Introduction to Programming Updated: 9/3/26 Link to Udacity subscription: Link. Link to Udacity course: Link. All rights of the (often wonderful) images found in these notes go to Udacity unless explicitly noted. We will learn the foundations of four of the most popular languages: HTML, CSS, Javascript, and Python. \[{ \underline{\textbf{I... Read more 09 Mar 2026 - less than 1 minute read
Note that \[{ \int _0 ^{1} \frac{1}{1+x ^2} \, dx = \frac{\pi}{4} . }\] This suggests studying inequalities of form \[{ \boxed{ \int _0 ^{1} \frac{P(x) ^2}{1 + x ^2} \, dx > 0 } }\] where ${ P(x) }$ is a polynomial, to get bounds on ${ \pi . }$ Ideally, we want ${ P(x) ^2 }$ to leave a constant remainder on dividing by ${ (1 + x ^2) . ... Read more 08 Mar 2026 - 2 minute read
The foundational idea behind money seems to be: Be of “value” to others, by involving in selling something. Earn “value” in return, by earning money. Some problems are: This “value” earned, namely money, is often unequally distributed. Value to other people isn’t the only measure of value. For eg, consider Scientific Disco... Read more 06 Mar 2026 - less than 1 minute read
Ref: “Linear Algebra and Matrices” by Helene Shapiro. Chapter on Simultaneous Triangularization. Math StackExchange answer here. Thm [Commuting matrices share an eigenvector] Let ${ A, B \in \mathbb{C} ^{n \times n} . }$ Suppose ${ AB = BA . }$ Then ${ A, B }$ share an eigenvector. Pf: Let ${ \lambda }$ be an eigenval... Read more 06 Mar 2026 - 2 minute read