ISBN 9780367180614
September 15, 2021 Forthcoming by Chapman and Hall/CRC
316 Pages 11 B/W Illustrations
Number Systems: A Path into Rigorous Mathematics aims to introduce number systems to an undergraduate audience in a way that emphasises the importance of rigour, and with a focus on providing detailed but accessible explanations of theorems and their proofs. The book continually seeks to build upon students' intuitive ideas of how numbers and arithmetic work, and to guide them towards the means to embed this natural understanding into a more structured framework of understanding.
The author’s motivation for writing this book is that most previous texts, which have complete coverage of the subject, have not provided the level of explanation needed for first-year students. On the other hand, those that do give good explanations tend to focus broadly on Foundations or Analysis and provide incomplete coverage of Number Systems.
Approachable for students who have not yet studied mathematics beyond school
Does not merely present definitions, theorems and proofs, but also motivates them in terms of intuitive knowledge and discusses methods of proof
Draws attention to connections with other areas of mathematics
Plenty of exercises for students, both straightforward problems and more in-depth investigations
Introduces many concepts that are required in more advanced topics in mathematics.
1. Introduction: The Purpose of this Book. 1.1. A Very Brief Historical Context. 1.2. The Axiomatic Method. 1.3. The Place of Number Systems within Mathematics. 1.4. Mathematical Writing, Notation and Terminology. 1.5. Logic and Methods of Proof. 2. Sets and Relations. 2.1. Sets. 2.2. Relations between Sets. 2.3. Relations on a Set. 3. Natural Number, N. 3.1. Peano's Axioms. 3.2. Addition of Natural Numbers. 3.3. Multiplication of Natural Numbers. 3.4. Exponentiation (Powers) of Natural Numbers. 3.5. Order in the Natural Numbers. 3.6. Bounded Sets in N. 3.7. Cardinality, Finite and Infinite Sets. 3.8. Subtraction: the Inverse of Addition. 4. Integers, Z. 4.1. Definition of the Integers. 4.2. Arithmetic on Z. 4.3. Algebraic Structure of Z. 4.4. Order in Z. 4.5 Finite, Infinite and Bounded Sets in Z. 5. Foundations of Number Theory. 5.1. Integer Division. 5.2. Expressing Integers in any Base. 5.3. Prime Numbers and Prime Factorisation. 5.4. Congruence. 5.5. Modular Arithmetic. 5.6. Zd as an Algebraic Structure. 6. Rational Numbers, Q. 6.1 Definition of the Rationals. 6.2. Addition and Multiplication on Q. 6.3. Countability of Q. 6.4. Exponentiation and its Inverse(s) on Q. 6.5. Order in Q. 6.6. Bounded Sets in Q. 6.7. Expressing Rational Numbers in any Base. 6.8. Sequences and Series. 7. Real Numbers, R. 7.1. The Requirements for our Next Number System. 7.2. Dedekind Cuts. 7.3. Order and Bounded Sets in R. 7.4 Addition in R. 7.5. Multiplication in R. 7.6. Exponentiation in R. 7.7. Expressing Real Numbers in any Base. 7.8. Cardinality of R. 7.9. Algebraic and Transcendental Numbers. 8. Quadratic Extensions I: General Concepts and Extensions of Z and Q. 8.1. General Concepts of Quadratic Extensions. 8.2. Introduction to Quadratic Rings: Extensions of Z. 8.3. Units in Z[√k]. 8.4. Primes in Z[√k]. 8.5. Prime Factorisation in Z[√k. 8.6. Quadratic Fields: Extensions of Q. 8.7. Norm-Euclidean Rings and Unique Prime Factorisation. 9. Quadratic Extensions II: Complex Numbers, C. 9.1. Complex Numbers as a Quadratic Extension. 9.2. Exponentiation by Real Powers in C: a First Approach. Geometry of C; the Principal Value of the Argument, and the Number π. 9.4. Use of the Argument to Define Real Powers in C. 9.5. Exponentiation by Complex Powers; the Number e. 9.6. The Fundamental Theorem of Algebra. 9.7. Cardinality of C. 10. Yet More Number Systems. 10.1. Constructible Numbers. 10.2. Hypercomplex Numbers. 11. Where Do We Go From Here? 11.1. Number Theory and Abstract Algebra. 11.2. Analysis. A. How to Read Proofs: The `Self-Explanation' Strategy.
Anthony Kay was a Lecturer in Mathematical Sciences at Loughborough University for 32 years up to his retirement in 2020. Although his research has been in applications of mathematics, he has taught a wide range of topics in pure and applied mathematics to students at all levels, from first year to postgraduate.
Copyright Year 2022
Available for pre-order. Item will ship after October 1, 2021
ISBN 9780367077952
October 1, 2021 Forthcoming by CRC Press
88 Pages 21 B/W Illustrations
This book deals with two important branches of mathematics, namely, logic and set theory. Logic and set theory are closely related and play very crucial roles in the foundation of mathematics, and together produce several results in all of mathematics.
The topics of logic and set theory are required in many areas of physical sciences, engineering, and technology. The book offers solved examples and exercises, and provides reasonable details to each topic discussed, for easy understanding.
The book is designed for readers from various disciplines where mathematical logic and set theory play a crucial role. The book will be of interested to students and instructors in engineering, mathematics, computer science, and technology.
1. Mathematical Logic. 2. Set Theory. 3. Relations. 4. Functions. 5. Cardinality of Sets.
Iqbal H. Jebril is Professor in the Department of Mathematics at Al-Zaytoonah University of Jordan, Amman, Jordan. He obtained his PhD from National University of Malaysia (UKM), Malaysia. His fields of research interest include Functional Analysis, Operator Theory and Fuzzy Logic. He has to his credit several prestigious Journal and Conference publications. He is also serving for several journals and conferences in different capacities.
Hemen Dutta obtained his M.Phil and Ph.D. both in Mathematics and also completed Post Graduate Diploma in Computer Application from Gauhati University, India. He is a regular teaching faculty member in the Department of Mathematics at Gauhati University, India. His current research interests include topics in nonlinear analysis and mathematical modelling. He is currently a regular and guest-editor of several SCI/SCIE indexed journals. He has also published several thematic issues in leading journals and books with reputed publishers.
Ilwoo Cho obtained his Ph.D. from University of Iowa, Department of Mathematics, USA. His major fields of research are Free Probability, Operator Theory, Operator Algebra, and Dynamical Systems. He is Full Professor of Saint Ambrose University, Department of Mathematics & Statistics, USA. He is also an Assistant Editor of Complex Analysis & Operator Theory and Mathematics Reviewer of American Mathematical Society.
Copyright Year 2022
Available for pre-order. Item will ship after November 23, 2021
ISBN 9781032100678
November 23, 2021 Forthcoming by Chapman and Hall/CRC
144 Pages 16 Color Illustrations
Diffusion Processes, Jump Processes, and Stochastic Differential Equations provides a compact exposition of the results explaining interrelations between di?usion stochastic processes, stochastic di?erential equations and the fractional in?nitesimal operators. The draft of this book has been extensively classroom tested by the author at Case Western Reserve University in a course that enrolled seniors and graduate students majoring in mathematics, statistics, engineering, physics, chemistry, economics and mathematical ?nance. The last topic proved to be particularly popular among students looking for careers on Wall Street and in research organizations devoted to ?nancial problems.
Quickly and concisely builds from basic probability theory to advanced topics
Suitable as a primary text for an advanced course in diffusion processes and stochastic differential equations
Useful as supplementary reading across a range of topics.
1. Random variables, vectors, processes and fields. 1.1. Random variables, vectors, and their distributions ? a glossary. 1.2. Law of Large Numbers and the Central Limit Theorem. 1.3. Stochastic processes and their finite-dimensional distributions. 1.4. Problems and Exercises. 2. From Random Walk to Brownian Motion. 2.1. Symmetric random walk; parabolic rescaling and related Fokker-Planck equations. 2.2 Almost sure continuity of sample paths. 2.3 Nowhere differentiability of Brownian motion. 2.4 Hitting times, and other subtle properties of Brownian motion. 2.5. Problems and Exercises. 3. Poisson processes and their mixtures. 3.1. Why Poisson process? 3.2. Covariance structure and finite dimensional distributions. 3.3. Waiting times and inter-jump times. 3.4. Extensions and generalizations. 3.5. Fractional Poisson processes (fPp). 3.6. Problems and Exercises. 4. Levy processes and the Levy-Khinchine formula: basic facts. 4.1. Processes with stationary and independent increments. 4.2. From Poisson processes to Levy processes. 4.3. Infinitesimal generators of Levy processes. 4.4. Selfsimilar Levy processes. 4.5. Properties of ?-stable motions. 4.6. Infinitesimal generators of ?-stable motions. 4.7. Problems and Exercises. 5. General processes with independent increments. 5.1. Nonstationary processes with independent increments. 5.2. Stochastic continuity and jump processes. 5.3. Analysis of jump structure. 5.4. Random measures and random integrals associated with jump processes. 5.5. Structure of general I.I. processes. 5.6. Problems and Exercises. 6. Stochastic integrals for Brownian motion and general Levy Processes. 6.1. Wiener random integral. 6.2. Ito's stochastic integral for Brownian motion. 6.3. An instructive example. 6.4. Ito's formula. 6.5. Martingale property of Ito integrals. 6.6. Wiener and Ito-type stochastic integrals for ?-stable motion and general Levy processes. 6.7. Problems and Exercises. 7. Ito stochastic differential equations. 7.1. Differential equations with random noise. 7.2. Stochastic differential equations: Basic theory. 7.3. SDEs with coefficients depending only on time. 7.4. Population growth model and other examples. 7.5. Ornstein-Uhlenbeck process. 7.6. Systems of SDEs and vector-valued Ito's formula. 7.7. Kalman-Bucy filter. 7.8. Numerical solution of stochastic differential equations. 7.9. Problems and Exercises. 8. Asymmetric exclusion processes and their scaling limits. 8.1. Asymmetric exclusion principles. 8.2. Scaling limit. 8.3. Other queuing regimes related to non-nearest neighbor systems. 8.4. Networks with multiserver nodes and particle systems with state-dependent rates. 8.5. Shock and rarefaction wave solutions for the Riemann problem for conservation laws. 8.6. Problems and Exercises. 9. Nonlinear diffusion equations. 9.1. Hyperbolic equations. 9.2. Nonlinear diffusion approximations. 9.3. Problems and Exercises
Wojbor A. Woyczy?ski earned his PhD in Mathematics in 1968 from Wroclaw
University, Poland. He moved to the U.S. in 1970, and since 1982, has been
the Professor of Mathematics and Statistics at Case Western Reserve University
in Cleveland, where he served as chairman of the department from 1982 to
1991, and from 2001 to 2002. He has held tenured faculty positions at Wroclaw
University, Poland, and at Cleveland State University, and visiting appointments
at Carnegie-Mellon University, and Northwestern University. He has also
given invited lecture series on short-term research visits at University
of North Carolina, University of South Carolina, University of Paris, Gottingen
University, Aarhus University, Nagoya University, University of Tokyo,
University of Minnesota, the National University of Taiwan, Taipei, Humboldt
University in Berlin, Germany, and the University of New South Wales in
Sydney. He is also (co)author and/or editor of more than fifteen books
on probability theory, harmonic and functional analysis, and applied mathematics,
and currently serves as a member of the editorial board of the Applicationes
Mathematicae, Springer monograph series UTX, and as a managing editor of
the journal Probability and Mathematical Statistics. His research interests
include probability theory, stochastic models, functional analysis and
partial differential equations and their applications in statistics, statistical
physics, surface chemistry, hydrodynamics and biomedicine in which he has
published about 200 research papers. He has been the advisor of more than
40 graduate students. Among other honors, in 2013 he was awarded Paris
Prix la Recherche, Laureat Mathematiques, for work on mathematical evolution
theory. He is currently Professor of Mathematics, Applied Mathematics and
Statistics, and Director of the Case Center for Stochastic and Chaotic
Processes in Science and Technology at Case Western Reserve University,
in Cleveland, Ohio, U.S.A.
Copyright Year 2022
Available for pre-order. Item will ship after November 11, 2021
ISBN 9780367486891
November 11, 2021 Forthcoming by Chapman and Hall/CRC
400 Pages 26 B/W Illustrations
The book is devoted to describing and applying methods of generalized and functional separation of variables used to find exact solutions of nonlinear PDEs. It also presents the direct method of symmetry reductions and its more general version. In addition, the authors describe the differential constraint method, which generalizes many other exact methods.
The presentation involves numerous examples of utilizing the methods to find exact solutions to specific nonlinear equations of mathematical physics. The equations of heat and mass transfer, wave theory, hydrodynamics, nonlinear optics, combustion theory, chemical technology, biology, and other disciplines are studied.
Particular attention is paid to nonlinear equations of a reasonably general form that depend on one or several arbitrary functions. Such equations are the most difficult to analyze. Their exact solutions are of significant practical interest as they are suitable to assess the accuracy of various approximate analytical and numerical methods.
The book contains much new material previously unpublished in monographs. It is intended for a broad audience of scientists, engineers, instructors, and students specializing in applied and computational mathematics, theoretical physics, mechanics, control theory, chemical engineering science, and other disciplines.
Individual sections of the book and examples are suitable for lecture courses on partial differential equations, equations of mathematical physics, and methods of mathematical physics, for delivering special courses, and for practical training.
1. Methods of Generalized Separation of Variables
2. Methods of Functional Separation of Variables
3. Direct Method of Symmetry Reductions. Weak Symmetries
4. Method of Differential Constraints
Andrei D. Polyanin, D.Sc., Ph.D., Professor, is a well-known scientist of broad interests and is active in various areas of mathematics, mechanics, and chemical engineering sciences. Professor Polyanin graduated with honors from the Department of Mechanics and Mathematics of Moscow State University in 1974. He received his Ph.D. degree in 1981 and D.Sc. degree in 1986 at the Institute for Problems in Mechanics of the Russian (former USSR) Academy of Sciences. Since 1975, Professor Polyanin has been working at the Institute for Problems in Mechanics of the Russian Academy of Sciences; he is also Professor of Mathematics at Bauman Moscow State Technical University and at National Research Nuclear University MEPhI. He is a member of the Russian National Committee on Theoretical and Applied Mechanics and of the Mathematics and Mechanics Expert Council of the Higher Certification Committee of the Russian Federation. Alexei I. Zhurov, Ph.D., is scientist in nonlinear mechanics, mathematical physics, computer algebra, biomechanics, and morphometrics. He graduated with honors from the Department of Airphysics and Space Research of the Moscow Institute of Physics and Technology in 1990. Since then has become a member of staff of the Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, where he received his PhD in mechanics and fluid dynamics in 1995 and has become a senior research scientist since 1999. Since 2001, he has joined Cardiff University as a research scientist in the area of biomechanics and morphometrics.
https://doi.org/10.1142/11927-vol1 | September 2021
Pages: 692
ISBN: 978-981-122-392-1 (hardcover)
USD 188.00
ISBN: 978-981-122-488-1 (softcover)
This volume is part of a reference set. To view or order the whole set, please visit the main page.
Yes, this is another Calculus book. However, I think it fits in a niche between the two predominant types of such texts. It could be used as a textbook, albeit a streamlined one ? it contains exposition on each topic, with an introduction, rationale, train of thought, and solved examples with accompanying suggested exercises. It could be used as a solution guide ? because it contains full written solutions to each of the hundreds of exercises posed inside. But its best position is right in between these two extremes. It is best used as a companion to a traditional text or as a refresher ? with its conversational tone, its "get right to it" content structure, and its inclusion of complete solutions to many problems, it is a friendly partner for students who are learning Calculus, either in class or via self-study.
Exercises are structured in three sets to force multiple encounters with each topic. Solved examples in the text are accompanied by "You Try It" problems, which are similar to the solved examples; the students use these to see if they're ready to move forward. Then at the end of the section, there are "Practice Problems": more problems similar to the You Try It problems, but given all at once. Finally, each section has Challenge Problems ? these lean to being equally or a bit more difficult than the others, and they allow students to check on what they've mastered.
My goal is to keep the students engaged with the text, and so the writing style is very informal, with attempts at humor along the way. Because we have large engineering and meteorology programs at my institution, and they make up the largest portion of our Calculus students; naturally, then, these sorts of STEM students are the target audience.
Prelude: Your Mathematics Pep Talk
Years of Work in One Chapter
Take It to the Limit
Embrace the Change
Abandon Hope All Ye Who Enter Here
Calculus Has Its Ups and Downs
The Best Mathematics Symbol Ever
Solutions to All Practice Problems
Solutions to All Challenge Problems
Index
Undergraduate students currently taking or refreshing themselves on Calculus.