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15 Sep 2011: chains Mj,K, t > 1,which have the same state space w MJ,K = (S,K, TJ,K ),but have product matrices J,K for transition functions, i.e.,chains in ... The tint is theta [ 1, an n(,K x. larray of integers,that is used as a sparse representation for the

15 Sep 2011: Corollary 3.6 (minimization of the expected weighted sum ofjob tardinesses). The tardiness of job j, Tj, is defined as max(Zj - Dj, 0). ... Suppose (A 1) and (A 4) hold, and D, : D2 _? 5 D, a.s. Then n minimizes E[ I=1 cj Tj]. Proof.

14 Nov 2011: 25. . Theorem 5.8. Suppose P is irreducible and recurrent.Then for all j I we have P(Tj < ) = 1.

7 Apr 2011: 14. Opposite page: TJ & Pat (pilots), Peter and me. This page, starting at the top going clockwise: Maasai hut, view of Maasai settlement from the plane, the clinic room with examination

18 Jul 2011: where zi = z(Ti). Alternatively, since. r(Ti, Tj) =. {x ei if i = jy/(x ek) if {i, j, k} = {1, 2, 3},. ... ρ(Ti, Tj) =. {αi if i = jb/αk if {i, j, k} = {1, 2, 3},.

29 Jul 2011: 1n1 00 αj. )m. ]. One computes easily that V j = qjn1/2t(Tj). ... Note that since t is analgebra homomorphism, the fact that the Tj commute implies that theoperators V j must also commute.

29 Jul 2011: BJ,TJ)) is a Borel andmaximal torus in I (resp. J). (This follows from the fact that anysmooth connected soluble subgroup of (resp. ... isogeny of I onto its image I, which is a semisimple algebraic group.Note that T/Fl = TI TJ and that B/Fl = BI BJ

15 Nov 2011: 3) Let Tj be the jth Chebychev polynomial. Suppose γj is a sequence of non-negative numberswith.

12 Jul 2011: Representation Theory. Lectured by S. Martin. Lent Term 2009, 2010, 2011. 1 Group Actions 1. 2 Linear Representations 3. 3 Complete Reducibility and Maschke’s Theorem 7. 4 Schur’s Lemma 10. 5 Character Theory 13. 6 Proofs and Orthogonality 17. 7

24 Jan 2011: tr](this means that σ is a product of disjoint cycles of length t1 > > tr where n = t1 tr,and some of the tj may be equal to 1.