A Characterisation of the Generalized Quadrangle Q (5, q) Using Cohomology

If a GQ S of order (s, s) is contained in a GQ S of order (s, s ²) as a subquadrangle, then for each point X of S\S the set of points of S collinear with X form an ovoid of S. Thas and Payne proved that if S= (4,q),q even, and is an elliptic quadric for each XS\S,thenS (5,q). In this paper we provide a single proof for the q odd and q even cases by establishing a link between the geometry involved and the first cohomology group of a related simplicial complex.     

Interval stability and interval covering property in finite posets

Given a finite ranked posetP, let (P) be the maximum size of a subset ofP such that no two elements of it belong simultaneously to some interval ofP and let (P) be the minimum number of intervals covering all elements ofP. We say thatP has the strong interval stability property (resp. the strong interval covering property) if for each subposetP induced by consecutive levels ofP, i.e.,P=P _(l)...P _(u), one has (P)=max{|P _(l)|, |P _(u)|} (resp. (P)=max{|P _(l)|, |P _(u)|}).We prove these properties for several classes of posets and discuss some general facts concerning the numbers (P) and (P), e.g., NP-completeness and min-max relations.     

Über konvergente Zerlegungen von Matrizen

LetA, M, N ben × n real matrices, letA=M–N, letA andM be nonsingular. LetMy0 implyNy0 (where the prime denotes the transpose). ThenAy0 impliesNy0 if and only if the spectral radius (M ^–1 N) ofM ^–1 N is less than one. This complements a result of Mangasarian, given in [1]. The same conclusions are true ifA, M, andN are replaced byA, M, andN respectively. The proof given here does not make use of the Perron-Frobenius theorem.

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Herrn Professor Dr. Johannes Weissinger zum 60. Geburtstag gewidmet     

Theorem of Bartle,Dunford, and Schwartz concerning vector measures

We show the existence, for an arbitrary vector measure: x (where X is a Banach space and gs is a-algebra of subsets of a set S) of a functional x X (X is the conjugate space of X) such that is absolutely continuous with respect to _x, _x (E)=(E)>, E gs.Translated from Matematicheskie Zametki, Vol. 7, No. 2, pp. 247–254, February, 1970.     

Weight functions on the Kneser graph and the solution of an intersection problem of Sali

LetX, Y be finite sets and suppose thatF is a collection of pairs of sets (F, G),FX,GY satisfying |FF|s, |GG|t and |FF|+|GG|s+t+1 for all (F, G),F, GF. Extending a result of Sali, we determine the maximum ofF.     

Trivialization of fans in planar ternary rings with rational prime field

Summary Making use of the intimate relations between real places and orderings, we continue studying the spaces of orderings of planar ternary rings (PTRs) with rational prime field. As in the classical case, given a preorderingS of such a PTRT, then the productA ^S of all natural place ringsA _p associated to the orderingsP X/S is itself a place ring ofT. In particular, ifS is a nontrivial fan ofT thenA ^s T. Thus L. Bröcker's celebrated theorem on the trivialization of fans also applies to our setting: For any fan Sof a PTRT with rational prime field there exists a place µ: T T {} such that the push down S:= (S)\{0,} of S is a trivial fan of T. Further, Bröcker's global stability formula and, by means of an approximation theorem, some classical, valuation theoretic characterizations of SAP-preorderings and fans also extend to PTRs with rational prime field.     

Isomorphismen von rechtseiträumen

Spaces called rectangular spaces were introduced in [5] as incidence spaces (P,G) whose set of linesG is equipped with an equivalence relation and whose set of point pairs P² is equipped with a congruence relation , such that a number of compatibility conditions are satisfied. In this paper we consider isomorphisms, automorphisms, and motions on the rectangular spaces treated in [5]. By an isomorphism of two rectangular spaces (P,G, , ) and (P,G, , ) we mean a bijection of the point setP onto P which maps parallel lines onto parallel lines and congruent points onto congruent points. In the following, we consider only rectangular spaces of characteristic 2 or of dimension two. According to [5] these spaces can be embedded into euclidean spaces. In case (P,G, , ) is a finite dimensional rectangular space, then every congruence preserving bijection ofP onto P is in fact an isomorphism from (P,G, , ) onto (P,G, , ) (see (2.4)). We then concern ourselves with the extension of isomorphisms. Our most important result is the theorem which states that any isomorphism of two rectangular spaces can be uniquely extended to an isomorphism of the associated euclidean spaces (see (3.2)). As a consequence the automorphisms of a rectangular space (P,G, , ) are precisely the restrictions (onP) of the automorphisms of the associated euclidean space which fixP as a whole (see (3.3)). Finally we consider the motions of a rectangular space (P,G, , ). By a motion of(P. G,, ) we mean a bijection ofP which maps lines onto lines, preserves parallelism and satisfies the condition((x), (y)) (x,y) for allx, y P. We show that every motion of a rectangular space can be extended to a motion of the associated euclidean space (see (4.2)). Thus the motions of a rectangular space (P,G, , ) are seen to be the restrictions of the motions of the associated euclidean space which mapP into itself (see (4.3)). This yields an explicit representation of the motions of any rectangular plane (see (4.4)).

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Herrn Professor Burau zum 85. Geburtstag gewidmet     

Some tilings of the plane whose singular points are uncountable and unbounded

Let I be a tiling of the plane such that for every tile T of I there correspond a tile T of I (not necessarily unique) and an integer k(T, T) (depending on T and T), k(T, T)>2, such that T meets T in k(T, T) connected components. Tiles T and T satisfying this condition are called associated tiles in I. Various properties concerning I and its singular points are obtained. First, it is not possible that every tile in I have a unique associated tile. In fact, there exist infinite families of tiles {F} {F _n:n1} such that F is the unique associated tile for every F _n. Next, if x is a singular point of I, then every neighborhood of x contains uncountably many singular points of I. Finally, the set of singular points of I is unbounded.     

On independent statistical decision problems and products of diffusions

Summary LetZ _t be a null recurrent diffusion on ^p with generatorG=(1/r)·(r ) for smooth positiver. This note constructs an independent recurrent diffusionZ _ton ¹ such that (Z _t, Z_t)is transient in ^p+1. This resolves negatively an old question in simultaneous estimation: Is there an admissible but not Bayes estimator(X) of the mean of a multivariate normal distribution for quadratic loss with the property: for every admissible (X), whereX is normal and independent ofX, (, ) remains admissible in the combined problem obtained by summing the component losses?Work supported by NSF at Mathematical Sciences Research Institute, Berkeley     

Inequalities in Products of Minors of Totally Nonnegative Matrices

Let _I,I be the minor of a matrix which corresponds to row set I and column set I. We give a characterization of the inequalities of the form _{I,I K,K J,J L,L} which hold for all totally nonnegative matrices. This generalizes a recent result of Fallat, Gekhtman, and Johnson.