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9062 results in Discrete Mathematics, Information Theory and Coding

Page 382 of 454

  • Additive Diophantine inequalities with mixed powers I

  • Jörg Brüdern
  • Journal:
    Mathematika / Volume 34 / Issue 2 / December 1987
    Published online by Cambridge University Press:
    26 February 2010, pp. 124-130
    Print publication:
    December 1987

  • In this paper we shall be concerned with the following problem. Let k1 ≤ k2 ≤…≤ ks be natural numbers, λ1,…, λs be nonzero real numbers, not all of the same sign. Is it then true that the values taken by

    at integer points (x1,…, xs) ∈ ℤk are dense on the real line, provided at least one of the ratios λi/λj, is irrational? We shall refer to this, for brevity, as the inequality problem for k1,…, ks. Optimistically one may conjecture that the inequality problem is true whenever

  • Radon-Nikodým compact spaces and fragmentability

  • Part of
  • I. Namioka
  • Journal:
    Mathematika / Volume 34 / Issue 2 / December 1987
    Published online by Cambridge University Press:
    26 February 2010, pp. 258-281
    Print publication:
    December 1987

  • While investigating Asplund spaces in [15], R. R. Phelps and the author noticed that weak* compact subsets of the duals of Asplund spaces (or equivalently, as it turned out, weak* compact subsets of dual Banach spaces with the Radon-Nikodým property) possessed many properties in common with weakly compact subsets of Banach spaces. The topological study of the spaces homeomorphic to the latter, the so-called Eberlein compact spaces, or EC spaces for short, had flourished and had already yielded a rich collection of results. Therefore it was natural to hope that a similar study of the former might also lead to interesting discoveries. In a series of letters with S. Fitzpatrick exchanged during the summer and the fall of 1981, we started to collect properties of compact spaces that are homeomorphic to weak* compact subsets of the duals of Asplund spaces, which we tentatively called “Asplund compact spaces“. However, as far as we are aware, Reynov's paper [16] is the first study in print of the topological properties of “Asplund compact spaces” or “compacta of RN type” as Reynov termed them.

  • ABOUT THIS BOOK

  • Alexander S. Kechris, Alain Louveau
  • Book:
    Descriptive Set Theory and the Structure of Sets of Uniqueness
    Published online:
    07 September 2010
    Print publication:
    27 November 1987, pp 17-20

  • Summary

    Since we view this book as addressed to both logicians or set theorists and analysts, we tried to keep the prerequisites from each field to a minimum, so we believe it is essentially self-contained modulo some basic material. As far as the analysis is concerned we assume only a familiarity with the fundamental results of real and complex analysis (complex methods are actually used rarely), and the elements of functional analysis, i.e. with topics covered in the first or second year of graduate studies. Typical references for them are Stromberg [1], Hewitt and Stromberg [1], Rudin [1J and Rudin [2, Chapters 1–4]. Also for becoming a little more familiar with the basics of harmonic analysis the reader can consult the first chapter of Katznelson [1J. From set theory the reader should be acquainted with its elements, including transfinite induction and ordinals. Some references here are Halmos [1], Enderton [1] or the first SO pages or so of Jech [11. Beyond that one needs some classical results in descriptive set theory concerning basic properties of Borel, analytic and coanalytic sets. These are all summarized in Chapter I V.I and can be found with detailed proofs in any standard text, like Kuratowski [1, Vol. I, §33–39] or Moschovakis [1, Ch. 1, 2].

    Except for this background, as a general rule we develop in detail all the material from analysis or descriptive set theory that we need. This necessitates the inclusion here of standard results from books or other references for which we claim no originality in either the context or the presentation, in particular there is no implication that results for which no specific credit or reference is given, usually because of lack of relevant information, are due to the authors.

  • Preface

  • Alexander S. Kechris, Alain Louveau
  • Book:
    Descriptive Set Theory and the Structure of Sets of Uniqueness
    Published online:
    07 September 2010
    Print publication:
    27 November 1987, pp vii-vii

  • Summary

    This book grew out of a set of notes prepared during the course of a joint Caltech-UCLA Seminar in Descriptive Set Theory and Harmonic Analysis, organized by the authors during the academic year 1985–86. We appreciate very much the help as well as the patience of the participants in this seminar.

    We are grateful to G. Debs, R. Dougherty, S. Jackson, R. Kaufman, R. Lyons, and J. Saint Raymond for many valuable comments and suggestions. The first author is indebted to S. Pichorides for introducing him to the subject of uniqueness for trigonometric series. We would like also to thank N. O'Connor for her efficiency, care and patience in typing the manuscript.

    The work of A. S. Kechris has been partially supported by NSF Gratnt DMS84-16349. A. Louveau has been supported by CNRS, France and by UCLA during his visit in the academic year 1985–86. He takes this opportunity to thank the Mathematics Department for its hospitality.

Page 382 of 454