Recently, nearly complete intersection ideals were defined by Boocher and Seiner to establish lower bounds on Betti numbers for monomial ideals (arXiv:1706.09866). Stone and Miller then characterized nearly complete intersections using the theory of …
We introduce a new homological dimension for finitely generated modules over a commutative local ring $R$, which is based on a complex derived from a free resolution $L$ of the residue field of $R$, and called $L$-dimension. We prove several …
Complete these exercises several months in advance of your anticipated research project with undergraduates. For example, if you are thinking of working with students over a summer, consider working on them between the fall and spring semesters.
The divisor sequence of an irreducible element (_atom_) $a$ of a reduced monoid $H$ is the sequence $(s_n)_{n \in \mathbb{N}}$ where, for each positive integer $n$, $s_n$ denotes the number of distinct irreducible divisors of $a^n$. In this work we …
We introduce a recursive decomposition algorithm for the Betti diagram of a complete intersection using the diagram of a complete intersection defined by a subset of the original generators. This alternative algorithm is the main tool that we use to …
We investigate decompositions of Betti diagrams over a polynomial ring within the framework of Boij-Soederberg theory. That is, given a Betti diagram, we determine if it is possible to decompose it into the Betti diagrams of complete intersections. …
We study the maximum likelihood degree (ML degree) of toric varieties, known as discrete exponential models in statistics. By introducing scaling coefficients to the monomial parameterization of the toric variety, one can change the ML degree. We …
We define three new pebbling parameters of a connected graph $ G $, the $ r $-, $ g $-, and $ u $-critical pebbling numbers. Together with the pebbling number, the optimal pebbling number, the number of vertices $ n $ and the diameter $ d $ of the …
In introductory calculus, most optimization problems begin with a continuous function. Departing from this trend, we describe a plausible micro-economic scenario with a discontinuous cost function. The case of multiple workers requires a more refined …
We investigate decompositions of Betti diagrams over a polynomial ring within the framework of Boij-Soederberg theory. That is, given a Betti diagram, we decompose it into pure diagrams. Relaxing the requirement that the degree sequences in such pure …