July 21, 2018 Saturday
Bedtime Story
Symmetrical Polynomial
Last night we had defined a symmetrical
polynomial as a polynomial with n variables where interchanging of any of the
variables results in the same polynomial.
We will see how this same statement can be
framed in the language of mathematics.
P is a symmetric polynomial if for any
permutation σ (small Greek letter sigma) of the subscripts 1, 2, … , n one has
P(Xσ(1), Xσ(2), … , Xσ(n)) = P(X1,
X2, … , Xn).
Abstract algebra is abstract and hence it
is inherently difficult to the minds of average apes since most minds even
though they do deal with abstractions (one can make a case for the greatest
non-dying abstraction that has infected the minds of human apes being the multitude
of gods that are representative of human apes bereft of all warts, moulds and
evil vices), do not go as far as needed in abstract algebra.
I want to tell you the story on how this
type of abstract algebra largely to do with the polynomials of higher roots came
to have such deep connections with the fundamental mathematics of symmetry as
it is something that I found very difficult to comprehend.
It was very hard as an average ape with an
even average unimaginative mind to visualize any kind of obvious or even
clandestine connection between algebra of polynomials to mathematics of
symmetry.
Being untrained in both only added to the
problem of the pedestrian mind that chance has bestowed me with.
Yet, even such a prosaic mind like mine is
capable of asking some interesting questions for after all a literal mind can
still be a curious mind.
These are some of the questions that the neurons
in my brain fired off.
What do polynomial equations have to do
with group theory?
What is group theory in abstract algebra?
What is the relation of group theory with
the mathematics of symmetry?
Who were the men who tackled these problems
and what was their motivation in doing so if any at all?
Many a times in my study for materials for
bedtime stories I have come across mathematicians who attempt to solve
mathematical problems for no sensible reason, at least not what would make
sense to most average apes.
In today’s modern world there exists
several substantial rewards (such as Fields medal, Abel Prize, Turing Award and
Millennium Prize by Clay Mathematics Institute to name a few) with serious
monetary benefits for path breaking mathematical discoveries in the Western
capitalistic democracies.
If these are near-impossible to achieve,
then there exists several posts of tenured professorships in various institutes
one whose support one can always fall back to maintain a decent way of life.
In overpopulated nations like India, China
and several others with similar demographic-socio-economic indices mathematicians
can aspire to have a fairly decent income (and even very high) as mathematical
tutors in private coaching centers thanks to the sheer need of the desperate
young people.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
Good night Mon Ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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