University Professor Uses Maple to Help Students Understand the Mathematics of Business - Maplesoft

User Case Study:
Northwestern University Professor Relies on Maple as a Valuable Research and Teaching Tool

Danny Abrams, researcher and professor at Northwestern University, wanted a research tool to analyze complex concepts, and improve learning of similar concepts for his students in the classroom.

Abrams was introduced to Maple as an undergraduate at the California Institute of Technology and quickly began to incorporate it into his work. He now uses it in most of his research projects, and to improve learning for his students.

Maple has supported Abrams’ research on a variety of topics and applications, allowing him to quickly complete calculations and study complex ideas. His students have also shown great improvement since he introduced Maple into his advanced differential equations class. Feedback has been positive.

No matter the type of work, having the right tools for the job is essential for optimizing efficiency and maintaining accuracy. When it comes to conducting research or teaching complex ideas, having the right tool is especially critical to create an optimal environment. For STEM subjects, due to the complex nature of the material, customized and powerful tools are required. For Professor Daniel M. Abrams at Northwestern University in Evanston, IL, Maple is one such tool to analyze complex concepts in his research and to improve learning for his students.

Abrams was introduced to Maple as an undergraduate at the California Institute of Technology. He worked with it in an applied physics class and liked it so much he has used it since, first as a graduate student and then as a researcher and teacher. Maple allows the user to move away from the mechanics of doing calculations and focus on the bigger mathematical picture, he said. “The simple answer as to why I like Maple is that it allows me to get calculations done quickly and easily,” he said. “I also like Maple’s animation features, specifically the implicit plots. Maple is far better than other tools on the market when it comes to handling animation.”

Abrams relies heavily on Maple in his research, which spans a variety of areas, including Coupled Oscillators, Mathematical Geoscience, and the Physics of Social Systems.


Coupled Oscillators
An oscillator is any entity that changes in a periodically repeating way, moving regularly between extreme values. Abrams’ research in this area looks at how oscillators can behave in surprising ways when connected in groups. One investigation examined the sway of London’s Millennial Bridge, which opened in 2000. The bridge had a slight sway, and as walking pedestrians began moving in lockstep to compensate for it, these natural vibrations were enhanced. This caused an even more considerable sway, resulting in a nearly two-year closure so engineers could find a solution to the problem.

Abrams’ research focused on applying ideas developed to describe the collective synchronization of biological oscillators, such as neurons and fireflies, to the bridge movements. He used Maple to model the phenomenon and demonstrate why it occurs. He modeled the bridge as a weakly damped and driven harmonic oscillator, and showed that, if people make slight adjustments to the way they walk just to stay upright, compensating for the natural bridge movements, that can explain both the bridge’s wobbling and the crowd synchronization dynamic that was observed. The model predicts both the synchronization timescale and the characteristic amplitude of the wobbles shown in the actual experiments. The research provided a more complete picture of what occurred on the Millennium Bridge, both in terms of the bridge dynamics and the crowd vibrations. The team’s findings can potentially be used to estimate the damping required to safeguard other bridges in the future.


Mathematical Geoscience
Abrams’ research in the area of Mathematical Geoscience uses simple mathematical models in Maple to provide insights into the behavior of complex geophysical systems. The focus is on environmental phenomena, including springs that carve out river-like channel networks, the shape of growing valleys, historical climate oscillations and statistical approaches to understanding topography. Abrams and his research colleagues use Maple to model theories surrounding occurrences such as seepage erosion and valley growth to examine how groundwater and springs impact soil mass and topographical features, with comparisons between similar features on Earth and Mars.

One such project examined the geometry of valley growth using field observations, laboratory experiments, analysis of a high-resolution topographic map, and mathematical theory to examine various phenomena. When groundwater emerges from a spring with sufficient intensity to remove sediment, it carves a valley into the landscape in a process known as “seepage erosion”, which causes the spring to migrate and results in a valley head with a characteristic rounded form. The team selected Liberty County, Florida, for their research, an area where networks of these seepage-driven channels are plentiful.

Abrams and his team conducted a three-dimensional ground-penetrating radar study to survey the water table in a segment of the channel network. The team examined the impact of various elements (rainfall, the channel network and subsurface heterogeneities) on the height of the water table, using Maple to test various components, such as the headward erosion rate of a channel tip and its proportion to the groundwater flux to the tip. Their findings suggest the distance to the nearest channel was the primary determinant of the water table’s shape. Using a conceptual model developed by Thomas Dunne, Abrams and his team used Maple to analyze the Florida channel network’s present and past development and confirm an explicit link between the dynamics of a channel network and its static structure. The team expects their findings will be useful for understanding mechanisms that produce such distinctive valley head shapes, and allow researchers to reconstruct past growth networks.

With the support of Maple, Abrams has published dozens of academic research papers, on a variety of topics. Other interesting research projects using Maple include understanding the connections between pattern formation and dimension/topology of a system or network of systems; using a mathematical model to explain why the USA has only two dominant political parties, while other countries with similar systems have more; and analyzing the optimal distribution of advertising investments in an efficient marketplace.


Maple in the Classroom
When it comes to his teaching, Abrams prefers Maple to competing tools, like Mathematica®, due to its user-friendly interface, better syntax and ease-of-use for beginners “Students are quite impressed with the Maple demonstrations I use in class,” he said. “My graduate students all have to learn Maple and they’re usually pretty quick to become proficient in it.”

In the past, Abrams taught an advanced differential equations class with multiple sections and used the opportunity to assess teaching methods. One section of the class was taught using traditional methods, such as blackboard diagrams and lectures, and few demonstrations. The other section was given demonstrations using Maple, which were then posted online for students to access and practice with outside of class. “The section of the class using Maple showed great improvement, with the students performing better on exactly the same exams,” he said. “Students appreciate using worksheets where they can engage with equations and solutions, and manipulate them directly.”

Whether using it for teaching, or to support his own analysis and investigative projects, Abrams uses Maple in almost all his work and has found it to be a valuable tool. “Just as when I was an undergrad, I find that it allows me to avoid the distractions of mechanical calculations, while forcing me to focus on the mathematical ‘big picture’."

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