Stefano Predelli
- Published in print:
- 2005
- Published Online:
- February 2006
- ISBN:
- 9780199281732
- eISBN:
- 9780191603204
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199281734.001.0001
- Subject:
- Philosophy, Philosophy of Language
This book discusses the views of meaning and truth implicit in the traditionalapproach to natural-language semantics. Its main thesis is that the philosophical presuppositions entailed by that ...
More
This book discusses the views of meaning and truth implicit in the traditionalapproach to natural-language semantics. Its main thesis is that the philosophical presuppositions entailed by that approach have often been misunderstood even by its foremost defenders, and that such a misunderstanding has fuelled an unjustified but increasingly popular sceptical attitude. This conclusion is defended by focusing on the distinction between the essential philosophical traits of traditional natural-language semantics, and certain accidental features allegedly characteristic of some of its typical applications to the study of particular utterances. As a result, the book puts forth claims about the logical profile of indexical expressions, about the semantic behaviour of attitude reports, and about phenomena such as discourse about fiction, approximation, and contextual dependence. In its central chapters, it challenges the adequacy of a token-reflexive approach to indexicality, and questions the urgency of the so-called contextualist view of communication.Less
This book discusses the views of meaning and truth implicit in the traditionalapproach to natural-language semantics. Its main thesis is that the philosophical presuppositions entailed by that approach have often been misunderstood even by its foremost defenders, and that such a misunderstanding has fuelled an unjustified but increasingly popular sceptical attitude. This conclusion is defended by focusing on the distinction between the essential philosophical traits of traditional natural-language semantics, and certain accidental features allegedly characteristic of some of its typical applications to the study of particular utterances. As a result, the book puts forth claims about the logical profile of indexical expressions, about the semantic behaviour of attitude reports, and about phenomena such as discourse about fiction, approximation, and contextual dependence. In its central chapters, it challenges the adequacy of a token-reflexive approach to indexicality, and questions the urgency of the so-called contextualist view of communication.
Mathew Penrose
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198506263
- eISBN:
- 9780191707858
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198506263.003.0002
- Subject:
- Mathematics, Probability / Statistics
This chapter derives some probabilistic results for later use. A dependency graph for a family of random variables is a deterministic graph with vertex-set identified with the random variables and ...
More
This chapter derives some probabilistic results for later use. A dependency graph for a family of random variables is a deterministic graph with vertex-set identified with the random variables and edges corresponding to dependencies between variables. Univariate and multivariate Poisson approximation theorems, and normal approximation theorems, are derived for sums of variables having a dependency graph using Stein's method. Random variables given as sums of martingale differences are also considered. For such variables, both concentration inequalities and a central limit theorem are given. Finally, a ‘de-Poissonization’ result is given which is to be used for extending central limit theorems for G(N(n),r) (with N(n) Poisson with mean n) to analogous results for G(n,r).Less
This chapter derives some probabilistic results for later use. A dependency graph for a family of random variables is a deterministic graph with vertex-set identified with the random variables and edges corresponding to dependencies between variables. Univariate and multivariate Poisson approximation theorems, and normal approximation theorems, are derived for sums of variables having a dependency graph using Stein's method. Random variables given as sums of martingale differences are also considered. For such variables, both concentration inequalities and a central limit theorem are given. Finally, a ‘de-Poissonization’ result is given which is to be used for extending central limit theorems for G(N(n),r) (with N(n) Poisson with mean n) to analogous results for G(n,r).
Andrea Braides
- Published in print:
- 2002
- Published Online:
- September 2007
- ISBN:
- 9780198507840
- eISBN:
- 9780191709890
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198507840.003.0009
- Subject:
- Mathematics, Applied Mathematics
Various types of approximations of free-discontinuity energies, in particular the Mumford-Shah functional, are given: the Ambrosio-Tortorelli approximation via elliptic functionals with an additional ...
More
Various types of approximations of free-discontinuity energies, in particular the Mumford-Shah functional, are given: the Ambrosio-Tortorelli approximation via elliptic functionals with an additional variable (interpreted as a damage parameter in a mechanical setting); convolution functionals (interpreted as non-local damage energies), in particular the Braides-Dal Maso approximation; and non-convex finite-difference schemes.Less
Various types of approximations of free-discontinuity energies, in particular the Mumford-Shah functional, are given: the Ambrosio-Tortorelli approximation via elliptic functionals with an additional variable (interpreted as a damage parameter in a mechanical setting); convolution functionals (interpreted as non-local damage energies), in particular the Braides-Dal Maso approximation; and non-convex finite-difference schemes.
Jean Zinn-Justin
- Published in print:
- 2007
- Published Online:
- January 2010
- ISBN:
- 9780199227198
- eISBN:
- 9780191711107
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199227198.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This book provides an elementary introduction to the notions of continuum limit and universality in statistical systems with a large number of degrees of freedom. The existence of a continuum limit ...
More
This book provides an elementary introduction to the notions of continuum limit and universality in statistical systems with a large number of degrees of freedom. The existence of a continuum limit requires the appearance of correlations at large distance, a situation that is encountered in second order phase transitions, near the critical temperature. In this context, the book emphasizes the role of gaussian distributions and their relations with the mean field approximation and Landau′s theory of critical phenomena. The book shows that quasi-gaussian or mean-field approximations cannot describe correctly phase transitions in three space dimensions. The book assigns this difficulty to the coupling of very different physical length scales, even though the systems we will consider have only local, that is, short range, interactions. To analyze the unusual situation, a new concept is required: the renormalization group, whose fixed points allow understanding the universality of physical properties at large distance, beyond mean-field theory. In the continuum limit, critical phenomena can be described by quantum field theories. In this framework, the renormalization group is directly related to the renormalization process; that is, the necessity to cancel the infinities that arise in straightforward formulations of the theory. The book discusses the renormalization group in the context of various relevant field theories. This leads to proofs of universality and to efficient tools for calculating universal quantities in a perturbative framework. Finally, the book constructs a general functional renormalization group, which can be used when perturbative methods are inadequate.Less
This book provides an elementary introduction to the notions of continuum limit and universality in statistical systems with a large number of degrees of freedom. The existence of a continuum limit requires the appearance of correlations at large distance, a situation that is encountered in second order phase transitions, near the critical temperature. In this context, the book emphasizes the role of gaussian distributions and their relations with the mean field approximation and Landau′s theory of critical phenomena. The book shows that quasi-gaussian or mean-field approximations cannot describe correctly phase transitions in three space dimensions. The book assigns this difficulty to the coupling of very different physical length scales, even though the systems we will consider have only local, that is, short range, interactions. To analyze the unusual situation, a new concept is required: the renormalization group, whose fixed points allow understanding the universality of physical properties at large distance, beyond mean-field theory. In the continuum limit, critical phenomena can be described by quantum field theories. In this framework, the renormalization group is directly related to the renormalization process; that is, the necessity to cancel the infinities that arise in straightforward formulations of the theory. The book discusses the renormalization group in the context of various relevant field theories. This leads to proofs of universality and to efficient tools for calculating universal quantities in a perturbative framework. Finally, the book constructs a general functional renormalization group, which can be used when perturbative methods are inadequate.
Nikolai Kopnin
- Published in print:
- 2001
- Published Online:
- January 2010
- ISBN:
- 9780198507888
- eISBN:
- 9780191709722
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198507888.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This book presents modern theory of nonstationary and nonequilibrium superconductivity. It deals with superconductors in external fields varying in time and studies transport phenomena in ...
More
This book presents modern theory of nonstationary and nonequilibrium superconductivity. It deals with superconductors in external fields varying in time and studies transport phenomena in superconductors. The book provides the microscopic theory based on the Green function formalism within the Bardeen, Cooper, and Schrieffer (BCS) theory. The method of quasiclassical Green functions is formulated for both stationary and nonequilibrium problems in the theory of superconductivity. Chapters 1 to 4 give an introduction to the Green function formalism in the BCS theory for clean materials and alloys. In next two chapters, the quasiclassical approximation is introduced and applied to some generic stationary problems such as the Ginzburg–Landau (GL) equations, critical magnetic fields, gapless superconductivity, d-wave superconductivity, bound states in the vortex core. Chapter 7 describes the quasiclassical method for layered superconductors. In Chapter 8 the nonstationary theory is formulated using both the method of analytical continuation and the Keldysh diagram technique. Next two chapters are devoted to the quasiclassical approximation and to generalized kinetic equations in nonstationary situations. Chapter 11 demonstrates how the GL model can be extended to nonstationary problems. A considerable part of the book is devoted to the vortex dynamics, which treats behaviour of type II superconductors when they carry electric currents in presence of a magnetic field. Chapters 12 to 15 deal with the dynamics of vortices. In Chapter 12, the time-dependent GL model is used to calculate the resistivity in the flux flow regime. Chapter 13 derives the forces acting on a moving vortex using the Green function formalism and applies the microscopic theory to the vortex dynamics in superconducting alloys. In Chapters 14 and 15 the vortex dynamics in clean superconductors is considered and the flux-flow conductivity, the vortex Hall effect, and the vortex mass are calculated.Less
This book presents modern theory of nonstationary and nonequilibrium superconductivity. It deals with superconductors in external fields varying in time and studies transport phenomena in superconductors. The book provides the microscopic theory based on the Green function formalism within the Bardeen, Cooper, and Schrieffer (BCS) theory. The method of quasiclassical Green functions is formulated for both stationary and nonequilibrium problems in the theory of superconductivity. Chapters 1 to 4 give an introduction to the Green function formalism in the BCS theory for clean materials and alloys. In next two chapters, the quasiclassical approximation is introduced and applied to some generic stationary problems such as the Ginzburg–Landau (GL) equations, critical magnetic fields, gapless superconductivity, d-wave superconductivity, bound states in the vortex core. Chapter 7 describes the quasiclassical method for layered superconductors. In Chapter 8 the nonstationary theory is formulated using both the method of analytical continuation and the Keldysh diagram technique. Next two chapters are devoted to the quasiclassical approximation and to generalized kinetic equations in nonstationary situations. Chapter 11 demonstrates how the GL model can be extended to nonstationary problems. A considerable part of the book is devoted to the vortex dynamics, which treats behaviour of type II superconductors when they carry electric currents in presence of a magnetic field. Chapters 12 to 15 deal with the dynamics of vortices. In Chapter 12, the time-dependent GL model is used to calculate the resistivity in the flux flow regime. Chapter 13 derives the forces acting on a moving vortex using the Green function formalism and applies the microscopic theory to the vortex dynamics in superconducting alloys. In Chapters 14 and 15 the vortex dynamics in clean superconductors is considered and the flux-flow conductivity, the vortex Hall effect, and the vortex mass are calculated.
Raymond L. Chambers and Robert G. Clark
- Published in print:
- 2012
- Published Online:
- May 2012
- ISBN:
- 9780198566625
- eISBN:
- 9780191738449
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566625.003.0011
- Subject:
- Mathematics, Probability / Statistics
Inference for non-linear population parameters develops model-based prediction theory for target parameters that are not population totals or means. The development initially is for the case where ...
More
Inference for non-linear population parameters develops model-based prediction theory for target parameters that are not population totals or means. The development initially is for the case where the target parameter can be expressed as a differentiable function of finite population means, and a Taylor series linearisation argument is used to get a large sample approximation to the prediction variance of the substitution-based predictor. This Taylor linearisation approach is then generalised to target parameters that can be expressed as solutions of estimating equations. An application to inference about the median value of a homogeneous population serves to illustrate the basic approach, and this is then extended to the stratified population case.Less
Inference for non-linear population parameters develops model-based prediction theory for target parameters that are not population totals or means. The development initially is for the case where the target parameter can be expressed as a differentiable function of finite population means, and a Taylor series linearisation argument is used to get a large sample approximation to the prediction variance of the substitution-based predictor. This Taylor linearisation approach is then generalised to target parameters that can be expressed as solutions of estimating equations. An application to inference about the median value of a homogeneous population serves to illustrate the basic approach, and this is then extended to the stratified population case.
A.M. Stoneham
- Published in print:
- 2001
- Published Online:
- September 2007
- ISBN:
- 9780198507802
- eISBN:
- 9780191709920
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198507802.003.0003
- Subject:
- Physics, Condensed Matter Physics / Materials
This chapter examines the key approximations in crystal lattice dynamics in a way that is particularly helpful for defect studies. It discusses the Born-Oppenheimer approximation, the adiabatic ...
More
This chapter examines the key approximations in crystal lattice dynamics in a way that is particularly helpful for defect studies. It discusses the Born-Oppenheimer approximation, the adiabatic approximations, the harmonic approximations, and the dipole approximation. It also details electron-phonon interaction, the Hellman-Feynman theorem, and electron-lattice coupling.Less
This chapter examines the key approximations in crystal lattice dynamics in a way that is particularly helpful for defect studies. It discusses the Born-Oppenheimer approximation, the adiabatic approximations, the harmonic approximations, and the dipole approximation. It also details electron-phonon interaction, the Hellman-Feynman theorem, and electron-lattice coupling.
Helmut Hofmann
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780198504016
- eISBN:
- 9780191708480
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198504016.003.0021
- Subject:
- Physics, Nuclear and Plasma Physics
This chapter begins by calculating the Wigner transform for the von Neumann equation for the one-body density operator. It shows how the Liouville equation follows in leading order in an expansion to ...
More
This chapter begins by calculating the Wigner transform for the von Neumann equation for the one-body density operator. It shows how the Liouville equation follows in leading order in an expansion to ℏ. Properties of this expansion and of the resulting equation are discussed with respect to their physical and practical importance. Semi-classical approximations to the collision term are described and interpreted in terms of relevant transition rates. In Born approximation, equations of Boltzmann-Uehling-Uhlenbeck (BUU)- or Landau-Vlasov-type are obtained. The relevance of the conservation laws for particle number, energy, and momentum is discussed. For relaxation processes to equilibrium, self-energies are introduced and the relaxation-time approximation to the collision term is presented. The physical meaning of self-energies is discussed, together with the formula for the leading-order dependence of their imaginary part on energy, chemical potential, and temperature.Less
This chapter begins by calculating the Wigner transform for the von Neumann equation for the one-body density operator. It shows how the Liouville equation follows in leading order in an expansion to ℏ. Properties of this expansion and of the resulting equation are discussed with respect to their physical and practical importance. Semi-classical approximations to the collision term are described and interpreted in terms of relevant transition rates. In Born approximation, equations of Boltzmann-Uehling-Uhlenbeck (BUU)- or Landau-Vlasov-type are obtained. The relevance of the conservation laws for particle number, energy, and momentum is discussed. For relaxation processes to equilibrium, self-energies are introduced and the relaxation-time approximation to the collision term is presented. The physical meaning of self-energies is discussed, together with the formula for the leading-order dependence of their imaginary part on energy, chemical potential, and temperature.
Eric Renshaw
- Published in print:
- 2011
- Published Online:
- September 2011
- ISBN:
- 9780199575312
- eISBN:
- 9780191728778
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199575312.001.0001
- Subject:
- Mathematics, Applied Mathematics, Mathematical Biology
The vast majority of random processes in the real world have no memory — the next step in their development depends purely on their current state. Stochastic realizations are therefore defined purely ...
More
The vast majority of random processes in the real world have no memory — the next step in their development depends purely on their current state. Stochastic realizations are therefore defined purely in terms of successive event-time pairs, and such systems are easy to simulate irrespective of their degree of complexity. However, whilst the associated probability equations are straightforward to write down, their solution usually requires the use of approximation and perturbation procedures. Traditional books, heavy in mathematical theory, often ignore such methods and attempt to force problems into a rigid framework of closed-form solutions.Less
The vast majority of random processes in the real world have no memory — the next step in their development depends purely on their current state. Stochastic realizations are therefore defined purely in terms of successive event-time pairs, and such systems are easy to simulate irrespective of their degree of complexity. However, whilst the associated probability equations are straightforward to write down, their solution usually requires the use of approximation and perturbation procedures. Traditional books, heavy in mathematical theory, often ignore such methods and attempt to force problems into a rigid framework of closed-form solutions.
Nikolai B. Kopnin
- Published in print:
- 2001
- Published Online:
- January 2010
- ISBN:
- 9780198507888
- eISBN:
- 9780191709722
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198507888.003.04
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter explains how to incorporate scattering by random impurity atoms into the general Green function formalism of the theory of superconductivity. The cross-diagram technique based on the ...
More
This chapter explains how to incorporate scattering by random impurity atoms into the general Green function formalism of the theory of superconductivity. The cross-diagram technique based on the averaging over random impurity positions is derived using the Born approximation for the scattering amplitude. Impurity self-energy is derived. Homogeneous state of an s-wave superconductor is considered.Less
This chapter explains how to incorporate scattering by random impurity atoms into the general Green function formalism of the theory of superconductivity. The cross-diagram technique based on the averaging over random impurity positions is derived using the Born approximation for the scattering amplitude. Impurity self-energy is derived. Homogeneous state of an s-wave superconductor is considered.
P. E. Mijnarends, Y. Kubo, B. Barbiellini, and A. Bansil
- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198501688
- eISBN:
- 9780191718045
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198501688.003.0008
- Subject:
- Physics, Atomic, Laser, and Optical Physics
This chapter covers the calculation of the electron momentum density within the Local Density Approximation (LDA) and the Local Spin Density Approximation (LDSA). The KKR-CPA method is discussed ...
More
This chapter covers the calculation of the electron momentum density within the Local Density Approximation (LDA) and the Local Spin Density Approximation (LDSA). The KKR-CPA method is discussed followed by sections on the linearized augmented plane wave (LAPW) and linearized muffin-tin orbitals (LMTO) methods. Band theory methods beyond the LDA, the self-interaction correction, the Lam-Platzman correction, the GW method, and Quantum Monte Carlo methods are described.Less
This chapter covers the calculation of the electron momentum density within the Local Density Approximation (LDA) and the Local Spin Density Approximation (LDSA). The KKR-CPA method is discussed followed by sections on the linearized augmented plane wave (LAPW) and linearized muffin-tin orbitals (LMTO) methods. Band theory methods beyond the LDA, the self-interaction correction, the Lam-Platzman correction, the GW method, and Quantum Monte Carlo methods are described.
Gary A. Glatzmaier
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691141725
- eISBN:
- 9781400848904
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691141725.003.0001
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter presents a model of Rayleigh–Bénard convection. It first describes the fundamental dynamics expected in a fluid that is convectively stable and in one that is convectively unstable, ...
More
This chapter presents a model of Rayleigh–Bénard convection. It first describes the fundamental dynamics expected in a fluid that is convectively stable and in one that is convectively unstable, focusing on thermal convection and internal gravity waves. Thermal convection and internal gravity waves are the two basic types of fluid flows within planets and stars that are driven by thermally produced buoyancy forces. The chapter then reviews the equations that govern fluid dynamics based on conservation of mass, momentum, and energy. It also examines the conditions under which the Boussinesq approximation simplifies conservation equations to a form very similar to that of an incompressible fluid. Finally, it discusses the key characteristics of the model of Rayleigh–Bénard convection.Less
This chapter presents a model of Rayleigh–Bénard convection. It first describes the fundamental dynamics expected in a fluid that is convectively stable and in one that is convectively unstable, focusing on thermal convection and internal gravity waves. Thermal convection and internal gravity waves are the two basic types of fluid flows within planets and stars that are driven by thermally produced buoyancy forces. The chapter then reviews the equations that govern fluid dynamics based on conservation of mass, momentum, and energy. It also examines the conditions under which the Boussinesq approximation simplifies conservation equations to a form very similar to that of an incompressible fluid. Finally, it discusses the key characteristics of the model of Rayleigh–Bénard convection.
Gary A. Glatzmaier
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691141725
- eISBN:
- 9781400848904
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691141725.003.0012
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter examines the effects of large variations in density with depth, that is, density stratification. It first describes anelastic models for 2D cartesian box and 2D cylindrical annulus ...
More
This chapter examines the effects of large variations in density with depth, that is, density stratification. It first describes anelastic models for 2D cartesian box and 2D cylindrical annulus geometries, using entropy and pressure as working thermodynamic variables or using temperature and pressure, for both convectively unstable and stable regions. In particular, it considers anelastic approximation and how to formulate the anelastic equations, as well as the anelastic form of mass conservation, momentum conservation with entropy as a variable, internal energy conservation with entropy as a variable, and temperature as a variable. It also discusses possible choices for a reference state, focusing on polytropes, before explaining modifications to the numerical method and presenting the numerical simulations using the anelastic model.Less
This chapter examines the effects of large variations in density with depth, that is, density stratification. It first describes anelastic models for 2D cartesian box and 2D cylindrical annulus geometries, using entropy and pressure as working thermodynamic variables or using temperature and pressure, for both convectively unstable and stable regions. In particular, it considers anelastic approximation and how to formulate the anelastic equations, as well as the anelastic form of mass conservation, momentum conservation with entropy as a variable, internal energy conservation with entropy as a variable, and temperature as a variable. It also discusses possible choices for a reference state, focusing on polytropes, before explaining modifications to the numerical method and presenting the numerical simulations using the anelastic model.
Karsten Matthies and Florian Theil
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199239252
- eISBN:
- 9780191716911
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199239252.003.0005
- Subject:
- Mathematics, Probability / Statistics, Analysis
In this chapter a novel, rigorous approach to analyse the validity of continuum approximations for deterministic interacting particle systems is discussed. The focus is on the Boltzmann–Grad limit of ...
More
In this chapter a novel, rigorous approach to analyse the validity of continuum approximations for deterministic interacting particle systems is discussed. The focus is on the Boltzmann–Grad limit of ballistic annihilation, a topic which has has received considerable attention in the physics literature. In this situation, due to the deterministic nature of the evolution, it is possible that correlations build up and the mean–field approximation by the Boltzmann equation breaks down. A sharp condition on the initial distribution, which ensures the validity of the Boltzmann equation is given, together with an example demonstrating the failure of the mean-field theory if the condition is violated.Less
In this chapter a novel, rigorous approach to analyse the validity of continuum approximations for deterministic interacting particle systems is discussed. The focus is on the Boltzmann–Grad limit of ballistic annihilation, a topic which has has received considerable attention in the physics literature. In this situation, due to the deterministic nature of the evolution, it is possible that correlations build up and the mean–field approximation by the Boltzmann equation breaks down. A sharp condition on the initial distribution, which ensures the validity of the Boltzmann equation is given, together with an example demonstrating the failure of the mean-field theory if the condition is violated.
Stefan Adams and Wolfgang König
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199239252
- eISBN:
- 9780191716911
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199239252.003.0008
- Subject:
- Mathematics, Probability / Statistics, Analysis
Bose–Einstein condensation predicts that, under certain conditions (in particular extremely low temperature), all particles will condense into one state. Some of the physical background is surveyed ...
More
Bose–Einstein condensation predicts that, under certain conditions (in particular extremely low temperature), all particles will condense into one state. Some of the physical background is surveyed in this chapter. The Gross–Pitaevskii approximation for dilute systems is also discussed. Variational problems appear here naturally, as the quantum mechanical ground state is of interest. In connection with positive temperature, related probabilistic models, based on interacting Brownian motions in a trapping potential, are introduced. Again, large deviation techniques are used to determine the mean occupation measure, both for vanishing temperature and large particle number.Less
Bose–Einstein condensation predicts that, under certain conditions (in particular extremely low temperature), all particles will condense into one state. Some of the physical background is surveyed in this chapter. The Gross–Pitaevskii approximation for dilute systems is also discussed. Variational problems appear here naturally, as the quantum mechanical ground state is of interest. In connection with positive temperature, related probabilistic models, based on interacting Brownian motions in a trapping potential, are introduced. Again, large deviation techniques are used to determine the mean occupation measure, both for vanishing temperature and large particle number.
Yvonne Choquet-Bruhat
- Published in print:
- 2008
- Published Online:
- May 2009
- ISBN:
- 9780199230723
- eISBN:
- 9780191710872
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199230723.003.0003
- Subject:
- Mathematics, Applied Mathematics
This chapter begins with a discussion of Newton's gravity law. It then covers general relativity, observations and experiments, Einstein's equations, field sources, Lagrangians, fluid sources, ...
More
This chapter begins with a discussion of Newton's gravity law. It then covers general relativity, observations and experiments, Einstein's equations, field sources, Lagrangians, fluid sources, Newtonian approximation, Minkowskian approximation, high-frequency gravitational waves, and coupled electromagnetic and gravitational waves.Less
This chapter begins with a discussion of Newton's gravity law. It then covers general relativity, observations and experiments, Einstein's equations, field sources, Lagrangians, fluid sources, Newtonian approximation, Minkowskian approximation, high-frequency gravitational waves, and coupled electromagnetic and gravitational waves.
David M. Paganin
- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198567288
- eISBN:
- 9780191717963
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567288.003.0002
- Subject:
- Physics, Atomic, Laser, and Optical Physics
This chapter considers the interactions of X-rays with matter. It opens by developing X-ray wave equations in the presence of scatterers, taking the Maxwell equations as a starting point. The ...
More
This chapter considers the interactions of X-rays with matter. It opens by developing X-ray wave equations in the presence of scatterers, taking the Maxwell equations as a starting point. The projection approximation is then discussed. The concept of a Green function, which is of immense importance in the formalism of X-ray scattering, is introduced. Equipped with this, an integral from of the X-ray wave equation is developed, approximate solutions to which are furnished by the famous first Born approximation. Second and higher-order Born approximations are also considered, heralding the transition from so-called kinematical diffraction to dynamical diffraction. Other subjects treated in the chapter include the Ewald sphere, the multislice approximation, the eikonal approximation, the link between refractive index and electron density, Compton scattering, photoelectric absorption, fluorescence, and the information content of scattered fields.Less
This chapter considers the interactions of X-rays with matter. It opens by developing X-ray wave equations in the presence of scatterers, taking the Maxwell equations as a starting point. The projection approximation is then discussed. The concept of a Green function, which is of immense importance in the formalism of X-ray scattering, is introduced. Equipped with this, an integral from of the X-ray wave equation is developed, approximate solutions to which are furnished by the famous first Born approximation. Second and higher-order Born approximations are also considered, heralding the transition from so-called kinematical diffraction to dynamical diffraction. Other subjects treated in the chapter include the Ewald sphere, the multislice approximation, the eikonal approximation, the link between refractive index and electron density, Compton scattering, photoelectric absorption, fluorescence, and the information content of scattered fields.
Andreas Kirsch and Natalia Grinberg
- 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
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 scattering problem is to determine the support D of the contrast from far field measurements. The chapter begins with a simple scattering model where the scatterers consists of a finite number of point scatterers. The inverse problem is to determine the locations of these point scatterers from the multistatic response matrix F, which is the discrete analog of the far field operator. In this situation, the Factorization Method is nothing else but the MUSIC-algorithm which is well known in signal processing. The chapter then formulates direct and inverse scattering problem for the scattering by an inhomogeneous medium, reformulates the direct problem as the Lippmann-Schwinger integral equation, and justifies the popular Born approximation. The chapter formulizes the far field operator and proves a characterization of D by the convergence of a Picard series which involves only known data derived from the far field operator. This characterization holds only if the frequency is not an eigenvalue of an unconventional eigenvalue problem of transmission type. The last section shows that there exist at most a quantifiable number of these values.Less
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 scattering problem is to determine the support D of the contrast from far field measurements. The chapter begins with a simple scattering model where the scatterers consists of a finite number of point scatterers. The inverse problem is to determine the locations of these point scatterers from the multistatic response matrix F, which is the discrete analog of the far field operator. In this situation, the Factorization Method is nothing else but the MUSIC-algorithm which is well known in signal processing. The chapter then formulates direct and inverse scattering problem for the scattering by an inhomogeneous medium, reformulates the direct problem as the Lippmann-Schwinger integral equation, and justifies the popular Born approximation. The chapter formulizes the far field operator and proves a characterization of D by the convergence of a Picard series which involves only known data derived from the far field operator. This characterization holds only if the frequency is not an eigenvalue of an unconventional eigenvalue problem of transmission type. The last section shows that there exist at most a quantifiable number of these values.
Jean-Frédéric Gerbeau, Claude Le Bris, and Tony Lelièvre
- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198566656
- eISBN:
- 9780191718014
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566656.003.0002
- Subject:
- Mathematics, Mathematical Physics
This chapter focuses on the modelling of one-fluid magnetohydrodynamics problems. The crucial point under consideration is the coupling between hydrodynamics phenomena and electromagnetic phenomena. ...
More
This chapter focuses on the modelling of one-fluid magnetohydrodynamics problems. The crucial point under consideration is the coupling between hydrodynamics phenomena and electromagnetic phenomena. From a mathematical viewpoint, the coupling induces a nonlinearity, additional to the nonlinearities already present in the hydrodynamics. A series of difficult, thus interesting, problems follow. With a reasonable amount of theoretical efforts, these problems can be dealt with. For instance, it can be shown that a system coupling the time-dependent incompressible Navier-Stokes equations with a simplified form of the Maxwell equations (the so-called low-frequency approximation) is well-posed when the electromagnetic equation is taken time-dependent, in parabolic form. In contrast, the same model is likely to be ill-posed when the electromagnetic equation is taken time-independent, in elliptic form.Less
This chapter focuses on the modelling of one-fluid magnetohydrodynamics problems. The crucial point under consideration is the coupling between hydrodynamics phenomena and electromagnetic phenomena. From a mathematical viewpoint, the coupling induces a nonlinearity, additional to the nonlinearities already present in the hydrodynamics. A series of difficult, thus interesting, problems follow. With a reasonable amount of theoretical efforts, these problems can be dealt with. For instance, it can be shown that a system coupling the time-dependent incompressible Navier-Stokes equations with a simplified form of the Maxwell equations (the so-called low-frequency approximation) is well-posed when the electromagnetic equation is taken time-dependent, in parabolic form. In contrast, the same model is likely to be ill-posed when the electromagnetic equation is taken time-independent, in elliptic form.
Rolf Niedermeier
- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198566076
- eISBN:
- 9780191713910
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566076.003.0014
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This chapter summarizes the currently known connections between fixed-parameter and polynomial-time approximation algorithmics. Important topics in this context are lower bound and impossibility ...
More
This chapter summarizes the currently known connections between fixed-parameter and polynomial-time approximation algorithmics. Important topics in this context are lower bound and impossibility results. In particular, polynomial-time approximation schemes are discussed in some detail, providing a link with parameterized complexity theory.Less
This chapter summarizes the currently known connections between fixed-parameter and polynomial-time approximation algorithmics. Important topics in this context are lower bound and impossibility results. In particular, polynomial-time approximation schemes are discussed in some detail, providing a link with parameterized complexity theory.
