Zero Eigenvalue Research Articles

Multipoint scatterers with bound states at zero energy

We study multipoint scatterers with zero-energy bound states in three dimensions. We present examples of such scatterers with multiple zero eigenvalue or with strong multipole localization of zero-energy bound states.

A full-pivoting algorithm for the Cholesky decomposition of two-electron repulsion and spin-orbit coupling integrals.

A significant reduction in the computational effort for the evaluation of the electronic repulsion integrals (ERI) in ab initio quantum chemistry calculations is obtained by using Cholesky decomposition (CD), a numerical procedure that can remove the zero or small eigenvalues of the ERI positive (semi)definite matrix, while avoiding the calculation of the entire matrix. Conversely, due to its antisymmetric character, CD cannot be directly applied to the matrix representation of the spatial part of the two-electron spin-orbit coupling (2e-SOC) integrals. Here, we present a computational strategy to achieve a Cholesky representation of the spatial part of the 2e-SOC integrals, and propose a new efficient CD algorithm for both ERI and 2e-SOC integrals. The proposed algorithm differs from previous CD implementations by the extensive use of a full-pivoting design, which allows a univocal definition of the Cholesky basis, once the CD δ threshold is made explicit. We show that 2δ is the upper limit for the errors affecting the reconstructed 2e-SOC integrals. The proposed strategy was implemented in the ab initio program Computational Emulator of Rare Earth Systems (CERES), and tested for computational performance on both the ERI and 2e-SOC integrals evaluation. © 2017 Wiley Periodicals, Inc.

A note on local integrability of differential systems

For an n-dimensional local analytic differential system x˙=Ax+f(x) with f(x)=O(|x|2), the Poincaré nonintegrability theorem states that if the eigenvalues of A are not resonant, the system does not have an analytic or a formal first integral in a neighborhood of the origin. This result was extended in 2003 to the case when A admits one zero eigenvalue and the other are non-resonant: for n=2 the system has an analytic first integral at the origin if and only if the origin is a non-isolated singular point; for n>2 the system has a formal first integral at the origin if and only if the origin is not an isolated singular point. However, the question of whether the system has an analytic first integral at the origin provided that the origin is not an isolated singular point remains open.

Algorithm 982

A method to compute explicit solutions of homogeneous triangular systems of first-order linear initial-value ordinary differential equations with constant coefficients is described. It is suitable for the limited case of well separated eigenvalues, or for multiple zero eigenvalues provided the entire column corresponding to a zero eigenvalue is zero. The solution for the case of constant inhomogeneity is described. The method requires only the computation of a constant matrix using a simple recurrence. Computing the solutions of the system from that matrix, for values of the independent variable, requires one to exponentiate only the diagonal of a matrix. It is not necessary to compute the exponential of a general triangular matrix. Although this work was motivated by a study of nuclear decay without fission or neutron absorption, which is used throughout as an example, it has wider applicability.

Sign-changing diagonal perturbations of Laplacian matrices of graphs

In this paper we provide sufficient conditions for positive (semi)definiteness of sign-changing diagonal perturbations of positive semidefinite difference operators and their matrix representations, the Laplacian matrices of graphs. Our estimates arise from the discrete version of the Poincaré inequality and essentially depend on the algebraic connectivity of the underlying graph, i.e., the second smallest eigenvalue of the graph Laplacian matrix. We generalize our results to positive semidefinite matrices with simple zero eigenvalue and illustrate our results by numerical experiments and discuss the optimality of our assumptions.

A robust bi-orthogonal/dynamically-orthogonal method using the covariance pseudo-inverse with application to stochastic flow problems

We develop a new robust methodology for the stochastic Navier–Stokes equations based on the dynamically-orthogonal (DO) and bi-orthogonal (BO) methods [1–3]. Both approaches are variants of a generalized Karhunen–Loève (KL) expansion in which both the stochastic coefficients and the spatial basis evolve according to system dynamics, hence, capturing the low-dimensional structure of the solution. The DO and BO formulations are mathematically equivalent [3], but they exhibit computationally complimentary properties. Specifically, the BO formulation may fail due to crossing of the eigenvalues of the covariance matrix, while both BO and DO become unstable when there is a high condition number of the covariance matrix or zero eigenvalues. To this end, we combine the two methods into a robust hybrid framework and in addition we employ a pseudo-inverse technique to invert the covariance matrix. The robustness of the proposed method stems from addressing the following issues in the DO/BO formulation: (i) eigenvalue crossing: we resolve the issue of eigenvalue crossing in the BO formulation by switching to the DO near eigenvalue crossing using the equivalence theorem and switching back to BO when the distance between eigenvalues is larger than a threshold value; (ii) ill-conditioned covariance matrix: we utilize a pseudo-inverse strategy to invert the covariance matrix; (iii) adaptivity: we utilize an adaptive strategy to add/remove modes to resolve the covariance matrix up to a threshold value. In particular, we introduce a soft-threshold criterion to allow the system to adapt to the newly added/removed mode and therefore avoid repetitive and unnecessary mode addition/removal. When the total variance approaches zero, we show that the DO/BO formulation becomes equivalent to the evolution equation of the Optimally Time-Dependent modes [4]. We demonstrate the capability of the proposed methodology with several numerical examples, namely (i) stochastic Burgers equation: we analyze the performance of the method in the presence of eigenvalue crossing and zero eigenvalues; (ii) stochastic Kovasznay flow: we examine the method in the presence of a singular covariance matrix; and (iii) we examine the adaptivity of the method for an incompressible flow over a cylinder where for large stochastic forcing thirteen DO/BO modes are active.

Characteristic Mode Formulation for Dielectric Coated Conducting Bodies

In this paper, characteristic mode (CM) formulations are developed from surface integral equation (SIE) for the modal analysis of dielectric coated conducting bodies. The electric field integral equation-Poggio, Miller, Chang, Harrington, Wu, and Tsai SIE is used for modeling the dielectric coated conducting bodies. By ensuring the field continuity on the interface boundary, two types of generalized eigenvalues equations are formulated to determine the resonant behavior of the dielectric coated conducting bodies. Following Poynting’s theorem, the resultant eigenvalues indicate the ratio of the imaginary and real part of the complex power for each CM. Zero eigenvalues indicate the resonance in dielectric coated conducting bodies. The corresponding modal fields provide with clear physical insights into the radiation/scattering mechanisms. Numerical results are presented to demonstrate the accuracy of the proposed CM formulations.

Zero‐zero‐Hopf bifurcation and ultimate bound estimation of a generalized Lorenz–Stenflo hyperchaotic system

This paper is devoted to the analysis of complex dynamics of a generalized Lorenz–Stenflo hyperchaotic system. First, on the local dynamics, the bifurcation of periodic solutions at the zero‐zero‐Hopf equilibrium (that is, an isolated equilibrium with double zero eigenvalues and a pair of purely imaginary eigenvalues) of this hyperchaotic system is investigated, and the sufficient conditions, which insure that two periodic solutions will bifurcate from the bifurcation point, are obtained. Furthermore, on the global dynamics, the explicit ultimate bound sets of this hyperchaotic system are obtained. Copyright © 2016 John Wiley & Sons, Ltd.

Opinion Consensus with Delay When the Zero Eigenvalue of the Connection Matrix is Semi-simple

Two or more groups with different initial opinions can develop into a consensus with appropriate inter- and intra group interactions and communication delay. Here we consider the case when zero is a semi-simple eigenvalue of the normalized connection matrix, and we expand the concept of consensus where opinions of all individual agents must converge to the common (consensus) value to the weak consensus where opinions are clustered and the consensus value is no longer a scalar constant but a vector in a certain consensus subspace. The linearity of the system enables us to adopt an argument of Atay, using the phase space decomposition in terms of the standard bilinear form for functional differential equations, to derive sharp conditions for the system to reach a weak consensus, and to calculate the consensus vector. This calculation also informs how the consensus value of the coupled systems depends on the consensus of the subgroups, initial conditions and communication delay. We also point out the challenge of extending these results to nonlinear versions of the opinion dynamical systems with delay.

Local phase synchronization and clustering for the delayed phase-coupled oscillators with plastic coupling

In this paper, we investigate the local synchronization problem for delayed phase-coupled oscillators with plastic coupling (also referred as a generalized Kuramoto model). By linearizing the phase-coupled oscillators system at a special solution, an approximation phase-coupled system with delay effects is deduced. Moreover, we find the multiplicity of semisimple zero eigenvalue for coupling strength coefficients matrix and the rank of Vandermonde matrix associated initial values are two sensitive factors. The former is closely related to the maximum number of clusters phase of linear coupled system, and the latter determines clusters phase synchronization. As results, both frequency synchronization criteria and phase synchronization criteria are established in this literature. Also, the final synchronization frequency and phase are formulated in terms of coupling strength, initial values and time delay.

Refined inertias of strongly connected orientations of the Petersen graph

The 18 non-isomorphic strongly connected orientations of the Petersen graph give rise to matrix patterns in which nonzero entries can be taken to be strictly positive, of arbitrary sign, or of fixed sign. The allowed refined inertias, in which the number of zero eigenvalues is split from others on the imaginary axis, are considered for each such pattern. Each nonnegative pattern is shown to have unique refined inertia determined by the number of required zero eigenvalues. For zero–nonzero patterns, a complete list of allowed refined inertias is given for each orientation. One particular sign pattern is presented that allows only two distinct refined inertias out of a possible 161 for a sign pattern of order 10.

Planar Pose Graph Optimization: Duality, Optimal Solutions, and Verification

Pose graph optimization (PGO) is the problem of estimating a set of poses from pairwise relative measurements. PGO is a nonconvex problem and, currently, no known technique can guarantee the computation of a global optimal solution. In this paper, we show that Lagrangian duality allows computing a globally optimal solution under conditions that are satisfied in most robotics applications and enables to certify optimality of a given estimate. Our first contribution is to frame planar PGO in the complex domain. This makes analysis easier and allows drawing connections with existing literature on unit gain graphs. The second contribution is to formulate and analyze the properties of the Lagrangian dual problem in the complex domain. Our analysis shows that the duality gap is connected to the number of zero eigenvalues of the penalized pose graph matrix. We prove that if this matrix has a single zero eigenvalue, then 1) the duality gap is zero, 2) the primal PGO problem has a unique solution (up to an arbitrary roto-translation), and 3) the primal solution can be computed by scaling an eigenvector of the penalized pose graph matrix. The third contribution is algorithmic: We leverage duality to devise and algorithm that computes the optimal solution when the penalized matrix has a single zero eigenvalue. We also propose a suboptimal variant when the zero eigenvalues are multiple. Finally, we show that duality provides computational tools to verify if a given estimate (e.g., computed using iterative solvers) is globally optimal. We conclude the paper with an extensive numerical analysis. Empirical evidence shows that, in the vast majority of cases (100% of the tests under noise regimes of practical robotics applications), the penalized pose graph matrix has a single zero eigenvalue; hence, our approach allows computing (or verifying) the optimal solution.

On the nullity of a family of tripartite graphs

Abstract The eigenvalues of the adjacency matrix of a graph form the spectrum of the graph. The multiplicity of the eigenvalue zero in the spectrum of a graph is called nullity of the graph. Fan and Qian (2009) obtained the nullity set of n-vertex bipartite graphs and characterized the bipartite graphs with nullity n − 4 and the regular n-vertex bipartite graphs with nullity n − 6. In this paper, we study similar problem for a class of tripartite graphs. As observed the nullity problem in tripartite graphs does not follow as an extension to that of the nullity of bipartite graphs, this makes the study of nullity in tripartite graphs interesting. In this direction, we obtain the nullity set of a class of n-vertex tripartite graphs and characterize these tripartite graphs with nullity n − 4. We also characterize some tripartite graphs with nullity n − 6 in this class.

The extremal graphs with respect to their nullity

The nullity of a graph G, denoted by $\eta(G)$ , is the multiplicity of the eigenvalue zero of its adjacency matrix. In this paper, we determine all graphs with $\eta(G)=n(G) - 2m(G) -c(G)$ , where $c(G) = e(G) - n(G) + \omega(G)$ , and $n(G)$ , $e(G)$ , $m( G )$ , and $\omega (G)$ are the vertex number, edge number, matching number, and the number of connected components of G, respectively.

Bounds for the Zeros of Polynomials

This paper discusses the relation between zeros of polynomials and eigenvalues of companion matrices,and some bounds for the zeros of polynomials are obtained.

Detecting the Shilnikov Scenario in a Hopf-Hopf Bifurcation with 1:3 Resonance

We investigate the behaviour of the primary solutions at a Hopf-Hopf interaction close to a 1:3 resonance. It turns out, that the secondary bifurcations from the primary periodic solution branches are governed by Duffing and Mathieu equations.By numerical path following a homoclinic orbit at a saddle node was detected, giving rise to the Shilnikov scenario. In order to understand the creation of homoclinic orbits, a continuation of that orbit was applied, which terminated at an equilibrium with a triple zero eigenvalue. The existence of different types of homoclinic and heteroclinic orbits in the vicinity of triple zero bifurcation points has already been established. A short discussion of the local bifurcations at the triple zero eigenvalue is given.

On Semidefiniteness of Signed Laplacians with Application to Microgrids

Abstract: The paper investigates the positive semidefiniteness of signed Laplacians. It is noted that a symmetric signed Laplacian defines a unique resistive electrical network, wherein the negative weights correspond to negative resistances. As such, the positive semidefiniteness of the signed Laplacians is equivalent to the passivity of the associated resistive networks. By utilizing n -port circuit theory, we obtain several equivalent conditions for signed Laplacians to be positive semidefinite with a simple zero eigenvalue. These conditions characterize a set of negative weights that maintain the semidefiniteness of the Laplacian. The results are used to analyze the small-disturbance angle stability of microgrids as an application.

Projection spectral analysis

This study investigates’ Projection Spectral Analysis’, which generalizes’ Principal or Independent Component Analysis’ by dealing with a non-symmetric square correlation or covariance matrix with multiplicities or singularities. This type of covariance matrix is decomposed into projections and nilpotents according to the spectral theorem. Projection spectral analysis solves a learning problem by reducing the dimension for multiple zero eigenvalues, and may be applied to a non-symmetric covariance with distinct eigenvalues. This method involves a sum-product of orthogonal projection operators and real distinct eigenvalues for a symmetric covariance, which makes it equivalent to principal component analysis. However, it becomes independent component analysis if the covariance is not symmetric.

On the periodic orbit bifurcating from one single non-hyperbolic equilibrium in a chaotic jerk system

This paper proposes a chaotic jerk system coexisting with only one non-hyperbolic equilibrium with one zero eigenvalue and a pair of complex conjugate eigenvalues. The system has no classical Hopf bifurcations and belongs to a newly category of chaotic systems. Based on the averaging theory, an analytic proof of the existence of zero-Hopf bifurcation is exhibited. Moreover, unstable periodic orbits from the zero-Hopf bifurcation are obtained. This approach may be useful to clarify chaotic attractors with non-hyperbolic equilibrium hidden behind complicated phenomena.

Mixed Spectral Element Method for 2D Maxwell's Eigenvalue Problem

AbstractIt is well known that conventional edge elements in solving vector Maxwell's eigenvalue equations by the finite element method will lead to the presence of spurious zero eigenvalues. This problem has been addressed for the first order edge element by Kikuchi by the mixed element method. Inspired by this approach, this paper describes a higher order mixed spectral element method (mixed SEM) for the computation of two-dimensional vector eigenvalue problem of Maxwell's equations. It utilizes Gauss-Lobatto-Legendre (GLL) polynomials as the basis functions in the finite-element framework with a weak divergence condition. It is shown that this method can suppress all spurious zero and nonzero modes and has spectral accuracy. A rigorous analysis of the convergence of the mixed SEM is presented, based on the higher order edge element interpolation error estimates, which fully confirms the robustness of our method. Numerical results are given for homogeneous, inhomogeneous, L-shape, coaxial and dual-inner-conductor cavities to verify the merits of the proposed method.