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29 Oct 2013: Which of the following sets of functions form a vector subspace of RR?(a) The set C of continuous functions.(b) The set {f C : |f(t)| 1 for all t ... i6=j Ui = {0}. (iii) The Bi are pairwise disjoint and their union B is a basis for U =. i Ui. Give an

6 Feb 2013: ii) If f ′(x) > 0 for all x (a, b) then f is increasing. ... iii) If f is strictly increasing then f ′(x) > 0 for all x (a, b).

15 Oct 2013: m,n) (p,q) = (m (1)n p,n q). 1. Let a = (1, 0) and b = (0, 1). ... 2. Describe the group Aut(G) of all self-isomorphisms of G. [You may use, andprove, that both the centre and commutator subgroups are characteristic sub-groups i.e.

31 Oct 2013: 4. Is it true that for all positive integers a, b, c, d we have (a, b)(c, d) = (ac, bd)? ... 6. The Fibonacci numbers F1, F2, F3,. are defined by: F1 = F2 = 1, and Fn = Fn1 Fn2 for all n > 2 (so eg.

21 Nov 2013: such that f(g(x)) = g(f(x)) for all x X. Show that g has a fixedpoint. ... x, y) for all x, y X with x 6= y, but such that f has no fixed point.

5 Feb 2013: By considering a suitable subgroup of thegroup G of all maps from Zp to itself of the form x 7 ax b, where a, b Zp with a 6= 0,show ... Without knowledge of what this group is, whyis it obvious that the group of all symmetries of the dodecahedron cannot

16 Oct 2013: m,n) (p,q) = (m (1)n p,n q). 1. Let a = (1, 0) and b = (0, 1). ... 2. Describe the group Aut(G) of all self-isomorphisms of G. [You may use, andprove, that both the centre and commutator subgroups are characteristic sub-groups i.e.

15 Feb 2013: Show that. there is a δ > 0 so that |a b| > δ for all a A and b B. ... Consider the subset Y Xconsisting of all functions f with |f(y)| 6 1.

29 Nov 2013: continuous function g: Bε(I) Mn such that g(A)2 = A for all A Bε(I). ... Show that f has a fixed point.(b) Suppose instead we assume only that for all x, y X at least one of the threedistances d(f(x),f(y)),

1 Nov 2013: Which of the following sets of functions form a vector subspace of RR?(a) The set C of continuous functions.(b) The set {f C : |f(t)| 1 for all t ... i6=j Ui = {0}. (iii) The Bi are pairwise disjoint and their union B is a basis for U =. i Ui. Give an

27 Nov 2013: and 18t 1 are all prime numbers. Prove that N is a Carmichael number. ... More generally, show. that if p and 2p 1 are both prime numbers, then N = p(2p 1) is a pseudoprime forprecisely half of all bases.

2 Oct 2013: We assume a < 1 r < c. (a) Find all the equivalent martingale measures for this model. ... b) For simplicity only, assume r = 0. Characterise all contingent claim with payoffY = (ya,yb,yb) at time 1 that can be replicated, that is for which there

2 Oct 2013: Show that(a) E[eθXf(X)] = eµθσ2θ2/2E[f(X θσ2)] for all real θ and suitable f;(b) for any nice function f, E[f(X)(X µ)] = σ2E[f ... 1 α2σ2) for all real α and β. (ii) Suppose that (X,Y ) has a jointly Gaussian distribution.

19 Jan 2013: for all w with |wf(a)| < δ,there is a unique g(w) such that f(g(w)) = w. ... 5) (i) Show that a bounded holomorphic function on D(0, 1) extendsholomorphically to all of D(0, 1).

7 Jan 2013: Show that there is nodecodable coding c such that all codewords have length 2 or less. ... Determine whether there are optimal binary codes with (a) all but onecodeword of the same length, or (b) each codeword a different length.

7 Jan 2013: 13 Prove the following results:(i) If K is a field containing F2, then (a b)2 = a2 b2 for all a,b K.(ii) If P F2[X] and K ... is a field containing F2, then P(a)2 = P(a2) for all a K.(iii) Let K be a field containing F2 in which X7 1 factorises into

4 Dec 2013: 4. Show that T B(H) is normal if and only if ‖Tx‖ = ‖Tx‖ for all x H. ... 8. Let S and T be compact normal operators on a complex Hilbert space withdim ES(λ) = dim ET (λ) for all λ.

15 Aug 2013: Wednesday, 14 August 2013. Focus to outline1. Write down all the words you need to. ... before b. Wednesday, 14 August 2013. Focus to outline1. Write down all the words you need to.

12 Sep 2013: emαm! preserves M. (b) For all b U(g) there exists i > 1 such that ad(reα)i. ... b 7 exp(λ(log b)) for all b B+. VERMA MODULES FOR IWASAWA ALGEBRAS ARE FAITHFUL 17.

11 Mar 2013: torsion point of itsJacobian, all such automorphisms are described by elements of H. ... is an isomorphism at all (gV ,gW , (a,b)) with D(a,b) 6= 0.