Discrete random bounds for general random variables and applications to reliability

We here propose some new algorithms to compute bounds for (1) cumulative density functions of sums of i.i.d. nonnegative random variables, (2) renewal functions and (3) cumulative density functions of geometric sums of i.i.d. nonnegative random variables. The idea is very basic and consists in bounding any general nonnegative random variable X   by two discrete random variables with range in hN$h\mathbb{N}$, which both converge to X as h goes to 0. Numerical experiments are lead on and the results given by the different algorithms are compared to theoretical results in case of i.i.d. exponentially distributed random variables and to other numerical methods in other cases.     

On a conjecture of ridge

The conjecture of Ridge on the numerical range of a shift of periodic weights is resolved in the affirmative, i.e., if the weights are nonzero, the numerical range of the corresponding shift is an open disc centered at the origin. The radius of the disc can be expressed as the Perron root of a nonnegative irreducible symmetric matrix. Some related results are obtained.

     

非负矩阵数值域的圆盘性质

本文利用图论方法并结合非负矩阵的有关经典结果,成功地给出了任意阶的非负方阵有以原点为中心的圆盘数值域的充要条件。     

Nonnegative solutions of an elliptic equation and Harnack ends of Riemannian manifolds

In this paper, we give a barrier argument at infinity for solutions of an elliptic equation on a complete Riemannian manifold. By using the barrier argument, we can construct a nonnegative (bounded, respectively) solution of the elliptic equation, which takes the given data at infinity of each end. In particular, we prove that if a complete Riemannian manifold has finitely many ends, each of which is Harnack and nonparabolic, then the set of bounded solutions of the elliptic equation is finite dimensional, in some sense. We also prove that if a complete Riemannian manifold is roughly isometric to a complete Riemannian manifold satisfying the volume doubling condition, the Poincaré inequality and the finite covering condition on each end, then there exists a nonnegative solution of an elliptic equation taking the given data at infinity of each end of the manifold. These results generalize those of Yau, of Donnelly, of Grigor'yan, of Li and Tam, of Holopainen, and of the present authors, but with the barrier argument at infinity that enables one to overcome the obstacle due to the nonlinearity of solutions. Received: 11 November 1999     

Proportionally modular diophantine inequalities

We study the sets of nonnegative solutions of Diophantine inequalities of the form with a,b and c positive integers. These sets are numerical semigroups, which we study and characterize.     

Critical metrics of the volume functional on three-dimensional manifolds

In this paper, we prove the three-dimensional $CPE$ conjecture with nonnegative Ricci curvature. Moreover, we establish rigidity theorems for three-dimensional compact, oriented, connected V-static metrics with nonnegative Ricci curvature. Finally, we obtain classification results on three-dimensional vacuum static space and Miao–Tam critical metric with nonnegative Ricci curvature.     

关于非紧流形上的Ricci流的一个注记

设（M^3,90）是非紧三维Riemann流形,其Ricci曲率非负,单射半径有正的下界,且当x→∞时数量曲率R（x）→0。则以（M^3,go）为初始值的Ricci流在M^3×[0,∞）上有长期解。这推广了马和朱最近的一个结果．在高维情形我们也有相应的结果,并且我们给Chau,Tam和Yu在Ktihler情形的类似定理一个新的证明。     

Finding the Maximum Eigenvalue of Essentially Nonnegative Symmetric Tensors via Sum of Squares Programming

Finding the maximum eigenvalue of a tensor is an important topic in tensor computation and multilinear algebra. Recently, for a tensor with nonnegative entries (which we refer it as a nonnegative tensor), efficient numerical schemes have been proposed to calculate its maximum eigenvalue based on a Perron–Frobenius-type theorem. In this paper, we consider a new class of tensors called essentially nonnegative tensors, which extends the concept of nonnegative tensors, and examine the maximum eigenvalue of an essentially nonnegative tensor using the polynomial optimization techniques. We first establish that finding the maximum eigenvalue of an essentially nonnegative symmetric tensor is equivalent to solving a sum of squares of polynomials (SOS) optimization problem, which, in its turn, can be equivalently rewritten as a semi-definite programming problem. Then, using this sum of squares programming problem, we also provide upper and lower estimates for the maximum eigenvalue of general symmetric tensors. These upper and lower estimates can be calculated in terms of the entries of the tensor. Numerical examples are also presented to illustrate the significance of the results.     

Linear and strong convergence of algorithms involving averaged nonexpansive operators

We introduce regularity notions for averaged nonexpansive operators. Combined with regularity notions of their fixed point sets, we obtain linear and strong convergence results for quasicyclic, cyclic, and random iterations. New convergence results on the Borwein–Tam method (BTM) and on the cyclically anchored Douglas–Rachford algorithm (CADRA) are also presented. Finally, we provide a numerical comparison of BTM, CADRA and the classical method of cyclic projections for solving convex feasibility problems.     

Proportionally Modular Diophantine Inequalities and Full Semigroups

A proportionally modular numerical semigroup is the set of nonnegative integer solutions to a Diophantine inequality of the type ax mod b ≤ cx. We give a new presentation for these semigroups and we relate them with a type of affine full semigroups. Next, we describe explicitly the minimal generating system for the affine full semigroups we are considering. As a consequence, we obtain generating systems for proportionally modular numerical semigroups and we exhibit several families of these semigroups in terms of their generators. Finally, we use the concept of fundamental gap to study when a proportionally modular numerical semigroup is symmetric and we propose some open problems.     

Strictly nonnegative tensors and nonnegative tensor partition

We introduce a new class of nonnegative tensors—strictly nonnegative tensors.A weakly irreducible nonnegative tensor is a strictly nonnegative tensor but not vice versa.We show that the spectral radius of a strictly nonnegative tensor is always positive.We give some necessary and su?cient conditions for the six wellconditional classes of nonnegative tensors,introduced in the literature,and a full relationship picture about strictly nonnegative tensors with these six classes of nonnegative tensors.We then establish global R-linear convergence of a power method for finding the spectral radius of a nonnegative tensor under the condition of weak irreducibility.We show that for a nonnegative tensor T,there always exists a partition of the index set such that every tensor induced by the partition is weakly irreducible;and the spectral radius of T can be obtained from those spectral radii of the induced tensors.In this way,we develop a convergent algorithm for finding the spectral radius of a general nonnegative tensor without any additional assumption.Some preliminary numerical results show the feasibility and effectiveness of the algorithm.     

Algorithmic copositivity detection by simplicial partition

We present new criteria for copositivity of a matrix, i.e., conditions which ensure that the quadratic form induced by the matrix is nonnegative over the nonnegative orthant. These criteria arise from the representation of the quadratic form in barycentric coordinates with respect to the standard simplex and simplicial partitions thereof. We show that, as the partition gets finer and finer, the conditions eventually capture all strictly copositive matrices. We propose an algorithmic implementation which considers several numerical aspects. As an application, we present results on the maximum clique problem. We also briefly discuss extensions of our approach to copositivity with respect to arbitrary polyhedral cones.     

Isospectral flow method for nonnegative inverse eigenvalue problem with prescribed structure

The nonnegative inverse eigenvalue problem is that given a family of complex numbers λ={λ₁,…,λ_n}, find a nonnegative matrix of order n with spectrum λ. This problem is difficult and remains unsolved partially. In this paper, we focus on its generalization that the reconstructed nonnegative matrices should have some prescribed entries. It is easy to see that this new problem will come back to the common nonnegative inverse eigenvalue problem if there is no constraint of the locations of entries. A numerical isospectral flow method which is developed by hybridizing the optimization theory and steepest descent method is used to study the reconstruction. Moreover, an error estimate of the numerical iteration for ordinary differential equations on the matrix manifold is presented. After that, a numerical method for the nonnegative symmetric inverse eigenvalue problem with prescribed entries and its error estimate are considered. Finally, the approaches are verified by the numerical test results.     

An alternating direction algorithm for matrix completion with nonnegative factors

This paper introduces an algorithm for the nonnegative matrix factorization-and-completion problem, which aims to find nonnegative low-rank matrices X and Y so that the product XY approximates a nonnegative data matrix M whose elements are partially known (to a certain accuracy). This problem aggregates two existing problems: (i) nonnegative matrix factorization where all entries of M are given, and (ii) low-rank matrix completion where nonnegativity is not required. By taking the advantages of both nonnegativity and low-rankness, one can generally obtain superior results than those of just using one of the two properties. We propose to solve the non-convex constrained least-squares problem using an algorithm based on the classical alternating direction augmented Lagrangian method. Preliminary convergence properties of the algorithm and numerical simulation results are presented. Compared to a recent algorithm for nonnegative matrix factorization, the proposed algorithm produces factorizations of similar quality using only about half of the matrix entries. On tasks of recovering incomplete grayscale and hyperspectral images, the proposed algorithm yields overall better qualities than those produced by two recent matrix-completion algorithms that do not exploit nonnegativity.     

Existence results for viscous polytropic fluids with vacuum

We consider the full Navier-Stokes equations for viscous polytropic fluids with nonnegative thermal conductivity. We prove the existence of unique local strong solutions for all initial data satisfying some compatibility condition. The initial density need not be positive and may vanish in an open set. Moreover our results hold for both bounded and unbounded domains.     

The common bisymmetric nonnegative definite solutions with extreme ranks and inertias to a pair of matrix equations

We in this paper consider the bisymmetric nonnegative definite solution with extremal ranks and inertias to a system of quaternion matrix equations AX = C, XB = D. We derive the extremal ranks and inertias of the common bisymmetric nonnegative definite solution to the system. The general expressions of the bisymmetric nonnegative definite solution with extremal ranks and inertias to the system mentioned above are also presented. In addition, we give a numerical example to illustrate the results of this paper.     

CVaR-constrained stochastic programming reformulation for stochastic nonlinear complementarity problems

We reformulate a stochastic nonlinear complementarity problem as a stochastic programming problem which minimizes an expected residual defined by a restricted NCP function with nonnegative constraints and CVaR constraints which guarantee the stochastic nonlinear function being nonnegative with a high probability. By applying smoothing technique and penalty method, we propose a penalized smoothing sample average approximation algorithm to solve the CVaR-constrained stochastic programming. We show that the optimal solution of the penalized smoothing sample average approximation problem converges to the solution of the corresponding nonsmooth CVaR-constrained stochastic programming problem almost surely. Finally, we report some preliminary numerical test results.     

矩阵不等式约束下矩阵方程AX=B的双对称解

本文讨论矩阵不等式CXD≥E 约束下矩阵方程AX=B的双对称解,即给定矩阵A,B,C,D和 E, 求双对称矩阵X, 使得AX=B 和 CXD≥E, 其中CXD≥E表示矩阵CXD-E非负.本文将问题转化为矩阵不等式最小非负偏差问题,利用极分解理论给出了求其解的迭代方法,并结合相关矩阵理论说明算法的收敛性.最后给出数值算例验证算法的有效性.     

Intervals of Totally Nonnegative and Related Matrices

We consider the class of the totally nonnegative matrices, i.e., the matrices having all their minors nonnegative, and intervals of matrices with respect to the chequerboard partial ordering, which results from the usual entrywise partial ordering if we reverse the inequality sign in all components having odd index sum. For these intervals we study the following conjecture: If the left and right endpoints of an interval are nonsingular and totally nonnegative then all matrices taken from the interval are nonsingular and totally nonnegative. We present a new class of the totally nonnegative matrices for which this conjecture holds true. Similar results for classes of related matrices are also given.     

Optimal matrix-segmentation by rectangles

We study the problem of decomposing a nonnegative matrix into a nonnegative combination of 0-1-matrices whose ones form a rectangle such that the sum of the coefficients is minimal. We present for the case of two rows an easy algorithm that provides an optimal solution which is integral if the given matrix is integral. An additional integrality constraint makes the problem more difficult if the matrix has more than two rows. An algorithm that is based on the revised simplex method and uses only very few Gomory cuts yields exact integral solutions for integral matrices of reasonable size in a short time. For matrices of large dimension we propose a special greedy algorithm that provides sufficiently good results in numerical experiments.