Related papers: On a Class of Harmonic-like numbers
Matrices over the dual numbers are considered. We propose an approach to classify these matrices up to similarity. Some preliminary results on the realization of this approach are obtained. In particular, we produce explicitly canonical…
What is ergonomic syntax for relations? In this first paper in a series of two, to answer the question we define regular calculi: a suitably structured functor from a category representing the syntax of regular logic to the category of…
In this article one builds a class of recursive sets, one establishes properties of these sets, and one proposes applications.
A hierarchy of matrix-valued polynomials which generalize the Jacobi polynomials is found. Defined by a Rodrigues formula, they are also products of a sequence of differential operators. Each class of polynomials is complete, satisfies a…
We define a class of spaces on which one may generalise the notion of compactness following motivating examples from higher-dimensional number theory. We establish analogues of several well-known topological results (such as Tychonoff's…
Harmonic numbers are significant in various branches of number theory. With the help of the digamma function, we prove ten conjectural series of Z.-W. Sun involving harmonic numbers. Several ones of them are also series expansions of…
Introducing the notion of a rational system of measure preserving transformations and proving a recurrence result for such systems, we give sufficient conditions in order a subset of rational numbers to contain arbitrary long arithmetic…
The harmonic numbers $H_n=\sum_{0<k\le n}1/k\ (n=0,1,2,\ldots)$ play important roles in mathematics. Let $p>3$ be a prime. With helps of some combinatorial identities, we establish the following two new congruences:…
Regular synchronization languages can be used to define rational relations of finite words, and to characterize subclasses of rational relations, like automatic or recognizable relations. We provide a systematic study of the decidability of…
Some years ago, the harmonic polynomial was introduced in order to understand better the harmonic topological index; for instance, it allows to obtain bounds of the harmonic index of the main products of graphs. Here, we obtain several…
This is a detailed and self-contained introduction to the real number system from a categorical perspective. We begin with the categorical definition of the natural numbers, review the Eudoxus theory of ratios as presented in Book V of…
A generalization of Hurwitz stable polynomials to real rational functions is considered. We establishe an analogue of the Hurwitz stability criterion for rational functions and introduce a new type of determinants that can be treated as a…
We define an extension of parity from the integers to the rational numbers. Three parity classes are found -- even, odd and `none'. Using the 2-adic valuation, we partition the rationals into subgroups with a rich algebraic structure. The…
In this paper, we introduce new classes of functions that extend the known classes of functions of complex variable, such as entire functions, meromorphic functions, rational functions and polynomial functions and take values in the set of…
Complex functions have multiple uses in various fields of study, so analyze their characteristics it is of extensive interest to other sciences. This work begins with a particular class of rational functions of a complex variable; over this…
For a real number $x$ and set of natural numbers $A$, define $x \ast A := \{ x a \bmod 1: a\in A\}\subseteq [0,1).$ We consider relationships between $x$, $A$, and the order-type of $x\ast A$. For example, for every irrational $x$ and…
We study expressive power of continuous logic in classes of (locally compact) groups. We also describe locally compact groups which are separably categorical structures.
In this paper we present algorithmic considerations and theoretical results about the relation between the orders of certain groups associated to the components of a polynomial and the order of the group that corresponds to the polynomial,…
We prove a classification of additive polynomial superfunctors, which allows us to compute some extensions of a superfunctor of the form $F \circ A$ where $F$ is a classical polynomial functor and $A$ is additive. We get a formula which…
We introduce a functional calculus with simple syntax and operational semantics in which the calculi introduced so far in the Curry-Howard correspondence for Classical Logic can be faithfully encoded. Our calculus enjoys confluence without…