一类用仿射几何构造的半对称图

图X是一个有限简单无向图,如果图X是正则的且边传递但非点传递,则称X是半对称图.主要利用仿射几何构造了一类2p~n阶连通p~4度的半对称图的无限族,其中p≥n≥11.     

利用有限仿射几何构作Cartesian认证码

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谈仿射性质在初等几何中的应用

利用高等几何中的仿射性质和平行投影方法,可解决大量初等几何问题,平行投影是比较容易理解也比较直观的一种投影方法,在初等几何中也适用。适当运用这种方法,可以在解决问题时带来事半功倍的效果,通过对三角形、平行四边形、椭圆的相关命题的证明,实例说明高等几何对初等几何的指导作用。     

仿射几何在轴测投影中的应用

本文利用仿射几何学建立起一套仿射理论，进而阐明轴测投的数学原理。     

齐次函数的一个仿射球定理

本文研究了相关齐次函数的仿射球定理.利用Hopf极大值原理,对任意给定的带凹性条件的初等对称曲率问题,获得了此类仿射球定理.特别地,这也给出了Deicke齐次函数定理的一个新证明.     

Clifford 代数,几何计算和几何推理

Clifford代数是一种深深根植于几何学之中的代数系统,被它的创始人称为几何代数．历史上,E．Cartan,R．Brauer,H．Weyl,C．Chevalley等数学大师都曾研究和应用过Clifford代数,对它的发展起了重要作用．近年来,Clifford代数在微分几何、理论物理、经典分析等方面取得了辉煌的成就,是现代理论数学和物理的一个核心工具,并在现代科技的各个领域,如机器人学、信号处理、计算机视觉、计算生物学、量子计算等方面有广泛的应用．本文主要介绍Clifford代数在几何计算和几何推理中的应用．作为一种优秀的描述和计算几何问题的代数语言,Clifford代数对于几何体,几何关系和几何变换有不依赖于坐标的、易于计算的多种表示,因而应用它进行几何自动推理,不仅使困难定理的证明往往变得极为简单,而且能够解决一些著名的公开问题,目前在国际上,几何自动推理已经成为Clifford代数的一个重要应用领域。     

反比例函数问题的几何直观

对于几何题,人们有画图的习惯;而对于非几何题,人们往往不会从几何直观人手去思考问题的解决方法．图形可以使我们对已知条件与结论之间的关系有更明确、更形象的了解,使问题的解决更加简单明了．函数不仅仅和方程、不等式等代数内容联系紧密;同时借助于平面直角坐标系,也和三角形、四边形建立起了紧密的联系．而反比例函数中k的几何意义具有非常好的几何直观,由此展开的几何联想也就愈加丰富了．笔者将从k的几何意义出发,探索反比例函数问题中的几何直观,并从几何直观去寻求问题解决的思路．     

广义η函数类(θ)与仿射代数的String函数

本文先刻划了广η对函数类（θ），并以Ｃ为例，给出了在（θ）中确定Ｓｔｒｉｎｇ函数的一种自然方法．     

仿射分形插值函数与纵向尺度因子

文中对一类仿射分形插值函数,用纵向尺度因子刻画了函数在特定值域分布的充分必要条件.     

射影几何对初等几何指导一例

１　引言射影几何是高等师范院校数学专业必修的一门基础课,开设此课的目的之一,是因它对初等几何有着广泛的指导作用．因此,尝试用射影几何的知识去解决初等几何的问题,倘能行通,再着力用射影的观点将原题推广,以求得出更为普遍的新命题,……．这就为改进和提高几何课的教学质量提供一条途径．２　一道赛题由香港数理教育学会主办的１９９８年初中数学竞赛,加试的三道解答题中第二题是如下一道平面几何题［１］．图１ＡＤＢＭＯＱＰＮＣ已知Ｐ为ＡＢＣＤ内一点,Ｏ为ＡＣ与ＢＣ的交点,Ｍ、Ｎ分别为ＰＢ、ＰＣ的中点,Ｑ为ＡＮ与…     

On Sylow Subgroups of Abelian Affine Difference Sets

An n-subsetD of a group G of order is called an affine difference set of G relativeto a normal subgroup N of G of order if the list of differences containseach element of G-N exactly once and no elementof N. It is a well-known conjecture that if Dis an affine difference set in an abelian group G,then for every prime p, the Sylow p-subgroupof G is cyclic. In Arasu and Pott [1], it was shownthat the above conjecture is true when . In thispaper we give some conditions under which the Sylow p-subgroupof G is cyclic.     

Planar Functions and Planes of Lenz-Barlotti Class II

Planar functions were introduced by Dembowski and Ostrom [4] to describe projective planes possessing a collineation group with particular properties. Several classes of planar functions over a finite field are described, including a class whose associated affine planes are not translation planes or dual translation planes. This resolves in the negative a question posed in [4]. These planar functions define at least one such affine plane of order 3^e for every e 4 and their projective closures are of Lenz-Barlotti type II. All previously known planes of type II are obtained by derivation or lifting. At least when e is odd, the planes described here cannot be obtained in this manner.     

Affine geometries and sets of equiorthogonal frequency hypercubes

Equiorthogonal frequency hypercubes are one particular generalization of orthogonal latin squares. A complete set of mutually equiorthogonal frequency hypercubes (MEFH) of order n and dimension d, using m distinct symbols, has (n − 1)^d/(m − 1) hypercubes. In this article, we prove that an affine geometry of dimension dh over 𝔽_m can always be used to construct a complete set of MEFH of order m^h and dimension d, using m distinct symbols. We also provide necessary and sufficient conditions for a complete set of MEFH to be equivalent to an affine geometry. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 435–441, 2000     

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.     

Affine Planes and Flag Linear Spaces

This paper deals with the construction of the flag linear space determined by any affine plane.     

On Embeddings of the Flag Geometries of Projective Planes in Finite Projective Spaces

The flag geometry =( ) of a finite projective plane of order s is the generalized hexagon of order (s, 1) obtained from by putting equal to the set of all flags of , by putting equal to the set of all points and lines of and where I is the natural incidence relation (inverse containment), i.e., is the dual of the double of in the sense of Van Maldeghem Mal:98. Then we say that is fully and weakly embedded in the finite projective space PG(d, q) if is a subgeometry of the natural point-line geometry associated with PG(d, q), if s = q, if the set of points of generates PG(d, q), and if the set of points of not opposite any given point of does not generate PG(d, q). Preparing the classification of all such embeddings, we construct in this paper the classical examples, prove some generalities and show that the dimension d of the projective space belongs to {6,7,8}.     

Classification of Embeddings of the Flag Geometries of Projective Planes in Finite Projective Spaces, Part 2

The flag geometry Γ=( ,  , I) of a finite projective plane Π of order s is the generalized hexagon of order (s, 1) obtained from Π by putting equal to the set of all flags of Π, by putting equal to the set of all points and lines of Π, and where I is the natural incidence relation (inverse containment), i.e., Γ is the dual of the double of Π in the sense of H. Van Maldeghem (1998, “Generalized Polygons,” Birkhäuser Verlag, Basel). Then we say that Γ is fully and weakly embedded in the finite projective space PG(d, q) if Γ is a subgeometry of the natural point-line geometry associated with PG(d, q), if s=q, if the set of points of Γ generates PG(d, q), and if the set of points of Γ not opposite any given point of Γ does not generate PG(d, q). In two earlier papers we have shown that the dimension d of the projective space belongs to {6, 7, 8}, that the projective plane Π is Desarguesian, and we have classified the full and weak embeddings of Γ (Γ as above) in the case that there are two opposite lines L, M of Γ with the property that the subspace U_L, M of PG(d, q) generated by all lines of Γ meeting either L or M has dimension 6 (which is automatically satisfied if d=6). In the present paper, we partly handle the case d=7; more precisely, we consider for d=7 the case where for all pairs (L, M) of opposite lines of Γ, the subspace U_L, M has dimension 7 and where there exist four lines concurrent with L contained in a 4-dimensional subspace of PG(7, q).     

Affine ovoids and extensions of generalized quadrangles

A set Δ of vertices of a generalized quadrangle of order (s, t) is said to be a hyperoval if any line intersects Δ in either 0, or 2 points. A hyperoval Δ is called an affine ovoid if |Δ|=2st. It is well known that μ-subgraphs in triangular extensions of generalized quadrangles are hyperovals. In the present paper we prove that ifS is a triangular extension forGQ(s, t) with totally regular point graph Γ such that μ=2st, thens is even, Γ is an τ-antipodal graph of diameter 3 with τ=1+s/2, and eithers=2, ort=s+2. Translated fromMatematicheskie Zametki, Vol. 68, No. 2, pp. 266–271, August, 2000.