This is part one of a two-volume book on real analysis and is intended for senior undergraduate students of mathematics who have already been exposed to calculus. The emphasis is on rigour and foundations of analysis. Beginning with the construction of the number systems and set theory, the book discusses the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several variable calculus and Fourier analysis, and then finally the Lebesgue integral. These are almost entirely set in the concrete setting of the real line and Euclidean spaces, although there is some material on abstract metric and topological spaces. The book also has appendices on mathematical logic and the decimal system. The entire text (omitting some less central topics) can be taught in two quarters of 25–30 lectures each. The course material is deeply intertwined with the exercises, as it is intended that the student actively learn the material (and practice thinking and writing rigorously) by proving several of the key results in the theory.
This gentle introduction to discrete mathematics is written for first and second year math majors, especially those who intend to teach. The text began as a set of lecture notes for the discrete mathematics course at the University of Northern Colorado. This course serves both as an introduction to topics in discrete math and as the "introduction to proof" course for math majors. The course is usually taught with a large amount of student inquiry, and this text is written to help facilitate this. Four main topics are covered: counting, sequences, logic, and graph theory. Along the way proofs are introduced, including proofs by contradiction, proofs by induction, and combinatorial proofs. The book contains over 360 exercises, including 230 with solutions and 130 more involved problems suitable for homework. There are also Investigate! activities throughout the text to support active, inquiry based learning. While there are many fine discrete math textbooks available, this text has the following advantages: It is written to be used in an inquiry rich course. It is written to be used in a course for future math teachers. It is open source, with low cost print editions and free electronic editions.
This book is an introduction to the language and standard proof methods of mathematics. It is a bridge from the computational courses (such as calculus or differential equations) that students typically encounter in their first year of college to a more abstract outlook. It lays a foundation for more theoretical courses such as topology, analysis and abstract algebra. Although it may be more meaningful to the student who has had some calculus, there is really no prerequisite other than a measure of mathematical maturity.
Fundamental arithmetic operations support virtually all of the engineering, scientific, and financial computations required for practical applications, from cryptography, to financial planning, to rocket science. This comprehensive reference provides researchers with the thorough understanding of number representations that is a necessary foundation for designing efficient arithmetic algorithms. Using the elementary foundations of radix number systems as a basis for arithmetic, the authors develop and compare alternative algorithms for the fundamental operations of addition, multiplication, division, and square root with precisely defined roundings. Various finite precision number systems are investigated, with the focus on comparative analysis of practically efficient algorithms for closed arithmetic operations over these systems. Each chapter begins with an introduction to its contents and ends with bibliographic notes and an extensive bibliography. The book may also be used for graduate teaching: problems and exercises are scattered throughout the text and a solutions manual is available for instructors.
Geared toward undergraduate and beginning graduate students, this study explores natural numbers, integers, rational numbers, real numbers, and complex numbers. Numerous exercises and appendixes supplement the text. 1973 edition.
This book explores arithmetic's underlying concepts and their logical development, in addition to a detailed, systematic construction of the number systems of rational, real, and complex numbers. 1956 edition.
Fully updated to Elixir 1.15, this authoritative bestseller reveals how Elixir tackles problems of scalability, fault tolerance, and high availability. Thousands of developers have learned to build applications in Elixir by using Saša Juric’s Elixir in Action. You’ll skip the programming basics or 101 introductions; this book builds on your existing knowledge to get you quickly writing real Elixir code. Along the way, you'll develop an appreciation for, and considerable skill in, functional and concurrent programming. Inside Elixir in Action, Third Edition you’ll find: Updates for Elixir 1.15 Elixir modules, functions, and type system Functional and concurrent programming Introduction to distributed system design Creating deployable releases Fully updated to Elixir 1.15, this book contains new coverage of working with application configuration and the latest OTP releases. It teaches you the underlying principles and functional concepts of Elixir, and how each piece fits into the bigger picture of building production-ready systems with Elixir, Erlang, and the OTP framework. Foreword by Francesco Cesarini. Purchase of the print book includes a free eBook in PDF and ePub formats from Manning Publications. About the technology With best-in-class fault tolerance and concurrency, a pragmatic approach to functional programming, and the power to operate at scale, Elixir is the perfect choice for mission-critical software. Start reading Elixir in Action, and you’ll quickly understand why Elixir creator José Valim says it “...tops the list” of Elixir books. About the book Elixir in Action, Third Edition teaches you how to create distributed applications and server-side systems using Elixir and the Erlang VM. This Third Edition from Elixir expert Saša Juric is fully updated to include the latest features of Elixir 1.15. In it, you’ll master the foundations of the language, discover how the OTP framework minimizes tedious boilerplate code, and explore numerous examples that ensure you’re learning hands-on. What's inside Elixir modules, functions, and type system Functional and concurrent programming Introduction to distributed system design Creating deployable releases About the reader For programmers comfortable with client/server applications. No experience with Elixir, Erlang, or functional programming required. About the author Saša Juric uses Elixir and Erlang to build fault-tolerant, scalable, highly concurrent systems. Technical editor on this book was Marius Butuc. Table of Contents PART 1 Functional Elixir 1 First steps 2 Building blocks 3 Control flow 4 Data abstractions PART 2 CONCURRENT ELIXIR 5 Concurrency primitives 6 Generic server processes 7 Building a concurrent system 8 Fault tolerance basics 9 Isolating error effects 10 Beyond GenServer PART 3 PRODUCTION 11 Working with components 12 Building a distributed system 13 Running the system
