## Sekai 🌐 🗺

Sekai is the kanji for 世界, meaning “the world”. That’s a great word because of the scale it designates.

## SCSS to SASS Converter

Background While porting Minimal Mistakes and Beautiful Hugo’s Staticman support to Introduction, I searched for “SCSS to SASS” on DuckDuckGo. The search engine returned results on “CSS to SASS/SCSS” or vice versa. The first theme has Staticman code in SCSS. That would fit into the third theme’s directory structure, which puts SASS files under assets/sass. However, SASS doesn’t have curly braces {}. I feared that after hours of tedious manual replacement, the code would fail to run. [Read More]

## Weak LLN Practice

My intended answer to a weak LLN problem on Math.SE. Problem: Suppose $(X_n)$ is a sequence of r.v’s satisfying $P(X_n=\pm\ln (n))=\frac{1}{2}$ for each $n=1,2\dots$. I am trying to show that $(X_n)$ satisfies the weak LLN. The idea is to show that $P(\overline{X_n}>\varepsilon)$ tends to 0, but I am unsure how to do so. My solution: As in the accepted answer in OP’s previous question https://math.stackexchange.com/q/3021650/290189, I’ll assume the independence of $(X_n)$. [Read More]

## Staticman Lab New Logos

StaticmanLab's new logo GitLab logo recreated from Wikimedia's logo by Darby under CC-BY-SA 4.0 and Staticman logo on GitHub by Erlen Masson under MIT. The old icon for Staticman Lab was made by GIMP from Staticman’s icon in PNG in the GitHub repo. Recently, I’ve found the SVG version of this icon. To serve customers better, I’ve recreated the logo from this SVG file so that the edges in the logo become sharper. [Read More]

## Solution to a $p$-test Exercise

I intended to answer Maddle’s $p$-test question, but T. Bongers has beaten me by two minutes, so I posted my answer here to save my work. The problem statement This is the sum: $$\sum\limits_{n=3}^\infty\frac{1}{n\cdot\ln(n)\cdot\ln(\ln(n))^p}$$ How do I tell which values of $p$ allow this to converge? The ratio test isn’t working out for me at all. Unpublished solution The integral test will do. \begin{aligned} & \int_3^{+\infty} \frac{1}{x\cdot\ln(x)\cdot\ln(\ln(x))^p} \,dx \\ &= \int_3^{+\infty} \frac{1}{\ln(x)\cdot\ln(\ln(x))^p} \,d(\ln x) \\ &= \int_3^{+\infty} \frac{1}{\ln(\ln(x))^p} \,d(\ln(\ln(x))) \\ &= \begin{cases} [\ln(\ln(\ln(x)))]_3^{+\infty} & \text{if } p = 1 \\ \left[\dfrac{[\ln(\ln(x))}{p+1}]^{p+1} \right]_3^{+\infty} & \text{if } p \ne 1 \end{cases} \end{aligned} When $p \ge 1$, the improper integral diverges. [Read More]

## Merge GitHub Pull Requests

Aim Merge a pull request. How? Let’s take Staticman PR 231 as an example. I would like to test it before commiting this merget to Heroku. $cd ~/staticman$ git branch -a * deploy dev master ... $git remote -v eduardoboucas https://github.com/eduardoboucas/staticman.git (fetch) eduardoboucas https://github.com/eduardoboucas/staticman.git (push) heroku https://git.heroku.com/staticman3.git (fetch) heroku https://git.heroku.com/staticman3.git (push) ...$ git pull eduardoboucas pull/231/head:deploy remote: Enumerating objects: 10, done. remote: Counting objects: 100% (10/10), done. [Read More]

## Nested Comments in Beautiful Hugo

Quick links A minimal demo site on GitLab (Source) Beautiful Hugo pull request 222 Pre-release notes for this pull request Motivation For the mathematical ones, please see my previous post. As a math student, it’s inefficient to reinvent the wheel like engineering students. Thanks to three existing examples, I had convinced myself that I could bring this to the theme Beautiful Hugo. Zongren’s Hexo theme (worked best) Made Mistakes Jekyll theme Network Hobo’s customization of Beautiful Hugo (inspired by the second one, but contains a logic error) [Read More]

## Interactive Blog on Static Web Host

Vision gain autonomy: freedom is the basis of moral actions. No freedom, no morality. transcend ourselves: change/improve our lives through free thoughts Goal Convert our free thoughts into free code. Free code allows users around the world to run and/or improve them. This would bring real enhancement to our tools. For example, beautiful math writing used to be a complicated process. A decade ago, this required the installation of a typesetting engine called $\LaTeX$. [Read More]

## Simplex Calculations for Stokes' Theorem

Oriented affine $k$-simplex $\sigma = [{\bf p}_0,{\bf p}_1,\dots,{\bf p}_k]$ A $k$-surface given by the affine function $$\sigma\left(\sum_{i=1}^k a_i {\bf e}_i \right) := {\bf p}_0 + \sum_{i=1}^k a_i ({\bf p}_i - {\bf p}_0) \tag{1},$$ where ${\bf p}_i \in \R^n$ for all $i \in \{1,\dots,k\}$. In particular, $\sigma({\bf 0})={\bf p}_0$ and for each $i\in\{1,\dots,k\}$, $\sigma({\bf e}_i)={\bf p}_i$. Standard simplex $Q^k := [{\bf 0}, {\bf e}_1, \dots, {\bf e}_k]$ A particular type of oriented affine $k$-simplex with the standard basis $\{{\bf e}_1, \dots, {\bf e}_k\}$ of $\R^k$. [Read More]