Graduation Year
Spring 2013
Document Type
Open Access Senior Thesis
Degree Name
Bachelor of Science
Department
Mathematics
Reader 1
Francis Edward Su
Reader 2
Michael E. Orrison
Reader 3
© 2013
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© 2013 Connor Ahlbach
Abstract
The Poincare-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 Poincare-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 Turzanski that proves the Poincare-Miranda theorem via the Steinhaus Chessboard Theorem, involving colorings of partitions of n-dimensional cubes. Then, we develop a new proof of the Poincare-Miranda Theorem that relies on a polytopal generalization of Sperner's Lemma of Deloera - Peterson - Su. Finally, we extend these discrete ideas to attempt to prove the existence of a zero with the boundary condition of Morales.
Recommended Citation
Ahlbach, Connor Thomas, "A Discrete Approach to the Poincare-Miranda Theorem" (2013). HMC Senior Theses. 47.
https://scholarship.claremont.edu/hmc_theses/47
