Search

22 Dec 2023: B.A. in Mathematics, 2008- 2011. • First class all three years.

8 Nov 2023: a) Compute dim SnV and dim ΛnV for all n. (b) Let g G and λ1,. , ... even. 7. Find all the characters of S5 obtained by inducing irreducible representations of S4.

4 Jun 2024: A sequence an iscalled subadditive if amn am an for all m,n. ... d(α(t),β(t)) 2δ d(α(L),β(L)). for all t [0,L]. 7. Let X be a proper metric space (i.e.

26 Feb 2024: a) Let L be a transverse collection of n lines in the hyperbolic plane, with n > 1.Then, there exists a number R = R(L) > 0 such any line intersecting all ... b) For each k > 0, there are finitely many π1Sg-orbits of transverse collections of klines in L

11 Jan 2024: 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 (. n mr m. )=. mj=0.

10 Apr 2024: The argument. All spaces of embeddings and so on are given by (realisation of)simplicial sets. ... Embt(D2, D4) Immt(D. 2, D4). is cartesian by [Las76, (a) and (b)] (Lashof supposes in this paper that the ambientdimension is 5, but for this particular

4 Jun 2024: homology spheres discussed in lectures. (a) Show that z is central in Gn, for all n. ... F({a1,. ,an}). Prove that F2 contains subgroupsisomorphic to Fn for all n 2.

24 Jan 2024: Which of (1)–(4)must be true? (1) If f is increasing then f ′(x) > 0 for all x (a,b).(2) If f ′(x) > 0 for all x (a,b) ... then f is increasing.(3) If f is strictly increasing then f ′(x) > 0 for all x (a,b).(4) If f ′(x) > 0 for all x (a,b)

11 Jan 2024: a) Show that, for all p (0, ) and all x (0, ),. P(|X| x) E(|X|p)xp. (b) Show that, for all β 0,P(X x) E(eβX)eβx. ... 3. Let X be a Poisson random variable of parameter λ (0, ). (a) By optimizing the estimate of Question 2(b) over β, show that, for

2 Jan 2024: Dic(Rd) 6' for all d 4. What is it then?2. Goal. ... 43d. Weiss ’22. For all d 0 and certain i > b(d 1)/2c there are.

19 Oct 2023: Find all invariant distributions of the transition matrix in Exercise 9 of Example Sheet 1. ... i < b, where 0 < λi, µi < 1 for all i, and λi µi = 1 for 1 i < b.Show that this process is time-reversible in equilibrium.

8 Feb 2024: Show that f is a polynomial, of degree k, if and only if there is aconstant M for which |f(z)| < M(1 |z|)k for all z. ... Show that if thereexists k such that |f(z)| |z|k for all z with |z| sufficiently large, then f is a rational function (i.e.

8 Feb 2024: Show that the sum of the residuesof f at all its poles equals zero. ... 6. Let p(z) = z5 z. Find all z such that |z| = 1 and Im p(z) = 0.

3 Nov 2023: a) Compute dim SnV and dim ΛnV for all n. (b) Let g G and λ1,. , ... even. 7. Find all the characters of S5 obtained by inducing irreducible representations of S4.

16 Nov 2023: 7. Find an injection from R2 to R. Is there an injection from the set of all real sequences to R? ... Is there an uncountable collection T of subsets of N such that A B isfinite for all distinct A,B T?

25 Nov 2023: 13. Show that T B(H) is positive if and only if 〈Tx,x〉> 0 for all x H. ... L(ab) = L(a)b, R(ab) = aR(b) and R(a)b = aL(b). Let M(A) be the set of all double centralizers for A.

12 Oct 2023: Write down its negation insymbolic form. (Here m, n, a, b should be understood as ranging over all natural numbers.). ... i) 1n = n 1 for all n;(ii) m1 = (m1)2 for all m > 1;.

11 Oct 2023: 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) =

20 Feb 2024: 𝑓 𝑗 (𝑥)𝑑𝑥 𝐸. 𝑓 (𝑥)𝑑𝑥. for all bounded measurable sets 𝐸 R𝑛 of positive measure. ... i) Show that 𝑥𝑖 𝑥 if and only if (𝑦,𝑥𝑖) (𝑦,𝑥) for all 𝑦 𝐻.

10 Oct 2023: Suppose thatF(,σ2) is increasing for all σ2 and F(m, ) is decreasing for all m. ... ψ(m,σ) = E[U(m σZ)].Show that(a) if U is increasing, then ψ(,σ) is increasing for all σ(b) if U is concave, then ψ concave, that