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13 May 2016: If φ : (Sn,) (B,B) , show that we can find a mapΦ : X E such that p Φ = φ π and Φ(Sn1 0) =. If we define φ =Φ|Sn11 : Sn1 ... F , check that the map [φ] 7 [φ] descends to a well-definedhomomorphism : πn(B,B) πn1(F,).

7 Oct 2016: Show that f : Q Q withf ( a. b) = b.

23 Oct 2016: φ(s,v) = α(s) r(n(s) cos v b(s) sin v). where n = n(s) and b = b(s) are the normal and binormal vectors of α. ... 10. Consider a surface of revolution parametrized by φ : (0, 2π) (a,b) R3, where.

23 Mar 2016: B = ( B ,. , B , e l ,. , e , ) and C = ( C 1 ,. , C , e ,. , e , ). T h e n i r a • 1,4,1,. ... Recall t h a t if B = ( B ,. , B, b l ,. , b,) is a E-algebra, then a value a of t y p e [¢Ix] interpreted at B I ,. , B

11 Nov 2016: u(0,x) =1. 1 x2. 1. 5. (Cauchy-Kovalevskaya Theorem for systems of ODEs) Suppose b > 0 and F : u0 (b,b)d Rd bereal analytic in a neighbourhood of ... u′(t) = F(t,u(t)), u(0) = u0 Rd,. on (a,a) for some a > 0 with u((a,a)) u0 (b,b)d.

27 Oct 2016: algebra. 1.2. 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 ... 1.8. Let B be a Borel subset of the interval [0, 1].

27 Oct 2016: Hint: Bolzano-Weierstrass.). 3. If A and B are subsets of Rn, let A B = {a b |a A,b B}. ... a) f(x) = sin x2 (b) f(x) = inf{|xn2| |n Z} (c) (sin x3)/(x 1).

29 Feb 2016: 8. By a selection function for a set a, we mean a function s: Pa Pa such that s(b) b for allb a, both inclusions being strict unless b is ... By considering congruence classes whose members are well-orderings, show that theHartogs ordinal of b is greater

23 Mar 2016: B = ( B ,. , B , e l ,. , e , ) and C = ( C 1 ,. , C , e ,. , e , ). T h e n i r a • 1,4,1,. ... Recall t h a t if B = ( B ,. , B, b l ,. , b,) is a E-algebra, then a value a of t y p e [¢Ix] interpreted at B I ,. , B

28 Nov 2016: T)the usual half-sum of B-positive roots. We define abstract Hecke algebras. ... Lλ,E = IndGB λ = {f E[G] | g G,b B,f(bg) = λ(b)f(g)}.

29 Apr 2016: In particular, this cocycle depends only on B, and not on any other choice. ... However, writing η = σB B andψ = tσB(f σf1), we have (η,ψ) H̃L and, by Lemma 1.7,. (

3 Feb 2016: Black holes in general relativity 3. hole region B of such perturbations of Kerr? ... J+(B) = B), in particular, they cannot reach large values of r.

25 Apr 2016: x(b)) = b, λ(b) 6 0 and f(x(b)) = φ(b) = inf{f(x) λ(b)T(h(x) b) :x Rn}. ... n. The second term is zero as well, because x(b) is feasible and thus(h(x(b)) b)k = 0 for k = 1,.

25 May 2016: are all valid,. (b) G is equivalent to H iffG(x,y) H(x,y). ... 13. (b) If F is j-separated, then so is FE for any E.

25 May 2016: are all valid,. (b) G is equivalent to H iffG(x,y) H(x,y). ... 13. (b) If F is j-separated, then so is FE for any E.

30 Apr 2016: t/j2s dt for W Œa;b! M:. Note that Es. s/ T. ... x;v/ and b W. ı0;ı0/! M adapted to. x;w/ we obtain.