From UPSC perspective, the following things are important :
Prelims level : Ramanujan and his theories
Mains level : Application of Ramanujan's theories
Due to the remarkable originality and power of Ramanujan’s genius, the ideas he created a century ago are now finding applications in diverse contexts. Two among these are signal processing and Black Hole physics.
- Examples of signals that are processed digitally include obvious ones like speech and music and more research-oriented ones such as DNA and protein sequences.
- All these have certain patterns that repeat over and over again and are called periodic patterns.
- For example, a DNA molecule is made up of 4 bases (Adenine Guanine, Thymine and Cytosine).
- Sometimes, a sequence, say AGT, keeps repeating several times in a region of the DNA.
- In real life, more complex repeating patterns may need to be identified as they bear significance to health conditions.
- So, in signal processing, one thing we are interested in is extracting and identifying such periodic information.
How is Ramanujan referred in signal processing?
- Identifying and separating the periodic portion is much like using a sieve to separate particles of different sizes.
- A mathematical operation akin to a sieve is used to separate out the periodic regions in the signal.
- Some of the best-known methods to extract periodic components in signals involve Fourier analysis.
- Using Ramanujan Sums for this process is much less known. “A Ramanujan Sum is a sequence like c(1), c(2), c(3) … This sequence itself repeats periodically…
- It was thought, by a number of authors before me, to be useful in identifying periodic components in signals, much the same as sines and cosines (trigonometry) are used in Fourier analysis.
Black Hole Physics
- Ramanujan’s interest in the number of ways one can partition an integer (a whole number) is famous.
- For instance, the integer 3 can be written as 1+1+1 or 2+1. Thus, there are two ways of partitioning the integer 3.
- As the integer to be partitioned gets larger and larger, it becomes difficult to compute the number of ways to partition it.
- The seemingly simple mathematical calculation is related to a very sophisticated method to reveal the properties of black holes.
Partition of integers
- Ramanujan related this counting problem to some special functions called “modular forms”.
- A modular form is symmetric, in the sense that it does not change, under a set of mathematical operations called “modular symmetry”.
- A geometric analogy for such a function would be a circle which does not change its shape under rotations [circular symmetry].
- Using this symmetry, Ramanujan and G.H. Hardy found a beautiful formula to compute the number of partitions of any integer.
Black Hole entropy
- A separate concept in physics, entropy, explains why heat flows from a hot body to a cold body and not the other way around.
- The mock theta functions of Ramanujan have come to play an important role in understanding the very quantum structure of space-time – in particular the quantum entropy of a type of Black Hole in string theory.
- Stephen Hawking showed that when we take into account quantum effects, a Black Hole is not quite black, it is rather like a hot piece of metal that is slowly emitting Hawking radiation.
- Thus, one can associate thermodynamic quantities like temperature and entropy to a Black Hole.