Gauss-Lucas Theorem

The Gauss-Lucas Theorem is an interesting theorem relating the roots of a polynomial P to the roots of its derivative P’.

Specifically, it states that the roots of P’ all lie within the convex hull of the roots of P.

Winning the Olympics

This video has an interesting showerthought about how to define who wins the olympics.

In short, rather than absolute number of medals (which favors large nations) or medals per capita (which favors small nations), a probabilistic-y combination of them is proposed (that’s obviously the technical term for it. Essentially …

SIR Models

This rolled up Twitter thread has a nice explanation of SIR (“suceptible”, “infected”, “recovered”) models of disease / processes.

Forecasting S-Curves

Forecasting s-curves (sigmoid, logistic, etc.) is hard.

This article has an easy explanation of why, and a conclusion that often predicting the specific s-curve given data is hard even up until one is already at the “flattening” portion of the curve.

“New” Objects in Problems

This 3b1b video points out a nice heuristic for how to find solutions to a general problem.

The “source” problem there is about randomly choosing points on a sphere and asking what the probability is that the induced tetrahedron contains the center of the sphere.

But the “algorithm” to find …

The Borsuk-Ulam Theorem and Stolen Necklaces

This 3b1b video overviews the Borsuk-Ulam Theorem, and its applications for the Necklace Splitting problem.

It’s a super nice connection between topology and combinatorics.