## HOUSTON JOURNAL OF MATHEMATICS

### Electronic Edition Vol. 25, No. 4, 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)

Contents

D. D. Anderson, Department of Mathematics, University of Iowa, Iowa City, IA 52242-1419 (dan-anderson@uiowa.edu), D. E. Dobbs, Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300 (dobbs@novell.math.utk.edu) and Bernadette Mullins, Department of Mathematics and Statistics, Youngstown State University, Youngstown, OH 44555-3302 (bmullins@math.ysu.edu).
The Primitive Element Theorem for Commutative Algebras, pp. 603-623.
ABSTRACT. Let R inside T be an extension of commutative rings (with the same 1). We say that the ring extension R inside T has FIP if the set of R-subalgebras of T is finite. If the ring extension R inside T has FIP, then T must be algebraic over R; if, in addition, R is a field, then T is a finite-dimensional R-vector space. If the ring extension R inside T has FIP and T is an integral domain, then either R and T are fields or T is an overring of R. If R is a perfect field, then the main result identifies four exhaustive cases which serve to characterize the condition that the ring extension R inside T has FIP. Considering extensions R inside T having FIP with T the quotient field of R amounts to studying integral domains R with only finitely many overrings. Such integral domains R are characterized as the semi-quasilocal i-domains of finite Krull dimension having only finitely many integral overrings. This property is interpreted further in case R is either integrally closed or a pseudo-valuation domain. Examples are given to illustrate the sharpness of the results.

Paul Centore, 1546 Route 12, GalesFerry, CT 06335, USA (centore@downcity.net).
Volume Forms in Finsler Spaces, pp. 625-640.
ABSTRACT. This paper considers two possible volume forms on a Finsler space and uses them to characterize Riemannian spaces and state a conditionwhich Berwald spaces must satisfy. The first form is Busemann's previously known volume form, and the second is the volume formarising from a Riemannian metric canonically associated to the Finsler metric. The first form always exceeds the second; they agree if and only if the Finsler manifold actually is Riemannian. In a Berwald space, the ``ratio" of the two forms is a constant.

Andrea Spiro, Dipartimento di Matematica, Università di Ancona, 60131 Ancona, Italy (spiro@popcsi.unian.it).
Chern's Orthonormal Frame Bundle of a Finsler Space, pp. 641-659.
ABSTRACT. The definitions of Chern's orthonormal frame bundle O(M,F) for a Finsler space M, with a real strongly convex Finsler metric F, and of non-linear connections on O(M,F), are given. It is also proved that O(M,F) admits a unique torsion-free non-linear connection and that this connection coincides with the non-linear Finsler connection introduced by S. S. Chern. This fact brings to a new interpretation of Chern's connection and to a simplified proof of the following theorem by Chern: the group of isometries of a Finsler space is a Lie group of dimension less or equal to n + n(n-1)/2, where n is the dimension of M.

R.Cowen and P. H.Fisher, Department of Mathematics, University of Botswana, Private Bag 0022, Gaborone, Botswana (cowenr@noka.ub.bw, fisherph@noka.ub.bw).
On Expanding Endomorphisms of the Circle II, pp. 661-666.
ABSTRACT. In this paper we give sufficient conditions for weak isomorphism of Lebesgue measure-preserving expanding endomorphisms of S1.

A Strong Completeness Condition in Uniform Spaces with Well Ordered Bases, pp. 667-678.
ABSTRACT. This papers deals with uniform spaces which admit a base linearly ordered by a regular uncountable cardinal k (i.e., k-metric spaces). In k-metric spaces, the completeness condition is not sufficient to ensure the existence of continuous selections for l.s.c. multivalued functions. Moreover, in contrast with the metric case, the hyperspace of a complete k-metric space is not necessarily complete.
A strengthening of the completeness condition can be obtained by requiring that the intersection of every chain of balls is non-empty. A subspace which satisfies this condition is said to be B-complete. We prove that the hyperspace of B-complete subsets of a complete k-metric space is complete in the Hausdorff uniformity. Furthermore, every l.s.c. function with B-complete values has a continuous selection. We also show that quite important classes of k-metric spaces are B-complete.

Anthony W. Hager, Mathematics Department, Wesleyan University, Middletown, CT 06457 (ahager@wesleyan.edu).
A note on alpha-Cozero-Complemented Spaces and alpha-Borel Sets, pp. 679-685.
ABSTRACT. Alpha is a regular cardinal,or infinity,and X is a compact Hausdorff space. It is shown that X is alpha-cozero-complemented iff each alpha-Borel set differs from an alpha-cozero set by a meager set. This implies that X is alpha-disconnected iff each alpha-Borel set differs from a clopen set by a meager set. The Boolean algebra of alpha-Borel sets modulo meager sets is considered as an extension of the clopen algebra (for Boolean X), and compared with other completions.

Yasunao Hattori, Department of Mathematics, Shimane University, Matsue, Shimane, 690-8504 Japan (hattori@math.shimane-u.ac.jp).
Finitistic Spaces and Dimension, pp. 687-696.
ABSTRACT. We shall consider two dimension-like properties on finitistic spaces. We shall prove that there is a universal space for the class of metrizable finitistic spaces of given weight. We shall also prove that a Pasynkov's type of factorization theorem for finitistic spaces.

Thelma West, Dept. of Mathematics, University of Southwestern Louisiana, Lafayette, LA 70504-1010.
Concerning the Spans of Certain Plane Separating Continua, pp. 697-708.
ABSTRACT. Let X be a plane separating continuum. Suppose C is a convex space contained in a bounded component of R2-X . It is shown that the span of the boundary of C is a lower bound for both the span and semispan of X . It is also shown that if a span of X is equal to the breadth of X and Y satisfies certain conditions relative to X then thatspan of X is an upper bound for the corresponding span of Y .

Grahame Bennett, Department of Mathematics, Indiana University, Bloomington, IN 47405-7106 (bennettg@indiana.edu).
An Inequality for Hausdorff Means, pp. 709-744.
ABSTRACT. We show that skyscrapers are possible even in cities like Meanie-apolis. This lofty assertion leads to a new class of elementary inequalities.

Narcisse Randrianantoanina, Department of Mathematics and Statistics, Miami University, Oxford, OH 45056 (randrin@muohio.edu).
Absolute Summing Operators on Non Commutative C*-algebras and Applications, pp. 745-756.
ABSTRACT. Let E be a Banach space that does not contain any copy of l1 and A be a non commutative C*-algebra. We prove that every absolutely summing operator from A into E* is compact, thus answering a question of Pel czynski. As application, we show that if G is a compact metrizable abelian group and Lambda is a Riesz subset of its dual then every countably additive A*-valued measure with bounded variation and whose Fourier transform is supported by Lambda has relatively compact range. Extensions of the same result to symmetric spaces of measurable operators are also presented.

Xiyu Liu and Baoqiang Yan, Shandong Normal University, Jinan, Shandong 250014, People's Republic of China, (Yliu@jn-public.sd.cninfo.net, yanbq@sdnu.edu.cn)
On the Structure of Solutions of a Class of Boundary Value Problems , pp. 757-768.
ABSTRACT. In a recent paper, "Combined effects of concave and convex nonlinearities in some problems, J. Functional Analysis, 122, No.4, (1994), 519--43", A. Ambrosetti, H. Brezis and C. Cerami studied the combined effects of concave and convex nonlinearities to a class of parameterized elliptic boundary value problems with nonlinear term as the sum of concave and convexpolynomials. They proved the existence of two positive solutions for small parameter by upper and lower solutions and variational techniques when the nonlinear term is subcritical.In that paper, they also indicated several interesting open problems. One of those is what the structure of the solutions is in the one-dimensional case. The purpose of the present paper is to study this problem. We give a different approachand a general setting of the problem. The main feature is the presence of nonlinearity having a sublinear and superlinear behavior. By applying topological methods on cones we will show the existence of a branch of solutions bifurcating from the origin that touches back. Thus we get the behavior of continua of the solution set. As applications, we discuss in detail a class of boundary value problems of ordinary differential equations. Some further structure results are obtained, and a partial a nswer is given to the question raised in the paper of A. Ambrosetti, H. Brezis and C. Cerami.

Porzio, Maria Michaela, Dipartimento di Matematica "G. Castelnuovo", Universite degli Studi di Roma "La Sapienza", P.le A. Moro, 2 -- 00185 Roma, Italy (porzio@mat.uniroma1.it).
Local regularity results for some parabolic equations, pp. 769-792.
ABSTRACT. In this paper we prove the local Ls regularity (where s depends on the summability of the data) for local "unbounded" weak solutions of a class of nonlinear parabolic equations including the p-Laplacian equation.