Chebyshev’s sum inequality is an important inequality and can be proved in various ways. Emile Picard proved it in the 1880s via concepts of centre of gravity. It can be proved easily once one has proved a more fundamental inequality. To see how read here.

11 February 2019:

The Math Stack Exchange seems to be a refuge for some arrogantly offensive types so students who want some illumination through that forum do so at their own risk. Don’t be surprised if someone humiliates you. And mathematicians wonder why many people don’t like mathematics! For my views on this forum, read this paper

21 April 2019:

How did Maxwell derive his famous velocity distribution law, one of the foundational elements of statistical mechanics? As you will see, he used geometry and functional equation concepts to derive the law in only a few lines. To see how the gaps are filled in read the detailed paper here.

11 July 2019:

Singular Value Decomposition (SVD) is an important part of machine learning algorithms and this article goes through the mechanics of SVD in a tutorial format. Download the article here.

A Powerpoint presentation can be accessed here.

27 August 2019:

The inequalities of Holder and Minkowski are fundamental to analysis and J E Littlewood devoted a whole book ( “Lectures on the Theory of Functions” ) to squeezing the mathematical pips out of them. To see how Littlewood proved these inequalities download this short article here.