Electronic Edition Vol. 30, No. 2, 2004

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



Tsiu--Kwen Lee, Department of Mathematics, National Taiwan University, Taipei 106, TAIWAN (tklee@math.ntu.edu.tw).
Posner's Theorem for (σ, τ)--Derivations and σ--Centralizing Maps, pp. 309- 320.
ABSTRACT. We give a characterization of mappings, which are additive modulo C, σ--centralizing on Lie ideals in a prime ring with extended centroid C, and then prove Posner's theorem for (σ, τ)--derivations on Lie ideals. These results give natural generalizations concerning derivations and centralizing mappings in prime rings.

Rosales, J. C., García-Sánchez, P. A., García-García, J. I., Universidad de Granada, Departamento de Álgebra, 18071 Granada, España (jrosales@ugr.es, pedro@ugr.es, jigg@ugr.es), and Branco, M. B., Universidade de Évora, Departamento de Matemática, 7000 Évora, Portugal (mbb@uevora.pt).
Saturated Numerical Semigroups, pp. 321-330.
ABSTRACT. We characterize those subsets of N that are saturated numerical  semigroups. We introduce the concept of SAT system of generators for a  saturated numerical semigroup, and this will enable us to arrange the set of all saturated numerical semigroups in a binary tree with no leaves.

David F. Anderson, Department of Mathematics, University of Tennessee, Knoxville, TN 37996, U. S. A. (anderson@math.utk.edu) and Ayman Badawi, Dept. of Math. & Stat., The American Univ. of Sharjah, P.O.Box 26666, United Arab Emirates (abadawi@ausharjah.edu ).
On φ-Prüfer Rings and φ-Bezout Rings, pp. 331-343.
ABSTRACT. The purpose of this paper is to introduce two new classes of rings that are closely related to the classes of Prüfer domains and Bezout domains. Let R be a commutative ring with 1 such that Nil(R) (the ideal of nilpotent elements of R) is a divided prime ideal of R, T(R) be the total quotient ring of R, and Z(R) be the set of zerodivisors of R. Then the map phi from T(R) into R_Nil(R) defined by phi(a/b) = a/b for every a in R and b in R\Z(R) is a ring homomorphism from T(R) into R_Nil(R), and phi restricted to R is also a ring homomorphism from R into R_Nil(R) given by phi(x) = x/1 for every x \in R. A nonnil ideal I of R is said to be phi-invertible if phi(I) is an invertible ideal of phi(R). If every finitely generated nonnil ideal of R is phi-invertible, then we say that R is a phi-Prüfer ring. Also, we say that R is a phi-Bezout ring if phi(I) is a principal ideal of phi(R) for every finitely generated nonnil ideal I of R. We show that the theories of phi-Prüfer and phi-Bezout rings resemble that of Prüfer and Bezout domains.

Shane P. Redmond Southeastern Louisiana University, Hammond, LA 70402, (sredmond@selu.edu).
Structure in the Zero-Divisor Graph of a Non-Commutative Ring, pp. 345-355.
ABSTRACT. In a manner analogous to the commutative case, the zero-divisor graph of a noncommutative ring R can be defined as the directed graph G. It has been shown that G is not a tournament if R is a finite ring with no nontrivial nilpotent elements and the graph has more than one vertex. This result is generalized to an arbitrary ring. This article also shows that G cannot be a network for a finite ring R. These results are used to determine which directed graphs on 1, 2, or 3 vertices can be realized as G. Finally, it is shown that for a finite ring R, G has an even number of directed edges.

Hetzel, Andrew J., Department of Mathematics, The University of Louisiana, 700 University Avenue, Monroe, LA 71209 (hetzel@ulm.edu).
Quasi-Going-up Rings, pp. 357-392.
ABSTRACT. We introduce and develop the theory of "quasi-going-up domains,"
a concept dual to going-down domains. By characterizing quasi-going-up domains as a particular type of going-down domain, we show that, in addition to Prüfer domains, the pseudo-valuation domains of Hedstrom and Houston are examples of quasi- going-up domains. We also define and develop the companion notions of "absolutely quasi-going-up domain" and "universally quasi-going-up domain." Both turn out to be examples of going-down domains and, in fact, the latter are precisely the i-domains of Papick. We conclude by defining and exploring "quasi-going-up rings," a generalization of quasi-going-up domains to the context of commutative rings with zero-divisors.

Burel, Jean-Marie, Lund University, 22100 Lund, Sweden (jean-marie.burel@math.lu.se).
Almost contact structures and harmonic maps with minimal fibres, pp. 393-411.
ABSTRACT. We study a class of maps between almost contact metric manifolds. We characterize harmonicity in terms of differential forms which allows one to construct minimal submanifolds. This new approach allows us to reduce the second order problem of harmonicity to a first order problem. In particular we show that any map submersive almost everywhere from a 3-manifold to a surface that commutes with the contact structure on the domain and the complex structure on the codomain can be rendered harmonic by a suitable choice of the metric.

Kristály, Alexandru and Varga, Csaba, Faculty of Mathematics and Informatics, Babes-Bolyai University, Cluj-Napoca, Romania (akristal@math.ubbcluj.ro), (csvarga@math.ubbcluj.ro) and Kozma, László, Institute of Mathematics and Informatics, University of Debrecen, Debrecen, Hungary (kozma@math.klte.hu).
The Dispersing of Geodesics in Berwald Spaces of Non-Positive Flag curvature, pp 413-420.
ABSTRACT. It is proved that a forward complete Berwald space of non-positive
flag curvature is a generalized Busemann's geodesic space of non-positive curvature:  the length of a median line in any geodesic triangle cannot succeed the half length of the corresponding side.

D. Shakhmatov, Department of Mathematical Sciences, Faculty of Science, Ehime University, Matsuyama 790--8577, Japan (dmitri@dpc.ehime-u.ac.jp),
M. Tkachenko, Departamento de Matemáticas, Universidad Autónoma Metropolitana, Av. San Rafael Atlixco 186, Del. Iztapalapa, C.P. 09340 Mexico D.F., Mexico (mich@xanum.uam.mx), and R. G. Wilson, Departamento de Matemáticas, Universidad Autónoma Metropolitana, Av. San Rafael Atlixco 186, Del. Iztapalapa, C.P. 09340 Mexico D.F., Mexico (rgw@xanum.uam.mx).
Transversal and T1-Independent Topologies, pp. 421-433.
ABSTRACT. A pair of T1 topologies on an infinite set is called T1-independent if their set-theoretic intersection is the cofinite topology, and transversal if their set-theoretic union generates the discrete topology. We show that every Hausdorff space admits a transversal compact Hausdorff topology. Then we apply Booth's Lemma to prove that no infinite set of cardinality less than continuum admits a pair of T1-independent Hausdorff topologies. This answers, in a strong form, a question posed by S. Watson in 1996. It is shown in ZFC that the remainder of the Stone-Cech compactification of the countable discrete set is a self T1-independent compact Hausdorff space, but the existence of self T1-independent compact Hausdorff spaces of cardinality continuum is both consistent with and independent of ZFC.

Proctor, C. Wayne, Department of Mathematics and Statistics, Stephen F. Austin State University, Nacogdoches, Texas 75962-3040 (proctor@math.sfasu.edu).
Continuously Ray Extendible Continua, pp. 435-450.
ABSTRACT. The class of all continuously ray extendible continua is defined and shown to contain all continua with zero span and to be a proper subcollection of the collection of Class W continua.

Song, Yan-Kui, Nanjing University, Nanjing 210093, P.R. China and Nanjing Normal University, Nanjing 210097, P.R. China (songyankui@email.njnu.edu.cn), and Shi, Wei-Xue, Nanjing University, Nanjing 210093, P.R. China (wxshi@nju.edu.cn).
Subspaces of Absolutely Star-Lindelöf Spaces, pp. 451-457.
ABSTRACT. In this paper, we consider two questions on absolute star-Lindelöfness and star-Lindelöfness: (1)  Whether every Tychonoff star-Lindelöf space can be embedded into some Tychonoff absolutely star-Lindelöf space as a closed subspace or as a closed Gd-subspace; (2) Characterize the Tychonoff star-Lindelöf spaces which can be embedded as regular closed subspaces into Tychonoff absolutely star-Lindelöf spaces. We proved that the answer to the first question is YES in general. For the second question we give a complete solution.

Krupski, Pawel, Mathematical Institute, University of Wroclaw, Pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland (krupski@math.uni.wroc.pl).
Families of Continua with the Property of Kelley, Arc Continua and Curves of Pseudo-Arcs, pp. 459-482.
ABSTRACT. It is shown that (1) the family of all continua in In, n>1, which have the property of Kelley (with the Hausdorff metric) is an absolute true Fσδ-set; (2) the family of all arc continua in In, n>2, is coanalytic complete; (3) the families of all arcs, circles, solenoids of pseudo-arcs and all Menger or Sierpinski curves of pseudo-arcs in cubes are Borel sets which are not Gδσ-set.

Yongge Tian, Department of Mathematics and Statistics, Queen's University, Kingston, Ontario, Canada K7L 3N6 (ytian@mast.queensu.ca).
Rank Equalities for Block Matrices and Their Moore-Penrose Inverses, pp. 483-510.
ABSTRACT. We present in this paper a variety of rank formulas for matrix expressions that involve Moore-Penrose inverses of block matrices, and use them to characterize various equalities for Moore-Penrose inverses of block matrices.

Stevo Stevic, Matematicki Institut Srpske Akademije Nauka, Knez Mihailova 35/I, 1000 Beograd, Serbia (sstevic@ptt.yu; sstevo@matf.bg.ac.yu ).
Weighted Integrals for Polyharmonic Type Functions, pp. 511-521.
ABSTRACT. We show that if a polyharmonic function on the unit ball belongs to the weighted Bergman space then all weighted derivations (of any positve order of derivations) belong to the related weighted Lebesgue space. Also, a similar result in the case of subharmonic type functions is proved.

F. Cabello and J.M.F. Castillo, Departamento de Matematicas, Universidad de Extremadura, Avenida de Elvas, 06071 Badajoz, Spain (fcabello@unex.es), (castillo@unex.es).
Uniform Boundedness and Twisted Sums of Banach Spaces, pp. 523-536.
ABSTRACT. We construct a Banach space X admitting an uncomplemented copy of l1 so that X/l1 = c0. To do that we study the uniform boundedness principles that arise when one considers exact sequences of Banach spaces; as well as several elements of homological algebra applied to the construction of nontrivial twisted sum of Banach spaces. The combination of both elements allows one to determine the existence of nontrivial twisted sums for almost all combinations of classical Banach spaces.

Damir Bakic and Boris Guljas, University of Zagreb, Department of Mathematics, 10000 Zagreb, Croatia (bakic@math.hr), (guljas@math.hr).
Extensions of Hilbert C*-modules, pp. 537-558.
ABSTRACT. Let V be a full Hilbert C*-module over a non unital C*-algebra. Denote by Vd the Hilbert C*-module over the multiplier algebra M(A) consisting of all adjointable maps from A to V. Then V can be naturally embedded in Vd   as an ideal submodule and restriction to V gives an isomorphism of C*-algebras of adjointable operators on Vd  and on V. The extended module Vd  is the completion of V with respect to a variant of strict topology and serves as the largest essential extension of V, thus can be regarded as the Hilbert C*-module version of the multiplier algebra. The extended module Vd   of the generalized Hilbert space over A is explicitly determined as a Hilbert C*-module of sequences in M(A) containing the generalized Hilbert space over M(A).

Vladimir Bolotnikov and Leiba Rodman, Department of Mathematics, College of William and Mary, P.O. Box 8795, Williamsburg, VA 23187-8795, USA (vladi@math.wm.edu; lxrodm@math.wm.edu).
Remarks on Interpolation in Reproducing Kernel Hilbert Spaces, pp. 559-576.
ABSTRACT. An interpolation problem in reproducing kernel Hilbert spaces is formulated and solved in terms of the minimal solution and the elements of an auxiliary reproducing kernel Hilbert space, subject to a norm constraint. The scheme is illustrated with two particular reproducing kernel Hilbert spaces: the Arveson space and the Bergman space, where the general theorem leads to convenient parametrizations of the solution sets.

David E. Edmunds, Centre for Mathematical Analysis and Its Applications, University of Sussex, Falmer, Brighton, BN1 9QH, U. K. and Ritva Hurri-Syrjanen, Department of Mathematics, P.O. Box 4, FIN-00014 University of Helsinki, Finland. Current address: University of Michigan, Department of Mathematics, Ann Arbor, MI 48103, USA (ritvahs@umich.edu)
Rellich's theorem in irregular domains, pp. 577-586.
ABSTRACT. We give geometric conditions on a domain which are sufficient for Rellich's theorem to hold on it. Our results complement earlier work on irregular domains, such as those satisfying a quasihyperbolic boundary condition and 'rooms and passages'.

Saïma Khenissy, Laboratoire Jacques-Louis Lions, Université Paris VI, 75252 Paris Cedex 05, France and Olivier Rey, Centre de Mathématiques de l'Ecole Polytechnique, 91128 Palaiseau Cedex, France (rey@math.polytechnique.fr).
A Criterion for Existence of Solutions to the Supercritical Bahri-Coron's Problem, pp. 587-613.
ABSTRACT. We consider a second order Dirichlet elliptic problem with slightly supercritical nonlinearity, on a smooth and bounded three dimensional domain W. We prove that nontriviality of the relative homology between the level sets of some function j in W¥ W, involving the Green's function and its regular part, implies the existence of a solution to the problem which blows up, as the nonlinearity goes to critical growth, at two points, which correspond to a critical point of j.