Electronic Edition Vol. 29, No. 1, 2003

Editors: H. Amann (Zürich), G. Auchmuty (Houston), D. Bao (Houston), H. Brezis (Paris), J. Damon (Chapel Hill), K. Davidson (Waterloo), C. Hagopian (Sacramento), R. M. Hardt (Rice), J. Hausen (Houston), J. A. Johnson (Houston), J. Nagata (Osaka), V. I. Paulsen (Houston), G. Pisier (College Station and Paris), S. W. Semmes (Rice)
Managing Editor: K. Kaiser (Houston)

Houston Journal of Mathematics


D.F. Anderson, Department of Mathematics, The University of Tennessee, Knoxville, Tennessee 37996 (anderson@math.utk.edu), G. Chang, Department of Mathematics, University of Incheon, Incheon 402-749 (whan@incheon.ac.kr) and J. Park, Department of Mathematics, Inha University, Incheon 402-751, Korea (jnpark@math.inha.ac.kr).
Generalized Weakly Factorial Domain, pp. 1-13.
ABSTRACT. In this paper, we study integral domains in which each nonzero prime ideal contains a primary element. We show that if R is an integrally closed domain, then each nonzero prime ideal of R[X] contains a primary element if and only if R[X] is an almost weakly factorial domain, if and only if R is an almost weakly factorial and almost GCD-domain. We also prove that if R is an almost weakly factorial Noetherian domain, then the integral closure of R is an almost factorial domain.

John Dauns, Tulane University, New Orleans, Louisiana 70118-5698 (dauns@tulane.edu) and Yiqiang Zhou, Memorial University of Newfoundland, St.John's, Newfoundland A1C 5S7, Canada (zhou@math.mun.ca)
Type dimension of modules and chain conditions, pp. 15-23.
ABSTRACT. For submodules A,B of a module M, a formula relating the type dimensions of M, A and M/A is derived and a formula for the type dimension of A+B is obtained. Modules satisfying both type descending and ascending chain conditions are characterized.

Emil Daniel Schwab, Department of Mathematical Sciences, University of Oradea, 3700 Oradea, Romania and Department of Mathematical Sciences, University of Texas at El Paso, El Paso, Texas, 79968 (sehwab@math.utep.edu).
On Triangular Categories, pp. 25-40.
ABSTRACT. The triangular categories defined and studied by P. Leroux [1] are particular Möbius categories. In this paper, considering a lattice-triangular category C, we study the basic properties of the "category of fractions" I(C) associated to C (the universal property and the representation theorem for I(C) are also presented). These properties are determined by the lattices of "C-morphisms having the same codomain", the lattices that also essentially determine the incidence algebra of the category C.
[1] Catégories triangulaires. Exemples, applications et problémes, Rapport de recherche, Univ. du Québec a Montr&eacuteal (1980), 72 pp.

Augusto Nobile, Louisiana State University, Department of Mathematics, Baton Rouge, LA 70803, USA (nobile@math.lsu.edu).
Singularities of Reflexive Sheaves, pp. 41-57.
ABSTRACT. Singularities of rank-two reflexive sheaves on smooth three-folds are studied. They must be isolated, and several numerical invariants are attached to them. It is shown that it is possible to eliminate the singularities by taking quadratic transformations and suitable transforms of the sheaf.

Y. Nikolayevsky, Department of Mathematics, Melbourne University, Parkville, 3052, Victoria, Australia (Y.Nikolayevsky@latrobe.edu.au)
Osserman Manifolds and Clifford structures , pp. 59-75.
ABSTRACT. A Riemannian manifold is called Osserman if the eigenvalues of its Jacobi operator are constant on the unit tangent bundle. R. Osserman conjectured that every such manifold is two-point homogeneous. A Riemannian manifold has a Clifford structure, if its curvature tensor can be expressed quadratically in terms of anticommuting almost Hermitian structures. We study the connection between the Osserman property and the existence of a Clifford structure and show, in particular, that a Riemannian manifold with a Clifford structure is two-point homogeneous, with few exceptions.

Duran, Carlos E., IVIC, matematicas, Aptdo 21827, Caracas 1020A, Venezuela (cduran@cauchy.ivic.ve).
Finsler Almost Blaschke manifolds, pp. 72-92.
ABSTRACT. A Riemannian (or Finslerian) manifold is said to be a Blaschke manifold if the injectivity radius and the diameter coincide. The Blaschke conjecture asserts that any Riemannian Blaschke manifold is isometric to a compact, rank one symmetric space. A manifold is said to be an almost-Blaschke manifold if the injectivity radius and the diameter are `almost equal' in a scale-independent sense. Given the Blaschke conjecture, it is natural to expect that a Riemannian almost Blaschke manifold is at least homeomorphic to a compact, rank one symmetric space. It has been shown by Duromeric (1984) that this is indeed the case under certain additional restrictions. The main result of this paper is the construction of examples of almost-Blaschke Finsler metrics on any product of real of complex Grassmann manifolds, thus showing that the rigidity related to the Blaschke condition goes beyond simple calculus of variations techniques which typically apply in a similar fashion in the Riemannian and Finslerian case. Several related problems are considered.

Caldas, Miguel, Departamento de Matemática Aplicada, Universidade Federal Fluminense, Niterói, Rio de Janeiro, CEP 24220-140, Brasil (gmamccs@vm.uff.br) and Jafari, Saeid, Department of Mathematics and Physics, Roskilde University, 4000 Roskilde, Denmark (sjafari@ruc.dk).
On Some Low Separation Axioms in Topological Spaces pp. 93-104.
ABSTRACT. We say that a subset A of a topological space X is D Lambdadelta-set if there are two (Lambda,delta)-open subsets (in the sense of Georgiou, Jafari and Noiri (2001))  U and V, such that A is the diference of U and V with U as a proper subset of X. If we replace open sets in the usual definition of Ti (i=0,1,2) with D Lambdadelta-set (resp. (Lambda,delta)-open ) we obtain new weak separation axioms. In this paper we study some of the characterizations and properties of these separation axioms. The implications of these axioms among themselves are also investigated. Moreover, the notions of Lambdadelta-Ri spaces (i=0,1) analogous to  Ri spaces (i=0,1) are presented and the necessary and sufficient conditions for a space to be   Lambdadelta-R0 are given.

E.E. Grace, Department of Mathematics, Arizona State University, Tempe, Arizona 85287-1804 (egrace@asu.edu) and E.J. Vought, Department of Mathematics and Statistics, California State University, Chico; Chico, California 95929-0525 (eeevought@worldnet.att.net).
Preservation of Properties of Continua by Refinable Maps pp. 105-112.
ABSTRACT. If f is a refinable map from a continuum X onto Y that is (1) an acyclic curve or (2) not an n-od, then Y is (1) an acyclic curve or (2) not an n-od, respectively. An example is given where X is hereditarily divisible by points and Y is not. A survey of research results concerning preservation of properties of continua by refinable maps and by the inverses of refinable maps is included.

R. Lowen and D. Vaughan Department of Mathematics and Computer Science, University of Antwerp, Antwerp, Belgium (rlow@ruca.ua.ac.be) and M. Sioen, Department of Mathematics, Vrije Universiteit Brussel, Brussels, Belgium (msioen@vub.ac.be).
Completing Quasi-metric Spaces -- an Alternative Approach , pp. 113--113.
ABSTRACT. We define a completion theory for approach spaces which satisfy only a very mild separation  property. The completion  agrees with the usual metric  completion  theory  for metric spaces. We can complete any quasi-metric space,  but remarkably the completion of a quasi-metric  space  need not be quasi-metric  and  can be a genuine approach space. The theory corresponds well with the strict completion of nearness spaces.

Amine Fawaz, The University of Texas of the Permian Basin, Department of Mathematics, 4901 East University, Odessa, TX 79762 (fawaz_a@utpb.edu).
A Note on Riemannian Flows on 3-Manifolds , pp. 137-147.
ABSTRACT. Let L be a Riemannian flow on a closed 3-dimensional manifold M and g a holonomy invariant metric on M. We give a geometric interpretation of the first Chern class of the normal bundle to L. We consider a projectable vector field X on M perpendicular to L and we define the residue of X at a singular point. We give an expression of the sum of the residues of X which in particular reduces to a curvature integral over M.

Salvador Hernandez, Universitat Jaume I, Departamento de Matematicas, 12071-Castellon, Spain (hernande@mat.uji.es)
Uniformly continuous mappings defined by isometries of spaces of bounded uniformly continuous functions , pp. 149-155.
ABSTRACT. Let μ X be a complete metrizable uniform space and let C*(μ X) denote its algebra of bounded uniformly continuous real-valued functions. In this paper we show that the metric structure of C*(μ X)determine the structure of μ X up to an isomorphism of uniform spaces. This result is applied to prove that if (G,R) is a topological group equipped with its associate right uniformity such that for every neighbourhood U of the neutral element there is a closed subgroup H contained in U with the space of left cosets (G/H,R/H) being metrizable, then G is a SIN group if and only if every left uniformly continuous real-valued function on G is right uniformly continuous.

Jack T. Goodykoontz, Jr., Department of Mathematics, West Virginia University, Morgantown, West Virginia 26506-6310} (jgoody@math.wvu.edu) and Choon Jai RheeDepartment of Mathematics, Wayne State University, Detroit, MI 48202} (rhee@math.wayne.edu).
The Hyperspace of Closed Connected Subsets of a Euclidean Space, pp. 157-162.
ABSTRACT. Let Rn denote the n-dimensional Euclidean space and C(Rn ) denote the hyperspace of closed connected subsets of Rn , with the Vietoris topology. The following results are obtained: (1) C(R1) is homeomorphic to R1×[0,1] and hence is locally compact; (2) If n≥2 and A∈ C(Rn ), then C(Rn ) is locally compact at A if and only if A is compact; (3) If n≥ 2, then C(Rn ) is not metrizable; (4) C(Rn ) is separable.

S. R. Foguel, Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel (foguel@math.huji.ac.il).
On disjoint Markov operators , pp. 163-171.
ABSTRACT. Let (X, Σ, λ) be a probability space, let P1, P2 be two Markov operators defined on L (X, Σ, λ) and assume that P1 and P2 are induced by transition probabilities denoted again by P1 and P2 respectively. For every subset β ⊂ X × X measurable with respect to the product σ-algebra Σ × Σ, put β' =( X × X)\ β and βx = { y ∈ X: (x, y) ∈ β}. Our main result is that the Markov operators P1 and P2 are disjoint iff there exists a Σ × Σ-measurable subset α of X × X such that P1(x, αx) = P2(x, α'x) = 0 λ-a.e.

P. Bandyopadhyay, Stat--Math Division, Indian Statistical Institute, 203, B. T. Road, Kolkata 700 108, India (pradipta@isical.ac.in), V. P. Fonf, Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel (fonf@cs.bgu.ac.il), B.-L. Lin, Department of Mathematics, The University of Iowa, Iowa City, IA 52242 USA (bllin@math.uiowa.edu) and M. Martin, Departamento de Analisis Matematico, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain (mmartins@ugr.es).
Structure of nested sequences of balls in Banach spaces , pp. 173-193.
ABSTRACT. In this paper, we study the structure of the union of unbounded nested sequences of balls, and use them to characterize some geometric properties of X*. We show that the union of an unbounded nested sequence of balls is a cone if the centers of the balls lie in a finite dimensional subspace. However, in general, such a union need not be a cone. In fact, examples can be constructed, up to renorming, in any infinite dimensional Banach space. We also study when such an union is the intersection of at most k half-spaces, and relate it with the number of extreme points of any face of the dual ball.

Juan Bes and Kit C. Chan, Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio 43403 (jbes@bgnet.bgsu.edu), (kchan@math.bgsu.edu).
Denseness of Hypercyclic Operators on a Frechet Space, pp. 195-206.
ABSTRACT. In 1969 Rolewicz raised the question whether every separable infinite dimensional Banach space admits a hypercyclic operator. This question was answered recently in the positive, and the result was generalized to the Frechet space case, in papers of Ansari, Bernal-Gonzalez, and Bonet and Peris. In the present paper, we further show that the hypercyclic operators on a separable infinite dimensional Frechet space form a dense subset of the algebra of continuous linear operators in the strong operator topology.

Aïssaoui, Noureddine, Ecole Normale Supérieure, B.P. 5206 Ben Souda, Fès, Maroc, (n.aissaoui@caramail.com)
Weighted strongly Nonlinear Potential theory pp. 207-230.
ABSTRACT. Let A be an N-function such that the Orlicz space LA becomes reflexive. Let W(m, A, ω) be the weighted Orlicz – Sobolev space and let L(m, A , ω ) be the weighted space of potentials. Then these two spaces coincide and have equivalent norms. An inequality between the inhomogeneous Riesz potential (and weighted Bessel potential) and the inhomogeneous maximal function of a positive function is established. On the other hand, weighted Bessel capacities are equivalent to different variational capacities that are associated with the norms in weighted Orlicz – Sobolev spaces.

Gang Li, Department of Mathematics,Yangzhou University, Yangzhou 225002, P. R. China (ligang@cimsl.yzu.edu.cn) and Jong Kyu Kim, Department of Mathematics, Kyungnam University, Masan 631-701, Korea, (jongkyuk@kyungnam.ac.kr).
Nonlinear Ergodic Theorems for Commutative Semigroups of Non-Lipschitzian Mappings in Banach Spaces, pp. 231-246.
ABSTRACT. In this paper, we study the ergodic theorems for commutative semigroups of asymptotically nonexpansive type mappings in a uniformly convex Banach space which satisfies Opial's condition.