# Notes

# Chapter 4: Systems Based on Numbers

## Section 1: The Notion of Numbershttps://www.wolframscience.com/nks/chap-4--systems-based-on-numbers--notes#sect-4-1--the-notion-of-numbers

## Section 2: Elementary Arithmetichttps://www.wolframscience.com/nks/chap-4--systems-based-on-numbers--notes#sect-4-2--elementary-arithmetic

Substitution systems [and digit sequences]

Multiplicative digit sequences

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

## Section 3: Recursive Sequenceshttps://www.wolframscience.com/nks/chap-4--systems-based-on-numbers--notes#sect-4-3--recursive-sequences

Computation of [recursive] sequences

Properties of [recursive] sequences

## Section 4: The Sequence of Primeshttps://www.wolframscience.com/nks/chap-4--systems-based-on-numbers--notes#sect-4-4--the-sequence-of-primes

## Section 5: Mathematical Constantshttps://www.wolframscience.com/nks/chap-4--systems-based-on-numbers--notes#sect-4-5--mathematical-constants

## Section 6: Mathematical Functionshttps://www.wolframscience.com/nks/chap-4--systems-based-on-numbers--notes#sect-4-6--mathematical-functions

Differential equations [for sine sums]

Musical chords [from sine sums]

## Section 7: Iterated Maps and the Chaos Phenomenonhttps://www.wolframscience.com/nks/chap-4--systems-based-on-numbers--notes#sect-4-7--iterated-maps-and-the-chaos-phenomenon

Exact iterates [in iterated maps]

Problems with computer experiments [in chaos theory]

Mathematical perspectives [on iterated map complexity]

Information content of initial conditions

Higher-dimensional generalizations [of iterated maps]

## Section 8: Continuous Cellular Automatahttps://www.wolframscience.com/nks/chap-4--systems-based-on-numbers--notes#sect-4-8--continuous-cellular-automata

Implementation [of continuous cellular automata]

History [of continuous cellular automata]

Properties [of addition continuous cellular automata]

## Section 9: Partial Differential Equationshttps://www.wolframscience.com/nks/chap-4--systems-based-on-numbers--notes#sect-4-9--partial-differential-equations

Ordinary differential equations

Origins of the [partial differential] equations

Existence and uniqueness [in PDEs]

Equation for the background [in my PDEs]