# Algebra Commons™

## All Articles in Algebra

1,267 full-text articles. Page 2 of 48.

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

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.

2021 University of Kentucky

#### Maximums Of Total Betti Numbers In Hilbert Families, Jay White

##### Theses and Dissertations--Mathematics

Fix a family of ideals in a polynomial ring and consider the problem of finding a single ideal in the family that has Betti numbers that are greater than or equal to the Betti numbers of every ideal in the family. Or decide if this special ideal even exists. Bigatti, Hulett, and Pardue showed that if we take the ideals with a fixed Hilbert function, there is such an ideal: the lexsegment ideal. Caviglia and Murai proved that if we take the saturated ideals with a fixed Hilbert polynomial, there is also such an ideal. We present a generalization of ...

2021 Claremont Colleges

#### On Properties Of Positive Semigroups In Lattices And Totally Real Number Fields, Siki Wang

##### CMC Senior Theses

In this thesis, we give estimates on the successive minima of positive semigroups in lattices and ideals in totally real number fields. In Chapter 1 we give a brief overview of the thesis, while Chapters 2 – 4 provide expository material on some fundamental theorems about lattices, number fields and height functions, hence setting the necessary background for the original results presented in Chapter 5. The results in Chapter 5 can be summarized as follows. For a full-rank lattice L ⊂ Rd, we are concerned with the semigroup L+ ⊆ L, which denotes the set of all vectors with nonnegative coordinates in L ...

2021 Colby College

#### Counting Conjugacy Classes Of Elements Of Finite Order In Compact Exceptional Groups, Qidong He

##### Honors Theses

Given a compact exceptional group $G$ and $m,s\in\mathbb{N}$, let $N(G,m)$ be the number of conjugacy classes of elements of order $m$ in $G$, and $N(G,m,s)$ the number of such classes whose elements have $s$ distinct eigenvalues. In string theory, the problem of enumerating certain classes of vacua in the string landscape can be rephrased in terms of the study of these quantities. We develop unified combinatorial algorithms based on Burnside's Lemma that can be used to compute both quantities for each of the five compact exceptional groups. Thus, we provide ...

The Complexity Of Symmetry, 2021 Claremont Colleges

#### The Complexity Of Symmetry, Matthew Lemay

##### HMC Senior Theses

One of the main goals of theoretical computer science is to prove limits on how efficiently certain Boolean functions can be computed. The study of the algebraic complexity of polynomials provides an indirect approach to exploring these questions, which may prove fruitful since much is known about polynomials already from the field of algebra. This paper explores current research in establishing lower bounds on invariant rings and polynomial families. It explains the construction of an invariant ring for whom a succinct encoding would imply that NP is in P/poly. It then states a theorem about the circuit complexity partial ...

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, 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 ...

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.

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 ...

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

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 ...

Colorings And Sudoku Puzzles, 2020 Murray State University

#### Colorings And Sudoku Puzzles, Katelyn D. May

Map colorings refer to assigning colors to different regions of a map. In particular, a typical application is to assign colors so that no two adjacent regions are the same color. Map colorings are easily converted to graph coloring problems: regions correspond to vertices and edges between two vertices exist for adjacent regions. We extend these notions to Shidoku, 4x4 Sudoku puzzles, and standard 9x9 Sudoku puzzles by demanding unique entries in rows, columns, and regions. Motivated by our study of ring and field theory, we expand upon the standard division algorithm to study Gr\"obner bases in multivariate polynomial ...

New Theorems For The Digraphs Of Commutative Rings, 2020 Indiana Wesleyan University

#### New Theorems For The Digraphs Of Commutative Rings, Morgan Bounds

The digraphs of commutative rings under modular arithmetic reveal intriguing cycle patterns, many of which have yet to be explained. To help illuminate these patterns, we establish a set of new theorems. Rings with relatively prime moduli a and b are used to predict cycles in the digraph of the ring with modulus ab. Rings that use Pythagorean primes as their modulus are shown to always have a cycle in common. Rings with perfect square moduli have cycles that relate to their square root.

Hamming Codes, 2020 University of Washington, Tacoma

#### Hamming Codes, Steve Mwangi, Sterling Quinn

##### Access*: Interdisciplinary Journal of Student Research and Scholarship

We will be looking into the application of Matrix Algebra in forming Hamming Codes. Hamming Codes are essential not just in the detection of errors, but also in the linear concurrent correction of these errors. The matrices we will use, will have entries that are binary units. Binary units are mathematically convenient, and their simplicity permits the representation of many open and closed circuits used in communication systems. The entries in the matrices will represent a message that is meant for transmission or reception, akin to the contemporary application of Hamming Codes in wireless communication. We will use Hamming (7 ...

Abstract Algebra: Theory And Applications, 2020 Stephen F Austin State University

#### Abstract Algebra: Theory And Applications, Thomas W. Judson

##### eBooks

Tom Judson's Abstract Algebra: Theory and Applications is an open source textbook designed to teach the principles and theory of abstract algebra to college juniors and seniors in a rigorous manner. Its strengths include a wide range of exercises, both computational and theoretical, plus many nontrivial applications. Rob Beezer has contributed complementary material using the open source system, Sage.

An HTML version on the PreText platform is available here.

The first half of the book presents group theory, through the Sylow theorems, with enough material for a semester-long course. The second-half is suitable for a second semester and presents ...