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29 Jan 2010: Show that f (x) = 0 for allx. 8. Let f : [a, b] R be bounded. ... 13. Let f : R R be a function which has the intermediate value property: If f (a) < c < f (b),then f (x) = c for some x between a and b.

29 Jan 2010: slide 12. Lemma j = isomn on HM(;)(") Z[ 12k!].(each component of complement is xedby some with "( ) = 1.)[D]ix(pr0#D [ [prk#D)(") 7!y2 2Hk1M (Uky ;k 1)(")'Hk1M ... E fyg)k;k 1)(")Eisenstein symbol (B)ny = 0, some n > 1[n]y 2 Hk1M ((E 0)k;k 1)(")?yoHk1M

2 Feb 2010: Show that f (x) = 0 for allx. 8. Let f : [a, b] R be bounded. ... 13. Let f : R R be a function which has the intermediate value property: If f (a) < c < f (b),then f (x) = c for some x between a and b.

1 Feb 2010: COMPLEX ANALYSIS EXAMPLES 1. Lent 2010 J. M. E. Hyland. These questions are a mix of questions from recent versions of the course together with some specificto my own take on ... 16. Let f be an entire function such that for some a C and r > 0, f takes

19 Jan 2010: Show thatf(x) = x for some x S1. 3. Which of the letters A,B,. ... 1,1) π1(S1,f(1)) when. (a) f(eiθ) = ei(θπ/2). (b) f(eiθ) = einθ for some n Z.

28 Mar 2010: Deduce that every finiteabelian group is the Galois group of some Galois extension K/Q. ... only if b = cnar for some c Kand r N with gcd(r, n) = 1.

24 Jan 2010: 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 minimal sufficient statistic T , and.

5 Nov 2010: 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.

13 Oct 2010: 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

4 Feb 2010: J : z 7 az bcz d. for some complex numbers a, b, c, d with ad bc = 1. ... For which choices of a, b, c, d is this map J aninvolution, that is J 2 = I?