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30 Nov 2011: 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.

30 Oct 2011: Hint: argue by contradiction.). (6) Let f : B B be a continuous map from the open disc B = {(x, y) R2 : x2 y2 < 1} into it-self. ... 1. (11) 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

2 Dec 2011: Find this solution to some reasonable accuracy using a pocket calculator or thecalculator on your computer (remember to work in radians!), and justify the claimed accuracy ofyour approximation. ... norm. Show, however, that T is a contraction under the

12 Nov 2011: 3.17. (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 ... Deduce that every finite abelian group is the Galois group of some Galois extension K/Q.[It is a long-standing unsolved problem (inverse Galois

16 Aug 2011: larger family Aand then randomly discard some of the sets in A to form a subfamily B. ... Then A = vf (W )Xfor some W U , X X W2 YW and W X B.

29 Jul 2011: Then Lemma 5(applied to the image of ψ/Fl) gives the desired map log : U Lie U ad V ′. Step 6: Some properties of G0(Fl). The pair ... k1γ(T(Fl)/Z0)γ1k = k′(T(Fl)/Z0)k′1. for some k′ B(Fl). Then (kk′)1γ lies in H and we deduce that Γ

25 Aug 2011: n1. maximising the number of intersecting subfamilies of size s. Theorem 2tells us that A = [n](>3) B for some B [n](2). ... Let U = {A [n](t) : B A for some B U} be the upper shadow of Uin layer t.

21 Dec 2011: So if we lift a1 to Z[X5] thena21c4 c6 = 2f for some f Z[X5]. ... Notice that if Φ′ = gΦfor some g = [A,B] with A,B GL5(K) then (Φ′) = (Φ) v(det g).

9 May 2011: First we construct some covariant columns. Let x1 = (a,b,c,d,e)T. If γ. ... Let (a : b : c : d : e) be the K-point on X(11) corresponding to (E,φ) for some.

29 Jul 2011: In the firstsection we recall some results on the Galois representations associatedto automorphic forms. ... If π is ramified, then as above we have for some t > 0.