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## 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

## The Calder´on-Zygmund Theory I: Ellipticity

*Brian Street*

### in Multi-parameter Singular Integrals. (AM-189)

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691162515
- eISBN:
- 9781400852758
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691162515.003.0001
- Subject:
- Mathematics, Analysis

This chapter discusses a case for single-parameter singular integral operators, where ρ is the usual distance on ℝn. There, we obtain the most classical theory of singular integrals, which is useful ... More

## Spectral Theory and Differential Operators

*David Edmunds and Des Evans*

- Published in print:
- 2018
- Published Online:
- September 2018
- ISBN:
- 9780198812050
- eISBN:
- 9780191861130
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198812050.001.0001
- Subject:
- Mathematics, Pure Mathematics

This book gives an account of those parts of the analysis of closed linear operators acting in Banach or Hilbert spaces that are relevant to spectral problems involving differential operators, and ... More

## Discrete Regularization

*Zhou Dengyong and Schülkopf Bernhard*

### in Semi-Supervised Learning

- Published in print:
- 2006
- Published Online:
- August 2013
- ISBN:
- 9780262033589
- eISBN:
- 9780262255899
- Item type:
- chapter

- Publisher:
- The MIT Press
- DOI:
- 10.7551/mitpress/9780262033589.003.0013
- Subject:
- Computer Science, Machine Learning

This chapter presents a systemic framework for learning from a finite set represented as a graph. Discrete analogues are developed here of a number of differential operators, and then a discrete ... More

## On Div-Curl for Higher Order

*Loredana Lanzani and Andrew S. Raich*

*Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, and Stephen Wainger (eds)*

### in Advances in Analysis: The Legacy of Elias M. Stein

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159416
- eISBN:
- 9781400848935
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159416.003.0011
- Subject:
- Mathematics, Numerical Analysis

This chapter produces a new class of differential operators of order k (where k is any given positive integer) that satisfy an appropriate analogue of a Gagliardo–Nirenberg inequality for functions ... More

## Fields

*Gerhard Oertel*

### in Stress and Deformation: A Handbook on Tensors in Geology

- Published in print:
- 1996
- Published Online:
- November 2020
- ISBN:
- 9780195095036
- eISBN:
- 9780197560792
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195095036.003.0006
- Subject:
- Earth Sciences and Geography, Geology and the Lithosphere

A scalar determined at every point in a given domain, analytically or otherwise, constitutes a scalar field. Vectors similarly determined constitute a vector field. The defining analytical ... More

## Eventual Quantities, Immediate Extensions, and Special Cuts

*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/9780691175423.003.0012
- Subject:
- Mathematics, Computational Mathematics / Optimization

This chapter deals with eventual quantities, immediate extensions, and special cuts. It first considers the behavior of eventual quantities before discussing Newton weight, Newton degree, and Newton ... More

## Metrics Spaces for the Matrix Groups

*Ulf Grenander and Michael I. Miller*

### in Pattern Theory: From representation to inference

- Published in print:
- 2006
- Published Online:
- November 2020
- ISBN:
- 9780198505709
- eISBN:
- 9780191916564
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198505709.003.0011
- Subject:
- Computer Science, Programming Languages

In this chapter the metric space structure of shape is developed. We do this by first studying the action of the matrix groups on the coordinate systems of shape. We ... More

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