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16 Dec 2015: possible area is an arc of a circle passing through A and B. ... 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.

25 Jun 2015: Then we may choose smooth functions ϕi : M [0, 1] such that supp(ϕi) int(Vi) and p(ν K) (. ϕi)1(1) and choose B =. B(A, ϕ1,. , ϕr) A ... Our choice of B guarantees that the linearcombination s =. ϕisi has s A.

22 Jan 2015: likelihood estimate of θ is (2a b)/(b 2c). 4. (a) Let X1,. ... b) For some n > 2, let X1,. ,Xn be iid with Xi Exponential(θ).

11 Feb 2015: 5. Let p : E B be a Serre fibration, e0 E, b0 = p(e0), F = p1(b0). ... Fi //. Ep //. p. B. {b0} // B B. and show using the naturality of the Serre spectral sequence that the compositions.

14 Oct 2015: 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].

25 Feb 2015: iii) Show that the map. H(A)iA H(M) [M] Hn(M). (iB) Hn(B). ... w(F), [Hint: sum of vector bundles is given by pulling backE F B B along the diagonal; relate and Th(E F) to Th(E) and Th(F)].

5 Feb 2015: 3. Two players A and B play a best of 5 set tennis match. ... Hint: Recall that H(A,B) = H(A|B) H(B).](b) Show that. H(I|Y,X) = 0H(Y |I,X) = P(I = 0)H(Y |I = 0,X)

4 Feb 2015: 3. Two players A and B play a best of 5 set tennis match. ... Hint: Recall that H(A,B) = H(A|B) H(B).](b) Show that. H(I|Y,X) = 0H(Y |I,X) = P(I = 0)H(Y |I = 0,X)

20 Nov 2015: 6. Let f : X Y. Let A, B X and C, D Y. ... For each of claims (a)–(f) below, give aproof or counter-example. (a) f(AB) = f(A)f(B) (b) f1(C D) = f1(C)f1(D)(c) f(AB) =

4 Feb 2015: possible area is an arc of a circle passing through A and B. ... 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.

4 Apr 2015: 25) ð(B) = B. Note that for any closed set B, if β is in B but not in ð(B), then β is in some Ky,mand therefore cannot attack ... Since ð(B) B,. x B and therefore f(x) B as B is f-closed.

17 Dec 2015: possible area is an arc of a circle passing through A and B. ... 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.

6 Apr 2015: On the other hand, given two well-orderings, (A, <A), (B, <B ), one is order-isomorphic to aninitial segment of the other: in short, any two well-orderings are comparable. ... 0, i〉 codes Oi;. 〈1, s〉 codes X r B if s codes B;.

4 Apr 2015: Then Z1(b) = {b} Zk1(a). An easy induction now shows that for each n > 0,Zn(b) Zkn(a). ... If %(w) = β ω, then w = a = Z(b), by 44 and 45.

4 Apr 2015: So for some b 2 a, B = (b). Butthen b 2 B () b 2 (b) () b =2 B, a ontradi tion. ... P(A) and will be a set, provably in M H.(iii) By Remark 019, (B) is neat in A.(iv) k(B) _B = _?410 LEMMA (B) =? () A = B.411 DEFINITION

4 Apr 2015: classes as objects to which such membership tests might be applied.If we write “a b” for “a is a member of b” and “a / b” for “a is not a ... for the particular casewhen y is B, B B iff B / B.

4 Apr 2015: aplicarse estos tests de pertenencia. Si escribimos “a b” para denotar que “a es un elemento de b” y “a 6 b” para “a no esmiembro de b”, podemos expresar el ... si y solo si y 6 y y, por tanto, para el caso particularen el que y es B,

11 Sep 2015: for some a,b K. The discriminant condition is 2b(a2 4b) 6= 0. ... We have div(f/x. 21) = 2b and div(f. †/x21) = 2b† with b b†.

4 Apr 2015: In both we shall have b. b for every b. 46 LEMMA Let f : X X be continuous, and let T be a non-empty well-founded tree with top-mostpoint. ... Examples of long delays in the Baire space. 48 DEFINITION Baire space, N , is {b | b : ω ω}; topologically it

11 Apr 2015: zB Φ df z (z B & Φ)zB Φ df z (z B = Φ)t = B df x (x t x B)B = t df x (x B x t)B ... h(u) =df(H(u) u. ) {(a,b)2 |a,b b u & a F(b)}.