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Chasing Fixed Points: Greedy Gremlin's Trade-Off |

Fixed points are points that a function doesn't change. But all fixed point theorems suffer from the same dilemma... In this video we compare different theorems and highlight the trade-off one has to accept in mathematics when generalizing results. Check out these videos by Dr. Trefor Bazett: on the proof of the Brouwer fixed point theorem:    • A beautiful combinatorical proof of t...   and the Banach fixed point theorem and fixed point iteration:    • What is cos( cos( cos( cos( cos( cos(...   Vsauce also has a great video on fixed points:    • Fixed Points   The results in the bonus part of the video were part of my Master's Thesis in Mathematics at the University of Innsbruck. Thesis: https://ulb-dok.uibk.ac.at/ulbtirolhs... University of Innsbruck: https://www.uibk.ac.at/en/ Department of Mathematics: https://www.uibk.ac.at/mathematik/ind... The presented research on nonexpansive functions on unbounded domains was published by Christian Bargetz, Simeon Reich, and Daylen Thimm (Paper: https://www.sciencedirect.com/science... ). The research was supported by the Austrian Science Fund (FWF): P 32523-N. The second author was partially supported by the Israel Science Foundation (Grant 820/17), the Fund for the Promotion of Research at the Technion (Grant 2001893) and by the Technion General Research Fund (Grant 2016723). References: S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundamenta Mathematicae., 3 (1922): 133–181. C. Bargetz, S. Reich, and D. Thimm, Generic properties of nonexpansive mappings on unbounded domains. Journal of Mathematical Analysis and Applications 526.1 (2023): 127179. F. S. de Blasi and J. Myjak. Sur la porosité de l’ensemble des contractions sans point fixe. C. R. Acad. Sci. Paris Sér. I Math., 308 (1989): 51–54. L. E. J. Brouwer, Über Abbildungen von Mannigfaltigkeiten. Mathematische Annalen, 71 (1911): 97–115. E. Rakotch. A note on contractive mappings. Proc. Amer. Math. Soc., 13:459–465, 1962. S. Reich, A. J. Zaslavski, The set of noncontractive mappings is σ-porous in the space of all nonexpansive mappings, C. R. Acad. Sci. Paris, Sér. I Math. 333 (6) (2001) 539–544. F. Strobin, Some porous and meager sets of continuous mappings, J. Nonlinear Convex Anal. 13 (2) (2012) 351–361.