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20 Nov 2015: 1. The square matrices A and B over the field F are congruent if B = PTAP for some invertible matrixP over F. ... 7. Let S be an nn real symmetric matrix with Sk = I for some k 1.

24 Jan 2015: C/Λ2 are conformally equivalent if and only if thelattices are related by Λ2 = λΛ1 for some λ C. ... 5. Show that complex tori C/〈1,τ1〉 and C/〈1,τ2〉 are analytically isomorphic if and only if τ2 =. (aτ1 b)/(cτ1 d), for some matrix(a bc d

28 Aug 2015: 7. (a) If f : [0, 1] [0, 1] is continuous, show there is some x [0, 1] withf(x) = x. ... For eachinteger n > 1, show that there is some x [0, 1] with f(x) = f(x 1.

22 Jan 2015: Xn be independent Poisson random variables, with Xi having meaniθ, for some θ > 0. ... b) For some n > 2, let X1,. ,Xn be iid with Xi Exponential(θ).

11 Feb 2015: Fi //. Ep //. p. B. {b0} // B B. and show using the naturality of the Serre spectral sequence that the compositions. ... for some monic polynomial p(x) with coefficients in H(B; Z). Homotopy Theory Example sheet 3.

24 Jan 2015: 8. By considering the singularity at or otherwise, show that any injective analytic map f : C Chas the form 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.

12 Nov 2015: b) All words w {a,b,c} consisting of some number of a’s (possiblynone), followed by some number of b’s (at least one), followed bysome number of c’s ... a) {anbn | n 0}. (b) {a2nb2n | n 0}. (c) {ww | w {0, 1}}.

13 Oct 2015: 5. The sum of some (not necessarily distinct) natural numbers is 100. ... 9. Find the convergents to the fraction 5744. Then prove that if x and y are integers such that57x 44y = 1, then x = 1744k and y = 57k 22 for some k

2 Feb 2015: C/Λ2 are conformally equivalent if and only if thelattices are related by Λ2 = λΛ1 for some λ C. ... 5. Show that complex tori C/〈1,τ1〉 and C/〈1,τ2〉 are analytically isomorphic if and only if τ2 =. (aτ1 b)/(cτ1 d), for some matrix(a bc d

14 Oct 2015: algebra. 1.2. Show that the following sets of subsets of R all generate the same σ-algebra:(a) {(a,b) : a < b}, (b) {(a,b] : a < b}, (c) {(,b] : b ... 1.9. Let (E,E,µ) be a measure space. Call a subset N E null if N B for some B E withµ(B) = 0.

19 Jan 2015: δ is abelian.(c) Assume that δ(g) = 1 for some g G. ... irreducible. 3 Find an example of a representation of some finite group over some field of charac-teristic p, which is not completely reducible.

30 Nov 2015: for some constant Kand an open subset U of Rm containing a, |F(t,x) F(t,y)| K‖xy‖ for all t [0, 1],x,y U. ... for some K and all t [a,b], x,y BR(x0).We showed in lecture that for each t0 [a,b], there is a unique f C([a,b];

2 Nov 2015: a) Both A, B are recursive. (b) Both A, B are r.e. ... b) Fix some computable ordering of Σ, and write wn for the nth word.

16 Oct 2015: 5. Let (fn) be a sequence of real-valued continuous functions on a closed, bounded interval [a,b], and. ... a) If Dn is the set of discontinuities of fn and D is the set of discontinuities of f, show that D n=1j=nDj.(b) Suppose that for some finite

2 Feb 2015: 8. By considering the singularity at or otherwise, show that any injective analytic map f : C Chas the form 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.

11 Mar 2015: in some quadratic field K L. Deduce that L has class number 1.15. ... If pOL = OLthen we can write xi =. aijxj for some aij p.] Deduce that if a and b are idealsin OK with aOL = bOL then a = b.

23 Nov 2015: 1. The square matrices A and B over the field F are congruent if B = PTAP for some invertible matrixP over F. ... 7. Let S be an nn real symmetric matrix with Sk = I for some k 1.

13 Nov 2015: Eo = {x X : B(x) E for some > 0}. ... 8. Use the Contraction Mapping Theorem to show that the equation cos x = x has a unique real solution.Find this solution to some reasonable accuracy using a calculator (remember to

27 Feb 2015: for some constant Kand an open subset U of Rm containing a, |F(t,x) F(t,y)| K‖xy‖ for all t [0, 1],x,y U. ... E is of the form (a,b), [a,b), (a,b] or [a,b] for some a,b with a b.

31 Aug 2015: 7. (a) If f : [0, 1] [0, 1] is continuous, show there is some x [0, 1] withf(x) = x. ... For eachinteger n > 1, show that there is some x [0, 1] with f(x) = f(x 1.