Category: News Clips

Richard Feynman’s Integral Trick

“I had learned to do integrals by various methods shown in a book that my high school physics teacher Mr. Bader had given me. [It] showed how to differentiate parameters under the integral sign — it’s a certain operation. It turns out that’s not taught very much in the universities; they don’t emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. [If] guys at MIT or Princeton had trouble doing a certain integral, [then] I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation for doing integrals, only because my box of tools was different from everybody else’s, and they had tried all their tools on it before giving the problem to me.” (Surely you’re Joking, Mr. Feynman!)

I read Surely You’re Joking, Mr. Feynman! dozens of times when I was a teenager, and I was always curious about exactly what this integration technique actually was. So I enjoyed reading this article about the Leibniz Integration Rule: https://medium.com/dialogue-and-discourse/richard-feynmans-integral-trick-e7afae85e25c

A Classical Math Problem Gets Pulled Into the Modern World

I enjoyed this article about how the solution of a pure mathematics problem from a century ago is finding an unlikely application now: https://www.quantamagazine.org/a-classical-math-problem-gets-pulled-into-the-modern-world-20180523\

From the introductory paragraphs:

Long before robots could run or cars could drive themselves, mathematicians contemplated a simple mathematical question. They figured it out, then laid it to rest — with no way of knowing that the object of their mathematical curiosity would feature in machines of the far-off future.

The future is now here. As a result of new work by Amir Ali Ahmadi and Anirudha Majumdar of Princeton University, a classical problem from pure mathematics is poised to provide iron-clad proof that drone aircraft and autonomous cars won’t crash into trees or veer into oncoming traffic.

“You get a complete 100-percent-provable guarantee that your system” will be collision-avoidant, said Georgina Hall, a final-year graduate student at Princeton who has collaborated with Ahmadi on the work.

The guarantee comes from an unlikely place — a mathematical problem known as “sum of squares.” The problem was posed in 1900 by the great mathematician David Hilbert. He asked whether certain types of equations could always be expressed as a sum of two separate terms, each raised to the power of 2.

Mathematicians settled Hilbert’s question within a few decades. Then, almost 90 years later, computer scientists and engineers discovered that this mathematical property — whether an equation can be expressed as a sum of squares — helps answer many real-world problems they’d like to solve.

“What I do uses a lot of classical math from the 19th century combined with very new computational math,” said Ahmadi.

5 Ways to go Beyond Recitation

Most students will encounter recitation in a math class during their academic career. How can math professors make the experience more meaningful? MAA Teaching Tidbits blog has 5 ways educators can enhance the student experience during recitation.

  1. Focus on getting students to do the work instead of doing it for them.
  2. Incorporate group work into your sessions.
  3. Get students to communicate what they understand to each other and to the class.
  4. Have students relate mathematics to their own experiences.
  5. Cultivate an environment where failure is ok and experimentation is encouraged.

Full article: http://maateachingtidbits.blogspot.com/2017/09/5-ways-to-go-beyond-recitation.html