On the Diameter of Finite Sidon Sets

We prove that the diameter of a Sidon set (also known as a Babcock sequence, Golomb ruler, or $B_2$ set) with $k$ elements is at least $k^2-b k^{3/2}-O(k)$ where $b\le 1.96365$, a comparatively large improvement on past results. Equivalently, a Sidon set with diameter $n$ has at most $n^{1/2}+0.98183n^{1/4}+O(1)$ elements. The proof is conceptually simple but very computationally intensive, and the proof uses substantial computer assistance. We also provide a proof of $b\le 1.99058$ that can be verified by hand, which still improves on past results. Finally, we prove that $g$-thin Sidon sets (aka $g$-Golomb rulers) with $k$ elements have diameter at least $g^{-1} k^2 - (2-\varepsilon)g^{-1}k^{3/2} - O(k)$, with $\varepsilon\ge 0.02g^{-2}$.Comment: 16 pages, 1 figur

Genetic protocols for DNA extraction from white-tailed deer cast antlers to confirm individuality

White-tailed deer (Odocoileus virginianus) are the most sought-after deer species in America. The antlers of mammals, such as deer, are one of the fastest regenerative tissues in the world and are grown and naturally cast every year. Research on cast antlers have been used for a variety of purposes including population comparisons and impacts of deer health due to climatic stressors. When investigating cast antlers, it is important to confirm individuality of match sets in addition to antlers of the same individual between years. Therefore, individuality must be confirmed genetically, and protocols must be developed and established to do so. Our objectives were to 1) establish a genetic protocol to harvest DNA from cast antlers using connective tissue, and 2) determine individuality from subsequent years and match sets. When fresh antlers are cast, they leave behind a viable connective tissue from which DNA can be extracted. The DNA was successfully extracted from the skin rings harvested from naturally cast antlers. This study developed viable methods to confirm individuality, which aid researchers and wildlife biologists in a better understanding of the white-tailed deer herd as they set management goals and harvest regulations