Search

7 Sep 2017: On the other hand, as the signature andEuler characteristic agree modulo 2, and the latter is multiplicative, it is true thatσ(E) σ(B) σ(F) mod 2 for all such ... More recently, Hambleton,Korzeniewski, and Ranicki [5, Theorem A] proved that σ(E) σ(B)

14 Feb 2017: dt2. t=0Hn(ϕt(M)) 0. for all vector fields X Cc (B; Rn1), which includes the case of locally area minimizing hypersur-faces. ... various valuesof p (valid in fact in all dimensions) and (iii) using a technique such as Moser iteration to arriveat the

14 Feb 2017: Set. A = {ω : ω An infinitely often}, B = {ω : ω An for all sufficiently large n}. ... Calculate the probability thatm given people will all be on the committee (a) directly, (b) using the inclusion-exclusion formula.Deduce that (. nmr m. )=. mj=0. (1

1 Mar 2017: Show that the sumof the residues of g at all its poles equals zero. ... Let p(z) = z5 z. Find all z such that |z| = 1 and Im p(z) = 0.

6 Dec 2017: 4. Find all bases for which 39 is an Euler pseudoprime. ... More generally, showthat if p and 2p1 are both prime numbers, then N = p(2p1) is a pseudoprimefor precisely half of all bases.

5 Feb 2017: every z C, |f(z) b| > ε.(iii) f = u iv and |u(z)| > |v(z)| for all z C.4. ... Showthat if there exists k such that |f(z)| |z|k for all z with |z| sufficiently large, then f is arational function (i.e.

13 Oct 2017: 6. Show that R(s,t) 6(st2s1. )for all s, t > 2. ... Suppose that (i) |A| = 3 for each A A; and (ii) |AB| = 1for all distinct A, B A.

14 Feb 2017: P(|X| x) E(|X|p)xp. (b) Show that, for all β 0,P(X x) E(eβX)eβx. ... a) By optimizing the estimate of Question 2(b) over β, show that, for all x λ,.

1 May 2017: 4. Let g : Rn R be convex. Show that for all real b the set {x : g(x) b} is convex.Suppose f : Rn R is convex, and for every b ... let. φ(b) = inf{f(x) : g(x) b}.Assuming φ(b) is finite for all b, show that the function φ is convex.

10 Jan 2017: 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

1 Mar 2017: Show that the sumof the residues of g at all its poles equals zero. ... Let p(z) = z5 z. Find all z such that |z| = 1 and Im p(z) = 0.

23 Feb 2017: b) Let τ be a stopping time for the filtration (Fn)n0 Show that 1l{τn1} isFn-measurable for all n 0. ... b) For simplicity only, assume r = 0. Characterise all contingent claims withpayout Y = f(S11 ) at time 1 that can be replicated, that is for which

8 Feb 2017: every z C, |f(z) b| > ε.(iii) f = u iv and |u(z)| > |v(z)| for all z C.4. ... Showthat if there exists k such that |f(z)| |z|k for all z with |z| sufficiently large, then f is arational function (i.e.

6 Oct 2017: 6. Are all numbers in the sequence 41, 43, 47, 53, 61, 71, 83, 97,. ... 9. Find the convergents to the fraction 15290. Find all integer solutions of 152x 90y = 2.

12 Oct 2017: 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) =

23 Oct 2017: 11. Show that A = {(xn) 1 | |xn| 1/n2 for all n} is a sequentially compact subset of(1,‖‖1), but that B = {(xn) 1 | |xn| 1/n for all n} ... Hint: show that Φ(I) contains all points of the form (a/2n,b/2n), wherea,b Z, 0 a,b 2n.).

7 Dec 2017: 4. Find all bases for which 39 is an Euler pseudoprime. ... More generally, showthat if p and 2p1 are both prime numbers, then N = p(2p1) is a pseudoprimefor precisely half of all bases.

10 Jan 2017: 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.

17 Nov 2017: 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?

23 Feb 2017: i) Suppose that X N(µ,σ2). Show that. (a) E[eθXf(X)] = eµθθ2σ2/2E[f(X θσ2)] for all θ R and suitable f,(b) E[f(X)(X µ)] = ... hence show inthe context of mean-variance analysis that, if all returns are jointly Gaussian, aninvestor maximizing