Notations We use the customary symbol s for sets of number s N C Z C Q C R C C, lettin g N** C Zd Q • • - indicate tf-tuples a = (a 1 , ... , a d ) o f suc h numbers.W e let (R d )+ denot e the a € R d wit h af. 0 for all i, an d we let (Q d )+ an d (ZJ)+ b e the corresponding subset s of Q d an d R d . W e notationally distinguis h th e dual vector space (R d )* fro m R d , lettin g ev ... , e d an d e x , ... , e d b e the canonical base s for R d an d (R d )*, respectively.W e give (Rd)* th e dua l ordering (Rd):={Z«/e,:«,o}, and we le t R d an d (R J )* have the dual norms ItE^JL = max la/l P a /e/||j = Ste/I- Wecall A , = { / € (R*)* : / 0 , \\f\\x = 1 } th e fundamental d-simplex. We say that a n r x s matri x \p = [fc f .A , K,.. £ R is positive i f fc f - 0 fo r all /, /, an d we then write y 0 . Equivalently , if we regard y a s a linear map Rs — • Rr, we have a 0 im- plies that if{a) 0 (a similar statement applie s to the dua l spaces).W e denote the transposed matrix by p* and we identify i t with a matrix (Rr)* —y (R*) * in the usual way. Finally we let GL(r , Z) denote the group of r x r matrices with entrie s in Z and de- terminant ± 1, and we let G\Jj, Z) + b e the correspondin g matrice s with entries in Z + . I n general y £ GL(r , Z) + doe s not impl y ip~ l G GL(r, Z) + . http://dx.doi.org/10.1090/cbms/046/02

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