We present necessary and sufficient conditions for the decomposition of the complete symmetric bipartite digraph into each of the orientations of a 4-cycle (in the cases for which such decompositions are not already known). We use these results to find optimal packings of the complete symmetric digraph with each of the orientations of a 4-cycle. Finally we give necessary and sufficient conditions for the existence of a decomposition of the complete symmetric digraph on v vertices with a hole of size w into each of the orientations of a 4-cycle.

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number of G. We study graphs having the same total domination number as their complements. In particular, we characterize the cubic graphs having this property. Also we characterize such graphs with total domination numbers equal to two or three, and we determine properties of the ones with larger total domination numbers.

Let Sn be the set of all permutations on [n]:= {1, 2,..., n}. We denote by κn the smallest cardinality of a subset A of Sn+1 that covers Sn, in the sense that each π ∈ Sn may be found as an order-isomorphic subsequence of some π′ in A. What are general upper bounds on κn? If we randomly select νn elements of Sn+1, when does the probability that they cover Sn transition from 0 to 1? Can we provide a fine-magnification analysis that provides the "probability of coverage" when νn is around the level given by the phase transition? In this paper we answer these questions and raise others.

We discuss methods for a priori selection of parameters to be estimated in inverse problem formulations (such as Maximum Likelihood, Ordinary and Generalized Least Squares) for dynamical systems with numerous state variables and an even larger number of parameters. We illustrate the ideas with an in-host model for HIV dynamics which has been successfully validated with clinical data and used for prediction and a model for the reaction of the cardiovascular system to an ergometric workload.

The nonlinear Schrödinger (NLS) equation is a classical field equation that describes weakly nonlinear wave-packets in one-dimensional physical systems. It is in a class of nonlinear partial differential equations (PDEs) that pertain to several physical and biological systems. In this project we apply a pseudo-spectral solution-estimation method to a modified version of the NLS equation as a means of searching for solutions that are solitons, where a soliton is a self-reinforcing solitary wave that maintains its shape over time. We use the pseudo-spectral method to determine whether cardiac action potential states, which are perturbed solutions to the Fitzhugh-Nagumo nonlinear PDE, create soliton-like solutions. We then use symmetry group properties of the NLS equation to explore these solutions and find new ones.

There are eight orientations of the complete graph on three vertices with a pendant edge, K3 ∪ {e}. Two of these are 3-circuits with a pendant arc and the other six are transitive triples with a pendant arc. Necessary and sufficient conditions are given for decompositions, packings, and coverings of the complete digraph with each of these eight orientations of K3 ∪ {e}.

Instant Insanity II is a 4 by 4 sliding tile puzzle designed by Philip Orbanes. The packaging indicates that there is a unique solution to the puzzle, up to rotations of the columns and permutations on the rows. However, a recent paper by Richmond and Young shows that there are in fact two solutions to the puzzle. This paper presents several attempts at “fixing” the puzzle to guarantee a unique solution. Of these, the only one that guaranteed a unique solution was removing a color to create a 3 by 3 puzzle.

In a 2011 paper by Beeler and Hoilman, the traditional game of peg solitaire is generalized to graphs in the combinatorial sense. One of the important open problems was to classify solvable trees. In this paper, we give necessary and sufficient conditions for the solvability for all trees with diameter four. We also give the maximum number of pegs that can be left on such a graph under the restriction that we jump whenever possible.

In our recent work in computational biology, our approach to modeling protein structures requires that each vertex be weighted by a vector of weights. This motivates the results we present here. We show that many graphical invariants can be generalized to vertex weights by replacing the maximum(minimum) cardinality of a vertex set with a sum over the vertex weights. We also generalize the Laplacian of a vertex-weighted graph to include vector-weighted vertices, including generalizing some spectral results.

The difference between two proportions, referred to as a risk difference, is a useful measure of effect size in studies where the response variable is dichotomous. Confidence interval methods based on a varying coefficient model are proposed for combining and comparing risk differences from multi-study between-subjects or within-subjects designs. The proposed methods are new alternatives to the popular constant coefficient and random coefficient methods. The proposed varying coefficient methods do not require the constant coefficient assumption of effect size homogeneity, nor do they require the random coefficient assumption that the risk differences from the selected studies represent a random sample from a normally distributed superpopulation of risk differences. The proposed varying coefficient methods are shown to have excellent finite-sample performance characteristics under realistic conditions.