Streaming Down The Stern-Brocot Tree: Finding And Expressing Solutions To Pell's Equation In Sl(2,Z), 2021 Jacksonville State University

#### Streaming Down The Stern-Brocot Tree: Finding And Expressing Solutions To Pell's Equation In Sl(2,Z), Marcus L. Shell

*Theses*

This paper explores and elaborates on a method of solving Pell’s equation as introduced by Norman Wildberger. In the first chapters of the paper, foundational topics are introduced in expository style including an explanation of Pell’s equation. An explanation of continued fractions and their ability to express quadratic irrationals is provided as well as a connection to the Stern-Brocot tree and a convenient means of representation for each in terms of 2×2 matrices with integer elements. This representation will provide a useful way of navigating the Stern-Brocot tree computationally and permit us a means of computing continued ...

Algebraic Structures And Variations: From Latin Squares To Lie Quasigroups, 2021 Northern Michigan University

#### Algebraic Structures And Variations: From Latin Squares To Lie Quasigroups, Erik Flinn

*All NMU Master's Theses*

In this Master's Thesis we give an overview of the algebraic structure of sets with a single binary operation. Specifically, we are interested in quasigroups and loops and their historical connection with Latin squares; considering them in both finite and continuous variations. We also consider various mappings between such algebraic objects and utilize matrix representations to give a negative conclusion to a question concerning isotopies in the case of quasigroups.

Symmetric Presentation Of Finite Groups, And Related Topics, 2021 California State University, San Bernardino

#### Symmetric Presentation Of Finite Groups, And Related Topics, Marina Michelle Duchesne

*Electronic Theses, Projects, and Dissertations*

We have discovered original symmetric presentations for several finite groups, including 2^{2}:^{.}(2^{4}:(2^{.}S_{3})), M_{11}, 3:(PSL(3,3):2), S_{8}, and 2^{.}M_{12}. We have found homomorphic images of several progenitors, including 2^{*18}:((6x2):6), 2^{*24}:(2^{.}S_{4}), 2^{*105}:A_{7}, 3^{*3}:_{m}(2^{3}:3), 7^{*8}:_{m}(PSL(2,7):2), 3^{*4}:_{m}(4^{2}:2^{2}), 7^{*5}:(2xA_{5}), and 5^{*6}:_{m}S_{5}. We have provided the isomorphism type of all of the finite images that we have discovered. We ...

Classification Of Cayley Rose Window Graphs, 2021 Presidency University, Kolkata

#### Classification Of Cayley Rose Window Graphs, Angsuman Das, Arnab Mandal

*Theory and Applications of Graphs*

Rose window graphs are a family of tetravalent graphs, introduced by Steve Wilson. Following it, Kovacs, Kutnar and Marusic classified the edge-transitive rose window graphs and Dobson, Kovacs and Miklavic characterized the vertex transitive rose window graphs. In this paper, we classify the Cayley rose window graphs.

Normality Properties Of Composition Operators, 2021 Taylor University

#### Normality Properties Of Composition Operators, Grace Weeks, Hallie Kaiser, Katy O'Malley

*Celebration of Scholarship 2021*

We explore two main concepts in relation to truncated composition matrices: the conditions required for the binormal and commutative properties. Both of these topics are important in linear algebra due to their connection with diagonalization.

We begin with the normal solution before moving onto the more complex binormal solutions. Then we cover conditions for the composition matrix to commute with the general matrix. Finally, we end with ongoing questions for future work.

C5 Is Almost A Fractalizer, 2021 Iowa State University

#### C5 Is Almost A Fractalizer, Bernard Lidicky, Connor Mattes, Florian Pfender

*Mathematics Publications*

We determine the maximum number of induced copies of a 5-cycle in a graph on n vertices for every n. Every extremal construction is a balanced iterated blow-up of the 5-cycle with the possible exception of the smallest level where for n=8, the Möbius ladder achieves the same number of induced 5-cycles as the blow-up of a 5-cycle on 8 vertices.

This result completes work of Balogh, Hu, Lidický, and Pfender [Eur. J. Comb. 52 (2016)] who proved an asymptotic version of the result. Similarly to their result, we also use the flag algebra method but we extend its ...

Some Model Theory Of Free Groups, 2021 The Graduate Center, City University of New York

#### Some Model Theory Of Free Groups, Christopher James Natoli

*Dissertations, Theses, and Capstone Projects*

There are two main sets of results, both pertaining to the model theory of free groups. In the first set of results, we prove that non-abelian free groups of finite rank at least 3 or of countable rank are not A-homogeneous. We then build on the proof of this result to show that two classes of groups, namely finitely generated free groups and finitely generated elementary free groups, fail to form A-Fraisse classes and that the class of non-abelian limit groups fails to form a strong A-Fraisse class.

The second main result is that if a countable group is elementarily ...

Total Differentiability And Monogenicity For Functions In Algebras Of Order 4, 2021 Politecnico di Milano

#### Total Differentiability And Monogenicity For Functions In Algebras Of Order 4, I. Sabadini, Daniele C. Struppa

*Mathematics, Physics, and Computer Science Faculty Articles and Research*

In this paper we discuss some notions of analyticity in associative algebras with unit. We also recall some basic tool in algebraic analysis and we use them to study the properties of analytic functions in two algebras of dimension four that played a relevant role in some work of the Italian school, but that have never been fully investigated.

A Gentle Introduction To Inequalities: A Casebook From The Fullerton Mathematical Circle, 2021 California State University, Fullerton

#### A Gentle Introduction To Inequalities: A Casebook From The Fullerton Mathematical Circle, Adam Glesser, Matt Rathbun, Bogdan Suceavă

*Journal of Math Circles*

Run for nearly a decade, the Fullerton Mathematical Circle at California State University, Fullerton prepares middle and high school students for mathematical research by exposing them to difficult problems whose solutions require only age-appropriate techniques and background. This work highlights one of the avenues of study, namely inequalities. We cover Engel's lemma, the Cauchy--Schwartz inequality, and the AM-GM inequality, as well as providing a wealth of problems where these results can be applied. Full solutions or hints, several written by Math Circle students, are given for all of the problems, as well as some commentary on how or when ...

Mathematical Magic: A Study Of Number Puzzles, 2021 Maryville College

#### Mathematical Magic: A Study Of Number Puzzles, Nicasio M. Velez

*Rose-Hulman Undergraduate Mathematics Journal*

Within this paper, we will briefly review the history of a collection of number puzzles which take the shape of squares, polygons, and polyhedra in both modular and nonmodular arithmetic. Among other results, we develop construction techniques for solutions of both Modulo and regular Magic Squares. For other polygons in nonmodular arithmetic, specifically of order 3, we present a proof of why there are only four Magic Triangles using linear algebra, disprove the existence of the Magic Tetrahedron in two ways, and utilizing the infamous 3-SUM combinatorics problem we disprove the existence of the Magic Octahedron.

The Beautiful Math Of Everything And You Included, 2020 University of Maryland, College Park

#### The Beautiful Math Of Everything And You Included, E. Ozie

*The STEAM Journal*

This a reflection on how there is beautiful math to everything. An author's interpretation of matrices and mechanics in its relationship to someone's identity.

Adaptive Analytics: It’S About Time, 2020 University of Central Florida

#### Adaptive Analytics: It’S About Time, Charles Dziuban, Colm Howlin, Patsy Moskal, Tammy Muhs, Connie Johnson, Rachel Griffin, Carissa Hamilton

*Current Issues in Emerging eLearning*

This article describes a cooperative research partnership among a large public university, a for-profit private institution and their common adaptive learning platform provider. The focus of this work explored adaptive analytics that uses data the investigators describe as metaphorical “digital learning dust” produced by the platform as a matter of course. The information configured itself into acquired knowledge, growth, baseline status and engagement. Two complimentary models evolved. The first, in the public university, captured end-of-course data for predicting success. The second approach, in the private university, formed the basis of a dynamic real-time data analytic algorithm. In both cases the ...

Behavior And Dynamics Of The Set Of Absolute Nilpotent And Idempotent Elements Of Chain Of Evolution Algebras Depending On The Time, 2020 V.I. Romanovskiy Institute of Mathematics, Tashkent

#### Behavior And Dynamics Of The Set Of Absolute Nilpotent And Idempotent Elements Of Chain Of Evolution Algebras Depending On The Time, Anvar Imomkulov

*Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences*

In this paper we construct some families of three-dimensional evolution algebras which satisfies Chapman-Kolmogorov equation. For all of these chains we study the behavior of the baric property, the behavior of the set of absolute nilpotent elements and dynamics of the set of idempotent elements depending on the time.

Using Desmos To Support Conceptual Understanding Of Quadratic Equations And Their Graphs, 2020 The College at Brockport: State University of New York

#### Using Desmos To Support Conceptual Understanding Of Quadratic Equations And Their Graphs, Shannon Bromley

*Education and Human Development Master's Theses*

When teaching secondary mathematics, there is often a struggle to distinguish true understanding versus the ability to follow a memorized procedure. This commonly occurs in Algebra I content of solving and graphing quadratic equations. Engaging students in Desmos interactive graphing learning activities has been shown to help deepen students’ understanding of such concepts. This curriculum project is designed to deepen students’ understanding and fluency of quadratic equations by including hands-on activities that support students discovering relationships and patterns between quadratic equations and their corresponding graphs. The unit plan aligns to the New York State Common Core State Standards.

On Tripartite Common Graphs, 2020 Jagiellonian University

#### On Tripartite Common Graphs, Andrzej Grzesik, Joonkyung Lee, Bernard Lidicky, Jan Volec

*Mathematics Publications*

A graph H is common if the number of monochromatic copies of H in a 2-edge-colouring of the complete graph is minimised by the random colouring. Burr and Rosta, extending a famous conjecture by Erdos, conjectured that every graph is common. The conjectures by Erdos and by Burr and Rosta were disproved by Thomason and by Sidorenko, respectively, in the late 1980s. Collecting new examples for common graphs had not seen much progress since then, although very recently, a few more graphs are verified to be common by the flag algebra method or the recent progress on Sidorenko's conjecture ...

Analytic Solutions For Diffusion On Path Graphs And Its Application To The Modeling Of The Evolution Of Electrically Indiscernible Conformational States Of Lysenin, 2020 Boise State University

#### Analytic Solutions For Diffusion On Path Graphs And Its Application To The Modeling Of The Evolution Of Electrically Indiscernible Conformational States Of Lysenin, K. Summer Ware

*Boise State University Theses and Dissertations*

Memory is traditionally thought of as a biological function of the brain. In recent years, however, researchers have found that some stimuli-responsive molecules exhibit memory-like behavior manifested as history-dependent hysteresis in response to external excitations. One example is lysenin, a pore-forming toxin found naturally in the coelomic fluid of the common earthworm *Eisenia fetida*. When reconstituted into a bilayer lipid membrane, this unassuming toxin undergoes conformational changes in response to applied voltages. However, lysenin is able to "remember" past history by adjusting its conformational state based not only on the amplitude of the stimulus but also on its previous its ...

On The Local Theory Of Profinite Groups, 2020 Western Michigan University

#### On The Local Theory Of Profinite Groups, Mohammad Shatnawi

*Dissertations*

Let *G* be a finite group, and *H* be a subgroup of G. The transfer homomorphism emerges from the natural action of G on the cosets of *H.* The transfer was first introduced by Schur in 1902 [22] as a construction in group theory, which produce a homomorphism from a finite group *G* into *H/H ^{'}* an abelian group where

*H*is a subgroup of

*G*and

*H'*is the derived group of

*H*. One important first application is Burnside’s normal

*p*-complement theorem [5] in 1911, although he did not use the transfer homomorphism explicitly to prove it ...

Sum Of Cubes Of The First N Integers, 2020 California State University, San Bernardino

#### Sum Of Cubes Of The First N Integers, Obiamaka L. Agu

*Electronic Theses, Projects, and Dissertations*

In Calculus we learned that Sum^{n}_{k=1} k = [n(n+1)]/2 , that Sum^{n}_{k=1} k^2 = [n(n+1)(2n+1)]/6 , and that Sum^{n}_{k=1} k^{3} = (n(n+1)/2)^{2}. These formulas are useful when solving for the area below quadratic or cubic function over an interval [a, b]. This tedious process, solving for areas under a quadratic or a cubic, served as motivation for the introduction of Riemman integrals. For the overzealous math student, these steps were replaced by a simpler method of evaluating antiderivatives at ...

Year 7 Students’ Interpretation Of Letters And Symbols In Solving Routine Algebraic Problems, 2020 International Islamic University Malaysia

#### Year 7 Students’ Interpretation Of Letters And Symbols In Solving Routine Algebraic Problems, Madihah Khalid Dr., Faeizah Yakop Dr., Hasniza Ibrahim

*The Qualitative Report*

In this study wefocused on one of the recurring issues in the learning of mathematics, which is students’ errors and misconceptions in learning algebra. We investigated Year 7 students on how they manipulate and interpret letters in solving routine algebraic problems to understand their thinking process. This is a case study of qualitative nature, focusing on one pencil and paper test, observation, and in-depth interviews of students in one particular school in Brunei Darussalam. The themes that emerged from interviews based on the test showed students’ interpretation of letters categorized as “combining” - which involved the combining of numbers during addition ...

The Name Tag Problem, 2020 Boise State University

#### The Name Tag Problem, Christian Carley

*Rose-Hulman Undergraduate Mathematics Journal*

The Name Tag Problem is a thought experiment that, when formalized, serves as an introduction to the concept of an orthomorphism of $\Zn$. Orthomorphisms are a type of group permutation and their graphs are used to construct mutually orthogonal Latin squares, affine planes and other objects. This paper walks through the formalization of the Name Tag Problem and its linear solutions, which center around modular arithmetic. The characterization of which linear mappings give rise to these solutions developed in this paper can be used to calculate the exact number of linear orthomorphisms for any additive group Z/nZ, which is ...