Mathematical Reasoning And The Inductive Process: An Examination Of The Law Of Quadratic Reciprocity, 2016 California State University - San Bernardino

#### Mathematical Reasoning And The Inductive Process: An Examination Of The Law Of Quadratic Reciprocity, Nitish Mittal

*Electronic Theses, Projects, and Dissertations*

This project investigates the development of four different proofs of the law of quadratic reciprocity, in order to study the critical reasoning process that drives discovery in mathematics. We begin with an examination of the first proof of this law given by Gauss. We then describe Gauss’ fourth proof of this law based on Gauss sums, followed by a look at Eisenstein’s geometric simplification of Gauss’ third proof. Finally, we finish with an examination of one of the modern proofs of this theorem published in 1991 by Rousseau. Through this investigation we aim to analyze the different strategies used ...

Development Of Utility Theory And Utility Paradoxes, 2016 Lawrence University

#### Development Of Utility Theory And Utility Paradoxes, Timothy E. Dahlstrom

*Lawrence University Honors Projects*

Since the pioneering work of von Neumann and Morgenstern in 1944 there have been many developments in Expected Utility theory. In order to explain decision making behavior economists have created increasingly broad and complex models of utility theory. This paper seeks to describe various utility models, how they model choices among ambiguous and lottery type situations, and how they respond to the Ellsberg and Allais paradoxes. This paper also attempts to communicate the historical development of utility models and provide a fresh perspective on the development of utility models.

Exploring Argumentation, Objectivity, And Bias: The Case Of Mathematical Infinity, 2016 University of Ontario Institute of Technology

#### Exploring Argumentation, Objectivity, And Bias: The Case Of Mathematical Infinity, Ami Mamolo

*OSSA Conference Archive*

This paper presents an overview of several years of my research into individuals’ reasoning, argumentation, and bias when addressing problems, scenarios, and symbols related to mathematical infinity. There is a long history of debate around what constitutes “objective truth” in the realm of mathematical infinity, dating back to ancient Greece (e.g., Dubinsky et al., 2005). Modes of argumentation, hindrances, and intuitions have been largely consistent over the years and across levels of expertise (e.g., Brown et al., 2010; Fischbein et al., 1979, Tsamir, 1999). This presentation examines the interrelated complexities of notions of objectivity, bias, and argumentation as ...

The Development Of Notation In Mathematical Analysis, 2016 Loyola Marymount University

#### The Development Of Notation In Mathematical Analysis, Alyssa Venezia

*Honors Thesis*

The field of analysis is a newer subject in mathematics, as it only came into existence in the last 400 years. With a new field comes new notation, and in the era of universalism, analysis becomes key to understanding how centuries of mathematics were unified into a finite set of symbols, precise definitions, and rigorous proofs that would allow for the rapid development of modern mathematics. This paper traces the introduction of subjects and the development of new notations in mathematics from the seventeenth to the nineteenth century that allowed analysis to flourish. In following the development of analysis, we ...

On The Conjugacy Problem For Automorphisms Of Trees, 2016 Boise State University

#### On The Conjugacy Problem For Automorphisms Of Trees, Kyle Douglas Beserra

*Boise State University Theses and Dissertations*

In this thesis we identify the complexity of the conjugacy problem of automorphisms of regular trees. We expand on the results of Kechris, Louveau, and Friedman on the complexities of the isomorphism problem of classes of countable trees. We see in nearly all cases that the complexity of isomorphism of subtrees of a given regular countable tree is the same as the complexity of conjugacy of automorphisms of the same tree, though we present an example for which this does not hold.

Counting Convex Sets On Products Of Totally Ordered Sets, 2015 Western Kentucky University

#### Counting Convex Sets On Products Of Totally Ordered Sets, Brandy Amanda Barnette

*Masters Theses & Specialist Projects*

The main purpose of this thesis is to find the number of convex sets on a product of two totally ordered spaces. We will give formulas to find this number for specific cases and describe a process to obtain this number for all such spaces. In the first chapter we briefly discuss the motivation behind the work presented in this thesis. Also, the definitions and notation used throughout the paper are introduced here The second chapter starts with examining the product spaces of the form {1; 2; : : : ;n} × {1; 2}. That is, we begin by analyzing a two-row by n-column ...

My Finite Field, 2015 Idaho State University

#### My Finite Field, Matthew Schroeder

*Journal of Humanistic Mathematics*

A love poem written in the language of mathematics.

Exploring A Generalized Partial Borda Count Voting System, 2015 Bard College

#### Exploring A Generalized Partial Borda Count Voting System, Christiane Koffi

*Senior Projects Spring 2015*

The main purpose of an election is to generate a fair end result in which everyone's opinion is gathered into a collective decision. This project focuses on Voting Theory, the mathematical study of voting systems. Because different voting systems yield different end results, the challenge begins with finding a voting system that will result in a fair election. Although there are many different voting systems, in this project we focus on the Partial Borda Count Voting System, which uses partially ordered sets (posets), instead of the linearly ordered ballots used in traditional elections, to rank its candidates. We introduce ...

On Some Min-Max Cardinals On Boolean Algebras, 2015 University of Colorado Boulder

#### On Some Min-Max Cardinals On Boolean Algebras, Kevin Selker

*Mathematics Graduate Theses & Dissertations*

This thesis is concerned with cardinal functions on Boolean Algebras (BAs) in general, and especially with min-max type functions on atomless BAs. The thesis is in two parts:

(1) We make use of a forcing technique for extending Boolean algebras.

elsewhere. Using and modifying a lemma of Koszmider, and using CH, we prove some general extension lemmas, and in particular obtain an atomless BA, *A* such that *f*(A) = s_{mm}(A) = *w* < *u*(A) = *w*_{1}.

(2) We investigate cardinal functions of min-max and max type and also spectrum functions on moderate products of Boolean algebras. We prove several ...

From Nonlinear Embedding To Graph Distances: A Spectral Perspective, 2015 University of Colorado Boulder

#### From Nonlinear Embedding To Graph Distances: A Spectral Perspective, Nathan D. Monnig

*Applied Mathematics Graduate Theses & Dissertations*

In this thesis, we explore applications of spectral graph theory to the analysis of complex datasets and networks. We consider spectral embeddings of general graphs, as well as data sampled from smooth manifolds in high dimension. We specifically focus on the development of algorithms that require minimal user input. Given the inherent difficulty in parameterizing these types of complex datasets, an ideal algorithm should avoid poorly-defined user-selected parameters.

A significant limitation of nonlinear dimensionality reduction embeddings computed from datasets is the absence of a mechanism to compute the inverse map. We address the problem of computing a stable inverse using ...

Morphological Operations Applied To Digital Art Restoration, 2014 University of Minnesota, Morris

#### Morphological Operations Applied To Digital Art Restoration, M. Kirbie Dramdahl

*Scholarly Horizons: University of Minnesota, Morris Undergraduate Journal*

This paper provides an overview of the processes involved in detecting and removing cracks from digitized works of art. Speciﬁc attention is given to the crack detection phase as completed through the use of morphological operations. Mathematical morphology is an area of set theory applicable to image processing, and therefore lends itself eﬀectively to the digital art restoration process.

The Mathematics Of The Card Game Set, 2014 Rhode Island College

#### The Mathematics Of The Card Game Set, Paola Y. Reyes

*Honors Projects Overview*

SET is a card game of visual perception. The goal is to be the first to see a SET from the 12 cards laid face up on the table. Each card has four attributes, which can vary as follows: 1. Shape: oval, squiggle, or diamond 2. Color: red, green, or blue 3. Number: the number of copies of each symbol can be 1, 2, or 3 4. Filling: solid, unfilled, stripped Each card has a unique combination, for a total of 34 = 81 different cards in a deck. A SET consist of three cards for which each of the four ...

Axioms Of Set Theory And Equivalents Of Axiom Of Choice, 2014 Boise State University

#### Axioms Of Set Theory And Equivalents Of Axiom Of Choice, Farighon Abdul Rahim

*Mathematics Undergraduate Theses*

Sets are all around us. A bag of potato chips, for instance, is a set containing certain number of individual chip's that are its elements. University is another example of a set with students as its elements. By elements, we mean members. But sets should not be confused as to what they really are. A daughter of a blacksmith is an element of a set that contains her mother, father, and her siblings. Then this set is an element of a set that contains all the other families that live in the nearby town. So a set itself can ...

Permutation Groups And Puzzle Tile Configurations Of Instant Insanity Ii, 2014 East Tennessee State University

#### Permutation Groups And Puzzle Tile Configurations Of Instant Insanity Ii, Amanda N. Justus

*Electronic Theses and Dissertations*

The manufacturer claims that there is only one solution to the puzzle Instant Insanity II. However, a recent paper shows that there are two solutions. Our goal is to find ways in which we only have one solution. We examine the permutation groups of the puzzle and use modern algebra to attempt to fix the puzzle. First, we find the permutation group for the case when there is only one empty slot at the top. We then examine the scenario when we add an extra column or an extra row to make the game a 4 × 5 puzzle or a ...

An Explicit Construction Of Kleinian Groups With Small Limit Sets, 2014 Selected Works

#### An Explicit Construction Of Kleinian Groups With Small Limit Sets, Andrew Lazowski

*Andrew Lazowski*

This paper provides an explicit construction of Kleinian groups that have small Hausdorff dimension of their limit sets. It is known that such groups exist and they can be constructed by results of Patterson. The purpose here is to work out the methods of calculation.

An Explicit Construction Of Kleinian Groups With Small Limit Sets, 2014 Sacred Heart University

#### An Explicit Construction Of Kleinian Groups With Small Limit Sets, Andrew Lazowski

*Mathematics Faculty Publications*

This paper provides an explicit construction of Kleinian groups that have small Hausdorff dimension of their limit sets. It is known that such groups exist and they can be constructed by results of Patterson. The purpose here is to work out the methods of calculation.

An Introduction To Set Theory And Topology, 2014 Washington University in St. Louis

#### An Introduction To Set Theory And Topology, Ronald C. Freiwald

*Books and Monographs*

These notes are an introduction to set theory and topology. They are the result of teaching a two-semester course sequence on these topics for many years at Washington University in St. Louis. Typically the students were advanced undergraduate mathematics majors, a few beginning graduate students in mathematics, and some graduate students from other areas that included economics and engineering. The usual background for the material is an introductory undergraduate analysis course, mostly because it provides a solid introduction to Euclidean space Rn and practice with rigorous arguments — in particular, about continuity. Strictly speaking, however, the material is mostly self-contained. Examples ...

A Systematic Martingale Construction With Applications To Permutation Inequalities, 2013 University of Pennsylvania

#### A Systematic Martingale Construction With Applications To Permutation Inequalities, Vladimir Pozdnyakov, John M. Steele

*Operations, Information and Decisions Papers*

We illustrate a process that constructs martingales with help from matrix products that arise naturally in the theory of sampling without replacement. The usefulness of the new martingales is illustrated by the development of maximal inequalities for permuted sequences of real numbers. Some of these inequalities are new and some are variations of classical inequalities like those introduced by A. Garsia in the study of rearrangement of orthogonal series.

Generalized Ordered Whist Tournaments For 6n+1 Players, 2013 Rhode Island College

#### Generalized Ordered Whist Tournaments For 6n+1 Players, Elyssa Cipriano

*Honors Projects Overview*

In this project, we worked to see if it would be possible to extend the idea of an ordered whist tournament to a generalized whist tournament on 6n or 6n + 1 players. We focused on tournaments where the players are divided into n games of size 6 each consisting of two teams of size 3. We aimed to balance the 3 occasions where the players meet as opponents.

Difference Sets In Non-Abelian Groups Of Order 256, 2013 University of Richmond

#### Difference Sets In Non-Abelian Groups Of Order 256, Taylor Applebaum

*Honors Theses*

This paper considers the problem of determining which of the 56092 groups of order 256 contain (256; 120; 56; 64) difference sets. John Dillon at the National Security Agency communicated 724 groups which were still open as of August 2012. In this paper, we present a construction method for groups containing a normal subgroup isomorphic to Z4 Z4 Z2 . This construction method was able to produce difference sets in 643 of the 649 unsolved groups with the correct normal subgroup. These constructions elimated approximately 90% of the open cases, leaving 81 remaining unsolved groups.