23 papers
Length density and numerical semigroups
Cole Brower, Scott Chapman, Travis Kulhanek +4
Length density is a recently introduced factorization invariant, assigned to each element of a cancellative commutative atomic semigroup , that measures how far the set of f…
Factorization length distribution for affine semigroups IV: a geometric approach to weighted factorization lengths in three-generator numerical semigroups
Stephan Ramon Garcia, Christopher O'Neill, Gabe Udell
For numerical semigroups with three generators, we study the asymptotic behavior of weighted factorization lengths, that is, linear functionals of the coefficients in the factoriza…
The fundamental theorem of finite fields: a proof from first principles
Anastasia Chavez, Christopher O'Neill
A mathematics student's first introduction to the fundamental theorem of finite fields (FTFF) often occurs in an advanced abstract algebra course and invokes the power of Galois th…
On length densities
Scott T. Chapman, Christopher O'Neill, Vadim Ponomarenko
For a commutative cancellative monoid , we introduce the notion of the length density of both a nonunit , denoted , and the entire monoid , denoted $\…
Factorization length distribution for affine semigroups III: modular equidistribution for numerical semigroups with arbitrarily many generators
Stephan Ramon Garcia, Mohamed Omar, Christopher O'Neill +1
For numerical semigroups with a specified list of (not necessarily minimal) generators, we describe the asymptotic distribution of factorization lengths with respect to an arbitrar…
Weighted Means of B-Splines, Positivity of Divided Differences, and Complete Homogeneous Symmetric Polynomials
Albrecht Boettcher, Stephan Ramon Garcia, Mohamed Omar +1
We employ the fact certain divided differences can be written as weighted means of B-splines and hence are positive. These divided differences include the complete homogeneous symm…