Functorial Semiotics for Creativity in Music and Mathematics
This book presents a new semiotic theory based upon category theory and applying to a classification of creativity in music and mathematics. It is the first functorial approach to mathematical semiotics that can be applied to AI implementations for creativity by using topos theory and its applications to music theory.Of particular interest is the generalized Yoneda embedding in the bidual of the category of categories (Lawvere) - parametrizing semiotic units - enabling a Čech cohomology of manifolds of semiotic entities. It opens up a conceptual mathematics as initiated by Grothendieck and Galois and allows a precise description of musical and mathematical creativity, including a classification thereof in three types. This approach is new, as it connects topos theory, semiotics, creativity theory, and AI objectives for a missing link to HI (Human Intelligence). The reader can apply creativity research using our classification, cohomology theory, generalized Yoneda embedding, and Java implementation of the presented functorial display of semiotics, especially generalizing the Hjelmslev architecture. The intended audience are academic, industrial, and artistic researchers in creativity.
Stochastic Analysis, Filtering, and Stochastic Optimization
This volume is a collection of research works to honor the late Professor Mark H.A. Davis, whose pioneering work in the areas of Stochastic Processes, Filtering, and Stochastic Optimization spans more than five decades. Invited authors include his dissertation advisor, past collaborators, colleagues, mentees, and graduate students of Professor Davis, as well as scholars who have worked in the above areas. Their contributions may expand upon topics in piecewise deterministic processes, pathwise stochastic calculus, martingale methods in stochastic optimization, filtering, mean-field games, time-inconsistency, as well as impulse, singular, risk-sensitive and robust stochastic control.
Mathematical Tools for Neuroscience
This book provides a brief but accessible introduction to a set of related, mathematical ideas that have proved useful in understanding the brain and behaviour. If you record the eye movements of a group of people watching a riverside scene then some will look at the river, some will look at the barge by the side of the river, some will look at the people on the bridge, and so on, but if a duck takes off then everybody will look at it. How come the brain is so adept at processing such biological objects? In this book it is shown that brains are especially suited to exploiting the geometric properties of such objects. Central to the geometric approach is the concept of a manifold, which extends the idea of a surface to many dimensions. The manifold can be specified by collections of n-dimensional data points or by the paths of a system through state space. Just as tangent planes can be used to analyse the local linear behaviour of points on a surface, so the extension to tangent spaces can be used to investigate the local linear behaviour of manifolds. The majority of the geometric techniques introduced are all about how to do things with tangent spaces.Examples of the geometric approach to neuroscience include the analysis of colour and spatial vision measurements and the control of eye and arm movements. Additional examples are used to extend the applications of the approach and to show that it leads to new techniques for investigating neural systems. An advantage of following a geometric approach is that it is often possible to illustrate the concepts visually and all the descriptions of the examples are complemented by comprehensively captioned diagrams.The book is intended for a reader with an interest in neuroscience who may have been introduced to calculus in the past but is not aware of the many insights obtained by a geometric approach to the brain. Appendices contain brief reviews of the required background knowledge in neuroscience and calculus.
Introductory Statistics
This textbook is a primer for students on statistics. It covers basic statistical operations, an introduction to probability, distributions and regression. The book is divided into a series of 10 chapters covering a basic introduction to common topics for beginners. The goal of the book is to provide sufficient understanding of how to organize and summarize datasets through descriptive and inferential statistics for good decision-making. A chapter on ethics also informs readers about best practices for using statistics in research and analysis. Topics covered: 1. Introduction to Statistics2. Summarizing and Graphing3. Basic Concepts of Probability4. Discrete Random Variables5. Continuous Random Variables6. Sampling Distributions7. Estimation8. Hypothesis Testing9. Correlation and Regression10. Ethics
Numerical and Engineering Analysis
The book is designed for use in a graduate program in Numerical Analysis that includes a basic introductory course and subsequent more specialized courses. The latter is envisaged to cover numerical linear algebra, the numerical solution of ordinary and partial differential equations, and perhaps additional topics related to complex analysis, multidimensional analysis, in particular optimization, and functional analysis and related functional equations.Viewed in this context, the first four chapters of our book could serve as a text for the basic introductory course on the Python program, andthe remaining chapters could provide a text for an advanced course on the numerical solution of ordinary differential equations. Therefore, the book breaks with tradition in that it no longer attempts to deal with all major topics of numerical mathematics. Those dealing with linear algebra and partial differential equations have developed into major fields of study that have attained a degree of autonomy and identity that justifies their treatment in separate books and separate courses on the graduate level. The term "Numerical Analysis" as used in this book, therefore, is to be taken in the narrow sense of the numerical analog of Mathematical Analysis, comprising such topics as machine arithmetic, the approximation of functions, approximate differentiation and integration, and the approximate solution of nonlinear equations and ordinary differential equations.This book aims to provide a good understanding of Numerical engineering analysis and its applications and optimization. The book begins with studying the concept of Python fundamentals for scientific computing. It then presents their applications in the different configurations shown in lucid detail.For more details, please visit https: //centralwestpublishing.com
The Generalized Riemann Hypothesis - Dirichlet L-functions
This book is the second of two books in a series by the author on the generalized Riemann hypothesis. The Euler-Maclaurin summation formula, the Borel integral summation method, the Euler reflection formula for the gamma function, and the result of the first book of this series are used to prove that all roots of Dirichlet L-functions with principal characters in the critical strip are identical to the roots of the Riemann zeta function, and therefore have real part equal to 1/2. Furthermore, the Euler-Maclaurin summation formula, the Borel integral summation method, bi-lateral integral transform representations of the partial sums of the Dirichlet L-functions with non-principal characters in the critical strip, and the generalized functional equation of the Dirichlet L-functions are used to prove that all roots of the Dirichlet L-functions with non-principal characters in the critical strip have real part equal to 1/2.
Scalar and Vector Risk in the General Framework of Portfolio Theory
This book is the culmination of the authors' industry-academic collaboration in the past several years. The investigation is largely motivated by bank balance sheet management problems. The main difference between a bank balance sheet management problem and a typical portfolio optimization problem is that the former involves multiple risks. The related theoretical investigation leads to a significant extension of the scope of portfolio theories. The book combines practitioners' perspectives and mathematical rigor. For example, to guide the bank managers to trade off different Pareto efficient points, the topological structure of the Pareto efficient set is carefully analyzed. Moreover, on top of computing solutions, the authors focus the investigation on the qualitative properties of those solutions and their financial meanings. These relations, such as the role of duality, are most useful in helping bank managers to communicate their decisions to the different stakeholders. Finally, bank balance sheet management problems of varying levels of complexity are discussed to illustrate how to apply the central mathematical results. Although the primary motivation and application examples in this book are focused in the area of bank balance sheet management problems, the range of applications of the general portfolio theory is much wider. As a matter of fact, most financial problems involve multiple types of risks. Thus, the book is a good reference for financial practitioners in general and students who are interested in financial applications. This book can also serve as a nice example of a case study for applied mathematicians who are interested in engaging in industry-academic collaboration.
Discrete-Time Semi-Markov Random Evolutions and Their Applications
Spreadsheet Problem Solving and Programming for Engineers and Scientists
This book provides a targeted and comprehensive resource essential to a full understanding of modern spreadsheet skills needed for engineering and scientific computations. Building on the authors' decades of experience teaching spreadsheets and programming, it is the ideal companion for all engineering courses and professional self-study.
A Comprehensive Textbook on Metric Spaces
This textbook provides a comprehensive course in metric spaces. Presenting a smooth takeoff from basic real analysis to metric spaces, every chapter of the book presents a single concept, which is further unfolded and elaborated through related sections and subsections. Apart from a unique new presentation and being a comprehensive textbook on metric spaces, it contains some special concepts and new proofs of old results, which are not available in any other book on metric spaces. It has individual chapters on homeomorphisms and the Cantor set. This book is almost self-contained and has an abundance of examples, exercises, references and remarks about the history of basic notions and results. Every chapter of this book includes brief hints and solutions to selected exercises. It is targeted to serve as a textbook for advanced undergraduate and beginning graduate students of mathematics.
Normal Forms and Stability of Hamiltonian Systems
This book introduces the reader to the study of Hamiltonian systems, focusing on the stability of autonomous and periodic systems and expanding to topics that are usually not covered by the canonical literature in the field. It emerged from lectures and seminars given at the Federal University of Pernambuco, Brazil, known as one of the leading research centers in the theory of Hamiltonian dynamics. This book starts with a brief review of some results of linear algebra and advanced calculus, followed by the basic theory of Hamiltonian systems. The study of normal forms of Hamiltonian systems is covered by Ch.3, while Chapters 4 and 5 treat the normalization of Hamiltonian matrices. Stability in non-linear and linear systems are topics in Chapters 6 and 7. This work finishes with a study of parametric resonance in Ch. 8. All the background needed is presented, from the Hamiltonian formulation of the laws of motion to the application of the Krein-Gelfand-Lidskii theory of stronglystable systems. With a clear, self-contained exposition, this work is a valuable help to advanced undergraduate and graduate students, and to mathematicians and physicists doing research on this topic.
Measure Theory and Integration
This textbook contains a detailed and thorough exposition of topics in measure theory and integration. With abundant solved examples and more than 200 problems, the book is written in a motivational and student-friendly manner. Targeted to senior undergraduate and graduate courses in mathematics, it provides a detailed and thorough explanation of all the concepts. Suitable for independent study, the book, the first of the three volumes, contains topics on measure theory, measurable functions, Lebesgue integration, Lebesgue spaces, and abstract measure theory.
Local Limit Theorems for Inhomogeneous Markov Chains
This book extends the local central limit theorem to Markov chains whose state spaces and transition probabilities are allowed to change in time. Such chains are used to model Markovian systems depending on external time-dependent parameters. The book develops a new general theory of local limit theorems for additive functionals of Markov chains, in the regimes of local, moderate, and large deviations, and provides nearly optimal conditions for the classical expansions, as well as asymptotic corrections when these conditions fail. Applications include local limit theorems for independent but not identically distributed random variables, Markov chains in random environments, and time-dependent perturbations of homogeneous Markov chains.The inclusion of appendices with background material, numerous examples, and an account of the historical background of the subject make this self-contained book accessible to graduate students. It will also be useful for researchers in probability and ergodic theory who are interested in asymptotic behaviors, Markov chains in random environments, random dynamical systems and non-stationary systems.
Optimization of Pharmaceutical Processes
Optimization of Pharmaceutical Processes presents contributions from leading authorities in the fields of optimization and pharmaceutical manufacturing. Formulated within structured frameworks, practical examples and applications are given as guidance to apply optimization techniques to most aspects of pharmaceutical processes from design, to lab and pilot scale, and finally to manufacturing. The increasing demand for better quality, higher yield, more efficient-optimized and green pharmaceutical processes, indicates that optimal conditions for production must be applied to achieve simplicity, lower costs and superior yield. The application of such methods in the pharmaceutical industry is not trivial. Quality of the final product is of major importance to human health and the need for deep knowledge of the process parameters and the optimization of the processes are imperative. The volume, which includes new methods as well as review contributions will benefit awide readership including engineers in pharmaceuticals, chemical, biological, to name just a few.
Stochastic Processes and Financial Mathematics
The book provides an introduction to advanced topics in stochastic processes and related stochastic analysis, and combines them with a sound presentation of the fundamentals of financial mathematics. It is wide-ranging in content, while at the same time placing much emphasis on good readability, motivation, and explanation of the issues covered. Financial mathematical topics are first introduced in the context of discrete time processes and then transferred to continuous-time models. The basic construction of the stochastic integral and the associated martingale theory provide fundamental methods of the theory of stochastic processes for the construction of suitable stochastic models of financial mathematics, e.g. using stochastic differential equations. Central results of stochastic analysis such as the It繫 formula, Girsanov's theorem and martingale representation theorems are of fundamental importance in financial mathematics, e.g. for the risk-neutral valuation formula (Black-Scholes formula) or the question of the hedgeability of options and the completeness of market models. Chapters on the valuation of options in complete and incomplete markets and on the determination of optimal hedging strategies conclude the range of topics. Advanced knowledge of probability theory is assumed, in particular of discrete-time processes (martingales, Markov chains) and continuous-time processes (Brownian motion, L矇vy processes, processes with independent increments, Markov processes). The book is thus suitable for advanced students as a companion reading and for instructors as a basis for their own courses.This book is a translation of the original German 1st edition Stochastische Prozesse und Finanzmathematik by Ludger R羹schendorf, published by Springer-Verlag GmbH Germany, part of Springer Nature in 2020. The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com) and in a subsequent editing, improved by the author. Springer Nature works continuously to further the development of tools for the production of books and on the related technologies to support the authors.
Durable-Strategies Dynamic Games
Durable strategies that have prolonged effects are prevalent in real-world situations. Revenue-generating investments, toxic waste disposal, long-lived goods, regulatory measures, coalition agreements, diffusion of knowledge, advertisement and investments to accumulate physical capital are concrete and common examples of durable strategies. This book provides an augmentation of dynamic game theory and advances a new game paradigm with durable strategies in decision-making schemes. It covers theories, solution techniques, and the applications of a general class of dynamic games with multiple durable strategies. Non-cooperative equilibria and cooperative solutions are derived, along with advanced topics including random termination, asynchronous game horizons, and stochastic analysis. The techniques presented here will enable readers to solve numerous practical dynamic interactive problems with durable strategies. This book not only expands the scope of applied dynamic game theory, but also provides a solid foundation for further theoretical and technical advancements. As such, it will appeal to scholars and students of quantitative economics, game theory, operations research, and computational mathematics."Not too many new concepts have been introduced in dynamic games since their inception. The introduction of the concept of durable strategies changes this trend and yields important contributions to environmental and business applications." Dusan M Stipanovic, Professor, University of Illinois at Urbana-Champaign "Before this book, the field simply did not realize that most of our strategies are durable and entail profound effects in the future. Putting them into the mathematical framework of dynamic games is a great innovative effort." Vladimir Turetsky, Professor, Ort Braude College "Durable-strategies Dynamic Games is trulya world-leading addition to the field of dynamic games. It is a much needed publication to tackle increasingly crucial problems under the reality of durable strategies." Vladimir Mazalov, Director of Mathematical Research, Russian Academy of Sciences & President of the International Society of Dynamic Games
Large Sample Techniques for Statistics
This book offers a comprehensive guide to large sample techniques in statistics. With a focus on developing analytical skills and understanding motivation, Large Sample Techniques for Statistics begins with fundamental techniques, and connects theory and applications in engaging ways.The first five chapters review some of the basic techniques, such as the fundamental epsilon-delta arguments, Taylor expansion, different types of convergence, and inequalities. The next five chapters discuss limit theorems in specific situations of observational data. Each of the first ten chapters contains at least one section of case study. The last six chapters are devoted to special areas of applications. This new edition introduces a final chapter dedicated to random matrix theory, as well as expanded treatment of inequalities and mixed effects models. The book's case studies and applications-oriented chapters demonstrate how to use methods developed from large sample theory in real world situations. The book is supplemented by a large number of exercises, giving readers opportunity to practice what they have learned. Appendices provide context for matrix algebra and mathematical statistics. The Second Edition seeks to address new challenges in data science.This text is intended for a wide audience, ranging from senior undergraduate students to researchers with doctorates. A first course in mathematical statistics and a course in calculus are prerequisites..
Statistical Modeling
Statistical Modeling provides an introduction to regression, survival analysis, and time series analysisfor students who have completed calculus-based coursesin probability and mathematical statistics.The book uses the R language to fit statistical models, conduct Monte Carlo simulation experiments, and generate graphics.Over 300 exercises at the end of the chapters make this an appropriate text for a class in statistical modeling.This book is an open educational resource.
Novel Mathematics Inspired by Industrial Challenges
This contributed volume convenes a rich selection of works with a focus on innovative mathematical methods with applications in real-world, industrial problems. Studies included in this book are all motivated by a relevant industrial challenge, and demonstrate that mathematics for industry can be extremely rewarding, leading to new mathematical methods and sometimes even to entirely new fields within mathematics.The book is organized into two parts: Computational Sciences and Engineering, and Data Analysis and Finance. In every chapter, readers will find a brief description of why such work fits into this volume; an explanation on which industrial challenges have been instrumental for their inspiration; and which methods have been developed as a result. All these contribute to a greater unity of the text, benefiting not only practitioners and professionals seeking information on novel techniques but also graduate students in applied mathematics, engineering, and related fields.
New Perspectives on the Theory of Inequalities for Integral and Sum
This book provides new contributions to the theory of inequalities for integral and sum, and includes four chapters. In the first chapter, linear inequalities via interpolation polynomials and green functions are discussed. New results related to Popoviciu type linear inequalities via extension of the Montgomery identity, the Taylor formula, Abel-Gontscharoff's interpolation polynomials, Hermite interpolation polynomials and the Fink identity with Green's functions, are presented. The second chapter is dedicated to Ostrowski's inequality and results with applications to numerical integration and probability theory. The third chapter deals with results involving functions with nondecreasing increments. Real life applications are discussed, as well as and connection of functions with nondecreasing increments together with many important concepts including arithmetic integral mean, wright convex functions, convex functions, nabla-convex functions, Jensen m-convex functions, m-convex functions, m-nabla-convex functions, k-monotonic functions, absolutely monotonic functions, completely monotonic functions, Laplace transform and exponentially convex functions, by using the finite difference operator of order m. The fourth chapter is mainly based on Popoviciu and Cebysev-Popoviciu type identities and inequalities. In this last chapter, the authors present results by using delta and nabla operators of higher order.
On Generalized Growth rates of Integer Translated Entire and Meromorphic Functions
The theory of entire and meromorphic functions is a very important area of complex analysis. This monograph aims to expand the discussion about some growth properties of integer translated composite entire and meromorphic functions on the basis of their (p, q, t)L -order and (p, q, t)L -type. This book presents six chapters. Chapter 1 introduces the reader to the preliminary definitions and notations. Chapter 2 and Chapter 3 discuss some results related to (p; q; t) L-th order and (p; q; t)L-th lower order of composite entire and meromorphic functions on the basis of their integer translation. Chapter 4 establishes some relations of integer translated composite entire and meromorphic functions based on their (p; q; t) L-th type and (p; q; t) L-th weak type. Chapter 5 deals with some results about (p; q; t) L-th order and (p; q; t) L-th type of composite entire and meromorphic functions on the basis of their integer translation. Chapter 6 focuses on some results about (p; q; t) L-th order and (p; q; t) L-th type of composite entire and meromorphic functions on the basis of their integer translation. This monograph will be very helpful for postgraduates, researchers, and faculty members interested in value distribution theorems in complex mathematical analysis.
Bayesian Methods for Statistical Analysis
Bayesian methods for statistical analysis is a book on statistical methods for analysing a wide variety of data. The book consists of 12 chapters, starting with basic concepts and covering numerous topics, including Bayesian estimation, decision theory, prediction, hypothesis testing, hierarchical models, Markov chain Monte Carlo methods, finite population inference, biased sampling and nonignorable nonresponse. The book contains many exercises, all with worked solutions, including complete computer code. It is suitable for self-study or a semester-long course, with three hours of lectures and one tutorial per week for 13 weeks.
An Optimization Primer
This richly illustrated book introduces the subject of optimization to a broad audience with a balanced treatment of theory, models and algorithms. Through numerous examples from statistical learning, operations research, engineering, finance and economics, the text explains how to formulate and justify models while accounting for real-world considerations such as data uncertainty. It goes beyond the classical topics of linear, nonlinear and convex programming and deals with nonconvex and nonsmooth problems as well as games, generalized equations and stochastic optimization.The book teaches theoretical aspects in the context of concrete problems, which makes it an accessible onramp to variational analysis, integral functions and approximation theory. More than 100 exercises and 200 fully developed examples illustrate the application of the concepts. Readers should have some foundation in differential calculus and linear algebra. Exposure to real analysis would be helpful but is not prerequisite.
Perturbed Semi-Markov Type Processes I
This book is the first volume of a two-volume monograph devoted to the study of limit and ergodic theorems for regularly and singularly perturbed Markov chains, semi-Markov processes, and multi-alternating regenerative processes with semi-Markov modulation. The first volume presents necessary and sufficient conditions for weak convergence for first-rare-event times and convergence in the topology J for first-rare-event processes defined on regularly perturbed finite Markov chains and semi-Markov processes. The text introduces new asymptotic recurrent algorithms of phase space reduction. It also addresses both effective conditions of weak convergence for distributions of hitting times as well as convergence of expectations of hitting times for regularly and singularly perturbed finite Markov chains and semi-Markov processes. The book also contains a comprehensive bibliography of major works in the field. It provides an effective reference for both graduate students as well as theoretical and applied researchers studying stochastic processes and their applications.
Statistics in the Public Interest
This edited volume surveys a variety of topics in statistics and the social sciences in memory of the late Stephen Fienberg. The book collects submissions from a wide range of contemporary authors to explore the fields in which Fienberg made significant contributions, including contingency tables and log-linear models, privacy and confidentiality, forensics and the law, the decennial census and other surveys, the National Academies, Bayesian theory and methods, causal inference and causes of effects, mixed membership models, and computing and machine learning. Each section begins with an overview of Fienberg's contributions and continues with chapters by Fienberg's students, colleagues, and collaborators exploring recent advances and the current state of research on the topic. In addition, this volume includes a biographical introduction as well as a memorial concluding chapter comprised of entries from Stephen and Joyce Fienberg's close friends, former students, colleagues, and other loved ones, as well as a photographic tribute.
Demand Forecasting for Executives and Professionals
This book surveys what executives who make decisions based on forecasts and professionals responsible for forecasts should know about forecasting. It discusses how individuals and firms should think about forecasting and guidelines for good practices. No prior knowledge is assumed in this book.
Data-Driven Seo with Python
Solve SEO problems using data science. This hands-on book is packed with Python code and data science techniques to help you generate data-driven recommendations and automate the SEO workload. This book is a practical, modern introduction to data science in the SEO context using Python. With social media, mobile, changing search engine algorithms, and ever-increasing expectations of users for super web experiences, too much data is generated for an SEO professional to make sense of in spreadsheets. For any modern-day SEO professional to succeed, it is relevant to find an alternate solution, and data science equips SEOs to grasp the issue at hand and solve it. From machine learning to Natural Language Processing (NLP) techniques, Data-Driven SEO with Python provides tried and tested techniques with full explanations for solving both everyday and complex SEO problems.This book is ideal for SEO professionals who want to take their industry skills to the next level and enhance their business value, whether they are a new starter or highly experienced in SEO, Python programming, or both. What You'll LearnSee how data science works in the SEO contextThink about SEO challenges in a data driven wayApply the range of data science techniques to solve SEO issuesUnderstand site migration and relaunches areWho This Book Is ForSEO practitioners, either at the department head level or all the way to the new career starter looking to improve their skills. Readers should have basic knowledge of Python to perform tasks like querying an API with some data exploration and visualization.
Improper Riemann Integrals
The scope of this book is the improper or generalized Riemann integral and infinite sum (series). The reader will study its convergence, principal value, evaluation and application to science and engineering.
Model-Based Clustering, Classification, and Density Estimation Using mclust in R
Model-based clustering and classification methods provide a systematic statistical approach to clustering, classification, and density estimation via mixture modeling. The model-based framework allows the problems of choosing or developing methods to be understood within the context of statistical modeling.
Climate, Chaos and COVID
Mathematical models are very much in the news now, as they are used to make decisions about our response to such vital areas as COVID-19 and climate change. Frequently, they are blamed for a series of dubious decisions, creating much concern amongst the general public. However, without mathematical models, we would have none of the modern technology that we take for granted, nor would we have modern health care, be able to forecast the climate, cook a potato, have electricity to power our home, or go into space.By explaining technical mathematical concepts in a way that everyone can understand and appreciate, Climate, Chaos and COVID: How Mathematical Models Describe the Universe sets the record straight and lifts the lid off the mystery of mathematical models. It shows why they work, how good they can be, the advantages and disadvantages of using them and how they make the modern world possible. The readers will be able to see the impact that the use of these models has on their lives, and will be able to appreciate both their power and their limitations.The book includes a very large number of both short and long case studies, many of which are taken directly from the author's own experiences of working as a mathematical modeller in academia, in industry, and between the two. These include COVID-19 and climate and how maths saves the whales, powers our home, gives us the material we need to live, and takes us into space.
Operator Theory and Analysis of Infinite Networks
This volume considers resistance networks: large graphs which are connected, undirected, and weighted. Such networks provide a discrete model for physical processes in inhomogeneous media, including heat flow through perforated or porous media. These graphs also arise in data science, e.g., considering geometrizations of datasets, statistical inference, or the propagation of memes through social networks. Indeed, network analysis plays a crucial role in many other areas of data science and engineering. In these models, the weights on the edges may be understood as conductances, or as a measure of similarity. Resistance networks also arise in probability, as they correspond to a broad class of Markov chains.The present volume takes the nonstandard approach of analyzing resistance networks from the point of view of Hilbert space theory, where the inner product is defined in terms of Dirichlet energy. The resulting viewpoint emphasizes orthogonality over convexity and provides new insights into the connections between harmonic functions, operators, and boundary theory. Novel applications to mathematical physics are given, especially in regard to the question of self-adjointness of unbounded operators.New topics are covered in a host of areas accessible to multiple audiences, at both beginning and more advanced levels. This is accomplished by directly linking diverse applied questions to such key areas of mathematics as functional analysis, operator theory, harmonic analysis, optimization, approximation theory, and probability theory.
Theory of Agglomerative Hierarchical Clustering
This book discusses recent theoretical developments in agglomerative hierarchical clustering. The general understanding of agglomerative hierarchical clustering is that its theory was completed long ago and there is no room for further methodological studies, at least in its fundamental structure. This book has been planned counter to that view: it will show that there are possibilities for further theoretical studies and they will be not only for methodological interests but also for usefulness in real applications. When compared with traditional textbooks, the present book has several notable features. First, standard linkage methods and agglomerative procedure are described by a general algorithm in which dendrogram output is expressed by a recursive subprogram. That subprogram describes an abstract tree structure, which is used for a two-stage linkage method for a greater number of objects. A fundamental theorem for single linkage using a fuzzy graph is proved, which uncovers several theoretical features of single linkage. Other theoretical properties such as dendrogram reversals are discussed. New methods using positive-definite kernels are considered, and some properties of the Ward method using kernels are studied. Overall, theoretical features are discussed, but the results are useful as well for application-oriented users of agglomerative clustering.
Causal Inference
A nontechnical guide to the basic ideas of modern causal inference, with illustrations from health, the economy, and public policy. Which of two antiviral drugs does the most to save people infected with Ebola virus? Does a daily glass of wine prolong or shorten life? Does winning the lottery make you more or less likely to go bankrupt? How do you identify genes that cause disease? Do unions raise wages? Do some antibiotics have lethal side effects? Does the Earned Income Tax Credit help people enter the workforce? Causal Inference provides a brief and nontechnical introduction to randomized experiments, propensity scores, natural experiments, instrumental variables, sensitivity analysis, and quasi-experimental devices. Ideas are illustrated with examples from medicine, epidemiology, economics and business, the social sciences, and public policy.
Turnpike Phenomenon and Symmetric Optimization Problems
Written by a leading expert in turnpike phenomenon, this book is devoted to the study of symmetric optimization, variational and optimal control problems in infinite dimensional spaces and turnpike properties of their approximate solutions. The book presents a systematic and comprehensive study of general classes of problems in optimization, calculus of variations, and optimal control with symmetric structures from the viewpoint of the turnpike phenomenon. The author establishes generic existence and well-posedness results for optimization problems and individual (not generic) turnpike results for variational and optimal control problems. Rich in impressive theoretical results, the author presents applications to crystallography and discrete dispersive dynamical systems which have prototypes in economic growth theory.This book will be useful for researchers interested in optimal control, calculus of variations turnpike theory and their applications, suchas mathematicians, mathematical economists, and researchers in crystallography, to name just a few.
Calculus
Calculus is important for first-year undergraduate students pursuing mathematics, physics, economics, engineering, and other disciplines where mathematics plays a significant role. The book provides a thorough reintroduction to calculus with an emphasis on logical development arising out of geometric intuition. The author has restructured the subject matter in the book by using Tarski's version of the completeness axiom, introducing integration before differentiation and limits, and emphasizing benefits of monotonicity before continuity. The standard transcendental functions are developed early in a rigorous manner and the monotonicity theorem is proved before the mean value theorem. Each concept is supported by diverse exercises which will help the reader to understand applications and take them nearer to real and complex analysis.
Regression
Now in its second edition, this textbook provides an applied and unified introduction to parametric, nonparametric and semiparametric regression that closes the gap between theory and application. The most important models and methods in regression are presented on a solid formal basis, and their appropriate application is shown through numerous examples and case studies. The most important definitions and statements are concisely summarized in boxes, and the underlying data sets and code are available online on the book's dedicated website. Availability of (user-friendly) software has been a major criterion for the methods selected and presented.The chapters address the classical linear model and its extensions, generalized linear models, categorical regression models, mixed models, nonparametric regression, structured additive regression, quantile regression and distributional regression models. Two appendices describe the required matrix algebra, as well as elements of probability calculus and statistical inference.In this substantially revised and updated new edition the overview on regression models has been extended, and now includes the relation between regression models and machine learning, additional details on statistical inference in structured additive regression models have been added and a completely reworked chapter augments the presentation of quantile regression with a comprehensive introduction to distributional regression models. Regularization approaches are now more extensively discussed in most chapters of the book.The book primarily targets an audience that includes students, teachers and practitioners in social, economic, and life sciences, as well as students and teachers in statistics programs, and mathematicians and computer scientists with interests in statistical modeling and data analysis. It is written at an intermediate mathematical level and assumes only knowledge of basic probability, calculus, matrix algebra and statistics.
Fiber Bundles
​This book presents a critical overview of statistical fiber bundle models, including existing models and potential new ones. The authors focus on both the physical and statistical aspects of a specific load-sharing example: the breakdown for circuits of capacitors and related dielectrics. In addition, they investigate some areas of open research.This book is designed for graduate students and researchers in statistics, materials science, engineering, physics, and related fields, as well as practitioners and technicians in materials science and mechanical engineering.
Quantifying Counterfactual Military History
Forces shaping human history are complex, but the course of history is undeniably changed on many occasions by conscious acts. These may be premeditated or responsive, calmly calculated or performed under great pressure. They may also be successful or catastrophic, but how are historians to make such judgements and appeal to evidence in support of their conclusions? Further, and crucially, how exactly are we to distinguish probable unrealized alternatives from improbable ones? This book describes some of the modern statistical techniques that can begin to answer this question, as well as some of the difficulties in doing so. Using simple, well-quantified cases drawn from military history, we claim that statistics can now help us to navigate the near-truths, the envelope around the events with which any meaningful historical analysis must deal, and to quantify the basis of such analysis. Quantifying Counterfactual Military History is intended for a general audience who are interested in learning more about statistical methods both in military history and for wider applications. Key Features: This book demonstrates how modern statistical techniques can measure the impact of counterfactual decisions. It examines the importance of counterfactual reasoning for both modern scholars and historical actors. It combines historical narrative, mathematical precision and data to create a straightforward presentation of both factual and counterfactual military history. It provides an original contribution to the debate over the validity and rigour of works of counterfactual history. It is written in a manner accessible to readers who have no formal training in History or Statistics.
Data Science in Practice
Data Science in Practice is the ideal introduction to data science. With or without math skills, here, you get the all-round view that you need for your projects. This book describes how to properly question data, in order to unearth the treasure that data can be. You will get to know the relevant analysis methods, and will be introduced to the programming language R, which is ideally suited for data analysis. Associated tools like notebooks that make data science programming easily accessible are included in this introduction. Because technology alone is not enough, this book also deals with problems in project implementation, illuminates various fields of application, and does not forget to address ethical aspects. Data Science in Practice includes many examples, notes on errors, decision-making aids, and other practical tips. This book is ideal as a complementary text for university students, and is a useful learning tool for those moving into more data-related roles.Key Features: Success factors and tools for all project phases Includes application examples for various subject areas Introduces many aspects of Data Science, from requirements analysis to data acquisition and visualization
Quantifying Counterfactual Military History
Forces shaping human history are complex, but the course of history is undeniably changed on many occasions by conscious acts. These may be premeditated or responsive, calmly calculated or performed under great pressure. They may also be successful or catastrophic, but how are historians to make such judgements and appeal to evidence in support of their conclusions? Further, and crucially, how exactly are we to distinguish probable unrealized alternatives from improbable ones? This book describes some of the modern statistical techniques that can begin to answer this question, as well as some of the difficulties in doing so. Using simple, well-quantified cases drawn from military history, we claim that statistics can now help us to navigate the near-truths, the envelope around the events with which any meaningful historical analysis must deal, and to quantify the basis of such analysis. Quantifying Counterfactual Military History is intended for a general audience who are interested in learning more about statistical methods both in military history and for wider applications.Key Features: This book demonstrates how modern statistical techniques can measure the impact of counterfactual decisions. It examines the importance of counterfactual reasoning for both modern scholars and historical actors. It combines historical narrative, mathematical precision and data to create a straightforward presentation of both factual and counterfactual military history. It provides an original contribution to the debate over the validity and rigour of works of counterfactual history. It is written in a manner accessible to readers who have no formal training in History or Statistics.
Numerical Techniques in MATLAB
In this book, various numerical methods are discussed in a comprehensive way. It delivers a mixture of theory, examples and MATLAB practicing exercises to help the students in improving their skills. The MATLAB codes are mentioned in the end of each topic. Throughout the text, a balance between theory, examples and programming is maintained.
