Topology

A mathematical gift, 2, interplay between topology, by Kenji Ueno, Koji Shiga, Shigeyuki Morita, Toshikazu Sunada

By Kenji Ueno, Koji Shiga, Shigeyuki Morita, Toshikazu Sunada

This e-book brings the wonder and enjoyable of arithmetic to the study room. It deals critical arithmetic in a full of life, reader-friendly type. incorporated are routines and plenty of figures illustrating the most techniques.

The first bankruptcy talks concerning the concept of trigonometric and elliptic features. It contains topics akin to strength sequence expansions, addition and multiple-angle formulation, and arithmetic-geometric capacity. the second one bankruptcy discusses numerous elements of the Poncelet Closure Theorem. This dialogue illustrates to the reader the belief of algebraic geometry as a style of learning geometric houses of figures utilizing algebra as a device.

This is the second one of 3 volumes originating from a sequence of lectures given by means of the authors at Kyoto college (Japan). it's appropriate for school room use for prime university arithmetic lecturers and for undergraduate arithmetic classes within the sciences and liberal arts. the 1st quantity is on the market as quantity 19 within the AMS sequence, Mathematical international. a 3rd quantity is approaching.

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Example text

To prove existence, define, for λ ∈ (0, 1], the operator Tλ x := (1 − λ )x + λ (Tx + y). Because λ = 0, it is clear that x is a fixed point of Tλ if and only if x = T x + y. Moreover, if x1 , x2 ∈ H, then Tλ x1 − Tλ x2 = (1 − λ )2 x1 − x2 2 ≤ (1 − λ ) x1 − x2 2 2 = Tλ x1 − Tλ x2 , Tλ x1 − Tλ x2 + λ 2 T x1 − T x2 2 +λ 2 2 T x1 − T x2 + 2λ (1 − λ ) Tx1 − T x2 , x1 − x2 2 ≤ [(1 − λ )2 + λ 2L2 ] x1 − x2 2 , where L is the Lipschitz constant of T . Next, observe that the factor (1 − λ )2 + λ 2 L2 , 1 as a function of λ , achieves its absolute minimum at λˆ = 1+L 2 , with (1 − λˆ )2 + λˆ 2L2 = L2 < 1.

P. 1007/978-1-4614-8893-4 2, © Springer Science+Business Media New York 2014 29 30 2 The Banach Fixed Point Theorem Several abstract tools have been developed to deal with problems of this kind; in this chapter, we begin with one of the most popular fixed point theorems in complete metric spaces—the contraction mapping theorem. 2 A Well-Known Case: Picard Method of Successive Approximations In the first chapter, we made extensive use of the well-known existence and uniqueness theorem for ordinary differential equations.

From our assumptions, the set {t ∈ [0, 1] : x(t) = 0} is finite; moreover, observe that if t0 < t1 are two consecutive zeros of x, then t1 t0 y (t)x(t) − x (t)y(t) y(t) dt = arctan x(t)2 + y(t)2 x(t) t1 = lim arctan t0 t→t1− y(t) y(t) − lim arctan . x(t) t→t0+ x(t) As the sign of x(t) between t0 and t1 is constant, we have three possible cases: y (t)x(t)−x (t)y(t) 1. If sgn(y(t0 )) = sgn(y(t1 )), then tt01 x(t)2 +y(t)2 dt = 0. 2. If, for example, y(t0 ) < 0 < y(t1 ) and x(t) > 0 on (t0 ,t1 ), then t1 t0 y (t)x(t) − x (t)y(t) dt = lim arctan u − lim arctan u = π .

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