Sekai 🌐 🗺

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

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

Minimal Jekyll Site with Static Comments

Setup Staticman v3 and Jekyll on GitHub Pages

Introduction This is the GitHub Pages version to my GitLab Pages with Staticman tutorial. I didn’t plan to test whether Staticman v3 work on GitHub since it’s proprietary. However, from Staticman issues #222 and #227, we know that the official server doesn’t respond to GET /v2/connnect/<USERNAME>/<REPONAME> To help others, I self-advertised my own Staticman API instance and the migration to GitLab pages. Unfortunately, nobody had managed to create a GitHub repo running on my API instance. [Read More]

Espace de trajectoires

Comparaison des références

Tribu produit source symbole engendrée par Prof $\Er{\OXT}$ $\mathcal{C}_0 = \Big\lbrace \lbrace f \in E^\Bbb{T} \mid f(t) \in B \rbrace \bigm\vert t \in \Bbb{T}, B \in \Er \Big\rbrace$ $\mathcal{C}_1 = \Big\lbrace \lbrace f \in E^\Bbb{T} \mid f(t_i) \in B_i \forall i \in \lbrace 1,\dots,n \rbrace \rbrace \newline \bigm\vert t_j \in \Bbb{T}, B_j \in \Er \forall j \in \lbrace 1,\dots,n \rbrace, n \in \N^* \Big\rbrace$ Meyre $\bigotimes_{t \in \Bbb{T}} \Er$ des cylindres $C = \prod_{t \in \Bbb{T}} A_t$ d’ensembles mesurables $A_t \in \Er$ de dimension finie $\card{\lbrace t \in \Bbb{T} \mid A_t \neq E \rbrace} < \infty$ Je trouve $\Er{\OXT}$ plus court à écrire, tandis que $\bigotimes_{t \in \Bbb{T}} \Er$ est plus flexible. [Read More]

Real Number Construction From Dedekind Cuts

A geometrically intuitive approach

Goal To gain a real understanding on real numbers. Analytical construction I “swallowed” the Compleness Axiom, then I worked on exercises on $\sup$ and $\inf$, and then the $\epsilon$-$\delta$ criterion for limits, before completing $\Q$ with Cauchy sequences. I’ve also heard about the completion of a metric space in a more general setting. My professor once said that it suffices to view this proof once throughout lifetime: the proof itself wasn’t very useful. [Read More]

Custom $\KaTeX$ Macros

More efficient math editing

Background Same as the last section in Beautiful Hugo Improvements. Goal To write math efficiently by automatically loading longer code with shorter macro code. For example, when I wrote Some Infinite Cardinality Identities, it would be ten times more quicker and efficient to type \card{C} than to write \mathop{\mathrm{card}}(C) all the time. Changes committed to my repo The current version of Beautiful Hugo is still using $\KaTeX$ v0.7, which doesn’t support macros in auto-rendering. [Read More]
Hugo  KaTeX  math 

Better Hugo ToC Fix

A JavaScript free way to improve default ToC

Background

I applied a fix to Hugo’s ToC ten days ago.

Drawbacks

To make the script non-render blocking, one has to place it in the footer. As a result, it takes about 0.2 seconds to remove the excess <ul> tag.

Solution

Thanks to Beej126’s Hugo template code, this site delivers table of contents processed by Hugo during GitLab’s continuous deployment.

Hugo 

Some Infinite Cardinality Identities

Working with infinite sets

Purpose This post aims at recapturing the main ideas of the formal proofs that I’ve read. It never tries to replace them. You may consult the references if you need any of them. Some notations Unless otherwise specified, all cardinalities here are infinite. Denote $\mathfrak{a} = \card{A}$, $\mathfrak{b} = \card{B}$ and $\mathfrak{i} = \card{I}$. Sum $\mathfrak{a} + \mathfrak{b} = \card{A \cup B}$ provided that $A \cap B =\varnothing$. Product $\mathfrak{a} \, \mathfrak{b} = \card{A \times B}$ Power $\mathfrak{a}^\mathfrak{i} = \card{A^I}$, where $A^I = \lbrace f \mid f: I \to A \rbrace$ denotes the set of functions from $I$ to $A$. [Read More]

Install Julia From Source

Custom built Julia from GitHub

Goal To get Julia installed as a normal user on RHEL 6. Motivation “Julia talks like Python but walks like C.” To do statistics more efficiently. The compiled binaries often contain install scripts which put files to shared folders under /usr. Consequently, they have to be run as sudo privileges. That drove me to start this lengthy Julia compilation. Installation Without sudo privileges, I’ve chosen to compile Julia from source. [Read More]
Linux  Julia