A Borsuk-Ulam Theorem for compact Lie group actions by Biasi C., de Mattos D.

By Biasi C., de Mattos D.

Show description

Read Online or Download A Borsuk-Ulam Theorem for compact Lie group actions PDF

Similar symmetry and group books

Abstract Harmonic Analysis: Structure and Analysis, Vol.2

This publication is a continuation of vol. I (Grundlehren vol. one hundred fifteen, additionally to be had in softcover), and includes a designated remedy of a few very important components of harmonic research on compact and in the neighborhood compact abelian teams. From the reports: "This paintings goals at giving a monographic presentation of summary harmonic research, way more entire and finished than any publication already present at the topic.

Travail, affection et pouvoir dans les groupes restreints (French Edition)

Les groupes restreints ont été et seront au coeur de nos vies pendant toute notre lifestyles et, pourtant, c'est depuis moins d'un siècle que chercheurs et théoriciens se penchent sur les rouages complexes de leur développement. Cet ouvrage suggest un modèle qui feel l. a. présence, dans tout groupe restreint, de trois zones dynamiques, les zones du travail, de l'affection et du pouvoir.

Extra resources for A Borsuk-Ulam Theorem for compact Lie group actions

Example text

Moreover, |˜ ρ(k)| ≤ 1 . In general, in the interval −π ≤ k ≤ π, the maximum ρ˜(k) = 1 is reached only for k = 0 except if ρ(q) is non-vanishing only on a subset of the form q = a + mb m ∈ Z. with a, b fixed and b > 1 , A simple example of such a situation, with a = 1, b = 2, is ρ(q) = 1 2 for q = ±1 . In a first analysis, we exclude this situation and comment on it later. Then, the Fourier series has the properties required to prove the central limit theorem. 7): ρ(Q )Pn (Q − Q ).

The benefit of considering this particular function, rather than the Fourier transform, is that the integrand is still a positive measure. The function Z(b) then is a generating function of the moments of the distribution, that is, of expectation values of monomials. Indeed, one recognizes, expanding the integrand in powers of the variables bk , the series ∞ Z(b) = =0 1 ! n bk1 bk2 . . bk xk1 xk2 . . xk . ,k =1 Expectation values can thus be obtained by differentiating the function Z(b) with respect to its arguments.

Let us point out that the proof of this convergence requires only weaker assumptions. Examples and counter-examples. (i) The distribution uniform on the segment [−1, +1] and vanishing outside, ρ(q) = 1 2 sgn(q + 1) − sgn(q − 1) , ⇒ q = 0 , q2 = 1 3 , (sgn is the sign function) is centred around zero and satisfies the hypotheses of the central limit theorem. Its Fourier transform is sin k 1 1 . dq eiqk = ρ˜(k) = 2 −1 k The generating function of cumulants has the expansion w(k) = ln(sin k/k) = − 61 k 2 − 4 1 180 k + O(k 6 ).

Download PDF sample

Rated 4.92 of 5 – based on 23 votes