Electronic Edition Vol. 29, No. 4, 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


Anderson, Daniel, Department of Mathematics, The University of Iowa, Iowa City, IA 52242 (dan-anderson@uiowa.edu) and Smith, Eric, Department of Mathematics, The University of Iowa, Iowa City, IA 52242, current address: Department of Mathematics, The University of Northern Iowa, Cedar Falls, IA 50613 (esmith@cns.uni.edu).
Weakly Prime Ideals, pp. 831-840.
ABSTRACT. Let R be a commutative ring with identity. We define a proper ideal P of R to be weakly prime if whenever a nonzero product ab is in P, then either a is in P or b is in P. For example, every proper ideal of a quasilocal ring (R,M) whose maximal ideal M squared is 0 is weakly prime. We show that a weakly prime ideal P that is not prime satisfies PP is 0, in fact, Pnil(R) is 0. A number of results concerning weakly prime ideals and examples of weakly prime ideals are given. We show that every proper (principal) ideal of R is a product of weakly prime ideals if and only if R is a finite direct product of Dedekind domains (locally factorial Krull domains) and SPIR's or (R,M) is a quasilocal ring whose maximal ideal M squared is 0.

Alfred Geroldinger, Institut für Mathematik, Karl-Franzens Universität, Heinrichstrasse 36, 8010 Graz, Austria (alfred.geroldinger@uni-graz.at) and Rüdiger Göbel, Universität Essen, Fachbereich 6, Mathematik und Informatik, 45117 Essen, Germany (r.goebel@uni-essen.de).
Half-Factorial Subsets in Infinite Abelian Groups, pp. 841-858.
ABSTRACT. Let G be an abelian group. A subset S of G is said to be half-factorial, if the block monoid over S is a half-factorial monoid. We show that every Warfield group has a half-factorial subset which generates the group in the monoid theoretical sense. In particular, this implies that for every Warfield group G there exists a half-factorial Dedekind domain whose divisor class group is isomorphic to G. We also provide torsion-free abelian groups with prescribed endomorphism ring (for any ring with free additive group) which have half-factorial generating sets but surely are not Warfield groups. The corresponding question about the existence of non totally projective abelian p-groups with a half-factorial set of generators remains open.

Clark, David M., SUNY at New Paltz, New Paltz, NY 12561, U.S.A. (clarkd@newpaltz.edu), Davey, Brian A., La Trobe University, Victoria 3086, Australia (B.Davey@latrobe.edu.au), Haviar, Miroslav, Matej Bel University, Ruzova 13, 974 01 Banska Bystrica, Slovak Republic (mhaviar@fpv.umb.sk), Pitkethly, Jane G., La Trobe University, Victoria 3086, Australia (janegp@alphalink.com.au), and Talukder, M. Rashed, La Trobe University, Victoria 3086, Australia (R.Talukder@latrobe.edu.au).
Standard Topological Quasi-varieties, pp. 859-887.
ABSTRACT. This study addresses a problem which lies at the confluence of algebra, topology and mathematical logic. It is motivated by the theory of natural dualities, which provides a tight connection between a quasi-variety A and a topological quasi-variety X. We introduce the notion of a standard topological quasi-variety and initiate a program of study to determine which choices of X are standard and which are not. We say that X is standard if, in an appropriate sense, there is a nice axiomatic description of its members which allows us to recognize them by looking only at their finite substructures.

Ovidiu, Munteanu, Transilvania University, Str. I. Maniu, 50, 2200, Brasov, Romania ( gh.munteanu@info.unitbv.ro).
Weitzenböck Formulas for Horizontal and Vertical Laplacians, pp. 889-900.
ABSTRACT. In this paper we study the horizontal and the vertical parts of the Laplace operator on the total space of a fiber bundle when an arbitrary nonlinear connection is given. We prove Weitzenböck formulas and vanishing theorems for the horizontal and vertical Laplacians and for their sum.
Unlike the Levi-Civita connection, the linear connection involved in our theorems preserves the horizontal and the vertical distributions.
We also study these operators for the normalized volume form.

Jorge Pérez, V. H.,  Instituto de Ciências Matemática e de Computação ICMC-USP Cx. Postal 668,  São Carlos - SP - Brazil, CEP 13560-970
( vhjperez@icmc.usp.br).
Polar Multiplicities and Equisingularity of Map Germs from C3 to C3, pp. 901-924.
ABSTRACT. Terence Gaffney (1992) showed that if some invariants associated to germs in a family ft from Cn to Cp   are constant along the parameter t, then the family is Whitney equisingular. The number of invariants involved depends on the dimensions n and p, and this number is large when n and p are large. It is then natural to ask:
Fixing a pair (n,p), what is the minimum number of invariants in Gaffney's Theorem that are necesary to ensure Whitney equisingularity of the family?. This question has been answered for the cases p=1, n not equal  3; n=p=2 and n=2, p=3. In this paper we deal with the case n=p=3. According to Gaffney's result, for the family ft  to be Whitney equisingular we require the constancy of 25 invariants. We reduce this number to 7 for corank 1 germs.

Jiling Cao, Department of Mathematical Science, Faculty of Science, Ehime University, 790-8577 Matsuyama, Japan (cao@sylow.math.sci.ehime-u.ac.jp), and Yankui Song, School of Mathematics and Computer Science, Nanjing Normal University, Nanjing 210097, China (songyankui@email.njnu.edu.cn).
Aquaro Number versus Absolute Star-Lindelöf Number, pp. 925-936.
ABSTRACT. In this paper, we discuss cardinal functions generated by or related to star covering properties of a topological space. In particular, we show that the Aquaro number of an absolutely star-Lindelöf Tychonoff space could be arbitrarily large; the extent and the absolute star-Lindelöf number of a discretely star-Lindelöf and pseudocompact Tychonoff space could be arbitrarily large simultaneously. These improve some recent results of Bonanzinga and Matveev, and also give answers to some of their questions.

Lutfi N. H. Kalantan, King Abdulaziz University, Department of Mathematics, P.O.Box 114641 Jeddah, 21381 Saudi Arabia (lkalantan@hotmail.com) and Nobuyuki Kemoto, Department of Mathematics, Faculty of Education, Oita University, Dannoharu, Oita, 870-1192, Japan (nkemoto@cc.oita-u.ac.jp)
Mild Normality in Products of Ordinals, pp. 937-947.
ABSTRACT. A space is said to be mildly normal (or κ-normal ) if every disjoint pair of regular closed sets are separated by disjoint open sets. In this paper, we will show:
(1) There is a compact linearly ordered topological space Y such that ω1×Y is not mildly normal.
(2) A×B is mildly normal whenever A and B are subspaces of ordinals.
(3) There is a subspace of ω12 which is not mildly normal.
(4) There is a closed subspace of ω1×(ω1+1) which is not mildly normal.

Zhongqiang Yang, Department of Mathematics, Shantou University, Shantou, Guangdong, 515063, China, P. R. (zqyang@stu.edu.cn).
Dieuodonné-Hahn-Tong Theorem for Complete Chains, pp. 949-960.
ABSTRACT. In this paper, we discuss which spaces satisfy the Dieudonné-Hahn-Tong Theorem for all complete chains, that is, for which spaces, does there exist a continuous insertion between lower and upper semicontinuous maps to any complete chain. We show that such spaces must be collectionwise normal and strongly zero-dimensional. But they may not be hereditarily normal. Moreover, all strongly zero-dimensional metrizable spaces and all ordinal number spaces have this property.

Hirsch, Morris W., University of California at Berkeley, Berkeley CA 94720 (mwhirsch@chorus.net). Permanent address: 7926 N. Hill Point Road, Cross Plains, WI 5328.
Common Fixed Points for Two Commuting Surface Homeomorphisms, pp. 961-981.
ABSTRACT. Let F and G denote orientation-preserving surface homeomorphisms that commute under composition. Conditions are found ensuring that the fixed point set of F contains a fixed or periodic point for G. Proofs are based on Brouwer's Plane Translation Theorem and the Cartwright-Littlewood Fixed Point Theorem.

Marsh, M.M., California State University, Sacramento, Sacramento, CA 95819 (mmarsh@csus.edu).
Covering Spaces, Inverse Limits, and Induced Coincidence Producing Mappings, pp. 983-992.
ABSTRACT. We introduce a notion of "covering coincidence" between mappings of covering spaces and use this theory to show that certain inverse limits of ANRs have the fixed point property (fpp). In particular, we show that each inverse limit on an even dimensional real projective space with essential bonding mappings has the fpp, answering a question of D.Bellamy.

Hatzenbuhler, James P., and Mattson, Don A., Minnesota State University Moorhead, Moorhead, MN 56563 (hatzenbu@mnstate.edu), (mattson@mnstate.edu).
Discrete, Zero- and Strongly Zero- Dimensional Remainders, pp. 993-1000.
ABSTRACT. The remainder of a Hausdorff compactification cX of a space X is the set cX-X. Relationships concerning remainders which are discrete, strongly zero-dimensional, zero-dimensional or totally disconnected are studied. In particular, if cX>dX in the natural ordering of compactifications, then results on whether dX-X can have one of these properties when cX-X has another are obtained.
Some special cases: When X has compact residue and is paracompact at infinity, then X has a zero-dimensional remainder iff X has a strongly zero-dimensional remainder. When the residue is finite, then X is rimcompact and paracompact at infinity iff X has a discrete remainder. Conditions are provided which characterize when spaces with compact residue are paracompact at infinity.

Ivansic, Ivan, University of Zagreb, FER, Unska 3, 10000 Zagreb, Croatia (ivan.ivansic@fer.hr), and Milutinovic, Uros, University of Maribor, PEF, Koroska 160, 2000 Maribor, Slovenia (uros.milutinovic@uni-mb.si).
Relative Embeddability into Lipscomb's 0-dimensional Universal Space, pp. 1001-1012.
ABSTRACT. Let S(t) be the generalized Sierpinski curve, which is naturally identified with the Lipscomb's space J(t). Then L0(t), the subspace of irrational points of S(t), is universal for 0-dimensional metric spaces of weight ≤ t. We prove that any embedding of a compact subspace of a 0-dimensional metric space of weight ≤ t into L0(t) can be extended to an embedding of the whole space.

Takuo Miwa, Department of Mathematics, Shimane University, Matsue 690-8504, Japan (miwa@math.shimane-u.ac.jp).
Fibrewise ANR in Stratifiable Maps, pp. 1013-1025.
ABSTRACT. Fibrewise General Topology or General Topology of Continuous Maps is concerned most of all in extending the main notions and results concerning spaces to continuous maps. In this paper, we study the theory of fibrewise ANR for the class of stratifiable maps which are maps of stratifiable spaces to a fixed stratifiable space B. Of course if B is the one-point space, the theory of fibrewise ANR for the class coincides with one of ANR (for the class of stratifiable spaces).

Tomas Dominguez, Department of Mathematical Analysis, University of Sevilla Apdo., 1160 41080 Sevilla, Spain (tomasd@us.es), Genaro Lopez, Department of Mathematical Analysis, University of Sevilla Apdo., 1160 41080 Sevilla, Spain (glopez@us.es), Henryk Hudzik, Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614, Poznan, Poland (hudzik@amu.edu.pl), M. Mastylo, Faculty of Mathematics and Computer Science, Adam Mickiewicz University, and Institute of Mathematics (Poznan Branch), Polish Academy of Sciences, Umultowska 87, 61-614, Poznan, Poland (mastylo@amu.edu.pl), and Brailey Sims, School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan NSW 2308, Australia (bsims@frey.newcastle.edu.au).
Complete Characterizations of Kadec-Klee Properties in Orlicz Spaces, pp. 1027-1044.
ABSTRACT. We study for Orlicz function spaces, equipped with either the Luxemburg norm or the Orlicz norm, the connection between the Kadec-Klee property for local convergence in measure, the Kadec-Klee property for global convergence in measure, and some properties of the Orlicz function which defines the space.

Krzysztof Stempak, Instytut Matematyki, Politechnika Wroclawska, Wyb. Wyspianskiego 27, 50-370 Wrocl aw, Poland (stempak@im.pwr.wroc.pl).
Uniform Two-Weight Norm Inequalities for Hankel Transform Partial Sum Operators, pp. 1045-1063.
ABSTRACT. Proved are two-weight, uniform with respect to the order of the involved Bessel function, norm inequalities for the Hankel transform partial sum operators. The proof heavily relies on uniform pointwise asymptotic estimates for the Bessel functions done by Barcelo and Cordoba. Also, a technique used earlier by Muckenhoupt in the (discrete) Laguerre case is applied. The conditions appearing in the main theorem are then proved to be necessary, except some singular cases. The result is applied to obtain uniform estimates for the partial sum operators of Fourier-Neumann expansions. This generalizes former results in this direction done by Barcelo and Cordoba, and Ciaurri, Guadalupe, Perez and Varona.

Sami Baraket and Lamia Ben Chaabane, Departement de Mathematiques, Faculte des Sciences de Tunis, Campus Universitaire 1060 Tunis, Tunisie (sami.baraket@fst.rnu.tn). The Wente Inequality on Weighted Sobolev Spaces, pp. 1065-1075.
ABSTRACT. In this work, we study the Wente problem on some weighted Sobolev spaces. Under suitable conditions on the weight, we prove some estimates about the best constant in Wente's inequality. In particular, we obtain the best constant for the radial case with special homogenous weights. Using these estimates, we show also some interesting gap phenomena..