On the Helly Number for Line Transversals to Parallel Triangles

We prove two results about the problem of finding the Helly number for line transversals to a family of parallel triangles in the plane: (1) If each three triangles of a family of parallel right triangles are intersected by an ascending (or a descending) line, then there is an ascending (or a descending) line that intersects all     

Helly Type Problems forSome Special Convex Polygons

In the combinatorial geometry of convex sets the question of how efficiently a family ofconvex sets can be pierced by points has led to various problems which may be regarded asextensions of the Helly-type problems. A family of sets is said to be n-pierceable (abbreviatedas n) if there exists a set of n points such that each member of the family contains at leastone of them. A family of sets is said to be nk: if every subfamily of size k or less is n. Thefamous Helly theorem in combinatorial …     

SPECTRAL RADIUSES OF THE GALTON-WATSON BRANCHING PROCESSES

§1IntroductionSuppose that(X,Px)is a Markov chain on a countable(or finite)state space E.Givenany x,y∈E,we say that y can be reached from x and write x y,if there is n≥1suchthat Px(Xn=y)>0.If x y and y z,then x z.The markov chain X is said to beirreducible if any two states can be reached from each other.(See[1]or[2]).If X isirreducible,then there is a number r,with0≤r≤1,such that lim supn→∞[Px(Xn=y)]n1=r forany x,y∈E.The number r is called the spectral radius of X(refer to[3]).…     

Inverse Functions of Number-Theoretic Functions (I)

If gf(x) =x for every x, then g is called a left inverse function of f and f is a right inverse function of g. If f is both left and right inverse function of g, then f and g are said to be mutually inverse to each other. We show that (§ 1) the following results hold. A function f has a left inverse if and only if f is univalent, a function g has a right inverse if and only if g is exhaustive, i. e., g takes every (natural) number as values. Hence f has both left and right inverse if and only if f is both univalent and exhaustive, i. e., f is a permutation on the domain of natural numbers. Let g_1 and g_2 be two left inverse functions of the function f. If for every left inverse g of f, we have $g_1(x) \leq g(x) \leq g_2(x)$, then g_1(x) is called the weak, and g_2(x) is the strong, left inverse function of f. Similarly we define the weak and the strong right inverse functions. We show that(§ 2) every strict increasing function f must possess weak and strong left inverse functions, and all of its left inverse functions must be exhaustive slow increasing (a function g(x) is slow increasing if and only if g(Sx) —Sg(x) =0, here s denotes the successor function). On the other hand, every exhaustive function g must possess weak and strong right inverse functions, and all of its right inverse functions must strict increasing. We show also that (§ 3): If f_1(x) and f_2(x) both take g(x) as their strong (weak) left inverse, then f_1(x)=f_2(x)(f_1(Sx)=f_2(Sx)). If g_1(x) and g_2(x) both take f(x) as their strong or weak right inverse, then g_1(x)=g_2(x). From these results we see that we may find a function from its strong (weak) left or right inverse function. Let there be f(c) \leq x 相似文献   

有限群的$p$-覆盖远离子群及$S$-拟正规嵌入子群

Let G be a finite group, p the smallest prime dividing the order of G and P a Sylow p-subgroup of G. If d is the smallest generator number of P, then there exist maximal subgroups P1, P2,..., Pd of P, denoted by Md(P) = {P1,...,Pd}, such that di=1 Pi = Φ(P), the Frattini subgroup of P. In this paper, we will show that if each member of some fixed Md(P) is either p-cover-avoid or S-quasinormally embedded in G, then G is p-nilpotent. As applications, some further results are obtained.     

RESEARCH ANNOUNCEMENTS Helly Type Problems for Some Special Convex Polygons

In the combinatorial geometry of convex sets the question of how efficiently a family of convex sets can be pierced by points has led to various problems which may be regarded as extensions of the Helly-type problems. A family of sets is said to be n-pierceable (abbreviated as Пn) if there exists a set of n points such that each member of the family contains at least one of them. A family of sets is said to be Пnk if every subfamily of size k or less is Пn. The famous Helly theorem in combinatorial geometry asserts that for finite families of convex sets in the plane П13 implies П1. In a recent paper by M. Katchalski and D. Nashtir[a] the following conjecture of Griinbaum[2] was mentioned again:     

不含相邻三角形的平面图的无圈边着色

An acyclic edge coloring of a graph G is a proper edge coloring such that there are no bichromatic cycles.The acyclic edge chromatic number of a graph G is the minimum number k such that there exists an acyclic edge coloring using k colors and is denoted by χ’ a(G).In this paper we prove that χ ’ a(G) ≤(G) + 5 for planar graphs G without adjacent triangles.     

SRB measures for pointwise hyperbolic systems on open regions Dedicated to the Memory of Professor Shantao Liao

A diffeomorphism f:M→M is pointwise partially hyperbolic on an open invariant subset N?M if there is an invariant decomposition TNM=E~u⊕E~c⊕E~ssuch that Dxf is strictly expanding on ■ and contracting on ■ at each x∈N.We show that under certain conditions f has unstable and stable manifolds,and admits a finite or an infinite u-Gibbs measureμ.If f is pointwise hyperbolic on N,thenμis a SinaiRuelle-Bowen (SRB) measure or an infinite SRB measure.As applications,we show that some almost Anosov diffeomorphisms and gentle perturbations of Katok’s map have the properties.     

Hausdorff dimension of quadratic rational Julia sets

Suppose a quadratic rational map has a Siegel disk and a parabolic fixed point. If the rotation number of the Siegel disk is an irrational of bounded type, then the Julia set of the map is shallow. This implies that its Hausdorff dimension is strictly less than two.     

ON QUINTIC K-3 SURFACES

If a non-normal quintic surface is birational to a K3 surface,then there are threepossibilities:either it is singular along a conic;or it is singular along two mutuallyintersecting lines;or it is singular along a line and has an isolated triple point outside theline.Conversely if a K3 surface contains a hyperelliptic curve of genus three with anode or simple cusp,then it is birational to a quintic surface of the first type mentionedabove.For the other two cases,the minimal models are also characterized.