Sekai 🌐 🗺

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

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]

Ultrafilters Are Maximal

Ultra filter A filer $\mathcal{F}$ containing either $Y$ or $Y^\complement$ for any $Y \subseteq X$. Two days ago, I spent an afternoon to understand Dudley’s proof of this little result. A filter is contained in some ultrafilter. A filter is an ultrafilter iff it’s maximal. At the first glance, I didn’t even understand the organisation of the proof! I’m going to rephrase it for future reference. [Read More]

2018-10-04 Seminar Notes

I jotted down only a few keywords that might be reusable. I didn’t understand any of the talks. Functional Data Analysis Goal: predict equipment temperature Tools: Fourier coefficients (trigo ones), followed by discretisation, min-error estimation, cross-validation 10-folds, $R^2$ adjusted ?, MAE, MSPE Comparison with non-functional data Tolérancement Thème : Traiter les incertitudes sur les dimensions des pièces de l’avion Objectif : établir une modélisation mathématiques construire un virtual twin de l’avion Outils : Modèle de variabilité Modèle d’assemblage $\text{airbus}: Y = \sum_{i = 1}^n a_iX_i? [Read More]

Measures Are Regular

Some remarks on constructing the $\sigma$-algebra

Problem To show that a measure $\mu$ defined on a metric space $(S,d)$ is regular. outer regularity: approximation by inner closed sets inner regularity: approximation by outer open sets Discussion Since this problem involves all borel sets $A \in \mathcal{B}(S)$, the direct way $\forall A \in \mathcal{B}(S), \dots$ won’t work. We have to use the indirect way: denote $$\mathcal{C} = \lbrace A \in \mathcal{B}(S) \mid \mathinner{\text{desired properties}} \dots \rbrace. [Read More]

Filters and Nets

Some basic examples

Motivation $\gdef\vois#1#2{\mathcal{V}_{#1}(#2)}$ Nets and filters are used for describing convergence in a non-metric space $X$. Denote the collection of (open) neighbourhoods of $x \in X$ by $\vois{X}{x}$. Definitions and examples Directed set A partially ordered set $I$ such that $$\forall i, j \in I: i \le j, \exists k \in I: k \ge j.$$ Net A function in $X^I$, where $I$ is a directed set. example: any sequence in $X^\N$ Convergence of nets to a point $x_i \to x$ if $$\forall A \in \vois{X}{x}, \exists j \in I: \forall k \ge j, x_k \in A. [Read More]

Cover Letter Organisation

A simple summary

I keep sentences below short and minimal for memory.

  1. Sender’s contact info at top-right hand corner, followed by receipent’s contact info left-aligned.
  2. “I’m …”, “apply for …, as advertised in …”
  3. Why apply? Link with the company(’s employee)
  4. Pastimes (all-rounded person), continual learning (for useful skills)
  5. Friendly, polite and to-the-point sign-off
  6. Signature followed by sender’s name