Neutrices and External Numbers

Neutrices and External Numbers
Author: Bruno Dinis
Publisher: CRC Press
Total Pages: 306
Release: 2019-07-03
Genre: Mathematics
ISBN: 1000012204
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Neutrices and External Numbers: A Flexible Number System introduces a new model of orders of magnitude and of error analysis, with particular emphasis on behaviour under algebraic operations. The model is formulated in terms of scalar neutrices and external numbers, in the form of an extension of the nonstandard set of real numbers. Many illustrative examples are given. The book starts with detailed presentation of the algebraic structure of external numbers, then deals with the generalized Dedekind completeness property, applications in analysis, domains of validity of approximations of solutions of differential equations, particularly singular perturbations. Finally, it describes the family of algebraic laws characterizing the practice of calculations with external numbers. Features Presents scalar neutrices and external numbers, a mathematical model of order of magnitude within the real number system. Outlines complete algebraic rules for the neutrices and external numbers Conducts operational analysis of convergence and integration of functions known up to orders of magnitude Formalises a calculus of error propagation, covariant with algebraic operations Presents mathematical models of phenomena incorporating their necessary imprecisions, in particular related to the Sorites paradox

Neutrices and External Numbers

Neutrices and External Numbers
Author: Bruno Dinis
Publisher: CRC Press
Total Pages: 342
Release: 2019-07-03
Genre: Mathematics
ISBN: 1000005380
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Neutrices and External Numbers: A Flexible Number System introduces a new model of orders of magnitude and of error analysis, with particular emphasis on behaviour under algebraic operations. The model is formulated in terms of scalar neutrices and external numbers, in the form of an extension of the nonstandard set of real numbers. Many illustrative examples are given. The book starts with detailed presentation of the algebraic structure of external numbers, then deals with the generalized Dedekind completeness property, applications in analysis, domains of validity of approximations of solutions of differential equations, particularly singular perturbations. Finally, it describes the family of algebraic laws characterizing the practice of calculations with external numbers. Features Presents scalar neutrices and external numbers, a mathematical model of order of magnitude within the real number system. Outlines complete algebraic rules for the neutrices and external numbers Conducts operational analysis of convergence and integration of functions known up to orders of magnitude Formalises a calculus of error propagation, covariant with algebraic operations Presents mathematical models of phenomena incorporating their necessary imprecisions, in particular related to the Sorites paradox

The Strength of Nonstandard Analysis

The Strength of Nonstandard Analysis
Author: Imme van den Berg
Publisher: Springer Science & Business Media
Total Pages: 415
Release: 2007-12-03
Genre: Mathematics
ISBN: 3211499059
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This book reflects the progress made in the forty years since the appearance of Abraham Robinson’s revolutionary book Nonstandard Analysis in the foundations of mathematics and logic, number theory, statistics and probability, in ordinary, partial and stochastic differential equations and in education. The contributions are clear and essentially self-contained.

Nonstandard Analysis in Practice

Nonstandard Analysis in Practice
Author: Francine Diener
Publisher: Springer Science & Business Media
Total Pages: 262
Release: 2012-12-06
Genre: Mathematics
ISBN: 364257758X
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This book introduces the graduate mathematician and researcher to the effective use of nonstandard analysis (NSA). It provides a tutorial introduction to this modern theory of infinitesimals, followed by nine examples of applications, including complex analysis, stochastic differential equations, differential geometry, topology, probability, integration, and asymptotics. It ends with remarks on teaching with infinitesimals.

Dynamic Control and Optimization

Dynamic Control and Optimization
Author: Tatiana V. Tchemisova
Publisher: Springer Nature
Total Pages: 244
Release: 2022-12-29
Genre: Mathematics
ISBN: 3031175581
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This book contains the revised selected papers of the International Conference on Dynamic Monitoring and Optimization, DCO 2021, held in Aveiro, Portugal, February 3-5, 2021. The papers present achievements in the most challenging areas of dynamic control, optimization and related topics, including recent results in nonlinear dynamic control systems, calculus of variations, sub-Riemannian geometry, conventional differential equations, control of PDE evolution, stochastic differential equations, the spread of acoustic waves in elastic media, dynamics in space-time, Nondegenerate abnormality, controllability, and the infimum gap phenomena in optimization and optimal control with state constraints.

Analysis with Ultrasmall Numbers

Analysis with Ultrasmall Numbers
Author: Karel Hrbacek
Publisher: CRC Press
Total Pages: 320
Release: 2014-12-01
Genre: Mathematics
ISBN: 149870266X
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Analysis with Ultrasmall Numbers presents an intuitive treatment of mathematics using ultrasmall numbers. With this modern approach to infinitesimals, proofs become simpler and more focused on the combinatorial heart of arguments, unlike traditional treatments that use epsilon-delta methods. Students can fully prove fundamental results, such as the

Singular Perturbations and Hysteresis

Singular Perturbations and Hysteresis
Author: Michael P. Mortell
Publisher: SIAM
Total Pages: 357
Release: 2005-01-01
Genre: Mathematics
ISBN: 9780898717860
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This book brings together many important recent developments in the analysis of singular perturbation and hysteresis phenomena in an accessible and reasonably comprehensive fashion. To bridge a gap between practitioners of these phenomena, the editors conducted a workshop in April 2002 at University College Cork to provide a forum for experts in both fields to share their interests and knowledge. For this book, the editors have compiled research from those practitioners in areas such as reacting systems, semiconductor lasers, shock phenomena in economic modeling, and fluid mechanics, all with an emphasis on hysteresis and singular perturbations.

Nonstandard Analysis, Axiomatically

Nonstandard Analysis, Axiomatically
Author: Vladimir Kanovei
Publisher: Springer Science & Business Media
Total Pages: 421
Release: 2013-03-09
Genre: Mathematics
ISBN: 366208998X
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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.

Mathematical Reviews

Mathematical Reviews
Author:
Publisher:
Total Pages: 984
Release: 2008
Genre: Mathematics
ISBN:
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