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 $$\forall A \in \vois{X}{x}, \exists j \in I: \forall k \ge j, x_k \in A.$$

• example: absolute convergence of series ($I$ is the collection of finite subsets of $\N$, finite sum $\Sigma \in \R^I$.)
• example: Riemann integral ($I$ is the collection of tagged partitions, the partial order doesn’t depend on tags, $\int \in \R^I$.)

Filter base

A nonempty collection $\mathcal{F} \subseteq \mathcal{P}(X) \setminus {\varnothing}$ such that $$\forall F,G \in \mathcal{F},\exists H \in \mathcal{F}: H \subseteq F \cap G.$$ (contains nonempty part of intersection)
Difference with topological basis: sets have to be nonempty here

Filter

A filter base $\mathcal{F}$ so that

• contains supersets: $\forall F \in \mathcal{F}, \forall G \supseteq F, G \in \mathcal{F}$
• contains intersection: $\forall F, G \in \mathcal {F}, F \cap G \in \mathcal{F}$

The image of a filter $\mathcal{F}$ under a function $f$ is also a filter, denoted by $f[[\mathcal{F}]]$.

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

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. To convince others that it’s also working on GitHub, I decided to create a minimal GitHub repo.

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.