具有常数红利边界的两类索赔相关风险模型数

研究常数红利边界下两类索赔相关的风险模型,两类索赔计数过程分别为独立的Poisson过程和广义Erlang(2)过程.利用分解Gerber-Shiu函数的方法,得到了Gerber-Shiu函数满足的积分-微分方程、边界条件、解析表达式及两类索赔额均服从指数分布时的破产概率表达式.     

关于无K—间隔的组合数

从排列在一条直线上的n个元素中选取m个元素,以f_k(n,m)表示任意两个被选元素的间隔均不为k之方式数。如果这n个元素排列在园周上,则相应的组合数以g_k(m,m)表示。关于这两类组合数,I.Kaplansky于1943年首先研究了k=0时的计数问题,J. Konvalina于1981年应用递归方法得到了k=1时的计数表达式。对于一般的自然数k,这一问题似乎更加复杂。     

哥德巴赫数

表为两个奇素数之和的偶数称为Goldbach数。很多数学工作者研究了对于怎样的数η(≥0),当h≥x~η时区间(x-h,x h]必含有Goldbach数。本文应用筛法余项的新的估计式证明了以下主要结果: 当h≥x~(245/5088)时,区间(x-h,x h]中必含有Goldbach数,这里x是充分大的正数。     

实二次域类数的一个同余式

在这篇短文里,我们要证明定理设p是一个奇素数.以h表实二次域Q(p~(1/2)的类数,而以表Q(p~(1/2))的基本单位,共中t,u是有理整数,Q是有理数域.则我们有同余式     

两类Stirling数的行列式表示

通过计算两个广义的范德蒙(Vandermonde)行列式,得到了第一类无符号Stirling数和第二类Stirling数的一种新的表示方法:用行列式来表示.     

高阶Bernoulli数与Stirling数的几个恒等式

利用第一、二类高阶Bernoulli数和二类Stirling数S1(n,k),S2(n,k)的定义.研究了二类高阶Bernoulli数母函数的幂级数展开,揭示了二类高阶Bernoulli数之间以及与第一类Stirling数S1(n,k)、第二类Stirling数S2(n,k)之间的内在联系,得到了几个关于二类高阶Bernoulli数和第一类Stirling数S1(n,k)、第二类Stirling数S2(n,k)之间有趣的恒等式.     

Goldbach数的例外集合(Ⅲ)

本文把能表成两个奇素数之和的偶数称为Goldbach数,以E(x)记作不超过x的非Goldbach数的数目,并且证明了E(x)=O(x~(0.95)     

关于幂数问题的一个Golomb猜想

本文用Pell方程的知识,否定了Golomb猜想2°,并且证明:任意一个数m(m≠0)均可真表示为两个幂数的差,且表法无限。     

一类实二次域类数的可除性

<正> 我们来证明 定理 设D=4q~(2n)+1是无平方因子正整数,其中n与q均为正整数,且q≥2,那么我们有: 1)n除尽实二次域Q(D~(1/2))的类数h(D),这里Q表有理数域;     

图的平面Turán数和平面anti-Ramsey数（英文）

在所有顶点数为n且不包含图G作为子图的平面图中,具有最多边数的图的边数称为图G的平面Turán数,记为ex_P(n,G)。给定正整数n以及平面图H,用T_n(H)来表示所有顶点数为n且不包含H作为子图的平面三角剖分图所组成的图集合。设图集合T_n (H)中的任意平面三角剖分图的任意k边染色都不包含彩虹子图H,则称满足上述条件的k的最大值为图H的平面anti-Ramsey数,记作ar_P(n,H)。两类问题的研究均始于2015年左右,至今已经引起了广泛关注。全面地综述两类问题的主要研究成果,以及一些公开问题。     

Applications of an Explicit Formula for the Generalized Euler Numbers

The authors establish an explicit formula for the generalized Euler NumbersE2n^（x）, and obtain some identities and congruences involving the higher＇order Euler numbers, Stirling numbers, the central factorial numbers and the values of the Riemann zeta-function.     

Weakly Computable Real Numbers

A real number x is recursively approximable if it is a limit of a computable sequence of rational numbers. If, moreover, the sequence is increasing (decreasing or simply monotonic), then x is called left computable (right computable or semi-computable). x is called weakly computable if it is a difference of two left computable real numbers. We show that a real number is weakly computable if and only if there is a computable sequence (x_s)_s of rational numbers which converges to x weakly effectively, namely the sum of jumps of the sequence is bounded. It is also shown that the class of weakly computable real numbers extends properly the class of semi-computable real numbers and the class of recursively approximable real numbers extends properly the class of weakly computable real numbers.     

Properties and Applications of the Reciprocal Logarithm Numbers

Via a graphical method, which codes tree diagrams composed of partitions, a novel power series expansion is derived for the reciprocal of the logarithmic function ln (1+z), whose coefficients represent an infinite set of fractions. These numbers, which are called reciprocal logarithm numbers and are denoted by A _k , converge to zero as k→∞. Several properties of the numbers are then obtained including recursion relations and their relationship with the Stirling numbers of the first kind. Also appearing here are several applications including a new representation for Euler’s constant known as Hurst’s formula and another for the logarithmic integral. From the properties of the A _k it is found that a term of ζ(2) cannot be eliminated by the remaining terms in Hurst’s formula, thereby indicating that Euler’s constant is irrational. Finally, another power series expansion for the reciprocal of arctangent is developed by adapting the preceding material.     

A p, q-Analogue of the Generalized Derangement Numbers

In this paper, we study the numbers D _n,k which are defined as the number of permutations σ of the symmetric group S _n such that σ has no cycles of length j for j ≤ k. In the case k = 1, D _n,1 is simply the number of derangements of an n-element set. As such, we shall call the numbers D _n,k generalized derangement numbers. Garsia and Remmel [4] defined some natural q-analogues of D _n,1, denoted by D _n,1(q), which give rise to natural q-analogues of the two classical recursions of the number of derangements. The method of Garsia and Remmel can be easily extended to give natural p, q-analogues D _n,1(p, q) which satisfy natural p, q-analogues of the two classical recursions for the number of derangements. In [4], Garsia and Remmel also suggested an approach to define q-analogues of the numbers D _n,k . In this paper, we show that their ideas can be extended to give a p, q-analogue of the generalized derangements numbers. Again there are two classical recursions for eneralized derangement numbers. However, the p, q-analogues of the two classical recursions are not as straightforward when k ≥ 2. Partially supported by NSF grant DMS 0400507.     

Sum-Difference Sequences and Catalan Numbers

 Let be a sequence of natural numbers > 1, and set . The sequence is called admissible if a _i divides for all i. It is known that the admissible sequences are counted by the Catalan numbers. We present a proof of this fact which, in turn, leads to some interesting combinatorial and number-theoretic questions.     

Sum-Difference Sequences and Catalan Numbers

 Let be a sequence of natural numbers > 1, and set . The sequence is called admissible if a _i divides for all i. It is known that the admissible sequences are counted by the Catalan numbers. We present a proof of this fact which, in turn, leads to some interesting combinatorial and number-theoretic questions. Received 12 May 1997; in revised form 9 June 1997     

Auslander-Reiten Numbers Games

Let \(\mathfrak {g}\) be a simple complex Lie algebra of types A _n , D _n , E _n , and Q a quiver obtained by orienting its Dynkin diagram. Let λ be a dominant weight, and E(λ) the corresponding simple highest weight representation. We show that the weight multiplicities of E(λ) may be recovered by playing a numbers game Λ _Q (λ), generalizing the well known Mozes game, constructing the orbit of λ under the action of the Weyl group W. The game board is provided by the Auslander-Reiten quiver Γ _Q of Q. The game moves are obtained by constructing Nakajima’s monomial crystal M(λ) directly out of Γ _Q . As an application, we consider Kashiwara’s parameterizations of the canonical basis. Let w ₀ be a reduced expression of the longest element w ₀ of W, adapted to a quiver Q of type A _n . We show that a set of inequalities defining the string (Kashiwara) cone with respect to w ₀, may be obtained by playing subgames of the numbers games Λ _Q (ω _i ) associated to fundamental representations.     

Simple Explicit Formulas for the Frame-Stewart Numbers

Several different approaches to the multi-peg Tower of Hanoi problem are equivalent. One of them is Stewart's recursive formula ¶¶ S (n, p) = min {2S (n₁, p) + S (n-n₁, p-1) | n₁, n-n₁ ? \mathbbZ⁺}. S (n, p) = min \{2S (n_1, p) + S (n-n_1, p-1)\mid n_1, n-n_1 \in \mathbb{Z}^+\}. ¶¶In the present paper we significantly simplify the explicit calculation of the Frame-Stewart's numbers S(n, p) and give a short proof of the domain theorem that describes the set of all pairs (n, n₁), such that the above minima are achieved at n₁.     

Partition Lattice q-Analogs Related to q-Stirling Numbers

We construct a family of partially ordered sets (posets) that are q-analogs of the set partition lattice. They are different from the q-analogs proposed by Dowling [5]. One of the important features of these posets is that their Whitney numbers of the first and second kind are just the q-Stirling numbers of the first and second kind, respectively. One member of this family [4] can be constructed using an interpretation of Milne [9] for S[n, k] as sequences of lines in a vector space over the Galois field F _q. Another member is constructed so as to mirror the partial order in the subspace lattice.     

Face Numbers of Scarf Complexes

Let A be a (d+1) × d real matrix whose row vectors positively span R ^d and which is generic in the sense of Bárány and Scarf [BS1]. Such a matrix determines a certain infinite d -dimensional simplicial complex Σ , as described by Bárány et al.[BHS]. The group Z ^d acts on Σ with finitely many orbits. Let f _i be the number of orbits of (i+1) -simplices of Σ . The sequence f=(f ₀ ,f ₁ , . . ., f _d-1 ) is the f -vector of a certain triangulated (d-1) -ball T embedded in Σ . When A has integer entries it is also, as shown by the work of Peeva and Sturmfels [PS], the sequence of Betti numbers of the minimal free resolution of k[x ₁ , . . . ,x _d+1 ]/I , where I is the lattice ideal determined by A . In this paper we study relations among the numbers f _i . It is shown that determine the other numbers via linear relations, and that there are additional nonlinear relations. In more precise (and more technical) terms, our analysis shows that f is linearly determined by a certain M -sequence , namely, the g -vector of the (d-2) -sphere bounding T . Although T is in general not a cone over its boundary, it turns out that its f -vector behaves as if it were. Received January 20, 1999.