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Trigonometry

June 22, 2012 Leave a comment

I have just realized that I haven’t posted a good puzzle here in a long time, so here it goes one on Trigonometry, that the average student of Calculus should be able to tackle: you can use anything you think it could help: derivatives, symmetry, periodicity, integration, summation, go to several variables, differential equations, etc

Prove that, for all real values x \in \mathbb{R}, it is

\sin x = x \cos\big(\tfrac{x}{2}\big) \cos\big(\tfrac{x}{4}\big) \cos\big(\tfrac{x}{6}\big) \cos\big(\tfrac{x}{8}\big) \dotsb \cos\big( \tfrac{x}{2n}\big) \dotsb

or, in a more compact notation,

\sin x = x \displaystyle{\prod_{n=1}^\infty \cos \big( \tfrac{x}{2n} \big)}

So you want to be an Applied Mathematician

September 16, 2011 10 comments

The way of the Applied Mathematician is one full of challenging and interesting problems. We thrive by association with the Pure Mathematician, and at the same time with the no-nonsense, hands-in, hard-core Engineer. But not everything is happy in Applied Mathematician land: every now and then, we receive the disregard of other professionals that mistake either our background, or our efficiency at attacking real-life problems.

I heard from a colleague (an Algebrist) complains that Applied Mathematicians did nothing but code solutions of partial differential equations in Fortran—his skewed view came up after a naïve observation of a few graduate students working on a project. The truth could not be further from this claim: we do indeed occasionally solve PDEs in Fortran—I give you that—and we are not ashamed to admit it. But before that job has to be addressed, we have gone through a great deal of thinking on how to better code this simple problem. And you would not believe the huge amount of deep Mathematics that are involved in this journey: everything from high-level Linear Algebra, Calculus of Variations, Harmonic Analysis, Differential Geometry, Microlocal Analysis, Functional Analysis, Dynamical Systems, the Theory of Distributions, etc. Not only are we familiar with the basic background on all those fields, but also we are supposed to be able to perform serious research on any of them at a given time.

My soon-to-be-converted Algebrist friend challenged me—not without a hint of smugness in his voice—to illustrate what was my last project at that time. This was one revolving around the idea of frames (think of it as redundant bases if you please), and needed proving a couple of inequalities involving sequences of functions in L_p—spaces, which we attacked using a beautiful technique: Bellman functions. About ninety minutes later he conceded defeat in front of the board where the math was displayed. He promptly admitted that this was no Fortran code, and showed a newfound respect and reverence for the trade.

It doesn’t hurt either that the kind of problems that we attack are more likely to attract funding. And collaboration. And to be noticed in the press.

Alright, so some of you are sold already. What is the next step? I am assuming that at his point you own your Calculus, Analysis, Probability and Statistics, Linear Programming, Topology, Geometry, Physics and you are able to solve most known ODEs. From here, as with any other field, my recommendation is to slowly build a Batman belt: acquire and devour a sequence of books and scientific articles, until you are very familiar with their contents. When facing a new problem, you should be able to recall from your Batman belt what technique could work best, in which book(s) you could get some references, and how it has been used in the past for related problems.

Following these lines, I have included below an interesting collection with the absolutely essential books that, in my opinion, every Applied Mathematician should start studying:

Read more…

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