Search

12 Mar 2008: If g is another such function then f(z) = g(z) exp h(z)for some entire function h. ... If g were another such function then g/f would be an entire function with no zeros and thereforeequal to exp h for some entire function h.

30 Nov 2008: Find the Taylor series of g(I H) about I. 9. 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 D2f exists at (a, b).Show that f is differentiable at (a, b).(b) Suppose instead that D1f

12 Aug 2008: vary in difficulty but cover some instructive points. 1. For what values of a and b does the system of simultaneous linear equations. ... What are the eigenvalues and eigenvectors?Show that if an endomorphism β of V commutes with α then β = p(α) for

27 Feb 2008: C = {M M(A) : ResAB(M) P(B)}.Since any P P(A) is a direct summand of An for some n > 0, the restrictionResAB(P) is a finitely generated projective ... It follows thatwe can find P,Q P(H) such that [P] [Q] = m[1] inside G0(H) for some m > 0depending only

5 Nov 2008: 1) History. [If you decide to include some history, keep it down to acouple of sentences. ... This is a mathematics exam.]. (2) Preliminary definitions. [You may decide to give some illustrativeexamples.].

2 Feb 2008: A′,B′) = (m11A N m12B N,m21A N m22B N). for some (M,N) G4(K) := GL2(K) GL4(K). ... Q = ξ2for some ξ F and F[x1,x2,x3,x4] a linear form.

14 Oct 2008: Show thatf(x) = x for some x S1. 3. Which of the letters A,B,. ... 1,1) π1(S1,f(1)) when. (a) f(eiθ) = ei(θπ/2). (b) f(eiθ) = einθ for some n Z.

19 Mar 2008: only if U = Y V for some V open in (X, dX ).(ii Show that A is closed in (Y, dY ) if and only if A = Y B for some B ... a contraction? Deduce that the differential equation. dy. dx= 2xy , with y = 1 when x = 0 ,. has a unique solution in some interval

4 Feb 2008: Show that f and g have the same splitting field if andonly if b = cmar for some c K and r N with gcd(r, m) = 1. ... Deduce that every finite abelian group is the Galois group of some Galois extensionK/Q.

8 Mar 2008: In some sense a topological group is an object were the the statement‘the neighbourhood of every point looks the same’ is (a) meaningful and (b)true. ... Theorem B. (Bochner’s theorem.) A continuous function f : G C ispositive definite if and only