Affinely Regular Polygons as Extremals of Area Functionals

For any convex n-gon P we consider the polygons obtained by dropping a vertex or an edge of P. The area distance of P to such (n−1)-gons, divided by the area of P, is an affinely invariant functional on n-gons whose maximizers coincide with the affinely regular polygons. We provide a complete proof of this result. We extend these area functionals to planar convex bodies and we present connections with the affine isoperimetric inequality and parallel X-ray tomography.     

A Higman-Haemers Inequality for Thick Regular Near Polygons

In this note we will generalize the Higman-Haemers inequalities for generalized polygons to thick regular near polygons.     

序为(3,3)的正则拟多边形

利用距离正则图的特征值方法,得到如下结论:设Γ是一个有序对为(3,3)正则拟多边形,如果d=r+1,则cd≠1,2,3.     

The Regular Near Polygons of Order (s, 2)

In this note we classify the regular near polygons of order (s, 2).     

Twisted Groups and Locally Toroidal Regular Polytopes

In recent years, much work has been done on the classification of abstract regular polytopes by their local and global topological type. Abstract regular polytopes are combinatorial structures which generalize the well-known classical geometric regular polytopes and tessellations. In this context, the classical theory is concerned with those which are of globally or locally spherical type. In a sequence of papers, the authors have studied the corresponding classification of abstract regular polytopes which are globally or locally toroidal. Here, this investigation of locally toroidal regular polytopes is continued, with a particular emphasis on polytopes of ranks and . For large classes of such polytopes, their groups are explicitly identified using twisting operations on quotients of Coxeter groups. In particular, this leads to new classification results which complement those obtained elsewhere. The method is also applied to describe certain regular polytopes with small facets and vertex-figures.

     

正则树的双变量色多项式研究

最近Klaus Dohmen等人提出新的双变量色多项式概念,对此,本文提出—个—般性的减边公式.通过反复运用该公式,可以方便求得任何简单图的双变量色多项式.由此减边公式,研究了一些特殊图和多分支图的双变量色多项式公式.本文还研究了由互不相连的多个子图都与某个顶点相连而成的图的双变量色多项式计算的删点公式以及简单图的双变量色多项式系数和问题.进而,本文提出—个新概念—正则树.利用这个减边公式,研究了正则树的双变量色多项式计算公式和—些性质,以及正则树整子图的双变量色多项式公式及其有关性质.     

超拟群的模糊正则关系

Regular relations play in hyperquasigroups theory a role analogous to congruences in semigroup theory.The aim of this paper is to introduce the concept of regular relations on hyperquasigroups and to investigate some related properties.Further,the notion of fuzzy regular relations on hyperquasigroups is introduced and some characterizations are discussed.     

Lower bounds for coverings of pairs by large blocks

Letnkt be positive integers, andX—a set ofn elements. LetC(n, k, t) be the smallest integerm such that there existm k-tuples ofX B ₁ B ₂,...,B _m with the property that everyt-tuple ofX is contained in at least oneB _i . It is shown that in many cases the standard lower bound forC(n, k, 2) can be improved (k sufficiently large,n/k being fixed). Some exact values ofC(n, k, 2) are also obtained.     

Affinely Prime Dynamical Systems

This paper deals with representations of groups by "affine" automorphisms of compact,convex spaces,with special focus on "irreducible" representations:equivalently "minimal" actions.When the group in question is PSL(2,R),the authors exhibit a oneone correspondence between bounded harmonic functions on the upper half-plane and a certain class of irreducible representations.This analysis shows that,surprisingly,all these representations are equivalent.In fact,it is found that all irreducible affine representations of this group are equivalent.The key to this is a property called "linear Stone-Weierstrass"for group actions on compact spaces.If it holds for the "universal strongly proximal space"of the group (to be defined),then the induced action on the space of probability measures on this space is the unique irreducible affine representation of the group.     

Gluing Affine 2-Manifolds with Polygons

Affine structures on surfaces are constructed by gluing polygons. The geometry of the affine surface depends on the shape of the polygon(s) and the particular gluing transformations used. The affine version of the Poincaré fundamental polygon theorem expresses the fundamental group and holonomy of the surface in terms of the gluing data. The theorem may be used to construct all complete affine structures on the 2-torus. The space of inequivalent holonomy representations of such structures is homeomorphic to R2.     

PMS-代数中的正则理想

本文提出伪补 Ｍ Ｓ 代数（简称 Ｐ Ｍ Ｓ 代数）中正则理想、正则同余关系的概念,研究正则理想与核理想、０ 理想之间的关系,讨论正则同余关系的性质,得到若干结果     

具有正则*-断面的正则半群的结构

设S是一个正则半群,如果存在一个S的子半群S~*及上的一元运算*满足条件：(1)(?)x∈S,x~*∈S~*∩V(x)；(2)(?)x∈S~*,(x~*)~*=x；(3)(?)x,y∈S,(x~*y)~*=y~*x~(**),(xy~*)~*=y~(xx)x~*则称S~*是S的一个正则*_-断面．本文刻画了具有正则*_-断面的正则半群的结构。     

Circle Numbers of Regular Convex Polygons

The circle number function is extended here to regular convex polygons. To this end, the radius of the polygonal circle is defined as the Minkowski functional of the circumscribed polygonal disc, and the arc-length of the polygonal circle is measured in a generalized Minkowski space having the rotated polar body as the unit disc.     

Monic integer Chebyshev problem

We study the problem of minimizing the supremum norm by monic polynomials with integer coefficients. Let denote the monic polynomials of degree with integer coefficients. A monic integer Chebyshev polynomial satisfies

and the monic integer Chebyshev constant is then defined by

This is the obvious analogue of the more usual integer Chebyshev constant that has been much studied.

We compute for various sets, including all finite sets of rationals, and make the following conjecture, which we prove in many cases.

Conjecture. Suppose is an interval whose endpoints are consecutive Farey fractions. This is characterized by Then

This should be contrasted with the nonmonic integer Chebyshev constant case, where the only intervals for which the constant is exactly computed are intervals of length 4 or greater.

     

强正则图的一些性质

文[3]给出了强正则图的概念及有关性质,本文在此基地上利用图的谱性质,得到了强正则图的又一些性质。     

一个广义Markov不等式（英）

This paper gives a generalized Markov inequality dx for every polynomial P of degree at most n provided that f′ is con tinuous and strictly increasing on [0,∞ ), where ‖?‖ denotes the uniform and T_n,stands for the n-th Chebyshev polynomial of the first kind.     

第二类Chebyshev多项式系数的绝对值和的卷积

本文主要的目的是利用广义Fibonacci多项式的生成函数及其偏导数来研究第二类Cheby-shev多项式卷积的计算,并给出—个有趣的计算公式.     

具有拟理想正则*-断面的正则半群

本文提出了具有正则*-断面正则半群的概念，所给出的例子表明具有拟理想正则*-断面的正则半群类真包含了具有拟理想逆断面的正则半群类和正则*-半群类；最后刻画了具有拟理想正则*-断面的正则半群的结构．     

Cyclic Ostrom Spreads

We call a transitive permutation group G stable if every non-trivial orbit B, of a point stabilizerG _O , is such that some element of G-G _O leaves invariant B {O}. We characterize all finite affine planes of order n that admit a collineation group G that acts, on the points of , as a stable 3/2-transitive permutation group of rank n+2 . The planes obtained are precisely what we have called cyclic Ostrom planes; these are translation planes whose associated spreads are obtained from the Desarguesian spread , of order n=p^r , by replacing a partial subspread of by another consisting of GF(p) subspaces that are unions of kern orbits. All André spreads, as well as many other spreads, are examples of cyclic Ostrom spreads.     

A Note on Regular Near Polygons

Let be a regular near polygon of order (s,t) with s>1 and t3. Let d be the diameter of , and let r:= max{i(c_i,a_i,b_i)=(c₁,a₁,b₁)}. In this note we prove several inequalities for . In particular, we show that s is bounded from above by function in t if We also consider regular near polygons of order (s,3).This work was partly supported by the Grant-in-Aid for Scientific Research (No 14740072), the Ministry of Education, Culture, Sports, Science and Technology, JapanThis work was partly done when the author was at the Com²MaC center at the Pohang University of Science and Technology. He would like to thank the Com²MaC-KOSEF for its support