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## Differential Polynomials

*Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven*

### in Asymptotic Differential Algebra and Model Theory of Transseries

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780691175423
- eISBN:
- 9781400885411
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175423.003.0005
- Subject:
- Mathematics, Computational Mathematics / Optimization

This chapter deals with differential polynomials. It first presents some basic facts about differential fields that are of characteristic zero with one distinguished derivation, along with their ... More

## Linear Differential Polynomials

*Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven*

### in Asymptotic Differential Algebra and Model Theory of Transseries

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780691175423
- eISBN:
- 9781400885411
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175423.003.0006
- Subject:
- Mathematics, Computational Mathematics / Optimization

This chapter introduces the reader to linear differential polynomials. It first considers homogeneous differential polynomials and the corresponding linear operators before proving various basic ... More

## Introduction and Overview

*Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven*

### in Asymptotic Differential Algebra and Model Theory of Transseries

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780691175423
- eISBN:
- 9781400885411
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175423.003.0001
- Subject:
- Mathematics, Computational Mathematics / Optimization

This book develops the algebra and model theory of the differential field of transseries, which are formal series in an indeterminate x > ℝ. is a field containing ℝ as a subfield and acquires the ... More

## Asymptotic Differential Algebra and Model Theory of Transseries

*Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven*

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780691175423
- eISBN:
- 9781400885411
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175423.001.0001
- Subject:
- Mathematics, Computational Mathematics / Optimization

Asymptotic differential algebra seeks to understand the solutions of differential equations and their asymptotics from an algebraic point of view. The differential field of transseries plays a ... More

##
*H*-Fields

*Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven*

### in Asymptotic Differential Algebra and Model Theory of Transseries

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780691175423
- eISBN:
- 9781400885411
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175423.003.0011
- Subject:
- Mathematics, Computational Mathematics / Optimization

This chapter considers H-fields, pre-differential-valued fields with a field ordering that interacts with the valuation and derivation. Axiomatizing this interaction yields the notion of a ... More

## Differential-Henselian Fields with Many Constants

*Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven*

### in Asymptotic Differential Algebra and Model Theory of Transseries

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780691175423
- eISBN:
- 9781400885411
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175423.003.0009
- Subject:
- Mathematics, Computational Mathematics / Optimization

This chapter considers differential-henselian fields with many constants. Here d-henselian includes having small derivation, so d-henselian valued differential fields with many constants are ... More

## Valued Differential Fields

*Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven*

### in Asymptotic Differential Algebra and Model Theory of Transseries

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780691175423
- eISBN:
- 9781400885411
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175423.003.0007
- Subject:
- Mathematics, Computational Mathematics / Optimization

This chapter deals with valued differential fields, starting the discussion with an overview of the asymptotic behavior of the function vsubscript P: Γ → Γ for homogeneous P ∈ K K{Y}superscript Not ... More

## Differential-Henselian Fields

*Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven*

### in Asymptotic Differential Algebra and Model Theory of Transseries

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780691175423
- eISBN:
- 9781400885411
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175423.003.0008
- Subject:
- Mathematics, Computational Mathematics / Optimization

This chapter discusses differential-henselian fields. Here K is a valued differential field with small derivation. An extension of K means a valued differential field extension of K whose derivation ... More

## The Newton Polynomial

*Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven*

### in Asymptotic Differential Algebra and Model Theory of Transseries

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780691175423
- eISBN:
- 9781400885411
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175423.003.0014
- Subject:
- Mathematics, Computational Mathematics / Optimization

This chapter focuses on the Newton polynomial based on assumption that K is a differential-valued field of H-type with asymptotic integration and small derivation. Here K is also assumed to be ... More

## Newtonian Differential Fields

*Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven*

### in Asymptotic Differential Algebra and Model Theory of Transseries

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780691175423
- eISBN:
- 9781400885411
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175423.003.0015
- Subject:
- Mathematics, Computational Mathematics / Optimization

This chapter deals with Newtonian differential fields. Here K is an ungrounded H-asymptotic field with Γ := v(Ksuperscript x ) not equal to {0}. So the subset ψ of Γ is nonempty and has no largest ... More

## Newtonianity of Directed Unions

*Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven*

### in Asymptotic Differential Algebra and Model Theory of Transseries

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780691175423
- eISBN:
- 9781400885411
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175423.003.0016
- Subject:
- Mathematics, Computational Mathematics / Optimization

This chapter considers the newtonianity of directed unions and proves an analogue of Hensel's Lemma for ω-free differential-valued fields of H-type: Theorem 15.0.1. Here K is an H-asymptotic field ... More

## Valued Abelian Groups

*Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven*

### in Asymptotic Differential Algebra and Model Theory of Transseries

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780691175423
- eISBN:
- 9781400885411
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175423.003.0003
- Subject:
- Mathematics, Computational Mathematics / Optimization

This chapter deals with valued abelian groups. It first introduces some terminology concerning ordered sets before discussing valued abelian groups and ordered abelian groups in more detail. Ordered ... More

## Field Equation of Nonlocal Gravity

*Bahram Mashhoon*

### in Nonlocal Gravity

- Published in print:
- 2017
- Published Online:
- July 2017
- ISBN:
- 9780198803805
- eISBN:
- 9780191842313
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198803805.003.0006
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

In extended general relativity (GR), Einstein’s field equation of GR can be expressed in terms of torsion and this leads to the teleparallel equivalent of GR, namely, GR||, which turns out to be the ... More

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