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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


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


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


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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