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Tony Phillips' Take on Math in the Media A monthly survey of math news |

This month's topics:

- UK apologizes to Alan Turing
- Markov chains for the Indus script
- Hiding behind Hermite polynomials
- Page-ranking the food chain

The *Daily Mail* reported on September 11, 2009 that U. K. Prime Minister Gordon Brown had issued a posthumous apology to "gay Enigma codebreaker Alan Turing." According to Ian Drury, the article's author, "Thirty thousand people had signed a petition seeking an apology for Mr Turing, who was credited by Winston Churchill with making the biggest single contribution to the Allied victory in World War II." Gordon Brown's statement is posted on the official Number 10 website of the Prime Minister's office. " ... The debt of gratitude he is owed makes it all the more horrifying ... that he was treated so inhumanely." It ends: "So on behalf of the British government, and all those who live freely thanks to Alan's work I am very proud to say: we're sorry, you deserved so much better."

The story was picked up that day on NPR's "All Things Considered." Robert Siegel interviewed John Graham-Cumming, who created the petition. Graham-Cumming sketched the background: "[Turing] is one of the great figures of mathematics and science of the 20th century. He did these amazing things to find the thing called the Turing machine, which is the underpinnings of all of computer science. He then worked in the thick of a war to break the Nazi enigma and other codes, which is supposed to have shortened the war by at least two years. And then after the war he worked on artificial intelligence before we even had computers that were powerful enough to do much addition." And the events of 1952: "... he reported a burglary at his home. And it transpired in the investigation that the burglar was related to Alan Turing's boyfriend. Because of that, Alan Turing's homosexuality came out. He was arrested. Tried and convicted. And then he was given the choice of prison or estrogen injections, a sort of cure for being gay." After a year of chemical castration, Turing killed himself. Graham-Cummings' reaction to the Prime Minister's statement: "When I read it I was very pleased to see this simple apology to this great man."

Markov chains for the Indus script

Samples of the Indus script: seals, and impressions from similar seals. Lengths about 1.5"; National Museum, Karachi. Image by J.M. Kenoyer, courtesy of Omar Khan at harappa.com.

The Indus civilization flourished in and around the Indus River valley between about 2600 and 1900 BC. Among the artifacts discovered beginning in our 19th century are over 3800 inscriptions in a script that has so far resisted decipherment. Most of the inscriptions are very short, around 5 characters, like the ones on the seals shown above; and we know almost nothing about the language that these symbols encode. There has been speculation that the inscriptions do not in fact correspond to messages in a natural language. This has been set to rest by a study reported on August 18, 2009 in the *Proceedings of the National Academy of Sciences* (**106**, 13685-13690; abstract). The authors, a team of six led by Rajesh Rao (Univ. of Washington) encoded the statistics of the corpus as a Markov chain: a set of states and transition probabilities.

This Markov chain generates words in a 2-character alphabet (A, B, #=space); *p*_{AA} is the probabilty that A will follow A, *p*_{AB} is the probabilty that B will follow A, etc.

In this case the states correspond to the characters in the script, and the transition probability from one state to another is the probability that the corresponding characters will appear, in that order, in an inscription. This analysis allowed the authors to compare the Indus script statistics with those from natural languages. They report: "our results indicate that the Indus script exhibits rich synactic structure and the ability to represent diverse content. both of which are suggestive of a linguistic writing system rather than a nonlinguistic symbol system."

An earlier article by the same team (*Science* **324** 1165) was reviewed by Brandon Klein in Wired Science with the title "Artificial Intelligence Cracks 4,000-Year-Old Mystery."

Hiding behind Hermite polynomials

*ScienceDaily* reported on August 17, 2009, that a group of University of Utah mathematicians had developed a new cloaking method: a way of making objects invisible to various forms of radiation. The research appears in *Physical Review Letters* **103** 073901, "Active Exterior Cloaking for the 2D Laplace and Helmholz Equations," by Fernando Guevara Vasquez, Graeme W. Milton and Daniel Onofrei, so it only covers 2-dimensional problems, although the authors "anticipate that the results extend to three dimensions and to the full Maxwell equations". Here I will describe their work on the Laplace equation, but I recommend a movie illustrating the Helmholz (wave equation) case.

In the Laplace case, the field to hide from is represented by a harmonic function; in the illustrations *f*(*x*,*y*) = *x*. This could be the electric potential generated by two oppositely charged wires parallel to the *y*-axis, in a medium with a homogeneous dielectric constant.

The potential function *f*(*x*,*y*) = *x*. The colors represent increasing voltage from blue to orange.

If an object of different dielectric constant is introduced, the potential undergoes long-range distortion.

An object (small black disc) with different dielectric constant distorts the potential. Expanded color coding of function values. The solid and dashed white, the red and the black curves are for reference to the next figure. Images courtesy of Fernando Guevara.

In this case cloaking is achieved by generating another potential field, say *c*(*x*,*y*), which is essentially zero outside of a neighborhood of the object and (almost) identical to minus the original field in a smaller neighborhood of the object. The authors use a high-degree Hermite polynomial *h*(*z*) as the main ingredient in the process. The next picture shows the magnitude of the 15th Hermite polynomial, interpreted as a complex-valued function of the complex variable *z*. The function takes 0 to 1 and 1 to zero, and at both points all its first fifteen derivatives are 0. Consequently it is very close to 0 on a neighborhood (solid white circle) of 1 and very close to 1 on a neighborhood (dashed white circle) of 0.

The magnitude of *h*(*z*), the fifteenth Hermite polynomial implemented as a function of a complex variable. The values are very close to 0 for *z* inside the solid white circle, and very close to 1 for *z* inside the dashed white circle. High-resolution image.

Now the authors compose the function 1-*h*(*z*) with the inversion *z* --> 1/*z* = *s* which maps the inside of the dashed circle to the outside of another dashed circle, and vice versa, and the inside of the solid circle to the inside of another solid circle including the point *z* = 1. The composed function is very close to 0 outside the dashed circle, and very close to 1 inside the solid one. Now make the original *f* the real part of an analytic function *F*(*s*), and consider the real part of the product *F*(*s*)*h*(*s*); this will be our cloaking potential: writing *s* = *x* + *iy*, it is *c*(*x*,*y*). Superimposing this potential on the original field will give a harmonic function which matches the original field *f* (to very close approximation) everywhere outside the dashed circle, but obliterates any trace of our object, tucked safely inside.

Cloaking: superimposing the potential constructed from *f* and the Hermite polynomial onto the original potential *f* creates a new field which matches *f* outside the dashed white circle but is very close to zero inside the solid white circle, where the object is hiding. High-resolution image.

Google answers your queries by returning the relevant pages which have the highest "page rank:" they have the most links from other sites which have the most links ... etc. The exact way this is accomplished is proprietary. Stefano Allesina (Santa Barbara) and Mercedes Pascual (Ann Arbor) published a paper in *PLoS Computational Biology* (September 4, 2009) showing how a similar algorithm could accurately determine which species are the most essential in a food chain. Their article was picked up by Hadley Leggett for Wired Science (9/4/09) and by John Platt for *Scientific American*'s 60 Second Science (9/3/09). Leggett quotes Allesina: "In PageRank, you're an important website if important websites point to you ... We took that idea and reversed it: Species are important if they support important species." As Leggett puts it: "grass is important because it's eaten by gazelles, and gazelles are important because they're eaten by lions." Platt: "So why is this advanced mathematics even necessary when looking at nature? Writing in the paper's abstract, the authors warned that 'because of their mutual dependence, the loss of a single species can cascade in multiple co-extinctions.' But food webs are so complex, it would take forever to go through all possible extinction scenarios without an algorithm like this."

Tony Phillips

Stony Brook University

tony at math.sunysb.edu