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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