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29 Jan 2021: 8. Let t1, t2,. be propositions such that, for every valuation v, there exists n withv(tn) = 1. ... Show that there is a proposition b, inwhich the only primitive propositions appearing are those that appear in both a and c,such that a b and b c.

19 Jan 2021: Show that φis isotopic to the identity. 8. Let α,β be a pair of simple closed curves on a surface S, such thati(α,β) = 1. ... Let α,β be astandard pair of simple closed curves on S such that i(α,β) = 1, andlet γ be the boundary curve.

29 Jan 2021: 8. Let t1, t2,. be propositions such that, for every valuation v, there exists n withv(tn) = 1. ... Show that there is a proposition b, inwhich the only primitive propositions appearing are those that appear in both a and c,such that a b and b c.

15 Oct 2021: b) Suppose that v V , with components vi and ω V with components ωj. ... Please send any corrections to c.warnick@warwick.ac.uk. 2. b) Fix x U and suppose that V ,W TxU are two vectors such that for any f

15 Oct 2021: Exercise 3.1. a) Suppose that {j. }1j=1 E () is a sequence such that j! ... f 2 L1(Rn). Fix > 0. Show thatthere exists f. 2 C10(Rn) such that.

15 Oct 2021: Define τ by:. U τ for all x U, there exists B β such that x B and B U. ... b) Fix a local base β̇. Show that a sequence (xn)n=1 is τ-Cauchy if and only if for anyB β̇ we can find an integer N such that.

23 Nov 2021: 13. Let S be a collection of subsets of N such that for every A,B S we have A B or B A.Can S be uncountable? ... Is there an uncountable collection T of subsets of N such that A B isfinite for all distinct A,B T?

26 Oct 2021: of B such thatπ1(Uα) is a disjoint union. ... H(B) H(E) such that π π! is multiplication by N.

16 Mar 2021: 7. Let R Q be a ring. If X is a 1-connected space such that H(X; R) = H(Sn; R),show that there is a map f : Sn X ... Hint: a) Show that such rings have Tor(,R) = 0;b) Consider the class of abelian groups A such that AR = 0.].

19 Jan 2021: i) Prove that fm is a group homomorphism. (ii) Let a,b Z be such that ab 1 modulo m. ... Deduce thatthere exists r > 0 such that. {n such that p | 2n 1} = rZ.

12 Oct 2021: The least such n iscalled the order of g.). (b) Show that there exists a positive integer n such that gn = e for all g G. ... 10. Let H and G be groups and let X G such that 〈X〉 = G.

29 Oct 2021: 2. Given (fi)iI an arbitrary collection of continuous functions [0, 1] R such thatsupiI |fi(x)| < at each x [0, 1], show that there is an interval [a,b] [0, ... 1] witha < b such that supx[a,b] supiI |fi(x)| <. 3. Let X be a closed subspace of 1. Assume

25 Nov 2021: Prove that A. contains an infinite set B such that no point of B is a convex combination of other points of B.

14 Oct 2021: 1. Calculate d = (a,b) and find integers r and s such that ra sb = d when. ... iii) Find constants c and d such that λ(a,b) 6 c log b d.

15 Oct 2021: 1t t > 0. a) Show that χ C(R). b) Show that there exists a function ψ C0 (Rn) such that. ... c) Give an example of a sequence {φj}j=1 C0 (Rn) such that.

8 Oct 2021: Make this assumption for the rest of the problem.(b) Why must there exist an equivalent measure Q such that 1. ... b) Show that. E(X1|Sn) =Snn. 9. Find three random variables M1,M2,M3 such that.

15 Oct 2021: Questions marked () are optional. 2. b) For all x 2 A, for all i 2 N there exists > 0 such that:. ... R such that f(x) = ˜f(x) for all x 2 (a, b).

8 Oct 2021: X. (ii) Cov(Y,f(X)) = Cov(X,Y )E[f ′(X)] for suitable f.(ii) Let Q be the equivalent probability measure such that dQ. ... 7. A contingent claim with time-1 payout Y is called attainable iff there exists a scalar aand portfolio b Rd such that Y = a b>S1

19 Jan 2021: for every x X there exists some B B such that x B; and. ... Leta,b G. Suppose that X has non-empty disjoint subsets U and V such that:.

29 Nov 2021: 1. Let T B(H) with |||T ||| < 1. Show that there is R B(H) such that R2 = 1 T. ... 11. Let T B(B) compact and λ 6 (σp(T) {0}). Show that T λ is bounded below.