By John R. Pierce

Starting with a glance on the energy and gear of sinusoidal waves, author John R. Pierce explores wave media and modes, part pace and team pace, vector and complicated illustration, strength and momentum, coupled modes and coupling among modes, polarization, diffraction, and radiation. References and an index look on the finish of the book.

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X j , T )dx j . ) (x j ,t)dx j . )) = Rn Ψi (xi , 0)m0 (xi )dxi . ). g(xi , v) . ) (x j , T )dx j . 13) j=i We want to show that a Nash equilibrium exists, made of identical feedbacks for all players. 14) and let vˆN (x, m, q) be the minimizer in the Hamiltonian. Set GN (x, m, q) = g(x, vˆN (x, m, q)). 15) Define finally ˆ hN (x, m) = ˆ ··· h x, 1 N−1 ∑ δxh N − 1 h=1 N−1 ∏ m(x j )dx j . 16) j=1 We consider the pair of functions (if it exists) uN (x,t), mN (x,t) the solution of the system ∂ uN + AuN = HN (x, mN , DuN ) ∂t ∂ mN + A∗ mN + div(GN (x, mN , DuN )mN ) = 0 ∂t − uN (x, T ) = hN (x, m(T )), mN (x, 0) = m0 (x).

18) ¯ r(t) = Y¯ (t) − P(t)X(t). 17) to exist. 12) involve P(t), so the condition will be expressed in terms of P(t) and thus is not easily checkable. 17) involves the data directly. Thus, simpler conditions of a solution can be obtained (see [10]) for a complete discussion. 17) is related to a nonsymmetric Riccati equation. 20) dΣ ¯ + A∗Σ − ΣBR−1B∗ Σ + Q + Q(I ¯ − S) = 0 + Σ(A + A) dt Σ(T ) = QT + Q¯ T (I − ST ). 21) with It follows that ¯ r(t) = (Σ(t) − P(t))X(t). 22) However Σ(t) − P(t) is not solution of a single equation.

32) We can similarly compute the difference EΦ(Ψ(x(t + ε ),t + ε ; T )) − Φ(EΨ(x(t + ε ),t + ε ; T )). )) − ∑ ai j (x) ij ˆ T t − ∑ ai j (x) ∑ ij kl ∂ 2F ∂ Ψk ∂ Ψl (x,t; s))ds ∂xj ∑ ∂ xk ∂ xl (Ψ(x,t; s)) ∂ xi kl ∂ 2Φ ∂ Ψk ∂ Ψl (Ψ(x,t; T )) (x,t; T )) + O(ε ). ∂ xk ∂ xl ∂ xi ∂ x j 28 4 The Mean Field Type Control Problems So we have the inequality V (x,t) ≤ EV (x(t + ε ),t + ε ) + ε f (x, v) + F(x) − ∑ ai j (x) ij ˆ T t − ∑ ai j (x) ∑ ij kl ∂ 2F ∂ Ψk ∂ Ψl (x,t; s)ds ∂xj ∑ ∂ xk ∂ xl (Ψ(x,t; s)) ∂ xi kl ∂ 2Φ ∂ Ψk ∂ Ψl (Ψ(x,t; T )) (x,t; T ) + O(ε ).

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