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25 Jun 2015: Foreach open set A Ψ(U )cs and each M A, we shall show that there is a smalleropen neighbourhood M B A Ψ(U )cs such that. ... Then we may choose smooth functions ϕi : M [0, 1] such that supp(ϕi) int(Vi) and p(ν K) (. ϕi)1(1) and choose B =. B(A, ϕ1

9 Feb 2015: 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.

20 Nov 2015: If A is the matrix of the first of these (say), find a non-singular matrix P such that PTAP is diagonalwith entries 1. ... 14. Let a1, a2,. , an be real numbers such that a1 an = 0 and a21 a2n = 1.

28 Aug 2015: p. k, where k is the smallest value ofn such that xn 6= yn. ... Hint:first find a subsequence (fni ) such that fni (x) converges for all x Q [0, 1].)Deduce that the closure of B in (C[0, 1], d) is compact.

26 Nov 2015: 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 12e(G).

27 Jan 2015: a). n=1. nenz on {z. 0 < r Re(z)}; (b) n=1. ... Can such a map ever be a Möbius trans-formation? 9 Calculateγ z sin z dz when γ is the straight line joining 0 to i.

26 Oct 2015: fn for V such that. ψ.  mi=1. xiei,. nj=1. yjfj. ... Suppose thatα: V W is a linear map such that α(Y ) Z.

11 Feb 2015: If X is a space such that H(X; R) = H(Sn; R), show thatthere is a map f : Sn X inducing an isomorphism on H(; R), and hence thatπi(X) ... Hint: a) Show that such rings have Tor(,R) = 0; b)Consider the class of abelian groups A such that AR = 0.].

9 Feb 2015: That is, for everya < b and every t there should exist x (a,b) such that f(x) = t. ... 14. Let f : R R be a function that has the intermediate value property: that is, iff(a) < c < f(b) then there exists x (a,b) such that f(x) =

21 Oct 2015: a) Let a G. Show that there is a positive integer n such that an = e, the identity element.(The least such positive n is the order of a.). ... b) Show that there exists a positive integer n such that an = e for all a G.(The least such positive n is the

2 Nov 2015: Fix j, let > 0 and choose a partition P = {a = a0 < a1 <. < an = b} such thatU(P,f) L(P,f) < j1. Let K = {k : Ej (ak,ak1) 6= }. Then Ej ... an = b} be any partition of [a,b] such thataj1 aj < δ, and let J = {j : [aj,aj1] F 6= }. Show that sup[aj,aj1] f

19 Feb 2015: Deducethe more general result that if f ′(a) 6= f ′(b) and z lies between f ′(a) and f ′(b), then thereexists c (a,b) such that f ′(c) = z. ... f(a) < 0 and f(b) > 0 then there exists c (a,b)such that f(c) = 0.

16 Oct 2015: 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 ... a) Must there exist a sequence (fn) of continuous functions on [0, 1] such that fn f uniformly on [0, 1]?(b)?

10 Feb 2015: 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.

20 Nov 2015: Can S be uncountable?Is there an uncountable family T PN such that AB is finite for all distinct A, B T? ... You are asked to supply them with a strategy such that,should they follow it, only finitely many of them will guess wrongly.

15 Oct 2015: 1. Calculate d = (a,b) and find integers r and s such that ra sb = d when. ... iii) Find constants c and d such that λ(a,b) 6 c log b d.

4 Mar 2015: Prove that f is. integrable. What is 10f(x)dx? 6. Let a < b and let f : [a,b] R be a Riemann integrable function such that f(x) 0for every ... 1, 1], such that f(x) = g(x) h(x) for every x.

13 Nov 2015: every polynomial p of degree n there is x0 [0, 1/n]such that |p(x)| C|p(x0)| for every x [0, 1].(b) If (X,d) is a metric ... That is to say, showthat there is no metric d on the set of continuous functions f : [a,b] R such that pointwise convergenceof

14 Oct 2015: algebra. 1.2. Show that the following sets of subsets of R all generate the same σ-algebra:(a) {(a,b) : a < b}, (b) {(a,b] : a < b}, (c) {(,b] : b ... an,bn] such that the Lebesgue measure of A4B(= (Ac B) (ABc)) is less than ε.

30 Nov 2015: Does it follow that thereexists c (a,b) such that f(b) f(a) = f′(c)(ba)? ... containing (x0,y0) and V containing F(x0,y0) = (x0, 0) such that F maps U bijectively to.

28 Oct 2015: has no solution with a, b Z. 14. Construct a function f : R R that takes every value on every interval — in other words,for every a < b and every c ... there is an x with a < x < b such that f(x) = c.

7 Mar 2015: n) is the output from a LFSR of length d. (c) The period of (xn)n>0 is the least integer r > 1 such that xnr = xn for all sufficiently largen. ... a) Show that N ϕ(N) < 3N. (b) Let k = (de 1)/ϕ(N).

26 Feb 2015: 7. Let R be an integral domain. The highest common factor (hcf) of non-zero elements a and b in R is anelement d in R such that d divides both ... ii) Show that can be identified with the set Fq {} in such a way that GL2(Fq) acts on as thegroup of

10 Feb 2015: a). n=1. nenz on {z. 0 < r Re(z)}; (b) n=1. ... Can such a map ever be a Möbius trans-formation? 9 Calculateγ z sin z dz when γ is the straight line joining 0 to i.

23 Jan 2015: b) The model is to be reparametrised in such a way that it can be specialised to (i)two parallel lines for the two groups, (ii) two lines with the same ... Z21 Z2np,. where the Zi are independent and Zi N(λi,σ2) for some λi such that.

23 Nov 2015: If A is the matrix of the first of these (say), find a non-singular matrix P such that PTAP is diagonalwith entries 1. ... 14. Let a1, a2,. , an be real numbers such that a1 an = 0 and a21 a2n = 1.

19 Jan 2015: to V4 V3; in this and the followingquestions, you are not being asked to find such an isomorphism explicitly). ... Here O(n) denotesthe subgroup of GLn(R) preserving the standard (positive definite) symmetric bilinear form.(b) Explicitly construct such a

31 Aug 2015: p. k, where k is the smallest value ofn such that xn 6= yn. ... Hint:first find a subsequence (fni ) such that fni (x) converges for all x Q [0, 1].)Deduce that the closure of B in (C[0, 1], d) is compact.

26 Oct 2015: fn for V such that. ψ.  mi=1. xiei,. nj=1. yjfj. ... Suppose thatα: V W is a linear map such that α(Y ) Z.

27 Feb 2015: Does there exist c (a,b)such tat f(b) f(a) = f′(c)(ba)?(v) Let f: Rn Rn be a C1 map with ‖Df(x)I‖ 1/2 for ... continuous function g: Bε(I) Mn such that g(A)2 = A for all A Bε(I).

9 Jan 2015: The measure ν can be written explicitly as follows. There is an integer b 1 such that ν(f) =S1fF dµ, where F(z) = 1 2. ... Let b denote the smallest value of n such that F.

18 Feb 2015: ri 1 such thati ri = r. For example, in the quaternionic case H = B the plectic reflex Galois group isisomorphic to (St n ΓtF ) (Srt n Γ. ... The cup productof two such classes is an element g1 g2 H2(Y, 2).

15 Sep 2015: kn) Zn such that Q(k1,. ,kn) = 0. We wish to consider thequestion: what is the probability that a random integral quadratic form in n variables is isotropic? ... For anyε > 0, we may find a nice distribution Dε that is a finite weighted average of box

7 Aug 2015: B/A. such that for any Banach B-module M there is a natural isomorphism. ... Then thereis a unique A-linear map. ψ : DerK(A) DerK(B). such that ψ(u) ϕ = ϕu for each u DerK(A).

20 May 2015: Let A be an nn matrix over F. The following are equivalent(i) there is a matrix B such that BA = In;. ... and we see easily that det B =. det Âij as required. Definition. Let A Matn(F). The adjugate matrix adj A is the element ofMatn(F) such that.

10 Dec 2015: c) Let Bsu be the variety of Borel subgroups B Ĝ such that s B and u B. ... a HB,K-module M V B such that the naturalmap M K Ql V B is an isomorphism.

2 Dec 2015: Let A be an nn matrix over F. The following are equivalent(a) there is a matrix B such that BA = In;(b) there is a matrix C such that AC = ... Thus (a) holds if and only if α is anisomorphism. Similarly (b) holds if and only if there exists γ such that

22 Sep 2015: If a < t < b, then t is a local maximum if and only if we canfind a u > 0 with u t a, b t such that f(s) f(t) ... iii) We say that f has a local minimum at s with a s b if wecan find a u > 0 such that, whenever a t b and |t s| u,

3 Apr 2015: Choosean element x R such that ρ(x) = 0. Then x = x, hence yx = yx for all y R. ... We fix for each such Q a choice of quaternion algebra BQover F, as follows:. •

10 Mar 2015: Hence such errors can bedetected. How do check digits work in ISBN-13? ... It is easy to check that each such sum has at least3 non-zero bits.