Ultrafilters Are Maximal
Posted on October 5, 2018
(Last modified on February 16, 2021)
| 2 minutes
| Vincent Tam
|
0 comment
- 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]
2018-10-04 Seminar Notes
Posted on October 4, 2018
(Last modified on February 16, 2021)
| 3 minutes
| Vincent Tam
|
0 comment
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 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”.
[Read More]
Measures Are Regular
Some remarks on constructing the $\sigma$-algebra
Posted on October 3, 2018
(Last modified on February 16, 2021)
| 3 minutes
| Vincent Tam
|
0 comment
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 $$\mathcal{C} = \lbrace A \in \mathcal{B}(S) \mid
\mathinner{\text{desired properties}} \dots \rbrace.$$ Show that
[Read More]
Filters and Nets
Some basic examples
Posted on October 3, 2018
(Last modified on February 16, 2021)
| 2 minutes
| Vincent Tam
|
0 comment
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 $f[[\mathcal{F}]]$.
[Read More]
Cover Letter Organisation
A simple summary
Posted on October 2, 2018
(Last modified on October 3, 2018)
| 1 minutes
| Vincent Tam
|
0 comment
I keep sentences below short and minimal for memory.
- Sender’s contact info at top-right hand corner, followed by receipent’s
contact info left-aligned.
- “I’m …”, “apply for …, as advertised in …”
- Why apply? Link with the company(’s employee)
- Pastimes (all-rounded person), continual learning (for useful skills)
- Friendly, polite and to-the-point sign-off
- Signature followed by sender’s name
Minimal Jekyll Site with Static Comments
Setup Staticman v3 and Jekyll on GitHub Pages
Posted on September 30, 2018
(Last modified on April 13, 2023)
| 2 minutes
| Vincent Tam
|
1 comment
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.
[Read More]
Espace de trajectoires
Comparaison des références
Posted on September 28, 2018
(Last modified on February 16, 2021)
| 2 minutes
| Vincent Tam
|
0 comment
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
Posted on September 27, 2018
(Last modified on February 16, 2021)
| 2 minutes
| Vincent Tam
|
0 comment
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.
[Read More]
Custom $\KaTeX$ Macros
More efficient math editing
Posted on September 27, 2018
(Last modified on April 13, 2023)
| 2 minutes
| Vincent Tam
|
3 comments
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.
[Read More]
Better Hugo ToC Fix
A JavaScript free way to improve default ToC
Posted on September 27, 2018
| 1 minutes
| Vincent Tam
|
0 comment
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.