Electronic Edition Vol. 28, No. 3, 2002

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), B. H. Neumann (Canberra), V. I. Paulsen (Houston), G. Pisier (College Station and Paris), S. W. Semmes (Rice)
Managing Editor: K. Kaiser (Houston)

Houston Journal of Mathematics


David E. Rush, Department of Mathematics, University of California, Riverside, California 92521 (rush@newmath.ucr.edu) and Laura J. Wallace,(wallace@csusb.edu).
Noetherian Maximal Spectrum and Coprimely Packed Localizations of Polynomial Rings, pp. 437-448.
ABSTRACT. A ring R is said to be coprimely packed if for every ideal I of R and for each set S of maximal ideals of R one has that I is a subset of the union of S only if I is a subset of some member of S.
A ring R is shown to have Noetherian maximal spectrum if and only if R(X) is coprimely packed.

M. Gehrke, Department of Mathematical Sciences New Mexico State University Las Cruces, NM 88003 USA (mgehrke@nmsu.edu) and H. A. Priestley, Mathematical Institute 24/29 Oxford OX1 3LB UK ( hap@maths.ox.ac.uk)
Non-canonicity of MV-algebras , pp. 449-455.
ABSTRACT. The variety MV of all MV-algebras is shown to be non-canonical in a strong sense. Specifically it is shown that the canonical extension of the Chang algebra, K2, is not an MV-algebra. As a consequence, no non-finitely generated variety of MV-algebras is canonical.

Nako Nachev, Department of Algebra, Plovdiv University, 4000 Plovdiv, Bulgaria (nachev@pu.acad.bg).
Isomorphism of Semisimple Twisted Group Algebras, pp. 457-464.
ABSTRACT. Let K be a field of characteristic different from 2. In this paper we give sufficient and necessary conditions for the isomorphism of semisimple twisted group algebras of cyclic 2-groups.

C. Watt, School of Mathematics, Trinity College Dublin, Dublin 2, Ireland (cwatt@maths1.kst.dit.ie).
Horizontal Complex Curves and Holomorphic Curvature , pp. 465-495.
ABSTRACT. Horizontal complex curves are defined on a complex Finsler manifold and are shown to coincide with those complex curves which realise the holomorphic curvature of the given Finsler metric. An existence and uniqueness theorem for horizontal complex curves is sketched and extensions of known characterisations of the Kobayashi metric are derived. The theory is illustrated for a family of metrics on the Euclidean open unit ball of a finite dimensional complex vector space.

Dusan Repovs, Institute of Mathematics, Physics and Mechanics, University of Ljubljana, Jadranska 19, P. O. Box 2964, Ljubljana, Slovenia 1001 (dusan.repovs@fmf.uni-lj.si and Pavel V. Semenov Department of Mathematics, Moscow City Pedagogical University, 2-nd Selskokhozyastvennyi pr. 4, Moscow,Russia 129226 (pavels@orc.ru ).
On Relative Approximation Theorems , pp. 497-509.
ABSTRACT. We consider an approach which reduces the solution of the approximation problem for a USC mapping to an appropriate selection-type restriction of the family of values of this mapping. As a corollary we prove that for every compact, not necessarily ANR, domain X of a USC convex-valued mapping H : XY and for every ε > 0 there exists δ > 0 such that every δ-approximation of H defined over a closed subset AX can be extended to an ε-approximation over the entire domain. Similar facts are established for selections and ε-selections of H over A.

Manuel Sanchis, Departament de Matematiques, Universitat Jaume I, Campus del Riu Sec s/n, 12071, Castelló, Spain (sanchis@mat.uji.es) and Angel Tamariz-Mascarúa, Universidad Nacional Autónoma de México, Facultad de Ciencias, Departamento de Matemáticas, 04510 México D.F., México (atamariz@servidor.unam.mx).
A note on p-bounded and quasi-p-bounded subsets, pp. 511-527.
ABSTRACT. We discuss the relationship between p-boundedness and quasi-p-boundedness in the realm of Generalized Linearly Ordered Spaces (GLOTS) for free ultrafilters p on the natural numbers. We show that p-pseudocompactness, p-compactness, and quasi-p-compactness are equivalent properties for GLOTS; that bounded subsets of a GLOTS are strongly bounded; and that C-compact subsets of a GLOTS are strongly C-compact. We also show that a topologically orderable group is locally precompact if and only if it is metrizable. For bounded subsets of a GLOTS, a version of the classical Glicksberg's theorem on pseudocompactness is obtained. Also we prove that there exists an ultrapseudocompact topological group which is not quasi-p-compact for any free ultrafilter p on the natural numbers. To see this example, p-pseudocompactness and p-compactness are investigated in the field of spaces of continuous functions defined on a space X with values in [0,1], and considered with the pointwise convergence topology: Cp(X,[0,1]); proving that ultracompactness, quasi-p-compactness and countable compactness (respectively, ultrapseudocompactness, quasi-p-pseudocompactness and pseudocompactness) are equivalent properties in these spaces.

Byung-Jay Kahng, Department of Mathematics, University of Kansas, Lawrence, KS 66045 (bjkahng@math.ukans.edu).
* -representations of a quantum Heisenberg group algebra , pp. 529-552.
ABSTRACT. In our earlier work, we constructed a specific non-compact quantum group whose quantum group structures have been constructed on a certain twisted group C*-algebra. In a sense, it may be considered as a ``quantum Heisenberg group C*-algebra''. In this paper, we will find, up to equivalence, all of its irreducible *-representations. We will point out the Kirillov type correspondence between the irreducible representations and the so-called dressing orbits. By taking advantage of its comultiplication, we will then introduce and study the notion of ``inner tensor product representations''. We will show that the representation theory satisfies a ``quasitriangular'' type property, which does not appear in ordinary group representation theory.

Leo B. Jonker and Ole A. Nielsen, Queen's University, Kingston ON K7L 3N6, Canada (leo@mast.queensu.ca, nielseno@post.queensu.ca).
An Example Concerning the Order-Two Density of a Set , pp. 553-563.
ABSTRACT. For each real number s strictly between 0 and 1 we construct a subset of the unit interval whose Hausdorff dimension is s, whose upper density (relative to the Hausdorff measure in that dimension) is almost everywhere equal to 1, and whose order-two density and lower density are both everywhere equal to 0.

Fred Richman, Florida Atlantic University, Boca Raton FL 33431 (richman@fau.edu), Douglas Bridges University of Waikato, Hamilton, New Zealand (d.bridges@math.canterbury.ac.nz) and Peter Schuster, Universität München, 80333 München (pschust@rz.mathematik.uni-muenchen.de).
Trace-class operators, pp. 565-583.
ABSTRACT. In this paper we give a direct definition of the von Neumann-Schatten classes of operators in terms of the existence of a certain supremum. The theory is developed without appeal to separability, or to the existence of an orthonormal basis, and without using countable choice or the law of excluded middle. We construct the singular values of compact operators, and characterize compact operators, and the von Neumann-Schatten classes, in terms of singular values.

N.J. Kalton, Department of Mathematics, University of Missouri-Columbia, Columbia, MO 65211 (nigel@math.missouri.edu) and A.E. Litvak, Department of Mathematics, Technion, Haifa, Israel, 32000 (alex@math.technion.ac.il) (current address: Department of Math. and Stat. Sc., University of Alberta, Edmonton, AB, Canada (alexandr@math.ualberta.ca).
Quotients of finite-dimensional quasi-normed spaces , pp. 585-598.
ABSTRACT. We study the existence of cubic quotients of finite-dimensional quasi-normed spaces, that is, quotients well isomorphic to k-dimensional cube for some k. We give two results of this nature. The first guarantees a proportional dimensional cubic quotient when the envelope is cubic; the second gives an estimate for the size of a cubic quotient in terms of a measure of non-convexity of the quasi-norm.

K. S. Ryu, Department of Mathematics, Han Nam University, Taejon 306-791, Korea (ksr@math.hannam.ac.kr) and S. C. Yoo, Jaeneung College, Incheon 401-714, Korea (yky3ng@mail.jnc.ac.kr).
The existence theorem and formula for an operator-valued function space integral over paths in abstract Wiener space , pp. 599-620.
ABSTRACT. In 1973, Kuelbs and LaPage showed the existence of analogy of Wiener measure over continuous paths in abstract Wiener space [1]. In 1992, Ryu found the formula, similar to the Wiener integration formula, for this measure [2]. In this article, we introduce the definition of scale-invariant weak integral and investigate some basic properties of this integral. We establish an existence theorem and formula for an operator-valued function space integral over continuous paths in an abstract Wiener space, which are represented by the scale-invariant weak integral. The work here is patterned to some extent on earlier work involving Wiener and Feynman path integrals but the present setting requires a number of new concepts and results.
[1] J. Kuelbs and R. LaPage, The law of the iterated Logarithm for Brownian Motion in a Banach space, Trans. Amer. Math. Soc., 185 (1973), 253-264.
[2] K.S. Ryu, The Wiener integral over paths in abstract Wiener space, J. Korean Math. Soc., Vol. 29, No. 2, (1992), 317-331.

B. Scarpellini, Mathematisches Institut der Universität, Rheinsprung 21, 4051 Basel, Switzerland (bscarpellini@hotmail.com).
On a family of large solutions of Navier-Stokes , pp. 621-647.
ABSTRACT. It is well known that large, smooth initial data give rise to local strong solutions of the Navier-Stokes equations, while small smooth initial data give rise to global strong solutions. Here we describe three classes of initial data w0 for which ||∇w0|| may be arbitrarily large, which give rise to global strong solutions.

Peihao Zhao, Department of Mathematics, Lanzhou University, Lanzhou, 730000, P. R. of China (zhaoph@lzu.edu.cn) and Chengkui Zhong.
Positive Solutions of Elliptic Equations Involving both Supercritical and Sublinear Growth , pp. 649-663.
ABSTRACT. Positive solutions of elliptic problem involving both supercritical and sublinear growth on the unit ball with Dirichlet boundary condition are analysed as their supremum norm tend to infinity. It is shown that they converge, uniformly away from the origin, as well as in H1, to the unique singular solution. Then we prove that there is only one solution when the sublinear term is small enough, this give a negative answer to the open problem in J.Funct.Anal.122(1994),519-543. And, an infinitely number of positive solutions has been obtained.