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Full-Text Articles in Discrete Mathematics and Combinatorics
On The Polytopal Generalization Of Sperner’S Lemma, Amit Harlev
On The Polytopal Generalization Of Sperner’S Lemma, Amit Harlev
HMC Senior Theses
We introduce and prove Sperner’s lemma, the well known combinatorial analogue of the Brouwer fixed point theorem, and then attempt to gain a better understanding of the polytopal generalization of Sperner’s lemma conjectured in Atanassov (1996) and proven in De Loera et al. (2002). After explaining the polytopal generalization and providing examples, we present a new, simpler proof of a slightly weaker result that helps us better understand the result and why it is correct. Some ideas for how to generalize this proof to the complete result are discussed. In the last two chapters we provide a brief introduction to …
Games For One, Games For Two: Computationally Complex Fun For Polynomial-Hierarchical Families, Kye Shi
Games For One, Games For Two: Computationally Complex Fun For Polynomial-Hierarchical Families, Kye Shi
HMC Senior Theses
In the first half of this thesis, we explore the polynomial-time hierarchy, emphasizing an intuitive perspective that associates decision problems in the polynomial hierarchy to combinatorial games with fixed numbers of turns. Specifically, problems in 𝐏 are thought of as 0-turn games, 𝐍𝐏 as 1-turn “puzzle” games, and in general 𝚺ₖ𝐏 as 𝑘-turn games, in which decision problems answer the binary question, “can the starting player guarantee a win?” We introduce the formalisms of the polynomial hierarchy through this perspective, alongside definitions of 𝑘-turn CIRCUIT SATISFIABILITY games, whose 𝚺ₖ𝐏-completeness is assumed from prior work (we briefly justify this assumption …