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Articles 1 - 15 of 15
Full-Text Articles in Algebra
Roots Of Quaternionic Polynomials And Automorphisms Of Roots, Olalekan Ogunmefun
Roots Of Quaternionic Polynomials And Automorphisms Of Roots, Olalekan Ogunmefun
Electronic Theses and Dissertations
The quaternions are an extension of the complex numbers which were first described by Sir William Rowan Hamilton in 1843. In his description, he gave the equation of the multiplication of the imaginary component similar to that of complex numbers. Many mathematicians have studied the zeros of quaternionic polynomials. Prominent of these, Ivan Niven pioneered a root-finding algorithm in 1941, Gentili and Struppa proved the Fundamental Theorem of Algebra (FTA) for quaternions in 2007. This thesis finds the zeros of quaternionic polynomials using the Fundamental Theorem of Algebra. There are isolated zeros and spheres of zeros. In this thesis, we …
“Product Development: Model Rockets As Toys”, Kelly W. Remijan
“Product Development: Model Rockets As Toys”, Kelly W. Remijan
Teacher Resources
No abstract provided.
"American Football: Field Goals And Quadratic Functions”, Kelly W. Remijan
"American Football: Field Goals And Quadratic Functions”, Kelly W. Remijan
Teacher Resources
No abstract provided.
"Crash Reconstruction: Stopping Distance”, Kelly W. Remijan
"Crash Reconstruction: Stopping Distance”, Kelly W. Remijan
Teacher Resources
No abstract provided.
Gröbner Bases And Systems Of Polynomial Equations, Rachel Holmes
Gröbner Bases And Systems Of Polynomial Equations, Rachel Holmes
All Graduate Theses, Dissertations, and Other Capstone Projects
This paper will explore the use and construction of Gröbner bases through Buchberger's algorithm. Specifically, applications of such bases for solving systems of polynomial equations will be discussed. Furthermore, we relate many concepts in commutative algebra to ideas in computational algebraic geometry.
Invariant Sum Defined In Terms Of Complex Multivariate Polynomial Given Degree, Matthew Niemiro '20
Invariant Sum Defined In Terms Of Complex Multivariate Polynomial Given Degree, Matthew Niemiro '20
Exemplary Student Work
We use a generalized version of arithmetic progressions to obtain a non- trivial everywhere-zero sum in terms of a complex univariate polynomial and its degree. We then remark on its generalization to multivariate polynomials.
On A Frobenius Problem For Polynomials, Ricardo Conceição, R. Gondim, M. Rodriguez
On A Frobenius Problem For Polynomials, Ricardo Conceição, R. Gondim, M. Rodriguez
Math Faculty Publications
We extend the famous diophantine Frobenius problem to a ring of polynomials over a field~k. Similar to the classical problem we show that the n = 2 case of the Frobenius problem for polynomials is easy to solve. In addition, we translate a few results from the Frobenius problem over ℤ to k[t] and give an algorithm to solve the Frobenius problem for polynomials over a field k of sufficiently large size.
Efficiently Representing The Integer Factorization Problem Using Binary Decision Diagrams, David Skidmore
Efficiently Representing The Integer Factorization Problem Using Binary Decision Diagrams, David Skidmore
All Graduate Plan B and other Reports, Spring 1920 to Spring 2023
Let p be a prime positive integer and let α be a positive integer greater than 1. A method is given to reduce the problem of finding a nontrivial factorization of α to the problem of finding a solution to a system of modulo p polynomial congruences where each variable in the system is constrained to the set {0,...,p − 1}. In the case that p = 2 it is shown that each polynomial in the system can be represented by an ordered binary decision diagram with size less than 20.25log2(α)3 + 16.5log2(α)2 + …
Algebra Tutorial For Prospective Calculus Students, Matthew Mckain
Algebra Tutorial For Prospective Calculus Students, Matthew Mckain
All Capstone Projects
Many undergraduate degrees require students to take one or more courses in calculus. Majors in mathematics, science, and engineering are expected to enroll in several rigorous calculus courses, but those majoring in social and behavioral sciences and business must also have some basic understanding of calculus. The goal of this project is to create a web-based tutorial that can be used by the GSU Mathematics faculty to reinforce the algebra skills needed for introductory or Applied Calculus. The tutorial covers the concepts of the slopes of lines, polynomial arithmetic, factoring polynomials, rational expressions, solving quadratic equations, linear and polynomial inequalities, …
Polynomial Factoring Algorithms And Their Computational Complexity, Nicholas Cavanna
Polynomial Factoring Algorithms And Their Computational Complexity, Nicholas Cavanna
Honors Scholar Theses
Finite fields, and the polynomial rings over them, have many neat algebraic properties and identities that are very convenient to work with. In this paper we will start by exploring said properties with the goal in mind of being able to use said properties to efficiently irreducibly factorize polynomials over these fields, an important action in the fields of discrete mathematics and computer science. Necessarily, we must also introduce the concept of an algorithm’s speed as well as particularly speeds of basic modular and integral arithmetic opera- tions. Outlining these concepts will have laid the groundwork for us to introduce …
Information-Preserving Structures: A General Framework For Quantum Zero-Error Information, Robin Blume-Kohout, Hui Khoon Ng, David Poulin, Lorenza Viola
Information-Preserving Structures: A General Framework For Quantum Zero-Error Information, Robin Blume-Kohout, Hui Khoon Ng, David Poulin, Lorenza Viola
Dartmouth Scholarship
Quantum systems carry information. Quantum theory supports at least two distinct kinds of information (classical and quantum), and a variety of different ways to encode and preserve information in physical systems. A system’s ability to carry information is constrained and defined by the noise in its dynamics. This paper introduces an operational framework, using information-preserving structures, to classify all the kinds of information that can be perfectly (i.e., with zero error) preserved by quantum dynamics. We prove that every perfectly preserved code has the same structure as a matrix algebra, and that preserved information can always be corrected. We …
Search Bounds For Zeros Of Polynomials Over The Algebraic Closure Of Q, Lenny Fukshansky
Search Bounds For Zeros Of Polynomials Over The Algebraic Closure Of Q, Lenny Fukshansky
CMC Faculty Publications and Research
We discuss existence of explicit search bounds for zeros of polynomials with coefficients in a number field. Our main result is a theorem about the existence of polynomial zeros of small height over the field of algebraic numbers outside of unions of subspaces. All bounds on the height are explicit.
Polynomial Extension Operators. Part I, Leszek Demkowicz, Jay Gopalakrishnan, Joachim Schöberl
Polynomial Extension Operators. Part I, Leszek Demkowicz, Jay Gopalakrishnan, Joachim Schöberl
Mathematics and Statistics Faculty Publications and Presentations
In this series of papers, we construct operators that extend certain given functions on the boundary of a tetrahedron into the interior of the tetrahedron, with continuity properties in appropriate Sobolev norms. These extensions are novel in that they have certain polynomial preservation properties important in the analysis of high order finite elements. This part of the series is devoted to introducing our new technique for constructing the extensions, and its application to the case of polynomial extensions from H½(∂K) into H¹(K), for any tetrahedron K.
Chinese Remainder Theorem And Its Applications, Jacquelyn Ha Lac
Chinese Remainder Theorem And Its Applications, Jacquelyn Ha Lac
Theses Digitization Project
No abstract provided.
The Solvability Of Polynomials By Radicals: A Search For Unsolvable And Solvable Quintic Examples, Robert Lewis Beyronneau
The Solvability Of Polynomials By Radicals: A Search For Unsolvable And Solvable Quintic Examples, Robert Lewis Beyronneau
Theses Digitization Project
This project centers around finding specific examples of quintic polynomials that were and were not solvable. This helped to devise a method for finding examples of solvable and unsolvable quintics.