Saturday, December 18, 2010

Ferdinand Georg Frobenius, new giant of the 'week'.

I cannot have enough of Grigory Perelman's picture, but people have been complaining: the meaning of 'week' is being undeservedly stretched.

The new giant, with the mandatory beard, is Ferdinand Georg Frobenius.
I stumbled upon him during week 16, more specifically, while reading about the amazing history of the Grandi series, where you will find Frobenius at the last paragraph of: http://en.wikipedia.org/wiki/History_of_Grandi's_series

Going deeper underground.

Mathematical analysis draws me in like a black hole sucking light, but everybody argues: "why the hell do you need this", help!!!

I have written about sqrt(2) more than once, and how it provided many motivations in the past.
It turns out, you can also use it to motivate interest in 'mathematical analysis'.

Take a right triangle with 2 sides of length 1, and place it on a 2D coordinate system as show in the figure. Now let us try to estimate the length of the hypotenuse using limits.
We will cut the hypotenuse into tiny segments, which is a standard practice of estimating lengths of curves, and compute the total length of the segments. If we do this we end up with the answer of 2 and not sqrt(2)! What is wrong with this?
But Jad! you say, what you just calculated is the Manhattan distance and not the length of the hypotenuse. And you would be right. But, can't we make the same claim about this not working for our summation of small rectangles for definite integrals? what exactly is the difference? and how can it be formalized? Mathematical analysis can tell you.
For more fun examples, I recommend page 3,4,5 from: http://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/week1.pdf

Another topic for this post is 'telescoping series'.
For discrete integrals, our current topic in the math project, sums of series becomes very important, it is also a brain enlarging topic and a lot of fun.
One technique for summing infinite series is called 'telescoping series'.
It works on sums of series where terms cancel each other out, and what is left is a finite and easy to calculate number.
The nice thing about this for me was that it ties to the discrete math toy problem of finding the sum of i's where i goes from 1 to n.
I have given a geometric proof of this I came up with in a previous post.
One can also do it by induction.

It turns out, it is also possible using telescoping as shown in the figure.
The interesting thing is that both induction and telescoping are very indirect, in that, if you wanted to find the answer to this problem, you would need intuition first, you would get the answer and THEN try to prove it.
The more we dig into math the more we see that this is how most of the important theories get discovered. This is one of the reasons why learning about this is great for one's brain.

Usually, one tries to proceed sequentially and directly from problem to solution, and that only works for the simpler problems. After a certain level, laziness is simply not an option anymore!

Richard Bellman had the following remark once:
"Each individual problem was an exercise in ingenuity, much like plane geometry. Change one small feature, and the structure of the solution was strongly altered. There was no stability!"

Thursday, December 16, 2010

Angular to linear speed

It is quite intuitive that the linear speed for a point rotating around the origin at distance R is wR if w is the angular speed.
Since we are very much concerned about proofs in our math project, here is a proof.
I found it interesting how such an intuitive equation actually passes through quite some transformations, most relevantly, it is the chain rule and the trigonometric derivatives that play a crucial role, a little bit less intuitive than one might think!

Tuesday, December 7, 2010

I found this very nice blog ...

the content of which I only barely understand, but that is exactly what we are working on ...

I also found this nice article:
"So in conclusion, we believe that the mental apparatus to perform "lightning fast" integer arithmetic calculations such as multiplication and division resides in us all, even though it is not normally accessible. The brain appears to perform something tantamount to arithmetic calculations (or analogously equipartitioning) for some unknown aspect of mental processing. The challenge now is to unravel which aspect."

Saturday, December 4, 2010

Who said that?

"Human reason, in one sphere of its cognition, is called upon to consider questions, which it cannot decline, as
they are presented by its own nature, but which it cannot answer, as they transcend every faculty of the mind.
It falls into this difficulty without any fault of its own. It begins with principles, which cannot be dispensed
with in the field of experience, and the truth and sufficiency of which are, at the same time, insured by
experience. With these principles it rises, in obedience to the laws of its own nature, to ever higher and more
remote conditions. But it quickly discovers that, in this way, its labours must remain ever incomplete,
because new questions never cease to present themselves; and thus it finds itself compelled to have recourse
to principles which transcend the region of experience, while they are regarded by common sense without
distrust. It thus falls into confusion and contradictions, from which it conjectures the presence of latent errors,
which, however, it is unable to discover, because the principles it employs, transcending the limits of
experience, cannot be tested by that criterion. The arena of these endless contests is called Metaphysic."