arXiv subjects

Lance Bryant

Publications and source records attributed to Lance Bryant.

5 recordsLinked to original sources

arXiv2018

Curriculum Guidelines for Undergraduate Programs in Data Science

The Park City Math Institute (PCMI) 2016 Summer Undergraduate Faculty Program met for the purpose of composing guidelines for undergraduate programs in Data Science. The group consisted of 25 undergraduate faculty from a variety of institutions in the U.S., primarily from the disciplines of mathematics, statistics and computer science. These guidelines are meant to provide some structure for institutions planning for or revising a major in Data Science.

stat.OT↗

arXiv2014

Position Vectors of Numerical Semigroups

We provide a new way to represent numerical semigroups by showing that the position of every Apéry set of a numerical semigroup $S$ in the enumeration of the elements of $S$ is unique, and that $S$ can be re-constructed from this "position vector." We extend the discussion to more general objects called numerical sets, and show that there is a one-to-one correspondence between $m$-tuples of positive integers and the position vectors of numerical sets closed under addition by $m+1$. We consider the problem of determining which position vectors correspond to numerical semigroups.

math.AC↗

arXiv2012

Maximal Denumerant of a Numerical Semigroup With Embedding Dimension Less Than Four

Given a numerical semigroup $S = < a_1, a_2,..., a_t>$ and $s\in S$, we consider the factorization $s = c_1 a_1 + c_2 a_2 +... + c_t a_t$ where $c_i\ge0$. Such a factorization is {\em maximal} if $c_1+c_2+...+c_t$ is a maximum over all such factorizations of $s$. We show that the number of maximal factorizations, varying over the elements in $S$, is always bounded. Thus, we define $\dx(S)$ to be the maximum number of maximal factorizations of elements in $S$. We study maximal factorizations in depth when $S$ has embedding dimension less than four, and establish formulas for $\dx(S)$ in this case.

math.AC↗

arXiv2012

The Maximal Denumerant of a Numerical Semigroup

Given a numerical semigroup S = and n in S, we consider the factorization n = c_0 a_0 + c_1 a_1 + ... + c_t a_t where c_i >= 0. Such a factorization is maximal if c_0 + c_1 + ... + c_t is a maximum over all such factorizations of n. We provide an algorithm for computing the maximum number of maximal factorizations possible for an element in S, which is called the maximal denumerant of S. We also consider various cases that have connections to the Cohen-Macualay and Gorenstein properties of associated graded rings for which this algorithm simplifies.

math.AC↗

arXiv2009

Goto Numbers of a Numerical Semigroup Ring and the Gorensteiness of Associated Graded Rings

The Goto number of a parameter ideal Q in a Noetherian local ring (R,m) is the largest integer q such that Q : m^q is integral over Q. The Goto numbers of the monomial parameter ideals of R = k[[x^{a_1}, x^{a_2},..., x_{a_ν}]] are characterized using the semigroup of R. This helps in computing them for classes of numerical semigroup rings, as well as on a case-by-case basis. The minimal Goto number of R and its connection to other invariants is explored. Necessary and sufficient conditions for the associated graded rings of R and R/x^{a_1}R to be Gorenstein are also given, again using the semigroup of R.

math.AC↗