Back to Reality, Back to Math

This was the plan: Linear Algebra --> Linear Programming --> Convex Programming (Semi-definite programming) --> Optimization, Error Bound Analysis     Especially in Graphs!

Thanks PatrickJMT for the hard work and easy and quick math reminders. L’Hospital’s Rule was just fun!

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PS:
After starting to read the book "Convex Optimization" by Stephen Boyd and Lieven Vandenberghe; I realized that convex optimization has lots of uses data mining, statistics, control theory, etc apart from the graph analysis topic that I am reading it for. It makes it even more exciting!

Keep my eyes in the sky, but my feet on the ground

Reality is we want to be there, but where we are, we have a number of steps before we can overcome all the limitations. Gustavo Stolovitzky (IBM Research)

http://www.youtube.com/watch?v=pKi30ai35mU

This did inspire me!

Virtual-Real Rehabiliation!

Rank Reduction

One of the interesting topics in linear algebra is matrix rank. It outlines ways that columns of a matrix correlate to each other. There are basic and free columns. and you can define free columns in terms of linear combination of basic columns. It highlighted a matter of data redundancy to me. So if you can define a column based on other columns, you won't need the content of those columns any more; just keep the coefficients that should be applied to basic columns and you are good to go! It seemed to be highly useful in matrix compression. An image which is a big matrix could be compressed like this by some matter of rank incrementation.

I succeeded in getting an algorithm to alter the rank. Unfortunately as it is not the current topic of my research, I was prohibited to work on this, deviate less and concentrate more on my research area. Anyway I found interesting relevant researches on matrix rank reduction, this is at the back of my mind for sometime to be used.

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Rashomon

Rashomon: One of the best movies I have seen in my life. Whose story is true? Which is to trust? How can you judge?

What would you do if you were involved in the case? If you were the samurai, or the samurai's wife or even the bandit Tajōmaru, or even the farmer witeness? That is a hard situation for each of these characters. Tell me what would you do.

I keep my faith in the man.

Achilles Heel!

When Achilles was a baby, it was foretold that he would die in battle from an arrow. Naturally, his mother Thetis did not want her son to die. So she took Achilles to the River Styx which was supposed to offer powers of invincibility and dipped his body into the water. But as Thetis had held Achilles by the heel, his heel was not washed over by the water of the magical river. Achilles grew up to be a man of war who survived many great battles. But one day, an arrow shot at him was lodged in his heel, killing him instantly.

I had heard Achilles heel several times but never knew what it was about.

Interesting Problems

The Hershey Bar problem:

Give an optimal algorithm to break an mxn Hershey bar into 1x1 pieces. At each step, you can choose a single rectangle of chocolate and crack it along one of its vertical or horizontal lines. A single crack counts one step. You are to make the fewest number of cracks.

David Gries's coffee can problem:

Given a can of black and white coffee beans, do the following: Pull out two beans: if both are the same color, replace them with a white bean. If the two are different, replace them with a black bean. What color is the last bean?

The Common Feature Among All Living Beings

What is pain?

Is it a feeling? (If so, what is a feeling?) Is it absolute? What makes it different between us, a crocodile, a tree, a microbe? Does a microbe really feel pain?

Is pain really a good thing or a bad thing?

I think one thing common among all creatures is the pain. Whether it is a feeling or inter-cellular reaction chemical reaction or inner-cellular or whatever. It seems to be existent among all living beings.

It is the sense of pain that causes a living being to continue living. I am definitely sure of myself that I didn't feel pain, I wouldn't have lived until now.

Distribution of Factor Count for Numbers

I extracted the frequency of the factors for numbers in the range [2,  10^6]. It was quiet interesting for me. Why is it that 2,688 numbers have 30 unique factors but there are ZERO numbers that have 32 factors?

Motivation:
While studying about randomized algorithms in wikipedia, one of the very first algorithms in this context is the Miller-Rabin primality test and an important statement in their algorithm is that:

"If n is composite then at least three-fourths of the natural numbers less than n are witnesses to its compositeness."[1,2]

But a number like 22 has only two numbers are witnesses of its compositeness (2 and 11). So what does that mean. I thought that it could be true for most numbers if we get to infinity, therefore I counted the number of factors (witness of compisiteness) of the numbers from two to one million. And as the mubers grow their factor count become less comparable to their magnitude. So what does that quote mean?

[1] . http://en.wikipedia.org/wiki/Randomized_algorithm
[2]. Dietzfelbinger, M. and Hagerup, T. and Katajainen, J. and Penttonen, M. "A Reliable Randomized Algorithm for the Closest-Pair Problem", Journal of Algorithms, Vol 25, Issue 1, 1997.

In the Beginning God!

Here, I start blogging about different topics, which would be mostly based on science, religion and philosophy. I am a Muslim but I am actively studying other religions too. Deep conceptual thoughts are just so appealing to me!

Some scripture books on alphabetical order:

Avesta starts with "In the name of God"
Quran starts with "In the Name of God the Most Compassionate, the Most Merciful",
Torah and hence Bible start with "In the Beginning God",

So, it looks like they pretty much have the same idea!

I hope you have a good time here!