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Archive for May 2010

Birthday Paradox

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We all have encountered coincidences of sort in our lives. For example, it happens that we keep on thinking about a particular person and suddenly we get a phone call or email or even a visit from that person. Similarly, we promise to meet someone at a particular time and venue but out of no where something emergency turns up and have to cancel the previously agreed meeting. A list of such coincidences can get very long. Mathematics also has a term for such coincidences and it is called Probability. There are so many things in Probability Theory that surprises us. And, one such is Birthday paradox.

In Probability Theory, ‘Birthday Paradox’ refers, that in a group containing certain randomly chosen people, there is a finite probability for some two people to have exactly the same birthday. For a group with 23 people, this probability is around 50%. This increases exponentially as the number people in the group increases. With as many as 80 people, the probability gets around 100%. This is of course, based on the idea that people are randomly chosen and each day is equally likely to become a birthday for someone.

For more details you may visit http://en.wikipedia.org/wiki/Birthday_problem.

Birthday paradox holds an important position in computer security also. It can be used to attack the hash functions. For example, let us say f is a hash function such as f(x) = y. With hash functions, it should be impossible to guess ‘x’ given ‘y’. In other words they are one way. Another characteristic is that it should be very difficult to find two inputs that yield same output i.e. it should be hard to find x1 and x2 such that f(x1) = y and f(x2) = y. This property is called collision resistance. But using birthday paradox it becomes relatively easier to find such collisions.

Today 16 May is my birthday. And, in facebook I have around 275 friends. Interesting thing is that I do not have a single friend whose birthday coincides with mine. So either some of them have entered incorrect birth date or hidden their birth date from display or for some reason facebook is not notifying me of this. Otherwise, based on birthday paradox I should have at least one friend with birthday today. Perhaps if I include the ‘near-miss effect’ then we can get some coincidences. For example, number of people with birthday separated by certain days. Such latitude will increase the probability of coincidence. But originally birthday paradox theory does not include this near miss effect.

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Written by prajwalan

May 16, 2010 at 1:13 pm

Music

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It is hard to find someone who doesn’t like any music at all. Music has many forms and varieties. They not only entice our ears but also refresh us and some even touch our soul and inspire us to do something particular. They reflect the culture, religion and history of the place from where they originate.  Hence music from different corners of the globe carry different sense, melody, harmony and notes.

Scientifically, music is an art form with sound as its medium [1]. Therefore, music is composed of sound waves vibrating with different frequencies and pitches [2]. One might wonder the difference between music and noise. While a particular sound pattern may be considered music by some, the same might be regarded as noise by others. However, scientists and researchers have attempted to point out the differences in mathematical terms. One source [3] mentions that “Music is sound with a discrete structure while noise is sound with a continuous structure.” Regardless of the mathematics behind music and noise, one thing is certain: music is what you find amusing and even noise can sometimes be musical to some.

As I write this article, I can hear chirping of birds outside my window. I find it melodious and I would consider it a music. But what about the birds? What are they trying to accomplish by making those sounds? Are they communicating? Or, are they just making some noise? I don’t believe that mathematics can explain that. They could be communicating or making noise, but I am finding it musical. So, noise or vocal communication can also be considered music. It all depends on the listener. There is a saying, beauty lies on the eyes of beholder. I would say something similar for the music also: “Musical value lies on the senses of perceiver”.

We have often heard that music has no language barriers. I believe this is true. Personally speaking, I listen to music from different parts of the world. It really does not matter what culture or religion or race the particular place belongs to.

Following are two links to music from Nepal.

Instrumental 1

Instrumental 2

You can visit the site http://www.pronepal.com/ for more music and songs from Nepal.

Please put links to some good music from your country or other countries you know in the comment below.

Written by prajwalan

May 13, 2010 at 4:01 pm