Math Colloquium

Each semester the Department of Mathematics & Physics offers colloquium talks on a variety of mathematical topics. The department also hosts the Advanced Research Initiative series, featuring a distinguished guest speaker giving two talks.

2025 Fall Math Colloquia

November 5 Smith Hall 621 & Virtual, 1:00pm

Chloe’ Napier (Marshall University)
Extensions in the Preprojective Algebras

Abstract: In 2001, Fomin and Zelevinsky introduced cluster algebras as a way to study total positivity. These objects lie within the intersection of algebra and combinatorics. One use of cluster algebras is that they can be used to describe many important algebraic spaces by associating a certain type of graph to them and then group its elements into n-tuples. Though these constructions can be very algebraic, in this work, we look at these spaces through a more combinatorial lense. To do this, we model objects of our space with tableaux and lattice paths. We are aiming to find a combinatorial description for when elements can be in the same n-tuple. In this talk, we will discuss the background of the Grassmannian and Richardson varieties, introduce the combinatorics for studying these varieties and provide answers to our overarching question.

October 16 Smith Hall 624 & Virtual, 5:00pm

Dr. Nick Wintz (Radford University)
Linear quadratic pursuit–evasion games on time scales

Abstract: In this paper, we unify and extend the linear quadratic pursuit-evasion games to dynamic equations on time scales. A mixed strategy for a single pursuer and evader is studied in two settings. In the open-loop setting, the corresponding controls are expressed in terms of a zero-input difference. In the closed-loop setting, the corresponding controls require a mixing feedback term when rewriting the system in extended state form. Finally, we offer a numerical simulation.

September 17 Smith Hall 154 & Virtual, 1:00pm

Dr. Stephen Deterding (Marshall University)
Epic Math Battle of History: Tartaglia, Cardano, and the Discovery of Imaginary Numbers

Abstract: In 1546 the mathematician Niccol’o Tartaglia accused Girolamo Cardano, a doctor and mathematician, of having published in one of his books a result that Tartaglia had shared with him under the condition that he never reveal it to anyone else. Thus began what is arguably the most infamous feud in all mathematics. Over the next year Tartaglia and Cardano’s pupil, Lodovico Ferrari, who stepped up to defend him, would write several open letters to each other arguing their position in this controversy. Ultimately, a lively debate would be held between Tartaglia and Ferrari to resolve their dispute.

From this backdrop of betrayal and feuding emerged some of the greatest mathematical discoveries, the cubic formula, the quartic formula, and imaginary numbers. In this talk, we will trace the history of the Tartaglia-Cardano controversy from its beginning, examine the mathematics that emerged as a result, and demonstrate how the controversy resulted in the discovery of imaginary numbers as an indispensable part of mathematics.

August 20 Smith Hall 516 & Virtual, 1:00pm

Dr. Shalmali Bandyopadhyay (University of Tennessee at Martin)
The Coexistence Paradox: When Bigger Isn’t Always Better in Fragmented Landscapes

Abstract: We analyse the diffusive Lotka-Volterra competition model in fragmented landscapes using Robin boundary conditions to capture realistic species responses at patch-matrix interfaces. Through eigenvalue analysis, sub-supersolution methods, and bifurcation theory, we establish existence and uniqueness of coexistence states across different competition regimes. Our key finding reveals that in semi strong competition cases, coexistence occurs only within finite patch size intervals—requiring patches that are neither too small nor too large—which fundamentally challenges classical results where larger patches always favour coexistence. Using computational methods for the asymmetric case, we demonstrate that this phenomenon emerges from dispersal-competition trade offs where one species excels at colonizing smaller patches while its competitor dominates direct resource competition, suggesting profound implications for conservation design in fragmented landscapes.

2025 Spring Math Colloquia

April 25 Smith Hall 518 & Virtual, 1:00pm

Dr. Andrej Gendiar (Slovak Academy of Sciences)
Entanglement in Tensor Networks

Abstract: Tensor Networks form a special class of variational quantum states widely used to study strongly correlated many-body systems. Originally developed from one-dimensional Matrix Product States—the foundation of the Density Matrix Renormalization Group—these methods have since been extended to compute ground and excited states in higher-dimensional systems.
In this work, we explore the mathematical foundations of Tensor Networks, emphasizing their connection to the maximization of quantum entanglement. Key principles are illustrated through simple, instructive examples. Leveraging the quantum-classical correspondence via imaginary time evolution, we analyze two equivalent representations of Tensor Networks:

1. the vertex representation, typically used in describing quantum states, and
2. the weight representation, familiar from classical statistical mechanics.

We examine two-dimensional classical spin models with equivalent partition functions under both representations. Although these two approaches yield identical observables and correlation functions, we find that the entanglement entropy is consistently lower in the vertex representation—across both finite and infinite systems. This reduction in entanglement holds universally, regardless of the spin-lattice model, the number of spin degrees of freedom, or the underlying lattice geometry. To understand this persistent discrepancy, we investigate various types of Tensor Network structures across a broad range of spin Hamiltonians and phase transition scenarios.

April 11 Smith Hall 518 & Virtual, 1:00pm

Dinh-Liem Nguyen (Kansas State University)
Direct reconstruction methods for inverse source and scattering problems with single-frequency data

Abstract: Inverse source and scattering problems involve determining an unknown object from indirect measurements associated with the object. These problems, which arise in a variety of applications such as nondestructive testing, radar, medical imaging, and geophysical exploration, have been an active research topic in the mathematics, engineering, and physics communities for several decades. However, they are highly ill-posed and nonlinear, posing significant challenges in developing efficient numerical methods for their solution.

In this talk, we will present our recent results in developing sampling type methods with novel imaging functions for solving inverse source and scattering problems using boundary data at a single frequency. Theoretical justifications and numerical simulations of the proposed method will be discussed. Our approach is non-iterative, computationally efficient, and straightforward to implement. This talk is based on joint work with Isaac Harris, Tran Lan, and Thu Le.

March 27 Smith Hall 518 & Virtual, 1:00pm

Anna Mummert (Marshall University)
A New Proof of the Pythagorean Theorems

Abstract: The Pythagorean Theorem has been proven in numerous ways throughout history, yet new insights continue to emerge. In 2023, high school students Jackson and Johnson presented a novel proof using a blend of geometry, algebra, and trigonometry. Their approach leverages geometric constructions, similarities, geometric series, and the Law of Sines to establish the theorem. In this talk will walk through their proof step by step, showcasing the power of creative reasoning in mathematical discovery. No advanced prerequisites are needed this talk will be accessible to all undergraduate students.

February 19 Smith Hall 516 & Virtual, 1:00pm

Chanaka Kottegoda (Marshall U)
Resurgence of Large Influenza A (H1N1) Outbreaks: Modeling the Interplay of Dynamic Infection, Loss of Immunity, and Vaccination Rates

Abstract: The 2009 epidemic outbreaks caused by the H1N1pdm09 influenza A virus strain had a profound global impact, resulting in millions of cases and hundreds of thousands of deaths worldwide. After the initial outbreaks, many regions experienced subsequent waves of H1N1, driven by environmental conditions and loss of immunity, among other elements. More dramatically, several locations witnessed large resurgent H1N1 outbreaks, and the underlying factors that contributed to such severe resurgence remain poorly understood. In this study, we investigate the factors contributing to large resurgent outbreaks of the H1N1pdm09 influenza A virus following the initial 2009 pandemic. Using a mechanistic model, we explore the roles of seasonal climate variations, vaccination coverage, and immunity loss due to viral mutations in driving subsequent outbreaks. The study provides insights into the complex interplay between these factors by identifying key parametric conditions for resurgence. Model simulations and historical data analysis from eight countries reveal oscillations in infection rates and immunity loss. The findings emphasize the importance of understanding these interactions to develop effective strategies for controlling H1N1 outbreaks.

2020 Fall Math Colloquia

September 16 Virtual, 4:00pm

Carl Mummert (Marshall University)
A case study in computable graph theory: König’s edge coloring theorem

Abstract Computable graph theory is a subfield of combinatorics and logic. A graph, in this sense, has a set of vertices (points), some of which are connected with edges. Graphs are applied to model many kinds of relationships in numerous disciplines. Computability theory uses a framework originally developed by Alan Turing to study the problems computers are theoretically able to solve. This talk will introduce computable graph theory using a specific theorem. König’s edge coloring theorem gives a way to assign colors to the edges of a graph using the least number of colors possible. I will introduce the theorem and many results about its computability theoretic properties.

No prior familiarity with graph theory or computability is needed, and this talk is aimed at a general audience.

October 21 Virtual, 4:00pm

Michael Otunuga (Marshall University)
Time Dependent Probability Density Function for Number of Infection in a Stochastic SIS Epidemic Model

Abstract: The closed-form time dependent probability density function of the number of infected individuals at a given time satisfying a stochastic Susceptible-Infected-Susceptible (SIS) epidemic model is derived and analyzed using the Fokker-Planck equation. The mean, median, variance, skewness and kurtosis of the distribution are obtained as a function of time. We study the effect of noise intensity on the distribution and later derive and analyze the effect of changes in the transmission and recovery rates of the disease. The result is applied using published Covid-19 data/parameters.

November 18 Virtual, 4:00pm

Michael Schroeder (Marshall University)
The null space of maximum density ASMs

Abstract: At some point in your life, you have been asked to “set it equal to zero and solve.” This is another such time. A large field of research is dedicated to computing the spectrum of a matrix, which in part asks the question,

If A is a matrix, for what vectors x does A x = 0?

The set of all solutions to this equation is the null space of A. In this talk, we look at a particular family of matrices called alternating sign matrices (or ASMs for short), which are matrices with only 0s, 1s, and -1s in it that satisfy a certain “alternating” condition.

We define and describe some properties of maximum density ASMs, then compute their null spaces. Surprisingly, some elementary graph theory was helpful to solve the problem. This research is joint work with Pauline van den Driessche, from the University of Victoria.

Having some background in linear algebra would be beneficial, but the only requirement is that, at some point in your life, you’ve been asked to set something equal to zero and solve.

2019 Fall Math Colloquia

September 18 Smith Hall 516, 4:00pm

Faith Hensley (Marshall University)
Extremal Numbrix Puzzles

Abstract: Numbrix is a puzzle in Parade magazine. The player is given a 9 × 9 grid with some integers between 1 and 81 filled in. The player then needs to fill in the rest of the integers between 1 and 81 so that consecutive integers appear in adjacent cells of the grid. Generalizing this puzzle we consider m × n grids with the entries being the integers between 1 and mn. We say that a set of clues defines a puzzle if there exists a unique solution given those clues. In 2018 Hanson and Nash find the maximum number of clues that fail to define an m × n puzzle for all m and n. I present our work on their conjecture concerning the minimum number of clues necessary to define a puzzle. This research is joint work Ashley Peper conducted as part of the 2019 REU program at Grand Valley State University.

Chloe’ Marcum (Marshall University)
Extended Schur Functions

Abstract: The set of quasisymmetric functions homogeneous of degree n form a polynomial
vector space, QSym, with multiple bases. Many of these bases can be generated
combinatorially using tableaux with various rules. The bases of QSym we are interested
in are the quasisymmetric Schur functions and the extended Schur functions.
Both generalize the symmetric Schur functions. A common question in linear algebra
is how to transition from one basis to another. We have an expansion of
extended Schur functions into quasisymmetric Schur functions for a family of indexing
compositions. We prove this expansion combinatorially by using a tree
whose leaves indicate the polynomials that appear in our expansion.

October 16 Smith Hall 516, 4:00pm

Logan Rose (Marshall University)
Modeling Malaria with Controls

Abstract: Malaria is a deadly parasitic disease that has been a major threat to public health for centuries, particularly in Sub Saharan Africa and South Asia. Although there are still 200 million cases reported yearly according to the CDC, there have been renewed efforts in recent decades to combat this illness. Mathematicians have created various models using differential equations to simulate the spread of malaria and to determine which control strategies are most effective in reducing the number of new infections. Many of these consider the interactions between humans and the Anopheles mosquito, the primary vector of the disease. This project considers the effectiveness of two popular malaria control strategies: using bed nets and releasing sterile mosquitoes. We used an eight-equation model that combines an SEIR (Susceptible-Exposed-Infected-Recovered) model for the human population, an SEI (Susceptible-Exposed-Infected) model for the vector population, and an additional equation for the sterile mosquito population. In addition, we applied the Next Generation Matrix method to our model to derive a formula for the basic reproduction number for infectious disease. Finally, we apply optimal control theory to find the optimal releasing strategies, and numerical simulations are presented for various cases.

Michael Waldeck (Marshall University)
Generalizing the Classical Construction for Complete Sets of Mutually Orthogonal Latin Squares

Abstract: Latin squares are a well established topic of study in Mathematics. A Latin square of size n is an n by n square where each number from 1 to n appears exactly once in each row and in each
column. Two Latin squares are orthogonal mates if, when the squares are superimposed, every
possible ordered pair of entries appears in the superimposed square. In the 18th century, Leonhard
Euler studied the question of finding a complete set of mutually orthogonal Latin squares (MOLS)
of a given size. Euler proved that for each n, a set of n by n MOLS can have no more than n – 1
squares. It has also been proved that, where n is a prime power, this upper bound is achieved
through the classical construction. The classical construction for a complete set of MOLS of size
n by n can be given by a formula M(i, j) = i + cj, where c is a nonzero element in the finite field of order n. We study the number of orthogonal mates and the graph of the mate relationship between a family of Latin squares based on a generalization of that formula. That is, when q is prime, we study the set of Latin squares of size q by q that can be given by a formula S(s, c)(i, j) = s(i) + cj,
where s mapping the finite field of order q to itself is a permutation of the field, c is a nonzero element in the finite field of order q, and the addition and multiplication are carried out in the finite field.

November 20 Smith Hall 530, 4:00pm

Carl Mummert (Marshall University)
How to Multiply Big Natural Numbers

Abstract: Big natural numbers are all around. They keep your data private on the internet, and scientists use them for high precision simulations. Adding big numbers is not very hard, but multiplying them is much more challenging. The problem of multiplying numbers efficiently is at the border between mathematics and computer science.

I will introduce several multiplication algorithms currently used in state of the art software. Then I will describe a breakthrough announced by two researchers in March 2019. They produced an algorithm that can multiply two natural numbers – of any size – with an efficiency we think is optimal. This talk is aimed at anyone with a background of college algebra or higher.

2018 Fall Math Colloquia

September 19
Elizabeth Niese (Marshall University)
The RSK algorithm and applications
Smith Hall 516, 4:00pm

Abstract: The Robinson-Schensted-Knuth (RSK) algorithm is a classical algorithm in algebraic combinatorics. It is a bijection between words and pairs of tableaux which has many interesting combinatorial properties. There are a number of distinct constructions equivalent to this algorithm, including the jeu-de-taquin and Viennot’s shadow lines construction. We will look at several of these constructions and their use in proofs of algebraic formulas.

October 17
Chloé Marcum (Marshall University)
Using polynomials to study knots
Smith Hall 516, 4:00pm

Abstract: A knot is any closed loop in space. Two knots may be the same knot even though they appear to be different. A local move is a change to a small portion of a knot and the rest of the knot is assumed to stay the same. A local move may or may not change the knot. An invariant is a characteristic of a knot that help us tell the difference between knots. In our research, we studied certain polynomials that are knot invariants. We studied the effect of certain local moves on Homflypt and Kauffman polynomials. As a consequence, we discovered some new properties of these invariants.

This research was completed in summer 2018 as part of a Research Experience for Undergraduates (REU) at St Mary’s College of Maryland. At the end of the talk, I will share general information about REUs as well as my experience at St. Mary’s.

Knot theory is an area of math that is understandable without a lot of math background. All students interested in math are encouraged to attend.

November 28
Raid Al-Aqtash (Marshall University)
Market Basket Analysis
Smith Hall 516, 4:00pm

Abstract: Market basket analysis is a data mining technique based upon using association rules to uncover the purchasing trends in large transaction datasets. The key factor here is to determine possible lists of items that are frequently sold together. In this talk, I will speak about association rules and how they can be used by large retail companies in market basket analysis. A real dataset will be provided in an application of association rules.

Keywords and phrases: data mining, basket analysis, beer and diapers, association rules, support, confidence, lift.

This will be a general audience talk, using a bit of information about proportions and percentages.

2018 Spring Math Colloquia

February 21
Michael Otunuga (Marshall University)
Global stability for a (2n+1)-dimensional HIV/AIDS epidemic model with treatments
Smith Hall 518, 4:00pm

Abstract: In this work, we derive and analyze a (2n+1)-dimensional deterministic differential equation modeling the transmission and treatment of HIV (Human Immunodeficiency Virus) disease. The model is extended to a stochastic differential equation by introducing noise in the transmission rate of the disease. A theoretical treatment strategy of regular HIV testing and immediate treatment with Antiretroviral Therapy (ART) is investigated in the presence and absence of noise. By defining R(0,n), R(t,n) and R(t,n) as the deterministic basic reproduction number in the absence of ART treatments, deterministic basic reproduction number in the presence of ART treatments and stochastic reproduction number in the presence of ART treatment, respectively, we discuss the stability of the infection-free and endemic equilibrium in the presence and absence of treatments by first deriving the closed form expression for R(0,n), R(t,n) and R(t,n). We show that there is enough treatment to avoid persistence of infection in the endemic equilibrium state if R(t,n)=1. We further show by studying the effect of noise in the transmission rate of the disease that transient epidemic invasion can still occur even if R(t,n)<1. This happens due to the presence of noise (with high intensity) in the transmission rate, causing R(t,n)>1. A threshold criterion for epidemic invasion in the presence and absence of noise is derived. Numerical simulation is presented for validation.

April 19
Carl Mummert (Marshall University)
The number TREE(3), and counting down in base infinity
Smith Hall 518, 4:00pm

Abstract: The motivation of this talk is a peculiar situation from computer science. In some cases, we know that a program will eventually stop, but we have no way to concretely describe or even bound the number of steps the program will take. For one such program, the number of steps is a number TREE(3) so large that there is no concrete way to describe it or bound it from above.

This talk will introduce TREE(3) and the related result known as Kruskal’s theorem. We will look at some simpler versions of the theorem, leading us to a “base infinity” number system. This system is like base ten, but each digit can be arbitrarily large. We will see that counting down to 1 from a base infinity number is not as easy as it sounds.

The work on base infinity numbers is joint research with mathematics major Samantha Colbert.

2017 Fall Math Colloquia

September 20
Matt Davis (Muskingum University)
Non-transitive dice: Constructions, Complications, and Questions
Smith Hall 509, 4:00pm

Abstract: Non-transitive dice have been a source of fascination for mathematicians for over 50 years. We are given a set of dice which are numbered in strange ways. Each player chooses a die, rolls it, and the higher roll wins. Our intuition suggests that in any set of dice, one is the “best”. However, it turns out that it is relatively easy to construct a set of dice which are non-transitive – where most dice are strong against some opponents and weak against others. In this talk we will look at lots of examples of these fascinating objects, aiming for a goal of a single construction that allows us to create a set of dice in any desired configuration. We will also talk briefly about the much harder problem of finding the most efficient way to create such a set of dice.

October 18
Skye Smith (Service Pump & Supply, Huntington WV)
Three Things I Wish I Had Known When I Was a Math Major
Smith Hall 509, 4:00pm

Abstract: Since graduating from Marshall University with an applied mathematics degree in 2014, I have used my degree in several various business roles. Each position has provided a new way to use my mathematics degree in a business setting and each role brought new lessons I wished I had considered throughout my time as an undergraduate student. In this presentation, I will discuss the three things I wish I had known while I was a mathematics major at Marshall University. Addressing these three observations will help guide mathematics students who are hoping to use their skill set in a business setting at a time when math minds are more important than ever to companies undergoing digital transformations and embracing the era of big data.

November 15
Avishek Mallick (Marshall University)
Statistical Modeling of Discrete/Count Data

Abstract: In this talk, I will introduce the idea of Statistical modeling, especially in context of count data. We will look at different facets of data fitting like estimation techniques and criterion for assessing goodness-of-fit. A substantial part of the talk will be about modeling inflated count data. We will be looking at lots of real world examples. This talk is intended for a general audience and thus should be appropriate for Mathematics undergraduate and graduate students.

2016 Spring Math Colloquia

January 26
Avishek Mallick (Marshall University)
A Look at Permuatation (a.k.a. Randomization) Tests

Abstract: Most of the standard statistical hypothesis testing procedures are based on the normality assumption, i.e., the data is normally distributed and also on few other assumptions. What happens if these assumptions do not hold? The answer lies in the Permutation testing approach also known as randomization or re-randomization tests. In this talk, I will discuss the randomization principle and also how these tests are impervious to complications that defeat other classical statistical significance testing techniques.

Keywords: Permutation testing, Statistical significance, Randomization, Distribution-free approach, P-values.

February 24
Carl Mummert (Marshall University)
Incompleteness in mathematics

Abstract: In 1931, Kurt Gödel published two theorems, known as the “incompleteness theorems,” which revolutionized the foundations of mathematics in much the same way that Einstein’s theory of relativity revolutionized the foundations of physics. In this talk, I will discuss what the incompleteness theorems say in plain language, and why they were so revolutionary. Along the way, we will encounter a few other few other major figures from the foundations of mathematics: Bertrand Russell, David Hilbert, Hermann Weyl, and L.E.J. Brouwer. No mathematical background beyond College Algebra is required for this talk.

March 7
Shubhabrata Mukherjee (University of Washington, Seattle)
Genetic analyses of late-onset Alzheimer’s Disease

Abstract: Genetic epidemiology holds great potential for personalized medicine and improved biological knowledge of disease pathogenesis. In this talk, I will introduce some basic concepts needed to understand, analyze, and interpret Genome-wide Association Studies (GWAS) data with late-onset Alzheimer’s disease (LOAD) as an example. I will briefly talk about extending these results to gene-wide and network/pathway-based analysis, which are complementary to GWASs.

March 8
Shubhabrata Mukherjee (University of Washington, Seattle)
Introduction to Genetic Epidemiology in GWAS era

Abstract: Alzheimer’s disease is the most common form of dementia. Genetics of late-onset Alzheimer’s disease (LOAD) is very complex. Genome-wide Association Studies (GWAS) are an important first step. Gene-based and network/pathways based analyses also contribute to the understanding of genetic determinants of LOAD. Thus, a three-component approach — GWAS, gene-based, and network-based analyses — is likely to best illuminate genetic determinants of a disease. I will explore the three component approach in this talk.

April 6
Anna Mummert (Marshall University)

2015 Fall Math Colloquia

September 2
Michael Schroeder (Marshall University)
Tournaments: Scheduling Them Fairly and More!

Abstract: In this talk, we will discuss round-robin tournaments — tournaments in which each team plays every other team. We’ll look at how to schedule games in a way so that, if there is only one playing surface, that no team is unreasonably burdened by playing games back-to-back. We’ll also talk about what the possible win-loss records are for teams which play in a round-robin tournament. We’ll relate this to complete graphs, score sequences, Landau’s inequalities, and matching sequencibility. A majority of this talk will be accessible to a general audience.

September 28
Nick Loehr (Virginia Tech)
Rook Theory 101

Abstract: Combinatorics is the mathematical theory of counting. In this talk, we introduce combinatorics through the subject of rook theory — which has nothing to do with chess! The first goal of rook theory is to count configurations of rooks on generalized chessboards so that no two rooks attack each other. The k’th rook number of a board counts the number of such ways to place k rooks on the board. It often happens that two different boards share the same rook numbers. This talk investigates when and why this occurs by using geometric and algebraic manipulations of boards and rook placements. We present theorems of Foata/Schutzenberger and Goldman/Joichi/White characterizing rook-equivalent boards in various ways. The techniques used illustrate three branches of modern combinatorics — enumerative combinatorics, bijective combinatorics, and algebraic combinatorics. No prior knowledge of counting (or chess) is required for this talk.

September 29
Nick Loehr (Virginia Tech)
Sweep Maps and Bounce Paths

Abstract: Mathematics is filled with open (unsolved) problems, ranging from deep foundational issues of physics, computation, geometry, and number theory to highly specialized research questions. This talk describes an open problem in algebraic combinatorics that can be stated and investigated with virtually no mathematical background, although the problem appears to be fiendishly challenging to solve. We define a family of maps on words, called “sweep maps.” A sweep map assigns a level to each letter in a word according to a simple rule, then sorts the letters according to their level. Surprisingly, although sweep maps act by sorting, they appear to be invertible: i.e., different input words are always sent to different output words by any given sweep map. The open problem is to prove the invertibility of all sweep maps, preferably by explicitly describing the inverse functions. We explain some known special cases of this problem using a model in which words are visualized using lattice paths. In some cases, we can pass from a lattice path to an associated “bounce path,” which provides the additional data needed to invert the sweep map. These bounce paths originally arose in the study of objects called q,t-Catalan polynomials. Many algorithms that have appeared in the q,t-Catalan literature over the last 20 years turn out to be particular instances of the sweep maps or their inverses. The sweep maps thus provide a simple unifying framework for understanding all of these algorithms.

October 21
Michael Otunuga (Marshall University)
Stochastic Modeling of Energy Commodity Spot Price Processes

Abstract: In this work, we undertake the study to shed light on world oil market and price movement, price balancing process and energy commodity behavior. We initiate the development of a stochastic model of energy commodity pricing. A system of stochastic model for dynamic of energy pricing process is proposed. Different methods for parameter estimation is discussed. In addition, by developing a Local Lagged Adapted Generalized Method of Moment (LLGMM) method, an attempt is made to compare the simulated estimates derived using LLGMM and other existing method. These developed results are applied to the Henry Hub natural gas, crude oil, coal and ethanol data sets.

November 4
Martha Yip (University of Kentucky)
Coloring: the Algebraic Way

Abstract: Suppose we want to color the countries on a map so that if two countries share a border, then they must be assigned different colors. What is the smallest number of colors that will do the job? The answer is the famous Four Color Theorem, which states that any map can be colored by at most four colors. Inspired by this problem, Birkhoff introduced the chromatic polynomial to study colorings of a graph. In this talk, we discuss some recent developments in the study of the chromatic polynomial; from a multivariable generalization known as the chromatic symmetric function, to a homology theory whose Euler characteristic recovers the chromatic symmetric function. No prior knowledge of any of the above terminology is assumed.

2015 Spring Math Colloquia

February 4
Elizabeth Niese (Marshall University)
What do trigonometry and combinatorics have to do with each other?

Abstract: Combinatorics is the study of how to count. Results from the field of combinatorics are used in probability and statistics, computer science, physics, and elsewhere. Trigonometry is the study of triangles and how their angles relate to side lengths. At first glance, these two topics seem quite unrelated. In this talk we will look at how some common trigonometric functions encode the number of up-down permutations.

February 17
David Cusick (Marshall University)
350 Years of Service … and Then Pffft!

Abstract: After more than three centuries as a well-known calculation tool, the slide rule was eclipsed by electronic calculators in the 1970s. Based on the “common” (base 10) logarithm, this analog device was state-of-the-art before its popularity collapsed. The elementary slide rule aided the approximation of products, quotients, powers and roots. A variety of other models could handle more sophisticated functions. Some rules are still in regular use even at the present time. Virtual rules exist, and there are some “apps” for that. This talk will just touch the basic theory and a few examples. Logarithm laws are well represented. If time permits, we can see a photo gallery. Everything should be readily accessible to those who are at and above pre-calculus.

March 4
Gregory Moses (Ohio University)
Clustering and Stability of Cyclic Solutions in the Cell Division Cycle of Yeast

Abstract: It has long been observed that in a large, well-mixed yeast culture (i.e. a bioreactor), yeast autonomous oscillations (such as dissolved oxygen percentages) are correlated with bud index, but only recently have researchers (biological or mathematical) attempted to analyze the cause of this correlation. They proposed a signaling model whereby a concentration of cells in one part of the cell division cycle might influence cells in other parts of the cycle through diffusible chemical products. We briefly introduce the model, then present an overview of work we have done analyzing the existence, stability and instability of periodic orbits under varying types of feedback. This is joint work with Nathan Breitsch.

March 27–28
MAA Ohio Section Meeting at Marshall University

April 14
Judy Day (University of Tennessee)
Modeling the host response to inhalation anthrax to uncover the mechanisms driving risk of disease.

Abstract: Bacillus anthracis, the causative agent of anthrax, can exist in the form of highly robust spores, making it a potential bio-terror threat. Once inhaled, the spores can germinate into vegetative bacteria capable of quick replication, leading to progressive disease and death. There is a critical need to better quantify the risk of disease from different inhalation exposure scenarios. Key to this effort is the use of mathematical and computational modeling to uncover the mechanisms driving risk. To this end, this presentation will discuss ongoing work on the development of models and methods that explore the host response to inhalation anthrax and provide insight into the mechanisms that drive the risk of disease. This is joint work is joint University of Tennessee graduate student Buddhi Pantha, and the NIMBioS Working Group on Modeling Anthrax Exposure.

April 15
Judy Day (University of Tennessee)
Determining the what, when, and how of therapeutic intervention strategies for controlling complex immune responses.

Abstract: Ideally, when challenged with a bacterial insult, a host orchestrates an immune response that, not only eliminates the offending pathogen, but also restores the host to homeostasis. However, due to the complex nature of the response, this is not always possible, especially in critically ill patients. Clearly, intervention is needed; however, determining the types of intervention that should be given, when they should be given, and in what amount remains a challenge. Computational modelling and control methodologies can provide fresh insight into this challenging biomedical problem and potentially offer techniques to answer these difficult questions. This talk will discuss computational control methodologies that are being explored to determine the what, when, and how of therapeutic intervention strategies for controlling complex immune responses.

2014 Fall Math Colloquia

September 3
Carl Mummert (Marshall University)
Is that a Prime Number?

Abstract: How can you tell whether a number is prime? Is 920,419,823 prime? I will talk about several methods for testing whether a number is prime, including the groundbreaking AKS algorithm published by Agrawal, Kayal, and Saxena in 2002, which was the first efficient general-purpose test for primes. Along the way we will see several topics from discrete mathematics that don’t seem very related to primes at first glance, such as polynomials and modular arithmetic. I will also talk about the history of the three Indian computer scientists behind the AKS algorithm.

September 17
Laura Adkins (Marshall University)
Interactive Regression Models with Centering

Abstract: Including interaction in a multiple regression model can increase its accuracy. Unfortunately, it also leads to multicollinearity, which violates the requirements of linear regression. I will discuss the use of centering to resolve this problem. I will then use an applied example to illustrate its use and explore its effectiveness.

October 1
Michael Schroeder (Marshall University)
Latin squares and their completions

Abstract
If you’ve even finished a Sudoku puzzle, then you’ve seen a “completed” Latin square. A Latin square is an array of symbols where each symbol appears exactly once in each row and each column. A solution to a Sudoku puzzle is a Latin square, with the additional constraint of having no repeated symbol in each $3\times 3$ block. Although Sudoku has only been popular for the past decade, mathematicians have been studying Latin squares since the 18th century, and have many applications including coding theory and experimental design. We will begin by discussing a brief exposition of Latin squares. Then we will go over some properties of Latin squares and some classical results identifying when partial Latin squares can be completed. To finish, we will talk about some recent results in the field. The beginning of the talk should be very accessible, and toward the end there will be some technical discussion.

October 15
Xue Gong (Ohio University)
Clustering and Noise-Induced Dispersion in Cell Cycle Dynamics (No Link to Abstract)

November 5
Richard Brualdi (University of Wisconsin–Madison)
The Gale-Berlekamp Light-Switching Problem and a Permutation Variation

Abstract: Consider an n by n array of light bulbs each controlled by a switch. Suppose there are also 2n other switches which allow one to simultaneously switch all the light bulbs in a row or all the light bulbs in a column. Now use the individual switches and turn some of the light bulbs on. With the row and column switches only, can one get all the lights in the off position? If not, how few on-lights are possible? This problem, its connections to coding theory, and a permutation variation is the subject of this talk.

November 6
Richard Brualdi (University of Wisconsin–Madison)
All Things Bruhat: Matrix Bruhat Decomposition, Complete Flags, Bruhat Order of Permutations, (0,1) and Integral Matrices, and Tournaments

Abstract: The title is a bit of an exaggeration, but we will discuss the topics it contains and various connections between them.

November 19
Bismark Oduro (Ohio University)
Designing Optimal Spraying Strategies for Controlling Re-infestation by Chagas Vectors

Abstract: Chagas disease is a major health problem in rural South and Central America where an estimated 8 to 11 million people are infected. It is a vector-borne disease caused by the parasite Trypanosoma cruzi, which is transmitted to humans mainly through the bite of insect vectors from several species of so-called “kissing bugs.” One of the control measures to reduce the spread of the disease is insecticide spraying of houses to prevent infestation by the vectors. However, re-infestation of homes by vectors has been shown to occur as early as four to six months after insecticide-based control interventions. In this talk, I will present re-infestation models that shed light on the effectiveness of the insecticide spraying. In particular, I will present both numerical explorations and some mathematical results. Comparison of the effectiveness of two spraying strategies namely; continuous and intermittent shows no statistically significant difference between the two strategies for small population size. For large population size, intermittent treatment is slightly more effective than continuous treatment. Another interesting result is the existence of a hysteresis-like phenomenon. This occurs when two different spraying rates lead to two different numbers of infested units at equilibrium. These results have potentially important implications for designing cost-effective control spraying strategies.

2014 Spring Math Colloquia

March 10
Jeffry L. Hirst (Appalachian State University)
Alan and Ada’s Theoretical Machines

Abstract: Ada Lovelace is often called “the first computer programmer,” though she died a century before the first general purpose computers were built. This claim is based on the notes she appended to her translation of Menabrea’s 1842 paper, Notions sur la Machine Analytique de M. Charles Babbage. Alan Turing is often called “the father of theoretical computer science” on the basis of his 1936 article On computable numbers, with an application to the Entscheidungsproblem. Lovelace’s notes describe Babbage’s Difference Engine and Babbage’s Analytical Engine, and Turing’s paper proves the existence of a Universal Turing Machine. This talk will compare the designs and capabilities of these early theoretical computing machines.

March 11
Jeffry L. Hirst (Appalachian State University)
Reverse Mathematics, Graphs, and Matchings

Abstract: How can we tell if two theorems are essentially the same? If we can prove that they are equivalent, then they are in some sense interchangeable. If our equivalence proof relies on a particularly small set of assumptions, then our claim of similarity is even stronger. This is the fundamental motivation of reverse mathematics, a program in the foundations of mathematics initiated by Harvey Friedman and Stephen Simpson. This talk will illustrate some results and techniques of the program.

April 9
JiYoon Jung (Marshall University)
The topology of chain selected complexes of a poset PDF

Abstract: For each composition ~c we show that the order complex of the poset of pointed set partitions Π• ~c is a wedge of β(~c) spheres of the same dimensions, where β(~c) is the number of permutations with descent composition ~c. Furthermore, the action of the symmetric group on the top homology is isomorphic to the Specht module SB where B is a border strip associated to the composition ~c. We also study the filter of pointed set partitions generated by knapsack integer partitions and show the analogous results on homotopy type and action on the top homology.
Let Λ be a sub semi-group of the natural numbers N. I am interested in filters of the partition lattice consisting of partitions where every block size belongs to Λ. Since Λ is closed under addition, these collections of partitions do form a filter in the partition lattice. In the case that Λ is generated by 〈a, d〉 where a and d are relatively prime, I conjecture that the action of the symmetric group on the top homology group of the associated order complexes is a direct sum of Specht modules. Precisely, I conjecture that ˜Hk(∆) is isomorphic to the direct sum of Specht modules ⊕~c Sβ(~c) where ~c = (c1, c2, . . . , ck+2) ranges over all compositions of n satisfying ci = a + mi · d for 0 ≤ mi < a.
This is joint work with Richard Ehrenborg.

April 11
Lingxing Yao (Case Western Reserve University)
Mathematical Modeling and Simulation for Biological Applications

Abstract: In this talk, I will go over some of my research projects in mathematical modeling and simulation for biological applications. I will focus on the analysis of a Hydrogel/Enzyme drug delivery oscillator, which involved mathematical modeling and dynamical analysis of models. It is well known that hydrogels are soft materials similar to natural tissue, and can be found in applications such as food additives, contact lens, cosmetic surgery, wound healing, and drug delivery devices. Since hydrogels are made of cross linked polymer networks that are capable of absorbing large amounts of water, they can be present in swollen or collapsed state. Under carefully designed physical and chemical conditions, we could observe volume phase transitions in polyelectrolyte hydrogels between those states, sometimes repeatedly. In this presentation, I will address the modeling of gel swelling, by combining the theory of the elastic, electrostatic and mixing energy in a model membrane made of polyelectrolyte hydrogel. Based on these energy principles, it is possible to propose the mathematical model system which is capable of explaining the oscillatory volume phase transition seen in a real membrane oscillator built in engineering laboratory for the purpose of hormone drug delivery. Some detailed analysis and results of the model system will be presented.

April 14
Stephen Flood (University of Connecticut – Waterbury)
Path, trees, and the computational strength of a packed Ramsey’s theorem

Abstract: Ramsey theory is a branch of combinatorics which provides results like the following: any large enough graph must either contain a large complete subgraph (all vertices connected) or a large independent set (no vertices connected). In this talk, we will introduce a â€œpacked” version of Ramsey’s theorem, due to Erdos and Galvin, which combines aspects of finite and infinite Ramsey theory. We will discuss the techniques used to extract these packed homogeneous sets, and we will study their strength using the tools of computability theory and reverse mathematics.

2013 Fall Math Colloquia

October 2
Lynne Yengulalp (University of Dayton)
Topological completeness

Abstract: I will start by reviewing the Baire Category Theorem for complete metric spaces. I will then talk about other classes of topological spaces for which the Baire Category Theorem holds. Such spaces have “generalized” completeness properties extending from the familiar local compactness property to the relatively new property of domain representability.

October 30
Roger Estep (Marshall University)
Filtered leapfrog time integration with enhanced stability properties

Abstract: The Leapfrog method for the solution of Ordinary Differential Equation initial value has been historically popular for several reasons. The method has second order accuracy, requires only one function evaluation per time step, and is non-dissipative. Despite the mentioned attractive properties, the method has some unfavorable stability properties. The absolute stability region of the method is only an interval located on the imaginary axis rather than a region in the complex plane. The method is only weakly stable and thus exhibits computational instability in long time integrations over intervals of finite length. In this work the use of filters is examined for the purposes of both controlling the weak instability and also enlarging the size of the absolute stability region of the method.

Robert Hughes (Marshall)
Agent-based modeling of pandemic influenza

Abstract: A striking characteristic of influenza pandemics is the multiple peaks of infection. For example, the United States has experienced two peaks of infection in each of the past four influenza pandemics, one peak during the summer months and a second peak during the typical flu season. In contrast, the number of infected individuals peaks only once during a seasonal flu. The mechanisms that cause the multiple peaks of infection during pandemic influenza seasons are not well understood. The goal of this project is to use agent-based modeling to investigate mechanisms that can generate two peaks of infection.
In this talk I will describe the susceptible-exposed-infectious-recovered (SEIR) agent-based model developed in Netlogo for simulating the 2009 H1N1 influenza pandemic. The incubation and infectiousness periods are drawn from gamma distributions. The model is calibrated by matching known average daily contacts and key epidemiological quantities, such as the basic reproduction number, the number of new infections generated from one infectious person at the beginning of the outbreak. Also, I will discuss the results of model simulations that include waning immunity, which is one potential mechanism for generating multiple peaks of infection.

November 18
Thomas Mathew (University of Maryland-Baltimore County)
The Assessment of Bioequivalence: A Statistical Overview

Abstract: The topic of bioequivalence deals with procedures for testing the equivalence of two drug products: typically, a generic drug and a brand name drug on the market. Bioequivalence testing consists of showing that the concentration of the active drug ingredient that enters the blood is similar for the two drugs. Area under the time-concentration curve, or the AUC, is usually used for this purpose, and the data are obtained based on cross-over designs. In the talk, the bioequivalence problem will be introduced, its history will be discussed, and examples will be provided. Statistical criteria that are used for bioequivalence testing, especially the criterion of average bioequivalence, will be discussed. Methodology for testing the hypotheses of average bioequivalence will be addressed. The emerging area of equivalence testing in the context of biosimilars will be briefly touched upon.

November 19
Thomas Mathew (University of Maryland-Baltimore County)
Methodology and Some Applications

Abstract: Standard likelihood based methods that are usually used to analyze data arising from a parametric model are typically accurate to the first order. Higher order inference procedures provide major improvements in accuracy, and are available for discrete as well as for continuous data. In the talk, two applications of higher order inference will be described. Both the applications deal with the computation of an upper tolerance limit: a limit that is expected to capture a specified proportion or more of a population with a given confidence level. The limit is constructed using a random sample, and the confidence level refers to the resulting sampling variability.
The first application that will be discussed is on the computation of tolerance limits under the logistic regression model for binary data. The data consist of binary responses, and upper tolerance limits are to be constructed for the number of positive responses in future trials corresponding to a fixed level of the covariates. The problem has been motivated by an application of interest to the U.S. Army, dealing with the testing of ballistic armor plates for protecting soldiers from projectiles and shrapnel, where the probability of penetration of the armor plate depends on covariates such as the projectile velocity, size of the armor plate, etc. The second application is on the computation of upper tolerance limits under a general mixed effects model with balanced or unbalanced data. Higher order inference procedures will be used to obtain accurate solutions in both the applications. Numerical results, examples and data analysis will also be reported.

2013 Spring Math Colloquia

January 30
Elizabeth Niese (Marshall)
A family of Catalan objects

Abstract: The Catalan numbers appear in many surprising places with over 200 known classes of objects counted by the Catalan numbers. Many of the objects counted by the Catalan numbers have connections to other areas of mathematics, such as algebra and geometry, and even to computer science. In this talk we will look at a particular class of objects, how they can be counted elegantly, and some of the connections between these objects and algebra.

February 20
Anna Mummert (Marshall)
Unit costs in optimal control of epidemics

Abstract: The cost of vaccinating an individual during an epidemic is not constant. It is assume that it is cheaper to vaccinate the first individuals and more expensive to vaccinate the last few individuals, due to logistics. In this talk, I will use mathematical modeling to compare the effects of different unit cost functions on the epidemic. I will describe a susceptible-exposed-infected-removed (SEIR) model of an epidemic, where susceptible individuals can be vaccinated and removed from the epidemic. Given a particular unit cost function for the vaccination, it is possible to determine the optimal vaccination rate that minimizes an associated “total cost” function, using the technique known as Pontryagin’s maximum principle. Different unit cost functions result in different optimal vaccination rates. Pontryagin’s maximum principle will be explained and several unit cost functions will be considered.

April 24
Carl Mummert (Marshall)
If 1+1=9, does 2+2=7?

Abstract: In mathematics, we often look at “if/then” relationships. It turns out there are several ways to interpret deceptively simple questions such as “If 1 + 1 = 9, does 2 + 2 = 7?”, which lead to surprising answers. I will explain how techniques of mathematical logic help us understand these questions and talk about recent undergraduate and graduate student research at Marshall in this area. This talk is aimed at undergraduates. Students who are taking algebra and calculus classes are particularly invited to attend.

2012 Spring Math Colloquia

January 25
Michael Schroeder (Marshall)
Cyclic Matching Sequencibility of Graphs

Abstract: Suppose you have 7 basketball teams in a round-robin tournament (each pair of teams play a game) and only one court. How can you schedule the games “fairly” for everyone? “Fair” is a vague term; when we say fair, we want all teams to have a maximal amount of down time between successive games (so no team plays two games in a row, for example). The answer to this question can be found using matching sequencibility.

Let G be a graph with m edges. Order the edges of G with the labels 1,2,…,m. The matching number of this ordering is the largest number d such that each consecutive d edges form a matching (don’t touch). The matching sequencibility (abbreviated ms) of G is the largest such d over all possible orderings of the edges. We will alter the premise a bit to talk about the cycle matching number of a graph ordering and define the cycle matching sequencibility (abbreviated cms) of a graph.

Feburary 22
Anna Mummert (Marshall)
Studying the recovery procedure for the time-dependent transmission rate in epidemic models

Abstract: In this talk I will discuss recent results on recovering the time-dependent transmission function for classical disease models given the disease incidence data. The recovery procedure is applied to a homogeneous population, meaning all individuals are equally likely to transmit the disease to any other individual. For a homogeneous epidemic model, there is one time-dependent transmission function. Also, the procedure is applied to a two population model, which has up to four distinct transmission functions. A two population model is appropriate for studying disease transmission in a heterogeneous population, for example, a population split into children and adults, who spread the disease differently among themselves. Determining the time-dependent transmission function that exactly reproduces disease incidence data can yield useful information about disease outbreaks, including a range potential values for the recovery rate of the disease and offers a method to test the “school year” hypothesis (seasonality) for disease transmission.

This will be a general audience talk. The talk will end with a discussion of several research projects that would be appropriate for a capstone project or a Master’s thesis.

March 7
Matthew Sedlock (Johns Hopkins University)
Percolation models

March 9
Avishek Mallick (University of New Hampshire)
Inferential procedures based on samples with non-detects from normal and related distributions

Abstract: In this presentation, I am going to talk about the problems of computing confidence interval, tolerance interval and prediction interval based on the samples with non-detectable values, i.e. Type I censored samples from Normal and related distributions. Firstly two types of imputation approach has been investigated: one based on the maximum likelihood estimates (MLEs) of the parameters, and the second uses some ad hoc estimates, that are particularly suitable for sample sizes that are small or moderately large. Monte Carlo simulation is used to investigate the performance of these procedures. For a given percentage of values below the DL, this proposed imputation approach exhibits excellent performance when the sample size is small to moderately large. However, as the sample size gets large, the ad hoc procedure performs poorly; but the MLE based procedure continues to perform reasonably well unless the sample size gets very large. However, the confidence levels can be calibrated so that the MLE based imputation approach continues to provide coverage probabilities close to the nominal level.

I will also talk about the inferential problems concerning the mean and the quantiles of a lognormal distribution based on a Type I censored sample. Here procedures based on generalized confidence intervals and modified signed log-likelihood ratio test (MSLRT) statistics are used. Performance of these two procedures along with that of the signed log-likelihood ratio test (SLRT) statistic is compared using Monte Carlo simulation. Based on numerical results, it is found that the generalized confidence interval and the MSLRT based confidence interval are both satisfactory for inference concerning a lognormal quantile, when the percentage of non-detects is fairly large, as large as 70%. However, the conclusion is not so clear cut for inference on the lognormal mean. In fact, this work shows that the routine application of the MSLRT must be approached with caution, the procedure may even give results that are less satisfactory compared to the SLRT based solution. A final point to note is that the generalized confidence interval idea is easier to understand and implement, especially for a practitioner, and it provides accuracy very similar to that of the MSLRT for estimating the lognormal quantiles. For each of the problems considered, the results the illustrated using practical examples.

March 12
Myung Soon Song (University of Pittsburgh)
An unconventional approach to likelihood of correlation matrices

Abstract: Numerical approximations are important research areas for dealing with complicated functional forms. Techniques for developing accurate and efficient calculation of combined likelihood functions in meta-analyses are studied. A multivariate numerical integration method for developing a better approximation of the likelihood of correlation matrices is studied. Analyses for (1) intercorrelations among Math, Spacial and Verbal scores in an SAT exam and (2) intercorrelations among Cognitive Anxiety, Somatic Anxiety and Self Confidence from Competitive State Anxiety Inventory (CSAI-2) are explored. Algorithms to evaluate likelihood and to find the MLE is developed. Comparison with two conventional methods (joint asymptotic weighted average and marginal asymptotic weighted average) is shown.

April 2
Sharad Silwal (Kansas State University)
Image quality assessment methods

Abstract: The central problem in our presentation is the following: Suppose we have an image and we want to find the best match to this reference image from a large database of images. The best way to tackle this problem is indeed to let human observers, the ultimate receivers of all image information, be the judge. However, due to the tedious and time-consuming nature of this problem, we would rather employ a computer algorithm which mimics the human visual system in recognizing similarity between images. In this presentation, we will describe some of the challenges of this problem and discuss some image quality assessment methods which attempt to address these challenges. In particular, we will see how tools from Statistics and Fourier Analysis can come into play in the development of such methods.

April 6
JiYoon Jung (University of Kentucky)
The topology of restricted partition posets PDF

2005 Spring Math Colloquia

February 22
Norah Esty (University of California – Berkeley)
Topological Properties of Orbit Sets for Groups of Homeomorphisms

Abstract: In this talk, I will introduce some dynamical systems by looking at some examples of the way a homeomorphism can iterate a point on the circle, from fixing points and having periodic orbits to creating dense orbits. I will go over the complete classification of orbit types given by the Poincare Classification Theorem, including the existence of homeomorphisms with an invariant Cantor set. Then I will discuss the analogy between the iterate set for a single homeo (corresponding to an action of the group Z) and the orbit set of more general groups of homeomorphisms. Sacksteder’s Theorem gives three possibilities for a particular group G. Sacksteder’s first case is the existence of a finite orbit, however, it does not give nay information about the orbit type of the remaining points. My work expands this case to examine what happens to all others point on the circle when the group has a common periodic set.

February 24
Elmas Irmak (Michigan State University)
Mapping Class Groups

Abstract: I will talk about the mapping class groups of compact, connected, orientable surfaces. I will discuss how combinatorics of different curve complexes on surfaces is used to study the algebraic structure of the mapping class groups.

March 3
Akhtar Khan (Michigan Technological University)
An inverse problem in elasticity

Abstract: In this talk I will discuss a simple inverse problem in elasticity. The talk will be focused on the issues such as: What is an inverse problem? What are the common approaches to solve an inverse problem? Why are inverse problems challenging? I will pose the inverse problem as an optimization problem and discuss several existing approaches. Numerical results will be given for the various approaches. Having established this background I will discuss some new results obtained by myself. Mainly I will show that the output least-squares approach for elliptic inverse problems, using a coefficient-dependent energy norm functional, results in a smooth, convex minimization problem. I will discuss several theoretical and numerical advantages of having a convex objective functional. The issue of dealing with the discontinuous coefficients will also be discussed.

April 8
Judith Silver
Conics in Projective Geometry

Abstract: The definition of a conic usually requires the concept of distance; but in projective geometry, conics are determined in a metric-free way. This talk will show how such conics are formed, using Geometer’s Sketchpad and some audience participation. All of the material should be accessible to students, as well as faculty.

April 22
Bonnie Shook
Topological Approaches to Fingerprint Identification

Abstract: In this talk, I will analyze the topology of different types of fingerprints in order to find a new tool to assist in computer identification. I will focus on the homology groups of the main types of fingerprints – loops, whorls, and arches – and their variations. I will examine if there are fundamental differences between the homological features of these types.

Nathan Cantrell
Cubical Homology in Medical Imaging

Abstract: Trabecular bone architecture is extremely important in early recognition of estrogen loss and osteoporosis, as loss of bone connectivity is one of the earliest signs of disease. Moreover, intense research is underway to quantify and understand the relationship between trabecular architecture and its mechanical properties for the benefit of both medicine and bio-engineering. Such a complex structure which appears as an evolutionary adaptation to the external forces acting on the bone is extremely difficult to examine quantitatively. Topology, however, may be able to reveal significant characteristics of this network. Is there a correlation between orientation and connectivity of the network and bone strength? The voxelized data produced by μ-MR or μ-CT is ideal for cubical homology, a relatively young homological method. In short, the technique uses the boundaries of cubical building blocks to expose these topological features and simplify certain information into an extremely finite amount of data. Cubical homology and computational abilities are now beginning to coalesce providing enormous implications for the cubical format of computer graphics and medical imaging.

April 29
Arthur Porter (Professor Emeritus, University of Toronto)
Manchester University’s Contributions to Analog and Digital Computing

Abstract: Dr. Porter will discuss the role Manchester University played in the development of computing and computer applications. His talk will include the following topics:

Analog and digital computer developments at the University of Manchester during 1932-1937 and 1946-1953 (a brief account).
Manchester University’s influence on 20th century physics and biological science and the pedagogical significance of “model” and “metaphor”.
Recruitment of British universities for service before and during World War II and the birth of radar, operations research, and computer code breaking (e.g. Engma).
Reflections on his debt to Vannevar Bush and Douglas Hartree.
The significance of the Marshall Differential Analyzer program.

2004 Fall Math Colloquia

September 24
Alfred Akinsete
The winning probability and ranking models for teams in soccer tournaments

Abstract: We discuss the statistical analysis of the 1993-2003 soccer results from the 20 teams in the FA Premier League (FAPLE). Using the Clarke-Norman (CLAMAN) model, we obtain the home advantage and strength of each team. The work focuses on ranking methodologies and develops new ranking models for teams in soccer tournaments. We show that the ranking, and the distance between any pair of teams prior to their encounter is equivalent to CLAMAN’s model. The winning probabilities of a team defeating other teams in the league are obtained. These probabilities form a ranking procedure, and provide a source for the performance evaluation of each team in the tournament. The results show that our ranking models are adequate, evidently from the 2003/2004 end-of-year ladder. The work leads to further investigation of stochastic modeling of soccer data.

October 8
Ralph Oberste-Vorth
From Chaos to Stability: Dynamic Equations Parameterized by Time Scales PDF

October 22
John L. Drost
What is the opposite of a prime number?

Abstract: Prime numbers have only two divisors, themselves and 1. The opposite pole would be a number that has a lot of divisors. To quantify this further, we say a positive integer is highly composite if it has more divisors than any smaller positive integer. The sequence of highly composite numbers starts 1, 2, 4 (3 divisors), 6 (4 divisors), 12 (6 divisors), … The concept is due to Ramanujan, but examples occur much further back, for example, Plato thought 5,040 a good number for the citizens of a city since it could be divided in so many ways.

November 5
Elizabeth Duke and Kelli Hall
Time Scale Calculus and Dynamical Systems

Abstract: In his 1988 Ph. D. dissertation, Stefan Hilger united continuous and discrete calculus under one calculus system: time-scale calculus. Employing the delta or Hilger derivative, time-scale calculus derives its beauty from the ability to perform on a “mixed” domain, which combines discrete and continuous data sets. Hilger’s calculus relies on the idea that we can use the same differentiation and integration systems and change only the time scale from discrete (or “isolated”) to continuous (or “dense”) and vice versa, rather than changing the calculus system. This talk examines the connection between Hilger’s work and the field of dynamical systems by exploring the possibilities for using time scales to shed light on the gap between solutions of the logistic equation in difference and differential calculus. We draw this parallel through an example, which models population with the logistic equation.

November 19
Christopher Johnson and Peter Saveliev
Topological Proteomics: Pure Mathematics in Life Sciences

Abstract: How can the 3-dimensional ‘folded’ structure of a protein be determined from the amino acid sequence? How can the function of a protein be determined from the 3-dimensional ‘folded’ structure? Concentrating mostly on the second question we apply techniques of algebraic topology to identify significant geometric features of the molecule. In this preliminary report we describe how we have been able to detect “tunnels” and “hands” normally used for holding a strand of DNA. A review of the necessary background material will be provided.

2004 Spring Math Colloquia

January 23
Yulia Dementieva
Statistical approaches to gene mapping

Abstract: Statistical analysis methods for gene mapping originated in counting recombinant and non-recombinant offspring, but have progressed to sophisticated approaches for the mapping of complex trait genes. The current statistical methods used in genetic mapping are model-dependent LOD score analysis, model independent sibpair analysis and association study. In genome-wide mapping studies, depending on the stage of process, a particular method could be chosen as a complement to others and the best overall design may include multiple techniques. For example, sibpair analysis may be used in the initial screening but association study is preferred for detailed analysis of specific genomic regions. In this talk, the speaker will present various approaches to gene mapping and discuss their statistical challenges.

February 6
Linda Hamilton
Robotics of the Mars Station program

Abstract: There will be an interactive session with a Marshall student, Juan Bueno, at the LEGO CITY at Nick J. Rahall, II Transportation Institute. Through modeling we may see first hand some of the difficulties with the real Mars Mission. Problem solving in the real world involves math and computer science. We in the Marshall Math department can get students interested in continuing their math education to be able to solve tomorrow’s problems. The Planetary Society’s Mars Station project is made possible through a partnership with the LEGO Company. The Planetary Society developed Mars Stations, a part of the Red Rover Goes to Mars project, in order to give people around the world a chance to enjoy the challenge and excitement of exploring another planet. By visiting and logging in to a Mars Station, you can remotely drive a LEGO rover across a Mars terrain, seeing through the rover’s Web camera “eye,” just like mission scientists use robotic rovers to sense and explore the surface of Mars.

February 19
Basant Karna (Baylor University)
Eigenvalue comparison for multipoint boundary value problems

Abstract: A brief history will be discussed concerning boundary value problems for ordinary differential equations. This will include circumstances under which eigenvalue problems arise, significance of smallest eigenvalues, and comparisons of smallest eigenvalues for comparable eigenvalue problems for second order boundary value problems. The technique, that will be used, will include positive cone theory and the signs of the Green’s functions. Some higher even order results will also be discussed.

February 24
John (Matt) Matthews (Duke University)
Granular Materials: An Introduction & Application to Hopper Flows

Abstract: Granular materials are a vital component of many industries, including the pharmaceutical, mining, and food industries. Equally important, however, are the mathematical and computational questions raised in the course of modeling these materials. This talk will serve as a gentle introduction to the field, covering a range of applications from Engineering and Physics and explaining how Mathematics has been brought to bear on these problems. A deeper examination of steady-state flows through hoppers follows, with a survey of some recent intriguing results that may have some direct application to the development better industrial hoppers.

February 27
Mohamed Elhamdadi (University of South Florida)
On knot invariants

Abstract: We will introduce an algebraic structure called a quandle and motivated by knot theory. We will give examples and show how quandles can distinguish between knots.

March 5
Scott Sarra
Scattered Data Approximation with Radial Basis Functions

Abstract: Over the last several decades radial basis functions (RBFs) have been found to be widely successful for the interpolation of scattered data. More recently, RBF methods have emerged as an important type of method for the numerical solution of partial differential equations. In this talk we will describe the RBF approximation method and discuss why it is successful when other methods fail. We will also examine some current RBF research topics.

April 2
John L. Drost
Arrow’s Theorem, or: Why we all just can’t get along

Abstract: In every election involving at least three choices, there must be a method of translating the individual voter’s preferences into a global or societal preference. More specifically, suppose every voter ranks his or her choices in some order. Then a societal preference must somehow reflect the will of the individual voters. For example, if every voter prefers choice x to choice y, then the society should also. Arrow’s theorem, proven by Kenneth Arrow in 1951, shows that there is no method of translating individual preferences into a societal preference that satisfies each of several seemingly reasonable requirements, other than a ‘dictatorship’ where one individual’s preferences are that of the entire society.

2003 Fall Math Colloquia

September 26
John L. Drost
Addition chains

Abstract: If n is a positive integer, an addition chain for n is a list of numbers beginning with 1 and ending with n so that each number is the sum of two previous numbers in the list. For example, an addition chain for 15 is 1, 2, 4, 8, 12, 14, 15, which has 6 links, but another is 1, 2, 3, 6, 12, 15 which has 5 links. Given n, what is the shortest chain for it?

October 10
Peter Saveliev
From slot machines to topology through calculus

Abstract: Consider the following problem: Suppose two gamblers are playing slot-machines and after a period of time they won $100 each. Is it true that there was a time interval during which each of them won exactly $50? Using Calculus we restate this problem in proper (topological) terms and then solve it. The talk is accessible to Calculus II students.

October 24
Kelli Hall
Escher’s tilings and ribbons

Abstract: Using combinatorial methods, M.C. Escher created repeating patterns of tilings with decorated squares, hoping to find every possible pattern. In this presentation, I will give an algebraic proof for his pictorial findings and then extend the mathematical approach to a few cases that involve ribbons.

November 7
Judith Silver
The Spherical Metric Project

Abstract: How do you describe a curve on a sphere? What do you use for a metric? The historical background of these questions will be discussed, with particular emphasis on work done by students at Marshall University. In particular it will be shown that the great circle distance in spherical coordinates does satisfy the definition of a metric. Students in Calculus II should be able to follow the discussion, but a Calc III background would be helpful.

December 5
Bonita Lawrence
Time Scales: A Snappy Link between Continuous Processes and Discrete Processes

Abstract: Since the time of Sir Isaac Newton, continuous time processes, or differential equations, have been studied and there are loads of lovely results for this class of dynamic equations. More recently, we have the development of discrete time processes, or difference equations. There are also many lovely results for this younger class of dynamic equations. In my talk I will discuss an interesting way that the two types of dynamic equations can be studied together, creating a nice generalization that offers us a new perspective on “differential equations.”

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