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Spectral Measures and Dynamics: Typical Behaviors (Latin American Mathematics Series)
by Moacir Aloisio Silas L. Carvalho César R. de OliveiraThis book convenes and deepens generic results about spectral measures, many of them available so far in scattered literature. It starts with classic topics such as Wiener lemma, Strichartz inequality, and the basics of fractal dimensions of measures, progressing to more advanced material, some of them developed by the own authors.A fundamental concept to the mathematical theory of quantum mechanics, the spectral measure relates to the components of the quantum state concerning the energy levels of the Hamiltonian operator and, on the other hand, to the dynamics of such state. However, these correspondences are not immediate, with many nuances and subtleties discovered in recent years.A valuable example of such subtleties is found in the so-called “Wonderland theorem” first published by B. Simon in 1995. It shows that, for some metric space of self-adjoint operators, the set of operators whose spectral measures are singular continuous is a generic set (which, for some, is exotic). Recent works have revealed that, on top of singular continuity, there are other generic properties of spectral measures. These properties are usually associated with a number of different notions of generalized dimensions, upper and lower dimensions, with dynamical implications in quantum mechanics, ergodicity of dynamical systems, and evolution semigroups. All this opens ways to new and instigating avenues of research.Graduate students with a specific interest in the spectral properties of spectral measure are the primary target audience for this work, while researchers benefit from a selection of important results, many of them presented in the book format for the first time.
Spectral Methods for Non-Standard Eigenvalue Problems
by Călin-Ioan GheorghiuThis book focuses on the constructive and practical aspects of spectral methods. It rigorously examines the most important qualities as well as drawbacks of spectral methods in the context of numerical methods devoted to solve non-standard eigenvalue problems. In addition, the book also considers some nonlinear singularly perturbed boundary value problems along with eigenproblems obtained by their linearization around constant solutions. The book is mathematical, poising problems in their proper function spaces, but its emphasis is on algorithms and practical difficulties. The range of applications is quite large. High order eigenvalue problems are frequently beset with numerical ill conditioning problems. The book describes a wide variety of successful modifications to standard algorithms that greatly mitigate these problems. In addition, the book makes heavy use of the concept of pseudospectrum, which is highly relevant to understanding when disaster is imminent in solving eigenvalue problems. It also envisions two classes of applications, the stability of some elastic structures and the hydrodynamic stability of some parallel shear flows. This book is an ideal reference text for professionals (researchers) in applied mathematics, computational physics and engineering. It will be very useful to numerically sophisticated engineers, physicists and chemists. The book can also be used as a textbook in review courses such as numerical analysis, computational methods in various engineering branches or physics and computational methods in analysis.
Spectral Methods in Chemistry and Physics
by Bernard ShizgalThis book is a pedagogical presentation of the application of spectral and pseudospectral methods to kinetic theory and quantum mechanics. There are additional applications to astrophysics, engineering, biology and many other fields. The main objective of this book is to provide the basic concepts to enable the use of spectral and pseudospectral methods to solve problems in diverse fields of interest and to a wide audience. While spectral methods are generally based on Fourier Series or Chebychev polynomials, non-classical polynomials and associated quadratures are used for many of the applications presented in the book. Fourier series methods are summarized with a discussion of the resolution of the Gibbs phenomenon. Classical and non-classical quadratures are used for the evaluation of integrals in reaction dynamics including nuclear fusion, radial integrals in density functional theory, in elastic scattering theory and other applications. The subject matter includes the calculation of transport coefficients in gases and other gas dynamical problems based on spectral and pseudospectral solutions of the Boltzmann equation. Radiative transfer in astrophysics and atmospheric science, and applications to space physics are discussed. The relaxation of initial non-equilibrium distributions to equilibrium for several different systems is studied with the Boltzmann and Fokker-Planck equations. The eigenvalue spectra of the linear operators in the Boltzmann, Fokker-Planck and Schrödinger equations are studied with spectral and pseudospectral methods based on non-classical orthogonal polynomials. The numerical methods referred to as the Discrete Ordinate Method, Differential Quadrature, the Quadrature Discretization Method, the Discrete Variable Representation, the Lagrange Mesh Method, and others are discussed and compared. MATLAB codes are provided for most of the numerical results reported in the book - see Link under 'Additional Information' on the the right-hand column.
Spectral Methods in Geodesy and Geophysics
by Christopher JekeliThe text develops the principal aspects of applied Fourier analysis and methodology with the main goal to inculcate a different way of perceiving global and regional geodetic and geophysical data, namely from the perspective of the frequency, or spectral, domain rather than the spatial domain. The word "methods" in the title is meant to convey that the transformation of a geophysical signal into the spectral domain can be applied for purposes of analysis as well as rapid computation. The text is written for graduate students; however, Chapters 1 through 4 and parts of 5 can also benefit undergraduates who have a solid and fluent knowledge of integral and differential calculus, have some statistical background, and are not uncomfortable with complex numbers. Concepts are developed by starting from the one-dimensional domain and working up to the spherical domain, which is part of every chapter. Many concepts are illustrated graphically with actual geophysical data primarily from signals of gravity, magnetism, and topography.
Spectral Methods Using Multivariate Polynomials On The Unit Ball (Chapman & Hall/CRC Monographs and Research Notes in Mathematics)
by Kendall Atkinson David Chien Olaf HansenSpectral Methods Using Multivariate Polynomials on the Unit Ball is a research level text on a numerical method for the solution of partial differential equations. The authors introduce, illustrate with examples, and analyze 'spectral methods' that are based on multivariate polynomial approximations. The method presented is an alternative to finite element and difference methods for regions that are diffeomorphic to the unit disk, in two dimensions, and the unit ball, in three dimensions. The speed of convergence of spectral methods is usually much higher than that of finite element or finite difference methods. Features Introduces the use of multivariate polynomials for the construction and analysis of spectral methods for linear and nonlinear boundary value problems Suitable for researchers and students in numerical analysis of PDEs, along with anyone interested in applying this method to a particular physical problem One of the few texts to address this area using multivariate orthogonal polynomials, rather than tensor products of univariate polynomials.
Spectral Spaces (New Mathematical Monographs #35)
by Max Dickmann Niels Schwartz Marcus TresslSpectral spaces are a class of topological spaces. They are a tool linking algebraic structures, in a very wide sense, with geometry. They were invented to give a functional representation of Boolean algebras and distributive lattices and subsequently gained great prominence as a consequence of Grothendieck's invention of schemes. There are more than 1,000 research articles about spectral spaces, but this is the first monograph. It provides an introduction to the subject and is a unified treatment of results scattered across the literature, filling in gaps and showing the connections between different results. The book includes new research going beyond the existing literature, answering questions that naturally arise from this comprehensive approach. The authors serve graduates by starting gently with the basics. For experts, they lead them to the frontiers of current research, making this book a valuable reference source.
Spectral Theory: Basic Concepts and Applications (Graduate Texts in Mathematics #284)
by David BorthwickThis textbook offers a concise introduction to spectral theory, designed for newcomers to functional analysis. Curating the content carefully, the author builds to a proof of the spectral theorem in the early part of the book. Subsequent chapters illustrate a variety of application areas, exploring key examples in detail. Readers looking to delve further into specialized topics will find ample references to classic and recent literature. Beginning with a brief introduction to functional analysis, the text focuses on unbounded operators and separable Hilbert spaces as the essential tools needed for the subsequent theory. A thorough discussion of the concepts of spectrum and resolvent follows, leading to a complete proof of the spectral theorem for unbounded self-adjoint operators. Applications of spectral theory to differential operators comprise the remaining four chapters. These chapters introduce the Dirichlet Laplacian operator, Schrödinger operators, operators on graphs, and the spectral theory of Riemannian manifolds. Spectral Theory offers a uniquely accessible introduction to ideas that invite further study in any number of different directions. A background in real and complex analysis is assumed; the author presents the requisite tools from functional analysis within the text. This introductory treatment would suit a functional analysis course intended as a pathway to linear PDE theory. Independent later chapters allow for flexibility in selecting applications to suit specific interests within a one-semester course.
Spectral Theory and Applications of Linear Operators and Block Operator Matrices
by Aref JeribiExamining recent mathematical developments in the study of Fredholm operators, spectral theory and block operator matrices, with a rigorous treatment of classical Riesz theory of polynomially-compact operators, this volume covers both abstract and applied developments in the study of spectral theory. These topics are intimately related to the stability of underlying physical systems and play a crucial role in many branches of mathematics as well as numerous interdisciplinary applications. By studying classical Riesz theory of polynomially compact operators in order to establish the existence results of the second kind operator equations, this volume will assist the reader working to describe the spectrum, multiplicities and localization of the eigenvalues of polynomially-compact operators.
Spectral Theory and its Applications
by Bernard HelfferBernard Helffer's graduate-level introduction to the basic tools in spectral analysis is illustrated by numerous examples from the Schrödinger operator theory and various branches of physics: statistical mechanics, superconductivity, fluid mechanics and kinetic theory. The later chapters also introduce non self-adjoint operator theory with an emphasis on the role of the pseudospectra. The author's focus on applications, along with exercises and examples, enables readers to connect theory with practice so that they develop a good understanding of how the abstract spectral theory can be applied. The final chapter provides various problems that have been the subject of active research in recent years and will challenge the reader's understanding of the material covered.
Spectral Theory and Mathematical Physics: STMP 2018, Santiago, Chile (Latin American Mathematics Series #254)
by Pablo Miranda Nicolas Popoff Georgi RaikovThis proceedings volume contains peer-reviewed, selected papers and surveys presented at the conference Spectral Theory and Mathematical Physics (STMP) 2018 which was held in Santiago, Chile, at the Pontifical Catholic University of Chile in December 2018. The original works gathered in this volume reveal the state of the art in the area and reflect the intense cooperation between young researchers in spectral theoryand mathematical physics and established specialists in this field. The list of topics covered includes: eigenvalues and resonances for quantum Hamiltonians; spectral shift function and quantum scattering; spectral properties of random operators; magnetic quantum Hamiltonians; microlocal analysis and its applications in mathematical physics. This volume can be of interest both to senior researchers and graduate students pursuing new research topics in Mathematical Physics.
Spectral Theory and Nonlinear Functional Analysis (Chapman & Hall/CRC Research Notes in Mathematics Series)
by null Julian Lopez-GomezThis Research Note addresses several pivotal problems in spectral theory and nonlinear functional analysis in connection with the analysis of the structure set of zeroes of a general class of nonlinear operators. Appealing to a broad audience, it contains many important contributions to linear algebra, linear functional analysis, nonlinear functional analysis, and topology. The author gives several applications of the abstract theory to reaction diffusion equations and systems. The results presented cover a thirty-year period and cut across a variety of mathematical fields.
Spectral Theory and Quantum Mechanics (Universitext)
by Mathieu LewinThis textbook presents the spectral theory of self-adjoint operators on Hilbert space and its applications in quantum mechanics. Based on a course taught by the author in Paris, the book not only covers the mathematical theory but also provides its physical interpretation, offering an accessible introduction to quantum mechanics for students with a background in mathematics. The presentation incorporates numerous physical examples to illustrate the abstract theory. The final two chapters present recent findings on Schrödinger’s equation for systems of particles. While primarily designed for graduate courses, the book can also serve as a valuable introduction to the subject for more advanced readers. It requires no prior knowledge of physics, assuming only a graduate-level understanding of mathematical analysis from the reader.
Spectral Theory and Quantum Mechanics
by Valter MorettiThis book pursues the accurate study of the mathematical foundations of Quantum Theories. It may be considered an introductory text on linear functional analysis with a focus on Hilbert spaces. Specific attention is given to spectral theory features that are relevant in physics. Having left the physical phenomenology in the background, it is the formal and logical aspects of the theory that are privileged. Another not lesser purpose is to collect in one place a number of useful rigorous statements on the mathematical structure of Quantum Mechanics, including some elementary, yet fundamental, results on the Algebraic Formulation of Quantum Theories. In the attempt to reach out to Master's or PhD students, both in physics and mathematics, the material is designed to be self-contained: it includes a summary of point-set topology and abstract measure theory, together with an appendix on differential geometry. The book should benefit established researchers to organise and present the profusion of advanced material disseminated in the literature. Most chapters are accompanied by exercises, many of which are solved explicitly.
Spectral Theory and Quantum Mechanics
by Valter MorettiThis book pursues the accurate study of the mathematical foundations of Quantum Theories. It may be considered an introductory text on linear functional analysis with a focus on Hilbert spaces. Specific attention is given to spectral theory features that are relevant in physics. Having left the physical phenomenology in the background, it is the formal and logical aspects of the theory that are privileged. Another not lesser purpose is to collect in one place a number of useful rigorous statements on the mathematical structure of Quantum Mechanics, including some elementary, yet fundamental, results on the Algebraic Formulation of Quantum Theories. In the attempt to reach out to Master's or PhD students, both in physics and mathematics, the material is designed to be self-contained: it includes a summary of point-set topology and abstract measure theory, together with an appendix on differential geometry. The book should benefit established researchers to organise and present the profusion of advanced material disseminated in the literature. Most chapters are accompanied by exercises, many of which are solved explicitly.
Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications (Monographs and Surveys in Pure and Applied Mathematics)
by null Janusz Mierczynski null Wenxian ShenProviding a basic tool for studying nonlinear problems, Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications focuses on the principal spectral theory for general time-dependent and random parabolic equations and systems. The text contains many new results and considers existing results from a fresh perspective.
A Spectral Theory for Simply Periodic Solutions of the Sinh-Gordon Equation (Lecture Notes in Mathematics #2229)
by Sebastian KleinThis book develops a spectral theory for the integrable system of 2-dimensional, simply periodic, complex-valued solutions u of the sinh-Gordon equation. Such solutions (if real-valued) correspond to certain constant mean curvature surfaces in Euclidean 3-space. Spectral data for such solutions are defined (following ideas of Hitchin and Bobenko) and the space of spectral data is described by an asymptotic characterization. Using methods of asymptotic estimates, the inverse problem for the spectral data is solved along a line, i.e. the solution u is reconstructed on a line from the spectral data. Finally, a Jacobi variety and Abel map for the spectral curve are constructed and used to describe the change of the spectral data under translation of the solution u. The book's primary audience will be research mathematicians interested in the theory of infinite-dimensional integrable systems, or in the geometry of constant mean curvature surfaces.
Spectral Theory of Infinite-Area Hyperbolic Surfaces
by David BorthwickThis text introduces geometric spectral theory in the context of infinite-area Riemann surfaces, providing a comprehensive account of the most recent developments in the field. For the second edition the context has been extended to general surfaces with hyperbolic ends, which provides a natural setting for development of the spectral theory while still keeping technical difficulties to a minimum. All of the material from the first edition is included and updated, and new sections have been added. Topics covered include an introduction to the geometry of hyperbolic surfaces, analysis of the resolvent of the Laplacian, scattering theory, resonances and scattering poles, the Selberg zeta function, the Poisson formula, distribution of resonances, the inverse scattering problem, Patterson-Sullivan theory, and the dynamical approach to the zeta function. The new sections cover the latest developments in the field, including the spectral gap, resonance asymptotics near the critical line, and sharp geometric constants for resonance bounds. A new chapter introduces recently developed techniques for resonance calculation that illuminate the existing results and conjectures on resonance distribution. The spectral theory of hyperbolic surfaces is a point of intersection for a great variety of areas, including quantum physics, discrete groups, differential geometry, number theory, complex analysis, and ergodic theory. This book will serve as a valuable resource for graduate students and researchers from these and other related fields. Review of the first edition: "The exposition is very clear and thorough, and essentially self-contained; the proofs are detailed. . . The book gathers together some material which is not always easily available in the literature. . . To conclude, the book is certainly at a level accessible to graduate students and researchers from a rather large range of fields. Clearly, the reader. . . would certainly benefit greatly from it. " (Colin Guillarmou, Mathematical Reviews, Issue 2008 h)
Spectral Theory of Localized Resonances and Applications
by Youjun Deng Hongyu LiuThis book is devoted to the spectral theory of localized resonances including surface plasmon/polariton resonances, atypical resonances, anomalous localized resonances and interior transmission resonances. Those resonance phenomena arise in different physical contexts, but share similar features. They form the fundamental basis for many cutting-edge technologies and applications including invisibility cloaking and super-resolution imaging. The book presents a systematic and comprehensive treatment on these resonance phenomena and the associated applications in a unified manner from a mathematical and spectral perspective, covering acoustic, electromagnetic and elastic wave scattering.The book can serve as a handy reference book for researchers in this field and it can also serve as a textbook or an inspiring source for postgraduate students who are interested in entering this field.
Spectral Theory of Nonautonomous Dynamical Systems and Applications
by Thai Son DoanThe main challenge in the study of nonautonomous phenomena is to understand the very complicated dynamical behaviour both as a scientific and mathematical problem. The theory of nonautonomous dynamical systems has experienced a renewed and steadily growing interest in the last twenty years, stimulated also by synergetic effects of disciplines which have developed relatively independent for some time such as topological skew product, random dynamical systems, finite-time dynamics and control systems. The book provides new insights in many aspects of the qualitative theory of nonautonomous dynamical systems including the spectral theory, the linearization theory, the bifurcation theory. The book first introduces several important spectral theorem for nonautonomous differential equations including the Lyapunov spectrum, Sacker-Sell spectrum and finite-time spectrum. The author also establishes the smooth linearization and partial linearization for nonautonomous differential equations in application part. Then the second part recalls the multiplicative ergodic theorem for random dynamical systems and discusses several explicit formulas in computing the Lyapunov spectrum for random dynamical systems generated by linear stochastic differential equations and random difference equations with random delay. In the end, the Pitchfork bifurcation and Hopf bifurcation with additive noise are investigated in terms of change of the sign of Lyapunov exponents and loss of topological equivalence. This book might be appealing to researchers and graduate students in the field of dynamical systems, stochastic differential equations, ergodic theory.
A Spectral Theory Of Noncommuting Operators
by Rongwei YangThe main goal of this book is to describe various aspects of the theory of joint spectra for matrices and linear operators. It is suitable for a graduate-level topic course in spectral theory and/or representation theory. The first three chapters can also be adopted for an advanced course in linear algebra. Centered around the concept of projective spectrum, the book presents a coherent treatment of fundamental elements from a wide range of mathematical disciplines, such as complex analysis, complex dynamics, differential geometry, functional analysis, group theory, and Lie algebras. Researchers and students, particularly those who aspire to gain a bigger picture of mathematics, will find this book both informative and resourceful.
Spectrum 8th Grade Math Workbook: Focused Practice for Math Mastery (Spectrum Series)
by Spectrum8th Grade Math Workbook for kids ages 13-14 <p> Support your child’s educational journey with the Spectrum grade 8 math workbook that teaches essential math skills to eighth graders. <p><p> Spectrum’s 8th grade math workbook is a great way for eighth graders to learn essential math skills such as learning Pythagorean Theorem, geometry, rational and irrational numbers, and more through a variety of problem-solving activities that are both fun AND educational! <p><p> Why You’ll Love This Math Book <p> Engaging and educational math for 8th grade students. “Using and rewriting exponents”, “solving word problems”, and “linear equations” are a few of the fun activities that incorporate math in everyday settings to help inspire learning. <p> Testing progress along the way. Pretests, posttests, a mid-test, final test, and an answer key are included in the 8th grade math workbook to help track your child’s progress along the way before moving on to new and exciting math lessons. <p><p> About Spectrum <p> For more than 20 years, Spectrum has provided solutions for parents who want to help their children get ahead, and for teachers who want their students to meet and exceed set learning goals—providing workbooks that are a great resource for both homeschooling and classroom curriculum.
Spectrum MathWorkbook, Grade 3
by Spectrum<p>The Spectrum Math Workbook for third grade features 160 pages of focused instruction and progressive practice to help students stay ahead in math and enhance problem-solving skills. <p>This standards-based workbook provides systematic and thought-provoking practice designed to increase in complexity for the following content areas: adding and subtracting to four-digit numbers, multiplying and dividing, fractions, perimeter and area, and graphs and line plots. With easy-to-follow instructions, tests, and an included answer key, parents and teachers can accurately monitor students’ progress. <p>The best-selling Spectrum series is a favorite of parents and teachers because it’s carefully designed to be both effective and engaging––the perfect building blocks for a lifetime of learning.</p>
The Spectrum of Hyperbolic Surfaces
by Nicolas BergeronThistext is an introduction to the spectral theory of the Laplacian oncompact or finite area hyperbolic surfaces. For some of thesesurfaces, called "arithmetic hyperbolic surfaces", theeigenfunctions are of arithmetic nature, and one may use analytictools as well as powerful methods in number theory to study them. Afteran introduction to the hyperbolic geometry of surfaces, with aspecial emphasis on those of arithmetic type, and then anintroduction to spectral analytic methods on the Laplace operator onthese surfaces, the author develops the analogy between geometry(closed geodesics) and arithmetic (prime numbers) in proving theSelberg trace formula. Along with important number theoreticapplications, the author exhibits applications of these tools to thespectral statistics of the Laplacian and the quantum uniqueergodicity property. The latter refers to the arithmetic quantumunique ergodicity theorem, recently proved by Elon Lindenstrauss. Thefruit of several graduate level courses at Orsay and Jussieu, The Spectrum of Hyperbolic Surfaces allows the reader to review an array of classical results andthen to be led towards very active areas in modern mathematics.
Speculation, Trading, and Bubbles (Kenneth J. Arrow Lecture Series)
by José A. ScheinkmanAs long as there have been financial markets, there have been bubbles—those moments in which asset prices inflate far beyond their intrinsic value, often with ruinous results. Yet economists are slow to agree on the underlying forces behind these events. In this book José A. Scheinkman offers new insight into the mystery of bubbles. Noting some general characteristics of bubbles—such as the rise in trading volume and the coincidence between increases in supply and bubble implosions—Scheinkman offers a model, based on differences in beliefs among investors, that explains these observations. Other top economists also offer their own thoughts on the issue: Sanford J. Grossman and Patrick Bolton expand on Scheinkman's discussion by looking at factors that contribute to bubbles—such as excessive leverage, overconfidence, mania, and panic in speculative markets—and Kenneth J. Arrow and Joseph E. Stiglitz contextualize Scheinkman's findings.
Speculation, Trading, and Bubbles
by Joseph E. Stiglitz Kenneth J. Arrow Sanford J. Grossman José A. Scheinkman Patrick BoltonThe history of financial markets is full of moments in which asset prices inflate far beyond their intrinsic value. These events are commonly called bubbles, and in this book, José A. Scheinkman and other top economists offer new explanations for this phenomenon.Scheinkman discusses some stylized facts concerning bubbles, such as high trading volume and the coincidence between bubbles' implosion and increases in supply, and he develops a model for bubbles based on differences in beliefs among investors that explains these observations. Sandy Grossman and Patrick Bolton offer commentaries on Scheinkman's work, investigating factors that contribute to bubbles, such as excessive leverage, overconfidence, mania, and panic in speculative markets. Kenneth J. Arrow and Joseph E. Stiglitz add introductory material contextualizing Scheinkman's findings.