Prof. Edward Aboufadel's Applied Mathematics Projects
This page provides an overview of applied mathematics projects, many that I have done with students, during my career. A significant portion of this work has been done at the Mathematics REU at Grand Valley State University. Starting in 2014, my projects are part of the Center for Applied Mathematics Projects (CAMP) at GVSU. Current areas of focus for CAMP are 3D printing, data analytics and visualization, wavelets, and applied mathematics competitions. Watch this page for updates.
I enjoy working on interesting applications of mathematics to various areas of science, medicine, and engineering. Many of the projects I have pursued utilize wavelets. As I learned about wavelets in the 1990's, I came across how the FBI uses wavelets, and an application of wavelets to denoising musical recordings. Since 2000, I have been working with students on other applications of wavelets. Wavelet-based filters are applied for basically two reasons: information rearrangement and information reduction. In some of these projects (particularly years 2000-2011), the use of wavelets is a key step in the method.
This is a list of the projects and the people involved, along with links to publications or videos, when available:
2017: Imitating the Shazam App With Wavelets. This project applies wavelets to analyze data from continuous glucose monitors (used by type-1 diabetics), incorporating a simplified version of the method behind the Shazam app. Errata. (No collaborators.)
2013-17: 3D Printing. Student Researchers: Melissa Sherman-Bennett and Sylvanna Krawczyk (2013), Lindsay Czap (2014), Samantha Law (2016-17), Zachary Ash (2016-17), along with further work done in 2014-17 by EFA, including a talk recorded at the 2016 SIAM Conference on Applied Mathematics Education.
2015: Visualizing Water Pollution Data Using Beck-Style Flow Path Maps. This entry (found at the bottom of the linked web page) was a Runner Up in the Visualizing Nutrients Challenge. Collaborator: Daniel P. Huffman.
2014: NASA-Nex Challenge. This entry was a winner of the "Ideation" stage of the Challenge. (No collaborators.)
2014: Ordinary Generating Functions of Context-Free Grammars. Student Researcher: Tanner Swett
2013: Fighting Fires in Siberia. Student Researcher: Beth Bjorkman. This entry won a 2nd place prize in the 2012-13 "Competitive Mathematical Game" sponsored by the la Fédération Française des Jeux Mathématiques.
2012: A Scoring System for Continuous Glucose Monitor Data. (No collaborators.)
2011: Identifying Potholes Using Smartphone Data and Wavelets. Student Researchers: Nathan Marculis and SaraJane Parsons (with assistance from Clark Bowman). This entry was one of the winners of the Innocentive Boston Pothole Challenge.
2009: Classification of Diabetic Patients Using Wavelets. Student Researchers: Robert Castellano and Derek Olsen. We also have their award-winning MathFest talk available.
2008: Creating Chuck Close-like Portraits Using Wavelets. Student Researchers: Clara Madsen and Sarah Boyenger Kaindl.
2004: Identifying Airplanes in Aerial Photographs Using Wavelets. Student Researchers: Kevin Brink and Drew Colthorp.
1997: Qualitative Analysis of a Singularly-Perturbed System of Differential Equations Related to the van der Pol Equations. (No collaborators, but thank you to my thesis advisor, Jane Smiley Cronin Scanlon.)
1996: Research in Differential Games. Student Researcher: David Szurley
1994: Applications Über Alles: Mathematics for the Liberal Arts. (No collaborators.)
About Wavelets ...
Here is some more information about wavelets: Wavelet analysis is a modification of Fourier analysis, where functions other than sine and cosine are used as the basis functions. In Fourier analysis, the goal is to decompose a function by thinking of it as a combination of trigonometric functions with different frequencies. This approach is very useful when dealing with periodic functions. However, if we wish to analyze non-periodic functions, then the focus on frequencies can lead to poor results. An example can be found in Figure 1, where the characteristic function on the time interval [0, p] is approximated by a sum of sines and cosines. Wavelet analysis is designed to better handle non-periodic functions, because of the focus on the time intervals where functions are defined.
Figure 1: An Example from Fourier analysis.
Figure 2 is a graph of the scaling function known as D4. Scaling functions play a role similar to the sine and cosine functions of Fourier analysis in that other wavelets in a wavelet family are generated from the scaling function. The scaling function is also called the father wavelet.
Figure 2: The D4 Scaling Function.
Some of this material is based upon work supported by the National Science Foundation under Grants No. DMS-0451254, DMS-0137264, DMS-9820221, DMS-1003993, and DMS-1262342. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).