On the theory of divergence-measure fields and its applications

Divergence-measure fields are extended vector fields, including vector fields inL ^p and vector-valued Radon measures, whose divergences are Radon measures. Such fields arise naturally in the study of entropy solutions of nonlinear conservation laws and other areas. In this paper, a theory of divergence-measure fields is presented and analyzed, in which normal traces, a generalized Gauss-Green theorem, and product rules, among others, are established. Some applications of this theory to several nonlinear problems in conservation laws and related areas are discussed. In particular, with the aid of this theory, we prove the stability of Riemann solutions, which may contain rarefaction waves, contact discontinuities, and/or vacuum states, in the class of entropy solutions of the Euler equations for gas dynamics.Dedicated to Constantine Dafermos on his 60^th birthday     

Curvature Estimates for Stable Minimal Hypersurfaces in R4 and R5

We obtain curvature estimates for certain stable minimalhypersurfaces in R ⁴ and R ⁵without using volume bounds. It follows that if M is acomplete stable minimal hypersurface in R ⁴ orR ⁵, then M is a hyperplane whenM intersects each extrinsic ball in, at most,N-components.     

Notes on sums of algebras, subdirect products and subdirectly irreducible algebras

The paper shows that certain sums of algebras admit a representation as subdirect products. As a corollary, one obtains a characterization of subdirectly irreducible algebras. Examples of modes show that the characterization gives a good basis for a detailed investigation of the subdirectly irreducible algebras. Received September 10, 1999; accepted in final form January 3, 2001.     

Residue Theorem and Theta Function Identities

In this paper we will use the residue theorem of elliptic functions to prove some theta function identities of Ramanujan. We also derive some new identities by this method.     

On the Problem of Describing Sequences of Best Trigonometric Rational Approximations

For a strictly decreasing sequence a_{n n=0} of nonnegative real numbers converging to zero, we construct a continuous 2-periodic function f such that R^T _n(f) = a_n, n=0,1,2,..., where R^T _n(f) are best approximations of the function f in uniform norm by trigonometric rational functions of degree at most n.     

On the Weierstrass Preparation Theorem

An analytic function of several variables is considered. It is assumed that the function vanishes at some point. According to the Weierstrass preparation theorem, in the neighborhood of this point the function can be represented as a product of a nonvanishing analytic function and a polynomial in one of the variables. The coefficients of the polynomial are analytic functions of the remaining variables. In this paper we construct a method for finding the nonvanishing function and the coefficients of the polynomial in the form of Taylor series whose coefficients are found from an explicit recursive procedure using the derivatives of the initial function. As an application, an explicit formula describing a bifurcation diagram locally up to second-order terms is derived for the case of a double root.     

A Quasi Fixed-Point Theorem for a Product of u.s.c. or o.l.s. Correspondences with an Economic Application

In the first place, we present a quasi fixed-point theorem for a correspondence defined on some infinite-dimensional locally convex topological vector space such that some variables have open lower sections and the other ones are upper semicontinuous.In the second place, we propose a direct application of the quasi fixed-point result in an economic model. More precisely, we prove a nonemptiness result of the core of an exchange economy with asymmetric information, a continuum of states and a finite number of commodities.     

基于分解测度的模糊积分及收敛定理(英文)

为了解决一些收敛定理 ,我们给出基于半环 ( [0 ,1 ], , )的伪可加分解测度的积分这种模糊积分被深入研究 .在给出这种积分的性质的基础上 ,我们得到一些收敛定理 ,它们是经典收敛定理的扩张 ,同时我们得到关于这种模糊测度的 Egorof定理     

The Boyer–Moore–Horspool heuristic with Markovian input

The Boyer–Moore–Horspool string‐matching heuristic is an algorithm for locating occurrences of a fixed pattern in a random text. Under the assumption that the text is an independently and identically distributed sequence of characters, the probabilistic behavior of this algorithm was investigated by Mahmoud, Smythe, and Régnier [Random Struct Alg 10 (1997), 169–186]. Here, we obtain similar results under the assumption that the text is generated by an irreducible Markov chain. A natural Markov renewal process structure is exploited to obtain the asymptotic behavior of the number of comparisons. Under suitable normalization, it is shown that a central limit theorem holds for the number of comparisons. The analysis is completely probabilistic and does not use the shift generating function. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 153–163, 2001     

Boundedness of Solutions in Asymmetric Oscillations via the Twist Theorem

In this paper we consider the boundedness of all the solutions for the equation x″ + ax ⁺−bx ⁻ = f(t) is a smooth 2π-periodic function, a and b are positive constants (a≠b). Received November 27, 1998, Accepted October 15, 1999