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26 Nov 2008: Provethat A contains an infinite set B such that no point of B is a convex combination of otherpoints of B. ... 6. Show that every graph G has a partition of its vertex-set as X Y such that the numberof edges from X to Y is at least 12 e(G).

6 Mar 2008: vertices into sets B andR such that the only edges lie between vertices in different sets.(i) Show that if any finite subgraph of a graph is bipartite, then so is ... Suppose that any such formula is equivalent modulo T to aquantifier-free formula.

27 Feb 2008: extensions. We show that the Cartan map from K-theory to G-theory of such. ... Regarding the assumption on B, we ask the following. Question. Let H be a finite dimensional Hopf algebra such that the Cartan mapc : K0(H) G0(H) is injective and let

4 Feb 2008: Let P be the set of all p Q such that 2 < p2 < 3. ... c) If A and B are closed subsets of R such that 1 A, 1 B and [1, 1] A B, thenA B 6=.

12 Aug 2008: 5. 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). ... fn for V such that when x =. m1xiei and y =. n1yjfj we. have ψ(x, y) =r. 1xkyk, where r is the rank of ψ.

19 Mar 2008: 2 ANALYSIS II EXAMPLES 3. 7. (i) Suppose that f : R2 R is such that D1f = f /x is continuous in a neighbourhood of(a, b), and D2f = f /y exists ... Show that f is differentiable at (a, b). (ii) Suppose that f : R2 R is such that D1f = f /x exists and is

28 Feb 2008: In this and the following questions, youare not being asked to find such an isomorphism explicitly.)Decompose V 23 , Λ. ... b) Explicitly construct such a representation ρ : SU (2) SO(3) by showing that SU (2)acts on the vector space of matrices of the

10 Mar 2008: Show that for any b C, there exists a sequence of points zn D(a,R) with zn 6= a such thatzn a and f(zn) b as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2.

19 Mar 2008: iii) Give an example of disjoint closed sets A and B in R2 with points an A and bn B such. ... 9. [Tripos IB 96401(b), modified] (i) Suppose that (X, d) is a nonempty complete metric space, andf : X X a continuous map such that, for any x, y X, the

17 Oct 2008: 11 fn 6 0. Is it possible to find such a sequence with |fn(x)| 1 for all n and for all x? ... On the other hand, show that if fn does notconverge uniformly to f then we can find a convergent sequence xm x in [a, b] such that fn(xn) 6 f

19 Mar 2008: ii) If fn does not converge uniformly, show that we can find a convergent sequence xn x in[a, b] such that fn(xn) does not converge to f (x). ... derivative’ limh0+ ((f (h) f (0))/h),and similarly for f ′(1)), but such that f ′ is unbounded on [0, 1

10 Mar 2008: Prove that the set S of points x [0, 1] such that x Aj for infinitely many j is dense.Must S be open? ... b) If f : (0, 1) R is continuous and bounded, prove that there exists a sequence of polynomialsQn such that Qn are uniformly bounded and Qn f

19 Mar 2008: n=1 = (x1, x2, x3, ) such that for each n, |xn| 1/n. ... In such circumstances, T is a bounded linear operator. (What is the moral of this equivalence?)Let B(E, F ) be the space of bounded linear operators equipped with the usual

28 Feb 2008: n}, let the non-negative numbers A(n)k and the“nodes” x(n)k [a, b] be given such that for each polynomial P , the error. ... such that for every f C([0, 1]), there exists a unique series.

14 Oct 2008: If so, howmany such ‘prime triplets’ are there?(b) Suppose that four out of five consecutive odd integers, greater than 10, are prime. ... But such a numberwould clearly be very interesting indeed. Therefore, all positive integers are interesting.

13 Nov 2008: a linear func-tional θ V such that θ(x) 6= θ(y).(b) Suppose that V is finite dimensional. ... If no such subspace exists,show that there exists a basis v, α(v),.

4 Feb 2008: Let a, b K such that thepolynomials f = xm a, g = xm b are irreducible. ... Show that G is solvable if and only if there is an i such that.

28 Aug 2008: anda distinguishedobject e ofL, such that thefollowing hold:. • (a o b) o c = a o (b o c) (associativity). • ... axbaandaxbb. • If m is such that vi a andm b thenm a x b.

28 Aug 2008: anda distinguishedobject e ofL, such that thefollowing hold:. • (a o b) o c = a o (b o c) (associativity). • ... axbaandaxbb. • If m is such that vi a andm b thenm a x b.

23 Feb 2008: 4) Let X be a space, x X, f : S1 X a map such that x is inthe image of f, and Y = X f B. ... 1) Showthat there is an element z C such that γ(z) and x′ in C′.