Electronic Edition
Vol. 25, No. 3, 1999

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


Allenby, R.B.J.T., Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, England (pmt6ra@leeds.ac.uk).
Some Remarks on the Upper Frattini Subgroup of a Generalized Free Product, pp. 399-403.
ABSTRACT. Prompted by a remark of M. K. Azarian, we introduce a new approach to the study of the upper and lower near Frattini subgroups of a generalized free product. The method applies more directly to the upper near Frattini subgroup and, consequently, answers questions left open by Azarian's investigations which centered mainly on the lower near Frattini subgroup.

Schmid, Jürg, Mathematisches Institut, Universität Bern, CH-3012 Bern, Switzerland (schmid@math-stat.unibe.ch).
Nongenerators, genuine generators and irreducibles, pp. 405-416.
ABSTRACT. We classify the elements of a (universal) algebra A with regard to the following properties: (i) a in A need not occur in any generating set for A, (ii) there is a generating set which must include a, or (iii) a must occur in all generating sets. Some relationships between these properties are established and illustrated, mainly by mono-unary algebras and distributive lattices.

Chin-Pi Lu, Department of Mathematics, University of Colorado, Denver, Colorado 80217-3364 U.S.A.
The Zariski Topology on the Prime Spectrum of a Module, pp. 417-432.
ABSTRACT. For any module M over a commutative ring R with identity, the prime spectrum Spec(M) of M is the collection of all prime submodules. We topologize Spec(M) with the Zariski topology, which is analogous to that for Spec(R), and investigate this topological space from the point of view of spectral spaces. For various types of modules M, we obtain conditions under which Spec(M) is a spectral space.
Addendum, posted February 13, 2009.
Proposition 5.2(3) and Proposition 6.3 of this paper are not correct as they stand. The correction of the propositions and full proofs of the corrected results can be found here.

Anderson, D. D., Department of Mathematics, The University of Iowa, Iowa City, IA 52242 U.S.A. (dan-anderson@uiowa.edu) and Zafrullah, M., Department of Mathematics, Scen 301, The University of Arkansas, Fayetteville, AR 72701 (kamla@compuserve.com).
Independent Locally-Finite Intersections of Localizations, pp. 433-452.
ABSTRACT. Let D be an integral domain and let F be a set of prime ideals of D. We say that D is an F-IFC domain if D is a locally finite intersection of localizations of D at the primes in F and if no two primes in F contain a common nonzero prime ideal. Examples of F-IFC domains include h-local domains, Noetherian domains in which grade-one primes have height one, and independent rings of Krull type. Using star operations we give several characterizations of F-IFC domains.

Birbrair, L. Departamento de Matemática, Universidade Federal do Ceará, Campus do Pici, Bloco 914, CEP. 60455-760, Fortaleza-CE, Brazil (lev@mat.ufc.br).
Local Bi-Lipschitz Classification of 2-Dimensional Semialgebraic Sets, pp. 453-472.
ABSTRACT. We present a combinatorial invariant for the problem of local bi-Lipschitz classification of 2-dimensional semialgebraic or subanalytic sets. We call it Hölder Complex. The invariant is complete in the following sense: two germs of 2-dimensional closed semialgebraic sets are bi-Lipschitz equivalent if and only if the corresponding Hölder Complexes are combinatorially equivalent.

Charatonik, J. J. , Charatonik, W. J., Omiljanowski, K., Mathematical Institute, University of Wroclaw pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland (jjc@hera.math.uni.wroc.pl, wjcharat@hera.math.uni.wroc.pl, komil@hera.math.uni.wroc.pl), and Prajs, J. R., Institute of Mathematics, University of Opole, ul. Oleska 48, 45-951 Opole, Poland (jrprajs@math.uni.opole.pl).
On Plane Arc-Smooth Structures, pp. 473-499.
ABSTRACT. Arc-structures on subspaces of the plane are studied in the paper. It is shown that each plane arc-smooth continuum admits an embedding in the plane such that its arc-smooth structure can be nicely extended to an arc-smooth structure on the whole plane. Using this it is proved that each plane arc-smooth continuum is a retract of the hyperspace of its closed subsets. Among several applications it is pointed out that each planar smooth dendroid admits a mean.

Valov, V., University of Swaziland, Private Bag 4, Kwaluseni, Swaziland (valov@realnet.co.sz).
Spaces of Bounded Functions, pp. 501-521.
ABSTRACT. We investigate some properties of a given topological space which are determined by the linear topological structure of all bounded continuous Banach space-valued functions on that space equipped with the compact open or poitwise convergence topology.

Foralewski, P. and Hudzik, H., Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Matejki 48/49, 60-769 Poznan, Poland (katon@amu.edu.pl), (hudzik@amu.edu.pl).
On Some Geometrical and Topological Properties of Generalized Calderón-Lozanovskii sequence spaces, pp. 523-542.
ABSTRACT. Generalized Calderon-Lozanovskii sequence spaces e_fi generated by a Musielak-Orlicz function fi and a Banach sequence lattice e are investigated. A regularity condition delta2e for fi is defined in such a way that it guarantee many positive properties of e_fi whenever e is also "good enough". There are considered the problems of order continuity and Fatou property of e_fi, some relationships between the norm and the modular near 0 and near 1, some embeddings between e_fi and their two subspaces, the Kadec-Klee property and the Kadec-Klee property with respect to coordinatewise convergence are characterized. Finally, necessary and sufficient conditions for various monotonicity properties are presented.

Caixing Gu, Department of Mathematics, California Polytechnic State University, San Luis Obispo, CA 93407 (cgu@calpoly.edu).
Finite Rank Products of Four Hankel Operators, pp. 543-561.
ABSTRACT. In this paper we characterize when the product of four Hankel operators is of finite rank. The charactization is in terms of a set of functional equations invloving the symbols of the Hankel operators. Given a permutation of four Hankel operators, the permuation product is the the product obtained by multiplying the operators in the order of the permuation. Using this characterization we show that if all permutation products of four Hankel operators have finite rank, then at least one of them is of finite rank. A rank formula is also given for the product of four Hankel operators, which extends the 1978 result of Axler, Chang and Sarason for the product of two Hankel operators.

Shujie Li, Institute of Mathematics, Academia Sinica, Beijing, China (lisj@math03.math.ac.cn) and Jiaquan Liu, Department of Mathematics, Peking University, Beijing, China.
Computations of Critical Groups at Degenerate Critical Point and Applications to Nonlinear Differential Equations with Resonance, pp. 563-582.
ABSTRACT. In this paper we study the critical groups at degenerate critical points. Together with new computations of the critical groups at infinity we obtain some abstract critical point theorems. As applications, we study the existence of nontrivial solutions of the elliptic boundary value problems and Hamiltonian systems.

Martio, O., Vuorinen, M. Department of Mathematics, P.O. Box 4 (Yliopistonkatu 5) FIN-00014 University of Helsinki, Finland (martio@cc.helsinki.fi, vuorinen@csc.fi) and V.M. Miklyukov Mathematics Department, Volgograd State University, 2 Prodolnaya 30, Volgograd 400062} (miklukov@math.vgu.tsaritsyn.su).
Critical points of A-solutions of quasilinear elliptic equations, pp. 583-601.
ABSTRACT. Critical points of solutions to degenerate elliptic equations in R,n n >= 2, consist of good (N-points) and bad points. Pseudoharmonic functions n=2, in the sense of Morse, have good points only. We give an estimate for the modulus of continuity of a generalized solution at an $N$-point. An analog of Sard's theorem is proved.