Other Mathematics Commons^™

Using Game Theory To Model Tripolar Deterrence And Escalation Dynamics, Grace Farson

Honors Theses

The study investigated how game theory can been utilized to model multipolar escalation dynamics between Russia, China, and the United States. In addition, the study focused on analyzing various parameters that affected potential conflict outcomes to further new deterrence thought in a tripolar environment.

A preliminary game theoretic model was created to model and analyze escalation dynamics. The model was built upon framework presented by Zagare and Kilgour in their work ‘Perfect Deterrence’. The model is based on assumptions and rules set prior to game play. The model was then analyzed based upon these assumptions using a form of mathematical …

Remotely Close: An Investigation Of The Student Experience In First-Year Mathematics Courses During The Covid-19 Pandemic, Sawyer Smith

Honors Theses

The realm of education was shaken by the onset of the COVID-19 pandemic in 2020. It had drastic effects on the way that courses were delivered to students, and the way that students were getting their education at the collegiate level. At the University of Nebraska – Lincoln, the pandemic dramatically changed the way that first-year mathematics courses looked for students. By Spring 2021, students had the opportunity to take their first-year math courses either in-person or virtually. This project sought to identify differences between the two methods of course delivery during the Spring 2021 semester, regarding interaction with peers …

Existence And Uniqueness Of Minimizers For A Nonlocal Variational Problem, Michael Pieper

Honors Theses

Nonlocal modeling is a rapidly growing field, with a vast array of applications and connections to questions in pure math. One goal of this work is to present an approachable introduction to the field and an invitation to the reader to explore it more deeply. In particular, we explore connections between nonlocal operators and classical problems in the calculus of variations. Using a well-known approach, known simply as The Direct Method, we establish well-posedness for a class of variational problems involving a nonlocal first-order differential operator. Some simple numerical experiments demonstrate the behavior of these problems for specific choices of …

Free Semigroupoid Algebras From Categories Of Paths, Juliana Bukoski

Department of Mathematics: Dissertations, Theses, and Student Research

Given a directed graph G, we can define a Hilbert space H_G with basis indexed by the path space of the graph, then represent the vertices of the graph as projections on H_G and the edges of the graph as partial isometries on H_G. The weak operator topology closed algebra generated by these projections and partial isometries is called the free semigroupoid algebra for G. Kribs and Power showed that these algebras are reflexive, and that they are semisimple if and only if each path in the graph lies on a cycle. We extend …

Exploring Pedagogical Empathy Of Mathematics Graduate Student Instructors, Karina Uhing

Department of Mathematics: Dissertations, Theses, and Student Research

Interpersonal relationships are central to the teaching and learning of mathematics. One way that teachers relate to their students is by empathizing with them. In this study, I examined the phenomenon of pedagogical empathy, which is defined as empathy that influences teaching practices. Specifically, I studied how mathematics graduate student instructors conceptualize pedagogical empathy and analyzed how pedagogical empathy might influence their teaching decisions. To address my research questions, I designed a qualitative phenomenological study in which I conducted observations and interviews with 11 mathematics graduate student instructors who were teaching precalculus courses at the University of Nebraska—Lincoln.

In the …

A Note On The Fine Structure Constant, Bilal Khan, Irshadullah Khan

CSE Technical Reports

We derive the numerical value of the fine structure constant in purely number-theoretic terms, under the assumption that in a system of charges between two parallel conducting plates, the Casimir energy and the mutual Coulomb interaction energy agree.

A Note On The Fine Structure Constant, Bilal Khan, Irshadullah Khan

Department of Sociology: Faculty Publications

We derive the numerical value of the fine structure constant $\alpha$ in purely number-theoretic terms, under the assumption that in a system of charges between two parallel conducting plates, the Casimir energy and the mutual Coulomb interaction energy agree.

A Note On The Fine Structure Constant, Bilal Khan, Irshadullah Khan

Department of Sociology: Faculty Publications

We derive the numerical value of the fine structure constant in purely number-theoretic terms, under the assumption that in a system of charges between two parallel conducting plates, the Casimir energy and the mutual Coulomb interaction energy agree.

Operator Algebras Generated By Left Invertibles, Derek Desantis

Department of Mathematics: Dissertations, Theses, and Student Research

Operator algebras generated by partial isometries and their adjoints form the basis for some of the most well studied classes of C*-algebras. Representations of such algebras encode the dynamics of orthonormal sets in a Hilbert space.We instigate a research program on concrete operator algebras that model the dynamics of Hilbert space frames.

The primary object of this thesis is the norm-closed operator algebra generated by a left invertible $T$ together with its Moore-Penrose inverse $T^\dagger$. We denote this algebra by $\mathfrac{A}_T$. In the isometric case, $T^\dagger = T^*$ and $\mathfrac{A}_T$ is a representation of the Toeplitz algebra. Of particular interest …

Supporting English Language Learners Inside The Mathematics Classroom: One Teacher’S Unique Perspective Working With Students During Their First Years In America, Amy Marie Fendrick

Research and Evaluation in Education, Technology, Art, and Design

Reflecting upon my personal experiences teaching mathematics to English Language Learners (ELL) in a public high school in Lincoln, Nebraska, this essay largely focuses on the time I spent as the only Accelerated Math teacher in my school building. From 2012 – 2017, I taught three different subjects at this high school: Advanced Algebra, Algebra, and Accelerated Math. This essay highlights why I chose to become a math and ELL teacher, as well as the challenges, issues, struggles, and successes I experienced during my time teaching. I focus on the challenges I faced teaching students who did not share my …

On Coding For Partial Erasure Channels, Carolyn Mayer

Department of Mathematics: Dissertations, Theses, and Student Research

Error correcting codes have been essential to the technology we use in everyday life in digital storage, wireless communication, barcodes, and much more. Different channel models are used for different types of communication (for example, if information is sent to one person or to many people) and different types of errors. Partial erasure channels were recently introduced to model applications in which some information remains after an erasure event. These remnants of information may be used to increase the chances of successful decoding. We introduce a new partial erasure channel in which partial erasures correspond to individual bit erasures in …

Essentials Of Structural Equation Modeling, Mustafa Emre Civelek

Zea E-Books Collection

Structural Equation Modeling is a statistical method increasingly used in scientific studies in the fields of Social Sciences. It is currently a preferred analysis method, especially in doctoral dissertations and academic researches. However, since many universities do not include this method in the curriculum of undergraduate and graduate courses, students and scholars try to solve the problems they encounter by using various books and internet resources.

This book aims to guide the researcher who wants to use this method in a way that is free from math expressions. It teaches the steps of a research program using structured equality modeling …

Design And Analysis Of Graph-Based Codes Using Algebraic Lifts And Decoding Networks, Allison Beemer

Department of Mathematics: Dissertations, Theses, and Student Research

Error-correcting codes seek to address the problem of transmitting information efficiently and reliably across noisy channels. Among the most competitive codes developed in the last 70 years are low-density parity-check (LDPC) codes, a class of codes whose structure may be represented by sparse bipartite graphs. In addition to having the potential to be capacity-approaching, LDPC codes offer the significant practical advantage of low-complexity graph-based decoding algorithms. Graphical substructures called trapping sets, absorbing sets, and stopping sets characterize failure of these algorithms at high signal-to-noise ratios. This dissertation focuses on code design for and analysis of iterative graph-based message-passing decoders. The …

The Impact Of Truncating Data On The Predictive Ability For Single-Step Genomic Best Linear Unbiased Prediction, Jeremy T. Howard, Thomas A. Rathje, Caitlyn E. Bruns, Danielle F. Wilson-Wells, Stephen D. Kachman, Matthew L. Spangler

Department of Animal Science: Faculty Publications

Simulated and swine industry data sets were utilized to assess the impact of removing older data on the predictive ability of selection candidate estimated breeding values (EBV) when using single-step genomic best linear unbiased prediction (ssGBLUP). Simulated data included thirty replicates designed to mimic the structure of swine data sets. For the simulated data, varying amounts of data were truncated based on the number of ancestral generations back from the selection candidates. The swine data sets consisted of phenotypic and genotypic records for three traits across two breeds on animals born from 2003 to 2017. Phenotypes and genotypes were iteratively …

Six Septembers: Mathematics For The Humanist, Patrick Juola, Stephen Ramsay

Zea E-Books Collection

Scholars of all stripes are turning their attention to materials that represent enormous opportunities for the future of humanistic inquiry. The purpose of this book is to impart the concepts that underlie the mathematics they are likely to encounter and to unfold the notation in a way that removes that particular barrier completely. This book is a primer for developing the skills to enable humanist scholars to address complicated technical material with confidence. This book, to put it plainly, is concerned with the things that the author of a technical article knows, but isn’t saying. Like any field, mathematics operates …

The Existence Of Solutions For A Nonlinear, Fractional Self-Adjoint Difference Equation, Kevin Ahrendt

Department of Mathematics: Dissertations, Theses, and Student Research

In this work we will explore a fractional self-adjoint difference equation which involves a Caputo fractional difference. In particular, we will develop a Cauchy function for initial value problems and Green's functions for several different types of boundary value problems. We will use the properties of those Green's functions and the Contraction Mapping Theorem to find sufficient conditions for when a nonlinear boundary value problem has a unique solution. We will also investigate the existence of nonnegative solutions for a nonlinear self-adjoint difference that have particular long run behavior.

Adviser: Allan Peterson

Sine, Cosine, And Tangent Table: 0 To 360 Degrees, Paul Royster

Department of Mathematics: Class Notes and Learning Materials

In helping with my high school student's math homework, I was astonished to find no trig tables in the 800-page textbook. I was further astonished to find no printable version online that extended beyond 90°.

While most smartphones will tell you the sine of an angle, they will not necessarily tell you the angle for which the sine is x. And since multiple angles may have the same sine (e.g. 59° and 121°), it seems useful to see the numerical progression of the functions in addition to their graphical representation.

Here is a printable sine-cosine-tangent table for all integer angle …

Applications Of Discrete Mathematics For Understanding Dynamics Of Synapses And Networks In Neuroscience, Caitlyn Parmelee

Department of Mathematics: Dissertations, Theses, and Student Research

Mathematical modeling has broad applications in neuroscience whether we are modeling the dynamics of a single synapse or the dynamics of an entire network of neurons. In Part I, we model vesicle replenishment and release at the photoreceptor synapse to better understand how visual information is processed. In Part II, we explore a simple model of neural networks with the goal of discovering how network structure shapes the behavior of the network.

Vision plays an important role in how we interact with our environments. To fully understand how visual information is processed requires an understanding of the way signals are …

Invariant Basis Number And Basis Types For C*-Algebras, Philip M. Gipson

Department of Mathematics: Dissertations, Theses, and Student Research

We develop the property of Invariant Basis Number (IBN) in the context of C*-algebras and their Hilbert modules. A complete K-theoretic characterization of C*- algebras with IBN is given. A scheme for classifying C*-algebras which do not have IBN is given and we prove that all such classes are realized. We investigate the invariance of IBN, or lack thereof, under common C*-algebraic construction and perturbation techniques. Finally, applications of Invariant Basis Number to the study of C*-dynamical systems and the classification program are investigated.

Adviser: David Pitts

Bioinformatic Game Theory And Its Application To Cluster Multi-Domain Proteins, Brittney Keel

Department of Mathematics: Dissertations, Theses, and Student Research

The exact evolutionary history of any set of biological sequences is unknown, and all phylogenetic reconstructions are approximations. The problem becomes harder when one must consider a mix of vertical and lateral phylogenetic signals. In this dissertation we propose a game-theoretic approach to clustering biological sequences and analyzing their evolutionary histories. In this context we use the term evolution as a broad descriptor for the entire set of mechanisms driving the inherited characteristics of a population. The key assumption in our development is that evolution tries to accommodate the competing forces of selection, of which the conservation force seeks to …

Transforming Precalculus Instruction: Evidence-Based Course Design, Wendy M. Smith

DBER Speaker Series

The UNL Mathematics Department has been focused on transforming precalculus instruction since 2012, with a goal of greater levels of student success. A short-term measure of student success is the passing rate (C or better), which has jumped from an average of 62% (2007-2011) to 80% for the past two falls. A longer-term measure of student success is recruiting and retaining undergraduates to STEM disciplines and careers. In this talk I will share specifics of the reform efforts (the who-what-when-where-why-and-how), and also share preliminary results from the research we have simultaneously been conducting into the reform efforts.

Clique Topology Reveals Intrinsic Geometric Structure In Neural Correlations, Chad Giusti, Eva Pastalkova, Carina Curto, Vladimir Itskov

Department of Mathematics: Faculty Publications

Detecting meaningful structure in neural activity and connectivity data is challenging in the presence of hidden nonlinearities, where traditional eigenvalue-based methods may be misleading. We introduce a novel approach to matrix analysis, called clique topology, that extracts features of the data invariant under nonlinear monotone transformations. These features can be used to detect both random and geometric structure, and depend only on the relative ordering of matrix entries. We then analyzed the activity of pyramidal neurons in rat hippocampus, recorded while the animal was exploring a 2D environment, and confirmed that our method is able to detect geometric organization using …

A Game-Theoretic Analysis Of The Nuclear Non-Proliferation Treaty, Peter Revesz

CSE Conference and Workshop Papers

Although nuclear non-proliferation is an almost universal human desire, in practice, the negotiated treaties appear unable to prevent the steady growth of the number of states that have nuclear weapons. We propose a computational model for understanding the complex issues behind nuclear arms negotiations, the motivations of various states to enter a nuclear weapons program and the ways to diffuse crisis situations.

The Neural Ring: Using Algebraic Geometry To Analyze Neural Codes, Nora Youngs

Department of Mathematics: Dissertations, Theses, and Student Research

Neurons in the brain represent external stimuli via neural codes. These codes often arise from stimulus-response maps, associating to each neuron a convex receptive field. An important problem confronted by the brain is to infer properties of a represented stimulus space without knowledge of the receptive fields, using only the intrinsic structure of the neural code. How does the brain do this? To address this question, it is important to determine what stimulus space features can - in principle - be extracted from neural codes. This motivates us to define the neural ring and a related neural ideal, algebraic objects …

Random Search Models Of Foraging Behavior: Theory, Simulation, And Observation., Ben C. Nolting

Department of Mathematics: Dissertations, Theses, and Student Research

Many organisms, from bacteria to primates, use stochastic movement patterns to find food. These movement patterns, known as search strategies, have recently be- come a focus of ecologists interested in identifying universal properties of optimal foraging behavior. In this dissertation, I describe three contributions to this field. First, I propose a way to extend Charnov's Marginal Value Theorem to the spatially explicit framework of stochastic search strategies. Next, I describe simulations that compare the efficiencies of sensory and memory-based composite search strategies, which involve switching between different behavioral modes. Finally, I explain a new behavioral analysis protocol for identifying the …

Development And Application Of Difference And Fractional Calculus On Discrete Time Scales, Tanner J. Auch

Department of Mathematics: Dissertations, Theses, and Student Research

The purpose of this dissertation is to develop and apply results of both discrete calculus and discrete fractional calculus to further develop results on various discrete time scales. Two main goals of discrete and fractional discrete calculus are to extend results from traditional calculus and to unify results on the real line with those on a variety of subsets of the real line. Of particular interest is introducing and analyzing results related to a generalized fractional boundary value problem with Lidstone boundary conditions on a standard discrete domain N_a. We also introduce new results regarding exponential order for functions on …