Suppose there exist an

n-tuple of strategies [mathematical expression not reproducible] and a trajectory [mathematical expression not reproducible] satisfying (2).

Let J be a distinct-integer

N-tuple ([j.sub.1], ..., [j.sub.N]), and let [mathematical expression not reproducible] be the system (11.5), where [[??].sub.p,jl] are the jl-th p-adic w-s motions (11.2), for all l = 1, ..., N.

Indeed the similar conclusions could be obtained for the

n-tuple [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we leave the proof for the readers.

An m-repeated low-density burst of length b(fixed) with weight w or less is an

n-tuple whose only non-zero components are confined to m distinct sets of b consecutive positions, the first component of each set is non-zero where each set can have at the most w non-zero components (w [less than or equal to] b), and the number of its starting positions in an

n-tuple is among the first n - mb + 1 positions.

of real

n-tuples or, in 'frequency domain', by component-wise multiplication of (complex)

n-tuples,

An

n-tuple ([a.sub.1], [a.sub.2], ...,[a.sub.n]) is the identity

n-tuple, if [a.sub.k] = +, for 1 [less than or equal to] k [less than or equal to] n, otherwise it is a non-identity

n-tuple.

Note that the ambiguity in choosing an

n-tuple for the weight [lambda] amounts to an integral translation of GT([lambda]), and hence does not affect its number of integral points.

We write m = [summation over (n/j=1)] [m.sub.j][e.sub.j], where [e.sub.j] is an

n-tuple with 1 in the jth position and 0's elsewhere, and we define 0 = (0, ..., 0).

Next, enumerate the elements of S and then, in the usual way, identify each subset of S with an

n-tuple of 0's and l's.

A voting pattern is an assignment of a strategy to each participant, in the form of an

n-tuple.

A CSP is a triple P = (X, D, C), where X is an

n-tuple of variables X = ([x.sub.1], [x.sub.2], ..., [x.sub.n]), D is a corresponding

n-tuple of domains D = ([D.sub.1], [D.sub.2], ..., [D.sub.n]) such that [x.sub.i] [member of] [D.sub.i], and C is a f-tuple of constraints C = ([C.sub.1], [C.sub.2],...., [C.sub.t]).

An

n-tuple ([a.sub.1], [a.sub.2], ..., [a.sub.n]) is symmetric, if [a.sub.k] = [a.sub.n-k+1], 1 [less than or equal to] k [less than or equal to] n.