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30 Aug 2019: a) Use suffix notation to prove that. a (b c) = c (a b). ... r (a b b c c a) = a (b c),.

30 Aug 2019: 2. (i) State and prove the Erdős–Ko–Rado Theorem. (ii) Two set systems, A and B, are called cross-intersecting if A B 6= whenever A Aand B B. ... f(r, s) elements such thatA B X 6= for all A A and B B.

30 Aug 2019: µ0ρ(B b b B),. b. t= B u u B. Consider solutions of the form u(s, φ, z, t) = ũ(s, z)eσtimφ, etc., where m is aninteger. ... Then verify that. B b b B = Jẑ b 12imJb,.

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: 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.

30 Aug 2019: H(,B) are natural in B B. Part III, Paper 22. 3. ... B(KB′,B)B(ρ. B′,B). B(B′,B). is a bijection. (iii) ρB : B KB is an isomorphism in B.

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: 3. Prove Scott’s Isomorphism Theorem for Lω1,ω. 4. (i) Let <A and <B be recursive (i.e. ... Must there exist a recursive isomorphism betweenthem? (ii) Find a natural deduction proof for ((((A B) A) A) B) B using.

30 Aug 2019: where A = A(t z) and B = B(t z) are non-zero smooth functions of t z. ... d) Show that the vacuum Einstein equation Rab = 0 reduces to A′′/A B′′/B = 0.

30 Aug 2019: Fix a number a > 1. Let (X, B, µ, T) be a measure preserving system, and letf L1(X, µ). ... Bwith 0 < µ(A) < ε, then for every k Z>0 and ε > 0, there is a set B B such that thesets B, T(B),. ,

30 Aug 2019: b) What is a formal group, and what is an isomorphism of formal groups? ... Show that N(B) < and that if N(B)/. B as B then rank E(Q) > 2.

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: b) Let Γ be a finitely generated group and H a subgroup of finite index. ... 4. 4. (a) State and prove Van Kampen’s Lemma. (b) Let P 〈a, b, c | [a, b], [b, c], [c, a]〉, the standard presentation of Z3.

30 Aug 2019: a) Use suffix notation to prove that. a (b c) = c (a b). ... r (a b b c c a) = a (b c),.

17 Jun 2019: Hence show that the line integral P2P1. F dx. along any path in R3 between points P1 = (0,a, 0) and P2 = (b,b,b) vanishes for anyvalues of a and ... b. The two-dimensional vector field G is defined at all points in R2 except (0, 0) by.

30 Aug 2019: Ag. //. a. (2). B. b. K′f′. // A′g′. // B′. ... A′p′. //. I′i′ //. B′. b. A p// I. i// B.

30 Aug 2019: Here N GL(n,C)is some neighbourhood of the identity matrix, W is some neighbourhood of the zero matrixin H(n) = {B Matr(n,C) : B = B}, Matr(n,C) ... You may also assume that exp maps a neighbourhood of zero inu(n) = {B Matr(n,C) : B = B}

30 Aug 2019: For each m denoteB(m) = [m, m]d Zd and B(m) = {(x1,. , ... a) Define the terms: voltage, current flow and effective resistance between a and b.

17 Jun 2019: a (b c). Given that the vector a b b c c a is orthogonal to the plane Ψ of the faceABC, obtain an equation for Ψ. ... b) (i) Prove that the real series. f(x) =. n=0. (1)n x2n.

30 Aug 2019: M). DefineC : Tp(M) Tp(M) , X 7 C(X) B(A(X)). Show that C is a tensor of rank (1,1), i.e. ... a multilinear map C : Tp(M) T. p (M) R,. and determine the components Cαβ of C in terms of the components of A and B.

30 Aug 2019: Write π(fa,b) C. under this identification explicitlyin terms of a and b. ... Hint: First check that fa,b fa′,b′ = faa′,x for some x depending ona, a′, b, b′, and that [fa,1, f1,b] = f1,b(1a1).

30 Aug 2019: 1. 4πρ0B B,. tB u B = B u, tθ u θ = ux, u = 0, B = 0,. ... b) Suppose the perturbations are independent of x and y and proportional toeσtikz.

30 Aug 2019: n,. b Gamma(c, 1). (a) Derive a Gibbs sampler for the posterior distribution of (θ1,. , ... θn, b). (b) Prove that the Markov chain on the parameter θ = (θ1,. ,

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.].

30 Aug 2019: B) B, (3). B. t= (u B) , (4). 2Φ = 4πGρ. ... R(t) = αĖaρb1tc. (3). Determine the values of the constants a, b, and c.

30 Aug 2019: 4π( B) B, (3). B. t= (u B) , (4). 2Φ = 4πGρ. ... t(u B) F = Qh,. without making assumptions about the spatial distribution of entropy s.

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)?

17 Jun 2019: a A there exists b B with f (b) = a and g(b) = h(a). ... b) Let k and m be positive integers, let n = km, let A1,. ,

17 Jun 2019: i) (a b) (c d),(ii) (a b) [(b c) (c a)]. ... b) Give the general solution for x and y of the simultaneous equations.

30 Aug 2019: b) Suppose that there is no non-zero function u C2,α() satisfying Lu = 0 in , u = 0on. ... question as a fixed point problem for a map T : B B whereB = {v C2,α() : v = 0 on and |v|2,α; 6 ǫ} for an appropriate choice ofǫ > 0.].

17 Jun 2019: 15C Linear Mathematics. Let A and B be n n matrices over C. ... b) After a proper time τ, measured in its own rest frame, particle A decays.

17 Jun 2019: You may assume that (ab) = a.b b.a (b.)a (a.)b. ]. Show that, in a two-dimensional flow, vortex lines keep their strength and movewith the fluid. ... another vortex ofcirculation Γ at the image point, distance b from the wall on the other side.

17 Jun 2019: φ = 1 on r = a for all θ,. φ = 2 on r = b for all θ. ... a) Prove that each pn, n > 1, has n distinct zeros in the interval (a,b).

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′?

17 Jun 2019: i) (A B) (B A) is finite and. (ii) there is a finite set F A B such that f(x) = g(x) for all x A B F. ... n. n. k=1. ak = a. (b) Suppose ak > 0 for every k.

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,

17 Jun 2019: You may assume. 2 14π|x|. = δ3(x) ,. and for fixed vectors a, b. ... Let b < min(a,r/C, 1/K). Show that there isa unique C1 function x : [b,b] [x0 r,x0 r] such that.

30 Aug 2019: that if A,B D,A B then we have B A D. ... 1. set is. In 1-D it is defined byµ((a,b]) = ba,.

17 Jun 2019: gi > (Pg)i. jI. pijgj > 0 i,(a). gi = 1 i B(b). ... c) gi > (P̃g)i i,. where. p̃ij =. {. pij (i / B),δij (i B).

17 Jun 2019: Show that if B is R-closed then c 6 b B implies c B.(b) Suppose that R satisfies the following condition. ... B C] = {a A |b 6 a (b B b C)}.

17 Jun 2019: Tie up your answers in separate bundles labelled A, B,. , H according to the. ... If themagnetic field is. B = b ( x, y, 2z ) ,. where b is a constant, find the induced current I, given that the total resistance of theloop is R.

17 Jun 2019: a′ (b b′) = 0. 15A MethodsLet L be the linear differential operator. ... bi =. b. a. w(x). n. j=1j 6=i. x cjci cj.

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

17 Jun 2019: Show that if B is R-closed then c 6 b B implies c B.(b) Suppose that R satisfies the following condition. ... B C] = {a A |b 6 a (b B b C)}.

17 Jun 2019: Tie up your answers in separate bundles labelled A, B,. , H according to the. ... b′ ei(k′xωt). ]. ,. where. e′ = e 2(n e)n ,. b′ = b 2(n b)n ,. k′ = k 2(n k)n.

17 Jun 2019: What is ptychography? b b. ca. 15 m. coherent x-ray beam (wavelength 150 pm).

17 Jun 2019: continuous? (b) Which of the following functions are uniformly continuous? Briefly justify youranswers. ... 21H Optimisation(a) Suppose that A Rmn and b Rm, with n > m.

17 Jun 2019: 7:. Read 6 items. > gender <- scan(,""). 1: b b b g g g. ... With this choice showthat in a general spacetime. Aa;b;b RacAc = 0.

17 Jun 2019: Tie up your answers in separate bundles labelled A, B,. , H according to the. ... A : A Z and f|B : B Z are continuous,show that f is continuous.

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.