September 12, 2005

Convergence (math please, forgive me)

This is probably the whole idea that made math simply beautiful. i'm not qualified enough to give a discourse on this topic, but it's just so brimming with possibilities that i cannot stop myself...
'disclaimer' tipe:
fine even if you listen or not, i like to keep rambling on, so this is not for morons who just can't appreciate the significance of maths. and neither is this for those who just can't keep up with my extremely irritating style of writing..(oops, i just lost my readership!)..

so what's with this convergence anyway huh??well, the first glimpse that i got of this function, was back in the days when i was preparing for the great IIT-JEE, when the class was introduced to Taylor's and McLaren's series... actually, i think, it's way before that... Remember mathematical induction? remember all those expressions for the nth term and stuff, sums of series types? well all those formulae are i think some kind of convergence... i vaguely remember there was a polynomial expansion for pi too. dunno. but definitely, there is an expression that evaluates to the value of 'e', the base of the natural logarithm, and closer the value for higher the degree of the polynomial you evaluate. so you see, the point of concern here is that the polynomial 'converges' onto the value of e for some n.. another instance, one you might relate with, is that of functions.. say f(x)=sin(x)/x. we know:
-1/x <= sin(x)/x <= 1/x; (since -1 <= sin(x) <= 1) we also know that (1/x) -> 0 as x->inf; and -1/x -> 0 as x -> inf; (the symbol '->' being approaches to ).. so we can clearly see, that
sin(x)/x -> 0 as x -> inf.

until now, if you've thought either "mamma" or "tty" or "abbeeyy" you really shouldn't be reading this, because this is all high school math till now. the reason i put up this post, beside the negative publicity that i'm sure it will create, is that this is probably the first that time that convergence knocked on our head and we didn't recognise. here, the function is "bounded" on both sides by convergent functions. 1/x cannever be zero. but we can make it as arbitrarily close as we want to. this is the idea of convergence. that in the near future, the hope that stability can be achieved. ganga and yamuna converge on the banks of allahabad. 1/x and -1/x can be thought of two rivers that are coming ever closer as we go to larger x, in search of the elusive Prayag. since sin(x)/x keeps bouncing between these two rivers, it also is bound to meet at prayag, namely, zero. but since nobody can plot out all the values of x (remember, the real number set is uncountably infinite) no one can say that it cannot reach 0.

these are only instances of convergence. like i said before, to me the word means the hope that stability will be achieved sometime later. the reason i chose this topic, is that this semester, especially after mla last sem, really feels converging to a point. although irregular, like sin(x)/x in the beginning, maybe the theory i read will someday achieve a closure of some kind. the hope that all different branches of mathematics or physical sciences will converge, is one kind of convergence, and such a Unification Theorem will achieve closure over knowledge gained from observations and experiments alone. i may not be blessed to live long enough to see this happen, but atleast the subjects that i read, if they converge on a topic, or on a set of ideas, that will more than be enough for me. my holy grail will have been achieved. already, there are a few basic ideas that are common to all the courses i've taken this semester. the idea that everything is a set, introduced first in mla, is now quite evident in moc. moreover, sets can have such deeper meanings that there are whole branches of mathematics dedicated to the study of one particular types of sets. Topology is a spinoff from Group Theory, i think, and Group Theory is but the study of specialised sets. a mathematician would have killed me at this point, for the sheer callousness with which I brush aside these holy and revered subjects.

point to note: Inner Direct Product and External Direct Product of groups is very similar to the internal and external products of two vectors. and since groups and vector spaces are both Algebraic Structures (note the capitals), probably there is some unifying theory for these Structures.

We were taught (a debatable point, i agree...) right in the first semester that matrices and systems of linear equations weren't all that different. now i see that even groups and vectors aren't all that different, and everybody knows vectors can be represented by matrices. you see, everything is coming closer... or did it start off like that? were matrices first discovered or Algebraic Structures? is math creation of useful tools to work with, like integrals and derivatives, or is it understanding the true nature (think fibonacci and golden rectangle) of Nature? i think these doubts can be classified as consequences of three essential 'dichotomies' - a dichotomy is a split - discussed in a journal in the library. infact i suppose this urge to post a blog on math is the direct effect of a rare occurence, namely my reading a journal in the library. this article discusses three fundamental dichotomies that can be found in math, and also in languages in general. well, more on that later...

Flotsam and Jetsam

LOTR enthusiasts will readily recognise this as a title of one of the chapters in the second book. Infact, that's where i looted it from. Flotsam and jetsam is the equivalent of debris, only flotsam and jetsam are the result of something drowning, whereas debris is the result of something exploding or falling apart. and why is this a title, no it's not the result of intensive GRE prep's like some of my colleagues are in right now, it's just a thought out of the blue that reflects my mood right now...

i suppose life is like a huge ocean. there are ebbs and tides, there are ups and downs. far away strangers might become friends, close knit 'coteries' might just break up. imagine a raft made of wood, slats tightly bound with unbreakable cords. or atleast, previously thought unbreakable. the moment you realise that you were mistaken, and that the cords can really break apart, is the moment you realise the real danger you are in. you might sink with only a wooden log to hold onto, if lucky, or not even that. what do you do? you make sure that they are bound extra tight, and that you have the sufficient skill to tighten them up if they slacken. what if you can't? you just make sure you don't do extra harm to these ties. all this is well and good, but what if you realised that the cords were indeed only mortal bonds, breakable by all means, and you realise this only after they break apart while you are in the middle of high seas? godsaveyou!!! something similar in line is going on right now in my mind. the friendships that i thought i was really a part of, the circles that i thought i belonged to, are long gone. people only talk to me if they think i'm intelligent enough to answer the few doubts they have. others just look away, or steer clear from me, OSTENSIBLY "sticking to their own thoughts", while all the time thinking "oh god, not him again". only now do i realise that the bonds that were tied need tending to, and need taking care of, if only to make sure they are tighter. and then there are people, survivors, who can make do with just one log, and then there are other groups either too selfish to share their raft or too many that their raft sinks due to sheer weight. where am i? what am i doing? do i even have a support? would i even know if i were drowning? dunno. a big Shunya. Cipher. Sifar. Zero. no naresh, you might prove to me that i am not an empty set, but i really don't think so.

just the other day i was actually counting my friends, still close to me. i know the very act is despicable, but one never knows. one always needs to take stock of one's company and one's self. taking stock of myself has never happened, can't happen, too tired, too fearful, so who are my real close friends is the more interesting question. and i wasn't really surprised that the set had a cardinality less than 10. i dunno if it's something i say, or it's somthing i do, but i do know that i'm a natural repellant for people in general. believe me, i have tried to improve on it. somehow, nothing works. being outright frank is probably not the right way, but sorry, i lack diplomacy. and if you don't have the guts to face the truth about yourself, probably you shouldn't have any friends at all. some might say who needs friends anyway. but there is always that something more that you get, if you believe that you are entitled to it, rather than just recieving it because you deserve it. imagine being stuck somewhere on the highway. you call up someone, get someone or the other somehow to help you out, and even your bike if possible. it's just that much more fun, if it's someone who accepts you as a friend. later you could even laugh about it. but imagine the guy who came came out of pure civic sense, with no emotional attachments, and even slight misgivings about the whole idea. he probably will curse you till the end of the day... you get the idea right?..

at a time when i should actually feel elated, i really don't know the reason why i'm so down.. it just could be the sympathy factor. or am i really just so boring that i put off people? i thought i was reasonably intelligent.. maybe infact i am low in IQ, or EQ, someQ.. donno...fine. gotta get outta this shit. there are people who care for me, and there are people that i care for. guess that should be enuff. life should go on..