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An Introduction to the Early Development of Mathematics
by Michael K. GoodmanAn easy-to-read presentation of the early history of mathematics Engaging and accessible, An Introduction to the Early Development of Mathematics provides a captivating introduction to the history of ancient mathematics in early civilizations for a nontechnical audience. Written with practical applications in a variety of areas, the book utilizes the historical context of mathematics as a pedagogical tool to assist readers working through mathematical and historical topics. The book is divided into sections on significant early civilizations including Egypt, Babylonia, China, Greece, India, and the Islamic world. Beginning each chapter with a general historical overview of the civilized area, the author highlights the civilization’s mathematical techniques, number representations, accomplishments, challenges, and contributions to the mathematical world. Thoroughly class-tested, An Introduction to the Early Development of Mathematics features: Challenging exercises that lead readers to a deeper understanding of mathematics Numerous relevant examples and problem sets with detailed explanations of the processes and solutions at the end of each chapter Additional references on specific topics and keywords from history, archeology, religion, culture, and mathematics Examples of practical applications with step-by-step explanations of the mathematical concepts and equations through the lens of early mathematical problems A companion website that includes additional exercises An Introduction to the Early Development of Mathematics is an ideal textbook for undergraduate courses on the history of mathematics and a supplement for elementary and secondary education majors. The book is also an appropriate reference for professional and trade audiences interested in the history of mathematics. Michael K. J. Goodman is Adjunct Mathematics Instructor at Westchester Community College, where he teaches courses in the history of mathematics, contemporary mathematics, and algebra. He is also the owner and operator of The Learning Miracle, LLC, which provides academic tutoring and test preparation for both college and high school students.
An Introduction to the Finite Element Method for Differential Equations
by Mohammad AsadzadehMaster the finite element method with this masterful and practical volume An Introduction to the Finite Element Method (FEM) for Differential Equations provides readers with a practical and approachable examination of the use of the finite element method in mathematics. Author Mohammad Asadzadeh covers basic FEM theory, both in one-dimensional and higher dimensional cases. The book is filled with concrete strategies and useful methods to simplify its complex mathematical contents. Practically written and carefully detailed, An Introduction to the Finite Element Method covers topics including: An introduction to basic ordinary and partial differential equations The concept of fundamental solutions using Green's function approaches Polynomial approximations and interpolations, quadrature rules, and iterative numerical methods to solve linear systems of equations Higher-dimensional interpolation procedures Stability and convergence analysis of FEM for differential equations This book is ideal for upper-level undergraduate and graduate students in natural science and engineering. It belongs on the shelf of anyone seeking to improve their understanding of differential equations.
Introduction to the Foundations of Applied Mathematics (Texts in Applied Mathematics #56)
by Mark H. HolmesThe objective of this textbook is the construction, analysis, and interpretation of mathematical models to help us understand the world we live in. Rather than follow a case study approach it develops the mathematical and physical ideas that are fundamental in understanding contemporary problems in science and engineering. Science evolves, and this means that the problems of current interest continually change. What does not change as quickly is the approach used to derive the relevant mathematical models, and the methods used to analyze the models. Consequently, this book is written in such a way as to establish the mathematical ideas underlying model development independently of a specific application. This does not mean applications are not considered, they are, and connections with experiment are a staple of this book. The book, as well as the individual chapters, is written in such a way that the material becomes more sophisticated as you progress. This provides some flexibility in how the book is used, allowing consideration for the breadth and depth of the material covered. Moreover, there are a wide spectrum of exercises and detailed illustrations that significantly enrich the material. Students and researchers interested in mathematical modelling in mathematics, physics, engineering and the applied sciences will find this text useful.The material, and topics, have been updated to include recent developments in mathematical modeling. The exercises have also been expanded to include these changes, as well as enhance those from the first edition.Review of first edition:"The goal of this book is to introduce the mathematical tools needed for analyzing and deriving mathematical models. … Holmes is able to integrate the theory with application in a very nice way providing an excellent book on applied mathematics. … One of the best features of the book is the abundant number of exercises found at the end of each chapter. … I think this is a great book, and I recommend it for scholarly purposes by students, teachers, and researchers." Joe Latulippe, The Mathematical Association of America, December, 2009
Introduction to the Foundations of Mathematics: Second Edition (Dover Books on Mathematics)
by Raymond L. WilderThis classic undergraduate text acquaints students with the fundamental concepts and methods of mathematics. In addition to introducing many historical figures from the 18th through the mid-20th centuries, it examines the axiomatic method, set theory, infinite sets, groups, intuitionism, formal systems, mathematical logic, and other topics. 1965 second edition.
Introduction to the Geometry of Complex Numbers (Dover Books on Mathematics)
by Howard Eves Roland DeauxGeared toward readers unfamiliar with complex numbers, this text explains how to solve the kinds of problems that frequently arise in the applied sciences, especially electrical studies. To assure an easy and complete understanding, it develops topics from the beginning, with emphasis on constructions related to algebraic operations.The three-part treatment begins with geometric representations of complex numbers and proceeds to an in-depth survey of elements of analytic geometry. Readers are assured of a variety of perspectives, which include references to algebra, to the classical notions of analytic geometry, to modern plane geometry, and to results furnished by kinematics. The third chapter, on circular transformations, revives in a slightly modified form the essentials of the projective geometry of real binary forms. Numerous exercises appear throughout the text.
An Introduction to the Kähler-Ricci Flow (Lecture Notes in Mathematics #2086)
by Sebastien Boucksom Philippe Eyssidieux Vincent GuedjThis volume collects lecture notes from courses offered at several conferences and workshops, and provides the first exposition in book form of the basic theory of the Kähler-Ricci flow and its current state-of-the-art. While several excellent books on Kähler-Einstein geometry are available, there have been no such works on the Kähler-Ricci flow. The book will serve as a valuable resource for graduate students and researchers in complex differential geometry, complex algebraic geometry and Riemannian geometry, and will hopefully foster further developments in this fascinating area of research. The Ricci flow was first introduced by R. Hamilton in the early 1980s, and is central in G. Perelman's celebrated proof of the Poincaré conjecture. When specialized for Kähler manifolds, it becomes the Kähler-Ricci flow, and reduces to a scalar PDE (parabolic complex Monge-Ampère equation). As a spin-off of his breakthrough, G. Perelman proved the convergence of the Kähler-Ricci flow on Kähler-Einstein manifolds of positive scalar curvature (Fano manifolds). Shortly after, G. Tian and J. Song discovered a complex analogue of Perelman's ideas: the Kähler-Ricci flow is a metric embodiment of the Minimal Model Program of the underlying manifold, and flips and divisorial contractions assume the role of Perelman's surgeries.
An Introduction to the Kolmogorov–Bernoulli Equivalence (SpringerBriefs in Mathematics)
by Gabriel Ponce Régis VarãoThis book offers an introduction to a classical problem in ergodic theory and smooth dynamics, namely, the Kolmogorov–Bernoulli (non)equivalence problem, and presents recent results in this field. Starting with a crash course on ergodic theory, it uses the class of ergodic automorphisms of the two tori as a toy model to explain the main ideas and technicalities arising in the aforementioned problem. The level of generality then increases step by step, extending the results to the class of uniformly hyperbolic diffeomorphisms, and concludes with a survey of more recent results in the area concerning, for example, the class of partially hyperbolic diffeomorphisms. It is hoped that with this type of presentation, nonspecialists and young researchers in dynamical systems may be encouraged to pursue problems in this area.
An Introduction to the Language of Category Theory (Compact Textbooks in Mathematics)
by Steven RomanThis textbook provides an introduction to elementary category theory, with the aim of making what can be a confusing and sometimes overwhelming subject more accessible. In writing about this challenging subject, the author has brought to bear all of the experience he has gained in authoring over 30 books in university-level mathematics. The goal of this book is to present the five major ideas of category theory: categories, functors, natural transformations, universality, and adjoints in as friendly and relaxed a manner as possible while at the same time not sacrificing rigor. These topics are developed in a straightforward, step-by-step manner and are accompanied by numerous examples and exercises, most of which are drawn from abstract algebra. The first chapter of the book introduces the definitions of category and functor and discusses diagrams, duality, initial and terminal objects, special types of morphisms, and some special types of categories, particularly comma categories and hom-set categories. Chapter 2 is devoted to functors and natural transformations, concluding with Yoneda's lemma. Chapter 3 presents the concept of universality and Chapter 4 continues this discussion by exploring cones, limits, and the most common categorical constructions - products, equalizers, pullbacks and exponentials (along with their dual constructions). The chapter concludes with a theorem on the existence of limits. Finally, Chapter 5 covers adjoints and adjunctions. Graduate and advanced undergraduates students in mathematics, computer science, physics, or related fields who need to know or use category theory in their work will find An Introduction to Category Theory to be a concise and accessible resource. It will be particularly useful for those looking for a more elementary treatment of the topic before tackling more advanced texts.
An Introduction to the Language of Mathematics
by Frédéric MynardThis is a textbook for an undergraduate mathematics major transition course from technique-based mathematics (such as Algebra and Calculus) to proof-based mathematics. It motivates the introduction of the formal language of logic and set theory and develops the basics with examples, exercises with solutions and exercises without. It then moves to a discussion of proof structure and basic proof techniques, including proofs by induction with extensive examples. An in-depth treatment of relations, particularly equivalence and order relations completes the exposition of the basic language of mathematics. The last chapter treats infinite cardinalities. An appendix gives some complement on induction and order, and another provides full solutions of the in-text exercises. The primary audience is undergraduate mathematics major, but independent readers interested in mathematics can also use the book for self-study.
An Introduction to the Mathematical Theory of Inverse Problems (Applied Mathematical Sciences #120)
by Andreas KirschThis book introduces the reader to the area of inverse problems. The study of inverse problems is of vital interest to many areas of science and technology such as geophysical exploration, system identification, nondestructive testing and ultrasonic tomography. The aim of this book is twofold: in the first part, the reader is exposed to the basic notions and difficulties encountered with ill-posed problems. Basic properties of regularization methods for linear ill-posed problems are studied by means of several simple analytical and numerical examples. The second part of the book presents two special nonlinear inverse problems in detail - the inverse spectral problem and the inverse scattering problem. The corresponding direct problems are studied with respect to existence, uniqueness and continuous dependence on parameters. Then some theoretical results as well as numerical procedures for the inverse problems are discussed. The choice of material and its presentation in the book are new, thus making it particularly suitable for graduate students. Basic knowledge of real analysis is assumed. In this new edition, the Factorization Method is included as one of the prominent members in this monograph. Since the Factorization Method is particularly simple for the problem of EIT and this field has attracted a lot of attention during the past decade a chapter on EIT has been added in this monograph as Chapter 5 while the chapter on inverse scattering theory is now Chapter 6.The main changes of this second edition compared to the first edition concern only Chapters 5 and 6 and the Appendix A. Chapter 5 introduces the reader to the inverse problem of electrical impedance tomography.
An Introduction to the Mathematical Theory of Inverse Problems (Applied Mathematical Sciences #120)
by Andreas KirschThis graduate-level textbook introduces the reader to the area of inverse problems, vital to many fields including geophysical exploration, system identification, nondestructive testing, and ultrasonic tomography. It aims to expose the basic notions and difficulties encountered with ill-posed problems, analyzing basic properties of regularization methods for ill-posed problems via several simple analytical and numerical examples. The book also presents three special nonlinear inverse problems in detail: the inverse spectral problem, the inverse problem of electrical impedance tomography (EIT), and the inverse scattering problem. The corresponding direct problems are studied with respect to existence, uniqueness, and continuous dependence on parameters. Ultimately, the text discusses theoretical results as well as numerical procedures for the inverse problems, including many exercises and illustrations to complement coursework in mathematics and engineering. This updated text includes a new chapter on the theory of nonlinear inverse problems in response to the field’s growing popularity, as well as a new section on the interior transmission eigenvalue problem which complements the Sturm-Liouville problem and which has received great attention since the previous edition was published.
Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Developments In Geomathematics Ser. #Volume 3)
by S. TwomeyIn this graduate-level monograph, S. Twomey, a professor of atmospheric sciences, develops the background and fundamental theory of inversion processes used in remote sensing — e.g., atmospheric temperature structure measurements from satellites—starting at an elementary level.The text opens with examples of inversion problems from a variety of disciplines, showing that the same problem—solution of a Fredholm linear integral equation of the first kind — is involved in every instance. A discussion of the reduction of such integral equations to a system of linear algebraic equations follows. Subsequent chapters examine methods for obtaining stable solutions at the expense of introducing constraints in the solution, the derivation of other inversion procedures, and the detailed analysis of the information content of indirect measurements. Each chapter begins with a discussion that outlines problems and questions to be covered, and a helpful Appendix includes suggestions for further reading.
Introduction to the Mathematics of Operations Research with Mathematica®
by Kevin J. HastingsThe breadth of information about operations research and the overwhelming size of previous sources on the subject make it a difficult topic for non-specialists to grasp. Fortunately, Introduction to the Mathematics of Operations Research with Mathematica®, Second Edition delivers a concise analysis that benefits professionals in operations research and related fields in statistics, management, applied mathematics, and finance.The second edition retains the character of the earlier version, while incorporating developments in the sphere of operations research, technology, and mathematics pedagogy. Covering the topics crucial to applied mathematics, it examines graph theory, linear programming, stochastic processes, and dynamic programming. This self-contained text includes an accompanying electronic version and a package of useful commands. The electronic version is in the form of Mathematica notebooks, enabling you to devise, edit, and execute/reexecute commands, increasing your level of comprehension and problem-solving.Mathematica sharpens the impact of this book by allowing you to conveniently carry out graph algorithms, experiment with large powers of adjacency matrices in order to check the path counting theorem and Markov chains, construct feasible regions of linear programming problems, and use the "dictionary" method to solve these problems. You can also create simulators for Markov chains, Poisson processes, and Brownian motions in Mathematica, increasing your understanding of the defining conditions of these processes. Among many other benefits, Mathematica also promotes recursive solutions for problems related to first passage times and absorption probabilities.
An Introduction to the Mathematics of Planning and Scheduling
by Geza Paul BottlikThis book introduces readers to the many variables and constraints involved in planning and scheduling complex systems, such as airline flights and university courses. Students will become acquainted with the necessity for scheduling activities under conditions of limited resources in industrial and service environments, and become familiar with methods of problem solving. Written by an expert author with decades of teaching and industry experience, the book provides a comprehensive explanation of the mathematical foundations to solving complex requirements, helping students to understand underlying models, to navigate software applications more easily, and to apply sophisticated solutions to project management. This is emphasized by real-world examples, which follow the components of the manufacturing process from inventory to production to delivery. Undergraduate and graduate students of industrial engineering, systems engineering, and operations management will find this book useful in understanding optimization with respect to planning and scheduling.
Introduction to the Maths and Physics of Quantum Mechanics
by Lucio PiccirilloIntroduction to the Maths and Physics of Quantum Mechanics details the mathematics and physics that are needed to learn the principles of quantum mechanics. It provides an accessible treatment of how to use quantum mechanics and why it is so successful in explaining natural phenomena. This book clarifies various aspects of quantum physics such as ‘why quantum mechanics equations contain “I”, the imaginary number?’, ‘Is it possible to make a transition from classical mechanics to quantum physics without using postulates?’ and ‘What is the origin of the uncertainty principle?’. A significant proportion of discussion is dedicated to the issue of why the wave function must be complex to properly describe our “real” world. The book also addresses the different formulations of quantum mechanics. A relatively simple introductory treatment is given for the “standard” Heisenberg matrix formulation and Schrodinger wave-function formulation and Feynman path integrals and second quantization are then discussed. This book will appeal to first- and second-year university students in physics, mathematics, engineering and other sciences studying quantum mechanics who will find material and clarifications not easily found in other textbooks. It will also appeal to self-taught readers with a genuine interest in modern physics who are willing to examine the mathematics and physics in a simple but rigorous way. Key Features: • Written in an engaging and approachable manner, with fully explained mathematics and physics concepts. <p class="MsoListParagraphCxSpMiddle" style="text-in
Introduction to the Maths and Physics of the Solar System
by Lucio PiccirilloThis book provides readers with an understanding of the basic physics and mathematics that governs our solar system. It explores the mechanics of our Sun and planets; their orbits, tides, eclipses and many other fascinating phenomena. This book is a valuable resource for undergraduate students studying astronomy and should be used in conjunction with other introductory astronomy textbooks in the field to provide additional learning opportunities. Features: Written in an engaging and approachable manner, with fully explained mathematics and physics concepts Suitable as a companion to all introductory astronomy textbooks Accessible to a general audience
Introduction to the Network Approximation Method for Materials Modeling
by Leonid Berlyand Alexander G. Kolpakov Alexei NovikovIn recent years the traditional subject of continuum mechanics has grown rapidly and many new techniques have emerged. This text provides a rigorous, yet accessible introduction to the basic concepts of the network approximation method and provides a unified approach for solving a wide variety of applied problems. As a unifying theme, the authors discuss in detail the transport problem in a system of bodies. They solve the problem of closely placed bodies using the new method of network approximation for PDE with discontinuous coefficients, developed in the 2000s by applied mathematicians in the USA and Russia. Intended for graduate students in applied mathematics and related fields such as physics, chemistry and engineering, the book is also a useful overview of the topic for researchers in these areas.
Introduction to the New Statistics: Estimation, Open Science, and Beyond
by Geoff Cumming Robert Calin-JagemanThis is the first introductory statistics text to use an estimation approach from the start to help readers understand effect sizes, confidence intervals (CIs), and meta-analysis (‘the new statistics’). It is also the first text to explain the new and exciting Open Science practices, which encourage replication and enhance the trustworthiness of research. In addition, the book explains NHST fully so students can understand published research. Numerous real research examples are used throughout. The book uses today’s most effective learning strategies and promotes critical thinking, comprehension, and retention, to deepen users’ understanding of statistics and modern research methods. The free ESCI (Exploratory Software for Confidence Intervals) software makes concepts visually vivid, and provides calculation and graphing facilities. The book can be used with or without ESCI. Other highlights include: - Coverage of both estimation and NHST approaches, and how to easily translate between the two. - Some exercises use ESCI to analyze data and create graphs including CIs, for best understanding of estimation methods. -Videos of the authors describing key concepts and demonstrating use of ESCI provide an engaging learning tool for traditional or flipped classrooms. -In-chapter exercises and quizzes with related commentary allow students to learn by doing, and to monitor their progress. -End-of-chapter exercises and commentary, many using real data, give practice for using the new statistics to analyze data, as well as for applying research judgment in realistic contexts. -Don’t fool yourself tips help students avoid common errors. -Red Flags highlight the meaning of "significance" and what p values actually mean. -Chapter outlines, defined key terms, sidebars of key points, and summarized take-home messages provide a study tool at exam time. -http://www.routledge.com/cw/cumming offers for students: ESCI downloads; data sets; key term flashcards; tips for using SPSS for analyzing data; and videos. For instructors it offers: tips for teaching the new statistics and Open Science; additional homework exercises; assessment items; answer keys for homework and assessment items; and downloadable text images; and PowerPoint lecture slides. Intended for introduction to statistics, data analysis, or quantitative methods courses in psychology, education, and other social and health sciences, researchers interested in understanding the new statistics will also appreciate this book. No familiarity with introductory statistics is assumed.
Introduction to the New Statistics: Estimation, Open Science, and Beyond
by Geoff Cumming Robert Calin-JagemanThis fully revised and updated second edition is an essential introduction to inferential statistics. It is the first introductory statistics text to use an estimation approach from the start and also to explain the new and exciting Open Science practices, which encourage replication and enhance the trustworthiness of research. The estimation approach, with meta-analysis (“the new statistics”), is exactly what’s needed for Open Science.Key features of this new edition include: Even greater prominence for Open Science throughout the book. Students easily understand basic Open Science practices and are guided to use them in their own work. There is discussion of the latest developments now being widely adopted across science and medicine. Integration of new open-source esci (Estimation Statistics with Confidence Intervals) software, running in jamovi. This is ideal for the book and extends seamlessly to what’s required for more advanced courses, and also by researchers. See www.thenewstatistics.com/itns/esci/jesci/. Colorful interactive simulations, including the famous dances, to help make key statistical ideas intuitive. These are now freely available through any browser. See www.esci.thenewstatistics.com/. Coverage of both estimation and null hypothesis significance testing (NHST) approaches, with full guidance on how to translate between the two. Effective learning strategies and pedagogical features to promote critical thinking, comprehension and retention Designed for introduction to statistics, data analysis, or quantitative methods courses in psychology, education, and other social and health sciences, researchers interested in understanding Open Science and the new statistics will also appreciate this book. No familiarity with introductory statistics is assumed.A comprehensive website offers data sets, key term flashcards, learning guides, and videos describing key concepts and demonstrating the use of esci. For instructors, there are guides for teaching the new statistics and Open Science, assessment exercises, question banks, downloadable slides, and more. Altogether, the website provides engaging learning resources for traditional or flipped classrooms. See www.routledge.com/cw/cumming.
An Introduction to the Philosophy of Mathematics
by Mark ColyvanThis introduction to the philosophy of mathematics focuses on contemporary debates in an important and central area of philosophy. The reader is taken on a fascinating and entertaining journey through some intriguing mathematical and philosophical territory, including such topics as the realism/anti-realism debate in mathematics, mathematical explanation, the limits of mathematics, the significance of mathematical notation, inconsistent mathematics and the applications of mathematics. Each chapter has a number of discussion questions and recommended further reading from both the contemporary literature and older sources. Very little mathematical background is assumed and all of the mathematics encountered is clearly introduced and explained using a wide variety of examples. The book is suitable for an undergraduate course in philosophy of mathematics and, more widely, for anyone interested in philosophy and mathematics.
Introduction to the Physics of Matter: Basic Atomic, Molecular, and Solid-State Physics (Undergraduate Lecture Notes in Physics)
by Nicola ManiniThis is the second edition of a well-received book. It provides an up-to-date, concise review of essential topics in the physics of matter, from atoms and molecules to solids, including elements of statistical mechanics. It features over 160 completely revised and enhanced figures illustrating the main physical concepts and the fundamental experimental facts, and discusses selected experiments, mainly in spectroscopy and thermodynamics, within the general framework of the adiabatic separation of the motions of electrons and nuclei. The book focuses on what can be described in terms of independent-particle models, providing the mathematical derivations in sufficient detail for readers to grasp the relevant physics involved. The final section offers a glimpse of more advanced topics, including magnetism and superconductivity, sparking readers’ curiosity to further explore the latest developments in the physics of matter.
Introduction to the Practice of Statistics
by David S. Moore George P. McCabe Bruce A. CraigWith this updated new edition, the market-leading Introduction to the Practice of Statistics (IPS) remains unmatched in its ability to show how statisticians actually work. Its focus on data analysis and critical thinking, step-by-step pedagogy, and applications in a variety of professions and disciplines make it exceptionally engaging to students learning core statistical ideas.
Introduction to the Practice of Statistics
by David S. Moore George P. McCabe Bruce A. CraigNow available with Macmillan’s new online learning tool Achieve, Introduction to the Practice of Statistics, 10th edition, prepares students for the application of statistics in the real world by using current examples and encouraging exploration into data analysis and interpretation. The text enforces statistical thinking by providing learning objectives and linked exercises to help students master core statistics concepts and think beyond the calculations. <p><p> Achieve for Introduction to the Practice of Statistics integrates outcome-based learning objectives and a wealth of examples with assessment in an easy-to-use interface. Students are provided with rich digital resources that solidify conceptual understanding, as well as homework problems with hints, answer-specific feedback, and a fully worked solution.
Introduction to the Practice of Statistics (3rd edition)
by David S. Moore George P. MccabeThis edition is an introductory text that focuses on data and on statistical reasoning. It is elementary in mathematical level, but conceptually rich in statistical ideas and serious in its aim to help students think about data and use statistical methods with understanding.
Introduction to the Practice of Statistics (Ninth Edition)
by David S. Moore George P. Mccabe Bruce A. CraigIntroduction to the Practice of Statistics (IPS) shows students how to produce and interpret data from real-world contexts—doing the same type of data gathering and analysis that working statisticians in all kinds of businesses and institutions do every day. With this phenomenally successful approach originally developed by David Moore and George McCabe, statistics is more than just a collection of techniques and formulas. Instead, students develop a systematic way of thinking about data, with a focus on problem-solving that helps them understand statistical concepts and master statistical reasoning.