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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.

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: 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).

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.

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.

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: 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.

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.

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.

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.

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).

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′).

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′?

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.

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,

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.

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 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)?

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-.

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.

17 Jun 2019: and B = UΛBU†,. where ΛA and ΛB are diagonal matrices and U is unitary. ... 4]. For such matrices A and B, using this result, or otherwise, show that.

17 Jun 2019: b) Calculate the slope of the path dy/dx at the point (12,. ... Notation: a b is equivalent to a b.]. (b) For vectors a, b, c and d, show that.

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.

17 Jun 2019: Use it to compute 3803 (mod 17). (b) The Fibonacci numbers F0, F1, F2,. ... n=1. fn(A) = and f(B) = B. [Here fn denotes the n-fold composite of f with itself.].

17 Jun 2019: A plane electromagnetic wave with a magnetic field B = B(t,x,z)ŷ, travelling inthe xz-plane at an angle θ to the x-direction, is incident on the interface ... An electron is at rest in the presence of a magnetic field B = (B, 0, 0), andexperiences an

30 Aug 2019: θn), θ̄ = (θ̄1,. , θ̄n) anti-commutingn-vectors n. i=1. dθ̄idθi exp(θ̄ B θ. )= ... µν Fµνwe are led to a quantum Lagrangian. Lq = 14Fµν Fµν b F µAµ 12ξ b b c̄ F.

30 Aug 2019: b) A volume of fluid V is released in a channel of unit width. ... b(z) ={b1, for z > 0;b1 b N20 z, for z < 0,.

20 Jul 2021: Write Pp for the correspondingprobability measure. (b) Define the percolation probability θ(p) and the critical probability pc. ... Show the following. (a) The process ξ is increasing in that ξAt ξBt if A B.(b) Moreover, ξ is additive in that ξABt =

30 Aug 2019: limn. µ(T nA A) = µ(A)2. holds for all A B. ... Suppose that for every ε > 0 and for every set A B, there is a number n Z>0and there is B B(ξn1.

30 Aug 2019: b) Find the smallest time t0 > 0 such that. eiHGt0|ψ〉 = eiα|w〉. ... 12. (|a〉|b〉 |b〉|a〉) = eiα|Φ〉. up to a global phase factor eiα.

30 Aug 2019: Define the PDF of the magnetic field P (B, t) =〈P̃ (B, t)〉. ... σim(t)P̃ (t)〉 =. dt′〈σim(t)σjn(t′)〉. 〈δP̃ (t)δσjn(t′). 〉. 2. Because of isotropy, P only depends on the absolute value B = |B|, so thenormalisation rule is.

17 Jun 2019: iv) Newton’s gravitational constant G;. (v) electric field E;. (vi) magnetic field B;. ... i) S1 = {f : A B | f is injective}.(ii) S2 = {g : B A | g is surjective}.(iii) S3 = {h : B B | h is bijective}.

30 Aug 2019: A retract diagram is a pair of maps. Ai B p A (1). ... such that pi = idA. (b) Show that in a retract diagram like (?), i : A B is the equaliser of the maps ipand idB : B B.

17 Jun 2019: Deduce from (b) that if e is an eigenvector of D with eigenvalue. ... two solutions when there are infinitely many. Consider the case. A =. . . a a b. b a a. a b a.

30 Aug 2019: n). Deduce (or prove otherwise), that if a, b A satisfy ab = ba, then. ... What does it mean to say that A is semisimple? Let X be a Banach space, of dimension greater than 1, and let B = B(X) bethe algebra of all bounded