Physical Sciences and Mathematics Commons^™

Structure For Regular Inclusions. I, David R. Pitts

Department of Mathematics: Faculty Publications

We give general structure theory for pairs (C,D) of unital C*- algebras where D is a regular and abelian C*-subalgebra of C.

When D is maximal abelian in C, we prove existence and uniqueness of a completely positive unital map E of C into the injective envelope I(D) of D such that EjD = idD; E is a useful replacement for a conditional expectation when no expectation exists. When E is faithful, (C,D) has numerous desirable properties: e.g. the linear span of the normalizers has a unique minimal C*- norm; D norms C; and isometric isomorphisms of norm-closed subalgebras lying …

Bridge Spectra Of Cables Of 2-Bridge Knots, Nicholas John Owad

Department of Mathematics: Dissertations, Theses, and Student Research

We compute the bridge spectra of cables of 2-bridge knots. We also give some results about bridge spectra and distance of Montesinos knots.

Advisors: Mark Brittenham and Susan Hermiller

Homological Characterizations Of Quasi-Complete Intersections, Jason M. Lutz

Department of Mathematics: Dissertations, Theses, and Student Research

Let R be a commutative ring, (f) an ideal of R, and E = K(f; R) the Koszul complex. We investigate the structure of the Tate construction T associated with E. In particular, we study the relationship between the homology of T, the quasi-complete intersection property of ideals, and the complete intersection property of (local) rings.

Advisers: Luchezar L. Avramov and Srikanth B. Iyengar

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 …

A Caputo Boundary Value Problem In Nabla Fractional Calculus, Julia St. Goar

Department of Mathematics: Dissertations, Theses, and Student Research

Boundary value problems have long been of interest in the continuous differential equations context. However, with the advent of new areas like Nabla Fractional Calculus, we may consider such problems in new contexts. In this work, we will consider several right focal boundary value problems, involving a Caputo fractional difference operator, in the Nabla Fractional Calculus context. Properties of the Green's functions for each of these boundary value problems will be investigated and, in the case of a particular boundary value problem, used to establish the existence of positive solutions to a nonlinear version of the boundary value problem.

Adviser: …

Stable Local Cohomology And Cosupport, Peder Thompson

Department of Mathematics: Dissertations, Theses, and Student Research

This dissertation consists of two parts, both under the overarching theme of resolutions over a commutative Noetherian ring R. In particular, we use complete resolutions to study stable local cohomology and cotorsion-flat resolutions to investigate cosupport.

In Part I, we use complete (injective) resolutions to define a stable version of local cohomology. For a module having a complete injective resolution, we associate a stable local cohomology module; this gives a functor to the stable category of Gorenstein injective modules. We show that this functor behaves much like the usual local cohomology functor. When there is only one non-zero local cohomology …

Cohen-Macaulay Dimension For Coherent Rings, Rebecca Egg

Department of Mathematics: Dissertations, Theses, and Student Research

This dissertation presents a homological dimension notion of Cohen-Macaulay for non-Noetherian rings which reduces to the standard definition in the case that the ring is Noetherian, and is inspired by the homological notion of Cohen-Macaulay for local rings developed by Gerko. Under this notion, both coherent regular rings (as defined by Bertin) and coherent Gorenstein rings (as defined by Hummel and Marley) are Cohen-Macaulay.

This work is motivated by Glaz's question regarding whether a notion of Cohen-Macaulay exists for coherent rings which satisfies certain properties and agrees with the usual notion when the ring is Noetherian. Hamilton and Marley gave …

Rigidity Of The Frobenius, Matlis Reflexivity, And Minimal Flat Resolutions, Douglas J. Dailey

Department of Mathematics: Dissertations, Theses, and Student Research

Let R be a commutative, Noetherian ring of characteristic p >0. Denote by f the Frobenius endomorphism, and let R^(e) denote the ring R viewed as an R-module via f^e. Following on classical results of Peskine, Szpiro, and Herzog, Marley and Webb use flat, cotorsion module theory to show that if R has finite Krull dimension, then an R-module M has finite flat dimension if and only if Tor_i^R(R^(e),M) = 0 for all i >0 and infinitely many e >0. Using methods involving the derived category, we show that one only needs vanishing for dim R +1 consecutive values of …

Conjugacy Geodesics In Coxeter Groups, Aaron Calderon

UCARE Research Products

Take a square and flip it over the vertical axis, rotate it 90 degrees counterclockwise and then flip it again over the vertical axis. This sequence is the same as a 90 degree clockwise rotation but takes more steps to demonstrate the same symmetry. In general, the question of when a sequence of symmetries has minimal length is hard to answer and is dependent on the chosen generating set (in our toy example, rotation by 90 degrees and reflection). By realizing sequences of symmetries as paths in a group's Cayley graph, the problem becomes one about the set of shortest …

The History And Applications Of Fibonacci Numbers, Cashous W. Bortner, Allan C. Peterson

UCARE Research Products

The Fibonacci sequence is arguably the most observed sequence not only in mathematics, but also in nature. As we begin to learn more and more about the Fibonacci sequence and the numbers that make the sequence, many new and interesting applications of the have risen from different areas of algebra to market trading strategies. This poster analyzes not only the history of Leonardo Bonacci, but also the elegant sequence that is now his namesake and its appearance in nature as well as some of its current mathematical and non-mathematical applications.

Oscillation Of Solution To Second-Order Half-Linear Delay Dynamic Equations On Time Scales, Hongwu Wu, Lynn Erbe, Allan Peterson

Department of Mathematics: Faculty Publications

This article concerns the oscillation of solutions to second-order half-linear dynamic equations with a variable delay. By using integral averaging techniques and generalized Riccati transformations, new oscillation criteria are obtained. Our results extend Kamenev-type, Philos-type and Li-type oscillation criteria. Several examples are given to illustrate our results.

Matched Metrics And Channels, Marcelo Firer, Judy L. Walker

Department of Mathematics: Faculty Publications

The most common decision criteria for decoding are maximum likelihood decoding and nearest neighbor decoding. It is well-known that maximum likelihood decoding coincides with nearest neighbor decoding with respect to the Hamming metric on the binary symmetric channel. In this work we study channels and metrics for which those two criteria do and do not coincide for general codes.

Knowledge And Tasks Connecting Elementary, Secondary, And Disciplinary Mathematics, Yvonne Lai

DBER Speaker Series

A well-prepared teacher should be able to help her students see mathematics as ideas that develop over time. Mathematics courses designed specifically for prospective secondary teachers aim for prospective teachers to see and find connections across elementary, secondary, and disciplinary mathematics, and beyond that to be able to use those connections in their future teaching. While there is broad agreement with these aims, there is also little consensus around how to carry them out. Two challenges in meeting these aims are identifying content that lends itself to such connections and designing tasks that can be used to engage with that …

R0 Analysis Of A Benthic-Drift Model For A Stream Population, Qihua Huang, Yu Jin, Mark A. Lewis

Department of Mathematics: Faculty Publications

One key issue for theory in stream ecology is how much stream flow can be changed while still maintaining an intact stream ecology, instream flow needs (IFNs); the study of determining IFNs is challenging due to the complex and dynamic nature of the interaction between the stream environ- ment and the biological community. We develop a process-oriented benthic-drift model that links changes in the flow regime and habitat availability with population dynamics. In the model, the stream is divided into two zones, drift zone and benthic zone, and the population is divided into two interacting compartments, individuals residing in the …

Math 433: Nonlinear Optimization—A Peer Review Of Teaching Project Benchmark Portfolio, Adam Larios

UNL Faculty Course Portfolios

My intention in this portfolio is to highlight various approaches to teaching higher-level mathematics with a programming component that I tried in a course on nonlinear optimization. There is a particular focus on students with little or no previous programming experience. Several case studies are done using "pre- and post-course" surveys which examined items such as student confidence in programming, and particular programming skills. Sample examples and sample homeworks are presented and discussed. Also presented are several materials I designed to lead students into programming and shed light on certain problems. These Materials assume some basic mathematical reasoning and knowledge, …

Characterizing Mathematics Graduate Student Teaching Assistants’ Opportunities To Learn From Teaching, Yvonne Lai, Wendy Smith, Nathan Wakefield, Erica R. Miller, Julia St. Goar, Corbin M. Groothuis, Kelsey M. Wells

Department of Mathematics: Faculty Publications

Exemplary models to inform novice instruction and the development of graduate teaching assistants (TAs) exist. What is missing from the literature is the process of how graduate students in model professional development programs make sense of and enact the experiences offered. A first step to understanding TAs’ learning to teach is to characterize how and whether they link observations of student work to hypotheses about student thinking and then connect those hypotheses to future teaching actions. A reason to be interested in these connections is that their strength and coherence determine how well TAs can learn from experiences. We found …

Comparing The Growth Of The Prime Numbers To The Natural Numbers, Michael A. Brilleslyper, Nathan Wakefield, A. J. Wallerstein, Bradley Warner

Department of Mathematics: Faculty Publications

We define a new method of measuring the rate of divergence for an increasing positive sequence of integers. We introduce the growth function for such a sequence and its associated growth limit. We use these tools to study the divergence rate for the natural numbers, polynomial and exponential-type sequences, and the prime numbers. We conclude with a number of open questions concerning general properties and characterizations of growth functions and the set of possible growth limits.

Simple Adaptive Control For Positive Linear Systems With Applications To Pest Management, Chris Guiver, Christina Edholm, Yu Jin, Markus Mueller, Jim Powell, Richard Rebarber, Brigitte Tenhumberg, Stuart Townley

Department of Mathematics: Faculty Publications

Pest management is vitally important for modern arable farming, but models for pest species are often highly uncertain. In the context of pest management, control actions are naturally described by a nonlinear feedback that is generally unknown, which thus motivates a robust control approach. We argue that adaptive approaches are well suited for the management of pests and propose a simple high-gain adaptive tuning mechanism so that the nonlinear feedback achieves exponential stabilization. Furthermore, a switched adaptive controller is proposed, cycling through a set of given control actions, that also achieves global asymptotic stability. Such a model in practice allows …