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17 Jun 2019: has a potential satisfying (). What is the cosmological significance of the constant B? ... b) Let K = Q(d), with d 6= 0, 1 and d square-free.

17 Jun 2019: b) Verify the Leibniz rule{fg, h} = f{g, h} g{f, h}. ... µ(AB) 6 ε,. where AB denotes the symmetric difference (A B) (B A).Show that there does not exist a Borel set B contained in [0, 1] such that, for

17 Jun 2019: Prove that X A is biholomorphic to X B. (b) Let Z C2 be the algebraic curve defined by the vanishing of the polynomialp(z,w) = w2 z3 z2 z. ... b) Let (Xt, t > 0) be a Markov chain on I = {0, 1,. }

17 Jun 2019: A B) = B ( A) A ( B) ;. (A B) = (B )A B( A) (A )B A( B). ... i) B has no real eigenvalues;. (ii) C is not diagonalisable over C;.

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.

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

17 Jun 2019:  ,find the scalar a, vector p and symmetric traceless tensor B such that. ... b) Let A be an mn matrix, m > n, such that rankA = n.

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: V. ( B) B dV = 0. 3/I/2A Methods. Write down the wave equation for the displacement y(x,t) of a stretched stringwith constant mass density and tension. ... 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

17 Jun 2019: b) Let σ1,σ2,. ,σk Σ where k > |Q|, and let p Q. ... b) Compute the fundamental group of X (giving your answer as a well-knowngroup).

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: F = aTr(Q2) b(Tr(Q2))2 K. 2|.Q|2. where b and K are positive constants, and. ... F =a. 2λ2. b. 4λ4. K. 2. ((λ).θ)θ λ(.θ)θ λ(θ.)θ λ.

11 Nov 2019: write. A B = {a b : a A, b B},. then A B K.(b) Show that, if Kn K, and. ... i) by assuming b B and using your result from part (a),.

30 Aug 2019: b) Explain (without proof) how one can construct an orthonormal wavelet from anMRA. ... 3. 2. (a) Let B := {gm}mN be an orthonormal basis of a Hilbert space H.

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] 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-.

19 Jun 2019: b = b[-1]. }. }. res = c(res, if(empty(b)) a else b). ... else {. h = floor(l/2). a = x[1:h]. b = x[-(1:h)].

16 Oct 2019: iii) Now take ǫ(n) = an b for positive constants a and b. ... Let T : X X be a bounded operator. Prove that T(B) B implies T(B) B.(b) Let P be the vector space of all polynomials in one variable with

17 Jun 2019: i) f(x) = ea|x|,. and (ii) g(x) ={. 1 , |x| 6 b0 , |x| > b. ... 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.

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)?