关于丢番图方程(ax~m±1)/(ax±1)=y~n与(ax~m±1)/(ax±1)=y~n+1

本文证明了方程(1.4)没有x是一个n次完全幂的整数解(a,x,y,m,n),从而推广了乐茂华的结论:方程(1.1)没有x是一个n次完全幂的整数解(x,y,m,n),并有条件的得到了方程(1.5)的全部解.     

On the Minimal Nonnegative Solution of Nonsymmetric Algebraic Riccati Equation

We study perturbation bound and structured condition number about the minimal nonnegative solution of nonsymmetric algebraic Riccati equation, obtaining a sharp perturbation bound and an accurate condition number. By using the matrix sign function method we present a new method for finding the minimal nonnegative solution of this algebraic Riccati equation. Based on this new method, we show how to compute the desired M-matrix solution of the quadratic matrix equation X^2 - EX - F = 0 by connecting it with the nonsymmetric algebraic Riccati equation, where E is a diagonal matrix and F is an M-matrix.     

Existence and uniqueness of solutions of reaction–convection equations with non-Lipschitz nonlinearity

In this work we study single balance law _ut+∇⋅Φ(u)=f(u)$u_t+\mathrm{\nabla }\cdot \Phi (u)=f(u)$ with bounded initial value, and find that there may exist maximal and minimal solutions, if f(u)$f(u)$ is not Lipschitz continuous at u=0$u=0$. We also show that comparison principle is valid for such solutions, and the solutions may blow up or not under certain conditions. It is determined by the strength of source supply, as well as the competition between the source and flux.     

Traveling wave solutions in a diffusive system with two preys and one predator

This paper is concerned with the traveling wave solutions in a diffusive system with two preys and one predator. By constructing upper and lower solutions, the existence of nontrivial traveling wave solutions is established. The asymptotic behavior of traveling wave solutions is also confirmed by combining the asymptotic spreading with the contracting rectangles. Applying the theory of asymptotic spreading, the nonexistence of traveling wave solutions is proved.     

Nonlinear force density method for the form-finding of minimal surface membrane structures

We develop an alternative approach for the form-finding of the minimal surface membranes (including cable membranes) using discrete models and nonlinear force density method. Two directed weighted graphs with 3 and 4-sided regional cycles, corresponding to triangular and quadrilateral finite element meshes are introduced as computational models for the form-finding problem. The triangular graph model is closely related to the triangular computational models available in the literature whilst the quadrilateral graph uses a novel averaging approach for the form-finding of membrane structures within the context of nonlinear force density method. The viability of the mentioned discrete models for form-finding are studied through two solution methods including a fixed-point iteration method and the Newton–Raphson method with backtracking. We suggest a hybrid version of these methods as an effective solution strategy. Examples of the formation of certain well-known minimal surfaces are presented whilst the results obtained are compared and contrasted with analytical solutions in order to verify the accuracy and viability of the suggested methods.     

一种可压缩流动的高阶加权紧致非线性格式(WCNS)的加速收敛方法

以高超声速粘性绕流的数值模拟为例,研究LU-SGS、高斯-赛德尔点松弛、线松弛以及GMRES等隐式求解方法在空间项采用高阶精度格式WCNS离散时的收敛性,并对GMRES(generalized minimal residual)方法中的子迭代影响作了对比计算.结果表明,采用准确的解析雅克比矩阵的点、线松弛的收敛速度优于LU-SGS,以线松弛为预处理的GMRES算法具有良好的收敛特性.     

Collapsing and the convex hull property in a soap film capillarity model

Soap films hanging from a wire frame are studied in the framework of capillarity theory. Minimizers in the corresponding variational problem are known to consist of positive volume regions with boundaries of constant mean curvature/pressure, possibly connected by “collapsed” minimal surfaces. We prove here that collapsing only occurs if the mean curvature/pressure of the bulky regions is negative, and that, when this last property holds, the whole soap film lies in the convex hull of its boundary wire frame.