一类极限循环连分式的加速收敛因子

本文获得了一类极限循环连分式的加速收敛因子,证明了它们具有良好的加速收敛性质.     

递推数列极限的初等求法和收敛渐近性

借助实例介绍一些非线性递推数列，特别是分式线性递推数列极限的初等求法。就一般分式线性递推数列，明确其收敛渐近性，并通过相关推论展示其应用。     

连分式 数列与极限

本文给出了连分式展开式分子、分母的递推关系,推导了递推数列的产生函数.由产生函数的渐近展开式,得到了连分式的极限值.     

几类迭代数列的收敛性

从一个简单的线性迭代数列出发,引出倒数迭代数列和分式迭代数列的极限结论.这三类迭代数列是常见的,得到的结论有较强的实用性.     

一类收敛数列极限公式的证明及应用

本文运用Ｓｔｏｌｚ定理证明了一类收敛数列的极限公式，并得到文［１］、［２］的结果．     

一道数列极限的多种解法

利用数学分析中关于数列极限的定义、收敛数列的性质及数列极限存在的条件,介绍一道数列极限问题的多种解法.     

关于Wallis数列的连分式估计（英文）

给出了Wallis数列的连分式表达式.基于获得的结果,建立了Wallis数列的一个双边不等式.     

一个新的连分式算法及其收敛性

本文利用连分式插值,得到了一个新的一维搜索方法——连分式算法.用此算法,每迭代一次,只需计算三个点的函数值;在计算连分式插值式的每个系数时,只需一次除法.因此,数值稳定性较好.本文还证明了此算法的收敛性,收敛速度较快,收敛阶近似1.8393.按效能指标E=P~(1/μ)评价,此算法是一个较好的局部一维搜索方法.如果用此法于不精确的一维搜索,因只需计算三个点的函数值,故它是一个较好的、不精确的一维搜索方法,同时也是解超越方程的一个新算法.数值例子表明,它确实有效.     

Levin变换应用于欧拉常数数列及连分式数列改进算法的加速收敛

利用Lu等人通过连分式修正更快收敛的欧拉常数数列及其相关余项式,进一步采用Levin变换进行二次加速,特别是在克服舍入误差的情况下,就能更有效地计算出欧拉常数的高精度数值结果.     

用特征值刻画分式线性迭代数列的渐近性态

利用分式线性迭代数列系数矩阵的特征值,刻画这类数列的敛散性及收敛速度.     

Estimation of the measure of the image of the ball

Under study is the class of ring Q-homeomorphisms with respect to the p-module. We establish a criterion for a function to belong to the class and solve a problem that stems from M. A. Lavrentiev [1] on the estimation of the measure of the image of the ball under these mappings. We also address the asymptotic behavior of these mappings at a point.     

On the distributions of the ratios of the extreme roots to the trace of the Wishart matrix

In this paper, the authors cosider the derivation of the exact distributions of the ratios of the extreme roots to the trace of the Wishart matrix. Also, exact percentage points of these distributions are given and their applications are discussed.     

On the rational approximation of the sum of the reciprocals of the Fermat numbers

Let $\mathcal{G}(z):=\sum_{n\geqslant0} z^{2^{n}}(1-z^{2^{n}})^{-1}$ denote the generating function of the ruler function, and $\mathcal {F}(z):=\sum_{n\geqslant} z^{2^{n}}(1+z^{2^{n}})^{-1}$ ; note that the special value $\mathcal{F}(1/2)$ is the sum of the reciprocals of the Fermat numbers $F_{n}:=2^{2^{n}}+1$ . The functions $\mathcal{F}(z)$ and $\mathcal{G}(z)$ as well as their special values have been studied by Mahler, Golomb, Schwarz, and Duverney; it is known that the numbers $\mathcal {F}(\alpha)$ and $\mathcal{G}(\alpha)$ are transcendental for all algebraic numbers α which satisfy 0<α<1. For a sequence u, denote the Hankel matrix $H_{n}^{p}(\mathbf {u}):=(u({p+i+j-2}))_{1\leqslant i,j\leqslant n}$ . Let α be a real number. The irrationality exponent μ(α) is defined as the supremum of the set of real numbers μ such that the inequality |α?p/q|<q ^?μ has infinitely many solutions (p,q)∈?×?. In this paper, we first prove that the determinants of $H_{n}^{1}(\mathbf {g})$ and $H_{n}^{1}(\mathbf{f})$ are nonzero for every n?1. We then use this result to prove that for b?2 the irrationality exponents $\mu(\mathcal{F}(1/b))$ and $\mu(\mathcal{G}(1/b))$ are equal to 2; in particular, the irrationality exponent of the sum of the reciprocals of the Fermat numbers is 2.     

On the approximation of positive operators and the behaviour of the spectra of the approximants

LetT be a positive linear operator on the Banach latticeE and let (S _n ) be a sequence of bounded linear operators onE which converge strongly toT. Our main results are concerned with the question under which additional assumptions onS _n andT the peripheral spectra (S _n ) ofS _n converge to the peripheral spectrum (T) ofT. We are able to treat even the more general case of discretely convergent sequences of operators.     

Refinement of the asymptotics of the power of the quantile test

One investigates the asymptotic properties of the quantile test, similar to the properties of the Pearson's chi-square test of fit.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 153, pp. 5–15, 1986.The author is grateful to D. M. Chibisov for useful remarks.