Some Quicker Approximations and Inequalities of the Wallis Ratio by Continued Fraction

In this paper, we present a new formula for approximating the Wallis ratio. This new formula is very fast in comparison with other classical or recently discovered asymptotic series. Next, inequalities related to this new formula and asymptotic series are established. Finally, for demonstrating the superiority of our new series, some numerical computations are provided.     

常息力更新场合有限时间破产概率对负相依索赔额的不敏感性

设索赔来到过程为具有常数利息力度的更新风险模型.在索赔额分布为负相依的次指数分布假定下,建立了有限时间破产概率的一个渐近等价公式.所得结果显示,在独立同分布索赔额情形,有限时间破产概率的有关渐近等价公式,在负相依场合依然成立.这表明有限时间破产概率对于索赔额的负相依结构是不敏感的.     

Wallis公式的精致化

证明了∫_0~(π/2)sin~αxdx(α→+∞)渐进展开式的存在性,并给出了展开式系数的递推公式,进而得到了((2n-1)!!)/((2n)!!)/((2n)!!)的渐进展开式和精致化的Wallis公式.在精致化的Wallis公式中取到1/n~m项,对π的逼近精度可达O(1/(n~(m+1))),比原来的Wallis公式的精度大为提高.     

Boundary voltage perturbations caused by small conductivity inhomogeneities nearly touching the boundary

We establish an asymptotic expansion of the steady-state voltage potentials in the presence of a diametrically small conductivity inhomogeneity that is nearly touching the boundary. Our asymptotic formula extends those already derived for a small inhomogeneity far away from the boundary and is expected to lead to very effective algorithms, aimed at determining location and certain properties of the shape of a small inhomogeneity that is nearly touching the boundary based on boundary measurements. Viability of the asymptotic formula is documented by numerical examples.     

The first moment of Salié sums

The main objective of this article is to study the asymptotic behavior of Salié sums over arithmetic progressions. We deduce from our asymptotic formula that Salié sums possess a bias towards being positive. The method we use is based on the Kuznetsov formula for modular forms of half-integral weight. Moreover, in order to develop an explicit formula, we are led to determine an explicit orthogonal basis of the space of modular forms of half-integral weight.     

Asymptotics of matrix integrals and tensor invariants of compact Lie groups

In this paper we give an asymptotic formula for a matrix integral which plays a crucial role in the approach of Diaconis et al. to random matrix eigenvalues. The choice of parameter for the asymptotic analysis is motivated by an invariant-theoretic interpretation of this type of integral. For arbitrary regular irreducible representations of arbitrary connected semisimple compact Lie groups, we obtain an asymptotic formula for the trace of permutation operators on the space of tensor invariants, thus extending a result of Biane on the dimension of these spaces.

     

Librations and ground-state splitting in a multidimensional double-well problem

We derive an asymptotic formula for the splitting of the lowest eigenvalues of the multidimensional Schrödinger operator with a symmetric double-well potential. Unlike the well-known formula of Maslov, Dobrokhotov, and Kolosoltsov, the obtained formula has the form A(h)e^?S/h(1 + o(1)), where S is the action on a periodic trajectory (libration) of the classical system with the inverted potential and not the action on the doubly asymptotic trajectory. In this expression, the principal term of the pre-exponential factor takes a more elegant form. In the derivation, we merely transform the asymptotic formulas in the mentioned work without going beyond the framework of classical mechanics.     

A new continued fraction approximation and inequalities for the Gamma function via the Tri-gamma function

The main objective of this paper is to propose a product approximation for the Gamma function in the form of continued fraction, via the Tri-gamma function. The importance of this new formula is that the convergence of the corresponding asymptotic series is faster than other recently discovered asymptotic series, and some new inequalities related to the new formula are also established.     

Asymptotic behavior of quantum walks on the line

This paper gives various asymptotic formulae for the transition probability associated with discrete time quantum walks on the real line. The formulae depend heavily on the ‘normalized’ position of the walk. When the position is in the support of the weak-limit distribution obtained by Konno (2005) [5], one observes, in addition to the limit distribution itself, an oscillating phenomenon in the leading term of the asymptotic formula. When the position lies outside of the support, one can establish an asymptotic formula of large deviation type. The rate function, which expresses the exponential decay rate, is explicitly given. Around the boundary of the support of the limit distribution (called the ‘wall’), the asymptotic formula is described in terms of the Airy function.     

Sum of the Dirichlet <Emphasis Type="Italic">L</Emphasis>-functions over nontrivial zeros of another Dirichlet <Emphasis Type="Italic">L</Emphasis>-function

A. Fujii, and later J. Steuding, considered an asymptotic formula for the sum of values of the Dirichlet L-function taken at the nontrivial zeros of another Dirichlet L-function. Here we improve the error term of this asymptotic formula.     

An Explicit Formula for the Coefficients in Laplace’s Method

Laplace’s method is one of the fundamental techniques in the asymptotic approximation of integrals. The coefficients appearing in the resulting asymptotic expansion arise as the coefficients of a convergent or asymptotic series of a function defined in an implicit form. Due to the tedious computation of these coefficients, most standard textbooks on asymptotic approximations of integrals do not give explicit formulas for them. Nevertheless, we can find some more or less explicit representations for the coefficients in the literature: Perron’s formula gives them in terms of derivatives of an explicit function; Campbell, Fröman and Walles simplified Perron’s method by computing these derivatives using an explicit recurrence relation. The most recent contribution is due to Wojdylo, who rediscovered the Campbell, Fröman and Walles formula and rewrote it in terms of partial ordinary Bell polynomials. In this paper, we provide an alternative representation for the coefficients that contains ordinary potential polynomials. The proof is based on Perron’s formula and a theorem of Comtet. The asymptotic expansions of the gamma function and the incomplete gamma function are given as illustrations.     

中矩形公式与梯形公式的注记

将数值积分中的中矩形公式与梯形公式推广到两个函数的情形,并讨论了中间值的渐近性质.     

关于Simpson公式的两个注记

对Simpson公式的积分余项作出渐近估计,并给出了Simpson公式反问题的一个结果.     

An asymptotic solution of a second-order linear differential equation

This paper discusses an asymptotic formula for solutions of a second-order linear differential equation. The asymptotic formula will enable us to provide information about the distribution of eigenvalues for the case of nonexistence of continuous spectrum in a singular Sturm-Liouville type boundary value problem. The result can be regarded as a partial generalization of those obtained by E. C. Titchmarsh and C. G. C. Pitts.     

随机映射图的连接概率

本文将计算随机映射图的给定顶点集的任意分类的连接概率及其极限性质,导出随机映射图的连通分支个数的分布与渐进分布．     

ON THE GENERAL k-TH KLOOSTERMAN SUMS AND ITS FOURTH POWER MEAN

§1. Introduction Let q ≥3 be a positive integer. For any integers m, n and k , we de?ne the generalk-th Kloosterman sums S(m,n,k,χ;q) as follows: q …     

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.     

关于广义Dedekind和的2k次均值

本文利用 Ｄｉｒｉｃｈｌｅｔ Ｌ－函数的均值定理,研究广义 Ｄｅｄｅｋｉｎｄ和的渐近性质,并给出一个有趣的渐近公式．     

保费随机复合二项模型的Gerber-shiu折现罚金函数

考虑保费随机收取的复合二项模型.得到了其Gerber-shiu折现罚金函数满足的递推公式,瑕疵更新方程及其渐近解,并且通过构造一个相关的复合几何分布函数,得到了这个更新方程的解析解.相应的也得到了一些相关精算量的渐近表示和分布函数,如破产前瞬时盈余分布的渐近解,导致破产的索赔额的分布函数.