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23 Mar 2009: As R is a PID we have (a, b) = d for some d R. ... So. i=1. (ai) = (b). 1. for some b R. So b (ak) for some k, and then.

15 Oct 2009: k=1 uk(x0) converges at some point x0 [a, b] and. (ii) the series of derivatives. ... Then for x [a, b] we can write g(x) g(x0) = g′(ξ)(x x0) with some ξ [x0, x] [a, b] (or ξ [x, x0] [a, b]), using the

28 Jan 2009: Proposition 1.2 Isometries of E2. An orientation preserving isometry of E2 is:(a) The identity.(b) A translation.(c) A rotation about some point c E2. ... a) The identity.(b) A translation.(c) A rotation about some line. (d) A screw rotation, that is a

30 Nov 2009: 4) If the map f of Question 3 is a conformal equivalence, show that F(z) = λz µfor some λ C. ... 5) Show that complex tori C/〈1,τ1〉 and C/〈1,τ2〉 are analytically isomorphic if andonly if τ2 = (aτ1 b)/(cτ1 d), for some matrix.

23 Mar 2009: a = qb r. for some q, r R with r = 0 or d(r) < d(b).For example. ... N by. N(a b. 2. )= a2 2b2. (=. a b22) (a, b Z).

7 Dec 2009: 5. (i) Suppose that f : R2 R is such that D1f = f/x is continuous in some open ball around(a,b), and D2f = f/y exists at (a,b). ... 11. Show that there is a continuous square-root function on some neighbourhood of I in Mn;that is, show that there is an

4 Mar 2009: Xn be independent Poisson random variables, with Xi having parameteriθ for some θ > 0. ... b) For some n > 2, let X1,. ,Xniid Exp(θ). Find a real-valued sufficient statistic T, and.

23 Mar 2009: b = uac for some unit u. Then g̃ = uf k̃ and sog = ag̃ = f uak̃ and so f|g in R[X]. ... f (X) = af̃ (X) for some a R and primitive f̃ R[X], and so.

19 Oct 2009: f(z) = az b, for some a C and b C. ... conformally equivalent). (10) Define an equivalence relation on C by z w iff z = 2sw for some s Z.Show that the quotient space R = C/ has the natural structure of

8 Dec 2009: 5. (i) Suppose that f : R2 R is such that D1f = f/x is continuous in some open ball around(a,b), and D2f = f/y exists at (a,b). ... 11. Show that there is a continuous square-root function on some neighbourhood of I in Mn;that is, show that there is an

5 Mar 2009: Show that if f has no zeros in B then f ′/f has an antiderivative on some openset containing B. ... Suppose that f is holomorphic on some open disc containing {|z| 1} and satisfies |f (z)| < 1when |z| = 1.

11 Nov 2009: K′ L, then L = K(x) for some x L. ... 5. (i) Let f K(X). Show that K(X) = K(f) if and only if f = (aX b)/(cX d)for some a, b, c, d K with ad bc

8 Oct 2009: δ is abelian.(c) Assume that δ(g) = 1 for some g G. ... irreducible. 3 Given any prime p. Find an example of a representation of some finite group oversome field of characteristic p, which is not completely reducible.

10 Oct 2009: 5. The sum of some (not necessarily distinct) natural numbers is 100. ... Prove that if x and y are integers such that 57x 44y = 1 , then x = 17 44k and y = 57k 22 for some k Z.

2 Dec 2009: limz. znf (z) = c,. prove that f (z0) = 0 for some z0 C. ... a) g(z) = z for some z S1. (b) g(z) = z for some z S1.

19 Feb 2009: Could there be some. other Euclidean function φ making Z[3] into a Euclidean domain? ... of elements a and b exists and can be writtenas d = ra sb for some r,s R. [

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.

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.

12 Aug 2009: for some 0 a b with a b = s. If a = b = 0, then v(F) 1, contrary tohypothesis. ... MINIMISATION AND REDUCTION 21. Lemma 4.4. If some pair (a,b) with 0 a b is admissible for F then at leastone of the pairs (0, 0), (0, 1), (1,