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11 Sep 2013: 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

2 Dec 2013: Show that f and g havethe same splitting field if and only if b = cmar for some c K and r N with gcd(r,m) = 1. ... Deduce that every finite abelian groupis the Galois group of some Galois extension K/Q.

29 Nov 2013: 7. Let f: R2 R and (a,b) R2.(a) Suppose that D1f exists and is continuous in some open ball around (a,b), and that. ... b) Suppose instead that D1f exists and is bounded on some open ball around (a,b), and.

11 Sep 2013: 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.

23 Oct 2013: 4. Let (fn) be a sequence of real-valued continuous functions on a closed, bounded interval [a,b], and. ... f. Prove that there is some subinterval [a,b] of [0, 1] with a < b on which f is bounded.

19 Sep 2013: 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

11 Feb 2013: 3. An element r of a ring R is nilpotent if rn = 0 for some n. ... 6. Let A and B be elements of the power-set ring P(S), for some set S.

24 Jan 2013: 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).

25 Feb 2013: domain. Could there be some other Euclidean function φ making Z[3] into a Euclidean. ... Does it follow that the ring R is either a field or a polynomial ringF [X] for some field F?

19 Sep 2013: 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.

19 Jan 2013: f(z) =a(z z0) bc(z z0) d. ,. for some a,b,c,d,z0 C. ... Nm,n=0. am,nf(z)mg(z)n 0, for some am,n C. Hence show that any two meromorphic functions f,g on a complextorus C/Λ are ‘algebraically

6 Feb 2013: By considering f (x) = g(x) x, orotherwise, show that there is some c in [0, 1] such that g(c) = c. ... 13. Let f : R R be a function that has the intermediate value property: if f (a) < c < f (b)then c = f (x) for some x between a and b.

20 Feb 2013: By using a second-order mean value theorem for f , show that φ′(x) = f ′′(θx)/2 for some. ... there is some c with a < c < b and f ′(c) = z.

26 Feb 2013: Thefinal three require some Galois Theory. 13. Let K be a number field. ... If pOL = OLthen we can write xi =. aijxj for some aij p.] Deduce that if a and b are ideals.

11 Jun 2013: Again this mypersonal view, some of my colleagues disagree.). 14 What is STEP? ... Part B) and the probability (Part C) and some will havedone neither.

13 Aug 2013: of (99) and equation (101) intoterm 2 leading to some cancellations. ... holds for some constant C if and only if. B := supa<x<b.

20 Feb 2013: For every isomorphism ι : Ql = C, the image of the residual representation rι(π′) contains SL2(Flb),up to conjugation, for some b a. • ... This contradiction shows that rι(π′) must in fact have residual imagecontaining SL2(Flb), some b a, and

12 Sep 2013: for some deformable R-algebra A, [3, Corollary 3.7] tells us that Gr(B) can becomputed as follows:. ... i. i! (b) = 0 for some i > 1. by Lemma 4.1(b).

21 Apr 2013: 4. We can get round this in the usual way(cf. part (b) proof of [11, 5.2]): take X = X′ = X(3) for some auxiliary integerN 3, and define ... If p 1 mod 3, then for some αp Zp (ordp(αp) = 2).

3 Apr 2013: element. Example. We illustrate some of the concepts introduced so far in the case where G is a split adjoint groupof type A2r. ... We begin with some generalconsiderations, and reduce to a case by case calculation using the invariant degrees of G.