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*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 *delta _{2}^{e}*
for

**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,