Independent Eigenvectors Research Articles

An accurate, robust and efficient convection-pressure flux splitting scheme for compressible Euler flows

The popular Roe's flux difference splitting scheme has high resolution for various types of discontinuities but it fails to satisfy the entropy condition and will encounter different forms of instability while calculating shock waves, such as the post-shock oscillation of the slowly moving shock waves and the carbuncle phenomenon of the multidimensional strong shock waves. In the present work, a simple flux difference splitting scheme based on the idea of convection-pressure splitting is proposed. With application of the Toro-Vázquez splitting procedure, the flux of the 3D Euler equations is split into the convection part and the pressure part. Due to having a complete set of linearly independent eigenvectors, the pressure subsystem can be solved by the conventional flux difference splitting method. The weakly hyperbolic convection subsystem, however, does not have a complete set of linearly independent eigenvectors and thus is solved by the flux difference splitting scheme based on the generalized eigenvectors [1]. In addition, a strategy that combines the THINC reconstruction with the BVD algorithm is employed on the numerical dissipation term of the convection flux to further improve the resolution for contact discontinuities. A comprehensive range of numerical simulations for classical 1D, 2D and 3D benchmark test problems demonstrate the good performance of the proposed scheme.

Classifications of surface waves in magneto-electro-elastic materials

The present study investigates the classifications of surface waves in magneto-electro-elastic materials. First, the standard eigenrelation, relating to the generalized displacement and the generalized tractions, is derived from the governing equations and constitutive relations. The number of distinct eigenvalues and the number of independent eigenvectors of the matrix N(v), which appears in the standard eigenrelation, play a crucial role in determining the surface wave solutions. However, this issue is often overlooked by most researchers. Therefore, our focus in this paper is to address this oversight by providing expressions for the solutions in various scenarios, considering the number of distinct eigenvalues and independent eigenvectors of the matrix N(v). Subsequently, we classify the surface waves in magneto-electro-elastic materials based on the aforementioned eigenvalue and eigenvector quantities of the 10 × 10 real matrix N(v). By considering the simple and semisimple properties, as well as the ten types of degenerate cases of the real matrix N(v), we identify a total of fifteen distinct types of surface waves within this classification.

A Study of the Eigenvalues of the Matrix Of Distance Reciprocals in K[r, n-r] And The Cycle C_n

This paper Deals with the complete bipartite graph K(r, n-r) and the cycle . The matrix of concern is the matrix B which is the (n, n) matrix and whose non zero entries are the reciprocals of the non zero entries of the distance matrix D. A complete characterization of the spectrum of B and a set of n independent eigenvectors of B will be presented. Two special cases will be mentioned, namely the star K(1, n-1) and the graph K(2, n-2). We will also look at the case of infinite graph, i. e if the size n grows big while r stays finite. Finally, some numerical data will be presented. As for the cycle, we present the complete set of eigenvalues of the matrix B.

Two Kinds of Laplacian Spectra and Degree Kirchhoff Index of the Weighted Corona Networks

Recently, the study related to network has aroused wide attention of the scientific community. Many problems can be usefully represented by corona graphs or networks. Meanwhile, the weight is a vital factor in characterizing some properties of real networks. In this paper, we give complete information about the signless Laplacian spectra of the weighted corona of a graph G1 and a regular graph G2 and the complete information about the normalized Laplacian spectra of the weighted corona of two regular graphs. The corresponding linearly independent eigenvectors of all these eigenvalues are also obtained. The spanning trees’ total number and the degree Kirchhoff index of the weighted corona graph are computed.

Gaussian Regularization of the Pseudospectrum and Davies’ Conjecture

AbstractA matrix is diagonalizable if it has a basis of linearly independent eigenvectors. Since the set of nondiagonalizable matrices has measure zero, every is the limit of diagonalizable matrices. We prove a quantitative version of this fact conjectured by E. B. Davies: for each , every matrix is at least ‐close to one whose eigenvectors have condition number at worst , for some depending only on n. We further show that the dependence on δ cannot be improved to for any constant .Our proof uses tools from random matrix theory to show that the pseudospectrum of A can be regularized with the addition of a complex Gaussian perturbation. Along the way, we explain how a variant of a theorem of Śniady implies a conjecture of Sankar, Spielman, and Teng on the optimal constant for smoothed analysis of condition numbers. © 2021 Wiley Periodicals, Inc.

Repeated eigenvalues and their derivatives of structural vibration systems with general nonproportional viscous damping

Repeated vibration modes often occur in practice in structural vibration systems with general nonproportional viscous damping. However, the topic remains perhaps the least understood in vibration analysis. Some researchers have suggested that the inclusion of nonproportional viscous damping renders a system defective in the case of repeated eigenvalues, while others have assumed that a complete set of linearly independent eigenvectors can always be found regardless of the forms and magnitudes of viscous damping. This paper seeks to first establish that for light non-proportional viscous damping, a system with repeated eigenvalues generally does not become defective based on perturbation theory and realistic numerical examples since rigorous theoretical proof is believed to be difficult. Once non-defectiveness is confirmed, a method for computing the eigen derivatives with repeated eigenvalues in the case of general viscous damping is developed. Effect of mode truncation on numerical accuracy has been discussed. When the magnitude of non-proportional viscous damping becomes considerably high however, it is possible for a damped system to become defective when repeated modes occur. This can have profound practical implications where very high damping is desired as in the cases of vibration suspension and absorber designs, since system defectiveness can lead to major difficulties in the applications of conventional vibration analyses.

How Does the Planck Scale Affect Qubits?

Gedanken experiments in quantum gravity motivate generalised uncertainty relations (GURs) implying deviations from the standard quantum statistics close to the Planck scale. These deviations have been extensively investigated for the non-spin part of the wave function, but existing models tacitly assume that spin states remain unaffected by the quantisation of the background in which the quantum matter propagates. Here, we explore a new model of nonlocal geometry in which the Planck-scale smearing of classical points generates GURs for angular momentum. These, in turn, imply an analogous generalisation of the spin uncertainty relations. The new relations correspond to a novel representation of SU(2) that acts nontrivially on both subspaces of the composite state describing matter-geometry interactions. For single particles, each spin matrix has four independent eigenvectors, corresponding to two 2-fold degenerate eigenvalues ħ±(ħ+β)/2, where β is a small correction to the effective Planck’s constant. These represent the spin states of a quantum particle immersed in a quantum background geometry and the correction by β emerges as a direct result of the interaction terms. In addition to the canonical qubits states, |0⟩=|↑⟩ and |1⟩=|↓⟩, there exist two new eigenstates in which the spin of the particle becomes entangled with the spin sector of the fluctuating spacetime. We explore ways to empirically distinguish the resulting "geometric" qubits, |0′⟩ and |1′⟩, from their canonical counterparts.

An accurate and shock-stable genuinely multidimensional scheme for solving the Euler equations

In numerical simulations of multidimensional high Mach flows, the conventional low-diffusion upwind schemes built on the dimensional splitting method will encounter the shock instability which is mainly manifested as the infamous carbuncle phenomena. A linear stability analysis reveals that the shock instability is triggered by the unattenuated perturbations in the transverse direction of the flow field. In the present work, an accurate and shock-stable genuinely multidimensional numerical scheme based on the Toro-Vázquez splitting method is presented. Different from the conventional one-dimensional upwind schemes that just consider the waves propagating along the direction normal to an interface, the proposed multidimensional scheme whose multidimensional properties are achieved by calculating multidimensional numerical fluxes at each corner of an interface also takes into account the waves traveling along the transverse direction. For obtaining these multidimensional numerical fluxes at the corner, a simple multidimensional upwind method is used to solve the weakly hyperbolic convection subsystem and the pressure subsystem whose flux Jacobian has a complete set of linearly independent eigenvectors is calculated by a multidimensional HLLEM scheme. Based on Balsara’s framework for constructing multidimensional schemes, the total numerical flux across an interface is obtained by using the Simpson’s rule to assemble the conventional one-dimensional numerical flux with the multidimensional numerical fluxes at the corner. A series of benchmark test problems validate the accuracy, robustness and efficiency of the proposed multidimensional scheme.

The geometry connectivity of hypergraphs

Let G be a k-uniform hypergraph, and LG be its Laplacian tensor. The β(G) denotes the maximum number of linearly independent nonnegative eigenvectors of LG corresponding to the eigenvalue 0. In this paper, β(G) is called the geometry connectivity of G. We show that the number of connected components of G equals the geometry connectivity β(G).

Use of Jordan forms for convection-pressure split Euler solvers

In this study, convection-pressure split Euler flux functions which contain weakly hyperbolic convective subsystems are analyzed. A system of first-order partial differential equations (PDEs) is said to be weakly hyperbolic if the corresponding flux Jacobian does not contain a complete set of linearly independent (LI) eigenvectors. Thus, the application of existing flux difference splitting (FDS) based schemes, which depend heavily on both eigenvalues and eigenvectors, are non-trivial to such systems. In the case of weakly hyperbolic systems, a required set of LI eigenvectors can be constructed through the addition of generalized eigenvectors by utilizing the theory of Jordan canonical forms. Once this is achieved for a weakly hyperbolic convective subsystem, an upwind solver can be constructed in the splitting framework.In the present work, the above approach is used for developing two new schemes. The first scheme is based on the Zha–Bilgen type splitting while the second is based on the Toro–Vázquez splitting. Both the schemes are tested on various benchmark problems in one-dimension (1-D) and two-dimensions (2-D). The concept of generalized eigenvectors based on Jordan forms is found to be useful in dealing with the weakly hyperbolic parts of the considered Euler systems.

Perturbation theory for Hermitian quadratic eigenvalue problem – damped and simultaneously diagonalizable systems

The main contribution of this paper is a novel approach to the perturbation theory of a structured Hermitian quadratic eigenvalue problems (λ2M+λD+K)x=0. We propose a new concept without linearization, considering two structures: general quadratic eigenvalue problems (QEP) and simultaneously diagonalizable quadratic eigenvalue problems (SDQEP). Our first two results are upper bounds for the difference |∥X2*MX˜1∥F2−∥X2*MX1∥F2|, and for ∥X2*MX˜1−X2*MX1∥F, where the columns of X1=[x1,…,xk] and X2=[xk+1,…,xn] are linearly independent right eigenvectors and M is positive definite Hermitian matrix. As an application of these results we present an eigenvalue perturbation bound for SDQEP. The third result is a lower and an upper bound for ∥sinΘ(X1,X˜1)∥F, where Θ is a matrix of canonical angles between the eigensubspaces X1 and X˜1,X1 is spanned by the set of linearly independent right eigenvectors of SDQEP and X˜1 is spanned by the corresponding perturbed eigenvectors. The quality of the mentioned results have been illustrated by numerical examples.

Modal analysis of a simple cyclic system

Cyclic systems have double modes with identical natural frequencies. Using the theory of circulants, independent eigenvectors can be exactly determined without slightly perturbing the eigenvalues. The paper presents the contribution of the double modes to the frequency response functions (FRFs) of cyclic systems with proportional damping. The mode indicator functions UMIF and CoMIF, based on the left singular vectors of a matrix encompassing all available FRFs as columns, are applied to simulated noise-free data. It is shown that they can reveal double modes even using FRFs from single point excitation.

A class of upwind methods based on generalized eigenvectors for weakly hyperbolic systems

In this article, a class of upwind schemes is proposed for systems, each of which yields an incomplete set of linearly independent eigenvectors. The theory of Jordan canonical forms is used to complete such sets through the addition of generalized eigenvectors. A modified Burgers’ system and its extensions generate δ,δ′, δ″,⋯,δn waves as solutions. The performance of flux difference splitting-based numerical schemes is examined by considering various numerical examples. Since the flux Jacobian matrix of pressureless gas dynamics system also produces an incomplete set of linearly independent eigenvectors, a similar framework is adopted to construct a numerical algorithm for a pressureless gas dynamics system.

Solving the general joint block diagonalization problem via linearly independent eigenvectors of a matrix polynomial

SummaryIn this paper, we consider the exact/approximate general joint block diagonalization (GJBD) problem of a matrix set ( p ≥ 1), where a nonsingular matrix W (often referred to as a diagonalizer) needs to be found such that the matrices W HAiW 's are all exactly/approximately block‐diagonal matrices with as many diagonal blocks as possible. We show that the diagonalizer of the exact GJBD problem can be given by W = [x1,x2,…,xn]Π, where Π is a permutation matrix and xi's are eigenvectors of the matrix polynomial , satisfying that [x1,x2,…,xn] is nonsingular and where the geometric multiplicity of each λi corresponding with xi is equal to 1. In addition, the equivalence of all solutions to the exact GJBD problem is established. Moreover, a theoretical proof is given to show why the approximate GJBD problem can be solved similarly to the exact GJBD problem. Based on the theoretical results, a three‐stage method is proposed, and numerical results show the merits of the method.

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.

Four independent eigenvectors and their properties for the problem of four-potential acoustic SH-wave propagation incorporating gravitational phenomena

Four independent eigenvectors and their properties for the problem of four-potential acoustic SH-wave propagation incorporating gravitational phenomena

Modal Analysis of Fluid Flows: An Overview

Modal Analysis of Fluid Flows: An Overview

Nodal domains of eigenvectors for 1-Laplacian on graphs

The eigenvectors for graph 1-Laplacian possess some sort of localization property: On one hand, the characteristic function on any nodal domain of an eigenvector is again an eigenvector with the same eigenvalue; on the other hand, one can pack up an eigenvector for a new graph by several fundamental eigencomponents and modules with the same eigenvalue via few special techniques. The Courant nodal domain theorem for graphs is extended to graph 1-Laplacian for strong nodal domains, but for weak nodal domains it is false. The notion of algebraic multiplicity is introduced in order to provide a more precise estimate of the number of independent eigenvectors. A positive answer is given to a question raised in Chang (2016) [3], to confirm that the critical values obtained by the minimax principle may not cover all eigenvalues of graph 1-Laplacian.

Double criticality and the two-way Boussinesq equation in stratified shallow water hydrodynamics

Double criticality and its nonlinear implications are considered for stratified N–layer shallow water flows with N = 1, 2, 3. Double criticality arises when the linearization of the steady problem about a uniform flow has a double zero eigenvalue. We find that there are two types of double criticality: non-semisimple (one eigenvector and one generalized eigenvector) and semi-simple (two independent eigenvectors). Using a multiple scales argument, dictated by the type of singularity, it is shown that the weakly nonlinear problem near double criticality is governed by a two-way Boussinesq equation (non-semisimple case) and a coupled Korteweg-de Vries equation (semisimple case). Parameter values and reduced equations are constructed for the examples of two-layer and three-layer stratified shallow water hydrodynamics.

Principal vectors of a nonlinear finite-dimensional eigenvalue problem

In a finite-dimensional linear space, consider a nonlinear eigenvalue problem analytic with respect to its spectral parameter. The notion of a principal vector for such a problem is examined. For a linear eigenvalue problem, this notion is identical to the conventional definition of principal vectors. It is proved that the maximum number of linearly independent eigenvectors combined with principal (associated) vectors in the corresponding chains is equal to the multiplicity of an eigenvalue. A numerical method for constructing such chains is given.