Other Mathematics Commons^™

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 …

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 …

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 …

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 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

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 …

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 …