Nonnegative Eigenvalues Research Articles

Eigenvalues for the Steklov problem via Ljusternic–Schnirelman principle

Abstract This paper deals with the existence of nondecreasing sequence of nonnegative eigenvalues for the systems div ( a ( x ) | ∇ u | p - 2 ∇ u ) = b ( x ) | u | p - 2 u in Ω , | ∇ u | p - 2 ∂ u ∂ n = λ c ( x ) | u | p - 2 u on ∂ Ω , by using the Ljusternic–Schnirelman principle, where Ω is a bounded domain in RN(N ⩾ 2).

Eigenvalue problem for p-Laplacian with mixed boundary conditions

Eigenvalue problem for p-Laplacian with mixed boundary conditions is concerned on a bounded domain. The existence of nonnegative eigenvalues are obtained by using the Lusternik-Schnirelman principle. Boundedness of eigenfunctions is obtained by using the Moser iteration. The simplicity and isolation of the first eigenvalue are proved. The existence of the second eigenvalue is also illustrated.

Weighted Dirichlet spaces of harmonic functions on the real hyperbolic ball

Let 𝔹 n denote the unit ball in ℝ n , n ≥ 2, with boundary S, and let τ, ∇ h and Δ h denote the volume measure, gradient and Laplacian with respect to the hyperbolic metric on 𝔹 n . In the article we consider the weighted Dirichlet space 𝒟γ and Hardy space ℋ p , p ≥ 1, of hyperbolic harmonic functions on 𝔹 n for which D γ(f) and ‖ f ‖ p are finite, where and One of the main results of the article is the following theorem. THEOREM Let f be a hyperbolic harmonic function on 𝔹 n . (a) If f ∈ 𝒟γ for some γ, (n − 3) < γ ≤ (n − 1) (0 < γ ≤ 1 when n = 2), then f ∈ ℋ p for all p, 0 < p ≤ 2(n − 1)/γ, with where C 1 is a positive constant independent of f. (b) If f ∈ ℋ p for some p, 1 < p ≤ 2, then f ∈ 𝒟γ for all γ ≥ 2(n − 1)/p with where C 2 is a positive constant independent of f. In this article, we also prove two results concerning the integrability of |∇ h f | of eigenfunctions f of Δ h with nonnegative eigenvalues and prove the Littlewood–Paley inequalities for hyperbolic harmonic functions on 𝔹 n .

A Novel Strategy of Nonnegative-Matrix-Factorization-Based Polarimetric Ship Detection

In this letter, a new strategy based on nonnegative matrix factorization (NMF) is proposed for polarimetric ship detection. This method utilizes the sparse feature of nonnegative eigenvalues, and the sparse degree is proposed to be estimated from the histogram which can reveal the sparse distribution of eigenvalues. Combining the nonnegative and sparse features, the NMF-based ship detection method can be implemented flexibly and efficiently. It has been carried out on the C-band quad polarimetric synthetic aperture radar (PolSAR) and dual PolSAR ocean data sets to validate its effectiveness. Unlike a constant-false-alarm-rate detector, the NMF method does not depend on target size and therefore offers improved detection performance under low-signal-to-clutter-ratio conditions.

Model-Based Decomposition of Polarimetric SAR Covariance Matrices Constrained for Nonnegative Eigenvalues

Model-based decomposition of polarimetric radar covariance matrices holds the promise that specific scattering mechanisms can be isolated for further quantitative analysis. In this paper, we show that current algorithms suffer from a fatal flaw in that some of the scattering components result in negative powers. We propose a simple modification that ensures that all covariance matrices in the decomposition will have nonnegative eigenvalues. We further combine our nonnegative eigenvalue decomposition with eigenvector decomposition to remove additional assumptions that have to be made before the current algorithms can be used to estimate all the scattering components. Our results are illustrated using Airborne Synthetic Aperture Radar data and show that current algorithms typically overestimate the canopy scattering contribution by 10%-20%.

Penalized versions of the Newman-Girvan modularity and their relation to normalized cuts andk-means clustering

Two penalized-balanced and normalized-versions of the Newman-Girvan modularity are introduced and estimated by the non-negative eigenvalues of the modularity and normalized modularity matrix, respectively. In this way, the partition of the vertices that maximizes the modularity can be obtained by applying the k-means algorithm for the representatives of the vertices based on the eigenvectors belonging to the largest positive eigenvalues of the modularity or normalized modularity matrix. The proper dimension depends on the number of the structural eigenvalues of positive sign, while dominating negative eigenvalues indicate an anticommunity structure; the balance between the negative and the positive eigenvalues determines whether the underlying graph has a community, anticommunity, or randomlike structure.

Theory of sparse random matrices and vibrational spectra of amorphous solids

A random matrix approach is used to analyze the vibrational properties of amorphous solids. We investigated a dynamical matrix M=AA^T with non-negative eigenvalues. The matrix A is an arbitrary real NxN sparse random matrix with n independent non-zero elements in each row. The average values <A_{ij}>=0 and dispersion <A_{ij}^2>=V^2 for all non-zero elements. The density of vibrational states g(w) of the matrix M for N,n >> 1 is given by the Wigner quarter circle law with radius independent of N. We argue that for n^2 << N this model can be used to describe the interaction of atoms in amorphous solids. The level statistics of matrix M is well described by the Wigner surmise and corresponds to repulsion of eigenfrequencies. The participation ratio for the major part of vibrational modes in three dimensional system is about 0.2 - 0.3 and independent of N. Together with term repulsion it indicates clearly to the delocalization of vibrational excitations. We show that these vibrations spread in space by means of diffusion. In this respect they are similar to diffusons introduced by Allen, Feldman, et al., Phil. Mag. B 79, 1715 (1999) in amorphous silicon. Our results are in a qualitative and sometimes in a quantitative agreement with molecular dynamic simulations of real and model glasses.

On Open Problems of Nonnegative Inverse Eigenvalues Problem

In this paper, we give solvability conditions for three open problems of nonnegative inverse eigenvalues problem (NIEP) which were left hanging in the air up to seventy years. It will offer effective ways to judge an NIEP whether is solvable.

Further developments of Furuta inequality of indefinite type

A selfadjoint involutive matrix J endows Cn with an indefinite inner product [·, ·] given by [x,y] := 〈Jx,y〉 , x,y ∈ Cn. We study matrix inequalities for J -selfadjoint matrices with nonnegative eigenvalues. Namely, Furuta inequality of indefinite type is revisited. Characterizations of the J -chaotic order and of the J -relative entropy are obtained via Furuta inequality. The parallelism between the definite versions of the inequalities on Hilbert spaces and the corresponding indefinite versions on Krein spaces is pointed out. Mathematics subject classification (2010): 47B50, 47A63, 15A45.

Eigenvalue problems for the p-Laplacian

We study nonlinear eigenvalue problems for the p -Laplace operator subject to different kinds of boundary conditions on a bounded domain. Using the Ljusternik–Schnirelman principle, we show the existence of a nondecreasing sequence of nonnegative eigenvalues. We prove the simplicity and isolation of the principal eigenvalue and give a characterization for the second eigenvalue.

Enumeration of perfect matchings of a type of Cartesian products of graphs

Let G be a graph and let Pm ( G ) denote the number of perfect matchings of G. We denote the path with m vertices by P m and the Cartesian product of graphs G and H by G × H . In this paper, as the continuance of our paper [W. Yan, F. Zhang, Enumeration of perfect matchings of graphs with reflective symmetry by Pfaffians, Adv. Appl. Math. 32 (2004) 175–188], we enumerate perfect matchings in a type of Cartesian products of graphs by the Pfaffian method, which was discovered by Kasteleyn. Here are some of our results: 1. Let T be a tree and let C n denote the cycle with n vertices. Then Pm ( C 4 × T ) = ∏ ( 2 + α 2 ) , where the product ranges over all eigenvalues α of T. Moreover, we prove that Pm ( C 4 × T ) is always a square or double a square. 2. Let T be a tree. Then Pm ( P 4 × T ) = ∏ ( 1 + 3 α 2 + α 4 ) , where the product ranges over all non-negative eigenvalues α of T. 3. Let T be a tree with a perfect matching. Then Pm ( P 3 × T ) = ∏ ( 2 + α 2 ) , where the product ranges over all positive eigenvalues α of T. Moreover, we prove that Pm ( C 4 × T ) = [ Pm ( P 3 × T ) ] 2 .

Spectral theory of copositive matrices

Let A ∈ R n× n . We provide a block characterization of copositive matrices, with the assumption that one of the principal blocks is positive definite. Haynsworth and Hoffman showed that if r is the largest eigenvalue of a copositive matrix then r ⩾ ∣ λ∣, for all other eigenvalues λ of A. We continue their study of the spectral theory of copositive matrices and show that a copositive matrix must have a positive vector in the subspace spanned by the eigenvectors corresponding to the nonnegative eigenvalues. Moreover, if a symmetric matrix has a positive vector in the subspace spanned by the eigenvectors corresponding to its nonnegative eigenvalues, then it is possible to increase the nonnegative eigenvalues to form a copositive matrix A′, without changing the eigenvectors. We also show that if a copositive matrix has just one positive eigenvalue, and n − 1 nonpositive eigenvalues then A has a nonnegative eigenvector corresponding to a nonnegative eigenvalue.

Enumeration of perfect matchings of graphs with reflective symmetry by Pfaffians

The Pfaffian method enumerating perfect matchings of plane graphs was discovered by Kasteleyn. We use this method to enumerate perfect matchings in a type of graphs with reflective symmetry which is different from the symmetric graphs considered in [J. Combin. Theory Ser. A 77 (1997) 67, MATCH—Commun. Math. Comput. Chem. 48 (2003) 117]. Here are some of our results: (1) If G is a reflective symmetric plane graph without vertices on the symmetry axis, then the number of perfect matchings of G can be expressed by a determinant of order | G|/2, where | G| denotes the number of vertices of G. (2) If G contains no subgraph which is, after the contraction of at most one cycle of odd length, an even subdivision of K 2,3, then the number of perfect matchings of G× K 2 can be expressed by a determinant of order | G|. (3) Let G be a bipartite graph without cycles of length 4 s, s∈{1,2,…}. Then the number of perfect matchings of G× K 2 equals ∏(1+ θ 2) m θ , where the product ranges over all non-negative eigenvalues θ of G and m θ is the multiplicity of eigenvalue θ. Particularly, if T is a tree then the number of perfect matchings of T× K 2 equals ∏(1+ θ 2) m θ , where the product ranges over all non-negative eigenvalues θ of T and m θ is the multiplicity of eigenvalue θ.

Graphs with fourth Laplacian eigenvalue less than two

In this paper, all connected graphs with the fourth largest Laplacian eigenvalue less than two are determined, which are used to characterize all connected graphs with exactly three Laplacian eigenvalues no less than two. Moreover, we determine bipartite graphs such that the adjacency matrices of their line graphs have exactly three nonnegative eigenvalues.

An artificially upstream flux vector splitting scheme for the Euler equations

A new method is proposed to split the flux vector of the Euler equations by introducing two artificial wave speeds. The direction of wave propagation is adjusted by these two wave speeds. If they are set to be the fastest wave speeds in two opposite directions, the method leads to the HLL approximate Riemann solver devised by Harten, Lax and van Leer, which indicates that the HLL solver is a vector flux splitting scheme as well as a Godunov-type scheme. A more accurate scheme that resolves 1D contact discontinuity is further proposed by carefully choosing two wave speeds so that the flux vector is split to two simple flux vectors. One flux vector comes with either non-negative or non-positive eigenvalues and is easily solved by one-side differencing. Another flux vector becomes a system of two waves and one, two or three stationary discontinuities depending on the dimension of the Euler equations. Numerical flux function for multi-dimensional Euler equations is formulated for any grid system, structured or unstructured. A remarkable simplicity of the scheme is that it successfully achieves one-sided approximation for all waves without recourse to any matrix operation. Moreover, its accuracy is comparable with the exact Riemann solver. For 1D Euler equations, the scheme actually surpasses the exact solver in avoiding expansion shocks without any additional entropy fix. The scheme can exactly resolve stationary 1D contact discontinuities, and it avoids the carbuncle problem in multi-dimensional computations. The robustness of the scheme is shown in 1D test cases designed by Toro, and other 2D calculations.

New hierarchy for the Liouville equation, irreversibility and Fokker-Planck-like structures

The issue of irreversibility is revisited for a closed system formed by N classical non-relativistic particles inside a volume Ω, interacting through two-body potentials, for large N and Ω. The classical phase-space distribution function f, multiplied by suitable Hermite polynomials and integrated over all momenta, yields new moments. The Liouville equation and the initial distribution fin imply a new non-equilibrium linear infinite hierarchy for the moments. That hierarchy differs from the BBGKY one for distribution functions and displays some suggestive Fokker-Planck-like structures. A physically motivated ansatz for fin (which introduces statistical assumptions), used by previous authors, is chosen. All moments of order n ≥ n0 are expressed in terms of those of order n0 — 1 and of fin. The properties of the Fokker-Planck-like structures (hermiticity, non-negative eigenvalues) allow for implementing a natural long-time approximation in the hierarchy, so as to introduce relaxation to equilibrium and irreversibility, consistently with the hydrodynamical balance equations. Further (more restrictive) assumptions and approximations lead to new irreversible models, generalizing non-trivially the Fokker-Planck equation. They are described through a truncated hierarchy of linear equations for moments of order n ≤ n0 — 1 (n0 being finite). The connections with Brownian particle dynamics and Fluid Dynamics are analyzed, for consistency.

A SPLITTING THEOREM FOR KÄHLER MANIFOLDS WHOSE RICCI TENSORS HAVE CONSTANT EIGENVALUES

It is proved that a compact Kähler manifold whose Ricci tensor has two distinct constant non-negative eigenvalues is locally the product of two Kähler–Einstein manifolds. A stronger result is established for the case of Kähler surfaces. Without the compactness assumption, irreducible Kähler manifolds with Ricci tensor having two distinct constant eigenvalues are shown to exist in various situations: there are homogeneous examples of any complex dimension n ≥ 2 with one eigenvalue negative and the other one positive or zero; there are homogeneous examples of any complex dimension n ≥ 3 with two negative eigenvalues; there are non-homogeneous examples of complex dimension 2 with one of the eigenvalues zero. The problem of existence of Kähler metrics whose Ricci tensor has two distinct constant eigenvalues is related to the celebrated (still open) conjecture of Goldberg [24]. Consequently, the irreducible homogeneous examples with negative eigenvalues give rise to complete Einstein strictly almost Kähler metrics of any even real dimension greater than 4.

Global Bounds of Schrödinger Heat Kernels with Negative Potentials

We obtain global in time bounds for the heat kernel G of the Schrödinger operator L=−Δ+V. The potential V satisfies V(x)∼−C/d(x)b near infinity with b∈(0, ∞). The result can be described as follows. Suppose L is positive and b=2. Then G=G(x, t; y, 0) has a global upper bound which is a standard Gaussian times a polynomial function of x, y, t. However when L is not nonnegative and negative eigenvalues exist G=G(x, t; 0, 0) is bounded below by a standard Gaussian times ectwithc>0. In other words, the growth rate of G(x, t; 0, 0) is comparable with the heat kernel of −Δ−c, regardless how fast V decays near infinity.

Graphs with a Small Number of Nonnegative Eigenvalues

In this paper, by means of computer checking, all simple graphs with at most two nonnegative eigenvalues, and all minimal simple graphs with exactly two (respectively, three) nonnegative eigenvalues are determined.

Modifying the inertia of matrices arising in optimization

Applications in constrained optimization (and other areas) produce symmetric matrices with a natural block 2 × 2 structure. An optimality condition leads to the problem of perturbing the (1,1) block of the matrix to achieve a specific inertia. We derive a perturbation of minimal norm, for any unitarily invariant norm, that increases the number of nonnegative eigenvalues by a given amount, and we show how it can be computed efficiently given a factorization of the original matrix. We also consider an alternative way to satisfy the optimality condition based on a projection approach. Theoretical tools developed here include an extension of Ostrowski's theorem on congruences and some lemmas on inertias of block 2 × 2 symmetric matrices.