Niches for Publishable Undergraduate Research - Comments

[Dr. Courtney]

Dr. Courtney submitted a new PF Insights post

Niches for Publishable Undergraduate Research

Continue reading the Original PF Insights Post.

 

[dkotschessaa]

Fantastic idea for an article. Would love to see if anybody has similar ideas for mathematics research, (although I would have liked to see it about 4 years ago when I was trying to do undergrad research). I have some ideas but little time to elaborate them now.

 

[houlahound]

Good idea when it would be scheduled might be practical problem IMO.

 

[Dr. Courtney]

Fantastic idea for an article. Would love to see if anybody has similar ideas for mathematics research, (although I would have liked to see it about 4 years ago when I was trying to do undergrad research). I have some ideas but little time to elaborate them now.

When I worked at the Air Force Academy, the Dept of Mathematical Sciences took a broad view on what fulfilled their "senior research" course requirements. I mentored a couple of their senior projects - one was working toward a three dimensional (two inputs, one output) cumulative probability function for standard weight in fish that would allow computing a standard weight equation with fewer data points than existing methods; another was an applied method to better understand rocket motor performance. I'm fairly certain that the committee would have approved just about any applied project a senior wanted to do that was heavy in math, regardless of the application.

A lot of very applied math also pops up in the Math category at ISEF-affiliated state and regional science fairs. Projects that might otherwise compete in Animal Sciences, Environmental Science, Biomedical and Health Sciences, etc. often compete in Mathematics if the data analysis is sufficiently intricate. Last year they added a new "Computational Biology and Bioinformatics" category to provide a place for projects that often competed in Mathematics.

When we work with students in brainstorming student research projects, we usually touch on a variety of projects that one can group under the general category of "Numerical Experiments." The idea of these is that the student formulates a hypothesis on which computational or analysis approach might be better for some situation, with a clear definition of what better means in the context: faster, more accurate, simpler, etc. Then one conducts a numerical experiment to test the hypothesis. The student may create data using a spreadsheet with known good inputs plus added Gaussian noise, and then conduct the numerical experiments with different analysis techniques trying to recover the known good inputs.

There are also lots of questions in chaos and dynamics of complex systems that can be answered with numerical experiments that simply numerically integrate the system of ODEs describing the system. Computers are fast enough these days that the numerical integration can be done with Runge-Kutta methods, so the numerical experiments are a matter of exploring phase space to determine how much of it is chaotic and varying system parameters to observe the transition from order to chaos as the knobs are turned. As a physicist, I tend to find Hamiltonian systems the most interesting, but I'm sure real math professors can suggest a variety of interesting non-Hamiltonian systems.

 

[Drakkith]

Nice article! We just had Senior Design Day here at the U of A, where all the seniors present the projects they've been working on all year. I'm not sure how many of these would qualify as "Research", but there were definitely some interesting projects. I talked for a bit with one of the seniors whose project was to estimate the size of particles in a solution by looking at their Brownian motion. It appeared to be pretty successful.

 

[houlahound]

I have started tutoring a high school student on a gifted & talented program.

I am going to get him to investigate the design and performance of a yagi antenna.

Plenty of things to optimise but I can't think of anything novel about it.

Any ideas. Will also post on its own thread to get ideas.

 

[Dr. Courtney]

I have started tutoring a high school student on a gifted & talented program.

I am going to get him to investigate the design and performance of a yagi antenna.

Plenty of things to optimise but I can't think of anything novel about it.

Any ideas. Will also post on its own thread to get ideas.

My son often points out (sometimes with sarcasm) that the path to novelty in well researched areas is adding qualifiers. With over 28k google scholar hits on Yagi antenna, I suspect that may be the case here. The trick with achieving novelty through qualifiers is not going so far off into the weeds that the project still addresses something interesting to readers who actually care. My son's sarcasm in practice usually suggests that a given case has added qualifiers in a way that renders the idea uninteresting.

In the Yagi antenna case, the recent trends are to smaller wavelengths (microwaves and mm waves), antenna arrays, and printed antennas. Smaller wavelengths add both computational and experimental challenges as designs become more sensitive to dimensional variations and lack of precision in realizing a design in practice since some dimensions can no longer be considered small relative to the operational wavelength. An additional challenge is that as wavelengths decrease, the costs of accurate equipment for performing experiments tends to rise. The cost and complexity of modeling software also tends to increase.

One approach for a student new to the field of Yagi antennas (or RF in general) might be to test whether some existing products meet their claimed specifications rather than trying to jump in and improve existing designs. There are several 2.4 GHz Yagi antennas on the market for extended range Wi-Fi applications. If the right equipment is available, it shouldn't be too hard to design some simple experiments to determine whether or not a few models meet their specifications, and (if not) how far out of spec different models are. Pointing out problems in existing work is often lower hanging fruit for people entering a new field than coming up with new, better designs.