Dedekind和与连分数展开的几个结果

一、引言令k,h为正整数,k/h=[a_0;a_1,a_2…a_(t-1)]为简单连分数展开。若t为偶数则称为典型展开,在此时定义。     

Dedekind和的一些性质

木文用初等方法证明了Ｄｅｄｅｋｉｎｄ和的几个重要性质．主要结果是简化了Ｋｎｏｐｐ等式的证明并建立了一个类似的公式．运用同样的方法，还给出了Ｄｅｄｅｋｉｎｄ和互反律的另一个初等证明．     

层次分析正互反矩阵的Hadamard乘积分析法

定义了标准形 ～ 型 ,并指出任一正互反矩阵可唯一分解为任一种标准形和一个一致性矩阵的Hadamard乘积 .     

衡量互反矩阵一致性的一种新指标

本文针对ＡＨＰ中利用几何平均值来估计各因素权向量的方法提出了与之匹配的一种新的衡量互反矩阵一致性的指标，并且制定了相应的衡量标准，同时将其与利用最大特征值所建立的指标入标准进行了比较与讨论。     

Gould-Hsu-Carlitz反演与Rogers-Ramanujan恒等式(Ⅰ)

本文将论述通过q-级数互反关系证明经典分拆恒等式的一般方法。应用Carlitz给出的Gould-Hsu反演的q-模拟,作者将建立一个重要的和式变换定理。作为例证:结合Jacobi三重积恒等式及组合计算技巧,给出Rogers-Ra-manujan恒等式一个新的简单推证。     

正互反判断矩阵元素与优先权重新的逻辑关系

揭示了一种含参数的正互反判断矩阵元素与优先权重新的逻辑关系,并对参数的含义作了解释.得出了四个重要结论.通过算例与Saaty提出的层次分析法做了比较,说明了该方法的合理、科学性.     

Y2=X3-p3上的无穷阶点

首先用(12)上的模函数参数化椭圆曲线y²=x³+1,然后利用虚二次域Q((-p)^1/2)上的类域论和Shimura互反律构造了当p≡7(mod24)为一素数时Y²=X³-p³上的一个无穷阶点.     

指数型有理插值与q-级数互反关系

成立. 显然,在上式中取q=1,便退化为Could-Hsu反演公式.在[2—5]中曾应用后者构造插值级数,并对其中一类广义牛顿插值级数进行系统的研究.作者在此基础上应用(1.3)构造指数型插值函数. 首先引进q差分算子△_q,定义     

互反判断阵比对值对满意一致性的影响

满意一致性是互反判断阵可否作为决策依据的指标.对CR>1的互反判断阵,一般是通过比对值的修正,使之达到满意一致性要求.利用多元隐函数偏导数及行列式基本性质,可以得出各比对值变化对满意一致性的影响比例.在已知影响比例情况下,比对值如何修正、修正多少就有了理论上的依据.     

套汇问题研究

把套汇问题转化为网络规划中找负回路问题,用F loyd算法解决了套汇问题.讨论了避免套汇兑换率矩阵必须是正互反矩阵的结论,以及给出了兑换率矩阵调整为正互反矩阵的方法.     

Summation of Some Trigonometric and Schlömilch Series

In this paper we consider trigonometric series in terms of the Riemann zeta function and related functions of reciprocal powers. The obtained closed form formulas we apply to the evaluation of the Riemann zeta function and related functions of reciprocal powers. One can establish recursive relations for them and relations between any two of those functions. These closed formulas enable us also to find sums of some Schlömilch series. We give an example which shows how the convergence of a trigonometric series can be accelerated by applying Krylov's method and our formula (7).     

On the summation of Schlömilch's series

ABSTRACT

Schlömilch's series is named after the German mathematician Oscar Xavier Schlömilch, who derived it in 1857 as a Fourier series type expansion in terms of the Bessel function of the first kind. However, except for Bessel functions, here we consider an expansion in terms of Struve functions or Bessel and Struve integrals as well. The method for obtaining a sum of Schlömilch's series in terms of the Bessel or Struve functions is based on the summation of trigonometric series, which can be represented in terms of the Riemann zeta and related functions of reciprocal powers and in certain cases can be brought in the closed form, meaning that the infinite series are represented by finite sums. By using Krylov's method we obtain the convergence acceleration of the trigonometric series.     

Generalizing the Reciprocal Logarithm Numbers by Adapting the Partition Method for a Power Series Expansion

Recently, a novel method based on the coding of partitions was used to determine a power series expansion for the reciprocal of the logarithmic function, viz. z/ln (1+z). Here we explain how this method can be adapted to obtain power series expansions for other intractable functions. First, the method is adapted to evaluate the Bernoulli numbers and polynomials. As a result, new integral representations and properties are determined for the former. Then via another adaptation of the method we derive a power series expansion for the function z ^s /ln  ^s (1+z), whose polynomial coefficients A _k (s) are referred to as the generalized reciprocal logarithm numbers because they reduce to the reciprocal logarithm numbers when s=1. In addition to presenting a general formula for their evaluation, this paper presents various properties of the generalized reciprocal logarithm numbers including general formulas for specific values of s, a recursion relation and a finite sum identity. Other representations in terms of special polynomials are also derived for the A _k (s), which yield general formulas for the highest order coefficients. The paper concludes by deriving new results involving infinite series of the A _k (s) for the Riemann zeta and gamma functions and other mathematical quantities.     

分母为平方因子的二项式系数倒数级数

根据一个已知级数,利用反正弦积分与多对数的结果,用积分-裂项法给出分母为1个平方因子,平方因子与1个,2个,3个奇因子乘积的二项式系数倒数级数.利用反三角函数与反双曲函数关系给出分母为平方因子的交错二项式系数倒数级数.所给出二项式系数倒数级数的和式是函数形式.并给出分母含有平方因子的二项式系数倒数数值级数恒等式.     

Perron fractions : an algorithm for computing the Padé table

A set of rhombus rules is given for generating the coefficients of the Perron fractions whose convergents form sawtooth sequences in the Padé table of a normal series. The coefficients of the Perron fractions for the reciprocal series are also derived, without finding the reciprocal series, and hence the Padé table for the series is obtained.     

Stress moments of nearly touching spheres in low Reynolds number flow

If two spheres are nearly touching, and the flow around them is governed by the Stokes equations, the integral moments of the surface stress are singular functions of the gap width. The method used previously to calculate the singular terms in the zeroth moment (the force) and the antisymmetric first moment (the couple) is extended here to calculate the singular terms in the symmetric first moment (the stresslet) for motions perpendicular to the line of centres. It is shown that the reciprocal theorem requires unexpected relations between the newly found singularities and ones found previously. It is also shown that the singular terms can be used to improve the rate of convergence of series expressions for the stresslets. The series expressions then become valid for all separations of the spheres.     

Mean characteristics of Gibbsian point processes

This paper deals with the derivation of an exact expression of mean characteristics of planar global Gibbsian point processes having pair potential functions. The method is analogous to that of the Mayer expansion of grand partition functions, i.e., the reciprocal of the normalizing constant of Gibbsian distribution (well-known in statistical physics). The explicit infinite series expansion of a logarithm of a class of mean quantities with respect to the activity parameter z is derived and the expression of its coefficients is given. The validity of this expansion for a range of z is also shown. Examples of mean characteristics to which this expansion can be applied are given. Finally, a simple numerical example is given in order to show the usage of this expansion as a numerical approximant of mean characteristics.     

一个条件收敛级数重排项后的和

研究了以自然数倒数所构成的一个典型交错级数重排项后所得级数的收敛性及求和问题．证明了当其正项和负项均按由小到大的顺序排列后,每出现r个正项后面接t个负项的排列所得到的级数收敛,并利用幂级数求得了重排项后级数的和．     

Harmonic numbers at half integer values

Half integer values of harmonic numbers and reciprocal binomial coefficients sums are investigated in this paper. Closed-form representations and integral expressions are developed for the infinite series.     

A combinatorial proof of a result for permutation pairs

In this paper, a direct combinatorial proof is given of a result on permutation pairs originally due to Carlitz, Scoville, and Vaughan and later extended. It concerns showing that the series expansion of the reciprocal of a certain multiply exponential generating function has positive integer coefficients. The arguments may then be applied to related problems, one of which concerns the reciprocal of the exponential series for Fibonacci numbers.