Sekai 🌐 🗺

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

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.

  • only if: let $\mathcal{F}$ be an ultrafilter contained in another filter $\mathcal{G}$. If $\mathcal{F}$ isn’t maximal, let $Y \in \mathcal{G} \setminus \mathcal{F}$. Since $\mathcal{F}$ is an ultrafilter, either $Y \in \mathcal{F}$ or $Y^\complement \in \mathcal{F}$. By construction of $Y$, only the later option is possible, so $Y^\complement \in \mathcal{G}$ by hypothesis, but this contradicts our assumption $Y \in \mathcal{G}$: $\varnothing = Y \cap Y^\complement \in \mathcal{G}$, which is false since $\mathcal{G}$ is a filter.

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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 $\text{airbus}: Y = \sum_{i = 1}^n a_iX_i?$
    • Notion de risque … calculs des coefficients de convolution

SVM

  • Multiclass vs structual, hidden Markov model
  • Plan for this year:
    • apply structual SVM for real SVM
    • apply structual SVM for deep neural network

Auxiliary information

  • auxiliary function given in one partition
  • auxiliary function given in mutiple partitions
  • bootstrap
  • law of iterated logarithms
  • Kullback–Leibler distance
  • convergence: Donsker class, var, covar
  • ranking ration method: convergence to Gaussian process, entropy conditions, Telegrandś inequality
    • weak convergence: KMT, Berthet-Maison
    • strong convergence: ?
      • consequences: Berry-Essen bound, bias & variance estimation of ranking ration method

Euler scheme SDE

I could only write “Toeplitz tape operator”.

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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.

  1. outer regularity: approximation by inner closed sets
  2. 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 $$\mathcal{C} = \lbrace A \in \mathcal{B}(S) \mid \mathinner{\text{desired properties}} \dots \rbrace.$$ Show that

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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 $$\forall A \in \vois{X}{x}, \exists j \in I: \forall k \ge j, x_k \in A.$$
  • 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 $f[[\mathcal{F}]]$.

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

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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.

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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.

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.

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

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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