A Factorization Theorem of Characteristic Polynomials of Convex Geometries

A convex geometry is a closure system whose closure operator satisfies the anti-exchange property. As is described in Sagan’s survey paper, characteristic polynomials factorize over nonnegative integers in several situations. We show that the characteristic polynomial of a 2-tight convex geometry K factorizes over nonnegative integers if the clique complex of the nbc-graph of K is pure and strongly connected. This factorization theorem is new in the sense that it does not belong to any of the three categories mentioned in Sagan’s survey. Received September 25, 2005     

Applications of Finite Fields to Combinatorics and Finite Geometries

We survey some applications of finite fields to finite geometries in part A and to combinatorics and error-correcting codes in parts B and C.     

Morphisms of projective geometries and semilinear maps

Morphisms between projective geometries are introduced; they are partially defined maps satisfying natural geometric conditions. It is shown that in the arguesian case the morphisms are exactly those maps which in terms of homogeneous coordinates are described by semilinear maps. If one restricts the considerations to automorphisms (collineations) one recovers the so-called fundamental theorem of projective geometry, cf. Theorem 2.26 in [2].Supported by a grant from the Fonds National Suisse de la Recherche Scientifique.     

The co-existence of elations and homologies of prime order

N. Krier and J. C. D. S. Yaqub have proved that if a projective plane admits an involutory homology and an involutory elation, then does not belong to the Lenz-Barlotti class I₁, and belongs to the class I₂. In this paper, we find the classification of projective planes having a homology of orderp and an elation of orderq, wherep andq are primes.This is based on a part of the doctoral dissertation of A. Solai Raju. The work was supported by a Senior Research Fellowship of the CSIR, India.     

Metric and Periodic Lines in Real Inner Product Space Geometries

The lines of euclidean and hyperbolic geometries are characterized as metric lines in the sense of Blumenthal–Menger, and the lines of spherical and elliptic geometry as periodic ones. In euclidean geometries there do not exist periodic lines.     

Pasch- oder Veblen-Axiom?

Das Pasch-Axiom wird mit dem manchmal ebenso bezeichneten Veblen-Axiom verglichen; dessen Herkunft wird untersucht.     

A characterization ofL n(K) as a permutation group

In 1972 M. O'Nan proved thatL _n (q),h 3; can be characterized as a doubly-transitive groupG on a finite set , whereG _a has an Abelian normal subgroup acting not semi-regularly on -a. In the Main Theorem we show that a similar statement holds if is infinite. Our result implies O'Nan's theorem.This paper is part of the author's Ph.D. thesis written under supervision of Prof. F. G. Timmesfeld.     

Projective planar spaces

A local condition on a planar space is given which is sufficient for its points, lines and planes to be the points, the lines and some subspaces of a projective space.     

Applications of coordinatization in modular lattice theory: The legacy of J. von Neumann

This paper reviews current work based on von Neumann's coordinization theorem.     

Semi-classical projective planes over half-ordered fields

This paper concerns a generalization of Moulton planes constructed by J. Jakóbowski. We consider those planes over ordered fields and solve the isomorphism and collineation problem posed inGeom. Dedicata 42 (1992), 243–253.Dedicated to Prof. H. Salzmann on the occasion of his 65th birthday     

Faces of faces of the tridiagonal Birkhoff polytope

The tridiagonal Birkhoff polytope, , is the set of real square matrices with nonnegative entries and all rows and columns sums equal to 1 that are tridiagonal. This polytope arises in many problems of enumerative combinatorics, statistics, combinatorial optimization, etc. In this paper, for a given a p-face of , we determine the number of faces of lower dimension that are contained in it and we discuss its nature. In fact, a 2-face of is a triangle or a quadrilateral and the cells can only be tetrahedrons, pentahedrons or hexahedrons.     

Blocking Subspaces By Lines In PG(<Emphasis Type="Italic">n,q</Emphasis>)

This paper studies the cardinality of a smallest set of t-subspaces of the finite projective spaces PG(n, q) such that every s-subspace is incident with at least one element of , where 0 t < s n. This is a very difficult problem and the solution is known only for very few families of triples (s, t, n). When the answer is known, the corresponding blocking configurations usually are partitions of a subspace of PG(n, q) by subspaces of dimension t. One of the exceptions is the solution in the case t = 1 and n = 2s. In this paper, we solve the case when t = 1 and 2s < n 3s-3 and q is sufficiently large.     

Blockings-dimensional subspaces by lines inPG(2s,q)

We investigate sets of lines inPG(2s,q) such that everys-dimensional subspace contains a line of this set. We determine the minimum number of lines in such a set and show that there is only one type of such a set with this minimum number of lines.     

The diameter of the acyclic Birkhoff polytope

In this work we give an interpretation of vertices and edges of the acyclic Birkhoff polytope, T_n=Ω_n(T), where T is a tree with n vertices, in terms of graph theory. We generalize a recent result relatively to the diameter of the graph G(T_n).     

Surjections on Grassmannians preserving pairs of elements with bounded distance

Let m and k be two fixed positive integers such that m>k?2. Let V be a left vector space over a division ring with dimension at least m+k+1. Let G_m(V) be the Grassmannian consisting of all m-dimensional subspaces of V. We characterize surjective mappings T from G_m(V) onto itself such that for any A,B in G_m(V), the distance between A and B is not greater than k if and only if the distance between T(A) and T(B) is not greater than k.     

Maximal arcs in Desarguesian planes of odd order do not exist

Forq an odd prime power, and 1<n<q, the Desarguesian planePG(2,q) does not contain an(nq–q+n,n)-arc.Supported by Italian M.U.R.S.T. (Research Group onStrutture geometriche, combinatoria, loro applicazioni) and G.N.S.A.G.A. of C.N.R.     

On the products of cross-ratios on diagonals of polygons

The purpose of this paper is to establish a conjecture of B. Grünbaum, which states that in every n-polygon P in the plane, n 5, some diagonals intersect in a pattern that defines a new n-polygon (P), such that the product of the cross-rations on the diagonals of P is equal to the product of the corresponding cross-ratios on the diagonals of (P).     

Face counting on an Acyclic Birkhoff polytope

In this paper we present some algorithms allowing an exhaustive account on the number of edges and faces of the acyclic Birkhoff polytope.     

Linear spaces with many projective planes

We assume that in a linear space there is a non-empty set M of points with the property that every plane containing a point of M is a projective plane. In section 3 an example is given that in general is not a projective space. But if M can be completed by two points to a generating set of P, then is a projective space.