A New Cover of the 3-Local Geometry of Co1

The 3-local geometry of the sporadic simple group Co₁ has been known to have a cover with a flag-transitive automorphism group which is a nonsplit extension of an elementary Abelian 2-group of rank 24 (the Leech lattice modulo 2) by Co₁. It was conjectured that was simply connected. We disprove this conjecture by constructing a double cover of . The automorphism group of is of the shape . However, it is not isomorphic to the involution centralizer of the Monster sporadic simple group.     

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.     

Eine Variante des Fano-Axioms fü angeordnete Ebenen

An ordered plane is an incidence structure ( ) with an order function , which satisfies the axioms (G), (V) and (S), but no continuation--axiom is required. Points a, b E are said to be in distinct sides of a line iff and in the same side if , respectively. For any lines , and we prove that if b,c are in the same side of line A and a,c are in the same side of B , then a and b are in distinct sides of C. As conclusions we deduce that is harmonic and that in each complete quadrangle the intersection points of the diagonals are never collinear, which is known as the axiom of Fano. So the Fano-axiom holds in each ordered plane, and also in those with boundary points.     

Dimension de Hausdorff de la Nervure

For a separable Hilbert space E whose dimension is 2 and for an open subset of E, not empty and different from E, let be the set of all points of which have at least two projections on the close set E\, and let be the set of all the centres of the open balls contained in and which are maximal for inclusion. We show that the Hausdorff dimension dim_H( ) of may be any real value s such that 0sdim E; we also show that can be chosen so that is everywhere dense in and so that we have dim_H( )=1.Associons à un ouvert d'un espace de Hilbert séparable E de dimension 2, non vide et distinct de E, l'ensemble des points de admettant plusieurs projections sur le fermé E\, et l'ensemble des centres des boules ouvertes inclues dans et maximales pour l'inclusion. Nous montrons d'une part que la dimension de Hausdorff dim_H( ) de peut prendre toute valeur réelle s telle que 0sdim E, et d'autre part qu'on peut choisir de sorte que soit dense dans et qu'on ait dim_H( )=1.     

Levi Classes Generated by Nilpotent Groups

Let be a class of all groups G for which the normal closure (x) ^G of every element x belongs to a class . is a Levi class generated by . Let and ₀ be classes of finitely generated nilpotent groups and of torsion-free, finitely generated, nilpotent groups, respectively. We prove that and , and so and . It is shown that quasivarieties and are closed under free products, and that each contains at most one maximal proper subquasivariety. It is also proved that is closed under free products if so is .     

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.     

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.     

Spin Models and Strongly Hyper-Self-Dual Bose-Mesner Algebras

We introduce the notion of hyper-self-duality for Bose-Mesner algebras as a strengthening of formal self-duality. Let denote a Bose-Mesner algebra on a finite nonempty set X. Fix p X, and let and denote respectively the dual Bose-Mesner algebra and the Terwilliger algebra of with respect to p. By a hyper-duality of , we mean an automorphism of such that for all ; and is a duality of . is said to be hyper-self-dual whenever there exists a hyper-duality of . We say that is strongly hyper-self-dual whenever there exists a hyper-duality of which can be expressed as conjugation by an invertible element of . We show that Bose-Mesner algebras which support a spin model are strongly hyper-self-dual, and we characterize strong hyper-self-duality via the module structure of the associated Terwilliger algebra.     

On the Admissibility of Some Linear and Projective Groups in Odd Characteristic

A bijective mapping defined on a finite group G is complete if the mapping defined by , , is bijective. In 1955 M. Hall and L. J. Paige conjectured that a finite group G has a complete mapping if and only if a Sylow 2-subgroup of G is non-cyclic or trivial. This conjecture is still open. In this paper we construct a complete mapping for the projective groups PSL and PGL(2,q),q odd. As a consequence, we prove that in odd characteristic the projective groups PGL(n,q GL , admit a complete mapping.     

Determinants in Triangulated Categories

For a small triangulated category , Bass's K ₁ group is described, and the theorem of the heart is proved. We define the determinant map from to Neeman's , and we compute this map when is the derived category of an Abelian category .     

The Spectrum of the C*-Algebra of Pseudodifferential Boundary Value Problems on a Manifold with Smooth Edges

The C ^*-algebra generated by the operators of pseudodifferential boundary value problems on a manifold with smooth closed disjoint edges and boundary is studied. The operators act in the space L ₂( ) L ₂( ). The goal of this paper is to describe all (up to an equivalence) irreducible representations of the algebra Bibliography: 12 titles.     

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 .     

Majorants and Extreme Points of Unit Balls in Bernstein Spaces

The Bernstein space B ^p () (1 $$ " align="middle" border="0"> 0) is the set of functions from L ^p( ) having Fourier transforms (in the sense of generalized functions) with supports in the compact segment [- , ]. Every function f has an analytic continuation onto the complex plane, which is an entire function of exponential type . The spaces B ^p ()\, are conjugate Banach spaces. Therefore, the closed unit ball in B ^p () has a rich set of extreme (boundary) points: coincides with the weakly ^* closed convex hull of its extreme points. Since, for 1< p< , B ^p () is a uniformly convex space, only the balls and have nontrivially arranged sets of extreme points. In this paper, in terms of zeros of entire functions, we obtain necessary and sufficient conditions of extremeness for functions from .     

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 Properties of Rectangular Hyperbolas

This paper improves an old theorem about a rectangular hyperbola : its centre lies on the pedal circle of any point on with respect to any triangle inscribed in . We also prove that an analogous result holds for Cevian circles. These results are used to obtain new characterisations of the Feuerbach, Jarabek, and Kiepert hyperbolas of a triangle.     

New Proof of the Semmes Inequality for the Derivative of the Rational Function

In the open disk of the complex plane, we consider the following spaces of functions: the Bloch space ; the Hardy--Sobolev space ; and the Hardy--Besov space . It is shown that if all the poles of the rational function R of degree n, , lie in the domain , then , where and depends only on . The second of these inequalities for the case of the half-plane was obtained by Semmes in 1984. The proof given by Semmes was based on the use of Hankel operators, while our proof uses the special integral representation of rational functions.     

Commutants and derivation ranges

In this paper we obtain some results concerning the set , where is the closure in the norm topology of the range of the inner derivation _A defined by _A (X) = AX – XA. Here stands for a Hilbert space and we prove that every compact operator in is quasinilpotent if A is dominant, where is the closure of the range of _A in the weak topology.     

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.     

Hilbert Geometry for Strictly Convex Domains

We prove in this paper that the Hilbert geometry associated with a bounded open convex domain in R ⁿ whose boundary is a ² hypersuface with nonvanishing Gaussian curvature is bi-Lipschitz equivalent to the n-dimensional hyperbolic space H ⁿ . Moreover, we show that the balls in such a Hilbert geometry have the same volume growth entropy as those in H ⁿ .     

Fonctions Séparément Finement Surharmoniques

Nous montrons que toute fonction séparément finement surharmonique sur un ouvert de la topologie produit _{n_1}×s× _{n_k} des topologies fines des espaces R ⁿ ₁,. . ., R ⁿ _k, _{n_1}×s× _{n_k}-localement bornée inférieurement est finement surharmonique dans . On en déduit que toute fonction séparément finement harmonique, _{n_1}×s× _{n_k}-localement bornée sur est finement harmonique dans .Separately Finely Superharmonic Functions Abstract.We prove that every separately finely surperharmonic function on an open set in R ⁿ ₁×s×R ⁿ _k for the product _{n_1}×s× _{n_k} of the fine topologies on the spaces R ⁿ ₁,. . ., R ⁿ _k, _{n_1}×s× _n-_klocally lower bounded, is finely superharmonic in . We then deduce that every separateltly finely harmonic function _{n_1}×s× _{n k}-locally bounded in is finely harmonic.