Moody T. Chu and Gene H. Golub
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198566649
- eISBN:
- 9780191718021
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566649.003.0006
- Subject:
- Mathematics, Applied Mathematics
Every inverse eigenvalue problem has a natural generalization to a least squares formulation, which sometimes does carry significant purposes in applications. The least squares approximation can be ...
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Every inverse eigenvalue problem has a natural generalization to a least squares formulation, which sometimes does carry significant purposes in applications. The least squares approximation can be applied to either the spectral constraint or the structural constraint. This chapter highlights some of the main notions when considering a least squares inverse problem, and describes a hybrid lift and projection method.Less
Every inverse eigenvalue problem has a natural generalization to a least squares formulation, which sometimes does carry significant purposes in applications. The least squares approximation can be applied to either the spectral constraint or the structural constraint. This chapter highlights some of the main notions when considering a least squares inverse problem, and describes a hybrid lift and projection method.
Moody T. Chu and Gene H. Golub
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198566649
- eISBN:
- 9780191718021
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566649.003.0007
- Subject:
- Mathematics, Applied Mathematics
This chapter shows that the problems of computing least squares approximations for various types of real and symmetric matrices subject to spectral constraints share a common structure. A general ...
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This chapter shows that the problems of computing least squares approximations for various types of real and symmetric matrices subject to spectral constraints share a common structure. A general framework by using the projected gradient method is described. A broad range of applications, including the Toeplitz inverse eigenvalue problem, the simultaneous reduction problem, and the nearest normal matrix approximation, are discussed.Less
This chapter shows that the problems of computing least squares approximations for various types of real and symmetric matrices subject to spectral constraints share a common structure. A general framework by using the projected gradient method is described. A broad range of applications, including the Toeplitz inverse eigenvalue problem, the simultaneous reduction problem, and the nearest normal matrix approximation, are discussed.
Kevin S. McCann
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691134178
- eISBN:
- 9781400840687
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691134178.003.0010
- Subject:
- Biology, Ecology
This chapter examines the basic assumptions of classic food web theory. It first considers the classic whole-community approach, which assumes that any specific matrix represents a sample from a ...
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This chapter examines the basic assumptions of classic food web theory. It first considers the classic whole-community approach, which assumes that any specific matrix represents a sample from a “statistical universe” of interaction strengths for a given set of n species. It then describes some matrix approaches to see if context-dependent techniques can be applied to matrix theory, along with the simple graphical techniques of Gershgorin discs employed as an intuitive approach to eigenvalues. It argues that there are some rather intriguing “gravitational-like” properties of Gershgorin discs for some important biologically motivated matrices. The chapter proceeds by discussing some classic whole-matrix results that highlight the connections between the stability of lower-dimensional modules and whole food webs. Finally, it shows how the ideas derived from classic whole-system matrix approaches generally agree with the results of modular theory.Less
This chapter examines the basic assumptions of classic food web theory. It first considers the classic whole-community approach, which assumes that any specific matrix represents a sample from a “statistical universe” of interaction strengths for a given set of n species. It then describes some matrix approaches to see if context-dependent techniques can be applied to matrix theory, along with the simple graphical techniques of Gershgorin discs employed as an intuitive approach to eigenvalues. It argues that there are some rather intriguing “gravitational-like” properties of Gershgorin discs for some important biologically motivated matrices. The chapter proceeds by discussing some classic whole-matrix results that highlight the connections between the stability of lower-dimensional modules and whole food webs. Finally, it shows how the ideas derived from classic whole-system matrix approaches generally agree with the results of modular theory.
Alfonso Novales, Emilio Domínguez, Javier J. Pérez, and Jesús Ruiz
- Published in print:
- 2001
- Published Online:
- November 2003
- ISBN:
- 9780199248278
- eISBN:
- 9780191596605
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199248273.003.0004
- Subject:
- Economics and Finance, Macro- and Monetary Economics
Discusses the main issues involved in practical applications of solution methods that have been proposed for rational expectations models, based on eigenvalue–eigenvector decompositions. It starts by ...
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Discusses the main issues involved in practical applications of solution methods that have been proposed for rational expectations models, based on eigenvalue–eigenvector decompositions. It starts by reviewing how a numerical solution can be derived for the standard deterministic Cass–Koopmans–Brock–Mirman economy, pointing out the relevance of stability conditions. Next the general structure used to solve linear rational expectations models, and its extension to nonlinear models, is summarized. The solution method is then applied to Hansen's (1985) model of indivisible labour, and comparisons with other solution approaches are discussed. It is then shown how the eigenvalue–eigenvector decomposition can help to separately identify variables of a similar nature (as is the case when physical capital and inventories are inputs in an aggregate production technology), and how the solution method can be adapted to deal with endogenous growth models.Less
Discusses the main issues involved in practical applications of solution methods that have been proposed for rational expectations models, based on eigenvalue–eigenvector decompositions. It starts by reviewing how a numerical solution can be derived for the standard deterministic Cass–Koopmans–Brock–Mirman economy, pointing out the relevance of stability conditions. Next the general structure used to solve linear rational expectations models, and its extension to nonlinear models, is summarized. The solution method is then applied to Hansen's (1985) model of indivisible labour, and comparisons with other solution approaches are discussed. It is then shown how the eigenvalue–eigenvector decomposition can help to separately identify variables of a similar nature (as is the case when physical capital and inventories are inputs in an aggregate production technology), and how the solution method can be adapted to deal with endogenous growth models.
Gary E. Bowman
- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780199228928
- eISBN:
- 9780191711206
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199228928.003.0006
- Subject:
- Physics, Condensed Matter Physics / Materials
This chapter introduces the matrix-mechanics formulation of quantum mechanics, emphasizing both calculational techniques and conceptual understanding. Parallels between matrix mechanics and ordinary ...
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This chapter introduces the matrix-mechanics formulation of quantum mechanics, emphasizing both calculational techniques and conceptual understanding. Parallels between matrix mechanics and ordinary vectors and matrices are extensively utilized. Starting with the representation of ordinary vectors as rows or columns of numbers, the scalar product is discussed, followed by the transformation of vectors by matrices, as illustrated by rotations. The vector representation of quantumstates, the inner product of two such states, and the matrix representation of operators are then introduced. The simple forms assumed in matrix mechanics by a basis state, and by an operator, when either is written in its eigenbasis, are discussed, as are the specific forms of adjoint, Hermitian, and unitary operators. The chapter concludes with a brief exposition of eigenvalue equations in matrix mechanics.Less
This chapter introduces the matrix-mechanics formulation of quantum mechanics, emphasizing both calculational techniques and conceptual understanding. Parallels between matrix mechanics and ordinary vectors and matrices are extensively utilized. Starting with the representation of ordinary vectors as rows or columns of numbers, the scalar product is discussed, followed by the transformation of vectors by matrices, as illustrated by rotations. The vector representation of quantumstates, the inner product of two such states, and the matrix representation of operators are then introduced. The simple forms assumed in matrix mechanics by a basis state, and by an operator, when either is written in its eigenbasis, are discussed, as are the specific forms of adjoint, Hermitian, and unitary operators. The chapter concludes with a brief exposition of eigenvalue equations in matrix mechanics.
Klaus Boehmer
- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780199577040
- eISBN:
- 9780191595172
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199577040.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Nonlinear elliptic problems play an increasingly important role in mathematics, science and engineering, and create an exciting interplay. Other books discuss nonlinearity by a very few important ...
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Nonlinear elliptic problems play an increasingly important role in mathematics, science and engineering, and create an exciting interplay. Other books discuss nonlinearity by a very few important examples. This is the first and only book, proving in a systematic and unifying way, stability and convergence results and methods for solving nonlinear discrete equations via discrete Newton methods for the different numerical methods for all these problems. The proofs use linearization, compact perturbation of the coercive principal parts, or monotone operator techniques, and approximation theory. This is examplified for linear to fully nonlinear problems (highest derivatives occur nonlinearly) and for the most important space discretization methods: conforming and nonconforming finite element, discontinuous Galerkin, finite difference and wavelet methods. The proof of stability for nonconforming methods employs the anticrime operator as an essential tool. For all these methods approximate evaluation of the discrete equations, and eigenvalue problems are discussed. The numerical methods are based upon analytic results for this wide class of problems, guaranteeing existence, uniqueness and regularity of the exact solutions. In the next book, spectral, mesh‐free methods and convergence for bifurcation and center manifolds for all these combinations are proved. Specific long open problems, solved here, are numerical methods for fully nonlinear elliptic problems, wavelet and mesh‐free methods for nonlinear problems, and more general nonlinear boundary conditions. Adaptivity is discussed for finite element and wavelet methods with totally different techniques.Less
Nonlinear elliptic problems play an increasingly important role in mathematics, science and engineering, and create an exciting interplay. Other books discuss nonlinearity by a very few important examples. This is the first and only book, proving in a systematic and unifying way, stability and convergence results and methods for solving nonlinear discrete equations via discrete Newton methods for the different numerical methods for all these problems. The proofs use linearization, compact perturbation of the coercive principal parts, or monotone operator techniques, and approximation theory. This is examplified for linear to fully nonlinear problems (highest derivatives occur nonlinearly) and for the most important space discretization methods: conforming and nonconforming finite element, discontinuous Galerkin, finite difference and wavelet methods. The proof of stability for nonconforming methods employs the anticrime operator as an essential tool. For all these methods approximate evaluation of the discrete equations, and eigenvalue problems are discussed. The numerical methods are based upon analytic results for this wide class of problems, guaranteeing existence, uniqueness and regularity of the exact solutions. In the next book, spectral, mesh‐free methods and convergence for bifurcation and center manifolds for all these combinations are proved. Specific long open problems, solved here, are numerical methods for fully nonlinear elliptic problems, wavelet and mesh‐free methods for nonlinear problems, and more general nonlinear boundary conditions. Adaptivity is discussed for finite element and wavelet methods with totally different techniques.
Klaus Böhmer
- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780199577040
- eISBN:
- 9780191595172
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199577040.003.0005
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Nonconforming FEMs avoid the strong restrictions of conforming FEMs. So discontinuous ansatz and test functions, approximate test integrals, and strong forms are admitted. This allows the proof of ...
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Nonconforming FEMs avoid the strong restrictions of conforming FEMs. So discontinuous ansatz and test functions, approximate test integrals, and strong forms are admitted. This allows the proof of convergence for the full spectrum of linear to fully nonlinear equations and systems of orders 2 and 2m. General fully nonlinear problems only allow strong forms and enforce new techniques and C1 FEs. Variational crimes for FEs violating regularity and boundary conditions are studied in ℝ2 for linear and quasilinear problems. Essential tools are the anticrime transformations. The relations between the strong and weak form of the equations allow the usually technical proofs for consistency. Due to the dominant role of FEMs, numerical solutions for five classes of problems are only presented for FEMs. Most remain valid for the other methods as well: vari-ational methods for eigenvalue problems, convergence theory for monotone operators (quasilinear problems), FEMs for fully nonlinear elliptic problems and for nonlinear boundary conditions, and quadrature approximate FEMs. We thus close several gaps in the literature.Less
Nonconforming FEMs avoid the strong restrictions of conforming FEMs. So discontinuous ansatz and test functions, approximate test integrals, and strong forms are admitted. This allows the proof of convergence for the full spectrum of linear to fully nonlinear equations and systems of orders 2 and 2m. General fully nonlinear problems only allow strong forms and enforce new techniques and C1 FEs. Variational crimes for FEs violating regularity and boundary conditions are studied in ℝ2 for linear and quasilinear problems. Essential tools are the anticrime transformations. The relations between the strong and weak form of the equations allow the usually technical proofs for consistency. Due to the dominant role of FEMs, numerical solutions for five classes of problems are only presented for FEMs. Most remain valid for the other methods as well: vari-ational methods for eigenvalue problems, convergence theory for monotone operators (quasilinear problems), FEMs for fully nonlinear elliptic problems and for nonlinear boundary conditions, and quadrature approximate FEMs. We thus close several gaps in the literature.
Bas C. van Fraassen
- Published in print:
- 1991
- Published Online:
- November 2003
- ISBN:
- 9780198239802
- eISBN:
- 9780191597466
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198239807.003.0008
- Subject:
- Philosophy, Philosophy of Science
Von Neumann's unification of Schroedinger's and Heisenberg's formalisms came with an interpretation of quantum theory involving two principles. The first is that assertions about the values of ...
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Von Neumann's unification of Schroedinger's and Heisenberg's formalisms came with an interpretation of quantum theory involving two principles. The first is that assertions about the values of observables are equivalent to assertions about the quantum‐mechanical state of the system. This is sometimes known as the ’eigenvalue–eigenstate link’, since it equates an observable having a value with the system being in an eigenstate of that observable. The second is his Projection Postulate—i.e. the postulate that during measurement there is a ’collapse of the wave packet’. It is argued that the theory does not force these principles on us, and that there are severe difficulties in this interpretation, despite also its more recent defences.Less
Von Neumann's unification of Schroedinger's and Heisenberg's formalisms came with an interpretation of quantum theory involving two principles. The first is that assertions about the values of observables are equivalent to assertions about the quantum‐mechanical state of the system. This is sometimes known as the ’eigenvalue–eigenstate link’, since it equates an observable having a value with the system being in an eigenstate of that observable. The second is his Projection Postulate—i.e. the postulate that during measurement there is a ’collapse of the wave packet’. It is argued that the theory does not force these principles on us, and that there are severe difficulties in this interpretation, despite also its more recent defences.
Bas C. van Fraassen
- Published in print:
- 1991
- Published Online:
- November 2003
- ISBN:
- 9780198239802
- eISBN:
- 9780191597466
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198239807.003.0009
- Subject:
- Philosophy, Philosophy of Science
The world of quantum theory is not deterministic; however, the quantum theory of an isolated system describes its state as evolving deterministically.How can these two points be reconciled? The modal ...
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The world of quantum theory is not deterministic; however, the quantum theory of an isolated system describes its state as evolving deterministically.How can these two points be reconciled? The modal interpretation (of which a number of variants have been developed) answers this as follows. An observable may have a value even if the system is not in the corresponding eigenstate of that observable. The state of the system only constrains the possible values the observable can have, and (at least under certain conditions) the probabilities that it has those values. Hence, the eigenvalue–eigenstate link is violated. There is no collapse; the state of an isolated system always evolves in accordance with the Schroedinger equation. Hence the values of observables can change indeterministically, though this stochastic process is constrained by the deterministically evolving quantum state. Terminology varies but in this book the quantum state is then called the dynamical state, and a summary of the values of the observables pertaining to the system is called the value state. Since not all observables are mutually compatible, they will not all have simultaneous sharp values, but for each there is a minimum Borel set of values as its current range. The specific modal interpretation developed here is the Copenhagen Variant of the Modal Interpretation, which places holistic constraints on values of observables pertaining to parts of composite systems.Less
The world of quantum theory is not deterministic; however, the quantum theory of an isolated system describes its state as evolving deterministically.How can these two points be reconciled? The modal interpretation (of which a number of variants have been developed) answers this as follows. An observable may have a value even if the system is not in the corresponding eigenstate of that observable. The state of the system only constrains the possible values the observable can have, and (at least under certain conditions) the probabilities that it has those values. Hence, the eigenvalue–eigenstate link is violated. There is no collapse; the state of an isolated system always evolves in accordance with the Schroedinger equation. Hence the values of observables can change indeterministically, though this stochastic process is constrained by the deterministically evolving quantum state. Terminology varies but in this book the quantum state is then called the dynamical state, and a summary of the values of the observables pertaining to the system is called the value state. Since not all observables are mutually compatible, they will not all have simultaneous sharp values, but for each there is a minimum Borel set of values as its current range. The specific modal interpretation developed here is the Copenhagen Variant of the Modal Interpretation, which places holistic constraints on values of observables pertaining to parts of composite systems.
Oliver Johns
- Published in print:
- 2005
- Published Online:
- January 2010
- ISBN:
- 9780198567264
- eISBN:
- 9780191717987
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567264.003.0010
- Subject:
- Physics, Atomic, Laser, and Optical Physics
A number of interesting mechanical systems have one or more essentially stable equilibrium configurations. When disturbed slightly, they vibrate about equilibrium in characteristic patterns called ...
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A number of interesting mechanical systems have one or more essentially stable equilibrium configurations. When disturbed slightly, they vibrate about equilibrium in characteristic patterns called normal modes. This chapter presents the Lagrangian theory of these small vibrations for the simple case of systems with a finite number of degrees of freedom. The theory has wide application. For example, the normal mode oscillations of crystalline solids underlie both the overtone structure of a church bell and the definition of phonons in solid state physics. A similar formalism leads to photons as the quanta of modes of the electromagnetic field. This chapter defines equilibrium points in the configuration space of a mechanical system and discusses how to find them, along with small coordinates, normal modes, generalised eigenvalue problem, stability, initial conditions, energy of small vibrations, single mode excitations, and zero-frequency modes.Less
A number of interesting mechanical systems have one or more essentially stable equilibrium configurations. When disturbed slightly, they vibrate about equilibrium in characteristic patterns called normal modes. This chapter presents the Lagrangian theory of these small vibrations for the simple case of systems with a finite number of degrees of freedom. The theory has wide application. For example, the normal mode oscillations of crystalline solids underlie both the overtone structure of a church bell and the definition of phonons in solid state physics. A similar formalism leads to photons as the quanta of modes of the electromagnetic field. This chapter defines equilibrium points in the configuration space of a mechanical system and discusses how to find them, along with small coordinates, normal modes, generalised eigenvalue problem, stability, initial conditions, energy of small vibrations, single mode excitations, and zero-frequency modes.
Jeffrey Alan Barrett
- Published in print:
- 2001
- Published Online:
- March 2012
- ISBN:
- 9780199247431
- eISBN:
- 9780191697661
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199247431.003.0004
- Subject:
- Philosophy, Philosophy of Science, Metaphysics/Epistemology
This chapter begins by discussing several strategies for interpreting the global state. Next, it explains that the bare theory is simply the standard von Neumann–Dirac formulation of quantum ...
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This chapter begins by discussing several strategies for interpreting the global state. Next, it explains that the bare theory is simply the standard von Neumann–Dirac formulation of quantum mechanics with the standard interpretation of states (the eigenvalue-eigenstate link) without the collapse postulate — hence, bare. Subsequently, it examines several suggestive properties of the bare theory that tell what an observer would report concerning his experience in various measurement situations if the theory were true. It then explains that all measurement interactions induce an imperfect correlation between the state of an observer's most immediately accessible physical records and the observed quantity of the object system without disturbing the property being measured. This chapter also differentiates ordinary and disjunctive records, experiences, and beliefs. Lastly, it discusses several problems encountered with the bare theory.Less
This chapter begins by discussing several strategies for interpreting the global state. Next, it explains that the bare theory is simply the standard von Neumann–Dirac formulation of quantum mechanics with the standard interpretation of states (the eigenvalue-eigenstate link) without the collapse postulate — hence, bare. Subsequently, it examines several suggestive properties of the bare theory that tell what an observer would report concerning his experience in various measurement situations if the theory were true. It then explains that all measurement interactions induce an imperfect correlation between the state of an observer's most immediately accessible physical records and the observed quantity of the object system without disturbing the property being measured. This chapter also differentiates ordinary and disjunctive records, experiences, and beliefs. Lastly, it discusses several problems encountered with the bare theory.
John von Neumann
Nicholas A. Wheeler (ed.)
- Published in print:
- 2018
- Published Online:
- September 2018
- ISBN:
- 9780691178561
- eISBN:
- 9781400889921
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691178561.003.0003
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter defines Hilbert space, which furnishes the mathematical basis for the treatment of quantum mechanics. This is done within the context of an equation introduced in the previous chapter, ...
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This chapter defines Hilbert space, which furnishes the mathematical basis for the treatment of quantum mechanics. This is done within the context of an equation introduced in the previous chapter, and which have accordingly the same meaning in the “discrete” function space FsubscriptZ of the sequences xsubscriptv (ν = 1, 2, . . .) and in the “continuous” Fsubscript Greek Capital Letter Omega of the wave functions φ(q₁, . . . , qₖ) (q₁, . . . , qₖ run through the entire state space Ω). In order to define abstract Hilbert space, this chapter takes as a basis the fundamental vector operations af, f ± g, (f, g).Less
This chapter defines Hilbert space, which furnishes the mathematical basis for the treatment of quantum mechanics. This is done within the context of an equation introduced in the previous chapter, and which have accordingly the same meaning in the “discrete” function space FsubscriptZ of the sequences xsubscriptv (ν = 1, 2, . . .) and in the “continuous” Fsubscript Greek Capital Letter Omega of the wave functions φ(q₁, . . . , qₖ) (q₁, . . . , qₖ run through the entire state space Ω). In order to define abstract Hilbert space, this chapter takes as a basis the fundamental vector operations af, f ± g, (f, g).
G. F. Roach, I. G. Stratis, and A. N. Yannacopoulos
- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691142173
- eISBN:
- 9781400842650
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691142173.003.0004
- Subject:
- Mathematics, Applied Mathematics
This chapter presents rigorous mathematical results concerning the solvability and well posedness of time-harmonic problems for complex electromagnetic media, with a special emphasis on chiral media. ...
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This chapter presents rigorous mathematical results concerning the solvability and well posedness of time-harmonic problems for complex electromagnetic media, with a special emphasis on chiral media. It also presents some results concerning eigenvalue problems in cavities filled with complex electromagnetic materials. The chapter also studies the behaviour of the interior domain problem for a chiral medium in the limit of low chirality. Next, it presents some comments related to the well posedness and solvability of exterior problems. Finally, using an appropriate finite-dimensional space and the variational formulation of the discretised version of the original boundary value problem, this chapter obtains numerical methods for the solution of the Maxwell equations for chiral media.Less
This chapter presents rigorous mathematical results concerning the solvability and well posedness of time-harmonic problems for complex electromagnetic media, with a special emphasis on chiral media. It also presents some results concerning eigenvalue problems in cavities filled with complex electromagnetic materials. The chapter also studies the behaviour of the interior domain problem for a chiral medium in the limit of low chirality. Next, it presents some comments related to the well posedness and solvability of exterior problems. Finally, using an appropriate finite-dimensional space and the variational formulation of the discretised version of the original boundary value problem, this chapter obtains numerical methods for the solution of the Maxwell equations for chiral media.
Hans-Peter Eckle
- Published in print:
- 2019
- Published Online:
- September 2019
- ISBN:
- 9780199678839
- eISBN:
- 9780191878589
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780199678839.003.0013
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics, Condensed Matter Physics / Materials
This chapter introduces the Heisenberg model, a fully quantum mechanical model that describes the magnetism of localized magnetic moments. The one-dimensional version of the Heisenberg model, the ...
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This chapter introduces the Heisenberg model, a fully quantum mechanical model that describes the magnetism of localized magnetic moments. The one-dimensional version of the Heisenberg model, the Heisenberg quantum spin chain, provides a good picture of magnetic materials that belong to a class of insulating magnetic materials where the interaction of the magnetic moments in one particular direction is much larger than in the perpendicular directions, and which can be described with high accuracy as quasi- one-dimensional magnets. A detailed description of the Heisenberg quantum spin chain is followed by a discussion of its various special cases, in particular the special case of the anisotropic Heisenberg quantum spin chain, the so-called XXZ quantum spin chain. It considers the solution of eigenvalue problem of this quantum spin and leads to Bethe’s conjecture for the wave function.Less
This chapter introduces the Heisenberg model, a fully quantum mechanical model that describes the magnetism of localized magnetic moments. The one-dimensional version of the Heisenberg model, the Heisenberg quantum spin chain, provides a good picture of magnetic materials that belong to a class of insulating magnetic materials where the interaction of the magnetic moments in one particular direction is much larger than in the perpendicular directions, and which can be described with high accuracy as quasi- one-dimensional magnets. A detailed description of the Heisenberg quantum spin chain is followed by a discussion of its various special cases, in particular the special case of the anisotropic Heisenberg quantum spin chain, the so-called XXZ quantum spin chain. It considers the solution of eigenvalue problem of this quantum spin and leads to Bethe’s conjecture for the wave function.
Hans-Peter Eckle
- Published in print:
- 2019
- Published Online:
- September 2019
- ISBN:
- 9780199678839
- eISBN:
- 9780191878589
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780199678839.003.0016
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics, Condensed Matter Physics / Materials
The Bethe ansatz can be generalized to problems where particles have internal degrees of freedom. The generalized method can be viewed as two Bethe ansätze executed one after the other: nested Bethe ...
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The Bethe ansatz can be generalized to problems where particles have internal degrees of freedom. The generalized method can be viewed as two Bethe ansätze executed one after the other: nested Bethe ansatz. Electronic systems are the most relevant examples for condensed matter physics. Prominent electronic many-particle systems in one dimension solvable by a nested Bethe ansatz are the one-dimensional δ-Fermi gas, the one-dimensional Hubbard model, and the Kondo model. The major difference to the Bethe ansatz for one component systems is a second, spin, eigenvalue problem, which has the same form in all cases and is solvable by a second Bethe ansatz, e.g. an algebraic Bethe ansatz. A quantum dot tuned to Kondo resonance and coupled to an isolated metallic ring presents an application of the coupled sets of Bethe ansatz equations of the nested Bethe ansatz.Less
The Bethe ansatz can be generalized to problems where particles have internal degrees of freedom. The generalized method can be viewed as two Bethe ansätze executed one after the other: nested Bethe ansatz. Electronic systems are the most relevant examples for condensed matter physics. Prominent electronic many-particle systems in one dimension solvable by a nested Bethe ansatz are the one-dimensional δ-Fermi gas, the one-dimensional Hubbard model, and the Kondo model. The major difference to the Bethe ansatz for one component systems is a second, spin, eigenvalue problem, which has the same form in all cases and is solvable by a second Bethe ansatz, e.g. an algebraic Bethe ansatz. A quantum dot tuned to Kondo resonance and coupled to an isolated metallic ring presents an application of the coupled sets of Bethe ansatz equations of the nested Bethe ansatz.
Rüdiger Verfürth
- Published in print:
- 2013
- Published Online:
- May 2013
- ISBN:
- 9780199679423
- eISBN:
- 9780191758485
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199679423.003.0004
- Subject:
- Mathematics, Applied Mathematics, Numerical Analysis
We establish a posteriori error estimates for the discretisation of linear elliptic equations departing from a general abstract framework which accentuates the generic equivalence of the error to the ...
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We establish a posteriori error estimates for the discretisation of linear elliptic equations departing from a general abstract framework which accentuates the generic equivalence of the error to the associated residual and highlights that this is a structural property of the variational problem which is completely independent of the discretisation. Thus, the derivation of a posteriori error estimates amounts to the task of establishing computable bounds for dual norms of residuals. In particular, we address the following topics: robustness of the a posteriori error estimates with respect to parameters inherent in the variational problem and with respect to anisotropy of the meshes or of the differential equation, mixed finite element methods for saddle-point problems, and differential equations of higher order and non-conforming methods. Finally, we give a general convergence proof for a generic adaptive algorithm.Less
We establish a posteriori error estimates for the discretisation of linear elliptic equations departing from a general abstract framework which accentuates the generic equivalence of the error to the associated residual and highlights that this is a structural property of the variational problem which is completely independent of the discretisation. Thus, the derivation of a posteriori error estimates amounts to the task of establishing computable bounds for dual norms of residuals. In particular, we address the following topics: robustness of the a posteriori error estimates with respect to parameters inherent in the variational problem and with respect to anisotropy of the meshes or of the differential equation, mixed finite element methods for saddle-point problems, and differential equations of higher order and non-conforming methods. Finally, we give a general convergence proof for a generic adaptive algorithm.
Rüdiger Verfürth
- Published in print:
- 2013
- Published Online:
- May 2013
- ISBN:
- 9780199679423
- eISBN:
- 9780191758485
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199679423.003.0005
- Subject:
- Mathematics, Applied Mathematics, Numerical Analysis
We present a posteriori error estimates for nonlinear stationary problems departing from an abstract result which establishes the basic equivalence of error and residual. The main tool is the ...
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We present a posteriori error estimates for nonlinear stationary problems departing from an abstract result which establishes the basic equivalence of error and residual. The main tool is the implicit function theorem along with modifications for bifurcation and turning points. We apply the abstract results to quasilinear elliptic equations and the stationary incompressible Navier–Stokes equations. The nonlinear approach also enables us to efficiently treat linear elliptic eigenvalue problems.Less
We present a posteriori error estimates for nonlinear stationary problems departing from an abstract result which establishes the basic equivalence of error and residual. The main tool is the implicit function theorem along with modifications for bifurcation and turning points. We apply the abstract results to quasilinear elliptic equations and the stationary incompressible Navier–Stokes equations. The nonlinear approach also enables us to efficiently treat linear elliptic eigenvalue problems.
Chun Wa Wong
- Published in print:
- 2013
- Published Online:
- May 2013
- ISBN:
- 9780199641390
- eISBN:
- 9780191747786
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199641390.003.0002
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Linear algebra of matrices is discussed in the context of rotations in space. Determinants are used to find solutions of simultaneous linear equations, to invert matrices and to find matrix ...
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Linear algebra of matrices is discussed in the context of rotations in space. Determinants are used to find solutions of simultaneous linear equations, to invert matrices and to find matrix eigenvalues. The linear wave equation quadratic in the infinitesimal generators of space and time translations is derived. Rotation operators and matrix groups are introduced.Less
Linear algebra of matrices is discussed in the context of rotations in space. Determinants are used to find solutions of simultaneous linear equations, to invert matrices and to find matrix eigenvalues. The linear wave equation quadratic in the infinitesimal generators of space and time translations is derived. Rotation operators and matrix groups are introduced.
Stefan Thurner, Rudolf Hanel, and Peter Klimekl
- Published in print:
- 2018
- Published Online:
- November 2018
- ISBN:
- 9780198821939
- eISBN:
- 9780191861062
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198821939.003.0004
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Understanding the interactions between the components of a system is key to understanding it. In complex systems, interactions are usually not uniform, not isotropic and not homogeneous: each ...
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Understanding the interactions between the components of a system is key to understanding it. In complex systems, interactions are usually not uniform, not isotropic and not homogeneous: each interaction can be specific between elements.Networks are a tool for keeping track of who is interacting with whom, at what strength, when, and in what way. Networks are essential for understanding of the co-evolution and phase diagrams of complex systems. Here we provide a self-contained introduction to the field of network science. We introduce ways of representing and handle networks mathematically and introduce the basic vocabulary and definitions. The notions of random- and complex networks are reviewed as well as the notions of small world networks, simple preferentially grown networks, community detection, and generalized multilayer networks.Less
Understanding the interactions between the components of a system is key to understanding it. In complex systems, interactions are usually not uniform, not isotropic and not homogeneous: each interaction can be specific between elements.Networks are a tool for keeping track of who is interacting with whom, at what strength, when, and in what way. Networks are essential for understanding of the co-evolution and phase diagrams of complex systems. Here we provide a self-contained introduction to the field of network science. We introduce ways of representing and handle networks mathematically and introduce the basic vocabulary and definitions. The notions of random- and complex networks are reviewed as well as the notions of small world networks, simple preferentially grown networks, community detection, and generalized multilayer networks.
D. E. Edmunds and W. D. Evans
- Published in print:
- 2018
- Published Online:
- September 2018
- ISBN:
- 9780198812050
- eISBN:
- 9780191861130
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198812050.003.0002
- Subject:
- Mathematics, Pure Mathematics
The geometric quantities entropy numbers, approximation numbers and n-widths are defined for compact linear maps, and connections with the analytic entities eigenvalues and essential spectra ...
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The geometric quantities entropy numbers, approximation numbers and n-widths are defined for compact linear maps, and connections with the analytic entities eigenvalues and essential spectra discussed. The celebrated inequality of Weyl between the approximation numbers and eigenvalues is established in the general context of Lorentz sequence spaces. Also included are an axiomatic approach to s-numbers, a discussion of non-compact maps, and the Schmidt decomposition theory for compact linear operators in Hilbert spaces.Less
The geometric quantities entropy numbers, approximation numbers and n-widths are defined for compact linear maps, and connections with the analytic entities eigenvalues and essential spectra discussed. The celebrated inequality of Weyl between the approximation numbers and eigenvalues is established in the general context of Lorentz sequence spaces. Also included are an axiomatic approach to s-numbers, a discussion of non-compact maps, and the Schmidt decomposition theory for compact linear operators in Hilbert spaces.
