Extensions of a result of Elekes and R\'onyai

Many problems in combinatorial geometry can be formulated in terms of curves or surfaces containing many points of a cartesian product. In 2000, Elekes and R\'onyai proved that if the graph of a polynomial contains $cn^2$ points of an $n\times n\times n$ cartesian product in $\mathbb{R}^3$, then the polynomial has the form $f(x,y)=g(k(x)+l(y))$ or $f(x,y)=g(k(x)l(y))$. They used this to prove a conjecture of Purdy which states that given two lines in $\mathbb{R}^2$ and $n$ points on each line, if the number of distinct distances between pairs of points, one on each line, is at most $cn$, then the lines are parallel or orthogonal. We extend the Elekes-R\'onyai Theorem to a less symmetric cartesian product. We also extend the Elekes-R\'onyai Theorem to one dimension higher on an $n\times n\times n\times n$ cartesian product and an asymmetric cartesian product. We give a proof of a variation of Purdy's conjecture with fewer points on one of the lines. We finish with a lower bound for our main result in one dimension higher with asymmetric cartesian product, showing that it is near-optimal.Comment: 23 page

The Voltammetric Study of the Reduction of Tetraalkylammonium Perchlorate by Fe(TPP)\u3csup\u3e2-\u3c/sup\u3e

Tetraalkylammonium ions react with Fe(TPP)2− to form Fe(TPP)(R)− and trialkylamine. The tetrabutylammonium cation was verified to be the source of the alkyl group in the product, Fe(TPP)(R)−, by using (1H5C2)3(2H5C2)N− as the cation and 2H NMR. The reaction of Fe(TPP)2− with Bu4N− was monitored by cyclic voltammetry and thin layer spectroelectrochemistry. The activation parameters were measured, and were most consistent with an electron transfer (ET) mechanism. The rate of the reaction of tetramethyl and tetraethylammonium ions with Fe(TPP)2− was also examined. The rate constant decreased significantly as the carbon chain length decreased, which was also consistent with an ET mechanism

Assessment of Polycyclic Aromatic Hydrocarbon Contamination of Breeding Pools Utilized by the Puerto Rican Crested Toad, Peltophryne lemur.

Habitat preservation and management may play an important role in the conservation of the Puerto Rican crested toad, Peltophryne lemur, due to this species' small geographic range and declining native wild population. Bioavailable water concentrations of Polycyclic Aromatic Hydrocarbon (PAH) contaminants within breeding pools at 3 sites were established using Passive Sampling Devices (PSDs) and gas chromatography-mass spectrometry (GC/MS). A more diverse population of PAH analytes were found in higher concentrations at the breeding site that allowed direct vehicular access, but calculated risk quotients indicated low risk to toad reproduction associated with the current PAH analyte levels

Simultaneous Arithmetic Progressions on Algebraic Curves

A simultaneous arithmetic progression (s.a.p.) of length k consists of k points (x_i, y_\sigma(i)), where x_i and y_i are arithmetic progressions and \sigma is a permutation. Garcia-Selfa and Tornero asked whether there is a bound on the length of an s.a.p. on an elliptic curve in Weierstrass form over Q. We show that 4319 is such a bound for curves over R. This is done by considering translates of the curve in a grid as a graph. A simple upper bound is found for the number of crossings and the 'crossing inequality' gives a lower bound. Together these bound the length of an s.a.p. on the curve. We then use a similar method to extend the result to arbitrary real algebraic curves. Instead of considering s.a.p.'s we consider k^2/3 points in a grid. The number of crossings is bounded by Bezout's Theorem. We then give another proof using a result of Jarnik bounding the number of grid points on a convex curve. This result applies as any real algebraic curve can be broken up into convex and concave parts, the number of which depend on the degree. Lastly, these results are extended to complex algebraic curves.Comment: 11 pages, 6 figures, order of email addresses corrected 12 pages, closing remarks, a reference and an acknowledgment adde