HOUSTON JOURNAL OF MATHEMATICS

Electronic Edition Vol. 32, No. 4, 2006

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), W. B. Johnson (College Station), J. Nagata (Osaka), V. I. Paulsen (Houston), , S.W. Semmes (Rice)
Managing Editor: K. Kaiser (Houston)

Houston Journal of Mathematics

Enochs, Edgar, University of Kentucky at Lexington, Kentucky, KY 40506-0027 (enochs@ms.uky.edu) and Estrada, Sergio, Universidad de Murcia, Campus del Espinardo, Espinardo (Murcia), 30100 Spain, (sestrada@ual.es)
The structure of compact coGalois groups, pp. 961-970.
ABSTRACT. The coGalois groups of torsion free covers were defined in E. Enochs, J.R. García Rozas, O.M.G. Jenda and L. Oyonarte: Compact coGalois groups. Math. Proc. Cambridge Phil. Soc. (2) 128 (2000), 233-244,  and were shown to have a natural topology. In that paper there was also given a necessary and sufficient condition on an abelian group in order that the coGalois group of its torsion free cover be compact. However the question of the structure of these compact groups was left open. In this paper we give a complete description of these compact groups. We also give a complete description of the coGalois groups associated with the torsion free cover of divisible groups.

McCasland, R.L., University of Edinburgh, Edinburgh EH8 9LE, Scotland UK (rmccasla@inf.ed.ac.uk), Moore, M.E., University of Texas at Arlington, Arlington, Texas 76019-0408 USA (moore@uta.edu), and Smith, P.F., University of Glasgow, Glasgow G12 8QW, Scotland UK (pfs@maths.gla.ac.uk).
Subtractive Bases of Zariski Spaces, pp. 971-983.
ABSTRACT. The notion of a subtractive basis for a Zariski Space is defined and examined. Such bases provide a means of generating Zariski Spaces, which exploits both the algebraic and topological-type properties of these spaces. In particular, it is shown that for every finitely-generated module M over a commutative ring with identity, then the Zariski Space of M has a (finite) subtractive basis. Moreover, it is shown that, provided M has the additional property of being a radical module (i.e., rad 0 = 0), then every subtractive basis of the Zariski Space of M corresponds to a direct sum decomposition of M. For such a module M, it then follows that every direct summand A of M must have the following property: if B is a submodule of M and rad B = A, then B = A.

Axtell, Michael, Wabash College, Crawfordsville, IN 47933 (axtellm@wabash.edu), Stickles, Joe, Dept. of Mathematics, Millikin University, Decatur IL 62522 and Warfel, Joseph, Wabash College, Crawfordsville, IN 47933.
Zero-Divisor Graphs of Direct Products of Commutative Rings, pp. 985-994.
ABSTRACT. We recall several results of zero divisor graphs of commutative rings. We then examine the preservation of diameter and girth of the zero divisor graph of direct products of commutative rings.

Peter, Ioan Radu, Department of Mathematics, Technical University of
Cluj-Napoca, G. Baritiu, nr. 15, RO-3400 Cluj-Napoca, Romania (Ioan.Radu.Peter@math.utcluj.ro). 
On the Morse Index theorem where the ends are submanifolds in Finsler geometry, pp. 995-1009.
ABSTRACT. In this paper we prove two cases of the Morse Index Theorem for Finsler manifolds. In the first case one of the endpoint of the geodesic is fixed, the other is variable in a submanifold. The second theorem deals with the case of two variable endpoints in submanifolds of a Finsler manifold.

Huff, Robert, Saint Louis University, Mathematics  and CS Department, 221 N. Grand Blvd, Saint Louis, MO 63103 (rhuff2@slu.edu).
Flat structures and the triply periodic minimal surfaces C(H) and tC(P), pp. 1011-1027.
ABSTRACT. The C(H) and tC(P) surfaces are one-parameter families of embedded, triply periodic minimal surfaces, with the former derived by A. Schoen and the latter derived by Fischer and Koch. Existence proofs of these surfaces using the conjugate Plateau method have been given by Karcher. In this paper, an alternative proof is presented using flat structures on the surfaces. This proof uses a different fundamental region than Karcher, and the geometry of this region makes clear a network of straight lines on the surfaces.

Tetsuya Hosaka, Department of Mathematics, Utsunomiya University, Mine-machi, Utsunomiya 321-8505, Japan (hosaka@cc.utsunomiya-u.ac.jp).
A class of rigid Coxeter groups, pp. 1029-1036.
ABSTRACT. In this paper, we give a new class of rigid Coxeter groups. Radcliffe have proved that right-angled Coxeter groups are rigid. The main theorem of this paper is an extension of this result.

Grace, E.E., Department of Mathematics, Arizona State University, Tempe, Arizona 85287-1804 (egrace@asu.edu) and  Vought, E.J., Department of Mathematics and Statistics, California State University, Chico; Chico, California  95929-0525 (eeevought@worldnet.att.net).
Plane Continua without disjoint subcontinua with nonvoid interiors , pp. 1037-1045.
ABSTRACT.  It is proved that every plane continuum, in which each two subcontinua with nonvoid interiors intersect, is not aposyndetic and semilocally connected at the same point. It is a corollary that any such continuum has a weak cutpoint.

Daniel, Dale, Lamar University, Beaumont, TX 77710-0047 (daniel@math.lamar.edu).
 On Metrizability of Images of Ordered Compacta , pp. 1047-1059.
ABSTRACT. We consider those Hausdorff spaces that are the continuous image of some compact ordered space. Utilizing a theorem of Treybig, we characterize those continuous images of compact ordered spaces that are metrizable. In doing so, we give a ''best possible" metrization theorem for separable images of compact ordered spaces. In particular, a Hausdorff space that is the continuous image of some compact ordered space is metrizable if and only it is separable and may be embedded as a G-delta subset of some locally connected continuum. We also obtain some corollaries and related results. .

Nitta, Shin-ichi.,Shijonawate-gakuen Junior College,Osaka,Japan (nitta@jc.shijonawate-gakuen.ac.jp),and Yoshioka,Iwao, Department of Mathematics, Okayama University,Okayama,Japan (yoshioka@math.okayama-u.ac.jp).
Nagata Spaces and wN-spaces which are preserved by Quasi-Perfect Maps, pp. 1061-1076.
ABSTRACT. G.Ying and C.Good (2001) described that Lutzer's example (1971) asserts that neither Nagata spaces nor wN-spaces are necessarily preserved under quasi-perfect maps. We introduce some classes of spaces which are contained in the class of wN-spaces and show that these classes are invariant under quasi-perfect maps or closed almost-open maps, but not preserved by closed map nor open finite-to-one maps. We also show that in the realm of paracompact spaces, these new classes coincide with the class of M-spaces in the sense of Morita, and study conditions for metrizability of these classes.

Chatyrko, Vitalij, Linkoping University, 581 83 Linkoping, Sweden (vitja@mai.liu.se), Hattori, Yasunao, Shimane University, Matsue, Shimane, 690-8504 Japan
(hattori@riko.shimane-u.ac.jp), and Ohta, Haruto, Shizuoka University, Ohya, Shizuoka, 422-8529 Japan (echohta@ipc.shizuoka.ac.jp).
Partitions of spaces by locally compact subspaces, pp. 1077-1091.
ABSTRACT. In this article, we shall discuss the possibility of different presentations of topological spaces as unions or partitions of locally compact subspaces.
For a space X, let lc(X) (resp. lcd(X)) denote the minimum cardinality of a cover (resp. partition) of X by locally compact subspaces. We prove:
(1) For every finite or countably infinite cardinal n, there exists a subspace X of the real line such that lc(X)=n; (2) for every Hausdorff space X, if lc(X) is at most countable, then lc(X)=lcd(X); (3) if X is a topologically complete, nowhere locally compact, Hasudorff space, then lc(X) is uncountable; (4) if a perfectly normal space X is covered by finitely many subpaces, each of which is locally compact with at most one exception, then the covering dimension of X coincides with the maximum of the dimensions of those subspaces. It is open whether there is an example in ZFC of a Hausdorff space X such that lc(X)<lcd(X).  

Dijkstra, Jan J., Vrije Universiteit, Amsterdam, The Netherlands (dijkstra@cs.vu.nl).
A homogeneous space that is one-dimensional but not cohesive, pp. 1093-1099.
ABSTRACT. A separable metric space is called cohesive if every point of the space has a neighbourhood that fails to contain nonempty clopen subsets of the space. A topological group is cohesive if and only if it is not zero-dimensional. We present a homogeneous, one-dimensional, almost zero-dimensional space that is not cohesive.
We also show that a complete homogeneous space is cohesive if and only if it is not zero-dimensional.

Antonio Peláez, Instituto de Matemáticas, UNAM, Circuito Exterior, Ciudad Universitaria, México D. F., C. P. 04510, MEXICO, (pelaez@matem.unam.mx)
Generalized inverse limits, pp. 1107-1119.
ABSTRACT. Recently, Professor W. Mahavier defined an inverse limit of a sequence of closed subsets of the unit square. We use that definition to construct, for a given positive integer m, an inverse limit of a sequence of closed subsets of the unit square having dimension equal to m. We extend Professor Mahavier's definition and define an inverse sequence of closed subsets and its generalized inverse limit. In this case, the closed sets are not necessarily contained in the unit square. We also extend some results obtained by Professor Mahavier in his paper "Inverse limits with subsets of [0,1]× [0,1], Topology and Its Applications, 141(2004), 225-231" and we give a condition to induce maps between those new spaces.

MacCluer, Barbara D., University of Virginia, Charlottesville, VA 22904 (bdm3f@virginia.edu) and Pons, Matthew A., University of Virginia, Charlottesville, VA 22904 (map6h@virginia.edu).
Automorphic Composition Operators on Hardy and Bergman Spaces in the Ball, pp.1121-1132.
ABSTRACT. We determine which composition operators with automorphism symbol are essentially normal on the Hardy and Bergman space of the ball. To do this, we first show that modulo the compact operators, the product of an automorphic composition operator followed by its adjoint is a Toeplitz operator with positive symbol.

Jupiter, Daniel, Texas A & M Health Science Center, Temple, TX 76504 (djupiter@tamu.edu), and  Redett, David,  Indiana University - Purdue University Fort Wayne, Fort Wayne, IN 46805 (redettd@ipfw.edu).
Invariant Subspaces of RL¹, pp. 1133-1138.
ABSTRACT. In this note we extend D. Singh and A. A. W. Mehanna's invariant subspace theorem for RH¹ (the real Banach space of analytic functions in H¹ with real Taylor coefficients) to the simply invariant subspaces of RL¹ (the real Banach space of functions in L¹ with real Fourier coefficients.

Mykhaylyuk, Volodymyr, Chernivtsi National University str. Kotsjubyn'skogo 2, Chernivtsi, 58012 Ukraine (mathan@chnu.cv.ua) and Popov, Mykhaylo, Chernivtsi National University str. Kotsjubyn'skogo 2, Chernivtsi, 58012 Ukraine (popov@chv.ukrpack.net).
On "weak" embeddings of L₁, pp. 1139-1152.
ABSTRACT. We study the relationships between some notions of "weak" embeddings (semi-embeddings, sign-embeddings, etc.) of the space of the Lebesgue summable functions on the unit segment. First kind of problems - which "weak" embeddings are automatically other ones - are simple and we answer them completely. The questions of the second kind - which "weak" embeddabilities imply others - are difficult and some of them remain still open.

Kahng, Byung-Jay, Department of Mathematics and Statistics, Canisius College, Buffalo, NY 14208 (kahngb@canisius.edu ).
Quantum double construction in the C*-algebra setting of certain Heisenberg-type quantum groups, pp. 1153-1189.
ABSTRACT. We carry out the ``quantum double construction'' of the specific quantum groups we constructed earlier, namely, the ``quantum Heisenberg group algebra'' and its dual, the ``quantum Heisenberg group''. Our approach is by constructing a suitable multiplicative unitary operator, retaining the C*-algebra framework of locally compact quantum groups. We also discuss the dual of the quantum double and the Haar weights on both of these double objects. Towards the end, a construction of a (quasitriangular) quantum universal R-matrix is given.

Pol, Roman, Department of Mathematics, Banacha 2, 02-097 Warsaw, Poland (pol@mimuw.edu.pl) .
Evaluation maps on products of separable metrizable spaces are Borel, pp. 1191-1196.
ABSTRACT. Given a completely regular space X we denote by C(X) the space of continuous real - valued functions on X, and let e : X×C(X) → R, e(x,f) = f(x), be the evaluation map. The function space C(X) is considered with the pointwise topology. We prove the result stated in the title, and give an example of a compact separable space X such that the evaluation map e on X is Borel measurable, but the preimage of some open set under e is not a countable union of closed sets, answering two questions by M.R.Burke, Borel measurability of separately continuous functions II, Top. Appl. 134 (2003), 154 - 188.

García-Falset, Jesus, Departament de Matemàtiques , Universitat de València, Dr Moliner 50, 46100 Burjassot, València, Spain (garciaf@uv.es), and Reich,  Simeon,  Department of Mathematics, the Technion-Israel Institute of Technology, 32000 Haifa, Israel (sreich@tx.technion.ac.il).
Zeroes of accretive operators and the asymptotic behavior of nonlinear semigroups, pp. 1197-1225.
ABSTRACT. We study necessary and sufficient conditions for an accretive operator which satisfies the range condition to have a zero. We obtain, in particular, a characterization of this property for m-accretive operators in L1. We also study the asymptotic behavior of nonexpansive semigroups in L1 and then apply our results to certain initial value problems.

De-xiang Ma, College of Information Science and Engineering, Shandong University of Science and Technology, Qingdao, Shandong Province 266510, China (madexiang@sohu.com),  Weigao-Ge, Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China (gew@bit.edu.cn), and Xuegang-Chen, Department of Mathematics, North China Electric  Power university, Beijing 102206, China.
New results on periodic solutions for a p-Laplacian Lienard equation with a deviating argument, pp. 1227-1239.
ABSTRACT. The existence of periodic solutions for Lienard equation with a deviating has been studied extensively. Generally, restricting conditions are imposed on the value of the deviating function. In this paper, we study the existence of periodic solutions for the p-laplacian Lienard equation with a deviating argument. We give no restriction on the deviating function, which is the significance of the paper. Our result is based on a new lemma which makes it possible to use Mawhin's continuation theorem as a tool. The methods we used to estimate a priori bounds on periodic solutions are also new.

Pigong Han, Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, P. R. China (pghan@amss.ac.cn)
The Effect of the Domain Topology on the Number of Positive Solutions of an Elliptic System Involving Critical Sobolev Exponents, pp. 1241-1257.
ABSTRACT. In this paper, we consider the Dirichlet problem for an elliptic system of two equations involving the critical Sobolev exponents. We first establish the concentration-compactness principle for elliptic systems; then by the variational method and applying Lusternik-Schnirelmann theory, we study the effect of the domain topology on the number of positive solutions.