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11 Nov 2004: 2. Let x1,. , xn be real numbers such that. ni=1 xi = 0 and. ... Prove that there are constants a and b such thatP1(x) = P2(ax b) for every x.7.

19 Oct 2004: Suppose p is anattracting fixed point for f. Show that there is a neighbourhood U of p such that the forward orbitof every point in U converges to p. ... 11. The Haar measure of a compact metrisable topological group G is the unique probabilitymeasure µ

14 Oct 2004: 3. Let K be a compact subset of R2 and let F be a closed subset of R2 such that KF =. Give two proofs that there exists δ > 0 such that ... to start by defining F A to be the set of all x such thatd(x, Ac) > max{d(x, Bc), d(x, Cc)}, and similarly for

16 Nov 2004: Corollary 39. If f and g are differentiable functions (a,b) R withf ′(t) = g′(t) for all a < t < b then we can find a constant c such thatf(t) = ... with γ(a1) = a2 and γ(b1) = b2 such that r1(t) =r2(γ(t)).

13 Nov 2004: p, so there are at least p1n4. pairs ((a′, b′), (a′′, b′′)) such that a′b′ = a′′b′′ (i.e. ... belonging to the same Xλ).For every (a, b) A A and every such pair, we get a collinear triple((a, b), (a a′, b b′′), (a a′′,

28 May 2004: An example is given by the Higman 4-group, see [23]. We do notknow if Theorem B is still true for such groups. ... The dimension of such a cell is the sumof the dimensions of the corresponding cells, of course.

20 Sep 2004: And yet there doesn't seem to be such a fact of the matter. ... Similarly, if B={x:Q(x)} then the statement "A cap B = emptyset" means nothing more than that there is no x such that P(x) and Q(x).

21 Oct 2004: B such that. j=0 ajzj converges for all |z| < δ and. ... A function f : B is said to be analytic on if there exists anf ′ : B such that, for all z.

14 Jun 2004: Theorem B. Let M be a closed oriented surface of negative Gaussian curvature.There exists λ0 > 0 such that the longitudinal KAM-cocycle of φ. ... unique 1-form ψ on SM (the connection form) such that. ψ(V ) = 1.

5 Oct 2004: pn log p B′n.(vi)There exists a constant B such that π(n) Bn(log n)1. ... ii) Find A and B such that g(s) = Aζ(s)B2sζ(s) for all <s > 1.