...whatever, say zN, will appear in the development with the coefficient unity, for any positive integer N can be written in one and only one way in the form of a sum of powers of 2. We have, then, if z | < 1 , • (16) F(z) = l + z + z2+ ... which can also...

...whatever, say zy, will appear in the development with the coefficient unity, for any positive integer N can be written in one and only one way in the form of a sum of powers of 2. We have, then, if z | < 1 , (16) F(z) = l + z + z2 + ... +z"+ ••• =...

...whatever, say zy, will appear in the development with the coefficient unity, for any positive integer N can be written in one and only one way in the form of a sum of powers of 2. We have, then, if z | < 1 , (16) 1-z which can also be very easily obtained...

...the following series of primary points (5) - ¿(г) =g<^ _. tSv - 'sv-l Further every primary point x can be written in one and only one way in the form x = — — — ~ ' * * g'l O*SO*2v —1 9*2v ' where s0, «t, *2, ...*»— i, *2v are integers such...

...an independent family and (2) SDÎ = It is easy to see that ЯП = £ © SW» if and only if every x can be written in one and only one way in the form £r, , x, € Ш, . PROPOSITION 1. Let (ÜTÍ, | v £ N\ be an independent family of ЧИ-submoduks...

...is equivalent to one and only one of the units £, (/ = l, 2, . . . , /?2) is to say that every unit can be written in one and only one way in the form ±a*£,(l - aa)s'(l a2a)s2 ...(l — a^aYf. This gives a complete description of the cyclotomic units....

...following. THEOREM 21. Let U and W be complementary snbspaces of vector space V. Then every coset of W in V can be written in one and only one way in the form u + W with u in U, and this correspondence between U and V/W is an isomorphism of vector spaces. In...

...support) implies X, = 0 for all t б Т ; to say that (л,)(ет is a basis of E means that every xe E can be written in one and only one way in the form x = 2* £t<z(; for all / e T, Ç, is then called the component (or coordinate) of x of index t with...

...is said to be the free product n* (AIA € 5} of the set 5 of subgroups if each g GG, where g * 1, can be written in one and only one way in the form xe A e S,A.#A.+ l . Note that a free group is a free product of a set of infinite cyclic subgroups....

...multiplicative, ie, that f(mn) = f(m)f(n) whenever gcd(m, n) = 1. To see this, we note that any divisor dlmn can be written (in one and only one way) in the form d 1 . d 2 , where diem, d 2 In. Since gcd(d 1 , d 2 ) = 1, we have ço(d) = p(di)p(d 2 ), because of...