Quick links

1. A minimal demo site on GitLab (Source)
2. Beautiful Hugo pull request 222
3. Pre-release notes for this pull request

Motivation

For the mathematical ones, please see my previous post.

As a math student, it’s inefficient to reinvent the wheel like engineering students. Thanks to three existing examples, I had convinced myself that I could bring this to the theme Beautiful Hugo.

• Zongren’s Hexo theme (worked best)

• Made Mistakes Jekyll theme

• Network Hobo’s customization of Beautiful Hugo (inspired by the second one, but contains a logic error)

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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 $\LaTeX$. Thanks to freely available scripts like MathJax and $\KaTeX$, it’s now possible to write math viewable by any modern web browser by writing the content in the middle.

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 $\R^k$.

$$ Q^k := \left\{ \sum_{i=1}^k a_i {\bf e}_i \Biggm| \forall i \in \{1,\dots,k\}, a_i \ge 0, \sum_{i=1}^k a_i = 1 \right\} $$

Note that an oriented affine $k$-simplex $\sigma$ has parameter domain $Q^k$.

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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

$$ \lVert f \rVert_{\rm Lip} = \sup_{x \ne y} \frac{|f(x) - f(y)|}{d(x, y)}. $$

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.

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. En effet, c’est une définition universaire sur des ensembles, selon une question sur la trace sur Math.SE.

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. (Say, poured some liquid into an empty cup and take the temperature.)

Background

I’ve installed Ubuntu 18.04 on my new laptop.

Problem

The /var partition was too small. The system complained that only 200 MB was left.

Solution

1. Rebooted with my live USB.
2. Opened GParted.
3. Moved /tmp partition to the left and grew it to 6 GB.
4. Grew /var partition to 16 GB
5. Click

Results

Details [TL;DR]

Here’s the GParted generated log.

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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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”.

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