Mathematics Commons^™

An Introduction To A Rigorous Definition Of Derivative, Dave Ruch

Analysis

No abstract provided.

Bolzano On Continuity And The Intermediate Value Theorem, Dave Ruch

Analysis

No abstract provided.

A Comparison Of Stimulus Presentation Methods In Temporal Discrimination Testing, Eavan M. Mcgovern, John Butler, Ines Beiser, Laura Williams, Brendan Quinlivan, Shruti Narasiham, Rebecca Beck, Sean O'Riordan, Richard B. Reilly, Michael Hutchinson

Articles

The temporal discrimination threshold (TDT) is the shortest time interval at which an individual detects two stimuli to be asynchronous (normal  =  30-50 ms). It has been shown to be abnormal in patients with disorders affecting the basal ganglia including adult onset idiopathic focal dystonia (AOIFD). Up to 97% of patients have an abnormal TDT with age- and sex-related penetrance in unaffected relatives, demonstrating an autosomal dominant inheritance pattern. These findings support the use of the TDT as a pre-clinical biomarker for AOIFD. The usual stimulus presentation method involves the presentation of progressively asynchronous stimuli; when three sequential stimuli are …

A Mathematical System For Human Implantable Wound Model Studies, Salomonsky Paul-Michael, Rebecca Segal

Mathematics and Applied Mathematics Publications

In this work, we present a mathematical model, which accounts for two fundamental processes involved in the repair of an acute dermal wound. These processes include the inflammatory response and fibroplasia. Our system describes each of these events through the time evolution of four primary species or variables. These include the density of initial damage, inflammatory cells, fibroblasts and deposition of new collagen matrix. Since it is difficult to populate the equations of our model with coefficients that have been empirically derived, we fit these constants by carrying out a large number of simulations until there is reasonable agreement between …

R-Hopfian And L-Co-Hopfian Abelian Groups, Brendan Goldsmith, Katao Gong

Articles

The notions of Hopfian and co-Hopfian groups are well known in both non-commutative and Abelian group theory. In this work we begin a systematic investigation of natural generalizations of these concepts and, in the case of Abelian p-groups, give a complete characterization of the generalizations in terms of the original concepts.

Tropical Derivation Of Cohomology Ring Of Heavy/Light Hassett Spaces, Shiyue Li

HMC Senior Theses

The cohomology of moduli spaces of curves has been extensively studied in classical algebraic geometry. The emergent field of tropical geometry gives new views and combinatorial tools for treating these classical problems. In particular, we study the cohomology of heavy/light Hassett spaces, moduli spaces of heavy/light weighted stable curves, denoted as $\calm_{g, w}$ for a particular genus $g$ and a weight vector $w \in (0, 1]^n$ using tropical geometry. We survey and build on the work of \citet{Cavalieri2014}, which proved that tropical compactification is a \textit{wonderful} compactification of the complement of hyperplane arrangement for these heavy/light Hassett spaces. For $g …

Combinatorial Polynomial Hirsch Conjecture, Sam Miller

HMC Senior Theses

The Hirsch Conjecture states that for a d-dimensional polytope with n facets, the diameter of the graph of the polytope is at most n-d. This conjecture was disproven in 2010 by Francisco Santos Leal. However, a polynomial bound in n and d on the diameter of a polytope may still exist. Finding a polynomial bound would provide a worst-case scenario runtime for the Simplex Method of Linear Programming. However working only with polytopes in higher dimensions can prove challenging, so other approaches are welcome. There are many equivalent formulations of the Hirsch Conjecture, one of which is the …

Alice In Wonderland For G4g13, Jeremiah Farrell, Emmanuelle Malte Salvatore, Todd Wilk Estroff

Scholarship and Professional Work - LAS

Each of the ten different letters in the title is used exactly three times to form the words in the circles. Martin Gardner's famous work The Annotated Alice was first published in 1960 and we honor him in this essay.

Geometric Structures On Lie Algebras, Sabrina C. Walker

Theses & Honors Papers

We investigate and classify the Lorentzian geometries on a variety of finite-dimensional Lie algebras. For a given Lie algebra, we first classify Lorentzian scalar products up to a notion of equivalence determined by a change of basis induced by an automorphism (or symmetry) of the underlying Lie algebra. We then use the classification of Lorentzian scalar products to find and classify those scalar products which correspond to algebraic Ricci solitons ( or algebraic Ricci soliton structures for the Lie algebra). A complete classification of Lorentzian scalar products and algebraic Ricci soliton structures is provided for a number of Lie algebras …

The Poynting–Robertson Effect In The Newtonian Potential With A Yukawa Correction, Ioannis Haranas, Omiros Ragos, Ioannis Gkigkitzis, Ilias S. Kotsireas, Connor Martz, Sheldon Van Middekoop

Physics and Computer Science Faculty Publications

We consider a Yukawa-type gravitational potential combined with the Poynting-Robertson effect. Dust particles originating within the asteroid belt and moving on circular and elliptic trajectories are studied and expressions for the time rate of change of their orbital radii and semimajor axes, respectively, are obtained. These expressions are written in terms of basic particle parameters, namely their density and diameter. Then, they are applied to produce expressions for the time required by the dust particles to reach the orbit of Earth. For the Yukawa gravitational potential, dust particles of diameter 10^-3 m in circular orbits require times of the …

Sensitivity Analysis Of Wolf Restoration In Yellowstone Nation Park Using Omnivory Models, Derek Fields

Theses, Dissertations and Capstones

In the ever-changing world of ecology, species survival often depends on approximations and measurements taken by biologists. These approximations help to ensure and predict the future of that given species. Our ecological community of interest involves wolves, elk, and berry producing shrubs within Yellowstone National Park. We use two different systems of ordinary differential equations, each increasing in complexity to model our community. In each model the predator (wolves) and consumers (elk) compete for a common resource, berry producing shrubs. We call this consumption of resources, from more than one trophic level, omnivory. We approximate each system with parameter values …

Slopes Of Modular Forms And The Ghost Conjecture, John Bergdall, Robert Pollack

Mathematics Faculty Research and Scholarship

We formulate a conjecture on slopes of overconvergent p" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; border: 0px; font-family: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: normal; vertical-align: baseline; display: inline-table; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">p-adic cusp forms of any p" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; border: 0px; font-family: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: normal; vertical-align: baseline; display: inline-table; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">p-adic weight in …

Problems In Relating Various Tasks And Their Sample Solutions To Bloom’S Taxonomy, Torsten Lindstrom

The Mathematics Enthusiast

In this paper we analyze sample solutions of a number of problems and relate them to their level as prescribed by Bloom’s taxonomy. We relate these solutions to a number of other frameworks, too. Our key message is that it remains insufficient to analyze written forms of these tasks. We emphasize careful observations of how different students approach a solution before finally assessing the level of tasks used.

We take the arithmetic series as our starting point and point out that the objective of the discussion of the examples here in no way is to indicate an optimal way towards …

A Brief Look At The Evolution Of Modeling Hyperbolic Space, Shelby Frazier

The Mathematics Enthusiast

Now considered one of the greatest discoveries of mathematical history, hyperbolic geometry was once laughed at and deemed insignificant. Credited most to Lobachevsky and Bolyai, they independently discovered the subject by proving the negation of Euclid’s Fifth Postulate but neither lived long enough to see their work receive any validation. Several models have been constructed since its acceptance as a subject. I’m going to discuss three graphical models: the Klein Model, the Upper Half Plane Model and the Poincaré Disk Model. I will also discuss and construct three tape and paper models: the Annular Model, the Polyhedral Model and the …

Z2-Orbifolds Of Affine Vertex Algebras And W-Algebras, Masoumah Abdullah Al-Ali

Electronic Theses and Dissertations

Vertex algebras arose in conformal field theory and were first defined axiomatically by Borcherds in his famous proof of the Moonshine Conjecture in 1986. The orbifold construction is a standard way to construct new vertex algebras from old ones. Starting with a vertex algebra V and a group G of automorphisms, one considers the invariant subalgebra V^G (called G-orbifold of V), and its extensions. For example, the Moonshine vertex algebra arises as an extension of the Z₂-orbifold of the lattice vertex algebra associated to the Leech lattice.

In this thesis we consider two problems. First, …

Topic Supervised Non-Negative Matrix Factorization, K. Macmillan, James D. Wilson

Mathematics

Topic models have been extensively used to organize and interpret the contents of large, unstructured corpora of text documents. Although topic models often perform well on traditional training vs. test set evaluations, it is often the case that the results of a topic model do not align with human interpretation. This interpretability fallacy is largely due to the unsupervised nature of topic models, which prohibits any user guidance on the results of a model. In this paper, we introduce a semi-supervised method called topic supervised non-negative matrix factorization (TS-NMF) that enables the user to provide labeled example documents to promote …

Statistical Modeling Of The Default Mode Brain Network Reveals A Segregated Highway Structure, P. E. Stillman, James D. Wilson, M. J. Denny, B. A. Desmarais, Shankar Bhamidi, S. J. Cranmer, Zhong-Lin Lu

Mathematics

We investigate the functional organization of the Default Mode Network (DMN) – an important subnetwork within the brain associated with a wide range of higher-order cognitive functions. While past work has shown the whole-brain network of functional connectivity follows small-world organizational principles, subnetwork structure is less well understood. Current statistical tools, however, are not suited to quantifying the operating characteristics of functional networks as they often require threshold censoring of information and do not allow for inferential testing of the role that local processes play in determining network structure. Here, we develop the correlation Generalized Exponential Random Graph Model (cGERGM) …

Soil-Transmitted Helminthiasis And Schistosomiasis In Children Of Poor Families In Leyte, Philippines: Lessons For Disease Prevention And Control, Harvy Joy Liwanag, Jhanna Uy, Janis Ruth Gatchalian, Betty De La Calzada, Justine Alessandra Uy, Manuel Dayrit, Ramil T. Bataller

Mathematics Faculty Publications

Objective

Neglected tropical diseases (NTDs) continue to be a public health problem in the Philippines. We assessed the association of soil-transmitted helminthiasis (STH) and schistosomiasis with selected health-related and socioeconomic variables in four villages in Leyte, Philippines.

Methods

Stool specimens from 418 adults and 533 of their children from 209 families were examined through the Kato-Katz technique.

Results

STH and schistosomiasis were present in 64.6% and 12.5%, respectively, of study participants. Analysis through the generalized linear mixed model revealed a number of associations between infection in parents and their children. Findings indicate that years of disease prevention and control efforts …

Hamiltonian Model For Coupled Surface And Internal Waves In The Presence Of Currents, Rossen Ivanov

Articles

We examine a two dimensional fluid system consisting of a lower medium bounded underneath by a flatbed and an upper medium with a free surface. The two media are separated by a free common interface. The gravity driven surface and internal water waves (at the common interface between the media) in the presence of a depth-dependent current are studied under certain physical assumptions. Both media are considered incompressible and with prescribed vorticities. Using the Hamiltonian approach the Hamiltonian of the system is constructed in terms of ’wave’ variables and the equations of motion are calculated. The resultant equations of motion …

Some Results On Almost Ricci Solitons And Geodesic Vector Fields., Ramesh Sharma

Mathematics Faculty Publications

We show that a compact almost Ricci soliton whose soliton vector field is divergence-free is Einstein and its soliton vector field is Killing. Next we show that an almost Ricci soliton reduces to Ricci soliton if and only if the associated vector field is geodesic. Finally, we prove that a contact metric manifold is K-contact if and only if its Reeb vector field is geodesic.

Management Of Invasive Allee Species, David Chan, C. M. Kent, D. M. Johnson

Mathematics and Applied Mathematics Publications

In this study, we use a discrete, two-patch population model of an Allee species to examine different methods in managing invasions. We first analytically examine the model to show the presence of the strong Allee effect, and then we numerically explore the model to test the effectiveness of different management strategies. As expected invasion is facilitated by lower Allee thresholds, greater carrying capacities and greater proportions of dispersers. These effects are interacting, however, and moderated by population growth rate. Using the gypsy moth as an example species, we demonstrate that the effectiveness of different invasion management strategies is context-dependent, combining …

Spectral Properties Of A Sequence Of Matrices Connected To Each Other Via Schur Complement And Arising In A Compartmental Model, Evan Haskell, Vehbi Emrah Paksoy

Mathematics Faculty Articles

We consider a sequence of real matrices An which is characterized by the rule that A_n−1 is the Schur complement in A_n of the (1,1) entry of A_n, namely −en, where en is a positive real number. This sequence is closely related to linear compartmental ordinary differential equations. We study the spectrum of A_n. In particular,we show that An has a unique positive eigenvalue λ_n and {λ_n} is a decreasing convergent sequence. We also study the stability of A_n for small n using the Routh-Hurwitz criterion.

P3: What Determines The Shape Of A Cloud?, William Calabrase

Undergraduate Research Posters 2017

Current climate models and weather forecasts suffer due to an uncertainty associated with the behavior of clouds, which directly impact the energy exchange between the earth and the Sun. This impact is determined in part by the shape of the clouds, thereby making the study of what affects cloud shape an area of interest. To characterize the shape of cumulus clouds we study the behavior of the cloud overlap ratio, or the ratio between the average cloud fraction and projected cloud cover. In this study, we used a high resolution computer model to 1) determine how the cloud overlap ratio …

P2: Reconciling Linear Measurements Of Fractal Cloud Structures, Nicholas Barron

Undergraduate Research Posters 2017

Clouds are a large unknown in meteorological predictions. Most of the issue can be derived from the odd shape of clouds. So, in order to correct the measurements of clouds, a thorough investigation of fractal cloud structures must be performed. Using the results from this study, a reconciliation method can then be constructed and applied to linear measurements of clouds.

Improving The Development Of The I-Chart For Use In Biopharmaceutical Manufacturing Operation, Kimberly Schveder

Undergraduate Research Posters 2017

The Shewhart control chart is a statistical tool used by pharmaceutical companies, as well as chemical and other batch manufacturers, to help detect errors in the manufacturing process and ensure control of product quality. One particular type of control chart is the I-chart. The average run length (ARL) statistic of the I-chart can easily be determined when output from the manufacturing process is normally distributed with known population parameters. This paper investigates the impact on the ARL statistic when the I-chart is based on mean and standard deviation estimates obtained from small sample sizes of less than 50 batches. The …

P1: How High Does The Lower Atmosphere Go?, Vladimir Sworski, Justin Flaherty

Undergraduate Research Posters 2017

The Atmospheric Boundary Layer (ABL), consisting of the bottom few kilometers of the troposphere, is a region with strong mixing of moisture and winds. This region's activity has a large impact on weather and climate models. In this study, we use a high resolution computer model: Large Eddy Simulation (LES). Statistics produced require a strong understanding of the height of the ABL. The purpose of this study was to create a method for determining this height accurately and consistently, as previous models demonstrated significant error.

Rational Number Operations On The Number Line, Christopher Yakes

The Mathematics Enthusiast

The Common Core State Standards make clear that teachers should use the number line to represent integer operations in Grades 6, 7 and 8, continuing the development of number line understanding begun in earlier grades. In this paper, we will investigate the basics of the colored chip and number line models for representing integers and their operations. We will then draw connections between the two models. Finally, we will argue that the number line model is an important one that should be taught alongside or in lieu of the chip model in order to deepen students’ understanding operations with integers, …

Perihelion Precession In The General Theory Of Relativity, Charles G. Torre

Tutorials on... in 1 hour or less

This is a relatively quick and informal sketch of a demonstration that general relativistic corrections to the bound Kepler orbits introduce a perihelion precession. Any decent textbook on the general theory of relativity will derive this result. My analysis aligns with that found in the good old text "Introduction to General Relativity", by Adler, Bazin and Schiffer.

On The Definition Of Linear Independence, Yonah Cherniavsky, Artour Mouftakhov

The Mathematics Enthusiast

We discuss a certain very common flaw in the definition of linear independence, which is one of the most important concepts taught in any college or university course of Linear Algebra. This note may be useful to lecturers and students which teach and study Linear Algebra of any level and like the mathematically rigorous approach.

Numberlines: Hockey Line Nicknames Based On Jersey Numbers, Egan J. Chernoff

The Mathematics Enthusiast

The purpose of this article, in general, is to expound Chernoff’s (2016) notion of numberlines, that is, hockey line nicknames based on jersey numbers. The article begins with a brief discussion of the history of hockey line nicknames, which allows for the parsing of numberlines and quasi-numberlines (nicknames based on numbers associated with hockey players). Focusing, next, on jersey number restrictions for the National Hockey League (NHL), a repeated calculation of the number of possible numberlines winnows down the number from a theoretical upper bound to a practical upper bound. Moving beyond the numbers, the names of natural numbers – …