On the twisted q-Euler numbers and polynomials associated with basic q-l-functions

One of the purposes of this paper is to construct the twisted q-Euler numbers by using p-adic invariant integral on Zp in the fermionic sense. Moreover, we consider the twisted Euler q-zeta functions and q-l-functions which interpolate the twisted q-Euler numbers and polynomials at a negative integer.     

An application of rosenthal's moment inequality to the strong law of large numbers

We apply a moment inequality of H.P. Rosenthal to get various generalizations of a theorem of Brunk-Chung-Prohoroff. In particular, we show that the strong law hols provided $\text{∑}\text{n=1}\text{∞}\text{n}{}^{\mathrm{?2p}}\text{E(X}{}_{\mathrm{n}}\text{}{}^{\mathrm{2p}}\text{) < ∞}$ and $\text{∑}\text{n=1}\text{∞}\text{n}{}^{\mathrm{?2?p}}\text{(E(}\text{X}\text{n}\text{2}\text{))}{}^{\mathrm{p}}\text{< ∞}$ for one p ? 1.     

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．     

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 .     

Optimally conditioned scaled ABS algorithms for linear systems

Using a strict bound of Spedicato to the condition number of bordered positive-definite matrices, we show that the scaling parameter in the ABS class for linear systems can always be chosen so that the bound of a certain update matrix is globally minimized. Moreover, if the scaling parameter is so chosen at every iteration, then the condition number itself is globally minimized. The resulting class of optimally conditioned algorithms contains as a special case the class of optimally stable algorithms in the sense of Broyden.This work was done in the framework of research supported by MPI, Rome, Italy, 60% Program.     

The Distribution of Condition Numbers of Rational Data of Bounded Bit Length

We prove that rational data of bounded input length are uniformly distributed (in the classical sense of H. Weyl, in [42]) with respect to the probability distribution of condition numbers of numerical analysis. We deal both with condition numbers of linear algebra and with condition numbers for systems of multivariate polynomial equations. For instance, we prove that for a randomly chosen n\times n rational matrix M of bit length O(n ⁴ log n) + log w , the condition number k(M) satisfies k(M) ≤ w n ^5/2 with probability at least 1-2w ^-1 . Similar estimates are established for the condition number μ_ norm of M. Shub and S. Smale when applied to systems of multivariate homogeneous polynomial equations of bounded input length. Finally, we apply these techniques to estimate the probability distribution of the precision (number of bits of the denominator) required to write approximate zeros of systems of multivariate polynomial equations of bounded input length. March 7, 2001. Final version received: June 7, 2001.