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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


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