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
B. Banaschewski and J.J.C. Vermeulen, University of Cape
Town, Rondebosch, South Africa.
On the Booleanization of a Finite Distributive Lattice,
pp. 537- 544.
ABSTRACT.
It is shown that there is no functor on the category of finite distributive
lattices
L which assigns to each L the Boolean algebra of its
elements which are equal to their double pseudocompletement. In addition,
several other results are derived from this, including the fact, obtained via
Priestly Duality, that there is no functor on the category of finite partially
ordered sets X taking each X to the set MaxX of maximal
elements. The basic tool here is the observation that there is no covariant
endofunctor T of the category of finite sets for which T0 = 1
and TE = E for all E not equal to 0 .
C.J. Maxson, Department of Mathematics, Texas A&M University, College
Station, TX 77843 (cjmaxson@math.tamu.edu).
Forcing Linearity Numbers for
Nonsingular Modules Over Semiprime Goldie Rings , pp. 545-551.
ABSTRACT.
In this paper we complete the determination, initiated by Albrecht and Hausen,
of the forcing linearity numbers of nonsingular modules over semiprime Goldie
rings.
Lutz Strüngmann,
University of Essen, 45117 Essen, Germany (lutz.struengmann@uni-essen.de).
On the existence of rings
with prescribed endomorphism group, pp. 553-557.
ABSTRACT.
If R is a unital ring, then the left multiplications by elements of R obviously
form endomorphisms of the additive group of R. In fact they form a group direct
summand of the endomorphism ring and if the complement is trivial, then these
rings are called E-rings which are well-studied. In this note we show how
results on realizing rings as endomorphism rings of abelian groups can be
carried over to results on endomorphism groups of rings. In particular we
construct in Goedel's universe (V=L) almost-free commutative unital rings S
which have a given suitable ring R as the complement of the left
multiplications. Furthermore, a lot of ZFC results follow without "almost-free".
Finally, we show that in ZFC there exists an almost-free ring R of minimal
uncountable cardinality such that the endomorphism ring of R is isomorphic to
the direct sum of the integers and R itself.
Dobbs, David E., University of Tennessee, Knoxville, TN 37996-1300
(dobbs@math.utk.edu), and Picavet, G., Universite Blaise Pascal (Clermont
II), 63177 Aubiere Cedex,France (Gabriel.Picavet@math.univ-bpclermont.fr).
Weak Baer Going-Down Rings,
pp. 559-581.
ABSTRACT.
Let A be a commutative ring with identity. A is said to be a going-down ring
(resp., universally going-down ring) if A/P is a going-down domain, (resp., a
universally going-down domain) for each P in Spec(A). Also, A is said to be an
EGD ring (resp., EUGD ring) if the inclusion map from A to B satisfies GD
(resp., is universally going-down) for each overring B of A. The concept of
going-down ring (resp., universall going-down ring) is not equivalent to the
concept of EGD ring (resp., EUGD ring). A is a going-down ring (resp.,
universally going-down ring) if and only if the weak Baer envelope of the
associated reduced ring of A is a going-down ring (resp., a universally
going-down ring). A weak Baer ring is a going-down ring (resp., a universally
going-down ring) if and only if it is an EGD (resp., EUGD) ring. The weak Baer
going-down (resp.,universally going-down) rings are characterized as the EGD
(resp., EUGD) rings whose total quotient ring is von Neumann regular. A weak
Baer ring A is a universally going-down ring if and only if A' (the integral
closure of A) is a Prufer ring such that the inclusion map from A to A' is
universally going-down. Transfer results include the facts that if a commutative
faithfully flat A-algebra B is a weak Baer going-down (resp., universally
going-down) ring, then A is also a weak Baer going-down (resp., universally
going-down) ring.
Tsiu--Kwen Lee, Department of Mathematics, National Taiwan
University, Taipei 106, Taiwan (tklee@math.ntu.edu.tw) and Tsai-Lien Wong,
Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung
804, Taiwan (tlwong@math.nsysu.edu.tw)
On Armendariz Rings, pp.
583-593.
ABSTRACT.
In this paper we are concerned with the connections among (weak) Armendariz
rings, reduced rings and semiprime right Goldie rings. We construct certain
Armendariz rings and prove that a semiprime right Goldie ring is weak Armendariz
if and only if it is a reduced ring.
H.H. Brungs,
Department of Mathematical Sciences, University of Alberta,Edmonton,
Alberta T6G 2G1, Canada (hbrungs@math.ualberta.ca), H. Marubayashi,
Department of Mathematics, Naruto University of Education, Naruto, 772-8502,
Japan (marubaya@naruto-u.ac.jp) and A. Ueda, Department of Mathematics,
Shimane University, Matsue, 690-8504, Japan} (ueda@math.shimane-u.ac.jp.
A Classification of primary
ideals of Dubrovin valuation rings, pp. 595-608.
ABSTRACT.
Let R be a Dubrovin valuation ring of a simple Artinian ring
Q, that is, R is a Bezout order in Q and R/J(R) is
simple Artinian. Primary ideals of R are classified by using prime
segments, the orders of ideals, and the concept of divisorial ideals. There is a
special class of primary ideals of R which does not occur in commutative
valuation rings, even in P.I. Dubrovin valuation rings. Some examples of total
valuation rings containing primary ideals in this special class are given.
G. P. Leonardi, Dipartimento di Matematica, Università di Trento,
38050 Povo-Trento, Italy (gippo@science.unitn.it), and S. Rigot,
Université de Paris-Sud, Mathématiques, Bâtiment 425, 91405 Orsay cedex,
France (Severine.Rigot@math.u-psud.fr).
Isoperimetric sets on Carnot
groups,
pp. 609-637.
ABSTRACT.
We prove the existence of isoperimetric sets in any Carnot group, that is, sets
minimizing the intrinsic perimeter amoung all measurable sets with prescribed
Lebesgue measure. We also show that, up to a null set, these isoperimetric sets
are open, bounded, their boundary is Ahlfors-regular and they satisfy the
condition B. Furthermore, in the particular case of the Heisenberg group, we
prove that any reduced isoperimetric set is a domain of isoperimetry. All these
properties are satisfied with implicit constants that depend only on the
dimension of the group and on the prescribed Lebesgue measure.
Johann Davidov,
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad.
G.Bonchev Str. Bl.8, 1113 Sofia, Bulgaria (jtd@math.bas.bg).
Almost contact metric
structures and twistor spaces, pp. 639-674.
ABSTRACT.
Relations between the twistor spaces of even- and odd-dimensional Riemannian
manifolds are studied in this paper (which can be considered as a continuation
of the author‘s previous article
Twistorial examples of almost contact metric manifolds, HJM 28 (2002),
711-740). Several examples are considered and it is shown how the established
relations can be used to describe the twistor space of certain manifolds. Some
properties of two natural almost contact metric structures on the twistor space
of an odd-dimensional Riemannian manifold are considered in order to obtain
examples of almost contact metric manifolds that obey various geometric
properties.
Shihshu Walter Wei, Department of Mathematics, University of Oklahoma,
601 Elm Avenue, Room 423, Norman, OK 73019-0315{wwei@ou.edu}.
The Structure of Complete Minimal
Submanifolds in Complete Manifolds of Nonpositive Curvature, pp. 675-689.
ABSTRACT.
We provide a topological obstruction for a complete submanifold with a specific
uniform bound involving Ricci curvature to be minimally immersed in any complete
simply-connected manifold of nonpositive sectional curvature. We prove that such
minimal submanifolds of dimension greater than two have only one topological
end. The proof uses the Liouville theorem for bounded harmonic functions on
minimal submanifolds of this sort due to Yau (Ann. Scient. Ec. Norm.Sup. (8)
(1975)), and also adapts a technique of Cao-Shen-Zhu (Math. Res.
Lett.(4),(1997)) to show the existence of nonconstant bounded harmonic functions
based on the Sobolev inequality of Hoffman-Spruck (Comm. Pure and Applied Math.
(XXVII), (1989)). This extends the above result of Yau. The same phenomena occur
in a wider class of n-submanifolds with bounded mean curvature in an Ln
sense. By improving the techniques in Cao-Shen-Zhu, one can obtain the
topological conclusion in the intrinsic settings. These generalize and unify the
structure theorems in the extrinsic settings. A number of examples sharply limit
these weakening. For dimension n=2, a simple proof of a Conjecture of
Schoen-Yau is given with weakened hypotheses.
James Perry, Department of Mathematics, Mary Washington College,
Fredericksburg, VA 22401 and Stephen Lipscomb, Department of Mathematics,
Mary Washington College, Fredericksburg, VA 22401 (slipscom@mwc.edu).
The Generalization of
Sierpinski's Triangle that lives in 4-space, pp. 691- 710.
ABSTRACT.
In 1975-76, a generalization L(A) of the unit interval (a generalization of the
quotient construction of identifying adjacent endpoints in Cantor's space)
provided nonseparable analogues of the classical Nobeling (1930) and Urysohn
(1925) metric-space imbedding theorems. At first, L(A) was known only as a
one-dimensional metrizable topological space --- separable for finite A, and of
weight cardinality(A) otherwise. By 1992, L(A) was imbedded into Hilbert's
space. With the induced geometry, the space L(A) was exposed as the closure of a
``web-complex,'' structured pastings of various n-webs n = 0,1,2,... An n-web
L({0,1,2,...,n}) resides (naturally) in an n-simplex as the attractor of an
iterated function system, the 2-web and 3-web being the well-known Sierpinski
triangle and 3D-gasket. Since the 2-web cannot be viewed (imbedded with fractal
dimension preserved) in a line, and since the 3-web cannot be viewed in a plane,
the obvious conjecture was that we cannot view the 4-web in 3-space. Here,
however, we construct a (fractal-dimension)-preserving isotopy of linear
transformations that moves the 4-web into 3-space. As a corollary, we show that
for each n greater than 3, the n-web may be viewed in (n-1)-space.
W. Makuchowski, University of Opole, Institute of Mathematics and
Informatics, Oleska 48, 45-052 Opole, Poland (mak@math.uni.opole.pl).
On Local Connectedness at
a Subcontinuum and Smoothness of Continua, pp. 711- 716.
ABSTRACT.
In this paper a number of results on local connectedness in the hyperspace of
subcontinua of metric continua are obtained as the corollaries of the following
Theorem 2:
Let X be a Hausdorff continuum, let A∈C(X) and X be locally connected at
the subcontinuum A. If B∈C(X) and A⊂B, then the hyperspace C(X) is strongly
locally arcwise connected at the point B.
Among others we generalize the
following result of J.J. Charatonik and W.J. Charatonik: a metric continuum
having the property of Kelley is smooth if it is locally connected at some
point.
Luis Miguel García-Raffi, Salvador Romaguera and Enrique A.
Sánchez Pérez, Departamento de Matemática Aplicada, Universidad Politécnica
de Valencia, 49071 Valencia, Spain (lmgarcia@mat.upv.es), (sromague@mat.upv.es)
easancpe@mat.upv.es
On Hausdorff asymmetric normed
linear spaces, pp. 717-728.
ABSTRACT.
An asymmetric norm is a nonnegative and subadditive positively homogeneous
function q defined on a linear space. An asymmetric normed linear space is a
pair (X,q) such tha X is a linear space and q is an asymmetric norm on X In this
paper we characterize those asymmetric linear spaces whose induced topology is
Hausdorff. We also find a decomposition of asymmetric normed linear spaces as
direct sums of a Hausdorff asymmetric normed linear space and a non Hausdorff
one under reasonable conditions.
Christopher Mouron Department of Mathematics and Computer Sciences,
Hendrix College, Conway, AR 72032 (mouron@grendel.hendrix.edu).
Rotations of hereditarily
decomposable circle-like continua
, pp. 729-736.
ABSTRACT.
An hereditarily decomposable circle-like continuum, not homeomorphic to the
circle, that admits arbitrarily small periodic homeomorphisms semiconjugate to
arbitrarily small rigid rotations at the level of the tranche decomposition to
the circle is constructed.
Edward A. Azoff, The University of Georgia, Athens, GA 30602
(azoff@alpha.math.uga.edu),
Eugen J. Ionascu, Columbus State University, 4225 University Avenue,
Columbus, GA 31907 (ionascu_eugen@colstate.edu),
David R. Larson and Carl M. Pearcy, Texas A&M University, College
Station, TX 77843 (David.Larson@math.tamu.edu) , (Carl.Pearcy@math.tamu.edu).
Direct Paths of Wavelets ,
pp. 737-756.
ABSTRACT.
We associate a von Neumann algebra with each pair of complete wandering vectors
for a unitary system. When this algebra is nonatomic, there is a
norm--continuous path of a simple nature connecting the original pair of
wandering vectors. We apply this technique to wavelet theory and compute the
above von Neumann algebra in some special cases. Results from selection theory
and ergodic theory lead to nontrivial examples where both atomic and nonatomic
von Neumann algebras occur.
Pavol Quittner, Institute of Applied Mathematics, Comenius University,
Mlynska dolina, 84228 Bratislava, Slovakia (quittner@fmph.uniba.sk).
Continuity of the blow-up time
and a priori bounds for solutions in superlinear parabolic problems, pp.
757-799.
ABSTRACT.
We prove a priori bounds for solutions of superlinear parabolic problems on
bounded and unbounded spatial domains. In these bounds, the norm of the solution
at time t can be bounded by a constant which depends only on the norm of the
initial condition and the distance from t to the maximal existence time of the
solution. Using these estimates we show that the maximal esistence time depends
continuously on the initial condition. The nonlinearities in the equations are
subcritical and they may be nonlocal. Our proofs are based on energy,
interpolation and maximal regularity estimates. Optimality of our results and
some open problems are discussed.
Manuel Delgado and Antonio Suárez, Dpto. Ecuaciones
Diferenciales y Análisis Numérico, 41012, Univ. of Sevilla, Spain
(delgado@numer.us.es), (suarez@numer.us.es).
Positive solutions for the
degenerate logistic indefinite superlinear problem: the slow diffusion case ,
pp. 801-820.
ABSTRACT.
In this work we study the existence, stability and multiplicity of the positive
steady-states solutions of the degenerate logistic indefinite superlinear
problem. By an adequate change of variable, the problem is transformed into an
elliptic equation with concave and indefinite convex nonlinearities. We use
singular spectral theory, the Leray-Schauder degree, bifurcation and monotony
methods to obtain the existence results, and fixed point index in cones and a
Picone identity to show the multiplicity results and the existence of a unique
positive solution linearly asymptotically stable.
Florica-Corina Cirstea, Victoria University of Technology, Melbourne
City MC, Victoria 8001, Australia (florica@sci.vu.edu.au) and Vicentiu
Radulescu,, University of Craiova, 1100 Craiova, Romania (vicrad@yahoo.com).
Solutions with Boundary
Blow-up for a Class of Nonlinear Elliptic Problems, pp. 821-829.
ABSTRACT.
We study the existence of blow-up boundary solutions for the nonlinear logistic
equation with absorption and non-negative variable potential. It is established
a necessary and sufficient condition for the existence of these solutions. This
condition is formulated in terms of the first eigenvalue of the Laplace operator
with Dirichlet boundary conditions on the null set of the potential term, which
plays a crucial role in our analysis. Our framework includes the critical case
that corresponds to a weight vanishing on the boundary. The proofs rely
essentially on the Maximum Principle for elliptic equations, as well as on a
result of Alama and Tarantello (1996) related to the Dirichlet boundary value
problem for the logistic equation.