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## Operator spaces

*David P. Blecher and Christian Le Merdy*

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

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

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526599.003.0001
- Subject:
- Mathematics, Pure Mathematics

This chapter presents some background results about operator spaces and establishes some notations which will be used throughout this book. It can serve as a mini-course on the basics of operator ... More

## Basic theory of operator modules

*David P. Blecher and Christian Le Merdy*

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

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

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526599.003.0003
- Subject:
- Mathematics, Pure Mathematics

This chapter discusses the basics of operator modules. Topics covered include Hilbert modules, operator modules over operator algebras, two module tensor products, module maps, module map extension ... More

## MULTI-DIMENSIONAL EXPANSION BASES

*George Em Karniadakis and Spencer J. Sherwin*

### in Spectral/hp Element Methods for Computational Fluid Dynamics

- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198528692
- eISBN:
- 9780191713491
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528692.003.0003
- Subject:
- Mathematics, Numerical Analysis

This chapter considers the extension of the one-dimensional formulation to two and three dimensions by the development of expansion bases in standard regions such as triangles or rectangles in two ... More

## Derived Categories of Coherent Sheaves

*D. Huybrechts*

### in Fourier-Mukai Transforms in Algebraic Geometry

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780199296866
- eISBN:
- 9780191711329
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199296866.003.0003
- Subject:
- Mathematics, Geometry / Topology

The discussion of the previous chapter is applied to the derived category of the abelian category of coherent sheaves. The Serre functor is introduced, and particular spanning classes are ... More

## Wavelets on General Domains

*Karsten Urban*

### in Wavelet Methods for Elliptic Partial Differential Equations

- Published in print:
- 2008
- Published Online:
- May 2009
- ISBN:
- 9780198526056
- eISBN:
- 9780191712340
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526056.003.0008
- Subject:
- Mathematics, Applied Mathematics, Mathematical Finance

The construction of wavelets on general domains is performed in three steps. This chapter starts by introducing the construction of scaling functions and wavelets on bounded univariate intervals. ... More

## Selfadjointness criteria

*David P. Blecher and Christian Le Merdy*

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

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

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526599.003.0007
- Subject:
- Mathematics, Pure Mathematics

This chapter discusses several interesting criteria which force an operator algebra to be selfadjoint. Such results are in some sense ‘negative’ in nature, showing that certain results or themes ... More

## Entangled states

*J. C. Garrison and R. Y. Chiao*

### in Quantum Optics

- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780198508861
- eISBN:
- 9780191708640
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198508861.003.0007
- Subject:
- Physics, Atomic, Laser, and Optical Physics

A state of two distinguishable particles is “separable” if the wavefunction is a product of single-particle wavefunctions. Schrödinger introduced the term “entangled” to describe a two-particle state ... More

## Basic Tools For Quantum Mechanics

*Robert Alicki and Mark Fannes*

### in Quantum Dynamical Systems

- Published in print:
- 2001
- Published Online:
- February 2010
- ISBN:
- 9780198504009
- eISBN:
- 9780191708503
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198504009.003.0002
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter reminds, without entering into details, the main mathematical concepts and results relevant for finite system quantum mechanics. The basic postulates single out a Hilbert space of wave ... More

## Composite Systems, Interaction, and Measurement

*Bas C. van Fraassen*

### in Quantum Mechanics: An Empiricist View

- Published in print:
- 1991
- Published Online:
- November 2003
- ISBN:
- 9780198239802
- eISBN:
- 9780191597466
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198239807.003.0007
- Subject:
- Philosophy, Philosophy of Science

The three main issues in the foundations of quantum theory are measurement, non‐locality, and identical particles (quantum statistics). All of these pertain to the relations between different parts ... More

## Preliminaries to Main Results

*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/9780691153551.003.0009
- Subject:
- Mathematics, Analysis

This chapter presents a number of results and notions that will be used in subsequent chapters. In particular, it considers the concept of regular differentiability and the lemma on deformation of ... More

## Further 2-Dimensional Categorical Structures

*Niles Johnson and Donald Yau*

### in 2-Dimensional Categories

- Published in print:
- 2021
- Published Online:
- February 2021
- ISBN:
- 9780198871378
- eISBN:
- 9780191914850
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198871378.003.0012
- Subject:
- Mathematics, Geometry / Topology

In this chapter, further 2-dimensional categorical structures are presented and discussed. These include monoidal bicategories, as one-object tricategories, along with braided monoidal bicategories, ... More

## Preliminaries

*Jayme Vaz and Roldão da Rocha*

### in An Introduction to Clifford Algebras and Spinors

- Published in print:
- 2016
- Published Online:
- August 2016
- ISBN:
- 9780198782926
- eISBN:
- 9780191826146
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198782926.003.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

In this chapter, a review of the main concepts, examples, and results from linear algebra are presented, as these are essential for developing the subsequent ideas in this book. As the concept of ... More

## Monoidal Categories

*Chris Heunen and Jamie Vicary*

### in Categories for Quantum Theory: An Introduction

- Published in print:
- 2019
- Published Online:
- January 2020
- ISBN:
- 9780198739623
- eISBN:
- 9780191802584
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198739623.003.0001
- Subject:
- Mathematics, Mathematical Physics, Applied Mathematics

A monoidal category is a category equipped with extra data, describing how objects and morphisms can be combined in parallel. This chapter introduces the theory of monoidal categories, including ... More

## Can an ICS Architecture Meet the Systematicity and Productivity Challenges?

*Brian P. McLaughlin*

### in The Architecture of Cognition: Rethinking Fodor and Pylyshyn's Systematicity Challenge

- Published in print:
- 2014
- Published Online:
- September 2014
- ISBN:
- 9780262027236
- eISBN:
- 9780262322461
- Item type:
- chapter

- Publisher:
- The MIT Press
- DOI:
- 10.7551/mitpress/9780262027236.003.0002
- Subject:
- Philosophy, Philosophy of Mind

After spelling out the systematicity challenge to connectionism, I examine Paul Smolensky and Géraldine Legendre's attempt to meet the challenge by appeal to a kind of architecture that they label ... More

## Algebra of vector bundles

*Clifford Henry Taubes*

### in Differential Geometry: Bundles, Connections, Metrics and Curvature

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780199605880
- eISBN:
- 9780191774911
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199605880.003.0004
- Subject:
- Mathematics, Geometry / Topology, Mathematical Physics

Any linear operation that can be performed to generate a new vector space from a given set of initial vector spaces can be done fibre-wise with an analogous set of vector bundles to generate a new ... More

## Cohomological Group Theory

*Graham Ellis*

### in An Invitation to Computational Homotopy

- Published in print:
- 2019
- Published Online:
- November 2019
- ISBN:
- 9780198832973
- eISBN:
- 9780191871375
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198832973.003.0004
- Subject:
- Mathematics, Computational Mathematics / Optimization, Geometry / Topology

This chapter introduces some of the basic ingredients of cohomological group theory and describes datatypes and algorithms for implementing them on a computer. These are illustrated using computer ... More

## Monoidal 2-Categories

*Chris Heunen and Jamie Vicary*

### in Categories for Quantum Theory: An Introduction

- Published in print:
- 2019
- Published Online:
- January 2020
- ISBN:
- 9780198739623
- eISBN:
- 9780191802584
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198739623.003.0008
- Subject:
- Mathematics, Mathematical Physics, Applied Mathematics

Monoidal 2-categories are higher-dimensional versions of monoidal categories, allowing a more expressive syntax that plays an important role in modern mathematics. We explore their two-dimensional ... More

## Basics

*Chris Heunen and Jamie Vicary*

### in Categories for Quantum Theory: An Introduction

- Published in print:
- 2019
- Published Online:
- January 2020
- ISBN:
- 9780198739623
- eISBN:
- 9780191802584
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198739623.003.0009
- Subject:
- Mathematics, Mathematical Physics, Applied Mathematics

This book assumes familiarity with some basic ideas from category theory, linear algebra and quantum computing. This self-contained chapter gives a quick summary of the essential aspects of these ... More

## Disentangling via Tensor Products and Ordered Supports

*Gerald W. Johnson, Michel L. Lapidus, and Lance Nielsen*

### in Feynman's Operational Calculus and Beyond: Noncommutativity and Time-Ordering

- Published in print:
- 2015
- Published Online:
- September 2015
- ISBN:
- 9780198702498
- eISBN:
- 9780191772160
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198702498.003.0003
- Subject:
- Mathematics, Mathematical Physics, Numerical Analysis

More involved types of disentanglings are considered in this chapter. We discuss the situation where the Banach space X can be written as a direct sum X=Y⊕Z, for some Banach spaces Y and Z, and each ... More

## Categories for Quantum Theory: An Introduction

*Chris Heunen and Jamie Vicary*

- Published in print:
- 2019
- Published Online:
- January 2020
- ISBN:
- 9780198739623
- eISBN:
- 9780191802584
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198739623.001.0001
- Subject:
- Mathematics, Mathematical Physics, Applied Mathematics

Monoidal category theory serves as a powerful framework for describing logical aspects of quantum theory, giving an abstract language for parallel and sequential composition and a conceptual way to ... More

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