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## Structure of trajectories of complex-matrix eigenvalues in the Hermitian-non-Hermitian transition

Bohigas, O.; De Carvalho, J. X.; Pato, Mauricio Porto
Fonte: AMER PHYSICAL SOC; COLLEGE PK Publicador: AMER PHYSICAL SOC; COLLEGE PK
Tipo: Artigo de Revista Científica
ENG
Relevância na Pesquisa
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The statistical properties of trajectories of eigenvalues of Gaussian complex matrices whose Hermitian condition is progressively broken are investigated. It is shown how the ordering on the real axis of the real eigenvalues is reflected in the structure of the trajectories and also in the final distribution of the eigenvalues in the complex plane.; CAPESCOFECUB; CAPES/COFECUB; CNPq; CNPq; FAPESP; FAPESP

## Bounds for the signless laplacian energy

Abreu, N.; Cardoso, D.M.; Gutman, I.; Martins, E.A.; Robbiano, M.
Tipo: Artigo de Revista Científica
ENG
Relevância na Pesquisa
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The energy of a graph G is the sum of the absolute values of the eigenvalues of the adjacency matrix of G. The Laplacian (respectively, the signless Laplacian) energy of G is the sum of the absolute values of the differences between the eigenvalues of the Laplacian (respectively, signless Laplacian) matrix and the arithmetic mean of the vertex degrees of the graph. In this paper, among some results which relate these energies, we point out some bounds to them using the energy of the line graph of G. Most of these bounds are valid for both energies, Laplacian and signless Laplacian. However, we present two new upper bounds on the signless Laplacian which are not upper bounds for the Laplacian energy. © 2010 Elsevier Inc. All rights reserved.; FCT; FEDER/POCI 2010; CNPq; PQ-305016/2006–2007; Serbian Ministry of Science; No. 144015G; Mecesup 2 UCN 0605; Fondecyt-IC Project 11090211

## Structural breaks in dynamic factor models

DI SALVATORE, ANTONIETTA
EN
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In this thesis we analysed the problem of a single structural change occurring at some unknown data in multivariate time series. Our results rely on the assumption that the multivariate time series is generated by a dynamic factor model. Dynamic factor analysis is a very rich methodology which can be extended in many way to get a closer approximation to complex economic reality. They attempt to capture the correlation structure of a large number of original variables with a small set of common factors, in order to reduce the dimensionality of the vector space of the original variables. Three di_erent type of breaks has been analysed: the break in the mean level, the break in the factor loadings and the break in the factor moments. For each of them we suggest a model and therefore we focus the attention on the population and sample moments. When a break occurs, the data-generating process is not stationary anymore. The break in level affects the first moment of the process but the variance is still stationary whereas the other break types affect the second order moments. Furthermore we showed that the estimates are always affected by the break. Given these preliminary results we are interested in the Fourier transform of the estimated variance covariance matrices. For Geweke (1977) we know that all variation in the observed data may be decomposed into variance across frequencies using spectral techniques and...

## Bounds on the extreme generalized eigenvalues of Hermitian pencils

Fargues, Monique P.
Tipo: Relatório
EN_US
Relevância na Pesquisa
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We present easily computable bounds on the extreme generalized eigenvalues of Hermitian pencils (R,B) with finite eigenvalues and positive definite B matrices. The bounds are derived in terms of the generalized eigenvalues of the subpencil of maximum dimension contained in (R,B). Known results based on the generalization of the Gershgorin theorem and norm inequalities are presented and compared to the proposed bounds. It is shown that the new bounds compare favorably with these known results; they are easier to compute, require less restrictions on the properties of the pencils studied, and they are in an average sense tighter than those obtained with the norm inequality bounds

## Eigenstructure of nonstationary factor models

Peña, Daniel; Poncela, Pilar
Tipo: Trabalho em Andamento Formato: application/pdf
Relevância na Pesquisa
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In this paper we present a generalized dynamic factor model for a vector of time series which seems to provide a general framework to incorporate all the common information included in a collection of variables. The common dynamic structure is explained through a set of common factors, which may be stationary or nonstationary, as in the case of cornmon trends. AIso, it may exist a specific structure for each variable. Identification of the nonstationary I(d) factors is made through the cornmon eigenstructure of the generalized covariance matrices, properly normalized. The number of common trends, or in general I(d) factors, is the number of nonzero eigenvalues of the above matrices. It is also proved that these nonzero eigenvalues are strictIy greater than zero almost sure. Randomness appears in the eigenvalues as well as the eigenvectors, but not on the subspace spanned by the eigenvectors.

## Homogenización de autovalores en operadores elípticos cuasilineales; Eigenvalue homogenization for quasilinear elliptic operators

Salort, Ariel Martín
Tipo: info:eu-repo/semantics/doctoralThesis; tesis doctoral; info:eu-repo/semantics/publishedVersion Formato: application/pdf
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Distintos problemas clásicos de vibraciones mecánicas son modelados con ecuaciones diferenciales, y las frecuencias de vibración corresponden a los autovalores de éstas. Estructuras tales como columnas, placas, membranas o cuerdas, obedecen distintas clases de problemas elípticos (el sistema de ecuaciones de la elasticidad, el laplaciano, el bilaplaciano, ecuaciones de Sturm Liouville). Estos operadores han sido muy estudiados y se conocen numerosas propiedades de sus autovalores, ver por ejemplo los trabajos clásicos de Courant, Hormander, Timoshenko, Titchmarsh, Weinstein [CoHi53, Hor68, Hor07, Ti46] entre otros. Durante el siglo XX, la teoría no lineal generó nuevas herramientas y problemas, y los autovalores son interpretados en este contexto como un parámetro de bifurcación, correponden a valores críticos para los cuales una estructura puede deformarse, colapsar o salir de equilibrio (buckling, bending). Podemos citar como ejemplo los trabajos de Antman, Browder, Berger, y Amann [Am72, An83, Be68, Br65]. En los últimos años, los nuevos materiales han creado nuevos desafíos. En particular, cuando se consideran mezclas de dos o más materiales se van obteniendo mejores propiedades específicas, y gracias a estas mejores características los materiales heterogéneos reemplazan a los homogéneos. Particularmente...

## The largest eigenvalues of sample covariance matrices for a spiked population: diagonal case

Féral, Delphine; Péché, Sandrine
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We consider large complex random sample covariance matrices obtained from "spiked populations", that is when the true covariance matrix is diagonal with all but finitely many eigenvalues equal to one. We investigate the limiting behavior of the largest eigenvalues when the population and the sample sizes both become large. Under some conditions on moments of the sample distribution, we prove that the asymptotic fluctuations of the largest eigenvalues are the same as for a complex Gaussian sample with the same true covariance. The real setting is also considered.

## Inference on Eigenvalues of Wishart Distribution Using Asymptotics with respect to the Dispersion of Population Eigenvalues

Sheena, Yo; Takemura, Akimichi
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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In this paper we derive some new and practical results on testing and interval estimation problems for the population eigenvalues of a Wishart matrix based on the asymptotic theory for block-wise infinite dispersion of the population eigenvalues. This new type of asymptotic theory has been developed by the present authors in Takemura and Sheena (2005) and Sheena and Takemura (2007a,b) and in these papers it was applied to point estimation problem of population covariance matrix in a decision theoretic framework. In this paper we apply it to some testing and interval estimation problems. We show that the approximation based on this type of asymptotics is generally much better than the traditional large-sample asymptotics for the problems.

## Estimates for the higher order buckling eigenvalues in the unit sphere

Huang, Guangyue; Li, Xingxiao; Qi, Xuerong
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We consider the higher order buckling eigenvalues of the following Dirichlet poly-Laplacian in the unit sphere $(-\Delta)^p u=\Lambda (-\Delta) u$ with order $p(\geq2)$. We obtain universal bounds on the $(k+1)$th eigenvalue in terms of the first $k$th eigenvalues independent of the domains. In particular, for $p=2$, our result is sharp than estimates on eigenvalues of the buckling problem obtained by Wang and Xia.; Comment: This article has been submitted for publication on 12, August

## Maass cusp forms for large eigenvalues

Then, H.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We investigate the numerical computation of Maass cusp forms for the modular group corresponding to large eigenvalues. We present Fourier coefficients of two cusp forms whose eigenvalues exceed r=40000. These eigenvalues are the largest that have so far been found in the case of the modular group. They are larger than the 130millionth eigenvalue.; Comment: 24 pages, 7 figures, 3 tables

## PT symmetry and necessary and sufficient conditions for the reality of energy eigenvalues

Bender, Carl M.; Mannheim, Philip D.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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Despite its common use in quantum theory, the mathematical requirement of Dirac Hermiticity of a Hamiltonian is sufficient to guarantee the reality of energy eigenvalues but not necessary. By establishing three theorems, this paper gives physical conditions that are both necessary and sufficient. First, it is shown that if the secular equation is real, the Hamiltonian is necessarily PT symmetric. Second, if a linear operator C that obeys the two equations [C,H]=0 and C^2=1 is introduced, then the energy eigenvalues of a PT-symmetric Hamiltonian that is diagonalizable are real only if this C operator commutes with PT. Third, the energy eigenvalues of PT-symmetric Hamiltonians having a nondiagonalizable, Jordan-block form are real. These theorems hold for matrix Hamiltonians of any dimensionality.; Comment: 11 pages, no figures

## Eigenvalues of the Derangement Graph

Ku, Cheng Yeaw; Wales, David B.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We consider the Cayley graph on the symmetric group Sn generated by derangements. It is well known that the eigenvalues of this grpah are indexed by partitions of n. We investigate how these eigenvalues are determined by the shape of their corresponding partitions. In particular, we show that the sign of an eigenvalue is the parity of the number of cells below the first row of the corresponding Ferrers diagram. We also provide some lower and upper bounds for the absolute values of these eigenvalues.; Comment: 26 pages

## Perturbation of eigenvalues of matrix pencils and optimal assignment problem

Akian, Marianne; Bapat, Ravindra; Gaubert, Stephane
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.68%
We consider a matrix pencil whose coefficients depend on a positive parameter $\epsilon$, and have asymptotic equivalents of the form $a\epsilon^A$ when $\epsilon$ goes to zero, where the leading coefficient $a$ is complex, and the leading exponent $A$ is real. We show that the asymptotic equivalent of every eigenvalue of the pencil can be determined generically from the asymptotic equivalents of the coefficients of the pencil. The generic leading exponents of the eigenvalues are the "eigenvalues" of a min-plus matrix pencil. The leading coefficients of the eigenvalues are the eigenvalues of auxiliary matrix pencils, constructed from certain optimal assignment problems.; Comment: 8 pages

## A Contour-integral Based Method for Counting the Eigenvalues Inside a Region in the Complex Plane

Yin, Guojian
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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In many applications, the information about the number of eigenvalues inside a given region is required. In this paper, we propose a contour-integral based method for this purpose. The new method is motivated by two findings. There exist methods for estimating the number of eigenvalues inside a region in the complex plane. But our method is able to compute the number of eigenvalues inside the given region exactly. An appealing feature of our method is that it can integrate with the recently developed contour-integral based eigensolvers to help them detect whether all desired eigenvalues are found. Numerical experiments are reported to show the viability of our new method.

## Estimates for eigenvalues of $\mathfrak L$ operator on Self-Shrinkers

Cheng, Qing-Ming; Peng, Yejuan
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.68%
In this paper, we study eigenvalues of the closed eigenvalue problem of the differential operator $L$, which is introduced by Colding and Minicozzi in [4], on an $n$-dimensional compact self-shrinker in ${R}^{n+p}$. Estimates for eigenvalues of the differential operator $L$ are obtained. Our estimates for eigenvalues of the differential operator $L$ are sharp. Furthermore, we also study the Dirichlet eigenvalue problem of the differential operator $L$ on a bounded domain with a piecewise smooth boundary in an $n$-dimensional complete self-shrinker in ${R}^{n+p}$. For Euclidean space ${R}^{n}$, the differential operator $L$ becomes the Ornstein-Uhlenbeck operator in stochastic analysis. Hence, we also give estimates for eigenvalues of the Ornstein-Uhlenbeck operator.; Comment: 21 pages, a final version, accepted for publication in Communications in Contemporary Mathematics

## Graphs with many valencies and few eigenvalues

van Dam, Edwin R.; Koolen, Jack H.; Xia, Zheng-jiang
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.68%
Dom de Caen posed the question whether connected graphs with three distinct eigenvalues have at most three distinct valencies. We do not answer this question, but instead construct connected graphs with four and five distinct eigenvalues and arbitrarily many distinct valencies. The graphs with four distinct eigenvalues come from regular two-graphs. As a side result, we characterize the disconnected graphs and the graphs with three distinct eigenvalues in the switching class of a regular two-graph.

## A unified approach for large deviations of bulk and extreme eigenvalues of the Wishart ensemble

Melo, Adolfo Camacho; Castillo, Isaac Pérez
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.68%
Within the framework of the Coulomb fluid picture, we present a unified approach to derive the large deviations of bulk and extreme eigenvalues of large Wishart matrices. By analysing the statistics of the shifted index number we are able to derive a rate function $\Psi (c, x)$ depending on two variables: the fraction $c$ of eigenvalues to the left of an infinite energetic barrier at position $x$. For a fixed value of $c$, the rate function gives the large deviations of the bulk eigenvalues. In particular, in the limits $c\to 0$ or $c\to 1$ it is possible to extract the left and right deviations of the smallest and largest eigenvalues, respectively. Alternatively, for a fixed value $x$ of the barrier, the rate function provides the large deviations of the shifted index number. All our analytical findings are compared with Metropolis Monte Carlo simulations, obtaining excellent agreement.; Comment: 16 pages, 7 figures, 25 references

## On the eigenvalues of the spatial sign covariance matrix in more than two dimensions

Dürre, Alexander; Tyler, David E.; Vogel, Daniel
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We gather several results on the eigenvalues of the spatial sign covariance matrix of an elliptical distribution. It is shown that the eigenvalues are a one-to-one function of the eigenvalues of the shape matrix and that they are closer together than the latter. We further provide a one-dimensional integral representation of the eigenvalues, which facilitates their numerical computation.

## Certain upper bounds on the eigenvalues associated with prolate spheroidal wave functions

Osipov, Andrei
Tipo: Artigo de Revista Científica