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30 Aug 2019: Denote the maximal normal p-subgroup of G by Op(G), and the maximal normalp′-subgroup (i.e. ... of order coprime to p) by Op′ (G). Let H = G/Op′ (G).

17 Jun 2019: Find by vector methods. (a) the angle between the faces OP R and OQR [5]. ... b) the angle between the vector OP and the normal to the face P QR [5].

30 Aug 2019: 4 Let M be a smooth n-dimensional manifold. (a) Let M̂ be the set of pairs (p, op), where p M and op is one of the twoorientations of TpM. ... op), where p Uand op is the orientation of TpM determined by ωp.

30 Aug 2019: b) Let C be a locally small category, and let X : C op Set be a presheaf on X. ... b) Given a presheaf X : C op Set on a locally small category C , show that thefollowing are equivalent.

30 Aug 2019: Use the exact sequences 0 1P OP(1)n1 OP 0, and 0 OX (d) 1P|X.

30 Aug 2019: i) Let f (X) be a monic polynomial in op[X] and suppose that the reductionmod p factors as f̃ (X) = φ1(X)φ2(X) where φ1, φ2 in ... ii) Suppose Pe || p and p | e. Show that if α OP then TrKP/kp (α) pp.

30 Aug 2019: Show that F may berepresented as the colimit in [C, Set] of a diagram of shape (1 F)op whose vertices arerepresentable functors. ... iii) (1 F)op is filtered. (iv) F is expressible as a (small) filtered colimit of representable functors.

19 Jun 2019: ID Time Op Age Sex Intervention Centre1 192 0 39 1 0 02 101 1 46 1 0 03 197 1 28 0 0 04 66 1 49 0 0 05 ... APPLIED STATISTICS [TURN OVER. 6. ID = Patient’s identification numberTime = Waiting time for operation (in days)Op = Operation done?

30 Aug 2019: Let L be a linear operator on Hilbert space H of finite dimension.Define |L| and define the trace norm ‖L‖1 and operator norm ‖L‖op of L.If L ... has polar decomposition L = U|L| where U is a unitary operator on H, showthat if Z is a linear

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

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

30 Aug 2019: op, Set]. 5. Let T be a geometric theory over a signature Σ.

30 Aug 2019: iii) (1 F)op is filtered. (iv) F is expressible as a filtered colimit of representable functors.

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

17 Jun 2019: NATURAL SCIENCES TRIPOS Part IA. Wednesday, 13 June, 2012 9:00 am to 12:00 pm. MATHEMATICS (2). Before you begin read these instructions carefully:. The paper has two sections, A and B. Section A contains short questions and carries. 20 marks in

17 Jun 2019: The point P on the line is defined so that OP = QR.As the point Q moves around the circle, the point P traces out a curve C.

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

30 Aug 2019: Ai = |0〉Ai 〈0|Ai projects on her i-th. qubit) and to send the result to Bob, and then for Alice and Bob to carry out further op-erations.

30 Aug 2019: 2. |F[Kh](u)|2du)1/2. = OP. (. M̃hhn. ). Part III, Paper 209 [TURN OVER. ... supx[M/2,M/2]. |b̂T (x, h) b(x)| 6 Rhα OP(. 1Th. ). [ Hint: You may use without proof the Itô isometry and that the invariant measure µ isunique, its

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.].