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Nonstandard Analysis and Generalized Functions  [PDF]
Robert A. Herrmann
Mathematics , 2004,
Abstract: This application of nonstandard analysis utilizes the notion of the highly-staturated enlargement. These nonstandard methods clarify many aspects of the theory of generalized functions (distributions).
Nonstandard Analysis in Topology  [PDF]
Sergio Salbany,Todor Todorov
Mathematics , 2011,
Abstract: We present Nonstandard Analysis by three axioms: the {\em Extension, Transfer and Saturation Principles} in the framework of the superstructure of a given infinite set. We also present several applications of this axiomatic approach to point-set topology. Some of the topological topics such as the Hewitt realcompactification and the nonstandard characterization of the sober spaces seem to be new in the literature on nonstandard analysis. Others have already close counterparts but they are presented here with essential simplifications.
Nonstandard Analysis of Graphs  [PDF]
F. Javier Thayer
Mathematics , 2001,
Abstract: This paper shows certain classes of metric spaces characterized by volume growth properties of balls can viewed as graphs with infinitesimal edges. Our approach is based on nonstandard analysis.
The unreasonable effectiveness of Nonstandard Analysis  [PDF]
Sam Sanders
Mathematics , 2015,
Abstract: The aim of this paper is to highlight a hitherto unknown computational aspect of Nonstandard Analysis. In particular, we provide an algorithm which takes as input the proof of a mathematical theorem from `pure' Nonstandard Analysis, i.e. formulated solely with the nonstandard definitions (of continuity, integration, differentiability, convergence, compactness, et cetera), and outputs a proof of the associated effective version of the theorem. Intuitively speaking, the effective version of a mathematical theorem is obtained by replacing all its existential quantifiers by functionals computing (in a specific technical sense) the objects claimed to exist. Our algorithm often produces theorems of Errett Bishop's Constructive Analysis. We shall work in Nelson's syntactic approach to Nonstandard Analysis, called internal set theory, and treat theorems up to the Stone-Weierstrass theorem. Notable results are that applying our algorithm to theorems involving nonstandard compactness, we rediscover, depending on the formulation of the latter, either totally boundedness, the preferred notion of compactness in constructive and computable analysis, or equivalences from the foundational program Reverse Mathematics. Finally, we establish that a theorem of Nonstandard Analysis has the same computational content as its `highly constructive' Herbrandisation. Thus, we establish an `algorithmic two-way street' between so-called hard and soft analysis, i.e.\ between the worlds of numerical and qualitative mathematics.
Multiplicity of nontrivial solutions for elliptic equations with nonsmooth potential and resonance at higher eigenvalues  [PDF]
Leszek Gasi'nski,Dumitru Motreanu,Nikolaos S Papageorgiou
Mathematics , 2006,
Abstract: We consider a semilinear elliptic equation with a nonsmooth, locally \hbox{Lipschitz} potential function (hemivariational inequality). Our hypotheses permit double resonance at infinity and at zero (double-double resonance situation). Our approach is based on the nonsmooth critical point theory for locally Lipschitz functionals and uses an abstract multiplicity result under local linking and an extension of the Castro--Lazer--Thews reduction method to a nonsmooth setting, which we develop here using tools from nonsmooth analysis.
Nonstandard Analysis: Its Creator and Place  [PDF]
S. S. Kutateladze
Mathematics , 2013,
Abstract: This is a biographical sketch and tribute to Abraham Robinson (1918-1974) on the 95th anniversary of his birth with a short discussion of the place of nonstandard analysis in the present-day mathematics.
Weak Subdifferential in Nonsmooth Analysis and Optimization  [PDF]
?ahlar F. Meherrem,Refet Polat
Journal of Applied Mathematics , 2011, DOI: 10.1155/2011/204613
Abstract: Some properties of the weak subdifferential are considered in this paper. By using the definition and properties of the weak subdifferential which are described in the papers (Azimov and Gasimov, 1999; Kasimbeyli and Mammadov, 2009; Kasimbeyli and Inceoglu, 2010), the author proves some theorems connecting weak subdifferential in nonsmooth and nonconvex analysis. It is also obtained necessary optimality condition by using the weak subdifferential in this paper. 1. Introduction Nonsmooth analysis had its origins in the early 1970s when control theorists and nonlinear programmers attempted to deal with necessary optimality conditions for problems with nonsmooth data or with nonsmooth functions (such as the pointwise maximum of several smooth functions) that arise even in many problems with smooth data, convex functions, and max-type functions. For this reason, it is necessary to extend the classical gradient for the smooth function to nonsmooth functions. The first such canonical generalized gradient was the generalized gradient introduced by Clarke in his work [1]. He applied this generalized gradient systematically to nonsmooth problems in a variety of problems. But the nonconvex basic or limiting normal cone to closed sets and the corresponding subdifferential of lower semicontinuous extended-real-valued functions satisfying these requirements were introduced by Mordukhovich at the beginning of 1975. The corresponding subdifferential is called Morduchovich subdifferential. The initial motivation came from the intention to derive necessary optimality conditions for optimal control problems with endpoint geometric constraints by passing to the limit from free endpoint control problems, which are much easier to handle. This was published in [2]. Let us remark also that Clarke’s normal cone is the closed convex closure of Mordukhovich normal cone [2]. Multifunctions (set-valued maps) naturally appear in various areas of nonlinear analysis, optimization, control theory, and mathematical economics. In Aubin and Frankowska's book [3] and in Mordukhovich's book is an excellent introduction to the theory of multifunctions. Coderivatives are convenient derivative-like objects for multifunctions and were introduced by Mordukhovich [2] motivated by applications to optimal control (see [4] for more discussions on the motivations and the relationship among coderivatives and other derivative-like objects for multifunctions). They are defined via “normal cones” to the graph of the multifunctions. Approximate and geometric subdifferentials are introduced by Ioffe in
A Note On Go Endgame And Nonstandard Analysis  [PDF]
An Huang
Mathematics , 2009,
Abstract: This paper has been withdrawn by the author, since the relation mentioned in the paper between nonstandard analysis and games is probably useless.
A Proof of the Ergodic Theorem using Nonstandard Analysis  [PDF]
Tristram de Piro
Mathematics , 2014,
Abstract: The following paper follows on from work by Kamae, and gives a rigorous proof of the Ergodic Theorem, using nonstandard analysis.
Nonstandard Analysis - A Simplified Approach  [PDF]
Robert A. Herrmann
Mathematics , 2003,
Abstract: In this monograph, nonstandard characteristics for many notions from real analysis are obtained and applied. However, only two simple types of atomic formula are used and almost all of the characteristics are shown to hold for a simple ultrapower styled structure generated by any free ultrafilter over the natural numbers.
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