Electronic Edition Vol. 23, No. 2, 1997


Editors: G. Auchmuty (Houston), H. Brezis (Paris), S.S. Chern (Berkeley), J. Damon (Chapel Hill), L.C. Evans (Berkeley), R.M. Hardt (Rice), J.A. Johnson (Houston), A. Lelek (Houston), J. Nagata (Osaka), B. H. Neumann (Canberra), V. Paulsen (Houston), G. Pisier (College Station and Paris), R. Scott (Houston), S.W. Semmes (Rice), K. Uhlenbeck (Austin)
Managing Editor: K. Kaiser (Houston)


Lu, Chin-Pi, University of Colorado, Denver, CO 80217-3364.
Unions of Prime Submodules, pp. 203-213.
ABSTRACT. A proper submodule P of a module M over a ring R is said to be prime if re is in P for r in R and e in M implies that either e is in P or r is in P:RM. In this paper we investigate the following two topics which are related to unions of prime submodules:
i) The Prime Avoidance Theorem for modules and
ii) S-closed subsets of modules.

Repovs, D., University of Ljubljana, 1001 Ljubljana, Slovenia, Semenov, P.V., Moscow State Pedagogical University, 119882 Moscow, Russia, and Scepin, E.V., Steklov Mathematical Institute, 117966 Moscow, Russia.
Topologically Regular Maps with Fibers Homeomorphic to a One-Dimensional Polyhedron, pp. 215-229.
ABSTRACT. We introduce the concept of a topologically regular map as a map with homeomorphic fibers, whose multivalued inverse map is continuous with respect to the Frechet metric. We prove that every topologically regular map between compact metric spaces with fibers homeomorphic to some one-dimensional polyhedron is a locally trivial bundle.
In our proof two nontrivial facts are used. First one is Michael's Selection theorem for maps with convex but nonclosed values. Second one is that the restriction of a locally 1-soft map onto ``small'' closed neighborhoods, is an open monotone map. Here we need Anderson's map of the universal Menger curve onto the Hilbert cube.

Dontchev, Julian, University of Helsinki, PL 4, Yliopistonkatu 15, 00014 Helsinki, Finland (dontchev@cc.helsinki.fi), Ganster, Maximilian, Graz University of Technology, Steyrergasse 30, A-8010 Graz, Austria (ganster@weyl.math.tu-graz.ac.at), and Rose, David, Southeastern College of the Assemblies of God, 1000 Longfellow Boulevard, Lakeland, Florida 33801-6099 (darose@secollege.edu).
Alpha-Scattered Spaces II, pp. 231-246.
ABSTRACT. A topological space X is scattered if the only perfect or equivalently crowded subset of X is the empty set. Crowded sets are sometimes called dense-in-themselves and scattered sets are known as dispersed, zerstreut or clairseme.
For a topological space X, the Cantor-Bendixson derivative D(X) is the set of all non-isolated points of X. It is a fact that the Cantor-Bendixson derivative of every scattered space is nowhere dense. That D(X) is nowhere dense is equivalent to the assumption that I(X), the sets of all isolated points of X, is dense in X. The spaces satisfying this last condition are precisely the spaces whose alpha-topologies are scattered. These spaces are called alpha-scattered. The concept was recently used to show that in alpha-scattered spaces the notions of submaximal spaces and alpha-spaces coincide and thus a recent result of Arhangel'skii and Collins was improved. Topological spaces with countable Cantor-Bendixson derivative have been also recently considered. Such spaces are called d-Lindelof.
The aim of this paper is to continue the study of scattered and alpha-scattered spaces. The following three open problems are left:
(1) When is an omega-scattered space alpha-scattered?
(2) When is a sporadic space alpha-scattered? When do the two concepts coincide?
(3) How are alpha-scattered and C-scattered spaces related?

Putinar, Mihai, University of California, Santa Barbara, CA 93106.
Spectral Sets and Scalar Dilations, pp. 247-265.

Hashimoto, Takahiro, Ehime University, 790-77 Matsuyama-shi, Japan (taka@math.sci.ehime-u.ac.jp), and Otani, Mitsuharu, Waseda University, 169 Tokyo, Japan (otani@mn.waseda.ac.jp).
Nonexistence of Weak Solutions of Nonlinear Elliptic Equations in Exterior Domains, pp. 267-290.
ABSTRACT. The nonexistence of nontrivial solution is discussed for some quasilinear elliptic equations for the case where a domain is exterior of bounded and starshaped domain. It should be noted that because of the degeneracy of the equation, the nontrivial solutions of the equation are not twice differentiable. Therefore there arises the necessity of discussing the nonexistence of solutions in a framework of weak solutions. When the domain is bounded, the second author introduced a ``Pohozaev-type inequality'' valid for a class of weak solutions, which is effective enough for discussing the nonexistence. We here give an exterior-domain version of Pohozaev-type inequality, whence some results on the nonexistence of solutions are derived. These suggest that concerning the existence and nonexistence of nontrivial solutions, there may exist an interesting duality between the interior problems and the exterior problems for starshaped domains.

Tshikuna-Matamba, T., Institut Superieur Pedagogique, B.P. 282-Kanaga, Zaire.
On the Structure of the Base Space and the Fibres of an Almost Contact Metric Submersion, pp. 291-305.

Seubert, S., Bowling Green State University, Bowling Green, OH 43403-0221 (sseuber@math.bgsu.edu).
Reducing Subspaces of Compressed Analytic Toeplitz Operators, pp. 307-327.

Poon, Chi-Cheung, National Chung Cheng University, Chiayi 621, Taiwan.
On the Heat Equation for Harmonic Maps into Round Cones, pp. 329-340.

Fister, Renee, Murray State University, Murray, KY 42071 (kfister@math.mursuky.edu).
Optimal Control of Harvesting in a Predator-Prey Parabolic System, pp. 341-355.

Jung, Michael, Technische Universitat Berlin, 10623 Berlin, Germany (mjung@math.tu-berlin.de).
Evolution Families and Generators of Semigroups, pp. 357-383.
ABSTRACT. In this article we will examine properties of evolution families and their generators. We start out with the generator of a semigroup and by perturbation obtain results on non-autonomous, abstract Cauchy problems. Some results on non-linear problems are also obtained. It is namely the (Z*)-condition, which will provide us with results and which we will relate to the (Z)-condition given by W. Desch and W. Schappacher. Some examples are given in the last section.

Morayne, M., and Ryll-Nardzewski, C. .
Errata to Superpositions with Differences of Semicontinuous Functions, p. 384.
(Vol. 22, No. 4, 1996, pp. 719-735.)

Garity, Dennis J., Jubran, Isa S., and Schori, Richard M. .
Corrigenda to a Chaotic Embedding of the Whitehead Continuum, pp. 385-390.
(Vol. 23, No. 1, 1997, pp. 33-44.)

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