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For any sequence with an algebraic generating function and thus for any nested sequence the n th element can always be expressed in terms of hypergeometric functions.
By adding a suitable constant to each element one can then arrange in such cases for the whole spectrum to be flat.
Growth rates [in substitution systems]
The total number of elements of each color that occur at each step in a neighbor-independent substitution system can be found by forming the matrix m where m 〚 i, j 〛 gives the number of elements of color j + 1 that appear in the block that replaces an element of color i + 1 .
Implementation [of cellular automaton state networks]
One can represent a network by a list such as {{1 2}, {0 3, 1 2}, {0 3, 1 1}} where each element represents a node whose number corresponds to the position of the element, and for each node there are rules that specify to which nodes arcs with different values lead.
But in strings where each element is considered equally important, no such layout is possible.
And after a few earlier attempts, Yang–Mills theories were introduced in 1954 by extending the notion of a phase factor to an element of an arbitrary non-Abelian group.
If one starts from the single-element set {1} then applying Union , Intersection and Complement one always gets either {} or {1} .
Bitwise optimizations [of cellular automata]
The C program above stores each cell value in a separate element of an integer array.
For a fair fraction of values of m , the hash codes obtained from this scheme change whenever any element of list is changed.
Given a list of blocks such as {{1, 1}, {0}} each element of Flatten[list] can be thought of as a state in a finite automaton or a Markov process (see page 1084 ).