Brauer groups are not characterized by Ulm invariants

Two important invariants of a fieldF are its Brauer groupB(F) and its character groupX(F). IfF is countable, these are countable abelian torsion groups, and so are determined by their Ulm invariants. We show here that Ulm’s invariants do not determine Brauer groups or character groups of uncountable fields. An essential tool, which is entirely group theoretic in nature, is a fact about ultraproducts of torsion groups. Supported in part by NSF Grant No. DMS-8500883. Supported in part by NSA Grant No. MDA904-85-H-0014. Supported in part by NSF Grant No. DMS-8500929.     

On the connected components of the space of line transversals to a family of convex sets

Let ℒ be the space of line transversals to a finite family of pairwise disjoint compact convex sets in ℝ³. We prove that each connected component of ℒ can itself be represented as the space of transversals to some finite family of pairwise disjoint compact convex sets. The research of J. E. Goodman was supported in part by NSF Grant DMS91-22065 and by NSA Grant MDA904-92-H-3069. R. Pollack's research was supported in part by NSF Grant CCR91-22103 and by NSA Grant MDA904-92-H-3075. The research of R. Wenger was supported in part by NSA Grant MDA904-93-H-3026 and by the NSF Regional Geometry Institute (Smith College, July 1993) Grant DMS90-13220.     

On aschbacher’s localC(G; T) theorem

Aschbacher’s localC(G; T) theorem asserts that ifG is a finite group withF*(G)=O ₂(G), andTεSyl₂(G), thenG=C(G; T)K(G), whereC(G; T)=〈N _G (T ₀)|1≠T ₀ charT〉 andK(G) is the product of all near components ofG of typeL ₂(2 ⁿ ) orA ₂ ⁿ ₊₁. Near components are also known asχ-blocks or Aschbacher blocks. In this paper we give a proof of Aschbacher’s theorem in the case thatG is aK-group, i.e., in the case that every simple section ofG is isomorphic to one of the known simple groups. Our proof relies on a result of Meierfrankenfeld and Stroth [MS] on quadratic four-groups and on the Baumann-Glauberman-Niles theorem, for which Stellmacher [St2] has given an amalgam-theoretic proof. Apart from those results, our proof is essentially self-contained. For John Thompson Supported in part by NSF grant #DMS 89-03124, by DIMACS, an NSF Science and Technology Center, funded under contract STC-88-09648, and by NSA grant #MDA-904-91-H-0043. Prof. Gorenstein died on August 26, 1992.     

The lattice of flats and its underlying flag matroid polytope

LetM be a matroid andF the collection of all linear orderings of bases ofM, orflags ofM. We define the flag matroid polytope Δ(F). We determine when two vertices of Δ(F) are adjacent, and provide a bijection between maximal chains in the lattice of flats ofM and certain maximal faces of Δ(F). Supported in part by NSA grant MDA904-95-1-1056.     

The Covering Radius of R(1,9) in R(4,9)

We prove that the covering radius of the Reed-Muller code R(1, 9) in R(4, 9) is 240, not exceeding the quadratic bound.Research supported by NSA Grant MDA 904-93-H-3025     

On the universal embedding of the near-hexagon forU 4(3)

A new proof that the universal embedding of the near-hexagon geometry ofU ₄(3) over F₂ is 21 dimensional. The proof avoids use of modular representation theory.Partially supported by NSA grant MDA 904-93-H-3039.     

Filtrations, mixed complexes, and cyclic homology in mixed characteristic

We describe an approach to computing cyclic homology suited to ground rings of characteristicp or mixed characteristic. We illustrate the method by computing HC for coordinate rings of affine plane curves and for maximal orders of semisimple algebras.Partially supported by NSF Grant No. DMS-8807203 and NSA Grant No. MDA 904-92-H-3026.     

A nonexistence result for abelian menon difference sets using perfect binary arrays

A Menon difference set has the parameters (4N ² ,2N ² -N, N ² -N). In the abelian case it is equivalent to a perfect binary array, which is a multi-dimensional matrix with elements ±1 such that all out-of-phase periodic autocorrelation coefficients are zero. Suppose that the abelian group contains a Menon difference set, wherep is an odd prime, |K|=p , andp ^j–1 (mod exp (H)) for somej. Using the viewpoint of perfect binary arrays we prove thatK must be cyclic. A corollary is that there exists a Menon difference set in the abelian group , where exp(H)=2 or 4 and |K|=3, if and only ifK is cyclic.This work is partially supported by NSA grant # MDA 904-92-H-3057 and by NSF grant # NCR-9200265. The author thanks the Mathematics Department, Royal Holloway College, University of London for its hospitality during the time of this researchThis work is partially supported by NSA grant # MDA 904-92-H-3067     

Embeddings ofC(Δ) andL 1[0, 1] in Banach lattices

It is proved that ifE is a separable Banach lattice withE′ weakly sequentially complete,F is a Banach space andT:E→F is a bounded linear operator withT′F′ non-separable, then there is a subspaceG ofE, isomorphic toC(Δ), such thatT _G is an isomorphism, whereC(Δ) denotes the space of continuous real valued functions on the Cantor discontinuum. This generalizes an earlier result of the second-named author. A number of conditions are proved equivalent for a Banach latticeE to contain a subspace order isomorphic toC(Δ). Among them are the following:L ¹ is lattice isomorphic to a sublattice ofE′;C(Δ)′ is lattice isomorphic to a sublattice ofE′; E contains an order bounded sequence with no weak Cauchy subsequence;E has a separable closed sublatticeF such thatF′ does not have a weak order unit. The research of both authors was partially supported by the National Science Foundation, NSF Grant No MPS 71-02839 A04.     

Clique coverings of the edges of a random graph

The edges of the random graph (with the edge probabilityp=1/2) can be covered usingO(n ²lnlnn/(lnn)²) cliques. Hence this is an upper bound on the intersection number (also called clique cover number) of the random graph. A lower bound, obtained by counting arguments, is (1–)n ²/(2lgn)².Research supported in part by ONR Grant N00014-85K0570 and by NSA/MSP Grant MDA904-90-H-4011.     

The Cluster Function of Single-Valued Functions

Given a single-valued function f between topological spaces X and Y, we interpret the cluster set C(f;x) as a multivalued function F=C(f;⋅) associated to f – the cluster function of f. For appropriate metrizable spaces X and Y, we characterize cluster functions C(f;⋅) among arbitrary set-valued functions F and show that every cluster function F=C(f;⋅) admits a selection h of Baire class 2 such that F=C(h;⋅). Mathematics Subject Classifications (2000) Primary: 54C50, 54C60; secondary: 26A21, 54C65.This research was partially supported by DFG Grant RI 1087/2.     

Asymptotic theory for estimators under random censorship

Summary The product limit estimator of an unknown distributionF is represented as aU-statistic plus an error of the ordero(1/n). Using this, the maximum likelihood estimator of the specific risk rate in the time interval [0,M], is shown to admit a two term Edgeworth expansion. This risk rate for a specific cause of death is defined as the ratio of the probability of death, due to that particular cause, in the time interval [0,M], to the mean life time of an individual up to that time pointM. Similar expansions for the bootstrapped statistics are used to show that the bootstrap distribution, of the studentized estimator of the risk rate, approximates the sampling distribution better than the corresponding normal distribution.Research supported in part by NSA Grant MDA 904-90-H-1001 and by NSF Grant DMS-9007717     

Some remarks on a question of Strassen

Summary Strassen's original functional law of the iterated logarithm for partial sums and Brownian motion examined convergence and clustering in the sup-norm. Here we address what happens if we use the much larger H-norm. We provide the answer to a query which appeared at the end of Strassen's original paper, and also present several contrasting results which are shown to be essentially best possible.Supported in part by NSA Grant MDA-904-93-H-3033Supported in part by NSF Grant DMS-9400024     

Regulus-free Spreads of PG(3,q)

An old conjecture of Bruck and Bose is that every spread of =PG(3,q) could be obtained by starting with a regular spread and reversing reguli. Although it was quickly realized that this conjecture is false, at least forq even, there still remains a gap in the spaces for which it is known that there are spreads which are regulus-free. In several papers Denniston, Bruen, and Bruen and Hirschfeld constructed spreads which were regulus-free, but none of these dealt with the case whenq is a prime congruent to one modulo three. This paper closes that gap by showing that for any odd prime powerq, spreads ofPG(3,q) yielding nondesarguesian flag-transitive planes are regulus-free. The arguments are interesting in that they are based on elementary linear algebra and the arithmetic of finite fields.Dedicated to Hanfried Lenz on the occasion of his 80th birthdayThis work was partially supported by NSA grant MDA 904-95-H-1013.This work was partially supported by NSA grant MDA 904-94-H-2033.     

The continuous spectrum of the relative trace formula for GL(3) over a quadratic extension

LetE/F be a quadratic extension of number fields,G the group GL(3,E) regarded as an algebraic group overF andU a quasi-split unitary group in three variables. Let alsoϑ be a generic character of a maximal unipotent subgroupN ofG. We derive an explicit expression for the integral whereK _cont is the continuous part of the kernel attached to a smooth function of compact support onG(A). In particular, we prove that this expression is absolutely convergent. The result can be used to show that a cuspidal representation ofG contains a vectorφ such thatεφ(u)du≠0 if and only if it is a base change from a representation of GL(3,F). Partially supported by NSF Grant DMS-91-01637.     

Three classes of smooth Banach submanifolds in B(E,F)

Let E, F be two Banach spaces, and B(E, F), Φ(E, F), SΦ(E, F) and R(E,F) be the bounded linear, Fredholm, semi-Frdholm and finite rank operators from E into F, respectively. In this paper, using the continuity characteristics of generalized inverses of operators under small perturbations, we prove the following result Let ∑ be any one of the following sets {T ∈ Φ(E, F) IndexT =const, and dim N(T) = const.}, {T ∈ SΦ(E, F) either dim N(T) = const. ＜ ∞ or codim R(T) = const.＜ ∞} and {T ∈ R(E, F) RankT =const.＜∞}. Then ∑ is a smooth submanifold of B(E, F) with the tangent space TA∑ = {B ∈ B(E,F) BN(A) (∪) R(A)} for any A ∈ ∑. The result is available for the further application to Thom's famous results on the transversility and the study of the infinite dimensional geometry.     

Growth of transcendental entire functions on algebraic varieties

LetX be a complete intersection algebraic variety of codimensionm>1 in ℂ ^m+n . We define the notion of (p,q)-order and (p,q)-K-type for transcendental entire functionsfεO(ℂ ^m+n ) whereK is a non-pluripolar compact subset of ℂ ^m+n . Further, we consider the analogues of (p,q)-order and (p,q)-K-type inO(X). We discuss the series expansions of the functions inO(X) in terms of an orthogonal basis in a Hilbert spaceL ²(X, μ), where μ is a capacitary extremal measure onK. Author is grateful to the NSA for partial support during the period of this research.     

The absolute Galois group of a Hilbertian PRC field

We determine the absolute Galois group of a countable Hilbertian P (seudo) R(eal) C(losed) fieldP of characteristic 0. This group turns out to be real-free, determined up to isomorphism by the topological space of orderings ofP. Examples of such fieldsP are the proper finite extensions of the field of all totally real numbers. Part of this work was done while the authors were fellows of the Institute for Advanced Studies in Jerusalem Supported by NSA grant MDA 14776 and BSF grant 87-00038 Supported by NSA grant MDA 904-89-H-2028     

Some computations of algebraic cycle homology

We compute the algebraic cycle homology for codimension 1 cycles on a variety over a perfect field; our computation agrees with Nart's computation of Bloch's higher Chow groups for codimension 1 cycles. We interpret algebraic cycle homology in terms of sheaves for Voevodsky's h-topology and use this to adapt a recent result of Suslin-Voevodsky: we establish for a complex variety that algebraic cycle homology with Z/n coefficients is naturally isomorphic to singular homology with Z/n coefficients.Partially supported by the NSF and NSA Grant #MDA904-90-H-4006.     

Upper Bounds for the Laplacian Graph Eigenvalues

We first apply non-negative matrix theory to the matrix K = D A, where D and A are the degree-diagonal and adjacency matrices of a graph G, respectively, to establish a relation on the largest Laplacian eigenvalue λ1 (G) of G and the spectral radius p(K) of K. And then by using this relation we present two upper bounds for λ1(G) and determine the extremal graphs which achieve the upper bounds.