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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.

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) =

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

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

27 Feb 2018: 71). Find an element of OK with norm 2a 3b 5 for some. ... a,b 0. Hence prove that the class group of K is cyclic and find its order.

13 Nov 2018: 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

12 Nov 2018: Assume that thereexists some decision rule δ0 such that R(δ0,θ) < for all θ Θ. ... Show thatthe sample mean X̄n is inadmissible for quadratic risk, but that it is still minimax.What happens if Θ = [a,b] for some 0 < a < b <? 9. Let X be

23 Nov 2018: Suppose thatsup. [a,b]BR(x0)‖F‖ R(b a)1 and that ‖F(t,x) F(t,y)‖ K‖x y‖ for some. ... 9. (Some useful properties of the operator norm). Recall that the operator norm on the.

8 Oct 2018: 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 R}. ... 10. Let (E,E,µ) be a measure space. Call a subset N E null if N B for some B E withµ(B) = 0.

26 Oct 2018: Suppose for some θ0 in the interior of Θ and every ε > 0small enough we have S(θ0 ε) < 0 < S(θ0 ε), and that Sn has exactly one zero θ̂nfor ... 1. 2. a) Assuming that Φ is one to one, show that the maximum likelihood estimator of φin the model

25 Sep 2018: 2. Consider the three equations in six unknowns given by Ax = b, where. ... that x Rn is an extreme point of the set{x Rn : x 0,Ax b} if and only if for some z Rm,.

6 Nov 2018: a) All words w {0, 1} consisting of either the string 01 repeated some number oftimes (possibly none), or the string 010 repeated some number of times (possiblynone). ... b) All words w {a,b,c} consisting of some number of a’s (possibly none), followed

19 Jan 2018: setU, and µ(U) < for some non-empty open set U. ... 58. Let X,Y be Banach spaces and S,T B(X,Y ). Assume that T is invertible andfor some δ < 1 we have ‖SxTx‖ 6 δ‖Tx‖ for all x

16 Nov 2018: r > 019 or of the form {z C | aRe(z) bIm(z) = c}{}for some a,b,c R with (a,b) 6= (0, 0)20. ... Thus by the discussionabove gfg1 extends z 7 azb0za = z b/a for some a,b C{0}.