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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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