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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.

22 Oct 2015: Proof. Suppose it is: χA is computably total. Find a, b N such thatGraph({a}) A and Graph({b}) / A. ... Obtain computable g such that:. g(n) = if Graph({n}) A then b else a.

7 Feb 2015: Must there also be a unique maximal (akamaximum) S R such that S is transitive? ... And must there be a uniquemaximal (maximum) S R such that S is symmetrical?

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).

11 Jul 2015: not concerned about the axiom of choice, this is equivalent to requiring thatfor every a A there is an element b B such that a Q b.) The second is to ... Suppose b1 = sat(X1) andb2 = sat(X2) are members of B such that b1 C b2.

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