Speculation. Limited by language, lack of knowledge, etc. 1) Is there a central cause for the World? If yes, They are God. If yes, what are Their characteristics? If yes, what again explains Their existence? 2) Human history is effectively observe and exploit. We lack many fundamental understandings. Read more 08 Nov 2025 - less than 1 minute read
Updated: 27/10/25 Ref: “Foundations of Applied Mathematics, Vol 2” by Humpherys, Jarvis, Evans. Consider the optimization problem \[{ {\begin{aligned} \text{minimize} \quad &\, f(x) \\ \text{subject to} \quad &\, g _1 (x) \leq 0, \\ &\, \quad \vdots \\ &\, g _m (x) \leq 0 \end{aligned}} }\] where ${ f ,}$ ${ g _i }$s are co... Read more 26 Oct 2025 - 7 minute read
Ref: “Convex Optimization” by Boyd, Vandenberghe. “Convex Optimization” by Boyd. Lecture-2. Link to lecture: Lecture. [Dual cones] Note that cones induce inequalities. Link to Stackexchange post: Link. Consider ${ \mathbb{R} ^n . }$ Let ${ K \subseteq \mathbb{R} ^n }$ be a cone. Consider the dual space \[{ (\mathbb{R} ^n)^{\ast} := \l... Read more 23 Oct 2025 - 3 minute read
Updated: 21/10/25 Ref: “Introduction to Mathematical Statistics” by Hogg, McKean, Craig. “Statistics for Mathematicians” by Panaretos. [Hypothesis Testing] Consider a random variable ${ X }$ with density ${ f(x; \theta) , }$ ${ \theta \in \Omega . }$ Suppose we think ${ \theta \in \Omega _0 }$ or ${ \theta \in \Omega _1 , }$ where ${ \... Read more 20 Oct 2025 - 6 minute read
Consider the plane ${ \mathbb{R} ^2 . }$ Consider the origin ${ (0, 0) . }$ Consider the natural error generating process: The random error vector is ${ \mathcal{E} = (\varepsilon _1, \varepsilon _2) . }$ The components ${ \varepsilon _1, \varepsilon _2 }$ are independent and identically distributed. The distribution of ${ \mathcal{E} }$... Read more 08 Oct 2025 - 1 minute read