Electronic Edition
Vol. 26, No. 2, 2000

Editors: H. Amann (Zürich), G. Auchmuty (Houston), D. Bao (Houston), H. Brezis (Paris), S. S. Chern (Berkeley), 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)


B.A. Davey and M. Haviar, La Trobe University Bundoora, Victoria 3083 Australia (B.Davey@latrobe.edu.au), (M.Haviar@latrobe.edu.au).
A Schizophrenic Operation which aids the Efficient Transfer of Strong Dualities , pp. 215-222.
ABSTRACT. We show that, in many cases, if DB and MB are finite algebras which generate the same quasi-variety CCD, then a strong duality for CCD based on DB may be transferred to a strong duality for CCD based on MB by the addition some endomorphisms of MB and just one further partial operation. This additional operation exhibits the schizophrenia so typical of the theory of natural dualities. We show how the result may be applied to yield an efficient strong duality in the case when MB is a distributive lattice, a semilattice or an abelian group.

H. P. Goeters, Department of Mathematics, Auburn University, Auburn, AL 36849-5310 (goetehp@mail.auburn.edu), and W. J. Wickless , Mathematics Department, University of Connecticut, Storrs, CT 06268 (wjwick@UConnVM.UConn.edu).
Relative Injectivity and Equivalence Theorems , pp. 223-239.
ABSTRACT. Two subgroups, H and K, of an abelian group G are said to be equivalent when there is an automorphism of G sending H onto K.. Here we will consider equivalence theorems for torsion-free reduced abelian groups of finite rank.
Hill wondered if a homogeneous group G with the type of the integers satisfying the simple test for the equivalence of pure subgroups must be free as an abelian group. The first author investigated Hill's problem in before and considered the homogeneous groups G with the following property: whenever H and K are pure subgroups of G and f : H --->K is an isomorphism, there is an automorphism of G restricting to f on H. It was established that the homogeneous group G has this isomorphism lifting property precisely when G is quasi-pure injective. In particular, G must be free as a module over the center of its endomorphism ring but G need not be free as an abelian group, thus answering Hill's query in the negative.
In this article we will classify the torsion-free abelian groups of finite rank which have the isomorphism lifting property lips defined above. Unlike the homogeneous situation, we show that groups having lips need not be quasi-pure injective qpi, but qpi groups have the lips property. A notion related to the lips condition is the mteps condition; a group G has the minimal test for the equivalence of pure subgroups( mteps), if any two isomorphic pure subgroups of G are equivalent in G. We characterize the groups with mteps below, showing in particular the only circumstance when a homogeneous group G with mteps fails to be qpi is when rank(G) = 3 and when the set of primes where pG is not equal to G excludes an infinite set of primes.

H. P. Goeters Department of Mathematics, Auburn University, Auburn, AL 36849-5310 (goetehp@mail.auburn.edu), and Bruce Olberding, Department of Mathematics, Northeast Louisiana University, Monroe, LA 71209 (maolberding@alpha.nlu.edu).
On the Multiplicative Properties of Submodules of the Quotient Field of an Integral Domain , pp. 241-254.
ABSTRACT. The notions of cancellation and invertiblity for ideals are generalized to submodules of the quotient field of an integral domain.

M. Crampin , Department of Applied Mathematics, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK (M.Crampin@open.ac.uk).
The second variation formula in Lagrange and Finsler geometry , pp. 255-275.
ABSTRACT. A relatively straightforward derivation is given of the second variation formula for an arbitrary problem in the calculus of variations, which leads to a covariant form of the formula. The relation between this formulation of the Lagrangian second variation formula, and the corresponding formula in Finsler geometry, is investigated. In particular, it is shown that it is not necessary to invoke a connection in order to derive the second variation formula in Finsler geometry; and that if one does use a connection, each of the four standard Finslerian connections produces the same result.

Bell, Steven R. Department of Mathematics, Purdue University, West Lafayette, IN 47906-1395 (bell@math.purdue.edu).
A Riemann surface attached to domains in the Plane and Complexity in Potential Theory , pp. 277-297.
ABSTRACT. We prove that if either of the Bergman or Szegö kernel functions associated to a multiply connected domain D in the plane is an algebraic function, then there exists a compact Riemann surface R such that D is a domain in R and such that a long list of classical domain functions associated to D extend to R as single valued meromorphic functions. Because the field of meromorphic functions on a compact Riemann surface is generated by just two functions, it follows that all the classical domain functions associated to D are rational combinations of just two functions of one variable. This result gives rise to some very interesting questions in potential theory and conformal mapping. We discuss how it may yield information about complexity in potential theory in a much more general context.

Kunzi, Hans-Peter A., Department of Mathematics, University of Berne, Sidlerstrasse 5, CH-3012 Berne, Switzerland (kunzi@math-stat.unibe.ch) and Losonczi, Attila, Alfred Renyi Institute of Mathematics, Hungarian Academy of Sciences, Realtanoda u. 13-15, H-1364 Budapest, Hungary (losonczi@math-inst.hu).
On Some Cardinal Functions Related to Quasi-uniformities, pp. 299-313.
ABSTRACT. We show that if a topological space possesses two (distinct) compatible quasi-uniformities, then it admits at least exp c nontransitive quasi-uniformities. We also prove that if a quasi-uniform space (X,W) has a subspace A and an entourage V such that either {V(x): x in A} or {V-1(x): x in A} does not have a subcollection of cardinality smaller than k covering A, then there are at least exp(exp k) quasi-uniformities belonging to the quasi-proximity class of W. Finally we show that if the quasi-proximity class P(W) of a quasi-uniformity W contains more than one quasi-uniformity and its coarsest member is transitive, then there are at least exp c transitive quasi-uniformities belonging to the quasi-proximity class P(W). (Here to avoid typesetting problems 2k is written exp k where k is an infinite cardinal.)

Galindo, Jorge, Departamento de Matemáticas, Universitat Jaume I, 12071-Castellón, Spain. (jgalindo@mat.uji.es).
Structure and Analysis on Nuclear Groups, pp. 314-334.
ABSTRACT. Nuclear groups form a class of topological Abelian groups closed under the most common operations which contains LCA groups and additive groups of nuclear locally convex spaces. In this paper we attempt to clarify the structure of these groups by giving a representation theorem. This is used to show that  many properties satisfied by LCA groups and nuclear locally convex spaces are also enjoyed by nuclear groups. Bounded subsets, the existence of interpolation sets, transmission of compactness to the  Bohr topology and Pontryagin duality for nuclear groups are studied.

Elzbieta Pol,Institute of Mathematics, University of Warzaw, Banacha 2 , 02-097 Warzaw, Poland (pol@mimuw.edu.pl)
Two examples of Perfectly Normal Spaces , pp. 335-341.
ABSTRACT. We construct a perfectly normal space X locally homeomorphic to the irrationals such that for some fixed-point free homeomorphism its Cech-Stone extension has a fixed point. This example is related to an example of Good of a normal space of this kind and a recent construction of van Hartskamp and van Mill. We give also an example of a perfectly normal space X with ind(X)=1 no compactification of which has small transfinite dimension - a modification of a normal space with these properties constructed by Charalambous. In both our examples we apply a construction of perfectly normal spaces given by E.Pol and R.Pol in 1979.

John Akeryod and Elias G. Saleeby Department of Mathematics, University of Arkansas, Fayetteville, AR 72701, USA (jakeroyd@comp.uark.edu), (esaleeby@comp.uark.edu).
Sampling and the Closure of the Polynomials in a Weighted Hardy Space , pp. 343-360.
ABSTRACT. In this paper, we use a recently developed collection of measures to obtain sampling results for a class of Banach spaces. Each of these Banach spaces is the closure of the polynomials in a certain weighted Hardy space of the slit-disk. We then give a representation theorem for the Hilbert space case that involves the classical Paley-Wiener space Ephi2.

E. Guentner, Department of Mathematical Sciences, IUPUI, 402 N. Blackford St., Indianapolis, IN 46202-3216 (eguentner@math.iupui.edu) .
Wick Quantization and Asymptotic Morphisms, pp. 361-375.
ABSTRACT. The E-theory of A. Connes and N. Higson provides a new realization of K-homology based on the notion of asymptotic morphisms. In this note we begin to develop the idea that through this realization K-homology becomes a receptacle for topological invariants of quantization schemes. We study the example of the Wick quantization of the complex plane. We show that it determines an element of the E-theory group of the complex plane and by constructing an explicit homotopy we show that this element is equal to the one determined by the Cauchy-Riemann operator.

V. Karunakaran and N.V. Kalpakam Mathematics, Madurai Kamaraj University, Madurai - 625 021, India.
Boehmians Representing Measures, pp. 377-386.
ABSTRACT. If X=[fn÷n] is a locally integrable Boehmian it is proved that whenever the sequence fn is bounded in L1- norm around a neighbourhood of origin then X is represented locally by a measure. This result generalizes a well known theorem due to Piotr Mikusinski and Mourad Tighiourat. We also deduce that if fn's are bounded in the Lp- norm in a neighbourhood of the origin then also the Boehmian is represented by a measure. For p>1 we show that the measure is absolutely continuous with respect to the Lebesgue measure. An example, to show that if fn's are not bounded in any Lp for p>1 but bounded in L1 hen the corresponding measure can be singular with respect to the Lebesgue measure, is also given.

Hyeong-Ohk Bae, Department of Mathematics, Hannam University 133 Ojeong-dong, Daeduk-gu 306-791 Taejon, Republic of Korea ( hobae@math.hannam.ac.kr) and Hi Jun Choe Department of Mathematics Korea Advanced Institute of Science and Technology (KAIST) Gusong-dong 373-1, Yousong-gu 305-701 Taejon, Republic of Korea (ch@math.kaist.ac.kr).
Existence of Weak Solutions to a Class of Non-Newtonian Flows, pp. 387-408 .
ABSTRACT. We show that there exist weak solutions to a class of non-Newtonian flows for the periodic domain. Galerkin approximation, an W1,r+2 compactness theorem, and Korn type inequalities are main ingredients for the proof of the existence of weak solutions. Moreover, we estimate the Hausdorff dimension of the set of singular times for the weak solutions.

George O. Golightly , Rt. 5, Box 276, Jacksonville, TX 75766
Laurent Series From Entire Functions , pp. 409-416 .
ABSTRACT. Certain series stemming from entire functions are investigated.