2}}\end{array}$ (3.4)
and $\displaystyle{\mathcal{L}}_{-1}:w\mapsto-w.$
###### Lemma 3.2.
Let $\displaystyle{\mathcal{Q}}_{\operatorname{\varepsilon}}$ be a prepared
family $\displaystyle(i.e.$ such that
$\displaystyle{\mathcal{Q}}_{\operatorname{\varepsilon}}^{\circ 2}$ has the
form (3.1)$\displaystyle)$ unfolding a codimension one resonant diffeomorphism
$\displaystyle{\mathcal{Q}}$ with multiplier equal to $\displaystyle-1,$ and
let $\displaystyle{\mathcal{Q}}_{0,\operatorname{\varepsilon}}$ be its formal
normal form, with same canonical parameter
$\displaystyle\operatorname{\varepsilon}.$ Then, for any $\displaystyle
N\in{\bf N}^{*}$ there exists a real family of germs of diffeomorphisms
$\displaystyle f_{\operatorname{\varepsilon}}$ tangent to the identity such
that:
$\begin{array}[]{lll}{\mathcal{Q}}_{\operatorname{\varepsilon}}\circ
f_{\operatorname{\varepsilon}}-f_{\operatorname{\varepsilon}}\circ{\mathcal{Q}}_{0,\operatorname{\varepsilon}}=O(x^{N+1}(\operatorname{\varepsilon}\pm
x^{2})^{N+1}).\end{array}$ (3.5)
###### Proof.
The proof is a slight modification of Theorem 6.2 in [12], being given that
the preparation of the family of diffeomorphisms is slightly different as
well. ∎
Any germ of family of holomorphic diffeomorphisms
$\displaystyle{\mathcal{Q}}_{\operatorname{\varepsilon}}:({\bf C},0)\to({\bf
C},0)$ verifying the hypotheses of Lemma 3.2, is said to be “admissible”.
###### Theorem 3.3.
Let $\displaystyle{\mathcal{Q}}_{\operatorname{\varepsilon}}:({\bf
C},0)\to({\bf C},0)$ be a real analytic family in the class of admissible
germs of families, with coefficients $\displaystyle
c_{k}(\operatorname{\varepsilon})$ depending analytically on the canonical
parameter $\displaystyle\operatorname{\varepsilon},$ and such that
$\displaystyle
2c_{2}(\operatorname{\varepsilon})^{2}+c_{3}(\operatorname{\varepsilon})(1+c_{1}(\operatorname{\varepsilon})^{2})\neq
0$ for all $\displaystyle\operatorname{\varepsilon}\in V.$ Then the second
iterate $\displaystyle{\mathcal{Q}}_{\operatorname{\varepsilon}}^{\circ 2}$ is
the monodromy of an elliptic generic family (1.3) of order one.
## 4\. Proof of Theorem 3.3.
The proof is achieved in several steps.
### 4.1. Family of abstract manifolds.
By Lemma 3.2, $\displaystyle{\mathcal{Q}}_{\operatorname{\varepsilon}}$
decomposes as
$\begin{array}[]{lll}{\mathcal{Q}}_{\operatorname{\varepsilon}}&=&(id+g_{\operatorname{\varepsilon}})\circ{\mathcal{Q}}_{0,\operatorname{\varepsilon}}\\\
&=&{\mathcal{Q}}_{0,\operatorname{\varepsilon}}\circ(id+\widehat{g}_{\operatorname{\varepsilon}}),\end{array}$
(4.1)
where $\displaystyle g_{\operatorname{\varepsilon}}$ and
$\displaystyle\widehat{g}_{\operatorname{\varepsilon}}:={\mathcal{Q}}_{0,\operatorname{\varepsilon}}^{\circ-1}\circ
g_{\operatorname{\varepsilon}}\circ{\mathcal{Q}}_{0,\operatorname{\varepsilon}}$
are $\displaystyle(N+1)$-flat in $\displaystyle x$ at the origin:
$\displaystyle
g_{\operatorname{\varepsilon}}(x),\widehat{g}_{\operatorname{\varepsilon}}(x)=O(x^{N+1}(\operatorname{\varepsilon}\pm
x^{2})^{N+1})),$ for a large integer $\displaystyle N\in{\bf N}.$
We recall that the standard monoidal map endows the blow-up space with
coordinates $\displaystyle(X,y)$ and $\displaystyle(x,Y),$ and the transition
between them is noted $\displaystyle\varphi.$
Let $\displaystyle v_{0,\operatorname{\varepsilon}}^{1}$ be the formal normal
form given by the pullback of (3.2) in $\displaystyle(X,y)$ variables (with
linear part $\displaystyle 2iX\frac{\partial}{\partial
X}+(\operatorname{\varepsilon}-i)y\frac{\partial}{\partial y})$ and let
$\displaystyle{\mathcal{F}}_{v_{0}^{1}}$ be its foliation on the product
$\displaystyle{\bf C}^{*}\times{\bf D}_{y},$ where $\displaystyle{\bf D}_{y}$
is the standard unit disk of the $\displaystyle y$ axis. Consider the region
$\displaystyle\widetilde{K}_{1}=\Big{\\{}\widetilde{X}\in Cov({\bf
C}^{*}):-\pi/4<\arg(\widetilde{X})<2\pi+\pi/4\Big{\\}}$
in the covering space $\displaystyle Cov({\bf C}^{*})$ of the exceptional
divisor, see Figure 3.
Figure 3. The domain of $\displaystyle\widetilde{X}$ in the covering space
$\displaystyle Cov({\bf C}^{*}).$
The pullback of $\displaystyle v_{0,\operatorname{\varepsilon}}^{1}$ by the
covering map $\displaystyle\pi_{1}:\widetilde{K}_{1}\times{\bf D}_{y}\to{\bf
C}^{*}\times{\bf D}_{y},$ defines a field
$\displaystyle\widetilde{v}_{\operatorname{\varepsilon}}^{1}(\widetilde{X},w)$
and a foliation $\displaystyle\widetilde{{\mathcal{F}}}_{v^{1}}$ on the
product $\displaystyle\widetilde{M}=\widetilde{K}_{1}\times{\bf D}_{y}.$ The
leaves of $\displaystyle\widetilde{{\mathcal{F}}}_{v^{1}}$ around the _flaps_
$\begin{array}[]{lll}S^{\prime}_{1}&=&\\{\widetilde{X}^{\prime}\in\widetilde{K}_{c}:-\pi/4<\arg(\widetilde{X}^{\prime})<\pi/4\\}\\\
S_{1}&=&\\{\widetilde{X}\in\widetilde{K}_{1}:2\pi-\pi/4<\arg(\widetilde{X})<2\pi+\pi/4\\}\end{array}$
are identified by means of a _sealing map_
$\displaystyle\Upsilon_{\operatorname{\varepsilon}}:S^{\prime}_{1}\times{\bf
D}_{y}\to S_{1}\times{\bf C},$ which preserves the first coordinate and
respects $\displaystyle\widetilde{{\mathcal{F}}}_{v^{1}}.$ It is constructed
as follows. For small values of $\displaystyle y,$ the holonomy map
$\displaystyle h_{\operatorname{\varepsilon},X}:\\{X\\}\times{\bf
D}_{y}\to\\{1\\}\times{\bf D}_{y}$ along the leaves of
$\displaystyle{\mathcal{F}}_{v_{0}^{1}}$ is covered by two holonomy maps,
$\displaystyle
h_{\operatorname{\varepsilon},\widetilde{X}^{\prime}}:\\{\widetilde{X^{\prime}}\\}\times{\bf
D}_{y}\to\Sigma^{\prime}\times{\bf D}_{y}$ and $\displaystyle
h_{\operatorname{\varepsilon},\widetilde{X}}:\\{\widetilde{X}\\}\times{\bf
D}_{y}\to\Sigma\times{\bf D}_{y}$ along the leaves of
$\displaystyle\widetilde{{\mathcal{F}}}_{v^{1}}.$ The holonomies is negatively
(resp. positively) oriented and noted $\displaystyle
h_{\operatorname{\varepsilon}}^{-}$ (resp. $\displaystyle
h_{\operatorname{\varepsilon}}^{+}),$ when $\displaystyle Im(X)>0$ (resp.
$\displaystyle Im(X)<0).$ The convention:
$\begin{array}[]{lll}\lim_{\widetilde{X}\to\widetilde{1}}h_{\operatorname{\varepsilon},\widetilde{X}}^{+}=id\end{array}$
(4.2)
will be taken into account as well. Then
$\displaystyle\Upsilon_{\operatorname{\varepsilon}}(\widetilde{X}^{\prime},y)=(\widetilde{X},\Delta_{\operatorname{\varepsilon}}(\widetilde{X}^{\prime},y)),$
where
$\begin{array}[]{lll}\Delta_{\operatorname{\varepsilon}}(\widetilde{X}^{\prime},y)=(h_{\operatorname{\varepsilon},\widetilde{X}}^{+})^{\circ-1}\circ(id+g_{\operatorname{\varepsilon}})\circ
h_{\operatorname{\varepsilon},\widetilde{X}^{\prime}}^{+}(y),\end{array}$
(4.3)
with $\displaystyle\pi_{1}(\widetilde{X}^{\prime})=\pi_{1}(\widetilde{X}).$
The map $\displaystyle\Upsilon_{\operatorname{\varepsilon}}$ is well defined
and real analytic on its image for $\displaystyle r>0$ small, and it depends
analytically on the parameter. Thus, it may be analytically extended to a
larger domain
$\displaystyle\\{\widetilde{X}\in\widetilde{K}_{1}:-\pi/4<\arg(\widetilde{X})<\pi\\}\times{\bf
D}_{y}.$
Around a region of the covering of the $\displaystyle(x,Y)$ chart, things are
naturally defined by means of the transition $\displaystyle\varphi.$ In
particular, the family
$\displaystyle\operatorname{\raisebox{6.83331pt}{\scalebox{1.0}[-1.0]{$\displaystyle\Upsilon$}}}_{\operatorname{\varepsilon}}=\varphi^{*}\Upsilon_{\operatorname{\varepsilon}}$
is a sealing map
$\displaystyle\operatorname{\raisebox{6.83331pt}{\scalebox{1.0}[-1.0]{$\displaystyle\Upsilon$}}}_{\operatorname{\varepsilon}}(\widetilde{Y}^{\prime},z)=(\widetilde{Y},\nabla_{\operatorname{\varepsilon}}(\widetilde{Y}^{\prime},z))$
with
$\begin{array}[]{lll}\nabla_{\operatorname{\varepsilon}}(\widetilde{Y}^{\prime},z)=(\ell_{\operatorname{\varepsilon},\widetilde{Y}}^{+})^{\circ-1}\circ(id+\widehat{g}_{\operatorname{\varepsilon}})\circ\ell_{\operatorname{\varepsilon},\widetilde{Y}^{\prime}}^{+}(z),\end{array}$
(4.4)
and the transition $\displaystyle\varphi$ defines a field
$\displaystyle\widetilde{v}_{\operatorname{\varepsilon}}^{2}(x,\widetilde{Y}),$
and foliation $\displaystyle\widetilde{{\mathcal{F}}}_{v^{2}}$ on the product
$\displaystyle\widetilde{N}:=\varphi^{*}\widetilde{M}$ (the latter endowed
with complex coordinates $\displaystyle(x,\widetilde{Y})$ defined in the
natural way). Here, $\displaystyle\ell_{\operatorname{\varepsilon}}^{\pm}$ are
the holonomies along the leaves of
$\displaystyle\widetilde{{\mathcal{F}}}_{v^{2}}.$ The map
$\displaystyle\operatorname{\raisebox{6.83331pt}{\scalebox{1.0}[-1.0]{$\displaystyle\Upsilon$}}}_{\operatorname{\varepsilon}}$
is real analytic on its image.
As the sealing
$\displaystyle(\Upsilon_{\operatorname{\varepsilon}},\operatorname{\raisebox{6.83331pt}{\scalebox{1.0}[-1.0]{$\displaystyle\Upsilon$}}}_{\operatorname{\varepsilon}})$
is canonically defined on the divisor, it defines a sealing family noted
$\displaystyle\Gamma_{\operatorname{\varepsilon}}:\widetilde{{\mathcal{M}}}\to\widetilde{{\mathcal{M}}},$
where $\displaystyle\widetilde{{\mathcal{M}}}$ is the pullback of
$\displaystyle(\widetilde{M},\widetilde{N})$ by the inverse of the monoidal
map. The vector fields
$\displaystyle(\widetilde{v}_{\operatorname{\varepsilon}}^{1},\widetilde{v}_{\operatorname{\varepsilon}}^{2})$
and foliations
$\displaystyle(\widetilde{{\mathcal{F}}}_{v^{1}},\widetilde{{\mathcal{F}}}_{v^{2}})$
induce a vector field
$\displaystyle\widetilde{v}_{\operatorname{\varepsilon}}$ and a foliation
$\displaystyle\widetilde{{\mathcal{F}}}_{\operatorname{\varepsilon}}$ on
$\displaystyle\widetilde{{\mathcal{M}}},$ and the coordinates on the latter
are $\displaystyle(x,y).$ Moreover,
$\displaystyle\Gamma_{\operatorname{\varepsilon}}$ is a germ of real analytic
family of diffeomorphisms that preserves the transversal fibers
$\displaystyle\Sigma_{\mu}=\\{x=\mu y,\mu\in{\bf C}^{*}\\},$ and
$\displaystyle(\Gamma_{\operatorname{\varepsilon}})_{*}\widetilde{v}_{\operatorname{\varepsilon}}=\widetilde{v}_{\operatorname{\varepsilon}},$
so that $\displaystyle\Gamma_{\operatorname{\varepsilon}}$ respects
$\displaystyle\widetilde{{\mathcal{F}}}_{\operatorname{\varepsilon}}.$ Then,
the quotient
$\displaystyle{\mathcal{M}}_{\operatorname{\varepsilon}}=\widetilde{{\mathcal{M}}}/\Gamma_{\operatorname{\varepsilon}}$
is well defined and the vector field
$\displaystyle\widetilde{v}_{\operatorname{\varepsilon}}$ induces a vector
field $\displaystyle v_{\operatorname{\varepsilon}}$ and a foliation
$\displaystyle{\mathcal{F}}_{\operatorname{\varepsilon}}$ on
$\displaystyle{\mathcal{M}}_{\operatorname{\varepsilon}}.$ The leaves of this
foliation project without critical points on the base $\displaystyle{\bf
C}^{*}\times{\bf D}_{y}$ in the $\displaystyle(X,y)$ chart $\displaystyle(i.e$
are transversal to all lines $\displaystyle\\{X=const.\\}),$ and hence the
loop generating the fundamental group of $\displaystyle{\bf C}^{*}\times{\bf
D}_{y}$ defines the holonomy map of the quotient foliation
$\displaystyle{\mathcal{F}}_{\operatorname{\varepsilon}}$ on
$\displaystyle{\mathcal{M}}_{\operatorname{\varepsilon}}$ (for the cross
section $\displaystyle\Sigma),$ referred to as the semi-monodromy.
###### Proposition 4.1.
The monodromy $\displaystyle\Sigma\to\Sigma$ of the field $\displaystyle
v_{\operatorname{\varepsilon}}$ along the leaves of
$\displaystyle{\mathcal{F}}_{\operatorname{\varepsilon}}$ coincides with
$\displaystyle{\mathcal{Q}}_{\operatorname{\varepsilon}}.$
###### Proof.
The holonomy $\displaystyle
h_{\operatorname{\varepsilon},\widetilde{1}}:\\{\widetilde{1}^{\prime}\\}\times{\bf
D}_{y}\to\\{\widetilde{1}\\}\times{\bf D}_{y}$ of
$\displaystyle\widetilde{v}_{\operatorname{\varepsilon}}^{1}$ coincides, by
construction, with the normal form
$\displaystyle{\mathcal{Q}}_{0,\operatorname{\varepsilon}}$ on
$\displaystyle\widetilde{M}.$ Then, in $\displaystyle(x,y)$ variables, the
image of the point $\displaystyle(x,x)\in\Sigma$ under the holonomy of
$\displaystyle\widetilde{v}_{\operatorname{\varepsilon}}$ (for the section
$\displaystyle\Sigma)$ is given by
$\displaystyle({\mathcal{Q}}_{0,\operatorname{\varepsilon}}(x),{\mathcal{Q}}_{0,\operatorname{\varepsilon}}(x))\in\Sigma.$
In addition,
$\begin{array}[]{lll}\Gamma_{\operatorname{\varepsilon}}({\mathcal{Q}}_{0,\operatorname{\varepsilon}}(x),{\mathcal{Q}}_{0,\operatorname{\varepsilon}}(x))&=&(\Delta_{\operatorname{\varepsilon}}(1,{\mathcal{Q}}_{0,\operatorname{\varepsilon}}(x)),\Delta_{\operatorname{\varepsilon}}(1,{\mathcal{Q}}_{0,\operatorname{\varepsilon}}(x)))\\\
&=&((id+g_{\operatorname{\varepsilon}})\circ{\mathcal{Q}}_{0,\operatorname{\varepsilon}}(x),(id+g_{\operatorname{\varepsilon}})\circ{\mathcal{Q}}_{0,\operatorname{\varepsilon}}(x))\\\
&=&({\mathcal{Q}}_{\operatorname{\varepsilon}}(x),{\mathcal{Q}}_{\operatorname{\varepsilon}}(x))\in\Sigma,\end{array}$
where the second equality comes after (4.2). ∎
### 4.2. Integrability on $\displaystyle
H_{\operatorname{\varepsilon}}({\mathcal{M}}_{\operatorname{\varepsilon}})$
In $\displaystyle(\widetilde{X},y)$ coordinates, we introduce a smooth _real_
nonnegative cutoff function $\displaystyle\chi$ _depending only on the
argument of_ $\displaystyle\widetilde{X}:$
$\chi(\arg\widetilde{X})=\left\\{\begin{array}[]{lll}1,\quad\arg\widetilde{X}\in(-\pi/4,\pi/4],\\\
0,\quad\arg\widetilde{X}^{\prime}\in(\pi,2\pi+\pi/4].\end{array}\right.$
An “identification map”
$\displaystyle\widetilde{H}_{\operatorname{\varepsilon}}^{1}$ is defined on
$\displaystyle\widetilde{M}:$
$\begin{array}[]{lll}\widetilde{H}_{\operatorname{\varepsilon}}^{1}:(\widetilde{X},y)\mapsto(\widetilde{X},y+\chi(\arg\widetilde{X})\\{\Delta_{\operatorname{\varepsilon}}(\widetilde{X},y)-y\\}),\end{array}$
(4.5)
for $\displaystyle c_{1},c_{2}$ the monoidal map in charts (2.1). Notice that
$\displaystyle\widetilde{H}_{\operatorname{\varepsilon}}^{1}|_{S^{\prime}_{1}\times{\bf
D}_{y}}\equiv(id_{X},\Delta_{\operatorname{\varepsilon}})$ and
$\displaystyle\widetilde{H}_{\operatorname{\varepsilon}}^{1}|_{S_{1}\times{\bf
D}_{y}}\equiv(id_{X},id_{y}),$ and so this map respects the sealing
$\displaystyle\Upsilon_{\operatorname{\varepsilon}}.$
In $\displaystyle(x,y)$ variables the function $\displaystyle\chi$ yields a
real smooth map $\displaystyle\widehat{\chi}(x,y)=\chi(\arg(x/y))$ which
depends only on the argument of the quotient $\displaystyle x/y,$ and the blow
down of (4.5) in $\displaystyle(x,y)$ coordinates equips the target space with
coordinates $\displaystyle(z,w):$
$\begin{array}[]{lll}(z,w)=\widetilde{H}_{\operatorname{\varepsilon}}(x,y)\\\
=(x+\widehat{\chi}(x,y)\\{\nabla_{\operatorname{\varepsilon}}\circ
c_{2}^{-1}(x,y)-x\\},w+\widehat{\chi}(x,y)\\{\Delta_{\operatorname{\varepsilon}}\circ
c_{1}^{-1}(x,y)-y\\}).\end{array}$ (4.6)
By definition, $\displaystyle\widetilde{H}_{\operatorname{\varepsilon}}$
induces an “identification family” in the quotient:
$\displaystyle
H_{\operatorname{\varepsilon}}:{\mathcal{M}}_{\operatorname{\varepsilon}}\to{\bf
C}^{2}.$
For every fixed $\displaystyle\operatorname{\varepsilon},$ the latter is a
real analytic diffeomorphism which endows the target space with an almost
complex structure induced from the standard complex structure on
$\displaystyle{\mathcal{M}}_{\operatorname{\varepsilon}},$ as shown later. In
addition, it depends analytically on the parameter. If the function
$\displaystyle g$ in (4.1) is $\displaystyle(N+1)$-flat at $\displaystyle
x=y=0,$ then (4.6) is infinitely tangent to the origin:
###### Proposition 4.2.
The maps (4.3) and (4.4) admit the asymptotic estimates
$\begin{array}[]{lll}|\Delta_{\operatorname{\varepsilon}}\circ
c_{1}^{-1}(x,y)-y|&=&O(|x|^{\frac{N}{2}(1-\operatorname{\varepsilon}_{2})}|y|^{\frac{N}{2}(1+\operatorname{\varepsilon}_{2})+1})\\\
|\nabla_{\operatorname{\varepsilon}}\circ
c_{2}^{-1}(x,y)-x|&=&O(|x|^{\frac{N}{2}(1+\operatorname{\varepsilon}_{2})+1}|y|^{\frac{N}{2}(1-\operatorname{\varepsilon}_{2})})\end{array}$
(4.7)
in the bidisk $\displaystyle{\bf D}_{x}\times{\bf D}_{y},$ where
$\displaystyle\operatorname{\varepsilon}=\operatorname{\varepsilon}_{1}+i\operatorname{\varepsilon}_{2}.$
###### Proof.
In $\displaystyle(\widetilde{X},y)$ variables, the following estimate for the
holonomy map $\displaystyle
h_{\operatorname{\varepsilon},\widetilde{X}}:\\{\widetilde{X}\\}\times{\bf
D}_{w}\to\\{\widetilde{1}\\}\times{\bf C}$ is well known:
$\begin{array}[]{lll}e^{-M|\lambda(\operatorname{\varepsilon})(\widetilde{X}-1)|-\frac{\operatorname{\varepsilon}_{1}\arg\widetilde{X}}{2}}|\widetilde{X}|^{\frac{1-\operatorname{\varepsilon}_{2}}{2}}|y|\leq|h_{\operatorname{\varepsilon},\widetilde{X}}(y)|\leq
e^{M|\lambda(\operatorname{\varepsilon})(\widetilde{X}-1)|-\frac{\operatorname{\varepsilon}_{1}\arg\widetilde{X}}{2}}|\widetilde{X}|^{\frac{1-\operatorname{\varepsilon}_{2}}{2}}|y|,\end{array}$
where $\displaystyle M=M(\widetilde{X},y)<\infty$ is a positive constant
depending on a bound for the nonlinear part of the foliation along the segment
with endpoints $\displaystyle\widetilde{X},1,$ and
$\displaystyle\lambda(\operatorname{\varepsilon})=(\operatorname{\varepsilon}-i)/2i$
is the ratio of eigenvalues in $\displaystyle(X,y)$ chart. By (4.1),
$\displaystyle
h_{\operatorname{\varepsilon},\widetilde{X}}^{-1}\circ(id+g)\circ
h_{\operatorname{\varepsilon},\widetilde{X}^{\prime}}=h_{\operatorname{\varepsilon},\widetilde{X}}^{-1}\circ(h_{\operatorname{\varepsilon},\widetilde{X}^{\prime}}+g\circ
h_{\operatorname{\varepsilon},\widetilde{X}^{\prime}})=id+O(|\widetilde{X}|^{\frac{N}{2}(1-\operatorname{\varepsilon}_{2})}|y|^{N+1}).$
In $\displaystyle(x,Y)$ coordinates, the estimate is obtained by symmetry.
Since $\displaystyle x=\widetilde{X}y$ and $\displaystyle y=\widetilde{Y}x,$
the conclusion follows. ∎
###### Corollary 4.3.
The family $\displaystyle H_{\operatorname{\varepsilon}}$ is tangent to the
identity.
The pullback of the complex structure on
$\displaystyle{\mathcal{M}}_{\operatorname{\varepsilon}}$ by the map
$\displaystyle H_{\operatorname{\varepsilon}}^{-1}$ is an almost complex
structure defined by the pullback of the $\displaystyle(1,0)$-subbundle on
$\displaystyle{\mathcal{M}}_{\operatorname{\varepsilon}},$ which is spanned by
$\begin{array}[]{lll}\widetilde{\zeta}_{1,\operatorname{\varepsilon}}=dz=d(x+\widehat{\chi}\cdot\\{\nabla_{\operatorname{\varepsilon}}\circ
c_{2}^{-1}-x\\}),\\\
\widetilde{\zeta}_{2,\operatorname{\varepsilon}}=dw=d(y+\widehat{\chi}\cdot\\{\Delta_{\operatorname{\varepsilon}}\circ
c_{1}^{-1}-y\\}),\end{array}$ (4.8)
on
$\displaystyle\widetilde{H}_{\operatorname{\varepsilon}}(\widetilde{{\mathcal{M}}}).$
The forms $\displaystyle d(\nabla_{\operatorname{\varepsilon}}\circ
c_{2}^{-1})$ and $\displaystyle d(\Delta_{\operatorname{\varepsilon}}\circ
c_{1}^{-1})$ are holomorphic on their domains and
$\displaystyle\widetilde{\zeta}_{1,\operatorname{\varepsilon}}$ and
$\displaystyle\widetilde{\zeta}_{2,\operatorname{\varepsilon}}$ have two
different sectorial representatives:
$\left.\begin{array}[]{lll}\widetilde{\zeta}_{1,\operatorname{\varepsilon}}=\left\\{\begin{array}[]{lll}\zeta_{1,\operatorname{\varepsilon}}^{0}=dx,&|\arg
x-\arg y-13\pi/8|<5\pi/8,\\\
\zeta_{1,\operatorname{\varepsilon}}^{1}=d(\nabla_{\operatorname{\varepsilon}}\circ
c_{2}^{-1}),&|\arg x-\arg y|<\pi/4,\end{array}\right.\\\ &\\\
\widetilde{\zeta}_{2,\operatorname{\varepsilon}}=\left\\{\begin{array}[]{lll}\zeta_{2,\operatorname{\varepsilon}}^{1}=d(\Delta_{\operatorname{\varepsilon}}\circ
c_{2}^{-1}),&|\arg x-\arg y|<\pi/4\\\
\zeta_{2,\operatorname{\varepsilon}}^{0}=dy,&|\arg x-\arg
y-13\pi/8|<5\pi/8,\end{array}\right.\end{array}\right.$ (4.9)
so that
$\displaystyle\zeta_{1,\operatorname{\varepsilon}}^{1}=\Gamma_{\operatorname{\varepsilon}}^{*}\zeta_{1,\operatorname{\varepsilon}}^{0}$
and
$\displaystyle\zeta_{2,\operatorname{\varepsilon}}^{1}=\Gamma_{\operatorname{\varepsilon}}^{*}\zeta_{2,\operatorname{\varepsilon}}^{0}.$
Thus they yield forms $\displaystyle\zeta_{1,\operatorname{\varepsilon}}$ and
$\displaystyle\zeta_{2,\operatorname{\varepsilon}}$ on
$\displaystyle{\mathcal{M}}_{\operatorname{\varepsilon}}.$ The almost complex
structure induced on $\displaystyle
H_{\operatorname{\varepsilon}}({\mathcal{M}}_{\operatorname{\varepsilon}})\subset{\bf
C}^{2}$ by the complex structure on
$\displaystyle{\mathcal{M}}_{\operatorname{\varepsilon}}$ is defined by the
two forms
$\begin{array}[]{lll}\omega_{1,\operatorname{\varepsilon}}=(H_{\operatorname{\varepsilon}}^{-1})^{*}\zeta_{1,\operatorname{\varepsilon}},\quad\quad\omega_{2,\operatorname{\varepsilon}}=(H_{\operatorname{\varepsilon}}^{-1})^{*}\zeta_{2,\operatorname{\varepsilon}}.\end{array}$
(4.10)
###### Lemma 4.4.
Let $\displaystyle\delta$ be a small positive number with
$\displaystyle|\operatorname{\varepsilon}|<\delta.$ If $\displaystyle\alpha$
and $\displaystyle\beta$ are the orders of flatness in $\displaystyle x$ and
$\displaystyle y$ (resp. $\displaystyle y$ and $\displaystyle x)$ of the
difference $\displaystyle\omega_{1,\operatorname{\varepsilon}}-dx$ (resp.
$\displaystyle\omega_{2,\operatorname{\varepsilon}}-dy),$ then the form
$\displaystyle\omega_{1,\operatorname{\varepsilon}}$ (resp.
$\displaystyle\omega_{2,\operatorname{\varepsilon}})$ can be extended as
$\displaystyle dx$ (resp. $\displaystyle dy)$ along the $\displaystyle x$-axis
(resp. $\displaystyle y$-axis) until the order $\displaystyle\alpha$ if the
number $\displaystyle N$ in (4.1) is sufficiently large so as to ensure
$\begin{array}[]{lll}N>\max\left\\{\frac{2(\alpha-1)}{1-\delta},\frac{2\beta}{1-\delta}\right\\}.\end{array}$
(4.11)
###### Proof.
By (4.10), it suffices to study the difference
$\displaystyle\widetilde{H}_{\operatorname{\varepsilon}}(x,y)-(x,y)=(\widehat{\chi}(x,y)\\{\nabla_{\operatorname{\varepsilon}}\circ
c_{2}^{-1}(x,y)-x\\},\widehat{\chi}(x,y)\\{\Delta_{\operatorname{\varepsilon}}\circ
c_{1}^{-1}(x,y)-y\\}).$
The definition of $\displaystyle\xi$ yields
$\begin{array}[]{lll}\left|\frac{\partial^{i+j}\widehat{\chi}}{\partial
x^{p}\partial\overline{x}^{q}\partial
y^{r}\partial\overline{y}^{s}}\right|<C^{\underline{st}}\cdot\frac{{\mathbf{M}}_{i+j}}{|x|^{i}|y|^{j}}\end{array}$
(4.12)
for all $\displaystyle i=p+q\in{\bf N},$ $\displaystyle j=r+s\in{\bf N}$ and
$\displaystyle{\mathbf{M}}_{i+j}:=\max_{\begin{subarray}{c}0\leq k\leq i+j\\\
\theta\in I\end{subarray}}|\chi^{(k)}(\theta)|$ with $\displaystyle
I=[-\pi/4,2\pi+\pi/4].$ To lighten the notation, put $\displaystyle
f(x,y)=\nabla_{\operatorname{\varepsilon}}\circ c_{2}^{-1}(x,y)-x.$
Proposition 4.2 implies that for all $\displaystyle k,l\in{\bf N},$ there
exists a real constant $\displaystyle L=L(N,\alpha,\beta)>0$ such that
$\begin{array}[]{lll}\left|\frac{\partial^{\alpha+\beta}(\widehat{\chi}\cdot
f)}{\partial x^{p}\partial\overline{x}^{q}\partial
y^{r}\partial\overline{y}^{s}}\right|\leq
L\cdot|x|^{\frac{N}{2}(1+\operatorname{\varepsilon}_{2})+1-\alpha}\cdot|y|^{\frac{N}{2}(1-\operatorname{\varepsilon}_{2})-\beta},\end{array}$
(4.13)
for $\displaystyle\alpha=p+q$ and $\displaystyle\beta=r+s.$ Hence, if
$\displaystyle|\operatorname{\varepsilon}|<\delta<<1$ and the order
$\displaystyle N$ of $\displaystyle g_{\operatorname{\varepsilon}}$ satisfies
(4.11) then the left hand side of (4.13) tends to zero uniformly in
$\displaystyle|x|<1,$ and thus
$\displaystyle\omega_{1,\operatorname{\varepsilon}}$ and $\displaystyle dx$
coincide until the order $\displaystyle\alpha$ along the $\displaystyle
x$-axis. The assertion for the difference
$\displaystyle\omega_{2,\operatorname{\varepsilon}}-dy$ follows by duality. ∎
The set $\displaystyle
H_{\operatorname{\varepsilon}}({\mathcal{M}}_{\operatorname{\varepsilon}})\subset{\bf
C}^{2}$ does not contain the axes of coordinates: its closure is
$\displaystyle C^{\infty}$-diffeomorphic to a closed neighborhood of the
origin of $\displaystyle{\bf C}^{2}.$ Lemma 4.4 shows that the almost complex
structure generated by (4.10) on $\displaystyle
H_{\operatorname{\varepsilon}}({\mathcal{M}}_{\operatorname{\varepsilon}})$
can be extended as $\displaystyle\omega_{1,\operatorname{\varepsilon}}=dx$
along the $\displaystyle x$-axis, and as
$\displaystyle\omega_{2,\operatorname{\varepsilon}}=dy$ along the
$\displaystyle y$-axis, until a well-defined order. This almost complex
structure is integrable. Indeed,
$\displaystyle\omega_{1,\operatorname{\varepsilon}}$ is obtained from the
pullback of $\displaystyle\zeta_{1,\operatorname{\varepsilon}}$ and since the
forms $\displaystyle d(\nabla_{\operatorname{\varepsilon}}\circ c_{2}^{-1})$
and $\displaystyle d(\Delta_{\operatorname{\varepsilon}}\circ c_{1}^{-1})$ are
holomorphic on their domains and $\displaystyle\widehat{\chi}$ is of class
$\displaystyle C^{\infty},$ $\displaystyle
d\widetilde{\zeta}_{1,\operatorname{\varepsilon}}$ contains no forms of type
$\displaystyle(0,2).$ By symmetry, the same holds for $\displaystyle
d\widetilde{\zeta}_{2,\operatorname{\varepsilon}}.$ If $\displaystyle L^{1,0}$
is the span of the forms
$\displaystyle\omega_{1,\operatorname{\varepsilon}},\omega_{2,\operatorname{\varepsilon}},$
then this integrability condition holds for $\displaystyle L^{1,0}$ on the
surface $\displaystyle
H_{\operatorname{\varepsilon}}({\mathcal{M}}_{\operatorname{\varepsilon}}),$
and by continuity it remains valid after extension until the axes. Hence, for
each $\displaystyle\operatorname{\varepsilon}\in V$ the Newlander-Nirenberg
Theorem ensures the existence of a smooth chart
$\displaystyle\widetilde{\Lambda}_{\operatorname{\varepsilon}}=\widetilde{\Lambda}_{\operatorname{\varepsilon}}(z,w),$
$\begin{array}[]{lll}\widetilde{\Lambda}_{\operatorname{\varepsilon}}=(\widetilde{\xi}_{\operatorname{\varepsilon}}^{1},\widetilde{\xi}_{\operatorname{\varepsilon}}^{2}):\widetilde{H}_{\operatorname{\varepsilon}}(\widetilde{{\mathcal{M}}})\to{\bf
C}^{2},\end{array}$ (4.14)
which is holomorphic in the sense of the almost complex structure (4.8). It
induces, in turn, the germ of a family of smooth charts
$\begin{array}[]{lll}\Lambda_{\operatorname{\varepsilon}}=(\xi_{\operatorname{\varepsilon}}^{1},\xi_{\operatorname{\varepsilon}}^{2}):{\mathbf{B}}(r)\subset
H_{\operatorname{\varepsilon}}({\mathcal{M}}_{\operatorname{\varepsilon}})\to{\bf
C}^{2}\end{array}$ (4.15)
in the quotient, where $\displaystyle{\mathbf{B}}(r)$ is a small ball around
the origin. This chart is, by definition, holomorphic in the sense of the
extended almost complex structure (4.10).
###### Theorem 4.5.
The germ of smooth charts $\displaystyle\Lambda_{\operatorname{\varepsilon}}$
respects the real foliation, is tangent to the identity at the origin, and
depends analytically on the parameter.
###### Proof.
In order to show that the chart respects the real foliation, it suffices to
prove that $\displaystyle\widetilde{\Lambda}_{\operatorname{\varepsilon}}$ is
real, namely, it sends $\displaystyle\\{z=\overline{w}\\}\simeq{\bf R}^{2}$
into $\displaystyle{\bf R}^{2}\subset{\bf C}^{2}$ when
$\displaystyle\operatorname{\varepsilon}\in{\bf R}.$
The family of diffeomorphisms (4.6) is analytic with respect to the structure
(4.8). It follows that, modulo a linear combination,
$\begin{array}[]{lll}dz&=&dx+e_{1,\operatorname{\varepsilon}}^{1}d\overline{x}+e_{2,\operatorname{\varepsilon}}^{1}d\overline{y}\\\
dw&=&dy+e_{1,\operatorname{\varepsilon}}^{2}d\overline{x}+e_{2,\operatorname{\varepsilon}}^{2}d\overline{y},\end{array}$
where the coefficients are computed in terms of
$\displaystyle\chi,\Delta,\nabla$ and its derivatives, and satisfy:
$\begin{array}[]{lll}\overline{e_{j,\overline{\operatorname{\varepsilon}}}^{k}(\overline{y},\overline{x})}&=&e_{k,\operatorname{\varepsilon}}^{j}(x,y),\quad
j,k\in\\{1,2\\}.\end{array}$ (4.16)
This is because $\displaystyle{\bf R}^{2}$ is itself invariant under (4.6):
$\displaystyle H_{\operatorname{\varepsilon}}(\\{x=\overline{y}\\})\subset{\bf
R}^{2}$ when the parameter is real. By Proposition 4.2, $\displaystyle
e_{j,\operatorname{\varepsilon}}^{k}=\frac{o(1)}{1+o(1)},$ yielding:
$\begin{array}[]{lll}e_{j,\operatorname{\varepsilon}}^{k}(0,0)=0,\quad
j,k\in\\{1,2\\}.\end{array}$ (4.17)
Suppose that the image
$\displaystyle\widetilde{H}_{\operatorname{\varepsilon}}({\mathcal{M}})$
contains a small bidisk $\displaystyle{\bf D}_{s}\times{\bf D}_{s},$ and write
$\displaystyle
G_{\operatorname{\varepsilon}}:=\widetilde{H}_{\operatorname{\varepsilon}}^{-1}.$
Consider the pullback
$\displaystyle{\mathfrak{a}}_{j,\operatorname{\varepsilon}}^{k}=G_{\operatorname{\varepsilon}}^{*}(e_{j,\operatorname{\varepsilon}}^{k}):{\bf
D}_{s}\times{\bf D}_{s}\to{\bf C}^{2}$ given by
$\begin{array}[]{lll}{\mathfrak{a}}_{j,\operatorname{\varepsilon}}^{k}(z,w)=G_{\operatorname{\varepsilon}}^{*}(e_{j,\operatorname{\varepsilon}}^{k})(z,w)\equiv
e_{j,\operatorname{\varepsilon}}^{k}(G_{\operatorname{\varepsilon}}(z,w)),\quad
j,k=1,2,\ \ \operatorname{\varepsilon}\in V,\end{array}$
for $\displaystyle(z,w)\in{\bf D}_{s}\times{\bf D}_{s}.$ By (4.16) the
collection $\displaystyle{\mathfrak{a}}_{j,\operatorname{\varepsilon}}^{k}$
satisfies again:
$\begin{array}[]{lll}\overline{{\mathfrak{a}}_{j,\overline{\operatorname{\varepsilon}}}^{k}(\overline{w},\overline{z})}&=&{\mathfrak{a}}_{k,\operatorname{\varepsilon}}^{j}(z,w),\end{array}$
(4.18)
and by (4.17),
$\displaystyle{\mathfrak{a}}_{j,\operatorname{\varepsilon}}^{k}(0,0)=0.$
_Notation._ We will write $\displaystyle z^{1}=z,\ \ z^{2}=w,$ and:
$\begin{array}[]{lll}\partial_{j}=\frac{\partial}{\partial
z^{j}},\quad\overline{\partial}_{j}=\frac{\partial}{\partial\overline{z^{j}}},\quad
j=1,2.\end{array}$
A complex valued function $\displaystyle\xi$ such that:
$\begin{array}[]{lll}\overline{\partial}_{j}\xi-({\mathfrak{a}}_{j}^{1}\partial_{1}\xi+{\mathfrak{a}}_{j}^{2}\partial_{2}\xi)=0,\quad
j=1,2\end{array}$ (4.19)
is called (cf. [7]) holomorphic with respect to the given almost complex
structure. Instead of considering the new coordinates (4.14) as solutions to
(4.19) and functions of $\displaystyle(z,w)$ and their complex conjugates, the
coordinates $\displaystyle(z,w)$ are supposed to be functions of (4.14) and
their complex conjugates. Inasmuch as it suffices to study only the real
character of the chart
$\displaystyle\widetilde{\Lambda}_{\operatorname{\varepsilon}},$ the tildes on
the chart
$\displaystyle(\widetilde{\xi}_{\operatorname{\varepsilon}}^{1},\widetilde{\xi}_{\operatorname{\varepsilon}}^{2})$
are dropped from now on.
_Notation._ The holomorphic and antiholomorphic dual differentials are:
$\begin{array}[]{lll}d_{j,\operatorname{\varepsilon}}=\frac{\partial}{\partial\xi_{\operatorname{\varepsilon}}^{j}},\quad\overline{d}_{j,\operatorname{\varepsilon}}=\frac{\partial}{\partial\overline{\xi_{\operatorname{\varepsilon}}^{j}}},\quad
j=1,2.\end{array}$ (4.20)
It is known (cf. [8], pp. 445) that for every
$\displaystyle\operatorname{\varepsilon}\in V,$ the map $\displaystyle
G_{\operatorname{\varepsilon}}$ from $\displaystyle{\bf D}_{s}\times{\bf
D}_{s}\subset{\bf C}^{2}$ to the almost complex manifold
$\displaystyle\widetilde{{\mathcal{M}}}$ is holomorphic if and only if its
coordinates $\displaystyle(z,w)=G_{\operatorname{\varepsilon}}^{*}(x,y)$
satisfy the differential equations
$\begin{array}[]{lll}\overline{d}_{j,\operatorname{\varepsilon}}z^{k}+{\mathfrak{a}}_{m,\operatorname{\varepsilon}}^{k}\overline{d}_{j,\operatorname{\varepsilon}}\overline{z}^{m}=0,\quad
j,k=1,2.\end{array}$ (4.21)
In such a case, (4.19) yields:
$\begin{array}[]{lll}\overline{d}_{j,\operatorname{\varepsilon}}\xi_{\operatorname{\varepsilon}}^{p}&=&\partial_{k}\xi_{\operatorname{\varepsilon}}^{p}\overline{d}_{j,\operatorname{\varepsilon}}z^{k}+\overline{\partial}_{k}\xi_{\operatorname{\varepsilon}}^{p}\overline{d}_{j,\operatorname{\varepsilon}}\overline{z}^{k}\\\
&=&\partial_{k}\xi_{\operatorname{\varepsilon}}^{p}\\{\overline{d}_{j,\operatorname{\varepsilon}}z^{k}+{\mathfrak{a}}_{m,\operatorname{\varepsilon}}^{k}\overline{d}_{j,\operatorname{\varepsilon}}\overline{z}^{m}\\}\\\
&=&0\end{array}$ (4.22)
for $\displaystyle j=1,2.$ Notice that the replacement of (4.21) in the term
after the first equality of (4.22), yields:
$\begin{array}[]{lll}\overline{d}_{j,\operatorname{\varepsilon}}\xi_{\operatorname{\varepsilon}}^{p}=\overline{d}_{j,\operatorname{\varepsilon}}\overline{z}^{k}\\{\overline{\partial}_{k}\xi_{\operatorname{\varepsilon}}^{p}-{\mathfrak{a}}_{k,\operatorname{\varepsilon}}^{i}\partial_{i}\xi_{\operatorname{\varepsilon}}^{p}\\},\quad
p=1,2.\end{array}$
Thus the parametric Cauchy-Riemann equations
$\displaystyle\overline{d}_{j,\operatorname{\varepsilon}}\xi_{\operatorname{\varepsilon}}^{p}=0$
are equivalent to the system (4.19) if $\displaystyle z,w$ satisfy (4.21) with
the matrix
$\displaystyle[\overline{d}_{j,\operatorname{\varepsilon}}\overline{z}^{k}]$
non-singular for all $\displaystyle\operatorname{\varepsilon}$ in the
symmetric neighborhood $\displaystyle V.$ We find real solutions to (4.21).
Denote by $\displaystyle T^{1},T^{2}$ the integral operators
$\begin{array}[]{lll}T^{1}f(z,w)=\frac{1}{2i\pi}\iint_{|\tau|<\rho}\frac{f(\tau,w)}{z-\tau}d\overline{\tau}d\tau,\\\
T^{2}f(z,w)=\frac{1}{2i\pi}\iint_{|\tau|<\rho}\frac{f(z,\tau)}{w-\tau}d\overline{\tau}d\tau,\\\
\end{array}$ (4.23)
with $\displaystyle\rho>0$ fixed and $\displaystyle f=f(z,w)$ has suitable
differentiability properties and, eventually, depends on additional complex
coordinates. A short calculation shows that if $\displaystyle f_{1},f_{2}$ are
as above and $\displaystyle
f_{1}(z,w)=\overline{f_{2}(\overline{w},\overline{z})},$ then
$\begin{array}[]{lll}T^{1}f_{1}(z,w)=\overline{T^{2}f_{2}(\overline{w},\overline{z})}.\end{array}$
(4.24)
The non-linear differential system corresponding to (4.21) is given by the
integral equation (cf. [8]):
$\begin{array}[]{lll}z^{k}(\xi_{\operatorname{\varepsilon}}^{1},\xi_{\operatorname{\varepsilon}}^{2})&=&\xi_{\operatorname{\varepsilon}}^{k}+\mathbf{TF}^{k}[z,w](\xi_{\operatorname{\varepsilon}}^{1},\xi_{\operatorname{\varepsilon}}^{2})-\mathbf{TF}^{k}[z,w](0,0),\
\ k=1,2\end{array}$ (4.25)
where
$\begin{array}[]{lll}\mathbf{TF}^{k}:=T^{1}f_{1k}+T^{2}f_{2k}-\frac{1}{2}\left\\{T^{1}\overline{d}_{1,\operatorname{\varepsilon}}T^{2}f_{2k}+T^{2}\overline{d}_{2,\operatorname{\varepsilon}}T^{1}f_{1k}\right\\},\quad
k=1,2\\\ \end{array}$
are the Nijenhuis-Woolf operators, and
$\begin{array}[]{lll}f_{jk}(z,w)(\xi^{1},\xi^{2})(\operatorname{\varepsilon})=-({\mathfrak{a}}_{1,\operatorname{\varepsilon}}^{k}(z,w)\overline{d}_{j,\operatorname{\varepsilon}}\overline{z}+{\mathfrak{a}}_{2,\operatorname{\varepsilon}}^{k}(z,w)\overline{d}_{j,\operatorname{\varepsilon}}\overline{w}),\
\ i,j\in\\{1,2\\}.\end{array}$
###### Corollary 4.6.
For every $\displaystyle(z,w)$ in a neighborhood of the origin and for every
initial value
$\displaystyle(\xi_{\operatorname{\varepsilon}}^{1},\xi_{\operatorname{\varepsilon}}^{2}),$
the Nijenhuis-Woolf operators are related through:
$\begin{array}[]{lll}\mathbf{TF}^{1}[z,w](\xi^{1},\xi^{2})(\operatorname{\varepsilon})=\overline{\mathbf{TF}^{2}[\overline{w},\overline{z}](\overline{\xi^{2}},\overline{\xi^{1}})(\operatorname{\varepsilon})}\end{array}$
(4.26)
when the parameter is real.
###### Proof.
This is plain consequence of (4.18), the definition of the dual differentials
(4.20) and property (4.24) on the maps $\displaystyle f_{jk}.$ ∎
The pair of coordinates
$\displaystyle(\xi_{\operatorname{\varepsilon}}^{1},\xi_{\operatorname{\varepsilon}}^{2})$
is referred to as the initial value of (4.25). For
$\displaystyle\operatorname{\varepsilon}=0,$ the system (4.25) is solved by
means of a Picard iteration process (fixed point Theorem) which converges in a
small ball $\displaystyle{\mathbf{B}}(r_{0})$ of radius $\displaystyle
r_{0}>0$ around the origin of $\displaystyle(z,w)$ coordinates (cf. [7],[8]).
It turns out that for all $\displaystyle|\operatorname{\varepsilon}|$ small
and fixed, any solution to (4.25) is well defined on
$\displaystyle{\mathbf{B}}(r),$ with $\displaystyle r=r_{0}/2.$ Moreover, if
$\displaystyle r$ is small enough, then the solution $\displaystyle(z,w)$ is
unique:
###### Lemma 4.7.
[7] For $\displaystyle r$ sufficiently small, and
$\displaystyle\operatorname{\varepsilon}\in V$ fixed, the integral system
(4.25) admits a unique solution $\displaystyle(z,w)$ satisfying also (4.21)
and such that the parametric transformation
$\displaystyle\widetilde{\Lambda}_{\operatorname{\varepsilon}}^{\circ-1}$ from
the
$\displaystyle(\xi_{\operatorname{\varepsilon}}^{1},\xi_{\operatorname{\varepsilon}}^{2})$
coordinates to $\displaystyle(z,w)$ coordinates has non-vanishing Jacobian.
###### Proposition 4.8.
The chart (4.14) respects the real foliation and is tangent to the identity.
###### Proof.
Let
$\displaystyle(\xi_{\operatorname{\varepsilon}}^{1},\xi_{\operatorname{\varepsilon}}^{2})$
be the initial value and $\displaystyle(z,w)$ be the solution to (4.25). If
the initial condition satisfies
$\displaystyle\xi_{\operatorname{\varepsilon}}^{1}=\overline{\xi_{\operatorname{\varepsilon}}^{2}},$
then Corollary 4.6 leads to
$\begin{array}[]{lll}\overline{z^{k}}&=&\xi_{\operatorname{\varepsilon}}^{k}+\mathbf{TF}^{k}[\overline{w},\overline{z}](\xi_{\operatorname{\varepsilon}}^{1},\xi_{\operatorname{\varepsilon}}^{2})-\mathbf{TF}^{k}[\overline{w},\overline{z}](0,0),\
\ k=1,2\end{array}$
and the unicity of the solution carries $\displaystyle z=\overline{w}.$ Since
$\displaystyle\widetilde{\Lambda}_{\operatorname{\varepsilon}}$ has non-
vanishing Jacobian, it is a local isomorphism if $\displaystyle r>0$ is small.
By (4.25), the chart
$\displaystyle\widetilde{\Lambda}_{\operatorname{\varepsilon}}$ is tangent to
the identity at the origin. ∎
Inasmuch as
$\displaystyle\Gamma_{\operatorname{\varepsilon}},H_{\operatorname{\varepsilon}}$
and (4.14) respect the real foliation, the chart
$\displaystyle\Lambda_{\operatorname{\varepsilon}}$ is real when
$\displaystyle\operatorname{\varepsilon}$ is real, and is clearly tangent to
the identity at the origin. This concludes the proof of Theorem 4.5. ∎
_End of the proof of Theorem 3.3._ The composition
$\displaystyle\vartheta_{\operatorname{\varepsilon}}=\Lambda_{\operatorname{\varepsilon}}\circ
H_{\operatorname{\varepsilon}}:H_{\operatorname{\varepsilon}}^{-1}({\mathbf{B}}(r))\to{\bf
C}^{2}$ between complex analytic manifolds is honestly biholomorphic. The
closure
$\displaystyle{\mathcal{W}}:=\overline{\vartheta_{\operatorname{\varepsilon}}(H_{\operatorname{\varepsilon}}^{-1}({\mathbf{B}}(r)))}$
contains the origin in its interior. It remains to check that the family of
vector fields defined on $\displaystyle{\mathcal{W}}$ by the pushforward
$\displaystyle{\mathbf{v}}_{\operatorname{\varepsilon}}=(\vartheta_{\operatorname{\varepsilon}})_{*}v_{\operatorname{\varepsilon}}$
is orbitally equivalent to a generic family unfolding a weak focus with formal
normal form (3.2).
###### Proposition 4.9.
The quotient of the eigenvalues of
$\displaystyle{\mathbf{v}}_{\operatorname{\varepsilon}}$ is equal to
$\displaystyle\frac{\operatorname{\varepsilon}+i}{\operatorname{\varepsilon}-i}.$
###### Proof.
By Theorem 4.5, the components
$\displaystyle{\mathbf{P}}_{\operatorname{\varepsilon}},{\bf
Q}_{\operatorname{\varepsilon}}$ of
$\displaystyle{\mathbf{v}}_{\operatorname{\varepsilon}}({\mathbf{x}},{\mathbf{y}})={\mathbf{P}}_{\operatorname{\varepsilon}}({\mathbf{x}},{\mathbf{y}})\frac{\partial}{\partial{\mathbf{x}}}+{\bf
Q}_{\operatorname{\varepsilon}}({\mathbf{x}},{\mathbf{y}})\frac{\partial}{\partial{\mathbf{y}}}$
are related through (1.4) as well, and then the eigenvalues of the vector
field $\displaystyle{\mathbf{v}}_{\operatorname{\varepsilon}}$ are complex
conjugate. We call them
$\displaystyle\tau(\operatorname{\varepsilon}),\overline{\tau(\operatorname{\varepsilon})},$
with
$\displaystyle\tau(\operatorname{\varepsilon})=a(\operatorname{\varepsilon})+ib(\operatorname{\varepsilon})$
and $\displaystyle
a(\operatorname{\varepsilon}),b(\operatorname{\varepsilon})$ depend
analytically on $\displaystyle\operatorname{\varepsilon}$ small and are real
on $\displaystyle\operatorname{\varepsilon}\in{\bf R}.$ In the
$\displaystyle({\mathbf{X}},{\mathbf{y}})$ chart of the blow-up,
$\displaystyle{\mathbf{v}}_{\operatorname{\varepsilon}}$ gives rise to a
family of equations of the form:
$\begin{array}[]{lll}\dot{\mathbf{X}}&=&(\tau-\overline{\tau}){\mathbf{X}}+...\\\
\dot{\mathbf{y}}&=&\overline{\tau}{\mathbf{y}}+...\end{array}$
with Poincaré map (cf. [6])
$\displaystyle{\mathcal{P}}_{\operatorname{\varepsilon}}({\mathbf{y}})=\exp\left(2i\pi\left(\frac{2\overline{\tau}}{\tau-\overline{\tau}}\right)\right){\mathbf{y}}+...,$
while in $\displaystyle({\mathbf{x}},{\mathbf{Y}})$ coordinates,
$\displaystyle{\mathbf{v}}_{\operatorname{\varepsilon}}$ gives rise to the
system:
$\begin{array}[]{lll}\dot{\mathbf{x}}&=&\tau{\mathbf{x}}+...\\\
\dot{\mathbf{Y}}&=&(\overline{\tau}-\tau){\mathbf{Y}}+...\end{array}$
with Poincaré map
$\displaystyle{\mathcal{P}}_{\operatorname{\varepsilon}}({\mathbf{x}})=\exp\left(-2i\pi\left(\frac{2\tau}{\overline{\tau}-\tau}\right)\right){\mathbf{x}}+...$
(computed on the cross section $\displaystyle{\mathbf{x}}={\mathbf{y}}).$ It
is easily seen that:
$\displaystyle\mu(\operatorname{\varepsilon})=\exp\left(2i\pi\left(\frac{2\overline{\tau}}{\tau-\overline{\tau}}\right)\right)=\exp\left(-2i\pi\left(\frac{2\tau}{\overline{\tau}-\tau}\right)\right)=\exp\left(2i\pi\left(\frac{\tau+\overline{\tau}}{\tau-\overline{\tau}}\right)\right),$
where
$\displaystyle{\mathcal{P}}_{\operatorname{\varepsilon}}^{\prime}(0)=\mu(\operatorname{\varepsilon}).$
On the other hand,
$\displaystyle\mu(\operatorname{\varepsilon})=\exp(2\pi\operatorname{\varepsilon})$
by the preparation (3.1). Thus,
$\displaystyle
2\pi\operatorname{\varepsilon}=2\pi\frac{2a(\operatorname{\varepsilon})}{2b(\operatorname{\varepsilon})}+2i\pi
m,$
for some $\displaystyle m\in{\bf N}.$ This means that
$\begin{array}[]{lll}\frac{a(\operatorname{\varepsilon})}{b(\operatorname{\varepsilon})}=\operatorname{\varepsilon}-im.\end{array}$
(4.27)
Inasmuch as $\displaystyle
a(\operatorname{\varepsilon}),b(\operatorname{\varepsilon})$ are real on
$\displaystyle\operatorname{\varepsilon}\in{\bf R},$ the equation (4.27)
implies that $\displaystyle m=0,$ and the conclusion follows. ∎
## 5\. Acknowledgements.
I am grateful to Christiane Rousseau for suggesting the problem and
supervising this work, and to Colin Christopher and Sergei Yakovenko for
helpful discussions. I am indebted to an unknown referee as well, for
suggesting many important comments concerning the previous version of the
paper.
## References
* [1] W. Arriagada. Characterization of the unfolding of a weak focus and modulus of analytic classification. PhD thesis, Université de Montréal, (2010).
* [2] M. Berthier, D. Cerveau, and A. Lins Neto. Sur les feuilletages analytiques réels et le problème du centre. J. Differential Equations, 131, No. 2: 244–266, (1996).
* [3] Y. Ilyashenko. In the theory of normal forms of analytic differential equations, divergence is the rule and convergence the exception when the bryuno conditions are violated. Moscow University Mathematics Bulletin, 36: 11–18, (1981).
* [4] Y. Il’yashenko. Nonlinear Stokes phenomena, volume 14. Advances in Soviet mathematics, Amer. Math. Soc., Providence RI, (1993).
* [5] Y. Il’yashenko and S. Yakovenko. Lectures on analytic differential equations, volume 86. Graduate Studies in Mathematics, Amer. Math. Soc.,Providence RI, (2008).
* [6] J.F.-Mattei and R. Moussu. Holonomie et intégrales premières. Ann. Scient. Éc. Norm. Sup., 4e série, 13: 469–523, (1980).
* [7] A. Newlander and L. Nirenberg. Complex analytic coordinates in almost complex manifolds. Annals of Mathematics, 65, No. 3, (May 1957).
* [8] A. Nijenhuis and W. Woolf. Some integration problems in almost-complex and complex manifolds. Annals of Mathematics, 77, No. 3, (May 1963).
* [9] R. Pérez-Marco and J.-C. Yoccoz. Germes de feuilletages holomorphes à holonomie prescrite. S.M.F., Astérisque, 222: 345–371, (1994).
* [10] J. Martinet-J.P. Ramis. Classification analytique des équations différentielles non linéaires résonnantes du premier ordre. Ann. Scient. Éc. Norm. Sup., 4e série, 16: 571–621, (1983).
* [11] C. Rousseau. Normal forms for germs of analytic families of planar vector fields unfolding a generic saddle-node or resonant saddle. Nonlinear dynamics and evolution equations, of Fields Inst. Commun., 48: 227–245, (2006).
* [12] C. Rousseau. The moduli space of germs of generic families of analytic diffeomorphisms unfolding of a codimension one resonant diffeomorphism or resonant saddle. J. Differential Equations, 248: 1794–1825, (2010).
* [13] C. Rousseau and C. Christopher. Modulus of analytical classification for the generic unfolding of acodimension one resonant diffeomorphism or resonant saddle. Annales de l’Institut Fourier, 57: 301–360, (2007).
|
arxiv-papers
| 2010-04-01T19:41:55 |
2024-09-04T02:49:09.417072
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Waldo Arriagada-Silva",
"submitter": "Waldo Arriagada",
"url": "https://arxiv.org/abs/1004.0219"
}
|
1004.0313
|
# A hybrid decision approach for the association problem in heterogeneous
networks
Salah Eddine Elayoubi Orange Labs
38-40 Rue du General Leclerc
92130 Issy-Les-Moulineaux, France
salaheddine.elayoubi@orange-ftgroup.com Eitan Altman INRIA Sophia Antipolis
10 route des Lucioles
06902 Sophia Antipolis, France
Eitan.Altman@sophia.inria.fr Majed Haddad, Zwi Altman Orange Labs
38-40 Rue du General Leclerc
92130 Issy-Les-Moulineaux, France
{majed.haddad,zwi.altman}@orange-ftgroup.com
###### Abstract
The area of networking games has had a growing impact on wireless networks.
This reflects the recognition in the important scaling advantages that the
service providers can benefit from by increasing the autonomy of mobiles in
decision making. This may however result in inefficiencies that are inherent
to equilibria in non-cooperative games. Due to the concern for efficiency,
centralized protocols keep being considered and compared to decentralized
ones. From the point of view of the network architecture, this implies the co-
existence of network-centric and terminal centric radio resource management
schemes. Instead of taking part within the debate among the supporters of each
solution, we propose in this paper hybrid schemes where the wireless users are
assisted in their decisions by the network that broadcasts aggregated load
information. We derive the utilities related to the Quality of Service (QoS)
perceived by the users and develop a Bayesian framework to obtain the
equilibria. Numerical results illustrate the advantages of using our hybrid
game framework in an association problem in a network composed of HSDPA and 3G
LTE systems.
## I Introduction
In order to handle the growing wireless traffic demand, operators are often
faced with the need to install new base stations. This could result in
splitting cells into smaller ones, or in having several base stations covering
the same cell. The second option may be preferred when the traffic has high
variability (in time and space) in which case it may be advantageous to have
the possibility to allocate resources from both base stations to any point in
the cell. This flexibility comes at a cost of having to include an access
control that takes the proper association decisions for the mobiles, that of
deciding to which base station (BS) to connect. To achieve efficient use of
the resources, these decisions should be based not only on the current system
state but also on expected future demand which may interact with traffic
assigned in the present.
We wish to avoid completely decentralized solutions of the association problem
in which all decisions are taken by the mobiles, due to well known
inefficiency problems that may arise when each mobile is allowed to optimize
its own utility. This inefficiency is inherent to the non-cooperative nature
of the decision making. On the other hand, we wish to delegate to the mobiles
a large part in the decision making in order to alleviate the burden from the
base stations.
The association schemes actually implemented are fully centralized: the
operator tries to maximize his utility (revenue) by assigning the users to the
different systems [1]-[3]. However, distributed RRM mechanisms are gaining in
importance: Users may be allowed to make autonomous decisions in a distributed
way. This has lead to game theoretic approaches to the association problems in
wireless networks, as can be found in [4]-[8]. The potential inefficiency of
such approaches have been known for a long time. The term ”The Tragedy of the
Commons” has been frequently used for this inefficiency [9]; it describes a
dilemma in which multiple individuals acting independently in their own self-
interest can ultimately destroy a shared limited resource even when it is
clear that it is not in anyone’s long term interest for this to happen.
We propose in this paper association methods that combine benefits from both
decentralized and centralized design. Central intervention is needed during
severe congestion periods. At those instants, we assume that the mobiles
follow the instructions of the base stations. Otherwise the association
decision is left to the mobiles, who make the decision based on aggregated
state information from the base stations. The decision making is thus based on
partial information that is signaled to the mobiles by the base station. A
central design aspect is then for the base stations to decide how to aggregate
information which then determines what to signal to the users. Note that this
decision making at the BS can be viewed as a mechanism design problem, or as a
Bayesian game.
## II Problem statement
### II-A System description
We consider a network composed of $S$ systems operated by the same operator.
Even if the model we develop is applicable to different kinds of situations,
we will focus on the more realistic and cost effective case where the operator
uses the same cell sites to deploy the new system (e.g. 3G LTE), while keeping
the old ones (e.g. HSDPA). As, in each cell, there are different radio
conditions following the position of the user regarding the cell site, the
peak throughput that can be obtained by the user connected to system $s$, if
served alone by a cell, differs following his position in the cell, as
illustrated in Figure 1 for a cell served by HSDPA and 3G LTE. For simplicity,
we consider that there are $N$ classes of radio conditions and that users with
radio condition $n$ have a peak rate $D_{n}^{s}$ if connected to system $s$.
The network state is then defined by the vector: ${\bf
M}=(M_{1}^{1},...,M_{N}^{1},...,M_{1}^{S},...,M_{N}^{S})$, $M_{n}^{s}$ being
the number of users with radio condition $n$ connected to system $s$.
Figure 1: LTE and HSDPA peak throughputs for different user positions.
We assume that the network broadcasts a partial load information $l$ ($1\leq
l\leq L$), e.g., an aggregated load information indicating for each system if
it is in low, medium, or high load state. An example of this load information
is described in Figure 2 for a network composed of HSDPA and LTE systems.
Figure 2: Aggregated load information.
### II-B Policy definition
As stated before, users are only aware of the load information $l$ sent by the
network. Their policies are then based on this information. Let ${\bf P}$ be a
policy defined by the actions taken by mobiles in the different load
conditions. ${\bf P}$ is a $N\times L$ matrix whose element $P_{nl}$ is equal
to $s$ if class-$n$ users connect to system $s$ when the network broadcasts
information $l$.
Let $\mathcal{A}$ be the space of feasible states, $\mathfrak{P}$ be the set
of all possible policies and let $\mathfrak{L}$ be the set of load
information. An assignment $f:\displaystyle\mathcal{A}\rightarrow\mathfrak{L}$
specifies for each network state ${\bf M}$ the corresponding load information
$f({\bf M})$. On the other hand, when the load information is equal to $l$ and
the policy is ${\bf P}$, we can determine the system to which users of class
$n$ will connect by the value $P_{nl}$. As an example, knowing the function
$f(.)$ and the policy ${\bf P}$, if the network is in state ${\bf M}$, a class
$n$ user will connect to system $P_{nl}$, where $l=f({\bf M})$.
There are some important remarks to keep in mind when speaking about policies.
The first is that we suppose that a user connected to a system stays within it
until the end of his communication in order to avoid vertical handovers and
their signaling overhead. Furthermore, even if the decision is distributed,
all users will have the same policy and learn together how to enhance it.
However, a policy change will occur after an observation time, long enough to
insure that the steady state of the network has been reached. Note also that
users can connect to a system only if there is room in it, otherwise they are
directed by the network to an available system or blocked if all systems are
saturated.
## III Utilities
We analyze a system offering streaming calls. The goal of a streaming user is
to achieve the best throughput, knowing that the different codecs allow a
throughput between an upper (best) $T_{max}$ and a lower (minimal) $T_{min}$
bounds. His utility is thus expressed by the quality of the streaming flow he
receives, which is in turn closely related to his throughput. Indeed, a
streaming call with a higher throughput will use a better codec offering a
better video quality. This throughput depends not only on the peak throughput,
but also on the evolution of the number of calls in the system where the user
decides to connect. Note that a user that cannot be offered this minimal
throughput in neither of the available systems is blocked in order to preserve
the overall network performance.
### III-A Steady state analysis
#### III-A1 Instantaneous throughput
The instantaneous throughput obtained by a user in a system depends on the
state of the system. The throughput of a user with radio condition class $n$
connected to system $s$ is given by:
$t_{n}^{s}({\bf M})=\min\left[D_{n}^{s}\frac{G({\bf
M})}{\sum_{m=1}^{N}\sum_{r=1}^{S}M_{m}^{r}},T_{max}\right]$ (1)
where $G({\bf M})$ is the scheduler gain. Note here that the admission control
will insure that $t_{n}^{s}({\bf M})\geq T_{min}$ by blocking new arrivals.
The space of feasible states $\mathcal{A}$ is thus the set of all states ${\bf
M}$ where this constraint is ensured:
$\frac{\sum_{m=1}^{N}\sum_{r=1}^{S}M_{m}^{r}}{G({\bf
M})}\leq\frac{T_{min}}{D_{n}^{s}},\forall n,s|M_{n}^{s}>0$ (2)
#### III-A2 Steady state probabilities
The throughput achieved by a user depends on the number of ongoing calls. This
latter is a random variable whose evolution is governed by the arrival and
departure processes. We assume that the arrival process of new connections
with radio condition $n$ is Poisson with rate $\lambda_{n}$. Each arriving
user makes a streaming connection whose duration is exponentially distributed
with parameter $1/\mu$.
Within the space of feasible states $\mathcal{A}$, transitions are due to:
* •
Arrivals of users of radio condition $n$. Let ${\cal G}_{n}^{s}({\bf M})$
denote the state of the system if we add one mobile of radio conditions $n$ to
system $s$: ${\cal G}_{n}^{H}({\bf
M})=(M_{1}^{1},...,M_{N}^{1},...,M_{1}^{s},...,M_{n}^{s}+1,...,M_{N}^{s},...,M_{1}^{S},...,M_{N}^{S})$.
The transition from state ${\bf M}$ to ${\cal G}_{n}^{s}({\bf M})$ happens if
the policy implies that system $s$ is to be chosen for the load information
corresponding to state ${\bf M}$, and if the state ${\cal G}_{n}^{s}({\bf M})$
is an admissible state. The corresponding transition rate is thus equal to:
$q({\bf M},{\cal G}_{n}^{s}({\bf M})|{\bf P})=\lambda_{n}\cdot I_{P_{n,f({\bf
M})}=s}\cdot I_{{\cal G}_{n}^{s}({\bf M})\in\mathcal{A}}$ (3)
where $I_{C}$ is the indicator function equal to 1 if condition $C$ is
satisfied and to 0 otherwise.
* •
Departures of users of radio condition $n$. Let ${\cal D}_{n}^{s}({\bf M})$
denote the state with one less mobile of class $(n,s)$. The transition from
state ${\bf M}$ to ${\cal D}_{n}^{s}({\bf M})$ is equal to:
$q({\bf M},{\cal D}_{n}^{s}({\bf M})|{\bf P})=M_{n}^{s}\cdot\mu\cdot
I_{M_{n}^{s}>0}$ (4)
The transition matrix $\textbf{Q}({\bf P})$ of the Markov process is written
for each policy ${\bf P}$ knowing that its diagonal element is:
$q({\bf M},{\bf M}|{\bf P})=-\sum_{n=1}^{N}\sum_{s=1}^{S}(q({\bf M},{\cal
D}_{n}^{s}({\bf M})|{\bf P})+q({\bf M},{\cal G}_{n}^{s}({\bf M})|{\bf P}))$
(5)
The steady-state distribution is then obtained by solving:
$\left\\{\begin{array}[]{l}{\bf\Pi}({\bf P})\cdot\textbf{Q}({\bf P})=0\\\
{\bf\Pi}({\bf P})\cdot{\bf e}=1;\end{array}\right.$ (6)
${\bf\Pi({\bf P})}$ being the vector of the steady-state probabilities
$\pi({\bf M}|{\bf P})$ under policy ${\bf P}$ and ${\bf e}$ is a vector of
ones.
Once the vector ${\bf\Pi}$ is obtained, the global performance indicators can
be calculated, e.g., the blocking rate of class-$n$ calls knowing that the
load information is equal to $l$:
$b_{n}(l|{\bf P})=\frac{\sum_{{\bf M}\in\mathcal{A};{\cal G}_{n}^{s}({\bf
M})\notin\mathcal{A},\forall s\in[1,S]}\pi({\bf M}|{\bf P})}{\sum_{{\bf
M}\in\mathcal{A};f({\bf M})=l}\pi({\bf M}|{\bf P})}$ (7)
In this equation, we consider as blocked all calls that arrive in states where
both systems are saturated, i.e., where $t_{n}^{s}({\bf M})<T_{min}$, $\forall
j\in{H,L}$. We also obtain the overall blocking rate:
$b({\bf
P})=\sum_{n=1}^{N}\frac{\lambda_{n}}{\sum_{m=1}^{N}\lambda_{m}}\sum_{{\bf
M}\in\mathcal{A};{\cal G}_{n}^{s}({\bf M})\notin\mathcal{A},\forall
s\in[1,S]}\pi({\bf M}|{\bf P})$ (8)
### III-B Transient analysis
The steady-state analysis described above is not sufficient to describe the
utility of the users as the throughput obtained by a user at his arrival is
not a sufficient indication about the quality of his communication because of
the dynamics of arrivals/departures. In order to obtain the utility, we modify
the Markov chain in order to allow tracking mobiles during their connection
time. For users of radio condition $n$ connected to system $s$, only states
where there is at least one user $(n,s)$ are considered. The calculation is as
follows:
1. 1.
Introduce absorbing states $A_{n}^{s}$ corresponding to the departure of
mobiles that have terminated their connections. Additional transitions are
thus added between ${\bf M}$ and $A_{n}^{s}$ with rate equal to:
$\tilde{q}_{n}^{s}({\bf M},A_{n}^{s})=\mu\cdot I_{M_{n}^{s}>0}$ (9)
The transitions to the neighboring states with one less user are then modified
accordingly by subtracting $\mu$ from the original transition rates defined in
equation (4):
$\tilde{q}_{n}^{s}({\bf M},{\cal D}_{n}^{s}({\bf M})|{\bf
P})=(M_{n}^{s}-1)\cdot\mu\cdot I_{M_{n}^{s}>0}$ (10)
The remaining transition rates remain equal to the original transitions:
$\tilde{q}_{n}^{s}({\bf M},{\cal G}_{n^{\prime}}^{s^{\prime}}({\bf M})|{\bf
P})=q({\bf M},{\cal G}_{n^{\prime}}^{s^{\prime}}({\bf M})|{\bf
P}),\quad\forall n^{\prime},s^{\prime}$
and
$\tilde{q}_{n}^{s}({\bf M},{\cal D}_{n^{\prime}}^{s^{\prime}}({\bf M})|{\bf
P})=q({\bf M},{\cal D}_{n^{\prime}}^{s^{\prime}}({\bf M})|{\bf
P}),\quad\forall n^{\prime},s^{\prime}\neq s$
2. 2.
Define matrix $\tilde{{\bf Q}}_{n}^{s}$ of elements $\tilde{q}_{n}^{s}({\bf
M},{\bf M}^{\prime})$ defined above and with diagonal elements as in equation
(5):
$\tilde{q}_{n}^{s}({\bf M},{\bf M}|{\bf P})=q({\bf M},{\bf M}|{\bf P})$
Under policy ${\bf P}$ , the volume of information $I_{n}^{s}({\bf M}|{\bf
P})$ sent by system $s$ users subject to radio conditions $n$ starting from
state ${\bf M}$ is then equal to the volume of information sent between state
${\bf M}$ and the absorbing state $A_{n}^{s}$. These values can be calculated
by solving the set of linear equations for all states ${\bf M}$:
$\sum\tilde{q}_{n}^{s}({\bf M},{\bf M}^{\prime}|{\bf P})I_{n}^{s}({\bf
M}^{\prime}|{\bf P})=-t_{n}^{s}({\bf M})$ (11)
knowing that $I_{n}^{s}(A_{n}^{s})=0$.
3. 3.
The utility of a class-$n$ user that has found the network in state ${\bf M}$
and chosen to connect to system $s$ is the volume of information sent starting
from state ${\cal G}_{n}^{s}({\bf M})$. Recall that ${\cal G}_{n}^{s}({\bf
M})$ is defined as the state with one more class-$n$ call connected to system
$s$:
$u_{n}^{s}({\bf M}|{\bf P})=I_{n}^{s}({\cal G}_{n}^{s}({\bf M})|{\bf P})$ (12)
## IV Optimality, game and control
In this section, we use the utilities of users that we obtained above to
derive the association policies. We first search for the optimal policy, i.e.
the policy that maximizes the global utility of the network. Nevertheless, as
it is not realistic to consider that the users will seek the global optimum,
we show how to find the policy that corresponds to the Nash equilibrium,
knowing that users will try to maximize their individual utility. We will next
show how the operator can control, by sending appropriate load information,
the equilibrium of its wireless users to maximize its own utility.
### IV-A Optimality
#### IV-A1 Global utility
When a global optimum is sought, it is important to maximize the QoS of all
users. The global utility function can be written as:
$\displaystyle U({\bf
P})=\sum_{n=1}^{N}\frac{\lambda_{n}}{\sum_{i=1}^{N}\lambda_{i}}\sum_{l\in\mathfrak{L}}[(1-b_{n}(l|{\bf
P}))\times$ (13) $\displaystyle\sum_{{\bf M}|f({\bf
M})=l}u_{n}^{(P_{n,l})}({\bf M}|{\bf P})\pi({\bf M}|{\bf P})]$
knowing that $P_{n,l}\in[1,S]$ is the system where new users of class-$n$
connect when they receive the load information $l$ and have the policy ${\bf
P}$.
Note that, in this utility, we consider not only the QoS of accepted users
(throughput), but also the blocking rate as the aim is also to maximize the
number of accepted users. We also weight the users with different radio
conditions with their relative arrival rates.
#### IV-A2 Optimal policy
Knowing the utility in equation (13), the optimal policy is the one among all
possible policies that maximizes this utility:
${\bf P}^{*}=\arg{\hbox{$\underset{{\bf P}}{\max}\,$}}U({\bf P})$ (14)
### IV-B Equilibrium
#### IV-B1 Individual utility
If the aim is to maximize the individual utility, users of different radio
conditions are interested by maximizing the QoS they obtain given the load
information broadcast by the network. The utility that a class $n$ user might
obtain if he chooses system $s$ when the load information is $l$, while all
other users follow policy ${\bf P}$ is then:
$U_{nl}^{s}({\bf P})=\frac{\sum_{{\bf M}|f({\bf M})=l}u_{n}^{s}({\bf M}|{\bf
P})\pi({\bf M}|{\bf P})}{\sum_{{\bf M}|f({\bf M})=l}\pi({\bf M}|{\bf P})}$
(15)
#### IV-B2 Nash equilibrium
A policy ${\bf P}^{*}$ corresponds to a Nash equilibrium if, for all radio
conditions and all load information, the individual utility obtained when
following ${\bf P}^{*}$ is the largest possible utility under ${\bf P}^{*}$.
Mathematically, this can be expressed by the following inequality for all
radio conditions $n\in[1,N]$ and all load information $l\in[1,L]$:
$U_{nl}^{(P^{*}_{n,l})}({\bf P}^{*})\geq U_{nl}^{s}({\bf P}^{*}),\forall
s\in[1,S]$ (16)
### IV-C Control
In the previous section, we derived the policy that corresponds to the Nash
equilibrium for a game where players are the wireless users that aim at
maximizing their utility. However, there is another dimension of the problem
related to the information sent by the network and corresponding to the
different load information. Motivated by the fact that the network may guide
users to an equilibrium that optimizes its own utility if he chooses the
adequate information to send, we introduce a control problem, that can also be
modeled as a game between the base station and its users. At the core lies the
idea that introducing a certain degree of hierarchy in non-cooperative games
not only improves the individual efficiency of all the users but can also be a
way of reaching a desired trade-off between the global network performance at
the equilibrium and the requested amount of signaling.
More formally, the way of aggregating the loads in the broadcast information
(expressed by the function $f(.)$) is inherent to the previous analysis. In
particular, the utilities of individual users, calculated in equation (15), is
function of $f(.)$:
$U_{nl}^{s}({\bf P}|f)=\frac{\sum_{{\bf M}|f({\bf M})=l}u_{n}^{s}({\bf M}|{\bf
P})\pi({\bf M}|{\bf P})}{\sum_{{\bf M}|f({\bf M})=l}\pi({\bf M}|{\bf P})}$
This leads the wireless users to a Nash equilibrium that depends on the way
the network aggregates the load information:
${\bf P}^{*}={\bf P}^{*}(f)$
The control problem is thus defined as the maximization of the utility of the
network by tuning the function $f(.)$. If the aim of the operator is to
maximize its revenues by maximizing the acceptance ratio, the optimal solution
is:
$f^{*}=\arg{\hbox{$\underset{f}{\max}\,$}}\frac{1}{b({\bf P}^{*}(f))}$ (17)
with blocking defined as in equation (8).
## V Results
For illustration, we consider the case of a network composed of HSDPA and 3G
LTE systems. Users are classified between users with good radio conditions (or
cell center users) and users with bad radio conditions (or cell edge users).
The network sends aggregated load information as shown in Figure 2 with the
following thresholds: $[H_{1}=0.3,H_{2}=0.7,L_{1}=0.3,L_{2}=0.7]$, meaning
that a system is considered as highly loaded if its load exceeds $0.7$ and as
low-loaded if its load is below $0.3$.
We also consider a streaming service where users require a minimal throughput
of 1 Mbps and can profit from throughputs up to 2 Mbps in order to enhance
video quality ($D_{min}=1Mbps$ and $D_{max}=2Mbps$). We consider an offered
traffic that varies from 1 to 10 Erlangs and obtain numerically the
equilibrium points.
### V-A Equilibrium
We focus on the Nash equilibrium when wireless users aim at maximizing their
individual utility. For comparison purposes, we study three different
association approaches:
* •
Hybrid decision approach: The proposed hybrid scheme where users receive
aggregated load information and aim at maximizing their individual utility. We
illustrate the global utility corresponding to the Nash equilibrium policy.
* •
Peak rate maximization approach: This is a simple association scheme where
users do not have any information about the load of the systems. They connect
to the system offering them the best peak rate:
$s^{*}=\arg{\hbox{$\underset{s}{\max}\,$}}D_{n}^{s}$
Note that this peak rate can be known by measuring the quality of the
receiving signal.
* •
Instantaneous rate maximization approach: The network broadcasts ${\bf M}$,
the exact numbers of connected users with different radio conditions. Based on
this information and on the measured signal strength, the wireless users
estimate the throughput they will obtain in both systems. A new user with
radio condition $n$ will then connect to the system $s^{*}$ offering him the
best throughput:
$s^{*}=\arg{\hbox{$\underset{s}{\max}\,$}}\frac{D_{n}^{s}}{1+\sum_{m=1}^{N}\sum_{r=1}^{S}M_{m}^{r}}$
Note that this scheme is not realistic as the network operator will not
divulge the exact number of connected users in each system and each position
of the cell.
We plot in Figure 3 the global utility for the three cases. This global
utility is the one defined in equation (13) and expressed in Mbits, as users
are interested in maximizing the information they send during their transfer
time.
As intuition would expect, the results show that the peak rate maximization
approach has the worst performance as a system that offers the largest peak
throughput may be highly-loaded, resulting in a bad QoS. However, a surprising
result is that the hybrid scheme, based on partial information, is comparable
and even outperforms the full information scheme when traffic increases. This
is due to the fact that streaming users will have relatively long sessions,
visiting thus a large number of network states; knowing the instantaneous
throughput at arrival will not bring complete information about the QoS during
the whole connection. On the contrary, the proposed hybrid, game theoretic,
approach aims at maximizing the QoS during the connection time.
Figure 3: Global utility.
### V-B Control
We now turn to the second stage of our problem, where the network tries to
control the users’ behavior by broadcasting appropriate information, expected
to maximize its utility while individual users maximize their own utility. We
plot in Figure 4 the blocking rate for different ways of aggregating load
information, obtained when users follow the policy corresponding to Nash
equilibrium. In this figure, we plot the results for three cases: the optimal
thresholds (in red stars) and two other sets of thresholds. We can observe
that the utility of the network (expressed in the acceptance rate) varies
significantly depending on the load information that is broadcast. Such an
accurate modeling of the control problem is a key to understand the actual
benefits brought by the proposed hybrid decision approach.
Figure 4: Blocking rate for different broadcast load information; a vector of
thresholds $[H_{1},H_{2},L_{1},L_{2}]$ means that system $s$ will be
considered as highly loaded if its load exceeds $s_{2}$ and as low-loaded if
its load is below $s_{1}$ ($s=H$ for HSDPA and $L$ for LTE).
## VI Conclusion
In this paper, we studied hybrid association schemes in heterogeneous
networks. By hybrid schemes we mean distributed decision schemes assisted by
the network, where the wireless users aim at maximizing their own utility,
guided by information broadcast by the network about the load of each system.
We first show how to derive the utilities of flows that are related to the QoS
they receive under the different association policies. We then derive the
policy that corresponds to the Nash equilibrium. Finally, we show how the
operator, by sending appropriate information about the state of the network,
can optimize its own utility. The proposed hybrid decision approach for the
association problem can reach a good trade-off between the global network
performance at the equilibrium and the requested amount of signaling.
## Acknowledgment
This work was supported by the ANR project WiNEM.
## References
* [1] E. Stevens-Navarro, Yuxia Lin, V. W. S. Wong, ”An MDP-Based Vertical Handoff Decision Algorithm for Heterogeneous Wireless Networks”, IEEE Transactions on In Vehicular Technology, , Vol. 57, No. 2. (2008), pp. 1243-1254.
* [2] D. Kumar, E. Altman, J.-M. Kelif, ”Globally Optimal User-Network Association in an 802.11 WLAN and 3G UMTS Hybrid Cell”, in: Proc. of 20th International Teletraffic Congress (ITC 20), Ottawa, Canada, Selected for Plenary Presentation, June 17-21, 2007.
* [3] S. Horrich, S-E. Elayoubi and S. Ben Jamaa, ”On the impact of mobility and joint RRM policies on a cooperative WiMAX/HSDPA network”, IEEE WCNC 2008, Las Vegas, April 2008.
* [4] Ercetin Ozgur, ”Association Games in IEEE 802.11 Wireless Local Area Networks”, IEEE transactions on wireless communications, 2008, vol. 7 (1), no12, pp. 5136-5143.
* [5] Srinivas Shakkottai, Eitan Altman, and Anurag Kumar, ”Multihoming of Users to Access Points in WLANs: A Population Game Perspective”, IEEE Journal on Selected Areas in Communications, Vol. 25, No. 6, 2007 , August 2007.
* [6] S. Shakkottai, E. Altman and A. Kumar, ”The Case for Non-cooperative Multihoming of Users to Access Points in IEEE 802.11 WLANs”, IEEE Infocom, 2006.
* [7] Libin Jiang, Shyam Parekh and Jean Walrand, ”Base Station Association Game in Multi-cell Wireless Networks”,
* [8] D. Kumar, E. Altman, J.-M. Kelif. User-Network Association in an 802.11 WLAN & 3G UMTS Hybrid Cell: Individual Optimality, in: Proc. of IEEE Sarnoff Symposium, Princeton, NJ, USA, April 30 - May 2, 2007\.
* [9] Garrett Hardin, ”The Tragedy of the Commons,” Science, 162(1968):1243-1248.
|
arxiv-papers
| 2010-04-02T10:39:06 |
2024-09-04T02:49:09.429643
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Salah Eddine Elayoubi, Eitan Altman, Majed Haddad, Zwi Altman",
"submitter": "Majed Haddad",
"url": "https://arxiv.org/abs/1004.0313"
}
|
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