Open Journal of mathematical Sciences (OMS)

In this paper, we prove some new formulas in the enumeration of labelled \(t\)-ary trees by path lengths. We treat trees having their edges oriented from a vertex of lower label towards a vertex of higher label. Among other results, we obtain counting formulas for the number of \(t\)-ary trees on \(n\) vertices in which there are paths of length \(\ell\) starting at a root with label \(i\) and ending at a vertex, sink, leaf sink, first child, non-first child and non-leaf. For each statistic, the average number of these reachable vertices is obtained for any random \(t\)-ary tree.

In this paper, improved and generalized version of Ostrowski’s type inequalities is established. The parameters used in the peano kernels help us to obtain previous results. The obtained bounds are then applied to numerical integration.

We show new integral representations for dilogarithm and trilogarithm functions on the unit interval. As a consequence, we also prove (1) new integral representations for Apéry, Catalan constants and Legendre \(\chi\) functions of order 2, 3, (2) a lower bound for the dilogarithm function on the unit interval, (3) new Euler sums.

We clarify some arguments concerning Jefimenko’s equations, as a way of constructing solutions to Maxwell’s equations, for charge and current satisfying the continuity equation. We then isolate a condition on non-radiation in all inertial frames, which is intuitively reasonable for the stability of an atomic system, and prove that the condition is equivalent to the charge and current satisfying certain relations, including the wave equations. Finally, we prove that with these relations, the energy in the electromagnetic field is quantised and displays the properties of the Balmer series.

In this paper, we derive the explicit expressions for single and product moments of generalized order statistics from Pareto-Rayleigh distribution using hypergeometric functions. Also, some interesting remarks are presented.

Local convergence of a family of sixth order methods for solving Banach space valued equations is considered in this article. The local convergence analysis is provided using only the first derivative in contrast to earlier works on the real line using the seventh derivative. This way the applicability is expanded for these methods. Numerical examples complete the article.

A q-rung orthopair fuzzy matrix (q-ROFM), an extension of the Pythagorean fuzzy matrix (PFM) and intuitionistic fuzzy matrix (IFM), is very helpful in representing vague information that occurs in real-world circumstances. In this paper we define some algebraic operations, such as max-min, min-max, complement, algebraic sum, algebraic product, scalar multiplication \((nA)\), and exponentiation \((A^n)\). We also investigate the algebraic properties of these operations. Furthermore, we define two operators, namely the necessity and possibility to convert q-ROFMs into an ordinary fuzzy matrix, and discuss some of their basic algebraic properties. Finally, we define a new operation(@) on q-ROFMs and discuss distributive laws in the case where the operations of \(\oplus_{q}, \otimes_{q}, \wedge_{q}\) and \(\vee_{q}\) are combined each other.

There are many possible definitions of derivatives, here we present some and present one that we have called generalized that allows us to put some of the others as a particular case of this but, what interests us is to determine that there is an infinite number of possible definitions of fractional derivatives, all are correct as differential operators each of them must be properly defined its algebra. We introduce a generalized version of fractional derivative that extends the existing ones in the literature. To those extensions it is associated a differentiable operator and a differential ring and applications that shows the advantages of the generalization. We also review the different definitions of fractional derivatives and it is shown how the generalized version contains the previous ones as a particular cases.

New Hadamard type inequalities for a class of \(s\)-Godunova-Levin functions of the second kind for fractional integrals are obtained. These new estimates extend and generalize some existing results for the \(Q\)-class and \(P\)-class functions. The generalized case for the Katugampola fractional integrals are also given.

For any non-square integer multiplier \(k\), there is an infinity of triangular numbers multiple of other triangular numbers. We analyze the congruence properties of indices \(\xi\) of triangular numbers multiple of triangular numbers. Remainders in congruence relations \(\xi\) modulo \(k\) come always in pairs whose sum always equal \((k-1)\), always include 0 and \((k-1)\), and only 0 and \((k-1)\) if \(k\) is prime, or an odd power of a prime, or an even square plus one or an odd square minus one or minus two. If the multiplier \(k\) is twice the triangular number of \(n\), the set of remainders includes also \(n\) and \((n^{2}-1)\) and if \(k\) has integer factors, the set of remainders include multiples of a factor following certain rules. Algebraic expressions are found for remainders in function of \(k\) and its factors, with several exceptions. This approach eliminates those \(\xi\) values not providing solutions.