Properties Of Eigenvalues Research Articles

Computation of Spectral Parameter of Discontinuous Dirac Systems with a Gaussian Multiplier

We aim in this paper to apply a sinc-Gaussian technique to compute the eigenvalues of a Dirac system which has a discontinuity at one point and contains a spectral parameter in all boundary conditions. We establish the needed properties of eigenvalues of our problem. The error of this method decays exponentially in terms of the number of involved samples. Therefore the accuracy of the new technique is higher than the classical sinc-method. Numerical worked examples with tables and illustrative figures are given at the end of the paper.

Asymptotic properties of eigenvalues and eigenfunctions of a Sturm-Liouville problem with discontinuous weight function

In this paper, we extend some spectral properties of regular Sturm-Liouville problems to those which consist of a Sturm-Liouville equation with discontinuous weight at two interior points together with spectral parameter-dependent boundary conditions. By modifying some techniques of [C. T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 77 (1977) 293-308; O. Sh. Mukhtarov and M. Kadakal, Some spectral properties of one Sturm-Liouville type problem with discontinuous weight, Siberian Mathematical Journal, 46 (2005) 681-694], we give an operator-theoretic formulation for the considered problem and obtain asymptotic formulas for the eigenvalues and eigenfunctions. 2010 Mathematics Subject Classification: 34L20; 35R10

Financial networks model based on random matrix

Random matrix theory is applied to study the correlation between different financial correlation coefficient matrices in the financial field. Correlation coefficient matrix is a key factor for constructing a network. In this paper we relate the random matrix theory to the network construction to study the financial networks model in terms of the random matrix. We select the stock data of Shanghai stock market, and divide them into four stages. We discuss the statistical properties of eigenvalues in financial correlation coefficient matrix and random matrix based on the random matrix theory, and improve the existing denoising method to construct the correlation coefficient matrix and to make it more suitable for building financial networks. After that we can build the financial network model. Then we analyze and compare the original financial network, the denoising financial network and the noise financial network in terms of the random matrix theory and the key node of networks. It is found that the primary important information is still in the original network, and the noise information corresponds to the information which the nodes of small degree in the original network include. Finally we analyze the topological structure of the financial networks, such as the minimum spanning tree, the motif and community structure. We also find that the topological properties of the improved financial networks are more remarkable and the topological structure is more compact.

Tate conjecture for products of Fermat varieties over finite fields

We prove under some assumptions that the Tate conjecture holds for products of Fermat varieties of different degrees. The method is to use a combinatorial property of eigenvalues of geometric Frobenius acting on l-adic etale cohomology.

<i>F</i> Functional and the First Eigenvalue for Quasi-Einstein Metrics

In this paper, we deal with the monotonicity properties of the first eigenvalue of the Laplacian operator. Firstly, by using the monotonicity formula of theFfunctional, we derive a monotonicity formula of the first eigenvalue of the Laplacian operator. Based on this, we also prove an exponential decreasing property of the first eigenvalue.

Fredholm's alternative breaks the confinement of electromagnetic waves

The recovery of information from the so called electromagnetic evanescent waves seems to be a very well explained item. Nevertheless, the travelling waves that becomes from the evanescent waves emerge from integral or differential equations that are very different to those describing the conventional ones. Indeed, we can say that the two kinds of solutions, the travelling and evanescent waves represent a mutually discriminating problem in which we cannot have simultaneous validity of both kinds of equations even they represents the physical evolution of a the same system. But if we can describe our system with a Fredholm's equation we can relate both situations through the properties of the Fredholm's eigenvalue. When the Fredholm's eigenvalue has its values into certain range then Fredholm's equation describes a normal travelling spectrum; otherwise, we are in the presence of another type of equation with abnormal or special behavior. In this work, we analyze the so-named Fredholm's alternative, which enables us to describe the change of positive refraction index-like conditions of broadcasting media to negative refraction index-like conditions. We also sketch some general conditions for the Fredholm's eigenvalue in order to establish general rules for the breaking of the waves’ confinement.

Sequential method for speech segmentation based on Random Matrix Theory

Speech segmentation to covariance-stationary regions is of great interest in various fields of speech processing. This study presents a novel sequential segmentation algorithm relying on Random Matrix Theory (RMT). The proposed approach utilises an RMT-derived test statistic checking stationarity of sample covariance matrices. Utilising the statistical properties of the sample eigenvalues recently investigated in the RMT literature, a new expression for the decision threshold is derived in the non-asymptotic case where the number of samples is comparable with the dimension. This derivation is of great help to detect very short non-stationary intervals where the performance of most segmentation methods based on Autoregressive models degrades. Furthermore, based on the results of previous step, a new segmentation procedure is proposed. This procedure is applied to both synthetic and real-world speech data. Comparing the obtained simulation results with the state-of-the-art segmentation algorithms, the proposed approach has demonstrated good performance with extremely lower computational cost.

Asymptotics for Products of Characteristic Polynomials in Classical β-Ensembles

We study the local properties of eigenvalues for the Hermite (Gaussian), Laguerre (Chiral), and Jacobi β-ensembles of N×N random matrices. More specifically, we calculate scaling limits of the expectation value of products of characteristic polynomials as N→∞. In the bulk of the spectrum of each β-ensemble, the same scaling limit is found to be $e^{p_{1}}{}_{1}F_{1}$ , whose exact expansion in terms of Jack polynomials is well known. The scaling limit at the soft edge of the spectrum for the Hermite and Laguerre β-ensembles is shown to be a multivariate Airy function, which is defined as a generalized Kontsevich integral. As corollaries, when β is even, scaling limits of the k-point correlation functions for the three ensembles are obtained. The asymptotics of the multivariate Airy function for large and small arguments is also given. All the asymptotic results rely on a generalization of Watson’s lemma and the steepest descent method for integrals of Selberg type.

The Lower/Upper Bound Property of Approximate Eigenvalues by Nonconforming Finite Element Methods for Elliptic Operators

This paper is a complement of the work (Hu et al. in arXiv:1112.1145v1[math.NA], 2011), where a general theory is proposed to analyze the lower bound property of discrete eigenvalues of elliptic operators by nonconforming finite element methods. One main purpose of this paper is to propose a novel approach to analyze the lower bound property of discrete eigenvalues produced by the Crouzeix---Raviart element when exact eigenfunctions are smooth. In particular, under some conditions on the triangular mesh, it is proved that the Crouzeix---Raviart element method of the Laplace operator yields eigenvalues below exact ones. Such a theoretical result explains most of numerical results in the literature and also partially answers the problem of Boffi (Acta Numerica 1---120, 2010). This approach can be applied to the Crouzeix---Raviart element of the Stokes eigenvalue problem and the Morley element of the buckling eigenvalue problem of a plate. As a second main purpose, a new identity of the error of eigenvalues is introduced to study the upper bound property of eigenvalues by nonconforming finite element methods, which is successfully used to explain why eigenvalues produced by the rotated $$Q_1$$ Q 1 element of second order elliptic operators (when eigenfunctions are smooth), the Adini element (when eigenfunctions are singular) and the new Zienkiewicz-type element of fourth order elliptic operators, are above exact ones.

Impulsive spatial control of invading pests by generalist predators

We model the conditions for pest eradication in a reaction-diffusion system made of a prey and a generalist predator through spatial impulsive control within a bounded domain. The motivating example is the control of the invasive horse chestnut leafminer moth through the yearly destruction of leaves in autumn, in which both the pest and its parasitoids overwinter. The model is made of two integro-partial differential equations, the integral portion describing the within-year immigration from the whole domain. The problem of pest eradication is strongly related to some appropriate eigenvalue problems. Basic properties of the principal eigenvalues of these problems are derived by using of Krein-Rutman's theorem and of comparison results for parabolic equations with non-local terms. Spatial control of the pest can be achieved, if one of these principal eigenvalues is large enough, at an exponential rate. This is true without and with parasitoids, the latter case being of course more rapid. We discuss the possible implementation of these results to the leafminer invasion problem and discuss complementary methods.

Quantum Classical Correspondence in Nonintegrable Systems with Continuous Parameter

We study quantum classical correspondence in nonintegrable systems with a continuous parameter. In a stadium billiard with an aspect ratio as a parameter, we show that the slope of the parametric motion of eigenvalues is mainly due to the first derivative of periodic orbit length with respect to the parameter. We also present a viewpoint for scarred wavefunctions in a continuous change of the systems. Since the discovery of scars 1 > in a stadium billiard/> a number of authors have studied properties of scars numerically in various systems.3> Several theoretical studies4 >,s> have tried to explain the extra accumulation of the wave function on an unstable periodic orbit (PO). Although these works seemed to confirm the existence of scars both numerically and theoretically, the semiclassical theory for individual eigenstates in nonintegrable systems has not been given yet. The difficulty in the investigation for the individual eigenstates is due to the strong interaction with other states. The origin of the repulsive interaction has been studied by many authors, and was related to chaotic behavior in the classical motion. 6> When we change a parameter in the Hamiltonian, it is well known that eigenvalues of nonintegrable systems show a number of avoided crossings due to the interaction. The definite representation for the interaction was given by the level dynamics,?> and the motion of levels was shown to be described by a classical Hamiltonian with complete integrability. The curvature of levels, i.e. the second derivative of eigenvalues with respect to the parameter, have been introduced to characterize the parametric property of eigenvalues, and the expressions for the large curvature tail of the distribution have been derived. 8> The universal behavior in large curvatures is checked in various systems by numerical calculations. 9 > On the other hand, the nonuniversal behavior of small curvatures was discovered numerically. 10> Zakrzewski and Delande11> suggest that the discrepancy in small curvatures can be used to classify the degree of scarring in different systems. In this paper, we consider the relation between classical PO's and the parametric motion of eigenvalues of the stadium billiard with an aspect ratio as a parameter. In § 2, we study properties of PO's when we change the parameter continuously. We concentrate on the continuity of the PO's and calculate the first derivative of the

Spectral properties of Google matrix of Wikipedia and other networks

We study the properties of eigenvalues and eigenvectors of the Google matrix of the Wikipedia articles hyperlink network and other real networks. With the help of the Arnoldi method we analyze the distribution of eigenvalues in the complex plane and show that eigenstates with significant eigenvalue modulus are located on well defined network communities. We also show that the correlator between PageRank and CheiRank vectors distinguishes different organizations of information flow on BBC and Le Monde web sites.

Low memory and low complexity iterative schemes for a nonsymmetric algebraic Riccati equation arising from transport theory

We reconsider Newton’s method and two fixed-point methods for finding the minimal positive solution of a nonsymmetric algebraic Riccati equation arising from transport theory. We rewrite the subproblem of the Newton and fixed-point iterative schemes into an equivalent form with some special structure. By the use of the particular structure of the subproblem, we then present low memory and low complexity versions of these iterative methods with a factored alternating-direction-implicit iteration. Some properties of eigenvalues for iterative coefficient matrices in solving the subproblem are derived and the convergence of the proposed methods is established. Numerical experiments show that the new iterative schemes are highly efficient to obtain the minimal positive solution. The proposed low memory and low complexity Newton’s method is particularly efficient for solving large scale Riccati equation arising from transport theory.

Lower Bounds for Eigenvalues of the Stokes Operator

AbstractIn this paper, we propose a condition that can guarantee the lower bound property of the discrete eigenvalue produced by the finite element method for the Stokes operator. We check and prove this condition for four nonconforming methods and one conforming method. Hence they produce eigenvalues which are smaller than their exact counterparts.

Criticality of viscous Hamilton–Jacobi equations and stochastic ergodic control

This paper deals with nonlinear additive eigenvalue problems for viscous Hamilton–Jacobi equations which appear in stochastic ergodic control. Certain qualitative properties of principal eigenvalues and associated eigenfunctions are studied. Such analysis plays a key role in studying the recurrence and transience of feedback diffusions for the corresponding stochastic control problems. Our results can be regarded as a nonlinear extension of the criticality theory for Schrödinger operators with decaying potentials.

Fractional Sturm–Liouville problem

In this paper, we define some Fractional Sturm–Liouville Operators (FSLOs) and introduce two classes of Fractional Sturm–Liouville Problems (FSLPs) namely regular and singular FSLP. The operators defined here are different from those defined in the literature in the sense that the operators defined here contain left and right Riemann–Liouville and left and right Caputo fractional derivatives. For both classes we investigate the eigenvalue and eigenfunction properties of the FSLOs. In the class of regular FSLPs, we discuss two types of FSLPs. As an operator for the class of singular FSLPs, we introduce a Fractional Legendre Equation (FLE) and discuss its solution. It is shown that the Legendre Polynomials resulting from an FLE are the same as those obtained from the integer order Legendre equation; however, the eigenvalues of the two equations differ. Using the Legendre integral transform we demonstrate some applications of our results by solving two fractional differential equations, one ordinary and the other partial. It is our hope that this paper will initiate new research in the area of FSLPs and many of its variations.

Geometrical Structure of Laplacian Eigenfunctions

We summarize the properties of eigenvalues and eigenfunctions of the Laplace operator in bounded Euclidean domains with Dirichlet, Neumann or Robin boundary condition. We keep the presentation at a level accessible to scientists from various disciplines ranging from mathematics to physics and computer sciences. The main focus is put onto multiple intricate relations between the shape of a domain and the geometrical structure of eigenfunctions.

Faber–Krahn type inequality for unicyclic graphs

The Faber–Krahn inequality is related to the smallest nonzero eigenvalue of the Laplacian operator with Dirichlet boundary condition on a bounded domain in ℝ n . In this article, we investigate some properties of the first Dirichlet eigenvalue of unicyclic graphs. These results are used to show that the Faber–Krahn type inequality also holds for unicyclic graphs with a given graphic unicyclic degree sequence with minor conditions.

A note on unimodular eigenvalues for palindromic eigenvalue problems

We consider the occurrence of unimodular eigenvalues for palindromic eigenvalue problems associated with the matrix polynomial where A i *=A n−i with M* ≡ M T, M H or . From the properties of palindromic eigenvalues and their characteristic polynomials, we show that eigenvalues are not generically excluded from the unit circle, thus occurring quite often, except for the complex transpose case when P n is complex and M* ≡ M T. This behaviour is observed in numerical simulations and has important implications on several applications such as the vibration of fast trains, surface acoustic wave filters, stability of time-delay systems and crack modelling.

On the rhotrix eigenvalues and eigenvectors

The concept of rhotrix eigenvector eigenvalue problem (REP) was introduced by Aminu (Int. J. Math. Educ. Sci. Technol. 41:98–105, 2010). As an extension to this, we have presented in this article some properties of rhotrix eigenvalues and eigenvectors considering the numerous applications of matrix eigenvector eigenvalue problem in areas of Applied Mathematics. A classic problem in linear algebra, the diagonalization problem in terms of rhotrices (RDP) is also introduced.