一个新的数论函数及其均值性质

对于给定的自然数q,我们利用最小公倍数定义一个新的数论函数g(n)=[q,n]/q.本文的主要目的是研究δm(g(n))的均值性质,并利用解析方法得到这个函数的几个渐近公式.     

一个新的Smarandache函数及其均值

对任意正整数n≥3,我们定义算术函数C(n)为最大的正整数m≤n-2使得n |Cnm=n!/m!·(n-m)!.即就是C(n)=max{m:m≤n-2,n|Cnm},并规定C(1)=C(2)=1.本文的主要目的是利用初等及解析方法研究这一函数的均值分布问题,并给出几个有趣的均值公式及渐近式.     

关于F.Smarandache函数S(mn)的渐近性质

设m≥2为给定的整数,n为任意正整数.本文的主要目的是利用初等方法研究著名的F.Smarandache函数S(mn)当n→∞时的渐近性质,并给出一个较强的渐近公式.     

一个新的数论函数及其它的值分布

本文利用初等方法研究了函数V(n)的值分布性质,并给出两个较强的渐近公式.     

一个类似广义Dedekind和S_2(h,n,k)的二次均值

本文的主要目的是利用 Dirichlet L -函数的均值研究 n为奇数时 ,类似广义 Dedekind和 S2 ( h,n,k)的二次均值 ,得到了两个有趣的渐近公式 .     

一个包含Smarandache函数的复合函数

对任意正整数n,著名的Smarandache函数S(n)定义为最小的正整数m使得n|m!,或者S(n)=min{m∶n|m!,m∈N}.而函数Z(n)定义为最小的正整数k使得n≤k(k 1)/2,即就是Z(n)=min{k:n≤k(k 1)/2}.本文的主要目的是利用初等及解析方法研究复合函数S(Z(n))的均值,并给出一个较强的渐近公式.     

Smarandache函数的值分布性质

对于给定的自然数n,著名的Smarandache函数S(n)定义为S(n)=min{m: m∈N,n|m!|.本文主要目的是利用初等方法研究S(n)的值分布性质,并给出了一些有趣的渐近公式．     

关于函数S(x)及S_*(x)的渐近性质

对于任意实数x∈(1,∞),定义S(x)=m in{m∈N∶x m!};x∈[1,∞),S*(x)=m ax{m∈N∶m!x}.主要目的是研究这两个函数的渐近性质,并给出了它们的渐近公式.     

关于函数S(x)及S*(x)的渐近性质

对于任意实数x∈(1,∞),定义S(x)=min{m∈Nx≤m!};x∈[1,∞),S*(x)=max{m∈Nm!≤x}.主要目的是研究这两个函数的渐近性质,并给出了它们的渐近公式.     

具有扩散的n斑块生态系统的渐近周期解

研究了具有渐近周期系数的两种群扩散竞争系统,该系统由n个斑块组成,其中一种群可以在n个斑块之间扩散,而另一种群在一个斑块中,不能扩散.结合运用Liapunov函数,得到该系统唯一存在全局渐近稳定的渐近周期解的条件.     

一类整函数零点的渐近公式

借助Rouché定理、留数定理及渐近分析的方法,给出了整函数f(z)=zmsinz-α(0≠α∈R,m∈Z+)零点的渐近公式及渐近迹.这种方法也适用于其它整函数的零点估计.     

On Asymptotics for the Signless Noncentral q-Stirling Numbers of the First Kind

In this paper, we derive asymptotic formulas for the signless noncentral q -Stirling numbers of the first kind and for the corresponding series. The signless noncentral q -Stirling numbers of the first kind appear as coefficients of a polynomial of q -number [ t ] _q , expressing the noncentral ascending q -factorial of t of order m and noncentrality parameter k . In this paper, we have two main purposes. The first is to give an expression by which we obtain the asymptotic behavior of these coefficients, using the saddle point method . The second main purpose is to derive an asymptotic expression for the signless noncentral q -Stirling of the first kind series by using the singularity analysis method. We then apply our first formula to provide asymptotic expressions for probability functions of the number of successes in m trials and of the number of trials until the occurrence of the n th success in sequences of Bernoulli trials with varying success probability which are both written in terms of the signless noncentral q -Stirling numbers of the first kind. In addition, we present some numerical calculations using the computer program MAPLE indicating that our expressions are close to the actual values of the signless noncentral q -Stirling numbers of the first kind and of the corresponding series even for moderate values of m .     

The central limit theorem for a certain class of random fields

The asymptotic normal distribution of a sum of dependent random functions of m variables defined on the positive part of the integral lattice is established by the method of moments.Translated from Matematicheskie Zametki, Vol. 14, No. 4, pp. 549–558, October, 1973.     

Smarandache幂函数的均值

对于给定的自然数n,Smarandache幂函数SP(n)定义为SP(n)=min{m: n|mm,m∈N}．本文研究了这个函数的均值分布性质,并利用解析方法得到了Smarandache幂函数的一个较强的均值公式．     

关于非齐次m阶马氏信源的渐近均分割性

本文研究非齐次m阶马氏信源的渐近均分割性，首先我们得到关于此种信源m 1元函数的一类强极限定理，作为推论，得到关于任意非齐次m阶马氏信源状态和熵密度的几个极限定理，最后得到一类非齐次m阶马氏信源的渐近均分割性。     

Asymptotic Approximations for Radial Spheroidal Wavefunctions with Complex Size Parameter

Radial spheroidal wavefunctions are functions of four variables, usually denoted by m , n , x , and γ, the last of which is known as the size parameter. This parameter becomes complex when the problem of scattering of a sound pulse by a spheroid is treated using a Laplace transform with respect to time together with the method of separation of variables. Several asymptotic approximations, involving modified Bessel functions, are developed and analyzed.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">m</Emphasis></Subscript> Extemal Polynomials Associated with Generalized Jacobi Weights

Asymptotic estimations of the Christoffel type functions for Lm extremal polynomials with an even integer m associated with generalized Jacobi weights are established. Also, asymptotic behavior of the zeros of the Lm extremal polynomials and the Cotes numbers of the corresponding Turan quadrature formula is given.     

On a Modification of Olver’s Method: A Special Case

We consider the asymptotic method designed by Olver (Asymptotics and special functions. Academic Press, New York, 1974) for linear differential equations of the second order containing a large (asymptotic) parameter \(\Lambda \): \(x^my''-\Lambda ^2y=g(x)y\), with \(m\in \mathbb {Z}\) and g continuous. Olver studies in detail the cases \(m\ne 2\), especially the cases \(m=0, \pm 1\), giving the Poincaré-type asymptotic expansions of two independent solutions of the equation. The case \(m=2\) is different, as the behavior of the solutions for large \(\Lambda \) is not of exponential type, but of power type. In this case, Olver’s theory does not give many details. We consider here the special case \(m=2\). We propose two different techniques to handle the problem: (1) a modification of Olver’s method that replaces the role of the exponential approximations by power approximations, and (2) the transformation of the differential problem into a fixed point problem from which we construct an asymptotic sequence of functions that converges to the unique solution of the problem. Moreover, we show that this second technique may also be applied to nonlinear differential equations with a large parameter.     

The divergence of spectral decompositions connected with homogeneous elliptic operators

In this work an asymptotic formula is obtained for the means of the Riesz spectral function of a homogeneous elliptic operator of arbitrary order m≥2 with constant coefficients. Conditions are obtained under which localization (and convergence) of the means of the Riesz spectral decompositions of functions from the Hölder class is absent.