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## Reliable eigenvalues of symmetric tridiagonals

Fonte: SIAM
Publicador: SIAM

Tipo: Artigo de Revista Científica

Publicado em /12/2011
ENG

Relevância na Pesquisa

66%

For the eigenvalues of a symmetric tridiagonal matrix T, the most accurate algorithms deliver approximations which are the exact eigenvalues of a matrix whose entries differ from the corresponding entries of T by small relative perturbations. However, for matrices with eigenvalues of different magnitudes, the number of correct digits in the computed approximations for eigenvalues of size smaller than ‖T‖₂ depends on how well such eigenvalues are defined by the data. Some classes of matrices are known to define their eigenvalues to high relative accuracy but, in general, there is no simple way to estimate well the number of correct digits in the approximations. To remedy this, we propose a method that provides sharp bounds for the eigenvalues of T. We present some numerical examples to illustrate the usefulness of our method.; FEDER (Programa Operacional Factores de Competitividade); FCT (Projecto PEst-C/MAT/UI0013/2011

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## Bounds for the signless laplacian energy

Fonte: Elsevier
Publicador: Elsevier

Tipo: Artigo de Revista Científica

ENG

Relevância na Pesquisa

56.03%

#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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## Upper Bounds for Randic Spread

Fonte: Mathematical Chemistry Monographs
Publicador: Mathematical Chemistry Monographs

Tipo: Artigo de Revista Científica

ENG

Relevância na Pesquisa

55.75%

The Randi´c spread of a simple undirected graph G, sprR(G), is equal to the maximal
difference between two eigenvalues of the Randi´c matrix, disregarding the spectral radius [Gomes
et al., MATCH Commun. Math. Comput. Chem. 72 (2014) 249–266]. Using a rank-one
perturbation on the Randi´c matrix of G it is obtained a new matrix whose matricial spread
coincide with sprR(G). By means of this result, upper bounds for sprR(G) are obtained.

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## Lower bounds for eigenvalues of self-adjoint problems

Fonte: PubMed
Publicador: PubMed

Tipo: Artigo de Revista Científica

Publicado em /11/1979
EN

Relevância na Pesquisa

55.87%

The equation y″ + [λ - q(x)]y = 0 on (0, ∞) or (-∞, ∞), in which q(x) → ∞ as x → ∞ or x → ± ∞, has a complete set of eigenfunctions with discrete eigenvalues {λn}n=0∞. We derive an inequality that contains λn, by using a quick and elementary method that does not employ a comparison theorem or assume anything special. Explicit lower bounds for λn can often be easily obtained, and three examples are given. The method also gives respectable lower bounds for λn in the classical Sturm—Liouville case.

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## Desigualdades universais para autovalores do operador poli-harmônico; Universal bounds for eigenvalues of the polyharmonic operator

Fonte: Universidade Federal de Goiás; BR; UFG; Mestrado em Matemática; Ciências Exatas e da Terra
Publicador: Universidade Federal de Goiás; BR; UFG; Mestrado em Matemática; Ciências Exatas e da Terra

Tipo: Dissertação
Formato: application/pdf

POR

Relevância na Pesquisa

65.97%

#Variedades Riemannianas#Cotas universais#Desigualdade tipo-Yang#Operador poli-harmônico#Riemannian manifolds#Universal bounds#Yang-type inequality#Polyharmonic operator#CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA

In this work, we study eigenvalues of polyharmonic operators on compact Riemannian manifolds with boundary (possibly empty). Here, we bring in a universal inequality for the eigenvalues of the polyharmonic operator on compact
domains in an Euclidean space Rn. This inequality controls the kth eigenvalue by the lower eigenvalues, independently of the particular geometry of the domain. Besides, a inequality we present covers the important Yang inequality on eigenvalues of the Dirichlet Laplacian. Finally, we introduce universal inequalities for eigenvalues of polyharmonic operator on compact domains in a unit n-sphere Sn. NOTE: Programs do not copy or copy errors with certain symbols, formulas, formatting, etc ..., n of Rn and Sn are overwritten. View all content by clicking pdf - dissertation at the bottom of the screen.; Neste trabalho, estudamos autovalores do operador poli-harmônico em variedades Riemannianas compactas com fronteira ( possivelmente vazia ). Aqui, apresentamos uma desigualdade universal para os autovalores do operador poliharmônico em domínios compactos no Espaço Euclidiano Rn. Esta desigualdade controla o k-ésimo autovalor pelos autovalores menores, independentemente da geometria particular do domínio. Além disso...

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## Relative perturbation theory for diagonally dominant matrices

Fonte: Society for Industrial and Applied Mathematics
Publicador: Society for Industrial and Applied Mathematics

Tipo: info:eu-repo/semantics/publishedVersion; info:eu-repo/semantics/article

Publicado em /10/2014
ENG

Relevância na Pesquisa

55.97%

#Accurate computations#Diagonally dominant matrices#Diagonally dominant parts#Eigenvalues#Inverses#Linear systems#Relative perturbation theory#Singular values#Eigenvalues and eigenfunctions#Inverse problems#Matrix algebra

In this paper, strong relative perturbation bounds are developed for a number of linear algebra problems involving diagonally dominant matrices. The key point is to parameterize diagonally dominant matrices using their off-diagonal entries and diagonally dominant parts and to consider small relative componentwise perturbations of these parameters. This allows us to obtain new relative perturbation bounds for the inverse, the solution to linear systems, the symmetric indefinite eigenvalue problem, the singular value problem, and the nonsymmetric eigenvalue problem. These bounds are much stronger than traditional perturbation results, since they are independent of either the standard condition number or the magnitude of eigenvalues/singular values. Together with previously derived perturbation bounds for the LDU factorization and the symmetric positive definite eigenvalue problem, this paper presents a complete and detailed account of relative structured perturbation theory for diagonally dominant matrices.; This research was partially supported by the Ministerio de Economía y Competitividad of Spain under grant MTM2012-32542.

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## Sharp Bounds for Eigenvalues and Multiplicities on Surfaces of Revolution

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/04/1996

Relevância na Pesquisa

55.82%

We find sharp upper bounds for the multiplicities and the numerical values of
all the distinct eigenvalues on a surface of revolution diffeomorphic to the
sphere.; Comment: LaTeX2e, 8 pages

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## New Probabilistic Bounds on Eigenvalues and Eigenvectors of Random Kernel Matrices

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/02/2012

Relevância na Pesquisa

56.04%

Kernel methods are successful approaches for different machine learning
problems. This success is mainly rooted in using feature maps and kernel
matrices. Some methods rely on the eigenvalues/eigenvectors of the kernel
matrix, while for other methods the spectral information can be used to
estimate the excess risk. An important question remains on how close the sample
eigenvalues/eigenvectors are to the population values. In this paper, we
improve earlier results on concentration bounds for eigenvalues of general
kernel matrices. For distance and inner product kernel functions, e.g. radial
basis functions, we provide new concentration bounds, which are characterized
by the eigenvalues of the sample covariance matrix. Meanwhile, the obstacles
for sharper bounds are accounted for and partially addressed. As a case study,
we derive a concentration inequality for sample kernel target-alignment.

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## Lower bound estimates for eigenvalues of the Laplacian

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/08/2012

Relevância na Pesquisa

55.78%

For an $n$-dimensional polytope $\Omega$ in $\mathbb{R}^{n}$, we study lower
bounds for eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. In
the asymptotic formula on the average of the first $k$ eigenvalues, Li and Yau
(1983) obtained the first term with the order $k^{\frac2n}$, which is optimal.
The next landmark goal is to give the second term with the order $k^{\frac1n}$
in the asymptotic formula. For this purpose, Kova\v{r}\'{\i}k, Vugalter and
Weidl (2009) have made an important breakthrough in the case of dimension 2. It
is our purpose to study the $n$-dimensional case for arbitrary dimension $n$.
We obtain the second term in the asymptotic sense.; Comment: 15 pages

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## Two-sided bounds for eigenvalues of differential operators with applications to Friedrichs', Poincar\'e, trace, and similar constants

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

55.91%

We present a general numerical method for computing guaranteed two-sided
bounds for principal eigenvalues of symmetric linear elliptic differential
operators. The approach is based on the Galerkin method, on the method of a
priori-a posteriori inequalities, and on a complementarity technique. The
two-sided bounds are formulated in a general Hilbert space setting and as a
byproduct we prove an abstract inequality of Friedrichs'-Poincar\'e type. The
abstract results are then applied to Friedrichs', Poincar\'e, and trace
inequalities and fully computable two-sided bounds on the optimal constants in
these inequalities are obtained. Accuracy of the method is illustrated on
numerical examples.; Comment: Extended numerical experiments and minor corrections of the previous
version. This version has been accepted for publication by SIAM J. Numer.
Anal

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## Extrinsic Bounds for Eigenvalues of the Dirac Operator

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 13/05/1998

Relevância na Pesquisa

55.9%

We derive upper eigenvalue bounds for the Dirac operator of a closed
hypersurface in a manifold with Killing spinors such as Euclidean space,
spheres or hyperbolic space. The bounds involve the Willmore functional.
Relations with the Willmore inequality are briefly discussed. In higher
codimension we obtain bounds on the eigenvalues of the Dirac operator of the
submanifold twisted with the spinor bundle of the normal bundle.; Comment: 24 pages, LaTeX2e. to appear in Ann. Glob. Anal. Geom

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## Universal bounds for eigenvalues of a buckling problem II

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

55.79%

#Mathematics - Differential Geometry#Mathematical Physics#Mathematics - Spectral Theory#35P15, 58G25, 53C42

In this paper, we investigate universal estimates for eigenvalues of a
buckling problem. For a bounded domain in a Euclidean space, we give a positive
contribution for obtaining a sharp universal inequality for eigenvalues of the
buckling problem. For a domain in the unit sphere, we give an important
improvement on the results of Wang and Xia [J. Funct. Anal. 245(2007),
334-352].; Comment: 21 pages, a final version to appear in Trans. Amer. Math. Soc

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## Upper and lower bounds for eigenvalues of the clamped plate problem

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 29/01/2012

Relevância na Pesquisa

55.82%

In this paper, we study estimates for eigenvalues of the clamped plate
problem. A sharp upper bound for eigenvalues is given and the lower bound for
eigenvalues in [10] is improved.; Comment: 16 pages

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## Guaranteed Lower and upper bounds for eigenvalues of second order elliptic operators in any dimension

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/06/2014

Relevância na Pesquisa

66.04%

In this paper, a new method is proposed to produce guaranteed lower bounds
for eigenvalues of general second order elliptic operators in any dimension.
Unlike most methods in the literature, the proposed method only needs to solve
one discrete eigenvalue problem but not involves any base or intermediate
eigenvalue problems, and does not need any a priori information concerning
exact eigenvalues either. Moreover, it just assumes basic regularity of exact
eigenfunctions. This method is defined by a novel generalized Crouzeix-Raviart
element which is proved to yield asymptotic lower bounds for eigenvalues of
general second order elliptic operators, and a simple post-processing method.
As a byproduct, a simple and cheap method is also proposed to obtain guaranteed
upper bounds for eigenvalues, which is based on generalized Crouzeix-Raviart
element approximate eigenfunctions, an averaging interpolation from the the
generalized Crouzeix-Raviart element space to the conforming linear element
space, and an usual Rayleigh-Ritz procedure. The ingredients for the analysis
consist of a crucial projection property of the canonical interpolation
operator of the generalized Crouzeix-Raviart element, explicitly computable
constants for two interpolation operators. Numerics are provided to demonstrate
the theoretical results.

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## The Bounds for Eigenvalues of Normalized and Signless Laplacian Matrices

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/08/2014

Relevância na Pesquisa

55.93%

In this paper, we obtain the bounds of the extreme eigenvalues of a
normalized and signless Laplacian matrices using by their traces. In addition,
we determine the bounds for k-th eigenvalues of normalized and signless
Laplacian matrices.; Comment: 5 pages

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## Note on Bounds for Eigenvalues using Traces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 30/08/2014

Relevância na Pesquisa

55.81%

We show that various old and new bounds involving eigenvalues of a complex n
x n matrix are immediate consequences of the inequalities involving variance of
real and complex numbers.; Comment: 13 pages

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## Perturbation bounds of eigenvalues of Hermitian matrices with block structures

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 31/08/2010

Relevância na Pesquisa

55.88%

We derive new perturbation bounds for eigenvalues of Hermitian matrices with
block structures. The structures we consider range from a standard 2-by-2 block
form to block tridiagonal and tridigaonal forms. The main idea is the
observation that an eigenvalue is insensitive to componentwise perturbations if
the corresponding eigenvector components are small. We show that the same idea
can be used to explain two well-known phenomena, one concerning extremal
eigenvalues of Wilkinson's matrices and another concerning the efficiency of
aggressive early deflation applied to the symmetric tridiagonal QR algorithm.; Comment: 12 pages

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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

Relevância na Pesquisa

55.92%

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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## The Lower Bounds for Eigenvalues of Elliptic Operators --By Nonconforming Finite Element Methods

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

66.08%

The aim of the paper is to introduce a new systematic method that can produce
lower bounds for eigenvalues. The main idea is to use nonconforming finite
element methods. The general conclusion herein is that if local approximation
properties of nonconforming finite element spaces $V_h$ are better than global
continuity properties of $V_h$, corresponding methods will produce lower bounds
for eigenvalues. More precisely, under three conditions on continuity and
approximation properties of nonconforming finite element spaces we first show
abstract error estimates of approximate eigenvalues and eigenfunctions.
Subsequently, we propose one more condition and prove that it is sufficient to
guarantee nonconforming finite element methods to produce lower bounds for
eigenvalues of symmetric elliptic operators. As one application, we show that
this condition hold for most nonconforming elements in literature. As another
important application, this condition provides a guidance to modify known
nonconforming elements in literature and to propose new nonconforming elements.
In fact, we enrich locally the Crouzeix-Raviart element such that the new
element satisfies the condition; we propose a new nonconforming element for
second order elliptic operators and prove that it will yield lower bounds for
eigenvalues. Finally...

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## Universal Bounds for Eigenvalues of the Polyharmonic Operators

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 11/10/2009

Relevância na Pesquisa

55.88%

We study eigenvalues of polyharmonic operators on compact Riemannian
manifolds with boundary (possibly empty). In particular, we prove a universal
inequality for the eigenvalues of the polyharmonic operators on compact domains
in a Euclidean space. This inequality controls the $k$th eigenvalue by the
lower eigenvalues, independently of the particular geometry of the domain. Our
inequality is sharper than the known Payne-P\'olya-Weinberg type inequality and
also covers the important Yang inequality on eigenvalues of the Dirichlet
Laplacian. We also prove universal inequalities for the lower order eigenvalues
of the polyharmonic operator on compact domains in a Euclidean space which in
the case of the biharmonic operator and the buckling problem strengthen the
estimates obtained by Ashbaugh. Finally, we prove universal inequalities for
eigenvalues of polyharmonic operators of any order on compact domains in the
sphere.; Comment: 30 pages

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