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Application of Ramanujan’s work in signal processing and Black hole physics

Date: 23 December 2019 Tags: Miscellaneous

Issue

Ramanujan’s  mathematicsdone over a hundred years ago, finds applications today in areas other than pure mathematics, which were not even established during his time. Two among these are signal processing and Black Hole physics.

 

Background

Srinivasa Ramanujan was an Indian mathematician whose contributions to the theory of numbers include pioneering discoveries of the properties of the partition function.

His papers were published in English and European journals, and in 1918 he was elected to the Royal Society of London.

 

Details

Signals processing

  • 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 a 4 bases (Adenine Guanine, Thymine and Cytosine).

  • 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.

  • A mathematical operation similar 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.

  • Further research showed that using these patterns gave a method that worked better than Fourier analysis when the region of periodicity is short.

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.

  • The seemingly simple mathematical calculation is related to a very sophisticated method to reveal the properties of black holes.

  • 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 results of Ramanujan and Hardy on partitions and the former’s subsequent work on what are called mock theta functions have come to play an important role in understanding the very quantum structure of spacetime , in particular the quantum entropy of a type of Black Hole in string theory.