Physical Sciences and Mathematics Commons^™

Perfect Matchings Of Trimmed Aztec Rectangles, Tri Lai

Department of Mathematics: Faculty Publications

We consider several new families of subgraphs of the square grid whose matchings are enumerated by powers of several small prime numbers: 2, 3, 5, and 11. Our graphs are obtained by trimming two opposite corners of an Aztec rectangle. The result yields a proof of a conjecture posed by Ciucu. In addition, we reveal a hidden connection between our graphs and the hexagonal dungeons introduced by Blum.

Ideal Containments Under Flat Extensions And Interpolation On Linear Systems In P2, Solomon Akesseh

Department of Mathematics: Dissertations, Theses, and Student Research

Fat points and their ideals have stimulated a lot of research but this dissertation concerns itself with aspects of only two of them, broadly categorized here as, the ideal containments and polynomial interpolation problems.

Ein-Lazarsfeld-Smith and Hochster-Huneke cumulatively showed that for all ideals I in k[Pⁿ], I^(mn) ⊆ I^m for all m ∈ N. Over the projective plane, we obtain I⁽⁴⁾< ⊆ I². Huneke asked whether it was the case that I⁽³⁾ ⊆ I². Dumnicki, Szemberg and Tutaj-Gasinska show that if I is the saturated homogeneous radical ideal of the 12 …

Antichains And Diameters Of Set Systems, Brent Mckain

Department of Mathematics: Dissertations, Theses, and Student Research

In this thesis, we present a number of results, mostly concerning set systems that are antichains and/or have bounded diameter. Chapter 1 gives a more detailed outline of the thesis. In Chapter 2, we give a new short proof of Kleitman's theorem concerning the maximal size of a set system with bounded diameter. In Chapter 3, we turn our attention to antichains with bounded diameter. Šileikis conjectured that an antichain of diameter D has size at most (n/D/2). We present several partial results towards the conjecture.

In 2014, Leader and Long gave asymptotic bounds on the size of …

Stable Cohomology Of Local Rings And Castelnuovo-Mumford Regularity Of Graded Modules, Luigi Ferraro

Department of Mathematics: Dissertations, Theses, and Student Research

This thesis consists of two parts:

1) A bimodule structure on the bounded cohomology of a local ring (Chapter 1),

2) Modules of infinite regularity over graded commutative rings (Chapter 2).

Chapter 1 deals with the structure of stable cohomology and bounded cohomology. Stable cohomology is a $\mathbb{Z}$-graded algebra generalizing Tate cohomology and first defined by Pierre Vogel. It is connected to absolute cohomology and bounded cohomology. We investigate the structure of the bounded cohomology as a graded bimodule. We use the information on the bimodule structure of bounded cohomology to study the stable cohomology algebra as a trivial extension …

Existence And Rapid Convergence Results For Nonlinear Caputo Nabla Fractional Difference Equations, Xiang Liu, Baoguo Jia, Lynn Erbe, Allan Peterson

Department of Mathematics: Faculty Publications

This paper is concerned with finding properties of solutions to initial value problems for nonlinear Caputo nabla fractional difference equations. We obtain existence and rapid convergence results for such equations by use of Schauder’s fixed point theorem and the generalized quasi-linearization method, respectively. A numerical example is given to illustrate one of our rapid convergence results.

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.

Languages, Geodesics, And Hnn Extensions, Maranda Franke

Department of Mathematics: Dissertations, Theses, and Student Research

The complexity of a geodesic language has connections to algebraic properties of the group. Gilman, Hermiller, Holt, and Rees show that a finitely generated group is virtually free if and only if its geodesic language is locally excluding for some finite inverse-closed generating set. The existence of such a correspondence and the result of Hermiller, Holt, and Rees that finitely generated abelian groups have piecewise excluding geodesic language for all finite inverse-closed generating sets motivated our work. We show that a finitely generated group with piecewise excluding geodesic language need not be abelian and give a class of infinite non-abelian …

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 …

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 …

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 …