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25995 results in Abstract analysis

Page 1129 of 1300

  • 101 - A little group theory

  • T. W. Körner, University of Cambridge
  • Book:
    Fourier Analysis
    Published online:
    05 June 2014
    Print publication:
    28 January 1988, pp 506-508

  • Summary

    In previous chapters we looked at some of the uses of the multiplicative group of the nth roots of unity. Under the name of the cyclic group of order n, it furnishes the simplest example of an Abelian group whilst under the name of the additive group of the integers modulo n, it forms a simple but very useful tool in number theory. In this chapter we introduce another kind of Abelian group which also plays an important role in number theory.

    Let ℤn be the set of integers mod n. (More formally ℤn is the collection of equivalence classes [a] = {b:b ≡ a mod n}.) Recall that b and c are said to be coprirne if they have no common factors. We make an obvious remark.

    Lemma 101.1. (i) if a and n are coprirne so are a + nm and n.

    Proof. If q divides n and a + nm then q divides n and a = a + nm – nm.

    Thus we can define

    G = {[a]∈ℤn:a and n are coprirne}

    without ambiguity.

    In the same vein we have the following results.

    Lemma 101.1. (ii) If [a1] = [a2] and [b1] = [b2] then [a1b1] = [a2b2].

    (iii) If [a], [b]∈G then [ab] ∈ G.

    Proof. (ii) Since a1 ™ a2, b1 ™ b2 ≡ 0 mod n we have

    a1b1 – a2b2 = (a1 – a2)b1 + a2(b1 – b2) ≡ 0 mod n,

    as required.

  • 90 - La Famille Picard va á Monte Carlo

  • T. W. Körner, University of Cambridge
  • Book:
    Fourier Analysis
    Published online:
    05 June 2014
    Print publication:
    28 January 1988, pp 467-470

  • Summary

    In Chapter 88 we saw how the study of analytic maps of Brownian motion lead to the ‘little’ theorem of Casorati and Weierstrass.

    Theorem 90.1.If f: ℂ → ℂ is analytic and non-constant then the range off is dense in ℂ.

    In this chapter we show how the additional ideas of Chapter 89 lead to th e deeper ‘little’ theorem of Picard.

    Theore m 90.2.If f: ℂ → ℂ is analytic and non-constant then the range off omits at most one point of ℂ.

    In other words we can choose w0∊ℂ such that for each w ≠ w0 the equation f(z) = w has a solution. If f(z) = exp z then we already know that the result is true with w0 = 0. If f is a polynomial then the fact that every non-constant polynomial has a root shows that the range of f is ℂ. But Picard's theorem says that any arbitrarily chosen analytic function such as f(z) = z sin z + exp(z2 + exp z) has range ℂ or ℂf{w0}. This theorem is one of the glories of nineteenth-century mathematics, a jewel ‘five words long that on the stretched forefinger of all time sparkles forever’. And the discovery by Burgess Davis that this result could be proved using Brownian motion was a triumph for modern probability theory.

Page 1129 of 1300