In this paper, we prove an arithmetic identity derived by Farkas [3]. We use generalized eta products and their logarithmic derivatives to determine the arithmetic identity.

The Dorroh extension is typically applied to embed a ring without unity into a ring containing unity. However, de Alwis investigated the ring resulting from applying this extension to the ring of integers. We show that the Dorroh extension of any ring with unity is isomorphic to a direct product of rings. Using this isomorphism we are able to verify the results of de Alwis and extend them to the Dorroh extension of any ring with unity. Also, given any ring $R$, we give conditions under which an ideal of the Dorroh extension $\mathbb{Z} * R$ can be expressed as a product (in the extension) of an ideal in $\mathbb{Z}$ and an ideal in $R$.

We provide formulas for the chromatic polynomials of the Cartesian product of a path on two vertices with a path on $n$ vertices, and a cycle on three vertices with a path on $n$ vertices. The result for the Cartesian product of paths involves the trinomial coefficients. We define a new sequence of numbers, $CP(n,i)$, which are part of the result for the chromatic polynomial of the Cartesian product of the three cycle and the path. In both cases the chromatic polynomial is factored in the form $\lambda (\lambda -1)$.

During the course of studying freely falling bodies, Galileo observed a wonderful property that is unique to the sequence of positive odd integers. In this paper, we present and explore a generalized variation of this classical result. We exhibit some properties of our variation and leave the reader with some open questions for possible further exploration.

We present a class of "nice" $n\times n$ Markov probability transition matrices and infinitesimal generators whose limiting (steady state) probabilities are proportional to the first $n$ Fibonacci numbers. We extend this model to other sequences and discover some curious matrix and sequence relationships.

Proposed model geometries are described for six types of flattened sickled erythrocytes. These new types occur with some frequency in blood smears of patients with sickle cell disease. The six types of cells are designated as five-pointed star, witch, hypocycloid, deltoid, astroid, and semi-circular. Formulae are provided for the volume and surface area of each type and a figure showing the profile of each is included. Measurements of actual sickled cells in vitro could be used to find values for the volume and surface area of each type of cell by using parameter values for the appropriate model geometry. These would give close approximations that could be useful in clinical therapies and laboratory investigations for sickle cell anemia. Surface area to volume ratios can also be found to a close approximation for each cell type.

A heuristic model of the motion of sperm in the vicinity of an egg is developed and analyzed. The model is based on differential equations and stochastic processes. Numerical simulations are performed.