Mathematics Commons^™

Time Series Data Mining: A Retail Application Using Sas Enterprise Miner, Daniel Hebert

Honors Projects in Mathematics

Modern technologies have allowed for the amassment of data at a rate never encountered before. Organizations are now able to routinely collect and process massive volumes of data. A plethora of regularly collected information can be ordered using an appropriate time interval. The data would thus be developed into a time series. With such data, analytical techniques can be employed to collect information pertaining to historical trends and seasonality. Time series data mining methodology allows users to identify commonalities between sets of time-ordered data. This technique is supported by a variety of algorithms, notably dynamic time warping (DTW). This mathematical …

Limit Theorems For Random Billiard Models, Timothy Chumley

All Theses and Dissertations (ETDs)

The central objects of study in this dissertation are random billiard systems. These are Markov chain systems with general state spaces derived from deterministic billiard systems by selecting one or more dynamical variables and replacing them with random variables with fixed probability distributions. In particular, we study two specific model systems; one which models gas diffusion through cylindrical channels whose walls have a microscopic structure, and another which models a minimalistic heat engine. The main results for the gas diffusion model, a series of probabilistic limit theorems, allow us to express transport characteristics such as mean exit times of the …

Sorting Permutations With Finite-Depth Stacks, Timothy Goodrich, Will Olson, Ruyue Yuan

Symposium on Research and Creative Expression (SORCE)

Sorting organizes information for optimal usage, and our work examines the mathematics behind sorting with stacks. In 1968, Donald Knuth showed that a permutation is sortable in an infinite-depth stack if and only if it avoids the pattern 231; Knuth also enumerated these permutations. Twenty-five years later, Julian West extended these ideas to permutations sortable with 2 consecutive stacks. We continue this work by limiting the stack(s) to a finite depth. In particular, we completely characterize permutations sortable through a single finite-depth stack and derive a handy enumeration formula. We also apply our pattern characterization and enumeration techniques to permutations …

Simulation Modeling And Analysis Of Coal Shipping Operations, Raymond Finzel, Timothy Goodrich, Graeme Roberts

Symposium on Research and Creative Expression (SORCE)

Computer simulations are increasingly powerful and realistic models for complex real-world scenarios, and our project applies this technology to model a coal transportation case study. Given a baseline scenario of fourteen carriers transporting coal from three U.S. locations to four international locations, we optimize operations in terms of product flow, time required for shipments, and total operation costs. Implementing the case study's factors into modular code, we introduce several potential changes to current operations and develop specific scenarios. Further, in analyzing these scenarios we test for robustness and sensitivity, by changing values such as demand and bad weather occurrences, and …

Optimizing The Allocation Of Vaccines In The Presence Of Multiple Strains Of The Influenza Virus, Abby Kenyon, Ana Eveler, Tayler Grashel, Jessica Richardson

Symposium on Research and Creative Expression (SORCE)

During the annual flu season, multiple strains of the influenza virus are often present within a population. It is a significant challenge for health care administrators to determine the most effective allocation of two different vaccines to combat the various strains when treating the public. We employ a mathematical model, a system of differential equations, to find a strategy for vaccinating a population in order to minimize the number of infected individuals. We consider various strengths of transmission of the disease, availability of vaccine doses, vaccination rates, and other model parameters. This research may lead to more effective health care …

Investigating Prerequisite Grade Requirements In The Calculus Sequence, Michelle Kleckner, Lexi Paradine, Katie Merkling, Mindy Capaldi

Symposium on Research and Creative Expression (SORCE)

The Valparaiso University Mathematics and Computer Science Department has been debating a shift from a D- to a C- in the calculus sequence prerequisite grade requirement. We first researched policies at other colleges and universities. Next, we gathered data on students' self-predicted grades by distributing pre- and post-semester surveys and measuring the accuracy of the results in comparison to the actual grade data. Using the surveys, we also associated students' expectations with their majors. In the end, we investigated whether or not prerequisite grades can predict subsequent course performance. Statistical analysis of the past four years of math student data …

On The (Co)Homology Of Non-Commutative Toroidal Orbifold, Safdar Quddus

All Theses and Dissertations (ETDs)

Noncommutative torus algebra was studied in the early 80's as a fundamental example of noncommutative geometry. Connes calculated its cyclic and Hochschild cohomology. In this thesis, we study noncommutative toroidal orbifolds generated by actions of finite subgroups of S L(2,) on a noncommutative torus algebra.

In the first part, we calculate the Hochschild and cyclic homology of Γ for all finite subgroups Γ S L(2,). In the second part we analyse the cohomology of these algebras and compute the Chern-Connes pairing between the elements of and explicit cocycles discovered in our calculations. In the third part we discuss some …

A Perspective On Multiple Waves Of Influenza Pandemics, Anna Mummert, Howard Weiss, Li-Pong Long, Jose´ M. Amigo´, Xiu-Feng Wan

Mathematics Faculty Research

Background: A striking characteristic of the past four influenza pandemic outbreaks in the United States has been the multiple waves of infections. However, the mechanisms responsible for the multiple waves of influenza or other acute infectious diseases are uncertain. Understanding these mechanisms could provide knowledge for health authorities to develop and implement prevention and control strategies. Materials and Methods: We exhibit five distinct mechanisms, each of which can generate two waves of infections for an acute infectious disease. The first two mechanisms capture changes in virus transmissibility and behavioral changes. The third mechanism involves population heterogeneity (e.g., demography, geography), where …

"Integration Of Math And Music In The Secondary Classroom", Brian O'Neill

Honors Theses

The disciplines of mathematics and music seem worlds apart at first glance. Harmonious connections can inevitably be created if a deeper appreciation is lent to these stereotypically dissimilar subjects. "Integration of Mathematics and Music in the Secondary Classroom" is a quadratic function unit that utilizes music to aid in teaching mathematical concepts. The unit consists of a compilation of traditional rote mathematics and three main inquiry lessons: Problems Without Polyrhythm, Ma-Thematics, and The Undertones of Overtones. The unique approach of inquiry allows students to construct meaningful learning through a curriculum that is driven by their own mathematical questions. In addition, …

A Look At Multi-Decompositions Of Complete Graphs Into Graph Pairs Of Order 4, Yizhe Gao, Daniel Roberts, Faculty Advisor

John Wesley Powell Student Research Conference

Firstly, a graph G consists of a vertex set V (G), and an edge set E (G) of endpoints which relate two vertices with each edge. Also, a decomposition of a graph is a list of subgraphs such that each edge appears in exactly one subgraph in the list. In the field ofgraph theory, graph decomposition is an active field of research. One type of decomposition is graph pairs. A graph pair is a pair of graphs on the same vertex set whose union is the complete graph. Abueida and Daven studieddecompositions of complete graphs into graph-pairs of order four. …

Frames And Spline Framelets, Tung Nguyen, Nahee Kim, Tian-Xiao He, Faculty Advisor

John Wesley Powell Student Research Conference

Frames can be seen as a generalization of orthonormal bases, which not only maintain useful characteristics of orthonormal bases but also allow more flexibility in applications. The applications of frames include communication and image processing, as its characteristic inherited from orthonormal bases helps speed up the transmitting and processing time while its additional flexibility adds to frames the ability to reconstruct lost information. In this project, we study the construction of a class of tight frames in Euclidean spaces. Also, we use Fourier transforms and the techniques of Multi-Resolution Analysis (MRA) in wavelet analysis to investigate a class of tight …

A Comprehensive, Multi-Scale Dynamical Model Of Erbb Receptor Signal Transduction In Human Mammary Epithelial Cells, Tomáš Helikar, Naomi Kochi, Bryan Kowal, Manjari Dimri, Mayumi Naramura, Srikumar M. Raja, Vimla Band, Hamid Band, Jim A. Rogers

Mathematics Faculty Publications

The non-receptor tyrosine kinase Src and receptor tyrosine kinase epidermal growth factor receptor (EGFR/ErbB1) have been established as collaborators in cellular signaling and their combined dysregulation plays key roles in human cancers, including breast cancer. In part due to the complexity of the biochemical network associated with the regulation of these proteins as well as their cellular functions, the role of Src in EGFR regulation remains unclear. Herein we present a new comprehensive, multi-scale dynamical model of ErbB receptor signal transduction in human mammary epithelial cells. This model, constructed manually from published biochemical literature, consists of 245 nodes representing proteins …

Sub-Riemannian Heat Kernels And Mean Curvature Flow Of Graphs, Luca Capogna, Giovanna Citti, Cosimo Senni Guidotti Magnani

Mathematics Sciences: Faculty Publications

We introduce a sub-Riemannian analogue of the Bence-Merriman-Osher algorithm (Merriman et al., 1992 [42]) and show that it leads to weak solutions of the horizontal mean curvature flow of graphs over sub-Riemannian Carnot groups. The proof follows the nonlinear semi-group theory approach originally introduced by L.C. Evans (1993) [27] in the Euclidean setting and is based on new results on the relation between sub-Riemannian heat flows of characteristic functions of subgraphs and the horizontal mean curvature of the corresponding graphs.

Periodic Modules Over Gorenstein Local Rings, Amanda Croll

Department of Mathematics: Dissertations, Theses, and Student Research

It is proved that the minimal free resolution of a module M over a Gorenstein local ring R is eventually periodic if, and only if, the class of M is torsion in a certain Z[t,t^{-1}] associated to R. This module, denoted (R), is the free Z[t,t^{-1}]-module on the isomorphism classes of finitely generated R-modules modulo relations reminiscent of those defining the Grothendieck group of R. The main result is a structure theorem for J(R) when R is a complete Gorenstein local ring; the link between periodicity and torsion stated above is a corollary.

Advisor: Srikanth Iyengar

Byu Computational Number Theory Research Group, Dr. Paul Jenkins

Journal of Undergraduate Research

The computational number theory group has clearly been a success, with students writing papers, giving conference presentations, winning prestigious national fellowships, and going off to strong PhD programs in mathematics. The weekly seminar has helped students learn presentation skills and learn more mathematics, and has provided a convenient entry point for new students interested in research. It is anticipated that the group will continue into the foreseeable future, and that students will continue to benefit from the allocated resources.

Disconjugacy And Differentiation For Solutions Of Boundary Value Problems For Second-Order Dynamic Equations On A Time Scale, Jeffrey W. Lyons

Mathematics Colloquium Series

On the specific time scale—given as integer multiples of a fixed, positive real number h—and under certain conditions, solutions of a nonlinear second-order dynamic equation with conjugate boundary conditions are differentiated with respect to the boundary values and delta differentiated with respect to the boundary points. Lyons will also present two corollaries of the result.

Tracial Rokhlin Property And Non-Commutative Dimensions, Qingyun Wang

All Theses and Dissertations (ETDs)

This dissertation focuses on finite group actions with the tracial Rokhlin property and the structure of the corresponding crossed products. It consists of two major parts. For the first part, we study several different aspects of finite group actions with certain versions of the Rokhlin property. We are able to give an explicit characterization of product-type actions with the tracial Rokhlin property or strict Rokhlin property. We also show that, in good circumstances, the actions with the tracial Rokhlin property are generic.

In the second portion of this dissertation, we introduce the weak tracial Rokhlin property for actions on non-simple …

Regularity For Solutions To Parabolic Systems And Nonlocal Minimization Problems, Joe Geisbauer

Department of Mathematics: Dissertations, Theses, and Student Research

The goal of this dissertation is to contribute to both the nonlocal and local settings of regularity within the calculus of variations. We provide analogues of higher differentiability results in the context of Besov spaces for minimizers of nonlocal functionals. We also establish the Holder continuity of solutions to a system of parabolic partial differential equations.

Advisor: Mikil Foss

The Schwinger Action Principle And Its Applications To Quantum Mechanics, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

No abstract provided.

Single Exhaled Breath Metabolomic Analysis Identifies Unique Breathprint In Patients With Acute Decompensated Heart Failure, Michael A. Samara, W.H. Wilson Tang, Frank Cikach Jr., Zeynep Gul, Lily Tranchito, Kelly M. Paschke, Jamie Viterna, Yuping Wu, Daniel Laskowski, Raed A. Dweik

Mathematics and Statistics Faculty Publications

Acute decompensated heart failure (ADHF) is the most common indication for hospital admission, particularly in the elderly, yet the identification of those with impending decompensation using conventional clinical methods is unreliable and frequently leaves insufficient lag time for therapeutic interventions (1). Exhaled breath constitutes a complex mixture of hundreds of volatile organic compounds (VOCs) that could potentially be used as a safe and noninvasive method of diagnostic and therapeutic monitoring (2). Previous research studies have identified elevated acetone, pentane, and nitric oxide levels in exhaled breath in the setting of HF correlated with disease severity (3–5). Selected ion-flow tube mass-spectrometry …