Character Formulas for a Class of Simple Restricted Modules over the Simple Lie Superalgebras of Witt Type

Let F be an algebraically closed field of prime characteristic, and W(m, n, 1) be the simple restricted Lie superalgebra of Witt type over F, which is the Lie superalgebra of superderivations of the superalgebra ■(m; 1) ■∧(n), where ■(m; 1) is the truncated polynomial algebra with m indeterminants and ∧(n) is the Grassmann algebra with n indeterminants. In this paper, the author determines the character formulas for a class of simple restricted modules of W(m, n, 1) with atypical weights of type Ⅰ.     

On 3-Submanifolds of S^3 Which Admit Complete Spanning Curve Systems

Let M be a compact connected 3-submanifold of the 3-sphere S~3 with one boundary component F such that there exists a collection of n pairwise disjoint connected orientable surfaces S = {S_1, ···, S_n} properly embedded in M, ?S = {?S_1, ···, ?S_n}is a complete curve system on F. We call S a complete surface system for M, and ?S a complete spanning curve system for M. In the present paper, the authors show that the equivalent classes of complete spanning curve systems for M are unique, that is, any complete spanning curve system for M is equivalent to ?S. As an application of the result,it is shown that the image of the natural homomorphism from the mapping class group M(M) to M(F) is a subgroup of the handlebody subgroup Hn.     

A linear estimation to the number of zeros for Abelian integrals in a kind of quadratic reversible centers of genus one

In this paper, using the method of Picard-Fuchs equation and Riccati equation, we consider the number of zeros for Abelian integrals in a kind of quadratic reversible centers of genus one under arbitrary polynomial perturbations of degree $n$, and obtain that the upper bound of the number is $2\left[{(n+1)}/{2}\right]+$ $\left[{n}/{2}\right]+2$ ($n\geq 1$), which linearly depends on $n$.     

ON THE EQUATION □φ=｜↓△φ｜^2 IN FOUR SPACE DIMENSIONS

This paper considers the following Cauchy problem for semilinear wave equations in n space dimensions □φ=F(δφ),φ(0,x)=f(x),δtφ(0,x)=g(x),whte □=δt^2-△ is the wave operator,F is quadratic in δεφ with δ=(δt,δx1,…,δxn).The minimal value of s is determined such that the above Cauchy problem is locally wellposed in H^s.It turns out that for the general equation s must satisfy s>max(n/2,n+5/4).This is due to Ponce and Sideris (when n=3)and Tataru (when n≥5).The purpose of this paper is to supplement with a proof in the case n=2,4.     

ON THE EQUATION $\square\phi =|\nabla \phi |^2$ IN FOUR SPACE DIMENSIONS

This paper considers the following Cauchy problem for semilinear wave equations in $n$ space dimensions $$\align \square\p &=F(\partial\p ),\\p (0,x)&=f(x),\quad \partial_t\p (0,x)=g(x), \endalign$$ where $\square =\partial_t^2-\triangle$ is the wave operator, $F$ is quadratic in $\partial\p$ with $\partial =(\partial_t,\partial_{x_1},\cdots ,\partial_{x_n})$. The minimal value of $s$ is determined such that the above Cauchy problem is locally well-posed in $H^s$. It turns out that for the general equation $s$ must satisfy $$s>\max\Big(\frac{n}{2}, \frac{n+5}{4}\Big).$$ This is due to Ponce and Sideris (when $n=3$) and Tataru (when $n\ge 5$). The purpose of this paper is to supplement with a proof in the case $n=2,4$.     

有限伪反射群的相对不变式的 Poincar\''e级数

Let F be a field with characteristic 0, V = Fn the n-dimensional vector space over F and let G be a finite pseudo-reflection group which acts on V . Let χ ： G→ F＊ be a 1- dimensional representation of G. In this article we show that χ（g） = （detg）α（0 ≤ α ≤ r - 1）, where g ∈ G and r is the order of g. In addition, we characterize the relation between the relative invariants and the invariants of the group G, and then we use Molien’s Theorem of invariants to compute the Poincar′e series of relative invariants.     

Upper bounds for the associated number of zeros of Abelian integrals for two classes of quadratic reversible centers of genus one

In this paper, by using the method of Picard-Fuchs equation and Riccati equation, we study the upper bounds for the associated number of zeros of Abelian integrals for two classes of quadratic reversible centers of genus one under any polynomial perturbations of degree $n$, and obtain that their upper bounds are $3n-3$ ($n\geq 2$) and $18\left[\frac{n}{2}\right]+3\left[\frac{n-1}{2}\right]$ ($n\geq 4$) respectively, both of the two upper bounds linearly depend on $n$.     

Upper Bound of the Number of Zeros for Abelian Integrals in a Kind of Quadratic Reversible Centers of Genus One

By using the methods of Picard-Fuchs equation and Riccati equation, we study the upper bound of the number of zeros for Abelian integrals in a kind of quadratic reversible centers of genus one under polynomial perturbations of degree $n$. We obtain that the upper bound is $7[(n-3)/2]+5$ when $n\ge 5$, $8$ when $n=4$, $5$ when $n=3$, $4$ when $n=2$, and $0$ when $n=1$ or $n=0$, which linearly depends on $n$.     

有限局部环上对称矩阵的计数定理(I)

设$A_{n}(R)$是有限局部环$Z/p^{k}Z$上$n$阶对称矩阵的集合, 这里$n\geq 2$. $p$是大于$2$素数, $p\equiv1({\rm mod}4)$ 且$k>1$. 通过确定有限局部环$Z/p^{k}Z$上对称矩阵的标准型, 计算出$A_{n}(R)$在线性群${\rm GL}_{n}(R)$作用下的轨道数, 从而计算出由特定对称矩阵确定的正交群的阶以及与特定对称矩阵在同一轨道的对称矩阵的阶.     

关于图中存在不含给定k-因子的[a,b]-因子的度条件

设$1\leq a<b, 0\leq k$是整数. 设$G$是一个含有$k$-因子$Q$且阶为$|G|$的图. 设\delta(G)$表示$G$的最小度, 且$\delta(G)\geq a+k$. 如果$Q$连通, 设$\varepsilon=k$, 否则设$\varepsilon=k+1$.证明:当$b\geq a+\varepsilon-1$时, 如果对$G$的任意两个不相邻的点$x$和$y$都有max$\{d_G(x),d_G(y)\}\geq {\rm max}\{{{a|G|} \over {a+b}},{{(|G|+(a-1)(2a+b+\varepsilon-2))} \over {b+1}}\}+k$, 那么$G$有一个$[a, b]$-因子$F$ 使得 $E(F)\cap E(Q)=\emptyset$. 这个度条件是最佳的, 条件$b\geqa+\varepsilon-1$不能去掉. 进一步,得到图存在含给定$k$-因子的$[a, b]$-因子的度条件.     

Heegaaed分解有不交曲线性质的一个充分条件

In the paper,we give two conditions that the Heegaard splitting admits the disjoint cnrve property.The main result is that for a genus g(g≥2)strongly irreducible Heegaard splitting(C1,C2;F),let Di be an essential disk in Ci,i=1,2,satisfying(1)at least one of (の)D4 and (の)D2 is separating in F and |(の)D1 (∩)(の)D2|≤ 2g-1;or(2)both (の)D1 and (の)D2 are non-separating in F and |(の)D1 (∩)(の)D2|≤ 2g-2,then(C1,C2;F)has the disjoint curve property.     

Triangulated Structures Induced by Triangle Functors

Given a triangle functor F : A → B, the authors introduce the half image hIm F,which is an additive category closely related to F. If F is full or faithful, then hIm F admits a natural triangulated structure. However, in general, one can not expect that hIm F has a natural triangulated structure. The aim of this paper is to prove that hIm F admits a natural triangulated structure if and only if F satisfies the condition(SM). If this is the case, hIm F is triangle-equivalent to the Verdier quotient A/Ker F.     

有限维模李超代数$W(m,n,l,\underline{t})$的$\Theta$-型导子与导子超代数

本文总设$F$是$p>2$的域,我们在域$F$上构造了有限维模李超代数$W(m,n,l,\underline{t})$, 定义了$W(m,n,l,\underline{t})$的$\Theta$-型导子,进而确定了它导子超代数.     

矩阵环的半交换子环

称环R是半交换的,如果对任意a∈R,rR(a)是R的理想．若n≥2,则任意具有单位元的环R上的n阶上三角矩阵环不是半交换环．我们证明了reduced环上的上三角矩阵环的一类特殊子环是半交换环．     

Illuminating Disjoint Line Segments in the Plane

It is shown that a set of $n$ disjoint line segments in the plane can always be illuminated by $\lfloor (n+1)/2\rfloor$ light sources, improving an earlier bound of $\lfloor 2n/3\rfloor$ due to Czyzowicz et al. It is also shown that $\lfloor 4(n+1)/5 \rfloor$ light sources are always sufficient and sometimes necessary to illuminate the free space and both sides of $n$ disjoint line segments for every $n\geq 2$.     

关于同余式$\sigma(n)\equiv 1\pmod n$

Let k ≥ 2 be an integer, and let σ(n) denote the sum of the positive divisors of an integer n. We call n a quasi-multiperfect number if σ(n) = kn + 1. In this paper, we give some necessary properties of them.     

ON ONE CONJECTURE OF R. S. SINGH

In 1979 R. S. Singh(Ann. Statist, 1979, p. 890) made a conjecture concerning the convergence rate of EB estimates of the parameter θ in an one-dimensional continuous exponential distribution family, under the square loss function, the prior distribution family being confined to a bounded interval. The conjecture asserts that the rate cannot reach o(1/n) or even O(1/n). In this article, the weaker part of this conjecture(i. e. the o(1/n) part) is shown to be correct.     

LUSTERNIK-SCHNIRELMANN CATEGORY AND EMBEDDING FINITE COVERING MAPS, PRINCIPAL G-BUNDLES INTO BUNDLES

The author proves several embedding theorems for finite covering maps,principal G-bundies into bundles.The main results are 1. Let π：E→X be a finite covering map, and X a connected locally path-connected paracompact space. If cat X≤k, then the finite covering space π:E→X can be embedded into the trivial real k-plane bundle. 2. Let π：E→X be a principal G-bundle over a paracompact space. If there exists a linera action of Gon F(F=R or C)and cat X≤k ,then π：E→X can be embedded into ξ1 … ξn for any F-vector bundles ξi,i=1,…k.     

Dynamics of a Function Related to the Primes

Let n = p1p2 ··· pk, where pi(1 ≤ i ≤ k) are primes in the descending order and are not all equal. Let Ωk(n) = P(p1 + p2)P(p2 + p3) ··· P(pk-1+ pk)P(pk+ p1), where P(n) is the largest prime factor of n. Define w0(n) = n and wi(n) = w(wi-1(n)) for all integers i ≥ 1. The smallest integer s for which there exists a positive integer t such thatΩs k(n) = Ωs+t k(n) is called the index of periodicity of n. The authors investigate the index of periodicity of n.     

关于同余式$\sigma(n)\equiv 1\pmod n$, II

Let k ≥ 2 be an integer, and let a（n） denote the sum of the positive divisors of an integer n. We call n a quasi-multiperfect number if a（n） = kn ＋ 1. In this paper, we give some necessary properties of quasi-multiperfect numbers with four different prime divisors.