arXiv subjects

Brian Wissman

Publications and source records attributed to Brian Wissman.

3 recordsLinked to original sources

arXiv2024

Families of numerical semigroups and a special case of the Huneke-Wiegand conjecture

The Huneke-Wiegand conjecture is a decades-long open question in commutative algebra. García-Sánchez and Leamer showed that a special case of this conjecture concerning numerical semigroup rings $\Bbbk[Γ]$ can be answered in the affirmative by locating certain arithmetic sequences within the numerical semigroup $Γ$. In this paper, we use their approach to prove the Huneke-Wiegand conjecture in the case where $Γ$ is generated by a generalized arithmetic sequence and showcase how visualizations can be leveraged to find the requisite arithmetic sequences.

math.AC↗

arXiv2017

Power domination and zero forcing

The power domination number arises from the monitoring of electrical networks and its determination is an important problem. Upper bounds for power domination numbers can be obtained by constructions. Lower bounds for the power domination number of several families of graphs are known, but they usually arise from specific properties of each family and the methods do not generalize. In this paper we exploit the relationship between power domination and zero forcing to obtain the first general lower bound for the power domination number. We apply this bound to obtain results for both the power domination of tensor products and the zero-forcing number of lexicographic products of graphs. We also establish results for the zero forcing number of tensor products and Cartesian products of graphs.

math.CO↗

arXiv2017

Minimal presentations of shifted numerical monoids

A numerical monoid is an additive submonoid of the non-negative integers. Given a numerical monoid $S$, consider the family of "shifted" monoids $M_n$ obtained by adding $n$ to each generator of $S$. In this paper, we examine minimal relations among the generators of $M_n$ when $n$ is sufficiently large, culminating in a description that is periodic in the shift parameter $n$. We explore several applications to computation, combinatorial commutative algebra, and factorization theory.

math.AC↗