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22 Jan 2009: 6) Let f : B B be a continuous map from the open disc B = {(x, y) R2 : x2 y2 < 1} into it-self. ... 10) Suppose f : R R has the intermediate value property that if a, b R, f (a) < c < f (b),then f (y) = c for some y between a and b.

21 Oct 2009: 6) Let f : B B be a continuous map from the open disc B = {(x, y) R2 : x2 y2 < 1} into it-self. ... 10) Suppose f : R R has the intermediate value property that if a, b R, f (a) < c < f (b),then f (y) = c for some y between a and b.

12 Feb 2009: t2(x, t)| M for some fixed. M (0, ). Let F (t) = g(x, t) dx. ... f (x)ϕ′′(x) dx = 0. for each smooth function ϕ : R R vanishing outside some compact interval.

27 Nov 2009: 1. The square matrices A and B over the field F are congruent if B = P T AP for some invertible matrixP over F. ... 7. Let S be an n n real symmetric matrix with Sk = I for some k 1.

5 Nov 2009: b) to illustrate some of the ways in which P-H is stronger than FiniteRamsey (tho’ the proof of independence from PA is not covered);. ... So, for some large j, consider the initialsegment consisting of the first j elements of X.

21 Jan 2009: Fix x g and f V. By [3, Proposition 1.4], some B-linear combinationof the derivations ρ(x)[p. ... By Borel’s fixed point theorem [4, Chapter III,Theorem 10.4], X(uk) contains a B-stable line 〈x〉 for some nonzero x gk.

1 Oct 2009: What isa good shape for a research star? Some careless generalisations can be made. ... Beyondthese, there some who have particular roles. (a) Martin Hyland (b) James Norris.

11 Nov 2009: I shall try to give some of them. You maydisagree with some or all of what I say. ... 2) When in doubt, display. Most mathematicians include some veryshort formulae in their text.

4 Aug 2009: c δ,c δ) [a,b] for some δ > 0Observe that f(c h) g(c) 0 (as c is a maximum point)So, if h > 0 ... Then. f(b) nj=0. f(j)(a)j! (ba)j = f(n1)(c′)(n 1)! (ba)n1. for some c′ with a < c′ < b.

9 Oct 2009: on the connection. These approaches have to some extent been subsumed by Paul. ... power domain (Lehmann 1976). Some time after Lehmann’s work and independently of.