Handbook of Mathematics

Index

© Copyright 2015.

Index

A-basis, 316A-module, 316Ab-category, 585Ab-enriched (symmetric) monoidal category, 586Ab el lemma, 121, 453Ab el theorem , 129Ab el-Po isson , 487Ab elian category, 584, 591Ab elian group , 42 , 43 , 82, 84, 88 , 89, 103 , 474 , 475, 516, 517, 521, 584 , 590 ,

599Ab elian integral, 437Ab elian sem igroup , 43 , 516Ab elian variety, 513Ab elian ization , 315Abraham-Shaw , 748Abso lute complement, 8, 12Abso lute consistency, 5Abso lute extrema, 45Abso lute frequency, 620Abso lute geom etry, 149 , 151, 152Abso lute gom etry

models 1,2,3,4, 151Abso lute homology groups , 291Abso lute Hurew icz theorem , 315Abso lute maximum, 670Abso lute valuation, 79Abso lute value , 145

in the fi eld o f rationals, 145non-A rch im edean, 145

Abso lute value o f a rational numb er, 62Abso lutely convergent series , 324Abso lutely flat ring, 526Absorbing, 33Absorbing element, 12 , 33Absorbing on the left, 33Absorbing on the right, 33Absorption , 8 , 9Absorption laws, 12Abstract algebra, 263 , 509Abstract algebraic variety, 513Abstract category, 38 , 598Abstract R iemann surface , 432Abstract simplicial complex, 295 , 534Abstract space, 370Abstract variety, 513Acceptance region , 617Accumulation point, 26, 195 , 198Action -angle variab le, 695Acw ise connectedness, 546Acyclic carrier, 304Acyclic complex, 290Acyclic cover, 296Acyclic cover o f a simplicial complex, 296Acyclic free cha in complex, 303Acyclic free cha in complex theorem , 303Acyclic graph , 260Acyclic m odel, 303Acyclic m odels, 303Acyclic ob ject, 556, 557Acyclic reso lution , 559Acyclic sheaf, 559Acyclic space , 287Aczel’s constructive Zerm elo-Fraenkel (CZF), 16Addition and multip lication in N, 52Addition in N, 52Addition in Q, 61Addition in Z, 58Additive ab elian group , 82Additive asso ciativ ity, 43, 516Additive category, 585Additive commutativ ity, 43 , 516Additive functor, 303 , 597Additive group, 103Additive group of integers modu lo n, 87Additive identity, 42 , 43 , 516Additive inverse, 42 , 43, 516Additive notation, 32, 39additively and multiplicatively stab le, 40Additively stab le, 40Additiv ity axiom , 553Adequate , 297Adequate sub complex, 297Adherent po int, 195 , 198, 219 , 674Adherent po int of a fi lter, 223Adic topo logy, 533 , 569Adjective

autologica l, 13heterologica l, 13

Adjo int, 607Adjo int endomorphism , 137

Adjo int functor, 302Adjo int functors, 595Adjo int functors (adjunction), 595Adjo int map, 243Adjo int of a linear map, 243Adjo int op erator, 243, 375Adjo int pair (o f functors), 311Adjo intness , 589Adjunction, 595Adjunction isomorph ism , 596Adjunction o f indeterm inates , 120Adm issible op en set, 306Adm issible param eter, 393Adm issible param eterization, 393Affi ne algebraic set, 513Affi ne bijection , 169Affi ne combination, 278Affi ne connection , 417, 418Affi ne coord inates system , 190Affi ne endomorphism , 191Affi ne form , 191Affi ne geom etry, 155Affi ne group , 168Affi ne indep endent, 278Affi ne indep endent set, 278Affi ne isom etry, 187

anti-d isp lacem ent, 187disp lacem ent, 187

Affi ne map, 168, 169

Affi ne map in R3 , 186Affi ne morphism , 564Affi ne n-cha in , 293Affi ne n-space , 270Affi ne ne ighborhood , 569Affi ne orthogonal symmetry, 168Affi ne plane, 155Affi ne plane of inc idence, 155Affi ne schem e, 270, 513 , 563, 566Affi ne set, 277Affi ne set (convex set), 277Affi ne space, 189, 277 , 531

affi ne subspace, 190canon ical structure, 190vectoria l structure, 190

Affi ne space (a lgebra ic geom etry), 531Affi ne space (ana lytic geom etry), 531Affi ne subspace, 190Affi ne subspace intersection , 190Affi ne symmetry, 168Affi ne transformation, 169Affi ne variety, 264 , 513, 531Affi ne Y-schem e, 564, 570Affi nes map, 190Affi nity, 168, 169Affi x, 69, 71Afraimovich -Sh iln ikov theorem , 739Aggregate demand function , 677Agument pred icate, 3AIC and SC criteria , 647AIC criteria, 647AIC criterion , 632AIC(p), 647Aitken estim ator, 633Akaike criterion (A IC), 640Akaike function , 632Akaike in formation criterion, 632Akaike-Schwarz, 629Akaike-Schwarz criteria , 647Alemb ert theorem , 224Aleph , 33Aleph-0 , 33 , 35, 37Aleph-1 , 33 , 35, 48Alexander-Veb len coro llary, 296Alexander-W hitney theorem , 319Alexandroff compactifi cation, 210Alexandroff extension , 210Alexandroff theorem , 225Algebra , xvii, 81, 133 , 264

asso ciative, 133commutative , 133of Banach, 442

σ-a lgebra (Σ-algebra, Ω-algebra), 511Algebra o f po lynom ia ls , 107Algebra o f sets , 8A lgebra o f the d isk , 460Algebra ic basis, 244Algebra ic category, 593Algebra ic closure , 69 , 77 , 124, 531Algebra ic closure o f a fi eld, 69, 78

7 7 9

Index

Algebraic closure of C, 77A lgebraic complex numbers over Q, 117A lgebraic curve, 150 , 336A lgebraic dim ension, 212A lgebraic dual, 238A lgebraic element, 530A lgebraic elements of a fi eld , 122A lgebraic extension , 123A lgebraic extension o f a fi eld, 69, 531A lgebraic function , 335, 436A lgebraic fundamenta l group , 568A lgebraic geometry, 509A lgebraic K -theory, 269 , 562A lgebraic numb er, 78, 117A lgebraic numb er fi eld , 78A lgebraic op eration, 8, 510A lgebraic op erations in sets, 8A lgebraic point, 437A lgebraic problem , 129A lgebraic ram ifi cation , 433A lgebraic relation, 335 , 336A lgebraic representation of a complex number, 70A lgebraic set, 513A lgebraic signature, 511A lgebraic space, 513A lgebraic structure , 38 , 81 , 512, 514, 582

o f integers, 64o f rationa ls , 65o f real numbers, 66

A lgebraic structure o f R, 321A lgebraic surface, 150A lgebraic system , 265 , 280, 510A lgebraic topo logy, xvii, 247, 270 , 509A lgebraic variety, 264 , 513A lgebraically closed, 77A lgebraically closed fi eld , 78 , 113, 531 , 569A lgebras for a monad , 593A lgebrization, 129A lmost everywhere convergence , 488A lternating algebra , 137A lternating b ilinear form , 449A lternating form , 138A lternating group , 87, 138A lternating map, 138A lternating series, 324A lternative hypothesis H1, 617A ltitude, 166Ambient manifold, 544Amenab le group, 18Amnestic functor, 595Ampère law , 364Amplitude of cycle, 734Amplitude sp ectrum , 662Analy sis , xviii, 321Analy sis o f auto corre lation function, 643Analy sis o f residuals, 646Analy sis o f sho cks, 647Analy sis o f variance, 617Analy sis o f variance (ANOVA), 627Analy sis s itus, xv iiAnaly tic, 548Analy tic at, 548Analy tic continuation , 421, 427, 441 , 457Analy tic continuations along d iff erent paths, 427Analy tic function, 333 , 415, 426, 433 , 548Analy tic function at a p oint, 548Analy tic function on interior o f domain , 548Analy tic functions, 421Analy tic geom etry, xvii, 130, 149 , 181

rea l or complex, 150Analy tic m an ifold , 540Analy tic representation , 172Analy tic set, 560Analy tic space, 560Analy tic subspace, 560Analy tic variety, 513Andronov-Hopf b ifurcation , 735, 748Andronov-Hopf theorem , 735Andronov-Pontryag in theorem , 724Andronov-W itt theorem , 719Angle , 160, 163

acute , 160obtuse, 160o f op en ing , 184refl ex, 160right, 160stra ight, 160

Angle m easure, 191Angle m easure on a regu lar patch , 400Angle m easurem ent, 161Angle trisection, 75, 130Angle -preserv ing transformation , 438Annihilating elem ent, 12Annulus, 195, 428 , 447, 450, 464Anosov, 705Anosov automorph ism , 703Anosov d iff eomorph ism , 705, 732ANOVA equation , 627ANOVA tab le, 617

for s imple regress ion model, 627Antecedent, 30 , 47 , 344, 620Anti-adjo int, 243Anti-m atched lines, 153Anti-m atch ing, 152Anti-m atch ing theorem , 153Anti-paired lines , 153Anti-se lf-adjo int op erator, 243Antiautomorphism , 562Anticon formal mapping, 438Antiderivative, 349 , 352, 353

Antidual, 405Antidual basis , 405Antihomomorphism , 562Antilinear map, 241Antinomy, 13

semantic or syntactic, 13Antip odal m ap , 294Antipode, 271, 294Antisymmetric function, 138Antisymmetric m ap , 138Antisymmetric multilinear form , 138Antisymmetrization, 138Antisymmetrization operator, 138Antisymmetry, 7Aperiod ic so lution , 722Apollon ius circle, 167Apollon ius defin ition of a circle , 168Applied mathematics, xviiiApproximation , 331Approximation by convo lution, 491Approximation by interpo lation po lynom ial, 340Approximation by po lynom ials

on a compact of Cn, 236on a compact of Rn, 236

Approximation by trigonometric po lynom ials, 236Approximation m ethods, 354Approximation of vector functions, 236Approximation theory, 338AR (autoregress ive), 644AR model, 644AR(p), 634, 644Arc, 196 , 204Arc length, 358Arc length of curve, 358Arc length parameterization , 499Arc of a curve, 392, 393ARCH (AR conditional heteroscedastic ity), 637ARCH model, 637ARCH test, 637ARCH(q), 652Archim edean prop erty, 52 , 67A rchim edean totally ordered commutative fi eld , 66A rchim edean totally well ordered field, 78A rchim edean va lued field , 146Arcw ise -connected , 545, 546Arcw ise -connected component, 204Arcw ise -connected set, 196, 204Arcw ise -connected space, 204 , 226, 545Area function , 168Area m easure on a regu lar patch , 400Area o f ellip tic triang le, 179Area prob lem , 328Areo lar function , 365Areo lar m easure, 365ARFIMA (AR fractionally integrated MA), 651ARFIMA process , 651, 652ARFIMA(p,d ,q ), 651, 652Argand-Cauchy plane, 71A rgand-Gauss p lane, 71A rgument, 2 , 74, 455 , 499Argument of a complex number, 74A rgument princip le, 476ariety (un iversa l a lgebra), 513ARIMA (AR integrated MA), 644ARIMA model, 644ARIMA process, 645Arithm etic, 51A rithm etic function , 141

Euclidean , 142Arity, 510 , 511ARMA (autoregress ive moving average), 645ARMA model, 645ARMA(p,q), 645ARMAX, 646Arnold, 687 , 695Arnold tongue, 742Arnold tongues, 740Arrangement, 613Arrangements w ith rep etitions, 614Arrangements w ithout rep etition , 614Arrow , 576Arrowsm ith-P lace, 708 , 712Articulation vertex, 262Artin braid group , 539Artin ian ring, 525Artin ian set, 35A scend ing chain cond ition (ACC), 523Ascoli theorem , 236 , 442Asso ciate, 309Asso ciative law , 41Asso ciative magma, 41 , 515Asso ciativ ity, 8, 9A sso ciativ ity isomorph ism , 587Asso ciator, 586Astro id , 336Asymptote, 334Asymptotic b ehavior o f Fourier co effi cients, 655Asymptotic line , 403Asymptotic point, 381Asymptotic right triangle, 178Asymptotical stab ility, 719Asymptotically stab le, 687Atlas , 413—415 , 542 , 687Atlas o f class Cr (or Cr -atlas), 414Atom , 2, 10Atom of a lattice , 10Atom ic formula, 16Attach ing cell, 299Attach ing map, 298 , 299, 535Attracting or rep elling basin , 747

780

Index

Attractor, 715, 724Attractor in sense of Ruelle (1981), 733Attractor in the sense o f M ilnor, 715augm entation, 303Augmentation homomorph ism , 553Augmentation preserv ing, 303Augmented complex, 296Augmented D ickey-Fuller test (ADF), 644Augmented singu lar complex, 292Aureum theorem , 144Autocorrelation function, 643, 725Autocorrelation function o f sampling , 643Autocorrelation o f errors, 633Autocorrelation test o f errors , 639Autofunctor, 595Autolog ical adjective, 13Automorph ism , 33 , 38 , 99, 483 , 520, 598Automorph ism group , 105Automorph ism set, 105Automorph ism s of C, S2,D, 483Autonomous d ifferentia l equation , 716Autonomous linear differentia l equation , 716Autonomous system , 693Autoregressive linear model, 639Axial vector, 184AxioM

of countable choice, 19Axiom , 2 , 5 , 9

of extensionality, 17A,A* (bisector), 159A1,A2,A3 (affi ne inc idence plane), 155C (connection), 154D (completion) Dedekin , 160functional , 14H (hyperb olic), 154Hausdorff , 219M1,M2,M3 (metric p lane) Absolute geo., 151of Cartesian product, 12of cho ice , 14 , 16 , 17of comprehension , 14 , 16of denumerab le cho ice , 19of dep endent choice, 19of existence, 14of extensionality, 10, 11, 14, 16of foundation, 16 , 17of induction, 50of infin ity, 14, 16, 17of pairing, 11, 16, 17of regularity, 16, 17of rep lacem ent, 16, 17of separation , 11 , 16of sing letons, 10, 14of subsets, 16of Tarski, 16of the empty set, 14of the pa ir, 11of the p ower set, 16 , 17of the set of parts o f a set, 11of the set of the parts of a set, 14of the sum of set, 17of the sum set, 16of the unordered pa ir, 16, 17of union , 11, 14, 16, 17P (po lar trila teral), 154P1,P2,P3 (pro jective incidence p lane), 155P5,P5’ (Fano), 158R (rectangle), 153schema of collection, 17schema of replacem ent, 17schema of restricted comprehension , 17schema of sp ec ifi cation, 17T0,T1,T2,T3,T4, 206T2, 199

Axiom (postu late), 5Axiom of b isector (A ),(A*), 159Axiom of Cartesian pro duct, 12Axiom of cho ice, 14, 16, 22, 25, 32 , 579Axiom of comprehension , 14, 16Axiom of connection (C), 154Axiom of countab le cho ice, 19Axiom of Dedekin (D), 160Axiom of denumerab le choice, 19Axiom of dep endent cho ice, 19Axiom of existence , 14Axiom of extensiona lity, 10, 11 , 14 , 16 , 17Axiom of extensiona lity (for maps), 30Axiom of foundation , 16, 17, 578Axiom of infin ity, 14, 16, 17 , 37Axiom of pairing, 11 , 16 , 17Axiom of po lar trilatera l (P ), 154Axiom of regu larity, 16 , 17Axiom of rep lacem ent, 16 , 17Axiom of revealed preferences, 760Axiom of separation , 11, 16, 31Axiom of s ingletons, 10, 14Axiom of subsets , 16Axiom of the empty set, 14Axiom of the pa ir, 11Axiom of the power set, 16, 17Axiom of the set of parts o f a set, 11Axiom of the set of the parts o f a set, 14Axiom of the sum of set, 17Axiom of the sum set, 16Axiom of the unordered pair, 16 , 17Axiom of un ion, 11, 14, 16 , 17Axiom schema of co llection , 17Axiom schema of rep lacem ent, 17Axiom schema of restricted comprehension, 17Axiom schema of sp ecifi cation , 17Axiomatic construction

of number system , 78Ax iomatic m ethod , 5, 149 , 150Ax iomatic set theory, 12—15, 577Ax iomatic sy stem , 3Ax iom s, 1

Euclid , 149of Zerm elo-Fraenkel set theory, 16of ZFC , 16separation, 205 , 206T0,T1,T2,T3,T4, 206

Ax iom s of categories, 590Ax iom s of closed sets , 218Ax iom s of Zerm elo -Fraenke l set theory, 16Ax is o f curvature, 396Ax is refl ection, 159Axonometric pro jection , 174

b-adic number, 68Bézout co effi cients , 60 , 111Bézout identity, 60Bézout theorem , 60Bachmann, 151, 157Bachmann metric plane, 151Back face, 318Backward elim ination , 632Baire category theorem , 232, 724Baire function, 368, 369Baire prop erty, 232Baire space, 232Baker map, 732Ball, 50Banach algebra, 24 , 442, 494Banach fixed p oint, 372Banach fixed-p oint theorem , 372Banach functional, 20Banach space , 20 , 21 , 237, 241, 371 , 489Banach theorem , 241 , 391Banach-A laoglu theorem , 442Banach-Ste inhaus theorem , 241, 371Banach-Tarski paradox, 8, 17Barratt-W hitehead lemma, 292Barrow rule, 499Barycenter, 190 , 252, 278, 675Barycentric coord inates, 190 , 252, 278Barycentric subdiv ision, 293Barycentric subdiv ision o f an affi ne n-s implex , 293Base, 560Base for a topo log ical space, 537Base line, 157Base of fib er space, 562Base scheme, 563Base space, 306, 560 , 574Base ten numera l system , 56Base-10 numera l system , 56Basepo int, 214Basepo int preserv ing map, 272Basic dynam ics of d iff eomorphism s, 687Basic variable, 670Basin of attraction , 715, 724, 733Basis , 102, 199 , 405 , 519Basis for a top ology, 308Basis of a K -vector subspace, 108Basis of a top ology, 199Basis of neighborhoods, 200 , 218, 222Basis of op en sets, 218Bauer, 370Bayes theorem , 619Behaviors of second-order linear system s, 716

diagram of b ehaviors, 717interpretation of e igenvalues , 717

Belote , 615Bernoulli inequality, 323Bernoulli numb er, 337Bernoulli shift m ap , 725Bernste in p olynom ia l, 338Bessel inequality, 244, 653Best approximation, 338, 394Best contact, 394Beth ’s definability theory, 5Betti group , 257, 550Betti number, 257 , 412Bezout identity, 143Bezout re lation, 120Bezout theorem , 60Bezout theorem (for polynom ia ls), 111B i-ratio , 171B ianch i identity, 419B icategory, 585, 593b icategory, 264B iclosed monoida l category, 586B icomplete category, 585B iconnected graph , 262B idua l, 239 , 405B idua l of a norm ed vector space , 239B ienaym é-Chebychev inequality, 620B ienaym é-T chebychev inequa lity, 620B ifunctor, 264, 585, 586 , 595B ifunctor (binary functor, i.e. 2-ary ), 595B ifurcation , 733, 734 , 748

Andronov-Hopf, 735 , 748Andronov-Hopf sup ercritical, 749cusp , 712fl ip , 748fold, 748Hopf, 733Hopf sub critical, 712Hopf sup ercritica l, 712node-sadd le, 733of fixed points, 733of p eriod ic orbits, 734p eriod-doub ling , 748

781

Index

p itch fork, 733sadd le node, 718 , 748subcritical, 734subharmonic, 734supercritical, 733transcritical, 733

B ifurcation d iagram , 713 , 735, 736 , 749, 751B ifurcation d iagram of Lorenz model, 751B ifurcation of doub le p eriod ic orbits, 736B ifurcation value , 734B ifurcations in Morse Smale system s

g loba l b ifurcation , 737sadd le-node separatrix d isappearance, 738sadd le-node to p eriod ic orbit, 738

lo cal bifurcation nearby periodic orb itsb ifurcation of doub le p eriod ic orbits, 736centre manifold theorem for maps, 736creation o f a torus, 737p eriod -doub ling (fl ip bifurcation), 737

lo cal bifurcations nearby steady states , 734Bogdanov-Takens b ifurcation , 736breaking sym etry, 736centre manifold theorem , 734cusp bifurcation , 735genera lized Hopf b ifurcation , 736Hopf bifurcation , 735sadd le-node, transcritica l bifurcations, 734

B ifurcations o f fixed po ints , (Cod-1), 733subcritical b ifurcations, 734supercritical bifurcations (normal form )

Hopf, 733p itchfork, 733sadd le-node, 733transcritical (stab ility exchange), 733

B ifurcations theory in Morse Smale system s, 734B ig éta le s ite, 568, 569B iggest elem ent, 9B iharmonic osc illa tor, 695B iholomorph ic map, 438B iholomorph ism , 482 , 483B iholomorph ism ,(con formal map), 482B ijection, 33, 37 , 520

inverse, 36B ijective homomorphism , 520B ijective homomorphism (bimorph ism ), 520B ijective map, 29 , 31 , 519B ijective map defined via compositions, 30B ijective morph ism , 33B ijective morph ism (isomorphism ), 33B ijective sets, 29B ijectiv ity of Galo is m ap , 127B ilinear extension, 103B ilinear form , 406B ilinear function, 304B imorph ism , 520 , 598B inary functor, 264B inary numeral system , 56B inary op eration , 265, 510B inary relation , 7, 13 , 27 , 28

antisymmetric, 28asymmetric, 28b iective (or biunivo ca l), 28doub ly surjective, 28injective, 28left surjective, 28prop erties , 28refl exive, 28right surjective, 28symmetric, 28tota l, 28transitive, 28univo cal, 28

B inom ial co effi cient, 90, 613B inom ial d istribution , 617 , 621B inom ial equation, 129B inom ial expansion, 90B inom ial formula , 92B inom ial series , 90 , 333B inom ial theorem , 90 , 92B inormal vector, 394B ipartite G raph (or b igraph), 259B ip eriod ic dynam ics, 722B ipo int, 161B ipo lar rectangu lar impulse, 663B iquadratic recipro c ity theorem , 145B irationa l transformation , 150B irkhoff ergod ic theorem , 725B irkhoff HSP theorem , 514B irkhoff prim e ideal theorem , 23B irkhoff theorem , 514, 718B irth o f an invariant torus, 748B isector, 152B isector axiom (A),(A*), 159B ite vector, 32B ivalence princip le, 1B ivalent logic, xviiB lakers-M assey theorem , 315B laschke condition , 464B laschke products, 468B lo ck, 262B lo ck-Guckenheim er-M isiuriew icz theorem , 733B low ing-up , 701

d irectiona l, 702po lar, 701

BLUE, 634, 635BLUE (Best Linear Unbiased Estim ator), 629B lue sky catastrophe, 746 , 748Bodganov, 712Bogdanov, 712Bogdanov, Takens, 712

Bogdanov-Takens b ifurcation, 736Bolya i, 169Bolzano-Weierstrass prop erty, 210Bolzano-Weierstrass theorem , 322 , 323Boolean algebra, 19Boolean algebras, 9Boolean lattice , 9, 10, 618Boolean prim e idea l theorem (BPI), 19, 23, 24Boolean ring, 10 , 93Bordant, 373 , 547Bordant manifolds, 547Bord ism , 263, 373 , 549Bord ism group, 547Borel measure, 725Borel σ-algebra , 618Borel set, 618 , 725Borel sigma-algebra , 618, 725Borel-Carathéodory inequality, 467Borel-Leb esgue, 223Borel-Leb esgue prop erty, 223Borsuk theorem , 473Borsuk-U lam , 294Borsuk-U lam theorem , 320Boundary, 286, 294 , 547Boundary conditions, 391

D irich let, 391Neumann, 391Robin, 391

Boundary map, 553 , 558Boundary morph ism s, 597Boundary o f a singular n -simple , 286Boundary operator, 286 , 534, 550Boundary p oint, 195, 294Boundary va lue problem , 378Bounded , 21Bounded distance, 217Bounded from ab ove, 46Bounded function , 21Bounded lattice , 23Bounded linear functiona l, 19, 21Bounded linear op erator, 372Bounded part, 209, 216Bounded tile, 209Bounded variation , 653Bouquet o f circles , 298Bouquet o f circles (rose), 298Bourbaki, 205Bourbakist conception o f mathematics, 37Boussinesq, 749Box-counting d im ension , 730Box-counting d im ension prop erties, 730Box-Jenkins method , 645Box-P ierce statistic, 643B ra, 409B rachisto chrone, 372, 373B racket o f two vector fi elds, 416B raid, 539B raid group, 539B raid index, 539B raided category, 587B raided monoida l category, 587B raiding, 587B ranch , 423 , 431B ranch cut, 423, 431B ranch po int, 422 , 431, 433B reaking sym etry, 736B reit-W igner curve, 663B reusch -Godfrey test, 634B roomhead-K ing , 416B rouwer, 4 , 212, 215, 472 , 674, 677B rouwer degree , 413B rouwer fixed point theorem , 215, 677B rouwer theorem , 214 , 472B rownian motion , 651Bubble sort, 87Bundle, 153, 263 , 268, 561Bundle (category), 561Bundle (topology), 561Bundle rank, 269 , 562Buniakow ski inequa lity, 191Bura li-Forti, 13Bura li-Forti paradox, 49By recurrence, 5

C is algebraically c losed , 115C∞ partition of unity sub ord inate to covering , 499C-ana lytic function , 453, 457

C-C function , 423C-C function , 423C-C function , 423C-C-function , 423 , 425C-d iff erentiab le function , 424, 452 , 548, 549C∞-m an ifold, 541C∞-structure, 541C∞ partition of unity, 495, 500C∗-a lgebra, 562Cr-m an ifold, 687Caeteris paribus, 647Cah , 175Calcu lation

o f geodesic curvature, 401o f normal curvature, 401

Calcu lus and ana lys is, xviiiC alcu lus of variations, 372Cancellable, 40Cancellable elem ent, 33, 40Cancella tion prop erty, 40Canon ical basis , 191, 443Canon ical basis o f the K -vector subspace , 108Canon ical euclidean norm , 191

782

Index

Canonical fl abby reso lution, 559Canonical fl asque resolution , 559Canonical form of a circle map , 740Canonical in jection , 30, 520Canonical map , 30 , 201, 563Canonical morphism , 87Canonical pro jection , 223 , 599, 603Canonical representation of s imple arc, 395Canonical reso lution, 559Canonical sca lar product in vector space Rn, 191Canonical structure , 190Canonical surjection , 29 , 34Canonical topo logy, 609Cantor, 13, 37, 47, 66Cantor comb, 730Cantor diagonal method , 32, 34Cantor diagonalization , 33Cantor function , 740Cantor native language, 35Cantor sequence, 49Cantor set, 706 , 709, 730 , 731Cantor theorem , 31, 37, 228Cantor’s transfin ite numbers , 48Capacity d imension, 730Cap ital p er un it o f lab or, 760Carathéodory theorem , 485Cardan formula, 74Cardan method, 128Card ina l, 10, 33 , 34 , 48

addition, 34arithm etic, 34comparison , 34exponentiation , 34multip lication, 34of a set, 36of fin ite or infin ite set, 34of rea ls, 37of the set of rea ls, 35op erations, 34

Card ina l number, 10 , 33Card ina l of a group , 82Card ina lity, 33Card ina lity 1, 33Card ina lity 2, 33Carnap, 1Carrier (universe), 510Carrier function, 303Carrier set, 510Carrier set (underly ing set), 510Cartan , 138, 205Cartesian c losed category, 588Cartesian geom etry, 130Cartesian product, 13 , 27Cartesian representation of a complex number, 70Cascade of p eriod -doublings, 738Casorati-Weierstrass theorem , 464Cat, 584Cata lan numbers, 32Catastroph ic instab ility, 749Categorica l d irect sum , 603Categorica l product, 603Categorica l sum , 603Categories o f rational functions, 335Category, 38 , 256, 265, 270 , 272, 576 , 577, 581 , 582, 598

by E ilenb erg-Mac Lane , 576by M ac Lane, Buchsbaum , G rothend ieck , Heller, 577

Category Ab (AbGrp), 583Category AbGrp (or Ab), 583Category asso c iated w ith ordered set, 589Category Cat, 583Category Comp, 289, 290Category CPO , 583Category D iff , 583Category D iffb , 583Category F ields , 583Category G rp (Groups, G roup), 583Category Heyt, 583Category hoTop , 583Category isomorph ism , 594Category M an, 583Category M anb, 583Category M on, 583Category o f (Eilenb erg -M oore) algebras, 594Category o f all étale morph ism s, 568Category o f all abstract simplicial complexes and simplic ial m aps, 295Category o f chain complexes , 589, 592Category o f functors, 586Category o f schem es, 270, 563Category o f sheaves o f sets, 551, 569Category o f sm all categories , 584Category PoSet, 583Category PreO rd, 583Category R ings, 583Category Sets, 583Category theory, 265, 575Category Top , 583Category Vec , 583Category VecC, 583Category VecR, 583Category w ith a generator, 591Catenary ring, 525Catenary schem e, 564Cathetus, 178Cauchy convergence criterion , 323Cauchy criterion , 228 , 244Cauchy fi lter, 231Cauchy form of Taylor remainder, 332Cauchy formula, 425 , 456

for a d isk (d irect demonstration), 459Cauchy formula (Homology form ), 475

Cauchy formula (Homotopy form ), 471Cauchy inequalities, 457 , 486Cauchy integral

formula, 425prop erty, 425

Cauchy integral formula, 425 , 429Cauchy integral prop erty, 425Cauchy integral theorem , 425, 429, 660Cauchy integration criterion, 355Cauchy kernel, 460Cauchy kernel o f un it d isk , 460Cauchy mean-va lue theorem , 331Cauchy principal va lue, 661Cauchy rational sequence, 69Cauchy real sequence, 69Cauchy sequence, 21, 66 , 67 , 146, 372

rational, 69real, 69

Cauchy theorem , 456 , 474Cauchy theorem (Homology version), 474, 475Cauchy theorem (Homotopy version), 470, 474Cauchy tran form , 459Cauchy transform , 453Cauchy transform of a Radon measure, 453Cauchy transform of Borel m easure, 459Cauchy-Goursat theorem , 453, 461Cauchy-Kowalewska theorem , 389Cauchy-Pompeiu formula , 456, 489Cauchy-R iemann cond itions, 421 , 424Cauchy-R iemann equation , 457Cauchy-Schwarz inequa lity, 191Causality in the sense o f Granger, 647Causality in the sense o f Sims, 647Cavalieri, 357Cavalieri princip le, 357Cavalieri theorem , 357Cayley-Menger determ inant, 251, 255Cconnectedness prop erties of topolog ica l spaces, 538Cech cohomology, 550 , 558, 559Cech cohomology group , 558Cech complex, 550Ceiling, 61Ceiling function , 61Cell, 534, 585Cell (a lgebra ic topology), 534Cell (geom etry), 534Cellu lar, 301 , 535Cellu lar approxim ation , 302Cellu lar approxim ation theorem , 302Cellu lar chain complex, 301Cellu lar chain complex lemma, 301Cellu lar chain complex of a fi ltration , 301Cellu lar fi ltration, 301Cellu lar homology, 301Cellu lar map, 301Cellu lar space , 301Cellu lar theory, 287Center, 83 , 717Center (centre), 83Center manifold , 700, 748Center manifold theorem , 700Center o f the shea f, 157Centering param eter, 666Centra l ang le, 163Centra l lim it theorem , 617Centra l pro jection , 170

Centra l pro jections in P3 (R), 170

Centra l pro jections in R3, 170Centra l shea f, 157Centre , 83 , 381Centre manifold , 748Centric, 156Centric p ersp ective, 156Ceteris paribus, 647Cevian , 166CH , 15Chain, 21 , 36 , 44 , 45, 260 , 285Chain complex, 289 , 533, 550 , 589Chain complex of fin ite typ e, 318Chain equ iva lence, 290, 303Chain equ iva lent, 290Chain group, 301Chain homotop ic, 290Chain homotop ic maps, 290Chain homotopy, 290Chain map, 290Chain o f an ordered set, 36Chain o f fi elds, 129 , 130Chaitin , 3Change o f adm iss ib le param eter, 393Change o f basins , 747Change o f basis, 405Change o f coord inate system , 187Change o f coord inates, 414 , 542Change o f lo ca l coordinates , 541Change o f param eterization , 499Change o f variab le, 353 , 380Change o f variab les , 357Chaos, 714 , 732

(DO ) density o f orb its, 732(SD IC), 732(TT) topolog ica l transitiv ity, 732

Chaos in the sense of D evaney, 732Chaos in the sense of L i-Yorke, 732Chaos theory, xviiiChaotic attractor, 732Chaotic dynam ical system , 732Chaotic sy stem s in the sense of D evaney, 732Character of a p oint, 538Character of a topolog ical space, 538

783

Index

Characteristic, 94, 530Characteristic class o f a fibration, 320Characteristic equation , 385Characteristic function , 32Characteristic m ap , 300, 535Characteristic of a ring , 94Characteristic polynom ia l, 135Characteristic root, 133Characteristic subspace, 135Characteristic value, 133Characterization

o f s imply connected op en sets, 485Characterization of an attractor, 730

fixed po int: Lyapunov exp . s ign (-,-,-) , 730lim it cyc le: Lyapunov exp . sign (0 ,-,-) , 730strange attractor: Lyap . sign (+ ,0 ,-) , 730Torus: Lyapunov exp . s ign (0,0,-) , 730

Characterization of fi e lds by ideals, 101Chart, 413, 541 , 542, 687Chasles relations, 75Chebychev inequality, 620

Chebyshev (T chebychev, T cheb icheff , C eby sev , Tschebyscheff , Cheb ishev),75 , 339

Chebyshev approximation, 669Chebyshev d istance, 220Chebyshev norm , 338, 340Chebyshev norm (uniform norm ), 338Chebyshev p olynom ia ls, 75 , 116, 338Chebyshev p olynom ia ls o f the fi rst k ind , 116Chebyshev p olynom ia ls o f the second kind, 116Chi-squared distribution, 617Chi-squared test, 617 , 622Chillingworth, 687Chinese congruence , 97Chinese hypothesis , 97Chinese remainder theorem , 97Chinese theorem , 97Choice function , 16 , 17 , 19Choice o f optim al model, 631 , 632Chorda l d istance, 423Chorda l theorem , 167Chow test, 635Christoff el symbol o f the fi rst k ind , 418Christoff el symbols , 417Christoff el symbols o f the second kind, 418Church , 3C ircle, 163 , 184, 185C ircle diff eomorph ism , 692C ircle diff eomorph isms w ith rational numb er, 692C ircle m ap , 742C ircle of curvature, 177 , 396, 398C ircle of curvature for an ellipse, 397C ircle quadrature, 75, 131C ircle squaring , 75 , 131C ircu it, 260, 297C ircu lar average, 491C ircu lar function , 174, 336C ircu lar p ermutation , 91C ircum center, 167C ircum circ le, 166C ircum rad ius, 167

Ck-atlas, 541C lass of a lgebras, 594C lass of generators for a category, 581C lass of groups, 317C lass V , 15 , 578C lasses o f quadrics , 188C lassical harmonic ana lysis, 487C lassifi cation

o f quadrila terals and triangles, 170C lassifi cation of closed surfaces, 255C lassifi cation of fi elds , 125C lassifi cation of m etric p lanes, 155C lassifi cation of patch po ints , 402C lassifi cation of p eriod ic orbits, 719C lassifi cation of rea l functions, 328C lassifi cation of valued fields, 145C lassify ing morph ism , 581C lique , 262C lop en set, 537C losed 1-form , 316C losed arc, 196C losed ba ll, 216C losed curve, 214 , 545C losed curve contractility, 546C losed d isk , 71, 196C losed Eulerian cha in , 260C losed form , 448C losed graph theorem , 241, 372C losed immersion , 564, 571closed immersion , 570C losed Jordan curve, 197C losed k-ce ll, 294C losed manifold , 542closed manifold, 547C losed manifold (compact boundaryless m an ifold), 547C losed monoidal category, 586C losed orb it, 722C losed part, 216C losed path, 279C losed set, 26, 195 , 198C losed subschem e, 564C losed surface, 255

classifi cation , 255C losed surfaces , 255C losed under, 520, 521C losed -b ounded , 197C losed -b ounded prop erty, 197C losure , 24 , 27, 42, 195 , 216, 520 , 521

o f a connected part, 226

C losure , 674C losure (abstract a lgebra), 520, 521C losure of valued fields, 146C losure op erator, 102C losure theorem s, 155C luster po int, 26 , 195, 220

o f a sequence, 322 , 323, 341, 423C luster po int o f a map, 221C luster po int o f a sequence, 221Cn[a, b] space , 371Co-cone, 601Co-unit, 320Co-unit o f a graded co-ring, 320Cobordant, 547Cobordant manifo ld , 547Cobordism , 373 , 547Cobordism class , 547Cobordism group , 547Coboundary, 317, 534Coboundary map, 555Coboundary map (diff erentia l), 555, 556Coboundary op erator, 534, 550Coboundary op erator (codiff erential), 558Cobound ing, 373co chain, 317Cochain complex, 534 , 555, 556Cochrane-O rcutt m ethod , 635Cocomplete category, 305, 585Cocone, 601Coconstant morph ism , 599Cocycle, 317 , 534Codiff erential, 558, 559Codim ension , 699, 712 , 734Codim ension of a bifurcation , 736Codim ension -1 , 733Codomain, 14, 29 , 30 , 38 , 519Coeffi cient group, 302Coeffi cient o f degree k, 108Coeffi cient o f determ ination , 627, 629Coeffi cients , 304Coequalizer, 305, 588 , 604Coexact sequence, 313Cofactor, 106Cofibration , 314Cofinal, 15, 23, 578Cofinality, 23Cogroup ob ject, 310Cohen-M acaulay ring, 525Cohen-M acaulay schem e, 564Coherence cond itions, 593Coherence prop erty, 587Coherent shea f, 553Cohomology, 263 , 316, 476 , 549, 556Cohomology (of top ologica l space), 550Cohomology (w ith values in sheaf ), 550Cohomology class, 317Cohomology functor, 597Cohomology group , 263, 316, 317 , 476, 550Cohomology groups, 568Cohomology ring, 318 , 319, 550Cohomology theory, 263 , 476, 509Cohomology w ith compact support, 571Cohort data, 625Coinc ident transformation, 161Cointegrated series, 648Cointegration, 626 , 648Cointegration ana lysis, 648Cointegration rank, 649Cointegration relation , 650Cointegration relation test, 650Cointegration test, 648Cointegration theory, 626Cointegration vector, 648Cokerne l, 599 , 605Colim it, 305, 602Collection of m orph isms, 38 , 598Collectiv izing prop erty, 11Collectiv izing relation , 8Collinear, 152Collinear vectors, 181Collinearity, 181Collineation , 151, 156 , 171

orthogona l, 151pro jective, 149

Collineation o f a pro jective plane, 156Collision , 748Coloring , 214, 261Comatrix , 106Comb space , 203Combination (or se lection), 613Combination w ith rep etitions, 615Combinations w ithout rep etition, 615Combinatorial ana lys is, 54 , 613Combinatorial homotopy, 254Combinatorial theory, 613Combinatorial topology, 250 , 254Combinatorics, 254 , 613Comeagre set, 232Comma category, 580, 584Comma derivative , 417Common divisor , 57Common multip le, 58Commutative Abelian groups, 89Commutative and un ita l ring , 522Commutative and un itary ring, 522Commutative Banach algebra , 24Commutative d iagram , 30, 273, 600Commutative fi eld structure, 64Commutative group, 82, 89 , 475Commutative group (Abelian group), 82

784

Index

Commutative group for the composition law , 156Commutative ring, 522Commutative ring w ith identity, 522Commutative ring w ith unity, 522Commutative sem igroups, 92Commutative un it ring , 10 , 64

w ith no div isor of zero, 43, 516Commutativ ity, 8 , 9Commutator, 315 , 502Commutator subgroup, 315Comonad, 593Comonad (cotrip le), 593Compact, 197Compact b oundaryless manifo ld , 547Compact b oundaryless manifo ld (c losed manifo ld), 547Compact invariant chaotic set, 732Compact manifold, 541, 547Compact op erator, 375Compact p olyhedra l pa ir, 302Compact set, 26, 197 , 208, 209 , 715Compact space, 26, 78, 208, 209 , 223, 538

by adjunction of a po int (at infin ity), 78product, 224

Compact subset, 197Compact support, 288, 668Compact support theorem , 288Compact-op en topo logy, 209, 309

Compact-op en topo logy on XY , 309Compactifi cation, 78 , 209, 210 , 422, 571

of the p lane, 78Compactifi cation o f C, 421, 422Compactness, 208Compactum , 210Comparability, 21 , 44 , 579Comparable d istances, 217Comparison of Fourier and Laplace transform s, 663Comparison of topolog ical spaces, 200Comparison of topolog ies, 199, 218Compass, 129Compatib ility, 414

of a map w ith top ologica l structures, 414Compatib ility o f charts , 414Compatib ility o f charts o f an atlas, 432Compatib le chart, 541Compatib le charts, 542Complement, 8, 12 , 13Complement formula , 468, 477Complementary angles, 162Complementary set, 195Complementation , 9Complemented lattice, 9 , 23Complete algebraic variety, 513Complete axiomatic set theory, 13Complete category, 305, 585Complete fi eld , 322Complete graph , 259Complete lattice, 128Complete lo ca l ring , 569Complete metric space, 67, 215 , 228, 230complete normed linear space, 21Complete normed space, 20Complete normed vector space, 21, 237 , 241, 371Complete pre-H ilb ert space, 137, 371Complete quadrang le , 158Complete quadrila teral, 158Complete space, 422Complete un iform space, 231Completed lo cal ring, 569Completely regular space, 207Completion, 78, 160, 215 , 528

of a m etric space, 229Completion (abstract algebra), 528Completion of a metric space, 229Completion of a ring, 528Completion of m etric space , 215Completion of pre-H ilb ert space , 242Completion of un iform space, 231Completion pro cess , 78Complex, 289Complex analysis , 421Complex analytic function, 548Complex analytic function (holomorph ic function), 548Complex analytic funtion, 548Complex analytic m an ifold, 540Complex con jugate , 71 , 76Complex continuation, 421Complex curvilinear integra l, 425Complex derivab ility, 421Complex d ifferentiability, 424, 548 , 549Complex e igenvalue , 686Complex entire series , 426Complex exponential, 76, 454Complex exponential function , 72, 76Complex Fourier co effi cients , 656Complex function , 423Complex Hahn-Banach theorem , 25Complex infinity, 78Complex line bund le , 561Complex m easure, 442Complex numb er, 69

addition, 69calcu lation ru les , 76multip lication, 69normal representation , 76

Complex p lane, 71, 216 , 717homeomorph ic to the sphere, 78

Complex p lane (z-p lane, Gauss p lane, A rgand-Gauss plane, Argand-Cauchyp lane), 71

Complex pro jective n -space, 299Complex representation of Fourier series, 654

Complex representation of Fourier sum , 656Complex sequence, 423Complex sphere , 78Complex structure , 433Complex vector bundle, 269 , 561Complex vector space, 20Complex-differentiable, 549Complex-differentiable (holomorph ic, analytic), 549Complex-va lued function , 421Composition

of refl ections, 153Composition law , 29, 30

asso ciative, 30external, 38 , 43, 516interna l, 38, 39, 41, 42 , 515

Composition o f differentiab le functions, 330Composition o f functions, 344Composition o f homotheties, 165Composition o f maps, 30Composition o f refl ections, 153Composition o f relations, 29Composition o f rotations, 41, 515Composition op erators, 493Composition ru le , 330Compositum , 162, 530Computation costs o f Fourier co effi cients, 656Computation of complex Fourier co effi cients , 656Comultip lication, 312Concave function , 677Concentrated probab ility m easure, 725Concept o f convergence, 38Concept o f correlation , 625Concrete category, 582, 584Concrete R iemann surface , 432Concurrent lines, 152Condensation po int, 195Cond itiona l distribution , 618Cond itiona l exp ectation , 618Cond itiona l forecast, 630Cond itiona l probab ility, 615, 618Cond itiona lly compact set, 375Cond itions o f order (o f identifi ab ility), 642Cond itions o f rank, 642Cone, 184, 276 , 601Cone construction, 288Cone of revo lution, 172, 184 , 185Cone over X , 276Confidence interval, 616, 629

of the variance of error, 629Confidence level, 616Configuration, 156Conformal map, 438 , 482Conformal mapping , 421, 438 , 482Conformal transformation , 421 , 438, 482

complex p lane, R iemann sphere, 438from C to C, 439

from C to C, 439from the interior un it d isk to itself , 439of simply connected domain , 439 , 440

Conformally equ iva lent, 482Congruence , 29 , 143, 151 , 161 , 273Congruence modulo an ideal, 143Congruence theorem s, 163Congruence transformation , 149Congruent, 143Congruent figures, 9Congruent sets, 160Conic, 185 , 186Conic section , 156 , 185Conic section directrix , 185Conics, 184Conjecture , 2, 5Conjugacy, 690, 723Conjugacy class , 86Conjugacy of d iff eomorphism s, 690Connected, 196 , 538, 539Connected acyclic graph , 260Connected compact set, 214Connected component, 203 , 226, 262 , 545Connected graph , 260Connected Hopf algebra, 320Connected set, 195, 196Connected sinp licial complex, 297Connected space, 197, 202 , 203, 225, 545Connected subset, 203Connectedness , 195Connectedness (connectiv ity), 539Connecting homomorph ism , 291 , 554, 560Connecting morphism , 597Connecting morphism (boundary morph ism ), 597Connection , 417Connection co effi cient, 417, 418Connection curvature , 417Connection geodesic, 417Connection on a diff erentiab le manifold, 417Connection torsion , 417Connections, 1Connections of prop ositions, 1Connectives

and, 1equ ivalent, 1implies, 1nand , 1non-equ iva lent, 1nor, 1not, 1or, 1xor, 1

Connectiv ity, 196Connectiv ity numb er, 256 , 257, 261 , 412

785

Index

Connectiv ity pair, 262Conservative Ham iltonian system , 694Consistency, 3Constant, 581, 610Constant map, 29Constant morph ism , 599Constant path , 279, 297Constant return to scale, 758Constant shea f, 552Constant symbols, 511Constraint, 670, 671Constraints, 681Constructib ility by straightedge and compass, 130Construction by stra ightedge and compass, 129Construction o f C, 69Construction o f an n-sided po lygon , 131Construction o f C, 69Construction o f N, 62Construction o f Number System , xv iiC onstruction o f Q, 64Construction o f R, 66Construction o f Z, 59 , 64Construction prob lem , 129Constructions by straightedge and compass, 162Constructive set theory, 16Contact configurations, 394 , 395Contiguous simplicial maps, 296Continuation, 308Continuation of a path, 308Continuity, 325 , 329 , 423

g loba l, 200lo cal, 200

Continuity at a p oint, 50Continuity of a sequence, 204Continuity of complex functions, 423Continuous b ijection , 194Continuous deformation , 401, 469Continuous d istribution, 620Continuous extension o f a function , 326Continuous function , 325Continuous geom etry, 193Continuous invariant, 193 , 195, 197Continuous linear form , 238Continuous linear functional, 21Continuous linear map, 237Continuous logarithm , 472Continuous map, 50, 193, 194 , 200, 219Continuous multilinear map, 239Continuously d iff erentiab le curves

exp licit representation , 358parametric representation , 359

Continuum , 35 , 49Continuum hypothesis, 15, 35 , 49 , 578Contour, 428 , 429Contour integral, 429 , 430Contour integral , 430Contour integration , 429Contour w ind ing numb er, 430Contracted curvature tensor, 409Contractib ility, 249Contractib le , 249, 275Contractib le lo op , 546Contractib le space, 275 , 305Contractile arcw ise connected space, 249Contractile space, 249Contractility, 545Contracting homotopy, 291 , 303Contracting homotopy of complex, 291Contraction, 229 , 696Contraction mapping, 229Contraction of volum e, 722Contraged ient matrix , 406Contragred ient representation , 406Contrap osition , 54Contravariant, 408Contravariant functor, 265, 273, 594Contravariant Hom functor, 273Contravariant index, 407, 409Contravariant tensor, 408 , 410Contravariant tensor a lgebra , 407Contravariant valency, 408Contravariant vector, 410Control, 670Control of pro cess , 685Control theory, xv iii, 669Convention o f summation , 405Convergence

o f a sequence, 50in term s of balls , 50

simple, 234uniform , 234

Convergence criterion , 323, 340

Convergence in C, 423Convergence in norm , 488Convergence o f a fi lter base, 205Convergence o f a sequence, 50, 234Convergence o f a sequence in term s of balls , 50Convergence o f Fourier series, 487

A -P convergence, 487a lm ost everywhere convergence, 488convergence in norm , 488weak∗ convergence, 488

Convergent estim ator, 626Convergent infin ite product, 324Convergent product, 324Convergent sequence , 66 , 67 , 322Convergent sequence o f functions, 327Conversion formulas, 661Convex, 196, 275Convex combination, 675

Convex enve lop e, 252 , 480Convex function , 678, 679Convex functional, 20Convex hull, 250, 255 , 278Convex optim ization problem , 684Convex part, 252Convex programm ing , 670, 673Convex set, 26, 277 , 674

prop erties , 675separation , 676

Convex set and convex function , 679Convex set spanned by X , 278Convexity, 196 , 252, 674Convexity in optim ization prob lem s, 674Convolution , 487, 488, 495 , 497, 662 , 668Convolution of d istributions, 668Convolution operation , 460Coord inate ball, 542Coord inate chart, 269 , 541, 542Coord inate curve, 398Coord inate curve on a patch , 398Coord inate domain , 542Coord inate function , 544Coord inate neighb orhood, 542Coord inate neighb orhoods, 315Coord inate neighb ourhood , 544Coord inate neighb ourhood (coord inate patch), 541Coord inate patch , 542Coord inate p lane, 157Coord inate ring, 531 , 532Coord inate transformation , 414 , 542Coord inates, 157

in affi ne planes, 157in pro jection p lanes, 157

Coord inates of a tensor, 406Cop lanar, 164Cop lanar vector, 181Cop lanarity, 181Coproduct, 299 , 591, 603Coproduct (d irect sum ), 591Coproduct o f ob jects , 310Coresolution , 558Corestriction , 30Coretraction , 598Coriolis, 360Correction of abnormal values , 630Correction of heteroscedasticity, 635, 636Correlation , 149, 156, 625Correlation co effi cient, 625Correlation dim ension, 731Correlation integral, 731Correlation of a pro jective p lane, 156Correlation theory, 625Correlogram , 643Correspondence, 677Correspondence b etween d iff erential complex and null sequence , 522, 534Correspondence b etween manifolds , 414Cosecant, 175Coset, 84 , 526, 599Cosine, 175Cospan, 601Cotangent, 175Cotriple, 593Countab ility axiom s, 211 , 212Countab le , 3, 18, 211, 542Countab le basis , 211, 539Countab le everywhere dense part, 236Countab le lo ca l basis, 543Countab le product, 217Countab le set, 18, 34, 78, 211Countab ly compact set, 211Countab ly infin ite, 18, 34, 211Countab ly infin ite = Denumerab le, 18, 211Countab ly infin ite set, 18Countab ly m etacompact space, 211Countab ly paracompact space, 211Counterclo ckw ise, 175Counting, 37Counting number, 33 , 62Counting o f poles, 476Counting o f zeros , 476Coupling o f frequency, 722 , 726, 734Cousin prob lem , 481Covariance Cov(x, y), 625Covariant, 408Covariant derivative, 417 , 418Covariant functor, 265, 273 , 594Covariant Hom functor, 273Covariant index, 407, 409Covariant left-exact functor, 557Covariant tensor, 406 , 408, 410Covariant tensor algebra, 407Covariant tensor of order 1 , 406Covariant va lency, 408Covariant vector, 409Cover, 26, 197 , 208, 296, 306 , 415Cover of a s implicia l complex, 296Covering , 26 , 193, 197, 208 , 306, 415Covering fam ily, 568 , 607Covering homotopy lemma, 306Covering homotopy prop erty, 562Covering homotopy theorem , 306Covering map, 574Covering map (covering pro jection), 574Covering pro jection , 306 , 574Covering sieve, 607 , 608Covering space, 204 , 306, 574Covering space o f a topo logical space X , 306Covering transformation, 307Covering transformation (deck transformation), 307

786

Index

Cr -atlas, 414Cr -structure, 541Cram er Formula , 107Creation of a torus, 737Crisp set, 16Criterion o f identifi ability o f the model, 642Critica l (or rejection) reg ion, 617Cross pro duct, 13, 182Cross-cap , 412Cross-caps, 256Cross-sectiona l data, 624 , 633Cross-sectiona l m odel, 631 , 634Crosscap, 300Crossword dictionary order, 518CRS property, 207Cube dup lication, 130Cubic parab ola, 395Cumulative d istribution, 620Cumulative d istribution function, 620 , 621Cumulative d istribution function (c.d.f ), 616Cumulative h ierarchy of sets , 15 , 578Cup product, 318—320 , 550Curl, 316Curl theorem , 348 , 412Curry-Yorke model (C -Y ), 741Curvature, 177, 393 , 394, 398, 415 , 418

of R iemann, 415Curvature formula , 395Curvature line, 403Curvature of a curve , 419Curvature of a p lane curve, 397Curvature of a regular patch, 401Curvature of a R iemannian space, 419Curvature of a surface, 419Curvature of circle, 394Curvature of helix , 394Curvature vector, 393, 394Curve, 214 , 215, 392

of Jordan , 445regu lar, 445

Curve length , 358

Curve length in R3 (or R2), 358Curve tors ion, 394Curve w ith iso lated po int, 336

Curves in R3 , 392Curvilinear abscissa , 359, 393Curvilinear coord inates , 343, 398Curvilinear integral, 360, 411 , 445

case o f continuous paths, 468of a grad ient, 361of rea l functions, 361prop erties, 445

Cusp , 702, 736Cusp and Generalized Hopf bifurcations, 712Cusp b ifurcation , 712, 735Cusp b ifurcation (Bogdanov, Takens), 712Cusp singu larity, 702CUSUM statistic, 631CUSUM test (CUmulative SUM ), 631CUSUM-SQ statistic, 631Cut, 65Cut p oint, 262CW complex, 298, 300CW complexes, 298 , 300CW decomposition , 300 , 535CW space, 300, 535CW subcomplex, 300CW -complex, 535CW -complexes, 196Cycle, 86, 260 , 285, 474 , 534Cyclic decomposition , 136Cyclic decomposition theorem , 136Cyclic Galo is group, 128Cyclic group, 87 , 257Cyclic multip licative group , 114Cyclic ob ject, 303Cyclic order, 160Cyclic order o f equ ivalence classes, 151Cyclic order re lation , 160Cyclic p ermutation , 91Cyclic subbasis, 136Cyclic subgroup , 73Cyclic subgroup of T, 73Cyclic subgroup of unit circ le, 73Cyclic subspace, 136Cyclic variab le, 695Cyclic vector, 136Cyclically ordered , 160Cyclically reduced word , 519Cycloid, 722Cyclotom ic equation , 131Cyclotom ic po lynom ial, 116, 125Cyclotom ic po lynom ial o f order k, 125

δ-functor, 557d-stable, 753d-stable matrix , 757Damped oscillation , 663Darboux-R ibaucour thriedron, 401Daughter wavelet, 665De l’Hospital rules, 334De M oivre numb er, 73De Rham cohomology, 316, 364 , 412, 449 , 501Decim al comma, 68Decim al expansion , 34 , 67

p eriodic, 68o f rational numb er, 68

term inating , 68Decim al numera l sy stem , 56Decim al numera l sy stem (base ten , base -10, denary), 56

Decim al po int, 68Deck transformation , 307Decomposition into irreducib le elements, 141Decomposition into prim e ideals, 142Decomposition into simple e lem ents , 335Decompostion into simple elem ents, 118Dedekind, 65Dedekind ring , 142Deduction theorem , 4Deductive system , 3Definable set, 512Definiendum , 5Definiens, 5Definite herm itian form , 136Definite positive herm itian form , 242Definition, 5Definition by comprehension , 11Definition by recurrence, 5Definition o f an attractor, 733Definitions of chaos, 732Deformation , 569 , 573Deformation retract, 277, 573Degeneracy cond itions, 699 , 702Degree, 117, 283 , 293Degree o f a continuous map, 293Degree o f a lo op, 503Degree o f a map, 472Degree o f a mapping , 413Degree o f a polynom ia l, 109Degree o f a vertex, 260Degree o f an a lgebraic numb er, 117Degree o f an e lem ent in a graded ring, 318Degree o f antip odal m ap , 304Degree o f freedom , 617 , 622Degrees of freedom , 629Del, 343 , 347Del op erator, 348Delta o f D irac, 668Demonstration , 5 , 6Demonstration by the absurd , 5Dence tra jectory, 722Denjoy integra l, 369Denjoy theorem , 692Dense, 24, 198 , 211, 542 , 702Dense morph ism , 572Dense orb it, 709Dense part, 221

of a topolog ica l space, 219Dense set, 65, 198Dense va luation , 145Density o f orb its (DO), 733Density o f rational function , 475Denumerability, 33—35

of the set of rationa ls, 34Denumerable, 3 , 18, 211Denumerable set, 18 , 34 , 53 , 54, 78, 211Dependence o f criteria (p opulation), 623Dependent variable, 617 , 685Depth , 525Derivab ility, 329Derivab le function , 329Derivate , 329Derivation , 114Derivation of d istributions, 496 , 668Derivation operator, 526Derivation operator (d iff erentiation), 526Derivation rule, 329Derivation rules, 392Derivative , 119Derivative o f a rational fraction , 119Derivative o f cumulative d istribution function , 620Derived category, 592Derived functor, 556 , 557, 597Derived po lynom ial, 114, 121Derived set, 369Derived structure, 38Desargues and Pappus-Pascal theorem s, 155Desargues theorem , 155Descartes, 4Descartes folium , 335Descend ing cha in cond ition (DCC), 523Descriptive geom etry, 150, 173 , 174Destination , 598Destruction o f a torus, 739Detection o f error auto correlation , 633

causes, 633detection , 633

Breusch-Godfrey test, 634Durbin -Watson test, 634

Detection o f heteroscedasticity, 635Detection tests of multico llinearity, 632Determ inant, 105 , 133, 717

calcu lation ru les , 106Determ inant ca lcu lation ru les, 106 , 344Determ ination

of argument, 455of logarithm , 455

Determ ination of a m easure , 350Determ ination of lag number, 646Determ ination of lag numbers , 640Determ in istic walk , 644Devaney, 732Developab le surface, 404Deviation, 230 , 685Dff erential form

change o f variab le, 410D iagonal approxim ation , 319D iagonal morphism , 565D iagonalization , 133, 134D iagram , 273 , 601

787

Index

D iagram of the pure rotation and its lift, 692D ice, 259, 615D ickey-Fuller

augmented , 644D ickey-Fuller test (DF), 644D iff eomorphism , 542, 546, 686 , 691

hyperbolic linear, 692D iff eomorphism and period ic p oints , 692D iff eomorphism of the c ircle, 687D iff eomorphism s and flow s, 686D iff erence, 8, 11D iff erence fi lter order, 644D iff erentiab ility, 329 , 342

complex, 424 , 549rea l, 424 , 549

D iff erentiab le 2-chain , 506D iff erentiab le atlas, 414 , 415, 542D iff erentiab le concave function , 678D iff erentiab le convex function , 678D iff erentiab le function , 329D iff erentiab le manifo ld , 414, 415 , 513, 540, 697D iff erentiab le op erator, 372D iff erentiab le p-form , 410D iff erentiab le structure, 540, 541D iff erential, 331, 410, 555D iff erential (cob oundary map), 555D iff erential 1-form , 443D iff erential 2-form , 449D iff erential algebra, 526D iff erential calcu lus, xv ii, 328, 341D iff erential calcu lus in Rn, 341D iff erential complex , 522, 534D iff erential Equation , 377D iff erential equation , xvii

(linear) homogeneous, 382exact, 379 , 380exp licit form , 378implicit, 381initia l cond itions, 378initia l-value prob lem , 378o f second order, 381order, 378structurally stable (robustness), 723w ith one variable, 378w ith separated variab les, 379

D iff erential equation system , 383D iff erential fi eld , 526D iff erential form , 316 , 409, 410

closed form , 448exact form , 448in the plane, 448on a d iff erentiab le manifold , 410on an open subset of C, 498

D iff erential form of degee 1, 443D iff erential form s of degree 2, 449D iff erential geom etry, xvii, 392 , 415

curves, 392D iff erential graded algebra, 527D iff erential k-form s, 138D iff erential man ifo ld , 686D iff erential modu le , 522, 523 , 527D iff erential op erator, 374, 415 , 668D iff erential p-form , 316D iff erential ring, 526D iff erential sheaf, 553D iff erential topo logy, 415D iff erentially stab le (d -stab le), 753D iff erentially stab le matrix , 757D iff erentiation , 708D iff erentiation of degree n , 289D iffusion model, 637, 638D igraph , 600D ilation , 165D ilation param eter, 666D im ension , 294D im ension axiom , 287 , 302, 312 , 549, 553 , 597D im ension axiom theorem , 312 , 317D im ension invariance , 212D im ension of a simplicia l complex, 294D im ension of the measure µ, 731D im ension theory, 212D im ensions, 730

D im ensions through invariant m easures, 731correlation d im ension , 731d im ension of a m easure , 731genera lized dim ension , 731in formation d im ension , 731Lyapunov d im ension , 731

Local Hausdorff d im . (Douady-O esterlé), 731m etric dim ensions

Box-counting dim ension or capacity, 730fractals, 730Haussdorff dim ension , 730self-sim ilarity, 731

D im etric pro jection, 174D in i theorem , 234D iophantine equation , 144D irac, 668 , 727D irac δ function and d istribution, 659D irac mass, 496D irac m easure, 725D irect demonstration , 5D irect fam ily, 535D irect fam ily (of algebra ic system s), 536D irect fam ily (of sets), 535D irect homothety, 165

D irect isom etries in R3 , 164D irect isom etry, 160D irect lim it, 266, 536, 601 , 602D irect lim it (fi ltered co lim it), 591

D irect lim it (inductive lim it), 601D irect lim it (of algebra ic system s), 536D irect lim it (of sets), 535D irect product, 266, 375, 519 , 524, 602D irect product of m odu les, 102D irect product of subgroups, 89D irect product of subgroups (commutative group), 89D irect sim ilarity, 165, 452D irect sum , 133, 266 , 290, 318 , 376, 519 , 591, 602D irect system , 536 , 559, 601D irect system in a category, 601D irected angle, 75D irected graph , 259, 582 , 600D irected graph (d igraph), 600D irected graph (or digraph), 259D irected line segm ent, 150D irected order, 205, 579D irected order (fi ltrant order), 579D irected preorder, 535D irected segm ent, 150D irected set, 23, 205 , 535, 579D irected straight-line, 150D irectedness, 259D irection curves, 404D irectiona l b low ing-up, 702D irectiona l derivate, 342D irections of principal curvature , 403D irectly congruent, 160D irectrix , 185, 186D irichlet box princ ip le , 53D irichlet cond itions, 655D irichlet Pigeonho le princip le, 53D irichlet problem , 487 , 492D isappearance of saddle-node separatrix , 738D isconnected space , 202, 203D isconnectiv ity, 203D iscontinu ity, 194, 326D iscrete category, 582D iscrete complex Fourier co effi cients, 656D iscrete complex Fourier transformation, 656D iscrete d istance, 220D iscrete dynam ica l sy stem , 686, 715, 723

invariant manifo lds, 723Hadamard-Perron theorem , 723separatrix surfaces , 723transverse homoclinic points, 723

p eriod ic orbits, 723prop erties of ω-lim it set, 723topolog ica l conjugacy of d iscrete system s, 723typ es o f equilib rium p oints (Steady states), 723

D iscrete Haar wave let transformation , 666D iscrete Lap lace transformation , 664D iscrete random variab le, 620D iscrete space, 226D iscrete topology, 198D iscrete valuation , 145D iscrete wavelet transformation (DWT), 666D isjoint sets, 26D isk, 71D isk w ith holes, 450 , 452D isplacem ent, 158, 186D issipative baker map, 732D issipative system

transitions stemm ing from linear stab ility loss, 718D istance, 66, 76 , 184, 219 , 220

b etween two parts of a metric space, 220Hamm ing , 220Hausdorff , 220M inkowski, 220on a norm ed vector space , 220p-ad ic, 220sem i, 220u ltra -metric, 220

D istance (in a norm ed vector space), 237D istance function, 727D istance on a space of functions, 217D istinction b etween εt and et, 628D istingu ished subgroup , 88D istingu ished triangle, 591D istributed lag models, 639D istributed parameter system , 739D istribution, 391 , 496, 659 , 668, 726

leptokurtic, 644normal, 644

D istribution of lagsfin ite , 640infin ite, 640

D istribution of order 0 , 496D istribution of order 1 , 496D istribution of zeros, 463D istribution theory, 654, 668D istributions, 616D istributions and subharmonic functions, 492D istributive lattice, 23D istributive law , 42, 516D istributiv ity, 8, 9D ivergence, 316, 347, 348D ivergence formula , 489D ivergence of a vector fi eld, 347D ivergence theorem , 348, 364, 412D iv idend, 64D iv ides, 95D iv isib ility, 55 , 141

a d iv ides b , 141a is a d iv isor o f b, 141a is a factor o f b , 141b is a multip le o f a, 141b is d iv isible by a, 141in a fi e ld , 145

D iv isib ility in N, 55

788

Index

Divis ib ility in Z, 55D ivis ib ility in polynom ia l ring , 120D ivis ib ility theory, 145D ivis ib le , 141D ivis ib le group , 317D ivisor, 64, 141 , 146, 147D ivisor function , 435Domain, 14, 29, 30 , 38 , 93 , 422, 502, 512 , 519Domain of defin ition , 29, 38, 119 , 519Domain of variation , 581Dom inant co effi cient, 109Dom inant morphism , 572Dom inant term , 109Dom inant term of a po lynom ial, 109Dot product, 21 , 182Douady-O esterlé theorem , 731Douady-O esterlé theorem (for d iff . eq .), 731Doub le cone, 311Doub le integral, 355, 356Doub le va lue princip le, 1Doub le-angle formula , 177Doub ling the cub e, 75Doub ly p eriod ic function , 436, 437Doub ly sto chastic, 616DS pro cess, 644

w ith drift, 644w ithout drift, 644

DS pro cess (random walk model), 644DS pro cesses (D ifference Stationary), 644Dual

algebraic, 238topo logical, 238

Dual basis, 238, 405Dual category, 583, 584Dual space, 19 , 21 , 139, 370 , 405, 408Dual space functor, 273Dual space o f V (V ∗=Hom(V, k)), 408Dual un iversa l co effi cients theorem , 318Dual vector space, 105, 405Duality princ ip le , 671Duffi ng attractor, 709, 710Duffi ng system , 710Dummy index, 39Dummy variab le, 630Dupin indicatrix , 403 , 419, 420Durb in -Watson test, 639Durb in -Watson test (DW ), 634Dyad ic wave let transformation , 666Dynam ical model, 648Dynam ical system , xviii, 684 , 686, 714

conservative, 715dissipative, 715volume-contracting (or d issipative), 715volume-preserv ing (or conservative), 715

E-cyc lic ob ject, 303ε-ne ighborhood , 199Eccentricity, 185Edge, 258Edge connectiv ity, 262Edge numb er, 263Edge of simplicial complex, 297Edge path, 297Edge path group, 297Edge-preserv ing b ijection , 258Eigen decomposition, 133Eigend irection, 189Eigenfunction m ethod , 390Eigenspace

stab le , 708unstable, 708

Eigensubspace, 133Eigenvalue, 133 , 385, 722

Eigenvalue ana lysis of the matrix eTR, 717Eigenvalue prob lem , 133 , 134Eigenvector, 385Eilenb erg , 575Eilenb erg criterion , 474Eilenb erg -M ac Lane, 576Eilenb erg -M oore a lgebras, 594Eilenb erg -M oore a lgebras (of a monad), 594Eilenb erg -M oore category, 594Eilenb erg -Steenrod axiom s, 549 , 553Eilenb erg -Steenrod theorem , 302Eilenb erg -Steenrod theorem extension , 302Eilenb erg -Z ilb er theorem , 305Einhe it, 42, 515Einstein summation , 343, 405Einstein summation convention, 405, 407Eisenste in series, 435Elastic transformation , 193Elation , 156Elem entary i-chain, 504Elem entary symmetric function, 113Elem entary topos, 608Ellip se, 185Ellip se circum ference, 358Ellip tic, 158Ellip tic function, 436Ellip tic function order, 436Ellip tic geom etry, 179Ellip tic invariant, 435Ellip tic m etric plane, 154Ellip tic p lane , 152Ellip tic po int, 402, 403 , 419Ellip tic right triang le , 180Ellip tic umbilic, 403Embed a metric p lane in a pro jective p lane, 157Embedded hypersurface , 544Embedded m-d im ensiona l tori, 720Embedded submanifo ld , 544

Embedd ing, 66, 416, 520 , 544, 594of m etric p lanes in pro jective planes, 157

Embedd ing (imbedding), 544Embedd ing of Q in R, 66Embedd ing space, 416Embedd ing Whitney theorem , 416Empirica l co eff cicient, 625Empirica l estim ator o f the variance , 627Empirica l t-student, 625Empty map, 30Empty set, 7End of a path , 279End of edge o f simplicial complex, 297Endoarrow , 600Endofunctor, 593Endogenous variab le, 625, 626Endomorph ism , 33 , 134, 520 , 598Endomorph ism (of algebraic system), 520Endomorph ism eigenvector, 189Endomorph ism reduction , 134Engle, 648Engle-G ranger, 649Engle-G ranger algorithm , 648Enough H-in jectives, 555Enough injective ob jects, 555Enough injectives, 555—557Enough pro jectives , 556Enriched category, 588Enriched category identity, 588Enriched functor, 597Entering variab le, 669Entire function, 433 , 457Entire ring , 43 , 93, 516Entire series , 132, 328 , 428, 453 , 457

convergence sets, 440of severa l variab les , 440w ith nonnull radius, 333

Entire series expansion, 421 , 426, 457of rational function , 426

Entire series expansion of c ircu lar functions, 337Entourages, 230Entrop ies

m etric entropy, 727topo logical entropy, 727

Entropy, 727Kolmogorov-Sina i, 727m etric, 727

Enumerable, 18, 211Enumeration, 37Envelop e

convex, 480upper, 230

Envelop e o f ho lomorphy, 480Epigraph , 349Epim enides, 13Epimorphism , 38, 520 , 598, 599Epimorphism (ep i), 264Epimorphism (ep ic), 598Epimorphism (or ep i), 598 , 606Equal sets, 7 , 14Equality of two pa irs, 27Equality sign , 3Equality test o f variances, 636Equalizer, 305 , 604Equation, 106Equation o f 3rd degree

reso lution, 128Equation o f 4th degree

reso lution, 129Equation o f a straight line, 157Equation o f line, 183Equation o f second degree, 186

in two coordinates, 186Equation o f tangent p lane, 399Equation o f the fi rst degree, 157Equation so lvable by rad ica ls , 129Equation w ith determ inant, 184Equation w ith p oint and normal, 184Equation w ith vector product, 184Equicontinu ity, 235Equicontinuous, 235

fam ily of m aps, 235Equicontinuous fam ily, 235Equicontinuous fam ily o f m aps, 235Equidecomposable, 17, 18Equilateral hyperbo lic function, 178Equilib rium in a d istribution economy, 677Equilib rium p oint o f typ e (m ,k), 719Equipo llent, 29, 33Equipo llent bipoints, 161Equipotence, 33, 34, 36Equipotent, 29 , 33Equipotent set, 34Equivalence, 594Equivalence class, 28 , 201, 276Equivalence in a category, 273Equivalence of categories , 266, 594 , 596Equivalence of flows, 692Equivalence of lo ops, 445Equivalence of paths, 445Equivalence relation, 18 , 28 , 64 , 67, 201 , 276, 415Equivalent absolute va lues, 145Equivalent arcs , 414Equivalent covering spaces , 307, 309Equivalent generating system , 200Equivalent lo ca l fam ilies , 711Equivalent norms, 237Equivalent norms (in a norm ed vector space), 237Equivalent representations o f Fourier integrals, 661Equivalents sets (equipotent sets, 47

789

Index

Equivariant map, 740Ergod ic dynam ical system , 725Ergod ic measure, 725Ergod ic theory, 725Error, 626Error correction model (ECM), 648Error εt , 626Error o f approximation, 332Error o f sp ecifi cation, 626Essentia l extension , 555Essentia l m onomorph ism , 555Essentia l singu larity, 421, 464Essentia l submodule , 555Essentia l supremum , 391Essentia lly surjective, 594 , 595Essentia lly un ique, 57Estim ation m ethod of d

aggregate variance , 652aggregated abso lute value , 652p eriodogram , 652R/S, 652variance o f the res idua ls , 652wavelets, 652Whittle, 652

Estim ation m ethods of non linear models, 638Estim ation of param eters , 626Estim ation pro cedures for error correlation , 634Estim ator, 626

BLUE, 629, 634 , 635o f fractional integration param eter, 652o f Hurst exponent, 652

Estim ator o f co effi cients , 626Estim ator o f G enera lized Least Squares, 633Estim ator o f the variance o f the error, 627

Et(X), 568, 569

Et/X, 569Etalé space , 610Etale, 568Etale cohomology, 568, 570 , 571Etale cohomology group, 570Etale cover, 567, 571 , 610Etale fundamenta l group, 568Etale map, 568Etale map of schemes, 610Etale morphism , 568 , 569Etale neighborhood , 568Etale presheaf, 610Etale presheaf on a schem e, 610 , 611Etale sheaf, 610Etale site, 568Etale site of a schem e, 610, 611Etale top ology, 568Etale top ology (a lgebra ic geom etry), 568Etale top os, 568 , 610Etale X -schem e, 568Euclid a lgorithm , 58, 111Euclid axiom s, 149Euclid lemma, 56, 57, 60, 95Euclid lemma (for po lynom ials), 111Euclid lemma in Z, 60Euclid theorem , 56 , 169Euclidean algorithm , 142Euclidean distance , 220Euclidean div is ion, 55, 59 , 109Euclidean div is ion algorithm , 110Euclidean div is ion of p olynom ials, 109Euclidean function , 142

monotone , 142Euclidean geometry, 149, 150, 160Euclidean group E(n), 163Euclidean metric, 153 , 193, 199, 409Euclidean moves, 163Euclidean plane, 154Euclidean polynom ia l d iv ision , 120Euclidean ring, 142 , 143Euclidean space, 211Euclidean transformation, 161Euclidean/non-Euclidean m etric plane, 153Euler affi nity, 169Euler approxim ation of Duffi ng attractor, 709Euler characteristic, 296 , 413Euler characteristic (of a genus), 412Euler characteristic (of a topolog ical space), 412Euler constant, 338Euler criterion , 144Euler d iff erential equation , 373Euler formula, 76 , 296Euler formulas , 74 , 654Euler function, 143, 338Euler line, 166 , 167Euler product formula, 58Euler theorem , 96 , 167, 403Euler totient, 57, 96Euler totient function, 57, 96Euler’s d isp lacem ent theorem , 186Euler-Cauchy method, 387Euler-Lagrange theorem , 83, 84Euler-Po incaré characteristic, 296, 413Euler-Po incaré characteristic o f (X ,E), 301Euler-Po incaré characteristic o f a polyhedron , 296Euler-Po incaré characteristic o f K , 296Eulerian chain , 260Eulerian cycle, 260Eulerian function , 651Eulerian graph , 260Eulerian multiplie r, 379Eulerian path, 260Eulerian trail, 260Eulerian walk , 260Euclidean div is ion in Z, 59

Evaluation map, 309Even and odd polynom ials , 109Even po lynom ial, 109Evenly covered op en set, 306Event, 615, 618Evolute, 397Evolute o f a curve, 396 , 397Evolute o f a plane curve, 398Evolute o f an arc, 397Evolvent, 396Exact differentia l equation , 379 , 380Exact differentia l form , 380Exact form , 448Exact functor, 595Exact homology sequence , 291Exact sequence, 290 , 313, 509 , 554, 605Exact sequence (acyc lic complex ), 290Exact triang le, 291Exact triang le theorem , 291Exactness axiom , 302 , 553Exactness of a sequence, 290Exchange ru les , 3Excision , 292Excision 1, 292Excision 2, 292Excision axiom , 302, 553, 597Excision coro llary, 301Excision theorem , 296, 317Excluded third princip le, 2Exclusion restrictions, 642Exhaustion sequence, 458Exhaustion sequence of compact sets, 458Existence of covering space, 308Existence of nu ll elem ent, 9Existential quantifi er, 10Exogenous variab le , 625Expansion , 696Expansion of 1/Γ, 468Expansion of 1/ζ, 467Expectation , 616, 621Expectation (of a random variab le), 616Expectation of the error, 626Expected va lue, 616Experim ent, 616Exp la ined variability (SSE), 627Exp lanatory variab le , 617, 626Exp losion , 746Exponent, 68

irrational, 68rationa l, 68

Exponentia l, 72, 336Exponentia l abso lute value , 145Exponentia l Fourier transformation , 661Exponentia l function, 72, 336, 454Exponentia l functor, 608Exponentia l law , 309Exponentia l ob ject, 580, 586Exponentia l series o f matrices, 384Express ion

identica l, 2satisfi ab le , 2

Ext functor, 558Extended complex p lane, 421, 422

Extended complex p lane (C, d), 422

Extended complex p lane C, 421Extended Euclid a lgorithm , 112Extended rea l line, 223 , 422

compact, 223Extended rea l number line, 217Extend ing by linearity, 285Extension , 573

o f absolute va lues, 146o f natura l numbers , 64o f sem iring of natural numb ers, 63

Extension fi eld , 65, 69, 122, 531Extension fi eld degree , 122Extension fi eld of Q, 65Extension fi eld of R, 65, 69, 422Extension of a fi eld , 122Extension of a map, 29, 573Extension of a sem igroup, 91Extension of fi rst-order pred icate calcu lus, 3Extension of Q, 67Extension of Q to R, 65Extension of R to C, 65Extension theorem , 228Extensional definition, 11Exterior, 214Exterior algebra , 137—139, 318

o f a vector space , 139o f alternating multilinear forms, 139

Exterior algebra (G rassmann algebra), 137Exterior ang le, 162Exterior ang le of a polygon, 162Exterior ca lcu lus, 138Exterior derivative, 316, 364 , 450Exterior derivative of a 1-form , 450Exterior d iff erential, 410Exterior d iff erential form , 138Exterior d iff erentiation , 410Exterior d iff erentiation of d iff erential form s, 138Exterior normal, 489Exterior po int, 195 , 198Exterior power, 316Exterior product, 138—140 , 497

o f two antisymmetric multilinear form s, 138un iversal prop erties, 140

Exterior product (wedge product), 139External center o f s im ilitude, 166External composition law , 43 , 516

790

Index

Externa l d irect sum , 524Externa l homothetic center, 166Extracted fam ily, 31Extracted sequence, 221Extraction o f a root nth , 77Extraord inary cohomology theory, 302Extrema, 373Extrema under constra ints, 346Extreme va lue theorem , 209Extremum, 209Extrinsic curvature, 418

F-acyc lic, 557F-acyc lic ob ject, 556F-algebra, 594F-distribution, 617F-homomorphism , 521F-model set, 303F-pro jective n -space , 299F-test, 617Face, 294, 303Factor, 141Factor ring, 43 , 526Factor through , 571, 586Factorab le polynom ia l, 120Factorial ring , 124, 142Factorization , 57Factorization theorem , 57Faith ful functor, 266, 594Faith fully flat module, 565False prop osition, 1Fam ily, 31Fam ily canonica lly asso ciated w ith a set, 31Fam ily o f e lem ents , 31Fam ily o f parts of a set, 32Fam ily o f p olynom ia ls w ith graded degrees, 109Fam ily o f sets, 13, 31Fano axiom , 158Fano axiom (P5),(P5’), 158Farrar-G lauber test, 632Fast Fourier transformation (FFT), 656Fast wave let transformation (FWT), 666Fatou theorem , 488Fber, 268Fber bund le, 268, 560Feasib le polyhedron , 671Feasib le region , 669, 670Feasib le region of optim ization prob lem , 671Feasib le va lue, 671Feasib le va lue of optim ization prob lem , 671Feedback effect, 647Feigenbaum attractor, 715Fermat conjecture, 96Fermat equation , 144Fermat infin ite descent, 52Fermat last theorem , 96Fermat little theorem , 96Fermat m ethod of infinite descent, 52Fermat number, 131, 148Fermat principle of infin ite descent, 52Fermat theorem , 93 , 96Ferrari m ethod, 129Feuerbach circle, 167F ib er, 53, 275 , 306, 520 , 560, 569F ib er (algebraic geom etry), 560F ib er (topo logy), 560F ib er bundle, 562F ib er of a map, 53, 560F ib er of cover, 306F ib er product, 314, 604F ib er space, 268, 562F ib onacc i numbers , 741F ibration , 268 , 314, 562F ibration (Hurew icz fib er space), 314F ibre bund le, 562F ibre product, 265 , 266F ibre square , 266FIEGARCH, 652Fie ld , 38 , 43, 81, 92, 98, 143, 360 , 516

left, 79noncommutative, 79

F ie ld ad junction , 122F ie ld characteristic , 125, 126F ie ld extension , 122F ie ld homomorphism , 99, 521F ie ld line, 348F ie ld of complex numb ers, 422

algebraically closed , 77F ie ld of complex numb ers is a lgebraically closed, 115F ie ld of fractions, 99 , 117F ie ld of fractions (fraction fi e ld , fi eld o f quotients , quotient fi eld), 117F ie ld of fractions o f a ring, 98F ie ld of fractions o f R , 98F ie ld of paralle l vectors, 417F ie ld of quotients, 99, 100 , 117F ie ld of rational numb ers, 64F ie ld of residue classes, 145F ie ld theory, 81, 514FIGARCH (fractionally integrated GARCH), 652FIGARCH (p,d,q), 652F ilter, 19, 205 , 208, 222 , 223 , 543

Cauchy, 231determ ined by a sequence, 231Fréchet, 222of neighborhoods, 222smallest, 222

F ilter base, 22 , 204, 205 , 209, 543F ilter bases , 222F iltered colim it, 591F iltered set, 23, 535F iltrant category, 589

F iltration , 301F ina l top ology, 201F initary op eration , 510, 511F inite Ab elian group , 89F inite distribution o f lags, 640F inite extension , 122F inite fi eld , 125 , 126F inite Fourier transformation, 657F inite Galo is extension , 127

prop erties, 127F inite Galo is extension o f Q, 131F inite graph , 259F inite group , 82F inite Laplace transformation , 656F inite morphism , 566Finite presentation , 569F inite product, 217F inite rank operator, 375F inite separable fi eld extension , 569F inite separable k(y)-schem e, 569F inite set, 5 , 53F inite simple extension, 123F inite supp ort, 40, 108F inite typ e , 318, 566F initely based cha in complex, 297F initely generated , 295F initely generated idea l, 529F initely generated module , 566F initely generated morphism , 566Finitely presented, 297F initely presented a lgebra , 529F initely presented R -algebra , 529F insler m an ifo ld , 540F irst axiom of countability, 538, 543F irst category, 232F irst countab le , 211, 542F irst countab le basis , 211F irst countab le space, 211F irst Cousin prob lem , 481First de Rham cohomology vector space, 501F irst difference model, 635F irst fundamenta l form , 400

of a sphere, 400patch theory, 400

F irst homology group, 475F irst quasi-d iff erences, 634F irst-countab le , 538F irst-countab le space, 211 , 212F irst-difference fi lter, 644F irst-order differentia l equation , 378F irst-order pred icate ca lculus, 2F irst-order pred icate ca lculus w ith identity, 3F isher law , 627F ive lemma, 291Fixed line, 173F ixed po int, 30 , 86 , 173, 687 , 730F ixed po int o f a pro jective collineation , 173F ixed po int theorem , 229Fixed-po int theorems, 677F ixed-po int theorems (B rouwer and Kakutan i), 674F labby sheaf, 559F lag, 150, 159F lasque reso lution (flabby resolution), 559F lasque shea f, 559F lasque shea f (flabby sheaf ), 559F lat manifo ld , 418F lat module , 565F lat morphism , 565, 569F lat point, 419F lat unram ifi ed morph ism , 569Floor, 61F loor function , 61, 624F loquet characteristic multiplier, 717, 746F loquet matrix , 717 , 722, 734F loquet theorem , 699, 716F loquet theory, 717, 734F loquet theory in d issipative system s, 717F low , 714, 715F low of a d iff erential equation, 714F low on the torus, 693F low -b ox, 691F low -b ox theorem , 691Flow s and d iff erential equations, 689Foca l parameter, 185Focus, 185, 186Fold ing map, 310Folium , 335Force, 360Force along a curve , 361Forced p endu lum , 747Forecast interval, 631Forecast re liab ility, 631Forecasting error, 627Forest, 260 , 261Forgetfu l functor, 273 , 597Formal proof, 4Formal system , 4Formally étale, 569 , 570Formally étale morphism s of schem es, 570Formally étale=Formally smo oth+Formally unram ifi ed , 570Formally smooth , 570Formally unram ifi ed , 570Formula o f a parallelogram , 242Formula o f the m ed ian, 242Fortuitous corre lation, 625Forward regress ion , 633Foundations of m athematics, 1Four-co lor prob lem , 261Fourier, 389Fourier co effi cient, 442, 457 , 460, 493, 653

791

Index

Fourier co effi cients, 244 , 656determ ination for symmetric functions, 655

symmetry of the fi rst k ind , 655symmetry of the fourth kind, 655symmetry of the second kind , 655symmetry of the third kind, 655

determ ination w ith numerical m ethods, 655Fourier cosine transformation , 657, 661Fourier expansion , 654Fourier integra l, 477 , 661Fourier integra l in complex representation , 661Fourier series , 435, 436 , 487, 653 , 654

complex representation, 654imp ortant prop erties, 654

Fourier series and Fourier integrals, 655Fourier series and transform

usual w ritings, 654Fourier series convergence, 653Fourier series expansion, 457

complex, 436rea l, 436three d iff erent form s, 655

Fourier series integration , 653Fourier sine transformation, 657 , 661Fourier sum , 654Fourier transform , 390, 653 , 654

continuous writing , 654d iscrete writing, 654three continuous writings, 654

Fourier transformable , 661Fourier transformation , 495, 657, 661Fourier transformation (defin ition), 661Fourier-P lancherel transformation , 654Fppf cover, 567Fp qc cover, 567Fréchet fi lter, 222, 543Frénet basis , 394, 397 , 398Frénet formulas, 394 , 397Frénet trihedra l, 394Fracta l dim ensional, 730Fracta ls, 730Fraction fi eld , 99, 117Fractional B rownian motion , 651Fractional Gaussian pro cess, 651Fractional ideal, 146Fractional integration param eter, 651Fram es, 666Frechet d iff erential, 372Fredholm alternative, 377, 390Fredholm determ inant, 390Fredholm integral equation, 390Fredholm operator, 376Fredholm theorem , 374, 390Fredholm theorem s, 390Fredholm theory, 390Free A-module, 316Free ab e lian , 295Free Ab elian group , 518, 519Free algebra, 518 , 523Free asso ciative algebra , 523Free chain complex, 303Free commutative a lgebra , 518 , 523Free functor w ith basis in M, 303Free group , 103, 254, 297 , 298, 502 , 518, 519Free group of rank m , 298Free group w ith basis, 297Free homotop ic lo ops, 502Free homotopy, 274Free ideal ring, 523Free ideal ring (fi r), 523Free mobility, 159Free module , 102, 524Free module (homological a lgebra), 524Free monoid, 518 , 523Free ob ject, 519, 580Free ob ject (un iversal algebra), 519Free orthogonal system , 243Free part, 102Free product, 297Free R -module, 518 , 523Free reso lution, 556Free ring, 524Free variab le, 2Free Z -module, 519Frege, 4, 14Frequency lo ck ing, 741 , 744Frequency sp ectrum , 662Frequency table, 620Freudenthal susp ension theorem , 315Frob en ius m ethod , 417Frob en ius theorem , 417From Torus to chaos, 739Front face , 318Frontier, 294Frontier p oint, 195, 198 , 219Frontier set, 195Fubin i formula , 54Fulcrum , 190Full functor, 594Full sub category, 264 , 281, 593Fully fa ith fu l functor , 594Function , 14 , 29

harmonic, 425Lagrange, 673logarithm ic barrier, 674one-to -one , 29p ena lty, 673subharmonic

nontriv ia l, 491test, 668

totient, 68Function from R to Rm, 341Function from Rn to R, 342Function from Rn to Rm, 344Function from Rn to Rn, 344Function Γ, 459Function ζ, 458Function of d iff erentiab le rea l variab le , 329Function ℘, 434, 437Function preserv ing structure, 576Function space, 233 , 309Function symbols , 511Function theory, xvii, 421Function theory (complex ana lys is), xviiFunctional, 19 , 20, 370 , 391Functional analysis , xvii, 370Functional axiom , 14Functions, 29Functions of severa l variab les, 440Functor, 256, 257 , 264, 265 , 272, 273, 576 , 594Functor categories, 576Functor category, 586Functor Hn, 271Functor in a lgebraic topo logy, 257Functor Sp ec , 572Functoria l, 587Functoria l isomorph ism , 587Functoria l m etho d, 256Functoria l variab le, 3Functoria lity, 587Fundamenta l (homotopy) group , 250Fundamenta l group , 247 , 248, 250 , 255, 280

o f a connected polyhedron , 254Fundamenta l group oid, 280Fundamenta l groups of p olyhedra , 297Fundamenta l homotopy groups , 256Fundamenta l m atrix , 385

complex, 385rea l, 385

Fundamenta l ne ighborhood , 306Fundamenta l ne ighborhood of base space, 306Fundamenta l relations

normal, geodesic curvature , curvature, 401Fundamenta l relations b etween χn,χg ,χ, 401

Fundamenta l sequence, 66Fundamenta l set, 8Fundamenta l structure, 37 , 38Fundamenta l system , 383Fundamenta l system of neighb orhoods, 200, 221, 230 , 234Fundamenta l tensor, 407, 418Fundamenta l theorem

for homogeneous linear diff . equations, 716for non-homogeneous linear equations, 716theory o f patches, 405

Fundamenta l theorem of a lgebra , 77, 283 , 284Fundamenta l theorem of arithm etic, 57Fundamenta l theorem of arithm etic (for p olynom ia ls), 111Fundamenta l theorem of curve theory, 395Fundamenta l theorem of integra l ca lcu lus, 349, 352Fundamenta l theorem of pro jective geom etry, 156Fundamenta l theorem of R iemannian geom ., 418Fuzzy set theory, 16Fuzzy set:, 16

G -equ ivariant, 307G -isomorph ism , 308G -map, 307G -module homomorph ism , 521G -paradoxical set, 8G -set, 306G -space, 306, 562Gödel, 3 , 37Gödel completeness theorem , 3Gödel incompleteness theorem , 3Gödel second incompleteness theorem , 3Gabor transformation , 666Galerk in m ethod , 749Galo is fi eld , 43 , 126, 516Galo is group, 126Galo is group of a polynom ia l, 128Galo is m ap , 126Galo is theory, 38 , 81, 126 , 514Galton board , 621Galtonian curve, 621Gambling , 615Gamma function , 337, 651GARCH(p,q), 652GARMA process, 651Gateaux diff erential, 372Gauss and M ainardi-C odazzi relations, 405Gauss complex p lane, 76Gauss curvature , 403Gauss formula, 501Gauss formulas, 404Gauss function , 622Gauss lemma, 144Gauss lemma (for p olynom ia ls), 111Gauss plane, 422Gauss relations, 405Gauss theorem , 57, 60, 348Gauss theorem in Z, 60Gauss trihedron , 404Gauss-Newton algorithm , 638Gaussian curvature, 403 , 419, 420Gaussian curvature of a surface, 420Gaussian e lim ination , 133Gaussian white no ise, 643Gavrilov-Sh iln ikov, 748, 749GCH, 15Gegenbauer, 651Gegenbauer frequency, 652

792

Index

General equation of degree n, 129General equilib rium , 677General equilib rium (econom ics), 677General form of dynam ic model, 684General form of state -space model, 685General L ie group , 132General linear group , 131, 172 , 416General linear group GLn(q), 172, 191, 417General linear group GLn(R), 172, 417General linear model, 628General m onic polynom ia l, 128General orthogona l group GOn(q), 191General orthogona l group GOn(q, F ), 172 , 191General p osition, 278General quantifi er, 2, 10General set theory, 15General term , 240General topology, 193General va lency, 408Generalization o f Cauchy criterion , 228Generalization o f VECM , 649Generalized ARFIMA process, 651Generalized continuum hypothesis (GCH), 15Generalized derivative, 391Generalized d im ension , 731Generalized elem ent, 581Generalized Euclidean space, 21, 137Generalized function, 391 , 659Generalized function (or d istribution), 391Generalized functions, 20Generalized Hopf bifurcation , 712, 713 , 736Generalized Hopf singu larity, 713Generalized integra l, 354 , 355Generalized Lagrang ian, 681Generalized lagrang ian, 679Generalized least squares (GLS), 633Generalized sequence, 205, 206 , 322Generating fam ily, 89Generating part, 102Generating set, 519Generating system , 199Generator, 84 , 297, 519 , 524, 581 , 591, 609Generator , 95Generator o f an ideal, 95Generator o f an ideal , 111Generator part, 168Generators of an abe lian group , 285Generatrix , 400Generatrix lines , 404Generic elem ent, 581 , 610Generic prop erty, 724Generic prop erty (or typ ical prop erty), 724Gentzen-typ e , 2Genus, 256Genus theorem , 145Geodesic, 179, 372 , 373, 401, 417Geodesic curvature, 401Geodesic line, 177, 401Geometric arc, 392Geometric complex, 251, 255Geometric congruence, 152Geometric construction , 129Geometric fib er, 560 , 569Geometric point, 560Geometric realization , 295Geometric sequence, 322Geometric series, 323Geometry, xvii, 149

absolute, 149, 151analytic, 149, 181axiom C (connection), 154axiom D (completion) Dedekin , 160axiom H (hyperb olic), 154axiom P (po lar trila teral), 154axiom R (rectangle), 153axiom s A,A∗ (b isector), 159axiom s A1,A2,A3 (affi ne), 155axiom s M1,M2,M 3 (m etric p lane), 151axiom s P1,P2 ,P3 (pro jective), 155axiom s P5,P5 ’ (Fano), 158descriptive, 150 , 173, 174ellip tic, 149, 179Euclidean, 150hyperb lo lic, 176hyperbo lic, 149 , 152Lobachevsky, 152Lobachevsky -Bolyai-Gauss, 149, 152non-Euclidean , 149ord inary, 150parabo lic, 149pro jective , 149spherica l, 179

Gereno lemniscate , 336Germ , 551Germ of an analytic set, 560G iraud theorem , 609G lesjer test, 636G lide , 162G lide refl ection , 162G loba l b ifurcation , 736 , 737, 746G loba l b ifurcations, 734G loba l continu ity, 200G loba l elem ent, 581, 610G loba l elem ent (global section , po int), 581 , 610G loba l homoclin ic b ifurcations, 738G loba l m aximum, 679G loba l m ean value, 491G loba l m in imum , 679G loba l P ro j, 572G loba l section, 558 , 563, 581, 610

G loba l Sp ec, 571G loba l stability, 759G loba lly stab le equilib rium , 757GLS estimator, 634 , 635G lu ing , 274, 547 , 606, 610G lu ing (concatenation , collation), 606G lu ing axiom , 551, 606 , 607G lu ing condition, 551, 606G lu ing lemma, 274Gnomonic pro jection , 179Gode, 559Godement resolution , 559Golbach con jecture, 1Goldbach con jecture, 148Golden ru le, 758Golden theorem (Gauss), 144Goldfe ld -Quandt test, 636Gollub-B enson , 726Gompertz model, 638Goodw in cyclica l grow th model, 761Goodw in model, 753G rade, 525G raded a lgebra, 263 , 476, 523G raded co-ring, 320G raded coring , 320G raded module, 263 , 522, 523G raded ring, 264 , 270, 318 , 476, 522 , 523G raded sheaf, 553G raded va luations, 110G rad iant, 489G rad ient, 316, 342 , 343G rad ient op erator, 412G rad ient theorem , 343, 412G raeco-Latin square, 613G raeff e m ethod, 341G rammar, 5G ranger, 647, 651G ranger representation theorem , 649G raph, 258, 299 , 560, 600G raph automorphism , 259G raph characteristic po lynom ial, 134G raph co loring, 261G raph coord inates , 539G raph cyc le (circu it), 260G raph edge, 258G raph isomorph im , 258G raph of a correspondence, 27G raph of a map, 27 , 30G raph of binary relation on a space, 299G raph of map, 29 , 519G raph representation , 258G raph theory, xvii, 258G raph vertices, 258G raph w ithout lo op and spindle, 259G raphica l integration , 355G raphica l pro jection, 174

anamorph ic pro jection , 174auxilia ry view , 174parallel pro jection, 174

axonometric pro jection, 174axonometric pro jection (d im etric), 174axonometric pro jection (isom etric), 174axonometric pro jection (trimetric), 174multiv iew s, 174multiv iew s (elevation), 174multiv iew s (floor p lane), 174multiv iew s (section), 174ob lique pro jection , 174ob lique pro jection (cab inet), 174ob lique pro jection (Cavalier), 174orthographic pro jection, 174

p ersp ective pro jection , 174G rassman manifold , 138G rassmann algebra , 137G rassmann coordinates, 138, 140G rassmannian , 138G ray code, 667G reat circle, 179G reatest common denom inator (GCD), 69G reatest common divisor, 57G reatest common divisor (gcd.), 120, 143G reatest common divisor in Z, 60G reatest element, 9, 45, 46G reatest lower bound, 9, 46G reen formula , 489, 501G reen theorem , 285, 316, 348 , 364G reen-R iemann formula , 363 , 411G reen-R iemann theorem , 363G relling , 13G rid , 171G rid search pro cedure (H ildreth -Lu), 635G ridd ing, 171G robman-Hartman theorem , 723G robman-Hartman theorem (for d iscrete sys.), 723G ronwall lemma, 754G ross substitutab ility assumption , 760G rothendieck, 269, 575G rothendieck ab elian category, 591G rothendieck cohomology, 550G rothendieck pretopology, 607G rothendieck theorem , 567, 570G rothendieck topo logy, 569 , 607G rothendieck topo logy axiom s, 607G rothendieck topos, 608G rothendieck trace formula, 571G rothendieck un iverse, 16 , 577G rothendieck un iverse U, 577G roup, 38 , 42 , 81 , 82, 515

Ab elian , 42, 516 , 517isom etry, 149 , 151

793

Index

Klein, 42o f rigid motions, 151sim iltude, 149transformation , 149

G roup by arrows, 590G roup commutator , 519G roup generators, 103 , 254G roup GLn(K) of nonsingu lar matrices, 106 , 173G roup homomorphism , 38, 84, 521G roup isomorphism , 85G roup kernel, 88G roup law , 82G roup ob ject, 310G roup of affi ne collineations G, 172G roup of automorph ism s, 38G roup of complex numb ers of m odulus 1, 72G roup of d isp lacem ents, 159G roup of fin ite typ e, 89G roup of invertible elements of a ring , 93G roup of isom etries, 187G roup of n -b oundaries, 290G roup of n -cycles, 290G roup of nonsingular matrices, 106G roup of orthogonal collineations, 173G roup of p eriods of ω, 503G roup of regular matrices, 106G roup of relative n -b oundaries , 291G roup of relative n -cycles, 291G roup of rig id motions, 151G roup of rotations, 41 , 515G roup order, 42G roup theory, 81, 82, 514G roupo id , 280 , 297, 582G roupo id category, 594G roupo id π(K), 297G roupo id theorem , 280G roups obtained by extension , 92Guckenhe im er, 732Guckenhe im er-Holm es, 699, 713Gysin sequence, 320

h-Cob ord ism , 548H -group , 311H-in jective , 555h-level, 312H -space, 283Hénon map, 696, 715Hénon quadratic m ap , 696Haar wave let, 666Hadamar formula , 453Hadamard matrices, 667Hadamard three circ les theorem , 491Hadamard-Cartan & Gromoll-M eyer theorem , 420Hadamard-Perron theorem , 720Hadamard-Perron theorem (for d iscrete sys.), 723Hahn-Banach separation theorem , 26Hahn-Banach theorem , 24 , 45 , 236, 238Hahn-Banach theorem (geom etric form ), 26Hairy ba ll theorem , 294Half-angle formulas, 177Half-ba ll, 230Half-cone, 184Half-exact functor, 595Half-p eriod ratio, 435Ham -sandw ich theorem , 320Ham ilton ru les , 79Ham iltonian , 694, 695, 716Ham iltonian chain, 260Ham iltonian circu it, 260Ham iltonian cycle, 260Ham iltonian flow , 694Ham iltonian function , 694Ham iltonian graph , 260Ham iltonian path, 260Ham iltonian system , 694 , 711, 724Ham iltonian system with n-degrees of freedom , 694Hamm ing d istance

b etween two finite sets , 220n-tuplets, 220

Hand le, 256 , 300, 432Hanke l transformation , 657Hardy space , 460, 489Harmonic, 726Harmonic analysis, 655Harmonic div is ion, 166 , 168Harmonic function , 425, 485Harmonic homology, 156, 158Harmonic ma jorant, 491Harmonic series, 323Harmonic upper-bound , 491Harmonicity and holomorphy, 485Harnack inequalities, 486Hartman-G robman, 697Hartman-G robman theorem (for diff eom .), 697Hartman-G robman theorem (for flow), 697Hartogs, 441Hartogs theorem , 494Hasse d iagram , 9 , 45Hasse D iagram (order diagram ), 45Hasse princip le, 79Hausdorff axiom , 219Hausdorff dim ension of the m easure µ, 731Hausdorff dim ension prop erties, 730Hausdorff distance, 220Hausdorff paradox, 17Hausdorff quotient space, 299Hausdorff separation axiom , 206Hausdorff space , 193, 199 , 206, 207 , 538, 539Haussdorf dim ension, 730Heat equation, 389 , 661Heavis ide, 668

Heaviside unit step function , 658Height, 166Heine , 223Heine theorem , 223, 224Helix , 393 , 396Henon map, 715Hered itarily fin ite set, 578Hered itary prop erty, 35Hered itary set, 14, 16 , 578Herm itian con jugate op erator, 375Herm itian form , 136 , 137, 242Herm itian inner product, 21, 136 , 375Herm itian inner product (inner product), 375Herm itian integral op erator, 377Herm itian norm , 137, 374 , 377Herm itian op erator, 375Herm itian sca lar product, 21 , 137Herm itian space , 21 , 137Herm itian symmetric sesqu ilinear form , 136Herm itian symmetry, 136

left sem ilinear, 136right linear, 136

Hesse form , 184Hessenberg, 155Hessian, 346Hessian matrix , 346Hetero c linic po int, 703Hetero c linic sadd le connection, 747Hetero logical ad jective, 13Heteroscedastic , 644Heteroscedastic ity, 633, 635Heteroscedastic ity), 643Hexadecim al numera l sy stem , 56Hexagon axiom , 587Hexagon identities, 587H igher aithm etic, 51H igher aithm etic (numb er theory), 51H igher D imensiona l A lgebra (HDA), 594H igher order partia l derivative , 344H igher-dim ensiona l categories , 576H igher-order pred icate ca lculus, 3H ilb ert basis, 244H ilb ert norm , 137H ilb ert p lane, 151H ilb ert prob lem , 3H ilb ert program , 4H ilb ert space, 21, 137 , 199, 211 , 241, 242, 371 , 489H ilb ert space of fin ite d im ension , 244H ilb ert-typ e, 2H ildreth -Lu method , 635H istogram , 615H jelm slev , 151H jelm slev theorem of p erp end iculars , 152, 153H jem slev, 157Hodge star, 412Hodge star op erator, 412Holes, 506, 549Holomorph ic C-C-function , 424 , 548, 549Holomorph ic continuation, 426Holomorph ic function , 333, 421 , 425, 426, 453 , 548

lo cal b ehavior, 477on an op en set o f S2 , 482

Holomorph ic function in an annu lus, 464Holomorph ically convex, 480Holomorph ically convex set, 480Holomorphy, 421, 424 , 440

equ iva lent formulations, 421extension of the concept, 422

Holomorphy of C-C-functions, 422

Holomorphy of C-C-functions, 422Hom functor, 595Hom -set, 582, 583Homeomorphic, 194Homeomorphic im age, 197Homeomorphic spaces, 219Homeomorphic top ologica l space , 200Homeomorphism , 193 , 194, 219 , 248, 271 , 538, 539, 546 , 687, 733Homeomorphism (topo logy), 538Homeomorphism onto its image, 545Homoclinic and hetero clinic orb its , 721Homoclinic lo op, 747

attracting , 747rep elling , 747

Homoclinic point, 702 , 703Homoclinic points, 709Homoclinic sadd le connection, 710 , 747Homoclinic sadd le connection theorem , 747Homoclinic tangle, 697, 709Homogeneity, 20Homogeneous coord inate, 532Homogeneous coord inates , 252Homogeneous e lem ent, 528Homogeneous function , 533Homogeneous ideal, 270, 528 , 533

Homogeneous integra l equation of 2nd kind , 374Homogeneous m ed ium , 661Homogeneous p olynom ial, 533Homogeneous system , 107Homogeneous transformation , 20Homograph ic equation, 184Homograph ic transformation , 439

classifi cation , 439composition , 439

Homography, 483Homological a lgebra, 263Homological equation, 698, 700Homologous cyc les , 474Homologous po ints , 166Homology, xvii, 156, 251 , 255, 263, 549Homology class , 287, 290

794

Index

Homology class of zn, 287Homology classes, 285Homology functor, 286 , 597Homology group, 251, 255 , 257, 263 , 287, 475 , 550, 749Homology group (Betti group), 257Homology of spheres, 292Homology sequence , 597Homology theory, 251, 255—257 , 263, 302 , 476, 509Homology theory (H,∂), 302Homomorph ic image, 510, 521Homomorph ism , 38, 81, 84 , 514, 521 , 598, 606Homomorph ism (abstract algebra), 521Homomorph ism (morphism ), 598Homomorph ism as cartes ian product, 521Homomorph ism of Banach a lgebras, 494Homomorph ism of complexes , 521Homomorph ism theorem , 88

applications, 88Homomorph ism theorem for modules, 102Homomorph ism theorem for rings, 100Homomorph ism s of an a lgebraic system , 590Homoscedasticity, 628Homothetic c ircles, 166Homothetic figures, 165Homothetic ratio , 165Homothetic triangle, 166Homothety, 165Homothety and point at infin ity, 165Homothety-refl ection, 166Homothety-rotation, 165Homotop ic, 247, 749Homotop ic continuous maps, 472Homotop ic maps in a basic sense, 249Homotop ic path edges, 297Homotop ic paths, 247, 502Homotop ic paths w ith fixed endpo ints, 247, 470Homotop ic topo logical space, 248Homotopy, xvii, 214 , 247, 248 , 254, 270, 274 , 279, 468 , 469, 472 , 562, 573

of continuous maps, 248of po lygonal chains, 254

Homotopy (fundamental group), 502Homotopy axiom , 288, 302 , 553, 597Homotopy axiom for pa irs, 292Homotopy axiom theorem , 288, 317Homotopy category, 274 , 305, 311 , 592Homotopy category (hTop), 311Homotopy class, 248 , 274Homotopy class (m apping class), 573Homotopy equivalence, 248, 249 , 274, 305Homotopy equivalence in homotopy category, 274Homotopy extension prop erty, 314Homotopy extension theorem , 301Homotopy function , 248Homotopy group, 247, 250 , 309, 312Homotopy identity, 283Homotopy invariance, 284Homotopy invariant, 573Homotopy lifting prop erty, 314, 573Homotopy of continuous maps, 248Homotopy of paths, 247Homotopy sequence of a fibration , 314Homotopy sequence of a pair, 314Homotopy sequence of a weak fibration , 315Homotopy set, 573Homotopy theory, 214, 270Hom set, 582 , 585Hooke law , 377Hopf a lgebra, 320Hopf b ifurcation , 711, 718 , 735

generalized , 713subcritica l, 712supercritica l, 712

Hopf b ifurcation (normal form ), 733Hopf b ifurcation theorem , 712Hopf fibration , 315Hopf recurrence theorem , 718Hopf s ingu larity, 711Hopf s ingu larity of typ e (p,+) or (p,−), 713Hopf s ingu larity of typ e p, 713Hopf theorem , 320Hopf-Landau model, 739Horizontal composition, 264 , 585Horner schema, 341Horner schem e, 109Horseshoe, 708Horseshoe d iff eomorphism , 703, 705, 708

canon ical example, 705dynam ics on symbol sequences, 707symbolic dynam ics for horsesho e d iff eo ., 707

Horseshoe map, 731Hosking , 651HSP theorem , 514hTop, 311Hurew icz fib er space, 314Hurew icz fib er space (fibration), 314Hurew icz homomorph ism , 315Hurew icz map, 289Hurew icz theorem , 288, 289 , 315Hurew icz theorem (1), 289Hurew icz theorem (2), 315Hurew icz theorem (absolute version), 315Hurew icz theorem (relative version), 315Hurst exp onent (H), 651Hurst exp onent estimator, 652Hurst statistic, 651Hurw itz theorem , 477Hyp ebo lic convergence , 650Hyp erbo la , 185, 186Hyp erbo lic, 158Hyp erbo lic attractor, 748

Hyperbo lic axiom (H), 154Hyperbo lic decreas ing, 651 , 652Hyperbo lic equ ilib rium point of typ e (m ,k), 723Hyperbo lic fixed p oint, 697, 747Hyperbo lic function , 177 , 337

hyperb olic cosecant, 177hyperb olic cosine, 177hyperb olic cotangent, 177hyperb olic secant, 177hyperb olic sine , 177hyperb olic tangent, 177

Hyperbo lic geom etry, 176Hyperbo lic invariant set, 708Hyperbo lic linear diff eomorph ism , 692, 696 , 697, 705Hyperbo lic linear flow , 696Hyperbo lic m etric p lane , 154Hyperbo lic m odel, 152Hyperbo lic non-wandering set, 709Hyperbo lic nonlinear fixed p oint, 697Hyperbo lic p eriodic orb it, 697Hyperbo lic plane, 152Hyperbo lic point, 402 , 403, 419Hyperbo lic R iemann surface , 415Hyperbo lic sadd le-p oint, 710Hyperbo lic segment, 176Hyperbo lic set, 708Hyperbo lic singular po int (for flow), 697Hyperbo lic singularity, 699Hyperbo lic structure , 708Hyperbo lic trygonometry, 178Hyperbo licity, 702Hypercomplex number, 35Hyperparallel, 152 , 154Hyperp lane, 717Hypersurface, 717Hypertetrahedron, 251 , 255Hypervo lum e, 251 , 255Hypoellipticity, 492Hypograph, 349Hypotenuse , 175, 178Hypothesis, 2 , 5Hypothesis H0, 625Hypothesis of normality of errors, 626Hypothesis of recurrence , 5, 39Hypothesis testing, 616 , 617, 622Hysteres is, 748 , 751

I-ad ic topology, 533i-b oundaries, 497i-b oundary, 506i-co cyc les , 501i-chains, 497 , 504i-chains w ith integra l co effi cients , 504i-cob oundaries , 501i-cycle, 90, 506I-depth , 525i-faces, 251 , 255i-th étale cohomology group , 571i-th cohomology group , 571i-th homology group, 556i-th right derived functor, 557Ideal, 22 , 43, 93, 100 , 142, 269 , 516, 523, 528Ideal e lem ent, 4, 149Ideal line, 157idea l ob ject, 4Ideal p lane, 157Ideal p lane of a m etric plane, 157Ideal p oint, 154, 157Ideal sheaf, 552Ideal sheaf (algebra ic geom etry), 552Ideal theory, 142Idempotence , 8Idempotent, 93Idempotent elem ent, 33Idempotent ring , 10Identically indep endently d istributed (i.i.d .), 643Identifi cation , 275, 299Identifi cation space, 574Identities for complex arguments, 177Identity, 3 , 576Identity e lem ent, 42 , 43 , 515, 516Identity functor, 273Identity map, 29 , 30 , 173, 219Identity matrix , 105Identity ob ject, 586if and on ly if (iff ), 5IGARCH (integrated GARCH), 652Image, 14 , 29, 30, 85, 290 , 519, 520Image of homomorph ism , 85Image of m orphism , 85Imaginary part, 76Imaginary part , 70Imaginary part of z, 70Immeasurable frequencies , 726Immersed submanifold, 544Immersion , 213, 546Immersion (a lgebraic geometry), 571Immersion (a lgebraic topo logy), 546Immersion theorem , 213Impact multp lier, 647Implic it d iff erential equation, 381Implic it d iff erentiation , 335Implic it function , 345Important prop erties of Fourier series , 654

asymptotic b ehavior, 655convergence o f function in m ean , Parseval equation , 654D irich let cond itions, 655least m ean squares error of a function , 654

Improper po int, 155Impulse function (D irac δ function), 659In b ijection w ith , 36, 47

795

Index

Inaccess ib le card inal, 15 , 578, 579Inadequacy of Zarisk i topology, 567Incidence , 150Incidence-preserv ing transformation , 157Incident, 152Inclus ion, 11Inclus ion function, 520 , 591Inclus ion functor, 584Inclus ion map, 520, 544, 599Incomplete basis theorem , 103Indefinite integra l, 349 , 352, 353

o f a lgebraic functions, 354o f rationa l functions, 353o f rparticu lar functions, 354

Indep endent events , 619Indeterm inates, 120Index, 430, 471Index change formula , 40Index gymnastics, 407Index of a cyc le, 474Index of a lo op , 473, 504Index of lo op w ith resp ect to a po int, 470Ind icator, 57Ind icator variab le, 630Ind icator variab le (or dummy variable), 630Ind irect demonstration, 5

Ind irect isom etries in R3 , 164Ind irect least squares (ILS), 642Ind irectly con formal mapping, 438Ind iv idual, 2Induced homomorph ism , 545Induced law , 32Induced map, 273 , 554Induced order, 38, 45Induced orientation, 286Induced top ology, 194, 199, 201 , 538, 543Induction , 35Induction axiom , 4 , 50 , 62Induction hypothesis , 5, 39Inductive defin ition , 5Inductive hyp othesis, 5Inductive lim it, 550, 552 , 601Inductive lim it (d irect lim it), 601Inductive order, 46Inductive ordered set, 47Inductive set, 47Inertia degree, 146Infimum , 46Infin ite cyc lic group , 296Infin ite descent, 52Infin ite d istribution of lags , 640Infin ite graph, 259Infin ite product, 324 , 467Infin ite product expansion , 467Infin ite set, 34Infin ite-d im ensional F-pro jective space, 299Infin ites im al calcu lus, xviiInfix form , 32Infl ection point, 334In formation d im ension , 731In itial cond ition problem , 386In itial cond itions, 378In itial morphism , 580In itial ob ject, 310, 580In itial prop erty, 580In itial segment, 48 , 65In itial topo logy, 201In itial-va lue prob lem , 378In jection , 310 , 520In jective arcs on manifo ld , 414In jective envelop e (in jective hu ll), 555In jective homomorphism , 520, 555In jective homomorphism (monomorphism ), 520In jective immersion , 544, 545In jective linear transformation, 546In jective map, 29, 31 , 519In jective module, 554In jective ob ject, 555 , 556In jective patch , 406In jective resolution , 555 , 556Inner product, 21 , 136, 137, 182 , 375Inner product space, 136 , 137Inner-product space, 21Innovation, 647Input variab le , 684Inscrib ed angle, 163Inscrib ed angles in a circle, 163Inscrib ed circle, 176Insertion function , 591Inset, 707, 709Insets and invariant manifold , 709Instability, 717

subharmonic, 734Instrum ental variable, 637Integer, 64

a lgebra ic structure, 64order structure, 64

Integer modulo n, 87Integer part, 61Integer part of a rationa l fraction, 118Integers

ca lcu lation ru les, 64Integrab ility of the logarithm , 463Integrab le system , 695Integral, 146Integral calcu lation

complem ent formula (m eromorphic embedd ing o f ζ) , 477Fourier integra l, 477o f a rational function w ithout real p oles, 477trigonometric inttegra l, 477

Integral calcu lus, xvii, 348Integral closure, 146Integral domain, 43, 93 , 99 , 100, 120 , 142, 457, 516 , 523

d iv isibility, 141Integral domain (entire ring), 43, 93 , 516Integral domain by prim e idea l, 100Integral equation , 374

(homogeneous) o f the second kind, 374(nonhomogeneous) of the second kind , 374classifi cation , 374

o f 1st kind , 374Integral equation o f the fi rst k ind, 374Integral equations of the second kind , 374Integral function , 352 , 457Integral kernel, 377Integral m ean-va lue, 351Integral of a 1 -form on a part of R, 445Integral of a d iff erential 2-form , 501Integral of a d iff erential form , 410

accord ing to a param eter, 411Integral of closed 1-form on a continuous path, 469Integral of surface, 362Integral on a continuous path , 469

Integral on a piecew ise-C1 path , 445Integral on an oriented curve, 447Integral op erator, 374Integral prop erty, 93, 98Integral scheme, 564Integral transformation, 656Integral transformations, 656

fin ite Fourier transformation , 657fin ite Lap lace transformation , 656for functions of severa l variab les, 657Fourier cosine transformation , 657Fourier sine transformation, 657Fourier transformation , 657, 661Hanke l transformation of order, 657Lap lace transformation , 656, 657Lap lace-Carson transformation , 657M ellin transformation, 657o f functions of one variab le, 656one-s ided Fourier transformation , 657Stie ltjes transformation, 657two-sided transformation, 656Walsh functions, 667wavelet transformation , 665Z-transformation , 664

Integrand , 349Integrating factor, 379Integration by parts, 353Integration m etho ds, 353Integration of 0 -form , 506Integration of 1 -form s, 506Integration of a C-C-function , 425Integration of series , 354Integration rule, 351Integration rules, 425Integration theorem s, 362Integrations term by term , 354Interior, 195, 214 , 216, 333 , 536, 537, 548Interior ang le , 162Interior ang le o f a po lygon , 162Interior p oint, 195, 198 , 219, 536 , 537, 674Interior p oint m ethods, 674Interior product, 139Interleave sequence, 216Intermediate fi eld , 122Intermediate value theorem , 203, 226, 326Interm ittency, 738 , 745

typ e 1, 718 , 746typ e 2, 746typ e 3, 746

Interna l (comp osition) laws of lattice, 9Interna l ang le b isector, 167Interna l center o f sim ilitude, 166Interna l comp osition law , 39, 41, 42, 515Interna l d irect sum , 525Interna l Hom functor, 586Interna l homothetic center, 166Interna lly homologous, 373Interpolation , 339Interpolation theory, 339Interpretation, 513Interpretation (log ic), 513Interpretation (model theory), 513Interpretation o f multip liers , 681Intersection , 8 , 9, 11, 13Intersection of a simple arc w ith a sphere, 396Intersection of neighborhoods, 195Intersection of op en sets, 195Intersection of sub complexes , 290Intput variable, 685Intrinsic curvature, 418Intrinsic geom etry, 400, 401 , 415Intrinsic param eter, 393Intrinsic prop erty, 197Intu itionism , 4Intu itionistic log ic , 4Intu itionistic propositional log ic , 4Intu itionistic Zermelo -Fraenke l (IZF), 16Invariance o f dim ension , 294Invariance o f domain, 294Invariance o f integral by homotopy, 469Invariant Cantor set, 692Invariant manifold , 697, 720 , 747Invariant manifold theorem , 697Invariant manifold theorem (for diffeomorph .), 697Invariant manifold theorem (for flow ), 697Invariant m easure, 19, 718 , 724, 727Invariant set, 690, 708

796

Index

Invariant set stability, 715Invariant sets , 715Invariant subgroup , 84 , 599Inverse, 32, 40Inverse edge path , 297Inverse element, 32 , 42 , 82, 86, 515Inverse function , 327 , 546Inverse function theorem , 390Inverse hom eomorph ism , 219Inverse im age, 29, 30, 498 , 519 , 520, 560Inverse im age (pre im age), 520Inverse im age of a 1-form by a map, 444Inverse im age of a 2-form by a map, 450Inverse Laplace transformation , 657Inverse map, 30Inverse mapping theorem , 568Inverse matrix , 106Inverse on the left, 32Inverse on the right, 32Inverse path, 279, 445Inverse relation, 29Inverse symbol, 30Inverse transformation, 666Invertib ility, 141, 344Invertib ility and continu ity, 327Invertib le, 40, 93 , 141Invertib le elem ent, 40, 141

in residue-class ring , 143Invertib le on the left, 32Invertib le on the right, 32Invertib le op erator, 372Invo lute, 396Invo lute of a curve, 396Invo lute of a p lane curve, 398Invo lute of an arc , 396Invo lution, 40 , 134, 161 , 240, 562Invo lutive function , 134 , 240Invo lutive map, 134, 240Invo lutive mapping , 134, 240Invo lutive transformation, 161Inward p erturbation , 747Irrational numb er, 68Irrational rotation number

(quasip eriodic b ehavior), 744

Irrationality o f√2, 65

Irreducib ility criteria, 121Irreducib ility o f polynom ia ls , 121Irreducib le, 141Irreducib le affi ne algebraic set, 531Irreducib le elem ent of a ring , 95Irreducib le form of a rational numb er, 62Irreducib le fraction, 68Irreducib le function, 530Irreducib le po lynom ial, 110, 120 , 121, 530Irreducib le prim e elem ent, 142Irreducib le schem e, 564Irrelevant ideal, 528is d iv isor of, 44is multip le o f, 44Iso clinal line , 379Iso cline, 379Isogonal m apping , 438Isolated logarithm ic s ingu larity, 433Isolated po int, 195 , 198Isolated singu lar p oint, 428Isolated singu lar solution , 381

asymptotic point, 381centre, 381nodal point (node), 381sadd le po int, 381

Isolated singu larities on k-sheets, 433Isolated singu larity, 464Isolated zeros, 462Isom etric patches, 400Isom etric pro jection, 174Isom etric regular patches, 400Isom etry, 18 , 151, 173, 193

canon ical transformation , 187glide, 149identity, 149refl ection , 149rotation , 149translation, 149

Isom etry group, 149, 151 , 173, 187Isom etry group Is(E ), 187

subgroup of d isp lacem ents Is+(E), 187

subgroup of d isp lacem ents Is−(E), 187Isomorph ic, 10 , 606Isomorph ic directed graphs, 600Isomorph ic functors (natura lly equivalent functors), 302Isomorph ic graphs, 259Isomorph ic groups, 85Isomorph ism , 33 , 38 , 81, 302 , 514, 520 , 598, 606Isomorph ism of groups, 85Isomorph ism of ordered sets, 47Isomorph ism theorem , 89Isomorph ism theorem for modules, 102Isomorph ism -dense sub category, 595Isop erim eter, 373Isotone map, 66Isotrop ic line, 158Isotrop ic vector, 137Isotropy group , 483Isotropy subgroup , 306ith face map, 286ith k-s implex, 550

J -shap ed d iagram , 601Jacob ian matrix , 343, 344 , 716Jacobson rad ica l, 529

Jarque-Bera test, 643Jensen formula, 463 , 492Johnson theorem , 494Joint entropy, 727Jordan arc, 214, 215 , 392, 393

adm iss ib le param eterization , 393Jordan areo lar measure, 365

countab le un ion of parts of R366of rectangles (possibly degenerated), 365problem , 366

Jordan basis, 135Jordan b lo ck, 135Jordan canonical form , 136Jordan canonical form theorem , 136Jordan Curve, 255Jordan curve, 197 , 214, 392, 445

closed , 197

Jordan curve in C, 423Jordan curve theorem , 214 , 294, 504Jordan domain, 485Jordan form , 698Jordan Lemma, 429Jordan loop , 444Jordan matrix , 135Jordan matrix decomposition , 135Jordan path , 444Jordan regu lar curve, 445Jordan theorem , 473 , 474Jordan-Brouwer separation theorem , 294Jordan-Brouwer theorem , 214Jou le, 360Joyeux, 651jth Betti number, 502Judge, 632, 634, 637

k(y)-schem e, 569k-algebra, 568k-connected graph , 261, 262k-face , 279k-jet, 701k-simplex, 251, 255k-ske leton, 300K-theory, 269 , 562K-vector subspace, 108Kähler differentia l, 527Königsb erg bridges, 258Künneth formula, 305Künneth formula for cohomology, 318Künneth theorem , 305Kakutani, 674, 677Kakutani fixed p oint theorem , 677KAM theorem , 724Kelley-M orse set theory (KM), 15Kelv in -Stokes theorem , 364Kep ler (approximation) m ethod, 355Kep ler folium , 336Kernel, 85 , 88, 133 , 161, 290 , 557, 599 , 605Kernel (in abstract algebra), 521Kernel for vector space , 521Kernel is a normal subgroup , 88Kernel o f derivation function of po lynom ials, 114Kernel o f group homomorph ism , 521Kernel o f homomorph ism , 85Kernel o f homomorph ism of d iff erentia l complexes, 522Kernel o f linear transformation , 521Kernel o f m agma homomorphism , 521Kernel o f m orphism , 85Kernel o f ring homomorph ism , 521Kernel pa ir, 588 , 599, 605Ket, 409Keynesian assumptions, 760K leene, 1K lein , 163K lein bottle, 255 , 412K lein group , 42K lein model, 149 , 152, 176K lein test, 632Knot, 197Knot w ithout crossing, 197Kolmogoroff axiom s, 618Kolmogorov-Arnold -M oser theorem , 724Kolmogorov-Sinai entropy, 727Kolmogorov-Sm irnov statistic, 631Kondrakov theorem , 391Korch curve, 731Koyck model (geom etric progression), 640Koyck transformation , 641Kripke-Platek set theory (KP), 15Kronecker, 4Kronecker delta, 343Kronecker formula , 413Kru ll dim ension, 526Kru ll topo logy, 533kth root o f a rea l, 69Kuhn-Tucker cond itions, 679

quantifi cation , 682Kuhn-Tucker multipliers

interpretation, 682un iqueness, 682

Kuhn-Tucker theorem , 682Kupka-Smale, 709Kuratowski graph , 263Kuratowski theorem , 263Kuratowski-Pontryag in theorem , 263Kurtosis test, 643

L2-space, 392L’Huilier theorem , 180L-adic étale cohomology, 570L-adic cohomology, 569 , 570L-automorphism , 126

797

Index

L-globally stable equ ilib rium , 757, 758α-lim it po int, ω-lim it po int, 690α-lim it set, ω-lim it set, 715L -lo ca lly stab le equilib rium , 757L -stab ility, 757L∞[a, b] space, 371Löwenhe im -Skolem theorem , 2Lagged variab le, 634Lagrange, 673Lagrange equation , 750Lagrange form of Taylor remainder, 332Lagrange formula, 395Lagrange interpo lating po lynom ial, 112Lagrange interpo lation , 339Lagrange interpo lation po lynom ial, 112Lagrange m ethod, 339Lagrange multip lier, 346, 634Lagrange multip lier (LM), 634Lagrange multip lier test, 630Lagrange po lynom ial, 112Lagrange stability, 718Lagrange theorem , 83, 86Lagrangian theorem , 681Lagrangien , 681Lamb ek, 577Lam inar phase , 738Lam inarization, 746Lap lace transformation , 656, 657

convergence, 657inverse Laplace transformation , 657ru les for the eva luation , 657transform of sp ecia l functions, 658

Lap lace-Carson transformation , 657Lap lacian vector, 347Large category, 568, 582 , 585Latera l classe, 84Latin square, 613Lattice, 9 , 22Lattice idea l, 23Lattice maximal ideal, 23Lattice non-triv ia l idea l, 23Lattice of p eriods of ω, 503Lattice of power set, 10Lattice prim e idea l, 23Lattice prop er ideal, 23Lattice theory, 9Lattices o f sets, 9Laurent series, 428 , 464, 664Laurent series expansion , 428Law of composition , 29 , 30Law of double negation, 2Law of excluded m idd le, 4Law of large numbers, 620Law of monotony, 63Law of non-contradiction , 2Laws of M organ , 2Lcal element, 581, 610Lead ing co effi cient, 109Lead ing co effi cient o f a polynom ia l, 109Lead ing monom ia l, 518Lead ing term , 109Lead ing term of a polynom ia l, 109Least common multip le, 58 , 121Least common multip le (lcm .), 143Least common multip le in Z, 60Least e lem ent, 45 , 46Least m ean squares error o f a function , 654Least squares m ethod

geom etric interpretation, 628 , 631Least upper bound , 9, 45 , 46 , 160Least upper bound theorem , 66Least-squares, 339Least-squares m ethod, 339Leaving variab le, 670Lebesgue diff erentiation theorem , 488Lebesgue integral, 367, 368Lebesgue integral prop erties, 368Lebesgue lemma, 469Lebesgue m easure, 365, 366 , 486Lebesgue-neglig ib le set, 368Ledrappier theorem , 731Leech lattice, 9Lefschetz fixed point theorem , 304Lefschetz number, 304 , 571Lefschetz theorem , 304Left ad joint, 593 , 607Left ad joint and right ad jo int functor, 302Left ad joint functor, 595Left and right d istributiv ity, 43, 516Left and right unitor, 586Left cance llab le elem ent, 33Left closed monoida l category, 587Left derived functor, 557Left exactness, 317Left fi eld, 79Left idea l, 22Left pro jective module, 554Left R -module M , 517Left reso lution , 556Left surjective and un ivo ca l relation , 29 , 519Left surjective relation , 29, 519Left translation , 82Left zero morph ism , 599Left-exact functor, 595Left-inverse , 32Left-inverse elem ent, 42, 515Left-invertible, 32Left-neutra l elem ent, 32, 42, 515Left-sh ift, 707Leg, 178

Legendre po lynom ials, 339Legendre symbol, 144Leibn iz criterion of convergence, 324Leibn iz formula , 114Leibn iz notation , 330Leibn iz ru le, 526Length , 447

o f a curve, 447o f a Jordan curve, 448o f a path, 447

Length measure, 415Length measure on a regular patch, 400Lens space , 301Leontief input-output model, 669Leptokurtic d istribution, 644Leray theorem , 296less than or equal to, 36, 44Level homotopy, 312Level of degeneracy (or co dim ension), 699Level set, 347, 520Lever, 190Levi-C iv ita, 407Levi-C iv ita connection , 418Lexicographic product, 49L i-Yorke, 732L ie a lgebra, 132L ie a lgebra of a L ie group, 132L ie bracket, 132, 698L ie group , 131, 417L ift, 268, 573 , 688, 691L ift (a lgebra ic topo logy), 573L ift (category theory), 573L ift (lifting), 573L ifted map, 740L ifting, 282 , 573L ifting criterion, 307L ifting criterion theorem , 307L ifting lemma, 282, 306L ifting of m aps, 216L ifting prop erty, 569 , 570, 573L im it, 220, 305 , 602L im it and cluster points of function w ith resp ect to a fi lter, 223L im it and co lim it, 305L im it cycle , 690, 713 , 717, 718, 720 , 724, 730 , 734L im it o f a fi lter, 223L im it o f a map, 221L im it o f a sequence, 221L im it o f function , 325L im it ord inal, 15 , 48, 578L im it p oint, 26 , 195L im it theorem , 620L im it va lue of function, 325L indelö f space , 212L ine, 163L ine ang les , 75L ine at infin ity, 157 , 173L ine bund le, 561L ine connectiv ity, 262L ine integra l, 343 , 412, 429L ine integra l of a vector fi e ld , 343, 412L ine of curvature, 403L inear a lgebra, 132 , 136

affi ne structure, 133d iagona lization and decomposition , 133Euclidean and Herm itian structure , 133linear structure, 133multi-linearity, 133

L inear c losure, 102L inear combinations, 102L inear correlation co effi cient, 625L inear d ifferentia l equation , 380, 716

autonomous, 716homogeneous, 716non-homogeneous, 716o f oder n , 382o f oder n (non-homogeneous), 383o f oder n w ith constant co effi cients, 383w ith p erio dic co effi cients, 716

L inear d ifferentia l op erator, 668L inear extension, 103L inear extension o f a partia l order, 45L inear form , 105, 238L inear function , 20L inear functional, 20L inear group", 240L inear indep endence , 102L inear interpo lation m ethod , 340L inear L ie group , 131L inear map, 103

composition , 105op erations, 104

L inear op erator, 20, 372L inear optim ization , 670L inear order, 45L inear programm ing , xviii, 669L inear stab ility loss in diss ipative system s, 718L inear topo logical space, 20L inear topo logy, 533L inear transformation, 20, 158L inear trend , 650L inear variety, 513L inearization, 685L inearization o f (statistical) nonlinear models , 637L inearization o f nonlinear model, 684, 685L inearly dep endent vectors, 181L inearly indep endent vectors, 181L inearly ordered set, 21 , 44L ines in shea f, 152L inked variab le, 2L iouville b oundedness theorem , 433

798

Index

Liouville theorem , 463, 465 , 695, 716Lipsch itz , 229Lipsch itz condition

global, 386lo ca l, 386

Little D esargues theorem , 155 , 157Littlewood theorem , 493Ljung-Box statistic, 643LM tes, 630LM test, 634Lobachevsky geometry, 152Lobachevsky-Bolya i-Gauss geom etry, 152Loca l basis, 543Loca l b ifurcation, 711Loca l b ifurcations, 734Loca l b ifurcations inearby steady states , 734Loca l b ifurcations nearby a periodic orb it, 736Loca l canon ica l parametrization, 432Loca l chart, 451, 708Loca l compactness and paracompactness, 540Loca l continu ity, 50, 51, 200

in terms of op en balls , 51in terms of sequences , 51

Loca l coord inate chart, 541Loca l coord inates , 413, 687Loca l cross section, 691Loca l d iff eomorphism , 454, 546Loca l elem ent (lo cal section), 581, 610Loca l embedding, 545Loca l extrema, 331, 332, 345Loca l extrema of functions from Rn to R, 345Loca l extremum , 346Loca l fam ily, 711Loca l hom eomorphism , 306, 413, 538Loca l m aximum , 45 , 679Loca l m inimum, 679Loca l presentation , 553Loca l representatives , 708Loca l ring , 526Loca l section , 563, 581 , 610Loca l stab ility, 758Loca l stab le manifold, 720Loca l structural stab ility, 703Loca l surface, 415 , 542Loca l system , 312Loca l system of groups, 312Loca l triv ia lization , 561Loca l-g loba l b ifurcation , 747Loca l-ringed space, 270Loca lity axiom , 551, 606Loca lization, 528Loca lization of a ring, 528Loca lization of an R -module, 528Loca lly 1-connected space, 308Loca lly arcw ise-connected space, 204 , 546Loca lly compact Hausdorff space , 299Loca lly compact space, 210 , 224, 422Loca lly connected space , 203, 227Loca lly constant sheaf, 552Loca lly exact form , 448Loca lly finite fam ily o f parts, 211, 496Loca lly finitely-presentab le fl at morphism , 569Loca lly invertib le function , 345Loca lly Noetherian scheme, 564Loca lly of fin ite typ e, 566 , 569Loca lly path connected, 277Loca lly pathw ise-connecte , 539Loca lly pathw ise-connected, 538Loca lly pathw ise-connected space, 196Loca lly quasi-fi nite morphism , 569Loca lly ringed space, 270, 552Loca lly small category, 582 , 583, 585Loca lly stab le equ ilib rium , 757Loca lly triv ial bundle, 268, 315, 561Location vector, 182Locking , 743Locus, 71 , 397, 531Logarithm , 336, 455Logarithm in C, 76Logarithm ic barrier function , 674Logarithm ic potentia l, 490Logic, xvii, 1, 512Logic of orders, 3logic of predicates by level, 3Logical connective, 1Logical constant, 3Logical propositions, 1logistic curve, 638Logistic m ap, 715 , 737Logistic m odel, 638Logistic m odel (Vehu lst or Pearl curve), 638Long exact sequence , 289, 554 , 605, 606Long exact sequence o f a pair axiom , 597Long exact sequence theorem , 317Long memory phenom ena, 652Long memory pro cess, 650

defined by auto corre lation , 650defined by sp ectral density, 651

Long range dep endence (LRD), 652Long itud inal data, 633Loop, 214, 430 , 444, 574Loop space, 311Lorentz curve, 663Lorentz tensor, 409Lorentz transformation , 409Lorenz model, 749Lorenz system , 715 , 716, 721Lotto, 615Lower (R iemann) integral, 350Lower bound, 46

Lower box-counting d imension, 730Lower capacity, 730Lower integral, 350Lower lim it of sequence, 323Lower po intw ise dim ension, 731Lower set, 23Lower triangu lar, 110Lower triangu lar matrix , 109Loxodrom e, 180

pseudospherica l, 180Lp-function , 392Lp-space, 392Lp[a, b] space, 371Lumped param eter model, 739Lune, 179Lusternik-Schn irelm ann theorem , 294Lyapunov, 687Lyapunov characteristic number, 728Lyapunov d im ension , 731Lyapunov exponent, 727Lyapunov function , 759Lyapunov stability, 718, 719

m -fold cover, 307m -sheeted covering space, 307m -simplex, 278Möbius function , 40Möbius inversion formula , 40 , 116Möbius strip , 412Möbius transformation, 439 , 483MA (moving average), 645MA model, 645MA(q), 645Mac Lane, 575, 589Mac Lane coherence theorem , 586 , 587Mac Lane introduction to categories, 589Mackey topo logy, 376Madaurin series , 90Magma, 41, 515, 520 , 521Main convergence criterion , 323Mainard i-Codazzi relations, 405Ma jor axis, 185Ma jorant, 45, 46, 53, 370 , 490Ma jorization , 324Malgrange theorem , 711Malgrange-Ehrenpreis theorem , 669Manifold , 300, 413—415, 513, 540 , 708Manifold w ith boundary, 417, 541Mantissa, 624Many-sorted language , 512Many-sorted logic, 512Many-to -one relation, 14 , 29Map, 9, 29, 38, 519

B ijective, 31bijective , 29 , 35 , 519contraction , 229fixed-p oint, 229in jective , 29 , 31 , 519of the fi rst return , 722surjective, 29, 31 , 36 , 519

Map fixed point, 229Map germ , 552Map lo cally o f fin ite presentation , 610Map ob ject, 580Map sink , 724Map winding numb er, 742Mapping cone, 303Mapping cy linder, 277Mapping fib er, 313, 314Mapping space, 233Maps, 9, 29Maps and structures for sets, 27Maps compatib le w ith structures (morphism s), 38Maps on unit circ le, 740Margina l optimality cond itions, 679Markov chain , 616, 623Markov pro cess, 616 , 623Mass po int, 190Match ing, 152Mathematica l exp ectation , 621Mathematica l induction, 49Mathematica l logic, xvii, 1Mathematica l programm ing, 673Mather theorem , 705Matrix , 103 , 104, 133Matrix d iagonalization , 134Matrix form , 628Matrix m in im al po lynom ial, 134Matrix of affi ne collineation , 172Maximal atlas, 542Maximal diff erentiab le atlas , 540Maximal element, 22 , 35 , 45, 46, 579Maximal fi lter, 19Maximal idea l, 22, 269 , 528, 569Maximal idea l (for lattices), 23Maximal sp ectrum , 528Maximal tree, 297Maximum element), 45Maximum likelihood , 635Maximum on a compact interval, 680Maximum princ ip le , 466, 482 , 486, 491Maxwell-Faraday equation , 364Mayer-V iétoris, 502Mayer-V ictoris coro llary, 301Mayer-V ietoris coro llary, 296Mayer-V ietoris sequence, 296Mayer-V ietoris theorem , 292Mayer-V ietoris theorem for reduced homology, 292Meagre, 232Meagre part, 232Meagre set, 232

799

Index

Mean curvature , 403, 419 , 421

M ean curvature o f a surface S2, 421M ean lag, 640M ean square contingency, 623M ean va lue inequa lity theorem (for po lynom ials), 115M ean va lue theorem , 331M ean va lue theorem s, 331M ean-va lue formula, 456M ean-va lue prop erty, 466M ean-va lue theorem , 332M easurability, 193M easurable function , 19, 367 , 368M easurable set, 18M easurable space , 18M easure, 18, 349, 442M easure concentrated on the attractor, 724M easure determ ination , 350M easure o f hyp ographs, 349M easure o f length , 448M easure o f regu lar surface, 359M easure o f surface, 359

approxim ation by lateral surfaces o f prim s, 359approxim ation by polyhedrons, 359

M easure o f unbounded parts , 367M easure theory, xvii, 365 , 368, 442M easurem ent, 160M edian , 166M ellin transformation, 657M eln ikov function , 710M eln ikov method , 710, 739M endelson, 1M enger, 213M enger n-arc theorem , 262M enger theorem , 262M ergelyan theorem , 479M eromorph ic continuation , 465M eromorph ic function, 430, 465M eromorph ic function on C, 434M eromorphy, 421

M eromorphy of C-C-functions, 422M ersenne numb er, 148M esh ing, 171M etacategory, 590M etagraph, 590 , 600M etamathematics, 4M etastab le chaos, 751M ethod of continu ity, 390M ethod of p ena lty functions, 673M ethod of proxy variables, 637M ethod of successive approxim ations, 390M ethod of sup erposition , 624M etric, 20 , 199

Euclidean , 199p-ad ic, 79

M etric d im ensions, 730M etric entropy, 727M etric geom etry on a regular patch, 401M etric p lane , 151

axioms M 1,M2,M3, 151M etric pro jective p lane, 158M etric space, 50, 193 , 199, 216 , 220, 227

countab le product, 217fin ite product, 217precompact, 229tota lly compact, 229uncountab le product, 217

M etric space dim ension , 213M etric subspace , 217M etric tensor, 406—408, 418M etric tensor (R iemannian m etric), 407M etric topology, 199M etrizab le space, 210M etrization, 193, 210M etrization theorems, 210, 211M eusnier theorem , 402M idpoint, 158M ilne pro cess, 388M ilnor, 715M in im al element, 10 , 45M in im al polynom ia l, 117, 122M in im al surface , 373M inkowski d istance, 220M inkowski inequality, 220 , 242M inkowski m etric , 409M inkowski norm , 178M inkowski tensor, 409M inor, 106M inor axis, 185M inor Hessian , 346M inorant, 46 , 370M inorization , 324M inus, 8M issp ecifi cation of m odel, 633M ittag-Leff er theorem , 481M ittag-Leffl er decomposition, 434M ixed product, 183M ixed structure , 37M ixed tensor, 408M ixing dynam ical system , 725M oëbius mesh ing, 171M oëbius strip , 255 , 400

lo cally orientable, 400M ode lo cking, 742M odel, 303M odel (in logic), 510M odel (Relation system ), 510M odel theory, 509 , 510M odel validation , 625M odel w ithout constant term , 626M odels of product life cycle , 638

M odels w ith errors on variab les , 637M odifi ed Jordan form , 758M odule, 38 , 43, 81, 101 , 103, 517

ca lculation rules, 101M odule homomorphism , 101 , 103M odule of Kähler differentia ls , 527M odule theory, 81, 101 , 514M odulo, 84M odulus, 71M odulus o f a complex numb er, 71 , 76M odus barbara , 2M odus ponens, 2, 5M odus to llens, 2, 5M oivre formula, 75M onad , 593M onad (trip le), 593M onad ic , 511M onad ic adjunction , 593M onge, 150, 173M onge-Ampère equation , 389 , 391M onic d iv isor, 110, 147M onic generator, 111M onic irreducib le po lynom ial, 121M onic p olynom ial, 36, 109 , 120, 134 , 563, 569M onic p olynom ial (un it po lynom ial, unitary po lynom ial), 36M onkey saddle, 402M onodromy, 427M onodromy group , 428M onodromy group of regular covering space, 308M onodromy matrix , 723M onodromy theorem , 306 , 427M onogen ic group , 83M onoid, 42 , 515, 517 , 520, 522 , 582, 593M onoid (in category theory), 593M onoid (sem igroup w ith identity), 590M onoid axioms, 520, 576 , 582M onoid homomorph ism , 521M onoida l category, 586 , 587M onoida l identity ob ject, 588M onoida l product, 586M onoida l structure, 588M onom ial, 518M onom ial a lgebra , 518M onomorph ism , 38 , 264, 520 , 598M onomorph ism (monic), 598M onomorph ism (or monic), 606M onotone arithm etic function , 141M onotony criteria, 323M onotony criterion , 141M onster p eaks, 650M onte-Carlo method , 624M ontel theorem , 458M oore, 2M oore-Sm ith sequence, 206M orera , 426, 461M orera theorem , 461M organ law , 9M organ laws, 2, 6 , 12M orphism , 33, 38, 576 , 598

b ijective, 38, 598surjective, 38, 598

M orphism lo cally of fin ite presentation , 566M orphism lo cally of fin ite typ e, 566M orphism of fin ite presentation , 566M orphism of fin ite typ e, 566M orphism of schem es, 270 , 563M orphism with fin ite fib ers, 566 , 567M orse surgery, 373M orse theory, 373M orse-K elley set theory (MK), 15M orse-Smale systems, 703, 724 , 748M other wavelet, 665, 666Multi-scale analysis , 666Multicellular domain , 441Multicollinearity, 631Multicollinearity problem s, 632Multifunctor, 595Multilinear form , 407Multilinear map, 239Multip le, 141Multip le correlation co effi cient, 627, 629Multip le integral, 355Multip le regression model, 628

D istinction b etween εt and et , 628estim ation of estim ators, 628estim ation of regress ion co effi cients, 628general linear model, 628

matrix form , 628geom etric interpretation, 631prop erties , 628prop erties of estim ators , 629statistica l tests, 629sto chastic assumptions, 628structural assumptions, 628variance ana lys is, 629, 630

Multip le ro ot, 112Multip le ro ots, 121Multip le structure, 37 , 38Multip le-va lued complex function, 78Multip le-va lued function , 78, 431Multip lication in N, 52Multip lication in Q, 61Multip lication in Z, 59Multip licative asso ciativ ity, 43 , 516Multip licative identity, 42, 43, 516Multip licative inverse, 42Multip licative notation , 32 , 39Multip licative set, 528Multip licative subset, 528Multip licatively stab le, 40, 93

800

Index

Multiplic ity, 462Multiplic ity o f a covering space , 307Multiplic ity o f a root, 112Multiplic ity o f a zero , 462Multiplic ity order, 134Multiplie r, 748Multiplie rs, 681Multiv iew orthographic pro jections, 174

n-ary op eration, 510n-ary re lation , 28n-ball, 271n-ba ll, 215n-boundary, 286, 287n-ce ll, 299, 534 , 585n-ce ll en, 299n-cha in , 286n-chart, 413n-coboundary, 317n-co chain, 317n-co cycle , 317n-cyc le, 86, 286, 287

n-dim ensional Ck-atlas, 541n-d im ensional Grothend ieck cohomology groups, 550n-d im ensional ho les , 550n-d imensiona l homotopy groups, 249n-d im ensional manifo ld , 540n-dim ensional R iemannian geometry, 415n-d isk , 271n-disk , 215n-d isk (or n-ball), 271n-F system (system w ith n degrees of freedom ), 694n-gon , 299n-manifold, 300 , 540n-order pred icate calcu lus, 3n-simple space , 313n-simplex, 286n-skeleton , 535n-sphere, 271n-th homology group, 550n-tim es continuously d iff erentiab le function , 330n-tim es d ifferentiable function , 330n-tuple, 12n-variab les function , 30Naïve sty le, 581Nabla, 343, 347 , 348Nagata-Sm irnov m etrization theorem , 212Naive set theory, 8, 13Napier analog ies, 180Napier number e, 323Napier ru les , 180Napp e, 188Natura l equ iva lence , 302Natura l exp onentia l, 336Natura l m easure , 725Natura l mumber, 62

op erations, 62Natura l number, 33 , 62

addition, 63asso ciativ ity, 63commutativ ity, 63D istributiv ity, 63monotony, 63neutral element, 63S implifi cation ru le , 63

multip lication, 63asso ciativ ity, 63commutativ ity, 63D istributiv ity, 63monotony, 63neutral element, 63

Natura l number ob ject, 581Natura l p ersp ective , 156Natura l topology, 193, 197—199 , 204, 413Natura l transformation , 267, 302, 576 , 600, 612Natura lly equ ivalent functors, 302Natura lly equ ivalent functors (isomorph ic functors), 302Natura lly isomorphic , 597Necessary and suffi cient condition , 5Necessary cond ition , 5Necessary cond ition (NC ), 5Negation , 8Negation rules, 3Negative auto correlation , 633Negative ly corre lated variab les, 625Negative ly invariant set, 690Negligib le part, 232Negligib le set, 366Neighborhood , 50 , 193, 194, 198 , 216ne ighborhood , 199Neighborhood axiom s, 197 , 218Neighborhood basis , 199, 200 , 543Neighborhood basis (lo ca l basis), 543Neighborhood fi lter, 222 , 543

trace, 222Neighborhood system , 193 , 538, 543Neighborhood system (ne ighborhood fi lter), 543Neighborhood system basis, 211Neil parabo la , 395Neimark-Sacker theorem , 737Nerve, 296 , 535, 559Nerve of a cover of a s implicia l complex, 296Nested intervals, 323Nested segm ents, 67Net, 206Neutra l elem ent, 32, 40, 42, 43, 64 , 82 , 86 , 515, 516Neutra l elem ent for ⊤, 40Neutra l line, 170Neutra l on the left, 32Neutra l p o int, 170Neutra lly stab le , 687

Nevan lina c lass, 463New Foundations (NFU ,NF), 16Newton , 14Newton sum s, 114Newton ’s second law , 360Newton-G regory method , 339Newton-Raphson method , 340Newton ·meter, 360Newton ian attraction , 347NF , 16NFU, 16Nilp otent, 93 , 135Nilp otent elem ent, 529Nilp otent endomorph ism , 135Nilp otent shea f of idea ls , 570Nilp otent shea f of idea ls„ 569Nilp otent upper triangular matrix , 135Nilrad ica l, 529Nine-point circ le, 166 , 167No m iddle third set, 730No-retraction theorem , 472Nodal po int, 381Node, 258, 381

attracting, 717center, 717sink, 717

Node (vertex), 258Node saddle bifurcation , 718No etherian, 35No etherian affi ne schem e, 564No etherian induction principle, 35No etherian ring , 142, 523No etherian scheme, 563Non denumerab ility, 34, 35Non denumerab ility of R, 35Non Euclidean m etric, 153Non param etric m odel, 685Non-a lgebraic functions, 336Non-A rchim edean va luation , 146Non-autonomous d iff erential equation, 716Non-b icontinuous, 194Non-connected space, 202Non-constructib ility theorem , 130Non-contrad iction

in an axiomatic system , 5Non-contrad iction prob lem , 4Non-degeneracy cond itions, 699Non-Euclidean geom etry, 155Non-Euclidean orthogonality, 158Non-Euclidean plane, 154Non-hyp erbo lic fixed p oints for p lanar fi e lds, 702Non-hyp erbo lic singular po int, 699Non-hyp erbo lic singularity, 699 , 701

of sadd le-node typ e, 701Non-incident center, 158Non-iso lated singu larity, 433Non-isotrop ic line, 158Non-orientab le surface, 255Non-principal fi lter, 19Non-strict ordering , 45Non-structured set, 47, 49Non-topolog ical m ap, 194Non-triv ia l idea l (for lattices), 23Non-wandering p oint, 690, 724Noncommutative domain , 518Noncommutative ring , 523Nondegenerate p oint, 315Nonhomogeneous integral equation o f the second kind, 374Nonhomogeneous system , 107Nonlinear diffeomorph ism , 697Nonlinear dynam ical system , 697Nonlinear dynam ical system , 717Nonlinear estimation m ethods, 638Non linear flow , 697Nonlinear model, 637Non linear optim ization , 673Non linear programm ing , xviii, 669, 670 , 673Non linearities, 742Nonmeasurab le set, 18Nonresidue, 144Nonsingu lar matrix , 105Nontriv ial subharmonic function , 491Norm , 20 , 146, 236, 708Norm of a complex numb er, 71Norm of the convergence in quadratic m ean , 244, 245Norm of the un iform convergence, 233 , 237, 244, 245 , 370, 371Norm of vector, 181Norm on a vector space, 236Normability, 20normable , 20Normable topolog ical vector space, 20Normal component, 401Normal curvature, 401 , 403, 419, 420Normal curvature o f a regu lar surface, 420Normal derivative, 489Normal d istribution, 616 , 617, 622Normal endomorph ism , 137Normal equation, 184Normal equations, 626, 628Normal factor, 89Normal form , 128, 296 , 734Normal form theorem (d ifferentia l equation), 698Normal form s for d iff eomorph ism s, 699Normal form s of b ifurcations, 733Normal monomorphism , 598Normal morph ism , 584, 598Normal param eterization , 393Normal param eterization of a circu lar helix , 393Normal p lane, 394Normal representation , 77

801

Index

of a complex number, 76Normal representation o f complex numbers , 76Normal section , 402Normal series, 89Normal space , 193, 206 , 207, 222Normal subgroup , 84 , 88 , 297, 308, 598 , 599Normal to a patch, 399Normal to a p lane , 164Normal tower, 89Normal vector, 397Normality hypothesis o f errors, 627Normality of errors, 626Normality tests, 643

Jarque-B era test, 643Skewness and Kurtosis tests, 643

Normalizer, 308Normally indep endently distributed (n.i.d .), 643Normally summable fam ily, 240Norm ed algebra, 24Norm ed linear space , 21Norm ed space, 20 , 21 , 370, 371Norm ed vector space, 21 , 370Norm ed vector spaces , 236North pole, 271Not top olog ica lly pa irw ise arc-connected , 196Notion of attractor, 724Nowhere dense, 232ntersection o f a simple arc w ith a plane, 396nth (singular) homology group , 287nth (singular) homology group of a space , 287nth cohomology group , 317nth cyclotom ic polynom ia l, 116nth homology group, 287 , 304nth homology group of a chain complex, 290nth homology group of a complex, 290nth prim itive roots of unity, 116nth relative cob omology group , 317nth relative homology group Hn(X,A), 291nth roots o f un ity, 131Nth un it roots , 656nth-rank tensor, 407

in m -dim ensiona l space, 407Null e lem ent, 9Null hyp othesis H0 , 617Null ob ject, 580Null sequence, 522, 534Null vector, 181Null-homotop ic, 274Null-homotop ic map, 274Nullary op eration , 265, 510Numéraire goods, 760Number

a lgebra ic, 78b-ad ic, 68Bernoulli, 337Betti, 257complex, 69connectiv ity, 257connectiv ity , 261edge , 263Fermat, 131, 148hypercomplex , 35irrational, 68Mersenne , 148p-ad ic , 78, 79prim e , 147transcendenta l, 78 , 122

Number of crosscaps, 300Number of edges, 296Number of elem ents o f a set, 7Number of faces , 296Number of hand les , 300Number of vertices, 296Number system

axiomatic construction, 78Number theory, xvii, 141Numerab le Cech cohomology, 559Numeration in base b, 56Numerica l harmonic ana lys is , 655

Fast Fourier transformation (FFT ), 656formulas for trigonom etric interp olation , 655, 656

Numerica l reso lution o f equation , 340

Ob, 576Ob ject, 38, 310 , 576 , 579, 598Ob jective function, 670, 671 , 681Observed residua l, 626Obstruction, 315Odd polynom ial, 109OLS estim ator, 633ω-ass ignm ent, 2One-form , 409, 410One-p oint compactifi cation , 78, 210 , 312, 422

o f complex numbers , 78One-re lator group , 298One-sector neo classic growth model, 757, 758One-s ided convolution , 662One-s ided Fourier transformation , 657One-s ided surface, 255One-s ided test, 617One-ta iled test, 617One-to -one correspondence, 33One-to -one function , 29One-to -one relation, 14 , 29Open affi ne cover, 571Open and closed maps, 537Open ba ll, 194, 197, 199 , 216Open chain , 260Open cover, 208 , 551, 607Open covering, 193 , 208, 499Open curve, 214

Open disk , 71 , 194Open elem entary cha in , 260Open Eulerian chain, 260Open ha lf-ball, 230Open image, 466Open immersion , 567 , 571Open interva l, 194Open manifo ld , 542Open map, 201, 537Open mapping theorem , 241 , 372, 472 , 477, 482 , 537Open morphism , 565Open part, 216Open q-simplex, 294Open set, 50, 193, 195 , 198, 199 , 536, 674Open set ax ioms, 218Open set top ology, 536Open subschem e, 564Open ing angle , 184Operad, 525Operand , 512Operation table, 41, 515Operations in Gauss complex p lane, 76Operations in N, 52Operations in Z, 58Operations on ordinals, 49Operations on sets, 30Operator, 20, 372 , 375

compact, 375d iff erentiab le, 372d iff erential, 374 , 377fin ite rank , 375Fredholm , 376Herm itian integral, 377integra l, 374 , 377invertib le, 372linear, 372, 416

Operator algebra , 375Operator set, 510Operator sp ectrum , 24Operator variab le , 3Opposite, 40, 64

o f a complex numb er, 76o f an irrational, 66

Opposite category, 265 , 584Optim al m ix of exp lanatory variab les, 631 , 632Optim ality cond ition , 669Optim ization , xviii, 669

generalization, 671normal form , 671

Optim ization theory, 669Optim ization w ith two variab les, 670Optimum w ith constra int

examples, 681geom etric interpretation, 681

Optimum w ithout constra int, 680O rb it, 306O rb it space , 309O rb ita l stab ility, 719O rder, 159, 579O rder and orientation, 159O rder contact, 396O rder diagram , 45O rder fi lter, 23O rder ideal, 23O rder isomorphic, 47O rder of a group, 82O rder of a po lynom ial , 110O rder of a root, 112O rder of integration , 648O rder of integration of time series, 648O rder of multiplic ity, 134O rder of ram ifi cation , 146O rder of the zero, 118O rder re lation, 7, 9 , 44

cyc lic, 160strict, 44

O rder re lation on Q, 61O rder re lation on Z, 59O rder structure, 38 , 44 , 63—65

antisymmetry, 44refl exiv ity, 44transitiv ity, 44

O rder structure o f R, 321O rder typ e, 47

comparison , 48O rdered d irected set, 589O rdered fie ld , 38O rdered set, 38, 44

lex icographic pro duct, 49O rdering relation , 7 , 9O rd inal, 47 , 48

fin ite , 48infin ite, 48

O rd inal c lasse, 49O rd inal comparison , 49O rd inal numb er, 33, 48O rd inary d iff erential equations

second-order, 717O rd inary geom etry, 150O rd inary least squares (OLS), 626, 644O rd inary m etric pro jective p lane , 158O rd inary sequence, 322O rientab le d iff erentiab le manifo ld , 416O rientab le regu lar curve , 447O rientab le surface, 255 , 400O rientation , 159, 286 , 363, 446O rientation o f a Jordan arc , 393O rientation o f a p lane, 150, 399O rientation o f a surface, 399, 400O rientation o f ∆n, 286

802

Index

Orientation -preserv ing , 159, 162, 173 , 187, 740Orientation -preserv ing (revers ing), 697Orientation -preserv ing diffeomorph ism , 692Orientation -preserv ing homeomorph ism , 688Orientation -reversing , 173Oriented angle, 75Oriented graph , 259Oriented plane, 159Oriented plane as topolog ical space, 159Oriented regu lar path, 446Oriented simplicial complex, 295Orientibility, 399Origin o f a path, 279Origin o f edge of simplic ia l complex, 297Orthocenter, 167Orthodrom e, 179 , 180Orthogonal, 242Orthogonal affi nity, 168Orthogonal axonometric m ap , 174Orthogonal basis , 137Orthogonal collineation , 165Orthogonal coord inate system , 173Orthogonal curvilinear coordinates, 343Orthogonal group , 131, 187Orthogonal group On, 172, 191Orthogonal measure, 479Orthogonal of a part of a pre-H ilb ert space , 242Orthogonal pro jection , 174 , 243Orthogonal system , 191 , 243, 666Orthogonal vectors, 191Orthogonal wavelet, 666Orthogonality, 21, 158 , 242

according to a herm itian form , 136Orthogonality condition , 400Orthograph ic pro jection, 174Orthonormal algebra ic basis, 244Orthonormal basis, 137Orthonormal coordinate system , 172, 182Orthonormal part of pre-H ilb ert space, 137Orthonormal system , 191, 243Oscilla tory, 686Osculating circ le, 394, 396 , 398Osculating plane, 394Osculating sphere, 396Ostrogradski formula, 363, 501Ostrogradsky formula, 411Ostrogradsky-Stokes formula , 363Ostrogradsky-Stokes theorem , 363Ostrow ski, 145Output variab le , 685Outset, 707, 709Outward perturbation , 747Over-harmonic, 493Over-harmonic function, 493Overlapping of regu lar patches, 399OX -m odu le, 565

p-adic abso lute va lue, 145p-ad ic d istance, 220p-ad ic m etric, 79p-adic number, 67p-ad ic number, 78 , 79p-ad ic top ology, 79p-ad ic valuation , 79p-adm issible op en set, 306p-function, 434p-linear antisymmetric form , 139p-value (statistics), 617Pair category, 584Pairs of harmonic points, 172Pairw ise d isjoint equivalence classes , 28Pairw ise d isjoint subsets , 28Pairw ise relatively prime, 57Palis -Smale theorem , 724Panel analy sis , 633Panel data, 625 , 633Pappus-Pasca l theorem , 155Parabola, 185, 186Parabolic fixed po int, 747Parabolic geom etry, 149Parabolic p oint, 402, 403 , 419Paracompact space, 208 , 211, 538Paradoxical decomposition , 8Paradoxical set, 8Paradoxical set w ith resp ect to the group G , 8Para llel affi ne subspace, 190Para llel axonom etric pro jection , 174Para llel lines, 154Para llel p ostu late, 149Para llel pro jection , 168Para llelepip ed , 191Param etering, 445Param eters of the model, 626Param etric representation , 393, 397 , 398Param etrization

of a curve , 445Param etrized arc , 392Parseval equa lity, 653Parseval equation, 655Parseval formula , 244Part, 11

bounded , 209denumerable, 54finite, 54stab le by a map, 30

Partial auto correlation function , 643Partial correlation , 631Partial derivative, 342 , 343Partial d ifferentia l equation , 388Partial fraction decomposition , 335 , 434, 659

of a rationa l function , 434

of the cotangent, 434of the function gamma Γ, 434

Partial graph , 258Partially ordered group, 512Partially ordered set, 9, 21, 45 , 512, 579Partially ordered set (Poset), 21, 45 , 579Partially ordered set (p oset), 9, 45Particu lar fi rst-order differentia l equations, 379Partition , 8, 9Partition o f un ity, 411 , 481, 495 , 499Partitions and covers, 32Partitions of an interva l, 349Pascal relation, 55Pascal triang le, 333Pascal’s arithm etical triangle, 55Patch , 415, 542 , 687Patch of class Cr , 398Patch p oint, 402

ellip tic point, 402hyperb olic po int, 402parabo lic po int, 402planar po int, 402

Path , 196 , 204, 260 , 276, 392, 444 , 499, 574regu lar, 444

Path class , 279Path component, 276Path connected , 276Path connected space , 276Path lifting prop erty, 574Path of integration, 660Path-connected , 546Path-connected (0-connected), 546Path-connected set, 196Pathw ise connected set, 196Pathw ise connectedness, 546Pathw ise-connected , 196, 538, 539 , 545, 546Pathw ise-connected (path-connected, 0-connected), 546Pathw ise-connected space, 196Paving, 447Peak of frequency, 650Peano , 4Peano axiom s, 49

5th , 5, 49Peano existence theorem , 386Pearl curve, 638Peixoto theorem (I) (for vector fi e ld , fl ow), 704Peixoto theorem (II) (for vector fi eld, fl ow), 704Peixoto theorem (II I) (for diff eomorph ism), 705Pentagon d iagram , 586Percentage d istribution, 620Perfect fi eld , 125, 530Perfect group , 315Perfect Reinhardt domain , 441Perfect set, 369Period bands, 436Period para llelogram s, 436Period three implies chaos theorem , 733Period-doub ling (fl ip b ifurcation), 737Period-doub ling (fl ip) b ifurcation , 748Periodic , 714Periodic complex function , 435Periodic function , 435 , 436

complex, 435doub ly, 436simply, 436

Periodic non-autonomous system , 693Periodic orb it, 710Periodic po int, 687Periodic po int of an Anosov diff eomorph ism , 705Periodic signal, 726Periodic so lution o f a system , 717Periods o f complex function , 435Permutation , 613Permutation cycle, 86, 90Permutation group , 86Permutation w ith rep etitions, 614Permutation w ithout rep etition , 614Perp end icular b isector, 151 , 153, 161, 167Perron integra l, 370Persp ective, 156Persp ective co llineation, 156 , 170Perturbation, 685

inward , 747outward , 747

Pesin formula , 728Peta ls, 298P faffi an differentia l form , 408P faffi an form , 408P faffi an form space , 408Phase d iagram , 715Phase p lane, 715Phase portrait, 715, 716Phase portrait o f the flow , 690Phase sp ectra, 662Phase-lo cking , 742Phase-space, 715Phi function , 57Phillip s and Perron tests , 644Phillip s curve , 638Physical depreciation rate, 760Physical m easure, 725P icard, 428P icard iteration , 387P icard-L indelö f theorem , 714P iecew ise d iff erentiab le functions, 659P iecew ise linear map, 295P iecew ise monotonic function , 351P iecew ise regu lar boundary, 501

P iecew ise-Ck path , 444P iecew ise-Cr geometric arc , 392

803

Index

Pigeonhole princip le, 53P ing-p ong lemma, 518P itch fork b ifurcation, 736P itch fork b ifurcation (normal form ), 733P ivot, 670P ivota l co lumn, 669P ivota l row , 670P lanar affi nities, 168P lanar curve, 394P lanar geom etric pro jection , 173P lanar graph , 259 , 261P lane, 150, 159P lane curve, 214 , 395, 397P lane lam ina , 412P lateau function, 495 , 500P laton ic solid s, 296P lucker coordinates , 138Poincaré, 4 , 138Poincaré Bendixson theorem , 720Poincaré circle m odel, 176Poincaré con jecture, 295Poincaré cut, 721Poincaré formula , 413Poincaré ha lf-p lane model, 177Poincaré lemma, 448 , 498Poincaré map, 416, 692 , 693, 695 , 721

for autonomous d iff erential equations, 721for non-autonomous d ifferentia l equations, 721ro les , 721

Poincaré model, 176Poincaré recurrence theorem (Poincaré return theorem), 718Poincaré section , 717, 721, 722Poincaré theorem , 412Poincaré-Bendixson theorem , 690Poincaré-Brouwer theorem , 412Poincaré-Cartan invariant, 695Point, 150

accumulation , 195adherent, 195boundary, 195cluster, 195 , 322, 323 , 341, 423exterior, 195frontier, 195hetero clin ic, 703homoclin ic , 703interior, 195, 537isolated , 195lim it, 195o f equilib rium , 714stationary, 714

Point at infin ity, 157 , 422Point-set topology, 193Pointed deformation, 313Pointed homotopy axiom , 312Pointed homotopy category, 281Pointed map, 272Pointed pair, 314Pointed pair homotopy, 314Pointed pair m ap, 314Pointed space, 538Pointed top ologica l space, 538Poisson bracket, 698Poisson distribution , 617, 621Poisson formula, 486Poisson integral, 487Poisson kernel, 486, 488Poisson stab ility, 718Poisson theory, 426Polar, 156, 184Polar b low ing up , 701Polar coord inates, 76, 687Polar form , 136, 137Polar p lane, 184Polar topo logy, 377Polar triangle, 154 , 180Pole , 118, 152 , 156, 158 , 184, 428, 429 , 464, 465Polish space, 232Polyad ic , 511Polyd isk , 440Polygon , 168, 289, 669Polygon area , 168Polygonal chain, 254Polygonal group , 254Polygons equal by ad junction , 169Polygons equal by decomposition , 168Polyhedra, 250 , 253

as topo logical subspaces, 250Polyhedral Formula, 412Polyhedron , 250 , 251, 255 , 294, 296Polyhedron in (Rp,Rp), p 3, 251Polynom ial

Bernstein , 338Polynom ial degree, 120Polynom ial equivalence, 69Polynom ial function , 112, 333Polynom ial function of lags (A lmon’s lags), 640Polynom ial interp olation, 339Polynom ial m ap, 112Polynom ial ring, 69, 119 , 120Polynom ial ro ots, 77, 112Polynom ials

Chebyshev, 338Legendre, 339 , 340

Polytop e, 168 , 250, 251, 255 , 534Polytop e edges, 251, 255Polytop e vertices, 251, 255Popu lation , 616, 622Popu lation variance , 616Porter-Hudak, 651Portmanteau test, 643

Poset, 9, 45Position vector, 182Positive auto corre lation , 633Positive herm itian form , 137, 242

and M inkow ski inequality, 137and Schwarz inequa lity, 137

Positive homogeneity, 20Positive measure, 442Positive sca lability, 20Positively correlated variables, 625Positively invariant set, 690Posterior d istribution , 618Posterior probab ility, 615 , 618Power, 33, 34Power functions w ith rational exponents, 336Power ob ject, 580, 609Power ob ject (in topos), 609Power of the continuum , 37Power set, 8, 13Power sp ectrum , 726Power theorem , 167Power-law , 652Prandtl number, 726, 749Pre-H ilb ert space, 21 , 136, 137, 242 , 370, 371Pre-H ilb ert space (inner product space), 136Pre-image, 521Pre-inner pro duct, 137Pre-lim it, 266, 602Preadd itive category, 585Precompact m etric space, 229Precomposition , 365Predecessor, 49 , 51 , 322Pred icate, 1Pred icate (w ith 3 argum ents), 2Pred icate ca lcu lus, 2Pred icate ca lcu lus theorem , 2Pred icate ca lcu lus theorem s, 3Pred icate symbols, 511Pred icate variab le , 3Pred icates , 2Preim age , 29 , 53, 194 , 519—521 , 560Preim age (inverse im age), 520Prelim it, 602Preorder, 23Preordered set, 579, 582Preordering, 579 , 582Presentation of a group, 297Presentation of an abe lian group , 285Preshea f, 267 , 606Preshea f (algebraic top ology), 551Preshea f (category theory), 551Preshea f of groups, 267, 551Presheaves, 267 , 551Primary decomposition , 136Prime e lem ent, 142Prime fie ld , 125, 126Prime ideal, 22 , 100, 142, 143 , 269, 528Prime numb er, 56 , 120

decomposition , 147Prime sp ectrum , 529Prime subfield, 125Primeva l, 2Prim itive, 352 , 353Prim itive elem ent, 123Prim itive function , 349 , 352Prim itive integral, 349Prim itive of a c losed form along a path , 469Prim itive of a d ifferentia l form , 498Prim itive po lynom ial, 121Prim itive po lynom ial irreducib le, 121Prim itive root, 125Prim itive root of un ity, 73Principal ana lytic set, 560Principal argum ent, 74, 76Principal bund le, 268, 561Principal curvature, 403 , 420Principal determ ination, 455

o f argum ent, 455o f the logarithm , 455

Principal determ ination o f the argument, 74Principal d irection , 188, 189Principal G -bund le, 561Principal ideal, 22 , 101, 528, 529 , 560Principal ideal domain , 101 , 528Principal ideal ring, 101 , 528, 560Principal normal, 373 , 393, 398Principal normal vector, 394Principal order fi lter, 23Principal order ideal, 23Principal part, 430, 465Principal ring, 95, 101 , 142, 528Principal symbol, 389Principal va lue, 499Princip ia M athematica, 16Princip ia mathematica , 14Princip le o f analytic continuation , 457 , 486Princip le o f excluded m idd le, 4Princip le o f induction , 5Princip le o f iso lated zeros, 462Princip le o f recurrence, 5, 49Princip le o f transfin ite induction, 47Prior distribution , 618Prior probab ility, 615, 618Probability, xviii, 613, 615 , 618Probability ca lcu lation , xviii, 615Probability charges , 26Probability density, 620 , 634Probability density function (p.d.f ), 615 , 620Probability function, 620Probability generating function (p .g .f ), 616

804

Index

Probab ility mass function (p .m .f ), 615, 620Probab ility m easure, 615 , 618Probab ility m easures on attractors , 724Probab ility sampling , 615Probab ility space, 615Probab le error (probability deviation), 617Prob lem of enumeration , 53Pro cess

ARCH , 652ARFIMA, 651ARMA, 652DS (D ifference Stationary), 644FIGARCH, 652GARCH

integrated , 652IGARCH, 652long m emory, 650of fi rst order, 644TS (Trend Stationary), 644

Process o f fi rst order, 644Process w ith memory, 633 , 643Product, 38, 603Product category, 583, 584Product group , 82Product law , 40Product m etric space, 217Product of a fam ily o f complete normed spaces, 237Product of a fam ily o f norm ed vector spaces, 237Product of edge paths, 297Product of ob jects, 310Product set, 13Product space , 202, 237Product structure , 38Product topo logy, 66 , 202, 237Product un iform structure , 237Products of compact spaces, 224Programming

convex, 673 , 674discrete , 673linear, 673, 674mathematical, 673nonlinear, 674quadratic , 673sto chastic , 673

Pro j, 572Pro j as schem e, 572Pro j as set, 572Pro j as topo logical space, 572Pro jection, 310 , 561Pro jection morph ism , 603Pro jection of fib er space, 562Pro jection theorem , 21, 26Pro jection theorem s, 242Pro jection to normal plane, 395Pro jection to oscu lating p lane , 395Pro jection to rectify ing plane, 395Pro jective algebraic set, 513 , 532Pro jective bi-ratio , 176Pro jective ccorre lation , 158Pro jective collineation , 156, 158 , 171, 173

Pro jective collineation in P 2 (R), 173Pro jective correlation , 156Pro jective general linear group PGLn(q), 172, 191Pro jective geometry, 149, 156Pro jective group , 149Pro jective map, 170, 171Pro jective metric, 158Pro jective module, 554Pro jective morph ism , 572Pro jective n-space, 270Pro jective n-space on a ring, 572Pro jective Nullstellensatz theorem , 533Pro jective ob ject, 555, 556Pro jective p ersp ective, 170Pro jective plane, 155 , 157, 412Pro jective plane of incidence, 155Pro jective polarity, 152 , 158Pro jective resolution, 556Pro jective schem e, 513 , 564Pro jective transformation, 156Pro jective variety, 513 , 532Proklos or Pro clus , 13Proo f, 6

auxilia ry hypothesis, 6by construction, 6by contradiction, 6by contrapositive, 6by descent, 6by d isjunction of cases , 6by exhaustion, 7by mathematica l induction, 6by recurrence, 6by transposition , 6combinatorial, 7constructive, 6direct, 6dissection, 7elementary, 6nonconstructive, 6two-column, 6visua l, 7w ithout words, 7

Proo f by mathematical induction , 5Proo f of exc ision , 293Prop er class, 16, 577 , 578, 582 , 585, 595Prop er d iv isor, 120, 141Prop er face, 294Prop er idea l, 22 , 528Prop er idea l (for lattices), 23Prop er idea l p o int, 157

Prop er map, 544Proper morph ism , 565Proper op en set, 309Prop er subgroup , 83Prop er subset, 7, 13Prop erties o f a category, 256Prop erties o f a functor, 257Prop erties o f an index , 471

dep endence w ith resp ect to a po int, 471invariance by homotopy, 471

Properties o f estimators, 626Prop erties o f lim it va lues , 323, 325Prop erties o f Tor, 304Prop erties preserved by topo logy change, 218Prop erty P, 65 , 66Propositiona l and pred icate Calcu lus, 2Propositiona l calcu lus, 2Propositiona l calcu lus theorems, 1, 2Propositiona l formula, 1Propositions, 1Proxy variab le, 637Pseudo-convex, 441Pseudo-random numb er, 624Pseudo-random numb ers, 616 , 623Pseudo-R iemannian m etric, 409Pseudo-tensor, 408Pseudoconvex, 441Pseudoconvex domain , 441Pseudometric , 220, 230Pseudometrics and d istances, 234Pseudosphere, 177Pseudospherica l loxodrome, 180Pseudotensor, 408pth exterior power, 316Pullback, 266 , 305, 314 , 364, 498 , 604 , 605, 607Pullback (diff erential), 365Pullback (fiber product), 604Pullback (pullback d iagram , fiber product, fibre product, fib ered product,

fib er square, fibre square, cartesian square, pu llback square), 604Pullback as fibre-product, 365Punctured p lane, 78Puppe sequence , 313Puppe sequence theorem , 313Pure im aginary number, 69Pure rotation , 691, 692Pure set, 14 , 16, 578Pure set (hered itary set), 14, 16 , 578Pure s inusoidal function , 726Purely categorica l, 581Purely random process, 633Pushforward , 19Pushforward (d ifferentia l), 365Pushforward m easure, 19Pushout, 266 , 305, 535, 605Pushout (fib ered coproduct, fib ered sum , co cartesian square, pushout square,

amalgamed sum ), 605Pythagoras theorem , 242Pythagorean theorem , 169, 175, 178Pythagorean triang le, 93Pythagorean trip lets , 144

χ2-d istribution , 617

χ2-test, 617, 622q-cell, 297q-cob oundary, 558q-co cha in , 559q-simplex, 558q-ske leton, 295q-th Cech cohomology, 559qth relative s implicia l homology group , 296qth simplicia l homology group , 295Quadrang le, 158Quadratic form , 678

definite negative , 678definite positive, 678sem i-defin ite negative, 678sem i-defin ite p ositive, 678

Quadratic programm ing, 670Quadratic recipro city law , 144Quadratic recipro city theorem , 144Quadratic res idue , 144Quadratic surface, 186Quadratic trend , 650Quadric, 186Quadrics, 188

classes, 188coincident p lane, 188ellip so id , 188ellip tic cone, 188ellip tic cylinder, 188ellip tic parabo lo id , 188genera l equation

princ ipal directions, 188hyperb olic cy linder, 188hyperb olic parab oloid, 188hyperb oloid of 1 sheet; 2 sheet, 188intersectiong p lanes, 188parabo lic cylinder, 188parallel planes, 188

Quadrila teral, 154Quantifi er, 2Quantifi ers , 2Quantization, 2Quasi-coherent sheaf, 553Quasi-coherent sheaf of ideals, 571Quasi-compact morph ism , 566Quasi-compact part, 208Quasi-compact space, 208, 538Quasi-d iff erence transformation , 635Quasi-d iff erences , 635

805

Index

Quasi-fi nite morphism , 566Quasi-inverse functor, 596Quasi-isomorphism , 534, 592Quasi-pro jective variety, 533Quasi-random number, 624Quasi-separated morph ism , 565Quasi-tensor category, 587Quasip eriodic function , 718 , 726Quasip eriodic signal, 726Quasip eriodic so lution, 722Quasip eriodicity, 734Quasipro jective variety, 513Quasitensor category, 586Quaternary relation , 161Quatern ion, 35, 79Quatern ionic pro jective n-space, 299Quotient, 64 , 109Quotient category, 273Quotient complex, 290 , 298Quotient fi e ld , 38, 99, 117Quotient group , 38 , 88 , 505Quotient map, 574Quotient module, 38Quotient ob ject, 580Quotient o f the euclidean d iv ision of po lynom ials, 109Quotient re lation , 38Quotient ring, 38, 43, 100 , 143, 516, 526Quotient ring (factor ring, residue class ring), 526Quotient ring (factor ring, residue-class ring), 43Quotient set, 28, 34Quotient space, 201 , 275, 574 , 575Quotient space (identifi cation space), 574Quotient structure, 38Quotient topology, 201 , 255, 275 , 574, 575

R2-R2-function , 423

R2-R-function, 423R -algebra , 523R -algebra ic structure homomorph ism , 521R-analytic function , 486R -B instab ility, 750R -B phenomenon, 750R -B prob lem , 751R-integrability, 349R-integrability o f m onoton ic function , 351R-integrable function , 349—351R -module, 517, 521R -module homomorphism , 536R -module M , 517α-representatives, 708R/S ana lysis, 651R/S ana lysis (Hurst statistic ), 651Rössler m odel, 729Rademacher functions (square wave), 668Radial, 491Radian , 182Radical, 129 , 529Radical idea l, 529Radical of a p olynom ial, 115Radical of idea l, 529Radical or power center theorem , 167Radius, 150Radius of convergence, 328Radius of convergence , 333Radius of convergence of series, 426Radius of curvature, 177, 394, 396Radius vector, 182Radon measure, 442, 446Raising to a power, 77Ral money sto ck p er un it o f labor, 760Ram ifi cation po int, 433 , 437Random , 616 , 623Random error (or no ise), 616Random in design o f exp erim ents, 616Random number generator, 616, 623Random numbers , 616, 623Random process , 642Random sample, 615Random variab le , 616, 620Random vector, 616Random walk, 616, 623Random walk model, 644Random with a un iform distribution, 623Random ize, 616Range, 14, 29, 519Rank, 546Rank of a differentia l m ap , 546Rank of a pure set, 15, 578Rank of an abelian group, 286Ratio o f affi nity, 168Ratio o f sim ilarity, 165Ratio o f sim ilitude, 165Ratio : chord length/circle diam eter, 174Rationa l domain, 43 , 516Rationa l exp ectations, 761Rationa l fraction, 119Rationa l function, 119 , 333, 334Rationa l Numb er

ca lcu lation ru les, 64Rationa l number, 64, 65

a lgebra ic structure, 65defined by a lignm ents, 65negative, 65order structure, 65

defau lt, 65positive, 65topolog ica l structure, 65

defau lt, 66Rationa l numbers , 60Rationa l or irrational ratio , 722

Rationa l power function , 336Rationa l rotation , 691Rationa l rotation number

(frequency lo cking), 744Ratios of d istances, 169Ray, 152, 159Rayleigh number, 749Rayleigh -B énard instab ility, 726Rayleigh -B énard phenom enon , 749Rayleigh -B enard instab ility (R -B ), 741Real ana lys is, xvii, 193, 198, 321Real ana lytic function , 548Real balances , 761Real d iff erentiab ility, 424 , 549Real function , 324

a lgebra ic op erations and composition, 325rationa l op erations and composition , 326

Real line, 216 , 225not compact, 223

Real line bund le, 561Real number, 65 , 66

a lgebra ic structure, 66Real part, 70, 76Real part of z , 70Real pro jective n -space, 299Recipro ca l circular function, 337Recipro ca l hyperb olic function, 337Recipro cation, 157Rectangle axiom (R), 153Rectangu lar impulse, 658Rectangu lar tiling, 447Rectifi able curve, 358Rectifi able path , 499Rectifi able set, 358Rectifi cation , 130Rectify ing plane, 394Recurrence , 47 , 51Recurrence equation, 639Recurrence hypothesis, 5Recurrence re lation , 39, 51Recurrence theorem , 51Recurs ive, 5Recurs ive defi nition, 5Recurs ive model, 625, 642Recurs ive regression , 631Recurs ive residual, 631Reduced algebra, 525Reduced closed edge path , 297Reduced edge path , 297Reduced equations, 641Reduced form , 128Reduced homology, 292Reduced homology groups, 292Reduced ring, 525Reduced schem e, 513, 564Reduced simplicial homology group , 296Reductio ad absurdum , 6Reduction theorem , 297Refinem ent, 567Reflection , 151, 161 , 168Reflection w ith resp ect to a p lane, 164Reflection -rotation , 165Reflection -translation, 162 , 164Reflexive norm ed vector space, 239Reflexiv ity, 7Region having no holes, 545Regress ion on the fi rst differences, 639Regu la fa ls i, 340Regu lar

curve, 445path , 444

Regu lar arc, 393Regu lar attractor, 724Regu lar b oundary, 500Regu lar card inal, 15 , 578Regu lar category, 588Regu lar covering space, 307Regu lar curve, 445Regu lar domain , 356Regu lar e lem ent, 33, 40Regu lar epim orphism , 588, 599Regu lar geom etric arc, 393Regu lar lo ca l ring, 526Regu lar map, 531Regu lar matrix , 105Regu lar matrix (nonsingular matrix), 105Regu lar monomorphism , 599Regu lar n -sided po lygon , 84Regu lar patch , 398

ang le m easure, 400area m easure , 400curvature, 401isom etric, 400length m easure, 400m etric geom etry, 401

Regu lar patches, 392 , 398, 399overlapp ing , 399

Regu lar path , 444Regu lar ring , 526Regu lar space, 193 , 206, 207 , 221Regu lar subharmonic function , 491Regu lar submanifold , 544Regu lar submanifold prop erty, 544Regu lar surface, 359

exp licit representation, 359param etric representation , 359

Regu lar T1-space , 207Regu lar va lue, 133Regu larity, 40Regu larity o f the add ition , 52

806

Index

Regu larity of the addition and multiplication in N, 52Regu larity of the multip lication, 55Regu larity of the multp lication , 52Reinhardt domain , 421 , 441Relam inarization , 746Relation, 2, 297

left surjective, 29, 519less than or equa l to, 36un ivo ca l, 29, 519

Relation o f homotopy, 573Relation symbols, 511Relationa l signature, 511Relationa l sy stem , 510Relations, xvii, 27, 29, 38 , 681

compatible, 38cond itions of compatib ility, 38

Relations and structures, xviiRelations b etween co effi cients and roots (of a p olynom ia l), 113Relative cob omology group , 317Relative complement, 8, 12Relative frequency, 620 , 624Relative hom eomorph ism , 299, 300Relative homology group, 291Relative homotopy, 279Relative homotopy group , 314Relative Hurew icz theorem , 315Relative slmplicial homology group, 296Relative Sp ec, 572Relative tensor, 408Relative topo logy, 197, 538Relatively compact, 480Relatively compact set, 375 , 458, 480Relatively compact set (cond itiona lly compact set), 375Relatively prim e, 57 , 143, 144Relatively prim e (coprim e), 111Remainder, 109Remainder of the euclidean div is ion o f p olynom ia ls, 109Remarkab le identity, 110Removable singu larity, 464Renam ing, 37Repeating decim al, 68Representable functor, 267, 597, 600Representation, 580Representation (representing ob ject), 580

Representation of a curve in R2, 397explicit representation , 397implicit representation , 397param etric representation , 397

Representation of a diffeomorph ism , 691Representation of line, 183

equation w ith point and normal, 184equation w ith vector product, 184homograph ic equation , 184vector equation w ith point and d irection, 183vector equation w ith two po ints, 183

Representation theorem , 243Representation theory, 509Representative, 28Representing ob ject, 580Reproducing kernel, 461Residual, 23, 626Residual et , 626Residue, 119 , 144, 430, 476 , 506Residue class , 143Residue class ring, 526Residue classes, 100Residue fi eld, 270, 526Residue theorem , 430 , 476Residue-class ring, 43, 143Reso lution, 555, 556 , 558Reso lution (abstract algebra), 558Reso lution (homolog ical a lgebra), 556Reso luton , 553Reso luton of d iff erential sheaf, 553Resonance, 698Resonant condition , 700Resonant eigenvalues, 698Restoring force, 648Restricted homotopy, 573Restriction, 30 , 573Restriction o f a map, 29 , 573Restriction ru le , 329Retract, 249, 271 , 573Retraction , 36, 271 , 573Return force, 648Return to scale, 758Revealed preferences, 760Reversal of axis, 173Reverse homothety, 165Reynolds number, 727R icc i ca lcu lus, 407R icc i curvature, 419R icc i tensor, 409, 419R ichardson theorem , 3R iemann, 4R iemann criterion, 350R iemann curvature , 415R iemann curvature tensor, 409R iemann geom etry, 415R iemann integral, 349, 350, 355 , 368

of functions o f several variables, 355

on measurab le domains of R2 in the Jordan sense, 356R iemann integral transformation into curvilinear integra l, 363R iemann mapping theorem , 483R iemann space , 407, 415R iemann sphere, 78 , 415, 422, 423 , 482R iemann sum , 350, 358 , 360R iemann surface , 78 , 415, 421, 423 , 431

abstract, 432concrete, 432

obtained by compactifi cation, 78of logarithm function , 432of sphere w ith handle, 432of square root function , 432topo logical structure , 432

R iemann tensor, 409, 415R iemann zeta function , 338R iemann-Christoff el curvature tensor, 409R iemann-Christoff el tensor, 409R iemann-integrab le function , 350, 368R iemann-Stieltjes integra l, 369R iemannian connection, 418R iemannian geometry, 413, 415R iemannian manifold , 407, 415 , 418, 540R iemannian m etric, 407 , 408, 418R iemannian space, 415R iemannian structure, 415, 708R ienhardt domain, 441R iesz decomposition theorem , 492R iesz representation theorem , 21 , 375, 442R iesz theorem , 375R ight ad joint, 589R ight ad joint functor, 595R ight angled triangle, 178R ight cancellable elem ent, 33R ight closed monoida l category, 587R ight derived functor, 556 , 557R ight ideal, 22R ight R -module M , 517R ight resolution, 556R ight resolution (coreso lution, reso lution), 558R ight triang le, 175 , 178R ight zero morph ism , 599R ight-exact functor, 595R ight-inverse, 32R ight-inverse elem ent, 42, 515R ight-invertib le , 32R ight-neutral elem ent, 32 , 42, 515R ig id motion , 161, 163R ig id ob ject, 588R ig id transformation, 161, 163R ig id transformation (rigid motion), 161

R ig id transformation in R3 , 164R ing, 38 , 42 , 81 , 92, 516

of integers , 63of po lynom ials, 69w ithout d iv isor o f zero, 43 , 516

R ing , 522R ing homomorph ism , 94 , 99, 521R ing homomorph ism (ring morphism ), 94R ing isomorph ism , 94R ing morph ism , 94R ing of congruence classes, 100R ing of fractions, 528R ing of integers, 63, 64

calcu lation ru les , 64R ing of integers modu lo n, 96R ing of matrices (n,n), 43 , 516R ing of polynom ia ls, 43 , 516R ing sp ectrum , 529R ing theory, 81 , 514R ing un it, 526R ing w ith a one, 522R ing w ith identity, 43, 516 , 522R ing w ith un ity, 43, 516 , 522R inged space, 513, 552R itt a lgebra , 527Rolle theorem , 331, 332Roof, 601Root extraction, 128Root nth of a complex number, 77Root of un ity, 73 , 131Root of un ity (de Moivre number), 73Roots o f a po lynom ial, 112, 121Roots o f a po lynom ial o f degree n , 121Roots or zeros o f a po lynom ial, 112Rose, 298Rose (bouquet of circles), 298Rotation, 41, 157, 162 , 164, 515

of a regu lar po lygon , 41, 515Rotation group, 41, 131 , 160, 191, 515Rotation number, 691, 734 , 740, 743Rotation of a vector fi eld , 413Rotation of ang le , 173Rotationa l, 347Rotations, 157Rouché theorem , 477, 504Rough set, 16Routh-Hurw itz conditions, 753Row sto chastic matrix , 623RSA (cryptosy stem ), 96Ruelle, 733Ruelle-Takens theory (R -T), 727 , 741Ruelle-Takens-Newhouse scenario (R -T -N), 727 , 739, 741Rule of contrap osition , 2Rule of m odus barbara, 2Rule of m odus ponens, 2Rule of m odus tollens, 2Ruled surface , 404Rules o f distributiv ity, 2Rules o f in ference, 1Runge, 479Runge approximation theorem , 479Runge theorem , 478Runge-Kutta method , 388Russel, 13Russell, 8, 14Russell paradox, 4Russell paradoxical set, 8Russell’s antinomy, 8

807

Index

Russe ll’s paradox, 8, 13, 14RVAR (Retricted Vector AutoRegressive), 647

σ-a lgebra, 18, 618σ-compact space, 212σ-fi e ld , 18, 618Sadd le, 717Sadd le connection , 712 , 746

hetero clin ic, 747homoclin ic , 747

Sadd le connection b ifurcation , 712Sadd le po int, 381, 673 , 694, 717Sadd le-connection , 710Sadd le-connection b ifurcation , 746Sadd le-node , 701, 748Sadd le-node (fold) b ifurcation , 748Sadd le-node and transcritical b ifurcations, 734Sadd le-node b ifurcation , 711, 748Sadd le-node b ifurcation (normal form), 733Sadd le-node p eriod ic orb it, 749Sadd le-node to a p eriodic-orb it, 738Sadd le-p o int, 710, 760Sadd le-typ e diff eomorph ism , 696Sam e cardinality, 33Sam e homotopy typ e , 274Sam e orientation, 286Sample , 615, 622Sample space, 615Sample variance , 622Sampling distribution , 615Sampling w ithout rep lacem ent, 615Sampling w itth rep lacem ent, 615Samuelson oscillator, 751—753SARIMA process, 645Satisfi able, 2Saturated set, 575Savings rate, 760SBR measure , 725SC criterion, 632, 647SC(p), 647Sca lar, 20Sca lar curvature, 409 , 419, 420Sca lar curvature of a R iemannian manifold, 420Sca lar density, 408Sca lar multip lication , 181Sca lar product, 21 , 182, 183 , 244

Sca lar product on the space l2(K), 244Sca lar square, 137Sca le parameter, 666Sca ling factor, 666Schauder, 370Schematization of homotopy groups, 250Schem e, 563Schem es, 269Schm idt orthogonalization pro cess, 243—245Schoenfl ies theorem , 215Schröder-B ernstein theorem , 34Schwartz, 391Schwartz Laurent, 668Schwartz theorem , 467Schwarz criterion , 632Schwarz criterion (SC), 640Schwarz function , 632, 640Schwarz inequality, 191, 242Schwarz lemma, 467Schwarz refl ection princ ip le , 461Scipion de l Ferro formula , 74Screw , 165Seasonal ad justm ent, 645Secant, 163, 175Secant of a circle , 163Second axiom of countab ility, 538, 543Second category, 232, 724Second countable topo logical space, 199Second countable topo logy, 199Second de Rham cohomology vector space, 502Second fundamenta l form , 401

patch theory, 401Second order diff erential equation, 381Second sample central moment, 622Second-axiom of countab ility, 212Second-countable, 211 , 538, 539 , 541, 542Second-countable basis, 211Second-countable space , 212, 543Second-order predicate calculus, 3Section , 563Section of a fiber bund le, 563Section of Po incaré , 722Segment, 67Segment d iv ision , 166Seifert, 412Seifert con jecture, 412Seifert-van Kampen theorem , 297 , 298Selection , 19Selector, 19Self-ad joint op erator, 243Self-con jugate subgroup, 599Self-homologous p oint, 166Self-s im ilarity, 652, 731Semantic antinomy, 13Semantic m etho d, 2Semantic or syntactic m ethod , 2Sem i-continuous, 349Sem i-continuous function

on Baire space, 233Sem i-cub ic parabola, 395Sem i-d istance, 220 , 230Sem i-m etric space, 227, 230

separated , 230Sem i-norm , 20, 236Sem i-simple algebra , 494, 530

Sem iadditive category, 585Sem igroup , 38 , 42, 62, 515 , 516, 520 , 522

o f natura l mumbers , 62Sem igroup homomorphism , 521Sem ilo ca lly 1-connected space , 308Sem ilo ca lly simply connected, 308Sem ilo ca lly simply connected space (sem ilo ca lly 1-connected space), 308Sem ima jor axis , 185Sem imetric, 230Sem im inor axis, 185Sensitive dep endence on initia l cond itions (SDIC ), 730, 732Sentential calcu lus, 2Sentential formula, 1Sentential variab le, 1Separable, 24, 211 , 542Separable algebra , 530Separable closure, 530Separable elem ent, 530Separable extension , 530Separable morphism , 599Separable po lynom ial, 113, 125 , 530Separable space , 24Separated morph ism , 565Separated parts , 202Separated preshea f, 608Separated schem e, 564Separated space, 538Separated space (Hausdorff space, T 2-space), 538Separately holomorph ic function , 494Separates po ints , 20Separating hyperplane theorem , 26Separation axiom , 193

T0 (Kolmogorov separation axiom ), 207T1, 207T2 (Hausdorff separation axiom ), 207T3 (regu larity axiom ), 207T4 (normality axiom), 207

Separation axioms, 205—207T0,T1,T2,T3,T4, 206 , 207

Separation o f convex sets, 676Separation o f p oints, 474Separatrices, 747Separatrix , 747Separatrix lo ops, 721Separatrix surface, 720 , 723Sequence, 30, 322

convergent, 67equ iva lent to zero, 66in a m etric space , 221interleave, 216monotone increasing , 66o f Cauchy, 66, 67

Sequence convergence, 204Sequence of complex numbers , 751Sequence of holomorph ic functions, 458, 477Sequential evolution law , 752Sequential space, 212Sequentially complete space , 231Series, 323Series expansion , 332, 426Series in norm ed vector spaces , 240Series of functions, 328Series w ith general term , 240Serration, 559Serre fib er space, 314Serre fib er space (weak fibration), 314Serre theorem , 314Sesquilinear form , 136 , 242Set, 7

at most denumerable, 35closed , 195connected, 196denumerab le , 35fin ite or infinite, 34, 53frontier, 195non-denumerable, 35op en , 50, 195totally ordered , 44

Set algebra , 8Set closure, 196Set d iff erence, 8, 11, 13Set d iff erences , 520Set d irect product, 13Set inclusion, 13Set m embersh ip , 10Set of accumulation po int, 195Set of adherent po ints (c losure), 195Set of a ll hereditarily fin ite sets, 578Set of a ll homogeneous prime ideals, 270Set of a ll homotopy classes o f p ointed pair maps, 314Set of a ll parts of, 7Set of a ll rotations

o f a regu lar n-sided po lygon , 42, 516Set of a ll sets, 13, 14Set of a ll the sets, 8Set of antecedents , 53 , 560Set of anti-herm itian matrices, 132Set of antisymmetric m atrices, 132Set of axiom s, 5Set of defin ition , 119 , 422

Set of d irect isom etries: E+(2), 163Set of d istributions, 496Set of ε-neighborho ods, 200Set of frontier p oints, 195Set of generators, 102Set of ind iv iduals , 3Set of inference rules (deductive reason ing), 5Set of interior points, 195Set of invertib le elem ents o f a ring, 93Set of m a jorants, 46

808

Index

Set o f null-trace matrices, 132Set o f parts of a set, 7, 31Set o f prim e ideals, 269Set o f sesqu ilinear form s, 136Set o f the parts o f, 14Set o f theorem s, 5Set o f topo logically equ iva lent p oints, 193Set o f topo logically non-equ ivalent po ints, 194Set theory, xvii, 7 , 12Set topology, 193Set w ith one element, 579Set w ith regu lar boundary, 450Set-theoretic d iff erence , 8, 12Set-theoretic topology, 193Sets and a lgebra ic structures , 81 , 514Sharkovskii, 733Sharkovskii ordering , 733Sharkovskii theorem , 733Sharkovskii theorem (1964), 733Sharkovsky, 733Sharkovsky theorem , 733Shea f, 153, 267, 551 , 606, 607Shea f (general topo logy), 551Shea f asso ciated to a presheaf, 268Shea f axiom s, 551 , 606Shea f cohomology, 558 , 570Shea f cohomology by Cech cohomology, 558Shea f cohomology by derived functor, 558Shea f of germ s, 513Shea f of germ s of ho lomorphic functions, 560Shea f of lines, 152 , 153, 157Shea f of p lanes, 152Shea f of rings, 551, 552Shea f on a site , 608Sheafification , 571 , 607Sheafification (sheaving), 607Shear, 168Sheaves, 267Sheaving , 607Sheet, 78 , 306Sheff er, 1Shem es, 269Sh ift op erator, 644Sh ifted endogenous variab le, 625Sh ifting parameter, 666Sh iln ikov theorem , 738Sh iln ikov-Turaev, 748 , 749Sh iln ikov-Turaev-Gavrilov , 748Sh ina i, 733Sh ina i theorem , 733Shock, 644, 647Short exact sequence, 554 , 605Short m emory pro cess, 650Short term Fourier transformation (STFT), 666Shortest path between two points, 373Shrinking , 249Shrinking of a closed curve to a point, 197Sierp inski carp et, 731Sierp inski gasket, 731Sieve, 607, 608Sieve o f E ratosthenes, 147Sigma algebra, 18, 618Sigma field , 18 , 618Signal analysis, 665Signal reconstruction , 665Signal synthesis, 665Signature, 138 , 510, 511, 513Signature (d isambiguation), 511Signature (in logic), 511Signature (of a knot), 512Signature (of a manifold), 512Signature (of a m etric), 512Signature (of a p ermutation), 511Signature (of a prime), 511Signature (of a structure), 511Signature (of a symmetric or Herm itian matrix), 511Signature (of an a lgebra ic system ), 511Signature (of quadratic form), 511Signature (typ e), 510Signature of a p ermutation , 87, 105Signature of a transposition , 87Sim ilar circles, 166Sim ilar figures , 165Sim ilar triang les, 166Sim ilarity, 149 , 165, 173, 193

direct, 452Sim ilarity (sim ilitude), 165Sim ilarity in geom etry, 165Sim ilarity typ e, 511Sim ilitude, 165Sim ilitude center, 166Sim ilitude group , 149Simple algebra , 530Simple auto correlation function, 643Simple cha in , 260Simple closed curve, 392Simple convergence, 234Simple curve, 214Simple dipath , 259Simple extension , 123Simple graph , 259Simple isolated singular po int, 428Simple isolated singularity, 433Simple iterative m etho d, 340Simple linear correlation co effi cient, 625Simple order, 45Simple rad ical extension, 129Simple regression model, 626

forecasting , 627geometric interpretation , 628

S imple root, 112, 121S imple root o f a polynom ia l, 121S implex, 190 , 251, 255, 277 , 534S implex algorithm , 669S implex m ethod, 669, 671S implex of vertices, 252S implex tab leau , 669S implexes , 251, 255 , 277S implexes in (Rp,Rp), p 3, 251S implices, 251 , 255, 277S implicia l approxim ation , 253 , 295S implicia l approxim ation theorem , 295S implicia l cohomology group, 319S implicia l complex, 250, 251, 255 , 294, 534S implicia l complex in (Rn,Rn), 253S implicia l homology, 251, 255, 257 , 263, 295 , 550S implicia l homology functor, 597S implicia l homology group , 295S implicia l k-chain , 550S implicia l m ap , 253, 295S implicia l Mapping , 251S implicia l m app ing, 263 , 550S implicia l sub complex, 535S implicia l sub complexes , 251, 255S implifi cation rule, 63S imply connected , 196 , 422, 545S imply connected (1-connected), 545S imply connected domain, 425S imply connected open set, 470S imply connected region , 197 , 202, 425 , 545S imply connected region (1-connected reg ion), 545S imply connected space , 197, 202 , 249, 282 , 545S imply ordered set, 21, 44S imply p eriod ic function, 436S impson (approxim ation) m ethod , 355S im s, 647S imultaneous congruences , 143S imultaneous equations model (SEM ), 641

just-identifi ed, 642over-identifi ed , 642under-identifi ed, 642

S inai-Bowen-Ruelle (SBR), 725S ine, 175S ine function , 174S ingle un it, 28S ingle-va lued function , 431S ingly p eriod ic function , 436S ingu lar card inal, 15, 578S ingu lar complex, 286S ingu lar homology, 263, 549 , 550S ingu lar m etric pro jective plane, 158S ingu lar n -co cha in , 317S ingu lar n -simplex, 286S ingu lar p oint, 428S ingu lar p oint o f algebraic curve, 336

cusp , 336isolated point, 336node, 336

S ingu lar p oint, index of a , 412S ingu lar solution , 381S ingu lar theory, 287S ingu lar va lues, 727S ingu lar va lues o f a matrix , 727S ingu larities of ana lytic functions, 433S ingu larity, 421

cusp , 702degenerate , 699hyperb olic, 699non-degenerate, 699non-hyp erbolic, 699of a complex function , 429of ana lytic functions, 433of holomorph ic functions, 428

S ingu larity o f ho lomorphic functions, 428S ink, 348, 717S inusoidal function w ith harmonics , 726S ite, 569 , 607, 608Skeleton , 254 , 260, 295 , 535Skew-symmetric, 138Skew-symmetrization , 138Skewness test, 643S lack variab le, 669S later regu larity condition, 673, 674S lice category, 585Smale theorem , 738Smale-B irkhoff theorem , 709Small étale site , 568Small category, 302, 568 , 582, 585Small circle, 179Small class, 16, 577, 578 , 585Small colim it, 585Small in itial variation, 717Small lim it, 585Small p erturbations, 702Small sets, 16, 577Smallest convex set, 250, 252 , 255Smallest elem ent, 9Smash product, 312Smash product o f pointed space, 312Sm ith normal form , 296Sm ith normal form theorem , 296Smooth embedding, 544 , 545Smooth function, 416, 546Smooth functions on manifolds, 546Smooth manifold, 540 , 541Smooth map, 546Smooth map (smooth function), 546Smooth maps b etween manifo lds, 547Smooth morph ism , 566Smooth schem e, 564

809

Index

Smooth structure , 541Smooth surface, 415Smooth unram ifi ed morph ism , 569Smooth variety, 569Smooth ly universal, 544Snake lemma, 554Sob olev space, 391Soh , 175Solow model (Pascal distribution), 641Solow -Swan model, 753 , 758Solution , 314

p eriod ic , 717stable, 686

Solvab le group, 89Solv ing equation ∂u/∂z = v, 480Sort, 512Sotomayor, 711Source, 348, 598 , 600, 717South p ole, 271Space

Baire , 232compact, 223complete, 230contractile , 249contractile arcw ise connected, 249Hausdorff , 199H ilb ert, 199

L2([a, b],K), 244

l2(K), 244

L2 , 392Lp, 392mapping (function), 233m etric, 50

complete, 67, 215 , 228o f 2π-p eriodic functions, 245o f Banach , 371o f Hausdorff , 199o f H ilb ert, 371pre-H ilb ert, 370sem i-metric

separated , 230sem im etric , 230Sobo lev, 391T2, 199topolog ica l, 50

tota lly disconnected, 226uniform

complete, 231Space average , 725Space Cn, 440Space d iagona l, 299

Space L2([a, b],K), 244

Space l2(K), 244Space o f 2π-p eriodic functions, 245Space o f a ll distributions, 496Space o f fi nite typ e , 318Space w ith no holes, 545Span (hu ll), 250 , 255Span„ 601Spec , 531Spec (as a set), 529Spec (as a shea f ), 529Spec (as a top ologica l space), 529Spec(R), 529Spec(Z), 529, 572Spec ia l chain , 506Spec ia l linear group , 131 , 417Spec ia l linear group SLn(q), 417Spec ia l linear group SLn(R), 417Spec ia l orthogonal group SOn, 172 , 191Spec ia l orthogonal group SOn(q), 191Spec ia l un itary group , 131Spectral energy, 654Spectral interpretation of Fourier transform , 662Spectral radius, 494Spectral radius formula , 494Spectral theorems, 24Spectral topology, 529Spectral value, 133Spectrum , 24, 269 , 466, 494 , 662

o f ap eriod ic signal, 727o f chaotic signal, 727o f p eriod ic signa l, 726

Spectrum of a b ounded operator, 24Spectrum of a ring , 270, 529Spectrum of function w ith f1/f2 irrational, 726Spectrum of function w ith f1/f2 rational, 726Spectrum of p eriod ic s igna l, 726Spectrum of po lynom ial ring, 270Spectrum of quasip eriod ic signa l

o f quasip eriod ic signa l, 726Speculation effect, 760Speed of convergence, 340Sphere , 184, 539 , 708Sphere model, 179Sphere o f curvature , 396Sphere theorem , 420Sphere w ith hand le, 432Spherica l arc, 396Spherica l geom etry, 179Spherica l p o int, 419Spherica l triang le , 180Sp inning diff eomorph ism , 709Sp inor, 360Sp iral, 717Sp iral source , 717Sp lit exact sequence, 303, 554Sp lit m onomorphism (coretraction), 598Sp lit p olynom ial, 113Sp litting fi eld, 124

Spring-mass system , 377O scillatory motion, 377

Spurious regression , 648Square matrix , 109Square matrix of order n , 109Square wave, 668Square-integrable p eriod ic functions, 245Squaring the circle, 75SS (sum of squares), 627SSE (sum of squares exp la ined), 627Ssingu larity, 441SSR (sum of squares of res iduals), 627SST (tota l sum of squares), 627Stab ility

asymptotically stable, 719Kolmogorov (invariant tori preservation in Ham iltonian), 718Lagrange stability, 718Lyapunov stab ility, 718orbital stab ility, 719Poisson stab ility, 718Structura l stability, 718

Stab ility Boundaries, 748Stab ility in the sense of Lyapunov, 757Stab ility of p eriod ic orbits, 719Stab ility of steady states , 719Stab ility prob lem of a p eriod ic solution , 717Stab ility test by recursive regress ion , 631Stab ility tests , 631Stab ility theory, 718Stab ilizer, 306Stab le

add itive ly, 40multip licative ly, 40

Stab le (unstable) eigenspace , 697Stab le fixed p oint

in the sense o f Lyapunov, 687Stab le in the sense of Lyapunov, 693Stab le part, 32Stab le part by a map, 30Stab le part for a law , 40Stage (or domain o f variation), 581Stage of defin ition , 581Stagew ise regress ion, 633Stalk , 267 , 552, 607Stalk (of a preshea f ), 552Stalk (of a shea f ), 552 , 607Standard étale, 569Standard étale morphism , 569Standard deviation , 616 , 621Standard function , 499Standard n-simplex, 286standard n-simplex, 271Standard normal d istribution , 617Standard Zarisk i cover, 567Star, 152, 294Star convex , 196Star-shap ed open set, 498State variable, 684 , 685State -space model, 684Static m odel, 648Stationarity, 643Stationarity tests

augm ented D ickey-Fu ller test (ADF), 644D ickey-Fuller test (DF), 644

Stationary Gaussian pro cess, 651Stationary series

(i.i.d .), 643(n .i.d .), 643w ith m emory, 643

Statistico f Jarque-Bera , 643

Statistical distribution , 620Statistical induction , 625Statistical in ference, 625Statistical methods, 622Statistical models, xviii, 624Statistical test, 629Statistics, xviii, 615Steady state, 714Step function , 350, 658

lower-b ound , 350upper bound, 350

Stepw ise regress ion, 633Stereograph ic pro jection , 78, 179 , 271, 422 , 482, 687Stereograph ic retraction , 312Stieltjes integral, 369, 375Stieltjes transformation, 657Stirling formula , 90Stirling numb er, 90Sto chastic, 643Sto chastic assumptions, 628Sto chastic m atrix , 616Sto chastic optim ization , 670Sto chastic pro cess, 616 , 623, 642Sto chastic programm ing , xviii, 669, 674Sto chastic variable, 616Stokes and Ostrogradsky formulas, 138Stokes formula, 411, 450

proof, 452Stokes formula for the rectangle, 411, 448Stokes theorem , 316, 363 , 411

closed form , 364exact form , 364general form , 363

Stokes theorem (in five d imension), 696Stokes theorem applications to tubular reg ion, 696Stone-Weierstrass theorem , 236 , 245Straight line, 150Straightedge, 129Straightedge and compass, 129

810

Index

Strange attractor, 724 , 730Strange attractor (Ruelle-Takens), 733Strange attractor in the sense of Ruelle-Takens, 733Strange attractors , 732Stranger, 111Stratifi ed sampling, 615Strict subset, 7Strictly concave function, 678Strictly convex function , 679strictly less than , 44Strong deformation retract, 301, 573Strong deformation retraction , 301Strong extrema, 373Strong induction , 51Strong law of large numbers (Kolmogorov), 620Strong recurrence, 51Strong Stable Manifold , 748Strong ly d ivergent sequence, 322Strong ly inaccessible card ina l, 15, 578Structural assumptions, 628Structural equations, 641Structural form of VAR representation , 646Structural stab ility, 702 , 703, 718Structural stab ility (robustness), 723Structural stab ility of linear system s, 703Structural stab ility of the lo cking, 744Structurally stable d iff erential equation, 723Structurally stable vector fi eld, 703Structure, 38, 511 , 512

algebraic, 81topo logical, 50

Structure (algebraic system ), 511Structure homomorph ism , 520Structure map, 593Structure morphism , 563Structure of lags , 640

Structure of Q, 67Structure of the set of rea ls , 38

Structure of the un iform convergence on FE , 234Structure sheaf, 270Structure sheaf , 552Structure-preserv ing bijection , 258Structure-preserv ing map, 545Structures, xvii, 37Structures (logic), 512Structures on R, 321Student law , 625 , 629Student ratio, 629Student t-d istribution, 617Student’s tab le, 625Sub-m ean-va lue prop erty, 466Subadditiv ity, 20Subalgebra, 375Subarc, 448Subbasis of a topology, 200Subcategory, 257 , 265, 272 , 584Subcomplex, 290 , 535Subcritical b ifurcations of fixed points, 734Subfield , 38, 69, 99Subgraph , 258Subgroup , 38, 83, 599Subgroup generated by a part, 83Subgroup generated by a set, 83Subgroup of Z, 59Subharmonic, 493Subharmonic b ifurcation, 718Subharmonic d istribution , 492Subharmonic function, 490, 491Subharmonic instab ility, 734Subharmonicity, 492Sub lattice , 22Sub linear function , 20Sub linear functional, 20Submanifo ld , 543Submanifo ld atlas, 544Submanifo ld prop erty, 544Submodule , 38 , 101Subob ject, 580Subob ject class ifi er, 581, 608Subproposition , 1Subring, 38, 93 , 99Subroup, 83Subscript notation, 30Subsequence, 221 , 322Subset, 7 , 11Subset relation , 13Subspace topology, 538Subspace topology (re lative topo logy, induced topo logy, trace topology), 538Substitution princ ip le, 289Substructure , 38 , 99Substructures, 99Substructures (subring , subfield), 99Subsystem of re lations, 38Successive derivatives, 330Successive integrations, 356Successive trials, 619Successor, 15 , 47, 48, 51, 322 , 578Suffi cient condition, 5Suffi cient condition (SC), 5Sum , 38Sum and product of complex numbers , 70Sum of sub complexes , 290Sum topo logical space, 202Sum topo logy, 202Summable fam ily, 240Supercritical Andronov -Hopf b ifurcation , 749Supergraph , 258Superp osab ility, 29 , 161Supplem entary ang les, 162Support, 40 , 108, 287

Support o f a distribution , 496Support o f a fam ily, 40Support o f a map, 443Support o f a m easure, 442Support o f distributions, 668Support o f i-cha in , 506Supremum , 45 , 46Surface, 255 , 398, 399

hyperb olic R iemann, 415lo ca l param etric representation , 414lo ca lly orientab le, 400of R iemann, 415Orientab ility and topolog ical invariant, 256orientab le, 400smooth , 415

Surface group , 298Surface integral, 360 , 362, 411Surface of absolute values of complex functions, 423Surface of revolution , 188 , 360, 402

ellip so id of revolution , 188hyperb oloid of revolution , 188parabo lo id of revolution , 188Quadrics of revolution , 188

Surface w ith boundary, 255

Surfaces in R3, 414Surgery, 373Surjection, 520Surjective embedd ing (isomorph ism ), 520Surjective homomorphism , 520Surjective homomorphism (ep im orph ism), 520Surjective map, 29 , 31 , 519Susp ension , 311, 692 , 693Susp ension o f a diff eomorph ism , 693Syllogism rule, 2Sylvester formulas , 618Symbol, 1Symbol

, 347 , 429

Symbol sequences , 707Symbolic dynam ics for horsesho e diff eo., 707Symbolic log ic , xviiSymmetric, 40Symmetric bilinear form , 406Symmetric bra ided monoida l category, 587Symmetric diff erence, 13, 33, 93Symmetric group , 86Symmetric monoida l category, 587, 588Symmetric sesquilinear form , 242Symmetrizab le, 40, 82Symmetry, 162Symplectic d iffeomorphism , 695Symplectic m anifold , 695Synchron ization of frequencies , 741Synchron ization of oscillators, 743Syntactic method , 2 , 3σ-system (Σ-system , Ω-system ), 511System of class representatives, 28System of cogenerators, 581System of congruences, 97System of d iff erential equations, 383, 684System of equations, 106System of fi rst-order d iff erential equations, 384System of generators , 581System of h igher-order d iff erential equations, 384System of linear equations, 107System of linear fi rst-order d iff . equations, 384System of propositiona l formulas (axioms), 2System of relations, 38System of structural equations, 641System of typ es, 14

T-algebra , 593t-distribution, 617T-p eriod ic , 714T0,T1,T2,T3,T4-axiom s, 206T0,T1,T2-axiom s, 206T1-space, 247T2-axiom , 199, 206T2-space, 199 , 206, 247T2-space (Hausdorff space), 207T3-axiom , 207T3-space (regular space), 207T4-axiom , 207Table of indefin ite integrals, 353Table of integration ru les, 356Table of prim itives, 353Table of random numbers , 624Table of variance ana lys is, 630Tables o f Fourier transformation , 661Tables o f random numbers, 623Takens, 712, 713 , 733Takens theorem , 416Tangent, 175 , 393Tangent angle, 163Tangent developab le, 396Tangent developab le of a curve , 396Tangent hyperp lane, 343Tangent linear map, 416Tangent of a c ircle, 163Tangent p lane, 399Tangent prob lem , 328 , 329Tangent space , 414—416, 689 , 708Tangent vector, 415 , 416Tangent vector to a surface, 399Tangential component, 401Tangle o f curves, 721Target, 600Tarski’s axiom , 16Tarski-G rothend ieck set theory (TG), 15Tauto logical propositional formula, 2Tauto logy, 1Taylor expansion , 701

811

Index

Taylor formula , 114Taylor p olynom ial, 332Taylor p olynom ial theorem , 332Taylor remainder, 332

Cauchy form , 332Lagrange form , 332

Taylor series , 333, 426

T chebychev (Chebyshev, T cheb icheff , C eby sev , Tschebyscheff , Cheb ishev),75 , 339

Tear, 194Tensor, 105 , 138, 405—407 , 415

relative, 408skew-symmetric r-tim es contravariant, 138

Tensor algebra , 406, 407Tensor ana lysis, 407 , 408Tensor bund le, 408Tensor ca lcu lus, 407Tensor category, 586Tensor contraction , 407 , 410Tensor density, 408

even , 408odd, 408

Tensor d iff erentiation , 407Tensor fi eld, 406—408Tensor of typ e (0, 1), 406Tensor of typ e (0, 2), 406Tensor of typ e (1, 1), 406Tensor of typ e (p, q), 406Tensor on a vector space, 408Tensor powers, 525Tensor powers and bra id ing , 525Tensor product, 407 , 586Tensor quantity, 407Tensor set, 407Tensor space, 407Tensor space of typ e (α, β), 407Tent map, 725Term of degree k, 108Term of degree n , 290Term inal morph ism , 580Term inal ob ject, 310, 580Term inal ob ject , 580Term inal prop erty, 580Ternary relation , 28 , 160, 161Test

Chow , 635Kurtosis (flatten ing), 643normality, 643Skewness (asymmetry ), 643

Test function, 668Tetragon , 154Tetrahedron , 251, 255Thales , 163Theorem

group of affi ne co llineations: G, 172Abel, 129about angles, 162acyclic free chain complex, 303add ition , product, quotient ru les, 330A fraim ovich-Shiln ikov, 739A lembert, 224A lembert-Gauss, 77A lexander-W hitney, 319A lexandroff , 225Andronov-Hopf, 735Andronov-Pontryag in , 724Andronov-W itt, 719anti-m atching , 153anticommutativ ity, 319any Euclidean ring is principal, 142any fin ite extension is algebra ic, 124Asco li, 236, 442Asco li (for a m etric space), 236Asco li (general case), 236aureum , 144Bézout, 60Bézout (for po lynom ials), 111Baire category, 232 , 724Banach, 241 , 391Banach fixed point, 372Banach-A laoglu , 442Banach-Steinhaus, 241 , 371Banach-Tarski paradox, 17basis of a vector space, 103Bayes, 619Bessel inequality, 244, 653b inom ial, 90 , 92b iquadratic recipro city, 145B irkhoff , 718B irkhoff (ergod ic), 725B irkhoff HSP, 514B irkhoff f, 514b isectors , 153B lakers-Massey, 315B lo ck-Guckenheim er-M isiuriew icz, 733Bolzano-Weierstrass, 322 , 323Bo olean prime ideal (BPI), 19 , 23, 24Borsuk, 473Borsuk-U lam , 294 , 320Brouwer, 214, 472Brouwer and Kakutan i fixed-point, 674Brouwer fixed p oint, 215, 677C is a lgebraically closed, 115Cantor, 31, 228Cantor-Schröder-Bernste in theorem , 36Carathéodory, 485Casorati-Weierstrass, 464Cauchy, 456Cauchy formula (homology form ), 475Cauchy formula (homotopy form ), 471

Cauchy integra l, 425 , 429Cauchy integration criterion , 355Cauchy theorem (homology vers ion), 475Cauchy theorem (homotopy vers ion), 470Cauchy-Goursat, 453, 461Cauchy-Kowalewska, 389Cauchy-Pomp eiu formula, 456Cayley-Ham ilton , 134center manifo ld , 700central lim it, 617Chinese, 97Chinese remainder, 97chordal, 167circle quadrature , 131closed graph theorem , 241, 372commutative group for composition law , 156compact of Rn, 197compact set, 209compact supp ort, 288compactness criterion o f m etrizab le space, 227completion o f a m etric space, 229Complex Hahn-Banach, 25composition rule, 330congruence, 163connected im age , 203connected space, 203constructib ility by straightedge-compass, 130convergence criterion, 340covering homotopy theorem , 306curl, 348, 412cyc lic decomposition , 136degree, 122Denjoy, 692derivab ility, 442Desargues (affi ne vers ion), 155Desargues (little), 157Desargues (pro jective vers ion), 155D iagona lization cond itions, 134d im ension axiom , 287 , 312, 317D in i, 234d ivergence, 364, 412d ivergence (Gauss), 348Douady-Oesterlé, 731Douady-Oesterlé (for diff . equations), 731dual un iversal co effi cients, 318E ilenb erg-Steenrod, 302E ilenb erg-Steenrod (extension), 302E ilenb erg-Z ilb er, 305entire series expansion and Taylor series, 333Euclid , 56 , 169Euclid lemma, 57, 60Euclid lemma in Z, 60Euclidean div is ion, 55, 59Euclidean div is ion in Z, 59Euler, 96, 167Euler criterion , 144Euler d isplacement, 186Euler-Lagrange, 83, 84exact triang le , 291exc is ion , 296 , 317extension, 228extreme va lue, 209factorization , 57Fatou , 488Fermat, 89, 93, 96Fermat (little), 89Fermat’s last, 93 , 96Fermat’s little, 96fi eld by two ideals, 101fi eld o f complex numb ers is a lgebraically closed, 115fi eld w ith characteristic zero, 125fin ite Abelian group , 89fin ite s imple extension, 123fin ite s imple extension (Stein itz), 123five lemma, 291fixed-po int, 229fixed-po int w ith param eter, 229F loquet, 699, 716flow-box, 691Fredholm , 374, 390Freudenthal susp ension , 315Frob en ius, 417Fundamental theorem of A lgebra , 77fundamental theorem of arithm etic, 57fundamental theorem of arithm etic (for po lynom ials), 111fundamental theorem of integral calcu lus, 352Gauss, 57, 60, 348Gauss (in Z), 60genus, 145G iraud, 609golden (Gauss), 144grad ient, 343, 412G ranger representation , 649G reen, 316 , 348, 364G reen-R iemann, 363G robman-Hartman, 723G robman-Hartman (for discrete syst.), 723G rothend ieck, 567 , 570groupoid , 280Gysin sequence, 320Hadamard three circles , 491Hadamard-Cartan & Gromoll-M eyer, 420Hadamard-Perron, 720Hadamard-Perron (for d iscrete system ), 723Hahn-Banach , 24 , 45 , 236, 238Hahn-Banach (geom etric form ), 26Hahn-Banach separation , 26hairy ba ll, 294, 412ham -sandw ich , 320Harnack inequalities, 486

812

Index

Hartman-G robman, 697Hartman-G robman (for d iff eomorphism ), 697Hartman-G robman (for flow), 697Hartogs, 494heights , 153Heine, 223, 224homo clinic sadd le connection, 747homomorphism , 88homothetic circles, 166homothetic triangles, 166homotopy axiom , 288, 317homotopy axiom for pa irs, 292homotopy extension, 301homotopy sequence o f a fibration , 314Homotopy sequence o f a pair, 314homotopy sequence o f a weak fibration , 315Hopf, 320Hopf b ifurcation, 712Hopf recurrence, 718HSP, 514Hurew icz, 289 , 315Hurew icz (1), 289Hurew icz (2), 315Hurew icz (abso lute version), 315Hurew icz (relative version), 315Hurw itz, 477immersion , 213insets and invariant manifold , 709integral domain by prim e ideal, 100integral m ean-va lue, 351interm ed iate value , 203, 226 , 326invariance o f d imension, 294invariance o f domain , 294invariance o f integral by homotopy, 470invariant manifold, 697invariant manifold (for d iff eomorphism ), 697invariant manifold (for flow ), 697inverse function, 390inverse mapping, 568irreducib le p olynom ia l, 121irreducib le prime e lem ent, 142isomorphism , 89Jensen formula , 463, 492Johnson , 494Jordan , 473, 474Jordan canon ical form , 136Jordan curve, 214 , 294, 504Jordan-Brouwer, 214Jordan-Brouwer separation , 294Künneth formula for cohomology, 318Künneth theorem , 305Kakutan i (fixed po int), 677Kelv in -Stokes, 364Kolmogorov-A rnold-Moser (KAM), 724Kondrakov, 391Kuhn-Tucker, 673 , 682Kuratowski, 263Kuratowski-Pontryag in , 263L’Huilie r, 180Löwenheim -Sko lem , 2Lagrange , 83 , 86Lagrange interpo lation po lynom ial, 112Lagrangian (genera lized), 681least upper b ound , 66Leb esgue d iff erentiation , 488Ledrapp ier, 731Lefschetz, 304Lefschetz fixed po int, 304Leray, 296lifting criterion , 307lifting lemma, 306lim it, 620linear extension , 103linear map and matrix asso c iated, 104Liouville , 77 , 463, 465 , 695, 716Liouville (vector version), 465Liouville b oundedness, 433Littlewood , 493long exact sequence, 317Lustern ik-Schn ire lm ann , 294Mac Lane coherence , 586, 587Malgrange, 711Malgrange-Ehrenpreis , 669Mather, 705Mayer-V ietoris , 292Mayer-V ietoris (for reduced homology), 292mean va lue, 331mean va lue inequality (for po lynom ials), 115medians, 153Menger, 262Menger n-arc theorem , 262Mergelyan , 479metrization, 210Meusn ier, 402M ittag-Leffer, 481modifi ed Jordan form , 758monic irreducible po lynom ial, 121monodromy, 427Monte l, 458Morera , 461Nagata-Sm irnov (m etrization), 212Neimark-Sacker, 737no-retraction , 472non constructib ility (straightedge comp.), 130normal form (d iff erential equation), 698numeration in base b , 56of well-ordering , 17op en mapping, 241 , 372, 472 , 477, 482, 537Ostrogradsky-Stokes, 363Ostrow ski, 145

Palis -Smale, 724Pappus-Pasca l (affi ne vers ion), 155Pappus-Pasca l (pro jective vers ion), 155Parseval formula , 244particu lar irreducible polynom ia l (Abel), 121Peano existence , 386Peixoto (I) (for vector fi e ld , fl ow), 704Peixoto (I I) (for vector fi eld, fl ow ), 704Peixoto (I I I) (for diffeomorph ism ), 705p eriod three implies chaos, 733p erp end icular b isector, 153p erp end iculars (H jelm slev), 152, 153Picard-L inde lö f, 714Poincaré, 412Poincaré Bend ix son , 720Poincaré recurrence, 718Poincaré recurrence (Poincaré return), 718Poincaré return , 718Poincaré-Bend ix son , 690Poincaré-Brouwer, 412polynom ia l d iv ision , 120power, 167prim ary decomposition, 136pro jection, 21, 26, 242pro jection on a complete convex set, 242pro jection on a complete vector subspace, 243Pro jective Nullstellensatz, 533propositional calcu lus , 1 , 2Puppe sequence , 313Pythagoras, 242Pythagorean , 169, 175 , 178quadratic recipro city, 144quadratic recipro city law , 144quadratic residue, 144rad ical or p ower center, 167recurrence, 51reduction , 297regu lar T1-space, 207representation, 243residue, 430 , 476restriction ru le , 329R icc i, 409R iemann con formal representation , 439R iemann criterion, 350R iemann mapping , 483R iemannian geom etry (fundamental th .), 418R iesz, 375R iesz decomposition , 492R iesz representation, 21, 375, 442Rolle, 331, 332roots and zeros of a po lynom ial, 112roots o f a polynom ia l, 112roots o f a polynom ia l o f degree n , 121Rouché, 477 , 504Runge , 478, 479Schoenfl ies , 215Schwartz, 467Se ifert-van Kamp en, 297, 298separability criterion for a po lynom ial, 125separating hyperplane, 26Serre, 314Sharkovskii, 733Sharkovskii -1964, 733Sharkovsky, 733Sh iln ikov, 738Sh ina i, 733signature of a p ermutation, 87signature of a transposition , 87sim ilar triang les , 166simple and multip le roots o f a po lynom ial, 112simplicial approximation, 295Smale, 738Smale-B irkhoff , 709Sm irnow -Nagata-B ing (fundamental th .), 211Sm ith normal form , 296sphere, 420Stokes, 316, 363 , 411Stokes (c losed form ), 364Stokes (exact form), 364Stokes (general form ), 363Stone-Weierstrass , 236Stone-Weierstrass , 245Stone-Weierstrass (for complex functions), 236Takens, 416Taylor polynom ia l, 332Thales, 163theory of patches (fundamenta l th .), 405three c ircles, 491T ietze, 297T ietze-U rysohn , 442T ikhonov, 209tota l probab ility, 619transport o f interva ls , 326Turan , 262 , 263Tychonoff , 22, 45, 209 , 224, 579Tychonov, 224undecidability, 3un ique factorization , 57un ique p olynom ial asso ciated w ith an algebraic element, 122un ique-factorization , 142un iqueness, 386un iqueness of the norm , 494un isolvence (for polynom ia l interpolations), 113un iversa l co effi cients for cohomology, 318un iversa l co effi cients for homology, 304Urysohn, 207, 211Urysohn (m etrization), 212van Kampen , 297 , 298variation o f constants formula, 716vector space norm , 371

813

Index

Vesentin i, 494Wedderburn , 114Weierstrass, 481Weierstrass convergence, 458Weierstrass product, 433Whitehead, 315Whitney, 415 , 416Whitney graph , 259W iener-Khinch in , 726W ilson, 98Yoneda lemma, 302Young (1), 731Young (2), 731Zerm elo, 36, 47zeros of a p olynom ia l, 112Zorn , 45, 47

Theorem of homothetic triang les , 166Theorem of sim ilar triang les , 166Theorem s ab out angles, 162Theory

o f approximation, 338o f cointegration, 626o f d ifferentia l equations, xv iio f interpo lation, 339

Theory of analytic functions, 421Theory of curves , 214Theory of d iff erential equations, xviiTheory of patches, 404

fundamental theorem , 404Theory of simplic ial homology, 257Theory of singu lar homology, 257Theory of typ es, 13, 16Three c ircles theorem , 491T i-space, 207T ietze theorem , 297T ietze -U rysohn theorem , 442T ikhonov space, 207T ikhonov theorem , 209T ile, 209, 355T iling, 447T im e average, 725T im e continuous system , 714T im e d iscrete system , 714T im e series, 642T im e series ana lys is (TSA), 642T im e-dep endent normal form , 699T im e-lag model, 625 , 639T im e-series, 624T im e-series cross-sectiona l data (TSCS), 633T ivialization o f line bund le on a torus, 269, 562Toa, 175Tob in model, 753Tob in monetary grow th model, 760Tolerance threshold , 622Topol. equ iva lence replaces top . con jugacy, 696Topological étale site, 610Topological and continuous invariants, 195Topological basis, 199, 308, 543

prop erties , 199Topological completion , 78Topological con jugacy, 692 , 696Topological con jugacy of d iscrete systems, 723Topological direct sum , 243Topological dual, 238Topological embedd ing , 544, 575Topological entropy, 727Topological equ ivalence, 416 , 696, 722 , 733Topological equ ivalence of d iff . equations, 722Topological graph , 258Topological group , 38, 66, 283Topological hom eomorph ism , 200Topological invariance , 248Topological invariance o f dim ension, 540Topological invariant, 193, 195Topological isomorphism , 200Topological K -theory, 269 , 562Topological linear space , 20Topological man ifo ld , 539, 540Topological man ifo ld are lo cally compact, 540Topological man ifo lds are paracompact, 540Topological map , 193, 194 , 200Topological n-d im ensional manifo ld , 541Topological n-m an ifold, 539Topological quotient space, 38Topological space , 27 , 50 , 197, 198 , 217 , 537

axiomatic defin ition , 198comparison , 200defined in term s of neighb orhoods, 197defined in term s of op en sets , 198example, 198in term s of neighb orhoods, 537in term s of op en sets, 537T0,T1,T2,T3,T4-axiom s (separation), 207tota lly disconnected, 226triangu lable, 253

Topological space in term s of neighb orhoods, 537Topological space in term s of op en sets , 537Topological structure , 38 , 50 , 65, 218

transfer, 201Topological structure o f Q, 65Topo logical structure o f R, 321Topological subspace, 38, 200 , 201Topological theorem of complex numbers, 78Topological transitiv ity (TT), 732Topological vector space, 20 , 38Topologically con jugate, 697, 723Topologically con jugate d iff eomorphism s, 690Topologically con jugate hyp erbolic linear diffeomorph ism s, 697Topologically equivalent, 194, 197 , 697, 722Topologically equivalent d istances , 217

Top ologica lly equivalent flow s, 692Top ologica lly equivalent sets, 194Top ologist’s sine curve , 204Top ology, xvii, 20, 27, 193 , 218

asso c iated w ith sem i-m etric structure, 230comparison

coarsest, finest, coarser, fi ner, 199induced , 199m etric, 199p-ad ic, 79relative , 197

Top ology change, 733Top ology engendered by maps, 201Top ology generated , 199Top ology generated by a fam ily of parts , 218Top ology of bands, 199 , 204Top ology of the complex plane, 472 , 497Top ology on a set, 538Top os, 569 , 608Tor, 304 , 560Tor group, 304Torsion, 304 , 394Torsion co effi cient, 257 , 296Torsion formula, 395Torsion group , 257Torsion product, 304Torsion shea f, 571Torsion tensor, 418Torus, 275 , 402

Torus T2 , 730Tota l curvature, 403Tota l degree , 117Tota l order, 9, 21, 36, 44, 46Tota l order relation , 34Tota l order structure of N, 63Tota l orthonormal fam ily, 244Tota l orthonormal system , 244, 245Tota l probab ility, 619Tota l probab ility theorem , 619Tota l space, 306, 560, 562 , 574Tota l variab ility (SST), 627Tota lity, 21, 44 , 579Tota lly disconnected top olog ica l space, 226Tota lly E-acyclic, 303Tota lly ordered set, 9 , 21 , 44—46Tota lly ordered set (Total order), 44Tota lly well ordered field

A rch imedean , 78Totient function , 57, 68, 96Trace , 201, 304 , 717Trace map, 432Trace topo logy, 538Traces, 194Tractrix , 177Tractro id , 177Transaction eff ect, 760Transcendanta l extension , 123Transcendenta l elem ents of a fi e ld , 122Transcendenta l function, 433Transcendenta l number, 78, 122Transcendenta l numbers , 117Transcritical b ifurcation (normal form ), 733Transcritical b ifurcations, 735Transfin ite induction , 47 , 49Transformation

ellip tic, 439hyp erbolic, 439loxodrom ic, 439parabolic , 439

Transformation group , 149Transformation matrix , 758Transition function, 414, 542Transition map, 414 , 542Transition map (transition function), 542Transition matrix , 623Transition probab ilities , 623Transitions to Chaos, 738

cascade o f p eriod-doub lings, 738destruction of a torus, 738, 739G lobal homoclinic bifurcation , 738interm ittency, 738

Transitive G -set, 306Transitive G -space, 306Transitiv ity, 7Transitiv ity ru le, 2Translation , 154, 161, 162 , 164Translation functor, 591Transm ission medium , 661Transp ose, 106Transp ose con jugate matrix , 136Transp osition, 86Transvection , 168Transversa l, 162Transverse homoclin ic p oint, 709, 723 , 738Trap ezium , 355Trap ezium (approxim ation) m ethod , 355Trap ezoidal formula , 656Tree, 260 , 297, 309Trees , 619Trefo il curvE , 336Trend, 650Tria l, 616Triang le , 154, 175Triang le axiom , 20Triang le centroid, 167Triang le o f signed Stirling numbers of the fi rst k ind , 90Triangu lable topo logical space, 253Triangu lar matrix , 135Triangu lar s implex, 255Triangu lar system , 642

814

Index

Triangu lated category, 591Triangu lated p olygon, 298Triangu lation, 250, 251 , 253, 255 , 294, 295Trigona lization , 135Trigonometric circle, 72Trigonometric function, 175, 178 , 455

cosecant, 178cosine, 178cotangent, 178hyperbo lic cos ine, 178hyperbo lic sine , 178hyperbo lic tangent, 178secant, 178sine, 178tangent, 178

Trigonometric identities, 178Trigonometric series , 653Trigonometrica l function, 174Trigonometry, 174

based on a circ le, 178based on a hyperbo la , 178

Trim etric pro jection , 174Triple, 593Triv ia l bund le, 269, 561Triv ia l d iv isor, 95 , 110Triv ia l group , 82, 89

Ablian , 89cyclic, 89

Triv ia l homomorph ism , 282Triv ia l m orph ism , 85Triv ia l ring , 523Triv ia l subgroup , 83Triv ia l top ology, 198Triv ia lization , 269 , 561True co effi cient, 625True proposition , 1Truncated Taylor series, 685Truncation , 312, 701Truth value, 1 , 2TS pro cess, 644TS pro cess (Trend Stationary), 644Tube lemma, 299Turan graph , 262Turan theorem , 262 , 263Turbulence burst, 718Turbulent phase, 738Turing, 3Twin prim es, 148Twist, 165Twisted K-theory, 562Twisted sphere, 155Two point compactifi cation o f R, 422Two-cycle, 86Two-sided convo lution, 662Two-sided idea l, 22Two-sided idea l (b ilatera l ideal), 100Two-sided test, 617Two-sided transformation , 656Two-stage least squares (2SLS), 642Two-tailed test, 617Two-value princip le, 1Two-valued logic, 1Two-valued principle, 4Tychonoff space, 207Tychonoff theorem , 22, 45, 209 , 224, 579Tychonov theorem , 224Typ e, 13, 510—512Typ e (sort), 512Typ e I error, 617Typ e II error, 617Typ e of algebra ic sy stem , 510Typ e theory, 13, 16Typ es of affi ne isom etry, 187Typ es of equilibrium point, 721Typical prop erty, 724

U-set, 579U-sm all, 579U-sm all set, 579Ultra-m etric d istance, 220Ultrafi lter, 19, 22, 209 , 222, 223, 543Ultrafi lter lemma, 19 , 24Ultram etric, 146Umbilic, 402Umbilic point, 402Umbilical p oint, 402 , 419Unary operation , 16, 265 , 510Unbiased estim ator, 617, 622 , 626Unbiased forcasting, 627Uncorrelated errors, 626Uncorrelated variables, 625Uncountable neglig ib le set, 367Uncountable product, 217Uncountable universe, 578Uncountably infin ite set, 18Undecidability theorem , 3Undecidable hypothesis , 35Underly ing set, 510 , 512Underly ing set (carrier set), 510Underly ing set (or domain), 512Underly ing space , 294Unfo ld ing, 711Uniform continu ity, 227, 327 , 424Uniform convergence , 234, 327Uniform convergence o f a sequence, 234Uniform medium , 661Uniform norm , 21 , 338Uniform norm (Chebyshev norm), 338Uniform space, 230Uniform space (Weil), 230Uniform structure , 230

Uniform topo logy, 230Uniform ity, 230Uniform ly approachable map, 338Uniform ly continuous function , 327Uniform ly continuous map, 227Uniform ly convergent sequence o f functions, 327Uniform ly equivalent distances, 217, 228Union , 8, 9, 13Union of op en balls , 50 , 199Union of op en sets, 195Union of regu lar patches, 399Unipolar rectangu lar impulse, 662Unique factorization theorem , 57Unique-factorization domain , 124Unique-factorization domain (UFD), 142Unique-factorization ring , 124, 142Unique-factorization theorem , 142Uniqueness of d istributive lattice, 9Uniqueness of the lim it, 206Uniqueness theorem , 386Unisolvence theorem , 113Unisolvence theorem (for polynom ia l interpo lations), 113Unit, 43 , 141, 516, 593Unit c ircle, 159 , 214, 717 , 748Unit c ircle of C, 72Unit e lem ent, 42 , 43 , 515, 516Unit jump, 658Unit law , 591Unit n -sphere, 539Unit ob ject, 586Unit p olynom ia l, 36, 109Unit ring, 43, 516 , 522Unit ring (ring w ith identity, ring w ith un ity), 522Unit ring (ring w ith identity, ring w ith un ity, ring w ith a one, unitary ring ,

un ita l ring), 522Unit root, 644Unit root test, 644Unit square, 65Unit tangent vector, 393 , 394Unit vector, 181, 399Unital Banach a lgebra, 24Unital ring, 43, 516Unitary d iv isor, 147Unitary group , 131Unitary po lynom ial, 36 , 109Unitary ring, 43 , 516, 522Unitary tile, 250Unitor, 586Unity, 42, 515Universal a lgebra, 264 , 509, 510Universal co effi cients , 304Universal co effi cients theorem for cohomology, 318Universal co effi cients theorem for homology, 304Universal cover, 574Universal covering space, 307 , 574Universal e lem ent, 9Universal m orph ism , 581Universal prop erties of d iagram s, 589Universal prop erty, 63 , 216, 231 , 302, 580 , 581

of completions, 216Universal prop erty of the emb edd ing , 91Universal quantifi er, 10Universal set, 8, 13 , 511Universally catenary ring, 525Universally closed morphism , 565Universally true , 2Universe U, 577Univo ca l relation , 29, 519Unknown error, 626Unram ifi ed morph ism , 565, 569Unsatisfi ab le formula , 2Unstab le in the sense of Lyapunov , 694Unstab le separatrix , 747Unstab le statistica l m ode l, 631Upper (R iemann) integral, 350Upper b ound , 45, 46, 53, 490Upper b ox-counting d im ension , 730Upper capacity, 730Upper envelop e , 230Upper integral, 350Upper lim it o f sequence, 323Upper p ointw ise d im ension , 731Upper sem i-continuous function, 233Upper sem i-continuous function (u .s.c.), 490Upper set, 23Upper triangu lar matrix , 109, 135Ure lem ent, 2, 16, 578Urysohn , 207, 211 , 213Urysohn metrization theorem , 212Usua l topology of Cn, 224Usua l topology of Rn, 224Utility maxim ization under budget constra int, 681UVAR (unrestricted vector AR), 647

V-valued presheaf, 606Valeur d ’adherence , 220, 228Valuation, 79, 142, 145

discrete or dense, 145Non-A rch imedean , 146

Valuation of a polynom ia l, 110Valuation of n at p , 57Valued field, 79, 146Van der Pol equation , 735Van der Pol oscilla tor, 747Van der Pol system , 747Van Kampen theorem , 297 , 298Vandermonde matrix , 113Van ishing line , 170Van ishing point, 170VAR (vector autoregressive), 646VAR model, 646

815

Index

VAR process, 646VAR(p), 646Variab ility of res idua ls (SSR), 627Variab le, 1

dep endent, 685dummy, 2free (in log ic), 2functorial, 3input, 685linked (in logic), 2output, 685propositiona l, 1sentenc ia l, 1state, 685

Variab le elem ent, 581Variab le to b e exp la ined , 626Variance, 617, 621Variance ana lys is, 630Variance decomposition , 647Variance o f the error, 626Variance-covariance matrix , 633Variate , 616Variation of constants formula, 716Variation of the argum ent along a path , 471Variational derivative, 372Variational equation , 719Variational method , 390Variety, 264, 513 , 529Variety (overview), 513Variety in a category, 264 , 593Variety o f algebras, 513Variety o f un iversa l a lgebras, 264 , 513, 593VARMA, 646VARMA (vector autoreg . moving average), 646Vect-enriched (symmetric) m onoida l category, 586Vector, 150, 161, 181Vector addition, 181Vector angles, 75Vector basis , 182Vector bundle, 268 , 269, 561Vector density, 408Vector equation o f polar p lane, 184Vector equation w ith 2 points, 183Vector equation w ith 3 points, 184Vector equation w ith point and d irection, 183 , 184Vector error correction model (VECM), 649Vector fi eld , 294 , 410, 411Vector function , 341Vector isom etry, 187Vector Liouville theorem , 465Vector of b its, 32Vector op erator, 343Vector product, 140, 182 , 183

anticommutativ ity law , 183d istributiv ity law , 183m ixed asso ciativ ity law , 183prop erties , 183

Vector resolute, 182Vector space, 20 , 38 , 43 , 44, 81, 101 , 103, 370 , 517

d im ension , 103norm ed , 370number o f elem ent, 103

Vector-va lued function , 341Vectoria l rotation, 175Vectoria l structure, 190Vehu lst curve, 638Venn (or Euler) representation , 7Venn d iagram , 7Versal un folding , 711Vertex ang le, 162Vertex connectiv ity, 262Vertex cut, 262Vertex m ethod , 669Vertex set, 294Vertical composition, 264 , 585Vesentin i theorem , 494V incu lum , 67, 68V io lation o f assumptions, 633V isua l exam ination of residua ls

v isual exam ination of res idua ls , 633VMA (vector moving average), 646VMA representation , 647Volume calculation , 357Volume formula , 357von Neumann hierarchy of sets , 14 , 578von Neumann integer, 37von Neumann integer zero, 37von Neumann method , 37von Neumann ordinals, 37von Neumann regu lar ring , 526von Neumann regu lar ring (absolutely flat ring), 526von Neumann universe, 14, 578von Neumann-Bernays-Göde l set theory (NBG), 15

Walras, 677Walras law , 677 , 759Walras model, 753Walrasian equ ilibrium , 759Walsh functions, 667Walsh systems, 667Wavelet fam ily, 666Wavelet moment, 666Wavelet order, 666Wavelet transformation , 665, 666Wavelets , 665Weak embedd ing, 544Weak extrema, 373Weak fibration, 314Weak fibration (Serre fib er space), 314Weak law of large numbers (Bernouilli), 620Weak topo logy, 300 , 535

Weak∗ convergence, 488Weakly embedded submanifo ld , 544Weakly inaccessible card ina l, 15, 578Wedderburn theorem , 114Wedge, 296Wedge of po inted spaces, 296Wedge product, 138, 139 , 497Wedge product (exterior product), 139Weierstrass approxim ation, 338Weierstrass convergence theorem , 458Weierstrass ellip tic function, 435Weierstrass ellip tic functions, 435Weierstrass eta -function , 435Weierstrass formula , 338Weierstrass functions, 435Weierstrass p -function , 435Weierstrass ℘-functions, 435Weierstrass product theorem , 433Weierstrass s igma-function , 434, 435Weierstrass theorem , 481Weierstrass zeta-function , 435Weight o f the topolog ica l space, 537, 543Weingarten formulas, 404Weingarten map, 419Weingarten surface, 419 , 420Well order, 36 , 46Well ordered set, 36 , 46Well ordering, 36, 46Well-founded order, 22 , 46, 579Well-ordered set, 22 , 579Well-ordering princip le, 46Well-ordering princip le (equ ivalent to axiom of choice), 22, 579Well-ordering theorem , 17Well-p owered category, 585Wey l lemma, 492White noise, 633, 644 , 651White noise pro cess, 643White ’s test, 636White -noise tests, 643Whitehead, 14Whitehead theorem , 315Whitney graph theorem , 259Whitney theorem , 415Whole number, xvii, 51 , 62W iener pro cess, 651W iener-Khinch in theorem , 726W ilson theorem , 98W ind ing numb er, 430W indow function , 666W indowed Fourier transformation (WFT), 666W intner-C onti condition, 716W ith no d iv isors o f zero , 43 , 516W ith no zero d iv isors, 43, 516Wolf, goat, cabbage prob lem , 614Word, 254word, 519Work done, 360Work-energy princip le, 360Wronskian, 382 , 383

Xet , 569

Y -scheme, 563Yoneda embedd ing, 612Yoneda Lemma, 267 , 600Yoneda lemma, 302 , 612Young theorem (1), 731Young theorem (2), 731

Z , 16z-p lane, 71z-transformable sequence, 664Z-transformation , 664Zarisk i cover, 567Zarisk i topo logy, 270, 529 , 530, 533 , 567Zarisk i topo logy on Pnk , 533Zerm elo and Zorn theorem s, 37Zerm elo axiom of choice, 14Zerm elo set theory, 15, 16Zerm elo theorem , 17 , 34 , 36Zerm elo-Fraenkel axiom s, 16, 22, 579Zerm elo-Fraenkel set theory, 15, 16 , 578Zero , 118Zero chain map, 290Zero complex, 290Zero element, 42 , 43 , 515, 516Zero inner product, 21Zero map, 556Zero morph ism , 599Zero ob ject, 310, 580Zero ob ject (nu ll ob ject), 580Zero vector, 521Zero-lo cus, 532Zeros and lo ca l extrema, 334Zeros of a p olynom ia l, 112Zeros of holomorph ic functions, 462Zeroth Betti number, 502Zeroth de Rham cohomology vector space, 502ZF, 15 , 16ZFC, 15, 16, 578ZFC axiom s, 17Zorn lemma, 14, 22, 25, 36, 238, 579Zorn theorem , 45

816