Index

Thierry VIALAR

HANDBOOK OFMATHEMATICS

Over a thousand Illustrations

2016

Index

A-basis, 427A-module, 426Ab-category, 792Ab-enriched (symmetric) monoidal category, 792Abel lemma, 163, 614Abel theorem, 174Abel-Poisson, 661Abelian category, 791, 799Abelian group, 56—58, 110, 114, 119, 120, 138, 643,

645, 700, 701, 707, 791, 798, 810Abelian integral, 591, 592Abelian semigroup, 58, 701Abelian variety, 696Abelianization, 426Abraham-Shaw, 1013Absolute complement, 11, 15Absolute consistency, 6Absolute extrema, 61Absolute frequency, 839Absolute geometry, 203, 206, 207Absolute gometry

models 1,2,3,4, 207Absolute homology groups , 394Absolute Hurewicz theorem, 426Absolute maximum, 906Absolute valuation, 106Absolute value, 196, 440

in the field of rationals, 197non-Archimedean, 197

Absolute value of a rational number, 83Absolutely convergent series, 439Absolutely flat ring, 713Absorbing, 43Absorbing element, 16, 44Absorbing on the left, 44Absorbing on the right, 44Absorbing state, 843Absorption, 11, 12Absorption laws, 16Abstract algebra, 357, 691Abstract algebraic variety, 696Abstract category, 51, 808Abstract Riemann surface, 585Abstract simplicial complex, 399, 724Abstract space, 501Abstract variety, 696Acceptance region, 835Accumulation point, 35, 266, 269Action-angle variable, 939Acwise connectedness, 739Acyclic carrier, 410Acyclic complex, 392Acyclic cover, 401Acyclic cover of a simplicial complex, 401Acyclic free chain complex, 409Acyclic free chain complex theorem, 409Acyclic graph, 353Acyclic model, 409Acyclic ob ject, 753, 754Acyclic resolution, 758Acyclic sheaf, 758Acyclic space, 389Aczel’s constructive Zermelo-Fraenkel (CZF), 21Addition and multiplication in N, 69Addition in N, 69Addition in Q, 81Addition in Z, 78Additive abelian group, 110

Additive associativity, 57, 700Additive category, 792Additive commutativity, 57, 700Additive functor, 410, 808Additive group, 138Additive group of integers modulo n, 117Additive identity, 56, 57, 700Additive inverse, 56, 57, 700Additive notation, 43, 52additively and multiplicatively stable, 54Additively stable, 54Additivity axiom, 750Adequate, 401Adequate subcomplex, 401Adherent point, 265, 269, 297, 911Adherent point of a filter, 303Adic topology, 723, 771Adjective

autological, 18heterological, 18

Adjoint, 820Adjoint endomorphism, 185Adjoint functor, 408Adjoint functors, 805Adjoint functors (adjunction), 805Adjoint map, 331Adjoint of a linear map, 331Adjoint operator, 331, 508Adjoint pair (of functors), 421Adjointness, 797Adjunction, 805Adjunction isomorphism, 805Adjunction of indeterminates, 162Admissible open set, 414Admissible parameter, 532Admissible parameterization, 532Affine algebraic set, 696Affine bijection, 230Affine combination, 376Affine connection, 565Affine coordinates system, 259Affine endomorphism, 260Affine form, 260Affine geometry, 211Affine group, 229Affine independent, 376Affine independent set, 376Affine isometry, 255, 256

anti-displacement, 255displacement, 255

Affine map, 229, 230Affine map in R3, 254Affine morphism, 764Affine n-chain, 396Affine n-space, 366Affine neighborhood, 771Affine orthogonal symmetry, 229Affine plane, 211Affine plane of incidence, 211Affine scheme, 365, 697, 763, 766Affine set, 376Affine set (convex set), 376Affine space, 258, 376, 720

affine subspace, 259canonical structure, 259vectorial structure, 259

Affine space (algebraic geometry), 720

1054

Index

Affine space (analytic geometry), 720Affine subspace, 259Affine subspace intersection, 259Affine symmetry, 229Affine transformation, 230Affine variety, 358, 696, 697, 720Affine Y-scheme, 764, 771Affines map, 260Affinity, 229, 230Affix, 93, 95Afraimovich-Shilnikov theorem, 1001Aggregate demand function, 915Agument predicate, 4AIC and SC criteria, 874AIC criteria, 875AIC criterion, 855AIC(p), 874Aitken estimator, 856Akaike criterion (AIC), 865Akaike function, 855Akaike information criterion, 855Akaike-Schwarz, 851Akaike-Schwarz criteria, 874Alembert theorem, 305Aleph, 44Aleph-0, 44, 46, 49Aleph-1, 44, 47, 64Alexander-Veblen corollary, 400Alexander-Whitney theorem, 431Alexandroff compactification, 285Alexandroff extension, 285Alexandroff theorem, 305Algebra, xix, 109, 180, 358

associative, 180commutative, 180of Banach, 599

σ-algebra (Σ-algebra, Ω-algebra), 693Algebra of polynomials, 145Algebra of sets, 11Algebra of the disk, 623Algebraic basis, 332Algebraic category, 802Algebraic closure, 92, 103, 166, 719Algebraic closure of a field, 92, 105Algebraic closure of C, 103Algebraic complex numbers over Q, 157Algebraic curve, 204, 455Algebraic dimension, 289Algebraic dual, 324Algebraic element, 719Algebraic elements of a field, 164Algebraic extension, 166Algebraic extension of a field, 92, 719Algebraic function, 454, 591Algebraic fundamental group, 769Algebraic geometry, 692Algebraic K-theory, 364, 761Algebraic number, 106, 157Algebraic number field, 106Algebraic operation, 11, 693Algebraic operations in sets, 11Algebraic point, 591Algebraic problem, 174Algebraic ramification, 585Algebraic relation, 454, 455Algebraic representation of a complex number, 94Algebraic set, 696Algebraic signature, 694Algebraic space, 697Algebraic structure, 50, 51, 109, 695, 697, 788

of integers, 85of rationals, 86of real numbers, 88

Algebraic structure of R, 435Algebraic surface, 204Algebraic system, 359, 379, 692

Algebraic topology, xix, 335, 366, 691Algebraic variety, 357, 696, 697Algebraically closed, 104Algebraically closed field, 105, 152, 719, 771Algebras for a monad, 803Algebrization, 174Almost everywhere convergence, 662Alternating algebra, 185Alternating bilinear form, 608Alternating form, 187Alternating group, 117, 187Alternating map, 187Alternating series, 439Alternative hypothesis H1, 835Altitude, 227Amalgamed sum, 818Ambient manifold, 737Amenable group, 24Amnestic functor, 805Ampère law, 493Amplitude of cycle, 993Amplitude spectrum, 894Analysis, xx, 435Analysis of autocorrelation function, 869Analysis of residuals, 873Analysis of shocks, 874Analysis of variance, 834Analysis of variance (ANOVA), 848Analysis situs, xixAnalytic, 742Analytic at, 742Analytic continuation, 570, 578, 597, 619Analytic continuations along different paths, 578Analytic function, 451, 562, 577, 586, 742Analytic function at a point, 742Analytic function on interior of domain, 742Analytic functions, 569Analytic geometry, xix, 175, 204, 247

real or complex, 204Analytic manifold, 732Analytic representation, 234Analytic set, 758Analytic space, 758Analytic subspace, 758Analytic variety, 696Andronov-Hopf bifurcation, 995, 1013Andronov-Hopf theorem, 995Andronov-Pontryagin theorem, 979Andronov-Witt theorem, 973Angle, 218, 222

acute, 218obtuse, 218of opening, 252reflex, 218right, 218straight, 218

Angle measure, 260Angle measure on a regular patch, 541Angle measurement, 219Angle trisection, 102, 175Angle-preserving transformation, 592Annihilating element, 16Annulus, 266, 579, 605, 610, 629Anosov, 953Anosov automorphism, 950Anosov diffeomorphism, 952, 953, 991ANOVA equation, 848ANOVA table, 834

for simple regression model, 848Antecedent, 40, 63, 467, 839Anti-adjoint, 331Anti-matched lines, 208Anti-matching, 208Anti-matching theorem, 209Anti-paired lines, 208Anti-self-adjoint operator, 331

1055

Index

Antiautomorphism, 762Anticonformal mapping, 592Antiderivative, 472, 473, 476, 479Antidual, 548Antidual basis, 548Antihomomorphism, 762Antilinear map, 328Antinomy, 17

semantic or syntactic, 17Antipodal map, 397Antipode, 367, 397Antisymmetric function, 186Antisymmetric map, 186Antisymmetric multilinear form, 186Antisymmetrization, 187Antisymmetrization operator, 186Antisymmetry, 10Aperiodic solution, 977Apollonius circle, 228Apollonius definition of a circle, 229Applied mathematics, xxApproximation, 449Approximation by convolution, 667Approximation by interpolation polynomial, 460Approximation by polynomials

on a compact of Cn, 321on a compact of Rn, 321

Approximation by trigonometric polynomials, 321Approximation methods, 480Approximation of vector functions, 321Approximation theory, 458AR (autoregressive), 871AR model, 871AR(p), 857, 871Arc, 267, 277Arc length, 485Arc length of curve, 485Arc length parameterization, 678Arc of a curve, 531ARCH (AR conditional heteroscedasticity), 861ARCH model, 861ARCH test, 861ARCH(q), 881Archimedean property, 70, 90Archimedean totally ordered commutative field, 88Archimedean totally well ordered field, 105Archimedean valued field, 198Arcwise-connected, 739Arcwise-connected component, 278Arcwise-connected set, 267, 277Arcwise-connected space, 277, 278, 308, 739Area function, 229Area measure on a regular patch, 541Area of elliptic triangle, 244Area problem, 445Areolar function, 494Areolar measure, 494ARFIMA (AR fractionally integrated MA), 879ARFIMA process, 879, 882ARFIMA(p,d,q), 879, 881Argand-Cauchy plane, 95Argand-Gauss plane, 95Arguesian plane, 211Argument, 3, 99, 616, 677Argument of a complex number, 99Argument principle, 646ariety (universal algebra), 696ARIMA (AR integrated MA), 871ARIMA model, 871ARIMA process, 872Arithmetic, 67Arithmetic function, 191

Euclidean, 192Arity, 692, 693ARMA (autoregressive moving average), 872ARMA model, 872

ARMA(p,q), 872ARMAX, 873Arnold, 928, 940Arnold tongue, 1004, 1005Arnold tongues, 1002Arrangement, 829Arrangements with repetitions, 831Arrangements without repetition, 831Arrow, 779Arrowsmith-Place, 957, 963Articulation vertex, 355Artin braid group, 730Artinian ring, 712Artinian set, 47Ascending chain condition (ACC), 709Ascoli theorem, 320, 598Associate, 418Associative law, 55Associative magma, 55, 699Associativity, 11, 12Associativity isomorphism, 794Associator, 793Astroid, 455Asymptote, 453Asymptotic behavior of Fourier coefficients, 885Asymptotic line, 545Asymptotic point, 516Asymptotic right triangle, 242Asymptotical stability, 972Asymptotically stable, 929Atlas, 559, 561, 734, 928Atlas of class Cr (or Cr-atlas), 560Atom, 2, 13Atom of a lattice, 13Atomic formula, 21Attaching cell, 404Attaching map, 403, 404, 725Attracting or repelling basin, 1012Attractor, 967, 980Attractor in sense of Ruelle (1981), 992Attractor in the sense of Milnor, 967augmentation, 410Augmentation homomorphism, 749Augmentation preserving, 410Augmented complex, 400Augmented Dickey-Fuller test (ADF), 870, 871Augmented singular complex, 395Aureum theorem, 196Autocorrelation function, 869, 981Autocorrelation function of sampling, 869Autocorrelation of errors, 856Autocorrelation test of errors, 864Autofunctor, 804Autological adjective, 18Automorphism, 44, 51, 134, 655, 705, 808, 809Automorphism group, 142Automorphism set, 142Automorphisms of C, S2,D, 655Autonomous differential equation, 968Autonomous linear differential equation, 968Autonomous system, 936Autoregressive linear model, 863Axial vector, 252AxioM

of countable choice, 25Axiom, 2, 6, 12

of extensionality, 22A,A* (bisector), 216A1,A2,A3 (affine incidence plane), 211C (connection), 209D (completion) Dedekin, 217functional , 18H (hyperbolic), 210Hausdorff, 298M1,M2,M3 (metric plane) Absolute geometry, 206of Cartesian product, 16

1056

Index

of choice, 19, 22, 23of comprehension, 18, 22of denumerable choice, 25of dependent choice, 26of existence, 18of extensionality, 13—15, 18, 22of foundation, 22, 23of induction, 66of infinity, 18, 22, 23of pairing, 14, 22of regularity, 22, 23of replacement, 22, 23of separation, 15, 22of singletons, 14, 18of subsets, 22of Tarski, 21of the empty set, 18of the pair, 14of the power set, 22, 23of the set of parts of a set, 15of the set of the parts of a set, 18of the sum of set, 22of the sum set, 22of the unordered pair, 22of union, 15, 18, 22P (polar trilateral), 210P1,P2,P3 (pro jective incidence plane), 212P5,P5’ (Fano), 215R (rectangle), 209schema of collection, 23schema of replacement, 23schema of restricted comprehension, 22schema of specification, 22T0,T1,T2,T3,T4, 280T2, 271

Axiom (postulate), 6Axiom of bisector (A),(A*), 216Axiom of Cartesian product, 16Axiom of choice, 19, 22, 29, 30, 33, 42, 784Axiom of comprehension, 18, 22Axiom of connection (C), 209Axiom of countable choice, 25Axiom of Dedekin (D), 217Axiom of denumerable choice, 25Axiom of dependent choice, 25, 26Axiom of existence, 18Axiom of extensionality, 13—15, 18, 22Axiom of extensionality (for maps), 40Axiom of foundation, 22, 23, 782Axiom of infinity, 18, 22, 23, 49Axiom of pairing, 14, 22Axiom of polar trilateral (P), 210Axiom of regularity, 22, 23Axiom of replacement, 22, 23Axiom of revealed preferences, 1029Axiom of separation, 15, 22, 41Axiom of singletons, 14, 18Axiom of subsets, 22Axiom of the empty set, 18Axiom of the pair, 14Axiom of the power set, 22, 23Axiom of the set of parts of a set, 15Axiom of the set of the parts of a set, 18Axiom of the sum of set, 22Axiom of the sum set, 22Axiom of the unordered pair, 22Axiom of union, 15, 18, 22Axiom schema of collection, 23Axiom schema of replacement, 23Axiom schema of restricted comprehension, 22Axiom schema of specification, 22Axiomatic construction

of number system, 105Axiomatic method, 6, 203, 204Axiomatic set theory, 16—18, 20, 781Axiomatic system, 4

Axioms, 2Euclid, 203of Zermelo-Fraenkel set theory, 22of ZFC, 22separation, 280T0,T1,T2,T3,T4, 280

Axioms of categories, 799Axioms of closed sets, 296Axioms of Zermelo-Fraenkel set theory, 22Axis of curvature, 535Axis reflection, 216Axonometric pro jection, 237

b-adic number, 91Bézout coefficients, 80, 150Bézout identity, 80Bézout theorem, 80B∗-algebra, 762Bachmann, 206, 213Bachmann metric plane, 206Back face, 430Backward elimination, 855Baire category theorem, 315, 979Baire function, 499Baire property, 315Baire space, 315Baker map, 990Ball, 67Banach algebra, 32, 599, 671Banach algebras with identity element, 32Banach fixed point, 504Banach fixed-point theorem, 504Banach functional, 27Banach space, 27, 28, 322, 327, 502, 663Banach theorem, 328, 529Banach-Alaoglu theorem, 599Banach-Steinhaus theorem, 327, 503Banach-Tarski paradox, 10, 23Barratt-Whitehead lemma, 395Barrow rule, 678Barycenter, 259, 342, 377, 912Barycentric coordinates, 259, 342, 377Barycentric subdivision, 396Barycentric subdivision of an affine n-simplex, 396Base, 759Base for a topological space, 728Base line, 214Base of fiber space, 761Base scheme, 763Base space, 413, 759, 777Base ten numeral system, 75Base-10 numeral system, 75Basepoint, 291Basepoint preserving map, 369Basic dynamics of diffeomorphisms, 928Basic variable, 905Basin of attraction, 967, 980, 992Basis, 138, 271, 548, 704Basis for a topology, 417Basis function, 900Basis of a K-vector subspace, 146Basis of a topology, 271Basis of neighborhoods, 272, 297, 302Basis of neighborhoods (equivalently, fundamental

system of neighborhoods), 272Basis of open sets, 297Bauer, 501Bayes theorem, 837Bean machine, 840Behaviors of second-order linear systems, 969

diagram of behaviors, 969interpretation of eigenvalues , 969

Belote, 832Bernoulli inequality, 437Bernoulli number, 456Bernoulli shift map, 980Bernstein polynomial, 458

1057

Index

Bessel inequality, 331, 883Best approximation, 459, 533Best contact, 533Beta-distribution, 844Beth’s definability theory, 7Betti group, 349, 745, 746Betti number, 349, 558Bezout identity, 194Bezout relation, 162Bezout theorem, 80Bezout theorem (for polynomials), 149Bi-ratio, 233Bianchi identity, 567Bicategory, 358, 792, 802Biclosed monoidal category, 793Bicomplete category, 792Biconnected graph, 355Bidual, 324, 548Bidual of a normed vector space, 324Bienaymé-Chebychev inequality, 838Bienaymé-Tchebychev inequality, 838Bifunctor, 358, 792, 793, 805Bifunctor (binary functor, i.e. 2-ary), 805Bifurcation, 992, 994, 1013

Andronov-Hopf, 995, 1013Andronov-Hopf supercritical, 1014cusp, 963flip, 1013fold, 1013Hopf, 993Hopf subcritical, 962Hopf supercritical, 962node-saddle, 993of fixed points, 992of periodic orbits, 993period-doubling, 1013pitchfork, 993saddle node, 971, 1013subcritical, 993subharmonic, 993supercritical, 992transcritical, 993

Bifurcation diagram, 964, 995, 996, 1014, 1017Bifurcation diagram of Lorenz model, 1017Bifurcation of double periodic orbits, 997Bifurcation value, 994Bifurcations in Morse Smale systems

global bifurcation, 998saddle-node separatrix disappearance, 999saddle-node to periodic orbit, 998

local bifurcation nearby periodic orbitsbifurcation of double periodic orbits, 997centre manifold theorem for maps, 997creation of a torus, 998period-doubling (flip bifurcation), 997

local bifurcations nearby steady states, 994Bogdanov-Takens bifurcation, 996breaking symetry, 996centre manifold theorem, 994cusp bifurcation, 995generalized Hopf bifurcation, 996Hopf bifurcation, 995saddle-node, transcritical bifurcations, 994

Bifurcations of fixed points, (Cod-1), 992subcritical bifurcations, 993supercritical bifurcations (normal form)

Hopf, 992pitchfork, 992saddle-node, 992transcritical (stability exchange), 992

Bifurcations theory in Morse Smale systems, 994Big étale site, 769, 770Biggest element, 12Biharmonic oscillator, 939Biholomorphic map, 592Biholomorphism, 655

Biholomorphism,(conformal map), 655Bijection, 44, 50, 705

inverse, 48Bijective homomorphism, 705Bijective homomorphism (bimorphism), 705Bijective map, 39, 41, 705Bijective map defined via compositions, 40Bijective morphism, 44Bijective morphism (isomorphism), 44Bijective sets, 39Bijectivity of Galois map, 171Bilinear extension, 139Bilinear form, 549Bilinear function, 411Bimorphism, 705, 808Binary functor, 358Binary numeral system, 75Binary operation, 359, 693Binary relation, 10, 17, 36, 37

antisymmetric, 37asymmetric, 37biective (or biunivocal), 37doubly surjective, 37injective, 37left surjective, 37properties, 37reflexive, 37right surjective, 37symmetric, 37total, 37transitive, 37univocal, 37

Binomial coefficient, 121, 829Binomial distribution, 835, 840Binomial equation, 174Binomial expansion, 121Binomial formula, 124Binomial series, 121, 452Binomial theorem, 121, 124Binormal vector, 533, 534Bipartite Graph (or bigraph), 351Biperiodic dynamics, 976Bipoint, 220Bipolar rectangular impulse, 896Biquadratic reciprocity theorem, 196Birational transformation, 204Birkhoff ergodic theorem, 981Birkhoff HSP theorem, 697Birkhoff prime ideal theorem, 31Birkhoff theorem, 697, 971Birth of an invariant torus, 1013Bisector, 207Bisector axiom (A),(A*), 216Bite vector, 42Bivalence principle, 1Bivalent logic, xixBlakers-Massey theorem, 426Blaschke condition, 629Blaschke products, 635Block, 355Block-Guckenheimer-Misiuriewicz theorem, 992Blowing-up, 948

directional, 948polar, 948

BLUE, 857, 859BLUE (Best Linear Unbiased Estimator), 850Blue sky catastrophe, 1011, 1013Bogdanov, 963Bogdanov, Takens, 963Bogdanov-Takens bifurcation, 996Bolyai, 230Bolzano-Weierstrass property, 285Bolzano-Weierstrass theorem, 437Boolean algebra, 26Boolean algebras, 12Boolean lattice, 12, 13, 836

1058

Index

Boolean prime ideal theorem (BPI), 26, 31, 32Boolean ring, 13, 125Bordant, 506, 741Bordant manifolds, 741Bordism, 356, 506, 744Bordism group, 741Borel measure, 981Borel σ-algebra, 836Borel set, 836, 981Borel sigma-algebra, 836, 981Borel-Carathéodory inequality, 633Borel-Lebesgue, 303Borel-Lebesgue property, 303Borsuk theorem, 641Borsuk-Ulam, 398Borsuk-Ulam theorem, 432Boundary, 266, 388, 398, 741Boundary conditions, 529

Dirichlet, 529Neumann, 529Robin, 529

Boundary map, 749, 755Boundary morphisms, 808Boundary of a singular n-simple, 388Boundary operator, 388, 724, 745Boundary point, 266, 398Boundary value problem, 512Bounded, 28Bounded distance, 294Bounded from above, 61Bounded from above real function, 440Bounded from below real function, 440Bounded function, 28Bounded lattice, 30Bounded linear functional, 26, 28Bounded linear operator, 503Bounded part, 284, 294Bounded real function, 440Bounded tile, 284Bounded variation, 884Bouquet of circles, 403Bouquet of circles (rose), 403Bourbaki, 279Bourbakist conception of mathematics, 50Boussinesq, 1015Box-counting dimension, 988Box-counting dimension properties, 989Box-Jenkins method, 872Box-Pierce statistic, 869Boxcar function, 440Bra, 554Brachistochrone, 504, 505Bracket of two vector fields, 563Braid, 730Braid group, 730Braid index, 730Braided category, 794Braided monoidal category, 794Braiding, 794Branch, 572, 583Branch cut, 572, 583Branch point, 571, 584, 585Breaking symetry, 996Breit-Wigner curve, 896Breusch-Godfrey test, 857Broadband spectrum with singular peaks, 982Broomhead-King, 562Brouwer, 5, 289, 292, 640, 911, 914, 915Brouwer degree, 558Brouwer fixed point theorem, 292, 915Brouwer theorem, 292, 640Brownian motion, 880Bubble sort, 117Bundle, 208, 356, 363, 759, 760Bundle (category), 760Bundle (topology), 759

Bundle rank, 364, 761Buniakowski inequality, 260Burali-Forti, 18Burali-Forti paradox, 65By recurrence, 7

C is algebraically closed, 155C∞ partition of unity subordinate to covering, 678C-analytic function, 614, 619

C-C function, 572C-C function, 572C-C function, 572C-C-function, 572, 575C-differentiable function, 574, 612, 743, 744C∞-manifold, 733C∞-structure, 733C∞ partition of unity, 672, 678C∗-algebra, 762Cr-manifold, 928Caeteris paribus, 874Cah, 239Calculation

of geodesic curvature, 543of normal curvature, 543

Calculus and analysis, xxCalculus of variations, 504Cancellable, 53Cancellable element, 43, 53Cancellation property, 53Canonical basis, 261, 600Canonical basis of the K-vector subspace, 146Canonical euclidean norm, 260Canonical flabby resolution, 757Canonical flasque resolution, 757Canonical form of a circle map, 1002Canonical injection, 40, 705Canonical map, 40, 274, 762Canonical morphism, 118Canonical pro jection, 303, 810, 816Canonical representation of simple arc, 534Canonical resolution, 757Canonical scalar product in vector space Rn, 260Canonical structure, 259Canonical surjection, 38, 39, 45Canonical topology, 823Cantor, 18, 50, 63, 88Cantor comb, 988Cantor diagonal method, 42, 45, 46Cantor diagonalization, 44Cantor function, 1002Cantor native language, 46Cantor sequence, 65Cantor set, 954, 959, 988, 989Cantor theorem, 41, 49, 310Cantor’s transfinite numbers, 64Capacity dimension, 988Capital per unit of labor, 1030Carathéodory theorem, 658Cardan formula, 100Cardan method, 173Cardinal, 13, 44, 45, 63

addition, 45arithmetic, 45comparison, 45exponentiation, 45multiplication, 45of a set, 48of finite or infinite set, 45of reals, 50of the set of reals, 46operations, 45

Cardinal number, 13, 44Cardinal of a group, 110Cardinality, 44Cardinality 1, 45Cardinality 2, 45

1059

Index

Carnap, 2Carrier (universe), 693Carrier function, 410Carrier set, 692Carrier set (underlying set), 692Cartan, 186, 279Cartesian closed category, 796Cartesian geometry, 175Cartesian product, 17, 36Cartesian representation of a complex number, 94Cascade of period-doublings, 999Casorati-Weierstrass theorem, 630Cat, 790Catalan numbers, 43Catastrophic instability, 1015Categorical direct sum, 816Categorical product, 816Categorical sum, 816Categories of rational functions, 454Category, 51, 85, 347, 359, 366, 368, 779—781, 787,

808by Eilenberg-Mac Lane, 780by Mac Lane, Buchsbaum, Grothendieck, Heller,

781Category Ab (AbGrp), 789Category AbGrp (or Ab), 789Category associated with ordered set, 796Category Cat, 789Category Comp, 392Category CPO, 789Category Diff, 789Category Diffb, 789Category Fields, 789Category Grp (Groups, Group), 789Category Heyt, 789Category hoTop, 788Category isomorphism, 804Category Man, 789Category Manb, 789Category Mon, 789Category of (Eilenberg-Moore) algebras, 803Category of all étale morphisms, 770Category of all abstract simplicial complexes and

simplicial maps, 399Category of chain complexes, 797, 801Category of functors, 792Category of schemes, 366, 763Category of sheaves of sets, 747, 770Category of small categories, 790Category PoSet, 789Category PreOrd, 789Category Rings, 789Category Sets, 788Category theory, 359, 779Category Top, 788Category Vec, 788Category VecC, 789Category VecR, 789Category with a generator, 799Catenary ring, 712Catenary scheme, 764Cathetus, 242Cauchy convergence criterion, 438Cauchy criterion, 310, 331Cauchy filter, 314Cauchy form of Taylor remainder, 450Cauchy formula, 575, 617, 618

for a disk (direct demonstration), 622Cauchy formula (Homology form), 645Cauchy formula (Homotopy form), 639Cauchy inequalities, 620, 660Cauchy integral

formula, 575property, 575

Cauchy integral formula, 575, 581Cauchy integral property, 575

Cauchy integral theorem, 575, 581, 892Cauchy integration criterion, 481Cauchy kernel, 623Cauchy kernel of unit disk, 623Cauchy mean-value theorem, 449Cauchy principal value, 894Cauchy rational sequence, 92Cauchy real sequence, 92Cauchy sequence, 28, 89, 198, 504

rational, 92real, 92

Cauchy theorem, 617, 643Cauchy theorem (Homology version), 643—645Cauchy theorem (Homotopy version), 638, 643Cauchy tranform, 623Cauchy transform, 614Cauchy transform of a Radon measure, 614Cauchy transform of Borel measure, 622Cauchy-Goursat theorem, 613, 626Cauchy-Kowalewska theorem, 526Cauchy-Pompeiu formula, 617, 664Cauchy-Riemann conditions, 570, 574Cauchy-Riemann equation, 619Cauchy-Schwarz inequality, 260Causality in the sense of Granger, 875Causality in the sense of Sims, 875Cavalieri, 484Cavalieri principle, 484Cavalieri theorem, 484Cayley-Menger determinant, 340, 346Cconnectedness properties of topological spaces, 729Cech cohomology, 744, 756, 757Cech cohomology group, 756Cech complex, 744Ceiling, 82Ceiling function, 82Cell, 724, 792Cell (algebraic topology), 724Cell (geometry), 724Cellular, 407, 725Cellular approximation, 408Cellular approximation theorem, 408Cellular chain complex, 407Cellular chain complex lemma, 407Cellular chain complex of a filtration, 407Cellular filtration, 407Cellular homology, 407Cellular map, 407Cellular space, 407Cellular theory, 389Center, 112, 969Center (centre), 112Center manifold, 946, 1014Center manifold theorem, 946Center of the sheaf, 214Centering parameter, 900Central angle, 222Central limit theorem, 835Central pro jection, 232Central pro jections in P3 (R), 232

Central pro jections in R3, 232Central sheaf, 214Centre, 112, 516Centre manifold, 1014Centric, 212Centric perspective, 212Ceteris paribus, 874Cevian, 227CH, 20Chain, 28, 48, 49, 59, 60, 352, 386Chain complex, 392, 723, 745, 797Chain complex of finite type, 429Chain equivalence, 393, 409Chain equivalent, 393Chain group, 407Chain homotopic, 393

1060

Index

Chain homotopic maps, 393Chain homotopy, 393Chain map, 392Chain of an ordered set, 48Chain of fields, 174, 175Chaitin, 3Change of admissible parameter, 532Change of basins, 1012Change of basis, 548Change of coordinate system, 256Change of coordinates, 560, 734Change of local coordinates, 733Change of parameterization, 678Change of variable, 478, 514Change of variables, 484Chaos, 965, 991

(DO) density of orbits, 991(SDIC), 991(TT) topological transitivity, 991

Chaos in the sense of Devaney, 991Chaos in the sense of Li-Yorke, 991Chaos theory, xxChaotic attractor, 991Chaotic behaviors, 982Chaotic dynamical system, 991Chaotic systems in the sense of Devaney, 991Character of a point, 728Character of a topological space, 728Characteristic, 128, 718Characteristic class of a fibration, 432Characteristic equation, 521Characteristic function, 42Characteristic map, 406, 725Characteristic of a ring, 128Characteristic polynomial, 182Characteristic root, 180Characteristic subspace, 182Characteristic value, 180Characterization

of simply connected open sets, 658Characterization of an attractor, 987

fixed point: Lyapunov exp. sign (-,-,-) , 988limit cycle: Lyapunov exp. sign (0,-,-) , 988strange attractor: Lyap. sign (+,0,-) , 988Torus: Lyapunov exp. sign (0,0,-) , 988

Characterization of fields by ideals, 136Chart, 559, 733, 734, 928Chasles relations, 101Chebychev inequality, 838

Chebyshev (or, Tchebychev, Tchebicheff, Cebysev,Tschebyscheff, Chebishev), 101, 459

Chebyshev approximation, 904Chebyshev distance, 299Chebyshev norm, 459, 461Chebyshev norm (uniform norm), 459Chebyshev polynomials, 101, 156, 459Chebyshev polynomials of the first kind, 156Chebyshev polynomials of the second kind, 156Chi-squared distribution, 834Chi-squared test, 834, 842Chillingworth, 928Chinese congruence, 131Chinese hypothesis, 131Chinese remainder theorem, 131Chinese theorem, 131Choice function, 22, 23, 26Choice of optimal model, 853, 855Chordal distance, 572Chordal theorem, 228Chow test, 859Christoffel symbol of the first kind, 565Christoffel symbol of the first kind (Connection co-

efficient), 565Christoffel symbols, 565Christoffel symbols of the first kind, 547

Christoffel symbols of the second kind, 547, 565Church, 3Circle, 222, 251, 252Circle diffeomorphism, 934Circle diffeomorphisms with rational number, 934Circle map, 1004Circle of curvature, 241, 536, 538Circle of curvature for an ellipse, 537Circle quadrature, 102, 176Circle squaring, 102, 176Circuit, 352, 402Circular average, 666Circular convolution, 900Circular function, 237, 456Circular permutation, 122Circumcenter, 228Circumcircle, 227Circumradius, 228

Ck-atlas, 732Class of algebras, 803Class of generators for a category, 786Class of groups, 429Class V, 20, 783Classes of quadrics, 257Classical harmonic analysis, 661Classification

of quadrilaterals and triangles, 231Classification of closed surfaces, 346Classification of fields, 169Classification of metric planes, 211Classification of patch points, 544Classification of periodic orbits, 973Classification of real functions, 445Classification of valued fields, 197Classifying morphism, 786Clique, 356Clopen set, 727Closed 1-form, 427Closed arc, 267Closed ball, 294Closed curve, 291, 739Closed curve contractility, 740Closed disk, 96, 267Closed Eulerian chain, 352Closed form, 607Closed graph theorem, 328, 503Closed immersion, 764, 773, 774closed immersion, 771Closed Jordan curve, 268Closed k-cell, 398Closed manifold, 734closed manifold, 741Closed manifold (compact boundaryless manifold),

741Closed monoidal category, 793Closed orbit, 977Closed part, 294Closed path, 378Closed set, 35, 266, 269Closed subscheme, 764Closed surface, 346

classification, 346Closed surfaces, 346Closed under, 706, 707Closed-bounded, 268Closed-bounded property, 268Closure, 32, 35, 57, 266, 294, 706, 707

of a connected part, 307Closure , 911Closure (abstract algebra), 706, 707Closure of valued fields, 198Closure operator, 138Closure theorems, 211Cluster point, 35, 266, 300

of a sequence, 437, 462, 572Cluster point of a map, 301

1061

Index

Cluster point of a sequence, 300Cn[a, b] space, 502Co-cone, 813Co-unit, 433Co-unit of a graded co-ring, 433Cobordant, 741Cobordant manifold, 741Cobordism, 506, 741Cobordism class, 741Cobordism group, 741Coboundary, 428, 724Coboundary map, 752Coboundary map (differential), 752, 753Coboundary operator, 724, 744Coboundary operator (codifferential), 756Cobounding, 506Cocartesian square, 818cochain, 428Cochain complex, 723, 752, 753Cochrane-Orcutt method, 858Cocomplete category, 413, 792Cocone, 813Coconstant morphism, 810Cocycle, 428, 724Codifferential, 756, 757Codimension, 944, 963, 994Codimension of a bifurcation, 996Codimension-1, 992Codomain, 19, 38, 40, 51, 704Coefficient group, 409, 750Coefficient of degree k, 146Coefficient of determination, 848, 850Coefficients, 411Coequalizer, 413, 796, 817Coexact sequence, 423Cofactor, 143Cofibration, 425Cofinal, 20, 31, 782Cofinality, 31Cogroup object, 418, 420Cohen-Macaulay ring, 712Cohen-Macaulay scheme, 764Coherence conditions, 802Coherence property, 794Coherent sheaf, 749Cohomology, 356, 426, 645, 744, 753Cohomology (of topological space), 744Cohomology (with values in sheaf ), 744Cohomology class, 428Cohomology functor, 808Cohomology group, 357, 428, 645, 746Cohomology groups, 768Cohomology ring, 430, 431, 745Cohomology theory, 357, 645, 692Cohomology with compact support, 773Cohort data, 845Coincident transformation, 219Cointegrated series, 876Cointegration, 846, 875, 876Cointegration analysis, 875Cointegration rank, 877Cointegration relation, 879Cointegration relation test, 878Cointegration test, 876Cointegration theory, 846Cointegration vector, 876Cokernel, 811, 818Colimit, 412, 814Collection of morphisms, 51, 808Collectivizing property, 15Collectivizing relation, 10Collinear, 208Collinear vectors, 247Collinearity, 247Collineation, 206, 213, 233

orthogonal, 206

projective, 203Collineation of a pro jective plane, 213Collision, 1013Coloring, 292, 354Comatrix, 143Comb space, 277Combination (or selection), 829Combination with repetitions, 832Combinations without repetition, 831Combinatorial analysis, 73, 829Combinatorial homotopy, 344Combinatorial theory, 829Combinatorial topology, 339, 344Combinatorics, 344, 829Comeagre set, 315Comma category, 786, 791Comma derivative, 565Common divisor , 76Common multiple, 77Commutative Abelian groups, 120Commutative and unital ring, 708Commutative and unitary ring, 708Commutative Banach algebra, 32Commutative diagram, 40, 369, 812Commutative field structure, 86Commutative group, 110, 120, 645Commutative group (Abelian group), 110Commutative group for the composition law, 213Commutative ring, 708Commutative ring with identity, 708Commutative ring with unity, 708Commutative semigroups, 124Commutative unit ring, 13, 85

with no divisor of zero, 57, 700Commutativity, 11, 12Commutator, 426, 682Commutator subgroup, 426Comonad, 802Comonad (cotriple), 802Compact, 268Compact boundaryless manifold, 741Compact boundaryless manifold (closed manifold),

741Compact invariant chaotic set, 991Compact manifold, 734, 741Compact operator, 507, 508Compact polyhedral pair, 409Compact set, 35, 268, 283, 285, 967Compact space, 35, 105, 283, 285, 303, 729

by adjunction of a point (at infinity), 105product, 304

Compact subset, 268Compact support, 389, 903Compact support theorem, 389Compact-open topology, 285, 418

Compact-open topology on XY , 418Compactification, 105, 284, 286, 571, 773

of the plane, 105Compactification of C, 570, 571Compactness, 283Compactum, 286Comparability, 29, 59, 783Comparable distances, 295Comparison of Fourier and Laplace transforms, 895Comparison of topological spaces, 272Comparison of topologies, 270, 296Compass, 174Compatibility, 560

of a map with topological structures, 560Compatibility of charts, 560Compatibility of charts of an atlas, 585Compatible chart, 733Compatible charts, 734Complement, 11, 15, 17Complement formula, 635, 648Complementary angles, 221

1062

Index

Complementary set, 266Complementation, 12Complemented lattice, 12, 30Complete algebraic variety, 696Complete axiomatic set theory, 17Complete category, 413, 792Complete field, 436Complete graph, 351Complete lattice, 172Complete local ring, 771Complete metric space, 89, 293, 310, 312complete normed linear space, 28Complete normed space, 27Complete normed vector space, 28, 322, 327, 502Complete pre-Hilbert space, 184, 502Complete quadrangle, 215Complete quadrilateral, 215Complete space, 571Complete uniform space, 314Completed local ring, 771Completely regular space, 282Completion, 105, 217, 293, 716

of a metric space, 311Completion (abstract algebra), 716Completion of a metric space, 311Completion of a ring, 715Completion of metric space, 293Completion of pre-Hilbert space, 329Completion of uniform space, 315Completion process, 105Complex, 392Complex analysis, 569Complex analytic function, 742Complex analytic function (holomorphic function),

742Complex analytic funtion, 742Complex analytic manifold, 732Complex conjugate, 95, 103Complex continuation, 570Complex curvilinear integral, 575Complex derivability, 570Complex differentiability, 573, 743Complex eigenvalue, 927Complex entire series, 576Complex exponential, 102, 615Complex exponential function, 97, 102Complex Fourier coefficients, 887Complex function, 572Complex Hahn-Banach theorem, 33Complex infinity, 105Complex line bundle, 760Complex measure, 598Complex number, 92, 93

addition, 92calculation rules, 103multiplication, 92normal representation, 103

Complex plane, 95, 294, 970homeomorphic to the sphere, 105

Complex plane (z-plane, Gauss plane, Argand-Gaussplane, Argand-Cauchy plane), 95

Complex power series, 576Complex power series (complex entire series), 576Complex pro jective n-space, 404Complex representation of Fourier series, 885Complex representation of Fourier sum, 887Complex sequence, 572Complex sphere, 105Complex structure, 585Complex vector bundle, 364, 760Complex vector space, 27Complex-differentiable, 743Complex-differentiable (holomorphic, analytic), 743Complex-valued function, 569Composition

of reflections, 208

Composition law, 38, 39associative, 39external, 50, 58, 701internal, 50, 52, 55, 56, 698, 699

Composition of differentiable functions, 447Composition of functions, 440, 466Composition of homotheties, 225Composition of maps, 39Composition of reflections, 208Composition of relations, 38Composition of rotations, 55, 699Composition operators, 669Composition rule, 447Compositum, 220, 719Computation costs of Fourier coefficients, 887Computation of complex Fourier coefficients, 887Computer-generated random number, 843Comultiplication, 421Concave function, 915Concentrated probability measure, 981Concept of convergence, 50Concept of correlation, 846Concrete category, 788, 790Concrete Riemann surface, 585Concurrent lines, 207Condensation point, 266Conditional distribution, 836Conditional expectation, 836Conditional forecast, 853Conditional probability, 832, 835, 837Conditionally compact set, 508, 651Conditions of order (of identifiability), 868Conditions of rank, 868Cone, 252, 373, 813Cone construction, 390Cone of revolution, 234, 252Cone over X, 373Confidence interval, 834, 851

of the variance of error, 851Confidence level, 834Configuration, 212Conformal map, 592, 593, 655Conformal mapping, 570, 592, 654Conformal transformation, 570, 592, 593, 654

complex plane, Riemann sphere, 593from C to C, 594

from C to C, 594from the interior unit disk to itself, 595of simply connected domain, 595

Conformally equivalent, 655Congruence, 38, 194, 206, 219, 369Congruence modulo an ideal, 194Congruence theorems, 222Congruence transformation, 204Congruent, 194Congruent figures, 12Congruent sets, 218Conic, 252, 253Conic section, 213, 252Conic section directrix, 253Conics, 251Conjecture, 2, 6Conjugacy, 933, 977Conjugacy class, 116Conjugacy of diffeomorphisms, 933Connected, 267, 729, 731Connected acyclic graph, 353Connected compact set, 291Connected component, 276, 277, 307, 355, 738Connected graph, 353Connected Hopf algebra, 433Connected set, 266, 267Connected sinplicial complex, 402Connected space, 268, 275, 276, 306, 738Connected subset, 276Connectedness, 267

1063

Index

Connectedness (connectivity), 731Connecting homomorphism, 394, 750, 758Connecting morphism, 808Connecting morphism (boundary morphism), 808Connection, 564Connection coefficient, 565Connection coefficient (Christoffel symbol of the

first kind), 565Connection curvature, 564Connection geodesic, 564Connection on a differentiable manifold, 564Connection torsion, 564Connections, 1Connections of propositions, 1Connectives

and, 1equivalent, 1implies, 1nand, 1non-equivalent, 1nor, 1not, 1or, 1xor, 1

Connectivity, 267Connectivity number, 347, 349, 354, 558Connectivity pair, 355Conservative Hamiltonian system, 938Consistency, 4Constant, 787, 824Constant function, 440Constant map, 39Constant morphism, 810Constant path, 378, 401Constant return to scale, 1026Constant sheaf, 748, 772Constant symbols, 694Constraint, 906Constraints, 920Construct, 790Constructibility by straightedge and compass, 175Construction by straightedge and compass, 174Construction of C, 92Construction of an n-sided polygon, 176Construction of C, 93Construction of N, 83Construction of Number System, xixConstruction of Q, 86Construction of R, 88Construction of Z, 78, 85Construction problem, 174Constructions by straightedge and compass, 221Constructive set theory, 21Contact configurations, 533, 535Contact element, 516Contiguous simplicial maps, 401Continuation, 416Continuation of a path, 416Continuity, 441, 446, 573

global, 273local, 273

Continuity at a point, 67Continuity of a sequence, 278Continuity of complex functions, 573Continuous bijection, 265Continuous deformation, 542, 637Continuous distribution, 839Continuous extension of a function, 442Continuous function, 441Continuous geometry, 263Continuous invariant, 264, 266, 268Continuous linear form, 324Continuous linear functional, 28Continuous linear map, 323Continuous logarithm, 641Continuous map, 67, 263, 264, 273, 298

Continuous multilinear map, 325Continuously differentiable curves

explicit representation, 485parametric representation, 486

Continuum, 46, 47, 65Continuum hypothesis, 20, 47, 65, 782Contour, 579, 580Contour integral, 580, 581Contour integral , 581Contour integration, 580Contour winding number, 581Contracted curvature tensor, 553Contractibility, 338Contractible, 338, 372Contractible loop, 739Contractible space, 372, 412Contractile arcwise connected space, 338Contractile space, 338Contractility, 739Contracting homotopy, 393, 409Contracting homotopy of complex, 393Contraction, 312, 941Contraction mapping, 312Contraction of volume, 977Contragedient matrix, 548Contragredient representation, 548Contraposition, 72Contravariant, 551Contravariant functor, 360, 370, 803Contravariant Hom functor, 370Contravariant index, 550, 554Contravariant tensor, 552, 554Contravariant tensor algebra, 551Contravariant valency, 551Contravariant vector, 554Control, 905Control of process, 925Control theory, xx, 904Convention of summation, 548Convergence

of a sequence, 67in terms of balls, 67

simple, 318uniform, 318

Convergence criterion, 438, 461

Convergence in C, 572Convergence in norm, 662Convergence of a filter base, 279Convergence of a sequence, 67, 318Convergence of a sequence in terms of balls, 67Convergence of Fourier series, 661

A-P convergence, 661almost everywhere convergence, 662convergence in norm, 662weak∗ convergence, 662

Convergence sets of entire series, 596Convergence sets of power series, 596Convergent estimator, 847Convergent infinite product, 439Convergent product, 439Convergent sequence, 89, 437Convergent sequence of functions, 443Conversion formulas, 894Convex, 267, 372Convex combination, 912Convex envelope, 342, 652Convex function, 916, 917Convex functional, 27Convex hull, 340, 346, 376Convex optimization problem, 924Convex part, 341Convex programming, 905, 910Convex set, 35, 376, 911

properties, 912separation, 913

Convex set and convex function, 917

1064

Index

Convex set spanned by X, 376Convexity, 267, 341, 911Convexity in optimization problems, 911Convolution, 661, 662, 671, 674, 890, 895, 903Convolution of distributions, 903Convolution operation, 623Coordinate ball, 734Coordinate chart, 364, 733, 734Coordinate curve, 539Coordinate curve on a patch, 539Coordinate domain, 734Coordinate function, 736Coordinate neighborhood, 734Coordinate neighborhoods, 425Coordinate neighbourhood, 736Coordinate neighbourhood (coordinate patch), 733Coordinate patch, 735Coordinate plane, 214Coordinate ring, 720, 721Coordinate transformation, 560, 734Coordinates, 213

in affine planes, 214in pro jection planes, 215

Coordinates of a tensor, 549Coplanar, 223Coplanar vector, 248Coplanarity, 247Coproduct, 404, 799, 815, 816Coproduct (direct sum), 799Coproduct of ob jects, 419Coresolution, 755Corestriction, 40Coretraction, 808Coriolis, 488Correction of abnormal values, 852Correction of heteroscedasticity, 859, 860Correlation, 203, 213, 846Correlation coefficient, 845Correlation dimension, 989Correlation integral, 989Correlation of a pro jective plane, 213Correlation theory, 845Correlogram, 869Correspondence, 915Correspondence between differential complex and

null sequence, 707, 724Correspondence between manifolds, 560Cosecant, 238Coset, 113, 713, 810Cosine, 239Cospan, 813Cotangent, 238Cotriple, 802Countability axioms, 287, 288Countable, 3, 24, 287, 735Countable basis, 287, 730Countable everywhere dense part, 321Countable local basis, 736Countable product, 296Countable set, 24, 25, 45, 106, 287Countably compact set, 287Countably infinite, 24, 45, 46, 287Countably infinite = Denumerable, 24, 287Countably infinite set, 24, 25Countably metacompact space, 287Countably paracompact space, 287Counterclockwise, 239Counting, 50Counting number, 44, 83Counting of poles, 646Counting of zeros, 646Coupling of frequency, 977, 982, 994Cousin problem, 652Covariance Cov(x, y), 845Covariant, 551Covariant derivative, 564, 566

Covariant functor, 360, 370, 803Covariant Hom functor, 370Covariant index, 550, 554Covariant left-exact functor, 755Covariant tensor, 549, 552, 554Covariant tensor algebra, 551Covariant tensor of order 1, 549Covariant valency, 551Covariant vector, 554Cover, 35, 268, 283, 401, 413, 561Cover of a simplicial complex, 401Covering, 35, 263, 268, 283, 413, 561Covering family, 770, 821Covering homotopy lemma, 414Covering homotopy property, 761Covering homotopy theorem, 414Covering map, 777Covering map (covering projection), 777Covering projection, 414, 777Covering sieve, 821Covering space, 278, 413, 414, 777Covering space of a topological space X, 413Covering transformation, 415Covering transformation (deck transformation), 415Cr-atlas, 560Cr-structure, 733Cramer Formula, 144Creation of a torus, 998Crisp set, 21Criterion of identifiability of the model, 868Critical (or rejection) region, 835Cross product, 17, 249Cross-cap, 558Cross-caps, 347Cross-sectional data, 845, 856Cross-sectional model, 853, 857Crosscap, 405Crossword dictionary order, 703CRS property, 282Cube duplication, 175Cubic parabola, 535Cumulative distribution, 839Cumulative distribution function, 839, 840Cumulative distribution function (c.d.f ), 834Cumulative distribution function (c.d.f.), 839Cumulative hierarchy of sets, 19, 783Cup product, 430, 432, 745Curl, 427Curl theorem, 472, 557Curry-Yorke model (C-Y), 1004Curvature, 241, 533, 538, 561, 562, 566

of Riemann, 562Curvature formula, 534Curvature line, 545Curvature of a curve, 566Curvature of a plane curve, 537Curvature of a regular patch, 542Curvature of a Riemannian space, 567Curvature of a surface, 566Curvature of circle, 533Curvature of helix, 533Curvature scalar, 553Curvature vector, 533Curve, 291, 292, 531

of Jordan, 602regular, 602

Curve length, 485Curve length in R3 (or R2), 485Curve torsion, 534Curve with isolated point, 455Curves in R3, 530Curvilinear abscissa, 486, 532Curvilinear coordinates, 465, 539Curvilinear integral, 487, 488, 555, 602

case of continuous paths, 635of a gradient, 489

1065

Index

of real functions, 489properties, 603

Cusp, 949, 995Cusp and Generalized Hopf bifurcations, 963Cusp bifurcation, 963, 995Cusp bifurcation (Bogdanov, Takens), 963Cusp singularity, 949CUSUM statistic, 853CUSUM test (CUmulative SUM), 853CUSUM-SQ statistic, 853Cut, 87Cut point, 355CW complex, 404, 406CW complexes, 404, 406CW decomposition, 406, 725CW space, 406, 725CW subcomplex, 406CW-complex, 725CW-complexes, 267Cycle, 116, 352, 386, 643, 724Cycle homologous to 0, 644Cyclic decomposition, 183Cyclic decomposition theorem, 183Cyclic Galois group, 173Cyclic group, 118, 349Cyclic multiplicative group, 153Cyclic ob ject, 410Cyclic order, 219Cyclic order of equivalence classes, 206Cyclic order relation, 218Cyclic permutation, 122Cyclic subbasis, 183Cyclic subgroup, 98Cyclic subgroup of T, 98Cyclic subgroup of unit circle, 98Cyclic subspace, 183Cyclic variable, 939Cyclic vector, 183Cyclically ordered, 218Cyclically reduced word, 704Cycloid, 977Cyclotomic equation, 177Cyclotomic polynomial, 156, 168Cyclotomic polynomial of order k, 168

δ-functor, 754d-stable, 1020d-stable matrix, 1026Damped oscillation, 896Darboux-Ribaucour thriedron, 542Daughter wavelet, 899De l’Hospital rules, 453De Moivre number, 98De Rham cohomology, 427, 493, 557, 608, 681De Rham theorem, 493Decimal comma, 92Decimal expansion, 46, 90

periodic, 91of rational number, 92

terminating, 91Decimal numeral system, 75Decimal numeral system (base ten, base-10, denary),

75Decimal point, 92Deck transformation, 415Decomposition into irreducible elements, 192Decomposition into prime ideals, 193Decomposition into simple elements, 453Decompostion into simple elements, 160Decreasing real function, 440Decryption, 131Dedekind, 87Dedekind ring, 193Deduction theorem, 5Deductive system, 3Definable set, 695Definiendum, 7

Definiens, 7Definite hermitian form, 184Definite positive hermitian form, 329Definition, 6, 7Definition by comprehension, 15Definition by recurrence, 7Definition of an attractor, 992Definitions of chaos, 991Deformation, 770, 775Deformation retract, 375, 775Degeneracy conditions, 944, 949Degree, 158, 383, 397Degree of a continuous map, 397Degree of a loop, 683Degree of a map, 640Degree of a mapping, 558Degree of a polynomial, 147Degree of a vertex, 352Degree of an algebraic number, 157Degree of an element in a graded ring, 430Degree of antipodal map, 411Degree of freedom, 834, 842Degrees of freedom, 851Del, 465, 471, 526Del (or nabla), 526Del operator, 472Delta of Dirac, 902Demonstration, 6, 7Demonstration by the absurd, 6Dence tra jectory, 977Denjoy integral, 500Denjoy theorem, 935Dense, 32, 270, 287, 735, 949Dense morphism, 774Dense orbit, 958Dense part, 301

of a topological space, 298Dense set, 87, 269Dense valuation, 197Density of orbits (DO), 991Density of rational function, 645Denumerability, 44—46

of the set of rationals, 46Denumerable, 3, 24, 287Denumerable set, 24, 45, 70, 72, 106, 287Dependence of criteria (population), 842Dependent variable, 834, 925Depth, 712Derivability, 445Derivable function, 446Derivate, 446Derivation, 153Derivation of distributions, 674, 903Derivation operator, 713Derivation operator (differentiation), 713Derivation rule, 446Derivation rules, 531Derivative, 160Derivative of a rational fraction, 160Derivative of cumulative distribution function, 839Derivative of zeta-function, 589Derived category, 801Derived functor, 753—755, 808Derived polynomial, 153, 163Derived set, 500Derived structure, 51Desargues and Pappus-Pascal theorems, 211Desargues theorem, 211, 212Descartes, 5Descartes folium, 454Descending chain condition (DCC), 709Descriptive geometry, 204, 236Destination, 809Destruction of a torus, 1000Detection of error autocorrelation, 856

causes, 856

1066

Index

detection, 856Breusch-Godfrey test, 857Durbin-Watson test, 856

Detection of heteroscedasticity, 859Detection tests of multicollinearity, 854Determinant, 142, 179, 969

calculation rules, 142Determinant calculation rules, 142, 467Determination

of argument, 616of logarithm, 616

Determination of a measure, 475Determination of lag number, 874Determination of lag numbers, 865Deterministic walk, 871Devaney, 991Developable surface, 546Deviation, 312, 926Dfferential form

change of variable, 554Diagonal approximation, 431Diagonal morphism, 764Diagonalization, 179, 181Diagram, 369, 813Diagram of the pure rotation and its lift, 934Dice, 351, 832Dickey-Fuller

augmented, 871Dickey-Fuller test (DF), 870Diffeomorphism, 734, 740, 927, 933

hyperbolic linear, 935Diffeomorphism and periodic points, 934Diffeomorphism of the circle, 929Diffeomorphisms and flows, 927Difference, 11, 15Difference filter order, 870Differentiability, 445, 463

complex, 573, 743real, 573, 743

Differentiable 2-chain, 688Differentiable atlas, 559—561, 734Differentiable concave function, 916Differentiable convex function, 916Differentiable function, 446Differentiable manifold, 560, 561, 697, 732, 942Differentiable operator, 503, 504Differentiable p-form, 555Differentiable structure, 732, 733Differential, 449, 554, 752Differential (coboundary map), 752Differential 1-form, 600Differential 2-form, 608Differential algebra, 714Differential calculus, xix, 444, 462Differential calculus in Rn, 462Differential complex, 707, 724Differential Equation, 511Differential equation, xx

(linear) homogeneous, 517exact, 514explicit form, 511implicit, 516initial conditions, 512initial-value problem, 512of second order, 516order, 511structurally stable (robustness), 979with one variable, 512with separated variables, 513

Differential equation system, 519Differential field, 714Differential form, 426, 553, 554

closed form, 606exact form, 606in the plane, 606on a differentiable manifold, 555

on an open subset of C, 676Differential form of degee 1, 599Differential forms of degree 2, 608Differential geometry, xx, 530, 562

curves, 531Differential graded algebra, 714Differential k-forms, 186Differential manifold, 927Differential module, 708, 709, 714Differential operator, 506, 562, 903Differential p-form, 427Differential ring, 713Differential sheaf, 749Differential topology, 562Differentially stable (d-stable), 1020Differentially stable matrix, 1026Differentiation, 957Differentiation of degree n, 392Diffusion model, 861, 862Digraph, 812Dilation, 225Dilation parameter, 900Dimension, 398Dimension axiom, 389, 409, 422, 744, 750, 808Dimension axiom theorem, 422, 428Dimension invariance, 289Dimension of a simplicial complex, 398Dimension of the measure µ, 989Dimension theory, 289Dimensions, 988

Dimensions through invariant measures, 989correlation dimension, 989dimension of a measure, 989generalized dimension, 989information dimension, 989Lyapunov dimension, 989

Local Hausdorff dim. (Douady-Oesterlé), 990metric dimensions

Box-counting dimension or capacity, 988fractals, 988Haussdorff dimension, 988self-similarity, 989

Dimetric pro jection, 237Dini theorem, 319Diophantine equation, 195Dirac, 902, 983Dirac δ function and distribution, 891Dirac mass, 673Dirac measure, 981Direct demonstration, 6Direct family, 726Direct family (of algebraic systems), 726Direct family (of sets), 726Direct homothety, 225Direct isometries in R3, 223Direct isometry, 218Direct limit, 360, 727, 813, 814Direct limit (filtered colimit), 799Direct limit (inductive limit), 813Direct limit (of algebraic systems), 726Direct limit (of sets), 726Direct product, 361, 509, 704, 710, 815Direct product of modules, 137Direct product of subgroups, 120Direct product of subgroups (commutative group),

120Direct similarity, 226, 613Direct sum, 180, 361, 393, 430, 509, 704, 799, 815Direct system, 726, 757, 813Direct system in a category, 813Directed angle, 101Directed graph, 351, 788, 811, 812Directed graph (digraph), 812Directed graph (or digraph), 351Directed line segment, 205Directed order, 279, 784

1067

Index

Directed order (filtrant order), 784Directed preorder, 726Directed segment, 205Directed set, 30, 31, 279, 726, 784Directed straight-line, 205Directedness, 351Direction curves, 546Directional blowing-up, 948Directional derivate, 464Directions of principal curvature, 544Directly congruent, 218Directrix, 252, 253Dirichlet box principle, 71Dirichlet conditions, 885Dirichlet Pigeonhole principle, 71Dirichlet problem, 661, 668Disappearance of saddle-node separatrix, 999Disconnected space, 276Disconnectivity, 276Discontinuity, 264, 441Discrete category, 788Discrete complex Fourier coefficients, 887Discrete complex Fourier transformation, 887Discrete distance, 299Discrete dynamical system, 927, 966, 978

invariant manifolds, 978Hadamard-Perron theorem, 978separatrix surfaces, 978transverse homoclinic points, 978

periodic orbits, 978properties of ω-limit set, 978topological conjugacy of discrete systems, 978types of equilibrium points (Steady states), 978

Discrete Haar wavelet transformation, 900Discrete Laplace transformation, 898Discrete random variable, 839Discrete space, 308Discrete topology, 270Discrete valuation, 197Discrete wavelet transformation (DWT), 900Disjoint sets, 35Disk, 96Disk with holes, 610, 612Displacement, 215, 254Dissipative baker map, 990Dissipative system

transitions stemming from linear stability loss,970

Distance, 88, 103, 251, 298, 299between two parts of a metric space, 298Hamming, 299Hausdorff, 299Minkowski, 299on a normed vector space, 299p-adic, 299semi, 299ultra-metric, 299

Distance (in a normed vector space), 322Distance function, 984Distance on a space of functions, 295Distinction between εt and et, 850Distinguished subgroup, 118Distinguished triangle, 800Distributed lag models, 864Distributed parameter system, 1000Distributed parameter system (or infinite-dimensional

system), 1000Distribution, 530, 673, 674, 891, 903, 982

leptokurtic, 870normal, 870

Distribution of lagsfinite, 865infinite, 865

Distribution of order 0, 673Distribution of order 1, 673Distribution of zeros, 628

Distribution theory, 884, 902Distributions, 834Distributions and subharmonic functions, 667Distributive lattice, 30Distributive law, 57, 700Distributivity, 11, 12Divergence, 427, 470, 471Divergence formula, 664Divergence of a vector field, 470Divergence theorem, 472, 494, 557Dividend, 86Divides, 128Divisibility, 74, 191

a divides b, 191a is a divisor of b, 191a is a factor of b, 191b is a multiple of a, 191b is divisible by a, 191in a field, 196

Divisibility in integral domain (entire ring), 128Divisibility in N, 74Divisibility in Z, 74Divisibility in polynomial ring, 162Divisibility theory, 196Divisible, 191Divisible group, 429Divisor, 86, 191, 199Divisor function, 589Domain, 19, 38, 40, 51, 125, 571, 681, 695, 704Domain of definition, 38, 51, 160, 704Domain of variation, 786Dominant coefficient, 146Dominant morphism, 774Dominant term, 146Dominant term of a polynomial, 146Dot product, 28, 249Douady-Oesterlé theorem, 990Douady-Oesterlé theorem (for diff. eq.), 990Double cone, 421Double integral, 481, 483Double value principle, 1Double-angle formula, 242Doubling the cube, 102Doubly periodic function, 590, 592Doubly stochastic, 834DS process, 870

with drift, 870without drift, 870

DS process (random walk model), 870DS processes (Difference Stationary), 870Dual

algebraic, 324topological, 324

Dual basis, 324, 548Dual category, 789, 790Dual space, 26, 28, 187, 501, 548, 551Dual space functor, 370Dual space of V (V ∗=Hom(V, k)), 551Dual universal coefficients theorem, 429Dual vector space, 141, 548Duality principle, 907Duffing attractor, 959, 961Duffing system, 961Dummy index, 53Dummy variable, 852Dupin indicatrix, 545, 567, 568Durbin-Watson test, 864Durbin-Watson test (DW), 856Dyadic wavelet transformation, 900Dynamical model, 876Dynamical system, xx, 924, 927, 965

conservative, 967dissipative, 967volume-contracting (or dissipative), 967volume-preserving (or conservative), 967

1068

Index

E-cyclic ob ject, 410ε-neighborhood, 271Eccentricity, 252, 253Edge, 350Edge connectivity, 355Edge number, 356Edge of simplicial complex, 401Edge path, 401Edge path group, 402Edge-preserving bijection, 350Eigen decomposition, 180Eigendirection, 258Eigenfunction method, 528Eigenspace

stable, 958unstable, 958

Eigensubspace, 179Eigenvalue, 179, 180, 521, 976

Eigenvalue analysis of the matrix eTR, 970Eigenvalue problem, 179, 181Eigenvector, 521Eilenberg, 779Eilenberg criterion, 643Eilenberg-Mac Lane, 780Eilenberg-Moore algebras, 803Eilenberg-Moore algebras (of a monad), 803Eilenberg-Moore category, 803Eilenberg-Steenrod axioms, 744, 749Eilenberg-Steenrod theorem, 409Eilenberg-Steenrod theorem extension, 409Eilenberg-Zilber theorem, 412Einheit, 56, 699Einstein summation, 465, 548Einstein summation convention, 548, 550Eisenstein series, 589Elastic transformation, 264Elation, 213Elementary i-chain, 685Elementary parallelogram, 587Elementary symmetric function, 153Elementary topos, 822Ellipse, 252, 253Ellipse circumference, 486Elliptic, 216Elliptic function, 590Elliptic function order, 591Elliptic geometry, 243Elliptic invariant, 589Elliptic metric plane, 210Elliptic plane, 207Elliptic point, 544, 545, 567Elliptic right triangle, 245Elliptic umbilic, 544Embed a metric plane in a pro jective plane, 213Embedded hypersurface, 737Embedded m-dimensional tori, 974Embedded submanifold, 737Embedding, 88, 562, 705, 738, 804

of metric planes in pro jective planes, 214Embedding (imbedding), 738Embedding (inclusion functor), 790Embedding of Q in R, 88Embedding space, 562Embedding Whitney theorem, 562Empirical coeffcicient, 846Empirical estimator of the variance, 848Empirical t-student, 846Empty map, 40Empty set, 9Encryption, 131End of a path, 378End of edge of simplicial complex, 401Endoarrow, 812Endofunctor, 802Endogenous variable, 845, 846Endomorphism, 44, 180, 705, 809

Endomorphism (of algebraic system), 705Endomorphism eigenvector, 258Endomorphism reduction, 180Engle, 875Engle-Granger, 877Engle-Granger algorithm, 876Enough H-injectives, 752Enough injective ob jects, 752Enough injectives, 752, 753, 755Enough projectives, 752Enriched category, 795Enriched category identity, 796Enriched functor, 807Entering variable, 905Entire function, 586, 619Entire ring, 57, 125, 700Entire series, 178, 444, 579, 614, 619

convergence sets, 596of several variables, 595with nonzero radius, 451

Entire series (power series), 178Entire series expansion, 619Entourages, 313Entropies

metric entropy, 984topological entropy, 984

Entropy, 984Kolmogorov-Sinai, 984metric, 984

Enumerable, 24, 287Enumeration, 50Envelope

convex, 652upper, 312

Envelope of holomorphy, 651Epigraph, 473Epimenides, 18Epimorphism, 51, 705, 808—810Epimorphism (epi), 358Epimorphism (epic), 809Epimorphism (or epi), 808, 819Equal sets, 9, 18Equality of two pairs, 36Equality sign, 4Equality test of variances, 859Equalizer, 413, 816Equation, 143Equation of 3rd degree

resolution, 173Equation of 4th degree

resolution, 173Equation of a straight line, 215Equation of line, 250Equation of second degree, 254

in two coordinates, 254Equation of tangent plane, 540Equation of the first degree, 214Equation solvable by radicals, 174Equation with determinant, 251Equation with point and normal, 251Equation with vector product, 250Equicontinuity, 320Equicontinuous, 320

family of maps, 320Equicontinuous family, 320Equicontinuous family of maps, 320Equidecomposable, 23, 24Equilateral hyperbolic function, 242Equilibrium in a distribution economy, 915Equilibrium point of type (m,k), 972Equipollent, 39, 44Equipollent bipoints, 220Equipotence, 44, 45, 48Equipotent, 39, 44Equipotent set, 45Equivalence, 804

1069

Index

Equivalence class, 37, 274, 373Equivalence in a category, 370Equivalence of categories, 360, 804, 806Equivalence of flows, 935Equivalence of loops, 602Equivalence of paths, 602Equivalence relation, 23, 37, 86, 89, 274, 373, 560Equivalent absolute values, 197Equivalent arcs, 560Equivalent covering spaces, 416, 417Equivalent generating system, 272Equivalent local families, 962Equivalent norms, 322Equivalent norms (in a normed vector space), 322Equivalent representations of Fourier integrals, 894Equivalents sets (equipotent sets, 63Equivariant map, 1002Ergodic dynamical system, 981Ergodic measure, 981Ergodic theory, 981Error, 847Error correction model (ECM), 875, 876Error εt, 847Error of approximation, 450Error of specification, 846Essential extension, 751Essential monomorphism, 751Essential singularity, 570, 630Essential submodule, 751Essential supremum, 529Essentially surjective, 804, 805Essentially unique, 76Estimation method of d

aggregate variance, 882aggregated absolute value, 882periodogram, 882R/S, 882variance of the residuals, 882wavelets, 882Whittle, 882

Estimation methods of nonlinear models, 863Estimation of parameters, 846Estimation procedures for error correlation, 857Estimator, 846

BLUE, 850, 857, 859of fractional integration parameter, 882of Hurst exponent, 882

Estimator of coefficients, 846Estimator of Generalized Least Squares, 856Estimator of the variance of the error, 848

Et(X), 769, 770

Et/X, 770Etalé space, 824Etale, 769Etale cohomology, 769, 771, 772Etale cohomology group, 771, 772Etale cover, 768, 772, 825Etale fundamental group, 769Etale map, 769Etale map of schemes, 825Etale morphism, 769, 770Etale neighborhood, 769Etale presheaf, 825Etale presheaf on a scheme, 825, 827Etale sheaf, 825Etale site, 769Etale site of a scheme, 825, 827Etale topology, 769, 770Etale topology (algebraic geometry), 769Etale topos, 769, 825Etale X-scheme, 769Euclid algorithm, 77, 150Euclid axioms, 203Euclid lemma, 75, 77, 80, 129Euclid lemma (for polynomials), 149Euclid lemma in Z, 80

Euclid theorem, 75, 230Euclidean algorithm, 192Euclidean distance, 299Euclidean division, 74, 79, 147Euclidean division algorithm, 148Euclidean division of polynomials, 147Euclidean function, 192

monotone, 192Euclidean geometry, 203, 204, 217Euclidean group E(n), 222, 223Euclidean metric, 209, 263, 271, 553Euclidean moves, 223Euclidean plane, 209Euclidean polynomial division, 162Euclidean ring, 192—194Euclidean space, 286Euclidean transformation, 219Euclidean vector norm, 979Euclidean/non-Euclidean metric plane, 209Euler affinity, 230Euler approximation of Duffing attractor, 959Euler characteristic, 400, 558Euler characteristic (of a genus), 558Euler characteristic (of a topological space), 558Euler constant, 458Euler criterion, 195Euler differential equation, 505Euler formula, 103, 400Euler formulas, 100, 885Euler function, 195, 457Euler line, 227Euler product formula, 77Euler theorem, 130, 227, 544Euler totient, 77, 129Euler totient function, 77, 129Euler’s displacement theorem, 254Euler-Cauchy method, 524Euler-Lagrange theorem, 112, 113Euler-Poincaré characteristic, 400, 558Euler-Poincaré characteristic of (X,E), 408Euler-Poincaré characteristic of a polyhedron, 401Euler-Poincaré characteristic of K, 400Eulerian chain, 353Eulerian cycle, 353Eulerian function, 880Eulerian graph, 353Eulerian multiplier, 514Eulerian path, 353Eulerian trail, 353Eulerian walk, 353Euclidean division in Z, 79Evaluation map, 418Even and odd polynomials, 146Even polynomial, 146Evenly covered open set, 413Event, 832, 836Evolute, 537Evolute of a curve, 536, 537Evolute of a plane curve, 538Evolute of an arc, 537Evolvent, 536Exact differential equation, 514Exact differential form, 514Exact form, 607Exact functor, 804Exact homology sequence, 393Exact sequence, 392, 423, 691, 750, 819Exact sequence (acyclic complex), 392Exact triangle, 394Exact triangle theorem, 394Exactness axiom, 409, 750Exactness of a sequence, 393Exchange rules, 3Excision, 395Excision 1, 395Excision 2, 395

1070

Index

Excision axiom, 409, 750, 808Excision corollary, 407Excision theorem, 400, 428Excluded third principle, 2Exclusion restrictions, 867Exhaustion sequence, 621Exhaustion sequence of compact sets, 621Existence of a root of a polynomiel, 162Existence of covering space, 416Existence of null element, 12Existential quantifier, 14Exogenous variable, 845Expansion, 941Expansion of 1/Γ, 634Expansion of 1/ζ, 634Expectation, 833, 840Expectation (of a random variable), 833Expectation of the error, 846Expected value, 833Experiment, 833Explained variability (SSE), 848Explanatory variable, 834, 846Explosion, 1010Exponent, 92

irrational, 92rational, 92

Exponential, 97, 456Exponential absolute value, 197Exponential Fourier transformation, 894Exponential function, 97, 456, 615Exponential functor, 822Exponential law, 418Exponential ob ject, 785, 794Exponential series of matrices, 520Expression

identical, 3satisfiable, 3

Ext functor, 755Extended complex plane, 570, 571

Extended complex plane (C, d), 571Extended complex plane C, 570Extended Euclid algorithm, 151Extended Gauss complex plane, 571Extended real line, 304, 571

compact, 304Extended real number line, 296Extending by linearity, 386Extension, 776

of absolute values, 198of natural numbers, 85of semiring of natural numbers, 85

Extension field, 88, 92, 164, 719Extension field degree, 164Extension field of Q, 88Extension field of R, 88, 93, 571Extension of a field, 164Extension of a map, 39, 776Extension of a semigroup, 122Extension of first-order predicate calculus, 4Extension of Q, 89Extension of Q to R, 87Extension theorem, 311Extensional definition, 14Exterior, 291Exterior algebra, 185—187, 430

of a vector space, 188of alternating multilinear forms, 187

Exterior algebra (Grassmann algebra), 185Exterior angle, 221Exterior angle of a polygon, 221Exterior calculus, 186Exterior derivative, 427, 493, 609Exterior derivative of a 1-form, 609Exterior differential, 554Exterior differential form, 186Exterior differentiation, 554

Exterior differentiation of differential forms, 186Exterior normal, 664Exterior point, 265, 269Exterior power, 427Exterior product, 186—188, 675

of two antisymmetric multilinear forms, 186universal properties, 188

Exterior product (wedge product), 187External center of similitude, 227External composition law, 58, 701External direct sum, 711External homothetic center, 227Extracted family, 41Extracted sequence, 300Extraction of a root nth, 104Extraordinary cohomology theory, 409Extrema, 505Extrema under constraints, 469Extreme value theorem, 285Extremum, 285Extrinsic curvature, 566

F-acyclic, 754F-acyclic ob ject, 753F-algebra, 803F-distribution, 834F-homomorphism, 706F-model set, 410F-projective n-space, 404F-test, 834Face, 398, 410Factor, 191Factor ring, 57, 713Factor through, 773, 793Factorable polynomial, 162Factorial ring, 167, 192, 193Factorization, 76Factorization theorem, 75Faithful functor, 360, 804Faithfully flat module, 765False proposition, 1Family, 41Family canonically associated with a set, 41Family of elements, 41Family of parts of a set, 42Family of polynomials with graded degrees, 147Family of sets, 17, 42Fano axiom, 215Fano axiom (P5),(P5’), 215Farrar-Glauber test, 854Fast Fourier transformation (FFT), 887Fast wavelet transformation (FWT), 900Fatou theorem, 662Fber, 363Fber bundle, 363, 759Feasible polyhedron, 907Feasible region, 904, 906Feasible region of optimization problem, 907Feasible value, 907Feasible value of optimization problem, 907Feedback effect, 875Feigenbaum attractor, 967Fermat conjecture, 130Fermat equation, 195Fermat infinite descent, 69Fermat last theorem, 130Fermat little theorem, 130Fermat method of infinite descent, 69Fermat number, 177, 201Fermat principle of infinite descent, 69Fermat theorem, 126, 130Ferrari method, 173Feuerbach circle, 227Fiber, 71, 373, 413, 705, 759, 770Fiber (algebraic geometry), 759Fiber (topology), 758

1071

Index

Fiber coproduct, 818Fiber of a map, 71, 758, 759Fiber of cover, 413Fiber product, 425, 817Fiber space, 363, 761Fibered coproduct, 818Fibered sum, 818Fibonacci numbers, 1003Fibration, 364, 424, 761Fibration (Hurewicz fiber space), 424Fibre bundle, 761Fibre product, 359, 361Fibre square, 361FIEGARCH, 881Field, 50, 57, 58, 109, 124, 132, 194, 487, 700

left, 107noncommutative, 107

Field adjunction , 165Field characteristic, 169Field extension, 164Field homomorphism, 134, 706Field line, 471Field of complex numbers, 571

algebraically closed, 104Field of complex numbers is algebraically closed,

155Field of fractions, 133, 134, 158Field of fractions (fraction field, field of quotients,

quotient field), 158Field of fractions of a ring, 133Field of fractions of R, 133Field of parallel vectors, 564Field of quotients, 133, 135, 158Field of rational numbers, 86Field of residue classes, 197Field theory, 110, 698Field with characteristic zero, 168FIGARCH (fractionally integrated GARCH), 881FIGARCH (p,d,q), 881Filter, 26, 278, 284, 302, 303, 736

Cauchy, 314determined by a sequence, 314Fréchet, 302of neighborhoods, 302smallest, 302

Filter base, 30, 278, 284, 736Filter bases, 302Filtered colimit, 799Filtered set, 30, 726Filtrant category, 796Filtration, 407Final topology, 273Finitary operation, 692, 693Finite Abelian group, 120Finite distribution of lags, 865Finite extension, 164Finite field, 168, 169Finite Fourier transformation, 888Finite Galois extension, 171

properties, 171Finite Galois extension of Q, 176Finite graph, 351Finite group, 110Finite Laplace transformation, 888Finite morphism, 766Finite presentation, 770Finite product, 295Finite rank operator, 508Finite separable field extension, 770Finite separable k(y)-scheme, 770Finite set, 6, 70Finite simple extension, 166Finite support, 53, 145Finite type, 429, 767Finitely based chain complex, 401Finitely generated, 400

Finitely generated ideal, 717Finitely generated module, 766Finitely generated morphism, 766Finitely presented, 402Finitely presented algebra, 717Finitely presented group, 404Finitely presented R-algebra, 717Finsler manifold, 732First axiom of countability, 728, 736First category, 315First countable, 287, 735First countable basis, 287First countable space, 288First Cousin problem, 652First de Rham cohomology vector space, 681First difference model, 858First fundamental form, 541

of a sphere, 541patch theory, 541

First group of Rham cohomology, 608First homology group, 645First quasi-differences, 858First-countable, 728First-countable space, 288First-difference filter, 870First-order differential equation, 512First-order predicate calculus, 3First-order predicate calculus with identity, 4Fisher law, 848Five lemma, 394Fixed line, 236Fixed point, 40, 116, 236, 928, 988Fixed point of a projective collineation, 236Fixed point theorem, 312Fixed-point theorems, 914Fixed-point theorems (Brouwer and Kakutani), 911Flabby sheaf, 757Flag, 205, 217Flasque resolution (flabby resolution), 758Flasque sheaf, 757Flasque sheaf (flabby sheaf ), 757Flat manifold, 565Flat module, 765Flat morphism, 765, 770Flat point, 567Flat unramified morphism, 770Floor, 82Floor function, 82, 844Floquet characteristic multiplier, 970, 1010Floquet matrix, 970, 976, 993Floquet theorem, 945, 969Floquet theory, 969, 993Floquet theory in dissipative systems, 970Flow, 965, 967Flow of a differential equation, 966Flow on the torus, 936Flow-box, 934Flow-box theorem, 934Flows and differential equations, 930Focal parameter, 253Focus, 253Folding map, 419Folium, 454Force, 487Force along a curve, 488Forced pendulum, 1012Forecast interval, 853Forecast reliability, 853Forecasting error, 848Forest, 353Forgetful functor, 370, 807Formal proof, 6Formal system, 5, 6Formally étale, 771Formally étale morphisms of schemes, 771

1072

Index

Formally étale=Formally smooth+Formally unram-ified, 771

Formally smooth, 771Formally unramified, 771Formula of a parallelogram, 329Formula of the median, 329Fortuitous correlation, 846Forward regression, 855Foundations of mathematics, 1Four-color problem, 353Fourier, 527Fourier coefficient, 598, 620, 623, 624, 670, 883Fourier coefficients, 332, 887

determination for symmetric functions, 885symmetry of the first kind, 885symmetry of the fourth kind, 886symmetry of the second kind, 885symmetry of the third kind, 885

determination with numerical methods, 886Fourier cosine transformation, 888, 894Fourier expansion, 885Fourier integral, 647, 893Fourier integral in complex representation, 893Fourier series, 589, 590, 661, 883, 885

complex representation, 885important properties, 885

Fourier series and Fourier integrals, 886Fourier series and transform

usual writings, 884Fourier series convergence, 883Fourier series expansion, 619

complex, 590real, 590three different forms, 886

Fourier series integration, 884Fourier sine transformation, 888, 894Fourier sum, 885Fourier transform, 528, 883, 884

continuous writing, 884discrete writing, 884three continuous writings, 885

Fourier transformable, 894Fourier transformation, 672, 888, 893Fourier transformation (definition), 894Fourier-Plancherel transformation, 884Fppf cover, 767Fpqc cover, 767Fréchet filter, 302, 736Frénet basis, 534, 538Frénet formulas, 534, 538Frénet trihedral, 533, 534Fractal dimensionality, 988Fractal dimensionality (Mandelbrot dimensionality),

988Fractals, 988Fraction field, 133, 158Fractional Brownian motion, 880Fractional Gaussian process, 880Fractional ideal, 199Fractional integration parameter, 880Frames, 900Frechet differential, 504Fredholm alternative, 510, 528Fredholm determinant, 527Fredholm integral equation, 527Fredholm operator, 510Fredholm theorem, 507, 527Fredholm theorems, 527Fredholm theory, 527Free A-module, 427Free abelian, 400Free Abelian group, 702, 704Free algebra, 702, 709Free associative algebra , 710Free chain complex, 409Free commutative algebra, 702, 709

Free functor with basis in M, 410Free group, 138, 345, 402, 403, 682, 703, 704Free group of rank m, 403Free group with basis, 402Free homotopic loops, 682Free homotopy, 371Free ideal ring, 710Free ideal ring (fir), 710Free mobility, 217Free module, 138, 710Free module (homological algebra), 710Free monoid, 702, 709Free object, 704, 785Free object (universal algebra), 704Free orthogonal system, 331Free part, 138Free product, 402Free R-module, 702, 710Free resolution, 753Free ring, 710Free variable, 3Free Z-module, 704Frege, 5, 18Frequency locking, 1003, 1007Frequency spectrum, 894Frequency table, 839Freudenthal suspension theorem, 426Frobenius method, 564Frobenius theorem, 564From Torus to chaos, 1000Front face, 430Frontier, 266, 398Frontier (or boundary), 266Frontier point, 266, 269, 297Frontier set, 266Fubini formula, 71Fulcrum, 259Full functor, 804Full subcategory, 358, 380, 802Fully faithful functor , 804Function, 19, 38

harmonic, 574Lagrange, 910logarithmic barrier, 910one-to-one, 39penalty, 910subharmonic

nontrivial, 667test, 902totient, 91

Function from R to Rm, 463Function from Rn to R, 463Function from Rn to Rm, 466Function from Rn to Rn, 467Function Γ, 622Function ζ, 621Function of differentiable real variable, 445Function ℘, 588, 592Function preserving structure, 780Function space, 317, 418Function symbols, 694Function theory, xx, 569Function theory (complex analysis), xxFunctional, 26, 27, 501, 529Functional analysis, xix, 501Functional axiom, 18Functions, 38Functions of several variables, 595Functor, 347, 348, 358, 359, 368, 370, 779, 780, 803Functor categories, 779Functor category, 792Functor Hn, 367Functor in algebraic topology, 348Functor Spec, 774Functorial, 794Functorial isomorphism, 794

1073

Index

Functorial method, 347Functorial variable, 4Functoriality, 794Fundamental (homotopy) group, 339Fundamental group, 336, 337, 339, 345, 379

of a connected polyhedron, 344Fundamental groupoid, 380Fundamental groups of polyhedra, 401Fundamental homotopy groups , 347Fundamental matrix, 521

complex, 522real, 522

Fundamental neighborhood, 413Fundamental neighborhood of base space, 413Fundamental relations

normal, geodesic curvature, curvature, 542Fundamental relations between χn,χg,χ, 542Fundamental sequence, 89Fundamental set, 11Fundamental structure, 50, 51Fundamental system, 518Fundamental system of neighborhoods, 272, 300,

312, 318Fundamental system of neighborhoods (Basis of neigh-

borhoods), 272Fundamental tensor, 550, 565Fundamental theorem

for homogeneous linear diff. equations, 968for non-homogeneous linear equations, 968theory of patches, 548

Fundamental theorem of algebra, 104, 383, 385Fundamental theorem of arithmetic, 76Fundamental theorem of arithmetic (for polynomi-

als), 149Fundamental theorem of curve theory, 534Fundamental theorem of integral calculus, 472, 476Fundamental theorem of pro jective geometry, 212Fundamental theorem of Riemannian geometry, 566Fuzzy set theory, 21Fuzzy set:, 21

G-equivariant, 416G-isomorphism, 416G-map, 416G-module homomorphism, 707G-paradoxical set, 10G-set, 414G-space, 414, 761Gödel, 3, 50Gödel completeness theorem, 3Gödel incompleteness theorem, 4Gödel second incompleteness theorem, 4Gabor transformation, 901Galerkin method, 1015Galois field, 58, 169, 170, 701Galois group, 170Galois group of a polynomial, 172Galois map, 170Galois theory, 51, 110, 170, 698Galton board, 840Galton board (quincunx, bean machine), 840Galtonian curve, 841Gambling, 832Gamma function, 457, 880Gamma-distribution, 844GARCH(p,q), 881GARMA process, 881Gateaux differential, 504Gauss and Mainardi-Codazzi relations, 547Gauss complex plane, 103Gauss coordinate system, 547Gauss curvature, 545, 547Gauss formula, 681Gauss formulas, 546, 547Gauss function, 841Gauss lemma, 196

Gauss lemma (for polynomials), 149Gauss parameters, 539Gauss plane, 571Gauss relations, 547Gauss staircase function, 440Gauss theorem, 77, 80, 472Gauss theorem in Z, 80Gauss trihedron, 546Gauss-Newton algorithm, 863Gaussian curvature, 545, 566, 568Gaussian curvature of a surface, 568Gaussian elimination, 179Gaussian white noise, 869Gavrilov-Shilnikov, 1013, 1014GCH, 20Gegenbauer, 881Gegenbauer frequency, 881General equation of degree n, 174General equilibrium, 915General equilibrium (economics), 914General form of dynamic model, 924General form of state-space model, 925General Lie group, 178General linear group, 177, 235, 563General linear group GLn(q), 235, 261, 563General linear group GLn(R), 235, 563General linear model, 849General monic polynomial, 173General orthogonal group GOn(q), 261General orthogonal group GOn(q, F ), 235, 261General position, 377General quantifier, 3, 14General set theory, 20General term, 326General topology, 263General valency, 551Generalization of Cauchy criterion, 310Generalization of VECM, 878Generalized ARFIMA process, 880Generalized continuum hypothesis (GCH), 20Generalized derivative, 530Generalized dimension, 989Generalized element, 786Generalized Euclidean space, 28, 184Generalized function, 530, 891Generalized function (or distribution), 530Generalized functions, 27Generalized Hopf bifurcation, 963, 964, 996Generalized Hopf singularity, 964Generalized integral, 480, 481Generalized Lagrangian, 920Generalized lagrangian, 918Generalized least squares (GLS), 856Generalized sequence, 279, 280, 436Generating family, 120Generating part, 138Generating set, 704Generating system, 272Generator, 113, 402, 704, 710, 786, 799, 800, 823Generator , 129Generator of an ideal, 129Generator of an ideal , 149Generator part, 229Generators of an abelian group, 386Generatrix, 542Generatrix lines, 546Generic element, 787, 824Generic property, 979Generic property (or typical property), 979Gentzen-type, 2Genus, 347Genus theorem, 196Geodesic, 243, 504, 505, 542, 564Geodesic curvature, 542, 543Geodesic line, 241, 543Geometric arc, 531

1074

Index

Geometric complex, 340, 346Geometric congruence, 208Geometric construction, 174Geometric fiber, 759, 770Geometric point, 759Geometric realization, 399Geometric sequence, 436Geometric series, 438Geometry, xix, 203

absolute, 203, 206analytic, 204, 247axiom C (connection), 209axiom D (completion) Dedekin, 217axiom H (hyperbolic), 210axiom P (polar trilateral), 210axiom R (rectangle), 209axioms A,A∗ (bisector), 216axioms A1,A2,A3 (affine), 211axioms M1,M2,M3 (metric plane), 206axioms P1,P2,P3 (pro jective), 212axioms P5,P5’ (Fano), 215descriptive, 204, 236elliptic, 203, 243Euclidean, 204hyperblolic, 240hyperbolic, 203, 207Lobachevsky, 207Lobachevsky-Bolyai-Gauss, 203, 207non-Euclidean, 203ordinary, 205parabolic, 203pro jective, 203spherical, 244

Gereno lemniscate, 455Germ, 747Germ of an analytic set, 758Giraud theorem, 823Glesjer test, 860Glide, 221Glide reflection, 221Global bifurcation, 996, 998, 1011Global bifurcations, 994Global continuity, 273Global element, 787, 824Global element (global section, point), 787, 824Global homoclinic bifurcations, 999Global maximum, 918Global mean value, 666Global minimum, 918Global Pro j, 774, 775Global section, 756, 762, 787, 824Global Spec, 774Global stability, 1028Globally stable equilibrium, 1025GLS estimator, 857, 859Gluing, 371, 741, 820, 825Gluing (concatenation, collation), 820Gluing axiom, 746, 747, 820, 821Gluing condition, 746, 820Gluing lemma, 371Gnomonic pro jection, 244Gode, 758Godement resolution, 757Golbach conjecture, 1Goldbach conjecture, 202Golden rule, 1027Golden theorem (Gauss), 196Goldfeld-Quandt test, 859Gollub-Benson, 983Gompertz model, 862Goodwin cyclical growth model, 1031Goodwin model, 1020Grade, 712Graded algebra, 357, 645, 709Graded co-ring, 433Graded coring, 433

Graded module, 357, 708, 709Graded ring, 357, 366, 430, 645, 708, 709Graded sheaf, 749Graded valuations, 148Gradiant, 664Gradient, 427, 464, 465Gradient operator, 557Gradient theorem, 465, 557Graeco-Latin square, 830Graeffe method, 462Grammar, 6Granger, 875, 879Granger representation theorem, 877Graph, 350, 404, 759, 811Graph automorphism, 351Graph characteristic polynomial, 181Graph coloring, 354Graph coordinates, 731Graph cycle (circuit), 352Graph edge, 350Graph isomorphim, 350Graph of a correspondence, 36Graph of a map, 36, 40Graph of binary relation on a space, 404Graph of map, 38, 705Graph representation, 350Graph theory, xix, 349Graph vertices, 350Graph without loop and spindle, 351Graphical integration, 481Graphical projection, 237

anamorphic pro jection, 237auxiliary view, 237parallel projection, 237

axonometric pro jection, 237axonometric pro jection (dimetric), 237axonometric pro jection (isometric), 237axonometric pro jection (trimetric), 237multiviews, 237multiviews (elevation), 237multiviews (floor plane), 237multiviews (section), 237oblique projection, 237oblique projection (cabinet), 237oblique projection (Cavalier), 237orthographic projection, 237

perspective pro jection, 237Grassman manifold, 186Grassmann algebra, 185Grassmann coordinates, 186, 188Grassmannian, 186Gray code, 902Great circle, 243Greatest common denominator (GCD), 92Greatest common divisor, 76Greatest common divisor (gcd.), 162, 194Greatest common divisor in Z, 80Greatest element, 12, 60, 61Greatest lower bound, 12, 62Greatest lower bound of real function, 440Green formula, 664, 681Green theorem, 386, 427, 472, 494Green-Riemann formula, 491, 556Green-Riemann theorem, 491Grelling, 18Grid, 234Grid search procedure (Hildreth-Lu), 858Gridding, 234Grobman-Hartman theorem, 978Grobman-Hartman theorem (for discrete sys.), 979Gronwall lemma, 1021Gross substitutability assumption, 1029Grothendieck, 365, 779Grothendieck abelian category, 799Grothendieck cohomology, 745Grothendieck pretopology, 821

1075

Index

Grothendieck theorem, 768, 772Grothendieck topology, 770, 821Grothendieck topology axioms, 821Grothendieck topos, 822Grothendieck trace formula, 773Grothendieck universe, 21, 781Grothendieck universe U, 781Group, 50, 51, 56, 109, 110, 695, 699

Abelian, 56, 700, 701isometry, 204, 207Klein, 56of rigid motions, 206similtude, 204transformation, 203

Group by arrows, 798Group commutator , 704Group generators, 138, 345Group GLn(K) of nonsingular matrices, 143, 235Group homomorphism, 51, 114, 706Group isomorphism, 114Group kernel, 119Group law, 110Group object, 418, 419Group of affine collineations G, 235Group of automorphisms, 51Group of complex numbers of modulus 1, 97Group of displacements, 216Group of finite type, 120Group of invertible elements of a ring, 125Group of isometries, 255Group of n-boundaries, 392Group of n-cycles, 392Group of nonsingular matrices, 143Group of orthogonal collineations, 236Group of periods of ω, 684Group of regular matrices, 143Group of relative n-boundaries, 394Group of relative n-cycles, 394Group of rigid motions, 206Group of rotations, 55, 699Group order, 56Group theory, 109, 110, 698Groupoid, 379, 380, 402, 788Groupoid category, 803Groupoid π(K), 402Groupoid theorem, 379Groups obtained by extension, 124Guckenheimer, 991Guckenheimer-Holmes, 944, 964Gysin sequence, 432

h-Cobordism, 742H-group, 420H-injective, 752h-level, 422H-space, 384Hénon map, 940, 967Hénon quadratic map, 940Haar wavelet, 900Hadamar formula, 614Hadamard matrices, 901Hadamard three circles theorem, 666Hadamard-Cartan & Gromoll-Meyer theorem, 568Hadamard-Perron theorem, 974Hadamard-Perron theorem (for discrete sys.), 978Hahn-Banach separation theorem, 35Hahn-Banach theorem, 32, 61, 321, 324Hahn-Banach theorem (geometric form), 35Hairy ball theorem, 398Half-angle formulas, 242Half-ball, 312Half-cone, 252Half-exact functor, 804Half-full functor, 807Half-period ratio, 589Ham-sandwich theorem, 432Hamilton rules, 107

Hamiltonian, 938, 939, 968Hamiltonian chain, 352Hamiltonian circuit, 353Hamiltonian cycle, 353Hamiltonian flow, 938Hamiltonian function, 938Hamiltonian graph, 353Hamiltonian path, 353Hamiltonian system, 938, 961, 979Hamiltonian system with n-degrees of freedom, 938Hamming distance

between two finite sets, 299n-tuplets, 299

Handle, 347, 405, 584Hankel transformation, 888Hardy space, 623, 663Harmonic, 982Harmonic analysis, 886Harmonic division, 226, 228Harmonic function, 574, 659Harmonic homology, 213, 215Harmonic majorant, 666Harmonic series, 438Harmonic upper-bound, 666Harmonicity and holomorphy, 659Harnack inequalities, 660Hartman-Grobman, 942Hartman-Grobman theorem (for diffeom.), 942Hartman-Grobman theorem (for flow), 942Hartogs, 597Hartogs theorem, 670Hasse diagram, 12, 13, 60Hasse Diagram (order diagram), 60Hasse principle, 107Hausdorff axiom, 298Hausdorff dimension of the measure µ, 989Hausdorff dimension properties, 988Hausdorff distance, 299Hausdorff paradox, 23Hausdorff quotient space, 404Hausdorff separation axiom, 281Hausdorff space, 263, 271, 280—282, 729, 730Haussdorf dimension, 988Heat equation, 527, 893Heaviside, 902Heaviside step function, 440Heaviside unit step function, 890Height, 227Heine, 303Heine theorem, 303, 304Helix, 533, 536Henon map, 967Hereditarily finite set, 782Hereditary property, 47Hereditary set, 19, 21, 782Hermitian conjugate operator, 508Hermitian form, 183, 184, 328Hermitian inner product, 28, 184, 508Hermitian inner product (inner product), 508Hermitian integral operator, 510Hermitian norm, 185, 506, 510Hermitian operator, 508Hermitian scalar product, 28, 185Hermitian space, 28, 184Hermitian symmetric sesquilinear form, 184Hermitian symmetry, 184

left semilinear, 184right linear, 184

Hesse form, 251Hessenberg, 211Hessian, 468, 469Hessian matrix, 468Heteroclinic point, 950Heteroclinic saddle connection, 1011Heteroclinic saddle-connection, 1011Heterological adjective, 18

1076

Index

Heteroscedastic, 870Heteroscedasticity, 855, 858, 859Heteroscedasticity), 870Hexadecimal numeral system, 75Hexagon axiom, 794Hexagon identities, 794Higher aithmetic, 67Higher aithmetic (number theory), 67Higher Dimensional Algebra (HDA), 803Higher order partial derivative, 466Higher-dimensional categories, 780Higher-order predicate calculus, 4Hilbert basis, 332, 333Hilbert norm, 184Hilbert plane, 206Hilbert problem, 4Hilbert program, 5Hilbert space, 28, 184, 271, 286, 328, 329, 502, 663Hilbert space of finite dimension, 332Hilbert-type, 2Hildreth-Lu method, 858Histogram, 832Hjelmslev, 206Hjelmslev theorem of perpendiculars, 207, 208Hjemslev, 213Hodge star, 557Hodge star operator, 557Holes, 688, 744Holomorphic C-C-function, 574, 743, 744Holomorphic continuation, 577Holomorphic function, 451, 570, 575, 576, 613, 742

local behavior, 647on an open set of S2, 654

Holomorphic function in an annulus, 629Holomorphically convex, 651Holomorphically convex set, 651Holomorphy, 570, 573, 595

equivalent formulations, 570extension of the concept, 570

Holomorphy of C-C-functions, 570

Holomorphy of C-C-functions, 570Hom functor, 805Hom-set, 788, 789Homeomorphic, 265Homeomorphic image, 268Homeomorphic spaces, 298Homeomorphic topological space, 272Homeomorphism, 263—265, 298, 337, 367, 729, 730,

739, 928, 992Homeomorphism (topology), 729Homeomorphism onto its image, 738Homoclinic and heteroclinic orbits, 975Homoclinic loop, 1011

attracting, 1012repelling , 1012

Homoclinic point, 949, 950Homoclinic points, 959Homoclinic saddle connection, 960, 1011, 1012Homoclinic saddle connection theorem, 1012Homoclinic tangle, 942, 959Homogeneity, 27Homogeneous coordinate, 721Homogeneous coordinates, 342Homogeneous element, 716Homogeneous function, 722Homogeneous ideal, 366, 716, 722Homogeneous integral equation of 2nd kind, 506Homogeneous medium, 893Homogeneous polynomial, 722Homogeneous system, 144Homogeneous transformation, 27Homographic equation, 250Homographic transformation, 594

classification, 594composition, 594

Homography, 655

Homological algebra, 357Homological equation, 943, 945Homologous cycles, 644Homologous points, 227Homology, xix, 213, 340, 346, 356, 744Homology class, 388, 392Homology class of zn, 388Homology classes, 386Homology functor, 387, 808Homology group, 340, 346, 349, 357, 388, 645, 745,

1015Homology group (Betti group), 349Homology of spheres, 396Homology sequence, 808Homology theory, 341, 346, 347, 349, 356, 409, 645,

691Homology theory (H, ∂), 409Homomorphic image, 693, 707Homomorphism, 51, 109, 113, 697, 706, 808, 809,

819Homomorphism (abstract algebra), 706Homomorphism (morphism), 809Homomorphism as cartesian product, 707Homomorphism of Banach algebras, 671Homomorphism of complexes, 707Homomorphism theorem, 119

applications, 119Homomorphism theorem for modules, 137Homomorphism theorem for rings, 135Homomorphisms of an algebraic system, 798Homoscedasticity, 850Homothetic circles, 227Homothetic figures, 225Homothetic ratio, 225Homothetic triangle, 226Homothety, 225Homothety and point at infinity, 225Homothety-reflection, 226Homothety-rotation, 226Homotopic, 335, 1015Homotopic continuous maps, 640Homotopic maps in a basic sense, 337Homotopic path edges, 401Homotopic paths, 335, 682Homotopic paths with fixed endpoints, 335, 637Homotopic topological space, 337Homotopy, xix, 290, 335, 337, 344, 367, 371, 378,

635, 637, 641, 761, 775of continuous maps, 337of polygonal chains, 344

Homotopy (fundamental group), 682Homotopy axiom, 390, 409, 749, 808Homotopy axiom for pairs, 395Homotopy axiom theorem, 390, 428Homotopy category, 371, 412, 420, 801Homotopy category (hTop), 420Homotopy class, 336, 371Homotopy class (mapping class), 775Homotopy equivalence, 337, 371, 412Homotopy equivalence in homotopy category, 371Homotopy extension property, 425Homotopy extension theorem, 407Homotopy function, 337Homotopy group, 336, 339, 418, 421Homotopy identity, 384Homotopy invariance, 384Homotopy invariant, 775Homotopy lifting property, 424, 776Homotopy of continuous maps, 337Homotopy of paths, 335Homotopy sequence of a fibration, 425Homotopy sequence of a pair, 424Homotopy sequence of a weak fibration, 425Homotopy set, 775Homotopy theory, 290, 367Homset, 788, 792

1077

Index

Hooke law, 511Hopf algebra, 433Hopf bifurcation, 962, 971, 995

generalized, 964subcritical, 962supercritical, 962

Hopf bifurcation (normal form), 992Hopf bifurcation theorem, 963Hopf fibration, 425Hopf recurrence theorem, 971Hopf singularity, 962Hopf singularity of type (p,+) or (p,−), 964Hopf singularity of type p, 964Hopf theorem, 433Hopf-Landau model, 1000Horizontal composition, 358, 792Horner schema, 461Horner scheme, 146Horseshoe, 957Horseshoe diffeomorphism, 950, 953, 957

canonical example, 953dynamics on symbol sequences, 955symbolic dynamics for horseshoe diffeo., 956

Horseshoe map, 990Hosking, 879, 881HSP theorem, 697hTop, 420Hurewicz fiber space, 424Hurewicz fiber space (fibration), 424Hurewicz homomorphism, 426Hurewicz map, 391Hurewicz theorem, 390, 391, 425Hurewicz theorem (1), 391Hurewicz theorem (2), 425Hurewicz theorem (absolute version), 426Hurewicz theorem (relative version), 426Hurst exponent (H), 880Hurst exponent estimator, 882Hurst statistic, 880Hurwitz theorem, 647Hypebolic convergence, 879Hyperbola, 252—254Hyperbolic, 216Hyperbolic attractor, 1013Hyperbolic axiom (H), 210Hyperbolic cosine, 242Hyperbolic decreasing, 880, 881Hyperbolic equilibrium point of type (m,k), 978Hyperbolic fixed point, 942, 1011Hyperbolic function, 241, 242, 457

hyperbolic cosecant, 241hyperbolic cosine, 241hyperbolic cotangent, 241hyperbolic secant, 241hyperbolic sine, 241hyperbolic tangent, 241

Hyperbolic geometry, 240Hyperbolic invariant set, 958Hyperbolic linear diffeomorphism, 935, 941, 942,

953Hyperbolic linear flow, 941Hyperbolic metric plane, 210Hyperbolic model, 207Hyperbolic non-wandering set, 958Hyperbolic nonlinear fixed point, 942Hyperbolic periodic orbit, 942Hyperbolic plane, 207Hyperbolic point, 544, 545, 567Hyperbolic Riemann surface, 562Hyperbolic saddle-point, 960Hyperbolic segment, 240Hyperbolic set, 957Hyperbolic sine, 242Hyperbolic singular point (for flow), 942Hyperbolic singularity, 944Hyperbolic structure, 957

Hyperbolic trygonometry, 242Hyperbolicity, 949Hypercomplex number, 47Hyperparallel, 207, 211Hyperplane, 970Hypersurface, 970Hypertetrahedron, 340, 346Hypervolume, 340, 346Hypoellipticity, 668Hypograph, 473Hypotenuse, 239, 242Hypothesis, 2, 6Hypothesis H0, 846Hypothesis of normality of errors, 847Hypothesis of recurrence, 7, 52Hypothesis testing, 834, 835, 841Hysteresis, 1013, 1017

I-adic topology, 723i-boundaries, 675i-boundary, 687i-cocycles, 681i-chains, 675, 685i-chains with integral coefficients, 685i-coboundaries, 681i-cycle, 122, 687I-depth, 712i-faces, 340, 346i-th étale cohomology group, 773i-th cohomology group, 773i-th homology group, 753i-th right derived functor, 754Ideal, 29, 31, 57, 126, 135, 193, 365, 700, 709, 716Ideal element, 5, 203Ideal line, 214ideal ob ject, 5Ideal plane, 213Ideal plane of a metric plane, 214Ideal point, 211, 214Ideal sheaf, 748Ideal sheaf (algebraic geometry), 748Ideal theory, 193Idempotence, 11Idempotent, 125Idempotent element, 43Idempotent ring, 13Identically independently distributed (i.i.d.), 869Identification, 373, 405Identification space, 777Identities for complex arguments, 242Identity, 4, 780Identity element, 56, 57, 699, 700Identity functor, 370Identity map, 39, 40, 236, 298Identity matrix, 142Identity object, 793if and only if (iff ), 7IGARCH (integrated GARCH), 881Image, 19, 38, 40, 114, 393, 704, 705Image of homomorphism, 114Image of morphism, 114Imaginary part, 103Imaginary part , 94Imaginary part of z, 94Immeasurable frequencies, 983Immersed submanifold, 737Immersion, 290, 740Immersion (algebraic geometry), 773Immersion (algebraic topology), 740Immersion theorem, 290Impact multplier, 874Implicit differential equation, 516Implicit differentiation, 454Implicit function, 467Important properties of Fourier series, 885

asymptotic behavior, 885

1078

Index

convergence of function in mean, Parseval equa-tion, 885

Dirichlet conditions, 885least mean squares error of a function, 885

Improper point, 212Impulse function (Dirac δ function), 890In bijection with, 48, 63Inaccessible cardinal, 20, 782, 783Inadequacy of Zariski topology, 768Incidence, 205Incidence-preserving transformation, 213Incident, 207Inclusion, 15Inclusion function, 705, 799Inclusion functor, 790Inclusion functor (embedding), 790Inclusion map, 705, 737, 810Incomplete basis theorem, 139Increasing real function, 440Indefinite integral, 472, 473, 477—479

of algebraic functions, 479of rational functions, 479of rparticular functions, 479

Independent events, 837Indeterminates, 162Index, 582, 638Index change formula, 53Index gymnastics, 550Index of a cycle, 644Index of a loop, 641, 684Index of loop with respect to a point, 638Indicator, 77Indicator variable, 852Indicator variable (or dummy variable), 852Indirect demonstration, 6Indirect isometries in R3, 223Indirect least squares (ILS), 868Indirectly conformal mapping, 592Individual, 2Induced homomorphism, 738Induced law, 43Induced map, 370, 750Induced order, 51, 60Induced orientation, 387Induced topology, 264, 271, 273, 729, 736Induction, 47Induction axiom, 5, 66, 83Induction hypothesis, 7, 52Inductive definition, 7Inductive hypothesis, 7Inductive limit, 745, 747, 813Inductive limit (direct limit), 813Inductive order, 62Inductive ordered set, 62Inductive set, 62Inertia degree, 199Infimum, 62Infinite cyclic group, 400Infinite descent, 69Infinite distribution of lags, 865Infinite graph, 351Infinite polynomial function, 444Infinite product, 439, 633Infinite product expansion, 634Infinite set, 45Infinite-dimensional F-pro jective space, 404Infinite-dimensional system, 1000Infinitesimal calculus, xixInfix form, 43Inflection point, 452Information dimension, 989Initial condition problem, 522Initial conditions, 512Initial morphism, 786Initial ob ject, 418, 784, 786Initial property, 786

Initial segment, 64, 87Initial topology, 273Initial-value problem, 512Injection, 419, 705Injective arcs on manifold, 560Injective envelope (injective hull), 751Injective homomorphism, 705, 752Injective homomorphism (monomorphism), 705Injective immersion, 737, 738Injective linear transformation, 740Injective map, 39, 41, 705Injective module, 750, 751Injective ob ject, 751—753Injective patch, 549Injective resolution, 752, 753Inner product, 28, 184, 185, 249, 508Inner product space, 184, 185Inner-product space, 28Innovation, 874Input variable, 925Inscribed angle, 222Inscribed angles in a circle, 222Inscribed circle, 239Insertion function, 799Inset, 955, 958Insets and invariant manifold, 958Instability, 970

subharmonic, 993Instrumental variable, 861Integer, 85

algebraic structure, 85order structure, 85

Integer modulo n, 117Integer part, 82Integer part of a rational fraction, 159Integers

calculation rules, 86Integrability of the logarithm, 628Integrable system, 939Integral, 199Integral calculation

complement formula (meromorphic embedding ofζ), 648

Fourier integral, 647of a rational function without real poles, 647trigonometric inttegral, 647

Integral calculus, xix, 472Integral closure, 199Integral domain, 57, 125, 134, 135, 162, 193, 619,

700, 709divisibility, 191

Integral domain (entire ring), 57, 125, 700Integral domain by prime ideal, 135Integral equation, 506, 507

(homogeneous) of the second kind, 506(nonhomogeneous) of the second kind, 507classification, 507of 1st kind, 506

Integral equation of the first kind, 506Integral equations of the second kind, 507Integral function, 477, 619Integral kernel, 511Integral mean-value, 476Integral of a 1-form on a part of R, 602Integral of a differential 2-form, 680Integral of a differential form, 555

according to a parameter, 555Integral of closed 1-form on a continuous path, 636Integral of surface, 490Integral on a continuous path, 636Integral on a piecewise-C1 path, 603Integral on an oriented curve, 605Integral operator, 506Integral property, 125, 133Integral scheme, 764Integral transformation, 887

1079

Index

Integral transformations, 887finite Fourier transformation, 888finite Laplace transformation, 888for functions of several variables, 888Fourier cosine transformation, 888Fourier sine transformation, 888Fourier transformation, 888, 893Hankel transformation of order, 888Laplace transformation, 888Laplace-Carson transformation, 888Mellin transformation, 888of functions of one variable, 888one-sided Fourier transformation, 888Stieltjes transformation, 888two-sided transformation, 888Walsh functions, 901wavelet transformation, 899Z-transformation, 897

Integrand, 472Integrating factor, 514Integration by parts, 478Integration methods, 478Integration of 0-form, 687Integration of 1-forms, 687Integration of a C-C-function, 575Integration of series, 480Integration rule, 475Integration rules, 575Integration theorems, 491Integrations term by term, 480Interior, 266, 291, 294, 451, 727, 742Interior angle, 221Interior angle of a polygon, 221Interior point, 265, 269, 297, 727, 911Interior point methods, 910Interior product, 187Interleave sequence, 293Intermediate field, 164, 165Intermediate value theorem, 276, 307, 442Intermittency, 999, 1009

type 1, 971, 1010type 2, 1011type 3, 1011

Internal (composition) laws of lattice, 12Internal angle bisector, 228Internal center of similitude, 227Internal composition law, 52, 55, 56, 698, 699Internal direct sum, 711, 712Internal Hom functor, 794Internal homothetic center, 227Internally homologous, 506Interpolation, 459Interpolation theory, 459, 460Interpretation, 696Interpretation (logic), 696Interpretation (model theory), 696Interpretation of multipliers, 921Intersection, 11, 15, 17Intersection of a simple arc with a sphere, 535Intersection of neighborhoods, 266Intersection of open sets, 266Intersection of subcomplexes, 393Interval of convergence, 444Intput variable, 925Intrinsic curvature, 566Intrinsic geometry, 542, 561Intrinsic parameter, 532Intrinsic property, 268Intuitionism, 5Intuitionistic logic, 5Intuitionistic propositional logic, 5Intuitionistic Zermelo-Fraenkel (IZF), 21Invariance of dimension, 398Invariance of domain, 398Invariance of integral by homotopy, 637Invariant by homeomorphism, 638

Invariant Cantor set, 935Invariant manifold, 942, 974, 1011Invariant manifold theorem, 942Invariant manifold theorem (for diffeomorph.), 942Invariant manifold theorem (for flow), 942Invariant measure, 25, 971, 980, 984Invariant set, 931, 957Invariant set stability, 967Invariant sets, 967Invariant subgroup, 114, 810Inverse, 43, 53Inverse edge path, 401Inverse element, 43, 56, 110, 116, 699Inverse function, 443, 447, 739Inverse function theorem, 529Inverse homeomorphism, 298Inverse image, 38, 40, 677, 705, 759Inverse image (preimage), 705Inverse image of a 1-form by a map, 601Inverse image of a 2-form by a map, 609Inverse Laplace transformation, 889Inverse map, 40Inverse mapping theorem, 768Inverse matrix, 143Inverse on the left, 43Inverse on the right, 43Inverse path, 378, 602Inverse relation, 38Inverse symbol, 40Inverse transformation, 900Inversion pole, 213Invertibility, 191, 467Invertibility and continuity, 443Invertible, 53, 125, 191Invertible element, 53, 191

in residue-class ring, 195Invertible on the left, 43Invertible on the right, 43Invertible operator, 504Involute, 536Involute of a curve, 536Involute of a plane curve, 538Involute of an arc, 536Involution, 53, 181, 219, 326, 762Involutive function, 181, 326Involutive map, 181, 326Involutive mapping, 181, 326Involutive transformation, 219Inward perturbation, 1012Irrational number, 91Irrational rotation number

(quasiperiodic behavior), 1008

Irrationality of√2, 87

Irreducibility criteria, 163Irreducibility of polynomials, 163Irreducibility theorem, 163Irreducible, 191Irreducible affine algebraic set, 720Irreducible element of a ring, 128Irreducible form of a rational number, 82Irreducible fraction, 90Irreducible function, 719Irreducible polynomial, 148, 162, 163, 719Irreducible prime element, 192Irreducible scheme, 764Irrelevant ideal, 716is divisor of, 59is multiple of, 59Isoclinal line, 513Isocline, 513Isogonal mapping, 593Isolated logarithmic singularity, 586Isolated point, 265, 266, 269Isolated singular point, 579Isolated singular solution, 516

asymptotic point, 516

1080

Index

centre, 516nodal point (node), 516saddle point, 516

Isolated singularities on k-sheets, 586Isolated singularity, 629Isolated zeros, 626Isometric patches, 542Isometric pro jection, 237Isometric regular patches, 542Isometric transformation, 542Isometry, 24, 206, 235, 264

canonical transformation, 255glide, 204identity, 204reflection, 204rotation, 204translation, 204

Isometry group, 204, 207, 235, 255Isometry group Is(E), 255

subgroup of displacements Is+(E), 255

subgroup of displacements Is−(E), 255Isomorphic, 13, 819Isomorphic directed graphs, 812Isomorphic functors (naturally equivalent functors),

408Isomorphic graphs, 351Isomorphic groups, 114Isomorphic vector spaces have same dimension, 139Isomorphism, 44, 51, 109, 409, 697, 705, 808, 809,

819Isomorphism of groups, 115Isomorphism of ordered sets, 63Isomorphism theorem, 120Isomorphism theorem for modules, 137Isomorphism-dense subcategory, 805Isoperimeter, 505Isotone map, 88Isotropic line, 216Isotropic vector, 184Isotropy group, 655Isotropy subgroup, 414ith face map, 388ith k-simplex, 745

J -shaped diagram, 813Jacobian matrix, 465—467, 969Jacobson radical, 717Jarque-Bera test, 870Jensen formula, 627, 668Johnson theorem, 671Joint entropy, 984Jordan arc, 291, 292, 531, 532

admissible parameterization, 532Jordan areolar measure, 494, 495

countable union of parts of R496of rectangles (possibly degenerated), 495problem, 496

Jordan basis, 182Jordan block, 183Jordan canonical form, 183Jordan canonical form theorem, 183Jordan Curve, 346Jordan curve, 268, 291, 531, 602

closed, 268

Jordan curve in C, 572Jordan curve theorem, 291, 398, 685Jordan domain, 658Jordan form, 944Jordan Lemma, 581Jordan loop, 602Jordan matrix, 182, 183Jordan matrix decomposition, 183Jordan path, 602Jordan regular curve, 602Jordan theorem, 642, 643Jordan-Brouwer separation theorem, 398

Jordan-Brouwer theorem, 292Joule, 488Joyeux, 879jth Betti number, 681Judge, 855, 858, 861

k(y)-scheme, 770k-algebra, 769k-connected graph, 354, 355k-face, 378k-jet, 947k-simplex, 340, 346k-skeleton, 406K-theory, 364, 761K-vector subspace, 145, 146Kähler differential, 714Königsberg bridges, 349Künneth formula, 412Künneth formula for cohomology, 429Künneth theorem, 412Kakutani, 911, 914, 915Kakutani fixed point theorem, 915KAM theorem, 980Kelley-Morse set theory (KM), 21Kelvin-Stokes theorem, 493Kepler (approximation) method, 481Kepler folium, 455Kernel, 114, 119, 180, 219, 393, 755, 811, 818Kernel (in abstract algebra), 707Kernel for vector space, 707Kernel is a normal subgroup, 119Kernel of derivation function of polynomials, 154Kernel of group homomorphism, 707Kernel of homomorphism, 114Kernel of homomorphism of differential complexes,

707Kernel of linear transformation, 707Kernel of magma homomorphism, 707Kernel of morphism, 114Kernel of ring homomorphism, 707Kernel pair, 796, 811, 818Ket, 554Keynesian assumptions, 1030Kleene, 2Klein, 222Klein bottle, 346, 558Klein group, 56Klein model, 203, 207, 240Klein test, 854Knot, 268, 269Knot without crossing, 269Kolmogoroff axioms, 836Kolmogorov-Arnold-Moser theorem, 980Kolmogorov-Sinai entropy, 984Kolmogorov-Smirnov statistic, 853Kondrakov theorem, 529Kondrakov theorem for Riemannian manifolds, 529Korch curve, 989Koyck model (geometric progression), 866Koyck transformation, 866Kripke-Platek set theory (KP), 20Kronecker, 6Kronecker delta, 465Kronecker formula, 558Krull dimension, 713Krull topology, 723kth root of a real, 92Kuhn-Tucker conditions, 918

quantification, 921Kuhn-Tucker multipliers

interpretation, 921uniqueness, 921

Kuhn-Tucker theorem, 921Kupka-Smale, 959Kuratowski graph, 356Kuratowski theorem, 356

1081

Index

Kuratowski-Pontryagin theorem, 356Kurtosis test, 870

L2-space, 530L’Huilier theorem, 245L-adic étale cohomology, 772L-adic cohomology, 770—772L-automorphism, 170L-globally stable equilibrium, 1026, 1027α-limit point, ω-limit point, 932α-limit set, ω-limit set, 967L-locally stable equilibrium, 1026L-stability, 1026L∞[a, b] space, 503Löwenheim-Skolem theorem, 3Lagged variable, 857Lagrange, 910Lagrange equation, 1015Lagrange form of Taylor remainder, 450Lagrange formula, 534Lagrange interpolating polynomial, 151Lagrange interpolation, 460Lagrange interpolation polynomial, 151Lagrange method, 460Lagrange multiplier, 469, 857Lagrange multiplier (LM), 857Lagrange multiplier test, 852Lagrange polynomial, 151Lagrange stability, 971Lagrange theorem, 112, 115Lagrangian theorem, 920Lagrangien, 920Lambek, 781Laminar phase, 999Laminarization, 1010Laplace operator, 526Laplace transformation, 888

convergence, 889inverse Laplace transformation, 889rules for the evaluation , 889transform of special functions, 890

Laplace transformation (Wagner or Laplace Carsonform), 889

Laplace-Carson transformation, 888Laplacian vector, 470Large category, 770, 788, 792Lateral classe, 113Latin square, 830Lattice, 12, 30Lattice ideal, 30Lattice maximal ideal, 30Lattice non-trivial ideal, 30Lattice of periods of ω, 684Lattice of power set, 13Lattice prime ideal, 30Lattice proper ideal, 30Lattice theory, 12Lattices of sets, 12Laurent series, 579, 629, 898Laurent series expansion, 580Law of composition, 38, 39Law of double negation, 2Law of excluded middle, 5Law of large numbers, 838Law of monotony, 84Law of non-contradiction, 2Laws of Morgan, 2Lcal element, 787, 824Leading coefficient, 146Leading coefficient of a polynomial, 146Leading monomial, 703Leading term, 146Leading term of a polynomial, 146Least common multiple, 77, 163Least common multiple (lcm.), 194Least common multiple in Z, 81

Least element, 61, 62Least mean squares error of a function, 885Least squares method

geometric interpretation, 849, 853Least upper bound, 12, 61, 217Least upper bound of real function, 440Least upper bound theorem, 88Least-squares, 459Least-squares method, 459Leaving variable, 905Lebesgue differentiation theorem, 663Lebesgue integral, 497—499Lebesgue integral properties, 499Lebesgue lemma, 636Lebesgue measure, 494, 496, 660Lebesgue-negligible set, 499Ledrappier theorem, 989Leech lattice, 12Lefschetz fixed point theorem, 410Lefschetz number, 411, 773Lefschetz theorem, 411Left adjoint, 802, 820Left adjoint and right adjoint functor, 408Left adjoint functor, 805Left and right distributivity, 57, 700Left and right unitor, 793Left cancellable element, 44Left closed monoidal category, 794Left derived functor, 755Left exactness, 428Left field, 107Left ideal, 29Left pro jective module, 751Left R-module M, 701Left resolution, 753Left surjective and univocal relation, 38, 704Left surjective relation, 38, 704Left translation, 111Left zero morphism, 810Left-exact functor, 804Left-inverse, 43Left-inverse element, 56, 699Left-invertible, 43Left-neutral element, 43, 56, 699Left-shift, 955Leg, 242Legendre polynomials, 459Legendre symbol, 196Leibniz criterion of convergence, 439Leibniz formula, 154Leibniz notation, 447Leibniz rule, 713Length, 605

of a curve, 605of a Jordan curve, 606of a path, 605

Length measure, 561Length measure on a regular patch, 541Lens space, 408Leontief input-output model, 904Leptokurtic distribution, 870Leray theorem, 401less than or equal to, 48, 59Level homotopy, 422Level of degeneracy (or codimension), 944Level set, 470, 705Lever, 259Levi-Civita, 551Levi-Civita connection, 566Lexicographic product, 66Li-Yorke, 991Lie algebra, 177Lie algebra of a Lie group, 178Lie bracket, 177, 943Lie group, 177, 563Lift, 364, 776, 929, 934

1082

Index

Lift (algebraic topology), 776Lift (category theory), 776Lift (lifting), 776Lifted map, 1002Lifting, 382, 776Lifting criterion, 415Lifting criterion theorem, 415Lifting lemma, 382, 414Lifting of maps, 293Lifting property, 770, 771, 776Limit, 300, 412, 814Limit and cluster points of function with respect to

a filter, 302Limit and colimit, 412Limit cycle, 933, 963, 970, 973, 980, 988, 993Limit function, 444Limit of a filter, 303Limit of a map, 300Limit of a sequence, 300Limit of function, 440Limit ordinal, 20, 65, 782Limit point, 35, 266Limit theorem, 838Limit value of function, 440Lindelöf space, 288Line, 222Line angles, 101Line at infinity, 214, 236Line bundle, 760Line connectivity, 355Line integral, 465, 557, 580, 581Line integral of a vector field, 465, 557Line of curvature, 545Linear algebra, 178, 183

affine structure, 179diagonalization and decomposition, 179Euclidean and Hermitian structure, 179linear structure, 179multi-linearity, 179

Linear closure, 138Linear combinations, 138Linear correlation coefficient, 845Linear differential equation, 514, 515, 968

autonomous, 968homogeneous, 968non-homogeneous, 968of oder n, 518of oder n (non-homogeneous), 519of oder n with constant coefficients, 518with periodic coefficients, 969

Linear differential operator, 902Linear extension, 139Linear extension of a partial order, 60Linear form, 141, 324Linear function, 27Linear functional, 27Linear group", 326Linear independence, 138Linear interpolation method, 461Linear Lie group, 177Linear map, 140

composition, 141operations, 140

Linear operator, 26, 27, 503Linear optimization, 906Linear order, 60Linear programming, xx, 904Linear stability loss in dissipative systems, 970Linear topological space, 27Linear topology, 723Linear transformation, 27, 215Linear trend, 879Linear variety, 696Linearization, 926Linearization of (statistical) nonlinear models, 861Linearization of nonlinear model, 925

Linearly dependent vectors, 248Linearly independent vectors, 248Linearly ordered set, 28, 59Lines in sheaf, 208Linked variable, 3Liouville boundedness theorem, 586Liouville theorem, 628, 631, 939, 968Lipschitz, 312Lipschitz condition

global, 523local, 523

Little Desargues theorem, 211, 215Littlewood theorem, 670Ljung-Box statistic, 869LM tes, 852LM test, 857Lobachevsky geometry, 207Lobachevsky-Bolyai-Gauss geometry, 207Local basis, 735Local bifurcation, 961Local bifurcations, 994Local bifurcations inearby steady states, 994Local bifurcations nearby a periodic orbit, 997Local canonical parametrization, 585Local chart, 611, 957Local compactness and paracompactness, 731Local continuity, 67, 273

in terms of open balls, 67in terms of sequences, 67

Local coordinate chart, 733Local coordinates, 559, 928Local cross section, 934Local diffeomorphism, 615, 740Local element (local section), 787, 824Local embedding, 738Local extrema, 448, 450, 468Local extrema of functions from Rn to R, 468Local extremum, 468Local family, 962Local homeomorphism, 414, 559, 730Local Lipschitz condition, 523Local maximum, 61, 918Local minimum, 918Local presentation, 749Local representatives, 957Local ring, 713Local section, 762, 787, 824Local stability, 1027Local stable manifold, 974Local structural stability, 950Local surface, 561, 735Local system, 422Local system of groups, 422Local trivialization, 759Local-global bifurcation, 1012Local-ringed space, 365Locality axiom, 747, 820Localization, 715Localization of a ring, 715Localization of an R-module, 715Locally 1-connected space, 417Locally arcwise-connected space, 278, 739Locally compact Hausdorff space, 405Locally compact space, 285, 305, 571Locally connected space, 277, 308Locally constant sheaf, 748Locally exact form, 607Locally finite family of parts, 287, 673Locally finitely-presentable flat morphism, 770Locally invertible function, 467Locally Noetherian scheme, 763Locally of finite type, 767, 771Locally path connected, 375Locally pathwise-connecte, 731Locally pathwise-connected, 729Locally pathwise-connected space, 267

1083

Index

Locally quasi-finite morphism, 770Locally ringed space, 365, 748Locally small category, 788, 789, 792Locally stable equilibrium, 1025Locally trivial bundle, 364, 425, 760Location vector, 248Locking, 1005Locus, 95, 537, 721Logarithm, 456, 616Logarithm in C, 103Logarithmic barrier function, 910Logarithmic potential, 665Logic, xix, 1, 695Logic of orders, 4logic of predicates by level, 4Logical connective, 1Logical constant, 4Logical propositions, 1logistic curve, 862Logistic map, 966, 998Logistic model, 862Logistic model (Vehulst or Pearl curve), 862Long exact sequence, 392, 750, 819Long exact sequence of a pair axiom, 808Long exact sequence theorem, 428Long memory phenomena, 882Long memory process, 879

defined by autocorrelation, 879defined by spectral density, 879

Long range dependence (LRD), 882Longitudinal data, 856Loop, 291, 581, 601, 777Loop space, 420Lorentz curve, 896Lorentz tensor, 553Lorentz transformation, 553Lorenz model, 1015Lorenz system, 966, 968, 975Lotto, 832Lower (Riemann) integral, 474Lower bound, 62Lower box-counting dimension, 988Lower capacity, 988Lower integral, 474Lower limit of sequence, 437Lower pointwise dimension, 989Lower set, 30, 31Lower triangular, 148Lower triangular matrix, 147Loxodrome, 246

pseudospherical, 246Lp-function, 530Lp-space, 530Lp[a, b] space, 502Lumped component model, 1000Lumped element model, 1000Lumped parameter model, 1000Lune, 244Lusternik-Schnirelmann theorem, 398Lyapunov, 929Lyapunov characteristic number, 985Lyapunov dimension, 989Lyapunov exponent, 984Lyapunov function, 1028Lyapunov stability, 971, 972

m-fold cover, 414m-sheeted covering space, 414m-simplex, 377Möbius function, 54Möbius inversion formula, 54, 156Möbius strip, 558Möbius transformation, 594, 656MA (moving average), 871MA model, 871MA(q), 871

Mac Lane, 779, 797Mac Lane coherence theorem, 793, 794Mac Lane introduction to categories, 797Mackey topology, 510Madaurin series, 121Magma, 55, 698, 706, 707Main convergence criterion, 438Mainardi-Codazzi relations, 547Major axis, 253Majorant, 61, 70, 501, 665Majorization, 439Malgrange theorem, 962Malgrange-Ehrenpreis theorem, 904Manifold, 406, 559, 561, 696, 731, 957Manifold with boundary, 563, 733Mantissa, 844Many-sorted language, 694Many-sorted logic, 694Many-to-one relation, 19, 38Map, 11, 38, 51, 704

Bijective, 41bijective, 39, 46, 705contraction, 312fixed-point, 312injective, 39, 41, 705of the first return, 977surjective, 39, 41, 48, 705

Map fixed point, 312Map germ, 747Map locally of finite presentation, 825Map object, 785Map sink, 980Map winding number, 1004Mapping cone, 409Mapping cylinder, 376Mapping fiber, 423, 425Mapping space, 317Maps, 11, 38Maps and structures for sets, 36Maps compatible with structures (morphisms), 51Maps on unit circle, 1002Marginal optimality conditions, 918Markov chain, 833, 843Markov process, 833, 843Mass point, 259Matching, 208Mathematical expectation, 840Mathematical induction, 66Mathematical logic, xix, 1Mathematical programming, 909Mather theorem, 953Matrix, 140, 179Matrix diagonalization, 181Matrix form, 849Matrix minimal polynomial, 181Matrix of affine collineation, 234Maximal atlas, 734Maximal differentiable atlas, 732Maximal element, 29, 47, 60, 61, 784Maximal filter, 26Maximal ideal, 29, 365, 716, 771Maximal ideal (for lattices), 30Maximal spectrum, 716Maximal tree, 402Maximum element), 60Maximum likelihood, 858Maximum on a compact interval, 918Maximum principle, 632, 655, 660, 666Maxwell-Faraday equation, 493Mayer-Viétoris, 682Mayer-Victoris corollary, 407Mayer-Vietoris corollary, 400Mayer-Vietoris sequence, 400Mayer-Vietoris theorem, 395Mayer-Vietoris theorem for reduced homology, 396Meagre, 315

1084

Index

Meagre part, 316Meagre set, 315Mean curvature, 545, 566, 569Mean curvature of a surface S2, 569Mean lag, 864Mean square contingency, 842Mean value inequality theorem (for polynomials),

155Mean value theorem, 448, 449Mean value theorems, 449Mean-value formula, 618Mean-value property, 632Mean-value theorem, 450Measurability, 263Measurable function, 25, 497, 498Measurable set, 25Measurable space, 25Measure, 25, 472, 598Measure concentrated on the attractor, 980Measure determination, 475Measure of hypographs, 474Measure of length, 606Measure of regular surface, 486Measure of surface, 486

approximation by lateral surfaces of prims, 486approximation by polyhedrons, 486

Measure of unbounded parts, 497Measure theory, xix, 494, 499, 598Measurement, 218Median, 227Mellin transformation, 888Melnikov function, 960, 961Melnikov method, 960, 1000Mendelson, 1Menger, 289Menger n-arc theorem, 355Menger theorem, 355Mergelyan theorem, 651Meromorphic continuation, 630Meromorphic function, 582, 630, 892Meromorphic function on C, 587Meromorphy, 570

Meromorphy of C-C-functions, 570Mersenne number, 201Meshing, 234Metacategory, 799Metagraph, 799, 811Metamathematics, 6Metastable chaos, 1017Method of continuity, 528Method of penalty functions, 910Method of proxy variables, 861Method of successive approximations, 528Method of superposition, 844Metric, 27, 271

Euclidean, 271p-adic, 106

Metric dimensions, 988Metric entropy, 984Metric geometry on a regular patch, 542Metric plane, 206

axioms M1,M2,M3, 206Metric pro jective plane, 215Metric space, 67, 263, 271, 294, 299, 308

countable product, 295finite product, 295precompact, 311totally compact, 311uncountable product, 296

Metric space dimension, 289Metric subspace, 295Metric tensor, 549, 550, 552, 565Metric tensor (Riemannian metric), 550, 552Metric topology, 271Metrizable space, 286Metrization, 263, 286

Metrization theorems, 286, 287Meusnier theorem, 543, 544Midpoint, 215Milne process, 525Milnor, 967Minimal element, 13, 61Minimal polynomial, 157, 164Minimal surface, 505Minkowski distance, 299Minkowski inequality, 299, 329Minkowski metric, 552, 553Minkowski norm, 242Minkowski tensor, 553Minor, 143Minor axis, 253Minor Hessian, 468Minorant, 62, 501Minorization, 439Minus, 11Misspecification of model, 856Mittag-Leffer theorem, 653Mittag-Leffler decomposition, 587Mixed product, 250Mixed structure, 50Mixed tensor, 552Mixing dynamical system, 981Moëbius meshing, 234Moëbius strip, 346, 541

locally orientable, 541Mode locking, 1004Model, 409Model (in logic), 692Model (Relation system), 692Model theory, 692, 693Model validation, 845Model without constant term, 847Models of product life cycle, 862Models with errors on variables, 861Modified Jordan form, 1027Module, 50, 58, 109, 136, 139, 701

calculation rules, 136Module homomorphism, 137, 140Module of Kähler differentials, 714, 715Module theory, 110, 137, 698Modulo, 113Modulus, 95Modulus of a complex number, 96, 103Modus barbara, 2Modus ponens, 2, 6Modus tollens, 2, 6Moivre formula, 100Monad, 802Monad (triple), 802Monadic, 693Monadic adjunction, 802Monge, 204, 236Monge-Ampère equation, 527, 529Monic divisor, 149, 199Monic generator, 149, 150Monic irreducible polynomial, 163Monic polynomial, 47, 146, 161, 162, 181, 762, 770Monic polynomial (unit polynomial, unitary poly-

nomial), 47Monkey saddle, 544Monodromy, 578Monodromy group, 579Monodromy group of regular covering space, 416Monodromy matrix, 978Monodromy theorem, 414, 578Monogenic group, 112Monoid, 55, 699, 702, 706, 708, 788, 802Monoid (in category theory), 802Monoid (semigroup with identity), 798Monoid axioms, 706, 780, 788Monoid homomorphism, 706Monoidal category, 792—794

1085

Index

Monoidal identity object, 796Monoidal product, 793Monoidal structure, 795Monomial, 703Monomial algebra , 703Monomorphism, 51, 358, 705, 808Monomorphism (monic), 809Monomorphism (or monic), 819Monotone arithmetic function, 192Monotony criteria, 437Monotony criterion, 192Monster peaks, 879Monte-Carlo method, 844Montel theorem, 621Moore, 2Moore-Smith sequence, 280Morera, 576, 625Morera theorem, 625Morgan law, 11Morgan laws, 2, 8, 16Morphism, 44, 51, 779, 808

bijective, 51, 808surjective, 51, 808

Morphism locally of finite presentation, 767Morphism locally of finite type, 766, 767Morphism of finite presentation, 766Morphism of finite type, 766Morphism of schemes, 366, 763Morphism with finite fibers, 767Morse surgery, 506Morse theory, 506Morse-Kelley set theory (MK), 21Morse-Smale systems, 950, 980, 1013Mother wavelet, 899, 900Multi-scale analysis, 900Multicellular domain, 596Multicollinearity, 853Multicollinearity problems, 855Multifunctor, 805Multilinear form, 551Multilinear map, 325Multiple, 191Multiple correlation coefficient, 848, 850Multiple integral, 481Multiple regression model, 849

Distinction between εt and et, 850estimation of estimators, 849estimation of regression coefficients, 849general linear model, 849

matrix form, 849geometric interpretation, 853properties, 849properties of estimators, 850statistical tests, 851stochastic assumptions, 850structural assumptions, 850variance analysis, 850, 851

Multiple root, 151Multiple roots, 162Multiple structure, 50Multiple-valued complex function, 105Multiple-valued function, 105, 583Multiplication in N, 69Multiplication in Q, 81Multiplication in Z, 78Multiplicative associativity, 57, 700Multiplicative identity, 56, 57, 695, 700Multiplicative inverse, 56Multiplicative notation, 43, 52Multiplicative set, 715Multiplicative subset, 715Multiplicatively stable, 54, 126Multiplicity, 627Multiplicity of a covering space, 414Multiplicity of a root, 151Multiplicity of a zero, 627

Multiplicity order, 180Multiplier, 1013Multipliers, 920Multiview orthographic pro jections, 237

n-ary operation, 692, 693n-ary relation, 37n-ball, 367n-ball, 292n-boundary, 388n-cell, 404, 724, 792n-cell en, 404n-chain, 387n-chart, 559n-coboundary, 428n-cochain, 428n-cocycle, 428n-cycle, 116, 388

n-dimensional Ck-atlas, 732n-dimensional Grothendieck cohomology groups, 745n-dimensional holes, 745n-dimensional homotopy groups, 338n-dimensional manifold, 732n-dimensional Riemannian geometry, 561n-disk, 367n-disk, 292n-disk (or n-ball), 367n-F system (system with n degrees of freedom), 938n-gon, 405n-manifold, 406, 732n-order predicate calculus, 4n-simple space, 422n-simplex, 387n-skeleton, 725n-sphere, 367n-th homology group, 745n-times continuously differentiable function, 447n-times differentiable function, 447n-tuple, 16n-variables function, 40Naïve style, 786Nabla, 465, 471, 472Nabla (or del), 526Nagata-Smirnov metrization theorem, 288Naive set theory, 10, 17Napier analogies, 245Napier number e, 437Napier rules, 245Nappe, 256Natural equivalence, 408Natural exponential, 456Natural logarithm as integral function, 477Natural measure, 981Natural mumber, 83

operations, 84Natural number, 44, 83

addition, 84associativity, 84commutativity, 84Distributivity, 84monotony, 84neutral element, 84Simplification rule, 84

multiplication, 84associativity, 84commutativity, 84Distributivity, 84monotony, 84neutral element, 84

Natural number ob ject, 786Natural perspective, 212Natural topology, 263, 269—271, 278, 559Natural transformation, 361, 408, 779, 780, 811,

827Naturally equivalent functors, 408

1086

Index

Naturally equivalent functors (isomorphic functors),408

Naturally isomorphic , 807Necessary and sufficient condition, 6Necessary condition, 6Necessary condition (NC), 7Negation, 11Negation rules, 3Negative autocorrelation, 856Negatively correlated variables, 845Negatively invariant set, 932Negatively oriented with respect to V, 604Negligible part, 316Negligible set, 495Neighborhood, 67, 263—265, 270, 271, 294Neighborhood axioms, 269, 297Neighborhood basis, 271, 272, 736Neighborhood basis (local basis), 736Neighborhood filter, 302, 736

trace, 302Neighborhood of a hyperbolic point, 544Neighborhood system, 263, 729, 736Neighborhood system (neighborhood filter), 736Neighborhood system basis, 288Neil parabola, 535Neimark-Sacker theorem, 998Nerve, 401, 725, 757Nerve of a cover of a simplicial complex, 401Nested intervals, 438Nested segments, 90Net, 280Neutral element, 43, 53, 56, 57, 85, 110, 116, 699,

700Neutral element for ⊤, 53Neutral line, 232Neutral on the left, 43Neutral point, 232Neutrally stable, 929Nevanlina class, 628New Foundations (NFU,NF), 21Newton, 18Newton sums, 153Newton’s second law, 488Newton-Gregory method, 460Newton-Raphson method, 461Newton·meter, 488Newtonian attraction, 470NF, 21NFU, 21Nilpotent, 125, 182Nilpotent element, 717Nilpotent endomorphism, 182Nilpotent sheaf of ideals, 771Nilpotent sheaf of ideals„ 770Nilpotent upper triangular matrix, 182Nilradical, 717Nine-point circle, 227No middle third set, 988No-retraction theorem, 640Nodal point, 516Node, 350, 516

attracting, 969center, 969sink, 969

Node (vertex), 350Node saddle bifurcation, 971Noetherian, 47Noetherian affine scheme, 763Noetherian induction principle, 47Noetherian ring, 193, 709Noetherian scheme, 763Non denumerability, 45, 46Non denumerability of R, 47Non Euclidean metric, 209Non parametric model, 926Non-algebraic functions, 456

Non-Archimedean valuation, 198Non-autonomous differential equation, 968Non-bicontinuous, 265Non-connected space, 276Non-constructibility theorem, 175Non-contradiction

in an axiomatic system, 6Non-contradiction problem, 5Non-degeneracy conditions, 944Non-Euclidean geometry, 211Non-Euclidean orthogonality, 215Non-Euclidean plane, 209Non-hyperbolic fixed points for planar fields, 949Non-hyperbolic singular point, 944Non-hyperbolic singularity, 944, 947

of saddle-node type, 947Non-incident center, 215Non-isolated singularity, 586Non-isotropic line, 216Non-orientable surface, 346Non-principal filter, 26Non-strict ordering, 60Non-structured set, 63, 65Non-topological map, 264, 265Non-trivial ideal (for lattices), 30Non-wandering point, 932, 980Noncommutative domain, 703Noncommutative ring, 709Nondegenerate point, 426Nonhomogeneous integral equation of the second

kind, 507Nonhomogeneous system, 145Nonlinear diffeomorphism, 942Nonlinear dynamical system, 942Nonlinear dynamical system , 970Nonlinear estimation methods, 863Nonlinear flow, 942Nonlinear model, 861Nonlinear optimization, 910Nonlinear programming, xx, 904, 905, 910Nonlinearities, 1004Nonmeasurable set, 24Nonresidue, 195Nonsingular matrix, 142Nontrivial subharmonic function, 667Norm, 27, 198, 321, 958Norm of a complex number, 95Norm of the convergence in quadratic mean, 333Norm of the uniform convergence, 317, 322, 333,

501, 502Norm of vector, 247Norm on a vector space, 321Normability, 27normable, 27Normable topological vector space, 27Normal component, 542Normal curvature, 542, 544, 566, 569Normal curvature of a regular surface, 569Normal derivative, 664Normal distribution, 834, 835, 841Normal endomorphism, 185Normal equation, 251Normal equations, 846, 849Normal factor, 120Normal form, 173, 401, 994Normal form theorem (differential equation), 944Normal forms for diffeomorphisms, 945Normal forms of bifurcations, 992Normal manifold, 493Normal monomorphism, 809Normal morphism, 791, 809Normal parameterization, 532Normal parameterization of a circular helix, 532Normal plane, 533Normal representation, 103

of a complex number, 103

1087

Index

Normal representation of complex numbers, 103Normal section, 543Normal series, 120Normal space, 263, 281, 282, 301Normal subgroup, 114, 119, 402, 416, 809, 810Normal to a patch, 539Normal to a plane, 223Normal tower, 120Normal vector, 538Normality hypothesis of errors, 847Normality of errors, 847Normality tests, 870

Jarque-Bera test, 870Skewness and Kurtosis tests, 870

Normalizer, 416Normally independently distributed (n.i.d.), 869Normally summable family, 327Normed algebra, 32Normed linear space, 28Normed space, 27, 28, 501, 502Normed vector space, 28, 502Normed vector spaces, 321North pole, 367Not topologically pairwise arc-connected, 268Notion of attractor, 980Nowhere dense, 315ntersection of a simple arc with a plane, 535nth (singular) homology group, 388nth (singular) homology group of a space, 388nth cohomology group, 428nth cyclotomic polynomial, 156nth homology group, 388, 411nth homology group of a chain complex, 392nth homology group of a complex, 392nth primitive roots of unity, 156nth relative cohomology group, 428nth relative homology group Hn(X,A), 394nth roots of unity, 177Nth unit roots, 887nth-rank tensor, 550

in m-dimensional space, 550Null element, 12Null hypothesis H0, 835Null ob ject, 785Null sequence, 707, 724Null vector, 247Null-homotopic, 371Null-homotopic map, 371Nullary operation, 359, 693Numéraire goods, 1029Number

algebraic, 106b-adic, 91Bernoulli, 456Betti, 349complex, 93connectivity, 349connectivity , 354edge, 356Fermat, 177, 201hypercomplex, 47irrational, 91Mersenne, 201p-adic , 106prime , 200transcendental, 106, 164

Number of crosscaps, 405Number of edges, 400Number of elements of a set, 10Number of faces, 400Number of handles, 405Number of vertices, 400Number system

axiomatic construction, 105Number theory, xix, 191Numerable Cech cohomology, 757

Numeration in base b, 74Numerical harmonic analysis, 886

Fast Fourier transformation (FFT), 887formulas for trigonometric interpolation, 886, 887

Numerical resolution of equation, 461

Ob, 780Object, 51, 418, 779, 784, 808Objective function, 906, 920Observed residual, 847Obstruction, 425, 426Odd polynomial, 146OLS estimator, 856ω-assignment, 3One-form, 554One-point compactification, 105, 285, 286, 421, 571

of complex numbers, 105One-relator group, 404One-sector neoclassic growth model, 1026One-sided convolution, 890, 895One-sided Fourier transformation, 888One-sided surface, 346One-sided test, 835One-tailed test, 835One-to-one correspondence, 44One-to-one function, 39One-to-one relation, 19, 38Open affine cover, 774Open and closed maps, 727Open ball, 265, 268, 271, 294Open chain, 352Open cover, 283, 746, 821Open covering, 263, 283, 678Open curve, 291Open disk, 96, 265Open elementary chain, 352Open Eulerian chain, 352Open half-ball, 312Open image, 632Open immersion, 767, 768, 773Open interval, 265Open manifold, 734Open map, 273, 727Open mapping theorem, 328, 503, 640, 647, 655,

727Open morphism, 766Open part, 294Open q-simplex, 398Open set, 66, 263, 266, 269—271, 727, 911Open set axioms, 296Open set topology, 727Open subscheme, 764Opening angle, 252Operad, 712Operand, 695Operation table, 55, 698Operations in Gauss complex plane, 103Operations in N, 69Operations in Z, 78Operations on ordinals, 66Operations on sets, 40Operator, 26, 27, 503, 507

compact, 507differentiable, 504differential, 506, 510finite rank, 508Fredholm, 510Hermitian integral, 510integral, 506, 510invertible, 504linear, 503, 563

Operator algebra, 507Operator norm, 979Operator set, 692Operator spectrum, 32Operator variable, 4Opposite, 53, 86

1088

Index

of a complex number, 103of an irrational, 88

Opposite category, 360, 790Optimal mix of explanatory variables, 853, 855Optimality condition, 904Optimization, xx, 904

generalization, 906normal form, 906

Optimization theory, 904Optimization with two variables, 906Optimum with constraint

examples, 920geometric interpretation, 920

Optimum without constraint, 919Orbit, 414Orbit space, 417Orbital stability, 972Order, 216, 784Order and orientation, 216Order contact, 536Order diagram, 60Order filter, 31Order ideal, 31Order isomorphic, 63Order of a group, 110Order of a polynomial , 148Order of a root, 151Order of integration, 876Order of integration of time series, 876Order of multiplicity, 180Order of ramification, 199Order of the zero, 159Order relation, 10, 12, 59

cyclic, 218strict, 59

Order relation on Q, 81Order relation on Z, 78Order structure, 50, 51, 59, 84, 85, 87

antisymmetry, 59reflexivity, 59transitivity, 59

Order structure of R, 435Order type, 63

comparison, 63Ordered directed set, 796Ordered field, 50Ordered set, 51, 59

lexicographic product, 66Ordering relation, 10, 12Ordinal, 63, 64

finite, 64infinite, 64

Ordinal classe, 65Ordinal comparison, 65Ordinal number, 45, 64Ordinary differential equations

second-order, 969Ordinary geometry, 205Ordinary least squares (OLS), 846, 870Ordinary metric pro jective plane, 215Ordinary sequence, 436Orientable differentiable manifold, 563Orientable regular curve, 605Orientable surface, 346, 541Orientation, 216, 217, 387, 491, 604Orientation of a Jordan arc, 532Orientation of a plane, 205, 540Orientation of a surface, 540, 541Orientation of ∆n, 387Orientation-preserving, 217, 220, 235, 255, 1002Orientation-preserving (reversing), 941Orientation-preserving diffeomorphism, 935Orientation-preserving homeomorphism, 929Orientation-reversing, 235Oriented angle, 101Oriented graph, 352

Oriented plane, 217Oriented plane as topological space, 217Oriented regular path, 604Oriented regular simple arc, 541Oriented simplicial complex, 399Orientibility, 540Origin of a path, 378Origin of edge of simplicial complex, 401Orthocenter, 227Orthodrome, 244, 246Orthogonal, 329Orthogonal affinity, 229Orthogonal axonometric map, 237Orthogonal basis, 184Orthogonal collineation, 225Orthogonal coordinate system, 236Orthogonal curvilinear coordinates, 465Orthogonal group, 177, 255Orthogonal group On, 235, 261Orthogonal measure, 650Orthogonal of a part of a pre-Hilbert space, 329Orthogonal pro jection, 237, 330Orthogonal system, 261, 331, 900Orthogonal vectors, 260Orthogonal wavelet, 900Orthogonality, 28, 216, 329

according to a hermitian form, 184Orthogonality condition, 541Orthographic pro jection, 237Orthonormal algebraic basis, 332Orthonormal basis, 185Orthonormal coordinate system, 235, 248Orthonormal part of pre-Hilbert space, 185Orthonormal system, 261, 331Oscillatory, 927Osculating circle, 533, 536, 538Osculating plane, 533Osculating sphere, 536Ostrogradski formula, 492, 681Ostrogradsky formula, 556Ostrogradsky-Stokes formula, 492Ostrogradsky-Stokes theorem, 492Ostrowski, 197Output variable, 925Outset, 955, 958Outward perturbation, 1012Over-harmonic, 669Over-harmonic function, 669Overlapping of regular patches, 540OX -module, 765

p-adic absolute value, 197p-adic distance, 299p-adic metric, 106p-adic number, 89p-adic number, 106p-adic topology, 106p-adic valuation, 106p-admissible open set, 414p-function, 588p-linear antisymmetric form, 188p-value (statistics), 835Pair category, 791Pairs of harmonic points, 234Pairwise disjoint equivalence classes, 37Pairwise disjoint subsets, 37Pairwise relatively prime, 76Palis-Smale theorem, 980Panel analysis, 856Panel data, 845, 856Pappus-Pascal theorem, 211, 212Parabola, 252, 253Parabolic fixed point, 1011Parabolic geometry, 203Parabolic point, 544, 545, 567Paracompact space, 283, 287, 729Paradoxical decomposition, 10

1089

Index

Paradoxical set, 10Paradoxical set with respect to the group G, 10Parallel affine subspace, 259Parallel axonometric projection, 237Parallel lines, 209Parallel postulate, 203Parallel pro jection, 229Parallelepiped, 261Parametering, 602Parameters of the model, 846Parametric representation, 531, 537—539Parametrization

of a curve, 602Parametrized arc, 531Parseval equality, 883Parseval equation, 885Parseval formula, 332Part, 15

bounded, 284denumerable, 72finite, 72stable by a map, 40

Partial autocorrelation function, 869Partial correlation, 854Partial derivative, 464Partial differential equation, 525Partial fraction decomposition, 453, 587, 891

of a rational function, 587of the cotangent, 588of the function gamma Γ, 588

Partial graph, 350Partially ordered group, 695Partially ordered set, 12, 29, 60, 695, 784Partially ordered set (Poset), 29, 60, 784Partially ordered set (poset), 12, 60Particular first-order differential equations, 513Partition, 10, 12Partition of unity, 555, 652, 672, 678Partitions and covers, 42Partitions of an interval, 473Pascal relation, 74Pascal triangle, 452Pascal’s arithmetical triangle, 74Patch, 561, 735, 928Patch of class Cr, 538Patch point, 544

elliptic point, 544hyperbolic point, 544parabolic point, 544planar point, 544

Path, 267, 277, 352, 374, 531, 601, 677, 777regular, 601

Path class, 378Path component, 374Path connected, 374Path connected space, 374Path lifting property, 777Path of integration, 892Path-connected, 739Path-connected (0-connected), 739Path-connected set, 267Pathwise connected set, 267Pathwise connectedness, 739Pathwise-connected, 267, 729, 731, 739Pathwise-connected (path-connected, 0-connected),

739Pathwise-connected space, 267Paving, 605Peak of frequency, 879Peano, 5Peano axioms, 66

5th, 7, 66Peano existence theorem, 523Pearl curve, 862Peixoto theorem (I) (for vector field, flow), 951Peixoto theorem (II) (for vector field, flow), 951

Peixoto theorem (III) (for diffeomorphism), 953Pentagon diagram, 793Percentage distribution, 839Perfect field, 168, 718Perfect group, 426Perfect Reinhardt domain, 596Perfect set, 500Period bands, 590Period parallelograms, 590Period three implies chaos theorem, 992Period-doubling (flip bifurcation), 997Period-doubling (flip) bifurcation, 1013Periodic , 966Periodic complex function, 589Periodic function, 589, 590

complex, 589doubly, 589simply, 590

Periodic non-autonomous system, 936Periodic orbit, 960Periodic point, 928Periodic point of an Anosov diffeomorphism, 953Periodic signal, 982Periodic solution of a system, 970Periods of complex function, 589Permutation, 829Permutation cycle, 116, 121, 122Permutation group, 115, 116Permutation with repetitions, 831Permutation without repetition, 831Perpendicular bisector, 205, 208, 219, 227Perron integral, 500Perspective, 212Perspective collineation, 213, 232Perturbation, 926

inward , 1012outward , 1012

Pesin formula, 985Petals, 403Pfaffian differential form, 551Pfaffian form, 551Pfaffian form space, 551Phase diagram, 968Phase plane, 968Phase portrait, 968Phase portrait of the flow, 932Phase spectra, 895Phase-locking, 1004Phase-space, 967Phi function, 77Phillips and Perron tests, 871Phillips curve, 862Physical depreciation rate, 1030Physical measure, 981Picard, 580Picard iteration, 523Picard-Lindelöf theorem, 966Piecewise differentiable functions, 890Piecewise function, 440Piecewise linear map, 399Piecewise monotonic function, 476Piecewise regular boundary, 680

Piecewise-Ck path, 601Piecewise-Cr geometric arc, 531Pigeonhole principle, 71Ping-pong lemma, 703Pitchfork bifurcation, 996Pitchfork bifurcation (normal form), 992Pivot, 905Pivotal column, 905Pivotal row, 905Planar affinities, 229Planar curve, 534Planar geometric pro jection, 236Planar graph, 351, 354Plane, 205, 217

1090

Index

Plane curve, 291, 534, 537Plane lamina, 558Plateau function, 672, 678Platonic or regular solid, 340Platonic solids, 400Plucker coordinates, 186Poincaré, 5, 186Poincaré Bendixson theorem, 973Poincaré circle model, 240Poincaré conjecture, 399Poincaré cut, 976Poincaré formula, 558Poincaré half-plane model, 240Poincaré lemma, 607, 677Poincaré map, 563, 935—937, 940, 975

for autonomous differential equations, 975for non-autonomous differential equations, 976roles, 976

Poincaré model, 240Poincaré recurrence theorem (Poincaré return the-

orem), 971Poincaré section, 970, 976Poincaré theorem, 557Poincaré-Bendixson theorem, 933Poincaré-Brouwer theorem, 557Poincaré-Cartan invariant, 940Point, 205

accumulation, 266adherent, 265boundary, 266cluster, 266, 437, 462, 572exterior, 265frontier, 266heteroclinic, 950homoclinic, 950interior, 265, 727isolated, 265limit, 266of equilibrium, 966stationary, 966

Point at infinity, 214, 571Point-set topology, 263Pointed deformation, 423Pointed homotopy axiom, 422Pointed homotopy category, 380Pointed map, 369Pointed pair, 424Pointed pair homotopy, 424Pointed pair map, 424Pointed space, 729Pointed topological space, 729Poisson bracket, 943Poisson distribution, 834, 841Poisson formula, 660Poisson integral, 661Poisson kernel, 660, 662Poisson stability, 971Poisson theory, 576Polar, 213, 252Polar blowing up, 948Polar coordinates, 103, 928Polar form, 184Polar plane, 252Polar topology, 510Polar triangle, 210, 245Pole, 159, 207, 213, 215, 252, 579, 580, 630Polish space, 315Polyadic, 693Polydisk, 596Polygon, 229, 392, 904Polygon area, 229Polygonal chain, 344Polygonal group, 344Polygons equal by adjunction, 230Polygons equal by decomposition, 230Polyhedra, 339, 343

as topological subspaces, 339Polyhedral Formula, 558Polyhedron, 339, 340, 346, 398, 400Polyhedron in (Rp,Rp), p 3, 341Polyhedrons, 339Polynomial

Bernstein, 458Polynomial degree, 161Polynomial division, 162Polynomial equivalence, 92Polynomial function, 151, 439, 440, 452Polynomial function of lags (Almon’s lags), 865Polynomial function of order n, 439Polynomial interpolation, 460Polynomial map, 151Polynomial ring, 92, 161Polynomial roots, 104, 151Polynomials

Chebyshev, 459Legendre, 459, 461

Polytope, 229, 340, 346, 724Polytope edges, 340, 346Polytope vertices, 340, 346Population, 833, 841Population variance, 834Porter-Hudak, 881Portmanteau test, 870Poset, 12, 60Position vector, 248Positive autocorrelation, 856Positive hermitian form, 184, 329

and Minkowski inequality, 184and Schwarz inequality, 184

Positive homogeneity, 27Positive measure, 598Positive scalability, 27Positively correlated variables, 845Positively invariant set, 932Positively oriented with respect to V, 604Posterior distribution, 836Posterior probability, 832, 836Power, 44, 45Power functions with rational exponents, 455Power object, 785, 824Power object (in topos), 824Power of the continuum, 50Power series, 178, 444, 451, 579, 595, 596, 614Power series (entire series), 178, 444, 451, 595, 614Power series expansion, 451, 456—458, 480, 515, 521,

525, 570, 576, 577, 596, 597, 614, 619, 624, 626,635, 742, 892, 897

of rational function, 577Power series expansion coincides with Taylor series,

451Power series expansion of circular functions, 456Power series of several variables, 595Power set, 10, 17Power spectrum, 982Power theorem, 228Power-law, 882Prandtl number, 983, 1015Pre-Hilbert space, 28, 183—185, 329, 502Pre-Hilbert space (inner product space), 184Pre-image, 707Pre-inner product, 185Pre-limit, 360, 814Preadditive category, 792Precompact metric space, 311Precomposition, 494Predecessor, 65, 68, 436Predicate, 1Predicate (with 3 arguments), 3Predicate calculus, 2Predicate calculus theorem, 3Predicate calculus theorems, 3Predicate symbols, 694

1091

Index

Predicate variable, 4Predicates, 2Preimage, 38, 71, 264, 705, 707, 759Preimage (inverse image), 705Prelimit, 814Preorder, 31Preordered set, 783, 788Preordering, 783, 788Presentation of a group, 402Presentation of an abelian group, 386Presheaf, 362, 819Presheaf (algebraic topology), 746Presheaf (category theory), 746Presheaf of groups, 362, 746Presheaves, 362, 746Primary decomposition, 183Prime element, 192Prime field, 168, 170Prime ideal, 29, 135, 136, 193, 194, 365, 716Prime number, 75, 162

decomposition, 199Prime spectrum, 717, 718Prime subfield, 169Primeval, 2Primitive, 476, 479Primitive element, 165Primitive function, 472, 476, 477Primitive integral, 472Primitive of a closed form along a path, 636Primitive of a differential form, 677Primitive polynomial, 163Primitive polynomial irreducible, 163Primitive root, 168Primitive root of unity, 98Principal analytic set, 758Principal argument, 99, 103Principal bundle, 364, 760Principal curvature, 544, 545, 568Principal determination, 616

of argument, 616of the logarithm, 616

Principal determination of the argument, 99Principal direction, 257, 258Principal G-bundle, 760Principal ideal, 29, 136, 716, 718, 758Principal ideal domain, 136, 716Principal ideal ring, 136, 716, 758Principal normal, 505, 533, 538Principal normal vector, 533, 534Principal order filter, 31Principal order ideal, 31Principal part, 582, 630Principal ring, 129, 136, 193, 716Principal symbol, 526Principal value, 677Principia Mathematica, 21Principia mathematica, 18Principle of analytic continuation, 619, 659Principle of excluded middle, 5Principle of induction, 7Principle of isolated zeros, 626, 627Principle of recurrence, 7, 66Principle of transfinite induction, 62Prior distribution, 836Prior probability, 832, 836Probability, xx, 829, 832, 836Probability calculation, xx, 832Probability charges, 35Probability density, 839, 857Probability density function (p.d.f ), 832, 839Probability function, 839Probability generating function (p.g.f ), 834Probability mass function (p.m.f ), 832, 839Probability measure, 832, 836Probability measures on attractors, 980Probability sampling, 833

Probability space, 832Probable error (probability deviation), 835Problem of enumeration, 71Problem of initial conditions, 512Process

ARCH, 881ARFIMA, 879ARMA, 881DS (Difference Stationary), 870FIGARCH, 881GARCH

integrated, 881IGARCH, 881long memory, 879of first order, 870TS (Trend Stationary), 870

Process of first order, 870Process with memory, 856, 869Product, 51, 815Product category, 789, 790Product group, 111Product law, 54Product metric space, 295Product of a family of complete normed spaces, 322Product of a family of normed vector spaces, 322Product of edge paths, 401Product of ob jects, 419Product set, 17Product space, 275, 322Product structure, 51Product topology, 88, 275, 322Product uniform structure, 322Products of compact spaces, 304Programming

convex, 909, 910discrete, 909linear, 909, 910mathematical, 909nonlinear, 910quadratic, 909stochastic, 909

Proj, 774Proj as scheme, 774Proj as set, 774Proj as topological space, 774Projection, 419, 759Projection morphism, 816Projection of fiber space, 761Projection theorem, 28, 34Projection theorems, 330Projection to normal plane, 535Projection to osculating plane, 535Projection to rectifying plane, 535Projective algebraic set, 696, 721Projective bi-ratio, 240Projective ccorrelation, 215Projective collineation, 213, 215, 233, 236Projective collineation in P2 (R), 236Projective correlation, 213Projective general linear group PGLn(q), 235, 261Projective geometry, 203, 213Projective group, 204Projective map, 232, 233Projective metric, 215Projective module, 750Projective morphism, 774Projective n-space, 366Projective n-space on a ring, 775Projective Nullstellensatz theorem, 722Projective ob ject, 751, 752Projective perspective, 232Projective plane, 211—213, 558Projective plane of incidence, 212Projective polarity, 207, 215Projective resolution, 753Projective scheme, 697, 764

1092

Index

Pro jective transformation, 212Pro jective variety, 696, 697, 721Proklos or Proclus, 18Proof, 7

auxiliary hypothesis, 8by construction, 8by contradiction, 8by contrapositive, 8by descent, 7by disjunction of cases, 8by exhaustion, 9by mathematical induction, 7by recurrence, 8by transposition, 8combinatorial, 9constructive, 8direct, 7dissection, 9elementary, 8nonconstructive, 8two-column, 9visual, 9without words, 9

Proof by mathematical induction, 7Proof of excision, 397Proper class, 21, 781, 782, 788, 792, 805Proper divisor, 162, 191Proper face, 398Proper ideal, 29, 716Proper ideal (for lattices), 30Proper ideal point, 214Proper map, 737Proper morphism, 765Proper open set, 418Proper subgroup, 111Proper subset, 10, 17Properties of a category, 348Properties of a functor, 348Properties of an index, 639

dependence with respect to a point, 639invariance by homotopy, 639

Properties of estimators, 847Properties of limit values, 438, 441Properties of Tor, 411Properties preserved by topology change, 296Property P, 88Propositional and predicate Calculus, 2Propositional calculus, 2Propositional calculus theorems, 2Propositional formula, 1Propositions, 1Proxy variable, 861Pseudo-convex, 597Pseudo-dimension of an Abelian group, 643Pseudo-random number, 844Pseudo-random numbers, 833, 843Pseudo-Riemannian metric, 553Pseudo-tensor, 552Pseudoconvex, 597Pseudoconvex domain, 597Pseudometric, 299, 300, 313Pseudometrics and distances, 318Pseudosphere, 241Pseudospherical loxodrome, 246Pseudotensor, 552pth exterior power, 427Pullback, 361, 413, 425, 492, 677, 817, 818, 821Pullback (differential), 494Pullback (fiber product), 817Pullback (pullback diagram, fiber product, fibre prod-

uct, fibered product, fiber square, fibre square,cartesian square, pullback square), 817

Pullback as fibre-product, 494Punctured plane, 105Puppe sequence, 423Puppe sequence theorem, 423

Pure imaginary number, 93Pure rotation, 934, 935Pure set, 19, 21, 782, 783Pure set (hereditary set), 19, 21, 782, 783Pure sinusoidal function, 982Purely categorical, 786Purely random process, 856Pushforward, 25Pushforward (differential), 494Pushforward measure, 25Pushout, 361, 413, 725, 818Pushout (fibered coproduct, fibered sum, cocarte-

sian square, pushout square, amalgamed sum),818

Pushout square, 818Pythagoras theorem, 330Pythagorean theorem, 231, 238, 242Pythagorean triangle, 126Pythagorean triplets, 195

χ2-distribution, 834χ2-test, 834, 842q-cell, 401q-coboundary, 756q-cochain, 757q-simplex, 756q-skeleton, 399q-th Cech cohomology, 757qth relative simplicial homology group, 400qth simplicial homology group, 400Quadrangle, 215Quadratic form, 916

definite negative, 916definite positive, 916semi-definite negative, 916semi-definite positive, 916

Quadratic programming, 905Quadratic reciprocity law, 196Quadratic reciprocity theorem, 196Quadratic residue, 195Quadratic surface, 254Quadratic trend, 879Quadric, 254Quadrics, 256

classes, 257coincident plane, 256ellipsoid, 256elliptic cone, 256elliptic cylinder, 256elliptic paraboloid, 256general equation

principal directions, 257hyperbolic cylinder, 256hyperbolic paraboloid, 256hyperboloid of 1 sheet; 2 sheet, 256intersectiong planes, 256parabolic cylinder, 256parallel planes, 256

Quadrilateral, 209Quantifier, 3Quantifiers, 2Quantization, 2Quasi-coherent sheaf, 749Quasi-coherent sheaf of ideals, 774Quasi-compact morphism, 767Quasi-compact part, 283Quasi-compact space, 283, 284, 729Quasi-difference transformation, 858Quasi-differences, 858Quasi-finite morphism, 767Quasi-inverse functor, 806Quasi-isomorphism, 724, 801Quasi-projective variety, 722Quasi-random number, 844Quasi-separated morphism, 765Quasi-tensor category, 794

1093

Index

Quasiperiodic function, 971, 983Quasiperiodic signal, 982Quasiperiodic solution, 976Quasiperiodicity, 994Quasipro jective variety, 696Quasitensor category, 792Quaternary relation, 219Quaternion, 47, 107Quaternionic pro jective n-space, 404Quincunx, 840Quotient, 86, 147Quotient category, 369, 370Quotient complex, 393, 403Quotient field, 51, 133, 134, 158Quotient function, 440Quotient group, 51, 118, 119, 685Quotient map, 778Quotient module, 51Quotient ob ject, 785Quotient of the euclidean division of polynomials,

147Quotient relation, 51Quotient ring, 51, 57, 135, 194, 700, 713Quotient ring (factor ring, residue class ring), 713Quotient ring (factor ring, residue-class ring), 57Quotient set, 37, 38, 45Quotient space, 274, 372, 777, 778Quotient space (identification space), 777Quotient structure, 51Quotient topology, 274, 346, 372, 777, 778

R2-R2-function, 572R2-R-function, 572R-algebra, 710R-algebraic structure homomorphism, 706R-analytic function, 659R-B instability, 1015R-B phenomenon, 1015R-B problem, 1017R-integrability, 472R-integrability of monotonic function, 475R-integrable function, 472, 475R-module, 701, 702, 707R-module homomorphism, 726R-module M, 702α-representatives, 957R/S analysis, 880R/S analysis (Hurst statistic), 880Rössler model, 987Rademacher functions (square wave), 902Radial, 667Radian, 249Radical, 174, 717Radical ideal, 717Radical of a polynomial, 155Radical of ideal, 717Radical or power center theorem, 228Radius, 205Radius of convergence, 444Radius of convergence , 451Radius of convergence of series, 576Radius of curvature, 241, 533, 536Radius vector, 248Radon measure, 599, 604Raising to a power, 104Ral money stock per unit of labor, 1030Ramification point, 585, 591Random, 833, 843Random error (or noise), 833Random in design of experiments, 833Random number generator, 833, 843Random numbers, 833, 843Random process, 868Random sample, 832Random variable, 833, 839Random vector, 833

Random walk, 833, 843Random walk model, 870Random with a uniform distribution, 843Randomize, 833Range, 19, 38, 704Rank, 740Rank of a differential map, 740Rank of a pure set, 19, 783Rank of an abelian group, 387Rate of increase, 446, 447Ratio of affinity, 229Ratio of similarity, 225Ratio of similitude, 225Ratio: chord length/circle diameter, 238Rational domain, 58, 701Rational expectations, 1031Rational fraction, 160Rational function, 160, 440, 452Rational Number

calculation rules, 86Rational number, 86, 87

algebraic structure, 86defined by alignments, 87negative, 87order structure, 87

default, 87positive, 87topological structure, 87

default, 88Rational numbers, 81Rational or irrational ratio, 977Rational power function, 455Rational rotation, 934Rational rotation number

(frequency locking), 1007Ratios of distances, 230Ray, 207, 217Rayleigh number, 1015Rayleigh-Bénard instability, 983Rayleigh-Bénard phenomenon, 1015Rayleigh-Benard instability (R-B), 1004Real analysis, xix, 263, 270, 435Real analytic function, 742Real balances, 1031Real differentiability, 573, 743Real function, 439

algebraic operations and composition, 440rational operations and composition, 442

Real line, 294, 306not compact, 303

Real line bundle, 760Real number, 87, 88

algebraic structure, 88Real part, 94, 103Real part of z, 94Real projective n-space, 404Real valued sequence, 436Reciprocal, 447Reciprocal circular function, 457Reciprocal function, 447Reciprocal hyperbolic function, 457Reciprocation, 213Rectangle axiom (R), 209Rectangle function, 440Rectangular hyperbola, 242Rectangular impulse, 890Rectangular tiling, 605Rectifiable curve, 485Rectifiable path, 678Rectifiable set, 485Rectification, 176Rectifying plane, 533Recurrence, 62, 68Recurrence equation, 863Recurrence hypothesis, 7Recurrence relation, 52, 68

1094

Index

Recurrence theorem, 68Recursive, 7Recursive definition, 7Recursive model, 845, 867Recursive regression, 853Recursive residual, 853Reduced algebra, 712Reduced closed edge path, 402Reduced edge path, 402Reduced equations, 867Reduced form, 173Reduced homology, 395Reduced homology groups, 395Reduced ring, 712Reduced scheme, 696, 764Reduced simplicial homology group, 400Reductio ad absurdum, 8Reduction theorem, 401Refinement, 767Reflection, 206, 219, 229Reflection with respect to a plane, 223Reflection-rotation, 224Reflection-translation, 221, 224Reflexive normed vector space, 325Reflexivity, 10Region having no holes, 739Regression on the first differences, 864Regula falsi, 461Regular

curve, 602path, 601

Regular arc, 532Regular attractor, 980Regular boundary, 679Regular cardinal, 20, 782Regular category, 796Regular covering space, 415Regular curve, 602Regular domain, 483Regular element, 43, 53Regular epimorphism, 796, 811Regular geometric arc, 532Regular local ring, 713Regular map, 720Regular matrix, 142Regular matrix (nonsingular matrix), 142Regular monomorphism, 810Regular n-sided polygon, 113Regular patch, 538

angle measure, 541area measure, 541curvature, 542isometric, 542length measure, 541metric geometry, 542

Regular patches, 530, 539, 540overlapping, 540

Regular path, 601Regular ring, 713Regular solid, 340Regular space, 263, 281, 282, 301Regular subharmonic function, 667Regular submanifold, 737Regular submanifold property, 736Regular surface, 486

explicit representation, 487parametric representation, 486, 487

Regular T1-space, 282Regular tetrahedron, 340Regular topological space, 280Regular value, 180Regularity, 53Regularity of the addition, 69Regularity of the addition and multiplication in N,

69Regularity of the multiplication, 74

Regularity of the multplication, 70Reinhardt domain, 570, 596Relaminarization, 1010Relation, 3, 402

left surjective, 38, 704less than or equal to, 48, 49univocal, 38, 704

Relation of homotopy, 775Relation symbols, 694Relational signature, 694Relational system, 692Relations, xix, 36, 38, 51, 920

compatible, 51conditions of compatibility, 51

Relations and structures, xixRelations between coefficients and roots (of a poly-

nomial), 153Relative cohomology group, 428Relative complement, 11, 15Relative CW complex, 725Relative frequency, 839, 844Relative homeomorphism, 404, 406Relative homology group, 394Relative homotopy, 378Relative homotopy group, 424Relative Hurewicz theorem, 426Relative slmplicial homology group, 400Relative Spec, 774Relative tensor, 552Relative topology, 268, 729Relatively compact, 651Relatively compact set, 508, 621, 651Relatively compact set (Conditionally compact set),

651Relatively compact set (conditionally compact set),

508Relatively prime, 76, 195Relatively prime (coprime), 149Remainder, 147Remainder of the euclidean division of polynomials,

147Remarkable identity, 148Removable singularity, 629Renaming, 49Repeating decimal, 91Representable functor, 362, 807, 811Representation, 785Representation (representing ob ject), 785

Representation of a curve in R2, 537explicit representation, 537implicit representation, 537parametric representation, 537

Representation of a diffeomorphism, 933Representation of line, 250

equation with point and normal, 251equation with vector product, 250homographic equation, 250vector equation with point and direction, 250vector equation with two points, 250

Representation theorem, 330Representation theory, 691Representative, 38Representing ob ject, 785Reproducing kernel, 624Residual, 31, 847Residual et, 847Residue, 160, 195, 582, 645, 688Residue class, 194Residue class ring, 713Residue classes, 135Residue field, 366, 713Residue theorem, 582, 645Residue-class ring, 57, 194Resolution, 752, 753, 755Resolution (abstract algebra), 755Resolution (homological algebra), 753

1095

Index

Resoluton, 749Resoluton of differential sheaf, 749Resonance, 944Resonant condition, 946Resonant eigenvalues, 943Restoring force, 876Restricted homotopy, 775Restriction, 40, 776Restriction of a map, 39, 776Restriction rule, 446Retract, 338, 367, 775, 776Retraction, 48, 367, 776Return force, 876Return to scale, 1026Revealed preferences, 1029Reversal of axis, 235Reverse homothety, 225Reynolds number, 984Ricci calculus, 551Ricci curvature, 567Ricci tensor, 553, 567Ricci tensor (contracted curvature tensor), 553Richardson theorem, 3Riemann, 5Riemann criterion, 475Riemann curvature, 562Riemann curvature tensor, 553Riemann geometry, 562Riemann integral, 472—475, 481, 499

of functions of several variables, 481on measurable domains of R2 in the Jordan sense,

482Riemann integral transformation into curvilinear in-

tegral, 492Riemann mapping theorem, 656Riemann space, 551, 562Riemann sphere, 105, 562, 571, 572, 654Riemann sum, 475, 484, 485, 487, 488Riemann surface, 105, 562, 570, 572, 583

abstract, 585concrete, 585obtained by compactification, 105of logarithm function, 584of sphere with handle, 585of square root function, 584topological structure, 585

Riemann tensor, 553, 562Riemann zeta function, 458Riemann-Christoffel curvature tensor, 553Riemann-Christoffel tensor, 553Riemann-integrable function, 475, 499Riemann-Stieltjes integral, 500Riemannian connection, 565, 566Riemannian geometry, 559, 561Riemannian manifold, 551, 561, 562, 565, 732Riemannian metric, 550, 552, 565Riemannian space, 561Riemannian structure, 561, 958Rienhardt domain, 596Riesz decomposition theorem, 668Riesz representation theorem, 28, 508, 599Riesz theorem, 508Right adjoint, 797Right adjoint functor, 805Right angled triangle, 242Right cancellable element, 44Right closed monoidal category, 794Right derived functor, 753, 755Right ideal, 29Right R-module M, 701Right resolution, 753Right resolution (coresolution, resolution), 755Right triangle, 239, 242Right zero morphism, 810Right-exact functor, 804Right-inverse, 43

Right-inverse element, 56, 699Right-invertible, 43Right-neutral element, 43, 56, 699Rigid motion, 219, 223Rigid object, 795Rigid transformation, 219, 223Rigid transformation (rigid motion), 219

Rigid transformation in R3, 223Ring, 50, 57, 109, 124, 700

of integers, 85of polynomials, 92without divisor of zero, 57, 700

Ring , 708Ring homomorphism, 127, 134, 706Ring homomorphism (ring morphism), 127Ring isomorphism, 127Ring morphism, 127Ring of congruence classes, 135Ring of fractions, 715Ring of integers, 85

calculation rules, 85Ring of integers modulo n, 129Ring of matrices (n,n), 57, 700Ring of polynomials, 57, 700Ring spectrum, 717Ring theory, 110, 698Ring unit, 713Ring with a one, 708Ring with identity, 57, 700, 708Ring with unity, 57, 700, 708Ringed space, 696, 748Ritt algebra, 714Robustness of a dynamical model, 949Rolle theorem, 448, 450Roof, 813Root extraction, 173Root nth of a complex number, 103Root of unity, 98, 177Root of unity (de Moivre number), 98Roots of a polynomial, 151, 162Roots of a polynomial of degree n, 163Roots or zeros of a polynomial, 151Rose, 403Rose (bouquet of circles), 403Rotation, 55, 213, 220, 224, 699

of a regular polygon, 55, 699Rotation group, 55, 177, 218, 261, 699Rotation number, 934, 994, 1002, 1006Rotation of a vector field, 558Rotation of angle, 235Rotational, 470, 471Rotations, 213Rouché theorem, 647, 685Rough set, 21Routh-Hurwitz conditions, 1020Row stochastic matrix, 843RSA (cryptosystem), 130Ruelle, 992Ruelle-Takens theory (R-T), 984, 1003Ruelle-Takens-Newhouse (R-T-N), 983, 1000, 1003Ruelle-Takens-Newhouse scenario (R-T-N), 1000Rule of contraposition, 2Rule of modus barbara, 2Rule of modus ponens, 2Rule of modus tollens, 2Ruled surface, 546Rules of distributivity, 2Rules of inference, 2Runge, 650, 651Runge approximation theorem, 650Runge theorem, 649Runge-Kutta method, 524Russel, 18Russell, 10, 18Russell paradox, 5Russell paradoxical set, 10

1096

Index

Russell’s antinomy, 10Russell’s paradox, 10, 18RVAR (Retricted Vector AutoRegressive), 875

σ-algebra, 25, 836σ-compact space, 288σ-field, 25, 836Saddle, 969Saddle-connection, 960, 963, 1011

heteroclinic, 1011homoclinic, 1011

Saddle-connection bifurcation, 963, 1011Saddle-node, 947, 1013Saddle-node (fold) bifurcation, 1013Saddle-node and transcritical bifurcations, 994Saddle-node bifurcation, 962, 1014Saddle-node bifurcation (normal form), 992Saddle-node periodic orbit, 1014Saddle-node to a periodic-orbit, 998Saddle-point, 468, 469, 516, 910, 932, 938, 941, 960,

969, 1030Saddle-type diffeomorphism, 941Same cardinality, 45Same homotopy type, 371Same orientation, 387Sample, 832, 841Sample space, 833Sample variance, 841Sampling distribution, 833Sampling without replacement, 832Sampling witth replacement, 832Samuelson oscillator, 1018—1020SARIMA process, 872Satisfiable, 3Saturated set, 779Savings rate, 1030SBR measure, 981SC criterion, 855, 875SC(p), 874Scalar, 27Scalar curvature, 553, 567, 569Scalar curvature of a Riemannian manifold, 569Scalar density, 552Scalar multiplication, 247Scalar product, 28, 249, 332Scalar product on the space l2(K), 332Scalar square, 184Scale parameter, 900Scaling factor, 900Schauder, 501Schematization of homotopy groups, 339Scheme, 762Schemes, 365Schmidt orthogonalization process, 331—333Schoenflies theorem, 292Schröder-Bernstein theorem, 45Schwartz, 530Schwartz Laurent, 902Schwartz theorem, 633Schwarz criterion, 855Schwarz criterion (SC), 865Schwarz function, 855, 865Schwarz inequality, 260, 329Schwarz lemma, 633Schwarz reflection principle, 625Scipion del Ferro formula, 100Screw, 224Seasonal adjustment, 872Secant, 221, 238Secant of a circle, 222Second axiom of countability, 728, 736Second category, 315, 979Second countable topological space, 271Second countable topology, 271Second de Rham cohomology vector space, 681Second fundamental form, 542

patch theory, 542Second order differential equation, 516Second sample central moment, 841Second-axiom of countability, 289Second-countable, 287, 728, 730, 733, 735Second-countable basis, 288Second-countable space, 288, 735Second-order predicate calculus, 4Section, 762Section of a fiber bundle, 762Section of Poincaré, 976Segment, 90Segment division, 226Seifert, 557Seifert conjecture, 557Seifert-van Kampen theorem, 402, 403Selection, 26Selector, 26Self-adjoint operator, 331Self-conjugate subgroup, 810Self-homologous point, 227Self-similarity, 882, 989Semantic antinomy, 18Semantic method, 2, 3Semantic or syntactic method, 3Semi-continuous, 473Semi-continuous function

on Baire space, 316Semi-cubic parabola, 535Semi-distance, 299, 300, 312Semi-metric space, 308, 312

separated, 313Semi-norm, 27, 322Semi-simple algebra, 671, 719Semiadditive category, 792Semigroup, 50, 55, 57, 83, 699, 700, 706, 708

of natural mumbers, 83Semigroup homomorphism, 706Semilocally 1-connected space, 417Semilocally simply connected, 417Semilocally simply connected space (semilocally 1-

connected space), 417Semimajor axis, 253Semimetric, 312Semiminor axis, 253Sensitive dependence on initial conditions (SDIC),

987, 991Sentential calculus, 2Sentential formula, 2Sentential variable, 1Separability criterion for a polynomial, 168Separable, 32, 287, 735Separable algebra, 719Separable closure, 719Separable element, 719Separable extension, 719Separable morphism, 810Separable polynomial, 152, 168, 719Separable space, 32Separated morphism, 765Separated parts, 276Separated presheaf, 822Separated scheme, 763Separated space, 729Separated space (Hausdorff space, T2-space), 729Separately holomorphic function, 670Separates points, 27Separating hyperplane theorem, 35Separation axiom, 263

T0 (Kolmogorov separation axiom), 282T1, 282T2 (Hausdorff separation axiom), 282T3 (regularity axiom), 282T4 (normality axiom), 282

Separation axioms, 280, 282T0,T1,T2,T3,T4, 280—282

1097

Index

Separation of convex sets, 913Separation of points, 643Separatrices, 1011Separatrix, 1011Separatrix loops, 975Separatrix surface, 974, 978Sequence, 39, 436

convergent, 89equivalent to zero, 89in a metric space, 300interleave, 293monotone increasing , 89of Cauchy, 89

Sequence convergence, 278Sequence of complex numbers, 1018Sequence of holomorphic functions, 620, 647Sequential evolution law, 1019Sequential space, 288Sequentially complete space, 314Series, 438Series expansion, 449, 576Series in normed vector spaces, 326Series of functions, 444Series with general term, 326Serration, 757Serre fiber space, 425Serre fiber space (weak fibration), 425Serre fibration, 761Serre theorem, 425Sesquilinear form, 184, 328Set, 9, 695

at most denumerable, 46closed, 266connected, 267denumerable, 46finite or infinite, 45, 71frontier, 266non-denumerable, 46open, 66, 266totally ordered, 59

Set algebra, 11Set closure, 267Set difference, 11, 15, 17Set differences, 705Set direct product, 17Set inclusion, 17Set membership, 13Set of accumulation point, 265Set of adherent points (closure), 265Set of all hereditarily finite sets, 782Set of all homogeneous prime ideals, 366Set of all homotopy classes of pointed pair maps,

424Set of all parts of, 10Set of all rotations

of a regular n-sided polygon, 56, 700Set of all sets, 17, 18Set of all the sets, 10Set of antecedents, 71, 759Set of anti-hermitian matrices, 178Set of antisymmetric matrices, 178Set of axioms, 6Set of complex Borel measures, 663Set of definition, 160, 571Set of direct isometries: E+(2), 223Set of distributions, 673Set of ε-neighborhoods, 272Set of frontier points, 265Set of generators, 138Set of individuals, 4Set of inference rules (deductive reasoning), 6Set of interior points, 265Set of invertible elements of a ring, 125Set of majorants, 61, 62Set of neighborhoods, 269Set of null-trace matrices, 178

Set of parts of a set, 10, 41Set of prime ideals, 365Set of sesquilinear forms, 184Set of the parts of, 18Set of theorems, 6Set of topologically equivalent points, 264Set of topologically non-equivalent points, 264Set theory, xix, 9, 16Set topology, 263Set with one element, 784Set with regular boundary, 610Set-theoretic difference, 11, 15Set-theoretic topology, 263Sets and algebraic structures, 109, 697Sharkovskii, 992Sharkovskii ordering, 992Sharkovskii theorem, 992Sharkovskii theorem (1964), 992Sharkovsky, 992Sharkovsky theorem, 992Sheaf, 208, 362, 746, 820, 821Sheaf (general topology), 746Sheaf associated to a presheaf, 363Sheaf axioms, 747, 820Sheaf cohomology, 756, 772Sheaf cohomology by Cech cohomology, 756Sheaf cohomology by derived functor, 756Sheaf of germs, 696Sheaf of germs of holomorphic functions, 758Sheaf of lines, 208, 214Sheaf of planes, 208Sheaf of rings, 747, 748Sheaf on a site, 822Sheafification, 773, 820Sheafification (sheaving), 820Shear, 229Sheaves, 362Sheaving, 820Sheet, 105, 413Sheffer, 1Shemes, 365Shift operator, 870Shifted endogenous variable, 845Shifting parameter, 900Shilnikov theorem, 999Shilnikov-Turaev, 1013, 1014Shilnikov-Turaev-Gavrilov, 1013Shinai, 992Shinai theorem, 992Shock, 871, 874Short exact sequence, 750, 819Short memory process, 879Short term Fourier transformation (STFT), 901Shortest path between two points, 505Shrinking, 338Shrinking of a closed curve to a point, 268Shuffling, 830Sierpinski carpet, 989Sierpinski gasket, 989Sieve, 821, 822Sieve of Eratosthenes, 200Sigma algebra, 25, 836Sigma field, 25, 836Sign function, 440Signal analysis, 899Signal reconstruction, 899Signal synthesis, 899Signature, 186, 693, 697Signature (disambiguation), 694Signature (in logic), 694Signature (of a knot), 694Signature (of a manifold), 694Signature (of a metric), 694Signature (of a permutation), 694Signature (of a prime), 694Signature (of a structure), 693

1098

Index

Signature (of a symmetric or Hermitian matrix),694

Signature (of an algebraic system), 693Signature (of quadratic form), 694Signature (type), 693Signature of a permutation, 117, 142Signature of a transposition, 117Similar circles, 226Similar figures, 226Similar triangles, 226Similarity, 204, 225, 236, 264

direct, 613Similarity (similitude), 225Similarity in geometry, 225Similarity type, 694Similitude, 225Similitude center, 227Similitude group, 204Simple algebra, 719Simple autocorrelation function, 869Simple chain, 352Simple closed curve, 531Simple convergence, 318Simple curve, 291Simple dipath, 351Simple extension, 165Simple graph, 351Simple isolated singular point, 579Simple isolated singularity, 586Simple iterative method, 461Simple linear correlation coefficient, 845Simple order, 60Simple radical extension, 174Simple regression model, 846, 847

forecasting, 848geometric interpretation, 849

Simple root, 151, 163Simple root of a polynomial, 163Simplex, 259, 340, 346, 376, 724Simplex algorithm, 905Simplex method, 904, 905, 907Simplex of vertices, 342Simplex tableau, 905Simplexes, 340, 346, 376Simplexes in (Rp,Rp), p 3, 341Simplices, 340, 346, 376Simplicial approximation, 343, 344, 399Simplicial approximation theorem, 399Simplicial cohomology group, 432Simplicial complex, 340, 341, 346, 398, 724Simplicial complex in (Rn,Rn), 342Simplicial homology, 340, 346, 349, 357, 399, 745Simplicial homology functor, 808Simplicial homology group, 400Simplicial k-chain, 745Simplicial map, 343, 399Simplicial Mapping, 340Simplicial mapping, 357, 745Simplicial subcomplex, 725Simplicial subcomplexes, 340, 346Simplification rule, 84Simply connected, 267, 571, 739Simply connected (1-connected), 739Simply connected domain, 575Simply connected open set, 638Simply connected region, 268, 276, 575, 739Simply connected region (1-connected region), 739Simply connected space, 268, 276, 338, 382, 739Simply ordered set, 28, 59Simply periodic function, 590Simpson (approximation) method, 481Sims, 875Simultaneous congruences, 194Simultaneous equations model (SEM), 866

just-identified, 867over-identified, 867

under-identified, 867Sinai-Bowen-Ruelle (SBR), 981Sine, 239Sine function, 238Single unit, 37Single-valued function, 583Singly periodic function, 590Singular cardinal, 20, 782Singular complex, 387, 388Singular contact element, 516Singular homology, 357, 744, 745Singular metric pro jective plane, 215, 216Singular n-cochain, 428Singular n-simplex, 387Singular point, 579Singular point of algebraic curve, 455

cusp, 455isolated point, 455node, 455

Singular point, index of a, 558Singular solution, 516Singular theory, 389Singular values, 984Singular values of a matrix, 984Singularities of analytic functions, 586Singularity, 570, 597

cusp, 949degenerate, 944hyperbolic, 944non-degenerate, 944non-hyperbolic, 944of a complex function, 580of analytic functions, 586of holomorphic functions, 579

Singularity of holomorphic functions, 579Sink, 471, 969Sinusoidal function with harmonics, 982Site, 770, 821, 822Skeleton, 345, 353, 399, 725Skew-symmetric, 186Skew-symmetrization, 187Skewness test, 870Slack variable, 905Slater regularity condition, 910Slice category, 791Smale theorem, 999Smale-Birkhoff theorem, 959Small étale site, 769Small category, 408, 769, 787, 788, 792Small circle, 243Small class, 21, 781, 782, 792Small colimit, 792Small initial variation, 970Small limit, 792Small perturbations, 949Small sets, 21, 782Smallest convex set, 340, 341, 346Smallest element, 12Smash product, 421Smash product of pointed space, 421Smith normal form, 401Smith normal form theorem, 401Smooth embedding, 737, 738Smooth function, 563, 740Smooth functions on manifolds, 740Smooth manifold, 732, 733Smooth map, 740Smooth map (smooth function), 740Smooth maps between manifolds, 740Smooth morphism, 766Smooth scheme, 763Smooth structure, 733Smooth surface, 561Smooth unramified morphism, 770Smooth variety, 771Smoothly universal, 737

1099

Index

Snake lemma, 750Sobolev space, 529Soh, 239Solow model (Pascal distribution), 866Solow-Swan model, 1020, 1026Solution, 425

periodic, 970stable, 927

Solvable group, 120, 121Solving equation ∂u/∂z = v, 652Sort, 695Sotomayor, 962Source, 471, 809, 811, 812, 969South pole, 367Space

Baire, 315compact, 303complete, 312contractile, 338contractile arcwise connected, 338Hausdorff, 271Hilbert, 271L2([a, b],K), 333

l2(K), 332

L2, 530Lp, 530mapping (function), 317metric, 67

complete, 89, 293, 310of 2π-periodic functions, 333of Banach, 502of Hausdorff, 271of Hilbert, 502pre-Hilbert, 502semi-metric

separated, 313semimetric, 312Sobolev, 529T2, 271topological, 66

totally disconnected, 307uniform

complete, 314Space average, 981Space Cn, 595Space diagonal, 404Space L2([a, b],K), 333

Space l2(K), 332Space of 2π-periodic functions, 333Space of all distributions, 674Space of finite type, 429Space with no holes, 739Span (hull), 340, 346Span„ 813Spec, 720Spec (as a set), 718Spec (as a sheaf ), 718Spec (as a topological space), 718Spec(R), 717Spec(Z), 718, 774Special chain, 687Special linear group, 177, 563Special linear group SLn(q), 563Special linear group SLn(R), 563Special orthogonal group SOn, 235, 261Special orthogonal group SOn(q), 261Special unitary group, 177Spectral energy, 884Spectral interpretation of Fourier transform, 894Spectral radius, 32, 671Spectral radius formula, 671Spectral theorems, 32Spectral topology, 717Spectral value, 180Spectrum, 32, 365, 631, 671, 894

of aperiodic signal, 983of chaotic signal, 983of periodic signal, 983

Spectrum of a bounded operator, 32Spectrum of a ring, 365, 717Spectrum of function with f1/f2 irrational, 983Spectrum of function with f1/f2 rational, 983Spectrum of periodic signal, 983Spectrum of polynomial ring, 366Spectrum of quasiperiodic signal

of quasiperiodic signal, 983Speculation effect, 1030Speed of convergence, 461Sphere, 251, 731, 957Sphere model, 243Sphere of curvature, 536Sphere theorem, 568Sphere with handle, 584Spherical arc, 536Spherical geometry, 244Spherical point, 567Spherical triangle, 245Spinning diffeomorphism, 959Spinor, 487Spiral, 969Spiral source, 969Split exact sequence, 409, 750Split monomorphism (coretraction), 808Split polynomial, 152Splitting field, 167Spring-mass system, 511

Oscillatory motion, 511Spurious regression, 876Square matrix, 147Square matrix of order n, 147Square wave, 902Square-integrable periodic functions, 333Squaring the circle, 102SS (sum of squares), 848SSE (sum of squares explained), 848SSR (sum of squares of residuals), 848SST (total sum of squares), 848Stability

asymptotically stable, 972Kolmogorov (invariant tori preservation in Hamil-

tonian), 972Lagrange stability, 971Lyapunov stability, 971orbital stability, 972Poisson stability, 971Structural stability, 971

Stability Boundaries, 1013Stability in the sense of Lyapunov, 1025Stability of periodic orbits, 973Stability of steady states, 972Stability problem of a periodic solution, 969Stability test by recursive regression, 853Stability tests, 853Stability theory, 971Stabilizer, 414Stable

additively, 54multiplicatively, 54

Stable (unstable) eigenspace, 941Stable fixed point

in the sense of Lyapunov, 929Stable in the sense of Lyapunov, 937Stable part, 43Stable part by a map, 40Stable part for a law, 54Stage (or domain of variation), 786Stage of definition, 786Stagewise regression, 855Stalk, 362, 747, 748, 820Stalk (of a presheaf ), 747Stalk (of a sheaf ), 747, 820

1100

Index

Standard étale, 770Standard étale morphism, 771Standard deviation, 834, 840Standard function, 678Standard n-simplex, 387standard n-simplex, 367Standard normal distribution, 835Standard Zariski cover, 768Star, 208, 398Star convex, 267Star-shaped open set, 677State variable, 925State-space model, 925Static model, 876Stationarity, 869Stationarity tests

augmented Dickey-Fuller test (ADF), 870Dickey-Fuller test (DF), 870

Stationary Gaussian process, 880Stationary series

(i.i.d.), 869(n.i.d.), 869with memory, 869

Statisticof Jarque-Bera, 870

Statistical distribution, 839Statistical induction, 845Statistical inference, 845Statistical methods, 841Statistical models, xx, 844Statistical test, 851Statistics, xx, 832Steady state, 966Step function, 474, 890

lower-bound, 474upper bound, 474

Stepwise regression, 855Stereographic projection, 105, 243, 244, 367, 571,

654, 928Stereographic retraction, 422Stieltjes integral, 500, 508Stieltjes transformation, 888Stirling formula, 121Stirling number, 121Stochastic, 869Stochastic assumptions, 850Stochastic matrix, 834Stochastic optimization, 905Stochastic process, 833, 843, 868Stochastic programming, xx, 904, 910Stochastic variable, 833Stokes and Ostrogradsky formulas, 186Stokes formula, 492, 556, 610

proof, 611Stokes formula for the rectangle, 556, 606Stokes theorem, 427, 492, 556

closed form, 493exact form, 493general form, 492

Stokes theorem (in five dimension), 940Stokes theorem applications to tubular region, 940Stone-Weierstrass theorem, 321, 333Straight line, 205Straightedge, 174Straightedge and compass, 174Strange attractor, 980, 988Strange attractor (Ruelle-Takens), 992Strange attractor in the sense of Ruelle-Takens, 992Strange attractors, 991Stranger, 149Stratified sampling, 832Strict subset, 10Strictly concave function, 917Strictly convex function, 917Strictly decreasing real function, 440Strictly increasing real function, 440

strictly less than, 59Strong deformation retract, 406, 776Strong deformation retraction, 406Strong extrema, 505Strong induction, 68Strong law of large numbers (Kolmogorov), 838Strong recurrence, 68Strong Stable Manifold, 1014Strongly divergent sequence, 437Strongly inaccessible cardinal, 20, 782Structural assumptions, 850Structural equations, 867Structural form of VAR representation, 873Structural stability, 949, 950, 971Structural stability (robustness), 979Structural stability of linear systems, 950Structural stability of the locking, 1008Structurally stable differential equation, 979Structurally stable vector field, 951Structure, 51, 693, 695

algebraic, 109topological, 66

Structure (algebraic system), 693Structure homomorphism, 705Structure map, 803Structure morphism, 763Structure of lags, 865

Structure of Q, 89Structure of the set of reals, 50Structure of the uniform convergence on FE , 318Structure sheaf, 365, 748Structure sheaf of a topological space, 748Structure-preserving bijection, 350Structure-preserving map, 738Structures, xix, 50Structures (logic), 695Structures on R, 435Student law, 846, 851Student ratio, 851Student t-distribution, 835Student’s table, 846Sub-mean-value property, 632Subadditivity, 27Subalgebra, 508Subarc, 606Subbasis of a topology, 272Subcategory, 348, 359, 369, 790Subcomplex, 392, 725Subcritical bifurcations of fixed points, 993Subfield, 51, 93, 133Subgraph, 350Subgroup, 51, 111, 810Subgroup generated by a part, 112Subgroup generated by a set, 112Subgroup of Z, 79Subharmonic, 669Subharmonic bifurcation, 971Subharmonic distribution, 667Subharmonic function, 665, 667Subharmonic instability, 993Subharmonicity, 668Sublattice, 30Sublinear function, 27Sublinear functional, 27Submanifold, 736Submanifold atlas, 737Submanifold property, 736Submodule, 51, 136Subobject, 785Subobject classifier, 786, 822Subproposition, 1Subring, 51, 126, 133Subroup, 112Subscript notation, 39Subsequence, 300, 437Subset, 10, 15

1101

Index

Subset relation, 17Subspace topology, 729Subspace topology (relative topology, induced topol-

ogy, trace topology), 729Substitution principle, 391Substructure, 51, 133Substructures, 133Substructures (subring, subfield), 133Subsystem of relations, 51Successive derivatives, 447Successive integrations, 483Successive trials, 837Successor, 20, 63, 65, 68, 436, 782Sufficient condition, 6Sufficient condition (SC), 7Sum, 51Sum and product of complex numbers, 94Sum of subcomplexes, 393Sum topological space, 275Sum topology, 275Summable family, 327Supercritical Andronov-Hopf bifurcation, 1014Supergraph, 350Superposability, 38, 219Superposition principle, 886Supplementary angles, 221Support, 53, 145, 389Support of a distribution, 674Support of a family, 53Support of a map, 599Support of a measure, 598Support of distributions, 903Support of i-chain, 687Supremum, 61Surface, 346, 538, 540

hyperbolic Riemann, 562local parametric representation, 560locally orientable, 541of Riemann, 562Orientability and topological invariant, 347orientable, 541smooth, 561

Surface group, 404Surface integral, 487, 490, 556Surface of absolute values of complex functions, 573Surface of revolution, 256, 487, 544

ellipsoid of revolution, 256hyperboloid of revolution, 256paraboloid of revolution, 256Quadrics of revolution, 256

Surface with boundary, 346Surfaces in R3, 560Surgery, 506Surjection, 705Surjective embedding (isomorphism), 705Surjective homomorphism, 705Surjective homomorphism (epimorphism), 705Surjective map, 38, 39, 41, 705Suspension, 420, 421, 935, 936Suspension of a diffeomorphism, 936Syllogism rule, 2Sylvester formulas, 836Symbol, 1Symbol

, 471, 581

Symbol sequences, 955Symbolic dynamics for horseshoe diffeo., 956Symbolic logic, xixSymmetric, 53Symmetric bilinear form, 549Symmetric braided monoidal category, 794Symmetric difference, 17, 44, 125Symmetric group, 116Symmetric monoidal category, 795Symmetric sesquilinear form, 328Symmetrizable, 53, 110Symmetry, 221

Symplectic diffeomorphism, 938Symplectic manifold, 939Synchronization of frequencies, 1003Synchronization of oscillators, 1005Syntactic method, 2, 3σ-system (Σ-system, Ω-system), 693System of class representatives, 38System of cogenerators, 786System of congruences, 131System of differential equations, 519, 924System of equations, 143System of first-order differential equations, 519System of generators, 786System of higher-order differential equations, 520System of linear equations, 144System of linear first-order diff. equations, 520System of neighborhoods, 269System of propositional formulas (axioms), 2System of relations, 51System of structural equations, 866System of types, 18

T-algebra, 803t-distribution, 835T-periodic , 966T0,T1,T2,T3,T4-axioms, 280T0,T1,T2-axioms, 280T1-space, 335T2-axiom, 271, 281T2-space, 271, 281, 335T2-space (Hausdorff space), 282T3-axiom, 281T3-space (regular space), 282T4-axiom, 282Table of indefinite integrals, 479Table of integration rules, 482Table of primitives, 479Table of random numbers, 844Table of variance analysis, 851Tables of Fourier transformation, 894Tables of random numbers, 843Takens, 963, 964, 992Takens theorem, 562Tangent, 238, 532Tangent angle, 222Tangent developable, 536Tangent developable of a curve, 536Tangent hyperplane, 465Tangent linear map, 563Tangent of a circle, 222Tangent plane, 539Tangent problem, 445Tangent space, 560, 561, 563, 931, 957Tangent vector, 560, 563Tangent vector to a surface, 540Tangential component, 542Tangle of curves, 976Target, 811, 812Tarski’s axiom, 21Tarski-Grothendieck set theory (TG), 21Tautological propositional formula, 3Tautology, 2Taylor expansion, 947Taylor formula, 154Taylor polynomial, 449, 450Taylor polynomial theorem, 450Taylor remainder, 449, 450

Cauchy form, 450Lagrange form, 450

Taylor series, 451, 576

Tchebychev (or, Chebyshev, Tchebicheff, Cebysev,Tschebyscheff, Chebishev), 101, 459

Tear, 264Temporal average, 987Tensor, 141, 186, 548—550, 562

relative, 552

1102

Index

skew-symmetric r-times contravariant, 186Tensor algebra, 549, 551Tensor analysis, 550, 551Tensor bundle, 551Tensor calculus, 551Tensor category, 792Tensor contraction, 550, 554Tensor density, 552

even, 552odd, 552

Tensor differentiation, 550Tensor field, 550, 551Tensor of type (0, 1), 549Tensor of type (0, 2), 549Tensor of type (1, 1), 549Tensor of type (p, q), 549Tensor on a vector space, 551Tensor powers, 712Tensor powers and braiding, 712Tensor product, 550, 792, 793Tensor quantity, 550Tensor set, 551Tensor space, 551Tensor space of type (α, β), 551Tent map, 980Term of degree k, 146Term of degree n, 392Terminal morphism, 786Terminal ob ject, 418, 784Terminal ob ject , 786Terminal property, 786Ternary relation, 37, 218, 219Test

Chow, 859Kurtosis (flattening), 870normality, 870Skewness (asymmetry), 870

Test function, 902, 903Tetragon, 209Tetrahedron, 340, 346Thales, 222Theorem

group of affine collineations: G, 235Abel, 174about angles, 221acyclic free chain complex, 409addition, product, quotient rules, 447Afraimovich-Shilnikov, 1001Alembert, 305Alembert-Gauss, 104Alexander-Whitney, 431Alexandroff, 305Andronov-Hopf, 995Andronov-Pontryagin, 979Andronov-Witt, 973anti-matching, 209anticommutativity, 431any Euclidean ring is principal, 193any finite extension is algebraic, 166Ascoli, 320, 598Ascoli (for a metric space), 320Ascoli (general case), 320aureum , 196Bézout, 80Bézout (for polynomials), 149Baire category, 315, 979Banach, 328, 529Banach fixed point, 504Banach-Alaoglu, 599Banach-Steinhaus, 327, 503Banach-Tarski paradox, 23basis of a vector space, 139Bayes, 837Bessel inequality, 331, 883binomial, 121, 124biquadratic reciprocity, 196

Birkhoff, 971Birkhoff (ergodic), 981Birkhoff HSP, 697Birkhofff, 697bisectors, 208Blakers-Massey, 426Block-Guckenheimer-Misiuriewicz, 992Bolzano-Weierstrass, 437Boolean prime ideal (BPI), 26, 31, 32Borsuk, 641Borsuk-Ulam, 398, 432Brouwer, 292, 640Brouwer and Kakutani fixed-point, 911Brouwer fixed point, 292, 915C is algebraically closed, 155Cantor, 41, 310Cantor-Schröder-Bernstein theorem, 48Carathéodory, 658Casorati-Weierstrass, 630Cauchy, 617Cauchy formula (homology form), 645Cauchy formula (homotopy form), 639Cauchy integral, 575, 581Cauchy integration criterion, 481Cauchy theorem (homology version), 645Cauchy theorem (homotopy version), 638Cauchy-Goursat, 613, 626Cauchy-Kowalewska, 526Cauchy-Pompeiu formula, 617Cayley-Hamilton, 180center manifold, 946central limit, 835Chinese, 131Chinese remainder, 131chordal, 228circle quadrature, 176closed graph theorem, 328, 503commutative group for composition law, 213compact of Rn, 268compact set, 284compact support, 389compactness criterion of metrizable space, 309completion of a metric space, 311Complex Hahn-Banach, 33composition rule, 447congruence, 222connected image, 276connected space, 276constructibility by straightedge-compass, 175convergence criterion, 461covering homotopy theorem, 414curl, 472, 557cyclic decomposition, 183De Rham, 493degree, 164Denjoy, 935derivability, 598Desargues (affine version), 211Desargues (little), 215Desargues (pro jective version), 212Diagonalization conditions, 181dimension axiom, 389, 422, 428Dini, 319divergence, 494, 557divergence (Gauss), 472Douady-Oesterlé, 990Douady-Oesterlé (for diff. equations), 990dual universal coefficients, 429Eilenberg-Steenrod, 409Eilenberg-Steenrod (extension), 409Eilenberg-Zilber, 412entire series expansion and Taylor series, 451Euclid, 75, 230Euclid lemma, 77, 80Euclid lemma in Z, 80Euclidean division, 74, 79

1103

Index

Euclidean division in Z, 79Euler, 130, 227Euler criterion, 195Euler displacement, 254Euler-Lagrange, 112, 113exact triangle, 394excision, 400, 428existence of a root of a polynomial, 162extension, 311extreme value, 285factorization, 75Fatou, 662Fermat, 120, 126, 130Fermat (little), 120Fermat’s last, 126, 130Fermat’s little, 130field by two ideals, 136field of complex numbers is algebraically closed,

155field with characteristic zero, 168finite Abelian group, 120finite simple extension, 166finite simple extension (Steinitz), 166five lemma, 394fixed-point, 312fixed-point with parameter, 312Floquet, 945, 969flow-box, 934Fredholm, 507, 527, 528Freudenthal suspension, 426Frobenius, 564Fundamental theorem of Algebra, 104fundamental theorem of arithmetic, 76fundamental theorem of arithmetic (for polyno-

mials), 149fundamental theorem of integral calculus, 476Gauss, 77, 80, 472Gauss (in Z), 80genus, 196Giraud, 823golden (Gauss), 196gradient, 465, 557Granger representation, 877Green, 427, 472, 494Green-Riemann, 491Grobman-Hartman, 978Grobman-Hartman (for discrete syst.), 979Grothendieck, 768, 772groupoid, 379Gysin sequence, 432Hadamard three circles, 666Hadamard-Cartan & Gromoll-Meyer, 568Hadamard-Perron, 974Hadamard-Perron (for discrete system), 978Hahn-Banach, 32, 61, 321, 324Hahn-Banach (geometric form), 35Hahn-Banach separation, 35hairy ball, 398, 557ham-sandwich, 432Harnack inequalities, 660Hartman-Grobman, 942Hartman-Grobman (for diffeomorphism), 942Hartman-Grobman (for flow), 942Hartogs, 670heights, 209Heine, 303, 304homoclinic saddle connection, 1012homomorphism, 119homothetic circles, 227homothetic triangles, 226homotopy axiom, 390, 428homotopy axiom for pairs, 395homotopy extension, 407homotopy sequence of a fibration, 425Homotopy sequence of a pair, 424homotopy sequence of a weak fibration, 425

Hopf, 433Hopf bifurcation, 963Hopf recurrence, 971HSP, 697Hurewicz, 391, 425Hurewicz (1), 391Hurewicz (2), 425Hurewicz (absolute version), 426Hurewicz (relative version), 426Hurwitz, 647immersion, 290insets and invariant manifold, 958integral domain by prime ideal, 136integral mean-value, 476intermediate value, 276, 307, 442invariance of dimension, 398invariance of domain, 398invariance of integral by homotopy, 637invariant manifold, 942invariant manifold (for diffeomorphism), 942invariant manifold (for flow), 942inverse function, 529inverse mapping, 768irreducible polynomial, 163irreducible prime element, 192isomorphism, 120Jensen formula, 627, 668Johnson, 671Jordan, 642, 643Jordan canonical form, 183Jordan curve, 291, 398, 685Jordan-Brouwer, 292Jordan-Brouwer separation, 398Künneth formula for cohomology, 429Künneth theorem, 412Kakutani (fixed point), 915Kelvin-Stokes, 493Kolmogorov-Arnold-Moser (KAM), 980Kondrakov, 529Kondrakov (for Riemannian manifolds), 529Kuhn-Tucker, 909, 921Kuratowski, 356Kuratowski-Pontryagin, 356L’Huilier, 245Löwenheim-Skolem, 3Lagrange, 112, 115Lagrange interpolation polynomial, 151Lagrangian (generalized), 920least upper bound, 88Lebesgue differentiation, 663Ledrappier, 989Lefschetz, 411Lefschetz fixed point, 410Leray, 401lifting criterion, 415lifting lemma, 414limit, 838linear extension, 139linear map and matrix associated, 140Liouville, 104, 628, 631, 939, 968Liouville (vector version), 631Liouville boundedness, 586Littlewood, 670locally Lipschitzian, 523long exact sequence, 428Lusternik-Schnirelmann, 398Mac Lane coherence, 793, 794Malgrange, 962Malgrange-Ehrenpreis , 904Mather, 953Mayer-Vietoris, 395Mayer-Vietoris (for reduced homology), 396mean value, 448, 449mean value inequality (for polynomials), 155medians, 209Menger, 355

1104

Index

Menger n-arc theorem, 355Mergelyan, 651metrization, 286Meusnier, 543, 544Mittag-Leffer, 653modified Jordan form, 1027monic irreducible polynomial, 163monodromy, 578Montel, 621Morera, 625Nagata-Smirnov (metrization), 288Neimark-Sacker, 998no-retraction, 640non constructibility (straightedge comp.), 175normal form (differential equation), 944numeration in base b, 74of well-ordering, 23open mapping, 328, 503, 640, 647, 655, 727Ostrogradsky-Stokes, 492Ostrowski, 197Palis-Smale, 980Pappus-Pascal (affine version), 211Pappus-Pascal (projective version), 212Parseval formula, 332particular irreducible polynomial (Abel), 163Peano existence, 523Peixoto (I) (for vector field, flow), 951Peixoto (II) (for vector field, flow), 951Peixoto (III) (for diffeomorphism), 953period three implies chaos, 992perpendicular bisector, 208perpendiculars (Hjelmslev), 207, 208Picard-Lindelöf, 966Poincaré, 557Poincaré Bendixson, 973Poincaré recurrence, 971Poincaré recurrence (Poincaré return), 971Poincaré return, 971Poincaré-Bendixson, 933Poincaré-Brouwer, 557polynomial division, 162power, 228power series expansion and Taylor series, 451primary decomposition, 183pro jection, 28, 34, 330pro jection on a complete convex set, 330pro jection on a complete vector subspace, 330Pro jective Nullstellensatz, 722propositional calculus , 2Puppe sequence, 423Pythagoras, 330Pythagorean, 231, 238, 242quadratic reciprocity, 196quadratic reciprocity law, 196quadratic residue, 195radical or power center, 228recurrence, 68reduction, 401regular T1-space, 282representation, 330residue, 582, 645restriction rule, 446Ricci, 553Riemann conformal representation, 595Riemann criterion, 475Riemann mapping, 656Riemannian geometry (fundamental th.), 566Riesz, 508Riesz decomposition, 668Riesz representation, 28, 508, 599Rolle, 448, 450roots and zeros of a polynomial, 151roots of a polynomial, 151roots of a polynomial of degree n, 163Rouché, 647, 685Runge, 649—651

same dimension of isomorphic vector spaces, 139Schoenflies, 292Schwartz, 633Seifert-van Kampen, 402, 403separability criterion for a polynomial, 168separating hyperplane, 35Serre, 425Sharkovskii, 992Sharkovskii -1964, 992Sharkovsky, 992Shilnikov, 999Shinai, 992signature of a permutation, 117signature of a transposition, 117similar triangles, 226simple and multiple roots of a polynomial, 151simplicial approximation, 399Smale, 999Smale-Birkhoff, 959Smirnow-Nagata-Bing (fundamental th.), 287Smith normal form, 401sphere, 568Stokes, 427, 492, 556Stokes (closed form), 493Stokes (exact form), 493Stokes (general form), 492Stone-Weierstrass, 321Stone-Weierstrass , 333Stone-Weierstrass (for complex functions), 321Takens, 562Taylor polynomial, 450Thales, 222theory of patches (fundamental th.), 548three circles, 666Tietze, 402Tietze-Urysohn, 599Tikhonov, 284total probability, 837total probability law, 837transport of intervals, 442Turan, 356Tychonoff, 29, 61, 284, 304, 784Tychonov, 304undecidability, 3unique factorization, 75unique polynomial associated with an algebraic

element, 164unique-factorization, 193uniqueness, 523uniqueness of the norm, 671unisolvence (for polynomial interpolations), 152universal coefficients for cohomology, 429universal coefficients for homology, 411Urysohn, 282, 286Urysohn (metrization), 288van Kampen, 402, 403variation of constants formula, 968vector space norm, 502Vesentini, 671Wedderburn, 153Weierstrass, 653Weierstrass convergence, 620Weierstrass product, 587Whitehead, 426Whitney, 562Whitney graph, 352Wiener-Khinchin, 982Wilson, 133Yoneda lemma, 408Young (1), 989Young (2), 989Zermelo, 48, 62zeros of a polynomial, 151Zorn, 60, 62

Theorem of homothetic triangles, 226Theorem of similar triangles, 226

1105

Index

Theorems about angles, 221Theory

of approximation, 458of cointegration, 846of differential equations, xixof interpolation, 460

Theory of analytic functions, 569Theory of curves, 291Theory of differential equations, xixTheory of patches, 546

fundamental theorem, 546Theory of simplicial homology, 349Theory of singular homology, 349Theory of types, 17, 18, 21Three circles theorem, 666Ti-space, 282Tietze theorem, 402Tietze-Urysohn theorem, 599Tikhonov space, 282Tikhonov theorem, 284Tile, 284, 481Tiling, 605Time average, 981Time continuous system, 965Time discrete system, 965Time series, 868Time series analysis (TSA), 868Time-dependent normal form, 945Time-lag model, 845, 863Time-series, 845Time-series cross-sectional data (TSCS), 856Tivialization of line bundle on a torus, 364, 761Toa, 239Tobin model, 1020Tobin monetary growth model, 1030Tolerance threshold, 842Topol. equivalence replaces top. conjugacy, 941Topological étale site, 825Topological and continuous invariants, 266Topological basis, 271, 417, 735

properties, 271Topological completion, 106Topological conjugacy, 935, 941Topological conjugacy of discrete systems, 978Topological direct sum, 330Topological dual, 324Topological embedding, 737, 778Topological entropy, 984Topological equivalence, 563, 941, 977, 992Topological equivalence of diff. equations, 977Topological graph, 350Topological group, 50, 88, 384Topological homeomorphism, 272Topological invariance, 336Topological invariance of dimension, 731Topological invariant, 263, 264, 266Topological isomorphism, 272Topological K-theory, 364, 761Topological linear space, 27Topological manifold, 730, 731Topological manifold are locally compact, 731Topological manifolds are paracompact, 731Topological map, 263, 265, 272, 273Topological n-dimensional manifold, 733Topological n-manifold, 731Topological quotient space, 51Topological space, 35, 66, 269, 270, 296, 727, 728

axiomatic definition, 270comparison, 272defined in terms of neighborhoods, 269defined in terms of open sets, 270example, 270in terms of neighborhoods, 727in terms of open sets, 727T0,T1,T2,T3,T4-axioms (separation), 282totally disconnected, 307

triangulable, 343Topological space in terms of neighborhoods, 728Topological space in terms of open sets, 728Topological structure, 50, 51, 66, 87, 296

transfer, 273Topological structure of Q, 87Topological structure of R, 435Topological subspace, 51, 272, 273Topological theorem of complex numbers, 105Topological transitivity (TT), 991Topological vector space, 27, 50Topologically conjugate, 942, 977Topologically conjugate diffeomorphisms, 933Topologically conjugate hyperbolic linear diffeomor-

phisms, 942Topologically equivalent, 265, 268, 941, 977Topologically equivalent distances, 295Topologically equivalent flows, 935Topologically equivalent sets, 264Topologist’s sine curve, 277Topology, xix, 27, 35, 263, 296

associated with semi-metric structure, 312comparison

coarsest, finest, coarser, finer, 270induced, 271metric, 271p-adic, 106relative, 268

Topology change, 992Topology engendered by maps, 273Topology generated, 272Topology generated by a family of parts, 296Topology of bands, 272, 278Topology of the complex plane, 640, 675Topology on a set, 729Topos, 770, 822Tor, 411, 758Tor group, 411Torsion, 411, 534Torsion coefficient, 349, 401Torsion formula, 534Torsion group, 349Torsion of the circular helix, 534Torsion product, 411Torsion sheaf, 773Torsion tensor, 565Torus, 372, 544Torus T2, 988Total curvature, 545Total degree, 158Total order, 12, 28, 49, 59, 62Total order relation, 45Total order structure of N, 84Total orthonormal family, 332Total orthonormal system, 332, 333Total probability, 837Total probability law, 837Total probability law theorem, 837Total probability theorem, 837Total space, 413, 759, 761, 777Total variability (SST), 848Totality, 29, 59, 783Totally disconnected topological space, 307Totally E-acyclic, 410Totally ordered set, 12, 28, 59, 60, 62Totally ordered set (Total order), 59Totally regular ring, 762Totally well ordered field

Archimedean , 105Totient function, 77, 91, 129Trace, 274, 411, 969Trace map, 585Trace topology, 729Traces, 264Tractrix, 241Tractroid, 241

1106

Index

Transaction effect, 1030Transcendantal extension, 166Transcendental elements of a field, 164Transcendental entire function, 586Transcendental function, 586Transcendental number, 106, 164Transcendental numbers, 157Transcritical bifurcation (normal form), 992Transcritical bifurcations, 995Transfinite induction, 62, 66Transformation

elliptic, 594hyperbolic, 594loxodromic, 594parabolic, 594

Transformation group, 203Transformation matrix, 1027Transition function, 560, 734Transition map, 560, 734Transition map (transition function), 734Transition matrix, 843Transition probabilities, 843Transitions to Chaos, 999

cascade of period-doublings, 999destruction of a torus, 999, 1000Global homoclinic bifurcation, 999intermittency, 999

Transitive G-set, 414Transitive G-space, 414Transitivity, 10Transitivity rule, 2Translation, 209, 219, 220, 224Translation functor, 800Transmission medium, 893Transpose, 142Transpose conjugate matrix, 184Transposition, 116Transvection, 229Transversal, 221Transverse homoclinic point, 959, 978, 999Trapezium, 481Trapezium (approximation) method, 481Trapezoidal formula, 887Tree, 353, 402, 417Trees, 837Trefoil curvE, 455Trend, 879Trial, 833Triangle, 210, 239Triangle axiom, 27Triangle centroid, 227Triangle function, 440Triangle of signed Stirling numbers of the first kind,

122Triangulable topological space, 343Triangular matrix, 182Triangular simplex, 346Triangular system, 867Triangulated category, 800Triangulated polygon, 403Triangulation, 340, 341, 343, 346, 398, 399Trigonalization, 182Trigonometric circle, 97Trigonometric function, 238, 242, 616

cosecant, 242cosine, 242cotangent, 242hyperbolic cosine, 242hyperbolic sine, 242hyperbolic tangent, 242secant, 242sine, 242tangent, 242

Trigonometric identities, 242Trigonometric series, 883Trigonometrical function, 237

Trigonometry, 237based on a circle, 242based on a hyperbola, 242

Trimetric pro jection, 237Triple, 802Trivial bundle, 364, 760Trivial divisor, 128, 148Trivial group, 111, 120

Ablian, 120cyclic, 120

Trivial homomorphism, 382Trivial morphism, 114Trivial ring, 709Trivial subgroup, 112Trivial topology, 270Trivialization, 364, 760True coefficient, 846True proposition, 1Truncated Taylor series, 926Truncation, 422, 947Truth value, 1, 3TS process, 870TS process (Trend Stationary), 870Tube lemma, 405Turan graph, 356Turan theorem, 356Turbulence burst, 971Turbulent phase, 999Turing, 3Twin primes, 202Twist, 224Twisted K-theory, 761Twisted sphere, 211Two point compactification of R, 571Two-cycle, 116Two-dimensional manifold, 559Two-sided convolution, 895Two-sided ideal, 29Two-sided ideal (bilateral ideal), 135Two-sided test, 835Two-sided transformation, 888Two-stage least squares (2SLS), 868Two-tailed test, 835Two-value principle, 1Two-valued logic, 1Two-valued principle, 5Tychonoff space, 282Tychonoff theorem, 29, 61, 284, 304, 784Tychonov theorem, 304Type, 18, 692—695Type (sort), 695Type I error, 835Type II error, 835Type of algebraic system, 692Type theory, 17, 18, 21Types of affine isometry, 256Types of equilibrium point, 975Typical property, 979

U-set, 784U-small, 784U-small set, 784UFD, 167Ultra-metric distance, 299Ultrafilter, 26, 30, 284, 302, 303, 736Ultrafilter lemma, 26, 31Ultrametric, 198Umbilic, 544Umbilic point, 544Umbilical point, 544, 567Unary operation, 22, 359, 693Unbiased estimator, 834, 841, 847Unbiased forcasting, 848Uncorrelated errors, 846Uncorrelated variables, 845Uncountable negligible set, 497Uncountable product, 296

1107

Index

Uncountable universe, 782Uncountably infinite set, 24Undecidability theorem, 3Undecidable hypothesis, 47Underlying set, 692, 695Underlying set (carrier set), 692Underlying set (or domain), 695Underlying space, 398Unfolding, 962Uniform continuity, 309, 443, 573Uniform convergence, 318, 444Uniform convergence of a sequence, 318Uniform medium, 893Uniform norm, 28, 459Uniform norm (Chebyshev norm), 459Uniform polyhedron, 340Uniform space, 312, 313Uniform space (Weil), 313Uniform structure, 313Uniform topology, 313Uniformity, 313Uniformly approachable map, 458Uniformly continuous function, 443Uniformly continuous map, 309Uniformly convergent sequence of functions, 443Uniformly equivalent distances, 295, 310Union, 11, 17Union of open balls, 67, 271Union of open sets, 266Union of regular patches, 540Unipolar rectangular impulse, 894Unique factorization theorem, 75Unique-factorization domain, 167Unique-factorization domain (UFD), 167, 192, 193Unique-factorization ring, 167, 192, 193Unique-factorization theorem, 193Uniqueness of distributive lattice, 12Uniqueness of the limit, 280Uniqueness theorem, 523Unisolvence theorem, 152Unisolvence theorem (for polynomial interpolations),

152Unit, 57, 191, 700, 802Unit circle, 217, 291, 970, 1013Unit circle of C, 97Unit element, 56, 57, 699, 700Unit jump, 890Unit law, 799Unit n-sphere, 731Unit ob ject, 793Unit polynomial, 47, 146Unit ring, 57, 700, 708Unit ring (ring with identity, ring with unity), 708Unit ring (ring with identity, ring with unity, ring

with a one, unitary ring, unital ring), 708Unit root, 870Unit root test, 870Unit square, 87Unit step function, 440Unit tangent vector, 532, 534Unit vector, 247, 539Unital Banach algebra, 32Unital ring, 57, 700Unitary divisor, 199Unitary group, 177Unitary polynomial, 47, 146Unitary ring, 57, 700, 708Unitary tile, 339Unitor, 793Unity, 56, 699Universal algebra, 358, 691, 692Universal coefficients, 411Universal coefficients theorem for cohomology, 429Universal coefficients theorem for homology, 411Universal cover, 777Universal covering space, 415, 777

Universal element, 12Universal morphism, 786Universal properties of diagrams, 797Universal property, 85, 293, 315, 408, 785, 786

of completions, 293Universal property of the embedding, 123Universal quantifier, 14Universal set, 10, 11, 17, 18, 693Universality condition for structure extension, 85Universally catenary ring, 712Universally closed morphism, 765Universally true, 3Universe U, 781Univocal relation, 38, 704Unknown error, 847Unramified morphism, 766, 770Unsatisfiable formula, 3Unstable in the sense of Lyapunov, 938Unstable separatrix, 1011Unstable statistical model, 853Upper (Riemann) integral, 474Upper bound, 61, 70, 665Upper box-counting dimension, 988Upper capacity, 988Upper envelope, 312Upper integral, 474Upper limit of sequence, 437Upper pointwise dimension, 989Upper semi-continuous function, 317Upper semi-continuous function (u.s.c.), 665Upper set, 30, 31Upper triangular matrix, 147, 182Urelement, 2, 21, 782Urysohn, 282, 286, 289Urysohn metrization theorem, 288Usual topology of Cn, 305Usual topology of Rn, 305Utility maximization under budget constraint, 920UVAR (unrestricted vector AR), 875

V-valued presheaf, 820Valeur d’adherence, 300, 311Valuation, 106, 193, 196—198

discrete or dense, 197Non-Archimedean, 198

Valuation of a polynomial, 148Valuation of n at p, 76Valued field, 106, 198Van der Pol equation, 995Van der Pol oscillator, 1012Van der Pol system, 1012Van Kampen theorem, 402, 403Vandermonde matrix, 152Vanishing line, 232Vanishing point, 232VAR (vector autoregressive), 873VAR model, 873VAR process, 874VAR(p), 873Variability of residuals (SSR), 848Variable, 1

dependent, 925dummy, 3free (in logic), 3functorial, 4input, 925linked (in logic), 3output, 925propositional, 1sentencial, 1state, 925

Variable element, 786Variable to be explained, 846Variance, 834, 840Variance analysis, 851Variance decomposition, 875

1108

Index

Variance of the error, 846Variance-covariance matrix, 856Variate, 833Variation of constants formula, 968Variation of the argument along a path, 638Variational derivative, 504Variational equation, 972Variational method, 528Variety, 358, 696, 697, 718Variety (overview), 696Variety in a category, 358, 802Variety of algebras, 697Variety of universal algebras, 358, 697, 802VARMA, 873VARMA (vector autoreg. moving average), 873Vect-enriched (symmetric) monoidal category, 792Vector, 205, 220, 247Vector addition, 247Vector angles, 101Vector basis, 249Vector bundle, 363, 364, 760Vector density, 552Vector equation of polar plane, 252Vector equation with 2 points, 250Vector equation with 3 points, 251Vector equation with point and direction, 250, 251Vector error correction model (VECM), 877Vector field, 398, 554, 556Vector function, 462Vector isometry, 255Vector Liouville theorem, 631Vector of bits, 42Vector operator, 465Vector product, 188, 249, 250

anticommutativity law, 250distributivity law, 250mixed associativity law, 250properties, 250

Vector resolute, 248Vector space, 27, 50, 58, 109, 136, 139, 501, 695,

701dimension, 139normed, 501number of element, 139

Vector spaces of finite dimension, 139Vector-valued function, 462Vectorial rotation, 238Vectorial structure, 259Vehulst curve, 862Venn (or Euler) representation, 9Venn diagram, 9Versal unfolding, 962Vertex angle, 221Vertex connectivity, 355Vertex cut, 355Vertex method, 904Vertex set, 398Vertical composition, 358, 792Vesentini theorem, 671Vinculum, 90, 91Violation of assumptions, 855Visual examination of residuals

visual examination of residuals, 856VMA (vector moving average), 874VMA representation, 874Volume calculation, 483Volume formula, 483von Neumann hierarchy of sets, 19, 783von Neumann integer, 49von Neumann integer zero, 49von Neumann method, 49von Neumann ordinals, 49von Neumann regular ring, 713von Neumann regular ring (absolutely flat ring),

713von Neumann universe, 19, 783

von Neumann-Bernays-Gödel set theory (NBG), 21

Walras, 915Walras law, 915, 1029Walras model, 1020Walrasian equilibrium, 1029Walsh functions, 901Walsh systems, 901Wavelet family, 900Wavelet moment, 900Wavelet order, 900Wavelet transformation, 899, 900Wavelets, 899Weak embedding, 737Weak extrema, 505Weak fibration, 425Weak fibration (Serre fiber space), 425Weak law of large numbers (Bernouilli), 838Weak topology, 406, 725Weak∗ convergence, 662Weakly embedded submanifold, 737Weakly inaccessible cardinal, 20, 782Wedderburn theorem, 153Wedge, 401Wedge of pointed spaces, 401Wedge product, 185, 187, 675Wedge product (exterior product), 187Weierstrass approximation, 458Weierstrass convergence theorem, 620Weierstrass elliptic function, 589Weierstrass elliptic functions, 589Weierstrass eta-function, 589Weierstrass formula, 458Weierstrass functions, 588Weierstrass p-function, 589Weierstrass ℘-functions, 589Weierstrass product theorem, 587Weierstrass sigma-function, 587, 588Weierstrass theorem, 653Weierstrass zeta-function, 589Weight of the topological space, 728, 736Weingarten formulas, 546, 547Weingarten map, 566Weingarten surface, 566, 568Well order, 48, 62Well ordered set, 48, 62Well ordering, 48, 62Well-founded order, 29, 62, 784Well-ordered set, 29, 784Well-ordering principle, 62Well-ordering principle (equivalent to the axiom of

choice), 29, 784Well-ordering theorem, 23Well-powered category, 791Weyl lemma, 668White noise, 856, 871, 879White noise process, 869White’s test, 860White-noise tests, 869Whitehead, 18Whitehead theorem, 426Whitney graph theorem, 352Whitney theorem, 562Whole number, xix, 67, 83Wiener process, 880Wiener-Khinchin theorem, 982Wilson theorem, 133Winding number, 582Window function, 901Windowed Fourier transformation (WFT), 901Wintner-Conti condition, 968With no divisors of zero, 57, 700With no zero divisors, 57, 700Wolf, goat, cabbage problem, 830Word, 345, 704Work done, 487

1109

Index

Work-energy principle, 488Wronskian, 517, 518

Xet, 770

Y-scheme, 763Yoneda embedding, 828Yoneda Lemma, 361, 811Yoneda lemma, 408, 827Young theorem (1), 989Young theorem (2), 989

Z, 22z-plane, 95z-transformable sequence, 897Z-transformation, 897Zariski cover, 767, 768Zariski topology, 365, 717, 718, 722, 768Zariski topology on Pnk , 722Zermelo and Zorn theorems, 49Zermelo axiom of choice, 19Zermelo set theory, 20, 22Zermelo theorem, 23, 45, 48Zermelo-Fraenkel axioms, 22, 29, 784Zermelo-Fraenkel set theory, 19, 22, 783Zero, 159Zero chain map, 392Zero complex, 392Zero element, 56, 57, 699, 700Zero inner product, 28Zero map, 753Zero morphism, 810Zero object, 418, 785Zero object (null ob ject), 785Zero vector, 707Zero-locus, 721Zeros and local extrema, 452Zeros of a polynomial, 151Zeros of holomorphic functions, 626Zeroth Betti number, 681Zeroth de Rham cohomology vector space, 681ZF, 20, 22ZFC, 19, 22, 782, 783ZFC axioms, 22Zorn lemma, 19, 29, 33, 48, 324, 784Zorn theorem, 60

1110