Algebra Commons^™

Application Of Symplectic Integration On A Dynamical System, William Frazier

Electronic Theses and Dissertations

Molecular Dynamics (MD) is the numerical simulation of a large system of interacting molecules, and one of the key components of a MD simulation is the numerical estimation of the solutions to a system of nonlinear differential equations. Such systems are very sensitive to discretization and round-off error, and correspondingly, standard techniques such as Runge-Kutta methods can lead to poor results. However, MD systems are conservative, which means that we can use Hamiltonian mechanics and symplectic transformations (also known as canonical transformations) in analyzing and approximating solutions. This is standard in MD applications, leading to numerical techniques known as symplectic …

On Rings Of Invariants For Cyclic P-Groups, Daniel Juda

Graduate Theses and Dissertations

This thesis studies the ring of invariants R^G of a cyclic p-group G acting on k[x_1,\ldots, x_n] where k is a field of characteristic p >0. We consider when R^G is Cohen-Macaulay and give an explicit computation of the depth of R^G. Using representation theory and a result of Nakajima, we demonstrate that R^G is a unique factorization domain and consequently quasi-Gorenstein. We answer the question of when R^G is F-rational and when R^G is F-regular.

We also study the a-invariant for a graded ring S, that is, the maximal graded degree of the top local cohomology module of S. …

Relation Algebras, Idempotent Semirings And Generalized Bunched Implication Algebras, Peter Jipsen

Mathematics, Physics, and Computer Science Faculty Articles and Research

This paper investigates connections between algebraic structures that are common in theoretical computer science and algebraic logic. Idempotent semirings are the basis of Kleene algebras, relation algebras, residuated lattices and bunched implication algebras. Extending a result of Chajda and Länger, we show that involutive residuated lattices are determined by a pair of dually isomorphic idempotent semirings on the same set, and this result also applies to relation algebras. Generalized bunched implication algebras (GBI-algebras for short) are residuated lattices expanded with a Heyting implication. We construct bounded cyclic involutive GBI-algebras from so-called weakening relations, and prove that the class of weakening …

Six Septembers: Mathematics For The Humanist, Patrick Juola, Stephen Ramsay

Zea E-Books Collection

Scholars of all stripes are turning their attention to materials that represent enormous opportunities for the future of humanistic inquiry. The purpose of this book is to impart the concepts that underlie the mathematics they are likely to encounter and to unfold the notation in a way that removes that particular barrier completely. This book is a primer for developing the skills to enable humanist scholars to address complicated technical material with confidence. This book, to put it plainly, is concerned with the things that the author of a technical article knows, but isn’t saying. Like any field, mathematics operates …

A Game Of Monovariants On A Checkerboard, Linwood Reynolds

Student Scholar Showcase

Abstract: Assume there is a game that takes place on a 20x20 checkerboard in which each of the 400 squares are filled with either a penny, nickel, dime, or quarter. The coins are placed randomly onto the squares, and there are to be 100 of each of the coins on the board. To begin the game, 59 coins are removed at random. The goal of the game is to remove each remaining coin from the board according to the following rules: 1. A penny can only be removed if all 4 adjacent squares are empty. That is, a penny cannot …

Session A-3: Three-Act Math Tasks, Lindsey Herlehy

Professional Learning Day

Participants will engage in a Three-Act Math task highlighting the application of properties of geometrical figures. Developed by Dan Meyer, an innovative and highly regarded mathematics instructor, Three-Act Math tasks utilize pedagogical skills that elicit student curiosity, collaboration and questioning. By posing a mathematical problem through active storytelling, this instructional approach redefines real-world mathematics and clarifies the role that a student plays in the learning process. Participants will be given multiple resources where they can access Three-Act Math tasks appropriate for upper elementary grades through Algebra and Geometry courses.

On A Class Of Quaternionic Positive Definite Functions And Their Derivatives, Daniel Alpay, Fabrizio Colombo, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we start the study of stochastic processes over the skew field of quaternions. We discuss the relation between positive definite functions and the covariance of centered Gaussian processes and the construction of stochastic processes and their derivatives. The use of perfect spaces and strong algebras and the notion of Fock space are crucial in this framework.

The Mathematics Of Superoscillations, Yakir Aharonov, Fabrizio Colombo, Irene Sabadini, Daniele C. Struppa, Jeff Tollaksen

Mathematics, Physics, and Computer Science Faculty Articles and Research

In the past 50 years, quantum physicists have discovered, and experimentally demonstrated, a phenomenon which they termed superoscillations. Aharonov and his collaborators showed that superoscillations naturally arise when dealing with weak values, a notion that provides a fundamentally different way to regard measurements in quantum physics. From a mathematical point of view, superoscillating functions are a superposition of small Fourier components with a bounded Fourier spectrum, which result, when appropriately summed, in a shift that can be arbitrarily large, and well outside the spectrum. Purpose of this work is twofold: on one hand we provide a self-contained survey of the …

Specifications Grading In A First Course In Abstract Algebra, Mike Janssen

Faculty Work Comprehensive List

Specifications grading offers an alternative to more traditional, points-based grading and assessment structures. In place of partial credit, students are assessed pass/fail on whether or not they have achieved the learning outcomes being assessed on a given piece of work according to certain specifications, with limited opportunities for revision of non-passing work. This talk will describe the learning outcomes and specifications grading system I used in my Fall 2016 abstract algebra course, as well as student responses.

Toric Ideals, Polytopes, And Convex Neural Codes, Caitlin Lienkaemper

HMC Senior Theses

How does the brain encode the spatial structure of the external world?

A partial answer comes through place cells, hippocampal neurons which

become associated to approximately convex regions of the world known

as their place fields. When an organism is in the place field of some place

cell, that cell will fire at an increased rate. A neural code describes the set

of firing patterns observed in a set of neurons in terms of which subsets

fire together and which do not. If the neurons the code describes are place

cells, then the neural code gives some information about the …

The Combinatorics Of Modified Macdonald Polynomials, Jacob Rodeheffer

Theses, Dissertations and Capstones

This paper is concerned with finding a combinatorial Schur expansion of the modified Macdonald polynomials. We will use the Robinson–Schensted–Knuth (RSK) algorithm to obtain the result for a limited set of partition shapes, and we will use Foata’s bijection to extend the result to conjugate shapes. We will explore possibilities for modifying the RSK algorithm in such a way that it could be applied to obtain the result for more general shapes.

Arithmetic | Algebra Homework, Samar Elhitti, Ariane Masuda, Lin Zhou

Open Educational Resources

Arithmetic | Algebra Homework book is a static version of the WeBWork online homework assignments that accompany the textbook Arithmetic | Algebra for the developmental math courses MAT 0630 and MAT 0650 at New York City College of Technology, CUNY.

Classifying The Jacobian Groups Of Adinkras, Aaron R. Bagheri

HMC Senior Theses

Supersymmetry is a theoretical model of particle physics that posits a symmetry between bosons and fermions. Supersymmetry proposes the existence of particles that we have not yet observed and through them, offers a more unified view of the universe. In the same way Feynman Diagrams represent Feynman Integrals describing subatomic particle behaviour, supersymmetry algebras can be represented by graphs called adinkras. In addition to being motivated by physics, these graphs are highly structured and mathematically interesting. No one has looked at the Jacobians of these graphs before, so we attempt to characterize them in this thesis. We compute Jacobians through …

Study Guide For Cuny Elementary Algebra Final Exam (Ceafe), Olen Dias, William Baker, Amrit Singh

Open Educational Resources

No abstract provided.

Crossed Product Algebras Over Dihedral Field Extensions, Kaitlyn Lee

Undergraduate Honors Thesis Collection

Let F be the field of fractions of R, a ring of power series with coefficients in some field. Let K/F be a finite Galois extension, and assume the integral closure S of R in K is also a power series ring.

In this paper, we consider a construction from ring theory called a crossed product algebra and an associated function f: G × G →K called a cocycle that is used to define a multiplication operation for the crossed product algebra. Consider a crossed product algebra whose cocycle f takes values in S. We give a proof concerning which …

Elementary Construction Of Lusztig’S Canonical Basis, Peter Tingley

Mathematics and Statistics: Faculty Publications and Other Works

In this largely expository article we present an elementary construction of Lusztig’s canonical basis in type ADE. The method, which is essentially Lusztig’s original approach, is to use the braid group to reduce to rank two calculations. Some of the wonderful properties of the canonical basis are already visible: that it descends to a basis for every highest weight integrable representation, and that it is a crystal basis.

Neutrosophic Perspectives: Triplets, Duplets, Multisets, Hybrid Operators, Modal Logic, Hedge Algebras. And Applications (Second Extended And Improved), Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

This book is part of the book-series dedicated to the advances of neutrosophic theories and their applications, started by the author in 1998. Its aim is to present the last developments in the field. For the first time, we now introduce:

— Neutrosophic Duplets and the Neutrosophic Duplet Structures;

— Neutrosophic Multisets (as an extension of the classical multisets);

— Neutrosophic Spherical Numbers;

— Neutrosophic Overnumbers / Undernumbers / Offnumbers;

— Neutrosophic Indeterminacy of Second Type;

— Neutrosophic Hybrid Operators (where the heterogeneous t-norms and t-conorms may be used in designing neutrosophic aggregations);

— Neutrosophic Triplet Loop;

— Neutrosophic Triplet …

Neutrosophic Triplet Groups And Their Applications To Mathematical Modelling, Florentin Smarandache, W.B. Vasantha Kandasamy, Ilanthenral K.

Branch Mathematics and Statistics Faculty and Staff Publications

The innovative notion of neutrosophic triplet groups, introduced by Smarandache and Ali in 2014-2016, happens to yield the anti-element and neutral element once the element is given. It is established that the neutrosophic triplet group collection forms the classical group under product for Zn, for some specific n. However the collection is not even closed under sum. These neutrosophic triplet groups are built using only modulo integers or Cayley tables. Several interesting properties related with them are defined. It is pertinent to record that in Zn, when n is a prime number, we cannot get a neutral element which can …

Fast Multipole Method Using Cartesian Tensor In Beam Dynamic Simulation, He Zhang, He Huang, Rui Li, Jie Chen, Li-Shi Luo

Mathematics & Statistics Faculty Publications

The fast multipole method (FMM) using traceless totally symmetric Cartesian tensor to calculate the Coulomb interaction between charged particles will be presented. The Cartesian tensor based FMM can be generalized to treat other non-oscillating interactions with the help of the differential algebra or the truncated power series algebra. Issues on implementation of the FMM in beam dynamic simulations are also discussed. © 2017 Author(s).

On P-Adic Fields And P-Groups, Luis A. Sordo Vieira

Theses and Dissertations--Mathematics

The dissertation is divided into two parts. The first part mainly treats a conjecture of Emil Artin from the 1930s. Namely, if f = a_1x_1^d + a_2x_2^d +...+ a_{d^2+1}x^d where the coefficients a_i lie in a finite unramified extension of a rational p-adic field, where p is an odd prime, then f is isotropic. We also deal with systems of quadratic forms over finite fields and study the isotropicity of the system relative to the number of variables. We also study a variant of the classical Davenport constant of finite abelian groups and relate it to the isotropicity of diagonal …