Electronic Edition Vol. 31, No. 1, 2005

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


Wickless, William, University of Connecticut, Storrs, CT 06269 (wickless@math.uconn.edu).
Multi-Isomorphism for Quotient Divisible Groups, pp. 1-20.
ABSTRACT. Abelian groups A, B are called multi-isomorphic if An is isomorphic to Bn for all natural numbers n>1. In a series of papers, K. O'Meara and C. Vinsonhaler have studied this notion for torsion-free groups of finite rank (tffr ). We study multi-isomorphism in the class of quotient divisible (qd) abelian groups. A reduced abelian group G is qd if it contains a finite rank free subgroup F such that G/F is a divisible torsion group. Multi-isomorphism for qd groups shares some properties with those discovered by O'Meara and Vinsonhaler in the tffr case, but also has some interesting differences.

Coy L. May and Jay Zimmerman, Department of Mathematics, Towson University, 8000 York Road, Towson, Maryland 21252 ( cmay@towson.edu), (jzimmerman@towson.edu).
The Groups of Strong Symmetric Genus 4, pp. 21-35.
ABSTRACT. Let G be a finite group. The strong symmetric genus is the minimum genus of any Riemann surface on which G acts preserving orientation. The groups of strong symmetric genus 3 or less have been classified. Here we classify the groups of strong symmetric genus four. There are exactly ten such groups; eight of these are automorphism groups of regular maps of genus 4.
We also consider non-abelian p-groups that have an element of maximal possible order. We complete the determination of the strong symmetric genus of each p-group with this property. Conversely, the non-abelian 2-groups of even positive strong symmetric genus have an element of maximum possible order. Further, we establish that for an odd prime p, the strong symmetric genus of a non-abelian p-group is congruent to one modulo a power of p.

Tetsuya Hosaka, Department of Mathematics, Utsunomiya University, Utsunomiya, 321-8505, Japan (hosaka@cc.utsunomiya-u.ac.jp).
Strong reflection rigidity of Coxeter systems of dihedral groups, pp. 37-41.
ABSTRACT. In this paper, we study strong rigidity and strong reflection rigidity of Coxeter systems of dihedral groups. We show that the Coxeter system of the dihedral group Dn of order 2n is strongly reflection rigid if and only if n∈{2,3,4,6}, and that the dihedral Coxeter group Dn is strongly rigid if and only if n∈{3,4}.

Florian Kainrath, Institut f&uumlr Mathematik, Karl-Franzens-Universität Graz, Heinrichstrasse 36, A-8010 Graz, Austria (florian.kainrath@uni-graz.at).
Elasticity of Finitely Generated Domains, pp. 43-64.
ABSTRACT. Let R be an atomic domain. Then every non unit a of R may be written as a product of irreducibles. Let L(a) resp. l(a) denote the longest resp. shortest length of all such factorizations of a. The elasticity of R is defined as the supremum of all quotients L(a)/l(a), where a runs through all non units of R. In this paper we characterize those finitely generated domains having finite elasticity.

Chatham, R. Douglas, Morehead State University, Morehead, KY 40351 (d.chatham@moreheadstate.edu), and Dobbs, David E., University of Tennessee, Knoxville, TN 37996-1300 (dobbs@math.utk.edu).
On Open Ring Pairs Of Commutative Rings, pp. 65-74.
ABSTRACT. If T is an integral commutative extension of a ring R such that R is an open ring, R[a, b] is a going-down ring for each a, b in T and T is semiquasilocal, then each ring contained between R and T is an open ring. An example is given to show that the "semiquasilocal" hypothesis cannot be deleted. If T is a commutative ring containing a ring R such that R[a, b] is an open ring for each a, b in T, then (R, T) is an INC-pair (equivalently, a residually algebraic pair).

Hedayat, Sina, University of Kerman, Kerman, Iran, and Nekooei, Reza, University of Kerman, Kerman, Iran (rnekooei@mail.uk.ac.ir).
Characterization of Prime Submodules of a Finitely Generated Free Module over a PID, pp. 75-85.
ABSTRACT. In this paper we discuss properties of a set of prime submodules of a finitely generated free module F over a UFD and characterize the prime submodules of a free module of finite rank over a PID. Finally; we obtain the radical of a cyclic submodule of F and the radical of a submodule of a rank 2 free module over a PID.

Daciberg Goncalves, Departamento de Matematica - IME - USP, Caixa Postal 66.281 - CEP 05311-970, São Paulo- SP, Brasil (dlgoncal@ime.usp.br) , and João Peres Vieira, Departamento de Matematica -IGCE - UNESP, Caixa Postal 178 - CEP 13500-230, Rio Claro - SP, Brasil (jpvieira@rc.unesp.br).
Free Actions of Abelian P-Groups on the N -Torus , pp 87-101.
ABSTRACT. In this work we make some contributions to the theory of actions of abelian p-groups on the n-Torus Tn. Set H ≈ (Zpk1)h1 × (Zpk2)h2 × … × (Zpkr)hr, r ≥ 1, k1 ≥ k2 ≥ … ≥ kr ≥ 1, p prime. Suppose that the group H acts freely on Tn and the induced representation on Π1(Tn) ≈ Zn is faithful and has first Betti number b. We show that the numbers n, p, b, ki and hi, (i=1,…,r) satisfy some relation. In particular, when H ≈ (Zp)h, the minimum value of n is  Φ(p)+b when b ≥ 1. Also when H ≈ Zpk1 × Zp the minimum value of n is Φ(pk1)+p-1+b for b ≥ 1. Here Φ denotes the Euler function.

Juan Climent Vidal, and Juan Soliveres Tur, Universidad de Valencia, Departamento de Lógica y Filosofía de la Ciencia, E-46071 Valencia, Spain. (juan.b.climent@uv.es), (juan.soliveres@uv.es).
The Completeness Theorem for Monads in Categories of Sorted Sets, pp. 103-129.
ABSTRACT. Birkhoff's completeness theorem of equational logic asserts the coincidence of the model-theoretic and proof-theoretic consequence relations. Goguen and Meseguer, giving a sound and adequate system of inference rules for finitary many-sorted equational deduction, generalized the completeness theorem of Birkhoff to the completeness of finitary many-sorted equational logic and provided simultaneously a full algebraization of finitary many-sorted equational deduction. In this paper, after defining the concepts of equational class and equational theory for a monad in a category of sorted sets and the concept of projective limit-compatible congruence on a category, we prove that the lattice of product-compatible congruences on the dual of the Kleisli category of a monad in a category of sorted sets is identical to the lattice of equational theories for the same monad. In this way we obtain a completeness theorem for monads in categories of sorted sets, and therefore independent of any explicit syntactical representation of the relevant concepts, that generalizes the completeness theorem of Goguen-Meseguer and provides a full categorization of many-sorted equational deduction.

Xiaohuan Mo, LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, People's Republic of China (moxh@pku.edu.cn).
On the Flag Curvature of a Finsler Space with Constant S-Curvature , pp. 131-144.
ABSTRACT. By establishing the intrinsic relation between the S-curvature and the flag curvature we investigate the Finsler spaces with constant S-curvature. In particular, we show that any Finsler space of scalar curvature and constant S-curvature is a constant curvature space if its dimension ≥3

Ronald A. Walton, 1671 Via Rancho, San Lorenzo, CA 94580 (rwsailor@mac.com).
A Symmetric Hyperbolic Structure for Isentropic Relativistic Perfect Fluids, pp. 145-160.
ABSTRACT. The dependent variables of an isentropic, relativistic perfect fluid can be consolidated into a generalized velocity vector field equal to the fluid's relativistic velocity vector field divided pointwise by the value of the fluid's specific enthalpy. The equations of motion for an isentropic, relativistic perfect fluid then become a quasilinear, first order, symmetric system of partial differential equations, equivalent to local energy and momentum conservation in the fluid. Furthermore, these equations of motion are symmetric hyperbolic wherever the material density of the fluid is positive and the speed of sound in the fluid does not exceed the speed of light. Two applications of these equations are presented. First, the characteristic hypersurfaces of an isentropic, relativistic perfect fluid are proven to consist of (1) timelike hypersurfaces generated by the fluid streamlines and (2) nonspacelike, conical hypersurfaces defined by the propagation of sound waves. And second, the equations of motion for a self-gravitating, isentropic, relativistic perfect fluid are proven to be equivalent to a quasilinear, first order, symmetric hyperbolic system if the spacetime coordinates are constrained to be harmonic.

Andrew Przeworski Dept. of Mathematics , University of Texas, Austin, TX 78712 (prez@math.utexas.edu).
Balls in Hyperbolic 3-Manifolds, pp. 161-171.
ABSTRACT. We show that in a closed orientable hyperbolic 3-manifold, any maximal embedded tube of radius r contains a ball of a certain radius. We then use the fact that most closed orientable hyperbolic 3-manifolds contain tubes of radius (log 3)/2 to provide a universal lower bound on the radius of the ball.

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).
The pointed version of Lipscomb's embedding theorem, pp. 173-192.
ABSTRACT. Let S(t) be the generalized Sierpinski curve, which is naturally identified with Lipscomb's space J(t).
Then for any n-dimensional metric space X of weight t there is an embedding of X into Ln(t), where Ln(t) is the subset of S(t)n+1 of all points having at least one so called irrational coordinate. Here we prove that this embedding may be chosen in such a way that its value at a certain point (the base point) is given in advance. In fact, we prove a stronger result that the values of the embedding may be given in advance at any finite set of points of X.

Yun Ziqiu, Department of Mathematics, Suzhou University, 215006 P.R.China (yunziqiu@public1.sz.js.cn).
On Closed Mappings, pp. 193-197.
ABSTRACT. In the present paper, some sufficient conditions for closed mappings to be peripherally compact are given and a problem of Y. Tanaka and C. Liu is answered.

Gauld, David, Department of Mathematics, University of Auckland, PB 92019, Auckland, New Zealand (gauld@math.auckland.ac.nz), and Mynard, Frédéric, Department of Mathematical Sciences, Georgia Southern University, 203 Georgia Ave. Room 3038, Statesboro, GA 30460-8093 (fmynard@georgiasouthern.edu ).
Metrisability of Manifolds in Terms of Function Spaces, pp.199-214.
ABSTRACT. Both internal and external new criteria of metrisability of a topological manifold are obtained.
The external ones involve topological properties of the space of real-valued continuous functions over the manifold, endowed either with the topology of pointwise convergence or with the compact-open topology.

Arhangel'skii, Alexander, Ohio University, Athens, Ohio, 45701 (arhangel@math.ohiou.edu).
Quotients with respect to Locally Compact Subgroups, pp. 215-226.
ABSTRACT. We establish that, for any locally compact subgroup H of a topological group G, the natural quotient mapping of G onto the quotient space G/H is locally perfect, that is, the restriction of it to the closure of some open set is an open mapping with compact fibers. We derive from this that many important topological invariants are transfered from the space G/H to G when H is locally compact.

Mou, Lei, Mathematics Department, Capital Normal University, Beijing, 100037 China (Current address: Faculty of Education, Shizuoka University, Ohya, Shizuoka, 422-8529 Japan), and Ohta, Haruto, Faculty of Education, Shizuoka University, Ohya, Shizuoka, 422-8529 Japan (echohta@ipc.shizuoka.ac.jp).
Sharp Bases and Mappings, pp. 227-238.
ABSTRACT. Let S be the class of spaces with a sharp base in the sense of B. Alleche, A. V. Arhangel'skii and J. Calbrix (2000). A map f is called boundedly finite-to-one (resp. k-to-one) if there is a natural number k such that each fiber of f consists of at most (resp. exactly) k many points. Answering a question asked by C. Good, R. W. Knight and A. M. Mohamad (2002), we prove: (1) The image of a space in S under a perfect map or an open finite-to-one map is not necessarily in S, but every open boundedly finite-to-one image of a space in S is in S. (2) The preimage of a space in S under an open closed boundedly finite-to-one map is not necessarily in S, but every open k-to-one preimage of a space in S is in S.

Miroslaw Sobolewski, Instytut Matematyki, Faculty of Mathematics, Informatics and Mechanics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland (msobol@mimuw.edu.pl).
A Curve with the Fixed Point Property whose Cylinder admits a Fixed Point Free Map, pp. 239-253.
ABSTRACT. We construct a curve, i.e., a one-dimensional metric continuum, which has the fixed point property but its product by the interval admits a fixed point free mapping. This answers a question by R.H. Bing.

Patton, Linda J., Mathematics Department, California Polytechnic State University, San Luis Obispo, CA 93402 (lpatton@calpoly.edu), and Robbins, Marian E., Mathematics Department, Bellarmine University, 2001 Newburg Road, Louisville, KY 40205 (mrobbins@bellarmine.edu).
Composition Operators that are M-isometries, pp. 255-265.
ABSTRACT. The results in this paper show that on many Hilbert spaces of analytic functions, including Hardy spaces on the ball and polydisc and standard weighted Bergman spaces on the disk, the only m-isometric composition operators are isometries. Necessary conditions for the symbols of m-isometric composition operators on Dirichlet space are also derived. Finally, it is shown that for symbols with fixed point in the unit disk, the associated composition operator has an m-isometric adjoint if and only if the symbol is an automorphism that fixes the origin.

Razvan Anisca, Department of Mathematical Sciences, Lakehead University, 955 Oliver Road, Thunder Bay, Ontario Canada, P7B 5E1(razvan.anisca@lakeheadu.ca), Adi Tcaciuc, Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1 (tcaciuc@math.ualberta.ca), and Nicole Tomczak-Jaegermann, Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1 (nicole@ellpspace.math.ualberta.ca).
Structure of normed spaces with extremal distance to the Euclidean space, pp. 267-283.
ABSTRACT. It is shown that a finite dimensional Banach space has the Euclidean distance of maximal order if and only if it contains a proportional dimensional subspace (and a quotient of a subspace) of a very special form.

Ignat, Radu, École Normale Supérieure, Paris, France (Radu.Ignat@ens.fr).
On an Open Problem about how to Recognize Constant Functions, pp. 285-304.
ABSTRACT. In this paper we generalize a result of Bourgain, Brezis and Mironescu about how to recognize constant functions; its motivation comes from the uniqueness of the lifting in some Sobolev spaces. We also answer to an open question raised by Brezis concerning the previous result.