Related papers: Valuation and Divisibility
A complete classification of unimodular valuations on the set of lattice polygons with values in the spaces of polynomials and formal power series, respectively, is established. The valuations are classified in terms of their behaviour with…
In this manuscript, fractal and fuzzy calculus are summarized. Fuzzy calculus in terms of fractal limit, continuity, its derivative, and integral are formulated. The fractal fuzzy calculus is a new framework that includes fractal fuzzy…
In this note, basing on a certain functional equation of the dilogarithm function, we establish nontrivial lower bounds for the $p$-adic valuation (where $p$ is a given prime number) of some type of rational numbers involving harmonic…
We present a method to prove the decidability of provability in several well-known inference systems. This method generalizes both cut-elimination and the construction of an automaton recognizing the provable propositions.
Data are invaluable. How can we assess the value of data objectively, systematically and quantitatively? Pricing data, or information goods in general, has been studied and practiced in dispersed areas and principles, such as economics,…
The bases of the theory of integrals for multidimensional differential systems are stated. The integral equivalence of total differential systems, linear homogeneous systems of partial differential equations, and Pfaff systems of equations…
This is a preliminary version of the Chapter 1 of a book "Computable Integrability"
The geometric and algebraic theory of valuations on cones is applied to understand identities involving summing certain rational functions over the set of linear extensions of a poset.
In this paper, we introduce a new method for calculating fractional integrals and differentials. The method involves an equation that we have obtained from infinite applied integration by parts. The equation works for special class of…
A Compact Introduction to Fractional Calculus is presented including basic definitions, fractional differential equations and special functions.
We discuss the dimensional characterization of the solutions space of a formally integrable system of partial differential equations and provide certain formulas for calculations of these dimensional quantities.
Classically domain theory is a rigourous mathematical structure to describe denotational semantics for programming languages and to study the computability of partial functions. Recently, the application of domain theory has also been…
This paper discusses some topics of enquiry concerning fractals, functions on them, and so on.
The role of geometrically infinitely divisible laws in renewal equations and superposition of renewal processes are explored here. Some examples are also discussed.
An introduction to geometric valuation theory is given. The focus is on classification results for $\operatorname{SL}(n)$ invariant and rigid motion invariant valuations on convex bodies and on convex functions.
We study the class of domains in which each w-ideal is divisorial, extending several properties of divisorial and totally divisorial domains to a much wider class of domains. In particular we consider PvMDs and Mori domains.
$\DeclareMathOperator{\IntR}{Int{}^\text{R}}$Integer-valued rational functions are a natural generalization of integer-valued polynomials. Given a domain $D$, the collection of all integer-valued rational functions over $D$ forms a ring…
This paper discusses the extension of the Prototype Verification System (PVS) sub-theory for rings, part of the PVS algebra theory, with theorems related to the division algorithm for Euclidean rings and Unique Factorization Domains that…
We give a formula and an estimation for the number of irreducible polynomials in two (or more) variables over a finite field.
We discuss without proof some recent advances in the theory of distributional boundary values, and give an example to show that the condition that a piecewise smooth domain has generic corners is necessary for the existence of boundary…