Education

Abstract

Long ago, I forgave Arthur Conan Doyle for making the greatest literary detective of all time a chemist and his arch villain a mathematician, but there was a logic behind it. After all, Sherlock Holmes was a man of action and intellect. He collected data before he interpreted it. His nemesis, Moriarty, is characterized as an organizer, acting by proxy and letting others be his eyes and ears. Doyle's fiction was a reflection of his time, but times have changed. There was a time when gathering data often required special skills, equipment, and a great deal of time. Now, inexpensive cameras and sensors are ubiquitous, as is the software to extract and manipulate data, and our two Education section contributions this issue exemplify this component of modern mathematical inquiry. The first is a piece about Bayesian inversion, a category of problems where a well-established model is available, but some internal parameters are unknown. The mathematical challenge is to map a noisy data set to the probability distributions of the parameters so that they best explain the data. The second contribution surveys models for describing angles of repose and free surfaces of granular materials. In both cases, the authors demonstrate the value of their ideas by comparing their mathematical results with simple, inexpensive, self-contained experiments. In “Estimating Parameters in Physical Models through Bayesian Inversion: A Complete Example” authors Moritz Allmaras, Wolfgang Bangerth, Jean Marie Linhart, Javier Polanco, Fang Wang, Kainan Wang, Jennifer Webster, and Sarah Zedler put forth a true team effort and walk the reader through an example of Bayesian inversion from start to finish. The probabilistic model is relatively simple, but as the title states, we face an inverse problem in trying to determine the parameters. The goal is to determine a probability density of the parameters given a set of measurements with known or assumed probability densities. In this case, the authors walk us through an example using data extracted from video footage of a falling object to determine the gravitational acceleration and the object's viscous drag coefficient. The inversion for a small number of parameters can be tackled directly with numerical integration, but as the authors note, larger numbers of parameters curse us with a greater number of dimensions, a topic beyond the scope of this introductory communication. Like any good detective story, things are not as they seem. The authors find that either their assumptions are a bit off or gravity is weaker in College Station than in other towns in Texas (or anywhere else in the world). To resolve the discrepancy, the authors propose an additional measurement error that produces a more believable set of parameter distributions, highlighting one of the main points of the article: No amount of data will help unless we carefully choose where and how we take measurements. In the second Education section offering, authors Rachael Gordon-Wright and Pierre A. Gremaud explore free surfaces of granular materials. Unlike motions of simple gases and liquids which in most circumstances can be explained adequately by the Navier--Stokes equations, continuum models of granular materials pose a greater challenge. As the authors suggest, a mountain of potatoes may have a different shape than a pile of sugar grains. In their article “Granular Free Surfaces,” they limit the discussion to steady free surfaces, a rich topic because unlike a fluid, granular materials can sustain a shear stress while at rest and so there are many possible free surfaces. For these problems, the relevant mathematical structure is not a differential equation but rather a differential inclusion. For linear slopes, a direct calculation will reveal the angle of repose if one knows a few internal parameters. Next, the authors put a “spin” on the problem by exploring free surfaces in rotating cylinders. Finally, they compare their mathematical results with experimental observation and in this case find qualitative agreement. While we feature two very different contributions in this issue, they share some important features beyond the use of home-brew data collection. Both articles would fit well with an upper division undergraduate or introductory graduate modeling course. Both hint at the challenges posed to mathematicians when theoretical predictions provide qualitative but not quantitative agreement. Finally, like all great detective stories, both articles describe a great problem, round up the usual theoretical suspects, and bring us to a satisfying resolution.