HOUSTON JOURNAL OF MATHEMATICS

Electronic Edition Vol. 23, No. 3, 1997

Editors: G. Auchmuty (Houston), H. Brezis (Paris), S.S. Chern (Berkeley), J. Damon (Chapel Hill), L.C. Evans (Berkeley), R.M. Hardt (Rice), J.A. Johnson (Houston), A. Lelek (Houston), J. Nagata (Osaka), B. H. Neumann (Canberra), V. Paulsen (Houston), G. Pisier (College Station and Paris), R. Scott (Houston), S.W. Semmes (Rice)
Managing Editor: K. Kaiser (Houston)

Wang, Huaxiong, University of Haifa, Haifa, 31905, Israel.
On Characters of Semirings, pp. 391-405.
ABSTRACT. A character of a semiring R is a morphism of semirings from R to the boolean semiring B. In this paper, we investigate various properties of characters of semirings. The main result is : If R is a commutative semiring which is not a ring then there exists a morphism of semirings from R to the boolean semiring. Applying this result, we prove that the Hilbert basis theorem fails in the semirings which are not the rings; the Krull dimension of the polynomial semiring R[t] is infinite for any commutative semiring R which is not a ring. Rational series with coefficients in a commutative semiring are also studied, and we prove that if R is a quasi-positive commutative semiring then the support of any rational series with coefficients in R is a rational language.

Mo, Xiaokang, University of Kansas, Lawrence, KS 66045 (mo@math.ukans.edu).
Bianchi Permutability by Moving Frames, pp. 407-420.
ABSTRACT. We present a new approach to the Bianchi permutability of surface transformations by moving frames. We give a simple proof to the classical permutability theorem of Bianchi for W-congruence. We then prove a new permutability theorem for Chern-Terng's affine Backlund transformation. Our method is based on Frobenius theorem and some algebraic symmetries of Cartan's structure equations.

Paternain, Gabriel P., Centro de Matematica, Facultad de Ciencias, Eduardo Acevedo 1139, Montevideo CP 11200, Uruguay (gabriel@cmat.edu.uy).
Finsler Structures on Surfaces With Negative Euler Characteristic, pp. 421-426.
ABSTRACT. We show that real analytic Landsberg structures and real analytic K-basic structures on closed oriented surfaces of negative Euler characteristic must be Riemannian.

Boeckx, E., Katholieke Universiteit Leuven, Leuven, Belgium (Eric.Boeckx@wis.kuleuven.ac.be), and Vanhecke, L., Katholieke Universiteit Leuven, Leuven, Belgium (Lieven.Vanhecke@wis.kuleuven.ac.be).
Characteristic Reflections on Unit Tangent Sphere Bundles, pp. 427-448.
ABSTRACT. In Sasakian geometry a locally phi-symmetric space is defined by a certain curvature condition, which has several equivalent geometric interpretations. In this paper we extend the notion of a locally phi-symmetric space to the broader class of contact metric manifolds using reflections with respect to characteristic curves. We show that the only phi-symmetric unit tangent sphere bundles are those of spaces of constant curvature. For this class of spaces we study some further geometric properties relating to the Ricci curvature tensor and to Jacobi operators.

Guaschi, John, Laboratoire de Mathematiques Emile Picard, Universite Paul Sabatier, 31062 Toulouse Cedex 4, France, and Llibre, Jaume, Universitat Autonoma de Barcelona, 08193 Bellaterra (Barcelona), Spain.
Orders and Periods of Algebraically-Finite Surface Maps, pp. 449-483.
ABSTRACT. Let M be a compact, connected, orientable surface of genus g without boundary, and let map f from M to M be a continuous map such that all the eigenvalues of the induced action on rational homology are roots of unity. We present an algorithm to compute as a function of g: firstly, all the algebraic orders; secondly, the potential periods; and thirdly, the potential least periods. We apply the algorithm up to genus 5 for degree -1 and up to genus 3 for degree +1. Also, we give results on the algebraic orders and potential periods for arbitrary genus. For instance, if g>2 then the map has a period less than or equal to 2g-2. Moreover, for degree -1, the algebraic orders are always even, and if g>1 is even then the set of algebraic orders for the surfaces of genus g and g+1 are equal. We improve some of these results in the particular case that f is a finite order homeomorphism.

Loveland, L. D., Utah State University, Logan, UT 84322-3900 (ldl@sunfs.math.use.edu), and Loveland, S. M., University of Alaska, Anchorage, AK 99508 (afsml@UAA.Alaska.edu).
Planar Sets That Lines Hit Twice, pp. 485-497.
ABSTRACT. This paper shows the relationship between two different two-point intersection properties of sets in the plane - sets with the double midset property and sets with a generalized Mazurkiewicz property. Many unsolved problems are stated along with proving that path connected sets with the generalized Mazurkiewicz property are simple closed curves. The paper closes with a proof that the simple analogue of a Mazurkiewicz set in three-space does not exist and with results related to such sets in three space.

Cole, Brian J., Brown University, Providence, RI 02912 (Brian_Cole@brown.edu), and Wermer, John, Brown University, Providence, RI 02912 (John_Wermer@brown.edu).
Boundaries of Interpolation Bodies, pp. 499-527.
ABSTRACT. Let (a_j,b_j), j in [1,n], be n points in the open unit bidisk in C². Let D be the interpolation body consisting of all points w = (w₁,..., w_n) in Cⁿ such that there exists a bounded analytic function F on the bidisk with infinity norm of F less than or equal to 1 and F(a_j,b_j) = w_j. We show that there exists a real non-constant polynomial on Cⁿ = R²ⁿ which vanishes on bd(D), and that bd(D) has a dense, relatively open subset which is a real analytic manifold. In the proofs we use a result on contractive representations of the bidisk algebra by operators on n-dimensional Hilbert space as well as the Tarski-Seidenberg theorem on semi-algebraic sets.

Octavio, Alfredo, IVIC, Caracas, Venezuela (aoctavio@ivic.ivic.ve), Kosiek, Marek, Uniwersytet Jagiellonski, Krakow, Poland (mko@@im.uj.edu.pl).
Representations of H^infty(D^N) and Absolute Continuity for N-tuples of Contractions, pp. 529-537.
ABSTRACT. In this paper we study a condition, introduced by Apostol, that implies the existence of a functional calculus for N-tuples of operators. We show that this condition can be written as the sequence of powers of each member of the N-tuple converging to 0 in a certain topology. We prove that if the N-tuple satisfies von Neumann's inequality, then this condition is equivalent to the (joint) absolute continuity of the N-tuple.

Alpay, D., Ben-Gurion University of the Negev, POB 653, 84105 Beer-Sheva, Israel, Bolotnikov, V., and Loubaton, Ph., Unite de Formation E.E.A., Universite de Marne-la-Vallee, 93166 Noisy-le-Grand Cedex, France.
On Interpolation for Hardy Functions in a Certain Class of Domains Under Moment Type Constraints, pp. 539-571.
ABSTRACT. We use reproducing kernel methods to characterize H₂ functions satisfying certain second order conditions. As an application we get a description of all solutions of the left--sided residue interpolation problem under the second order constraints.