The age old problem of Navier- Stoke’s Equations and Von Neumann Machines!

Millennium Problems : These are very big deal when it comes to Mathematics. But, for some obscure reason, One of them is not very popular among Engineers, even though one of the last standing problems in classical physics – Navier–Stokes existence and smoothness (one of the millennium problems) is of major consequence to Fluid Dynamicists, Aerospace/Mechanical/Chemical Engineers and Many others alike. I still don’t know why is that the case. Maybe the recruiting agents at undergraduate level do not want to scare away students who are already bog down by assignments and homework based on the ‘Solved’ mysteries of last century. I agree, even ‘Euler’ equations – simplified form of N-V equations, will require a significant amount mathematical prowess to tame that beast ‘Computationally’ (leave aside analytically) …

Something interesting happened in February this year, Dr.Terence Tao from UCLA, has a new paper out. (Title Reads : Finite Time Blowup for an Averaged Three-Dimensional Navier-Stokes Equation). His new idea is making quite a buzz around but as most of them say, it just a new direction to the approach, but no way closer to resolving the problem. Next stop is to study the consequences of this paper and ( the one involving finite time blowup of Euler Equations), but that would be a summer Project altogether. I would be seeking help from some colleagues who have a formal training in Logic Gates. Next are the Von Neumann Machines. Still this cross pollination of different ideas from distant branches of Engineering might just pay off for the next generation wave of researchers … Yes it is as weird as Indian Hip Hop – Beware the MC !

Blowup :

Now, just to learn what it means to ‘Blow up’ a solution, We would need a whole lot of machinery of Numerical Methods and Linear Algebra. But in short means exactly what it sounds – Reaching non physical values of Physical entities ( or just infinities). Let the melody ring in your ear over and over again …

Computational Vs Theoretical Vs Experiments !!!!

This is how it feels sometimes:

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On ‘origin of the most important mathematical symbol in the history of mathematics …’

inpursuitoftheunknown

An equation derives its power from a simple source. It tells us that two calculations, which appear different, have the same answer. The key symbol is the equals sign, =. The origins of most mathematical symbols are either lost in the mists of antiquity, or are so recent that there is no doubt where they came from. The equals sign is unusual because it dates back more than 450 years, yet we not only know who invented it, we even know why. The inventor was Robert Recorde, in 1557, in The Whetstone of Witte. He used two parallel lines (he used an obsolete word gemowe, meaning ‘twin’) to avoid tedious repetition of the words ‘is equal to’. He chose that symbol because ‘no two things can be more equal’. Recorde chose well. His symbol has remained in use for 450 years.

— Ian Stuart,  In Pursuit of the Unknown: 17 Equations That Changed the World

Refer : Brain Pickings Article on this !

Quote of the day – Michael Atiyah

The mathematical problems that have been solved or techniques that have arisen out of physics in the past have been the lifeblood of mathematics — Sir Michael Atiyah

Exponential Re-distribution

Y axis (Confidence Level) : Confidence level of an average graduate student in the class to include the word ‘We can suffice from’ while doing the mathematical derivations in home-assignment

X-axis : The Count of the the phrase ‘maybe we can’ , which the mathematics professor uses while performing the prototype derivations in the class which were well accepted and cited in scientific community (definite skeptical nature implied)

Absurdity Exemplified!!

Concept of Absolute Randomness is self involving as one tries to get a practical meaning out of it. I was always intrigued by the efforts of mathematicians and programmers who try to codify the randomness of the system. For instance, random number generators. Even random digit generators are also pseudo random in nature due to their inherent artificial formulation. For years, many bright minds have been scratching their heads to handle the problem which is sort of a self-contradiction in itself. Imagine finding a coherent pattern which will eventually evolve ( as you proceed iteratively ) and you will find yourself staring at a ultimate series of numbers (called a ‘cipher-text’ if you happen to work for a intelligence agency) which can never be decoded to get the information out of it. It can be compared to a effort to understand the very mind and  work-ethic of the ‘creator’, who according to many does not prefer the use of theory of probability at his disposal to determine the sequence of  events in  the universe. Will get back to this after a short detour.

Many efforts towards finding a way to describe the randomness in the universe, by defining set of equations which will give a plausible rather conforming explanation to every observation made, appear to be futile in a sense ( not to discredit the progress which has been made for few decades).The very idea of defining the system in distinct mathematical equations is a attempt to find a symmetry( symmetry-one of the inherent and likable forms of pattern in mathematics) in this random unsymmetrical surrounding.

Why is it so important to define a mathematical equalities for describing the  dependent events? Mathematics has been a very cherished language for  physicists to communicate and mathematical equivalence now appears to be existing and continues to evolve in the nature like a dominant gene suppressing every recessive trait in its path. To name a few, Cauchy–Schwarz inequality, Minkowski inequality, Chebyshev’s inequality ,there are inequalities still surviving and finding their bread and butter in the mathematical world through just being unique and by the inability of mathematicians to avoid them. Notable amongst them, the forerunner of quantum mechanics, Heisenberg’s Uncertainty Principle has enjoyed much of the attention.

Who doesn’t like beautiful objects with inherent symmetry! Aesthetic and ergonomic symmetry remains one of the major marketing strategies to sell a product. Many famous t-shirt brands have synthetic embedded ‘Celtic Knots’ on them.

Well not many would disagree, but Audrey still epitomizes the word ‘beautiful’, emphasizing bilateral symmetry in human grimace (not to mention Roman Holiday!! how wonderful actress she was!!).

Self-repeating randomness is not new to the mathematical world. I am referring to the fractals. Mandelbrot set is a well known example of fractals**. I tried to simulate fractals in spare time, like a program written to create a fractal for defined co-ordinate set (but  still not working!! ). Some of the Mandelfract generators are really fun to watch and simulate provided you have enough computational resources to follow them as deep as you desire. But fact remains same,these are also half baked cookies!! not random enough!!

Refer this : http://homepage.ntlworld.com/daniel.freeland/index.html for nice and subtly created fractals.

Recently (February, 2010 to be precise **), a group of physicists manage to produce a true random computational number generator. A significant cornerstone itself. Work certified by the Bell’s theorem,which is basically a inequality in mathematical sense. According to them,such a strong form of randomness generation is impossible classically and possible in quantum systems only. An another attempt to violate the basic implications of Nature! Mathematical inequalities are inevitable. So rather than avoiding it,how about pursuing them. Progress through ‘Finding the inequalities in Nature’ ….that can easily be the future title of my book!!!

If we try to connect it to the ‘unscriptedness’ of human lives,we can easily make out some disadvantages of being told what to do or being guided to be told what to do. Very idea of the ‘creator’ watching our actions and dictating my day-today activities like a puppet master is kind a disappointing. Keeping the philosophical school of thought and religious views aside. Imagine a source code, knowing that it will constantly deny your application in an emergency, would you still be taking a leap of faith and applying just in case it will fail and grant ‘one’ out of a million rejections it has produced? These other set of lesser probabilities will contribute towards what we interpret as ‘Hope’. This randomness brings a great joy in our trivial lives if we choose to pay attention to them. That too if you are in charge of the ‘’strings’’!!! Unscripted life is the most valuable gift humans have received from its ‘’creator’’ … again if we choose to believe there is one…so I would say ‘creator’ in more general way like from ‘randomness of nature’ otherwise ‘scriptedness’ or ”mathematical equalities” would have been ‘continuous’ as in saying ‘perpetually available’ in nature. I found this analogy engaging in mathematical sense. Lets hope that this randomness remains unbounded,which will eventually give us a unprecedented drive to attain the great mathematical milestones yet to be seen!!!

May Chaos prevail!!

Peace!!!

**Futher reading please refer

http://www.nature.com/nature/journal/v464/n7291/full/nature09008.html

http://www.cs.princeton.edu/~wayne/mandel/gallery/

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