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19 Mar 2008: 2 ANALYSIS II EXAMPLES 3. 7. (i) Suppose that f : R2 R is such that D1f = f /x is continuous in a neighbourhood of(a, b), and D2f = f /y exists ... Show that f is differentiable at (a, b). (ii) Suppose that f : R2 R is such that D1f = f /x exists and is

21 Nov 2016: If A is the matrix of the first of these (say), find a non-singular matrix P such that PTAP is diagonalwith entries 1. ... 15. Let a1, a2,. , an be real numbers such that a1 an = 0 and a21 a2n = 1.

14 Feb 2024: 8. Find two totally ordered sets such that neither is isomorphic to a subset of the other.Can you find three such sets? ... Is there a countable ordinal α such thatα is the αth fixed point?

28 Oct 2022: 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

6 Dec 2021: the unique V τ such that V A and, if W τ and. ... Proof. The proof, which follows the standard pattern for such things, is givenon page 69.

14 May 2020: a) Showthat the unique minimiser is x = 0.(b) Find constants 0 < m < M and L > 0 such that m f ′′(x) M and |f ′′′(x)| L forall x.(c) Apply ... 8. Given constants b1, b2 such that b1 eb2 0 use the Lagrangian method to.

13 Feb 2017: Such a c is called a fixed point of g.Give an example of a bijection h: [0, 1] [0, 1] with no fixed point. ... Thatis, for every a < b and every y there exists x (a,b) such that f(x) = y.Can such a function be continuous at any point?

26 Oct 2016: 4. (a) Show that if x 6= y are vectors in the finite dimensional vector space V , then there is a linearfunctional θ V such that θ(x) 6= ... fn for V such that. ψ.  mi=1. xiei,. nj=1. yjfj.

19 Jan 2017: 7. Let R be an integral domain. The greatest common divisor (gcd) of non-zero elements a andb in R is an element d in R such that d divides both ... b) Show that may be identified with the set Fq{} in such a way that GL2(Fq) acts on as the group of

13 Oct 2022: 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.

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.

18 Feb 2010: abelian category of “mixed motives”. The ultimate definition of such a category remains to be found. ... φ : IK T. Here T/K is a number field and there exist integers a, b with ab = w such that.

28 Aug 2015: p. k, where k is the smallest value ofn such that xn 6= yn. ... Hint:first find a subsequence (fni ) such that fni (x) converges for all x Q [0, 1].)Deduce that the closure of B in (C[0, 1], d) is compact.

30 Mar 2022: In 7 we prove that such orbits admit integral representatives away from smallprimes. ... We have a Gm-action on Vgiven by λ v = λv and there is a unique Gm-action on B such that π is Gm-equivariant.

28 Oct 2016: 7. A map π : E B is called a covering map if there is an open cover {Uα} of B such that π1(Uα)is a disjoint union. ... H(B) H(E) such that π π! is multiplication by N.

20 Nov 2020: 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. 2. 9. Find three random variables M1,M2,M3 such that.

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,

12 Aug 2022: Then there is a point y / E such. that B(y,r) E 6= whenever r > 0. ... Since Uj1 isopen we can find a δj1 > 0 with δj1 δj/4 such that B(xj1,δj1) Uj1.

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.

27 Nov 2019: 12. Let S be a collection of subsets of N such that for every A, B S we have A B orB A. ... otherwords, for every a < b and every c there is an x with a < x < b such that f(x) = c.