Related papers: Fraction space revisited
The theory of fractional calculus in the complex plane was not built with a specific application in mind. The main obstacle to application was the difficulty with obtaining analytic continuations of fractional derivatives and integrals. It…
We provide a formal introduction into the classic theorems of general topology and its axiomatic foundations in set theory. Starting from ZFC, the exposition in this first part includes relation and order theory as well as a construction of…
All constructive methods employed in modern mathematics produce only countable sets, even when designed to transcend countability. We show that any constructive argument for uncountability -- excluding diagonalization techniques --…
We present a first-order theory of sequences with integer elements, Presburger arithmetic, and regular constraints, which can model significant properties of data structures such as arrays and lists. We give a decision procedure for the…
The compactness phenomenon is one of the featured aspects of structuralism in mathematics. In simple and broad words, a compactness property holds in a structure if a related property is satisfied by sufficiently many substructures of that…
Abstrct: In this note, by considering fractionally linear functions over a finite field and consequently developing an abstract sequence, we study some of its properties.
More often than not, recently popular structuralist interpretations of physical theories leave the central concept of a structure insufficiently precisified. The incipient causal sets approach to quantum gravity offers a paradigmatic case…
A reasonably complete theory of the approximation of an irrational by rational fractions whose numerators and denominators lie in prescribed arithmetic progressions is developed in this paper. Results are both, on the one hand, from a…
We develop a novel formal theory of finite structures, based on a view of finite structures as a fundamental artifact of computing and programming, forming a common platform for computing both within particular finite structures, and in the…
This article presents the most interesting philosophical issues as they arise in causal set theory. The first concerns the apparent disappearance of spacetime at the fundamental level. It shows how the looming empirical incoherence is…
There exists a particular subset of algebraic power series over a finite field which, for different reasons, can be compared to the subset of quadratic real numbers. The continued fraction expansion for these elements, called…
When a series of measurements is performed with increasingly coarse (or increasingly fine) precision, consecutive observations seem to be erratically distributed at first, and then organize themselves into cycles and patterns. The patterns,…
Compositionality is a key property for dealing with complexity, which has been studied from many points of view in diverse fields. Particularly, the composition of individual computations (or programs) has been widely studied almost since…
Partial algebras and datatypes are discussed with the use of signatures that allow partial functions, and a three-valued short-circuit (sequential) first order logic with a Tarski semantics. The propositional part of this logic is also…
The recent discoveries of new forms of quantum statistics require a close look at the under-lying Fock space structure. This exercise becomes all the more important in order to provide a general classification scheme for various forms of…
This paper presents three new families of fractional Sobolev spaces and their accompanying theory in one-dimension. The new construction and theory are based on a newly developed notion of weak fractional derivatives, which are natural…
This talk presents foundations of mathematics as a historically variable set of principles appealing to various modes of human intuition and devoid of any prescriptive/prohibitive power. At each turn of history, foundations crystallize the…
Starting from a small number of well-motivated axioms, we derive a unique definition of sums with a noninteger number of addends. These "fractional sums" have properties that generalize well-known classical sum identities in a natural way.…
Fractional calculus is the calculus of differentiation and integration of non-integer orders. In a recently paper (Annals of Physics 323 (2008) 2756-2778), the Fundamental Theorem of Fractional Calculus is highlighted. Based on this…
Partiality is a natural phenomenon in computability that we cannot get around. So, the question is whether we can give the areas where partiality occurs, that is, where non-termination happens, more structure. In this paper we consider…