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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: 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.

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|.

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: 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.

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.

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

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.

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) 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.

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: 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).

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: 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) 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.