几类数列通项的矩阵求法

本利用矩阵给出了几类数列的通项公式的求法，把数列通项公式的求法转化为矩阵幂的计算，思路简单、计算简便，并能判别其敛散性。     

组合计数的群论与计算机方法

本文综述组合数学和图论中解决计数问题的群论与计算机方法及其最新发展。传统的计数方法得到有限的计数公式或递推公式等，然而许多复杂的问题很难得到有限的表达式，即使能得到，公式也往往非常复杂。由于计算机技术的发展不仅使复杂的计数公式有了实际意义，而且可以设计恰当的计算方法进行数值计算，使计数问题有更为广阔的发展领域。另一方面，为了计算不同构的图或组合结构，最有效的方法是群论方法，因此把群论方法与计算机方     

三角形面积的华丽转身

计算一个三角形的面积，一般可用三角形的面积公式来完成．但是，由于问题的设定所限，有时并非面积公式能轻易所为．此时，就有必要跳出公式的束缚，让三角形面积来一个华丽转身，通过适当地转换来计算求取．本文就几个常见的转换途径作简要介绍，供同学们参考．     

对角双线性时间序列的近似三阶矩结构和双谱密度

三阶矩结构和双谱是分析和研究非线性时间序列的重要方法。但对于双线性时间序列来说，计算两者的精确值是困难的。因此，寻找比较好的近似值就非常有必要。本文针对一般的对角双线性时间序列，给出了计算近似三阶矩和双谱密度的公式。仿真计算表明，这些公式是有效的。     

空盘公式前乘法

为将简捷乘算法统一于十字相乘公式之中，进而运用多种方法，简化运算过程，提高计算速度，向大家介绍新法——空盘公式前乘法，抛砖引玉，仅供参改。     

一对一错号排列的简明公式

对于一对一错号排列问题：有编号为1，2，…，n的n个球，将其装入编号为1，2，…，n的n个金中，每盒装1个球，且球与盒的编号不同，求不同的装球方法种数S。文［1］给出了如下一个递推公式：利用该公式计算S。时，需首先依次逐一求出SI，JZ，S3，…，S。－l的值，笔者认为，当n较大时，其计算相当复杂．下面利用集合思想方法和容斥原理来推导该问题的一个较为简明的计算公式．设n个球任意放入n个盘中，且每盒装1个球的所有不同放法组成全集I，其中第i个球恰放入第i盘中的放法组成集合A。（i—1，2，…，n），显然A。MI．又用符号IAI…     

一类RNA二级结构的计数

多核苷酸的二级结构可视为一类顶点标号平面图，通常通过枚举每类RNA二级结构图的各种子图来计算其递推公式。本文作者给出了限制端环长度的RNA二级结构的递推公式，并运用隐式估计法计算它的渐近值。     

积分第一中值定理中的ξ在数值积分上的应用

根据积分第一中值定理的中间点ξ的渐近性质推导出一种单节点数值求积公式，证明余项的表达式，进行数值实验，此求积公式还适于瑕积分的数值计算。     

三角形中线长度计算面积公式的简证

文[1]利用余弦定理及三角形面积公式推导出三角形中线长度计算面积公式：如果m，n，P分别是△ABC三边上的中线，那么     

对三重积分f(x,y,z)dxdy方法的一些看法

关于三重积分的计算在[1]中给出了以下公式［2」中作者对此作了探讨。究竟在什么条件下，使用公式（1）能简化三重积分的计算，本人就此问题提出一些自己的看法。笔者认为用公式（1）所简化三重积分的计算应满足以下二个条件：（1）人x，y，z）中至少缺二个变量，即人x，y，z）一人x）或人工，y，z）。人y）或f（，y，）一八）；（2）若缺的变量为x，y，则对于积分区域D的Z截面风的面积应该很容易计算（实际上应是初等数学的结果）；对于缺变量Z，Z或。，Z的情形，相应的截面A，民的面积应很容易计算。例1计算三重积分Illxdxdydz，其中D…     

Recursive formulas for multinomial probabilities with applications

Recursive formulas are provided for computing probabilities of a multinomial distribution. Firstly, a recursive formula is provided for computing rectangular probabilities which include the cumulative distribution function as a special case. These rectangular probabilities can be used to provide goodness-of-fit tests for the cell probabilities. The probability that a certain cell count is the maximum of all cell counts is also considered, which can be used to assess the probability that the maximum cell count corresponds to the cell with the maximum probability. Finally, a recursive formula is provided for computing the probability that the cell counts satisfy a certain ordering, which can be used to assess the probability that the ordering of the cell counts corresponds to the ordering of the cell probabilities. The computational intensity of these recursive formulas is linear in the number of cells, and they provide opportunities for calculating probabilities that would otherwise be computationally challenging. Some examples of the applications of these formulas are provided.     

Simple formulas for the expected costs in the newsboy problem: An educational note

This educational note presents closed-form (or near closed-form) formulas for computing the expected cost and the optimal expected cost of the newsboy problem. The formulas are sufficiently easy to be used in undergraduate OR and MBA operations-management courses.     

Generalized Differentiation of a Class of Normal Cone Operators

This paper investigates generalized differentiation of normal cone operators to parametric smooth-boundary sets in Asplund spaces. We obtain formulas for computing the Fréchet and Mordukhovich coderivatives of such normal cone operators. We also give several examples to illustrate how the formulas can be used in practical calculations and applications.     

On numerical realization of the function of the hereditary operator : PMM vol. 42, no. 6, 1978, pp. 1115–1122

The problem of numerical realization of a function of the hereditary operator acting on some function of time is considered. Laplace transformations are used for the operators with kernels of the Rabotnov and Rzhanitsyn type to obtain formulas which reduce the problem in question to that of computing a quadrature. When the variable assumes large values, the formulas become asymptotic equations with an estimable error of approximation.     

Construction of fundamental solutions of two-dimensional problems of the nonstationary theory of elasticity

We propose a method of constructing the images of the fundamental solutions in the space of the Laplace transform with respect to time, leading to simple formulas. The method is illustrated using three dynamical problems: planar deformation for an anisotropic body; flexural vibrations of an anisotropic plate; and vibrations of a shallow isotropic shell of arbitrary Gaussian curvature. Quadrature formulas are given for computing the values of the fundamental solutions. We give a new interpretation and a new method of computing the values of the special functions used in the construction of singular solutions in problems of the static theory of shells. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 86–92.     

Computation of Gauss-type quadrature formulas

Gaussian formulas for a linear functional L (such as a weighted integral) are best computed from the recursion coefficients relating the monic polynomials orthogonal with respect to L. In Gauss-type formulas, one or more extraneous conditions (such as pre-assigning certain nodes) replace some of the equations expressing exactness when applied to high-order polynomials. These extraneous conditions may be applied by modifying the same number of recursion coefficients. We survey the methods of computing formulas from recursion coefficients, methods of obtaining recursion coefficients and modifying them for Gauss-type formulas, and questions of existence and numerical accuracy associated with those computations.     

On computing signatures of coherent systems

It is difficult to compute the signature of a coherent system with a large number of components. This paper derives two basic formulas for computing the signature of a system which can be decomposed into two subsystems (modules). As an immediate application, we obtain the formula for computing the signature of systemwise redundancy in terms of the signatures of the original system and the backup one. The formula for computing the signature of a componentwise redundancy system is also derived. Some examples are given to illustrate the power of the main results.     

Numerical integration using sparse grids

We present new and review existing algorithms for the numerical integration of multivariate functions defined over d-dimensional cubes using several variants of the sparse grid method first introduced by Smolyak [49]. In this approach, multivariate quadrature formulas are constructed using combinations of tensor products of suitable one-dimensional formulas. The computing cost is almost independent of the dimension of the problem if the function under consideration has bounded mixed derivatives. We suggest the usage of extended Gauss (Patterson) quadrature formulas as the one‐dimensional basis of the construction and show their superiority in comparison to previously used sparse grid approaches based on the trapezoidal, Clenshaw–Curtis and Gauss rules in several numerical experiments and applications. For the computation of path integrals further improvements can be obtained by combining generalized Smolyak quadrature with the Brownian bridge construction. This revised version was published online in August 2006 with corrections to the Cover Date.     

Rational circular complex centered forms

A discussion is given of the problem of computing including estimates of the range of a complex rational function over a circular complex interval. For this purpose, rational circular complex centered forms are defined. Explicit formulas are given for the first few forms and these formulas are used to prove that forms of higher order are an improvement over the forms of lower order. The forms are furthermore shown to be quadratically convergent.A semi-centered form is also discussed. This form is shown to be quadratically convergent depending on some conditions on the coefficients of the polynomials defining the complex rational function.Finally, a number of numerical examples are given showing the improvements obtained using the circular centered forms as compared to simple circular complex rational function estimations.     

COMPUTATION OF THE E(s2) VALUES OF SOME E(s2) OPTIMAL SUPERSATURATED DESIGNS

1 IntroductionAn rerun design for m two-level faCtors is saturated if n = m 1. Such designs haveminimum number of runs for estimating all the main effects when the interactions are negligible,and are useful for screening experiments in the initial stage of an investigation where the primarygoal is to identify the few active faCtors from a large number of potential faCtors. And whelln < in 1, such designs are called supersaturated designs, which provide more flexibility andcost saving. No…