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21 Oct 2015: 3. Show that the set of functions on R of the form f(x) = ax b, where a and b are realnumbers and a 6= 0, forms a group under composition ... b) Show that there exists a positive integer n such that an = e for all a G.(The least such positive n is the

25 Feb 2015: that 〈Sq(x) y, [M]〉 = 〈x Sq1(y) v, [M]〉 for all x,y H(M). ... i) Show that Sqi : Hni(M) Hn(M) = F2 is zero for all i > 0.

12 Nov 2015: b) All words w {a,b,c} consisting of some number of a’s (possiblynone), followed by some number of b’s (at least one), followed bysome number of c’s ... c) All words w {0, 1} which contain a 1 somewhere in the last 5positions.

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 ... Show that. (a) C is uncountable and has Lebesgue measure 0,(b) for all x [0, 1], the limit F(x) =

30 Nov 2015: for some K and all t [a,b], x,y BR(x0).We showed in lecture that for each t0 [a,b], there is a unique f C([a,b]; ... tt0F(s,f(s)) ds, t [a,b]. Assuming that F. extends to all of [a,b]Rn as a continuous function, show that this f is in fact the

16 Feb 2015: Part II Algebraic Geometry. Example Sheet III, 2015(For all questions, assume k is algebraically closed. ... 22 x. 33 k[x0,. ,x3]. Find the dimension of the tangent space at all the.

4 Mar 2015: In particular,. prove that L(ab) = L(a) L(b) for all positive a and b. ... 2)n cos(θx)dx. Prove that θ2In =. 2n(2n1)In1 4n(n1)In2 for all n 2, and hence that θ2n1In(θ) = n!(Pn(θ) sin θ Qn(θ) cos

5 Feb 2015: Find an optimal binarycode. Determine whether there are optimal binary codes with either (a) all but one codeword ofthe same length, or (b) each codeword a different length. ... Show that in the case where all of the odds are equal this maximum andthe

4 Feb 2015: Find an optimal binarycode. Determine whether there are optimal binary codes with either (a) all but one codeword ofthe same length, or (b) each codeword a different length. ... Show that in the case where all of the odds are equal this maximum andthe

19 Jan 2015: b) Find all 1-dimensional representations of G.(c) Let ψ : Fp C be a non-trivial 1-dimensional representation of the cyclic group. ... d) Prove that the collection of representations constructed in (b) and (c) gives a com-plete list of all irreducible

20 Nov 2015: An 6= holds for all n. Must it bethat. n=1 An 6=? ... Can S be uncountable?Is there an uncountable family T PN such that AB is finite for all distinct A, B T?

2 Nov 2015: 1. Quickies: (a) Describe all continuous functions f : [0, 1] Rn satisfying ‖ 10f‖ =. 10‖f‖. (b) Show that two norms ‖ ‖, ‖ ‖′ on a vector space V are Lipschitz equivalent if and only ... denote the vector space of all bounded Riemann

23 Feb 2015: Wii =1. aiV (µi){g′(µi)}2,. (you need not specify i(σ2), and you may assume /βjσ2 = /σ2βj for all j). ... c) How do the expressions in (a) and (b) simplify when g(µi) is the canonical linkfunction?

19 Jan 2015: 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 and let ... constituent of IndGKχ, then all |G : K| conjugates of χ are constituents of ResGKφ.

10 Feb 2015: Show that f is a polynomial, of degree k, if and only if there is a constantM for which |f(z)| < M(1 |z|)k for all z.(ii) Show that ... Show that if there existsk such that |f(z)| |z|k for all z with |z| sufficiently large, then f is a rational function

23 Nov 2015: 3. (i) Show that the function ψ(A,B) = tr(ABT ) is a symmetric positive definite bilinear form on the spaceMatn(R) of all nn 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

31 Aug 2015: for every x X, there is an open neighborhood U of x such thatf(y) = f(x) for all y U. ... 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.

27 Apr 2015: Show that f is continuous ifand only if f(A) f(A) for all A X. ... Show thatthis is a topology, that all points of R are closed with respect to it, but thatthe topology is not Hausdorff.

15 Oct 2015: a) The set C of continuous functions. (b) The set {f C : |f(t)| 1 for all t [0, 1]}.(c) The set {f C : f(t) 0 as t }.(d) ... i Ui. Give an example where Ui Uj = {0} for all i 6= j, yet U1.

27 Feb 2015: continuous function g: Bε(I) Mn such that g(A)2 = A for all A Bε(I). ... b) = inf (γ), wherethe inf is taken over all continuous γ: [0, 1] E with γ(0) = a, γ(1) = b.