Publications & Preprints
Peer-reviewed journal articles, conference proceedings, and mathematical research preprints on nonnegative matrix theory, polynomials, and inverse eigenvalue problems.
📚 Journal Articles & Conference Papers
Polynomials that preserve nonnegative matrices of order two
Abstract: A known characterization for entire functions that preserve all nonnegative matrices of order two is shown to characterize polynomials that preserve nonnegative matrices of order two. Equivalent conditions are derived and used to prove that P₃ ⊂ P₂, which was previously unknown. A new characterization is given for polynomials that preserve nonnegative circulant matrices of order two.
Polynomials that preserve nonnegative matrices
Abstract: In further pursuit of a solution to the celebrated nonnegative inverse eigenvalue problem, Loewy and London (1978/1979) posed the problem of characterizing all polynomials that preserve all nonnegative matrices of a fixed order. If Pₙ denotes the set of all polynomials that preserve all n-by-n nonnegative matrices, then it is clear that polynomials with nonnegative coefficients belong to Pₙ. However, it is known that Pₙ contains polynomials with negative entries. In this work, novel results for Pₙ with respect to the coefficients of the polynomials belonging to Pₙ are established. Along the way, a generalization for the even-part and odd-part are given and shown to be equivalent to another construction that appeared in the literature. Implications for further research are discussed.
Would gamers collaborate given the opportunity
Abstract: Understanding player preference and behavioral tendency in the presence of collaborative opportunities is fundamental to building cooperative video games. This paper presents an ongoing study that approaches the subject by posing a simple question: when opportunities are provided, would players choose to cooperate? The work analyzes existing well-established cooperative game design patterns and identifies effective attributes of game mechanics that are characterized by the patterns. Small multiplayer games that focused on each of the attributes are built where in each case the players have the options of collaborating or completing the tasks individually.
📝 Research Preprints & Reports
The NIEP is solvable by reality and finitely many polynomial inequalities
Abstract: The nonnegative inverse eigenvalue problem (NIEP) asks for necessary and sufficient conditions for a list of complex numbers to be the spectrum of a nonnegative matrix. In this paper, the NIEP is shown to be solvable by the reality condition (spectrum equal to its conjugate) as well as by a finite union and intersection of polynomial inequalities. It is also shown that the symmetric NIEP (SNIEP) and real NIEP (RNIEP) form semi-algebraic sets and can therefore be solved just by a finite union and intersection of polynomial inequalities. An overview of ideas are given in how tools from real algebraic geometry may be applied to the NIEP and related sub-problems.
Properties of the cone of polynomials of fixed degree that preserve nonnegative matrices
Abstract: As was detailed by Loewy and London (1978/79), the cone of polynomials that preserve the nonnegativity of matrices may play an important role in the solution to the nonnegative inverse eigenvalue problem. In this paper, we start by showing the cone generated by polynomials of degree greater than or equal to 2n that preserve nonnegative matrices of order n is non-polyhedral. Next, a question posed by Loewy (2023) about how negative the center term can be in a degree 2n polynomial is answered. We extend this to show that a polynomial that preserves nonnegative matrices of order n can have its largest term, in absolute value, be arbitrarily negative with the remaining coefficients being one. We conclude by exploring properties of the measure of the cone when restricted to the unit sphere and by proving initial bounds of that volume.
Polynomials that preserve nonnegative monomial matrices
Abstract: A recently-established necessary condition for polynomials that preserve the class of entrywise nonnegative matrices of a fixed order is shown to be necessary and sufficient for the class of nonnegative monomial matrices. Along the way, we provide a formula for computing an arbitrary power of a monomial matrix and a formula for computing the polynomial of a nonnegative monomial matrix.