Partial Sums of Generating Function Coefficients

Now that I've begun to blog a little Mathematics, I suddenly find it difficult hold myself from not blogging. I started digging my amateurish works in mathematics and I found some worth blogging. Lemme start with one on the Exponential Generating Functions.

Well, Generating functions are ubiquitous in Mathematics. There are many forms of generating functions available at a mathematician's arsenal depending on which area of mathematics he's working. If I have to give my favorites, I would go for the Ordinary generating functions (OGFs) and the Exponential generating functions (EGFs) mostly because of their versatility in Combinatorics. Apart from these two, I find Dirichlet's generating functions (DGFs) and Lambert's generating functions (LGFs) of Number theory fascinating.

The idea of this post nothing innovative or original but began when I noticed something in the OGFs and wondered how to do the same for EGFs. Let's move on to that 'something'...

The Ordinary generating function of a sequence $a_n$ is $G(a_n;x)=f(x)=\sum_{n=0}^{\infty} a_n x^n$.
Therefore, we find $[x^n]f(x)=a_n$.

Let's define a new sequence, $b_n=\sum_{r=0}^na_r$. Let's denote the OFG of this new sequence by $G(b_n;x)=g(x)$. How would one find $g(x)$ given $f(x)$?

$\begin{align}
g(x)&=b_0+b_1x+b_2x^2+\cdots\\
&=a_0+(a_0+a_1)x+(a_0+a_1+a_2)x^2+\cdots\\
&=(a_0+a_1x+a_2x^2+\cdots)+(a_0+a_1x+a_2x^2+\cdots)x+(a_0+a_1x+a_2x^2+\cdots)x^2+\cdots\\
\end{align}$

Recognizing $a_0+a_1x+a_2x^2+\cdots$ as the OGF of $f(x)$, we get
$\begin{align}
g(x)&=f(x)+xf(x)+x^2f(x)+\cdots\\
&=(1+x+x^2+\cdots)f(x)\\
&=\frac{f(x)}{1-x}
\end{align}$

Hence $g(x)$, whose OGF coefficients are the sums of $f$'s OGF coefficients, can be found by simply diving $f(x)$ by $1-x$. Let's apply it in an example..

$x(1-x)^{-2}=x+2x^2+3x^3+\cdots=\sum_{n=0}^{\infty}nx^n$, and
$x(1-x)^{-3}=\binom{2}{2}x+\binom{3}{2}x^2+\binom{4}{2}x^3=\sum_{n=0}^{\infty}\binom{n+1}{2}x^n$ which is indeed true.

In short, $\displaystyle\sum_{r=0}^n[x^r]f(x)=[x^n](\frac{f(x)}{1-x})$.

Many would have known of this result. There's absolutely nothing new about it and any material you find on the internet would give you this one. But the question that led me to this post is how would this 'translate' to the EGFs?

Let $ f = \sum_{i=0}^\infty a_i \dfrac{x^i}{i!}$
Then $D^k f = \sum_{i=k}^\infty a_i \dfrac{x^i}{i!}$ where $D \equiv \dfrac{d}{dx}$

Summing for all $k \geq 0$,
$f + Df + D^2f + D^3f + ... = \sum_{i=0}^\infty a_i^{\infty} \dfrac{x^i}{i!}$ where $a_n^\infty = \sum_{j=n}^\infty a_j$

Denoting the RHS by $y$, we can write the above as,
$(1-D)^{-1}f=y$ (ie) $(1-D)y=f$

Solving it with Mathematica gives,
$y(x)= c e^x - e^x\int e^{-x}f(x)dx$ (c being an arbitrary constant)

Hence $y$, whose EGF coefficients are the partial sums of $f's$ EGF coefficients, has an integral expression of the above form.

Unlike the OGF version, we have
$\sum_{r=0}^nr![x^r]f(x)=a_0+a_1+a_2+\cdots+a_n=a_0^{\infty}-a_{n+1}^{\infty}=0![x^0]y(x)-(n+1)![x^{n+1}]y(x)$

In light of the above formulae, we can just set $c$ to be $0$ in the expression for $y$. That is we can re-write the formulae as,

$y(x)= -e^x\int e^{-x}f(x)dx$

In my opinion, this formulae is more subtle than the one we saw for the OGFs and is not at all obvious at first sight. Let's see what the formula gives for some functions..

$xe^x=1.\dfrac{x}{1!}+2.\dfrac{x^2}{2!}+3.\dfrac{x^3}{3!}+\cdots=\sum_{n=0}^{\infty}n \dfrac{x^n}{n!}$

$\sum_{r=0}^nr![x^r]xe^x=1+2+3+\cdots+n$

Using the formulae above, $y(x)=-e^x\int e^{-x}xe^xdx=-e^x\int xdx=\dfrac{-x^2e^x}{2}=-\dfrac{x^2}{2}-\dfrac{x^3}{2.1!}-\dfrac{x^4}{2.2!}-\cdots$

It is also easy to note that, $n![x^n]\dfrac{-x^2e^x}{2}=-\dbinom{n}{2}$

Hence,$0![x^0]\dfrac{-x^2e^x}{2}-(n+1)![x^{n+1}]\dfrac{-x^2e^x}{2}=-\dbinom{0}{2}+\dbinom{n+1}{2}=\dbinom{n+1}{2}$ as expected.

Though these results produce compact expressions for the partial sums, we have to note that they do not always give closed form solutions. For example, one may be tempted to use EGF of $(1-x)^{-1}$ to find a formulae for sum of the first $n$ factorials. But soon we find that the resulting integral in the formula above is not that easy to evaluate to give a close from solution. Nevertheless, these expressions come in handy at situations unexpected and as such would prove to surprisingly useful.

As always, I've enjoyed blogging and the joy has only doubled when it is about Mathematics. Hope you did not feel that you've wasted your time. I'll try to keep blogging at my present pace and see ya soon.


Until then
Yours Aye,
Me

Pólya's Enumeration Theorem with Applications - Part I

Hi All, its been a long time again and I thought I would not be blogging again anytime soon. But recently, I came across what can arguably called one of the most powerful and beautiful theorem in Combinatorics, The Pólya Enumeration Theorem. When I started reading about it and applied it to solve certain enumeration problems I just couldn't stop but wonder this marvelously simple theorem. Maybe, had Gauss discovered it he would have called it 'The Remarkable Theorem in Combinatorics'.

Pólya's Enumeration Theorem's usefulness comes from its ability to solve a great many variety of counting problems which would otherwise be too hard to solve or would require a customized approach that give you insights for that particular problem but cannot be generalized to very many other class of problems. The theorem uses the fundamental concepts of Group Theory, especially the permutation groups.

After surfing through a lot of material in the web to clearly understand the theorem I stumbled upon the PDF titled 'Counting and Coloring with Symmetry - A presentation of Pólya's Enumeration Theorem with Applications' by Amanda Noel Bjorge. This PDF was easy to follow because it explains every step of all the proofs with examples which makes it easier for the reader to follow. Of course, you've got to have some understanding of basic group theory and this PDF takes it from there. Yet another material I found very useful is Polya's Counting theory by Mollee Huisinga.

Now one may ask, If I have this PDF what is the point in creating a post about the same theorem? After all am an amateur and what more could I possibly add to an already well written thesis? Well, most of all the files available on the internet seems to explain the theorem in a more academic way. What I intend to do is to present the theorem to myself and maybe, add a few things which a casual google search may not reveal. So I suggest one should first read the file above for it will make following this post a lot easier and interesting.

To start with, I assume you've developed an understanding about the permutation groups and their corresponding pattern index. In particular, the cycle index formula for the Symmetric group is extremely interesting and required a bit of thought before I completely understood it. The four basic permutation groups that appear frequently in counting problems are the Symmetric group, the Alternating group, the Cyclic group and  the Dihedral group. Of these, the most easiest to understand are the last two. Mainly because they have a geometric description and can be visualized as placing beads or balls in a regular n-gon.

So when the question is about n-bead necklace discounting rotational symmetries, you understand that the answer has to do with Cyclic group and when it is about discounting both rotational and reflectional symmetries, you use the dihedral group.

To begin with, let's look at a simple example:

Example 1: How many 4-bead necklaces can formed with $m$ different colored beads discounting rotational symmetries?

Strictly speaking (or blogging), we don't need Pólya's Enumeration Theorem to find this. Burnside's Lemma will suffice. To solve this we first need the cycle index of the Cyclic group of degree 4, conventionally denoted as $Z_{C_4}$.

$Z_{C_4}(x_1,x_2,x_3,x_4)=\frac{1}{4}(x_1^4+x_2^2+2x_4)$

We substitute $m$ for all $x_i$'s, which gives us, $Z_{C_4}(m,m,m,m)=\frac{1}{4}(m^4+m^2+2m)$

Say you have three different colored beads, substitute $m =3$ in the above expression and you will have  $Z_{C_4}(3,3,3,3)=24$

Example 2: But what if I have one White bead, one Black bead and two Red beads and I want the number of 4-bead necklaces discounting rotational symmetries?

Here's where Pólya comes to the rescue. Plainly, the theorem says let $x_i=W^i+B^i+R^i$ and the coefficient of $WBR^2$ will give you the required answer. With a little help from Mathematica, we find

$Z_{C_4}(W+B+R, W^2+B^2+R^2, W^3+B^3+R^3, W^4+B^4+R^4)=B^4 + B^3 R + 2 B^2 R^2 + B R^3 + R^4 + B^3 W + 3 B^2 R W +$
$3 B R^2 W + R^3 W + 2 B^2 W^2 + 3 B R W^2 + 2 R^2 W^2 + B W^3 +
 R W^3 + W^4$

Hence you can form $3$ 4-bead necklaces with the above restrictions. Though the theorem doesn't show how these three necklaces will be, with little thought we find them to be brww, bwrw and rbww. Note that the first and third arrangements are different as you cannot transform one into the other using rotations.

The same logic applies to the Dihedral group as well. Just find the cycle index of the Dihedral group and follow the same procedure.

But what of the Symmetric and Alternating groups? We visualized the Cyclic and Dihedral groups with n-gons and applied them to n-bead necklaces. Well, I cannot find an equivalent way for visualizing the Symmetric and Alternating groups. But as for application, we can use the Symmetric groups in n-ball Urns. Both these groups helps us in solving urn problems where order doesn't matter.

Example 3: Say if we have 4 Urns and balls of $m$ different colors. In how many ways can you put one ball in each urn?

Now if the order matters, then this is one the most elementary questions in Combinatorics. Each urn has $m$ choices. Hence the answer is $m^4$. But the Symmetric group doesn't take order into account. So Let's see what Burnside's Lemma gives. We first need $Z_{S_4}$.

$Z_{S_4}(x_1,x_2,x_3,x_4)=\frac{1}{24}(x_1^4+6x_1^2x_2+3x_2^2+8x_1x_3+6x_4)$

One could easily see that $Z_{S_4}(m,m,m,m)=(m^4+6m^3+11m^2+6m)/24$. For $m=3$, this gives $Z_{S_4}(3,3,3,3)=15$

Let's finish applying the Pólya's Enumeration theorem with three colors - White, Black and Red.

$Z_{S_4}(W+B+R, W^2+B^2+R^2, W^3+B^3+R^3, W^4+B^4+R^4)=B^4 + B^3 R + B^2 R^2 + B R^3 + R^4 + B^3 W + B^2 R W +$
$B R^2 W +R^3 W + B^2 W^2 + B R W^2 + R^2 W^2 + B W^3 + R W^3 + W^4$

These are the 15 combinations that we got from Burnside's Lemma. Note that bbbr and brbb are different when you take order into account whereas they are counted as same when order doesn't matter.

Now you maybe tempted to disregard Symmetric groups as uninteresting in Pólya's Enumeration theorem. But they come in a remarkable fashion when we try to understand the Alternating groups. The 'definition' of Alternating groups, which am about to give, is one of the high points of this discussion. In fact, I cannot find any material in the internet which clearly spells this application.

Example 4: Let's take same 4-urn problem. But now we want is the number of ways you can put one ball in each urn such that no urns have balls of the same color?

Clearly, if you have 4 urns and only three different colored balls, you cannot do it. But if you have 4 different colored balls, you can do it in 1 way. In fact, the answer to the question is too simple. Its $\binom{m}{4}$ when you have $m$ different colored balls. Let's see whether Burnside's Lemma gives the same answer. But first we need the cycle index. But of what? The Alternating group? The Symmetric group? No. Its the difference between the cycle index of the Alternating group and the Symmetric group.

Equation (14) of the PDF gives a succinct way of calculating this.

$Z_{A_n}-Z_{S_n}=Z_{S_n}(x_1,-x_2,x_3,-x_4,...)$, which we will denote by $Z_{-S_n}(x_1,x_2,x_3,x_4,...)$ for convenience. Hence we see that the Alternating group comes in a disguised form to solve a special class of constraint namely 'no two are alike'.

$Z_{A_4}-Z_{S_4}(x_1,x_2,x_3,x_4)=Z_{S_4}(x_1,-x_2,x_3,-x_4)=Z_{-S_4}(x_1,x_2,x_3,x_4)=\frac{1}{24}(x_1^4-6x_1^2x_2+3x_2^2+8x_1x_3-6x_4)$

So in case of $m$ colors, we have $Z_{S_4}(m,-m,m,-m)=Z_{-S_4}(m,m,m,m)=(m^4-6m^3+11m^2-6m)/24=\binom{m}{4}$

Note that $Z_{-S_4}(m,m,m,m)=0$ for $m<4$.

Pólya's Enumeration Theorem for the same gives,

$Z_{-S_4}(x_1,x_2,x_3,x_4)=B G R W + B G R Y + B G W Y + B R W Y + G R W Y$, where $x_i=W^i+B^i+R^i+G^i+Y^i$.

We have used five colors for this example because using anything less than four colors will only give us $0$.

Frankly, we don't see very good use of the last example as a source of solving problems. But I see the power of Symmetric and Alternating polynomials are apparent when we start using compositions of the permutation groups. See Example 6.1 and Figure 8 in the PDF to see what I mean.

The same problem also gives us further scope to understand our notion of the Alternating group. In the said Example the author has used the Symmetric group because he was looking for the number of arrangements of 2 necklaces without regard to order. Hence the three arrangements are {bbbb, wwbb}, {bbbb, wbwb}, and {bwbb, bwbb}. The third arrangement repeats the same necklace twice. But what if we only want the arrangements where no two necklaces are alike. Here we would be using our idea of Alternating group along with the Symmetric group.

Using $Z_{-S_2}$ and $Z_{C_4}$ we get,

$Z_{-S_2[C_4]}(W+B,...,W^8+B^8)=B^7 W + 2 B^6 W^2 + 3 B^5 W^3 + 3 B^4 W^4 + 3 B^3 W^5 + 2 B^2 W^6 + B W^7$

Now we see that, of the 15 arrangements of 2 necklaces each with 4 beads, there are 2 with a total combination of 6 black beads and 2 white beads such that the two necklaces are different from each other.

To sum up the permuatation groups, we give the following:

• Cyclic group: Discounts rotational symmetry.
• Dihedral group: Discounts rotational and relfectional symmetry.
• Symmetric group: Arrangements not taking order into account.
• Alternating group: Arrangements not taking order into account and no two patterns are alike.

We'll continue with the other applications of the theorem in this post. Write ya later.



Until then
Yours Aye
Me

#JokarforLife

I think this is my first post in IIMC. A lot of things have changed. Night outs which were once a delight has become an exhaustive affair. Eating barely a meal a day making it up with noodles and rolls has become a regular diet. With less than about four hours sleep a day, Calcutta has literally turned me into a Zombie. I always feel like am in half-sleep mode barely able to concentrate on anything, subjects in particular. But even amidst all these, the best part is that I enjoy the remarkable, unforgettable experience that am going through at this phase of my life. I would be eternally grateful for IIMC for giving me the best and probably the brightest part of my student life.

I've to accept the fact that am ruthlessly unfair to put the blame entirely on the institution. I know am not very good at managing time. When I really should be making presentations, I play FIFA. The times that should have allocated to projects have been allocated to watching Game of Thrones (or Sherlock). Most important of all, when I really should be sleeping, I end up not just doing that. Result - Wake up ten minutes before the first class (usally relying on Wingies), almost not get ready for the class, sleep-run to the Academic block and sleep-wonder why the heck is the Marketing prof at the dias for a Fin class.

Speaking of the Academic Block, I must really be writing the Profs and the courses. Being a Mechy, I really had no idea of how vast, deep and beautiful the other areas of science were. Every subject that we study here (mind you, am only a first semester-er) is artistic in its own sense. Or atleast the Profs make it that way. Though at times I feel there's nothing new to learn, it's the structured and organized thought process that makes the subjects fascinating. But even amidst my adoration for the courses, I keep wondering why marks never cross the class average. Probably, I guess I'll figure that out before I leave IIMC.

There was time after I gave the CAT when I kept thinking whether the effort I put in for preparation is worth it. Whether leaving a 10L job and joining an IIM will make any sense. Now I say with great pride and no regrets, it worth every moment of that hardwork. Because the institution, along with the knowledge you garner here and the network you earn, keeps reminding you constantly that it'll make sure it gives not what you need but what you deserve. I guess one has to come here and live this place to understand what I mean.

Yours Aye
Me

A tribute to the Little Master

What does it feel to be a great man? How does it feel it to carry the burden of an entire nation's expectations? What it means to play like an extraordinary sportsman and live to be an exceptional Human being? What does it take to define longevity, expand the realm of Humbleness and be the definition of 'down-to-earth'? Is it an ordinary feat to manage the fans who are ready to kneel before you in a sense of admiration and awe? And, above all, how does it feel to be worshipped to as a God?

You must have correctly guessed that am writing about 'Bharat Ratna' Sachin Tendulkar, a man who is an epitome of Greatness in every sense of the word. I've been a fan of him for as long as I know. For an atheist like me, He is a God. Coz my sense of Godness does not simply come from divine entity that created the world kinda stuff. Nor does it mean to be Omnipotent or Everlasting. Its in doing what you do with utmost perfection. I could write a lot about my passion for the Little Master.

He is the one reason I started watching Cricket. For me, he gave Cricket an art form. He taught me what does the word 'timing' means. He taught me what 'elegance' means. He taught me what 'Humility' means. Most of all, he taught me redefine the notion of 'God'. Just watching him at the crease gives me a great sense of pleasure and satisfaction. That little statured man with his willow has won millions of hearts by doing what he does best. And that, for me, is Godness.

For someone like me, who started watching cricket, the game is no more. There are no more on-the-field-aggresion. There are no more nail bitting moments. As much as I love to watch him play, I pray to keep him off the batting end. I just want him on the field. Nothing more. The mere presence of him would take me off and that is something I'm gonna miss for the rest of my life. Maybe that is why I was in tears when he gave his last speech at his home ground. I just could'nt control myself. With every word he said, it was just..

For the fellow worshippers, I give here his speech that would linger in our ears, beat in rhythm with our hearts in the days to come. The least I could do for the man who has been a truly Spectacular Sportsperson the world has ever witnessed.

"All my friends.. Settle down.. Let me talk.. I'll get more and more emotional.. My life between twenty two yards for twenty four years, it's hard to believe that wonderful journey is coming to an end. But, I would like to take this oppotunity to thank all who've played an important role in my life. Also, for the first time in my life I'm carrying this list to remember all the names in case I forget someone. I hope you understand... It's getting a little difficult to talk but I'll manange..

The most important person in my life and the I missed him since 1999 passed-away my father. Without his guidance I don't think I would be standing here in front of you. He gave me freedom at the age of eleven and told me that "Chase your dreams, but make sure you don't find shortcuts. Path might me difficult but don't give up" and I've simply followed his instructions. Above all, he told me to be a nice human being which I've continued to so. I've tried my best. Everytime I've done something special, whenever I've showed my bat, it was my father. So I miss him today.

My mother.. I don't know how she managed such a naughty child like me... I was not easy to manage.. She must be extremely patient.. Uh.. For a mother, the most important thing that her child remains safe and healthy and fit.. And thats what she was most bothered about.. and worried about... She took care of me for the last twenty four years that I've played for India but even before that she started praying for me... The day I started playing cricket, she just prayed and prayed and prayed... And I think her prayers and blessing have given me the strength to go and then perform.. So big thank you to my mother for all the sacrifices..

In my school days for four years I stayed with my Uncle and my aunt because my schools was quiet far from my home.. and they treated me like their son.. My aunt after having had a hard day's play, I would be half-asleep... and she would be feeding me food so that I could go and play again tomorrow... I cant forget those moments.. I'm like their son and I'm glad that continue to be the same way..

My eldest brother Nithin and his family has always encouraged me.. My eldest brother doesnt like to talk much.. But the one thing he always told me was "Whatever you do, I know you would give your hundred percent.. and I've full confidence and faith in you"..His encouragement meant a lot to me..  My sister Savitha and her family was no different.. The first cricket bat of my life was presented to me by my sister. It was a Kashmir Willow bat... But that is where the journey began... She is one of those many who continue to fast when I bat.. So thank you very much..

Ajith,my brother, and what do I talk about him? I don't know really.. uh.. We've lived this dream together.. He was the one who sacrificed his career for my cricket. He spotted the spark in me and it all started from the age of eleven when he took me Acheraker sir, my coach, and from there on my life changed. You would find it hard to believe that even last night he called me and discussing my dismissal... Knowing that there was remote chance of batting again.. But just that the..habit which we've developed, that the rapport that we've developed for.. since my birth. It has continued and it'll continue.. Maybe even when I'm not playing cricket, we'll still be discussing technique...Various thing we agreed upon, my technique and so many technical things which I didn't agree with him.. We've had arguments and disagreements.. But in the end, when I look back at all those things, If that had'nt happend in my life, I would've been a lesser cricketer..

The most beautiful thing happened to me in 1990 when I met my wife Anjali.. Those were special years and it has continued and it'll always continue that way.. I know Anjali, being a doctor,.. there was a wonderful career in front of her..uh.. when we decided to have a family.. Anjali took the initiative to step back and say "You continue with your cricket and I'll take the responsibility of the family".. uh... Without that I dont think I would have been able to play cricket free and without any stress.. Thanks for bearing with all my faults, all my frustrations.. and all sorts of rubbish that I've spoken.. I normally do.. Thanks for bearing with me.. and always staying by my side through the ups and downs.. You're the best partnership I've had in my life..

Then the two precious diamonds of my life.. Sarah and Arjun.. They've already grown up.. My daughter is sixteen, my son is fourteen.. Time has flown by... I wanted to spend so much time with them..on special occasions like their Birthdays, their Annual days, their Sports day, goin on holidays.. Whatever... I missed out on all those things.. Thanks for your understanding.. Both of you have been so, so special to me.. You cannot imagine..  I promise you for fourteen years and sixteen years I've not spent enough time for both of you.. But the next sixteen years or even beyond that everything's for you...

My in laws Anand Mehta and Anubam.. both have been so, so supportive, loving, caring.. uh..  I've discussed on various things in life generally with them.. and taken their advice.. you know its so important to have a strong family to be always with you, guiding you.. Before you start clapping, the most important thing they did was allowing me to marry Anjali.. So, thank you very much...

In the last twenty four years, I've played for India, I've made new friends but before that I've had friends from my childhood.. They all have had terrific contribution right from.. as and when I call them to come and bowl to me in the nets, they've left all their work aside and come and helped me.. Be it joining me on holidays or having discussion about cricket, when I was little stressed and wanting to find a solution so that I could perform better.. All those moments, my friends were with me.. Even whenever I was injured, I would wake up in the morning because I could not sleep I thought my career was over because of injuries.. that's when my friends have woken up at three'o clock in the morning to drive with me, and just make me believe that your career is not over.. Life would be incomplete without all those friends.. Thanks for being there for me...

My cricket career started when I was eleven.. The turning point of my career was when my brother took me Acheraker Sir, my Coach.. I was extremely delighted to see him up in the stands.. Normally he sits infront of the television and he watches all the games that I play.. when I was eleven, twelve.. those were the day when I used to hop back on his scooter and play a lot of practice matches in a day.. First half of the innings I would be batting on Shivaji Park.. the Second Park, some other match at Haazad Mehta.. Sir would taking me all over Mumbai to make sure that I got match practice.. On a lighter note, in the last twenty nine years, Sir has never said well played to me because he thought I would get complacent and I would stop working hard... Maybe he can push his luck and wish me now.. I'm done on my career and because there are no more matches Sir in my life.. I would still be seeing cricket and Cricket would always stay in my heart.. But you've had immense contribution in my life and so Thank you very much..

My cricket for Mumbai started right here on this ground.. Mumbai Cricket Associsation.. which is so dear to me.. I remember landing from New Zealand at four'o clock in the morning and turning up for a game at eight'o clock here.. Just because I want to be part of Mumbai cricket.. Not that anyone forced me or Mumbai Cricket Association pressurised me to be here.. but that was for the love of Mumbai Cricket.. Thank you Very much... The president is here.. Thank you very much along with your team for taking care of me and looking after my cricket..

The dream was obviously to play for India.. and that's where my association with BCCI started.. BCCI was fantastic right from my debut.. Believing in my ability, selecting me in my squad at the age of sixteen was a big step.. So thanks to all the selectors for having faith in me..and BCCI for giving me the freedom to express myself out in the middle.. Things would've been different if you've not been behind me and I really appreciate your support.. specially when I was injured.. you were right with me and making sure that all the treatments were taken care of and I got fit and fine and played for India...

The journey has been special.. For the last twenty four years, I've played with many senior cricketers.. and even before that there were many senior cricketers whom I watched on television.. they inspired me.. to play cricket and play the right way... Thanks so much to all those senior cricketers.. Unfortuantely, I've not been able to play with them.. But I'm highly (something) for all their achievements, all their contributions.. You see on the mega-screen Rahul, Laxman, and.. Sourav.. Anil is not here... and my team-mates right here in front of me... You're like my family away from home.. I've had some wonderful times with you.. uh.. It's gonna be difficult not to be part of the dressing room.. Sharing those special moments.. All the coaches for their guidance..It's been special for me.. I know when M S Dhoni presented me with the two hundredth test match cap on Day one morning.. I had a.. brief message for the team.. I would like to repeat that... I just feel that all of us are so, so fortunate and proud to be part of Indian cricket team... serving the nation.. Knowing all of you guys I know you'll continue to serve the nation in the right spirit, in the right values.. I believe we've been the lucky ones to be chosen by the almighty to serve this wonderful sport... Each generation gets this opportunity to really take care of this sport and serving to the best of our ability.. I've full faith in you that you'll continue to serve this nation in the right spirit to the best of your ability and bring all the laurels to our country... All the very best...

I would be failing in my duty, If I didn't thank all the doctors, the physios, the trainers who put this difficult body together to back on the field and be able to play... The amount of injury that I've had in my career, I don't know how you managed to keep me fit but without your special efforts.. it would never have happened... The doctors have met me at weird hours... I mean I've called them... From Mumbai to Chennai... From Mumbai to Delhi.. wherever.. they've just taken the next flight.. they have left their work, they've come and treated me which has allowed me to play... So a big thank you to all three of you... for keeping me in good shape...

My dear friend Late Mark Mascarenhas.. My first manager.. We unforunately lost him in a car accident in 2001.. But he was such a well wisher of Cricket, my cricket, specially Indian Cricket... He was so passionate... He understood what it takes to represent a nation.. and gave me all the space to go our and express myself and never pressurised me to do this ad or promotion.. whatever my sponsors demanded.. He has taken care of all that and today I miss him.. Thank you Mark for all your contributions.. My current management team for repeating what Mark has done... when we signed the contract... exactly told them what I want from them and what I was to be representing India... They've understood that and respected that.. So thank you very much (current management team)...

Someone who has worked closely with me for fourteen years my Manager Vinod Naidu... He is more like my family... and all sacrifices, spending time away from his family for my work has been special.. So big thank you to your family as well.. for giving so much time... for my work to go on..

In my School days when I performed well.. the media backed me a lot... They continue to do that till this morning... Thank you so much to all the media... for supporting me and appreciating my performances.. It surely had a positive effect on me... Thank you so much for all the Photographers as well.. Those wonderfully captured moments will stay with me the for the rest of my life.... So to all the Photographers, a big Thank you...

I know my speech is getting a bit too long but this is the last thing I want to say... I want to thank all the people here who've flown in from various parts of the world.. and supported me endlessly.. whether I scored a zero or I scored hundred plus.. whatever... your support was so dear to me and it meant a lot to me... Whatever you've done for me, I know so many guys who've fasted for me, prayed for me, done all sorts of things for me.. You know, without all that, life wouldn't have been like this for me... I want to thank you from the bottom of my heart and also say that time has flown by rather quickly.. But the memories that you've left with me, be always be with me for ever and ever... Especially "Sachin!! Sachin!!"... that'll reverberate in my ears till I stop breathing... Thank you very much... Good Bye..."

Hi Guys N Gals,
Well, this being my first post and all, I just dont know where to begin. Maybe an Introduction would be good, you kinda get to know more about me. And thats one reason I am doing this blog. Wanted to share my interests, my thoughts, my one-liners, my favourite movies, songs, my et cetera, etc. Well, having said that, that justifies why 'The Introduction' is the more pertinent of all. Am in no hurry to introduce my self, so I'm gonna postpone Me until the next post.

If you ever feel that am trying to be too smart, get the hell outta of here coz its my Blog and I will make it my way. Oooov.. Calm down. Jus kidding fellas. If you ever feel that way, Plz.. Plz... Plz....Guys, bear with me N Gals, bare with me (No Offence Gals.. Kidding again). Write ya soon.


Until then
Yours Aye,
Me