In many fields, advanced education seems useful only if one intends to pursue those specific fields.

With this setup, a network consisting of just one node is {{1, 1}} and a 1D array of n nodes can be obtained with CyclicNet[n_] := RotateRight[ Table[Mod[{i - 1, i + 1}, n] + 1, {i, n}]] With above connections represented as 1 and the below connections as 2 , the node reached by following a succession s of connections from node i is given by Follow[list_, i_, s_List] := Fold[list 〚 #1 〛 〚 #2 〛 &, i, s] The total number of distinct nodes reached by following all possible succession of connections up to length d is given by NeighborNumbers[list_, i_Integer, d_Integer] := Map[Length, NestList[Union[Flatten[list 〚 # 〛 ]] &, Union[list 〚 i 〛 ], d - 1]] For each such list the rules for the network system then specify how the connections from node i should be rerouted.

Known as sequency order, it has the property that each row involves one more change of color than the previous row.

Of the sequences on page 594 , (a) through (d) as well as (f) fail every single one of the tests, (e) fails only the serial test, while (g) and (h) pass all the tests.

And in addition, with sufficient effort, I believe one should be able to construct an automated system that will allow many universality proofs of this general kind to be found almost entirely by computer (compare page 810 ).

There are Ceiling[a/2] + Ceiling[2 a/3] - (a + 1) solutions, the one with smallest x being {Mod[2 a + 2, 3] + 1, 2 Floor[(2a + 2)/3] - (a + 2)} .

The so-called free semigroup has no relations and thus no rules, so that all strings of generators correspond to distinct elements, and the Cayley graph is a tree like the ones shown on page 196 .

As it happens, in 1973, as one of my earliest computer programs, I created a simulation of essentially the same kind of system (see page 17 ).

In the 1800s more detailed analogies began to emerge, sometimes as offshoots of the field of morphology named by Johann Goethe , and sometimes with mathematical interpretations, and in 1917 D'Arcy Thompson published the first edition of his book On Growth and Form which used mathematical methods—mostly from analytical geometry—to discuss a variety of biological processes, usually in analogy with ones in physics.

Black regions represent yin and white ones yang.