Chaotic Numbers of Complete Bipartite Graphs and Tripartite Graphs

In this paper, we study the chaotic numbers of complete bipartite graphs and complete tripartite graphs. For the complete bipartite graphs, we find closed-form formulas of the chaotic numbers and characterize all chaotic mappings. For the complete tripartite graphs, we develop an algorithm running in O(n ⁴ ₃) time to find the chaotic numbers, with n ₃ the number of vertices in the largest partite set.Research supported by NSC 90-2115-M-036-003.The author thanks the authors of Ref. 6, since his work was motivated by their work. Also, the author thanks the referees for helpful comments which made the paper more readable.     

A multiplication theorem for the Lerch zeta function and explicit representations of the Bernoulli and Euler polynomials

A multiplication theorem for the Lerch zeta function ?(s,a,ξ) is obtained, from which, when evaluating at s=−n for integers n?0, explicit representations for the Bernoulli and Euler polynomials are derived in terms of two arrays of polynomials related to the classical Stirling and Eulerian numbers. As consequences, explicit formulas for some special values of the Bernoulli and Euler polynomials are given.     

The Remak height for units

We investigate the values of the Remak height, which is a weighted product of the conjugates of an algebraic number. We prove that the ratio of logarithms of the Remak height and of the Mahler measure for units αof degree d is everywhere dense in the maximal interval [d/2(d-1),1] allowed for this ratio. To do this, a “large” set of totally positive Pisot units is constructed. We also give a lower bound on the Remak height for non-cyclotomic algebraic numbers in terms of their degrees. In passing, we prove some results about some algebraic numbers which are a product of two conjugates of a reciprocal algebraic number. This revised version was published online in June 2006 with corrections to the Cover Date.     

递归序列与Bernoulli数的一个恒等式

先给出射影函数的迭代与2阶线性递归序列的关系，进而得到此递归序列与Bernoulli数的一个恒等式。     

罗尔中值定理的一些新证法

给出了罗尔微分中值定理的三种新的证明方法,其中第二种很简便的方法仅依赖于大家熟知的Heine-Borel有限覆盖定理.由此可见罗尔微分中值定理可以是实数的完备性的直接推论.     

混合序列部分和的强大数定律

设{Xn，n≥1）是ρ^-混合序列．利用随机变量的截尾方法和ρ^-混合序列的三级数定理这一工具研究了ρ^-混合序列的性质。得到了矩条件下ρ^-混合序列的—类强极限定理和强大数定律。并给出了一些简单应用。推广了若干经典的强大数定律．     

Nonconforming h-p spectral element methods for elliptic problems

In this paper we show that we can use a modified version of the h-p spectral element method proposed in [6,7,13,14] to solve elliptic problems with general boundary conditions to exponential accuracy on polygonal domains using nonconforming spectral element functions. A geometrical mesh is used in a neighbourhood of the corners. With this mesh we seek a solution which minimizes the sum of a weighted squared norm of the residuals in the partial differential equation and the squared norm of the residuals in the boundary conditions in fractional Sobolev spaces and enforce continuity by adding a term which measures the jump in the function and its derivatives at inter-element boundaries, in fractional Sobolev norms, to the functional being minimized. In the neighbourhood of the corners, modified polar coordinates are used and a global coordinate system elsewhere. A stability estimate is derived for the functional which is minimized based on the regularity estimate in [2]. We examine how to parallelize the method and show that the set of common boundary values consists of the values of the function at the corners of the polygonal domain. The method is faster than that proposed in [6,7,14] and the h-p finite element method and stronger error estimates are obtained.     

Fuzzy Bayesian system reliability assessment based on exponential distribution

The Bayesian system reliability assessment under fuzzy environments is proposed in this paper. In order to apply the Bayesian approach, the fuzzy parameters are assumed as fuzzy random variables with fuzzy prior distributions. The (conventional) Bayesian estimation method will be used to create the fuzzy Bayes point estimator of system reliability based on Exponential distribution by invoking the well-known theorem called “Resolution Identity” in fuzzy sets theory. On the other hand, we also provide the computational procedures to evaluate the membership degree of any given Bayes point estimate of system reliability. In order to achieve this purpose, we transform the original problem into a nonlinear programming problem. This nonlinear programming problem is then divided into four subproblems for the purpose of simplifying computation. Finally, the subproblems can be solved by using any commercial optimizers, e.g., GAMS or LINGO (LINDO).     

随机狄里克莱级数在收敛右半平面上的增长性和值分布

本文利用随机变量序列的强大数定律 ,研究了随机变量序列 {Xn}在独立 (可不同分布 )情形下的性质 ,并得到当随机狄里克莱级数 ∑∞n =1anXne-λns 满足(ⅰ )limn ∞nλn =D <∞ ;(ⅱ ) limn ∞ln｜an｜λn =0 等条件时的增长性以及值分布 .     

Transitive fuzzy orderings of fuzzy numbers

In this paper we introduce a generalization of the Baas-Kwakernaak index by replacing the min operation in their definition by a t-norm. Some properties of the thus defined induced fuzzy ordering are established. In particular, it is shown that restrictions of the induced fuzzy ordering on some special classes of fuzzy numbers are reflexive fuzzy orders.