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## Joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 10/09/2012

Relevância na Pesquisa

26.6%

#Mathematics - Classical Analysis and ODEs#Mathematical Physics#15A52, 33C45, 33E17, 42C05, 60K35, 62E15

The density function for the joint distribution of the first and second
eigenvalues at the soft edge of unitary ensembles is found in terms of a
Painlev\'e II transcendent and its associated isomonodromic system. As a
corollary, the density function for the spacing between these two eigenvalues
is similarly characterized.The particular solution of Painlev\'e II that arises
is a double shifted B\"acklund transformation of the Hasting-McLeod solution,
which applies in the case of the distribution of the largest eigenvalue at the
soft edge. Our deductions are made by employing the hard-to-soft edge
transitions to existing results for the joint distribution of the first and
second eigenvalue at the hard edge \cite{FW_2007}. In addition recursions under
$a \mapsto a+1$ of quantities specifying the latter are obtained. A Fredholm
determinant type characterisation is used to provide accurate numerics for the
distribution of the spacing between the two largest eigenvalues.; Comment: 26 pages, 1 Figure, 2 Tables

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## Nonpositive Eigenvalues of the Adjacency Matrix and Lower Bounds for Laplacian Eigenvalues

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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Let $NPO(k)$ be the smallest number $n$ such that the adjacency matrix of any
undirected graph with $n$ vertices or more has at least $k$ nonpositive
eigenvalues. We show that $NPO(k)$ is well-defined and prove that the values of
$NPO(k)$ for $k=1,2,3,4,5$ are $1,3,6,10,16$ respectively. In addition, we
prove that for all $k \geq 5$, $R(k,k+1) \ge NPO(k) > T_k$, in which $R(k,k+1)$
is the Ramsey number for $k$ and $k+1$, and $T_k$ is the $k^{th}$ triangular
number. This implies new lower bounds for eigenvalues of Laplacian matrices:
the $k$-th largest eigenvalue is bounded from below by the $NPO(k)$-th largest
degree, which generalizes some prior results.; Comment: 23 pages, 12 figures

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## Domain dependence of eigenvalues of elliptic type operators

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 09/03/2012

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#Mathematics - Spectral Theory#Mathematics - Analysis of PDEs#Mathematics - Optimization and Control#35P05, 47A75, 49R50, 47A55

The dependence on the domain is studied for the Dirichlet eigenvalues of an
elliptic operator considered in bounded domains. Their proximity is measured by
a norm of the difference of two orthogonal projectors corresponding to the
reference domain and the perturbed one; this allows to compare domains that
have non-smooth boundaries and different topology. The main result is an
asymptotic formula in which the remainder is evaluated in terms of this
quantity. As an application, the stability of eigenvalues is estimated by
virtue of integrals of squares of the gradients of eigenfunctions for elliptic
problems in different domains. It occurs that these stability estimates imply
well-known inequalities for perturbed eigenvalues.

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## Fine asymptotic behavior in eigenvalues of random normal matrices: Ellipse Case

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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#Mathematical Physics#Mathematics - Classical Analysis and ODEs#Mathematics - Probability#15A52 (Primary), 60B20, 33C45, 42C05 (Secondary)

We consider the random normal matrices with quadratic external potentials
where the associated orthogonal polynomials are Hermite polynomials and the
limiting support (called droplet) of the eigenvalues is an ellipse. We
calculate the density of the eigenvalues near the boundary of the droplet up to
the second subleading corrections and express the subleading corrections in
terms of the curvature of the droplet boundary. From this result we
additionally get the expected number of eigenvalues outside the droplet. We
also obtain the asymptotics of the kernel and found that, in the bulk, the
correction term is exponentially small. This leads to the vanishing of certain
Cauchy transform of the orthogonal polynomial in the bulk of the droplet up to
an exponentially small error.; Comment: 39 pages, 5 figures. Extended version: Theorem 1.2, Theorem 1.4,
Section 6 and Section 7.3 are new

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## Random matrices: Localization of the eigenvalues and the necessity of four moments

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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Consider the eigenvalues $\lambda_i(M_n)$ (in increasing order) of a random
Hermitian matrix $M_n$ whose upper-triangular entries are independent with mean
zero and variance one, and are exponentially decaying. By Wigner's semicircular
law, one expects that $\lambda_i(M_n)$ concentrates around $\gamma_i \sqrt n$,
where $\int_{-\infty}^{\gamma_i} \rho_{sc} (x) dx = \frac{i}{n}$ and
$\rho_{sc}$ is the semicircular function.
In this paper, we show that if the entries have vanishing third moment, then
for all $1\le i \le n$
$$\E |\lambda_i(M_n)-\sqrt{n} \gamma_i|^2 = O(\min(n^{-c}
\min(i,n+1-i)^{-2/3} n^{2/3}, n^{1/3+\eps})) ,$$ for some absolute constant
$c>0$ and any absolute constant $\eps>0$. In particular, for the eigenvalues in
the bulk ($\min \{i, n-i\}=\Theta (n)$), $$\E |\lambda_i(M_n)-\sqrt{n}
\gamma_i|^2 = O(n^{-c}). $$
\noindent A similar result is achieved for the rate of convergence.
As a corollary, we show that the four moment condition in the Four Moment
Theorem is necessary, in the sense that if one allows the fourth moment to
change (while keeping the first three moments fixed), then the \emph{mean} of
$\lambda_i(M_n)$ changes by an amount comparable to $n^{-1/2}$ on the average.
We make a precise conjecture about how the expectation of the eigenvalues vary
with the fourth moment.; Comment: 19 pages...

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## All Real Eigenvalues of Symmetric Tensors

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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This paper studies how to compute all real eigenvalues of a symmetric tensor.
As is well known, the largest or smallest eigenvalue can be found by solving a
polynomial optimization problem, while the other middle eigenvalues can not. We
propose a new approach for computing all real eigenvalues sequentially, from
the largest to the smallest. It uses Jacobian SDP relaxations in polynomial
optimization. We show that each eigenvalue can be computed by solving a finite
hierarchy of semidefinite relaxations. Numerical experiments are presented to
show how to do this.

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## An Upper Bound to the Marginal PDF of the Ordered Eigenvalues of Wishart Matrices

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 30/12/2011

Relevância na Pesquisa

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Diversity analysis of a number of Multiple-Input Multiple-Output (MIMO)
applications requires the calculation of the expectation of a function whose
variables are the ordered multiple eigenvalues of a Wishart matrix. In order to
carry out this calculation, we need the marginal pdf of an arbitrary subset of
the ordered eigenvalues. In this letter, we derive an upper bound to the
marginal pdf of the eigenvalues. The derivation is based on the multiple
integration of the well-known joint pdf, which is very complicated due to the
exponential factors of the joint pdf. We suggest an alternative function that
provides simpler calculation of the multiple integration. As a result, the
marginal pdf is shown to be bounded by a multivariate polynomial with a given
degree. After a standard bounding procedure in a Pairwise Error Probability
(PEP) analysis, by applying the marginal pdf to the calculation of the
expectation, the diversity order for a number of MIMO systems can be obtained
in a simple manner. Simulation results that support the analysis are presented.; Comment: 6 pages, 2 figures

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## Finite-temperature chiral condensate and low-lying Dirac eigenvalues in quenched SU(2) lattice gauge theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 18/09/2008

Relevância na Pesquisa

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The spectrum of low-lying eigenvalues of overlap Dirac operator in quenched
SU(2) lattice gauge theory with tadpole-improved Symanzik action is studied at
finite temperatures in the vicinity of the confinement-deconfinement phase
transition defined by the expectation value of the Polyakov line. The value of
the chiral condensate obtained from the Banks-Casher relation is found to drop
down rapidly at T = Tc, though not going to zero. At Tc' = 1.5 Tc = 480 MeV the
chiral condensate decreases rapidly one again and becomes either very small or
zero. At T < Tc the distributions of small eigenvalues are universal and are
well described by chiral orthogonal ensemble of random matrices. In the
temperature range above Tc where both the chiral condensate and the expectation
value of the Polyakov line are nonzero the distributions of small eigenvalues
are not universal. Here the eigenvalue spectrum is better described by a
phenomenological model of dilute instanton - anti-instanton gas.; Comment: 8 pages RevTeX, 5 figures, 2 tables

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## General Relativity in terms of Dirac Eigenvalues

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 13/12/1996

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26.6%

The eigenvalues of the Dirac operator on a curved spacetime are
diffeomorphism-invariant functions of the geometry. They form an infinite set
of ``observables'' for general relativity. Recent work of Chamseddine and
Connes suggests that they can be taken as variables for an invariant
description of the gravitational field's dynamics. We compute the Poisson
brackets of these eigenvalues and find them in terms of the energy-momentum of
the eigenspinors and the propagator of the linearized Einstein equations. We
show that the eigenspinors' energy-momentum is the Jacobian matrix of the
change of coordinates from metric to eigenvalues. We also consider a minor
modification of the spectral action, which eliminates the disturbing huge
cosmological term and derive its equations of motion. These are satisfied if
the energy momentum of the trans Planckian eigenspinors scale linearly with the
eigenvalue; we argue that this requirement approximates the Einstein equations.; Comment: 6 pages, RevTex

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## Numerical Investigation of Graph Spectra and Information Interpretability of Eigenvalues

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 24/01/2015

Relevância na Pesquisa

26.6%

#Computer Science - Information Theory#Mathematics - Dynamical Systems#Mathematics - Spectral Theory

We undertake an extensive numerical investigation of the graph spectra of
thousands regular graphs, a set of random Erd\"os-R\'enyi graphs, the two most
popular types of complex networks and an evolving genetic network by using
novel conceptual and experimental tools. Our objective in so doing is to
contribute to an understanding of the meaning of the Eigenvalues of a graph
relative to its topological and information-theoretic properties. We introduce
a technique for identifying the most informative Eigenvalues of evolving
networks by comparing graph spectra behavior to their algorithmic complexity.
We suggest that extending techniques can be used to further investigate the
behavior of evolving biological networks. In the extended version of this paper
we apply these techniques to seven tissue specific regulatory networks as
static example and network of a na\"ive pluripotent immune cell in the process
of differentiating towards a Th17 cell as evolving example, finding the most
and least informative Eigenvalues at every stage.; Comment: Forthcoming in 3rd International Work-Conference on Bioinformatics
and Biomedical Engineering (IWBBIO), Lecture Notes in Bioinformatics, 2015

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## Parametrization of the quark mixing matrix involving its eigenvalues

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 29/09/2002

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A parametrization of the $3\times 3$ Cabibbo-Kobayashi-Maskawa matrix, $V$,
is presented in which the parameters are the eigenvalues and the components of
its eigenvectors. In this parametrization, the small departure of the
experimentally determined $V$ from being moduli symmetric (i.e.
$|V_{ij}|=|V_{ji}|$) is controlled by the small difference between two of the
eigenvalues. In case, any two eigenvalues are equal, one obtains a moduli
symmetric $V$ depending on only three parameters. Our parametrization gives
very good fits to the available data including CP-violation. Our value of $\sin
2\beta\approx 0.7$ and other parameters associated with the ` unitarity
triangle' $V_{11}V_{13}^{*}+V_{21}V_{23}^{*}V_{31}V_{33}^{*}=0$ are in good
agreement with data and other analyses.; Comment: Latex, 11 pages, no figures

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## The Distribution of the Largest Non-trivial Eigenvalues in Families of Random Regular Graphs

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

26.6%

Recently Friedman proved Alon's conjecture for many families of d-regular
graphs, namely that given any epsilon > 0 `most' graphs have their largest
non-trivial eigenvalue at most 2 sqrt{d-1}+epsilon in absolute value; if the
absolute value of the largest non-trivial eigenvalue is at most 2 sqrt{d-1}
then the graph is said to be Ramanujan. These graphs have important
applications in communication network theory, allowing the construction of
superconcentrators and nonblocking networks, coding theory and cryptography. As
many of these applications depend on the size of the largest non-trivial
positive and negative eigenvalues, it is natural to investigate their
distributions. We show these are well-modeled by the beta=1 Tracy-Widom
distribution for several families. If the observed growth rates of the mean and
standard deviation as a function of the number of vertices holds in the limit,
then in the limit approximately 52% of d-regular graphs from bipartite families
should be Ramanujan, and about 27% from non-bipartite families (assuming the
largest positive and negative eigenvalues are independent).; Comment: 23 pages, version 2 (MAJOR correction: see footnote 7 on page 7: the
eigenvalue program unkowingly assumed the eigenvalues of the matrix were
symmetric...

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## Distribution of Eigenvalues in Non-Hermitian Anderson Model

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 22/07/1997

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We develop a theory which describes the behaviour of eigenvalues of a class
of one-dimensional random non-Hermitian operators introduced recently by Hatano
and Nelson. Under general assumptions on random parameters we prove that the
eigenvalues are distributed along a curve in the complex plane. An equation for
the curve is derived and the density of complex eigenvalues is found in terms
of spectral characteristics of a ``reference'' hermitian disordered system.
Coexistence of the real and complex parts in the spectrum and other generic
properties of the eigenvalue distribution for the non-Hermitian problem are
discussed.; Comment: 6 pages (LaTeX)

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## Weinberg Eigenvalues and Pairing with Low-Momentum Potentials

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 05/09/2007

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The nonperturbative nature of nucleon-nucleon interactions evolved to low
momentum has recently been investigated in free space and at finite density
using Weinberg eigenvalues as a diagnostic. This analysis is extended here to
the in-medium eigenvalues near the Fermi surface to study pairing. For a fixed
value of density and cutoff Lambda, the eigenvalues increase arbitrarily in
magnitude close to the Fermi surface, signaling the pairing instability. When
using normal-phase propagators, the Weinberg analysis with complex energies
becomes a form of stability analysis and the pairing gap can be estimated from
the largest attractive eigenvalue. With Nambu-Gorkov Green's functions, the
largest attractive eigenvalue goes to unity close to the Fermi surface,
indicating the presence of bound states (Cooper pairs), and the corresponding
eigenvector leads to the self-consistent gap function.; Comment: 16 pages, 9 figures

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## Bounds on positive interior transmission eigenvalues

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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The paper contains lower bounds on the counting function of the positive
eigenvalues of the interior transmission problem when the latter is elliptic.
In particular, these bounds justify the existence of an infinite set of
interior transmission eigenvalues and provide asymptotic estimates from above
on the counting function for the large values of the wave number. They also
lead to certain important upper estimates on the first few interior
transmission eigenvalues. We consider the classical transmission problem as
well as the case when the inhomogeneous medium contains an obstacle.; Comment: We corrected inaccuracies cost by the wrong sign in the Green formula
(17). In particular, the sign in the definition of \sigma was changed

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## A Survey on the Eigenvalues Local Behavior of Large Complex Correlated Wishart Matrices

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 16/09/2015

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The aim of this note is to provide a pedagogical survey of the recent works
by the authors ( arXiv:1409.7548 and arXiv:1507.06013) concerning the local
behavior of the eigenvalues of large complex correlated Wishart matrices at the
edges and cusp points of the spectrum: Under quite general conditions, the
eigenvalues fluctuations at a soft edge of the limiting spectrum, at the hard
edge when it is present, or at a cusp point, are respectively described by mean
of the Airy kernel, the Bessel kernel, or the Pearcey kernel. Moreover, the
eigenvalues fluctuations at several soft edges are asymptotically independent.
In particular, the asymptotic fluctuations of the matrix condition number can
be described. Finally, the next order term of the hard edge asymptotics is
provided.; Comment: 29 pages; 7 figures; to be published in the "Proceedings of the
Journ{\'e}es MAS 2014"

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## On the distribution of linear combinations of eigenvalues of the Anderson model

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 31/08/2008

Relevância na Pesquisa

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Probabilistic estimates on linear combinations of eigenvalues of the one
dimensional Anderson model are derived. So far only estimates on the density of
eigenvalues and of pairs were found by Wegner and by Minami. Our work was
motivated by perturbative explorations of the Nonlinear Schroedinger Equation,
where linear combinations of eigenvalues are the denominators and evaluation of
their smallness is crucial.; Comment: 12 pages, 0 figures

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## Acceleration of the Arnoldi method and real eigenvalues of the non-Hermitian Wilson-Dirac operator

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 07/04/2011

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In this paper, we present a method for the computation of the low-lying real
eigenvalues of the Wilson-Dirac operator based on the Arnoldi algorithm. These
eigenvalues contain information about several observables. We used them to
calculate the sign of the fermion determinant in one-flavor QCD and the sign of
the Pfaffian in N=1 super Yang-Mills theory. The method is based on polynomial
transformations of the Wilson-Dirac operator, leading to considerable
improvements of the computation of eigenvalues. We introduce an iterative
procedure for the construction of the polynomials and demonstrate the
improvement in the efficiency of the computation. In general, the method can be
applied to operators with a symmetric and bounded eigenspectrum.; Comment: 11 pages, 6 figures

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## Eigenvalues of a nonlinear ground state in the Thomas-Fermi approximation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 21/06/2008

Relevância na Pesquisa

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We study a nonlinear ground state of the Gross-Pitaevskii equation with a
parabolic potential in the hydrodynamics limit often referred to as the
Thomas--Fermi approximation. Existence of the energy minimizer has been known
in literature for some time but it was only recently when the Thomas-Fermi
approximation was rigorously justified. The spectrum of linearization of the
Gross-Pitaevskii equation at the ground state consists of an unbounded sequence
of positive eigenvalues. We analyze convergence of eigenvalues in the
hydrodynamics limit. Convergence in norm of the resolvent operator is proved
and the convergence rate is estimated. We also study asymptotic and numerical
approximations of eigenfunctions and eigenvalues using Airy functions.

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## On split graphs with four distinct eigenvalues

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/05/2014

Relevância na Pesquisa

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It is a well-known fact that a graph of diameter $d$ has at least $d+1$
eigenvalues. Let us call a graph \emph{$d$-extremal} if it has diameter $d$ and
exactly $d+1$ eigenvalues. Such graphs have been intensively studied by various
authors. %Much attention has been devoted to the study of graphs that are
extremal with respect to this relation: \emph{i.e} have diameter $d$ and
exactly $d+1$ distinct eigenvalues.
A graph is \emph{split} if its vertex set can be partitioned into a clique
and a stable set. Such a graph has diameter at most $3$. We obtain a complete
classification of the connected bidegreed $3$-extremal split graphs. We also
show how to construct certain families of non-bidegreed $3$-extremal split
graphs.

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