Epicureans
and atomism, 876
and complexity, 861
and contents of space, 1028
Epicurus (Greece, 341-270 BC)
and randomness, 967
Epicycles
as examples of models, 992
Epistemology
and computational irreducibility, 1132
and Gödel's Theorem, 1167
implications for, 1196
and models vs. reality, 365, 991
EPR (Einstein-Podolsky-Rosen) experiment, 1064
epsilon_0 (transfinite ordinal), 1162
and unprovability in arithmetic, 1163
epsilon-delta arguments, 1177
Equal ( == )
calculus of expressions, 1170
in multivalued logic, 1175
non-universality of, 1174
truth tables for, 802
Equality testing
randomized methods in, 1192
Equational logic
automated proofs in, 1158
proofs in, 1172
Equational systems
vs. modus ponens, 1151
operator systems as, 801
see also Operator systems
Equations
antidiffusion, 923
axioms as, 1153
computation based on, 731
and concept of purpose, 1185
as constraints, 940
and defining mathematics, 860
as defining numbers, 916
in different axiom systems, 812
difficulties in models with, 368
difficulties of solving, 1129
in history of science, 44
integer
see Diophantine equations
linear, 940
ordinary differential, 922
partial differential, 161-164
sequence, 1141
for sequences, 944, 1141
for snowflakes, 372
of state and gravity, 1053
for strings, 944, 1141
time vs. space in, 1032
undecidability of integer, 787
see also Burger's equation
see also Differential equations
see also Diffusion equation
see also Diophantine equations
see also Dirac equation
see also Einstein equations
see also Functional equation
see also Hodgkin-Huxley equation
see also KdV equation
see also Klein-Gordon equation
see also KPZ equation
see also Kuramoto-Sivashinsky equation
see also Laplace equation
see also Lorenz equations
see also Maxwell's equations
see also Navier-Stokes equations
see also Nonlinear Schrödinger equation
see also Partial differential equations
see also Polynomial equations
see also Quartic equations
see also Quintic equations
see also Schrödinger equation
see also Sine-Gordon equation
see also Transcendental equations
see also van der Pol equation
see also Wave equation
Equiangular spirals
generated from curvature, 418
in shells, 1008, 1008'
Equidistribution
and concatenation sequences, 913
of digit sequences, 912
of sequences, 903
statistical test for, 1084
in three-squares problem, 910
Equilibrium
approach to, 449
in chemical reactions, 984
in early universe, 1055
in Ising model, 982
and quantum measurement, 1063
in random networks, 1038
in self-gravitating systems, 1021
in thermodynamic systems, 450
Equivalence
between logic expressions, 775
in context-free languages, 1138
of NP-complete problems, 1145
in operator systems, 801
Principle of Computational
see Principle of Computational Equivalence
topological of manifolds, 1051
undecidability in problems of, 1138
see also Universality
Equivalence operations
and confluence, 1036
Equivalence principle, 530, 1047
and gravitational energy, 1054
and particle production, 1062
Equivalential calculus, 803, 1170
possible models of, 805
theorems in, 813
Eratosthenes (Egypt, 276-195 BC)
and prime sieve, 909, 909'
Erector sets
and history of CAs, 876
Erf (error function)
in diffusion equation, 1024
Ergodic theory
history of, 1020
Ergodicity
in dynamic spin systems, 982
of hard sphere gas, 1022
and Lissajous figures, 917
and rule 37R, 457
in search process, 347
vs. Second Law, 1020
Ermentrout, G. Bard (USA, 1954- )
and patterns on shells, 1012
ERNIE (randomness generator), 969
Erosion
and landscape structure, 1001
in mathematical morphology, 1077
Error-correcting codes, 1101
and data compression, 1069
found by searches, 1193
and Hadamard matrices, 1073
and quantum computers, 1148
and sphere packings, 986
and Voronoi diagrams, 987
Error-diffusion algorithm, 1077
Errors of measurement
and history of statistics, 1082
and origin of randomness, 967
Escapements
characteristic shapes of, 1183
Escher, Maurits C. (Netherlands, 1898-1972)
and ornamental art, 874
Esperanto (artificial language), 1103
Espresso, 1097
Essential incompleteness, 1159
Essential singularities
in QED perturbation theory, 1060
Essential undecidability, 1159
Estimation
of entropies, 959
in statistics, 1083
Ethane, 1194
Ether (physics)
atoms as knots in, 1044
as content of space, 1028
discrete model for, 1027
history of, 1024
and theory of light, 1041
Ethernet
randomization in, 1192
Ethics
and free will, 750
and responsibility, 1136
Euclid (?Egypt, ~300 BC)
and axioms for geometry, 1154
and foundations of math, 1149
and math in science, 859
and mathematical induction, 1152
and nature of space, 1028
and perfect numbers, 911
and primes, 909
theorem network of, 1176
and theory of lunes, 873
Euclidean geometry
systematic proofs in, 1177
Euclid's algorithm
and ContinuedFraction, 915
and GCD, 1093
as iterative algorithm, 1141
running time of, 1147
|  | Euler, Leonhard (Switzerland/Russia/Germany, 1707-1783)
and continued fractions, 914, 915
and a cubic Diophantine equation, 1165
and differential geometry, 1009
and equations for geodesics, 1049
and Fermat's Last Theorem, 1166
and number theory, 878
and perfect numbers, 911
and primes, 909
Euler circuits
and polynomial time, 1146
EulerGamma (Euler's constant)
and Egyptian fractions, 915
and growth of DivisorSigma, 910
as not from rational integral, 916
as transcendental number, 912
EulerPhi (totient)
and cycles in shift rules, 963
and cyclic multiplication, 950
and encoding of lists, 1120
and GCD pattern, 1093
and RSA cryptography, 1090
and shift register periods, 975
Euler's formula
for planar networks, 1038
and spherical networks, 1049
Europa
and extraterrestrial life, 1179
Evaluation schemes
and recursive sequences, 906
Evens and Odds game, 1105
Event horizons
and black hole formation, 1053
and particle production, 1062
and time and computation, 1130
Events
causal network of, 488
Evil
and free will, 1135
Evoked potentials, 1098
Evolution theory
summary of relations to, 14
see also Biological evolution
Exact solutions, 1133
for additive CAs, 610, 951
and computational irreducibility, 1132
for dimer problem, 959
for hard hexagon model, 959
for Ising model, 982
for iterated map periodic points, 955
for Kondo model, 1057
lack of for 3-body problem, 972
for logistic map, 1098
and math functions, 610
for multiway systems, 1168
to Navier-Stokes equations, 996
for shift map, 919
for Turing machines, 1144
see also Closed forms
Excess (flicker) noise, 969
see also 1/f noise
Excitable media, 1013
Excluded blocks
from 1D constraints, 941
in cellular automata, 278, 958
in code 20, 958
Excluded middle
law of in logic, 817, 1158
Exclusion principle (in quantum theory), 1059
Executive toys
and chaos theory, 1183
plasma-filled globes as, 995
Exhaustive search, 217, 343
and cryptanalysis, 604
and mechanism vs. purpose, 838
for optimal programs, 842
Existence of solutions
to constraints, 940
to Navier-Stokes equations, 997
in PDEs, 923
Exists
in predicate logic, 1151
and r.e. sets, 1139
Exogenous randomness, 301-314
in cellular automata, 947
Exoskeleton in arthropods, 1011
Exp (exponential function)
and canonical ensemble, 982
continued fractions for, 144
difficulty of evaluating, 1134
digits of values of, 141
in Gaussian distribution, 976
as non-computable, 1128
and path integrals, 1061
Expand
Boolean analog of, 1095
Expansion of universe, 1055
and arrow of time, 1021
and cosmological term, 1052
and Einstein equations, 1053
and network evolution, 530
and particle production, 1062
and relativity history, 1048
and thermodynamics, 455
Experience
and perception, 547, 634
Experimental design
and Hadamard matrices, 1073
Experimental mathematics
foundations of, 793
and foundations of math, 821
history of, 899
practical methods in, 1090
summary of relations to, 14
this book as work of, 863
see also Computer experiments
Experiments
and computational irreducibility, 748
crucial for this book, 23-39
detecting fraud in, 1184
experience of modelling and, 991
and learning from this book, 856
problems with chaos, 919
repeatable randomness in, 326, 976
and thermodynamic initial conditions, 444
ExpIntegralEi (exponential integral)
and Egyptian fractions, 915
Explicitness
in design of Mathematica, 1177
importance of, 880
Exploratory computer experiments, 46
Exponent
of groups, 1172
Exponential distribution
in laws for human behavior, 1014
of mobile automaton periods, 887
Exponential divergence
in chaos theory, 921
Exponential growth
in biology, 1010
in combinators, 713, 1123
in Makanin's algorithm, 1141
of modes in PDEs, 988
in multiway systems, 207
in primitive recursion, 908
in proof sizes, 1155
in substitution systems, 82, 890
in Turing machines, 762, 1144
Exponential system
periods in, 257
Export
and image compression, 1074
Expressions
number of Nand, 1175
numbering of, 1170
representing integers, 916
representing numbers, 143
structure of symbolic, 896
in symbolic systems, 102
EXPTIME (exponential time), 1143
Extended objects in quantum field theory, 1057
ExtendedGCD
and linear congruential cryptanalysis, 1089
and linear Diophantine equations, 944, 1164
Extensive quantities
in cellular automata, 1022
Extrapolation of data, 549
Extrasolar planets, 1179
Extraterrestrial intelligence, 822
history of, 1180
messages to send, 1189
and perception, 550
perception by, 635
search for, 1188
Extraterrestrial life, 825, 1179
Extreme value distribution
as general result, 977
for random walks, 978
Extrinsic curvature
and Einstein equations, 1052
and initial value general relativity, 1053
Extrinsic randomness, 301-314
Eyes
appearance of data to, 577
of flies, 385
randomness in saccades of, 1011, 1192
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