Digital Commons Network^™

Understanding And Advancing College Students' Mathematical Reasoning Using Collaborative Argumentation, Rachel Kay Heili

MSU Graduate Theses

This study explored students’ mathematical reasoning skills and offered supports to advance them through a collaborative argumentation framework in a college intermediate algebra class. The goals of this study were to make observations about student reasoning, identify specific actions to address those observations, and document student growth in reasoning as a result of those actions. An iterative analysis, mixed method study was conducted in which the researcher engaged students in responding to questions that required conceptual understandings using a collaborative argumentation framework as a tool to identify and code components of their responses—claim, evidence, and reasoning. After coding and analyzing …

On Covering Groups With Proper Subgroups, Collin B. Moore

MSU Graduate Theses

In this paper, we explore groups that can be expressed as a union of proper subgroups. Using “covering number” to denote the minimal number of proper subgroups required to cover a group, we explore the nature of groups with covering numbers 3 and 4, while also finding covering numbers for p-groups, dihedral, and generalized dihedral groups.

Geometric Dissections, Daniel Robert Martin

MSU Graduate Theses

In the study of geometry, the notion of dissection and its mechanics are occasionally over-looked. We consider and trace the history and theorems surrounding geometric dissections in both recreational and academic mathematics. We explore the important advancements in this particular topic from antiquity through the nineteenth and early twentieth centuries. We conclude with an exploration of the Banach-Tarski paradox

On The Chromatic Numbers Of Subgroup Lattices, Jacob C. Miles

MSU Graduate Theses

In this thesis we investigate the chromatic number of the Hasse diagram of a subgroup lattice. We combine results of Bollobás and Tůma to show that there exist infnite groups whose subgroup lattices have arbitarily high chromatic numbers. We show that fnite supersolvable groups have bipartite subgroup lattices but that CLT and non-solvable groups may not have bipartite subgroup lattices. Lastly, we give a preliminary argument suggesting that there are an infnite number of non-solvable groups whose subgroup lattices are bipartite.

Dot Product Bounds In Galois Rings, David Lee Crosby

MSU Graduate Theses

We consider the Erdős Distance Conjecture in the context of dot products in Galois rings and prove results for single dot products and pairs of dot products.

Finite Groups In Which The Number Of Cyclic Subgroups Is 3/4 The Order Of The Group, James Alexander Cayley

MSU Graduate Theses

Let $G$ be a finite group, c(G) denotes the number of cyclic subgroups of G and α(G) = c(G)/|G|. In this thesis we go over some basic properties of alpha, calculate alpha for some families of groups, with an emphasis on groups with α(G) = 3/4, as all groups with α(G) > 3/4 have been classified by Garonzi and Lima (2018). We find all Dihedral group with this property, show all groups with α(G) = 3/4 have at least |G|/2-1 involutions, and discuss existing work by Wall (1970) and Miller (1919) classifying all such groups.

Elliptic Curves And Their Practical Applications, Henry H. Hayden Iv

MSU Graduate Theses

Finding rational points that satisfy functions known as elliptic curves induces a finitely-generated abelian group. Such functions are powerful tools that were used to solve Fermat's Last Theorem and are used in cryptography to send private keys over public systems. Elliptic curves are also useful in factoring and determining primality.

On The Hamiltonicity Of Subgroup Lattices, Nicholas Charles Fleece

MSU Graduate Theses

In this paper we discuss the Hamiltonicity of the subgroup lattices of

different classes of groups. We provide sufficient conditions for the

Hamiltonicity of the subgroup lattices of cube-free abelian groups. We also

prove the non-Hamiltonicity of the subgroup lattices of dihedral and

dicyclic groups. We disprove a conjecture on non-abelian p-groups by

producing an infinite family of non-abelian p-groups with Hamiltonian

subgroup lattices. Finally, we provide a list of the Hamiltonicity of the

subgroup lattices of every finite group up to order 35 barring two groups.

On Elliptic Curves, Montana S. Miller

MSU Graduate Theses

An elliptic curve over the rational numbers is given by the equation y² = x³+Ax+B. In our thesis, we study elliptic curves. It is known that the set of rational points on the elliptic curve form a finitely generated abelian group induced by the secant-tangent addition law. We present an elementary proof of associativity using Maple. We also present a relatively concise proof of the Mordell-Weil Theorem.

Proper Sum Graphs, Austin Nicholas Beard

MSU Graduate Theses

The Proper Sum Graph of a commutative ring with identity has the prime ideals as vertices, with two ideals adjacent if their sum is a proper ideal. This thesis expands upon the research of Dhorajia. We will cover the groundwork to understanding the basics of these graphs, and gradually narrow our efforts into the minimal prime ideals of the ring.

Transitioning From The Abstract To The Concrete: Reasoning Algebraically, Andrea Lynn Martin

MSU Graduate Theses

Why are students not making a smooth transition from arithmetic to algebra? The purpose of this study was to understand the nature of students’ algebraic reasoning through tasks involving generalizing. After students’ algebraic reasoning had been analyzed, the challenges they encountered while reasoning were analyzed. The data was collected through semi-structured interviews with six eighth grade students and analyzed by watching recorded interviews while tracking algebraic reasoning. Through data analysis of students’ algebraic reasoning, three themes emerged: 1) it was possible for students to reach stage two (informal abstraction) and have an abstract understanding of the mathematical pattern even if …

The Game Of Cops And Robbers On Planar Graphs, Jordon S. Daugherty

MSU Graduate Theses

In graph theory, the game of cops and robbers is played on a finite, connected graph. The players take turns moving along edges as the cops try to capture the robber and the robber tries to evade capture forever. This game has received quite a bit of recent attention including several conjectures that have yet to be proven. In this paper, we restricted our attention to planar graphs in order to try to prove the conjecture that the dodecahedron graph is the smallest planar graph, in terms of vertices, that has cop number three. Along the way we discuss several …

Groups Satisfying The Converse To Lagrange's Theorem, Jonah N. Henry

MSU Graduate Theses

Lagrange’s theorem, which is taught early on in group theory courses, states that the order of a subgroup must divide the order of the group which contains it. In this thesis, we consider the converse to this statement. A group satisfying the converse to Lagrange’s theorem is called a CLT group. We begin with results that help show that a group is CLT, and explore basic CLT groups with examples. We then give the conditions to guarantee either CLT is satisfied or a non-CLT group exists for more advanced cases. Additionally, we show that CLT groups are properly contained between …

The Frenet Frame And Space Curves, Catherine Elaina Eudora Ross

MSU Graduate Theses

Essential to the study of space curves in Differential Geometry is the Frenet frame. In this thesis we generate the Frenet equations for the second, third, and fourth dimensions using the Gram-Schmidt process, which allows us to present the form of the Frenet equations for n-dimensions. We highlight several key properties that arise from the Frenet equations, expound on the class of curves with constant curvature ratios, as well as characterize spherical curves up to the fourth dimension. Methods for generalizing properties and characteristics of curves in varying dimensions should be handled with care, since the structure of curves often …

Elements Of Functional Analysis And Applications, Chengting Yin

MSU Graduate Theses

Functional analysis is a branch of mathematical analysis that studies vector spaces with a limit structure (such as a norm or inner product), and functions or operators defined on these spaces. Functional analysis provides a useful framework and abstract approach for some applied problems in variety of disciplines. In this thesis, we will focus on some basic concepts and abstract results in functional analysis, and then demonstrate their power and relevance by solving some applied problems under the framework. We will give the definitions and provide some examples of some different spaces (such as metric spaces, normed spaces and inner …

Survey Of Lebesgue And Hausdorff Measures, Jacob N. Oliver

MSU Graduate Theses

Measure theory is fundamental in the study of real analysis and serves as the basis for more robust integration methods than the classical Riemann integrals. Measure theory allows us to give precise meanings to lengths, areas, and volumes which are some of the most important mathematical measurements of the natural world. This thesis is devoted to discussing some of the major proofs and ideas of measure theory. We begin with a study of Lebesgue outer measure and Lebesgue measurable sets. After a brief discussion of non-measurable sets, we define Lebesgue measurable functions and the Lebesgue integral. In the last chapter …

Ridge Regression And Lasso Estimators For Data Analysis, Dalip Kumar

MSU Graduate Theses

An important problem in data science and statistical learning is to predict an outcome based on data collected on several predictor variables. This is generally known as a regression problem. In the field of big data studies, the regression model often depends on a large number of predictor variables. The data scientist is often dealing with the difficult task of determining the most appropriate set of predictor variables to be employed in the regression model. In this thesis we adopt a technique that constraints the coefficient estimates which in effect shrinks the coefficient estimates towards zero. Ridge regression and lasso …

The Average Measure Of A K-Dimensional Simplex In An N-Cube, John A. Carter

MSU Graduate Theses

Within an n-dimensional unit cube, a number of k-dimensional simplices can be formed whose vertices are the vertices of the n-cube. In this thesis, we analyze the average measure of a k-simplex in the n-cube. We develop exact equations for the average measure when k = 1, 2, and 3. Then we generate data for these cases and conjecture that their averages appear to approach n^k/2 times some constant. Using the convergence of Bernstein polynomials and a k-simplex Bernstein generalization, we prove the conjecture is true for the 1-simplex and 2-simplex cases. We then develop a generalized formula for …

Affine And Projective Planes, Abraham Pascoe

MSU Graduate Theses

In this thesis, we investigate affine and projective geometries. An affine geometry is an incidence geometry where for every line and every point not incident to it, there is a unique line parallel to the given line. Affine geometry is a generalization of the Euclidean geometry studied in high school. A projective geometry is an incidence geometry where every pair of lines meet. We study basic properties of affine and projective planes and a number of methods of constructing them. We end by prov- ing the Bruck-Ryser Theorem on the non-existence of projective planes of certain orders.

Cayley Graphs Of Groups And Their Applications, Anna Tripi

MSU Graduate Theses

Cayley graphs are graphs associated to a group and a set of generators for that group (there is also an associated directed graph). The purpose of this study was to examine multiple examples of Cayley graphs through group theory, graph theory, and applications. We gave background material on groups and graphs and gave numerous examples of Cayley graphs and digraphs. This helped investigate the conjecture that the Cayley graph of any group (except Z_2) is hamiltonian. We found the conjecture to still be open. We found Cayley graphs and hamiltonian cycles could be applied to campanology (in particular, to the …

Residues, Bernoulli Numbers And Finding Sums, Mohammed Saif Alotaibi

MSU Graduate Theses

A large number of infinite sums, such as , cannot be found by the methods of real analysis. However, many of them can be evaluated using the theory of residues. In this thesis we characterize several methods of summations using residues, including methods integrating residues and the Bernoulli numbers. In fact, with this technique we derive some summation formulas for particular Finite Sums and Infinite Series that are difficult or impossible to solve by the methods of real analysis.

Support Vector Machine And Its Application To Regression And Classification, Xiaotong Hu

MSU Graduate Theses

Support Vector machine is currently a hot topic in the statistical learning area and is now widely used in data classification and regression modeling. In this thesis, we introduce the basic idea for support vector machine, its application in the classification area including both linear and nonlinear parts, and the idea of support vector regression contains the comparison of loss functions and the usage of kernel function. Two real life examples, which are taken from R package, are also provided for both classification and regression part respectively, talking about classification of glass type and prediction for Ozone pollution.

Solving Boundary Value Problems On Various Domains, Ibraheem Otuf

MSU Graduate Theses

Domain-sensitivity is a hallmark in the realm of solving boundary value problems in partial differential equations. For example, the method used in solving a boundary value problem on an finite cylindrical domain is very different from one that arises from a rectangular domain. The difference is also reflected in the types of functions employed in the processes of solving these boundary value problems, as are the mathematical tools utilized in deriving an analytic solution. In this thesis, we solve an important class of partial differential equations with boundary conditions coming from various domains, such as the n dimensional cube, circles, …

When There Is A Unique Group Of A Given Order And Related Results, Haya Ibrahim Binjedaen

MSU Graduate Theses

It is well-known that any group whose order is a prime number must be cyclic, that is there is only one group of that order up to isomorphism. This is also the case for some non-prime orders, for example there is only one group of order 15 up to isomorphism. This thesis provides the necessary background material to completely characterize those n for which these is a unique group of order n, namely when n and the Euler phi function of n are relatively prime. We also determine for which n there are exactly two groups of order n up …

On The Number Of Distinct Cyclic Subgroups Of A Given Finite Group, Joseph Dillstrom

MSU Graduate Theses

In the study of finite groups, it is a natural question to consider the number of distinct cyclic subgroups of a given finite group. Following an article by M. Tarnauceanu in the American Mathematical Monthly, we consider arithmetic relations between the order of a finite group and the number of its cyclic subgroups. We classify several infinite families of finite groups in this fashion and expand upon an open problem posed in the article.

A Geometric Approach To Ramanujan's Taxi Cab Problem And Other Diophantine Dilemmas, Zachary Kyle Easley

MSU Graduate Theses

In 1917, the British mathematician G.H. Hardy visited the Indian mathematical genius Ramanujan in the hospital. The number of the taxicab Hardy arrived in was 1729. Ramanujan immediately recognized this as the smallest positive integer that can be expressed as the sum of two cubes in two essentially different ways. In this thesis, we use properties of conics and elliptic curves to investigate this problem, its generalization to fourth powers, and a Diophantine equation involving the distance of a point from the vertices of a regular tetrahedron (the latter extends work of Christina Bisges).

A Cycle Generating Function On Finite Local Rings, Tristen Kirk Wentling

MSU Graduate Theses

We say a function generates a cycle if its output returns the initial value for some number of successive applications of . In this thesis, we develop a class of polynomial functions for finite local rings and associated functions . We show that the zeros of one are precisely the fixed points of the other and that every ring element is either one of these fixed points or is in a cycle of fixed length equal to the order of 2 in the associated group of units. Particular emphasis is given to rings of integers modulo the square of a …

Aspects Of Fourier Analysis On Euclidean Space, Joseph William Roberts

MSU Graduate Theses

The field of Fourier analysis encompasses a vast spectrum of mathematics and has far reaching applications in all STEM fields. Here we introduce and study the Fourier transform and Fourier series on Euclidean space. After defining the Fourier transform, establishing its basic properties, and presenting some classical results we looked into what impact the smoothness of a function has on the growth and integrability of its Fourier transform. This endeavor also involved a brief study of Bessel functions and interpolation of operators. Having established several results indicating that the behavior of a function's Fourier transform is largely dictated by the …

Vitis Gene Expression Profiling Using Mixed Models, Yin Yin

MSU Graduate Theses

The purpose of this thesis is to analyze gene expression in grapevine under different treatments using a mixed linear statistical model. The experiment involves two Vitis species (V. vinifera Cabernet sauvignon and V. aestivalis Norton) and applies two different treatments to them (inoculation with Erysiphe necator conidiospores and mock inoculation). There are three biological replicates measured at each of the following six assigned time points: 0, 4, 8, 12, 24, and 48 hours. By setting up split-plot model for the data, statistical hypotheses concerning gene expressions, especially gene expressions in terms of treatment effect, are tested. The result of the …

Convergence And The Lebesgue Integral, Ryan Vail Thomas

MSU Graduate Theses

In this paper, we examine the theory of integration of functions of real variables. Background information in measure theory and convergence is provided and several examples are considered. We compare Riemann and Lebesgue integration and develop several important theorems. In particular, the Monotone Convergence Theorem and Dominated Convergence Theorem are considered under both pointwise convergence and convergence in measure.