Weight enumerators of Reed-Muller codes from cubic curves and their duals

Let $\mathbb{F}_q$ be a finite field of characteristic not equal to $2$ or $3$. We compute the weight enumerators of some projective and affine Reed-Muller codes of order $3$ over $\mathbb{F}_q$. These weight enumerators answer enumerative questions about plane cubic curves. We apply the MacWilliams theorem to give formulas for coefficients of the weight enumerator of the duals of these codes. We see how traces of Hecke operators acting on spaces of cusp forms for $\operatorname{SL}_2(\mathbb{Z})$ play a role in these formulas.Comment: 19 pages. To appear in "Arithmetic, Geometry, Cryptography, and Coding Theory" (Y. Aubry, E. W. Howe, C. Ritzenthaler, eds.), Contemp. Math., 201

Counting Numerical Semigroups

A numerical semigroup is an additive submonoid of the natural numbers with finite complement. The size of the complement is called the genus of the semigroup. How many numerical semigroups have genus equal to $g$? We outline Zhai's proof of a conjecture of Bras-Amor\'os that this sequence has Fibonacci-like growth. We now know that this sequence asymptotically grows as fast as the Fibonacci numbers, but it is still not known whether it is nondecreasing. We discuss this and other open problems. We highlight the many contributions made by undergraduates to problems in this area.Comment: 14 pages. To appear in American Mathematical Monthly special issue on research by undergraduate

Distribution of enzymes cleaving pyridine nucleotides in animal tissues

A number of enzymes capable of splitting the pyridine nucleotides have been found in animal tissues. A portion of the DPNase (1) that hydrolyzes the nicotinamide ribose bond of DPN and TPN was reported in the microsomes of rat liver (1). The DPN pyrophosphorylase, first described by Kornberg (2), has been found by Hogeboom and Schneider to be largely localized in the nucleus (3). In previous investigations (4) an enzyme from pigeon liver which splits DPNH and not DPN at the pyrophosphate linkage was described. The present communication deals with the intracellular distribution of enzymes from various species that attack the pyridine coenzymes at the pyrophosphate linkage. The distribution and properties of DPNases from different species and tissues will also be presented

Time, Distance, and Epic Memory in The Tempest

Senior Project submitted to The Division of Languages and Literature of Bard College