Sekai 🌐 🗺

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

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, I’ll assume the independence of $(X_n)$. [Read More]

Staticman Lab New Logos

StaticmanLab's new logoGitLab 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 Unpublished solution The integral test will do.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 (fetch) eduardoboucas (push) heroku (fetch) heroku (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]

La norme lipschitzienne est complète

Dans l’article de Robert Fortet et Edith Mourier en 1953, une distance entre deux mesures de probabilité sur un espace métrique est définie. De nos jours, je trouve la façon dont ils l’ont écrit assez difficile à comprendre. Je suis plus à l’aise avec $\sup$ que “b.s.” que désigne “borne supérieure”. Ils se sont servi de $M[f]$ pour $\lVert f \rVert_{\rm Lip}$, où $$\lVert f \rVert_{\rm Lip} = \sup_{x \ne y} \frac{|f(x) - f(y)|}{d(x, y)}. [Read More]

Mesurabilité des réalisations trajectorielles

$X: \omega \mapsto X(\cdot, \omega) \in \mathcal{M}((\Omega, \mathcal{A}), (\CO(\Bbb{T},\R), \Bor{\CO}))$

Notations Supposons toutes les notations dans Espace de trajectoires. Problématique La mesurabilité de l’application dans le sous-titre est basée sur l’égalité suivante. $$\Bor{\R}{\OXT} \cap \CO = \Bor{\CO}$$ J’ai passé quatres heures pour comprendre pourquoi ça entraîne la mesurabilité ? pourquoi l’égalité elle-même est vraie ? Réponses Mesurabilité de la trace sur $\CO$ de $\Bor{\R}{\OXT}$ A la première lecture, je ne connaisais même pas la définition de la trace d’une tribu sur un emsemble. [Read More]

What are Dataframes?

Understand dataframes from a non-example

Motivation The books that I read in the past didn’t explain what a dataframe meant. Definition Dataframe A table of data in which the values of each observed variable is contained in the same column. Counterexample I’ve difficulty in reading long lines of text like the above definition, so let’s illustrate this definition with a counterexample. We have carried out repeated experiments with four types of things and obtaine some data. [Read More]