Algebra Commons^™

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 …

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.

A New Approach To The Q-Conjugacy Character Tables Of Finite Groups, Ali Moghani

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we study the Q-conjugacy character table of an arbitrary finite group and introduce a general relation between the degrees of Q-conjugacy characters with their corresponding reductions. This could be accomplished by using the Hermitian symmetric form. We provide a useful technique to calculate the character table of a finite group when its corresponding Qconjugacy character table is given. Then, we evaluate our results in some useful examples. Finally, by using GAP (Groups, Algorithms and Programming) package, we calculate all the dominant classes of the sporadic Conway group Co₂ enabling us to find all possible the integer-valued …

On Submoduloids Of A Moduloid On Nexus, Reza Kamranialiabad, Abbas Hasankhani, Masoud Bolourian

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, the submoduloid of a moduloid on nexus that is generated by a subset, cyclic submoduloid and bounded sets are defined and the properties of structures on it are investigated. Also, the fractions of a moduloid on nexus are defined and shown to be isomorphic with a moduloid on nexus.

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. …

The Lattice Of Intuitionistic Fuzzy Topologies Generated By Intuitionistic Fuzzy Relations, Soheyb Milles

Applications and Applied Mathematics: An International Journal (AAM)

We generalize the notion of fuzzy topology generated by fuzzy relation given by Mishra and Srivastava to the setting of intuitionistic fuzzy sets. Some fundamental properties and necessary examples are given. More specifically, we provide the lattice structure to a family of intuitionistic fuzzy topologies generated by intuitionistic fuzzy relations. To that end, we study necessary structural characteristics such as distributivity, modularity and complementary of this lattice.

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 …

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 the endpoints a and b. From my recollection, a former instructor informed us to do the value of memorizing these formulas. …

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 …

On The Global Operator And Fueter Mapping Theorem For Slice Polyanalytic Functions, Daniel Alpay, Kamal Diki, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we prove that slice polyanalytic functions on quaternions can be considered as solutions of a power of some special global operator with nonconstant coefficients as it happens in the case of slice hyperholomorphic functions. We investigate also an extension version of the Fueter mapping theorem in this polyanalytic setting. In particular, we show that under axially symmetric conditions it is always possible to construct Fueter regular and poly-Fueter regular functions through slice polyanalytic ones using what we call the poly-Fueter mappings. We study also some integral representations of these results on the quaternionic unit ball.

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 demonstrated …

Colorings And Sudoku Puzzles, Katelyn D. May

Rose-Hulman Undergraduate Mathematics Journal

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 rings. …

New Theorems For The Digraphs Of Commutative Rings, Morgan Bounds

Rose-Hulman Undergraduate Mathematics Journal

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, 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 (7,4) …

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 …

Some Generalizations Of Classical Integer Sequences Arising In Combinatorial Representation Theory, Sasha Verona Malone

Masters Theses & Specialist Projects

There exists a natural correspondence between the bases for a given finite-dimensional representation of a complex semisimple Lie algebra and a certain collection of finite edge-colored ranked posets, laid out by Donnelly, et al. in, for instance, [Don03]. In this correspondence, the Serre relations on the Chevalley generators of the given Lie algebra are realized as conditions on coefficients assigned to poset edges. These conditions are the so-called diamond, crossing, and structure relations (hereinafter DCS relations.) New representation constructions of Lie algebras may thus be obtained by utilizing edge-colored ranked posets. Of particular combinatorial interest are those representations whose corresponding …

Numerical Computations Of Vortex Formation Length In Flow Past An Elliptical Cylinder, Matthew Karlson, Bogdan Nita, Ashwin Vaidya

Department of Mathematics Facuty Scholarship and Creative Works

We examine two dimensional properties of vortex shedding past elliptical cylinders through numerical simulations. Specifically, we investigate the vortex formation length in the Reynolds number regime 10 to 100 for elliptical bodies of aspect ratio in the range 0.4 to 1.4. Our computations reveal that in the steady flow regime, the change in the vortex length follows a linear profile with respect to the Reynolds number, while in the unsteady regime, the time averaged vortex length decreases in an exponential manner with increasing Reynolds number. The transition in profile is used to identify the critical Reynolds number which marks the …

Length Neutrosophic Subalgebras Of Bck/Bci-Algebras, Florentin Smarandache, Young Bae Jun, Madad Khan, Seok-Zun Song

Branch Mathematics and Statistics Faculty and Staff Publications

the notion of (i, j, k)-length neutrosophic subalgebras in BCK/BCI-algebras is introduced, and their properties are investigated. Characterizations of length neutrosophic subalgebras are discussed by using level sets of interval neutrosophic sets. Conditions for level sets of interval neutrosophic sets to be subalgebras are provided.

On Product Of Smooth Neutrosophic Topological Spaces, Florentin Smarandache, Kalaivani Chandran, Swathi Sundari Sundaramoorthy, Saeid Jafari

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper, we develop the notion of the basis for a smooth neutrosophic topology in a more natural way. As a sequel, we define the notion of symmetric neutrosophic quasi-coincident neighborhood systems and prove some interesting results that fit with the classical ones, to establish the consistency of theory developed. Finally, we define and discuss the concept of product topology, in this context, using the definition of basis.

Spectral Sequences For Almost Complex Manifolds, Qian Chen

Dissertations, Theses, and Capstone Projects

In recent work, two new cohomologies were introduced for almost complex manifolds: the so-called J-cohomology and N-cohomology [CKT17]. For the case of integrable (complex) structures, the former cohomology was already considered in [DGMS75], and the latter agrees with de Rham cohomology. In this dissertation, using ideas from [CW18], we introduce spectral sequences for these two cohomologies, showing the two cohomologies have natural bigradings. We show the spectral sequence for the J-cohomology converges at the second page whenever the almost complex structure is integrable, and explain how both fit in a natural diagram involving Bott-Chern cohomology and the Frolicher spectral sequence. …

A Generic Implementation Of Fast Fourier Transforms For The Bpas Library, Colin S. Costello

Electronic Thesis and Dissertation Repository

In this thesis we seek to realize an efficient implementation of a generic parallel fast Fourier transform (FFT). The FFT will be used in support of fast multiplication of polynomials with coefficients in a finite field. Our goal is to obtain a relatively high performing parallel implementation that will run over a variety of finite fields with different sized characteristic primes. To this end, we implement and compare two Cooley-Tukey Six-Step fast Fourier transforms and a Cooley-Tukey Four-Step variant against a high performing specialized FFT already implemented in the Basic Polynomial Algebra Subprograms (BPAS) library. We use optimization techniques found …

Math Active Learning Lab: Math 98 Notebook, Gwennie Byron, Michele Iiams, Department Of Mathematics, University Of North Dakota

Open Educational Resources

This course notebook has been designed for students of Math 98 (Intermediate Algebra) at the University of North Dakota. It has been designed to help you get the most out of the ALEKS resources and your time.

• Topics in the Notebook are organized by weekly learning module.
• Space for notes from ALEKS learning pages, e-book and videos directs you to essential concepts.
• Examples and “You Try It” problems have been carefully chosen to help you focus on these essential concepts.
• Completed Notebook is an invaluable tool when studying for exams.

Universal Localizations Of Certain Noncommutative Rings, Tyler B. Bowles

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

A common theme throughout algebra is the extension of arithmetic systems to ones over which new equations can be solved. For instance, someone who knows only positive numbers might think that there is no solution to x + 3 = 0, yet later learns x = -3 to be a feasible solution. Likewise, when faced with the equation 2x = 3, someone familiar only with integers may declare that there is no solution, but may later learn that x = 3/2 is a reasonable answer. Many eventually learn that the extension of real numbers to complex numbers unlocks solutions …

Math Active Learning Lab: Math 103 College Algebra Notebook, Gwennie Byron, Department Of Mathematics, University Of North Dakota

Open Educational Resources

This course notebook has been designed for students of Math 103 (College Algebra) at the University of North Dakota. It has been designed to help you get the most out of the ALEKS resources and your time.

• Topics in the Notebook are organized by weekly learning module.
• Space for notes from ALEKS learning pages, e-book and videos directs you to essential concepts.
• Examples and “You Try It” problems have been carefully chosen to help you focus on these essential concepts.
• Completed Notebook is an invaluable tool when studying for exams.

A General Model Of Neutrosophic Ideals In Bck/Bci-Algebras Based On Neutrosophic Points, Florentin Smarandache, Hashem Bordbar, Rajab Ali Borzooei, Young Bae Jun

Branch Mathematics and Statistics Faculty and Staff Publications

More general form of (∈, ∈∨q)-neutrosophic ideal is introduced, and their properties are investigated.

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 …

Exact And Strongly Exact Filters, M. A. Moshier, A. Pultr, A. L. Suarez

Mathematics, Physics, and Computer Science Faculty Articles and Research

A meet in a frame is exact if it join-distributes with every element, it is strongly exact if it is preserved by every frame homomorphism. Hence, finite meets are (strongly) exact which leads to the concept of an exact resp. strongly exact filter, a filter closed under exact resp. strongly exact meets. It is known that the exact filters constitute a frame FiltE(L) somewhat surprisingly isomorphic to the frame of joins of closed sublocales. In this paper we present a characteristic of the coframe of meets of open sublocales as the dual to the frame of strongly exact filters FiltsE(L).

On Pseudo-Spectral Factorization Over The Complex Numbers And Quaternions, Daniel Alpay, Fabrizio Colombo, Izchak Lewkowicz, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

This paper is a continuation of the research of our previous work[5] and considers quaternionic generalized Carathéodory functions and the related family of generalized positive functions. It is addressed to a wide audience which includes researchers in complex and hypercomplex analysis, in the theory of linear systems, but also electric engineers. For this reason it includes some results on generalized Carathéodory functions and their factorization in the classic complex case which might be of independent interest. An important new result is a pseudo-spectral factorization and we also discuss some interpolation problems in the class of quaternionic generalized positive functions.

A Real World Example Of Solving A Quadratic Equation In Movie Cgi, Cynthia J. Huffman Ph.D.

Faculty Submissions

It is important to expose students to the beauty and usefulness of mathematics. Since computer graphics are familiar to most students due to video games and movies, they make a great source for motivating topics in mathematics. This activity shows students an application of solving quadratic equations to computing the line of sight to spherical objects in computer graphics.