Mathematics Commons^™

Determining The Idealizers Of Principal Monomial Ideals Over A Rational Normal Curve, Perla A. Maldonado Cortez

Mathematics & Statistics ETDs

Given an ideal J generated by an element of the form sm1 tm2 , where
m1 ≥ 2 and m2 ≥ 0, we illustrate how to compute the idealizer I(J) over the ring
of the rational normal curve of degree n and we give a formula for it using the
graded pieces of the sets of differential operators.

The Zariski-Riemann Space As A Universal Model For The Birational Geometry Of A Function Field, Giovan Battista Pignatti Morano Di Custoza

Dissertations, Theses, and Capstone Projects

Given a function field $K$ over an algebraically closed field $k$, we propose to use the Zariski-Riemann space $\ZR (K/k)$ of valuation rings as a universal model that governs the birational geometry of the field extension $K/k$. More specifically, we find an exact correspondence between ad-hoc collections of open subsets of $\ZR (K/k)$ ordered by quasi-refinements and the category of normal models of $K/k$ with morphisms the birational maps. We then introduce suitable Grothendieck topologies and we develop a sheaf theory on $\ZR (K/k)$ which induces, locally at once, the sheaf theory of each normal model. Conversely, given a sheaf …

The Examination Of The Arithmetic Surface (3, 5) Over Q, Rachel J. Arguelles

Electronic Theses, Projects, and Dissertations

This thesis is centered around the construction and analysis of the principal arithmetic surface (3, 5) over Q. By adjoining the two symbols i,j, where i² = 3, j² = 5, such that ij = -ji, I can produce a quaternion algebra over Q. I use this quaternion algebra to find a discrete subgroup of SL₂(R), which I identify with isometries of the hyperbolic plane. From this quaternion algebra, I produce a large list of matrices and apply them via Mobius transformations to the point (0, 2), which is the center of my Dirichlet domain. This …

John Horton Conway: The Man And His Knot Theory, Dillon Ketron

Electronic Theses and Dissertations

John Horton Conway was a British mathematician in the twentieth century. He made notable achievements in fields such as algebra, number theory, and knot theory. He was a renowned professor at Cambridge University and later Princeton. His contributions to algebra include his discovery of the Conway group, a group in twenty-four dimensions, and the Conway Constellation. He contributed to number theory with his development of the surreal numbers. His Game of Life earned him long-lasting fame. He contributed to knot theory with his developments of the Conway polynomial, Conway sphere, and Conway notation.

Varieties Of Nonassociative Rings Of Bol-Moufang Type, Ronald E. White

All NMU Master's Theses

In this paper we investigate Bol-Moufang identities in a more general and very natural setting, \textit{nonassociative rings}.

We first introduce and define common algebras. We then explore the varieties of nonassociative rings of Bol-Moufang type. We explore two separate cases, the first where we consider binary rings, rings in which we make no assumption of it's structure. The second case we explore are rings in which, $2x=0$ implies $x=0$.

Counting The Moduli Space Of Pentagons On Finite Projective Planes, Maxwell Hosler

Senior Independent Study Theses

Finite projective planes are finite incidence structures which generalize the concept of the real projective plane. In this paper, we consider structures of points embedded in these planes. In particular, we investigate pentagons in general position, meaning no three vertices are colinear. We are interested in properties of these pentagons that are preserved by collineation of the plane, and so can be conceived as properties of the equivalence class of polygons up to collineation as a whole. Amongst these are the symmetries of a pentagon and the periodicity of the pentagon under the pentagram map, and a generalization of …

Some Contributions To Free Probability And Random Matrices., Sukrit Chakraborty Dr.

Doctoral Theses

No abstract provided.

Some Topics In Leavitt Path Algebras And Their Generalizations., Mohan R. Dr.

Doctoral Theses

The purpose of this section is to motivate the historical development of Leavitt algebras, Leavitt path algebras and their various generalizations and thus provide a context for this thesis. There are two historical threads which resulted in the definition of Leavitt path algebras. The first one is about the realization problem for von Neumann regular rings and the second one is about studying algebraic analogs of graph C ∗ -algebras. In what follows we briefly survey these threads and also introduce important concepts and terminology which will recur throughout.

College Algebra Through Problem Solving (2021 Edition), Danielle Cifone, Karan Puri, Debra Maslanko, Ewa Stelmach

Open Educational Resources

This is a self-contained, open educational resource (OER) textbook for college algebra. Students can use the book to learn concepts and work in the book themselves. Instructors can adapt the book for use in any college algebra course to facilitate active learning through problem solving. Additional resources such as classroom assessments and online/printable homework is available from the authors.

A Study On Approximations Of Totally Acyclic Complexes, Tyler Dean Anway

Mathematics Dissertations - Archive

Let $R$ be a commutative local ring to which we associate the subcategory $\Ktac(R)$ of the homotopy category of $R$-complexes, consisting of totally acyclic complexes. Further suppose there exists a surjection of Gorenstein local rings $Q \xrightarrowdbl{\varphi} R$ such that $R$ can be viewed as a $Q$-module with finite projective dimension. Under these assumptions, Bergh, Jorgensen, and Moore define the notion of approximations of totally acyclic complexes. In this dissertation we make extensive use of these approximations and define several novel applications. In particular, we extend Auslander-Reiten theory from the category of $R$-modules over a Henselian Gorenstein ring and show …

Train Algebra, Mary Soon Lee

Journal of Humanistic Mathematics

No abstract provided.

Factoring: Difference Of Squares, Thomas Lauria

Open Educational Resources

This lesson plan will explain how to factor basic difference of squares problems

Quantum Symmetries In Noncommutative Geometry., Suvrajit Bhattacharjee Dr.

Doctoral Theses

No abstract provided.

Representations Of Principal W-Algebra For The Superalgebra Q(N) And The Super Yangian Yq(1), Elena Poletaeva, Vera Serganova

School of Mathematical & Statistical Sciences Faculty Publications

We classify irreducible representations of finite W-algebra for the queer Lie superalgebra Q(n) associated with the principal nilpotent coadjoint orbits. We use this classification and our previous results to obtain a classification of irreducible finite-dimensional representations of the super Yangian YQ(1).

Oer Curve Fitting Applied To Easter Island Stone Foundations, Cynthia Huffman Ph.D.

Faculty Submissions

In this activity, curve fitting is applied to drone pictures of ruins of stone foundations of the traditional houses (hare paenga) on the island of Rapa Nui. The free mathematics application GeoGebra (geogebra.org) is used, but the activity can be adapted to other technology, such as Desmos (desmos.com). The activity can be used as a teacher demonstration or completed by students, individually or in small groups, with access to computers.

Quantum Markov Maps: Structureand Asymptotics., Vijaya Kumar U. Dr.

Doctoral Theses

No abstract provided.

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 …

Geometric Properties Of Representation Varieties For An Elementary Abelian Group Of Rank 2 In Positive Characteristic, Eric Johnson

Graduate Research Theses & Dissertations

This dissertation’s motivation is the exploration of the irreducible components of Repn(kG), the affine variety whose points are n-dimensional representations of a finite group G over a field k. We let G = Z/pZ×Z/pZ and assume k is algebraically closed with char(k) = p > 0. In this case there is an isomorphism of affine varieties φ : Repn (kG) → C nil 1 (n) where C nil 1 (n) = {(x, y) Mn(k)×Mn(k) | x p = y p = xy−yx = 0}. Hence, for an irreducible component X of Repn (kG), φ(X) is an irreducible component of C nil …

Zn Orbifolds Of Vertex Operator Algebras, Daniel Graybill

Electronic Theses and Dissertations

Given a vertex algebra V and a group of automorphisms of V, the invariant subalgebra V^G is called an orbifold of V. This construction appeared first in physics and was also fundamental to the construction of the Moonshine module in the work of Borcherds. It is expected that nice properties of V such as C2-cofiniteness and rationality will be inherited by V^G if G is a finite group. It is also expected that under reasonable hypotheses, if V is strongly finitely generated and G is reductive, V^G will also be strongly finitely generated. This is an analogue …

Oer Ellipses And Traditional Rapanui Houses On Easter Island, Cynthia Huffman Ph.D.

Faculty Submissions

This worksheet activity is appropriate for secondary students in a class studying conic sections or students in a college algebra class. The first part of the activity gives an algebraic review of ellipses with exercises while the second part finds the equation of an ellipse corresponding to a Rapanui boat house foundation.

Classical And Quantum Integrability: A Formulation That Admits Quantum Chaos, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

The concept of integrability of a quantum system is developed and studied. By formulating the concepts of quantum degree of freedom and quantum phase space, a realization of the dynamics is achieved. For a quantum system with a dynamical group G in one of its unitary irreducible representative carrier spaces, the quantum phase space is a finite topological space. It is isomorphic to a coset space G=R by means of the unitary exponential mapping, where R is the maximal stability subgroup of a fixed state in the carrier space. This approach has the distinct advantage of exhibiting consistency between classical …

Oer Indoor Ellipse Multicultural (Easter Island) Activity, Cynthia Huffman Ph.D.

Faculty Submissions

This activity would fit in with a secondary or college algebra class studying conic sections, in particular ellipses, and gives students a multicultural hands-on application of the definition of an ellipse, while tracing out a scale model of the foundation of a hare paenga (boat house) from prehistoric Easter Island (Rapa Nui)..

Oer Outdoor Ellipse Multicultural (Easter Island) Activity, Cynthia Huffman Ph.D.

Faculty Submissions

This activity would fit in with a secondary or college algebra class studying conic sections, in particular ellipses, and gives students a multicultural hands-on application of the definition of an ellipse, while tracing out a full-scale model of the foundation of a hare paenga (boat house) from prehistoric Easter Island (Rapa Nui)..

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 …

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 …

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.

College Algebra Notes And Exercises (Gcsu), Rabia Shahbaz, Janice Alves

Mathematics Ancillary Materials

Developed as part of a Round 13 Mini-Grant, these updated supplementary materials for Stitz-Zeager Open Source Mathematics and the LibGuides Open Course for College Algebra at GCSU include notes and exercises on equations, inequalities, functions, polynomial and rational functions, and exponential and logarithmic functions are included in one .zip file.

Studies On Polynomial Rings Through Locally Nilpotient Derivations., Nikhilesh Dasgupta Dr.

Doctoral Theses

No abstract provided.

“Product Development: Model Rockets As Toys”, Kelly W. Remijan

Teacher Resources

No abstract provided.