Frege, Gottlob, 111, 127, 140, 146, 780–82, 929, 931–34
Frege system, 593–94
Freiman’s theorem, 717–18
Frey, Gerhard, 692
Frobenius, Ferdinand Georg, 514, 560, 783–84, 785
Frobenius map, 731
full parametric statistical models, 924–25
fullerenes, 831–32
function field of an algebraic variety, see varieties, function field of
function spaces, 29, 210–13, 239, 253–54
function-classes, 621–22
functional equation for the Riemann zeta function, 214, 229, 337, 356–57, 729–32
functional equations, 200–202, 455–56
functions, 10–11, 29, 51, 121, 125, 128, 144, 151, 184, 930–31
functors, 166–67
fundamental discriminants, 320, 324
fundamental groups, 166, 221, 225, 310, 385–88, 390, 416–17, 436, 441–42, 446, 517, 786
fundamental parallelogram, 413
fundamental quadratic form, 320–21
fundamental solutions, 187, 473
fundamental theorem of algebra, 81, 100, 147, 224, 275, 294, 326, 386, 640, 695, 698–99
fundamental theorem of arithmetic, 57, 104, 191, 221, 283, 332–33, 336, 699–700
fundamental theorem of calculus, 37, 175, 177, 179, 290, 651, 742, 758
fundamental theorem of exponential generating functions, 557
fundamental theorem of numerical analysis, 610
Funk-Hecke formula, 297
Gabo, Naum, 948–49
Galois, Évariste, 50, 81–82, 101–2, 104, 331, 709–10, 766, 767–68
Galois correspondence, 310, 710
Galois groups, 28, 191, 213, 331, 709, 761, 783
gamma function, the, 213–14, 290, 337
Gaudin distribution, 359
Gauss, Carl Friedrich, 42, 81, 88, 90–91, 100–101, 103–4, 137, 255, 269, 292, 315, 318, 320, 326–28, 335, 341, 349–50, 353, 356, 605, 609, 692, 699, 714, 718–19, 755–57, 760–62, 764, 929
Gauss sums, 328
Gauss-Bonnet theorem, 219, 394, 684
Gauss-Cramér model, 335–36, 341–43
Gaussian distribution, see normal distribution
Gaussian elimination, 606–7
Gaussian function, 293, 528, 705
Gaussian integers, 104, 108, 221, 318–19, 325–26, 719, 756
Gaussian quadrature, see quadrature, Gaussian
Gaussian unitary ensemble, 359
Gauss’s circle problem, 801
Gel’fond-Schneider theorem, 223
Gelfand-Naimark theorem, 519
genealogical tree, 655–56
general linear group, 29, 39, 161, 230–31, 429
general relativity, see relativity, general
generalized continuum hypothesis, 624, 629–30, 633
generalized eigenvectors, 224
generalized functions, 185
generalized inverse of a matrix, 836
generalized solutions, 187, 475–77
generating functions, 62, 214–15, 228, 304–5, 552, 555, 557, 753, 807
generic absoluteness, 633–34
generic extension, 626
generic real numbers, 625–26
genetic algorithm, 834
Gentzen, Gerhard, 151
genus, 54, 215, 370, 399, 411; of an algebraic curve, 721, 730
geodesics, 47, 92, 248, 312, 446, 488
geometric classification problems, 408–9
geometric distribution, 263
geometric group theory, 443–45
geometric Langlands program, 538
geometric mean, 703
geometric multiplicity, 225
geometric proof systems, 594
geometric structures on manifolds, 401, 406–7
geometrization conjecture, 281, 402–3, 406, 440
geometry, 1–2, 4–5, 38–47, 58, 83–97, 130, 136–39
giant component of a random graph, 573, 660
Girard, Albert, 81
global maximum principle for the heat equation, 217
global optimization, 834–35
Gödel, Kurt, 151–56, 624, 639, 701, 703, 819
Gödel’s completeness theorem, 623, 638, 819
Gödel’s incompleteness theorem, 141, 623, 629, 634, 638, 640, 700–702, 819
Gödel’s second incompleteness theorem, 623, 627, 701, 819
Goldbach’s conjecture, 69–70, 343–44, 346, 362, 715, 745, 751, 803–4; ternary, 715
Goldberg Variations (Bach), 939
golden ratio, 193, 222, 316–17, 320
Google, see search engines
Gordan, Paul, 103, 144, 769, 772, 788, 800
gradient, 34, 180, 239, 255–57
Gram-Schmidt orthogonalization, 607
graphs, 196, 198, 215, 245, 564, 645, 660, 697, 725, 846; bipartite, 157–58, 199, 565; coloring of, 564–65, 697; complete, 564, 725; edge-coloring of, 565; of groups, 442; minors of, 725–26; planar, 564, 697, 725; regular, 196
gravitational radiation, 491
gravitational waves, 490
great circle, 40
great wave of translation, 234–35
greatest common divisor, 107, 321
greedy algorithm, 245–46, 565, 873
Greek mathematics, 131–33
Green, George, 760
Green-Tao theorem, 344, 346, 570, 958
grim strategy, 904–5
Gromoe, Mikhail, 199, 299, 444, 447
Gromov’s filling theorem, 445–46
Gromov’s nonsqueezing theorem, 298–99
Gromov’s polynomial-growth
Gromoe-Witten invariants, see invariants, Gromoe-Witten
Grothendieck, Alexander, 285, 373, 394, 724, 731–32, 974, 1001, 1006
group C*-algebras, 520
group algebra, 274
groups, 19–20, 27–29, 39, 53, 66–68, 102, 204, 208, 221, 229, 248–49, 252, 260, 273, 277, 279, 284, 296, 636–37; actions of, 297, 309–10, 420–21, 424–25; axioms for, 20, 636; direct products of, 23–24, 436; free products of, 436–38; generators of, 433–34; of Lie type, 162, 688; presentations of, 436; residually finite, 439; simple, 26, 59, 261, 439, 687–88; solvable, 447, 710; of transformations, 39–41, 93, 419–20, 441
Grundlagen der Arithmetik (Frege), 127, 781
Grundlagen der Geometric (Hilbert), 138
GUE conjecture, 359
Hölder’s inequality, 704
Haar measure, 425
Hadamard, Jacques, 63, 155, 338, 356, 468, 715, 790–91, 963–65
Haken, Wolfgang, 117, 142, 563, 698
Hales, Thomas, 58
half-plane model of hyperbolicgeometry, see hyperbolic geometry, half-plane model of
Hall’s theorem, 566
halting problem, 638–39, 707–8
Hamilton, Richard, 281, 406, 714
Hamilton, William Rowan, 81–82, 105, 276, 299, 567, 765
Hamilton cycle, 567
Hamilton’s equations, 216, 286–87, 298–99
Hamilton’s least action principle, 478–79, 524, 525, 527, 541
Hamilton-Jacobi equations, 463–64, 466
Hamiltoninns, 215–16, 286, 524–27, 540, 542
Hamming, Richard, 879, 881, 981
Hardy, Godfrey Harold, 62–64, 346, 797–98, 807–8
Hardy-Littlewood k-tuple conjecture, 804
Hardy-Littlewood maximal inequality, 452, 455
Hardy-Weinberg law, 798
Harish-Chandra, 428–29
harmonic analysis, 205, 448–55, 859. See also Fourier analysis
harmonic functions, 283, 503, 652–53
harmonic map flow, 218
harmonic oscillators, 526, 542
harmonic polynomials, 296–97
Harnack inequality, 217–18
Hasse principle, see local-to-global principle, of Hasse
Hausdorff, Felix, 140, 184, 792, 794
Hausdorff dimension, see dimension, Hausdorff
Hausdorff topological space, 302, 633
Hausdorff-Young inequality, 454, 705
Hayter, Stanley William, 953
heat equation, 34–35, 216–19, 406, 456, 458–59, 470, 473, 478, 654
heat kernel, 218–19
Hecke, Erich, 730
Heine-Borel theorem, 168
Heisenberg equation, 287
Heisenberg group, 702
Heisenberg uncertainty principle, 207, 272, 306, 453, 513, 527
Helly’s theorem, 571
Hermitian matrices, see matrices, Hermitian
Hermitian operators, see operators, Hermitian
Hermitian structure, 163
heuristic evidence, 361–62
Higher Arithmetic, The (Davenport), 315
highest common factor, 107, 191–92
Higman, Graham, 437–39
Higman’s embedding theorem, 439–40
Hilbert, David, 50, 83, 95, 103, 110, 113, 125, 128, 138–46, 151–54, 358, 489, 511, 619, 623, 700–701, 707, 719–20, 726, 751, 788–89, 790, 804, 961–62
Hilbert spaces, 189, 211, 219–21, 240–41, 253–54, 270, 295, 423, 511, 513, 526, 540, 705, 799
Hilbert’s basis theorem, 144, 303
Hilbert’s Nullstellensatz, 594, 640, 642, 703; arithmetic, 365; effective, 371–72; weak, 371, 703
Hilbert’s problems, 789; tenth, 50, 110–11, 113, 639–40, 707–8, 821; twelfth, 720; thirteenth, 815; seventeenth, 822
Hirsch conjecture, 289
Hirzebruch, Friedrich, 394, 683, 724
History of Algebraic Geometry (Dieudonné), 377
HNN extensions, 437–38, 440, 442–43
Hodge, William Vallance Douglas, 816–17
Hodgkin-Huxley equations, 844
hole argument, the, 489
Holmgren’s theorem, 471
holomorphic functions, 37–38, 144, 205, 213, 228, 252, 282–83, 300, 307–8, 723, 804
holonomy, 163–64
homeomorphisms, 40, 302, 383, 387, 412, 497
homeostasis, 838
HOMFLY polynomial, 225–27
homogeneous coordinates, 267
homogeneous polynomials, 296–97
homological mirror symmetry conjecture, 536
homology groups, 182, 221, 389–92, 694–95
homomorphisms, 27–28, 165, 284, 801; ring, 284, 330, 801
homotopic loops, 385
homotopy groups, 221, 309, 385–87; of spheres, 389–92, 395
Hopf algebra, 273–74
Hopf link, 225–26
horocycle, 90
household maximization problem, 897–99
Householder method, 608
Hurwitz’s theorem, 278
hyperbolic geometry, 41–43, 47, 90–91, 208–9, 281, 283, 401, 447, 728–29; Beltrami’s disk model of, 92, 93; half-plane model of, 41; hyperboloid model of, 42; Poincaré’s disk model of, 42, 47, 94, 209, 728, 786
hyperbolic groups, 447–48
hyperbolic manifolds, see manifolds, hyperbolic
hyperbolic PDEs, see partial differential equations, hyperbolic
hyperbolic structure, 401–2
hyperbolicity conjecture, 508
hyperelliptic curves, 370
hyperfunctions, 186
hypergeometric sequences, 992
i (square root of -1), 18, 56, 284
ibn al-Haytham, 86
ibn Qurra, Th
bit, 86
ideal class group, 221–22, 322–23
ideals, 58, 221, 284–85, 322, 376, 378, 642, 719
idempotents, 240
identities, technique for proving, 261
image compression, 848–62
In Artem Analyticem Isagoge (Viète), 99, 737
inaccessible cardinals, see cardinals, inaccessible
inadmissible operator, 919
inclusion-exclusion principle, 345, 560
indefinite integrals, 175
independent random variables, 265–67
independent set: of elements of a matroid, 244–45; of vectors, 22, 158; of vertices in a graph, 564
index: of a continuous map, 695; of an elliptic equation, 682–83; of a fixed point, 695; of a Fredholm operator, 520; of a Toeplitz operator, 521
Indian mathematics, 79, 98, 192, 320, 736
indirect proofs of existence, 71, 117, 143–44, 149
induction: principle of, 16, 152, 258, 592, 638, 701, 741, 787, 998
ineffective proofs, see effective and ineffective proofs
inequalities, 3, 123, 126, 703–6
infinite cluster in percolation, 662–63
infinite prime, 720
infinite series, 120–21, 123, 193
infinite sets, 124, 127–28, 143–44, 148–49, 165, 167, 170–71, 616–17, 620, 623
infinite-dimensional vector spaces, 5, 22, 29–30
infinitesimals, 119–22, 128–29, 640, 770, 823, 934
infinities, 118–19, 127–28, 616, 744, 778
initial data, 459
initial value problem, 459, 464–67
inner models, 624, 629, 631–32
inner products, 185, 189, 219–20, 240, 268, 278, 301, 704
Institutiones Calculi Differentialis (Euler), 120–21
integers, 17, 82, 127, 170, 254, 284, 377, 570, 636, 992
integral delay equation, 346
integral domain, 377
integral equations, 510–11
integration, 35–37, 51, 175–80, 247, 450, 686, 744; Lebesgue, 247, 686, 796; numerical, 292, 609; Riemann, 247, 796
interactive proof systems, 596–97
interesting numbers, 261
interior methods for linear programming, 614
intermediate value theorem, 49, 67, 124, 384, 693
International Congress of Mathematicians, 110, 145, 619, 783, 789, 811, 950, 961, 966
Internet search engines, see search engines
intersecting family, 569
intersection numbers, 189, 383–85, 391–93
intuitionism, 116, 148–51, 799
invariant subspaces, 422, 425, 427, 515, 689
invariants, 53–54, 103, 221, 225, 370, 383–87, 395, 404, 407, 410–12, 442, 695, 788, 800; cardinal, 627; discrete, 370–71, 411–12; Gromov-Witten, 419, 533–34, 537; of elliptic equations, 682; Seiberg-Witten, 404, 407, 419; theory of, 103, 144, 550, 773, 789, 800
inverse problems: in additive number theory, 717; in chemistry, 829, 836–37; in spectral theory, 472
inverse-Gamma distribution, 926
inverses, 13; of functions, 10, 202; of matrices, 174; of operators, 294; under multiplication, 276–78, 284
involution principle, 556
irrational numbers, 18, 80, 222–23, 315–17, 328, 710
irrational rotation algebra, 520
irrationality of π, 193
irreducible element of a ring, 318–19
irreducible polynomial, 102, 328, 710, 888
Islamic mathematics, 79–80, 86–87, 98, 133–34
isomers, 830–31
isomorphism problem: for graphs, 584, 596; for groups, 436
isomorphisms, 27–28, 165–67, 202, 408, 411, 645, 801
isoperimetric inequalities, 210, 445–46, 670, 672–73, 676, 679, 705–6
iteration, 107, 112–14, 190, 244, 495
Itô, Kiyoshi, 655
Jacobi, Carl Gustav Jacob, 727, 762–63, 766
Jacobian, 418
James-Stein estimator, see estimators, statistical
Jensen, Ronald, 624
Jensen’s inequality, 704
Jones, Vaughan, 518
Jordan, Camille, 777
Jordan curve theorem, 777, 799
Jordan normal form, 223–25, 285
Journal des Mathématiques Pures et Appliquées, 102
Journal für die reine und angewandte Mathematik, see Crelle’s Journal
J-stability, 506
just intonation, 936–37
K-theory, 227, 394–95, 521–22, 683
Kähler manifolds, 163, 297, 300
Kac-Moody algebras, 234
Kakutani fixed point theorem, 694, 901
Kant, Immanuel, 93, 136, 928–29
Kaplan-Meier curves, 923–24
Kasparov, Gennadi, 522
Kauffman polynomial, 227
KdV equation, 235–38, 471, 481
Kepler conjecture, 58
kernel: heat, 218–19; integral, 29, 239, 791
kernel of a homomorphism, 28, 284
key management, 887
Khinchin, Alexander Yakovlevich, 814
Khinchin’s inequality, 705
Kirchhoff formula, 474
Kleene, Stephen, 111–12
Klein, Christian Felix, 92–93, 137, 209, 324, 327, 777–78, 782–83, 794
Klein bottle, 279, 388, 400, 441
Klein-Gordon equation, 456, 478
Kleinian groups, 209
Kloosterman sum, 347
Kneser conjecture, 569
Koch snowflake, 184
Koebe, Paul, 210
Kolmogorov, Andrei Nikolaevich, 453, 648, 677, 687, 793, 795, 814–16
Korteweg, Diederik, 235, 237, 799
Kovalevskaya, Sofya (Sonya), 125, 784–85
Kronecker, Leopold, 143–44, 146, 315, 327–28, 330–31, 773–74
Kronecker-Weber theorem, 720, 774
Krylov subspace iterations, 608–9
Kummer, Ernst Eduard, 81, 692, 719, 767
Kuratowski’s theorem, 725
La Géométrie (Descartes), 100, 739–40, 742
Lagrange, Joseph Louis, 101, 122, 136, 193, 268, 554, 570, 636, 727, 751–52
Lagrange inversion theorem, 554, 752
Lagrange multipliers, 256–57, 865–67, 870, 899
Lambert, Johann, 81, 87–88, 193
Landau, Edmund, 63
Lang’s conjecture, see Mordell-Lang conjecture
Langlands, Robert, 69
Langlands program, 69, 191, 331, 419, 429–31
Laplace, Pierre-Simon, 552, 752–54
Laplace equation, 35, 125, 283, 291, 296, 456–58, 468, 478
Laplace-Beltrami operator, 218, 296, 472
Laplacian, 34, 206–7, 217–18, 239, 287, 296–97, 312, 426, 456–57, 459–60, 477
large-cardinal axioms, 627–34
lattices, 59–60, 227, 250–52, 318, 324, 330, 412–13, 530; hexagonal, 228, 415, 663; square, 415
law of large numbers, 753, 906
Lax equivalence theorem, 611
Lax-Wendroff formula, 611
least action principle, see Hamilton’s least action principle
least upper bound axiom, 758
Lebesgue, Henri, 182, 628, 686, 795, 796–97
Lebesgue differentiation theorem, 455
Lebesgue measure, 247, 628, 686, 796
Lebesgue spaces, 211
Leech lattice, 59–60, 227–28, 549
Lefschetz fixed point theorem, 695
left coset of a subgroup, 26, 421
Legendre, Adrien-Marie, 88, 104, 714, 754–55
Legendre polynomials, 291–92, 297, 609
Legendre’s equation, 291
Leibniz, Gottfried Wilhelm, 118–19, 134, 743–45, 746, 935
lemmas, 73
length, 31, 57, 183–84, 220, 246–47, 307
length spaces, 444
Lennard-Jones clusters, 835–36
Les Méthodes Nouvelles de la Mécanique Céleste (Poincaré), 786
Lévy’s aresine law, 650
Lewy operator, 471
L-functions, 228–29, 316–17, 339–41, 345, 347–48, 381, 812; Dirichlet, 228–29, 284, 339–40, 345, 347–48, 689, 715, 764; of elliptic curves, 229, 347–48, 381, 685–86
Liber Abbaci (Fibonacci), 99
Lie, Sophus, 137, 230, 777–78 232–34
Lie algebras, 231–32, 234, 273–74, 427, 541, 544, 778, 794; classical, 234; semisimple, 232–33; simple, 232–34
Lie groups, 161, 229–32, 234, 240, 272–73, 277, 279, 298, 402, 425, 428, 778, 794; classical, 161, 234; linear, 230; semisimple, 713; simple, 232
lifting a path, 309
limit groups, 448
limit ordinal, 258, 617–18, 620, 624
limits, 30–32, 122–26, 168–69, 200–201, 254, 258
linear algebra, 103
linear approximation, 33, 37, 109
linear equations, 48–49
linear feedback shift register (LFSR), 888
linear functionals, 176, 185, 188, 212
linear groups, 161
linear maps, 28–30, 33, 37, 49, 51, 174, 219, 223, 239, 255, 276, 294
linear operators, 216, 239–41, 294–97, 511
linear programming, 288, 612–13
linear wave equation, 611
linearization, 470
link-route incidence matrix, 863
linking numbers, 385
Liouville, Joseph, 38, 71, 81, 222, 293, 766–67
Liouville’s theorem: in complex analysis, 38, 723–24, 766; in mechanics, 766; on transcendence, 294, 299, 711
Littlewood, John Edensor, 346, 797–98, 803–5, 859, 963
Littlewood-Paley theory, 804, 859
Lobachevskii, Nicolai Ivanovich, 42, 89–92, 137, 759–60
local connectedness, 505; of the Mandelbrot set, 508
local-search algorithms, 875
local-to-global principles, 167–68; of Hasse, 241–43, 685
locality, 543
locally symmetric manifolds, see manifolds, locally symmetric
logarithms, 80–81, 202, 290, 523
logic, 6, 13–16, 140, 634–39, 819, 931–33; Ω-, 634; Aristotelean, 932; first-order, 259, 314, 448, 621–22, 623, 636, 700–701; propositional, 153
logical connectives, 13–14, 621, 635
logical consequences, 637–38
long multiplication, 106–7, 170, 204, 349–50
Lorentz gauge, 490
Lorentzian geometry, 43–44, 402, 478, 484, 487–89
Lorenz attractor, 496
loss function, 918
Lovász local lemma, 574
Löwenheim-Skolem theorem, 622, 624–25, 806
lower-triangular matrices, see matrices, lower-triangular
machine epsilon, 606
Maclaurin, Colin, 121
Mac Lane, Saunders, 167
major arcs, 346
majority function, 588
Mandelbrot set, 244, 505–9; hyperbolic components of, 507
manifolds, 4, 5, 44–46, 47, 57, 244, 258, 281–82, 300, 396–408, 794; complex, 163, 191, 300; differentiable, 45; four-dimensional, 388, 403–4, 440–41; hyperbolic, 401, 712; locally symmetric, 712–13; nonorientable, 384, 399–400; orientable, 163, 384, 399–400; simply connected, 281, 388, 403, 714; smooth, 396–400, 403; symplectic, 297–301; three-dimensional, 280, 388, 401–3, 441, 714; topological, 45, 397–400, 404. See also Calabi-Yau manifolds, Kahler manifolds, Lorentzian manifold, Riemannian manifolds
manipulatorics, 555–56
mapping class group, 418
Margulis, Gregori, 197–98, 269, 713
market completeness, 912
market efficiency principle, 910
market equilibrium, 901
Markov chain, 596
Markov process, 647, 649, 653, 655
Martin’s maximum (MM), 633
Martin-Löf thesis, 116
martingale, 652, 912–13; problem for Brownian motion, 652
mathematical collaboration, 1001
Mathematical Physics (Courant and Hilbert), 809
mathematical physics, 7–8
Mathematische Annalen, 93, 153, 782, 800, 817, 822
Mathieu, Émile Léonard, 688, 776–77
matrices, 28–30, 33, 49, 174, 223, 240; diagonalizable, 223–24; Hermitian, 240; invertible, 174–75; lower-triangular, 607; nilpotent, 224; orthogonal, 240; permutation, 423, 607; self-adjoint, 240; similar, 174; skew-Hermitian, 231; stochastic, 694; symmetric, 240, 511; symplectic, 298; transpose of, 240; unipotent, 430; unitary, 240, 271, 277, 511; upper-triangular, 607
matrix multiplication, see multiplication, of matrices
matroids, 244–46
maximal function, 455
maximal operator, 452
maximal torus, 430
maximum principle for the Laplace equation, 475
maximum-likelihood estimate, 924
Maxwell’s equations, 479, 484, 490, 525
McKay, John, 60
mean of a random variable, 265–66
meantone temperament, 937
measurable cardinals, see cardinals, measurable
measurable sets, 128, 158, 247, 627–29, 631–32
measure problem, the, 628–29
measurement problem, 269
measures, 246–47, 628, 815; probability, 264. See also Haar measure, Lebesgue measure
Méchanique Analitique (Lagrange), 751
Mellin transform, 214
melodic retrograde and inversion, 938–40
memorylessness, 265
meromorphic continuation, 228–29
meromorphic functions, 213, 723–24
Mersenne, Marin, 936
Mersenne primes, 353
Mesopotamian mathematics, 77–78, 733
metamathematics, 152, 154, 622
metastable states, 829
method of characteristics, 236
method of exhaustion, 132, 735
metric spaces, 46, 172, 181, 220, 247–48, 253, 302
Meyer, Y., 861–62
microlocal analysis, 477
millionaires’ problem, 602
Mills-Robbins-Rumsey determinant, 997–98
Milman, Vitali, 675–76
Milnor-Švarc lemma, 444
minimal polynomial, 225, 328, 330
minimal surface equation, 312, 457
minimal surfaces, 312, 534, 670, 832–33
minimum connector problem, 245
minimum spanning tree problem (MSTP), 872–75
Minkowski, Hermann, 330, 484, 487, 672, 789–90
Minkowski space, 43, 268, 402, 457, 478, 484–85, 487
Minkowski’s inequality, 704
minor arcs, 346
mirror symmetry, 69, 164, 190, 523–24, 529–32, 534, 537–38
Möbius, August Ferdinand, 759
Möbius function, 345
Möbius inversion, 561
Möbius strip, 384, 392–93, 399–400, 759, 950, 979–80
Möbius transformations, 208–9, 415–16
model spaces for geometric structures, 402
model theory, 6, 645, 814, 822
models, 621–22, 636, 639, 645, 822; of set theory, 248–49, 806; of ZFC, 620–21, 623–27, 629–30
Moderne Algebra (van der Waerden), 105, 824
modes in quantum field theory, 532, 543
modular arithmetic, 249–50
modular automorphism group, 517
modular elliptic curve, 692
modular forms, 251–52, 268, 419, 692, 724, 807–8
modular functions, 347, 545, 549
moduli spaces, 191, 252, 370–71, 408–19, 711–13
modulus of a complex number, 19, 276
modus ponens, 700
molecular dynamics, 836, 841–42
momentum operator, 286
Monge-Ampère equation, 462
Monier-Rabin theorem, 351
monochromatic subsets, 567, 802
monoidal category, 275
monotone circuits, see Boolean circuits, monotone
Monstrous Moonshine conjecture, 60, 548–49
Mordell conjecture, the, 117, 382, 681, 722
Mordell-Lang conjecture, 382, 722
Mordell-Weil theorem, 190
Morlet, J., 861
Mostow, George, 713
Mostow’s rigidity theorem, 712–13
multiplication, 275–78, 284, 306–7, 635–36, 638; of ideal classes, 323; of ideals, 322; of integers, 65, 586; of matrices, 29, 65, 67, 272, 277–78, 591
multiplication operators, 239–40, 294–95, 511–12, 519
multiplicative sequence, 228
multiplicity of a solution, 366
multiplier of a fixed point, 499
multistep methods for numerical solution of ODEs, 609
Music for Strings, Percussion, and Celesta of Bartók, 940
Musical Offering, The (Bach), 939–40
naive set theory, see set theory, naive
nanoporous architectures, 834
Nash equilibrium, 694, 901, 982
Nash’s theorem, 364
natural numbers, 17, 258; theory of, 638
natural proofs in computational complexity, 589
natural transformations, 167
Navier-Stokes equations, 193–96, 477
n-body problem, 493, 726–28, 764
NC, see complexity classes
n-category, 167
negation, 15
negative numbers, 17, 81, 126 networks, 862–71
Neumann boundary conditions, 458, 469
neural networks, 844
Newton, Isaac, 87, 100, 109, 118–20, 134, 136, 493–94, 609, 612, 726, 742–43, 827
Newton-Raphson method, 109–10
Newton’s law of gravitation, 493
Newton’s method, 494–95, 498, 509, 612–13
Newton’s second law of motion, 194, 286, 311, 493, 524, 726
nilpotent groups, 444, 447, 702
nilpotent matrices, see matrices, nilpotent
no arbitrage principle, 910
Noether, Emmy, 82, 104, 525, 800–801
Noether’s principle, 479, 525, 540
noise, 878
non-Euclidean geometry, 84, 88–94, 832, 946
noncollision singularities in the n-body problem, 727
noncommutative algebraic topology, 523
noncommutative geometry, 57, 272, 522–23
nonconstructible real numbers, 629
nonconstructible sets, 629
nonconstructive arguments, 157
nonconstructive proofs, 693
nondetermined game, 631
nonlinear approximation scheme, 856
nonlinear Poisson equation, 312
nonmeasurable sets, 158, 627–28, 796, 802
nonorientable manifolds, see manifolds, nonorientable
nonparametric approach to statistical modeling, 922–24
nonperturbative phenomena in physics, 530
nonpositively curved groups, 447
nonrigorous arguments, 68–71
nonstandard analysis, 128, 823
nonstandard models of arithmetic, 702, 822
norm residue symbol, 719
normal distribution, 51, 214, 262, 266–67, 647–50, 654, 678, 797
normal number, 262
normal operators, see operators, normal
normal subgroup, 26
normed division algebra, 278
normed spaces, 185, 210, 252–54
norms, 210–11, 253, 278, 319, 321, 704, 810; of quadratic integers, 319, 321, 323
NOT gates, 584
notional prices, 899
NP, see complexity classes
NP-complete problems, 68, 271–72, 583–87, 596, 874
NP versus co NP problem, 582, 593
null hypotheses, 923–24
Nullstellensatz, see Hilbert’s Nullstellensatz
number field sieve, see sieves
number fields, 254–55, 329, 730
number systems, 16–19, 77–83, 104, 278, 984
number theory, 4. See also additive number theory, algebraic number theory, analytic number theory, combinatorial number theory, computational number theory
numeracy, 983–91
numerical analysis, 109, 604–15; integration, 292, 609; linear algebra, 606–9
numerical evidence, 69–70
numerical instability, 606, 610
n-vector model, 70
o-minimal structures, 644
objective functions, 255–57, 288–89, 613, 835, 866
objective prior distributions, 926
objects in a category, 185–87, 536–37
observables, 286–87, 513, 540–42
octahedron, n-dimensional, 53
octatonic scale, 939
octonions, 278–79
odd functions, 204
odd permutations, see permutations
one-step methods for numerical solution of ordinary differential equations, 610
one-time key, 890
one-way functions, 598–601, 890
one-way hash function, 893–95
operator algebras, 240, 510–23
operators, 295, 450, 526; compact, 519; essentially normal, 521; Fredholm, 520–22; Hermitian, 240, 295, 540; normal, 240, 518; self-adjoint, 511, 513; Toeplitz, 521; trace-class, 515, 523; unitary, 513–14, 689, 691
optimization, 255–56, 612–14, 865–70
Opus Restitutae Mathematicae Analyseos, seu Algebra Nova (Viète), 738
orbifolds, 257–58, 367, 371, 534–35
order: of a group element, 67; of a permutation, 260
ordered fields, 643
ordinals, 145, 258, 616–22, 624–27, 629, 779; countable, 624–26
ordinary differential equations (ODEs), 51–52, 464–65, 609–11
orientable manifolds, see manifolds, orientable
orthogonal arrays, 173
orthogonal groups, 39, 230, 232
orthogonal maps, 240
orthogonal projections, see projections, orthogonal
orthogonality: of Legendre polynomials, 292; of spherical harmonics, 296–97; of trigonometric functions, 308; of wavelets, 852–54
orthonormal basis, 212, 220, 240, 296, 423
overdetermined system of equations, 459
P, see complexity classes
P versus BPP problem, 595, 601
P versus NP problem, 69, 170, 580–81, 585, 591, 598–600, 713–14, 874
PageRank, 877
pairings, 189
Paley, R. E. A. C., 572, 804, 812, 859
palindromic numbers, 75
parabolic equations, see partial differential equations, parabolic
paradifferential calculus, 477
paradoxes, 145–46