Abstract. In this paper we prove several generalizations of the Banach fixed point theorem for partial metric spaces (in the sense of O'Neill) given in [14], obtaining as a particular case of our results the Banach fixed point theorem of Matthews ([12]), and some well-known classical fixed point theorems when the partial metric is, in fact, a metric.2000 AMS Classification: 54H25, 54E50, 54E99, 68Q55.

CONTENTS 1. Introduction 2. Families of Curves of Genus 1 3. Families of Curves of Genus 2 4. Degenerate Cases 5. Conclusion 6. Some Recent Information Acknowledgements Electronic Availability References

I would like to express my thanks to the Canadian Mathematical Society for inviting me to present this lecture. I would also like to express my appreciation to C.J. Smyth for numerous helpful conversations during his visit this year at the University of British Columbia.This paper follows reasonably closely the outline of the lecture presented in Ottawa. More details are given here though and a number of proofs which would not be otherwise accessible have been added as Appendices. The attentive reader will soon realize the appropriateness of the title.

Abstract. An explicit formula is derived for the logarithmic Mahler measure m(P) of P(x, y) = p(x)y− q(x), where p(x) and q(x) are cyclotomic. This is used to find many examples of such polynomials for which m(P) is rationally related to the Dedekind zeta value ζ F (2) for certain quadratic and quartic fields.

A special case of the theorem of Marcinkiewicz states that if T is a linear operator which satisfies the weak-type conditions (p, p) and (q,q), then T maps Lr continuously into itself for any r with p < r < q. In a recent paper (5), as part of a more general theorem, Calderόn has characterized the spaces X which can replace Lr in the conclusion of this theorem, independent of the operator T. The conditions which X must satisfy are phrased in terms of an operator S(σ) which acts on the rearrangements of the functions in X.One of Calderόn's results implies that if X is a function space in the sense of Luxemburg (9), then X must be a rearrangement-invariant space.

The purpose of this paper is to investigate conditions under which the Hilbert transform defines a bounded linear operator from a given function space into itself. The spaces with which we deal have the property of rearrangement-invariance which is defined in §1. This class of spaces includes the Lebesgue, Orlicz, and Lorentz spaces.

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