Gromov Hyperbolicity of Riemann Surfaces

We study the hyperbolicity in the Gromov sense of Riemann surfaces. We deduce the hyperbolicity of a surface from the hyperbolicity of its ＂building block components＂. We also prove the equivalence between the hyperbolicity of a Riemann surface and the hyperbolicity of some graph associated with it. These results clarify how the decomposition of a Riemann surface into Y-pieces and funnels affects the hyperbolicity of the surface. The results simplify the topology of the surface and allow us to obtain global results from local information.     

On the number of generators needed for free profinite products of finite groups

We provide lower estimates on the minimal number of generators of the profinite completion of free products of finite groups. In particular, we show that if C ₁, ..., C _n are finite cyclic groups then there exists a finite group G which is generated by isomorphic copies of C ₁, ..., C _n and the minimal number of generators of G is n. The first author’s research is partially supported by NSF grant DMS-0701105. The second author’s research is partially supported by OTKA grant T38059 and the Magyary Zoltán Postdoctoral Fellowship.     

On strictly ergodic models which are not almost topologically conjugate

Answering a question raised by Glasner and Rudolph (1984) we construct uncountably many strictly ergodic topological systems which are metrically isomorphic to a given ergodic system (X, ℬ,μ, T) but not almost topologically conjugate to it. This paper is part of the second author’s Ph.D. thesis, written under the supervision of Professor A. Bellow of the Department of Mathematics, Northwestern University. The author is grateful for her encouragement and advice. We acknowledge B. Weiss for helpful comments.     

On the supercyclicity and hypercyclicity of the operator algebra

Let B（X） be the operator algebra for a separable infinite dimensional Hilbert space H, endowed with the strong operator topology or ＊-strong topology. We give sufficient conditions for a continuous linear mapping L ： B（X） →B（X） to be supercyclic or ,-supercyclic. In particular our condition implies the existence of an infinite dimensional subspace of supercyclic vectors for a mapping T on H. Hypercyclicity of the operator algebra with strong operator topology was studied＇ by Chan and here we obtain an analogous result in the case of ＊-strong operator topology.     

The distance between homotopy classes of <Emphasis Type="Italic">S</Emphasis><Superscript>1</Superscript>-valued maps in multiply connected domains

Certain Sobolev spaces of S ¹-valued functions can be written as a disjoint union of homotopy classes. The problem of finding the distance between different homotopy classes in such spaces is considered. In particular, several types of one-dimensional and two-dimensional domains are studied. Lower bounds are derived for these distances. Furthermore, in many cases it is shown that the lower bounds are sharp but are not achieved. The first author’s work of was supported in part by NSF grant 0503887. The second author’s research of was supported by G.S. Elkin research fund.     

Upper bounds for growth in the Ryll-Nardzewski function of anω-categorical,ω-stable theory

We establish a sharp upper bound for growth in the sequences _k (T):=the number ofk-types consistent withT, forT ω-categorical andω-stable. This paper forms part of the author’s doctoral dissertation written at the University of Maryland under the direction of David W. Kueker.     

Extending the factorization principle to hypergeometric series of general form

It is shown that the formulas of operator factorization of hypergeometric functions obtained in the author’s previous works can be extended to hypergeometric series of the most general form. This generalization does not make the technical apparatus of the factorization method more complicated. As an example illustrating the practical effectiveness of the formulas obtained in the paper, we analyze transformation properties of the Horn seriesG ₃, whose structure is typical for general hypergeometric functions. It is shown that Erdélyi’s transformation formula relating the seriesG ₃ to the Appell functionF ₂, contains erroneous expressions in the arguments ofG ₃. The correct analog of Erdélyi’s formula is found, and some new transformations of the seriesG ₃ are presented. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 573–581, April, 2000.     

Bounds on Codes Derived by Counting Components in Varshamov Graphs

We are interested in improving the Varshamov bound for finite values of length n and minimum distance d. We employ a counting lemma to this end which we find particularly useful in relation to Varshamov graphs. Since a Varshamov graph consists of components corresponding to low weight vectors in the cosets of a code it is a useful tool when trying to improve the estimates involved in the Varshamov bound. We consider how the graph can be iteratively constructed and using our observations are able to achieve a reduction in the over-counting which occurs. This tightens the lower bound for any choice of parameters n, k, d or q and is not dependent on information such as the weight distribution of a code. This work is taken from the author’s thesis [10]     

Twisted Semigroup Crossed Products and Twisted Toeplitz Algebras of Ordered Groups

Let г^＋ be the positive cone of a totally ordered abelian group г, and σa cocycle in г. We study the twisted crossed products by actions of г＋ as endomorphisms of C^＊-algebras, and use this to generalize the theorem of Ji.     

Irreducible actions and faithful actions of hopf algebras

LetH be a Hopf algebra acting on an algebraA. We will examine the relationship betweenA, the ring of invariantsA ^H, and the smash productA # H. We begin by studying the situation whereA is an irreducibleA # H module and, as an application of our first main theorem, show that ifD is a division ring then [D : D ^H]≦dimH. We next show that prime rings with central rings of invariants satisfy a polynomial identity under the action of certain Hopf algebras. Finally, we show that the primeness ofA # H is strongly related to the faithfulness of the left and right actions ofA # H onA. The first author was supported by the Faculty Research and Development Fund of the College of Liberal Arts and Sciences and the University Research Council at DePaul University and NSF Grant No. 8521704. He also wishes to thank Ben Gurion University for its hospitality, where much of this work was done. The second and third authors were supported by the Fund for Basic Research administered by the Israel Academy of Sciences and Humanities. Part of this author’s contribution is included in her Ph.D. thesis at Ben Gurion University under the supervision of the second author.     

Path-closed sets

Given a digraphG = (V, E), call a node setT ⊆V path-closed ifv, v′ εT andw εV is on a path fromv tov′ impliesw εT. IfG is the comparability graph of a posetP, the path-closed sets ofG are the convex sets ofP. We characterize the convex hull of (the incidence vectors of) all path-closed sets ofG and its antiblocking polyhedron inR ^v , using lattice polyhedra, and give a minmax theorem on partitioning a given subset ofV into path-closed sets. We then derive good algorithms for the linear programs associated to the convex hull, solving the problem of finding a path-closed set of maximum weight sum, and prove another min-max result closely resembling Dilworth’s theorem.     

Theories of linear order

LetT be a complete theory of linear order; the language ofT may contain a finite or a countable set of unary predicates. We prove the following results. (i) The number of nonisomorphic countable models ofT is either finite or 2^ω. (ii) If the language ofT is finite then the number of nonisomorphic countable models ofT is either 1 or 2^ω. (iii) IfS ₁(T) is countable then so isS _n(T) for everyn. (iv) In caseS ₁(T) is countable we find a relation between the Cantor Bendixon rank ofS ₁(T) and the Cantor Bendixon rank ofS _n(T). (v) We define a class of modelsL, and show thatS ₁(T) is finite iff the models ofT belong toL. We conclude that ifS ₁(T) is finite thenT is finitely axiomatizable. (vi) We prove some theorems concerning the existence and the structure of saturated models. Most of the results in this paper appeared in the author’s Master of Science thesis which was prepared at the Hebrew University under the supervision of Professor H. Gaifman.     

Minor and minimum cycle bases of a 3-connected planar graph

In this paper, we prove that if any set of ｜E（G）｜- ｜V（G）｜ ＋ 1 facial cycles of a 3-connected planar graph G embedded in the plane doesn＇t form a minimum cycle base of G, then any minimum cycle base of G contains a separating cycle, and G has a minor isomorphic to T6, where T6 is the graph obtained from the complete graph K6 by deleting a path with four edges.     

A predual ofl 1 which is not isomorphic to aC(K) space

We give an example of a Banach spaceX such that (i)X ^* is isometric tol ₁, (ii)X is isometric to a subspace ofC(ω^θ) and (iii)X is not isomorphic to a complemented subspace of anyC(K) space. This is a part of the first author’s Ph. D. Thesis prepared in the Hebrew University of erusalem under the supervision of the second author.     

Weyl’s Theorem and Perturbations

In the present paper we examine the stability of Weyl’s theorem under perturbations. We show that if T is an isoloid operator on a Banach space, that satisfies Weyl’s theorem, and F is a bounded operator that commutes with T and for which there exists a positive integer n such that Fⁿ is finite rank, then T + F obeys Weyl’s theorem. Further, we establish that if T is finite-isoloid, then Weyl’s theorem is transmitted from T to T + R, for every Riesz operator R commuting with T. Also, we consider an important class of operators that satisfy Weyl’s theorem, and we give a more general perturbation results for this class.     

Characterization of groups with generalized chernikov periodic part

A characterization of groups with generalized Chernikov periodic part is obtained first in the class of groups without elements of order 2 and then without this restriction. Two author’s theorems for periodic groups are generalized to mixed groups. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 270–275, February, 2000.     

Minimal models of theories of one function symbol

IfT ₁ is a theory of one function symbol andT ₁ has a minimal model which is not prime, thenT ₁ has 2^No non-isomorphic minimal models. This paper is part of the author’s doctoral dissertation prepared at the Hebrew University under the supervision of Professor Haim Gaifman.     

Generalized frobenius groups. II

A pair (G, K) in whichG is a finite group andK a normal nontrivial proper subgroup ofG is said to be an F2-pair (a Frobenius type pair) if |C _G (x)|=|C _G/K (xK)| for allx∈G\K. A theorem of Camina asserts that in this case eitherK orG/K is ap-group or elseG is a Frobenius group with Frobenius kernelK. The structure ofG will be described here under certain assumptions on the Sylowp-subgroups ofG. This author’s research was partially supported by the Technion V.P.R. fund — E.L.J. Bishop research fund. This author’s research was partially supported by the MPI fund.     

Tensor products, multiplications and Weyl’s theorem

Tensor productsZ=T ₁⊗T ₂ and multiplicationsZ=L _{T 1} R _{T 2} do not inherit Weyl’s theorem from Weyl’s theorem forT ₁ andT ₂. Also, Weyl’s theorem does not transfer fromZ toZ*. We prove that ifT _i,i=1, 2, has SVEP (=the single-valued extension property) at points in the complement of the Weyl spectrumσ _w(T_i) ofT _i, and if the operatorsT _i are Kato type at the isolated points ofσ(T_i), thenZ andZ* satisfy Weyl’s theorem.     

A Note on the Browder＇s and Weyl＇s Theorem

Let T be a Banach space operator, E（T） be the set of all isolated eigenvalues of T and π（T） be the set of all poles of T. In this work, we show that Browder＇s theorem for T is equivalent to the localized single-valued extension property at all complex numbers λ in the complement of the Weyl spectrum of T, and we give some characterization of Weyl＇s theorem for operator satisfying E（T） = π（T）. An application is also given.