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13 Dec 2017: Show that E|B0 'E|B1 as follows. (a) For each b B show there is an open neighborhood Vb of b such that E|VbI istrivial. ... b) Choose a finite cover V = V1,. ,Vm of B such that E|VbI is trivial.

7 Sep 2017: On the other hand, as the signature andEuler characteristic agree modulo 2, and the latter is multiplicative, it is true thatσ(E) σ(B) σ(F) mod 2 for all such ... More recently, Hambleton,Korzeniewski, and Ranicki [5, Theorem A] proved that σ(E) σ(B)

14 Feb 2017: Λ > 0 there is a constant C thatdepends only on Λ such that if n 5 and if M is a stable minimal hypersurface of B with 0 Mand Hn(M) ... Such an estimate cannot hold for stablehypersurfaces when n 7, nor can it hold without stability even when n = 2).

5 Feb 2017: Show that for any b C, there exists a sequence of points zn D(a,R)with zn 6= a such that zn a and f(zn) b as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2.[A much harder theorem of Picard says that in any neighbourhood of an essential

13 Oct 2017: Calculate d = (a,b) and find integers r and s such that ra sb = d when. ... iii) Find constants c and d such that λ(a,b) 6 c log b d.

13 Feb 2017: Such a c is called a fixed point of g.Give an example of a bijection h: [0, 1] [0, 1] with no fixed point. ... Thatis, for every a < b and every y there exists x (a,b) such that f(x) = y.Can such a function be continuous at any point?

5 Feb 2017: Show that there exist unique complexnumbers A, B such that for every z C, T(z) = Az Bz̄. ... Can such a map everbe a Möbius transformation? 9. Let f : U C be a holomorphic function where U is an open set (you may assume f isalso C1).

22 Oct 2017: b) If (fn) does not converge uniformly, show that there is a convergent sequence (xm) x [0, 1] such that (fn(xn)) does not converge to f(x). ... a) The space Cb(R) of bounded continuous functions on R.(b) The space C0(R) of continuous functions f : R R

10 Mar 2017: Provethat A contains an infinite set B such that no point of B is a convex combination of otherpoints of B. ... 6. Show that every graph G has a partition of its vertex-set as X Y such that the numberof edges from X to Y is at least 12e(G).

11 Oct 2017: ii) Let α be such that |K(α) : K| is odd. ... and let β = α4.Find. rationals a, b, c such that β = a bα cα2.

8 Feb 2017: Show that for any b C, there exists a sequence of points zn D(a,R)with zn 6= a such that zn a and f(zn) b as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2.[A much harder theorem of Picard says that in any neighbourhood of an essential

19 Jan 2017: 7. Let R be an integral domain. The greatest common divisor (gcd) of non-zero elements a andb in R is an element d in R such that d divides both ... b) Show that may be identified with the set Fq{} in such a way that GL2(Fq) acts on as the group of

8 Feb 2017: Show that there exist unique complexnumbers A, B such that for every z C, T(z) = Az Bz̄. ... Can such a map everbe a Möbius transformation? 9. Let f : U C be a holomorphic function where U is an open set (you may assume f isalso C1).

10 Mar 2017: Show that for every. closed subinterval I [a,b] of positive length and every ε > 0 there existsa closed subinterval J I of positive length such that supJ f infJ f < ... 0. sinxx. dx = π2. 12. A function f : [a,b] R is said to have bounded variation if

23 Feb 2017: 7. Let R be an integral domain. The greatest common divisor (gcd) of non-zero elements a andb in R is an element d in R such that d divides both ... b) Show that may be identified with the set Fq{} in such a way that GL2(Fq) acts on as the group of

17 Nov 2017: Can S be uncountable?Is there an uncountable family T PN such that AB is finite for all distinct A, B T? ... You are asked to supply them with astrategy such that, should they follow it, only finitely many of them will guess wrongly.

23 Oct 2017: b) If (fn) does not converge uniformly, show that there is a convergent sequence (xm) x [0, 1] such that (fn(xn)) does not converge to f(x). ... a) The space Cb(R) of bounded continuous functions on R.(b) The space C0(R) of continuous functions f : R R

13 Oct 2017: Calculate d = (a,b) and find integers r and s such that ra sb = d when. ... iii) Find constants c and d such that λ(a,b) 6 c log b d.

8 Nov 2017: Show that there is a divergentseries. n=1 yn with yn > 0 for all n, such that yn/xn 0. ... a < x < b such that f(x) = c.

25 Jul 2017: By Lemma 1, we can find A B with|A| = 1 and colours ca (0 6 a 6 k) such that if x is in the span of A andhas pattern ... By Lemma 1, there exist a subset A B with |A| = 1 and colours cπ(π Π′) such that if x is in the span of A and has pattern π