20 research outputs found

    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum

    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum

    Mathematical Surprises

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    This is open access book provides plenty of pleasant mathematical surprises. There are many fascinating results that do not appear in textbooks although they are accessible with a good knowledge of secondary-school mathematics. This book presents a selection of these topics including the mathematical formalization of origami, construction with straightedge and compass (and other instruments), the five- and six-color theorems, a taste of Ramsey theory and little-known theorems proved by induction. Among the most surprising theorems are the Mohr-Mascheroni theorem that a compass alone can perform all the classical constructions with straightedge and compass, and Steiner's theorem that a straightedge alone is sufficient provided that a single circle is given. The highlight of the book is a detailed presentation of Gauss's purely algebraic proof that a regular heptadecagon (a regular polygon with seventeen sides) can be constructed with straightedge and compass. Although the mathematics used in the book is elementary (Euclidean and analytic geometry, algebra, trigonometry), students in secondary schools and colleges, teachers, and other interested readers will relish the opportunity to confront the challenge of understanding these surprising theorems

    PI – NOT JUST AN ORDINARY NUMBER

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    Abstract Algebra : An Introductory Course

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    This book is intended for students encountering the beautiful subject of abstract algebra for the first time. My goal here is to provide a text that is suitable for you, whether you plan to take only a single course in abstract algebra, or to carry on to more advanced courses at the senior undergraduate and graduate levels. Naturally, I wish to encourage you to study the subject further and to ensure that you are prepared if you do so. At many universities, including my own, abstract algebra is the first serious proof-based course taken by mathematics majors. While it is quite possible to get through, let us say, a course in calculus simply by memorizing a list of rules and applying them correctly, without really understanding why anything works, such an approach would be disastrous here. To be sure, you must carefully learn the definitions and the statements of theorems, but that is nowhere near sufficient. In order to master the material, you need to understand the proofs and then be able to prove things yourself. This book contains hundreds of problems, and I cannot stress strongly enough the need to solve as many of them as you can. Do not be discouraged if you cannot get all of them! Some are very difficult. But try to figure out as many as you can. You will only learn by getting your hands dirty

    Impossibilities proved by Galois theory

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    I studied three areas in my paper:1. The basic field theory needed to prove the impossibility of three goals of Ruler and Compass construction. Here I found the results flowed rather easily. I found that the level of material needed to prove these problems was, for the most part, not out of reach for any undergraduate mathematics student.2. I studied the step into Galois theory and the necessary material needed to prove the impossibility of solving the general Quintic with radicals. This was the biggest step. The jump from the basic field theory I studied in 1. to the Galois theory needed for 2. is extreme. I spent most of my time studying this material so that I may understand it well enough to present it.3. I studied the rational distance problem, specifically how its arguments are extremely similar to those of the problems in 1. and 2. I found that the main argument lies in drawing an equality between the geometry of the problem, and then using this relation to prove equivalent field extensions. This can only happen for some polygons, and that is the conclusion in the paper.More generally, I found that independent study, no matter how expository and non novel, in the area you're interested in is extremely fascinating. I was motivated and intrigued to continue studying well beyond the normal hours, and certainly this project has cemented in me the will to pursue a Ph.D in mathematics

    Mathematics and Its Applications, A Transcendental-Idealist Perspective

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    This monograph offers a fresh perspective on the applicability of mathematics in science. It explores what mathematics must be so that its applications to the empirical world do not constitute a mystery. In the process, readers are presented with a new version of mathematical structuralism. The author details a philosophy of mathematics in which the problem of its applicability, particularly in physics, in all its forms can be explained and justified. Chapters cover: mathematics as a formal science, mathematical ontology: what does it mean to exist, mathematical structures: what are they and how do we know them, how different layers of mathematical structuring relate to each other and to perceptual structures, and how to use mathematics to find out how the world is. The book simultaneously develops along two lines, both inspired and enlightened by Edmund Husserl’s phenomenological philosophy. One line leads to the establishment of a particular version of mathematical structuralism, free of “naturalist” and empiricist bias. The other leads to a logical-epistemological explanation and justification of the applicability of mathematics carried out within a unique structuralist perspective. This second line points to the “unreasonable” effectiveness of mathematics in physics as a means of representation, a tool, and a source of not always logically justified but useful and effective heuristic strategies

    Tarski's geometry modelled in Mizar computerized proof assistant

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