Attractors for a Three-Dimensional Thermo-Mechanical Model of Shape Memory Alloys

Abstract In this note, we consider a Frémond model of shape memory alloys. Let us imagine a piece of a shape memory alloy which is fixed on one part of its boundary, and assume that forcing terms, e.g., heat sources and external stress on the remaining part of its boundary, converge to some time-independent functions, in appropriate senses, as time goes to infinity. Under the above assumption, we shall discuss the asymptotic stability for the dynamical system from the viewpoint of the global attractor. More precisely, we generalize the paper [12] dealing with the one-dimensional case. First, we show the existence of the global attractor for the limiting autonomous dynamical system; then we characterize the asymptotic stability for the non-autonomous case by the limiting global attractor. * Project supported by the MIUR-COFIN 2004 research program on “Mathematical Modelling and Analysis of Free Boundary Problems”.     

The Neumann problem for some degenerate elliptic equations

In the paper we study the equation L _u = f, where L is a degenerate elliptic operator, with Neumann boundary condition in a bounded open set μ. We prove existence and uniqueness of solutions in the space H(μ) for the Neumann problem.     

EXISTENCE OF PERIODIC SOLUTIONS OF SYSTEM OF DIFFERENCE EQUATIONS

This paper studies the nonautonomous nonlinear system of difference equationsΔx(n)=A(n)x(n)+f(n,x(n)),n∈Z,(*) where x(n)∈R~N,A(n)=(a_(ij)(n))N×N is an N×N matrix,with a-(ij)∈C(R,R) for i,j= 1,2,3,...,N,and f=(f_1,f_2,...,f_N)~T∈C(R×R~N,R~N),satisfying A(t+ω)=A(t),f(t+ω,z)=f(t,z) for any t∈R,(t,z)∈R×R~N andωis a positive integer.Sufficient conditions for the existence ofω-periodic solutions to equations (*) are obtained.     

共振下n维Lienard型方程的2π周期解

本文在跨共振点条件下证明n维Lienard型方程存在2π周期解.     

A possible counterexample to well posedness of entropy solutions and to Godunov scheme convergence

A particular case of initial data for the two-dimensional Euler equations is studied numerically. The results show that the Godunov method does not always converge to the physical solution, at least not on feasible grids. Moreover, they suggest that entropy solutions (in the weak entropy inequality sense) are not well posed.

     

On an Application of Newton's Method to Nonlinear Operators with w-Conditioned Second Derivative

We present a new approach to study the convergence of Newton's method in Banach spaces, which relax the conditions appearing in the usual studies. The approach is based on the bound required for the second derivative of the operator involved. An application to a nonlinear integral equation is presented.     

On a class of new and practical performance indexes for approximation of fold bifurcations of nonlinear power flow equations

Efficient measurement of the performance index (the distance of a loading parameter from the voltage collapse point) is one of the key problems in power system operations and planning and such an index indicates the severity of a power system with regard to voltage collapse. There exist many interesting methods and ideas to compute this index. However, some successful methods are not yet mathematically justified while other mathematically sound methods are often proposed directly based on the bifurcation theory and they require the initial stationary state to be too close to the unknown turning point to make the underlying methods practical.This paper first gives a survey of several popular methods for estimating the fold bifurcation point including the continuation methods, bifurcation methods and the test function methods (Seydel's direct solution methods, the tangent vector methods and the reduced Jacobian method) and discuss their relative advantages and problems. Test functions are usually based on scaling of the determinant of the Jacobian matrix and it is generally not clear how to determine the behaviour of such functions. As the underlying nonlinear equations are of a particular type, this allows us to do a new analysis of the determinants of the Jacobian and its submatrices in this paper. Following the analysis, we demonstrate how to construct a class of test functions with a predictable analytical behaviour so that a suitable index can be produced. Finally, examples of two test functions from this class are proposed. For several standard IEEE test systems, promising numerical results have been achieved.     

Numerical solutions of the shallow water equations with discontinuous bed topography

A simple scheme is developed for treatment of vertical bed topography in shallow water flows. The effect of the vertical step on flows is modelled with the shallow water equations including local energy loss terms. The bed elevation is denoted with z_b^‐ for the left and z_b⁺ for the right values at each grid point, hence exactly representing a discontinuity in the bed topography. The surface gradient method (SGM) is generalized to reconstruct water depths at cell interfaces involving a vertical step so that the fluxes at the cell interfaces can accurately be calculated with a Riemann solver. The scheme is verified by predicting a surge crossing a step, a tidal flow over a step and dam‐break flows on wet/dry beds. The results have shown good agreements compared with analytical solutions and available experimental data. The scheme is efficient, robust, and may be used for practical flow calculations. Copyright © 2002 John Wiley & Sons, Ltd.     

用变形全息透镜实现变形分数傅里叶变换

利用一发散点光源和一束倾斜平行光制作出在两个垂直方向上具有不同焦距的变形全息透镜,且用此全息元件实现了变形分数傅里叶变换。变形全息透镜两个方向上的焦距随全息记录条件和再现条件而变化,分析了用这种全息透镜实现变形分数傅里叶变换的条件。实验表明,用该元件能够方便地实现任意级次的变形分数傅里叶变换。     

图有分数因子的度条件

本文研究了图有分数因子的度条件,得到了下面的结果：令k(?)1是一个整数,G是一个连通的n阶图,n(?)4k-3且最小度δ(G)(?)k,若对于每一对不相邻的顶点u,v∈V(G)都有max{d_G(u),d_G(v)}(?)n/2,则G有分数k-因子．并指出该结果在一定意义上是最好可能的。