## 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

## Diffusion MRI in Epilepsy

*Rajkumar Munian Govindan and Harry T. Chugani*

### in Neuroimaging in Epilepsy

- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780195342765
- eISBN:
- 9780199863617
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195342765.003.0006
- Subject:
- Neuroscience, Disorders of the Nervous System

Diffusion MRI facilitates the measurement of water diffusion in specific spatial directions and is highly sensitive to microstructural changes in neural tissue. This chapter provides an overview of ... More

## 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

## 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

## GRAVITATIONAL WAVE EXTRACTION

*Miguel Alcubierre*

### in Introduction to 3+1 Numerical Relativity

- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199205677
- eISBN:
- 9780191709371
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199205677.003.0008
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Gravitational waves are one of the most important physical phenomena associated with the presence of strong and dynamic gravitational fields, and as such they are of great interest in numerical ... 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

## MATERIAL DEFORMATION

*Nasr M. Ghoniem and Daniel D. Walgraef*

### in Instabilities and Self-Organization in Materials: Volume I: Fundamentals of Nanoscience, Volume II: Applications in Materials Design and Nanotechnology

- Published in print:
- 2008
- Published Online:
- May 2008
- ISBN:
- 9780199298686
- eISBN:
- 9780191720222
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199298686.003.0002
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter discusses the main elements of continuum theory, which will be utilized further in later sections dealing with a variety of applications. First, a basic review of vector and tensor ... More

## Stress and elastic constants

*Erich H. Kisi and Christopher J. Howard*

### in Applications of Neutron Powder Diffraction

- Published in print:
- 2008
- Published Online:
- January 2009
- ISBN:
- 9780198515944
- eISBN:
- 9780191705663
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198515944.003.0011
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter highlights a different but significant application of neutron diffraction in the measurement of engineering strains. It begins by defining the stress and strain tensors, and their ... More

## SPACE AND TIME PROPERTIES

*R. E. Raab and O. L. de Lange*

### in Multipole Theory in Electromagnetism: Classical, quantum, and symmetry aspects, with applications

- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198567271
- eISBN:
- 9780191717970
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567271.003.0003
- Subject:
- Physics, Atomic, Laser, and Optical Physics

Polar and axial vectors and tensors are defined in terms of their behaviour under coordinate transformations. The effect of the time-reversal operator on kets and matrix elements is considered, ... More

## A Systems Approach to the Aging Brain: Neuroanatomic Changes, Their Modifiers, and Cognitive Correlates

*Naftali Raz and Kristen M. Kennedy*

### in Imaging the Aging Brain

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780195328875
- eISBN:
- 9780199864836
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195328875.003.0004
- Subject:
- Neuroscience, Techniques, Development

Success in diagnosing and treating age-related brain disease depends on understanding normative and optimal aging of the brain. Neuroimaging studies of healthy adults reveal differential brain ... 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

## Classification of Mumford-Tate Subdomains

*Mark Green, Phillip Griffiths, and Matt Kerr*

### in Mumford-Tate Groups and Domains: Their Geometry and Arithmetic (AM-183)

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691154244
- eISBN:
- 9781400842735
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691154244.003.0008
- Subject:
- Mathematics, Analysis

This chapter develops an algorithm for determining all Mumford-Tate subdomains of a given period domain. The result is applied to the classification of all complex multiplication Hodge structures (CM ... 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

## Lorentz Geometry

*Yvonne Choquet-Bruhat*

### in General Relativity and the Einstein Equations

- Published in print:
- 2008
- Published Online:
- May 2009
- ISBN:
- 9780199230723
- eISBN:
- 9780191710872
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199230723.003.0001
- Subject:
- Mathematics, Applied Mathematics

This chapter presents a survey of the basic definitions of Riemannian and Lorentzian differential geometry used in this book. The first nine sections use the simplest formulations, in local ... More

## Tensors in special relativity

*Ta-Pei Cheng*

### in Relativity, Gravitation and Cosmology: A Basic Introduction

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199573639
- eISBN:
- 9780191722448
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199573639.003.0012
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

Tensors in a general coordinate system are introduced. When a tensor is expanded in terms of a set of basis (or inverse basis) vectors, the coefficients of expansion are its contravariant (or ... More

## Tensors in general relativity

*Ta-Pei Cheng*

### in Relativity, Gravitation and Cosmology: A Basic Introduction

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199573639
- eISBN:
- 9780191722448
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199573639.003.0013
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

Differentiation of tensor components in a curved space must be handled with extra care. By adding another term (related to Christoffel symbols) to the ordinary derivative operator, we can form a ... More

## Mobility of Frenkel Excitons

*Vladimir M. Agranovich*

### in Excitations in Organic Solids

- Published in print:
- 2008
- Published Online:
- January 2009
- ISBN:
- 9780199234417
- eISBN:
- 9780191715426
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199234417.003.0014
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This chapter discusses the different regimes of mobility of Frenkel excitons. The diffusion equation and diffusion-related quantities (diffusion coefficient/length, diffusion constant, exciton mean ... More

## Field Quantization and Vacuum Energy in the Presence of Boundaries

*M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko*

### in Advances in the Casimir Effect

- Published in print:
- 2009
- Published Online:
- September 2009
- ISBN:
- 9780199238743
- eISBN:
- 9780191716461
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199238743.003.0003
- Subject:
- Physics, Condensed Matter Physics / Materials, Atomic, Laser, and Optical Physics

From the standpoint of quantum field theory, the Casimir effect is related to the vacuum polarization that arises in quantization volumes restricted by boundaries or in spaces with nontrivial ... More

## Polaritons: Excitonic States Taking Account of Retardation

*Vladimir M. Agranovich*

### in Excitations in Organic Solids

- Published in print:
- 2008
- Published Online:
- January 2009
- ISBN:
- 9780199234417
- eISBN:
- 9780191715426
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199234417.003.0004
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This chapter presents the microscopical theory of polaritons, being coherent superposition of one exciton and one transverse photon state. The dispersion of polaritons including the effect of spatial ... More