To compute the double

cosets for [[GAMMA].sub.0](p)[W.sub.p] with [mathematical expression not reproducible] with a,b,c,d [member of] Z such that ad - bcp = 1, [+ or -] [[beta].sub.m] [gamma][W.sub.p] [[beta].sub.n]

By definition, [OMEGA] consists of one and just one point from each distinct

coset [[gamma]] = [LAMBDA] + [gamma], i.e., [OMEGA] [intersection]([LAMBDA] + [gamma]) consists of a single point in [OMEGA].

In a

coset diagram for the action of PGL(2, Z) on PL([F.sub.q]), if a vertex v is fixed by w, then the vertex (v)t is fixed by [w.sup.*].

continue sampling for the

cosets of the -[A.sub.0][R.sub.i] +(H(id)-H(id*))TG,

For each such subset, called a

coset, its members differ by multiples of the "modulus" r.

Treble dodging minor methods: ringing the

cosets, on six bells.

Then B is fractal if and only if there exists a family [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of unital and fractal *-homomorphisms [W.sub.t] : B [right arrow] [C.sub.t] from B into unital [C.sup.*]-algebras [C.sup.t] such that, for every sequence {[B.sub.n]} [member of] B, the following equivalence holds: The

coset {[B.sub.n]} + B [intersection] N is invertible in B/(B [intersection] N) if and only if [W.sub.t] {[B.sub.n]} is invertible in [C.sub.t] for all t [member of] [T.sub.0].

The twelve elements of CUBE which map the tetrahedra onto each other form a

coset of A.

Furthermore, in order to resolve the scaling problems of Wyler's expression raised by Robertson, Gilmore showed why it is essential to use dimensionless volumes by setting the throat sizes of the Anti de Sitter hyperboloids to r = 1, because this is the only choice for r where all elements in the bounded domains are also

coset representatives, and therefore, amount to honest group operations.

The 4-cyclotomic

coset of modulo n is the set [C.sup.(4).sub.s] = {s, 4s, [4.sup.2]s,..., [4.sup.k-1]s}(modn), where k is the smallest positive integer such that [4.sup.k]s [equivalent to] s(modn).

(2) An affine (from the Latin, affinis, "connected with") subspace of a vector space (sometimes called a linear manifold) is a

coset of a linear subspace.