The principle of Least/Stationary action remains central in modern physics and mathematics, being applied in thermodynamics, fluid mechanics, the theory of relativity, quantum mechanics, particle physics, and string theory.
But adding new gestures to the game reduces the odds of getting a tie but increases the complexity of the game. One popular five-weapon expansion is “rock-paper-scissors-Spock-lizard”, invented by Sam Kass and Karen Bryla.
What looks like a drip irrigation system on a farm land is actually a close up of your hands when sweating.
The reason why this visualization is fascinating is because modern techniques (like full field optical coherence tomography (FF-OCT) )in Forensics can extract sweat pores information from the fingerprint offering more information about the perpetrator crucial to the investigation.
hits only the palm and not the third(ring) finger, there will be a significant
reduction in the total ‘snap’ sound. Try it out!
When you get tired of snapping your fingers, try cracking your knucles. The sharp sound that you hear is caused by the collapse of a cavitation bubble (see image above) in the sinuovial fluid present at the joint.
Fingers, shadows and the transit of Venus
If you are wondering how on earth can all of these words be possibly related, then try answering this question:
When you bring two of your fingers closer in the back drop of a light source, long before your fingers actually touch, the edges magically seem to touch each other.
When one is solving problems on the two dimensional plane and you are using polar coordinates, it is always a challenge to remember what the velocity/acceleration in the radial and angular directions () are. Here’s one failsafe way using complex numbers that made things really easy :
From the above expression, we can obtain and
From this we can obtain and with absolute ease.
Something that I realized only after a mechanics course in college was done and dusted but nevertheless a really cool and interesting place where complex numbers come in handy!
If one remembers this particular episode from the popular sitcom ‘Friends’ where Ross is trying to carry a sofa to his apartment, it seems that moving a sofa up the stairs is ridiculously hard.
But life shouldn’t be that hard now should it?
The mathematician Leo Moser posed in 1966 the following curious mathematical problem: what
is the shape of largest area in the plane that can be moved around a
right-angled corner in a two-dimensional hallway of width 1? This question became known as the moving sofa problem, and is still unsolved fifty years after it was first asked.
The most common shape to move around a tight right angled corner is a square.
And another common shape that would satisfy this criterion is a semi-circle.
what is the largest area that can be moved around?
Well, it has been
conjectured that the shape with the largest area that one can move around a corner is known as “Gerver’s
sofa”. And it looks like so:
Wait.. Hang on a second
sofa would only be effective for right handed turns. One can clearly
see that if we have to turn left somewhere we would be kind of in a tough
Prof.Romik from the University of California, Davis has
proposed this shape popularly know as Romik’s ambidextrous sofa that
solves this problem.
Although Prof.Romik’s sofa may/may not be the not the optimal solution, it is definitely is a breakthrough since this can pave the way for more complex ideas in mathematical analysis and more importantly sofa design.
Once when lecturing in class Lord Kelvin used
the word ‘mathematician’ and then interrupting himself asked his class:
’Do you know what a mathematician is?’
Stepping to his blackboard he
wrote upon it the above equation.
Then putting his finger on what he had written, he turned to
his class and said, ‘A mathematician is one to whom that is as obvious
as that twice two makes four is to you.’
** Two interesting ways to arrive at the Gaussian Integral
Woah… The backlash that Lord Kelvin got after this post was just phenomenal.
There are many ways to obtain this integral (click here to know about other methods) , but here are two interesting ways to arrive at the Gaussian Integral which you may/may not have seen and may/may not be easy to follow.
Gamma Function to the rescue
If you know about factorials (5!= 184.108.40.206.1), you know that they make sense only for integers.
But Gamma function
extends this to non-integers values. This integral form allows you to
calculate factorial values such as (½)!, (¾)! and so on.
The same can be used to evaluate the Gaussian Integral as follows:
Differentiating under the Integral sign
In this technique known as ‘Differentiating under the integral sign’, you choose an integral whose boundary values are easy integrals to evaluate.
Here I(0) and I(∞); and differentiate with respect to a parameter β instead of the variable x to obtain the result.