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30 Aug 2019: Suppose that if a A and b B then (o(a), o(b)) = 1. ... Show that thereexists a generalised character ϑ of G with ϑ(a) = 1 if a A and ϑ(b) = 0 if b B.

30 Aug 2019: u B),. B = 0. The flow is such that all physical quantities are independent of t and z, when referred toCartesian coordinates (x,y,z). ... 1µ0ρ. B B ν2v,. (. t 2Ax. y. )B 2ABx ey v B = B v η2B,.

30 Aug 2019: ρ. t+ u ρ = ρ u ,. p. t+ u p = γp u ,. ρ. (ut. u u). = ρΦ p 1µ0. ( B) B ,. Bt. = (u B) ,. B = 0 ,. 2Φ = 4πGρ. ... Σ =. ρ dz. (b) Assuming that the perturbations are also isothermal, derive the linearized equationsgoverning displacements of

30 Aug 2019: u. t+ uu. )= ρΦ p 1. µ0(B) B ,. B. ... t= E ,. B = µ0J , B = 0. under a Galilean transformation to a frame of reference moving with uniform relativevelocity v.

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

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(.

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: You may assume that div (a b) = b. curl a a. ... a (a. )b a div b b div a.]. Paper 64.

30 Aug 2019: The region y < 0 is insulating. A line current J cos ωt flows in a wire parallel to thez-axis and located at x = 0, y = b (b > 0). ... b) Suppose that the frequency ω is large, so that (η/ω)1/2 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: 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.

17 Jun 2019: where r, σ and b are positive constants and (x,y,z) R3. ... where a and b are constants, µ is a parameter and a 6= 0.

17 Jun 2019: Show that:. (a) A a = B b = C c = 1,. ... b) A b = A c = 0,. (c) A, B and C are non-coplanar.

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: 2. The dyadic rationals { a2b| a Z, b Z0} form a subring of Q. ... Why is 2 irreducible? Suppose 2 = ab where a and b are non-units.

30 Aug 2019: B. t+ u B = B u η2B,. where η is the magnetic diffusivity. ... There is also an additional body force µ10 B B (theLorentz force) on the right hand side of the momentum equation.

30 Aug 2019: ut. u u 2 u = Ψ 1µ0ρ. B B ν2u. ... and the induction equation. Bt. u B = B u η2B.

19 Jun 2019: 2. 1 Consider the optimization problem. Minimize f (x). subject to h(x) = b. ... Define the function φ by. φ(b) = infxX. {f (x) : h(x) = b}.

17 Jun 2019: anV RT. }[6]. where a,b,n,R are constants, find ( UV )T. ... firsttime. [6]. (b) In another trial counters are now drawn without being replaced.

17 Jun 2019: 1]. (b) B =. (. cos θ sin θsin θ cos θ. ). ... r (a b b c c a) = a (b c).

17 Jun 2019: Deduce from (b) thatlims1. (s 1)ζ(s) = 1. A4/10 Number Theory. ... Consider thecases where (a) g′(t) has one simple zero at t = t0 with A < t0 < B; (b) g′(t) has morethan one simple zero in A < t < B; and (c)

30 Aug 2019: b) Let R be the roots of so5. Draw them, and identify a set of simple roots and positiveroots. ... Draw the crystals forB B and B B. Write down the highest weights of each of the representations that occur in thesetensor products.

19 Jun 2019: 40%. (b) Describe and compare hierarchical and partitioning methods for clustering of sam-ples. ... theoutput distribution is specified by B = b(j,k).

17 Jun 2019: 10]. 5C. A, B and C are three points with position vectors a = a i, b = b j and c = c krespectively, where a, b and c are positive scalars. ... a) Sketch f (x). [4]. (b) Find the Fourier expansion of f (x).

19 Jun 2019: STATISTICAL THEORY. Attempt FOUR questions, not more than TWO of which should be from Section B. ... 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;.

30 Aug 2019: Define what it means for g to be (a) solvable and (b)semisimple. ... Decompose B B intoirreducibles. Paper 3. 3. 5 Let g = sl2, and V be a respresentation of g.

17 Jun 2019: r. (rφ. r. ). 1r22φ. θ2= 0. In the region between two concentric cylinders of radii a and b (b > a > 0), φ(r,θ) satisfies2φ = 0. ... By separationof variables or otherwise find φ in the region a < r < b.

17 Jun 2019: 7:. Read 6 items. > gender <- scan(,""). 1: b b b g g g. ... b) Consider a body with three unequal moments of inertia, I3 < I2 < I1.

30 Aug 2019: b) Show that the normaliser of G in Sym() is isomorphic to the semi-direct productof G by its automorphism group. ... With addition definedby symmetric difference,. A B = (A B) (B A),.

30 Aug 2019: 4 Let p be a prime and suppose that A, B Z/pZ are sets with pα 6 |A| 6 p1α and|B| > pβ. ... Show that there is some b B for which |A b A| > |A|1cα,β , where cα,β > 0.

30 Aug 2019: field B = (B(z), 0, 0). ... You may assume that. curl(a b) = a div b b div a (b.)a (a.)b. ].

17 Jun 2019: 4]. where a = |a|, b = |b| and c = |c|, and A, B and C are the internal angles of the triangle. ... b) Consider an arbitrary tetrahedron with four faces f1, f2, f3 and f4.

30 Aug 2019: Then A has a subset B of cardinality at least cn such that thedifference set B B has cardinality at most Cn, where c and C are positive constantsthat depend on ... iii) If G is the group ZN , and if n > θN, then 2A 2A contains an arithmeticprogression

30 Aug 2019: µ(B) =. B. dp dθ. for B (0,) [0, 2π). A random countable set Φ R2 is defined to consist of allintersections of pairs of lines in Π.

30 Aug 2019: Pp(|C0| > n) 6 exp(an). for all n > 2.(ii) Deduce from the result in (i) that if p > 1/2 then there is a constant b = b(p) > 0.

30 Aug 2019: Solve the word problem for A/〈[g,h]〉A. (ii) Let N = 〈a,b : [a,b,a] = 1 = [a,b,b]〉. ... Assuming that the form ambn[a,b]k is unique, solve the word problem for N.

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: k(X, C) over B. Assume that Xb = π1(b) is a compact Kähler manifold for all b B. ... F pHk Hk, whose fibres over b B are (F pHk)b =F pHk(Xb, C).

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.

30 Aug 2019: Forevery f B, denote by mf : B B the operator of point–wise multiplication by f :. mf : g 7 f g. (i) Compute the composition rule of n mf for every ... al(x)ul(x)dx,. for some c R and some a1,. , am1 B.

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: 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.

30 Aug 2019: b) Suppose that is an open set in C with {z : z > 0}. ... h 0. as h 0, show that A = A′, B = B′.(ii) Let E be a finite set.

30 Aug 2019: C′ = BU n. and determine the constant B as a multiple of A. ... You may assume that bb. t dt. (1 t)2=. 2b(1 b)2.

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.

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?

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′).