### Tony Phillips' Take Blog on Math Blogs

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# This month's topics:

## The math of the lasso

"The Flat Loop trick is described mathematically as static solutions in a rotating frame of an inextensible string or elastic rod held at a distance R from the axis." Here the rope has length $s_1+s_2$, with a loop of length $s_2$ terminating in a much smaller loop (the honda) which can slide along the cord. The cowboy's hand describes a planar circle of radius $R$ with angular frequency $\Omega$. Image adapted from Brun, Ribe and Audoly, Proc. R. Soc. A 470 no. 2171.

"The Math of Whips, Chains and Ropes" was James Gorman's attempt to put a louche spin, so to speak, on An introduction to the mechanics of the lasso, published on the Proceedings of the Royal Society A by Pierre-Thomas Brun, Neil Ribe and Basile Audoly (at EPFL-Lausanne, Orsay and Paris-6, respectively). Writing on the Science webpage of the New York Times, Gorman tells us how this work grew out of Brun's thesis (his advisors were Ribe and Audoly) and how Brun approached the topic: "First, he learned basic trick roping, because, he said, reality is important in applied math. He also read The Lasso: A Rational Guide to Trick Roping, by Carey Bunks. He recruited a professional roper from Disneyland Paris. And he built a robotic arm that could do a simple trick." That trick is the "flat rope trick", illustrated above. (The professional roper is Jesus Garcilazo, who performs that trick and other amazing ones in the video that goes with Gorman's posting). Along with deriving and solving the equations for the movement of the lasso, the article discovers some useful facts for those who would like to learn how the "flat loop" is done. Gorman: "it is important to start with a big enough loop to do the trick. [From the article: $s_2 \approx 3s_1$]. They also discovered, by analyzing video frame by frame, that the hand is actually in a different position in relation to the loop than ropers think it is -- not leading the loop but traveling right along in phase with it." Some additional hints are given in the article. One of them can also be gleaned from careful inspection of Garcilazo's performance: "To avoid accumulating twist in the rope, the cowboy constantly rolls the rope between his thumb and forefinger while spinning it." The other concerns $\Omega$. " [A] minimal ... frequency of ... 1.4 turns per second, or 84 r.p.m., is required to produce Flat Loops. A key to success is therefore to pick a good angular velocity."

## Lifesaving statistics

Bayseian statistics helped save a fisherman's life. That's the grabber in a New York Times article by F. D. Flam, published in the "Science" section on September 30, 2014. Here's how the story starts: "Statistics may not sound like the most heroic of pursuits. But if not for statisticians, a Long Island fisherman might have died in the Atlantic Ocean after falling off his boat early one morning last summer. The man owes his life to a once obscure field known as Bayesian statistics--a set of mathematical rules for using new data to continuously update beliefs or existing knowledge." Flam gives a quick survey of the difference between Bayesian statistics and garden-variety, "frequentist" statistics. "The essence of the frequentist technique is to apply probability to data. If ... a coin .. came up heads nine times out of 10, a frequentist would calculate the probability of getting such a result with an unweighted coin. The answer (about 1 percent) is not a direct measure of the probability that the coin is weighted; it's a measure of how improbable the nine-in-10 result is--a piece of information that can be useful" in evaluating the hypothesis that the coin is loaded. "By contrast, Bayesian calculations go straight for the probability of the hypothesis, factoring in not just the data from the coin-toss experiment but any other relevant information--including whether you have previously seen your friend use a weighted coin." We are not told exactly how that other data is "factored in."

Flam refers to Andrew Gelman (Statistics, Columbia) to explain how frequentist evaluation of experimental results almost unavoidably leads to problems ("Even if scientists always did the calculations correctly--and they don't, he argues--accepting everything with a $p$-value of 5 percent means that one in 20 'statistically significant' results are nothing but random noise.") and gives an example of a pubished and appealingly counterintuitive study that Gelman reanalyzed à la Bayes: "the study's statistical significance evaporated." Another example comes from the celebrated "Monty Hall" problem. Flam sets the stage, then gives the Bayesian explanation:

• "A Bayesian calculation would start with one-third odds that any given door hides the car, then update that knowledge with the new data: Door No. 2 had a goat. The odds that the contestant guessed right--that the car is behind No. 1--remain one in three. Thus, the odds that she guessed wrong are two in three. And if she guessed wrong, the car must be behind Door No. 3. So she should indeed switch."

After some more examples, and a reference to Uri Simonsohn (Psychology, Penn), who "said he had looked into Bayesian statistics and concluded that if people misused or misunderstood one system, they would do just as badly with the other," we get back to the fisherman, John Aldridge. He was alone on watch when he fell off his boat, "sometime from 9 p.m. on July 24 [2013] to 6 the next morning," some 40 miles offshore. The Coast Guard implemented SAROPS, their Search and Rescue Optimal Planning System, organized along Bayesian lines. "Over the next few hours, searchers added new information--on prevailing currents, places the search helicopters had already flown and some additional clues found by the boat's captain. The system could not deduce exactly where Mr. Aldridge was drifting, but with more information, it continued to narrow down the most promising places to search." And they found him. "Even in the jaded 21st century, it was considered something of a miracle."

Tony Phillips
Stony Brook University
tony at math.sunysb.edu