Numb3rs

Feedback?

Sun, 01 Nov 2009 00:00:00 -0400

No shows for November yet, so no new blogs -- see the October ones.

In the absence of any new Numb3rs math, I'd like to get feedback from readers of this blog, especially related to the math or non math of the show.

Send to: bridger@neu.edu,

Your blogmeister

Mathless and unfocussed

Sat, 31 Oct 2009 00:00:00 -0400

Last night's episode, as you may have noticed, had no mathematics in it. I was hoping that it would at least be exciting after a spooky beginning, but no -- it petered out into the usual foot and car chases. The one moment of tension, when Charlie and Floyd (see below) are confronted by the death-ray drone, ends with the plane inexplicably simply flying away. Wow, what a fizzle.

Which brings me to the new character, a Pentagon "agent" named Floyd Mayborne, who may replace Larry as the show's annoying character. He seems destined, like Larry, to appear out of nowhere to invade Charlie and Amita's privacy, while voicing cryptic and self-promoting speeches. He mumbles references to fraud perpetrated by unnamed government contractors, aided by higher ups in Pentagon administration -- all so hush-hush that even his cellphone is invisible.

Here is where the show cops out. What the writers should be saying is that this kind fraud has permeated the (unfortunately) very real Strategic Defense Initiative (SDI), also known as Reagan's "Star Wars" program. This has been a multi-billion dollar boondoggle for defense contractors in which scientific opposition has been muzzled and test results have been faked. In fact, Star Wars is all about weapons systems that are pure science fiction: e.g. nuclear-bomb-fueled laser cannons, death rays and anti-missile missiles. These either never existed and most scientists believe never will, or haven't come close to working. While probably no one has been murdered, the diversion of funds to this wasteful program has, I'm sure, been responsible for shortfalls in programs that would actually have saved lives.

Too bad the show couldn't have actually been a little more explicit.

Let's see what Wolfram will come up with next week.

Your (disappointed) Blogmeister

Matching sounds

Sat, 24 Oct 2009 00:00:00 -0400

Charlie claims that he can create a "sound signature" for a car that identifies that car uniquely -- or nearly uniquely. I'm not entirely convinced, because the sound a car emits varies a lot with conditions: the speed of the engine, the incline of the road, the condition of the muffler etc. It may even depend on the ambient temperature and the kind of gas. However, I am not an automotive expert, so I won't deny the possibility.

How would it work?

Well, we've already discussed ways of identifying people via Face Recognition Algorithms (1/19/06). The idea is to find certain quantifiable features of a human face, and create a "face vector". The face vectors for a known database are then compared with a new face vector to try to find the best possible match, using a technique called Singular Value Decomposition or SVD -- a very important and useful technique from linear algebra. See, also, the blog on Text Recognition Algorithms (12/16/06).

So, for sounds, one needs to find quantifiable features to make into a "sound vector". One possibility is to use Fourier analysis: a way of breaking a complex sound into its component frequencies. These are automatically expressed as a vector of numbers (the "weights" or proportions of each frequency in the sample sound) obtained via computer using the Fast Fourier Transform (FFT), a very efficient computer algorithm. Now, SVD can be used to compare the sound vector of interest with those obtained from other cars on the road. I don't know how fast this can be done, and how accurately one can determine which car sound vector matches best the one that comes from the car we're searching for.

A somewhat more modern version of this idea is to use wavelet analysis to break down a sound into certain simpler components called wavelets (instead of Fourier analysis which is based on simple sine and cosine waves). Some of the background computer screens in the episode seem to show wavelet decomposition.

As I said, I don't know exactly how feasible this is in the case of real-time auto-identification. If any reader out there has experience with this kind of application, let me know.

(If you want more details on Fourier transforms, Singular Value Decomposition, and Wavelet theory, you can find lots of good articles on the Web. Wikipedia is a good place to start: it's usually pretty dependable on technical topics.)

Your blogmeister

Moisture, desiccants, and differential equations

Thu, 22 Oct 2009 00:00:00 -0400

The most interesting math on last Friday's show involved Charlie's analysis of the drying agent or desiccant (silica gel is probably the most common one). Charlie points out that the way the desiccant works is not simply linear. This means the it is not true, for example, that it takes three times the time to remove three times the amount of water. At first the desiccant is very dry and removes water quickly according to some linear formula like: amount of water removed in time t is approximately, say, -2t, with time in days and liquid in liters for example. This formula then expresses a rate, namely:

Water is removed at the rate of -2 liters per day

(We use the negative sign to indicate removal.)

However, as the desiccant removes more and more water, its ability to absorb liquid decreases, so the rate at which water is being removed gets smaller -- in this case less negative. Thus, the rate at which water is being removed by the desiccant starts off at some negative constant, -2 say, then becomes less negative. As the desiccant reaches its maximum capacity, the rate approaches 0. Here is a possible plot for the rate of water removal as time goes by:

Note this "S"-shaped curve. Certain curves of this type are called logistic curves. I discussed such curves here. The curve above doesn't tell how much water there is, just how fast its being removed at time t. That is, it's giving the rate of change of water with time, not the actual water itself. It is the plot of a differential equation. The rate of change of a quantity W (water) say, with time t, is denoted dW/dt by mathematicians. This rate of change, in real life, depends usually on many quantities. In this case, it depends on time as well as humidity and temperature and the chemical nature of the particular desiccant etc. To simplify matters, we are looking at just the relationship

dW/dt = f(t).

and the plot above is a plot of f(t).

But we want to know how W depends on t. In other words, we need W = F(t) for some function F(t). This is what it means to solve the differential equation. In general, it is very hard to solve differential equations. Mostly, it is done numerically using computers. Here is the graph of a possible solution to the differential equation pictured above:

(To get these plots, I made up a simple differential equation to get the first plot, which was produced by the powerful mathematical program Maple. Then I used Maple to solve and plot the solution.)

Once you have the graph of the solution, you can work backwards: given a particular amount of water remaining, you can use the plot to see at which time that amount of water was present. For example, according to the plot above, there were about 2 liters of water remaining after about 1 1/3 hours.

Your blogmeister

A forgettable episode

Sat, 10 Oct 2009 00:00:00 -0400

Last night's episode was pretty much a loser.

There was almost no mathematics, and what there was contained a careless mistake, caught by, among others, blog readers P.J. Campbell and J. Simonoff. In describing the "game" of Russian Roulette, Larry says the following: "The chances of the gun going off are 1 out of 6, or 6-to-1 odds." The probability is indeed 1 out of 6, since there is exactly 1 bullet in one of the 6 chambers. However, the odds are 1 to 5: 1 shot to 5 clicks.

(There was an even worse error last year; see the blog Triple shootout: Charlie goofs (12/21/08) .)

I'm afraid that the mathematical quality of the show has been in a slow but definite decline over the past few seasons -- probably dating from the time that the Wolfram people took over all the mathematical content and, I think, editing. (Which was about the time, I might add, that my wife and I stopped vetting scripts for the show.)

However, there have been other things. There is much more explicit and graphic violence, for example, and a lot of the common junk that makes most TV series undistinguished: car chases, shootings, gratuitous sex. (Did we really need the lap-dancing scenes in last Friday's show?) At one time about 5 years ago I defended the show, in a mathematics newsletter, as good family fare; I don't think I could honestly say the same any more.

Larry, who was always the Polonius (or Nestor) of the show, spouting dressed up platitudes and tired self-promoting baloney, has now become truly insufferable. Let's hope he is written out summarily.

I don't think I'm going to be able to take much more of the new Numb3rs, especially with the demands of Friday's crossword and playoff baseball (though I fear the Red Sox will soon be out of it).

Your blogmeister