Mathematics Commons^™

Graph Centers, Hypergraph Degree Sequences, And Induced-Saturation, Sarah Lynne Behrens

Department of Mathematics: Dissertations, Theses, and Student Research

The center of a graph is the set of vertices whose distance to other vertices is minimal. The centralizing number of a graph G is the minimum number of additional vertices in any graph H where V(G) is the center of H. Buckley, Miller, and Slater and He and Liu provided infinite families of graphs with each centralizing number. We show the number of graphs with each nonzero centralizing number grows super-exponentially with the number of vertices. We also provide a method of altering graphs without changing the centralizing number and give results about the centralizing …

Tame Filling Functions And Closure Properties, Anisah Nu'man

Department of Mathematics: Dissertations, Theses, and Student Research

Let G be a group with a finite presentation P = such that A is inverse- closed. Let f : N[1/4] → N[1/4] be a nondecreasing function. Loosely, f is an intrinsic tame filling function for (G;P) if for every word w over A* that represents the identity element in G, there exists a van Kampen diagram Δ for w over P and a continuous choice of paths from the basepoint * of Δ to points on the boundary of Δ such that the paths are steadily moving outward as measured by f. The isodiametric function (or intrinsic diameter function) …

Local And Nonlocal Models In Thin-Plate And Bridge Dynamics, Jeremy Trageser

Department of Mathematics: Dissertations, Theses, and Student Research

This thesis explores several models in continuum mechanics from both local and nonlocal perspectives. The first portion settles a conjecture proposed by Filippo Gazzola and his collaborators on the finite-time blow-up for a class of fourth-order differential equations modeling suspension bridges. Under suitable assumptions on the nonlinearity and the initial data, a finite-time blowup is demonstrated as a result of rapid oscillations with geometrically growing amplitudes. The second section introduces a nonlocal peridynamic (integral) generalization of the biharmonic operator. Its action converges to that of the classical biharmonic as the radius of nonlocal interactions---the ``horizon"---tends to zero. For the corresponding …

Some Relations Between The Caputo Fractional Difference Operators And Integer-Order Differences, Baoguo Jia, Lynn Erbe, Allan Peterson

Department of Mathematics: Faculty Publications

In this article, we are concerned with the relationships between the sign of Caputo fractional differences and integer nabla differences. In particular, we show that if N -1 < v < N. f : Na -N + 1 -> R, va * f(t) > O, for t - Na +1 and N-1f(a) > 0, then N -1 f(t) > 0 for t- Na +1, then va* f(t) > 0, for each t - Na +1. As applications of these two results, we get that if 1 < vR, va*f(t) > 0 for t - Na +1 and f(a) > f(a-1), then f(t) is an increasing function for t- Na -1. Conversely if 0 < vR and f is an increasing function for t - Na, then va*f(t) > 0, for each t - Na +1. …

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 …

Knörrer Periodicity And Bott Periodicity, Michael K. Brown

Department of Mathematics: Dissertations, Theses, and Student Research

The main goal of this dissertation is to explain a precise sense in which Knörrer periodicity in commutative algebra is a manifestation of Bott periodicity in topological K-theory. In Chapter 2, we motivate this project with a proof of the existence of an 8-periodic version of Knörrer periodicity for hypersurfaces defined over the real numbers. The 2- and 8-periodic versions of Knörrer periodicity for complex and real hypersurfaces, respectively, mirror the 2- and 8-periodic versions of Bott periodicity in KU- and KO-theory. In Chapter 3, we introduce the main tool we need to demonstrate the compatibility between Knörrer …

Extremal Results For The Number Of Matchings And Independent Sets, Lauren Keough

Department of Mathematics: Dissertations, Theses, and Student Research

This dissertation answers several questions in extremal graph theory, each concerning the maximum or minimum number of certain substructures a graph can have, given that it must satisfy certain properties. In recent years there has been increased interest in such problems, which are extremal problems for "counting" parameters of graphs. The results in this dissertation focus on graphs that have n vertices and e edges and 3-uniform hypergraphs that have n vertices and e edges.

We first observe in the preliminaries chapter that for graphs with a fixed number of vertices and edges there is a threshold graph attaining the …

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

Discipline-Based Education Research Group 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.

Betti Sequences Over Local Rings And Connected Sums Of Gorenstein Rings, Zheng Yang

Department of Mathematics: Dissertations, Theses, and Student Research

This thesis consists of two parts:

1) Polynomial growth of Betti sequences over local rings (Chapter 2),

2) Connected sums of Gorenstein rings (Chapter 3).

Chapter 1 gives an introduction for the two topics discussed in this thesis.

The first part of the thesis deals with modules over complete intersections using free resolutions. The asymptotic patterns of the Betti sequences of the finitely generated modules over a local ring R reflect and affect the singularity of R. Given a commutative noetherian local ring and an integer c, sufficient conditions and necessary conditions are obtained for all Betti sequences …

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 …

The Interplay Between Wnt Mediated Expansion And Negative Regulation Of Growth Promotes Robust Intestinal Crypt Structure And Homeostasis, Huijing Du, Qing Nie, William R. Holmes

Department of Mathematics: Faculty Publications

The epithelium of the small intestinal crypt, which has a vital role in protecting the underlying tissue from the harsh intestinal environment, is completely renewed every 4–5 days by a small pool of stem cells at the base of each crypt. How is this renewal controlled and homeostasis maintained, particularly given the rapid nature of this process? Here, based on the recent observations from in vitro “mini gut” studies, we use a hybrid stochastic model of the crypt to investigate how exogenous niche signaling (from Wnt and BMP) combines with auto-regulation to promote homeostasis. This model builds on the sub-cellular …

Teac 451p: Learning And Teaching Principles And Practices (Secondary Mathematics)—A Peer Review Of Teaching Project Benchmark Portfolio, Lorraine Males

UNL Faculty Course Portfolios

The goal of my peer review portfolio was to better understand how to improve students' learning of how to teach secondary mathematics in reform-oriented ways. Most students that pursue admission into the Secondary Mathematics Teacher Education Program have little to no experience learning mathematics in reform-oriented ways. These preservice teachers (PSTs) were “successful” in mathematics courses in middle and high school, most of them taking honors or accelerated courses. However, many of these PSTs did not have opportunities to engage as active participants in their own learning and develop complex cognitive skills and processes, the focus of reform-oriented instruction. This …

Toric Varieties, Monoid Schemes And Cdh Descent, Guillermo Cortiñas, C. Haesemeyer, Mark E. Walker, Charles Weibel

Department of Mathematics: Faculty Publications

We give conditions for the Mayer–Vietoris property to hold for the algebraic K-theory of blow-up squares of toric varieties and schemes, using the theory of monoid schemes. These conditions are used to relate algebraic K-theory to topological cyclic homology in characteristic p. To achieve our goals, we develop many notions for monoid schemes based on classical algebraic geometry, such as separated and proper maps and resolution of singularities.

Crosscap Number: Handcuff Graphs And Unknotting Number, Anne Kerian

Department of Mathematics: Dissertations, Theses, and Student Research

Many types of invariants are used in the study of knots. Some are based on polynomials, some are purely algebraic, and some have their origins in geometry. One of the best known geometric knot invariants is the genus of a knot. A closely related but lesser-known invariant, crosscap number, was first introduced by Bradd Evans Clark in 1978. This thesis primarily concerns crosscap number two knots. Starting with a list of knots found to have crosscap number two by Burton and Ozlen using a linear programing approach, we verify, though are unable to expand, this list using a computer search. …

Integrating Path-Dependent Functionals On Yeh-Wiener Space, Ian Pierce, David Skough

Department of Mathematics: Faculty Publications

Denote by Ca,b(Q) the generalized two-parameter Yeh-Wiener space with associated Gaussian measure. We investigate several scenarios in which integrals of functionals on this space can be reduced to integrals of related functionals over an appropriate single-parameter generalized Wiener space Cˆa,ˆb[0, T ]. This extends some interesting results of R. H. Cameron and D. A. Storvick.

Enumeration Of Tilings Of Quartered Aztec Rectangles, Tri Lai

Department of Mathematics: Faculty Publications

We generalize a theorem of W. Jockusch and J. Propp on quartered Aztec diamonds by enumerating the tilings of quartered Aztec rectangles. We use subgraph replacement method to transform the dual graph of a quartered Aztec rectangle to the dual graph of a quartered lozenge hexagon, and then use Lindstr¨om-Gessel- Viennot methodology to find the number of tilings of a quartered lozenge hexagon.

Von Neumann Algebras And Extensions Of Inverse Semigroups, Allan P. Donsig, Adam H. Fuller, David R. Pitts

Department of Mathematics: Faculty Publications

In the 1970s, Feldman and Moore classified separably acting von Neumann algebras containing Cartan MASAs using measured equivalence re- lations and 2-cocycles on such equivalence relations. In this paper, we give a new classification in terms of extensions of inverse semigroups. Our approach is more algebraic in character and less point-based than that of Feldman-Moore. As an application, we give a restatement of the spectral theorem for bimodules in terms of subsets of inverse semigroups. We also show how our viewpoint leads naturally to a description of maximal subdiagonal algebras.

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

School of Computing: 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.

A Mentoring Program For Inquiry-Based Teaching In A College Geometry Class, Nathaniel Miller, Nathan Wakefield

Department of Mathematics: Faculty Publications

This paper describes a mentoring program designed to prepare novice instructors to teach a college geometry class using inquiry-based methods. The mentoring program was used in a medium-sized public university with approximately 12,000 undergraduate students and 1,500 graduate students. The authors worked together to implement a mentoring program for the first time. One author was an associate professor and experienced using inquiry-based learning. The other author was a graduate student in mathematics education. During the course of the year the graduate student first observed and then taught a college level inquiry-based geometry course for pre-service teachers. This article describes the …

Boundary Value Problems Of Nabla Fractional Difference Equations, Abigail M. Brackins

Department of Mathematics: Dissertations, Theses, and Student Research

In this dissertation we develop the theory of the nabla fractional self-adjoint difference equation,

∇_a^ν(p∇y)(t)+q(t)y(ρ(t)) = f(t),

where 0 < ν < 1.We begin with an introduction to the nabla fractional calculus. In the second chapter, we show existence and uniqueness of the solution to a fractional self-adjoint initial value problem. We find a variation of constants formula for this fractional initial value problem, and use the variation of constants formula to derive the Green's function for a related boundary value problem. We study the Green's function and its properties in several settings. For a simplified boundary value problem, we show that the Green's function is nonnegative and we find its maximum and the maximum of its integral. For a boundary value problem with generalized boundary conditions, we find the Green's function and show that it is a generalization of the first Green's function. In the third chapter, we use the Contraction Mapping Theorem to prove existence and uniqueness of a positive solution to a forced self-adjoint fractional difference equation with a finite limit. We explore modifications to the forcing term and modifications to the space of functions in which the solution exists, and we provide examples to demonstrate the use of these theorems.

Advisers: Lynn Erbe and Allan Peterson

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 …

Algebraic Properties Of Ext-Modules Over Complete Intersections, Jason Hardin

Department of Mathematics: Dissertations, Theses, and Student Research

We investigate two algebraic properties of Ext-modules over a complete intersection R of codimension c. Given an R-module M, Ext(M,k) can be viewed as a graded module over a polynomial ring in c variables with an action given by the Eisenbud operators. We provide an upper bound on the degrees of the generators of this graded module in terms of the regularities of two associated coherent sheaves. In the codimension two case, our bound recovers a bound of Avramov and Buchweitz in terms of the Betti numbers of M. We also provide a description of the differential graded (DG) R-module …

Results On Edge-Colored Graphs And Pancyclicity, James Carraher

Department of Mathematics: Dissertations, Theses, and Student Research

This thesis focuses on determining when a graph with additional structure contains certain subgraphs, particularly circuits, cycles, or trees. The specific problems and presented results include a blend of many fundamental graph theory concepts such as edge-coloring, routing problems, decomposition problems, and containing cycles of various lengths. The three primary chapters in this thesis address the problems of finding eulerian circuits with additional restrictions, decomposing the edge-colored complete graph K_n into rainbow spanning trees, and showing a 4-connected claw-free and N(3,2,1)-free graph is pancyclic.

Adviser: Stephen G. Hartke

Combinatorial And Algebraic Coding Techniques For Flash Memory Storage, Kathryn A. Haymaker

Department of Mathematics: Dissertations, Theses, and Student Research

Error-correcting codes are used to achieve reliable and efficient transmission when storing or sending information across a noisy channel. This thesis investigates a mathematical approach to coding techniques for storage devices such as flash memory storage, although many of the resulting codes and coding schemes can be applied in other contexts. The main contributions of this work include the design of efficient codes and decoding algorithms using discrete structures such as graphs and finite geometries, and developing a variety of strategies for adapting codes to a multi-level setting.

Information storage devices are prone to errors over time, and the frequency …

A Generalization Of Aztec Diamond Theorem, Part I, Tri Lai

Department of Mathematics: Faculty Publications

We consider a new family of 4-vertex regions with zigzag boundary on the square lattice with diagonals drawn in. By proving that the number of tilings of the new regions is given by a power 2, we generalize both Aztec diamond theorem and Douglas’ theorem. The proof extends an idea of Eu and Fu for Aztec diamonds, by using a bijection between domino tilings and non-intersecting Schr¨oder paths, then applying Lindstr¨om-Gessel-Viennot methodology.

A Simple Proof For The Number Of Tilings Of Quartered Aztec Diamonds, Tri Lai

Department of Mathematics: Faculty Publications

We get four quartered Aztec diamonds by dividing an Aztec diamond region by two zigzag cuts passing its center. W. Jockusch and J. Propp (in an unpublished work) found that the number of tilings of quartered Aztec diamonds is given by simple product formulas. In this paper we present a simple proof for this result.

Downhill Domination In Graphs, Teresa W. Haynes, Stephen T. Hedetniemi, Jessie D. Jamieson, William B. Jamieson

Department of Mathematics: Faculty Publications

A path π = (v₁, v₂, . . . , v_k+1) iun a graph G = (V, E) is a downhill path if for every i, 1 ≤ i ≤ k, deg(v_i) ≥ deg(v_i+1), where deg(vi) denotes the degree of vertex v_i ∈ V. The downhill domination number equals the minimum cardinality of a set S ⊆ V having the property that every vertex v ∈ V lies on a downhill path originating from some vertex in S …

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 …

Closure And Homological Properties Of (Auto)Stackable Groups, Ashley Johnson

Department of Mathematics: Dissertations, Theses, and Student Research

Let G be a finitely presented group with Cayley graph Γ. Roughly, G is a stackable group if there is a maximal tree T in Γ and a function φ, defined on the edges in Γ, for which there is a natural ‘flow’ on the edges in Γ\T towards the identity. Additionally, if graph (φ), which consists of pairs (e; φ(e)) for e an edge in Γ, forms a regular language, then G is autostackable. In 2011, Brittenham and Hermiller introduced stackable groups in [4], in part, as a means …