Electrical hum

If you hear electrical equipment humming, it’s probably at a pitch of about 60 Hz since that’s the frequency of AC power, at least in North America. In Europe and most of Asia it’s a little lower at 50 Hz. Here’s an audio clip in a couple formats: wav, mp3.

The screen shot above comes from a tuner app taken when I was around some electrical equipment. The pitch sometimes registered at A# and sometimes as B, and for good reason. In a previous post I derived the formula for converting frequencies to musical pitches:

h = 12 log(P / C) / log 2.

Here C is the pitch of middle C, 261.626 Hz, P is the frequency of your tone, and h is the number of half steps your tone is above middle C. When we stick P = 60 Hz into this formula, we get h = -25.49, so our electrical hum is half way between 25 and 26 half-steps below middle C. So that’s between a A# and a B two octaves below middle C.

For 50 Hz hum, h = -28.65. That would be between a G and a G#, a little closer to G.

Update: So why would the frequency of the sound match the frequency of the electricity? The magnetic fields generated by the current would push and pull parts, driving mechanical vibrations at the same frequency.

Programming languages and magic

In the context of programming languages, “magic” is often a pejorative term for code that does something other than what it appears to do.

Programmers seem to have a love/hate relationship with magic. Even people who say that don’t like magic (e.g. because it’s hard to debug) end up using it. The Haskell community prides itself on having a transparent language with no magic, and yet monads are slightly magical. The whole purpose of a monad is to hide explicit data flow, though in a principled way. Haskell’s do notation is more magical, and templates are even more magical still. (However, I do hear some Haskellers express disdain for templates.)

People who like magic tend to use the word “automagic” instead. It means about the same thing as “magic” but with a positive connotation.

To conclude with a couple sweeping generalizations, magic fans tend to be tool-oriented (such as Microsoft developers) while magic detractors tend to be language-oriented (such as Haskell developers ).

Update: Someone asked me on Twitter about the difference between abstraction and magic. I’d say abstraction hides details, but magic is actively misleading or ironic.

 

Go anywhere in the universe in two years

Here’s a totally impractical but fun back-of-the-envelope calculation from Bob Martin.

Suppose you have a space ship that could accelerate at 1 g for as long as you like. Inside the ship you would feel the same gravity as on earth. You could travel wherever you like by accelerating at 1 g for the first half of the flight then reversing acceleration for the second half of the flight. This approach could take you to Mars in three days.

If you could accelerate at 1 g for a year you could reach the speed of light, and travel half a light year. So you could reverse your acceleration and reach a destination a light year away in two years. But this ignores relativity. Once you’re traveling at near the speed of light, time practically stops for you, so you could keep going as far as you like without taking any more time from your perspective. So you could travel anywhere in the universe in two years!

Of course there are a few problems. We have no way to sustain such acceleration. Or to build a ship that could sustain an impact with a spec of dust when traveling at relativistic speed. And the calculation ignores relativity until it throws it in at the end. Still, it’s fun to think about.

Update: Dan Piponi gives a calculation on G+ that addresses the last of the problems I mentioned above, sticking relativity on to the end of a classical calculation. He does a proper relativistic calculation from the beginning.

If you take the radius of the observable universe to be 45 billion light years, then I think you need about 12.5 g to get anywhere in it in 2 years. (Both those quantities as measured in the frame of reference of the traveler.)

If you travel at constant acceleration a for time t then the distance covered is c^2/a (cosh(a t/c) – 1) (Note that gives the usual a t^2/2 for small t.)

Distribution of a range

Suppose you’re drawing random samples uniformly from some interval. How likely are you to see a new value outside the range of values you’ve already seen?

The problem is more interesting when the interval is unknown. You may be trying to estimate the end points of the interval by taking the max and min of the samples you’ve drawn. But in fact we might as well assume the interval is [0, 1] because the probability of a new sample falling within the previous sample range does not depend on the interval. The location and scale of the interval cancel out when calculating the probability.

Suppose we’ve taken n samples so far. The range of these samples is the difference between the 1st and the nth order statistics, and for a uniform distribution this difference has a beta(n-1, 2) distribution. Since a beta(a, b) distribution has mean a/(a+b), the expected value of the sample range from n samples is (n-1)/(n+1). This is also the probability that the next sample, or any particular future sample, will lie within the range of the samples seen so far.

If you’re trying to estimate the size of the total interval, this says that after n samples, the probability that the next sample will give you any new information is 2/(n+1). This is because we only learn something when a sample is less than the minimum so far or greater than the maximum so far.

 

 

Elementary vs Foundational

Euclid’s proof that there are infinitely many primes is simple and ancient. This proof is given early in any course on number theory, and even then most students would have seen it before taking such a course.

There are also many other proofs of the infinitude of primes that use more sophisticated arguments. For example, here is such a proof by Paul Erdős. Another proof shows that there must be infinitely many primes because the sum of the reciprocals of the primes diverges. There’s even a proof that uses topology.

When I first saw one of these proofs, I wondered whether they were circular. When you use advanced math to prove something elementary, there’s a chance you could use a result that depends on the very thing you’re trying to prove. The proofs are not circular as far as I know, and this is curious: the fact that there are infinitely many primes is elementary but not foundational. It’s elementary in that it is presented early on and it builds on very little. But it is not foundational. You don’t continue to use it to prove more things, at least not right away. You can develop a great deal of number theory without using the fact that there are infinitely many primes.

The Fundamental Theorem of Algebra is an example in the other direction, something that is foundational but not elementary. It’s stated and used in high school algebra texts but the usual proof depends on Liouville’s theorem from complex analysis.

It’s helpful to distinguish which things are elementary and which are foundational when you’re learning something new so you can emphasize the most important things. But without some guidance, you can’t know what will be foundational until later.

The notion of what is foundational, however, is conventional. It has to do with the order in which things are presented and proved, and sometimes this changes. Sometimes in hindsight we realize that the development could be simplified by changing the order, considering something foundational that wasn’t before. One example is Cauchy’s theorem. It’s now foundational in complex analysis: textbooks prove it as soon as possible then use it to prove things for the rest of course. But historically, Cauchy’s theorem came after many of the results it is now used to prove.

Related: Advanced or just obscure?

Rudyard Kipling and applied math

This evening something reminded me of the following line from Rudyard Kipling’s famous poem If:

… If all men count with you, but none too much …

It would be good career advice for a mathematician to say “Let all areas of math count with you, but none too much.” This warns against dismissing something offhand because you’re sure you’ll never use it, and becoming so fond of something that it becomes a solution in search of a problem.

The same applies to technology: Let all technologies count with you, but none too much.

Related posts:

 

Timid medical research

Cancer research is sometimes criticized for being timid. Drug companies run enormous trials looking for small improvements. Critics say they should run smaller trials and more of them.

Which side is correct depends on what’s out there waiting to be discovered, which of course we don’t know. We can only guess. Timid research is rational if you believe there are only marginal improvements that are likely to be discovered.

Sample size increases quickly as the size of the effect you’re trying to find decreases. To establish small differences in effect, you need very large trials.

If you think there are only small improvements on the status quo available to explore, you’ll explore each of the possibilities very carefully. On the other hand, if you think there’s a miracle drug in the pipeline waiting to be discovered, you’ll be willing to risk falsely rejecting small improvements along the way in order to get to the big improvement.

Suppose there are 500 drugs waiting to be tested. All of these are only 10% effective except for one that is 100% effective. You could quickly find the winner by giving each candidate to one patient. For every drug whose patient responded, repeat the process until only one drug is left. One strike and you’re out. You’re likely to find the winner in three rounds, treating fewer than 600 patients. But if all the drugs are 10% effective except one that’s 11% effective,  you’d need hundreds of trials with thousands of patients each.

The best research strategy depends on what you believe is out there to be found. People who know nothing about cancer often believe we could find a cure soon if we just spend a little more money on research. Experts are more sanguine, except when they’re asking for money.

Commutative diagrams in LaTeX

There are numerous packages for creating commutative diagrams in LaTeX. My favorite, based on my limited experience, is Paul Taylor’s package. Another popular package is tikz-cd.

To install Paul Taylor’s package on Windows, I created a directory called localtexmf, set the environment variable TEXINPUTS to its location, and copied diagrams.sty file in that directory.

Here are a couple examples, diagrams used in the definition of product and coproduct.

And here’s the LaTeX to produce the diagrams.

\begin{diagram}
& & X & & \\
& \ldTo^{f_1} & \dDashto_f & \rdTo^{f_2} & \\
A & \lTo_{\pi_1} & A\times B & \rTo_{\pi_2} & B \\
\end{diagram}

\begin{diagram}
& & X & & \\
& \ruTo^{f_1} & \uDashto_f & \luTo^{f_2} & \\
A & \rTo_{i_1} & A\oplus B & \lTo_{i_2} & B \\
\end{diagram}

For much more information, see the package page.

The mean of the mean is the mean

There’s a theorem in statistics that says

E( \bar{X} ) = \mu

You could read this aloud as “the mean of the mean is the mean.” More explicitly, it says that the expected value of the average of some number of samples from some distribution is equal to the expected value of the distribution itself. The shorter reading is confusing since “mean” refers to three different things in the same sentence. In reverse order, these are:

  1. The mean of the distribution, defined by an integral.
  2. The sample mean, calculated by averaging samples from the distribution.
  3. The mean of the sample mean as a random variable.

The hypothesis of this theorem is that the underlying distribution has a mean. Lets see where things break down if the distribution does not have a mean.

It’s tempting to say that the Cauchy distribution has mean 0. Or some might want to say that the mean is infinite. But if we take any value to be the mean of a Cauchy distribution — 0, ∞, 42, etc. — then the theorem above would be false. The mean of n samples from a Cauchy has the same distribution as the original Cauchy! The variability does not decrease with n, as it would with samples from a normal, for example. The sample mean doesn’t converge to any value as n increases. It just keeps wandering around with the same distribution, no matter how large the sample. That’s because the mean of the Cauchy distribution simply doesn’t exist.

Quintic root

Here’s a curious result I ran across the other day. Suppose you have a quintic equation of the form z x5x – 1 = 0. (It’s possible to reduce a general quintic equation to this form, known as Bring-Jerrard normal form.) There is no elementary formula for the roots of this equation, but the following infinite series does give a root as a function of the leading coefficient z:

\sum_{n=0}^\infty {5n \choose n} \frac{z^n}{4n+1}

One reason this is interesting is that the series above has a special form that makes it a hypergeometric function of z. You can read more about it here.

I could imagine situations where having such an expression for a root is useful, though I doubt the series would be much use if you just wanted to find the roots of a fifth degree polynomial numerically. Direct application of something like Newton’s method would be much simpler.