HOUSTON JOURNAL
OF
MATHEMATICS

Electronic Edition Vol. 27, No. 2, 2001

Editors: H. Amann (Z├╝rich), 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), G. Pisier (College Station and Paris), S. W. Semmes (Rice)
Managing Editor: K. Kaiser (Houston)

Contents

Zanardo,Paolo, Dept. of Mathematics, University of Padova, 35131 Padova, Italy (pzanardo@math.unipd.it).
A Classical Result on Maximal Valuation Domains Revisited, pp. 237-245.
ABSTRACT. We prove that a non linearly compact valuation domain R admits a proper immediate extension S. This is the main point of Kaplansky's classical result that a valuation domain R is linearly compact if and only if it is maximal. In fact, we see by a counterexample that Kaplansky's original proof, as well as later versions of it, do not show that R and the constructed extension S have the same residue field.

Llibre, Jaume, Departament de Matematiques, Universitat Autonoma de Barcelona, 08193-Bellaterra, Barcelona, Spain (jllibre@mat.uab.es) , and Xiang Zhang, Department of Mathematics, Nanjing Normal University, Nanjing 210097, P. R. China. (xzhang@pine.njnu.edu.cn).
Topological Phase Portraits of Planar Semi-Linear Quadratic Vector Fields , pp. 247-296.
ABSTRACT. In this paper we solve completely the topological classification of the phase portraits for a class of semi-linear quadratic vector fields, i.e. vector fields such that its first component is a homogeneous polynomial of degree 1 and its second component is an arbitrary polynomial of degree 2 without constant term. As a corollary of our results we answer the problem proposed by Ye Yanqian at the end of Section 2 of his book Qualitative Theory of Polynomial Differential Systems, Shanghai Scientific and Technical Publishers, Shanghai, 1995 (in Chinese). Moreover, we prove that quadratic systems of class (I) in the Chinese classification of quadratic systems have exactly 50 different topological phase portraits, which corrects the result that such quadratic systems have only 47 different topological phase portraits (see Theorem 12.3 of the previous book).

Magnani, Valentino,Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy (magnani@cibs.sns.it)
Differentiability and Area formula on stratified Lie groups , pp. 297-323.
ABSTRACT. We prove the Area Formula for Lipschitz maps between stratified nilpotent Lie groups. The main tool is the a.e. differentiablity of Lipschitz maps, proved by P. Pansu in Ann. of Math. '89. We extend this result to the case of measurable domains with non trivial technical modifications. A suitable notion of jacobian is given for differential maps, called G-linear maps, finding relations with the classical definition of jacobian.

Zweck, John W. ,Department of Mathematics, University of Nevada, Reno, Nevada.
The Stiefel-Whitney Spark , pp. 325-351.
ABSTRACT. In this paper deRham currents and geometric measure theory are used to study the mod2 and integer Stiefel-Whitney classes. The paper is part of a larger program of Harvey and his collaborators to develop a theory of primary and secondary characteristic currents.
In a previous paper we proved that, in the case of the integer Stiefel-Whitney classes, to each "atomic" collection of sections of a real vector bundle there is associated a linear dependency current, with rectifiability properties, which is supported on the linear depenency set of the collection. The integer cohomology class of a linear dependency current is an integer Stiefel-Whitney class.
In this paper a locally Lebesgue integrable current, called the Stiefel-Whitney spark, is associated to each atomic collection of sections. The Stiefel-Whitney spark satisfies the local current equation that its exterior derivative is equal to the linear dependency current. This equation is the natural analogue for the integer Stiefel-Whitney classes of Harvey and Lawson's local Gauss-Bonnet-Chern formula for the Euler class. (A similar current equation is derived for the mod~2 Stiefel-Whitney classes.) Consequently, the Stiefel-Whitney spark plays the same role for the Stiefel-Whitney class that Chern's spherical potential (or transgression) plays for the Euler class.
An explicit local formula is derived for the Stiefel-Whitney spark, analogous to Chern's formula for the spherical potential. Furthermore, the Stiefel-Whitney spark yields a natural generalization of Eells' method of representing Stiefel-Whitney classes by pairs of forms with singularities.

Fernandez-Lopez, Manuel,Universidade de Santiago de Compostela, Facultade de Matemáticas, 15706 Santiago de Compostela, Spain (manfl@zmat.usc.es).
Geodesic Transformations in Quaternionic Geometry , pp. 353-376.
ABSTRACT. In this paper we study partially conformal geodesic transformations with respect to submanifolds in quaternionic manifolds. We show that non-isometrical ones only exist when the submanifold reduces to a point or is a real hypersurface. We study both cases separately getting necessary and sufficient conditions for their existence, which are expressed in terms of the Jacobi operators and their covariant derivatives. As an application we use these transformations to obtain new characterizations of quaternionic space forms and provide a classification of all the partially conformal geodesic transformations occurring in them.

Gil-Medrano O., Departamento de Geometria y Topologi, Facultade de Matematicas, Universidad de Valencia, 46100 Burjassot, Valencia, Spain (olga.gil@uv.es), Gonzalez-Davila, J.C., Departamento de Matematica Fundamental, Seccion de Geometria y Topologia, Universidad de La Laguna, la Laguna, Spain (jcgonza@ull.es), and Vanhecke, L., Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, 3001 Leuven, Belgium (lieven.vanheck@wis.kuleuven.ac.be).
Harmonic and Minimal Invariant Unit Vector Fields on Homogeneous Riemannian Manifolds, pp. 377-409.
ABSTRACT. We consider unit vector fields on homogeneous Riemannian manifolds (M = G/G0,g) which are G-invariant. We derive a criterion for the minimality and for the harmonicity of such vector fields by means of the infinitesimal models which correspond to (locally) homogeneous spaces and which are determined by using homogeneous structures. This leads to the construction of a lot of new examples of unit vector fields which are minimal or harmonic or which determine a harmonic map from (M,g) into its unit tangent sphere bundle equipped with the Sasaki metric. For several cases we obtain the complete list of such vector fields, in particular for low dimensions.

Duda, E. and Fernandez, H.V., University of Miami, Coral Gables, Fl.
Span and Plane Separating Continua , pp. 411-422.
ABSTRACT. A lower bound for the span of a plane separating continua is computed.

John R. Martin , Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, SK. S7N 5E6 Canada (math@sask.usask.ca).
Factors of Compact Absolute Fixed Point Sets , pp. 423-430.
ABSTRACT. A space X is an absolute fixed point set for a class Q of topological spaces (or an AFS(Q)-space) if X is a Q-space and whenever X is embedded as a closed subset of a Q-space Z, then X is the fixed point set of a self-mapping of Z. For many classes Q, it is shown that a compact AFS(Q)-space X cannot have a noncontractible ANR(Q)-space or a nonmetrizable generalized arc as a factor, and X can be expressed as a product of finite-dimensional metric spaces if and only if X is homeomorphic to a cube or is a finite-dimensional AR-space. An example is given which shows that a product of a cube with a contractible and locally contractible compactum need not be an AFS(Q)-space.

Dow, Alan , Department of Mathematics, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3, (dowa@yorku.ca) and Hart, Klaas Pieter, Faculty of Information Technology and Systems, TU Delft, Postbus 5031, 2600 GA Delft, the Netherlands, (k.p.hart@its.tudelft.nl).
Hereditary indecomposability and the Intermediate Value Theorem , pp. 431-438.
ABSTRACT. We study compact spaces X for which the ring C*(X) of bounded real-valued continuous functions satisfies the Intermediate Value Theorem. In this context the theorem says that, given a polynomial p over the ring C*(X) and two elements u and v of the ring such that p(u)<=0<=p(v) one can find an element w between inf{u,v} and sup{u,v} such that p(w)=0. Our main result characterizes hereditarily indecomposable spaces in term of the Intermediate Value Theorem for a limited class of polynomials. The question remains what topological properties of X characterize the full Intermediate Value Theorem for C*(X).

Cuervo M.T., Duda, E. and Fernandez H.V., University of Miami, Coral Gables, Fl.
Upper Semicontinuous Continuum Valued Functions and Spans of Continua, pp. 439-444.
ABSTRACT. Upper semicontinuous functions are used to prove that for a plane continuum homeomorphic to a closed disk, the symmetric span is equal to the symmetric span of its boundary. For a plane continuum the symmetric span is shown to be equal to the symmetric span of its outer boundary. Also continua in Rn irreducible with respect to span are shown to have empty interior.

Rodriguez-Lopez, Jesus, and Romaguera, Salvador
Reconciling Proximal Convergence with Uniform Convergence in Quasi-Metric Spaces, pp. 445-459.
ABSTRACT. Denote by l the lower quasi-pseudo-metric on R, i.e. l(x,y)=max{x-y,0}, for all x,y in R. The family of all lower semicontinuous real valued functions on a topological space X is denoted by SC(X). We prove that if (X,d) is a quasi-pseudo-metric space, then the proximal topology induced by the quasi-pseudo-metric space (XxR, d-1 x l ) agrees with the topology of uniform convergence on SC(X) if and only if every member of SC(X) is quasi-uniformly continuous. Some variants of this result are also obtained. In particular, the coincidence on SC(X) between the topology of uniform convergence and the Hausdorff quasi-pseudo-metric topology induced by d-1 x l and by d x l , respectively, is discussed. Our results extend to the nonsymmetric case well-known theorems by G. Beer and S. Naimpally, respectively.

Martinez, Teresa, Departamento de Matematicas, Universidad Autonoma de Madrid, 28049 Madrid, Spain.
Uniform Convexity in Terms of Martingale H1 and BMO spaces , pp. 461-478.
ABSTRACT. Martingale type and cotype properties of a Banach space were introduced and characterized by Pisier and are equivalent to super-reflexivity. In this paper we prove new characterizations of these properties in terms of inequalities between BMO spaces of B-valued martingales. The main tool in order to obtain these new characterizations is a theory of vector-valued martingale transform operators, which arise as a natural generalization of Burkholder's martingale transforms. Moreover, these techniques allow us to obtain as easy corollaries of the general theory the already known characterizations of martingale type and cotype properties.

Glass Miller, V., and Miller, T.L. Dept. of Mathematics, Mississippi State University, Mississippi State, MS 39762 (vivien@math.msstate.edu, miller@math.msstate.edu)
On the approximate point spectrum of the Bergman space Cesaro operator , pp. 479-494.
ABSTRACT. We identify the spectrum of the Cesaro operator on the Bergman spaces Lap(D) for p>1 and the approximate point spectrum is given for p> 2. In the case p>=4, we give a growth condition on the resolvent and obtain as a consequence that Cesaro operator has Bishop's property beta.