Number Of Negative Eigenvalues Research Articles

Triangle-free signed graphs with small negative inertia index

Let Γ=(G,σ) be a signed graph with underlying graph G and sign function σ:E(G)→{+,−}. The number of negative eigenvalues (including multiplicities) of Γ, denoted by n(Γ), is called the negative inertia index of Γ. In this paper, we characterize the signed graphs Γ with n(Γ)=1, and the triangle-free signed graphs Γ with n(Γ)=2, respectively.

Characterizing the negative inertia index of connected graphs in terms of their girth

Suppose that a connected graph G has at least one cycle and let g be the length of the shortest cycle in G, which is called the girth of G. In this paper, we consider relationship between the girth of G and the number of negative eigenvalues (including multiplicities) of the adjacency matrix of G, known as negative inertia index of G and denoted by i−(G). We prove that i−(G)≥⌈g2⌉−1. Furthermore, all extremal graphs corresponding to i−(G)=⌈g2⌉−1 and i−(G)=⌈g2⌉ are characterized, respectively.

A new SOCP relaxation of nonconvex quadratic programming problems with a few negative eigenvalues

We present a new second order cone programming (SOCP) relaxation of nonconvex quadratic programs with a few negative eigenvalues (NQP-r-NE) by employing the difference of convex (DC) decomposition and simultaneous matrix diagonalization together. The auxiliary variables are bounded from above by one more convex quadratic constraint in the proposed SOCP relaxation than that in the classical SOCP relaxation provided by Kim and Kojima (2001). We prove that the proposed SOCP relaxation is strictly tighter than the classical SOCP relaxation under certain circumstances, especially when there exists one constraint matrix being positive definite in the primal problem. Three types of numerical experiments including the large-scale NQP-r-NE problem with the number of negative eigenvalues r≤20, the optimal spectrum sharing problem in MIMO cognitive radio networks and the two-trust-region subproblem are provided to illustrate that the proposed SOCP relaxation could achieve a better lower bound than that of the classical SOCP relaxation. Moreover, when the number of variables is larger than 250, the proposed SOCP relaxation shows its great superiority both in the bound quality and computing efficiency compared to the classical SOCP relaxation.

Multi-pulse edge-localized states on quantum graphs

We construct the edge-localized stationary states of the nonlinear Schrödinger equation on a general quantum graph in the limit of large mass. Compared to the previous works, we include arbitrary multi-pulse positive states which approach asymptotically a composition of N solitons, each sitting on a bounded (pendant, looping, or internal) edge. We give sufficient conditions on the edge lengths of the graph under which such states exist in the limit of large mass. In addition, we compute the precise Morse index (the number of negative eigenvalues in the corresponding linearized operator) for these multi-pulse states. If N solitons of the edge-localized state reside on the pendant and looping edges, we prove that the Morse index is exactly N. The technical novelty of this work is achieved by avoiding elliptic functions (and related exponentially small scalings) and closing the existence arguments in terms of the Dirichlet-to-Neumann maps for relevant parts of the given graph. We illustrate the general results with three examples of the flower, dumbbell, and single-interval graphs.

Eigenvalues of zero divisor graphs of principal ideal rings

In this paper, we first study zero divisor graphs over finite chain rings. We determine their rank, determinant, and eigenvalues using reduction graphs. Moreover, we extend the work to zero divisor graphs over finite commutative principal ideal rings using a combinatorial method, finding the number of positive eigenvalues and the number of negative eigenvalues, and finding upper and lower bounds for the largest eigenvalue. Finally, we characterize all finite commutative principal ideal rings such that their zero divisor graphs are complete and compute the Wiener index of the zero divisor graphs over finite commutative principal ideal rings.

The clustering of critical points in the evolving cosmic web

ABSTRACT Focusing on both small separations and baryonic acoustic oscillation scales, the cosmic evolution of the clustering properties of peak, void, wall, and filament-type critical points is measured using two-point correlation functions in ΛCDM dark matter simulations as a function of their relative rarity. A qualitative comparison to the corresponding theory for Gaussian random fields allows us to understand the following observed features: (i) the appearance of an exclusion zone at small separation, whose size depends both on rarity and signature (i.e. the number of negative eigenvalues) of the critical points involved; (ii) the amplification of the baryonic acoustic oscillation bump with rarity and its reversal for cross-correlations involving negatively biased critical points; (iii) the orientation-dependent small-separation divergence of the cross-correlations of peaks and filaments (respectively voids and walls) that reflects the relative loci of such points in the filament’s (respectively wall’s) eigenframe. The (cross-) correlations involving the most non-linear critical points (peaks, voids) display significant variation with redshift, while those involving less non-linear critical points seem mostly insensitive to redshift evolution, which should prove advantageous to model. The ratios of distances to the maxima of the peak-to-wall and peak-to-void over that of the peak-to-filament cross-correlation are ${\sim} \sqrt{2}$ and ${\sim} \sqrt{3}$, respectively, which could be interpreted as the cosmic crystal being on average close to a cubic lattice. The insensitivity to redshift evolution suggests that the absolute and relative clustering of critical points could become a topologically robust alternative to standard clustering techniques when analysing upcoming surveys such as Euclid or Large Synoptic Survey Telescope (LSST).

Radiomic features based on Hessian index for prediction of prognosis in head-and-neck cancer patients

This study demonstrated the usefulness of radiomic features based on the Hessian index of differential topology for the prediction of prognosis prior to treatment in head-and-neck (HN) cancer patients. The Hessian index, which can indicate tumor heterogeneity with convex, concave, and other points (saddle points), was calculated as the number of negative eigenvalues of the Hessian matrix at each voxel on computed tomography (CT) images. Three types of signatures were constructed in a training cohort (n = 126), one type each from CT conventional features, Hessian index features, and combined features from the conventional and index feature sets. The prognostic value of the signatures were evaluated using statistically significant difference (p value, log-rank test) to compare the survival curves of low- and high-risk groups. In a test cohort (n = 68), the p values of the models built with conventional, index, combined features, and clinical variables were 2.95 times 10–2, 1.85 times 10–2, 3.17 times 10–2, and 1.87 times 10–3, respectively. When the features were integrated with clinical variables, the p values of conventional, index, and combined features were 3.53 times 10–3, 1.28 times 10–3, and 1.45 times 10–3, respectively. This result indicates that index features could provide more prognostic information than conventional features and further increase the prognostic value of clinical variables in HN cancer patients.

Exponential Stability of Gyroscopically Stabilized Multi-Degree-of-Freedom Unstable Potential Systems Using Linear Damping

This paper shows that -degree-of-freedom generic unstable potential systems, whose potential matrices have an even number of negative eigenvalues and which are gyroscopically stabilized, can always be made exponentially stable through the use of an uncountably infinite number of indefinite damping matrices. A step-by-step methodology is provided for gyroscopic stabilization of such unstable potential systems that guarantees their exponential stability through the simultaneous use of positive and negative velocity feedback. Dissipative damping and positive velocity feedback are shown to constructively cooperate to bring about such stability. In contrast to the well-known Kelvin–Tait–Chetaev paradigm, which states that gyroscopically stabilized unstable potential systems are always made unstable in the presence of the slightest dissipation of energy, the paper points to a new paradigm, which states that the proper simultaneous dissipation and infusion of energy into generic gyroscopically stabilized unstable potential systems guarantees their exponential stability. What is meant by “generic” is explicitly stated, and such generic systems are shown to include most real-life -degree-of-freedom unstable potential systems.

About the Spectral Properties of One Three-Partial Model Operator

We investigate the structure of the essential spectrum of one three particle model operator H. We prove the existence of negative eigenvalues of the operator H and obtain the estimate for a number of negative eigenvalues of the operator H.

Spectrum of a Class of Difference Operators with Indefinite Weights

In this study, we use analytical methods and Sylvester inertia theorem to research a class of second order difference operators with indefinite weights and coupled boundary conditions. The eigenvalue problem with sign-changing weight has lasted a long time. The number of eigenvalues and the number of sign changes of the corresponding eigenfunctions of discrete equations under different boundary conditions are mainly studied. For the discrete Sturm-Liouville problems, similar conclusions about the properties of eigenvalues and the number of sign changes of the corresponding eigenfunctions are obtained under different boundary conditions, such as periodic boundary conditions, antiperiodic boundary conditions and separated boundary conditions etc. The purpose of this paper is to extend the similar conclusion to the coupled boundary conditions, which is of great significance to the perfection of the theory of the discrete Sturm-Liouville problems. We came to the following conclusions: first, the eigenvalues of the problem are real and single, the number of the positive eigenvalues is equal to the number of positive elements in the weight function, and the number of negative eigenvalues is equal to the number of negative elements in the weight function. Second, under some conditions, we obtain the sign change of the eigenfunction corresponding to the j-th positive/negative eigenvalue.

Spectral and analytic properties of some non-local Schrödinger operators and related jump processes

We discuss recent developments in the spectral theory of non-local Schrodinger operators via a Feynman-Kac-type approach. The processes we consider are subordinate Brownian motion and a class of jump Levy processes under a Kato-class potential. We discuss some explicitly soluble specific cases, and address the spatial decay properties of eigenfunctions and the number of negative eigenvalues in the general framework of the processes we introduce.

A low-dimensional SDP relaxation based spatial branch and bound method for nonconvex quadratic programs

In this paper, we propose a novel low-dimensional semidefinite programming (SDP) relaxation for convex quadratically constrained nonconvex quadratic programming problems. This new relaxation is derived by simultaneous matrix diagonalization method under the difference of convex decomposition scheme. The highlight of the relaxation is the low dimensionality of the positive semidefinite constraint, which only depends on the number of negative eigenvalues in the objective function. We prove that a mixed SOCP and SDP relaxation appeared in the literature is equivalent to the proposed relaxation, while the proposed relaxation has fewer constraints. We also provide conditions under which the proposed relaxation is as tight as the classical SDP relaxation and provides an optimal value for the original problem. For general cases, a spatial branch-and-bound algorithm is designed for finding a global optimal solution. Extensive numerical experiments support that the proposed algorithm outperforms two cutting-edge algorithms when the problem size is medium or large and the number of negative eigenvalues in the nonconvex objective function is relatively small.

A NOTE ON VARIANTS OF ZERO FORCING

A small improvement is made to the zero-forcing variants defined by Butler, Grout, and Hall (2015) for matrices with a given number of negative eigenvalues, resulting in a better value for the Barioli-Fallat tree and one negative eigenvalue.

On the sum of the k largest eigenvalues of graphs and maximal energy of bipartite graphs

Let G be a graph of order n. Also let λ1≥λ2≥⋯≥λn be the eigenvalues of graph G. In this paper, we present the following upper bound on the sum of the k(≤n) largest eigenvalues of G in terms of the order n and negative inertia θ (the number of negative eigenvalues): ∑i=1kλi≤n2(θ+1)(θ+θ(kθ+k−1)) with equality holding if and only if G≅nK1 or G≅pK‾t(n=pt,p≥2) with k=1 (where pK‾t is the complete p-partite graph of p t vertices with all partition sets having size t). Moreover, we obtain a sharp upper bound on the energy of bipartite graphs with given order n and rank r. We also propose an open problem as follows:Characterize all connected d-regular bipartite graphs of order n with 5 distinct eigenvalues and rank r=2n2(4d−n)2, where d>n4.Finally we give a partial solution to this problem.

Eigenvalue Bounds for a Class of Schrödinger Operators in a Strip

This paper is concerned with the estimation of the number of negative eigenvalues (bound states) of Schrödinger operators in a strip subject to Neumann boundary conditions. The estimates involve weighted L1 norms and Lln⁡L norms of the potential. Estimates involving the norms of the potential supported by a curve embedded in a strip are also presented.

Spectrum theory of second-order difference equations with indefinite weight

In this paper, we study the spectrum structure of second-order difference operators with sign-changing weight. We apply the Sylvester inertia theorem to show that the spectrum consists of real and simple eigenvalues; the number of positive eigenvalues is equal to the number of positive elements in the weight function, and the number of negative eigenvalues is equal to the number of negative elements in the weight function. We also show that the eigenfunction corresponding to the j -th positive/negative eigenvalue changes its sign exactly j-1 times.

High-Frequency Covariance Matrix Estimation Using Price Durations

We propose a price duration based covariance matrix estimator using high frequency transactions data. The effect of the last-tick time-synchronisation methodology, together with effects of important market microstructure components is analysed through a comprehensive Monte Carlo study. To decrease the number of negative eigenvalues produced by the non positive-semi-definite (psd) covariance matrix, we devise an average covariance estimator by taking an average of a wide range of duration based covariance matrix estimators. Empirically, candidate covariance estimators are implemented on 19 stocks from the DJIA. The duration based covariance estimator is shown to provide comparably accurate estimates with smaller variation compared with competing estimators. An out-of-sample GMV portfolio allocation problem is studied. A simple shrinkage technique is introduced to make the sample matrices psd and well-conditioned. Compared to competing high-frequency covariance matrix estimators, the duration based estimator is shown to give more stable portfolio weights and higher Sharpe ratios while maintaining comparably low portfolio variances.

The Maslov index and the spectra of second order elliptic operators

We consider second order elliptic differential operators on a bounded Lipschitz domain Ω. Firstly, we establish a natural one-to-one correspondence between their self-adjoint extensions, with domains of definition containing in H1(Ω), and Lagrangian planes in H1/2(∂Ω)×H−1/2(∂Ω). Secondly, we derive a formula relating the spectral flow of the one-parameter families of such operators to the Maslov index, the topological invariant counting the signed number of conjugate points of paths of Lagrangian planes in H1/2(∂Ω)×H−1/2(∂Ω). Furthermore, we compute the Morse index, the number of negative eigenvalues, in terms of the Maslov index for several classes of the second order operators: the θ→-periodic Schrödinger operators on a period cell Q⊂Rn, the elliptic operators with Robin-type boundary conditions, and the abstract self-adjoint extensions of the Schrödinger operators on star-shaped domains. Our work is built on the techniques recently developed by B. Booß-Bavnbek, K. Furutani, and C. Zhu, and extends the scope of validity of their spectral flow formula by incorporating the self-adjoint extensions of the second order operators with domains in the first order Sobolev space H1(Ω). In addition, we generalize the results concerning relations between the Maslov and Morse indices quite recently obtained by G. Cox, J. Deng, C. Jones, J. Marzuola, A. Sukhtayev and the authors. Finally, we describe and study a link between the theory of abstract boundary triples and the Lagrangian description of self-adjoint extensions of abstract symmetric operators.

Spectral Theory for Schr\xf6dinger Operators with varvec{delta }-Interactions Supported on Curves in varvec{mathbb {R}^3}

The main objective of this paper is to systematically develop a spectral and scattering theory for self-adjoint Schrödinger operators with delta -interactions supported on closed curves in mathbb {R}^3. We provide bounds for the number of negative eigenvalues depending on the geometry of the curve, prove an isoperimetric inequality for the principal eigenvalue, derive Schatten–von Neumann properties for the resolvent difference with the free Laplacian, and establish an explicit representation for the scattering matrix.

Copositivity and Sparsity Relations Using Spectral Properties

A standard quadratic optimization problem consists in minimizing a quadratic form over the standard simplex. This problem has a closer connection with copositive optimization, where the copositivity appears in local optimality conditions. In this paper, we establish a relationship between the sparsity of a solution of the standard quadratic optimization problem, from one hand, and the copositivity of the Hessian matrix of the Lagrangian from other hand. More precisely, if the Hessian matrix is copositive we prove that the number of zero components, of a local minimizer associated with a standard quadratic optimization problem, is greater or equal to the number of negative eigenvalues counting multiplicities. Moreover, we show that if the number of zero components of a local minimizer is equal to the number of negative eigenvalues of the Hessian matrix, then the strict complementarity condition is satisfied and the critical cone is a linear subspace.