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.

  [Read More]

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

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.