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Full-Text Articles in Physical Sciences and Mathematics
The Mean Sum Of Squared Linking Numbers Of Random Piecewise-Linear Embeddings Of $K_N$, Yasmin Aguillon, Xingyu Cheng, Spencer Eddins, Pedro Morales
The Mean Sum Of Squared Linking Numbers Of Random Piecewise-Linear Embeddings Of $K_N$, Yasmin Aguillon, Xingyu Cheng, Spencer Eddins, Pedro Morales
Rose-Hulman Undergraduate Mathematics Journal
DNA and other polymer chains in confined spaces behave like closed loops. Arsuaga et al. \cite{AB} introduced the uniform random polygon model in order to better understand such loops in confined spaces using probabilistic and knot theoretical techniques, giving some classification on the mean squared linking number of such loops. Flapan and Kozai \cite{flapan2016linking} extended these techniques to find the mean sum of squared linking numbers for random linear embeddings of complete graphs $K_n$ and found it to have order $\Theta(n(n!))$. We further these ideas by inspecting random piecewise-linear embeddings of complete graphs and give introductory-level summaries of the ideas …
Motion Planning Algorithm In A Y-Graph, David Baldi
Motion Planning Algorithm In A Y-Graph, David Baldi
Rose-Hulman Undergraduate Mathematics Journal
We present an explicit algorithm for two robots to move autonomously and without collisions on a track shaped like the letter Y. Configuration spaces are of practical relevance in designing safe control schemes for automated guided vehicles. The topological complexity of a configuration space is the minimal number of continuous instructions required to move robots between any initial configuration to any final one without collisions. Using techniques from topological robotics, we calculate the topological complexity of two robots moving on a Y-track and exhibit an optimal algorithm realizing this exact number of instructions given by the topological complexity.