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Notes
Chapter 4
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Notes
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] "calculus"