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Wednesday, December 31, 2008

hi there!! Happy New 2009!

Hey there!

3rd semesters over matey!! Seems like weve got plenty to do!!

Twas a very successful semester for me- I finished off the ten seasons of FRIENDS- although a meagre first viewing only! But 20X24X10=4800 mins=30 hrs of popcorn popping was not ALL I did!! I also passed a chemistry course in spectroscopy (having flunked the midsems by one point- but made up good time to claim 85% in the final assessment), passed a physics course in Quantum Mechanics which was over-the-line mathematical, loved a complex analysis course, proved myself a SENSITIVE scientist wannabe (humanities course in “Indian Heritage in sciences” baby!), and watched many more movies of my kind.

I also grew a goatee and shaved it, twice, notwithstanding my status as a bovine creature. There was also a great deal of growing up, yknow, philosophically and emotionally; but the present stand is to put all the growing up behind the linen!! We do need to grow sometime soon, mind our own business and manners- but give it a break grumpy; we aren’t 20 yet!!

New year is coming up, and the resolutions I take grow increasingly stupid!! Ahh , the time to start all over again!! We do need that now, don’t we? I wish I had a bunch of Donald Shimoda quotes to throw at you, but then do you know at all who he is?

Ayn Rand does write beautifully and she climbed into my hallowed unrankable authors list with Tolkien, Rowling, Richard Bach, Erich Segal- after I read only one chapter of “Atlas Shrugged” (talk of me being biased from the hype she has around people- but the factor is that she satisfied all my expectations- she could have let me down, like Rowling in “Deathly Hallows”- but AR didnt).

VARIABLE CHANGE from movie ‘21’

And last of all the insight into variable change that my musings revealed to me only this morning. You see, in the movie 21, the protagonist who goes by the name of Ben is a brilliant young student who gets into a ‘card counting’ club in MIT, and the subsequent repercussions on his lowly life. In my opinion, it is not a timeless piece at all, but enjoyable like all crap in this world is. There however Kevin Spacey stars as Miki Rosen, the MIT prof with a helluva gift of oratory and the leader of this card counting team who go into Vegas at weekends to beat the game of Blackjack using logic and cool head. So, Miki asks Ben an apparent paradox, also known as ‘Monty Hall Paradox’ which I will now proceed to explain.

Miki tells Ben to imagine ( I am trying my hands at my own version of Kevin Spacey’s inimitable style in this monologue) that he is in a game show; the host offers him three doors behind one of which is “brand new sports car” and two goats behind the others. Ben chooses door ‘1’ in this choice. Before declaring the result however the host spices things up when he undoes door ‘3’ behind which there is a goat. Then Miki asks Ben if he would switch his choice to ‘2’ or not.

Heres the key- Ben chooses to swich to door ‘2’.

This decision, he says, is propelled by cold hard logic, because the host knows everything and the unlocking of door ‘3’ changes everything; ‘1’ has a 33% percent chance of having the car, while ‘2’ has 66% odds of having the car and hence “thank-you-very-much for that extra 33%”, he says.

People are very confused by this explanation, as was I, because apparently the unlocking of the door ‘3’ redistributes the odds to 50:50 between ‘1’ and ‘2’. BUT what I forgot, and most people do, is that the choice to unlock the door ‘3’ by the host is a conscious choice, and in doing so he seals the odds to one-thirds and two-thirds respectively.

Lets examine the 33% case when the car is originally behind ‘1’; if Ben switches then, then he loses. Note that the host would not show ‘1’ and would not show the door with the car.

For the 66% case that the car is either behind ‘2’ or ‘3’, the host eliminates ‘3’ and now if one imagines that the car cannot be behind ‘1’ which IS our assumption here, one is 100% certain that it is behind ‘2’. If the host uncovers either ‘2’ or ‘3’, the response is the same, make the switch for a 66% chance of win.

Summing up quite, at the beginning:

‘1’=> 33%

‘2’ or ‘3’ => 66%

The host eliminates ‘3’, one is left with,

‘1’=> 33%

‘2’=> 66%

This required quite some thinking, but still people convince themselves that this is actually not a correct explanation at all; it requires one to keep track of the abstraction long enough to see the answer.


Saturday, July 19, 2008

The Curse of the Evil Prof.

Tuesdays have always been pretty hectic for me, at least ever since I entered college. The morning, as usual, was fully packed with classes on all slots, & after lunch, I head for my first physics lab. I'd always heard from my coaching teachers, the names of the experiments I was about to encounter & also done some reading about the practicals in IIT (Chetan Bhagat's 'Five Pt. Someone'!!lol). So, I was pretty confident. I've been forging readings ever since I went to my first science lab in school, so my confidence level was further boosted. Even made fun of a guy who said that my 1st experiment was to be in a dark room & he was totally frightened, after the experiment, to go to any dark room. That the room was cursed, the experiment was cursed!

So, I went inside, feeling how much I'd love doing all my experiments in there itself, for there was air-conditioning in the room! My first experiment-'Grating Spectrometer'. My instructor for the experiment- Prof. S.N. Bhatia, a name none of us from P4/2 batch would ever forget. I had known it to be a difficult experiment with sure shot time wastage on the set-up part (the exp. in itself is very small, only the setting up of the apparatus is time consuming), the first time. But, I had that the experiment is somehow completed in the given 3 hrs. Huh, were they all bluffing when they said the last line? It took me 2hrs to identify that I was to reach even till the Schuster's method. After 10 more minutes wasted, where the Prof. was shouting at me for being such a lousy bastard, good for nothing, bribed entry into the IIT, he came to check out my status.

I've always felt, though never actually done, that I have every right to shout & abuse, at least twice as much as he did, a Prof. if he himself is unable to do an experiment of which he is the instructor. Whatsay? I don't have even 30mins to do the experiment, & he says, "Maybe there is a problem here." What 'maybe'?? :-o You were trying for the past 20mins. There has to be a problem. Pair me up with someone else. Help, help!! This experiment has 6marks contribution in the total. Somebody plz call 911! Somebody do something. Oh God, I wanted to do well in this sem at any cost and this prof., he has made me a pauper before I could say 'any cost'! He has screwed up my 1st experiment and with that my confidence level…

Well, that's how I felt. But, the prof. never did. I was still stuck up with the same apparatus, but was asked to bypass that step, for then. How incredulous…

But, then someone had called 911. Oh no, it wasn't 911, it was a Godsend, my guardian angel in the form of a lab assistant. He came, and with him, returned my life, which was going out of the Dark Room, like the light from the high powered bulbs, on our tables, on the click of a button. He came really as a savior, a messiah, and said, "Bhatia ko jaane de, fir paanch minute me tera Schuster's khatam kar doonga. Tu bas uske baad apna readings jaldi se le lena. Yeh Bhatia bada hi ghatia hai!". And we both stuck to our word. I took only 30mins extra time & completed my experiment, and got real poor marks, but in my own P4/2, it was an average score.

This was when I realized how that superb air-conditioning became the DARK ROOM, all in capitals, when it became the dungeon, the a/c & the darkness felt like I was in a drawer of the mortuary of AIIMS, and this was when I decided never to return to that room again, never to go back to that death chamber…

Thank God, I came out alive!

PS: I did go back for the last 2 experiments and actually cracked them, & got an overall AB in the course. Maybe someone had broken the 'curse of the evil prof.'!!

OSL- The “Coolest” Place After Midnight!!

I'd never oppose any of you who say, "hostel life is the best", that you'd give anything to come back to the hostel. But, this article is for some of us, you may call spoilt brats, who are ready to forgo the best place… It is for those of us, who want to find the coolest place, literally, and not the best place. Being a Delhite, for me living without A/C is like a nightmare (I can assure you coz I have lived that way too), and Bombay, I'd say is much worse than Delhi, with the unbearable humidity combined with equally abysmal temperature conditions, which at some point even crossed that of Delhi! The only solution was to find an air-conditioned environment. And this is an article about that night, when I along with Twish, or Charsi, as we call him, made that extra effort for the greater good. It is about that night when the two of us, without even caring about the exam the day after went out and searched for THE HEAVEN, for the place which could satisfy the three basic necessities of ours, "the spoilt brats", A/C, WiFi, and Plug point!

We had been to the library once, but found out that even there the A/C is switched off at 9pm. 9 PM! That's precisely when a normal IITian even starts to think that he has books which should be utilized for studying rather than as laptop stands for the HP and Dell laptops! That's when we, after the 1st course of dinner, think that we should plan the day's study! That's when it all starts and they end it then…why? L. Why should we then go to the liby, a place so far off, when the same condition is available even in the hostel. Maybe there is a bit of noise, with the 5.1 Bose speakers shouting for us to come and dance to those foot tapping tunes of some Punjabi singer, but then of there is no A/C why the hell will we travel so far? Just to get tired and spoil the "study mood", and sit in the staff canteen? We needed a place…a place where we can do whatever we want to do, study, play, chat, and everything, and also should have A/C. Comp Centre was an option, so we went there on that unforgettable night, loaded with enthusiasm and our laptop bags, just to be left high and dry…that place was nothing more than a teaser! You go inside, you have A/C, a nice shaant environment, but internet…you wont be able to connect your laptops though…we even tried using the lan wires there just to realize that we hadn't achieved anything, but surely had plugged out a couple of comps from the internet! And just outside what we found was no A/C, but a nice WiFi environment, and yeah, here also we had shaanti! A real teaser…high and dry…

From there, dejected though we are, we still do not give up on our search for that heaven of a place, and hop relentlessly, from MB to GG to KReSIT to where not, to only find everywhere that we, like the wildlife of today, are now in a place which ain't our natural environment. Our natural environment and our basic necessities were no longer available. We realized the bitter truth, that though we are in one of the top institutes of the world, we can't find even a single place that'd suit our comfort, and where we could study without that physical and mental discomfort caused but the unbearable weather. on our way back to the hostel, to become the laughing stock of the whole wing, we stop at the most unlikely of places-the Old Software Lab. It wasn't like we had never been there, I used to literally live there during our CS101 days. I used to study there, sleep there, and used to come back to the hostel only for the morn brushing and bath! But the moment they banned our logins there(after the CS101 got over), I never stepped back into that lab. I was standing there today, way past midnight(I think it was around 2.45 or 3 in the morn), waiting for Twish to switch on his WiFi and confirm that we as usual weren't in the right place. But, as all of you bollywood watchers would have realized, we struck gold this time, or as I can write now, with the rising inflation and the price of crude, we struck oil! We had everything we needed right there, in front of us. This was it…we had done it. This was our very own Terebethia, and that night, the 2 of us had found that magic rope and forged the bridge to Terebethia!

Every night since then, we were there unfailingly, after midnight, playing typeracer or NFS Carbon, or chatting or watching Chak De, & also, I forgot, studying during the free time, for our ongoing endsems!

Monday, December 31, 2007

Period three implies CHAOS!





note on the title:this is the exact title to the 1975 paper by Yorke, an applied mathematician - who is credited with bringing back Lorenz' seminal work.Who was Lorenz?Read on!



We can relate pretty well to our intermediate school differential equations. And though they did not take as much toll as indefinite integration did - it all boiled down to some really frightening (and at times transparent) expressions that had to be moulded into a solvable form we knew of and follow the old techniques.
But we have heard of seldom-encountered-but-know-they-are-there type of UNSOLVABLE differential equations that cannot be solved. So? Some differential equations cannot be solved. We can take that in our stride right? Try this! Only some differential equations can be solved, a wider variety - infinite for all we know - are unsolvable. Its a bit like being at rest with the idea that our rational number system being infinitely dense and that makes the irrational numbers exceptions- and you get a proof out of the blue that irrational numbers are more dense on the number line, or metric space-technically speaking.
Since its inception in 1600+something , calculus has provided a novel way for studying continuous variations- infact it arose as a tool for physics (no offence to the mathematicians)- and has been the 'quantum leap' for sudy and modelling of physical systems in nature, otherwise known as PHYSICS.
An etymological observation here- quantum is a very small quantity right? so how does a 'big' leap translate to 'quantum' leap?- or is it a reference to the enormous paradigm shift the 20th century physics community had to undergo in order to accomodate the most successful theory of their time.
Returning to the main stream of thought, though the chaotic behaviour of systems (i am going to ask you to go by the 'feel' of the word here - a random unpredictable behaviour that refuses to settle down to any semblance of predictabilty) seems disorderly, it is actually deterministic-- that is you can predict, without any approximations, the outcome of some system. The science of 'Chaos'(the term was popularised by James A Yorke, an applied mathematician who is said to have rediscovered Lorenz, in his publication 'period three implies chaos') is essentially classical- there is no relativity or quantum mechanics involved here.
but people back then were being paid to find order in systems! why study disorder?

But considering its simplicity and comprehensibiliy, this science started producing papers only in the late 70s and 80s becase of the stagnantic (no such word- you DO get the sense!) set of invisible rules laid down by the scientific community-> to produce a paper that was RADICAL and DYNAMIC DEPARTURE FROM ORTHODOX was suicide! To get a grant and publish papers you had to, and still have to stay away from being too original.
You have to model physical, biological or chemical systems with the existing mathematical tools- who dares to explore systems that are random and defy modelling (read predictability). Regarding the unsolvable diferential equations we were talking of, imagine a system of two or three such equations that are coupled (meaning that in a system of x,y and z, all three can appear in any form, linear or nonlinear,in each expression.) - we are trying to gauge the amount of complexity that can be built in around these systems.these system of differential equations can, however, be approximately solved using computing algorithms. so we can get a stream of data with varying time in which given a set of initial conditions mark the starting points and the system of equations mould the path of the variables. this is all the insight we need to get into the story of Edward Lorenz





"Lorenz was born in West Hartford, Connecticut. He studied mathematics at both Dartmouth College in New Hampshire and Harvard University in Cambridge, Massachusetts. During World War II, he served as a weather forecaster for the United States Army Air Corps. After his return from the war, he decided to study meteorology. Lorenz earned two degrees in the area from the Massachusetts Institute of Technology where he later was a professor for many years."--wikipedia. for all our purposes, edward lorenz was a meteorologist. during his days as weather forecaster he built a very primitive model of the climate system of earth, with a view to predict local weather. though the work was not very successful in terms of scientific design, it became famous in his department, the system of twelve interwoven differential equations churned out a stream of data on a long roll of paper (remember that this was around 1960, give or take one year on each side).people would bet on what the weather would be that day- windy, or sunny or wet. one day, so the story dictates, he wanted to start midway instead of starting allover again - putting in the same initial conditions and wait for an hour to go back where he left- instead he looked up the data in the previous sheet and fed it to the computer.this data seems to have survived for storytellers- he put in an approximation."One day in 1961, he wanted to see a particular sequence again. To save time, he started in the middle of the sequence, instead of the beginning.He entered the number off his printout and left to let it run. When hecame back an hour later, the sequence had evolved differently. Instead of the same pattern as before, it diverged from the pattern, ending upwildly different from the original. Eventually he figured out what happened. The computer stored the numbers to six decimal places in itsmemory. To save paper, he only had it print out three decimal places. In the original sequence, the number was .506127, and he had only typedthe first three digits, .506. "- http://library.thinkquest.org/3120/old_htdocs.1/text/fraz1.txt .




Although i would like to think that there was more than one variable in the calculations. this illustration does its job pretty well though, it shows that the pattern, or sequence if you please, was very dependant on the initial conditions. For practical purposes, an experimentalist prides himself if he can determine a physical quantity to the third decimal place. Here the absence of the fourth, fifth, and sixth decimals had erased all similarities to the original sequences within three-four cycles. What lorenz aptly demonstrated after this was that long term weather prediction was impossible. To do so he stripped down his system of equations for convection to their bare essence to three eqations.afterwards it was discovered that his equations mimicked a water-wheel- "At the top, water drips steadily into containers hanging on the wheel's rim. Each container drips steadily from a small hole. If the stream of water is slow, the top containers never fill fast enough to overcome friction, but if the stream is faster, the weight starts to turn the wheel. The rotation might become continuous. Or if the stream is so fast that the heavy containers swing all the way around the bottom and up the other side, the wheel might then slow, stop, and reverse its rotation, turning first one way and then the other. " (James Gleick, Chaos - Making A New Science, pg. 29).This is a system of equations of the same family, or perhaps the same set that lorenz used, you can try plotting the behaviour of this system at home.




(i)Dx=10*(y-x)



(ii)Dy=-x*z+28*x-y



(iii)Dz=x*y-(8/3)*z



for starters, D=> d/dt(operator);x*y='x' multiplied by 'y'
Whenever physicists encounter such a system they try to think of a quantity that is conserved, or in special cases, show predictable behaviour. Think of the simple harmonic equation D**2(x)=-w*x['**'=>raised to the power; here D**2=>d/dt(d/dt)]. in "phase space"- in which all possible states of the system is representable- this translates into a circle.Please note that lorenz was not aware of this terminology because this field of mathematics came in 1971. Say for turbulent flow in a cylindrical pipe, the elements of water that collide with the cylindrical wall, act as if it is attracted by the centre of the cross section of the pipe- THAT, thus, is the attractor in this system.Note that the elements that dont collide but are bound attractively to the surrounding elements also behave as if they are attracted, albiet by some different force law than inverse square. in lorenz' system, the attractor came out in the form of a double spiral.see picture.so although the momentary behaviour of the particle was seemingly random, in the long term, it was sketched out in the form of a ever-continuing, never intersecting double-spiral.sort of gives you insight into the butterfly effect- "The flapping of a single butterfly's wing today produces a tiny change in the state of the atmosphere. Over a period of time, what the atmosphere actually does diverges from what it would have done. So, in a month's time, a tornado that would have devastated the Indonesian coast doesn't happen. Or maybe one that wasn't going to happen, does. (Ian Stewart, Does God Play Dice? The Mathematics of Chaos, pg. 141)"
The bottomline in this first phase of insight into chaotic systems was that certain systems behave too sensitively to their their initial conditions and that Lorenz' work established that long-term-weather forecasting was impossible. Now comes the sadder part of it, his paper was published in a Swiss Meterological Journal that lay in obscurity waiting to be rediscovered.
The next story that defined birth of this new science was the work of Robert May, an ecologist in the 1970s. When one stands in the shoes of an ecologist and tries to model the population of, say fish in a pond, one inadvertedly realizes that the simplest model would be one in which there is unlimited food and the population of one year depends linearly on that of the previous year.i.e P(n+1)=a*P(n);P(n)=>population of the nth year.Now if you start taking into account the total food being constant,and the survival for existence, you arrive at :


P(n+1)=a*P(n)*[b-P(n)]


If we want to capture the essence of the equation in , x(next)=r*x(present)*[1-x(present)]- this is the logistic difference equation that can be used to model a biological system in a closed ecological system.Believe me when i say that this is only an approximation, and is probably arrived on by hit and trial rather than some deep insight into the factors of demography. Certainly, this is an accepted recurrence relation only because it yields the results, i.e. it can mimic, within permissible errors, the population of certain ecological systems like that of fish in a pond,instead of being the the factor for the population variation. The least we can do to explore this field of the logistic equation is to study the patterns that appear for certain values of r, or the characteristics of the demographic pattern that evolves with r:(courtesy wikipedia-'logistic map'): When r is between 0 and 1, the population heads to zero, independent of the initial population.


When r is between 1 and 2, the population quickly stabilizes on the value (r-1)/r.



When r is between 2 and 3, the population again stabilizes on (r-1)/r but oscillates around the value for a while.The rate of convergence is linear, but at r=3, the rate is excruciatingly slow - less than linear even.
With r between 3 and 3+(6)**.5[~3.45], the population oscillates between two values dependant on r.



With r within 3.45 and 3.54(approx), the population oscillates between 4 values forever.





With increase of r from 3.54 onwards the the population oscillates between 8 values then 16,32,64..... and on.The period of r with the same no of oscillations decreases rapidly, with the ratio between two such intervals approaching delta=4.669.... the Feigenbaum constant- whose story we shall come in with later. Now we can throw in the term period doubling cascade- think of infinitely branching arteries.


Beyond 3.57, most values exhibit chaotic behaviour, but again some values of r are there that show non-chaotic behaviour. Like 1+(8)**.5[~3.83], these are islands of stability. In fact the bifurcation map is a fractal, if you zoom in on certain areas they seem 'similar' to the whole map again.

Perhaps we got in too deep into this mathematical system than we intended to, but this isn't even starting to get technical.Robert May thus, showed how complex and beautifully simple systems like these can be at the same time.

The third story has to be that of Yorke who with Li proved that any one-dimensional system which exhibits a regular cycle of period three will also display regular cycles of every other length as well as completely chaotic orbits in 1975.Yorke's paper was named "Period three implies Chaos"- where he coined the term for use as we know it today.
For those attracted to abstract objects/theories of beauty, Smale's work on the mathematics of chaos was the first attempt that popularised the field of chaos to physics. His work can be, at its very basic simplification, be visualised by amazingly simple means. Consider a bar, bend it into a horseshoe shape and fold the whole thing overall again. the points that were initially distinctly apart originally, come together. The action of folding, however can represented by the difference equations operating on the variables. That's pretty much all a layman, like myself, can make of Smale's work.

But there is still one matter of importance in these regions and that is an insight into the true nature of chaos as we should know it. Chaos is NOT random, as the word sems to imply. What a chaotic system IS that it is very sensitive to initial conditions. So any approximations anywhere shifts the data wide off course. So once we have the precise set of initial conditions, chaotic systems ARE deterministic.
note: i attended a lecture by prof. gadagkar of iisc bangalore, on behaviour of insects, and there he said that 'simple algorithms CAN produce complex behaviour if some feedback relation is inbuilt'-- remember smale and his shape-bending difference equations...... poetic isnt it?

Friday, November 02, 2007

IISER up next!

well well!!
post the iit exams this is going to be my first meaningful gibberish(_note that i can be quite comprehensible when i want to!).
sisters and brothers of the kvpy community! i am at iiser kolkata!
here we are taught, as we were all informed during the summer camp of last year, all the four disciplines of science... pcmb that is.
but what we are actually aiming at is fusing these lines together, and promoting the scientific community as a whole.(this was my version of the iiser brochure documents!)
now for the real picture, iiser kolkata has no hostel and campus of its own and we are the first of many future batches(technically 2nd, but egotising is an important skill these days!).
the hostel is pretty far from the institute and we reach there by bus.
the highlight though is that we have no restrictions on the noise level we make at hostels and ,this is important, no protocols for punishment.
so everything you do is a trend setter (applause!)
apart from that , all you get here is a bunch of marginally intelligent (modesty?maybe!) people packed up, and voila!!
the effects are quite satisfying!!
we have a fab physics course, lapping up the intricacies of mechanical systems in general and exploring whats been wrong with the high school matter.
the computing course(read mathematics practical) is complementing the same course- we learnt python and gnuplot and together they help us model novel mechanical systems and see how they behave-- visualization that is!
the chemistry and biology take their toll -- i am not intersted in either but still its nowhere near physics in fabricational capacity.
but our biology department is exceptional and contains both good human beings and pedagogues(they usually are.) and there is an overall interest in biology because of them.
the maths is the boring now, but i intend to take it up afterwards-- we are doing the cornerstones of calculus now.
i guess that it got a tad tooo long, but you have to give in to egoTism (egoism with a T for talk, that is) sometime!
thats it.
about other kvampyiians,
sunita panda is studying mech. in nit trichy,
abhijit awadhiya is in iiserpune or niser,
jyothi v nair (bangalore camp) is in iiser kol
......... uh thats all,
and i have material for a post or two now!