Handbook of Mathematics.
Ebook published on September, 2023
Hardcover published on August, 2023
Softcover published on September, 2023
Format: 15.5x22 cm, 1132 pages, revised
ISBN 978-2-9551990-5-3 : Hardcover
ISBN 978-2-9551990-4-6 : Softcover
ISBN 978-2-9551990-3-9 : EBook
Author: Thierry Vialar
Publisher: HDBOM

TABLE OF CONTENTS
PDF version:

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 ℚ

4.8. Isomorphism of ℚ

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 ℤ_n

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 M_n,m(K)

5.4. Dual vector spaceL(V,K)

5.5. Composition of linear maps

5.6. Rings L(V, V ) and M_n,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 GL_n(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,..,X_p] 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 ℝ^p

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,ℝ^p

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 (Advanced)

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 (ℝ^p,ℜ^p), p≤3

2.4. Polyhedron in (ℝ^p,ℜ^p), p≤3

2.5. Convexity and simplexes in (ℝ^p,ℜ^p)

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 (Advanced)

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, Power 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 ℝ^m

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 ℝ^m

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 D_f

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. L^p[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, L^p-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. General concept

7.3. Convention of summation

7.4. Dual, bidual vector spaces

7.5. Basis, change of basis

7.6. Tensors

7.7. Coordinates of a tensor

7.8. Tensors and injective patches

7.9. Tensor calculus

7.10. Tensor algebra

7.11. Tensor analysis

7.12. Tensor bundle

7.13. Tensor on a vector space

7.14. Tensor density

7.15. Covariant tensor

7.16. Metric tensor

7.17. Riemann tensor

7.18. Minkowski metric

7.19. Ricci tensor

7.20. 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. Power 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. Power series of several variables

16.4. Analytic continuation (singularities)

16.5. Continuity property

16.6. Remark about continuations

Chap.21. Complex Analysis (Advanced)

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. Power 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 h^p

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