Multiple Eigenvalues Research Articles

Asymptotic expansions for approximate eigenvalues of integral operators with nonsmooth kernels of multiplicity $m>1$

We consider an integral operator with a kernel of the Green's function type. We prove the existence of asymptotic expansion of an eigenvalue of multiplicity $m>1$, when the integral operator is approximated by the iterated Galerkin operator. This enables us to use the Richardson extrapolation to increase the order of convergence of the eigenvalue. We consider a numerical example to illustrate our theoretical results.

Periodic solutions for the one-dimensional fractional Laplacian

In this paper we are concerned with the construction of periodic solutions of the nonlocal problem (−Δ)su=f(u) in R, where (−Δ)s stands for the s-Laplacian, s∈(0,1). We introduce a suitable framework which allows, by means of regularity, to link the searching of such solutions into the existence of the ones of a semilinear problem in a suitable Hilbert space. Then by a bifurcation theory from eigenvalues of odd multiplicity and also variational method that avoid the constant solutions we get existence theorems which are lately enlightened with the analysis of some examples. In particular, multiplicity results for generalized Benjamin-Ono equation are obtained.

Exceptional points for resonant states on parallel circular dielectric cylinders

Exceptional points (EPs) are special parameter values of a non-Hermitian eigenvalue problem where eigenfunctions corresponding to a multiple eigenvalue coalesce. In optics, EPs are associated with a number of counterintuitive wave phenomena and have potential applications in lasing, sensing, mode conversion, and spontaneous emission processes. For open photonic structures, resonant states are complex-frequency solutions of the Maxwell’s equations with outgoing radiation conditions. For open dielectric structures without material gain or loss, the eigenvalue problem for resonant states can have EPs, since it is non-Hermitian due to radiation losses. For applications in nanophotonics, it is important to understand EPs for resonant states on small finite dielectric structures consisting of conventional dielectric materials. To achieve this objective, we study EPs of resonant states on finite sets of parallel infinitely long circular dielectric cylinders with subwavelength radii. For systems with two, three, and four cylinders, we develop an efficient numerical method for computing EPs, present examples for second- and third-order EPs, and highlight their topological features. Our work provides insight to understanding EPs on more complicated photonic structures, and can be used as a simple platform to explore applications of EPs.

Minimal Laplacian controllability problems of threshold graphs

This study is concerned with the minimal controllability problem of a connected threshold graph following the Laplacian dynamics. The goal is to find the minimum number of controllers and a small set of vertices for the controllers to connect to render the graph Laplacian controllable. A simple algorithm is provided to generate a spanning set of orthogonal Laplacian eigenvectors of the graph from a straightforward computation on its Laplacian matrix. A necessary and sufficient condition for the graph to be Laplacian controllable is then proposed. The condition suggests that the minimum number of controllers to make a connected threshold graph Laplacian controllable is the maximum multiplicity of Laplacian eigenvalues of the graph, and this minimum can be achieved using a binary control matrix. If a controller can be connected to one vertex only, the minimum number is the difference between the number of vertices in the graph and the number of vertices with different degrees. The condition also implies that the controllers ensuring the Laplacian controllability should be connected to the vertices with repeating degrees to break the symmetry of the network topology. Several examples are provided to illustrate the authors' results.

A multi-parameter ultrasonic evaluation of mean grain size using optimization

An evaluation model based on multiple ultrasonic parameters was developed to control mean absolute error between an actual measured value and a model calculation value. Superalloy GH4169 was used to validate the presented model. A multi-dimensional ultrasonic parameter set was built for each sample by ultrasonic and nonlinear ultrasonic test and parameter calculation. The feature scale of different parameters was restricted to the same range using normalization. The dimensionality of the parameter set was descended by a correlation metric with grain size. A new one-dimensional ultrasonic characteristic parameter was comprised of the select normalized set of ultrasonic parameters by quadratic polynomial, exponential, Gauss kernel and linear mapping functions with undetermined coefficients. The relationship between the new ultrasonic multiple eigenvalues mapping parameter and estimated grain size was expressed using a linear fitting function. An optimization problem aimed at minimizing the mean absolute error between the estimated and quantitative measured grain size was solved using evolutionary algorithms to obtain coefficients for the mapping and fitting functions. These calculations were used to generate four full multi-parameter ultrasonic evaluation models, which were benchmarked against three single parameter evaluation models (i.e. velocity, attenuation coefficient, and backscattering). The performance analysis and verification using leave-one-out cross-validation demonstrate the utility of the MUE model through its high precision, good stabilization, super anti-interference and practicality.

Note on a nonlocal Sturm–Liouville problem with both right and left fractional derivatives

In this paper, we research the geometric multiplicity of eigenvalues for a nonlocal Sturm–Liouville eigenvalue problem. To this end, we study the uniqueness of solutions for a nonlocal Sturm–Liouville problem under some initial value conditions.

On the Spectrum of Differential Operators Under Riemannian Coverings

For a Riemannian covering p :M_{2} rightarrow M_{1}, we compare the spectrum of an essentially self-adjoint differential operator D_{1} on a bundle E_{1} rightarrow M_{1} with the spectrum of its lift D_{2} on p^{*}E_{1} rightarrow M_{2}. We prove that if the covering is infinite sheeted and amenable, then the spectrum of D_{1} is contained in the essential spectrum of any self-adjoint extension of D_{2}. We show that if the deck transformations group of the covering is infinite and D_{2} is essentially self-adjoint (or symmetric and bounded from below), then D_{2} (or the Friedrichs extension of D_{2}) does not have eigenvalues of finite multiplicity and in particular, its spectrum is essential. Moreover, we prove that if M_{1} is closed, then p is amenable if and only if it preserves the bottom of the spectrum of some/any Schrödinger operator, extending a result due to Brooks.

A globally convergent method to compute the real stability radius for time-delay systems

This paper presents a novel algorithm to compute the real stability radius for a linear delay system of retarded type with multiple delays. The real stability radius is the distance to instability, measured as the minimal real-valued perturbation that renders the system unstable. Our method is based on characterizing this distance to instability as the inverse of the global maximum of a real structured singular value function. We develop a criss-cross type algorithm that globally converges to this maximum, and whose convergence rate seems to be superlinear and sometimes quadratic in numerical experiments. The algorithm exploits that the intersections of these singular value functions with constant functions can be written as purely imaginary eigenvalues of certain delay eigenvalue problems (DEP) with positive and negative delays. This is an extension of the well-known linear case (without delays) where this results in algebraic eigenvalue problems. In addition, a novel numerical solver to compute all the imaginary eigenvalues of this DEP is also presented. It combines an approximation using spectral discretizations and an automatic procedure to determine the required number of discretization points. Finally, due to the presence of multiple eigenvalues at the maximum, these approximations are corrected with a block-Newton algorithm for nonlinear eigenvalue problems.

A unified definition and computation of Laplacian spectral distances

Laplacian spectral kernels and distances (e.g., biharmonic, heat diffusion, wave kernel distances) are easily defined through a filtering of the Laplacian eigenpairs. They play a central role in several applications, such as dimensionality reduction with spectral embeddings, diffusion geometry, image smoothing, geometric characterisations and embeddings of graphs. Extending the results recently derived in the discrete setting [38,39] to the continuous case, we propose a novel definition of the Laplacian spectral kernels and distances, whose approximation requires the solution of a set of inhomogeneous Laplace equations. Their discrete counterparts are equivalent to a set of sparse, symmetric, and well-conditioned linear systems, which are efficiently solved with iterative methods. Finally, we discuss the optimality of the Laplacian spectrum for the approximation of the spectral kernels, the relation between the spectral and Green kernels, and the stability of the spectral distances with respect to the evaluation of the Laplacian spectrum and to multiple Laplacian eigenvalues.

Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients

Let q ≥ 2 be a positive integer and let ( a j ) , ( b j ) , and ( c j ) (with j a non-negative integer) be three given C -valued and q-periodic sequences. Let A ( q ) : = A q − 1 ⋯ A 0 , where A j is as is given below. Assuming that the “monodromy matrix” A ( q ) has at least one multiple eigenvalue, we prove that the linear scalar recurrence x n + 3 = a n x n + 2 + b n x n + 1 + c n x n , n ∈ Z + is Hyers-Ulam stable if and only if the spectrum of A ( q ) does not intersect the unit circle Γ : = { w ∈ C : | w | = 1 } . Connecting this result with a recently obtained one it follows that the above linear recurrence is Hyers-Ulam stable if and only if the spectrum of A ( q ) does not intersect the unit circle.

Stability of nonlinear dynamical system motion under constantly acting perturbations

We consider the motion of a mechanical system with several degrees of freedom near zero of the phase space of states under conditions of constantly acting small perturbations. The generalized forces are represented in the dynamic equations by homogeneous forms of the first and the third degree with respect to the phase coordinates and by small time functions characterizing the constantly acting perturbations. It is assumed that there are no multiple eigenvalues of the matrix of the system linear part. For a definitely positive quadratic Lyapunov function, we define a differential inequality with Riccati differential comparison equation, together with an exponential differential inequality that is integrable in quadratures. When solving Riccati quadratic differential inequality, we assume one particular solution of Riccati equation to be known. As a result of integrating in quadratures of the exponential differential inequalities, the estimate of transient processes in the finite domain of the phase coordinates is obtained.

On the multiplicity of the second eigenvalue of the Laplacian in non simply connected domains – with some numerics –

We revisit an interesting example proposed by Maria Hoffmann-Ostenhof, the second author and Nikolai Nadirashvili of a bounded domain in R 2 for which the second eigenvalue of the Dirichlet Laplacian has multiplicity 3. We also analyze carefully the first eigenvalues of the Laplacian in the case of the disk with two symmetric cracks placed on a smaller concentric disk in function of their size.

The curl operator on odd-dimensional manifolds

We study the spectral properties of curl, a linear differential operator of first order acting on differential forms of appropriate degree on an odd-dimensional closed oriented Riemannian manifold. In three dimensions, its eigenvalues are the electromagnetic oscillation frequencies in vacuum without external sources. In general, the spectrum consists of the eigenvalue 0 with infinite multiplicity and further real discrete eigenvalues of finite multiplicity. We compute the Weyl asymptotics and study the ζ-function. We give a sharp lower eigenvalue bound for positively curved manifolds and analyze the equality case. Finally, we compute the spectrum for flat tori, round spheres, and 3-dimensional spherical space forms.

On multiplicities of eigenvalues of a boundary value problem on a snowflake graph

We find a vector majorizing any ordered vector of multiplicities of eigenvalues of a spectral problem generated by the Stieltjes string recurrence relations on a snowflake graph, i.e. a graph obtained by attaching edges to pendant vertices of a star graph.

Solve the linear quaternion-valued differential equations having multiple eigenvalues

The theory of two-dimensional linear quaternion-valued differential equations (QDEs) was recently established {see the work of Kou and Xia [Stud. Appl. Math. 141(1), 3–45 (2018)]}. They observed some profound differences between QDEs and ordinary differential equations. Also, an algorithm to evaluate the fundamental matrix by employing the eigenvalues and eigenvectors was presented in the work of Kou and Xia [Stud. Appl. Math. 141(1), 3–45 (2018)]. However, the fundamental matrix can be constructed providing that the eigenvalues are simple. If the linear system has multiple eigenvalues, how to construct the fundamental matrix? In particular, if the number of independent eigenvectors might be less than the dimension of the system, that is, the numbers of the eigenvectors are not enough to construct a fundamental matrix, how to find the “missing solutions”? The main purpose of this paper is to answer this question.

Error estimates of the quadrature finite element method with biquadratic finite elements for elliptic eigenvalue problems in the square domain

A positive definite eigenvalue problem for second-order self-adjoint elliptic differential operator defined on a square domain in the plane with homogeneous Dirichlet boundary condition is considered. This problem has a nondecreasing sequence of positive eigenvalues of finite multiplicity with a limit point at infinity. To the sequence of eigenvalues, there corresponds an orthonormal system of eigenfunctions. The original differential eigenvalue problem is approximated by the finite element method with numerical integration and Lagrange biquadratic finite elements. Error estimates for approximate eigenvalues and eigenfunctions are established.

Spectrum Sensing Using Multiple Large Eigenvalues and Its Performance Analysis

Cognitive radio (CR) is a promising technology to address the challenge of spectrum scarcity due to the massive number of objects in the Internet of Things (IoT). Equipping IoT objects with CR capability can also alleviate interference situations and achieve seamless connectivity in IoT. This paper deals with CR spectrum sensing and proposes a new eigenvalue-based detector by exploiting the summation of multiple large eigenvalues of the covariance matrix of received signals. By analyzing the distribution of the sum of the dependent large eigenvalues, we derive an approximate but explicit expression for the theoretical performance of the proposed detector. The theoretical analysis of the proposed detector is validated and its superior performance is demonstrated with real world signals. It is shown that the proposed detector outperforms the existing eigenvalue-based detectors and is more robust against noise uncertainty.

Transmission of Higher Order Solitons Created by Optical Multiplexing

The nonlinear Fourier transform (NFT) is a promising tool to linearize the inherently nonlinear optical fiber channel. The NFT transforms a time-domain signal into the continuous and the discrete spectrum. The discrete spectrum is composed of an arbitrary number of complex valued discrete eigenvalues and their associated amplitudes. These discrete eigenvalues relate to solitons, which maintain their shape or return to it in an oscillating manner, while passing through the optical channel. Higher order solitons consisting of multiple eigenvalues are complex pulses, which are created and demodulated by sophisticated digital signal processing (DSP) leading to demanding hardware requirements. This paper shows a way to work with higher order solitons in a wavelength division multiplexing such as fashion by using optical-electrical signal processing and presents boundaries of this method. Optical-electrical signal processing decreases the required electrical and electro-optical hardware specifications substantially and enables to use a simplified DSP. The proposed creation method is subsequently employed to transmit higher order solitons consisting of five QPSK modulated eigenvalues. Furthermore, the optical-electrical processing is benchmarked against the Darboux transformation, which creates higher order solitons purely numerically. The results show that for a fifth-order soliton transmission the proposed method can significantly reduce the hardware requirements and DSP complexity.

Polynomial (chaos) approximation of maximum eigenvalue functions

This paper is concerned with polynomial approximations of the spectral abscissa function (the supremum of the real parts of the eigenvalues) of a parameterized eigenvalue problem, which are closely related to polynomial chaos approximations if the parameters correspond to realizations of random variables. Unlike in existing works, we highlight the major role of the smoothness properties of the spectral abscissa function. Even if the matrices of the eigenvalue problem are analytic functions of the parameters, the spectral abscissa function may not be everywhere differentiable, even not everywhere Lipschitz continuous, which is related to multiple rightmost eigenvalues or rightmost eigenvalues with multiplicity higher than one. The presented analysis demonstrates that the smoothness properties heavily affect the approximation errors of the Galerkin and collocation-based polynomial approximations, and the numerical errors of the evaluation of coefficients with integration methods. A documentation of the experiments, conducted on the benchmark problems through the software Chebfun, is publicly available.

The location of classified edges due to the change in the geometric multiplicity of an eigenvalue in a tree

Abstract Given a combinatorially symmetric matrix A whose graph is a tree T and its eigenvalues, edges in T can be classified in four categories, based upon the change in geometric multiplicity of a particular eigenvalue, when the edge is removed. We investigate a necessary and sufficient condition for each classification of edges. We have similar results as the case for real symmetric matrices whose graph is a tree. We show that a g-2-Parter edge, a g-Parter edge and a g-downer edge are located separately from each other in a tree, and there is a g-neutral edge between them. Furthermore, we show that the distance between a g-downer edge and a g-2-Parter edge or a g-Parter edge is at least 2 in a tree. Lastly we give a combinatorially symmetric matrix whose graph contains all types of edges.