Copyright © 2009. All rights reserved

book details


HANDBOOK OF

MATHEMATICS

Published on July 13, 2015

XVIII, 840 p. Paperback/Softcover.
ISBN-10: 2322009679
ISBN-13: 978-2-322-00967-1
EAN: 9782322009671
Thierry Vialar (PhD).
BoD 2015.

Language: English
Appendices: 14 p.
Index: 37 p.
Illustrations: Thousands illustrations and pictures.

Format A4 (21 cm x 4.8 cm x 29.7 cm - 8.27 in x 1.89 in × 11.7 in).
Format: Two-column format. Fontsize 8.
Weight (for paperback version): 2.4 kgs, or 5.3 pounds.
Typesetting: LateX, Scientific Word, MacKichan Software
Paper: 90 grs.
Refer to the website: HDBOM.COM

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table of contents (level 1)

Introduction
Part I. Foundations of Mathematics.
Chap.1. Mathematical Logic.
  • Section 1. Propositions, Connections
  • Section 2. Propositional and Predicate Calculus
  • Section 3. Extension of First-order Predicate Calculus
  • Section 4. Formal System
  • Section 5. Demonstrations and Definitions
Chap.2. Set Theory.
  • Section 1. Basic Concepts
  • Section 2. Set Algebra
  • Section 3. Lattice Theory
  • Section 4. Foundations of Sets
  • Section 5. Problems of Set Theory
  • Section 6. Zermelo-Fraenkel Set Theory (ZFC)
  • Section 7. Banach-Tarski paradox and ZF
  • Section 8. Hahn-Banach Theorems
Chap.3. Relations and Structures.
  • Section 1. Relations
  • Section 2. Maps, Functions
  • Section 3. Families
  • Section 4. Laws of Composition
  • Section 5. Power, Cardinal, Denumerability
  • Section 6. Cardinals
  • Section 7. Structures
  • Section 8. Algebraic Structures
  • Section 9. Order Structure
  • Section 10. Ordinals
  • Section 11. Topological Structures
Chap.4. Arithmetic.
  • Section 1. Set of Natural Numbers ℕ
  • Section 2. Denumerability (Counting)
  • Section 3. Divisibility
  • Section 4. Integers ℤ
  • Section 5. Rational Numbers ℚ
  • Section 6. A General Exercise
Chap.5. Construction of Number System.
  • Section 1. Semigroup of Natural Numbers
  • Section 2. Ring of Integers
  • Section 3. Field of Rational Numbers
  • Section 4. Real Numbers
  • Section 5. Complex Numbers
  • Section 6. Synthesis, Generalization
Part II. Algebra.
Chap.6. Algebra
  • Section 1. Introduction
  • Section 2. Group Theory
  • Section 3. Rings and Fields
  • Section 4. Modules, Vector Spaces
  • Section 5. Linear Maps and Matrices
  • Section 6. Equation and System of Equations
  • Section 7. Algebra of Polynomials
  • Section 8. Polynomials over ℝ and ℂ
  • Section 9. Fractions and Rational Functions
  • Section 10. Polynomial Rings
  • Section 11. Field Extensions
  • Section 12. Prime fields, Finite fields
  • Section 13. Galois Theory
  • Section 14. Galois Theory Application
  • Section 15. Lie Groups
  • Section 16. Linear Algebra
  • Section 17. Eigenvalue, Eigensubspace
  • Section 18. Hermitian Form, Pre-Hilbert Space
  • Section 19. Exterior Algebra of Vector Space
Part III. Number Theory.
Chap.7. Number Theory
  • Section 1. Divisibility in an integral domain
  • Section 2. Diophantine Equations and Residues
  • Section 3. Absolute Value, Valuation
  • Section 4. Prime Numbers
Part IV. Geometry.
Chap.8. Geometry
  • Section 1. Contruction, and Overview
  • Section 2. Fundamental Concepts
  • Section 3. Absolute geometry
  • Section 4. Euclidean and non Euclidean Metrics
  • Section 5. Affine and Projective Planes
  • Section 6. Collineations and Correlations
  • Section 7. Ideal Plane, Coordinates
  • Section 8. Projective Metric
  • Section 9. Order and Orientation
  • Section 10. Angles and Measurements
  • Section 11. Rigid Transformations
  • Section 12. Similarity in Geometry
  • Section 13. Affine Maps
  • Section 14. Projective maps
  • Section 15. Analytic Representations
  • Section 16. Descriptive Geometry
  • Section 18. Hyperbolic Geometry
  • Section 19. Elliptic Geometry
Part V. Analytic Geometry.
Chap.9. Analytic Geometry
  • Section 1. Vector spaces V3
  • Section 2. Scalar, Vector and Mixed products
  • Section 3. Line and Plane Equations
  • Section 4. Sphere and Conics
  • Section 5. Displacement, Affine map in R3
  • Section 6. Quadrics
  • Section 7. Affine Space on Rn
Part VI. Topology.
Chap.10. Topology
  • Section 1. Overview
  • Section 3. Basic Topological Notions of ℝp
  • Section 4. Definition of a Topological Space
  • Section 5. Metric Space, Basis, Neighborhood Basis
  • Section 6. Topological Map, Topological Subspace
  • Section 7. Quotient, Product and Sum Spaces
  • Section 8. Connectedness, Arc-connected space
  • Section 9. Sequence Convergence and Filter Base
  • Section 10. Separation Axioms
  • Section 11. Compactness
  • Section 12. Metrization
  • Section 13. Dimension Theory
  • Section 14. Theory of Curves
  • Section 15. Completion
Chap.11. Topology II
  • Section 1. Introduction
  • Section 2. Prerequisites, Reminders
  • Section 3. Distances and Metric Spaces
  • Section 4. Limits and Cluster Points
  • Section 5. Compact and locally compact spaces
  • Section 6. Connected spaces
  • Section 7. Metric and Semi-metric Spaces
  • Section 8. Baire Spaces
  • Section 9. Mapping Spaces (or Function Spaces)
  • Section 10. Normed Vector Spaces
  • Section 11. Hilbert Spaces
Part VII. Algebraic Topology.
Chap.12. Algebraic Topology
  • Section 1. Homotopy
  • Section 2. Polyhedra
  • Section 3. Fundamental Group of a Connected Polyhedron
  • Section 4. Surfaces
  • Section 5. Homology Theory
  • Section 6. Graph Theory
  • Section 7. Bundles
Chap.13. Algebraic Topology II
  • Section 1. Introduction
  • Section 2. Some Topological Notions
  • Section 3. Simplexes
  • Section 4. The Fundamental Group
  • Section 5. Singular Homology
  • Section 6. Long Exact Sequences
  • Section 7. Excision
  • Section 8. Simplicial Complexes
  • Section 9. CW Complexes
  • Section 10. Natural Transformations
  • Section 11. Covering Spaces
  • Section 12. Homotopy Groups
  • Section 13. Cohomology
Part VIII. Analysis.
Chap.14. Real Analysis
  • Section 1. Structures on ℝ
  • Section 2. Sequences, Series
  • Section 3. Real Functions
Chap.15. Differential Calculus
  • Section 1. Overview
  • Section 2. Functions of Differentiable Real Variable
  • Section 3. Mean Value Theorems
  • Section 4. Expansion In Series
  • Section 5. Rational Functions
  • Section 6. Algebraic Functions
  • Section 7. Non-algebraic Functions
  • Section 8. Approximation Theory
  • Section 9. Interpolation Theory
  • Section 10. Numerical Resolution of Equations
  • Section 11. Differential Calculus in ℝn
Chap.16. Integral Calculus
  • Section 1. Overview
  • Section 2. Riemann Integral
  • Section 3. Integration Rules, R-integrable Functions
  • Section 4. Primitive Functions, Indefinite Integrals
  • Section 5. Integration Methods, Series Integration
  • Section 6. Approximation Methods, Generalized Integrals
  • Section 7. Riemann Integral of Functions of Several Variables
  • Section 8. Successive Integrations, Change of Variables
  • Section 9. Riemann Sums and Applications
  • Section 10. Curvilinear Integral, Surface Integral
  • Section 11. Field, force, work-done
  • Section 12. Integration Theorems
  • Section 13. Jordan Areolar Measure, Lebesgue Measure
  • Section 14. Measurable Functions, Lebesgue Integral
Chap.17. Functional Analysis
  • Section 1. Abstract Spaces
  • Section 2. Differentiable Operators
  • Section 3. Calculus of Variations
  • Section 4. Integral Equations
  • Section 5. Compact Operators
Chap.18. Differential Equations
  • Section 1. Classic Differential Equations
  • Section 2. First-order Differential Equations
  • Section 3. Second-order Differential Equations
  • Section 4. Linear n-order Differential Equations
  • Section 5. Systems of Differential Equations
  • Section 6. Existence and Uniqueness Theorems
  • Section 7. Numerical Methods
  • Section 8. Partial Differential Equations
Chap.19. Differential Geometry
  • Section 1. Curves in R3
  • Section 2. Plane Curves
  • Section 3. Regular Patches, Surfaces
  • Section 4. First Fundamental Form
  • Section 5. Second Fundamental Form, Curvature
  • Section 6. Fundamental Theorem
  • Section 7. Tensors
  • Section 8. Tensors II
  • Section 9. Differential Forms
  • Section 10. Manifolds, Riemannian Geometry
  • Section 11. Complements of Differential Geometry
Chap.20. Function Theory (Complex Analysis)
  • Section 1. Introduction
  • Section 2. Complex Numbers, Compactification
  • Section 3. Sequences and Complex Functions
  • Section 4. Holomorphy
  • Section 5. Cauchy Integral Theorem and Formulas
  • Section 6. Entire Series Expansion
  • Section 7. Analytic Continuation
  • Section 8. Singularities, Laurent Series
  • Section 9. Meromorphy, Residues
  • Section 10. Riemann Surfaces
  • Section 11. Entire Functions
  • Section 12. Meromorphic functions on C
  • Section 13. Periodic functions
  • Section 14. Algebraic functions
  • Section 15. Conformal transformations
  • Section 16. Functions of several variables
Chap.21. Complex Analysis II
  • Section 1. Introduction
  • Section 2. Prerequisites
  • Section 3. Recall on Functions of Complex Variables
  • Section 4. Curvilinear Integral
  • Section 5. Differential Forms in the Plane
  • Section 6. Holomorphic Functions I
  • Section 7. Holomorphic Functions II
  • Section 8. Homotopy
  • Section 9. Topology of the Complex Plane
  • Section 10. Cauchy Theorem (Homology Version)
  • Section 11. Residues
  • Section 12. Runge Theorem, Applications
  • Section 13. Conformal Mapping
  • Section 14. Harmonic Functions
  • Section 15. Subharmonic Functions
  • Section 16. (A1) Convolution, Partition of Unity
  • Section 17. (A2) Distributions
  • Section 18. (A3) Topology of the Complex Plane II
Part IX. Category Theory.
Chap.22. Areas Involved in Category Theory
  • Section 1. Areas Involved
  • Section 2. Algebraic System, Universal Algebra
  • Section 3. Signatures
  • Section 4. Structures
  • Section 5. Interpretations
  • Section 6. Varieties
  • Section 7. Algebraic Structures
  • Section 8. Maps and Structure Homomorphisms
  • Section 9. Algebraic Structure Morphisms and Kernels
  • Section 10. Ring Theory
  • Section 11. Complexes
  • Section 12. Direct limits
  • Section 13. Topological spaces
  • Section 14. Manifolds
  • Section 15. Homology, Cohomology
  • Section 16. Sheaf Theory
  • Section 17. Bundles
  • Section 18. Scheme Theory
  • Section 19. Homotopy
Chap.23. Category Theory
  • Section 1. Introduction
  • Section 2. Universe
  • Section 3. Objects
  • Section 4. Categories
  • Section 5. Functors
  • Section 6. Morphisms
  • Section 7. Graphs and Diagrams
  • Section 8. Limits and Products
  • Section 9. Pullbacks and Pushouts
  • Section 10. Kernels and Cokernels
  • Section 11. Exact Sequences
  • Section 12. Sheaves
  • Section 13. Grothendieck topology, site, sieve
  • Section 14. Topos
  • Section 15. Étale
  • Section 16. Yoneda Lemma, Yoneda Embedding
Part X. Probability and Statistics.
Chap.24. Probability
  • Section 1. Combinatorial Analysis
Chap.25. Probability Calculation, Statistics
  • Section 1. Introduction
  • Section 2. Event, Probability
  • Section 3. Statistical Distribution, Cumulative Distribution
  • Section 4. Statistical Methods
  • Section 5. Statistical Models
Part XI. Applied Mathematics.
Chap.26. Miscellaneous
  • Section 1. Fourier Series and Fourier Transform
  • Section 2. Integral transformations
  • Section 3. Distribution Theory
Chap.27. Optimization
  • Section 1. Introduction
  • Section 2. Linear Optimization
  • Section 3. Simplex Method
  • Section 4. Other Optimizations
  • Section 5. Convex sets, concave and convex functions
  • Section 6. Marginal Optimality Conditions
Chap.28. Dynamical Systems
  • Section 1. Dynamical Systems
  • Section 2. Chaos Theory
  • Section 3. Example in Physics
  • Section 4. Examples in Economics
Appendices
Symbols, Tables
  • Mathematical Symbols
  • Transformation Tables
  • Statistical Tables
Bibliography
Index


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table of contents (level 2)

Introduction
Part I. Foundations of Mathematics.
Chap.1. Mathematical Logic.
  • Section 1. Propositions, Connections
    • 1.1. Logical propositions, truth value
    • 1.2. Connection of propositions, logical connectives
    • 1.3. Propositional calculus theorems
  • Section 2. Propositional and Predicate Calculus
    • 2.1. Predicates, quantifiers
    • 2.2. First-order predicate calculus
    • 2.3. Semantic or syntactic methods: ω assignments, or set of axioms
    • 2.4. Theorems of predicate calculus
  • Section 3. Extension of First-order Predicate Calculus
    • 3.1. First-order predicate calculus with identity
    • 3.2. Higher-order predicate calculus
    • 3.3. Intuitionism
  • Section 4. Formal System
    • 4.1. Non contradiction problem, and Hilbert program
    • 4.2. Formal system
    • 4.3. Axiomatic method
  • Section 5. Demonstrations and Definitions
    • 5.1. Demonstrations
    • 5.2. Proof by mathematical induction (by recurrence)
    • 5.3. Definition
    • 5.4. Methods of proof
Chap.2. Set Theory.
  • Section 1. Basic Concepts
    • 1.1. Subset, set of the parts of a set
    • 1.2. Paradoxical set construction
    • 1.3. Russell paradoxical set
  • Section 2. Set Algebra
    • 2.1. Algebraic operations in sets
    • 2.2. Maps
    • 2.3. Partition
  • Section 3. Lattice Theory
    • 3.1. Lattices, lattices of sets
    • 3.2. Lattices and order relations
    • 3.3. Lattices and Boolean rings
  • Section 4. Foundations of Sets
    • 4.1. Set membership, elements
    • 4.2. Definition by comprehension
    • 4.3. Constructors
  • Section 5. Problems of Set Theory
    • 5.1. Basic definitions and context
    • 5.2. Basic concepts of set theory
    • 5.3. Antinomies, theory of types
    • 5.4. Axioms of set theory
    • 5.5. Von Neumann universe
    • 5.6. Continuum hypothesis and set theory
    • 5.7. Axiomatic set theory
  • Section 6. Zermelo-Fraenkel Set Theory (ZFC)
    • 6.1. Zermelo-Fraenkel axioms
    • 6.2. Comments on ZFC axioms
  • Section 7. Banach-Tarski paradox and ZF
    • 7.1. Banach-Tarski paradox
    • 7.2. Countable, measure
    • 7.3. Hausdorff paradox
    • 7.4. Axioms ADC and ACC
    • 7.5. Banach-Tarski paradox and AC, ADC, ZF, ZFC
  • Section 8. Hahn-Banach Theorems
    • 8.1. Prerequisites
    • 8.2. Hahn-Banach theorem
    • 8.3. Hahn-Banach separation theorem
Chap.3. Relations and Structures.
  • Section 1. Relations
    • 1.1. Cartesian product, relations
    • 1.2. Properties of binary relations
    • 1.3. Equivalence relation, quotient set
    • 1.4. Composition of relations
    • 1.5. Inverse relation
  • Section 2. Maps, Functions
    • 2.1. Relations, functions and maps
    • 2.2. Definition of a map
    • 2.3. Particular maps
    • 2.4. Composition of maps
    • 2.5. Inverse map
    • 2.6. Maps and operations on sets
    • 2.7. Commutative diagram
    • 2.8. n-variables functions
    • 2.9. Maps and graphs
    • 2.10. Images and antecedents
    • 2.11. Set F(E,F) of maps from E to F
  • Section 3. Families
    • 3.1. Family of elements of a set
    • 3.2. Family of sets
    • 3.3. Family of parts of a set
  • Section 4. Laws of Composition
    • 4.1. General vocabulary, notation
    • 4.2. Application to set-calculation
  • Section 5. Power, Cardinal, Denumerability
    • 5.1. Number, equipotence and cardinality
    • 5.2. Power, or cardinal
    • 5.3. Finite and infinite sets
    • 5.4. Operations on cardinals
    • 5.5. Cardinal comparison
    • 5.6. Denumerability, non denumerability
  • Section 6. Cardinals
    • 6.1. Induction (noetherian induction)
    • 6.2. Equipotence
    • 6.3. Finite and infinite cardinals
  • Section 7. Structures
    • 7.1. Fundamental structures
    • 7.2. Multiple structures
    • 7.3. System of relations
    • 7.4. Derived structures
    • 7.5. Structures, maps, morphisms
    • 7.6. Isomorphisms
  • Section 8. Algebraic Structures
    • 8.1. Internal composition law (general)
    • 8.2. Internal composition law
    • 8.3. Associativity, associative law
    • 8.4. Semigroup or Monoid
    • 8.5. Neutral element
    • 8.6. Inverse of an element
    • 8.7. Group
    • 8.8. Ring, integral domain (entire ring)
    • 8.9. Field
    • 8.10. External composition law
    • 8.11. Module, vector space
  • Section 9. Order Structure
    • 9.1. Comparability of sets
    • 9.2. Order relation, ordered set
    • 9.3. Contruction of order structures
    • 9.4. Totally ordered set
    • 9.5. Partially ordered set
    • 9.6. Order diagram
    • 9.7. Induced order
    • 9.8. Greatest element,maximal element
    • 9.9. Upper bound, least upper bound (supremum)
    • 9.10. Lower bound, greatest lower bound (infimum)
    • 9.11. Well ordered set
    • 9.12. Zermelo and Zorn theorems
  • Section 10. Ordinals
    • 10.1. Isomorphisms of ordered sets
    • 10.2. Order type
    • 10.3. Comparison of order type
    • 10.4. Ordinals
    • 10.5. Finite ordinals
    • 10.6. Infinite ordinals
    • 10.7. Burali-Forti paradox
    • 10.8. Ordinal comparison
    • 10.9. Ordinal classes
    • 10.10. Operations on ordinals
    • 10.11. Spotting of elements of a set
    • 10.12. Transfinite induction
  • Section 11. Topological Structures
    • 11.1. Topological space
    • 11.2. Metric space
    • 11.3. Continuous maps
    • 11.4. Particular topological structures
Chap.4. Arithmetic.
  • Section 1. Set of Natural Numbers ℕ
    • 1.1. Order relation and natural numbers
    • 1.2. Recurrence (induction)
    • 1.3. Addition and multiplication in ℕ
  • Section 2. Denumerability (Counting)
    • 2.1. Finite sets, denumerable sets
    • 2.2. Combinatorial analysis
  • Section 3. Divisibility
    • 3.1. Euclidean division, numeration
    • 3.2. Primes, integer factorization
    • 3.3. GCD, LCM, Euclid algorithm
  • Section 4. Integers ℤ
    • 4.1. Operations
    • 4.2. Subgroups of ℤ, divisibility in ℤ
  • Section 5. Rational Numbers ℚ
  • Section 6. A General Exercise
Chap.5. Construction of Number System.
  • Section 1. Semigroup of Natural Numbers
    • 1.1. Construction of ℕ
    • 1.2. Operations
    • 1.3. Order structures
  • Section 2. Ring of Integers
    • 2.1. Question of extension
    • 2.2. Construction of ℤ
    • 2.3. Algebraic structure of ℤ
    • 2.4. ℤ extension of ℕ
    • 2.5. Order structure in ℕ
    • 2.6. Calculation rules in ℤ
  • Section 3. Field of Rational Numbers
    • 3.1. Construction of ℚ
    • 3.2. Algebraic structure of ℚ
    • 3.3. ℚ smallest field containing ℤ
    • 3.4. Order structure in ℚ
    • 3.5. Topological structure of ℚ
  • Section 4. Real Numbers
    • 4.1. Default of order structure of ℚ
    • 4.2. Construction of ℝ
    • 4.3. Embedding of ℚ in ℝ
    • 4.4. Algebraic structure of ℝ
    • 4.5. Default of the topological structure of ℚ
    • 4.6. Construction of the completed space of ℚ
    • 4.7. Structure of
    • 4.8. Isomorphism of
    • 4.9. Nested segments
    • 4.10. Decimal expansion
    • 4.11. Representation of reals
    • 4.12. Rational and irrational exponents
  • Section 5. Complex Numbers
    • 5.1. Construction of ℂ
    • 5.2. Construction of ℂ (other presentation)
    • 5.3. Cartesian representation
    • 5.4. Complex plane (Gauss plane)
    • 5.5. Conjugation
    • 5.6. Modulus
    • 5.7. Square roots
    • 5.8. Group of complex numbers of modulus 1
    • 5.9. Root of unity (de Moivre number)
    • 5.10. Trigonometric representation of complex numbers
    • 5.11. Arguments of complex numbers
    • 5.12. nth roots of complex numbers
    • 5.13. Applications to trigonometry
    • 5.14. Some geometric applications
    • 5.15. Convergence in ℂ (recalls)
    • 5.16. Complex exponential
    • 5.17. Exponential, logarithm in ℂ
    • 5.18. Geometrical representation of complex numbers
    • 5.19. Operations in the Gauss complex plane
    • 5.20. Algebraic closure of ℂ
    • 5.21. D’Alembert-Gauss theorem
    • 5.22. Others properties of ℂ
    • 5.23. Topological theorem of complex numbers
    • 5.24. Riemann sphere, compactification
  • Section 6. Synthesis, Generalization
    • 6.1. Axiomatic construction of number system
    • 6.2. Univocal characterization of natural numbers
    • 6.3. Algebraic numbers and transcendental numbers
    • 6.4. p-adic numbers
    • 6.5. Quaternions
Part II. Algebra.
Chap.6. Algebra
  • Section 1. Introduction
    • 1.1. Sets and algebraic structures
    • 1.2. Group theory
    • 1.3. Ring theory
    • 1.4. Field theory
    • 1.5. Galois theory
    • 1.6. Module theory
    • 1.7. Vector space theory
  • Section 2. Group Theory
    • 2.1. Group, group properties
    • 2.2. Subgroups
    • 2.3. Monogenic groups
    • 2.4. Lateral classes, cosets
    • 2.5. Homomorphisms
    • 2.6. Isomorphic groups
    • 2.7. Permutation groups
    • 2.8. Symmetric groups
    • 2.9. Additive group of integers modulo n
    • 2.10. Quotient group, distinguished subgroup
    • 2.11. Group of congruence classes modulo n
    • 2.12. Homomorphism theorem
    • 2.13. Applications of Homomorphism theorem
    • 2.14. Solvable groups
    • 2.15. Permutation cycles
    • 2.16. Extension of a semigroup
  • Section 3. Rings and Fields
    • 3.1. Ring, calcularion rules
    • 3.2. Subrings, ideals, homomorphisms
    • 3.3. Characteristic of a ring
    • 3.4. Divisibility in integral domain (entire ring)
    • 3.5. Ring of integers modulo n
    • 3.6. Fields
    • 3.7. Substructures (subring, subfield)
    • 3.8. Ring homomorphisms
    • 3.9. Ring and field homomorphisms
    • 3.10. Automorphisms
    • 3.11. Field of fractions
    • 3.12. Quotient field
    • 3.13. Quotient ring, ideals
    • 3.14. Homomorphism theorem for rings
    • 3.15. Ring of congruence classes ℤn
    • 3.16. Characterization of integral domains by prime ideals
    • 3.17. Characterization of fields by ideals
    • 3.18. Principal ideals, principal rings
  • Section 4. Modules, Vector Spaces
    • 4.1. Submodules
    • 4.2. Module theory related to commutative groups and ideals
    • 4.3. Module homomorphisms
    • 4.4. Homomorphism theorem for modules
    • 4.5. Quotient module
    • 4.6. Isomorphism theorem for modules
    • 4.7. Direct product of modules
    • 4.8. Linear closure, generating part
    • 4.9. Linear independence, basis
    • 4.10. Vector spaces
    • 4.11. Vector spaces of finite dimension
    • 4.12. Properties of vector spaces of finite dimension
  • Section 5. Linear Maps and Matrices
    • 5.1. Linear maps
    • 5.2. Linear maps in vector spaces of finite dimension and matrices
    • 5.3. Vector spaces L(V, V') and Mn,m(K)
    • 5.4. Dual vector spaceL(V,K)
    • 5.5. Composition of linear maps
    • 5.6. Rings L(V, V ) and Mn,n(K)
    • 5.7. The group of automorphisms Aut(V, V )
    • 5.8. Regular or nonsingular matrix
    • 5.9. Determinant of a matrix
    • 5.10. Determinant calculation rules
    • 5.11. Group GLn(K) of regular or nonsingular matrices
  • Section 6. Equation and System of Equations
    • 6.1. Equation and resolution
    • 6.2. System of linear equations
    • 6.3. Cramer Formula
    • 6.4. Homogeneous system
    • 6.5. Nonhomogeneous system
  • Section 7. Algebra of Polynomials
    • 7.1. Construction and axioms
    • 7.2. Elementary calculation rules
    • 7.3. Arithmetic properties of polynomials
    • 7.4. Polynomial functions and roots
    • 7.5. Derived polynomials
  • Section 8. Polynomials over ℝ and ℂ
    • 8.1. D’Alembert-Gauss theorem application
    • 8.2. Cyclotomy
    • 8.3. Cyclotomic polynomials
    • 8.4. Chebyshev polynomials
    • 8.5. Algebraic numbers
  • Section 9. Fractions and Rational Functions
    • 9.1. The field K(X) of rational fractions
    • 9.2. Arithmetic properties of K(X)
    • 9.3. Rational functions
  • Section 10. Polynomial Rings
    • 10.1. Construction of intermediate rings, adjunction
    • 10.2. Polynomial, polynomial rings
    • 10.3. Degree of a polynomial
    • 10.4. Ring R[X1,..,Xp] of polynomials
    • 10.5. Divisibility in the ring K[X] of polynomials over a commutative field K
    • 10.6. Euclidean polynomial division
    • 10.7. Roots of a polynomial
    • 10.8. Multiple roots
    • 10.9. Irreducibility criteria
    • 10.10. Irreducibility over ℚ
    • 10.11. Algebraic elements of a field
  • Section 11. Field Extensions
    • 11.1. Extension of a field
    • 11.2. Finite extensions
    • 11.3. Construction of an intermediate field, adjunction to a field
    • 11.4. Simple extension
    • 11.5. Finite simple extensions
    • 11.6. Algebraic extensions
    • 11.7. Algebraic closure
    • 11.8. Splitting field of a polynomial
    • 11.9. Separable polynomial, perfect field
  • Section 12. Prime fields, Finite fields
    • 12.1. Prime field
    • 12.2. Prime subfield of a field K
    • 12.3. Characteristic of a field
    • 12.4. Finite fields
    • 12.5. Existence of finite fields
  • Section 13. Galois Theory
    • 13.1. Galois group, Galois map
    • 13.2. Bijectivity of the Galois map
    • 13.3. Finite Galois extension, fundamental theorem
    • 13.4. Polynomial criterion for finite Galois extensions
    • 13.5. Properties of finite Galois extensions
    • 13.6. Galois group of a polynomial
    • 13.7. Galois group of the general monic polynomial
    • 13.8. Galois group of the polynomial Xn-1∈ℚ[X]
    • 13.9. Galois group of a finite field
  • Section 14. Galois Theory Application
    • 14.1. Resolution of equations by radicals
    • 14.2. Resolution condition
    • 14.3. Abel theorem
    • 14.4. Geometric construction and algebrization
  • Section 15. Lie Groups
    • 15.1. Introduction
    • 15.2. Linear Lie groups
    • 15.3. Lie algebra of linear Lie group
    • 15.4. General Lie groups
    • 15.5. Lie algebra of a Lie group
    • 15.6. Exponential map
    • 15.7. Lie group with a given Lie algebra
    • 15.8. First Lie theorem example
  • Section 16. Linear Algebra
    • 16.1. Scope
    • 16.2. Problematics
    • 16.3. Content of linear algebra
  • Section 17. Eigenvalue, Eigensubspace
    • 17.1. Basic definitions
    • 17.2. Endomorphism reduction
    • 17.3. Diagonalization
    • 17.4. Trigonalization
    • 17.5. Characteristic subspace
    • 17.6. Jordan matrix
    • 17.7. Jordan canonical form theorem
  • Section 18. Hermitian Form, Pre-Hilbert Space
    • 18.1. Hermitian symmetry
    • 18.2. Hermitian symmetric sesquilinear form
    • 18.3. Hermitian form
    • 18.4. Orthogonality according to hermitian form
    • 18.5. Positive hermitian form
    • 18.6. Pre-Hilbert spaces
    • 18.7. Normal endomorphism
  • Section 19. Exterior Algebra of Vector Space
    • 19.1. Antisymmetric function
    • 19.2. Alternating map
    • 19.3. Alternating group
    • 19.4. Exterior product (wedge product)
    • 19.5. Interior product
    • 19.6. Exterior algebra of alternating multilinear forms
    • 19.7. p-linear antisymmetric forms in finite dimension
    • 19.8. Exterior algebra of alternating multilinearforms (in finite dimension)
    • 19.9. Exterior algebra of a (finite dimension) vector space
    • 19.10. Vector product, exterior product
    • 19.11. Universal properties of exterior product
Part III. Number Theory.
Chap.7. Number Theory
  • Section 1. Divisibility in an integral domain
    • 1.1. Overview
    • 1.2. Notion of divisibility
    • 1.3. Arithmetic functions
    • 1.4. Monotony criterion
    • 1.5. Prime element, factorial ring
    • 1.6. Theory of ideals
    • 1.7. GCD and LCM
    • 1.8. Bezout’s identity: Linear representation of GCD
    • 1.9. Residue class and residue-class ring
    • 1.10. Simultaneous congruences
    • 1.11. Invertible elements in the residue-class ring
  • Section 2. Diophantine Equations and Residues
    • 2.1. Diophantine equations
    • 2.2. Residues
  • Section 3. Absolute Value, Valuation
    • 3.1. Divisibility in a field
    • 3.2. Absolute values
    • 3.3. Absolute values of the field ℚ
    • 3.4. Absolute values of the field K(X) of rational fractionsover K
    • 3.5. Valuation
    • 3.6. Closure of valued fields
    • 3.7. Extension of absolute values
    • 3.8. Norms in finite algebraic extensions
    • 3.9. Extensions of discrete absolute values on finite algebraic extensions
    • 3.10. Divisors
    • 3.11. Application of valuation theory to quadratic fields
  • Section 4. Prime Numbers
    • 4.1. Infinitude of prime numbers
    • 4.2. Table of primes
    • 4.3. Properties of π
    • 4.4. Fermat numbers
    • 4.5. Mersenne numbers
    • 4.6. Unsolved problems of number theory
Part IV. Geometry.
Chap.8. Geometry
  • Section 1. Contruction, and Overview
    • 1.1. The axiomatic method
    • 1.2. Euclid’s axiom of parallelism
    • 1.3. Absolute geometry
    • 1.4. Projective geometry, and transformation groups
    • 1.5. Analytic geometry
    • 1.6. Descriptive geometry
    • 1.7. Axiomatic construction of geometry
  • Section 2. Fundamental Concepts
    • 2.1. Intuitive analysis of concepts
  • Section 3. Absolute geometry
    • 3.1. Metric plane
    • 3.2. The Models
    • 3.3. Absolute geometry properties
    • 3.4. Hjelmslev theorem of perpendiculars
    • 3.5. Theorems about triangles and trilaterals
    • 3.6. Anti-matching theorem
  • Section 4. Euclidean and non Euclidean Metrics
    • 4.1. Rectangle axiom
    • 4.2. Axiom of connection
    • 4.3. Axiom of polar trilateral
    • 4.4. Hyperbolic axiom
    • 4.5. Triangle in different spaces
    • 4.6. Classification of metric planes
  • Section 5. Affine and Projective Planes
    • 5.1. Affine planes
    • 5.2. Projective planes
    • 5.3. Metric, affine, projective planes
  • Section 6. Collineations and Correlations
    • 6.1. Projective transformations
    • 6.2. Collineations
    • 6.3. Correlations
  • Section 7. Ideal Plane, Coordinates
    • 7.1. Rotations
    • 7.2. Ideal plane of a metric plane
    • 7.3. Coordinates in affine planes
    • 7.4. Coordinates in projective planes
  • Section 8. Projective Metric
    • 8.1. Fano’s axiom
    • 8.2. Ordinary and singular metric projective planes
  • Section 9. Order and Orientation
    • 9.1. Bisector axiom
    • 9.2. Orientation
    • 9.3. Oriented plane as topological space
    • 9.4. Completion
  • Section 10. Angles and Measurements
    • 10.1. Angles
    • 10.2. Order relation on angles
    • 10.3. Cyclic order relation
    • 10.4. Angle measurements
  • Section 11. Rigid Transformations
    • 11.1. Isometries, invariants
    • 11.2. Reflections
    • 11.3. Translations
    • 11.4. Rotations
    • 11.5. Reflections-translations
    • 11.6. Symmetries
    • 11.7. Theorems about angles
    • 11.8. Angles, lines and circle
    • 11.9. Congruence theorems
    • 11.10. Euclidean group E(n)
    • 11.11. Rigid transformations in ℝ3
    • 11.12. Examples of direct and indirect isometries in ℝ3
    • 11.13. Reflection with respect to a plane
    • 11.14. Translations
    • 11.15. Rotations
    • 11.16. Reflections-Translations
    • 11.17. Reflections-Rotations
    • 11.18. Screws
  • Section 12. Similarity in Geometry
    • 12.1. Homotheties
    • 12.2. Similarities (similitudes)
    • 12.3. Homothetic triangles and division of segment
    • 12.4. Similar triangles and circles
    • 12.5. Similitude center
    • 12.6. Euler line, Nine-point circle
    • 12.7. Power and chordal theorems
    • 12.8. Apollonius circle
  • Section 13. Affine Maps
    • 13.1. Affinity, transvection (shear)
    • 13.2. Polygon area
    • 13.3. Affine maps
    • 13.4. Right triangle properties
    • 13.5. Classification of quadrilaterals and triangles
  • Section 14. Projective maps
    • 14.1. Central projections in ℝ3
    • 14.2. Central projections in ℙ3(ℝ)
    • 14.3. Perspective collineations
    • 14.4. Projective collineation
    • 14.5. Collineation characterization
    • 14.6. Meshings and griddings
    • 14.7. Pairs of harmonic points
    • 14.8. Cone of revolution
  • Section 15. Analytic Representations
    • 15.1. Matrix of affine collineation
    • 15.2. Orthonormal coordinate system
    • 15.3. Isometries
    • 15.4. Similarites
    • 15.5. Others affine bijections
    • 15.6. Projective collineation in ℙ2(ℝ)
  • Section 16. Descriptive Geometry
    • 16.1. Overview
  • Section 17. Trigonometry
    • 17.1. Trigonometrical functions
    • 17.2. Properties about triangles
  • Section 18. Hyperbolic Geometry
    • 18.1. Klein model (length of hyperbolic segment)
    • 18.2. Poincaré circle model (angle measure)
    • 18.3. Poincaré half-plane model
    • 18.4. Pseudosphere
    • 18.5. Curvature
    • 18.6. Hyperbolic functions
    • 18.7. Two types of trigonometry
    • 18.8. About triangles
  • Section 19. Elliptic Geometry
    • 19.1. The sphere model
    • 19.2. Lunes, triangles, areas
    • 19.3. Polar triangles
    • 19.4. Calculations about triangles
    • 19.5. Napier’s analogies
    • 19.6. Orthodrome and loxodrome
Part V. Analytic Geometry.
Chap.9. Analytic Geometry
  • Section 1. Vector spaces V3
    • 1.1. Analytic geometry
    • 1.2. Vectors and operations
    • 1.3. Linear dependence (independence)
    • 1.4. Coordinate system and basis
    • 1.5. Vector basis
  • Section 2. Scalar, Vector and Mixed products
    • 2.1. Remark about products
    • 2.2. Scalar product
    • 2.3. Vector product
    • 2.4. Mixed product
  • Section 3. Line and Plane Equations
    • 3.1. Line and plane representations
    • 3.2. Distances
  • Section 4. Sphere and Conics
    • 4.1. Sphere, conics
    • 4.2. Ellipse, parabola, hyperbola
    • 4.3. 2nd degree equation in two coordinates
  • Section 5. Displacement, Affine map in ℝ3
    • 5.1. Displacement
    • 5.2. Endomorphism
    • 5.3. Affine maps in ℝ3
    • 5.4. Affine isometries
    • 5.5. Coordinate system change
    • 5.6. Types of affine isometry
  • Section 6. Quadrics
    • 6.1. General equation
    • 6.2. Principal directions
    • 6.3. Classes of quadrics
    • 6.4. Endomorphism eigenvectors
  • Section 7. Affine Space on ℝn
    • 7.1. Affine space
    • 7.2. Canonical structure
    • 7.3. Vectorial structure
    • 7.4. Affine subspace
    • 7.5. Affine coordinates system
    • 7.6. Parallelism
    • 7.7. Affine subspace intersection
    • 7.8. Affine subspace generated by E
    • 7.9. Barycenter
    • 7.10. Affines maps
    • 7.11. Image of an affine subspace by an affine map
    • 7.12. Affine endomorphisms
    • 7.13. Affine forms
    • 7.14. Extension to ℝn
    • 7.15. Canonical scalar product in the vector space ℝn
    • 7.16. Canonical euclidean norm
    • 7.17. Angle measure, orthogonal vectors
    • 7.18. Orthogonal system
    • 7.19. Isometries in ℝ3
    • 7.20. Parallelepiped
Part VI. Topology.
Chap.10. Topology
  • Section 1. Overview
  • Section 2. Homeomorphism notion
    • 2.1. Elastic transformation
    • 2.2. Neighborhood
    • 2.3. Continuous maps
    • 2.4. Homeomorphism
  • Section 3. Basic Topological Notions of ℝp
    • 3.1. Homeomorphisms of ℝ,ℝ2,ℝ3
    • 3.2. Adherent, exterior, interior, isolated, accumulation, frontier points
    • 3.3. Open, interior, closed, frontier sets, and closure
    • 3.4. Topological and continuous invariants
    • 3.5. Connected sets
    • 3.6. Connected set of ℝ,ℝ2,ℝ3
    • 3.7. Compact sets in ℝ,ℝp
    • 3.8. Intrinsic properties
  • Section 4. Definition of a Topological Space
    • 4.1. In terms of neighborhoods
    • 4.2. In terms of open sets
    • 4.3. Examples of topological spaces
    • 4.4. Comparison of topologies
  • Section 5. Metric Space, Basis, Neighborhood Basis
    • 5.1. Metric spaces
    • 5.2. Basis of a topology
    • 5.3. Properties of bases
    • 5.4. Equivalent generating systems
    • 5.5. Subbasis of a topology
    • 5.6. Basis of neighborhoods
  • Section 6. Topological Map, Topological Subspace
    • 6.1. Comparison of topological spaces
    • 6.2. Local and global continuities
    • 6.3. Topology generated by the maps
    • 6.4. Induced topology, topological subspace
  • Section 7. Quotient, Product and Sum Spaces
    • 7.1. Quotient space and topology
    • 7.2. Product space and topology
    • 7.3. Sum topology
  • Section 8. Connectedness, Arc-connected space
    • 8.1. Connected space
    • 8.2. Connected component
    • 8.3. Locally connected space
    • 8.4. Arcwise-connected space
    • 8.5. Arcwise-connected component
    • 8.6. Locally arcwise-connected space
    • 8.7. Covering space
  • Section 9. Sequence Convergence and Filter Base
    • 9.1. Convergence of a sequence
    • 9.2. Continuity of a sequence
    • 9.3. Filter base and convergence
    • 9.4. Filter base comparison
    • 9.5. Generalization of a sequence
  • Section 10. Separation Axioms
    • 10.1. Hausdorff space, T0,T1,T2-axioms
    • 10.2. Regular space, T3 axiom
    • 10.3. Normal space, T4 axiom
    • 10.4. Completely regular space
    • 10.5. Synthesis
  • Section 11. Compactness
    • 11.1. Quasi-compact and compact spaces
    • 11.2. Compact sets of ℝn
    • 11.3. Compact-open topology
    • 11.4. Bolzano-Weierstrass property
    • 11.5. Locally compact space
    • 11.6. One point compactification, Alexandroff compactification
    • 11.7. Compactification
    • 11.8. Compactum
  • Section 12. Metrization
    • 12.1. Metrization theorems
    • 12.2. Countability axioms
  • Section 13. Dimension Theory
    • 13.1. Algebraic dimension
    • 13.2. Metric space dimension endowed with a countable basis
    • 13.3. Immersion theorem
  • Section 14. Theory of Curves
    • 14.1. Jordan arc
    • 14.2. Jordan curve
    • 14.3. Curves
  • Section 15. Completion
    • 15.1. Completion of metric space
    • 15.2. Universal property
Chap.11. Topology II
  • Section 1. Introduction
  • Section 2. Prerequisites, Reminders
    • 2.1. Metric spaces
    • 2.2. Topological spaces
    • 2.3. Continuous maps
  • Section 3. Distances and Metric Spaces
    • 3.1. Notion of distance
    • 3.2. Examples of distance
    • 3.3. Distance, semi-distance, pseudometric
  • Section 4. Limits and Cluster Points
    • 4.1. Sequences in a topological space
    • 4.2. Limit and cluster point of a map
    • 4.3. Filters
  • Section 5. Compact and locally compact spaces
    • 5.1. Compactness, properties
    • 5.2. Compact spaces and continuous maps
    • 5.3. Products of compact spaces
    • 5.4. Locally compact spaces
  • Section 6. Connected spaces
    • 6.1. Definitions, examples
    • 6.2. Properties
    • 6.3. Connected components
    • 6.4. Arcwise-connectedness
    • 6.5. Locally connected spaces
  • Section 7. Metric and Semi-metric Spaces
    • 7.1. Topology of metric spaces
    • 7.2. Uniform continuity and metric spaces
    • 7.3. Cauchy sequences, complete spaces
    • 7.4. Fixed-point theorem
    • 7.5. Semi-metric spaces and uniform spaces
    • 7.6. Complete uniform spaces
  • Section 8. Baire Spaces
    • 8.1. Baire property
    • 8.2. Examples of Baire spaces
    • 8.3. Continuous and semi-continuous functions on Baire space
    • 8.4. Some applications
  • Section 9. Mapping Spaces (or Function Spaces)
    • 9.1. Simple and uniform convergence
    • 9.2. Other uniform structures on mapping spaces
    • 9.3. Equicontinuous families
    • 9.4. Stone-Weierstrass theorem
  • Section 10. Normed Vector Spaces
    • 10.1. Norm on a vector space
    • 10.2. Continuous linear maps
    • 10.3. Continuous multilinear map
    • 10.4. Series in a normed vector space
    • 10.5. Some results on Banach spaces
  • Section 11. Hilbert Spaces
    • 11.1. Sesquilinear forms, Hermitian forms
    • 11.2. Pre-Hilbert and Hilbert spaces
    • 11.3. Completion of pre-Hilbert space
    • 11.4. Orthogonality
    • 11.5. Projection theorems
    • 11.6. Dual of a Hilbert space
    • 11.7. Orthogonal systems, Hilbert bases
    • 11.8. Some Hilbert spaces and bases
Part VII. Algebraic Topology.
Chap.12. Algebraic Topology
  • Section 1. Homotopy
    • 1.1. Homotopy of paths
    • 1.2. Operations on the classes C
    • 1.3. Homotopy groups, fundamental group
    • 1.4. Topological invariance
    • 1.5. Homotopy of continuous maps
    • 1.6. Simply connected spaces
    • 1.7. Contractile spaces
    • 1.8. Shrinking, contractibility
    • 1.9. n-dimensional homotopy groups
    • 1.10. Schematization of homotopy groups
  • Section 2. Polyhedra
    • 2.1. Polyhedra as topological subspaces
    • 2.2. Polytopes, simplicial complexes
    • 2.3. Simplexes in (ℝp,ℜp), p≤3
    • 2.4. Polyhedron in (ℝp,ℜp), p≤3
    • 2.5. Convexity and simplexes in (ℝp,ℜp)
    • 2.6. Simplicial complex in (ℝn,ℜn)
    • 2.7. Polyhedra in (ℝn,ℜn)
    • 2.8. Triangulable spaces
    • 2.9. Triangulation
    • 2.10. Simplicial maps
    • 2.11. Simplicial approximations
    • 2.12. Combinatorial topology
  • Section 3. Fundamental Group of a Connected Polyhedron
    • 3.1. Polygonal chains
    • 3.2. Homotopy of polygonal chains, polygonal groups
    • 3.3. Free group of finite type
    • 3.4. Construction of the fundamental group with a free group of finite type
  • Section 4. Surfaces
    • 4.1. Closed surfaces, surfaces with boundary
    • 4.2. Classification of closed surfaces
    • 4.3. Orientability and topological invariance
    • 4.4. Connectivity number
  • Section 5. Homology Theory
    • 5.1. Functorial method
    • 5.2. Functors in algebraic topology
    • 5.3. Theory of simplicial homology
    • 5.4. Theory of singular homology
  • Section 6. Graph Theory
    • 6.1. Introduction
    • 6.2. Definition of a graph
    • 6.3. Graph representation, topological graph
    • 6.4. Subgraph, supergraph
    • 6.5. Isomorphy of graphs
    • 6.6. Graph automorphism
    • 6.7. Directedness
    • 6.8. Specific graphs
    • 6.9. Whitney graph theorem
    • 6.10. Degree of a vertex
    • 6.11. Chains, paths, cycles
    • 6.12. Connected graphs
    • 6.13. Trees, skeleton
    • 6.14. Four-color problem
    • 6.15. Planar graphs
    • 6.16. k-connected graph, connectivity number
    • 6.17. Menger theorem
    • 6.18. Turan theorem
    • 6.19. Kuratowski theorem
  • Section 7. Bundles
    • 7.1. Homology, cohomology
    • 7.2. Algebraic variety
    • 7.3. Variety
    • 7.4. Variety in a category
    • 7.5. Bicategory, morphism
    • 7.6. Functor, bifunctor
    • 7.7. Variety of universal algebras
    • 7.8. Category, functor, fibre product
    • 7.9. Sheaves, presheaves
    • 7.10. Bundle, fiber bundle
    • 7.11. Fiber space, fibration
    • 7.12. Principal bundle
    • 7.13. Vector bundle
    • 7.14. Trivialization
    • 7.15. K-theory
    • 7.16. Schemes
Chap.13. Algebraic Topology II
  • Section 1. Introduction
    • 1.1. Purpose of algebraic topology
    • 1.2. Some standard notations
    • 1.3. Brouwer fixed point theorem
    • 1.4. Categories and functors
  • Section 2. Some Topological Notions
    • 2.1. Homotopy
    • 2.2. Convexity, contractibility, cones
    • 2.3. Paths and path connectedness
  • Section 3. Simplexes
    • 3.1. Affine spaces
    • 3.2. Affine maps
  • Section 4. The Fundamental Group
    • 4.1. The fundamental groupoid
    • 4.2. Functor π1
    • 4.3. π1(S1)
    • 4.4. Dependence on the basepoint
    • 4.5. Homotopy invariance
    • 4.6. π1(ℝ) = 0, π1(S1) = ℤ
  • Section 5. Singular Homology
    • 5.1. Holes, and Green’s theorem
    • 5.2. Free abelian groups
    • 5.3. Singular complex, homology functors
    • 5.4. Dimension axiom, compact supports
    • 5.5. Homotopy axiom
    • 5.6. Hurewicz theorem (first version)
  • Section 6. Long Exact Sequences
    • 6.1. The category Comp
    • 6.2. Exact homology sequences
    • 6.3. Reduced homology
  • Section 7. Excision
    • 7.1. Excision, Mayer-Vietoris theorem
    • 7.2. Homology of spheres
    • 7.3. Barycentric subdivision
    • 7.4. Proof of excision
    • 7.5. Applications
  • Section 8. Simplicial Complexes
    • 8.1. Definitions
    • 8.2. Simplicial approximation
    • 8.3. Abstract simplicial complexes
    • 8.4. Simplicial homology
    • 8.5. Contrast with singular homology
    • 8.6. Calculations
    • 8.7. Fundamental groups of polyhedra
    • 8.8. The van Kampen theorem
  • Section 9. CW Complexes
    • 9.1. Hausdorff quotient spaces
    • 9.2. Attaching cells
    • 9.3. Attaching cells and homology
    • 9.4. CW complexes
    • 9.5. Cellular homology
  • Section 10. Natural Transformations
    • 10.1. Definitions
    • 10.2. Eilenberg-Steenrod axioms
    • 10.3. Chain equivalences
    • 10.4. Acyclic models
    • 10.5. Lefschetz fixed point theorem
    • 10.6. Tensor product
    • 10.7. Universal coefficients
    • 10.8. Eilenberg-Zilber theorem
    • 10.9. Künneth theorem and formula
    • 10.10. Homotopy categories and equivalence
    • 10.11. Limits and colimits
  • Section 11. Covering Spaces
    • 11.1. Definitions
    • 11.2. Unique path lifting property
    • 11.3. Covering transformations
    • 11.4. Existence of a covering space
    • 11.5. Orbit spaces
  • Section 12. Homotopy Groups
    • 12.1. Function spaces
    • 12.2. Group objects (cogroup objects)
    • 12.3. Loop space, suspension
    • 12.4. Homotopy groups
    • 12.5. Exact sequences
    • 12.6. Fibration
    • 12.7. Hurewicz theorem (general versions)
    • 12.8. Freudenthal suspension theorem
    • 12.9. Blakers-Massey theorem
  • Section 13. Cohomology
    • 13.1. Differential forms (recall)
    • 13.2. Cohomology groups
    • 13.3. Universal coefficients theorems (for cohomology)
    • 13.4. Künneth formula for cohomology
    • 13.5. Cohomology rings
    • 13.6. Calculations
Part VIII. Analysis.
Chap.14. Real Analysis
  • Section 1. Structures on ℝ
    • 1.1. Overview
    • 1.2. Algebraic structure of ℝ
    • 1.3. Order structure of ℝ
    • 1.4. Topological structure of ℝ
  • Section 2. Sequences, Series
    • 2.1. Sequences
    • 2.2. Generalized sequence
    • 2.3. Convergent sequences
    • 2.4. Subsequence
    • 2.5. Cluster point of sequence
    • 2.6. Bolzano-Weierstrass theorem
    • 2.7. Upper and lower limits of sequence
    • 2.8. Monotony criteria
    • 2.9. Properties of limit values
    • 2.10. Nested intervals
    • 2.11. Cauchy convergence criterion
    • 2.12. Series
    • 2.13. Criteria for series with positive terms
    • 2.14. Alternating series
    • 2.15. Absolutely convergent series
    • 2.16. Calculation rules for convergent series
    • 2.17. Infinite products
  • Section 3. Real Functions
    • 3.1. Definition
    • 3.2. Cases of real function
    • 3.3. Algebraic operations and composition
    • 3.4. Limit value of function
    • 3.5. Infinite limits
    • 3.6. Properties of limit values
    • 3.7. Continuous function, continuity
    • 3.8. Discontinuity, non continuity
    • 3.9. Rational operations and composition
    • 3.10. Properties of continuous functions
    • 3.11. Continuous extension of a function
    • 3.12. Invertibility and continuity
    • 3.13. Uniform continuity
    • 3.14. Sequences of functions, uniform convergence
    • 3.15. Series of functions, entire series
Chap.15. Differential Calculus
  • Section 1. Overview
  • Section 2. Functions of Differentiable Real Variable
    • 2.1. Tangent problem
    • 2.2. Differentiability, derivability
    • 2.3. Differentiability, continuity
    • 2.4. Derivation rule
    • 2.5. Composition of differentiable functions
    • 2.6. Differentiation of reciprocal functions
    • 2.7. Successive derivatives
  • Section 3. Mean Value Theorems
    • 3.1. Local extrema, Rolle theorem
    • 3.2. Mean value theorem
    • 3.3. Differential
  • Section 4. Expansion In Series
    • 4.1. Taylor polynomial and remainders
    • 4.2. Application to local extremum
    • 4.3. Taylor series
    • 4.4. Analytic functions
    • 4.5. Binomial series
  • Section 5. Rational Functions
    • 5.1. Polynomial functions
    • 5.2. Zeros and local extrema
    • 5.3. Inflection point
    • 5.4. Polynomial function limits
    • 5.5. Rational functions
    • 5.6. Asymptotes
    • 5.7. De l’Hospital rules
    • 5.8. Partial fraction decomposition
  • Section 6. Algebraic Functions
    • 6.1. Algebraic relations and functions
    • 6.2. Implicit differentiation
    • 6.3. Power functions with rational exponents
    • 6.4. Algebraic curve
  • Section 7. Non-algebraic Functions
    • 7.1. Exponential function, logarithm
    • 7.2. Circular functions
    • 7.3. Reciprocal circular functions
    • 7.4. Hyperbolic functions
    • 7.5. Reciprocal hyperbolic functions
    • 7.6. Gamma function
  • Section 8. Approximation Theory
    • 8.1. Introduction
    • 8.2. Best approximations
    • 8.3. Least-squares method
  • Section 9. Interpolation Theory
    • 9.1. Introduction
    • 9.2. Lagrange method
    • 9.3. Newton-Gregory method
    • 9.4. Approximations by interpolation polynomials
  • Section 10. Numerical Resolution of Equations
    • 10.1. Simple iterative methods
    • 10.2. Newton-Raphson method
    • 10.3. Method of linear interpolation, "regula falsi"
    • 10.4. Horner scheme
    • 10.5. Graeffe method
  • Section 11. Differential Calculus in ℝn
    • 11.1. Preamble
    • 11.2. Properties of ℝn
    • 11.3. Vector-valued function
    • 11.4. Example of a function from ℝ2 to ℝ
    • 11.5. Functions from ℝ to ℝm
    • 11.6. Functions from ℝn to ℝ
    • 11.7. Differentiability
    • 11.8. Directional derivates
    • 11.9. Partial derivatives
    • 11.10. Gradients
    • 11.11. Gradient theorem
    • 11.12. Tangent hyperplane
    • 11.13. Higher order partial derivatives
    • 11.14. Functions from ℝn to ℝm
    • 11.15. Invertibility and reciprocal function
    • 11.16. Functions from ℝn to ℝn, invertibility
    • 11.17. Implicit functions
    • 11.18. Local extrema of functions from ℝn to ℝ
    • 11.19. Hessian
    • 11.20. Extrema with constraints, Lagrange multiplier
    • 11.21. Comment about extremum with constraints
    • 11.22. Level set, divergence, rotational, and Laplacian vector
    • 11.23. Operations on gradient, divergence, rotational
    • 11.24. Gauss, Green and curl theorems
    • 11.25. Nabla
Chap.16. Integral Calculus
  • Section 1. Overview
  • Section 2. Riemann Integral
    • 2.1. Hypograph, epigraph
    • 2.2. Partitions of an interval
    • 2.3. Measure of hypographs
    • 2.4. Step functions
    • 2.5. Upper and lower integrals
    • 2.6. Riemann integral
    • 2.7. Riemann sum, Riemann integral
  • Section 3. Integration Rules, R-integrable Functions
    • 3.1. Integration rules
    • 3.2. R-integrable functions
    • 3.3. Mean-value of the integral
    • 3.4. Fundamental theorem of integral calculus
  • Section 4. Primitive Functions, Indefinite Integrals
    • 4.1. Existence conditions of primitive functions
    • 4.2. Calculation methods of primitive functions
    • 4.3. Indefinite integrals
  • Section 5. Integration Methods, Series Integration
    • 5.1. Integration methods
    • 5.2. Table of primitives
    • 5.3. Integration of series
  • Section 6. Approximation Methods, Generalized Integrals
    • 6.1. Generalized integrals
    • 6.2. Cauchy integration criterion
    • 6.3. Graphical integration
    • 6.4. Trapezium, Kepler and Simpson approximation methods
  • Section 7. Riemann Integral of Functions of Several Variables
    • 7.1. Construction of Riemann integral
    • 7.2. Riemann integral on tiles of ℝ2
    • 7.3. Riemann integral on measurable domains of ℝ2 in the Jordan sense
    • 7.4. Volumes and Riemann integral
    • 7.5. Riemann integral of functions of n variables (n ≥ 3)
  • Section 8. Successive Integrations, Change of Variables
    • 8.1. Integration in practice
    • 8.2. Regular domain, double integral
    • 8.3. Volume calculation
    • 8.4. Change of variables
  • Section 9. Riemann Sums and Applications
    • 9.1. Riemann sums
    • 9.2. Curve length in ℝ3 (or ℝ2)
    • 9.3. Arc length
    • 9.4. Measure of arbitrary surfaces
    • 9.5. Measure of regular surface
    • 9.6. Problem of surfaces of revolution
  • Section 10. Curvilinear Integral, Surface Integral
  • Section 11. Field, force, work-done
    • 11.1. Curvilinear integral
    • 11.2. Curvilinear integral of real functions
    • 11.3. Curvilinear integral of a gradient
    • 11.4. Integral of a surface
  • Section 12. Integration Theorems
    • 12.1. Preliminary
    • 12.2. Riemann integral with two variables represented by a curvilinear integral along the frontier of Df
    • 12.3. Green-Riemann theorem and formula
    • 12.4. Ostrogradsky-Stokes theorem and formula
    • 12.5. Overview of Stokes theorem
  • Section 13. Jordan Areolar Measure, Lebesgue Measure
    • 13.1. Preliminary
    • 13.2. Jordan areolar measure
    • 13.3. Problems of Jordan areolar measure
    • 13.4. Lebesgue measure
    • 13.5. Measure of unbounded parts
  • Section 14. Measurable Functions, Lebesgue Integral
    • 14.1. Measurable functions, measurability
    • 14.2. Measurable functions and continuous functions
    • 14.3. Lebesgue integral definition
    • 14.4. Properties of Lebesgue integral
    • 14.5. Lebesgue integral, Riemann integral
    • 14.6. Denjoy integral
    • 14.7. Stieltjes integral
    • 14.8. Perron integral
Chap.17. Functional Analysis
  • Section 1. Abstract Spaces
    • 1.1. Introduction
    • 1.2. Vector spaces
    • 1.3. Normed spaces
    • 1.4. Pre-Hilbert spaces
    • 1.5. Banach spaces and Hilbert spaces
    • 1.6. Cn[a, b] spaces
    • 1.7. Lp[a, b] space
    • 1.8. L[a, b] space
    • 1.9. Some results on Banach spaces
  • Section 2. Differentiable Operators
    • 2.1. Bounded linear operators
    • 2.2. Differentiable operators
    • 2.3. Frechet differential
    • 2.4. Invertible operators
    • 2.5. Gateaux differential
    • 2.6. Banach fixed-point theorem
  • Section 3. Calculus of Variations
    • 3.1. Introduction
    • 3.2. Four variational problems
    • 3.3. Euler differential equation
    • 3.4. Strong and weak extrema
    • 3.5. Morse theory
  • Section 4. Integral Equations
    • 4.1. Differential and integral operators
    • 4.2. Integral equations
    • 4.3. Integral equations of the second kind
  • Section 5. Compact Operators
    • 5.1. Compact operators
    • 5.2. Finite rank operators
    • 5.3. Hermitian operators
    • 5.4. Fredholm operator
    • 5.5. Differential and integral operators
    • 5.6. Fredholm alternative
Chap.18. Differential Equations
  • Section 1. Classic Differential Equations
    • 1.1. An introduction
    • 1.2. Notion of differential equation
    • 1.3. Initial-value problem, initial conditions, boundary value problem
    • 1.4. Questions on differential equation resolution
  • Section 2. First-order Differential Equations
    • 2.1. First-order differential equations with one variable
    • 2.2. Particular first-order differential equations
    • 2.3. Implicit first order differential equations
    • 2.4. Isolated singular solutions
    • 2.5. Singular solutions
  • Section 3. Second-order Differential Equations
    • 3.1 Second order differential equations
    • 3.2. General resolution of linear differential equations of second-order
    • 3.3. Resolution method of linear homogeneous equations
    • 3.4. Particular solution of non-homogeneous equation
    • 3.5. Linear differential equations of second-order with constant coefficients
  • Section 4. Linear n-order Differential Equations
    • 4.1. Linear homogeneous n-order differential equations
    • 4.2. Linear homogeneous n-order differential equations with constant coefficients
    • 4.3. Linear non-homogeneous n-order differential equations
  • Section 5. Systems of Differential Equations
    • 5.1. Example
    • 5.2. Systems of first-order differential equations
    • 5.3. Systems of higher order differential equations
    • 5.4. Systems of linear first-order differential equations
    • 5.5. Systems of homogeneous linear first order differential equations
    • 5.6. Systems of homogeneous linear first-order differential equations with constant coefficients
    • 5.7. Eigenvalues, eigenvectors
    • 5.8. Fundamental matrix
    • 5.9. Non-homogeneous system of linear first-order differential equations
    • 5.10. Problem of initial conditions
  • Section 6. Existence and Uniqueness Theorems
    • 6.1. Preamble
    • 6.2. Existence theorem
    • 6.3. Uniqueness theorem
    • 6.4. Extension
  • Section 7. Numerical Methods
    • 7.1. Introduction
    • 7.2. Numerical methods
    • 7.3. Euler-Cauchy method
    • 7.4. Runge-Kutta method
    • 7.5. Milne process
  • Section 8. Partial Differential Equations
    • 8.1. Introduction
    • 8.2. Partial differential equations
    • 8.3. Cauchy-Kowalewska theorem
    • 8.4. Heat equation
    • 8.5. Fredholm theorems
    • 8.6. Sobolev spaces
    • 8.7. L2-space, Lp-space
Chap.19. Differential Geometry
  • Section 1. Curves in ℝ3
    • 1.1. Introduction
    • 1.2. Curves in differential geometry
    • 1.3. Orientation of a Jordan arc
    • 1.4. Regular geometric arcs
    • 1.5. Intrinsic parameter, and curvilinear abscissa
    • 1.6. Tangent, unit tangent vector
    • 1.7. Curvature, curvature vector, principal normal
    • 1.8. Circle of curvature, osculating plane
    • 1.9. Binormal vector, Frénet trihedral
    • 1.10. Frénet formulas
    • 1.11. Curve torsion
    • 1.12. Torsion and curvature formulas
    • 1.13. Fundamental theorem of curve theory
    • 1.14. Canonical representation of simple arc
    • 1.15. Contact configurations
    • 1.16. Spherical arcs
    • 1.17. Helix
    • 1.18. Involutes of an arc
    • 1.19. Evolutes of an arc
  • Section 2. Plane Curves
    • 2.1. Plane curves
    • 2.2. Representation of a curve in ℝ2
    • 2.3. Curvature of a plane curve
    • 2.4. Frénet basis
    • 2.5. Osculating circle
    • 2.6. Involutes and evolutes of a plane curve
  • Section 3. Regular Patches, Surfaces
    • 3.1. Introduction
    • 3.2. Concept of regular patch
    • 3.3. Curvilinear coordinates, and coordinate curves on a regular patch
    • 3.4. Curves drawn on regular patch
    • 3.5. Tangent plane and normal to a patch
    • 3.6. Notion of surface
    • 3.7. Orientation of a surface
  • Section 4. First Fundamental Form
    • 4.1. Length measure on a regular patch
    • 4.2. Angle measure on a regular patch
    • 4.3. Area measure on a regular patch
    • 4.4. Isometric regular patches, and intrinsic geometry
  • Section 5. Second Fundamental Form, Curvature
    • 5.1. Curvature of a regular patch
    • 5.2. Normal curvature, geodesic curvature
    • 5.3. Calculation of geodesic curvature
    • 5.4. Calculation of normal curvature
    • 5.5. Classification of patch points
    • 5.6. Umbilics
    • 5.7. Principal curvature
    • 5.8. Dupin indicatrix
    • 5.9. Curvature lines
    • 5.10. Asymptotic lines
    • 5.11. Mean curvature, total curvature
    • 5.12. Ruled surfaces, developable surfaces
  • Section 6. Fundamental Theorem
    • 6.1. Introduction
    • 6.2. Gauss formulas, Weingarten formulas
    • 6.3. Gauss and Mainardi-Codazzi relations
    • 6.4. Fundamental theorem
  • Section 7. Tensors
    • 7.1. Preamble
    • 7.2. Convention of summation
    • 7.3. Dual, bidual vector spaces
    • 7.4. Basis, change of basis
    • 7.5. Tensors
    • 7.6. Coordinates of a tensor
    • 7.7. Tensors and injective patches
  • Section 8. Tensors II
    • 8.1. General concept
    • 8.2. Tensor calculus
    • 8.3. Tensor algebra
    • 8.4. Tensor analysis
    • 8.5. Tensor bundle
    • 8.6. Tensor on a vector space
    • 8.7. Tensor density
    • 8.8. Covariant tensor
    • 8.9. Metric tensor
    • 8.10. Riemann tensor
    • 8.11. Minkowski metric
    • 8.12. Ricci tensor
    • 8.13. Scalar curvature
  • Section 9. Differential Forms
    • 9.1. Definition
    • 9.2. One-form
    • 9.3. Change of variables in differential form
    • 9.4. Exterior differentiation
    • 9.5. Fundamental relation
    • 9.6. Differential forms, vector field
    • 9.7. Integral of differential form
    • 9.8. Differential forms on differentiable manifold
    • 9.9. Integral of a differential form on a manifold
    • 9.10. Integral of a differential form according to a parameter
    • 9.11. Curvilinear integral
    • 9.12. Surface integral
    • 9.13. Stokes formula application
    • 9.14. Stokes formula
    • 9.15. Stokes theorem
    • 9.16. Line integral
    • 9.17. Poincaré theorem
    • 9.18. Rotation of a vector field
  • Section 10. Manifolds, Riemannian Geometry
    • 10.1. Introduction
    • 10.2. 2-dimensional manifold
    • 10.3. Chart, atlas
    • 10.4. Differentiable manifold
    • 10.5. Surfaces in ℝ3
    • 10.6. Correspondence between manifolds
    • 10.7. Injective arcs on manifold, and tangent spaces
    • 10.8. Riemannian manifold, and Riemannian geometry
  • Section 11. Complements of Differential Geometry
    • 11.1. Whitney theorem
    • 11.2. Takens theorem
    • 11.3. Tangent space
    • 11.4. Manifold with boundary
    • 11.5. Frobenius theorem
    • 11.6. Connection
    • 11.7. Covariant derivative
    • 11.8. Christoffel symbols
    • 11.9. Torsion tensor
    • 11.10. Riemannian manifold
    • 11.11. Flat manifold
    • 11.12. Levi-Civita connection
    • 11.13. Curvature
    • 11.14. Principal curvature
    • 11.15. Gaussian curvature
    • 11.16. Normal curvature
    • 11.17. Scalar curvature
    • 11.18. Mean curvature
Chap.20. Function Theory (Complex Analysis)
  • Section 1. Introduction
  • Section 2. Complex Numbers, Compactification
    • 2.1. Field of complex numbers
    • 2.2. Compactification of ℂ
  • Section 3. Sequences and Complex Functions
    • 3.1. Complex sequences
    • 3.2. Complex functions
    • 3.3. Continuity
    • 3.4. Uniform continuity
  • Section 4. Holomorphy
    • 4.1. Real differentiability
    • 4.2. Complex differentiability
    • 4.3. Cauchy-Riemann conditions
    • 4.4. Harmonic functions
  • Section 5. Cauchy Integral Theorem and Formulas
    • 5.1. Introduction
    • 5.2. Complex curvilinear integral
    • 5.3. Cauchy integral property
    • 5.4. Cauchy integral formulas
  • Section 6. Entire Series Expansion
    • 6.1 Series expansion
  • Section 7. Analytic Continuation
    • 7.1. Holomorphic continuation
    • 7.2. Analytic continuation
  • Section 8. Singularities, Laurent Series
    • 8.1. Singularity of a holomorphic function
    • 8.2. Laurent series
    • 8.3. Contour, contour integral
    • 8.4. Contour integration
  • Section 9. Meromorphy, Residues
    • 9.1. Meromorphic function
    • 9.2. Residue
    • 9.3. Residue theorem
  • Section 10. Riemann Surfaces
    • 10.1. Introduction
    • 10.2. Riemann surfaces
    • 10.3. Abstract Riemann surfaces
    • 10.4. Concrete Riemann surfaces
    • 10.5. Local canonical parametrization
    • 10.6. Complex structure
    • 10.7. Analytic functions
  • Section 11. Entire Functions
    • 11.1. Entire functions
  • Section 12. Meromorphic functions on ℂ
    • 12.1. Partial fraction decomposition
    • 12.2. Weierstrass functions
  • Section 13. Periodic functions
    • 13.1. Periods of complex functions
    • 13.2. Simply periodic functions
    • 13.3. Doubly periodic functions
  • Section 14. Algebraic functions
    • 14.1. Algebraic functions
    • 14.2. Abelian integrals
  • Section 15. Conformal transformations
    • 15.1. Introduction
    • 15.2. Conformal transformation
    • 15.3. Conformal transformation from Ĉ to Ĉ
    • 15.4. Conformal transformation from ℂ to ℂ
    • 15.5. Classification of homographic transformations
    • 15.6. Conformal transformation from the interior unit disk to itself
    • 15.7. Conformal transformation of a simply connected domain
    • 15.8. Conformal transformation from a domain to another
  • Section 16. Functions of several variables
    • 16.1. Space ℂn
    • 16.2. Holomorphy
    • 16.3. Entire series of several variables
    • 16.4. Analytic continuation (singularities)
    • 16.5. Continuity property
    • 16.6. Remark about continuations
Chap.21. Complex Analysis II
  • Section 1. Introduction
  • Section 2. Prerequisites
  • Section 3. Recall on Functions of Complex Variables
    • 3.1. Continuity, Differentiability
  • Section 4. Curvilinear Integral
    • 4.1. Differential forms of degee 1
    • 4.2. Paths, loops, curves
    • 4.3. Curvilinear integral
    • 4.4. Orientation
    • 4.5. Length
  • Section 5. Differential Forms in the Plane
    • 5.1. Exact forms, closed forms
    • 5.2. Differential forms of degree 2
    • 5.3. Stokes formula
  • Section 6. Holomorphic Functions I
    • 6.1. Definitions, examples
    • 6.2. Usual functions
    • 6.3. Cauchy theorem and formula
    • 6.4. Entire series expansion
    • 6.5. Cauchy inequalities, applications
    • 6.6. Cauchy formula for a disk (direct demonstration)
    • 6.7. Cauchy transform of Borel measure
    • 6.8. Cauchy kernel, A(D) and H2(D) spaces
  • Section 7. Holomorphic Functions II
    • 7.1. Primitives, logarithms
    • 7.2. Morera theorem
    • 7.3. Cauchy-Goursat theorem
    • 7.4. Zeros of holomorphic functions
    • 7.5. Laurent series
    • 7.6. Isolated singularities, and meromorphic functions
    • 7.7. Liouville theorem
    • 7.8. Maximum principle
    • 7.9. Schwarz lemma
    • 7.10. Infinite products
  • Section 8. Homotopy
    • 8.1. Recall
    • 8.2. Curvilinear integral (case of continuous paths)
    • 8.3. Homotopy
    • 8.4. Index of loop with respect to a point
  • Section 9. Topology of the Complex Plane
    • 9.1. Prerequisite
    • 9.2. Degree of a map defined on a circle
    • 9.3. Brouwer theorem, open mapping theorem
    • 9.4. Homotopy, extensions, continuous logarithms
    • 9.5. Jordan theorem
    • 9.6. Separation of points
  • Section 10. Cauchy Theorem (Homology Version)
    • 10.1. Recall
    • 10.2. Cycles and homology
    • 10.3. Cauchy theorem (Homology version)
    • 10.4. Density of rational functions
    • 10.5. Homology, cohomology
  • Section 11. Residues
    • 11.1. Residue theorem
    • 11.2. Calculation of a residue
    • 11.3. Counting of zeros and poles
    • 11.4. Integral calculations (examples)
  • Section 12. Runge Theorem, Applications
    • 12.1. Introduction
    • 12.2. Runge theorem
    • 12.3. Envelope of holomorphy
    • 12.4. Solving equation ∂u/∂z = v
    • 12.5. Cousin problem
    • 12.6. Mittag-Leffer theorem
    • 12.7. Weierstrass theorem
  • Section 13. Conformal Mapping
    • 13.1. Riemann sphere
    • 13.2. Holomorphic functions on an open set of S2
    • 13.3. Illustrations : The automorphisms of ℂ, S2, D
    • 13.4. Riemann mapping theorem
    • 13.5. Characterizations of simply connected open sets
    • 13.6. Carathéodory theorem, and Jordan domain
  • Section 14. Harmonic Functions
    • 14.1. Definitions, properties
    • 14.2. Harmonicity and holomorphy
    • 14.3. Principle of analytic continuation (analyticity)
    • 14.4. Maximum principle
    • 14.5. Poisson formula
    • 14.6. Cauchy inequalities
    • 14.7. Harnack inequalities
    • 14.8. Poisson integral
    • 14.9. Dirichlet problem
    • 14.10. Convergence of Fourier series (Abel-Poisson)
    • 14.11. Spaces hp
    • 14.12. Green formula
  • Section 15. Subharmonic Functions
    • 15.1. Definitions, properties
    • 15.2. Maximum principle
    • 15.3. Harmonic majorant, global mean value (properties)
    • 15.4. Circular average, three circles theorem
    • 15.5. Integrability
    • 15.6. Approximation by convolution
    • 15.7. Distributions and subharmonic functions
    • 15.8. Subharmonicity (examples)
  • Section 16. (A1) Convolution, Partition of Unity
    • 16.1. Convolution
    • 16.2. Plateau functions, partition of Unity
  • Section 17. (A2) Distributions
    • 17.1. Definition, examples
    • 17.2. Algebraic operations
    • 17.3. Restriction
    • 17.4. Derivation
    • 17.5. Support of a distribution
    • 17.6. Convolution
  • Section 18. (A3) Topology of the Complex Plane II
    • 18.1. Elements of linear algebra and differential calculus
    • 18.2. Differential forms on an open subset Ω of ℂ
    • 18.3. Partition of unity
    • 18.4. Regular boundaries
    • 18.5. Integration of differential 2-forms (and Stokes formula)
    • 18.6. Homotopy (Fundamental group)
    • 18.7. Integration of closed 1-forms along continuous paths
    • 18.8. Index of a loop
    • 18.9. Homology, i-chains
    • 18.10. Residues
    • 18.11. Symbols/Notations of (A3)
Part IX. Category Theory.
Chap.22. Areas Involved in Category Theory
  • Section 1. Areas Involved
    • 1.1. Abstract algebra (Areas)
    • 1.2. Homological algebra (Areas)
    • 1.3. Representation theory (Areas)
    • 1.4. Universal algebra (Areas)
    • 1.5. Algebraic topology (Areas)
    • 1.6. Homology theory (Areas)
    • 1.7. Algebraic geometry (Areas)
    • 1.8. Model theory (Areas)
  • Section 2. Algebraic System, Universal Algebra
    • 2.1. Algebraic system
    • 2.2. Universal algebra
    • 2.3. Algebraic operation (n-ary operation)
    • 2.4. Arity, polyadic
  • Section 3. Signatures
    • 3.1. Signature (of an algebraic system)
    • 3.2. Signature (of a structure)
    • 3.3. Signature (logic)
    • 3.4. Signature (disambiguation)
    • 3.5. Many-sorted logic (logic)
    • 3.6. Type, sort (logic)
  • Section 4. Structures
    • 4.1. Structures
    • 4.2. Structures (logic)
    • 4.3. Algebraic structure (abstract algebra)
  • Section 5. Interpretations
    • 5.1. Definable set (logic)
    • 5.2. Interpretation (logic)
    • 5.3. Interpretation (model theory)
  • Section 6. Varieties
    • 6.1. Variety (overview)
    • 6.2. Algebraic variety
    • 6.3. Variety of algebras (universal algebra)
    • 6.4. Variety (universal algebra)
    • 6.5. Birkhoff HSP theorem (universal algebra)
  • Section 7. Algebraic Structures
    • 7.1. Sets and algebraic structures
    • 7.2. Group theory (abstract algebra)
    • 7.3. Ring theory (abstract algebra)
    • 7.4. Field theory (abstract algebra)
    • 7.5. Galois theory (abstract algebra)
    • 7.6. Module theory (abstract algebra)
    • 7.7. Vector space theory
    • 7.8. Internal composition law
    • 7.9. Group
    • 7.10. Ring
    • 7.11. Field
    • 7.12. External composition law
    • 7.13. Module
    • 7.14. Abelian group
    • 7.15. Monoid (group theory)
    • 7.16. R-module (homological algebra)
    • 7.17. Free algebraic structures
    • 7.18. Free Abelian group (free ℤ-module)
  • Section 8. Maps and Structure Homomorphisms
    • 8.1. Map (left surjective, univocal)
    • 8.2. Surjection (onto and not one-to-one)
    • 8.3. Injection (one-to-one and not onto)
    • 8.4. Bijection (one-to-one and onto)
    • 8.5. Illustrations (one-to-one, onto)
    • 8.6. Image, preimage (image inverse)
    • 8.7. Inclusion map (canonical injection)
    • 8.8. Structure homomorphisms
    • 8.9. Endomorphism (of algebraic system)
  • Section 9. Algebraic Structure Morphisms and Kernels
    • 9.1. Closure (abstract algebra)
    • 9.2. Magma (abstract algebra)
    • 9.3. Monoid (abstract algebra)
    • 9.4. Semigroup (abstract algebra)
    • 9.5. R-algebraic structure
    • 9.6. R-module (abstract algebra)
    • 9.7. Homomorphisms (abstract algebra)
    • 9.8. Kernels (abstract algebra)
  • Section 10. Ring Theory
    • 10.1. Ring (ring theory)
    • 10.2. Ring with unity (ring theory)
    • 10.3. Graded ring (ring theory)
    • 10.4. Graded algebra
    • 10.5. Graded module (homological algebra)
    • 10.6. Noetherian ring (ring theory)
    • 10.7. Ascending chain condition (ACC)
    • 10.8. Descending chain condition (DCC)
    • 10.9. Integral domain (entire ring)
    • 10.10. Free algebra (ring theory)
    • 10.11. Free module (homological algebra)
    • 10.12. Direct product (abstract algebra)
    • 10.13. External direct sum, internal direct sum
    • 10.14. Reduced ring (ring theory)
    • 10.15. Reduced algebra
    • 10.16. Tensor powers, and braiding
    • 10.17. Operad (abstract algebra)
    • 10.18. Artinian ring (ring theory)
    • 10.19. Catenary ring (ring theory)
    • 10.20. Cohen-Macaulay ring (ring theory)
    • 10.21. Depth (ring theory)
    • 10.22. Local ring (ring theory)
    • 10.23. Ring unit (ring theory)
    • 10.24. Regular ring (ring theory)
    • 10.25. Regular local ring (ring theory)
    • 10.26. Krull dimension (ring theory)
    • 10.27. Residue field (ring theory)
    • 10.28. Quotient ring
    • 10.29. Von Neumann regular ring
    • 10.30. Differential ring
    • 10.31. Differential field
    • 10.32. Differential algebra
    • 10.33. Differential graded algebra
    • 10.34. Ritt algebra
    • 10.35. Differential module (homological algebra)
    • 10.36. Kähler differential (algebraic geometry)
    • 10.37. Module of Kähler differentials
    • 10.38. Multiplicative set
    • 10.39. Localization (abstract algebra)
    • 10.40. Completion (abstract algebra)
    • 10.41. Maximal spectrum
    • 10.42. Proper ideal (ring theory)
    • 10.43. Ideal, prime ideal (ring theory)
    • 10.44. Maximal ideal (ring theory)
    • 10.45. Irrelevant ideal (ring theory)
    • 10.46. Principal ideal and ring
    • 10.47. Principal ideal domain and ring
    • 10.48. Homogeneous element (ring theory)
    • 10.49. Homogeneous ideal (ring theory)
    • 10.50. Finitely generated ideal (ring theory)
    • 10.51. Finitely presented algebra (ring theory)
    • 10.52. Radical (disambiguation)
    • 10.53. Jacobson radical (ring theory)
    • 10.54. Nilradical (ring theory)
    • 10.55. Radical of ideal (ring theory)
    • 10.56. Radical ideal (ring theory)
    • 10.57. Nilpotent element
    • 10.58. Spectrum of a ring (Spec(R))
    • 10.59. Prime spectrum (Spec)
    • 10.60. Perfect field (field theory)
    • 10.61. Characteristic (algebra)
    • 10.62. Separable algebra
    • 10.63. Simple algebra
    • 10.64. Semi-simple algebra
    • 10.65. Irreducible polynomial (algebra)
    • 10.66. Separable extension (field theory)
    • 10.67. Separable polynomial (field theory)
    • 10.68. Separable element
    • 10.69. Separable closure
    • 10.70. Compositum (field theory)
    • 10.71. Algebraic element (field theory)
    • 10.72. Algebraic extension of a field
    • 10.73. Extension field (field theory)
    • 10.74. Algebraic closure (field theory)
    • 10.75. Algebraically closed field
    • 10.76. Regular map (algebraic geometry)
    • 10.77. Affine space (analytic geometry)
    • 10.78. Affine space (algebraic geometry)
    • 10.79. Affine variety (algebraic geometry)
    • 10.80. Coordinate ring (algebraic geometry)
    • 10.81. Locus (geometry)
    • 10.82. Zero-locus
    • 10.83. Zero set
    • 10.84. Projective space
    • 10.85. Projective variety
    • 10.86. Quasi-projective variety
    • 10.87. Homogeneous polynomial
    • 10.88. Homogeneous function
    • 10.89. Homogeneous ideal (and graded ring)
    • 10.90. Linear topology
    • 10.91. Adic topology
  • Section 11. Complexes
    • 11.1. Chain complex (homological algebra)
    • 11.2. Cochain complex (homological algebra)
    • 11.3. Differential complex
    • 11.4. Null sequence (homological algebra)
    • 11.5. Quasi-isomorphism (homological algebra)
    • 11.6. Boundary operator
    • 11.7. Coboundary operator
    • 11.8. Coboundary
    • 11.9. Cycle, cocycle
    • 11.10. Simplex
    • 11.11. Simplicial complex
    • 11.12. Abstract simplicial complex
    • 11.13. Polytope (geometry)
    • 11.14. Cell (geometry)
    • 11.15. Cell (algebraic topology)
    • 11.16. Subcomplex (algebraic topology)
    • 11.17. Skeleton (algebraic topology)
    • 11.18. CW-complex (algebraic topology)
    • 11.19. Nerve (algebraic topology)
  • Section 12. Direct limits
    • 12.1. Directed set (set theory)
    • 12.2. Direct family (of sets)
    • 12.3. Direct limit (of sets)
    • 12.4. Direct family (of algebraic systems)
    • 12.5. Direct limit (of algebraic systems)
    • 12.6. Direct system (homological algebra)
    • 12.7. Direct limit (homological algebra)
  • Section 13. Topological spaces
    • 13.1. Open set (topology)
    • 13.2. Clopen set (topology)
    • 13.3. Interior (topology)
    • 13.4. Interior point (topology)
    • 13.5. Open map (topology)
    • 13.6. Open mapping theorem
    • 13.7. Topological space (topology)
    • 13.8. Topology on a set
    • 13.9. Subspace topology
    • 13.10. Pointed topological space (topology)
    • 13.11. Paracompact space (topology)
    • 13.12. Quasi-compact space (topology)
    • 13.13. Separated space (topology)
    • 13.14. Hausdorff space (topology)
    • 13.15. Homeomorphism (topology)
    • 13.16. Local homeomorphism (topology)
    • 13.17. Braid (topology)
    • 13.18. Braid group (topology)
  • Section 14. Manifolds
    • 14.1. Topological manifold
    • 14.2. Manifold
    • 14.3. Differentiable manifold
    • 14.4. Differentiable structure
    • 14.5. Smooth manifold
    • 14.6. Manifold with boundary
    • 14.7. Compact manifold
    • 14.8. Closed manifold
    • 14.9. Open manifold
    • 14.10. Atlas (topology)
    • 14.11. Differentiable atlas
    • 14.12. Maximal atlas (topology)
    • 14.13. Coordinate chart (topology)
    • 14.14. Transition map (topology)
    • 14.15. Coordinate patch (differential geometry)
    • 14.16. Countable, separable, dense
    • 14.17. Second-countable space
    • 14.18. Topological basis, local basis
    • 14.19. Neighborhood system (topology)
    • 14.20. Neighborhood basis (local basis)
    • 14.21. Filter base (topology)
    • 14.22. Filter (topology)
    • 14.23. Ultrafilter
    • 14.24. Free ultrafilter
    • 14.25. Submanifold (topology)
    • 14.26. Regular submanifold
    • 14.27. Immersed submanifold
    • 14.28. Embedded submanifold
    • 14.29. Weakly embedded submanifold
    • 14.30. Smoothly universal
    • 14.31. Proper map
    • 14.32. Topological embedding (topology)
    • 14.33. Embedding
    • 14.34. Induced homomorphism
    • 14.35. Connected component (topology)
    • 14.36. Connected space (topology)
    • 14.37. Totally disconnected space
    • 14.38. Simply connected (1-connected)
    • 14.39. Simply connected space
    • 14.40. Arcwise-connected space
    • 14.41. Locally arcwise-connected space
    • 14.42. Pathwise and arcwise connectedness
    • 14.43. Contractible loops
    • 14.44. Immersion (algebraic topology)
    • 14.45. Rank (of differential map)
    • 14.46. Diffeomorphism (topology)
    • 14.47. Local diffeomorphism (topology)
    • 14.48. Smooth map (smooth function)
    • 14.49. Smooth functions on manifolds
    • 14.50. Smooth maps between manifolds
    • 14.51. Cobordism (algebraic topology)
    • 14.52. h-Cobordism (algebraic topology)
    • 14.53. Analytic functions (analysis)
    • 14.54. Complex differentiability (analysis)
    • 14.55. Regular function (functional analysis)
  • Section 15. Homology, Cohomology
    • 15.1. Homology (algebraic topology)
    • 15.2. Cohomology (algebraic topology)
    • 15.3. Cohomology (of topological space)
    • 15.4. Cohomology (with values in sheaf )
    • 15.5. Cohomology ring (algebraic topology)
    • 15.6. Cup product (algebraic topology)
    • 15.7. Singular homology (algebraic topology)
    • 15.8. Simplicial homology (algebraic topology)
    • 15.9. Simplicial mapping
    • 15.10. Homology group
    • 15.11. Cohomology group
    • 15.12. Betti group
  • Section 16. Sheaf Theory
    • 16.1. Presheaf (algebraic topology)
    • 16.2. Presheaf (category theory)
    • 16.3. Sheaf (basic definition)
    • 16.4. Sheaf (general topology)
    • 16.5. Sheaf of rings
    • 16.6. Germ
    • 16.7. Stalk (of a sheaf )
    • 16.8. Ringed space
    • 16.9. Locally ringed space
    • 16.10. Ideal sheaf (algebraic geometry)
    • 16.11. Structure sheaf
    • 16.12. Constant sheaf
    • 16.13. Coherent sheaf
    • 16.14. Quasi-coherent sheaf
    • 16.15. Graded sheaf
    • 16.16. Differential sheaf and resoluton
    • 16.17. Eilenberg-Steenrod axioms
    • 16.18. Snake lemma (homological algebra)
    • 16.19. Exact sequence (homological algebra)
    • 16.20. Split exact sequence
    • 16.21. Projective module
    • 16.22. Injective module
    • 16.23. Injective envelope (injective hull)
    • 16.24. Essential extension
    • 16.25. Essential monomorphism
    • 16.26. Injective object (category theory)
    • 16.27. Enough injectives
    • 16.28. Projective object (category theory)
    • 16.29. Enough projectives
    • 16.30. Resolution (homological algebra)
    • 16.31. Injective resolution
    • 16.32. Projective resolution
    • 16.33. Acyclic object (homological algebra)
    • 16.34. Derived functor (homological algebra)
    • 16.35. Ext functor (homological algebra)
    • 16.36. Resolution (abstract algebra)
    • 16.37. Sheaf cohomology
    • 16.38. Cech cohomology (algebraic topology)
    • 16.39. Flasque sheaf (homological algebra)
    • 16.40. Canonical resolution
    • 16.41. Godement resolution
    • 16.42. Acyclic resolution
    • 16.43. Acyclic sheaf
    • 16.44. Tor (homological algebra)
    • 16.45. Analytic space
  • Section 17. Bundles
    • 17.1. Fiber (topology)
    • 17.2. Fiber (algebraic geometry)
    • 17.3. Geometric point (algebraic geometry)
    • 17.4. Geometric fiber (algebraic geometry)
    • 17.5. Fiber bundle (topology)
    • 17.6. Bundle (topology)
    • 17.7. Bundle (category)
    • 17.8. Vector bundle
    • 17.9. Line bundle
    • 17.10. Principal bundle
    • 17.11. Trivial bundle
    • 17.12. Locally trivial bundle
    • 17.13. Trivialization
    • 17.14. Fiber space (algebraic topology)
    • 17.15. Fibration (algebraic topology)
    • 17.16. Covering homotopy property
    • 17.17. G-space (topology)
    • 17.18. Bundle rank (topology)
    • 17.19. K-theory
    • 17.20. C*-algebra
    • 17.21. Antiautomorphism (group theory)
    • 17.22. Involution (general)
    • 17.23. Monic polynomial (algebra)
    • 17.24. Canonical map (set theory)
    • 17.25. Section of a fiber bundle
    • 17.26. Local section
    • 17.27. Global section
  • Section 18. Scheme Theory
    • 18.1. Scheme (field theory)
    • 18.2. Morphism of schemes
    • 18.3. Category of schemes
    • 18.4. Y-scheme, structure morphism
    • 18.5. Affine scheme (algebraic topology)
    • 18.6. Noetherian scheme (algebraic geometry)
    • 18.7. Separated scheme
    • 18.8. Noetherian affine scheme
    • 18.9. Smooth scheme (algebraic geometry)
    • 18.10. Reduced scheme
    • 18.11. Irreducible scheme
    • 18.12. Integral scheme
    • 18.13. Open subscheme (algebraic geometry)
    • 18.14. Closed subscheme (algebraic geometry)
    • 18.15. Projective scheme
    • 18.16. Catenary scheme
    • 18.17. Cohen-Macaulay scheme
    • 18.18. Affine morphism, affine Y -scheme
    • 18.19. Diagonal morphism
    • 18.20. Separated morphism
    • 18.21. Proper morphism
    • 18.22. Quasi-separated morphism
    • 18.23. Flat morphism (algebraic geometry)
    • 18.24. Flat module (homological algebra)
    • 18.25. Faithfully flat module
    • 18.26. Open morphism (closed morphism)
    • 18.27. Unramified morphism
    • 18.28. Smooth morphism (algebraic geometry)
    • 18.29. Finite morphism (algebraic geometry)
    • 18.30. Finitely generated module
    • 18.31. Morphism of finite type
    • 18.32. Morphism locally of finite type
    • 18.33. Morphism of finite presentation
    • 18.34. Morphism locally of finite presentation
    • 18.35. Quasi-compact morphism
    • 18.36. Quasi-finite morphism
    • 18.37. Morphism with finite fibers
    • 18.38. Cover (topology)
    • 18.39. Refinement
    • 18.40. Zariski cover
    • 18.41. Fppf cover (algebraic geometry)
    • 18.42. Fpqc cover (algebraic geometry)
    • 18.43. Ètale cover (algebraic geometry)
    • 18.44. Zariski topology (algebraic geometry)
    • 18.45. Zariski cover (algebraic topology)
    • 18.46. Inadequacy of Zariski topology
    • 18.47. Ètale fundamental group
    • 18.48. Ètale map
    • 18.49. Ètale neighborhood
    • 18.50. Ètale topology (algebraic geometry)
    • 18.51. Ètale morphism (algebraic geometry)
    • 18.52. Standard étale morphism
    • 18.53. Formally étale morphisms of schemes
    • 18.54. Ètale cohomology (algebraic geometry)
    • 18.55. Ètale cohomology groups
    • 18.56. l-adic étale cohomology
    • 18.57. l-adic cohomology (general)
    • 18.58. Immersion (algebraic geometry)
    • 18.59. Closed immersion (algebraic geometry)
    • 18.60. Global Spec (algebraic geometry)
    • 18.61. Spec(ℤ)
    • 18.62. Dominant morphism
    • 18.63. Dense morphism
    • 18.64. Projective morphism
    • 18.65. Proj (algebraic geometry)
    • 18.66. Global Proj (algebraic geometry)
  • Section 19. Homotopy
    • 19.1. Homotopy (algebraic topology)
    • 19.2. Deformation (algebraic topology)
    • 19.3. Deformation retract
    • 19.4. Strong deformation retract
    • 19.5. Retract (algebraic topology)
    • 19.6. Retraction (algebraic topology)
    • 19.7. Restriction, extension
    • 19.8. Lift (category theory)
    • 19.9. Lift (algebraic topology)
    • 19.10. Lifting property
    • 19.11. Homotopy lifting property
    • 19.12. Path lifting property
    • 19.13. Covering space (algebraic topology)
    • 19.14. Covering projection
    • 19.15. Covering map
    • 19.16. Universal covering space
    • 19.17. Universal cover (algebraic topology)
    • 19.18. Path (topology)
    • 19.19. Loop (topology)
    • 19.20. Quotient topology (topology)
    • 19.21. Quotient map (topology)
    • 19.22. Quotient space (topology)
    • 19.23. Saturated set (topology)
Chap.23. Category Theory
  • Section 1. Introduction
    • 1.1. Significant dates
    • 1.2. Category theory
    • 1.3. Object, category, functor, morphism, natural transformation
    • 1.4. Current approach of a category
    • 1.5. Original definitions of a category (by Mac Lane, Eilenberg, Grothendieck, Lambek)
  • Section 2. Universe
    • 2.1. Universe 𝕌
    • 2.2. Inaccessible cardinal
    • 2.3. Class
    • 2.4. Proper class
    • 2.5. Small class
    • 2.6. Urelement
    • 2.7. Pure set (hereditary set)
    • 2.8. Hereditarily finite set
    • 2.9. Preorder set (and small category)
    • 2.10. Totally ordered set
    • 2.11. Partially ordered set
    • 2.12. Well-ordered set
    • 2.13. Directed order
    • 2.14. Directed ordered set
    • 2.15. Zorn lemma
    • 2.16. Maximal element
    • 2.17. Greatest element
    • 2.18. 𝕌-set, 𝕌-small set
    • 2.19. Set {pt} with one element
  • Section 3. Objects
    • 3.1. Object
    • 3.2. Initial object, terminal object
    • 3.3. Zero object
    • 3.4. Subobject, quotient object
    • 3.5. Free object
    • 3.6. Exponential object
    • 3.7. Universal property
    • 3.8. System of generators
    • 3.9. Class of generators for a category
    • 3.10. Subobject classifier
    • 3.11. Natural numbers object
    • 3.12. Generalized element, global element
  • Section 4. Categories
    • 4.1. Category
    • 4.2. Small and large categories
    • 4.3. 𝕌-category
    • 4.4. Essentially 𝕌-small category
    • 4.5. Category of small categories
    • 4.6. Dual category
    • 4.7. Product category
    • 4.8. Subcategory
    • 4.9. Precategory
    • 4.10. Opposite category
    • 4.11. Concrete category
    • 4.12. Abelian category
    • 4.13. Pair category
    • 4.14. Comma category
    • 4.15. Slice category
    • 4.16. Well-powered category
    • 4.17. Bicategory
    • 4.18. Ab-category
    • 4.19. Preadditive category
    • 4.20. Additive category
    • 4.21. Complete category
    • 4.22. Small and large categories
    • 4.23. Functor category
    • 4.24. Tensor category
    • 4.25. Factor through
    • 4.26. Tensor product
    • 4.27. Operad
    • 4.28. Monoidal category
    • 4.29. Closed monoidal category
    • 4.30. Braided monoidal category
    • 4.31. Symmetric monoidal category
    • 4.32. Enriched category
    • 4.33. Cartesian closed category
    • 4.34. Regular category
    • 4.35. Category associated with ordered set
    • 4.36. Filtrant category
    • 4.37. Category of chain complexes
    • 4.38. Mac Lane's introduction to categories
    • 4.39. Metacategory
    • 4.40. Grothendieck abelian category
    • 4.41. Triangulated category
    • 4.42. Homotopy category
    • 4.43. Derived category
    • 4.44. Monoid (in category theory)
    • 4.45. Variety (in a category)
    • 4.46. Algebraic category
    • 4.47. Monad
    • 4.48. Algebras for a monad (T-algebra)
    • 4.49. Eilenberg-Moore algebras (of a monad)
    • 4.50. Class of algebras
    • 4.51. F-algebra
  • Section 5. Functors
    • 5.1. Functor
    • 5.2. Faithful functor
    • 5.3. Embedding
    • 5.4. Equivalence
    • 5.5. Category isomorphism
    • 5.6. Full functor
    • 5.7. Autofunctor
    • 5.8. Exact functor
    • 5.9. Essentially surjective
    • 5.10. Amnestic functor
    • 5.11. Hom functor
    • 5.12. Bifunctor
    • 5.13. Multifunctor
    • 5.14. Adjoint functors (adjunction)
    • 5.15. Quasi-inverse functor
    • 5.16. Half-full functor
    • 5.17. Forgetful functor
    • 5.18. Representable functor
    • 5.19. Enriched functor
    • 5.20. Additive functor
    • 5.21. Derived functor
    • 5.22. Homology functor
    • 5.23. Conservative functor
  • Section 6. Morphisms
    • 6.1. Morphisms
    • 6.2. Homomorphism (morphism)
    • 6.3. Monomorphism (monic)
    • 6.4. Epimorphism (epic)
    • 6.5. Isomorphism
    • 6.6. Endomorphism
    • 6.7. Automorphism
    • 6.8. Normal monomorphism
    • 6.9. Inclusion map
    • 6.10. Subgroup, coset
    • 6.11. Normal subgroup
    • 6.12. Abelian group
    • 6.13. Canonical projection and epimorphism
    • 6.14. Zero morphism
    • 6.15. Separable morphism
    • 6.16. Regular monomorphism
    • 6.17. Regular epimorphism
    • 6.18. Kernel, cokernel
    • 6.19. Kernel pair
    • 6.20. Natural transformation
  • Section 7. Graphs and Diagrams
    • 7.1. Metagraph
    • 7.2. Directed graph
    • 7.3. Isomorphic directed graphs
    • 7.4. Commutative diagram
    • 7.5. Span, cospan
    • 7.6. Diagram, cone, cocone
  • Section 8. Limits and Products
    • 8.1. Direct system in a category
    • 8.2. Direct limit, direct system
    • 8.3. Limit, colimit
    • 8.4. Prelimit, direct limit
    • 8.5. Direct product, direct sum, and functors
    • 8.6. Direct product, direct sum
    • 8.7. Product, coproduct
    • 8.8. Direct sum is not necessarily coproduct
    • 8.9. Equalizer
    • 8.10. Coequalizer
  • Section 9. Pullbacks and Pushouts
    • 9.1. Pullback (fiber product)
    • 9.2. Pushout (fiber coproduct)
  • Section 10. Kernels and Cokernels
    • 10.1. Kernel (as morphism)
    • 10.2. Cokernel (as morphism)
    • 10.3. Kernel (as pullback)
    • 10.4. Cokernel (as pushout)
    • 10.5. Kernel pair (as pulback square)
  • Section 11. Exact Sequences
    • 11.1. Exact sequence
    • 11.2. Short exact sequence
    • 11.3. Long exact sequence
  • Section 12. Sheaves
    • 12.1. Presheaf
    • 12.2. Sheaf
    • 12.3. Gluing axiom
    • 12.4. Sheafification (sheaving)
    • 12.5. Stalk
    • 12.6. Resolution of a sheaf
  • Section 13. Grothendieck topology, site, sieve
    • 13.1. Grothendieck topology
    • 13.2. Grothendieck pretopology
    • 13.3. Covering sieve
    • 13.4. Sieve
    • 13.5. Site
  • Section 14. Topos
    • 14.1. Topos (topoi)
    • 14.2. Power object (in topos)
    • 14.3. Global element
  • Section 15. Ètale
    • 15.1. Étalé space
    • 15.2. Ètale sheaf
    • 15.3. Topological étale site
    • 15.4. Ètale site of a scheme
    • 15.5. Ètale presheaf
    • 15.6. Sheaf on étale site
    • 15.7. Ètale topos
    • 15.8. Smooth and étale maps (of schemes)
    • 15.9. Topological grounds for sites
  • Section 16. Yoneda Lemma, Yoneda Embedding
    • 16.1. Yoneda lemma
    • 16.2. Yoneda embedding
Part X. Probability and Statistics.
Chap.24. Probability
  • Section 1. Combinatorial Analysis
    • 1.1. Introduction
    • 1.2. Scope and examples of combinatorial analysis
    • 1.3. Permutation without repetition
    • 1.4. Permutation with repetitions
    • 1.5. Arrangements without repetition
    • 1.6. Arrangements with repetitions
    • 1.7. Combinations without repetition
    • 1.8. Combination with repetitions
Chap.25. Probability Calculation, Statistics
  • Section 1. Introduction
  • Section 2. Event, Probability
    • 2.1. Notion of event
    • 2.2. Intersection and union
    • 2.3. Conditional probabilities
    • 2.4. Independent events
    • 2.5. Total probability
    • 2.6. Trees, successive trials
    • 2.7. Probability and combinatorial analysis
    • 2.8. Law of large numbers, limit theorem
  • Section 3. Statistical Distribution, Cumulative Distribution
    • 3.1. Relative frequency
    • 3.2. Random variable
    • 3.3. Cumulative distribution function
    • 3.4. Expectation, variance, standard deviation
    • 3.5. Binomial distribution
    • 3.6. Poisson distribution
    • 3.7. Normal distribution
  • Section 4. Statistical Methods
    • 4.1. Population, sample
    • 4.2. Hypothesis testing
    • 4.3. χ2 test
    • 4.4. Dependence of two criteria
    • 4.5. Random numbers
    • 4.6. Random number, Markov chain, random walk
    • 4.7. Monte-Carlo method
  • Section 5. Statistical Models
    • 5.1. Notion of Model
    • 5.2. Simple regression model
    • 5.3. Multiple regression model
    • 5.4. Multicollinearity, and choice of optimal model
    • 5.5. Violation of assumptions
    • 5.6. Nonlinear models
    • 5.7. Time-lag models
    • 5.8. Simultaneous equations models (SEM)
    • 5.9. Time series analysis
    • 5.10. VAR modeling
    • 5.11. Cointegration and model with error correction
    • 5.12. Long memory processes
    • 5.13. Processes developed from the ARFIMA process
    • 5.14. The estimation of the integration parameter d in ARFIMA(p,d,q) process
Part XI. Applied Mathematics.
Chap.26. Miscellaneous
  • Section 1. Fourier Series and Fourier Transform
    • 1.1. Bessel inequality
    • 1.2. Parseval equality
    • 1.3. Trigonometric series
    • 1.4. Fourier series
    • 1.5. Fourier series convergence
    • 1.6. Fourier series integration
    • 1.7. Fourier transform
    • 1.8. Usual writings of the Fourier series and transform
    • 1.9. Complex representation of Fourier series
    • 1.10. Important properties of Fourier series
    • 1.11. Determination of coefficients for symmetric functions
    • 1.12. Forms of Fourier series expansions
    • 1.13. Determination of coefficients by numerical methods
    • 1.14. Fourier series and Fourier integrals
    • 1.15. Numerical harmonic analysis
  • Section 2. Integral transformations
    • 2.1. General definition
    • 2.2. Laplace transformation
    • 2.3. Fourier transformation
    • 2.4. Z-transformation
    • 2.5. Wavelet transformation
    • 2.6. Walsh functions
  • Section 3. Distribution Theory
    • 3.1. Definition of a distribution
    • 3.2. Derivation of distributions
    • 3.3. Multiplication
    • 3.4. Distribution support
    • 3.5. Convolution of distributions
    • 3.6. Application to partial differential equations with constant coefficients
    • 3.7. Use of elementary solutions
Chap.27. Optimization
  • Section 1. Introduction
  • Section 2. Linear Optimization
    • 2.1. Optimization with two variables
    • 2.2. Generalization, normal form
  • Section 3. Simplex Method
    • 3.1. Principle and example
    • 3.2. Normal form optimization by simplex method
    • 3.3. General case
    • 3.4. Duality, normal form
  • Section 4. Other Optimizations
    • 4.1. Introduction
    • 4.2. Nonlinear programming
    • 4.3. Stochastic programming
    • 4.4. Interior point methods
  • Section 5. Convex sets, concave and convex functions
    • 5.1. Convexity properties in the optimization problems
    • 5.2. Neighborhood, interior point, adherent point
    • 5.3. Convex sets
    • 5.4. Properties of convex sets
    • 5.5. Separation of convex sets
    • 5.6. Fixed-point theorems (Brouwer, Kakutani)
    • 5.7. Concave and convex functions
    • 5.8. Differentiable concave and convex functions
    • 5.9. Convex sets and concave functions
  • Section 6. Marginal Optimality Conditions
    • 6.1. Introduction
    • 6.2. Maximum of one-variable function (definitions)
    • 6.3. Optimum without constraint
    • 6.4. Examples of optimum with constraints
    • 6.5. Generalized Lagrangian theorem
    • 6.6. Kuhn-Tucker necessary conditions
    • 6.7. Application
    • 6.8. Proof of generalized Lagrangian theorem
    • 6.9. Sufficient conditions of optimality
Chap.28. Dynamical Systems
  • Section 1. Dynamical Systems
    • 1.1. Systems of differential equations
    • 1.2. State-space models, and linearization of nonlinear models
    • 1.3. Diffeomorphisms and flows
    • 1.4. Local properties of flows and diffeomorphisms
    • 1.5. Structural stability, hyperbolicity, homoclinic points
    • 1.6. Local bifurcations
  • Section 2. Chaos Theory
    • 2.1. Recalls on dynamical systems
    • 2.2. Invariant sets
    • 2.3. Phase space, flows
    • 2.4. Linear differential equations
    • 2.5. Floquet theory
    • 2.6. Stability theory
    • 2.7. Invariant manifolds
    • 2.8. Poincaré map
    • 2.9. Topological equivalence of differential equations
    • 2.10. Discrete dynamical systems
    • 2.11. Structural stability (robustness)
    • 2.12. Notion of attractor
    • 2.13. Probability measures on attractors
    • 2.14. Entropies
    • 2.15. Lyapunov exponents
    • 2.16. Dimensions
    • 2.17. Chaos and strange attractors
    • 2.18. Definitions of Chaos and attractors
    • 2.19. Chaos in one-dimensional maps
    • 2.20. Introduction to bifurcations
    • 2.21. Bifurcations theory in Morse Smale systems
    • 2.22. Transitions to Chaos
    • 2.23. Illustrations of Ruelle-Takens quasiperiodic route to Chaos, and Landau Tn tori
    • 2.24. Synchronization of oscillators
    • 2.25. The intermittencies
    • 2.26. Saddle connections, blue sky catastrophes
  • Section 3. Example in Physics
    • 3.1. Lorenz model
  • Section 4. Examples in Economics
    • 4.1. Samuelson oscillator
    • 4.2. Models of Solow-Swan, Walras, Tobin, Goodwin
Appendices
Symbols, Tables
  • Mathematical Symbols
  • Transformation Tables
  • Statistical Tables
Bibliography
Index




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