Download or read book The Theory of Finitely Additive Probability Using Nonstandard Analysis written by Thomas Quinton Sibley and published by . This book was released on 1980 with total page 314 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Nonstandard Analysis written by Leif O. Arkeryd and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 374 pages. Available in PDF, EPUB and Kindle. Book excerpt: 1 More than thirty years after its discovery by Abraham Robinson , the ideas and techniques of Nonstandard Analysis (NSA) are being applied across the whole mathematical spectrum,as well as constituting an im portant field of research in their own right. The current methods of NSA now greatly extend Robinson's original work with infinitesimals. However, while the range of applications is broad, certain fundamental themes re cur. The nonstandard framework allows many informal ideas (that could loosely be described as idealisation) to be made precise and tractable. For example, the real line can (in this framework) be treated simultaneously as both a continuum and a discrete set of points; and a similar dual ap proach can be used to link the notions infinite and finite, rough and smooth. This has provided some powerful tools for the research mathematician - for example Loeb measure spaces in stochastic analysis and its applications, and nonstandard hulls in Banach spaces. The achievements of NSA can be summarised under the headings (i) explanation - giving fresh insight or new approaches to established theories; (ii) discovery - leading to new results in many fields; (iii) invention - providing new, rich structures that are useful in modelling and representation, as well as being of interest in their own right. The aim of the present volume is to make the power and range of appli cability of NSA more widely known and available to research mathemati cians.
Download or read book Nonstandard Methods in Functional Analysis written by Siu-Ah Ng and published by World Scientific. This book was released on 2010 with total page 339 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the early 1960s, by using techniques from the model theory of first-order logic, Robinson gave a rigorous formulation and extension of Leibniz'' infinitesimal calculus. Since then, the methodology has found applications in a wide spectrum of areas in mathematics, with particular success in the probability theory and functional analysis. In the latter, fruitful results were produced with Luxemburg''s invention of the nonstandard hull construction. However, there is still no publication of a coherent and self-contained treatment of functional analysis using methods from nonstandard analysis. This publication aims to fill this gap.
Download or read book An Introduction to Measure Theory written by Terence Tao and published by American Mathematical Soc.. This book was released on 2021-09-03 with total page 206 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a graduate text introducing the fundamentals of measure theory and integration theory, which is the foundation of modern real analysis. The text focuses first on the concrete setting of Lebesgue measure and the Lebesgue integral (which in turn is motivated by the more classical concepts of Jordan measure and the Riemann integral), before moving on to abstract measure and integration theory, including the standard convergence theorems, Fubini's theorem, and the Carathéodory extension theorem. Classical differentiation theorems, such as the Lebesgue and Rademacher differentiation theorems, are also covered, as are connections with probability theory. The material is intended to cover a quarter or semester's worth of material for a first graduate course in real analysis. There is an emphasis in the text on tying together the abstract and the concrete sides of the subject, using the latter to illustrate and motivate the former. The central role of key principles (such as Littlewood's three principles) as providing guiding intuition to the subject is also emphasized. There are a large number of exercises throughout that develop key aspects of the theory, and are thus an integral component of the text. As a supplementary section, a discussion of general problem-solving strategies in analysis is also given. The last three sections discuss optional topics related to the main matter of the book.
Download or read book Scientific and Technical Aerospace Reports written by and published by . This book was released on 1985 with total page 1076 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Nonstandard Analysis for the Working Mathematician written by Peter A. Loeb and published by Springer. This book was released on 2015-08-26 with total page 485 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Nonstandard Analysis Axiomatically written by Vladimir Kanovei and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 421 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the aftermath of the discoveries in foundations of mathematiC's there was surprisingly little effect on mathematics as a whole. If one looks at stan dard textbooks in different mathematical disciplines, especially those closer to what is referred to as applied mathematics, there is little trace of those developments outside of mathematical logic and model theory. But it seems fair to say that there is a widespread conviction that the principles embodied in the Zermelo - Fraenkel theory with Choice (ZFC) are a correct description of the set theoretic underpinnings of mathematics. In most textbooks of the kind referred to above, there is, of course, no discussion of these matters, and set theory is assumed informally, although more advanced principles like Choice or sometimes Replacement are often mentioned explicitly. This implicitly fixes a point of view of the mathemat ical universe which is at odds with the results in foundations. For example most mathematicians still take it for granted that the real number system is uniquely determined up to isomorphism, which is a correct point of view as long as one does not accept to look at "unnatural" interpretations of the membership relation.
Download or read book Victoria Symposium on Nonstandard Analysis written by A. Hurd and published by Springer. This book was released on 2006-11-15 with total page 361 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Probabilistic Analysis and Related Topics written by A. T. Bharucha-Reid and published by Elsevier. This book was released on 2014-05-10 with total page 220 pages. Available in PDF, EPUB and Kindle. Book excerpt: Probabilistic Analysis and Related Topics, Volume 2 focuses on the integrability, continuity, and differentiability of random functions, as well as functional analysis, measure theory, operator theory, and numerical analysis. The selection first offers information on the optimal control of stochastic systems and Gleason measures. Discussions focus on convergence of Gleason measures, random Gleason measures, orthogonally scattered Gleason measures, existence of optimal controls without feedback, random necessary conditions, and Gleason measures in tensor products. The text then elaborates on an introduction to nonstandard analysis and hyperfinite probability theory, including applications to stochastic processes, conversion from nonstandard to standard measure spaces, and an introduction to nonstandard analysis. The text examines stochastic matrices, ergodic Markov chains, and measures on semigroups, as well as limit theorems for convolution products of probability measures on completely simple semigroups; ergodicity of Markov chains and probability measures on semigroups; and limits of convolutions in groups and semigroups. The selection is a dependable source of data for mathematicians and researchers interested in the general theory of random functions.
Download or read book Nonstandard Analysis and Vector Lattices written by Semën Samsonovich Kutateladze and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nonstandard methods of analysis consist generally in comparative study of two interpretations of a mathematical claim or construction given as a formal symbolic expression by means of two different set-theoretic models: one, a "standard" model and the other, a "nonstandard" model. The second half of the twentieth century is a period of significant progress in these methods and their rapid development in a few directions. The first of the latter appears often under the name coined by its inventor, A. Robinson. This memorable but slightly presumptuous and defiant term, non standard analysis, often swaps places with the term Robinsonian or classical non standard analysis. The characteristic feature of Robinsonian analysis is a frequent usage of many controversial concepts appealing to the actual infinitely small and infinitely large quantities that have resided happily in natural sciences from ancient times but were strictly forbidden in modern mathematics for many decades. The present-day achievements revive the forgotten term infinitesimal analysis which reminds us expressively of the heroic bygones of Calculus. Infinitesimal analysis expands rapidly, bringing about radical reconsideration of the general conceptual system of mathematics. The principal reasons for this progress are twofold. Firstly, infinitesimal analysis provides us with a novel under standing for the method of indivisibles rooted deeply in the mathematical classics.
Download or read book Ergodicity of Markov Processes via Nonstandard Analysis written by Haosui Duanmu and published by American Mathematical Society. This book was released on 2021-12-09 with total page 114 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.
Download or read book Transactions of the Seventh Prague Conference on Information Theory Statistical Decision Functions Random Processes and of the 1974 European Meeting of Statisticians written by J. Kozesnik and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 577 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Prague Conferences on Information Theory, Statistical Decision Functions, and Random Processes have been organized every three years since 1956. During the eighteen years of their existence the Prague Conferences developed from a platform for presenting results obtained by a small group of researchers into a probabilistic congress, this being documented by the increasing number of participants as well as of presented papers. The importance of the Seventh Prague Conference has been emphasized by the fact that this Conference was held jointly with the eighth European Meeting of Statisticians. This joint meeting was held from August 18 to 23, 1974 at the Technical University of Prague. The Conference was organized by the Institute of Information Theory and Automation of the Czechoslovak Academy of Sciences and was sponsored by the Czechoslovak Academy of Sciences, by the Committee for the European Region of the Institute of Mathematical Statistics, and by the International As sociation for Statistics in Physical Sciences. More than 300 specialists from 25 countries participated in the Conference. In 57 sessions 164 papers (including 17 invited papers) were read, 128 of which are published in the present two volumes of the Transactions of the Conference. Volume A includes papers related mainly to probability theory and stochastic processes, whereas the papers of Volume B concern mainly statistics and information theory.
Download or read book The Place of Probability in Science written by Ellery Eells and published by Springer Science & Business Media. This book was released on 2010-06-08 with total page 396 pages. Available in PDF, EPUB and Kindle. Book excerpt: Science aims at the discovery of general principles of special kinds that are applicable for the explanation and prediction of the phenomena of the world in the form of theories and laws. When the phenomena themselves happen to be general, the principlesinvolved assume the form of theories; and when they are p- ticular, they assume the form of general laws. Theories themselves are sets of laws and de nitions that apply to a common domain, which makes laws indispensable to science. Understanding science thus depends upon understanding the nature of theories and laws, the logical structure of explanations and predictions based upon them, and the principles of inference and decision that apply to theories and laws. Laws and theories can differ in their form as well as in their content. The laws of quantum mechanics are indeterministic (or probabilistic), for example, while those of classical mechanics are deterministic (or universal) instead. The history of science re ects an increasing role for probabilities as properties of the world but also as measures of evidential support and as degrees of subjective belief. Our purpose is to clarify and illuminate the place of probability in science.
Download or read book Probability Theory and Applications written by and published by Walter de Gruyter GmbH & Co KG. This book was released on 2020-05-18 with total page 820 pages. Available in PDF, EPUB and Kindle. Book excerpt: No detailed description available for "Probability Theory and Applications".
Download or read book Nonstandard Analysis and Its Applications written by Nigel Cutland and published by Cambridge University Press. This book was released on 1988-09-30 with total page 365 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook is an introduction to non-standard analysis and to its many applications. Non standard analysis (NSA) is a subject of great research interest both in its own right and as a tool for answering questions in subjects such as functional analysis, probability, mathematical physics and topology. The book arises from a conference held in July 1986 at the University of Hull which was designed to provide both an introduction to the subject through introductory lectures, and surveys of the state of research. The first part of the book is devoted to the introductory lectures and the second part consists of presentations of applications of NSA to dynamical systems, topology, automata and orderings on words, the non- linear Boltzmann equation and integration on non-standard hulls of vector lattices. One of the book's attractions is that a standard notation is used throughout so the underlying theory is easily applied in a number of different settings. Consequently this book will be ideal for graduate students and research mathematicians coming to the subject for the first time and it will provide an attractive and stimulating account of the subject.
Download or read book Mathematical Reviews written by and published by . This book was released on 2004 with total page 796 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Infinity Causation and Paradox written by Alexander R. Pruss and published by Oxford University Press. This book was released on 2018-07-26 with total page 248 pages. Available in PDF, EPUB and Kindle. Book excerpt: Infinity is paradoxical in many ways. Some paradoxes involve deterministic supertasks, such as Thomson's Lamp, where a switch is toggled an infinite number of times over a finite period of time, or the Grim Reaper, where it seems that infinitely many reapers can produce a result without doing anything. Others involve infinite lotteries. If you get two tickets from an infinite fair lottery where tickets are numbered from 1, no matter what number you saw on the first ticket, it is almost certain that the other ticket has a bigger number on it. And others center on paradoxical results in decision theory, such as the surprising observation that if you perform a sequence of fair coin flips that goes infinitely far back into the past but only finitely into the future, you can leverage information about past coin flips to predict future ones with only finitely many mistakes. Alexander R. Pruss examines this seemingly large family of paradoxes in Infinity, Causation and Paradox. He establishes that these paradoxes and numerous others all have a common structure: their most natural embodiment involves an infinite number of items causally impinging on a single output. These paradoxes, he argues, can all be resolved by embracing 'causal finitism', the view that it is impossible for a single output to have an infinite causal history. Throughout the book, Pruss exposits such paradoxes, defends causal finitism at length, and considers connections with the philosophy of physics (where causal finitism favors but does not require discretist theories of space and time) and the philosophy of religion (with a cosmological argument for a first cause).
