We introduce ideas from geometric group theory related to boundaries of groups. We consider the visual boundary of a free abelian group, and show that it is an uncountable set with the trivial topology.

Suppose that $P\left(x\right)$, $Q\left(x\right)\in \mathbb{Z}\left[x\right]$ are two relatively prime polynomials, and that $P\left(x\right)∕Q\left(x\right)={\sum }_{n=0}^{\infty }{a}_n{x}^n$ has the property that $a_n\in \mathbb{Z}$ for all $n$. We show that if $Q\left(1∕\alpha \right)=0$, then $\alpha$ is an algebraic integer. Then, we show that this result can be used to provide a solution to Problem 11213(b) of the American Mathematical Monthly (2006).

We define a generalization of the chromatic number of a graph $G$ called the $k$-clique-relaxed chromatic number, denoted ${\chi }^{\left(k\right)}\left(G\right)$. We prove bounds on ${\chi }^{\left(k\right)}\left(G\right)$ for all graphs $G$, including corollaries for outerplanar and planar graphs. We also define the $k$-clique-relaxed game chromatic number, ${\chi }_g^{\left(k\right)}\left(G\right)$, of a graph $G$. We prove ${\chi }_g^{\left(2\right)}\left(G\right)\le 4$ for all outerplanar graphs $G$, and give an example of an outerplanar graph $H$ with ${\chi }_g^{\left(2\right)}\left(H\right)\ge 3$. Finally, we prove that if $H$ is a member of a particular subclass of outerplanar graphs, then ${\chi }_g^{\left(2\right)}\left(H\right)\le 3$.

In referendum elections, voters are frequently required to register simultaneous yes/no votes on multiple proposals. The separability problem occurs when a voter’s preferred outcome on a proposal or set of proposals depends on the known or predicted outcomes of other proposals in the election. Here we investigate cost-consciousness as a potential cause of nonseparability. We develop a mathematical model of cost-consciousness, and we show that this model induces nonseparable preferences in all but the most extreme cases. We show that when outcome costs are distinct, cost-conscious electorates always exhibit both a weak Condorcet winner and a weak Condorcet loser. Finally, we show that preferences consistent with our model of cost-consciousness are rare in randomly generated electorates. We then discuss the implications of our work and suggest directions for further research.

For $\theta \in \left[0,2\pi \right)$ and $\lambda >1$, consider the matrix $h=\left(\genfrac{}{}{0ex}{}{\lambda }{0}\genfrac{}{}{0ex}{}{0}{0}\right)$ and the rotation matrix $R_{\theta }$. Let $W_n\left(\theta \right)$ denote some product of $m$ instances of $R_{\theta }$ and $n$ of $h$, with the condition $m\le ϵn$ ($0<ϵ<1$). We analyze the measure of the set of $\theta$ for which $\parallel W_n\left(\theta \right)\parallel \ge {\lambda }^{\delta n}$ ($0<\delta <1$). This can be regarded as a model problem for the Bochi–Fayad conjecture.

We define the concept of continuous $p$-frames ($cp$-frames) for Banach spaces, generalizing discrete $p$-frames. We prove that under certain conditions the direct sum of a finite number of $cp$-frames is again a $cp$-frame. We obtain equivalent conditions for duals of $cp$-Bessel mappings and show existence and uniqueness of duals of independent $cp$-frames. Lastly we discuss perturbation of these frames.

We provide a variety of results concerning the problem of determining maximal vectors $g$ such that the Diophantine system $Mx=g$ has no solution: conditions for the existence of $g$, conditions for the uniqueness of $g$, bounds on $g$, determining $g$ explicitly in several important special cases, constructions for $g$, and a reduction for $M$.

The celebrated Gauss–Bonnet formula has a nice generalization to surfaces with densities, in which both arclength and area are weighted by positive functions. Surfaces with densities, especially when arclength and area are weighted by the same factor, appear throughout mathematics, including probability theory and Perelman’s recent proof of the Poincaré conjecture.