Bondy的泛圈图定理的改进

记G=(V,E)是简单图,1971年Bondy得到O re条件下的泛圈图的著名结果:若2连通n阶图G的不相邻的任两点x、y均有d(x) d(y)≥n,则G是泛圈图或G=Kn/2,n/2.这里进一步研究条件d(x) d(y)≥n-1,得到:若2连通n阶图G的不相邻的任两点x、y均有d(x) d(y)≥n-1,则G是泛圈图或G∈{K(Cn 1)/2∨G(n-1)/2,Kn/2,n/2}.本文作者得知最近国际著名权威专家Ho lton等人也得到完全相同的结果,但本证明更简捷.     

关于Win猜想的部分结果

<正> 本文假定G=(V,E)是2n个点的简单图,我们用C[U]表示点集U的导出子图,用d(x)表示G中点x的次,d_H(x)表示G的子图H中点x的次.其余符号见[3]. 给定非负整数k,若图G中每一对不相邻的顶点u和ν,都有d(u)+d(ν)≥2n+k,则称G为Ore k-型图.S.Win给出下述猜想: 若G是Ore k-型图,则G有k+2个1-因子.其中k≤2n-4.     

有向D—回路

G为有向图,μ是G的一个有向回路,如果G的每条弧至少有一端在μ上,就称μ为G的有向D-回路,本文主要结果为 定理1 设G为强连通有向1-图,n阶,(n≥7),无环,对于G的任一条弧(x,y),有 d~-(x) d~ (y)≥ n-3.那么G含有向D-回路. 定理2 设G为强连通有向1-图,n阶(n≥6),无环,对于G的任一条弧(x,y),有 d(x) d(y)≥2n-3.那么G含有向D-回路.     

汉米尔顿图泛圈性的奥尔型条件

本文所说的图是简单图,未定义的术语见[1,2].n 阶图 G,n≥3,若有长为 n 的圈,则说 G 是汉米尔顿图;若对每个 k,3≤k≤n,G 含有长为 k 的圈,则说 G 是泛圈图.定理1.在 n 阶图 G 中,若对任何点对 x,y∈V(G),xy(?)E(G),都有 d(x)+d(y)≥n,则 G 是汉米尔顿图.     

关于Veldman等人的一个猜想

本文证明了下列源于Veldman等人的一个猜想;设G是阶为n≥13的1—tough图且对所有独立集x,y,z有d(x) d(y) d(z)≥(3n-14)/2,则G是哈米顿的。     

[a,b]-对等图的范-型条件

既是[a,b]-覆盖又是[a,b]-消去的图称为[a,b]-对等图.设1≤aan+1a+b,则G为[a,b]-对等图.给出了一个图是[a,b]-对等图的关于范-型条件及邻域并的若干充分条件,并指出定理中的条件在一定意义上是最好可能的.     

有向S—回路和有向S—迹

本文给出了有向生成回路和有向生成迹的几个充分条件。 设G为强连通1—图,无环,n阶。对于G的任两不相邻顶点x,y。①如果 d(x)+d(y)≥2n+1。那么对于G的任两顶点u和v,或有u-v有向S-迹,或有v-u有向S-迹。②如果 d(x)+d(y)≥2n-3。那么G含有向S-回路。     

图的上可嵌入性与圈中顶点度

本文利用非上可嵌入图的充要条件,结合圈中顶点最大度与图的上可嵌入性之间的关系,得到了下两个结果:(1)设G是2-边连通简单图,若对G中任意圈G,存在点x∈C满足,d(x)>|V(G)|/3 1,则图G是上可嵌入的,且不等式的下界是不可达的.(2)设G={x,y;E}为简单二都图,且是2-边连通的. |x|=m,|Y|=n(m,n≥3),若对G中任意圈C,存在点x∈C且x∈X满足d(x)>n/3 1,则图G是上可嵌入的,且不等式的下界是不可达的.     

带弦圈的最小2宽直径

设k为正整数,G是简单k连通图.图G的k宽直径,dk(G),是指最小的整数ι使得对任意两不同顶点x,y∈V(G),都存在k条长至多为ι的内部不交的连接x和y的路.用C(n,t)表示在圈Gn上增加t条边所得的图.定义h(n,t):min{d2(C(n,t))}.本文给出了h(n,2)=[n/2].而且,给出了当t较大时h(n,t)的界.     

一类新的泛圈图

本文所说的图都是简单无向图。未定义的术语和记号参见[2]。设 G=(V,E)的 n 阶图(n≥3),若 G 中含有 Hamilton 圈,则称 G 是 H-图。若G 中含有从3到 n 的所有长度的圈,则称 G 为泛圈图。如下两个定理是众所周知的。定理1 (Ore,1960)。若在 n 阶图 G 中,有uv(?)E(G)(?)d(u) d(v)≥n,则 G 是 H-图。     

Two-interval Sturm-Liouville operators in modified Hilbert spaces

By modifying the inner product in the direct sum of the Hilbert spaces associated with each of two underlying intervals on which the Sturm-Liouville equation is defined, we generate self-adjoint realizations for boundary conditions with any real coupling matrix whose determinant is positive. This contrasts with the usual theory which requires the coupling matrix to have determinant one.     

On conditions for optimality of a class of nondifferentiable functions

The purpose of this paper is to derive first-order necessary conditions for optimality of a class of nondifferentiable functions. The first-order necessary conditions for optimality for the minimax function and thel ₁-function can be considered as special cases of the present method. Furthermore, the optimality conditions obtained are used to obtain threshold values for the controlling parameters of a class of exact penalty functions.     

Regularity Conditions for the Linear Separation of Sets

In this paper, we study the linear separation between a set and a convex cone. We introduce the concepts of regularity and total regularity of the separation with respect to a face of the cone and we give theorems characterizing them.     

一类泛函微分方程解的振动定理

给出了二阶泛函数微分方程 x"(t)+f(t,x(g(t,x(t)))=0 t≥ t_0 其中 f(t,u)(?)C([t_0,∞)×R,R),f(t,0)=0 和 g(t,v)(?)C([t_0,∞)×R,R),(?)(t,v)=∞的一切解均为振动的必要条件。     

Structural interpretation and derivation of necessary and sufficient conditions for delay moments in FIFO multiserver queues

Scheller-Wolf [12] established necessary and sufficient conditions for finite stationary delay moments in stable FIFO GI/GI/s queues that incorporate the interaction between service time distribution, traffic intensity (ρ) and the number of servers in the queue. These conditions can be used to show that when the service time has finite first but infinite αth moment, s slow servers can give lower delays than one fast server. In this paper, we derive an alternative derivation of these moment results: Both upper bounds, that serve as sufficient conditions, and lower bounds, that serve as necessary conditions are presented. In addition, we extend the class of service time distributions for which the necessary conditions are valid. Our new derivations provide a structural interpretation of the moment bounds, giving intuition into their origin: We show that FIFO GI/GI/s delay can be represented as the minimum of (s − k) i.i.d. GI/GI/1 delays, when ρ satisfies k < ρ < k+1. AMS Subject Classification 60K25     

The convection-diffusion equation for a finite domain with time varying boundaries

A solution is developed for a convection-diffusion equation describing chemical transport with sorption, decay, and production. The problem is formulated in a finite domain where the appropriate conservation law yields Robin conditions at the ends. When the input concentration is arbitrary, the problem is underdetermined because of an unknown exit concentration. We resolve this by defining the exit concentration as a solution to a similar diffusion equation which satisfies a Dirichlet condition at the left end of the half line. This problem does not appear to have been solved in the literature, and the resulting representation should be useful for problems of practical interest.

Authors of previous works on problems of this type have eliminated the unknown exit concentration by assuming a continuous concentration at the outflow boundary. This yields a well-posed problem by forcing a homogeneous Neumann exit, widely known as Danckwerts condition. We provide a solution to that problem and use it to produce an estimate which demonstrates that Danckwerts condition implies a zero concentration at the outflow boundary, even for a long flow domain and a large time.     

Heat content asymptotics with singular initial temperature distributions

We study the heat content asymptotics with either Dirichlet or Robin boundary conditions where the initial temperature exhibits radial blowup near the boundary. We show that there is a complete small-time asymptotic expansion and give explicit geometrical formulas for the first few terms in the expansion.     

Smooth and Nonsmooth Lipschitz Controls for a Class of Vector Differential Equations

This paper is devoted to the study of a class of control problems associated to a nonlinear second-order vector differential equation with pointwise state constraints. The control is realized via a function of the state. We extend the results of Akkouchi, Bounabat, and Goebel to vector differential equations and furthermore consider the more general case. Under proper conditions, we prove the existence of optimal controls in the class of Lipschitz functions and obtain an optimality condition which looks somehow like the Pontryagin maximum principle for a smooth optimal control function. For a nonsmooth optimal control function, we derive a suboptimality condition by means of the Ekeland variational principle.Communicated by M. J. BalasThis work was supported by 985 Project of Jilin University. The author thanks Professor Yong Li for valuable suggestions. He also thanks Professor M. J. Balas and the anonymous referees for their comments.     

Dynamic frictionless contact of a nonlinear beam with two stops

In this work, we consider mathematical and numerical approaches to a dynamic contact problem with a highly nonlinear beam, the so-called Gao beam. Its left end is rigidly attached to a supporting device, whereas the other end is constrained to move between two perfectly rigid stops. Thus, the Signorini contact conditions are imposed to its right end and are interpreted as a pair of complementarity conditions. We formulate a time discretization based on a truncated variational formulation. We prove the convergence of numerical trajectories and also derive a new form of energy balance. A fully discrete numerical scheme is implemented to present numerical results.