A Proof Of A Generalization Of Niven's Theorem Using Algebraic Number Theory, 2021 University of Maryland, College Park
A Proof Of A Generalization Of Niven's Theorem Using Algebraic Number Theory, Caroline Nunn
Rose-Hulman Undergraduate Mathematics Journal
Niven’s theorem states that the sine, cosine, and tangent functions are rational for only a few rational multiples of π. Specifically, for angles θ that are rational multiples of π, the only rational values of sin(θ) and cos(θ) are 0, ±½, and ±1. For tangent, the only rational values are 0 and ±1. We present a proof of this fact, along with a generalization, using the structure of ideals in imaginary quadratic rings. We first show that the theorem holds for the tangent function using elementary properties of Gaussian integers, before extending the approach to other imaginary quadratic rings. We …
Equisingular Approximation Of Analytic Germs, 2021 The University of Western Ontario
Equisingular Approximation Of Analytic Germs, Aftab Yusuf Patel
Electronic Thesis and Dissertation Repository
This thesis deals with the problem of approximating germs of real or complex analytic spaces by Nash or algebraic germs. In particular, we investigate the problem of approximating analytic germs in various ways while preserving the Hilbert-Samuel function, which is of importance in the resolution of singularities. We first show that analytic germs that are complete intersections can be arbitrarily closely approximated by algebraic germs which are complete intersections with the same Hilbert-Samuel function. We then show that analytic germs whose local rings are Cohen-Macaulay can be arbitrarily closely approximated by Nash germs whose local rings are Cohen- Macaulay and …
(R1466) Ideals And Filters On A Lattice In Neutrosophic Setting, 2021 University of M’sila
(R1466) Ideals And Filters On A Lattice In Neutrosophic Setting, Lemnaouar Zedam, Soheyb Milles, Abdelhamid Bennoui
Applications and Applied Mathematics: An International Journal (AAM)
The notions of ideals and filters have studied in many algebraic (crisp) fuzzy structures and used to study their various properties, representations and characterizations. In addition to their theoretical roles, they have used in some areas of applied mathematics. In a recent paper, Arockiarani and Antony Crispin Sweety have generalized and studied these notions with respect to the concept of neutrosophic sets introduced by Smarandache to represent imprecise, incomplete and inconsistent information. In this article, we aim to deepen the study of these important notions on a given lattice in the neutrosophic setting. We show their various properties and characterizations, …
Linear Algebra For Computer Science, 2021 Singapore Management University
Linear Algebra For Computer Science, M. Thulasidas
Research Collection School Of Computing and Information Systems
This textbook introduces the essential concepts and practice of Linear Algebra to the undergraduate student of computer science. The focus of this book is on the elegance and beauty of the numerical techniques and algorithms originating from Linear Algebra. As a practical handbook for computer and data scientists, LA4CS restricts itself mostly to real fields and tractable discourses, rather than deep and theoretical mathematics.
Symmetric Representations Of Finite Groups And Related Topics, 2021 California State University, San Bernardino
Symmetric Representations Of Finite Groups And Related Topics, Connie Corona
Electronic Theses, Projects, and Dissertations
In this thesis, we have presented our discovery of original symmetric presentations of a number of non-abelian simple groups, including several sporatic groups, linear groups, and classical groups.
We have constructed, using our technique of double coset enumeration, J2, M12, J1, PSU(3, 3):2, M11, A10, S(4,3), M22:2, PSL(3, 4), S6, 2:S5, 2:PSL(3, 4) as homomorphic images of the involutory progenitors 2*32:(25:A5), 2*110: PSL(2, 11), 2*5:A5, 3*4:D8, 2*110:PSL(2, 11), …
A Study In Applications Of Continued Fractions, 2021 California State University, San Bernardino
A Study In Applications Of Continued Fractions, Karen Lynn Parrish
Electronic Theses, Projects, and Dissertations
This is an expository study of continued fractions collecting ideas from several different sources including textbooks and journal articles. This study focuses on several applications of continued fractions from a variety of levels and fields of mathematics. Studies begin with looking at a number of properties that pertain to continued fractions and then move on to show how applications of continued fractions is relevant to high school level mathematics including approximating irrational numbers and developing new ideas for understanding and solving quadratics equations. Focus then continues to more advanced applications such as those used in the studies of number theory …
Finite Groups In Which The Number Of Cyclic Subgroups Is 3/4 The Order Of The Group, 2021 Missouri State University
Finite Groups In Which The Number Of Cyclic Subgroups Is 3/4 The Order Of The Group, James Alexander Cayley
MSU Graduate Theses
Let $G$ be a finite group, c(G) denotes the number of cyclic subgroups of G and α(G) = c(G)/|G|. In this thesis we go over some basic properties of alpha, calculate alpha for some families of groups, with an emphasis on groups with α(G) = 3/4, as all groups with α(G) > 3/4 have been classified by Garonzi and Lima (2018). We find all Dihedral group with this property, show all groups with α(G) = 3/4 have at least |G|/2-1 involutions, and discuss existing work by Wall (1970) and Miller (1919) classifying all such groups.
Algorithmic Correspondence For Relevance Logics, Bunched Implication Logics, And Relation Algebras Via An Implementation Of The Algorithm Pearl, 2021 University of the Witwatersrand, Johannesburg
Algorithmic Correspondence For Relevance Logics, Bunched Implication Logics, And Relation Algebras Via An Implementation Of The Algorithm Pearl, Willem Conradie, Valentin Goranko, Peter Jipsen
Mathematics, Physics, and Computer Science Faculty Articles and Research
The non-deterministic algorithmic procedure PEARL (acronym for ‘Propositional variables Elimination Algorithm for Relevance Logic’) has been recently developed for computing first-order equivalents of formulas of the language of relevance logics LR in terms of the standard Routley-Meyer relational semantics. It succeeds on a large class of axioms of relevance logics, including all so called inductive formulas. In the present work we re-interpret PEARL from an algebraic perspective, with its rewrite rules seen as manipulating quasi-inequalities interpreted over Urquhart’s relevant algebras, and report on its recent Python implementation. We also show that all formulae on which PEARL succeeds are canonical, i.e., …
Unary-Determined Distributive ℓ -Magmas And Bunched Implication Algebras, 2021 Chapman University
Unary-Determined Distributive ℓ -Magmas And Bunched Implication Algebras, Natanael Alpay, Peter Jipsen, Melissa Sugimoto
Mathematics, Physics, and Computer Science Faculty Articles and Research
A distributive lattice-ordered magma (dℓ-magma) (A,∧,∨,⋅) is a distributive lattice with a binary operation ⋅ that preserves joins in both arguments, and when ⋅ is associative then (A,∨,⋅) is an idempotent semiring. A dℓ-magma with a top ⊤ is unary-determined if x⋅y=(x⋅⊤∧y)∨(x∧⊤⋅y). These algebras are term-equivalent to a subvariety of distributive lattices with ⊤ and two join-preserving unary operations p, q. We obtain simple conditions on p, q such that x⋅y=(px∧y)∨(x∧qy) is associative, commutative, idempotent and/or has an identity element. This generalizes previous results on the structure of doubly idempotent semirings and, in the case when the …
Generalized Grassmann Algebras And Applications To Stochastic Processes, 2021 Chapman University
Generalized Grassmann Algebras And Applications To Stochastic Processes, Daniel Alpay, Paula Cerejeiras, Uwe Kaehler
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this paper, we present the groundwork for an Itô/Malliavin stochastic calculus and Hida's white noise analysis in the context of a supersymmetry with Z3-graded algebras. To this end, we establish a ternary Fock space and the corresponding strong algebra of stochastic distributions and present its application in the study of stochastic processes in this context.
The Structure Of Finite Commutative Idempotent Involutive Residuated Lattices, 2021 Chapman University
The Structure Of Finite Commutative Idempotent Involutive Residuated Lattices, Peter Jipsen, Olim Tuyt, Diego Valota
Mathematics, Physics, and Computer Science Faculty Articles and Research
We characterize commutative idempotent involutive residuated lattices as disjoint unions of Boolean algebras arranged over a distributive lattice. We use this description to introduce a new construction, called gluing, that allows us to build new members of this variety from other ones. In particular, all finite members can be constructed in this way from Boolean algebras. Finally, we apply our construction to prove that the fusion reduct of any finite member is a distributive semilattice, and to show that this variety is not locally finite.
Rigid Connections On The Projective Line With Elliptic Toral Singularities, 2021 Louisiana State University and Agricultural and Mechanical College
Rigid Connections On The Projective Line With Elliptic Toral Singularities, Alisina Azhang
LSU Doctoral Dissertations
We generalize two studies of rigid $G$-connections on $\pp$ which have an irregular singularity at origin and a regular singularity at infinity with unipotent monodromy: one is the work of Kamgarpour-Sage which classifies rigid homogeneous Coxeter $G$-connections with slope $\frac{r}{h}$, where $h$ is the Coxeter number of $G$, and the other is the work of Chen, which proves the existence of rigid homogeneous elliptic regular $G$-connections with slope $\frac{1}{m}$, where $m$ is an elliptic number for $G$. In our work, similar to Chen, we look for rigid homogeneous elliptic regular $G$-connections, but we allow the slope to have a numerator …
Conjunctive Join-Semilattices, 2021 Louisiana State University at Baton Rouge
Conjunctive Join-Semilattices, Charles N. Delzell, Oghenetega Ighedo, James J. Madden
Mathematics, Physics, and Computer Science Faculty Articles and Research
A join-semilattice L with top is said to be conjunctive if every principal ideal is an intersection of maximal ideals. (This is equivalent to a first-order condition in the language of semilattices.) In this paper, we explore the consequences of the conjunctivity hypothesis for L, and we define and study a related property, called “ideal conjunctivity,” which is applicable to join-semilattices without top. Results include the following: (a) Every conjunctive join-semilattice is isomorphic to a join-closed subbase for a compact T1-topology on max L, the set of maximal ideals of L, and under weak hypotheses …
Categorical Aspects Of Graphs, 2021 Western University
Categorical Aspects Of Graphs, Jacob D. Ender
Undergraduate Student Research Internships Conference
In this article, we introduce a categorical characterization of directed and undirected graphs, and explore subcategories of reflexive and simple graphs. We show that there are a number of adjunctions between such subcategories, exploring varying combinations of graph types.
College Algebra Through Problem Solving (2021 Edition), 2021 CUNY Queensborough Community College
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.
Beurling-Lax Type Theorems And Cuntz Relations, 2021 Chapman University
Beurling-Lax Type Theorems And Cuntz Relations, Daniel Alpay, Fabrizio Colombo, Irene Sabadini, Baruch Schneider
Mathematics, Physics, and Computer Science Faculty Articles and Research
We prove various Beurling-Lax type theorems, when the classical backward-shift operator is replaced by a general resolvent operator associated with a rational function. We also study connections to the Cuntz relations. An important tool is a new representation result for analytic functions, in terms of composition and multiplication operators associated with a given rational function. Applications to the theory of de Branges-Rovnyak spaces, also in the indefinite metric setting, are given.
Linear Algebra For Computer Science, 2021 Singapore Management University
Linear Algebra For Computer Science, M. Thulasidas
Research Collection School Of Computing and Information Systems
This book has its origin in my experience teaching Linear Algebra to Computer Science students at Singapore Management University. Traditionally, Linear Algebra is taught as a pure mathematics course, almost as an afterthought, not fully integrated with any other applied curriculum. It certainly was taught that way to me. The course I was teaching, however, had a definite pedagogical objective of bringing out the applicability and the usefulness of Linear Algebra in Computer Science, which is nothing but applied mathematics. In today’s age of machine learning and artificial intelligence, Linear Algebra is the branch of mathematics that holds the most …
Free Complexes Over The Exterior Algebra With Small Homology, 2021 University of Nebraska-Lincoln
Free Complexes Over The Exterior Algebra With Small Homology, Erica Hopkins
Department of Mathematics: Dissertations, Theses, and Student Research
Let M be a graded module over a standard graded polynomial ring S. The Total Rank Conjecture by Avramov-Buchweitz predicts the total Betti number of M should be at least the total Betti number of the residue field. Walker proved this is indeed true in a large number of cases. One could then try to push this result further by generalizing this conjecture to finite free complexes which is known as the Generalized Total Rank Conjecture. However, Iyengar and Walker constructed examples to show this generalized conjecture is not always true.
In this thesis, we investigate other counterexamples of …
Algorithms Related To Triangle Groups, 2021 Louisiana State University and Agricultural and Mechanical College
Algorithms Related To Triangle Groups, Bao The Pham
LSU Doctoral Dissertations
Given a finite index subgroup of $\PSL_2(\Z)$, one can talk about the different properties of this subgroup. These properties have been studied extensively in an attempt to classify these subgroups. Tim Hsu created an algorithm to determine whether a subgroup is a congruence subgroup by using permutations \cite{hsu}. Lang, Lim, and Tan also created an algorithm to determine if a subgroup is a congruence subgroup by using Farey Symbols \cite{llt}. Sebbar classified torsion-free congruence subgroups of genus 0 \cite{sebbar}. Pauli and Cummins computed and tabulated all congruence subgroups of genus less than 24 \cite{ps}. However, there are still some problems …
Injective And Projective Semimodules Over Involutive Semirings, 2021 Chapman University
Injective And Projective Semimodules Over Involutive Semirings, Peter Jipsen, Sara Vanucci
Mathematics, Physics, and Computer Science Faculty Articles and Research
We show that the term equivalence between MV-algebras and MV-semirings lifts to involutive residuated lattices and a class of semirings called involutive semirings. The semiring perspective leads to a necessary and sufficient condition for the interval [d,1] to be a subalgebra of an involutive residuated lattice, where d is the dualizing element. We also import some results and techniques of semimodule theory in the study of this class of semirings, generalizing results about injective and projective MV-semimodules. Indeed, we note that the involution plays a crucial role and that the results for MV-semirings are still true for …