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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

10 Jan 2017: to V4 V3; in this and the followingquestions, you are not being asked to find such an isomorphism explicitly). ... Here O(n) denotesthe subgroup of GLn(R) preserving the standard (positive definite) symmetric bilinear form.(b) Explicitly construct such a

12 Oct 2017: x E such that x Am for all large enough m N, and lim sup Anis the subset of those x E such that x belongs to Am for infinitely many ... 1. 2. 1.7. A subset E R is called Jordan measurable if for every > 0 there are two finite unions ofintervals A =. n1

22 Nov 2017: If A is the matrix of the first of these (say), find a non-singular matrix P such that PTAP is diagonalwith entries 1. ... 13. Let a1, a2,. , an be real numbers such that a1 an = 0 and a21 a2n = 1.

13 Mar 2017: 2. for a > 0and b R, prove that such inequality cannot hold when p (2, ]. Exercise 4. Consider f L1(R) that is differentiable almost everywhere with f′ L1(R), does ... and a limit function f L2(U) such that ‖fϕ(n) f‖L2() 0 as n.

24 Apr 2017: 2) Let (Rn,d) denote Euclidean n-space. If P,Q,R are points in Rn such that. ... For any sequence (gn) Y such that gn g X, show that g Y.

5 Dec 2017: 2) Convert the following CFG to CNF:. S aSbb | T, T bTaa | S |. (3) Give a CFG for each of the following CFL’s, and then transform each such CFG intoCNF:. ... b) Show that the language L := {anbncn | n 1} is not a CFL.

13 Nov 2017: b) By replacing A in the construction of B with a suitably-chosen set, construct a setC N such that neither C nor NC are r.e. ... Use this to show there exists some m N such that Wm has exactly m elements.

8 Nov 2017: tiplicativity of the norm, we can find ξ Q( 3a) such that NQ( 3a)/Q(ξ) = b. ... Algorithm 3.6. (Legendre-type algorithm for solving cubic norm equations). Input: A pair of positive integers (a,b) such that b is a norm for Q( 3a)/Q.

17 Apr 2017: If A, B, U are subspaces of a vector space V such that U = (A U) (B U)then (A U) (B U) = (AB) U. ... 19. (2.4.11) The main inductive step is the following: let n = a b, and V = WnV be anobject of Crk such that gr.

26 Jul 2017: We choose φ1 such that. φ1(e1) = (0, 0), φ1(eσi ) = (1 σ. ... y4 such that. 4i=1. xibij(y1,. ,y4) V for all j = 1,. ,

7 Mar 2017: Lemma 2.1. There exist ξ,η,a1,a2,a3,a4,a6 Z[aij] such that(i) We have c4 = b. ... Lemma 3.1. There exist ξ,η,a1,a2,a3,a4,a6 Z[sijk] such that(i) We have c4 = b.

11 Sep 2017: b) If ι′ : Y X is another such inclusion and ιOY = ι′OY ′ as coadmissibleÙDX-modules, then Y = Y ′. 1.3. Structure of this paper. As in our previous paper [2] ... Lemma. Suppose that f1,. ,fr B are such that L fi A for each 1 6 i 6 rand there

15 Mar 2017: Let A be an nn matrix over F. The following are equivalent(a) there is a matrix B such that BA = In;(b) there is a matrix C such that AC = ... Thus (a) holds if and only if α is anisomorphism. Similarly (b) holds if and only if there exists γ such that

13 Sep 2017: X! 1. T X. , for X 2 C,. such that the following conditions hold:. • ... f,g. to be the unique 2-cell. such that. g. f. g f//.