From ced231c4808134d144962edf8df1dd2ab66948cd Mon Sep 17 00:00:00 2001 From: Evan Chen Date: Tue, 14 Nov 2023 15:57:30 -0500 Subject: [PATCH] style: clear trailing whitespace --- flake.nix | 4 ++-- patch-asy.sty | 8 ++++---- references.bib | 1 - tex/alg-NT/frobenius.tex | 2 +- tex/frontmatter/title-embellishments.tex | 2 +- tex/measure/martingale.tex | 3 +-- tex/rep-theory/characters.tex | 2 +- 7 files changed, 10 insertions(+), 12 deletions(-) diff --git a/flake.nix b/flake.nix index cb1afb16..581dbe66 100644 --- a/flake.nix +++ b/flake.nix @@ -14,8 +14,8 @@ inherit (pkgs.texlive) scheme-medium asymptote wrapfig amsmath ulem hyperref capt-of - latexmk biber xpatch - tkz-graph tikz-cd xcolor todonotes + latexmk biber xpatch + tkz-graph tikz-cd xcolor todonotes mdframed mathtools braket multirow prerex cleveref wasysym stmaryrd diff --git a/patch-asy.sty b/patch-asy.sty index 853d8b1a..2727b71b 100644 --- a/patch-asy.sty +++ b/patch-asy.sty @@ -7,15 +7,15 @@ %% asy-latex.dtx (with options: `pkg') %% ____________________________ %% The ASYMPTOTE package -%% +%% %% (C) 2003 Tom Prince %% (C) 2003-2010 John Bowman %% (C) 2010 Will Robertson -%% +%% %% Adapted from comment.sty -%% +%% %% Licence: GPL2+ -%% +%% \ProvidesPackage{asymptote} [2012/08/25 v1.27 Asymptote style file for LaTeX] \def\Asymptote{{\tt Asymptote}} diff --git a/references.bib b/references.bib index 8697fef1..b6461a0a 100644 --- a/references.bib +++ b/references.bib @@ -148,7 +148,6 @@ @unpublished{ref:gorin year={2018}, url={https://www.mit.edu/~txz/links.html}, } - // Notes used in passing @unpublished{ref:covering_all_we_know, diff --git a/tex/alg-NT/frobenius.tex b/tex/alg-NT/frobenius.tex index 9daa7026..24018846 100644 --- a/tex/alg-NT/frobenius.tex +++ b/tex/alg-NT/frobenius.tex @@ -600,7 +600,7 @@ \section{Frobenius elements control factorization} To do this, suppose $S = \{ \alpha_1,\alpha_2,\dots, \alpha_n\}$ are the roots of $f$ (distinct roots since $f$ is irreducible over $\QQ$). We let $\Frob_{\kP}$ act on $S$. This splits $S$ into orbits $S_1$, $S_2$, $\dots$, $S_k$. Construct polynomials $f_i$ with coefficients in $E$ having roots exactly the elements of $S_i$. This forms a factorization of $f$ over $E$, say \[ f = f_1f_2 \dots f_k. \] -We claim that this in fact induces a factorization of $f \pmod p$. +We claim that this in fact induces a factorization of $f \pmod p$. To see this, consider the images of these polynomials $f_i$ under the quotient $\OO_K \to \OO_K/\kP$, denote them by $\overline{f_i}$. Then since $p$ is unramified, we know that the decomposition group $D(\kP|p)$ is isomorphic to the Galois group $\mathcal{G} = \Gal((\OO_E/\kP) / (\ZZ/p\ZZ))$. Thus $\Frob_{\kP}$ corresponds to the generator $\sigma$ of $\mathcal{G}$. It is not hard to believe that the action of $\Frob_{\kP}$ on the roots of $f$ is the same as that of $\sigma$ on the roots of $ \overline{f}$. Since the roots of $f_i$ form an orbit under the action of $\Frob_{\kP}$, we see that the roots of $\overline{f_i}$ form an orbit under the action of $\sigma$ and hence under the action of $\mathcal{G}$. It is now a standard fact of Galois theory that $\overline{f_i}$ is an irreducible polynomial over $\FF_p$ (since it is fixed by $\mathcal{G}$), thus the claim is proved. Now we just need to observe that the roots of $f$ correspond to the cosets of $H$, this will be established later. diff --git a/tex/frontmatter/title-embellishments.tex b/tex/frontmatter/title-embellishments.tex index eca5c233..6fa1f9c9 100644 --- a/tex/frontmatter/title-embellishments.tex +++ b/tex/frontmatter/title-embellishments.tex @@ -19,7 +19,7 @@ \url{https://ko-fi.com/evanchen/} \end{minipage} \end{center} - + \vfill { \small diff --git a/tex/measure/martingale.tex b/tex/measure/martingale.tex index 88c9526e..b7b6ed7f 100644 --- a/tex/measure/martingale.tex +++ b/tex/measure/martingale.tex @@ -580,7 +580,7 @@ \section{\problemhead} Finally let $\tau \colon \Omega \to \{1, 2, \dots\}$ be a stopping time such that $\mathbb E[\tau] < \infty$, such that the event $\tau = n$ depends only on $X_1$, \dots, $X_n$. - + Prove that \[ \EE[X_1 + X_2 + \dots + X_\tau] = \mu \EE[\tau]. \] \end{problem} @@ -621,4 +621,3 @@ \section{\problemhead} (Incidentally, using the law of large numbers could work too.) \end{hint} \end{problem} - diff --git a/tex/rep-theory/characters.tex b/tex/rep-theory/characters.tex index d7262622..7ae85767 100644 --- a/tex/rep-theory/characters.tex +++ b/tex/rep-theory/characters.tex @@ -494,7 +494,7 @@ \section{Examples of character tables} Here, $C_G(g) = \left\{ x \in G : xg = gx \right\}$ is the centralizer of $g$. \begin{hint} - Construct two square $r \times r$ matrices $A$ and $B$ + Construct two square $r \times r$ matrices $A$ and $B$ such that $AB$ is the identity by the first orthogonality. Then use $BA$ to prove the second orthogonaliy relation. \end{hint}