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17 Jun 2019: Denote the order of x in Fp by Op(x). Further suppose Φ(N) divides 2. ... Further, prove that the. number of x satisfying Op(xb) 6= Oq(xb) is > Φ(N)/2.

19 Jun 2019: Find θ̂n and provefrom first principles that θ̂n is consistent. By considering the distribution function ofn(θ θ̂n)/θ, show that θ̂n = θ op(n.

17 Jun 2019: Show that the centre P of Trsatisfies. OP = r|b c|a |c a|b |a b|c.

30 Aug 2019: iii) (1 F )op is filtered, where 1 denotes a one-element set.

19 Jun 2019: 0 = U(θ̂n) = U(θ0) j(θ0)(θ̂n θ0) op(n1/2),. as n , where j(θ) is the observed information matrix at θ.].

30 Aug 2019: Part III, Paper 111. 3. 4 Let G be a finite group and let f : G Mn(C) be a function such that ‖f(x)‖op 6 1for every x G,

30 Aug 2019: Let. : Set - [O(S)op, Set]. be the functor assigning to a set A the presheaf A with constant value A.

17 Jun 2019: Denote the order of x in Fp by Op(x). Further suppose Φ(N) divides 2. ... Further, prove that the. number of x satisfying Op(xb) 6= Oq(xb) is > Φ(N)/2.

30 Aug 2019: Find θ̂n and provefrom first principles that θ̂n is consistent. By considering the distribution function ofn(θ θ̂n)/θ, show that θ̂n = θ op(n1/2) as n.

30 Aug 2019: iii) (1 F)op is filtered. [You may assume the result that filtered colimits commute with finite limits in Set.].