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15 Aug 2012: In advanceof stating this theorem, we present some further notation. Let C1,C2,. ... πD be a collection of site-disjoint green paths of Γ1, each joining some.

15 Aug 2012: for some functions a and b which are slowly varying as p pc. ... Potts, R. B. (1952). Some generalized order-disorder transformations. Proceedings of the Cam-.

15 Aug 2012: Theprobability that this construction ultimately includes some vertex of B dependstherefore only on the marginal probabilities. ... If Un Vn in D(n), we may pick an oriented path π of D(n) joining some.

28 Feb 2012: where α = 12(1. 71). Find an element of OK with norm 2a 3b 5 for some. ... Hint: Use that some power of a is principal.]. The following extra questions are on cyclotomic fields.

26 Oct 2012: or(b)is of not less than five years’ standing from admission to his or her first degree of some other university and has been admitted (i) to some office in ... case the Degree Committee have allowed a candidate to submit material in some other language

5 Nov 2012: 3.17. (i) Let f K(X). Show that K(X) = K(f) if and only if f = (aX b)/(cX d)for some a, b, c, d K with ad ... Deduce that every finite abelian group is the Galois group of some Galois extension K/Q.[It is a long-standing unsolved problem (inverse Galois

11 Apr 2012: More generally, for A, B Zdwe write A B (and similarly for the other relations) if a b for some a A and b B. ... directed edge from some x to x ed liesin no more than one of πa,b , πu,v.

6 Jun 2012: large number of individual members each interacting with their environment in some way. ... sequence of interactions.). In this course we consider mostly approach (A), but include some of approach (B).

24 Jul 2012: exp(iNαj) λj| < ǫ. for each 1 j M. Our construction requires some preliminary results. ... 14. 5 Some simple geometry of numbers. We need the following extension of Theorem 35.

8 Nov 2012: X R which satisfies (for some positivenumbers‖B‖, γ):. • ... whereP is any strictly positive polynomial (i.e.P (x) θ > 0 for some realθ.