Index
A
Abelian groups, 517
Absorption, neutron, 2 , 3 , 13 , 21 , 22 , 24 , 25
Acceleration, 167 , 168 -171, 174
Active material, 18 , 21 , 22 , 24 , 25 , 34
ADAM, 431 , 432 -433, 436 , 444
Additive processes, 1
Adiabatic equation, 124
Algebra, 144 ;
Boolean, 348 -349, 446 , 479 , 480 , 481 , 482 , 491 , 494 , 498 , 518 , 520 ;
pattern, 492 , 494 , 496 (see also Relational structures);
polyadic, 480 ;
projective, 480 -481, 496 , 520 ;
relation, 492 -496
American Philosophical Society, xii
Amino acids, 466 , 469 , 470 , 471 , 472
analogies between, 513 , 518 ;
and complexity, 518 ;
criteria for, 514 ;
distance measures, 483 , 485 , 489 ;
transformations preserve, 523
Appell, Paul-Emile, 516 n, 530
Arithmetic:
complexity, 445 -463;
significance, 483
Artificial intelligence, 466
Automatic plotting devices, 301 , 303 . See also Oscilloscope
Automorphism, 492
Autonomous systems, 344
B
Baire categories, 350
Banach, Stefan, ix , x , 510 , 513
Bednarek, Alexander R., xii , xiv , 477 , 482 , 485 , 516
Behavior:
chaotic, 192 ;
convergence, 192 , 197 , 199 , 200 , 201 -202, 211 , 213 -217, 218 , 224 -225, 227 , 233 , 272 , 305 -306;
ergodic, xvi , 13 , 132 , 140 , 143 , 155 -162, 192 , 293 , 294 , 348 ;
of gas, 123 -129;
limiting, 189 , 190 , 192 , 193 , 198 , 201 , 207 -209, 221 , 227 , 296 (see also Oscillation);
pathological, 358 ;
qualitative, 131 ;
Bell, George, 527
Bendixson, I., 344
Bernouillian, formulas, 2 , 183 , 345 , 349
Beyer, William A., xiii , 399 , 445 , 465 , 469 , 482
Billowing, 126 -129
Binary reaction systems, 194 -286, 294 -296;
as commutative, 196 ;
convergence behavior of, 199 , 200 , 201 -202, 211 , 213 -217, 218 , 224 -225,
form stability in, 210 -213;
four-variable, 189 ;
as non-associative, 196 -197;
as non-commutative, 212 -213;
and ordinary differential equations, 225 -226;
oscillation in, 207 , 208 , 209 , 211 , 212 , 222 , 224 , 225 , 227
Biology:
distance in, 527 -528;
mathematics/metrics applied to, ix , xiii , xiv , 465 -475;
molecular, xiii
Birkhoff, George D., 157
Boltzmann, Ludwig, 162 ;
equation of, 18 , 417 , 418 -419, 422 -423;
H-function of, 419 -420, 421 , 427 ;
on kinetic energy, 417 , 418 -419
Boole, see Algebra, Boolean
Borel sets, 86
Borsuk, K., 517
Boundary:
in growth pattern, 382 , 383 ;
of vacuum, 123 , 125 , 126 -127
Boundary points:
double, 199 , 202 , 208 , 226 ;
fixed, 200 , 201 , 202 , 221 -222, 224 , 226 ;
periodic, 201 ;
Brain, xii , 532 . See also Nervous system
Branching processes, xi , 1 -15, 37 , 520 . See also Multiplicative processes
Broken-linear transformations, 139 , 151 , 152 , 161 , 301 , 345 -349, 350 , 351 , 352 -353, 372
Brouwer, L.E., fixed point theorem of, 201 , 202 , 330 n, 491
Brute force approaches, xvi , 121 , 304 , 305 -306
Burks, Arthur, 379
C
Carlson, Bengt, 34
Cartesian coordinates, 213 -215
Centre National de Recherche (France), xvi
Cenzer, Douglas, 517
Chain:
rule, 333
Classical mechanics, 122
Clifford, A.H., 492
Cluster:
analysis, 473 ;
of sets, 518 ;
Coding problems, 470
Collisions, 17 , 18 , 19 , 20 , 22 , 24 -25, 190 , 195 -196, 418
Combinatorics, xii
Complexity:
and analogy, 518 ;
arithmetic, 445 -463;
calculated, 447 -448;
complement, 448 ;
conditional, 460 ;
and entropy, 445 , 446 , 460 -461, 482 ;
mathematized, 518 -521;
for modulo a prime integer, 445 , 448 , 519 ;
of rational numbers, 522 , 523 ;
of real numbers, 522 -523;
Composition, 478 , 480 -482, 495 ;
functional and relational, 479 -492;
in integration, 483 ;
and projection, 480 .
See also Iteration
Compound pendulum problem, 144
Computers:
cathode ray tubes in, 128 -129;
combinatorial systems on, xvi ;
complexity calculated on, 447 ;
at Los Alamos, 122 (see also MANIAC);
mathematical research aided by, xv , xvi , 121 -138;
memory of, 132 ;
nonlinear transformations on, xvi , 139 -154, 297 -377;
Conjugacy, 294 , 295 , 345 , 348 , 349 , 351
Continuum mechanics, xiv , 121 , 122 -123
Convergence, 192 , 197 , 351 ;
in binary reaction systems, 199 , 200 , 201 -202, 211 , 213 -217, 218 , 224 -225, 227 , 233 , 272 ;
brute force applied to, 305 -306;
to fixed point, 4 , 305 , 308 , 309 -310;
under iteration, 37 ;
to limit set, 306 , 321 , 345 ;
linear, 337 ;
Frobenius-Perron theorem of, 218 ;
region of, 211 n, 222 , 308 -309, 340 ;
uniform, 82 -83
Cooper, Leon, N., 484 -485
Cooper, Ralph, 447
Cosmic rays, 131
Crestey, M., 492
Criticality, 18 , 19 , 20 -21, 25 -26;
below-, 65 -84;
generating function in, 9 -10;
just-, 68 , 70 , 71 , 72 , 74 , 76 , 77 ;
strong ratio theorem for, 106 -118;
sub-, 4 , 7 , 9 -10, 65 -84, 118 ;
super-, 4 -5, 6 -7, 37 -38, 49 -60, 106 -118
Cubic transformations, 139 , 141 , 144 , 149 , 161 , 300 , 306 , 308 ;
modified, 326 -343;
three-variable, 293 , 298 , 299 , 309 -311, 312 , 315 -321, 326 -343, 355 -356
Curve, closed. See Limit sets, Class I
D
Darwin, Charles, 430
and criticality, 4 , 5 , 73 -74;
distribution of, 73 -74;
fixed point, 43 , 44 , 85 , 105 ;
in growth pattern, 379 , 380 , 381 , 382 -384, 385 ;
probability of, 4 , 5 , 7 , 67 , 73 -74, 105
Déblais et ramblais problem, 516 n, 530
Decomposition, 97 -98, 104 , 106
Density distribution, 127 , 130
Difference equations, 121 , 300 , 326 ;
linear, 323 ;
Differential equations, 344 -345;
approximating, 413 -414;
hyperbolic, 478 ;
nonlinear, 300 ;
ordinary, 225 -226;
parabolic, 478 ;
partial, 12 , 121 , 122 -123, 140 , 160 ;
Discrimination/discernment, 20 , 526 . See also Recognition
Diseases, contagious, 400
Displacement, 20 , 139 , 140 , 145 , 151 , 153 , 160 , 161 , 338 n. See also Perturbation
Dissimilarity/difference, 465 -466, 467 -470, 473 . See also Distance
Distance (metric), 85 , 98 , 514 , 523 ;
analogy/similarity measured by, 483 , 485 , 489 ;
calculated, 469 -470, 473 , 487 -488, 489 ;
defined, 468 -469, 470 -471, 472 , 486 , 491 n, 515 , 516 , 527 , 528 , 530 ;
betwen DNA code sequences, 526 -528;
Hamming, 527 ;
Hausdorff, 341 , 468 -469, 471 , 472 , 482 , 488 , 516 , 518 , 529 , 530 ;
Markov, 472 ;
perihelion, 22 ;
properties of, 526 ;
pseudo-, 466 ;
for recognition, 526 -533;
semi, 467 ;
between sets/classes, 466 , 468 -469, 485 , 486 , 492 , 516 -517, 518 , 529 , 530 -531;
DNA, 467 , 469 -470, 471 , 472 , 514 , 526 -528
Dog-bone pattern, 382
Domb, C., 400
Dynamical systems:
ergodic, behavior of, xvi , 155 -162;
flow in, 156 -157;
phase space in, 156
Dynamics, 129
Dyson, Freeman, 163
E
Eddington, A. S., 446
Efficiency, theory of, 19 , 20
Einstein, Albert, 417 , 422 -423
Electrons, 38
Endomorphism, 492
Energy:
equipartition of, 139 , 140 -142, 143 , 159 , 160 , 161 -162, 186 , 187 , 218 ;
via fusion, 155 ;
gravitational, xiii , 185 -188;
kinetic, 125 , 130 , 141 -142, 146 , 154 , 160 , 166 , 185 , 186 , 187 , 188 , 417 , 418 -419;
Maxwellian distribution of, 186 -187;
potential, 130 , 141 -142, 146 , 160 ;
transformation of, 144
ENIAC, xiii , 19 , 20 -21, 25 -26
Entropy, 122 ;
complexity and 445 , 446 , 460 -461, 482
Epidemics, 37
Equilibrium, approach to, 156 , 161 , 162
Ergodic:
average, 349 ;
behavior, xvi , 131 , 132 , 140 , 143 , 155 -162, 192 , 293 , 294 , 348 ;
motion, 144 ;
theorem, 143 , 155 -158, 399 , 409 -410;
transformations, 155 , 158 , 186
Euclidian:
space, 39 , 54 , 156 , 217 , 293 , 298 , 515 , 517 , 523 , 526
Eulerian variable, 125 -126
Evans, Trevor, 495
PM, 438 -440;
Everett, C.J., xi , xii , 1 , 37 , 163 , 188 , 417
Evolution:
Darwinian, 430 ;
in mathematics, 522 ;
via mutations, 287 , 430 -431, 432 -433, 434 -442, 443 -444;
rate/development of, 429 -430, 432 , 434 -442, 443 , 468
Evolutionary trees, 466 , 469 , 470 , 471 , 472 , 473
Explosion:
external, 163 -177;
history-dependent, 402 -405;
nuclear, 163 -177, 179 -180, 182 -183;
velocity of, 165 -166
F
Faber, Vance, 480
Feller, W., 400
Fermi, Enrico, x , xii , 139 , 156
Fine-structure, 357
Fission, xi , 19 , 21 , 22 , 24 -25;
bombs, 164 -165;
Fixed points, 6 , 8 , 37 , 40 , 42 -45, 198 , 337 , 338 , 361 , 368 , 372 , 481 ;
attractive, 202 , 294 , 299 , 308 , 321 , 330 -333, 345 , 348 ;
boundary (see Boundary points);
Brouwer on, 201 , 202 , 330 n, 491 ;
for continuous function, 4 ;
convergence to, 4 , 305 , 308 -309, 310 ;
equation, 234 ;
inside gap, 358 ;
interior, 199 , 200 , 201 , 202 -207, 208 , 210 -211, 212 , 213 , 215 , 222 , 223 , 224 , 225 , 226 , 314 , 328 ;
invariant points as, 294 -299;
iteration behavior of, 221 , 327 ;
limit points as, 43 ;
limit sets as, 330 ;
nodal, 199 , 200 , 201 , 213 , 222 ;
non-attractive, 225 ;
repellent, 202 , 294 , 299 , 316 , 328 , 330 -333, 342 , 345 , 354 , 359 ;
in supercritical case, 38
Flow, 481 -482;
ergodic, 159 ;
Liouville, 186 ;
volume-preserving, 156 -157, 159
Flux, 176
Ford, Kenneth, W., 188
Formal systems, 521
Form stability, 210 -213
Fourier, Baron Joseph, 159 ;
series of, 122 , 139 , 141 , 143 , 160 , 161 , 490
Fox, R.H., 500
France, research in, xvi
Fréchet spaces, 497
Frisch, Otto, 12
Fusion, 155
G
Gases, 123 -129
Genealogies/genealogical systems, 490 -491, 496 , 520 ;
intervals in, 85 ;
measure theory of, 85 , 86 -91, 95 ;
of multiplicative processes, 85 , 92 -105;
space of, 98 -105
General Dynamics, 163
Generating function, 6 , 40 -41, 61 , 62 , 64 , 74 , 76 , 79 ;
geometrical factors in, 12 , 13 -14;
iteration of, 2 -5, 7 , 9 -14;
monotonic, 7 ;
and probability distribution, 38 , 39 , 75 , 81 ;
in sub-critical systems, 9 -10;
for time sum, 11
Generating transformation, 37 , 38 -39, 42 -45, 49 , 65 -66, 69 , 70 , 74 , 75 , 79 , 84 , 92 , 100 , 104 , 105
Generation, xi , 2 -15, 38 , 494 -496;
asexual/nonsexual, 429 , 430 , 431 , 432 -433, 444 ;
via collision, 190 ;
as continuous, 2 ;
probability in, 2 -3, 8 , 10 , 12 , 13 ;
quadratic functions in, 190 ;
random, 2 ;
sexual, 429 , 430 , 431 , 434 -442, 443 , 520 ;
in subcritical case, 9
Geometry:
recursively studied, 379 -386;
of vertex, 60
Gibbs, G. W., 162
Gluskin, L.M., 492
Goad, Walter, 526 -527
Good, I. J., 446
Graphs. See Genealogies/genealogical systems
Gravitating systems, 129 -131
Gravity, xiii , 124 , 174 , 185 -188
Growth pattern: xiii ;
conflict for survival in, 379 -380, 381 , 384 -385;
death in, 379 , 380 , 381 , 382 -384, 385 ;
dogbone, 382 ;
self-replicating states in, 384 ;
three-dimensional, 380 , 381 , 385 -386;
two-dimensional, 380 , 381 -382
H
Haar, Alfred ter, 155
Hamilton, William R., equation of, 156 , 158 , 186
Hamming, R.W., 470 ;
distance, 527
Harlow, Frank, 121
Harris, T.E., 37
Hartigan, J.A., 473
Hausdorff, Felix, distance, 341 , 468 -469, 471 , 472 , 482 , 488 , 516 , 518 , 529 , 530
Hawkins, David, xi , 1 , 38 , 65
Hendry, Barbara, 535
Hessians, 38 , 40 -42. See also Moments, second
Heuristic studies, 121 -138
History-dependent processes, 399 -410;
explosions, 402 -405
Holonomic systems, 122
Homeomorphisms, 293 , 294 , 295 , 349 , 498 , 517
Homogeneous transformations. See Quadratic transformations, homogeneous
Homomorphisms, 500 , 518 , 521 , 522 ;
Hydrodynamics/hydrodynamical problems, 19 , 20 , 121 , 122 -126;
magneto-, 131 ;
neighbor relations in, 124 -125, 126 ;
time intervals in, 126
Hydrogen, 164
Hyers, D.H., 517
Hypergeometric function, 414
I
IBM 704, 197
IBM 7090, 309
IBM 7094, 442
IBM STRETCH, 303 n, 304 n, 311 , 323 , 354 , 356 , 357 n
Inertia, 130
Information retrieval, 483 -484
Instability, 123 -126, 339 , 342 , 343 . See also Stability
Integer:
-differential equations (see Differential equations, partial);
modulo a prime, 445 , 448 , 519 ;
Intercontinental ballistic missiles, 163 , 170
Interior points. See Fixed points interior
Intervals, 85 , 86 , 87 , 88 , 90 , 95 , 96 , 97 , 100 , 102 , 103
Isomorphism, 480 , 492 , 493 , 517 ;
and flow, 482 ;
non-product, 498 ;
product, 481 -482, 497 , 498 , 499 ;
convergence under, 37 ;
and ergodic behavior, 293 , 294 ;
of generating function, 2 -5, 7 , 9 -14;
of nonlinear transformations, 189 ;
parameters of, 126 ;
of quadratic transformations, 191 -192, 197 -198;
of transformations, 84 -85, 345 , 349 .
See also Composition
J
Jacobians/jacobian matrix, 37 , 38 -39, 308 , 323 ;
value of, 202 , 206 , 215 , 330 , 331 , 341 n
Jardine, N., 467
Jetting, 128
JOHNNIAC, xiii
K
Kepler, Johannes, 186 , 187 -188
Kolmogorov, A., 460
Kronecker, Karl Hugo, 158 ;
Kronecker-Weyl theorem, 192 , 218
L
Lagrange, J. L., 122 , 124 , 141
Laplace, Pierre Simon, 3
Larson, Jean, 480
Lebesgue, Henri, measure, 217 , 350 n
Lebesgue-Stieltjes, measure, 217
Limit, 192 ;
ergodic, 217 -218, 350 , 352 ;
expectation, 81 ;
function, 82 ;
periodic, 207 -209;
point, 4 , 45 -48, 313 , 314 ;
sets (see Limit sets)
Limiting:
behavior, 189 , 190 , 192 , 193 , 198 , 201 , 207 -209, 221 , 227 , 296 (see also Oscillation);
distribution, 9
of associated transformations, 338 -340;
brute force method to find, 304 , 305 -306;
Class I (closed curve), 315 -317, 322 , 324 , 339 , 340 , 343 , 347 ;
Class II, 317 -319, 322 , 328 ;
Class III (pseudo-periods), 319 , 322 , 328 -330, 339 , 341 , 342 , 343 , 357 , 358 ;
Class IV, 319 -321, 322 -323, 326 , 337 , 339 , 340 , 341 , 342 , 348 ;
convergence to, 306 , 321 , 345 ;
defined, 313 -314;
finite, 294 , 306 , 310 , 314 , 322 , 324 , 333 n, 338 -340, 345 ;
fixed points as, 330 ;
gap, 358
infinite, 314 -321, 322 , 324 , 326 , 328 , 336 , 339 , 340 ;
one-dimensional, 317 -319;
pathological, 299 , 301 , 330 , 340 , 358 ;
periodic, 317 , 333 -334, 338 , 339 , 340 , 341 , 342 , 357 , 358 ;
for quadratic transformations, 323 -326;
structure problem and, 314 ;
three-dimensional, 302
Linear transformations, 216 , 217 , 218 ;
piece-wise, 301 . See also Broken-linear transformations
Lipschitz constant, 108 -109, 112
Looping coefficients, 133
Los Alamos National Laboratory, ix , x , xii -xiv, 179
Lwów school of mathematics, x , 509 , 510 . See also Banach, Stefan
M
McKinsey, J. C. C., 480 , 492 , 494
Magill, K.D., Jr., 492 , 493 , 494
Magnetic field/force, 131 -133, 159 , 164 -165, 176
Manhattan Project, x
MANIAC I, xiii , 139 , 140 , 158 -159
MANIAC II, 307 -308, 310 , 353 , 354 , 355 n, 357 n
Many-body problems, xvi , 187 -188
Marczewski, Edward, 471
Margoliash, E., 466 , 527 -528
Markoff operators, 57
Markov, Andrey Andreevich:
distance, 472 ;
Mass, 165 -166;
constant, 172 -174;
ratios, 163 , 164 , 169 -170, 172 , 174
Mathematics:
applied, ix , xiii , xvi , 465 -475;
in biology, xiii , xvi , 465 -475;
computer aided, 121 -138;
evolution in, 522 ;
experimental, 297 ;
graph of, 490 -491;
Matijasevic, J. V., 461
Mating:
preferential, 438 , 439 , 444 ;
random, 434 , 436 , 438 , 439 , 440 , 444 ;
Matrix:
arbitrary, 57 ;
first moment, 37 , 64 , 69 , 70 , 72 , 74 , 105 ;
positive, 55 , 56 , 60 , 66 , 67 , 68 , 144 ;
supercritical, 56 -57
Mauldin, R. Daniel, 509
Maxwell, James Clark, 186 -187
Mazur, S., 511
Measure/measure theory, 86 -91;
and dissimilarity/difference, 465 -466, 467 -468, 469 -470, 473 ;
intervals in, 85 , 86 , 87 , 88 , 90 , 95 , 96 , 97 , 100 , 102 , 103 ;
invariant, 217 ;
and neighborhoods, 98 , 99 , 100 , 103 ;
points in, 86 ;
properties of, 446 ;
of sets, 86 , 87 , 88 , 89 , 90 -91;
of similarity, 465 ;
Mechanics. See Classical mechanics; Continuum mechanics; Statistical mechanics
Meiosis, 521
Memory:
built, 489 -490;
computer, 132 ;
nervous system/brain and, 478 , 484 -485, 486 , 489 , 490 , 532 ;
and recognition, 483 , 485 -490, 526 , 529 , 530 , 531 -532, 533
Mendel, Gregor, 194 n, 287 n, 289 n
Menzel, Mary Tsingou, xiii , 189
Mesons, 38
for molecular taxonomy, 469 -473;
non-Euclidean, 125 ;
in pattern recognition, 465 ;
sequence, 472 -473;
set, 526 (see also Space, Euclidean);
space, 466 , 467 , 468 -469, 472 , 528 ;
transitivity, 143 , 157 -158, 186 (see also Behavior, ergodic);
ultra, 468 .
See also Distance
Metropolis, Nicholas, 483
Miller, D. D., 492
Mitosis, 430 , 491 , 520 , 521
Mixing, 123 , 142 , 143 , 159 , 161
Moments:
combinatorial, 6 ;
first, 2 , 5 , 7 , 37 , 38 -39, 64 , 65 , 69 , 70 , 72 , 74 , 105 ;
and generating function, 5 , 11 -12;
properties of, 123 , 129 -130, 131 ;
second, 2 , 5 , 15 , 38 , 40 -42;
Montague, J.S., 492
Monte Carlo method, ix , xi , xvi , 17 , 18 -36, 402 , 406 , 477 -478, 530
Morphism, 492 . See also Homeomorphism; Homomorphism; Isomorphism
Multiplicative processes, xi , 1 -15, 37 -119;
branching and, 520 ;
as continuous, 2 ;
Mutation:
evolution via, 287 , 430 -431, 432 -433, 434 -442, 443 -444;
N
Neighborhoods, 98 , 99 -100, 102 , 103 , 491
Nervous system, 478 , 484 -485, 486 , 489 , 490 , 532
Neumann, John von, ix , x , xii , xv , 17 , 18 -33, 125 , 127 , 157 , 409
Neutron:
absorption, 2 , 3 , 13 , 21 , 22 , 24 , 25 ;
active material of, 18 , 21 , 22 , 24 , 25 , 34 ;
collision. 17 , 18 , 19 , 20 , 22 , 24 -25, 190 , 195 -196, 418 ;
cross-section of, 19 ;
density distribution of, 127 , 130 ;
fission, xi , 19 , 21 , 22 , 24 -25, 35 ;
flux, 176 ;
heating, 183 ;
linearly extrapolated path of, 22 -24;
mean free path of, 15 , 20 , 130 ;
parent, 13 ;
scattering. ix , 21 , 22 , 24 , 25 , 35 , 400 ;
slower-down material of, 18 , 21 , 22 , 24 , 25 , 34 , 35 ;
sojourn time of, 143 , 157 , 350 , 352 ;
tamper material of, 18 , 21 , 22 , 24 , 25 , 34 ;
velocity, 18 -19, 21 -22, 34 , 35
Nonlinearity/nonlinear transformations, 293 -377;
broken-linear (see Broken-linear transformations);
computer study of, xii , xvi , 139 -154, 297 -377;
cubic (see Cubic transformations);
difference equations, 191 -192, 344 -345;
differential equations, 300 , 344 -345;
displacement in, 139 , 140 , 145 , 151 , 153 , 160 , 161 ;
ergodic behavior of, 140 ;
iterations of, 189 ;
polynomial, 345 ;
quadratic (see Quadratic transformations);
time in study of, 140
Normalization, 192 -193
Norris, E.N., 495
Nuclear:
constant, 2 ;
explosion, 163 -177, 179 -180, 182 -183;
propulsion, 163 -177, 179 -184
Numbers:
complicated, 445 , 460 , 519 ;
large, 8 -9;
p-adic, 468 ;
prime, 409 , 445 , 460 , 461 , 519 ;
random, 355 n;
rational/real, 522 -523
O
Oscillation, 198 , 199 , 200 , 201 , 207 , 208 , 209 , 211 , 222 , 224 , 225 , 227 , 296 , 343
Oscilloscope, 301 , 307 , 310 , 319 , 326
Ostrowski, A., 330 , 333 , 345
Oxtoby, John, 158
P
Parallel computations, 477 , 478 -508
Particles:
cascades of, 37 ;
elementary, xvi ;
and radiation 417 , 418 , 422 .
See also Neutron
Pasta, John, xii , xiii , 121 , 139 , 156
Pathological:
behavior, 358 ;
limit sets, 299 , 301 , 330 , 340 , 358 ;
systems, 221
Pattern:
algebras, 492 , 494 , 496 (see also Relational structures);
of growth (see Growth pattern);
recognition, xiii , 465 , 466 , 478 , 483 , 485 -490, 531 ;
Period (periodicity), 294 :
accidental or false, 353 -354;
attractive, 314 , 316 , 333 -334, 354 ;
fortuitous, 353 -354;
non-attractive, 342 ;
of order k, 311 , 314 , 316 , 322 , 333 -334, 338 , 340 , 354 n, 356 , 357 -358;
pseudo (see Limit sets, Class III)
Periodic:
belt, 358 ;
limit, 207 -209;
limit sets, 317 , 333 -334, 338 , 339 , 340 , 341 , 342 , 357 , 358 ;
points, 201
Perturbation, 20 , 337 -338, 341 , 342 , 343
Piece-wise linear transformations. See Broken-linear transformations
Pitt, H.R., 409
Planck-Einstein-Tolman treatment, 417
Plasma, 165
Plemmons, R.J., 492
Points:
corner, 353 ;
exceptional, 350 -351;
limit, 4 , 43 , 45 -48, 313 , 314 ;
non-periodic, 350 ;
periodic, 201 ;
set defined by (see Limit sets).
See also Boundary points; Fixed points
Pólya, George, urn scheme of, 400 , 401 , 402 , 403 , 410
Polynomial, 461 -462;
transformations, 293 , 297 , 300 , 319 , 345 , 348 , 352 n
Probability, xii ;
and branching processes, 37 ;
of death/mortality, 4 , 5 , 7 , 67 , 73 -74, 105 ;
and generating function, 38 , 39 , 75 , 81 ;
in generation, 2 -3, 8 , 10 , 12 , 13 ;
transition, 85
Procreation. See Generation
Product:
isomorphism, 481 -482 497 , 498 , 499 ;
Projection, 480 -481, 498 , 520
Propellant/propulsion:
acceleration in, 167 , 168 -171, 174 ;
air as, 176 ;
external, 163 -177;
gravity as, xiii ;
hydrogen as, 164 ;
internal, 179 -181;
kinetic energy in, 166 ;
magnetic field in, 164 -165, 176 ;
positioning of, 176 -177;
temperature, 165 ;
velocity in, 165 -166, 168 , 173 , 182 , 183
Protein, 466 , 469 , 470 , 471 , 472
Pseudo-periods. See Limit sets, Class III
Q
Quadratic function, 190
Quadratic transformations, 139 , 142 , 144 , 147 , 149 , 161 , 189 -291;
fourvariable, 298 , 299 , 302 , 311 , 313 , 323 -326;
homogeneous, 191 -192, 218 , 225 , 286 -291, 294 -296 (see also Binary reaction systems);
iteration in, 191 -192, 197 -198;
limit sets for, 323 -326;
three-variable, 189 , 218 -221, 300 n, 304 -305.
See also Broken-linear transformations
R
Rademacher, Hans Adolph, 122 , 490
Random:
ergodic theorem, 399 , 409 -410;
history-dependent processes, 399 -410;
mating, 434 , 436 , 438 , 439 , 440 , 444 ;
processes, xi , 129 , 399 -410;
procreation, 2 ;
walk, 399 -405
Rationals, binary, 351
Ratios, 61 -64;
strong, 106 -118
Reaction systems. See Binary reaction systems
Recognition, 525 ;
discrimination compared to, 526 ;
distance as tool for, 526 -533;
memory and, 483 , 485 -490, 526 , 529 , 530 , 531 -532, 533 ;
pattern, xiii , 465 , 466 , 478 , 483 , 485 -490, 531 ;
of two-dimensional objects, 526 , 528
Recursion, 5 -6, 11 , 79 , 379 -397
Reichert, T.A., 471
Reines, F., 164
Relation:
algebras, 492 -496;
theory, 496
elational structures, 480 -492;
semigroups of, 481 , 492 -496;
topological, 494
Richtmyer, Robert D., xii , 17 , 34 -36
Rockets. See Propellant/propulsion; Space vehicles
Rotations, irrational, 192
S
Scattering, ix , 21 , 22 , 24 , 25 , 35 , 400 . See also Monte Carlo method
Schein, B.M., 492
Schrandt, Robert, xiii , 379 , 399 , 429 , 466 , 485 , 531
Schwarz, S., 492
Schwarz inequality, 108
Scottish Book, preface to, 509 -512
Sellers, Peter, 527
Sets:
bounded, 54 ;
closed, 54 , 60 , 99 , 102 , 103 ;
clusters of, 518 ;
distances between, 466 , 468 -469, 485 , 486 , 492 , 516 -517, 518 , 529 , 530 -531;
of exceptional points, 350 -351;
finite (see Period);
inner point of, 54 ;
limit (see Limit sets);
lower tree of p, 351 ;
measurable, 86 , 87 , 88 , 89 , 90 -91, 94 , 99 , 100 ;
metric (see Space, Euclidean);
non-void, 54 ;
null, 100 ;
Shannon, Claude, E., 461
Sharp, David, x
Sherwood Project, 155
Shrinking operator, 330 n
Sibson, R., 467
Similarities. See Analogy
Simmons, L.M., x
Smith, Temple, 465
Sneath, P. H. A., 466
Sojourn time, 143 , 157 , 350 , 352
Sokal, R.R., 466
Solitons, 139
Space:
Euclidean, 39 , 54 , 156 , 217 , 293 , 298 , 515 , 517 , 523 , 526 ;
flow in, 156 ;
of genealogies, 98 -105;
metric, 466 , 467 , 468 -469, 472 , 528 ;
transformations of, 126 , 143 , 144 , 157 , 517 , 523 ;
variables of, 122
Space vehicles:
chemical, 164 -176;
gravity affects, xiii , 185 -188;
kinetic energy of, 185 , 186 , 187 , 188 ;
multi-stage, 164 .
See also Propellant/propulsion
form, 210 -213.
See also Instability
Statics, 129
Statistical mechanics, 142 -143, 155 , 159 , 161 , 187
Steady state solution, 420 -421, 425
Stein, Myron L., xiii , 445 , 465 , 482
Stein, Paul, R., xii , xiii , 189 , 293 , 409
Steiner, Jakob, 528
Stieltjes, T. J., 217
Stirling number, 298
Strong ratio theorem, 106 -118
Subbiah, S., 495
Substitution, 7
Supercriticality, 4 , 6 , 7 , 37 -38, 49 -60, 106 -118. See also Criticality
Survival of the fittest, 430
Synergesis, 301 n
T
Tamper, 18 , 21 , 22 , 24 , 25 , 165 -166, 170 ;
tuballoy, 34
Taxonomy:
metric spaces in, 466 , 467 , 468 -469, 472 , 528 ;
molecular, 465 , 467 , 469 -473
Taylor, Theodore, 44 , 47 , 69 , 72 , 77 , 79 , 163
Thermalization. See Mixing
Thermodynamical systems, 417
Time:
-dependent solutions, 421 -422, 425 -427;
as function of problems, 122 , 125 , 129 -130, 140 , 143 , 144 , 145 ;
intervals, 122 , 123 , 156 -157;
particle, 83 -84;
Transformations:
analogy preserved by, 523 ;
associated, 335 , 337 , 338 -341, 342 ;
billowing, 126 -129;
bounded, 293 ;
broken-linear (see Broken-linear transformations);
conjugacy of, 294 , 295 , 345 , 348 , 349 , 351 ;
cubic (see Cubic transformations);
of energy, 144 ;
equivalent, 295 ;
fixed points of, 42 -45, 198 , 481 ;
generating (see Generating transformations);
homogeneous (see Quadratic transformations, homogeneous);
inequivalent, 295 -296, 298 , 309 , 311 ;
inverse, 354 -355;
iterated, 84 -85, 191 -192, 197 -198, 345 , 349 ;
linear, 216 , 217 , 218 , 301 ;
many-to-one, 294 ;
modified, 337 , 339 , 359 -361, 368 ;
multivalued, 481 ;
mutations as, 471 -472;
nonlinear (see Nonlinearity/nonlinear transformations);
one-dimensional, 294 , 301 , 317 , 349 , 355 -359;
polynomial, 293 , 297 , 300 , 319 , 345 , 348 , 352 n;
quadratic (see Quadratic transformations);
of space, 126 , 143 , 144 , 157 , 517 , 523
Transitivity, metric, 143 , 157 -158, 186 . See also Behavior, ergodic
Trees:
evolutionary, 466 , 467 , 470 , 471 , 472 , 473 ;
lower, of p, 351 ;
Turbulence, 123 , 139 , 159 , 161
Turing machine, 521
U
Unperturbed state, 20 , 342 . See also Perturbation
V
Valuation theory, 468
Velocity, 18 -19, 21 -22, 34 , 35 ;
of explosion, 165 -166;
gravity affects, 174 ;
of propellant, 165 -166, 168 , 173 , 182 , 183 ;
Vertex, geometric, 60
Vibrating string calculations, xii , 141 , 142 , 144 , 146 -154, 160 -161
Viking/V-2, 176 -177
von Neumann. See Neumann, John von
W
Walk, random, 399 -405;
self-avoiding, 399 , 400 , 404
Walsh. J.L., 490
Westinghouse, 163
Whyte, L. L., 511
Wilks, S., 20
Wolfram, S., 379
Wong, A.K.C., 471
Wyler, A., 446
Z
Zarecki, K.A., 492