Physical Sciences and Mathematics Commons^™

Unique Factorization In The Rings Of Integers Of Quadratic Fields: A Method Of Proof, Zachary Warren

Senior Honors Theses

It is a well-known property of the integers, that given any nonzero a ∈ Z, where a is not a unit, we are able to write a as a unique product of prime numbers. This is because the Fundamental Theorem of Arithmetic (FTA) holds in the integers and guarantees (1) that such a factorization exists, and (2) that it is unique. As we look at other domains, however, specifically those of the form O(√D) = {a + b√D | a, b ∈ Z, D a negative, squarefree integer}, we find that …

Figure-Ground Perception: A Poem Proof, Richard Delaware

Journal of Humanistic Mathematics

This is a proof, in poetic form, of a bit of real analysis, more specifically involving the topology of accumulation points, that exploits the human optical phenomenon of figure-ground perception. Sometimes it is not a change in content, but a snap shift in point of view that yields a proof.

Incomplete? Or Indefinite? Intuitionism On Gödel’S First Incompleteness Theorem, Quinn Crawford

The Yale Undergraduate Research Journal

This paper analyzes two natural-looking arguments that seek to leverage Gödel’s first incompleteness theorem for and against intuitionism, concluding in both cases that the argument is unsound because it equivocates on the meaning of “proof,” which differs between formalism and intuitionism. I argue that this difference explains why “proof” has definite extension for the formalist but not for the intuitionist. Sections 1-3 summarize various philosophies of mathematics and Gödel’s result. Section 4 argues that, because the Gödel sentence of a formal system is provable to the intuitionist, they are neither aided nor attacked by Gödel’s first incompleteness theorem. Section 5 …

Mathematical Rigor From Within, Lowell Abrams

Journal of Humanistic Mathematics

There is a certain feel that is unique to the rarefied context of rigorous mathematics. These poems constitute an exploration of my experience of mathematical rigor when I am in the midst of exercising my skills as a research mathematician.

Student Use Of Mathematical Content Knowledge During Proof Production, Chelsey Lynn Van De Merwe

Theses and Dissertations

Proof is an important component of advanced mathematical activity. Nevertheless, undergraduates struggle to write valid proofs. Research identifies many of the struggles students experience with the logical nature and structure of proofs. Little research examines the role mathematical content knowledge plays in proof production. This study begins to fill this gap in the research by analyzing what role mathematical content knowledge plays in the success of a proof and how undergraduates use mathematical content knowledge during proofs. Four undergraduates participated in a series of task-based interviews wherein they completed several proofs. The interviews were analyzed to determine how the students …

Enthymemathical Proofs And Canonical Proofs In Euclid’S Plane Geometry, Abel Lassalle, Marco Panza

MPP Published Research

Since the application of Postulate I.2 in Euclid’s Elements is not uniform, one could wonder in what way should it be applied in Euclid’s plane geometry. Besides legitimizing questions like this from the perspective of a philosophy of mathematical practice, we sketch a general perspective of conceptual analysis of mathematical texts, which involves an extended notion of mathematical theory as system of authorizations, and an audience-dependent notion of proof.

Finding Spanning Trees In Strongly Connected Graphs With Per-Vertex Degree Constraints, Samuel Benjamin Chase

Computer Science and Software Engineering

In this project, I sought to develop and prove new algorithms to create spanning trees on general graphs with per-vertex degree constraints. This means that each vertex in the graph would have some additional value, a degree constraint d. For a spanning tree to be correct, every vertex v_i in the spanning tree must have a degree exactly equal to a degree constraint d_i. This poses an additional constraint on what would otherwise be a trivial spanning tree problem. In this paper, two proofs related to my studies will be discussed and analyzed, leading to my algorithm …

One = Zero, Eric John Gofen

Journal of Humanistic Mathematics

In this paper, I use Mathematics in addition to the three most pure sciences --- Physics, Chemistry, and Rap --- to prove that 1=0. The argument uses The Ideal Gas Law, Ohm's Law, the Definitions of Power and Velocity in addition to indefinite integrals, simple mathematical operations, and the 99 Problems Law. The intuition-crushing result can be applied to all branches of mathematics and sciences and will likely go down as one of the greatest discoveries of all time.

Does Content Matter In An Introduction-To-Proof Course?, Milos Savic

Journal of Humanistic Mathematics

Introduction-to-proof courses are becoming more prevalent in mathematics departments as more recognize the need to support students while they transition from courses focused on computation (such as calculus) to proof-intensive courses (such as real analysis). In such introduction courses, there are some common proving techniques to teach (induction, contradiction, and contraposition to name a few), but the content varies from institution to institution. This note adds to the discussion on content in such courses by analyzing two prior studies, one using a coding scheme designed to illuminate step-by-step justifications in a proof, and the other focused on interviews with course …

Every Minute Of Your Life Has Been Interesting, Susan D'Agostino

Journal of Humanistic Mathematics

In this short paper, we prove that every minute of your life has been interesting. We also provide four exercises intended to solidify understanding of this result, including one exercise related to the torturously boring family road trip you took as a child.

Reading Between The Lines: Verifying Mathematical Language, Tristan Johnson

Honors Theses

A great deal of work has been done on automatically generating automated proofs of formal statements. However, these systems tend to focus on logic-oriented statements and tactics as well as generating proofs in formal language. This project examines proofs written in natural language under a more general scope of mathematics. Furthermore, rather than attempting to generate natural language proofs for the purpose of solving problems, we automatically verify human-written proofs in natural language. To accomplish this, elements of discourse parsing, semantic interpretation, and application of an automated theorem prover are implemented.

A Beautiful Proof By Induction, Lars-Daniel Öhman

Journal of Humanistic Mathematics

The purpose of this note is to present an example of a proof by induction that in the opinion of the present author has great aesthetic value. The proof in question is Thomassen's proof that planar graphs are 5-choosable. I give a self-contained presentation of this result and its proof, and a personal account of why I think this proof is beautiful.

A secondary purpose is to more widely publicize this gem, and hopefully make it part of a standard set of examples for examining characteristics of proofs by induction.

An Analysis Of Charles Fefferman's Proof Of The Fundamental Theorem Of Algebra, Kyle O. Linford

Senior Honors Theses and Projects

Many peoples' first exploration into more rigorous and formalized mathematics is with their early explorations in algebra. Much of their time and effort is dedicated to finding roots of polynomials-a challenge that becomes more increasingly difficult as the degree of the polynomials increases, especially if no real number roots exist. The Fundamental Theorem of Algebra is used to show that there exists a root, particularly a complex root, for any nth degree polynomial. After struggling to prove this statement for over 3 centuries, Carl Friedrich Gauss offered the first fairly complete proof of the theorem in 1799. Further proofs of …

Inverting The Transition-To-Proof Classroom, Robert Talbert

Funded Articles

In this paper, we examine the benefits of employing an inverted or “flipped” class design in a Transition-to-Proof course for second-year mathematics majors. The issues concomitant with such courses, particularly student acquisition of “sociomathematical norms” and self-regulated learning strategies, are discussed along with ways that the inverted classroom can address these issues. Finally, results from the redesign of a Transition-to-Poof class at the author’s university are given and discussed.

The Cosm Newsletter

The COSM Newsletter (2008-2018)

• Laura Regassa, GSTA 2015 Teacher of the Year
• The Department of Biology receives the Kay T. Reissing Terrestrial Snail Collection
• Biology Professor Invited to United Nations Meeting in Rome
• Charles and Sandra Chandler Research Scholarship Program
• Biology Student Kym Coley
• Announcing a New Scholarship in Biology
• Biology Professor coauthors environmental science textbook
• Biology faculty member makes research presentation in Bali, Indonesia
• Chemistry students attend NOBCChE Conference
• Graduate student gives talk at UGA
• Georgia Southern Scientists Map Wassaw Sound
• Graduates go to sea
• Department News
• Student News
• Professor Xiao-Jun Wang appointed as Editor
• Nanotechnology Computational group conducts research
• Updates from the …

Pruebas Entimemáticas Y Pruebas Canónicas En La Geometría Plana De Euclides, Marco Panza, Abel Lassalle Casanave

MPP Published Research

Dado que la aplicación del Postulado I.2 no es uniforme en Elementos, ¿de qué manera debería ser aplicado en la geometría plana de Euclides? Además de legitimar la pregunta misma desde la perspectiva de una filosofía de la práctica matemática, nos proponemos esbozar una perspectiva general de análisis conceptual de textos matemáticos que involucra una noción ampliada de la teoría matemática como sistema de autorizaciones o potestades y una noción de prueba que depende del auditorio.

Since the application of Postulate I.2 in the Elements is not uniform, one could wonder in what way should it be applied in Euclid’s …

Reasoning & Proof In The Hs Common Core, Laurie O. Cavey

Laurie O. Cavey

No abstract provided.

Using Formative Assessment To Enhance Student Performance On Geometric Proof Writing, Benjamin Hargrave

LSU Master's Theses

The purpose of this research is to uncover best practices to create competent proof writers. Studies have shown the best setting to do this is in the high school geometry classroom. Throughout a yearlong study of geometry, students were exposed to theorems and their demonstrations. Despite constant exposure, students were still unable to produce their own proof of propositions. The questions then became how can an educator provide critical feedback that encourages student reasoning and develops logical argumentation skills? With the goal in mind, twenty-five students enrolled in a geometry course at Baton Rouge High School in Baton Rouge, Louisiana …

Using Writing Assignments In High School Geometry To Improve Students' Proof Writing Ability, Amanda Choppin Mcallister

LSU Master's Theses

The Common Core State Standards encourage the use of writing in mathematics classrooms. This study was designed to create a template for high school teachers to use in a geometry class to improve students’ proof writing ability. The students enrolled in the class were asked to complete journal and expository writing assignments throughout the course. The assignments were scored with a rubric. To assess if growth was made in proof writing, the students were all given a test four times throughout the school year. All four tests were assessed using the same rubric. We provide evidence that the template was …

Efficient Non-Interactive Range Proof, Duncan Wong, Willy Susilo, Yi Mu, Tsz Hon Yuen, Guomin Yang, Q.X. Huang

Dr Guomin Yang

No abstract provided.

Proof-Of-Knowledge Of Representation Of Committed Value And Its Applications, Willy Susilo, Yi Mu, Man Ho Allen Au

Professor Yi Mu

We present a zero-knowledge argument system of representation of a committed value. Specifically, for commitments C = Commit1(y), D = Commit2(x), of value y and a tuple x = (x1, . . . , xL), respectively, our argument system allows one to demonstrate the knowledge of (x, y) such that x is a representation of y to bases h1, . . . , hL. That is, y = hx11 · · · hxLL . Our argument system is zero-knowledge and hence, it does not reveal anything such as x or y. We note that applications of our argument system are …

Proof-Of-Knowledge Of Representation Of Committed Value And Its Applications, Willy Susilo, Yi Mu, Man Ho Allen Au

Dr Man Ho Allen Au

We present a zero-knowledge argument system of representation of a committed value. Specifically, for commitments C = Commit1(y), D = Commit2(x), of value y and a tuple x = (x1, . . . , xL), respectively, our argument system allows one to demonstrate the knowledge of (x, y) such that x is a representation of y to bases h1, . . . , hL. That is, y = hx11 · · · hxLL . Our argument system is zero-knowledge and hence, it does not reveal anything such as x or y. We note that applications of our argument system are …

Prove It!, Kenny W. Moran

Journal of Humanistic Mathematics

A dialogue between a mathematics professor, Frank, and his daughter, Sarah, a mathematical savant with a powerful mathematical intuition. Sarah's intuition allows her to stumble into some famous theorems from number theory, but her lack of academic mathematical background makes it difficult for her to understand Frank's insistence on the value of proof and formality.

Proof-Of-Knowledge Of Representation Of Committed Value And Its Applications, Willy Susilo, Yi Mu, Man Ho Allen Au

Faculty of Informatics - Papers (Archive)

We present a zero-knowledge argument system of representation of a committed value. Specifically, for commitments C = Commit1(y), D = Commit₂(x), of value y and a tuple x = (x₁, . . . , x_L), respectively, our argument system allows one to demonstrate the knowledge of (x, y) such that x is a representation of y to bases h₁, . . . , h_L. That is, y = h^x1₁ · · · h^xL_L . Our argument system is zero-knowledge and hence, it does not reveal anything …

Slides: Keynote: Living With Limits In The West, Alice Madden

Shifting Baselines and New Meridians: Water, Resources, Landscapes, and the Transformation of the American West (Summer Conference, June 4-6)

Presenter: Alice Madden, Colorado House Majority Leader and President, Western Progress

8 slides

The Separation Of Facts And Values, Arthur Kantrowitz

RISK: Health, Safety & Environment (1990-2002)

Dr. Kantrowitz maintains that much modern pessimism derives from failure to separate what is from what ought to be and urges that scientific conflicts be resolved as value neutrally as possible.

Review Of: Dorothy J. Howell, Scientific Literacy And Environmental Policy- The Missing Prerequisite For Sound Decision Making (Quorum Books 1992), Diane M. Albert

RISK: Health, Safety & Environment (1990-2002)

Review of: Dorothy J. Howell, Scientific Literacy and Environmental Policy- The Missing Prerequisite for Sound Decision Making (Quorum Books 1992). Acknowledgements, bibliography, epilogue, index, introduction. LC 91- 36028; ISBN 0-89930-616-0. [181 pp. Cloth $45.00. One Madison Ave., New York, NY 10010.1

Proof In Law And Science, David H. Kaye

Journal Articles

This article addresses proof in both science and law. Both disciplines utilize proof of facts and proof of theories, but for different purposes and, consequently, in different ways. Some similarities exist, however, in how both disciplines use a series of premises followed by a conclusion to form an argument, and thus constitute a logic. This article analyzes the ways in which legal logic and scientific logic differ. Finding facts in law involves the same logic but quite different procedures than scientific fact-finding. Finding, or rather constructing, the law is also very different from scientific theorizing. But such differences do not …