Caleb M. Shor, Ph.D.

Box H-5156

Western New England University

1215 Wilbraham Road

Springfield, MA 01119

Hello! I am a Professor of Mathematics at Western New England University in Springfield, Massachusetts, where I teach in the undergraduate and masters programs. Each summer, I serve as Director of PROMYS for Teachers at Boston University.

My research interests over the years have been in number theory, algebraic geometry, and coding theory. In recent years, I have focused more on problems involving numerical semigroups. Check out my Publications page for more information.

I have grown increasingly interested in using WeBWorK with the college courses that I teach. I have created problem sets for Calculus III, Differential Equations, and a course that we call Vector Calculus & Fourier Series. I have also created course notes for these courses. I am making all of this material available in Github. Check out my Repositories (or, currently, my repository) to see what’s available.

This site is new as of May 2024. I hope to add more content/information soon.

If you want to get in touch, please email me at caleb.shor@wne.edu.

selected publications

Characterizations of numerical semigroup complements via Apéry sets

T. Alden Gassert, and Caleb M. Shor

Semigroup Forum, 2019

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In this paper, we generalize the work of Tuenter to give an identity which completely characterizes the complement of a numerical semigroup in terms of its Apéry sets. Using this result, we compute the \(m\)th power Sylvester and alternating Sylvester sums for free numerical semigroups. Explicit formulas are given for small \(m\).
@article{GassertShor19, author = {Gassert, T. Alden and Shor, Caleb M.}, title = {Characterizations of numerical semigroup complements via {A}p{\'e}ry sets}, journal = {Semigroup Forum}, year = {2019}, volume = {98}, number = {1}, pages = {31--47}, issn = {1432-2137}, doi = {10.1007/s00233-018-9935-4}, url = {https://doi.org/10.1007/s00233-018-9935-4}, }
Equidistribution of numerical semigroup gaps modulo \(m \)

Caleb M. Shor

Discrete Mathematics, 2022

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For a positive integer \(m\), a finite set of integers is said to be equidistributed modulo \(m \) if the set contains an equal number of elements in each congruence class modulo \(m \). In this paper, we consider the problem of determining when the set of gaps of a numerical semigroup \(S \) is equidistributed modulo \(m \). Of particular interest is the case when the nonzero elements of an Apéry set of \(S \) form an arithmetic sequence. We explicitly describe such numerical semigroups \(S\) and determine conditions for which the sets of gaps of these numerical semigroups are equidistributed modulo \(m \).
@article{Shor22, author = {Shor, Caleb M.}, title = {Equidistribution of numerical semigroup gaps modulo \( m \)}, journal = {Discrete Mathematics}, volume = {345}, number = {10}, year = {2022}, url = {https://doi.org/10.1016/j.disc.2022.112995}, doi = {10.1016/j.disc.2022.112995}, }
On the residues of rounded fractions with a common numerator

N. Dent, and C. Shor

Journal of Integer Sequences, 2024

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The proportion of integers \[ ⌊n/1⌋, ⌊n/2⌋, …, ⌊n/n⌋\]that are odd is asymptotically \( \log2 \). If instead of the floor function, one uses the nearest-integer function, the proportion drops to \( \pi/2-1 \). In this work, we prove these facts and, more generally, give an integral formula for the proportion of \[ ⌊(n-ν)/1+α⌋, ⌊(n-ν)/2+α⌋, …, ⌊(n-ν)/n+α⌋\]that are congruent to \(r \) modulo \(m \). We conclude by linking this problem to the Dirichlet divisor and Gauss circle problems, and counting lattice points in other quadratic regions.
@article{DentShor2024, author = {Dent, N. and Shor, C.}, title = {On the residues of rounded fractions with a common numerator}, journal = {Journal of Integer Sequences}, volume = {27}, year = {2024}, pages = {Article 24.2.5}, }