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Special Functions and their Application
by Bipin Singh Koranga Sanjay Kumar Padaliya Vivek Kumar NautiyalSpecial functions are mathematical functions that have established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. This short text gives clear descriptions and explanations of the Gamma function, the Probability Integral and its related functions, Spherical Harmonics Theory, The Bessel function, Hermite polynomials and Laguerre polynomials. Each chapter finishes with a description of how the function is most commonly applied and a set of examples for the student to work through.
Special Functions in Fractional Calculus and Engineering (Mathematics and its Applications)
by H. M. Srivastava R. K. Pandey Harendra SinghSpecial functions play a very important role in solving various families of ordinary and partial differential equations as well as their fractional-order analogs, which model real-life situations. Owing to the non-local nature and memory effect, fractional calculus is capable of modeling many situations which arise in engineering. This book includes a collection of related topics associated with such equations and their relevance and significance in engineering. Special Functions in Fractional Calculus and Engineering highlights the significance and applicability of special functions in solving fractional-order differential equations with engineering applications. This book focuses on the non-local nature and memory effect of fractional calculus in modeling relevant to engineering science and covers a variety of important and useful methods using special functions for solving various types of fractional-order models relevant to engineering science. This book goes on to illustrate the applicability and usefulness of special functions by justifying their numerous and widespread occurrences in the solution of fractional-order differential, integral, and integrodifferential equations.This book holds a wide variety of interconnected fundamental and advanced topics with interdisciplinary applications that combine applied mathematics and engineering sciences, which are useful to graduate students, Ph.D. scholars, researchers, and educators interested in special functions, fractional calculus, mathematical modeling, and engineering.
Special Functions in Physics with MATLAB
by Wolfgang SchweizerThis handbook focuses on special functions in physics in the real and complex domain. It covers more than 170 different functions with additional numerical hints for efficient computation, which are useful to anyone who needs to program with other programming languages as well. The book comes with MATLAB-based programs for each of these functions and a detailed html-based documentation. Some of the explained functions are: Gamma and Beta functions; Legendre functions, which are linked to quantum mechanics and electrodynamics; Bessel functions; hypergeometric functions, which play an important role in mathematical physics; orthogonal polynomials, which are largely used in computational physics; and Riemann zeta functions, which play an important role, e.g., in quantum chaos or string theory. The book’s primary audience are scientists, professionals working in research areas of industries, and advanced students in physics, applied mathematics, and engineering.
Special Functions of Mathematical (Applied and Numerical Harmonic Analysis)
by Willi Freeden Martin GuttingSpecial functions enable us to formulate a scientific problem by reduction such that a new, more concrete problem can be attacked within a well-structured framework, usually in the context of differential equations. A good understanding of special functions provides the capacity to recognize the causality between the abstractness of the mathematical concept and both the impact on and cross-sectional importance to the scientific reality. The special functions to be discussed in this monograph vary greatly, depending on the measurement parameters examined (gravitation, electric and magnetic fields, deformation, climate observables, fluid flow, etc.) and on the respective field characteristic (potential field, diffusion field, wave field). The differential equation under consideration determines the type of special functions that are needed in the desired reduction process. Each chapter closes with exercises that reflect significant topics, mostly in computational applications. As a result, readers are not only directly confronted with the specific contents of each chapter, but also with additional knowledge on mathematical fields of research, where special functions are essential to application. All in all, the book is an equally valuable resource for education in geomathematics and the study of applied and harmonic analysis. Students who wish to continue with further studies should consult the literature given as supplements for each topic covered in the exercises.
Special Functions, Partial Differential Equations, and Harmonic Analysis
by Constantine Georgakis Alexander M. Stokolos Wilfredo UrbinaThis volume of papers presented at the conference in honor of Calixto P. Calderón by his friends, colleagues, and students is intended to make the mathematical community aware of his important scholarly and research contributions in contemporary Harmonic Analysis and Mathematical Models applied to Biology and Medicine, and to stimulate further research in the future in this area of pure and applied mathematics.
Special Integrals (University Texts in the Mathematical Sciences)
by Abhishek MishraTargeted to upper-undergraduate and graduate students of mathematics, this book discusses special integrals and their applications in finding certain series sums. It starts with the differentiation and the integration methods for summing a series that is applied to find the sum of various binomial and trigonometrical series. It also discusses methods to find the sum of series involving the variables having exponents in integral or fractional powers of 2. Complex variables are freely used to derive several theorems, which result in several special integrals and series sums. Bessel coefficients, Bessel functions, and their various generalizations are also discussed in the book. As a particular case of generalized Bessel functions, pseudo-exponential functions are defined, and their properties are studied in the book. Broadly divided into two parts—Part 1 and Part 2—the book has six chapters in Part 1, whereas Part 2 has six chapters on solutions to the problems in Part 1. To understand the topics in the book, the minimum prerequisites are the knowledge of calculus, complex analysis, and Fourier series.
Special Integrals of Gradshteyn and Ryzhik: the Proofs - Volume I (Chapman & Hall/CRC Monographs and Research Notes in Mathematics)
by Victor H. MollA Guide to the Evaluation of IntegralsSpecial Integrals of Gradshetyn and Ryzhik: The Proofs provides self-contained proofs of a variety of entries in the frequently used table of integrals by I.S. Gradshteyn and I.M. Ryzhik. The book gives the most elementary arguments possible and uses Mathematica to verify the formulas. Readers discover the beau
Special Integrals of Gradshteyn and Ryzhik: the Proofs - Volume II
by Victor H. MollA Guide to the Evaluation of IntegralsSpecial Integrals of Gradshetyn and Ryzhik: the Proofs provides self-contained proofs of a variety of entries in the frequently used table of integrals by I.S. Gradshteyn and I.M. Ryzhik. The book gives the most elementary arguments possible and uses Mathematica to verify the formulas. You will discover the bea
Special Matrices and Their Applications in Numerical Mathematics: Second Edition
by Miroslav FiedlerThis revised and corrected second edition of a classic book on special matrices provides researchers in numerical linear algebra and students of general computational mathematics with an essential reference.Author Miroslav Fiedler, a Professor at the Institute of Computer Science of the Academy of Sciences of the Czech Republic, Prague, begins with definitions of basic concepts of the theory of matrices and fundamental theorems. In subsequent chapters, he explores symmetric and Hermitian matrices, the mutual connections between graphs and matrices, and the theory of entrywise nonnegative matrices. After introducing M-matrices, or matrices of class K, Professor Fiedler discusses important properties of tensor products of matrices and compound matrices and describes the matricial representation of polynomials. He further defines band matrices and norms of vectors and matrices. The final five chapters treat selected numerical methods for solving problems from the field of linear algebra, using the concepts and results explained in the preceding chapters.
Special Needs and Drug Education (nasen spotlight)
by Richard IvesTaking drugs is complex and there are concerns about the best ways of addressing drugs issues in schools - particularly for pupils with special educational needs. Many teachers are worried about discussing drugs with their pupils. They fear that they know too little and that some of their pupils may know too much. They also worry that talking about drugs to naive children may raise their interest. Yet the government expects all pupils to receive drug education and requires all schools to have a drug policy. It has ambitious targets in reducing the use of drugs by young people. This book aims to help teachers of pupils with special educational needs to assess what their contribution should be and identify what the particular issues associated with their pupils are. It will help schools to: create or revise a drugs policy plan a program of study deliver drug education appropriate to their pupils deal with drug related incident.
Special Relativity
by Valerio FaraoniThis book offers an essential bridge between college-level introductions and advanced graduate-level books on special relativity. It begins at an elementary level, presenting and discussing the basic concepts normally covered in college-level works, including the Lorentz transformation. Subsequent chapters introduce the four-dimensional worldview implied by the Lorentz transformations, mixing time and space coordinates, before continuing on to the formalism of tensors, a topic usually avoided in lower-level courses. The book's second half addresses a number of essential points, including the concept of causality; the equivalence between mass and energy, including applications; relativistic optics; and measurements and matter in Minkowski spacetime. The closing chapters focus on the energy-momentum tensor of a continuous distribution of mass-energy and its covariant conservation; angular momentum; a discussion of the scalar field of perfect fluids and the Maxwell field; and general coordinates. Every chapter is supplemented by a section with numerous exercises, allowing readers to practice the theory. These exercises constitute an essential part of the textbook, and the solutions to approximately half of them are provided in the appendix.
Special Relativity for the Enthusiast
by Thomas StrohmThis textbook introduces special relativity with a focus on a profound understanding of the physics behind the theory. The main part of the book is targeted to undergraduates, for physics education, for undergraduate students in natural sciences in general, and even to interested laypersons. To serve these target groups, the book uses only basic mathematics and, in contrast to many other introductions to special relativity, the book is based on a pedagogical approach that relies on geometry and space-time diagrams to make the surprising predictions of the theory particularly clear. Special relativity is a geometric theory, and space-time diagrams are an efficient and easily understandable way to comprehend its implications. The textbook, however, is also suitable for advanced students and enthusiasts that already learned the basics of the special theory of relativity and want to know more. Special digression sections provide plenty of interesting material. Carefully selected problems with solutions and in-depth explanations for all key experiments help deepen the knowledge.
Special Relativity in General Frames: From Particles to Astrophysics
by Éric GourgoulhonSpecial relativity is the basis of many fields in modern physics: particle physics, quantum field theory, high-energy astrophysics, etc. This theory is presented here by adopting a four-dimensional point of view from the start. An outstanding feature of the book is that it doesn't restrict itself to inertial frames and to considering accelerated and rotating observers. It is thus possible to treat physical effects such as the Thomas precession or the Sagnac effect in a simple yet precise manner. In the final chapters, more advanced topics like tensorial fields in spacetime, exterior calculus and relativistic hydrodynamics are addressed. In the last, brief chapter the author gives a preview of gravity and shows where it becomes incompatible with Minkowsky spacetime. Well illustrated and enriched by many historical notes, this book also presents many applications of special relativity, ranging from particle physics (accelerators, particle collisions, quark-gluon plasma) to astrophysics (relativistic jets, active galactic nuclei), and including practical applications (Sagnac gyrometers, synchrotron radiation, GPS). In addition, the book provides some mathematical developments, such as the detailed analysis of the Lorentz group and its Lie algebra. The book is suitable for students in the third year of a physics degree or on a masters course, as well as researchers and any reader interested in relativity. Thanks to the geometric approach adopted, this book should also be beneficial for the study of general relativity. "A modern presentation of special relativity must put forward its essential structures, before illustrating them using concrete applications to specific dynamical problems. Such is the challenge (so successfully met!) of the beautiful book by Éric Gourgoulhon." (excerpt from the Foreword by Thibault Damour)
Special Relativity: An Introduction with 200 Problems and Solutions (Undergraduate Lecture Notes in Physics)
by Michael TsamparlisThis textbook develops Special Relativity in a systematic way and offers the unique feature of having more than 200 problems with detailed solutions to empower students to gain a real understanding of this core subject in physics. This new edition has been thoroughly updated and has new sections on relativistic fluids, relativistic kinematics and on four-acceleration. The problems and solution section has been significantly expanded and short history sections have been included throughout the book.The approach is structural in the sense that it develops Special Relativity in Minkowski space following the parallel steps as the development of Newtonian Physics in Euclidian space. A second characteristic of the book is that it discusses the mathematics of the theory independently of the physical principles, so that the reader will appreciate their role in the development of the physical theory.The book is intended to be used both as a textbook for an advanced undergraduate teaching course in Special Relativity but also as a reference book for the future. In that respect it is linked to an online repository with more than 200 problems, carefully classified according to subject area and solved in detail, providing an independent problem book on Special Relativity.
Special Relativity: For Inquiring Minds (Undergraduate Lecture Notes in Physics)
by Yury DeshkoThis textbook introduces the special theory of relativity at a level which is accessible to undergraduate students and even high school students with a strong foundation in algebra. The presentation emphasizes clean algebraic and geometrical methods, visualized with plenty of illustrations, resulting in a textbook that is modern and serious yet accessible. Replete with many solved exercises and copious spacetime diagrams, this book will help students develop relativistic intuition when encountering the subject for the first time. The emphasis on geometric methods, combined with the pedagogically appealing k-calculus approach, makes this book ideal for a self-contained course on special relativity or as supplementary reading for modern physics courses. It will also appeal to high schoolers with a strong math background who want to get ahead.
Special Sciences and the Unity of Science
by Shahid Rahman Olga Pombo John Symons Juan Manuel TorresScience is a dynamic process in which the assimilation of new phenomena, perspectives, and hypotheses into the scientific corpus takes place slowly. The apparent disunity of the sciences is the unavoidable consequence of this gradual integration process. Some thinkers label this dynamical circumstance a 'crisis'. However, a retrospective view of the practical results of the scientific enterprise and of science itself, grants us a clear view of the unity of the human knowledge seeking enterprise. This book provides many arguments, case studies and examples in favor of the unity of science. These contributions touch upon various scientific perspectives and disciplines such as: Physics, Computer Science, Biology, Neuroscience, Cognitive Psychology, and Economics.
Special Topics in Structural Dynamics & Experimental Techniques, Volume 5: Proceedings of the 38th IMAC, A Conference and Exposition on Structural Dynamics 2020 (Conference Proceedings of the Society for Experimental Mechanics Series)
by David S. EppSpecial Topics in Structural Dynamics & Experimental Techniques, Volume 5: Proceedings of the 38th MAC, A Conference and Exposition on Structural Dynamics, 2020, the fifth volume of eight from the Conference brings together contributions to this important area of research and engineering. The collection presents early findings and case studies on fundamental and applied aspects of Structural Dynamics, including papers on:Analytical MethodsEmerging Technologies for Structural DynamicsEngineering ExtremesExperimental TechniquesFinite Element TechniquesGeneral Topics
Special Topics in Structural Dynamics & Experimental Techniques, Volume 5: Proceedings of the 40th IMAC, A Conference and Exposition on Structural Dynamics 2022 (Conference Proceedings of the Society for Experimental Mechanics Series)
by Matt Allen Sheyda Davaria R. Benjamin DavisSpecial Topics in Structural Dynamics & Experimental Techniques, Volume 5: Proceedings of the 40th MAC, A Conference and Exposition on Structural Dynamics, 2022, the fifth volume of nine from the Conference brings together contributions to this important area of research and engineering. The collection presents early findings and case studies on fundamental and applied aspects of Structural Dynamics, including papers on:Analytical MethodsEmerging Technologies for Structural DynamicsEngineering ExtremesExperimental TechniquesFinite Element Techniques
Specialization of Quadratic and Symmetric Bilinear Forms
by Thomas Unger Manfred KnebuschThe specialization theory of quadratic and symmetric bilinear forms over fields and the subsequent generic splitting theory of quadratic forms were invented by the author in the mid-1970's. They came to fruition in the ensuing decades and have become an integral part of the geometric methods in quadratic form theory. This book comprehensively covers the specialization and generic splitting theories. These theories, originally developed for fields of characteristic different from 2, are explored here without this restriction. In addition to chapters on specialization theory, generic splitting theory and their applications, the book contains a final chapter containing research never before published on specialization with respect to quadratic places and will provide the reader with a glimpse towards the future.
Species Sensitivity Distributions in Ecotoxicology
by Glenn W. Suter II Leo Posthuma Theo P. TraasIn spite of the growing importance of Species Sensitivity Distribution models (SSDs) in ecological risk assessments, the conceptual basis, strengths, and weaknesses of using them have not been comprehensively reviewed. This book fills that need. Written by a panel of international experts, Species Sensitivity Distributions in Ecotoxicology reviews
Spectra and Normal Forms (SpringerBriefs in Mathematics)
by Claudia Valls Luís BarreiraThis book presents the reader with a streamlined exposition of the notions and results leading to the construction of normal forms and, ultimately, to the construction of smooth conjugacies for the perturbations of tempered exponential dichotomies. These are exponential dichotomies for which the exponential growth rates of the underlying linear dynamics never vanish. In other words, its Lyapunov exponents are all nonzero. The authors consider mostly difference equations, although they also briefly consider the case of differential equations. The content is self-contained and all proofs have been simplified or even rewritten on purpose for the book so that all is as streamlined as possible. Moreover, all chapters are supplemented by detailed notes discussing the origins of the notions and results as well as their proofs, together with the discussion of the proper context, also with references to precursor results and further developments. A useful chapter dependence chart is included in the Preface. The book is aimed at researchers and graduate students who wish to have a sufficiently broad view of the area, without the discussion of accessory material. It can also be used as a basis for graduate courses on spectra, normal forms, and smooth conjugacies.The main components of the exposition are tempered spectra, normal forms, and smooth conjugacies. The first two lie at the core of the theory and have an importance that undoubtedly surpasses the construction of conjugacies. Indeed, the theory is very rich and developed in various directions that are also of interest by themselves. This includes the study of dynamics with discrete and continuous time, of dynamics in finite and infinite-dimensional spaces, as well as of dynamics depending on a parameter. This led the authors to make an exposition not only of tempered spectra and subsequently of normal forms, but also briefly of some important developments in those other directions. Afterwards the discussion continues with the construction of stable and unstable invariant manifolds and, consequently, of smooth conjugacies, while using most of the former material.The notion of tempered spectrum is naturally adapted to the study of nonautonomous dynamics. The reason for this is that any autonomous linear dynamics with a tempered exponential dichotomy has automatically a uniform exponential dichotomy. Most notably, the spectra defined in terms of tempered exponential dichotomies and uniform exponential dichotomies are distinct in general. More precisely, the tempered spectrum may be smaller, which causes that it may lead to less resonances and thus to simpler normal forms. Another important aspect is the need for Lyapunov norms in the study of exponentially decaying perturbations and in the study of parameter-dependent dynamics. Other characteristics are the need for a spectral gap to obtain the regularity of the normal forms on a parameter and the need for a careful control of the small exponential terms in the construction of invariant manifolds and of smooth conjugacies.
Spectral Action in Noncommutative Geometry (SpringerBriefs in Mathematical Physics #27)
by Michał Eckstein Bruno IochumWhat is spectral action, how to compute it and what are the known examples? This book offers a guided tour through the mathematical habitat of noncommutative geometry à la Connes, deliberately unveiling the answers to these questions.After a brief preface flashing the panorama of the spectral approach, a concise primer on spectral triples is given. Chapter 2 is designed to serve as a toolkit for computations. The third chapter offers an in-depth view into the subtle links between the asymptotic expansions of traces of heat operators and meromorphic extensions of the associated spectral zeta functions. Chapter 4 studies the behaviour of the spectral action under fluctuations by gauge potentials. A subjective list of open problems in the field is spelled out in the fifth Chapter. The book concludes with an appendix including some auxiliary tools from geometry and analysis, along with examples of spectral geometries.The book serves both as a compendium for researchers in the domain of noncommutative geometry and an invitation to mathematical physicists looking for new concepts.
Spectral Analysis of N-Body Schrödinger Operators at Two-Cluster Thresholds (Mathematical Physics Studies)
by Erik Skibsted Xue Ping WangThis book provides a systematic study of spectral and scattering theory for many-body Schrödinger operators at two-cluster thresholds. While the two-body problem (reduced after separation of the centre of mass motion to a one-body problem at zero energy) is a well-studied subject, the literature on many-body threshold problems is sparse. However, the authors’ analysis covers for example the system of three particles interacting by Coulomb potentials and restricted to a small energy region to the right of a fixed nonzero two-body eigenvalue. In general, the authors address the question: How do scattering quantities for the many-body atomic and molecular models behave within the limit when the total energy approaches a fixed two-cluster threshold? This includes mapping properties and singularities of the limiting scattering matrix, asymptotics of the total scattering cross section, and absence of transmission from one channel to another in the small inter-cluster kinetic energy region. The authors’ principal tools are the Feshbach–Grushin dimension reduction method and spectral analysis based on a certain Mourre estimate. Additional topics of independent interest are the limiting absorption principle, micro-local resolvent estimates, Rellich- and Sommerfeld-type theorems and asymptotics of the limiting resolvents at thresholds. The mathematical physics field under study is very rich, and there are many open problems, several of them stated explicitly in the book for the interested reader.
Spectral Analysis of Nonlinear Elastic Shapes
by James F. DoyleThis book concerns the elastic stability of thin-walled structures — one of the most challenging problems facing structural engineers because of its high degree of nonlinearity — and introduces the innovative approach of using spectral analysis of the shapes and the stiffness to gain insights into the nonlinear deformations. The methodology greatly facilitates correlating the shape changes with the stiffness changes. Professor Doyle also develops specific computer procedures that complement finite element methods so that the ideas and methods are applicable to general structural problems. Basic validity of the procedures is established using key archetypal problems from buckling/post-buckling of columns, arches, curved plates, and cylindrical shells, all worked out in significant detail. The book is ideal for a wide variety of structural engineers, particularly those in aerospace and civil fields. Researchers in computational mechanics also find a rich source of new ideas for post-processing data from nonlinear analyses.
Spectral Approach to Transport Problems in Two-Dimensional Disordered Lattices: Physical Interpretation And Applications (Springer Theses)
by Evdokiya Georgieva KostadinovaThis book introduces the spectral approach to transport problems in infinite disordered systems characterized by Anderson-type Hamiltonians. The spectral approach determines (with probability one) the existence of extended states for nonzero disorder in infinite lattices of any dimension and geometry. Here, the author focuses on the critical 2D case, where previous numerical and experimental results have shown disagreement with theory. Not being based on scaling theory, the proposed method avoids issues related to boundary conditions and provides an alternative approach to transport problems where interaction with various types of disorder is considered.Beginning with a general overview of Anderson-type transport problems and their relevance to physical systems, it goes on to discuss in more detail the most relevant theoretical, numerical, and experimental developments in this field of research. The mathematical formulation of the innovative spectral approach is introduced together with a physical interpretation and discussion of its applicability to physical systems, followed by a numerical study of delocalization in the 2D disordered honeycomb, triangular, and square lattices. Transport in the 2D honeycomb lattice with substitutional disorder is investigated employing a spectral analysis of the quantum percolation problem. Next, the applicability of the method is extended to the classical regime, with an examination of diffusion of lattice waves in 2D disordered complex plasma crystals, along with discussion of proposed future developments in the study of complex transport problems using spectral theory.