Mathematics Commons^™

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

On The Superabundance Of Singular Varieties In Positive Characteristic, Jake Kettinger

Department of Mathematics: Dissertations, Theses, and Student Research

The geproci property is a recent development in the world of geometry. We call a set of points Z\subseq\P_k^3 an (a,b)-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point P to a plane is a complete intersection of curves of degrees a and b. Examples known as grids have been known since 2011. Previously, the study of the geproci property has taken place within the characteristic 0 setting; prior to the work in this thesis, a procedure has been known for creating an (a,b)-geproci half-grid for 4\leq a\leq b, but it was not …

Odd Covers Of Graphs, Calum Buchanan, Alexander Clifton, Eric Culver, Jiaxi Nie, Jason O'Neill, Puck Rombach, Mei Yin

Mathematics: Faculty Scholarship

Given a finite simple graph 𝐺, an odd cover of 𝐺 is a collection of complete bipartite graphs, or bicliques, in which each edge of 𝐺 appears in an odd number of bicliques, and each nonedge of 𝐺 appears in an even number of bicliques. We denote the minimum cardinality of an odd cover of 𝐺 by 𝑏2⁡(𝐺) and prove that 𝑏2⁡(𝐺) is bounded below by half of the rank over 𝔽2 of the adjacency matrix of 𝐺. We show that this lower bound is tight in the case when 𝐺 is a bipartite graph and almost tight when 𝐺 …

The Future Is Now In Twisted Coil Polymer Actuators (Tcpa), Ryan Ronquillo

Electronic Theses and Dissertations

This thesis aimed to fabricate and test twisted coiled polymer actuators (TCPA) to understand the mechanical and thermal aspects of this artificial muscle fiber. The purpose of this thesis was to find a linear relationship using the LVDT sensor, fabricating TCPA fibers, and interpreting the data. The project tested whether nylon/polymer could be used as a better artificial muscle fiber.

This research accomplished three goals: (1) designing and fabricating a system capable of creating supercoiled muscle fibers consistently, (2) calibrating the Linear Variable Differential Transformer (LVDT) and Core, and (3) analyzing/interpreting the data of the Twisted Coiled Polymer Actuators (TCPA) …

Partitions Of R^N With Maximal Seclusion And Their Applications To Reproducible Computation, Jason Vander Woude

Department of Mathematics: Dissertations, Theses, and Student Research

We introduce and investigate a natural problem regarding unit cube tilings/partitions of Euclidean space and also consider broad generalizations of this problem. The problem fits well within a historical context of similar problems and also has applications to the study of reproducibility in randomized computation.

Given $k\in\mathbb{N}$ and $\epsilon\in(0,\infty)$, we define a $(k,\epsilon)$-secluded unit cube partition of $\mathbb{R}^{d}$ to be a unit cube partition of $\mathbb{R}^{d}$ such that for every point $\vec{p}\in\R^d$, the closed $\ell_{\infty}$ $\epsilon$-ball around $\vec{p}$ intersects at most $k$ cubes. The problem is to construct such partitions for each dimension $d$ with the primary goal of minimizing …

Gordian Distance And Complete Alexander Neighbors, Ana Wright

Department of Mathematics: Dissertations, Theses, and Student Research

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 …

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 …

Fractal Newton Methods, Ali Akgül, David E. Grow

Mathematics and Statistics Faculty Research & Creative Works

We introduce fractal Newton methods for solving (Formula presented.) that generalize and improve the classical Newton method. We compare the theoretical efficacy of the classical and fractal Newton methods and illustrate the theory with examples.

Fully Decoupled Energy-Stable Numerical Schemes For Two-Phase Coupled Porous Media And Free Flow With Different Densities And Viscosities, Yali Gao, Xiaoming He, Tao Lin, Yanping Lin

Mathematics and Statistics Faculty Research & Creative Works

In this article, we consider a phase field model with different densities and viscosities for the coupled two-phase porous media flow and two-phase free flow, as well as the corresponding numerical simulation. This model consists of three parts: a Cahn-Hilliard-Darcy system with different densities/viscosities describing the porous media flow in matrix, a Cahn-illiard-Navier-Stokes system with different densities/viscosities describing the free fluid in conduit, and seven interface conditions coupling the flows in the matrix and the conduit. Based on the separate Cahn-Hilliard equations in the porous media region and the free flow region, a weak formulation is proposed to incorporate the …

An Integer Garch Model For A Poisson Process With Time-Varying Zero-Inflation, Isuru Panduka Ratnayake, V. A. Samaranayake

Mathematics and Statistics Faculty Research & Creative Works

A serially dependent Poisson process with time-varying zero-inflation is proposed. Such formulations have the potential to model count data time series arising from phenomena such as infectious diseases that ebb and flow over time. The model assumes that the intensity of the Poisson process evolves according to a generalized autoregressive conditional heteroscedastic (GARCH) formulation and allows the zero-inflation parameter to vary over time and be governed by a deterministic function or by an exogenous variable. Both the expectation maximization (EM) and the maximum likelihood estimation (MLE) approaches are presented as possible estimation methods. A simulation study shows that both parameter …

Effects Of Factors On The Market Price Of The Shares Using Design Of Experiment, Amir Ahmad Dar, Mohammad Shahfaraz Khan, Imran Azad, Tanveer Ahmad Tarray, N. Anuradha, Qaiser Farroq Dar

Applied Mathematics & Information Sciences

When the cost of capital, dividends and the price of the share at the beginning is known, Modigliani and Miller’s model can be used to estimate the price of the share at the end of the period. A design of experiment (Taguchi’s orthogonal array) is used in order to investigate the impact of three parameters on the price of the share at the end of the period. The main aim of this research article is to find which parameter is more significant on the price of the share at the end of the period. Taguchi’s methodology of design of the …

Why Inverse Layers In Pavement? Why Zipper Fracking? Why Interleaving In Education? A General Explanation, Edgar Daniel Rodriguez Velasquez, Aaron Velasco, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, if we split our efforts into two disconnected chunks, we get better results: a pavement is stronger if instead of a single strengthening layer, we place two parts of this layer separated by no-so-strong layers; teaching is more effective if instead of concentrating a topic in a single time interval, we split it into two parts separated in time, etc. In this paper, we provide a general explanation for all these phenomena.

Gradient Estimates And The Fundamental Solution For Higher-Order Elliptic Systems With Lower-Order Terms, Ariel E. Barton, Michael J. Duffy Jr.

Mathematical Sciences Faculty Publications and Presentations

We establish the Caccioppoli inequality, a reverse Hölder inequality in the spirit of the classic estimate of Meyers, and construct the fundamental solution for linear elliptic differential equations of order 2m with certain lower order terms.

Long-Time Stability For An Imex Discretization Of The 1d Fujita Equation, Victoria Luongo

Honors College Theses

We study an efficient time-stepping scheme for the 1D Fujita equation that is implicit for the linear terms but explicit for the nonlinear terms. We analyze the long-time stability of the scheme for varying parameter values, which reveal parameter value regimes in which the method is stable. We provide numerical results that illustrate the theory and show the analytically derived stability conditions are sufficient to achieve long-time stability results.

Fast -- Asymptotically Optimal -- Methods For Determining The Optimal Number Of Features, Saied Tizpaz-Niari, Luc Longpré, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In machine learning -- and in data processing in general -- it is very important to select the proper number of features. If we select too few, we miss important information and do not get good results, but if we select too many, this will include many irrelevant ones that only bring noise and thus again worsen the results. The usual method of selecting the proper number of features is to add features one by one until the quality stops improving and starts deteriorating again. This method works, but it often takes too much time. In this paper, we propose …

Framing How We Think About Curves, Megan Sattler

Mathematical Sciences Undergraduate Honors Theses

A frame defines a basis at a point in R^n, and we can frame a curve by placing one at every point along it. This thesis investigates adapted framed curves in R^2 and R^3 in which they are used to provide information about how a curve twists and turns. We derive differential information from the frames that describe how they change and consequently how the curve changes. We also deduce special properties of each framing and discuss how the differential information suffices to describe the shape of curves.

Let’S Make Patterns!: Symmetric Rubik’S Cube Permutations, Danny Anderson

Theses/Capstones/Creative Projects

Rubik’s cubes are well-known for having several different patterns, or permutations, that can be made from them. Meanwhile, cubes generally display a wide variety of symmetries. Naturally, these ideas can be combined to form a notion of "symmetric Rubik's cube patterns." The goal of this paper is to find an algorithm that can produce all of the permutations that display symmetries.

A Study In The Freeness Of Finitely Generated Anp-Modules Upon Restriction To Principal Subalgebras, Luke Manford Flattery

Mathematics Dissertations - Archive

We are interested in quantitative information on the freeness of modules over a truncated polynomial ring when restricting to subalgebras generated by a linear form. After investigating the structure of the truncated polynomial ring, subalgebras generated by a linear form, and corresponding vector spaces, we construct a generic representation and discuss its connection to a certain affine space. We quantify the abundance of freeness of modules using a certain variety called the rank variety. For any possible dimension we construct a module whose rank variety has that dimension. Finally, we define another variety, called the module variety, and show that …

Mechanism Of Hairpin Vortex Formation By Liutex, Yifei Yu

Mathematics Dissertations - Archive

Turbulence is still a mystery for human after more than one century’s development of fluid dynamics. Hairpin vortex formation is regarded as an essential process for a laminar flow transition to the turbulent flow. A new correct third generation vortex identification method, Liutex, was proposed in 2018, which can represent local rotation direction and reveal the local angular speed correctly. Using this powerful tool, the mechanism of hairpin vortex formation is re-examined. This dissertation (1) explains the mechanism of hairpin vortex formation by solving Orr-Sommerfeld equation using Chebyshev spectrum method (2) observes the DNS result of flat plate boundary layer …

Likelihood Inference For Flexible Cure Models With Interval Censored Data, Jodi Treszoks

Mathematics Dissertations - Archive

Models for survival data with a surviving fraction, known as cure rate models, play a vital role in survival analysis. Due to the improvement of intervening methodologies, some subjects are seen to be immune permanently. While cure rate models have been studied extensively in the recent literature with a standard assumption of right-censored data, in many applied settings, such as recidivism studies or medical studies where the event of interest is not immediately harmful, continuous observation of a subject is impracticable. We call lifetime data generated with discrete follow-up times as interval-censored. In this thesis, we extend several existing cure …