Wednesday, November 30, 2005

She is always a Oommen to me !

Well I came across this poem while searching for my name "Sumant" on yahoo. I haven't written this poem, the guys who I think wrote this are here
I loved it so I wanted to share this masterpiece with all of you. Read and Enjoy this Billy Joel Parody and I am sure this will remind you of your teacher in your school.

She can begin your day with a casual good morning
She can ruin your day if she catches you yawning
You thought that you’d lost her when you left 7B
Yet she drones on and on,
Yes she’s always Miss Oommen to me.
She’ll ask you a question when you least expect one
Before you can mumble she’ll ask you the next one
“Off to detention” she’ll say, full off glee
Althought she's quite married
She’s always Miss Oommen to me.

Oh – she’s not very cool. And she’s nothing like hot.
She’s got grades on her mind.
Oh – she’s in every school. And she’s in every class.
She’s not one of her kind.
She knows all the answers to all of the questions
She clearly remembers all your transgressions
She’ll never even let you leave class for a pee
She acts like Pol Pot
But she’s always Miss Oommen to me.
Oh – she’s out for your blood. She can blade if she wants.
To the power of nine.
Oh – she’s in every school. And she’s in every class.
She’s not one of her kind.
She's frequently cruel
And she’s suddenly crueler.
She’ll smack your knuckles with a heavy steel ruler.
She’ll frighten you now and for eternity
Her friends call her Liz,
But she’s always Miss Oommen to me.

Tuesday, November 29, 2005

New words Shingle and Wadi

Its been a long time I posted new words, So here are some I learned today at readers digest website.
Shingle - Large, coarse, waterworn gravel; as, crescents of shingle beach created by sediments from the sea. Norwegian singel gave the following definitions
1. A thin oblong piece of material, such as wood or slate, that is laid in overlapping rows to cover the roof or sides of a house or other building.
2. Informal. A small signboard, as one indicating a professional office: After passing the bar exam, she hung out her shingle.
3. A woman's close-cropped haircut.

I discovered that there are shingle style homes too. Here are some more links to shingle style homes.

I liked the 2nd informal definition so I will coin my sentence around that
Her Shingle outside her home cum office was enveloped with a thick layer of dust after yesterday's twister.

Here is one more Shingles is also a disease that causes rashes and other types of illness.

Wadi- Streambed that is dry except in the rainy season; as, Our unit was camped by the wadi. Arabic.

The gave the following definition
Inflected forms: pl. wa·dis also wa·dies1a. A valley, gully, or streambed in northern Africa and southwest Asia that remains dry except during the rainy season. b. A stream that flows through such a channel. 2. An oasis.

Merriam Webster defines it as
: the bed or valley of a stream in regions of southwestern Asia and northern Africa that is usually dry except during the rainy season and that often forms an oasis : GULLY, WASH2 : a shallow usually sharply defined depression in a desert region

Thus we see that the word is more apt to Desert like area. So here is my take on Wadi
The site of Wadi in that loansome desert finally lifted his spirit and he renewed his efforts of making a big sign to attract any aircraft flying over.

Trip to California

Yesterday I finally booked my Amtrak Ticket to goto California. I will be starting my train journey on 18th December at 3 O clock and will be reaching San Jose on 20th around 21 hrs. Hopefully I will have tons of photograph to show you all of the scenery. This trip has been in my mind ever since I saw the "transcontinental railroad" documentary. Here are some wonderful links you can read more about the genesis of Transcontinental railway.

In between we have covered Riemann Integration. I learned some new terms like Partition and Refinements. A Partition is just an ordered set, whereas a refinement of a set is a superset of that set. For example if P is an ordered set then the refinement of P is a set Q and it is a superset of P. There are several lemmas the gist of those is as you add more points to your partition your ares gets sandwiched into the lower and upper areas with less number of points. The main consideration for Riemann integrability is when the lower sum becomes equal to the upper sum. So we have two main theorems.

1. A bounded function f is integrable on [a,b] if and only if for every epsilon, there exists a partition Peps of [a,b] such that

U(f,Peps) - L(f,Peps) <>

The second main theorem is

2. If f is continuous on [a,b] then it is integrable

*****The work on my white board shows about some of the terms and their definition in context of Riemann Integration*****

Friday, November 25, 2005

Black Friday

Today I got up early in the morning to witness the Black Friday rush. I had nothing particular in my mind to buy. So I got up a little late around 6:00 am for that kind of occasion when everybody else gets up at 4:00 am in the morning. I rang Mitsumi around 6:30 am and she was still sleeping so I went alone. The first and the only place on my list was Best buy as I thought if I could get a cheaper 1 Gb memory stick I might buy it. Those selling for $19 were already sold out. Then I went to Bed and Bath to look for comforters but I didn't like any. So I came back home and decided to goto boat dock thinking I could see the campus lake being frozen. But it turned out in spite of -5 centigrade outside the water wasn't yet frozen. But it did provide some wonderful shots. Then on way back I visited Mohanji and we reviewed analysis . Today I also tried a mix lentil of 3 different varieties. Hopefully it will taste good. Sumant at 11:19 am

Tuesday, November 22, 2005

Thanksgiving break update

Today Mohanji and I covered more stuff on Analysis. We covered the Cauchy Criteria for both series and sequences of functions. It almost resembles the one for sequences. One good thing about uniform convergence is -
If there are sequences of functions converging uniformly the resulting function is continuous.
If there are sequences of partial sums of functions converging uniformly the resulting function is continuous. I will post the maple output of both shortly. The example for sequence of functions converging uniformly is 1/(n(x^2+1)). The sequence of functions converge to f(x) = 0.
The example for partial series of sequence of functions converging uniformly can be given by a slight modification of the above function 1/(n^2(x^2+1)). Note that this function has the highest value at x =0 and when n = 1. So its bounded. A discussion with Gayan revealed that the sum of the series 1/n^2 is Pi^2/6. We were not sure how to prove it that it well be bell shaped. But definitely its not difficult to prove that its going to be uniformly convergent.
Yesterday's fruitful discussion with Mohanji had me cleared that if sequence of functions converge uniformly to a function than that function is continuous and when its uniformly convergent its easy to show that epsilon/3 style proof.
Discussion with Kathleen over 319 was good. Now I have all the study problems marked and we were able to solve couple of them. So the goal for tomorrow should be to do more of 319 and wrap up with the proof of Wierstrass M test. Which makes use of comparison test. That if every term of a sequence is less than corresponding term of another sequence and if that other sequence converges than the former sequence converge. The comparison is easy to proof. It can be proved in two ways. One using Cauchy convergence and other using the Sigma notation and making use of bounded monotone sequence.

Monday, November 21, 2005

Thanksgiving holiday and Analysis !

Well this figure should tell you that I am indeed working on my Analysis class. This is a Maple output for a sequence of function I drew to convince myself that sequence of functions fn(x) =x^(1+1/(2n-1)) converges uniformly to function f(x) = abs(x). But note that the function itself f(x) = abs(x) is itself not differentiable.
Today I tried to prove the L,Hospital rule 0/0 proof. Still have to do the infinite/infinite proof. Other than that I now have some idea of uniform convergence and pointwise convergence. I am done with the proof that if sequence of continuous functions are converging uniformly to a function then that function is continuous. If they converge pointwise than we can't really say that function is continuous and an easy example for it is x^n. Which converges to 0 in [0,1) and to 1 at 1. There is also this function 1/(n*(x^2+1)) which converges uniformly. It was great to know that Mohanji has written a book back in Nepal. I think I will collaborate with him someday to write a book ! Tomorrow I should also be studying Abstract algebra. Lets see how much I am able to cover that stuff. It should be fun.

Wednesday, November 16, 2005

Analysis Rocks these days

Ah, I am so happy now :). Today I was able to understand lot of things including the proof of "Intermediate Value Theorem", "Extreme Value Theorem", "Interior Extremum Theorem", "Rolle's Theorem", "Mean Value Theorem". The key to understanding all these was the good amount of time I spent understanding the "Continuity", "Limit proofs" and the relation to Compactness and its theorems. So there are 4 equivalent ways to state the same thing that a function has a limit at any pt c, or a function is continuous at a pt c. The first one is the definition, the second one is delta epsilon, the third one is topological definition and the fourth one is sequence definition. Also these two proofs makes the life much simple.
1. On a compact set a continuous function has a compact image.
2. On a connected set a continuous function has a connected image.
The Proof of Extreme Value Theorem follows from the compactness.
The Extreme Value theorem says that a continuous function on a compact domain has a maximum and minimum. Since the domain of the function is compact it implies the range is also compact. Thus the range is closed and bounded. Therefore it has a max and a min.
The Rolle's theorem proof is just the application of Interior Extremum Theorem which says that if a function is differentiable in the open interval then it has a max and a min and the derivative is zero there.
The proof of Mean Value theorem follows if we subtract the equation of the line from the function and differentiate.
The proof of Intermediate Value theorem was done by choosing an epsilon such that it gives a contradiction for continuity. (LOL) I am not sure anybody reading this last line will understand this, what I mean by that. But if you been studying this proof for sometime. It will definitely makes sense. Otherwise check the proof on Wikipedia. Its pretty good.
Finally if you are wondering about what is written on the white board at my desk, well its a proof of if domain is compact then range is also compact !

Wednesday, November 09, 2005

Honda's GoldWing

Honda's new edition of GoldWing touring bike has air bag for the safety of its rider in case of collision. Read the full story here. Being a pedal power enthusiast naturally I have inclination for bikes also, the recent innovation and luxury in the bike certainly has pepped my interest into these new fancy machines equipped with music system and GPS to rival the comfort of car. However its the feeling of air passing by your face and body that makes bike riding so much exhilarating and ability to go where cars cannot go that make me bike riding more appealing but that doesn't mean that my resolution of doing a Manali to Leh trip on bicycle is going to be replace with any such fangled machine. Its going to be just paddle power.

Good day today

Today has been good. I collected my quiz and previous homeworks from Dr. James Phegley. It was the first time I went to his office (ok, I was there once before to turn in homework just after the class). But it was the first time I really talked to him and he is really nice person to talk to. I was surprised that he remembered my name and quickly shuffled through the pile of papers to get my homework and quizzes. I haven't been able to make to his class early morning and so was afraid that he might be thinking that I am taking his class too lightly. So the good thing is he recognizes me and know my grades and said I won't have problem if I continue to do well in forthcoming quizzes.
Then I went to check out with other Professor Dr. Yucas regarding my performance in Abstract Algebra. Well I love this class and making progress but its not reflecting in the test and I know the reason also. I am taking more time on proving the theorems rigorously to my satisfaction than just cramming up the result and move ahead. I guess I am doing this because I plan to take a second course in Abstract algebra and after knowing that quote "We prove not to prove it but to understand it". So a better foundation is needed and taking 452 alongside makes you realize that how much a proof reveals about the theorem.
Yesterday I was finally able to understand the proof of "Division algorithm" from the wikipedia attachment and it makes so much sense.
Today also Dr. Budzban postponed the take home final to next Friday and I am so glad that now I will have a breather to prepare for the test on Monday, attend the party at Mary's place on Saturday and pick up more theorems in Math 452 and 319 this weekend. Before I sign off, wish you all a happy Veterans day, enjoy the looong weekend :)

Tuesday, November 08, 2005

My staple diet these days !

This is just to document what my staple food has become these days. I usually cook rice and lentil and eat along with any of these pre cooked vegetables some of which are shown in pic here. And there is always some pickle by Sanjeev Kapoor's Khana Khazana. Also showing is Cranberry Juice to which I am hooked now. The taste is sublime, its bit pungent and I like it that way.
Today was good. I finally figured out the relation between the existence of limit and its equivalence to sequence definition. Compared to the regular definition which says that for all epsilon >0 there exists a delta > 0 such that whenever abs(x-a) < delta there exists abs(f(x)-l) < epsilon.

The very first thing one wonders is what this definition is trying to convey.

Actually its talking about how we define a limit of a function. A function f(x) has a limit l, iff for any epsilon neighborhood around l, we can find a delta neighborhood around c, such that the function on any of the value x in the delta neighborhood gets mapped into the epsilon neighborhood around l. Imp things to note that A functional limit is defined only for limit points. So if a point is isolated, we are not concerned. Actually while talking about continuity one can see that its always continuous at isolated pt. So now what is the relation between the limit of a function and sequence ? Now re read the definition it says "delta neighborhood around c", what that means is c is a limit point. So the functional limit can be stated in terms of sequence. and this is how it is stated.
A function

Sunday, November 06, 2005

Sumant first on Google, Once Again !!

No matter if all other search engine list your web site at top, if google doesn't it than it doesn't feel good. I am glad my blog "Daily Chaos" lists as the top blog when you search on Yahoo. If you search for Sumant than also its the third result on Yahoo. My website on Tripod made it to google aftersome 4 years it was around. It ranked as #1 for search on Sumant for some time but recently it went down to #5 today its back again at number 1 and here is the proof. I don't know how long it will remain there. But certainly it feels good. There are some more Sumant on the net and I am proud that most of them are doing well !!
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