Related papers: A New Operator for Egyptian Fractions
The class of Parikh word representable graphs were recently introduced. In this work, we further develop its general theory beyond the binary alphabet. Our main result shows that this class is equivalent to the class of bipartite…
In an earlier paper we introduced the notion of 'bifurcating continued fractions' in a heuristic manner. In this paper a formal theory is developed for the 'bifurcating continued fractions'.
In this paper, we present a new numerical method to solve fractional differential equations. Given a fractional derivative of arbitrary real order, we present an approximation formula for the fractional operator that involves integer-order…
We find a polynomial in three variables whose values at nonnegative integers satisfy the Erd\H{o}s-Straus Conjecture. Although the perfect squares are not covered by these values, it allows us to prove that there are arbitrarily long…
Some mathematical models of applied problems lead to the need of solving boundary value problems with a fractional power of an elliptic operator. In a number of works, approximations of such a nonlocal operator are constructed on the basis…
The Operator axioms have produced new real numbers with new operators. New operators naturally produce new equations and thus extend the traditional mathematical models which are selected to describe various scientific rules. So new…
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…
We present an algorithm to compute all factorizations into linear factors of univariate polynomials over the split quaternions, provided such a factorization exists. Failure of the algorithm is equivalent to non-factorizability for which we…
This paper aims to introduce high school students to the intriguing world of continued fractions, a mathematical concept that provides a unique representation of numbers. The study focuses on the exploration and development of the…
A recently developed computational methodology for executing numerical calculations with infinities and infinitesimals is described in this paper. The developed approach has a pronounced applied character and is based on the principle `The…
In this article we present ways to evaluate certain sums, products and continued fractions using tools from the theory of elliptic functions. The specific results appear to be new, although similar ones can be found in the leterature; in…
Let p(n, k) denote the number of partitions of n into parts less than or equal to k. We show several properties of this function modulo 2. First, we prove that for fixed positive integers k and m, p(n,k) is periodic modulo m. Using this, we…
New formulas for the construction of Pythagorean triples and generalizations to equations of higher powers. Application of formulas to some problems, in particular Fermat's equation with n=4.
We investigate the possibilities to calculate vector partition functions by means of iterated partial fraction decomposition, as suggested by Beck (2004). Particularly, for an important type of families of rational functions, we describe an…
Any rational number can be factored into a product of several rationals whose sum vanishes. This simple but nontrivial fact was suggested as a problem on a maths olympiad for high-school students. We completely solve similar questions in…
This paper introduces a new structure of commutative semiring, generalizing the tropical semiring, and having an arithmetic that modifies the standard tropical operations, i.e. summation and maximum. Although our framework is combinatorial,…
Fractal sets, by definition, are non-differentiable, however their dimension can be continuous, differentiable, and arithmetically manipulable as function of their construction parameters. A new arithmetic for fractal dimension of polyadic…
Subtraction is a powerful technique for creating new bijections from old. Let's reinvent it! While we're at it, let's reinvent division as well.
Monads can be interpreted as encoding formal expressions, or formal operations in the sense of universal algebra. We give a construction which formalizes the idea of "evaluating an expression partially": for example, "2+3" can be obtained…
We prove a continued fraction expansion for the reciprocal of a certain $q$-series. All the specialists in the world are asked whether it is new or not.