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

Fonte: AMER PHYSICAL SOC; COLLEGE PK
Publicador: AMER PHYSICAL SOC; COLLEGE PK

Tipo: Artigo de Revista Científica

ENG

Relevância na Pesquisa

26.6%

#REAL MATRICES#STATISTICS#ENSEMBLES#UNITARY#SYSTEMS#CHAOS#PHYSICS, FLUIDS & PLASMAS#PHYSICS, MATHEMATICAL

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

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## Bounds for the signless laplacian energy

Fonte: Elsevier
Publicador: Elsevier

Tipo: Artigo de Revista Científica

ENG

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26.6%

#Graph spectrum#Laplacian energy#Laplacian graph spectrum#Signless Laplacian energy#Signless Laplacian spectrum#Absolute values#Adjacency matrices#Arithmetic mean#Eigenvalues#Energy of a graph#Graph spectra

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

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## Structural breaks in dynamic factor models

Fonte: La Sapienza Universidade de Roma
Publicador: La Sapienza Universidade de Roma

Tipo: Tese de Doutorado

EN

Relevância na Pesquisa

26.6%

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...

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## Bounds on the extreme generalized eigenvalues of Hermitian pencils

Fonte: Monterey, California. Naval Postgraduate School
Publicador: Monterey, California. Naval Postgraduate School

Tipo: Relatório

EN_US

Relevância na Pesquisa

26.6%

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

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## Eigenstructure of nonstationary factor models

Fonte: Universidade Carlos III de Madrid
Publicador: Universidade Carlos III de Madrid

Tipo: Trabalho em Andamento
Formato: application/pdf

Publicado em /12/1997
ENG

Relevância na Pesquisa

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#Cointegration and common factors#eigenvectors and eigenvalues#generalized covariance matrices#factor model#nonstationary I(d) factors#vector time series#Wiener processes#Estadística

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.

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

Fonte: Facultad de Ciencias Exactas y Naturales. Universidad de Buenos Aires
Publicador: Facultad de Ciencias Exactas y Naturales. Universidad de Buenos Aires

Tipo: info:eu-repo/semantics/doctoralThesis; tesis doctoral; info:eu-repo/semantics/publishedVersion
Formato: application/pdf

Publicado em //2012
SPA

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#P-LAPLACIAN#MONOTONE OPERATORS#HOMOGENIZATION#EIGENVALUES#RATE CONVERGENCE#G-CONVERGENCE#OSCILLATING INTEGRALS#P-LAPLACIANO#OPERADORES MONOTONOS#HOMOGENEIZACION#AUTOVALORES

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...

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/12/2008

Relevância na Pesquisa

26.6%

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.

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 18/04/2007

Relevância na Pesquisa

26.6%

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.

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 30/08/2009

Relevância na Pesquisa

26.6%

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

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## Maass cusp forms for large eigenvalues

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 23/05/2003

Relevância na Pesquisa

26.6%

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 09/02/2009

Relevância na Pesquisa

26.6%

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

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## Eigenvalues of the Derangement Graph

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 19/03/2008

Relevância na Pesquisa

26.6%

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/02/2004

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

26.68%

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.

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

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

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## Graphs with many valencies and few eigenvalues

Fonte: Universidade Cornell
Publicador: Universidade Cornell

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.

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 09/12/2015

Relevância na Pesquisa

26.68%

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.

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 20/06/2012

Relevância na Pesquisa

26.68%

#Mathematics - Functional Analysis#Mathematics - Classical Analysis and ODEs#Mathematics - Numerical Analysis#33E10, 34L15, 35S30, 42C10, 45C05, 54P05

Prolate spheroidal wave functions (PSWFs) play an important role in various
areas, from physics (e.g. wave phenomena, fluid dynamics) to engineering (e.g.
signal processing, filter design). One of the principal reasons for the
importance of PSWFs is that they are a natural and efficient tool for computing
with bandlimited functions, that frequently occur in the abovementioned areas.
This is due to the fact that PSWFs are the eigenfunctions of the integral
operator, that represents timelimiting followed by lowpassing.
Needless to say, the behavior of this operator is governed by the decay rate
of its eigenvalues. Therefore, investigation of this decay rate plays a crucial
role in the related theory and applications - for example, in construction of
quadratures, interpolation, filter design, etc.
The significance of PSWFs and, in particular, of the decay rate of the
eigenvalues of the associated integral operator, was realized at least half a
century ago. Nevertheless, perhaps surprisingly, despite vast numerical
experience and existence of several asymptotic expansions, a non-trivial
explicit upper bound on the magnitude of the eigenvalues has been missing for
decades.
The principal goal of this paper is to close this gap in the theory of PSWFs.
We analyze the integral operator associated with PSWFs...

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## Spectral and combinatorial properties of friendship graphs, simplicial rook graphs, and extremal expanders

Fonte: University of Delaware
Publicador: University of Delaware

Tipo: Tese de Doutorado

Relevância na Pesquisa

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Cioaba, Sebastian M.; Algebraic combinatorics is the area of mathematics that uses the theories and methods of abstract and linear algebra to solve combinatorial problems, or conversely applies combinatorial techniques to solve problems in algebra. In particular, spectral graph theory applies the techniques of linear algebra to study graph theory. Spectral graph theory is the study of the eigenvalues of various matrices associated with graphs and how they relate to the properties of those graphs. The graph properties of diameter, independence number, chromatic number, connectedness, toughness, hamiltonicity, and expansion, among others, are all related to the spectra of graphs. In this thesis we study the spectra of various families of graphs, how their spectra relate to their properties, and when graphs are determined by their spectra. We focus on three topics (Chapters 2-4) in spectral graph theory. The wide range of these topics showcases the power and versatility of the eigenvalue techniques such as interlacing, the common thread that ties these topics together. In Chapter 1, we review the basic definitions, notations, and results in graph theory and spectral graph theory. We also introduce powerful tools for determining the structure of a graph and its subgraphs using eigenvalue interlacing. Finally...

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