Functional Analysis
This textbook presents the principles of functional analysis in a clear and concise way. The first three chapters describe the general notions of distance, integral, and norm, as well as their relations. Fundamental examples are provided in the three chapters that follow: Lebesgue spaces, dual spaces, and Sobolev spaces. Two subsequent chapters develop applications to capacity theory and elliptic problems. In particular, the isoperimetric inequality and the P籀lya-Szegő and Faber-Krahn inequalities are proved by purely functional methods. The epilogue contains a sketch of the history of functional analysis in relation to integration and differentiation. Starting from elementary analysis and introducing relevant research, this work is an excellent resource for students in mathematics and applied mathematics. The second edition of Functional Analysis includes several improvements as well as the addition of supplementary material. Specifically, the coverage of advanced calculus and distribution theory has been completely rewritten and expanded. New proofs, theorems, and applications have been added as well for readers to explore.
Getting Started in Mathematical Life Sciences
This book helps the reader make use of the mathematical models of biological phenomena starting from the basics of programming and computer simulation. Computer simulations based on a mathematical model enable us to find a novel biological mechanism and predict an unknown biological phenomenon. Mathematical biology could further expand the progress of modern life sciences. Although many biologists are interested in mathematical biology, they do not have experience in mathematics and computer science. An educational course that combines biology, mathematics, and computer science is very rare to date. Published books for mathematical biology usually explain the theories of established mathematical models, but they do not provide a practical explanation for how to solve the differential equations included in the models, or to establish such a model that fits with a phenomenon of interest. MATLAB is an ideal programming platform for the beginners of computer science. This book starts from the very basics about how to write a programming code for MATLAB (or Octave), explains how to solve ordinary and partial differential equations, and how to apply mathematical models to various biological phenomena such as diabetes, infectious diseases, and heartbeats. Some of them are original models, newly developed for this book. Because MATLAB codes are embedded and explained throughout the book, it will be easy to catch up with the text. In the final chapter, the book focuses on the mathematical model of the proneural wave, a phenomenon that guarantees the sequential differentiation of neurons in the brain. This model was published as a paper from the author's lab (Sato et al., PNAS 113, E5153, 2016), and was intensively explained in the book chapter "Notch Signaling in Embryology and Cancer", published by Springer in 2020. This book provides the reader who has a biological background with invaluable opportunities to learn and practice mathematical biology.
A Modern Approach to Teaching an Introduction to Optimization
Optimization should be the science of making the best possible decisions. Making decisions is a virtually universal human activity encountered by professionals (in any field) or people in their everyday lives. You would think, then, that the study of making good decisions is a subject that should be taught broadly to students throughout engineering, the physical and social sciences, business, and policy. Yet today, "optimization" is widely taught as a mathematically sophisticated subject, often limited to graduate students in specialized fields. In operations research (or industrial engineering), "optimization" is equivalent to deterministic math programming, starting with linear programs (and the simplex algorithm), and then transitioning through integer linear programs and nonlinear programs. If you are in departments like electrical or mechanical engineering, optimization means teaching optimal control. And if you are in computer science, optimization today could be interpreted in the context of machine learning (such as fitting models to data) or as reinforcement learning. This book claims that the traditional style of teaching optimization is misguided and out of date. First, while the simplex algorithm is a powerful strategy for solving linear programs, the details of the simplex algorithm are completely inappropriate in an introductory course in optimization. Second, while linear programs are appropriate for solving many problems, they are only applicable to a tiny fraction of all decisions. Third, linear programs (along with integer and nonlinear programs) are static models for problems with (typically) vector-valued decisions. By contrast, most decisions are sequential since they are made periodically over time as new information is arriving. In addition, the vast majority of these decisions are scalar (possibly continuous or discrete). This book is designed for instructors (or potential instructors) looking to introduce the science of making good decisions to the broadest possible audience. It should also be of interest to anyone who has already had a traditional course in optimization of any type. The presentation is organized around a series of topics that suggest a fundamentally different approach to teaching "optimization" spanning both sequential decision problems (which offer the simplest problem settings) before transitioning to more complex vector-valued decisions. It also makes the case that most problems which are modeled as linear (or integer, or nonlinear programs) are actually methods for making decisions in a sequential setting. For this reason, these topics are introduced with much less emphasis on algorithms than is traditionally used, both in static and sequential settings.
Introducing Game Theory and Its Applications
Introducing Game Theory and its Applications presents an easy-to-read introduction to the basic ideas and techniques of game theory.
Divine Games
A game-theoretical analysis of interactions between a human being and an omnipotent and omniscient godlike being highlights the inherent unknowability of the latter's superiority. In Divine Games, Steven Brams analyzes games that a human being might play with an omnipotent and omniscient godlike being. Drawing on game theory and his own theory of moves, Brams combines the analysis of thorny theological questions, suggested by Pascal's wager (which considers the rewards and penalties associated with belief or nonbelief in God) and Newcomb's problem (in which a godlike being has near omniscience) with the analysis of several stories from the Hebrew Bible. Almost all of these stories involve conflict between God or a surrogate and a human player; their representation as games raises fundamental questions about God's superiority. In some games God appears vulnerable (after Adam and Eve eat the forbidden fruit in defiance of His command), in other games his actions seem morally dubious (when He subjects Abraham and Job to extreme tests of their faith), and in still other games He has a propensity to hold grudges (in preventing Moses from entering the Promised Land and in undermining the kingship of Saul). If the behavior of a superior being is indistinguishable from that of an ordinary human being, his existence would appear undecidable, or inherently unknowable. Consequently, Brams argues that keeping an open mind about the existence of a superior being is an appropriate theological stance.
Student Solutions Manual for Non Linear Dynamics and Chaos
This official Student Solutions Manual includes solutions to the odd-numbered exercises featured in the third edition of Steven Strogatz's classic text Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering.
Discrete Choice Analysis with R
This book is designed as a gentle introduction to the fascinating field of choice modeling and its practical implementation using the R language. Discrete choice analysis is a family of methods useful to study individual decision-making. With strong theoretical foundations in consumer behavior, discrete choice models are used in the analysis of health policy, transportation systems, marketing, economics, public policy, political science, urban planning, and criminology, to mention just a few fields of application. The book does not assume prior knowledge of discrete choice analysis or R, but instead strives to introduce both in an intuitive way, starting from simple concepts and progressing to more sophisticated ideas. Loaded with a wealth of examples and code, the book covers the fundamentals of data and analysis in a progressive way. Readers begin with simple data operations and the underlying theory of choice analysis and conclude by working with sophisticated models including latent class logit models, mixed logit models, and ordinal logit models with taste heterogeneity. Data visualization is emphasized to explore both the input data as well as the results of models. This book should be of interest to graduate students, faculty, and researchers conducting empirical work using individual level choice data who are approaching the field of discrete choice analysis for the first time. In addition, it should interest more advanced modelers wishing to learn about the potential of R for discrete choice analysis. By embedding the treatment of choice modeling within the R ecosystem, readers benefit from learning about the larger R family of packages for data exploration, analysis, and visualization.
Random Graphs and Complex Networks: Volume 2
Complex networks are key to describing the connected nature of the society that we live in. This book, the second of two volumes, describes the local structure of random graph models for real-world networks and determines when these models have a giant component and when they are small-, and ultra-small, worlds. This is the first book to cover the theory and implications of local convergence, a crucial technique in the analysis of sparse random graphs. Suitable as a resource for researchers and PhD-level courses, it uses examples of real-world networks, such as the Internet and citation networks, as motivation for the models that are discussed, and includes exercises at the end of each chapter to develop intuition. The book closes with an extensive discussion of related models and problems that demonstratemodern approaches to network theory, such as community structure and directed models.
An Introduction to Partial Differential Equations with MATLAB
The first two editions of An Introduction to Partial Differential Equations with MATLAB(R) gained popularity among instructors and students at various universities throughout the world. Plain mathematical language is used in a friendly manner to provide a basic introduction to partial differential equations (PDEs).Suitable for a one- or two-semester introduction to PDEs and Fourier series, the book strives to provide physical, mathematical, and historical motivation for each topic. Equations are studied based on method of solution, rather than on type of equation.This third edition of this popular textbook updates the structure of the book by increasing the role of the computational portion, compared to previous editions. The redesigned content will be extremely useful for students of mathematics, physics, and engineering who would like to focus on the practical aspects of the study of PDEs, without sacrificing mathematical rigor. The authors have maintained flexibility in the order of topics.In addition, students will be able to use what they have learned in some later courses (for example, courses in numerical analysis, optimization, and PDE-based programming). Included in this new edition is a substantial amount of material on reviewing computational methods for solving ODEs (symbolically and numerically), visualizing solutions of PDEs, using MATLAB(R)'s symbolic programming toolbox, and applying various schemes from numerical analysis, along with suggestions for topics of course projects.Students will use sample MATLAB(R) or Python codes available online for their practical experiments and for completing computational lab assignments and course projects.
Fundamentals of Real and Complex Analysis
The primary aim of this text is to help transition undergraduates to study graduate level mathematics. It unites real and complex analysis after developing the basic techniques and aims at a larger readership than that of similar textbooks that have been published, as fewer mathematical requisites are required. The idea is to present analysis as a whole and emphasize the strong connections between various branches of the field. Ample examples and exercises reinforce concepts, and a helpful bibliography guides those wishing to delve deeper into particular topics. Graduate students who are studying for their qualifying exams in analysis will find use in this text, as well as those looking to advance their mathematical studies or who are moving on to explore another quantitative science.Chapter 1 contains many tools for higher mathematics; its content is easily accessible, though not elementary. Chapter 2 focuses on topics in real analysis such as p-adic completion, Banach Contraction Mapping Theorem and its applications, Fourier series, Lebesgue measure and integration. One of this chapter's unique features is its treatment of functional equations. Chapter 3 covers the essential topics in complex analysis: it begins with a geometric introduction to the complex plane, then covers holomorphic functions, complex power series, conformal mappings, and the Riemann mapping theorem. In conjunction with the Bieberbach conjecture, the power and applications of Cauchy's theorem through the integral formula and residue theorem are presented.
Recent Trends and Future Challenges in Learning from Data
This book collects together selected peer-reviewed contributions presented at the European Conference on Data Analysis, ECDA 2022, held in Naples, Italy, September 14-16, 2022. Highlighting the role of statistics in discovering novel and interesting patterns in the era of big data, it follows the motto of the conference: "Avoiding drowning in the data: recent trends and future challenges in learning from data". The central focus is on multidisciplinary approaches to data analysis, classification, and the interface between computer science, data mining and statistics. Both methodological and applied topics are covered. The former includes supervised and unsupervised techniques with particular emphasis on advances in regression and clustering analysis and constructing composite indicators. The applications are mainly in risk analysis, biology, and education. The volume is organized into four main macro themes: methodological contributions in the social sciences and education, multivariate analysis methods for big data, innovative contributions for applications inspired by biology, and strategies for analyzing complex data in finance.
Modeling with Stochastic Programming
This is an updated version of what is still the only text to address basic questions about how to model uncertainty in mathematical programming, including how to reformulate a deterministic model so that it can be analyzed in a stochastic setting. This second edition has important extensions regarding how to represent random phenomena in the models (also called scenario generation) as well as a new chapter on multi-stage models. This text would be suitable as a stand-alone or supplement for a second course in OR/MS or in optimization-oriented engineering disciplines where the instructor wants to explain where models come from and what the fundamental modeling issues are. The book is easy-to-read, highly illustrated with lots of examples and discussions. It will be suitable for graduate students and researchers working in operations research, mathematics, engineering and related departments where there is interest in learning how to model uncertainty. Alan King is a Research Staff Member at IBM's Thomas J. Watson Research Center in New York. Stein W. Wallace is a Professor of Operational Research and head of Center for Shipping and Logistics at NHH Norwegian School of Economics, Bergen, Norway.
A Comprehensive Guide to Coding and Programming in Stata
This book is an introductory guide to programming and coding in Stata. Commonly encountered code in the field of medical statistics as well as the analyses of observational data are presented.This book covers loops and macros and then describes other commonly required coding commands.
An Introduction to Partial Differential Equations
This textbook is an introduction to the methods needed to solve partial differential equations (PDEs). Readers are introduced to PDEs that come from a variety of fields in engineering and the natural sciences. The chapters include the following topics: First Order PDEs, Second Order PDEs, Fourier Series, Separation of Variables, the Fourier Transform, and higher dimensional problems. Readers are guided through these chapters where techniques for solving first and second order PDEs are introduced. Each chapter ends with series of exercises to facilitate learning as well as illustrate the material presented in each chapter.
The Mathematics of Machine Learning
This book is an introduction to machine learning, with a strong focus on the mathematics behind the standard algorithms and techniques in the field, aimed at senior undergraduates and early graduate students of Mathematics. There is a focus on well-known supervised machine learning algorithms, detailing the existing theory to provide some theoretical guarantees, featuring intuitive proofs and exposition of the material in a concise and precise manner. A broad set of topics is covered, giving an overview of the field. A summary of the topics covered is: statistical learning theory, approximation theory, linear models, kernel methods, Gaussian processes, deep neural networks, ensemble methods and unsupervised learning techniques, such as clustering and dimensionality reduction. This book is suited for students who are interested in entering the field, by preparing them to master the standard tools in Machine Learning. The reader will be equipped to understand the main theoretical questions of the current research and to engage with the field.
Synthetic Aperture Radar (Sar) Data Applications
This carefully curated volume presents an in-depth, state-of-the-art discussion on many applications of Synthetic Aperture Radar (SAR). Integrating interdisciplinary sciences, the book features novel ideas, quantitative methods, and research results, promising to advance computational practices and technologies within the academic and industrial communities. SAR applications employ diverse and often complex computational methods rooted in machine learning, estimation, statistical learning, inversion models, and empirical models. Current and emerging applications of SAR data for earth observation, object detection and recognition, change detection, navigation, and interference mitigation are highlighted. Cutting edge methods, with particular emphasis on machine learning, are included. Contemporary deep learning models in object detection and recognition in SAR imagery with corresponding feature extraction and training schemes are considered. State-of-the-art neural network architectures in SAR-aided navigation are compared and discussed further. Advanced empirical and machine learning models in retrieving land and ocean information -- wind, wave, soil conditions, among others, are also included.
Analysis of Categorical Data with R
Analysis of Categorical Data with R, Second Edition presents a modern account of categorical data analysis using the R software environment. It covers recent techniques of model building and assessment for binary, multicategory, and count response variables and discusses fundamentals, such as odds ratio and probability estimation. The authors give detailed advice and guidelines on which procedures to use and why to use them.The second edition is a substantial update of the first based on the authors' experiences of teaching from the book for nearly a decade. The book is organized as before, but with new content throughout, and there are two new substantive topics in the advanced topics chapter--group testing and splines. The computing has been completely updated, with the "emmeans" package now integrated into the book. The examples have also been updated, notably to include new examples based on COVID-19, and there are more than 90 new exercises in the book. The solutions manual and teaching videos have also been updated.Features: Requires no prior experience with R, and offers an introduction to the essential features and functions of R Includes numerous examples from medicine, psychology, sports, ecology, and many other areas Integrates extensive R code and output Graphically demonstrates many of the features and properties of various analysis methods Offers a substantial number of exercises in all chapters, enabling use as a course text or for self-study Supplemented by a website with data sets, code, and teaching videos Analysis of Categorical Data with R, Second Edition is primarily designed for a course on categorical data analysis taught at the advanced undergraduate or graduate level. Such a course could be taught in a statistics or biostatistics department, or within mathematics, psychology, social science, ecology, or another quantitative discipline. It could also be used by a self-learner and would make an ideal reference for a researcher from any discipline where categorical data arise.
An Introduction to Traffic Flow Theory
This second edition of An Introduction to Traffic Flow Theory adds new material in several chapters related to advanced technologies including autonomy, the use of sensors and communications, and particularly congestion mitigation solutions that leverage connected and autonomous vehicles (CAVs). It also includes a new chapter that briefly outlines several mathematical analysis techniques commonly used in traffic flow theory, aiming to introduce students to some of the most frequently used tools available for traffic operational-related analysis. This new edition also includes several updates related to the most recent versions of the Highway Capacity Manual and the Green Book. This textbook is meant for use in advanced undergraduate/graduate level courses in traffic flow theory with prerequisites including two semesters of calculus, statistics, and an introductory course in transportation. The text would also be of interest to transportation professionals as a refresherin traffic flow theory or as a reference. Students and engineers of diverse backgrounds will find this text accessible and applicable to today's traffic issues.This text provides a comprehensive and concise treatment of the topic of traffic flow theory and includes several topics relevant to today's highway transportation system. It provides the fundamental principles of traffic flow theory as well as applications of those principles for evaluating specific types of facilities (freeways, intersections, etc.). Newer concepts of Intelligent transportation systems (ITS) and their potential impact on traffic flow are discussed. State-of-the-art traffic flow research, microscopic traffic analysis, and traffic simulation have significantly advanced and are also discussed in this text. Real-world examples and useful problem sets complement each chapter.
A Compact Capstone Course in Classical Calculus
This textbook offers undergraduates a self-contained introduction to advanced topics not covered in a standard calculus sequence. The author's enthusiastic and engaging style makes this material, which typically requires a substantial amount of study, accessible to students with minimal prerequisites. Readers will gain a broad knowledge of the area, with approaches based on those found in recent literature, as well as historical remarks that deepen the exposition. Specific topics covered include the binomial theorem, the harmonic series, Euler's constant, geometric probability, and much more. Over the fifteen chapters, readers will discover the elegance of calculus and the pivotal role it plays within mathematics. A Compact Capstone Course in Classical Calculus is ideal for exploring interesting topics in mathematics beyond the standard calculus sequence, particularly for undergraduates who may not be taking more advanced math courses. It would also serve as a useful supplement for a calculus course and a valuable resource for self-study. Readers are expected to have completed two one-semester college calculus courses.
Elliptic Equations: An Introductory Course
The aim of this book is to introduce the reader to different topics of the theory of elliptic partial differential equations by avoiding technicalities and complicated refinements. Apart from the basic theory of equations in divergence form, it includes subjects as singular perturbations, homogenization, computations, asymptotic behavior of problems in cylinders, elliptic systems, nonlinear problems, regularity theory, Navier-Stokes systems, p-Laplace type operators, large solutions, and mountain pass techniques. Just a minimum on Sobolev spaces has been introduced and work on integration on the boundary has been carefully avoided to keep the reader attention focused on the beauty and variety of these issues. The chapters are relatively independent of each other and can be read or taught separately. Numerous results presented here are original, and have not been published elsewhere. The book will be of interest to graduate students and researchers specializing in partial differential equations.
Linear and Nonlinear Non-Fredholm Operators
This book is devoted to a new aspect of linear and nonlinear non-Fredholm operators and its applications. The domain of applications of theory developed here is potentially much wider than that presented in the book. Therefore, a goal of this book is to invite readers to make contributions to this fascinating area of mathematics.First, it is worth noting that linear Fredholm operators, one of the most important classes of linear maps in mathematics, were introduced around 1900 in the study of integral operators. These linear Fredholm operators between Banach spaces share, in some sense, many properties with linear maps between finite dimensional spaces. Since the end of the previous century there has been renewed interest in linear - nonlinear Fredholm maps from a topological degree point of view and its applications, following a period of "stagnation" in the mid-1960s. Now, linear and nonlinear Fredholm operator theory and the solvability of corresponding equations both from the analytical and topological points of view are quite well understood.Also noteworthy is, that as a by-product of our results, we have obtained an important tool for modelers working in mathematical biology and mathematical medicine, namely, the necessary conditions for preserving positive cones for systems of equations without Fredholm property containing local - nonlocal diffusion as well as terms for transport and nonlinear interactions.
Partial Least Squares Regression
Partial least squares (PLS) regression is, at its historical core, a black-box algorithmic method for dimension reduction and prediction based on an underlying linear relationship between a possibly vector-valued response and a number of predictors.Through envelopes, much more has been learned about PLS regression, resulting in a mass of information that allows an envelope bridge that takes PLS regression from a black-box algorithm to a core statistical paradigm based on objective function optimization and, more generally, connects the applied sciences and statistics in the context of PLS. This book focuses on developing this bridge. It also covers uses of PLS outside of linear regression, including discriminant analysis, non-linear regression, generalized linear models and dimension reduction generally.Key Features: - Showcases the first serviceable method for studying high-dimensional regressions.- Provides necessary background on PLS and its origin.- R and Python programs are available for nearly all methods discussed in the book.This book can be used as a reference and as a course supplement at the Master's level in Statistics and beyond. It will be of interest to both statisticians and applied scientists.
Tidy Finance with Python
This textbook shows how to bring theoretical concepts from finance and econometrics to the data. Focusing on coding and data analysis with Python, we show how to conduct research in empirical finance from scratch.
Confidence Intervals for Discrete Data in Clinical Research
There is only one published book on confidence interval for clinical research. This book has a cookbook style with several examples and codes so that methods presented in the book can be implemented. The primary audience will be statisticians.
Nonlinear Dynamics and Chaos
The goal of this Third Edition is the same as previous editions: to provide a good foundation - and a joyful experience - for anyone who'd like to learn about nonlinear dynamics and chaos from an applied perspective.
Advanced Electromagnetic Wave Propagation Methods
This textbook provides a solid foundation into the approaches used in the analysis of complex electromagnetic problems and wave propagation. The techniques discussed are essential to obtain closed-form solutions or asymptotic solutions and meet an existing need for instructors and students in electromagnetic theory.
The Big Book of Real Analysis
This book provides an introduction to real analysis, a fundamental topic that is an essential requirement in the study of mathematics. It deals with the concepts of infinity and limits, which are the cornerstones in the development of calculus. Beginning with some basic proof techniques and the notions of sets and functions, the book rigorously constructs the real numbers and their related structures from the natural numbers. During this construction, the readers will encounter the notions of infinity, limits, real sequences, and real series. These concepts are then formalised and focused on as stand-alone objects. Finally, they are expanded to limits, sequences, and series of more general objects such as real-valued functions. Once the fundamental tools of the trade have been established, the readers are led into the classical study of calculus (continuity, differentiation, and Riemann integration) from first principles. The book concludes with an introduction to the studyof measures and how one can construct the Lebesgue integral as an extension of the Riemann integral. This textbook is aimed at undergraduate students in mathematics. As its title suggests, it covers a large amount of material, which can be taught in around three semesters. Many remarks and examples help to motivate and provide intuition for the abstract theoretical concepts discussed. In addition, more than 600 exercises are included in the book, some of which will lead the readers to more advanced topics and could be suitable for independent study projects. Since the book is fully self-contained, it is also ideal for self-study.
Introduction to Quantum Groups
This book introduces the reader to quantum groups, focusing on the simplest ones, namely the closed subgroups of the free unitary group.Although such quantum groups are quite easy to understand mathematically, interesting examples abound, including all classical Lie groups, their free versions, half-liberations, other intermediate liberations, anticommutation twists, the duals of finitely generated discrete groups, quantum permutation groups, quantum reflection groups, quantum symmetry groups of finite graphs, and more.The book is written in textbook style, with its contents roughly covering a one-year graduate course. Besides exercises, the author has included many remarks, comments and pieces of advice with the lone reader in mind. The prerequisites are basic algebra, analysis and probability, and a certain familiarity with complex analysis and measure theory. Organized in four parts, the book begins with the foundations of the theory, due to Woronowicz, comprising axioms, Haar measure, Peter-Weyl theory, Tannakian duality and basic Brauer theorems. The core of the book, its second and third parts, focus on the main examples, first in the continuous case, and then in the discrete case. The fourth and last part is an introduction to selected research topics, such as toral subgroups, homogeneous spaces and matrix models.Introduction to Quantum Groups offers a compelling introduction to quantum groups, from the simplest examples to research level topics.
Modern Discrete Probability
Providing a graduate-level introduction to discrete probability and its applications, this book develops a toolkit of essential techniques for analysing stochastic processes on graphs, other random discrete structures, and algorithms. Topics covered include the first and second moment methods, concentration inequalities, coupling and stochastic domination, martingales and potential theory, spectral methods, and branching processes. Each chapter expands on a fundamental technique, outlining common uses and showing them in action on simple examples and more substantial classical results. The focus is predominantly on non-asymptotic methods and results. All chapters provide a detailed background review section, plus exercises and signposts to the wider literature. Readers are assumed to have undergraduate-level linear algebra and basic real analysis, while prior exposure to graduate-level probability is recommended. This much-needed broad overview of discrete probability could serve as a textbook or as a reference for researchers in mathematics, statistics, data science, computer science and engineering.
Generating Functions in Engineering and the Applied Sciences
Generating function (GF) is a mathematical technique to concisely represent a known ordered sequence into a simple continuous algebraic function in dummy variable(s). This Second Edition introduces commonly encountered generating functions (GFs) in engineering and applied sciences, such as ordinary GF (OGF), exponential GF (EGF), as also Dirichlet GF (DGF), Lambert GF (LGF), Logarithmic GF (LogGF), Hurwitz GF (HGF), Mittag-Lefler GF (MLGF), etc. This book is intended mainly for beginners in applied science and engineering fields to help them understand single-variable GFs and illustrate how to apply them in various practical problems. Specifically, the book discusses probability GFs (PGF), moment and cumulant GFs (MGF, CGF), mean deviation GFs (MDGF), survival function GFs (SFGF), rising and falling factorial GFs, factorial moment, and inverse factorial moment GFs. Applications of GFs in algebra, analysis of algorithms, bioinformatics, combinatorics, economics, finance, genomics, geometry, graph theory, management, number theory, polymer chemistry, reliability, statistics and structural engineering have been added to this new edition. This book is written in such a way that readers who do not have prior knowledge of the topic can easily follow through the chapters and apply the lessons learned in their respective disciplines.
Fractal Patterns with MATLAB
This book presents the iterative beauty of fractals and fractal functions graphically with the aid of MATLAB programming. The fractal images generated using the MATLAB codes provide visual delight and highly encourage the fractal lovers for creative thinking. The book compiles five cutting-edge research chapters, each with state-of-the art fractal illustrations. It starts with the fundamental theory for the construction of fractal sets via the deterministic iteration algorithm. Incorporating the theoretical base, fractal illustrations of elementary fractal sets are provided with the explicit MATLAB code. The book gives examples of MATLAB codes to present the fractal surfaces. This book is contributed to all the research beginners as well as the professionals on the field of fractal analysis. As it covers basic fractals like Sierpinski triangle to advanced fractal functions with explicit MATLAB code, the presented fractal illustrations hopefully benefit even the non-field readers. The book is a useful course to all the research beginners on the fractal and fractal-related fields.
Partial Differential Equations
This textbook introduces the study of partial differential equations using both analytical and numerical methods. By intertwining the two complementary approaches, the authors create an ideal foundation for further study. Motivating examples from the physical sciences, engineering, and economics complete this integrated approach. A showcase of models begins the book, demonstrating how PDEs arise in practical problems that involve heat, vibration, fluid flow, and financial markets. Several important characterizing properties are used to classify mathematical similarities, then elementary methods are used to solve examples of hyperbolic, elliptic, and parabolic equations. From here, an accessible introduction to Hilbert spaces and the spectral theorem lay the foundation for advanced methods. Sobolev spaces are presented first in dimension one, before being extended to arbitrary dimension for the study of elliptic equations. An extensive chapter on numerical methods focuses onfinite difference and finite element methods. Computer-aided calculation with Maple(TM) completes the book. Throughout, three fundamental examples are studied with different tools: Poisson's equation, the heat equation, and the wave equation on Euclidean domains. The Black-Scholes equation from mathematical finance is one of several opportunities for extension. Partial Differential Equations offers an innovative introduction for students new to the area. Analytical and numerical tools combine with modeling to form a versatile toolbox for further study in pure or applied mathematics. Illuminating illustrations and engaging exercises accompany the text throughout. Courses in real analysis and linear algebra at the upper-undergraduate level are assumed.
Analytical Properties of Nonlinear Partial Differential Equations
Nonlinear partial differential equations (PDE) are at the core of mathematical modeling. In the past decades and recent years, multiple analytical methods to study various aspects of the mathematical structure of nonlinear PDEs have been developed. Those aspects include C- and S-integrability, Lagrangian and Hamiltonian formulations, equivalence transformations, local and nonlocal symmetries, conservation laws, and more. Modern computational approaches and symbolic software can be employed to systematically derive and use such properties, and where possible, construct exact and approximate solutions of nonlinear equations. This book contains a consistent overview of multiple properties of nonlinear PDEs, their relations, computation algorithms, and a uniformly presented set of examples of application of these methods to specific PDEs. Examples include both well known nonlinear PDEs and less famous systems that arise in the context of shallow water waves and far beyond. The book will beof interest to researchers and graduate students in applied mathematics, physics, and engineering, and can be used as a basis for research, study, reference, and applications.
Mathematical Methods for Life Sciences
This book introduces calculus, and other key mathematical methods, to students from applied sciences. Special attention is paid to real-world applications, and for every concept, many concrete examples are provided.
A Course in the Calculus of Variations
This book provides an introduction to the broad topic of the calculus of variations. It addresses the most natural questions on variational problems and the mathematical complexities they present.Beginning with the scientific modeling that motivates the subject, the book then tackles mathematical questions such as the existence and uniqueness of solutions, their characterization in terms of partial differential equations, and their regularity. It includes both classical and recent results on one-dimensional variational problems, as well as the adaptation to the multi-dimensional case. Here, convexity plays an important role in establishing semi-continuity results and connections with techniques from optimization, and convex duality is even used to produce regularity results. This is then followed by the more classical H繹lder regularity theory for elliptic PDEs and some geometric variational problems on sets, including the isoperimetric inequality andthe Steiner tree problem. The book concludes with a chapter on the limits of sequences of variational problems, expressed in terms of Γ-convergence.While primarily designed for master's-level and advanced courses, this textbook, based on its author's instructional experience, also offers original insights that may be of interest to PhD students and researchers. A foundational understanding of measure theory and functional analysis is required, but all the essential concepts are reiterated throughout the book using special memo-boxes.
The Joy of Statistics
The vast majority of statistics books delineate techniques used to analyze collected data. The Joy of Statistics is not one of these books. It consists of a series of 42 "short stories", each illustrating how statistical methods applied to data produce insight and solutions to the questions the data were collected to answer. Real-life and sometimes artificial data are used to demonstrate the often painless method and magic of statistics. In addition, the text contains brief histories of the evolution of statistical methods and a number of brief biographies of the most famous statisticians of the 20th century. Sprinkled throughout are statistical jokes, puzzles and traditional stories. The levels of statistical texts span a spectrum, from elementary to introductory to application to theoretical to advanced mathematical. The Joy of Statistics explores a variety of statistical applications using graphs and plots, along with detailed and intuitive descriptions, and occasionally a bit of 10th grade mathematics. Examples of a few of the topics included among these "short stories" are pet ownership, gambling games such as roulette, blackjack and lotteries, as well as more serious subjects such as comparison of African-American and white infant mortality risk, infant birth weight and maternal age, estimation of coronary heart disease risk and racial differences in Hodgkin disease. The statistical descriptions of these topics are in many cases accompanied by easy to understand explanations labelled "How It Works."
Modeling Reality with Mathematics
Simulating the behavior of a human heart, predicting tomorrow's weather, optimizing the aerodynamics of a sailboat, finding the ideal cooking time for a hamburger: to solve these problems, cardiologists, meteorologists, sportsmen, and engineers can count on math help. This book will lead you to the discovery of a magical world, made up of equations, in which a huge variety of important problems for our life can find useful answers.
Practical Guide to Applied Conformal Prediction in Python
Elevate your machine learning skills using the Conformal Prediction framework for uncertainty quantification. Dive into unique strategies, overcome real-world challenges, and become confident and precise with forecasting.Key Features: Master Conformal Prediction, a fast-growing ML framework, with Python applicationsExplore cutting-edge methods to measure and manage uncertainty in industry applicationsUnderstand how Conformal Prediction differs from traditional machine learningBook Description: In the rapidly evolving landscape of machine learning, the ability to accurately quantify uncertainty is pivotal. The book addresses this need by offering an in-depth exploration of Conformal Prediction, a cutting-edge framework to manage uncertainty in various ML applications.Learn how Conformal Prediction excels in calibrating classification models, produces well-calibrated prediction intervals for regression, and resolves challenges in time series forecasting and imbalanced data. Discover specialised applications of conformal prediction in cutting-edge domains like computer vision and NLP. Each chapter delves into specific aspects, offering hands-on insights and best practices for enhancing prediction reliability. The book concludes with a focus on multi-class classification nuances, providing expert-level proficiency to seamlessly integrate Conformal Prediction into diverse industries. With practical examples in Python using real-world datasets, expert insights, and open-source library applications, you will gain a solid understanding of this modern framework for uncertainty quantification.By the end of this book, you will be able to master Conformal Prediction in Python with a blend of theory and practical application, enabling you to confidently apply this powerful framework to quantify uncertainty in diverse fields.What You Will Learn: The fundamental concepts and principles of conformal predictionLearn how conformal prediction differs from traditional ML methodsApply real-world examples to your own industry applicationsExplore advanced topics - imbalanced data and multi-class CPDive into the details of the conformal prediction frameworkBoost your career as a data scientist, ML engineer, or researcherLearn to apply conformal prediction to forecasting and NLPWho this book is for: Ideal for readers with a basic understanding of machine learning concepts and Python programming, this book caters to data scientists, ML engineers, academics, and anyone keen on advancing their skills in uncertainty quantification in ML.
Advances in Optimization and Applications
This book constitutes the refereed proceedings of the 14th International Conference on Advances in Optimization and Applications, OPTIMA 2023, held in Petrovac, Montenegro, during September 18-22, 2023.The 21 full papers included in this book were carefully reviewed and selected from 68 submissions. They were organized in topical sections as follows: ​mathematical programming; global optimization; continuous optimization; discrete and combinatorial optimization; optimal control; game theory and mathematical economics; optimization in economics and finance; and applications.
Crowd Dynamics, Volume 4
This contributed volume explores innovative research in the modeling, simulation, and control of crowd dynamics. Chapter authors approach the topic from the perspectives of mathematics, physics, engineering, and psychology, providing a comprehensive overview of the work carried out in this challenging interdisciplinary research field. The volume begins with an overview of analytical problems related to crowd modeling. Attention is then given to the importance of considering the social and psychological factors that influence crowd behavior - such as emotions, communication, and decision-making processes - in order to create reliable models. Finally, specific features of crowd behavior are explored, including single-file traffic, passenger movement, modeling multiple groups in crowds, and the interplay between crowd dynamics and the spread of disease.Crowd Dynamics, Volume 4 is ideal for mathematicians, engineers, physicists, and other researchers working in the rapidly growing field of modeling and simulation of human crowds.
Parabolic Problems
Parabola is a mathematics magazine published by UNSW, Sydney. Among other things, each issue of Parabola has contained a collection of puzzles/problems, on various mathematical topics and at a suitable level for younger (but mathematically sophisticated) readers.Parabolic Problems: 60 Years of Mathematical Puzzles in Parabola collects the very best of almost 1800 problems and puzzles into a single volume. Many of the problems have been re-mastered, and new illustrations have been added. Topics covered range across geometry, number theory, combinatorics, logic, and algebra. Solutions are provided to all problems, and a chapter has been included detailing some frequently useful problem-solving techniques, making this a fabulous resource for education and, most importantly, fun!Features Hundreds of diverting and mathematically interesting problems and puzzles. Accessible for anyone with a high school-level mathematics education. Wonderful resource for teachers and students of mathematics from high school to undergraduate level, and beyond.
A Pen and Paper Introduction to Statistics
This book proposes to reverse the way statistics is taught, by starting with the introduction of linear models. Today, many statisticians know that the one unifying principle of statistical tests is that most of them are instances of linear models.
Spatial Analysis in Geology Using R
Empowers you to harness the potential of spatial analysis in addressing pressing geological challenges. Can also be used as a textbook either in introductory courses in spatial analysis or in advanced GIS and Digital mapping - comprehensive guide tailored to professionals at all levels.
The Evolution of the Vehicle Routing Problem
This book presents state-of-the-art research and practice in optimization routing, specifically the vehicle routing problem (VRP). Since its introduction in the late 1950s, the VRP has been a very significant area of research and practice in operations research. Vehicles are used to make deliveries and for pick-ups every day and everywhere. Companies such as Amazon, UPS, FedEx, and DHL use route optimization to reduce mileage, fuel use, number of trucks on the road, and carbon dioxide emissions. The authors compile and analyze 135 survey and review articles on vehicle routing topics published between 2005 and 2022 in an effort to make key observations about publication and trend history, summarize the overall contributions in the field, and identify trends in VRP research and practice. The authors have compiled published research on models, algorithms, and applications for specific areas, including: alternative and multiple objectives; arc routing and general routing; drones, last-mile delivery, and urban distribution; dynamic and stochastic routing; green routing; inventory routing; loading constraints; location-routing; multiple depots; pickup and delivery and dial-a-ride problems; rich and multi-attribute routing; routing over time; shipping; two-echelon, collaborative, and inter-terminal problems; specific variants, benchmark datasets, and software; and exact algorithms and heuristics. In addition, the book discusses how vehicle routing problems are among the most widely studied problems in combinatorial optimization due to the mathematical complexity and practical significance.
Tales of the Tail
"Tales of the Tail: Legendary Cats Throughout History" is an enchanting exploration into the feline realm, delving into the captivating stories of extraordinary cats that have left an indelible mark on human history. This whimsical tome unveils the rich tapestry of legends, folklore, and real-life anecdotes surrounding these enigmatic creatures, celebrating their mystique and the profound impact they have had on various cultures.Within the pages of "Tales of the Tail," readers will embark on a journey through time, discovering mythical cats that have been revered as divine entities, guardians, and symbols of good fortune. From ancient civilizations to modern times, the book weaves together narratives of cats that have transcended the ordinary, earning their place as revered beings in the annals of history.The tales within this collection are not limited to mere whimsy; they are intertwined with historical events, cultural beliefs, and the evolving roles that cats have played in societies around the world. Whether exploring the sacred cats of ancient Egypt, the mischievous Cheshire Cat of Wonderland fame, or the famed literary cat companions of renowned authors, each story is a testament to the profound connections between humans and their feline counterparts.Each chapter unfolds like a storybook, with vivid descriptions and captivating illustrations bringing these legendary cats to life. From the graceful and regal to the cunning and magical, "Tales of the Tail" pays homage to the diverse personalities and qualities that have made cats such beloved companions throughout the ages.This enchanting collection is a celebration of the enduring fascination with cats and their timeless allure. "Tales of the Tail" invites readers to immerse themselves in the magical world of legendary felines, where history, mythology, and the timeless charm of cats converge in a purrfectly delightful tapestry of tales.
Twisted Logic
Draws upon an array of popular and novel puzzles, problems, and paradoxes, and uses these to help understand and navigate our everyday world. Addresses some of the big questions. Aimed at all those interested in learning about weird and wonderful problems and paradoxes.
Interactive Epistemology
Robert J Aumann has received numerous prizes, including the Nobel Memorial Prize in Economic Sciences for 2005.With his 1976 paper, 'Agreeing to Disagree', Robert Aumann pioneered the subject of interactive epistemology: the study of what people know, and what they know about what others know. Since then, the discipline has burgeoned enormously. This book documents Aumann's work leading to the 1976 paper and his subsequent contributions to the discipline. The scientific controversies emanating from his work are also included.
