Related papers: Hypernatural Numbers as Ultrafilters
This thesis is intended to provide an account of the theory and applications of Operational Methods that allow the "translation" of the theory of special functions and polynomials into a "different" mathematical language. The language we…
A combinatorial methods are used to investigate some properties of certain generalized Stirling numbers, including explicit formula and recurrence relations. Furthermore, an expression of these numbers with symmetric function is deduced.
We introduce $\textit{Laver ultrafilters}$, namely ultrafilters $\mathcal{U}$ for which the associated Laver forcing $\mathbb{L}_{\mathcal{U}}$ has the Laver property. We give simple combinatorial characterisations of these ultrafilters,…
This paper develops an infinitesimal order of magnitude coupled with overflow technique that allows nonnumerical proofs of nondegenerate and degenerate inverse mapping theorems for mappings minimally regular at a point. This approach is…
Fine-tuning criteria are frequently used to place upper limits on the masses of superpartners in supersymmetric extensions of the standard model. However, commonly used prescriptions for quantifying naturalness have some important…
The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear…
The emergent field of probabilistic numerics has thus far lacked clear statistical principals. This paper establishes Bayesian probabilistic numerical methods as those which can be cast as solutions to certain inverse problems within the…
For real application and theoretical investigation of ordinary hypergraphs and non-ordinary hypergraphs, researchers need to establish standard rules and feasible operating methods. We propose a visualization tool for investigating…
Algebras of ultradifferentiable generalized functions are introduced. We give a microlocal analysis within these algebras related to the regularity type and the ultradifferentiable property.
In the present survey paper, we present several new classes of Hochster's spectral spaces "occurring in nature", actually in multiplicative ideal theory, and not linked to or realized in an explicit way by prime spectra of rings. The…
In this paper some new ways of generalizing perfect numbers are investigated, numerical results are presented and some conjectures are established.
We deliver a call to arms for probabilistic numerical methods: algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such…
Anomaly detection has many applications ranging from bank-fraud detection and cyber-threat detection to equipment maintenance and health monitoring. However, choosing a suitable algorithm for a given application remains a challenging design…
An assortments of `species' of families of subsets of a set and some of their properties are investigated, with an eye on the logic and limit role they may play as `relaxed parallels' to ultrafilters.
The paper first covers several properties of the extension of the divisibility relation to a set ${}^*\hspace{-0.5mm}N$ of nonstandard integers. After that, a connection is established with the divisibility in the Stone-\v{C}ech…
The purpose of this article is to present, in a simple way, an analytic approach to special numbers and polynomials. The approach is based on the derivative polynomials. The paper is, to some extent, a review article, although it contains…
Short nonstandard proofs are given for some results about infinite systems of equations in infinitely many variables.
We propose a method for the unsupervised clustering of hyperspectral images based on spatially regularized spectral clustering with ultrametric path distances. The proposed method efficiently combines data density and geometry to…
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…
In this article, we introduce combinatorial models for poly-Bernoulli polynomials and poly-Euler numbers of both kinds. As their applications, we provide combinatorial proofs of some identities involving poly-Bernoulli polynomials.