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17 Jun 2019: a) Find x and y. (b) Find a matrix M that is a rotation leaving T invariant such that Ma = b andMb = a. ... Find a transformation. u = f(x,y), v = g(x,y) ,. such that S is transformed into a rectangle in the (u,v) plane.

30 Aug 2019: in terms of the unknowns ψs and vg only. (iii) Consider a scatterer D with Dirichlet boundary conditions, and assume thatthere exists a function g(r̂) L2(S1) such that ... y(δ) such that ‖ y(δ) y ‖6 δ, with xn and x(δ)n the corresponding

30 Aug 2019: τλ), where A and B are constants.Show that the continuum spectral slope of the transmission spectrum for such anatmosphere in the optical is given by d log δλ/d log ... ii) Derive the condition for which this atmosphere does or does not have a

30 Aug 2019: kl for all k, l > 0 such that. (i) D is R-linear. ... Prove that if M is orientedthen HndR(M) is non-zero. (b) Now suppose that f : M M is a diffeomorphism such that f f = idM.

30 Aug 2019: The structure ofthe giant is such that its radius. R = f(L)M0.271. ,. ... 8M. Algol now has an orbital period of 3 days and masses M1 = 1 M and M2 = 3 M.Argue that it could well have formed via such conservative case A

17 Jun 2019: Let ψ : U U be a linear map such that b(ψ(x),y)b(x,ψ(y)) = 0for all x and y in U. ... f̂′′(λ) λ2f̂(λ) = µf̂(λ). (b) Prove that, for each n 0, there is a polynomial pn(x) of degree n, unique upto multiplication by a constant, such that.

30 Aug 2019: Show then that any set of nodes A {1,. , p}{k}such that Zk Z(A{k})c|ZA must have A mb(k). ... Assume that forsome c (0, 1) there exists φ > 0 such that for all b Rp with (1c)‖bN ‖1 6 (1c)‖bS‖1,we have.

30 Aug 2019: φ̃B(p), followed by a rescaling of the momentum coordinates and thefield such that. ... Define a suitable Renormalization Group transformation with scalefactors b and c in the spaces V and V‖ respectively such that the momentum coordinatesand the

17 Jun 2019: Find constantsα and β such that. xjTij = [( B) B]i. ... b) Let T be an invertible complex matrix. By considering the Jordan normal formof T show that there exists an invertible matrix P such that.

30 Aug 2019: d) What are the possible values of ξ0 such that the market has no arbitrage? ... Part III, Paper 211. 3. 3. A discrete-time Markov process (Xt)t>0 is called affine iff there exist (finite-valued)functions A and B such that.

30 Aug 2019: b) Let Pn be an n-dimensional polydisc. Consider α Ap,q(Pn), p > 1, such thatα = 0. ... c) Assume now that M is as in part (b) and that n = 2.

19 Jun 2019: 2 Suppose that B is a standard Brownian motion and that H is a locally boundedprevisible process such that. ... Using Girsanov’s theorem, find a new probabilitymeasure P, absolutely continuous with respect to P̃, such that B is a Brownian motionunder P

30 Aug 2019: 6 ϕ. in such that uk = V (uk1) in , uk = ψ on for eachk = 1,2,3,. , ... where u0 = ϕ. (c) Show that there is a constant C > 0 depending on ϕ,ψ,V such that the functionsuk as in (b) satisfy.

30 Aug 2019: on J such that ŵk(z) is a solution of the Taylor-Goldsteinequation corresponding to a neutral perturbation with c = 0. ... b) Identify the appropriate initial t = 0 and terminal t = T conditions for the adjointvariables for such a locally optimal

30 Aug 2019: 2. A′n Pn such that. U† (A1 A2. An)U = A′1 A. ... any such quantum process offers no computational time speed up overclassical computing (up to polynomial overheads).

17 Jun 2019: Briefly explain your answer. 6A Methods. If Tij is a second rank tensor such that biTijcj = 0 for every vector b and everyvector c, show that Tij = 0. ... Consider a polynomial p(z) = z4 az3 bz2 cz d, and let R be a positive realnumber such that |a|R3

30 Aug 2019: b) Show that there is a unique linear partial differential operator P such thatfor any open B(0,1) and any u Cm() and v Cmc () (i.e. ... b) We recall that an entropic solution to (2) is u L(R+ R) such that for allϕ C1c (R+ R; R+) all η : R R convex and

17 Jun 2019: x)F = tF. t. (b) The vector fields B and A satisfy B = A. ... Show that there is a scalar τ(s) (the torsion) such that.

17 Jun 2019: iii) Give an example of two matrices A,B with the same rank and the same minimaland characteristic polynomials such that there is no invertible matrix P with PAP1 = B. ... Now consider such a spherical triangle with the sides a,b replaced by λa,λb fora

30 Aug 2019: Suppose that thereexist a cyclic vector v V , g G and λ C, λ 6= 1, such that ρ(g)v = λv. ... s 1. (c) Prove that if λ E µ then there exists an Sn-invariant subspace Lλ,µ in thepermutation module Mλ such that Mλ = Mµ Lλ,µ.