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This book presents fixed point theory, one of the crucial tools in applied mathematics, functional analysis, and topology, which has been used to solve distinct real-world problems in computer science, engineering, and physics. The authors begin with an overview of the extension of metric spaces. Readers are introduced to general fixed-point theorems while comparing and contrasting important and insignificant metric spaces. The book is intended to be self-contained and serves as a unique resource for researchers in various disciplines.

This textbook provides an introduction to methods for solving nonlinear partial differential equations (NLPDEs). After the introduction of several PDEs drawn from science and engineering, readers are introduced to techniques to obtain exact solutions of NLPDEs. The chapters include the following topics: Nonlinear PDEs are Everywhere; Differential Substitutions; Point and Contact Transformations; First Integrals; and Functional Separability. Readers are guided through these chapters and are provided with several detailed examples. Each chapter ends with a series of exercises illustrating the material presented in each chapter. This Second Edition includes a new method of generating contact transformations and focuses on a solution method (parametric Legendre transformations) to solve a particular class of two nonlinear PDEs.

This book studies a category of mathematical objects called Hamiltonians, which are dependent on both time and momenta. The authors address the development of the distinguished geometrization on dual 1-jet spaces for time-dependent Hamiltonians, in contrast with the time-independent variant on cotangent bundles. Two parts are presented to include both geometrical theory and the applicative models: Part One: Time-dependent Hamilton Geometry and Part Two: Applications to Dynamical Systems, Economy and Theoretical Physics. The authors present 1-jet spaces and their duals as appropriate fundamental ambient mathematical spaces used to model classical and quantum field theories. In addition, the authors present dual jet Hamilton geometry as a distinct metrical approach to various interdisciplinary problems.

This book explores the relationships between music, the sciences, and mathematics, both ancient and modern, with a focus on the big picture for a general audience as opposed to delving into very technical details. The language of music is deciphered through the language of mathematics. Readers are shown how apparently unrelated areas of knowledge complement each other and in fact propel each other¿s advancement. The presentation as well as the collection of topics covered throughout is unique and serves to encourage exploration and also, very concretely, illustrates the cross- and multidisciplinary nature of knowledge. Inspired by an introductory, multidisciplinary course, the author explores the relationships between the arts, sciences, and mathematics in the realm of music. The book has no prerequisites; rather it aims to give a broad overview and achieve the integration of the three presented themes. Mathematical tools are introduced and used to explain various aspects of music theory, and the author illustrates how, without mathematics, music could not have been developed.

This book is intended for a first-semester course in calculus, which begins by posing a question: how do we model an epidemic mathematically? The authors use this question as a natural motivation for the study of calculus and as a context through which central calculus notions can be understood intuitively. The book¿s approach to calculus is contextual and based on the principle that calculus is motivated and elucidated by its relevance to the modeling of various natural phenomena. The authors also approach calculus from a computational perspective, explaining that many natural phenomena require analysis through computer methods. As such, the book also explores some basic programming notions and skills.

This book provides practical demonstrations of how to carry out definite integrals with Monte Carlo methods using Mathematica. Random variates are sampled by the inverse transform method and the acceptance-rejection method using uniform, linear, Gaussian, and exponential probability distribution functions. A chapter on the application of the Variational Quantum Monte Carlo method to a simple harmonic oscillator is included. These topics are all essential for students of mathematics and physics. The author includes thorough background on each topic covered within the book in order to help readers understand the subject. The book also contains many examples to show how the methods can be applied.

This book systematically establishes the almost periodic theory of dynamic equations and presents applications on time scales in fuzzy mathematics and uncertainty theory. The authors introduce a new division of fuzzy vectors depending on a determinant algorithm and develop a theory of almost periodic fuzzy multidimensional dynamic systems on time scales. Several applications are studied; in particular, a new type of fuzzy dynamic systems called fuzzy q-dynamic systems (i.e. fuzzy quantum dynamic systems) is presented. The results are not only effective on classical fuzzy dynamic systems, including their continuous and discrete situations, but are also valid for other fuzzy multidimensional dynamic systems on various hybrid domains. In an effort to achieve more accurate analysis in real world applications, the authors propose a number of uncertain factors in the theory. As such, fuzzy dynamical models, interval-valued functions, differential equations, fuzzy-valued differential equations, and their applications to dynamic equations on time scales are considered.

A problem factory consists of a traditional mathematical analysis of a type of problem that describes many, ideally all, ways that the problems of that type can be cast in a fashion that allows teachers or parents to generate problems for enrichment exercises, tests, and classwork. Some problem factories are easier than others for a teacher or parent to apply, so we also include banks of example problems for users. This text goes through the definition of a problem factory in detail and works through many examples of problem factories. It gives banks of questions generated using each of the examples of problem factories, both the easy ones and the hard ones. This text looks at sequence extension problems (what number comes next?), basic analytic geometry, problems on whole numbers, diagrammatic representations of systems of equations, domino tiling puzzles, and puzzles based on combinatorial graphs. The final chapter previews other possible problem factories.

This is the second part of our book on continuous statistical distributions. It covers inverse-Gaussian, Birnbaum-Saunders, Pareto, Laplace, central ,,,,2, ,,,,, ,,,,, Weibull, Rayleigh, Maxwell, and extreme value distributions. Important properties of these distribution are documented, and most common practical applications are discussed. This book can be used as a reference material for graduate courses in engineering statistics, mathematical statistics, and econometrics. Professionals and practitioners working in various fields will also find some of the chapters to be useful.Although an extensive literature exists on each of these distributions, we were forced to limit the size of each chapter and the number of references given at the end due to the publishing plan of this book that limits its size. Nevertheless, we gratefully acknowledge the contribution of all those authors whose names have been left out.Some knowledge in introductory algebra and college calculus is assumed throughout the book. Integration is extensively used in several chapters, and many results discussed in Part I (Chapters 1 to 9) of our book are used in this volume.Chapter 10 is on Inverse Gaussian distribution and its extensions. The Birnbaum-Saunders distribution and its extensions along with applications in actuarial sciences is discussed in Chapter 11. Chapter 12 discusses Pareto distribution and its extensions. The Laplace distribution and its applications in navigational errors is discussed in the next chapter. This is followed by central chi-squared distribution and its applications in statistical inference, bioinformatics and genomics. Chapter 15 discusses Student's ,,,, distribution, its extensions and applications in statistical inference. The ,,,, distribution and its applications in statistical inference appears next. Chapter 17 is on Weibull distribution and its applications in geology and reliability engineering. Next two chapters are on Rayleigh and Maxwell distributions and its applications in communications, wind energy modeling, kinetic gas theory, nuclear and thermal engineering, and physical chemistry. The last chapter is on Gumbel distribution, its applications in the law of rare exceedances.Suggestions for improvement are welcome. Please send them to rajan.chattamvelli@vit.ac.in.

This text provides an introduction to the applications and implementations of partial differential equations. The content is structured in three progressive levels which are suited for upper-level undergraduates with background in multivariable calculus and elementary linear algebra (chapters 1-5), first- and second-year graduate students who have taken advanced calculus and real analysis (chapters 6-7), as well as doctoral-level students with an understanding of linear and nonlinear functional analysis (chapters 7-8) respectively. Level one gives readers a full exposure to the fundamental linear partial differential equations of physics. It details methods to understand and solve these equations leading ultimately to solutions of Maxwell's equations. Level two addresses nonlinearity and provides examples of separation of variables, linearizing change of variables, and the inverse scattering transform for select nonlinear partial differential equations. Level three presents rich sources of advanced techniques and strategies for the study of nonlinear partial differential equations, including unique and previously unpublished results. Ultimately the text aims to familiarize readers in applied mathematics, physics, and engineering with some of the myriad techniques that have been developed to model and solve linear and nonlinear partial differential equations.

Computational analysis of natural science experiments often confronts noisy data due to natural variability in environment or measurement. Drawing conclusions in the face of such noise entails a statistical analysis. Parametric statistical methods assume that the data is a sample from a population that can be characterized by a specific distribution (e.g., a normal distribution). When the assumption is true, parametric approaches can lead to high confidence predictions. However, in many cases particular distribution assumptions do not hold. In that case, assuming a distribution may yield false conclusions. The companion book Statistics is Easy, gave a (nearly) equation-free introduction to nonparametric (i.e., no distribution assumption) statistical methods. The present book applies data preparation, machine learning, and nonparametric statistics to three quite different life science datasets. We provide the code as applied to each dataset in both R and Python 3. We also include exercises for self-study or classroom use.

Book V completes the discussion of the first four books by treating in some detail the analytic results in elliptic operator theory used previously. Chapters 16 and 17 provide a treatment of the techniques in Hilbert space, the Fourier transform, and elliptic operator theory necessary to establish the spectral decomposition theorem of a self-adjoint operator of Laplace type and to prove the Hodge Decomposition Theorem that was stated without proof in Book II. In Chapter 18, we treat the de Rham complex and the Dolbeault complex, and discuss spinors. In Chapter 19, we discuss complex geometry and establish the Kodaira Embedding Theorem.

This is an introductory book on continuous statistical distributions and its applications. It is primarily written for graduate students in engineering, undergraduate students in statistics, econometrics, and researchers in various fields. The purpose is to give a self-contained introduction to most commonly used classical continuous distributions in two parts. Important applications of each distribution in various applied fields are explored at the end of each chapter. A brief overview of the chapters is as follows. Chapter 1 discusses important concepts on continuous distributions like location-and-scale distributions, truncated, size-biased, and transmuted distributions. A theorem on finding the mean deviation of continuous distributions, and its applications are also discussed. Chapter 2 is on continuous uniform distribution, which is used in generating random numbers from other distributions. Exponential distribution is discussed in Chapter 3, and its applications briefly mentioned. Chapter 4 discusses both Beta-I and Beta-II distributions and their generalizations, as well as applications in geotechnical engineering, PERT, control charts, etc. The arcsine distribution and its variants are discussed in Chapter 5, along with arcsine transforms and Brownian motion. This is followed by gamma distribution and its applications in civil engineering, metallurgy, and reliability. Chapter 7 is on cosine distribution and its applications in signal processing, antenna design, and robotics path planning. Chapter 8 discusses the normal distribution and its variants like lognormal, and skew-normal distributions. The last chapter of Part I is on Cauchy distribution, its variants and applications in thermodynamics, interferometer design, and carbon-nanotube strain sensing. A new volume (Part II) covers inverse Gaussian, Laplace, Pareto, ,,,,2, T, F, Weibull, Rayleigh, Maxwell, and Gumbel distributions.

This book is intended for undergraduate students of Mathematics, Statistics, and Physics who know nothing about Monte Carlo Methods but wish to know how they work. All treatments have been done as much manually as is practicable. The treatments are deliberately manual to let the readers get the real feel of how Monte Carlo Methods work. Definite integrals of a total of five functions ,,,,(,,,,), namely Sin(,,,,), Cos(,,,,), e,,,,, loge(,,,,), and 1/(1+,,,,2), have been evaluated using constant, linear, Gaussian, and exponential probability density functions ,,,,(,,,,). It is shown that results agree with known exact values better if ,,,,(,,,,) is proportional to ,,,,(,,,,). Deviation from the proportionality results in worse agreement. This book is on Monte Carlo Methods which are numerical methods for Computational Physics. These are parts of a syllabus for undergraduate students of Mathematics and Physics for the course titled "e;Computational Physics."e;Need for the book: Besides the three referenced books, this is the only book that teaches how basic Monte Carlo methods work. This book is much more explicit and easier to follow than the three referenced books. The two chapters on the Variational Quantum Monte Carlo method are additional contributions of the book. Pedagogical features: After a thorough acquaintance with background knowledge in Chapter 1, five thoroughly worked out examples on how to carry out Monte Carlo integration is included in Chapter 2. Moreover, the book contains two chapters on the Variational Quantum Monte Carlo method applied to a simple harmonic oscillator and a hydrogen atom. The book is a good read; it is intended to make readers adept at using the method. The book is intended to aid in hands-on learning of the Monte Carlo methods.

The contents of this brief Lecture Note are devoted to modeling, simulations, and applications with the aim of proposing a unified multiscale approach accounting for the physics and the psychology of people in crowds. The modeling approach is based on the mathematical theory of active particles, with the goal of contributing to safety problems of interest for the well-being of our society, for instance, by supporting crisis management in critical situations such as sudden evacuation dynamics induced through complex venues by incidents.

This is an introductory book on discrete statistical distributions and its applications. It discusses only those that are widely used in the applications of probability and statistics in everyday life. The purpose is to give a self-contained introduction to classical discrete distributions in statistics. Instead of compiling the important formulas (which are available in many other textbooks), we focus on important applications of each distribution in various applied fields like bioinformatics, genomics, ecology, electronics, epidemiology, management, reliability, etc., making this book an indispensable resource for researchers and practitioners in several scientific fields. Examples are drawn from different fields. An up-to-date reference appears at the end of the book.Chapter 1 introduces the basic concepts on random variables, and gives a simple method to find the mean deviation (MD) of discrete distributions. The Bernoulli and binomial distributions are discussed in detail in Chapter 2. A short chapter on discrete uniform distribution appears next. The next two chapters are on geometric and negative binomial distributions. Chapter 6 discusses the Poisson distribution in-depth, including applications in various fields. Chapter 7 is on hypergeometric distribution. As most textbooks in the market either do not discuss, or contain only brief description of the negative hypergeometric distribution, we have included an entire chapter on it. A short chapter on logarithmic series distribution follows it, in which a theorem to find the kth moment of logarithmic distribution using (k-1)th moment of zero-truncated geometric distribution is presented. The last chapter is on multinomial distribution and its applications.The primary users of this book are professionals and practitioners in various fields of engineering and the applied sciences. It will also be of use to graduate students in statistics, research scholars in science disciplines, and teachers of statistics, biostatistics, biotechnology, education, and psychology.

Uncertainty is an inseparable component of almost every measurement and occurrence when dealing with real-world problems. Finding solutions to real-life problems in an uncertain environment is a difficult and challenging task. As such, this book addresses the solution of uncertain static and dynamic problems based on affine arithmetic approaches. Affine arithmetic is one of the recent developments designed to handle such uncertainties in a different manner which may be useful for overcoming the dependency problem and may compute better enclosures of the solutions. Further, uncertain static and dynamic problems turn into interval and/or fuzzy linear/nonlinear systems of equations and eigenvalue problems, respectively. Accordingly, this book includes newly developed efficient methods to handle the said problems based on the affine and interval/fuzzy approach. Various illustrative examples concerning static and dynamic problems of structures have been investigated in order to show the reliability and efficacy of the developed approaches.

The subject of fractional calculus has gained considerable popularity and importance during the past three decades, mainly due to its validated applications in various fields of science and engineering. It is a generalization of ordinary differentiation and integration to arbitrary (non-integer) order. The fractional derivative has been used in various physical problems, such as frequency-dependent damping behavior of structures, biological systems, motion of a plate in a Newtonian fluid, ,,,,,,,,I ,,,,? controller for the control of dynamical systems, and so on. It is challenging to obtain the solution (both analytical and numerical) of related nonlinear partial differential equations of fractional order. So for the last few decades, a great deal of attention has been directed towards the solution for these kind of problems. Different methods have been developed by other researchers to analyze the above problems with respect to crisp (exact) parameters.However, in real-life applications such as for biological problems, it is not always possible to get exact values of the associated parameters due to errors in measurements/experiments, observations, and many other errors. Therefore, the associated parameters and variables may be considered uncertain. Here, the uncertainties are considered interval/fuzzy. Therefore, the development of appropriate efficient methods and their use in solving the mentioned uncertain problems are the recent challenge.In view of the above, this book is a new attempt to rigorously present a variety of fuzzy (and interval) time-fractional dynamical models with respect to different biological systems using computationally efficient method. The authors believe this book will be helpful to undergraduates, graduates, researchers, industry, faculties, and others throughout the globe.

This book continues the material in two early Fast Start calculus volumes to include multivariate calculus, sequences and series, and a variety of additional applications. These include partial derivatives and the optimization techniques that arise from them, including Lagrange multipliers. Volumes of rotation, arc length, and surface area are included in the additional applications of integration. Using multiple integrals, including computing volume and center of mass, is covered. The book concludes with an initial treatment of sequences, series, power series, and Taylor's series, including techniques of function approximation.

This book introduces integrals, the fundamental theorem of calculus, initial value problems, and Riemann sums. It introduces properties of polynomials, including roots and multiplicity, and uses them as a framework for introducing additional calculus concepts including Newton's method, L'Hopital's Rule, and Rolle's theorem. Both the differential and integral calculus of parametric, polar, and vector functions are introduced. The book concludes with a survey of methods of integration, including u-substitution, integration by parts, special trigonometric integrals, trigonometric substitution, and partial fractions.

This book reviews the algebraic prerequisites of calculus, including solving equations, lines, quadratics, functions, logarithms, and trig functions. It introduces the derivative using the limit-based definition and covers the standard function library and the product, quotient, and chain rules. It explores the applications of the derivative to curve sketching and optimization and concludes with the formal definition of the limit, the squeeze theorem, and the mean value theorem.

Introduction to Statistics Using R is organized into 13 major chapters. Each chapter is broken down into many digestible subsections in order to explore the objectives of the book. There are many real-life practical examples in this book and each of the examples is written in R codes to acquaint the readers with some statistical methods while simultaneously learning R scripts.

The inverse obstacle scattering problem consists of finding the unknown surface of a body (obstacle) from the scattering ,,,,(,,,,;,,,,;,,,,), where ,,,,(,,,,;,,,,;,,,,) is the scattering amplitude, ,,,,;,,,, ,,,, ,,,,2 is the direction of the scattered, incident wave, respectively, ,,,,2 is the unit sphere in the and k > 0 is the modulus of the wave vector. The scattering data is called non-over-determined if its dimensionality is the same as the one of the unknown object. By the dimensionality one understands the minimal number of variables of a function describing the data or an object. In an inverse obstacle scattering problem this number is 2, and an example of non-over-determined data is ,,,,(,,,,) := ,,,,(,,,,;,,,,,, By sub-index 0 a fixed value of a variable is denoted.</p>It is proved in this book that the data ,,,,(,,,,), known for all ,,,, in an open subset of ,,,, determines uniquely the surface ,,,, and the boundary condition on ,,,,. This condition can be the Dirichlet, or the Neumann, or the impedance type.</p>The above uniqueness theorem is of principal importance because the non-over-determined data are the minimal data determining uniquely the unknown ,,,,. There were no such results in the literature, therefore the need for this book arose. This book contains a self-contained proof of the existence and uniqueness of the scattering solution for rough surfaces.</p>

This is an introduction to methods for solving nonlinear partial differential equations (NLPDEs). After the introduction of several PDEs drawn from science and engineering, the reader is introduced to techniques used to obtain exact solutions of NPDEs. The chapters include the following topics: Compatibility, Differential Substitutions, Point and Contact Transformations, First Integrals, and Functional Separability. The reader is guided through these chapters and is provided with several detailed examples. Each chapter ends with a series of exercises illustrating the material presented in each chapter.The book can be used as a textbook for a second course in PDEs (typically found in both science and engineering programs) and has been used at the University of Central Arkansas for more than ten years.

Book IV continues the discussion begun in the first three volumes. Although it is aimed at first-year graduate students, it is also intended to serve as a basic reference for people working in affine differential geometry. It also should be accessible to undergraduates interested in affine differential geometry. We are primarily concerned with the study of affine surfaces {which} are locally homogeneous. We discuss affine gradient Ricci solitons, affine Killing vector fields, and geodesic completeness. Opozda has classified the affine surface geometries which are locally homogeneous; we follow her classification. Up to isomorphism, there are two simply connected Lie groups of dimension 2. The translation group is Abelian and the ,,,,,,,, + ,,,, group\index{ax+b group} is non-Abelian. The first chapter presents foundational material. The second chapter deals with Type ,,,, surfaces. These are the left-invariant affine geometries on . Associating to each Type ,,,, surface the space of solutions to the quasi-Einstein equation corresponding to the eigenvalue ,,,,=-1$ turns out to be a very powerful technique and plays a central role in our study as it links an analytic invariant with the underlying geometry of the surface. The third chapter deals with Type ,,,, surfaces; these are the left-invariant affine geometries on the ,,,,,,,, + ,,,, group. These geometries form a very rich family which is only partially understood. The only remaining homogeneous geometry is that of the sphere ,,,,2. The fourth chapter presents relations between the geometry of an affine surface and the geometry of the cotangent bundle equipped with the neutral signature metric of the modified Riemannian extension.

This book gives a necessary and sufficient condition in terms of the scattering amplitude for a scatterer to be spherically symmetric. By a scatterer we mean a potential or an obstacle. It also gives necessary and sufficient conditions for a domain to be a ball if an overdetermined boundary problem for the Helmholtz equation in this domain is solvable. This includes a proof of Schiffer's conjecture, the solution to the Pompeiu problem, and other symmetry problems for partial differential equations. It goes on to study some other symmetry problems related to the potential theory. Among these is the problem of "e;invisible obstacles."e; In Chapter 5, it provides a solution to the Navier‒Stokes problem in ℝ³. The author proves that this problem has a unique global solution if the data are smooth and decaying sufficiently fast. A new a priori estimate of the solution to the Navier‒Stokes problem is also included. Finally, it delivers a solution to inverse problem of the potential theory without the standard assumptions about star-shapeness of the homogeneous bodies.

lt;p>Atherosclerosis is a pathological condition of the arteries in which plaque buildup and stiffening (hardening) can lead to stroke, myocardial infarction (heart attacks), and even death. Cholesterol in the blood is a key marker for atherosclerosis, with two forms: (1) LDL - low density lipoproteins and (2) HDL - high density lipoproteins. Low LDL and high HDL concentrations are generally considered essential for limited atherosclerosis and good health.</p><div><p>This book pertains to a mathematical model for the spatiotemporal distribution of LDL and HDL in the arterial endothelial inner layer (EIL, intima). The model consists of a system of six partial differential equations (PDEs) with the dependent variables</p></div><div><p>1. ,,,,(,,,,,,,,,): concentration of modified LDL</p></div><div><p>2. ,,,,,,,): concentration of HDL</p></div><div><p>3. ,,,,(,,,,,,,,,): concentration of chemoattractants</p></div><div><p>4. ,,,,(,,,,,,,,,): concentration of ES cytokines</p></div><div><p>5. ,,,,(,,,,,,,,,): density of monocytes/macrophages</p></div><div><p>6. ,,,,(,,,,,,,,,): density of foam cells</p></div><div><p>and independent variables</p></div><div><p>1. ,,,,: distance from the inner arterial wall</p></div><div><p>2. ,,,,: time</p></div><div><p>The focus of this book is a discussion of the methodology for placing the model on modest computers for study of the numerical solutions. The foam cell density ,,,,(,,,,,,,,,) as a function of the bloodstream LDL and HDL concentrations is of particular interest as a precursor for arterial plaque formation and stiffening.</p></div><p>The numerical algorithm for the solution of the model PDEs is the method of lines (MOL), a general procedure for the computer-based numerical solution of PDEs. The MOL coding (programming) is in R, a quality, open-source scientific computing system that is readily available from the Internet. The R routines for the PDE model are discussed in detail, and are available from a download link so that the reader/analyst/researcher can execute the model to duplicate the solutions reported in the book, then experiment with the model, for example, by changing the parameters (constants) and extending the model with additional equations.</p></div>

Differential Geometry is a wide field. We have chosen to concentrate upon certain aspects that are appropriate for an introduction to the subject; we have not attempted an encyclopedic treatment. Book II deals with more advanced material than Book I and is aimed at the graduate level. Chapter 4 deals with additional topics in Riemannian geometry. Properties of real analytic curves given by a single ODE and of surfaces given by a pair of ODEs are studied, and the volume of geodesic balls is treated. An introduction to both holomorphic and Khler geometry is given. In Chapter 5, the basic properties of de Rham cohomology are discussed, the Hodge Decomposition Theorem, Poincar duality, and the Knneth formula are proved, and a brief introduction to the theory of characteristic classes is given. In Chapter 6, Lie groups and Lie algebras are dealt with. The exponential map, the classical groups, and geodesics in the context of a bi-invariant metric are discussed. The de Rham cohomology of compact Lie groups and the Peter--Weyl Theorem are treated. In Chapter 7, material concerning homogeneous spaces and symmetric spaces is presented. Book II concludes in Chapter 8 where the relationship between simplicial cohomology, singular cohomology, sheaf cohomology, and de Rham cohomology is established. We have given some different proofs than those that are classically given and there is some new material in these volumes. For example, the treatment of the total curvature and length of curves given by a single ODE is new as is the discussion of the total Gaussian curvature of a surface defined by a pair of ODEs.