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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 Oct 2007: f(z) =a(z z0) bc(z z0) d. ,. for some a,b,c,d,z0 C. ... 1 < |z| < 2}, {0 < |z| < 1}, {0 < |z| < }. 8. (i) Let R and S be some Riemann surfaces, f : R S a continuous map, and p a pointin R.

3 Jun 2007: The questions carry equal weight. This is a Mock examination, intended to give you some idea of the sortof questions you will face in the proper Part III examination. ... The function f : DR C is analytic on some disc DR = {z C : |z| < R} withR > 1 and

21 Nov 2007: a) Suppose that D1f exists and is continuous in some open ball around (a, b), and that D2f exists at (a, b). ... Show that f is differentiable at (a, b). (b) Suppose instead that D1f exists and is bounded on some open ball around (a, b), and that for

21 Nov 2007: Questions 7–10 are more challenging than others and some parts certainly go beyond limitsof the examination. ... neous cubic polynomial in the generalized Weierstrass normal form XZ24Y 3AX2Y BX3,for some complex constants A, B, A3 27B2 6= 0.

20 Oct 2007: f(z) =a(z z0) bc(z z0) d. ,. for some a,b,c,d,z0 C. ... 1 < |z| < 2}, {0 < |z| < 1}, {0 < |z| < }. 8. (i) Let R and S be some Riemann surfaces, f : R S a continuous map, and p a pointin R.

12 Oct 2007: Show that if fn f uniformly and (xm) isa sequence of points in [a, b] with xm x then fn(xn) f (x). ... Prove that there is some subinterval [a, b] of [0, 1] with a < b on which f is bounded.

5 Oct 2007: b) Use (a) to show that every finitely presented group is the fundamental groupof some space. ... Freeness of a group action is as definedin lectures; some people call this property “free and properly discontinuous”.).

9 Mar 2007: Riordan. Basic Examples: straightforward material on some of the main definitions and theorems. ... Let B̃ be the (n1)m matrix obtained by deleting some row of B.

3 Jun 2007: u(z) = b log |z| Re a(z)for some b R and some analytic function a : A C. ... a) Show that Z(T) is a subgroup of Möb.(b) Find which groups (up to isomorphism) can arise as Z(T) for some Möbius transformation T.

25 Oct 2007: An interval is a set taking one of the forms (, ), (, b), (, b], [a, ), (a, ), [a, b], [a, b), (a, b]or (a, b) for some a, b R with a b. ... 15. Let V be a real vector space with a countably infinite basis; that is to say, there is some sequence e1, e2,e3

17 Oct 2007: 3. Let a,b,c be three positive integers. Are the following true or false? ... b) Now let F = Zp be the field of integers mod p, for some prime p, and suppose d dividesp 1.

21 Nov 2007: Questions 7–10 are more challenging than others and some parts certainly go beyond limitsof the examination. ... neous cubic polynomial in the generalized Weierstrass normal form XZ24Y 3AX2Y BX3,for some complex constants A, B, A3 27B2 6= 0.

17 Oct 2007: Hence x1 = x2 = = xk1, so theresult is true for n = k 1.(b) Clearly, some positive integers are interesting: for example, 6 is the smallest perfectnumber, 1729 is the smallest number ... Suppose some positive integers were not interesting: then, by the

11 Mar 2007: ordering <a—which is initialised to <K. We use a to discard some orderedpairs. ... abbreviate to <<a. We will weed out some ordered pairs from <<a to obtain<a.

21 Nov 2007: 1. The square matrices A and B over the field F are congruent if B = P T AP for some invertible matrixP over F. ... 8. Let S be a real symmetric matrix with Sk = I for some k 1.

3 Jun 2007: a) Show that Z(T) is a subgroup of Möb.(b) Find which groups (up to isomorphism)can arise as Z(T) for some Möbius transformation T. ... Show thatevery Möbius transformation is equal to exp A for some matrix A.

3 Jun 2007: a) {0}. (b) Zω1 = {nω1 : n Z} for some ω1 C {0}. ... Suppose that we did,then ω = xω1 yω2 for some x, y R.

5 Feb 2007: 7) Let R be a commutative ring with unit. For a,b R we say that a divides b (notation a|b) ifb = ac for some c R. ... If R is a domain, then a b (a|b b|a). We call an elementa RR irreducible if for any factorization a = bc one of b,c is a unit

3 Jun 2007: B |G(z)| = |F (z)| at some z R {zo} if, and only if, F = ωG for some complex number ω of unitmodulus. ... Hence Gw = T Gzo for some T Aut D. E G is injective.

3 Jun 2007: respectively. Any 1-form ω on can be written as adx bdy or λdz µdz for some 0-forms a,b,λ,µ.In particular, for f : C we have. ... The 1-form on D can be written as = adz bdzfor some functions a,b : D C.

3 Feb 2007: e =1. c2i(fh hf). To prove this proposition we need some preparations. ... Math. IHES 78 (1993) 163–185.[14] V. Guillemin, D. Kazhdan, Some inverse spectral results for negatively curved 2-manifolds,.

22 Jul 2007: There is some pointto this question, as the following example will make clear. ... We can alsoprove TI 6 TJ 6 TI as follows. Since I 6 J there is some axiom B in I which is not in J.

26 Nov 2007: In 6.2we give some algorithms for computing flex matrices, starting fromequations for C. ... Let m and n be coprime integers. Suppose thatn a2 (mod m) and m b2 (mod n) for some integers a and b.Then there is a k-isomorphism of

5 Apr 2007: First note the following. Lemma 5.1. T is open iff T (B(1)) B(ǫ) for some ǫ > 0.Proof. ... Since by Theorem 5.3, we have. T (B(1)) B(δ),for some δ > 0, it follows that.

18 Sep 2007: n1)(c′). (n 1)!(b a)n1. for some c′ with a < c′ < b. ... b. f (x) dx a. b. g(x) dx. 13 Some properties of the integral.

20 Apr 2007: If α has a diagonal matrix with respect to some basis we say that α isdiagonalisable. ... 9The reader is warned that some mathematicians reverse this convention and call xHa left coset.

19 May 2007: The next lemma gives us some information under weaker hypotheses than those of. ... with some probability p, whereas a graph with the weaker intersections property looks like.

27 Jun 2007: In this case, we cannotrecover g from dg up to isometry, unless some additional assumptions are imposedon g, see, e.g., [9]. ... dif-feomorphism f : M M, fixing M, and some function ϕ vanishing on M.Linearize this near f = Id and ϕ = 0.