What are Dataframes?

Understand dataframes from a non-example

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. [Read More]

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 $\mathcal{G}$. [Read More]

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? [Read More]

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. outer regularity: approximation by inner closed sets 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. [Read More]

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. [Read More]

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. [Read More]

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. [Read More]

Some Infinite Cardinality Identities

Working with infinite sets

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 $A$. [Read More]

$\pi$–$\lambda$ Theorem

Monotone Class Lemma

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. $$\mathcal{P} \subseteq \mathcal{D} \Longrightarrow \sigma(\mathcal{P}) \subseteq \mathcal{D}$$ [Read More]

CSB Theorem

A visual argument for CSB Theorem

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. [Read More]