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29 Mar 2011: Our pointof departure is formula 6 which says there is an external isomorphism between〈A,A〉 and 〈B,B〉.(. ... We have now established that if there is an external isomorphism between〈A,A〉 and 〈B,B〉 then there is precisely one.

30 Nov 2008: around (a, b), and that D2f exists at (a, b).Show that f is differentiable at (a, b).(b) Suppose instead that D1f exists and is bounded on some open ball ... around (a, b), and that for fixed xthe function y 7 f (x, y) is continuous.

20 May 2002: 15. p. 243, lines 6 and 9: replace "(., f((r - 1)b/N /- |B_r| (b/N)2))" by "(., f((r - 1)b/N) /- |B_r| (b/N)2)". ... the brackets are misplaced). 16. p. 247, line 6: delete "<= B b3 N(-2)" (I am not quite sure here.).

12 Oct 2017: 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 (E,E,µ) be a measure space. Call a subset N E null if N B for some B E withµ(B) = 0.

1 May 2017: 5. Consider the following problems:. (a) minimise c>x subject to Ax = b;. ... Let P() be the linear programming problem when the right-hand side b = (2, 5, 6)> is.

18 Jan 2005: Lïiì Õ Þòð;ñ;à Õ Ð5 Ðîbª5àÏTÐbÑÑEÑEÐàÒ Â5ËE»b«»b¿ª¥bÔwb¿ ¥ÂµwEÁmY«=ª¥«'á;Sû!YE»lª=5bi=T¥«=µ¥ ðSÔw}Á'TþwEÁÁEíYE=¥i¥w bªEÁmª¥@¥Âµ(ðÔs«=ÓªEüÖwE¿ª@ªÁm¥w¥Âµ¥w¥«=

19 Dec 2014: Let a = min{| A |: A is an infinite maximalalmost disjoint family of subsets of ω}, b = min{| B |: B is unbounded in (ωω,)},and d = min{| D |: D is dominating in ( ... i) Prove that 1 min{a, b, d} max{a, b, d} 20.

15 Feb 2013: 2. Suppose that A and B are disjoint, closed, bounded subsets of R2. ... Show that. there is a δ > 0 so that |a b| > δ for all a A and b B.

29 Jan 2010: 1/2 it is the subspace spanned by the functionφ: bk 7 δ(b) (for b B, k K), and the linear form in (1.2) can be nor-malised so ... hence b = c since (,) is symmetric. Likewise b = a = c hence a = b =c = 0.

20 Jan 2017: 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).