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Articles 1 - 22 of 22
Full-Text Articles in Physical Sciences and Mathematics
Using Difference-In-Differences Analysis And The Kocyk Geometric Lag Model To Estimate Aspects Of Carbon Tax Effectiveness In Nordic Countries, Kyle Riley
Honors Theses
This paper generally looks at the connections between carbon taxes and carbon emission levels in Nordic countries over a period from the 1960s to the early 2010s. Most of the existing literature on this topic looks at and finds that carbon taxes do have a significant impact upon carbon emissions levels in some countries while not in others. In many countries which have this policy there is not a significant impact that can be seen and there is a discussion as to why this might be the case and what needs to be done to fix these potential issues to …
Classification Of Torsion Subgroups For Mordell Curves, Zachary Porat
Classification Of Torsion Subgroups For Mordell Curves, Zachary Porat
Honors Theses
Elliptic curves are an interesting area of study in mathematics, laying at the intersection of algebra, geometry, and number theory. They are a powerful tool, having applications in everything from Andrew Wiles’ proof of Fermat’s Last Theorem to cybersecurity. In this paper, we first provide an introduction to elliptic curves by discussing their geometry and associated group structure. We then narrow our focus, further investigating the torsion subgroups of elliptic curves. In particular, we will examine two methods used to classify these subgroups. We finish by employing these methods to categorize the torsion subgroups for a specific family of elliptic …
Geometric Constructions, Origami, And Galois Theory, Julia Greene
Geometric Constructions, Origami, And Galois Theory, Julia Greene
Honors Theses
Geometric constructions using an unmarked straightedge and a compass have been studied for thousands of years. In these constructions, we can draw circles and lines starting with any two points, and we can create new points where they intersect. An n-gon is said to be constructible if can be constructed in a finite number of steps using these guidelines. We begin with constructions of several n-gons, and examine the field theory behind geometric constructions. Galois theory then provides a precise classification of which n-gons are constructible and which are not. Next is an exploration of origami construction, which examines a …
Approximation Of Continuous Functions By Artificial Neural Networks, Zongliang Ji
Approximation Of Continuous Functions By Artificial Neural Networks, Zongliang Ji
Honors Theses
An artificial neural network is a biologically-inspired system that can be trained to perform computations. Recently, techniques from machine learning have trained neural networks to perform a variety of tasks. It can be shown that any continuous function can be approximated by an artificial neural network with arbitrary precision. This is known as the universal approximation theorem. In this thesis, we will introduce neural networks and one of the first versions of this theorem, due to Cybenko. He modeled artificial neural networks using sigmoidal functions and used tools from measure theory and functional analysis.
Category Theory And Universal Property, Niuniu Zhang
Category Theory And Universal Property, Niuniu Zhang
Honors Theses
Category theory unifies and formalizes the mathematical structure and concepts in a way that various areas of interest can be connected. For example, many have learned about the sets and its functions, the vector spaces and its linear transformation, and the group theories and its group homomorphism. Not to mention the similarity of structure in topological spaces, as the continuous function is its mapping. In sum, category theory represents the abstractions of other mathematical concepts. Hence, one could use category theory as a new language to define and simplify the existing mathematical concepts as the universal properties. The goal of …
Galois Theory And The Quintic Equation, Yunye Jiang
Galois Theory And The Quintic Equation, Yunye Jiang
Honors Theses
Most students know the quadratic formula for the solution of the general quadratic polynomial in terms of its coefficients. There are also similar formulas for solutions of the general cubic and quartic polynomials. In these three cases, the roots can be expressed in terms of the coefficients using only basic algebra and radicals. We then say that the general quadratic, cubic, and quartic polynomials are solvable by radicals. The question then becomes: Is the general quintic polynomial solvable by radicals? Abel was the first to prove that it is not. In turn, Galois provided a general method of determining when …
Introduction To Computational Topology Using Simplicial Persistent Homology, Jason Turner, Brenda Johnson, Ellen Gasparovic
Introduction To Computational Topology Using Simplicial Persistent Homology, Jason Turner, Brenda Johnson, Ellen Gasparovic
Honors Theses
The human mind has a natural talent for finding patterns and shapes in nature where there are none, such as constellations among the stars. Persistent homology serves as a mathematical tool for accomplishing the same task in a more formal setting, taking in a cloud of individual points and assembling them into a coherent continuous image. We present an introduction to computational topology as well as persistent homology, and use them to analyze configurations of BuckyBalls®, small magnetic balls commonly used as desk toys.
The Axiom Of Choice In Topology, Ruoxuan Jia
The Axiom Of Choice In Topology, Ruoxuan Jia
Honors Theses
Cantor believed that properties holding for finite sets might also hold for infinite sets. One such property involves choices; the Axiom of Choice states that we can always form a set by choosing one element from each set in a collection of pairwise disjoint non-empty sets. Since its introduction in 1904, this seemingly simple statement has been somewhat controversial because it is magically powerful in mathematics in general and topology in particular. In this paper, we will discuss some essential concepts in topology such as compactness and continuity, how special topologies such as the product topology and compactification are defined, …
Choice Of Choice: Paradoxical Results Surrounding Of The Axiom Of Choice, Connor Hurley
Choice Of Choice: Paradoxical Results Surrounding Of The Axiom Of Choice, Connor Hurley
Honors Theses
When people think of mathematics they think "right or wrong," "empirically correct" or "empirically incorrect." Formalized logically valid arguments are one important step to achieving this definitive answer; however, what about the underlying assumptions to the argument? In the early 20th century, mathematicians set out to formalize these assumptions, which in mathematics are known as axioms. The most common of these axiomatic systems was the Zermelo-Fraenkel axioms. The standard axioms in this system were accepted by mathematicians as obvious, and deemed by some to be sufficiently powerful to prove all the intuitive theorems already known to mathematicians. However, this system …
Bitcoin Volatility And Currency Acceptance: A Time-Series Approach, Francis Rocco
Bitcoin Volatility And Currency Acceptance: A Time-Series Approach, Francis Rocco
Honors Theses
Virtual currencies emerged in 2009 as alternatives to traditional methods of payment, offering faster transaction speeds and increased privacy. The prime example of these currencies is Bitcoin. Prior literature in the past five years has generally predicted that bitcoin would fail to supplant an existing widely traded currency, but the volatility of the currency has been decreasing since then. I test Dowd and Greenaway’s (1993) currency acceptance model using recent data on Bitcoin, including Bitcoin volatility. This paper will show whether Bitcoin's ability to act as a store of value and its level of price volatility affect the number of …
An Investigation Of The Four Vertex Theorem And Its Converse, Rebeka Kelmar
An Investigation Of The Four Vertex Theorem And Its Converse, Rebeka Kelmar
Honors Theses
In the study of curves there are many interesting theorems. One such theorem is the four vertex theorem and its converse. The four vertex theorem says that any simple closed curve, other than a circle, must have four vertices. This means that the curvature of the curve must have at least four local maxima/minima. In my project I explore different proofs of the four vertex theorem and its history. I also look at a modified converse of the four vertex theorem which says that any continuous real- valued function on the circle that has at least two local maxima and …
Elliptic Curve Cryptology, Francis Rocco
Elliptic Curve Cryptology, Francis Rocco
Honors Theses
In today's digital age of conducting large portions of daily life over the Internet, privacy in communication is challenged extremely frequently and confidential information has become a valuable commodity. Even with the use of commonly employed encryption practices, private information is often revealed to attackers. This issue motivates the discussion of cryptology, the study of confidential transmissions over insecure channels, which is divided into two branches of cryptography and cryptanalysis. In this paper, we will first develop a foundation to understand cryptography and send confidential transmissions among mutual parties. Next, we will provide an expository analysis of elliptic curves and …
Primality Proving Based On Eisenstein Integers, Miaoqing Jia
Primality Proving Based On Eisenstein Integers, Miaoqing Jia
Honors Theses
According to the Berrizbeitia theorem, a highly efficient method for certifying the primality of an integer N ≡ 1 (mod 3) can be created based on pseudocubes in the ordinary integers Z. In 2010, Williams and Wooding moved this method into the Eisenstein integers Z[ω] and defined a new term, Eisenstein pseudocubes. By using a precomputed table of Eisenstein pseudocubes, they created a new algorithm in this context to prove primality of integers N ≡ 1 (mod 3) in a shorter period of time. We will look at the Eisenstein pseudocubes and analyze how this new algorithm works with the …
Reading Between The Lines: Verifying Mathematical Language, Tristan Johnson
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.
Noether's Theorem: Symmetry And Conservation, Tristan Johnson
Noether's Theorem: Symmetry And Conservation, Tristan Johnson
Honors Theses
A common calculus problem is to find an input that optimizes (maximizes or minimizes) a function. An extension of this problem is to find a function that optimizes an expression depending on the function. This paper studies how small (differentiable) variations of functions give us more information about expressions dependent on these functions. Specifically, Noether’s Theorem states that in a system of functions, each differential symmetry – or small variation where the system is invariant– constructs a conserved quantity. We will describe, interpret and prove Noether’s Theorem using techniques from linear algebra, differential geometry, and the calculus of variations. Furthermore, …
General Relativity And Differential Geometry, Harry Hausner
An Algebraic Approach To Number Theory Using Unique Factorization, Mark Sullivan
An Algebraic Approach To Number Theory Using Unique Factorization, Mark Sullivan
Honors Theses
Though it may seem non-intuitive, abstract algebra is often useful in the study of number theory. In this thesis, we explore some uses of abstract algebra to prove number theoretic statements. We begin by examining the structure of unique factorization domains in general. Then we introduce number fields and their rings of algebraic integers, whose structures have characteristics that are analogous to some of those of the rational numbers and the rational integers. Next we discuss quadratic fields, a special case of number fields that have important applications to number theoretic problems. We will use the structures that we introduce …
An Analysis Of Polynomials That Commute Under Composition, Samuel J. Williams
An Analysis Of Polynomials That Commute Under Composition, Samuel J. Williams
Honors Theses
It is well known that polynomials commute under addition and multiplication. It turns out that certain polynomials also commute under composition. In this paper, we examine polynomials with coefficients in the field of complex numbers that commute under composition (also referred to as “commuting polynomials”). We begin this examination by defining what it means for polynomials to commute under composition. We then introduce sequences of commuting polynomials and observe how the polynomials in these sequences (later defined as chains) along with other commuting polynomials relate to a concept called similarity. These observations allow us to better understand the qualities and …
Une Histoire De La Formation Mathématique En France: Les Réformes Et Les Philosophies De L’Enseignement Primaire Et Secondaire De 1420 Jusqu'À Aujourd’Hui, Rebecca Robinson
Une Histoire De La Formation Mathématique En France: Les Réformes Et Les Philosophies De L’Enseignement Primaire Et Secondaire De 1420 Jusqu'À Aujourd’Hui, Rebecca Robinson
Honors Theses
France has produced many illustrious mathematicians who have profoundly impacted mathematics as they are today. While Descartes, Cauchy, and Borel (among others) viewed math as a lifelong pursuit, they began their education in an elementary school classroom with everybody else. In this study, I explore mathematical reforms and governmental documents throughout history to show how the education system has grown to emphasize a strong mathematical curriculum for all students and have consulted many philosophical articles on both the importance of math in a student’s education as well as different views on the manner in which mathematics should be taught to …
The Calculus Of Variations, Erin Whitney
The Calculus Of Variations, Erin Whitney
Honors Theses
The Calculus of Variations is a highly applicable and advancing field. My thesis has only scraped the top of the applications and theoretical work that is possible within this branch of mathematics. To summarize, we began by exploring a general problem common to this field, finding the geodesic be-tween two given points. We then went on to define and explore terms and concepts needed to further delve into the subject matter. In Chapter 2, we examined a special set of smooth functions, inspired by the Calabi extremal metric, and used some general theory of convex functions in order to de-termine …
The History And Mathematics Behind The Construction Of The Islamic Astrolabe, Lyda P. Urresta
The History And Mathematics Behind The Construction Of The Islamic Astrolabe, Lyda P. Urresta
Honors Theses
In this paper, we examine the mathematical methods employed in the construction of the astrolabe, an ancient measuring device used to solve problems in the field of astronomy. Essentially, the astrolabe is a two dimensional representation of the heavens obtained by projecting the celestial sphere onto the plane. Though several different types of astrolabes exist, our primary focus is on the most popular deisgn, which is created by the stereographic projection of the celestial sphere onto the plane defined by the equator with the south pole as the projection point.
Factorization Of Primes Primes Primes: Elements Ideals And In Extensions, Peter J. Bonventre
Factorization Of Primes Primes Primes: Elements Ideals And In Extensions, Peter J. Bonventre
Honors Theses
It is often taken it for granted that all positive whole numbers except 0 and 1 can be factored uniquely into primes. However, if K is a finite extension of the rational numbers, and OK its ring of integers, it is not always the case that non-zero, non-unit elements of OK factor uniquely. We do find, though, that the proper ideals of OK do always factor uniquely into prime ideals. This result allows us to extend many properties of the integers to these rings. If we a finite extension L of K and OL of OK , we find that …