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29 Jan 2010: Suppose that f ((x y)/2) (f (x) f (y))/2 for all x, y [a, b].Prove that f is continuous on (a, b). ... Which of (1)–(4) must betrue? (1) If f is increasing then f ′(x) 0 for all x (a, b).(2) If f ′(x) 0 for all x (a, b) then

2 Feb 2010: Suppose that f ((x y)/2) (f (x) f (y))/2 for all x, y [a, b].Prove that f is continuous on (a, b). ... Which of (1)–(4) must betrue? (1) If f is increasing then f ′(x) 0 for all x (a, b).(2) If f ′(x) 0 for all x (a, b) then

18 Mar 2010: The questions are not all equally difficult. I welcome both comments and corrections which can be sent to m.hyland@dpmms.cam.ac.uk. ... Show that all roots lie inside the circle|z| = 3/2. (b) How many zeros does z4 12z 1 have in the annulus 2 < |z| < 3?

14 Oct 2010: Suppose that (i) |A| = 3 for each A A; and (ii) |A B| = 1for all distinct A, B A. ... Showthat there must be a rectangle, with sides parallel to the coordinate axes, all four of whose vertices are thesame colour.

22 Nov 2010: 3. Write down sentences (in the language of ZF) to express the assertions that, for any twosets A and B, the product A B and the set of all functions from ... 8. What is the cardinality of the set of all continuous functions from R to R?

16 May 2010: 3. Write down sentences (in the language of ZF) to express the assertions that, for any twosets A and B, the product A B and the set of all functions from ... 8. What is the cardinality of the set of all continuous functions from R to R?

3 Nov 2010: f (x)| < 2ǫ for all sufficiently large n. (11) Calculate the first five Chebychev polynomials. ... supx[a,b]. |xn p(x)| supx[a,b]. |xn q(x)|. for all polynomials q of degree n 1.

4 Feb 2010: 5. Find all of the Möbius transformations that commute with Mk for a fixed k. ... Are these all inversions? 9. How many square roots of a Möbius transformation are there?

30 Mar 2010: 1 Recall the character table of S4 from Sheet 2. Find all the characters of S5 induced fromthe irreducible characters of S4. ... 7 Let ρ : G GL(V ) be a representation of G of dimension d.(a) Compute the dimension of SnV and ΛnV for all n.(b) Let g G

1 Nov 2010: Is it Lipschitz equivalent. to the uniform norm? (b) Let R[0, 1] denote the vector space of all integrable functions on [0, 1]. ... Usingthe fact that all norms on a finite-dimensional space are Lipschitz equivalent, deduce that α iscontinuous.

21 Oct 2010: 3. Which of these are true for all natural numbersa , b and c? ... 6. In the sequence 41, 43, 47, 53, 61,. , each difference is two more than the previous one.Are all the numbers in the sequence prime?

18 Nov 2010: 1+. xnfor all n 1. Show. that (xn)n=1 converges, and determine its limit. ... 14. Is there an uncountable family S PN such that A B is finite for all distinct A, B S?

7 Feb 2010: 10. (i) Show that the set P(S) of all subsets of a given set S is a ring with respect to the operations ofsymmetric difference and intersection - the power-set ... 12. Let $ be a set of prime numbers. Write Z$ for the collection of all rationals m/n (in

19 Nov 2010: Deducethat the set GLn(R) of all invertible nn real matrices is an open subset of Mn(R). ... some r > 0 and a continuous functiong : B(I; r) Mn such that g(X)2 = X for all X B(I; r).Is it possible to define a square-root

26 Nov 2010: 3. (i) Show that the function ψ(A,B) = tr(ABT ) is a symmetric positive definite bilinear form on the spaceMatn(R) of all n n real matrices. ... sn forms a basis for Pn.(iii) For all 1 k n, sk spans the orthogonal complement of Pk1 in Pk.(iv) sk is an

27 Oct 2010: Show also that rk rk1 rk1 rk2. [Consider the restriction. of α to Im(αk).] Deduce that if, for some k 0, we have rk = rk1, then rk = rk+ for all ... Note that the sum of all columns of A has all entries equal.].

15 Oct 2010: 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 ... i Ui. Give an example where Ui Uj = {0} for all i 6= j, yet U1.

29 Apr 2010: f) Define a subset of the integers Z to be open either if it is empty or if for some k Zthe set S contains all integers k. ... b) Show f : X Y is continuous if and only if for all A X, f (cl(A)) cl(f (A)).Deduce that if f is surjective, the continuous

29 Jan 2010: Write µw = (µ2,µ1).The normalised1 induced representation is then. IndGB (µ) =. {. f : G k locally constant s.t.f(bg) = µ(b)δ(b)1/2f(g) for all b ... f(bg) = χ(b1)δf (b)f(g)for all g Gf and b =. (. b1 0 b2. ). Bf. . . . Then I(χ) is an

17 Dec 2010: 2.2. Properties of B. Express the categorical properties of M and hence P, discussed in Section 1, in terms of additional structure, and these properties can all be described in ... 0. The first step is to extend this result to all maps between separated