Math Musings

Navier-Stokes equations are about motion of fluids. Example of fluids being air and water.  Mathematicians and physicists believe that an explanation for both the breeze and the turbulence of fluids can be found through an understanding of solutions to the Navier-Stokes equations .  A fluid is something that you can assume to be a continuum- i.e. not made of discrete particles. To this end, we can already see that the Navier-stokes equations must be approximations, because fluids are made of atoms! The Navier-Stokes equations are, in essence, just Newton’s 2 nd law written in a form that is applicable to continuum bodies, rather than discrete objects. We know that Newton’s second law as F=ma , which can be written as Where v is the velocity. If we are dealing with a fluid, we don’t care about mass, we care about density- which for now we assume is constant-i.e., the fluid is incompressible. Therefore, just by taking the continuum form of Newton’s...

A rope burns non-uniformly for exactly one hour. How do you measure 45 minutes, given 2 such ropes Puzzle Answer

Computer scientists classify problems based on how hard they are to solve, for example P and NP complexity classes. P classes, Polynomial classes, are a polynomial function of the size of the inputs and are generally “easy” This means that when the size of the inputs grows exponentially then the time to solve them does not. Take the example of multiplication. If we keep multiplying with bigger and bigger digits, the processing time of the calculator does not really change to get the answer. This is largely due to the fantastic algorithm it uses. NP stands for non-deterministic polynomial time. Non deterministic means we don’t have a defined way to solve them and rely on heuristics like trial and error. For example, if we have to factorize a very large prime numbers, we will have to use trial and error to find the solution. The solving time would grow exponentially as the size of the prime number increases. In fact, a large part of our internet security is based on the fact t...

Euler is considered to be one of the greatest mathematician of the 18 th century and many believe that he was among the greatest of all time. The answer to the Basel problem is considered to be among his many gems. It is the sum of the series shown below and the answer is astounding! What is π doing there? Mathematician at the time summed up the series to the first 10, 100, and 1000 terms and the slow rate of convergence gave very little clue to what the answer could be.  Euler realized that he would have to find a function that would converge much faster. After some trial and error he found the function shown below whose power series expansion is shown.   So if we integrate the above expression from 0 to 1 This is what he was looking for! To evaluate the integral in the left, he broke it into two parts. Then he evaluated each of the integrals in turn. Then the next integral Put x=1-t When x=1/2, t=1/2 When ...

Enigma In 1918 , German engineer Arthur Scherbius applied for a patent for a machine that coded and decoded secret messages quickly and easily. Scherbius’s  Enigma  shifted letters similarly to a Caesar cipher, but Enigma constantly changed the order in which the letters were shifted. Such pattern changes mask the nature of the cipher by appearing random. Without knowledge of the algorithms generating the cipher patterns, code breaking methods require enormous amounts of computation. With around   possible configurations, the Enigma seemed a cryptographically safe bet. The Enigma worked quickly and accurately. While the Enigma had only modest commercial success, it became widely adopted by the German military. From the mid- until the end of World War II, Germany and its allies encrypted and decrypted important communications using various versions of the Enigma. Conquering Enigma During World War II, many of the world’s best problem solvers consid...

A prime number is a number that is divisible only by itself and 1. Some examples being  2, 3, 5, 7, 9, 11, 13.…   The great mathematician Euclid proved centuries ago that there are an infinite number of them continuing without pattern ad infinitum. However, their real importance is given by what is known as the Fundamental Theorem of Arithmetic. It states that every number can be uniquely expressed as a product of prime numbers. Let’s take the example of 364 = 2 × 2 × 7 × 13. This is the only way that 364 can be obtained by prime numbers! Therefore, they are the atoms or building blocks of numbers. But in 1859, the great German mathematician Bernhard Riemann hypothesized that the spacing of the primes logically follows from other numbers, now known as the “nontrivial zeros” of the Riemann zeta function. The Riemann zeta function takes inputs not only real numbers but also  complex numbers — meaning they have both “real” and “imaginary” components — and yields ...

Young Achiever: Gurugram teen wants to promote beauty of Math and its impact on everyday life Rohan Jha is the inaugural winner of Steven H Strogatz Prize for Math Communication conducted by National Museum of Mathematics in New York For Rohan Jha (16), the passion for  Math  was fuelled by a short video that he made at his architect mother’s request. It was the summer following his IXth grade at the Pathways School  Gurgaon , when Rohan made the short video on the Birch Swinnerton Dyer (BSD) Conjecture, a difficult millennium problem in Math, that changed his life’s course. His father Pankaj Jha, a startup owner in India who had worked at the Wall Street for over 20 years, sent the video to reputed mathematicians hoping one of them might respond with encouragement. Michael Harris, professor at Columbia university, wrote back saying that he would be delighted to have Rohan as a student when the time comes, while Benedict Gross, professor at Harvard Universit...