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26077 results in Abstract analysis

Page 1063 of 1304

  • On the frequency distribution of partial quotients of U-numbers

  • Part of
  • Edward B. Burger, Thomas Struppeck
  • Journal:
    Mathematika / Volume 40 / Issue 2 / December 1993
    Published online by Cambridge University Press:
    26 February 2010, pp. 215-225
    Print publication:
    December 1993

  • In Mahler's classification of complex numbers [10] (see [4]), a transcendental number ξ is called a U-number if there exists a fixed integer N ≥ 1 so that for all ω > 0, there exists a polynomial so that

    where the height h(f) = max {|α0|, |α1|, …, |αN|}. The number ξ is called a Um-number if the above holds for N = m but for no smaller value of N (examples and further details may be found in [9,1 and 2]). Thus the set of U1-numbers is precisely the set of Liouville numbers. In this paper we investigate the statistical behavior of the partial quotients of real U-numbers, in particular, U2-numbers. In addition, we demonstrate the existence of a U2-number with the property that if it is translated by any nonnegative integer and then squared, the result is a Liouville number. Related results involving badly approximate U2-numbers are also discussed.

  • Sums and differences of quartic norms

  • Part of
  • Henryk Iwaniec, Jacek Pomykala
  • Journal:
    Mathematika / Volume 40 / Issue 2 / December 1993
    Published online by Cambridge University Press:
    26 February 2010, pp. 233-245
    Print publication:
    December 1993

  • Let K be a number field of degree k > 1. We would like to know if a positive integer N can be represented as the sum, or the difference, of two norms of integral ideals of K. Suppose K/ℚ is abelian of conductor Δ. Then from the class field theory (Artin's reciprocity law) the norms are fully characterized by the residue classes modulo Δ. Precisely, a prime number p ∤ Δ (unramified in K) is a norm (splits completely in K), if, and only if,

    where k is a subgroup of (ℤ/Δℤ)* of index k. Accordingly we may ask N to be represented as the sum

    or the difference

    of positive integers a, b each of which splits completely in K. For N to be represented in these ways the following congruences

    must be solvable in α β є k, respectively. Moreover the condition

    must hold. Presumably the above local conditions are sufficient for (−) to have infinitely many solutions and for (+) to have arbitrarily many solutions, provided N is sufficiently large in the latter case.

  • A short algebraic proof of a theorem of Warfield

  • A. Laradji
  • Journal:
    Mathematika / Volume 40 / Issue 2 / December 1993
    Published online by Cambridge University Press:
    26 February 2010, pp. 275-277
    Print publication:
    December 1993

  • In this note we give a direct algebraic proof of a theorem of Warfield on algebraically compact modules. It is shorter than the one given by Azumaya in [1], in that it does not use the embedding of a module M into M** (where M* is the character Homz (M, Q/Z)).

  • Preface

  • T. W. Körner, University of Cambridge
  • Book:
    Exercises in Fourier Analysis
    Published online:
    05 August 2012
    Print publication:
    19 August 1993, pp ix-x

  • Summary

    My book Fourier Analysis has no exercises and, in my view, is complete without them. However, exercises are useful both to the teacher of a course and to the student who wishes to learn by doing. This supplement provides such exercises (the exercises are grouped by chapter, although not all chapters have exercises).

    The two remarks that follow are addressed to students using this book by themselves.

    (1) I have tried to produce exercises and not problems. You should find them more in the nature of a hill top walk than a rock climbing expedition. I have marked some of the easier questions with a minus sign to prevent you searching for non-existent subtleties. Very occasionally part of a question is marked by a plus sign to indicate that further reflection may be required.

    (2) Unless you intend to do all the questions, you should browse until you find a question or sequence of questions that interest you. You are more likely to pick up knowledge or technique from an exercise that interests you than from one that does not.

    The references to other books and papers which occur from time to time are intended to encourage further reading, and not as a complete record of any indebtedness to other sources. The Cambridge Tripos examinations of various years have been the largest single source of questions, but experts will recognise the influence of the texts of Helson, Katznelson, Rogosinski, Dym and McKean and many others. Experts will also recall the verses of Kipling.

Page 1063 of 1304