(Note that with finite-precision arithmetic, some exponentially small probability exists that truncation of numbers will lead to incorrect results.)

With r = 2 , it appears that if a solution exists, it must have length n + 4 or less. … To know that a solution exists in a particular case, it is sufficient just to exhibit it. … If only one color of element ever appears this is the complete condition for a solution—and for r = 2 solutions exist if Apply[Times, d] < 0 and are then of length at least Apply[Plus[##]/GCD[##]&, Abs[d]] .

It is known that for sufficiently large n a gap of any size must exist.

What the values for these blocks should be can be found by solving a system of linear equations; that a solution must exist can be seen by looking at the de Bruijn network (see page 941 ), with nodes labelled by size b + 2r - 1 blocks, and connections by value differences between size b blocks at the center of the possible size b + 2r blocks.

Diagrammatic and mechanical methods for minimizing simple logic expressions have existed since at least medieval times.

Before the input can have a chance of specifying meaningful action there are often all sorts of issues about whether variables in it refer to entities that can be considered to exist.

But long before this happens one runs into many alkanes that presumably exist, but apparently have never explicitly been studied.

For if one considers feeding m' as input to itself there is immediately no consistent answer to the question of whether m' halts—leading to the conclusion that in fact no machine m could ever exist in the first place.

But even given this it is still possible that a finite procedure can exist which decides whether a given statement is true, and indeed this happens in the theory of commutative groups (see note below ).

In quantum field theory particles of any mass can always in principle exist for short times in virtual form.