Here presented is a unified expression of Stirling numbers and their generalizations by using generalized factorial functions and generalized divided difference. Previous well-known extensions of Stirling numbers due to Riordan, Carlitz, Howard, Charalambides-Koutras, Gould-Hopper, Hsu-Shiue, Tsylova, Todorov, and Ahuja-Enneking are included as particular
cases of our generalization. Four algorithms for calculating the Stirling numbers and their generalizations based on our unified form are also given, which include two comprehensive algorithms using the characterization of Riordan arrays.

We give necessary and sufficient conditions to decompose \( \lambda \) copies, where necessarily \( \lambda \geq 2 \), of the complete graph \( K_v \), into so-called “2-petal”, “stem-infinity”, “barbell”, and “box-edge” graphs, all with four vertices and five edges.

The total chromatic number conjecture, which has appeared in a few hundred articles and in numerous books thus far, is now one of the classic mathematical unsolved problems. It appears that many authors coincidentally have attributed it to Professor M. Behzad and/or to Professor V.G. Vizing. Eventually, after four decades, Professor A. Soifer investigated the origin of this conjecture; published his findings in *The Mathematical Coloring Book* (2009); and stated that, “In my opinion this unquestionably merits the joint credit to Vizing and Behzad.” After checking all the arguments presented and the blames cited, I decided to investigate the controversy stated in this book on my own. My findings, which are presented in this report, specifically signify the following two points:

1. M. Behzad is the sole author of the Total Chromatic Number Conjecture.
2. The wrong referrals provided by numerous authors over the last forty-four years, to indicate Vizing’s authorship, must be brought to the attention of the authors and researchers, by appropriate means, as soon as possible.

Let \(\mathcal{G}_n\) be the set of all simple loopless undirected graphs on \(n\) vertices. Let \(T\) be a linear mapping, \(T : \mathcal{G}_n \rightarrow \mathcal{G}_n\) for which the independence number of \(T(G)\) is the same as the independence number for \(G\) for any \(G \in \mathcal{G}_n\). We show that \(T\) is necessarily a vertex permutation. Similar results are obtained for mappings preserving the matching number of bipartite graphs, the vertex cover number of undirected graphs, and the edge independence number of undirected graphs.

We will study the random perturbation on a linear differential equation as a nowhere differentiable function. The noise in the historical Langevin stochastic differential equation will be treated as a nowhere differentiable model for Brownian motion. A short introduction of Wiener process leading to It\^o’s calculus will be used in derivation of the mean and variance of the solutions to the Langevin Equation. Computational algorithms were developed and applied to study the numerical solutions to linear stochastic differential equations. Symbolic computation and simulation of a computer algebra system will be used to demonstrate the behavior of the solution to the Langevin Stochastic Differential Equation when the perturbation is density independent.

A bi-level balanced array (B-array) \( T \) with parameters \( (m, N, t) \) and index set \( \underline{\mu}’ = \{\mu_0, \mu_1, \ldots, \mu_t\} \) is a matrix with \( m \) rows, \( N \) columns, and with two elements (say, \( 0 \) and \( 1 \)) such that in every \( (t \times N) \)-submatrix \( T^* \) (clearly, there are \( \binom{m}{t} \) such submatrices) of \( T \), the following combinatorial condition is satisfied: every \( (t \times 1) \) vector \( \underline{\alpha} \) of \( T^* \) with \( i \) (\( 0 \leq i \leq t \)) ones in it appears the same number \( \mu_i \) (say) times. \( T \) is called a B-array of strength \( t \). Clearly, an orthogonal array (O-array) is a special case of a B-array. These combinatorial arrays have been extensively used in information theory, coding theory, and design of experiments. In this paper, we restrict ourselves to arrays with \( t = 4 \) and \( t = 6 \). We derive some inequalities involving \( m \) and \( \mu_i \), using the concept of coincidences amongst the columns of \( T \), which are necessary conditions for B-arrays to exist. We then use these inequalities to study the existence of these arrays and to obtain the bounds on the number of rows (also called constraints) \( m \), for a given value of \( \underline{\mu}’ \).