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And as a result, I had no choice but to study a very simplified version of the process in the book.
And of the 25 of these that are not trivially equivalent, it then turns out that the two given as (g) and (h) on the facing page can actually be proved as on the next two pages [ 810 , 811 ] to be axiom systems for logic—thus showing that in the end quite remarkable simplification can be achieved relative to ordinary textbook axiom systems.
Boole's work was progressively clarified and simplified, notably by Ernst Schröder , and by around 1900, explicit axiom systems for Boolean algebra were being given. Often they included most of the 14 highlighted theorems of page 817 , but slight simplifications led for example to the "standard version" of page 773 .
One example that allows easy visualization is a simplification of several common games known as nim.
I have always tried to read original writings—for I have often found that later characterizations drop elements crucial for my purposes, or recast history to simplify pedagogy.
But some simplified ordinary differential equations which potentially approximate various situations in fluid flow can be more amenable to analysis—and can exhibit the chaos phenomenon.
But while this axiom is convenient in simplifying work in set theory it has not been found generally useful in mathematics, and is normally considered optional at best.
And in general, the probabilities for all 8 possible combinations of 3 cells are given by
probs = Apply[Times, Table[IntegerDigits[8 - i, 2, 3], {i, 8}] /. {1 p, 0 1 - p}, {1}]
In terms of these probabilities the density at the next step in the evolution of cellular automaton with rule number m is then given by
Simplify[probs .
But the table below gives for example the actual algebraic formulas obtained in the case a = 4 after applying FullSimplify —and shows that these increase quite rapidly in complexity.
In 1962, however, Edward Lorenz did a computer simulation of a set of simplified differential equations for fluid convection (see page 998 ) in which he saw complicated behavior that seemed to depend sensitively on initial conditions—in a way that he suggested was like the map x FractionalPart[2x] .