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13 Oct 2006: b) Suppose that f is bounded. Prove that there exists M and n0 such that |fn(x)| < M for all. ... a,b] such that fn(xn) does not converge to f(x). 7. Which of the following functions f are (a) uniformly continuous, (b) bounded on [0,)?

10 Nov 2006: Does it follow that every open set must be. closed? Give an example of such a metric space. ... Show that thereexists a norm on C(I) such that the map sending f to y0.

28 Nov 2006: ii) Suppose that f : R2 R is such that D1f = f/x exists and is bounded near (a,b), and. ... is a function l : U M such that exp(l(A)) = A for all A U.

13 Mar 2006: a) LetG be a compact group. Show that there is a group homomorphismρ : G On(R) if and only if G has ann dimensional representation overR.(b) Explicitly construct such a ... subset ofH given by matrices such thatb = 0, and letφ : Fp C be a non-trivial

24 Nov 2006: Show that, for such a map,. dimH f (M ) 61α. ... Prove that the limit set for such a group is totallydisconnected.

7 Feb 2006: Prove that there exists such a reparametrization Γ̃ = Γffor which ‖dΓ̃/dt‖ is constant, namely l. ... 10) For a surface of revolution S, corresponding to a curve η : (a,b) R3 given byη(u) = (f(u),0,g(u)), where η is parametrized in such a way

30 Nov 2006: If A is the matrix of the first of these (say), find a non-singular matrix P such that P tAP is diagonalwith entries 1. ... 19. Let a1, a2,. , an be real numbers such that a1 an = 0 and a21 a.

13 Nov 2006: 1. 11. Show that if x 6= y are vectors in the finite dimensional vector space V , then there is a linear functionalθ V such that θ(x) 6= θ(y). ... If no such subspace exists,show that there exists a basis v,α(v),.

16 Feb 2006: for b1 6 y 6 b2.(b) Deduce that. R. f. x+ i. ... such piecewise continuously differentiable curves, andf : D C is an analytic function, then.

10 Oct 2006: How large a family A P[n] can we find such thatthe intersection of any two sets in A contains a cyclic translate (modulon) of B? ... Howlarge a family A P[n] can we find such that the intersection of any two setsin A contains a cyclic translate (modulo n)

4 May 2006: Prove that f has a real root. 17. Prove there is no continuous function f : [0, 1] R such that x Q f (x) 6 Q (whereQ denotes the rational numbers). ... 18. A continuous surjective map p : E B is a covering if every point b B has an openneighbourhood V

4 May 2006: b) Give a topology on R which is neither trivial nor discrete such that every open set isclosed (and vice-versa). ... a) Prove (R, TS ) is Hausdorff. (b) Prove (R, TS ) is not metrisable.

16 Feb 2006: 6. Let v and w be two non-zero complex numbers such that v/w is not real. ... Let (an)nZ be a sequence of complex numbers indexed by Z such that.

1 Jun 2006: i) There exist B GLn(k) and c k such that Φ1 = cBT Φ2B.(ii) The matrix B is uniquely determined up to scalars. ... Such a matrix B describes an endomorphism of H0(C, E). Wesaw in [12, 4] that E may be recovered from Φ(E).

22 Oct 2006: iii) For any two subsets A, B V (G) the number of pairs (x, y) A B such thatxy E(G) differs from p|A||B| by at most c2n2. ... number of pairs (x, y) A B such.

18 Sep 2006: finite subset S N+ suchthat, whenever S is k-coloured there exist positive integers a, x, a λx S allthe same colour, but such that there are no positive integers b, ... coloured,. 2. there are positive integers a, b, a λb S all the same colour, but

16 May 2006: Thus we have B Qn such that. • |B| = |A|;. ... be such that if G, H A then G H. contains a triangle.

9 Oct 2006: xj =. 4i=1 Bijx. ′i with det A det B 6= 0. We write G4 = GL2 GL4 for the group of all such transformations. ... Proof: Let φ be such a matrix, with vector of submaximal PfaffiansP = (p1,. ,

6 Apr 2006: varyingtogether. But this does not cause a problem—we may simply identify such. ... x pa} S will be calleda picture-line if there exist b,y F0 such that a Sb, x Sy, x a Syb,x 2a Sy2b,. ,

13 May 2006: By the non-degeneracy of the Weil pairing, all such homomorphisms arise in this. ... of K-group varieties ι : Λ A such that(i) ι preserves scalars, i.e.

20 Nov 2006: b) The set of all strings w such that every set of five consecutive charactersin w contains at least two 0’s;. ... 6. Let Σ be an alphabet and let B and C be subsets of Σ such that the emptystring is not in B.