For then one can start with an expression, convert it to standard form, then convert back to any expression that is equivalent. … Given a particular axiom system that one knows reproduces all equivalences for a given operator one can tell whether a new axiom system will also work by seeing whether it can be used to derive the axioms in the original system. … Since before 1920 it has been known that one way to disprove the validity of a particular axiom system is to show that with k > 2 truth values it allows additional operators (see page 805 ).

[No text on this page] The universal cellular automaton emulating one step in the evolution of the rule shown above, which involves next-nearest as well as nearest-neighbor cells.

Error-correcting codes In many information transmission and storage applications one needs to be able to recover data even if some errors are introduced into it. … Then—somewhat in analogy to retrieving closest memories—one can take a sequence of length n that one receives and find the codeword that differs from it in the fewest elements. … PM[s], 2], 2]], -s] A number of families of linear codes are known, together with a few nonlinear ones.

Halftoning In printed books like this one, gray levels are usually obtained by printing small dots of black with varying sizes. … To give the best impression of uniform gray, one must in general minimize features detected by the human visual system. One simple way to do this appears to be to use nested patterns like the ones below.

If every node has say 4 connections, then eventually one gets dendrimers that cannot realistically be constructed in 3D. But long before this happens one runs into many alkanes that presumably exist, but apparently have never explicitly been studied. The small unbranched ones (methane, ethane, propane, butane, pentane, etc.) are all well known, but ones with more complicated branching are decreasingly known.

To show that a particular problem like the halting problem is undecidable one typically argues by contradiction, setting up analogs of self-referential logic paradoxes such as "this statement is false". … 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. (To make the proof rigorous one must add another level of self-reference, say setting up m' to ask m whether a Turing machine will halt when fed its own description as input.)

First-order transitions occur when a system has two possible states, such as liquid and gas, and as a parameter is varied, which of these states is the stable one changes. … Note that one feature of first-order transitions is that as soon as the transition is passed, the whole system always switches completely from one state to the other. … On one side of the transition, a system is typically completely disordered.

One can take the original stem to extend from the point -1 to 0; the rule is then specified by the list b of complex numbers corresponding to the positions of the new tip obtained after one step.

And unless one limits the number of elements k it is in general undecidable whether a given axiom system will allow no more than a given set of forms. … And as an example of this one can consider commutative group theory. … (The groups can be written as products of cyclic ones whose orders correspond to the possible factors of n .)

Finding layouts [for networks] One way to lay out a network g so that network distances in it come as close as possible to ordinary distances in d -dimensional space, is just to search for values of the x[i, k] which minimize a quantity such as With[{n = Length[g]}, Apply[Plus, Flatten[(Table[Distance[g, {i, j}], {i, n}, {j, n}] 2 - Table[ Sum[(x[i, k] - x[j, k]) 2 , {k, d}], {i, n}, {j, n}]) 2 ]]] using for example FindMinimum starting say with x[1, _]  0 and all the other x[_, _]  Random[] . … One can imagine weighting different network distances differently, but usually I have found that equal weightings work best. If one ignores all constraints beyond network distance 1, then one is in effect just trying to build the network out of identical rigid rods.