41st Annual WKU Mathematics Symposium Schedule and Abstracts

Presenters: Eric Xing* (GA), Dr. Gongbo Liang (F)

Abstract: The performance of neural networks (NNs) on imaging-related tasks has improved dramatically within the last decade. Exciting results have been reported on various tasks, including disease diagnosis and autonomous driving. However, such results are usually based on the overall performance of NNs, such as accuracy or F1 score for classification tasks, which do not provide insight into the prediction forming mechanism. Specifically, NNs usually produce a relatively stable performance on the same task across multiple training trials. However, due to the black-box nature of NNs, the learned feature spaces could differ significantly between training trials. We believe that the uncertainty analysis of the learned feature space is equally important, if not more important, than the analysis of overall model performance. Through this work, we propose to evaluate the learned feature space using feature-attribution explanation methods in combination with computational analysis and clustering analysis methods. We apply the method to three popular convolutional neural network (CNN) architectures and the Vision Transformer (ViT) architecture. We find that the learned feature spaces are easily separable between different training trials of the same architecture with the same hyperparameter training setting. We also find that pre-training and transfer learning can be used to mitigate the inconsistency between training trails. Based on this finding, we propose an Averaged Training (AT) schema. We demonstrate the AT schema on the ViT model and show the method has effectively reduced the gaps between the learned feature space.

Presenters: DJ Price* (I)

Abstract: In the practice of software engineering, the main objective of the job is not necessarily always to write code. In large software systems, there can easily be hundreds of thousands of lines of code with only a handful of engineers to look at and work on the code. A mindset should be developed for the readability and scalability of code that is written and this should often (though not always) be prioritized over code that is sleek or full of assumptions. These principles will be illustrated in a broader context of other programming languages but specific attention will be given to the Mathematica programming environment.

Presenters: Logan Stewart* (GA), Trey Crouch* (U), Matthew Poynter (U), Ahmet Kaan Aydin (G), Dr. Ozkan Ozer (F)

Abstract: A one-dimensional wave equation, describing vibrations on a clamped-free string, is considered. The corresponding partial differential equation (PDE) is known to be fully controllable by a boundary feedback controller (force) applied at the tip of a string. However, its well-known space-discretized approximations (a system of difference differential equations) by Finite Differences or Finite Elements are not fully controllable without proper filtering of the numerical scheme. It is simply due to the artificial high-frequency vibrational modes caused by the blind use of these approximations. To avoid the discrepancy, an indirect filtering technique is adopted to retain the controllability, mimicking the PDE-counterpart. Moreover, an alternate order-reduced numerical scheme, utilizing a clever use of Finite Differences without filtering, is also introduced for comparison. Approximate solutions are built to where all control parameters can be controlled, as well as being able to set different types of initial conditions, such as sinusoidal, box-type, and sawtooth wave, with low or high frequencies. All parameters can be manipulated via a Mathematica program (called a Wolfram Demonstration). We were able to find that the results from the numerical schemes happened to match what was happening in the real world. All three approximation techniques are compared side-by-side in terms their computational costs during the presentation. This research is funded by KY NSF EPSCoR grant #3200002692-22-08.

Presenter: Nikhil Akula* (GA), Anish Penmecha* (GA)

Abstract: In this talk, we present a demo of our UNO card game. In this project, we created a single player mode, where the user plays against the computer and a multiplayer mode where the user can locally play UNO with other people. The UNO card game will allow human players along with a culmination of AI players. The UNO card game can be played by a minimum of 2 players and a maximum of 10 players. The user indicates how many players will be playing and our program deals out that many hands. We are going to show:

1. The heuristic programming of our UNO solver, where we showcase the computer playing against a human player. This talk includes several graphics generated using Mathematica.
2. We also show the UNO card generator, which can create any UNO card using only Mathematica graphics primitives.
   

Presenter: Sahil Chhabra* (GA), Dr. Huanjing Wang* (F)

Abstract: This simulation shows the motion of a ball on a curved ramp and the ball’s motion in the air after leaving the ramp. The shape of the ramp is controlled by the user. For the sake of simplicity, the only force on the ball that is taken into account is the force of gravity. Air resistance and friction are neglected. The total horizontal and vertical distance that the ball travels and the ball’s maximum height are calculated. The user can enter answers for these values and compare them to the actual distances traveled and height reached by the ball. To aid in the learning process, step-by-step solutions that explain how to determine each value can be displayed. This model can help its users visualize and conceptualize physics principles, namely kinematics and energy, at work in the real world using a ball on a ramp as an example.

Presenter: Matthew Poynter* (U), Logan Stewart (GA), Trey Crouch (U), Ahmet Kaan Aydin (G), Dr. Ozkan Ozer (F)

Abstract: We consider the problem of simulating vibrations in a perfectly bonded three-layer beam, consisting of stiff outer layers and a viscoelastic core layer. The core layer is allowed to have a shear. The vibrational interactions between the shear of the middle layer and the overall bending motion is governed by a system of Partial Differential equations (PDEs). The system is known to be uniformly observable by a suitable sensor design at the tip of the beam. However, to obtain numerical results for any particular system, the system of PDEs must first be discretized by known methods, such as Finite Differences, and these discretizations fail to preserve the desirable observability property, and typically fail to match physical models. To remedy this, we consider modifying the blind use of the Finite Difference method, by the addition of a numerical filtering (viscosity) term to the discretization. To demonstrate this technique, we built a Mathematica program that allows a user to manipulate initial conditions and relevant constants, including material properties, control parameters, and a choice of a variety of initial conditions, single box and pinch-type discontinuities, as well as sinusoidal, sawtooth- type, square-type, and triangle-type waveforms, with variable frequencies. The demonstrations and codes being used in the presentation will be submitted for publication at the Wolfram’ Demonstration Project website.
This project is sponsored via KY NSF EPSCoR grant #3200002692-22-08.

Presenter: Kaaustaaub Shankar* (GA), Matthew Pimienta* (GA)

Abstract: Using the surface of a genus two structure and topology, we were able to create unique version of minesweeper played on an octagon tiled with 45-67.5-67.5 isosceles triangles. Previous solving approaches based on normal Minesweeper treated the game as a constraint satisfaction problem by representing the game as a linear system of equations. In this version, the approach is modified to work on this new, seemingly more complex version of Minesweeper. After running some tests with the solver, we suspect that this version is more solvable than normal Minesweeper in that less guesses are needed. 

Presenter: Justin H. Mills* (G), Dr. Uta Ziegler (F)

Abstract: Artificial Neural Networks (ANN) have many use cases with both commercial and noncommercial applications so any increase in their efficiency would be of a major benefit. One way to gain this benefit is to optimize the topology of the ANN, which involves reducing the size while retaining acceptable levels of performance. This presentation will explain basic ANN concepts along with two methods of selecting promising starting topology, one method was published in 1994 and the other in 2020. If time permits the data sets and experiments which will be used to evaluate the methods will be covered. This presentation describes a work in progress.

Presenter: Hank Helmers* (GA), Dr. Huanjing Wang (F)

Abstract: With the popularization of digital marketing and services such as Amazon, online shopping has become an almost daily occurrence for many. In our research, we are exploring data collected during customer’s online-shopping sessions, to better predict customer’s purchases, and understand the data produced during online-shopping. The dataset we used contains 18 features, some of which include, Page Value, Exit Rate, Bounce Rate, etc. These features were collected during real customers’ online-shopping purchases and because of this, it may provide redundant information or have an adverse effect on the prediction model. Feature selection is a technique used to reduce a feature subset to an optimal size. Thus, using feature selection techniques would allow us to make an intelligent selection of the most influential features for our prediction, prior to building classification models, which may improve the result of whether a customer’s time on the website will end in a purchase. We utilized the Waikato Environment for Knowledge Analysis (WEKA) data mining tool in both the feature selection and classification algorithms to explore our data. The feature selection methods we tested include Information Gain, Pearson’s Correlation, Gain Ratio, ReliefF, and OneR Evaluations. In addition, three classifiers (Multilayer Perceptron, K-Nearest Neighbor, and Decision Tree) were tested and used to build classification models with the selected attributes to predict customer’s purchase intent. Results demonstrate that the Information Gain feature selection algorithm performed best and the ReliefF algorithm performed the worst. In addition, the model built with the Multilayer Perceptron classifier performed best. This leads us to recommend the use of Information Gain algorithm to select feature subsets and the Multilayer Perceptron classifier for building prediction models due to its time efficiency and accuracy with the selected features.

Presenter: Allen Lin* (GA), Dr. Sarah Tamnen (F)

Abstract: Circular symmetrization, a procedure first introduced by Pólya and Szegö, has applications to isoperimetric problems in various settings because it is known to reduce perimeter of a region. The original proof that this symmetrization procedure reduces perimeter uses calculus of variations. We present an elementary proof (without using calculus of variations) that circular symmetrization reduces the diameter of a shape in the Cartesian plane. We also present the images of certain shapes after circular symmetrization. This research was conducted at the Research Science Institute (RSI) during the summer of 2021.

Presenters: Dr. Jose Henrique Rodrigues* (P), Madhumita Roy (G)

Abstract: In this talk we shall consider the following semi-linear wave model:

Similar models with simpler nonlinear boundary terms have been already studied broadly whereas the generosity of our model is not only the presence of nonlinear damping but also the nonlinear boundary source f₁, which makes the problem heavily nonlinear and even more realistic. It is first shown that, the model is Hadamard well-posed, which allow us to establish an evolution operator in the phase space. In addition, we shall provide an answer regarding the long-term dynamics of the corresponding trajectories namely the compact global attractor.

Presenters: Dr. Lan Nguyen* (F)

Abstract: Given a real-valued exponential function 𝑓(𝑡) = exp(𝑎𝑡), then we think of its characteristics such as 𝑓′(𝑡) = 𝑎𝑓(𝑡), 𝑓(𝑡 + 𝑠) = 𝑓(𝑡)𝑓(𝑠) and the Taylor expansion of 𝑓(𝑡). Using these characteristics, we generalize the concept of exponential functions, when their range is in other spaces like complex plane, n-dimensional, or even infinitely dimensional spaces. That leads us to introduce the semigroup of linear operators. Applications to partial differential equations are also discussed.

Presenters: Ahmet Kaan Adyin* (G), Dr. Ozkan Ozer (F)

Abstract: The set of partial differential equations describing vibrations on a three-layer Mead-Marcus beam model, consisting of piezoelectric or elastic outer layers constraining a compliant viscoelastic layer, is considered. This model fully describes uniform transverse vibrations (bending) of the perfectly bonded beam as well as the shear due to the compliant layer. The eigenvalues, and in particular, the uniform gap among the eigenvalues of the model with hinged boundary conditions is proved to be written in terms of the ones of the single-layer standard Euler-Bernoulli beam model. Therefore, this comparison allows that the three-layer beam model can be considered as a perturbation of the single-layer beam model. A variation of Ingham’s inequality known as Haraux’s inequality is utilized to show the uniform observability result of the model with a single boundary sensor, which is, the sensor data allows to fully describe the motion on the beam. Next, space- discretized Finite-Difference approximations of the model are considered to mimic this behavior, yet it is shown that the approximated model is not able to retain the uniform observability due to high-frequency spurious eigenvalues generated by these approximations. To obtain a uniform observability result with the same sensor design, the spurious eigenvalues of the approximated model are filtered by the so-called Direct Fourier filtering method. After filtering, the approximated solution space uniformly converges to the whole infinite-dimensional solution space as the mesh parameter goes to zero. This research is funded by KY NSF EPSCoR grant #3200002692-22-08.

Presenter: Dr. Mikhail Khnner* (F), Lars Hebenstiel (U)

Abstract: Monolayer graphene is a 2D honeycomb lattice of carbon atoms. Bilayer graphene with a top layer rotated with respect to a bottom layer, or vice versa, typically forms a periodically corrugated surface called a Moiré superlattice. This surface presents a complex potential energy landscape for diffusion of deposited atoms or molecules, which can be exploited to assemble nanoclusters with well-defined positions and sizes. We constructed Moiré superlattices and corresponding potentials in Mathematica for various values of a twist angle and strain, and followed with a formulation of a nonlinear, fourth-order diffusion PDE for atoms deposited on a Moiré. Next, we determined a quasi-2D steady-state nanocluster distributions by analytically solving a set of ODE BVPs in one spatial dimension. Finally, we used Mathematica’s built-in solvers to compute solutions of a full, 2D nonlinear diffusion problem and determined the kinetics of approaching a steady-state distributions.

Presenter: Ahmet Kaan Adyin* (G), Dr. Ozkan Ozer (F)

Abstract: A perfectly bonded clamped-free three-layer beam, consisting of piezoelectric or elastic outer layers constraining a compliant viscoelastic layer, is considered. This model fully describes uniform transverse vibrations (bending) as well as the shear due to the compliant layer. On the contrary to the PDE model with hinged boundary conditions in the other talk; the eigenvalue-based Fourier techniques are not appropriate to spectrally analyze the PDE model with clamped-free boundary conditions. Therefore, the technique so-called the “multipliers" is utilized to show the uniform observability (sensor design) of the PDE model with a single boundary sensor, which is, the sensor data allows to fully describe the motion on the beam. For the approximations of the PDE, different from the classical use of Finite-Differences, an equivalent first order formulation is considered. This leads to an order-reduced Finite Difference approximations by the implementation of average operators using auxiliary middle nodes in the uniform meshing. By this novel approximation method, a new variation of the boundary equations is obtained. The energy of the approximated model is defined appropriately, and shown to be conservative mimicking the PDE model. Open problems will be discussed. This research is funded by KY NSF EPSCoR grant #3200002692-22-08.

Presenter: Dr. Mark Robinson* (F)

Abstract: In this presentation, several useful techniques of numerical integration are considered. These include two methods typically introduced in elementary calculus classes: the trapezoidal rule and Simpson's rule, for approximating the definite integral of a continuous function over a closed and bounded interval. An additional method considered is Gaussian quadrature, a highly accurate method for approximating definite integrals for which an explicit formula for the integrand is available. Gaussian quadrature involves computing a linear combination of function values selected in such a way as to maximize the degree of precision of the approximation formula. This method is examined both in the single-variable case and also for approximating multiple integrals.

Presenter: Dr. Jose Henrique Rodrigues* (P)

Abstract: In this talk we are going to consider a system of interactions in a 3D acoustic medium (bounded domain) and its structural wall (boundary). The dynamics in the acoustic medium are given by a linear wave equation with locally distributed boundary dissipation and coupling term, while the interaction happening on the structural wall (portion of the boundary) are given by a 2D Kirchhoff-Boussinesq plate equation, subject to linear dissipation. Our main goal is to establish the existence of global attractors for the corresponding acoustic- structural system.

Presenter: Alex Driehaus* (U), Dr. Ozkan Ozer (F)

Abstract: The partial differential equations (PDE) model governing the energy in an oscillating piezoelectric smart beam will be discussed, with an emphasis on closed-loop stabilization of the beam through the sensor placed at the tip of beam, reading out the tip velocity and the total current. Piezoelectric materials produce electrical energy when a mechanical force is applied. This property makes piezoelectric materials desirable for multiple purposes, including the use of quartz crystals in the production of some nano materials, and some electronic devices. Consider a beam of such a material. This beam clamped at one end is free at the other to oscillate longitudinally. Transverse oscillations are not of interest in this case, so the focus will be on longitudinal vibrations and the movement of charges within the beam itself. Due to the uniform observability property of the beam, proved in [Wilson, Ozer, GAPA, 2021], readings of tip velocity and the total current after a proper exposure time are sufficient to reconstruct the behavior of the entire beam. So, a controller (an actuator) at the free end of the beam can be designed by the so-called filtered Finite Difference method to generate desired stability behavior in the beam as a whole. The process of determining the optimal feedback gains and damping rate will be discussed, through the use of oscillations in a string. The practices used to model this situation will then be utilized for the discretized partial differential equation model of a piezoelectric beam. This research is funded by KY NSF EPSCoR grant #3200002692-22-08.

Presenter: Md Fayaz Ahamed* (G), Dr. Thomas Hagen (F)

Abstract: Blown film extrusion is the most common process to produce plastic films. Mathematical models for film blowing are given by nonlinear, coupled boundary value problems involving free parameters that have to be found as a part of the solution. The problem can be formulated with the liquid bubble treated within a quasi- cylindrical setting or via a thin-shell approximation. In the former formulation, stationary solutions can be found in near-explicit form. The latter formulation leads to a nonlinear system of first-order ODEs with boundary conditions that can only be satisfied for appropriate choices of the model parameters. In how far such choices can be made successfully is the central theme of this work.

Presenter: Rasika Mahawattege* (G), Dr. James Nutaro (I)

Abstract: Computer simulation of a system described by differential equations requires that some elements of the system be approximated by discrete quantities. There are two system aspects that can be made discrete; time and state. When time is discrete, the differential equation is approximated by a difference equation and solution is calculated at fixed points in time. When the state is discrete, the differential equation is approximated by a discrete event system. Here we introduce a novel second order state discretization method for the first order linear differential equations. This research was supported in part by an appointment with the National Science Foundation (NSF) Mathematical Sciences Graduate Internship (MSGI) Program sponsored by the NSF Division of Mathematical Sciences. This program is administered by the Oak Ridge Institute for Science and Education (ORISE) through an interagency agreement between the U.S. Department of Energy (DOE) and NSF. ORISE is managed for DOE by ORAU. All opinions expressed in this paper are the author's and do not necessarily reflect the policies and views of NSF, ORAU/ORISE, or DOE.