Denote by [[[beta]].sub.[alpha]] = [beta] + ([alpha]) the residue class modulo [alpha] containing [beta].
where [x + yl] is the residue class of x + yl modulo N([alpha]).
Then there exists k [member of] Z coprimeto n[p.sup.e] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the residue class in Z containing x modulo n[p.sup.e].
One can verify that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is defines an action (as a group) of [H.sub.[alpha]] on the additive group of [Z[i].sub.[alpha]], where x + yi, [i.sup.s] and (x + yi)[i.sup.s] are interpreted as their residue classes modulo [alpha].
In earlier writings, Xenakis notated the transpositional index of a residue class as an exponent applied to its modulus.
By combining individual residue classes (henceforth RCs), Xenakis transforms their pedantic monotony into an astonishingly powerful and varied vocabulary of collections featuring complex patterns of interference between the various units of modular repetition.
Since 10 is very nearly half of 21, every other iteration of the mod-10 residue classes will intersect with the mod-21 RCs in similar ways.
Recall an extension of a local field is said to be tamely ramified if the degree is prime to the characteristic of the residue class field, and the corresponding residue class field extension is separable.
Suppose L is a maximal subfield of [D.sub.t] with residue class field L.
Let L/k((t)) be a maximal subfield of D, with residue class field L.
Let [A.sub.0] be a squarefree integer from the residue class A mod M.
Thus, the congruence [n.sup.2] = [m.sup.g] mod [p.sup.2] puts n in one of two residue classes modulo [p.sup.2].