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1 Feb 2010: Find a conjugate harmonic functionv(x, y), that is, a function v such that u and v together satisfy the Cauchy-Riemann equations. ... Show that there is an analyticfunction H : C C such that h(z) = exp H(z) for all z.

27 Jan 2010: Use a suitableLagrange resolvent to find a Q(µ3) such that K = Q(ζ3, 3. ... Use (ii) to show that there exists b K such thatdet(σiσjQ1(b)) ̸= 0.

29 Jan 2010: The irreducibility of Sp(χ) for such a prime pin conjunction with a) implies that Φq,χ = 0. ... Since GẐ acts transitively on |MK |,. there exists a positive constant cn Q such that for every x |M|,.

12 Oct 2010: Suppose then that g is a. quartic with J = 0 such that z(g) = z(g). ... We now fix such a u and put αi = B(ei, u)where e1, e2, e3 K3 are the standard basis vectors.

22 Nov 2010: Let t = a! and d = t. By Ramsey’s theorem, there exists a positive integer b such that whenever[b](t) is ka-coloured, there exists a monochromatic subset of order ... d. Letu = b a 1 and v = u tr. By Ramsey’s theorem again, there exists a positive

18 Nov 2010: 14. Is there an uncountable family S PN such that A B is finite for all distinct A, B S? ... such that. (qn qn1)2 converges? A.G.Thomason@dpmms.cam.ac.uk - 1 - 18 November 2010.

29 Jan 2010: Therefore there are constants a, b, c k such that if v, v′ Spand w IndGB δ1/2, then. ... π′′ = I(1)0, then we can choose w such that HomGw (πw.

3 Nov 2010: Recall that the existence of such a mini-mizer was proved in lecture.). ... b) Choose numbers a1, b1 such that 0 < a1 < a < b < b1 < 1, and set, for x R and n = 1, 2, 3,. ,.

17 Dec 2010: such that. MIA A M/B. ll 1r MA oh_ MB. commuts up to natural isomorphism. ... For g : Y-, B in M/B we expect IIh 0 Y E M/A to be such that a fibre (II,, 0 Y)n is the modest set of sections

23 Feb 2010: 6. Let R be an integral domain. The highest common factor of non-zero elements a and b in R is an elementd in R such that d divides both a ... identified with the set Fq {} in such a way that GL2(Fq) acts on asthe group of Möbius transformations z 7 azb

29 May 2010: Let Γ Aut(X′) be a finite subgroup such that f factorsas X′ X′/Γ g X, where g is a radicial alteration. ... Then one knows that there exists a finite extension K′/K such that ρ(IK′) isunipotent.

19 Nov 2010: 5. (i) Suppose that f : R2 R is such that f/x is continuous in some open ball around (a,b),and f/y exists at (a,b). ... Show that f is differentiable at (a,b). (ii) Suppose that f : R2 R is such that f/x exists and is bounded near (a,b), and that fora

29 Jan 2010: W2 E = G. Q(1). and GrWj E = 0 for j > 0 or j < 2, such that. ... The reader who is sceptical of the general finiteness of groups such as IIIwill be reassured that we do not require equality in (b).).

28 Mar 2010: 4. Let F/K be a finite extension. Show that there is a unique intermediate fieldK L F such that L/K is separable and F/L is purely inseparable. ... Let a,b K such that the polynomials P(X) = Xn a and Q(X) = Xn b are irreducible.Show that P and Q have the

22 Jan 2010: a, b] be given such that for each polynomial P , the error. ... let fn : C befunctions such that fn converge uniformly on to a bounded function f : C.

17 Feb 2010: B(µp,νp) spanned by tensors pφ̂p such that. φ̂p belongs to Bs(µp,νp) for at least one p, and this is the unique invariant subspace of B(µ,ν) ... Then C is finite.Proof. Choose a prime p 7 such that GZp K.

19 Nov 2010: n}, let the non-negative numbers A(n)k and the. “nodes” x(n)k [a, b] be given such that for each polynomial P , the error. ... such that for every f C([0, 1]), there exists a unique series.

24 Nov 2010: Use a suitableLagrange resolvent to find a Q(µ3) such that K = Q(ζ3, 3. ... Use (ii) to show that there exists b K such thatdet(σiσjQ1(b)) ̸= 0.

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.

9 Nov 2010: Let a,b K such that the polynomials P(X) = Xn a and Q(X) = Xn b are irreducible. ... an i such that Gi = {1}.(iv) Let G be the group of invertible n n upper triangular matrices with entries in a.