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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.