Related papers: Cheap Non-standard Analysis and Computability
Cost functions provide a framework for constructions of sets Turing below the halting problem that are close to computable. We carry out a systematic study of cost functions. We relate their algebraic properties to their expressive…
The present article is a brief informal survey of computability logic --- the game-semantically conceived formal theory of computational resources and tasks. This relatively young nonclassical logic is a conservative extension of classical…
We apply methods of nonstandard mathematics in order to regard analytic geometry in a very different way. For example, complex spaces are seen to be the "standard part" of certain algebraic nonstandard schemes. We construct a category of…
In 2007, Terence Tao wrote on his blog an essay about soft analysis, hard analysis and the finitization of soft analysis statements into hard analysis statements. One of his main examples was a quasi-finitization of the infinite pigeonhole…
Nonstandard analysis is very complex, so finding a simple description of infinitesimal points will be useful. In this paper, ultrafilters as infinitesimal points in a topological space will be proposed, and some topological concepts is…
Short nonstandard proofs are given for some results about infinite systems of equations in infinitely many variables.
This note has two principal aims: to portray an essence of Non-Standard Analysis as a particular structure (which we call lim-rim), noting its interplay with the notion of ultrapower, and to present a construction of Non-Standard Analysis,…
In this paper, we develop Terence Tao's harmonic analysis method and apply it to restricted sumsets. The well known Cauchy-Davenport theorem asserts that if $A$ and $B$ are nonempty subsets of $Z/pZ$ with $p$ a prime, then $|A+B|\ge…
Topological Data Analysis is a recent and fast growing field providing a set of new topological and geometric tools to infer relevant features for possibly complex data. This paper is a brief introduction, through a few selected topics, to…
We introduce a notion of computable randomness for infinite sequences that generalises the classical version in two important ways. First, our definition of computable randomness is associated with imprecise probability models, in the sense…
I present a novel mathematical technique for dealing with the infinities arising from divergent sums and integrals. It assigns them fine-grained infinite values from the set of hyperreal numbers in a manner that refines the standard…
We give a number of formal proofs of theorems from the field of computable analysis. Many of our results specify executable algorithms that work on infinite inputs by means of operating on finite approximations and are proven correct in the…
Using nonstandard analysis (NSA), the proof of the Laplace's formula is given. The usage of NSA reduces the intricacy of taking limit, and the crude line of the proof would be clearly seen, compared to the done with the rigorous classical…
We give estimates for the convolution product of an arbitrary number of endlessly continuable functions. This allows us to deal with nonlinear operations for the corresponding resurgent series, e.g. substitution into a convergent power…
When quantitative models are used to support decision-making on complex and important topics, understanding a model's ``reasoning'' can increase trust in its predictions, expose hidden biases, or reduce vulnerability to adversarial attacks.…
This paper aims to build a new understanding of the nonstandard mathematical analysis. The main contribution of this paper is the construction of a new set of numbers, $\mathbb{R}^{\mathbb{Z}_< }$, which includes infinities and…
In the present paper, we propose a new axiomatic approach to nonstandard analysis and its application to the general theory of spatial structures in terms of category theory. Our framework is based on the idea of internal set theory, while…
In previous papers on this project a general static logical framework for formalizing and mechanizing set theories of different strength was suggested, and the power of some predicatively acceptable theories in that framework was explored.…
Computation with advice is suggested as generalization of both computation with discrete advice and Type-2 Nondeterminism. Several embodiments of the generic concept are discussed, and the close connection to Weihrauch reducibility is…
The goal of this paper consists of developing a new (more physical and numerical in comparison with standard and non-standard analysis approaches) point of view on Calculus with functions assuming infinite and infinitesimal values. It uses…