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In general, we have the next result. 4 If N = ML and 7r is the perm,utation of ZIN defined by 7r(a bM) = b aL, 0 < a < M, 0 < b < L, then P(70= P(N,L). Consider the set of N x N permutation matrices {P(N,L) LI NI . 10) We will describe the permutation matrices in this set in terms of the unit group U(N - 1) of Z/(N - 1). The unit group U(N -1) is given by U (N - 1) = {0 < T < N - 1 : (T - 1) = 1} . , N - 21. 3 Stride Permutations Define the permutation 71-7-, of Z/N by the two rules 72-(k) kT mod (N —1), 0 < k < N —1, 71-2-(N —1) = (N —1).

6 Parallel Implementation 51 Alternatively, we can use the identity A0 = (P(2M, 2) 0 /L)(im 0 A® h)(P(2M, M) 0 IL). Consider the factor im 0 A0 IN. As above, we can implement the action by M parallel computations of A0 IN. If MN processors are available, we can use the identity im 0 A 0 = P(2MN,2M)(1114N 0 A)P(2MN,N) or the identity im 0 A 0 = (Im P(2N,2))(ImN 0 A)(1m P(2N, N)) to compute Im A0 IN as MN paxallel computations of A. In this way, we naturally control the granularity of the parallel computation and fit the computation to granularity and to the number of available processors.

Every divisor of f(x) and g(x) in F[x] divides d(x). Equivalently, d(x) is the unique monic polynomial over F, which is a common divisor of f (x) and g(x) of maximal degree. We call d(x) the greatest common divisor of f(x) and g(x) over F and write d(x) = (f (x), g(x)). By the divisibility condition above, (f (x), g(x)) =- a(x) f (x) b(s)g(x), where a(x) and b(x) are polynomials over F . 22) for some polynomials ao(x) and bo(x) over F. Arguing as in section 2, we have the following corresponding results.

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Methods of modern mathematical physics by Ataraxiainc

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