Tribu produit

Je trouve $\Er{\OXT}$ plus court à écrire, tandis que $\bigotimes_{t \in \Bbb{T}} \Er$ est plus flexible.

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.

The basic arithmetic properties of $\R$, as an equivalence class of Cauchy sequences sharing the same limits, didn’t arouse our interests. That’s just an extension of its rational counterpart due to some arithmetic properties of limits.

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. It would be inconvenient to include the macros after invoking $\KaTeX$’s render function.

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.

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 $A$.

I've chosen $I$ instead of $B$ to express the index set because this reminds me of an array of $(a_i)_i$ indexed by $I$.

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. I was too lazy to get the dependencies fixed. I just compiled it without GFortran and pkg-config under the ~/src folder.

Introduction

This article records my errors and difficulties encountered on the first day I came across Red Hat Enterprise Linux 7 in my school’s laboratory, as a normal user without sudo privileges.

The login screen was gdm, and the desktop environment was GNOME. IBus was used as the input engine.

Packages installed

The principal goal is to install tools that I usually use on RHEL without sudo permissions. To do so, I’ve downloaded the executable binaries or source code of these packages. As I wanted to focus on my studies, I prefer downloading executable binaries.

Statement

Slogan version

$$\sigma = \pi + \lambda$$

$\sigma$	$\pi$	$\lambda$
“sum”	“product”	“limit”
universe	nonempty	universe
complement		complement
countable union	finite intersection	disjoint countable union

A $\sigma$-algebra is a $\pi$-system and a $\lambda$-system, and vice versa.

Wiki version

A $\lambda$-system is a synonym of a Dykin system.

$$\mathcal{P} \subseteq \mathcal{D} \Longrightarrow \sigma(\mathcal{P}) \subseteq \mathcal{D}$$

Given a $\pi$-system contained in a $\lambda$-system. Then the $\sigma$-algebra generated by the $\pi$-system is also contained in the $\lambda$-system.

[Read More]

Update: I suggest reading a newer tutorial for setting up your custom API server that works with GitHub Apps. The package maintainers suggest hosting your own API server.

Goal

To host an instance of Staticman v3 server on Heroku.

This post involves server-side setup of the commenting system. If you simply want to have a taste of this system on GitLab, you may try my demo GitLab Page.

I try to address some concerns about this API service in the introduction of this series to keep this page focused on the technical aspects of my customizations against staticman/dev branch.

Background

While making changes to a theme for a static site generator, I changed files under a Git submodule included in the repo for my blog. (e.g. themes/beautifulhugo) That’s ideal for local testing, but not for version control. As a result, I cloned the repo for the theme to a directory separate from the one for my bloge (say, ~/beautifulhugo), and commit the changes there, then performed a Git submodule update so as to make the workflow clean.