Archive

Bunch of my work that I think are worth sharing on the internet. Though I had to lower my standards upon realizing how little material of such quality I've created.

1. Lambda Calculus Notes. My notes from the UofT mentorship program about the lambda-calculus. Still needs some polishing and I also intend to make a lambda-calculus interpreter for this by using the definitions.

2. Lambda Calculus Presentation. The presentation I gave at for the UofT mentorship program.

3. Fibonacci Triangle Problem. My solution to an interesting number theory (?) problem involving a triangle of fibonacci numbers.

4. A proof of the additivity of the Riemann integral. Most textbooks which prove the property $$ \int_a^b f(x)\,dx = \int_a^c f(x)\,dx + \int_c^b f(x)\,dx $$ do it via upper and lower sums and the Darboux definition of the Riemann integral. Textbooks which don't cover the Darboux definition usually skip the proof of this fact. This is because a satisfying proof directly from Riemann sums has to deal with the subtle special case that $b - c$ and $c - a$ are incommensurable and so no two Riemann partitions on the intervals $[a,c]$ and $[c,b]$ will have equal equal subintervals. This file contains a short proof of this fact using Riemann partitions alone.