Medescape

Skepticism, Medicine and Science News

Gravity Fingers Mathematically Explained

Since I have my math exam tomorrow, it seems only natural I should write about something math related. 

When water soaks down into the ground, it does not do so evenly, but rather forms spikes known as gravity fingers (see image below). Though it is a well known phenomenon in fluid mechanics, no one has been able to explain, mathematically, why it happens. In a recently published paper however, mathematicians at MIT give a both simple and elegant explanation. They got the idea when one of the researchers observed that gravity fingers looked very much like water flowing down a window (when you look at the picture it looks really obvious, it was definitely the first think I though of), which is a well understood phenomenon. Then it was just a matter of taking the equations describing that and apply it to water movement in soil. 

The short explanation to this phenomenon is that in order for water to flow down a window or in soil, the surface tension of the water has to be overcome. This will cause the water to flow in a finger-like pattern because as water builds up in the fingers, the weight overcomes the surface tension. My understanding of the phenomenon is that small indentations in the bottom flow line are bound to form no matter what because the soil/window is not perfectly uniform (and can never be), and when this happens the flow rate at the indentations increase because of the extra weight, leading to gravity fingers. 

Gravity fingers

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December 14, 2008 Posted by | Math | , , , , , , , , , , , , | Leave a comment

Completely Random

Randomness is a tricky thing. It is easily found in the world around us, but if you want to actively generate, say, a random sequence of numbers, it suddenly becomes quite elusive. The more advanced hand-held calculators have a program that generates, seemingly, random numbers by using specific algorithms, but the fact is that this is not true randomness, but rather pseudorandomness. This means that although the number sequence generated may seem random to a person, a statistical analysis may reveal subtle patterns and skews in the data. For the applications of a pocket calculator, this is probably not a big deal. However, random numbers are a big part of modern cryptography, and are used in, for example, Internet banking. It would be inherently bad if your personal bank ID could be easily predicted by a large statistical analysis of ID-data, so true randomness would be an advantage for these applications.

One common pseudorandom number generator is the linear congruential generator (LCG). It uses the following algorithm to generate seemingly random numbers (click the link for a proper explanation of the equation): 

This is one of the oldest methods for generating random numbers, but it has several problems, and is subsequently not recommended when high-quality randomness is needed.  

A more recent algorithm is the Mersenne twister developed in 1997. It is by far more complicated than the LCG, but also produces a better result. It passes a lot of different tests for true randomness, including the Diehard tests and even some of the more rigorous TestU01 Crush randomness tests (go here for a brief description of these tests, and for the really daring, go here for a more….detailed document). 

To generate a truly random sequence of numbers, a hardware random number generator is required. These devices uses random processes in nature to generate randomness. These processes can range from radioactive decay, thermal noise, avalanche noise and atmospheric noise. These devices have been known to silently break over time though, and the true randomness of the numbers generated should therefore be tested from time to time. 

Another problem with these devices have been that they do not generate random numbers very fast. Typical existing devices only generate a random string of numbers with a speed of 10s-100s megabits per second. To deal with this problem, a new method for generating random numbers have recently been developed. According to the researchers, the method is capable of generating random numbers with a speed of up to 1.7 gigabits, which is 10 times faster than previous devices. The method was developed by researchers at Saitama University in Japan, and uses semiconductor lasers. The process itself is really quite simple, and consist of an external mirror that reflect some of the light back inside the laser. This feedback causes the light to oscillate randomly, and this is then converted into an AC current and further to a binary signal. In total, the process uses two of these lasers to produce a single, random number sequence. According to the researchers, Atsushi Uchida and Peter Davis, the system can be built into cryptographic systems for secure network links with very little extra cost.

November 24, 2008 Posted by | General Science, Math, Physics | , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

The 13 million-digit prime

The Great Internet Mersenne Prime Search (or GIMPS) just found a prime number with 13 million digits. Continue reading

September 28, 2008 Posted by | Math | , | Leave a comment