A RESULT ON WOODALL''''S CONJECTURE

The binding number of a simple graph G=(V,E),denoted by Bind(G),isdefined as Bind(G)-min{|г(X)|/|X|φ≠x V(G),г(X)≠V(G)}where г(X)={u|(u,v)∈E(G),v∈X}。 In [1] Woodall had shown that G contains a triangle if Bind (G)≥[1+5~(1/2)]/2≈     

The Largest Cardinals of Isomorphic Class of Topology-set-lattices on an Infinite Set

Let X be an infinite set,K={τ:τ is a topology on X},defineτ≌σ iff (f)(f is a lattice-isomorphism from τ to σ),for a given τ∈K we define τ={σ:σ∈K ＆σ≌τ}K(K)is an imcomparable class iff (τ∈K)(σ∈K)(τ≠→τ and σ are incomparable),M={K:K is an incomparable class}. Theorem 1 sup{|τ|:τ∈K}=max{|τ|:τ∈k}=|K|=2~2~[X]=exp(exp(|X|)). Theorem 2 sup{sup{|τ|:τ∈K}:KK ＆ K is an incomparable class}=sup     

图带宽和与其对偶超图带宽和的关系

设H=（E1,E2,…,Em）是集合X上的一个超图,一个1-1映射f:X→{1,2,…,|X|}称为H的一个标号,对H的任一标号f,BS（H,f）=∑(E∈H)max{|f(u)-f(v)|;u,v∈E}称为超图H的关于标号f的带宽和BS(H)=min{BS(H,f)|f是超图H的标号|}称为H的带宽和．论文研究图带宽和与其对偶超图的带宽和这两个参数间的关系．     

Asymptotic Property of Approximation to x~αsgnx by Newman Type Operators

The approximation of |x| by rational functions is a classical rational problem.This paper deals with the rational approximation of the function x~αsgnx, which equals |x| ifα=1.We construct a Newman type operator r_n(x) and show {|x~αsgnx-r_n(x)|}～Cn~(-(α/2)e-(2nα)~(1/2)), where C is a constant depending onα.     

空间情形下旋转体体积的计算

运用定积分中的元素法,给出了空间曲线绕空间直线旋转一周所成的旋转曲面与垂直于旋转轴的两个平面所围成的旋转体体积的计算公式:V=π(m2+n2+p2)23∫tt12{[p(y(t)-b)-n(z(t)-c)]2+[m(z(t)-c)-p(x(t)-a)]2+[n(x(t)-a)-m(y(t)-b)]2}m.x′(t)+n.y′(t)+p.z′(t)dt从而将平面图形的旋转体体积推广到了空间情形.     

ON INTERSECTIONS OF INDEPENDENT NONDEGENERATE DIFFUSION PROCESSES

Let X(1)= {X(1)(s), s ∈ R+ } and X(2)= {X(2)(t), t ∈ R+ } be two independent nondegenerate diusion processes with values in Rd. The existence and fractal dimension of intersections of the sample paths of X(1)and X(2)are studied. More generally, let E1, E2 ■(0, ∞) and F  Rd be Borel sets. A necessary condition and a suffcient condition for P{X(1)(E1) ∩ X(2)(E2) ∩ F = φ} 0 are proved in terms of the Bessel-Riesz type capacity and Hausdor measure of E1 ×E2 ×F in the metric space(R+ ×R+ ×Rd, ρ), where ρ is an unsymmetric metric defined in R+ × R+ × Rd. Under reasonable conditions, results resembling those of Browian motion are obtained.     

复有理型插值一致逼近函数及其导数

本文研究在单位圆周{|z| =1}上一致逼近函数f(z)及其导数,利用Hermite插值中的基函数建立复有理型插值,并证明它们在{|z| =1}上分别一致收敛于f(z)或f′(z) ,给出了收敛速度.     

Quasi-Normal Structure and Fixed Point of Nonlinear Mappings

Let (X,|| ||) be a Banach space. For $\Omega \subset X^*$ and $x\in X$ we introduce the following notations (p\geq 1 and n\in N) $|X|_{\Omega _p(n)}=sup{(\sum\limits_{f\in F} |f(x)|^p)^{1/p}:F \subset \Omega,|F|\leq n$ $|X|_{\Omega _\infty}=sup{|f(x)|:f\in \Omega}$ A convex subset E of X is said to have guasi-normal structure whenever there exists a norm 1 | on A which satisfies the following conditions; (i) E has norinal structure relative to the norm ||| |||. (ii) There exist $\Omega \subset X^*$, p\geq 1 and \theta \in (0,1], such that $|x|_{\Omega _p(2) \leq |||x||| \leq ||x||}$ for x\in E and |||x|||<||x|| implies $2^1/p |x|_\Omega_\infty \geq \theta ||x||+(1-\theta)|||x|||$ or (ii)' There exist \Omega \subset X^*,p\geq 1 and \alpha \in [1,4^1/p) such that for all x\in E, |x|_\Omega_\rho(4)\leq |||x|||,||x||=max{|||x|||,\alpha|x|_\Omega_\infty} and for any countable subset w of \Omega $sup{\sum\limits _{\delt\in w |f(x)|^p:x\in E}<+\infty$ We notice that a set with normal steucture must have quasi-normal structure and there exist sets without normal structure which quasi-normal structure. The main result of the present paper is as follows. Theorem. Let (X, || ||) be a Banach space, E a weak compact convex nonempty subset of X with quasi-normal structure. Let T be a mapping of E in to itself. If there exists a sequence {x_n} in any T-invariant convex subset of E such that $lim_{n\rightarrow \infty} ||x_n-x_n+1||=lim_{n\rightarrow \infty}||x_n-Tx_n||=0$ and $lim_{n\rightarrow \infty} ||y-x_n||=\delta(\bar co{x_n}),for y\in \bar co({c_n})$ limll2/-*?ll=3(coK}), for y€co({xa}), then the mapping T has a fixed point in E, In particular, if the mapping T satisfies $||Tx-Ty||\leq max{||x-y||,1/2(||x-Ty||+||y-Tx||)},for x,y\in E$ then the mapping T has a fixed point in E.     

重调和方程非平凡解的存在性

该文主要研究$R^N(N>4)$上重调和方程\begin{eqnarray*}\left\{\begin{array}{ll} \Delta^2 u+\lambda u=\overline{f}(x,u);\\ \lim\limits_{|x|\rightarrow\infty}u(x)=0;\\u\in{H^2}(R^N),\hspace{0.1cm}x\in{R^N } \end{array}\right.\end{eqnarray*}的非平凡解的存在性.为了便于研究,将方程转化为$R^N(N>4)$ 上带有扰动项的重调和方程\begin{eqnarray*}\left\{\begin{array}{ll} \Delta^2 u+\lambda u=f(u)+\varepsilon g(x,u);\\ \lim\limits_{|x|\rightarrow\infty}u(x)=0;\\u\in{H^2}(R^N),\hspace{0.1cm}x\in{R^N } .\end{array}\right.\end{eqnarray*}并运用扰动方法进行研究(其中$f(u)=\lim\limits_{|x|\longrightarrow \infty}\overline{f}(x,u),\varepsilon g(x,u)=\overline{f}(x,u)-f(u),\varepsilon$为任意小常数),证明了在适当条件下上述问题非平凡解的存在性.     

浅谈离散型随机变量的数学期望

当X为离散型随机变量时,如果X的取值是有限个,要求X的数学期量E(X),只要知道X的分布律就行了,但是在一些情况下,要求出X的分布律是非常困难和非常复杂的.有些时候,分布律求出来后,可按定义算出X的数学期望:E(X)一∑xipi.然而有时这个和比较难求.在以上两种情况下,我们可以利用数学期望的性质:E(X1+…+Xn)=E(X1)+…十E(Xn)把X分解为几个随机变量的和,而这几个随机变量的数学期望很容易求.一般当X表示的是与计数有关的随机变量时,大部分情形我们可以把它分解,并且是分解成0一1分布或两点分布的随机变量的和.下面通过几个例子来说明这种方法的应用.     

有限个互素因子链上幂GCD矩阵与幂LCM矩阵的行列式的整除性

设S={x1,x2,...,xn}是由n个不同的正整数组成的集合,并设a为正整数.如果一个n阶矩阵的第i行j列元素是S中元素xi和xj的最大公因子的a次幂(xi,xj)a,则称该矩阵为定义在S上的a次幂最大公因子(GCD)矩阵,用(Sa)表示;类似定义a次幂LCM矩阵[Sa].如果存在{1,2,...,n}上的一个置换σ使得xσ(1)|xσ(2)|···|xσ(n),则称S为一个因子链.如果存在正整数k,使得S=S1∪S2∪···∪Sk,其中每一个Si(1ik)均为一个因子链,并且对所有的1i=jk,Si中的每个元素与Sj中的每个元素互素,则称S由有限个互素因子链构成.本文中,设S由有限个互素的因子链构成,并且1∈S.我们首先给出幂GCD矩阵与幂LCM矩阵的行列式的公式,然后证明:如果a|b,则det(Sa)|det(Sb),det[Sa]|det[Sb],det(Sa)|det[Sb].最后我们指出:如果构成S的有限个因子链不互素,则此结论一般不成立.     

用泛函观点看LEBESGUE积分

从泛函分析观点来看Lebesgue积分,使得Lebesgue积分可以用泛函分析最简单最基本的方法独立导出．基本做法是将Riemann对于区间[0,1]上的连续函数的积分看成连续函数空间C[0,1]上的连续线性泛函,再将它“自然”延拓到C[0,1]在积分范数意义下的完备化空间,而这个完备化空间正是Lebesgue可积函数空间L1[0,1]．     

没有4-圈的平面图的BB-染色

设H为G的一个生成子图,(G,H)的一个BB-k染色是指一个映射f:V(G)→{1,2…,k},满足以下两条:(i)|f(u)-f(u)|≥1,uu∈E(G)\E(H).(ii)|f(u)-f(u)|≥2,uv∈E(H).定义(G,H)的BB-色数xb(G,H)为最小的整数k,使得(G,H)是BB-k可染的.本文证明了...     

折叠立方体图的邻点可区别全色数

简单图G的全染色是指对G的点和边都进行染色．称全染色为正常的如果没有相邻或关联元素染同一种颜色．简单图G=（VE）的正常全染色^称为它的邻点可区别全染色如果对任意两个相邻顶点u、v,有H（u）≠H（v）,其中H（u）={（u））U{^（uw）｜uw∈E（G））而H（v）={h（u）}U{h（vx）｜vx∈E（G））．G...     

路和圈多重联图的邻点可区别E-全染色

设G（V,E）是一个简单图,k是一个正整数,f是一个V（G）∪E（G）到{1,2,...,k}的映射.如果u,v∈E（G）,则f（u）=f（v）,f（u）=f（uv）,f（v）=f（uv）,C（u）=C（v）,其中C（u）={f（u）}∪{f（uv）｜uv∈E（G）}.称f是图G的邻点可区别E-全染色,称最小的数k为图G的邻点可区别E-全色数.讨论了路和圈的多重联图的邻点可区别E-全色数。     

单圈图的伴随多项式的极小根（英文）

引入伴随多项式是为了从补图的角度研究色多形式,图的伴随多项式的极小根可用于判定色等价图.β（G）表示图G的伴随多项式的极小根.n表示n个顶点的单圈图的集合.分别确定了具有max{β（G）｜G∈Ωn}和min{β（G）｜G∈Ωn}的所有单圈图.     

Morse指数与含Grushin算子的Liouville型定理

本文研究退化椭圆型方程-Δxu-(α+1)2|x|~(2α)Δyu=|u|~(p-1)u,(x,y)∈Rm×Rk和方程-Δxu-(α+1)2|x|~(2α)Δyu=|u|~(p-1)u,(x,y)∈Π的Liouville型定理,其中-Δx-(α+1)2|x|~(2α)Δy是Grushin算子,Π={(x,y)∈Rm×Rk:x10}或{(x,y)∈Rm×Rk:y10}.本文将证明,当1p(Q+2)/(Q-2)时,上述方程Morse指数有限的有界解只有零解,其中Q=m+(α+1)k为齐次空间的维数,因此,本文将Laplace方程的结果推广到含Grushin算子的方程.     

树的Hk-cordial性

如果可以给图G的边用集合（±1,±2,.. ,±k）中的元素标号,使得对G每个顶点u,其标号,即所有与其相邻的边的标号之和,都落在集合（±1,±2,.. ,±k）中,且Ie（i）-e（-i）I≤1和lu（i）-u（-i）1≤1,其中t心）和e（i）（1≤i≤k）分别是标号为i的顶点数和边数,那么就称该图G为Hk-cordial的．本文证明了除了尥以外,每棵树都是H3-cordial的．     

Compact and Loeb Hausdorff spaces in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {ZF}$\end{document} and the axiom of choice for families of finite sets

Given a set X, $\mathsf {AC}^{\mathrm{fin}(X)}$ denotes the statement: “$[X]^{<\omega }\backslash \lbrace \varnothing \rbrace$ has a choice set” and $\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )$ denotes the family of all closed subsets of the topological space $\mathbf {2}^{X}$ whose definition depends on a finite subset of X. We study the interrelations between the statements $\mathsf {AC}^{\mathrm{fin}(X)},$ $\mathsf {AC}^{\mathrm{fin}([X]^{<\omega })},$ $\mathsf {AC}^{\mathrm{fin} (F_{n}(X,2))},$ $\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$ and “$\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$has a choice set”. We show:

• (i) $\mathsf {AC}^{\mathrm{fin}(X)}$ iff $\mathsf {AC}^{\mathrm{fin}([X]^{<\omega } )}$ iff $\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$ has a choice set iff $\mathsf {AC}^{\mathrm{fin}(F_{n}(X,2))}$.
• (ii) $\mathsf {AC}_{\mathrm{fin}}$ ($\mathsf {AC}$ restricted to families of finite sets) iff for every set X, $\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$ has a choice set.
• (iii) $\mathsf {AC}_{\mathrm{fin}}$ does not imply “$\mathcal {K}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$ has a choice set($\mathcal {K}(\mathbf {X})$ is the family of all closed subsets of the space $\mathbf {X}$)
• (iv) $\mathcal {K}(\mathbf {2}^{X})\backslash \lbrace \varnothing \rbrace$ implies $\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$ but $\mathsf {AC}^{\mathrm{fin}(X)}$ does not imply $\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$.

We also show that “For every setX, “$\mathcal {K}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$has a choice set” iff “for every setX, $\mathcal {K}\big (\mathbf {[0,1]}^{X}\big )\backslash \lbrace \varnothing \rbrace$has a choice set” iff “for every product$\mathbf {X}$of finite discrete spaces,$\mathcal {K}(\mathbf {X})\backslash \lbrace \varnothing \rbrace$ has a choice set”.     

重模剩余类环的矩阵表示

对模m的剩余类环Zm上的多项式环Zm[x]中的任一n次（n≥1）首一多项式P,给出了重模剩余类环Zm[x]/（P）到Zm上的n阶全矩阵环Mn（Zm）的一类单同态,从而实现了Zm[x]/（P）的矩阵表示.若A为P的友矩阵,则Zm[x]/（P）的矩阵表示为{an-1An-1＋…＋a1A＋a0E｜ai∈Zm,0≤i≤n-1}...