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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.

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.

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.

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.

19 Jan 2021: Show that E has a subgroup isomorphic toE/M, and show that all such subgroups are conjugate. ... such that iApA iBpB = idM , pAiA = idA, pBiB = idB, pAiB = 0 andpBiA = 0.

15 Oct 2021: x = 0 then there exist constants s > 0, C > 0 and > 0 such that:. ... B to mean that there exists a universal constant C such that A  CB.

18 Oct 2021: In each case, describe the subsetof maximal ideals. 2. Given an example of a homomorphism of rings ϕ : A B such that the preimage of a maximal ideal is notmaximal. ... the topological space consisting of two points{p1,p2} such that {p1} is dense and {p2}

15 Oct 2021: b) Show that f is smooth and periodic. c) Show that there exists a constant cn depending only on n such that:. ... 2q. ) such that 0 andX. g2g = 1. b) Show that if u 2 D0(Rn) is periodic and , 0 are both as in part a), then.

15 Mar 2021: 1. |B|vB. d(v) > (α ε)|A|. Show there is B′ B with |B′| > ε|B| so that d(v) > α|A|, for all v B′. ... Prove that A. contains an infinite set B such that no point of B is a convex combination of other points of B.

30 Nov 2021: ϕα : E|Uα := π1(Uα) Uα M,. such that each. ϕβ ϕ1α : (Uα Uβ) M (Uα Uβ) M. ... π. : Hid(E) Ei,d Ei,d2 = Hi(B; b 7 Hd(π1(b))) = Hi(B).

18 Feb 2021: criterion. Let F : xCO Ñ Sets be a continuous functor such that Fpkq is a singleton.For A, B, C P xCO and maps α : A Ñ B and β : B ... It will be enough to find a P kerpGLnpB Iq Ñ GLnpkqq such thataρpΓqa1 GLnpBq, so we can assume that B I A.Now

30 Oct 2021: Then there exists ε > 0 such the for any t0 [a,b] the nth-order IVP (initial valueproblem). ... By Lindelöf–Picard (Theorem 3.10 in the lectures), there exists ε > 0 such that theIVP.

15 Oct 2021: b) Let f L2(U). For each γ > µ, denote by uγ the unique solution of (†). i) Show that for any γ > µ there exists wγ H10 (U) such that:. ... b) Show that there exists a constant C > 0, depending only on n, such that:.

19 Jan 2021: 1. Let T B(B) with |||T||| < 1. Show that there is U B(B) such that S2 = I T. ... 16. Let T L(2) defined by Tem =. n1 amnen where (amn)m,n=1 is such that.

21 Nov 2021: and B, there exist disjoint open sets U and V such that A Uand B V. ... Deducethat B′ is compact. 15. Show that there exist topological spaces X and Y with continuous bijections f : X Yand g : Y X such that X and Y are not homeomorphic.

21 May 2021: a) Show that the unique minimizer is x = 0. (b) Find constants 0 < m < M and L > 0 such that m f ′′(x) M and |f ′′′(x)| L for all x. ... Problem 8. Given constants b1, b2 such that b1 eb2 0 use the Lagrangian method to.

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.

12 Mar 2021: Local Rings and Tangent Spaces7. For each integer n 1, give an example of an algebraic variety X of dimension 1 such. ... Construct a morphism F : P1 P1 suchthat the following two conditions both hold: (1) for each negative entry bi in b, thereexists a

7 Oct 2021: 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 ... an,bn] such that the Lebesgue measure of A4B(= (Ac B) (ABc)) is less than ε.

12 Feb 2021: Show that for any b C, there exists a sequence of points zn D(a,R)with zn 6= a such that zn a and f(zn) b as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2.[A much harder theorem of Picard says that in any neighbourhood of an essential

18 Jan 2021: A Néron model for A is asmooth scheme A/R such that for all smooth schemes X/Rthere is a bijection HomR (X ,A) HomK (X, A). ... L contains a maximal torus T such that L/T = U isunipotent.Over K , T and U become isomorphic to products of Gm andGa,

21 Nov 2021: and B, there exist disjoint open sets U and V such that A Uand B V. ... Deducethat B′ is compact. 15. Show that there exist topological spaces X and Y with continuous bijections f : X Yand g : Y X such that X and Y are not homeomorphic.

26 Oct 2021: A = {x M : (xn) in A such that xn x} and A = {x M : r > 0 Dr(x) A}. ... Some more questions. 13. (a) A topological space is said to be second countable if there is a countable family Bof open sets such that every open set is a union

1 Nov 2021: Fix such an n and prove the theorem as follows. (a) Show directly that a nowhere-zero vector field on Sn induces a homotopy from the identityto the antipodal map. ... b) Use de Rham cohomology to prove that no such homotopy exists.

15 Oct 2021: Let a, b K such that thepolynomials f = xm a, g = xm b are irreducible. ... Show that G is solvable if and only if there is an i such that.

14 Oct 2021: y2= 0 (3). Construct a sequence of real analytic functions (uk)k=1 with uk : R2 R solving (4) such. ... u1. 1We write A B to mean that there exists a universal constant C such that A CB.

13 Oct 2021: foreach γ C there exist elements α(γ) A and β(γ) B, such that for all γ we have:. ... multiplication. An R-linear derivation OX,p R is an R-linearmap d : OX,p R such that d(fg) = d(f)g(p) f(p)d(g) for all f

22 Nov 2021: S aSbb | T, T bTaa | S | ǫ. (3) Give a CFG for each of the following CFL’s, and then transform each such CFG intoCNF (giving a justification for the transformation). ... b) Show that the language L := {anbncn | n 1} is not a CFL.

29 Nov 2021: Before we begin, let us define the notion of base. A base for a topology on a setX is a family B of open subsets of X such that every open ... such that x B U.Examples The family of open balls in a metric space is a base for the metrictopology.

14 Oct 2021: Show thatS1M = 0 if and only if there exists an s S such that sM = 0. ... ii) There exists nonzero elements e1,e2 A such that e1e2 = 0, e21 = e1, e22 = e2 and e1 e2 = 1.iii) A is isomorphic to A1 A2 for some rings

28 Oct 2021: A = {x M : (xn) in A such that xn x} and A = {x M : r > 0 Dr(x) A}. ... Some more questions. 13. (a) A topological space is said to be second countable if there is a countable family Bof open sets such that every open set is a union

15 Oct 2021: Incomponents we write B. j,γ,δ= ((bklj,γ,δ)) and c. γ,δ = (c. ... j, c such that the corresponding. series for u converges near x = 0.

4 Mar 2021: AC B,ρA ρB) D.5 For an inverse system (Ai,ρi) D such that limAi CO then. ... Example (Determinant). Fix a continuous character ψ : Γ O such that ψ mod λ det ρ̄.

27 Oct 2021: if any), and why? (b) By replacing A in the construction of B with a suitably chosen set, construct a setC N such that neither C nor N C are ... and such that Wn is recursive foreach n I but. nI Wn is not r.e.