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23 Feb 2018: 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.

12 Feb 2018: 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.

11 Nov 2018: 4. Let K be a field containing a primitive mth root of unity for some m > 1. ... Show that f andg have the same splitting field if and only if b = cmar for some c K and r N withgcd(r,m) = 1.

22 Oct 2018: 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

18 Jan 2018: 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.

9 Oct 2018: 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 Oct 2018: 5. Let (fn) be a sequence of real-valued continuous functions on a closed, bounded interval [a,b], and suppose. ... 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

9 Oct 2018: 4. Find the largest product of (some not necessarily distinct) natural numbers whose sum is 100. ... For each of claims (a)–(f) below, give aproof or counter-example. (a) f(AB) = f(A)f(B) (b) f1(C D) = f1(C)f1(D)(c) f(AB) =

23 Feb 2018: 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.

20 Nov 2018: The square matrices A and B over the field F are congruent if B = PTAP for some invertible matrixP over F. ... for some λ,µ,ν R (which should turn out in this example to be integers).Does there exist a linear transformation which reduces the pair of