A Topological Criterion for the Existence of Half-Bound States

The following theorem is proved: if (M⁴ⁿ⁺¹,g) is a completeRiemannian manifold and M is an oriented hypersurface partitioningM and with non-zero signature, then the spectrum of the Hodge–deRhamLaplacian is [0,]. This result is obtained by a new Callias-typeindex. This new formula links half-bound harmonic forms (thatis, nearly L² but not in L²) with the signature of .     

A Criterion for the Existence of Transversals of Set Systems

Nash-Williams [6] formulated a condition that is necessary andsufficient for a countable family A=(A_i)_iI of sets to have atransversal. In [7] he proved that his criterion applies alsowhen we allow the set I to be arbitrary and require only that_iJA_i=Ø for any uncountable JI. In this paper, we formulateanother criterion of a similar nature, and prove that it isequivalent to the criterion of Nash-Williams for any familyu. We also present a self-contained proof that if _iJA_i=Øfor any uncountable JI, then our condition is necessary andsufficient for the family u to have a transversal.     

向量值切触有理插值存在性的判别方法

在用广义Vandermonde行列式给出Hermite插值多项式的表达式的基础上,针对a<,i>=2(i=1,2,…,s)的情形给出向量值切触有理插值存在性问题有解的条件及表达式.     

一元切触有理插值存在性的判别方法

在用广义Vandermonde行列式给出Hermite插值多项式的表达式的基础上,分别针对iα=2,iα=3(i=1,2,…,s)的情形给出切触有理插值问题有解的条件及解的表达式.     

A Criterion for the t-Vertex Condition of Graphs

Combining two concepts of regularity of graphs, namely k-isoregularity and the t-vertex condition, a generalization of a classical result by Hestenes and Higman is presented. As an application it is shown that two infinite series of graphs constructed by Brouwer, Ivanov, and Klin which are not rank 3 satisfy the 5-vertex condition.     

A Larger Family of Planar Graphs that Satisfy the Total Coloring Conjecture

The article shrinks the Δ = 6 hole that exists in the family of planar graphs which satisfy the total coloring conjecture. Let G be a planar graph. If ${v_n^k}$ represents the number of vertices of degree n which lie on k distinct 3-cycles, for ${n, k \in \mathbb{N}}$ , then the conjecture is true for planar graphs which satisfy ${v_5^4 +2(v_5^{5^+} +v_6^4) +3v_6^5 +4v_6^{6^+} < 24}$ .     

二元向量值有理插值存在性的一种判别方法

文章给出了对于矩形网格上基于二元Newton插值公式的二元向量值有理插值存在性的充要条件.在存在的情况下,建立了具有显式表达式的不同于向量连分式的二元向量值有理插值函数,并且这种方法具有承袭性.最后给出的实例说明了这种算法的有效性.     

[a,b]-因子存在性的范-型条件

设G是一个图,a,b是整数且满足0≤a≤b.如果存在G的一个支撑子图F,使对任意的x∈V(G)有a≤d_F(x)≤b,则称F是G的一个[a,b]-因子.本文给出图中具有特定性质的[a,b]-因子的范-型条件.进一步指出这个结果是最好的.     

[a,b]-因子存在性的邻集条件

本文给出图中具有特定性质的[a,b]-因子的邻集条件,并指出这个结果是最好的.     

Existence of disjoint weakly mixing operators that fail to satisfy the Disjoint Hypercyclicity Criterion

Recently, Bès, Martin, and Sanders [11] provided examples of disjoint hypercyclic operators which fail to satisfy the Disjoint Hypercyclicity Criterion. However, their operators also fail to be disjoint weakly mixing. We show that every separable, infinite dimensional Banach space admits operators _T1,_T2,…,_TN$T_1,T_2,\dots ,T_N$ with N?2$N?2$ which are disjoint weakly mixing, and still fail to satisfy the Disjoint Hypercyclicity Criterion, answering a question posed in [11]. Moreover, we provide examples of disjoint hypercyclic operators _T1$T_1$, _T2$T_2$ whose corresponding set of disjoint hypercyclic vectors is nowhere dense, answering another question posed in [11]. In fact, we explicitly describe their set of disjoint hypercyclic vectors. Those same disjoint hypercyclic operators fail to be disjoint topologically transitive. Lastly, we create examples of two families of d-hypercyclic operators which fail to have any d-hypercyclic vectors in common.     

Subgroups of a Direct Product of Groups that Satisfy the Strong Frattini Argument

We give criteria to characterize subgroups of a direct product of finite solvable groups that satisfy the strong Frattini argument.     

广义指数型二分性一个判别法和有界解存在性

该文给出线性系统(dx/dt)=A(t)x具有广义指数型二分性一个充分条件，作为应用研究非线性扰动系统的有界解的存在性. 推广了［1］与［2］结果.     

A Sufficient Condition for the Existence of Large Empty Convex Polygons

Abstract. Let P be a set of points in general position in the plane. We say that P is k -convex if no triangle determined by P contains more than k points of P in the interior. We say that a subset A of P in convex position forms an empty polygon (in P ) if no point of P \ A lies in the convex hull of A . We show that for any k,n there is an N=N(k,n) such that any k -convex set of at least N points in general position in the plane contains an empty n -gon. We also prove an analogous statement in R ^d for each odd d≥ 3 .     

A Sufficient Condition for the Existence of Large Empty Convex Polygons

   Abstract. Let P be a set of points in general position in the plane. We say that P is k -convex if no triangle determined by P contains more than k points of P in the interior. We say that a subset A of P in convex position forms an empty polygon (in P ) if no point of P \ A lies in the convex hull of A . We show that for any k,n there is an N=N(k,n) such that any k -convex set of at least N points in general position in the plane contains an empty n -gon. We also prove an analogous statement in R ^d for each odd d≥ 3 .     

A Necessary and Sufficient Condition for the Existence of a Heterochromatic Spanning Tree in a Graph

We prove the following theorem. An edge-colored (not necessary to be proper) connected graph G of order n has a heterochromatic spanning tree if and only if for any r colors (1≤r≤n−2), the removal of all the edges colored with these r colors from G results in a graph having at most r+1 components, where a heterochromatic spanning tree is a spanning tree whose edges have distinct colors.     

一类满足熵增条件的流体力学方程守恒型格式

Lax,Wandtoff曾经证明:对于与守恒律方程组相容的守恒型差分格式,如果其差分解几乎处处有界收敛,那么极限函数是原方程组的一个弱解,并且提出了二阶精度的L-W格式.但是,一些数值计算表明,用二阶守恒型格式(如L-W格式及Mac Corma-ck格式),可能得到非物理解的计算结果.通常称满足熵条件的弱解为物理解.对     

A Countable Structure Does Not Have a Free Uncountable Automorphism Group

It is proved that the automorphism group of a countable structurecannot be a free uncountable group. The idea is that insteadof proving that every countable set of equations of a certainform has a solution, it is proved that this holds for a co-meagrefamily of appropriate countable sets of equations. This researchwas partially supported by the Israel Science Foundation.     

Some Results on Systems of Finite Sets That Satisfy a Certain Intersection Condition

Let S be a finite set, and fix K>2. Let F be a family of subsets of S with the property that whenever A₁,...,A_k are sets in F, not necessarily distinct, and A₁ ? ? ? A_k = ?, then A₁ ? ? ? A_k = S. We prove here that the maximum size of such a family is 2^|S|?1 + 1. If we require that the sets A₁,...,A_k be distinct, then the maximum size of F is again 2^|S|?1 + 1, provided that |S| ≥ log₂(K?2)+3.