We are starting a new segment on the blog where we recommend one or two books in Math or Physics that everyone can read.
You are absolutely welcome to share your comments and reviews here once you are done. Also, if you would like us to check a book out, do let us know too!
Have a good one!
“For someone who doesn’t surf in tropical waters and doesn’t go backcountry hiking and camping where bears are prevalent, it’s true: your odds are trillions-to-one that you’ll get both bitten by a bear and a shark in your lifetime. But behavior and risk-exposure matter. It’s not surprising news when someone gets bit by a snake: it happens about 8,000 times per year in the USA alone. It’s not surprising when a surfer gets bit by a shark; surfers are the most likely people to receive shark bites and it happens dozens of times a year. And it’s not surprising to encounter a hungry black bear in the back country woods. And finally, it’s not surprising to survive all of these, as it’s very uncommon for any of these encounters to be fatal.
Dylan is certainly an unusual case, but in every case, he put himself in the most at-risk group for these types of encounters.”
Earlier this week, it was reported that a young man named Dylan McWilliams was bitten by a shark while surfing in Hawaii. This wouldn’t be such a big deal on its own, but last year Dylan was bitten by a bear while camping in Colorado, and two years before was bitten by a snake. Is he just the unluckiest person on Earth, who overcame astronomically small odds to have all three of these things happen to him? Or are the odds, given his behavior and location and circumstances, far higher than a naive calculation would indicate?
One of my professors was asked the same question and let me paraphrase his response:
You give up on intuition when it gives up on you.
One of the many reasons why most of physics is deeply mathematical is because our intuition alone is unable to explain all the results that we observe in nature and when that happens, we rely on mathematical theories to shed light on the nature of reality
** The reason why this is awesome is because one does need to worry about the unit vectors in the r and theta directions, which makes the algebra so much more simpler
Its name reflects its pareidolic resemblance to classical depictions of Gautama Buddha, seated in a meditation pose with a forehead mark (tikka) and traditional topknot (ushnisha).
* Read more interesting things about Buddhabrot and Nebulabrot here
** Pareidolia is a psychological phenomenon in which the mind responds to an image or a sound, by perceiving a familiar pattern where none exists. Check out more examples of pareidolia here
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:
sofa would only be effective for right handed turns. One can clearly
see that if we were to turn left somewhere we would in kind of tough
Prof.Romik from the University of California 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 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.’
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.
If you know about factorials (5!= 126.96.36.199.1), 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:
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.
Have a nice Pi-Day! In memory of Stephen Hawking!