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E) v(x, t) = (x − t)2 . 2. Let f : R −→ R be any twice-differentiable function. Define u : R × R −→ R by u(x, t) := f (x − t), for all (x, t) ∈ R × R. Does u satisfies the (one-dimensional) Wave Equation ∂t2 u = △u? Justify your answer. 3. Let u(x, t) be as in 1(a) and let v(x, t) be as in 1(e), and suppose w(x, t) = 3u(x, t) − 2v(x, t). Conclude that w also satisfies the Wave Equation, without explicitly computing any derivatives of w. 4. Suppose u(x, t) and v(x, t) are both solutions to the Wave equation, and w(x, t) = 5u(x, t) + 2v(x, t).

10: Let a = 0 and b = 1. For all y ∈ R and x ∈ (0, 1), let fy (x) = F (x) = ∞ fy (x) dy −∞ −∞ Now, let β(y) = ∞ (a) ∞ = ✷ x|y|+1 . Thus, 1 + y4 x|y|+1 dy. 1 + y4 1 + |y| . Then 1 + y4 β(y) dy ∞ = −∞ −∞ 1 + |y| dy 1 + y4 (b) For all y ∈ R and all x ∈ (0, 1), and fn′ (x) = < ∞ (check this). |fy (x)| = 1 1 + |y| x|y|+1 < < = β(y), 1 + y4 1 + y4 1 + y4 (|y| + 1) · x|y| 1 + |y| < = β(y). 9 are satisfied, so we conclude that F ′ (x) = ∞ −∞ fn′ (x) dy = ∞ −∞ (|y| + 1) · x|y| dy. 8. 9 is also true if the functions fy involve more than one variable.

N 36 CHAPTER 2. 10: A conformal map preserves the angle of intersection between two paths. 11: The map f (z) = z 2 conformally identifies the quarter plane and the half-plane. 3 A linear map f : RD −→ RD is called conformal if it preserves the angles between vectors. Thus, for example, rotations, reflections, and dilations are all conformal maps. Let U, V ⊂ RD be open subsets of RD . A differentiable map f : U −→ V is called conformal if its derivative D f (x) is a conformal linear map, for every x ∈ U.

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Aubry-Mather theory

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