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LEAST SQUARES INVERSE EIGENVALUE PROBLEMS

Moody T. Chu and Gene H. Golub

in Inverse Eigenvalue Problems: Theory, Algorithms, and Applications

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 ... More

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

SPECTRALLY CONSTRAINED APPROXIMATION

Moody T. Chu and Gene H. Golub

in Inverse Eigenvalue Problems: Theory, Algorithms, and Applications

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 ... More

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

Classic Food Web Theory

Kevin S. McCann

in Food Webs (MPB-50)

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 ... More

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

Solving Nonlinear Rational Expectations Models by Eigenvalue–Eigenvector Decompositions

Alfonso Novales, Emilio Domínguez, Javier J. Pérez, and Jesús Ruiz

in Computational Methods for the Study of Dynamic Economies

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 ... More

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

Matrix Mechanics

Gary E. Bowman

in Essential Quantum Mechanics

This chapter introduces the matrix-mechanics formulation of quantum mechanics, emphasizing both calculational techniques and conceptual understanding. Parallels between matrix mechanics and ordinary ... More

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

Nonconforming finite element methods

Klaus Böhmer

in Numerical Methods for Nonlinear Elliptic Differential Equations: A Synopsis

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 ... More

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

Critique of the Standard Interpretation

Bas C. van Fraassen

in Quantum Mechanics: An Empiricist View

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 ... More

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

Modal Interpretation of Quantum Mechanics 1

Bas C. van Fraassen

in Quantum Mechanics: An Empiricist View

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 ... More

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

Small Vibrations About Equilibrium

Oliver Johns

in Analytical Mechanics for Relativity and Quantum Mechanics

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 ... More

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

The Bare Theory and Determinate Experience

Jeffrey Alan Barrett

in The Quantum Mechanics of Minds and Worlds

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 ... More

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

Abstract Hilbert Space

John von Neumann

Nicholas A. Wheeler (ed.)

in Mathematical Foundations of Quantum Mechanics: New Edition

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, ... More

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

Linear Elliptic Equations

Rüdiger Verfürth

in A Posteriori Error Estimation Techniques for Finite Element Methods

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 ... More

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

Nonlinear Elliptic Equations

Rüdiger Verfürth

in A Posteriori Error Estimation Techniques for Finite Element Methods

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 ... More

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

Well Posedness

G. F. Roach, I. G. Stratis, and A. N. Yannacopoulos

in Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics

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. ... More

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

The Anisotropic Heisenberg Quantum Spin Chain

Hans-Peter Eckle

in Models of Quantum Matter: A First Course on Integrability and the Bethe Ansatz

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 ... More

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

Electronic Systems

Hans-Peter Eckle

in Models of Quantum Matter: A First Course on Integrability and the Bethe Ansatz

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 ... More

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

Transformations, matrices and operators

Chun Wa Wong

in Introduction to Mathematical Physics: Methods & Concepts

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 ... More

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

Structured Populations and Games over Generations

John M. McNamara and Olof Leimar

in Game Theory in Biology: concepts and frontiers

The actions and state of an individual in one generation can affect the state of offspring in the next generation, and hence the ability of these offspring to leave offspring themselves. This chapter ... More

The actions and state of an individual in one generation can affect the state of offspring in the next generation, and hence the ability of these offspring to leave offspring themselves. This chapter deals with games in this multigenerational setting. Projection matrices are used to keep track of the state and number of descendants in successive years and generations. Invasion fitness is then defined in terms of the leading eigenvalue of the projection matrix. Simple examples illustrate these concepts and show how to apply them. Reproductive value is a function that measures how the ability to leave descendants in future generations depends on the current state. The Nash equilibrium condition is reformulated in terms of reproductive value maximization. This new criterion is used to justify the fitness proxy used in the analysis of sex allocation earlier in the book. The analysis is extended to the case where offspring may inherit aspects of their mother’s quality, with a focus on the question of whether high-quality mothers should produce sons or daughters. As a second application of reproductive value maximization, the co-evolution of female preference for a particular male trait and the trait itself is analysed, with the evolution of tail length in the widowbird as an illustrative application. Mean lifetime reproductive success is used as a fitness proxy in much of the book. Its use is finally justified in this chapter, where the fitness proxy is used to analyse the evolutionarily stable age of first reproduction in a population.Less

Vectors and Functions and Operators

Jochen Autschbach

in Quantum Theory for Chemical Applications: From Basic Concepts to Advanced Topics

This chapter introduces – briefly – vectors and functions and the similarities between them, some basic linear algebra concepts, operators (including the del and Laplace operators), eigenvalues and ... More

This chapter introduces – briefly – vectors and functions and the similarities between them, some basic linear algebra concepts, operators (including the del and Laplace operators), eigenvalues and eigenvectors &eigenfunctions, the scalar (dot) and vector (cross) product between two vectors, the scalar product between two functions, the concepts of normalization, orthogonality, and orthonormality. The concept of an operator is first introduced by considering the rotation and stretching or compression of a vector. It is then generalized to a mathematical prescription that changes a function into another function.Less