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16 May 2006: Write B for Bi(A). Then for the set-systems A and B, wehave |A| = |A||A+||A4A+| and |B| = |B||B+||B4B+|.Now, |B| |A| and |B+| |A+| (by the ... system B = Ak satisfies |B| = |A|, |B| |A| and Bi(B) = B forall i.

1 Jun 2006: A A A (1 A) and A (1 A) =. (ii) |A| = min{|B| : A B (1 B)}.(iii) |A| = max{|B| : B A,B (1 B) = }. Proof: (i) We may assume ... B : B Z/nZ, |B| = r 1,B (1 B) = }. In particular I(SecrΓn) is generated by a space of (r 1)-ics of dimen-sion β(r 1,n).

22 Oct 2006: Then there exist a A, b B and c C with ab = c.In particular, this is true if all of A, B and C have size greater than 2n8/9. ... Equivalently, there exist a A and b B suchthat ab = c, as claimed.

13 May 2006: Then thereis an isomorphism of Brauer-Severi diagrams defined over K,. C //. φ. Pn1. ψ. E|n.0|. // Pn1. with σ(φ)φ1 = τξσ. We lift ψ to a matrix B ... It is evident that yσ =σ(B)B1 is an element of ΘE projecting onto ξσ.

26 Apr 2006: 49:904[þ%Hü;490HS68P?ý?üxú40hø684 F úLP?ý?þ%?ýý70þ:AYý76a=b[Rc;dfe , }r6n.E.32g4h4? ... þ6.04ø[ÉÈLÊ$[,ûB2iË([%b [%b ¿ dfe.