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23 Mar 2007: Fix some subset B of [n]. How large a family A of subsets of [n] can we find such that the intersection of any two sets in A contains a cyclic ... Chung, Graham, Frankl and Shearer have proved that, in the case where B is a block of length t, we can do

20 Nov 2007: p’ and ‘q’ beingliterals—even tho’ A XOR B is not self-dual in general.). ... Show—by considering disjunctive normal forms or otherwise—that whenever A is a self-dual formula there is a formulaB such that A is logically equivalent to B B̂.

20 Jan 2007: Show that the curve joining A and B with length and such that together with AB bounds the largest possible area is an arc of a circle passingthrough A and B. ... Compute the Gaussian curvature and the mean curvature. 10. (i) Determine an equation for the

21 Nov 2007: 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 1.

12 Oct 2007: 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 ... b) Give an example of a sequence of uniformly continuous, real-valued functions (fn) on R such thatfn

12 Mar 2007: b) Explicitly construct such a representation ρ : SU2 SO3(R) by showing that SU2acts on the vector space of matrices of the form. ... by conjugation. Show that this subspace is isomorphic to R3, that (A,B) 7 tr(AB)is a positive definite non-degenerate

2 Mar 2007: Prove that there exists such a reparametrization Γ̃ = Γffor which ‖dΓ̃/dt‖ is constant, namely l. ... 9) 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

23 Feb 2007: Show that for anyb C, there exists a sequence of pointszn D(a, R) with zn 6= a such thatzn a andf (zn) b asn. ... 16 Let f be a meromorphic function onC such thatf (1/z) is also meromorphic.

16 Nov 2007: Can we find such a bijection f which isorder-preserving (that is, satisfies f(x) < f(y) whenever x < y)? ... Let S be a collection of subsets of N such that for every A, B S we have eitherA B or B A.

17 Oct 2007: But such a numberwould clearly be very interesting indeed. Therefore, all positive integers are interesting. ... 12. Find a positive integer k such that, for all positive integers n, the number n4 k iscomposite.

5 Feb 2007: b) Let N Zn be a submodule of rank n. Show that there is a matrix A Mn(Z) such thatA(Zn) = N and the index [Zn : N] of N in ... 8) An integral domain R is called a euclidian domain if there is a map N : R{0} Z>0 such thatfor all a,b R, b 6= 0, there are

15 Mar 2007: Assume there is θ OK such that 1,θ,. ,θn1 isan integral basis of OK. ... dK| (nn. n! )2 (π4. )n. Deduce that |dK| > 1 for every number field K 6= Q.(b) Show that there are constants A > 1, c > 1, such that for every

21 Nov 2007: If A is the matrix of the first of these (say), find a non-singular matrix P such that P T AP is diagonalwith entries 1. ... Show that r(P Q) is odd. 19. Let a1, a2,. , an be real numbers such that a1 an = 0 and a21 a2n = 1.

11 Mar 2007: Then there is asubset B A of size β such that all the unordered γ-tuples from it are in thesame piece of the partition. ... an1, ν〉 such that 〈a1. an1, ν, c〉 and 〈a1. an1, ν, c′〉 are coloureddifferently by.

7 Nov 2007: 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),.

8 May 2007: b) Give a topology on R which is neither trivial nor discrete such that every open set isclosed (and vice-versa). ... b) (x, y, z) (x′, y′, z′) x = x′, y = y′, z = 1 z′.

14 May 2007: Is there always an orientable such surface, i.e. one in which“turning clockwise” makes sense globally on the surface? ... b) What is the smallest genus (number of holes) of such a surface with boundary thetrefoil knot, which is the knot which lies on

29 Mar 2007: The intuitionistic correctness of wffs such as. 4. A (B C). ... B (A C) reveals an infinite family of terms such as.

18 Sep 2007: Then we can find an such that. f(z) =. n=. an(z b)n. ... i) The point b is a removable singularity if and only if we can find ananalytic function f̃ : C such that f(z) = f̃(z) for all z {b}.

18 Sep 2007: a) = g(b), then we can find a c (a, b)such that g′(c) = 0. ... Then there is a unique solution to the differential equationg′(t) = f (t) [t (a, b)] such that g(u) = c.