Related papers: Fraction space revisited
The recent trend in mathematics is towards a framework of abstract mathematical objects, rather than the more concrete approach of explicitly defining elements which objects were thought to consist of. A natural question to raise is whether…
The continued fraction expansion of an irrational number $\alpha$ is eventually periodic if and only if $\alpha$ is a quadratic irrationality. However, very little is known regarding the size of the partial quotients of algebraic real…
Farey sequence has been a topic of interest to the mathematicians since the very beginning of last century. With the emergence of various algorithms involving the digital plane in recent times, several interesting works related with the…
Continued fractions with prescribed structures on sequences of their partial quotients have been intensively studied in the literature. As far as an integer sequence, especially a randomly generated one is concerned, an attractive question…
Farey sequences, Stern-Brocot sequences, the Calkin-Wilf sequences are shown to be generated via almost identical second order recurrence relations. These sequences have combinatorial, computational, and geometric applications, and are…
We introduce two families of transcendental numbers which we call finite factorial (FF) and partially finite factorial (PFF) numbers respectively, with the former one being subfamily of the latter one. These numbers arise naturally from…
We present several philosophical ideas emerging from the studies of complex systems. We make a brief introduction to the basic concepts of complex systems, for then defining "abstraction levels". These are useful for representing…
In this paper, a Fourier series in fractional dimensional space is introduced for an arbitrarily periodic function $f(t;\alpha)$. We call it fractional Fourier series of the order $\alpha$. Extending the basis functions of the linear space…
Fractional supersymmetry denotes a generalisation of supersymmetry which may be constructed using a single real generalised Grassmann variable, $\theta = \bar{\theta}, \, \theta^n = 0$, for arbitrary integer $n = 2, 3, ...$. An explicit…
A new matrix operation based on inserting columns and rows, similarly to the mediant operation between fractions, gives rise to the Farey determinants matrix or, equivalently, the matrix of the numerators of the differences of Farey…
We report a rigorous theory to show the origin of the unexpected periodic behavior seen in the consecutive differences between prime numbers. We also check numerically our findings to ensure that they hold for finite sequences of primes,…
The aim of this paper is to present a didactical sequence that fosters the development of meanings related to fractions, conceived as numbers that can be placed on the number line. The sequence was carried out in various elementary school…
A new algebraic object is introduced - recurrent fractions, which is an n-dimensional generalization of continued fractions. It is used to describe an algorithm for rational approximations of algebraic irrational numbers. Some…
We formalize an existing computability-theoretic method of presenting first-order structures whose domains have the cardinality of the continuum. Work using these methods until now has emphasized their topological properties. We shift the…
Orthogonal arrays are arguably one of the most fascinating and important statistical tools for efficient data collection. They have a simple, natural definition, desirable properties when used as fractional factorials, and a rich and…
The area of fractional calculus (FC) has been fast developing and is presently being applied in all scientific fields. Therefore, it is of key relevance to assess the present state of development and to foresee, if possible, the future…
We explore a novel link between two seemingly disparate mathematical concepts: Egyptian fractions and fractals. By examining the decomposition of rationals into sums of distinct unit fractions, a practice rooted in ancient Egyptian…
We consider a family of integer sequences generated by nonlinear recurrences of the second order, which have the curious property that the terms of the sequence, and integer multiples of the ratios of successive terms (which are also…
In the paper, I considered construction of algebra of fractions of algebra with conjugation. I also considered algebra of polynomials and algebra of rational mappings over algebra with conjugation.
In the present article, a new method for the evaluation of fractional derivatives of arbitrary real order is proposed. Numerous but inequivalent formulations have been given in the past. Some of them exhibit unsatisfactory properties such…