Department: Mathematics ![[clear]](close-x.png "Remove this limiter")
25 matches in the database.
These are records: 1 - 25.
1.
Adams, Lynn I.
Classifying Triply-Invariant Subspaces.
Degree: MS, Mathematics, 2007, University of Akron
► Let p be a prime number and consider the vector space consisting…
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▼ Let p be a prime number and consider the vector space consisting of all p-by-p-by-pmatrices with entries taken from the field with p elements. We wish to construct, list, and describe all those subspaces that are simultaneously invariant under three particular linear transformations on this vector space. Even for small primes p, this is an extensive and difficult computational problem. Using an elaborate overall strategy based on concepts from linear algebra, we completely solve this problem for the prime p=2, and we have completed several cases of this problem for the prime p=3. This problem has connections with classification problems for certain subgroups of wreath product finite groups of prime-power order.
Advisors/Committee Members: Riedl, Jeffrey.
Subjects: Mathematics
Keywords: wreath product; linear transformations; invariant subspaces; finite group theory; three-dimensional matrices; finite vector spaces; finite fields
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2.
Auger, Joseph Thomas.
Orbits of the Dissected Polygons of the Generalized Catalan Numbers.
Degree: MS, Mathematics, 2011, University of Akron
► Catalan numbers occur regularly throughout many different areas of mathematics. They date…
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▼ Catalan numbers occur regularly throughout many different areas of mathematics. They date back all the way to Euler, and today there are literally hundreds of different realizations of the Catalan number. We can extend the idea of a Catalan number to a generalized Catalan number by adding and extra parameter. In particular, one realization of the generalized Catalan numbers is the number of ways to dissect a regular polygon with (p-1)k+2 sides into k disjoint p+1-gons using (k-1) non-intersecting diagonals. The action of the dihedral group partitions these dissections into some number of orbits. Our main goal of this paper is to find a way to count these orbits for given p and k. In order to do this, we introduce the notion of p-ary trees. We show that the number of diagonalizations of our polygon is equivalent to the number of p-ary trees with k source nodes. Finally, we use Burnside's Lemma as a way to count the number of orbits for a given tree core and extract formulas for the number of orbits for many tree cores that exist for large enough p and k.
Advisors/Committee Members: Cossey, James.
Subjects: Mathematics
Keywords: Catalan; Burnside's Lemma; trees; dissected polygons
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3.
Bissell, Stephanie.
The Effects of Quorum Sensing on the Phenotypes of Pseudomonas Aeruginosa Bacteria Cells Within a Biofilm.
Degree: MS, Mathematics, 2011, University of Akron
► Pseudomonas aeruginosa is an opportunistic, gram-negative bacteria that targets individuals with compromised…
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▼ Pseudomonas aeruginosa is an opportunistic, gram-negative bacteria that targets individuals with compromised immune systems. This deadly bacteria is extremely resistant to antibiotics due to its ability to utilize quorum sensing to alter the phenotypes of its cells. The model presented uses advection equations to describe the mixture of down-regulated, up-regulated, inert, and persister P. aeruginosa phenotypes within a biofilm. Diffusion equations are used to model soluble components of a biofilm, including the signaling molecules needed for quorum sensing. We assume that the conversion between down-regulated and up-regulated phenotypes is controlled by extracellular signaling molecules reaching a critical level. We also assume that the conversion between up-regulated and persister phenotypes depends on the concentration of nutrients within the biofilm. The work presented in this paper shows that by increasing the rates of conversion between down-regulated and upregulated phenotypes, we are able to decrease the population of down-regulated cells and increase the number of both up-regulated and persister cells in a growing biofilm. Altering the quorum sensing system within the bacteria cells will change the timing at which the different phenotypes reach equilibria but ultimately does not change the concentrations of the different cell phenotypes within a mature biofilm.
Advisors/Committee Members: Young, Gerald.
Subjects: Mathematics; Microbiology
Keywords: Pseudomonas aeruginosa; quorum sensing; biofilm; phenotype
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4.
Burkett, Shawn Tyler.
Conjugacy Class Sizes and Character Degrees in the Linear and Unitary Groups.
Degree: MS, Mathematics, 2012, University of Akron
► Finite simple groups are the building blocks for all finite groups. Among…
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▼ Finite simple groups are the building blocks for all finite groups. Among the simple groups, finite classical groups are the oldest and perhaps most important. The most familiar family of finite classical groups is the linear groups, which consist of invertible linear transformations of a finite-dimensional vector space over a finite field. Other families are unitary groups, symplectic groups, and orthogonal groups, which are groups of those transformations preserving a certain bilinear form on V. In this paper, we classify the conjugacy classes of small sizes up to a certain bound in the linear and unitary groups. Using the knowledge on conjugacy classes, we also study a problem on bounding the largest irreducible character degree in the simple linear groups over the field of two elements in terms of the smaller degrees.
Advisors/Committee Members: Nguyen, Hung.
Subjects: Mathematics
Keywords: classical groups; conjugacy class; unipotent characters; character degrees
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5.
Cerrone, Kathryn L.
TESSELLATIONS: LESSONS FOR EVERY AGE.
Degree: MS, Mathematics, 2006, University of Akron
► Tessellations are a mathematical concept which many elementary teachers use for interdisciplinary…
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▼ Tessellations are a mathematical concept which many elementary teachers use for interdisciplinary lessons between math and art. Since the tilings are used by many artists and masons many of the lessons in circulation tend to focus primarily on the artistic part, while overlooking some of the deeper mathematical concepts such as symmetry and spatial sense. The inquiry-based lessons included in this paper utilize the subject of tessellations to lead students in developing a relationship between geometry, spatial sense, symmetry, and abstract algebra for older students. Lesson topics include fundamental principles of tessellations using regular polygons as well as those that can be made from irregular shapes, symmetry of polygons and tessellations, angle measurements of polygons, polyhedra, three-dimensional tessellations, and the wallpaper symmetry groups to which the regular tessellations belong. Background information is given prior to the lessons, so that teachers have adequate resources for teaching the concepts. The concluding chapter details results of testing at various age levels.
Advisors/Committee Members: Saliga, Linda.
Keywords: tessellation; polyhedra; wallpaper groups
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6.
Cullion, Paul D.
3-Regularizing 3-semiFayers Partitions.
Degree: MS, Mathematics, 2012, University of Akron
► In this thesis, we study certain combinatorial questions about partitions motivated by…
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▼ In this thesis, we study certain combinatorial questions about partitions motivated by issues in the representation theory of the symmetric group. In particular, we will study the class of the 3-semiFayers partitions and their behavior with respect to 3-regularization. We will also show how the 3-regularization algorithm can be effectively reversed for this class of partitions.
Advisors/Committee Members: Cossey, James.
Subjects: Mathematics
Keywords: Symmetric group; Fayers; Young Diagrams
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7.
Felix, Christina M.
Classification of Doubly-Invariant Subgroups for p=2.
Degree: MS, Mathematics, 2008, University of Akron
► We consider the additive group of 2-by-2 matriceswith entries taken from the…
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▼ We consider the additive group of 2-by-2 matriceswith entries taken from the ring of integers modulo 4. We construct all those subgroups which are simultaneously invariant under two particular endomorphisms of this group, using an elaborate overall strategy based on concepts from linear algebra and group theory. We then calculate the orbits of these subgroups under a particular action of GL(2,2). This problem has connections with classification problems for certain subgroups of wreath product finite groups of prime-power order.
Advisors/Committee Members: Riedl, Jeffrey.
Subjects: Mathematics
Keywords: doubly-invariant subgroups
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8.
Garlow, Michael W.
An Elementary Classification of the Groups of Order 81.
Degree: MS, Mathematics, 2006, University of Akron
► The classification of finite, non-abelian groups is of interest, not only to…
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▼ The classification of finite, non-abelian groups is of interest, not only to algebraists, but to mathematicians in all fields. In general, little is known about non-abelian p-groups. However, cyclic extensions provide a convenient way to classify such groups. One particularly interesting set of p-groups is the groups of order p4. Much has been done to classify these groups, but little has been done using cyclic extensions to do so. In this paper, I will deal with the non-abelian groups of order 81. Along the way, many general results applicable for non-abelian groups of order p4 will be presented. A first step in using cyclic extensions for such a problem is to prove the Cyclic Extension Theorem.
Advisors/Committee Members: Adler, Jeffrey.
Subjects: Mathematics
Keywords: Finite p-groups; Classification; Elementary; Groups
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9.
Hafner, Jonathan H.
Evolving a Genetic Algorithm for Network Flow Maximization.
Degree: MS, Mathematics, 2012, University of Akron
► The problem of maximizing network flow has been approached over the decades…
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▼ The problem of maximizing network flow has been approached over the decades with a variety of techniques; genetic algorithms, however, have seen relatively little application to this problem. We chose to pursue a simple genetic algorithm in which the rate of mutation decreases in successive generations, in a fashion inspired by simulated annealing. To assist the determination of optimal parameters, we constructed a meta-genetic algorithm to evolve an efficient genetic algorithm for this task.
Advisors/Committee Members: Norfolk, Timothy.
Subjects: Artificial Intelligence; Computer Science; Mathematics
Keywords: network flow; genetic algorithms
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10.
Holik, Nicklos L III.
NONSTANDARD HULLS OF GROUPS.
Degree: MS, Mathematics, 2007, University of Akron
► The nonstandard hull construction, one of the most useful applications of Abraham…
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▼ The nonstandard hull construction, one of the most useful applications of Abraham Robinson's nonstandard analysis, is generalized to an arbitrary group. Some important connections between the classical construction as applied to vector spaces and groups are noted, and several useful results regarding this generalized construction are developed. The theory developed provides particularly interesting results when applied to certain discrete groups.
Advisors/Committee Members: Adler, Jeffery.
Subjects: Mathematics
Keywords: Nonstandard Analysis; Group Theory
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11.
Kaschner, Scott R.
Results and Examples Regarding Bifurcation with a Two-Dimensional Kernel.
Degree: MS, Mathematics, 2008, University of Akron
► Many problems in pure and applied mathematics entail studying the structure of…
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▼ Many problems in pure and applied mathematics entail studying the structure of solutions to F(x,y)=0, where F is a nonlinear operator between Banach spaces and y is a real parameter. A parameter value where the structure of solutions of F changes is called a bifurcation point. The particular method of analysis for bifurcation depends on the dimension of the kernel of DxF(0;λ), the linearization of F. The purpose of our study was to examine some consequences of a recent theorem on bifurcations with 2-dimensional kernels. This resent theorem was compared to previous methods. Also, some specific classes of equations were identified in which the theorem always holds, and an algebraic example was found that illustrates bifurcations with a 2-dimensional kernel.
Advisors/Committee Members: Wilber, J. Patrick.
Subjects: Mathematics
Keywords: bifurcation; two-dimensional kernel
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12.
Kreighbaum, Kevin M.
Combinatorial Problems Related to the Representation Theory of the Symmetric Group.
Degree: MS, Mathematics, 2010, University of Akron
► It is known that the set of p-regular partitions of n, the…
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▼ It is known that the set of p-regular partitions of n, the set of Γ0 partitions of n, and the set of Alperin weights of the symmetric group Sn all have equal size. We investigate the relationship between partitions and representations of the symmetric group in search of a natural bijection between the Alperin weights and the partitions in Γ0. Through this search, we were able to classify the abacus display of one-ladder partitions and also describe circumstances when the set of Γ0 partitions of n is in fact equal to the set of p-regular partitions of n.
Advisors/Committee Members: Cossey, James.
Subjects: Mathematics
Keywords: representation theory; symmetric group; alperin weights; partitions; p-regular
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13.
Lyons, Corey Francis.
The Γ0 Graph of a p-Regular Partition.
Degree: MS, Mathematics, 2010, University of Akron
► We investigate the p-regularization function from the set Γ0(n) to the set…
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▼ We investigate the p-regularization function from the set Γ0(n) to the set of p-regular partitions of n. This function induces a natural directed graph for a given p-regular partition λ, called the Γ0 graph of λ. We demonstrate how MATLAB can be used to help create the Γ0(n) graph and develop examples with complicated structure. Then we narrow our view to one-ladder partitions to see how the Γ0 graph can be arbitrarily complicated.
Advisors/Committee Members: Cossey, James.
Subjects: Mathematics
Keywords: p-regular partitions; partitions; symmetric group; abacus; Young diagram; S_n; p-regularization
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14.
Marko, Benjamin David.
Teaching Concepts Foundational to Calculus Using Inquiry and Technology.
Degree: MS, Mathematics, 2006, University of Akron
► There is currently a push in the state of Ohio to enhance…
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▼ There is currently a push in the state of Ohio to enhance the teaching of concepts foundational to Calculus in grades 8-12. Evidence of this came in the spring of 2005 when the Ohio Department of Education called for grant proposals to develop programs that are focused on topics that are foundations to Calculus. Further, the National Council of Teachers of Mathematics has stressed the effective use of technology and inquiry in their past and current standards. This thesis contains inquiry-based lessons that explore concepts that would traditionally be taught in the Calculus classroom, but can be used in lower level courses such as Algebra. The use of the graphing calculator is essential for each one of these lessons to allow the students the opportunity to explore more mathematical concepts in a less time consuming manner.
Advisors/Committee Members: Quesada, Antonio R.
Subjects: Mathematics
Keywords: Mathematics; Inquiry; Technology; Calculus
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15.
Maurer, Kendall Nicole.
Minimally Simple Groups and Burnside's Theorem.
Degree: MS, Mathematics, 2010, University of Akron
► William Burnside’s paqb theorem is a very important result in group theory,…
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▼ William Burnside’s paqb theorem is a very important result in group theory, which states that any group G of order paqb is solvable. An interesting fact about this theorem is that it was originally proven using techniques from character theory, another branch of algebra. In fact, it was about seventy years before a group-theoretic proof of Burnside’s theorem was developed through the work of Goldschmidt, Matsuyama,Bender, and other mathematicians. Their approach to proving the theorem was to show that, in essence, minimally simple groups of size paqb do not exist. Our purpose here is to use various techniques from the group-theoretic proof of Burnside’s theorem to establish and prove similar results about minimally simple groups G of arbitrary order.
Advisors/Committee Members: Cossey, James.
Subjects: Mathematics
Keywords: minimally simple groups; simple groups; groups; group theory; Burnside
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16.
McGough, Erin Patrick.
Analyzing the Relationship Between Player Personnel and Optimal Mixed Strategies in American Football.
Degree: MS, Mathematics, 2009, University of Akron
► The purpose of this paper is to explore how the optimal mix…
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▼ The purpose of this paper is to explore how the optimal mix of run and pass is affected by a change in player personnel in American football. To investigate this notion, we construct a model under the hypothesis that the offense has recently acquired a proven quarterback that will increase the production of the passing game. We then solve the game, and in the process construct Nash equilibrium functions that depend on the influence of the new quarterback. Lastly, as an example, we use empirical data and model results to examine how the addition of Jay Cutler impacts the mix of run and pass for the 2009 Chicago Bears.
Advisors/Committee Members: Young, Gerald.
Subjects: Mathematics
Keywords: game theory; football; optimal mix; free agency
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17.
Moses, Lawrenzo D.
Error Estimates for Entropy Solutions to Scalar Conservation Laws with Continuous Flux Functions.
Degree: MS, Mathematics, 2012, University of Akron
► Let f : R to R be a continuous flux function and…
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▼ Let f : R to R be a continuous flux function and let u be a solution to the inviscid scalar conservation law For each epsilon > 0, let u" be a solution to the associated viscous conservation law We establish the following error estimate for the solutions of these systems: for all t in (0, T), where TV(u_0) denotes the total variation of u_0 on R. This estimate generalizes the known result for the case of Lipschitz flux functions. Our result is useful for the method of vanishing viscosity and for other numerical methods.
Advisors/Committee Members: Nguyen, Truyen.
Subjects: Mathematics
Keywords: vanishing viscosity; partial differential equation; front tracking; hyperbolic; conservation law; entropy
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18.
Mott, Brittany Nicole.
Analysis of the Generalized Catalan Orbits.
Degree: MS, Mathematics, 2011, University of Akron
► The Catalan numbers are well-known in the field of combinatorics. This sequence…
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▼ The Catalan numbers are well-known in the field of combinatorics. This sequence of integers counts a variety of combinatorial objects including binary trees, triangulations of regular polygons, and paths on a square grid which lie below the diagonal. Of the hundreds of realizations of the Catalan numbers, the majority can be extended and considered under the scope of the generalized Catalan numbers. The generalized Catalan numbers consider a second parameter in addition to the single parameter of the regular Catalan numbers, consequently forming a grid of values instead of a single sequence of integers. Although a formula has been developed to determine the generalized Catalan number when given values for the two parameters, much is still unknown about the generalized Catalan numbers. This thesis will count the generalized Catalan orbits, defined to be the orbits of the combinatorial objects counted by the generalized Catalan numbers. Two realizations of the generalized Catalan numbers, p-ary trees and the dissection of regular polygons, will be used throughout our study of the generalized Catalan orbits. A variety of techniques will be used to create formulas which describe the number of generalized Catalan orbits in terms of the given parameters.
Advisors/Committee Members: Cossey, James.
Subjects: Mathematics
Keywords: Catalan Numbers; Generalized Catalan Numbers; Burnside's Lemma; Orbits; Dissection of polygons
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19.
Raies, Daniel N.
Counting the Faithful Irreducible Characters of Subgroups of the Iterated Regular Wreath Product.
Degree: MS, Mathematics, 2012, University of Akron
► Given a fixed odd prime p we start with the iterated regular…
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▼ Given a fixed odd prime p we start with the iterated regular wreath product group which we call P. We then consider a particular collection of subgroups of P. For each of those subgroups H we would like to determine the number of faithful irreducible characters of H of each degree. In the absence of a closed-form solution we have an intricate algorithm to calculate the desired data. This algorithm was implemented through hand calculation in the case when p=3 but those hand calculations are infeasible for larger values of p. We wrote a piece of software that confirms the hand calculations in the case when p=3 and implements the algorithm in the case when p=5. An analysis of the data in the case when p=3 suggests three conjectures which greatly improve upon this algorithm. The software verifies one of the conjectures and disproves the other two conjectures in the case when p=5.
Advisors/Committee Members: Riedl, Jeffrey.
Subjects: Mathematics
Keywords: character; character theory; wreath product; group; group theory; faithful character
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20.
Reisdorf, Stephen R.
Cellohedra.
Degree: MS, Mathematics, 2012, University of Akron
► The associahedron has been generalized to a great variety of combinatorial structures.…
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▼ The associahedron has been generalized to a great variety of combinatorial structures. In each example the convex polytope is found whose face poset is the same as a certain poset structure on the combinatorial structures. Here we find polytopes whose face poset models the containment order of certain order ideals of the face poset of a cell complex. This is progress in a program which asks which posets in general have their ideals modeled by convex polytopes.
Advisors/Committee Members: Forcey, Stefan.
Subjects: Mathematics
Keywords: associahedra; associahedron; graph associahedra; graph associahedron; pseudograph associahedron; cell complex; geometric combinatorics; polytope; polytopes
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21.
Shelton, Patty Jean.
ACHIEVEMENT AND ATTITUDES IN DEVELOPMENTAL MATHEMATICS.
Degree: MS, Mathematics, 2007, University of Akron
► The purpose of this study was two-fold: To test whether or not…
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▼ The purpose of this study was two-fold: To test whether or not the use of MathXL in Intermediate Algebra at The University of Akron impacted students' mathematics achievement in Intermediate Algebra and their subsequent mathematics courses and to test whether or not the use of technology changed students' attitude towards mathematics over the course of one semester, fall 2006.
Advisors/Committee Members: Saliga, Linda Marie.
Subjects: Mathematics
Keywords: Developmental Mathematics; Technology in Mathematics; Mathematics Attitudes
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22.
Smith, Michael M.
PRE-CALCULUS CONCEPTS FUNDAMENTAL TO CALCULUS.
Degree: MS, Mathematics, 2006, University of Akron
► Technology has transformed the mathematics curriculum. Instructional techniques are constantly evolving because…
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▼ Technology has transformed the mathematics curriculum. Instructional techniques are constantly evolving because of efforts to maximize the benefits of technology. To continue this process of enhancing the mathematics curriculum, this thesis will examine the following questions. What concepts, that are foundational to calculus, can be taught, with the assistance of the graphing calculator, at a level before calculus? How well do current precalculus textbooks incorporate these concepts? Finally, how well do practicing secondary mathematics teachers understand these concepts? To answer the first question, several concepts foundational to calculus were identified. Next, we examined how accessible these concepts were to secondary students. Finally, the concept list was narrowed to those that have been rarely emphasized in the secondary curriculum. This paper will address seven of these concepts. The goal of identifying these concepts is to promote the integration of them before calculus to enable students to make connections between pre-calculus (any course before calculus) and calculus. To answer the second question, twelve precalculus textbooks were examined to see how well they integrated these concepts. To accomplish this, a grading rubric was created to evaluate the textbooks. Then each textbook was reviewed and scored based on the rubric. To answer the third question, an assessment was given to a group of forty-one secondary mathematics teachers taking part in a continuing education workshop. The assessment was given at the beginning and at the end of the workshop. This was done for two reasons. The initial assessment was given to provide general information regarding how well teachers understood the topics that we propose are foundational to calculus. The post test was administered to determine the effectiveness of the workshop.
Advisors/Committee Members: Quesada, Antonio R.
Subjects: Mathematics
Keywords: Mathematics education; Precalculus; Foreshadowing calculus; Foundations of calculus; Mathematics curriculum; Secondary Mathematics
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23.
Stroup, David A.
Collatz’s Problem and Encoding Vectors.
Degree: MS, Mathematics, 2006, University of Akron
► The first chapter introduces the Collatz conjecture. The second chapter presents a…
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▼ The first chapter introduces the Collatz conjecture. The second chapter presents a brief literature survey. The third chapter presents some theorems and conjectures regarding encoding vectors for various moduli. Appendices include a presentation of numerical data, which serves as a concrete illustration of the findings in chapter 3, and a Java program for independent analysis of the Collatz conjecture.
Advisors/Committee Members: Norfolk, T.
Subjects: Mathematics
Keywords: Collatz conjecture; 3x+1 problem; Syracuse problem; Ulam Conjecture; hailstone numbers; Hasse algorithm
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24.
Tran, Vanthu Thy.
NEWTON'S METHOD AS A MEAN VALUE METHOD.
Degree: MS, Mathematics, 2007, University of Akron
► In this thesis, the relationships between fixed-point problems, relaxation methods and Newton's…
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▼ In this thesis, the relationships between fixed-point problems, relaxation methods and Newton's method were investigated. It was proven that Newton's method was a modified relaxation method, whose mean value parameter approached the optimal parameter for relaxation method. Also, under some conditions on the derivative of the function whose fixed point was sought, a convergent relaxation sequence that converged to a fixed point of the function was introduced.
Advisors/Committee Members: Hajjafar, Ali.
Subjects: Mathematics
Keywords: Newton's method; fixed point; relaxation method
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25.
Wojtasinski, Justyna Agata.
Classifying Triply-Invariant Subspaces for p=3.
Degree: MS, Mathematics, 2008, University of Akron
► Let p be a prime number. Consider the vector space consisting of…
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▼ Let p be a prime number. Consider the vector space consisting of all p-by-p-by-p arrays of numbers taken from the field with p elements. It is desirable to construct, list, and describe all those subspaces that are simultaneously invariant under three particular linear transformations on this vector space. Even for a small prime p, such as p = 3, this is an extensive computational problem. Using an elaborate strategy based on concepts from linear algebra, we were able to complete several cases of this problem for the prime p = 3. This problem is connected with a problem of classifying certain subgroups of wreath product finite groups of prime-power order.
Advisors/Committee Members: Riedl, Jeffrey.
Subjects: Mathematics
Keywords: triply-invariant subspaces
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