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18 Jan 2016: 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

12 Dec 2016: Bn(θ) =a. n+. b. n2+ O(n3). for some real numbers a,b. ... 5. Let f,g : R [0,) be bounded probability density functions such that f(x) Mg(x)for all x R and some constant M > 0.

27 Oct 2016: 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.

6 Jan 2016: 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.

18 Jan 2016: 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.

16 Nov 2016: This result is called Burnside’s Lemma, or Cauchy-Frobenius Lemma, or just the Count-ing Lemma.)Deduce that if G acts transitively and |X| > 1 then there must exist some ... Canyou also find some analogues considering subgroups of D2n? 2.

28 Oct 2016: Suppose for some θ0 in the interior of Θ and every ε > 0 small enough we haveS(θ0 ε) < 0 < S(θ0 ε), and that Sn has exactly one zero θ̂n ... model {f(,φ) : φ = Φ(θ) for some θ Θ} equals Φ(θ̂MLE).b) Now consider a mapping Φ that is not

1 Feb 2016: This strengthens Liouville’s theorem.)(ii) If for some a C and > 0, f never takes values in B(a; ), show that f is constant.(iii) If f = u iv and ... Show that f(1/n) 6= 1/(n 1) for some n Z>0. 9.

15 Feb 2016: Could there be some other Euclidean function φ making Z[3] into a Euclidean domain? ... iii) Show that if R is a PID, the gcd of elements a and b exists and can be written as rasbfor some r,s R.

29 Feb 2016: α for some α.(iv) Any limit ordinal can be written in the form α. ... Given a subset b Pα with more thanone element, consider the least β belonging to some but not all members of b.].