Region-Based Borsuk-Ulam Theorem and Wired Friend Theorem

TL;DR

This paper extends the Borsuk-Ulam Theorem to string-based regions on spheres, introducing a new theorem with applications in EEG pattern analysis and establishing a wired friend theorem in descriptive string theory.

Contribution

It introduces a string-based extension of the Borsuk-Ulam Theorem and a wired friend theorem, expanding topological results to string regions and their mappings.

Findings

Existence of antipodal strings with matching descriptions on spheres.

Strongly proximal continuous mappings preserve antipodal string relationships.

Application to EEG pattern evaluation demonstrates practical relevance.

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. 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 $\mathbb{R}^n$. This leads to a wired friend theorem in descriptive string theory. An application of strBUT is given in terms of the evaluation of Electroencephalography (EEG) patterns.