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13 Dec 2017: ALGEBRAIC TOPOLOGY (PART III) MICHAELMAS 2017. EXAMPLE SHEET 4. 1. Suppose π : E B is a real vector bundle and that f : B′ B. ... Show that E|B0 'E|B1 as follows. (a) For each b B show there is an open neighborhood Vb of b such that E|VbI istrivial.

30 Nov 2017: b) Deduce that χ(K) =. i0(1)i#{i-simplices of K}. 6. Let p : X̃ X be a finite-sheeted covering space, and h : |K| X a triangulation.Show that ... Show that the linear map A : R2 R2preserves the equivalence relation (a,b) (a′,b′) (a a′,b b′) Z2,

22 Oct 2017: Hint: Bolzano-Weierstrass.). 3. If A and B are subsets of Rn, let A B = {a b |a A,b B}. ... a) f(x) = sin x2 (b) f(x) = inf{|xn2| |n Z} (c) (sin x3)/(x 1).

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 ... 1.8. Let (E,E,µ) be a measure space. Call a subset N E null if N B for some B E withµ(B) = 0.

1 May 2017: 5. Consider the following problems:. (a) minimise c>x subject to Ax = b;. ... Let P() be the linear programming problem when the right-hand side b = (2, 5, 6)> is.

17 Nov 2017: 6. Let f : X Y. Let A, B X and C, D Y. ... For each of claims (a)–(f) below, give aproof or counter-example. (a) f(AB) = f(A)f(B) (b) f1(C D) = f1(C)f1(D)(c) f(AB) =

20 Jan 2017: likelihood estimate of θ is (2a b)/(b 2c). 4. (a) Let X1,. ... b) For some n > 2, let X1,. ,Xn be iid with Xi Exponential(θ).

23 Oct 2017: Hint: Bolzano-Weierstrass.). 3. If A and B are subsets of Rn, let A B = {a b |a A,b B}. ... a) f(x) = sin x2 (b) f(x) = inf{|xn2| |n Z} (c) (sin x3)/(x 1).

27 Jun 2017: For A, B £ co we write A s B iff ABe Fin. ... 1.1. PROPOSITION. K0 < cf(t) = t < cf(b) = b < cf(d) < d < 2Ko.

8 Nov 2017: P. α(P)). induces a map on H1’s, and so by the Kummer isomorphism gives us a grouphomomorphism u : A/(A)p B/(B)p. ... that we may solve the original norm equation. Theorem 3.7. If a,b B then Algorithm 3.6 takes O((logB)2) iterations.

11 Sep 2017: We will abuse notation and simply write x b tomean σ(x)(b) if x L and b B. ... b) Let ξ M[f]k Vk Uk map to zero in Mk.