等周问题的一个初等证明

本文把欧氏平面，半球面和非欧面之中，不含给定边界，含有给定边界和含有边界而且在其上给定端点这样三种等周问题、给以初等、统一的证明。其要点在于把它们的存在性和唯一性简明扼要地归结到下述初等引理，即一个给定凹边边长的四边形的面积以四顶共圆时为其唯一的极大     

Representation of convex geometries by circles on the plane

Convex geometries are closure systems satisfying the anti-exchange axiom. Every finite convex geometry can be embedded into a convex geometry of finitely many points in an $n$-dimensional space equipped with a convex hull operator, by the result of Kashiwabara et al. (2005). Allowing circles rather than points, as was suggested by Czédli (2014), may presumably reduce the dimension for representation. This paper introduces a property, the Weak 2 × 3-Carousel rule, which is satisfied by all convex geometries of circles on the plane, and we show that it does not hold in all finite convex geometries. This raises a number of representation problems for convex geometries, which may allow us to better understand the properties of Euclidean space related to its dimension.     

Transitive bislim geometries of gonality 3, Part II: The group theoretic cases

We consider point-line geometries having three points on every line, having three lines through every point (bislim geometries), and containing triangles. We classify such geometries under the hypothesis of the existence of a collineation group acting transitively on the point set.     

Berührstrukturen, die keine Kettenräume sind

Using sets of automorphisms of chain geometries over commutative algebras fulfilling certain transitivity conditions, a construction of touching structures is given, which are no chain spaces.     

Association schemes and hypergroups

In this paper, we investigate hypergroups which arise from association schemes in a canonical way; this class of hypergroups is called realizable. We first study basic algebraic properties of realizable hypergroups. Then we prove that two interesting classes of hypergroups (partition hypergroups and linearly ordered hypergroups) are realizable. Along the way, we prove that a certain class of projective geometries is equipped with a canonical association scheme structure which allows us to link three objects; association schemes, hypergroups, and projective geometries (see, Section 1.2 for details).     

Geometries of the Group PSL(2,11)

We determine all residually weakly primitive flag-transitive geometries for the groups PSL(2,11) and PGL(2,11). For the first of these we prove the existence by simple constructions while uniqueness, namely the fact that the lists are complete, relies on MAGMA programs. A central role is played by the subgroups Alt(5) in PSL(2,11). The highest rank of a geometry in our lists is four. Our work is related to various atlases of coset geometries.     

Induced layered clusters,hereditary mappings,and convex geometries

A method for structural clustering proposed by the authors is extended to the case when there are externally defined restrictions on the relations between sets and their elements. This framework appears to be related to order-theoretic concepts of the hereditary mappings and convex geometries, which enables us to give characterizations of those in terms of the monotone linkage functions.     

Geometrical constructions of class‐uniformly resolvable structure

We use arcs, ovals, and hyperovals to construct class‐uniformly resolvable structures. Many of the structures come from finite geometries, but we also use arcs from non‐geometric designs. Most of the class‐uniformly resolvable structures constructed here have block size sets that have not been constructed before. We construct CURDs with a variety of block sizes, including many with block sizes 2 and 4. In addition, these constructions give the first systematic way of constructing infinite families of CURDs with three block sizes. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:329‐344, 2011     

Categories of Projective Geometries with Morphisms and Homomorphisms

Projective geometries studied as Pasch geometries possess morphisms and homomorphisms. A homomorphic image of a projective geometry is shown to be projective. A projective geometry is shown to be Desarguesian iff it is a homomorphic image of a higher dimensional one, which in a sense is dual to the classical imbedding theorem. Semi-linear maps induce morphisms which are homomorphisms iff the associated homomorphisms of skewfields are isomorphisms. Projective geometries form categories with morphisms as well as homomorphisms and Desarguesian ones form a subcategory with Desarguesian homomorphisms.     

On the Generation of Some Embeddable GF(2) Geometries

The generating rank is determined for several GF(2)-embeddable geometries and it is demonstrated that their generating and embedding ranks are equal. Specifically, we prove that each of the two generalized hexagons of order (2, 2) has generating rank 14, that the central involution geometry of the Hall-Janko sporadic group has generating rank 28, and that the dual polar space DU(6,2) has generating rank 22. We also include a survey of all instances in which either the generating or embedding rank of an embeddable GF(2) geometry is known.     

Fermat Dynamics, Matrix Arithmetics, Finite Circles, and Finite Lobachevsky Planes

Congruences generalizing Fermat's little theorem are proved for the traces of powers of integer matrices. Their relations to Lobachevsky geometries over finite fields and combinatorics of the matrix squaring operation as well as to the corresponding Riemann surfaces with their Kepler cubes are discussed.     

Non-rigid parabolic geometries of Monge type

In this paper we study a novel class of parabolic geometries which we call parabolic geometries of Monge type. These parabolic geometries are defined by gradings such that their −1 component contains a nonzero co-dimension 1 abelian subspace whose bracket with its complement is non-degenerate. We completely classify the simple Lie algebras with such gradings in terms of elementary properties of the defining set of simple roots. In addition we characterize those parabolic geometries of Monge type which are non-rigid in the sense that they have nonzero harmonic curvatures in positive weights. Standard models of all non-rigid parabolic geometries of Monge type are described by under-determined ODE systems. The full symmetry algebras for these under-determined ODE systems are explicitly calculated; surprisingly, these symmetries are all just prolonged point symmetries.     

On Scattered Convex Geometries

A convex geometry is a closure space satisfying the anti-exchange axiom. For several types of algebraic convex geometries we describe when the collection of closed sets is order scattered, in terms of obstructions to the semilattice of compact elements. In particular, a semilattice Ω(η), that does not appear among minimal obstructions to order-scattered algebraic modular lattices, plays a prominent role in convex geometries case. The connection to topological scatteredness is established in convex geometries of relatively convex sets.     

Two-Weight Codes, Partial Geometries and Steiner Systems

Two-weight codes and projectivesets having two intersection sizes with hyperplanes are equivalentobjects and they define strongly regular graphs. We construct projective sets in PG(2m – 1,q) that have the sameintersection numbers with hyperplanes as the hyperbolic quadricQ⁺(2m – 1,q). We investigate these sets; we provethat if q = 2 the corresponding strongly regular graphsare switching equivalent and that they contain subconstituentsthat are point graphs of partial geometries. If m = 4the partial geometries have parameters s = 7, t = 8, = 4 and some of them are embeddable in Steinersystems S(2,8,120).     

USING FINITE GEOMETRIES TO CONSTRUCT 3-PBIB(2) DESIGNS AND 3-DESIGNS

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On proper secrets, (<Emphasis Type="Italic">t</Emphasis>, <Emphasis Type="Italic">k</Emphasis>)-bases and linear codes

This paper contains three parts where each part triggered and motivated the subsequent one. In the first part (Proper Secrets) we study the Shamir’s “k-out-of-n” threshold secret sharing scheme. In that scheme, the dealer generates a random polynomial of degree k−1 whose free coefficient is the secret and the private shares are point values of that polynomial. We show that the secret may, equivalently, be chosen as any other point value of the polynomial (including the point at infinity), but, on the other hand, setting the secret to be any other linear combination of the polynomial coefficients may result in an imperfect scheme. In the second part ((t, k)-bases) we define, for every pair of integers t and k such that 1 ≤ t ≤ k−1, the concepts of (t, k)-spanning sets, (t, k)-independent sets and (t, k)-bases as generalizations of the usual concepts of spanning sets, independent sets and bases in a finite-dimensional vector space. We study the relations between those notions and derive upper and lower bounds for the size of such sets. In the third part (Linear Codes) we show the relations between those notions and linear codes. Our main notion of a (t, k)-base bridges between two well-known structures: (1, k)-bases are just projective geometries, while (k−1, k)-bases correspond to maximal MDS-codes. We show how the properties of (t, k)-independence and (t, k)-spanning relate to the notions of minimum distance and covering radius of linear codes and how our results regarding the size of such sets relate to known bounds in coding theory. We conclude by comparing between the notions that we introduce here and some well known objects from projective geometry.      

三角几何的基本群(英)

利用Building理论获得一种计算某些三角几何的基本群的新的方法．这种方法能够容易地计算出无限多有限三角几何的基拓扑基本群．