Electronic Edition
Vol. 26, No. 1, 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. Bossard, Equipe d'Analyse, Universite Paris VI, Case 186, 4, place Jussieu, 75252 - Paris Cedex 05.
On a Problem of H. P. Rosenthal, pp. 1-15.
ABSTRACT. Let X be a non reflexive separable Banach space. H.P. Rosenthal associates with X an ordinal index defined on the elements of the bidual, and shows that Xcontains no isomorph of c0 iff for any element of the bidual which is not in the space the value of the index is countable. Using tools of analytic set theory and ordinal ranks on Baire class 1 functions, we prove that if X contains no isomorph of c0, then in some cases the ordinal index is uniformly bounded on the elements of the bidual which are not in the space by a countable ordinal. In particular it is true when X contains no isomorph of l1.

Sidney A. Morris, School of Mathematics, University of South Australia, Mawson Lakes, S.A. 5095, Australia, Peter Nickolas, School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia, and Vladimir Pestov, School of Mathematical and Computing Sciences, Victoria University of Wellington, PO Box 600, Wellington, New Zealand
Limit Laws for Wide Varieties of Topological Groups II, pp. 17-27.
ABSTRACT. A class of topological groups is a wide variety if it is closed under the formation of subgroups, products and continuous homomorphic images. Walter Taylor introduced limit laws as analogues for topological groups of algebraic laws for abstract groups, and proved a Birkhoff-style characterisation: a class is a wide variety if and only if it is the class of models for some set of algebraic laws and some perhaps proper class of limit laws. The class of wide varieties T(m) , for infinite cardinals m , has played a central role in the theory to date. A group is in T(m) if and only if each neighbourhood of its identity contains a normal subgroup of index strictly less than m . This paper contributes to our knowledge of the T(m) , and of the relation of other wide varieties to the T(m) . In particular, it is shown that the T(m) are definable by a set (rather than a proper class) of limit laws; indeed, the same is true of any wide subvariety of any T(m) . Further, the class of wide varieties lying in each T(m) is a set. On the other hand, it is also shown that there exists a proper class of wide varieties which do not lie in any T(m) , and constructions are given of certain families of such varieties, each defined by sets of particularly simple limit laws.

Gerhard Gierz, Department of Mathematics University of California at Riverside, Ca 92526 (gierz@math.ucr.edu), and Albert R. Stralka, Department of Mathematics University of California at Riverside, Ca 92526 (stralka@math.ucr.edu).
Quotients of Full Sublattices of Euclidean Space, pp. 29-53.
ABSTRACT. In this note, we show that every full sublattice of Euclidean space is a quotient of a sublattices for which primes and coprimes are closed subsets.

Mollov, Todor Zh., University of Plovdiv, 4000 Plovdiv, Bulgaria (mollov@ulcc.uni-plovdiv.bg) and Nachev, Nako A., University of Plovdiv, 4000 Plovdiv, Bulgaria (nachev@ulcc.uni-plovdiv.bg).
On the semisimple twisted group algebras of primary cyclic groups, pp. 55-66.
ABSTRACT. Let KtG be a twisted group algebra of a finite cyclic p-group G over a field K of characteristic different from p and of the second kind with respect to p. In this paper we have given, up to an isomorphism, the decomposition of KtG into a direct sum of fields, precising their type and multiplicity of appearance and specifying the multiplicative group U(KtG) of KtG. We have found a sufficient and necessary condition for the isomorphism of KtG and KtH as K-algebras, where G and H are finite cyclic p-groups of the same order.

Byung Gyun Kang , Department of Mathematics, POSTECH, Pohang, 790-600, South Korea.
When Are the Prime Ideals of the Localization R[X]T Extended from R , pp. 67-81.
ABSTRACT. Let R be an integrally closed integral domain, { Xi} a set of indeterminates over R, and T a multiplicatively closed subset of R[ { Xi}]. We prove the equivalence of the following statements: (1) Every prime ideal of R[{ Xi} ]T is extended from R. (2) Every ideal of R[{ Xi} ]T is extended from R. (3) Every principal ideal of R[{ Xi} ]T is extended from R. (4) There exists a Prüfer v-multiplication overring A of R such that R[{ Xi} ]T=Av, where Av is the Kronecker function ring of A with respect to the v-operation. The case when R is not integrally closed is also taken care of. Similar statements for rings with zero divisors are considered and their equivalence is established.

O. T. Alas, University of Sao Paulo, Caixa Postal 66281, 05315-970 Sao Paulo, Brazil (Alas@ime.usp.br), W.W. Comfort, Wesleyan University, Middletown CT 06459, (Wcomfort@wesleyan.edu), S. Garcia-Ferreira, Instituto de Matematicas, Ciudad Universitaria (UNAM), 04510 Mexico D.F., Mexico (garcia@servidor.unam.mx), M. Henriksen, Harvey Mudd College, Claremont CA 91711 (Henriksen@hmc.edu), R.G. Wilson, Departamento de Matematicas, Universidad Autonoma Metropolitana, Unidad Iztapalapa, 09340, Mexico D.F., Mexico (rgw@xanum.mx) and R.G. Woods, University of Manitoba, Winnipeg, Man. R3T 2N2 (Rgwoods@cc.umanitoba.ca).
When is |C(Xx Y)| = |C(X)| x |C(Y)|? , pp. 83-115.
ABSTRACT. Sufficient conditions on the Tychonoff spaces X and Y are found that imply that the equation in the title holds. Sufficient conditions on the Tychonoff space X are found that ensure that the equation holds for every Tychonoff space Y. A series of examples (some using rather sophisticated cardinal arithmetic) are given that witness that these results cannot be generalized much.

Jawad Sadek Northwest Missouri State University (jawads@mail.nwmissouri.edu)
Normal Limits and Star-Invariant Subspaces of Bounded Mean Oscillatio in Multiply Connected Domains, pp. 117-129.
ABSTRACT. Let D be a domain in the plain bounded by n+1 analytic Jordan curves. Let H2 be the usual Hardy class of analytic functions in D. Denote by BMOA the space of analytic functions of bounded mean oscillation in D and let K2 be the star-invariant subspace generated by an inner function phi in D. Let K* be the intersection of K2and BMOA. In this paper, we give a necessary and sufficient condition for the normal limit to exist at a point on the boundary of D for a function in K*.

Efton Park,Department of Mathematics Texas Christian University Fort Worth, Texas 76129 (e.park@tcu.edu)
Isometries of Unbounded Fredholm Modules over Manifolds , pp. 131-144 .
ABSTRACT. A self-adjoint first-order elliptic differential operator D acting on sections of a Hermitian vector bundle over a compact Riemannian manifold Mdetermines an unbounded Fredholm module over Mn(C(M)) for each positive integer n. We show that the group of automorphisms of Mn(C(M)) that respect the unbounded Fredholm module is a compact topological group in the topology of pointwise convergence. If D is an operator of Dirac type and we restrict to scalar functions, then this group is also a Lie group.

Joel D. Avrin Department of Mathematics, University of North Carolina at Charlotte, Charlotte, North Carolina 28223.
Flame propagation in Models of Complex Chemistry , pp. 145-163.
ABSTRACT. We consider models of laminar flames with Arrhenius kinetics in long thin tubes. In previous studies of a model of a one-step reaction of the form A--> B, we identified certain conditions imposed on the initial temperature that guarantee a rough sense of flame propagation resulting in complete asymptotic burning of the fuel. We now extend these studies to a model of a two-step reaction of the form A -->B --> C, and to some models of one-step reactions with multiple species. We again identify sufficient conditions on the initial temperature that guarantee a rough sense of flame propagation and complete asymptotic burning.For these results to hold we need to add some restrictions on other parameters for technical reasons, but we show that our results nonetheless apply to a wide range of cases.

Srdjan Petrovic, Department of Mathematics & Statistics, Western Michigan University, Kalamazoo, MI 49008-5152 (petrovic@wmich.edu).
An Extremal Problem in Interpolation Theory, pp. 165-181.
ABSTRACT. If z1,z2,...,zn are complex numbers in the open unit disk D and A1,A2,...,An are N-by-N matrices, let  F denote the family of analytic functions, bounded in  D, such that for each F in F, F(zk)=Ak, k=1,2,...,n. For z in D and F in F, let
 |F(z)|sp be the spectral radius of F(z). We consider the supremum of  |F(z)|sp over all z in D and the infimum of the last quantity when F ranges over all functions in F.  H. Bercovici has raised the question whether this infimum is attained. We will show that the answer is affirmative for N=2 and N=3, and we point out at the obstructions to generalize this result to the case N>3.

Porretta, Alessio, Universita di Roma I, P.le A. Moro 2, 00185 Roma, ITALY (porretta@mat.uniroma1.it).
Some remarks on the regularity of solutions for a class of elliptic equations with measure data, pp. 183-213 .
ABSTRACT. We deal with a class of Dirichlet problems in an open bounded subset of the N-euclidean space for second order nonlinear elliptic operators in divergence form of the type -div(a(x,u,Du)) where a(x,s,p) is a Caratheodory function monotone, coercive and with linear growth with respect to p, while no growth assumption from above is made on a(x,s,p) as a function of s. We consider the equation -div(a(x,u,Du))=f with Dirichlet boundary conditions assuming that f is a bounded Radon measure, proving the existence of a weak solution and some regularity results on the summability of both u and Du in the case that a(x,s,p), as a function of s, grows like s to the power m. In this latter case, we also prove the existence of a solution to the perturbated problem -div(a(x,u,Du)+H(x,u,Du)=f, where H(x,s,p) is a Caratheodory function satisfying a sign condition (H(x,s,p)s>0 except for s=0) and which has quadratic growth with respect to p and, as a function of s, it grows, for s large, at most like s to the power r, with r+1<m. This restriction in the link between m and r is in fact necessary, in order to have solutions for every measure f, as a consequence of some recent results on removable singularities by H. Brezis and L. Nirenberg.