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Connor Ahlbach

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A Discrete Approach to the Poincaré-Miranda Theorem

Francis Edward Su
Second Reader(s)
Michael E. Orrison


The Poincaré-Miranda Theorem is a topological result about the existence of a zero of a function under particular boundary conditions. In this thesis, we explore proofs of the Poincaré-Miranda Theorem that are discrete in nature—that is, they prove a continuous result using an intermediate lemma about discrete objects. We explain a proof by Tkacz and Turza\'nski that proves the Poincaré-Miranda Theorem via the Steinhaus Chessboard Theorem, involving colorings of partitions of $n$-dimensional cubes. Then, we develop another new proof that relies on a polytopal generalization of Sperner's Lemma of DeLoera–Peterson–Su. Finally, we extend these discrete ideas to prove the existence of a zero with the boundary condition of Morales, in dimension 2.


Combinatorial Analog of the Poincare Miranda Theorem

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