Note that if all blocks in a cyclic tag system with n blocks have lengths divisible by n , then one can tell in advance on which steps blocks will be added, and the overall behavior obtained must correspond to a neighbor-independent substitution system.

To construct a cellular automaton analog of DLA one can introduce gray as well as black and white cells, and then have the gray cells represent pieces of solid that have not yet become permanently attached to the main cluster.

(A better approach may be one similar to general relativity.)

Such objects have the property that they are not left invariant by 360° rotations, but only by 720° ones—a feature potentially fairly easy to reproduce with networks, perhaps even without definite integer dimensions.

(Since the development of wire rope in the 1870s precise designs have been used, including at least recently the 7×7×7 one shown last below.) … Games are normally based on definite rules, but are set up so that at each step they involve choosing one of many possibilities, either by skill or randomness. … From antiquity until about the 1500s the majority of mazes followed a small number of designs—most often based directly on the one shown on page 873 , or with subunits like it.

Before the example above one might have thought that primitive recursive functions would always have to show rather simple behavior. … But given an enumeration of primitive recursive functions (say ordered first by LeafCount , then with Sort ) in which the m th function is w[m] diagonalization (see page 1128 ) yields the function w[x][x] shown below which cannot be primitive recursive. … But by reducing the results modulo 2 one gets a function that does not grow—and has seemingly quite random behavior—yet is presumably again not primitive recursive.

Beginning as early as the 1700s, the foundations of statistical analysis have been vigorously debated, with a succession of fairly specific approaches being claimed as the only ones capable of drawing unbiased conclusions from data.

One example is finding buildings or machines from aerial reconnaissance images; another is finding boat or airplane wreckage on an ocean floor from sonar data.

The most common are ones based on repetition or iteration, classic examples being Euclid's algorithm for GCD (page 915 ), Newton's method for FindRoot and the Gaussian elimination method for LinearSolve .

(One can always set up the analog of types by having rules only for expressions whose heads have particular structures.)