The trajectory of fractional calculus has undergone several periods of intensive development, both in pure and applied sciences. During the last few decades fractional calculus has also been associated with the power law effects and its various applications. It is a natural to ask if fractional calculus, as a nonlocal calculus, can produce new results within the well-established field of Lie symmetries and their applications. In Lie Symmetry Analysis of Fractional Differential Equations the authors try to answer this vital question by analyzing different aspects of fractional Lie symmetries and related conservation laws. Finding the exact solutions of a given fractional partial differential equation is not an easy task, but is one that the authors seek to grapple with here. The book also includes generalization of Lie symmetries for fractional integro differential equations. Features Provides a solid basis for understanding fractional calculus, before going on to explore in detail Lie Symmetries and their applications Useful for PhD and postdoc graduates, as well as for all mathematicians and applied researchers who use the powerful concept of Lie symmetries Filled with various examples to aid understanding of the topics

Over the past decade, the field of image processing has made tremendous advances. One type of image processing that is currently of particular interest is "tomographic imaging," a technique for computing the density function of a body, or discontinuity surfaces of this function. Today, tomography is widely used, and has applications in such fields as medicine, engineering, physics, geophysics, and security. The Radon Transform and Local Tomography clearly explains the theoretical, computational, and practical aspects of applied tomography. It includes sufficient background information to make it essentially self-contained for most readers.

Introduction to Computational Models with Python explains how to implement computational models using the flexible and easy-to-use Python programming language. The book uses the Python programming language interpreter and several packages from the huge Python Library that improve the performance of numerical computing, such as the Numpy and Scipy modules. The Python source code and data files are available on the author's website. The book's five sections present: An overview of problem solving and simple Python programs, introducing the basic models and techniques for designing and implementing problem solutions, independent of software and hardware tools Programming principles with the Python programming language, covering basic programming concepts, data definitions, programming structures with flowcharts and pseudo-code, solving problems, and algorithms Python lists, arrays, basic data structures, object orientation, linked lists, recursion, and running programs under Linux Implementation of computational models with Python using Numpy, with examples and case studies The modeling of linear optimization problems, from problem formulation to implementation of computational models This book introduces the principles of computational modeling as well as the approaches of multi- and interdisciplinary computing to beginners in the field. It provides the foundation for more advanced studies in scientific computing, including parallel computing using MPI, grid computing, and other methods and techniques used in high-performance computing.

Sparked by demands inherent to the mathematical study of pollution, intensive industry, global warming, and the biosphere, Adjoint Equations and Perturbation Algorithms in Nonlinear Problems is the first book ever to systematically present the theory of adjoint equations for nonlinear problems, as well as their application to perturbation algorithms. This new approach facilitates analysis of observational data, the application of adjoint equations to retrospective study of processes governed by imitation models, and the study of computer models themselves. Specifically, the book discusses: Principles for constructing adjoint operators in nonlinear problems Properties of adjoint operators and solvability conditions for adjoint equations Perturbation algorithms using the adjoint equations theory for nonlinear problems in transport theory, quasilinear motion, substance transfer, and nonlinear data assimilation Known results on adjoint equations and perturbation algorithms in nonlinear problems This groundbreaking text contains some results that have no analogs in the scientific literature, opening unbounded possibilities in construction and application of adjoint equations to nonlinear problems of mathematical physics.

Modeling Tools for Environmental Engineers and Scientists enables environmental professionals, faculty, and students with minimal computer programming skills to develop computer-based mathematical models for natural and engineered environmental systems. The author illustrates how commercially available syntax-free authoring software can be adapted to create customized, high-level models of environmental phenomena in groundwater, soil, aquatic, and atmospheric systems, and in engineered reactors. This book includes a review of mathematical modeling and fundamental concepts such as material balance, reactor configurations, and fate and transport of environmental contaminants. It illustrates, using numerous examples, how mathematical and dynamic modeling software can be applied in analyzing and simulating natural and engineered environmental systems. The tools and examples included are applicable to a wide range of problems, both in the classroom and in the field.

An up-to-date survey of mathematical models of carcinogenesis, providing the most recent findings of cancer biology as evidence of the models, as well as extensive bibliographies of cancer biology and in-depth mathematical analyses for each of the models. May be used as a reference for courses on st

Impulsive differential equations have been the subject of intense investigation in the last 10-20 years, due to the wide possibilities for their application in numerous fields of science and technology. This new work presents a systematic exposition of the results solving all of the more important problems in this field.

The concept of derivatives of non-integer order, known as fractional derivatives, first appeared in the letter between L'Hopital and Leibniz in which the question of a half-order derivative was posed. Since then, many formulations of fractional derivatives have appeared. Recently, a new definition of fractional derivative, called the "fractional conformable derivative," has been introduced. This new fractional derivative is compatible with the classical derivative and it has attracted attention in areas as diverse as mechanics, electronics, and anomalous diffusion. Conformable Dynamic Equations on Time Scales is devoted to the qualitative theory of conformable dynamic equations on time scales. This book summarizes the most recent contributions in this area, and vastly expands on them to conceive of a comprehensive theory developed exclusively for this book. Except for a few sections in Chapter 1, the results here are presented for the first time. As a result, the book is intended for researchers who work on dynamic calculus on time scales and its applications. Features Can be used as a textbook at the graduate level as well as a reference book for several disciplines Suitable for an audience of specialists such as mathematicians, physicists, engineers, and biologists Contains a new definition of fractional derivative About the Authors Douglas R. Anderson is professor and chair of the mathematics department at Concordia College, Moorhead. His research areas of interest include dynamic equations on time scales and Ulam-type stability of difference and dynamic equations. He is also active in investigating the existence of solutions for boundary value problems. Svetlin G. Georgiev is currently professor at Sorbonne University, Paris, France and works in various areas of mathematics. He currently focuses on harmonic analysis, partial differential equations, ordinary differential equations, Clifford and quaternion analysis, dynamic calculus on time scales, and integral equations.

Image Processing and Acquisition using Python provides readers with a sound foundation in both image acquisition and image processing-one of the first books to integrate these topics together. By improving readers' knowledge of image acquisition techniques and corresponding image processing, the book will help them perform experiments more effectively and cost efficiently as well as analyze and measure more accurately. Long recognized as one of the easiest languages for non-programmers to learn, Python is used in a variety of practical examples. A refresher for more experienced readers, the first part of the book presents an introduction to Python, Python modules, reading and writing images using Python, and an introduction to images. The second part discusses the basics of image processing, including pre/post processing using filters, segmentation, morphological operations, and measurements. The second part describes image acquisition using various modalities, such as x-ray, CT, MRI, light microscopy, and electron microscopy. These modalities encompass most of the common image acquisition methods currently used by researchers in academia and industry. Features Covers both the physical methods of obtaining images and the analytical processing methods required to understand the science behind the images. Contains many examples, detailed derivations, and working Python examples of the techniques. Offers practical tips on image acquisition and processing. Includes numerous exercises to test the reader's skills in Python programming and image processing, with solutions to selected problems, example programs, and images available on the book's web page. New to this edition Machine learning has become an indispensable part of image processing and computer vision, so in this new edition two new chapters are included: one on neural networks and the other on convolutional neural networks. A new chapter on affine transform and many new algorithms. Updated Python code aligned to the latest version of modules.

The finite element and the boundary element methods are the two most important developments in numerical mathematics to occur in this century. Many engineering and mathematics graduate curricula now include a course in boundary element methods. Such a course must cover numerical methods, basic methodology to real problems, and interactive computer usage. Both theory and applications, necessary for applied courses, are available in this new textbook. An Introduction to Boundary Element Methods is logically organized and easy to read. The topics are carefully selected and meticulously presented. Applications are described for use in identifying potential problems and for heat transfer, diffusion equations, linear elasticity, water waves, ocean acoustics, acoustic scattering, aerodynamics, porous media, and simple laminar flows. More than 20 computer subroutines help develop and explain the computational aspect of the subject. Hundreds of figures, exercises, and solved examples supplement text and help clarify important information. The computer programs have been tested on some benchmark problems. Even in single precision the results are more accurate and better than those obtained from available Fortran programs.

This new work is an introduction to the numerical solution of the initial value problem for a system of ordinary differential equations. The first three chapters are general in nature, and chapters 4 through 8 derive the basic numerical methods, prove their convergence, study their stability and consider how to implement them effectively. The book focuses on the most important methods in practice and develops them fully, uses examples throughout, and emphasizes practical problem-solving methods.

A Computational Approach to Statistical Learning gives a novel introduction to predictive modeling by focusing on the algorithmic and numeric motivations behind popular statistical methods. The text contains annotated code to over 80 original reference functions. These functions provide minimal working implementations of common statistical learning algorithms. Every chapter concludes with a fully worked out application that illustrates predictive modeling tasks using a real-world dataset. The text begins with a detailed analysis of linear models and ordinary least squares. Subsequent chapters explore extensions such as ridge regression, generalized linear models, and additive models. The second half focuses on the use of general-purpose algorithms for convex optimization and their application to tasks in statistical learning. Models covered include the elastic net, dense neural networks, convolutional neural networks (CNNs), and spectral clustering. A unifying theme throughout the text is the use of optimization theory in the description of predictive models, with a particular focus on the singular value decomposition (SVD). Through this theme, the computational approach motivates and clarifies the relationships between various predictive models. Taylor Arnold is an assistant professor of statistics at the University of Richmond. His work at the intersection of computer vision, natural language processing, and digital humanities has been supported by multiple grants from the National Endowment for the Humanities (NEH) and the American Council of Learned Societies (ACLS). His first book, Humanities Data in R, was published in 2015. Michael Kane is an assistant professor of biostatistics at Yale University. He is the recipient of grants from the National Institutes of Health (NIH), DARPA, and the Bill and Melinda Gates Foundation. His R package bigmemory won the Chamber's prize for statistical software in 2010. Bryan Lewis is an applied mathematician and author of many popular R packages, including irlba, doRedis, and threejs.

Get First-Hand Insight from a Contributor to the Standard Model of Particle Physics Written by an award-winning former director-general of CERN and one of the world's leading experts on particle physics, Electroweak Interactions explores the concepts that led to unification of the weak and electromagnetic interactions. It provides the fundamental elements of the theory of compact Lie groups and their representations, enabling a basic understanding of the role of flavor symmetry in particle physics. Understand Conceptual Elements of the Theory of Elementary Particles The book begins with the identification of the weak hadronic current with the isotopic spin current, Yang-Mills theory, and the first electroweak theory of Glashow. It discusses spontaneous breaking of a global symmetry and a local symmetry, covering the Goldstone theorem, Brout-Englert-Higgs mechanism, and the theory of Weinberg and Salam. The author then describes the theory of quarks, quark mixing, the Cabibbo angle, the Glashow-Iliopoulos-Maiani (GIM) mechanism, the theory of Kobayashi and Maskawa, six quark flavors, and CP violation. Delve into Experimental Tests and Unresolved Problems The author goes on to explore some phenomenological topics, such as neutral current interactions of neutrinos and CP violation in the neutral K-meson system. He also highlights how flavor-changing neutral current processes have emerged as probes to reveal the presence of new phenomena at energies not yet accessible with particle accelerators. The book concludes with an explanation of the expected properties of the Higgs boson and the methods adopted for its search. The predictions are also compared with relevant experimental results. View the author's first book in this collection: Relativistic Quantum Mechanics: An Introduction to Relativistic Quantum Fields.

Perturbation theory is a powerful tool for solving a wide variety of problems in applied mathematics, a tool particularly useful in quantum mechanics and chemistry. Although most books on these subjects include a section offering an overview of perturbation theory, few, if any, take a practical approach that addresses its actual implementation Introduction to Perturbation Theory in Quantum Mechanics does. It collects into a single source most of the techniques for applying the theory to the solution of particular problems. Concentrating on problems that allow exact analytical solutions of the perturbation equations, the book resorts to numerical results only when necessary to illustrate and complement important features of the theory. The author also compares different methods by applying them to the same models so that readers clearly understand why one technique may be preferred over another. Demonstrating the application of similar techniques in quantum and classical mechanics, Introduction to Perturbation Theory in Quantum Mechanics reveals the underlying mathematics in seemingly different problems. It includes many illustrative examples that facilitate the understanding of theoretical concepts, and provides a source of ideas for many original research projects.

This book serves as a guide for discovering pathways to more efficient energy use. The first part of the book illustrates basic laws of energy conversion and principles of thermodynamics. Laws of energy conservation and direction of energy conversion are formulated in detail, and the types of thermodynamic processes are explained. Also included is the characterization of various types of real energy conversion. The second part of the book discusses types of energy conversion referred to as thermal-energy technologies. The advantages of the co-generation processes and devices operating within the Brayton direct cycle and their adaptively to household energetics are underlined.

This book covers applications of fractional calculus used for medical and health science. It offers a collection of research articles built into chapters on classical and modern dynamical systems formulated by fractional differential equations describing human diseases and how to control them. The mathematical results included in the book will be helpful to mathematicians and doctors by enabling them to explain real-life problems accurately. The book will also offer case studies of real-life situations with an emphasis on describing the mathematical results and showing how to apply the results to medical and health science, and at the same time highlighting modeling strategies. The book will be useful to graduate level students, educators and researchers interested in mathematics and medical science.

This book examines engineering and mathematical models for documenting and approving mechanical and environmental discharges. The author emphasizes engineering design considerations as well as applications to waste water and atmospheric discharges. Chapters discuss: the fundamentals of turbulent jet mixing, dilution concepts, and mixing zone concepts diffuser configurations and head loss calculations different modeling techniques and accepted models - discussed in detail with theoretical background, restrictions, input, output, and examples Lagrangian and the EPA UM 2-dimensional diffuser model the PLUMES interface Eulerian integral methods, EPA UDKHG 3-dimensional diffuser model, and PDSG surface discharge model empirical techniques, RSB diffuser model, the CORMIX family of models for both diffusers and surface discharge numerical methods with a discussion of shelf commercial models Gaussian atmospheric plume models Fundamentals of Environmental Discharge Modeling includes numerous case studies and examples for each model and problem.

The authors examine topics in modern physics and offer a unitary and original treatment of the fundamental problems of the dynamics of physical systems, as well as a description of the nuclear matter within a framework of general relativity. They show that some physical phenomena studied at two different resolution scales (e.g. microscale, cosmological scale), apparently with no connection between them, become compatible by means of the operational procedures, acting either as some "hidden" symmetries, or harmonic-type mappings. The book is addressed to the students, researchers and university/high school teachers working in the fields of mathematics, physics, and chemistry.

The exponential distribution is one of the most significant and widely used distribution in statistical practice. It possesses several important statistical properties, and yet exhibits great mathematical tractability. This volume provides a systematic and comprehensive synthesis of the diverse literature on the theory and applications of the expon

An introduction to general theories of stochastic processes and modern martingale theory. The volume focuses on consistency, stability and contractivity under geometric invariance in numerical analysis, and discusses problems related to implementation, simulation, variable step size algorithms, and random number generation.

Provides One Unified Formula That Gives Solutions to Several Types of GSEs Generalized Sylvester equations (GSEs) are applied in many fields, including applied mathematics, systems and control, and signal processing. Generalized Sylvester Equations: Unified Parametric Solutions presents a unified parametric approach for solving various types of GSEs. In an extremely neat and elegant matrix form, the book provides a single unified parametric solution formula for all the types of GSEs, which further reduces to a specific clear vector form when the parameter matrix F in the equations is a Jordan matrix. Particularly, when the parameter matrix F is diagonal, the reduced vector form becomes extremely simple. The first chapter introduces several types of GSEs and gives a brief overview of solutions to GSEs. The two subsequent chapters then show the importance of GSEs using four typical control design applications and discuss the F-coprimeness of a pair of polynomial matrices. The next several chapters deal with parametric solutions to GSEs. The final two chapters present analytical solutions to normal Sylvester equations (NSEs), including the well-known continuous- and discrete-time Lyapunov equations. An appendix provides the proofs of some theorems. The book can be used as a reference for graduate and senior undergraduate courses in applied mathematics and control systems analysis and design. It will also be useful to readers interested in research and applications based on Sylvester equations.

In recent years air-sea interaction has emerged as a subject in its own right, encompassing small-scale and large-scale processes in both air and sea. Air-Sea Interaction: Laws and Mechanisms is a comprehensive account of how the atmosphere and the ocean interact to control the global climate, what physical laws govern this interaction, and its prominent mechanisms. The topics covered range from evaporation in the oceans, to hurricanes, and on to poleward heat transport by the oceans. By developing the subject from basic physical (thermodynamic) principles, the book is accessible to graduate students and research scientists in meteorology, oceanography, and environmental engineering. It will also be of interest to the broader physics community involved in the treatment of transfer laws, and thermodynamics of the atmosphere and ocean.

Undergraduate engineering students need good mathematics skills. This textbook supports this need by placing a strong emphasis on visualization and the methods and tools needed across the whole of engineering. The visual approach is emphasized, and excessive proofs and derivations are avoided. The visual images explain and teach the mathematical methods. The book's website provides dynamic and interactive codes in Mathematica to accompany the examples for the reader to explore on their own with Mathematica or the free Computational Document Format player, and it provides access for instructors to a solutions manual. Strongly emphasizes a visual approach to engineering mathematics Written for years 2 to 4 of an engineering degree course Website offers support with dynamic and interactive Mathematica code and instructor's solutions manual Brian Vick is an associate professor at Virginia Tech in the United States and is a longtime teacher and researcher. His style has been developed from teaching a variety of engineering and mathematical courses in the areas of heat transfer, thermodynamics, engineering design, computer programming, numerical analysis, and system dynamics at both undergraduate and graduate levels. eResource material is available for this title at www.crcpress.com/9780367432768.

Localized Dynamics of Thin-Walled Shells focuses on localized vibrations and waves in thin-walled structures with variable geometrical and physical characteristics. It emphasizes novel asymptotic methods for solving boundary-value problems for dynamic equations in the shell theory, in the form of functions which are highly localized near both fixed and moving lines/points on the shell surface. Features First-of-its-kind work, synthesizing knowledge of the localization of vibrations and waves in thin-walled shells with a mathematical tool to study them Suitable for researchers working on the dynamics of thin shells and also as supplementary reading for undergraduates studying asymptotic methods Offers detailed analysis of wave processes in shells with varying geometric and physical parameters

Modelling with Ordinary Differential Equations: A Comprehensive Approach aims to provide a broad and self-contained introduction to the mathematical tools necessary to investigate and apply ODE models. The book starts by establishing the existence of solutions in various settings and analysing their stability properties. The next step is to illustrate modelling issues arising in the calculus of variation and optimal control theory that are of interest in many applications. This discussion is continued with an introduction to inverse problems governed by ODE models and to differential games. The book is completed with an illustration of stochastic differential equations and the development of neural networks to solve ODE systems. Many numerical methods are presented to solve the classes of problems discussed in this book. Features: Provides insight into rigorous mathematical issues concerning various topics, while discussing many different models of interest in different disciplines (biology, chemistry, economics, medicine, physics, social sciences, etc.) Suitable for undergraduate and graduate students and as an introduction for researchers in engineering and the sciences Accompanied by codes which allow the reader to apply the numerical methods discussed in this book in those cases where analytical solutions are not available