AM Mathematics AM3KV

AM Mathematics Mathematics AM Common subdivisions AM2 AM2 . Common subdivisions . . . . Logic AM2 GW * For mathematical forms of presentation, see AM3 4. * Add to AM2 numbers 2/9 from Auxiliary Schedule 1, AM2 L Agents with the adjustments indicated below. M . Machines in mathematics . . By level of presentation * The general class for machine computation is under * Use these distinctions only if they are helpful to the Operations (AM7 H2M). Use this class only if needed library concerned. to qualify concepts appearing earlier in the schedule 3EM . . . Collegiate level than AM7 H or for completely general works on the 3EO . . . Advanced level, research level subject. 3MC . . Aids to study N . . Computers in mathematics * For computer programs, see Operations P . . . Programs in mathematics AM7 H2M. . Conceptual agents 3MY . . Digests, abridgements S . . Symbols, notation in mathematics 3N . . Mathematical tables Forms of mathematical presentation * And similar compilations. * The primary use of the concepts below is in qualifying . . . By degree of accuracy specific mathematical topics (e.g. Groups - Word 3NK . . . . Figure tables problems ASA 3KR). Use the classes below on their * Four-figure, five-figure, etc. tables. own only for truly general studies of the concepts . . . By function involved. 3NL . . . . General tables, ready-reckoners AM3 4 . Mathematical presentation 3NM . . . . Conversion tables . . Properties 3NO . . . . Logarithmic & elementary function tables 5 . . . Errors 3NP . . . . Higher function tables 6 . . . . Error bounds 3X . . Mathematical recreations, mathematical puzzles, 9 . . Statements mathematical games A . . Theory, abstract theory * Exposition of the abstract principles of a topic. For 7 . . History of mathematics abstract mathematical theory as a method (and itself . . . Periods in history of mathematics the topic studied) see AM3 N. * Add to AM2 7 letters A/V from Auxiliary AX . . . Hypotheses Schedule 4B. B . . . Theories special to a context 8 . . . Places in history of mathematics * For use when qualifying a specific topic studied. * Add to AM2 8 letters A/Z from Auxiliary * See also Models AM3 L Schedule 2. * Add to -3C letters H/W in Auxiliary Schedule . . . . By cultural group AM1, e.g. -3CW B Potential theory 8BT . . . . . Ethnomathematics * An alternative (not recommended) to this division 9 . . Biography of mathematicians by concept is to arrange theories A/Z by name. * For names of individuals used to characterize D . . Axioms, postulates properties, etc., see AP2. * For axiomatic method, see Mathematical logic, 9B . . Relations with other subjects AM4 F. 9C . . . Comparison with other subjects E . . . Special axioms A . . . Philosophy of mathematics * Add to AM3 E letters B/H following AM5 3E so * For quasi-philosophical doctrines underlying far as applicable, e.g. -3EF Axioms of choice. mathematical methodology, see Foundations of F . . Theorems mathematics AM3 P. G . . Lemmas * Add to AM2 letters A/J following A in AA/AJ. H . . Formulae Some prominent concepts are given below for I . . Demonstrations convenience. J . . Proofs BA . . . . Schools of philosophy in mathematics * For proof theory as a subject, see AM4 P. * For philosophical schools in mathematical logic, see AM4 2BA. JKP . . Problems * Will generally be used only as a qualifier of another COP . . . . . Mathematical Platonism topic, e.g. in collections of problems for students. * See also Logicism AM3 R * For problem solving as a subject, see AM4 FV. GW . . . . . Logic KR . . . Word problems * See Mathematical logic AM4 KS . . . Incorrectly posed problems KV . . . Problems special to a context

1 AM3L AM4994 Mathematical logic

Mathematics AM Mathematics AM Forms of mathematical presentation Methodologies in general AM3 M . . Problems AM3 JKP Foundations of mathematics AM3 P . . . Problems special to a context AM3 KV Schools of thought . Inductive mathematics AM3 W AM3 L . . Models * Representations assisting interpretation of a subject. AM4 Mathematical logic * For model theory as a subject, see AM4 Q. * Usually treated as nearly synonymous with symbolic * For models as mathematical relations, see logic. But for general studies of symbolic logic per se in AM9 S3L. mathematical logic, see AM4 9Y. For symbolic logic in LX . . . Simulation general, see AL9 Y. * Probabilistic modelling. * As a subclass of mathematics the structure sought is one consistent with that in other subclasses of mathematics. The order of the mathematical concepts is therefore Methodologies in mathematics exactly the same. * But in order to make the purely logical part completely consistent with logic in general (AL) the notation has M . Methodologies in general been modified slightly. Division is directly by the N . . Abstract mathematical theory scheduled class AM/AW and does not use the provisions * As a subject; for theory as a form of presentation, of Auxiliary Schedule AM1. The effect of this is to see AM3 A. shorten classmarks both here (in AM4) and in AL3 3X (Mathematics of logic); but it does this at the expense of . . Philosophical methodologies dropping the facility in Auxiliary Schedule AM1 (at * Approaches to mathematics as a whole. -H/-W) for specifying types of mathematical concepts. In both AM4 and AL3 3X concepts are primarily logical ones and when Types of these are required they P . . . Foundations of mathematics, metamathematics are got by enumeration at the end of the relevant class; * The term ‘metamathematics’ is used loosely, e.g. AM4 QR/AM4 RET Types of (logical) models; sometimes being treated as a particular AM4 RH/RW Types of formal deductive systems. In development of Formalism, sometimes more the last example, division via Auxiliary Schedule AM1 narrowly still, as equivalent to proof theory. It is is utilized for some of the types. used here in its widest sense. * Add to AM4 numbers 2/8 following AM in AM2/AM8. * See also Philosophy of mathematics AM2 A. * Add to AM4 9 number 9 following AM at AM9 and . . . . Schools of thought letters M/W following A in AM/AW. A selection is * General studies only. A particular topic in logic given below of some prominent concepts. or mathematics as treated by a particular 2A . Philosophy of mathematical logic schoool goes with the topic. 2BA . . Schools of philosophy in mathematical logic R . . . . . Logicism, logicalists 2L . Agents * See also Mathematical Platonism, 2N . . Computers AM2 COP 2P . . . Programs S . . . . . Formalism 34 . Forms of mathematical presentation * For Metamathematics, see AM3 P and the * Many of these (e.g. statements) have a special note there. significance in logic and these are enumerated in their * For properties of formal languages, see appropriate positions. In case of doubt, the latter Mathematical logic, AM4 AD. should be preferred to the terms synthesized here...... Proof theory 3D . . Axioms in mathematical logic * See AM4 P * For axiomatic method see AM4 F...... Model theory 3M . Methodologies * See AM4 Q 3P . . Foundations of mathematical logic, T . . . . . Constructivism & intuitionism metamathematics in mathematical logic U ...... Constructivism * See also Proof theory AM4 P; Model theory * See also Recursion theory AM4 H3A AM4 Q. V ...... Intuitionism 3R . . . Logicism in mathematical logic * See also Intuitionist logic AM4 RF 3S . . . Formalism in mathematical logic V3D ...... Axioms 3T . . . Constructivism & intuitionism V3E ...... Constructability postulate, effective 5 . . . Set theory in mathematical logic constructability postulate 62 . . Mathematical methods by various characteristics W . . . . . Inductive mathematics 74 . Mathematical operations & processes * For mathematical induction see AM4 T. * See also Recursion theory AM4 H3A 8J . Mathematical relations 993 . . Forms 994 . . Mathematical relations by various characteristics

2 AM49MN Deductive logic in mathematics AM4GLM

Mathematics AM Foundations of mathematics AM3 P Methodologies in general AM3 M Mathematical logic AM4 Foundations of mathematics AM3 P Deductive logic in mathematics AM4 A Mathematical logic AM4 Logical properties AM4 AD Mathematical relations AM4 8J . Equivalence in mathematical logic AM4 BB . Mathematical relations by various characteristics AM4 994 . . Identity in mathematical logic AM4 BC

AM4 9MN Mathematical properties AM4 BE . Opposition 9PA Parts, elements & entities in mathematics BH . Truth 9R5 Mathematical systems, branches BH8 L . . Truth functions BV Properties special to a given context * For example, stability of models AM4 QBV. 9WZ Systems in mathematical logic * From this point onwards, the schedule for CL Logical operations mathematical logic is basically an expansion of that D . Reasoning for general logic (AL), with additional material . . Elements incorporating mathematical operations, relations, etc. DF . . . Fallacies and expanding particular classes (e.g. algorithms). DL . . . Paradoxes * It is important to note that, apart from the expanded DP . . . Contradictions classes, the following schedule is only a selection of EB . Analysis in mathematical logic the concepts in AL. This selection is designed to ED . Interpretation show the scope and structure of the class and to indicate the concepts particularly relevant to EF . Formalization mathematical logic.But all the detail in AL is EH . Implication available here and this class should be used in F . Axiomatics in mathematical logic conjunction with AL. * For axioms as statements (used in qualifying * Add to AM4 numbers 9X/9Y and letters A/W other concepts), see AM3 D. following AL in ALA/ALW. F3E . . Axioms 9X . Classical logic in mathematics F3F . . . Theorems * Classical logic in general is at AL9 X. F3H . . Formulae 9Y . Symbolic logic in mathematics FH . . Syntax, rules of syntax * For works considering symbolic logic as an FJ . . . Formation rules element in mathematical logic, rather than FK . . . Introduction rules synonymous with it. For symbolic logic in general FL . . . Elimination rules see AL9 Y. * Do NOT qualify this class, but treat apparent FN . . . Transformation rules divisions of it (e.g. axioms of symbolic logic) as FS . . Semantics (general) divisions of mathematical logic in general. FV . Problem solving G . . Algorithms in mathematical logic . . . Operations A . Deductive logic in mathematics * This is usually assumed. Locate here general works GF . . . . Axiomatics on the role of deductive logic per se in GFN . . . . . Transformation rules mathematics. . . . . Special to algorithms AD . . Logical properties GJ . . . . . Decision procedures * Most of these are unlikely to be used on their * Determining whether a proposition is own (i.e. have general works on them). They will logically true, i.e. proving by a usually feature as qualifiers of other concepts, demonstration. e.g. Propositions - Equivalence AM4 KBB. * For word problems see Groups AF . . . Effectiveness ASA 3KR. AG . . . Complexity ...... Properties AH . . . Consistency in mathematical logic GJA S ...... Enumerability AL . . . Independence GJA T ...... Decidability AM . . . Completeness in mathematical logic GJA U ...... Undecidability AM3 A . . . . Theories ...... Theorems AM3 F . . . . . Godel’s incompleteness theorem GJA U3F ...... Church’s theorem AQ . . . Constructability in mathematical logic GJA V ...... Truth tables AS . . . Enumerability GK . . . . . Solution & solvability AT . . . Decidability GL ...... Unsolvability AU . . . . Undecidability in mathematical logic GLM ...... Degrees of unsolvability, Turing BB . . . Equivalence in mathematical logic degrees BC . . . . Identity in mathematical logic

3 AM4GN AM4PH76 Deductive logic in mathematics

Mathematical logic AM4 Mathematical logic AM4 Deductive logic in mathematics AM4 A Deductive logic in mathematics AM4 A Logical operations AM4 CL Formal structures in mathematical logic AM4 JX . . . Operations . . . Propositions in mathematical logic AM4 K . . . . . Solution & solvability AM4 GK . . . . Mathematical relations AM4 K8J ...... Degrees of unsolvability AM4 GLM ...... Propositional functions AM4 K8L

AM4 GN . . . . . Calculation & calculability AM4 KAD . . . . Logical properties * For immediately calculable, see KBB . . . . . Equivalence Recursion AM4 H8L. KBC ...... Identity GP ...... Effectively computable KBE . . . . . Opposition GP3 A ...... Theory KBH . . . . . Truth GP3 C ...... Computability theory in algorithms . . . . Logical elements H . . . . . Recursion KL . . . . . Truth functional elements H3A ...... Theory KM . . . . . Operators in mathematical logic H3B B ...... Higher type recursion theory KN ...... Quantifiers H3B C ...... Church’s hypothesis KR . . . . . Terms, expressions in mathematical logic ...... By method L . . . . Propositions (narrowly), judgements H3C HD ...... Axiomatic recursion theory LM . . . . . Simple propositions ...... By property LQ . . . . . Compound propositions H3C MO ...... Abstract recursion theory LW ...... Truth function propositions H74 P ...... Operations & processes ME . . . Statements in mathematical logic H76 ...... Recursion analysis MG . . . Sentences in mathematical logic H8J ...... Relations MH . . . . Infinitely long sentences ...... Functions Systems H8L ...... Recursive functions, immediately MS . Formal systems in mathematical logic calculable functions * Any set of axioms and formal rules in some * What can or cannot be done by an specified formal language. ideal computer. MV . . Calculus of classes H8L LR ...... Generalized recursive functions * See also General logic ALM V; Intuitionist logic H8M TP ...... Primitive recursive functions AM4 RF H9R ...... Mathematical systems N . . . Propositional calculus in mathematical logic ...... Ordinals * Includes classical propositional calculus. H9R KD ...... Recursive ordinals O . . . Predicate calculus in mathematical logic HI ...... Logical elements * Includes classical predicate calculus. HJ ...... Hierarchies . . . . Types by interpretation HM . . . Algebraic forms of algorithms OP . . . . . One sorted logic * Usually assumed. HN . . . . Calculus of equations * General works only. HP . . . . Turing machine OQ . . . . . Many sorted logic HR . . . . Lamda calculus OS . . . . . First order calculus of classes HS . . . . Post algorithms OT . . . . . Higher order predicate calculus, higher * Of E.L. Post. order logic HV . . . . Markov algorithms * Of A.A. Markov. P . . Proofs in mathematical logic, proof theory * Syntactic study of formal systems by examination JX Formal structures in mathematical logic of the structure of the proofs within them. JXF S . Semantics * See also Inductive logic AM4 T; Constructive JY . Logical calculi in mathematical logic mathematics AM6 9 * Use for propositional and predicate calculus P8J . . . Mathematical relations together. P8X . . . . Functionals in proof theory . . Elements of logical calculi . . . Logical properties K . . . Propositions in mathematical logic PAG . . . . Complexity * Includes works on propositions, statements and sentences together. PAH . . . . Consistency * For the last two considered per se, see . . . Operations & processes AM4 ME. PED . . . . Interpretation K8J . . . . Mathematical relations PG . . . . Algorithms . . . . . Functions PH . . . . . Recursion K8L ...... Propositional functions ...... Analysis PH7 6 ...... Recursion analysis

4 AM4PQ Formal systems in mathematical logic AM53EC

Mathematical logic AM4 Methodologies in general AM3 M Deductive logic in mathematics AM4 A Foundations of mathematics AM3 P Formal systems in mathematical logic AM4 MS Mathematical logic AM4 Proofs in mathematical logic AM4 P . . Deductive logic in mathematics AM4 A . Operations & processes . . . . . Intuitionist logic AM4 RF . . . . . Recursion analysis AM4 PH7 6 ...... Intermediate logic, superintuitionist logic AM4 RFS . Types . . . . . Types of formal deductive systems AM4 PQ . . Normal form proofs * For the logic of particular conceptual PR . . . Cut-free segment proofs systems, see system, e.g. lattices, quantum PS . . Forcing proofs mechanics. * See also Model theory - Forcing AM4 QPS. AM4 RH ...... By mathematical characteristics Q Models in mathematical logic, model theory * Add to AM4 R letters H/W from * Semantic study of formal systems, via their models Auxiliary Schedule AM1. (i.e. set theoretic interpretations)...... By relation * See also Universal algebras ATA NI RLL ...... Equational logic Q3A . Theories ...... By property Q3C NA . . First order model theory RO2 F ...... Fuzzy logic . Methods ...... By system Q6R S . . Model theoretic algebra RRB ...... Logic on admissible sets . Logical properties RRR X ...... Partially ordered systems QBV . . Stability RRS ...... Algebraic logic in general . . Special operations * For Boolean algebra, see ARB X. QCN . . . Construction of models RSX ...... Categorical logic QCP . . . . Ultra products method * See also Topoi (model theory) QCQ . . . . . Indiscernibles, Ehrenfeucht-Mostowski AM4 RET models ...... By values . Subsystems SB ...... Two valued logic in general QK . . Propositions * Usually asumed. QKN . . . Quantifiers SD ...... Three or more valued logics, many QKN FL . . . . Elimination of quantifiers valued logics . Systems * See also Cybernetics Class 3. QP . . Proof theory SF ...... K-valued logics . . . Forcing T . . Inductive logic in mathematics, mathematical QPS . . . . Model theoretic forcing induction . Types of models * The term mathematical induction is usually . . By property equated with one prominent application - the proving of a theorem relating to natural numbers QR . . . Non-classical models (e.g. Peano’s axiom). But its applications are QT . . . Abstract models potentially wider. QV . . . Non-standard models V . . Other forms of arguments QV9 NLH . . . . Infinitesimal analysis (models) * From general logic so far as applicable. QX . . . Generic models * Add to AM4 letters V/W following AL in RB . . . Countable models ALV/ALW. RD . . . Other AM5 Set theory * E.g saturated models. * Sets themselves are not a purely mathematical . . By system concept, whether defined by enumeration of RES . . . Algebraic categorical models membership or by defining conditions (properties). RET . . . . Topoi, elementary topoi But set theory is a formal language which constitutes an integral part of mathematical foundations and is RF Intuitionist logic therefore located here. * See also Intuitionism AM3 V * For sets as mathematical structures, see ARB. RFN . Propositional calculus * For sets in general (non-mathematical applications) RFO . Predicate calculus see Logic, AL9 RB. RFS . Intermediate logic, superintuitionist logic . Hypotheses * Intermediate between classical and intuitionist 3AX . . Continuum hypothesis logic. 3D . Axioms 3EB . . Extensionality axiom 3EC . . Replacement axiom

5 AM53ED AM74L Methods in mathematics

Mathematics AM Mathematics AM Methodologies in general AM3 M Methodologies in mathematics . . . . Axioms AM5 3D . . Methods by relations, by property etc . . . . . Replacement axiom AM5 3EC . . . Approximation methods AM6 QR AM5 3ED . . . . . Regularity axiom . . Methods by mathematical branch used 3EF ...... Axioms of choice * For use in synthesis only, when a method derived 3EG ...... Axioms of determinacy from a particular branch of mathematics is applied 3EH . . . . . Infinity axiom to a particular topic, e.g. geometric methods in . . . . Properties solution of differential equations AWE ME 6TS. AN4 AH . . . . . Consistency in set theory General works on a method so derived nearly always go with the branch (e.g. geometric methods AN4 AL . . . . . Independence in set theory in general under geometry). AN4 AT . . . . . Decidability in set theory * Often, the name of a branch used adjectivally seems . . . . Types of set theory to specify a type of the thing it describes (e.g. HD . . . . . Axiomatic set theory geometric number theory) whereas it is in fact * Use AM5 X. describing only a method used; e.g. geometric M66 . . . . . Descriptive set theory number theory indicates not a type of number * Use AM5 YD. theory but number theory using geometric methods. P2 . . . . . Cantor set theory In such cases this position should be used to qualify X . . . . . Axiomatic set theory the concept in question (e.g. Number theory - ...... Special systems Methods - Geometric, ARJ 6TS). * Add to AM6 letters R/W following A in AR/AW. XXC ...... Zermelo-Fraenkel system, ZF system * A selection of frequently occuring methods is given XXE ...... Neumann-Bernays-Godel system, here for convenience. NBG system, Bernays-Godel AM6 RD . . . Combinatorial methods set theory RI . . . Arithmetic methods, numerical methods XXG ...... Morse & Kelley system, MK system RS . . . Algebraic methods XY ...... Others by name A/Z RU . . . . Homological methods * For example, Quine AM5 XYQ. RVK . . . . Cohomological methods YD . . . . . Descriptive set theory RVL . . . . Homotopy methods ...... Inductive logic ST . . . . Modular methods YD4 T ...... Inductive definition SX . . . . Categorical methods AM6 Methods in mathematics TB . . . . Matrix methods * Do not add numbers or letters from Auxiliary TS Schedule AM1 to AM6 alone - only to its divisions. . . . Geometric methods . Methods by mixed characteristics VE . . . . Trigonometric methods 2 . . Elementary methods VJ . . . Topological methods 3 . . Advanced methods W . . . Analytic methods * For advanced as a level of presentation, see . . . . Methods special to analysis AM2 3EO. * Add to AM6 letters X/Y following AW6. * 4 . . Classical methods Add to AM7 letters 2/49 following AW7, e.g. Sequences & series - Summability - Integral 5 . . Non-classical methods methods AWL EOC 4Y7 3. 6 . . Descriptive methods * Note that numbers 45/49 following AW7 are 7 . . Heuristic methods reserved for special methods; see introduction, 8 . . Algorithmic methods section 13.35, title 5 for an example. 9 . . Constructive methods X . . . Probalistic methods B . . Formal methods * Use AM7 4L. D . . Numerical analysis AM7 4L . . . Probalistic methods E . . Iterative methods, recursive methods * Add to AM7 4L numbers & letters 5/9, A/X . Methods by relations, by property etc following AX so far as applicable. * Add to AM6 letters H/Q from Auxiliary Schedule AM1. Some prominent examples are given here to Operations demonstrate application. * Mappings in the widest possible sense. For the normal KL . . Function theoretic methods and exact use of the term, use the position at AM8 K. MQG . . Non-standard methods * The main use of the terms below is in qualfying specific NJ . . Finite methods mathematical topics. In some cases, it is very unlikely QR . . Approximation methods that the term will reflect any literature on the concept in general, but in many cases it will. But note that only truly general works on the concepts go here.

6 AM74P Mathematics AM8HX

Mathematics AM Mathematics AM Operations Operations . Operations in general AM7 4P * Where the results of an operation are significant in the . . . Partition AM7 W literature (e.g. sums, products) these have been enumerated in the Elements facet (see Elements AM7 X . . . Ramification resulting... at APG et seq). A document considering AM8 2 . . . Extension both the operation and the result together is preferred in 32 . . . Surgery the Elements facet (see note preceding APG). 33 . . . Cobordism AM7 4P . Operations in general 4 . Operations special to a context 5 . . Testing (general) * For use primarily in statistics and probability. Processes 6 . . Analysis (general operations) * The main use of the terms below is in qualifying * In most general sesnse (as distinct from its usual specific mathematical topics. Only truly general specific sense at AW). treatment of the concepts will go here. 7 . . Synthesis 5P . Processes in general 8 . . Solution 6 . . Approximation * This represents the ‘presentational’ role of the * See also Approximations (expansions) AQR concept, used for simple qualification of a topic 63F . . . Approximation theorems (e.g. Differential equations - Solutions AM9 ME7 8). For problem-solving as a subject in . . . Processes its own right, see mathematical logic AM4 FV. 68B . . . . Convergence of approximations 9 . . Resolution 68B 35 . . . . . Convergence errors A . . Enumeration . . . Types by property B . . Definition 6OG . . . . Asymptotic approximation C . . Classification 7 . . Continuation D . . Construction 8 . . Variation E . . Substitution 9 . . Growth F . . Combination A . . Distribution * For combinatorial structures and analysis, B . . Convergence see ARD. C . . Divergence H . . Computation D . . Oscillation * For numerical computation see ARI 7H. E . . Interpolation * See also Algorithms AM4 G EX . . Superposition . . . Agents F . . Generation, generating H2M . . . . Machine computation GE . . Perturbation . . . Computation operations GJ . . Involution * Most of the operations below generate products GM . . Decomposition (sums, quotients, etc.). These are treated as GP . . Splitting elements and enumerated at APK/APW. When a document deals with the operation and its GS . . Separation product together, class under the latter, e.g. GV . . Valuation addition with sums. The primary use of this array H . . Optimization of operations is in simple qualification (e.g. HX . . Minimization Integers - Addition; Groups - Factorization). J . . . . Addition K . . . . Subtraction L . . . . Multiplication M . . . . . Complex multiplication N . . . . Division P . . . . Factorization Q . . . . Extraction of roots R . . . Differentiation S . . . . Differentiable * Property dependent on Differentiation. T . . . . Partial differentiation U . . . . Pseudo-differentiation V . . . Integration W . . . Partition

7 AM8J AM9G Mathematics

Mathematics AM Mathematics AM Processes Relations . . Minimization AM8 HX Mappings AM8 K . Functions AM8 L Relations . . . By property AM8 N9 * This is a composite facet in that most of its terms . . . . Polynomials reflect more than one broad principle of division at a AM8 NDX . . . . Exponential functions time. These terms (many of them prominent in the NXH . . . . Potential functions literature) reflect concepts in which it is difficult in practice to distinguish between the operation or process P2 . . . . Named after a mathematician concerned, or the end product or property of these. . . . By system Very often the same term is used to describe all three RI . . . . Arithmetic functions categories and no clear distinction is drawn between . . . Special types of functions them in the literature. X . . . . Functionals * The main use of the terms below is in qualifying YD . . . . Determinants mathematical topics (e.g. Lattices - Generalizations). * Function of a square matrix. Only truly general treatments of the concepts go here. YG . . . . Zeta functions AM8 J . Relation in general YH . . . . Hypergeometric functions * For general relations of one concept with another, AM9 3 . Forms e.g. categories and geometry ASX 8JT S. * Add to AM8 J numbers and letters 3/9, A/E, R/W in . . Methods Auxiliary Schedule AM1. 36T S . . . Geometry of forms JX . . Analogue . . Types of forms K . Mappings, functions (broadly) 3MC . . . Automorphic forms * For functions in the narrower and more usual sense, 3NA . . . Linear forms use AM8 L below. 3NC . . . Quadratic forms * See also Theory of functions (under Analysis) 3ND H . . . Forms of higher degree AW8 L3A. 3ND N . . . Forms in several variables . . Types by property 3ND P . . . Bilinear forms KJS . . . Differentiable mappings Relations arising from operations on structures KNV . . . Continuous mappings * Includes the operation/process itself, as no useful KOM . . . Conformal mappings distinction can normally be made. L . . Functions, operators (broadly) 4 . Transforms * The main use of this class is in qualifying concepts 5 . Transformations and branches other than from Analysis, e.g. 62 . . Rotation Algebraic number theory - Zeta functions, 64 . . Reflection ARM 8YG. A selection of frequently occurring 66 . . Translation functions used in qualifying specific topics outside Analysis proper is given here for convenience. 7 . . Deformation L3A . . . Theory of functions 8 . . . Torsion * Use Analysis (AW8 L3A) for the classical 9 . . Projection theory of functions; this location is for theory of AD . . . Dilatation functions in other disciplines. AF . . . Enlargement L74 P . . . Operations on functions AI . . Injection LAN . . . Properties of functions AK . . Surjection . . . Types of functions AM . . Bijection * Add to AM8 L letters H/L from Auxiliary AP . . Compression Schedule AM1. B . . Compactification * Add to AM8 letters M/W in Special Auxiliary C . . Topological transformations, homeomorphisms AM1. CX . . Diffeomorphisms * Some prominent examples are given below for convenience. D . Derivation * See also Special functions AWD XE E . Extensions . . . . By relation E3A . . Extensions theory LLR . . . . . Generalized functions F . Conjugates, adjoints LLS . . . . . Representation functions G . . Self-adjoints MC . . . . . Automorphic functions . . . . By property MY . . . . . Rational functions N4F . . . . . Transcendental functions N9 . . . . . Polynomials (functions)

8 AM9HE Mathematics AM9X

Mathematics AM Mathematics AM Relations Relations Relations arising from operations on structures Status relations and relations of magnitude . . Self-adjoints AM9 G . Equations AM9 L . . . Polynomial equations AM9 MN9 Status relations and relations of magnitude AM9 HE . Correspondence AM9 MNA . . . Linear equations HG . Similarity MNB . . . Non-linear equations I . Congruence MNC . . . Quadratic equations J . Equivalence MND A . . . Cubic equations K . Identity, identity relations MND C . . . Quartic equations * In general sense of a statement of equality, e.g. MND P . . . Bilinear equations identies of place figure trigonometry. For MND T . . . Mixed equations narrower sense implying the use of an operator to MND X . . . Exponential equations produce a special value (e.g. identity matrix) see N . Inequality, inequalities Entities AQU Y. NX . . Greater than L . Equations, equality NY . . Less than L3A . . Theory of equations P . . Inverse, inversion . . . Subsystems PY . . Reverse, reversion L3A FTB . . . . Matrices in the theory of equations Q . . Complement . . Types by method - operations QY . . Proportion, ratio * It is unlikely that the need will arise to specify Relations of structure and composition equations by the facets of Forms of R . Generalizations mathematical presentation or by Methodology (AM3/AM7 4N). But provision is made for it S . Representations here in case the need does arise. S3A . . Representation theory LH . . . By form of mathematical presentation * Main application is to groups. This position * Add to AM9 L letter H from Auxiliary takes only truly general works. Schedule AM1. S3L . . Models LI . . . By methodology * For model theory in general, see Mathematical * Add to AM9 L letter I from Auxiliary logic, AM4 Q. Schedule AM1. . . Types * Add to AM9 LJ numbers 2/4N following J * For asymptotic representations, see Asymptotic in Auxiliary Schedule AM1 (representing approximation AM8 6OG. AM72/AM74N), e.g. Recursive equations SJV . . . Integral representations AM9 LIE. SL9 . . . Projective representations LJ4 P . . . By operation SQX . . . Modular representations * Add to AM9 JL numbers and letters 4P/Y SXP 8A . . . Ordinary representations following J in Auxiliary Schedule AM1 with SXP 8B . . . Unitary representations the modification indicated at AM9 LJR Spatial relations, relations of location below. T . Packing and covering LJL . . . . Multiplicative equations TRR . . Lattice packing and covering LJR . . . . Differential equations U . . Packing * Use AM9 ME. Normal synthesis is interrupted here; it is resumed at URR . . . Lattice packing AM9 MJW. V . . Covering * Add to AM9 M letters E/ENG following VRR . . . Lattice covering AWE M. WI . Incidence * Add to AM9 M letters F/I following AW. WM . Immersions ME . . . . Differential equations WS . Suspensions MEN G . . . . . P-adic differential equations X . Embedding, imbedding MF . . . . Ordinary differential equations * Of a configuration in enveloping space. MH . . . Integral equations . . By embedded system . . Types by other operations etc * Add to AM9 Y letters R/W following A in * Normal synthesis by Auxiliary Schedule AM1 AR/AW, e.g. Banach space - Embedding - of is resumed here after the interruption at manifolds AWP P29 YUG. AM9 LJR. * Add to AM9 M letters JW/W from Auxiliary schedule AM1. MN9 . . . Polynomial equations

9 AMB AMN7V Properties in general

Mathematics AM Mathematics AM Relations Properties . Spatial relations, relations of location . . . By embedded system AMN Properties in general . Properties derived from other facets . Functional relations, relations of association * These are used only for the properties per se, NOT as AMB . . Homomorphisms, morphisms specifiers (defining Types). For example, Distributivity * Mappings from one system to a like system which as a lattice property (ARR AN8 A); modularity as a preserve structure. property of congruence in algebraic varieties Y . . . Isomorphisms (ATL 9ID 6X). But ‘distributive’ used as a specifier AMC . . . Automorphisms would be taken from Distribution as a Process; AMD . . . Endomorphisms ‘Modular’ as specifier would be taken from moduli as AME E . . . Epimorphisms an entity (e.g. modular lattices ARR QX). H . . . Monomorphisms * Since there is very little literature considering properties per se, a comprehensive listing here would N . . . Null morphisms be unwarranted. So only a brief selection is given AMF . . . Complexes (homomorphisms) below to show how each facet may provide such AMG . . Holomorphisms properties. AMH . . Meromorphisms . Properties derived from earlier facets AMI . . Holonomy * For properties derived from later facets see AP5/AP7. X . . Non-holonomy * Add to AMN numbers and letters 3/9, A/L following AMJ . . Homology AM. A selection of frequently occuring terms is given * Including homology and cohomology together. below for convenience and to show scope. AMK . . . Cohomology . . Properties derived from mathematical presentations 3B . . . . K-theory (cohomology) 3D . . . Axiomatic NV . . . . Continuous cohomology . . Properties derived from methodologies AML . . . Homotopy * Note that the methodologies used here exclude those Y . . . Isotopy which are themselves derived from other facets, e.g. algebraic properties are to be taken from the Systems and branches facet (see below for examples). Properties 3U . . . Constructive * There is relatively little literature on these properties per 4O . . . Predicate se (e.g. on uniqueness, linearity, complexity). So the 4T . . . Inductive main role of this facet is to provide for qualification of 62 . . . Elementary mathematical objects by their properties (e.g. Lattices - Distributivity) and for specification of them (i.e. using a 64 . . . Classical property to specify a type or species, e.g. elliptic 66 . . . Descriptive equations). In some cases both roles will appear in a 6B . . . Formal single index description, e.g. Geometry - Convex sets - . . Properties derived from operations Spherical convexity. Use the classes below on their own * For properties reflecting the products of operations only for truly general works on the property per se. (e.g. sums, produced by addition), see Properties * When properties per se are intended, the strict form is derived from entities and elements AP5/AP6. preferred (e.g. duality). But the adjectival form (e.g. 76 . . . Analytic dual) used in specifiers is far commoner; so this facet * See note under Analysis at AM7. uses it, and the property per se is implicit, e.g. bilinear 77 . . . Synthetic implies bilinearity. 7B . . . Definability * It should be noted that although some properties appear 7D . . . Constructability to be special to particular disciplines (e.g. elliptic to geometry) this is not necessarily the case. Many terms 7F . . . Combinatorial with strong disciplinary associations may well be 7J . . . Additive applicable in other areas (e.g. elliptic equations) and it is 7K . . . Subtractive for this purpose that they are included here. 7L . . . Multiplicative 7N . . . Divisibility 7P . . . Factorization properties 7Q . . . Radical 7R . . . Differential 7S . . . . Differentiable * Use AM7 S. 7T . . . . Partially differential 7U . . . . Pseudo-differential 7V . . . Integral

10 AMN86 Properties in general AN8P

Mathematics AM Mathematics AM Properties Properties Properties in general AMN Properties in general AMN Properties derived from earlier facets General special properties . Properties derived from operations . Characteristic AMT C . . Integral AMN 7V AMT E . Exceptional . Properties derived from processes L . Principal AMN 86 . . Approximate P . Primitive 87 . . Continuous AMU . Simple * Use ANV. AMV . . Semisimple 8A . . Distributive, distributivity AMW . Complex 8B . . Convergent AMX . . Almost complex 8C . . Divergent AMY . Rational 8D . . Oscillatory AN2 . . Diophantine 8GJ . . Involute, involuted Y . . Birational 8GM . . Decomposable, decomposed AN3 . Real 8GV . . Valuation (properties), valued Y . . Formally real . Properties derived from relations AN4 D . Ideal 8L . . Functional E . Imaginary 95 . . Transformation properties F . Transcendental 98 . . . Torsion (properties) By number of elements/operations involved 99 . . . Projective H . Discrete, discreteness 9D . . Derivative, derived * For Continuous, see ANV. * From derivation; cf. derivative as end-product of By sign differentiation. J . Positive 9F . . Conjugate, Adjoint K . . Non-positive 9G . . . Self-adjoint L . Negative 9I . . Congruent M . . Non-negative 9J . . Equivalent N . Zero 9L . . Equational O . . Non-zero 9P . . Inverse (properties) By value 9R . . Generalized, generalizing Q . Absolute value 9S . . Representational, representative R . Relative value 9V . . Covering properties S . Conditional 9WI . . Incidence (properties) By dimension B . . Homomorphic V . Dimensional, dimension C . . . Automorphic * See also Degree AN9 Y; Finiteness ANJ G . . Holomorphic AN5 . . Measure H . . Meromorphic X . . . Special measures Properties derived from later facets * For example, Baire categories. * See AP5. Y . . Large General special properties AN6 . . Small, low dimension * General in scope, but special to this facet in that they Y . . Dimension less than one (<1) are not derived from other facets. AN7 . . One dimensional, line, linear (dimensions), singular . By nature of other elements involved * For linear as a degree, see ANA; for singularities, see T . . Existence AQC G. U . . Uniqueness AN8 B . . . Non-linear (dimensional) AMO . Abstract D . . Two dimensional, plane (dimensional), planar AMP . Concrete (dimensional) AMQ B . Definite F . . Three dimensional, space (dimension), spatial D . Standard, neutral (dimension), solid (dimension) G . Non-standard H . . Higher dimensional, polydimensional, M . Mean, average multidimensional, n-dimensional P . Fundamental J . . Height AMR . Normal L . . Length AMS . Intrinsic N . . Width, breadth AMT C . Characteristic P . . Volume, capacity

11 AN8R ANXS Properties in general

Mathematics AM Mathematics AM Properties Properties Properties in general AMN Properties in general AMN By dimension By level of finiteness . . Volume AN8 P . . Infinitesimal ANL H

AN8 R . . Weight Special properties T . . Density * Special in scope as well as special to this facet (i.e. not . . Special forms of dimension derived from other facets). VB . . . Norms (dimension) Compositional etc properties By number of terms ANL N . Strong W . Monomial P . Poor, weak X . Binomial R . Open AN9 . Polynomial T . Closed By degree of terms ANM . Smooth * For dimensions (1-d, 2-d, etc.) see AN7. ANN . Regular Y . Order (degree of terms) ANO B . . Non-regular ANA . Linear, first order D . Uniform ANB . Non-linear (order) G . Homogeneous ANC . Quadratic, second order H . Heterogeneous, mixed AND A . Cubic, third order J . Symmetric, symmetry C . Quartic L . . Asymmetric H . Higher order N . Reflex, reflexivity I . . Infinite order T . Monotone, monotonic By number of variables ANP . Ordered K . With one variable ANQ B . . Partially ordered L . Binary L . . Linearly ordered M . Ternary R . Graded N . With several variables ANR . Compact, compactness By degree of variables NF . . Locally compact P . Bilinear ANS . Free, freeness Q . Multilinear * For Connected, see AOC H. By nature of variables ANT E . Amalgamated S . Homogeneous variables G . Coupled T . Mixed variables J . Complemented V . Symmetric variables L . Partial, partially X . Exponential variables, exponential N . Complete, completeness ANE C . Complex variables P . . Almost complete E . . One complex variable R . . Pre-complete H . . Several complex variables T . . Quasi-complete R . Real variables ANU . Entire By range of applicability Y . Perfect ANF . Local ANV . Continuous, continuity ANG . . P-adic * For Discrete, see AN4 H. ANH . . . Galois properties and theory ANW . . Completely continuous Y . Semilocal ANX B . . Discontinuous ANI . Universal, global D . Dual, duality By level of finiteness D3F . . Duality theorems ANJ . Finite, finiteness F . Periodic * Including related conditions. G . . Almost periodic Y . . Finite dimensional H . . Harmonic, potential ANK . Infinite K . . . Subharmonic * For infinite order see AND I. M . . . Biharmonic ANL D . . Infinite dimensional, infinite dimensions P . . . Polyharmonic F . . Infinitary Q . Cyclic H . . Infinitesimal R . Direct S . . Subdirect

12 ANXT Properties in general AP5P

Mathematics AM Mathematics AM Properties Properties Properties in general AMN Properties in general AMN Compositional etc properties Properties of space . . Subdirect ANX S . Affine AOL

Properties by performance AOM . Conformal ANX T . Qualitative behaviour X . . Pseudo-conformal V . Optimal, optimality AON . Metric X . Maximal AOO . . Metrizable * See note at AP6 2. AOP . Symplectic, simplicial Y . Minimal AOQ B . Riemannian * See note at AP6 3. D . . Pseudo-Riemannian ANY . Stable Properties of motion AO2 D . Wild R . Speed, rate F . Fuzzy AOR . Dynamic H . Exact AOS . Kinematic J . Invariant Properties derived from geometric figures L . Equivariant AOT . Plane (properties) N . Covariant Y . Square P . Contravariant AOU . Orthogonal S . Separable X . Orthonormal AO3 . Solvable AOV . Elliptic AO4 . . Non-solvable AOW . Parabolic Y . Summable, summability AOX . Hyperbolic AO5 . Nilpotent AOY B . Spherical AO6 C . Nil By name of mathematician E . Idempotent * Where exactly defined these should be subordinated By conformity to fundamental laws according to the concept defined; (e.g. Boolean algebra = AO7 . Associative Algebra of sets, and is thus subordinated to Sets, and not AO8 . . Non-associative to Types of algebras; Abelian groups = Commutative AO9 . Commutative, Abelian groups and is located as the latter. AOA . . Non-commutative, non-Abelian AP2 . Mathematician especially prominent in context * Spatial properties AP2 and AP3 are used for the two most prominent. AP4 accommodates any others, followed by first letter of AOB . Proximity name. AOC D . Neighbouring * If the ‘negative’ of a mathematician’s property is given, F . Neighbourhood allow for the ‘positive’ to precede it, e.g. AVL 8L H . Connectedness, connected Functions in analytic spaces; AVL 8P3 J . Shape Non-Archimedean functions (which allows for the use L . Interior of AVL 8P2 Archimedean functions - should these ever N . Exterior arise in this context). P . Geodesic AP4 . Others (A/Z) R . Curvature Properties derived from later facets * The first note following AMN applies here also. T . Concave * For properties derived from earlier facets, see AMN and V . . Pseudo-concave the explanatory notes there. AOD . Convex, convexity AP5 . Properties derived from elements AOE C . . Pseudo-convex * Derived properties should be distinguished carefully P . Parallel, parallelism from the originating structures (in the Elements and AOF . Biaxial components array), e.g. boundary properties should be Y . Direction distinguished from boundaries. AOG . . Asymptotic * Add to AP5 letters A/Y following AP. Properties of space * Add to AP6 numbers and letters 2/9, A/D following AQ. AOI . Euclidean F . . Bounded AOJ . Non-Euclidean L . . Difference AOK . Pseudo-Euclidean M . . Product Y . Affine & projective N . . Power AOL . Affine P . . Quotient

13 AP5Q APT3A Mathematics

Mathematics AM Mathematics AM Properties Properties Properties in general AMN . . Properties by other characteristics AP8 Properties derived from later facets . Properties derived from elements AP5 Parts of mathematical structures or systems . . Quotient AP5 P APA . Elements, components AP5 Q . . Residual * Elements within some containing whole, but more R . . Factor concrete in conception than properties per se. AP6 3 . . Maximal (extrema) . . Elements in mapping or function or operation * When implies extrema (AQ2). For the loose use of APC . . . Domain maximal to mean most and minimal to mean least, Y . . . Range use ANX X and ANX Y. APD . . . . Value 4 . . Minimal (extrema) APE . . . . . Initial value * See note above at AP6 3. * Use for initial value problems. 8Y . . Adjunctive APF . . . Boundary, boundaries 9 . . Connective * Use for boundary problems. . Properties derived from Entities 85 . . . . Boundary behaviour * Derived properties should be distinguished carefully AN . . . . Boundary properties from the originating structures (in the Entities array). DD . . . . Boundary value * Add to AP6 letters E/Y following AQ. * Use for boundary value problems. E . . Scalar . . Elements resulting from mapping or function or H . . Vector operation I . . Tensor * Most of these imply some operation. These will be N . . Spectral found in the Operations facet (AM7/8). But if a P . . Sequential document considers the operation and its resulting S . . Invariant (derived properties) element together, prefer here. See note preceding AM7 4P. T . . Variable * The order of elements is consistent with that of UY . . Identity (properties) operations, etc. from which they result; but the X . . Modularity notation is different in order to maximize brevity. AP7 . Properties derived from systems & branches APG . . . Solutions * Care is needed here to distinguish qualification of a Y . . . . Bounds concept by these properties per se from situations in APH . . . . Limits which the system represents a subsystem (see -F in Y Auxiliary Schedule AM1) or acts as the specifier of a . . . . . Limit cycles type of the concept concerned. For example, under the API . . . . Moment problems concept Algebras, the simple intersections of the APJ . . . Classes concept with the ‘Groups’ disguises three different APK . . . Sums relationships: group properties of algebras (which NXR . . . . Direct sums would use class AOX as a qualifier by properties); NXS . . . . . Subdirect sums groups of algebras (which would use -F from Auxiliary PN . . . . Exponential sums Schedule AM1 to represent this as a subsystem); group APL . . . Differences algebras (representing a type of algebra). APM * These distinctions are probably more difficult to make . . . Products at the broad level of ‘disciplines’ (e.g. distinguishing NS . . . . Free products topological properties of rings from topology of rings). NXR . . . . Direct products But at the more specific level of particular structures the NXS . . . . . Subdirect products distinctions are usually clearer. OCL . . . . Inner products AP8 Properties by other characteristics APN . . . . Powers, exponents * Use for any property which cannot be equated or nearly APO . . . Divisors equated with the various kinds of properties already APP . . . Quotients provided for. Y . . . Completions APQ . . . Residues AN . . . . Residual properties APR . . . Factors Y . . . . Unique factors APS . . . . Primes X . . . . . Semiprimes APT . . . Radicals, Roots 3A . . . . Radical theory

14 APTRKF Mathematics AQX

Mathematics AM Mathematics AM Parts of mathematical structures or systems Parts of mathematical structures or systems Elements resulting from mapping or function or operation Elements, components APA . Radicals APT . . . Projectives AQD Y . . Radical theory APT 3A Entities APT RKF . . Integral roots * Subsystems which may exist independently of any APU . Derivatives given containing system, i.e. act as quasi-systems or X . . Partial derivatives simple systems themselves. APV . Differentials . By dimension APW . Integrals AQE . . Scalars APX Elements resulting from other operations etc AQG . . . Eigenvalues, Eigenfunctions * That is, other than those enumerated above AQH . . Vectors (APA/APW). Y . . . Spinors * Add to APX numbers 7/9 and letters A/D from AQI . . Tensors Auxiliary Schedule AM1. Two examples are given . By form below to demonstrate their use. AQJ K . . Characters 9D . Derivations S . . Symbols (mathematical entities) AD . Endomorphisms AQK . . Coefficients * E.g. rings of endomorphisms ASM DXA D. Y . . Expressions Elements reflecting structure AQL . . . Polynomials APY . Structural elements, structures * For mathematical structures as systems, see ARA. Y . . Continua * See also Model theory AM4 Q AQM . . Spectra and series (together) X . . Richer structures AQN . . . Spectra, spectrum AQ2 . . Extrema, maxima and minima 3A . . . . Spectral theory AQ3 . . . Maxima 3B . . . . . Scattering theory AQ4 . . . Minima AQO . . . Sequences & series (general) AQ5 . . Conditions * See explanatory note preceding AWL EP on the treatment of these. E3 . . . Maximal conditions AQP . . . . Sequences (general), progressions (general) E4 . . . Minimal conditions * For types of sequences, use AQQ H/AQQ W. AQ6 . . . Chain conditions AQQ . . . . Series (general) AQ7 C . . . . Side conditions * For types of series, use AQQ H/AQQ W. E . . . . Necessary & sufficient conditions . . . . Specific types of sequences & series G . . . . . Necessary conditions, significant conditions * Add to AQQ letters H/W from Auxiliary AQ8 . . Additional structures Schedule AM1. Y . . Adjunctions MR . . . . . Normal series AQ9 . . Connections PN ...... Power series Elements reflecting space QN . . . . . Spectral sequences AQA . Spatial elements AQR . . . . . Approximations, expansions AQB . . Points ...... Methods AQC G . . . Singular points, singularities ...... Polynomials K . . . Co-ordinates 6N9 ...... Splines N . . . Fixed points . By performance P . . . Coincidence points AQS . . Invariants R . . . Critical points AQT . . Variables R3A . . . . Critical point theory AQU . Operators, linear operators AQD . . Spaces * For operators in the wider and looser sense of * Use for the general concept only. In most cases functions see AM8 L. where a specific type of space is concerned, the Y . . Identities preferred arrangement is to treat them as * For identity relations in general, see AM9 K and subsystems and to draw the detail from the the note there. Branches and Systems facets; e.g. Hilbert spaces in AQV . . Functors functional analysis AWO FUA P2. . . . Properties NOG . . . Homogeneous spaces BXD . . . . Duality of functors VJ . . . Topological spaces . Entities with particular values Y . . Projectives AQW . . Bases AQX . . Moduli

15 AR5 ARCA Mathematical systems

Mathematics AM Mathematics AM Parts of mathematical structures or systems Types of mathematical structures or systems . . Entities with particular values . . . Moduli AQX AR5 Mathematical systems, mathematical disciplines, branches of mathematics . . Subsystems * Any given system may be qualified by all * This class does not exist as a general class. But preceding facets (and if necessary specified by particular systems, branches, etc. below (AR/AW) them as to its types) as instructed in Auxiliary often have subsystems derived from other branches Schedule AM1. and systems. These are provided for via Auxiliary Schedule 1 (at -F) and are divided like AR/AW. . . Special parts ARA . Mathematical structures in general * This class does not exist as a general class. But * For literature on the nature of a mathematical particular systems, branches, etc. below (AP/AV) structure or system. may well display parts additional to those elements * For structural elements within systems, see and entities indicated above and defined by the APY. systems themselves. These are provided for via * For specific structures, see individual branches, Auxiliary Schedule AM1 (at -G). etc. below (AR/AW) Types of mathematical structures or systems * For model theory, see AM4 Q. * The facets at AM2/AQ are all used primarily to qualify particular mathematical concepts (e.g., Groups - ARB . . Sets Factorization ASA 7P; Finite groups - Automorphisms * As mathematical structures. ASD AC). 3A * To show the types (‘species’) of anything (i.e. to specify . . . Theory * them) Auxiliary Schedule AM1 provides (at H/W) For general set theory, see AM5. facilities for adding concepts from any facet. This gives, . . . Relations for example, types of groups by relation, such as 8L . . . . Functions automorphism groups, or types by property, such as . . . Elements commutative groups. E6 . . . . Chain conditions * Where a concept has types special to it (i.e. not derived . . . Subsystems synthetically from other facets), these are enumerated in FRI . . . . Arithmetic structures the schedule after types got synthetically. * For cardinal and ordinal numbers, see * The arrangement under any system is as follows: % ARK C/D. (Forms of mathematical presentation) % (Methodologies) FRX . . . . Algebraic structures, algebra of sets % (Operations) % (Processes) % (Relations) % (Properties) % (Elements & Entities) % (Subsystems) % . . . . . By property (Parts or subsystems special to the context) % (Types) % . FRX NTN ...... Complete algebras (Specified by previous facets, AM/AQ) % . (Specified by ...... Named systems branches or systems, AR/AW) % . (Special to the class FRX P2 ...... Boolean divided) % . . . * Enumerated under the class concerned. * For Boolean algebra use ARB X. * When types of anything are specified synthetically care * Normal synthesis by Auxiliary should be taken to distinguish the use of Methods to schedule AM1 is interrupted here; define a type and the use of Systems or Branches; e.g., if it is resumed at ARC A. ‘arithmetic groundfields’ reflects groundfields X ...... Boolean algebra characterized by the use of arithmetic methods rather than ...... Relations by arithmetic structure, the classmark would be ATQ IRI X9S ...... Representations of Boolean (where the first ‘I’ indicates a division of AM6 Methods - algebra see Auxiliary Schedule AM1) and not ATQ RI (where the ‘RI’ is taken directly from ARI Arithmetic). When in ...... Elements doubt, treat such cases as specification by System. XE6 ...... Chain conditions in Boolean algebra ...... Subsystems XFS M ...... Boolean rings ARC A . . . Other subsystems of sets * Normal synthesis is resumed here after its interruption at ARB FRX P2 * Add to ARC A letters P3/Q from Auxiliary Schedule AM1 and R/W following A in AR/AW.

16 ARCH Combinatorics ARHM

Mathematics AM Mathematics AM Mathematical systems AR5 Mathematical systems AR5 Mathematical structures in general ARA Mathematical structures in general ARA Sets ARB Combinatorics ARD . Other subsystems of sets ARC A Relations . Identity ARD 9K ARC H . Types of sets * Add to ARC letters H/W from Auxiliary ARD 9N . Inequalities Schedule AM1. 9P . . Inversion . . By operation 9P3 H . . . Inversion formulae JB . . . Definable sets 9T . Packing & covering . . By process 9TR R . . Lattice packing & covering KF . . . Generating sets Special combinatorial structures . . . . Special parts ARE C . Combinations & permutations KFG . . . . . Generators, generatrix D . . Permutations . . By property E . . Combinations . . . By special dimension G . Matroids N8U . . . . Fractals H . Hypergraphs * Sets with non-integral Hausdorff N . Finite geometries in combinatorics dimension. ARF . Designs & configurations * See also Generating sets ARC KF; * For design of experiment see Probability and Topological vector spaces statistics AXR. AWP FRC N8U; Hausdorff dimension . . Relations AWR YN5 B8U 9T . . . Packing & covering NP . . . Ordered sets . . Types . . . . Relations . . . By system NP9 HE . . . . . Correspondence TB . . . . Matrices as designs NP9 HEP 2 ...... Galois correspondence X . . . Tesselation & tiling NQB . . . . Partially ordered sets ARG . Graphs & maps NUY . . . Perfect sets 3A . . Graph theory O2F . . . Fuzzy sets . . Methods OP . . . Simplicial sets 6VJ . . . Topological graph theory . . By elements * See also Embedding AM9 X. PL . . . Difference sets . . Operations PLN UY . . . . Perfect difference sets 7A . . . Enumeration of graphs and maps X . . Subsets 7P . . . Factorization . . . Subsystems . . Relations XFR R . . . . Lattices of subsets 9T . . . Packing & covering . . Elements ARD Combinatorics, combinatorial structures, DF . . . Boundaries combinatorial analysis . . . . Demonstrations * Branch of mathematics concerned with the DF3 I . . . . . Four colour problem, chromatic theory computation of the number of different ways . . Parts special to graphs certain operations can be performed. GC . . . Nodes, vertices . Mathematical forms GE . . . Edges, faces of graphs 3A . . Combinatorial theory GH . . . Paths & circuits 5 . . Set theory GM . . . Extremities of graphs 53E F . . Axioms of choice . . . . Problems . Operations GM3 KP . . . . . Extremal problems 7A . . Enumeration . . Types of graphs * See also specific problems involving ARH C . . . Connected graphs (general) enumeration, e.g. generating functions, identities. E . . . . Trees 7W . . Partition G . . . Directed graphs (general), digraphs (general) * . Relations For networks, see ARH N. H . . . Undirected graphs (general) . . Functions L . . . Planar graphs 8L . . . Combinatorial functions M . . . . Maps as graphs 8LK F . . . Generating functions * Finite, connected, planar graphs. 9K . . Identity

17 ARHN ARJEQP2 Mathematical systems

Mathematics AM Mathematics AM Mathematical systems AR5 Mathematical systems AR5 Mathematical structures in general ARA Arithmetic ARI . . . . . Planar graphs ARH L . Parts special to arithmetic ...... Maps as graphs ARH M . . . . . Other systems ARI GM ARH N . . . . . Networks . Types of arithmetic * Directed graphs whose elements are . . By entity expressed as activities and events. ARI QX . . . Modular arithmetic P . . . . . Multigraphs Q ...... Transversable multigraphs * Bridges of Konigsberg, etc. ARJ Number theory, higher arithmetic, arithmetica . Methods * In this context, most of these are more helpfully ARI Arithmetic regarded as specifying types of number theory - * For arithmetic structures in general, see Sets see ARL. ARB FRI. . Processes . For particular kinds of users 8A . . Distribution 23C . . Rapid calculation arithmetic 8BQ X . . . Distribution modulo one 23C T . . Business arithmetic . Relations . Methods 8L . . Functions 62 . . Elementary arithmetic 8MC . . . Automorphic functions 6B . . Formal arithmetic . . . . Operations . . Types by property 8MC 7M . . . . . Complex multiplication * For modular arithmetic, see ARI QX. . . . . Properties 6MQ G . . . Non-standard arithmetic 8MC BH . . . . . Galois properties . Operations . . . By property . . Computation 8P2 . . . . Dirichlet functions in number theory 7H . . . Numerical computation . . . By entity 7J . . Addition 8QX . . . . Modular functions 7L . . Multiplication 8YG . . . Zeta function 7M . . . Complex multiplication 93 . . Forms 7N . . Division . . . Properties . Relations 93B G . . . . P-adic 9QY . . Proportion 93B GA3 . . . . . P-adic theory * For fractions, see ARK I. . . . Subsystems . Elements 93F SM . . . . Rings DN . . Powers 93F SMN I . . . . . Forms over global rings & fields DNM U . . . Simple powers 93F SV . . . . Fields DT . . Roots 93F SVN F . . . . . Forms over local fields DTR KF . . . Integral roots . . . Types . Entities 93M C . . . . Automorphic forms EP . . Progressions 93N C . . . . Quadratic forms . . . Types by systems 93N DL . . . . . Binary forms EPR I . . . . Arithmetic progressions 93N DLN C ...... Binary quadratic forms EPT S . . . . Geometric progressions 93N DP . . . . Bilinear forms . Parts special to arithmetic 93Q X . . . . Modular forms GC . . Numbers (general) * For number theory, see ARJ. 9I . . Congruences in number theory GE . . . Numeration systems, digital representations . Elements * For number systems, see ARK B. DK . . Sums . . . . By property DKP N . . . Exponential sums GG . . . . . Binary systems . Entities GH . . . . . Octal systems EP . . Progressions GJ . . . . . Decimal systems EP8 A . . . Distribution in progressions GK . . . . . Duodecimal systems EQ . . Series GL . . . . . Hexadecimal systems EQP 2 . . . Dirichlet series GM . . . . . Other systems

18 ARJF Number systems ARKV

Mathematics AM Mathematics AM Mathematical systems AR5 Mathematical systems AR5 Number theory ARJ Number theory ARJ Entities Number systems ARK B . . Dirichlet series ARJ EQP 2 Natural numbers ARK E

ARJ F Subsystems ARK F Integers * For number fields, see ARN RKV. . Operations FTS . Geometry F7J . . Addition of integers * For geometry of numbers use ARJ X. F7N . . Division of integers * Normal synthesis by Auxiliary Schedule . Relations AM1 is interrupted here; it is resumed at F9I . . Congruences ARJ Y. . Elements & entities X . Geometry of numbers FDQ . . Residues * See also Convex bodies theory, AUC 3A FDQ NC . . . Quadratic residues . . Relations FEJ K . . Characters X93 . . . Forms FEP . . Sequence of integers X93 NA . . . . Linear forms FEP 7J . . . Addition of sequences . . . . . Elements . . . Relations X93 NAD M ...... Products of linear forms FEP 8JX . . . . Analogues X93 NC . . . . Quadratic forms FEP 9R . . . . Generalizations X9T . . . Packing & covering . . Bases X9T RR . . . . Lattice packing & covering FEW JJ . . . Additive bases X9T RRE B . . . . . Points . Special types X9T RRE B8A ...... Lattice points distribution G . . Prime numbers . . Subsystems of geometry of numbers . . . Processes XFR R . . . Lattices G8A . . . . Distribution of primes XFR RFS A . . . . Groups G8A 3A . . . . . Distribution theory of primes . . . . . Automorphism . . . Types XFR RFS CMC . . . . . Automorphism groups of lattices GLR . . . . Generalized primes Y Other subsystems of number theory * Normal synthesis is resumed here after H Rational numbers interruption at ARJ FTS I . Fractions * Add to ARJ Y letters TT/W following A in J . . Proper fractions ATT/AW. K . . Improper fractions YX . Special subsytems L . . Continued fractions * For subsystems special to number theory, M Irrational numbers use ARK. N . Algebraic irrational numbers * Normal synthesis is interrupted here and is O . Transcendental numbers resumed at ARK Y. P Real numbers * For numeration systems (binary, decimal, etc.), see ARI GE. Q Complex numbers ARK B . . Number systems QXN . Imaginary numbers, purely imaginary numbers * For numeration systems (binary, decimal, QXR . Algebraic numbers etc.), see ARI GE. * For algebraic number theory, see ARM. C . . . Cardinal numbers R . Quaternions D . . . Ordinal numbers Other number systems * For recursive ordinals, see * Some numbers do not fit readily into the above Mathematical logic - Algorithms AM4 ‘evolutionary’ order. These are accommodated here H9R KD by drawing on the order of other facets. E . . . Natural numbers, counting numbers, S . Normal numbers whole numbers T . P-adic numbers * See also Algebraic number theory ARM V . Number fields, number domains * General works only; most of the literature relates relates to algebraic number fields and some writers treat the latter as synonymous with number fields in general. * See also Algebraic number fields ARN RKV

19 ARKY ARNDJ3COA Number theory

Mathematics AM Mathematics AM Mathematical systems AR5 Mathematical systems AR5 Number theory ARJ Number theory ARJ Other subsystems of number theory ARJ Y Diophantine methods in number theory ARL N2 . . . . Number fields ARK V . Processes . . . . Linear Diophantine approximations ARK Y Types of number theory ARL N28 6NA * Normal synthesis is resumed here after its interruption at ARJ YX. . Relations * Add to ARK Y letter H from Auxiliary ARL N29 L . . Diophantine equations Schedule AM1. N29 LJL . . . Multiplicative equations YI . By method N29 MNA . . . Linear equations * Use ARL for types defined by methods N29 MNC . . . Quadratic equations used. N29 MND A . . . Cubic equations * Normal synthesis is interrupted here; it is N29 MND C . . . Quartic equations resumed at ARO Y. N29 MND P . . . Bilinear equations N29 MND X . . . Exponential equations ARL . By method . . Inequalities * Add to ARL numbers and letters 2/9, A/Y N29 N . . . Diophantine inequalities following AM6. RS Algebraic 2 . . Elementary number theory * For algebraic number theory, use ARM. . . . Operations * Normal synthesis by Auxiliary Schedule AM1 is 27W . . . . Partition interrupted here; it is resumed at ARN Y. . . . Relations ARM Algebraic number theory 29I . . . . Congruences . Relations 29S . . . . Representations 8L . . Functions B . . Formal number theory 8YG . . . Zeta functions JJ . . Additive number theory 9E . . Extensions . . . Relations . Properties JJ9 3 . . . . Forms BF3 A . . Local number theory . . . . . Methods . Entities JJ9 36T S ...... Geometry of forms EL . . Polynomials . . . . . Types . Subsystems JJ9 3NA ...... Linear forms FRK T . . P-adic numbers JJ9 3NC ...... Quadratic forms FSV . . Fields JJ9 3ND H ...... Higher degree forms * For Fields in algebraic number theory use JJ9 3ND LNC ...... Binary quadratic forms ARN. * Normal synthesis by Auxiliary Schedule JJ9 3ND N ...... Forms in several variables AM1 is interrupted here; it is resumed at . . . Subsystems ARN X. . . . . Primes ARN . . Fields in algebraic number theory JJF RKG . . . . . Additive problems with primes 3A . . . General field theory JL . . Multiplicative number theory * See also Class field theory ARN DJ3 A * See Analytical number theory, ARO. . . . Operations N2 . . Diophantine methods in number theory 7X3 A . . . . Ramification theory . . . Processes . . . Elements N28 6 . . . . Diophantine approximations DJ . . . . Classes . . . . . Subsystems DJ3 A . . . . . Class field theory ...... Local fields DJ3 COA ...... Non-Abelian class field theory N28 6FS VNF ...... Diophantine approximation in local fields ...... Fixed fields N28 6FS VNF X ...... Approximation by numbers from fixed fields N28 6N7 ...... Approximation to one number . . . . . Types N28 6NA ...... Linear Diophantine approximations

20 ARNKL Number theory AROWX8RI

Mathematics AM Mathematics AM Mathematical systems AR5 Mathematical systems AR5 Number theory ARJ Number theory ARJ Fields in algebraic number theory ARN By method ARL Elements . . Subsystems . . . Non-Abelian class field theory ARN DJ3 COA ...... Cyclic fields, cyclic number fields ARN RKV NXQ Types . By relation ARN XF . . Other subsystems in algebraic number theory * Normal synthesis is resumed here after ARN KL . . Function fields interrruption at ARM FSV. . . . Theory * Add to ARN XF (for Subsystems) letters KL3 CRI . . . . Arithmetic theory of algebraic function SW/W following A in ASW/AW. fields XH . . Types of algebraic number theory . By property * Add to ARN X letters H/W in Auxiliary N3 . . Real fields Schedule AM1. . . . Forms Y Types of number theory by other methods N39 3 . . . . Forms over real fields * Normal synthesis is resumed here after its NC . . Quadratic fields interruption at ARL RS. NDC . . Quartic fields * Add to ARN Y letters RT/W in Auxiliary Schedule AM1. NF . . Local fields YW . Analytic . . . Properties * Use ARO for Analytic number theory. NFB H . . . . Galois properties * Normal synthesis is interrupted here; the array of . . . . . Cohomology methods defined by mathematical systems is NFB HAK ...... Galois cohomology concluded at ARO X and normal synthesis is NI . . Global fields resumed at ARO Y. NIB H . . . Galois properties ARO . Analytic number theory, multiplicative number . . . . Cohomology theory NIB HAK . . . . . Galois cohomology . . Operations NJ . . Finite fields 7W . . . Partition . . . Elements . . Relations NJD K . . . . Sums 8L . . . Functions NJD KND X . . . . . Exponential sums 8ND X . . . . Exponential functions . By structure 8P2 . . . . Dirichlet functions in analytic number RKV . . Algebraic number fields theory * See note at ARK V Number fields in 8RJ . . . . Number theoretic functions general. 8RJ EJK . . . . . Characters . . . Processes 8YG . . . . Zeta functions RKV 8A . . . . Distribution in number fields 93 . . . Forms * See also Distribution of primes 93N C . . . . Quadratic forms ARK G8A. 93Q X . . . . Modular forms . . . Relations . . Elements RKV 8JX . . . . Analogues in number fields DQ . . . Residues RKV 8YG . . . . Zeta functions of number fields DQD J8A . . . . Distribution of residue classes . . . . Forms . . Entities RKV 93 . . . . . Forms over number fields EO . . . Series & progressions . . . Elements EO8 A . . . . Distribution in series and progressions RKV DO . . . . Divisors EQ . . . . Series RKV DPY . . . . Completions EQP 2 . . . . . Dirichlet series in analytic number . . . Subsystems theory RKV FPS . . . . Primes . . Subsystems . . . . . Distribution FRK G . . . Primes RKV FPS 8A ...... Distribution of primes in number FRK GLR . . . . Generalized primes field FRK O . . . Transcendental numbers in analytic number . . . Types of algebraic number fields theory RKV NDA . . . . Cubic fields WX . Probabilistic number theory RKV NXQ . . . . Cyclic fields, cyclic number fields . . Relations WX8 L . . . Functions WX8 RI . . . . Arithmetic functions

21 AROY ARUOA Algebra

Mathematics AM Mathematics AM Mathematical systems AR5 Mathematical systems AR5 Number theory ARJ Ordered structures ARQ B . . Types of number theory by other methods ARN Y . Partially ordered systems ARR X ...... Arithmetic functions ARO WX8 RI ARS Algebra ARO Y . . Other types of number theory * Regarded primarily as a mathematical method. * Normal synthesis is resumed here after its * For algebras as mathematical structures see ARX. interruption at ARK YI. . Types by method * Add to ARO Y letters J/W from Auxiliary I2 . . Elementary Schedule AM1. * For elementary algebra, use ART. * Normal synthesis by Auxiliary Schedule ARQ B Ordered structures AM1 is interrupted here; it is resumed at * See also individual systems, e.g. ordered ART Y. semigroups ARY NP. ART . . Elementary algebra D . Ordered spaces 74P . . . Algebraic operations . . Types by property 7J . . . . Addition DNA . . . Ordered linear spaces 7N . . . . Division F . Semilattices . . . Relations . . Types by system 8L . . . . Functions FVJ . . . Topological semilattices 8MY . . . . . Rational functions ARR . Lattices 8YD . . . . . Determinants . . Mathematical presentation 9L . . . . Equations 3KR . . . Word problems 9L3 A . . . . . Theory of equations . . Relations ...... Elements 9R . . . Generalizations of lattices 9L3 ADT ...... Radical theory 9S . . . Representations of lattices ...... Subsystems . . Properties 9L3 AFT B ...... Matrices in theory of equations AN . . . Lattice properties 9QY . . . . Proportion AN8 A . . . . Distributivity Y . Types of algebra by other methods AN8 ANT N . . . . . Complete distributivity * Normal synthesis is resumed here after its . . Entities interruption at ARS I2. EB . . . Lattice points * For differential algebra, see ARW JR. * Add to ART Y numbers and letters 3/Q EBA N8A . . . . Distributivity of lattice points following AM6 so far as applicable. . . Subsystems . Types by methods derived from Systems FRB . . . Sets * Add to ART Y letters R/RU following A in FRC NQB . . . . Partially ordered sets AR/ARU. FSM . . . Rings YRU . . Homological methods FSS . . . . Ideals * For homological algebra use ARU. FTL . . . . Varieties of lattices * Normal synthesis by Auxiliary Schedule . . Types of lattices AM1 is interrupted here; it is resumed at . . . By relation ARW J. L9 . . . . Projective lattices ARU . . Homological algebra, homology (algebraic . . . By property methods) NS . . . . Free lattices . . . Elements NTJ . . . . Complemented lattices DJ . . . . Classes NTN . . . . Complete lattices DJL 9 . . . . . Projective classes OA . . . . Non-commutative lattices E7X . . . . Chain complexes P2 . . . . Dedekind lattices E7X O2H . . . . . Exact couples . . . By entity EDY . . . . Projectives QX . . . . Modular lattices . . . Entities QXN TJ . . . . . Complemented modular lattices EV . . . . Functors . . . By system EVL D . . . . . Derived functors TS . . . . Geometric lattices . . . Types by relation VJ . . . . Topological lattices * For Cohomology, Homotopy, etc. see ARV. X . Partially ordered systems, posets . . . Types by property OA . . . . Non-Abelian homological algebra

22 ARVK Semigroups ARYVJDX95

Mathematics AM Mathematical systems AR5 Mathematical systems AR5 Algebraic systems & structures ARX Algebra ARS Semigroups ARY . Types by methods derived from Systems Subsystems . . . Types by property . Sets ARY FRB . . . . Non-Abelian homological algebra ARU OA . . Generating sets & relations on semigroups ARY FRC KF . . . Special types of homological algebra * Add to ARV letters K/L following AM in ARY FSB . Subgroups of semigroups AMK/AML. FSS . Ideals of semigroups ARV K . . . . Cohomology (homological algebra) . . By property K3B . . . . . K-theory (homological algebra) FSS NXY . . . Semigroups with minimal ideals L . . . . Homotopy (homological algebra) FTL . Varieties . . . . . Elements Types of semigroups LE7 X ...... Chain complexes . By operation LE7 X3A ...... Homotopy theory of chain JL . . Multiplicative semigroups complexes . By relation RX . Types of algebra by methods from other LP . . Inverse semigroups, generalized groups systems . . . Relations * Add to ARV letters RX/W following A in LP9 S . . . . Representations of inverse ARX/AW. semigroups ARW J . Types of algebra by other characteristics LPA B . . . . Homomorphisms of inverse * Normal synthesis using Auxiliary Schedule is semigroups resumed here after its interrruption at LPA F . . . . . Complexes in inverse semigroups ART YRU. . . . Elements resulting from operations etc * Add to ARW letters J/W from Auxiliary LPD X95 . . . . Inverse semigroups of Schedule AM1 so far as applicable. transformations JR . . Differential algebra LPX . . . Generalized heaps JR9 ME . . . Differential equations . By property JR9 MEN G . . . . P-adic differential equations NA . . Linear semigroups PL . . Difference algebra . . Finite semigroups NJ . . . Semigroups with finiteness condition ARX Algebraic systems & structures NN . . Regular semigroups ARY . Semigroups NP . . Ordered semigroups . . Processes . . . Types by property 8GJ . . . Semigroups with involution NPN A . . . . Linearly ordered semigroups . . Relations NPR R . . . . Lattice ordered semigroups 8K . . . Mappings of semigroups NS . . Free semigroups 9I . . . Congruences of semigroups, congruences . . . Subsytems on semigroups NSF RCX . . . . Subsets of free semigroups 9J . . . Equivalence NUF RCX FRY . . . . . Semigroups of subsets of free 9S . . . Representations of semigroups semigroups 9X . . . Embedding of semigroups NXF . . Periodic semigroups AB . . . Homomorphisms of semigroups O5 . . Nilpotent semigroups AD . . . . Endomorphisms O6C . . Nil semigroups . . Elements O6E . . Idempotent semigroups DM . . . Products O9 . . Commutative semigroups DMN XS . . . . Subdirect product . By system DQ . . . Residues TB . . Matrix semigroups DQA N . . . . Residual properties of semigroups VJ . . Topological semigroups DT . . . Radicals . . . Elements resulting from operations etc DT3 A . . . . Radical theory . . . . Transformations . . . Elements resulting from operations etc VJD X95 . . . . . Topological semigroups of DX9 5 . . . . Semigroups of transformations transformations . . Subsystems * See also Topological semigroups FRB . . . Sets of topological spaces AVK VJF FRC KF . . . . Generating sets & relations on RYV J. semigroups

23 ARYX ASBQX Groups

Mathematics AM Mathematics AM Mathematical systems AR5 Mathematical systems AR5 Algebraic systems & structures ARX Algebraic systems & structures ARX Semigroups ARY Groups ASA . . By system Entities ...... Topological semigroups of transformations . . Derived series ASA EQL D ARY VJD X95 Subsytems . . Special types of semigroups ASA FRR . Lattices ARY X . . . Subsemigroups . . Properties . . . . Subsystems FRR AN . . . Lattice properties XFR R . . . . . Lattices of subsemigroups FTL . Varieties of groups G . Special subsystems ASA Groups * For subgroups, use ASB. . Mathematical presentation * Normal synthesis by Auxiliary Schedule AM1 is 3KR . . Word problems interrupted here; it is resumed at ASC. . Methods ASB . . Subgroups 68 . . Algorithmic methods in group theory . . . Processes 6RU . . Homological methods in group theory 86 . . . . Approximation of groups . Operations . . . Relations 7P . . Factorization 8J . . . . Relations between subgroups 7W . . Partition AD . . . . Endomorphisms . Processes . . . Elements 8GP . . Splitting of groups E5 . . . . Conditions for subgroups . Relations E5E 3 . . . . . Maximal conditions for subgroups 9K . . Identity relations in groups E5E 4 . . . . . Minimal conditions 9L . . Equations in groups . . . Entities 9R . . Generalizations of groups EQM R . . . . Normal series 9S . . Representation of groups . . . Subsystems 9S3 A . . . Representation theory of groups FRR . . . . Lattices of subgroups 9SL 9 . . . Projective representations FRR ABY . . . . . Lattice isomorphisms 9SQ X . . . Modular representations . . . Types of subgroups 9X . . Embedding . . . . By relation 9X3 F . . . Embedding theorems in group theory LI . . . . . Congruence subgroups AC . . Automorphisms . . . . By property . Properties MV . . . . . Semi-simple subgroups * For lattice properties, see ASA FRR AN. N4H . . . . . Discrete subgroups BF . . Local properties of groups NR . . . . . Compact subgroups BJ . . Finiteness conditions NV . . . . . Continuous subgroups . Elements NXX . . . . . Maximal subgroups DM . . Products O3 . . . . . Solvable subgroups . . . Types . . . . By entities DMN N . . . . Regular products QX . . . . . Modular subgroups DMN S . . . . Free products DMN XR . . . . Direct products DMN XRN TN . . . . . Complete direct products DX . . Elements resulting from operations etc DXA J . . . Homology of groups . Entities . . Characters EJK . . . Group characters EQ . . Series EQL D . . . Derived series

24 ASC Groups ASEO5

Mathematics AM Mathematical systems AR5 Mathematical systems AR5 Algebraic systems & structures ARX Algebraic systems & structures ARX Groups ASA Groups ASA By property Subsytems . . Subsystems . . . . . Modular subgroups ASB QX . . . Arithmetic structures ASD FRI ASC Types of groups . . Types of finite groups * Normal synthesis resumed here after its interruption . . . By relation at ASA G. ASD MC . . . . Finite automorphism groups * Add H/NJ from Auxiliary Schedule AM1. . . . By property . By methods MO . . . . Abstract finite groups I4 . . Classical groups . . . . . Relations . By operations MO9 S ...... Representations JL . . Multiplicative groups MO9 SNA ...... Linear representations . By relations MOA C ...... Automorphisms MC . . Automorphism groups . . . . . Subsytems . . . Subsystems MOF SB ...... Subgroups MCF RR . . . . Automorphism groups of lattices ...... Elements MCF SA . . . . Automorphism group of groups MOF SBD M ...... Products of subgroups . . . Types by property . . . By system MCN K . . . . Infinite automorphism groups RED . . . . Permutation groups MI . . Holonomy groups RED 6RD . . . . . Combinatorial methods MJ . . Homology groups . . . Special types MK . . Cohomology groups X . . . . Finite P-groups ML . . Homotopy groups ASE Types of groups by other properties . By property * Normal synthesis is resumed here after MQP . . Fundamental groups interruption at ASC NJ. MTP . . Primitive groups * Add to ASE letters NK/O9 from Auxiliary MU . . Simple groups Schedule AM1. NJ . . Finite NK . Infinite groups * For finite groups, use ASD . . Relations * Normal synthesis by Auxiliary Schedule AM1 NK9 S . . . Representations is interrupted here; it is resumed at ASE. NK9 SJV . . . . Integral representations ASD . . Finite groups NOJ . Symmetric groups . . . Operations . . Subsystem 7P . . . . Factorization NOJ FSB . . . Subgroups of symmetric groups . . . Relations NP . Ordered groups 9E . . . . Extensions of finite groups . . Types 9Q . . . . Complement NQB . . . Partially ordered groups 9QM R . . . . . Normal complements NQL . . . Linearly ordered groups 9S . . . . Representations of finite groups . . . . By system 9SJ V . . . . . Integral representations NQL RR . . . . . Lattice ordered groups 9SN G . . . . . P-adic representations NS . Free groups AC . . . . Automorphisms NTN . Complete groups AK . . . . Cohomology NXF . Periodic groups . . . Elements & Entities NXQ . Cyclic groups DY . . . . Structure O3 . Solvable groups DYM R . . . . . Normal structure . . Types . . . . Series O3L R . . . Generalized solvable groups EQM R . . . . . Normal series . . . . Elements . . . Subsystems O3L RDT . . . . . Radicals FRI . . . . Arithmetic structures O3N J . . . Finite solvable groups O4 . Non-solvable groups . . Subsystems O4F SB . . . Subgroups of non-solvable groups O5 . Nilpotent groups

25 ASEO9 ASHO9 Groups

Mathematics AM Mathematics AM Mathematical systems AR5 Mathematical systems AR5 Algebraic systems & structures ARX Algebraic systems & structures ARX Groups ASA Groups ASA Types of groups by other properties ASE Topology ASG VJ . Nilpotent groups ASE O5 ASH Topological groups ASE O9 . Commutative * Groups (e.g. the set of real numbers) which * For commutative groups use ASF. constitute a topological space in which * Normal synthesis by Auxiliary Schedule AM1 multiplication and inversion are continuous is interrupted here; it is resumed at ASG. operations. ASF . Commutative groups, abelian groups 3A . Topological group theory . . Methods . Relations 6RU . . . Homological methods 9S . . Representations of topological groups . . Operations 9X . . Embedding 7P . . . Factorization 9X3 F . . . Embedding theorems for topological . . Relations groups 8K . . . Mappings of Abelian groups AB . . Homomorphisms 9E . . . Extensions . Properties AC . . . Automorphisms BF . . Local properties of topological groups AD . . . Endomorphisms BJ . . Finiteness . . Elements * Including related conditions. DK . . . Sums . Elements DKN XS . . . . Subdirect sums DM . . Products . . . Elements resulting from operations etc . Subsystems DX9 E . . . . Group of extensions FSB . . Subgroups DXA C . . . . Group of automorphisms . Types of topological groups . . Types of Abelian groups . . By property . . . By relation NR . . . Compact topological groups L8 . . . . Torsion groups NRN F . . . . Locally compact topological groups . . . By property . . . . . Relations NDT . . . . Mixed Abelian groups NRN F9S ...... Representations of locally . . . . . Processes compact topological NDT 8GP ...... Splitting of mixed Abelian groups groups NJ . . . . Finite Abelian groups . . . . . Properties NR . . . . Compact Abelian groups NRN FBX D ...... Duality NRN F . . . . . Locally compact Abelian groups NRN FBX D3F ...... Duality theorems ASG Types of groups by other properties etc . . . . . Elements resulting from operations * Normal synthesis is resumed here after etc interruption at ASE O9. NRN FDX AC ...... Automorphism groups of locally * Add to ASG letters OA/VJ from Auxiliary compact topological Schedule AM1. groups By property . . . . . Subsystems P2 . Grothendieck groups ...... Algebras By system * For Algebraic groups, see Algebraic geometry NRN FFR X ...... Group algebras of locally ATH compact topological RI . Arithmetic groups groups TB . Matrix groups ...... Relations . . Subsystems NRN FFR X9S ...... Representations TBF SB . . . Subgroups of matrix groups O3 . . . Solvable topological groups . . . . Types O3L R . . . . Generalized solvable topological TBF SBL I . . . . . Congruence subgroups groups TS . Geometric groups O5 . . . Nilpotent topological groups VJ . Topology O5L R . . . . Generalized nilpotent topological * For Topological groups, use ASH. groups * Normal synthesis by Auxiliary Schedule AM1 O9 . . . Abelian topological groups is interrupted here; it is resumed at ASI.

26 ASI Groups ASKXNG

Mathematics AM Mathematics AM Mathematical systems AR5 Mathematical systems AR5 Algebraic systems & structures ARX Algebraic systems & structures ARX Groups ASA Groups ASA By system By relation . . . . Abelian topological groups ASH O9 . Transformation ASJ L5

ASI Types of groups by other systems ASK . Lie transformation groups * Normal synthesis is resumed here after . . Elements & entities interruption at ASG VJ. DP . . . Quotients * Add to ASI letters VK/W from Auxiliary ES . . . Invariants Schedule AM1. ES7 V . . . . Invariant integration . . . Subsytems Special types of groups FUA NOG . . . . Homogeneous spaces Y . Pseudo-groups FUA NOG NDV . . . . . Symmetric homogeneous . . Types of pseudo-groups by property spaces YNV . . . Continuous . . Types of Lie transformation groups * For Continuous pseudogroups (Lie groups) . . . By property use ASJ. MV . . . . Semisimple Lie groups * Normal synthesis by Auxiliary Schedule . . . . . Subsystems AM1 is interrupted here; it is resumed as ASK Y. MVF SB ...... Subgroups * See also Topological groups ASH. MVF SBN 4H ...... Discrete subgroups of ASJ . . . Lie groups, continuous pseudo-groups semisimple Lie . . . . Methods groups 6B . . . . . Formal Lie group methods MVF UAN OG ...... Homogeneous spaces of . . . . Relations semisimple Lie 8K . . . . . Mappings groups ...... Types by element MVF UAN OGN OJ ...... Symmetric homogeneous 8KP N ...... Exponential mappings spaces 9S . . . . . Representations N4H . . . . Discrete transformation groups 9SN A ...... Linear representations . . . . . Linear 9SN ANJ Y ...... Finite dimensional linear N4H NA ...... Discrete groups of linear representations fractional AK . . . . . Cohomology transformations AKN V ...... Continuous cohomology O3 . . . . Solvable Lie groups . . . . Properties . . . . . Subsystems BXD . . . . . Duality O3F SB ...... Subgroups BXD 3F ...... Theorems O3F SBN 4H ...... Discrete subgroups of . . . . Subsystems solvable Lie groups FSB . . . . . Subgroups O3F UAN OG ...... Homogeneous spaces FSB N4H ...... Discrete subgroups O5 . . . . Nilpotent Lie groups FSB N4H 97 ...... Deformations of discrete subgroups . . . . . Subsystems FSB NV ...... Continuous subgroups O5F UAN OG ...... Homogeneous spaces of FTA . . . . . Algebras nilpotent Lie groups FTC ...... Lie algebras of Lie groups X Lie groups by other relations etc . . . . Types of Lie groups * Normal synthesis by Auxiliary Schedule . . . . . By relation AM1 is resumed here after its interruption at ASJ L5. L5 ...... Transformation * Add to ASK X letters L6/W from * For transformation groups use ASK. Auxiliary Schedule AM1. * Normal synthesis by Auxiliary By property Schedule AM1 is interrupted here; it is resumed at ASK X. XMW . Complex Lie groups XN3 . Real Lie groups XNA . Linear Lie groups XNF . Local Lie groups XNG . P-adic Lie groups

27 ASKXNK ASNFSGP2 Rings

Mathematical systems AR5 Mathematics AM Algebraic systems & structures ARX Mathematical systems AR5 Groups ASA Algebraic systems & structures ARX . . . Pseudo-groups ASI Y Rings ASM . . . . Types of pseudo-groups by property Subsystems ...... P-adic Lie groups ASK XNG . Geometries over rings ASM FTT

ASK XNK ...... Infinite Lie groups Types of rings XNK FSB ...... Subgroups . By operation XNK FSB P2 ...... Cartan subgroups ASM JN . . Division rings XNR ...... Compact Lie groups XP2 ...... Cartan pseudogroups . By property Y . . . . Pseudogroups by other properties etc MTP . . Primitive rings * Normal synthesis by Auxiliary N9 . . Polynomial rings Schedule AM1 is resumed here after its interruption at ASI YNV. NP . . Ordered rings * Add to ASK Y letters NW/W from NQL . . . Linearly ordered rings Auxiliary Schedule AM1. NR . . Compact rings ASL G . . . Groupoids NRN F . . . Locally compact rings . . . . Relations NUY . . Perfect rings G8K . . . . . Mappings of groupoids O7 . . Associative . . . . Types * For Associative rings, use ASN. GQ8 . . . . . Groupoids with additional structure * Normal synthesis by Auxiliary Schedule AM1 is interrrupted here. It is resumed at ASR. Q . . . Quasi-groups ASN . . Associative rings . . . . Methods . . . Methods Q6R UML . . . . . Homotopy of quasi-groups 6RU . . . . Homological methods . . . . Entities . . . Relations QEU Y . . . . . Identities on quasi-groups 97 . . . . Deformation S . . . Loops 973 A . . . . . Deformation theory . . . . Relations 9D . . . . Derivation of rings S8K . . . . . Mappings of loops 9E . . . . Extension of rings 9S . . . . Representations of rings ASM Rings 9X . . . . Embedding of rings . Operations AC . . . . Automorphisms 7C . . Classification AD . . . . Endomorphisms . . . Methods AK . . . . Cohomology 7C6 RU . . . . Homological classification of rings AK3 B . . . . . K-theory . Processes . . . Properties 8GJ . . Rings with involution BXD . . . . Duality . Relations . . . Elements 9E . . Extensions DC . . . . Domains 9E3 A . . . Extension theory . . . . . Types by element 9K . . Identity relations in rings DCP RY ...... Unique factorization domains 9R . . Generalizations of rings DT . . . . Radicals AC . . . Automorphisms of rings E8 . . . . Additional structures . Elements . . . . . By property DXA D . . Rings of endomorphisms E8P 2 ...... Lie structures on associative rings E6 . . Rings with chain conditions E8P 3 ...... Jordan structures on associative rings . Subsystems . . . Subsystems . . Subsets FSA . . . . Groups of rings FRC X . . . Subrings FSG P2 . . . . . Grothendieck groups of rings FRY . . Semigroups FRY JL . . . Multiplicative semigroups of rings FTT . . Geometries over rings

28 ASNMU Rings ASPRIQQPNIB

Mathematical systems AR5 Mathematics AM Algebraic systems & structures ARX Mathematical systems AR5 Rings ASM Algebraic systems & structures ARX Associative rings ASN Rings ASM . Subsystems Non-associative rings ASO . . . Grothendieck groups of rings ASN FSG P2 . . Power associative rings ASO PN . Types of associative rings ASP Commutative rings . . By properties . Methods ASN MU . . . Simple rings 6RU . . Homological methods MUB H . . . . Galois theory of simple rings . Processes N4D . . . Ideal rings 8GV . . Valuations on commutative rings N4D MTL . . . . Principal ideal rings . Relations NJ . . . Finite rings 97 . . Deformations NJY . . . . Finite dimensional rings 9E . . Ring extensions NN . . . Regular rings 9E3 A . . . Extension theory NSN 4D . . . Free ideal rings . . . . Operations O5 . . . Nilpotent rings 9E7 X . . . . . Ramification O6C . . . Nil rings . . . . Elements . . By elements 9ED P . . . . . Quotients PP . . . Quotient rings AK . . Cohomology of commutative rings PR . . . Factor rings AK3 B . . . K-theory PS . . . Prime rings . Properties PSX . . . . Semiprime rings . . Galois properties Q6 . . . By chain conditions BH . . . Galois theory Q6P 2 . . . . Artinian rings BJ . . . Finiteness Q6P 3 . . . . Non-Artinian rings . Elements Q6P 3MU . . . . . Simple non-Artinian rings DT . . Radicals Q6P 4N . . . . Noetherian rings DT3 A . . . Radical theory . . . . . Subsystems E6 . . Chain conditions Q6P 4NF SS ...... Ideals in Noetherian rings EL . . Polynomials over commutative rings . . By systems . Subsystems RBX . . . Boolean rings FRW JR . . Differential algebra * See ARB XFS M . Types of commutative rings RY . . . Semigroup rings . . By property SA . . . Group rings MN9 K . . . With identity SAN J . . . . Group rings of finite groups MN9 KX . . . . Zero rings SAN K . . . . Group rings of infinite groups MN9 KY . . . . Non-zero rings, integral domains SX . . . Categorical rings * See also Division rings, ASM JN . . . Theory NF . . . Local commutative rings SX3 A . . . . Categorical ring theory NJ . . . Finite commutative rings TB . . . Matrix rings NN . . . Regular commutative rings TBS W . . . . Matrix rings over skew fields NNN F . . . . Regular local rings ASO Non-associative rings O7 . . . Associative commutative rings . Elements P2 . . . Dedekind rings DT3 C . . Radical theory . . By elements . Types by property PN . . . Powers P2 . . Lie rings PNO 7 . . . . Power associative commutative rings P3 . . Jordan rings . . By system P4M . . Mal’cev rings RI . . . Arithmetic rings . Types by elements . . . . By process PN . . Power associative rings RIK GV . . . . . Valuation rings . . . . By property RIK GVN 4H . . . . . Discrete valuation rings . . . . By entity RIQ QPN . . . . . Power series rings RIQ QPN IB ...... Formal power series rings

29 ASQ ASVNI Algebraic systems & structures

Mathematical systems AR5 Mathematics AM Algebraic systems & structures ARX Mathematical systems AR5 Rings ASM Algebraic systems & structures ARX By property Rings ASM . Commutative rings ASP . . . Orders AST Y ...... Formal power series rings ASP RIQ QPN IB ASU S Semifields . Types by systems ASQ . Non-commutative rings SVJ . . Topological semifields . . By property N4D . . . Non-commutative ideal rings N4D MTL . . . . Non-commutative principal ideal rings ASV Fields NF . . . Non-commutative local rings . Methods ASR Types of rings by other properties etc 6RU . . Homological methods * Normal synthesis by Auxiliary Schedule AM1 is . Processes resumed here after its interruption at ASM O7. 8GV . . Valuations * Add to ASR letters OB/W from Auxiliary . Relations Schedule AM1. 93 . . Forms . By element 9D . . Derivations in fields PRY . . Unique factorization rings 9E . . Extensions ASS . Ideals 9E8 GV . . . Extensions of valuations to a field extension . . Properties . . . Types of field extensions BJ . . . Finiteness 9EM R . . . . Normal extensions . . Types by property 9EN 4F . . . . Transcendental extensions NS . . . Free ideals 9EO 2S . . . . Separable extensions P2 . . . Lie ideals 9R . . Generalizations . . Types by elements 9R8 GV . . . Generalizations of valuations PS . . . Prime ideals 9X . . Embedding AST . Modules . Properties . . Theory BH . . Galois properties and theory 3B . . . Algebraic K-theory . . . Relations . . Property BH9 E . . . . Galois extensions BXD . . . Duality BHA K . . . . Galois cohomology . . Elements . Elements DT . . . Radicals in modules DJ . . Classes . . Subsystems DJ3 A . . . Class field theory FSP . . . Modules over commutative rings DJN F3A . . . Local class field theory . . Types of modules DJO A3A . . . Non-Abelian class field theory . . . By relations EL . . Polynomials L8 . . . . Torsion modules . Subsystems L9 . . . . Projective modules FRW JR . . Differential algebra in fields LAI . . . . Injective modules FRW PL . . Difference algebra in fields . . . By property . Types of fields NP . . . . Ordered modules . . By properties NS . . . . Free modules N3 . . . Real fields NUY . . . . Perfect modules N3Y . . . . Formally real fields O7 . . . . Associative modules NF . . . Local fields . . . . . Properties . . . . Operations O7B XD ...... Duality NF7 X3A . . . . . Ramification theory . . . By elements . . . . Relations PP . . . . Quotient modules NF9 3 . . . . . Forms over local fields . . . By system NF9 E . . . . . Extensions of local fields VJ . . . . Topological modules . . . . Properties . . . Special types of modules NFB G . . . . . P-adic analysis X . . . . Injective hull NFX . . . Fixed fields Y . Orders NI . . . Global fields

30 ASVNJ Categories ASXLL

Mathematics AM Mathematics AM Mathematical systems AR5 Mathematical systems AR5 Algebraic systems & structures ARX Algebraic systems & structures ARX Fields ASV Fields ASV By properties . . . . . Finite dimensional skew fields ASW NJY . Global fields ASV NI ASX Categories ASV NJ . Finite fields 3A . Category theory . . Relations . Methods NJ9 L . . . Equations over a finite field 6RS . . Categories & algebraic theory . . Properties 6RU . . . Homological algebra . . Entities 6RU OA . . . . Non-Abelian homological algebra NJE L . . . Polynomials over finite field 6RV L . . . Homotopical algebra NJE P . . . Sequences over a finite field 6RV LOA . . . . Non-Abelian homotopical algebra NJE QNA . . . . Linear sequences over a finite field 6TS . . Categories & geometric theory . . Subsystems . Operations NJF RB . . . Sets 7P . . Factorization NJF RCP L . . . . Difference sets . Relations NJF RCP LNU Y . . . . . Perfect difference sets 8JT S . . Categories & geometry NP . Ordered fields 9S . . Representations of categories OA . Non-commutative fields . Elements * See Skew fields ASW DT . . Radicals By systems E8Y . . Adjunctions VJ . Topological fields . . . Types by property . . Processes E8Y NLN . . . . Strong adjunctions VJ8 GV3 A . . . Valuation theory . Entities . . Types EV . . Functors VJN P . . . Ordered topological fields . . . Properties Special types EVB XD . . . . Duality of functors . Number fields . . . . Types by property * See Number theory, ARN RKV EVN LN . . . . . Strong functors ASW . Skew fields, non-commutative fields . Subsystems . . Relations . . Groups 9X . . . Embeddings in skew fields FSC ML . . . Homotopy groups in categories . . . . Types of embeddings by system . . Elements special to categories 9XS M . . . . . Rings G . . . Objects of categories 9XS MO7 ...... Embeddings of associative rings . . . . Elements 9XT A . . . . . Algebras GE8 . . . . . Additional structures 9XT AO7 . . . . . Embeddings of associative GE8 JL . . . . . Multiplicative structures on the objects algebras of categories . . Properties . . . . Types BH . . . Galois theory of skew fields GOP . . . . . Simplicial objects . . Subsystems . Types of categories FSA . . . Groups of a skew field . . By method FSC JL . . . . Multiplicative groups of a skew HXT . . . Inductive categories field . . By operation . . Types by property JJ . . . Additive categories NJY . . . Finite dimensional skew fields . . . . Relations JJ8 J . . . . . Additive relations KJ . . . Categories of relations KK . . . . Mappings KKN V . . . . . Categories of continuous mappings LL . . . . Equational categories

31 ASXNF ATANS Algebras

Mathematics AM Mathematics AM Mathematical systems AR5 Mathematical systems AR5 Algebraic systems & structures ARX Algebraic systems & structures ARX Categories ASX Algebras ATA . . By operation Relations . . . . Equational categories ASX LL . Linear equations ATA 9L . . By property Elements ASX NF . . . Local categories ATA DJ . Classes of algebras NI . . . Universal categories . . Types by property O9 . . . Abelian categories DJH L . . . Model classes of algebras . . . . Methods DJM TP . . . Primitive classes of algebras O96 RU . . . . . Homological algebra in Abelian Entities categories ES . Invariants . . . . Relations ES3 A . . Invariant theory O99 S . . . . . Representations Subsystems . . . . Elements FRI . Arithmetic of algebras O9D T . . . . . Radicals in Abelian categories FSX . Categories of algebras . . . . Types by relation FTL . Varieties of algebras O9L D . . . . . Derived categories . Special to algebras . . . . Types by property G . . Subalgebras O9O 2H . . . . . Exact categories P2 . . . . . Grothendieck categories Types of algebras . . . . Types by element . By operation QDV J . . . . . Categories of topological spaces JN . . Division algebras ...... Subsystems JNP 2 . . . Cayley algebras QDV JFR B ...... Sets . By property QDV JFR BB5 X ...... Baire categories MV . . Semi-simple algebras * Measure of sets in . . . Properties topological space. MVB 9Y . . . . Orders in semisimple algebras Y Vector spaces * For topological vector spaces, see AWP...... Subsystems . Subsystems MVB 9YF RI ...... Arithmetic problems of orders . . Skew fields N8H . . Multidimensional algebras YFS W . . . Vector spaces over skew fields NDQ . . Multilinear algebras . Types . . . Relations YN8 VB . . Normed spaces NDQ 93 . . . . Forms NDQ 93N C . . . . . Quadratic forms NDQ 93N DP . . . . . Bilinear forms ATA Algebras, linear algebras NI . . Universal algebras * Deal with linear equations, matrices, . . . Mathematical presentation determinants and other algebraic structures. * See also Vector spaces ASX Y NI3 KR . . . . Word problems . Methods . . . Relations 6RI . . Arithmetic in algebras NIA C . . . . Automorphism 6RU . . Homological methods NIA D . . . . Endomorphisms 6RV K . . Cohomology of algebras . . . Properties . . . Types by property NIB R . . . . Compactness 6RV KNF . . . . Local cohomology of algebras NIB RLL . . . . . Equational compactness 6RV L . . Homotopy . . . Entities . . . Subsystems NIE L . . . . Polynomials 6RV LFS A . . . . Homotopy groups . . . Subsystems . Relations NIF RR . . . . Lattices related to universal algebras 8K . . Linear mappings . . . Types by property 95 . . Linear transformations NIN LF . . . . Infinitary algebras 9L . . Linear equations NIP 82 . . . . Universal enveloping algebras NOJ . . Symmetric algebras NS . . Free algebras

32 ATAO7 Algebras ATCNQR

Mathematics AM Mathematics AM Mathematical systems AR5 Mathematical systems AR5 Algebraic systems & structures ARX Algebraic systems & structures ARX Algebras ATA Algebras ATA By property Algebras by system . Free algebras ATA NS . . Frechet algebras ATA VJP 4F

ATA O7 . Associative algebras Special types of algebras . . Relations ATB . Matrix algebras, matrices O79 7 . . . Deformation 3A . . Matrix theory O79 73A . . . . Deformation theory . . . Methods . . Subsystems 6E . . . . Iterative analysis O7F SS . . . Ideals . . Relations O7F SSP 2 . . . . Lie ideals of associative algebras . . . Linear mappings . . Types by property 8KN A . . . . Linear mappings & matrices O7N JY . . . Finite dimensional algebras 8YD . . . Determinants O7N QR . . . Graded algebras 8YD 9R . . . . Generalizations of determinants O7N S . . . Free algebras . . . Equations O7O 2S . . . Separable algebras 9L . . . . Matrix equations O8 . Non-associative algebras 9N . . . Inequalities . . Types by operation . . . Inversions O8J N . . . Non-associative division algebras 9P . . . . Matrix inversion . . Types by property . . Subsystems O8N JY . . . Finite dimensional non-associative algebras FRK F . . . Matrices of integers O9 . Commutative algebras FSM . . . Matrices over rings . . Properties FSM KL . . . Matrices over function rings O9B J . . . Finiteness conditions FSR X . . . Matrices over special rings . . Elements . . Types O9E 6 . . . Chain conditions in commutative algebras OTY . . . Square matrices OCN . Exterior algebras . . . . Other properties OCN P2 . . Gassman algebra OTY D8E . . . . . Trace (square matrices) P2 . By named mathematician ATC . Lie algebras * For Boolean algebra, see ARB X . . Methods * For Lie algebras, see ATC 6RU . . . Homological methods * For Banach algebras, see ATD B . . Relations * For Jordan algebras, see ATD J 9D . . . Derivations of Lie algebras P4C . . Clifford algebra 9R . . . Generalizations P5R . Factor algebras 9S . . . Representations * See also Factor spaces, AUA PA; Clifford algebra, ATA P4C . . . . Types by properties Algebras by element 9SN JY . . . . . Finite dimensional linear representation PL . Difference algebras of Lie algebras QH . Vector algebras 9X . . . Embeddings of Lie algebras QI . Tensor algebras AC . . . Automorphisms of Lie algebras QU . Operator algebras AD . . . Endomorphisms of Lie algebras * For C* algebra, see ATD BKG J. AJ . . . Homology of Lie algebras QUP 2 . . Neumann algebra, W* algebra AK . . . Cohomology of Lie algebras Algebras by system . . Elements TS . Geometric algebras DT . . . Radicals of Lie algebras VJ . Topological algebras DX9 D . . . Lie algebras of derivations VJP 2 . . Hopf algebras . . Types of Lie algebras . . . Methods . . . By operation VJP 26R U . . . . Homology of Hopf algebras JN . . . . Division algebras (Lie algebras) VJP 4F . . Frechet algebras . . . By property MU . . . . Simple Lie algebras NDL . . . . Binary Lie algebras NJY . . . . Finite dimensional Lie algebras NLD . . . . Infinite-dimensional Lie algebras NQR . . . . Graded Lie algebras

33 ATCNR ATHNADX95ES3A Algebraic geometry

Mathematical systems AR5 Mathematics AM Algebraic systems & structures ARX Mathematical systems AR5 Algebras ATA Algebraic systems & structures ARX . . . Lie algebras ATC Algebraic geometry ATG . . . . . By property Elements ...... Graded Lie algebras ATC NQR . . . Deformations of singularities ATG ECG 97

ATC NR ...... Compact Lie algebras Types of algebraic geometry O3 ...... Solvable Lie algebras . By relation O5 ...... Nilpotent Lie algebras ATG L9 . . Projective algebraic geometry . . . . . By entities . . . Methods QX ...... Modular Lie algebras L96 . . . . Projective techniques ATD B . . . Banach algebras . By property . . . . Types N2Y . . Birational geometry . . . . . With involution . . . Relations BKG J ...... B* algebra, C* algebra N2Y 8K . . . . Mappings J . . . Jordan algebras N2Y 8KM Y . . . . . Rational mappings . . . . Relations N2Y 95 . . . . Transformations J9S . . . . . Representations N2Y 953 L . . . . Models JAC . . . . . Automorphisms N2Y 953 LNX Y . . . . . Birational transformation of . . . . Elements & entities minimal models JDC . . . . . Domains . . . Elements ...... Types by property N2Y ECG . . . . Singularities JDC N3Y ...... Formally real domains . . . . . Operations JEU Y . . . . . Identities N2Y ECG 7C ...... Classification . . . . Types by property . By system JMT E . . . . . Exceptional Jordan algebras RY . . Semi-groups JMU . . . . . Simple Jordan algebras * For algebraic semigroups, use ATG Y JNL D . . . . . Infinite-dimensional Jordan algebras * Normal synthesis by Auxiliary JOA . . . . . Non-commutative Jordan algebras Schedule AM is interrupted here; it is resumed at ATI. Y . . Algebraic semi-groups in algebraic ATG Algebraic geometry geometry * Alternative (not recommended) is to locate with ATH . . Algebraic groups in algebraic geometry geometry at AUI TG. . . . Types . Relations . . . . By method 8YG . . Zeta functions IB . . . . . Formal groups 9V . . Coverings ...... Types by property AK . . Cohomology IBN G ...... P-adic analytic groups . . . Types by property . . . . By property AKM W . . . . Complex cohomology NA . . . . . Linear algebraic groups AKN F . . . . Local cohomology ...... Processes AKN G . . . . P-adic cohomology NA8 6 ...... Approximation . Properties NA8 63F ...... Approximation theorems BF . . Local algebra ...... Relations BF3 A . . . Local theory ...... Representations . . . Relations NA9 S ...... Linear representations BF9 7 . . . . Deformation ...... Elements BF9 73A . . . . . Local deformation theory NAD X ...... By relation . Elements ...... Transformations E9 . . Connections NAD X95 ...... Algebraic groups of ECG . . . Singularities transformations ECG 97 . . . . Deformations of singularities ...... Elements NAD X95 ES ...... Invariants NAD X95 ES3 A ...... Geometric theory of invariants

34 ATHNAFSV Algebraic geometry ATMFSMDXAD

Algebraic geometry ATG Mathematical systems AR5 By system Algebraic systems & structures ARX . Algebraic groups in algebraic geometry ATH Algebraic geometry ATG . . . . Linear algebraic groups ATH NA Families in algebraic geometry ATK EN . . . . . Elements . Entities ...... Geometric theory of invariants . . . Geometric invariants ATK ENE STS ATH NAD X95 ES3 A . Subsystems . . . . . Subsystems ATK ENF UA . . Spaces ATH NAF SV ...... Fields ENF UAQ X . . . Moduli spaces NAF SVN F ...... Algebraic groups over local fields . Types of families by system NAF SVN I ...... Algebraic groups over global ENT UU . . Families of curves fields FF Fibrations in algebraic geometry, algebraic NAF SVN J ...... Algebraic groups over finite fields fibrations NG . . . . P-adic groups FFU EN . Families of fibrations NGN A . . . . . Linear P-adic groups ATL Varieties, algebraic varieties NLD . . . . Infinite-dimensional algebraic groups . Methods O9 . . . . Commutative algebraic groups 4RL L . . Equational logic . . . . . Types by method . Relations O9I B ...... Commutative formal algebraic 8YG . . Zeta functions groups 9I . . Congruence OA . . . . Non-commutative algebraic groups 9IA N8A . . . Distributivity . . . . . Types by method 9ID 6X . . . Modularity OAI B ...... Non-commutative formal groups AK . . Cohomology of algebraic varieties ATI . Other algebraic structures AK3 B . . . K-theory * Normal synthesis is resumed here after its AKP 2 . . . Grothendiek cohomology interruption at ATG RY. . Elements * Add to ATI letters IY/X following AS. DMN TE . . Amalgamated product ATJ By geometric structure * Add to ATJ letters T/Y following AT. EB . . . Points * Add to ATK letters A/H following AU. EBM Y . . . . Rational points on algebraic varieties UU . Algebraic curves ECG . . . . Singular points of algebraic varieties . . Relations EDY . . Projectives UU9 V . . . Coverings . Entities . . Elements & entities EX . . Moduli of algebraic varieties UUE CG . . . Singular points of algebraic curves . Subsystems UUE X . . . Moduli of algebraic curves FRI . . Arithmetic problems of varieties . . Subsystems FRR . . . Lattices of varieties UUF RI . . . Arithmetic problems of algebraic curves . Types of varieties VS . Algebraic surfaces MY . . Rational varieties . . Elements & entities . . By named mathematician VSD J . . . Classes of algebraic surfaces P2 . . . Abelian varieties VSD J7C . . . . Classification * For Abelian varieties, use ATM. * Normal synthesis by Auxiliary Schedule VSE CG . . . Singular points of algebraic surfaces AM1 is interrupted here; it is resumed at VSE X . . . Moduli of algebraic surfaces ATN. . . Subsystems ATM . . . Abelian varieties & schemes VSF RI . . . Arithmetic problems of algebraic surfaces . . . . Operations VSF SGP 3 . . . Picard groups 7M . . . . . Complex multiplication . . Types by property . . . . Entities VSM Y . . . Rational surfaces EX . . . . . Moduli ATK EN Families in algebraic geometry . . . . Subsystems . Relations FRI . . . . . Arithmetic in Abelian varieties EN9 7 . . Deformations FSM . . . . . Rings . Elements ...... Elements by relation END Y . . Structure FSM DXA D ...... Rings of endomorphisms of . Entities Abelian varieties ENE S . . Invariants ENE STS . . . Geometric invariants

35 ATMFTT ATSSCLWIL9 Mathematical systems

Algebraic systems & structures ARX Mathematics AM Algebraic geometry ATG Mathematical systems AR5 Varieties ATL Algebraic systems & structures ARX . Types of varieties . . . Ground fields ATQ . . . . . Rings ATM FSM . . . . . Arithmetic ground fields ATQ RI ...... Rings of endomorphisms of Abelian varieties ATQ Y . . . Injectives ATM FSM DXA D

ATM FTT . . . . . Geometric structures ATS Geometry FTU UOV ...... Elliptic curves in Abelian . Methods varieties 62 . . Elementary geometry ...... Subsystems * Alternative (not recommended) to locating at FTU UOV FRI ...... Arithmetic in elliptic curves AUI 2 (where there is an explanatory note). FUA . . . . . Spaces 6RS . . Algebraic methods in geometry FUA NOG ...... Homogeneous spaces of Abelian 6RU . . Homological methods varieties 6RV L . . . Homotopy FUA NOG MTL ...... Principal homogeneous spaces 6RV L3A . . . . Abstract homotopy theory of Abelian varieties 6W . . Analytic methods in geometry ATN . Other types of varieties . Relations * Normal synthesis using Auxiliary 9N . . Geometric inequalities Schedule AM1 is resumed here, after its 9WI . . Incidence geometry interruption at ATL P2. AC . . Automorphisms * Add to ATN letters P3/W in Auxiliary . Properties Schedule AM1. BOG . . Homogeneous systems P3 . . Picard varieties BOJ . . Symmetry SA . . Group varieties . . Special properties . Special types D9H G . . . Similarity in geometry ATO B . . Subvarieties D9H G3F . . . . Interception theorem D . . Intersections . Elements . . . Types by property * For geometric objects and special structures DNT N . . . . Complete intersections (lines, planes, etc.) see ATT. F . . Schemes DF . . Boundaries . . . Types by system . . Elements resulting from relations FSA . . . . Group schemes . . . Inequalities FSA NJ . . . . . Finite group schemes DX9 N . . . . Geometry of inequalities ATP . . Cycles & subschemes DY . . Structure . . . Relations * For special structures and subsystems see 9J . . . . Equivalence ATT/AUH. 9JM Y . . . . . Rational equivalence in cycles E9 . . Connections . . . Elements . . . By spatial property DO . . . . Divisors E9O L . . . . Affine connections . . . Subsystems E9O M . . . . Conformal connections FSG P3 . . . . Picard groups in cycles . Subsystems . . . Types by property * Normal synthesis by Auxiliary Schedule AM1 NA . . . . Linear systems of cycles is interrupted here. Subsystem classmarks . . . . . Relations (H/W) are added directly to ATS, not to ATS F. Special (enumerated) subsystems follow at NA8 K ...... Mappings ATT. Types of geometry begin at AUH Y. NA8 KMY ...... Linear systems & rational Direct application of Auxiliary Schedule AM1 mappings resumes at AUL. ATQ Ground fields RB . . Sets (geometry) . Types by property * For convex sets, see AUC RB. N3 . . Real ground fields . . Groups NF . . Local ground fields SA . . Geometries over groups NG . . . P-adic ground fields SCI 4 . . . Classical groups NI . . Global ground fields SCL WI . . . Incidence groups NJ . . Finite ground fields SCL WIL 9 . . . . Projective incidence groups . Types by system . . Rings RI . . Arithmetic ground fields

36 ATSSM Geometry ATWJ

Mathematics AM Mathematics AM Mathematical systems AR5 Mathematical systems AR5 Geometry ATS Geometry ATS Subsystems One dimensional ATU J . Rings . Angles ATU W

ATS SM . Geometries over rings ATU Y Two dimensional ST . . Modular geometry ATV B . Plane, planes . Algebras . . Types TA . Geometries over algebras . . . By relation . . Subsystems BL9 . . . . Projective plane TAF UA . . . Spaces in geometries over algebras . . . . . Subsystems TAF UAN C . . . . Quadratic spaces BL9 FTU U ...... Curves on projective plane TAF UAN CMW . . . . . Complex quadratic spaces . . . By property TAF UAP 2 . . . . Hermitian spaces BOL . . . . Affine planes, euclidean planes . . . Manifolds BOL FTU U . . . . . Curves on affine plane TAF UG . . . . Geometries with algebraic manifold . . . By system structure BTT . . . . Plane figures (general) . . . Special figures D . . . . Faces Special subsystems F . . . . Polygons G . . . . . Quadrilaterals ATT . Geometric structures & figures, geometric J . . . . Circles & spheres objects * For spheres alone, see ATW T. OP . . Simplicial figures K . . . . . Conic sections, conics * The most elementary figure of a given * Plane sections of a circular surface. dimension. * For cones as solids, see ATW R. . By dimension L ...... Ellipse ATU E . . Points in geometry M ...... Parabola F . . . Geometric loci N ...... Hyperbola J . . One dimensional Q . . . . . Circle L . . . Lines R ...... Disc M . . . . Straight lines & planes (together) * Area bounded by circumference. * For planes alone, see ATV B. S . Surfaces N . . . . . Straight lines * See also Curves and surfaces treated together, P ...... Tangents ATU T. R . . . . Geodesics . . Types by property T . . . . Curves & surfaces SNR . . . Compact surfaces * Treated together. . . Types by system * For surfaces alone, see ATV S. SW . . . Analytic surfaces TNC . . . . . Second order curves & surfaces T . . Surface strips U . . . . . Curves Y Three dimensional ...... Projection ATW B . Solids, bodies U99 ...... Trace (curve projections) C . . Polyhedra (general) ...... Types * See also Polytopes ATY C ...... Generating COC T . . . Concave polyhedra UKF ...... Peano curves COD . . . Convex polyhedra UW ...... Analytic curves * Use ATW E...... Conics, conic sections E . . . Regular polyhedra, platonic solids, convex * See ATV K polyhedra W . . . Angles, goniometry * See also Convex bodies AUC TWB * See also Trigonometry AVE F . . . . Tetrahedra G . . . . Hexahedra, cubes, cuboids H . . . . Dodecahedra I . . . . Octahedra J . . . . Icosahedra

37 ATWP AUBOD Spaces in geometry

Mathematical systems AR5 Mathematics AM Geometry ATS Mathematical systems AR5 By dimension Geometry ATS . Three dimensional ATV Y Spaces in geometry AUA . . . . Regular polyhedra ATW E . . By property . . . . . Icosahedra ATW J . . . Non-Euclidean spaces AUA OJ

. . . . Special types AUA OKY . . . Affine & projective spaces * Reflecting a mixture of characteristics of . . . . Relations division. OKY 8K . . . . . Mappings of affine & projective spaces * Documents dealing jointly with any two . . . . Subsystems types should be classed under the first one OKY FTA . . . . . Affine & projective spaces over appearing, e.g. pyramids and cones would go under pyramids. algebras ATW P . . . . . Prisms OL . . . . Affine spaces Q . . . . . Pyramids OM . . . Conformal spaces R . . . . . Cones OMX . . . . Pseudo-conformal spaces S . . . . . Cylinders OS . . . Kinematic spaces T . . . . . Spheres, balls P2 . . . Eilenberg-Maclane spaces V . . . . . Others (A/Z) . . By element * E.g. solids of rotation. PA . . . Spaces with elements, factor spaces . . . . . Polytopes PAN A . . . . Spaces of linear elements * See ATY C PP . . . . Quotient spaces ATX . Four dimensional structures, n-dimensional Q9 . . . . Spaces with a connection structures, multidimensional structures Q9O I . . . . . Spaces with a Euclidean connection MW . . Complexes Q9O M . . . . . Spaces with a conformal connection MWO P . . . Simplicial complexes QH . . . . Vector spaces MWX . . . Cell complexes * Use ASX Y. MWY . . . . CW-complexes . . By system MX . . Almost complex structures SLS . . . Loop spaces ATY C . . Polytopes SW . . . Analytic spaces V Neighbourhoods * See Topology AVL . . Special types AUB A . . . Retract spaces AUA Spaces in geometry B . . . . Neighbourhood retract spaces * These normally imply 3-dimensional or higher C . . . Extensor spaces dimensional entities, but not necessarily (e.g. in topology). D . . . . Neighbouring extensor spaces . Types DN4 Q . . . . . Absolute neighbouring extensor spaces . . By method E . . . H-spaces I4 . . . Classical spaces F Subspaces . . By operation . Subsystems JC . . . Classifying spaces FFR R . . Lattices of subspaces . . By relation H Geometric structures by other characteristics * For structures by dimension, see ATU. L9 . . . Projective spaces * Add to AUB letters H/OD from Auxiliary * See also Affine and projective spaces Schedule AM1. together AUA OKY. NLH . Infinitesimal structures . . By property NP . Ordered structures MR . . . Normal spaces OCT . Concave structures N4Q . . . Absolute spaces . . Types by system N6Y . . . Spaces of dimension less than or equal to OCT TYC . . . Concave polytopes one OD . Convex N8H . . . N-dimensional spaces, multidimensional * For convex structure, use AUC. spaces * Normal synthesis by Auxiliary Schedule AM1 is NC . . . Quadratic spaces interrupted here; it is resumed at AUD OE. NOG . . . Homogeneous spaces OEP . . . Spaces with parallelism OEP N4Q . . . . Spaces with absolute parallelism OF . . . Biaxial spaces OJ . . . Non-Euclidean spaces

38 AUC Geometry AUGOQBFUAPANAQ9

Mathematics AM Mathematics AM Mathematical systems AR5 Mathematical systems AR5 Geometry ATS Geometry ATS Geometric structures by other characteristics AUB H G-structures AUE G . Convex AUB OD . . Almost quaternion G-structures AUE GRK RX AUC . Convex structures 3A . . Convex structures theory Associations of structures . . Relations * For loops, see Groups, ASL S. AUE J . Configurations 9N . . . Inequalities * See also Extremum problems AUC E2. K . . Figure arrangements 9V . . . Covering L . . Blocks . . Properties N . Families CD . . . Convexity P . Bundles * For Fibre and block bundles, see CDO X . . . . Hyperbolic convexity Fibre bundles AUF D. . . Elements Q . . Microbundles DF . . . Boundary S . Sheaves * For Convex surfaces see AUC TVS. AUF B . Fibres E2 . . . Extrema C . . Fibre spaces . . Types of convex structures by system D . . Fibre bundles . . . Sets . . . By entity RB . . . . Convex sets DQH . . . . Vector bundles RBE S . . . . . Invariants of convex sets E . . . Block bundles . . . Geometry F . Fibrations TUU . . . . Convex curves H . Foliations TVF . . . . Convex polygons J . Maps TVS . . . . Convex surfaces . . Types by property TWB . . . . Convex bodies * See also Solid bodies ATW B; Solid JNL RMX . . . Almost open maps geometry AUI 2FT WB M . Nets & webs together . . . . . Subsystems N . . Nets TWB FRR ...... Lattices & convex bodies P . . Webs TYC . . . . Convex polytopes . . . Methods AUD OE Geometric structures by other characteristics P6R S . . . . Algebraic questions * Normal synthesis is resumed here after its Q . Knots & links interruption at AUB OD. * Add to AUD letters OE/W form Auxiliary AUG . Manifolds Sxhedule AM1. * See also Topology of manifolds OEP . Structures with parallelism AVO G. RD . Combinatorial structures in geometry . . Types AUE G G-structures . . . By operation . Types by element/entity JS . . . . Differentiable manifolds GPM . . Product G-structures . . . By relations GPM X . . Almost product G-structures LR . . . . Generalized manifolds . Types by system . . . By property GRK R . . Quaternion G-structures OQB . . . . Riemannian spaces GRK RX . . Almost quaternion G-structures . . . . . Relations OQB 8KO M ...... Conformal mappings . . . . . Subsystems OQB FUA PA ...... Spaces with elements OQB FUA PAN A ...... Spaces with linear elements ...... Types by elements OQB FUA PAN AQ9 ...... Spaces with linear elements with a connection

39 AUGOQBNTN AUITG Geometries

Geometry ATS Mathematical systems AR5 Special subsystems Geometry ATS . . Manifolds AUG Geometries AUH Y . . . . By property Elementary geometry AUI 2 ...... Subsystems . Operations ...... Spaces with linear elements with a . . . Theory of geometric constructions AUI 27D 3A connection AUG OQB FUA PAN AQ9 . Subsystems AUI 2FT UW . . Angles ...... Types of Riemannian spaces * For Trigonometry, see AVE. AUG OQB NTN ...... Complete Riemannian spaces . . Planes . . . . By system 2FT VB . . . Plane geometry TX . . . . . Complex structures 2FT VQ . . . . Circle geometry TXM W ...... Manifolds with complex structures . . Solids ...... Elements 2FT WB . . . Solid geometry TXM WE9 ...... Connections on manifolds with . . . . Spheres complex structures 2FT WT . . . . . Spherical geometry UEG ...... Manifolds with G-structures 2FU A . . Spaces ...... Relations 2FU AN8 H . . . Elementary geometry in UEG AC ...... Automorphisms of manifolds multidimensional spaces with G-structures 4 Classical geometry, euclidean geometry AUH B . . . . Submanifolds * For Non-Euclidean geometries, see AVB OJ...... Types by property . Relations BQC R ...... Critical submanifolds 49I . . Congruence S . . Spectral sequences . Subsystems . . . Types by property 4FU AOI . . Euclidean spaces SP2 . . . . Eilenberg-Maclane spectral sequences . . . Relations Types of geometry 4FU AOI 9X . . . . Embedding . Types Y Geometries 4OI . . Euclidean geometries * The interruption of normal synthesis which began 4OK . . Pseudo-Euclidean geometries at ATS F is continued here. Synthesis by . . . Subsystems Auxiliary Schedule AM1 is resumed at AUL. 4OK FUA . . . . Pseudo-Euclidean spaces * Add to AUH Y numbers and letters following H 5 Non-classical geometry (general) in Auxiliary Schedule AM1 (introducing classes 6 Descriptive geometry AM3/AM5). . Subsystems AUI . By method 6FT UT . . Descriptive geometry of curves & surfaces * Add to AUI numbers and letters 2/W following I in Auxiliary Schedule AM1 (introducing . . . Processes classes AM6 2/AM6 W). 6FT UT8 F . . . . Generation 2 . . Elementary geometry . . . . . Types by property * Theoretically, all the special structures, etc. at 6FT UT8 FOS ...... Kinematic generation ATT/AUH could appear here, qualifying 6FU AOJ . . Descriptive geometry of Non-Euclidean elementary geometry by normal retroactive spaces synthesis. But in pratice most of them are . Types of descriptive geometry seldom, if ever, considered at this level. Class . . By property here only those works which reflect truly 6MO . . . Abstract descriptive geometry elementary geometry in method. * An alternative (not recommended) is to locate 6N8 H . . . Multidimensional descriptive geometry only general works on elementary geometry 6NB . . . Non-linear descriptive geometry at ATS 62 and distribute the subclasses 6OK Y . . . Affine & projective descriptive geometry under the objects, etc. concerned (e.g. plane QX Modular geometry geometry under Planes). RD Combinatorial geometry . . . Operations TG Algebraic geometry 27D . . . . Constructions * Alternative (not recommended) for libraries 27D 3A . . . . . Theory of geometric constructions wishing to locate this under geometry. It is preferred under algebra, at ATG. * If this option is taken, proceed as follows: Add to AUI T letters G/Q following AT at ATG/ATQ.

40 AUIW Geometries AUMEI

Mathematics AM Mathematics AM Mathematical systems AR5 Mathematical systems AR5 Geometry ATS Geometry ATS Geometries AUH Y Geometries AUH Y By method AUI Geometries by other methods, operations AUL . Algebraic geometry AUI TG * Add to AUL letters J7/JR from AUI W . Analytic methods Auxiliary Schedule AM1. * For Analytic geometry, use AUJ. Normal AUL JR . Differentiation synthesis is interruptedd here; it is resumed * For differential geometry, use AUM. at AUL. * Normal synthesis by Auxiliary Schedule AM1 is interrupted here; it AUJ . Analytic geometry is resumed at AUW. . . Subsystems FTL . . . Analytic geometry of algebraic AUM . Differential geometry varieties . . Methods FTL NA . . . . Analytic geometry of linear varieties 64 . . . Classical differential geometry FTU T . . . Analytic geometry of curves & surfaces . . . . Subsystems FTU UNC . . . . Curves of second order 64F TUL . . . . . Differential line geometry FTU URX . . . Algebraic curves 6RS . . . Algebraic methods FTU URX NDH . . . . Algebraic curves of higher order 6RS EI . . . . Tensor algebra FTV B . . . Planes . . . . . Elements FTV BOI . . . . Analytic geometry in the Euclidean 6RS EIE S ...... Invariants & concomitants plane of tensors FUA . . . Space 6W . . . Analytic methods FUA OI . . . . Euclidean * See also Vector analysis AUM * For Euclidean spaces, use AUK. EH; Tensor analysis AUM EI. Normal synthesis is interrupted here; . . Relations it is resumed at AUK X. 8K . . . Mappings AUK . . . . Euclidean spaces 93 . . . Forms . . . . . Subsystems 93O CN . . . . Exterior forms FTU N ...... Analytic geometry of straight . . . . . Methods lines 93O CN6 W ...... Calculus of exterior forms . . . . . Types of Euclidean spaces ...... Relations N8F ...... Three dimensional Euclidean 93O CN6 W9M E ...... Differential equations spaces ...... Relations ...... Subsystems 93O CN6 W9M E9E ...... Extensions of N8F FTU N ...... Straight lines systems of N8F FTV B ...... Planes differential N8F FTV S ...... Surfaces equations N8F FTV SNC ...... Surfaces of second order ...... Subsystems N8H ...... Analytic geometry of 93O CN6 WFT A ...... Differential algebras multidimensional . . . . . Types Euclidean spaces 93O CNJ R ...... Exterior differential forms X . . Other subsystems in analytic spaces . . Entities * Normal synthesis is resumed here after EH . . . Vectors & vector analysis its interruption at AUJ FUA OI. . . . . Subsystems * Add to AUK X letters UAO J/W in EHF UA . . . . . Spaces Auxiliary Schedule AM1. EHF UAQ 9 ...... Spaces with a connection Y . . Types of analytic geometry EHY . . . Spinors, spinor analysis . . . By system . . . . By system YTV B . . . . Plane analytic geometries EHY RS . . . . . Spinor algebra YTW B . . . . Solid analytic geometries . . . Tensors EI . . . . Tensor analysis, absolute AUL Geometries by other methods, operations differential calculus * Normal synthesis is resumed here after its * See also Differential calculus original interruption at ATS F and continued AW7 2. interruption at AUH Y/AUK.

41 AUMEIES AUNOM8KQBNXD Differential geometry

Mathematical systems AR5 Geometry ATS Geometry ATS Geometries AUH Y Geometries AUH Y Differential geometry AUM Differential geometry AUM Spaces in differential geometry AUN Entities Biaxial spaces AUN OF . . Tensor analysis, absolute differential calculus . . Surfaces AUN OFF TVS AUM EI AUN OI Euclidean spaces . . . Entities . Relations . . . . Invariants OI8 K . . Mappings AUM EIE S . . . . . Differential invariants & OI8 KQB . . . Pointwise mappings of Euclidean concomitants of tensors spaces . . . By system . Subsystems EIR S . . . . Tensor algebra OIF TUM . . Straight lines & planes OIF TUM UEN . . . Families of straight lines & planes Subsystems of differential geometry OIF TUT . . Curves & surfaces FTA . Differential algebras . . . Types by property OIF TUT NN . . . . Regular curves & surfaces OIF TUT NOB . . . . . Non-regular curves & surfaces FTT . Geometric structures OIF TUT UEN . . . . . Families of curves & surfaces . . Relations OIF TUU . . . . . Curves FTT 9E . . . Extensions of geometric objects OIF TUU N8D ...... Plane curves . . . Types by operation OIF TVS . . . . . Surfaces FTT JR . . . . Differential geometric objects OIF TVT ...... Surface strips . . . . . Relations OIF UFM . . Nets & webs FTT JR9 S ...... Representions of differential . . . Surfaces geometric objects OIF UFM FTV S . . . . Nets & webs on surfaces in FTU U . . Curves in differential geometry Euclidean space FTV S . . Surfaces . Types FTV SNX Y . . . Minimal surfaces OIN 8H . . Euclidean N-space FUA . . Spaces * For spaces in differential geometry, . . . Subsystems use AUN OIN 8HF TVS . . . . Surfaces * Normal synthesis using Auxiliary OIN 8HF TVS N8H . . . . . Higher dimension surfaces in Schedule AM1 is interrupted here; it Euclidean space is resumed at AUO. OJ Non-Euclidean spaces . Subsystems AUN . . Spaces in differential geometry OJF TUM . . Straight lines & planes . . . Types by property OJF TUM UEN . . . Families of straight lines & planes NA . . . . Linear spaces OJF TUT . . Curves & surfaces . . . . . Subsystems OJF TUT UEN . . . Families of curves & surfaces NAF SA ...... Groups OJF TUU . . Curves NAF SCM QP ...... Linear spaces with OJF TVT . . Surface strips fundamental groups OK Pseudo-Euclidean spaces OF . . . . Biaxial spaces . Relations * Differential geometry in biaxial OK8 K . . Mappings spaces. OK8 KQB . . . Pointwise mappings of pseudo- . . . . . Subsystems Euclidean space OFF TUM ...... Straight lines & planes . Subsystems OFF TUM UEN ...... Families of straight lines & OKF TUU . . Curves planes OM Conformal spaces in differential OFF TUM UEN 3A ...... Theory of families of geometry straight lines & . Relations planes OM8 K . . Mappings OFF TUU ...... Curves OM8 KQB . . . Pointwise mappings of conformal OFF TVS ...... Surfaces spaces OM8 KQB NXD . . . . Dual pointwise mappings of conformal spaces

42 AUNOMFTUU Differential geometry AUQOQB8K

Differential geometry AUM Mathematical systems AR5 Subsystems of differential geometry Geometry ATS . . . Types by property Geometries AUH Y . . . . Conformal spaces in differential geometry Differential geometry AUM AUN OM Webs . . . . . Relations . . . . Differential topological groups AUO FPF SHJ R ...... Dual pointwise mappings of conformal spaces AUO G Manifolds in differential geometry AUN OM8 KQB NXD * See also Differential topology of manifolds AVQ...... Subsystems AUN OMF TUU ...... Curves . Types by operation OMF TUU 3A ...... Conformal theory of curves GJS . . Differentiable OMF TVJ ...... Circles & spheres * For Differentiable manifolds use AUP. OMF TVJ UEN ...... Families of circles & * Normal synthesis by Auxiliary Schedule AM1 spheres is interrupted here; it is resumed at AUT (for other types of manifolds) and at AUU Y (for OMF TVT ...... Surface strips other subsystems in differential geometry). OMF TVT 3A ...... Conformal theory of surface strips OP . . . . Symplectic spaces AUP . . Differentiable manifolds in differential . . . . . Subsystems geometry, smooth manifolds in OPF TUM ...... Straight lines & planes differential geometry * See also Lie groups, ASJ; Smooth manifolds, OPF TUM UEN ...... Families of straight lines & AVO GNM Riemannian geometries AVD ONJ planes RP2. OPF TUM UEN 3A ...... Symplectic theory of . . . Methods families of 6Y . . . . Calculus of variations straight lines & . . . Relations planes 93 . . . . Forms OPF TUU ...... Curves 93O CN . . . . . Exterior differential forms on manifolds OPF TUU 3A ...... Symplectic theory of curves 9L . . . . Equations OPF TVS ...... Surfaces 9MG . . . . . Geometry of partial differential equations OPF TVS 3A ...... Symplectic theory of . . . Properties surfaces BF . . . . Local properties . . . Types of spaces by elements & . . . Elements systems DW . . . . Geometry of the integral PA . . . . Spaces of elements . . . Subsystems PAN A . . . . . Spaces of linear elements FSV . . . . Fields on manifolds SA . . . . . Groups FSV 9E . . . . . Extensions of fields on manifolds SCM QP ...... Spaces with fundamental . . . . Spaces groups FUA . . . . Manifolds by underlying spaces . . . . . Types by method AUO Other subsystems in differential geometry FUA I4 ...... Classical * Normal synthesis resumed after * For classical spaces, use AUQ. interruption at AUM FUA. * Normal synthesis by Auxiliary * Add to AUO letters B/G following AU Schedule AM1 is interrupted here; it is in AUB/AUG. resumed at AUR I5 (for other . Webs underlying spaces) and at AUS (for FP . . Geometry of webs other subsystems in differentiable . . . Subsystems manifolds). FPF SH . . . . Topological groups AUQ ...... Classical spaces (manifolds) FPF SHJ R . . . . . Differential topological groups ...... Types by relation L9 ...... Projective Euclidean spaces ...... Types by property OQB ...... Riemannian spaces ...... Relations OQB 8K ...... Mappings of Riemannian spaces

43 AUQOQBFSCMI AUSFHFUA7C Differential geometry

Differential geometry AUM Differential geometry AUM Manifolds in differential geometry AUO G Manifolds in differential geometry AUO G Types by method Subsystems . . Types by property . . Underlying spaces by other methods operations etc . . . . Relations AUR I5 . . . . . Mappings of Riemannian spaces . . . . Subsystems AUQ OQB 8K ...... Invariant infinitesimal structures in homogeneous spaces . . . . Subsystems AUR NOG FUB NLH O2J AUQ OQB FSC MI . . . . . Holonomy groups of . . . . Types by relation Riemannian spaces AUR NOG LWM . . . . . Manifolds immersed in OQB FTT . . . . . Objects in Riemannian spaces homogeneous spaces OQB FTT O2J ...... Invariant objects in . . By elements Riemannian spaces . . . Linear elements OQB FUG . . . . . Riemannian manifolds PAN A . . . . Differentiable manifolds in spaces OQB FUH B ...... Submanifolds of Riemannian with linear elements spaces . . . . . Elements OQB FUH BFT UU ...... Curves PAN AE8 ...... Additional structures in general OQB FUH BP2 ...... Einstein spaces . . . . . Subsystems . . . . Types of Riemannian spaces PAN AFU BNL H ...... Infinitesimal structures in a OQB NOJ . . . . . Symmetric Riemannian spaces space of linear OQD . . . . . Pseudo-Riemannian spaces elements . . Types of classical spaces by element PAN AFU HB ...... Submanifolds in spaces of Q9 . . . Classical spaces with connections linear elements Q9L 9 . . . . Spaces with a projective AUS Other subsystems of differentiable connection manifolds Q9O K . . . . Spaces with a pseudo-Euclidean * Normal synthesis is resumed after connection interruption at AUP FUA I4. Q9O L . . . . Spaces with an affine connection * Add to AUS letters BA/L following AU . . . . . Relations and letters UM/W following A. Q9O L8K ...... Mappings EG . G-structures on differentiable manifolds Q9O L8K OCP ...... Geodesic mappings . . Types by property . . . . . Types by property EGO P . . . Symplectic G-structures Q9O LNO J ...... Symmetric classical spaces EP . Bundles with affine EPT UP . . Tangent bundles connection FC . Fibre spaces Q9O P . . . . Spaces with a symplectic . . Elements connection FCE 9 . . . Connections in fibre spaces AUR I5 Underlying spaces by other methods . . . . Types by operation operations etc FCE 9L9 . . . . . Projective connections * Normal synthesis is resumed here after FCE 9NA . . . . . Linear connections interruption at AUP FUA I4. FCE 9NB . . . . . Non-linear connections * Add to AUR letters I5/Q from Auxiliary . . Subsystems Schedule AM1 and R/W following A in FCF SA . . . Groups AR/AW. FCF SCM I . . . . Holonomy groups of fibre spaces NOG . Homogeneous spaces FCF TT . . . Geometric objects . . Elements FCF TTF SV . . . . Fields of geometric objects in fibre NOG E9 . . . Connections in homogeneous spaces space FCF UHB . . . Submanifolds in fibre spaces NOG E9O 2J . . . . Invariant connections in FF . Fibrations on manifolds homogeneous spaces FH . Foliations on manifolds . . Subsystems . . Subsytems NOG FSC MI . . . Holonomy groups of homogeneous FHF UA . . . Spaces spaces FHF UA7 C . . . . Classifying spaces for foliations on NOG FUB NLH . . . Infinitesimal structure manifolds NOG FUB NLH O2J . . . . Invariant infinitesimal structures in homogeneous spaces

44 AUSX Differential geometry AUWL9

Geometries AUH Y Mathematical systems AR5 Differential geometry AUM Geometry ATS Manifolds in differential geometry AUO G Geometries AUH Y Types by operation Geometries by other methods, operations AUL . . Other subsystems of differentiable manifolds . . Other subsystems in differential geometry AUO AUS ...... Non-regular submanifolds ...... Classifying spaces for foliations on AUU XUH BNO B manifolds AUS FHF UA7 C AUU Y . . Other subsystems in differential geometry * Normal synthesis is resumed here after AUS X . . Types of differentiable manifolds interruption at AUP. * Normal synthesis resumed here after * Add to AUV letters UH/W following A. interruptions beginning at AUP FUA I4. * Add to AUS X letters H/W from AUV . . Types of differential geometry Auxiliary Schedule AM1. * See also Projective differential geometry AUT Types of manifolds by other operations etc AUX JR; % Intrinsic differential * Other than differentiable. geometry AVB MS; Affine differential * Normal synthesis is resumed here after geometry AVC OL; Metric differential interruption at AUO GJU. geometry AVD ON. * Add to AUT letters JV/Q from Auxiliary . . . By operation Schedule AM1 and R/W following A in J7 . . . . Synthetic differential geometry AR/AW. . . . By property MIX . Non-holonomic manifolds NF . . . . Local differential geometry . . Subsystems . . . . . Properties MIX FUA OI . . . Euclidean space NFC CR ...... Curvature MIX FUA OJ . . . Non-Euclidean space NFC N ...... Metrics MIX FUA OK . . . Pseudo-Euclidean space NFC NP2 ...... Lorentz metrics Types of manifolds by system . . . . . Subsystem TXM W . Complex structures NFF UHB ...... Local submanifolds * For manifolds with complex structures, NI . . . . Global differential geometry, integral use AUU. Normal synthesis is interrupted geometry here; it is resumed at AUU X...... Properties AUU . Manifolds with complex structures NIC CR ...... Curvature LWM . . Immersed manifolds . . . . . Subsystems . . . Subsystems NIF TUR ...... Geodesics LWM FTT . . . . Geometric objects NIF TUU ...... Curves LWM FTT MQP . . . . . Fundamental objects NIF TUU NXY ...... Minimal curves LWM FTT OCH . . . . . Geometric objects invariantly NIF UA ...... Spaces connected to immersed NIF UAN OJ ...... Symmetric spaces manifold NIF UFC ...... Fibre spaces OO . . Metrizable manifolds NIF UG ...... Manifolds . . . Subsystems NIF UGP 2 ...... Lorentz manifolds OOF TUR . . . . Geodesics NIF UHB ...... Submanifolds in the large OOF UHY . . . . Geometries AUW Types of geometries by other operations etc OOF UIP 2 . . . . . Minkowski geometries * Normal synthesis is resumed here after P2 . . Hermitian manifolds interruption at AUL JR. P3 . . Kahler manifolds * Add to AUW letters JV/L8 from Auxiliary X Types of manifolds by other characteristics Schedule AM1 (i.e. specifiers drawn from * Normal synthesis is resumed here after its AM7V/98). interruption at AUT TXM W. JV . Integration * Add to AUU X letters TYF/W in Auxiliary * For Integral geometry, see Global Schedule AM1. differential geometry AUV NI. XUH B . Submanifolds in differential geometry L9 . Projection . . Subsystems * For Projective geometry, use AUX. XUH BFT UT . . . Curves & surfaces * Normal synthesis by Auxiliary Schedule . . Types by property AM1 is interrupted here; it is resumed at XUH BNO B . . . Non-regular submanifolds AUY.

45 AUX AVBNXD Geometries

Mathematics AM Mathematics AM Mathematical systems AR5 Mathematical systems AR5 Geometry ATS Geometry ATS Geometries AUH Y Geometries AUH Y Types of geometries by other operations etc AUW Types of geometries by other relations AUY . Projection AUW L9 Types by property AUX . Projective geometry AUY MO . Abstract geometries . . Methods MOO EP . . Abstract geometries with parallelism 6RS . . . Algebraic theory of projective MS . Intrinsic geometry geometries . . Subsystems . . Relations MSF TVS . . . Intrinsic geometry of surfaces 9HE . . . Geometry of projective . . Types by operation correspondences MSJ R . . . Intrinsic differential geometry . . Subsystems NA . Linear geometry FRR . . . Lattices . . Types by relation FRR 3A . . . . Lattice theory NAL WI . . . Incidence FTL . . . Algebraic varieties * For linear incidence geometry, use AVA. FTL NA . . . . Geometry of linear varieties * Normal synthesis by Auxiliary Schedule . . . Spaces AM1 is interrupted here; it is resumed at FUA . . . . Projective spaces AVA X...... Subsystems AVA . . . Linear incidence geometry FUA FTO B ...... Algebraic subvarieties . . . . Relations ...... Geometries AC . . . . . Automorphism in linear incidence FUA FUJ ...... Analytic geometry in projective geometry spaces . . . . Properties FUA FUL J7 ...... Synthetic geometry in BXD . . . . . Duality projective spaces . . . . Subsystems . . . . . Types of projective spaces FUD OEP . . . . . Structures with parallelism FUA NJ ...... Finite projective spaces FUE J . . . . . Configurations ...... Subsystems FUE J3F ...... Configuration theorems FUA NJF RD ...... Combinatorial structures in . . . Types by property finite projective P2 . . . . Lie geometries spaces . . . . . Subsystems FUE J . . . . Configurations P2F TWT ...... Lie geometries of spheres . . Types of projective geometry by method X . . Other types of linear geometry IW . . . Analytic projective geometry * This resumes normal synthesis after the . . Types by operation interruption at AUY NAL WI. * Add to AVA X letters LWM/W from JR . . . Differential projective geometry Auxiliary Schedule AM1. . . . . Subsystems AVB Types of geometries by other properties etc JRF TUM . . . . . Straight lines & planes * This resumes normal synthesis after the JRF TUM UEN ...... Families of straight lines & interruption at AUY NAL W and partial planes resumption at AVA X. JRF TUT . . . . . Curves & surfaces * Add to AVB letters NB/OK from Auxiliary JRF TUT UEN ...... Families of curves & surfaces Schedule AM1. JRF TUU ...... Curves NB . Non-linear geometry JRF TVS ...... Surfaces . . Types by relation JRF TVT ...... Surface strips NBL WI . . . Non-linear incidence geometry JRF UG . . . . . Manifolds NJ . Finite geometry JRF UGM IX ...... Non-holonomic manifolds . . Subsystems AUY Types of geometries by other relations NJF TVB . . . Plane * Normal synthesis by Auxiliary Schedule NJF TVB L9 . . . . Projective plane AM1 is resumed here after its interruption at NJF UEL . . . Block designs AUW L9. . . Types by property * Add to AUY letters LAD/NA from Auxiliary NJN B . . . Finite non-linear geometries Schedule AM1 (i.e. specifiers drawn from NP . Ordered geometries 9AD/ANA). NV . Continuous geometries NXD . Dual geometries

46 AVBOF Geometries AVDTUW

Mathematics AM Mathematics AM Mathematical systems AR5 Mathematical systems AR5 Geometry ATS Geometry ATS Geometries AUH Y Geometries AUH Y Dual geometries AVB NXD Types of geometries by other properties etc AVB ...... Non-holonomic manifolds AVB OF Biaxial geometries AVC JRF UGM IX . Types by property AVD Types of geometries by other properties etc OFO V . . Elliptic biaxial geometry * Normal synthesis resumed here after OFO X . . Hyperbolic biaxial geometry interruption at AVB OL. OI Euclidean geometries * Add to AVD letters OM/W in Auxiliary OJ Non-Euclidean geometries Schedule AM1. . Subsystems Types of geometries by property . . Spaces OM . Conformal geometry OJF UA . . . Non-Euclidean spaces ON . Metric geometry . . . Relations . . Relations OJF UA9 X . . . . Embedding ON9 I . . . Congruence OK Pseudo-Euclidean geometry . . Types by operation . Subsystems ONJ R . . . Metric differential geometry OKF UA . . Pseudo-Euclidean spaces ONJ RP2 . . . . Riemannian geometries OL Affine ONJ RP3 . . . . Minkowski geometries * For Affine geometry, use AVC. ONO IJR . . . . Euclidean metric differential geometry * Normal synthesis by Auxiliary Schedule ONO J . . . . Non-Euclidean metric geometries AM1 is interrupted here; it is resumed at OP . Symplectic geometry AVD. . . Subsystems OPF UA . . . Symplectics spaces AVC Affine geometry . . Types . Methods OPL 9 . . . Projective symplectic geometry 6RS 3A . . Algebraic theory of affine geometry OPO L . . . Affine symplectic geometry . Properties OS . Kinematic geometry CS . . Affine kinematics . . Relations . Subsytems OS8 K . . . Kinematic mappings FTL . . Varieties . . Subsystems FTL NA . . . Geometry of linear varieties OSF TVB . . . Kinematic geometry on a plane FTT . . Geometric structures OSF UA . . . Kinematic geometry in space FTU T . . . Curves & surfaces OSF UAN 8H . . . . Multidimensional spaces FTU TNC . . . . Second order curves & surfaces OSF UAO J . . . . Non-Euclidean spaces FUA . . . Affine spaces OV . Elliptic geometries . . . . Subsystems * For Riemannian geometries, see AVD ONJ FUA FUJ . . . . . Analytic geometry in affine spaces RP2. FUA FUL J7 . . . . . Synthetic geometry in affine spaces OX . Hyperbolic geometries . . . . Types TUW . By angles FUA Q9 . . . . . Spaces with affine connection * For trigonometry use AVE. . Types by method * Normal synthesis using Auxiliary Schedule AM1 is interrupted here; it is resumed at IW . . Analytic affine geometry AVE Y . Types by operation JR . . Differential affine geometry . . . Subsystems JRF TUM . . . . Straight lines & planes JRF TUM UEN . . . . . Families of straight lines & planes JRF TUT . . . . Curves & surfaces JRF TUT UEN . . . . . Families of curves & surfaces JRF TUU . . . . . Curves JRF TVS . . . . . Surfaces JRF TVT ...... Surface strips JRF UG . . . . Manifolds JRF UGM IX . . . . . Non-holonomic manifolds

47 AVE AVKOIELYOT Topology

Mathematics AM Mathematics AM Mathematical systems AR5 Mathematical systems AR5 Geometry ATS Topology AVJ . . Types of geometries by property Entities . . . By angles AVD TUW . . Dimension type invariants AVJ ESN 4V

AVE . . . Trigonometry Subsystems . . . . Relations AVJ FRX . Algebraic systems 8L . . . . . Trigonometric functions . . Relations . . . . Systems by geometric structure FRX 9S . . . Topological representations TVB . . . . . Plane trigonometry FSX . . Categories TVB W ...... Plane analytic trigonometry FSX MQP . . . Fundamental categories TWB . . . . . Solid trigonometry FSX TWC . . . Polyhedral categories TWB W ...... Solid analytic trigonometry FSX TXM WY . . . . CW-complexes TWT ...... Spherical trigonometry . . . . . Operation Y . . Geometries by other systems FSX TXM WY7 D ...... Construction over * Normal synthesis is resumed after CW-complexes interruption at AVD TUW. * Add to AVE Y letters UY/W from Auxiliary Schedule AM1 so far as FTT . Geometric structures applicable. FUA . . Spaces * For Topology of spaces use AVK. * Normal synthesis by Auxiliary AVJ Topology Schedule AM1 is interrupted here; it is . Methods resumed at AVO. 6RS . . Algebraic techniques . Processes AVK . . Topology of spaces, point set topology 8GS . . Separation . . . Properties 8GS 3D . . . Separation axioms CCH . . . . Connectedness . Relations CCH X . . . . . Path connected, pathwise 8K . . Mappings connected, path spaces 8KN V . . . Continuous mappings . . . Types by relation 8KN V9R . . . . Generalizations of continuous LD . . . . Derived spaces mappings . . . Types by property 9V . . Coverings N6Y . . . . Spaces of dimension less than or . Properties equal to one . . Dimensions . . . . . Subsytems B4V . . Topological dimensions N6Y FTU U ...... Curves * See also Hausdorff dimension, AWR YN5 B8U NOD . . . . . Uniform spaces B4V 3A . . . Dimension theory NOD 3D ...... Axioms of uniform spaces BLD . . . Infinite dimensions NQB . . . . Partially ordered spaces BLD 3A . . . . Theory of infinite dimensions NR . . . . Compact spaces BR . . Compactness . . . . . Properties BRN F . . . Local compactness NRB 4V ...... Dimension CCH . . Connectedness NRB 4V3 A ...... Dimension theory of compact CCH NF . . . Local connectedness spaces CCH NFN 8H . . . . Higher dimensional local OB . . . . Proximity spaces, proximity connectedness topology CCJ . . Shape OB3 D . . . . . Axioms of uniform & proximity CCJ 3A . . . Shape theory spaces . Elements . . . . . Relations ECN . . Fixed points OB9 E ...... Extensions ECP . . Coincidence points OB9 ENR ...... Compact extensions . Entities OI . . . . Euclidean spaces ES . . Invariants . . . . . Entities ESN 4V . . . Dimension type invariants OIE LY ...... Continua OIE LYO T ...... Plane continua

48 AVKON Topology of spaces AVL8MGNEHDQNF

Mathematical systems AR5 Mathematics AM Topology AVJ Mathematical systems AR5 Topology of spaces AVK Topology AVJ Types by property Geometric structures AVJ FTT . Euclidean spaces AVK OI Topology of spaces AVK . . . . Plane continua AVK OIE LYO T Analysis AVK W

AVK ON . Metric spaces AVL Analytic spaces ON3 D . . Axioms of metric spaces . Processes . . Properties 87 . . Analytic continuation ONA N . . . Metric properties of metric spaces . Relations . . Types by property 8K . . Mappings ONN R . . . Compact metric spaces 8KM G . . . Holomorphic mappings . . . . Entities 8KM GAK . . . . Cohomology of holomorphic ONN REL Y . . . . . Continua mappings ONN TN . . . Complete metric spaces, complete 8L . . Functions spaces . . . Types by operation OO . . . Metrizable spaces 8LJ S . . . . Differentiable functions on Types of spaces by element analytic spaces PYX . Spaces with richer structures . . . Types by relation PYX B5 . . Measures 8MC . . . . Automorphic functions PYX B5X . . . Baire categories . . . . . Properties PYX X . . Sigma spaces 8MC BEC ...... Automorphic functions with complex variables 8MC BEE ...... Automorphic functions with Types of spaces by system one complex variable SLS . Loop spaces . . . . . Elements . . Relations 8MC DC ...... Domains SLS 9WS . . . Suspensions over loop spaces 8MC DCN OJ ...... Symmetric domains VJ . Topological spaces . . . . . Subsystems . . Operations 8MC FTT ...... Geometric structures VJ7 D . . . Construction of topological spaces 8MC FTV R ...... Automorphic functions in the . . Relations disc VJ9 B . . . Compactification 8MG . . . . Holomorphic functions VJ9 E . . . . Extensions . . . . . Processes VJ9 ENR . . . . . Compact extensions 8MG 86 ...... Approximation of holomorphic . . Subsystems functions VJF RB . . . Sets 8MG 863 F ...... Approximation theorems for . . . . Methods holomorphic VJF RB6 6 . . . . . Descriptive set theory of topological functions spaces . . . . . Elements VJF RY . . . Semigroups 8MG DF ...... Boundaries VJF RYD X . . . . Relations as elements 8MG DFN I ...... Global boundaries VJF RYD X95 . . . . . Topological semigroups of . . . . . Subsystems transformations of 8MG FRX ...... Algebras of holomorphic topological spaces functions . . Types of topological spaces by property . . . . . Types by Property VJN P . . . Ordered topological spaces 8MG NEC ...... Complex variables W . Analysis * For analytic spaces, use AVL. 8MG NEH ...... Holomorphic functions of * Normal synthesis by Auxiliary Schedule several complex AM1 is interrupted here; it is resumed at variables AVO...... Relations 8MG NEH 9S ...... Representation 8MG NEH 9SJ V ...... Integral representation ...... Elements 8MG NEH DQ ...... Residues 8MG NEH DQN F ...... Local theory

49 AVL8MGNEHFUEN AVLFUHB97 Topology of spaces

Topology of spaces AVK Mathematics AM Relations Mathematical systems AR5 . Functions AVL 8L Topology AVJ . . Types by relation Topology of spaces AVK . . . Holomorphic functions AVL 8MG Elements ...... Elements . . . Local singularities AVL ECG NF ...... Local theory AVL 8MG NEH DQN F Subsystems of analytic spaces . Sets ...... Subsystems AVL FRB . . Analytic sets AVL 8MG NEH FUE N ...... Families FRX . . Algebraic structures 8MG NEH FUE NMR ...... Normal families . . . Relations 8MH . . . Meromorphic functions FRX 97 . . . . Deformations of algebraic . . . . Subsystems structures 8MH FSV . . . . . Fields of meromorphic FRX 973 A . . . . . Analytic theory of functions deformations of . . Types by property etc algebraic structures 8OV . . . Elliptic functions FSI Y . Pseudo-groups 8P3 . . . Non-Archimedean functions FSI YAK . . Cohomology 8QX . . . Modular functions FTT . Geometric structures 93 . Forms FUB F . . Subspaces 93M C . . Automorphic forms FUE G . . G-structures . Deformations FUE G97 . . . Deformations of G-structures 97 . . Deformations of structures FUE P . . Bundles 9R . Generalizations of analytic spaces FUE PQH . . . Vector bundles AK . Cohomology FUE PQH MG . . . . Holomorphic vector bundles . . Relations FUF C . . Fibre spaces AK9 3MC . . . Automorphic forms & . . . Relations cohomology FUF C97 . . . . Deformations of fibre spaces . . Types by property . . . Types by property AKN F . . . Local cohomology FUF CMG . . . . Holomorphic fibre spaces Properties of analytic spaces . . . . . Elements BF . Local FUF CMG E9 ...... Holomorphic connections in BF3 A . . Local theory fibre spaces BXD . Duality FUF D . . . Fibre bundles BXD 3F . . Duality theorems FUF DMG . . . . Holomorphic fibre bundles CD . Convexity FUG . . Manifolds CDM G . . Holomorphic convexity . . . Types by property CDT T . . Geometric convexity FUG MW . . . . Complex manifolds Elements . . . . . Subsystems DC . Domains FUG MWF UHB ...... Submanifolds . . Types by property FUG MWF UHB N3 ...... Real submanifolds in DCO CV . . . Pseudo-concave domains complex manifolds DCO EC . . . Pseudo-convex domains FUG MX . . . . Almost complex manifolds ECG . Singularities . . . . . Methods . . Relations FUG MX6 W ...... Analytic study of almost ECG 97 . . . Deformation of singularities complex manifolds . . Types by property FUG NOG . . . . Homogeneous manifolds ECG NF . . . Local singularities FUG NOG MW . . . . . Homogeneous complex manifolds FUG OEC . . . . Pseudo-convex manifolds . . Submanifolds FUH B . . . Analytic submanifolds . . . . Relations FUH B97 . . . . . Deformations of submanifolds

50 AVLJS Topology of spaces AVNX

Mathematics AM Topology AVJ Mathematical systems AR5 Topology of spaces AVK Topology AVJ Types of spaces by system Topology of spaces AVK . . . By property Subsystems of analytic spaces ...... Complex homogeneous spaces . . . . . Deformations of submanifolds AVM NOG AVL FUH B97 ...... Kahler homogeneous spaces AVM NOG P2 Types of analytic spaces . By operation AVM NTN ...... Complete analytic spaces AVL JS . . Differentiable spaces NTN MG ...... Holomorphically complete . By property spaces MW . . Complex OD ...... Convex analytic spaces * For Complex spaces use AVM. ODM G ...... Holomorphically convex spaces Normal synthesis by Auxiliary AVN . . . Analytic spaces by other characteristics Schedule AM1 is interrupted here; it * Normal synthesis by Auxiliary is resumed at AVN. Schedule AM1 is resumed here after its AVM . . Complex spaces interruption at AVL MW. . . . Operations * Add to AVN letters MX/WT in 7V . . . . Integration on analytic spaces Auxiliary Schedule AM1. . . . Relations N3 . . . . Real analytic spaces 8K . . . . Mappings . . . . . Properties 8KM G . . . . . Holomorphic mappings N3B F ...... Local properties of real analytic 97 . . . . Deformation of complex spaces spaces . . . Subsystems N3B I ...... Global properties of real analytic FSA . . . . Groups spaces FSC MC . . . . . Automorphism groups NR . . . . Compact analytic spaces FSC MCD X ...... Elements by relation . . . . . Relations FSC MCD XAC ...... Complex spaces with a NR9 B ...... Compactification group of . . . . . Subsystems automorphisms NRF TVS ...... Compact surfaces FSC MCD XAC P2 ...... Hermitian symmetric . . . . . Types spaces NRN OG ...... Homogeneous FSC MCD XAC PP ...... Quotient spaces NRN OGM W ...... Compact complex homogeneous FSJ . . . . . Lie groups spaces ...... Transformations NTL . . . . Partially analytic spaces FSJ L5 ...... Complex Lie P3 . . . . Non-Archimedean analytic spaces transformational P4S . . . . Serre analytic spaces groups Other types of spaces in topology . . . . Varieties WX . Probabilistic spaces FTL . . . . . Complex spaces close to X . Special types * algebraic varieties Add to AVN X letters A/F following AUB, e.g. Extensor spaces AVN XC. . . . . Geometric structures FTV S . . . . . Complex surfaces ...... Elements FTV SEC G ...... Singular points . . . Types of complex spaces MP . . . . Concrete complex spaces N4V . . . . Complex spaces of one or two or three dimensions N4V FTV S . . . . . Complex surfaces NOG . . . . Complex homogeneous spaces . . . . . Subsystems NOG FUE P ...... Homogeneous bundles NOG FUE PQH ...... Homogeneous vector bundles . . . . . Types by property NOG P2 ...... Kahler homogeneous spaces

51 AVO AVOGNMFUFF Topology

Mathematics AM Mathematical systems AR5 Mathematical systems AR5 Topology AVJ Topology AVJ Topology of manifolds AVO G Subsystems Subsystems . . . . Special types AVN X . Groups AVO GFS A . . Fundamental groups AVO GFS CMQ P AVO Other topological subsytems & structures AVO GFU A . Spaces * Normal synthesis is resumed here (in . . Types by relation modified form) after its interruption at GFU ALV . . . Covering spaces AVJ FUA and AVK W. It is continued GFU ANX D . . . Duality spaces in completely normal form (using GFU ANX DP2 . . . . Poincare duality spaces Auxiliary Schedule AM1) at AVP Types of manifold (Types of topology). . By property * Add to AVO letters BH/H following GMT C . . Characteristic manifolds AU in AUB H/AUH and letters UI/W following A in AUI/AW. GNL D . . Infinite-dimensional manifolds BNA . Linear structures (topology) GNM . . Smooth manifolds * See also Differentiable manifolds BNA LWI . . Linear incidence structures (differential geometry) AUP; ES . Sheaves (topology) Piecewise linear manifolds AVO FJ . Maps (topology) GRD . . Relations . . . Operations FJ8 L . . . Functions GNM 7C . . . . Classification FJ8 N3 . . . . Real valued functions * See also Equivalence of FJ9 X . . . Embedding polyhedra . . Properties AVO GNM FTW C9J RD FJC CJ . . . Shape . . . Relations FJC CJ3 A . . . . Shape theory . . . . Mappings . . Subsystems GNM 8K . . . . . Smooth mappings FJF RB . . . Sets ...... Relations FJF RCP 2 . . . . Baire sets & functions GNM 8K9 3 ...... Forms . Manifolds GNM 8K9 3JR ...... Differential forms on smooth mappings ...... Elements G . Topology of manifolds GNM 8KE CG ...... Singularities of smooth . . Theorems mappings G3F P2 . . . Poincare conjecture GNM 8KE CR ...... Critical points of smooth . . Relations mappings GAL Y . . . Isotopy . . . Subsystems . . Elements . . . . Groups GEU . . . Operators GNM FSA . . . . . Groups acting on smooth . . . . Types by operation manifolds GEU JR . . . . . Differential operators on . . . . Geometric structures manifolds GNM FTW C . . . . . Polyhedra GEU JV . . . . . Integral operators on manifolds ...... Relations ...... Subsystems GNM FTW C9J ...... Equivalence GEU JVF SV ...... Fields GNM FTW C9J RD ...... Combinatorial GEU JVF SVQ H ...... Vector fields equivalence of GEU JVF SVQ H7V ...... Integration polyhedra GEU JVF SVQ I ...... Tensor fields . . . . . Spheres GEU JVF SVQ I7V ...... Integration GNM FTW T ...... Smooth structures on balls & . . . . Types by property spheres GEU OV . . . . . Elliptic operators on manifolds GNM FUE Q . . . . . Microbundles . . Subsystems . . . . . Fibre bundles GFS A . . . Groups GNM FUF D ...... Fibre bundles with smooth GFS CMQ P . . . . Fundamental groups manifolds as bases GNM FUF F . . . . . Fibrations of smooth surfaces

52 AVOGNMMW Topologies AVQAKO2L

Topology AVJ Mathematics AM Other topological subsytems & structures AVO Mathematical systems AR5 . . . By property Topology AVJ . . . . Smooth manifolds AVO GNM Types of topology . . . . . Subsystems ...... Fibrations of smooth surfaces AVP Topologies AVO GNM FUF F * Concepts from facets other than the Types facet . . . . . Types of smooth manifolds (H/W in Auxiliary Schedule AM1) should not normally be applied to AVP, but to Topology ...... By property AVJ as the general class. But very occasionally AVO GNM MW ...... Complex smooth manifolds a concept relates to the notion of a topological GNM MX ...... Almost complex smooth system per se, when it may be used to qualify manifolds AVP, as in the example below (AVP 74). GNM NLD ...... Manifolds of infinite 74P . Operations on topologies dimension GNM P2 ...... Kahler topology, kahlerian . Types of topology by method manifolds IRD . . Combinatorial topology ...... By elements . . . Elements GNM Q8 ...... Smooth manifolds with IRD DA . . . . Combinatorial elements additional structure . . . Subsystems GOX . . . . Hyperbolic manifolds IRD FUE S . . . . Sheaves GOX MW . . . . . Hyperbolic complex manifolds IRD FUF D . . . . Fibre bundles * See also Smooth manifolds AVO IRD FUF Q GNM; PL manifolds AVO GRD . . . . Knots GP2 . . . . Riemannian manifolds IRS . . Algebra * For Algebraic topology, use AVQ. Normal . . . By system synthesis by by Auxiliary Schedule AM1 is . . . . Combinatorics interrupted here; it is resumed at AVU. GRD . . . . Combinatorial manifolds, piecewise linear topology, PL manifolds AVQ . . Algebraic topology, analysis situ . . . . . Operation . . . Methods GRD 7C ...... Classification 6RU . . . . Applied homological algebra . . . . . Processes . . . . . Properties GRD 86 ...... Approximation 6RU BXD ...... Duality . . . . . Relations . . . Relations GRD ALY ...... Isotopy 8K . . . . Mappings . . . . . Subsystems . . . . . Operations GRD FSA ...... Groups acting on combinatorial 8K7 C ...... Classification of mappings manifolds . . . . . Relations GRD FTY V ...... Neighbourhoods 8K9 AP ...... Compression of mappings GRD FTY VNN ...... Regular neighbourhoods 8K9 E ...... Extension of mappings GRD FUF Q ...... Knots & links 8KA K ...... Cohomology GVJ . . . . Topological manifolds 8KA K3B ...... K-theory . . . . . Subsystems 8KA LY3 A ...... Isotopy theory GVJ FUE P ...... Bundles of topological manifolds AK . . . . Cohomology GVJ FUE Q ...... Microbundles AK3 B . . . . . K-theory cohomology GVJ FUF E ...... Block bundles . . . . . Operations GVJ FUH Y ...... Geometries on topological AK8 4 ...... Stable operations manifolds . . . . . Properties Types of topology AKB 4V ...... Dimension AKB 4V3 A ...... Dimension theory AKB XD ...... Duality AKB XD3 F ...... Duality theorems . . . . . Subsystems AKF UES ...... Sheaf cohomology . . . . . Types by property AKO 2L ...... Equivariant cohomology

53 AVQAL AVRVJFTAP2 Topologies

Topology AVJ Mathematics AM Topologies AVP Mathematical systems AR5 Algebraic topology, analysis situ AVQ Topology AVJ Relations Topologies AVP . Cohomology AVQ AK Algebraic topology, analysis situ AVQ . . . Equivariant cohomology AVQ AKO 2L Groups AVQ FSA

AVQ AL . Homotopy AVR Groups in algebraic topology AL3 A . . Homotopy theory . Types by method . . Operations I4 . . Classical groups AL7 9 . . . Homotopy resolutions . . . Subsystems AL7 D . . . Construction I4F SA . . . . Groups AL7 DNX D . . . . Dual constructions I4F SCM J . . . . . Homology & cohomology groups of . . Relations classical groups AL9 E . . . Homotopy extension properties . Types by relations AL9 J . . . Homotopy equivalence MJ . . Homology & cohomology groups . . Entities . . . Subsystems ALE V . . . Homotopy functors MJF UAP 2 . . . . Eilenberg-Maclane spaces . . Subsystems . . . Types by property ALF UA . . . Spaces MJN 7 . . . . Singular homology & cohomology ALF UAP 2 . . . . Eilenberg-Maclane spaces groups ALF UAS LS . . . . Loop spaces ML . . Homotopy groups ALF UAS LSN K . . . . . Infinite loop spaces . . . Operations . . Types by property ML7 4P . . . . Operations in homotopy groups ALN Y . . . Stable homotopy . . . Relations ALN Y3A . . . . Stable homotopy theory ML9 R . . . . Generalizations of homotopy groups Elements in algebraic topology . . . Entities DM . Product MLE SP2 . . . . Hopf invariants . . Relations . . . Subsystems DMA J . . . Homology of a product MLF TWT . . . . Homotopy group of spheres ECN . Fixed points . . . Types by properties ECP . Coincidence points MLN Y . . . . Stable homotopy groups ECR . Critical points . . . . . Subsystems ECR 3A . . Critical point theory MLN YFT WT ...... Stable homotopy groups of Subsystems spheres FRB . Sets . Types of groups by property . . Types by property NJ . . Finite groups FRC MW . . . Complex sets, complexes . . . Elements derived from relations FRC OP . . . Simplicial sets NJD XL5 . . . . Finite groups of transformations . . . . Operations OP . . Simplicial groups & semigroups FRC OP7 4P . . . . . Operations over simplicial sets . Types of groups by system . . . . Subsystems VJ . . Topological groups FRC OPF SCM L . . . . . Homotopy groups of simplicial . . . Operations sets VJ7 4P . . . . Operations on topological groups . . . . Types by property . . . Relations . . . . . Complexes VJA J . . . . Homology groups FRC OPM W ...... Simplicial complexes VJA L . . . . Homotopy groups . . Types by entity . . . Elements FRC QN . . . Spectrum VJE V . . . . Functors FRC QNO P . . . . Simplicial spectra VJE VMP . . . . . Concrete functors FSA . Groups . . . Subsystems * For Groups in algebraic topology, use VJF SA . . . . Groups AVR. VJF SCM L . . . . . Homotopy groups of topological * Normal synthesis by Auxiliary Schedule groups AM1 is interrupted here; it is resumed at VJF SGO CJ . . . . . Shape groups AVR Y. VJF TA . . . . Algebras VJF TAP 2 . . . . . Hopf algebras

54 AVRVJL5 Topologies AVSQUFSCMJ

Topology AVJ Mathematical systems AR5 Topologies AVP Topology AVJ Algebraic topology, analysis situ AVQ Topologies AVP Subsystems Algebraic topology, analysis situ AVQ . . . . Subsystems Categories AVR YSX ...... Hopf algebras ...... Homotopy groups AVR VJF TAP 2 AVR YSX MQP TXM WY4

. . . . Types of topological groups by AVR YUA Spaces relation * For spaces in algebraic topology, use AVS. Normal synthesis by Auxiliary AVR VJL 5 . . . . . Topological transformation Schedue AM1 is interrupted here; it is groups, groups of resumed at AVT. homeomorphisms AVS Spaces in algebraic topology ...... Types by operation . Subsystems VJL 5JS ...... Groups of differentiable FSA . . Groups transformations FSC MJ . . . Homology & cohomology groups of ...... Types by property spaces VJL 5NJ ...... Finite transformation FSC MJ9 3 . . . . Forms groups FSC MJ9 33D . . . . . Axioms of spaces VJL 5NR ...... Compact groups of . Types by operation homeomorphisms JL . . Spaces with multiplication, H-spaces ...... By system in algebraic topology VJL 5SJ ...... Lie groups (topological . . . Relations groups) JLA J . . . . Homology & Cohomology VJL 5SJ NR ...... Compact Lie groups JLA L . . . . Homotopy VJL 5SJ NRJ S ...... Compact Lie groups of JLA L3A . . . . . Homotopy theory differentiable . . . Subsystems transformations JLF TAP 2 . . . . Hopf algebras in H-spaces Y Other subsystems in algebraic . Types by property topology MP . . Concrete spaces * Normal synthesis by Auxiliary . . . Subsystems Schedule AM1 is resumed here after its interrruption at AVQ FSA. MPF SA . . . . Groups * Add to AVR Y letters SB/UA MPF SCM J . . . . . Homology & cohomology following A in ASB/AUA. groups in concrete YSX . Categories spaces YSX MQP . . Fundamental categories ...... Operations . . . Types by system MPF SCM J7H ...... Computation of homology YSX MQP TWC . . . . Polyhedral categories & cohomology YSX MQP TXM WY . . . . CW-complexes groups . . . . . Subsystems OI . . Euclidean spaces YSX MQP TXM WY1 ...... Groups . . . Relations * Full classmark is % OI9 X . . . . Embedding AVR YSX MQP TXM . . . Subsystems WYF SA. OIF TXM WOP . . . . Simplicial complexes YSX MQP TXM WY2 ...... Homology groups OIF TXM WOP 9X . . . . . Embedding of simplicial * Full classmark is % complexes AVR YSX MQP . Types by elements TXM WYF SCM J. QU . . Spaces with operators YSX MQP TXM WY3 ...... Cohomology groups * Full classmark is % . . . Elements AVR YSX MQP QUD M . . . . Product TXM WYF QUD MNO J . . . . . Symmetric product SCM K. . . . Subsystems YSX MQP TXM WY4 ...... Homotopy groups QUF SA . . . . Groups * Full classmark is % QUF SCM J . . . . . Homology & cohomology AVR YSX MQP groups in spaces with TYXM WYF SCM L. operators

55 AVT AVX Topologies

Mathematical systems AR5 Mathematics AM Topology AVJ Mathematical systems AR5 Topologies AVP Topology AVJ Types of topology by method Topologies AVP . . Other subsystems in algebraic topology AVR Y Types of topology by other methods AVU ...... Homology & cohomology groups in . . . Global analysis AVU WNI spaces with operators AVS QUF SCM J AVU Y Types of topology by operation * Add to AVU Y numbers and letters 4P/9, A/R AVT . . Other subsystems in algebraic topolopy following J in Auxiliary Schedule AM1 * Normal synthesis by Auxiliary (representing classes AM7 4 /AM7R). Schedule AM1 is resumed here after its YJR . Differential topology interrruption at AVR YUA. . . Subsystem * Add to AVT letters UB/W following A in AUB/AW. YJR FUG . . . Manifolds * For differential topology of manifolds, use UEQ . . . Microbundles AVV. Normal synthesis by Auxiliary UFC . . . Fibre spaces Schedule AM1 is interrupted here; it is . . . . Relations resumed at AVW. UFC 9R . . . . . Generalizations of fibre spaces AVV . . . Differential topology of manifolds UFC AJ . . . . . Homology & Cohomology of . . . . Operations fibre spaces 832 . . . . . Surgery UFC AJ3 B ...... K-theory (fibre spaces) 832 DX ...... Obstructions . . . . Subsystems 833 . . . . . Cobordism UFC FRC OP . . . . . Simplicial sets 833 MW ...... Complex cobordism UFC FRC OP3 A ...... Theory . . . . Processes UFD . . . Fibre bundles 86 . . . . . Approximations . . . . Relations 86N M ...... Smooth approximations UFD AL . . . . . Homotopy . . . . Relations . . . . Types by entity 8K . . . . . Mappings UFD QH . . . . . Vector space bundles 8KJ S ...... Differentiable mappings UFD QHD J ...... Classes 8KJ SEC G ...... Singularities of differentiable UFD QHD JNY ...... Stable classes mappings . . . . Types by system ALY . . . . . Isotopy UFD TWT . . . . . Sphere bundles . . . . Elements UG . . . Manifolds DY . . . . . Structures UGM J . . . . Homological manifolds DYJ R ...... Differential structures UHS . . . Spectral sequences ECG . . . . . Singularities . . . . Relations ECR . . . . . Critical points UHS AK . . . . . Cohomology . . . . Subsystems UHS AKL R ...... Generalized cohomology FUF D . . . . . Fibre bundles UHS DX . . . . Elements derived from relations AVW . . Other subsystems of differential topology UHS DX8 K . . . . . Mappings * Normal synthesis by Auxiliary Schedule UHS DX8 KNV ...... Spectral sequence of a AM1 is resumed here after its interruption at continuous mapping AVU YJR FUG. . . . . Types by property * Add to AVW letters UH/W following A in UHS P2 . . . . . Eilenberg-Maclane spectral AUH/AW. sequences X . . Types of differential topology * Add to AVW X letters H/W in Auxiliary UHS P3 . . . . . Serre spectral sequences Schedule AM1. X . . Types of algebraic topology AVX Types of topologies by other characteristics * Add to AVT X letters H/W in * Normal synthesis is resumed here after the first Auxiliary Schedule AM1. interruption of division into types at AVP IRS AVU Types of topology by other methods and later at AVU YJR FUG. * Normal synthesis is resumed here after its * Add to AVX letters JS/W in Auxiliary Schedule interruption at AVP IRS. AM1. * Add to AVU letters RT/W in Auxiliary Schedule AM1. W . Analytic topologies . . Types by property WNI . . . Global analysis (topologies)

56 AVXN6 Calculus AW87

Mathematics AM Mathematics AM Mathematical systems AR5 Mathematical systems AR5 Topology AVJ Analysis AW . . Types of topologies by other characteristics Methods special to analysis AVX . . Calculus of variations AW6 Y . . . . . Discontinuities AW6 YEC G AVX N6 . . . Low dimension topology . . . . Subsystems AW6 YY . . Operational calculus N6F SLS . . . . . Loops . . . Elements N6F SLS 3F ...... Loop theorems YYD G . . . . Solutions N6F TWT . . . . . Spheres YYD GDI . . . . . Moment problems N6F TWT 3F ...... Sphere theorem . . . Types by property . . . . . Manifolds YYI 4 . . . . Classical operational calculus N6F UG . . . . . Low dimension topology of YYP 2 . . . . Mikusinski’s calculus manifolds AW7 2 . . Differential calculus ...... Relations . . . Relations N6F UG9 WM ...... Immersions in low 28L . . . . Functions dimensions . . . . . Processes N6F UG9 X ...... Embeddings in low 28L 88 ...... Variation of functions dimensions ...... Relations ...... Subsystems 28L 888 K ...... Mappings N6F UGF SCM QP ...... Fundamental groups . . . Elements N6F UGF UFQ ...... Knots 2DU . . . . Derivatives . . . . Types of low dimension topology 2DV . . . . Differentials ON . . . . . Metric topology . . . Types * For absolute differential calculus, see Tensor analysis AUM EI. AW Analysis 3 . . Integral calculus AW3 H . Formulae . . . Elements HX . . Asymptotic formulae & expressions 3DW . . . . Integrals . Methods special to analysis . . . . . Subsystems AW6 X . . Calculus, infinitesimal calculus 3DW FUG ...... Manifolds Y . . . Calculus of variations, variational 3DW FUG OCR ...... Integrals over curved calculus manifolds . . . . Special theories . . . . . Types Y3C C . . . . . Catastrophe theory 3DW N8H ...... Multiple integrals * See also Discontinuities 44 . . Vector calculus AW6 YEC G . . . Entities . . . . Methods 44E U . . . . Operators of vector calculus Y6V J . . . . . Topological methods 45 . . Other analytical methods Y6W O . . . . . Functional analytic methods * Numbers 45/49 following AW7 may be ...... Elements used for special not otherwise given. Y6W OE7 G ...... Necessary conditions See introduction, section 13.35, title 5 . . . . Relations for an example. Y9L . . . . . Equations of the calculus of Operations in analysis variations R . Differentiation . . . . Properties . . Types Y9P . . . . . Inverse problems RMO . . . Abstract differentiation . . . . Elements RMO 3A . . . . Abstract differentiation theory YDG . . . . . Solutions of variational V . Integration problems Processes ...... Properties AW8 7 . Continuation YDG ANT ...... Existence * For analytic continuations, see under . . . . Entities analytic functions, AWD W87...... Singularities YEC G ...... Discontinuities * See also Catastrophe theory, AW6 Y3C C

57 AW8K AW8LIWRYNU9SQQPW Functions

Mathematics AM Mathematical systems AR5 Mathematical systems AR5 Analysis AW Analysis AW Functions AW8 L Processes Elements & Entities . Continuation AW8 7 . Integrals AW8 LDW . . . . Integrals of potential type Relations AW8 LDW WB AW8 K . Analytic mappings AW8 LEO . Series & sequences LEO C4Y . . Summability L . Functions Subsystems L3A . . Theory of functions LFT VS . Surfaces L3C ON . . Metric theory of functions LFT VSO CR . . Curved surfaces * Many of the operations below usually imply the use of metric theory. . . Operations Types of functions . By method L7R . . . Differentiation LIR S . . Algebraic & algebroid functions L7R 3A . . . . Theory of differentiation LIW . . Analytic functions (+general+), L7V . . . Integration real analytic functions L7V 3A . . . . Theory of integration * Analytical functions should not . . Processes be divided by Auxiliary L86 . . . Approximation Schedule AM1. The divisions of . . . . Types by entity Analytical functions L86 QL . . . . . Approximation by polynomials (AW8 LIW) are the same as ...... Elements those of analysis itself (AW) L86 QLD I ...... Moments between AW3 and AWT, L86 QLD I3A ...... Theory of moments including the various interruptions of normal synthesis L86 QLD U ...... Derivatives within Analysis...... Relations . . . Forms of presentation L86 QLD U9N ...... Inequalities for derivatives LIW 3FP 3 . . . . Picard theorem of polynomials . . . Processes ...... Extrema LIW 87 . . . . Analytic continuation L86 QLE 2 ...... Extremal properties of LIW 8E . . . . Interpolation polynomials . . . Relations L89 . . . Growth LIW J9R . . . . Generalizations of analytic L89 CQR . . . . Rate of growth of functions functions L8E . . . Interpolation . . . . . Subsystems . . Relations LIW J9R FTB ...... Analytic matrices L8K . . . Mappings LIW L . . . Other properties & elements . . Properties LIW LDF AN . . . . Boundary properties of LB5 . . . Measure analytic functions . . Elements & Entities LIW LDW . . . . Integrals LDF . . . Boundary LIW LDW P2 . . . . . Cauchy-type integrals LDF AN . . . . Boundary properties of functions . . . Entities LDU . . . Derivatives LIW LEQ PN . . . . Power series LDU NOJ . . . . Symmetric derivatives LIW N . . . Subsytems LDW . . . Integrals LIW NUA . . . . Spaces of a function . . . . Types by property LIW NY . . . Types of analytic functions LDW N7 . . . . . Singular integrals LIW RY . . . Types by other relations LDW P2 . . . . . Fourier integral LIW RYM C . . . . Automorphic functions ...... Properties LIW RYM H . . . . Meromorphic functions LDW P2C 4Y ...... Summability LIW RYN 4H . . . . Discrete analytic functions LDW P3 . . . . . Lebesgue integral LIW RYN U . . . . Entire functions LDW P4R . . . . . Riemann integral . . . . . Relations . . . . Types by system LIW RYN U9S ...... Representations . . . . . Harmonic functions LIW RYN U9S QQP W ...... Representations by LDW WB ...... Integrals of potential type series of integrals

58 AW8LIWSY Functions AW8N9OUP4LC

Analysis AW Mathematical systems AR5 Functions AW8 L Analysis AW Types of functions Functions AW8 L By method Types of functions . . Types by other relations AW8 LIW RY Types of functions by relation ...... Representations by series of integrals . . . Algebraic theory of generalized functions AW8 LIW RYN U9S QQP W AW8 LLR 6RS

AW8 LIW SY . . Types by other properties Types of functions by property LIW SYP 5F . . . Bounded functions AW8 MB . Homomorphic functions LIW TY . . Types by other properties other MG . Holomorphic functions elements etc * For Special functions, see AWD X. . . Special types of analytic functions MH . Meromorphic functions LIW U . . . Systems of functions MQG . Non-standard functions . . . . Properties MQG 3A . . Non-standard function theory LIW UBL T . . . . . Closedness MR . Normal functions LIW UBT N . . . . . Completeness problems MY . Rational functions . . . . Entities N3 . Real functions LIW UEW . . . . . Bases of systems of functions . . Elements . . . . Subsystems N3D W . . . Integrals LIW UFU EN . . . . . Families of functions N3D W3H . . . . Integral formulae LIW UFU ENN R ...... Compact families of functions . . Subsystems . . . . Types by property N3F UA . . . Spaces LIW UOU . . . . . Orthogonal systems of N3F UAN LD . . . . Functions on infinite-dimensional functions spaces ...... Entities N3F UAN LD6 W . . . . . Calculus of functions on infinite- LIW UOU EQ ...... Orthogonal series dimensional spaces ...... Processes N3F UEN . . . Families of functions LIW UOU EQ8 B ...... Convergence N3F UEN MR . . . . Normal families ...... Properties . . Types by method LIW UOU EQC 4Y ...... Summability N3I VE . . . Trigonometric functions LIW V . . . Quasi-analytic functions & pseudo- . . . . Relations analytic functions N3I VE9 N . . . . . Inequalities Types of functions by operation N3I W . . . Analytic functions LJS . Differentiable functions * See AW8 LIW . . Elements . . Types by property LJS DU . . . Derivatives N3M Y . . . Rational functions LJS DUX . . . . Partial derivatives N3N 7 . . . Singular functions . . . . . Relations N3N DN . . . Functions of several variables LJS DUX 9N ...... Inequalities between partial . . . . Properties derivatives N3N DNB V . . . . . Continuity Types of functions by relation . . Types by entity LLR . Generalized functions N3Q H . . . Vector functions . . Methods . . . . Methods LLR 6RS . . . Algebraic theory of generalized N3Q H6W . . . . . Calculus of vector functions functions N4F . Transcendental functions * See also Special functions (higher transcendental functions) AWD XE. N9 . Polynomial functions N9N DN . . Polynomial functions with several variables N9O U . . Orthogonal polynomial functions * See also Hypergeometric functions, AWD XKY H N9O UP4 C . . . Chebyshev functions N9O UP4 H . . . Hermite functions N9O UP4 J . . . Jacobi functions N9O UP4 LC . . . Laguerre functions

59 AW8N9OUP4LF AW9YR6RI Functions

Mathematical systems AR5 Functions AW8 L Analysis AW Types of functions Functions AW8 L Types of functions by property Types of functions . . Subsystems Polynomial functions AW8 N9 . . . Arithmetic AW9 FRI . . Laguerre functions AW8 N9O UP4 LC . . . . Continued fractions AW9 FRK L

AW8 N9O UP4 LF . . Legendre functions AW9 FTV S . . . Surfaces N9O UP4 LH . . Legrange functions FTV SP2 . . . . Riemannian surfaces of functions of NEC With complex variables a complex variable * For Functions of a complex variable, use . . Types AW9. Normal synthesis by Auxiliary MH . . . Meromorphic functions of a complex Schedule AM1 is interrupted here. Specification by other properties is resumed variable at AWA; normal synthesis by Auxiliary MR . . . Normal functions of a complex Schedule AM1 is resumed at AWD. variable * For functions of several complex variables NU . . . Entire functions of a complex variable see AW9 YH. . . Special types AW9 Functions of a complex variable, complex XB . . . Systems of functions of a complex analysis variable . Theory XD . . . . Univalent functions 3CV J . . Topological function theory XE . . . . Multivalent functions . Relations Y Functions by other variables 8K . . Mappings * Add to AW9 Y letters E/R following ANE. . . . Properties YH . Functions of several complex variables 8KD 7TS . . . . Geometric properties of mappings . . Processes 8KO M . . . Conformal mappings YH8 6 . . . Approximation of functions . Elements YH8 6MG . . . . Holomorphic approximation DC . . Domain . . Relations . . . Types by property YH9 3 . . . Forms DCM W . . . . Complex domains YH9 3MC . . . . Automorphic forms . . . . . Processes YH9 S . . . Representations DCM W86 ...... Approximation YH9 SJV . . . . Integral representations . . . . . Relations . . Properties DCM W9N ...... Inequalities in the complex YHC D . . . Convexity domain YHC DMG . . . . Holomorphic convexity DCM W9S ...... Representation in the complex YHC DN9 . . . . Polynomial convexity domain . . Elements DCM W9S OG ...... Asymptotic representation YHD C . . . Domains DCX . . . Special domains YHD CW . . . . Domains of analyticity DCX 8KO M . . . . Conformal mappings of special . . Types by relation domains YHM C . . . Automorphic functions of several DFD D . . Boundary values complex variables DV . . Differentials . . . . Elements . . . Subsystems YHM CDC . . . . . Domains DVF TVS . . . . Surfaces YHM CDC NOJ ...... Symmetric domains DVF TVS P2 . . . . . Differentials on Riemann surfaces YHM G . . . Holomorphic functions of several E2 . . Extrema complex variables . Entities . . . . Elements EQ . . Series YHM GDF AN . . . . . Boundary properties EQP 2 . . . Dirichlet series . . . . Entities EQP N . . . Power series YHM GEQ PN . . . . . Power series . . . . Elements . . Types by property EQP NDF AN . . . . . Boundary properties YHN U . . . Entire functions of several complex . Subsystems variables FRI . . Arithmetic YR . Functions of a real variable FRK L . . . Continued fractions YR6 RI . . Numerical methods

60 AWAXG Functions AWDXE

Mathematical systems AR5 Functions AW8 L Analysis AW Types of functions Functions AW8 L Types of functions by other properties Types of functions . . Subsystems Functions by other variables AW9 Y . . . Spaces AWB FUA . . Numerical methods AW9 YR6 RI . . . . Dirichlet spaces AWB FUA P3

Types of functions by other properties AWB FUG . . . Manifolds * Add to AWA letters F/XH following AN in FUG OQB . . . . Harmonic functions on Riemannian ANF/ANX H. manifolds AWA XG . Almost periodic functions . . Types of harmonic functions by property . . Processes * The interruption in normal synthesis at XG8 6 . . . Approximations of almost periodic AWA XH is continued here; normal functions synthesis is resumed at AWD. XG8 E . . . Interpolation NXK . . . Subharmonic . . Properties * Use AWC K. XGD 5Y . . . Structural properties of almost NXM . . . Biharmonic periodic functions * Use AWC M. . . Entities NXP . . . Polyharmonic XGE Q . . . Series of almost periodic functions * Use AWC P. XGE QP2 . . . . Fourier series of almost periodic AWC . . Special types functions * Add to AWC letters K/P following ANX...... Processes K . . . Subharmonic functions XGE QP2 8B ...... Convergence of Fourier series K3C N8D . . . . Two dimensional theory of almost periodic M . . . Biharmonic functions & equations functions M3C N8D . . . . Two dimensional theory . . Types by property M3C N8H . . . . Higher dimensional theory P XGM O . . . Abstract almost periodic functions . . . Polyharmonic & plurisubharmonic XH . Harmonic functions * For harmonic functions, use AWB. P3C N8D . . . . Two dimensional theory Normal synthesis is interrupted here; it is P3C N8H . . . . Higher dimensional theory resumed at AWD. Q Types of functions by other properties etc AWB . Harmonic functions, potential functions * Add to AWC letters Q/Y following ANX. 3A . . Potential theory AWD Other types of functions 3CN 4H . . . Discrete potential theory * Normal synthesis is resumed here after its initial 3CN 4Q . . . Absolute potential theory interruption at AW8 NEC and its final interruption at AWB NXK. 3CN 8D . . . Two dimensional potential theory * Add to AWD X letters NY/W from Auxiliary 3CN 8H . . . Higher dimensional theory Schedule AM1. . . Relations OD . Convex functions 9R . . . Generalizations of harmonic functions OV . Elliptic functions 9S . . . Representation of harmonic functions OYB . Spherical functions 9SJ V . . . . Integral representations Types of functions by element 9SJ V3C N8D . . . . . Two dimensional theory P82 . Kernel functions 9SJ V3C N8H . . . . . Higher dimensional theory . . Entities . . Properties P82 EQ . . . Series B5 . . . Harmonic measure P82 EQQ L . . . . Series of polynomials B8L . . . . Length XE Special functions, higher transcendental B8L Q2 . . . . . Extremal length functions B8L Q23 CN8 D ...... Two dimensional theory * A number of special functions often considered B8L Q23 CN8 H ...... Higher dimensional theory together as a body in the mathematical literature . . Elements and closely related to the special functions of DF8 5 . . . Boundary behaviour mathematical physics. DF8 53C N8D . . . . Two dimensional theory * The provision for functions in general at AM8L DFD D . . . Boundary value problems and the use of these to qualify specific subjects (e.g. Othogonal polynomials AW8 N9O U) DFD D3C N8H . . . . Higher dimensional theory means that many of these are distributed under . . Subsystems these specific subjects. FUA . . . Spaces FUA P3 . . . . Dirichlet spaces

61 AWDXE AWFBYPY Analysis

Mathematics AM Mathematics AM Mathematical systems AR5 Mathematical systems AR5 Analysis AW Analysis AW Relations Transforms AWE 4 . Functions AW8 L . . . Laplace transforms AWE 4JV P2 . . . Special functions AWD XE AWE L Equations in analysis * An alternative is provided below for libraries wishing to collect these special functions in one location. But a number of them are best . Types by operation located here in any case, and these are enumerated below. ME . . Differential equations AWD XF . . . . By individual name other than a . . . Special theories mathematician ME3 CC . . . . Chaos theory XFB . . . . . Beta functions . . . Elements XFC . . . . . Gamma functions MED L . . . . Differences XFG . . . . . Zeta functions MED LNJ . . . . . Finite differences * For general works this is an alternative to locating at AM8 YG. . . . Types by relation XFI . . . . . Theta functions MEK L . . . . Functional differential equations XG . . . . By name of mathematician MEK L3A . . . . . Theory * Alternative to locating general works on MEK L3C NXT ...... Qualitative theory each of these at AM8 P2. MEK L3C NY ...... Stability theory XGB . . . . . Bessel function MEK L3C OG ...... Asymptotic theory XGG . . . . . Gegenbauer function . . . . . Elements XGH . . . . . Hankel function MEK LDF DD ...... Boundary values XGM . . . . . Mathieu function XGN . . . . . Neumann function . . . Types by property XH . . . . By method by operation by property etc MEN G . . . . P-adic differential equations * Alternative to locating general works on MEN TL . . . . Partial differential equations the function under the subject. * For partial differential equations, * Add to AWD letters H/W from Auxiliary use AWG. Normal synthesis is Schedule AM1. For example: interrupted here; it is resumed at XKY H . . . . . Hypergeometric functions AWI. XOV . . . . . Elliptic functions MF . . . . Ordinary differential equations XX . Functionals * For ordinary differential equations, . . Types by system use AWF. Normal synthesis is XXW . . . Analytic functionals interrupted here; it is resumed at YD . Determinants AWI (for other equations) and YDK L . . Functional determinants AWJ (for other relations). AWF . . . . Ordinary differential equations 3CW . . . . . Analytic theory AWE Other relations in analysis . . . . . Methods * Normal synthesis is resumed here after its 6OG ...... Asymptotic methods interruption at AW8 NEC. * Add to AWE numbers and letters 3/ME following . . . . . Relations AM9 in AM93/AM9ME. 9N ...... Differential inequalities 4 . Transforms . . . . . Properties . . Types ANU ...... Uniqueness 4JV . . . Integral transforms ANU 3F ...... Uniqueness theorems 4JV N7 . . . . Singular integrals BXT ...... Qualitative 4JV N8H . . . . Multiple transforms BXT 3A ...... Qualitative theory 4JV P2 . . . . Laplace transforms ...... Elements BXT 3AE CG ...... Singular points BY ...... Stability ...... Processes BY8 GE ...... Pertubations BY8 GEN 7 ...... Singular perturbations ...... Types by property BYP Y ...... Structural stability

62 AWFCG Differential equations AWGEGBTN

Analysis AW Mathematical systems AR5 Equations in analysis AWE L Analysis AW Differential equations AWE ME Equations in analysis AWE L Properties Differential equations AWE ME . Stability AWF BY Ordinary differential equations AWF . . . Structural stability AWF BYP Y . . . . . Singular points AWF PCM WEC G

AWF CG . Asymptotic properties AWG Partial differential equations CG8 GE . . Perturbutations . Theory CG8 GEN 7 . . . Singular perturbations 3CT S . . Geometric theory Elements 3FP 2 . . Cauchy-Kowalewski theorems DE . Initial value . Methods DFD D . Boundary value problems 6W . . Analytic methods DFD DEN . . Spectra 6Y . . Variational methods DFD DEN 3C . . . Spectral theory . Processes DG . Solutions 8GE . . Perturbations . . Processes . Relations DG8 6 . . . Approximation of solutions . . Functions DG8 6RI . . . . Numerical approximations 8P2 . . . Hamiltonians . . Elements & entities 9N . . Differential inequalities DGD HY . . . Limit cycles 9P . . Inverse problems DGE R . . . Expansions of solutions . Properties DW . Integrals ANT . . Existence properties DWX . . First integrals ANU . . Uniqueness properties Entities BY . . Stability ER . Expansions . Elements ERO G . . Asymptotic expansions DFD D . . Boundary value problems Subsystems DG . . Solutions FUA . Spaces . . . Processes FUA P2 . . Differential equations in Banach spaces DG8 6 . . . . Approximations of solutions FUG . Manifolds DG8 6RI . . . . . Numerical approximations FUG 9L . . Equations on manifolds . . . Relations FUG DG . . Manifolds of solutions DG9 S . . . . Representations of solutions . . . Properties DG9 SJV . . . . . Integral representations FUG DGB Y . . . . Stability of manifolds . . . Properties Types of ordinary differential equations DGC G . . . . Asymptotic behaviour . By property DGD 2 . . . . Cauchy problem NA . . Linear ordinary differential equations . . . . . Subsystems . . . Properties DGD 2FR Y ...... Semi-groups related to the Cauchy NAA NT . . . . Existence problem NAA NT3 F . . . . . Existence theorems . . . Types by relation . . . Entities DGL R . . . . Generalized solutions NAE G . . . . Eigenvalues . . . . . Properties . . . Types by system DGL RAN T ...... Existence NAT B . . . . Matrix linear ordinary differential . . . Types by property equations DGN XF . . . . Periodic solutions NB . . Non-linear ordinary differential equations DGN XG . . . . . Almost periodic solutions . . . Elements . . . Types by entity NBD FDD . . . . Boundary value problems DGQ Q . . . . Series solutions NDI . . Differential equations of infinite order . Entities . By element EG . . Eigenfunctions PCM W . . Equations in the complex domain . . . Processes . . . Elements EG8 A . . . . Distribution of eigenvalues PCM WEC G . . . . Singular points EG8 AOG . . . . . Asymptotic distribution of eigenvalues . . . Properties EGB TN . . . . Completeness of eigenvalues

63 AWGEN3A AWHEG Differential equations

Analysis AW Equations in analysis AWE L Equations in analysis AWE L Differential equations AWE ME Differential equations AWE ME . Types by property Entities . . . . By property . Eigenfunctions AWG EG . . . . . Parabolic equations AWG OW . . . Completeness of eigenvalues AWG EGB TN ...... Second order parabolic equations AWG OWN C AWG EN3 A . Spectral theory ...... Boundary value problems AWG OWN CDF DD EN3 B . . Scattering theory EU . Operators AWG OWN DH ...... Higher order parabolic EUJ U . . Pseudo-differential operators equations . . . Elements ...... Elements EUJ UDE . . . . Initial value problems OWN DHD E ...... Initial value problems EUJ UDF DD . . . . Boundary value problems OWN DHD FDD ...... Boundary value problems Subsystems OX . . . . . Hyperbolic equations FUA . Spaces OXN A ...... First order hyperbolic equations FUA KL . . Function spaces OXN C ...... Second order hyperbolic FUA KL9 L . . . Equations on function spaces equations Types of partial differential equations ...... Elements . By relations OXN CDE ...... Initial value problems KL . . Functional differential equations OXN CDF DD ...... Boundary value problem . By property OXN DH ...... Higher order hyperbolic NA . . Linear partial differential equations, first equations order partial differential ...... Elements equations OXN DHD E ...... Initial value problems . . . Elements OXN DHD FDD ...... Boundary value problems NAD E . . . . Initial value problems . . . . Types by element or entity NAD FDD . . . . Boundary value problems PL . . . . . Difference equations NB . . Non-linear partial differential equations * For difference-partial . . . Elements differential equations see NBD E . . . . Initial value problems Difference equations, NBD FDD . . . . Boundary value problems AWI PLJ S. NC . . Second order partial differential . . . . . Coefficients equations QK ...... Equations with constant NDH . . Higher order partial differential equations coefficients OV . . Elliptic-type equations ...... Elements . . . Theory QKD E ...... Initial value OV3 CWB . . . . Potential theory . . . . . Operators . . . Elements QU ...... Operator differential equations OVD FDD . . . . Boundary value problems Y . Other types of differential equations * Add to AWG Y letters T/U following OVD G . . . . Solutions AM7. OVD GDF DD . . . . Boundary values of solution AWH Integral equations . . . Entities . Properties OVE N3A . . . . Spectral theory BXT . . Qualitative behaviour . . . Types by property BY . . Stability OVN B . . . . Non-linear elliptic-type equations BY3 A . . . Stability theory OVN C . . . . Second order elliptic-type equations CG . . Asymptotics . . . . . Methods . Elements OVN C6Y ...... Variational methods DG . . Solutions OVP 2 . . . . Laplace equations . . . Processes OVP 3 . . . . Poisson equations DG8 6 . . . . Approximation of solutions OW . . Parabolic equations DG8 6RI . . . . . Numerical approximations OWN B . . . Non-linear parabolic equations . Entities OWN C . . . Second order parabolic equations EG . . Eigenvalue problems . . . . Elements OWN CDE . . . . . Initial value problems OWN CDF DD . . . . . Boundary value problems

64 AWHJR Analysis AWK9MG

Mathematical systems AR5 Mathematics AM Analysis AW Mathematical systems AR5 Equations in analysis AWE L Analysis AW Types by operation Other relations in analysis AWE . . Entities ...... Elements . . . Eigenvalue problems AWH EG ...... Boundary value problems AWI PLJ RDF DD . . Types by operation AWI PLJ RDG ...... Solutions AWH JR . . . Integro-differential equations PLJ RDG NXF ...... Periodic solutions JRN 7 . . . . Singular integro-differential equations ...... Types JRN A . . . . Linear integro-differential equations PLJ T ...... Difference-partial differential JRN B . . . . Non-linear integro-differential equations equations . . Types by property ...... Types of difference equations by property MO . . . Abstract integral equations PLN A ...... Linear difference equations NA . . . Linear integral equations PLN B ...... Non-linear difference equations NAN 7 . . . . Singular integral equations . . . . . Elements PLN J ...... Finite difference equations AWJ Other relations in analysis NAN 7DF ...... Boundary problems * Normal synthesis is resumed here after its NAP 2 . . . . Volterra integral equations interruption at AWE MF. NB . . . . Non-linear integral equations * Add to AWJ number and letters 9N/AL in NBN 7 . . . . . Singular non-linear equations Auxiliary Schedule AM1...... Elements 9N . Inequalities NBN 7DF ...... Boundary problems . . Types by relations P2 . . . Fredholm integral equations 9NK L . . . Functional inequalities AWI Types of equations by other operations etc Properties in analysis * Normal synthesis by Auxiliary Schedule AM1 * Add to AWJ letters AN/BXV in Auxiliary is resumed here after its interruption at Schedule AM1. AWE MEN TL and AWE MF. BXV . Optimality * Add to AWI letters JW/W from Auxiliary * For Optimal control use AWK. Normal Schedule AM1. Note that -JW represents synthesis is interrupted here; it is resumed at AM7 W, which immediately follows AM7 V AWL. Integration (which specifies Integral equations). KL . Functional equations . . Types by method AWK . Optimality, optimization KLI 4 . . . Classical functional equations . . Methods . . Types by property 6QR . . . Approximation methods KLN A . . . Linear functional equations . . . Variational methods KLN DQ . . . Multilinear functional equations 6Y . . . . Control theory, optimal control . . Types by system . . . . . Special theories KLT B . . . Matrix functional equations 6Y3 B ...... Maximum principle, minimum Types of equations by property principle, Pontryagin’s OR . Dynamic systems of equations, dynamical principle systems . . Relations . . Subsystems 8L . . . Functions ORF VJ . . . Topological questions 8L3 KP . . . . Problems involving functional . . Types relations ORN B . . . Non-linear dynamical systems 9L . . . Equations Types of equations by elements 9L3 KP . . . . Problems involving equations PL . Difference equations 9MF . . . . Ordinary differential equations . . Types by operation 9MG . . . . Partial differential equations (optimal PLJ R . . . Differential difference equations control) PLJ R3C OG . . . . Asymptotic theory . . . . Properties PLJ RBY . . . . . Stability PLJ RBY 3A ...... Stability theory . . . . Elements PLJ RDF DD . . . . . Boundary value problems

65 AWKDG AWLEQP28BN4Q Analysis

Mathematics AM Mathematics AM Mathematical systems AR5 Mathematical systems AR5 Analysis AW Analysis AW Properties in analysis Other properties & elements/entities in analysis AWL . . Relations . . . . Partial differential equations AWK 9MG Entities in analysis AWL EO . Sequences & series in analysis . . Elements * See note under (Types) below for AWK DG . . . Solutions to optimal control problems explanation of the treatment of sequences . . . . Methods and series separately. DG6 Q7G . . . . . Necessary conditions methods . . Processes DG6 QR . . . . . Approximate methods, approximation EO8 B . . . Convergence methods EO8 BN4 Q . . . . Absolute convergence DG6 X ...... Finite difference methods EO8 BN4 S . . . . Conditional convergence DG6 Y ...... Linear programming methods . . Properties . . . . Properties EOC 4Y . . . Summability DGA NT . . . . . Existence problems for optimal . . . . Methods solutions EOC 4Y6 KL . . . . . Function-theoretic methods E5 . . . Conditions for optimality EOC 4Y6 TB . . . . . Matrix methods . . . . Properties EOC 4Y6 WO . . . . . Functional analytic methods E5B S . . . . . Freeness, free problems EOC 4Y7 3 . . . . . Integral methods ...... Types by property . . . . Types by property E5B SND K ...... Free problems in one independent EOC 4YN 4Q . . . . . Absolute summability variable EOC 4YN LN . . . . . Strong summability E5B SND N ...... Free problems in two or more . . Elements independent variables EOD M . . . Product E6 . . . . Chain conditions EOD MNK . . . . Infinite product E7E . . . . . Necessary & sufficient conditions . . Subsystems E7S . . . . States in analysis EOF RKL . . . Continued fractions E7S ET . . . . . State variables . . General types of sequences & series . . . Variables * The literature usually considers ET . . . . Control variables sequences and series (a sequence . . Subsystems expressed as a sum) together, although FUA . . . Spaces often only the series is significant. To avoid an unhelpful separation into three FUA MO . . . . Abstract spaces separate sequences (Types of sequences FUG . . . Manifolds and series jointly; Types of sequences; . . . . Properties Types of series) the arrangement is FUG B5 . . . . . Measure & integration modified as shown below. FUG B5T S ...... Geometric measure & integration EP . . . Sequences, progressions . . Types by property * For specific types of sequences, see NA . . . Linear optimal systems AWL EQH/W. AWL Other properties & elements/entities in analysis EQ . . . Series * Normal synthesis is resumed here after its * For specific types of series, see interrruption at AWJ BXV. AWL EQH/W. * Add to AWL letters BXX/EU from Auxiliary . . Specific types of sequences & series Schedule AM1. * Add to AWL EQ letters H/W from Auxiliary Schedule AM1. . . . Types by process & relation EQK C . . . . Divergent series EQK L . . . . Functional series & sequences . . . Types by property EQN 8H . . . . Multiple series & sequences EQP 2 . . . . Fourier series . . . . . Processes EQP 28B ...... Convergence EQP 28B N4Q ...... Absolute convergence of Fourier series

66 AWLEQP2C4Y Operators AWMFUANA

Analysis AW Mathematics AM Entities in analysis Mathematical systems AR5 Sequences & series in analysis AWL EO Analysis AW . . Types by property Entities in analysis . . . . Processes Approximations AWL ER ...... Absolute convergence of Fourier series . . Asymptotic approximations AWL ERO G AWL EQP 28B N4Q AWL EU Operators . . . . Properties EU3 A . Operator theory AWL EQP 2C4 Y . . . . . Summability . Types of operator by operation . . . . Elements EUJ R . . Differential operators EQP 2EK . . . . . Fourier coefficients EUJ T . . . Partial differential operators . . . . Types by property EUJ V . . Integral operators EQP 2N8 H . . . . . Multiple Fourier series . Types by relation ...... Properties EUL G . . Self-adjoint operators EQP 2N8 HC4 Y ...... Summability . . . Subsystems . . Types by element EUL GFU AP2 . . . . Self-adjoint operators in Hilbert EQP N . . . Power series spaces . . . . Processes . Types by property EQP N87 . . . . . Continuation EUN A . . Linear EQP N87 W ...... Analytic continuation * For linear operators, use AWM. Normal . . Types by system synthesis by Auxiliary Schedule AM1 is EQR I . . . Numerical series & sequences interrupted here; it is resumed at . . . . Processes AWM X. EQR I87 . . . . . Continuation AWM . . Linear operators EQT S . . . Geometric series . . . Methods EQV E . . . Trigonometric series 6YY . . . . Operational calculus of linear . . . . Properties operators EQV EAN U . . . . . Uniqueness problems . . . Relations . . . . Types by property 8L . . . . Functions EQV EN8 H . . . . . Multiple trigonometric series . . . . . Types by entity ER Approximations, expansions 8N3 QH ...... Vector functions & operator . Processes functions ER8 B . . Convergence . . . . Similarity ER8 BCQ R . . . Rate of convergence 9HG NA . . . . . Linear similarity ER8 E . . Interpolation . . . . Equivalence . Relations 9JN A . . . . . Linear equivalence . . Functions . . . . Equations . . . Rational 9L . . . . . Equations involving linear ER8 MY . . . . Approximations by rational operators functions ...... Problems . Elements 9L3 KS ...... Incorrectly posed problems ERE U . . Approximations by operators . . . Entities . Types by property EN . . . . Spectra ERM O . . Abstract approximations EN3 A . . . . . Spectral theory of linear operators ERM O3A . . Abstract approximation theory . . . . . Relations ERO G . . Asymptotic approximations EN9 E ...... Operators extensions . . . . . Properties EN9 P ...... Inverse problems . . . Subsystems FRY . . . . Semi-groups of linear operators FTA . . . . Algebras of linear operators FUA . . . . Spaces FUA NA . . . . . Linear spaces of operators

67 AWMFUANLD AWO8NVFTDB Operators

Mathematical systems AR5 Analysis AW Analysis AW Entities in analysis Entities in analysis . . Types by property Operators AWL EU . . . . Types by property Subsystems . . . . . Single linear operator AWM N7 . . Linear spaces of operators AWM FUA NA ...... Fredholm operators AWM N7P 4F

AWM FUA NLD . . Infinite-dimensional spaces AWM NW . . . . . Completely continuous operators FUA NLD NA . . . Infinite-dimensional linear spaces ...... Entities . . . . Relations NWE N3A ...... Spectral theory of completely FUA NLD NA9 L . . . . . Linear equations in infinite- continuous operators dimensional linear P5F . . . . . Bounded linear operators spaces PL . . . . . Difference operators FUA OD . . Convex spaces . . . . Types by element/entity FUA ODN F . . . Locally convex spaces QN . . . . . Spectral operators FUA P2 . . Linear operator in Hilbert spaces X . . Other types of operators FUA P2E N3A . . . Spectral theory of linear operators in * Normal synthesis is resumed here after its Hilbert spaces interruption at AWL EUN A. FUA P3 . . Linear operators in Banach spaces * Add to AWM X letters NB/W in Auxiliary FUA P3E N3A . . . Spectral theory of linear operators in Schedule AM1. Banach spaces XNB . . . Non-linear operators Types by property . . . . Relations LG . Self-adjoint linear operators XNB 9L . . . . . Equations involving non-linear . . Subsystems operators LGF UAP 2 . . . Self-adjoint linear operators in . . . . Types by property Hilbert spaces XNB NOT . . . . . Monotone operators MP . Concrete operators Y Other entities in analysis * . . Entities Add to AWM Y letters U/X following AMQ. AWN Subsystems in analysis MPE N . . . Spectrum * Add to AWN letters R/W following A in MPE R . . . Expansions AR/AW. MPE RQG . . . . Expansions in Eigenfunctions N7 . Single linear operator . . Entities Y Types of analysis * Add to AWN Y letters H/KL from Auxiliary N7E N . . . Spectrum Schedule AM1 N7E N3B . . . . Scattering theory YKL . Functional . . Subsystems * For functional analysis, use AWO. Normal N7F RB . . . Sets synthesis is interrupted here; it is resumed at N7F RBQ N . . . . Spectral sets AWR Y. N7F UBF . . . Subspaces N7F UBO 2J . . . . Invariant subspaces AWO . Functional analysis . . Types by property * See also Topology, TJ N7M R . . . Normal operators . . Methods . . . . Subsystems 6SX . . . Categorical methods N7M RFU A . . . . . Spaces . . Relations N7M RFU AP2 ...... Normal operators in Hilbert 8L . . . Functions spaces . . . . By relation N7N OJ . . . Symmetric operators 8LL R . . . . . Generalized functions N7N R . . . Compact operators . . . . By property N7P 2 . . . Volterra operators 8NV . . . . . Continuous functions N7P 2FU AP2 . . . . Volterra operators in Hilbert ...... Subsystems spaces 8NV FTA ...... Algebras N7P 2FU AP3 . . . . Volterra operators in Banach 8NV FTD B ...... Banach algebras of continuous spaces functions N7P 3 . . . Hermitian operators N7P 4F . . . Fredholm operators

68 AWO8X Analysis AWOFUAPMOCLFVJ

Mathematical systems AR5 Mathematical systems AR5 Analysis AW Analysis AW Types of analysis AWN Y Types of analysis AWN Y Relations Algebras AWO FTA . Functions AWO 8L . . Types by relation ...... Banach algebras of continuous functions . . . Banach algebras of continuous functions AWO 8NV FTD B AWO FTD BKN V

AWO 8X . Functionals . . Types by property . . Types by property AWO FTD BO9 . . . Commutative Banach algebras 8XN A . . . Linear functionals . . . . Relations . . . . Elements FTD BO9 9S . . . . . Representations of 8XN ADI . . . . . Moments commutative 8XN ADI 3A ...... Theory of moments Banach algebras . . Types of functionals by system FUA Spaces in functional analysis 8XW . . . Linear analytic functionals . Types by relation 94 . Transforms FUA KL . . Function spaces . . Types by property . . . Relations 94P 2 . . . Fourier transforms * For Functionals in linear 94P 28L LR . . . . Fourier transforms of generalized spaces see Linear spaces functions AWO FUA NA8 X. Properties . Types by property B5 . Measure FUA NA . . Linear spaces Elements . . . Relations DU . Derivatives FUA NA8 LLR . . . . Generalized functions DW . Functional integrals FUA NA8 NOG . . . . Homogeneous functions FUA NA8 NOG LR . . . . . Homogeneous generalized functions Subsystems in functional analysis FUA NA8 X . . . . Functionals in linear spaces FRB . Sets FUA NA8 XN4 J . . . . . Positive FRB X . . Boolean algebra in functional analysis FUA NA8 XN4 JMQ B ...... Positive definite . . . Relations functionals in FRB X9S . . . . Representations of Boolean algebra linear spaces FSA . Groups FUA P2 . . . Hilbert spaces . . Relations * For Banach spaces, see FSA 9S . . . Representations of groups AWP P2. FSA 9SN LD . . . . Infinite-dimensional representations FUA P85 . . . Normed linear spaces FSM . Rings . . . . Properties FSS . . Ideals FUA P85 BON . . . . . Reflexivity FST . . Modules FUA P85 BXD . . . . . Duality FST MR . . . Normal modules FUA P85 C4Y . . . . . Summability FST P2 . . . Banach modules . Types of spaces by element FTA . Algebras FUA PMO CL . . Inner product spaces, pre-Hilbert FTA G . . Subalgebras spaces FTA VJ . . Topological algebras . . . Elements . . . Relations FUA PMO CLD Y . . . . Structure FTA VJ9 S . . . . Representations . . . Subsystems FTC . . Lie algebras FUA PMO CLF VJ . . . . Topology . . . Relations FTC 9S . . . . Representations of Lie algebras FTC 9SN LD . . . . . Infinite-dimensional representations of Lie algebras FTD B . . Banach algebras . . . Types by relation FTD BKN V . . . . Banach algebras of continuous functions

69 AWOFUAVJ AWRFUA9MNBANT Analysis

Mathematics AM Analysis AW Mathematical systems AR5 Types of analysis AWN Y Analysis AW Subsystems in functional analysis Types of analysis AWN Y . . Types by system Types of spaces by element . . . . Topological linear spaces, topological vector . . . Topology AWO FUA PMO CLF VJ spaces AWP ...... Spaces of matrices AWP TB Types by system AWO FUA VJ . Topological spaces AWP Y . . . . Other types of topological spaces FUA VJN A . . Linear * Normal synthesis is resumed * For topological linear spaces use AWP. here after its interruption at Normal synthesis is interrupted here; it is AWO FUA VJN A. resumed at AWP Y. * Add to AWP Y letters NB/W in AWP . . Topological linear spaces, topological Auxiliary Schedule AM1. vector spaces AWQ A . . Other types of spaces in functional . . . Processes analysis * Normal synthesis is resumed here 8A . . . . Topological linear spaces of after its interruption at distributions AWO FUA VJN A. . . . Properties * Add to AWQ A letters VK/W from B4V . . . . Dimension Auxiliary Schedule AM1. B4V K6 . . . . . Approximate dimensions B Other subsystems in functional analysis BXD 3A . . . . Duality theory * Add to AWQ B letters UB/W following C4Y . . . . Summability A in AUB/AW. . . . Elements & entities H Types of functional analysis E2 . . . . Extremal problems of topological linear * Add to AWQ letters H/NA in Auxiliary spaces Schedule AM1. EW . . . . Bases in topological linear spaces NB . Non-linear * . . . Subsystems For Non-linear functional analysis, use AWR. Normal synthesis is interrupted FRB . . . . Sets here; it is resumed at AWR X. FRC N8U . . . . . Fractals (topological linear spaces) AWR . Non-linear functional analysis FTS . . . . Geometry of topological linear spaces . . Relations FUA . . . . . Spaces 8X . . . Non-linear functionals FUA LF ...... Conjugate spaces . . . . Methods ...... Relations 8X7 2 . . . . . Differential calculus for FUA LF8 X ...... Linear functionals of conjugate non-linear functionals spaces . . . . Entities FUB F . . . . . Subspaces 8XE U . . . . . Non-linear operators FUB FX ...... Retracts, retractions 8XE UNO T ...... Monotone operators FUC RB . . . . . Convex sets in linear spaces . . . . Subsystems . . . Types of topological linear spaces by 8XF VJ . . . . . Topological properties relation . . Entities KNV . . . . Spaces of continuous functions EU . . . Non-linear operators KW . . . . Spaces of analytic functions . . . . Methods . . . Types of topological linear spaces by EU6 X . . . . . Infinitesimal calculus for property non-linear operators OD . . . . Convex spaces . . . . Entities ODN F . . . . . Locally convex spaces EUE G . . . . . Eigenvalues of non-linear P2 . . . . Banach spaces operators P5F . . . . Bounded . . Subsystems P5F NF . . . . . Locally bounded topological linear FUA . . . Functional spaces spaces . . . . Relations P82 . . . . Kernel spaces FUA 9MN B . . . . . Non-linear equations P85 . . . . Normed spaces FUA 9MN B3K S ...... Incorrectly posed problems . . . Types by entity ...... Properties QO . . . . Spaces of sequences FUA 9MN BAN T ...... Existence . . . Types by system TB . . . . Spaces of matrices

70 AWRX Analysis AWT8LFRB

Mathematics AM Analysis AW Mathematical systems AR5 Types of analysis by other relations etc AWR Y Analysis AW . . Subsystems Types of analysis AWN Y . . . Groups AWS FSA . . Types of functional analysis AWQ H . . . . Subsystems ...... Existence . . . . . Algebras on groups AWS FSA FTA AWR FUA 9MN BAN T ...... Measure algebras on groups AWS FSA FTA N5 AWR X . . Types of functional analyis by other properties etc AWS FSA FUV NI . . . . . Integral geometry * Normal synthesis is resumed here after its . . . . Types of groups by property interruption at AWQ NB. FSF . . . . . Abelian groups (harmonic * Add to AWR X letters NC/W in Auxiliary analysis) Schedule AM1. FSF NR ...... Compact Y Types of analysis by other relations etc FSF NRN F ...... Locally compact Abelian * Normal synthesis is resumed here after its groups, LCA groups interruption at AWN YKL. * Add to AWR Y letters KX/NXH in Auxiliary ...... Relations Schedule AM1 (representing AM8X/ANXH in FSF NRN FA9 4 ...... Transforms on locally the schedule). compact Abelian YN5 . Measure, measure theory groups * See also Probability AWX FSF NRN FA9 4P2 ...... Fourier transforms . . Operations FUA . . . Spaces YN5 7V . . . Integration & disintegration of measures FUA NOG . . . . Homogeneous spaces . . Relations . . . . . Relations YN5 8L . . . Measurable functions FUA NOG 8L ...... Functions on homogeneous YN5 8YM . . . . Non-measurable functions spaces . . Properties . . Types of harmonic analysis by YN5 B4V . . . Dimensions property YN5 B8U . . . . Hausdorff dimension MO . . . Abstract harmonic analysis . . Types by property . . . . Methods YN5 O2J . . . Invariant measures MO6 SX . . . . . Categorical methods YN5 P2 . . . Lebesgue measures . . . . Relations YN5 P4H . . . Hausdorff measure MO8 OYB . . . . . Spherical functions YN5 P4I . . . Hausdorff distance, Hausdorff metric . . . . Subsystems . . Types by entity MOF SA . . . . . Groups & semi-groups YN5 QH . . . Vector valued measures ...... Operations YNX H . Harmonic analysis MOF SA7 7 ...... Synthesis * For Harmonic analysis, use AWS. Normal MOF SA7 7QN ...... Spectral synthesis on synthesis is interrupted here; it is resumed at groups & AWS Y. semigroups AWS . Harmonic analysis, spectral analysis Y Types of analysis by other properties etc . . Subsystems * Normal synthesis is resumed here after its FSA . . . Groups interruption at AWR YNX H. . . . . Methods * Add to AWS Y letters NXJ/QP2 in FSA 6O4 Y . . . . . Summability methods on groups & Auxiliary Schedule AM1. semi-groups YQP 2 . Fourier series . . . . Relations * For Fourier analysis, use AWT...... Functions on groups Normal synthesis is interrupted here; it is resumed at AWT Y. FSA 8L ...... Harmonic analysis of functions on AWT . Fourier analysis groups . . Processes ...... Types of functions 8E . . . Interpolation FSA 8NX G ...... Almost periodic functions on 8EV E . . . . Trigonometric interpolation groups . . Relations FSA 9S . . . . . Representations of groups & 8L . . . Functions semi-groups . . . . Subsystems . . . . Subsystems 8LF RB . . . . . Sets of functions FSA FTA . . . . . Algebras on groups ...... Properties FSA FTA N5 ...... Measure algebras on groups

71 AWT8LFRBBTN AWY Probability

Mathematical systems AR5 Mathematics AM Analysis AW . . . Operations . Types of analysis by other properties etc AWS Y . . . . Convergence of probability measures AWX 8B . . . Relations . . . . Functions AWT 8L . . . Relations . . . . . Subsystems ...... Properties AWX 8L . . . . Functions 8L8 H . . . . . Optimization AWT 8LF RBB TN ...... Completeness of sets of 8L8 HX . . . . . Minimization function 95 . . . . Transformations . . . . . Types by property . . . Properties 8N4 J ...... Positive D8 . . . . Special statistical properties 8N4 JMQ B ...... Positive definite functions * Add to AWX D8 letters B/Y following 8NX G ...... Almost periodic functions AXA in AXA B/AXA Y. 8NX GI4 ...... Classical almost periodic * Add to AWX D9 letters B/J following functions AXB in AXB B/AXB J. A selection of the more prominent concepts is given here for 94 . . . . Fourier transforms convenience. . . . Elements . . . . . Dependence & independence DI . . . . Moment problems D8L ...... Dependence DIV E . . . . . Trigonometric moment problems D8M ...... Independence . . . Entities D8X . . . . . Evidence EQ . . . . Series D9F . . . . . Chance . . . . . Types by property . . . Elements & entities EQO U ...... Series of orthogonal functions DH . . . . Limits EQO UAN U ...... Uniqueness for orthogonal series DI . . . . Moment problems EQO UBF ...... Localization for orthogonal ES . . . . Invariants series ET . . . . Variables . . . . . Types by system . . . Systems EQV E ...... Trigonometric series * Include probability theory and measures on ...... Properties mathematical structures EQV EC4 Y ...... Summability RD . . . . Combinatorial probability Y . Types of analysis by other properties etc RX . . . . Algebraic structures in probability * Normal synthesis is resumed here after its SA . . . . . Groups (probability) interruption at AWS YQP 2. * See also Fourier transforms, AWT94 * Add to AWT Y letters QP3/W in Auxiliary TS . . . . Geometric probability Schedule AM1. VJ . . . . Topological structures in probability VKO N . . . . . Metric spaces in probability AWX Probability VKV J . . . . . Topological spaces in probability * Most of the literature on this branch of W . . . . Analysis in probability mathematics refers to statistics and so the * See also Stochastic processes, AWU XN detailed schedule appears under class AX W6X . . . . . Calculus of probability Statistics and probability. W8L * A selection of concepts prominent in general . . . . . Functions in probability probability is given below to demonstrate the WB ...... Potential theory in probability scope of the schedule. . . . . Special systems * Division is like that for any other branch of * Add to AWU X letters G/X following AX. Mathematics, following Auxiliary Schedule XGT . . . . . Conditional probability AM1, but with the addition of some concepts XH . . . . . Random variables from AX. XI . . . . . Probability distributions . Theory XN . . . . . Stochastic processes 3A . . Abstract probability theory 3D . Axioms AWY Applied mathematics 3P . Foundations * Other than Statistics (AX). 4 . . Logical probability * The preferred arrangement is to subordinate the . Operations mathematics of a given subject (e.g. Physics) to the 86 . . Approximation subject. 8B . . Convergence of probability measures

72 AWY Statistics & probability AX6H

Applied mathematics AWY Applied mathematics AWY Statistics & probability AX * Two alternatives are provided: First, to locate Forms of mathematical presentation AX3 4 applications at the end of the class to which they refer; for this, the letter Y is reserved in Auxiliary Schedule AM1 - AX3 5 . Errors in descriptive statistics see the notes given there. Second, this location (AWY) is * Apply only to Descriptive statistics. In Mathematical an alternative for libraries wishing to keep all applied statistics use AXA B. mathematics together. If this option is taken, proceed as . Theory follows: A . . General theory of statistics & probability * Add to AWY numbers and letters 4/9, A/Z from the * Use only for works dealing with descriptive as well whole classification, e.g. AWY B Mathematics of as mathematical statistics. physics. * Each subject (where the hyphen represents the classmark) AX5 Descriptive statistics may be fully qualified by preveding facets, as follows: * Add to - the number 2 followed by numbers 2/9 from 7 . Data collection Auxiliary Schedule 1 (with any amendments shown at * The detailed schedule in K7 applies primarily to social AM2). sciences but may be used here so far as applicable. * Add to - the number 3 followed by numbers 3/9 following * Add to AX5 7 numbers & letters 8/9, A/V following AM. K7. * Add to - the number 3 followed by letters MB/WX 77 . . Survey design following A. * Interviewing, questionnaires, etc. * Further subdivisions of the subject field to which 78 . . . Confidentiality Mathematics is applied may be added as follows: 796 . . . Non-response * Add to - the number 4 followed by numbers 3/4 3/4 for 7B . . . Factor control any topical subdivisions of the class numbered 3/4. 7C . . . Sampling operation (narrowly) * Add to - the numbers 5/9 for any topical subdivisions of * For sampling distribution and theory see AXW. the class numbered 5/9. 7J . . . Observation * Add to - letters A/Z. 7JL . . . . Recording observations * For a worked example of this method, see Applied statistics AXY. 7L . . . . Questioning 7M . . . . . Interviewing 7N . . . . . Questionnaires AX . Statistics & probability 7X . Data processing AX2 . . Common subdivisions * Use AX2 M. * These are amended in the same way as for 8 . Presentation of data Mathematics. 8C . . Charts, graphical presentation, pictorial presentation * Add to AX2 numbers 2/9 from Auxiliary Schedule 1 with the adjustments given in AM2. 8E . . Tabulation 3N . . . Statistical tables 8H . . Index numbers 7 . . . History of statistics & probability 8P . Preservation of original data 9 . . . . Biography of statisticians 9 . Descriptive measures 9B . . . Relations with other subjects 9C . . Classification of data * See also Analysis of variance, AXT A . . . . Philosophy . . Agents & instruments in statistical operations AX6 . . Frequency distributions * Distributions derived from actual data serving purely M . . . Data processing in statistics descriptive function. For distribution theory & * The provision below expands that at AM2M to theoretical distributions, see AXI 73A. meet the increased need for detail in Statistics. * For variables in general, see AXG H. MG . . . . Machines & equipment in statistics 7 . . . Population characteristics & parameters N . . . . . Computers in statistics 9 . . . . Central tendency, averages, means in statistics P ...... Programs in statistics B . . . . . Clustering Q . . . . Special operations C . . . . . Arithmetic mean * Add to AX2 Q letters H/Y following K7X. D . . . . . Median * Add to AX2 R letters C/L following K7Y. E . . . . . Mode QC . . . . . Coding in statistics F . . . . . Geometric mean AX3 4 . . Forms of mathematical presentation * These apply mainly to mathematical statistics (AX7). G . . . . . Harmonic mean Use this position only for those few concepts which H . . . . . Logarithmic mean may be needed to qualify Descriptive statistics (AX5/6). * Add to AX3 numbers 5/9 & letters A/L following AM3 in AM35/39, AM3A/3L.

73 AX6J AX7ET Mathematical statistics

Statistics & probability AX Applied mathematics AWY Descriptive statistics AX5 Statistics & probability AX . . Frequency distributions AX6 Mathematical statistics AX7 . . . Population characteristics & parameters AX6 7 Methodologies . . . . Central tendency AX6 9 . . Calculus AX7 6X . . . . . Logarithmic mean AX6 H Operations AX6 J . . . . Variability AX7 7R . Differentiation K . . . . . Dispersion 7V . Integration * Most of the literature implies Probability 7W distributions; see AXI BK. . Partition * See also Analysis of variance, AXT L . . . . . Deviation Processes M ...... Standard deviation 86 . Approximation N ...... Standard error 8B . Convergence O ...... Large deviation 8E * See also Limit theorems, AXM 73F. . Interpolation P . . . . . Range in statistical populations 8EX . Superposition Q ...... Quartiles 8GM . Decomposition R ...... Extreme quartiles range 8H . Optimization S . . . . . Skewness 8HX . Minimization T . . . . . Kurtosis Relations X . . . Types of frequency distribution 8L . Functions in mathematical statistics * Add to AX6 X letters IJ/M following AX e.g. 8YJ . . Functions special to statistics Poisson frequency distribution AX6 XKL. * These are subordinated to specific problems - e.g. Y . . Other descriptive measures Markov processes - Transition functions AXO 78Y J; Analysis of experiments - Functions - Contrasts AXS 78Y H. AX7 Mathematical statistics, statistical mathematics 94 . Transforms * The complete mathematics schedule is available here if 95 . Transformations need be. In some cases, a concept has its own special 9E . Extensions significance and set of relationships in statistics and in 9L . Equations such cases this should be preferred to the concept as 9MP 3 derived from AM/AW, e.g. Decision procedures AXC J . . Planck’s equation (not AX7 4GJ); inequalities AXA V (not AX7 9N). 9N . Inequalities * Add to AX7 numbers & letters 3/9, A/W in Auxiliary * See also Statistical methods AXA V Schedule AM1, with details from AM/AW. 9QY . Proportion, ratio * Occasionally, a mathematical concept will need to be 9S . Representations qualified by a concept from the purely statistics class. In AB . Homomorphisms such cases the end of the classmark derived from AC . . Automorphisms Mathematics must be signalled by ’Z’ before adding the Properties statistics component - e.g. Discrete distributions - CG . Asymptotic Superposition - Decomposition AXJ K78 EXZ 8DE. CU (See also Introduction, Section 10.42). . Orthogonal . Forms of mathematical presentation Elements 3A . . Theory DC . Domain 3D . . Axioms DCY . Range 3F . . Theorems DF . Boundary 3L . . Models DG . Solutions 3LX . . . Simulation DH . . Limits . Methodologies EB . Points 3P . . Foundations ED . Spaces 4 . . . Mathematical logic Entities 4G . . . . Algorithms EE . Scalars 4GJ . . . . . Decision procedures EH . Vectors * See also Decision making AXC J. EK . Coefficients 6D . . Numerical analysis EN . Spectra 6RS . . Algebraic methods EP . Sequences, progressions 6TS . . Geometric methods EQ . Series 6VJ . . Topological methods ES . Invariants 6W . . Analytic methods ET . Variables 6X . . . Calculus

74 AX7F Statistical method AX8TX

Applied mathematics AWY Applied mathematics AWY Statistics & probability AX Statistics & probability AX Mathematical statistics AX7 Mathematical statistics AX7 Entities Systems, branches of mathematics . Variables AX7 ET . . . Bayesian theorem AX7 X3F

AX7 F Subsystems AX8 Statistical method * Add to AX7F letters R/W following A in AR/AW. 8 . Conceptual agents H Types * For use when any statistical concept is used in an * Auxiliary Schedule AM1 provides (via H/W) a set exceptional role as an agent. of specifiers for indicating particular types of * Add to AX88 numbers & letters 8/9, A/X following anything. AX in AX8/AXX, e.g. Regression - Estimation - * In it, letters H/Q introduce the earlier facets in Maximum likelihood - using Normal distribution AM3/AQ (Methodologies, Operations, Processes, AXU NDK 88 KP. etc.) when these are used as specifiers to signify 9 . . Special concepts as agents types (species), e.g. Spaces AX7 UN; Linear spaces * When special to a given context; see Estimators AX7 UN NA (in which the -NA is from ANA). AXD 89E for example. Systems, branches of mathematics A . Properties of operations RB . Sets * Use this provision only when qualifying the RD . Combinatorics Operations below (AX8D/AX9). RI . Arithmetic * Add to - (where hyphen represents the classmark of RJ . Number theory the operation) letters A/B following AX in AXA/AXB, e.g. Errors in scaling AX8 UFA B. RQB . Ordered structures . Operations RR . . Lattices * All operations in AX8D/AX9C may be qualified RS . Algebra retroactively by dropping the initial AX8 of SA . . Groups preceding classmarks; e.g. AX8 TF (Reliability - TB . . Matrices Tests). TB8 YD . . . Determinants D . . Analysis in statistics, statistical analysis in TS . Geometry general VJ . Topology * Use only for qualification of concepts filing W . Analysis before design & analysis (AXR) where it is W6X . . Calculus mainly applicable. W8L . . Functions DE . . . Decomposition WB3 A . . . Potential theory F . . Testing, tests WLE U . . Operators S . . Measurement WO . . Functional analysis * The following details are taken from K6S/K6Y. (social science research). Notation is modifed as WRY N5 . . Measure theory indicated. Further details may be obtained from WS . . Harmonic analysis, spectral analysis K6T/K6Y. WX . Probability . . . Properties * See AXG SAB . . . . Errors . Special systems SAI . . . . Accuracy X . . Bayesian statistics SR . . . . Scales & scaling * Hypotheses classified by experimenter’s confidence in them, modified as experiment ST . . . . Validity & reliability proceeds. T . . . . . Reliability, precision * See also Conditional probability, AXG T. TF ...... Tests of reliability X3F . . . Bayesian theorem TG ...... Test-retest TH ...... Equivalent forms TJ ...... Split halves test TV . . . . . Validity, relevance ...... Properties TVA I ...... Accuracy TVA Q ...... Bias TWE ...... External validity, empirical validity TWN ...... Internal validity TX ...... Construct validity, congruent validity

75 AX8UC AXCL Statistical method

Mathematical statistics AX7 Statistics & probability AX Statistical method AX8 Mathematical statistics AX7 Operations Statistical method AX8 . Measurement AX8 S Properties in statistical method AXA . . Properties . Dependence & independence AXA KY . . . . . Construct validity AX8 TX . . Independence AXA M . . Techniques & characteristics of measurement AXA N . . Laws of large numbers AX8 UC . . . Scaling, scales NP . . . Weak law UD . . . . Construction of scales NS . . . Strong law UF . . . . Scaling techniques P . Consistency in statistical methods UH . . . . Levels of measurement Q . Bias UJ . . . . . Discrete scales, discontinuous scales R . Unbiasedness UK ...... Non-metric scales S . Efficiency & inefficiency UL ...... Nominal scales, category scales * For Asymptotic efficiency, see Estimators AXD 89E AS UM ...... Ordinal scales, rank scales T . Robustness UMV ...... Ranking techniques U . Risk UQ . . . . . Continuous scales V . Inequality in statistical method UR ...... Metric scales W . Likelihood US ...... Interval scales X . Evidence UV ...... Ratio scales, absolute scales . . . . By number of variables AXB B . Expectation * Usually implies random variables. UX . . . . . One variable measurements D . Degrees of freedom V ...... Rating scales, intensity scales F . Chance W . . . . . Multivariable measurements G . Properties special to a context XB ...... Unidimensional scales * Classmarks -BG/-BY are reserved for properties YB ...... Multidimensional scales special to a particular context and which are YG . . . Other techniques therefore enumerated under that context. For * See Measures of stochastic processes AXN 8YG examples of their application see Hypothesis for example. testing AXF; Probability distributions AXI. AX9 C . Sampling operation * For samples as statistical models, see AXW. Types of methods D . Operations special to a context * Classmarks -9D/9U are reserved for operations . By distribution assumed special to a particular context, which are therefore AXC D . . Parametric methods enumerated under that context. For an example of its * Usually assumed. Use to qualify specific application, see AXR 9 Design of experiments. concepts only when explicitly distinguished AXA Properties in statistical method from non-parametric. * Many of the properties below have special (& possibly F . . Non-parametric methods, distribution-free unique) reference to particular operations, etc. (e.g. methods inference, test of significance). When they do refer to a F8S . . . Measurement particular problem they should, of course, be cited after F8U MV . . . . Ranking techniques that problem, e.g. Inference - Consistency. G . . . Order statistics B . Errors in statistical method . By technique * For Type I & II errors, see AXF AE. J . . Decision making C . . Control of errors J3A . . . Decision theory D . . Process errors * See also Bayesian probabilities AXG T7X G . . Probable errors . . . Special properties H . . Random errors JBJ . . . . Admissability I . Accuracy JBK . . . . Utility J . Sufficiency JBK 3A . . . . . Theory * See also Comparison of experiments AXR 9Q JBL . . . Loss K . Exchangeability JBM . . . . Regret KP . . Partial exchangeability K . . Change detection KY . Dependence & independence L . . Compound decision, multiple decision * Applicable primarily to Conditional probability AXG T. L . . Dependence M . . Independence

76 AXCN Inference AXFPV

Applied mathematics AWY Statistics & probability AX Statistics & probability AX Mathematical statistics AX7 Mathematical statistics AX7 Statistical method AX8 Statistical method AX8 Inference AXC N By technique Estimation AXD . Compound decision AXC L . . . Non-parametric methods AXE CF

AXC N Inference AXE G . . . Special intervals . Properties * See measure involved: e.g. confidence interval NAP . . Consistency for variance of normally distributed variables. NAR . . Unbiasedness AXF Tests of significance, hypothesis testing, statistical NAS . . Efficiency testing NAT . . Robustness . Theory AXD . Estimation, parameter estimation, prediction 73C A . . Asymptotic theory (estimation) . Properties * See also Expectation (probability) AXH BB. AB . . Errors 7 . . Mathematics AE . . . Type I error 73C A . . . Asymptotic theory AF . . . Type II error 73C F . . . Filtering theory AR . . Unbiasedness 74G . . . Algorithms AS . . Efficiency . . . Bayesian statistics BG . . Power in hypothesis testing 7XZ CN . . . . Bayesian inference * Probability of rejecting null hypothesis. . . Conceptual agents BG7 EN . . . Power spectra 89E . . . Estimators BH . . . Uniformly most powerful, UMP . . . . Properties BJ . . Discrimination 89E AS . . . . . Asymptotic efficiency CD . Parametric tests . . Properties * For specific parametric tests, see Types below. CF . Non-parametric tests AU . . . Risk DJ . Point estimation AV . . . Inequality * See also Tolerance intervals AXE DP . . Minimax test AW . . . Likelihood . Types . . Types of estimation . . By complexity * See also Bayesian inference AXD 7XZ CN GS . . . Simple tests F . . . Finite population process GT . . . . Individual tests, A/Z G . . . Prediction (estimation) GV . . . Composite tests * See also Correlation AXU E GW . . . . Individual tests, A/Z G7 . . . . Mathematics . . By method G7W S . . . . . Harmonic analysis H . . . Likelihood ratio test H . . . Underestimation * For maximum likelihoood, see Point J . . . Point estimation estimations, AXD K. . . . . Properties I . . . Sequential tests JAP . . . . . Consistency J . . . Other parametric tests (A/Z) JAR . . . . . Unbiasedness . . By sample JAS . . . . . Efficiency K . . . Two sample problem JCF . . . . Non-parametric methods L . . . K-sample problem . . . . Types . . By direction K . . . . . Maximum likelihood M . . . Direction test L . . . . . Least squares N . . . Non-directional tests, two sided tests, two M . . . . . Fiducial estimation tailed tests . . . . . Minimax test . . By hypothesis * See AXF DP. O . . . Hypothesis test AXE . . . Interval estimation, confidence intervals, * Add to AXF P letters K/V following K6I. confidence limits, tolerance intervals PK . . . . Working hypothesis . . . . Properties PP . . . . . Null hypothesis BD . . . . . Degrees of freedom PQ . . . . . Alternative hypotheses CF . . . . Non-parametric methods . . . . By propositions factor PV . . . . . Complex hypotheses, composite hypotheses

77 AXFPW AXGU Statistical probability

Mathematical statistics AX7 Applied mathematics AWY Statistical method AX8 Statistics & probability AX . Types of methods Mathematical statistics AX7 . . . Tests of significance AXF Statistical probability AXG . . . . . By hypothesis Properties ...... Complex hypotheses AXF PV . Chance AXG BF

AXF PW ...... Simple hypotheses Elements . . . . . By other non-purposive characteristics AXG H . Variables in statistical probability Q ...... Outlier tests I . . Treatment variables . . . . . By purpose * For the general class, see Design & analysis of R ...... Goodness of fit experiments AXQ RGI. Use this provision only when qualifying (e.g. Optimal designs - S ...... Tests of independence Treatment variables AXR SP GI). . . . Types of inference J . . Dependent variables U . . . . Abstract inference . . . Properties V . . . . Linear inference JBG . . . . Contingency W . . . . Simultaneous inference JBG 23N . . . . . Contingency tables . Design & analysis of experiments K . . Independent variables * See AXR. L . . Discrete variables, discontinuous variables Y Probability * Alternative (not recommended) for libraries wishing M . . Continuous variables to keep together all Probability. The preferred N . . Qualitative variables arrangement is to treat probability in general as a N8S R . . . Scales branch of mathematics, at AWX. N8U K . . . . Non-metric scales * If this option is taken, add to AXF Y in the same way O . . . Nominal variables, categorical data, as shown at AWX (i.e. using Auxiliary Schedule classification variables AM1). P . . . Ordinal variables, ranked data * The position at AWX would still be retained for use Q . . Quantitative variables as a qualifier of specific mathematical concepts in class AM/AW. Q8S R . . . Scales * See also Logical probability AWX 4. Q8U R . . . . Metric scales Q8U RS . . . . . Ordered metric scales R . . . Interval variables AXG Statistical probability S . . . Ratio variables * For probability in pure mathemetics, see AWX (or the alternative at AXF Y). . . Random variables * See AXH. 7 . Mathematics 7B5 . . Measure theory * As applied to statistics. Systems . . Elements T . Conditional probability 7DG . . . Solutions . . Bayesian statistics 7DH . . . . Limits (statistical probability) T7X . . . Bayesian probabilities . . Entities * See also Decision theory AXC J3A 7EN . . . Spectra (statistical probability) . . . Properties 7ES . . . Invariants (statistical probability) TAK Y . . . . Dependence & independence 7ET . . . Variables . . . Methods * See AXG H. TCN . . . . Inference 7W . . Analysis . . . . . Bayesian statistics 7W6 X . . . Calculus of probability TCN 7X ...... Bayesian theorem . Properties U . . Transition probability AKY . . Dependence & independence AX . . Evidence BF . . Chance

78 AXH Probability distributions AXIBX

Applied mathematics AWY Statistics & probability AX Statistics & probability AX Mathematical statistics AX7 Mathematical statistics AX7 Statistical probability AXG Statistical probability AXG Random variables AXH Conditional probability AXG T Types . Transition probability AXG U . . Point random variables AXH QB

AXH Random variables, variates AXI Probability distributions, statistical distributions 7 . Mathematics 7 . Mathematics . . Functions . . Theory 78L . . . Random functions 73A . . . Distribution theory . . Elements & entities * See also Monte Carlo methods, AXX X 7DK . . . Sums of random variables 73C A . . . . Asymptotic theory . . . . Integrations 73C C . . . . Characterization & structure theory 7DK 7V . . . . . Convolution 73C I . . . . Infinitely divisible laws 7EH . . . Vectors 73C S . . . . Stable laws 7EP . . . Sequences 73L . . Models 7EP 8B . . . . Convergence 73L X . . . Simulation . . Systems 73L YB . . . Urn models 7RS . . . Algebra 78L . . Functions 7SV . . . . Random fields * See also Properties of distributions, AXI BK 7W . . . Analysis 78P 2 . . . Hartree Fock functions 7WO . . . . Functional analysis 794 . . . Transforms of variates 7WP . . . . . Topological linear spaces 795 . . Transformations 7WP P2 ...... Banach spaces 79C X . . . Probit transformation . Properties 79L . . Equations BB . . Expected values 79M P3 . . . Planck’s equation BBA B . . . Errors 7W . . Analysis BBA G . . . . Probable errors 7WT . . . Fourier analysis . . Normality . Properties * See Normal distribution, AXK P. BK . . Statistical dispersion . Types * See also Variability AX6 J; Variance theory . . By statistical property AXT 73A GJ . . . Dependent real variables BKL . . . Dispersion functions GK . . . Independent real variables BL . . Homogeneity GK7 DK . . . . Sums BM . . Heterogeneity . . . . . Limits BN . . Shape of distributions GK7 DKD H ...... Limit theorem BP . . Moments in statistical probability H . . By mathematical characteristics BQ . . . Moment generating functions, MGF * Add to AXH letters H/W from Auxiliary BR . . . Moment about the mean Schedule AM1, e.g. Transformed Beta BS . . . Variance in statistical probability distributions AXK WB HL4 BSS . . . . Semi-invariants NOG . . . Homogeneous random variables BV . . Characteristic function QB . . . Point random variables * See also Fourier analysis AXI 7WT BX . . Density function, probability density function, PDF, distribution function, frequency function

79 AXJD AXN7EPDH Probability distributions

Mathematical statistics AX7 Mathematical statistics AX7 Statistical probability AXG Statistical probability AXG Random variables AXH Random variables AXH Probability distributions AXI Probability distributions AXI Properties . . . Univariate distributions AXK . Density function AXI BX . . . . . Others AXK X Types of probability distributions AXL . . . Multivariate distributions . By mode 78L . . . . Functions AXJ D . . Unicharacteristic distributions, unimodal 78L KGM . . . . . Decomposition function distributions 8S . . . . Measurement E . . Bimodal distributions 8SR . . . . . Scaling G . Conditional & non-conditional distributions H . . . . Random variables H . Marginal distributions H78 L . . . . . Random functions of several variables K . Discrete distributions HNO G ...... Homogeneous random variables K78 EX . . Superposition HQB ...... Point random variables K7P G . . Solutions JK . . . . Discrete distributions K7P H . . . Limits L . . . . . Multinomial distribution O . Continuous distributions M . . . . . Others (A/Z) . By number of variables O . . . . Continuous multivariate distributions AXK . . Univariate distributions P . . . . . Multivariate normal distribution 8S . . . Measurement Q . . . . . Others (A/Z) 8SR . . . . Scaling V . . . . Bivariate distributions JK . . . Discrete AXM . . . Limit distributions, limiting K . . . . Binomial distribution . . . . Theorems * For Multinomial distribution see 73F . . . . . Limit theorems Multivariate distributions, AXL L. * See also Law of large numbers KBX . . . . . Density functions AXM AN; Large deviations, AX6 O. KN . . . . . Negative binomials 73F XC ...... Central limit theorem L . . . . Poisson distribution 73F XE ...... Zero-one law MB . . . . Logarithmic series distribution 73F XM ...... De Moivre-Laplace theorem ME . . . . Contagious distribution . . . . Properties MG . . . . Exponential distribution AN . . . . . Laws of large numbers MJ . . . . Hypergeometric distribution ANS ...... Strong theorems N . . . . Others (A/Z) . . . . Types of limit distributions O . . . Continuous univariate distributions N . . . . . Asymptotic distributions P . . . . Normal distribution, gaussian distribution P79 5 . . . . . Transformation AXN Stochastic processes, stochastic chains Q . . . . Chi-square distribution 7 . Mathematics * By far the greater part of the literature is on . . Theory the use of these in tests of significance. 73A . . . General theory of stochastic processes S . . . . F distribution . . Methods T . . . . T distribution 76R D . . . Combinatorial methods V . . . . Gamma distribution . . Processes WB . . . . Beta distribution 786 . . . Approximation WF . . . . Log-normal distribution 78B . . . . Convergence WH . . . . Truncated distribution . . Functions WJ . . . . Mixed distribution * See Stochastic analysis AXN 7W WL . . . . Extreme value distribution . . Elements & entities WN . . . . Weibull distribution 7EP . . . Sequences WP . . . . Pearson distribution 7EP DH . . . . Limit theorems for stochastic processes WS . . . . Sequential distributions . . Systems X . . . . Others (A/Z) * For example, Cauchy distribution AXK XCA.

80 AXN7W Stochastic processes AXOT

Mathematical statistics AX7 Mathematical statistics AX7 Statistical probability AXG Statistical probability AXG Random variables AXH Random variables AXH Stochastic processes AXN Stochastic processes AXN Mathematics AXN 7 Inference AXN CN . Systems . . . Reversibility AXN UBS

. . Analysis Types of stochastic processes AXN 7W . . . Stochastic analysis AXN W . Ergodic processes 7W6 X . . . . Calculus AXO . . Markov processes 7W7 2DV . . . . . Stochastic differentials 7 . . . Mathematics 7W7 3DW . . . . . Stochastic integrals 78L . . . . Functions 7W8 L . . . Functions 78Y J . . . . . Transition functions . . . . Theory 78Y L . . . . . Green’s function 7W8 L3A . . . . . Random function theory 78Y M . . . . . Sample function 7WB . . . . . Harmonic functions 795 . . . . Transformations 7WB 3A ...... Potential theory . . . . Elements . . . Equations 7DF . . . . . Boundaries, general boundary 7WE L . . . . Stochastic equations conditions 7WE ME . . . . . Differential equations 7DF 3A ...... Boundary theory 7WH . . . . . Integral equations 7DF X ...... Martin boundary 7WO . . . Functional analysis 7VJ . . . . Topology 7WO FUA KL . . . . Function spaces 7VO ZXO . . . . . Topologies connected with Markov 8S Measurement processes 8SA B . Errors 7W . . . . Analysis 8SA H . . Random errors 7W8 L . . . . . Functions 8YJ . Induced measures ...... Potential theory . . Properties 7WB 3A ...... Probabilistic potential theory 8YJ BQ . . . Continuity 7WL EU . . . . . Operators 8YJ BR . . . Singularity 7WM XX ...... Infinitesimal & characteristic 8YK . Stopping times operators 8YL . Extreme values 7WO . . . . . Functional analysis 8YL 3B . . Asymptotic theory 7WO 8LN F ...... Local time functions CN Inference 7WO 8RL JJ ...... Additive functions DG . Prediction * See also Potential theory, DG3 A . . Prediction theory AXO 7WB 3A O . Time 8S . . Measurement P . Sample path 8S7 94 . . . Transformation of measures PQ . . Generalized sample paths . . Special properties S . Stochastic dynamical systems AU . . . Markov risk S76 D . . Numerical analysis BT . . . Strong Markov properties ST . . States in stochastic processes D . . Estimation STD . . . Estimation I . . Distributions SV . . Linear stochastic dynamical systems I7D H . . . Limits T . Ensembles I7D H3F . . . . Ergodic theorem U . Networks in statistical inference . . . Discrete . . Properties JK . . . . Markov processes with discrete UBS . . . Reversibility parameters JO . . . Continuous distributions Q . . Markov chains QU . . . Denumerable Markov chains R . . Birth & death processes RT . . . Logistic processes . . . . Transformation RT7 95 . . . . . Logit S . . . Birth processes, pure birth processes T . . Poisson processes

81 AXOU AXR9N Random variables

Mathematical statistics AX7 Applied mathematics AWY Statistical probability AXG Statistics & probability AX Random variables AXH Mathematical statistics AX7 Stochastic processes AXN Statistical probability AXG . . Ergodic processes AXN W . . . . Games theory AXP X . . . Poisson processes AXO T . . . . . Differential games AXP YD

AXO U . . . Gaussian processes AXP YF . . . . . Search games U7 . . . . Mathematics YH . . . . . Pursuit & evasion games U7D U . . . . . Derivatives AXQ B . . . . Queuing theory . . . . Properties C . . . . . Congestion UAL . . . . . Dependence D . . . . . Order in queuing theory UAL M ...... Asymptotic weakening of dependence F . . . . Renewal theory UBU . . . . . Sample function properties . . . . . Markov processes V . . . Branching process FO ...... Markov renewal processes, semi-Markov . . . . Birth & death processes * See AXO R G . . . . Inventory & storage VW . . . . Galton-Watson process H . . . . . Storage theory X . . . Polya process J . . . . Success runs AXP D . . . Diffusion process, epidemic processes K . . . . Interacting random processes E . . . . Non-linear diffusion process L . . . . Point processes, point random processes F . . . . Wiener process, brownian motion process M . . . . Traffic flows G . . . . Random walk N . . . . Innovation processes H . . . . . Jump processes P . . . . One dimensional series . . . Other Markov processes J . . . . Dirichlet forms R Statistical models, design & analysis of experiments K . . . . Markov processes with independent * See also Statistical method AX8 increments RGH . Variables L . . . . Hunt process RGI . . Treatment variables M . . . . Controlled Markov processes * Differences amongst effects observed by . . . . Semi-Markov processes experimenter. * See Renewal theory, AXQ F RGO . . Classification variables S . . Stationary processes AXR . Design of experiments, statistical design * See also Time series AXV * Do not use this classmark (AXR) in compounding. S7 . . . Mathematics Only specific design concepts are cited first, e.g. . . . . Theorems Optimal designs - Treatment variables S73 F . . . . . Ergodic theorems AXR SPG I. . . . . Processes . . Operations S78 E . . . . . Interpolation 9D . . . Randomization . . . Special processes 9E . . . . Random numbers ST . . . . Filtration 9F . . . Adjustment SV . . . . . Stopping times 9G . . . Sensitivity problems V . . Martingales 9H . . . Screening tests V7 . . . Mathematics 9J . . . Preference tests V78 B . . . . Convergence 9K . . . Response surfaces . . . . Limits 9L . . . Weighting V7D H . . . . . Martingale limit theory 9M . . . Replication Special probabilistic phenomena * Add to AXR 9M letters C/R following * A number of processes studied initially in a special K9C 6HW, e.g. Experimental units context (e.g. stores inventory) are then found to be AXR 9MC; Internal replication AXR 9MD. more widely applicable. A process which is special . . . . Properties to given application, goes with the latter. But these 9MB Y . . . . . Number of replications more generally applicable ones go here. 9MS . . . . Practical replication X . Games theory 9N . . . Confounding factors X7 . . Mathematics * Add to AXR 9N letters D/W following X7R J . . . Number theory K9C 7B, e.g. Matching AXR 9NG; YB . . Ill-posed problems Situational variables AXR 9NR. YC . . Dynamic non-cooperative games . . . . Randomization YD . . Differential games * See AXR 9D

82 AXR9P Analysis of experiments AXUK

Mathematical statistics AX7 Statistics & probability AX Statistical models AXQ R Mathematical statistics AX7 Design of experiments AXR Statistical models AXQ R . Operations Analysis of experiments AXS . . Confounding factors AXR 9N Multivariate analysis AXS S . . . Randomization . Canonical variables AXS U

AXR 9P . . Independent variables control AXS V . Factor analysis 9Q . . Combining tests, compounding tests, comparison W . Component analysis, element analysis tests X . Bivariate analysis 9R . . . Paired comparisons AXT Variance, analysis of variance 9S . . . . Tournaments 7 . Mathematics 9T . . . Multiple comparisons 73L . . Models 9U . . Operations special to a particular context 73L YC . . . Fixed effects model * See Sampling theory AXW 9V/X for example. 73L YE . . . Variance components model . Types of designs 73L YG . . . Mixed models * Add to AXR S letters E/G following K9C. Two EBD . Degrees of freedom major types are given here as examples. F . Tests of significance SE . . Experimental groups FO . . Hypothesis test SF . . Control groups FOP . . . Non-standard conditions SL . . Linear statistical models FOQ . . . . Failure of assumptions SN . . Log-linear statistical models GH . Variables data SP . . Optimal designs GO . . Categorical data SS . . Systematic designs GP . . Ranked data ST . . Repeated & sequential experiments RSY . Comparative designs SY . . Comparative designs, classical designs RWB . . Multicomparison procedures * See also Analysis of variance AXT RWD . . . Orthogonal comparisons T . . . Block designs RWF . . . Non-orthogonal comparisons U . . . . Randomized blocks, latin squares RWH . . . Students range, studentization UW . . . . Complete & incomplete blocks SS . Multivariate analysis V . . . Factorial designs, factorial arrangements . . Models VW . . . . With blocks factorial designs SS7 3L . . . Multivariate models WJ . . Representative designs V . Covariance, analysis of covariance WL . . Lattice designs AXU C Correlation & regression WN . . Mixed designs E . Correlation WP . . Field experiments * See also Prediction, AXD G WR . . Laboratory experiments E8S . . Measurement X . . Others (A/Z) E8S 7EK . . . Correlation coefficient EGN . . Qualitative variables AXS Analysis of experiments, statistical analysis EGN 7EK . . . Coefficients * Use only for general studies. In compounding, only EGO . . . Nominal variables specific analysis concepts (AXS/AXU) are cited first, EGO BG . . . . Contingency e.g. Variance analysis - Categorical data AXT GO. EGO BG2 3N . . . . . Contingency tables * For analysis in its looser and wider sense, see AX8 D. EGP . . . Ordinal variables 7 . Mathematics EGP 8UM . . . . Rank order 78H . . Optimization . . . . . Coefficients 78H X . . Minimization EGP 8UM 7EK ...... Rank order correlation coefficient 78L . . Functions EGQ . . Quantitative variables 78Y M . . . Contrasts . . . Coefficients RY . Univariate analysis EGQ 7EK . . . . Pearson product moment correlation S . Multivariate analysis, analysis of dispersion coefficient S7 . . Mathematics F . . Partial correlation S79 N . . . Inequalities G . . Autocorrelation T . . Discriminant analysis, discriminant classification H . . Serial correlation * See also Cluster samples, AXX F I . . Cross correlation TV . . . Discrete discriminant analysis J . . Spatial correlation U . . Canonical variables K . . Directional correlation * For canonical correlation, see AXU L

83 AXUL AXXH9YE Statistical models

Mathematical statistics AX7 Statistics & probability AX Statistical models AXQ R Mathematical statistics AX7 Analysis of experiments AXS Statistical models AXQ R . Correlation & regression AXU C Series design & analysis AXU Y . . Correlation AXU E . . Constituent movements . . . Directional correlation AXU K . . . Irregular fluctuations AXV VJ

AXU L . . . Canonical correlation AXV VN . . Stationary series * For Canonical variables, see AXS U * For stationary processes see AXP S N . . Regression VQ . . Non-stationary series N7 . . . Mathematics W . . Forecasting, prediction (forecasting) N7E L . . . . Polynomials * See also Estimation AXD O . . . Linear regression WV . . . Longterm forecasting. * Usually assumed. AXW Sampling theory, random sampling theory . . . . Hypothesis * Use of samples to estimate population parameters OFO . . . . . Linear hypothesis in regression and to determine magnitude of errors involved. P . . . Non-linear regression * See also Specific types of sampling distributions PV . . . . Curve fitting (e.g. F distribution AXK S). PW . . . . . Moving averages 3A . Theory PX . . . . . Least squares methods 8S . Measurement Q . . . Multiple regression analysis 8SR . . Scales Q78 L . . . . Functions 8T . . Precision Q78 YM . . . . . Contrasts . Operations Q78 YME K ...... Contrast coefficients 9V . . Acceptance . . . Autocorrelation analysis 9W . . Inspection * See Time series AXV UR 9X . . Process control S . Graphical analysis . Properties T . Longitudinal analysis BU . . Nature & number of units U . Nearest neighbour analysis CG . Order statistics UDH . . Underestimation CN . Inference V . Other types of analysis I . Sampling distributions Y Series design & analysis . Design AXV . Time series R . . Sampling design * See also Stochastic processes AXN S . Analysis 7 . . Mathematics SS . . Multivariate analysis 78L . . . Functions T . . Variance 794 . . . Transforms TRS Y . . . Comparative designs 7W . . . Analysis TRW H . . . . Studentization in sampling theory 7WS . . . . Harmonic analysis, spectral analysis UC . . Correlation & regression . . . . . Functions . Types of sampling 7WS 8L ...... Spectrum (time series), spectral X . . Small samples function, spectral density AXX A . . Probability samples 7WT . . . . Fourier analysis B . . Simple random samples O . . Markov processes C . . Stratified samples T . . Variance D . . . Proportional stratified samples T8F . . . Tests E . . Non-probability samples T8F X . . . . Runs EW . . . Quota samples UE . . Correlation EX . . . Judgement samples UH . . . Serial correlation F . . Cluster samples UN . . Regression FSS . . . Multivariate analysis UPV . . . Curve fitting . . . . Discriminant analysis UPW . . . . Moving averages FST . . . . . Cluster analysis UR . . . Autocorrelation regression G . . . . Multistage sample . . Constituent movements H . . Sequential sampling, sequential analysis VB . . . Longterm movements, secular trends . . . Special operations VE . . . Cyclical movements H9Y C . . . . Cumulative sum techniques VG . . . Seasonal variation H9Y E . . . . Optimal stopping VJ . . . Irregular fluctuations, random fluctuations * See also Stochastic processes - Stopping times AXN 8YH

84 AXXHI Statistics & probability AXYHOL

Statistics & probability AX Statistics & probability AX Mathematical statistics AX7 Applied statistics AXY . . Sampling theory AXW Applied statistics in health & medical sciences AXY HH . . . . Sequential sampling AXX H Clinical medicine & pathology AXY HN . . . . . Special operations Radiotherapy AXY HNS ...... Optimal stopping AXX H9Y E . Sampling theory AXY HNS 3W

AXX HI . . . . . Distributions AXY HOL Surgery HM ...... Limit distributions HMP ...... Non-central distributions . . . . . Types of sequential sampling I ...... Linear forms sampling J ...... Quadratic forms sampling . . . . Other types of sampling L . . . . . Sampling with unequal probability M . . . . . Multiphase sampling N . . . . . Double sampling O . . . . . Censored sampling P . . . . . Systematic sampling Q . . . . . Periodic sampling R . . . . . Work sampling S . . . . . Matched samples T . . . . . Panel data, panel samples X . . Monte Carlo methods * See also Numerical analysis AX7 6DA; Simulation AX7 3LX . . . Approximation X78 6X . . . . Stochastic approximation AXY Applied statistics * Alternative (not recommended) to locating under the specific subject wherever it is located in the general classification. * Add to AXY numbers 3/9 and letters A/Z from whole classification, e.g. Operations research AXY TQS. * Each subject (where the hyphen represents the classmark) may be fully qualified by preceding facets, as follows: * Add to - the number 2 followed by numbers 2/9 from Auxiliary Schedule 1 (with any amendments shown at AM2). * Add to - the number 3 followed by numbers and letters 3/9, A/X following AX. * Further subdivisions of the subject field to which Statistics is applied may be added as follows: * Add to - the number 4 followed by numbers 3/4 for any topical subdivision of the class numbered 3/4. * Add to - the numbers 5/9 for any topical subdivisions of the class numbered 5/9. * Add to - letters A/Z. * An example follows: HH . Applied statistics in health & medical sciences HH2 9 . . Medical statisticians HN . . Clinical medicine & pathology HN3 57 . . . Data collection HN3 UC . . . Correlation & regression HN9 . . . Internal medicine HN9 357 . . . . Data collection HNS . . . Radiotherapy HNS 3W . . . . Sampling theory

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