Properties Of Eigenvalues Research Articles

Some qualitative questions on the equation −div(a(x,u,∇u))=f(x, u)

In this article, we establish several applications of Picone's identity for the operator of the form−div(a(x,u,∇u)), such as Hardy type inequality, Sturmian comparison theorem, monotonicity property of the first eigenvalue, nonexistence of positive supersolutions and Caccioppoli inequality under certain conditions on a.

On discontinuous Dirac operator with eigenparameter dependent boundary and two transmission conditions

In this paper, we consider a discontinuous Dirac operator with eigenparameter dependent both boundary and two transmission conditions. We introduce a suitable Hilbert space formulation and get some properties of eigenvalues and eigenfunctions. Then we investigate the Green’s function, the resolvent operator, and some uniqueness theorems by using the Weyl function and some spectral data.

Spectral Analysis for a Singular Differential System with Integral Boundary Conditions

In this paper, by constructing a cone K1 × K2 in the Cartesian product space C[0, 1] × C[0, 1], and using spectral analysis of the relevant linear operator for the corresponding differential system, some properties of the first eigenvalue corresponding to the relevant linear operator are obtained, and the fixed-point index of nonlinear operator in the K1 × K2 is calculated explicitly and the existence of at least one positive solution or two positive solutions of the singular differential system with integral boundary conditions is established.

A stable Cox–Ingersoll–Ross model with restart

We consider a stable Cox–Ingersoll–Ross model in a domain D=[0,∞) which is killed at the boundary and which, while alive, jumps instantaneously at an exponentially distributed random time with intensity λ>0 to a new point, according to a distribution ν. From this new point, it repeats the above behavior independently of what has transpired previously. We give the infinitesimal generator of this new process and prove that the corresponding semigroup is compact and thus the spectrum of the generator consists exclusively of eigenvalues. We further show that the principal eigenvalue gives the exponential rate of decay of not exiting the domain by time t. Finally, we study some properties of the principal eigenvalue.

Eigenvalues, global bifurcation and positive solutions for a class of nonlocal elliptic equations

In this paper, we shall study global bifurcation phenomenon for the following Kirchhoff type problem: \begin{equation*} \begin{cases} \displaystyle -\bigg(a+b\int_\Omega \vert \nabla u\vert^2dx\bigg)\Delta u=\lambda u+h(x,u,\lambda) &\text{in } \Omega,\\ u=0 &\text{on }\Omega. \end{cases} \end{equation*} Under some natural hypotheses on $h$, we show that $(a\lambda_1,0)$ is a bifurcation point of the above problem. As an application of the above result, we shall determine the interval of $\lambda$, in which there exist positive solutions for the above problem with $h(x,u;\lambda)=\lambda f(x,u)-\lambda u$, where $f$ is asymptotically linear at zero and asymptotically 3-linear at infinity. To study global structure of bifurcation branch, we also establish some properties of the first eigenvalue for a nonlocal eigenvalue problem. Moreover, we provide a positive answer to an open problem involving the case $a=0$.

Hermitian Adjacency Matrix of Digraphs and Mixed Graphs

AbstractThe article gives a thorough introduction to spectra of digraphs via its Hermitian adjacency matrix. This matrix is indexed by the vertices of the digraph, and the entry corresponding to an arc from x to y is equal to the complex unity i (and its symmetric entry is ) if the reverse arc is not present. We also allow arcs in both directions and unoriented edges, in which case we use 1 as the entry. This allows to use the definition also for mixed graphs. This matrix has many nice properties; it has real eigenvalues and the interlacing theorem holds for a digraph and its induced subdigraphs. Besides covering the basic properties, we discuss many differences from the properties of eigenvalues of undirected graphs and develop basic theory. The main novel results include the following. Several surprising facts are discovered about the spectral radius; some consequences of the interlacing property are obtained; operations that preserve the spectrum are discussed—they give rise to a large number of cospectral digraphs; for every , all digraphs whose spectrum is contained in the interval are determined.

On the definition and the properties of the principal eigenvalue of some nonlocal operators

In this article we study some spectral properties of the linear operator LΩ+a defined on the space C(Ω¯) by:LΩ[φ]+aφ:=∫ΩK(x,y)φ(y)dy+a(x)φ(x) where Ω⊂RN is a domain, possibly unbounded, a is a continuous bounded function and K is a continuous, non-negative kernel satisfying an integrability condition.We focus our analysis on the properties of the generalised principal eigenvalue λp(LΩ+a) defined byλp(LΩ+a):=sup⁡{λ∈R|∃φ∈C(Ω¯),φ>0,such that LΩ[φ]+aφ+λφ≤0 in Ω}.We establish some new properties of this generalised principal eigenvalue λp. Namely, we prove the equivalence of different definitions of the principal eigenvalue. We also study the behaviour of λp(LΩ+a) with respect to some scaling of K.For kernels K of the type, K(x,y)=J(x−y) with J a compactly supported probability density, we also establish some asymptotic properties of λp(Lσ,m,Ω−1σm+a) where Lσ,m,Ω is defined by Lσ,2,Ω[φ]:=1σ2+N∫ΩJ(x−yσ)φ(y)dy. In particular, we prove thatlimσ→0⁡λp(Lσ,2,Ω−1σ2+a)=λ1(D2(J)2NΔ+a), where D2(J):=∫RNJ(z)|z|2dz and λ1 denotes the Dirichlet principal eigenvalue of the elliptic operator. In addition, we obtain some convergence results for the corresponding eigenfunction φp,σ.

Qualitative analysis of eigenvalues and eigenfunctions of one boundary value-transmission problem

The aim of this study is to investigate various qualitative properties of eigenvalues and corresponding eigenfunctions of one Sturm-Liouville problem with an interior singular point. We introduce a new Hilbert space and integral operator in it such a way that the problem under consideration can be interpreted as a spectral problem of this operator. By using our own approaches we investigate such properties as uniform convergence of the eigenfunction expansions, the Parseval equality, the Rayleigh-Ritz formula, the minimax principle, and the monotonicity of eigenvalues for the considered boundary value-transmission problem (BVTP).

A Note on Lower Bounds of Eigenvalues of Fourth Order Elliptic Operators on Local Quasi-Uniform Grids

Abstract This paper is a generalization and improvement of some recent results concerned with the lower bound property of eigenvalues produced by the Morley element of the biharmonic operator. Such an extension is of two fold. First, we generalize those results for the biharmonic operator to general fourth order elliptic operators; second, we extend them from quasi-uniform grids to local quasi-uniform grids. In particular, for the general case, we prove the asymptotic lower bound property of the discrete eigenvalues under a very weak saturation condition in terms of local meshsizes which can in general not be improved; for the special case with B = 0, we show a similar result without any assumption except the fineness of the mesh. These results can be regarded as a substantial improvement of a related result on regular meshes, including adaptive local refined meshes, for general fourth order elliptic operators, due to Yang, Li and Bi [http://arxiv.org/abs/1509.00566], while the analysis herein is completely different and largely simpler than that in the above paper.

Polynomial viscosity methods for multispecies kinematic flow models

Multispecies kinematic flow models are defined by systems of strongly coupled, nonlinear first‐order conservation laws. They arise in various applications including sedimentation of polydisperse suspensions and multiclass vehicular traffic. Their numerical approximation is a challenge since the eigenvalues and eigenvectors of the corresponding flux Jacobian matrix have no closed algebraic form. It is demonstrated that a recently introduced class of fast first‐order finite volume solvers, called polynomial viscosity matrix (PVM) methods [M. J. Castro Díaz and E. Fernández‐Nieto, SIAM J Sci Comput 34 (2012), A2173–A2196], can be adapted to multispecies kinematic flows. PVM methods have the advantage that they only need some information about the eigenvalues of the flux Jacobian, and no spectral decomposition of a Roe matrix is needed. In fact, the so‐called interlacing property (of eigenvalues with known velocity functions), which holds for several important multispecies kinematic flow models, provides sufficient information for the implementation of PVM methods. Several variants of PVM methods (differing in polynomial degree and the underlying quadrature formula to approximate the Roe matrix) are compared by numerical experiments. It turns out that PVM methods are competitive in accuracy and efficiency with several existing methods, including the Harten, Lax, and van Leer method and a spectral weighted essentially non‐oscillatory scheme that is based on the same interlacing property. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1265–1288, 2016

Graphs with reciprocal eigenvalue properties

In this paper, only simple graphs are considered. A graph G is nonsingular if its adjacency matrix A(G) is nonsingular. A nonsingular graph G satisfies reciprocal eigenvalue property (property R) if the reciprocal of each eigenvalue of the adjacency matrix A(G) is also an eigenvalue of A(G) and G satisfies strong reciprocal eigenvalue property (property SR) if the reciprocal of each eigenvalue of the adjacency matrix A(G) is also an eigenvalue of A(G) and they both have the same multiplicities. From the definitions property SR implies property R. Furthermore, for some classes of graphs (for example, trees), it is known that these properties are equivalent. However, the equivalence of these two properties is not yet known for any nonsingular graph. In this article, it is shown that these properties are not equivalent in general.

飞行器尾涡对的不稳定性建模

For aircrafts the wake vortex separation and the disturbance-induced vortex instability draw more and more concerns. A 1st-order approximation of the Biot-Savart law was used to build the induced motion model for arbitrarily multiple vortex pairs. Symmetric and antisymmetric modes of the induced motion of vortex pairs were deduced with the linear combination method. The properties of the modal matrix eigenvalues were obtained to describe instability of the multiple vortex pairs. The modes of 2 vortex pairs were used for validation of the proposed model, which was thereafter extended to 3 vortex pairs to give the instable modes corresponding to the modal matrices. The theoretical analysis and eigenvalue calculation show that, with the increase of the number of vortex filaments, the instability of the 3 vortex pairs will rise, and the amplification effects of the vortex system on the disturbance will highten.

Computationally Efficient Angle and Polarization Estimation in the Presence of Multipath Propagation Using Dual-Polarization Vector Sensor Array

This paper presents a computationally efficient angle and polarization estimation method for a mixture of uncorrelated and coherent sources using a dual-polarization vector sensor array. The uncorrelated sources are separated from the coherent sources on the basis of the modulus property of eigenvalues. The angles of the uncorrelated sources are estimated by employing rotational invariance and the associated polarization is obtained from the estimate of the uncorrelated array response matrix through elementwise division. For the distinguished coherent sources, two Hankel matrices are constructed from the elements of the estimated coherent array response matrix of each coherent group, from which two rotational-invariant submatrix pairs are extracted for estimating the coherent angles with a high precision. The least-square solution to the coherent polarization equation is derived for estimating the coherent polarization parameters. For each uncorrelated source and coherent group, the proposed method estimates the associated angle and polarization parameters separately, which avoids the need of 3D spectral search. In comparison with the existing methods, the simulation results show that the proposed method yields favorable performance in terms of computational efficiency and estimation accuracy.

On The Eigenvalue Distribution Of Adjacency Matrices For Connected Planar Graphs

Abstract This paper describes the previously unknown statistical distribution of adjacency matrix spectra for planar graphs, also known as spatial weights matrices, in terms of the following three readily available eigenvalue properties: extremes, rank orderings, and sums of powers. This distribution is governed by at most six parameters that, once known, allow accurate approximations of eigenvalues to be computed without resorting to numerical matrix methods applied on a case-by-case basis. Parameter estimates for illustrative real-world examples are obtained using nonlinear least squares regression techniques. Three conjectures are proposed, and graphical and trend results are reported for a diverse set of planar graph-based matrices.

Spectral properties of the temporal evolution of brain network structure.

The temporal evolution properties of the brain network are crucial for complex brain processes. In this paper, we investigate the differences in the dynamic brain network during resting and visual stimulation states in a task-positive subnetwork, task-negative subnetwork, and whole-brain network. The dynamic brain network is first constructed from human functional magnetic resonance imaging data based on the sliding window method, and then the eigenvalues corresponding to the network are calculated. We use eigenvalue analysis to analyze the global properties of eigenvalues and the random matrix theory (RMT) method to measure the local properties. For global properties, the shifting of the eigenvalue distribution and the decrease in the largest eigenvalue are linked to visual stimulation in all networks. For local properties, the short-range correlation in eigenvalues as measured by the nearest neighbor spacing distribution is not always sensitive to visual stimulation. However, the long-range correlation in eigenvalues as evaluated by spectral rigidity and number variance not only predicts the universal behavior of the dynamic brain network but also suggests non-consistent changes in different networks. These results demonstrate that the dynamic brain network is more random for the task-positive subnetwork and whole-brain network under visual stimulation but is more regular for the task-negative subnetwork. Our findings provide deeper insight into the importance of spectral properties in the functional brain network, especially the incomparable role of RMT in revealing the intrinsic properties of complex systems.

Higher-degree eigenvalue complementarity problems for tensors

In this paper, we introduce a unified framework of Tensor Higher-Degree Eigenvalue Complementarity Problem (THDEiCP), which goes beyond the framework of the typical Quadratic Eigenvalue Complementarity Problem for matrices. First, we study some topological properties of higher-degree cone eigenvalues of tensors. Based upon the symmetry assumptions on the underlying tensors, we then reformulate THDEiCP as a weakly coupled homogeneous polynomial optimization problem, which might be greatly helpful for designing implementable algorithms to solve the problem under consideration numerically. As more general theoretical results, we present the results concerning existence of solutions of THDEiCP without symmetry conditions. Finally, we propose an easily implementable algorithm to solve THDEiCP, and report some computational results.

Complete Stability of Neural Networks With Nonmonotonic Piecewise Linear Activation Functions

This brief studies the complete stability of neural networks with nonmonotonic piecewise linear activation functions. By applying the fixed-point theorem and the eigenvalue properties of the strict diagonal dominance matrix, some conditions are derived, which guarantee that such $n$ -neuron neural networks are completely stable. More precisely, the following two important results are obtained: 1) The corresponding neural networks have exactly $5^n$ equilibrium points, among which $3^n$ equilibrium points are locally exponentially stable and the others are unstable; 2) as long as the initial states are not equal to the equilibrium points of the neural networks, the corresponding solution trajectories will converge toward one of the $3^n$ locally stable equilibrium points. A numerical example is provided to illustrate the theoretical findings via computer simulations.

Computational method for transmission eigenvalues for a spherically stratified medium.

We consider a computational method for the interior transmission eigenvalue problem that arises in acoustic and electromagnetic scattering. The transmission eigenvalues contain useful information about some physical properties, such as the index of refraction. Instead of the existence and estimation of the spectral property of the transmission eigenvalues, we focus on the numerical calculation, especially for spherically stratified media in R3. Due to the nonlinearity and the special structure of the interior transmission eigenvalue problem, there are not many numerical methods to date. First, we reduce the problem into a second-order ordinary differential equation. Then, we apply the Hermite finite element to the weak formulation of the equation. With proper rewriting of the matrix-vector form, we change the original nonlinear eigenvalue problem into a quadratic eigenvalue problem, which can be written as a linear system and solved by the eigs function in MATLAB. This numerical method is fast, effective, and can calculate as many transmission eigenvalues as needed at a time.

Spectrum and Green’s Function of a Many-Interval Sturm–Liouville Problem

Abstract This article considers a Sturm–Liouville-type problem on a finite number disjoint intervals together with transmission conditions at the points of interaction. We introduce a new operator-theoretic formulation in such a way that the problem under consideration can be interpreted as a spectral problem for a suitable self-adjoint operator. We investigate some principal properties of eigenvalues, eigenfunctions, and resolvent operator. Particularly, by applying Green’s function method, it is shown that the problem has only point spectrum, and the set of eigenfunctions form a basis of the adequate Hilbert space.

Discrete limit and monotonicity properties of the Floquet eigenvalue in an age structured cell division cycle model.

We consider a cell population described by an age-structured partial differential equation with time periodic coefficients. We assume that division only occurs within certain time intervals at a rate [Formula: see text] for cells who have reached minimal positive age (maturation). We study the asymptotic behavior of the dominant Floquet eigenvalue, or Perron-Frobenius eigenvalue, representing the growth rate, as a function of the maturation age, when the division rate [Formula: see text] tends to infinity (divisions become instantaneous). We show that the dominant Floquet eigenvalue converges to a staircase function with an infinite number of steps, determined by a discrete dynamical system. This indicates that, in the limit, the growth rate is governed by synchronization phenomena between the maturation age and the length of the time intervals in which division may occur. As an intermediate result, we give a sufficient condition which guarantees that the dominant Floquet eigenvalue is a nondecreasing function of the division rate. We also give a counter example showing that the latter monotonicity property does not hold in general.