Let us recall that a function s : [a,b] [right arrow] X is called simple if there is a

finite sequence [mathematical expression not reproducible] of Lebesgue measurable sets such that [E.sub.m] [intersection] [E.sub.l] = [theta] for m [not equal to] l and [mathematical expression not reproducible], and in this case the Bochner integral of s is [mathematical expression not reproducible].

Thus, it is conjectured that every n [member of] <Z [union] I> using the modified Collatz conjecture (3a -1) + (3b -1)I; a, b [member of] Z \ {0} or 3a - 1 if b = 0 or (3b + 1)I if a = 0, has a

finite sequence which terminates at only one of the elements from the set B.

When A is a structure and [??] [subset or equal to] U is a

finite sequence of elements of U, then [tp.sub.U]([??]) denotes the type of [??] in A, where the subscript U is dropped when the structure is clear from the context.

We say that a

finite sequence d of nonnegative integers is bipartite graphic if d can be realized as the degree sequence of both parts of a bipartite simple graph.

The Fourier transform of the

finite sequence x = ([x.sub.0], [x.sub.1],..., [x.sub.N-1]) is the sequence x = ([x.sub.0], [x.sub.1],..., [x.sub.N-1]) = F (x), with

Two knots are equivalent (via Reidemeister moves) denoted by the symbol ~, if and only if (any of) their projections differ by a

finite sequence of Reidemeister moves [4].

Given a denumerable set S and an infinite denumerable class C of overlapping directed circuits (or directed cycles) with distinct points (except for the terminals) in S such that all the points of S can be reached from one another following paths of circuit edges; that is, for each two distinct points i and j of S there exists a

finite sequence [c.sub.1], ...

Furthermore, for every

finite sequence ([c.sub.k]),

We start with a

finite sequence of positive integers, say (1,1,2,2,1).

Flip a coin in the long run; then in the short run you will witness any

finite sequence of coin tosses whatsoever.

The group code C produced by (3.1) and (3.2) will not be controllable if there are two states s and s' such that s [not equal to] v([u.sub.n], v([u.sub.n-1], v([u.sub.n-2], ...,v([u.sub.2], v([u.sub.1], s')) ...))), for any

finite sequence [{[u.sup.i]}.sup.n.sub.i=1] of inputs.