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Department of Mathematics: Dissertations, Theses, and Student Research
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Articles 1 - 30 of 42
Full-Text Articles in Education
Positioning Undergraduate Learning Assistants In Instruction: A Case Study Of The La Role In Active Learning Mathematics Classrooms At The University Of Nebraska-Lincoln, Rachel Funk
Department of Mathematics: Dissertations, Theses, and Student Research
Research suggests learning assistant (LA) programs can be a change lever to support the institutionalization of active learning in postsecondary education. Some research suggests LAs offer unique benefits for STEM courses, independent from other change levers, but more research needs to be done to understand how LAs support active learning classrooms, specifically in mathematics. Research on mathematics instruction and the use of reform resources suggests that the successful implementation of reforms is impacted by perceptions individuals hold about that resource, such as the LA role. Yet, there is little research about the LA role in mathematics, particularly where the instructor …
Classroom Social Support: A Multiple Phenomenological Case Study Of Mathematics Graduate Teaching Assistants’ Decision Making In The Classroom, Brittany Johnson
Classroom Social Support: A Multiple Phenomenological Case Study Of Mathematics Graduate Teaching Assistants’ Decision Making In The Classroom, Brittany Johnson
Department of Mathematics: Dissertations, Theses, and Student Research
Research suggests that support offered by an instructor can have a significant impact on the student experience, both in terms of classroom performance and affective well-being. Research also suggests that there are different types of support that instructors can offer (e.g., emotional support, instrumental support, informational support, and appraisal support). Although such research suggests that students perceive and are affected by these different types of support in different ways, there does not appear to be research surrounding the decision-making process behind instructors offering the support or the extent to which social support is a priority for them in the classroom. …
Exploring Pedagogical Empathy Of Mathematics Graduate Student Instructors, Karina Uhing
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 …
High Cognitive Demand Examples In Precalculus: Examining The Work And Knowledge Entailed In Enactment, Erica R. Miller
High Cognitive Demand Examples In Precalculus: Examining The Work And Knowledge Entailed In Enactment, Erica R. Miller
Department of Mathematics: Dissertations, Theses, and Student Research
Historically, pass rates in undergraduate precalculus courses have been dismally low and the teaching practices and knowledge of university instructors have been understudied. To help improve teaching effectiveness and student outcomes in undergraduate precalculus courses, I have studied the cognitive demand of enacted examples. The purpose of this dissertation is to examine the pedagogical work and mathematical knowledge entailed in the enactment of high cognitive demand examples in a three-part study. To answer my research questions, I conducted classroom observations as well as pre- and post-observation interviews with seven graduate student instructors at a large public R1 university in the …
Boundary Value Problems For Discrete Fractional Equations, Khulud Alyousef
Boundary Value Problems For Discrete Fractional Equations, Khulud Alyousef
Department of Mathematics: Dissertations, Theses, and Student Research
In this dissertation we are interested in proving the existence of solutions for various fractional boundary value problems. Our technique will be to apply certain fixed point theorems. Also comparison theorems for fractional boundary problems and a so-called Liapunov inequality will be given.
Commutative Rings Graded By Abelian Groups, Brian P. Johnson
Commutative Rings Graded By Abelian Groups, Brian P. Johnson
Department of Mathematics: Dissertations, Theses, and Student Research
Rings graded by Z and Zd play a central role in algebraic geometry and commutative algebra, and the purpose of this thesis is to consider rings graded by any abelian group. A commutative ring is graded by an abelian group if the ring has a direct sum decomposition by additive subgroups of the ring indexed over the group, with the additional condition that multiplication in the ring is compatible with the group operation. In this thesis, we develop a theory of graded rings by defining analogues of familiar properties---such as chain conditions, dimension, and Cohen-Macaulayness. We then study the …
An Analysis Of Nonlocal Boundary Value Problems Of Fractional And Integer Order, Christopher Steven Goodrich
An Analysis Of Nonlocal Boundary Value Problems Of Fractional And Integer Order, Christopher Steven Goodrich
Department of Mathematics: Dissertations, Theses, and Student Research
In this work we provide an analysis of both fractional- and integer-order boundary value problems, certain of which contain explicit nonlocal terms. In the discrete fractional case we consider several different types of boundary value problems including the well-known right-focal problem. Attendant to our analysis of discrete fractional boundary value problems, we also provide an analysis of the continuity properties of solutions to discrete fractional initial value problems. Finally, we conclude by providing new techniques for analyzing integer-order nonlocal boundary value problems.
Adviser: Lynn Erbe and Allan Peterson
Prime Ideals In Two-Dimensional Noetherian Domains And Fiber Products And Connected Sums, Ela Celikbas
Prime Ideals In Two-Dimensional Noetherian Domains And Fiber Products And Connected Sums, Ela Celikbas
Department of Mathematics: Dissertations, Theses, and Student Research
This thesis concerns three topics in commutative algebra:
1) The projective line over the integers (Chapter 2),
2) Prime ideals in two-dimensional quotients of mixed power series-polynomial rings (Chapter 3),
3) Fiber products and connected sums of local rings (Chapter 4),
In the first chapter we introduce basic terminology used in this thesis for all three topics.
In the second chapter we consider the partially ordered set (poset) of prime ideals of the projective line Proj(Z[h,k]) over the integers Z, and we interpret this poset as Spec(Z[x]) U Spec(Z[1/x]) with an appropriate identification. …
The Weak Discrepancy And Linear Extension Diameter Of Grids And Other Posets, Katherine Victoria Johnson
The Weak Discrepancy And Linear Extension Diameter Of Grids And Other Posets, Katherine Victoria Johnson
Department of Mathematics: Dissertations, Theses, and Student Research
A linear extension of a partially ordered set is simply a total ordering of the poset that is consistent with the original ordering. The linear extension diameter is a measure of how different two linear extensions could be, that is, the number of pairs of elements that are ordered differently by the two extensions. In this dissertation, we calculate the linear extension diameter of grids. This also gives us a nice characterization of the linear extensions that are the farthest from each other, and allows us to conclude that grids are diametrally reversing.
A linear extension of a poset might …
Modeling And Mathematical Analysis Of Plant Models In Ecology, Eric A. Eager
Modeling And Mathematical Analysis Of Plant Models In Ecology, Eric A. Eager
Department of Mathematics: Dissertations, Theses, and Student Research
Population dynamics tries to explain in a simple mechanistic way the variations of the size and structure of biological populations. In this dissertation we use mathematical modeling and analysis to study the various aspects of the dynamics of plant populations and their seed banks.
In Chapter 2 we investigate the impact of structural model uncertainty by considering different nonlinear recruitment functions in an integral projection model for Cirsium canescens. We show that, while having identical equilibrium populations, these two models can elicit drastically different transient dynamics. We then derive a formula for the sensitivity of the equilibrium population to …
Systems Of Nonlinear Wave Equations With Damping And Supercritical Sources, Yanqiu Guo
Systems Of Nonlinear Wave Equations With Damping And Supercritical Sources, Yanqiu Guo
Department of Mathematics: Dissertations, Theses, and Student Research
We consider the local and global well-posedness of the coupled nonlinear wave equations
utt – Δu + g1(ut) = f1(u, v)
vtt – Δv + g2(vt) = f2(u, v);
in a bounded domain Ω subset of the real numbers (Rn) with a nonlinear Robin boundary condition on u and a zero boundary conditions on v. The nonlinearities f1(u, v) and f2(u, v) are with supercritical exponents …
Combinatorics Using Computational Methods, Derrick Stolee
Combinatorics Using Computational Methods, Derrick Stolee
Department of Mathematics: Dissertations, Theses, and Student Research
Computational combinatorics involves combining pure mathematics, algorithms, and computational resources to solve problems in pure combinatorics. This thesis provides a theoretical framework for combinatorial search, which is then applied to several problems in combinatorics. Some results in space-bounded computational complexity are also presented.
Covariant Representations Of C*-Dynamical Systems Involving Compact Groups, Firuz Kamalov
Covariant Representations Of C*-Dynamical Systems Involving Compact Groups, Firuz Kamalov
Department of Mathematics: Dissertations, Theses, and Student Research
Given a C*-dynamical system (A, G, σ) the crossed product C*-algebra A x σG encodes the action of G on A. By the universal property of A x σG there exists a one to one correspondence between the set all covariant representations of the system (A, G, σ) and the set of all *-representations of A x σG. Therefore, the study of representations of A x σG is equivalent to that of covariant representations of (A, G, σ).
We study induced covariant representations of systems involving compact groups. We prove that every irreducible (resp. factor) covariant …
On Morrey Spaces In The Calculus Of Variations, Kyle Fey
On Morrey Spaces In The Calculus Of Variations, Kyle Fey
Department of Mathematics: Dissertations, Theses, and Student Research
We prove some global Morrey regularity results for almost minimizers of functionals of the form u → ∫Ω f(x, u, ∇u)dx. This regularity is valid up to the boundary, provided the boundary data are sufficiently regular. The main assumption on f is that for each x and u, the function f(x, u, ·) behaves asymptotically like the function h(|·|)α(x), where h is an N-function.
Following this, we provide a characterization of the class of Young measures that can be generated by a sequence …
Global Well-Posedness For A Nonlinear Wave Equation With P-Laplacian Damping, Zahava Wilstein
Global Well-Posedness For A Nonlinear Wave Equation With P-Laplacian Damping, Zahava Wilstein
Department of Mathematics: Dissertations, Theses, and Student Research
This dissertation deals with the global well-posedness of the nonlinear wave equation
utt − Δu − Δput = f (u) in Ω × (0,T),
{u(0), ut(0)} = {u0,u1} ∈ H10 (Ω) × L 2 (Ω),
u = 0 on Γ × (0, T ),
in a bounded domain Ω ⊂ ℜ n with Dirichlét boundary conditions. The nonlinearities f (u) acts as a strong source, which is allowed to …
Hilbert-Samuel And Hilbert-Kunz Functions Of Zero-Dimensional Ideals, Lori A. Mcdonnell
Hilbert-Samuel And Hilbert-Kunz Functions Of Zero-Dimensional Ideals, Lori A. Mcdonnell
Department of Mathematics: Dissertations, Theses, and Student Research
The Hilbert-Samuel function measures the length of powers of a zero-dimensional ideal in a local ring. Samuel showed that over a local ring these lengths agree with a polynomial, called the Hilbert-Samuel polynomial, for sufficiently large powers of the ideal. We examine the coefficients of this polynomial in the case the ideal is generated by a system of parameters, focusing much of our attention on the second Hilbert coefficient. We also consider the Hilbert-Kunz function, which measures the length of Frobenius powers of an ideal in a ring of positive characteristic. In particular, we examine a conjecture of Watanabe and …
Homology Of Artinian Modules Over Commutative Noetherian Rings, Micah J. Leamer
Homology Of Artinian Modules Over Commutative Noetherian Rings, Micah J. Leamer
Department of Mathematics: Dissertations, Theses, and Student Research
This work is primarily concerned with the study of artinian modules over commutative noetherian rings.
We start by showing that many of the properties of noetherian modules that make homological methods work seamlessly have analogous properties for artinian modules. We prove many of these properties using Matlis duality and a recent characterization of Matlis reflexive modules. Since Matlis reflexive modules are extensions of noetherian and artinian modules many of the properties that hold for artinian and noetherian modules naturally follow for Matlis reflexive modules and more generally for mini-max modules.
In the last chapter we prove that if the Betti …
Groups And Semigroups Generated By Automata, David Mccune
Groups And Semigroups Generated By Automata, David Mccune
Department of Mathematics: Dissertations, Theses, and Student Research
In this dissertation we classify the metabelian groups arising from a restricted class of invertible synchronous automata over a binary alphabet. We give faithful, self-similar actions of Heisenberg groups and upper triangular matrix groups. We introduce a new class of semigroups given by a restricted class of asynchronous automata. We call these semigroups ``expanding automaton semigroups''. We show that this class strictly contains the class of automaton semigroups, and we show that the class of asynchronous automaton semigroups strictly contains the class of expanding automaton semigroups. We demonstrate that undecidability arises in the actions of expanding automaton semigroups and semigroups …
On A Family Of Generalized Wiener Spaces And Applications, Ian Pierce
On A Family Of Generalized Wiener Spaces And Applications, Ian Pierce
Department of Mathematics: Dissertations, Theses, and Student Research
We investigate the structure and properties of a variety of generalized Wiener spaces. Our main focus is on Wiener-type measures on spaces of continuous functions; our generalizations include an extension to multiple parameters, and a method of adjusting the distribution and covariance structure of the measure on the underlying function space.
In the second chapter, we consider single-parameter function spaces and extend a fundamental integration formula of Paley, Wiener, and Zygmund for an important class of functionals on this space. In the third chapter, we discuss measures on very general function spaces and introduce the specific example of a generalized …
Extremal Trees And Reconstruction, Andrew Ray
Extremal Trees And Reconstruction, Andrew Ray
Department of Mathematics: Dissertations, Theses, and Student Research
Problems in two areas of graph theory will be considered.
First, I will consider extremal problems for trees. In these questions we examine the trees that maximize or minimize various invariants. For instance the number of independent sets, the number of matchings, the number of subtrees, the sum of pairwise distances, the spectral radius, and the number of homomorphisms to a fixed graph. I have two general approaches to these problems. To find the extremal trees in the collection of trees on n vertices with a fixed degree bound I use the certificate method. The certificate is a branch invariant, …
Packings And Realizations Of Degree Sequences With Specified Substructures, Tyler Seacrest
Packings And Realizations Of Degree Sequences With Specified Substructures, Tyler Seacrest
Department of Mathematics: Dissertations, Theses, and Student Research
This dissertation focuses on the intersection of two classical and fundamental areas in graph theory: graph packing and degree sequences. The question of packing degree sequences lies naturally in this intersection, asking when degree sequences have edge-disjoint realizations on the same vertex set. The most significant result in this area is Kundu's k-Factor Theorem, which characterizes when a degree sequence packs with a constant sequence. We prove a series of results in this spirit, and we particularly search for realizations of degree sequences with edge-disjoint 1-factors.
Perhaps the most fundamental result in degree sequence theory is the Erdos-Gallai Theorem, characterizing …
Annihilators Of Local Cohomology Modules, Laura Lynch
Annihilators Of Local Cohomology Modules, Laura Lynch
Department of Mathematics: Dissertations, Theses, and Student Research
In many important theorems in the homological theory of commutative local rings, an essential ingredient in the proof is to consider the annihilators of local cohomology modules. We examine these annihilators at various cohomological degrees, in particular at the cohomological dimension and at the height or the grade of the defining ideal. We also investigate the dimension of these annihilators at various degrees and we refine our results by specializing to particular types of rings, for example, Cohen Macaulay rings, unique factorization domains, and rings of small dimension.
Adviser: Thomas Marley
The Theory Of Discrete Fractional Calculus: Development And Application, Michael T. Holm
The Theory Of Discrete Fractional Calculus: Development And Application, Michael T. Holm
Department of Mathematics: Dissertations, Theses, and Student Research
The author's purpose in this dissertation is to introduce, develop and apply the tools of discrete fractional calculus to the arena of fractional difference equations. To this end, we develop the Fractional Composition Rules and the Fractional Laplace Transform Method to solve a linear, fractional initial value problem in Chapters 2 and 3. We then apply fixed point strategies of Krasnosel'skii and Banach to study a nonlinear, fractional boundary value problem in Chapter 4.
Adviser: Lynn Erbe and Allan Peterson
Formalizing Categorical And Algebraic Constructions In Operator Theory, William Benjamin Grilliette
Formalizing Categorical And Algebraic Constructions In Operator Theory, William Benjamin Grilliette
Department of Mathematics: Dissertations, Theses, and Student Research
In this work, I offer an alternative presentation theory for C*-algebras with applicability to various other normed structures. Specifically, the set of generators is equipped with a nonnegative-valued function which ensures existence of a C*-algebra for the presentation. This modification allows clear definitions of a "relation" for generators of a C*-algebra and utilization of classical algebraic tools, such as Tietze transformations.
On The Betti Number Of Differential Modules, Justin Devries
On The Betti Number Of Differential Modules, Justin Devries
Department of Mathematics: Dissertations, Theses, and Student Research
Let R = k[x1, ..., xn] with k a field. A multi-graded differential R-module is a multi-graded R-module D with an endomorphism d such that d2 = 0. This dissertation establishes a lower bound on the rank of such a differential module when the underlying R-module is free. We define the Betti number of a differential module and use it to show that when the homology ker d/im d of D is non-zero and finite dimensional over k then there is an inequality rankR D ≥ 2n. This …
The Cohomology Of Modules Over A Complete Intersection Ring, Jesse Burke
The Cohomology Of Modules Over A Complete Intersection Ring, Jesse Burke
Department of Mathematics: Dissertations, Theses, and Student Research
We investigate the cohomology of modules over commutative complete intersection rings. The first main result is that if M is an arbitrary module over a complete intersection ring R, and if one even self-extension module of M vanishes then M has finite projective dimension. The second main result gives a new proof of the fact that the support variety of a Cohen-Macaulay module whose completion is indecomposable is projectively connected.
Applications Of Linear Programming To Coding Theory, Nathan Axvig
Applications Of Linear Programming To Coding Theory, Nathan Axvig
Department of Mathematics: Dissertations, Theses, and Student Research
Maximum-likelihood decoding is often the optimal decoding rule one can use, but it is very costly to implement in a general setting. Much effort has therefore been dedicated to find efficient decoding algorithms that either achieve or approximate the error-correcting performance of the maximum-likelihood decoder. This dissertation examines two approaches to this problem.
In 2003 Feldman and his collaborators defined the linear programming decoder, which operates by solving a linear programming relaxation of the maximum-likelihood decoding problem. As with many modern decoding algorithms, is possible for the linear programming decoder to output vectors that do not correspond to codewords; such …
Vanishing Of Ext And Tor Over Complete Intersections, Olgur Celikbas
Vanishing Of Ext And Tor Over Complete Intersections, Olgur Celikbas
Department of Mathematics: Dissertations, Theses, and Student Research
Let (R,m) be a local complete intersection, that is, a local ring whose m-adic completion is the quotient of a complete regular local ring by a regular sequence. Let M and N be finitely generated R-modules. This dissertation concerns the vanishing of Tor(M, N) and Ext(M, N). In this context, M satisfies Serre's condition (S_{n}) if and only if M is an nth syzygy. The complexity of M is the least nonnegative integer r such that the nth Betti number of M is bounded by a polynomial of degree r-1 for all sufficiently large n. We use this notion of …
Mathematical Modeling Of Optimal Seasonal Reproductive Strategies And A Comparison Of Long-Term Viabilities Of Annuals And Perennials, Anthony Delegge
Mathematical Modeling Of Optimal Seasonal Reproductive Strategies And A Comparison Of Long-Term Viabilities Of Annuals And Perennials, Anthony Delegge
Department of Mathematics: Dissertations, Theses, and Student Research
In 1954, Lamont Cole posed a question which has motivated much ecological work in the past 50 years: When is the life history strategy of semelparity (organisms reproduce once, then die) favored, via evolution, over iteroparity (organisms may reproduce multiple times in their lifetime)? Although common sense should dictate that iteroparity would always be favored, we can observe that this is not always the case, since annual plants are not only prevalent, but can dominate an area. Also, certain plant species may be perennial in one region, but annual in another. Thus, in these areas, certain characteristics must be present …
Properties Of The Generalized Laplace Transform And Transport Partial Dynamic Equation On Time Scales, Chris R. Ahrendt
Properties Of The Generalized Laplace Transform And Transport Partial Dynamic Equation On Time Scales, Chris R. Ahrendt
Department of Mathematics: Dissertations, Theses, and Student Research
In this dissertation, we first focus on the generalized Laplace transform on time scales. We prove several properties of the generalized exponential function which will allow us to explore some of the fundamental properties of the Laplace transform. We then give a description of the region in the complex plane for which the improper integral in the definition of the Laplace transform converges, and how this region is affected by the time scale in question. Conditions under which the Laplace transform of a power series can be computed term-by-term are given. We develop a formula for the Laplace transform for …