Questions concerning the factorizations of escalator numbers are considered. Specifically, we determine exactly when escalator numbers are nontrivial integral powers of rational numbers. We further bound the number of primes and thus the size of the largest prime dividing the numerator and denominator of an escalator number in terms of its position in an escalator number sequence.

A method is presented for determining barycentric coordinates of points of intersection of a line and a circle. The method is applied specifically to the Euler line, the line of the circumcenter and incenter, the Brocard axis, and several circles, including the circumcircle, incircle, nine-point circle, and Brocard circle. The method also applies to intersections of certain pairs of lines, harmonic conjugate pairs, and to centers of similitude of pairs of circles.

An infinite family of monogenic sextic fields with Galois group $A_4$ is exhibited.

In this paper, a technique is introduced for proving the product rule for the differentiation of the product of two functions in an undergraduate real analysis class. The general form of the product rule is proved using the sum rule and constant multiple rule as well as a special case of the product rule. This special case can be proved quite easily using only the definition of the derivative and the fact that a differentiable function is continuous.

The $F$ distributions are becoming increasingly prominent in several applied areas. In this paper, a new generalization of the $F$ distribution is introduced by compounding. It takes the form of the product of two $F$ pdf's. Various structural properties of this distribution are derived, including its cdf, moments, mean deviation about the mean, mean deviation about the median, entropy, asymptotic distribution of the extreme order statistics, maximum likelihood estimates and the Fisher information matrix.

In this paper we show that a paraquaternionic nearly Kähler manifold is necessarily a paraquaternionic Kähler manifold.

The basis number of a graph $G$ is defined to be the least integer $d$ such that there is a basis $\mathcal{B}$ of the cycle space of $G$ such that each edge of $G$ is contained in at most $d$ members of $\mathcal{B}$. MacLane [13] proved that a graph $G$ is planar if and only if the basis number of $G$ is less than or equal to 2. Ali [3] proved that the basis number of the strong product of a path and a star is less than or equal to 4. In this work, (1) We give an appropriate decomposition of trees. (2) We give an upper bound of the basis number of a cycle and a bipartite graph. (3) We give an upper bound of the basis number of a path and a bipartite graph. This is a generalization of Ali's result [3].