Math

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.)

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 $f[[\mathcal{F}]]$.

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.

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 $I$.

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]

This isn’t a substitute for books .

Reminder

• $A \preceq B$: $A$ can be “injected” into $B$.
• $A \sim B$: $A$ and $B$ share the same cardinality.
• $A \prec B$: $A$ can be “injected” into $B$, but it’s “smaller” than $B$.
• A finite set can be “counted” from one to some nonnegative integer.
• Infinite is the “antonym” of finite.

Wolf’s proof

When I first saw this proof in Robert S. Wolf’s Proof, Logic and Conjecture: The Mathematician’s Toolbox, I gave it up since it wasn’t as intuitive as the statement.