Honors Program Theses

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Open Access Honors Program Thesis

First Advisor

Adrienne Stanley

Abstract

This report starts out with a discussion of fractals. Learning about fractals will lead us to the explanation of what affine functions are and how they can be seen in nature. We will then move on to a mathematical discussion of how affine functions behave. We will see how preserving ratios of distances is significant in our process by looking at a relationship of relative distances. This equation will be further described in our first definition. We then move on to prove that we can state affinity in a new way. After proving the first lemma, we discuss our main results. We will first provide the definition of an additive function, as it will be used in several lemmas. From here, we want to prove that some function, named g, is additive. Once we prove this is true, we will use a different but similar function to show that g(x + 1) = g(x). This result, the triangle inequality, and the definition of bounded will all be used in the proof of our next step, which shows that g is bounded. After showing this conclusion, we will recall the Archimedean principle to help show that g, a bounded additive function, is identically equal to zero. This leads us to our only theorem and the last result. The theorem states that when f is a continuous affine function on the real numbers, then f is linear. This will conclude our report.

Year of Submission

2024

Department

Department of Mathematics

University Honors Designation

A thesis submitted in partial fulfillment of the requirements for the designation University Honors

Date Original

5-2024

Object Description

1 PDF (8 pages)

Language

en

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