The aim of this article is to record and add some references on the recent post by Gappy discussing a number of drawdown formulas, see the original post [1] ( here) - primarily to act as a keepsake for myself in the future. Some other useful references on this topic include [2,3,4] .

In this article, we implement Conway’s Game of Life, see [1] for more information. This is more of an exercise for myself to determine how to insert and play around with JavaScript directly within my Hugo setup.

Background #

For context, and for those of you unaware, Conway’s famous Game of Life, is a simple game that aims to emulate in an extremely basic sense, life. More specifically, with a few very simple rules underpinning the game, you can create an abundance of complexity, including a general purpose computer, and even the Game of Life itself, see here, or here for the Game of Life inside the Game of Life, inside the Game of Life! In more specific terms, it is a simple game that produces emergent behaviour.

This is a short summary of the Kelly Criterion, initially developed by J. Kelly Jr. in their 1956 manuscript [1] , which describes the optimal way in which to size bets as to obtain the greatest expected logarithmic returns in a betting game. Plenty of further reading is available on this topic, one useful resource is [2] , which is a short article on the Kelly Criterion from a course taught by T. Ferguson of UCLA in 2009. Ferguson also has a number of other interesting articles on his webpage, which you can find here. There also exist a plethora of other articles from the famous E. Thorp on the matter, see for example [3] . Note, for brevity, the notation used here aims to remain consistent with that of Thorp and Ferguson.

This is a generalisation of Q18 from 50 Challenging Problems in Probability.

Assume first that we have a fair coin, and thus the probability of getting a heads or a tails is $1/2$. Let $X_n(\omega) = (t_1(\omega),\dots,t_n(\omega))$ denote a possible realisation of a sequence of $n$ coin tosses from a random sampling $\omega$ from the space of all possible random samplings $\Omega$. Then, we wish to determine

This is an oldie, but a cute little trick to solving Fermat’s Little Theorem that I came across back when I was an undergraduate. This is a well known proof and can be found in Wikipedia¹, although I found this proof in an article posted by Fermat’s Library ².