Let $n$ a be positive integer. The Catalan matrix of order $n$, denoted by $\mathscr{C}_n[x]$, is a lower triangular Toeplitz matrix whose nonzero elements are expressions containing Catalan numbers and a real parameter $x$. In this paper, we investigate the multiplicative order of the cyclic group generated by $\mathscr{C}_n[x]$ (for $x\in\mathbb{N}$) modulo a positive integer $m$.

In this paper, we introduce the notion of weak connectedness ($Cl_{\mathcal{T}_{1}}$-$Cl_{\mathcal{T}_{2}}$-connectedness) and weak component ($Cl_{\mathcal{T}_{1}}$-$Cl_{\mathcal{T}_{2}}$-component) in bitopological spaces. Several properties of this notion are discussed.

In this paper, we extend the classical Hille Yosida Theorem to fractional semi-groups of operators on Banach spaces.

We introduce the notion of a smocked metric space and explore the balls and geodesics in a collection of different smocked spaces. We find their rescaled Gromov-Hausdorff limits and prove these tangent cones at infinity exist, are unique, and are normed spaces. We close with a variety of open questions suitable for advanced undergraduates, masters students, and doctoral students.

We present a new proof of a result of Knuth and Buckholtz concerning the period of the number of alternating congruences modulo an odd prime. The proof is based on properties of special functions, specifically the polylogarithm, Dirichlet eta and beta functions, and Stirling numbers of the second kind.

When mortality statistics are reported for fatal diseases, they reflect the ratio of the overall mortality within the target population which is impacted from the disease in question. Reporting overall mortality leads to erroneous predictions since cohorts with different frailties are impacted at different rates. In this paper, we study methods for predicting mortality under varying conditions with the goal of removing the impact of hidden heterogeneity from the parameter estimates.

Three methods for obtaining uniform sampling on the standard simplex are summarized and a new derivation of one method is provided, which only uses basic notions from calculus, difference equations, and differential equations.