Related papers: Meadow based Fracterm Theory
A \emph{fractal} is an object exhibiting complexity at arbitrarily small scales. In order to study and characterise fractals, one is often interested in quantifying how they fill up space on small scales. This gives rise to various notions…
Recently, we have proposed a new diffusive representation for fractional derivatives and, based on this representation, suggested an algorithm for their numerical computation. From the construction of the algorithm, it is immediately…
Fairness aware data mining (FADM) aims to prevent algorithms from discriminating against protected groups. The literature has come to an impasse as to what constitutes explainable variability as opposed to discrimination. This distinction…
We study an abstract notion of tree structure which lies at the common core of various tree-like discrete structures commonly used in combinatorics: trees in graphs, order trees, nested subsets of a set, tree-decompositions of graphs and…
An algebraic theory, sometimes called an equational theory, is a theory defined by finitary operations and equations, such as the theories of groups and of rings. It is well known that algebraic theories are equivalent to finitary monads on…
The AMR (Abstract Meaning Representation) formalism for representing meaning of natural language sentences was not designed to deal with scope and quantifiers. By extending AMR with indices for contexts and formulating constraints on these…
In this paper, we revisit the diffusive representations of fractional integrals established in \cite{diethelm2023diffusive} to explore novel variants of such representations which provide highly efficient numerical algorithms for the…
Metaphysical interpretations of set theory are either inconsistent or incoherent. The uses of sets in mathematics actually involve three distinct kinds of collections (surveyable, definite, and heuristic), which are governed by three…
A physical field has an infinite number of degrees of freedom since it has a field value at each location of a continuous space. Therefore, it is impossible to know a field from finite measurements alone and prior information on the field…
Formal grammars are extensively used in Computer Science and related fields to study the rules which govern production of a language. The use of these grammars can be extended beyond mere language production. One possibility is to view…
We present families of combinatorial classes described as trees with nodes that can carry one of two types of "flowers": integer partitions or integer compositions. Two parameters on the flowers of trees will be considered: the number of…
This chapter deals with error and uncertainty in data. Treats their measuring methods and meaning. It shows that uncertainty is a natural property of many data sets. Uncertainty is fundamental for the survival os living species, Uncertainty…
We define a partition of a reductive group into finitely many subsets, refining the partition of the group into strata. We state some conjectural properties of these subsets (called substrata) and verify them in some examples.
There are many resources useful for processing images, most of them freely available and quite friendly to use. In spite of this abundance of tools, a study of the processing methods is still worthy of efforts. Here, we want to discuss the…
Any rational number can be written as the sum of distinct unit fractions. In this survey paper we review some of the many interesting questions concerning such 'Egyptian fraction' decompositions, and recent progress concerning them.
We discuss a formal system of mathematics. We use it to construct the natural numbers.
There are many possible definitions of derivatives, here we present some and present one that we have called generalized that allows us to put some of the others as a particular case of this but, what interests us is to determine that there…
A macroscopic characterization of fractals showing up a structural transition from dense to multibranched growth is made using optical diffraction theory. Such fractals are generated via the numerical solution of the 2D Poisson and…
In this note we consider fractions of the form 1/m and their floating-point representation in various arithmetic bases. For instance, what is 1/7 in base 2005? And, what about 1/4? We give a simple algorithm to answer these questions. In…
Category theory has foundational importance because it provides conceptual lenses to characterize what is important in mathematics. Originally the main lenses were universal mapping properties and natural transformations. In recent decades,…