An extension of the incomplete beta function for negative integers

The incomplete beta function Bx(a,b) is defined for a,b>0 and 0<x<1. Its definition can be extended, by regularization, to negative non-integer values of a and b. In this paper we define the incomplete beta function Bx(a,b) for negative integer values of a and b. Further we prove that the function exists for m,n=0,1,2,… and all a and b.     

Defining the kth powers of the Dirac-delta distribution for negative integers

In [1], Koh and Kuan defined the powers of the Dirac-delta distribution for positive integers. Here we extend their definition for negative integers. Also, we give meaning to the distribution δ^−k₊.     

Heaviside公式的一种改进

针对一般积分变换教材中Heaviside公式的形式不容易被记忆和理解的情况,运用复变函数的相关知识对Heaviside公式的形式做进一步改进和说明.     

Distribution of a certain partition function modulo powers of primes

In this paper, we study a certain partition function a(n) defined by Σ _n≥0 a(n)q ⁿ := Π _n=1(1 − q ⁿ )⁻¹(1 − q ²ⁿ )⁻¹. We prove that given a positive integer j ≥ 1 and a prime m ≥ 5, there are infinitely many congruences of the type a(An + B) ≡ 0 (mod m ^j ). This work is inspired by Ono’s ground breaking result in the study of the distribution of the partition function p(n).     

The use of random weights for the training of multilayer networks of neurons with Heaviside characteristics

Artificial neural networks have, in recent years, been very successfully applied in a wide range of areas. A major reason for this success has been the existence of a training algorithm called backpropagation. This algorithm relies upon the neural units in a network having input/output characteristics that are continuously differentiable. Such units are significantly less easy to implement in silicon than are neural units with Heaviside (step-function) characteristics. In this paper, we show how a training algorithm similar to backpropagation can be developed for 2-layer networks of Heaviside units by treating the network weights (i.e., interconnection strengths) as random variables. This is then used as a basis for the development of a training algorithm for networks with any number of layers by drawing upon the idea of internal representations. Some examples are given to illustrate the performance of these learning algorithms.     

On the longest length of consecutive integers

Choose m numbers from the set {1, 2, …, n} at random without replacement. In this paper we first establish the limiting distribution of the longest length of consecutive integers and then apply the result to test randomness of selecting numbers without replacement.     

On sets of integers whose shifted products are powers

Let N be a positive integer and let A be a subset of {1,…,N} with the property that aa^′+1 is a pure power whenever a and a^′ are distinct elements of A. We prove that |A|, the cardinality of A, is not large. In particular, we show that |A|?(logN)^2/3(loglogN)^1/3.     

Polynomial interpolation and sums of powers of integers

In this note, we revisit the problem of polynomial interpolation and explicitly construct two polynomials in n of degree k + 1, P_k(n) and Q_k(n), such that P_k(n) = Q_k(n) = f_k(n) for n = 1, 2,…?, k, where f_k(1), f_k(2),…?, f_k(k) are k arbitrarily chosen (real or complex) values. Then, we focus on the case that f_k(n) is given by the sum of powers of the first n positive integers S_k(n) = 1^k + 2^k + ??? + n^k, and show that S_k(n) admits the polynomial representations S_k(n) = P_k(n) and S_k(n) = Q_k(n) for all n = 1, 2,…?, and k ≥ 1, where the first representation involves the Eulerian numbers, and the second one the Stirling numbers of the second kind. Finally, we consider yet another polynomial formula for S_k(n) alternative to the well-known formula of Bernoulli.     

On computing the dispersion function

A two-dimensional analogue of the well-known bisection method for root finding is presented in order to solve the following problem, related to the dispersion function of a set of random variables: given distribution functionsF ₁,...,F _n and a probabilityp, find an interval [a,b] of minimum width such thatF _i(b)–F _i(a ^–)p, fori=1,...,n.The author wishes to thank Dr. I. D. Coope, for helpful advice offered during the preparation of this paper, and the referee, whose comments contributed to a clearer presentation.     

A result on the composition of distributions

LetF be a distribution and letf be a locally summable function. The distributionF(f) is defined as the neutrix limit of the sequenceF _n (f), whereF _n (x) = F(x) * δ _n (x) andδ _n (x) is a certain sequence of infinitely differentiable functions converging to the Dirac delta-functionδ(x). The distribution (x^r)^−s is valuated forr, s = 1,2, ….     

On functions in p-adic BMO and the distribution of prime integers

We study the spaces BMOp of functions of bounded mean oscillation modeled on a p-adic martingale, and determine their relationship with the ordinary, continuous space BMO of functions of bounded mean oscillation. Somewhat surprisingly, these results are related to information about the distribution of primes.     

关于Smarandache函数的一个问题

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!.即就是S(n)=min{m:m∈N,n|m!}.令PS(n)表示区间[1,n]中S(n)为素数的正整数n的个数.在一篇未发表的文献中,J. Castillo建议我们研究当n→∞时,比值PS(n)/n的极限存在问题.如果存在,确定其极限.本文的主要目的是利用初等方法研究这一问题,并得到彻底解决!即就是证明该极限存在且为1.     

关于Smarandache函数的奇偶性

对任意正整数n,著名的F. Smarandache函数S(n)定义为最小的正整数m使得n│m!.即就是S(n)=min{m:m∈N,n│m!}.令OS(n)表示区间[1,n]中S(n)为奇数的正整数n的个数;ES(n)表示区间[1,n]中S(n)为偶数的正整数n的个数.在文[2]中,Kenichiro Kashihara建议我们研究极限limn→∞ES(n)/OS(n)的存在问题.如果存在,确定其极限,本文的主要目的是利用初等方法研究这一问题,并得到彻底解决!即就是证明该极限存在且为零.     

On the shape of the mean residual lifetime function

Some general properties of the mean residual life (MRL) function are studied. The analysis is based on the shape of the corresponding failure rate. The conditions under which the failure rate and the reciprocal to the MRL function have asymptotically equivalent behaviour as t→∞ are discussed. The simplest non‐monotone shapes of the functions under consideration (bathtub and upside down bathtub) are also considered. The MRL functions for mixtures of distributions are described via the corresponding conditional probability density functions. The direct proportional model of mixing is characterized and some asymptotic results on the shape of the mixture MRL are obtained. Some simple examples are given. Copyright © 2002 John Wiley & Sons, Ltd.     

The Frobenius problem, sums of powers of integers, and recurrences for the Bernoulli numbers

In the Frobenius problem with two variables, one is given two positive integers a and b that are relative prime, and is concerned with the set of positive numbers NR that have no representation by the linear form ax+by in nonnegative integers x and y. We give a complete characterization of the set NR, and use it to establish a relation between the power sums over its elements and the power sums over the natural numbers. This relation is used to derive new recurrences for the Bernoulli numbers.     

On the Gerber-Shiu discounted penalty function for subexponential claims

We obtain the asymptotics of the Gerber-Shiu discounted penalty function in the classical Lundberg model. We cosider claims from a class of subexponential distributions and find the asymptotics as the initial surplus x tends to infinity. The main term of the discounted penalty function ψ(x, δ) has different expressions in the cases where the interest rate δ > 0 and where δ = 0. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 598–605, October–December, 2006.     

On the number of primitive representations of integers as sums of squares

Formulas for the number of primitive representations of any integer n as a sum of k squares are given, for 2 ≤ k ≤ 8, and for certain values of n, for 9 ≤ k ≤ 12. The formulas have a similar structure and are striking for their simplicity. Dedicated to Richard Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—11E25; Secondary—05A15, 33E05.