40th Annual WKU Mathematics Symposium Abstracts and Recordings

Time: 9:45-10:10

Presenters: Donald J. Price and Dr. Ahmet Ozkan Ozer

Abstract: We consider Finite Difference semi-discretizations (approximations) of infamous one-dimensional wave and beam equations. The corresponding partial-differential-equation (PDE) counterparts are all known to be controllable by one boundary controller applied at the tip of a string or a beam. However, semi-discretized models are not stabilizable without filtering which does not mimic the behavior of PDEs. Therefore, dynamic boundary controllers together with an indirect filtering are introduced. We show

1. The boundary feedback stabilization of a vibrating string utilizing the filtered finite difference method.

2. We also show boundary feedback stabilization of a vibrating beam for two different models: Euler-Bernoulli type and Rayleigh type.

These solutions were built to fit the equations in a custom manner such that all control parameters and even initial equations with high and low frequencies may be manipulated via a Mathematica program. The demonstrations and codes being used in the presentation have been published at the Wolfram’ Demonstration Project website:Demo 1 andDemo 2.

This project is sponsored via KY NSF.

Time: 10:15-11:40

Presenters: Emma Moore, Dr. Ahmet Ozkan Ozer

Abstract: In control theory, a system is exactly observable if, when you measure a certain system property for a finite time, you are able to distinguish any two different initial states on a string. With partial differential equations (PDEs), like the one-dimensional wave equation, there are infinitely many eigenvalues, which makes the PDE infinite-dimensional, so exact observability of vibrations on a string is an infinite-dimensional problem. In practice, however, system properties are observed by the sensors, and sensors can only observe finitely many vibrational modes. Therefore, sensor design for the observed property has to be done for a numerical approximation of the PDE, which makes it a finite dimensional system. However, exact observability does not hold for well-known numerical approximation techniques like Finite Difference Method (FDM) and Finite Element Method (FEM). The major issue we encounter in trying to show the exact observability of the approximated system is that the uniform gap property of eigenvalues is lost, which is not the case for the infinite-dimensional system. This uniform gap ensures exact vibrational observations by the sensors. Without it, sensors cannot distinguish one vibrational mode from another. This happens because the deficiencies of the numerical techniques cause artificial, high-frequency eigenvalues, which results in the uniform gap approaching zero as the mesh parameter goes to zero.

This talk will focus on

1. the lack of observability for the FEM space-approximation of the wave equation

2. a remedy, the so-called “direct filtering technique,” which enables us to prove exact observability.

The proofs are based on the so-called “discrete multipliers technique.” This project is sponsored via KY NSF and WKU RCAP.

Time: 10:45-11:10

Presenters: Ryan St. Clair and Dr. Ahmet Ozkan Ozer

Abstract: In this project, full spectral analysis is performed for the continuous and discrete versions of the one-dimensional PDE models for the wave equation with Dirichlet-Neumann and Neumann-Neumann boundary conditions. Space-discretized Finite Difference Method is used for the discrete model. Our research shows that the discretized model for each type of boundary condition lacks the gap property in eigenvalues, particularly for the high-frequency vibrational modes, as the discretization parameter tends to zero. However, this is not the case for the continuous model.

Time: 11:15-11:40

Presenter: Wilson Horner, Dr. Ahmet Ozkan Ozer

Abstract:In control theory, the time it takes to receive a signal after it is sent is referred to as the observation time. For certain types of materials, the observation time to receive a wave signal differs depending on a variety of factors, such as material density, flexibility, etc. Suppose we have a strongly coupled wave equations describing the vibrations on a piezoelectric beam of length L. Wave equations come with different wave speeds c_1 and c_2. The minimal observation time is proven to satisfy is T>T_1=2L/(min{c_1,c_2}). Next, we discretize the spacial variable for the coupled wave equations via Finite Differences. We find out that for this particular discretization, the observability time approaches infinity which does not mimic the continuous case. To resolve this issue, a filtering technique known as direct filtering is applied, and a (non-optimal) observation time T>T_1>T_2 is obtained.

Time: 9:45-10:10

Presenters: Maggie Jones & Gloria Huang

Abstract:Biosystem Balancer is an ecosystem simulation game that models predator-prey interactions within a four-species food chain in four various biomes. Predator-prey interactions are implemented through a four-equation differential equation model. Our program teaches the user about the model and how altering interactions between species affects the populations of other species in the ecosystem. The user’s goal in our program is to find the correct starting populations for each species to prevent overpopulation/extinction in a species. Finding the correct starting populations would result in an ecosystem at equilibrium in which all four populations stay constant over time or a balanced biosystem.

Time: 10:15-10:40

Presenters: Gloria Huang, Diksha Satish, Dr. Richard Schugart

Abstract: The treatment of chronic wounds has long been a challenge to medical professionals and presents an economic burden to healthcare globally. To aid the healing response of chronic wounds, four differential equations were previously used to model the relationships between matrix metalloproteinases (MMP-1), their inhibitors (TIMP-1), and the extracellular matrix (ECM) using averaged patient data during the diabetic foot-ulcer-healing process. In this work, the mathematical model is employed and analyzed with individual data from the sixteen patients over twelve weeks. A mixed-effects modeling approach was used to curve fit the mathematical model. Mixed-effects modeling was implemented in MONOLIX, where its Stochastic Approximation Expectation-Maximization Algorithm was used to calculate population parameters from the patient data. The generated population parameter values were then used as a starting point to calculate individual parameters specific to each patient’s data. Through MONOLIX, all sixteen patients were successfully curve-fit to generate a more individualized patient response. To study parameter identifiability, the three-dimensional geometry of parameter space was visualized for all pairwise combinations of parameters of all patients to see the effect on the biological system. This approach plots two parameters against the sum of squared errors to generate a three-dimensional graph. The Delayed Rejection Adaptive Metropolis algorithm (DRAM) in MATLAB was then utilized to confirm geometry results. These results indicate that the parameters in the TIMP equation most influence the healing response of the wound.

Time: 10:45-11:10

Presenters: Ruby Mason and Olivia Sharp

Abstract: Various public computer programs have been developed utilizing the standard SIR differential equation model to display the elemental spread of an epidemic; however, an extended version of the SIR equations applied to a hands-on educational experience has not been found in research. This project includes a broadened SIR equation used to model a variety of epidemics dependent on multiple user-defined variables. This mathematical system was implemented in Wolfram Mathematica to create a pedagogical program. This disease spread simulator is intended for use by secondary and high school students receiving an introduction to epidemiology. The program does not produce scientifically accurate data such that it should be used to predict real-life situations or for advanced epidemiological work. Instead, it provides estimations sufficient to allow students to identify causal relationships between the parameters they input and the outcome of their epidemic. The developed model is presented through a user-friendly interface with corresponding visuals and informational prompts. As a learning tool, it delivers a thorough learning experience suitable for the target audience and assists in making fundamental epidemiology concepts more accessible.

Time: 11:15-11:40

Presenters: Ferhan M. Atici and Ngoc Nguyen

Abstract: In this talk, we introduce the h-discrete and h-discrete fractional representation of a pharmacokinetics-pharmacodynamics (PK-PD) model describing tumor growth and anticancer effects in continuous time considering a time scale hN_0, where h>0. Since the measurements of the drug concentration in plasma were taken hourly, we consider h=1/24 and obtain the model in discrete time (i.e. hourly). We then continue with fractionalizing the -discrete nabla operator in the h-discrete model to obtain the model as a system of nabla h-fractional difference equations. To solve the fractional h-discrete system analytically we state and prove some theorems in the theory of discrete fractional calculus. After estimating and getting confidence intervals of the model parameters, we compare residual squared sum values of the models in one table. Our study shows that the new introduced models provide fitting as good as the existing models in continuous time.

Time: 9:45-10:10

Presenter: Claus Ernst

Abstract: Using a simple class of knots called rational knots we discuss the relationship of these knots with the concept of continued fractions in number theory.

Time: 10:15-10:40

Presenters: Benjamin Handshoe and Calvin Higdon

Abstract: Incoherent nullification is an operation on a knot diagram that simplifies the knot. The simplified knots diagrams are called the "children" of the knot diagram. The presentation will explain this operation and our how to perform it on many knots using the computer program Mathematica. Some examples of what will be shown are as follows: relationsbetween knots and their children by incoherent nullification, the frequencies of knots resulting from nullification, and an analysis of the number of unique children a knot has by incoherent nullification.

Time: 10:45-11:10

Presenter: Harrison Gover

Abstract: We describe several ways of how a knot can curve when drawn on paper. We will define several terms such “spiral” and “curly” to describe how the drawing of the knot can curve and prove some results.

Time: 11:15-11:40

Presenters: Donald J Price and Nathan Gillispie

Abstract: A knot can be defined as a closed, non-self-intersecting curve that is embedded in 3 dimensions. Knots are traditionally viewed as a projection into 2 dimensions, such as drawings. One notable exception of this is the KnotPlot software which allows a user to view knots in 3 dimensions. We display work towards a solution using an Oculus headset and the Unity Game Engine that will allow users to view knots in 3D on a computer and/or in Virtual Reality.

Time: 9:45-10:10

Presenter: Mikhail Khenner

Abstract: Electric currents are routinely used in materials science and nanoscience to structure surfaces of conducting solid films. I will present and discuss a model, its analysis and computation, where a surface is a binary crystal. Thus the dynamics of a surface shape evolution and the dynamics of a composition evolution are coupled. The model is based on a pair of highly nonlinear parabolic PDEs of orders 4 and 2.

Time: 10:15-10:40

Presenter: Ethan Kelly

Abstract: In this paper are obtained an implementation and graphical results, from a series expansion of a previously discovered travelling-wave voltage function, described by a solution to the Hodgkin-Huxley action potential equations. Using the Mathematica-based analysis of the previously derived voltage function, there are obtained well-shaped graphs that behave as solitons under interactions with other waves. Calculations, using the aforementioned voltage function with numerical values for the system parameters, indicate that the theoretically predicted travelling-wave potential solution matches the experimentally observed action potential of the neuron; these results have been demonstrated herein

Time: 10:45-11:10

Presenter: Shreeya Arora & Bruce Kessler

Abstract: Partisan gerrymandering is the manipulation of district boundaries in order to establish an unfair political advantage for a particular party or group. The goal of this research project is to develop code to evaluate the current districting of a state using computational mathematics. We developed a random walk model using Wolfram Mathematica that generates thousands of district plans. The new districts continue to meet criteria regarding population bounds and precinct adjacencies. These plans are then analyzed to find normal distributions. In order to validate the algorithm, a model state, “Pentucky,” made up of 6 districts and 100 precincts, was used. Two disparate initial district plans were used as inputs for the algorithm; a consistent winning-party average resulted from both district plans. This concept is being further applied with the 3,692 precincts within Kentucky. Using registration statistics, the algorithm will generate a distribution of districting plans with hopes to guarantee equal voter representation.

Time: 11:15-11:40

Presenter: Mark P. Robinson, Western Kentucky University

Abstract: Newton's method and variations of it have proven useful in solving many problems in nonlinear optimization. An introduction to these methods is provided in this presentation.

Time: 2:30-2:55

Presenters: Mina Ryumae, Dr. Uta Ziegler

Abstract: The two-person, zero sum, high strategy game Dots and Boxes has a large computational burden, which causes difficulty in using brute force to consider possible board configurations that increase exponentially with the board size. The challenge of the game is introduced when a player is forced to take another move after scoring, which creates a vast number of player move sequences. The Monte Carlo Tree Search (MCTS) algorithm is a machine learning approach that builds a partial game tree of board positions and their moves deemed to be important to the game, rather than using pruning techniques or searching the entire game tree. A large number of simulations is run for each board configuration added to the tree to collect statistical data (such as the number of times each move possible in that configuration was tested and the move’s overall benefit). Each simulation uses data pre-collected to select moves and updates the data after finding the simulation results.

This work focuses on transpositions, that is, board configurations that can be reached through multiple move sequences, which are quite common in many games. Specifically, the aim is to improve the win/loss/draw ratio and efficiency of the MCTS player by affecting the amount and quality of information learned through simulations. The basic MCTS selection process uses the benefit of the move for selection calculations. The proposed selection process modifies the basic process by using the benefit of resulting board configurations after a move was taken for selection calculations. A framework was developed which allows players utilizing various selection processes to play directly against one another. Win/loss/draw ratio and efficiency comparisons between players using the basic and proposed selection processes will be presented.

Time: 3:00-3:25

Presenters: Brian Mazzoni, K. Tate Thomson, and Shane Mathew Palmquist

Abstract: The dedicated work of Prof. Dr. Georg Cantor is of significant importance to mathematics as it clearly demonstrates that differing infinities exist, and infinity is far more than just a concept. However, his work, like all other works in mathematics and science, is relative and subject to interpretation. While his work offers a unique and critical perspective on the subject of infinities especially regarding alephs and omegas, other perspectives are possible and equally as valid and impactful. This work builds upon Cantor’s foundation offering a new and somewhat differing perspective on the subject of infinity. A new and differing set of numbers referred to as lemniscate numbers is proposed. However, this new set of proposed numbers is directly connected back to Cantor’s paradise to help unify mathematics for future contributions. The results are surprising, yet simple, and will have a significant impact on numerical mathematics now and in the coming future. This work helps to bridge the divide between our current understanding of infinity and actual infinity by offering an approach which is closer to mathematical truth than presently exists. It is hoped that this work will serve as a catalyst for future mathematical developments impacting many areas of knowledge requiring greater numerical understanding such as applied mathematics, science, engineering, and technology.

Time: 3:30-3:55

Presenter: Allen Lin

Abstract. We generalize the results of the alternating Dirichlet series first examined by Junesung Choi. Specifically, we examine its analytical continuation and its special values. We also connect values of the Choi Dirichlet series to values of another Dirichlet series. Finally, we connect this last class of Dirichlet series to Dirichlet L-functions, ultimately obtaining values of the alternating Dirichlet series.

Time: 4:00-4:25

Presenters: K. Tate Thompson, Cameron Jackson, and Shane Mathew Palmquist

Abstract: Numbers are foundational concepts and principles in mathematics. They allow us to measure, compare, quantify, and analyze phenomenon in engineering, applied sciences, business and more. This paper examines the work of two undergraduate civil engineering students who explore what a number is in mathematics, examining the meaning and application. Number types and groups of known finite numbers are explored. Based on this investigation, potentially new numbers are engineered and presented to fill the gap in understanding that exists within our number line system. The results are surprising and serve as a means of helping to unify and clarify numerical mathematics in the wake of the work by Kurt Gödel and his incompleteness theorems. In addition, this work serves as a beacon to help illuminate new possibilities for future developments in applied mathematics. A student assessment of this research project presented.

Time: 2:30-2:55

Presenter: Lukun Zheng

Abstract: In this paper, we study the modelling and estimation of dependence structure among foreign exchange rate markets using the returns of U.S. dollar-based exchange rates (Japanese Yen, Euro, Australian dollars and the British Pounds against US dollars). A new approach is proposed based on a mixed copula model, which can capture various interdependence patterns among exchange rates. The returns of each exchange rate are filtered using an ARMA-GARCH filtration to remove serial dependence. A two-stage algorithm is then utilized for model estimation: First, the marginal distribution of each individual filtered returns is estimated non-parametrically; second, the parameters in the mixed copula model are estimated using maximum likelihood estimation method (MLE). A goodness-of-fit test is conducted to evaluate the performance of the estimated model. The empirical results reveal strong interdependence among these major exchanges rates markets, which provides important implications for various stakeholders in the market.

Time: 3:00-3:25

Presenters: Andrew Davison, Morgan Samples

Abstract: In the card game Love Letter, players make several assumptions about the contents of their opponents’ hand through a combination of human reasoning and clear probability. The information gathered in this process is not definite, but is often vital to playing the game well. We developed a version of Love Letter in Mathematica whose AI players tackle the use of indefinite data about the opposing players by weighing definite possibilities with inferences made by the computer player. We present our final product, which along with stunning graphics, can often outplay a human competitor.

Time: 3:30-3:55

Presenters: Ahmet Ozkan Ozer, Natasha Gerstenschlager, Lukun Zheng

Abstract: Calculus I is the most fundamental courses for college level science, technology, engineering, and mathematics (STEM)-intended majors. In the fall of 2010, there were over 300,000 students taking Calculus I across the USA, and the US colleges and universities are struggling with high attrition and failure rates in calculus. In fact, the recent National Study of College Calculus, conducted by the Mathematical Association of America with support from the National Science Foundation, it is observed that success rates in Calculus I are very low for traditional students, and it is even lower for non-traditional students: women, students from underrepresented minorities, economically disadvantaged students, and first-generation college students. In the same report, it is underlined that the retention rate is fairly low in calculus courses. Students also express concerns about the insufficiency of information covered in the course and the lack of opportunities to practice learned pedagogy. Therefore, students struggle in practicing complex steps of calculus.

Over the course of last two years, over 700 copyrighted instructional and tutorial videos have been developed and some have been published at the Dr. Ozer's YouTube Channel: MathbyDrOzz. The production of these videos was the first phase of transitioning to traditional lecturing to a video-platform by the worked-out content and cutting-edge video recording & editing technologies. These videos are exposed to experimental and control groups. Based on the analysis for the collected data it is observed that there is a significant long-term effect (about four months) in student success. The talk will focus on the production, design, development, and implementation aspects of these videos.

Time: 4:00-4:25

Presenter: Richard Schugart

Abstract: In this talk, I will honor Dr. Kessler by presenting calculus word problems that I have written and used in either my class or on a test. The problems show off our departing department head’s talents in problem solving and music. *Ode does not stand for ordinary differential equation.

Time: 2:30-2:55

Presenters: Natalie Heath & Raychel Kool

Abstract: A tangram puzzle is a type of dissection puzzle consisting of seven polygon pieces. To "solve" a tangram puzzle, you follow an outline given to you and attempt to recreate it with the pieces you are given. Our Mathematica program allows a user to solve such puzzles, in two distinct ways: one, using the standard set of tangram pieces (two small triangles, one medium triangle, two large triangles, one square, and one parallelogram) to solve preset puzzle designs; two, using our Create-Your-Own interface to first draw an outline, then allowing our program to split the user's created puzzle into 7 randomized shapes. To ensure that the user's Create-Your-Own puzzle would split in a viable manner, we used a combination of planar and linear intersections to determine at what point and on what slope a line would "cut" their original piece into seven smaller pieces.

Time: 3:00-3:25

Presenters: Starr May and Amariah Street

Abstract: Sampling is one of the most useful tools in electronic music production, but it is limited as a digital instrument because of the short length associated with samples that are re-pitched higher. Using Mathematica and the Fast Fourier Transform, we created a software synthesizer that ameliorates this issue. The synthesizer operates in two main modes, as a sampler synthesizer and as an eight-oscillator sample approximator. The program uses the Fast Fourier Transform to find the eight most influential sinusoids within a given sample and recreates the wave in a way that can be interpreted by the synthesizer and manipulated in more useful ways than regular sampling allows for.

Time: 3:30-3:55

Presenters: Ferhan M. Atici andS. Chang

Abstract: In this talk, we present an introduction to fractional h-discrete calculus. The fractional h-difference and the fractional h-sum operators are given in the sense of the established Riemann-Liouville definition. We continue with introducing linear fractional order h-difference equations, where the order of the equation is any non-integer positive real number. We obtain the general solution of the fractional order equation by means of Mittag-Leffler type functions. This talk includes several illustrations generated using Mathematica-12.

Time: 4:00-4:25

Presenters: Sachi Barnaby and Lola Obielodan

Abstract: In this project, we created a random music generator that writes sheet music following the rules of musical composition. This music generator helps musicians practice sight-reading. Sight-reading is a skill where musicians play unfamiliar music as accurately as they can on the first try. By being a better sight-reader, musicians are able to learn their music more efficiently. The first step was to represent melodies as a sequence of notes in Mathematica, where each note has a pitch and a length. Algorithms were developed to generate a random sequence of pitch and length values which were then modified to fit counterpoint rules of music. User-selected values for difficulty guided the generated random sequences, and a third-party software was used to write the sheet music for these melodies. The program operates as an engaging tool for musicians to improve their sight-reading skills.