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## Operator Algebras and Their Modules: An operator space approach

*David P. Blecher and Christian Le Merdy*

- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198526599
- eISBN:
- 9780191712159
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526599.001.0001
- Subject:
- Mathematics, Pure Mathematics

This book presents the general theory of algebras of operators on a Hilbert space, and the modules over such algebras. The new theory of operator spaces is presented early on and the text assembles ... More

## BROWNIAN ROUGH PATHS

*Terry Lyons and Zhongmin Qian*

### in System Control and Rough Paths

- Published in print:
- 2002
- Published Online:
- September 2007
- ISBN:
- 9780198506485
- eISBN:
- 9780191709395
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198506485.003.0004
- Subject:
- Mathematics, Probability / Statistics

This chapter proves that the dyadic polygonal approximations to a Brownian paths, with probability 1, a Cauchy sequence of rough paths, and in this way, the chapter proves a result first established ... More

## Smoothness, Convexity, Porosity, and Separable Determination

*Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav*

### in Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.003.0003
- Subject:
- Mathematics, Analysis

This chapter shows how spaces with separable dual admit a Fréchet smooth norm. It first considers a criterion of the differentiability of continuous convex functions on Banach spaces before ... More

## Gâteaux differentiability of Lipschitz functions

*Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav*

### in Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.003.0002
- Subject:
- Mathematics, Analysis

This chapter presents the main results on Gâteaux differentiability of Lipschitz functions by recalling the notions of the Radon-Nikodým property (RNP) and null sets. The discussion focuses not only ... More

##
*Ε*-Fr ´Echet Differentiability

*Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav*

### in Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.003.0004
- Subject:
- Mathematics, Analysis

This chapter treats results on ε-Fréchet differentiability of Lipschitz functions in asymptotically smooth spaces. These results are highly exceptional in the sense that they prove almost Frechet ... More

## THE PME AS AN ABSTRACT EVOLUTION EQUATION. SEMIGROUP APPROACH

*Juan Luis Vázquez*

### in The Porous Medium Equation: Mathematical Theory

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198569039
- eISBN:
- 9780191717468
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198569039.003.0010
- Subject:
- Mathematics, Mathematical Physics

This chapter addresses the problem of the construction of solutions of the GPME by viewing it as an abstract evolution equation, or more precisely as an ordinary differential equation with values in ... More

## From linear to nonlinear equations, fundamental results

*Klaus Böhmer*

### in Numerical Methods for Nonlinear Elliptic Differential Equations: A Synopsis

- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780199577040
- eISBN:
- 9780191595172
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199577040.003.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Chapter 1 demonstrates, for the simple mechanical example of a bent rod, the change in character from linear to nonlinear regimes. This is followed by several examples for different types of ... More

## Unavoidable Porous Sets and Nondifferentiable Maps

*Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav*

### in Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.003.0014
- Subject:
- Mathematics, Analysis

This chapter discusses Γₙ-nullness of sets porous “¹at infinity” and/or existence of many points of Fréchet differentiability of Lipschitz maps into n-dimensional spaces. The results reveal a ... More

## Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

*Joram Lindenstrauss, David Preiss, and Jaroslav Tier*

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.001.0001
- Subject:
- Mathematics, Analysis

This book makes a significant inroad into the unexpectedly difficult question of existence of Fréchet derivatives of Lipschitz maps of Banach spaces into higher dimensional spaces. Because the ... More

##
Γ-Null and Γ*N*-Null Sets

*Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav*

### in Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.003.0005
- Subject:
- Mathematics, Analysis

This chapter introduces the notions of Γ-null and Γₙ-null sets, which are σ-ideals of subsets of a Banach space X. Γ-null set is key for the strongest known general Fréchet differentiability ... More

## Fr ´Echet Differentiability Except For Γ-Null Sets

*Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav*

### in Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.003.0006
- Subject:
- Mathematics, Analysis

This chapter gives an account of the known genuinely infinite dimensional results proving Fréchet differentiability almost everywhere except for Γ-null sets. Γ-null sets provide the only notion of ... More

## Fr ´Echet Differentiability of Vector-Valued Functions

*Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav*

### in Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.003.0013
- Subject:
- Mathematics, Analysis

This chapter shows that if a Banach space with a Fréchet smooth norm is asymptotically smooth with modulus o(tⁿ logⁿ⁻¹(1/t)) then every Lipschitz map of X to a space of dimension not exceeding n has ... More

## Examples of diamonds

*Peter Scholze and Jared Weinstein*

### in Berkeley Lectures on p-adic Geometry: (AMS-207)

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.003.0015
- Subject:
- Mathematics, Geometry / Topology

This chapter assesses some interesting examples of diamonds. So far, the only example encountered is the self-product of copies of Spd Qp. The chapter first studies this self-product. It is useful to ... More

## Elements of Geometry of Balls in Banach Spaces

*Kazimierz Goebel and Stanislaw Prus*

- Published in print:
- 2018
- Published Online:
- April 2019
- ISBN:
- 9780198827351
- eISBN:
- 9780191866265
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198827351.001.0001
- Subject:
- Mathematics, Pure Mathematics

One of the subjects of functional analysis is classification of Banach spaces depending on various properties of the unit ball. The need of such considerations comes from a number of applications to ... 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

## Lp Spaces

*Fon-Che Liu*

### in Real Analysis

- Published in print:
- 2016
- Published Online:
- January 2017
- ISBN:
- 9780198790426
- eISBN:
- 9780191831676
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198790426.003.0006
- Subject:
- Mathematics, Analysis

LP spaces are the most interesting examples of Banach spaces and play a salient role in modern analysis. In this chapter, basic features of LP spaces are studied; in particular, their dual spaces are ... More

## Computational Anatomy: Hypothesis Testing on Disease

*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/9780198505709.003.0018
- Subject:
- Computer Science, Programming Languages

Computational Anatomy (CA) is the study of human anatomy I ∈ I = Iα ◦G, an orbit under groups of diffeomorphisms of anatomical exemplars Iα ∈ I. The observable images ... More

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