Developing a Mathematics Capstone Experience at The U.S. Air Force Academy

There are many details to consider when designing a capstone experience. Expectations of students, faculty workload, resources, logistics, and timing all play a part in the value of this endeavor. We discuss the experience of creating a research capstone experience from scratch including the components of our current course as well as lessons learned. Supplementary materials are available for this article. Go to the publisher’s online edition of PRIMUS to view the supplemental file

Minimal generating sets of ideals and torsion in the tensor product of I and (,R)I(\u27-1)

Let (R, m) be a commutative one-dimensional local Noetherian domain. Assume the integral closure R is a discrete valuation ring and the residue fields of R and R are the same. Let Rx be the maximal ideal of R and let I be a non-principal ideal of R. A question, motivated by the work of Auslander in the early 1960\u27s, is whether the torsion submodule of I\ \otimes\sb{R}\ I\sp{-1} is always non-zero. We investigate this question by examining the stronger condition \mu\sb{R}(I)\mu\sb{R}(I\sp{-1})\u3e\mu\sb{R}(II\sp{-1}) (where $\mu$ denotes the number of generators). We will show that if both m and I are generated by powers of x then this strict inequality holds whenever R has maximal embedding dimension, embedding dimension two or multiplicity at most seven. Without the assumption that m and I are generated by powers of x, we will prove the strict inequality holds whenever R has multiplicity at most four

Developing a Mathematics Capstone Experience at The U.S. Air Force Academy

There are many details to consider when designing a capstone experience. Expectations of students, faculty workload, resources, logistics, and timing all play a part in the value of this endeavor. We discuss the experience of creating a research capstone experience from scratch including the components of our current course as well as lessons learned. Supplementary materials are available for this article. Go to the publisher’s online edition of PRIMUS to view the supplemental file