从高斯公式到格林公式和牛顿-莱布尼茨公式

本文讨论高等数学课程中,高斯公式、格林公式和牛顿-莱布尼兹公式之间的内在联系,指出格林公式和牛顿-莱布尼茨公式可以分别看作一维和二维欧氏空间中的高斯公式.实际上,n维欧氏空间中的高斯公式可以看作微积分基本定理在高维欧氏空间中的表述形式.利用高斯公式还可以导出定积分、二重积分和任意n重积分的分部积分公式.     

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.     

Ramanujan formula for the generalized Stirling approximation

The aim of this paper is to improve Ramanujan’s formula for approximation of the factorial function, starting from Burnside’s formula in contradistinction with the classical formula that starts from Stirling’s formula.     

Bijective proofs of formulae for the number of standard Yound tableaux

We give a bijective proof of a formula due independently to Frobenius and Young for the number of standard Young tableau of shape λ for λ any partition. Frame, Robinson, and Thrall derived their hook formula for the number of standard Young tableau from the Frobenius-Young formula. As a corollary to our bijective proof of the Frobenius-Young formula, we also give a bijective proof of the Frame-Robinson-Thrall hook formula.     

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.     

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.     

解读巴塞尔协议Ⅱ中内部评级法关于资本要求的计算

鉴于新巴塞尔协议中内部评级法关于资本要求的计算缺乏透明度,本文在归纳总结内部评级法关于资本要求计算的基础上,对资本要求计算的理论依据和统计含义进行了描述和分析。本文指出,资本要求计算的核心内容是Vasicek公式,它可以被解释为单一借款者的条件违约概率,或者单一个充分分散化贷款组合的风险价值。最后,我们对内部评级法关于资本要求计算中存在的一些问题作了简单讨论。     

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.     

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.     

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.     

Cotes数值求积公式的校正

本文研究了Cotes数值求积公式代数精度的问题,给出了Cotes求积公式余项"中间点"的渐进性定理.利用该定理得到了改进的Cotes求积公式,并证明了改进后的Cotes求积公式比原来的公式具有较高的代数精度.     

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.     

Cauchy-type integral formula and Plemelj formula of hypermonogenic functions on unbounded domains

In this paper, we discuss the Cauchy-type integral formula of hypermonogenic functions on unbounded domains in real Clifford analysis, then we extend the Plemelj formula and Cauchy–Pompeiu formula of hypermonogenic functions on bounded domains to unbounded domains. We also deal with the Green-type formula on unbounded domains and get several important corollaries.     

Lead time demand for simple exponential smoothing: an adjustment factor for the standard deviation

A new simple formula is found to correct the underestimation of the standard deviation for total lead time demand when using simple exponential smoothing. The traditional formula for the standard deviation of lead time demand is to multiply the standard deviation for the one-period-ahead forecast error (estimated by using the residuals) by the square root of the number of periods in the lead time. It has been shown by others that the traditional formula significantly underestimates variation in the lead time demand when the mean of the process is somewhat changing and simple exponential smoothing is appropriate. This new formula allows one to see readily the significant size of the underestimation of the traditional formula and can easily be implemented in practice. The formula is derived by using a state-space model for simple exponential smoothing.     

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.     

The operator remainder in the Euler-Maclaurin formula

We give a formula for the remainder term in the operator-theoretic formulation of the Euler-Maclaurin formula, using Laplace transforms. There is also a related formula for the partial sums.Keywords and phrases: Euler-Maclaurin sum formula, remainder term, Laplace transform, Bernoulli polynomial.Dedicated to Professor Janos Aczél on his 60th birthday     

Combining dependent evidential bodies that share common knowledge

We establish a formula for combining dependent evidential bodies that are conditionally independent given their shared knowledge. Markov examples are provided to illustrate various aspects of our combination formula, including its practicality. We also show that the Dempster–Shafer formula and the conjunctive rule of the Transferable Belief Model can be recovered as special cases of our combination formula.     

On solving general algebraic equations by integrals of elementary functions

We obtain an integral formula for a solution to a general algebraic equation. In this formula the integrand is an elementary function and integration is carried out over an interval. The advantage of this formula over the well-known Mellin formula is that the integral has a broader convergence domain. This circumstance makes it possible to describe the monodromy of a solution for trinomial equations.