**Author**: Imme van den Berg

**Publisher:**Springer

**ISBN:**3540478108

**Category:**Mathematics

**Page:**192

**View:**6032

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# Search Results for: nonstandard-asymptotic-analysis-lecture-notes-in-mathematics

**Author**: Imme van den Berg

**Publisher:** Springer

**ISBN:** 3540478108

**Category:** Mathematics

**Page:** 192

**View:** 6032

This research monograph considers the subject of asymptotics from a nonstandard view point. It is intended both for classical asymptoticists - they will discover a new approach to problems very familiar to them - and for nonstandard analysts but includes topics of general interest, like the remarkable behaviour of Taylor polynomials of elementary functions. Noting that within nonstandard analysis, "small", "large", and "domain of validity of asymptotic behaviour" have a precise meaning, a nonstandard alternative to classical asymptotics is developed. Special emphasis is given to applications in numerical approximation by convergent and divergent expansions: in the latter case a clear asymptotic answer is given to the problem of optimal approximation, which is valid for a large class of functions including many special functions. The author's approach is didactical. The book opens with a large introductory chapter which can be read without much knowledge of nonstandard analysis. Here the main features of the theory are presented via concrete examples, with many numerical and graphic illustrations. N

**Author**: Vladimir Kanovei,Michael Reeken

**Publisher:** Springer Science & Business Media

**ISBN:** 366208998X

**Category:** Mathematics

**Page:** 410

**View:** 5123

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.

**Author**: Nigel Cutland

**Publisher:** Cambridge University Press

**ISBN:** 052135109X

**Category:** Mathematics

**Page:** 346

**View:** 3188

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.

**Author**: Imme van den Berg,Vitor Neves

**Publisher:** Springer Science & Business Media

**ISBN:** 3211499059

**Category:** Mathematics

**Page:** 421

**View:** 4176

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.

**Author**: Taras Kudryk

**Publisher:** N.A

**ISBN:** 9789667148461

**Category:** Generalized spaces

**Page:** 255

**View:** 9065

**Author**: Philippe G. Ciarlet,Jacques-Louis Lions

**Publisher:** North-Holland is

**ISBN:** 9780444899286

**Category:** Computers

**Page:** 778

**View:** 2722

*Research memorandum*

**Author**: N.A

**Publisher:** N.A

**ISBN:** N.A

**Category:** Economic research

**Page:** N.A

**View:** 4990

**Author**: O. E. Barndorff-Nielsen,D. R. Cox

**Publisher:** Chapman and Hall/CRC

**ISBN:** N.A

**Category:** Mathematics

**Page:** 252

**View:** 5633

The use in statistical theory of approximate arguments based on such methods as local linearization (the delta method) and approxi mate normality has a long history. Such ideas play at least three roles. First they may give simple approximate answers to distributional problems where an exact solution is known in principle but difficult to implement. The second role is to yield higher-order expansions from which the accuracy of simple approximations may be assessed and where necessary improved. Thirdly the systematic development of a theoretical approach to statistical inference that will apply to quite general families of statistical models demands an asymptotic formulation, as far as possible one that will recover 'exact' results where these are available. The approximate arguments are developed by supposing that some defining quantity, often a sample size but more generally an amount of information, becomes large: it must be stressed that this is a technical device for generating approximations whose adequacy always needs assessing, rather than a 'physical' limiting notion. Of the three roles outlined above, the first two are quite close to the traditional roles of asymptotic expansions in applied mathematics and much ofthe very extensive literature on the asymptotic expansion of integrals and of the special functions of mathematical physics is quite directly relevant, although the recasting of these methods into a probability mould is quite often enlightening.
*Secțiunea Matematică*

**Author**: N.A

**Publisher:** N.A

**ISBN:** N.A

**Category:** Mathematics

**Page:** N.A

**View:** 1358

*Proceedings of a Conference held in Luminy, France, March 5-10, 1990*

**Author**: Eric Benoit

**Publisher:** Springer

**ISBN:** 3540464719

**Category:** Mathematics

**Page:** 222

**View:** 616

**Author**: Nigel Cutland,Mauro Di Nasso,David A. Ross

**Publisher:** A K Peters Ltd

**ISBN:** 9781568812915

**Category:** Mathematics

**Page:** 248

**View:** 5339

This book is a collection of peer-reviewed papers from a conference on Nonstandard Methods and Applications in Mathematics (NS2002) that was held in Pisa, Italy. The papers address nonstandard analysis, which is one of the great achievements of modern applied mathematical logic. They focus on its important philosophical achievement of providing a sound mathematical basis for using infinitesimals in analysis, and they show how this methodology is now well established as a tool for both research and teaching.
*a practical guide with applications*

**Author**: Robert Lutz,Michel Goze

**Publisher:** Springer

**ISBN:** N.A

**Category:** Mathematics

**Page:** 261

**View:** 3749

**Author**: American Mathematical Society

**Publisher:** N.A

**ISBN:** N.A

**Category:** Global analysis (Mathematics)

**Page:** 3920

**View:** 5288

*Asymptotic Solutions and Statistical Problems*

**Author**: J.M. Burgers

**Publisher:** Springer Science & Business Media

**ISBN:** 940101745X

**Category:** Mathematics

**Page:** 174

**View:** 4657

Since the 'Introduction' to the main text gives an account of the way in which the problems treated in the following pages originated, this 'Preface' may be limited to an acknowledgement of the support the work has received. It started during the pe riod when I was professor of aero- and hydrodynamics at the Technical University in Delft, Netherlands, and many discussions with colleagues ha ve in:fluenced its devel opment. Oftheir names I mention here only that ofH. A. Kramers. Papers No. 1-13 ofthe list given at the end ofthe text were written during that period. Severa! ofthese were attempts to explore ideas which later had to be abandoned, but gradually a line of thought emerged which promised more definite results. This line began to come to the foreground in pa per No. 3 (1939}, while a preliminary formulation ofthe results was given in paper No. 12 (1954}. At that time, however, there still was missing a practica! method for manipulating a certain distribution function of central interest. A six months stay at the Hydrodynamics Laboratories ofthe California Institute of Technology, Pasadena, California (1950-1951}, was supported by a Contract with the Department of the Air F orce, N o. AF 33(038}-17207. A course of lectures was given during this period, which were published in typescript under the title 'On Turbulent Fluid Motion', as Report No. E-34. 1, July 1951, of the Hydrodynamics Laboratory.

**Author**: Lawrence Sirovich

**Publisher:** Springer Science & Business Media

**ISBN:** 1461264022

**Category:** Mathematics

**Page:** 306

**View:** 5503

These notes originate from a one semester course which forms part of the "Math Methods" cycle at Brown. In the hope that these notes might prove useful for reference purposes several additional sections have been included and also a table of contents and index. Although asymptotic analysis is now enjoying a period of great vitality, these notes do not reflect a research oriented course. The course is aimed toward people in applied mathematics, physics, engineering, etc., who have a need for asymptotic analysis in their work. The choice of subjects has been largely dictated by the likelihood of application. Also abstraction and generality have not been pursued. Technique and computation are given equal prominence with theory. Both rigorous and formal theory is presented --very often in tandem. In practice, the means for a rigorous analysis are not always available. For this reason a goal has been the cultivation of mature formal reasoning. Therefore, during the course of lectures formal presentations gradually eclipse rigorous presentations. When this occurs, rigorous proofs are given as exercises or in the case of lengthy proofs, reference is made to the Reading List at the end.
*Theory, Applications, and Open Problems*

**Author**: Jiming Jiang

**Publisher:** CRC Press

**ISBN:** 1351645595

**Category:** Mathematics

**Page:** 252

**View:** 5756

Large sample techniques are fundamental to all fields of statistics. Mixed effects models, including linear mixed models, generalized linear mixed models, non-linear mixed effects models, and non-parametric mixed effects models are complex models, yet, these models are extensively used in practice. This monograph provides a comprehensive account of asymptotic analysis of mixed effects models. The monograph is suitable for researchers and graduate students who wish to learn about asymptotic tools and research problems in mixed effects models. It may also be used as a reference book for a graduate-level course on mixed effects models, or asymptotic analysis.

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