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## Shape sensitivity analysis of the eigenvalues of the Reissner-Mindlin system

Buoso, Davide; Lamberti, Pier Domenico
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.68%
We consider the eigenvalue problem for the Reissner-Mindlin system arising in the study of the free vibration modes of an elastic clamped plate. We provide quantitative estimates for the variation of the eigenvalues upon variation of the shape of the plate. We also prove analyticity results and establish Hadamard-type formulas. Finally, we address the problem of minimization of the eigenvalues in the case of isovolumetric domain perturbations. In the spirit of the Rayleigh conjecture for the biharmonic operator, we prove that balls are critical points with volume constraint for all simple eigenvalues and the elementary symmetric functions of multiple eigenvalues.; Comment: Preprint version of a paper accepted for publication in SIAM Journal on Mathematical Analysis

## Tree simplification and the 'plateaux' phenomenon of graph Laplacian eigenvalues

Saito, Naoki; Woei, Ernest
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.68%
We developed a procedure of reducing the number of vertices and edges of a given tree, which we call the "tree simplification procedure," without changing its topological information. Our motivation for developing this procedure was to reduce computational costs of graph Laplacian eigenvalues of such trees. When we applied this procedure to a set of trees representing dendritic structures of retinal ganglion cells of a mouse and computed their graph Laplacian eigenvalues, we observed two "plateaux" (i.e., two sets of multiple eigenvalues) in the eigenvalue distribution of each such simplified tree. In this article, after describing our tree simplification procedure, we analyze why such eigenvalue plateaux occur in a simplified tree, and explain such plateaux can occur in a more general graph if it satisfies a certain condition, identify these two eigenvalues specifically as well as the lower bound to their multiplicity.

## Estimates on the number of eigenvalues of two-particle discrete Schr\"odinger operators

Albeverio, Sergio; Lakaev, Saidakhmat N.; Abdullaev, Janikul I.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.68%
Two-particle discrete Schr\"{o}dinger operators $H(k)=H_{0}(k)-V$ on the three-dimensional lattice $\Z^3,$ $k$ being the two-particle quasi-momentum, are considered. An estimate for the number of the eigenvalues lying outside of the band of $H_{0}(k)$ via the number of eigenvalues of the potential operator $V$ bigger than the width of the band of $H_{0}(k)$ is obtained. The existence of non negative eigenvalues below the band of $H_{0}(k)$ is proven for nontrivial values of the quasi-momentum $k\in \T^3\equiv (-\pi,\pi]^3$, provided that the operator H(0) has either a zero energy resonance or a zero eigenvalue. It is shown that the operator $H(k), k\in \T^3,$ has infinitely many eigenvalues accumulating at the bottom of the band from below if one of the coordinates $k^{(j)},j=1,2,3,$ of $k\in \T^3$ is $\pi.$; Comment: 12 pages

## Preconditioned eigensolvers for large-scale nonlinear Hermitian eigenproblems with variational characterizations. II. Interior eigenvalues

Szyld, Daniel B.; Vecharynski, Eugene; Xue, Fei
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.68%
We consider the solution of large-scale nonlinear algebraic Hermitian eigenproblems of the form $T(\lambda)v=0$ that admit a variational characterization of eigenvalues. These problems arise in a variety of applications and are generalizations of linear Hermitian eigenproblems $Av\!=\!\lambda Bv$. In this paper, we propose a Preconditioned Locally Minimal Residual (PLMR) method for efficiently computing interior eigenvalues of problems of this type. We discuss the development of search subspaces, preconditioning, and eigenpair extraction procedure based on the refined Rayleigh-Ritz projection. Extension to the block methods is presented, and a moving-window style soft deflation is described. Numerical experiments demonstrate that PLMR methods provide a rapid and robust convergence towards interior eigenvalues. The approach is also shown to be efficient and reliable for computing a large number of extreme eigenvalues, dramatically outperforming standard preconditioned conjugate gradient methods.

## Free convolution with a semi-circular distribution and eigenvalues of spiked deformations of Wigner matrices

Capitaine, Mireille; Donati-Martin, Catherine; Féral, Delphine; Février, Maxime
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.68%
We investigate the asymptotic behavior of the eigenvalues of spiked perturbations of Wigner matrices when the dimension goes to infinity. The entries of the Hermitian Wigner matrix have a distribution which is symmetric and satisfies a Poincar\'e inequality. The perturbation matrix is a deterministic Hermitian matrix whose spectral measure converges to some probability measure with compact support. We assume that this perturbation matrix has a fixed number of fixed eigenvalues (spikes) outside the support of its limiting spectral measure whereas the distance between the other eigenvalues and this support uniformly goes to zero as the dimension goes to infinity. We establish that only a particular subset of the spikes will generate some eigenvalues of the deformed model which will converge to some limiting points outside the support of the limiting spectral measure. This phenomenon can be fully described in terms of free probability involving the subordination function related to the additive free convolution of the limiting spectral measure of the perturbation matrix by a semi-circular distribution. Note that up to now only finite rank perturbations had been considered (even in the deformed GUE case).

## Semicontinuity of Eigenvalues under Flat Convergence in Euclidean Space

Portegies, Jacobus W
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.68%
Recall that Federer-Fleming defined the notion of flat convergence of submanifolds of Euclidean space to solve the Plateau problem. Here we prove the upper semicontinuity of Neumann eigenvalues of the submanifolds when they converge in the flat sense without losing volume. With an additional condition on the boundaries of the submanifolds we prove the Dirichlet eigenvalues are semicontinuous as well. We show this additional boundary condition is necessary as well as the condition that the volumes converge to the volume of the limit submanifold. As an application of our theorems we see that the Dirichlet and Neumann eigenvalues of a sequence of surfaces with a common smooth boundary curve approaching the solution to the Plateau problem are upper semicontinuous. This work is built upon Fukaya's study of the metric measure convergence of Riemannian manifolds. One may recall that Cheeger-Colding proved continuity of the eigenvalues when manifolds with uniform lower Ricci curvature bounds converge in the metric measure sense. While they obtain continuity, here, we produce an example demonstrating that continuity is impossible to obtain with our weaker hypothesis.; Comment: 13 pages, 3 figures

## Explicit eigenvalues of certain scaled trigonometric matrices

Sra, Suvrit
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.68%
In a very recent paper "\emph{On eigenvalues and equivalent transformation of trigonometric matrices}" (D. Zhang, Z. Lin, and Y. Liu, LAA 436, 71--78 (2012)), the authors motivated and discussed a trigonometric matrix that arises in the design of finite impulse response (FIR) digital filters. The eigenvalues of this matrix shed light on the FIR filter design, so obtaining them in closed form was investigated. Zhang \emph{et al.}\ proved that their matrix had rank-4 and they conjectured closed form expressions for its eigenvalues, leaving a rigorous proof as an open problem. This paper studies trigonometric matrices significantly more general than theirs, deduces their rank, and derives closed-forms for their eigenvalues. As a corollary, it yields a short proof of the conjectures in the aforementioned paper.; Comment: 7 pages; fixed Lemma 2, tightened inequalities

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

Hu, Jun; Ma, Rui
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.68%
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.

## Eigenvalues of harmonic almost submersions

Loubeau, E.; Slobodeanu, R.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.68%
Maps between Riemannian manifolds which are submersions on a dense subset, are studied by means of the eigenvalues of the pull-back of the target metrics, the first fundamental form. Expressions for the derivatives of these eigenvalues yield characterizations of harmonicity, totally geodesic maps and biconformal changes of metric preserving harmonicity. A Schwarz lemma for pseudo harmonic morphisms is proved, using the dilatation of the eigenvalues and, in dimension five, a Bochner technique method, involving the Laplacian of the difference of the eigenvalues, gives conditions forcing pseudo harmonic morphisms to be harmonic morphisms.; Comment: 29 pages

## Outlier eigenvalues for deformed i.i.d. random matrices

Bordenave, Charles; Capitaine, Mireille
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.68%
We consider a square random matrix of size N of the form A + Y where A is deterministic and Y has iid entries with variance 1/N. Under mild assumptions, as N grows, the empirical distribution of the eigenvalues of A+Y converges weakly to a limit probability measure \beta on the complex plane. This work is devoted to the study of the outlier eigenvalues, i.e. eigenvalues in the complement of the support of \beta. Even in the simplest cases, a variety of interesting phenomena can occur. As in earlier works, we give a sufficient condition to guarantee that outliers are stable and provide examples where their fluctuations vary with the particular distribution of the entries of Y or the Jordan decomposition of A. We also exhibit concrete examples where the outlier eigenvalues converge in distribution to the zeros of a Gaussian analytic function.; Comment: Introduction developed and minor corrections

## The WKB approximation of semiclassical eigenvalues of the Zakharov-Shabat problem

Kim, Yeongjoh; Lee, Long; Lyng, Gregory D.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.68%
We numerically compute eigenvalues of the non-self-adjoint Zakharov--Shabat problem in the semiclassical regime. In particular, we compute the eigenvalues for a Gaussian potential and compare the results to the corresponding (formal) WKB approximations used in the approach to the semiclassical or zero-dispersion limit of the focusing nonlinear Schroedinger equation via semiclassical soliton ensembles. This numerical experiment, taken together with recent numerical experiments [17,18], speaks directly to the viability of this approach; in particular, our experiment suggests a value for the rate of convergence of the WKB eigenvalues to the true eigenvalues in the semiclassical limit. This information provides some hint as to how these approximations might be rigorously incorporated into the asymptotic analysis of the singular limit for the associated nonlinear partial differential equation.; Comment: 21 pages, 4 figures

## Density of Positive Eigenvalues of the Generalized Gaussian Unitary Ensemble

Bouali, Mohamed
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.68%
We compute exact asymptotic of the statistical density of random matrices belonging to the Generalized Gaussian orthogonal, unitary and symplectic ensembles such that there no eigenvalues in the interval $[\sigma, +\infty[$. In particular, we show that the probability that all the eigenvalues of an $(n\times n)$ random matrix are positive (negative) decreases for large $n$ as $\sim exp[-\beta\theta(\alpha)n^2]$ where the Dyson index $\beta$ characterizes the ensemble, $\alpha$ is some extra parameter and the exponent $\theta(\alpha)$ is a function of $\alpha$ which will be given explicitly. For $\alpha=0$, $\theta(0)= (\log 3)/4 = 0.274653...$ is universal. We compute the probability that the eigenvalues lie in the interval $[\sigma,+\infty[$ with $(\sigma>0,\; {\rm if}\;\alpha>0)$ and $(\sigma\in\mathbb R,\; {\rm if }\;\alpha=0)$. This generalizing the celebrated Wigner semicircle law to these restricted ensembles. It is found that the density of eigenvalues generically exhibits an inverse square-root singularity at the location of the barriers. These results generalized the case of Gaussian random matrices ensemble studied in \cite{D}, \cite{S}.; Comment: arXiv admin note: text overlap with arXiv:0801.1730 by other authors

## Inequalities among eigenvalues of different self-adjoint discrete Sturm-Liouville problems

Zhu, Hao; Shi, Yuming
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.68%
In this paper, inequalities among eigenvalues of different self-adjoint discrete Sturm-Liouville problems are established. For a fixed discrete Sturm-Liouville equation, inequalities among eigenvalues for different boundary conditions are given. For a fixed boundary condition, inequalities among eigenvalues for different equations are given. These results are obtained by applying continuity and discontinuity of the n-th eigenvalue function, monotonicity in some direction of the n-th eigenvalue function, which were given in our previous papers, and natural loops in the space of boundary conditions. Some results generalize the relevant existing results about inequalities among eigenvalues of different Sturm-Liouville problems.; Comment: 32 pages, 5 figures

## On the eigenvalues of certain Cayley graphs and arrangement graphs

Chen, Bai Fan; Ghorbani, Ebrahim; Wong, Kok Bin
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.68%
In this paper, we show that the eigenvalues of certain classes of Cayley graphs are integers. The (n,k,r)-arrangement graph A(n,k,r) is a graph with all the k-permutations of an n-element set as vertices where two k-permutations are adjacent if they differ in exactly r positions. We establish a relation between the eigenvalues of the arrangement graphs and the eigenvalues of certain Cayley graphs. As a result, the conjecture on integrality of eigenvalues of A(n,k,1) follows.; Comment: 12 pages, final version

## Descriptor approach for eliminating spurious eigenvalues in hydrodynamic equations

Manning, M. Lisa; Bamieh, B.; Carlson, J. M.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.68%
We describe a general framework for avoiding spurious eigenvalues -- unphysical unstable eigenvalues that often occur in hydrodynamic stability problems. In two example problems, we show that when system stability is analyzed numerically using {\em descriptor} notation, spurious eigenvalues are eliminated. Descriptor notation is a generalized eigenvalue formulation for differential-algebraic equations that explicitly retains algebraic constraints. We propose that spurious eigenvalues are likely to occur when algebraic constraints are used to analytically reduce the number of independent variables in a differential-algebraic system of equations before the system is approximated numerically. In contrast, the simple and easily generalizable descriptor framework simultaneously solves the differential equations and algebraic constraints and is well-suited to stability analysis in these systems.; Comment: 13 pages, 1 figure, revised for submission to SIAM Sci. Comp., moved background information to appendices

## Inequalities for eigenvalues of the weighted Hodge Laplacian

Chen, Daguang; Zhang, Yingying
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.68%
In this paper, we obtain "universal" inequalities for eigenvalues of the weighted Hodge Laplacian on a compact self-shrinker of Euclidean space. These inequalities generalize the Yang-type and Levitin-Parnovski inequalities for eigenvalues of the Laplacian and Laplacian. From the recursion formula of Cheng and Yang \cite{ChengYang07}, the Yang-type inequality for eigenvalues of the weighted Hodge Laplacian are optimal in the sense of the order of eigenvalues.; Comment: 19 pages,any comments are welcome

## On the distribution of eigenvalues of Maass forms on certain moonshine groups

Jorgenson, Jay; Smajlović, Lejla; Then, Holger
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.68%
In this paper we study, both analytically and numerically, questions involving the distribution of eigenvalues of Maass forms on the moonshine groups $\Gamma_0(N)^+$, where N>1$is a square-free integer. After we prove that$\Gamma_0(N)^+$has one cusp, we compute the constant term of the associated non-holomorphic Eisenstein series. We then derive an "average" Weyl's law for the distribution of eigenvalues of Maass forms, from which we prove the "classical" Weyl's law as a special case. The groups corresponding to N=5 and N=6 have the same signature; however, our analysis shows that, asymptotically, there are infinitely more cusp forms for$\Gamma_0(5)^+$than for$\Gamma_0(6)^+$. We view this result as being consistent with the Phillips-Sarnak philosophy since we have shown, unconditionally, the existence of two groups which have different Weyl's laws. In addition, we employ Hejhal's algorithm, together with recently developed refinements from [31], and numerically determine the first 3557 of$\Gamma_0(5)^+$and the first 12474 eigenvalues of$\Gamma_0(6)^+\$. With this information, we empirically verify some conjectured distributional properties of the eigenvalues.; Comment: A version with higher resolution figures can be downloaded from http://www.maths.bris.ac.uk/~mahlt/research/JST2012a.pdf

## On the eigenvalues of some nonhermitian oscillators

Fern/'andez, Francisco M.; Garcia, Javier
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.68%
We consider a class of one-dimensional nonhermitian oscillators and discuss the relationship between the real eigenvalues of PT-symmetric oscillators and the resonances obtained by different authors. We also show the relationship between the strong-coupling expansions for the eigenvalues of those oscillators. Comparison of the results of the complex rotation and the Riccati-Pad\'{e} methods reveals that the optimal rotation angle converts the oscillator into either a PT-symmetric or an Hermitian one. In addition to the real positive eigenvalues the PT-symmetric oscillators exhibit real positive resonances under different boundary conditions. They can be calculated by means of the straightforward diagonalization method. The Riccati-Pad\'e method yields not only the resonances of the nonhermitian oscillators but also the eigenvalues of the PT-symmetric ones.

## Analysis of structural correlations in a model binary 3D liquid through the eigenvalues and eigenvectors of the atomic stress tensors

Levashov, Valentin A.
Tipo: Artigo de Revista Científica