The Fourier Transform, explained in one sentence. If, like me, you struggled to understand the Fourier Transformation when you first learned about it, this succinct one-sentence colour-coded explanation from Stuart Riffle probably comes several years too late: Stuart provides a more detailed explanation here. This is the formula for the Discrete Fourier Transform, which converts sampled signals (like a digital sound recording) into the frequency domain (what tones are represented in the sound, and at what energies?). It's the mathematical engine behind a lot of the technology you use today, including mp3 files, file compression, and even how your old AM radio stays in tune.
The daunting formula involves imaginary numbers and complex summations, but Stuart's idea is simple. Imagine an enormous speaker, mounted on a pole, playing a repeating sound. The speaker is so large, you can see the cone move back and forth with the sound. Mark a point on the cone, and now rotate the pole. AltDevBlog: Understanding the Fourier Transform. Maths 98: Why do buses come in Markov chains? Career advice. Advice is what we ask for when we already know the answer but wish we didn’t. (Erica Jong) Here is my collection of various pieces of advice on academic career issues in mathematics, roughly arranged by the stage of career at which the advice is most pertinent (though of course some of the advice pertains to multiple stages).
Disclaimer: The advice here is very generic in nature; I don’t pretend to have any sort of “silver bullet” that will solve all career issues. You will of course need to evaluate many factors, contexts, and needs specific to your own situation, as well as employing a healthy dose of common sense, before making any important career decisions. I would in particular recommend discussing such decisions with your advisor if you have one, as he or she will be familiar with your situation and will likely be able to provide pertinent advice.
I am also (slowly) in the process of gathering my thoughts on time management from the perspective of a research mathematician. Accurately computing running variance. The most direct way of computing sample variance or standard deviation can have severe numerical problems. Mathematically, sample variance can be computed as follows. The most obvious way to compute variance then would be to have two sums: one to accumulate the sum of the x's and another to accumulate the sums of the squares of the x's. If the x's are large and the differences between them small, direct evaluation of the equation above would require computing a small number as the difference of two large numbers, a red flag for numerical computing.
The loss of precision can be so bad that the expression above evaluates to a negative number even though variance is always positive. See Comparing three methods of computing standard deviation for examples of just how bad the above formula can be. There is a way to compute variance that is more accurate and is guaranteed to always give positive results. This better way of computing variance goes back to a 1962 paper by B. Jdc.