On the Galois module structure over CM-fields

In this paper we make a contribution to the problem of the existence of a normal integral basis. Our main result is that unramified realizations of a given finite abelian group Δ as a Galois group Gal (N/K) of an extensionN of a givenCM-fieldK are invariant under the involution on the set of all realizations of Δ overK which is induced by complex conjugation onK and by inversion on Δ. We give various implications of this result. For example, we show that the tame realizations of a finite abelian group Δ of odd order over a totally real number fieldK are completely characterized by ramification and Galois module structure.     

Structure of augmentation quotients of finite homocyclic abelian groups

Let G be a finite abelian group and its Sylow p-subgroup a direct product of copies of a cyclic group of order p~r,i.e.,a finite homocyclic abelian group.LetΔ~n (G) denote the n-th power of the augmentation idealΔ(G) of the integral group ring ZG.The paper gives an explicit structure of the consecutive quotient group Q_n(G)=Δ~n(G)/Δ~(n 1)(G) for any natural number n and as a consequence settles a problem of Karpilovsky for this particular class of finite abelian groups.     

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     

The bernoulli property for expansive ℤ2 actions on compact groups

We show that an expansive ℤ² action on a compact abelian group is measurably isomorphic to a two-dimensional Bernoulli shift if and only if it has completely positive entropy. The proof uses the algebraic structure of such actions described by Kitchens and Schmidt and an algebraic characterisation of theK property due to Lind, Schmidt and the author. As a corollary, we note that an expansive ℤ² action on a compact abelian group is measurably isomorphic to a Bernoulli shift relative to the Pinsker algebra. A further corollary applies an argument of Lind to show that an expansiveK action of ℤ² on a compact abelian group is exponentially recurrent. Finally an example is given of measurable isomorphism without topological conjugacy for ℤ² actions. Supported in part by N.S.F. grant No. DMS-88-02593 at the University of Maryland and by N.S.F. grant No. DMS-91-03056 at Ohio State University.     

On the fundamental group of real toric varieties

LetX (Δ) be the real toric variety associated to a smooth fan Δ. The main purpose of this article is: (i) to determine the fundamental group and the universal cover ofX (Δ), (ii) to give necessary and sufficient conditions on Δ under which π₁(X(Δ)) is abelian, (iii) to give necessary and sufficient conditions on Δ under whichX(Δ) is aspherical, and when Δ is complete, (iv) to give necessary and sufficient conditions forC _Δ to be aK (π, 1) space whereC _Δ is the complement of a real subspace arrangement associated to Δ.     

Commuting elements and spaces of homomorphisms

This article records basic topological, as well as homological properties of the space of homomorphisms Hom(π,G) where π is a finitely generated discrete group, and G is a Lie group, possibly non-compact. If π is a free abelian group of rank equal to n, then Hom(π, G) is the space of ordered n–tuples of commuting elements in G. If G = SU(2), a complete calculation of the cohomology of these spaces is given for n = 2, 3. An explicit stable splitting of these spaces is also obtained, as a special case of a more general splitting. Alejandro Adem was partially supported by the NSF and NSERC. Frederick R. Cohen was partially supported by the NSF, grant number 0340575.     

The moduli space of curves of genus two covering elliptic curves

We consider the moduli spaceS _n of curvesC of genus 2 with the property:C has a “maximal” mapf of degreen to an elliptic curveE. Here, the term “maximal” means that the mapf∶C→E doesn't factor over an unramified cover ofE. By Torelli mapS _n is viewed as a subset of the moduli spaceA ₂ of principally polarized abelian surfaces. On the other hand the Humbert surfaceH _Δ of invariant Δ is defined as a subvariety ofA ₂(C), the set of C-valued points ofA ₂. The purpose of this paper is to releaseS _n withH _Δ.     

Duality of Fourier type with respect to locally compact Abelian groups

It is shown that a Banach space X has Fourier type p with respect to a locally compact abelian group G if and only if the dual space X′ has Fourier type p with respect to G if and only if X has Fourier type p with respect to the dual group of G. This extends previously known results for the classical groups and the Cantor group to the setting of general locally compact abelian groups. Supported by DFG grant Hi 584/2-2. Partially supported by a DAAD-grant A/02/46571.     

The linear arboricity of graphs

Alinear forest is a forest in which each connected component is a path. Thelinear arboricity la(G) of a graphG is the minimum number of linear forests whose union is the set of all edges ofG. Thelinear arboricity conjecture asserts that for every simple graphG with maximum degree Δ=Δ(G), . Although this conjecture received a considerable amount of attention, it has been proved only for Δ≦6, Δ=8 and Δ=10, and the best known general upper bound for la(G) is la(G)≦⌈3Δ/5⌉ for even Δ and la(G)≦⌈(3Δ+2)/5⌉ for odd Δ. Here we prove that for everyɛ>0 there is a Δ₀=Δ₀(ɛ) so that la(G)≦(1/2+ɛ)Δ for everyG with maximum degree Δ≧Δ₀. To do this, we first prove the conjecture for everyG with an even maximum degree Δ and withgirth g≧50Δ. Research supported in part by Allon Fellowship, by a Bat Sheva de Rothschild grant, by the Fund for Basic Research administered by the Israel Academy of Sciences and by a B.S.F. Bergmann Memorial grant.     

On the first Hochschild cohomology group of an algebra

Let A≡KΔ /I be a factor of a path algebra. We develop a strategy to compute dim H ¹(A), the dimension of the first Hochschild cohomology group of A, using combinatorial data from (Δ,I). That allows us to connect dim H ¹(A) with the rank and p-rank of the fundamental group π₁(Δ,I) of (Δ,I). We get explicit formulae for dim H ¹(A), when every path in Δ parallel to an arrow belongs to I or when I is homogeneous. Received: 12 April 1999 / Revised version: 9 October 2000     

On two Hamilton cycle problems in random graphs

We study two problems related to the existence of Hamilton cycles in random graphs. The first question relates to the number of edge disjoint Hamilton cycles that the random graph G _n,p contains. δ(G)/2 is an upper bound and we show that if p ≤ (1 + o(1)) ln n/n then this upper bound is tight whp. The second question relates to how many edges can be adversarially removed from G _n,p without destroying Hamiltonicity. We show that if p ≥ K ln n/n then there exists a constant α > 0 such that whp G − H is Hamiltonian for all choices of H as an n-vertex graph with maximum degree Δ(H) ≤ αK ln n. Research supported in part by NSF grant CCR-0200945. Research supported in part by USA-Israel BSF Grant 2002-133 and by grant 526/05 from the Israel Science Foundation.     

Upper bounds for the diameter and height of graphs of convex polyhedra

Let Δ(d, n) be the maximum diameter of the graph of ad-dimensional polyhedronP withn-facets. It was conjectured by Hirsch in 1957 that Δ(d, n) depends linearly onn andd. However, all known upper bounds for Δ(d, n) were exponential ind. We prove a quasi-polynomial bound Δ(d, n)≤n ^{2 logd+3}. LetP be ad-dimensional polyhedron withn facets, let ϕ be a linear objective function which is bounded onP and letv be a vertex ofP. We prove that in the graph ofP there exists a monotone path leading fromv to a vertex with maximal ϕ-value whose length is at most . This research was supported in part by a BSF grant, by a GIF grant, and by ONR-N00014-91-C0026.     

Noded Teichmüller spaces

LetG be a finitely generated Kleinian group and let Δ be an invariant collection of components in its region of discontinuity. The Teichmüller spaceT(Δ,G) supported in Δ is the space of equivalence classes of quasiconformal homeomorphisms with complex dilatation invariant underG and supported in Δ. In this paper we propose a partial closure ofT(Δ,G) by considering certain deformations of the above hemeomorphisms. Such a partial closure is denoted byNT(Δ,G) and called thenoded Teichmüller space ofG supported in Δ. Some concrete examples are discussed. Partially supported by Projects Fondecyt 1030252, 1030373, 1040333, Projects UTFSM 12.05.21, 12.05.23 and by grant of the University of Bergen.     

Density results in the Δ20 e-degrees

We show that the Δ⁰ ₂ enumeration degrees are dense. We also show that for every nonzero n-c. e. e-degree a, with n≥ 3, one can always find a nonzero 3-c. e. e-degree b such that b < a on the other hand there is a nonzero ωc. e. e-degree which bounds no nonzero n-c. e. e-degree. Received: 13 June 2000 / Published online: 3 October 2001     

Combined perturbation bounds: Ⅱ. Polar decompositions

In this paper, we study the perturbation bounds for the polar decomposition A= QH where Q is unitary and H is Hermitian. The optimal (asymptotic) bounds obtained in previous works for the unitary factor, the Hermitian factor and singular values of A are σ2r||△Q||2F ≤ ||△A||2F,1/2||△H||2F ≤ ||△A||2F and ||△∑||2F ≤ ||△A||2F, respectively, where ∑ = diag(σ1, σ2,..., σr, 0,..., 0) is the singular value matrix of A and σr denotes the smallest nonzero singular value. Here we present some new combined (asymptotic)perturbation bounds σ2r ||△Q||2F 1/2||△H||2F≤ ||△A||2F and σ2r||△Q||2F ||△∑ ||2F ≤||△A||2F which are optimal for each factor. Some corresponding absolute perturbation bounds are also given.     

On a graph packing conjecture by Bollobás,Eldridge and Catlin

Two graphs G ₁ and G ₂ of order n pack if there exist injective mappings of their vertex sets into [n], such that the images of the edge sets are disjoint. In 1978, Bollobás and Eldridge, and independently Catlin, conjectured that if (Δ(G ₁) + 1)(Δ(G ₂) + 1) ≤ n + 1, then G ₁ and G ₂ pack. Towards this conjecture, we show that for Δ(G ₁),Δ(G ₂) ≥ 300, if (Δ(G ₁) + 1)(Δ(G ₂) + 1) ≤ 0.6n + 1, then G ₁ and G ₂ pack. This is also an improvement, for large maximum degrees, over the classical result by Sauer and Spencer that G ₁ and G ₂ pack if Δ(G ₁)Δ(G ₂) < 0.5n. This work was supported in part by NSF grant DMS-0400498. The work of the second author was also partly supported by NSF grant DMS-0650784 and grant 05-01-00816 of the Russian Foundation for Basic Research. The work of the third author was supported in part by NSF grant DMS-0652306.     

Strong Morita equivalence of higher-dimensional noncommutative tori. II

We show that two C*-algebraic noncommutative tori are strongly Morita equivalent if and only if they have isomorphic ordered K ₀-groups and centers, extending N. C. Phillips’s result in the case that the algebras are simple. This is also generalized to the twisted group C*-algebras of arbitrary finitely generated abelian groups. This research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada, held by George A. Elliott.     

Verifying Huppert’s Conjecture for <Superscript>2</Superscript><Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">q</Emphasis><Superscript>2</Superscript>)

Let G be a finite group and cd(G) be the set of irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd(G) = cd(H), then G ≅ H×A, where A is an abelian group. In this paper, we verify the conjecture for the twisted Ree groups ² G ₂(q ²) for q ² = 3^2m + 1, m ≥ 1. The argument involves verifying five steps outlined by Huppert in his arguments establishing his conjecture for many of the nonabelian simple groups.     

Graded metrics adapted to splittings

Homogeneous graded metrics over split ℤ₂-graded manifolds whose Levi-Civita connection is adapted to a given splitting, in the sense recently introduced by Koszul, are completely described. A subclass of such is singled out by the vanishing of certain components of the graded curvature tensor, a condition that plays a role similar to the closedness of a graded symplectic form in graded symplectic geometry: It amounts to determining a graded metric by the data {g, ω, Δ′}, whereg is a metric tensor onM, ω 0 is a fibered nondegenerate skewsymmetric bilinear form on the Batchelor bundleE → M, and Δ′ is a connection onE satisfying Δ′ω = 0. Odd metrics are also studied under the same criterion and they are specified by the data {κ, Δ′}, with κ ∈ Hom (TM, E) invertible, and Δ′κ = 0. It is shown in general that even graded metrics of constant graded curvature can be supported only over a Riemannian manifold of constant curvature, and the curvature of Δ′ onE satisfiesR ^Δ′ (X,Y)² = 0. It is shown that graded Ricci flat even metrics are supported over Ricci flat manifolds and the curvature of the connection Δ′ satisfies a specific set of equations. 0 Finally, graded Einstein even metrics can be supported only over Ricci flat Riemannian manifolds. Related results for graded metrics on Ω(M) are also discussed. Partially supported by DGICYT grants #PB94-0972, and SAB94-0311; IVEI grant 95-031; CONACyT grant #3189-E9307.