By applying an algorithm of Stallings regarding separability of elements in a free group, we give an alternative approach to that of Osborne and Zieschang in describing all primitive elements in the free group of rank 2. As a result, we give a proof of a classical result of Nielsen, used by Osborne and Zieschang in their work, that the only automorphisms of $F_2$ that act trivially on the abelianization are those defined by conjugation. Finally, we compute the probability that a Whitehead graph in rank 2 contains a cut vertex. We show that this probability is approximately $1∕{\ell }^2$, where $\ell$ is the number of edges in the graph.

We provide examples of perimeter-minimizing tilings of the plane by convex pentagons and examples of perimeter-minimizing tilings of certain small flat tori.

We apply discrete time optimal control theory to the mathematical modeling of pest control. Two scenarios: biological control and the combination of pesticide and biological control are considered. The goal is maximizing the “valuable” population, minimizing the pest population and the cost to apply the control strategies. Using the extension of Pontryagin’s maximum principle to discrete system, the adjoint systems and the characterization of the optimal pest controls are derived. Numerical simulations of various cases are provided to show the effectiveness of our methods.

Using the double-cut-and-join (DCJ) model for genome rearrangement we use combinatorial techniques to analyze the distribution of genomes under DCJ distance. We present an exponential generating function for the number of genomes that are maximally distant from a given genome and provide a formula for the number of genomes that are any given distance from an arbitrary starting genome.

Cellular motility is an important function in many cellular processes. Among the key players in cellular movement are transmembrane receptor proteins called integrins. Through the development of a mathematical model we investigate the dynamic relationship between integrins and other molecules known to contribute to initial cellular movement such as extracellular ligands and intracellular adhesion proteins called talin. Gillespie’s stochastic simulation algorithm was used for numerical analysis of the model. From our stochastic simulation, we found that most activity in our system happens within the first five seconds. Additionally we found that while ligand-integrin-talin complexes form fairly early in the simulation, they soon disassociate into ligand-integrin or integrin-talin complexes, suggesting that the former tertiary complex is less stable than the latter two complexes. We also discuss our theoretical analysis of the model and share results from our sensitivity analysis, using standardized regression coefficients as measures of output sensitivity to input parameters.

Following a procedure outlined by Kang, we view the generalized eigenspaces, known as root spaces, of the infinite dimensional Kac–Moody algebra $E_{10}$ as generalized eigenspaces for representations of the finite dimensional special linear algebra $A_9$. Then, using the combinatorial representation theory of the special linear Lie algebras, we determine the dimensions of certain root spaces in $E_{10}$.

François Jaeger presented the two-variable Kauffman polynomial of an unoriented link $L$ as a weighted sum of HOMFLY-PT polynomials of oriented links associated with $L$. Murakami, Ohtsuki and Yamada (MOY) used planar graphs and a recursive evaluation of these graphs to construct a state model for the $sl\left(n\right)$-link invariant (a one-variable specialization of the HOMFLY-PT polynomial). We apply the MOY framework to Jaeger’s work, and construct a state summation model for the $SO\left(2n\right)$ Kauffman polynomial.

In this paper, we seek to understand the behavior of dynamical systems that are perturbed by a parameter that changes discretely in time. If we impose certain conditions, we can study certain embedded systems within a hybrid system as time-homogeneous Markov processes. In particular, we prove the existence of invariant measures for each embedded system and relate the invariant measures for the various systems through the flow. We calculate these invariant measures explicitly in several illustrative examples.