Abstracts

(Most talks will be held on the 3rd floor of the Martin Luther King Jr. Student Union, times subject to change)

Title: Ask Not What Mathematics Can Do For Biology - Ask What Biology Can Do For Mathematics

Abstract:During the last decade or so Genetic Algorithms (GA) have been routinely used in high-dimensional optimization problems. As they get implemented in various problems, GA natural selection and evolution tools, such as methods of mating, rates of mutation keep getting modified by users. Most, if not all of these modifications aim to increase efficiency at the expense of available computing resources. We propose a modification that reduces the computations needed without losing efficiency.

Abstract: In this paper, I examine some of the bases for the apprehension felt by many academics in the humanities towards their mathematical colleagues and mathematics itself. Starting with a brief historical recapitulation of the common origins of philosophy and mathematics, I show how the 'Continental' philosophical tradition has become accustomed to defusing the dangers of mathematics for its own disciplinary (and institutional) claims by annexing, on a rhetorical and political basis, mathematics' epistemological foundations.

Moving on to more recent interactions between mathematics and the humanities, we will see how the failed interactions between literary theory and the natural sciences have caused a foreboding directed towards mathematics as well as the sciences. I will argue that attempts to transfer wholesale concepts or methods from mathematics or the sciences to literary (or 'critical') theory are unjustified epistemologically, and at the heart of the enmity between humanities and the sciences.

I will finally propose that a valid and profitable interaction should be developed on the basis of the models of questioning of the disciplines about their own underpinnings. Mathematics will be shown to have been far more efficient at recognizing its own foundational limits than 'critical' theory, while some commonplace forms of social constructionism in the humanities have been responsible for limiting the very reach of literary and philosophical studies. A (pedagogical) tripartite classification of disciplinary verities, distinguishing between mathematical certitude, scientific theories, and literary or philosophical assertions, will be proposed as a guide to reform the literary scholar's understanding.

Title: Mathematical Modeling of the Dynamic Exchange of Solutes in a Prototype Hemodialyzer

Edward Boamah, Ph. D
Department of Mathematics & Computer Science
Blackburn College
Carlinville, IL

Abstract: Hemodialysis (HD) is one type of procedure for eliminating toxic chemicals and infusing bicarbonate in patients with end-stage renal disease (ESRD). Research and development in the hemodialyzer industry have, hitherto, depended mostly on empirical evidence to optimize HD therapy. This is often costly and involves numerous clinical trials. Researchers have used ‘black-box’ input-output ordinary differential equation models to study the dynamic exchange of solutes during HD. In the era of advances in contemporary HD therapy, these models are over-simplistic. We have developed a comprehensive mathematical model to describe the dynamic exchange process of solutes in a prototype hemodialyzer. The model, which is represented by a coupled set of transport equations, delineates the blood and dialyzate compartments of the hemodialyzer, and includes bicarbonate-buffering reaction in the blood channel and bicarbonate replenishment mechanism in the dialyzate. A numerical solution of the model gives solute concentrations at various distances along the blood and dialyzate channels at different times. This modeling exercise has allowed us to examine some physical mechanisms of the hemodialyzer, some of which include key fiber properties, ultrafiltration rate, blood and dialyzate flow rates and their effects on the dynamic exchange of solutes during HD therapy

Abstract: TEX and LATEX are well known tools for typesetting text, with particular strengths when mathematics is involved. A mathematics course which emphasizes proof techniques and mathematical writing is a natural place for students to be introduced to LATEX. One impediment to its use, however, is the fact that the required software is not as widely available as more mainstream and less capable “office” suites. We will demonstrate how a very capable collection of TEX-related programs can be assembled and placed on a USB flash memory device. Armed with such a USB device, a student can create and typeset LATEX documents anywhere they have access to a PC — no installation of special software is needed.

AbstractThree questions will be considered. First, what mathematics did seventeenth century immigrants to the American colonies study? Second, what mathematics did the immigrants use? Third, to what extent were the answers to the first two questions influenced by what went on in the “mother countries”?

Title: Laboratory Projects for a Second Semester of Biocalculus

Timothy D. Comar

Abstract:The presentation will highlight several computer laboratory projects used in the second semester biocalculus course at Benedictine University. Examples of projects presented will include an application of life tables using (approximations of) improper integrals, a matrix model for age-structured populations, and difference equation models for host-parasitoid interactions. In addition to providing useful examples which require students to apply mathematics to biological models, there are several benefits derived from these projects. The students develop skills using computational software to analyze biological problems. The computer implementation of these models enables students to observe dynamic behavior graphically and readily allows students to see how the behavior of a model changes by changing parameter values. These projects can be used in other mathematics courses as well.

Title: Balls on a Sphere – Polyhedra as Solutions to a Minimization Problem

Howard Dwyer

Abstract:The regular polyhedra can be circumscribed by a sphere, with the center of the sphere coincident with the center of the polyhedron; the vertices of a regular polyhedron can be thought of as points on a sphere. Inspection of the tetrahedron and octahedron suggest that the vertex points are positioned so that they are as far away from each other as possible – that the location of the vertices may represent an equilibrium position for points which are mutually repelling. To explore this notion, a small computer program has been constructed which places "charged balls" at random on the surface of a sphere and then allows them to migrate across the surface, driven by the repelling forces, until an equilibrium position is reached. Faces and edges are then added to construct a convex polyhedron.

In this talk, we will demonstrate the program and discuss the computational geometry problems involved. Some of the results are what you would expect, and some will surprise you. This material is accessible to a general audience.

AbstractVirtual knots are a generalization of classical knots that was first introduced by Louis Kauffman in 1996. In virtual knot theory, we consider tying knots around surfaces instead of in 3-dimensional space. We will cover some basic topics from knot theory and virtual knots. This talk will be accessible to undergraduates.

Title:Consecutive integers with the same number of principal divisors

Roger B. Eggleton
Mathematics Dept., Illinois State University

Abstract The principal divisors of an integer n are its maximal prime-power divisors, so p^a is a principal divisor of n if p is prime, a > 1, and n is divisible by p^a but not by p^(a+1). For brevity, we say n has rank r if it has exactly r principal divisors. For instance, 360 = (2^3)(3^2)5 so 360 has rank 3, and its principal divisors are 5, 8, and 9. Every positive integer is uniquely the product of its principal divisors (Fundamental Theorem of Arithmetic).

We are interested in runs of k > 2 consecutive integers which all have the same rank. The runs of rank 1 integers include 2, 3, 4, 5 and 7, 8, 9, but every other run has size 2. Are there infinitely many rank 1 runs of size 2? The runs of rank 2 integers include 33, 34, 35, 36 and 38, 39, 40. What is the longest possible run of rank 2 integers? Are there infinitely many rank 2 runs of size 2? Are there runs of rank r integers when r > 2?

Many such questions can be asked, and surprisingly, a few of them can be answered, or almost answered! The talk will look at a number of the results that I have been able to prove (with Jason Kimberley and Jim MacDougall, of the University of Newcastle), and some surprisingly large numbers will be included.

Abstract:This presentation will summarize findings of an investigation into the extent to which prospective middle-school mathematics specialists in Illinois know and understand linear and quadratic equations, linear inequalities, and quadratic functions.

Abstract: There are many ways of calculating the famous number PI. Some of the ways are based on hard computer calculating programs, some on different probabilistic models, like the ``Buffon Model’’. In my talk, I will give a very simple deterministic model of calculating PI that includes just two billiard balls and one reflecting obstacle (a wall) located on one side from the balls. All the hits in the system are elastic.

To prove this result, I will involve into consideration the notion of a configuration and a face space of a dynamical system and a very interesting non-trivial arithmetic identity. All my proofs are elementary (based only on the high school mathematics) and all necessary notions will be introduced during the talk.

Title: Braids, Cables, and Cells: an intersection of Mathematics, Computer Science, and Art

Joshua Brandon Holden
Assoc. Professor

Abstract: The mathematical study of braids combines aspects of topology and group theory to study mathematical representations of one-dimensional strands in three-dimensional space. These strands are also sometimes viewed as representing the movement through a time dimension of points in two-dimensional space. On the other hand, the study of cellular automata usually involves a one- or two-dimensional grid of cells which evolve through a time dimension according to specified rules. This time dimension is often represented as an extra spacial dimension.

Therefore, it seems reasonable to ask whether rules for cellular automata can be written in order to produce depictions of braids. The ideas of representing both strands in space and cellular automata have also been explored in many artistic media, including drawing, sculpture, knitting, crochet, and macramé, and we will touch on some of these.

Abstract: Galois once stated that given a polynomial f it would be difficult for him to tell you what the Galois group for f was.. Indeed this is a difficult task. My talk will focus on the basics of Galois groups concerning quartic polynomials and two simple results will be presented to aid in the identification of a quartic polynomial’s Galois group via observation.

Abstract: Mathematics teachers are sometimes regarded as interchangeable parts. Nothing is further from the truth; we all have varying skill levels, pet interests and quirks. We need to let our personalities and interests come out as we teach. Two aids that can help are humor and history.

This talk will feature methods of integrating history and humor into everyday topics with the objective of making the attendant mathematics more accessible to students. Cartoons, photos, quotations, jokes, songs and poetry are just some of the ways which will be illustrated. The use of humor and history as bridges to mathematical concepts will be emphasized

Not everyone can or should use all of these techniques, but everyone should at least consider using some of them in an effort to foster better student engagement in our classes. Given its topics, it is hoped that the talk will both instruct and entertain.

Jerzy Kocik
Mathematics Department
Southern Illinois University at Carbondale, IL

Abstract: Somewhat unexpected connections between algebra and geometry, realized in a single figure of an Apollonian gasket. The presentation includes a number of colorful illustrations.

Abstract:A graph labeling is "distinguishing" if no nontrivial automorphism (i.e., symmetry) of the graph preserves it. We begin the project of counting such labelings. This is an extension of a problem originally motivated by Frank Rubin's "blind man's keys". We pursued it as part of a 2007 REU.

Abstract:Many problems about conics are easier to solve when considered in the projective plane. Homogeneous coordinates (rather than Cartesian) are used, allowing the application of tools from linear algebra. Come and learn the basics of projective geometry and how to use them to write a /very/ brief /Mathematica/ program to draw a conic through five points.

Paul M. Musial
Chicago State University
Department of Mathematics and Computer Science

Abstract: Real Analysis is a famously abstract subject. Analysts over time develop mental images to help concretize the abstract notions of analysis. We will discuss some metaphors that will help students understand concepts such as neighborhood, upper and lower Riemann sums, and Lebesgue measure.

Abstract: Presentation or discussion on topics concerning teacher preparation issues in the state. It is hoped that this would be a time where ISMAA and IMTE members could get together to talk about common issues.

Abstract: Graph drawing considers ways to nicely imbed graphs on the plane or other surfaces. The first thing that one tries to avoid is having two edges cross. When that is not possible, one tries to minimize the number of crossings in a drawing. Our approach to this problem has been through rotation systems, that is, the clockwise ordering of edges at each vertex.

During this talk everything will be introduced from scratch; no knowledge of graph theory will be assumed. The talk will include some remarkably easy algorithmic proofs of classic theorems such as the Hanani-Tutte Theorem:

If a graph can be drawn in the plane such that every two edges cross an even number of times, then it can be redrawn with no crossings at all. Joint work with Marcus Schaefer and Daniel Stefankovic.

Dr. Euguenia Peterson & Dr. M. Vali Siadat
Department of Mathematics, Richard J. Daley College, Chicago, Illinois

Abstract: We will present the findings of our comprehensive research study on the effects of implementation of formative and summative assessments on achievement and retention of students in elementary algebra classes at Richard J. Daley College. The methodology used was the Keystone method which is based on the centrality of student learning and continuous adjusting of the teaching practices. This method incorporates frequent quizzing, feedback, and reteaching in order for students to attain mastery. Using experimental/control group design it was found that approximately after four months, students (N=222) who were in the Keystone group achieved significantly higher scores on the final examination than the students in the control group (1352). The higher improvements in student performance were achieved in conjunction with higher retention rates. The results obtained on internal summative assessments were also highly correlated to the external examination scores.

Abstract: Calculus III is a course where visual and analytical ways of thinking happily meet. Unfortunately, many students are content to just applying formulas and algorithms, thus missing on the intuition that visualization of concepts develops. In this talk I will present an attempt to jump start the process of spatial visualization in my students. "A plane cutting a cube" is a hands on activity that my class performs during the first week of Calculus III. I hope that this presentation will generate discussions and ideas on how to improve visual perception and representation of calculus concepts.

Abstract We extend the classical notion of Trace to a class of transcendental numbers by introducing a distance on the set D of finite subsets of distinct complex numbers. For each transcendental number in the class, we also define an analytic function on C\K (K compact) for which the coefficient of z is the generalized trace.

Abstract: Memory is a curious thing. To remember important things, people use a variety of tools. We write notes to ourselves or on our hands, send messages to ourselves on our own message system (e-mail, voice mail), create mnemonics (PEMDAS), and even may tie a string on our finger. In my calculus classes, I use keywords to trigger student memory of basic concepts and techniques in the subject. In this talk I’ll share my favorite calculus keywords (e.g., in the box, the trio, etc.), the mathematical concepts they represent, and how students use these keyword-concept pairs in solving problems.

Title: Parametric and Nonparametric Biased Sampling Corrections in Romalea microptera

Andrew Thurman
Department of Mathematics
Illinois State University

Abstract:In sampling studies, the sample is said to be length biased when an individual’s sample inclusion probability is proportional to some measure. Estimators used under the assumption of random sampling will be both biased and inconsistent. When the measurement causing length bias follows a known parametric distribution, we assume the sample is length biased and find unbiased and consistent estimators of the population parameters. When the distribution is unknown, we propose a method of bootstrapping to create a sample in which the random sampling estimators are valid. We approach both methods in sampling the eastern lubber grasshopper, Romalea microptera.

Title: Differences between STEM Students and Non-STEM Students
at Community Colleges

Tingxiu Wang, Professor of Mathematics, Oakton Community College
David Smith, Senior Lecturer of Psychology, Northwestern University
Gloria Liu, Coordinator, Center for Promoting STEM, Oakton Community
College
Joe Kotowski, Professor of Engineering, Oakton Community College
David Rudden, Manager, Institutional Research, Oakton Community College
Bob Sompolski, Professor of Mathematics and Computer Science, Oakton
Community College

Abstract: Through surveys and basic statistical analysis, this research investigates differences between STEM and non-STEM students, impacts of STEM activities and incentives, and consequently ways to encourage non-STEM students to become interested in STEM and pursue a STEM degree. Based on the findings on these differences, we offer suggestions how to support non-STEM students so that they can develop interests in STEM and pursue a degree in STEM.

Abstract: This hands on presentation will give participants a chance to explore the pedagogical tools offered by the newest technology. Interesting problems solved in multiple representations can address the different learning styles of students and provide deeper understanding of the mathematics.

This would be particularly appropriate for teachers of the course for pre-service secondary math teachers and those teaching developmental algebra.

Richard J. Wilders
Marie and Bernice Gantzert Professor in the Liberal Arts and Sciences
Professor and Chair of Mathematics
Special Assistant to the President
North Central College

Abstract: This presentation will introduce several worksheets created by he presenter to help students visualize the concepts in multivariable calculus. These worksheets will be available for participants at the presenter's website. The presentation will include basic instruction in using Maple components; a tool which allows instructors to set up worksheets which make graphing and other Maple functions much easier for students.

AbstractThe computer scientists and mathematicians have been using the idea of grouping or clustering of raw data either to study the nature of these data or search through the data for particular information. Lately, the focus has been on combining information from several clustering solution to produce a superior clustering solution, known as cluster ensemble problem. Although several methods have been proposed in the past to ensemble cluster, none of them are lossless, i.e. the original clusters cannot be constructed back once the ensemble is created. The method proposed below reduces the cluster ensemble problem to bipartite graphs partitioning problem while preserving all the original information.

Abstract We will discuss the history and properties of a Latin square, including what it means to have a pair of orthogonal Latin squares. We will touch on some common applications for Latin squares, specifically addressing the construction of orthogonal pairs and the famous thirty-six officer problem, in which the existence of an orthogonal pair of order six is challenged.

Title: Calibration Methods for Achieving Stereoscopic Vision with Cheap Cameras

Abstract It is well known how to assimilate the data from two cameras into a three dimensional view, but most methods use a pinhole ideal approximation for the cameras involved. Because pinhole cameras are very expensive, this project explores how to use methods from linear algebra and computer science to filter the data from inexpensive cameras so as the approximate the pinhole case, and then construct a three dimensional image from the two views.

Title: Knot Theory and How it Applies to DNA Studies:
What has your DNA in knots?

Abstract:If one were to say "imagine a knot," what would spring to mind? One may think of a knot in their hair, a phone cord, or a rope. Many would not think about the mathematical applications to the tangles that we experience. This presentation will give the audience a taste of knot theory by explaining the definition of a knot, knot equivalence, Reidemeister moves, and tricolorability. The talk will then use this background to explain applications of knot theory in DNA studies.

Abstract:An alpha-regular conformation of a knot (or link) K is a polygonal embedding of K in space such that all edges have the same length and all angles between adjacent edges are equal to alpha. The alpha-regular stick number of K is the minimum number of sticks required to construct an alpha-regular conformation of K. We construct alpha-regular conformations of (2n; 2)-torus links, where alpha = arccos(-1/3). These conformations provide good upper bounds for alpha-regular stick numbers and, in some cases, realize alpha-regular stick

Abstract:Morley’s Theorem states that: The three points of intersection of the adjacent trisectors of the angles of any triangle form an equilateral triangle. We have created an interactive exercise to demonstrate Morley’s Theorem using the geometry software Cinderella. We will discuss the proof of this theorem and how this can be incorporated into the high school classroom.

Title: Analysis of the Effects of Origin and Relevant Baseline Medical Conditions

Abstract: The Platelet Glycoprotein IIb/IIIa in Unstable Angina: Receptor Suppression Using Integrilin Therapy Trial, referred to as the PURSUIT Trial, was a multi-national, double-blind study involving 9,382 participants that measured the effects integrilin had on preventing death or myocardial infarction in a diverse sample group. There were many novel methodologies applied to the trial that yielded a wealth of data. The PURSUIT Trial was the first clinical trial of its magnitude to enroll a proportionately high percentage of women and was innovative in that the decision for a patient to undergo percutaneous revascularization was left to the treating physician as opposed to clinical trial protocol. Chief among the ingenuities, however, was expanding the trial globally. This triggered an influx research questions, some of which have yet to be answered.

One such question seeks to find the effects a person’s global origin and relevant baseline medical characteristics have on predicting survival.. More specifically, it is the purpose of this research project to conclude that a person’s medical history and global location provide a strong statistical model with which survival for certain time periods can be predicted. Analysis of the data was conducted using the Statistical Analysis Software (SAS). The end product is a statistical regression model with which, given a particular set of baseline characteristics, a patient’s probability of experiencing death or a myocardial infarction within 30 and 90 days can be calculated.

The focus of the 60 minute presentation would be to introduce both the biology and statistics necessary to understand the analysis of the PURSUIT data set. The bulk of the presentation would be the discussion of the multiple logression regression model that was used in the analysis and the results obtained from the model. Emphasis will be further placed on the implications the results have clinical practices today.

Abstractt: I will discuss a host-parasite model due to Leslie and Gower. The key feature of the dynamics of this model is that there is a unique ositive locally stable equilibrium. We illustrate the dynamics of this model using a dynamic Excel spreadsheet.

Abstract: Our main interest lies in exploring isomorphisms of elliptic curves. In particular, we will look at certain properties regarding families of curves in a field. These properties arise naturally for fields with certain characteristics.

Abstract:The Nine Point Circle Theorem states: If _ABC is any triangle, then the midpoints of the sides of _ABC, the feet of the altitudes of _ABC, and the midpoints of the segments joining the orthocenter of _ABC to the three vertices of _ABC all lie on a single circle.We have developed an interactive exercise to illustrate the Nine Point Circle Theorem using the geometric software Cinderella. We discussed the proof of this theorem and how this activity can be incorporated into the high school classroom environment.

Abstract: I will be discussing two classical discrete models for host parasitoid interactions. One model is the Nicholson Bailey model in which the number of encounters between a host and a parasitoid are distributed randomly and follow the Poisson distribution. In turns out that the nontrivial equilibrium of the this model is unstable, which is rarely observed in nature. I also discuss a modification of the Nicholson Bailey model known as the the Negative Binomial Model, in which the nontrivial equilibrium is stable. I will also demonstrate dynamic Excel implementations of these models.

Abstract::This talk discusses pattern databases, how to build a pattern database for the Rubik's cube puzzle, and how to use this to solve the puzzle itself.

Abstract:This interdisciplinary project is a creative repetition of part of an experiment as described in “Hierarchical Structures Induce Long-range Dynamical Correlations in Written Texts,” published in the May 2006 Proceedings of the National Academy of Science by Alvarez-Lacalle et al. When reading a written text, the reader is focused upon a particular window of attention, containing the words just read. By looking at the co-occurrence of words in each window, it is noted that certain groups of words arise consistently together. These groups of words are called the concepts of the text, and their strength can be quantitatively measured. The meaning of the text can be better understood by considering the principal concept to represent the meaning of the text as a whole, while the lesser concepts can represent the secondary meanings of the text. A Java program was used to convert the text into a format suitable for analysis. Using the mathematics software MatLab, principal component analysis that involved advanced topics in statistics and linear algebra was run on the formatted text to extract the meaningful patterns of words in the text. This research may lead to greater understanding regarding how the structure of writing uses the reader’s memory and to some interesting applications.

Abstract I will discuss the dynamics of a modification of the Nicholson Bailey host parasitoid model due to Beddington, Free, and Lawton. A significant aspect of this model is that the growth of the host population is assumed to be density dependent in the absence of the parasitoid. Depending on certain parameter values, this model can exhibit a wide range of dynamical behavior. We illustrate the dynamics using an Excel implementation of the model.

Abstract:We will describe variations of one-predator, two prey systems that arise as systems of impulsive differential equations and present results about premanence and the dynamics of these systems. Such systems are important in integrated pest management strategies.