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30 Aug 2019: Definition. The vector space IndGHV is defined to be the space with basis givenby the abstract symbols tj vi where t1,. , ... is given by g (tj vi) := ts hvi where ts is the unique coset representative suchthat gtj tsH, and h = t1s gtj.

17 Jun 2019: for j 6= i. Prove that the random vectors Yj AjX are independent, and thatY (Y T1 ,. ,Y TJ )T has a multivariate normal distribution.[ Hint: Random vectors are independent if

30 Aug 2019: Deduce that for every n 1,. Tn =d1j=1. cn(j)Tj. 2 Define the Weierstrass -function (z) associated to a lattice Λ.

30 Aug 2019: the space of splines of degree k1 spanned by the B-splines (Nj )nj=1on a knot sequence = (tj )nkj=1 such that tj < tjk. ... Prove that‖Qf‖C[tj ,tj1] ck‖f‖C[tj1k,tjk] ,. hence derive that‖f Qf‖ = O(|t|k) f Ck[a, b],.

30 Aug 2019: Justify your answers. Suppose now that, in addition to the delay Dj(yj), users of link j incur a traffic-dependent toll Tj(yj), and suppose each user selects a route ... Show that it is possible to choose the functions Tj() sothat the equilibrium flow

30 Aug 2019: i)B (t). (c) tj and tk are times of consecutive events (tj < tk) in a relative survival dataset.Describe the behaviour of F̂E (t) for tj < t < tk. ... Describe an example of a situation whereF̂E (tK ) might be greater than F̂E (tj ). END OF PAPER.

30 Aug 2019: Justify your answers. Suppose now that, in addition to the delay Dj (yj ), users of link j incur a traffic-dependent toll Tj(yj ), and suppose each user selects a route

30 Aug 2019: sequence = (tj ). nkj=1 such that tj < tjk.

30 Aug 2019: where Si1, Si2 correspond respectively to the swimming times of the strong swimmer andthe not so strong swimmer in the ith pair, for 1 6 i 6 3, and Tj corresponds ... a) Show that for j = 4,. 8, Tj has probability density function.

30 Aug 2019: State the dual Cauchy identity. Let F (t) =. j>0 fj tj be a formal power series, where f0 = 1.