Electronic Edition Vol. 29, No. 2, 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


G. Grätzer, Department of Mathematics, University of Manitoba, Winnipeg, MB R3T 2N2, Canada (gratzer@ms.umanitoba.ca), E.T. Schmidt, Mathematical Institute of the Budapest University of Technology and Economics, Müegyetem rkp. 3, H-1521 Budapest, Hungary (schmidt@math.bme.hu) and K. Thomsen, Department of Mathematics, University of Manitoba Winnipeg, MB R3T 2N2, Canada (kurt_thomsen@hotmail.com).
Congruence Lattices of Uniform Lattices, pp. 247-263.
ABSTRACT. A lattice is uniform, if for any congruence, any two congruence classes are of the same size. A classical result of R. P. Dilworth represents a finite distributive lattice as the congruence lattice of a finite lattice. We show that this finite lattice can be constructed as a finite uniform lattice.

Takahiro Sudo, Department of Mathematical Sciences, Faculty of Science, University of the Ryukyus, Nishihara-cho, Okinawa 903-0213, JAPAN (sudo@math.u-ryukyu.ac.jp ).
Group C*-algebras of Some Connected Lie Groups with Stable Rank One, pp. 265-280.
ABSTRACT. We show that group C*-algebras of some connected Lie groups have stable rank one, connected stable rank one and general stable rank one.

Bang-Yen Chen, Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, USA (bychen@math.msu.edu).
Constant-Ratio Space-Like Submanifolds in Pseudo-Euclidean Space, pp. 281-294.
ABSTRACT. A submanifold of a pseudo-Euclidean space is said to be of constant-ratio if the ratio of the length of the tangential and normal components of its position vector function is constant. Submanifolds of constant-ratio relate closely to the notion of convolution of Riemannian manifolds and to a problem in physics concerning the motion in a central force field which obeys the inverse-cube law originated from Newton in 1679. The purpose of this paper is to completely classify constant-ratio space-like submanifolds in pseudo-Euclidean space.

Richard H. Escobales, Jr., Department of Mathematics, Canisius College, Buffalo, NY 14208 (escobalr@canisius.edu).
Integrability Criteria and Vector-Bundle Valued Cohomology for Foliations, pp. 295-311.
ABSTRACT. We first study a flow F on a closed, connected, n-dimensional, Riemannian manifold (M, g). We assume that the mean curvature one-form κ associated with F is closed. We show that this naturally induces a flat Bott-type connection D on V, the distribution tangent to the flow F. We observe that in fact this connection D depends only on the basic real cohomology class [κ]B. Then the natural exterior derivative d' associated with this Bott-type connection on V-valued differential forms has the property that (d')2 = 0, and so one think of a cohomology of V-valued differential forms, H*(M,V), canonically determined by [κ]B. We show that H, the distribution orthogonal to V in TM with respect to the metric g, is integrable if and only if a certain non-trivial cohomology class exists in H1(M, V). Hence, in the integrable case, H1(M, V) is not 0. We discuss an analogue of this result for distribution orthogonal to a foliation of leaf dimension p greater than or equal to 2. For the foliation F itself, we have results provided the Bott connection B on H is flat. B is flat on H provided F is bundle-like with respect to the Riemannian g on M and the base spaces of the local Riemannian submersions which define the bundle-like foliation F are flat. Under these assumptions on F, H1(M, H) is not 0. In particular, when F is codimension-one foliation on M which is bundle-like with respect to the metric g, we show that H1(M,H) is not equal to 0.

Alejandro Illanes and Likin C. Simón Instituto de Matemáticas, UNAM, Circuito Exterior, Ciudad Universitaria, México 04510, D.F. MÉXICO (illanes@matem.unam.mx, lsimon@matem.unam.mx).
Means with Special Properties, pp. 313-324.
ABSTRACT. Let X be a metric continuum. A mean is a continuous function m: X× X → X such that m(x,x)=x and m(x,y)=m(y,x) for every x, y ∈ X. In this paper we study means with additional properties, namely, we consider confluent, monotone and open means. We give examples and we include some open questions.

Zhongqiang Yang, Department of Mathematics, Shantou University, Shantou, Guangdong, 515063, China P.R. (zqyang@stu.edu.cn) and Katsuro Sakai, Institute of Mathematics, University of Tsukuba, Tsukuba, 305-8571, Japan (sakaiktr@sakura.cc.tsukuba.ac.jp).
The Space of Limits of Continua in the Fell Topology, pp. 325-335.
ABSTRACT. By Cld(X), we denote the hyperspace of non-empty closed sets of a locally compact metrizable space X with the Fell topology. Let Cont(X) be its subspace consisting of all continua and Cont(X) the closure of Cont(X) in Cld(X). It is proved that if X is connected, locally connected, non-compact and has no free arcs, then the pair (Cont(X), Cont(X)) is homeomorphic to the pair (Q × [0,1] - Z × {0}, Q × (0,1]), where Q is the Hilbert cube and Z is a Z-set in Q which is homeomorphic to EX the space of ends (i.e., the remainder of the Freudenthal compactification of X).

Katsuya Yokoi, Department of Mathematics, Interdisciplinary faculty of Science and Engineering, Shimane University, Matsue, 690-8504, Japan (yokoi@math.shimane-u.ac.jp).
Bubbly Continua and Homogeneity,  pp. 337-343.
ABSTRACT. We show that an n-dimensional homogeneous ANR continuum which is cyclic in dimension n is an n-bubble. As a consequence, we obtain that no compact subset of such a space X, which is acyclic in dimension n-1, separates X.
This is a partial answer to a problem of Bing and Borsuk.

Yan-Kui Song, Department of Mathematics, Nanjing University, Nanjing, 210093 P. R. China and Department of Mathematics, Nanjing Normal University, Nanjing, 210097 P. R. China (songyankui@email.njnu.edu.cn)
Spaces with Large Extent and Large Star-Lindelöf Number, pp. 345-352.
ABSTRACT. In this paper, we prove the following statements: (1) For every regular uncountable cardinal κ, there exists a centered-Lindelöf, Tychonoff space X such that St-l(X)≥κ. (2) For every uncountable cardinal κ, there exists a discretely star-Lindelöf, Tychonoff space X with the property (a) such that e(X)≥κ . (3) For every infinite cardinal κ, there exists a discretely star-Lindelöf, pseudocompact, Tychonoff space X such that e(X)≥κ. The statement (1) answers negatively three questions of Matveev and Bonanzinga on spaces with large extent and star-Lindelöf number.

Kaori Yamazaki, Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan (kaori@math.tsukuba.ac.jp).
Extending Point-Finite Partitions of Unity, pp. 353-359.
ABSTRACT. We prove that: (1) A subspace A of a space X is Pω (point-finite)-embedded in X if and only if every countable point-finite cozero-set cover of A can be extended to a countable point-finite cozero-set cover of X. (2) For a space X of second category and its dense subspace A of first category (in itself), there exists a countable point-finite open cover of A which can not be extended to a point-finite open cover of XA. These results imply that the rationals Q of the Michael line RQ is not Pω (point-finite)-embedded in RQ which answers a question of J. Dydak in: Extension Theory: the interface between set-theoretic and algebraic topology, Topology Appl. 74 (1996), 225-258.

Sophia Zafiridou, Department of Mathematics, University of Patras, 26500 Patras, Greece (zafeirid@math.upatras.gr).
Rim-Scattered Space, Rim-type of a Space, Containing Continuum, pp. 361-369.
ABSTRACT. We prove that for every ordinal α = β +m (where β is a limit ordinal or 0 and m is a positive integer) and for every k=0,..., m+min {β,1}-1 there is no containing (planar) continuum of rim-type ≤α +k for the family of all (planar) spaces having a basis B of open sets such that for every U∈B: (α ) the α-derivative of Bd(U) is empty and (β ) the set Bd(U) has a compactification with the α +k-derivative empty.

Alexander E. Richman, Department of Mathematics, Bucknell University, Lewisburg, PA 17837 (arichman@bucknell.edu).
Composition Operators with Complex Symbol having Subnormal Adjoint, pp. 371-384.
ABSTRACT. Following earlier work by the author on the Bergman space and work by C. Cowen and T. Kriete on the Hardy and Dirichlet spaces, it has been believed that in order for Cφ* to be subnormal, the Taylor series for φ must contain only real-valued coefficients. Assuming the linear fractional form which is the only possible form for a variety of analytic function spaces including those mentioned above, we identify those with non-real coefficients for which Cφ* may be subnormal on the weighted Bergman space A2α of the unit disk for each α. Furthermore, we prove that certain of these actually give rise to subnormal operators. Additionally, using similar techniques when the coefficients are real, we establish a necessary condition which, as α becomes large, approaches a long known necessary condition of Cowen due to spectral radius considerations.

Zhe Dong, Institute of Mathematics, Fudan University, Shanghai 200433, China  and Shijie Lu, Department of Mathematics, Zhejiang University, Hangzhou 310027, China.
The Hyperspace of Closed Connected Subsets of a Euclidean Space, pp. 385-392.
ABSTRACT. In this paper, we study module isomorphisms between weakly closed T(N)-modules, and obtain the following result: suppose that U, V are weakly closed T(N)-modules and that Φ: UV is a module isomorphism, then U= V and there exists a non-zero number λ such that Φ(T)=λ T, for all T in U.

C. L. García and W. B. Johnson, Department of Mathematics, Texas A&M University, College Station, TX 77843, U.S.A (clgarcia@itam.mx), (johnson@math.tamu.edu)
Power Type Uniform Convexity of X via p-asymptotic Uniform Convexity of Lr(X),  pp. 393-402.
ABSTRACT. We show that if Lr(X), 1 < r < ∞, has an asymptotically uniformly convex renorming of power type then X admits a uniformly convex norm of power type.

Thayer, F. Javier, 9112 Decatur Ave S, Bloomington, MN 55438 (jt@mitre.org).
Nonstandard Analysis of Graphs, pp. 403-436.
ABSTRACT. This paper shows certain metric length spaces characterized by volume growth properties of balls can viewed as graphs with infinitesimal edges. Our approach is based on nonstandard analysis.

V. Karunakaran, School of Mathematics, Madurai Kamaraj University, Madurai - 625 021, India. vkarun_mku@yahoo.co.in and R. Vembu, SBK College, Aruppukottai - 626 101, India. ( msrvembu@yahoo.co.in)
Hilbert Transform on Periodic Boehmians, pp. 437-452.
ABSTRACT. The theory of Hilbert transform on periodic functions and on periodic distributions is well known. In this paper we shall extend this theory to a suitable Boehmian space and identify a subspace of this Boehmian space on which the Hilbert transform becomes a one-to-one continuous linear map. We shall also construct examples of Boehmians which admit Hilbert transform in our sense, but do not represent periodic distributions.

Tom Hadfield, Department of Mathematics, University College, Cork, Republic of Ireland (T.Hadfield@ucc.ie).
Noncommutative Geometry of the Discrete Heisenberg Group, pp. 453-481.
ABSTRACT. Motivated by the search for new examples of ``noncommutative manifolds'', we study the noncommutative geometry of the group C*-algebra of the three dimensional discrete Heisenberg group. We present a unified treatment of the K-homology, cyclic cohomology and derivations of this algebra, placing it squarely within the framework of Connes' noncommutative geometry.

J. López-Gómez, Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040--Madrid, Spain. (Lopez_Gomez@mat.ucm.es).
Coexistence and Meta-Coexistence for Competing Species, pp. 483-536.
ABSTRACT. In this paper we analyze the dynamics of a family of competing species models where one of the species grows in the presence of finitely many refuges according to a logistic law. Basically, our results show how the model behaves like a superlinear indefinite problem for a single equation ( cf. H. Amann and J. Lopez-Gomez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Diff. Eqns. 146 (1998), 336-374.). As a result of the presence of refuges, for certain ranges of values of the parameters involved in the formulation of the model, the dynamics of its classical positive solutions is regulated by a metacoexistence state. By a metacoexistence state it is meant a solution couple consisting of a metasolution coupled with a classical regular solution. 
Roughly speaking, metasolutions are continuous extensions by infinity of large solutions (cf. C. Bandle and M. Marcus, Large solutions of semilinear elliptic equations: Existence, uniqueness and asymptotic behavior, J. D'Analysis Math. 58(1991), 9-24.) that regulate the dynamics of the positive solutions of a semilinear elliptic equation or system.
Metasolutions have been introduced and studied  by J. Garcia-Melian, R Gomez-Renasco, J. Lopez-Gomez and J. C. Sabina de Lis (Arch. Rat. Mech. Anal. 145 (1998)), R. Gomez-Renasco and J. Lopez-Gomez (Nonl. Anal. TMA 48 (2002) ), J. Lopez-Gomez ( El. J. Diff. Eqns. Conf. 05 (2000)) and in the dissertation of R. Gomez-Renasco (Universidad de La Laguna, Tenerife, February 1999.)