广义调和级数及其推广应用研究

创建了广义调和级数和广义欧拉常数函数的概念,得到了广义调和级数和广义欧拉常数一系列十分重要的性质,得到了完全和公式,近似夹公式,讨论了广义欧拉常数与黎曼和的关系.利用复数的知识,找到了一种研究级数∑n=1∞b0+b1n+…+brnr/a0+a1n+…+aknk敛散性和求和的全新思路和方法—广义调和级数法.     

ζ（3）的级数公式

给出ζ（3）的计算公式ζ（3）=π^2/7[1-4∑n=1^∞ζ（2n）/（2n＋1）（2n＋2）2^2n].     

Precise Rates in the Law of Iterated Logarithm for the Moment of I.I.D. Random Variables

Let{X,Xn;n≥1} be a sequence of i,i.d, random variables, E X = 0, E X^2 = σ^2 〈 ∞.Set Sn=X1＋X2＋…＋Xn,Mn=max k≤n│Sk│,n≥1.Let an=O（1/loglogn）.In this paper,we prove that,for b〉-1,lim ε→0 →^2（b＋1）∑n=1^∞ （loglogn）^b/nlogn n^1/2 E{Mn-σ（ε＋an）√2nloglogn}＋σ2^-b/（b＋1）（2b＋3）E│N│^2b＋3∑k=0^∞ （-1）k/（2k＋1）^2b＋3 holds if and only if EX=0 and EX^2=σ^2〈∞.     

论新常数μ、θ和新公式π=1/2e^θ^＊

引入一个新常数μ,它是调和级数与1nn,欧拉常数γ、1/2n之差的尾项的级数和．由于新常数μ和欧拉常数γ都同调和级数与Inn之差有关,因此可再定义一个新常数θ=1＋γ＋2μ,它和圆周率π、自然对数的底e之间可组成公式π=1/2e^＊．     

负相依样本平滑移动过程的完全收敛性

设{Yi;－∞＜i＜∞}为一负相伴的同分布随机变量序列，{ai；－∞＜i＜∞}为绝对可和的实数序列。本文在适当的条件下。证明了平滑移动过程{∑k=1^n∑i=-∞^∞ai kYi/n^1/t;n≥1}的完全收敛性。     

Precise rates in the law of the iterated logarithm under dependence assumptions

Let {X, X1, X2,...} be a strictly stationaryφ-mixing sequence which satisfies EX = 0,EX^2（log2{X}）^2〈∞and φ（n）=O（1/log n）^Tfor some T〉2.Let Sn=∑k=1^nXk and an=O（√n/（log2n）^γ for some γ〉1/2.We prove that limε→√2√ε^2-2∑n=3^∞1/nP（｜Sn｜≥ε√ESn^2log2n＋an）=√2.The results of Gut and Spataru （2000） are special cases of ours.     

负相伴样本平滑移动过程的完全收敛性

设{Y，Yi，-∞＜i＜∞}为一负相伴同分布随机变量序列，{ai，-∞＜i＜∞}绝对可和的实数序列，本文在适当的条件下，证明了平滑移动过程{∑k=1^n∑i=-∞^∞ai k Yi/n^1/t,n≥1}的完全收敛性．所得的结果改进了[1]中的定理1．     

The precise asymptotics of the complete convergence for moving average processes of <Emphasis Type="Italic">m</Emphasis>-dependent <Emphasis Type="Italic">B</Emphasis>-valued elements

Let {εt;t ∈ Z} be a sequence of m-dependent B-valued random elements with mean zeros and finite second moment. {a3;j ∈ Z} is a sequence of real numbers satisfying ∑j=-∞^∞｜aj｜ 〈 ∞. Define a moving average process Xt = ∑j=-∞^∞aj＋tEj,t ≥ 1, and Sn = ∑t=1^n Xt,n ≥ 1. In this article, by using the weak convergence theorem of { Sn/√ n _〉 1}, we study the precise asymptotics of the complete convergence for the sequence {Xt; t ∈ N}.     

整体退火遗传算法的几乎处处强收敛性

本文通过鞍论来分析整体退火遗传算法收敛的特性，得出整体退火遗传算法几乎处处强收敛的条件∑n=1^∞e^-δ,Tn＜ ∞，并且给出其收敛率0（1/N N0 （2-Cn^-^N0-mn^-^N0)e^(△-d)/Tm)。     

On the Order of Magnitude of the Divisor Function

Let D be an increasing sequence of positive integers, and consider the divisor functions： d（n, D） =∑d｜n,d∈D,d≤√n1, d2（n,D）=∑[d,δ]｜n,d,δ∈D,[d,δ]≤√n1, where [d,δ]=1.c.m.（d,δ）. A probabilistic argument is introduced to evaluate the series ∑n=1^∞and（n,D） and ∑n=1^∞and2（n,D）.     

A general result on precise asymptotics for linear processes of positively associated sequences

Let {εt; t ∈ Z^＋} be a strictly stationary sequence of associated random variables with mean zeros, let 0〈Eε1^2〈∞ and σ^2=Eε1^2＋1∑j=2^∞ Eε1εj with 0〈σ^2〈∞.{aj;j∈Z^＋} is a sequence of real numbers satisfying ∑j=0^∞｜aj｜〈∞.Define a linear process Xt=∑j=0^∞ ajεt-j,t≥1,and Sn=∑t=1^n Xt,n≥1.Assume that E｜ε1｜^2＋δ′〈 for some δ′〉0 and μ（n）=O（n^-ρ） for some ρ〉0.This paper achieves a general law of precise asymptotics for {Sn}.     

Complete moment and integral convergence for sums of negatively associated random variables

For a sequence of identically distributed negatively associated random variables {Xn; n ≥ 1} with partial sums Sn = ∑i=1^n Xi, n ≥ 1, refinements are presented of the classical Baum-Katz and Lai complete convergence theorems. More specifically, necessary and sufficient moment conditions are provided for complete moment convergence of the form ∑n≥n0 n^r-2-1/pq anE（max1≤k≤n｜Sk｜^1/q-∈bn^1/qp）^＋〈∞to hold where r 〉 1, q 〉 0 and either n0 = 1,0 〈 p 〈 2, an = 1,bn = n or n0 = 3,p = 2, an = 1 （log n） ^1/2q, bn=n log n. These results extend results of Chow and of Li and Spataru from the indepen- dent and identically distributed case to the identically distributed negatively associated setting. The complete moment convergence is also shown to be equivalent to a form of complete integral convergence.     

ON THE COMMUTATIVITY OF MULTIPLE SERIES AND BASIC SEQUENCE

Let X be a Banach space and {e_j}_(j=1)~∞ be a sequence in X. The author showsthat {e_j}_(j=1)~∞ is a basic sequence if and only if ∑_(n=1)~∞, r_nα_(nj) converges for every j≥1 and∑_(n=1)~∞ r_n ∑_(j=1)~∞, α_(nj)e_j=∑_(j=1)~∞,(∑_(n=1)~∞ r_nα_(nj))e_j holds for every choice of scalar variables{α_(nj)} such that ∑_(j=1)~∞ α_(nj)e_j converges for each n≥1 and any choice of scalar variables{r_n} such that ∑_(n=1)~∞ ∑_(j=1)~∞, r_nα_(nj)e_j converges. Moreover, some applications about theresult are given.     

三个数列的收敛速度

利用不等式形式对e-〔1＋1/n〕^n,〔1＋1/n〕^（n＋1）-e,e-n∑k=0 1/k!进行了估计,给出了数列〔〔1＋1/n〕^n〕,〔〔1＋1/n〕^（n＋1）〕,{n∑k=0 1/k!}收敛于e的速度.     

幂级数∑n=0^∞（2n）!／（n!）^2（1／2）^2的收敛性讨论

利用stirling公式和阿拉伯判别法可证级数∑n=0^∞(2n)!/(n!)^2(1/2)^2发散，但其相应的交错级数条件收敛．     

Precise rates in the law of the logarithm for the moment convergence in Hilbert spaces

Let （X, Xn; n ≥1） be a sequence of i.i.d, random variables taking values in a real separable Hilbert space （H, ｜｜·｜｜） with covariance operator ∑. Set Sn = X1 ＋ X2 ＋ ... ＋ Xn, n≥ 1. We prove that, for b 〉 -1,
lim ε→0 ε^2（b＋1） ∞ ∑n=1 （logn）^b/n^3/2 E{｜｜Sn｜｜-σε√nlogn}=σ^-2（b＋1）/（2b＋3）（b＋1） B｜｜Y｜^2b＋3
holds if EX=0,and E｜｜X｜｜^2（log｜｜x｜｜）^3bv（b＋4）〈∞ where Y is a Gaussian random variable taking value in a real separable Hilbert space with mean zero and covariance operator ∑, and σ^2 denotes the largest eigenvalue of ∑.     

<Emphasis Type="Italic">q</Emphasis>-Besselian Frames in Banach Spaces

In this paper, we introduce the concepts of q-Besselian frame and （p, σ）-near Riesz basis in a Banach space, where a is a finite subset of positive integers and 1/p＋1/q = 1 with p 〉 1, q 〉 1, and determine the relations among q-frame, p-Riesz basis, q-Besselian frame and （p, σ）-near Riesz basis in a Banach space. We also give some sufficient and necessary conditions on a q-Besselian frame for a Banach space. In particular, we prove reconstruction formulas for Banach spaces X and X^＊ that if {xn}n=1^∞ C X is a q-Besselian frame for X, then there exists a p-Besselian frame {y＆＊}n=1^∞ belong to X^＊ for X^＊ such that x = ∑n=1^∞ yn^＊（x）xn for all x ∈ X, and x^＊ =∑n=1^∞ x^＊（xn）yn^＊ for all x^＊ ∈ X^＊. Lastly, we consider the stability of a q-Besselian frame for the Banach space X under perturbation. Some results of J. R. Holub, P. G. Casazza, O. Christensen and others in Hilbert spaces are extended to Banach spaces.     

The Law of Iterated Logarithm of Rescaled Range Statistics for AR（1） Model

Let {Xn,n ≥ 0} be an AR（1） process. Let Q（n） be the rescaled range statistic, or the R/S statistic for {Xn} which is given by （max1≤k≤n（∑j=1^k（Xj - ^-Xn）） - min 1≤k≤n（∑j=1^k（ Xj - ^Xn ））） /（n ^-1∑j=1^n（Xj -^-Xn）^2）^1/2 where ^-Xn = n^-1 ∑j=1^nXj. In this paper we show a law of iterated logarithm for rescaled range statistics Q（n） for AR（1） model.     

关于i=n∑n ai＋bi/1〈c型数列放缩的一点想法

i=n∑n ai＋bi/1〈c（c为常数）型的数列不等式的推导，可以使用数学归纳法来证明，但在实际解决中，有时难以实现从n=k到n-k＋1的过渡；也可以考虑将原命题强化为i=n∑n ai＋bi/1〈c-f（n）/1（其中f（n）与n有关，且f（n）〉O）的形式。     

A note on the Hoffman-Wielandt theorem for generalized eigenvalue problems

Let {A, B} and {C, D} be diagonalizable pairs of order n, i.e., there exist invertible matrices P, Q and X, Ysuchthat A = P∧Q, B = PΩQ, C =XГY, D= X△Y, where
∧ = diag（α1, α2, …, αn）, Ω= diag（βl, β2, …βn）,
Г=diag（γ1,γ2,…,γn）, △=diag（δl,δ2,…,δn）.
Let ρ（（α,β）, （γ,δ））=｜αδ-βγ｜/√｜α｜^2＋｜β｜^2√｜γ｜^2＋｜δ｜^2.In this paper, it will be proved that there is a permutation τ of {1,2,... ,n} such that
n∑i=1[ρ（（αi,βi）,（γτ（i）,δτ（i）））]^2≤n[1-1/κ^2（Y）κ^2（Q）（1-d2F（Z,W）/n）],
where κ（Y） = ｜｜Y｜｜2｜｜Y^-1｜｜2,Z= （A,B）,W= （C, D） and dF（Z,W） = 1/√2｜｜Pz＊ -Pw＊｜｜F.