- On visualizing music, by Claudia Clark

"Visualizing Music Mathematically,"*Forbes*, 29 July 2016 - On Jim Papadopoulos’s bicycle math, by Rachel Crowell

""The bicycle problem that nearly broke mathematics,"*Nature*, 20 July 2016 - On the 2016 IMO, by Mike Breen

"U.S. students win prestigious International Math Olympiad—for second straight year,"*The Washington Post*, 18 July 2016 - On puzzles inspired by Ramanujan, by Annette Emerson

"Three Puzzles Inspired by Ramanujan,"*Quanta Magazine*, 14 July 2016 - On Artur Avila, by Rachel Crowell

"The Brazilian Genius Trying to Get Non-Mathematicians Interested in Maths,"*Vice*, 4 July 2015

**On visualizing music, by Claudia Clark**

In this article, University of Florida professor of mathematics **Kevin Knudson** explains to the general reader how a piece of music can be described and visualized using mathematics. He starts by noting that a typical pop song has distinct parts, like verses and a chorus, each of which has certain chord progressions. In addition, various instruments, including voices, are used at different times. But to "see" a song mathematically, Knudson describes one method, which was presented by Duke University PhD student **Chris Tralie** and mathematics professor **Paul Bendich** in their 2015 paper, "Cover Song Identification with Timbral Shape Sequences." He explains how the authors analyze songs--which are, after all, simply waves--using "some commonly used features in music analysis called 'timbral features,' the 'Mel-Frequency Cepstral Coefficients,' and a feature set called 'chroma' which gives information about notes and chord." These features can be measured along small pieces of the graph, ultimately resulting in a "cloud of points in a 59-dimensional Euclidean space." Then, the authors use topological data analysis to "develop some novel methods for organizing a point cloud into clusters based on [the] local dimension," notes Knudson. "Finally, to visualize, you project the data into 3-dimensional space using the first three principal components of the data" (which involves principle component analysis).

*Image*: Principal Component Analysis of an eight-beat block from the hook of Robert Palmer's "Addicted To Love" with a window size of .05 seconds; cool colors indicate windows towards the beginning of the block, and hot colors indicate windows towards the end, courtesy of Christopher J. Tralie and Paul Bendich.

See "Visualizing Music Mathematically," by Kevin Knudson. *Forbes*, 29 July 2016.

*--- Claudia Clark*

**On Jim Papadopoulos’s bicycle math, by Rachel Crowell**

This article describes Jim Papadopoulos's obsession with bikes, which goes beyond riding them and even his previous competition in recreational races. Papadopoulos, 62, is an assistant teaching professor of mechanical and industrial engineering at Northeastern University. (Photo courtesy Northeastern University.)

For decades, Papadopoulos has been trying to use mathematics to explain how bicycles work. He pursued this interest in greater depth when he was hired as a postdoc at Cornell by his friend Andy Ruina. The Cornell Bicycle Research Project was formed, a project that garnered support from bicycle companies to study different aspects of bicycle performance. Papadopoulos set forth with the goal of writing equations that describe the stability of certain bikes over others. According to the article, he found that, "No single variable... could account for self-stability," of bicycles.

The Bicycle Research Project came to a halt when Ruina couldn't support Papadopoulos's work any longer, and Papadopoulos couldn't raise enough support from bicycle companies. Papadopoulos was first-author on one paper related to the mathematics behind bicycles before he held teaching and industry jobs. Then in 2003, Ruina called him. Arend Schwab of Delft University of Technology in the Netherlands was visiting Ruina's lab at Cornell and wanted to revitalize the research on bike stability. With Schwab's collaboration, Papadopoulos and Ruina unearthed answers to the long-held question of why bikes can move forward with stability, without a rider, findings that were published in the Royal Society and *Science*. The three researchers, along with Jaap Meijaard, who is now an engineer at University of Twente in the Netherlands, also published a set of bicycle equations in the *Proceedings of the Royal Society of London A* that agreed with Papadopoulos's previously derived but unpublished equations.

Video: TMS bicycle, Andy Ruina explains how bicycles balance.

Now at Northeastern University, Papadopoulos has moved on to studying a new problem: what causes some bikes to wobble at high speeds and what modifications can be made to bicycles to remedy the wobble.

See "The bicycle problem that nearly broke mathematics," by Brendan Borrell, *Nature*, 20 July 2016.

*--- Rachel Crowell* (Posted 8/11/16)

**On the 2016 IMO, by Mike Breen**

The U.S. International Mathematical Olympiad (IMO) team won first place in the IMO for the second straight year. Korea finished seven points behind the U.S. and China was third. All six U.S. team members earned gold medals in the competition. In an article in The Washington Post, Valerie Strauss talks with U.S. coach Po-Shen Loh, Carnegie Mellon University, about the team and the competition. The online article links to a video about the MAA's summer program for the team and other top scorers in the American Mathematics Competition, and includes three of the six problems from the 2016 IMO exam. The competition took place in Hong Kong. Next year's IMO will take place in Rio de Janeiro, July 12-24. Three of the U.S. team members, Ankan, Michael, and Ashwin, are former *Who Wants to Be a Mathematician* contestants. (*Photo*, left to right: Ankan Bhattacharya, Allen Liu, Ashwin Sah, Michael Kural, Yuan Yao, Junyao Peng; courtesy of the Mathematical Association of America/Carnegie Mellon University.)

See "U.S. students win prestigious International Math Olympiad — for second straight year," by Valerie Strauss, *The Washington Post*, 18 July 2016.

--- *Mike Breen *

**On puzzles inspired by Ramanujan, by Annette Emerson**

"Ramanujan indeed had preternatural insights into infinity: he was a consummate bridge builder between the finite and the infinite, finding ways to represent numbers in the form of infinite series, infinite sums and products, infinite integrals, and infinite continued fractions, an area in which, in the words of Hardy, his mastery was 'beyond that of any mathematician in the world.'" Article author Mutalik goes on to explore some simple infinite forms with some puzzles inspired by Ramanujan and "using nothing more than middle school algebra." One puzzle sets out to prove an equation involving an infinite nested radical, another uses basic algebra to prove that the famous golden ratio, phi, is equal to the infinite continued fraction illustrated in the article, and the third is a word problem with a background story involving Ramanujan: "A certain street has between 50 and 500 houses in a row, numbered 1, 2, 3, 4, … consecutively. There is a certain house on the street such that the sum of all the house numbers to the left side of it is equal to the sum of all the house numbers to its right. Find the number of this house."

Mutalik also poses, "What if Ramanujan had modern calculating tools?," "Where do you think Ramanujan’s results came from?," and "How would 21st-century mathematics be different had Ramanujan lived a life of normal length?" the authjor invites solutions in the comments section after the article.

See "Three Puzzles Inspired by Ramanujan," by Pradeep Mutalik, *Quanta Magazine*, 14 July 2016.

*--- Annette Emerson*

**On Artur Avila, by Rachel Crowell**

In 2014, **Artur Avila **became the first Latin American to win the Fields Medal. He received it for his contributions to the field of dynamical systems. This article calls him, "an evangelical of sorts," because he is, "an example, an inspiration and a role model for the young people living in places with no history in the field of mathematics." Avila is Director of Research at the (CNRS) Centre National de la Recherche Scientifique in Paris. He studies chaotic systems. Avila told *VICE*, "Mathematicians find pleasure, one that is almost artisanal, in getting our hands dirty with abstract dirt."

He noted that many non-mathematicians perceive math as something that is stagnant. They do not see math the way he sees it--as a changing discipline that is a necessary part of growing and evolving societies.

Avila started his master's degree at the Brazilian Institute for Pure and Applied Math (IMPA) while he was still attending São Bento, a conservative school that forced students to complete courses in religion. He rebelled against those courses by intentionally flunking them. During that time, Avila was socially isolated. He spent his free time focused on mathematics, which led to him earning a doctoral degree by age 21. When Avila was younger, he solved a problem in the interval exchange field with the help of a mentor. The problem had been unsolved for over thirty years.

Avila told *Vice *that he hasn't decided what he will focus on next, but that he tends to work on several research projects at the same time, "each at its own pace."

See "The Brazilian Genius Trying to Get Non-Mathematicians Interested in Maths," by Mattias Max, Translation by Thiago "Índio" Silva. *Vice*, 4 July 2015. (The article originally appeared on *VICE Brasil* and was republished on 4 July in *Vice UK*.)

--- *Rachel Crowell *(Posted 7/18/16)

Math Digest Archives
|| 2010 || 2009 ||
2008 || 2007 || 2006 || 2005 || 2004 || 2003 || 2002 || 2001 || 2000 || 1999 || 1998 || 1997 || 1996 || 1995
Click here for a list of links to web pages of publications covered in the Digest. |