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), 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.