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- W133035569 abstract "In this chapter we consider the problem of finding zero points of a continuous function f from the n-dimensional symmetric cube C n into ℝn.It will be shown in a constructive way that there exists a zero point of f in C n if for every ∈ bd(C n), f (−x) ≠ α f (−x) for all α > 0. It will be argued that this result is so strong that it implies several powerful fixed point theorems, including Borsuk-Ulam’s theorem and Brouwer’s fixed point theorem. We will introduce two algorithms to compute zero points of the continuous function. One is an integer labeling algorithm and the other a vector labeling algorithm. The integer labeling algorithm will lead to a constructive proof for Tucker’s theorem. The algorithm we introduce here is a modification of the algorithms developed by Todd and Wright [1980], Freund and Todd [1981], and van der Laan [1984] in the sense that the modified algorithm can start with an arbitrarily chosen point in C n, whereas the existing methods can only start with the point 0n. The modified algorithm will be built upon a new triangulation of C n, called the AS-triangulation.KeywordsFixed Point TheoremLabel FunctionConstructive ProofVector LabelTucker TheoremThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves." @default.
- W133035569 created "2016-06-24" @default.
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- W133035569 date "1999-01-01" @default.
- W133035569 modified "2023-09-27" @default.
- W133035569 title "The Computation of Antipodal Fixed Points" @default.
- W133035569 doi "https://doi.org/10.1007/978-1-4757-4839-0_10" @default.
- W133035569 hasPublicationYear "1999" @default.
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