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Ne jetons pas les morceauxDe nos coeurs aux porceauxPerdons pas notre latinAu profit des pantinsParlons pas la langue des DieuxPour les balourds, les fesse-mathieuLes paltoquets ni les bobechesLes foutriquets ni les pimbechesNi pour la femme . . .

G.B.

My bibliographic research was greatly simplified by "A thousand papers in ModelTheory and Algebra", a list compiled by Wilfrid Augustine Hodges ; I shalltake the opportunity to thank him for the critical reading which he gave themanuscript of this text.

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Index of Notation

Not seeing anything better to do, I have arranged all symbols in order oftheir appearance in the text.

a=(al , . .. ,an ) n-tuple 1.1Sp(R , R' ) p-isomorphisms from R to R' 1.1Sw(R , R' ), SOI(R,R') w-isomorphisms, a -isomorphisms 1.1(a, R) "'p (b, R') p-equivalence 1.1R -< R' elementary extension 1.1iii concatenation 1.1

f( x) formula 2.10 , V, /\, --+, +-+ boolean connect ives 2.1\1,:3 quantifiers 2.1S(f) sub formulas 2.1QR(f) quantifier rank 2.1FV(f) free variables 2.1(3!x) "there exists exactly one" 2.1R F f(a), R ~ f(a) satisfaction 2.3, 2.4A F I, 9 F f (semantic) consequence 2.3

IIuSdU ultraproduct 4.1SU ultrapower 4.1(J) elopen set in the space of theories 4.2T(M) complete diagram of M 4.2

430 Index of Notation

L(M) language of the diagram of M 4.2

L H , f H, T(H) Henkinizations 4.3

SI(T),Sn(T) spaces of types 5.1SI (A), Sn(A), S[(A) spaces of types with parameters 5.1T(A) "diagram" of a set of parameters 5.1f(x,a) formula with parameters 5.1(J(x,a» = {p : p F f(x,a)} elopen set 5.1

k[X] polynomial ring 6.1k(a) field generated 6.1x',x", .. . ,x(n) derivation 6.2oP/oX, oP/ox(n), .. . partial derivatives 6.2P* polynomial with derived coefficients 6.2K[X]d ring of differential polynomials 6.2K(a)d differential field generated 6.2I(P) prime differential ideal of the minimalpolynomial P 6.2RD dimension rank 6.2

N,w set of natural numbers 7.1(Vy ~ x), (3y ~ x) bounded quantifiers 7.6~o , ~n, En , IIn Kleene hierarchy 7.6VEn, VIIn En truth, IIn truth 7.6Pr(x, y) proof predicate 7.10Cons(A) consistency 7.11

A + B , A x B sum and product of chains 8.1ord(A) ordinal of A 8.1No = N,w 8.1ZF Zermelo-Fraenkel set theory 8.2card(A) ,IAI cardinality of A 8.32A,21< powerset and its cardinality 8.3K+ successor cardinal 8.3No =w,N1 =wl,N2 , • . • ,No<, '" 8.3Jo = w,:11 = 2W

, :12 , • •. ,:10<' . . . 8.3K A, K <A cardinal exponentiation 8.3K +.x, K x.x cardinal sum and product 8.3Dedfx), dedfx) number of cuts 8.3

M, N models 9.1A, B sets of parameters 9.1ITI, ILl cardinality of a language 9.1

0'. , (3", .. ./'i"A ,M, · · ·

ordinals 9.1cardinals 9.1

Index of Notation 431

H(T) space of Henkin enumerations 10.1Ao first 0'. terms of a construction 10.4E(I) rich space constructed over 1 10.6

llpdU, pU ultraproduct, ultrapower of types 11.1(M,dp) structure of "definition" of a type 11.1,13',2

In, 1>. axioms of the independence property 12.4

p ~ q,p rv q fundamental order 13.1Ao(T) minimum of the spectrum 13.2/'i,(T) , /'i,n(T) supremum of decreasing ordinal sequencesin the fundamental order 13.2

(3(p) bound of a type 15.1

SF1(A), SFn(A), SFo(A) space of strong types 16.1Teq, Leq,Meq jump to the imaginary 16.1G(SF1(A)/A) Galois group of strong types 16.6G(Aalg/A) Galois group of the algebraic closure 16.6

R(p) rank 17.1O'.R ordinal supremum of a rank 17.1RU Lascar's U rank 17.1RC Shelah's continuous rank 17.2R(T) maximum of a continuous rank 17.2RM(p) , RM(f) Morley rank 17.3aT = aRM 0'. of the Morley rank 17.3R( ,f, w), R( .z ,2) infinite division rank 17.4R( .i.2), R( ,~, 2) dichotomy rank 17.4

tp(a/A) type of a over A 18.2R(tp(a/A)) , R(a/A) rank of the type of a/A 18.2jM elements of M satisfying j 18.2Kalg , K m in , Kdc algebraic, minimal, and differential closures 18.3M(1) Ehrenfeucht model 18.5

Po in a sequence, the type of ao over Ao 19.10'.{3 ordinal exponentiation 19.20'. + (3 natural sum 19.2p ~D q domination order D 19.4

432 Index of Notation

P ~R q realization (or Rudin-Keisler) order R 20.1M(ii) prime model over M U iii} 20.1Pit "foundation" type over ii 20.2prof(p) depth 20.3

Index

A-automorphism, 183D-equivalent, 388D-order, 388I-valued ultrametric space, 86M-special sequence, 268R-equivalent, 394U rank, 334Doo formula, 124Don set , 125IIn formula, 125IIn set, 125En Choice, 128En formula, 124En set, 125No, 163No, 173a-equivalence, 11a-isomorphism, 11,6-function, 117a., 173oo-equivalence, 11, 57, 58eo-isomorphism, 13, 35x-atomic, 366x-oompact, 186x-dense, 365

II:-homogeneous, 186x-lsolated, 365x-resplendent, 193, 296, 317x-saturated, 186x-universal, 186w,163w-categorical, 209, 210w-isomorphism,3w-saturated, 57w-stable, 285WI-categorical, 413I-regular type, 397n-ary relation, 1n-rich ultrametric space, 223, 286n-tuple, 1p-equivalence, 4p-isomorphism, 2, 35Los' Theorem, 41O-dimensional space, 81

compact, 48

absolute type, 56algebra

Boolean, 79polyadic, 82

434 Index

Tarski-Lindenbaum, 82algebraic (over a field), 66

transitivity of, 69algebraic closure, 215, 321, 359algebraic group, 358algebraic logic, 82algebraically closed field, 64, 66,

149, 286, 312algorithm, 130antithesis, 25arithmetic, 116, 231, 286

standard model, 116arithmetic set, 124arity of a relation, 1atom, 82

in fundamental order, 279atomic, 207, 355atomic Boolean algebra, 82atomic formula , 17atomically algebraic, 357atomically saturated, 357atomless Boolean algebra, 83axiom, 25axiom of choice, 164

equivalents of, 166axiom of countable choice, 169axiom of dependent choice, 169axiom of ultrafilters, 39, 170axiomatization, 25

back-and-forth method, 2, 11, 58,59

Baire, property of, 204base, of a filter, 39basis, 381, 407Bernays-Godel set theory, 166Bernstein's theorem, 171Beth's Theorem, 185binary relation, 6Boolean algebra, 79, 286

atomic,82atomless, 83

Boolean combination, 24Boolean formula, 18Boolean order, 79

Boolean ring, 78Boolean symbols, 17Boolean-independent, 176bound of a type, 301, 302bound variable, 18bounded multiplicity condition of

rank, 333.bounded quantifier, 124bounding of zeros, 100bracketing, 16branch, 144branch (of a tree), 144Buechler's lemma, 390

canonical diagonal embedding, 42canonical parameter, 321, 402Cantor normal form, 378Cantor rank, 285, 326, 341, 343Cantor's Theorem, 171Cantor-Bendixson rank, 12Cantorian rank, 341cardinal, 172

product, 174regular, 177, 178singular, 177, 178, 291successor , 172, 178sum , 174

cardinal arithmetic, 172, 174cardinality, 2, 171, 172

of a language, 36categorical, 209, 370, 372categorical (in a cardinal) , 109chain, 7, 262

colored , 266cut in, 265dense, 8, 44discrete, 8of elementary extensions, 28, 36of saturated models, 295

chain of saturated models, 318character (of a unary relation) , 5characterist ic (of a field), 65Chinese Remainder Theorem, 117choice, 128, 164, 166

countable, 169

dependent, 169chronically resplendent, 201Church's thesis, 131class of types in fundamental

order, 277clopen set, 43, 81closed

in a construction, 212, 218closed formula, 18closure

algebraic, 321, 359differential, 76, 359minimal, 357, 359rational, 321

code, 122coded set, 122cofinal, 177cofinality, 177coheir, 243

duality with heir , 243coherent ring, 96, 287collection, 137colored chain, 266comaximal son, 301combinatorics, 117, 118comeager, 205common elementary extension, 46,

47compact O-dimensional space, 48compactness, 57, 206Compactness theorem, 43, 44companion, 62, 96complete theory, 26, 42-44completeness, 148complexity (of a formula) , 17complexity (of a term), 33computable, 130computably enumerable, 130conjunction, 17connective, 17consecutivity, 109consequence, 25, 44, 146, 148consistency sentence, 152consistent, 25, 43, 227constant (in differential field), 359

Index 435

constant symbol, 33constrained ideal , 76constructed, 212constructed model , 351constructible, 364constructible topology, 70construction, 212continuity, 100continuous rank, 337continuum hypothesis, 172contradictory, 25, 48countable, 172cut (in a chain), 265

definable, 265

decidable, 130decidable theory, 148Dedekind, 175deep theory, 417definable, 110, 184, 195, 230

explicitly, 184implicitly, 184with parameters, 184

definable cut, 265definable type, 306definition of a type, 230degree of computability, 134dense, 204dense order, 8, 228, 244, 278, 286denumerable, 172derivation, 71derivative (of a topological space),

12diagonal embedding, 42diagonalization, 154diagram, 45

free, 45dichotomy property, 235dichotomy rank, 346differential closure, 76, 359differential equation

linear, 360differential field, 71differential ideal, 72Differential Nullstellensatz, 76

436 Index

differential polynomial, 71minimal, 73order of, 72

differential ring, 71differentially closed field, 71, 286,

358differentially transcendental, 74dimension, 326, 381, 403dimension rank, 74dimensional, 296dimensional order property, 416dimensional theory, 403dimensional type, 402Diophantine set, 126discrete order, 8, 110, 286Disjoint Consistency Lemma, 191disjunction, 17distance, 86, 218distance type, 90, 266divisible sequence, 250domination (of types, 387dope, 414

Ehrenfeucht, 11Ehrenfeucht game, 11Ehrenfeucht model, 368Ehrenfeucht set, 246elementary class, 96elementary embedding, 4elementary equivalence, 4, 35elementary extension, 4, 27, 32, 35

chain of, 28, 36common, 46, 47

elementary local isomorphism, 3elementary restriction, 4elimination of imaginaries, 321,

323elimination of quantifiers, 60, 67embedding, 2

diagonal, 42elementary, 4

end extension, 231endpoints (of a chain), 8enumerable, 130enumeration

Henkin, 50equality, 49equation, 65equivalence relation, 6, 45, 286

finite, 309equivalence, elementary, 4, 35equivalent formulas, 20equivalent in fundamental order,

278, 280Euclidean division, 112exchange lemma, 383existentially closed, 94explicit definition, 184exponential, 118extension

elementary, 4, 27, 32, 35end, 231of a type, 225

extension condition of rank, 333

factorization domain, 71false, 33fat, 205field

algebraically closed, 64, 66, 149,286, 312

characteristic of, 65differential, 71differentially closed, 71, 286ordered,98real closed, 98, 101

field of definition, 325filter, 38, 80

Frechet, 38of groups, 93of primitive subgroups, 287

filter base, 39filter prebase, 38finite equivalence relation, 309Finite Equivalence Relation

Theorem, 310finite structure, 44finite type, 35finite type module, 96finite-dimensional theory, 413

finit ely axiomatizable, 200finitely axiomatizable theory, 44finitely presented module, 96first-order language, 114forking, 75, 94

symmet ry of, 220, 245, 302, 303t ra nsit ivity of, 303

forking son, 303formula , 15, 17, 122

atomic, 17Boolean (quantifier-free) , 18closed , 18complexity of, 17equivalent , 20prenex ,23primitive, 91st rongly minimal, 371

found ation (of a model) , 407found ation rank, 334Frechet filter , 38Frafsse rank, 3, 11Fraisse's back-and-forth method ,

11Fraisse's t heorem, 24, 32free digram , 45free vari able, 18functi on symbol, 33fund amental order , 277

at om in , 279class of types in , 277completeness of, 278equivalence in, 278, 280

fundamental system, 359

Godelcompleteness theorem , 148first Incomplet eness Theorem,

151second Incompleteness Theorem ,

152Oodel ,B-function, 117Galois correspondence, 330Galois group, 362Galois theory, 185, 329, 358game

Index 437

Ehrenfeucht , 11generalized continuum hypothesis ,

172generated (substructure ), 35good (set or function ), 138good definition , 306good model, 196

Hausdorff 's ax iom, 166Hausdorff 's theorem, 176height , 144heir , 94, 225, 243

duality with coheir, 243strong, 228

Henkin enumerat ion, 50, 143,205Henkin tree, 143Henkin 's method , 47, 50, 142Henkin, Leon , 47heredity condition of rank, 333Hilber t 's program , 155homogeneous , 186

idealconstrained, 76differenti al , 72field of definit ion of, 325pr ime, 65, 66

ideal of equat ions, 65imaginary element, 318

eliminat ion of, 321, 323implicit definition , 184incompleteness

Godel 's First Theorem , 151Godel's Second Theorem, 152philosophical implicat ions of,

153inconsistent , 25increasing sequence of par ameters ,

374independence prop er ty, 249, 258,

297independent over a model, 245independent over a set, 304independent sequence of

par ameters, 375

438 Index

indiscernible sequence, 246, 353similar, 246

indiscernible sequence ofparameters, 374

indivisible, 375indivisible sequence, 250induction, 135~o- , En - , IIn - , 136transfinite, 12, 163

induction, definition by, 129infinitary definition of a type, 239,

247infinite i-branching rank, 345infinite i-twofold branching rank,

346initial segment, 160interpretable, 1l0, 184, 195, 196,

286inverse chain, 10isolated point, 44isolated type, 76isomorphic ultrapowers, 47isomorphism, 1, 32, 35

local , 2, 35isomorphism condition of rank, 333isosceles polygon , 86

jump-free rank, 334

Konig's Lemma, 144Konig 's Theorem, 179Karpian family, 12kind (of a strongly regular

dimensional type), 406Kleene hierarchy, 126Kolchin envelope, 364Kolchin extension, 362Kolchin formula , 359Kuratowski 's axiom, 167

Lowenheim 's theorem, 28, 175Lowenheim-Skolem theorem, 45language, 33

first and second order, 114Lascar decomposition, 400

Lascar inequalities, 378, 380Lascar rank, 334Lascar symmetry lemma, 380Lascar's technical lemma, 389lexicographic product, 10, 377limit ordinal, ll , 163limit type, 250, 375linear chain , 8linear differential equation, 360local isomorphism, 2, 35

elementary, 3local monotonicity, 100local rank, 345

main gap, 417MatijaceviC's Theorem, 126maximal son, 301meager, 205mean type, 375membership relation, 119minimal arithmetic, 134minimal closure , 215, 357, 359minimal model , 78, 215minimal polynomial, 66, 73model ,25

x-resplendent, 317w-saturated, 57atomic,207constructed, 214, 351Ehrenfeucht, 368good , 196homogeneous, 186minimal, 78, 215prime, 69, 208, 209, 214, 221,

222, 343, 352, 354, 365recursively saturated, 197resplendent, 191, 200, 294saturated, 186, 291, 295, 316,

318,365universal, 186

model companion, 62, 96model complete, 96model completion, 62,67, 97module, 90, 287

existentially closed, 94

finite type, 96finitely presented, 96

monotonicitylocal , 100

Morleytheorem on Vaught's conjecture,

211Morley degree, 341Morley rank, 341Morley sequence, 247, 248, 314,

375Morley's Theorem on categorical

theories, 372motive, 411multiplicity, 305multirelation, 31

natural sum , 378negation, 17Neumann's Lemma, 91New Foundations, 166nonforking, 75, 94nonforking son, 303nonstandard element, 111nonstandard model, 141nonstandard model of arithmetic,

116normal,354nullary relation, 32, 42Nullstellensatz

differential, 76Hilbert's, 68

omit a formula, 277omitted type, 56omitting types theorem, 48, 206Open Mapping Theorem, 220, 314order, 262

Boolean, 79cut in, 265dense, 8, 44, 228, 244, 278, 286discrete, 8, 110, 286fundamental, 277of a differential polynomial, 72stratified, 287

Index 439

total (linear), 8well, 160

order of a language, 114order property, 272ordered field, 98

real closed, 101ordering a set (by a formula), 270ordinal, 11, 161, 163

finite and infinite, 163limit, 11, 163natural sum of, 378product of, 160regular, 177successor, 11, 163sum of, 160Von Neumann, 162

orientation group, 411orthogonal, 370orthogonal types, 384, 397, 401

package, 206, 212, 218parameter, 56, 184

canonical, 321, 402separation of, 360, 364

parameter separation theorem, 261partial order, 79Peano arithmetic, 135perfect , 344Picard- Vessiot envelope, 364Picard- Vessiot extension, 364Pillay's Lemma, 399Pleonasm theorem, 146polyadic algebra, 82polygon, 86polygonal type, 90, 267polynomial, 65

differential, 71power of types, 388prebase, of a filter , 38predecessor, 8prenex form, 23, 25preorder

defined by formula, 271Presburger axioms, 115presentation, 96

440 Index

prime, 208prime differential ideal , 72prime ideal, 65, 66prime model , 69, 89, 208, 209, 214,

221, 343, 352, 354, 365non-uniqueness of, 222

primitive element theorem, 331primitive formula, 91primitive recursive, 132primitive subgroup, 92, 287

filter of, 287product

lexicographic, 377of types, 390

product of cardinals, 174product of integers, 116product of ordinals, 160profinite group, 329proof, 48, 147proof theory, 148property of Baire, 204provable, 48, 147, 148provably ~n , 138provably total function , 138pseudo-elementary class, 196pseudo-limit type, 90, 267pure type, 56

quantifier, 17bounded and unbounded, 124existential, 17universal, 17

quantifier elimination, 60, 67quantifier rank, 18quantifier-free, 18

Ramsey's Theorem, 273Ramsey's theorem

finite , 273infinite version , 274

rankI -dichotomy, 346bounded multiplicity condition,

333Cantor, 285, 326, 341, 343

Cantor-Bendixson, 12Cantorian, 341continuous, 337dimension, 74extension condition, 333foundation , 334Fraisse, 3heredity condition, 333infinite I-branching, 345infinite I-twofold branching, 346isomorphism condition, 333jump-free, 334Lascar (U), 334local,345Morley, 341notions of, 332of quantifier, 150quantifier, 18Shelah,338

rational closure, 215, 321real closed field, 98, 101real closure, 103realization order, 394realized type, 56recursion, definition by, 129recursion, transfinite, 164recursive, 130recursively axiomatizable, 148recursively enumerable, 130recursively saturated, 197regular cardinal, 177, 178regular ordinal, 177regular type, 386, 390relation, 1

O-ary, 32, 42equivalence, 6, 45, 286unary, 45, 217, 285with empty domain, 33with empty universe, 42

relation symbol, 33represent a formula, 277resplendent, 191, 200, 201resplendent model, 294Ressayre

theorem on constructed models,214

theorem on recursive saturation,198

restriction, elementary, 4rich ultrametric space, 87, 218,

266,286ring

Boolean, 78differential, 71

root, 144Rudin-Keisler order, 394Ryll-Nardzewski theorem, 209

satisfaction, 1, 19, 123satisfaction predicate, 198saturated, 186, 291, 316saturated model , 365

chain of, 295, 318second-order language, 114semantics, 19sentence, 18separant, 72separation of parameters theorem,

360, 364sequence, 246, 374

M -special, 268divisible, 250increasing, 374independent, 375indiscernible, 246, 374indivisible, 250Morley, 247, 248, 314, 375special, 268totally indiscernible, 247, 248

set theory, 166Shelah degree, 338Shelah rank, 338side of satisfiability, 265signature, 31, 33similar sequence, 246similarity type, 32, 33singular cardinal, 177, 178, 291Skolem envelope, 368Skolem function, 367

Index 441

Skolemization, 368son, 225

comaximal, 301forking, 303maximal, 301nonforking, 75, 303special, 239

special sequence, 268special son, 239spectrum of stability, 283stability spectrum, 283stable, 233, 284, 307

in a cardinal, 235stable theory, 260standard element, 112standard model of arithmetic, 116stationary type, 305Stone space, 43, 82Stone's theorem, 81stratified order, 287strict order property, 272strong heir, 228strong type, 309strongly x-homogeneous, 187strongly minimal formula, 365, 371strongly regular type, 397structure, 31

finite, 44with empty universe, 42

subformula, 17subgroup

primitive, 287substructure, 34

generated by set, 35of finite type, 35

successor, 8successor cardinal, 172, 178successor function, 108successor ordinal, 11, 163sum of cardinals, 174sum of integers, 111sum of ordinals, 10, 160superstable, 284, 296, 336Svenonius's theorem, 184symbol,33

442 Index

symmetry of forking, 220, 245,302, 303

Tarski's test , 36Tarski's theorem, 123Tarski-Lindenbaum algebra, 82term , 33

complexity of, 33theorem, 25Theorem of the Bound, 302theory, 25

A-categorical, 370w-categorical, 372w-stable, 285(recursively) axiomatizable, 148complete, 26, 42-44decidable, 148deep , 417dimensional, 403finitely axiomatizable, 44stable, 233, 235, 260, 284superst able, 296totally transcendental, 285, 342unstable, 285

thesis, 25topology

O-dimensional, 81const ruct ible, 70of Stone space, 43, 82Zariski ,70

total order, 8totally indiscernible sequence, 247,

248, 375totally indiscernible set , 247totally transcendental, 285, 342transcendence basis, 109transcendental

differentially, 74over a field, 66

transfinite induction, 163transfinite recursion, 164transitive set , 120, 162transitivity of algebraicity, 69transitivity of forking, 303tree, 144

branch of, 144finitely branching, 144height in, 144root of, 144

tree of formulas, 346true, 33truth, 130, 133tuple, 1Turing machine , 131Two-Cardinal Theorem , 370Tykhonoff theorem, 170type, 55, 183

x-isolated, 365f-regular, 397absolute or pure, 56coheir , 243definable, 230, 306dimensional , 402, 406distance, 266end extension , 231extension or son of, 225finite , 35heir , 243heir of, 225in dense order , 228in rich ultrametric space, 266infinitary definition of, 239isolated, 76limit, 250omitted,56omitting, 206polygonal, 267power of, 388pseudo-limit, 267realized, 56regular, 386, 390stable, 233, 307stationary, 305strong, 309strongly regular, 397, 406superstable, 336

ultrafilter, 39, 81, 170, 176, 297,416

axiom of, 39

ultrametric inequality, 86ultrametric space, 86

n-rich, 223rich, 218, 266, 286

ultrap ower, 42, 227isomorphic,47

ultraproduct , 40, 227unary relation , 45, 217, 285unbounded quantifier , 124Uniform Definability Theorem, 257universal, 186universal 1.:n set, 134universe

empty, 42of a relation, 1of a st ruct ure , 33

unst able, 285

variable, 17bound, 18free, 18

Index 443

Vaught pair , 370Vaught 's conject ure, 211

weakly x-homogeneous , 186Weierst rass function, 362weight , 381, 382Weil's Theorem, 325well-order , 160witness, 48, 50word, 16Wronskian , 360

Zariski topology, 70Zermelo's theorem, 164, 166Zermelo-Fraenkel set theory, 164Zorn's lemma, 167

Universitext (continued)

Meyer: Essential Mathematics for Applied FieldsMinesIRichman/Ruitenburg: A Coursein Constructive AlgebraMoise: Introductory Problems Coursein AnalysisandTopologyMorris: Introduction to GameTheoryPoizat: A CourseIn ModelTheory: An Introduction to Contemporary MathematicalLogic

Polster: A Geometrical PictureBookPorterlWoods: Extensions and Absolutes of HausdorffSpacesRadjaviIRosenthal: Simultaneous TriangularizationRamsaylRichtmyer: Introduction to Hyperbolic GeometryReisel:Elementary Theoryof MetricSpacesRickart: Natural Function AlgebrasRotman: GaloisTheoryRubeUColliander: Entireand Meromorphic FunctionsSagan: Space-Filling CurvesSamelson: Noteson Lie AlgebrasSchiff: Normal FamiliesShapiro: Composition Operators and Classical Function TheorySimonnet: Measures and ProbabilitySmith: PowerSeriesFroma Computational Pointof ViewSmoryski: Self-Reference and ModalLogicStillwell: Geometry of SurfacesStroock: An Introduction to the Theoryof LargeDeviationsSunder: An Invitation to von Neumann AlgebrasTondeur: Foliations on Riemannian ManifoldsVVong:VVeylTransformsZhang: Matrix Theory: BasicResults and TechniquesZong: SpherePackingsZong: StrangePhenomena in Convexand Discrete Geometry