So to avoid this one can consider systems that are more like circuits in which any element can get data from any other.

The element at position n in the first sequence discussed above can however be obtained in about Log[n] steps using ((IntegerDigits[#3 + Quotient[#1, #2], 2] 〚 Mod[#1, #2] + 1 〛 &)[n - (# - 2)2 # - 1 - 2, #, 2 # - 1 ]&)[NestWhile[# + 1&, 0, (# - 1)2 # + 1 < n &]] where the result of the NestWhile can be expressed as Ceiling[1 + ProductLog[1/2(n - 1)Log[2]]/Log[2]] Following work by Maxim Rytin in the late 1990s about k n+1 digits of a concatenation sequence can be found fairly efficiently from k/(k - 1) 2 - (k - 1) Sum[k (k s - 1) ((1 + s - s k)/(k - 1)) (1/((k - 1) (k s - 1) 2 ) - k/((k - 1) (k s + 1 - 1) 2 ) + 1/(k s + 1 - 1)), {s, n}] Concatenation sequences can also be generated by joining together digits from other representations of numbers; the picture below shows results for the Gray code representation from page 901 .

But it turns out that the standard methods used (such as finite difference and finite element) involve extremely similar computations for initial and for boundary value problems, leaving no trace of the significant differences between these cases that are so obvious in the discrete systems that we discuss in most of this book.

Here TMCompile creates a program segment for each element of the Turing machine rule, and TMToRM resolves addresses and links the segments together.

With the release of Mathematica in 1988, mathematical experiments began to emerge as a standard element of practical mathematical pedagogy, and gradually also as an approach to be tried in at least some types of mathematical research, especially ones close to number theory.

If f has an identity element, so that f[1, i]  i for all i , and has inverses, so that f[i, j]  1 for some j , then the system is a group.

In example (a), the canonical form is all elements black; in (b) it is a single black element, and in (c) all elements are black, except the last one, which is white if there were any initial white elements.

With completely random input, the probability that the length b subsequence which begins at element n is a repeat of a previous subsequence is roughly 1 - (1 - 2 -b ) n - 1 .

But as mentioned in the note above, it is not complete—since for example it cannot establish the theorem a  b which states that a group has just one element.

The word problem then asks if a given product of such generators is equal to the identity element.