Search Results

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

Linear Algebra and the Dirac Notation

Phillip Kaye, Raymond Laflamme, and Michele Mosca

in An Introduction to Quantum Computing

We assume the reader has a strong background in elementary linear algebra. In this section we familiarize the reader with the algebraic notation used in quantum ... More

We assume the reader has a strong background in elementary linear algebra. In this section we familiarize the reader with the algebraic notation used in quantum mechanics, remind the reader of some basic facts about complex vector spaces, and introduce some notions that might not have been covered in an elementary linear algebra course. The linear algebra notation used in quantum computing will likely be familiar to the student of physics, but may be alien to a student of mathematics or computer science. It is the Dirac notation, which was invented by Paul Dirac and which is used often in quantum mechanics. In mathematics and physics textbooks, vectors are often distinguished from scalars by writing an arrow over the identifying symbol: e.g a⃗. Sometimes boldface is used for this purpose: e.g. a. In the Dirac notation, the symbol identifying a vector is written inside a ‘ket’, and looks like |a⟩. We denote the dual vector for a (defined later) with a ‘bra’, written as ⟨a|. Then inner products will be written as ‘bra-kets’ (e.g. ⟨a|b⟩). We now carefully review the definitions of the main algebraic objects of interest, using the Dirac notation. The vector spaces we consider will be over the complex numbers, and are finite-dimensional, which significantly simplifies the mathematics we need. Such vector spaces are members of a class of vector spaces called Hilbert spaces. Nothing substantial is gained at this point by defining rigorously what a Hilbert space is, but virtually all the quantum computing literature refers to a finite-dimensional complex vector space by the name ‘Hilbert space’, and so we will follow this convention. We will use H to denote such a space. Since H is finite-dimensional, we can choose a basis and alternatively represent vectors (kets) in this basis as finite column vectors, and represent operators with finite matrices. As you see in Section 3, the Hilbert spaces of interest for quantum computing will typically have dimension 2n, for some positive integer n. This is because, as with classical information, we will construct larger state spaces by concatenating a string of smaller systems, usually of size two. Less

Introductory Quantum Algorithms

Phillip Kaye, Raymond Laflamme, and Michele Mosca

in An Introduction to Quantum Computing

In this chapter we will describe some of the early quantum algorithms. These algorithms are simple and illustrate the main ingredients behind the more useful and ... More

In this chapter we will describe some of the early quantum algorithms. These algorithms are simple and illustrate the main ingredients behind the more useful and powerful quantum algorithms we describe in the subsequent chapters. Since quantum algorithms share some features with classical probabilistic algorithms, we will start with a comparison of the two algorithmic paradigms. Classical probabilistic algorithms were introduced in Chapter 1. In this section we will see how quantum computation can be viewed as a generalization of probabilistic computation. We begin by considering a simple probabilistic computation. Figure 6.1 illustrates the first two steps of such a computation on a register that can be in one of the four states, labelled by the integers 0, 1, 2, and 3. Initially the register is in the state 0. After the first step of the computation, the register is in the state j with probability p0,j . For example, the probability that the computation is in state 2 after the first step is p0,2. In the second step of the computation, the register goes from state j to state k with probability qj,k. For example, in the second step the computation proceeds from state 2 to state 3 with probability q2,3. Suppose we want to find the total probability that the computation ends up in state 3 after the second step. This is calculated by first determining the probability associated with each computation ‘path’ that could end up at the state 3, and then by adding the probabilities for all such paths. There are four computation paths that can leave the computation in state 3 after the first step. The computation can proceed from state 0 to state j and then from state j to state 3, for any of the four j ∊ {0, 1, 2, 3}. The probability associated with any one of these paths is obtained by multiplying the probability p0,j of the transition from state 0 to state j, with the probability qj,3 of the transition from state j to state 3. Less

Other Applications of Neural Networks to Quantum Mechanical Problems

Lionel Raff, Ranga Komanduri, Martin Hagan, and Satish Bukkapatnam

in Neural Networks in Chemical Reaction Dynamics

The usual method for solving the vibrational Schrödinger equation to obtain molecular vibrational spectra and the associated wave functions generally involves the ... More

The usual method for solving the vibrational Schrödinger equation to obtain molecular vibrational spectra and the associated wave functions generally involves the expansion of the vibrational wave function, ψk(y), in terms of a linear combination of a set of basis functions. Less

Demographic Models and Branching Processes

C. Y. Cyrus Chu

in Title Pages

All models describing the dynamic pattern of human population have two common features. First, the human population is usually divided into several types, ... More

All models describing the dynamic pattern of human population have two common features. First, the human population is usually divided into several types, and second, each type has a type-specific stochastic reproduction rate. The traditional literature of demography has been dominated by the age-specific models of Lotka (1939) and Leslie (1945,1948), where the type refers to the age of an individual and the type-specific reproduction rates refer to the age-specific vital rates in a life table, It has been shown that, mathematically, these age-specific models can be analyzed in a more general framework, namely, the multitype branching process. Most demography researchers, however, do not bother to pursue properties of the general branching process. They prefer to follow Lotka’s (1939) age-specific renewal equation approach in proceeding with their analysis because that renewal equation is technically convenient, whereas the steady-state and dynamic properties of a general branching process are usually much more difficult to derive. Although the analytical convenience of the age-specific models has facilitated the research on age-related topics, it also tends to obscure the fact that the age-specific model is merely a special kind of branching process. When female fertility becomes a decision variable of the family and the fertility-related family decision problems expand, these age-specific models are often unworkable. Despite the difficulties inherent in applying the traditional age-specific models to these decision dimensions, researchers still hesitate to go back to the general, but more difficult, branching process for solutions. This is perhaps why, as we mentioned in chapter 1, the demand-side theory of demography has not made much progress in describing the macro aggregate pattern of the population. In this chapter, I separate the discussion into the age-specific branching process and general branching processes. I show that the steady states and ergodic properties of these models can both be established under some regularity conditions. Although the material in this chapter is mostly a reorganization of previously established mathematical results, I believe that my summary is systematic and will be helpful to most readers. All the results summarized will be used in later chapters, but aspects of branching processes that are irrelevant to our purposes will not be discussed. Less

Income-Specific Population Models: Steady States and Comparative Dynamics

C. Y. Cyrus Chu

in Population Dynamics: A New Economic Approach

I mentioned in chapter 1 that the standard new household economics model of fertility, derived and modified by Becker (1960), DeTray (1973), Willis (1973), ... More

I mentioned in chapter 1 that the standard new household economics model of fertility, derived and modified by Becker (1960), DeTray (1973), Willis (1973), and later followers, emphasized the parental choices and tradeoffs between the quantity and the quality of their children. As Becker (1960) pointed out, one motivation for the new household economics approach to fertility decisions is to construct a demand-side household-decision structure to replace Malthus’s out-of-date supply-side population theory. The fertility decision theory along these lines has been called by Schultz (1981, 1988) and Dasgupta (1995) the demand-side demography theory. One difference between the demand-side demography theory and the classical Malthusian theory is that the former approach emphasized the static decision of a micro agent, whereas the Malthusian theory described the macro dynamic pattern of the population. Thus, from a theoretical point of view, the development of the demand-side demography lacks a macro dynamic counterpart. In this chapter I shall establish a macro dynamic population theory based on a fairly general version of Becker’s and others’ static setup of fertility demand. Once we shift our focus to the household fertility decision, it is natural that the household economic variables that affect female fertility decisions, such as her wages, family income, or the opportunity cost of babysitting, will become important explanatory variables of aggregate demographic patterns. Given that the fluctuation of mortality is no longer significant in recent years and that human fertility decisions are largely affected by the above-mentioned household economic variables, then in order to explain the aggregate pattern of population movement, it is natural to classify people by these economic variables rather than by ages. This is another motivation for the derivation of a non-age-specific stable population theory. As we focus upon the macro dynamic implications of Becker’s micro static fertility decision model, it is convenient to ignore sex differences and suppress the age structure of a person by assuming that everyone lives two periods, young and old. This is very much the same as the one-sex Samuelsonian (1958) overlapping-generation model: individuals who remain in the parental household are called young; they become old when they form their own families. Less

Lineage Extinction and Inheritance Patterns

C. Y. Cyrus Chu

in Population Dynamics: A New Economic Approach

Perhaps the most basic biological instincts of all creatures are to survive and to produce offspring. In ancient times, poor hygienic environment and ... More

Perhaps the most basic biological instincts of all creatures are to survive and to produce offspring. In ancient times, poor hygienic environment and occasional widespread epidemics obviously gave people strong reasons to worry about the possible extinction of their own lineage. But to transform such a worry into a mathematical problem, it is helpful if the upper class of the society, which has the ability and the leisure to think about the problem on an abstract level, also feels the possibility of such an extinction. Indeed, this was the case in eighteenth-century western Europe. The development of the theory of branching processes in fact started with the calculation of the probability of family surname extinction. Mode (1971) argued that one of the reasons for the decay of family names was that “physical comfort and intellectual capacity were necessarily accompanied by a diminution in fertility.” This statement, that parents choose to have fewer children because they want increase their enjoyment of life, seems to be a more suitable characterization of the argument of Becker’s (1991) contemporary household economics. Others, such as Chu and Lee (1994), argued that it was the scourges of war and famine that were responsible for the major rises of mortality in ancient history. Whatever the cause of lineage extinction, as a large proportion of family surnames continued to die out, Francis Galton (1873), one of the founders of the theory of branching processes, presented his concern with lineage extinction on an abstract level: The problem of family surname extinction concerns part (i) of problem 4001; part (ii) is about the distribution of surnames (types), which was the focus of chapter 4 of this book. Problem 4001 did not attract much attention until Agner Krarup Erlang became interested in this problem because his mother’s surname, Krarup, was about to become extinct. Erlang arrived at the solution below. First, we remove the restriction in problem 4001 that a man can have at most five sons, and let pk be the probability of having k surviving male children. 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

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