A quadric model for Klingenberg chain spaces

Consider a finite-dimensional algebra with involution over a commutative local ring. The chain geometry over this algebra is a Klingenberg chain space. We embed this structure into a projective Klingenberg space, such that the points are identified with points of a quadric and the chains with plane sections.     

On linear spaces and matroids of arbitrary cardinality

In this paper, we study linear spaces of arbitrary finite dimension on some (possibly infinite) set. We interpret linear spaces as simple matroids and study the problem of erecting some linear space of dimension n to some linear space of dimension n + 1 if possible. Several examples of some such erections are studied; in particular, one of these erections is computed within some infinite iteration process.Dedicated to the memory of Gian-Carlo Rota     

Spinor spaces of Clifford algebras

In this paper, we study the structures of Clifford algebras. We represent the pinor and spinors spaces as subspaces of Clifford algebras. With suitable bases of the Clifford algebras, we construct isomorphisms between Clifford algebras and matrix algebras. In doing these we develop some spinor calculus.     

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.     

Self-dual sets of type (m, n) in projective spaces

In this paper we introduce and analyze the notion of self-dual k-sets of type (m, n). We show that in a non-square order projective space such sets exist only if the dimension is odd. We prove that, in a projective space of odd dimension and order q, self-dual k-sets of type (m, n), with , are of elliptic and hyperbolic type, respectively. As a corollary we obtain a new characterization of the non-singular elliptic and hyperbolic quadrics.     

Complete caps in projective spaces PG (n, q)

A computer search in the finite projective spaces PG(n, q) for the spectrum of possible sizes k of complete k-caps is done. Randomized greedy algorithms are applied. New upper bounds on the smallest size of a complete cap are given for many values of n and q. Many new sizes of complete caps are obtained.     

Pseudo unitary conformal groups and Clifford algebras for standard pseudo hermitian spaces

This paper, self-contained, deals with pseudo-unitary spin geometry. First, we present pseudo-unitary conformal structures over a 2n-dimensional complex manifold V and the corresponding projective quadrics for standard pseudo-hermitian spaces H_p,q. Then we develop a geometrical presentation of a compactification for pseudo-hermitian standard spaces in order to construct the pseudo-unitary conformal group of H_p,q. We study the topology of the projective quadrics and the “generators” of such projective quadrics. Then we define the space S of spinors canonically associated with the pseudo-hermitian scalar product of signature (2ⁿ⁻¹, 2ⁿ⁻¹). The spinorial group Spin U(p,q) is imbedded into SU(2ⁿ⁻¹, 2ⁿ⁻¹). At last, we study the natural imbeddings of the projective quadrics      

Klingenberg chain spaces

We propose a definition ofKlingenberg chain space, motivated by examples that are obtained from the action of the linear group on the projective line over an algebra over a local ring.     

A characterization of linear spaces based on the number of transversals

We characterize a class of linear spaces by the property that through any point outside two disjoint, but non-parallel lines there is at most one transversal.     

Relation between metric spaces and Finsler spaces

In a connected Finsler space Fn=(M,F) every ordered pair of points p,q∈M determines a distance ?F(p,q) as the infimum of the arc length of curves joining p to q. (M,?F) is a metric space if Fn is absolutely homogeneous, and it is quasi-metric space (i.e. the symmetry: ?F(p,q)=?F(q,p) fails) if Fn is positively homogeneous only. It is known the Busemann-Mayer relation , for any differentiable curve p(t) in an Fn. This establishes a 1:1 relation between Finsler spaces Fn=(M,F) and (quasi-) metric spaces (M,?F).We show that a distance function ?(p,q) (with the differentiability property of ?F) needs not to be a ?F. This means that the family {(M,?)} is wider than {(M,?F)}. We give a necessary and sufficient condition in two versions for a ? to be a ?F, i.e. for a (quasi-) metric space (M,?) to be equivalent (with respect to the distance) to a Finsler space (M,F).     

Splittings of Banach spaces induced by Clifford algebras

Let be an infinite-dimensional Hilbert space of density character . By representing as a module over an appropriate Clifford algebra, it is proved that possesses a family of proper closed nonzero subspaces such that

Analogous results are proved for spaces and for and () when is an arbitrary nonzero Banach space.

     

A Birkhoff theorem for partial algebras via completion

An equational theory (a Birkhoff theorem) for functorial partial algebras is established via the corresponding theory for functorial total algebras.This work was done with partial support of the DFG (BRD).Presented at the European Colloquium of Category Theory, Tours, France, 25–31 July 1994.     

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.     

Chain homogeneous Souslin algebras

Assuming Jensen's principle ?⁺ we construct Souslin algebras all of whose maximal chains are pairwise isomorphic as total orders, thereby answering questions of Koppelberg and Todor?evi?.     

Miquel Sätze in Minkowski—Ebenen I

Summary For the coordinization of axiomatically given nonlinear incidence structures, the theorem of Miquel acts as a powerful tool. In the literature there exist many different versions of this theorem, especially in the most complicated area of Minkowski geometries. In three papers (this one and the following parts II, III) all possibilities to formulate Miquelian theorems are discussed. This part I and the first half of part II mainly deal with conditions which are necessary to get interesting results. Having done this troublesome job we are able to present a big number of Miquelian theorems which can be used to coordinize Minkowski geometries, too. At the end of part III we present a peculiar version of a nontrivial theorem of Miquelian type. Up to now this is the only one which is not equivalent to the other interesting theorems of Miquel.

     

Miquel Sätze in Minkowski-Ebenen III

In this paper we continue the investigations of [10], [11]. We are interested in Miquel configurations having six, seven, or eight pairwise different points. Using the results of the previous papers we are able to present a lot of Miquelian theorems which can be used to coordinatize Minkowski geometries. At the end we consider a very special theorem of this kind. Up to now this is the only one which is not equivalent to the other interesting theorems of Miquel.

In dieser Arbeit werden Ergebnisse aus der Habilitationsschrift des Autors vorgestellt.     

Generalized affine chain geometries

The affine chain geometry over a group with a partial fibration into subgroups and a certain involution is introduced. This concept generalizes the affine trace of the chain geometry over an associative algebra. We study the geometric properties of these geometries and give examples.Dedicated to Professor Helmut Mäurer on the occasion of his 60th birthday     

Symmetric designs and geometroids

A-setS in a symmetric 2-(v, k, ) design is a subset which every block meets in 0, 1 or points such that for any point ofS there is a unique block meetingS at that point only. Ovoids in three-dimensional projective spaces are examples of-secs. It is shown that if has a-set then is a geometroid withv=u ²+u+1 andk=u+1, whereu–1. The cases whenu is–1, and+1 are investigated and some open problems discussed.     

The monad of probability measures over compact ordered spaces and its Eilenberg-Moore algebras

The probability measures on compact Hausdorff spaces K form a compact convex subset PK of the space of measures with the vague topology. Every continuous map of compact Hausdorff spaces induces a continuous affine map extending f. Together with the canonical embedding associating to every point its Dirac measure and the barycentric map β associating to every probability measure on PK its barycenter, we obtain a monad (P,ε,β). The Eilenberg-Moore algebras of this monad have been characterised to be the compact convex sets embeddable in locally convex topological vector spaces by Swirszcz [T. Swirszcz, Monadic functors and convexity, Bul. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 22 (1974) 39-42].We generalise this result to compact ordered spaces in the sense of Nachbin [L. Nachbin, Topology and Order, Von Nostrand, Princeton, NJ, 1965. Translated from the 1950 monograph “Topologia e Ordem” (in Portugese). Reprinted by Robert E. Kreiger Publishing Co., Huntington, NY, 1967]. The probability measures form again a compact ordered space when endowed with the stochastic order. The maps ε and β are shown to preserve the stochastic orders. Thus, we obtain a monad over the category of compact ordered spaces and order preserving continuous maps. The algebras of this monad are shown to be the compact convex ordered sets embeddable in locally convex ordered topological vector spaces.This result can be seen as a step towards the characterisation of the algebras of the monad of probability measures on the category of stably compact spaces (see [G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, D.S. Scott, Continuous Lattices and Domains, Encyclopedia Math. Appl., vol. 93, Cambridge University Press, 2003, Section VI-6]).     

Injective spaces via adjunction

The work of the present author and his coauthors over the past years gives evidence that it may be useful to regard each topological space as a kind of enriched category, by interpreting the convergence relation x→x between ultrafilters and points of a topological space X as arrows in X. Naturally, this point of view opens the door to the use of concepts and ideas from enriched Category Theory for the investigation of topological spaces. Topological theories introduced by the author provide a convenient general setting for appropriately transferring these concepts and ideas to the world of topological spaces and some other geometric objects such as approach spaces. Using tools like adjunction and the Yoneda lemma, we show that the cocomplete spaces are precisely the injective spaces, and they are algebras for a suitable monad on . This way we obtain enriched versions of known results about injective topological spaces and continuous lattices.