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30 Aug 2019: Starting from the Ricci identity. Xa;dc Xa;cd = R abcd Xb ,. show that for a Killing vectorξa;b. ;b Rab ξb = 0. ... JS = 116π. Sξa;b d2 Σab , (). where Σab is the proper area element on S.

30 Aug 2019: Suppose that B is a closed unital subalgebra of A, and that b B. ... Suppose that a, b and b2 a2 are positive. Show that b a is positive.

30 Aug 2019: For the particular choice. ψ(x) = min{b, max{x,b}}, b < :. (i) Find the asymptotic variance V (ψ, Φ), where Φ is the standard normal distri-bution;. ... ii) Verify that the estimator is B-robust, by determining an explicit bound on theinfluence

30 Aug 2019: ut. (u )u = 1ρr. p ρρr. g 1ρr. ( B) B ν2u,. ... ψ(x,z,τ) =pλ. πsin. πx. λsin πz a(τ),. Θ(x,z,τ) = cosπx. λsin πz b(τ),.

17 Jun 2019: 5]. [Note that a b a b. ]. (b) Write down the equation for a sphere S given that its centre is at position vector aand its radius is p > 0. ... 2(b a) r = p2 q2 b2 a2 ,where a = |a| and b = |b|.

17 Jun 2019: 5]. 5B (a) Let A and B be n n Hermitian matrices (A = A†, B = B† ) with distincteigenvalues. ... 5]. (b) If C is a unitary matrix and A is Hermitian show that(C1AC.

30 Aug 2019: 3 Let Ta,b denote the first hitting time of the line a bs by a standard Brownianmotion, where a > 0 and < b < and let Ta = Ta,0 represent the first ... 2θ or otherwise, derive an expression. for E(eθTa,b. )for each b, < b <. Hence, or otherwise, show

21 Aug 2019: Suppose that A and B arenn Hermitian matrices such that U = A iB. ... Hence, or otherwise, expand. (i) (a b) (c d) ,. (ii) (a b) [(b c) (c a)].

17 Jun 2019: Let A and B be 33 matrices representing linear mappings α and β. ... 6. 11A Vector Calculus. Prove that. (a b) = a b b a (b )a (a )b.

4 Aug 2022: b) By considering the dual 1-forms, show that on U we have. ... Let A be a connection on a principal G-bundle π : P B.(b) Define the horizontal distribution H and curvature F of A, and show that H is.