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.     

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-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.     

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     

Hypersurfaces of restricted type in Minkowski space

A submanifold M _n ^r of Minkowski space is said to be of restricted type if its shape operator with respect to the mean curvature vector is the restriction of a fixed linear transformation of to the tangent space of M _n ^r at every point of M _n ^r . In this paper we completely classify hypersurfaces of restricted type in . More precisely, we prove that a hypersurface of is of restricted type if and only if it is either a minimal hypersurface, or an open part of one of the following hypersurfaces: S ^k × , S _k ¹ × , H ^k × , S _n ¹ , H ⁿ , with 1kn–1, or an open part of a cylinder on a plane curve of restricted type.This work was done when the first and fourth authors were visiting Michigan State University.Aangesteld Navorser N.F.W.O., Belgium.     

Towards a Characterization of Self-Similar Tilings in Terms of Derived Voronoï Tessellations

In this paper, a technique for analyzing levels of hierarchy in a tiling of Euclidean space is presented. Fixing a central configuration P of tiles in , a `derived Voronoï' tessellation _P is constructed based on the locations of copies of P in . A family of derived Voronoï tilings is formed by allowing the central configurations to vary through an infinite number of possibilities. The family will normally be an infinite one, but we show that for a self-similar tiling it is finite up to similarity. In addition, we show that if the family is finite up to similarity, then is pseudo-self-similar. The relationship between self-similarity and pseudo-self-similarity is not well understood, and this is the obstruction to a complete characterization of self-similarity via our method. A discussion and conjecture on the connection between the two forms of hierarchy for tilings is provided.     

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.     

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.     

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.     

Topological and bornological characterisations of ideals in von Neumann algebras: I

Suppose is a von Neumann algebra on a Hilbert space and is any ideal in . We determine a topology on , for which the members of that are to norm continuous are exactly those in ; and a bornology on such that the elements of which map the unit ball to an element of , equivalently those members of that are norm to bounded, are exactly those in . This is achieved via analogues of the notions of injectivity and surjectivity in the theory of operator ideals on Banach spaces.     

Absolute Continuity of the Spectrum of a Periodic Schrödinger Operator

We prove the absolute continuity of the spectrum of the Schrödinger operator in , , with periodic (with a common period lattice ) scalar and vector potentials for which either , , or the Fourier series of the vector potential converges absolutely, , where is an elementary cell of the lattice , for , and for , and the value of is sufficiently small, where and otherwise, , and .     

On posets with isomorphic interval posets

Let be a partially ordered set, Int the system of all (nonempty) intervals of partially ordered by the set-theoretical inclusion . We are interested in partially ordered sets with Int isomorphic to Int . We are going to show that they correspond to couples of binary relations on A satisfying some conditions. If is a directed partially ordered set, the only with Int isomorphic to Int are corresponding to direct decompositions of ( denotes the dual of . The present results include those presented in the paper [11] by V. Slavík. Systems of intervals, particularly of lattices, have been investigated by many authors, cf. [1]–[11].     

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.     

Operator inequalities and construction of Krein spaces

Let be a Hilbert space. A continuous positive operatorT on uniquely determines a Hilbert space which is continuously imbedded in and for which with the canonical imbedding . A Kreîn space version of this result, however, is not valid in general. This paper provides a necessary and sufficient condition for that a continuous selfadjoint operatorT uniquely determines a Kreîn space ( ) which is continuously imbedded in and for which with the canonical imbedding .     

Bernstein-Type Theorems and Uniqueness Theorems

Let be an entire function of finite type with respect to finite order and let be a subset of an open cone in a certain n-dimensional subspace (the smaller , the sparser ). We assume that this cone contains a ray 0} \right\}$$ " align="middle" border="0"> . It is shown that the radial indicator of at any point may be evaluated in terms of function values at points of the discrete subset . Moreover, if tends to zero fast enough as over , then this function vanishes identically. To prove these results, a special approximation technique is developed. In the last part of the paper, it is proved that, under certain conditions on and , which are close to exact conditions, the function bounded on is bounded on the ray.     

Sharp Conditions for the CLT of Linear Processes in a Hilbert Space

In this paper we study the behavior of sums of a linear process associated to a strictly stationary sequence with values in a real separable Hilbert space and are linear operators from H to H. One of the results is that satisfies the CLT provided are i.i.d. centered having finite second moments and . We shall provide an example which shows that the condition on the operators is essentially sharp. Extensions of this result are given for sequences of weak dependent random variables under minimal conditions.     

Trajectory and Global Attractors of Three-Dimensional Navier--Stokes Systems

We construct the trajectory attractor of a three-dimensional Navier--Stokes system with exciting force . The set consists of a class of solutions to this system which are bounded in , defined on the positive semi-infinite interval of the time axis, and can be extended to the entire time axis so that they still remain bounded-in- solutions of the Navier--Stokes system. In this case any family of bounded-in- solutions of this system comes arbitrary close to the trajectory attractor . We prove that the solutions are continuous in t if they are treated in the space of functions ranging in . The restriction of the trajectory attractor to , , is called the global attractor of the Navier--Stokes system. We prove that the global attractor thus defined possesses properties typical of well-known global attractors of evolution equations. We also prove that as the trajectory attractors and the global attractors of the -order Galerkin approximations of the Navier--Stokes system converge to the trajectory and global attractors and , respectively. Similar problems are studied for the cases of an exciting force of the form depending on time and of an external force rapidly oscillating with respect to the spatial variables or with respect to time .     

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 .     

Beispiele starrer,topologischer Faserungen des reellen projektiven 3-Raums

A spread of a projective 3-space is said to be rigid (German: starr) if the only collineation of leaving invariant is the identity; it is called nearly rigid if there are only finitely many collineations of this kind. A spread of real projective 3-space is called topological if the associated translation plane in the sense of André (or Bruck and Bose) is a topological plane; it is then a 4-dimensional translation plane (abbreviated: 4-dtp) in the terminology of Betten. is rigid if and only if every collineation of the associated 4-dtp fixes the translation line pointwise. In 1977 D. Betten asked for such 4-dtps and termed them rigid. If is nearly rigid, the collineation group of the associated 4-dtp is 5-dimensional.In the present paper, examples of rigid and nearly rigid 4-dtps are constructed. The central tool is the method of crosswise tacking together two topological spreads of along a common regulus, which yields two further topological spreads. In a first step, this method when applied to known spreads produces nearly rigid spreads. Rigid spreads are then obtained by iteration of the method; the simplest example is composed of parts of four elliptic linear line congruences. The rigidness of a spread of is proved by arguments from projective differential geometry applied to the image ( ) under Klein's correspondence from line geometry.     

K-Theory for the Banach Algebra of Operators on James''s Quasi-reflexive Banach Spaces

We prove that the K-groups of the Banach algebra of bounded, linear operators on the pth James space , where 1 < p < , are given by and . Moreover, for each Banach space and each non-zero, closed ideal contained in the ideal of inessential operators, we show that and . This enables us to calculate the K-groups of for each Banach space which is a direct sum of finitely many James spaces and -spaces.