Open Journal of Mathematical Sciences (OMS)

Squares of odd index Fibonacci polynomials are used to define a new function \(\Phi\left(10^{n}\right)\) to approximate the number \(\pi\left(10^{n}\right)\) of primes less than \(10^{n}\). Multiple of 4 index Fibonacci polynomials are further used to define another new function \(\Psi\left(10^{n}\right)\) to approximate the number \(\Delta\left(\pi\left(10^{n}\right)\right)\) of primes having \(n\) digits and compared to a third function \(\Psi’\left(10^{n}\right)\) defined as the difference of the first function \(\Phi\left(10^{n}\right)\) based on odd index Fibonacci polynomials. These three functions provide better approximations of \(\pi\left(10^{n}\right)\) than those based on the classical \(\left(\frac{x}{log\left(x\right)}\right)\), Gauss’ approximation \(Li\left(x\right)\), and the Riemann \(R\left(x\right)\) functions.

We show that Euler’s relation and the Taxi-Cab relation are both solutions of the same equation. General solutions of sums of two consecutive cubes equaling the sum of two other cubes are calculated. There is an infinite number of relations to be found among the sums of two consecutive cubes and the sum of two other cubes, in the form of two families. Their recursive and parametric equations are calculated.

This study introduces theorems concerning matrix products, which delineate the transformations of sequences or series into other sequences or series, ensuring either the preservation of limits or the guarantee of convergence. Previous literature has explored the properties of matrices facilitating transformations between sequences, series, and their combinations, with detailed insights available in references [1,2,3].

The concept of weak UP-algebras (shortly wUP-algebra) is an extension of the notion of UP-algebras introduced in 2021 by Iampan and Romano. In this report, an effective extension of a (weak) UP-algebra to a wUP-algebra is created. In addition to the previous one, the concept of atoms in wUP-algebras is introduced and their important properties are registered. Finally, the concept of wUP-filters in wUP-algebras was introduced and its connections with other substructures in wUP-algebras were analyzed.

In normed spaces, Birkhoff orthogonality and isosceles orthogonality can be used to characterize space structures, and many scholars have introduced geometric constants to quantitatively describe the relationship between these two types of orthogonality. This paper introduces a new orthogonal relationship – Skew orthogonality – and proposes a new geometric constant to measure the “distance” of difference between skew orthogonality and Birkhoff orthogonality in normed spaces. In the end, we provide some examples of specific spaces.

This corrigenda makes seven corrections to D. Raske, “The Galerkin method and hinged beam dynamics,” Open J. of Math. Sci. 2023, 7, 236-247.

The main goal of this brief article is to provide an elementary proof of Sun’s six conjectures on Apéry-like sums involving ordinary harmonic numbers.

In this paper, we develop a new application of the Laplace transform method (LTM) using the series expansion of the dependent variable for solving fractional logistic growth models in a population as well as fractional prey-predator models. The fractional derivatives are described in the Caputo sense. To illustrate the reliability of the method some examples are provided. The results reveal that the technique introduced here is very effective and convenient for solving fractional-order nonlinear differential equations.

The generation of coefficients of terms of positive and negative powers of \(n\) and \(-n\) of Kifilideen trinomial theorem as the terms are progress is stressful and time-consuming which the same problem is identified with coefficients of terms of binomial theorem of positive and negative powers of \(n\) and \(-n\). This slows the process of producing the series of any particular trinomial expansion. This study established Kifilideen coefficient tables for positive and negative powers of \(n\) and \(-n\) of the Kifilideen trinomial theorem and other developments based on matrix and standardized methods. A Kifilideen theorem of matrix transformation of the positive power of \(n\) of trinomial expression in which three variables \(x,y\), and \(z\) are found in parts of the trinomial expression was originated. The development would ease evaluating the trinomial expression’s positive power of \(n\). The Kifilideen coefficient tables are handy and effective in generating the coefficients of terms and series of the Kifilideen expansion of trinomial expression of positive and negative powers of \(n\) and \(-n.\)

The Taylor expansion is a widely used and powerful tool in all branches of Mathematics, both pure and applied. In Probability and Mathematical Statistics, however, a stronger version of Taylor’s classical theorem is often needed, but only tacitly assumed. In this note, we provide an elementary proof of this measurable Taylor’s theorem, which guarantees that the interpolating point in the Lagrange form of the remainder can be chosen to depend measurably on the independent variable.