Index
Bold numbers indicate pages that may be particularly helpful, usually because they contain definitions or detailed discussions.
3SAT, 582–83
4-potential, 490
Ω-logic, see logic, Ω-
abacuses, 106
Abel, Niels Henrik, 20, 50, 81, 101, 123–24, 331, 709, 760–62
Abelian field extensions, 720
Abelian groups, 20, 190, 255, 274, 285, 323, 761
Abelian varieties, see varieties, Abelian
absolute convergence, see convergence, absolute
abstract algebra, 82, 95–106, 800–801
abstraction in mathematics, 20, 55–56, 96, 539
Ackermann function, 112
Acta Mathematica, 784
action principle, see Hamilton’s least action principle
actions of a group, see groups, actions of
actions of a physical system, 287, 311, 478, 524–26, 541
Adams formulas, 609–10
adaptive algorithms, 614
additive number theory, 715–18
adjacency matrix of a graph, 198, 571
adjoint, 172, 179, 186, 188, 212, 240, 272, 277
advanced encryption standard, 889–90
affine algebraic groups, 429
affine buildings, 162
affine geometry, 39–40
Airy function, 449
Al-
üs
, Na
ir al-Dn, 86–87
Al-Karaj, 98
Al-Khw
rizm, Abu Ja’far Muhammad ibn Müs, 79–80, 98–99, 106, 133, 736–37, 986
al-Kitb al-mukhtaar f
isb al-jabr wa’1-muqbala (al-Khw
rizm), 98, 106, 133, 736
Aldous’s theorem, 656
Alexander polynomial, the, 225–27
algebra, 1–4, 57–58, 80, 95–106, 539
Algebra (Bombelli), 737
algebraic closure, 642
algebraic curves, 190, 367, 381, 392, 721
algebraic functions, 241
algebraic geometry, 5, 285, 363–72
algebraic integrals, 726
algebraic multiplicity, 225
algebraic number theory, 4, 315–32; algebraic integers, 254, 315, 317–19, 324–30; algebraic numbers, 171, 222, 241–42, 315, 325–30, 616, 641, 779
algebraic structure on a surface, 411
algebraic topology, 40, 383–96, 801
algebraically closed fields, 640–42
algebras, 105, 172, 239–40, 272
algorithms, 50, 65, 68, 71–73, 106–17, 436, 575–77, 579, 707, 871–72
alternating groups, 61, 261, 279, 688
alternating knots, 227
AM-GM inequality, 703–4
amalgamated free products, 437, 442
amicable numbers, 747
A-model, 531–34
analysis, 2–3, 5–6, 30–38, 118, 122–23, 125, 127–28, 136, 138
Analyst, The (Berkeley), 120
analytic continuation, 38
analytic formulas for special values of L-functions, 316–17, 323
analytic number theory, 4, 332–48
analytic philosophy, 928–35
Andrews, George, 997–98
Apollonius, 735–36
Appel, Kenneth, 117, 142, 563, 698
approximate algorithms, 874
approximate counting, 595
approximating square roots, 110
approximation by polynomials, 253
approximation by rational numbers, 192, 222, 315–16, 710
approximation method in computational complexity, 587
approximation schemes, 852–53, 856
a priori estimates, 474–75
arbitrage, 911–12
arboreal group theory, 442
Archimedean property, 636
Archimedes, 79, 97, 108, 132, 609, 734–35
arguments of complex numbers, 19
Aristotelean logic, see logic, Aristotelean
Aristotle, 83, 86, 151, 931–33
arithmetic circuits, 589–91
arithmetic geometry, 372–83
Arithmetica (Diophantus), 97–99
Arithmetica Universalis (Newton), 100, 136
Arrow’s theorem, 982
arrows, 166
Ars Conjectandi (Jacob Bernoulli), 746
Ars Magna (Cardano), 99, 134, 737
Artin, Emil, 161, 720, 730, 812–13
Artin zeta functions, 730
Artin’s reciprocity law, see reciprocity, Artin’s law of
Asian option, 914
Ask, see search engines
associative law, 13, 105, 272, 278, 301, 323, 892
asymptotically stable orbit, 495
Atiyah-Singer index theorem, 219, 460, 521, 681–84, 725
Atkinson’s theorem, 520–21
atonal set theory, 942
attaching maps, 441
attracting basin, 501–2
automatic differentiation, 613
automorphic forms, 191, 252, 419
automorphisms, 27–28, 29, 412, 709
average-case complexity, 603
avoidable properties, 626, 633
axiom of choice, 145, 147–48, 157–58, 159, 314, 619–21, 623–24, 626–28, 684
axiom of comprehension, 157
axiom of extensionality, 620–21
axiom of foundation, 620
axiom of infinity, 620–21
axiom of regularity, 620–21
axiom of replacement, 620–22
axiom of separation, 620
axiom of union, 620–21
axiomatic approach: to mathematics, 84, 128, 138–40, 145, 152; to probability, 793, 795, 815
Babbage, Charles, 111
Babylonian mathematics, 96
Bach, J. S., 938–40
Baire category theorem, 633
Baker-Campbell-Hausdorff formula, 232
Banach, Stefan, 254, 809–11, 813
Banach algebras, 172, 202, 211, 239; commutative, 307
Banach spaces, 172, 188, 210, 239–40, 252–54, 270, 294, 799, 810
Banach-Tarski paradox, 158, 684–85, 813
bandlimited signals, 860
Barban-Davenport-Halberstam theorem, 341
barrier option, 913
Bartók, Béla, 940
base space, 392
basic feasible solutions, 288
basis: of a matroid, 245–46; of a topology, 302; of a vector space, 21–22, 28, 30, 223
basis states, 270
Baum-Connes conjecture, 522
Bayesian analysis, 159–60, 753, 920, 926–27
Begriffsschrift (Frege), 140
Beltrami operator, 296
Beltrami, Eugenio, 92
Berkeley, George, 120
Bernoulli, Daniel, 747
Bernoulli, Jacob I, 746
Bernoulli, Johann I, 746
Bernoulli, Nicolaus I, 120, 747
Bernoulli distribution, 263, 267
Bernoulli numbers, 395
Betti number, 731
Bézout’s lemma, 114
Bianchi identities, 488
bicharacteristic curves, 463, 466–67
Bieberbach conjecture, 803
bifurcation set, 505
big bang, 492
biholomorphic equivalence, 728
bijective proofs, 555–56
binary symmetric channel, 879, 883
binomial distribution, 263, 266–67
biological fluid dynamics, 843
biorthogonal wavelet bases, 855
bipartite graphs, see graphs, bipartite
birational equivalence, 722
Birch-Swinnerton-Dyer conjecture, 229, 381, 685–86
Birkhoff’s ergodic theorem, 691, 803
black holes, 491
Black-Scholes equation, 655, 910, 912–13
block designs, 172–73
B-model, 531–34
Bochner identity, 474
Bolyai, Jnos, 42, 89–92, 137, 762
Bolzano-Weierstrass theorem, 124, 144, 147, 168, 758, 771
Bombelli, Rafael, 81, 104, 317, 737
Boolean algebra, 770
Boolean circuits, 584–89; monotone, 586–87
bootstrap argument, 475
Borcherds, Richard, 60, 548–49
Borel sets, 247, 628, 631–32, 801
Borsuk’s antipodal theorem, 978
Borsuk’s problem, 677
Bott periodicity theorem, 227, 394, 682–83
Böttcher maps, 502–5
bound states, 472
bound variables, see free and bound variables
boundary conditions, 217, 458–59, 467, 469
bounded-depth circuits, 587–88
Bourbaki, Nicolas, 823–25
BPP, see complexity classes, 595
Braess’s paradox, 866–68
braid groups, 160–61, 274, 388
branching processes, 658–59
breakdown of solutions to partial differential equations, 194–96, 481
breaking time, 237
Bride Stripped Bare by her Bachelors, Even (The Large Glass), The (Duchamp), 947
Brouwer, Luitzen Egbertus Jan, 116, 142, 148–51, 153, 155, 181, 799–800
Brouwer’s fixed point theorem, 693–96, 800, 901
Brown-Douglas-Fillmore theorem, 521
Brownian excursion, 656–57
Brownian motion, 218, 647–56, 910–11
Brun sieve, the, see sieves
Brunn-Minkowski inequality, 672–73, 705; reverse form of, 676
brute-force search, 59, 62–63, 580, 840, 873, 883
buildings, 161–62
Burali-Forti paradox, 145, 779
Burgers equation, 236, 463, 467, 467
Burnside problem, 68, 438. 785
Burnside’s lemma, 560
butterfly effect, 496
C*-algebras, 57, 172, 227, 313, 518–20, 522–23
Calabi conjecture, 164
Calabi-Yau manifolds, 163–65, 190, 530–34
calculus, 118–19, 122–24, 128, 134–36, 743–44, 770, 934
calculus of variations, 65, 310–13, 478–79, 908
cancelation law, 250
cancrizans canon, 939–40
canonical inner models, 629
canonical transformations, 298–99
Cantor, Georg, 71, 81, 116, 125, 127, 144–46, 155, 171, 183, 222, 616–19, 623, 629, 632, 634, 703, 778–80
Cantor’s diagonal argument, 171, 779
CAR algebra, 519
Cardano, Girolamo, 101, 104, 133–34, 737
cardinal exponentiation, 618, 630
cardinal invariants, see invariants, cardinal
cardinality, 165, 616–19, 622, 626
cardinals, 145, 165, 616–19, 779; inaccessible, 627–29, 632, 702; measurable, 628–32; regular, 627–29; singular, 630; supercompact, 630, 632–33; uncountable, 626–29; weakly compact, 628; Woodin, 632
Cardy’s formula, 666
Carleson’s theorem, 453, 686–87
Carmichael numbers, 351
Cartan, Élie Joseph, 232, 713, 794–95
Cartan subalgebra, 233–34
Cartesian coordinates, 21, 739
Cartesian product, 618
Casorati-Weierstrass theorem, 771
casus irreducibilis of the cubic, 737
Catalan conjecture, 360
Catalan’s constant, 150
category theory, 6, 165–67, 275, 417, 536, 801
Cauchy, Augustin-Louis, 102, 122–24, 147, 459, 560, 569, 758–59, 760, 791
Cauchy problem, 235–37, 459, 468–69
Cauchy’s residue theorem, 202, 337
Cauchy’s theorem, 38
Cauchy-Davenport theorem, 569
Cauchy-Hadamard theorem, 791
Cauchy-Kovalevskaya theorem, 464, 467–68
Cauchy-Riemann equations, 37, 459–60
Cauchy-Schwarz inequality, 220, 268, 704–5
Cayley, Arthur, 82, 92, 103, 105, 110, 498, 509, 768–69, 772–73, 831
Cayley graphs, 443, 445, 447, 702
Cayley numbers, see octonions
Cayley’s graph theorem, 772
Cayley’s theorem, 422
Cayley-Dickson construction, 278–79
Cayley-Hamilton theorem, 329
cell complex, 441
cellular automata, 836
central limit theorem, 207, 266–67, 335, 648, 678, 687, 919
character tables, 429–30
characteristic classes, 393, 411
characteristic coordinates, 236
characteristic curves, 236, 462–63, 466
characteristic hypersurfaces, 466–67, 469
characteristic of a field, 640
characteristic polynomial, 224–25, 294, 329
characters: of Abelian groups, 189, 207, 295–96, 308, 339, 426, 428; of group representations, 423–26, 428, 430, 783, 785; in phylogenetics, 846. See also Dirichlet characters
Chebotaryov density theorem, 783
Chebyshev, Pafnuty, 771
Chebyshev polynomials, 293, 297, 771
chemical informatics, 836
chemical topology, 830–31
Chern classes, 394
Chevalley groups, 688
chiral algebra, 544
chirality, 979
choice number of a graph, 574
choice sequence, 150
chromatic index of a graph, 565
chromatic number of a graph, 564, 566
Church, Alonzo, 50, 111–12, 577, 707, 816
Church’s thesis, 113
circle method, 346–47, 797, 804, 807
circle of fifths, 938
circuits, see Boolean circuits
class field theory, 243, 268, 720, 812–13
class numbers, 255, 322–24, 340–41
classical computation, 269, 271–72
classification, 52–54, 56, 232, 252, 408, 411; of finite simple groups, 141, 252, 429, 687–89; of Lie algebras, 161–62, 232–34
Clenshaw-Curtis quadrature, see quadrature, Clenshaw-Curtis
Clifford, William Kingdon, 780
cliques in graphs, 564, 573, 586–87
closed-form solutions, 51, 766
coarse moduli spaces, 415–16
cofinality, 629
Cohen, Paul, 141, 155, 624–27, 703, 780, 819
Cohen-Lenstra heuristics, 324
cohomologg, 189, 221, 384, 389, 391–94, 411, 523, 531, 732
collapse of the polynomial-time hierarchy, 585
combinatorial geometry, 570–71
combinatorial number theory, 569–70
combinatorics, 6–7, 562–63; algebraic, 561; extremal, 215, 563–72; probabilistic, 572–74
communication channel, 879
communication complexity, 589
commutant of a set of operators, 515
commutative diagrams, 166, 274
commutative law, 13, 82, 105, 179, 278, 284, 301, 323, 519, 770
commutator, 231, 287, 444, 526, 542
compactification, 168–69, 267, 721; one-point, 169; Stone-
ech, 169
compactness, 167–69, 303, 398, 639–40, 645
complement of a set, 188
complete graphs, see graphs, complete
completeness: of an axiomatic system, 139, 153, 637–39; in computer science, 170; of a metric space, 220, 254, 514, 696; of a normed space, 810; of the real numbers, 144, 636
complex analysis, 37, 282–83, 337–38, 758, 775
complex cobordism, 395
complex manifolds, see manifolds, complex
complex numbers, 18–19, 81–82, 102, 105, 201–2, 275–78, 284–85, 296, 317–18, 328, 640–41, 698, 737
complex orientation, 163–64
complex structures, 300, 411–13, 417, 816
complex systems, 838
complexity, see computational complexity
complexity classes, 169–70; BPP, 595; co NP, 582, 584; EXP, 580–81, 595; NC, 170; NP, 170, 446, 580–83, 595–96, 598–99; P, 579–81, 595, 713; PSPACE, 170, 597
complexity of algorithms, 578
composition: of braids, 160; of morphisms, 165–66, 536–37; of operators, 240, 294, 515; of permutations, 259–60; of symmetries, 20, 277, 420, 484
comprehension principle, 145
computable functions, 112–13, 577, 816, 821
computational chemistry, 830
computational complexity, 114, 575–604
computational fluid dynamics, 611
computational number theory, 348–62
computer memory, 114, 169–70, 578, 597, 848–49, 980
computer-assisted proofs, 142, 496, 575, 698, 972
concatenation of paths, 176–77, 221, 401
conditional probability, 159
Condorcet’s paradox, 982–83
conformal equivalence, 209, 282, 411
conformal field theory, 543–45
conformal invariance, 654, 665
conformal structure on a surface, 209, 411
conformal vector, 546
conic sections, 43, 365, 735–36, 739, 743
Conics (Apollonius), 735
conjectures, 69–70, 76, 142, 335, 349–60, 381, 722, 957
conjugacy (in group theory), 26; classes, 422–26, 428–31; conjugacy problem, the, 436; conjugate subgroups, 421
conjugates: algebraic, 319, 329–31; complex, 19, 276, 278, 710
connectedness, 198, 230, 245, 303, 309, 383–85, 398, 504–8, 564, 573, 660–64
Connes, Alain, 517, 522–23, 956–57
co NP, see complexity classes
conservation laws, 236, 286, 479, 486, 488, 525, 540, 800
conservative extensions, 154
consistency, 139–40, 145–46, 153, 622–23, 625, 639, 701, 819; of the continuum hypothesis, 155, 624, 780; of Euclidean geometry, 789; of Peano arithmetic, 702; of ZFC, 629, 702
consistency strength, 629; lower bound, 629; upper bound, 629
constant-curvature metric, 92, 281, 712, 728, 775
constrained optimization, 256–57
constructible set theory, 623–26, 629, 819
construction of regular 17-gon, 101, 327
constructive proofs, 143–44, 149, 157
constructivism, 116, 157–59; in art, 948–50
contextual definition, 933–34
Continental philosophy, 929
continued fractions, 192–93, 315–17, 326; for tangent function, 193
continuous functions, 32–33, 123, 144, 151, 168, 211, 301–2
continuum hypothesis, 145, 155, 618, 623–27, 629, 632, 634, 703, 780, 802, 819; independence of, 703
contour function of a tree, 655–56
contractible spaces, 309, 387, 388, 442
contraction mapping theorem, 696
convergence, 31–33, 109–10, 123, 126, 168–69, 254, 452; absolute, 334; almost everywhere, 452–53, 687, 815; in distribution, 650; in probability, 266; quadratic, 110, 612; superlinear, 612; uniform, 124, 126, 211; weak, 186–87
convexity, 72, 288, 671, 675, 696, 704–5, 790
convolution, 203, 207, 213, 303–4, 306–7, 450
Conway, J. H., 59, 227, 268, 549
Conway group, 59
coordinate charts, 45, 47, 181, 279, 282–83, 396–98, 401
coordinate ring, 376–78
coprime integers, 107
corollaries, 74
correlation function, 528
correlation length in percolation, 663
cosine function, see trigonometric functions
count, 273
countable additivity, 247, 628
countable chain condition (CCC), 632–33
countable models, 625–26, 645–46
countable sets, 71, 157, 170–72, 223, 617, 619, 623, 779
counterexamples, 69, 121, 124–26
Courant, Richard, 808–9
Cours d Analyse (Cauchy), 758
covariant 2-tensors, 485
cover of a topological space, 310
Cox regression model, 925
Coxeter, Harold Scott MacDonald (“Donald”), 53, 950–51
Cramér, Harald, 335
Cramer’s rule, 329
creation and annihilation operators, 528, 542
Crelle’s Journal, 91, 125, 761, 774
crisis in foundations, 142–56
critical exponents, 659, 663–65, 668
critical phenomena, 657–58
critical points, 310
critical probability, 658, 660, 662–64
critical temperature, 666–68
Critique of Pure Reason (Kant), 137
crossing probability, 665
crystallographic point groups, 828
cube, n-dimensional, 53; discrete, 197
cubic equations, 81, 98–99, 101, 326, 708, 737
cubism, 946
cuneiform texts, 96
Curie-Weiss model, 668
curl, 180
curvature, 42, 92, 172, 280, 311, 388, 394, 670; Ricci, 218, 280–81, 406–7, 488; scalar, 280; Weyl, 489
cut rule, 593
cybernetics, 812
cycle decomposition of permutations, 260, 558
cyclic cohomology, 523
cyclotomic fields, 254
cylinder, 734
d’Alembert, Jean Le Rond, 35, 121, 136, 749–50
d’Alembert’s solution to the wave equation, 236
d’Alembertian, 35, 457, 460, 478, 490
Dalí, Salvador, 951
Darboux, Gaston, 125, 777, 794
Darboux’s theorem, 300
Das Kontinuum (Weyl), 149
data encryption standard (DES), 889
de Granges, Louis, 804
de Gennes, Pierre-Gilles, 70
de la Vallée Poussin, Charles-Jean, 63, 338, 356, 686, 715, 792
de Rham cohomology, 175, 177, 179
De Thiende (Stevin), 738
decay of particles, 528
decidability, 638, 640, 643, 645, 813
decimal notation, 30, 79–80, 106, 171, 242, 738, 986
decision problems, 269–70, 577–79, 581
decoherence, 271
decomposition of a finite set, 551
Dedekind, Julius Wilhelm Richard, 104, 127, 138, 143–45, 241, 729–70, 776
Dedekind zeta functions, 730
definable real numbers, 146
definable sets, 624, 627, 631, 643–44
definitions, 74, 84, 146–47, 149
deformation theory, 418
degree, 410; of an algebraic number, 328; of a continuous function, 388, 694–95; of a number field, 329
Dehn, Max, 435–36
Dehn functions, 445–47
Delaunay triangulation, 830
Deligne, P., 347, 729, 732, 808
depth of a circuit, 587–88
derandomization, 601
derangements, 560
derivative of a set of real numbers, 618
derivatives pricing, 910–14
Desargues, Girard, 945; Brouillon Project of, 945
Descartes, René, 81, 100, 134–35, 739–40, 955
descriptive set theory, 631–32
designs, 172–73
determinacy, 159, 630–34; axiom of, 159, 631
determinants, 39, 103, 174–75, 277, 420, 514, 590–91, 641
determined system of equations, 459
Deuring-Heilbronn phenomenon, 340–41
deviation principles, 673–76, 679
diagonalization, 206, 223, 297
diagrams in Greek mathematics, 131, 134, 137, 139
diffeomorphism, 298
difference set, 715
differentiable manifolds, see manifolds, differentiable
differential equations, 51–52, 297, 455, 523–24, 609–11; linear, 51. See also ordinary differential equations, partial differential equations, stochastic differential equations
differential forms, 175–80, 189, 273, 300
differential geometry, 44–46
differential operators, 456–57, 478
differentiation, 30, 33–34, 36–37, 45, 51, 65, 74, 122, 125, 144, 177, 179, 186, 239, 255–56, 282, 397, 450
Diffie-Hellman protocol, 891–92
digital signatures, 893–94
dihedral group, 24, 420–21, 424
dimension, 52, 56, 180–84, 367, 516–17, 724; algebraic, 367; codimension, 391; cohomological, 182; fractional, 184; Hausdorff, 184, 508, 793; homological, 182–83; inductive, 181–82; of a manifold, 396; topological, 184, 367; of a vector space, 22
dimension argument, 571
Diophantine equations, 50–51, 111, 373–75, 378, 692, 706–8, 720, 722
Dirac, Paul, 542
Dirac distribution, 186, 473, 542–43
Dirac equation, 460
direct products of groups, see groups, direct products of
direct sums and products, 24
Dirichlet, Peter Gustav Lejeune, 124, 143, 229, 305, 339, 686, 689, 764–65, 775–76
Dirichlet L-functions, see L-functions, Dirichlet
Dirichlet boundary conditions, 458, 469, 654
Dirichlet characters, 339, 764
Dirichlet principle, 125–26, 475, 789
Dirichlet problem, 458, 476, 653, 764
Dirichlet series, 228–29
Dirichlet summation operators, 451, 453
Dirichlet’s class number formula, 340
Dirichlet’s theorem, 689
Dirichlet’s unit theorem, 255
discrepancy theory, 574
discrete logarithm problem, 892–93
discrete mathematics, see combinatorics
discrete subgroups, 402
discrete topology, 302
discrete-time stochastic process, 649
discretization, 203
discriminant, 320–21, 323, 788, 800; of a binary quadratic form, 320; of a number field, 330; of a polynomial, 317; of an elliptic curve, 347
disk model of hyperbolic geometry, see hyperbolic geometry
dispersive PDEs, see partial differential equations, dispersive
Disquisitiones Arithmeticae (Gauss), 101, 103, 315, 320, 756, 761, 763–64
Disquisitiones Generales Circa Superficies Curvas (Gauss), 756
distance, 31, 41–43, 46, 220, 248, 253, 280, 671
distributed computation, 603, 877–78
distributions, 184–87, 190, 211, 456, 475, 542, 544
distributive law, 20, 284, 770
diversifiable risk, 915–16
divisibility, 118, 242, 249, 807
domain of a function, 11
domain of attraction, 110
Douady rabbit, 500
double contradiction, method of, 132, 134
double cover, 277
double digest problem, 839
downsampling, 860
drift of a Brownian motion, 654, 911–12
duality, 177, 185–90, 212, 274–75, 288; of convex bodies, 189; of groups, 189; of linear spaces, 185, 188, 212; Pontryagin, 189, 205–6
Duchamp, Marcel, 947–48
Dvoretzky’s theorem, 675–76
dynamic replication, 912
dynamical systems: continuous, 494; discrete, 495
dynamics, 5–6, 190, 493–504, 506–10, 576, 713, 728, 766, 802–3; holomorphic, 497–509; topological, 495
Dynkin diagrams, 233–34
e, 71, 81, 200–201, 222–23, 748, 773; transcendence of, 773
effective and ineffective proofs, 117, 722
efficiency of a proof system, 593
efficient computation, 197, 579, 872–74, 883–85
Egyptian fractions, 77–78
eigenfunctions, 206–7, 217, 297, 306, 511
eigenvalues and eigenvectors, 30, 198, 206, 223–25, 240, 294–97, 608, 694, 876–77; eigenvalue decomposition, 608; eigenvalue problem, 472
Einstein, Albert, 83, 95, 153–54, 218, 483–93, 647–48
Einstein constraint equations, 490
Einstein equations, 164, 460, 470, 483, 489–93; vacuum, 489, 491
Eisenstein series, 251–52
elementary functions, 293, 726, 766
Éléments de Géométrie (Legendre), 88
Éléments de Mathématique (Bourbaki), 823
elements of a set, 9
Elements of Algebra (Euler), 104
Elements (Euclid), 84–88, 96, 98, 107–8, 118, 130–31, 133–34, 733–34, 762, 928
elliptic curves, 51, 190–91, 252, 347, 356, 370–71, 380–82, 412–14, 685, 692, 721, 730–31, 892; group law of, 355, 721, 892; use in factoring, 355–56
elliptic functions, 241, 293, 724, 773, 949
elliptic modular function, 60, 324–25
elliptic PDEs, see partial differential equations, elliptic
elliptic regularity theorem, 682
encoding and decoding functions, 880–83
Encyclopédie (d’Alembert), 107
entropy function, 882
Entscheidungsproblem, 113, 707
equal temperament, 937
equation of a circle, 374
equipotentials, 502–6
equivalence: of binary quadratic forms, 320–22; of physical theories, 523–25, 529–30, 532
equivalence relations and equivalence classes, 12, 25, 40, 185, 221, 252
Erdós, Paul, 64, 338, 342, 351, 359, 361, 572, 627, 660, 802
Erdós-Ko-Rado theorem, 569
ergodic theorems, 299, 512–13, 689–91
Erlanger Programm, 93, 777, 782
error function, 293
error-correcting codes, 173, 364, 575, 598, 881–86, 981
Escher, Maurits Cornelis, 950–51
Essai sur la Théorie des Nombres (Legendre), 754
essential supremum, 211
estimates, 62–63, 72, 200, 474–75, 714–17, 916–21, 924; asymptotic, 335
estimators, statistical, 916–21
Euclid, 74, 84–85, 96–97, 107, 118, 132, 689, 734
Euclidean algorithm, 107–8, 114–15, 191–92, 353, 378, 700
Euclidean geometry, 39, 44, 83–84, 87–94, 139, 208, 283, 401, 789
Euclidean structure, 401–2
Euler, Leonhard, 81, 104, 120–21, 228, 261, 290, 333–34, 348, 555, 692, 718–19, 727–29, 746, 747–49, 751
Euler characteristic, 53, 215, 219, 393, 684
Euler equation, 193–96
Euler product, 228–29, 283, 336, 340, 347
Euler vector field, 427
Euler’s formula, 697
Euler’s identity, 748
Euler’s totient function, 250
Euler-Lagrange equations, 65, 311–12, 478–79, 489, 748, 751, 908, 977
evaluation, operation of, 184–86, 378
even functions, 204
even permutations, see permutations
events in probability, 265
evolution maps, 299
examples, importance of, 1001–2
exchange axiom, 244–45
excluded middle, law of, 149, 157, 799–800
existence of solutions, 48, 51, 510
exotic options, 914
EXT, see complexity classes
explicit constructions, 197, 574; of expanders, 197–99; strong, 197
explicit formula for counting primes, 337, 344
exploration process, 664
exponential distribution, 265
exponential function, 30, 193, 199–202, 223, 232, 265, 308, 746
exponential generating functions, 557–59
exponential varieties, see varieties, exponential
exponential-time algorithm, 349–50, 355, 580, 874
extended real line, 169
external rays, 504–6
extremal problems, 64–66;in combinatorics, 215, 562–72, 865–70
factorization of integers, 271, 353–56, 583, 590; into primes, 699–700
factors in von Neumann algebra theory, 516–18
faithful actions, 420
fast Fourier transform, the, 65, 202–4, 271
Fatou, Pierre, 498
Fatou sets, 501–2
fault-tolerant networks, 198
feasible set, 288, 613, 898–900
feedbacks, 900
Feistel cipher, 889
Feit, Walter, 785
Fejér summation operators, 452
Ferguson, Helaman, 952
Fermat, Pierre, 100, 104, 134, 268, 325, 353, 380, 691–92, 740–41
Fermat equation, 50, 111, 117, 254, 681, 722
Fermat’s last theorem, 51, 69, 104, 111, 141, 191, 229, 243, 252, 255, 347, 359–60, 364, 380, 562, 691–93, 764, 820
Fermat’s little theorem, 55, 250, 350–52, 355
Fermat-Catalan conjecture, 360–61
Feynman, Richard, 527–28, 541, 1007
Feynman-Kac formula, 218
Fibonacci, see Leonardo of Pisa
Fibonacci numbers, 115, 222, 249–50, 316, 737
fields, 18, 20–21, 23–25, 27–28, 102, 254, 284, 317–18, 329; extensions of, 21, 28, 102, 254
fields (in physics), 525–26, 542–43
figure eight knot, 225–26
filtering, 860
finite simple groups, 26, 59, 261, 439, 687–89; search for, 783, 785
finitely generated groups, 67, 438–39, 443–44, 685
finitely presented groups, 434–36, 439–48
finitism, 152
first-order logic, see logic, first-order
fixed field, 710
fixed point, 495, 499, 559–60, 693, 731–32; attracting, 499; indifferent, 499; repelling, 499; super-attracting, 499
fixed point theorems, 693–96, 732, 799, 901
flat metrics, 711–12
Flatland: A Romance of Many Dimensions (Abbott), 946–47
floating-point arithmetic, 605–6
flow in networks, 864–70
fluid dynamics, 193–96, 847; biological, 843; computational, 611
forbidden minors, 725
forcing, 624–30, 632–33, 703; iterated, 627
forcing axioms, 632–34
formal languages, 621–23, 635–37
formal power series, 546, 552, 556
formalism, doctrine of, 153–55
formalization of mathematics, 16, 74, 111, 138, 140, 152
formulas, 140, 151, 153, 259, 582, 592, 621–24, 635–37, 640, 642, 700; atomic, 621; Boolean, 588
Foundations of Algebraic Geometry (Weil), 731, 820
Foundations of Geometry (Hilbert), 789
Foundations of Probability Theory (Kolmogorov), 815
four-color theorem, 117, 142, 563, 696–98
four-dimensional manifolds, see manifolds, four-dimensional Fourier, Jean-Baptiste Joseph, 216, 755 Fourier analysis, 220, 261–62, 295–97, 425, 457
Fourier coefficients, 202–3, 205, 212, 262, 451, 454; phase of, 859 Fourier series, 124, 305–6, 451, 511, 686–87
Fourier transforms, 186, 189, 203, 204–8, 214, 236, 239, 274, 306, 450, 453–54, 457, 473–74, 523–24, 859; discrete, 203, 590, 611; inversion formula for, 206, 306, 426; non-Abelian, 274, 424–25, 429–30. See also fast Fourier transform, the
fractal sets, 31, 57, 110, 184, 244, 496, 498, 502, 509
Fredholm, Ivar, 511, 520, 791–92
Fredholm operators, see operators, Fredholm
free Abelian group, 390
free and bound variables, 15–16, 635–36
free Burnside group, 438
free group, 387–88, 390, 433–34, 437, 440, 447, 685
free products of groups, see groups, free products of