*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

*Contents*

**Ismail M. Idris, ** Department of Mathematics, Faculty of Science,
Ain-Shams University, Cairo 11566, Egypt; *current address: *Mathematics
Department, Faculty of Science, UAE university, P.O.Box 17551, Al-Ain, United
Arab Emirates (ismail.idris@uaeu.ac.ae).

*-Value Functions of *-Rings,
pp. 1-9.

ABSTRACT.
The notion of a *-value function on a noncommutative ring with involution is
studied. The results obtained generalize those of valuations in the case of a
commutative ring.

**Thomas M. McCall, ** **Charles J. Parry,** Department of
Mathematics, Virginia Tech, Blacksburg, VA 24061 (Parry@math.vt.edu), and **
Romona R. Ranalli,** Department of Mathematics, 3900 University
Boulevard, University of Texas at Tyler, Tyler, TX 757999.

The 2-Class Group of Certain
Number Fields, pp. 11-26.

ABSTRACT.
In this article we describe a method for determining the structure of the 2 -
class group of a bicyclic biquadratic extension of an arbitrary number field
with odd class number. We also determine all imaginary octic fields of type
(2,2,2) having class number less than or equal 16 or prime class number.

**Vadim Ponomarenko,** Department of Mathematics, Trinity University, 715
Stadium Drive, San Antonio, Texas 78212-7200 (vadim@trinity.edu).

Reduction of Jump Systems,
pp. 27-33.

ABSTRACT.
A jump system is a set of integer lattice points satisfying an exchange axiom.
We discuss an operation on lattice points, called reduction, that preserves the
jump system two-step axiom. We use reduction to prove a weakened version of a
matroid conjecture by Rota.

**Young Suk Choi **and **Young Jin Suh,** Department of Mathematics,
Kyungpook National University, Taegu, 702-701, KOREA (yjsuh@bh.knu.ac.kr) and
**Jung-Hwan Kwon,** Department of Mathematics Eduacation, Taegu University,
702-701, Korea (jhkwon@biho.daegu.ac.kr).

On Chern Type Problems in
Space-Like Complex Submanifolds of an Indefinite Complex Hyperbolic Space,
pp. 35-54.

ABSTRACT.
In this paper, we introduce a kind of Chern type problem in an n-dimensional
complete space-like submanifold of an (n+p)-dimensional indefinite complex
hyperbolic space of constant holomorphic sectional curvature c with signature
(2p, 2(n+p)). Moreover, we give a *best possible* estimation for the norm
of the second fundamental form of a complex quadric Q^n immersed in indefinite
complex hyperbolic space with signature (2,2(n+1)).

**W. Gu, **1250 N. Dartmouth Ave., Claremont, CA 91711} (gu@math.hmc.edu),
and **Christopher Pries, ** 340 E. FootHill Blvd., Claremont, CA 91711
(cpries@hmc.edu).

Examples of Cayley 4-Manifolds,
pp. 55-87.

ABSTRACT.
We determine several families of so-called Cayley 4-dimensional manifolds in the
real Euclidean 8-space. Such manifolds are of interest because Cayley
4-manifolds and Cayley 4-cycles in Calabi-Yau 4-folds and Spin(7) holonomy
manifolds are supersymmetric cycles that are candidates for representations of
fundamental particles in String Theory. Moreover, some of the examples of Cayley
manifolds discovered in this paper may be modified to construct explicit
examples in our current search for new holomorphic invariants for Calabi-Yau
4-folds and for the further development of mirror symmetry.

We apply the classic results of Harvey and Lawson to find Cayley manifolds which
are graphs of functions from the set of quaternions to itself. We consider
graphs which are invariant under the action of three dimensional subgroups of
Spin(7) which fix the quaternions as a subgroup of the Cayley numbers. Spin(7)
is a subgroup of SO(8) which preserves the Cayley form. Systems of ODEs and PDEs
are derived and solved, some special cases of a classic theorem of Harvey and
Lawson are investigated, and theorems aiding in the classification of all such
manifolds described here are proven. Several families of interesting Cayley
4-dimensional manifolds are discovered. Some of them are novel.

**Sophia Zafiridou, ** Department of Mathematics, University of Patras,
26500 Patras, Greece (zafeirid@math.upatras.gr).

Planar Rim-Scattered
Compactifications of Planar Spaces, pp. 89-97.

ABSTRACT.
We prove that a space *X * admits a planar rim-scattered compactification
iff there exists a homeomorphism
*h* of *X *into the plane such that *h(X)* is nowhere dense in
the plane and each point of the closure of *h(X) *in the plane is contained
in the interior of an arbitrarily small disk, whose boundary intersects *h(X)*
in a set with a scattered closure in the plane.

**Gerardo Acosta, **Instituto de Matematicas, Circuito Exterior, Ciudad
Universitaria, Area de la Investigacion cientifica, Mexico, D. F., 04510, MEXICO
(gacosta@math.unam.mx).

On Smooth Fans and Unique
Hyperspaces,
pp. 99-115.

ABSTRACT.
In this paper we show that if *X * is a smooth fan and *Y *is a fan
such that the hyperspaces of subcontinua *C(X) *and *C(Y)* are
homeomorphic, then *X* and *Y *are homeomorphic. This is a
generalization of a result by C. Eberhart and S. B. Nadler, Jr.

**Michael E. Taylor, ** Math. Dept., Univ. of North Carolina Chapel Hill,
NC 27599-3250 (met@math.unc.edu).

Fourier Series and Lattice Point
Problems, pp. 117-135.

ABSTRACT.
Here we study functions on the torus satisfying a thin-shell Fourier series
estimate, a phenomenon that arose in the study of pointwise convergence of
Fourier series in previous work of the author. We consider classes of functions
with conormal singularities, and see that the geometry of the singular set
influences the nature of thin-shell estimates. This analysis brings in certain
non-isotropic lattice point estimates, and these estimates in turn lead to
variants of stationary phase methods, investigated by studying Schrodinger
equations with singular initial data.

**Steven M. Seubert, **Bowling Green State University, Bowling Green, OH,
43403-0221 (sseuber@bgnet.bgsu.edu).

Semigroups of Analytic Toeplitz
Operators on H^{2}, pp. 137-145.

ABSTRACT.
Using a result of D. Suarez, we show that a closed operator *R* having
domain *D(R)* dense in the Hardy space H^{2} of the open unit disc
and commuting with the standard unilateral shift *S* on *D(R)* is
given by an unbounded analytic Toeplitz operator R = T_{C} having symbol
C from the Nevanlinna class N^{+}.

Using this result, we show that any collection of bounded linear operators R_{t}
on H^{2} commuting with *S* defines a C_{0}-semigroup if
and only if there exists a function *C* analytic and having real part
bounded above on the open unit disc for which R_{t }= T_{etC}
for all nonnegative numbers t.

**David W. Kribs, **Department of Mathematics and Statistics, University
of Guelph, Guelph, ON, CANADA N1G 1M8 (dkribs@uoguelph.ca).

Non-Selfadjoint operator algebras
generated by weighted shifts on Fock space, pp. 147-169.

ABSTRACT.
Non-commutative multi-variable versions of weighted shifts arise naturally as
`weighted' left creation operators acting on Fock space. We investigate the weak
operator topology closed algebras they generate. The unweighted case yields
non-commutative analytic Toeplitz algebras. The commutant can be described in
terms of weighted right creation operators when the weights satisfy a condition
specific to the non-commutative setting. We prove these algebras are reflexive
when the eigenvalues for the adjoint algebra include an open set in complex
n-space, and provide a new elementary proof of reflexivity for the unweighted
case. We compute eigenvalues for the adjoint algebras in general, finding
geometry not present in the single variable setting. Motivated by this work, we
obtain general information on the spectral theory for non-commuting n-tuples of
operators.

**Taskinen, Jari **, University of Joensuu, FIN 80101 Joensuu, Finland
(jari.taskinen@joensuu.fi).

On the Continuity of Bergman
and Szego Projections, pp. 171-190.

ABSTRACT.
We study the triplet of function spaces, call them H, h, and L, of analytic,
harmonic and measurable functions on the open unit disk of the complex place.
The following facts hold: the Bergman projection is continuous from L onto H,
the Szego projection is continuous from h onto H, and harmonic conjugation is an
isomorphism on h. We show that these spaces are in a sense the smallest
extensions of the classical Banach space of bounded analytic functions (and
related spaces) which have the above mentioned property.

**J. Pecaric, ** Faculty of Textile Technology, University of Zagreb,
Pierottieva 6, 10000 Zagreb, Croatia (pecaric@hazu.hr), **J. Micic, **
Electrical Engineering Department, Polytechnic of Zagreb, Konavoska 2, 10000
Zagreb, Croatia (jmicic@public.srce.hr), and **Y. Seo, **Tennoji Branch,
Senior Highschool, Osaka Kyoiku University, Tennoji, Osaka 543-0054, Japan
(yukis@cc.osaka-kyoiku.ac.jp).

Inequalities Between Operator
Means Based on the Mond-Pecaric Method, pp. 191-207.

ABSTRACT.
As a continuation of the previous paper by J. Micic, J. Pecaric ,
and Y. Seo, *
Complementary inequalities to inequalities of Jensen and Ando based on Mond-
Pecaric method*, Linear Alg. Appl., **318** (2000), 87--107, we show
further general complementary inequalities to operator inequalities on a
positive linear map associated with two operator means.

**Raymond Mortini,** Université de Metz, F-57045 Metz, France (
mortini@poncelet.sciences.univ-metz.fr).

Maximal Gleason parts and
support sets for trivial points, pp. 209-218.

ABSTRACT. The paper is concerned with the topological
space S of trivial points in the algebra of bounded analytic functions in the
open unit disk. It is shown that every point in the closure of the set of
trivial points outside the fiber M_{1 }has a maximal Gleason part
within M_{1}._{ }Also, every point in M_{1}
which lies in the closure of E\ M_{1} for an interpolation set E
of trivial points has maximal support.

**Alec Matheson,**Department of Mathematics, Lamar University, Beaumont TX
77710 (matheson@math.lamar.edu).

Isometries into Function
Algebras, pp. 219-230.

ABSTRACT.
The isometries from a uniform algebra into another come in two distinct types.
The Type 1 isometries are associated with a set of uniqueness in the Shilov
boundary of the range, and are completely described for certain algebras of
analytic functions. Some more or less concrete examples of Type 2 isometries are
also given.

**Yiqiang Liu** and **Yifeng Xue ** (xyf63071@public9.sta.net.cn),
Department of Mathematics, East China University of Science and Technology,
Shanghai 200237, P.R. China and **Fahui Zhai**, Qingdao Chemical Technology
College, Qingdao 266042, P.R. China.

The closures of (U+K)-orbits of
certain essentially normal models, pp. 231-244.

ABSTRACT.
Let Ω be a simply connected analytic Cauchy domain and μ be a measure on ∂Ω
equivalent to the arc length measure on ∂ Ω. Let M(Ω, μ) be the multiplication
operator on the Hilbert space of all μ--integrable functions on ∂Ω which are
analytic in Ω. Let M(i) be the direct sum of i M(Ω, μ) s' and M[j] be the direct
sum of j M(Ω*, μ) s'. In this paper, we determine the closure of (U+K)-orbit of
the operator M(i)⊕ M[j] with i and j finte. This solves the problem presented by
M. Dost\'al in his Ph.D thesis.

**Espínola, Rafael,** Departamento de Analisis Matematico, Universidad de
Sevilla, Sevilla, 41-080 Spain (espinola@us.es), and
**Wisnicki, Andrzej,** Department of Mathematics, Maria Curie - Sklodowska
University, 20-031 Lublin, Poland (awisnic@golem.umcs.lublin.pl),and **Wosko,
Jacek**, Department of Mathematics, Maria Curie - Sklodowska University,
20-031 Lublin, Poland (jwosko@golem.umcs.lublin.pl).

On a Unified Study of Relative
Chebyshev Radii and Hausdorff Measures of Noncompactness, pp. 245-257.

ABSTRACT.
The paper is concerned with the notion of the Lifschitz modulus introduced by
Wisnicki and Wosko (1996) and its relationship with both relative Chebyshev
radii and Hausdorff measures of noncompactness.

**Metcalfe, Jason, **Georgia Institute of Technology, School of
Mathematics, Atlanta, GA 30332-0160 (metcalfe@math.gatech.edu
).

Global Existence for
Semilinear Wave Equations Exterior to Nontrapping Obstacles, pp. 259-281.

ABSTRACT.
In this paper, we prove the existence of global small amplitude solutions to
semilinear wave equations with quadratic nonlinearities exterior to a
nontrapping obstacle. This generalizes the work of Hayashi in a domain
exterior to a ball and of Shibata and Tsutsumi in spatial dimensions greater
than or equal to 6.

**Wan Se Kim, **
Department of Mathematics, Hanyang University, Seoul 133-791, KOREA
(wanskim@hanyang.ac.kr).

Multiple Existence of Periodic
Solutions for Semilinear Parabolic Equations with Large Source, pp.283-295.

ABSTRACT.
Multiple existence of solutions for the Dirichlet-periodic boundary value
problem of semilinear parabolic equations is discussed.

**Ravi P. Agarwal,** Department of Mathematical Sciences, Florida
Institute of Technology, Melbourne, FL 32901--6975, U.S.A. (agarwal@fit.edu) and **
Donal O'Regan, **Department of Mathematics, National University of Ireland,
Galway, Ireland.

Semipositone Dirichlet Boundary
Value Problems with Singular Nonlinearities , pp. 297-308.

ABSTRACT.
The existence of positive solutions to semipositone singular problems is
discussed in this paper. Our analysis relies on a cone fixed point theorem.