具有四个不同特征值的竞赛矩阵

设A是n阶竞赛矩阵，k是非负整数。文[3]刻划了恰好有三个不同特征值的n阶竞赛矩阵，文[4]刻划了恰好有四个不同特征值并且0作为一个一重特征值的n阶竞赛矩阵。在这篇文章中我们主要研究了两个问题：（1）讨论当k是A的特征值时A的性质。（2）刻划恰好有四个不同特征值并且k作为一个一重特征值的全部n阶竞赛矩阵。     

含谱参数高次幂的Zakharov—Shabat特征值问题的迹公式

本文研究x∈(-∞,∞)上的特征值问题:这里非负整数l和k均小于整数n,且k≤n-l;ζ为复参数;i=(-1)~(1/2);诸位势q_j(x)和r_j(x)为复值函数。 当n=1,(1)恰为Zakharov-Shabat特征值问题;当n=2,l=1,q_o(x)=r_o(x)=0,(1)恰为KaupNewell特征值问题。 本文给出(1)的散射问题的Jost解、散射量和色散关系,在此基础上导出Zakhazov-Faddeev型的迹恒等式。     

ζ(k)的部分和五阶和式的计算

u1,u2…是独立、同分布于(0,1)区间上均匀分布的随机变量.本文证明了1-u1u2…uk的n-1阶矩(n≥1)是以调和数的部分和ζn(r)=∑j=1n 1/jr,r≥1为变元的指数型完全Bell多项式,因此Riemann-Zeta函数ζ(k),k≥2能够被展开成第一类无符号Stirling数s(n,k)的级数,从而计算出与ζn(r)有关的全部6个五阶和式.它们都是ζ(5)与ζ(2)ζ(3)的有理组合.     

Automorphisms of a Class of Finite p-groups with a Cyclic Derived Subgroup

Let p be an odd prime,and let k be a nonzero nature number.Suppose that nonabelian group G is a central extension as follows1→G'→G→Z_(p~k)×…×Z_(p~k),where G'≌Z_(p~k),and ζG/G' is a,direct factor of G/G'.Then G is a central product of an extraspecial p~kgroup E and ζG.Let |E|=p~((2n+1)k) and |ζG|=p~((m+1)k).Suppose that the exponents of E and ζG are p~(k+l) and p~(k+r),respectively,where 0≤l,r≤k.Let Aut_(G') G be the normal subgroup of Aut G consisting of all elements of Aut G which act trivially on the derived subgroup G',let Aut_(G/ζG,ζG) G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on the center ζG and let Aut_(G/ζG,ζG/G') G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on ζG/G'.Then(ⅰ) The group extension 1→Aut G'→Aut G→Aut G'→1 is split.(ⅱ) Aut_(G') G/Aut_(G/ζG,ζG) G≌G_1 × G_2,where Sp(2n-2,Z_(p~k))■H≤G_1≤Sp(2n,Z_(p~k)),H is an extraspecial p~k-group of order p~((2n-1)k) and(GL(m-1,Z_(p~k))■Z_(p~k)~((m-1))■Z_(p~k)~((m))≤G_2≤GL(m,Z_(p~k))■Z_(p~k)~((m)).In particular,G_1=Sp(2n-2,Z~(p~k))■ H if and only if l=k and r=0;G_1=Sp(2n,Z_(p~x)) if and only if l≤r;G_2=(GL(m-1,Z_(p~k))■ Z_(p~k)~((m-1))■ Z_(p~k)~((m)) if and only if r=k;G_2=GL(m,Z_(p~k))■Z_(p~k)((m)) if and only if r=0.(ⅲ) Aut_(G') G/Aut_( G/ζG,ζG/G') G≌G_1 × G_3,where G_1 is defined in(ⅱ);GL(ml,Z_(p~k))■ Z_(p~k)~((m-1))≤G_3 ≤GL(n,Z_(p~k)).In particular,G_3=GL(m-1,Z_(p~k))■ Z_(p~k)~((m-1)) if and only if r=k;G_3=GL(m,Z_(p~k)) if and only if r=0.(ⅳ) Ant_(G/ζG,ζG/G') G≌ Aut_(G/ζG,ζG/G') G■ Z_(p~k)~((m)),If m=0,then Ant_(G/ζG,ζG/G') G=Inn G≌Z_(p~k)~((2n));If m 0,then Ant_(G/ζG,ζG/G') G≌Z_(p~k)~((2nm))×Z_(p~(k-r))~((2n)),and Aut_(G/ζG,ζG) G/Inn G≌Z_(p~k)~((2n(m-1))× Z_(p~(k-r))~((2n)).     

向量组线性关系式的求法及应用

本给出n维向量中一组向量α1,α2,……，αm的线性关系式k1α1 k2α2 …… kmαm=0中系数k1,k2，……，km的求解方法。这一方法可用于求向量ζ在基ε1,ε2,……，εm下的坐标；两组基的过渡矩阵以及求解非齐次线性方程组的通解三个方面的应用。     

关于带重力项一维渗流方程解的自由边界的C^k正则性的一点注记

本文研究带重力项的一维渗流方程 u_t=(u~m)_(xx)+(u~n)_x,m>1,n>1Cauchy问题解的自由边界的正则性.正如我们所知,此退化方程解的显著特征是满足有限传播速度:当初值u_0(x)具有紧支集时,自由边界x=ζ_i(t),(i=1,2)是两条Lipschitz连续曲线.本文进一步研究指出:当n≥m对压力v=m/(m-1)u~(m-1)有ζ'_1(t)=-limv_x(x,t),t∈(0,∞),且对ζ_1(t)的任何移动部份Γ是C~1正则的;当n-1/m-1≥k,k为正整数,则微商(1≤2l+j≤k)在Γ的每一侧附近是有界的;特别当n-1/m-1=k,则任意阶微商(l≥0,j≥0)在Γ的每一侧附近有界,从而v在Γ的每一侧是C~∞的。 本文只考虑i=1的情形,至于i=2可类似地加以考虑。     

等差数列的一个充要条件

定理　数列 {an}为等差数列的充要条件为 :对任意整数 k,当 m 1≤ k≤ n - 1时 ,恒有等式 :( n - k) am ( k - m) an =( n - m) ak,其中 m,n∈ N且 n >m≥ 1 .证明　 (必要性 )设数列 {an}为等差数列 ,公差为 d,则　 an =am ( n - m) d,于是对任意正整数 m,n,k有　 ( n - k) am ( k - m) an= ( n - k) am ( k - m) [am ( n - m) d]= ( n - m) [am ( k - m) d]=( n - m) ak.由于正整数 m,n,k的任意性 ,故当 m 1≤ k≤ n - 1时 ,等式仍然成立 .(充分性 )若对任意正整数 k都有等式( n - k) am ( k - m) an =( n - m) ak,( 1…     

The maximum and minimum degrees of random bipartite multigraphs

In this paper the authors generalize the classic random bipartite graph model, and define a model of the random bipartite multigraphs as follows:let m = m（n） be a positive integer-valued function on n and ζ（n,m;{pk}） the probability space consisting of all the labeled bipartite multigraphs with two vertex sets A ={a₁,a₂,...,a_n} and B = {b₁,b₂,...,b_m}, in which the numbers t_ai,b_j of the edges between any two vertices a_i∈A and b_j∈ B are identically distributed independent random variables with distribution P{t_ai,b_j=k}=pk,k=0,1,2,...,where pk ≥0 and ∞Σk=0 pk=1. They obtain that X_c,d,A, the number of vertices in A with degree between c and d of G_n,m∈ζ（n, m;{pk}） has asymptotically Poisson distribution, and answer the following two questions about the space ζ（n,m;{pk}） with {pk} having geometric distribution, binomial distribution and Poisson distribution, respectively. Under which condition for {pk} can there be a function D（n） such that almost every random multigraph G_n,m∈ζ（n,m;{pk}） has maximum degree D（n）in A? under which condition for {pk} has almost every multigraph G（n,m）∈ζ（n,m;{pk}） a unique vertex of maximum degree in A?     

圆锥曲线直径的一个有趣性质

与圆的直径相仿,经过有心圆锥曲线中心的弦叫做圆锥曲线直径,经研究,它有如下一个有趣的统一性质:定理AB是经过圆锥曲线x2m+y2n=1(mn≠0,m,n不同时为负)中心的弦,P是圆锥曲线上异于A,B外的任意一点,PA,PB的斜率分别为k1,k2,则k1k2=-nm(当m=n>0时,圆锥曲线是圆;当m>0,n>0,m≠n时,圆锥曲线是椭圆;当m和n异号时,圆锥曲线是双曲线).     

一个新的素数筛选法

在初等教论中,历来只知道艾氏(Eratos-thenes)素数筛法。本文给出一种新的素数筛选程序,它依赖于如下命题。定理 (张文亮)2n 1为(奇)素数的充分必要条件是n≠(2k 1)m k(n,m,k∈N)。证明如果2n 1为合数,则必为二奇数之积,即有m,k∈N,使得2n 1=(2m 1)·(2k 1),则 n=(2k 1)m k反之,如果对某m,k∈N,使得n=(2k 1)m k,则 2n 1=2[(2k 1)m k] 1 =(2m 1)(2n 1)为合数,因此2n 1为(奇)素数的充要条件是:对任何m,k∈N,自然数n≠(2k 1)m k 定理表明,当n跑遍N={s|s≠(2k 1)·m k,s、m、k∈N}时,2n 1遍历奇素数集,     

A REPRESENTATION FORMULA RELATED TO SCHR(O)DINGER OPERATORS

Schr(o)dinger operator is a central subject in the mathematical study of quantum mechanics.Consider the Schrodinger operator H = -△ V on R, where △ = d2/dx2 and the potential function V is real valued. In Fourier analysis, it is well-known that a square integrable function admits an expansion with exponentials as eigenfunctions of -△. A natural conjecture is that an L2 function admits a similar expansion in terms of "eigenfunctions" of H, a perturbation of the Laplacian (see [7], Ch. Ⅺ and the notes), under certain condition on V.     

On a class of self-similar processes with stationary increments in higher order Wiener chaoses

We study a class of self-similar processes with stationary increments belonging to higher order Wiener chaoses which are similar to Hermite processes. We obtain an almost sure wavelet-like expansion of these processes. This allows us to compute the pointwise and local Hölder regularity of sample paths and to analyse their behaviour at infinity. We also provide some results on the Hausdorff dimension of the range and graphs of multidimensional anisotropic self-similar processes with stationary increments defined by multiple Wiener–Itô integrals.     

On a class of Riemann surfaces

It is considered the class of Riemann surfaces with dimT1 = 0, where T1 is a subclass of exact harmonic forms which is one of the factors in the orthogonal decomposition of the spaceΩH of harmonic forms of the surface, namely The surfaces in the class OHD and the class of planar surfaces satisfy dimT1 = 0. A.Pfluger posed the question whether there might exist other surfaces outside those two classes. Here it is shown that in the case of finite genus g, we should look for a surface S with dimT1 = 0 among the surfaces of the form Sg\K , where Sg is a closed surface of genus g and K a compact set of positive harmonic measure with perfect components and very irregular boundary.     

CONTENTS OF VOLUME 30

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Instructions for contributions

正Applied Mathematics-A Journal of Chinese Universities,Series B(Appl.Math.J.Chinese Univ.,Ser.B)is a comprehensive applied mathematics journal jointly sponsored by Zhejiang University,China Society for Industrial and Applied Mathematics,and Springer-Verlag.It is a quarterly journal with     

Journal of Mathematical Research with Applications Guide for Authors

正Journal overview:Journal of Mathematical Research with Applications(JMRA),formerly Journal of Mathematical Research and Exposition(JMRE)created in 1981,one of the transactions of China Society for Industrial and Applied Mathematics,is a home for original research papers of the highest quality in all areas of mathematics with applications.The target audience comprises:pure and applied mathematicians,graduate students in broad fields of sciences and technology,scientists and engineers interested in mathematics.     

Staircase transportation problems with superadditive rewards and cumulative capacities

A cumulative-capacitated transportation problem is studied. The supply nodes and demand nodes are each chains. Shipments from a supply node to a demand node are possible only if the pair lies in a sublattice, or equivalently, in a staircase disjoint union of rectangles, of the product of the two chains. There are (lattice) superadditive upper bounds on the cumulative flows in all leading subrectangles of each rectangle. It is shown that there is a greatest cumulative flow formed by the natural generalization of the South-West Corner Rule that respects cumulative-flow capacities; it has maximum reward when the rewards are (lattice) superadditive; it is integer if the supplies, demands and capacities are integer; and it can be calculated myopically in linear time. The result is specialized to earlier work of Hoeffding (1940), Fréchet (1951), Lorentz (1953), Hoffman (1963) and Barnes and Hoffman (1985). Applications are given to extreme constrained bivariate distributions, optimal distribution with limited one-way product substitution and, generalizing results of Derman and Klein (1958), optimal sales with age-dependent rewards and capacities.To our friend, Philip Wolfe, with admiration and affection, on the occasion of his 65th birthday.Research was supported respectively by the IBM T.J. Watson and IBM Almaden Research Centers and is a minor revision of the IBM Research Report [6].