(1) F is said to be of strongly bounded variation (BV) on E if the number [mathematical expression not reproducible] is finite, where the supremum is taken over all

finite sequences {[[c.sub.i], [d.sub.i]] of nonoverlapping intervals that have endpoints in E.

It is conjectured every n [member of] <Z [union] I> using the modified form of Collatz conjecture has a

finite sequence which terminates at one and only element from the set A or B according as (3a + 1) + (3b + 1)I formula is used or (3a - 1) + (3b -1)I formula is used respectively.

If A [subset or equal to] M, then x [member of] M is definable from A iff there exist an L-formula [phi](v, [??]) and a

finite sequence [??] [subset or equal to] A such that x is the unique element v of M with M |= [phi](v, [??]).

Being given two

finite sequences x = ([x.sub.0], [x.sub.1],..., [x.sub.N-1]) and y = ([y.sub.0], [y.sub.1],..., [y.sub.N-1]) of the same of the same length N, is called discrete circular convolution of period N the sequence z = x [cross product] y = ([z.sub.0], [z.sub.1],..., [z.sub.N-1]), where

(f) As in (d), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for every

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

(1) Here the author shows how things change if you are allowed the option to quit early: Subject to some mild conditions, you should essentially always accept a sufficiently long

finite sequence of good bets.

We describe state transitions as follows: We use the notation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to denote that the

finite sequence of actions [Beta]' takes the system from state [N, L] to state [N', L'].

No

finite sequence of Yablo sentences is paradoxical.

Informally, the difference between a task and a super-task is that a task consists of a

finite sequence of actions, while a super-task consists of an infinite sequence of actions.

Observe that the positions of the multiples of 7 can be obtained by successively adding the elements of the

finite sequence 12,7,4,7,4,7,12,3.

In the other case, where [beta] < [[alpha].sup.-], from ([P.sub.3] : [[alpha].sup.-]), there exists a

finite sequence ([[gamma].sub.0], ..., [[gamma].sub.n]) [member of] [[OMEGA].sup.n+1] satisfying