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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.

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 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

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