A Candle in the Dark

A look on science, politics, religion and events

Archive for September 2007

Final Lecture

leave a comment »

Dr Randy Pausch is a 46 year old professor of Computer Science at Carnegie Mellon University. He’s got terminal pancreatic cancer and expects to live for around 3 months. Here’s his final lecture hosted on Google videos, “How to Live Your Childhood Dreams“.

The whole lecture is really worth watching. It’s moving, funny, witty, powerful and inspiring.

Here’s a short part of the lecture.

Written by parseval

September 23, 2007 at 7:56 am

Posted in death, people, videos

Newton’s method and fractals

with one comment

Many of you would have come across the Newton-Raphson iterative technique of finding the root of a single non-linear equation, f(x)=0 in high-school.

The iterative method of solution is essentially given by,

x_{k+1}= x_{k} - \frac{f(x_k)}{f'(x_k)}, \quad k=0,1,2,....

Newton’s method can also be extended to find solutions to a set of simultaneous non-linear equations for functions of many variables.

For example, let’s say you need to find a solution to the equations,

\begin{cases} f_1(x_1,x_2,...,x_n)=0 \\ f_2(x_1,x_2,...,x_n)=0 \\ ... \\ f_n(x_1,x_2,...,x_n)=0 \end{cases}

with an initial guess, \bf{x} = (x^0_1,x^0_2,...,x^0_n)

The recursion which is used to find the solution is now given by

\bf{x}^{(k+1)} = \bf{x}^{(k)} - [\bf{J}(\bf{x}^{(k)}]^{-1} f(\bf{x}^{(k)}), \quad k=0,1,2,...

That is,

\left(\begin{array}{c} x^{(k+1)}_1 \\ x^{(k+1)}_2 \\ . \\x^{(k+1)}_n \end{array} \right) = \left( \begin{array}{c} x^{(k)}_1 \\ x^{(k)}_2 \\ . \\x^{(k)}_n \end{array} \right) - \left( \begin{array}{cccc} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & . & \frac{\partial f_1}{\partial x_n} \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & . & \frac{\partial f_2}{\partial x_n} \\ . & . & . & . \\ \frac{\partial f_n}{\partial x_1} & \frac{\partial f_n}{\partial x_2} & . & \frac{\partial f_n}{\partial x_n} \end{array} \right)^{-1} \left(\begin{array}{c} f_1(\bf{x}^{(k)}) \\ f_2(\bf{x}^{(k)}) \\ . \\ f_n(\bf{x}^{(k)}) \end{array}\right)

Note that you’ll have to evaluate the Jacobian matrix at \bf{x} = \bf{x^{(k)}} during each iteration. For a proof of when the method converges, and some modifications to make it converge faster, check the references which are given at the end of this post.

There’s something interesting which happens when you apply this technique to an equation in the complex plane. For a complex number z=x+iy, any polynomial equation f(z) = 0 can be written as two equations in the variables x and y. For example, if f(z) = z^2 + 1 = 0, then,

(x+iy)^2 + 1 = 0

(x^2 -y^2 + 1) + i(2xy) = 0,

which means

f(x,y)=x^2-y^2+1=0 and g(x,y)=2xy=0

Also, note that for every (x,y) we have a corresponding point in the complex plane.

Now consider this question. Let’s say we have a polynomial equation z^n - 1 = 0. Given an arbitrary initial guess of the root to this equation

a) Will the algorithm converge to give a solution for that particular initial guess?
b) If so, how long does it take to converge to a solution for that particular initial guess?

An attempt to answer to these questions leads to fractal patterns. In the color scheme I’ve used, the bluer regions correspond to where the initial guesses converge faster. The yellow/red regions is where the initial guesses converge slowly or do not converge.

z^3+1=0,

nis3.png

z^4+1=0

nis4.png

z^5+1=0

nis5.png

As you can see, while some initial guesses converge without much fuss (the basin regions), others can show remarkable behavior (guesses near the “boundary”) and can lead to a number of intricate patterns for various functions. I’ve linked to some websites which contain a gallery of such images for various functions. Some of them are truly remarkable!

References and External Links
A Class of Methods for Solving Nonlinear Simultaneous Equations, C. G. Broyden, Mathematics of Computation, Vol. 19, No. 92 (Oct., 1965), pp. 577-593
– An Introduction to Numerical Analysis, Endre Suli and David Mayers, Cambridge University Press,Pg 104-125
Fractals from Newton’s Method
Newton Basins
Newton’s method and Newton basin fractals
Newton’s Method gallery
Another Newton’s Method Gallery

Written by parseval

September 20, 2007 at 1:23 pm

Posted in mathematics

The power of faith

with 9 comments

Recently, the Archaeological Survey of India released a perfectly normal statement.

The Archaeological Survey of India on Wednesday asserted in the Supreme Court that there was no evidence to prove that Ramar Sethu/Adam’s bridge was man-made. It is a natural formation made up of shoals/sand bars, which are possessed of their particular shape and form due to several millennia of wave action and sedimentation.

That’s a perfectly logical and valid statement based on current experimental observations and scientific reasoning. In fact, they even highlighted why they don’t think it’s man-made.

“The existence of human remains, whether in the form of bones etc. or in the form of other artefacts, is primary to prove archaeologically the existence and veracity of a historical fact. No such human remains have been discovered at the site of the formation known as Adam’s bridge.

And from another interview,

Explaining the bridge’s geological history, he said both the Palk Strait and the GoM were once part of the Cauvery basin, which was formed during the separation of India and Antarctica about 70 million years ago during the `Gondwana period.’ They were combined till a ridge was formed in the region owing to thinning of earth’s crust. The development of this ridge augmented the coral growth in the region. “The coral cover acted as a `sand trapper’ leading to the formation of Rameswaram Island,” Dr. Ramanujam said.

Then, the ASI added this

In its affidavit filed in the Sethusamudram case, the ASI said “the petitioners [Subramanian Swamy and others] while seeking relief [not to damage Ramar Sethu] have primarily relied upon the contents of the Valmiki Ramayana, the Ramcharitmanas by Tulasidas and other mythological texts, which admittedly form an important part of ancient Indian literature, but which cannot be said to be historical record to incontrovertibly prove the existence of the characters or the occurrence of the events, depicted therein.”

which was predictably followed by much outrage for daring to offend the sacred beliefs and accusations that the affidavit amounts to “pouring contempt on crores of Hindus”, and “hurting the religious sentiments of millions”.

The government has set in motion the process of questioning religious beliefs. We will launch a nationwide movement if it does not withdraw immediately this blasphemous submission questioning the very existence of Lord Ram

Ack, how dare the government apply science to “question” our sacred beliefs? How arrogant of the government to presume that observation, logic and reason can dare to question our most cherished and mightiest of faiths, which forms an integral part our glorious culture and heritage?

What else, but the blind power of faith, would make people disregard scientific evidence and continue believing in whatever silly personal fantasy which pleases them?

On a more important note, I’d like to use this situation to highlight something else. In cases such as this, when some scientists claim that the bridge is indeed man-made (although, not remotely related to the Ramayana), how does one know what the current scientific consensus is? Herein lies the importance of peer-reviewed work, which ensures that only unbiased, scientific facts are published.

References
No evidence to prove Ramar Sethu is man-made – The Hindu
BJP mulls movement on Lord Ram – Hindustan Times
It’s not a man-made structure

Written by parseval

September 13, 2007 at 9:34 pm

Posted in politics, religion

The curious incident of the man and the reactor

with 4 comments

Written by professor Levenspiel1 (Do check out his page on Dinosaurs and Elephants!)

The curious incident of the man and the reactor

HOLMES: You say he was last seen tending this vat …

SIR BOSS: You mean “overflow stirred tank reactor,” Mr Holmes

HOLMES: You must excuse my ignorance of your particular technical jargon, Sir Boss

SIR BOSS: That’s all right; however, you must find him, Mr. Holmes. Imbit was a queer chap; always staring into the reactor, taking deep breaths and licking his lips, but he was our very best operator. Why, since he left, our conversion of googliox has dropped from 80% to 75%.

HOLMES (tapping the side of the vat idly): By the way, what goes on in the vat?

SIR BOSS: Just an elementary second order reaction, between ethanol and googliox, if you know what I mean. Of course, we maintain a large excess of alcohol, about 100 to 1 and …

HOLMES (interrupting): Intriguing, we checked every possible lead in town and found not a single clue.

SIR BOSS (wiping away the tears): We’ll give the old chap a raise – about twopence per week- if only he’ll come back.

DR WATSON: Pardon me, but may I ask a question?

HOLMES: Why certainly, Watson

WATSON: What is the capacity of this vat, Sir Boss?

SIR BOSS: A hundred imperial gallons, and we always keep it filled to the brim. That is why we call it an overflow reactor. You see, we are running at full capacity – profitable operation, you know.

HOLMES: Well, my dear Watson, we must admit we are stumped, for without clues deductive powers are of no avail.

WATSON: Ahh, but there is where you are wrong, Holmes. (Then, turning to the manager): Imbit was a largish fellow – say about 18 stones – was he not?

SIR BOSS: Why yes, but how did you know?

HOLMES (with awe): Amazing, my dear Watson!

WATSON: Why it’s quite elementary, Holmes. We have all the clues necessary to deduce what happened to the happy fellow. But first of all, would someone fetch me some dill?

With Sherlock Holmes and Sir Boss impatiently waiting, Dr. Watson casually leaned against the vat, slowly and carefully filled his pipe, and – with the keen sense of the dramatic – lit it. There our story ends.

(i) What momentous revelation was Dr. Watson planning to make, and how did he arrive at this conclusion?

and best of all,

(ii) Why did he never make it?

😀

References
[1]- Chemical Reaction Engineering by Octave Levenspiel, 3rd edn, Pg 117-118. Excerpt taken under fair use

Written by parseval

September 9, 2007 at 10:35 pm

Posted in fun, science

Curve fitting

with 3 comments

From a study published in Science1,

The dashed line is a quadratic polynomial (given by y = -0.516 + 1.08x - 0.23x^2) which gave the best fit to the points.

Can you spot the duck?

(via David Harrison)

References
[1]- Competitive Interactions Between Neotropical Pollinators and Africanized Honey Bees, Science, 15 September 1978: Vol. 201. no. 4360, pp. 1030 – 1032

Written by parseval

September 3, 2007 at 10:29 pm

Posted in fun, science

Science stuff – I

leave a comment »

Here’s a bunch of fascinating posts from some really good science blogs.

(i) Microfluidic chips for imaging behaviour & neuronal activity
(ii) More on microfluidics

From the Neurophilosophy blog. Mo talks about a new study which uses microfluidics devices to look at how the behaviour of a worm is related to it’s neural activity.

(iii) Interface Between 2 Insulators Becomes Superconducting

ZapperZ posts a link to a study which shows that the two dimensional interface between two insulators can be a superconducting. Fancy that!

(iv) A Tiny Arthropod Tells A Tectonic Story

The Zooillogix team reports on a study by a bunch of Harvard biologists who looked at the DNA of mites, and then linked that to plate tectonics and the shape of the Pangaea landmass!

Written by parseval

September 1, 2007 at 8:58 pm

Posted in science