反函数的原函数公式及其应用

在本文第一部分给出了反函数原函数公式一个简单的证明，第二部分列举了几个用反函的原函数公式求积的例子，最后用此公式给出了Ｙｏｕｎｇ不等式一个证明，此证明不必借助于定积分的几何意义和图形面积的几何直观，从而是数学意义上更为严格的证明。     

二重积分变量代换公式证明的探讨

在一般教材中二重积分变量代换公式的证明通常采用几何的方法,也有部分数学分析教材给与了严格的分析证明,但证明不便直观的几何说明.本文对二重积分变量代换公式进行了一些探讨,给出了一个较简洁直观的分析证明方法.     

广义Taylor公式的几个简单证明方法

<正> 微分中值定理在高等数学中具有重要的理论意义。其中Taylor中值定理更具有一般性。文[1]给出了两个广义Taylor公式,文[2]对其中一个公式给出一个简单证明方法。本文对文[1]的两个公式再给出三个简单的证明方法。     

Krattenthaler反演公式的简短证明

为了提出研究Lagrange反演关系的统一方法,Krattenthaler建立了算子方法,并由此得到一个广泛的反演关系:Gould-Hsu-Carliz-Krattenthaler反演公式.本文利用Lagrange插值公式给出它的一个简短证明,进一步地证明在Lagrange插值的意义下,该反演关系是唯一性的.     

WZ方法与一个二项式级数的部分和公式

本文基于WZ理论给出了Peter Paule与Carsten Schneider的一篇文章中的一个二项式级数的部分和公式的新证明,并且发现他们的文章中所给的另一个二项式级数的部分和公式实际上是错误的,我们给出了其相应的一个正确公式.     

Cauchy型积分高阶导数公式的一个证明

<正> 在现行《复变函数论》教材中,Cauchy 型积分的高阶导数公式一般都不证明,最多仅仅指出证明的方法——数学归纳法.而用数学归纳法证明比较繁.下面介绍一个较简单的证明方法(主要取材于 J.B.Conway,Function of One Complex variable).这个方法不仅使 Cauchy 型积分的高阶导数公式得到圆满的证明,而且使 Cauchy 积分的高阶导数公式作为它的特殊情形而得到证明.下面就来证明这个公式.     

复函数微分中值公式的注记

本文给出了复函数的一个一般性的微分中值公式,获得了该公式的“中间值”的渐近性质,给出了该性质的简洁证明.     

从等周不等式谈Green公式的一个应用

Green公式给出了平面图形面积与曲线积分的关系,利用这种关系给出了平面曲线等周不等式的一个简捷证明,并给出了平面图形面积的一个近似计算公式以及两个教学计算实例.     

关于矩阵代数中几个公式的证明

矩阵代数中有关伴随矩阵的几个公式,其证明方法通常都是考虑所给矩阵是否奇异的情形,分别证明所要求的结果,然后综合表成一个公式,这种证明方法显得比较繁琐。本文介绍一种简洁而统一的处理方法,避免了上述证明方法的缺点,而且这种方法还可应用于证明矩阵的其他一些命题。     

关于典范体积形式的一个公式

本文给出了一个典范体积（Canonical volume)形式公式的代数证明。     

Wolfe线搜索下一类混合共轭梯度法的全局收敛性

本文给出了一个新的共轭梯度公式,新公式在精确线搜索下与DY公式等价,并给出了新公式的相关性质.结合新公式和DY公式提出了一个新的混合共轭梯度法,新算法在Wolfe线搜索下产生一个下降方向,并证明了算法的全局收敛性,并给出了数值例子.     

Configuration spaces,bistellar moves,and combinatorial formulae for the first Pontryagin class

The paper is devoted to the problem of finding explicit combinatorial formulae for the Pontryagin classes. We discuss two formulae, the classical Gabrielov-Gelfand-Losik formula based on investigation of configuration spaces and the local combinatorial formula obtained by the author in 2004. The latter formula is based on the notion of a universal local formula introduced by the author and on the usage of bistellar moves. We give a brief sketch for the first formula and a rather detailed exposition for the second one. For the second formula, we also succeed to simplify it by providing a new simpler algorithm for decomposing a cycle in the graph of bistellar moves of two-dimensional combinatorial spheres into a linear combination of elementary cycles.     

The best quadrature formula based on Hermite information for the class <Emphasis Type="Italic">KW</Emphasis><Superscript>2</Superscript> [<Emphasis Type="Italic">a,b</Emphasis>]

The best quadrature formula has been found in the following sense: for a function whose norm of the second derivative is bounded by a given constant and the best quadrature formula for the approximate evaluation of integration of that function can minimize the worst possible error if the values of the function and its derivative at certain nodes are known. The best interpolation formula used to get the quadrature formula above is also found. Moreover, we compare the best quadrature formula with the open compound corrected trapezoidal formula by theoretical analysis and stochastic experiments.     

A new Newton-like iterative method for roots of analytic functions

A new Newton-like iterative formula for the solution of non-linear equations is proposed. To derive the formula, the convergence criteria of the one-parameter iteration formula, and also the quasilinearization in the derivation of Newton's formula are reviewed. The result is a new formula which eliminates the limitations of other methods. There is now no need to first ensure a good initial approximation to the root, complex roots are found without necessarily starting from a complex formulation of the iteration formula, and the convergence is faster. The rate of convergence is discussed, and examples given.     

Longitudinal electric conductivity in a quantum plasma with a variable collision frequency in the framework of the Mermin approach

In the framework of the Mermin approach, we obtain formulas for the longitudinal electric conductivity in a quantum collisional plasma with a collision frequency depending on the momentum. We use a kinetic equation in the momentum space in the relaxation approximation. We show that as the Planck constant tends to zero, the derived formula transforms into the corresponding formula for a classical plasma. We also show that as the frequency of collisions between plasma particles tends to zero (the plasma transforms into a collisionless plasma), the derived formula transforms into the well-known Klimontovich-Silin formula for the collisionless plasma. We show that if the collision frequency is constant, then the derived formula for the permittivity transforms into the well-known Mermin formula.     

A de bruijn-type formula for enumerating patterns of partitions

We develop a formula for enumerating the partitions of a set up to the action of a group. We give some special cases of this formula and illustrate its use with an example. Some comments are made relating the formula to an earlier formula.This work was supported in part by NSF Contract GP-43010.     

The best quadrature formula based on Hermite information for the class KW~2[a,b]

The best quadrature formula has been found in the following sense:for afunction whose norm of the second derivative is bounded by a given constant and thebest quadrature formula for the approximate evaluation of integration of that function canminimize the worst possible error if the values of the function and its derivative at certainnodes are known.The best interpolation formula used to get the quadrature formula aboveis also found.Moreover,we compare the best quadrature formula with the open compoundcorrected trapezoidal formula by theoretical analysis and stochastic experiments.     

用单位变化率定义和认识费雪效应公式

通过分析利用连续复利解释费雪效应公式中存在的问题,利用单位变化率概念给出了费雪效应关系新的表达式,并证明了所给公式与通常教材中的费雪效应公式等价.     

A study over the general formula of regression sum of squares in multiple linear regression

In this study, in addition to the formula of regression sum of squares (SSR) in linear regression, a general formula of SSR in multiple linear regression is given. The derivations of the formula presented are given step by step. This new formula is proposed for estimation of the SSR in multiple linear regression. By using this formula, the researcher can find easily SSR and so the researcher can compose easily the table of variance analysis to interpret the regression made.