During the late 1800’s, James Clerk Maxwell and Ludwig Boltzmann developed a differential equation that predicted how gaseous material distributes itself in space and how it reacts to changes in things like temperature, pressure or velocity. The only problem: for 140 years, solutions to the equation could only be found for gases that were perfect equilibrium. Enter Philip Gressman and Robert Strain of the University of Pennsylvania. Using recently developed mathematical techniques, they were able to describe solutions to the equation under any conditions. As this article in the Time of India notes, now it’s possible to “describe the location of gas molecules probabilistically and predict the likelihood that a molecule will reside at any particular location and have a particular momentum at any given time in the future.”

# Category: mathematical modeling

## The Ninth Annual Year in Ideas

The NY Times Magazine annually publishes its The Year in Ideas issue, devoted entirely to “the most clever, important, silly and just plain weird innovations … from all corners of the thinking world.” A surprising number of these ideas are based on a study or research article or something similar that employs some bit of mathematical and/or statistical analysis. The ones I’ve listed below are chosen as being the ones that most prominently feature mathematics ideas, or feature mathematics and/or mathematicians centrally. Listed alphabetically:

- Black Quarterbacks Are Underpaid by Jason Zengerle describes the statistical analysis of two economists, David J. Berri and Rob Simmons, who discovered that in the NFL black quarterbacks are typically paid much less than white quarterbacks. Their analysis goes farther, however, and notes that the apparent cause is not necessarily racism. Instead, the NFL quarterback rating statistic is the culprit. NFL contracts are often based on hitting certain statistical levels, and for quarterbacks the statistic used is often the QB rating. Since QB rating fails to count rushing yards at all–something that black quarterbacks typically excel at–black quarterbacks are typically ‘discounted’, QB rating-wise.
- Forensic Polling Analysis visits a topic seen already here in this blog: the suspicious polling numbers of the polling firm Strategic Vision LLC. You can visit that entry or the Times article for more info.
- In a blow to meritocracy-lovers everywhere, another entry notes that Random Promotions, rather than merit-based ones, can actually produce better businesses (and typically do, at least in simulations). The article by Clive Thompson describes a study done by a trio of Italian scientists in which the researchers created a virtual 160-person company and then tried out various different promotion schemes within the company, with the aim of seeing which scheme improved the company’s productivity the most. Promoting on merit turned out to be a lousy idea (at least for the company as a whole) while promoting at random turned out to be the top strategy. In the middle was the curious idea of alternately promoting the best and then the worst employees. The fact that the mixed best/worst strategy outperformed the merit strategy is yet another example of Parrando’s Paradox, a phenomenon first identified by game theory.
- Massively Collaborative Mathematics features the first mathematical theorem proved by a ‘collective mind’, if you will. In January 2009, Timothy Gowers, one of the top mathematicians in the field, proposed on his blog that the mathematical community, as a whole–or at least that portion that knew and read his blog–attack a long-standing unsolved problem in mathematics known as the Density Hales-Jewett Theorem. Contributors ranged from eminent mathematicians to high school teachers, and hundreds of thousands of words worth of ideas were eventually proposed, developed, discarded, combined, and so forth. Gowers had initially set the bar low, hoping this ‘Polymath’ project would result in “anything that could count as genuine progress toward an understanding of the problem.” Instead, six weeks later the problem was completely solved. A paper detailing the result, authored by D.H.J. Polymath, has been submitted to a leading journal.
- Finally, the (alphabetically) last idea listed, “Zombie-Attack Science,” features a story that appeared on this blog previously. See that entry, or the Times article, of course, for details.

## Darwin Plays Game Theory—and Wins

Game theory is a branch of mathematics that explores how people (or entities made up of people, like organizations) make decisions. This article from Discover magazine shows one of the first instances of game theory being applied to animal behavior. The animals in question are ravens, and in 2002 Sasha Dall, a mathematical ecologist at the University of Exeter in England, used game theory to explain why young ravens scout for carrion by themselves but then recruit other birds to join the feast.

Even more impressively, Dall’s model predicted that ravens would likely employ another strategy, one that had never been observed in ravens: gang foraging, where a large group of birds scavenge together. The article describes what happened when scientists looked to see if Dall was correct:

Behavioral ecologist Jonathan Wright of the Norwegian University of Science and Technology discovered this very behavior in the field. He tracked ravens in North Wales by implanting carcasses with different-colored beads that the birds ingested and later coughed up. Analysis of the beads indicated that ravens in some roosts were searching, eating, and benefiting together, just as Dall anticipated.

## Scientists set sights on invisibility cloaks

A couple of articles, at the online sites for CNN and FoxNews, feature the work of Graeme Milton, a mathematics professor at the University of Utah. People have long imagined materials (like Harry Potter’s invisibility cloak) that would let light ‘pass through’ or ‘bend around’ any object that was cloaked, thus rendering the object invisible. In fact, that dream seems a long way off. However, Milton’s work shows that an alternative idea–cancelling out light waves coming towards an object, much like noise cancellation devices filter out noise–is entirely possible.

Milton’s results are purely mathematical at this point; as he says, “We just do the math and hope other people do the experiments.” But the research represents a new approach to cloaking, one that has wider applicability than to just light waves. “Results from the study demonstrate that it is conceivable to build cloaking devices that generate waves to create a quiet zone to protect oil rigs against incoming tsunami waves, or to create vibrations to neutralize incoming seismic waves from an earthquake.”

## Math Mimics Hard-to-Heal Wounds

This brief article from US News & World Report describes a new mathematical model of ischemic wounds. Ischemic wounds are wounds that do not get as much blood flow as normal wounds, and they affect six and a half million Americans each year. The model, developed by Avner Friedman of Ohio State, includes factors that mimic the actions of healing agents like white blood cells, capillary sprouts, blood-vessel-forming proteins and oxygen concentrations, and is the first to accurately predict healing times for these kind of wounds. The hope is that models like this are “the start of something that could give valuable insight to the wound healing problem in the future.”

## Plugging Holes in the Science of Forensics

This NY Times article discusses the recent National Academy of Sciences (NAS) report that found “serious problems” with many of the forensic ‘sciences’ practiced in crime labs today. Most of these problems are of the mathematical variety: with the exception of DNA evidence, much of the type of research done on the forensic sciences is geared towards finding information at the crime scene, and not geared towards statistically checking how valid and reliable these finds really are in identifying the perpetrator. A foretaste of this came a few years back, when the forensic science of bullet lead analysis (first used to convict Lee Harvey Oswald of the murder of John F. Kennedy) was abandoned by the FBI after an NAS report found it lacking as well.

The article goes on to describe the efforts of various scientists to “refine [the] mathematical tools” being used to put the other forensic sciences on the same level as DNA.

## New Music Software Predicts The Hits

NPR’s Morning Edition takes a look at Hit Song Science, a new software program that purports to predict whether a pop song will be a hit or not–and seems to do a fairly good job of it. A study by the Harvard Business School found that the algorithms worked 8 out of 10 times. A sample quote from the Morning Edition broadcast:

Many of us like to believe that there’s a little magic behind the making of a hit single. Take a song like “I Gotta Feeling” by The Black Eyed Peas. That’s a good song, judging by sales: It’s on top of the Billboard pop chart. David Meredith, CEO of Music Intelligence Solutions, says there’s no magic in that; it’s math. His software, called Hit Song Science, gave the song a hit score of 8.9 out 10.

The transcript of the piece is here; the page also contains a link to the audio of the piece itself.

## Word Association Mystery Revisited

This article (Spanish-language) in La Nacion describes work whose roots lie in an old project of Sigmund Freud and Carl Jung, among others. Both of those luminaries spent considerable time and effort on the ‘word association problem’: trying to divine when and why certain words were associated with each other in many people’s minds. Mariano Sigman, Martin Elias and Flavia Bonomo of the University of Buenos Aires applied mathematics to the problem, using a huge corpus of text from newspapers and books to develop a metric which determines how “close” and “far” different words typically are from each other. The associations given by their metric appear to do a pretty good job mirroring the responses people give when given a word and asked to say what pops into their head.

## Best Science Visualization Videos of 2009

Wired magazine presents the best science visualization videos of 2009:

The Department of Energy honored 10 of this year’s best scientific visualizations with its annual SciDAC Vis Night awards, at the Scientific Discovery through Advanced Computing conference (SciDAC) in June. Researchers submitted visualizations to the contest, and program participants voted on the best of the best. From earthquakes to jet flames, this gallery of videos and images show how beautiful (and descriptive) visual data can be.

All of these videos are of course essentially illustrations of mathematical models, models that are so complex that just making the individual frames of the videos requires heavy-duty mathematics and heavy-duty computational power.

## Credit Card Use Is Ripe for Data Mining

Credit card companies are increasingly using statistical models to guess which of their customers are most likely to default, to not care about rate increases, to incur late fees, and things like that. This radio segment, which appeared on the daily “Marketplace” segment of American Public Radio, describes some of the issues surrounding the practice.