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.     

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.     

一个新的数值积分公式

本文主要构造了一个数值积分公式,进一步研究了数值积分公式中的中间点的渐进性,基于中间点的渐进性质,得到了该数值积分公式的校正公式,最后给出数值试验,数值试验表明该公式的精度非常高。     

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.     

柯特斯校正公式及其误差估计

提出了一类计算定积分的高精度柯特斯校正公式,通过两种方法进行了推导,给出了它的复化公式及其加速公式,并得到了它们的误差估计和收敛阶.数值实验验证了复化柯特斯校正公式及其加速公式的高效性.     

Modification of the Euler quadrature formula for functions with a boundary-layer component

The Euler quadrature formula for the numerical integration of functions with a boundary-layer component on a uniform grid is investigated. If the function under study has a rapidly growing component, the error can be significant. A uniformly accurate quadrature formula is constructed by modifying the Hermite interpolation formula so that the resulting one is exact for the boundary-layer component. An analogue of the Euler formula that is exact for the boundary-layer component is constructed. It is proved that the resulting composite quadrature formula is third-order accurate in space uniformly with respect to the boundary-layer component and its derivatives.     

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.     

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.     

A Diophantine problem of Frobenius in terms of the least common multiple

The Diophantine Problem of Frobenius is to find a formula for the least integer not representable as a nonnegative linear form of positive integers. A reduction formula for the Diophantine Problem of Frobenius is presented. The formula can be applied whenever there are common divisors of the coefficients except for the whole set of them. The reduction formula is expressed in terms of the least common multiple of the coefficients. For some classes of coefficients this formula gives an exact answer for the problem of Frobenius, and these classes are fully characterized in the paper.     

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.     

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.     

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.     

On the Integration by Parts and Characterization of a Fractional Diffusion Process北大核心CSCD

本文研究分数扩散过程和其分部积分公式的关系.首先利用Bismut方法给出拉回公式,进而得到分数扩散过程的分部积分公式。反过来,证明了分数扩散过程可由其分部积分公式唯一刻画.     

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.     

Stratonovich and Ito Stochastic Taylor Expansions

The Stratonovich stochastic Taylor formula for diffusion processes is stated and proved. It has a simpler structure and is a more natural generalization of the deterministic Taylor formula than the Ito stochastic Taylor formula.     

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 three-point formula for numerical quadrature of oscillatory integrals with variable frequency

A simple three-point formula is constructed for the evaluation of general oscillatory integrals.A rigorous derivation of the local error term is presented, and the implications to high frequency oscillations are discussed.Simple examples given include integrals with variable frequency for which the usual Filon formula would be inappropriate. For cases where Filon's formula is appropriate, the new formula appears to be computationally more efficient.The main application of the formula is to an example chosen from a class of integrals arising in the theory of water waves on a sloping beach. Comparison with exact results is possible from the work of Stoker [16] for a case which, whilst special in the physical sense, does not simplify the integral involved.In all cases the implementation of the formula is as straightforward as the implementation of the ordinary Simpson Rule.     

Stable local trace formula

Langlands program is the central concern of modern number theory. In general, we think that the main tool for studying this program is the trace formula. Arthur-Selberg’s trace formula is successful in solving the endoscopic case of Langlands program. Arthur stabilized the geometric side of the local trace formula in the general case, but did not stabilize the general spectral. This paper aims to solve the general case by different method in the Archimedean case, by directly stabilizing the spectral side of the local trace formula, and obtains the stable local trace formula.     

Numerical calculation of fourier integrals with cubic splines

The most used formula for calculation of Fourier integrals is Filon's formula which is based on approximation of the function by a quadratic in each double interval. In order to obtain a better approximation we use the cubic spline fit. The method is not restricted to equidistant points, but the final formulas are only derived in this case. Test computations show that the spline formula may be superior to Filon's formula.