Showing posts with label Publications. Show all posts
Showing posts with label Publications. Show all posts

### Telescoping continued fractions

Krishnan Rajkumar (JNU) and I have a new preprint -- on telescoping continued fractions. I have written previously on telescoping and several times on continued fractions, but this one is unique. I don't think anyone has tried to combine the two ideas. We apply it to find lower bounds for the remainder term for Stirling's formula. Ultimately, we discovered a nice new technique, proved several things required to make it work, but were unable to take it to its natural conclusion (so far!). The preprint (now available at arxiv) has several conjectures.

The pic is taken on June 9, 2022, in the Blue Door Cafe, Khan Market where we revised the paper. (That day I showed symptoms of COVID, my second time.)

Here is a link to the preprint.

Here is the abstract:

Title: Telescoping continued fractions for the error term in Stirling's formula
Authors: Gaurav Bhatnagar, Krishnan Rajkumar
Categories: math.CA math.NT

In this paper, we introduce telescoping continued fractions to find lower
bounds for the error term $r_n$ in Stirling's approximation $n! =\sqrt{2\pi}n^{n+1/2}e^{-n}e^{r_n}.$ This improves lower bounds given earlier by Cesàro (1922), Robbins (1955), Nanjundiah (1959), Maria (1965) and Popov (2017). The expression is in terms of a continued fraction, together with an algorithm to find successive terms of this continued fraction. The technique we introduce allows us to experimentally obtain upper and lower bounds for a sequence of convergents of a continued fraction in terms of a difference of two continued fractions.

Here is a talk introducing the method. I presented in the Ashoka Math Colloquium on November 2, 2021. It has an overview of the technique. The talk was made keeping undergraduate students in mind, so there is something here which is quite accessible. In particular I have outlined Robbins' approach at the beginning of the talk.

### Prime number conjectures from the Shapiro class structure

The first of hopefully many joint projects with my childhood friend Hartosh Singh Bal. For many reasons this has been a most exciting collaboration. For one thing, Hartosh and I have been discussing mathematical ideas since Class 11 in Modern School. So it was good to work on something which will lead to something new. For another, Shapiro was Hartosh's number theory Professor at NYU. And for three more reasons, you will have to look at the last section of this paper.

Here is the abstract:
The height $H(n)$ of  $n$, introduced by  Pillai in 1929,  is the  smallest positive integer $i$ such that the  $i$th iterate of Euler's totient function at $n$ is $1$.  H.  N. Shapiro (1943) studied the structure of the set of all numbers at a height. We provide a formula for the height function thereby extending a result of Shapiro. We list steps to generate numbers of any height which turns out to be a useful way to think of this construct. In particular, we extend some results of Shapiro regarding the largest odd numbers at a height. We present some theoretical and computational evidence to show that $H$ and its relatives are closely related to the important functions of number theory, namely $\pi(n)$ and the $n$th prime $p_n$. We conjecture formulas for $\pi(n)$ and $p_n$ in terms of the height function.
Here is a link to a reprint of the paper.

Prime number conjectures from the Shapiro class structure (with Hartosh Singh Bal),
UPDATE (Feb 14, 2020): The paper has appeared in INTEGERS: Electronic Journal of Combinatorial Number Theory (Volume 20), #A11, 23pp.

From left to right: Sonit, Hartosh, me, Punya in 1983 or so

### An infinite family of Borwein-type + - - conjectures

Another collaboration with Michael Schlosser written to celebrate the 80th birthday of Professor George Andrews. The paper's opening paragraph talks about a very interesting event that took place  in a conference celebrating George's birthday in June 2018.

The so-called Borwein conjectures, due to Peter Borwein (circa 1990), were  popularized by Andrews. The first of these concerns the expansion of finite products of the form
$$(1-q)(1-q^2)(1-q^4)(1-q^5)(1-q^7)(1-q^8)\cdots$$
into a power series in $q$ and the sign pattern displayed by the coefficients.  In June 2018, in a conference at Penn State celebrating Andrews' 80th birthday,  Chen Wang, a young Ph.D. student studying at the University of Vienna, announced that he has vanquished the first of the Borwein conjectures. In this paper, we  propose another set of Borwein-type conjectures.  The conjectures here are consistent with the first two Borwein conjectures as well as what is known about their refinement proposed by Andrews. At the same time, they do not appear to be very far from these conjectures in form and content.
Our first conjecture considers products of the form
$$\prod_{i=0}^{n-1} (1-q^{3i+1}) (1-q^{3i+2}) \prod_{j=1}^m \prod_{i=-n}^{n-1} (1-p^jq^{3i+1})(1- p^jq^{3i+2}) .$$
These are motivated by theta products.

Here is a link to a preprint of the paper.
A partial theta function Borwein conjecture, by Gaurav Bhatnagar and Michael Schlosser.

UPDATE (September 16, 2019). The paper has been accepted to appear in the Andrews 80 Special Issue in the Annals of Combinatorics.

Here is a picture from a trip to Hong Kong for an OPSF meeting in June 2017.  From left to right: Heng Huat Chan (Singapore), Michael Schlosser (Vienna), Hjalmar Rosengren (Sweden), Shaun Cooper (New Zealand), me. A special team of Special Functions people from around the world!

### Orthogonal polynomials associated with continued fractions

My first joint paper with Professor Mourad Ismail. This has been a most interesting collaboration. Mourad taught me what to do on the sidelines of a series of meetings around the world. Most of these meetings were organized by the OPSF activity group of SIAM, one of the most interesting and diverse group of mathematicians and physicists. It began with a couple of meetings in Maryland in July 2016. The next one was in Hong Kong in July 2017, followed by a week long visit of Mourad to Austria (Oct 2017), where he gave me an exclusive, one-on-one, tutorial. Next we met in a summer school on $q$-series in Tianjin university in July-Aug 2018. Finally, we finished up things when I met him in Baltimore at the joint AMS meetings (Jan 2019), followed by a weekend trip to Orlando, right after visiting UF in Gainesville.  This work was presented in Baltimore in a special session on continued fractions.

I hope it is the first in a series on Orthogonal Polynomials. There is much to learn and much to do.

Here is a link to the preprint on ArXiv.

***

Orthogonal polynomials associated with a continued fraction of Hirschhorn

Gaurav Bhatnagar and Mourad E. H. Ismail

Abstract

We study orthogonal polynomials associated with a continued fraction due to Hirschhorn.
Hirschhorn's continued fraction contains as special cases the famous Rogers--Ramanujan continued fraction and two of Ramanujan's generalizations. The orthogonality measure of the set of
polynomials obtained has an absolutely continuous component. We find generating functions, asymptotic formulas, orthogonality relations, and the Stieltjes transform of the measure. Using standard generating function techniques, we show how to obtain formulas for the convergents of Ramanujan's continued fractions, including a formula that Ramanujan recorded himself as Entry 16 in Chapter 16 of his second notebook.

***
Here is a picture of Mourad with me in Tianjin (July-Aug 2018).

The picture below is the conference group photo from Hong Kong (July 2017).

Mourad is seated in the front row second from the left. Many of the leading lights of the OPSF world are in this picture.

### A bibasic Heine transformation formula

While studying chapter 1 of Andrews and Berndt's Lost Notebook, Part II, I stumbled upon a bibasic Heine's transformation. A special case is Heine's 1847 transformation. Other special cases include an identity of Ramanujan (c. 1919), and  a 1966 transformation formula of Andrews. Eventually, I realized that it follows from a Fundamental Lemma given by Andrews in 1966. Still, I'm happy to have rediscovered it. Using this formula one can find many identities proximal to Ramanujan's own $_2\phi_1$ transformations.

And of course, the multiple series extensions (some in this paper, and others appearing in another paper) are all new.

Here is a preprint.

Here is a video of a talk I presented at the Alladi 60 Conference. March 17-21, 2016.

Update (November 10, 2018). The multi-variable version has been accepted for publication in the Ramanujan Journal. This has been made open access. It is now available online, even though the volume and page number has not been decided yet. The title is: Heine's method and $A_n$ to $A_m$ transformation formulas.

Here is a reprint.
--

UPDATE (Feb 11, 2016). This has been published. Reference (perhaps to be modified later): A bibasic Heine transformation formula and Ramanujan's $_2\phi_1$ transformations, in Analytic Number Theory, Modular Forms and q-Hypergeometric Series, In honor of Krishna Alladi's 60th Birthday, University of Florida, Gainesville, Mar 2016,  G. E. Andrews and F. G. Garvan (eds.), 99-122 (2017)

The book is available here. The front matter from the Springer site.

--

UPDATE (June 16, 2016).  The paper has been accepted to appear in: Proceedings of the Alladi 60 conference held in Gainesville, FL. (Mar 2016), K. Alladi, G. E. Andrews and F. G. Garvan (eds.)

### The determinant of an elliptic, Sylvesteresque matrix

My second determinant project with Christian Krattenthaler.

The determinant of the Sylvester matrix corresponding to the polynomials
$x^2+2s_1x+s_1^2 = (x+s_1)^2$
and
$x^3+3 s_2 x^2 +3s_2^2 x + s_2^3 = (x+s_2)^3$
is given by
$\det \begin{pmatrix} 1 & 2s_1 & s_1^2 & 0 & 0\\ 0 & 1& 2s_1 & s_1^2 & 0\\ 0 & 0 &1 & 2s_1 & s_1^2 \\ 1 & 3s_2 & 3s_2^2 & s_2^3 & 0\\ 0& 1 & 3s_2 & 3s_2^2 & s_2^3 \\ \end{pmatrix} = (s_1-s_2)^6.$

The determinant is $0$ when $s_1$ and $s_2$ are both $1$. In general, if the determinant of a Sylvester matrix is $0$, then this indicates that the two polynomials have a common root.

Here is an abstract of our paper.

We evaluate the determinant of a matrix whose entries are  elliptic hypergeometric terms and whose form is reminiscent of Sylvester matrices. In particular, it generalizes the determinant evaluation above. A hypergeometric determinant evaluation of a matrix of this type has appeared in the context of approximation theory, in the work of  Feng, Krattenthaler and Xu.  Our determinant evaluation is an elliptic extension of their evaluation, which has two additional parameters (in addition to the base $q$ and nome $p$ found in elliptic hypergeometric terms).  We also extend the evaluation to a formula transforming an elliptic determinant into a multiple of another elliptic determinant. This transformation has two further parameters. The proofs of the determinant evaluation and the transformation formula require an elliptic determinant lemma due to Warnaar, and the application of two $C_n$ elliptic formulas that extend Frenkel and Turaev's $_{10}V_9$ summation formula and $_{12}V_{11}$ transformation formula, results due to Warnaar, Rosengren, Rains, and Coskun and Gustafson.

This paper has been published in Sigma. Here is a link:
The determinant of an elliptic Sylvesteresque matrix (with Christian Krattenthaler), SIGMA, 14 (2018), 052, 15pp.

I presented this paper in Combinatory Analysis 2018, a conference in honor of George Andrews' 80th birthday conference. Here is a picture from Andrews' talk. (The picture inside the picture is of Freeman J. Dyson.)

Next I expect to present the same paper in a Summer Research Institute on $q$-series in the University of Tianjin, China.

A long version (with lots of background information) was presented in our "Arbeitsgemeinschaft "Diskrete Mathematik" (working group in Discrete Mathematics) Seminar, TU-Wien and Uni Wien, on Tuesday, June 5, 2018.

### How to discover the exponential function

Another article on the "How to discover/guess/prove/..." series written for a high school audience. The basic idea is to find a function whose derivative is itself, and to find the power series which satisfies this. Then messing with it to guess it must be the exponential function. No proofs, in fact, it is outrageously un-rigourous.  I hope the editor allows it.

I try to include only the most beautiful items, and state facts which I feel every high school student should know, even if they doesn't appear formally in the syllabus.

Update (Nov 2017). The article was published in the November issue of At Right Angles. A nice surprise was Shailesh Shirali's companion article which gives some graphical intuition to complement the algebraic computations in my article. Here is the link to a reprint

Abstract

If a function is such that its derivative is the function itself, then what would it be? Some interesting mathematical objects  appear while trying to answer this question, including a power series, the irrational number $e$ and the exponential function $e^x$. The article ends with a beautiful formula that  connects $e$, $\pi$, the complex number $i=\sqrt{-1}$, $1$ and $0$.

### WP Bailey Lemmas (Elliptic, multivariable)

After many many years, Michael Schlosser and I wrote another joint paper. We first collaborated in 1995-96 when both of us were Ph.D. students or shortly thereafter. Our joint work was part of his thesis, and published in Constructive Approximation. This time around, I was his post-doc in Vienna from Feb 1, 2016 to Feb 28, 2017.

The picture was taken in Strobl, a favorite place for small meetings and conferences for Krattenthaler's group in the University of Vienna.

In this paper, we give multivariable extensions (over root systems) of the elliptic well-poised (WP) Bailey Transform and Lemma. In the classical (i.e. dimension = 1) case, this work was done by Spiridonov, who in turn extended the work of Andrews and Bailey. It is Andrews' exposition which we found very useful while finding generalizations. We used  previous $q$-Dougall summations due to Rosengren, and Rosengren and Schlosser, and found a few of our own along with some new elliptic Bailey $_{10}\phi_9$ transformation formulas, extending some fundamental formulas given in the classical case by Frenkel and Turaev in 1997. Along the way, we discovered a nice trick to generalize the theorem of my advisor, Steve Milne, that  I had named "Fundamental Theorem of $U(n)$ series" in my thesis.

Hopefully, there will be many more collaborative ventures in the near future.

Update (Mar 22, 2018): The paper has been published. Here is the reference and Link:
G. Bhatnagar and  M.J. Schlosser, Elliptic well-poised Bailey transforms and lemmas on root systems, SIGMA, 14 (2018), 025, 44pp.

### Spiral Determinants

We consider Spiral Determinants of the kind
$$\text{det}\left( \begin{matrix} {16}&{15}&{14}&{13}\\ {5}&{4}&{3}&{12}\\ {6}&1&{2}&{11}\\ {7}&{8}&{9}&{10} \end{matrix} \right)$$
and
$$\text{det} \left( \begin{matrix} {17}&{16}&{15}&{14}&{13}\\ {18}&{5}&{4}&{3}&{12}\\ {19}&{6}&1&{2}&{11}\\ {20}&{7}&{8}&{9}&{10}\\ {21}&{22}&{23}&{24}&{25} \end{matrix} \right)$$
where the entries spiral out from the center. Christian Krattenthaler, who is one of the greatest experts on determinants, tells the story of how he came across such determinants and how he went about discovering the formulas for such determinants. The preprint is available on arxiv.

I have wanted to work with Christian ever since my Ph.D. days, when I tried to generalize a matrix inversion due to him. Finally, we have a joint paper. This also means that my Erdos number has come down from 4 to 3.

The picture above is from Christian's course on "Bijections" which I had an opportunity to attend in the University of Vienna during the period October 2016 to January 2017.

Update: April 26, 2017  The paper has been accepted and will appear in Linear Algebra and its Applications. Here is a preprint on arxiv.
Update: May 10, 2017. The paper is published online. The reference is:
G. Bhatnagar and C. Krattenthaler, Spiral Determinants, Linear Algebra Appl., 529 (2017) 374-390.
Here is a link to the publisher's site: https://www.sciencedirect.com/science/article/pii/S0024379517302719

### Analogues of a Fibonacci-Lucas Identity

Recently, in 2014, Sury published a Fibonacci-Lucas  identity in the Monthly. It turned out that the identity had appeared earlier (as Identity 236 in Benjamin and Quinn's book: Proofs that count: The art of combinatorial proof). When I tried to prove it using my usual telescoping method, I found its connection with one of the oldest Fibonacci identities due to Lucas in 1876. I also found many generalizations and analogous identities for other Fibonacci type sequences and polynomials. This  small paper has been accepted in the Fibonacci Quarterly.

Here is a link to a preprint: Analogues of a Fibonacci-Lucas Identity

Update: Its has appeared. The ref is: Analogues of a Fibonacci-Lucas identity, Fibonacci Quart., 54 (no. 2) 166-171,  (2016)

I use the approach of my earlier paper on Telescoping: In praise of an elementary identity of Euler.

I am pleased, because I have thought of getting a paper in the Fibonacci Quarterly since I was in high school, and feel lucky I found something they found acceptable!

### How to Discover the Rogers-Ramanujan Identities

Dec 22, 2012: It is Ramanujan's 125th birthday, but how many of his famous identities do you know? Here we examine a method to conjecture two very famous identities that were conjectured by Ramanujan, and later found to be known to Rogers.

Here is a link:  How to Discover the Rogers-Ramanujan Identities.

This was presented to a some high school math teachers in a conference. I tried to write it in a way that it could be understood by a motivated high school student.

Update (May 26, 2015): The article has been published. Here is a reference. Resonance, 20 (no. 5), 416-430, (May 2015).

Update (January 18, 2014): This article has been accepted for publication in Resonance, a popular science magazine aimed at the undergraduate level.

### Of Art and Math: A series of articles with Punya Mishra

 Right Angle: One of the many ambigrams made by Punya Mishra that appear in this series of articles appearing in "At Right Angles". All ambigrams are copyright Punya Mishra and cannot be used without permission.
Punya and I are writing a series of articles on the subject of ambigrams. All the ambigrams are made by Punya. For this series, he has been making many new ambigrams, which communicate mathematical ideas. Already, in the space of working on a few articles, it looks like he has made the largest number of mathematical ambigrams.

Here is a longer blog entry from Punya's blog, about this series of articles. His blog has further links to his amazing ambigrams.

Dec, 2015. I presented Punya's and my work in TIME 2015, in Baramati, Maharashtra in my talk: On Punya Mishra's Mathematical Ambigrams. This was the seventh edition of TIME, which stands for 'Technology and innovation in Math education'.

Mar, 2015. The fourth article is on Paradoxes. It is part 1 of 2 articles on this topic. Here we consider what TRUE and FALSE mean in the context of mathematics. Its an introduction to math philosophy. Again, it has many interesting ambigrams.

Feb 2015. The Michigan State Museum has launched an exhibit entitled "Deep Play: Creativity in Math and Art through Visual Wordplay." Check out: the exhibitions web-page

July 2014. The third article on Self-similarity. This one has some amazing ambigrams, and a graphic of the binary pascal's triangle I made many years ago.

Mar, 2014. The second article is on Introducing Symmetry. I think Punya outdid himself in some of the ambigrams here. The ambigram for sin (which is periodic, a sin wave, an odd function) and inverse (modeled on a hyperbola) and exp-log were my favorites. But this month's  puzzle ambigram is mind-blowing too.

Nov, 2013. The first article has come out. It is: Introducing Ambigrams. There is a hidden message in the article. See if you can find it.

### How to prove Ramanujan's q-Continued Fractions

 The photograph of a page in Ramanujan's Lost Notebook where he expanded a ratio of two series in terms of three continued fractions. These three are among the continued fractions explained in this paper.

Its the 125th year of Ramanujan's birth, but how many of his formulas do you know? Here is an opportunity to get familiar with 9 of Ramanujan's continued fraction formulas. These include the three continued fractions that appear in the Lost Notebook in the above photograph.

Abstract:
By using Euler's approach of using Euclid's algorithm to expand a power series into a continued fraction, we show how to derive Ramanujan's $q$-continued fractions in a systematic manner.

A Preprint of this expository paper is now available from arXiv. The latest version fixes a typo. The final version appears in this book. You may wish to buy/access the entire volume from the AMS, its really an amazing piece of work.

Update (Sept 7, 2018): I presented this topic in IISER, Mohali, after adding a few ideas from the recent joint work with Mourad Ismail. Here is the presentation.

Update (December 20, 2014): Published in Contemporary Mathematics: Ramanujan 125, K. Alladi, F. Garvan, A. J. Yee (eds.)  627, 49-68 (2014)

Update (July 17, 2013): Accepted for publication in the proceedings of Ramanujan 125, where I presented this paper.

### How to discover 22/7 and other rational approximations to $\pi$

A short article written for a magazine that caters to high-school students. The basic idea is to use continued fractions found using Euclid's algorithm, and then to chop off the continued fraction to get rational approximations. Written at a high-school level. Some of the material was already present in my book Maths Concepts.

### In Praise of an Elementary Identity of Euler

After many years, a new math paper. Its mostly a survey of my favorite identities, but has some new identities too. The new results have been checked (as typeset in the paper) using Maxima.  I have tried to write the first few sections so that  anyone can read and appreciate it.
I would appreciate any comments, typos, etc.

Update (March 16, 2011):
Presentation from: Georgia Southern q-Series conference, March 15. Here is a link.

Keywords:
Telescoping, Fibonacci Numbers,  Pell Numbers, Derangements, Hypergeometric Series, Fibonacci Polynomials,  q-Fibonacci Numbers,  q-Pell numbers, Basic Hypergeometric Series, q-series, Binomial Theorem, q-Binomial Theorem, Chu--Vandermonde sum, q-Chu--Vandermonde sum, Pfaff--Saalschutz sum, q-Pfaff--Saalschutz sum, q-Dougall summation, very-well-poised 6 phi 5 sum, Generalized Hypergeometric Series, WZ Method

### On Returning To Desh

Book Review: A Long Day’s Night, by Pradip Ghosh, Srishti Publishers and Distributors (2002) Rs. 295/- Reissued by Rupa (2009). Available on Amazon.

As a grad student, whenever I was with a few of my friends, three pegs down, discussing this and that—work, courses, the Buddha (I mean, of course, the advisor), teaching, research, the job scene, or whatever—we often brought up the topic of returning to desh.

We talked about the systems in oosa, and the lack of them in desh; of the good times we had in the hostel, the fundu guys we grew up with (now all in the US) and the Ajit jokes they had invented. Those who had “worked” in India were determined not to return, the others were not so sure. The nature of ABCDs was discussed, and horrors of having ABCDs as children listed. (The children were yet to come: most of us were not even married.)

I read a book recently that reminded me of these discussions.

The book is “A Long Day’s Night”, written by Professor Pradip Ghosh—formerly of IIT-Kanpur, and currently in the University of Hawaii. It is about being a research scientist in India. One day in the life of Professor Virendra Chauhan, our hero, should convince most people that the probability of returning to India AND hoping to have a research career—without considerable misery—is vanishingly small.

***

The setting of the story is the campus of an engineering school. IITK junta might find it somewhat familiar. The story is about an experimental research scientist who has bought a major piece of equipment for the lab, and found that it never met the specs. An engineer from the company is visiting to solve all problems, and the story is all about the day of this visit.

It was a long day. The visiting engineer first tried to convince Professor Chauhan that the company had made no mistakes, and the equipment meets the specs. But he dodged these arguments and constrained the engineer to admit the lapses of the company. Work began on the equipment.

And then they began facing other difficulties, all of which had to be overcome in a few hours, since the engineer had to take a flight back in the evening. Professor Chauhan doggedly refused to allow the visitor any excuse to leave the task undone. He even undertook a trip to the city in the hot afternoon, to get work done by a capable machinist.

You will have to read the book to find out whether the piece of equipment worked or not, and for the reasons why the faulty equipment was bought in the first place.

And about the night that followed the long day.

***

Pradip Ghosh’s book is just delectable. Once I began the book, I couldn’t stop, and as soon as I finished it, I just HAD to re-read the end.

I found myself really respecting the character of Professor Chauhan. So much so, that I feel calling him Virendra in this review might offend him, though I know he is a big enough guy to know that addressing him by his first name has nothing to do with the respect I have for him. This shows the skill of the author, in creating such an impression about his characters.

***

Reading about Professor Chauhan’s hot, tiring day, I came out with a lot of respect for those who are able to do good work under our desi system. I bailed out after only one year in an Indian research institution—and that too when my work was theoretical, only requiring pencil, paper, TeX and Mathematica. And here was this scientist who goes on and on. Even after buying the equipment his research requires, he is unable to make it work for years. Still, he goes on.

I have no doubt that most scientists working in Indian institutions face these difficulties. It is remarkable that so many still have the energy to continue doing good work.

***

The book also has advice for budding scientists—about choosing an area of study, an advisor, a research problem, etc., etc. Gyan dispensed by the wise Professor Chauhan, while having chai with his US bound students.

There are many other issues there that I have not described, but you may find interesting. Consider the story of Harjinder Singh, a colleague of Professor Chauhan:

Harjinder Singh, on the other hand, was a totally different kind of character. An electrical engineer specializing in circuits, thoroughly domesticated with a family of wife, two daughters, and a son, involved and astute in family matters, socially conscious, professionally active, almost perpetually unhappy about the milieu in which he lived. He struggled in a set-up in which he found that it was the system that determined and limited his professional accomplishment and not his innate and acquired abilities. He rejected, he rebelled, but totally because of social and economic constraints, could not kick and leave the system. Beyond this anguish, which was not limited to Harjinder but to many colleagues of Virendra, Harjinder was a person of much sensitivity.

***
Harjinder once told Virendra that he wanted to study history formally, but there were social forces that worked against it. Relatives, even his father, first asserted and then ruled that he should study engineering, because of better job prospects. He did well in the entrance examination to a regional engineering college, and that sealed his fate forever.
How many Harjinders do you know? I can identify many from my own friends.

***

After reading the book, you may wonder if anyone will want to return to desh. But still people do. Why do they return? Why did I return?

People keep asking me why I returned to India. Other people whom I meet, who have studied or worked in the US and have returned, are also asked this question all the time.

I really don’t know how to answer this question. Sometimes I answer: I wonder, man, I wish I had more sense. Sometimes I throw it right back and say: Why do you wish to stay in another country? Many times I just stay quiet, and leave the question unanswered.

There is only one thing that I am sure of. In our long discussions with other desi friends, we always missed the point.

***

Here is an extract from Eric Segal’s The Class explaining the urge to return home. In the words of Professor Finley, discussing Odysseus’ decision to return home from the enchanted isle of the nymph Calypso:
"Imagine our hero is offered an unending idyll with a nymph who will remain
forever young. Yet he forsakes it all to return to a poor island and a woman who, Calypso explicitly reminds him, is fast approaching middle age, which no cosmetic can embellish. A rare, tempting proposition, one cannot deny. But what is Odysseus’ reaction?”

“Goddess, I know that everything you say is true and that clever Penelope is no match for your face and figure. But she is after all a mortal and you divine and ageless. Yet despite all this I yearn for home and for the day of my returning.”

“Here,” he said, at a whisper that was nonetheless audible in the farthest corner, “is the quintessential message of the Odyssey…”

A thousand pencils poised in readiness to transcribe the crucial words to come.

“In, as it were, leaving an enchanted—and one must presume pleasantly tropical—isle to return to the cold winter winds of, shall we say, Brookline, Massachusetts, Odysseus forsakes immortality for—identity.”

***
All in all, I don’t think Professor Chauhan is unhappy in India, despite facing long days as a research scientist. This is where he belongs. He enjoys the natural beauty around him, and enjoys the company of his students and colleagues. This is evident from the author’s description of the world around him. The author is silent about the happiness that Professor Chauhan’s family brings him. But they are there. As the author says: “His is a mixed lot, like everybody else’s.”

***

I think that is enough about Professor Chauhan. Lets let him be for now.

But maybe the next occasion where you are with friends, three pegs down, discussing this and that—the invasion of Iraq, black and white, colonialism, the linear or cyclic nature of time—you will bring up A Long Day’s Night for discussion.

Book Review: A Long Day’s Night, by Pradip Ghosh, Srishti Publishers and Distributors (2002) Rs. 295/- Reissued by Rupa (2009). Available on Amazon.

### From the Diary of a Netizen

My day began at my daddy’s tea stall at 6 am and the big guard gave me fifty paisa as a tip. I bought a chocolate toffee from the corner shop on the other side of the orange office building.

***

The children from the school near my house laughed at me because my shorts were torn. I hid behind the wall and aimed a stone at the boys, and ran into my house.

***

I am nine years old. I don’t go to school because I have to help my father and elder brother run the tea and cigarette stall. Today I was able to give change—to someone buying cigarettes—without asking my brother or father. You don’t have to go to school to be able to do arithmetic, I guess.

***

While playing cricket, our new ball went over the top of the wall, right into the manicured garden of the office complex. A tall auntie with a long sharp nose wearing a saree gave it back.

***

Today a strange man came and distributed packets full of a pink liquid. My old man drank one too many, fought with a friend, and got a black eye in the bargain. The man promised to bring something for my mother and me next time. My brother told me that he is a politician.

***

Today three people came and made a hole in the wall of the orange building. They have put a computer there. Guddi, Raju and I went to see what we can do with it. It’s a little TV screen with pictures on it. They showed us a little black square on the side. By moving our fingers on this square we can move an arrow on the screen. It feels like it is made of thick rubber.

***

My father got drunk and beat up my mother. They fought so loudly that I went away and ate at Bina and Guddu’s house.
***

Ramu, the big bully in our colony, will never bother us again. Today we made so much fun of him when he slipped and fell into the big pile of cow-dung lying in the middle of the ground. He tried to catch and beat us up but we ran away.

***

We moved our cricket playing so that the ball does not break the TV screen. Renu Aunty came to ask whether we like the computer. Raju said he liked to play games on it. Guddi did not say anything.

***

My mother hung clothes to dry and my shirt flew over the roof. She held up my two-year old sister so she could clamber up the asbestos roof to bring back my shirt. She nearly fell down but mommy caught her in time.

***

Today I somehow shut down the computer. An uncle from inside re-started it. We closed it again and asked the guard to re-start the computer. I showed everyone how it could be done. For some reason, Vivek Uncle was in a smiling mood that day.

***

There are many aunties and uncles in this office building. They all come in Maruti cars, and hang around the front of the building drinking coffee or tea. The guard told me they have a machine which can make tea.

***

Sanju Bhaiyya, who lives near my house and goes to work, made a picture in the computer. He knows a lot about the computer. But all I wanted to do was to play with Mickey Mouse. In the evening some older people came and asked us to show them how to play with the computer. But they left soon after, because they could not understand anything.

***

My mother’s sister came over from the village, with my mausa and Tejali. They will live with us until they find a room for themselves. My father and mausaji had to sleep outside in the cold.

***

Something went wrong with the computer last evening and Raju cut the touch pad and spoiled it. Somebody complained to his father who gave him a thrashing. I too cried that night.

***

Today another politician came in an auto-rickshaw and gave pouches of a red drink to my father and his friends. He gave us children small flags and we ran all the way behind the auto until it turned the corner. My father and his friends laughed and shouted boisterously all night.

***

Today, an uncle came and took our photograph. He asked Guddi whether she knew how to use the computer. I told him that she is a girl, and not too interested. He asked my name. The next day Raju, Guddi and I were on the front page of a newspaper. Vivek Uncle and Renu Aunty were very excited, but Guddi thought her hair was not looking too good.

**
Vivek uncle put a page in Hindi on the computer. It had stuff written in Hindi, but no pictures. I closed it and went back to playing the Tarzan game on Mickey’s site. We found some Hindi film songs on the internet and some movies.
***

Today was the market day of the week. A naked beggar snaked his way through the crowded bazaar. I saw the Aunty with the long nose drive her Esteem through the crowd, trying to avoid the crawling beggar on the road.

***

Some foreigners came to see us and talk to us. They were shown round by an uncle wearing a suit. I showed them how I wrote my name in English on the computer. They were quite surprised.

***

Ran into some uniformly dressed schoolchildren again. This time their jokes did not bother me much.

The characters and events mentioned above are a creation of the author’s imagination.

However, they have been inspired by conversations with colleagues conducting an experiment in Minimally Invasive Education, at a slum adjoining the NIIT Corporate office, in Kalkaji, New Delhi. Children who live in this slum were given access to an Internet Kiosk. This experiment received wide media exposure, following the front page headline: Rajender Ban Gaya Netizen (by Parul Chandra, The Times Of India, May 12, 1999).