Mathematics Commons^™

Interpolation Problems And The Characterization Of The Hilbert Function, Bryant Xie

Mathematical Sciences Undergraduate Honors Theses

In mathematics, it is often useful to approximate the values of functions that are either too awkward and difficult to evaluate or not readily differentiable or integrable. To approximate its values, we attempt to replace such functions with more well-behaving examples such as polynomials or trigonometric functions. Over the algebraically closed field C, a polynomial passing through r distinct points with multiplicities m1, ..., mr on the affine complex line in one variable is determined by its zeros and the vanishing conditions up to its mi − 1 derivative for each point. A natural question would then be to consider …

An Explicit Construction Of Sheaves In Context, Tyler A. Bryson

Dissertations, Theses, and Capstone Projects

This document details the body of theory necessary to explicitly construct sheaves of sets on a site together with the development of supporting material necessary to connect sheaf theory with the wider mathematical contexts in which it is applied. Of particular interest is a novel presentation of the plus construction suitable for direct application to a site without first passing to the generated grothendieck topology.

Pairings In A Ring Spectrum-Based Bousfield-Kan Spectral Sequence, Jonathan Toledo

Dissertations, Theses, and Capstone Projects

Bousfield and Kan traditionally formulated their homotopy spectral sequence over a simplicial set X resolved with respect to a ring R. By considering an adequate category of ring spectra, one can take a ring spectrum E, create from it a functor of a triple on the category of simplicial sets, and build a cosimplicial simplicial set EX. The homotopy spectral sequence can then be formed over such cosimplicial spaces by a similar construction to the original. Pairings can be established on these spectral sequences, and, for nice enough spaces, these pairings on the E₂-terms coincide with certain …

Asymptotic Stability Of Solitary Waves For The 1d Nls With An Attractive Delta Potential, Satoshi Masaki, Jason Murphy, Jun Ichi Segata

Mathematics and Statistics Faculty Research & Creative Works

We Consider the One-Dimensional Nonlinear Schrödinger Equation with an Attractive Delta Potential and Mass-Supercritical Nonlinearity. This Equation Admits a One-Parameter Family of Solitary Wave Solutions in Both the Focusing and Defocusing Cases. We Establish Asymptotic Stability for All Solitary Waves Satisfying a Suitable Spectral Condition, Namely, that the Linearized Operator Around the Solitary Wave Has a Two-Dimensional Generalized Kernel and No Other Eigenvalues or Resonances. in Particular, We Extend Our Previous Result [35] Beyond the Regime of Small Solitary Waves and Extend the Results of [19, 29] from Orbital to Asymptotic Stability for a Suitable Family of Solitary Waves.

Modeling And A Domain Decomposition Method With Finite Element Discretization For Coupled Dual-Porosity Flow And Navier–Stokes Flow, Jiangyong Hou, Dan Hu, Xuejian Li, Xiaoming He

Mathematics and Statistics Faculty Research & Creative Works

In This Paper, We First Propose and Analyze a Steady State Dual-Porosity-Navier–Stokes Model, Which Describes Both Dual-Porosity Flow and Free Flow (Governed by Navier–Stokes Equation) Coupled through Four Interface Conditions, Including the Beavers–Joseph Interface Condition. Then We Propose a Domain Decomposition Method for Efficiently Solving Such a Large Complex System. Robin Boundary Conditions Are Used to Decouple the Dual-Porosity Equations from the Navier–Stokes Equations in the Coupled System. based on the Two Decoupled Sub-Problems, a Parallel Robin-Robin Domain Decomposition Method is Constructed and Then Discretized by Finite Elements. We Analyze the Convergence of the Domain Decomposition Method with the Finite …

Computing Brain Networks With Complex Dynamics, Anca R. Radulescu

Biology and Medicine Through Mathematics Conference

No abstract provided.

Gap Junctions And Synchronization Clusters In The Thalamic Reticular Nuclei, Anca R. Radulescu, Michael Anderson

Biology and Medicine Through Mathematics Conference

No abstract provided.

Modelling Illiquid Stocks Using Quantum Stochastic Calculus: Asymptotic Methods, Will Hicks

Journal of Stochastic Analysis

No abstract provided.

Errata: Continuous Lattices And Domains, Jimmie D. Lawson

Seminar on Continuity in Semilattices

No abstract provided.

Movie Recommender System Using Matrix Factorization, Roland Fiagbe

Data Science and Data Mining

Recommendation systems are a popular and beneficial field that can help people make informed decisions automatically. This technique assists users in selecting relevant information from an overwhelming amount of available data. When it comes to movie recommendations, two common methods are collaborative filtering, which compares similarities between users, and content-based filtering, which takes a user’s specific preferences into account. However, our study focuses on the collaborative filtering approach, specifically matrix factorization. Various similarity metrics are used to identify user similarities for recommendation purposes. Our project aims to predict movie ratings for unwatched movies using the MovieLens rating dataset. We developed …

G-Coatomic Modules, Ahmed H. Alwan

Al-Bahir Journal for Engineering and Pure Sciences

Let R be a ring and M be a right R-module. A submodule of is said to be g-small in , if for every submodule , with implies that . Then is a g-small submodule of . We call g-coatomic module whenever and then . Also, is called right (left) g-coatomic ring if the right (left) -module (R) is g-coatomic. In this work, we study g-coatomic modules and ring. We investigate some properties of these modules. We prove is g-coatomic if and only if each is g-coatomic. It is proved that if is a g-semiperfect ring with , then …

Constructing Spanning Sets Of Affine Algebraic Curvature Tensors, Stephen J. Kelly

Rose-Hulman Undergraduate Mathematics Journal

In this paper, we construct two spanning sets for the affine algebraic curvature tensors. We then prove that every 2-dimensional affine algebraic curvature tensor can be represented by a single element from either of the two spanning sets. This paper provides a means to study affine algebraic curvature tensors in a geometric and algebraic manner similar to previous studies of canonical algebraic curvature tensors.

A Note On The Involutive Concordance Invariants For Certain (1,1)-Knots, Anna Antal, Sarah Pritchard

Rose-Hulman Undergraduate Mathematics Journal

A knot K is a smooth embedding of the circle into the three-dimensional sphere; two knots are said to be concordant if they form the boundary of an annulus properly embedded into the product of the three-sphere with an interval. Heegaard Floer knot homology is an invariant of knots introduced by P. Ozsváth and Z. Szabó in the early 2000's which associates to a knot a filtered chain complex CFK(K), which improves on classical invariants of the knot. Involutive Heegaard Floer homology is a variant theory introduced in 2015 by K. Hendricks and C. Manolescu which additionally considers a chain …

Gordian Distance And Complete Alexander Neighbors, Ana Wright

Dissertations, Theses, and Student Research Papers in Mathematics

We call a knot K a complete Alexander neighbor if every possible Alexander polynomial is realized by a knot one crossing change away from K. It is unknown whether there exists a complete Alexander neighbor with nontrivial Alexander polynomial. We eliminate infinite families of knots with nontrivial Alexander polynomial from having this property and discuss possible strategies for unresolved cases.

Additionally, we use a condition on determinants of knots one crossing change away from unknotting number one knots to improve KnotInfo’s unknotting number data on 11 and 12 crossing knots. Lickorish introduced an obstruction to unknotting number one, which proves …

Modelling Nuclear Weapon Effects In Wargaming Using Monte Carlo Simulations, Tyler Guetzke, Alexander Withenbury, Zachary Dugger

West Point Research Papers

The United States Army’s interpretation of nuclear weapon effects needs change and modernization. Wargaming exercises are commonplace in today’s military, however, despite the growing threat of non-strategic nuclear weapons (NSNW), little has been done to inform battlefield commanders on their true effects. Our research seeks to develop a tool for commanders to easily interpret quantifiable effects of a NSNW. Utilizing Monte Carlo simulation, we are developing a new methodology to analyze NSNW effects. Our model allows a commander to calculate the expected unit strength following a NSNW strike which will aid in their operational decision making ability. The Monte Carlo …

Formula 101 Using 2022 Formula One Season Data To Understand The Race Results, Christopher Garcia, Oliver Lopez

Student Scholar Symposium Abstracts and Posters

The reason why I am interested in Formula One is that my friend showed me what Formula One was all about. It became interesting to see the action of the sport, including the battles the drivers have during the race and how fast they go through a corner. Also, when qualifying comes around, they push their car to the absolute limit to gain a few seconds off their opponents. The drivers only in the top 10 receive points from the winner getting 25 points, the last driver in the top 10 getting 1 point, and those below the top ten …

A Frobenius-Schur Extension For Real Projective Representation, Levi Gagnon‐Ririe

All Graduate Theses and Dissertations

Many problems in physics have explicit mathematical descriptions. This thesis aims to provide the mathematical tools for a particular problem in physics, that of Quantum Mechanical symmetries. In essence, we extend the known mathematics to a more general setting and provide a wider view of Real projective representation theory. The work done in this thesis contributes to the subfield of mathematics known as representation theory and to the subfield of physics concerned with time reversal symmetry.

Intrinsic Tame Filling Functions And Other Refinements Of Diameter Functions, Andrew Quaisley

Dissertations, Theses, and Student Research Papers in Mathematics

Tame filling functions are quasi-isometry invariants that are refinements of the diameter function of a group. Although tame filling functions were defined in part to provide a proper refinement of the diameter function, we show that every finite presentation of a group has an intrinsic tame filling function that is equivalent to its intrinsic diameter function. We then introduce some alternative filling functions—based on concepts similar to those used to define intrinsic tame filling functions—that are potential proper refinements of the intrinsic diameter function.

Adviser: Susan Hermiller and Mark Brittenham

Investigating The Effect Of Greediness On The Coordinate Exchange Algorithm For Generating Optimal Experimental Designs, William Thomas Gullion

All Graduate Theses and Dissertations

Design of Experiments (DoE) is the field of statistics concerned with helping researchers maximize the amount of information they gain from their experiments. Recently, researchers have been turning to optimal experimental designs instead of classical/catalog experimental designs. One of the most popular algorithms used today to generate optimal designs is the Coordinate Exchange (CEXCH) Algorithm. CEXCH is known to be a greedy algorithm, which means it tends to favor immediate, locally best designs instead of globally optimal designs. Previous research demonstrated that this tradeoff was efficacious in that it reduced the cost of a single run of CEXCH and allowed …

Automorphisms Of A Generalized Quadrangle Of Order 6, Ryan Pesak

Undergraduate Honors Theses

In this thesis, we study the symmetries of the putative generalized quadrangle of order 6. Although it is unknown whether such a quadrangle Q can exist, we show that if it does, that Q cannot be transitive on either points or lines. We first cover the background necessary for studying this problem. Namely, the theory of groups and group actions, the theory of generalized quadrangles, and automorphisms of GQs. We then prove that a generalized quadrangle Q of order 6 cannot have a point- or line-transitive automorphism group, and we also prove that if a group G acts faithfully on …