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17 Jun 2019: Find constantsα and β such that. xjTij = [( B) B]i. ... i) the union of all sets in B is X;. (ii) if x B1 B2 for two sets B1 and B2 in B, then there is a set B B withx

30 Aug 2019: You do notneed to write a complete proof.). (d) Let A = 〈a, a′ | aa′a1(a′)1〉 and let B = 〈b, b′ | bb′b1(b′)1〉 and consider the freeproduct. ... A B = 〈a, a′, b, b′ | aa′a1(a′)1, bb′b1(b′)1〉. Let T be the graph with a

30 Aug 2019: A B| |A B| > |A| |B|. (iii) For A, B P(n), write. ... A B ={. A B : A A, B B}. Show that|A B||B A| > |A||B|.

30 Aug 2019: Bt. ui B = B ui, (3). where ui and un are the velocities of the ions and of the neutral particles, respectively; alldiffusion terms have been neglected; µin is the ... In this approximation, the magnetic field obeys. Bt. = c4πen. [( B) B].

30 Aug 2019: 2 Consider a pressure-balancing magnetic field satisfying. ( B) B = µ0p. ... where B = B(r,θ), p = p(r,θ) in spherical polar co-ordinates (r,θ,φ).

30 Aug 2019: µ0( B) B ,. B. t= (u B) ,. B = 0 ,. 2Φ = 4πGρ. 1. (a) Formulate the equations governing the steady, spherically symmetric, adiabatic flowof an unmagnetized, non-self-gravitating perfect gas in ... form. u = v(z,t) axey , B = B(z,t) , ρ = ρ(z,t) ,.

30 Aug 2019: instructed to do so by the Invigilator. 2. 1 Consider a Parker wave dynamo model for a mean magnetic field B = B(x, t)eφ A(x, t)eφ in the ... Defining A = A Ã(τ ), B = B B̃(τ ), where the overbar indicates an averageover the fast (τ ) timescale,

30 Aug 2019: Π 1. µ0B B,. B. t+ u B = B u,. ... u = 0,. B = 0. Explain the meaning of Π in the first equation, and write down an appropriate setof boundary conditions for this system.

19 Jun 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: µ0( B) B ,. B. t= (u B) ,. B = 0 ,. 2Φ = 4πGρ. Part III, Paper 314. 3. 1. (a) In magnetohydrodynamics, the cross-helicity contained in a fixed volume V isdefined as. ... The basic state is in magnetostaticequilibrium, with density ρ(z), pressure p(z)

19 Jun 2019: b) Let X be a random variable such that EX2 < and E(X | G. ... n0. b) Let(Xn. )n0 be a positive supermartingale. For fixed a, b, b > a > 0, we put N0 = 1.

19 Jun 2019: b) there exists a positive random variable ν, P (ν > 0) = 1, with E ‖νA1‖ < suchthat A0 = E (νA1). ... 2θ, or otherwise, derive an. expression for E(eθTa,b. )for each b, < b <. Hence, or otherwise, show that, for t > 0,. P (Ta,b 6 t) = e2abΦ(.

30 Aug 2019: ii) Let A and B be finite multisets of subsets of [n] with A > B, i.e. ... 4log |B|6 |A B||A A||B B|. END OF PAPER. Part III, Paper 13.

19 Jun 2019: 6]. 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 ... 2θ or otherwise, derive an expression. for E(eθTa,b. )for each b, < b <. [8]. Hence, or otherwise,

30 Aug 2019: B) B. B. t= (u B). 2Φ = 4πGρ. You may assume that for any vectors C and D. ... kB. ρ (. 1. 2|u|2 Φ. ). =Rω. µ0ρeφ (( B) B) , and.

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]. 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.

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.

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)].

12 Jul 2023: c) Briefly define the parallel transport map PAγ : Eγ(0) Eγ(1) associated to a pathγ : [0, 1] B. ... Assuming that µ(b) End(Eb) is anisomorphism for some b B, show that it is an isomorphism for all 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.

30 Aug 2019: n(h)dh = B exp(bh)dh ,. where B, b are parameters which can be expressed in terms of the mean draft hm, thecutoff h0, and the total number of ridges ... On the assumption that ridged ice is composed ofpressure ridges of congruent triangular shape, with

30 Aug 2019: b) Let X be a random variable such that EX2 < and E(X | G. ... n0. b) Let(Xn. )n0 be a positive supermartingale. For fixed a, b, b > a > 0, we put N0 = 1.

30 Aug 2019: b) there exists a positive random variable ν, P (ν > 0) = 1, with E ‖νA1‖ < suchthat A0 = E (νA1). ... 2θ, or otherwise, derive an. expression for E(eθTa,b. )for each b, < b <. Hence, or otherwise, show that, for t > 0,. P (Ta,b 6 t) = e2abΦ(.

17 Jun 2019: Each section B question has a number and a letter (for example, 11S). ... x = a (b x)b b x. Show that. x b =a b.

17 Jun 2019: Use suffix notation to expand (a b) (b c).Let a, b, c be given unit vectors in R3, and let m = (a b) (b c) (c a). ... Show that f is continuous at x0. (b) State the Mean Value Theorem.

30 Aug 2019: b) Define the coaction of SLq(2) on kq[x, y] and compute explicitly 4x2y. ... Define (by stating its universal property) what is meant by the universal measuringcoalgebra P (A, B).

17 Jun 2019: Each section B question has a number and a letter (for example, 11S). ... 5]. (b) B and C are real non-zero 3 3 matrices and satisfy the equation.

19 Jun 2019: 5 Statistical Genetics. Consider a genetic locus with two alleles, a and b. ... i) Under Hardy-Weinberg equilibrium, what are the expected frequencies ofindividuals with genotypes a/a, a/b and b/b respectively, in terms of p?

30 Aug 2019: b) Write down the matrix Ax = (Nj (xi)), and evaluate the norm ‖A1‖. ... x ti). Prove also that. (b) b. a. Mi(t)dt = 1, (c)n.

30 Aug 2019: State two furtherconditions satisfied by v and B. Show that there exist plane-wave solutions of the form. ... B = Re{B̃(t) exp [ik(t)x]. },. and obtain equations governing the evolution of ṽ, B̃ and k.

30 Aug 2019: Katona Theorem and one using cyclic orderings. Let A, B [n](r), where r 6 n/2. ... Show that if |A|, |B| >(n1r1. ). then there exist. A A and B B with A B =. Part III, Paper 12. 3. 3. State and prove the vertex-isoperimetric inequality in

30 Aug 2019: f1ḟ = Tr(F 1Ḟ ). (c) If A, B B(H) with [A, B] trace–class, prove that eAeBeAeB lies in G(H)with. ... Whena = d = 0, b = 1 and c = 1, find a corresponding operator V.

17 Jun 2019: Each section B question has a number and a letter (for example, 2Y). ... 2]. (b) The probability density function for a continuous random variable X is.

17 Jun 2019: Show that if a = a′, b = b′ and c = c′, thenthere is a hyperbolic isometry taking ABC to A′B′C′. ... Is there always such an isometry ifinstead the triangles have one angle the same and a = a′, b = b′?

30 Aug 2019: Illustrate how to solvethe word problem for G, giving examples. You may assume that [b, a] = [a, b]1, [ab, c] = [a, c]b[b, c], and [a, bc] = [a, ... 4. (i) Let A and B be isomorphic subgroups of a finite group G.

17 Jun 2019: A B) = B ( A) A ( B). [6]. (ii) State the divergence theorem, and use it to show that for a scalar field a(x,y,z) andvector field ... b) the integrand F has no explicit dependence on x, F = F(y,y′).

30 Aug 2019: b) Show that. dλi 1. 2fijkλ. j λk = 0,. where the constants fijk should be determined. ... Part III, Paper 55. 3. 3. Define a principal bundle (π, P, B, G).

4 Aug 2022: y b. x= 0,. where b = b(x,y,t), ψ = ψ(x,y,t), 1 is a constant parameter and. ... 10]. (c) State Parseval’s identity and use the results from part (b) to evaluate.

17 Jun 2019: Must there exist a A and b B such that d(a, b) 6 d(x, y)for all x A and y B? ... B = b[r (kẑ lŷ)ẑ r px̂(ŷ r) nẑ(x̂ r)],. where b 6= 0, r = (x, y, z) and x̂, ŷ, ẑ are unit vectors in the x,

20 Jul 2021: Let M be an A-module, and let MB denote the B-module withunderlying group M and multiplication bm = ϕ(b)m for b B, m M. ... Show that Supp s is a closed subset of X. (b) Let Z X be a closed subset of X.

30 Aug 2019: Prove that. (a) if ξ Irr(T |θ) then ξG Irr(G|θ);(b) the map ξ 7 ξG is a bijection of Irr(T |θ) onto Irr(G|θ). ... Assume K = A B {1}. Prove that there exists a generalised character ξ of K such thatξ(a) = 1 for all a A and ξ(b) = 0 for all b B.

17 Jun 2019: A, B] AB BA. Show that [A, A] = 0; [A, B] = [B, A]; and [A, λB] = λ[A, B] for λ C. ... Use it to show that if f : (a, b) R is a differenti-.

3 Aug 2022: Find a, b, and the corresponding value of E, in terms of l. ... The weights b1,. ,bn and nodes. c1,. ,cn are chosen so that the Gaussian quadrature formula for a function f C[a,b] b.

30 Aug 2019: in particular B = B(0, 1). ... B(x0,ρ)|u|2dx 6 C4ρ. 12α(. ‖u‖2L2(B) ‖f‖2L2(B). ). , B(x0, ρ) B.

30 Aug 2019: Prove that if (X, B, µ, T) is a weak mixing system then. ... What are the ergodic components of the circle rotation (R/Z, B, m, R1/5)?