Thursday, March 17, 2005

Happy St. Patrick's Day !! I wish I had green glasses too !! Today is Matt's birthday too. What a day to have a birthday ! It morning now and I as you can see the day is bright and sunny and the picture is taken right in front of my room at ambassador hall. I am excited today because I am going to learn Cauchy Intergral formulae ! C you folks take care and drink in moderation ! Posted by Hello

Wednesday, March 16, 2005

I love Lou Dobbs

If you been watching news on American channels like CNN you would agree that they are just obsessed with people. I was thinking about what they have shown in past 6 months. It was election time and the whole day there was just a news of George Bush and Senator Kerry. After that there was Scot Peterson who killed his wife, Then there was this guy who also killed his wife and lied about his life that he was going to Med School, after that there was Tsunami Crisis in Asia, then BTK killer,Release of Martha Stewart, after that there was a shootout in Atlanta Court house, Michael Jackson's trial and his appearence in court house wearing Jamies and today back again to Scot Peterson finally going for lethal injection . Its amusing that the amount of time thier coverge spends on talking about people. I like Larry king however his talent is being wasted in interviewing people who have already been talked about the whole day. Is it anything to do with scarcity of news in America or their marketing head is dumb to think that people only see news channel to know what is going on with others life. They can make anybody a celebrity They spend too much time on news which has little significance to the life of the people considering their reach and potential. They have some great programs like Next which should get more room, also they have some good documentries. Talking about documentries BBC rules and I miss that.

CNN has some great anchors like Anderson Cooper, Paula Zhan to some real characters like "Lou Dobbs". I love his show to me he is the funniest newscaster around because he makes you believe that he has never even taken a high school class in economics. The three main axis of his program called "Lou Dobbs tonight" are "Red star rising" which talks about growing might of China, "Illegal aliens" which focusses on Mexican workers and threat to USA and a "daily internet poll" besides some equally enthusiastic guest like him. I wish he could send me his book he has written. The poll always has a rating of above 95 percent approval because of you know if you have ever taken a stat course of the kind of bias it generates. You cannot stop China from growing. It has a leadership which has clear goal in its mind where to steer China in the coming years. The funny thing about people like LouDobbs is they still live in the medival era philosophy that other countries cannot grow and there has to be a supreme power to rule the world. Internet has dramatically changed the work equation worldwide and the salaries in developed countries are taking a plunge because of cheaper availability of workers in other countries. Few years down the line these countries will have competitors in the form of other developed countries and they will have to think like what USA needs to do today.

There are equally hilarious reports I read in Newspaper like Timesofindia which are always making dismal comments about the progress of USA. They should remember that the size of country does matter in the long run and if you can educate your workforce your economy will be self sustainning and both of these things are in favour of US. It continues to get the best people because of its size and does a good job in educating its people. The second generation Indian students are lot more successful in US which should convince anyone about the first rate education system who continually to harp on the indian system of education . India like many other countries is now heavily depended on US universities to get quality graduates to feed its economy and US also needs these people to push its economy. US has clear advantage in terms of the best education system in the world and in knowledge economy is its ticket to futher economic prosperity and ofcourse with a fertile land like its it will continue to be a major agri business force in the world. US is a great country and no doubt anyone will fell in love with this country. Wonderful warm people who think for others and considerate and thanks to Lou Dobbs it makes it wonderful.

Saturday, March 12, 2005

Day 1 of Spring break

Well its day1 of the summer break. Yesterday I partied with David and Tomei at Quatros and today it was Andy and Mitsu. We went to a place in Marion called Pioneers Cafe. Andy had been to this place before and it was his idea to have a lunch there. I liked the place with thier waitresses dressed in traditional outfits which you can only see in movies of bygone era. Everything at that place could hold your attention. They had RC instead of Pepsi or Coke. Instead of the glass it was served in a kind of bottle. Look closely at my cola. The glass for water was my favorite. It was a small cute one. The food was scrumptious and thats why we ate like pigs. Morover the waitress was very jolly and by the time we were ready to live we were stuffed to our necks. I like the southern US the people are so warm and friendly and yes the biker groups on their big harleys and leather outfits. Don't ask me when I am going to buy a harley for myself and cruise to Dakota. It should be fun ! isn't it ? I can visulalize my frail body riding a 1100cc bike with dark glasses and saying Yo Man!! Posted by Hello
Me goofing in my room ! Folks, spring break has started and its snowing outside. Is it spring break or chill break. Holidays are sweet !! Posted by Hello

Tuesday, March 08, 2005

Analysis of Real and Complex Numbers Analysis

This is a work in progress, I tried to copy paste it from my other blog but it destroyed all the formatting and coloring. But here it is as of now. Certainly it will undergo changes. However at present its not in bad shape.

The study of analysis means studying infite. Sequence Sequence are function whose domain is natural number and range is real number or complex number depending on the context. Real Number History and Challenges of Complex Number Origin of Complex Numbers does present a challenge to anybody. Most of us are introduced to the complex numbers in our grade 8 when we learn to solve quadratic equation and we encounter sqrt(-ve number). I am currently taking complex analysis so I am fortunate to read some books and broswe through websites, so here is an easy way to understand these mysterious, imaginary numbers in a concrete way. Historically five people whose name comes up have an interesting story. These people are Del Ferro,Fior,Tartaglia,Cardano, Ferrari and Bombelli. Del Ferro was the first person who is known to have come up with the solution of the cubic equation, since he didn't published much infact none of his publication has survived. The legend is he revealed his work to one of his pupil called Fior. However he revealed only one part of the solution. Not realizing this Fior become overconfident in his mathematical ability and met Tartaglia in a mathematical duel. Tartaglia solved all the 30 problems posed by Fior and established himself a force in mathematics from being a simple self taught teacher. History doesn't pain Tartaglia in a very good light. He is said to have not published his results so that he could won more mathematics competition. Then arrives Cardano a person who was an illegitimate, genius mathematician. Whose story is from rags to riches and then dying in ignomity. He was an Italian who tried several times to gain entry into influential mathematical circle but couldn't gain it until he proved his mettle by writing books and excelling in treating his patients. Once he arrived in the mathematical scence he was a force to reckon with. He persuaded Tartaglia to reveal his formula for cubic equation but Tartaglia entered into a secret pact with him not to reaveal it to other. Cardano had a terrific protege whose name was Ferrari and they discovered that its not Tartaglia who first discovered the formula for solving cubic equation but it was Del Ferro. So Del Ferro thought his promise to Tartaglia was not binding to him anymore and he published the result in his celebrated text Art magna. Tartaglia was infuriated and challenged him for mathematical duel. After much deleberations Tartaglia finally agreed to take on Ferrari, Cardano's protege. Much to his chagrin he finds him very much familiar with the nuances of mathematics and Tartaglia fleds the very same day. Thus Ferrari comes out of the shadow of Cardano. He goes on to find the general solution of fourh power equation and lived on a happy life unlike Cardano. Who suffers ignominity because of his son's act to poison his wife and his name in that conspiracy. His son was decapitated. The other son is equally incompetent and that tears apart Cardano. Finally comes Bombelli who after looking at the formula sees what all had previously overlooked and dismissed as unimportant and that is Bombelli's wild thought. Well today I also got to know that there are numbers called Quaternions which are superset of Complex Numbers.

As complex numbers are defined for 2 dimensions. Quaternions are defined for four dimensions. Studying complex analysis is very much similar to any other analysis class and infact some of the theorems like Bolzanno Wierstrauss, Heine Borel do pop up in complex analysis.

Application of complex Numbers There are lot of places where complex numbers are used. One is while studying Boundary Value Problems in Heat flow. For a simple disc if we keep it heated at the boundary the temperature at the centre is the average value at the boundary. For example if 3/4 of the boundary of the disc is at 0 degree and other quarter is at 100 degree then the tempeature at the centre of the disc is 25 degree centigrade. Complex numbers also find application in things like finding the nth root fot eh equation. One thing where complex numbers rule is the Fundamental theorem of Algebra. Here is a list of common place where we can use complex numbers. Complex Number Advantage of Complex numbers Riemann's Contribution Zeta Function, Riemann Sphere Riemann is known for extending the concept of Zeta function from real to complex number. Zeta function was discovered by Euler and it gives a relation between prime number and the infinite series. For real numbers it looks like Zeta(s) = Summation (1/(n^s)) where s is any real number > = 1. Note for s =<>

Cauchy's Contribution Cauchy and Riemann have affected the development of Complex number so much. Cauchy Cauchy's Theorem for Analiticity Cauchy's theorem for Analiticity says that

Laplace Equation Laplace equation is a fundamental eq in many fields of physics and engineering and it gives a test for very important harmonic function. If we have a function U(x,y) then its harmonic if we can find Uxx+Uyy = 0. Note here Uxx means second partial with respect to x and Uyy is second partial with respect to y Topology to know for Complex Numbers The first thing is we will define Neighbor in set notation and from there all other definitions of Interior pt, Exterior pt, Boundary pt, Open Set, Close set etc will follow.

Neighborhood Its very easy to define neighbor in terms of delta. N(a,delta) ={x abs(x-a) <>Interior Point The interior point is all points Open Set Usually open set is denoted by letter G. It includes all the points in the domain except the boundary points. In set notation it will be that we can find a delta Neighborhood > 0 such that all the points in the neighbor are subset of the set.

Closed Set Closed set enclose the boundary also. Other definition of close set which maks more sense is that it contains all the limit points inside the set. So if we have a converging sequence like {1/n, n belongs to N}, we can see that this sequence is bounded between 0 and 1. However the set is not closed because it doesn't contains zero the limit point of the sequence 1/n. (If you may be wondering about the series 1/n, then yes its not convering, 1/n is a diverging series. What it means is if you do sum like 1+1/2+1/3+1/4+1/5... that sum will goto infinity when n tends to infinity) but as far as sequence 1/n concerns its converging.

Connection Simply Connected A set is simply connected if you can draw a polyonal line from one pt to another without going outside the domain ie each point on the line is inside the domain. Multiply Connected If there are holes in the domain set then there are multiple ways you can go from one pt to another and this is called multiply connected. Compact Set Set which are bounded and closed are called Compact set. Thats a Heine Borel theorem. This gives an easy way to say that all open sets are not compact. However if some one ask you to define Compact set. Then you got to know what is a Cover.

A cover is basically a union of sets. So a cover looks like {{}.{},{},{} ...} and its CoverC = {UAi, where Ai is elesment of CoverC, U stands for union}. Cover is basically an open set. Coming back to our original thought a set is said to be Compact set if every cover has a finite subcover. So what is a finite subcover ? A finite subcover is something which covers the set To prove why the interval (0,1) is not compact By Heine Borel theorem we can say that since its not close its not compact. Think of some subcovers for (0,1) for ex (-1,.5) U(.5,1,5) is one subcover. Now the definition says that every subcover has to have a finite subcover. We can actually come up with a cover for example {1/n,n} Why set [0,1] is compact. The first thing we note that its close and second thing we note that its bounded so its compact. Now we list two famous theorem of intervals. Note both these theorems tell that if we have nested interval or Rectangle or we can extend to any other geometrical figure than we will have a unique pt which will be common to all the other figures.

Completness Theorem The completness theorem of real number says that any non empty set of real number which is bounded above has a Supremum Nested Interval Theorem This is a concequence of completness theorem of R. This theorem says that if we have a sequence of interval i1,i2,i3... 1. Each interval i(n+1) nested inside i(n) 2. As n-> infinity, length of interval -> 0 The intersection of all the nested interval is NOT AN EMPTY SET

Bolzanno Wierstrauss Theorem This states that every bounded sequence contains a converging subsequence. This can be proved using nested interval theorem. Convergence of a sequence A sequence an (note its a subscript n) converges to a point a if for every epsilon there exists a number n* such that for all n > n* ,an -a <> 0 Every convergent sequence is bounded. By bounded we mean there exists a natural number M such that M > an for all n belonging to Real

Cauchy Sequence A sequence is said to be Cauchy sequence if for every epsilon there exists a number n* belonging to N such that for all m,n > n* am-an <>Every convergent sequence is a Cauchy Sequence. Cauchy sequence is bounded. Infact every Cauchy sequence is convergent and bounded.

Bounded Monotone Sequence Every bounded monotone sequence is convergent. Which includes both increasing and decreasing sequence.

Nested Rectangle Theorem This theorem says that if we have a sequence of nested rectangles r1, r2... 1. Each rectangle r(n+1) nested inside r(n) 2. As n-> infinity, length of diagonal -> 0 Then there will be a only one pt in the rectangle and it will be common to all the rectangles.

Hine Borel's Theorem Hine Borel's Theorem links the infinite pts in complex domain to something manageable. The statement of this theorem is "Every close and bounded set is compact". Compactness is not easy to define but using Heine Borel's theorem it becomes so simple. Note Heine and Borel are two persons not one.

Compactness is defined in terms of cover. Where cover is a union of sets.

Jordon Curve Well Jordon curve also has an interesting history. The theorem is so obvious yet its general proof is not so easy. Infact at the time of its statement there were not enough mathematical tools to prove it. No doubt the proof given by the Jordon turned out to be wrong ! Later its proof was given, The theorem says that a "simple close curve" divides the plane into two classes. What we mean by simple close curve is "imagine you have a rubber band, now you can strech it in any number of directions and you still get a polygon curve, like ellipse, pentagon, hexagon etc. However the edges shouldn't cross for example you cannot make a shape 8 where the edges are overlapping. Let us call them class A (pts insides the close curve) and class B(points outside the close curve). Jordon's theorem says that any point in class A can be connected with the points by a polygonal line without needing to cross the boundary of that class.
Convergence Limit, Supremum, Infimum To have a limit means all the point in the neighbourhood converge to the limit. That is where delta epsilon definition comes in Continuity of Complex Number We define continuity of complex number in the context

Analytic function
What are analytic functions ? Analytic functions are differentiable in every pt of their open domain (G) and we say its analytic at a pt then it means it is analytic in some neighborhood of that pt.

Integration in Complex Plane Integration in complex plane is bit different. For example integral using Cauchy's integral we can integrate at a point by just integration on the boundary of a close curve. Cauchy's integral formula simplifies integrating at a point if the function is analytic at that point.

Monday, March 07, 2005

New Words, Postprandial, Hoosegaw and Gamboge

Here are three new words with thier meanings

Postprandial -> After meal, usually dinner
Hoosegaw-> Jail
Gamboge -> A reddish yellow color, also tree sap ie gum

My take on these words
My way of warding off a postprandial drowsiness is taking a icy cold shower !
The fan of Martha Stewart hadn't had to wait for her release from penetentiary, her hoosegaw beamed an hour of her routine every day.
Her gamboge colored T shirt stoodout in the crowd of students all dressed in maroon.

The 2 sentence on Martha Stewart I coined is ficticious. However lets analyze her. Isn't it empowering for Martha that channel like CNN hosted so many shows on her. Before she was leaving for jail her interview with Larry King. Her release from prison was broadcasted live. Her private jet trotted the screen for more than 15 minutes. Then her first day in her mansion in Newyork was captured with papparazi everywhere and she had to complain of too much noise from the choppers flying above. Hour long program with Paula Zahn. Its like her company getting a full throttle publicity. She already has big contracts to her coffer (Even her coiffeur was also on CNN) and I am very sure her book on her life in prison will surely be newyork times #1 best seller. After becoming a celebrity going to "Hoosegaw" is certainly a career enhancing move !

Sunday, March 06, 2005

Holding a vintage book

An hour ago I had in my hands a book published and binded in 1888. My curosity began when I saw the book in a strange custom fit box lying on the table next to which I was studying. My initial guess was that it to be some expensive reference manual. The book was published by some Putnam publisher. After reading that it was published in 1888, torrent of thoughts engulfed my mind. Who was the author ? How he must have been dressed up ? Who was the person who printed it ? Did he ever think that it will be held by someone in year 2005 ? I was amused and happy at the same time. A strange feeling. A feeling of sense of achievement. A really good feeling :)

Other than that got more insight into Analysis. Each time I read its like learning something new and I wonder when I can summarize all that I learn in a single sheet of paper and yet it takes so long to understand the subtle points. Perhaps this is what we call discovering. Feeling now a bit drowsy and thats a telltale sign that I should move now and retire to sleep.


Thursday, March 03, 2005

Robinson Crusoe is Available for free

Well why we like internet so much. One of the reason is it is like an Oracle. You are curious about something, just punch in some keys and viola you have the answer. I am euphoric to discover that my favorite book Robinson Crusoe is on the internet for free. I haven't checked the project Guttenberg site recently but i believe it should be there too.
Today I add three more words to my vocabulary. These are Consternation, Demiurge and Dissed. The last word was a source of consternation to me because i couldn't find it in my desktop dictionary. But a search on google pooped up so many references. My curosity about this word stemmed after reading this sentence "Theologians explain that Adam and Eve were denied immortality and expelled from Eden because they dissed the Demiurge". Well that was a neat sentence the only thing is it has a negative meaning i.e to show disrepect. So here I coin one sentence. "Never to diss anybody". I am beginning to realize that the origin of the word might be from dissent. Which obviously mean to show disregard, disapproval or protest against. By the way Demiurge is pronounced sthg like demi+urge. Funny when i read the pronunciation i thought it was like Demi-oorge. Well thats wrong its plain and simple Demi-urge. The meaning is skilled person, or productive or a powerful personality. This was a sentence on my word list website "Without Jamie, our charismatic demiurge, nothing would get done around here". So to be a demiurge is a empowering title. Next time you meet somebody address him as demiurge and see the effect it has on that person. Well if you call me demiurge I will be a surely elated ;)
I was just going to signoff that this web page blew up on my screen. I had put a quick search for consternation and this website on the whole adventures of "Lewis and clark" came up. I have seen this movie in Imax in chicago's Addler planetarium if I remember correctly. They are legends and the movie is breathtaking thier adventure to map the united states is simply an exhilarating experience.
My dad sent me this pic of mine when I was around 5 yrs old. I like this pic with my Grandpa. Who was a person of intense curosity. He loved to travel and had travelled all over India and neighboring countries. He had businesses too and probably never cared much about that. He believed in enjoying life and learning. He enjoyed reading and it was always fun to go back to his place and read the wonderful stories book in the library (his son and my mom had created for the children). I still vividly remember when I first read Robinson Crusoe lying on the big wooden desk in the patio. He was magnanimous and loved to help people in need. The village school was one of the other beneficiary of his philanthropy. I believe his stories had a strong influence on me. I remember talking to him in the early year 1991 when he visited us in Delhi and we were seriously considering installing solar panels to light the home in his village. He had asthama and he told me about the magic cure he had in early 1960s after a serious bout with Asthama. A strict Vegetarian he lived for 76 years of age. His elder brother who was also a strict vegetarian lived for 87 yrs. Our summer holidays were mostly spend at his house and then at my other grandpa's house. That was one of the strong early childhood experience I still treasure. Looking closely the village life without electricity, spending time reading books and discussing with elders. Sweet times. Ah sorry for getting nostalgic. Posted by Hello

Wednesday, March 02, 2005

Till today

Its 2nd March today and I shouldn't forget that tommorow is Suman Didi's birthday. She should have been back from California, I did send her several messages on yahoo but she is yet to respond to any. Well there are other things which are happenning. I am pumping with excitement to learn so much interesting stuff. In the past few days had a small party with Sailesh, Manoj and others. Moday I went to Neckers and Alisson answered most of my doubts. Tommorow I have to meet Jessica to complete the remainning stuff of complex integrals, thank god finally she got a new phone. Forgot the appointment with Kathleen hopefully I will be able to finish off those problems by today. Did play badminton last friday. Still have to respond to so many emails I been having from Subhasis, Sidharth, Junghee, Mary, Satya and Ayako. I wonder they will start considering me a recluse. Meanwhile did learn some new words to keep myself happy. Two words which I especially liked are "Dromedary" the one humped camel, the other one was "Pistolero". A person who is a hired assassin. Still having some troble with the Outlook express as it is not easily allowing me to send the messages to hotmail though receiving message is smooth !
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