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11 Sep 2015: for some a,b K. The discriminant condition is 2b(a2 4b) 6= 0. ... for some ξ3 K.Let L = K(. b/ξ1) and write f(l,m) = κ(l εm)(l εm) where κ K and.

22 Sep 2015: a. (1 exp(at). )exp at. = B exp at Ca. for some constant B. ... for some constant B. 106. If h(t) = C exp at for all t,.

10 Jul 2015: ω there is some n P N with xxiyi n P B.If t,u P B we write t u if there exist m n,x1. ... 26. (a’) Φ1 P Uηδλ. (b’) Θ P U¤ηδλ ñ Θ ¤ Φ1.By (a) we have χ1 tppL,l1q for some l1 : L Ñ U and tppLq ηδλ.Define a

7 Aug 2015: blow-ups.By Lemma 3.4(b), there exists some a > 0 such that πaτOY τ](OX) τOY.Applying Γ(X ,) produces inclusions πaO(Y) O(X) O(Y) inside ... X). Bysymmetry, we also obtain πbO(Y) O(X ′) O(Y) for some b > 0.

20 May 2015: i) = 0 for some λi F not all zero.Suppose that λk 6= 0 and let B be the matrix with ith column ei for i 6= k and. ... vector spaces. Definition. We say matrices A and B are similar (or conjugate) if B = P1APfor some invertible matrix P.

29 Nov 2015: M (n) = 3M (n/2) kn. for some constantk that covers the additions, subtractions and overheads. ... The mostit can do is to give up neater and easier ways of describing some of these.

2 Dec 2015: tmin T distinct such that t0 =. mi=1 λiti for some λi F. ... LINEAR ALGEBRA 15. Suppose first thatki=1 λifi = 0 for some λi F.

11 Jul 2015: Let n be leastsuch that g[m] <Q f[n] for some m. ... Notice that some authors (e.g., Forster [2] and Marcone [9]) require that a block Bown a unique initial segment of any given X [. B]ω. This is the notion of

10 Dec 2015: We now give some indications on the proof, concentrating for definiteness on the more difficult case,n = 8. ... are regular algebraic up to twist(and which is solved in some cases in 6).

9 Jan 2015: Try some plots. Do they enlighten you? i <- 1:3 ; ttax <- t(tax). ... These pairwise plots result in Figures 4.1 and 4.2 respectively, and show some separation betweenthe three groups.