Blocking-sets in infinite projective and affine spaces

In every projective or affine space of infinite order there exist blockingsets. Moreover, we prove that every bijective map which preserves blockingsets of a fixed level 1 is a collineation.     

Combinatorial analysis of proofs in projective and affine geometry

The axioms of projective and affine plane geometry are turned into rules of proof by which formal derivations are constructed. The rules act only on atomic formulas. It is shown that proof search for the derivability of atomic cases from atomic assumptions by these rules terminates (i.e., solves the word problem). This decision method is based on the central result of the combinatorial analysis of derivations by the geometric rules: The geometric objects that occur in derivations by the rules can be restricted to those known from the assumptions and cases. This “subterm property” is proved by permuting suitably the order of application of the geometric rules. As an example of the decision method, it is shown that there cannot exist a derivation of Euclid’s fifth postulate if the rule that corresponds to the uniqueness of the parallel line construction is taken away from the system of plane affine geometry.     

Combinatorial analysis of proofs in projective and affine geometry

The axioms of projective and affine plane geometry are turned into rules of proof by which formal derivations are constructed. The rules act only on atomic formulas. It is shown that proof search for the derivability of atomic cases from atomic assumptions by these rules terminates (i.e., solves the word problem). This decision method is based on the central result of the combinatorial analysis of derivations by the geometric rules: The geometric objects that occur in derivations by the rules can be restricted to those known from the assumptions and cases. This “subterm property” is proved by permuting suitably the order of application of the geometric rules. As an example of the decision method, it is shown that there cannot exist a derivation of Euclid’s fifth postulate if the rule that corresponds to the uniqueness of the parallel line construction is taken away from the system of plane affine geometry.     

Designs having the parameters of projective and affine spaces

Two constructions are described that yield an improved lower bound for the number of 2-designs with the parameters of PG _d (n, q), and a lower bound for the number of resolved 2-designs with the parameters of AG _d (n, q).     

Integral geometry in constant curvature Lorentz spaces

We extend classical Cauchy formulas and Crofton formulas to the Lorentz-Minkowski space, and to constant curvature Lorentz spaces.     

Integral automorphisms of affine spaces over finite fields

A permutation of the point set of the affine space \({{\mathrm{AG}}}(n,q)\) is called an integral automorphism if it preserves the integral distance defined among the points. In this paper, we complete the classification of the integral automorphisms of \({{\mathrm{AG}}}(n,q)\) for \(n\ge 3\).     

On the geometry in projective limits of Hilbert spaces

In this paper we consider the concept of orthogonality with respect to infinitely many inner products. We describe geometric properties related to this concept of orthogonality in certain Köthe sequence spaces (power series spaces), spaces of holomorphic functions in one and several variables and spaces of infinitely differentiable functions. The methods are required from a mixture of functional analysis (theory of bases), theory of functions of one complex variable, Fourier analysis and interpolation theory.     

On the axiomatics of projective and affine geometry in terms of line intersection

By providing explicit definitions, we show that in both affine and projective geometry of dimension >- 3, considered as first-order theories axiomatized in terms of lines as the only variables, nd the binary line-intersection predicate as primitive notion, non-intersection of two ines can be positively defined in terms of line-intersection.     

Integral geometry and the Gauss-Bonnet theorem in constant curvature spaces

We give an integral-geometric proof of the Gauss-Bonnet theorem for hypersurfaces in constant curvature spaces. As a tool, we obtain variation formulas in integral geometry with interest in its own.

     

Integral point sets in higher dimensional affine spaces over finite fields

We consider point sets in the m-dimensional affine space where each squared Euclidean distance of two points is a square in Fq. It turns out that the situation in is rather similar to the one of integral distances in Euclidean spaces. Therefore we expect the results over finite fields to be useful for the Euclidean case.We completely determine the automorphism group of these spaces which preserves integral distances. For some small parameters m and q we determine the maximum cardinality I(m,q) of integral point sets in . We provide upper bounds and lower bounds on I(m,q). If we map integral distances to edges in a graph, we can define a graph Gm,q with vertex set . It turns out that Gm,q is strongly regular for some cases.