String-Based Borsuk-Ulam Theorem

Abstract

This paper introduces a string-based extension of the Borsuk-Ulam Theorem (denoted by strBUT). A string is a region with zero width and either bounded or unbounded length on the surface of an $n$-sphere or a region of a normed linear space. In this work, an $n$-sphere surface is covered by a collection of strings. For a strongly proximal continuous function on an $n$-sphere into $n$-dimensional Euclidean space, there exists a pair of antipodal $n$-sphere strings with matching descriptions that map into Euclidean space ${\mathbb{R}}^n$. Each region $M$ of a string-covered $n$-sphere is a worldsheet (denoted by $\mbox{wsh}M$). For a strongly proximal continuous mapping from a worldsheet-covered $n$-sphere to ${\mathbb{R}}^n$, strongly near antipodal worldsheets map into the same region in ${\mathbb{R}}^n$. An application of strBUT is given in terms of the evaluation of Electroencephalography (EEG) patterns.

14 pages, 8 figures. arXiv admin note: substantial text overlap with arXiv:1605.02987