From 99f57be39e284d75279f782dd6073341b1425651 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Fri, 6 Mar 2026 14:55:54 -0500 Subject: [PATCH] Housekeeping. --- .gitignore | 1 + document.tex | 16 ++++++++-------- preamble.sty | 18 ++---------------- refs.bib | 9 +++++++++ spec.toml | 14 ++++++++++++++ src/dg/derivative/sets.tex | 8 +++----- src/fa/tvs/completion.tex | 2 ++ src/measure/measure/index.tex | 1 - src/topology/uniform/metric.tex | 1 + 9 files changed, 40 insertions(+), 30 deletions(-) create mode 100644 spec.toml diff --git a/.gitignore b/.gitignore index 79114f6..5b78f8f 100644 --- a/.gitignore +++ b/.gitignore @@ -13,3 +13,4 @@ codebook.toml *.log *.out *.pdf +./spec.db diff --git a/document.tex b/document.tex index 722a05d..8adcd41 100644 --- a/document.tex +++ b/document.tex @@ -1,17 +1,17 @@ -\documentclass{report} +% \documentclass{report} \usepackage{amssymb, amsmath, hyperref} \usepackage{preamble} \begin{document} +\input{./src/cat/index} +\input{./src/topology/index} +\input{./src/fa/index} +\input{./src/measure/index} +\input{./src/dg/index} +\input{./src/process/index} -\input{./src/cat/index.tex} -\input{./src/topology/index.tex} -\input{./src/fa/index.tex} -\input{./src/measure/index.tex} -\input{./src/dg/index.tex} - -\bibliographystyle{abbrv} % We choose the "plain" reference style +\bibliographystyle{alpha} % We choose the "plain" reference style \bibliography{refs} % Entries are in the refs.bib file diff --git a/preamble.sty b/preamble.sty index 9585830..b567d48 100644 --- a/preamble.sty +++ b/preamble.sty @@ -1,5 +1,5 @@ -\NeedsTeXFormat{LaTeX2e} -\ProvidesPackage{jerrylicious}[Jerry's Tex Mess] +% \NeedsTeXFormat{LaTeX2e} +% \ProvidesPackage{jerrylicious}[Jerry's Tex Mess] \RequirePackage{amsthm,amssymb,amsfonts,amsmath} @@ -8,28 +8,19 @@ % ------------- Block Environmets --------------- - -\makeatletter - -\theoremstyle{plain} \newtheorem{theorem}[subsection]{Theorem} \newtheorem{proposition}[subsection]{Proposition} \newtheorem{lemma}[subsection]{Lemma} -\theoremstyle{definition} \newtheorem{definition}[subsection]{Definition} \newtheorem{example}[subsection]{Example} % \newtheorem{exercise}[subsection]{Exercise} % \newtheorem{situation}[subsection]{Situation} -\theoremstyle{remark} \newtheorem{rem}[subsection]{Remark} \newtheorem{remark}[subsection]{Remark} % \newtheorem{remarks}[subsection]{Remarks} -\numberwithin{equation}{subsection} - -\makeatother % ------------- References -------------- @@ -202,10 +193,6 @@ \newcommand{\normrd}[1]{\norms{#1}{\real^d}} \newcommand{\normrdn}[1]{\normns{#1}{\real^d}} -% for variation -\newcommand{\var}{\text{var}} - - % Category Shenanigans \newcommand{\mf}[1]{\mathfrak{#1}} \newcommand{\catc}{\mathfrak{C}} @@ -217,7 +204,6 @@ \newcommand{\ppi}{\frac{\partial}{\partial x^i}} \newcommand{\ppj}{\frac{\partial}{\partial y^j}} \newcommand{\ppip}{\ppi\bigg\vert_p} -\renewcommand{\part}[2]{\frac{\partial{#1}}{\partial{#2}}} \newcommand{\vf}{\mathfrak{X}} \newcommand{\grad}[1]{\text{grad}\paren{#1}} \newcommand{\Grad}[1]{\text{Grad}\paren{#1}} diff --git a/refs.bib b/refs.bib index 60718f4..f7eebb3 100644 --- a/refs.bib +++ b/refs.bib @@ -94,3 +94,12 @@ isbn={978-3-319-57117-1}, doi={10.1007/978-3-319-57117-1} } +@book{Cartan, + title={Differential Calculus}, + author={Cartan, H.}, + isbn={9780901665140}, + series={International studies in mathematics}, + url={https://books.google.ca/books?id=ZFgZAQAAIAAJ}, + year={1983}, + publisher={Hermann} +} diff --git a/spec.toml b/spec.toml new file mode 100644 index 0000000..81aa901 --- /dev/null +++ b/spec.toml @@ -0,0 +1,14 @@ +database = "spec.db" +document = "document.tex" +siteTitle = "Garden" + +[compiler] +compileAll = false +redoTags = false +indirectReferences = true + +[website] +font = "roboto" +primaryColour = "blue" +neutralColour = "grey" +searchLimit = 16 diff --git a/src/dg/derivative/sets.tex b/src/dg/derivative/sets.tex index fd2da4f..4972e89 100644 --- a/src/dg/derivative/sets.tex +++ b/src/dg/derivative/sets.tex @@ -82,7 +82,6 @@ \label{proposition:chain-rule-sets-conditions} Let $E$, $F$, $G$, be TVSs over $K \in \RC$ with $F, G$ being separated. If $\sigma \subset B(E)$ and $\tau \subset B(F)$ correspond to the following families of sets on $E$ and $F$: \begin{enumerate} - \item Finite sets. \item Compact sets. \item Bounded sets. \end{enumerate} @@ -96,9 +95,7 @@ \begin{proof} (1): Let $A \in \sigma$ and $U \in \cn_G(0)$. Since $T$ is continuous, there exists $V \in \cn_F(0)$ such that $T(V) \subset U$. Since $r \in \mathcal{R}_\sigma(E; G)$, there exists $t > 0$ such that $r(sA)/s \in V$ for all $s \in (0, t)$. In which case, $T \circ r(sA)/s \in U$ for all $s \in (0, t)$. - (2): Firstly, for any $A \subset E$ finite, $(T + r)(A)$ is also finite, so the claim holds directly in the case of finite sets. - - To show that $s \circ (T + r) \in \mathcal{R}_\sigma(E; G)$, it is sufficient to show that for every $A \in \sigma$, $\seq{t_n} \subset \real_{> 0}$ with $t_n \downto 0$ as $n \to \infty$, and $\seq{a_n} \subset A$, + (2): To show that $s \circ (T + r) \in \mathcal{R}_\sigma(E; G)$, it is sufficient to show that for every $A \in \sigma$, $\seq{t_n} \subset \real_{> 0}$ with $t_n \downto 0$ as $n \to \infty$, and $\seq{a_n} \subset A$, \[ \limv{n} \frac{1}{t_n}s \circ (T + r)(t_na_n) = \limv{n}\frac{1}{t_n}s\braks{t_n\paren{Ta_n + \frac{r(t_na_n)}{t_n}}} = 0 \] @@ -112,13 +109,14 @@ \[ \limv{n}\frac{1}{t_n}s\braks{t_n\paren{Ta_n + \frac{r(t_na_n)}{t_n}}} = 0 \] - \end{proof} \begin{remark} +\label{remark:differentiation-bornology} + In \autoref{definition:differentiation-small}, the system $\sigma$ can be chosen based on the bornology of $E$, and the definition of small-ness depends exclusively on $\sigma$. As such, there is an apparent disconnect between differentiation and the topology of the domain. Consider for example a Hilbert space equipped with its norm and weak topology. The norm itself is differentiable with respect to both topologies, because the bounded sets coincide. Moreover, the data for differentiability needs to only come from a neighbourhood of $0$ in the norm topology. As such, a function may be differentiable even if its domain is too small to have an interior. diff --git a/src/fa/tvs/completion.tex b/src/fa/tvs/completion.tex index db8c58d..2b184a4 100644 --- a/src/fa/tvs/completion.tex +++ b/src/fa/tvs/completion.tex @@ -31,5 +31,7 @@ \end{proof} \begin{remark}[{{\cite[Section I.1]{SchaeferWolff}}}] +\label{remark:hausdorff-completion-field} + The Hausdorff completion works in general with arbitrary valuated fields. Though the completion yields a TVS over the completion of the field, the field need not to be complete. \end{remark} diff --git a/src/measure/measure/index.tex b/src/measure/measure/index.tex index 1c5c587..834b740 100644 --- a/src/measure/measure/index.tex +++ b/src/measure/measure/index.tex @@ -9,6 +9,5 @@ \input{./radon.tex} \input{./outer.tex} \input{./lebesgue-stieltjes.tex} -\input{./radon.tex} \input{./riesz.tex} \input{./kolmogorov.tex} diff --git a/src/topology/uniform/metric.tex b/src/topology/uniform/metric.tex index ce65a6e..aae8b57 100644 --- a/src/topology/uniform/metric.tex +++ b/src/topology/uniform/metric.tex @@ -155,6 +155,7 @@ The axioms of uniform spaces strongly resembles working in a metric space. In fa \end{proof} \begin{remark} +\label{remark:pseudometric-proof} Unfortunately, it was hard to access sources for this proof online, so I could not provide a specific citation. As such, I followed an online PDF given by the link: https://krex.k-state.edu/server/api/core/bitstreams/1bdf2b14-3b5a-4962-a589-93ee1998950c/content \end{remark}