On inverse spectrum problems for normal nonnegative matrices

For a set σ with n complex numbers, some sufficient conditions are found for σ to be the spectrum of an n ×n normal (entrywise) nonnegative (positive) matrix. After proving a fundamental theorem and introducing the companion set σ' of σ which consists of real numbers, we prove that if σ' satisfies any known sufficient conditions for a real set to be the spectrum of a nonnegative matrix introduced by Suleimanova, Perfect, Salzmann and Kellogg respectively, then σ is the spectrum of an n×n normal nonnegative matrix.     

On inverse spectrum problems for normal nonnegative matrices

For a set σ with n complex numbers, some sufficient conditions are found for σ to be the spectrum of an n ×n normal (entrywise) nonnegative (positive) matrix. After proving a fundamental theorem and introducing the companion set σ′ of σ which consists of real numbers, we prove that if σ′ satisfies any known sufficient conditions for a real set to be the spectrum of a nonnegative matrix introduced by Suleimanova, Perfect, Salzmann and Kellogg respectively, then σ is the spectrum of an n×n normal nonnegative matrix.     

Nonnegative matrices having nonnegative Drazin pseudoinverses

Necessary and sufficient conditions for nonnegative matrices having nonnegative Drazin pseudoinverses are obtained. A decomposition theorem which characterizes the class of all nonnegative matrices with nonnegative Drazin pseudoinverses is proved, thus answering a question raised by several people. It is also shown that if a row (or column) stochastic matrix has a nonnegative Drazin pseudoinverse A(^d), then A(^d) is some power of A. These results extend known results for nonnegative group-monotone matrices.     

具两个时滞项的微分方程的稳定性

本文考虑了一类具两个时滞项的微分方程的稳定性，其中一个时滞项的系数非负，另一个时滞项的系数非正．当非负系数恒为零时，本文所得结论改进了Yorke等提出的3/2稳定性定理的相应结论，当非负系数项的时滞为零，本文不同于已有文献用Liapunov函数或泛函法将时滞项作为干扰处理，而是反过来利用时滞项让方程稳定．     

Characterizations of linear Volterra integral equations with nonnegative kernels

We first introduce the notion of positive linear Volterra integral equations. Then, we offer a criterion for positive equations in terms of the resolvent. In particular, equations with nonnegative kernels are positive. Next, we obtain a variant of the Paley-Wiener theorem for equations of this class and its extension to perturbed equations. Furthermore, we get a Perron-Frobenius type theorem for linear Volterra integral equations with nonnegative kernels. Finally, we give a criterion for positivity of the initial function semigroup of linear Volterra integral equations and provide a necessary and sufficient condition for exponential stability of the semigroups.     

一类二阶脉冲微分方程三点边值问题三个非负解的存在性

研究一类二阶脉冲微分方程三点边值问题三个非负解的存在性,利用Leggett-Williams不动点定理得到了方程三个非负解存在的充分条件.     

一类随机时滞递归神经网络的指数稳定性(英文)

本文研究了一类随机时滞递归神经网络的指数稳定性问题.利用非负鞅收敛定理和Lyapunov泛函的方法,获得了这类神经网络矩指数稳定性的新的代数准则,所给代数准则简单易用.一个具体实例用来说明稳定性判别准则的应用.     

Convergence dynamics of stochastic reaction–diffusion recurrent neural networks with continuously distributed delays

Convergence dynamics of reaction–diffusion recurrent neural networks (RNNs) with continuously distributed delays and stochastic influence are considered. Some sufficient conditions to guarantee the almost sure exponential stability, mean value exponential stability and mean square exponential stability of an equilibrium solution are obtained, respectively. Lyapunov functional method, M-matrix properties, some inequality technique and nonnegative semimartingale convergence theorem are used in our approach. These criteria ensuring the different exponential stability show that diffusion and delays are harmless, but random fluctuations are important, in the stochastic continuously distributed delayed reaction–diffusion RNNs with the structure satisfying the criteria. Two examples are also given to demonstrate our results.     

Eigenvectors and ratio limit theorems for Markov chains and their relatives

Associated to classes of countable discrete Markov chains or, more generally, column-finite nonnegative infinite matrices, and a finite subset of the state space, is a dimension group. In many cases, this dimension group gives information about the nonnegative eigenvectors of the process. Moreover, the study of the nonnegative eigenvectors is, equivalent to the traces on an analytic one parameter family of dimension groups. We pay particular attention to the case that there is at most one nonnegative eigenvector per eigenvalue, giving a number of sufficient conditions. Using the techniques developed here, we also show that under a reasonable set of conditions (principle among them that there be just one nonnegative eigenvector for the spectral radius), a (one-sided) ratio limit theorem holds. Supported in part by an operating grant from NSERC (Canada) and an Isaac Walton Killam Fellowship (Canada Council).     

A quadrangle comparison theorem and its application to soul theory for Alexandrov spaces

We shall derive two sufficient conditions for complete finite-dimensional Alexandrov spaces of nonnegative curvature to be contractible. One of the new technical tools used in our proof is a quadrangle comparison theorem inspired by Perelman.     

The Perron-Frobenius Theorem Revisited

An extension of the Perron-Frobenius Theorem is presented in the much more general setting of indecomposable semigroups of nonnegative matrices. Many features of the original theorem including the existence of a fixed positive vector, a block-monomial form, and spectral stability properties hold simultaneously for these semigroups. The paper is largely self-contained and the proofs are elementary. The classical theorem and some related results follow as corollaries.     

多时滞随机神经网络的稳定性

通过构建李雅普偌夫函数的方法和利用半鞅收敛定理对一类随机时滞神经网络的全局指数稳定进行了分析,提出了易于判定随机时滞神经网络几乎必然指数稳定性新的代数判据,推广了[1]中的主要结论.     

Almost Schur lemma for manifolds with boundary

In this paper, we prove the almost Schur theorem, introduced by De Lellis and Topping, for the Riemannian manifold M of nonnegative Ricci curvature with totally geodesic boundary. Examples are given to show that it is optimal when the dimension of M is at least 5. We also prove that the almost Schur theorem is true when M is a 4-dimensional manifold of nonnegative scalar curvature with totally geodesic boundary. Finally we obtain a generalization of the almost Schur theorem in all dimensions only by assuming the Ricci curvature is bounded below.     

Positive solutions for a singular fractional boundary value problem

We investigate the existence of positive solutions for a system of Riemann-Liouville fractional differential equations, supplemented with uncoupled nonlocal boundary conditions which contain various fractional derivatives and Riemann-Stieltjes integrals, and the nonlinearities of the system are nonnegative functions and they may be singular at the time variable. In the proof of our main theorems, we use the Guo-Krasnosel'skii fixed point theorem.     

Positive Periodic Solution of a Periodic Cooperative Model with State Dependent Delays

By using the continuation theorem of coincidence degree theory, sufficient conditions are obtained for the existence of positive periodic solution for the periodic cooperative system with state dependent delays where r i , a i , c i , i = 1, 2 are positive continuous y -periodic functions, b i , i = 1, 2 are nonnegative continuous y -periodic functions, and ij , i = 1, 2; j = 1, 2, 3 are nonnegative continuous functions and y -periodic with respect to their first argument.     

Smoothness of isotopy for symplectic pairs

Let M be a 2n-dimensional smooth manifold with a symplectic pair which is a pair of closed 2-forms of constant ranks with complementary kernel foliations. Similar to Moser's stability theorem for symplectic forms, one desires to establish a stability theorem for symplectic pairs. Some sufficient and necessary conditions are obtained by Bande, Ghiggini and Kotschick. In this article, we consider a technical problem relating to the stability theorem. To complete the proof of the stability theorem for symplectic pairs, we verify the smoothness of the isotopy which is ignored in the literature. The Hodge theory for Riemannian foliation is crucial to our discussion.     

New eigenvalue inclusion sets for tensors

Two new eigenvalue inclusion sets for tensors are established. It is proved that the new eigenvalue inclusion sets are tighter than that in Qi's paper “Eigenvalues of a real supersymmetric tensor”. As applications, upper bounds for the spectral radius of a nonnegative tensor are obtained, and it is proved that these upper bounds are sharper than that in Yang's paper “Further results for Perron–Frobenius theorem for nonnegative tensors”. And some sufficient conditions of the positive definiteness for an even‐order real supersymmetric tensor are given. Copyright © 2012 John Wiley & Sons, Ltd.     

EQUIVALENCE BETWEEN NONNEGATIVE SOLUTIONS TO PARTIAL SPARSE AND WEIGHTED l1-NORM MINIMIZATIONS

Based on the range space property (RSP), the equivalent conditions between nonnegative solutions to the partial sparse and the corresponding weighted l₁-norm minimization problem are studied in this paper. Different from other conditions based on the spark property, the mutual coherence, the null space property (NSP) and the restricted isometry property (RIP), the RSP-based conditions are easier to be verified. Moreover, the proposed conditions guarantee not only the strong equivalence, but also the equivalence between the two problems. First, according to the foundation of the strict complementarity theorem of linear programming, a sufficient and necessary condition, satisfying the RSP of the sensing matrix and the full column rank property of the corresponding sub-matrix, is presented for the unique nonnegative solution to the weighted l₁-norm minimization problem. Then, based on this condition, the equivalence conditions between the two problems are proposed. Finally, this paper shows that the matrix with the RSP of order k can guarantee the strong equivalence of the two problems.     

On elliptic problems with indefinite superlinear boundary conditions

In this paper we prove the existence of positive solutions to some nondivergent elliptic equations with indefinite nonlinear boundary conditions. The proof is based on a new Liouville-type theorem about the nonnegative solutions to some canonical indefinite elliptic equations, which is also proved in this paper by the method of moving planes.