diff --git a/tex/H113/quotient.tex b/tex/H113/quotient.tex index 6cdf07e1..2603b76d 100644 --- a/tex/H113/quotient.tex +++ b/tex/H113/quotient.tex @@ -572,10 +572,9 @@ \section{(Digression) The first isomorphism theorem} and no one has any clue what's going on, because no one has any clue what a normal subgroup actually should look like. -Other sources like to also write the so-called first isomorphism theorem.\footnote{ - There is a second and third isomorphism theorem. - But four years after learning about them, - I \emph{still} don't know what they are. +Other sources like to also write the so-called first isomorphism theorem.\footnote{There + is a second and third isomorphism theorem. + But four years after learning about them, I \emph{still} don't remember what they are. So I'm guessing they weren't very important.} It goes like this. \begin{theorem} diff --git a/tex/alg-NT/dedekind.tex b/tex/alg-NT/dedekind.tex index 6513fc5a..ac09c711 100644 --- a/tex/alg-NT/dedekind.tex +++ b/tex/alg-NT/dedekind.tex @@ -306,8 +306,8 @@ \section{Unique factorization works} Section 3 of \cite{ref:ullery} does a nice job of explaining it. When we proved the fundamental theorem of arithmetic, the basic plot was: \begin{enumerate}[(1)] - \ii Show that if $p$ is a rational prime\footnote{ - Note that the kindergarten definition of a prime is + \ii Show that if $p$ is a rational prime\footnote{Note + that the kindergarten definition of a prime is that ``$p$ isn't the product of two smaller integers''. This isn't the correct definition of a prime: the definition of a prime is that $p \mid bc$ diff --git a/tex/cats/functors.tex b/tex/cats/functors.tex index 18826062..241aa333 100644 --- a/tex/cats/functors.tex +++ b/tex/cats/functors.tex @@ -656,9 +656,8 @@ \section{(Optional) The Yoneda lemma} Let $\AA$ be the category of finite sets whose arrows are bijections between sets. For $A \in \AA$, let $F(A)$ be the set of \emph{permutations} of $A$ and - let $G(A)$ be the set of \emph{orderings} on $A$.\footnote{ - A permutation is a bijection $A \to A$, - and an ordering is a bijection $\{1, \dots, n\} \to A$, + let $G(A)$ be the set of \emph{orderings} on $A$.\footnote{A permutation + is a bijection $A \to A$, and an ordering is a bijection $\{1, \dots, n\} \to A$, where $n$ is the size of $A$.} \begin{enumerate}[(a)] \ii Extend $F$ and $G$ to functors $\AA \to \catname{Set}$. diff --git a/tex/complex-ana/holomorphic.tex b/tex/complex-ana/holomorphic.tex index 3e94fcbd..1c8407f3 100644 --- a/tex/complex-ana/holomorphic.tex +++ b/tex/complex-ana/holomorphic.tex @@ -652,8 +652,8 @@ \section{Holomorphic functions are analytic} \begin{dproblem}[Maximums Occur On Boundaries] Let $f \colon U \to \CC$ be holomorphic, let $Y \subseteq U$ be compact, - and let $\partial Y$ be boundary\footnote{ - The boundary $\partial Y$ is the set of points $p$ + and let $\partial Y$ be boundary\footnote{The boundary $\partial Y$ + is the set of points $p$ such that no open neighborhood of $p$ is contained in $Y$. It is also a compact set if $Y$ is compact. } of $Y$. diff --git a/tex/complex-ana/meromorphic.tex b/tex/complex-ana/meromorphic.tex index 0c18b154..47c38441 100644 --- a/tex/complex-ana/meromorphic.tex +++ b/tex/complex-ana/meromorphic.tex @@ -296,8 +296,9 @@ \section{Winding numbers and the residue theorem} The proof from here is not really too impressive -- the ``work'' was already done in our statements about the winding number. \begin{proof} - Let the poles with nonzero winding number be $p_1, \dots, p_k$ (the others do not affect the sum).\footnote{ - To show that there must be finitely many such poles: recall that all our contours $\gamma \colon [a,b] \to \CC$ + Let the poles with nonzero winding number be $p_1, \dots, p_k$ + (the others do not affect the sum).\footnote{To show + that there must be finitely many such poles: recall that all our contours $\gamma \colon [a,b] \to \CC$ are in fact bounded, so there is some big closed disk $D$ which contains all of $\gamma$. The poles outside $D$ thus have winding number zero. Now we cannot have infinitely many poles inside the disk $D$, for $D$ is compact and the diff --git a/tex/frontmatter/advice.tex b/tex/frontmatter/advice.tex index 24f7e3a9..e4d42b11 100644 --- a/tex/frontmatter/advice.tex +++ b/tex/frontmatter/advice.tex @@ -91,9 +91,8 @@ \section{Paper} \\ \scriptsize Image from \cite{img:read_with_pencil} \end{center} You are not God. -You cannot keep everything in your head.\footnote{ - See also \url{https://blog.evanchen.cc/2015/03/14/writing/} - and the source above.} +You cannot keep everything in your head.\footnote{See also + \url{https://blog.evanchen.cc/2015/03/14/writing/} and the source above.} If you've printed out a hard copy, then write in the margins. If you're trying to save paper, grab a notebook or something along with the ride.