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

Witte, N. S.; Bornemann, F.; Forrester, P. J.
Tipo: Artigo de Revista Científica
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26.6%
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

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

Charles, Zachary B.; Farber, Miriam; Johnson, Charles R.; Kennedy-Shaffer, Lee
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.6%
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

## Domain dependence of eigenvalues of elliptic type operators

Tipo: Artigo de Revista Científica
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26.6%
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.

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

Lee, Seung-Yeop; Riser, Roman
Tipo: Artigo de Revista Científica
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26.6%
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

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

Tao, Terence; Vu, Van
Tipo: Artigo de Revista Científica
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26.6%
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...

## All Real Eigenvalues of Symmetric Tensors

Cui, Chun-Feng; Dai, Yu-Hong; Nie, Jiawang
Tipo: Artigo de Revista Científica
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26.6%
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.

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

Park, Hong Ju; Ayanoglu, Ender
Tipo: Artigo de Revista Científica
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26.6%
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

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

Buividovich, P. V.; Luschevskaya, E. V.; Polikarpov, M. I.
Tipo: Artigo de Revista Científica
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26.6%
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

## General Relativity in terms of Dirac Eigenvalues

Landi, Giovanni; Rovelli, Carlo
Tipo: Artigo de Revista Científica
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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

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

Zenil, Hector; Kiani, Narsis A.; Tegnér, Jesper
Tipo: Artigo de Revista Científica
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26.6%
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

## Parametrization of the quark mixing matrix involving its eigenvalues

Chaturvedi, S.; Gupta, Virendra
Tipo: Artigo de Revista Científica
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26.6%
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

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

Miller, Steven J.; Novikoff, Tim; Sabelli, Anthony
Tipo: Artigo de Revista Científica
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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...

## Distribution of Eigenvalues in Non-Hermitian Anderson Model

Goldsheid, Ilya Ya.; Khoruzhenko, Boris A.
Tipo: Artigo de Revista Científica
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26.6%
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)

## Weinberg Eigenvalues and Pairing with Low-Momentum Potentials

Ramanan, S.; Bogner, S. K.; Furnstahl, R. J.
Tipo: Artigo de Revista Científica
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26.6%
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

## Bounds on positive interior transmission eigenvalues

Lakshtanov, Evgeny; Vainberg, Boris
Tipo: Artigo de Revista Científica
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26.6%
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

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

Hachem, Walid; Hardy, Adrien; Najim, Jamal
Tipo: Artigo de Revista Científica
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26.6%
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"

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

Fishman, Shmuel; Krivolapov, Yevgeny; Soffer, Avy
Tipo: Artigo de Revista Científica
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26.6%
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

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

Bergner, G.; Wuilloud, J.
Tipo: Artigo de Revista Científica
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26.6%
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

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

Gallo, Clément; Pelinovsky, Dmitry
Tipo: Artigo de Revista Científica
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.