Gaurav Bhatnagar

Guessing the Binomial Theorem

2011-09-16T04:31:00.000-07:00

Here is a way to guess the Binomial Theorem for any index.

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A q-analog of the Gamma Function

2011-09-16T04:10:00.000-07:00

I have begun reading Bruce Berndt's "Ramanujan's Notebooks", Part III. Here is a small morsel from Ramanujan's table: Entry 1(ii) of Chapter 16 of his Notebooks. Its a discovery proof of the limit of the q-Gamma function, as q goes to 1. In my humble opinion, this is easier than the usual proof (due to Gosper) which appears in Gasper and Rahman.

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Precalculus by Askey and Wu

2011-08-03T02:16:00.000-07:00

Many years back, Professor Richard Askey sent me hard copies of some notes he had made, with a supplement by H. Wu. I think these notes are just wonderful, and am placing them here.

Review of Pre-Calculus by Richard Askey and H. Wu
Precalculus - Further Notes by Richard Askey

Punya - A nearly Palindromic poem

2011-04-27T06:16:00.000-07:00

Punya

Love

your palindromes,

everything

the poetry,

ambigrammed symmetry.

An inspiration-

your blog.

Hope you keep up with

your slog.

the perspiration:

programmed asymmetry,

Ambi-poettary,

anything

but palindromes!

love

This was inspired by a blog post by Punya, about an 8th grader who loved his palindromic poetry. The associated facebook page attracted some comments admiring Punya, including a little palindromic poem by me.

While I am not an 8th grader anymore, I do feel many times that I am still in 12th grade.
So, I thought of a

nearly-palindromic poem.

So near a palindromic poem, yet far from it. The first of its kind. Enjoy. Or not.

Identities and Mathematical Intuition: Talk in DPS - Dwarka to DPS Math Teachers

2011-04-25T03:58:00.000-07:00

On April 18th, I gave a talk on Identities to Delhi Public School (DPS) Math teachers attending a training conference/workshop. The teachers were from DPSs all over the country and teach in senior school (XIth-XIIth).

The overall idea of the talk was to organize information about identities according to the three kinds of mathematical intuition I have spoken about earlier. The three kinds of mathematical intuition are: Symbolic, graphical or physical intuition, and structural intuition. These are motivated by the following quote:

…some mathematicians are more endowed with the talent of making pictures, others with that of juggling symbols and yet others with the ability of picking a flaw in an argument.

~Gian Carlo Rota

Sunil Mittal

2011-03-08T19:24:00.000-08:00

A schoolboy, named Sunil Mittal,
What goes through
his adolescent mind?

Here's a clue:

The movies playing in his brain
and the color of his uniform,
are both the same.

They are Blue!

Sunil is a friend from modern school. This one came up on FB as a comment on a discussion.

In Praise of an Elementary Identity of Euler

2011-02-09T09:21:00.000-08:00

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.

Update (June 11, 2011): The paper is published by the Electronic J. of Combinatorics, Vol 18 (2), P13 44 pp. Download.

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

My Mathematical Forefathers

2011-02-06T22:30:00.000-08:00

From time to time, I look at the Mathematics Genealogy Project, and search for my own mathematical tree. I was happy to note that I am a direct descendant of Gauss and of Leibnitz. What I noticed today, was that I am a mathematical cousin of Saroj Malik, my teacher in Hindu College, who taught me abstract algebra and elementary number theory. We branch out at Gauss.

Here is the complete list of my mathematical forefathers.

Friedrich Leibnitz
Jakob Thomasius
Otto Mencke
Johann Christoph Wichmannshausen
Christian August Hausen
Abraham Gotthelf Kästner
Johann Friedrich Pfaff
Carl Friedrich Gauß
Christoph Gudermann
Karl Theodor Wilhelm Weierstraß
Leo Königsberger
Georg Alexander Pick
Charles Loewner
Adriano Mario Garsia
Stephen Carl Milne
Gaurav Bhatnagar

Math Problem Book - Grade 6

2010-08-02T00:52:00.000-07:00

Here is a book made from some problem sets I used to give Tejasi and her friends in my garage. The explanations have been added later on. Most of the kids who took these problem sets benefited...in the sense that they began doing well in exams. Here is a collection of the problem sets for grade 6.

Click here to download/view Math Problem Book - Grade 6

Experience Mathematics

2010-08-01T23:41:00.000-07:00

A book I wrote long ago. Recently, I re-edited it based on comments made by Professor Dick Askey. Looking for a publisher, but meanwhile here it is for my friends and their kids...

Click here to view the PDF File

The q-disease

2010-06-04T00:59:00.000-07:00

Special Functions,
by Andrews, Askey and Roy.
Here's a belated review,
and a thank you.

~*~*~*~*~*~

Beauty in mathematics,
said Polya,
is seeing the truth
without effort.

Everything
in The Book
is as elegant,
as could be.

Everything
as simple,
as effortless,
as should be.

Everything
as beautiful,
as it is.

~*~*~*~*~*~

The Book
is an introduction.
A gateway.

Enticing,

pulling you in,

inviting you
to enter.

And try a problem
on your own.

Or find
your own proof
of a classic theorem.

~*~*~*~*~*~

And then,
suddenly,
you experience
the heady rush,
that comes
when you get
an idea.

Or
the disappointment,
when
the idea
doesn't work.

And then again --
another idea.
And another
intoxicating rush.

~*~*~*~*~*~

Or perhaps
you wish
to derive
a q-identity.

And experience
the peaceful flow
found in
symbolic calculation.

The meditative calm
which comes
when you know the next step,
without knowing why
its the next step.
When things work out
almost by themselves.

And the wave of
satisfaction,
when the RHS
becomes
equal to
the LHS.
Suddenly,
and magically,
at the last step.

A Warning:
This book has been known to

infect people with

the q-disease.

~*~*~*~*~*~

Enter at your own risk.
This is The Book.
And this gateway--
this is
The Gateway.
Once you are in,
it is impossible to return.

The q-disease
has no cure.

Not if you have been infected
by Special Functions,
written by
Andrews, Askey and Roy.

~*~*~*~*~*~

Book Review:

Special Functions,

George E. Andrews, Richard Askey and Ranjan Roy,

Encyclopedia of Mathematics and its Applications,

Cambridge University Press (1999).

GB's Math Poettary
Written on June 4, 2010, Professor Dick Askey's Birthday.

Design II

2009-05-14T05:00:00.001-07:00

Design II
Ambigram by punyamishra

A great design:
Everything fits in nicely
into one complete whole.

Not a hair out of place,
and not one thing
more
than what is
needed.

The form
and the function,
made for each other.

GB #32

Paradox

2009-04-05T08:11:00.001-07:00

Paradox
Ambigram by punyamishra

All Cretans are liars
said Epiminedes,
a Cretan,
a classic Paradox.

If Epiminedes tells the truth
then he must be lying.
And if he is lying,
he is telling the truth.

Sounds like a
contradiction,
doesn't it?
But a paradox is
different from
a contradiction.

Contradictions
are not allowed:
because
P and Not P
implies any Q.
In other words,
a contradiction
implies
any proposition.

"The Pope and I are two".
said Russell,
"If 2 = 1, then
the Pope and I are one."
Thus went Russell's proof
that he was the Pope.

But a paradox
is different,
its more valuable.
It increases
the size
of your universe.

Russell's paradox
showed the gaps
in set theory
and set the stage
for the next
leap forward.

Paradoxes
are the stuff
of which
giant leaps
are made.

GB #31

Watson-Crick

2009-04-04T18:25:00.001-07:00

Watson-Crick
Ambigram by punyamishra

Watson and Crick
show
what fun it is
to be
a scientist.

What fun it is
to discover
something new.

What fun it is
to compete
with the best
and win.

Watson and Crick
discovered
the secret of life
itself.
GB #30

Internet

2009-02-27T22:37:00.000-08:00

Internet
Ambigram by punyamishra

The Internet
inking pacts
across the world.

Linking
and connecting
all humanity.

Shakily
connecting
all the dots
into one
continuous whole.

GB #29

Internet II

2009-02-27T22:20:00.000-08:00

Internet II
Ambigram by punyamishra

Small knots
woven together
become a net.
Flexible, stable,
and very strong.

Small computers
inter-connected
become the Net.

Unleashing
the power
of communication,
of creativity,
and
of community.

Small individuals,
linked together
become the
Internet.

GB #28

Douglas R Hofstadter

2009-02-20T23:58:00.000-08:00

Sides-reversed-is
Ambigram by punyamishra

Douglas R Hofstadter
sides reversed is
Retdatsfoh R Salgoud
sides reversed is
Douglas R Hofstadter
...
is one Strange Loop.

Hofstadter wrote
Godel, Escher and Bach:
An eternal golden braid.

A personal review follows.

Four young boys
growing up
the Modern way:
Gaurav and Hartosh,
Punya
and GB.

Like other 16 year olds
just focussed on
having fun.

Fun,
defined for them
by Hofstadter
with wordplay,
Palindromes,
Anagrams, Ambigrams,
and paradoxical
self-referential statements.

Many years went by,
in the pursuit of fun,
doing what they please,
each in their own way.

Until,
like everyone else,
they grew up
and became serious.

Each in their own way,
they did what
they had to do;
they became what
they had to become:
Consultant, Journalist,
Professor,
and Manager.

And thats what
they are
in this life.

Their one
and only one
life.

Only one life?

Is it ever
enough
for anyone
to live
just one life?

Memories of
a fun-filled youth
don't just die:
They bring in
A Certain Ambiguity -
Ambigrams,
and not to forget,
Ambi-Poettary.

Back to Hofstadter
and the book review.

Just one life
is certainly not enough
for Hofstadter's
Godel, Escher and Bach.

But do not fear.
Perhaps one day
a Gardner will come
to annotate GEB,
and give us a chance,
to figure it all out
in just one life-time.

GB # 27

Dog

2009-02-14T19:16:00.000-08:00

Dog
Originally uploaded by punyamishra

A dog,
wags his tail,
seeks attention.
And licks your face,
without asking for
your permission.

Comes up close
and becomes
personal.

And becomes
your best friend.

GB #26

God

2009-02-14T07:40:00.000-08:00

God
Originally uploaded by punyamishra

GOD
looks the same
whichever way
you look at it.

Look down from the top
or up from bottom --
and find GOD.

Look in the mirror
and see GOD.

Just like the 0,
symbol for
nothingness.

But still,
required,
to count from
one to infinity.

GB #25

I Love My India

2009-02-13T21:46:00.001-08:00

Tejasi made this using My Paint Free, on my iPhone.

Trees

2009-02-13T21:44:00.001-08:00

Birds

2009-02-13T21:40:00.001-08:00

Flowers

2009-02-13T21:38:00.001-08:00

Math Poettary - Infinite

2009-02-12T23:47:00.000-08:00

Check out the post here.

A brief explanation

The post mentions the sphere as a "one-point compactification" of the (complex) plane (by adding a point at infinity). The property of the sphere being compact somehow makes it a little closer to being "finite" and therefore easier to handle. But to understand more precisely what all this means you need to take a good course in Complex Analysis or Topology.

When studying complex analysis, I thought that the theorems are simpler, more beautiful, and closer to the finite case than analogous theorems in Real Analysis. I don't know whether that is due to the relationship with the sphere, but I suspect it is so.

Here is an example: You know that a polynomial p(x) with real coefficients (and a finite degree) can be written as a product of factors of the form (x-a) where a is a zero of the polynomial. (The root a is of-course a complex number). Turns out, under certain conditions, we can write a function (which can be viewed as an infinite series) as an (infinite) product of its zeros. For example, consider this formula:

Euler's Product:

(The formula above taken from Wikipedia's entry on the Wallis Product.)

The formula looks nicer if you replace x by (pi)x. Then the expression on the left has zeros at +1, +2, +3, ... and -1, -2, -3, .... And on the right you get factors of the form (1-x/n)(1+x/n) which is zero for x = +n and -n.
In fact, the way we write the product is something to do with making the product "converge" (or make sense).

This formula is definitely something I will write about one day. I think I need to pick up a complex analysis book again...its been too long...and have almost forgotten the beautiful stuff I used to see everyday.

Darwin

2009-02-12T21:30:00.000-08:00

Darwin
Ambigram by punyamishra

"Beauty in mathematics,"
says Polya,
"is seeing the truth
without effort."

Polya's dictum applies
to Science
as well.

Darwin explained
nature's bounty --
from simplicity
emerged complexity,
adapting by competing.

Darwin explained
so much, so simply,
so beautifully.

GB #24