EDUCATION

• Ph.D., operations research, Princeton University, Princeton, New Jersey, 2001.
• M.A., operations research, Princeton University, Princeton, New Jersey, 1999.
• M.S., mathematics, Texas A&M University, College Station, Texas, 1997.
• B.S., mathematics, Samford University, Birmingham, Alabama, 1994.

COURSES TAUGHT

• High School:
  • C++ Programming
• Undergraduate:
  • Contemporary Mathematics / Math for the Liberal Arts
  • College Algebra / Precalculus
  • Elementary Statistics
  • Calculus I, II, III
  • Integrated Calculus II
  • Discrete Mathematics
  • Modeling Seminar
  • Linear Algebra
  • Differential Equations
  • Operations Management
  • Optimization
  • Probability Theory
  • Mathematical Modeling
  • Senior Seminar
• Graduate:
  • Management Science

ARTICLES

1. "Some Fixed-Point Results for the Dynamic Assignment Problem," with Warren Powell. Annals of Operations Research, 124: 15-33, 2003.
   • In the process of testing the adaptive approximation algorithm described in the following paper, we discovered that for certain problem classes, when the optimal solution is input to our algorithm, the algorithm outputs the optimal solution in the next iteration. This means that the optimal solution is a fixed point under our algorithm for these problem classes. This paper contains the proofs of this for two of these classes (provided the optimal solution is unique).
2. "The Dynamic Assignment Problem," with Warren Powell. Transportation Science, 38 (4): 399-419, 2004. (c) Informs
   • In this paper we introduce the dynamic assignment problem, which involves the solution of assignment problems over time with evolving information on the availability of resources and tasks, contributions for assigning them, and restrictions on the availability of assignments. The paper includes 1) a formal model of the problem, 2) an adaptive approximation strategy for problems with relatively small state spaces based on the values of resources, tasks, and resource-task pairs that significantly outperforms greedy heuristics, and 3) an extension of this strategy, based on aggregations of resources and tasks, to problems with larger state spaces. (This paper represents the bulk of the work from my Ph. D. dissertation.)
3. "The Humble Sum of Remainders Function." Mathematics Magazine, 78 (4): 300-305, 2005.
   • Given an integer n, define the sum of remainders function such that when n is input to the function, the output is the sum of the the remainders obtained by dividing n successively by 1, 2, ..., n. This paper describes two uses of the sum of remainders function: 1) It provides a simple alternative characterization of perfect numbers, and 2) together with the more famous sum of divisors function, it helps give an expression for the sums of powers of the first n positive integers.
4. "The k-Binomial Transforms and the Hankel Transform," with Laura Steil. Journal of Integer Sequences, 9 (1): Article 06.1.1, 2006.
   • We give a new proof of the invariance of the Hankel transform under the binomial transform of a sequence. Our method of proof leads to three variations of the binomial transform; we call these the k-binomial transforms. We give a simple means of constructing these transforms via a triangle of numbers. We show how the exponential generating function of a sequence changes after our transforms are applied, and we use this to prove that several sequences in the On-Line Encyclopedia of Integer Sequences are related via our transforms. In the process, we prove three conjectures in the OEIS. Addressing a question of Layman, we then show that the Hankel transform of a sequence is invariant under one of our transforms, and we show how the Hankel transform changes after the other two transforms are applied. Finally, we use these results to determine the Hankel transforms of several integer sequences. (This is an extension of Laura's undergraduate senior thesis; she is now a graduate student at the University of Kentucky.)
5. "The Euler-Maclaurin Formula and Sums of Powers." Mathematics Magazine, 79 (1): 61-65, 2006.
   • This paper uses the Euler-Maclaurin formula for approximating a sum by an integral to prove that the limiting value of a certain expression related to the power sum is 1/(e-1).
6. "Some Inversion Formulas for Sums of Quotients," with Natalio Guersenzvaig. Crux Mathematicorum, 32 (1): 39-43, 2006.
   • This paper proves some identities related to summing the quotients of an integer n from two different directions. The results generalize a problem posed in a previous issue of Crux.
7. "Fibonacci Identities via the Determinant Sum Property." College Mathematics Journal, 37 (4): 286-289, 2006.
   • We use the sum property for determinants of matrices to give a three-stage proof of an identity involving Fibonacci numbers. Cassini's and d'Ocagne's Fibonacci identities are obtained at the ends of stages one and two, respectively. Catalan's Fibonacci identity is also a special case.
8. "Quadratic Residues and the Frobenius Coin Problem." Mathematics Magazine, 80 (1): 64-67, 2007.
   • An odd prime p has (p-1)/2 quadratic residues mod p, and for relatively prime p and q there are (p-1)(q-1)/2 non-representable Frobenius numbers. We show that there is a relationship between the non-representable Frobenius numbers of p and q and the quadratic residues of p that accounts for the presence of (p-1)/2 in both expressions.
9. "Combinatorial Sums and Finite Differences." Discrete Mathematics, 307 (24): 3130-3146, 2007.
   • This paper shows how the binomial transform of a sequence is related to the binomial transform of the finite difference of the sequence. This relationship can be used to give fairly quick derivations of some known binomial sum identities. The approach is then modified so that it applies to alternating binomial sums, aerated binomial sums, and to sums involving signed and unsigned Stirling numbers of the first kind. The paper concludes with a generalization of the results for binomial sums and unsigned Stirling numbers of the first kind.
10. "Counting Functions and Finite Differences," with Natalio Guersenzvaig. Integers, 7: Article A59, 2007.
    • Any increasing function p(d) on the natural numbers has an associated counting function pi(n) that yields the number of inputs d for which p(d) <= n. In this article we derive three formulas that relate a sequence to its finite difference sequence by way of counting functions and the technique of summation by parts. We demonstrate our formulas by using them to produce several identities for Fibonacci numbers and binomial coefficients.
11. "Symmetric Functions, Pascal Matrices, and Stirling Matrices," with Andy Zimmer. Linear Algebra and Its Applications, 428 (4): 1127-1134, 2008.
    • We consider lower-triangular matrices consisting of symmetric polynomials, and we show how to factorize and invert them. Since binomial coefficients and Stirling numbers can be represented in terms of symmetric polynomials, these results contain factorizations and inverses of Pascal and Stirling matrices as special cases. This work generalizes that of several other authors on Pascal and Stirling matrices. (Andy did his part of the work as an undergraduate at the University of Puget Sound.)
12. "A Generalized Recurrence for Bell Numbers." Journal of Integer Sequences, 11 (2): Article 08.2.5, 2008.
    • We give a combinatorial proof of a result that generalizes the two most well-known expressions for Bell numbers.
13. "Staircase Rook Polynomials and Cayley's Game of Mousetrap." European Journal of Combinatorics, to appear.
    • We determine the rook polynomials for two special classes of card decks arising in the game of Mousetrap, introduced by Arthur Cayley 150 years ago.

OTHER PUBLICATIONS

• Solution to Problem 2826, with P. Bornsztein, et al. Crux Mathematicorum, 30 (3): 178-179, April 2004.
• Solution to Problem 2834, with C. Curtis. Crux Mathematicorum, 30 (3): 186, April 2004.
• Solution to Problem 1, 32nd Austrian Mathematics Olympiad, Crux Mathematicorum, 31 (6): 376, October 2005.
• Problem 1737, Mathematics Magazine, 79 (1): 67, February 2006.