Electronic Edition Vol. 27, No. 4, 2001

Editors: H. Amann (Zürich), G. Auchmuty (Houston), D. Bao (Houston), H. Brezis (Paris), 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), V. I. Paulsen (Houston), G. Pisier (College Station and Paris), S. W. Semmes (Rice)
Managing Editor: K. Kaiser (Houston)

Houston Journal of Mathematics


Bill Yankosky, North Carolina Wesleyan College, Rocky Mount NC 27804 (byankosky@ncwc.edu).
On Nilpotent Extensions of Lie algebras, pp. 719-724.
ABSTRACT. In the extension of groups 0→L→C→M→0, one finds that C need not be nilpotent. J. Schafer (1995) has shown that the nilpotency of C depends on M, L and a homomorphism from L into Out(M). In this paper we prove the group theoretical results of Schafer also hold for Lie algebras. We also show that two extensions C and C' can have different nilpotency classes. The paper concludes with examples illustrating these concepts.

Ayman Badawi, Dept. of Mathematics, Birzeit University, Box 14, Birzeit, West Bank, Palestine, via Israel (abring@birzeit.edu).
On phi-Chained Rings and phi-Pseudo-Valuation Ring, pp. 725-736.
ABSTRACT. Let R be a commutative ring with 1 such that Nil(R) is a divided prime ideal of R. Then R is called a phi-chained ring if for every nonnilpotent x,y in R either x divides y or y divides x. Also, R is called a phi-pseudo-valuation ring if for every nonnilpotent x,y in R either x divided y or y divides xm for each nonunit m in R. We show that a quasilocal ring R with maximal ideal M containing a nonzerodivisor of R is a phi-pseudo-valuation ring iff M:M is a phi-chained ring. We show that a phi-pseudo-valuation ring is a pullback of a phi-chained ring. Also, we show that for each positive integer n there is a phi-chained ring of Krull dimension n that is not a chained ring.

Coy L. May and Jay Zimmerman, Department of Mathematics, Towson University, Baltimore, Maryland 2125 (cmay@towson.edu, jzimmerman@towson.edu).
The Group of Symmetric Euler Characteristic -3, pp. 737-752.
ABSTRACT. Let G be a finite group.  The symmetric Euler characteristic χ(G) is the maximal Euler characteristic of any surface X (orientable or non-orientable) on which G acts. The groups of symmetric Euler characteristic χ ≥ -2 have been classified.  We show that S5 is the unique group with Euler characteristic -3. Two related parameters are the symmetric genus σ and the symmetric crosscap number τ. We consider some basic relations among the three parameters and also determine the symmetric crosscap number τ (Zn x Zn), when n is even. The quantity τ -2= σ is a measure of the difference between the orientable genus action and the non-orientable genus action.  We present a series of examples that show this quantity can be either positive or negative and have arbitrarily large magnitude in either case.

Mehri Akhavan-Malayeri, Department of Mathematical Sciences Az-Zahra University Vanak, Tehran 19834 Iran, (makhavan@karun.ipm.ac.ir).
Commutator length and square length of the wreath, pp. 753-756.
ABSTRACT. The object of this note is to show that if G is any group and W is the wreath product of G by the infinite cyclic group then every element of W' is a product of at most three commutators, and every element of W2 is a product of at most seven squares in W.

Costantino Delizia and Chiara Nicotera, Universita degli Studi di Salerno, Dipartimento di Matematicae Informatica, Via S. Allende, I-84081 Baronissi (SA), ITALY (deliziac@matna2.dma.unina.it, nicotera@matna2.dma.unina.it).
On residually finite groups with an Engel condition on infinite subsets, pp. 757-761.
ABSTRACT. In this paper we prove that if G is a finitely generated residually finite group and every infinite subset of G contains different elements x,y such that [x,y,y]=1, then G/Z2(G) is finite.

Mestdag, Tom and Sarlet, Willy, Department of Mathematical Physics and Astronomy, Ghent University, B-9000 Ghent, Belgium, (tom.mestdag@rug.ac.be), (willy.sarlet@rug.ac.be).
The Berwald-type connection associated to time-dependent second-order differential equations, pp. 763-797.
ABSTRACT. We investigate the notions of a connection of Finsler type and of Berwald type on the first jet bundle J1E of a manifold E which is fibred over the real numbers. Such connections are associated to a given horizontal distribution on the bundle p: J1EE, which in particular may come from a time-dependent system of second-order ordinary differential equations. In order to accomodate three existing constructions of a Berwald-type connection for a second-order system, we first introduce equivalence classes of connections of Finsler and Berwald type. By exploring the differences between the existing models in more depth, we come to a new construction which in many respects can be regarded as giving an optimal representative of the class of Berwald-type connections. We briefly enter into two related matters: one is the definition of connections of the type of Cartan, Chern-Rund and Hashiguchi when a metric tensor field is given; the other one is the potential effect of the newly acquired insights on the theory of derivations on forms along the projection p.

Lopez, Rafael, Departamento de Geometria y Topologia Universidad de Granada 18071 Granada, Spain.
Cyclic surfaces of constant Gauss curvature, pp. 799-805.
ABSTRACT. A cyclic surface in Euclidean three-space is a surface foliated by pieces of circles, that is, it is generated by a smooth uniparametric family of pieces of circles. Surfaces of revolution are the best known examples of cyclic surfaces. In the eighteenth century, Euler proved that the catenoid is the only minimal surface of revolution. In 1860s Riemann found a family of embedded minimal surfaces foliated by circles in parallel planes. At the same time, Enneper proved that in a minimal cyclic surface, the foliating planes must be parallel. As a consequence of Euler, Riemann and Enneper's works, we have that the catenoid and Riemann minimal examples are the only minimal cyclic surfaces in Euclidean space. A century later, Nitsche studied in 1989 cyclic surfaces with nonzero constant mean curvature and he proved that the only such surfaces are the surfaces of revolution discovered by Delaunay in 1841. In this paper we prove that a cyclic surface in Euclidean three-space and with nonzero constant Gauss curvature is a surface of revolution. In the case that the Gauss curvature vanishes on the surface, then the foliating planes must be parallel and we obtain explicit parametrisations of all such surfaces.

M. Crampin, Department of Applied Mathematics, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK (crampin@btinternet.com).
The Morse index theorem for general end conditions , pp. 807--821.
ABSTRACT. This paper is devoted to proving a version of the Morse index theorem which applies to an extremal of a first-order Lagrangian whose Hessian with respect to the velocity variables is positive definite, when the end conditions are general; that is to say, the endpoints of the extremal are just required to lie in submanifolds. It is shown that the Morse index in this situation is equal to the sum of the number of focal points of one end submanifold, and the index of a certain symmetric bilinear form. The corresponding result in the Riemannian case, which is generalized here, is due to Kalish (Trans. Amer. Math. Soc. 308 (1988) 341-348).

Florin P. Boca, School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF2 4YH, UK (BocaFP@cardiff.ac.uk) and Alexandru Zaharescu, Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, Bucharest 70700, Romania.
On a class of subgroups of R associated with subsets of prime numbers, pp. 823-844.
ABSTRACT. In this paper we define and study a natural topology on the space of representable subgroups. The structure of the homeomorphisms of this space and the cohomology of a certain natural sheaf are being investigated. A stronger version of the separation property is derived as a corollary of the vanishing of the first cohomology group on certain open sets.

Edwin Duda , University of Miami, PO Box 249085, Coral Gables, FL 33124-4250.
Confluent Maps and Spans, pp. 845-849.
ABSTRACT. Let f be a mapping of X to Y such that f(X)=Y, where X and Y are metric continua. If the product map of X × X onto Y ×Y is confluent and the span of X is zero then the span of Y is zero.

Vladimir V. Tkachuk, Universidad Autónoma Metropolitana, 09340 México D.F., México (vova@xanum.uam.mx) and Wilson, Richard G., Universidad Autónoma Metropolitana, 09340 México D.F., México (rgw@xanum.uam.mx).
Weaker connected Hausdorff topologies on spaces with countable network., pp. 851-860.
ABSTRACT. We prove that a disconnected Hausdorff space with a countable network has a weaker connected Hausdorff topology if and only if it is not H-closed. This solves in a strong form problems 3.2 and 3.3 of a paper by Tkachenko, Tkachuk, Uspensky and Wilson (Commentationes Mathematicae Universitatis Carolinae 37:4, 1996, 825-841). As a corollary we show that a non-compact metric space of weight not exceeding the cardinality of continuum, has a weaker second countable connected Hausdorff topology.

Kaori, Yamazaki, , Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan, (kaori@math.tsukuba.ac.jp).
C*-embeddings and P- or M-embeddings on product spaces , pp. 861-872.
ABSTRACT. It is established that for a P-embedded closed subset A of a normal P-space X and a paracompact Sigma-space Y, AxY is C*-embedded in XxY iff it is P-embedded in XxY iff it is M-embedded in XxY. The former equivalence affirmatively answers a problem in an earlier paper. A theorem that for an M-embedded subset A of a space X and a sigma-locally compact paracompact space Y AxY is C*-embedded in XxY iff it is M-embedded in XxY is also proved. This develops the corresponding results for C- or P-embeddings in the earier paper.

Keiji Izuchi , Department of Mathematics, Niigata University, Niigata 950-2181, Japan (izuchi@scux.sc.niigata-u.ac.jp).
Trivial Points in the Maximal Ideal Space of H III , pp. 873-881.
ABSTRACT. It is proved that there exists a trivial point in the maximal ideal space of H, excluding those in the Shilov boundary, which is not contained in the closure of any union sets of countable non-trivial Gleason parts except the open unit disk.

Leon Brown, Wayne State University, Detroit, MI, 4820 (lbrown@math.wayne.edu) and Jawad Sadek Northwest Missouri State University, Maryville, MO, 64468 (jawads@mail.nwmissouri.edu).
Invariant Subspaces in the Space of Analytic Functions of Bounded Mean Oscillation, pp. 883-886.
ABSTRACT. In this note we present a new proof that the proper invariant subspaces in BMOA for the shift operator are of the form I·BMOA ∩ BMOA where I is an inner function.

Subiman Kundu, Department of Mathematics, Indian Institute of Technology, New Delhi-110016, INDIA(skundu@maths.iitd.ernet.in) and A. B. Raha, Stat-Math Unit, Indian Statistical Institute, 203 Barrackpore Trunk Road, Calcutta-700035, INDIA (abraha@isical.ac.in).
The Stone-Weierstrass Theorem and Dini's Theorem : An Insight pp. 887-895.
ABSTRACT. The classical Stone-Weierstrass theorem and the Dini's theorem have motivated the study of topological spaces for which the contentions of these theorems are true. Stone-Weierstrass property and Dini properties are thus introduced. Characterizations and inter-relationship of these properties have been obtained. The study reveals a new characterization of the Stone-Weierstrass property.

G. G. Sirotkin, Department of Mathematics, Indiana University - Purdue University at Indianapolis, Indianapolis, IN 46202 (syrotkin@math.iupui.edu).
New properties of Lebesgue-Bochner Lp(Ω, Σ, μ; X) spaces pp. 897-906.
ABSTRACT. In this paper we investigate the relation between the geometrical structures of the unit ball of a Banach space X and of the Lebesgue-Bochner function space Lp(μ, X), p in (1,∞). Also, for a new class of Banach spaces, properly containing uniformly convex and uniformly smooth spaces, we show that the spaces in this class possess the normal structure.

Bonilla, A., Dept. Análisis Matemático, Universidad de La Laguna, 38271 La Laguna (Tenerife), Spain (abonilla@ull.es)and Calderón-Moreno, M.C., Dept. Análisis Matemático, Universidad de Sevilla, 41080 Sevilla, Spain (mccm@cica.es).
On universality of composition operators in several variables pp. 907-918.
ABSTRACT. In this paper we characterize the universality of a sequence of composition operators generated by automorphisms of the N-dimensional unit polydisc DN, on Hardy spaces of DN. In addition, we provide suitable conditions for the universality of partial derivative-composition operators in certain spaces X of holomorphic functions in DN and the N-dimensional unit ball. Our theorems improve or extend earlier ones due to Bourdon and Shapiro, Herzog, León, Bernal and the authors, among others.