Spreads and group divisible designs

By removing the components of at-spread of a finite projective spacePG(d, q) from each hyperplane ofPG(d, q), the blocks of a regular group divisible design are obtained We characterize geometrict-spreads as thoset-spreads which are such that the dual of is also a group divisible design.     

Transvection free groups and invariants of polynomial tensor exterior algebras

Let :GGl(n, ) be a representation of a finite groupG over a field such that the ring of invariants is a polynomial algebra . It is known that in the nonmodular case (i.e., when the order of the group is not divisible by the characteristic of ), the invariants ofG acting on the tensor product of a polynomial and an exterior algebra are given by ,d denoting the exterior derivative. We show that in the modular case, the ring of invariants in is of this form if and only if is a polynomial algebra and all pseudoreflections in (G) are diagonalizable.     

A Necessary and Sufficient Condition for the Existence of Some Ternary [n, k, d] Codes Meeting the Greismer Bound

Let k and d be any integers such that k 4 and . Then there exist two integers and in {0,1,2} such that . The purpose of this paper is to prove that (1) in the case k 5 and (,) = (0,1), there exists a ternary code meeting the Griesmer bound if and only if and (2) in the case k 4 and (,) = (0,2) or (1,1), there is no ternary code meeting the Griesmer bound for any integers k and d and (3) in the case k 5 and , there is no projective ternary code for any integers k and such that 1k-3, where and for any integer i 0. In the special case k=6, it follows from (1) that there is no ternary linear code with parameters [233,6,154] , [234,6,155] or [237,6,157] which are new results.     

Rings of regular functions on nilpotent orbits and their covers

Summary LetG be a complex semisimple algebraic group with Lie algebra . Let be a nilpotentG-orbit, its ring of regular functions. We derive a formula for as aG-module and prove some partial results on a cover of . We then relate this formula to various existing multiplicity formulas forK-types in Harish-Chandra bimodules ofG.Supported by National Science Foundation Grant DMS-8505550     

Minihypers and Linear Codes Meeting the Griesmer Bound: Improvements to Results of Hamada, Helleseth and Maekawa

This article improves results of Hamada, Helleseth and Maekawa on minihypers in projective spaces and linear codes meeting the Griesmer bound.In [10,12],it was shown that any -minihyper, with , where , is the disjoint union of points, lines,..., -dimensional subspaces. For q large, we improve on this result by increasing the upper bound on non-square, to non-square, square, , and (4) for square, p prime, p<3, to . In the case q non-square, the conclusion is the same as written above; the minihyper is the disjoint union of subspaces. When q is square however, the minihyper is either the disjoint union of subspaces, or the disjoint union of subspaces and one subgeometry . For the coding-theoretical problem, our results classify the corresponding codes meeting the Griesmer bound.     

The Inverse Problem of the Calculus of Variations for Sixth- and Eighth-order Scalar Ordinary Differential Equations

On the equation manifold of the 2nth-order scalar ordinary differential equation, n3,

Homogeneous algebraic varieties defined by Jordan pairs

Every Jordan pair defines an algebraic varietyX containing as a dense open subset.X is projective (affine) if and only if is separable (radical). The Picard group ofX is generated by the irreducible factors of the generic norm of . If is separable then the automorphism group ofX is the projective group of .     

Spectral-Null Codes and Null Spaces of Hadamard Submatrices

Codes of length 2 ^m over {1, -1} are defined as null spaces of certain submatrices of Hadamard matrices. It is shown that the codewords of all have an rth order spectral null at zero frequency. Establishing the connection between and the parity-check matrix of Reed-Muller codes, the minimum distance of is obtained along with upper bounds on the redundancy of . An efficient algorithm is presented for encoding unconstrained binary sequences into .     

On equiaffine planes

In an equiaffine plane of characteristic 2, let denote the group generated by the affine reflections with axes in a parallel pencil . The centre of is the group of the translations in the direction , and the quotient group is isomorphic to the group generated by the affine reflections in the direction . Let * . Then is the product of at most two affine reflections unless is a glide reflection in which case three affine reflections are needed. In general, the affine reflections may be required to belong to . We also show that the equiaffine group of is the commutator subgroup of the affine group.This paper contains the results of the author's doctoral dissertation submitted to the University of Toronto and written under the supervision of Professor Peter Scherk.     

The Oscillator Character Formula,for isometry groups of split forms in deep stable range

Let and be a reductive dual pair of the type mentioned in the title, with the smaller member. Let and be unitary representations of , which occur in Howe's correspondence. We express the distribution character of in terms of the character of via an explicit integral kernel operator.Oblatum 4-I-1995Research partially supported by the¹UMK Grant 514-M, and the²NSF Grant DMS 9204488.     

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 .     

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.     

Sur l'homotopie rationnelle des espaces fonctionnels

Let X be a nilpotent space such that it exists k1 with H^p (X,) = 0 p > k and H^k (X,) 0, let Y be a (m–1)-connected space with mk+2, then the rational homotopy Lie algebra of Y^X (resp. is isomorphic as Lie algebra, to H^* (X,) (_* (Y) ) (resp.⁺ (X,) (_* (Y) )). If X is formal and Y -formal, then the spaces Y^X and are -formal. Furthermore, if dim _* (Y) is infinite and dim H^* (Y,Q) is finite, then the sequence of Betti numbers of grows exponentially.     

On Complete Arcs Arising from Plane Curves

We point out an interplay between -Frobenius non-classical plane curves and complete -arcs in . A typical example that shows how this works is the one concerning an Hermitian curve. We present some other examples here which give rise to the existence of new complete -arcs with parameters and being a power of the characteristic. In addition, for q a square, new complete -arcs with either and or and are constructed by using certain reducible plane curves.     

Hyperplane Sections of Kantor's Unitary Ovoids

The projective plane is embedded as a variety of projective points in , where M is a nine dimensional -module for the groupG=GL(3,q ²). The hyperplane sections of thisvariety and their stabilizers in the group G aredetermined. When q 2 (mod 3) one such hyperplanesection is a member of the family of Kantor's unitary ovoids.We furtherdetermine all sections whereD has codimension two in M and demonstratethat these are never empty. Consequences are drawn for Kantor'sovoids.     

Optimal Subcodes of Second Order Reed-Muller Codes andMaximal Linear Spaces of Bivectors of Maximal Rank

There are exactlytwo non-equivalent [32,11,12]-codes in the binaryReed-Muller code which contain and have the weight set {0,12,16,20,32}. Alternatively,the 4-spaces in the projective space over the vector space for which all points have rank 4 fall into exactlytwo orbits under the natural action of PGL(5) on .     

Change of variables in a multiple integral for local fields

A proof of the formula for locally compact fields andC ¹-isomorphisms :UV, whereU andV are open subsets of , was never published. In this paper we give two short proofs, one of them is a more elementary variant of the other.     

Skew-symmetric derivations of a complex Lie algebra

Let be a complex Lie algebra, its underlying real Lie algebra, a real form of and ·, · the euclidean product induced by the real part of an hermitian inner product on . Let aut be the Lie algebra of skew-symmetric derivations of . We give necessary and sufficient conditions to ensure that aut is composed of skew-hermitian derivations. As an application, we study holomorphy in large subgroups of isometries of Lie groups.     

On the Nonexistence of q-ary Linear Codes of Dimension Five

There do not exist codes over the Galois field GF attaining the Griesmer bound for for andfor for .