Electronic Edition Vol. 29, No. 3, 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


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.