Let G = 10, 112. Design and implement an algorithm for the visualization of the harmonic...

 

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Let G = 10, 112. Design and implement an algorithm for the visualization of the harmonic measure on JG for any initial value x € G and to solve the corresponding Dirichlet problem. a) Consider a d-dimensional random vector X whose distribution is symmetric with respect to μ€ Rd, and a function f: Rd → R such that f(X) is square-integrable. Show that 0² (Y)</0² (Y) Cov(f(X), f(2μ-X)) <0 for Y = f(x) and Ỹ = fg (X). b) Suppose that d = 1. Show that o² (Ỹ) < 0² (Y), if f is strictly monotone. Let G = 10, 112. Design and implement an algorithm for the visualization of the harmonic measure on JG for any initial value x € G and to solve the corresponding Dirichlet problem. a) Consider a d-dimensional random vector X whose distribution is symmetric with respect to μ€ Rd, and a function f: Rd → R such that f(X) is square-integrable. Show that 0² (Y)</0² (Y) Cov(f(X), f(2μ-X)) <0 for Y = f(x) and Ỹ = fg (X). b) Suppose that d = 1. Show that o² (Ỹ) < 0² (Y), if f is strictly monotone.

Answer rating: 100% (QA)

a We can prove that oY 0Y by using the definition of cova... View the full answer