A Helly-type theorem for simple polygons

Let be a family of simple polygons in the plane. If every three (not necessarily distinct) members of have a simply connected union and every two members of have a nonempty intersection, then {P:P in } . Applying the result to a finite family of orthogonally convex polygons, the set {C:C in } will be another orthogonally convex polygon, and, in certain circumstances, the dimension of this intersection can be determined.Supported in part by NSF grant DMS-9207019.     

Bounds for the lattice point enumerator

For the lattice point enumerator of a lattice and a convex body K we give bounds in terms of the intrinsic volumes of K and of minimal determinants of . The intrinsic volumes are the normalized Minkowski quermassintegrals and the minimal determinants are analogous functionals of .     

Sechzehndimensionale lokalkompakte Translationsebenen,deren Kollineationsgruppe G2 enthält

New results of Salzmann and Hubig say that a 16-dimensional (locally) compact topological projective plane in which the group of continuous collineations has dimension 40 is a translation plane. It is therefore important to determine all 16-dimensional locally compact translation planes with dim 40. From previous work of the author ([10]), it is known that such a plane is either the classical octonion plane, or dim =40 and contains a subgroup isomorphic to the compact exceptional group G₂, but no larger compact simple subgroup. In the present paper, all planes satisfying the latter property more generally with dim 38 are explicitly determined. Together with the classification of all 16-dimensional locally compact translation planes in which contains Spin(7) given by the author in [8], one thus knows all 16-dimensional locally compact translation planes with containing G₂ and dim 38. Via suitable Baer subplanes, the classification makes use of analogous results for 8-dimensional planes ([7]).

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Meinem verehrten Lehrer Helmut Salzmann zum 60. Geburtstag     

Invariant differental forms on compact nilmanifolds

Let N=G/ be a compact nilmanifold, G a connected, simply connected, nilpotent Lie group with its discrete subgroup and Lie algebra . Let I* ( ) denote the invariant differential forms on .If I* ( ) H* ( ) is an injective map, then G is abelian and N is a torus. Furthermore, N has a formal minimal model. If N is an even-dimensional compact nilmanifold, it has a Kähler structure and invariant symplectic structure if and only if I* ( ) H* ( ) is injective.     

Tissus linéaires et théorèmes d'algébrisation de type Abdel-inverse et Reiss-inverse

A d-web in ( ,0) is given by d complex analytic foliations of codimension one in ( ,0) which are in general position. A d-web in ( ,0) is linear if all the leaves are (pieces of) hyperplanes in and is algebraic if it is associated, by duality, to a nondegenerate algebraic curve in of degree d. We characterize linear webs in ( ,0). We give explicit conditions under which a linear d-web in ( ,0) is algebraic and we obtain equations for in this case. Some related problems are discussed and some questions are posed.     

Sub-Riemannian homogeneous spaces in dimensions 3 and 4

Let (M, , g) be a sub-Riemannian manifold (i.e. M is a smooth manifold, is a smooth distribution on M and g is a smooth metric defined on ) such that the dimension of M is either 3 or 4 and is a contact or odd-contact distribution, respectively. We construct an adapted connection on M and use it to study the equivalence problem. Furthermore, we classify the 3-dimensional sub-Riemannian manifolds which are sub-homogeneous and show the relation to Cartan's list of homogeneous CR manifolds. Finally, we classify the 4-dimensional sub-Riemannian manifolds which are sub-symmetric.     

Sylow subgroups of finitary linear groups

In this paper we describe the structure and the conjugacy classes of Sylow p-subgroups of FGL(V, ), the group of finitary -automorphisms of the -vector space V.The Author is member of the GNSAGA.     

On the dimension of the cartesian product of relations and orders

It is known that for incidence structures and , max , wheref dim stands for Ferrers relation. We shall show that under additional assumptions on and , both bounds can be improved. Especially it will be shown that the square of a three-dimensional ordered set is at least four-dimensional.     

Symmetric planes with non-classical tangent translation planes

We construct symmetric planes associated with an arbitrary locally compact connected nearfield . If is a proper nearfield, i.e. {;;}, then the tangent translation plane of this symmetric plane is not classical. All previously known examples of symmetric planes have classical tangent translation planes.Herrn Professor Dr. H. Salzmann zum 65. Geburtstag gewidmet     

Hankel operators on the weighted Bergman spaces with exponential type weights

Let denote the closed subspace of consisting of analytic functions in the unit disk . For certain class of subharmonic , the Hankel operatorH _b on with symbol is studied. Criteria for boundedness and compactness of such kind of Hankel operators are presented.R. Rochberg's research was partially supported by a grant from the National Science Foundation.     

Discussing Knarr's 2-Surface of \mathbb{R} 4 which Generates the First Single Shift Plane

The generating line of the first single shift plane (cf. [11, p. 435]) is a 2-surface of ⁴ which we call the the affine part of Knarr's surface. We compute all affinities leaving invariant. After embedding ⁴ into PG(4, ) we calculate the uniquely determined projective closure _Kn of . Using a suitable projection we transform questions on Knarr's surface to questions on Cayley's surface in PG(3, ). In this way we determine all planes carrying 1-dimensional algebraic varieties of _Kn . We exhibit all automorphic collineations of _Kn .     

Isometric representations of some quotients ofH ∞ of an annulus

Let denote an annulus,E a finite subset of with at least three elements, and the ideal of functions in which vanish at the points ofE. The quotient does not have a completely isometric representation on a finite dimensional Hilbert space. This complements a result of [11] which implies that the quotient has an isometric representation on a Hilbert space of dimension twice the cardinality ofE.     

Notes on Brown-McCoy-radicals for Γ-near-rings

The Brown-McCoy radical is known to be an ideal-hereditary Kurosh-Amitsur radical in the variety of zerosymmetric near-rings. We define the Brown-McCoy and simplical radicals, and , respectively, for zerosymmetric -near-rings. Both and are ideal-hereditary Kurosh-Amitsur radicals in that variety. IfM is a zerosymmetric -near-ring with left operator near-ringL, it is shown that , with equality ifM has a strong left unity. is extended to the variety of arbitrary near-rings, and and are extended to the variety of arbitrary -near-rings, in a way that they remain Kurosh-Amitsur radicals. IfN is a near-ring andA N, then , with equality ifA if left invariant.     

Groupe modulaire, fractions continues et approximation diophantienne en caractéristique p

The aim of this paper is to give a geometric interpretation of the continued fraction expansion in the field of formal Laurent series in X ^–1 over , in terms of the action of the modular group on the Bruhat–Tits tree of , and to deduce from it some corollaries for the diophantine approximation of formal Laurent series in X ^–1 by rational fractions in X.     

Sub-weakly Embedded Singular and Degenerate Polar Spaces

We show that every sub-weak embedding of any singular (degenerate or not) orthogonal or unitary polar space of non-singular rank at least 3 in a projective space PG , a commutative field, is the projection of a full embedding in some subspace PG of PG , where PG contains PG and is a subfield of . The same result is proved in the symplectic case under the assumption that the field over which the polarity is defined is perfect if the characteristic is 2 and if each secant line of the embedded polar space contains exactly two points of . This completes the classification of all sub-weak embeddings of orthogonal, symplectic and unitary polar spaces (singular or not; degenerate or not) of non-singular rank at least 3 and defined over a commutative field , where in the characteristic 2 case is perfect if the polar space is symplectic and the degree of the embedding is 2.     

On a theorem of Seidel and Walsh

Given a sequence ( _n ) _n in with there are functions such that , is a dense subset of , and the set of functions with this property is residual in . We will show that in and some related Banach spaceX there are functionsf with is dense in , and we will give a sufficient condition when the set of such functions is residual inX.     

Zonotopal Subdivisions of Cyclic Zonotopes

The cyclic zonotope (n, d) is the zonotope in ^d generated by any n distinct vectors of the form (1, t, t ²,..., t ^d–1). It is proved that the refinement poset of all proper zonotopal subdivisions of (n, d) which are induced by the canonical projection : (n, d) (n, d), in the sense of Billera and Sturmfels, is homotopy equivalent to a sphere and that any zonotopal subdivision of (n, d) is shellable. The first statement gives an affirmative answer to the generalized Baues problem in a new special case and refines a theorem of Sturmfels and Ziegler on the extension space of an alternating oriented matroid. An important ingredient in the proofs is the fact that all zonotopal subdivisions of (n, d) are stackable in a suitable direction. It is shown that, in general, a zonotopal subdivision is stackable in a given direction if and only if a certain associated oriented matroid program is Euclidean, in the sense of Edmonds and Mandel.     

Finitely generated relatively universal varieties of Heyting algebras

Given distinct varieties and of the same type, we say that is relatively -universal if there exists an embedding :K from a universal categoryK such that for every pairA, B ofK-objects, a homomorphismf:A B has the formf=_g for someK-morphismg:A B if and only if Im(f) . Finitely generated relatively -universal varieties of Heyting algebras are described for the variety of Boolean algebras, the variety generated by a three element chain, and for the variety generated by the four element Boolean algebra with an added greatest element.Dedicated to the memory of Alan DayPresented by J. Sichler.The support of the NSERC is gratefully acknowledged.     

SU4 (ℂ) als Kollineationsgruppe in sechzehndimensionalen lokalkompakten Translationsebenen

We study 16-dimensional locally compact translation planes in which, for an affine point o, the stabilizer of the affine collineation group contains a subgroup locally isomorphic to SU₄ (). If has only one affine fixed point o, then it is shown that either the plane is the classical Moufang plane over the Cayley numbers, or else must be normal in the stabilizer and has dimension at most 37. This also comprises the proof of the fact that if contains a subgroup locally isomorphic to SU₄() × SL₂() then the plane is the classical Cayley plane. The case that has more affine fixed points in dealt with as well; then, except for a well-known family of planes admitting Spin₇() as a group of collineations, has dimension at most 34.     

Complétude des métriques lorentziennes de\mathbb{T}^2 et difféomorphismes du cercle

Let and the foliations by the null geodesics of some lorentzian metricg on the torus . We analyse how geodesic completeness properties ofg are related to the dynamics of and .