Discovering Dynamical Systems Through Experiment and Inquiry differs from most texts on dynamical systems by blending the use of computer simulations with inquiry-based learning (IBL). IBL is an excellent tool to move students from merely remembering the material to deeper understanding and analysis. This method relies on asking students questions first, rather than presenting the material in a lecture. Another unique feature of this book is the use of computer simulations. Students can discover examples and counterexamples through manipulations built into the software. These tools have long been used in the study of dynamical systems to visualize chaotic behavior. We refer to this unique approach to teaching mathematics as ECAP-Explore, Conjecture, Apply, and Prove. ECAP was developed to mimic the actual practice of mathematics in an effort to provide students with a more holistic mathematical experience. In general, each section begins with exercises guiding students through explorations of the featured concept and concludes with exercises that help the students formally prove the results. While symbolic dynamics is a standard topic in an undergraduate dynamics text, we have tried to emphasize it in a way that is more detailed and inclusive than is typically the case. Finally, we have chosen to include multiple sections on important ideas from analysis and topology independent from their application to dynamics.

Wavelet Analysis: Basic Concepts and Applications provides a basic and self-contained introduction to the ideas underpinning wavelet theory and its diverse applications. This book is suitable for master's or PhD students, senior researchers, or scientists working in industrial settings, where wavelets are used to model real-world phenomena and data needs (such as finance, medicine, engineering, transport, images, signals, etc.). Features: Offers a self-contained discussion of wavelet theory Suitable for a wide audience of post-graduate students, researchers, practitioners, and theorists Provides researchers with detailed proofs Provides guides for readers to help them understand and practice wavelet analysis in different areas

This book is directed toward the numerical integration (solution) of a system of partial differential equations (PDEs) that describes a combination of chemical reaction and diffusion, that is, reaction-diffusion PDEs. The particular form of the PDEs corresponds to a system discussed by Alan Turing and is therefore termed a Turing model.Specifically, Turing considered how a reaction-diffusion system can be formulated that does not have the usual smoothing properties of a diffusion (dispersion) system, and can, in fact, develop a spatial variation that might be interpreted as a form of morphogenesis, so he termed the chemicals as morphogens.Turing alluded to the important impact computers would have in the study of a morphogenic PDE system, but at the time (1952), computers were still not readily available. Therefore, his paper is based on analytical methods. Although computers have since been applied to Turing models, computer-based analysis is still not facilitated by a discussion of numerical algorithms and a readily available system of computer routines.The intent of this book is to provide a basic discussion of numerical methods and associated computer routines for reaction-diffusion systems of varying form. The presentation has a minimum of formal mathematics. Rather, the presentation is in terms of detailed examples, presented at an introductory level. This format should assist readers who are interested in developing computer-based analysis for reaction-diffusion PDE systems without having to first study numerical methods and computer programming (coding).The numerical examples are discussed in terms of: (1) numerical integration of the PDEs to demonstrate the spatiotemporal features of the solutions and (2) a numerical eigenvalue analysis that corroborates the observed temporal variation of the solutions. The resulting temporal variation of the 2D and 3D plots demonstrates how the solutions evolve dynamically, including oscillatory long-term behavior.In all of the examples, routines in R are presented and discussed in detail. The routines are available through this link so that the reader can execute the PDE models to reproduce the reported solutions, then experiment with the models, including extensions and application to alternative models.

The book presents qualitative results for different classes of fractional equations, including fractional functional differential equations, fractional impulsive differential equations, and fractional impulsive functional differential equations, which have not been covered by other books. It manifests different constructive methods by demonstrating how these techniques can be applied to investigate qualitative properties of the solutions of fractional systems. Since many applications have been included, the demonstrated techniques and models can be used in training students in mathematical modeling and in the study and development of fractional-order models.

This book is intended for students of computational systems biology with only a limited background in mathematics. Typical books on systems biology merely mention algorithmic approaches, but without offering a deeper understanding. On the other hand, mathematical books are typically unreadable for computational biologists. The authors of the present book have worked hard to fill this gap. The result is not a book on systems biology, but on computational methods in systems biology. This book originated from courses taught by the authors at Freie Universitat Berlin. The guiding idea of the courses was to convey those mathematical insights that are indispensable for systems biology, teaching the necessary mathematical prerequisites by means of many illustrative examples and without any theorems. The three chapters cover the mathematical modelling of biochemical and physiological processes, numerical simulation of the dynamics of biological networks and identification of model parameters by means of comparisons with real data. Throughout the text, the strengths and weaknesses of numerical algorithms with respect to various systems biological issues are discussed. Web addresses for downloading the corresponding software are also included.

Without sacrificing scientific strictness, this introduction to the field guides readers through mathematical modeling, the theoretical treatment of the underlying physical laws and the construction and effective use of numerical procedures to describe the behavior of the dynamics of physical flow.
The book is carefully divided into three main parts:
- The design of mathematical models of physical fluid flow;
- A theoretical treatment of the equations representing the model, as Navier-Stokes, Euler, and boundary layer equations, models of turbulence, in order to gain qualitative as well as quantitative insights into the processes of flow events;
- The construction and effective use of numerical procedures in order to find quantitative descriptions of concrete physical or technical fluid flow situations.
Both students and experts wanting to control or predict the behavior of fluid flows by theoretical and computational fluid dynamics will benefit from this combination of all relevant aspects in one handy volume.

The main objective of the book is to highlight the modeling of magnetic particles with different shapes and magnetic properties, to provide graduate students and young researchers information on the theoretical aspects and actual techniques for the treatment of magnetic particles in particle-based simulations. In simulation, we focus on the Monte Carlo, molecular dynamics, Brownian dynamics, lattice Boltzmann and stochastic rotation dynamics (multi-particle collision dynamics) methods. The latter two simulation methods can simulate both the particle motion and the ambient flow field simultaneously. In general, specialized knowledge can only be obtained in an effective manner under the supervision of an expert. The present book is written to play such a role for readers who wish to develop the skill of modeling magnetic particles and develop a computer simulation program using their own ability. This book is therefore a self-learning book for graduate students and young researchers. Armed with this knowledge, readers are expected to be able to sufficiently enhance their skill for tackling any challenging problems they may encounter in future.

Mankind now faces even more challenging environment- and health-related problems than ever before. Readily available transportation systems facilitate the swift spread of diseases as large populations migrate from one part of the world to another. Studies on the spread of the communicable diseases are very important. This book, Mathematical Population Dynamics and Epidemiology in Temporal and Spatio-Temporal Domains, provides a useful experimental tool for making practical predictions, building and testing theories, answering specific questions, determining sensitivities of the parameters, forming control strategies, and much more. This volume focuses on the study of population dynamics with special emphasis on the migration of populations and the spreading of epidemics among human and animal populations. It also provides the background needed to interpret, construct, and analyze a wide variety of mathematical models. Most of the techniques presented in the book can be readily applied to model other phenomena, in biology as well as in other disciplines.

Finite element analysis has become the most popular technique for studying engineering structures in detail. It is particularly useful whenever the complexity of the geometry or of the loading is such that alternative methods are inappropriate. The finite element method is based on the premise that a complex structure can be broken down into finitely many smaller pieces (elements), the behaviour of each of which is known or can be postulated. These elements might then be assembled in some sense to model the behaviour of the structure. Intuitively this premise seems reasonable, but there are many important questions that need to be answered. In order to answer them it is necessary to apply a degree of mathematical rigour to the development of finite element techniques. The approach that will be taken in this book is to develop the fundamental ideas and methodologies based on an intuitive engineering approach, and then to support them with appropriate mathematical proofs where necessary. It will rapidly become clear that the finite element method is an extremely powerful tool for the analysis of structures (and for other field problems), but that the volume of calculations required to solve all but the most trivial of them is such that the assistance of a computer is necessary. As stated above, many questions arise concerning finite element analysis. Some of these questions are associated with the fundamental mathematical formulations, some with numerical solution techniques, and others with the practical application of the method. In order to answer these questions, the engineer/analyst needs to understand both the nature and limitations of the finite element approximation and the fundamental behaviour of the structure. Misapplication of finite element analysis programs is most likely to arise when the analyst is ignorant of engineering phenomena.

This second edition provides a broad range of methods and concepts required for the analysis and solution of equations which arise in the modeling of phenomena in the natural, engineering, and applied mathematical sciences. It may be used productively by both undergraduate and graduate students, as well as others who wish to learn, understand, and apply these techniques. Detailed discussions are also given for several topics that are not usually included in standard textbooks at this level of presentation: qualitative methods for differential equations, dimensionalization and scaling, elements of asymptotics, difference equations and several perturbation procedures. Further, this second edition includes several new topics covering functional equations, the Lambert-W function, nonstandard sets of periodic functions, and the method of dominant balance. Each chapter contains a large number of worked examples and provides references to the appropriate books and literature.

This book presents an in-depth treatment of various mathematical aspects of electromagnetism and Maxwell's equations: from modeling issues to well-posedness results and the coupled models of plasma physics (Vlasov-Maxwell and Vlasov-Poisson systems) and magnetohydrodynamics (MHD). These equations and boundary conditions are discussed, including a brief review of absorbing boundary conditions. The focus then moves to well-posedness results. The relevant function spaces are introduced, with an emphasis on boundary and topological conditions. General variational frameworks are defined for static and quasi-static problems, time-harmonic problems (including fixed frequency or Helmholtz-like problems and unknown frequency or eigenvalue problems), and time-dependent problems, with or without constraints. They are then applied to prove the well-posedness of Maxwell's equations and their simplified models, in the various settings described above. The book is completed with a discussion of dimensionally reduced models in prismatic and axisymmetric geometries, and a survey of existence and uniqueness results for the Vlasov-Poisson, Vlasov-Maxwell and MHD equations. The book addresses mainly researchers in applied mathematics who work on Maxwell's equations. However, it can be used for master or doctorate-level courses on mathematical electromagnetism as it requires only a bachelor-level knowledge of analysis.

Simulation-Based Engineering and Science (SBE&S) cuts across disciplines, showing tremendous promise in areas from storm prediction and climate modeling to understanding the brain and the behavior of numerous other complex systems.In this groundbreaking volume, nine distinguished leaders assess the latest research trends, as a result of 52 site visits in Europe and Asia and hundreds of hours of expert interviews, and discuss the implications of their findings for the US government.The authors conclude that while the US remains the quantitative leader in SBE&S research and development, it is very much in danger of losing that edge to Europe and Asia.Commissioned by the National Science Foundation, this multifaceted study will capture the attention of Fortune 500 companies and policymakers.Distinguished contributors: Sharon C Goltzer, University of Michigan, Ann Arbor, USA Sangtae Kim, Morgridge Institute for Research, USA Peter T Cummings, Vanderbilt University, USA and Oak Ridge National Laboratory, USA Abhijit Deshmukh, Texas A&M University, USA Martin Head-Gordon, University of California, Berkeley, USA George Em Karniadakis, Brown University, USA Linda Petzold, University of California, Santa Barbara, USA Celeste Sagui, North Carolina State University, USA Masanobu Shinozuka, University of California, Irvine, USA

The problem of motion of extended bodies in General Relativity is notorious for its analytical difficulty, but at the same time highly relevant for comparison of theoretical predictions with modern precision measurements in relativistic astrophysics and cosmology. Its one of the most important topics in General Relativity and its application to astrophysics.
Equations of Motion in General Relativity focuses attention on two aspects of equations of motion in general relativity: the motion of extended bodies (stars) and the motion of small black holes. The objective is to offer a guide to prospective researchers into these areas of general relativity and to point out open questions and topics that are ripe for further development. It is over forty years since a text on this subject was published and in that time the research area of equations of motion in general relativity has undergone extraordinary development, stimulated by the discovery of the binary neutron star PSR 1913+16 in 1974 (which was the first isolated gravitating system found in which general relativity plays a fundamental role in describing theoretically its evolution), and more recently by the advent of kilometre size interferometric gravitational wave detectors which are expected to detect gravitational waves produced by coalescing binary neutron stars.

In many parts of the world, groundwater resources are under increasing threat from growing demands, wasteful use, and contamination. To face the challenge, good planning and management practices are needed. A key to the management of groundwater is the ability to model the movement of fluids and contaminants in the subsurface. The purpose of this book is to construct conceptual and mathematical models that can provide the information required for making decisions associated with the management of groundwater resources, and the remediation of contaminated aquifers. The basic approach of this book is to accurately describe the underlying physics of groundwater flow and solute transport in heterogeneous porous media, starting at the microscopic level, and to rigorously derive their mathematical representation at the macroscopic levels. The well-posed, macroscopic mathematical models are formulated for saturated, single phase flow, as well as for unsaturated and multiphase flow, and for the transport of single and multiple chemical species. Numerical models are presented and computer codes are reviewed, as tools for solving the models. The problem of seawater intrusion into coastal aquifers is examined and modeled. The issues of uncertainty in model input data and output are addressed. The book concludes with a chapter on the management of groundwater resources. Although one of the main objectives of this book is to construct mathematical models, the amount of mathematics required is kept minimal.

This book covers an interdisciplinary approach for understanding mathematical modeling by offering a collection of models, solved problems related to the models, the methodologies employed, and the results using projects and case studies with insight into the operation of substantial real-time systems. The book covers a broad scope in the areas of statistical science, probability, stochastic processes, fluid dynamics, supply chain, optimization, and applications. It discusses advanced topics and the latest research findings, uses an interdisciplinary approach for real-time systems, offers a platform for integrated research, and identifies the gaps in the field for further research. The book is for researchers, students, and teachers that share a goal of learning advanced topics and the latest research in mathematical modeling.

Mathematical Modelling in Motor Neuroscience: State of the Art and Translation to the Clinic, Gaze Orienting Mechanisms and Disease, Volume 249, the latest release in the Progress in Brain Research series, highlights new advances in the field, with this new volume presenting interesting chapters on a variety of topics, including Sequential Bayesian updating, Maps and Sensorimotor Transformations for Eye-Head Gaze Shifts: Role of the Midbrain Superior Colliculus, Modeling Gaze Position-Dependent Opsoclonus, Eye Position-Dependent Opsoclonus in Mild Traumatic Brain Injury, Saccades in Parkinson's disease -- hypometric, slow, and maladaptive, Brainstem Neural Circuits for Fixation and Generation of Saccadic Eye Movements, and much more.

The eigenvalue densities in various matrix models in quantum chromodynamics (QCD) are ultimately unified in this book by a unified model derived from the integrable systems. Many new density models and free energy functions are consequently solved and presented. The phase transition models including critical phenomena with fractional power-law for the discontinuities of the free energies in the matrix models are systematically classified by means of a clear and rigorous mathematical demonstration. The methods here will stimulate new research directions such as the important Seiberg-Witten differential in Seiberg-Witten theory for solving the mass gap problem in quantum Yang-Mills theory. The formulations and results will benefit researchers and students in the fields of phase transitions, integrable systems, matrix models and Seiberg-Witten theory.

The theory of water waves is most varied and is a fascinating topic. It includes a wide range of natural phenomena in oceans, rivers, and lakes. It is mostly concerned with elucidation of some general aspects of wave motion including the prediction of behaviour of waves in the presence of obstacles of some special configurations that are of interest to ocean engineers. Unfortunately, even the apparently simple problems appear to be difficult to tackle mathematically unless some simplified assumptions are made. Fortunately, one can assume water to be an incompressible, in viscid and homogeneous fluid. The linearised theory of water waves is based on the assumption that the amplitude of the motion is small compared to the wave length. If rotational motion is assumed, then the linearised theory of water waves is essentially concerned with solving the Laplace equation in the water region together with linearised boundary condition. There are varied classes of problems that have been/are being studied mathematically in the literature within the framework of linearised theory of water waves for last many years. Scattering by obstacles of various geometrical configurations is one such class of water wave problems. This book is devoted to advanced mathematical work related to water wave scattering. Emphasis is laid on the mathematical and computational techniques required to study these problems mathematically. The book contains nine chapters. The first chapter is introductory in nature. It includes the basic equations of linearised theory for a single layer fluid, a two-layer fluid, solution of dispersion equations, and a general idea on scattering problems and the energy identity in water with a free surface. Chapter 2 is concerned with wave scattering involving thin rigid plates of various geometrical configurations, namely, plane vertical barriers or curved barriers, inclined barriers, horizontal barrier, and also thin elastic vertical plate. For the horizontal case, the barrier is submerged below an ice-cover modelled as a thin elastic plate floating on water. Chapter 3 discusses wave scattering by a rectangular trench by using Galerkin technique. Chapter 4 involves wave scattering by a dock by using Carleman singular integral equation followed by reduction to Riemann-Hilbert problems. Chapter 5 involves several wave scattering problems involving discontinuities at the upper surface of water by using the Wiener-Hopf technique, by reduction to Carleman singular integral equations. Chapter 6 considers scattering by a long horizontal circular cylinder either half immersed or completely submerged. In chapter 7, some important energy identities are derived for scattering problems in a single-layer and also in a two-layer fluid. Chapter 8 is concerned with wave scattering in a two-layer fluid by a thin vertical plate and by a long horizontal circular cylinder submerged in either of the two layers. Chapter 9 is the final chapter which considers a number of wave scattering problems in a single-layer or a two-layer fluid with variable bottom topography by using a simplified perturbation analysis It is hoped that this book will be useful to researchers on water waves. The several wave scattering problems presented in the book are mostly based on the research work carried out by the authors and their associates.

International migration has huge economic, social and political consequences and as a result there is growing demand for evidence-based policies. This has long been unfeasible due to inaccuracies in migration data, caused, for example, by different data sources employing different definitions and different collection methods. However, in recent years methods involving the use of models have been developed to derive reliable estimates of migration. International Migration in Europe draws together modern statistical approaches to provide a unified treatment of these methods, allowing a consistent and dependable set of migration flows to be developed.

The book:

Uses European-based real data to demonstrate migration estimation and modelling tools, with worldwide applications.

Explores the modelling techniques from both the frequentist and Bayesian perspectives.

Discusses vital concepts in detail, such as missing data and collection methods.

Includes chapters dedicated to the key areas of modelling asylum flows and migration forecasting.

Is authored by the leading experts working in migration data and models.

Is supported by a supplementary website featuring further reading material.

International Migration in Europe is primarily aimed at academics, undergraduate and graduate students in the fields of demography, statistics, geography, sociology, economics and political science. However, this is also an important book for government statisticians responsible for publishing migration data, as well as analysts working on migration-related topics such as planning and demography at all levels, from local to international.

A comprehensive, step-by-step introduction to wavelets in statistics.

What are wavelets? What makes them increasingly indispensable in statistical nonparametrics? Why are they suitable for "time-scale" applications? How are they used to solve such problems as denoising, regression, or density estimation? Where can one find up-to-date information on these newly "discovered" mathematical objects? These are some of the questions Brani Vidakovic answers in Statistical Modeling by Wavelets. Providing a much-needed introduction to the latest tools afforded statisticians by wavelet theory, Vidakovic compiles, organizes, and explains in depth research data previously available only in disparate journal articles. He carefully balances both statistical and mathematical techniques, supplementing the material with a wealth of examples, more than 100 illustrations, and extensive references—with data sets and S-Plus wavelet overviews made available for downloading over the Internet. Both introductory and data-oriented modeling topics are featured, including:

• Continuous and discrete wavelet transformations.
• Statistical optimality properties of wavelet shrinkage.
• Theoretical aspects of wavelet density estimation.
• Bayesian modeling in the wavelet domain.
• Properties of wavelet-based random functions and densities.
• Several novel and important wavelet applications in statistics.
• Wavelet methods in time series.

  Accessible to anyone with a background in advanced calculus and algebra, Statistical Modeling by Wavelets promises to become the standard reference for statisticians and engineers seeking a comprehensive introduction to an emerging field.

A typical subsystem found in almost all aircraft and space vehicles consists of beam, plate and/or shell elements attached to each other in a rigid or flexible manner. Due to limitations on their weights, the elements themselves must be highly flexible, and due to limitations on their initial configuration (i.e., before deployment), those aggregates often have to contain several links so that the substructure may be unfolded or telescoped once it is deployed. The defining philosophy of this monograph is that in order to understand completely the dynamic response of such a complex elastic structure, it is not sufficient to consider only its global motion but also necessary to take into account the flexibility of individual elements and the interaction and transmission of elastic effects such as bending, torsion, and axial deformations at junctions where members are connected to each other. Therefore, the purposes of this monograph are: to derive distributed parameter models of the transient behavior of some or all of the state and interval variables which describe the dynamic response of multiple-link flexible structures such as trusses, frames, robot arms, solar panels, antennae and deformable mirrors, based on the principles of continuum mechanics and under reasonable constraints on the geometry of the admissible deformations; to provide rigorous mathematical analyses of the resulting models; and to develop control theoretic properties of multiple-link flexible structures based on the control theoretic properties of the models. The modelling and analysis of these complicated and realistic structural configurations should be of interest to a diverse group of applied mathematicians, structural, aeronautical, aerospace, and mechanical engineers and to advanced graduate students working on such problems.

This book introduces several mathematical models in assembly line balancing based on stochastic programming and develops exact and heuristic methods to solve them. An assembly line system is a manufacturing process in which parts are added in sequence from workstation to workstation until the final assembly is produced. In an assembly line balancing problem, tasks belonging to different product models are allocated to workstations according to their processing times and precedence relationships among tasks. It incorporates two features, uncertain task times, and demand volatility, separately and simultaneously, into the conventional assembly line balancing model. A real-life case study related to the mask production during the COVID-19 pandemic is presented to illustrate the application of the proposed framework and methodology. The book is intended for graduate students who are interested in combinatorial optimizations in manufacturing with uncertain input.

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Quantitative Methods for Finance and Investments" ensures that readers will gain a reasonable degree of comfort and proficiency in applying elementary mathematics to financial analysis in a variety of areas. All of the methodology in this book is geared toward the development, implementation, and analysis of financial models to solve problems encountered by both finance students and practitioners. The book:
analyzes theoretical and practitioner-oriented models long with the mathematics required to construct them
presents the most essential mathematical techniques and their applications to financial analysis
provides dozens of practical applications, examples, and end-of-chapter exercises with detailed solutions
demonstrates key spreadsheet applications of the mathematical models in chapter appendices
emphasizes practical applications of modeling technique.

Field Programmable Gate Arrays (FPGAs) are currently recognized as the most suitable platform for the implementation of complex digital systems targeting an increasing number of industrial electronics applications. They cover a huge variety of application areas, such as: aerospace, food industry, art, industrial automation, automotive, biomedicine, process control, military, logistics, power electronics, chemistry, sensor networks, robotics, ultrasound, security, and artificial vision. This book first presents the basic architectures of the devices to familiarize the reader with the fundamentals of FPGAs before identifying and discussing new resources that extend the ability of the devices to solve problems in new application domains. Design methodologies are discussed and application examples are included for some of these domains, e.g., mechatronics, robotics, and power systems.

Modelling is now an accepted part in the understanding, prediction and planning of environmental strategies. Perfect for undergraduate students and non-specialist readers, Modelling Coastal and Marine Processes (2nd Edition) offers an introduction into how coastal and marine models are constructed and used.The mathematics, statistics and numerical techniques used are explained in the first few chapters, making this book accessible to those without a high-level maths background. Later chapters cover modelling sea bed friction, tides, shallow sea dynamics, and ecosystem dynamics. Importantly, there is also a chapter on modelling the impact of climate change on coastal and near shore processes.New to this revised edition is a chapter on tides, tsunamis and the prediction of sea level, and additional material on the new application of the numerical techniques: flux corrected transport, finite volumes and adaptive grids to coastal and marine modelling.