Navigation: main courses, side dishes, dessert, aged wine.

Additive Number/Group Theory (MSC Classification 20D60/20M14/11P99):

Polynomial Irreducibility via Shifts (with R. Lycan). To appear in Involve.

Length Density and Numerical Semigroups (with C. Brower, S. Chapman, T. Kulhanek, J. McDonough, C. O'Neill, and V. Pavlyuk). Combinatorial and Additive Number Theory: CANT 2021, Springer Proceedings in Mathematics and Statistics 2022.

Factorization of Perplex Integers American Mathematical Monthly 129 (6) 2022, pp. 576-581.

On Length Densities (with C. Chapman and C. O'Neill). Forum Mathematicum 34 (2) 2022, pp. 293-306.

Complementary Numerical Sets (with M. Guhl, J. Juarez, R. Rechkin, and D. Singhal). INTEGERS 22 (#A17) 2022, pp. 1-13.

Squarefree Divisor Complexes of Certain Numerical Semigroup Elements (with J. Autry, P. Graves, J. Loucks, C. O'Neill, and S. Yih). Involve 14 (1) 2021, pp. 1-9.

Elasticity in Apery Sets (with J. Autry, T. Gomes, C. O'Neill). American Mathematical Monthly 127 (8) 2020, pp. 744-749.

On Numerical Semigroups with Almost-Maximal Genus (with J. Arroyo, J. Autry, C. Crandall, J. Lefler). The PUMP Journal of Undergraduate Research (3) 2020, pp. 62-67.

Membership and Elasticity in Certain Affine Monoids (with J. Autry). Pi Mu Epsilon journal 15 (1) 2019.

Augmented Hilbert Series of Numerical Semigroups (with J. Glenn, C. O'Neill, and B. Sepanski). INTEGERS: The Electronic Journal of Number Theory (19) 2019, #A32.

Numerical Semigroups on Compound Sequences (with C. Kiers and C. O'Neill). Communications in Algebra 44 (9) 2016, pp. 3842-3852.

Arithmetic of Congruence Monoids (with A. Fujiwara, J. Gibson, M. Jenssen, D. Montealegre, and A. Tenzer). Communications in Algebra 44 (8) 2016, pp. 3407-3421.

On the Set of Catenary Degrees of Finitely Generated Cancellative Commutative Monoids (with C. O'Neill, R. Tate, and G. Webb). International Journal of Algebra and Computation 26 (3) 2016, pp. 565-576.

Arithmetic of Semigroup Semirings Ukrainian Mathematical Journal 67 (2) 2015, pp. 213-229.

Accepted Elasticity in Local Arithmetic Congruence Monoids (with L. Crawford, J. Steinberg, M. Williams). Results in Mathematics 66 (1) 2014, pp. 227-245.

Arithmetic-Progression-Weighted Subsequence Sums (with D. Grynkiewicz, A. Philipp). Israel Math Journal 193 (1) 2013, pp. 359-398.

Irreducible Factorization Lengths and the Elasticity Problem Within N (with M. Jenssen, D. Montealegre). American Mathematical Monthly 120 (4) 2013, pp. 322-328.

Two Semigroup Elements Can Commute With Any Positive Rational Probability (with N. Selinski). College Mathematics Journal 43 (4) 2012, pp. 334-336.

Diversity in Inside Factorial Monoids (with U. Krause, J. Maney). Czechoslovak Mathematics Journal 62 (3) 2012, pp. 811-827.

Diversity in Monoids (with J. Maney). Czechoslovak Mathematics Journal 62 (3) 2012, pp. 795-809.

The Multi-Dimensional Frobenius Problem (with J. Amos, I. Pascu, E. Trevino, Y. Zhang). Involve 4 (2) 2011, pp. 187-197.

On Image Sets of Integer-Valued Polynomials (with S. Chapman). Journal of Algebra (348) 2011, pp. 350-353.

Number Theory of Matrix Semigroups (with D. Adams, R. Ardila, N. Baeth, D. Hannasch, A. Kosh, H. McCarthy, R. Rosenbaum). Journal of Linear Algebra and its Applications (434) 2011, pp. 694-711.

Determining Integer-Valued Polynomials From Their Image Proceedings of the CIRM Meetings 2 (2) 2010, pp. 51-52.

Distinct Solution to a Linear Congruence (with D. Adams). Involve 3 (3) 2010, pp. 345-347.

Numerical Semigroups From Open Intervals (with R. Rosenbaum). Involve 3 (3) 2010, pp. 341-344.

Bifurcus Semigroups and Rings (with D. Adams, R. Ardila, D. Hannasch, A. Kosh, H. McCarthy, R. Rosenbaum). Involve 2 (3) 2009, pp. 351-356.

Invariant Polynomials and Minimal Zero Sequences (with B. Finklea, T. Moore, Z. Turner), Involve 1 (2) 2008, pp. 159-165. Also available are the associated big table (in Excel format), and software.

The Frobenius Number of Geometric Sequences (with D. Ong). INTEGERS: The Electronic Journal of Number Theory (8) 2008, #A33.

Robert Gilmer's Contributions to the Theory of Integer-Valued Polynomials (with S. Chapman, W. Smith) "Multiplicative Ideal Theory in Commutative Algebra", Springer, New York, 2006, pp. 109-122.

The Extraction Degree of Cale Monoids (with K. Cervello, D. Terry, L. Zhu), Semigroup Forum 72 (1) 2006, pp. 149-158.

The Catenary and Tame Degree in Finitely Generated Commutative Cancellative Monoids (with S. Chapman, P.A. Garcia-Sanchez, D. Llena, J.C. Rosales), Manuscripta Mathematica (120) 2006, pp. 253-264.

Minimal Zero Sequences of Finite Cyclic Groups Integers: The Electronic Journal of Combinatorial Number Theory (4) 2004, article A24.

Various Other Papers:

Wolstenholme's Congruence for Lucas sequences (with R. Lycan, J.P. Zendejas). Under review.

Construction of an Endgame Rulebook for Sylver Coinage Using Trees of Numerical Semigroups (with M. Bras-Amoros, G. Moskowitz). Under review.

A Further Self-Improvement of the Cauchy-Schwarz Inequality (with R. Farhadian, H. Jafari). To appear in the American Mathematical Monthly.

A Simple Self-Improvement of the Cauchy-Schwarz Inequality (with R. Farhadian). To appear in Mathematical Gazette.

A Generalization of Bonse's Inequality (with R. Farhadian). To appear in American Mathematical Monthly.

More on the Euler Limit for e (with R. Farhadian). The Mathematical Gazette 108 (572) July 2024, pp. 329-331.

A Geometric mean-Arithmetic mean Ratio Limit (with R. Farhadian). The Mathematical Gazette 108 (572) July 2024, pp. 334-335.

A Statistical Proof of Chebyshev's Sum Inequality (with R. Farhadian). American Mathematical Monthly 131 (3) 2024, p. 224.

Indeterminate Exponentials Without Tears (with R. Farhadian). The Mathematical Gazette 108 (571) March 2024, pp. 146--148.

The Elliptical Case of an Odds Inversion Problem (with K. Hilmer, A. Jin, R. Lycan). Involve 16 (3) 2023, pp. 431-452.

Wald's Identity vs. Tail Sum Formula (with R. Farhadian). American Mathematical Monthly 130 (6) 2023, p. 540.

Sums of Two Generalized Tetrahedral Numbers (with R. Lycan). The PUMP Journal of Undergraduate Research (5) 2022, pp. 206-217.

The Golden Supercircle (with R. Farhadian). The Mathematical Intelligencer 2022. https://doi.org/10.1007/s00283-022-10236-4.

Summation Using an Expected Value Identity (with R. Farhadian). American Mathematical Monthly 129 (10) 2022, p. 923.

Odds Inversion Problem With Replacement (with K. Hilmer, R. Lycan). American Mathematical Monthly 129 (9) 2022, p.885.

Sums of Generalized Octahedral Numbers. American Mathematical Monthly 129 (7) 2022, p. 677.

A Heuristic Approach to the Game of Sylver Coinage (with G. Moskowitz). Advances in Computer Games 2021 conference proceedings, Lecture Notes in Computer Science vol. 13262, pp. 61--70.

Sums of Generalized Tetrahedral Numbers. American Mathematical Monthly 129 (5) 2022, p. 474.

Applying Iterating Mapping to the No-Three-in-a-Line Problem (with C. Brower). Involve 15 (1) 2022, pp. 69-74.

Pollock's Generalized Tetrahedral Numbers Conjecture. American Mathematical Monthly 128 (10) 2021, p.921.

On the Monotonicity of the Number of Positive Entries in Nonnegative Five Element Matrix Powers (with M. Rodriguez Galvan and J. Sabio). Involve 14 (4) 2021, pp. 703-721.

A Theorem on Indifference Graphs (with K. Hilmer and R. Pinchasi). The PUMP Journal of Undergraduate Research (4) 2021, pp. 161-167.

A Cauchy-Schwarz Type Inequality for Differences (with R. Farhadian) American Mathematical Monthly 128 (2) 2021, p. 150.

Eigenvalues of the Sum and Product of Anticommuting Matrices (with L. Selstad) Involve 13 (4) 2020, pp. 625-632.

The Power Mean Test (with T. Kulhanek). The PUMP Journal of Undergraduate Research (3) 2020, pp. 68-71.

Surprises in Knockout Tournaments (with T. Kulhanek). Mathematics Magazine 93 (3) 2020, pp. 193-199.

On the monotonicity of the number of positive entries in nonnegative four element matrix powers (with C. Brower). The PUMP Journal of Undergraduate Research (3) 2020, pp. 26-36.

On the Monoid of Monic Binary Quadratic Forms (with J. Dimabayao, O. Tigas). Miskolc Mathematical Notes 20 (2) 2019, pp. 863-871.

Primes in (0,n] vs. (n,m] (with F. Huppe). American Mathematical Monthly 126 (10) 2019, p.942.

Dilated Floor Functions that Commute Sometimes (with T. Kulhanek, J. McDonough). The PUMP Journal of Undergraduate Research (2) 2019, pp. 107-117.

Subsums of the Harmonic Series (with B. Lubeck). American Mathematical Monthly 125 (4) 2018, pp. 351-355.

Asymptotic Formula for (1+1/x)^x, Revisited American Mathematical Monthly 122 (6) 2015, p. 587.

Fibonacci Nim and a Full Characterization of Winning Moves (with C. Allen). Involve 7 (6) 2014, pp. 807-822.

Multivariate analysis of functional metagenomes (E. Dinsdale, R. Edwards, B. Bailey, I. Tuba, S. Akhter, K. McNair, R. Schmieder, N. Apkarian, M. Creek, E. Guan, M. Hernandez, C. Isaacs, C. Peterson, T. Regh, V. Ponomarenko). Frontiers in Statistical Genetics and Methodology 41 (4) 2013.

How to Play the One-Lie Renyi-Ulam Game (with R. Ellis, C. Yan). Discrete Mathematics 308 (2008), pp. 5805-5808.

Characterizing Pseudodendrimers. I. Graph Representations of Pseudodendrimers Formed by Enhancements to Propagation of Linear Units (with D. Johnson, J. Wang). Macromolecular Theory and Simulations 18 (8) 2007, pp. 727-731.

Geometry of Jump Systems (with V. Lyubashevsky, C. Newell), Rocky Mountain Journal of Mathematics, Volume 35 (5) 2005, pp. 1675-1688.

The Renyi-Ulam Pathological Liar Game with a Fixed Number of Lies (with R. Ellis and C. Yan), Journal of Combinatorial Theory, Series A, Volume 112 (2) November 2005, pp. 238-336.

The Convergence of Difference Boxes (with A. Behn, C. Kribs Zaleta) American Mathematical Monthly 112 (5) May 2005, pp. 426-439.

Jump Systems and Laminated Manhattan Sets (with J. Cuomo, N. Nwasokwa) Australasian Journal of Combinatorics (31) February 2005, pp. 135-144.

Faa di Bruno's Formula and Nonhyperbolic Fixed Points of One-Dimensional Maps (corrected) International Journal of Mathematics and Mathematical Sciences (29) 2004, pp. 1543-1549.

Reduction of Jump Systems Houston Journal of Mathematics 30 (1) 2004, pp. 27-33.

DRILL (Depository of Repetitive Internet-based probLems and Lessons) Journal of Online Mathematics and Applications (3) 2003.

Stability of Hyperbolic and Non Hyperbolic Fixed Points of One-Dimensional Maps (with F. Dannan, S. Elaydi) Journal of Difference Equations and Applications 9 (5) 2003, pp. 449-457.

Mimeomatroids Ars Combinatoria (66) 2003, pp. 103-107.

Pseudo-mathematical pieces:

Grades Are Not Money American Mathematical Monthly 128 (6) 2021, p. 524.

The United States of America Prime Math Horizons 28 (4) 2021, p. 19.

The Mathematical Association of America Prime American Mathematical Monthly 127 (7) 2020, p. 672.

Proof Techniques for the New Millennium American Mathematical Monthly 127 (5) 2020, p. 443.

Count to Six in Six Languages (with M. Kutzman) American Mathematical Monthly 126 (8) 2019, p.747.

The Color of Pi MAA FOCUS 39 (5) 2019, p. 6.

Solution to the Hardest Jumbles (with B. LeTourneau) American Mathematical Monthly 126 (5) 2019, p. 479.

The Hardest Jumbles (with B. LeTourneau) American Mathematical Monthly 126 (5) 2019, p. 436.

Help with the Jumble American Mathematical Monthly 126 (1) 2019, p. 77.

A Base Conversion Surprise American Mathematical Monthly 122 (5) 2015, pp. 366, 402.

Anagrams of "American Mathematical Monthly" American Mathematical Monthly 121 (3) 2014, p. 243.

Most Active Mathematical Fields, in the American Mathematical Monthly (with R. Devitt-Ryder) American Mathematical Monthly 121 (2) 2014, p. 166.

Most Active Mathematical Fields (with R. Devitt-Ryder) American Mathematical Monthly 121 (2) 2014, p. 119.

Growth Rate of Calculus Textbooks American Mathematical Monthly 120 (9) 2013, p.786.

Famous American Mathematical Monthly Authors American Mathematical Monthly 120 (5) 2013, p. 408.

The Music of Pi American Mathematical Monthly 119 (8) 2012, p. 711. (associated sound file)

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