The proofs of the vast majority of theorems are presented whilst pedagogical considerations are given priority
Preliminary discussion of each major topic via examples and diagrams helps ease the reader into the appropriate context for the topic
Examples follow most major results to improve comprehension and chapter-end exercises help with the refinement of proof and calculation skills
Real Analysis and Infinity presents the essential topics for a first course in real analysis with an emphasis on the role of infinity in all of the fundamental concepts. After introducing sequences of numbers, it develops the set of real numbers in terms of Cauchy sequences of rational numbers, and uses this development to derive the important properties of real numbers like completeness. The book then develops the concepts of continuity, derivative, and integral, and presents the theory of infinite sequences and series of functions.
Topics discussed are wide-ranging and include the convergence of sequences, definition of limits and continuity via converging sequences, and the development of derivative. The proofs of the vast majority of theorems are presented and pedagogical considerations are given priority to help cement the reader's knowledge.
Preliminary discussion of each major topic is supplemented with examples and diagrams, and historical asides. Examples follow most major results to improve comprehension, and exercises at the end of each chapter help with the refinement of proof and calculation skills.
Preface
1:Manifestations of Infinity: An Overview
2:Sets, Functions, Logic and Countability
3:Sequences and Limits
4:The Real Numbers
5:Infinite Series of Constants
6:Differentiation and Continuity
7:Integration
8:Infinite Sequences and Series of Functions
Appendix: Cantor's Construction: Additional Detail
Appendix: Discontinuity in a Space of Functions
References and Further Reading
Hardback
Published: 31 March 2022 (Estimated)
320 Pages | 109 black and white illustrations
234x156mm
ISBN: 9780192895622
Oxford Handbooks in Physics
An accessible introduction to long-standing 20th and 21st centuries scientific controversies
Technical and scientific content is kept to a minimum to ensure a wide readership
Draws on the wide-ranging expertise of authors from diverse backgrounds including historians of physics, physicists, and philosophers of physics
Crucial to most research in physics, as well as leading to the development of inventions such as the transistor and the laser, quantum mechanics approaches its centenary with an impressive record. However, the field has also long been the subject of ongoing debates about the foundations and interpretation of the theory, referred to as the quantum controversy.
This Oxford Handbook offers a historical overview of the contrasts which have been at the heart of quantum physics for the last 100 years. Drawing on the wide-ranging expertise of several contributors working across physics, history, and philosophy, the handbook outlines the main theories and interpretations of quantum physics. It goes on to tackle the key controversies surrounding the field, touching on issues such as determinism, realism, locality, classicality, information, measurements, mathematical foundations, and the links between quantum theory and gravity.
This engaging introduction is an essential guide for all those interested in the history of scientific controversies and history of quantum physics. It also provides a fascinating examination of the potential of quantum physics to influence new discoveries and advances in fields such quantum information and computing.
Allows as much time as it takes to establish confident facility with fundamental ideas
Explicitly and accessibly describes the processes by which solutions are discovered, as well as how those discoveries are refined into logically secure mathematical arguments
Incorporates worked examples and exercises supported by partial solutions or hints to scaffold understanding and reinforce learning
Includes a companion website with solutions to the problems in the text
Complex analysis, more than almost any other undergraduate topic in mathematics, runs the full pure/applied gamut from the most subtle, difficult, and ingenious proofs to the most direct, hands-on, engineering-based applications. This creates challenges for the instructor as much as for the very wide range of students whose various programmes require a secure grasp of complex analysis. Its techniques are indispensable to many, but skill in the use of a mathematical tool is hazardous and fallible without a sound understanding of why and when that tool is the right one to pick up. This kind of understanding develops only by combining careful exploration of ideas, analysis of proofs, and practice across a range of exercises.
Integration with Complex Numbers: A Primer on Complex Analysis offers a reader-friendly contemporary balance between idea, proof, and practice, informed by several decades of classroom experience and a seasoned understanding of the backgrounds, motivation, and competing time pressures of today's student cohorts. To achieve its aim of supporting and sustaining such cohorts through those aspects of complex analysis that they encounter in first and second-year study, it also balances competing needs to be self-contained, comprehensive, accessible, and engaging - all in sufficient but not in excessive measures. In particular, it begins where most students are likely to be, and invests the time and effort that are required in order to deliver accessibility and introductory gradualness.
1:Background Part A: set, sequence, series
2:What are complex numbers?
3:Background Part B: real functions and their limits
4:Complex functions
5:Background Part C: real integration
6:Paths in the complex plane
7:Cauchy's theorem(s)
8:Taylor's theorem
9:Residues
10:Reality from complexity
11:The repair shop for broken promises
Hardback
Published: 14 March 2022 (Estimated)
256 Pages
234x156mm
ISBN: 9780192846075
Paperback
ISBN: 9780192846433
A meticulously researched history on the development of American mathematics in the three decades following World War I
As the Roaring Twenties lurched into the Great Depression, to be followed by the scourge of Nazi Germany and World War II, American mathematicians pursued their research, positioned themselves collectively within American science, and rose to global mathematical hegemony. How did they do it? The New Era in American Mathematics, 1920?1950 explores the institutional, financial, social, and political forces that shaped and supported this community in the first half of the twentieth century. In doing so, Karen Hunger Parshall debunks the widely held view that American mathematics only thrived after European emigres fled to the shores of the United States.
Drawing from extensive archival and primary-source research, Parshall uncovers the key players in American mathematics who worked together to effect change and she looks at their research output over the course of three decades. She highlights the educational, professional, philanthropic, and governmental entities that bolstered progress. And she uncovers the strategies implemented by American mathematicians in their quest for the advancement of knowledge. Throughout, she considers how geopolitical circumstances shifted the course of the discipline.
Examining how the American mathematical community asserted itself on the international stage, The New Era in American Mathematics, 1920?1950 shows the way one nation became the focal point for the field.
The yearfs finest mathematical writing from around the world
This annual anthology brings together the yearfs finest mathematics writing from around the world?and you donft need to be a mathematician to enjoy the pieces collected here. These essays?from leading names and fresh new voices?delve into the history, philosophy, teaching, and everyday aspects of math, offering surprising insights into its nature, meaning, and practice, and taking readers behind the scenes of todayfs hottest mathematical debates.
Here, Viktor Blasjo gives a brief history of glockdown mathematicsh; Yelda Nasifoglu decodes the politics of a seventeenth-century play in which the characters are geometric shapes; and Andrew Lewis-Pye explains the basic algorithmic rules and computational procedures behind cryptocurrencies. In other essays, Terence Tao candidly recalls the adventures and misadventures of growing up to become a leading mathematician; Natalie Wolchover shows how old math gives new clues about whether time really flows; and David Hand discusses the problem of gdark datah?information that is missing or ignored. And there is much, much more.
A definitive proof of global nonlinear stability of Minkowski space-time as a solution of the Einstein-Klein-Gordon equations
Series:
Annals of Mathematics Studies (AMS-213)
This book provides a definitive proof of global nonlinear stability of Minkowski space-time as a solution of the Einstein-Klein-Gordon equations of general relativity. Along the way, a novel robust analytical framework is developed, which extends to more general matter models. Alexandru Ionescu and Benoit Pausader prove global regularity at an appropriate level of generality of the initial data, and then prove several important asymptotic properties of the resulting space-time, such as future geodesic completeness, peeling estimates of the Riemann curvature tensor, conservation laws for the ADM tensor, and Bondi energy identities and inequalities.
The book is self-contained, providing complete proofs and precise statements, which develop a refined theory for solutions of quasilinear Klein-Gordon and wave equations, including novel linear and bilinear estimates. Only mild decay assumptions are made on the scalar field and the initial metric is allowed to have nonisotropic decay consistent with the positive mass theorem. The framework incorporates analysis both in physical and Fourier space, and is compatible with previous results on other physical models such as water waves and plasma physics.