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Submodular set functions are undoubtedly among the most important building blocks of combinatorial optimization. Somewhat surprisingly, continuous counterparts of such functions have also appeared in an analytic line of research where they…
Based on rectangle theory of formal concept and set covering theory, the concept reduction preserving binary relations is investigated in this paper. It is known that there are three types of formal concepts: core concepts, relative…
This paper continues a series in which we study deficiencies in previously published works concerning fixed point assertions for digital images.
Much of statistics relies upon four key elements: a law of large numbers, a calculus to operationalize stochastic convergence, a central limit theorem, and a framework for constructing local approximations. These elements are…
This thesis concerns embeddings and self-embeddings of foundational structures in both set theory and category theory. The first part of the work on models of set theory consists in establishing a refined version of Friedman's theorem on…
Subadditive set functions play a pivotal role in computational economics (especially in combinatorial auctions), combinatorial optimization or artificial intelligence applications such as interpretable machine learning. However, specifying…
The Arithmetic Fourier Transform is a numerical formulation for computing Fourier series and Taylor series coefficients. It competes with the Fast Fourier Transform in terms of speed and efficiency, requiring only addition operations and…
Hyperspaces form a powerful tool in some branches of mathematics: lots of fractal and other geometric objects can be viewed as fixed points of some functions in suitable hyperspaces - as well as interesting classes of formal languages in…
A new interpretation of zeta functions is given for F1-schemes which do not satisfy Soul\'e's condition. Functional equations for reductive groups are computed and a new definition of zeta functions attached to more general counting…
The Additive Transform of an arithmetic function represents a novel approach to examining the interplay between multiplicative arithmetic function and additive functions. This transform concept introduces a method to systematically generate…
This article presents a reformulation of the Theory of Functional Connections: a general methodology for functional interpolation that can embed a set of user-specified linear constraints. The reformulation presented in this paper exploits…
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…
Let $\mathcal{A}$ denote a real, $n$-dimensional, unital, associative algebra.This paper provides an introductory exposition of calculus over $\mathcal{A}$. An $\mathcal{A}$-differentiable function is one for which the differential is…
In this paper,we develop a novel representation of the zeta function expressed as the limiting difference between two structured double sums. This approach leads to a new and elegant identity involving maximum functions and additive terms,…
We consider the connection of functional decompositions of rational functions over the real and complex numbers, and a question about curves on a Riemann sphere which are invariant under a rational function.
In the context of the complex-analytic structure within the unit disk centered at the origin of the complex plane, that was presented in a previous paper, we show that singular Schwartz distributions can be represented within that same…
All components of complements of discriminant varieties of simple real function singularities are explicitly listed. New invariants of such components (for not necessarily simple singularities) are introduced. A combinatorial algorithm…
Pseudodifferential operators of several variables are formal Laurent series in the formal inverses of $\partial_1, ..., \partial_n$ with $\partial_i = d$ $1 \leq i \leq n$. As in the single variable case, Lax equations can be constructed…
By considering the prime zeta function, the author intended to demonstrate in that the Riemann zeta function zeta(s) does not vanish for Re(s)>1/2, which would have proven the Riemann hypothesis. However, he later realised that the proof of…
Analyzing in detail the analytic continuation of the Riemann zeta function we are able to generate several new identities which may be useful for application in physics and mathematics.