Saturday, June 18, 2011

So much to learn...

Our trip to Italy combined some work and fun. After giving my colloquium in Rome, we drove north, stopping in Orvieto, Luca, Florence, Pisa and San Gimignano to see the sights. Then, we drove south to Cetraro (the site of my conference), stopping in Caserta for the night to break up the long drive. In Cetraro, I spent the week at the NOMA conference, where I gave a talk and interacted with many colleagues. Our trip ended with a drive up to Rome to catch our plane back to the US, with a stop in Pompeii to visit the amazing ancient city.

Upon getting back, the work that had piled up during my absence hit me like a wrecking ball. While sitting at my desk writing and responding to zillions of emails, I noticed a delightful message from World Scientific press stating, "We are pleased to introduce a new title that may interest you." Usually, I reflexively delete such emails, but this time I glanced at the title, "Fractional Calculus." Immediately, my mind raced back to my high school days when I was learning calculus concurrently with learning to drive a car.

I had noticed that the second derivative was represented as the square of the derivative operator; so, when taking a differential geometry class at a local university (driving there with my new license), I asked the professor if there was any meaning to a fractional derivative - that is, the derivative operator to a fractional power. He pondered the question for a moment and answered "yes." He then described how one would approach the problem. I was intrigued, but the issue had never come up again, until today.

With a little thought about the basics, the idea of a fractional derivative is quite simple. For example, consider the derivative operator to the 1/2 power. When operating twice, it should give the familiar first derivative. Using this fact allows one to develop the mathematics of fractional calculus. Similarly, a negative power can be expressed as an integral. Even complex powers are possible.

This got me very excited. I was tempted to drop everything to study the topic in great detail. Beginning with the Wikipedia page on fractional calculus, I skimmed over the introductory material that described how the gamma function is an important part of fractional calculus as factorials are to integer calculus. This makes sense given that the gamma function reduces to factorials for the integers; and in the rest of the interval connects the points between the factorial.

I read further, and found that one can formulate fractional calculus using the Laplace transform. Ironically, I had gotten intrigued by the Laplace transform when teaching statistical mechanics last semester. At that time, I had received the Dover book catalog, a treasure trove of inexpensive math and physics books. As I skimmed through the catalogue, I saw a book titled, "The Laplace Transform." I could not resist buying it. I also bought a companion book that teaches mathematically deficient people like me how to do mathematical proofs. Sadly, while those two books sat on my desk for months, I only had enough time to read a few pages.

This fact stifled my temptation to buy the book on fractional calculus. But, I was reminded of all the beautiful and wondrous topics that remain to be learned. Though I do not have the time to chew over all the profound topics that have been produced by the human mind, I take pleasure in the privilege of mastering a microscopic fraction of it all, and having the opportunity to contribute an infinitesimal fraction.

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