By Alex Bellos

The area of maths can look mind-boggling, beside the point and, let's face it, dull. This groundbreaking booklet reclaims maths from the geeks.

Mathematical principles underpin as regards to every thing in our lives: from the incredible geometry of the 50p piece to how chance will help win in any on line casino. looking for strange mathematical phenomena, Alex Bellos travels around the globe and meets the world's quickest psychological calculators in Germany and a startlingly numerate chimpanzee in Japan.

Packed with interesting, eye-opening anecdotes, Alex's Adventures in Numberland is an exciting cocktail of background, reportage and mathematical proofs that would go away you awestruck.

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For any neighborhood O of K ¯p , if ap − a ¯p H 1 (0,π) is small enough then K∞ ⊂ O . e. 18)) can be chosen the same for all ap ∈ N . 3 and its proof, we have K0 = ΠV1 K∞ . 1 for a ¯p , they satisfy property (P) also for ap in a small neighborhood of a ¯p . 1 satisfying q ≤ q1 < . . < qM . 1 By assumption π b(x)v q H = Ω v(t, x)q H(t, x) dt dx = 0 , b(x) 0 ∀b(x) ∈ H 1 (0, π) T and therefore, setting H(t, x) := h(t + x) − h(t − x), q v(t, x)q H(t, x) dt = η(t + x) − η(t − x) T h(t + x) − h(t − x) dt = 0 , ∀x ∈ [0, π] .

21) is verified. 1 Let q > p be an integer. There exist bi (x) ∈ H 1 (0, π), qi ∈ N, qi ≥ q, i = 1, . . 1. 1 below, which is proved in the next subsection. 1 Let q > p. Let v, H ∈ V be analytic and v have minimal period 2π. Then b(x)v q H = 0 , ∀ q ≥ q , q ∈ N, ∀ b(x) ∈ H 1 (0, π) =⇒ H = 0. 1. ∀v1 ∈ K0 there exists a finite set of nonlinearities {bi (x)uqi , i = 1, . . , N } with qi ≥ q > p, qi ∈ N, such that {∇Φi (v1 ) , i = 1, . . , N } span the whole V1 . 1 with v = v1 + v2 (v1 ), H = h1 + ∂v1 v2 [h1 ] = 0.

You, KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Comm. Math. Phys. 211, 2000, no. 2, 497–525. [12] W. Craig, Probl`emes de petits diviseurs dans les ´equations aux d´eriv´ees partielles, Panoramas et Synth`eses, 9, Soci´et´e Math´ematique de France, Paris, 2000. [13] W. Wayne, Newton’s method and periodic solutions of nonlinear wave equation, Comm. Pure and Appl. Math, vol. XLVI, 1409-1498, 1993. [14] W. Wayne, Nonlinear waves and the 1 : 1 : 2 resonance, Singular limits of dispersive waves, 297–313, NATO Adv.

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