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6 Jul 2011: 1. Pedley TJ, Kessler JO (1992) Hydrodynamic phenomena in suspensions of swimmingmicroorganisms.

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

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},.

27 Jul 2011: J(p, q) TJ. (p, q) T. p. q. 2 (2.19). (p, q) T.

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

1 Sep 2011: More details on the problems relevant to k;p can befound in [1],[2],[5].Recall that Nj(t) := (tjk tj) [tj; tj1; : : :; tjk] ( t)k1+ ;so that suppNj = (tj; tjk); ... degree p such thatsuppMj;p = (tj; tjp1); Z tjp1tj Mj;p(x)dx = 1;and by!

1 Sep 2011: 2. Condition number and related constantsLet fN̂jg be the B-spline basis of order k on a knot sequence t = (tj), tj < tjk,normalized with respect to the Lp-norm (1 ... p 1), i.e.,N̂j(x) = (k=(tjk tj))1=pNj(x);where fNjg is the B-spline basis which

1 Sep 2011: x) : 1(yi1) = g(yi1); 01(yi1) = g0(yi1+);ki1(x) : ki1(yi) = g(yi); 0ki1(yi) = g0(yi);and for j such that Tj = Tj ,j(x) : 8><>: (r)j ... Hence 2 S(A0i;1); A0i 2 Xj: Tj=Tj 1:Since i(x) = maxf(x); p(x)g;we change into p at most once in each T j.

1 Sep 2011: The subintervals of and their lengthes will be denoted by. Ij := (tj, tj1), |hj| := tj1 tj. ... Nj = 1, (1.1.1). Mj (x) =k. tjk tjNj(x),. tjk. tj.