Asymptotic Differential Algebra and Model Theory of Transseries 1st Edition by Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven ISBN 9780691175423 069117542X

Asymptotic Differential Algebra and Model Theory of Transseries 1st Edition by Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven – Ebook PDF Instant Download/Delivery: 9780691175423 ,069117542X
Full download Asymptotic Differential Algebra and Model Theory of Transseries 1st Edition after payment

Product details:

ISBN 10: 069117542X
ISBN 13: 9780691175423
Author: Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven

Asymptotic Differential Algebra and Model Theory of Transseries 1st Edition Table of contents:

1 Some Commutative Algebra

1.1 The Zariski Topology and Noetherianity

1.2 Rings and Modules of Finite Length

1.3 Integral Extensions and Integrally Closed Domains

1.4 Local Rings

1.5 Krull’s Principal Ideal Theorem

1.6 Regular Local Rings

1.7 Modules and Derivations

1.8 Differentials

1.9 Derivations on Field Extensions

2 Valued Abelian Groups

2.1 Ordered Sets

2.2 Valued Abelian Groups

2.3 Valued Vector Spaces

2.4 Ordered Abelian Groups

3 Valued Fields

3.1 Valuations on Fields

3.2 Pseudoconvergence in Valued Fields

3.3 Henselian Valued Fields

3.4 Decomposing Valuations

3.5 Valued Ordered Fields

3.6 Some Model Theory of Valued Fields

3.7 The Newton Tree of a Polynomial over a Valued Field

4 Differential Polynomials

4.1 Differential Fields and Differential Polynomials

4.2 Decompositions of Differential Polynomials

4.3 Operations on Differential Polynomials

4.4 Valued Differential Fields and Continuity

4.5 The Gaussian Valuation

4.6 Differential Rings

4.7 Differentially Closed Fields

5 Linear Differential Polynomials

5.1 Linear Differential Operators

5.2 Second-Order Linear Differential Operators

5.3 Diagonalization of Matrices

5.4 Systems of Linear Differential Equations

5.5 Differential Modules

5.6 Linear Differential Operators in the Presence of a Valuation

5.7 Compositional Conjugation

5.8 The Riccati Transform

5.9 Johnson’s Theorem

6 Valued Differential Fields

6.1 Asymptotic Behavior of vP

6.2 Algebraic Extensions

6.3 Residue Extensions

6.4 The Valuation Induced on the Value Group

6.5 Asymptotic Couples

6.6 Dominant Part

6.7 The Equalizer Theorem

6.8 Evaluation at Pseudocauchy Sequences

6.9 Constructing Canonical Immediate Extensions

7 Differential-Henselian Fields

7.1 Preliminaries on Differential-Henselianity

7.2 Maximality and Differential-Henselianity

7.3 Differential-Hensel Configurations

7.4 Maximal Immediate Extensions in the Monotone Case

7.5 The Case of Few Constants

7.6 Differential-Henselianity in Several Variables

8 Differential-Henselian Fields with Many Constants

8.1 Angular Components

8.2 Equivalence over Substructures

8.3 Relative Quantifier Elimination

8.4 A Model Companion

9 Asymptotic Fields and Asymptotic Couples

9.1 Asymptotic Fields and Their Asymptotic Couples

9.2 H-Asymptotic Couples

9.3 Application to Differential Polynomials

9.4 Basic Facts about Asymptotic Fields

9.5 Algebraic Extensions of Asymptotic Fields

9.6 Immediate Extensions of Asymptotic Fields

9.7 Differential Polynomials of Order One

9.8 Extending H-Asymptotic Couples

9.9 Closed H-Asymptotic Couples

10 H-Fields

10.1 Pre-Differential-Valued Fields

10.2 Adjoining Integrals

10.3 The Differential-Valued Hull

10.4 Adjoining Exponential Integrals

10.5 H-Fields and Pre-H-Fields

10.6 Liouville Closed H-Fields

10.7 Miscellaneous Facts about Asymptotic Fields

11 Eventual Quantities, Immediate Extensions, and Special Cuts

11.1 Eventual Behavior

11.2 Newton Degree and Newton Multiplicity

11.3 Using Newton Multiplicity and Newton Weight

11.4 Constructing Immediate Extensions

11.5 Special Cuts in H-Asymptotic Fields

11.6 The Property of λ-Freeness

11.7 Behavior of the Function ω

11.8 Some Special Definable Sets

12 Triangular Automorphisms

12.1 Filtered Modules and Algebras

12.2 Triangular Linear Maps

12.3 The Lie Algebra of an Algebraic Unitriangular Group

12.4 Derivations on the Ring of Column-Finite Matrices

12.5 Iteration Matrices

12.6 Riordan Matrices

12.7 Derivations on Polynomial Rings

12.8 Application to Differential Polynomials

13 The Newton Polynomial

13.1 Revisiting the Dominant Part

13.2 Elementary Properties of the Newton Polynomial

13.3 The Shape of the Newton Polynomial

13.4 Realizing Cuts in the Value Group

13.5 Eventual Equalizers

13.6 Further Consequences of ω-Freeness

13.7 Further Consequences of λ-Freeness

13.8 Asymptotic Equations

13.9 Some Special H-Fields

14 Newtonian Differential Fields

14.1 Relation to Differential-Henselianity

14.2 Cases of Low Complexity

14.3 Solving Quasilinear Equations

14.4 Unravelers

14.5 Newtonization

15 Newtonianity of Directed Unions

15.1 Finitely Many Exceptional Values

15.2 Integration and the Extension K(x)

15.3 Approximating Zeros of Differential Polynomials

15.4 Proof of Newtonianity

16 Quantifier Elimination

16.1 Extensions Controlled by Asymptotic Couples

16.2 Model Completeness

16.3 ΛΩ-Cuts and ΛΩ-Fields

16.4 Embedding Pre-ΛΩ-Fields into ω-Free ΛΩ-Fields

16.5 The Language of ΛΩ-Fields

16.6 Elimination of Quantifiers with Applications

A Transseries

B Basic Model Theory

B.1 Structures and Their Definable Sets

B.2 Languages

B.3 Variables and Terms

B.4 Formulas

B.5 Elementary Equivalence and Elementary Substructures

B.6 Models and the Compactness Theorem

B.7 Ultraproducts and Proof of the Compactness Theorem

B.8 Some Uses of Compactness

B.9 Types and Saturated Structures

B.10 Model Completeness

B.11 Quantifier Elimination

B.12 Application to Algebraically Closed and Real Closed Fields

B.13 Structures without the Independence Property

Bibliography

List of Symbols

Index

People also search for Asymptotic Differential Algebra and Model Theory of Transseries 1st Edition:

difference between linear algebra and differential equations
    
difficult differential equations
    
asymptotic behavior differential equation
    
asymptotically stable differential equations
    
asymptotic algebra

Tags: Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven, Asymptotic Differential, Transseries, Algebra