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Articles 1  30 of 91
FullText Articles in Set Theory
Patterns, Symmetries, And Mathematical Structures In The Arts, Sarah C. Deloach
Patterns, Symmetries, And Mathematical Structures In The Arts, Sarah C. Deloach
University Honors Program Theses
Mathematics is a discipline of academia that can be found everywhere in the world around us. Mathematicians and scientists are not the only people who need to be proficient in numbers. Those involved in social sciences and even the arts can benefit from a background in math. In fact, connections between mathematics and various forms of art have been discovered since as early as the fourth century BC. In this thesis we will study such connections and related concepts in mathematics, dances, and music.
Properties Of Functionally Alexandroff Topologies And Their Lattice, Jacob Scott Menix
Properties Of Functionally Alexandroff Topologies And Their Lattice, Jacob Scott Menix
Masters Theses & Specialist Projects
This thesis explores functionally Alexandroff topologies and the order theory asso ciated when considering the collection of such topologies on some set X. We present several theorems about the properties of these topologies as well as their partially ordered set.
The first chapter introduces functionally Alexandroff topologies and motivates why this work is of interest to topologists. This chapter explains the historical context of this relatively new type of topology and how this work relates to previous work in topology. Chapter 2 presents several theorems describing properties of functionally Alexandroff topologies ad presents a characterization for the functionally Alexandroff topologies ...
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 biologicallyinspired 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.
An Alternative Almost Sure Construction Of Gaussian Stochastic Processes In The L2([0,1]) Space, Kevin Chen
An Alternative Almost Sure Construction Of Gaussian Stochastic Processes In The L2([0,1]) Space, Kevin Chen
Honors Projects
No abstract provided.
Computable Reducibility Of Equivalence Relations, Marcello Gianni Krakoff
Computable Reducibility Of Equivalence Relations, Marcello Gianni Krakoff
Boise State University Theses and Dissertations
Computable reducibility of equivalence relations is a tool to compare the complexity of equivalence relations on natural numbers. Its use is important to those doing Borel equivalence relation theory, computability theory, and computable structure theory. In this thesis, we compare many naturally occurring equivalence relations with respect to computable reducibility. We will then define a jump operator on equivalence relations and study proprieties of this operation and its iteration. We will then apply this new jump operation by studying its effect on the isomorphism relations of wellfounded computable trees.
Group Theoretical Analysis Of Arbitrarily Large, Colored Square Grids, Brett Ehrman
Group Theoretical Analysis Of Arbitrarily Large, Colored Square Grids, Brett Ehrman
Student Scholar Showcase
In this research, we examine n x n grids whose individual squares are each colored with one of k distinct colors. We seek a general formula for the number of colored grids that are distinct up to rotations, reflections, and color reversals. We examine the problem using a group theoretical approach. We define a specific group action that allows us to incorporate Burnside’s Lemma, which leads us to the desired general results
Experience Of A NoyceStudent Learning Assistant In An InquiryBased Learning Class, Melissa Riley
Experience Of A NoyceStudent Learning Assistant In An InquiryBased Learning Class, Melissa Riley
Student Research and Creative Activity Fair
This presentation refers to an undergraduate course called introduction to abstract mathematics at the University of Nebraska at Omaha. During the academic year 20172018, undergraduate, mathematics student Melissa Riley was a Noycestudent learning assistant for the Inquiry Based Learning (IBL) section of the course. She assisted the facultyincharge with all aspects of the course. These included: materials preparation, class organization, teamwork, class leading, presentations, and tutoring. This presentation shall address some examples of how the IBL approach can be used in this type of class including: the structure of the course, the activities and tasks performed by the students, learning ...
Some Intuition Behind Large Cardinal Axioms, Their Characterization, And Related Results, Philip A. White
Some Intuition Behind Large Cardinal Axioms, Their Characterization, And Related Results, Philip A. White
Theses and Dissertations
We aim to explain the intuition behind several large cardinal axioms, give characterization theorems for these axioms, and then discuss a few of their properties. As a capstone, we hope to introduce a new large cardinal notion and give a similar characterization theorem of this new notion. Our new notion of near strong compactness was inspired by the similar notion of near supercompactness, due to Jason Schanker.
TutteEquivalent Matroids, Maria Margarita Rocha
TutteEquivalent Matroids, Maria Margarita Rocha
Electronic Theses, Projects, and Dissertations
We begin by introducing matroids in the context of finite collections of vectors from a vector space over a specified field, where the notion of independence is linear independence. Then we will introduce the concept of a matroid invariant. Specifically, we will look at the Tutte polynomial, which is a welldefined twovariable invariant that can be used to determine differences and similarities between a collection of given matroids. The Tutte polynomial can tell us certain properties of a given matroid (such as the number of bases, independent sets, etc.) without the need to manually solve for them. Although the Tutte ...
Elementary Set Theory, Richard P. Millspaugh
Elementary Set Theory, Richard P. Millspaugh
Open Educational Resources
This text is appropriate for a course that introduces undergraduates to proofs. The material includes elementary symbolic logic, logical arguments, basic set theory, functions and relations, the real number system, and an introduction to cardinality. The text is intended to be readable for sophomore and better freshmen majoring in mathematics.
The source files for the text can be found at https://github.com/RPMillspaugh/SetTheory
Selective Strong Screenability, Isaac Joseph Coombs
Selective Strong Screenability, Isaac Joseph Coombs
Boise State University Theses and Dissertations
Screenability and strong screenability were both introduced some sixty years ago by R.H. Bing in his paper Metrization of Topological Spaces. Since then, much work has been done in exploring selective screenability (the selective version of screenability). However, the corresponding selective version of strong screenability has been virtually ignored. In this paper we seek to remedy this oversight. It is found that a great deal of the proofs about selective screenability readily carry over to proofs for the analogous version for selective strong screenability. We give some examples of selective strongly screenable spaces with the primary example being Pol ...
The Structure Of Models Of SecondOrder Set Theories, Kameryn J. Williams
The Structure Of Models Of SecondOrder Set Theories, Kameryn J. Williams
All Dissertations, Theses, and Capstone Projects
This dissertation is a contribution to the project of secondorder set theory, which has seen a revival in recent years. The approach is to understand secondorder set theory by studying the structure of models of secondorder set theories. The main results are the following, organized by chapter. First, I investigate the poset of Trealizations of a fixed countable model of ZFC, where T is a reasonable secondorder set theory such as GBC or KM, showing that it has a rich structure. In particular, every countable partial order embeds into this structure. Moreover, we can arrange so that these embedding preserve ...
Distributive Lattice Models Of The Type C OneRowed Weyl Group Symmetric Functions, William Atkins
Distributive Lattice Models Of The Type C OneRowed Weyl Group Symmetric Functions, William Atkins
Murray State Theses and Dissertations
We present two families of diamondcolored distributive lattices – one known and one new – that we can show are models of the type C onerowed Weyl symmetric functions. These lattices are constructed using certain sequences of positive integers that are visualized as ﬁlling the boxes of onerowed partition diagrams. We show how natural orderings of these onerowed tableaux produce our distributive lattices as sublattices of a more general object, and how a natural coloring of the edges of the associated order diagrams yields a certain diamondcoloring property. We show that each edgecolored lattice possesses a certain structure that is associated with ...
Introduction To Game Theory: A Discovery Approach, Jennifer Firkins Nordstrom
Introduction To Game Theory: A Discovery Approach, Jennifer Firkins Nordstrom
Linfield Authors Book Gallery
Game theory is an excellent topic for a nonmajors quantitative course as it develops mathematical models to understand human behavior in social, political, and economic settings. The variety of applications can appeal to a broad range of students. Additionally, students can learn mathematics through playing games, something many choose to do in their spare time! This text also includes an exploration of the ideas of game theory through the rich context of popular culture. It contains sections on applications of the concepts to popular culture. It suggests films, television shows, and novels with themes from game theory. The questions in ...
Transfinite Ordinal Arithmetic, James Roger Clark
Transfinite Ordinal Arithmetic, James Roger Clark
All Student Theses
Following the literature from the origin of Set Theory in the late 19th century to more current times, an arithmetic of finite and transfinite ordinal numbers is outlined. The concept of a set is outlined and directed to the understanding that an ordinal, a special kind of number, is a particular kind of wellordered set. From this, the idea of counting ordinals is introduced. With the fundamental notion of counting addressed: then addition, multiplication, and exponentiation are defined and developed by established fundamentals of Set Theory. Many known theorems are based upon this foundation. Ultimately, as part of the conclusion ...
Joint Laver Diamonds And Grounded Forcing Axioms, Miha Habič
Joint Laver Diamonds And Grounded Forcing Axioms, Miha Habič
All Dissertations, Theses, and Capstone Projects
In chapter 1 a notion of independence for diamonds and Laver diamonds is investigated. A sequence of Laver diamonds for κ is joint if for any sequence of targets there is a single elementary embedding j with critical point κ such that each Laver diamond guesses its respective target via j. In the case of measurable cardinals (with similar results holding for (partially) supercompact cardinals) I show that a single Laver diamond for κ yields a joint sequence of length κ, and I give strict separation results for all larger lengths of joint sequences. Even though the principles get strictly ...
Classification Of VertexTransitive Structures, Stephanie Potter
Classification Of VertexTransitive Structures, Stephanie Potter
Boise State University Theses and Dissertations
When one thinks of objects with a significant level of symmetry it is natural to expect there to be a simple classification. However, this leads to an interesting problem in that research has revealed the existence of highly symmetric objects which are very complex when considered within the framework of Borel complexity. The tension between these two seemingly contradictory notions leads to a wealth of natural questions which have yet to be answered.
Borel complexity theory is an area of logic where the relative complexities of classification problems are studied. Within this theory, we regard a classification problem as an ...
The Classification Problem For Models Of Zfc, Samuel Dworetzky
The Classification Problem For Models Of Zfc, Samuel Dworetzky
Boise State University Theses and Dissertations
Models of ZFC are ubiquitous in modern day set theoretic research. There are many different constructions that produce countable models of ZFC via techniques such as forcing, ultraproducts, and compactness. The models that these techniques produce have many different characteristics; thus it is natural to ask whether or not models of ZFC are classifiable. We will answer this question by showing that models of ZFC are unclassifiable and have maximal complexity. The notions of complexity used in this thesis will be phrased in the language of Borel complexity theory.
In particular, we will show that the class of countable models ...
Combinatorial Polynomial Hirsch Conjecture, Sam Miller
Combinatorial Polynomial Hirsch Conjecture, Sam Miller
HMC Senior Theses
The Hirsch Conjecture states that for a ddimensional polytope with n facets, the diameter of the graph of the polytope is at most nd. This conjecture was disproven in 2010 by Francisco Santos Leal. However, a polynomial bound in n and d on the diameter of a polytope may still exist. Finding a polynomial bound would provide a worstcase scenario runtime for the Simplex Method of Linear Programming. However working only with polytopes in higher dimensions can prove challenging, so other approaches are welcome. There are many equivalent formulations of the Hirsch Conjecture, one of which is the Combinatorial ...
The Density Topology On The Reals With Analogues On Other Spaces, Stuart Nygard
The Density Topology On The Reals With Analogues On Other Spaces, Stuart Nygard
Boise State University Theses and Dissertations
A point x is a density point of a set A if all of the points except a measure zero set near to x are contained in A. In the usual topology on ℝ, a set is open if shrinking intervals around each point are eventually contained in the set. The density topology relaxes this requirement. A set is open in the density topology if for each point, the limit of the measure of A contained in shirking intervals to the measure of the shrinking intervals themselves is one. That is, for any point x and a small enough interval ...
Mathematical Reasoning And The Inductive Process: An Examination Of The Law Of Quadratic Reciprocity, Nitish Mittal
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, Timothy E. Dahlstrom
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, Ami Mamolo
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, Alyssa Venezia
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, Kyle Douglas Beserra
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, Brandy Amanda Barnette
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 tworow by ncolumn ...
My Finite Field, Matthew Schroeder
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, Christiane Koffi
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 MinMax Cardinals On Boolean Algebras, Kevin Selker
On Some MinMax 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 minmax 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 minmax 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, Nathan D. Monnig
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 poorlydefined userselected 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 ...