A Candle in the Dark

A look on science, politics, religion and events

Archive for March 2007

Banach Contraction Theorem

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Today, I came across a nice theorem which, If I understood correctly, proves the existence and uniqueness of points of a self map in a metric space, such that the map of that point is itself. The formal statement goes as follows.

Let be a non-empty closed subset of a complete metric space and be a map. Let be such that . Then for each , the iterative sequence converges to a fixed point x, which is unique.

To prove this, one needs to first show that
(i) for any . This shows that as
(ii) as . That is, is a Cauchy sequence in X. One can prove this with judicious use of the triangle inequality. Further, since is a complete metric space, also converges to the same in

Using the above, one can show that , and that this is unique.

Written by parseval

March 28, 2007 at 6:10 pm

Posted in mathematics

Happy Pi Day

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Apart from being Albert Einstein’s birth anniversary, today also happens to be, for obvious reasons, Pi Day! So, happy Pi Day everyone!

Written by parseval

March 14, 2007 at 1:57 pm

Posted in fun, mathematics

Optical Illusions

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These are some really neat optical illusions. Many optical illusions are beautiful, astonishing and powerful. I think they show how easy it is to fool the visual system and the brain.

Same color illusion

In the above image, look carefully at the squares marked as A and B. One appears white and the other appears black, right?

Amazingly, both the squares have the same color! Don’t believe it?

To verify that the color’s are the same, try cutting the two squares and place them side by side in paint or photoshop. Or, you could use the colorpicker tool to check that the colors are the same.

There’s an explanation of why this illusion works at the MIT site.

Scintillating Grid Illusion

This is a famous one. You’ll see the sudden appearance of black dots at the crossings. Also, if you keep your eye fixed on a particular intersection, no dots appear there.

I wasn’t able to find any explanation on how this illusion actually works.

Peripheral drift illusion

Click on the image to get a larger version

The ‘rotating snake’ illusion is one of my favorites. The ‘rotation’ of the wheels occurs relative to your eye movements. It’s thought that asymmetric luminance at the periphery of the vision causes this illusion.

Click on the image to get a larger version of the illusion.

Afterimage illusion

Keep concentrating on the sign at the center of the circle.

Do you see the rotating green spot?

Fraser’s Spiral

Actually, there are no spirals!
The intersecting black arcs appear to form a spiral, but they are in fact concentric circles.

Zollner_illusion

Parallel or not?

External Resources

70 Optical Illusions & Visual Phenomena by Michael Bach
Optical Illusions, Wikipedia

Written by parseval

March 8, 2007 at 8:00 pm

Posted in fun, physics

What a sad waste

with 3 comments

Today, there was a small article which was unobtrusively placed in a tiny section of the daily paper which, not surprisingly, caught my attention.

When US engineer Paul Roediger visited the Jagannath Temple in Puri, he literally didn’t know what he was getting into!

At Jagannath Temple, there’s an age old ‘tradition’ which allows only Hindus, Jains, Sikhs and Buddhists born in the Indian subcontinent to enter. So, when Paul Roediger was caught wandering inside, the local priests were predictably outraged. So much so, that they decided to bury the food which was to feed 7000 devotes, as it had been ‘defiled’ by the ‘corrupting’ presence of the foreigner.1

Once sufficient hue and cry was raised, the local police took Roediger into custody. A three hour round of questioning later, they realized that there was no law to ‘punish’ Roediger, but still allowed him to leave only after he paid a fine of Rs 209 to finance the ‘cleaning ritual’.2

Contrast this, with the fairly recent starvation deaths in Orrisa.3 At least they could have given away the food to the numerous hungry people below the poverty line who can’t afford a decent meal, instead of burying it? 4

Notes

[2] – Sources close to the author of this post, indicate that the cleaning ritual is based on a simple, yet marvelous theory.
Now, it is currently well known that the brains of engineers and scientists are a source of those atheistic subatomic particles, corruptrons. In those of subcontinental origin, corruptrons are mysteriously blocked by the large amount of melanin in the skin. However, in ‘foreigners’, especially Americans and Europeans, the lack of melanin means that corruptrons are freely emitted. These free corruptrons have the unique property of interacting and negating the ‘holy fields’ which permeate temples.
This is when priests effectively employ the ‘cleaning ritual’. Through the effective use of mumbo jumbo, priests have the ability to create powerful ‘faith’ particles, which are the anti-particles to the corruptrons. Whenever a faith particle meets with a corruptron, it annihilates the other in a flash of irrational logic, thus restoring the holy field.

[4] – Unfortunately, the corruptrons trigger an irreversible decay which leads to the emission of more corruptrons when they interact with food. This is why the food must be buried.

Written by parseval

March 3, 2007 at 10:23 am

Posted in religion

Richard Dawkins

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I got a copy of The God Delusion by Richard Dawkins recently, and I must say that I’ve enjoyed reading it.

The way Dawkins cuts through the theological rhetoric and gets to the heart of a proposition is enjoyable to read. His claim , that all religions are irrational, is aggressively illustrated with logic and lovely wit as he dissects the various arguments for God’s existence. It certainly one of the better books I’ve read.

Written by parseval

March 1, 2007 at 10:36 pm