Related papers: Non-standard Nonstandard Analysis and the computat…
Kohlenbach's proof mining program deals with the extraction of effective information from typically ineffective proofs. Proof mining has its roots in Kreisel's pioneering work on the so-called unwinding of proofs. The proof mining of…
The aim of this paper is to highlight a hitherto unknown computational aspect of Nonstandard Analysis pertaining to Reverse Mathematics (RM). In particular, we shall establish RM-equivalences between theorems from Nonstandard Analysis in a…
In this paper, we highlight a new computational aspect of Nonstandard Analysis relating to higher-order computability theory. In particular, we prove that the Gandy-Hyland functional equals a primitive recursive functional involving…
Over extended systems of finite type arithmetic, we utilize a formal representation of the outer measure to define a translation which allows for the systematic formalization of probabilistic statements. As a main result, this translation…
The Loeb measure is one of the cornerstones of Nonstandard Analysis. The traditional development of the Loeb measure makes use of saturation and external sets. Inspired by [13], we give meaning to special cases of the Loeb measure in the…
This is a Research and Instructional Development Project from the U. S. Naval Academy. In this monograph, the basic methods of nonstandard analysis for n-dimensional Euclidean spaces are presented. Specific rules are deveoped and these…
We introduce constructive and classical systems for nonstandard arithmetic and show how variants of the functional interpretations due to Goedel and Shoenfield can be used to rewrite proofs performed in these systems into standard ones.…
This commentary considers non-standard analysis and a recently introduced computational methodology based on the notion of \G1 (this symbol is called \emph{grossone}). The latter approach was developed with the intention to allow one to…
Non standard analysis is an area of Mathematics dealing with notions of infinitesimal and infinitely large numbers, in which many statements from classical analysis can be expressed very naturally. Cheap non-standard analysis introduced by…
As suggested by the title, it has recently become clear that theorems of Nonstandard Analysis (NSA) give rise to theorems in computability theory (no longer involving NSA). Now, the aforementioned discipline divides into classical and…
We examine the convergence properties of sequences of nonnegative real numbers that satisfy a particular class of recursive inequalities, from the perspective of proof theory and computability theory. We first establish a number of results…
This paper is part of the general project of proof mining, developed by Kohlenbach. By "proof mining" we mean the logical analysis of mathematical proofs with the aim of extracting new numerically relevant information hidden in the proofs.…
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…
This paper considers the problem of testing whether there exists a non-negative solution to a possibly under-determined system of linear equations with known coefficients. This hypothesis testing problem arises naturally in a number of…
This paper considers the problem of testing whether there exists a solution satisfying certain non-negativity constraints to a linear system of equations. Importantly and in contrast to some prior work, we allow all parameters in the system…
Someone knowledgeable in nonstandard analysis may get the feeling that in the nonlinear theory of generalized functions, too often one works directly on the nets and spends effort to obtain results that should be clear from general…
In 1960s, Dana Scott gave a recursion theoretic characterization of standard systems of countable non-standard models of arithmetic, i.e., collections of sets of standard natural numbers coded in non-standard models. Later, Knight and Nadel…
These lecture notes, to be completed in a later version, offer a short and rigorous introduction to Nostandard Analysis, mainly aimed to reach to a presentation of the basics of Loeb integration, and in particular, Loeb measures. The…
Within the framework of generalized noncontextuality, we introduce a general technique for systematically deriving noncontextuality inequalities for any experiment involving finitely many preparations and finitely many measurements, each of…
In these lecture notes we present an introduction to non-standard analysis especially written for the community of mathematicians, physicists and engineers who do research on J. F. Colombeau' theory of new generalized functions and its…