The arithmetic of the natural numbers $\mathbb{N}$ can be extended to arithmetic operations on planar binary trees. This gives rise to a noncommutative arithmetic theory. In this exposition, we describe this arithmetree, first defined by Loday, and investigate prime trees.

We describe a population logistic model exposed to a mild life-long sexually transmitted disease, that is, without significant increased mortality among infected individuals and providing no immunity/recovery. We then modify this model to include groups isolated from sexual contact and analyze their potential effect on the dynamics of the population. We are interested in how the isolated class may curb the growth of the infected group while keeping the healthy population at acceptable levels. In particular, we analyze the connection between vertical transmission and isolation from reproduction on the long term behavior of the disease. A comparison with similar effects caused by vaccination and quarantine is also provided.

Let $Q$ be a finite set of points in the plane. For any set $P$ of points in the plane, $S_Q\left(P\right)$ denotes the number of similar copies of $Q$ contained in $P$. For a fixed $n$, Erdős and Purdy asked for the maximum possible value of $S_Q\left(P\right)$, denoted by $S_Q\left(n\right)$, over all sets $P$ of $n$ points in the plane. We consider this problem when $Q=△$ is the set of vertices of an isosceles right triangle. We give exact solutions when $n\le 9$, and provide new upper and lower bounds for $S_{△}\left(n\right)$.

We examine the stability properties of a predictor-corrector implementation of a class of implicit linear multistep methods. The method has recently been described in the literature as suitable for the efficient integration of stiff systems and as having stability regions similar to well known implicit methods. A more detailed analysis reveals that this is not the case.

We study condensed zero-divisor graphs (those whose vertices are equivalence classes of zero-divisors of a ring $R$) having exactly five vertices. In particular, we determine which graphs with exactly five vertices can be realized as the condensed zero-divisor graph of a ring. We provide the rings for the graphs which are possible, and prove that the rest of graphs can not be realized via any commutative ring. There are 34 graphs in total which contain exactly five vertices.

By a result of the second author, the Connes embedding conjecture (CEC) is false if and only if there exists a self-adjoint noncommutative polynomial $p\left(t_1,t_2\right)$ in the universal unital $C^{\ast }$-algebra $\mathcal{A}=\langle t_1,t_2:t_j=t_j^{\ast },0<t_j\le 1for1\le j\le 2\rangle$ and positive, invertible contractions $x_1,x_2$ in a finite von Neumann algebra $\mathcal{ℳ}$ with trace $\tau$ such that $\tau \left(p\left(x_1,x_2\right)\right)<0$ and ${Tr}_k\left(p\left(A_1,A_2\right)\right)\ge 0$ for every positive integer $k$ and all positive definite contractions $A_1,A_2$ in $M_k\left(ℂ\right)$. We prove that if the real parts of all coefficients but the constant coefficient of a self-adjoint polynomial $p\in \mathcal{A}$ have the same sign, then such a $p$ cannot disprove CEC if the degree of $p$ is less than $6$, and that if at least two of these signs differ, the degree of $p$ is $2$, the coefficient of one of the $t_i^2$ is nonnegative and the real part of the coefficient of $t_1{t}_2$ is zero then such a $p$ disproves CEC only if either the coefficient of the corresponding linear term $t_i$ is nonnegative or both of the coefficients of $t_1$ and $t_2$ are negative.

In 1998, Filipponi and Hart introduced many Zeckendorf representations of Fibonacci, Lucas and mixed products involving two variables. In 2008, Artz and Rowell proved the simplest of these identities, the Fibonacci product, using tilings. This paper extends the work done by Artz and Rowell to many of the remaining identities from Filipponi and Hart’s work. We also answer an open problem raised by Artz and Rowell and present many Zeckendorf representations of mixed products involving three variables.

We generalize the concepts of even and odd functions in the setting of complex-valued functions of a complex variable. If $n>1$ is a fixed integer and $r$ is an integer with $0\le r<n$, we define what it means for a function to have type $rmodn$. When $n=2$, this reduces to the notions of even ($r=0$) and odd ($r=1$) functions respectively. We show that every function can be decomposed in a unique way as the sum of functions of types-0 through $n-1$. When the given function is differentiable, this decomposition is compatible with the differentiation operator in a natural way. We also show that under certain conditions, the type $r$ component of a given function may be regarded as a real-valued function of a real variable. Although this decomposition satisfies several analytic properties, the decomposition itself is largely algebraic, and we show that it can be explained in terms of representation theory.