A knight’s tour is a sequence of knight’s moves such that each square on the board is visited exactly once. An Aztec diamond is a square board of size $2n$ where triangular regions of side length $n-1$ have been removed from all four corners.

We show that the existence of knight’s tours on Aztec diamonds cannot be proved inductively via smaller Aztec diamonds, and explain why a divide-and-conquer approach is also not promising. We then describe two algorithms that aim to efficiently find knight’s tours on Aztec diamonds. The first is based on random walks, a straightforward but limited technique that yielded tours on Aztec diamonds for all $n\ne 22$ apart from $n=17,21$. The second is a path-conversion algorithm that finds a solution for all $n\le 100$. We then apply the path-conversion algorithm to random graphs to test the robustness of our algorithm. Online supplements provide source code, output and more details about these algorithms.

We have created and analyzed a model for kleptoparasitic interactions when individuals decide on the level of aggression in which they want to engage in the contest over a resource item. The more aggressive each individual is relative to an opponent, the higher are the chances of winning the item, but also the higher is the cost of the interaction for that individual. We consider a general class of cost functions and show that for any parameter values, i.e., for any maximal potential level of aggression of the individuals, any value of the resource and any type of the cost function, there is always a unique Nash equilibrium. We identify four possible kinds of Nash equilibria and give precise conditions for when they occur. We find that nonaggressive behavior is not a Nash equilibrium even when the cost function is such that aggressive behavior yields lower payoffs than avoiding the conflict altogether.

We provide results for the exponential dominating numbers and total exponential dominating numbers of a family of triangular grid graphs. We then prove inequalities for these numbers and compare them with inequalities that hold more generally for exponential dominating numbers of graphs.

Given a graph $G$, the tree cover number of the graph, denoted $T\left(G\right)$, is the minimum number of vertex disjoint simple trees occurring as induced subgraphs that cover all the vertices of G. This graph parameter was introduced in 2011 as a tool for studying the maximum positive semidefinite nullity of a graph, and little is known about it. It is conjectured that the tree cover number of a graph is at most the maximum positive semidefinite nullity of the graph.

In this paper, we establish bounds on the tree cover number of a graph, characterize when an edge is required to be in some tree of a minimum tree cover, and show that the tree cover number of the $d$-dimensional hypercube is 2 for all $d\ge 2$.

A matrix completion problem asks whether a partial matrix composed of specified and unspecified entries can be completed to satisfy a given property. This work focuses on determining which patterns of specified and unspecified entries correspond to partial matrices that can be completed to solve three different matrix equations. We approach this problem with two techniques: converting the matrix equations into linear equations and examining bases for the solution spaces of the matrix equations. We determine whether a particular pattern can be written as a linear combination of the basis elements. This work classifies patterns as admissible or inadmissible based on the ability of their corresponding partial matrices to be completed to satisfy the matrix equation. Our results present a partial or complete characterization of the admissibility of patterns for three homogeneous linear matrix equations.

The problem of characterizing maximal non-Hamiltonian graphs may be naturally extended to characterizing graphs that are maximal with respect to nontraceability and beyond that to $t$-path traceability. We show how $t$-path traceability behaves with respect to disjoint union of graphs and the join with a complete graph. Our main result is a decomposition theorem that reduces the problem of characterizing maximal $t$-path traceable graphs to characterizing those that have no universal vertex. We generalize a construction of maximal nontraceable graphs by Zelinka to $t$-path traceable graphs.

Boolean (BL) systems and coupled ordinary differential equations (ODEs) are popular models for studying biological networks. BL systems can be set up without detailed reaction mechanisms and rate constants and provide qualitatively useful information, but they cannot capture the continuous dynamics of biological systems. On the other hand, ODEs are able to capture the continuous dynamic features of biological networks and provide more information on how the activities of components depend on other components and parameter values. However, a useful coupled ODE model requires details about interactions and parameter values. The introduction of the relationships between the two types of models will enable us to leverage their advantages and better understand the target network systems. In this paper, we investigate the relations between the conditions of the existence of limit cycles in ODE networks and their homologous discrete systems. We prove that for a single feedback loop, as long as the corresponding governing functions of the homologous continuous and discrete systems have the same upper and lower asymptotes, the limit cycle borne via Hopf bifurcation corresponds to the cycle of the discrete system. However, for some coupled feedback loops, besides having the same upper and lower asymptotes, parameters such as the decay rates also play crucial roles.

In this paper we study the brachistochrone problem in an inverse-square gravitational field on the unit disk. We show that the time-optimal solutions consist of either smooth strong solutions to the Euler–Lagrange equation or weak solutions formed by appropriately patched together strong solutions. This combination of weak and strong solutions completely foliates the unit disk. We also consider the problem on annular domains and show that the time-optimal paths foliate the annulus. These foliations on the annular domains converge to the foliation on the unit disk in the limit of vanishing inner radius.

This paper presents the steady-state solutions and traveling wave solutions for a spatially distributed PDE version of the spruce budworm model. The ODE (undistributed) model has been used in practical scenarios to model the outbreaks of the spruce budworm in forest environments, alongside the study of concepts involving fixed points and bifurcations in introductory differential equations courses. This study represents the spread of an outbreak from one end of a forest to the other. Numerical simulations are conducted using spectral methods.

For a given integer $k$, general necessary and sufficient conditions for the existence of integer solutions to an equation of the form $x^2+y^2=z^2-k$ are given. It is shown that when there is a solution, there are infinitely many solutions. An elementary method for finding the solutions, when they exist, is described.

In a recent paper, Frechette and Locus examined and found expressions for the infinite product $D_m\left(q\right):={\prod }_{t=1}^{\infty }\left(1-q^{mt}\right)∕\left(1-q^t\right)$ in terms of products of $q$-series of the Rogers–Ramanujan type coming from Hall–Littlewood polynomials, when $m\equiv 0,1,2\left(mod4\right)$. These $q$-series were originally discovered in 2014 by Griffin, Ono, and Warnaar in their work on the framework of the Rogers–Ramanujan identities. Extending this framework, Rains and Warnaar also recently discovered more $q$-series and their corresponding infinite products. Frechette and Locus left open the case where $m\equiv 3\left(mod4\right)$. Here we find such an expression for the infinite products for $m\equiv 3\left(mod4\right)$ by making use of the new $q$-series obtained by Rains and Warnaar.