Sunday, June 19, 2022

Dr. Sneh Raj - Sneh Mausi - Bua - Daadi - is no more.



My Mausi, Dr. Sneh Raj is no more. She died at age 91. She went on her own terms. She said she did not wish any tubes or artificial means of respiration (and such) and her family respected her wishes. All her life she has battled the circumstances she found herself in cheerfully and with enthusiasm. She decided when it was time not to battle any more. 

She made a family and community across three countries -- India, US and Canada. 

When I first landed in Columbus, she was there to pick me up. When Tejasi was born, she was the Daadi available to look after us and to teach me how to wrap Tejasi in a blanket and hold her properly. And was available in every visit to the US and every marriage in India. Her empathy, sympathy and good humor was legendary -- she shared herself generously with all of us. 

On her 90th birthday, her daughter, Madhulika Raj, together with Madhulika Agarwal, made a book of wishes for her. The picture above is the cover. Click here to  download a copy. I had written there what I wanted to say to her. 

Here is an obituary published in the Akron Beacon Journal.

Mausi was the youngest of three sisters. After them three brothers followed. It is difficult to accept that all three sisters -- Kusum Mausi, Mummy and Sneh Mausi, left so soon after each other. All three sisters in their own way were forces of nature. Their force multiplied because of each other. The three Shaktis of this family. 

Their passing is the end of an era. 

Wednesday, April 06, 2022

Telescoping continued fractions

Krishnan Rajkumar (JNU) and I have a new preprint -- on telescoping continued fractions. I have written previously on telescoping and several times on continued fractions, but this one is unique. I don't think anyone has tried to combine the two ideas. We apply it to find lower bounds for the remainder term for Stirling's formula. Ultimately, we discovered a nice new technique, proved several things required to make it work, but were unable to take it to its natural conclusion (so far!). The preprint (now available at arxiv) has several conjectures. 

The pic is taken on June 9, 2022, in the Blue Door Cafe, Khan Market where we revised the paper. (That day I showed symptoms of COVID, my second time.)

Here is a link to the preprint.

Here is the abstract:

Title: Telescoping continued fractions for the error term in Stirling's formula
Authors: Gaurav Bhatnagar, Krishnan Rajkumar
Categories: math.CA math.NT

In this paper, we introduce telescoping continued fractions to find lower
bounds for the error term $r_n$ in Stirling's approximation \[ n! =\sqrt{2\pi}n^{n+1/2}e^{-n}e^{r_n}.\] This improves lower bounds given earlier by Ces├áro (1922), Robbins (1955), Nanjundiah (1959), Maria (1965) and Popov (2017). The expression is in terms of a continued fraction, together with an algorithm to find successive terms of this continued fraction. The technique we introduce allows us to experimentally obtain upper and lower bounds for a sequence of convergents of a continued fraction in terms of a difference of two continued fractions.

Here is a talk introducing the method. I presented in the Ashoka Math Colloquium on November 2, 2021. It has an overview of the technique. The talk was made keeping undergraduate students in mind, so there is something here which is quite accessible. In particular I have outlined Robbins' approach at the beginning of the talk.