New integer pairs for Hjelmslev planes

Associated with every finite projective Hjelmslev plane is an invariant pair(t, r); t is the order of the Hjelmslev plane andr is the order of the underlying projective plane. The aim of this paper is to give some new constructions of Hjelmslev planes with an invariant pair (t, 2). First we construct a PH-plane with the invariant pair (20, 2). Using this, 16 more invariant pairs (t, 2) witht 1000 are obtained. In all, we thus obtain 17 new PH-planes with invariant pairs (t, 2),t 1000.     

Order and topology in projective Hjelmslev planes

The concept of an ordered projective Hjelmslev plane was intuitively introduced by Hjelmslev in Einleitung in die allgemeine Kongruenglehre ([9], [10]).This paper is concerned with formalizing and examing preorderings and orderings for projective Hjelmslev planes. In addition we show that orderings generated topologies of the point and line sets which render the plane a topological Hjelmslev plane ([19], [13]). These planes — unlike the ordinary ordered planes ([18]) — are, due to the existence of infinitesimals, non-archimedian, non-compact and disconnected with the neighbour classes as certain quasi-components.The authors gratefully acknowledge the support of the Natural Sciences and Engineering Research Council of Canada.     

The classification of compact punctally cohesive Desarguesian projective Klingenberg planes

It is shown that for a compact Desarguesian projective Klingenberg plane P with incidence structure P=#x2119;, , I and neighbour relation , where two distinct points always lie on some line, exactly one of the following holds: P is a non-discrete connected or totally disconnected ordinary projective plane with =id, P is a finite projective plane with =id, P is a finite projective Hjelmslev plane with id, or P is a non-discrete totally disconnected ordinary projective plane with id.Dedicated to H. Salzmann on his 60th birthdayThe author wishes to thank the Natural Sciences and Engineering Research Council of Canada (NSERC) for their financial assistance in the writing of this paper.     

Lotketten in metrischen Ebenen

In [3] F. Bachmann defined chains of perpendiculars for a geometric structure which is endowed with a set of lines and a symmetric orthogonality relation on this set, and he studied these chains in the group plane of a Hjelmslev group. All Bachmann groups (the groups which are the main subject of [1]) are Hjelmslev groups, and the theory of Hjelmslev groups includes also Hjelmlev's theory of Allgemeine Kongruenzlehre. In a Hjelmslev group, the chains of perpendiculars are closely related to the subgroup which is generated by the points. In this paper we establish some results concerning chains of perpendiculars mainly with regard to Bachmann groups.     

Polarities in n-uniform projective Hjelmslev planes

In [9] the author has studied polarities in finite 2-uniform projective Hjelmslev planes. The present paper deals with polarities in finite n-uniform projective Hjelmslev planes (n 2).The author's research was supported by IWONL grant no. 840037.     

On the theorems of Drake and Lenz

A generalization and an improvement of the results of Drake and Lenz on the constructions of projective Hjelmslev planes are obtained. Using this, some new series of invariant pairs including 4 new pairs (t, 2),t 1000, for projective Hjelmslev planes are obtained.     

Affine Barbilian structures

W.Leissner has characterized, by geometric axioms, the affine Barbilian planes over a Z-ring (i.e, a ring with 1 such that ab=1 ba=1) [10].The aim of the present paper is to characterize correspondingly the affine Barbilian planes over an arbitrary ring with 1. First we shall deal with the translation Barbilian planes, which generalize Leissner's parallelodromic planes [11]. The paper concludes with a study of the kernel of the translation Barbilian plane.Here, the terms affine Barbilian structure and affine Barbilian plane are used in a more general sense than in [10] and [11]. Also, the definitions of translation and parallelodromy are slightly different from those in [10] and [11], insomuch that the invariance of the non-neighbour relation is not postulated any more, this being a consequence in a translation Barbilian plane.In H.J.Arnold's geometry of rings, for any two distinct points, there exists a smallest line incident with them [1]. This property, assumed only for the non-neighbour pairs of points, will replace the usual postulate that two non-neighbour points are incident with exactly one line. Thus, ideas of D.Barbilian [2] and H. J.Arnold [1] are combined with methods of affine ring-geometry due to J.Hjelmslev [5], [6], W.Klingenberg [7], [8], [9], H.Lüneburg [12], W.Benz [3], [4], W.Leissner [10], [11], and others. Many parts of the proofs in [10] and [11] could be used here almost unchanged, under relaxed assumptions.     

On the γ-Interior and γ-Closure of a Set

For a set X, let : exp X exp X satisfy A B whenever A B X. In [4], -open subsets of X, -interior iA and -closure cA of A X have been defined. The purpose of the present paper is to show that, under suitable conditions on , explicit formulas furnish iA and cA.     

Preordered uniform Hjelmslev planes

In Order and topology in projective Hjelmslev planes, the authors introduced the notions of preorder and order on projective Hjelmslev planes (PH-planes); cf. [3: 3.7, 5.3]. They also showed that in both ordered and preordered uniform PH-planes singular segments are uniquely determined by their endpoints. In this note, we show that the notions of order and preorder actually coincide on uniform PH-planes.The author gratefully acknowledges the support of the Natural Sciences and Engineering Research Council of Canada.     

On representing sylvester-gallai designs

A misstated conjecture in [3] leads to an interesting (1, 3) representation of the 7-point projective plane inR ⁴ where points are represented by lines and planes by 3-spaces. The corrected form of the original conjecture will be negated if there is a (1, 3) representation of the 13-point projective plane inR ⁴ but that matter is not settled.     

On the abundance of chaotic behavior for generic one-parameter families of maps

In this paper, we want to show the abundance of chaotic systems with absolutely continuous probability measures in the generic regular family with perturbable points. More precisely, we prove that iff _a:I I, a P is a regular family satisfying some conditions described in the next section, then there exists a Borel set P of positive Lebesgue measure such that for everya ,f _a admits an absolutely continuous invariant probability measure w.r.t. the Lebesgue measure. The idea of proof in this paper, as compared with that shown in [1] and [7], follows a similar line.Supported by the NSFC and the National 863 Project.     

Sharp Large Deviation Estimates for the Stochastic Heat Equation

We consider the family {X , 0} of solution to the heat equation on [0,T]×[0,1] perturbed by a small space-time white noise, that is _t X = X +b({X })+({X }) . Then, for a large class of Borelian subsets of the continuous functions on [0,T]×[0,1], we get an asymptotic expansion of P({X }A) as 0. This kind of expansion has been handled for several stochastic systems, ranging from Wiener integrals to diffusion processes.     

On an assertion of Dembowski

In his Finite Geometries, Dembowski asserted that the class of uniform projective Hjelmslev planes of order q and a certain class of symmetric divisible partial designs coincide. We give a counterexample for q=2 and prove the validitiy of Dembowski's assertion for all q3.     

∑II∑ threshold formulas

A II formula has the form, where eachL is either a variable or a negated variable. In this paper we study the computation of threshold functions by II formulas. By combining the proof of the Fredman-Komlós bound [5, 10] and a counting argument, we show that fork andn large andkn/2, every II formula computing the threshold functionT _k ⁿ has size at least exp . Fork andn large andkn ^2/3, we show that there exist II formulas for computingT _k ⁿ with size at most exp .     

Hyper-reguli

The concept of a regulus in three dimensional projective space is generalized to a hyper-regulus in projective space of dimension 2d–1. The most obvious examples come from the André planes. We find other examples related to generalized André translation planes and consider possibilities for extending partial hyper-reguli to hyper-reguli.Dedicated to Professor Adriano Barlotti on his 70th birthday.     

On the limit superior of analytic sets

. , A ₀,A ₁,— - lim supA _j - H, . , - - . , , ; , , . - . - .     

Normalformen für Bewegungen in hyperbolischen Räumen

If K is an Euclidean field and f a symmetric bilinear form of index 1, there is for each element O₄ (K,f) an orthogonal decomposition of the corresponding metric vector space V₄(K,f) into — invariant subspaces U_i such that dim U_i 3. Thus it is possible to determine nomal form for and to decompose the characteristic polynomial X into irreducible factors.     

Halbordnungstopologien in Projektiven Ebenen

The order topology of an ordered projective plane leads to a topological projective plane in which intersection and joining are continuous operations (c.f. O.WYLER [5]).Here continuity properties in projective planes with half order topology derived from a SPERNER'sche Ordnungsfunktion are analysed. Central projections are continuous and planes which are connected with respect to the half order topology are (fully) ordered and therefore topological planes.

---

---

Herrn Prof. Dr. WERNER BURAU zum 70. Geburtstag     

Projective Modules and Complete Intersections

Let A be a Noetherian ring of dimension n and P be a projective A module of rank n having trivial determinant. It is proved that if n is even and the image of a generic element g P^* is a complete intersection, then [P] = [Q A] in K₀(A) for some projective A module Q of rank n – 1. Further, it is proved that if n is odd, A is Cohen–Macaulay and [P] = [Q A] in K₀(A) for some projective A module Q of rank n – 1, then P has a unimodular element.     

On the automorphism groups of β-derived planes

This paper contains some general results about the automorphism group H of PG(3,q) leaving a -derived spread F invariant. The main result asserts that H contains a cyclic subgroups M^*, whose order divides (q+1)/2. In particular, H is not trivial. In terms of translation planes this means that the translation complement of any -derived plane always admits some collineations which are not dilatations.Research partially supported by M.P.I. (Research project Strutture Geometriche Combinatorie e loro Applicazioni).