Chapter 4: Systems Based on Numbers

Section 1: The Notion of Numbers

Implementation of digit sequences Gray code A note for mathematicians [about numbers] History of numbers History of digit sequences

Section 2: Elementary Arithmetic

Substitution systems [and digit sequences] Digit counts Negative bases Non-power bases Multiplicative digit sequences Powers of three in base 2 Leading digits [in numbers] Powers of 3/2 General powers [of numbers] Multiples of irrational numbers Relation [of powers] to substitution systems Other uniformly distributed sequences Implementation [of 3/2 system] Reconstructing initial conditions [in the 3n+1 problems] A reversible [3n+1 problem] system Reversal-addition systems History [of reversal-addition systems] Digit reversal Iterated run-length encoding Digit count sequences Iterated bitwise operations

Section 3: Recursive Sequences

Recurrence relations Ackermann functions  Computation of [recursive] sequences Properties of [recursive] sequences History [of recursive sequences] Primitive recursive functions Ulam sequences

Section 4: The Sequence of Primes

History of primes Finding primes Decimation systems Divisors Results about primes History of number theory Tables of primes Numbers of primes Relative primes Properties [of number theoretic sequences] Trapezoidal primes Other integer functions Spectra [of number theoretic sequences] Perfect numbers Iterated aliquot sums

Section 5: Mathematical Constants

Digits of pi Rational numbers Digit sequence properties [Computing] square roots Nested digit sequences Concatenation sequences Specially constructed transcendental numbers Runs of digits [in numbers] Leading digits [in numbers] Continued fractions History [of continued fractions] Egyptian fractions Nested radicals Digital slope representation Representations for integers Operator representations Number classification

Section 6: Mathematical Functions

Mathematical functions Lissajous figures Two sine functions Differential equations [for sine sums] Musical chords [from sine sums] Three sine functions Substitution systems [and sine sums] Many sine functions FM synthesis Zeta function

Section 7: Iterated Maps and the Chaos Phenomenon

History of iterated maps History of chaos theory Exact iterates [in iterated maps] Problems with computer experiments [in chaos theory] Mathematical perspectives [on iterated map complexity] Information content of initial conditions Smooth iterated maps Higher-dimensional generalizations [of iterated maps] Distribution of chaotic behavior Lyapunov exponents Chaos in nature [Iterated maps from] bitwise operations

Section 8: Continuous Cellular Automata

Implementation [of continuous cellular automata] History [of continuous cellular automata] Properties [of addition continuous cellular automata] Additive [continuous cellular automaton] rules Probabilistic cellular automata

Section 9: Partial Differential Equations

Ordinary differential equations Klein–Gordon equation Origins of the [partial differential] equations Nonlinearity [in PDEs] [PDEs in] higher dimensions Singular behavior [in PDEs] Existence and uniqueness [in PDEs] Field equations Equation for the background [in my PDEs] Numerical analysis [of PDEs] Implementation [of my PDEs] [PDEs involving] different powers Other PDEs

Section 10: Continuous Versus Discrete Systems

History [of continuous versus discrete mathematics] [The word]

From Stephen Wolfram: A New Kind of Science [citation]