Topology

2410 Submissions

[3] viXra:2410.0154 [pdf] submitted on 2024-10-25 17:23:04

A Constructive Proof and Algorithm for the 2D Brouwer Fixed-Point Theorem with Surjective Mapping

Authors: Ruohan Yang
Comments: 12 Pages.

This article investigates the two-dimensional Brouwer Fixed-Point Theorem within the context of a surjective continuous transformation function ( f(x) ). This function can be interpreted as defining a continuous vector field. In this framework, each point in the disk is mapped to another point via a specific vector associated with the continuous transformation, thereby establishing a coherent vector field. This function can be interpreted as defining a continuous vector field. In this framework, the vector field can be decomposed two vector fields. Instead of proving the existence of fixed point directly, the article aim to focus on prove the vector fields always has intersection where at this point, the vector fields has opposite directiond and same norm.The paper also provide the programming experiment which further verifies the proof.
Category: Topology

[2] viXra:2410.0153 [pdf] submitted on 2024-10-25 17:21:54

A Generalized Contructive Proof for Brouwer Fixed-Point Theorem on D^2 and D^3

Authors: Ruohan Yang
Comments: 7 Pages.

This article present a constructive proof by analyzing decompositions of continuous vector field. The original proof of Brouwer's theorem relies on a contradiction argument, which, while effective, does not offer a constructive method for locating the fixed point. Through projecting arbitrary vector field the basis of the vector field, it can be proved there exists zero points on both of the basis. The article will also generalize the proof from 2D to 3D dimensions. The method is also valid under surjective and scaling map.
Category: Topology

[1] viXra:2410.0112 [pdf] submitted on 2024-10-19 02:40:00

Analysis on the Topology of Problem Spaces

Authors: Theophilus Agama
Comments: 7 Pages. This paper introduces an analysis on the topology of problem spaces.

We develop the analysis of the theory of problem and their solution spaces. We adapt some classical concepts in functional analysis to study problems and their corresponding solution spaces. We introduce the notion of compactness, density, convexity, boundedness, amenability and the interior. We examine the overall interplay among these concepts in theory.
Category: Topology