# Education Commons™

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Science and Mathematics Education

Dissertations, Theses, and Student Research Papers in Mathematics

Articles 1 - 30 of 39

## Full-Text Articles in Education

#### High Cognitive Demand Examples In Precalculus: Examining The Work And Knowledge Entailed In Enactment, Erica R. Miller

##### Dissertations, Theses, and Student Research Papers in Mathematics

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 Aug 2012

#### Boundary Value Problems For Discrete Fractional Equations, Khulud Alyousef

##### Dissertations, Theses, and Student Research Papers in Mathematics

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 Aug 2012

#### Commutative Rings Graded By Abelian Groups, Brian P. Johnson

##### Dissertations, Theses, and Student Research Papers in Mathematics

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 ...

Aug 2012

#### Prime Ideals In Two-Dimensional Noetherian Domains And Fiber Products And Connected Sums, Ela Celikbas

##### Dissertations, Theses, and Student Research Papers in Mathematics

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 ...

An Analysis Of Nonlocal Boundary Value Problems Of Fractional And Integer Order, Christopher Steven Goodrich Aug 2012

#### An Analysis Of Nonlocal Boundary Value Problems Of Fractional And Integer Order, Christopher Steven Goodrich

##### Dissertations, Theses, and Student Research Papers in Mathematics

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

The Weak Discrepancy And Linear Extension Diameter Of Grids And Other Posets, Katherine Victoria Johnson Jul 2012

#### The Weak Discrepancy And Linear Extension Diameter Of Grids And Other Posets, Katherine Victoria Johnson

##### Dissertations, Theses, and Student Research Papers in Mathematics

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 ...

Jun 2012

#### Modeling And Mathematical Analysis Of Plant Models In Ecology, Eric A. Eager

##### Dissertations, Theses, and Student Research Papers in Mathematics

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 changes ...

Apr 2012

#### Systems Of Nonlinear Wave Equations With Damping And Supercritical Sources, Yanqiu Guo

##### Dissertations, Theses, and Student Research Papers in Mathematics

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 representing strong sources, while g1(ut) and g2(vt) act as damping. It is well-known that the ...

Combinatorics Using Computational Methods, Derrick Stolee Mar 2012

#### Combinatorics Using Computational Methods, Derrick Stolee

##### Dissertations, Theses, and Student Research Papers in Mathematics

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.

Jul 2011

#### Covariant Representations Of C*-Dynamical Systems Involving Compact Groups, Firuz Kamalov

##### Dissertations, Theses, and Student Research Papers in Mathematics

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 ...

May 2011

#### On A Family Of Generalized Wiener Spaces And Applications, Ian Pierce

##### Dissertations, Theses, and Student Research Papers in Mathematics

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 ...

May 2011

#### On Morrey Spaces In The Calculus Of Variations, Kyle Fey

##### Dissertations, Theses, and Student Research Papers in Mathematics

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 of functions {fj} uniformly bounded in the Morrey space Lp, λ ...

Homology Of Artinian Modules Over Commutative Noetherian Rings, Micah J. Leamer May 2011

#### Homology Of Artinian Modules Over Commutative Noetherian Rings, Micah J. Leamer

##### Dissertations, Theses, and Student Research Papers in Mathematics

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 ...

May 2011

#### Global Well-Posedness For A Nonlinear Wave Equation With P-Laplacian Damping, Zahava Wilstein

##### Dissertations, Theses, and Student Research Papers in Mathematics

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 have, in some cases, a super-supercritical exponent. Under suitable restrictions on the parameters and with careful analysis involving the theory of monotone operators, we prove the existence and ...

Groups And Semigroups Generated By Automata, David Mccune May 2011

#### Groups And Semigroups Generated By Automata, David Mccune

##### Dissertations, Theses, and Student Research Papers in Mathematics

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 ...

Hilbert-Samuel And Hilbert-Kunz Functions Of Zero-Dimensional Ideals, Lori A. Mcdonnell May 2011

#### Hilbert-Samuel And Hilbert-Kunz Functions Of Zero-Dimensional Ideals, Lori A. Mcdonnell

##### Dissertations, Theses, and Student Research Papers in Mathematics

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 ...

Apr 2011

#### Packings And Realizations Of Degree Sequences With Specified Substructures, Tyler Seacrest

##### Dissertations, Theses, and Student Research Papers in Mathematics

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 ...

Extremal Trees And Reconstruction, Andrew Ray Apr 2011

#### Extremal Trees And Reconstruction, Andrew Ray

##### Dissertations, Theses, and Student Research Papers in Mathematics

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 ...

Annihilators Of Local Cohomology Modules, Laura Lynch Apr 2011

#### Annihilators Of Local Cohomology Modules, Laura Lynch

##### Dissertations, Theses, and Student Research Papers in Mathematics

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.

Apr 2011

#### The Theory Of Discrete Fractional Calculus: Development And Application, Michael T. Holm

##### Dissertations, Theses, and Student Research Papers in Mathematics

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 Mar 2011

#### Formalizing Categorical And Algebraic Constructions In Operator Theory, William Benjamin Grilliette

##### Dissertations, Theses, and Student Research Papers in Mathematics

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 Jan 2011

#### On The Betti Number Of Differential Modules, Justin Devries

##### Dissertations, Theses, and Student Research Papers in Mathematics

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 relates to a problem of Buchsbaum ...

Dec 2010

#### The Cohomology Of Modules Over A Complete Intersection Ring, Jesse Burke

##### Dissertations, Theses, and Student Research Papers in Mathematics

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 Aug 2010

#### Applications Of Linear Programming To Coding Theory, Nathan Axvig

##### Dissertations, Theses, and Student Research Papers in Mathematics

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 Jul 2010

#### Vanishing Of Ext And Tor Over Complete Intersections, Olgur Celikbas

##### Dissertations, Theses, and Student Research Papers in Mathematics

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 ...

#### Mathematical Modeling Of Optimal Seasonal Reproductive Strategies And A Comparison Of Long-Term Viabilities Of Annuals And Perennials, Anthony Delegge

##### Dissertations, Theses, and Student Research Papers in Mathematics

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 ...

Jan 2010

#### Properties Of The Generalized Laplace Transform And Transport Partial Dynamic Equation On Time Scales, Chris R. Ahrendt

##### Dissertations, Theses, and Student Research Papers in Mathematics

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 ...

#### A Computational Study Of The Effects Of Temperature Variation On Turtle Egg Development, Sex Determination, And Population Dynamics, Amy L. Parrott

##### Dissertations, Theses, and Student Research Papers in Mathematics

Climate change and its effects on ecosystems is a major concern. For certain animal species, especially those that exhibit what is known as temperature-dependent sex determination (TSD), temperature variations pose a possibly serious threat (Valenzuela and Lance, 2004). In these species, temperature, and not chromosomes, determines the sex of the animal (Valenzuela and Lance, 2004). It is conceivable therefore, that if the temperature changes to favor only one sex, then dire consequences for their populations could occur. In this dissertation, we examine possible effects that climate change may have upon Painted Turtles (Chrysemys picta), a species with TSD. We investigate ...

Aug 2009

#### Fan Cohomology And Equivariant Chow Rings Of Toric Varieties, Mu-Wan Huang

##### Dissertations, Theses, and Student Research Papers in Mathematics

Toric varieties are varieties equipped with a torus action and constructed from cones and fans. In the joint work with Suanne Au and Mark E. Walker, we prove that the equivariant K-theory of an affine toric variety constructed from a cone can be identified with a group ring determined by the cone. When a toric variety X(Δ) is smooth, we interpret equivariant K-groups as presheaves on the associated fan space Δ. Relating the sheaf cohomology groups to equivariant K-groups via a spectral sequence, we provide another proof of a theorem of Vezzosi and Vistoli: equivariant K ...

Modeling And Analysis Of Biological Populations, Joan Lubben Jul 2009

#### Modeling And Analysis Of Biological Populations, Joan Lubben

##### Dissertations, Theses, and Student Research Papers in Mathematics

Asymptotic and transient dynamics are both important when considering the future population trajectory of a species. Asymptotic dynamics are often used to determine whether the long-term trend results in a stable, declining or increasing population and even provide possible directions for management actions. Transient dynamics are important for estimating invasion speed of non-indigenous species, population establishment after releasing biocontrol agents, or population management after a disturbance like fire. We briefly describe here the results in this thesis.

(1) We consider asymptotic dynamics using discrete time linear population models of the form n(t + 1) = An(t) where A is a ...