Existence of solutions to an anisotropic phase‐field model

We consider an anisotropic phase‐field model for the isothermal solidification of a binary alloy due to Warren–Boettinger ( Acta. Metall. Mater. 1995; 43 (2):689). Existence of weak solutions is established under a certain convexity condition on the strongly non‐linear second‐order anisotropic operator and Lipschitz and boundedness assumptions for the non‐linearities. A maximum principle holds that guarantees the existence of a solution under physical assumptions on the non‐linearities. The qualitative properties of the solutions are illustrated by a numerical example. Copyright © 2003 John Wiley & Sons, Ltd.     

Augmented Lagrangian algorithms for linear programming

We introduce new augmented Lagrangian algorithms for linear programming which provide faster global convergence rates than the augmented algorithm of Polyak and Treti'akov. Our algorithm shares the same properties as the Polyak-Treti'akov algorithm in that it terminates in finitely many iterations and obtains both primal and dual optimal solutions. We present an implementable version of the algorithm which requires only approximate minimization at each iteration. We provide a global convergence rate for this version of the algorithm and show that the primal and dual points generated by the algorithm converge to the primal and dual optimal set, respectively.     

Global linear convergence of a path-following algorithm for some monotone variational inequality problems

We consider a primal-scaling path-following algorithm for solving a certain class of monotone variational inequality problems. Included in this class are the convex separable programs considered by Monteiro and Adler and the monotone linear complementarity problem. This algorithm can start from any interior solution and attain a global linear rate of convergence with a convergence ratio of 1 ?c/√m, wherem denotes the dimension of the problem andc is a certain constant. One can also introduce a line search strategy to accelerate the convergence of this algorithm.     

Chip-Firing Games on Directed Graphs

We consider the following (solitary) game: each node of a directed graph contains a pile of chips. A move consists of selecting a node with at least as many chips as its outdegree, and sending one chip along each outgoing edge to its neighbors. We extend to directed graphs several results on the undirected version obtained earlier by the authors, P. Shor, and G. Tardos, and we discuss some new topics such as periodicity, reachability, and probabilistic aspects.Among the new results specifically concerning digraphs, we relate the length of the shortest period of an infinite game to the length of the longest terminating game, and also to the access time of random walks on the same graph. These questions involve a study of the Laplace operator for directed graphs. We show that for many graphs, in particular for undirected graphs, the problem whether a given position of the chips can be reached from the initial position is polynomial time solvable.Finally, we show how the basic properties of the probabilistic abacus can be derived from our results.     

On affine scaling and semi-infinite programming

We consider an extension of the affine scaling algorithm for linear programming problems with free variables to problems having infinitely many constraints, and explore the relationship between this algorithm and the finite affine scaling method applied to a discretization of the problem.This material is based on research supported by Air Force Office of Scientific Research Grant AFOSR 89-0410.     

局部对称共形平坦黎曼流形中具有平行平均曲率向量的子流形

本文把[1]的结论推广到了环绕空间是局部对称共形平坦的情形,即获得了:设M~是局部对称共形平坦黎曼流形N~+p(p>1)中具有平行平均曲率向量的紧致子流形,如果则M~位于N~+p的全测地子流形N~+1中。其中S,H分别是M~的第二基本形式长度的平方和M~的平均曲率,T_C、t_c分别是N~+p的Ricci曲率的上、下确界,K是N~+p的数量曲率。     

带权值的模糊多目标线性规划

本文提出了求解一般多目标性规划问题 (MOL P)的带权值的模糊多目标线性规划方法 .证明了在权值都大于零的条件下 ,与 (MOLP)原问题对应的带权值的模糊多目标线性规划问题的最优解为模糊有效解 ,从而为原问题的有效解 ,并作了实例验证 .     

AN ALGORITHM FOR JORDAN CANONICAL FORM OF A QUATERNION MATRIX

In this paper, we first introduce a concept of companion vector, and study the Jordan canonical forms of quaternion matrices by using the methods of complex rep resentation and companion vector, not only give out a practical algorithm for Jordan canonical form J of a quaternion matrix A, but also provide a practical algorithm for corresponding nonsingular matrix P with P-1AP = J.     

An approximation problem of the finite rank operator in Banach spaces

By the method of geometry of Banach spaces, we have proven that a bounded linear operator in Banach space is a compact linear one iff it can be uniformly approximated by a sequence of the finite rank bounded homogeneous operators, which reveals the essence of the counter example given by Enflo.     

Curvature Computations of 2-Manifolds in $\mathbb{R}^k$

In this paper, we provide simple and explicit formulas for computing Riemannian curvatures, mean curvature vectors, principal curvatures and principal directions for a 2-dimensional Riemannian manifold embedded in IR^k with k ≥ 3.