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Neutrices And External Numbers

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Neutrices And External Numbers

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Author : Bruno Dinis
language : en
Publisher: CRC Press
Release Date : 2019-07-03

Neutrices And External Numbers written by Bruno Dinis and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-07-03 with Mathematics categories.

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

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Author : Bruno Dinis
language : en
Publisher: CRC Press
Release Date : 2019-07-03

The Strength Of Nonstandard Analysis

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Author : Imme van den Berg
language : en
Publisher: Springer Science & Business Media
Release Date : 2007-12-03

The Strength Of Nonstandard Analysis written by Imme van den Berg 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 2007-12-03 with Mathematics categories.

Nonstandard Analysis enhances mathematical reasoning by introducing new ways of expression and deduction. Distinguishing between standard and nonstandard mathematical objects, its inventor, the eminent mathematician Abraham Robinson, settled in 1961 the centuries-old problem of how to use infinitesimals correctly in analysis. Having also worked as an engineer, he saw not only that his method greatly simplified mathematically proving and teaching, but also served as a powerful tool in modelling, analyzing and solving problems in the applied sciences, among others by effective rescaling and by infinitesimal discretizations. This book reflects the progress made in the forty years since the appearance of 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

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Author : Francine Diener
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Nonstandard Analysis In Practice written by Francine Diener 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 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

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Author : Tatiana V. Tchemisova
language : en
Publisher: Springer Nature
Release Date : 2022-12-29

Dynamic Control And Optimization written by Tatiana V. Tchemisova and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-12-29 with Mathematics categories.

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

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Author : Karel Hrbacek
language : en
Publisher: CRC Press
Release Date : 2014-12-01

Analysis With Ultrasmall Numbers written by Karel Hrbacek and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-12-01 with Mathematics categories.

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

Nonstandard Analysis

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Author : Karl Kuhlemann
language : en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date : 2024-12-16

Nonstandard Analysis written by Karl Kuhlemann and has been published by Walter de Gruyter GmbH & Co KG this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-12-16 with Mathematics categories.

Currently, nonstandard analysis is barely considered in university teaching. The author argues that nonstandard analysis is valuable not only for teaching, but also for understanding standard analysis and mathematics itself. An axiomatic approach wich pays attention to different language levels (for example, in the distinction between sums of ones and the natural numbers of the theory) leads naturally to a nonstandard theory. For motivation historical ideas of Leibniz can be taken up. The book contains an elaborated concept that follows this approach and is suitable, for example, as a basis for a lecture-supplementary course. The monograph part presents all major approaches to nonstandard analysis and discusses logical, model-theoretic, and set-theoretic investigations to reveal possible mathematical reasons that may lead to reservations about nonstandard analysis. Also various foundational positions as well as ontological, epistemological, and application-related issues are addressed. It turns out that the one-sided preference for standard analysis is justified neither from a didactic, mathematical nor philosophical point of view. Thus, the book is especially valuable for students and instructors of analysis who are also interested in the foundations of their subject.

Summable Spaces And Their Duals Matrix Transformations And Geometric Properties

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Author : Feyzi Başar
language : en
Publisher: CRC Press
Release Date : 2020-02-04

Summable Spaces And Their Duals Matrix Transformations And Geometric Properties written by Feyzi Başar and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-02-04 with Mathematics categories.

The aim of Summable Spaces and Their Duals, Matrix Transformations and Geometric Properties is to discuss primarily about different kinds of summable spaces, compute their duals and then characterize several matrix classes transforming one summable space into other. The book also discusses several geometric properties of summable spaces, as well as dealing with the construction of summable spaces using Orlicz functions, and explores several structural properties of such spaces. Each chapter contains a conclusion section highlighting the importance of results, and points the reader in the direction of possible new ideas for further study. Features Suitable for graduate schools, graduate students, researchers and faculty, and could be used as a key text for special Analysis seminars Investigates different types of summable spaces and computes their duals Characterizes several matrix classes transforming one summable space into other Discusses several geometric properties of summable spaces Examines several possible generalizations of Orlicz sequence spaces

Glider Representations

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Author : Frederik Caenepeel
language : en
Publisher: CRC Press
Release Date : 2019-11-05

Glider Representations written by Frederik Caenepeel and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-11-05 with Mathematics categories.

Glider Representations offer several applications across different fields within Mathematics, thereby motivating the introduction of this new glider theory and opening numerous doors for future research, particularly with respect to more complex filtration chains. Features • Introduces new concepts in the Theory of Rings and Modules • Suitable for researchers and graduate students working in this area, and as supplementary reading for courses in Group Theory, Ring Theory, Lie Algebras and Sheaf Theory • The first book to explicitly outline this new approach to gliders and fragments and associated concepts

Lattice Point Identities And Shannon Type Sampling

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Author : Willi Freeden
language : en
Publisher: CRC Press
Release Date : 2019-10-28

Lattice Point Identities And Shannon Type Sampling written by Willi Freeden and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-10-28 with Technology & Engineering categories.

Lattice Point Identities and Shannon-Type Sampling demonstrates that significant roots of many recent facets of Shannon's sampling theorem for multivariate signals rest on basic number-theoretic results. This book leads the reader through a research excursion, beginning from the Gaussian circle problem of the early nineteenth century, via the classical Hardy-Landau lattice point identity and the Hardy conjecture of the first half of the twentieth century, and the Shannon sampling theorem (its variants, generalizations and the fascinating stories about the cardinal series) of the second half of the twentieth century. The authors demonstrate how all these facets have resulted in new multivariate extensions of lattice point identities and Shannon-type sampling procedures of high practical applicability, thereby also providing a general reproducing kernel Hilbert space structure of an associated Paley-Wiener theory over (potato-like) bounded regions (cf. the cover illustration of the geoid), as well as the whole Euclidean space. All in all, the context of this book represents the fruits of cross-fertilization of various subjects, namely elliptic partial differential equations, Fourier inversion theory, constructive approximation involving Euler and Poisson summation formulas, inverse problems reflecting the multivariate antenna problem, and aspects of analytic and geometric number theory. Features: New convergence criteria for alternating series in multi-dimensional analysis Self-contained development of lattice point identities of analytic number theory Innovative lattice point approach to Shannon sampling theory Useful for students of multivariate constructive approximation, and indeed anyone interested in the applicability of signal processing to inverse problems.

Neutrices And External Numbers

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