Mathematics Commons^™

Detecting Finite Flat Dimension Of Modules Via Iterates Of The Frobenius Endomorphism, Douglas J. Dailey, Srikanth B. Iyengar, Thomas Marley

Department of Mathematics: Faculty Publications

It is proved that a module M over a Noetherian ring R of positive characteristic p has finite flat dimension if there exists an integer t > 0 such that Tor R (M, fe R) = 0 for t < i< t + dim R and infinitely many e. This extends results of Herzog, who proved it when M is finitely generated. It is also proved that when R is a Cohen-Macaulay local ring, it suffices that the Tor vanishing holds for one e > logp e(R) is the multiplicity of R.

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 …

A Framework For Predicting Impacts On Ecosystem Services From (Sub)Organismal Responses To Chemicals, Valery E. Forbes, Chris J. Salice, Bjorn Birnir, Randy J.F. Bruins, Peter Calow, Virginie Ducrot, Nika Galic, Kristina Garber, Bret C. Harvey, Henriette Jager, Andrew Kanarek, Robert Pastorok, Steve F. Railsback, Richard Rebarber, Pernille Thorbek

Department of Mathematics: Faculty Publications

Protection of ecosystem services is increasingly emphasized as a risk-assessment goal, but there are wide gaps between current ecological risk-assessment endpoints and potential effects on services provided by ecosystems. The authors present a framework that links common ecotoxicological endpoints to chemical impacts on populations and communities and the ecosystem services that they provide. This framework builds on considerable advances in mechanistic effects models designed to span multiple levels of biological organization and account for various types of biological interactions and feedbacks. For illustration, the authors introduce 2 case studies that employ well-developed and validated mechanistic effects models: the inSTREAM individual-based …

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 …

Teac 308: Teaching Mathematics In The Elementary School–A Peer Review Of Teaching Project Benchmark Portfolio, Amanda Thomas

UNL Faculty Course Portfolios

This portfolio outlines four aspects of the peer review of teaching project, which focused on TEAC 308: Teaching Mathematics in the Elementary School. The first aspect was the explicit articulation of student learning objectives drawn from the National Council of Teachers of Mathematics Principles to Actions: Ensuring Mathematical Success for All (2014). During this project, the nine objectives were aligned with instructional opportunities and assignments with those objectives. In addition to defining these objectives, the portfolio describes analysis of three components: student progress toward developing and demonstrating productive beliefs about aspects of teaching and learning mathematics, student progress across three …

An Early Semester Mastery Activity And Intervention In First-Year Calculus, Allan P. Donsig, Nathan Wakefield

Department of Mathematics: Faculty Publications

Success in first-year mathematics courses is essential for students to pursue STEM careers, including teaching careers. We investigate a mastery activity given during the first two weeks of a first-year calculus course at the research site. Previous work showed a model using this activity in College Algebra, together with ACT and high school rank, was predictive of student success in precalculus. Here we do a similar analysis for such an activity in calculus, including an intervention for students who do not complete the activity. We also investigate the intervention’s effectiveness. These results show that the early mastery activity, especially when …

Gene Expression Noise Enhances Robust Organization Of The Early Mammalian Blastocyst, William R. Holmes, Nabora Soledad Reyes De Mochel, Qixuan Wang, Huijing Du, Tao Peng, Michael Chiang, Olivier Cinquin, Ken Cho, Qing Nie

Department of Mathematics: Faculty Publications

A critical event in mammalian embryo development is construction of an inner cell mass surrounded by a trophoectoderm (a shell of cells that later form extraembryonic structures). We utilize multi-scale, stochastic modeling to investigate the design principles responsible for robust establishment of these structures. This investigation makes three predictions, each supported by our quantitative imaging. First, stochasticity in the expression of critical genes promotes cell plasticity and has a critical role in accurately organizing the developing mouse blastocyst. Second, asymmetry in the levels of noise variation (expression fluctuation) of Cdx2 and Oct4 provides a means to gain the benefits of …

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 …

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 …

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

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

Discipline-Based Education Research Group 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 …

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 …

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 …

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 …

Unique Pseudo-Expectations For C∗-Inclusions, David R. Pitts, Vrej Zarikian

Department of Mathematics: Faculty Publications

Given an inclusion D⊆C of unital C ∗ -algebras (with common unit), a unital completely positive linear map Φ of C into the injective envelope I(D) of D which extends the inclusion of D into I(D) is a pseudo-expectation. Pseudo-expectations are generalizations of conditional expectations, but with the advantage that they always exist. The set PsExp(C,D) of all pseudo-expectations is a convex set, and when D is Abelian, we prove a Krein–Milman type theorem showing that PsExp(C,D) can be recovered from its set of extreme points. In general, PsExp(C,D) is not a singleton. However, there are large and natural classes …

Comparison Theorems And Asymptotic Behavior Of Solutions Of Discrete Fractional Equations, Baoguo Jia, Lynn Erbe, Allan Peterson

Department of Mathematics: Faculty Publications

Consider the following n-th order nabla and delta fractional difference equations

rn r (a)x(t) = c(t)x(t), t 2 Na+1, x(a) > 0.

and

Va+v-1x(t) = c(t)x(t + v - 1), t 2 Na, x(a + n - 1) > 0

We establish comparison theorems by which we compare the solutions x(t) of (*) and (**) with the solutions of the equations rn r(a)x(t) = bx(t) and Dn a+v-1x(t) = bx(t + v -1), respectively, where b is a constant. We obtain four asymptotic results, one of them extends the recent result [F. M. Atici, P. W. Eloe, Rocky Mountain J. Math. 41(2011), …

Systems Of Parameters And The Cohen–Macaulay Property, Katharine Shultis

Department of Mathematics: Dissertations, Theses, and Student Research

Let R be a commutative, Noetherian, local ring and M a finitely generated R-module. Consider the module of homomorphisms Hom_R(R/a,M/bM) where b [subset of] a are parameter ideals of M. When M = R and R is Cohen-Macaulay, Rees showed that this module of homomorphisms is isomorphic to R/a, and in particular, a free module over R/a of rank one. In this work, we study the structure of such modules of homomorphisms for a not necessarily Cohen-Macaulay R-module M.

The Strict Higher Grothendieck Integral, Scott W. Dyer

Department of Mathematics: Dissertations, Theses, and Student Research

This thesis generalizes A. Grothendieck’s construction, denoted by an integral, of a fibered category from a contravariant pseudofunctor, to a construction for n- and even ∞-categories. Only strict higher categories are considered, the more difficult theory of weak higher categories being neglected. Using his axioms for a fibered category, Grothendieck produces a contravariant pseudofunctor from which the original fibered category can be reconstituted by integration. In applications, the integral is often most efficient, constructing the fibered category with its structure laid bare. The situation generalizes the external and internal definitions of the semidirect product in group theory: fibration is …