## The MUSIC algorithm and scattering by an inhomogeneous medium

*Andreas Kirsch and Natalia Grinberg*

### in The Factorization Method for Inverse Problems

- Published in print:
- 2007
- Published Online:
- September 2008
- ISBN:
- 9780199213535
- eISBN:
- 9780191707629
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199213535.003.0004
- Subject:
- Mathematics, Applied Mathematics

This chapter examines the case of a penetrable scatterer with an index of refraction that can be space-dependent and is assumed to be different from the constant background index. The inverse ... More

## The factorization method for Maxwell's equations

*Andreas Kirsch and Natalia Grinberg*

### in The Factorization Method for Inverse Problems

- Published in print:
- 2007
- Published Online:
- September 2008
- ISBN:
- 9780199213535
- eISBN:
- 9780191707629
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199213535.003.0005
- Subject:
- Mathematics, Applied Mathematics

This chapter examines the time-harmonic Maxwell's equations for isotropic and non-magnetic but possibly inhomogeneous media. It begins by formulating the direct scattering problem for arbitrary ... More

## Functional Analysis and Abstract Error Estimates

*Peter Monk*

### in Finite Element Methods for Maxwell's Equations

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198508885
- eISBN:
- 9780191708633
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198508885.003.0002
- Subject:
- Mathematics, Numerical Analysis

This chapter presents the basic functional analysis and abstract error estimates used in the book. A summary of the relevant theory of linear variational problems, compactness, and the Fredholm ... More

## Sobolev Spaces, Vector Function Spaces and Regularity

*Peter Monk*

### in Finite Element Methods for Maxwell's Equations

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198508885
- eISBN:
- 9780191708633
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198508885.003.0003
- Subject:
- Mathematics, Numerical Analysis

This chapter presents the basic functional analysis and abstract error estimates used in the book. A summary of the relevant theory of linear variational problems,compactness, and the Fredholm ... More

## Finite Elements Methods for the Cavity Problem

*Peter Monk*

### in Finite Element Methods for Maxwell's Equations

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198508885
- eISBN:
- 9780191708633
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198508885.003.0007
- Subject:
- Mathematics, Numerical Analysis

This chapter concerns the use of edge finite element methods to approximate the cavity problem with mixed boundary data. It provides two proofs of convergence of the discrete solution. First under ... More

## Introduction to Eigenanalysis

*Gidon Eshel*

### in Spatiotemporal Data Analysis

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691128917
- eISBN:
- 9781400840632
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691128917.003.0004
- Subject:
- Environmental Science, Environmental Studies

Eigenanalysis and its numerous offsprings form the suite of algebraic operations most important and relevant to data analysis, as well as to dynamical systems, modeling, numerical analysis, and ... More

## Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188)

*Christopher D. Sogge*

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691160757
- eISBN:
- 9781400850549
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691160757.001.0001
- Subject:
- Mathematics, Numerical Analysis

Based on lectures given at Zhejiang University in Hangzhou, China, and Johns Hopkins University, this book introduces eigenfunctions on Riemannian manifolds. The book gives a proof of the sharp Weyl ... More

## Operators in quantum mechanics

*Victor Galitski, Boris Karnakov, Vladimir Kogan, and Victor Galitski, Jr.*

### in Exploring Quantum Mechanics: A Collection of 700+ Solved Problems for Students, Lecturers, and Researchers

- Published in print:
- 2013
- Published Online:
- December 2013
- ISBN:
- 9780199232710
- eISBN:
- 9780191774973
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199232710.003.0001
- Subject:
- Physics, Atomic, Laser, and Optical Physics

Mathematical formalism of quantum mechanics is closely connected with the theory of linear operators. A key quantum-mechanical principle is that physical quantities (observables) are represented by ... More

## A Sturm-Liouville Equation: The Time-Independent One-Dimensional Schrödinger Equation

*John A. Adam*

### in Rays, Waves, and Scattering: Topics in Classical Mathematical Physics

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691148373
- eISBN:
- 9781400885404
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691148373.003.0023
- Subject:
- Mathematics, Applied Mathematics

This chapter examines the mathematical properties of the time-independent one-dimensional Schrödinger equation as they relate to Sturm-Liouville problems. The regular Sturm-Liouville theory was ... More

## Invariant subspaces and Jordan form

*Leiba Rodman*

### in Topics in Quaternion Linear Algebra

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691161853
- eISBN:
- 9781400852741
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691161853.003.0005
- Subject:
- Mathematics, Algebra

This chapter starts by introducing the notion of root subspaces for quaternion matrices. These are basic invariant subspaces, and the chapter proves in particular that they enjoy the Lipschitz ... More

## On Ecalle-Hakim 's theorems in holomorphic dynamics

*Marco Arizzi and Jasmin Raissy*

*Araceli Bonifant, Mikhail Lyubich, and Scott Sutherland (eds)*

### in Frontiers in Complex Dynamics: In Celebration of John Milnor's 80th Birthday

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159294
- eISBN:
- 9781400851317
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159294.003.0016
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter provides detailed proofs for the results by M. Hakim regarding the dynamics of germs of biholomorphisms tangent to the identity of order k + 1 ≥ 2 and fixing the origin. One of the main ... More

## The Spectral Structure of TN Matrices

*Shaun M. Fallat and Charles R. Johnson*

### in Totally Nonnegative Matrices

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691121574
- eISBN:
- 9781400839018
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691121574.003.0006
- Subject:
- Mathematics, Applied Mathematics

This chapter highlights the spectral structure of TN matrices. By “spectral structure” this chapter refers to facts about the eigenvalues and eigenvectors of matrices in a particular class. In view ... More

## Quantum Theory

*Frank S. Levin*

### in Surfing the Quantum World

- Published in print:
- 2017
- Published Online:
- November 2017
- ISBN:
- 9780198808275
- eISBN:
- 9780191846014
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198808275.003.0009
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

The subject of Chapter 8 is the fundamental principles of quantum theory, the abstract extension of quantum mechanics. Two of the entities explored are kets and operators, with kets being ... More

## Linear, age-structured models and their long-term dynamics

*Louis W. Botsford, J. Wilson White, and Alan Hastings*

### in Population Dynamics for Conservation

- Published in print:
- 2019
- Published Online:
- November 2019
- ISBN:
- 9780198758365
- eISBN:
- 9780191818301
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198758365.003.0003
- Subject:
- Biology, Biodiversity / Conservation Biology, Biomathematics / Statistics and Data Analysis / Complexity Studies

The chapter describes age-structured models that are linear (i.e. without density dependence). Like simple (non-age-structured) linear models they eventually either increase to infinity or decrease ... More

## Age-structured models: Short-term transient dynamics

*Louis W. Botsford, J. Wilson White, and Alan Hastings*

### in Population Dynamics for Conservation

- Published in print:
- 2019
- Published Online:
- November 2019
- ISBN:
- 9780198758365
- eISBN:
- 9780191818301
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198758365.003.0004
- Subject:
- Biology, Biodiversity / Conservation Biology, Biomathematics / Statistics and Data Analysis / Complexity Studies

Linear age-structured models eventually grow geometrically, and reach a stable age distribution (as in Chapter 3). This chapter describes what happens before “eventually.” That is, it describes the ... More

## Coupled Oscillations—Two Degrees of Freedom

*Mohammad Samiullah*

### in A First Course in Vibrations and Waves

- Published in print:
- 2015
- Published Online:
- October 2015
- ISBN:
- 9780198729785
- eISBN:
- 9780191804113
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198729785.003.0003
- Subject:
- Physics, Condensed Matter Physics / Materials

The motion of systems with two coupled degrees of freedom is studied in this chapter. The ideas of linear systems and normal modes are explored in detail. Both algebraic and matrix methods for ... More

## Precursory Vehicle Modelling

*David J. N. Limebeer and Matteo Massaro*

### in Dynamics and Optimal Control of Road Vehicles

- Published in print:
- 2018
- Published Online:
- April 2019
- ISBN:
- 9780198825715
- eISBN:
- 9780191864636
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198825715.003.0004
- Subject:
- Physics, Condensed Matter Physics / Materials

Chapter 4 considers a number of precursory car and bicycle models, as well as an important oscillatory phenomenon commonly referred to as shimmy. The chapter begins by considering a classical ... More

## A Primer on Empirical Methodology

*G. Andrew Karolyi*

### in Cracking the Emerging Markets Enigma

- Published in print:
- 2015
- Published Online:
- June 2015
- ISBN:
- 9780199336623
- eISBN:
- 9780190232047
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199336623.003.0003
- Subject:
- Economics and Finance, International, Development, Growth, and Environmental

This chapter focuses on the empirical methodology—principal components analysis (PCA)—used to build each of the six risk indicators across the 33 emerging and 24 developed markets featured in the ... More

## The physical properties of crystals and their description by tensors

*Christopher Hammond*

### in The Basics of Crystallography and Diffraction

- Published in print:
- 2015
- Published Online:
- August 2015
- ISBN:
- 9780198738671
- eISBN:
- 9780191801938
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198738671.003.0014
- Subject:
- Physics, Crystallography: Physics, Condensed Matter Physics / Materials

The chapter starts with a general introduction of the topics covered here. If provides an ‘everyday’ example or analogy of a tensor property and a description of suffix notation. The chapter then ... More

## Single-species dynamics

*Tim Coulson and H. Charles J. Godfray*

### in Theoretical Ecology: Principles and Applications

- Published in print:
- 2007
- Published Online:
- November 2020
- ISBN:
- 9780199209989
- eISBN:
- 9780191917370
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780199209989.003.0006
- Subject:
- Environmental Science, Applied Ecology

What determines the densities of the different species of plants, animals, and micro-organisms with which we share the planet, why do their numbers fluctuate and ... More