Real Analysis Through Modern Infinitesimals
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Real Analysis Through Modern Infinitesimals
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Author : Nader Vakil
language : en
Publisher: Cambridge University Press
Release Date : 2011-02-17
Real Analysis Through Modern Infinitesimals written by Nader Vakil and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-02-17 with Mathematics categories.
A coherent, self-contained treatment of the central topics of real analysis employing modern infinitesimals.
Lectures On The Hyperreals
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Author : Robert Goldblatt
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Lectures On The Hyperreals written by Robert Goldblatt and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
An introduction to nonstandard analysis based on a course given by the author. It is suitable for beginning graduates or upper undergraduates, or for self-study by anyone familiar with elementary real analysis. It presents nonstandard analysis not just as a theory about infinitely small and large numbers, but as a radically different way of viewing many standard mathematical concepts and constructions. It is a source of new ideas, objects and proofs, and a wealth of powerful new principles of reasoning. The book begins with the ultrapower construction of hyperreal number systems, and proceeds to develop one-variable calculus, analysis and topology from the nonstandard perspective. It then sets out the theory of enlargements of fragments of the mathematical universe, providing a foundation for the full-scale development of the nonstandard methodology. The final chapters apply this to a number of topics, including Loeb measure theory and its relation to Lebesgue measure on the real line. Highlights include an early introduction of the ideas of internal, external and hyperfinite sets, and a more axiomatic set-theoretic approach to enlargements than is usual.
Infinitesimal
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Author : Amir Alexander
language : en
Publisher: Simon and Schuster
Release Date : 2014-07-03
Infinitesimal written by Amir Alexander and has been published by Simon and Schuster this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-07-03 with Science categories.
On August 10, 1632, five leading Jesuits convened in a sombre Roman palazzo to pass judgment on a simple idea: that a continuous line is composed of distinct and limitlessly tiny parts. The doctrine would become the foundation of calculus, but on that fateful day the judges ruled that it was forbidden. With the stroke of a pen they set off a war for the soul of the modern world. Amir Alexander takes us from the bloody religious strife of the sixteenth century to the battlefields of the English civil war and the fierce confrontations between leading thinkers like Galileo and Hobbes. The legitimacy of popes and kings, as well as our modern beliefs in human liberty and progressive science, hung in the balance; the answer hinged on the infinitesimal. Pulsing with drama and excitement, Infinitesimal will forever change the way you look at a simple line.
Stochastic Calculus With Infinitesimals
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Author : Frederik S. Herzberg
language : en
Publisher: Springer
Release Date : 2012-11-06
Stochastic Calculus With Infinitesimals written by Frederik S. Herzberg and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-11-06 with Mathematics categories.
Stochastic analysis is not only a thriving area of pure mathematics with intriguing connections to partial differential equations and differential geometry. It also has numerous applications in the natural and social sciences (for instance in financial mathematics or theoretical quantum mechanics) and therefore appears in physics and economics curricula as well. However, existing approaches to stochastic analysis either presuppose various concepts from measure theory and functional analysis or lack full mathematical rigour. This short book proposes to solve the dilemma: By adopting E. Nelson's "radically elementary" theory of continuous-time stochastic processes, it is based on a demonstrably consistent use of infinitesimals and thus permits a radically simplified, yet perfectly rigorous approach to stochastic calculus and its fascinating applications, some of which (notably the Black-Scholes theory of option pricing and the Feynman path integral) are also discussed in the book.
Infinitesimal Analysis
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Author : E.I. Gordon
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-14
Infinitesimal Analysis written by E.I. Gordon and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-14 with Mathematics categories.
Infinitesimal analysis, once a synonym for calculus, is now viewed as a technique for studying the properties of an arbitrary mathematical object by discriminating between its standard and nonstandard constituents. Resurrected by A. Robinson in the early 1960's with the epithet 'nonstandard', infinitesimal analysis not only has revived the methods of infinitely small and infinitely large quantities, which go back to the very beginning of calculus, but also has suggested many powerful tools for research in every branch of modern mathematics. The book sets forth the basics of the theory, as well as the most recent applications in, for example, functional analysis, optimization, and harmonic analysis. The concentric style of exposition enables this work to serve as an elementary introduction to one of the most promising mathematical technologies, while revealing up-to-date methods of monadology and hyperapproximation. This is a companion volume to the earlier works on nonstandard methods of analysis by A.G. Kusraev and S.S. Kutateladze (1999), ISBN 0-7923-5921-6 and Nonstandard Analysis and Vector Lattices edited by S.S. Kutateladze (2000), ISBN 0-7923-6619-0
Real And Abstract Analysis
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Author : E. Hewitt
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Real And Abstract Analysis written by E. Hewitt and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
This book is first of all designed as a text for the course usually called "theory of functions of a real variable". This course is at present cus tomarily offered as a first or second year graduate course in United States universities, although there are signs that this sort of analysis will soon penetrate upper division undergraduate curricula. We have included every topic that we think essential for the training of analysts, and we have also gone down a number of interesting bypaths. We hope too that the book will be useful as a reference for mature mathematicians and other scientific workers. Hence we have presented very general and complete versions of a number of important theorems and constructions. Since these sophisticated versions may be difficult for the beginner, we have given elementary avatars of all important theorems, with appro priate suggestions for skipping. We have given complete definitions, ex planations, and proofs throughout, so that the book should be usable for individual study as well as for a course text. Prerequisites for reading the book are the following. The reader is assumed to know elementary analysis as the subject is set forth, for example, in TOM M. ApOSTOL'S Mathematical Analysis [Addison-Wesley Publ. Co., Reading, Mass., 1957], or WALTER RUDIN'S Principles of M athe nd matical Analysis [2 Ed., McGraw-Hill Book Co., New York, 1964].
Introduction To Infinitesimal Analysis
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Author : Oswald Veblen
language : en
Publisher: Trieste Publishing
Release Date : 2017-09-17
Introduction To Infinitesimal Analysis written by Oswald Veblen and has been published by Trieste Publishing this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-09-17 with History categories.
Trieste Publishing has a massive catalogue of classic book titles. Our aim is to provide readers with the highest quality reproductions of fiction and non-fiction literature that has stood the test of time. The many thousands of books in our collection have been sourced from libraries and private collections around the world.The titles that Trieste Publishing has chosen to be part of the collection have been scanned to simulate the original. Our readers see the books the same way that their first readers did decades or a hundred or more years ago. Books from that period are often spoiled by imperfections that did not exist in the original. Imperfections could be in the form of blurred text, photographs, or missing pages. It is highly unlikely that this would occur with one of our books. Our extensive quality control ensures that the readers of Trieste Publishing's books will be delighted with their purchase. Our staff has thoroughly reviewed every page of all the books in the collection, repairing, or if necessary, rejecting titles that are not of the highest quality. This process ensures that the reader of one of Trieste Publishing's titles receives a volume that faithfully reproduces the original, and to the maximum degree possible, gives them the experience of owning the original work.We pride ourselves on not only creating a pathway to an extensive reservoir of books of the finest quality, but also providing value to every one of our readers. Generally, Trieste books are purchased singly - on demand, however they may also be purchased in bulk. Readers interested in bulk purchases are invited to contact us directly to enquire about our tailored bulk rates.
3000 Years Of Analysis
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Author : Thomas Sonar
language : en
Publisher: Springer Nature
Release Date : 2020-12-27
3000 Years Of Analysis written by Thomas Sonar and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-12-27 with Mathematics categories.
What exactly is analysis? What are infinitely small or infinitely large quantities? What are indivisibles and infinitesimals? What are real numbers, continuity, the continuum, differentials, and integrals? You’ll find the answers to these and other questions in this unique book! It explains in detail the origins and evolution of this important branch of mathematics, which Euler dubbed the “analysis of the infinite.” A wealth of diagrams, tables, color images and figures serve to illustrate the fascinating history of analysis from Antiquity to the present. Further, the content is presented in connection with the historical and cultural events of the respective epochs, the lives of the scholars seeking knowledge, and insights into the subfields of analysis they created and shaped, as well as the applications in virtually every aspect of modern life that were made possible by analysis.
Nonstandard Analysis For The Working Mathematician
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Author : Peter A. Loeb
language : en
Publisher: Springer
Release Date : 2015-08-26
Nonstandard Analysis For The Working Mathematician written by Peter A. Loeb and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-08-26 with Mathematics categories.
Starting with a simple formulation accessible to all mathematicians, this second edition is designed to provide a thorough introduction to nonstandard analysis. Nonstandard analysis is now a well-developed, powerful instrument for solving open problems in almost all disciplines of mathematics; it is often used as a ‘secret weapon’ by those who know the technique. This book illuminates the subject with some of the most striking applications in analysis, topology, functional analysis, probability and stochastic analysis, as well as applications in economics and combinatorial number theory. The first chapter is designed to facilitate the beginner in learning this technique by starting with calculus and basic real analysis. The second chapter provides the reader with the most important tools of nonstandard analysis: the transfer principle, Keisler’s internal definition principle, the spill-over principle, and saturation. The remaining chapters of the book study different fields for applications; each begins with a gentle introduction before then exploring solutions to open problems. All chapters within this second edition have been reworked and updated, with several completely new chapters on compactifications and number theory. Nonstandard Analysis for the Working Mathematician will be accessible to both experts and non-experts, and will ultimately provide many new and helpful insights into the enterprise of mathematics.
Introduction To Infinitesimal Analysis
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Author : Oswald Veblen
language : en
Publisher: Legare Street Press
Release Date : 2022-10-27
Introduction To Infinitesimal Analysis written by Oswald Veblen and has been published by Legare Street Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-10-27 with categories.
This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work is in the "public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.