The Brouwer’s Fixed Point Theorem is one of the most well known and important existence principles in mathematics. Since the theorem and its many equivalent formulations or extensions are powerful tools in showing the existence of solutions for many problems in pure and applied mathematics, many scholars have been studying its further extensions and applications.
The Brouwer Theorem itself gives no information about the location of fixed points. However, effective ways have been developed to calculate or approximate the fixed points. Such techniques are important in various applications including calculation of economic equilibria.
Because Brouwer Fixed Point Theorem has a significant role in mathematics, there are many generalizations and proofs of this theorem. In this paper, we will try to show several proves of Brouwer Fixed Point Theorem. First, let’s take a look at Brouwer Theorem from real world illustrations. There are several real world examples, and we will take in consideration few of them.
Inhaltsverzeichnis (Table of Contents)
- ON BROUWER FIXED POINT THEOREM
- Brouwer Fixed Point Theorem on R¹
- Combinatorial: Sperner's Lemma
- Induction proof for N = 1.
- Three cases of the induction proof for N + 1
- Complete triangles with different labels inside the polygon (Sperner's Lemma)
- Sperner's Lemma implies Brouwer's Fixed Point Theorem
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
This paper aims to explore the Brouwer Fixed Point Theorem, a fundamental principle in mathematics with broad applications in various fields. It aims to demonstrate the theorem's significance and present several proofs for its validity. The paper will cover different aspects of the theorem, including its real-world applications, combinatorial proofs, and connections to other mathematical concepts.
- Brouwer Fixed Point Theorem and its significance in mathematics
- Real-world illustrations of Brouwer's Theorem
- Various proofs of Brouwer's Theorem
- Sperner's Lemma and its relationship to Brouwer's Theorem
- The concept of fixed points and their implications
Zusammenfassung der Kapitel (Chapter Summaries)
- The paper begins by providing a real-world illustration of Brouwer's Fixed Point Theorem on R¹, using examples involving folded sheets of paper, decks of cards, and identical disks. This section aims to provide an intuitive understanding of the theorem's core concept.
- The paper then delves into a combinatorial proof of Brouwer's Theorem using Sperner's Lemma. It introduces Sperner's labeling, a method of assigning colors to vertices in a triangulation, and presents a simple proof of Sperner's Lemma in a one-dimensional space.
- The paper further explores Sperner's Lemma in a two-dimensional space, using polygons and triangulations. It demonstrates the relationship between the number of complete triangles and the number of 1-2 edges on the boundary of the polygon.
- The paper concludes by showing how Sperner's Lemma implies Brouwer's Fixed Point Theorem, demonstrating the connection between these two fundamental concepts.
Schlüsselwörter (Keywords)
The primary focus of this paper lies on the Brouwer Fixed Point Theorem, its various proofs, and its connection to Sperner's Lemma. The paper explores real-world examples of the theorem, including its applications in economics. The paper further delves into the concept of fixed points, their implications, and their significance in different branches of mathematics. Key terms associated with this work include Brouwer Fixed Point Theorem, Sperner's Lemma, fixed point property, combinatorial proofs, triangulation, and real-world applications.
Frequently Asked Questions
What is the Brouwer Fixed Point Theorem?
It is a fundamental existence principle in mathematics stating that for any continuous function mapping a compact convex set to itself, there is at least one point that remains unchanged (a fixed point).
How can the theorem be illustrated in the real world?
A common example is stirring a cup of coffee; the theorem implies that at least one point on the surface of the liquid will be in the exact same spot it was before the stirring began.
What is Sperner's Lemma?
Sperner's Lemma is a combinatorial result about labeled triangulations of a simplex. It is famously used to provide a non-analytical proof of the Brouwer Fixed Point Theorem.
Does the Brouwer Theorem help locate the fixed point?
The theorem itself only guarantees the existence of a fixed point. However, specific algorithms and techniques have been developed to approximate these points for practical use.
What are the applications of this theorem in economics?
It is crucial for proving the existence of economic equilibria, such as in the Nash equilibrium in game theory or general equilibrium models in market analysis.
- Quote paper
- Duli Pllana (Author), 2018, Reflection on Brouwer's Fixed Point Theorem, Munich, GRIN Verlag, https://www.grin.com/document/428448
