From 2df49f63b4e5095ff3fbe31d61a78302981cbe95 Mon Sep 17 00:00:00 2001
From: Chris Wells
Date: Sun, 8 Feb 2026 20:34:13 -0600
Subject: Making sure things compile properly
---
app/Main.hs | 31 ++++--
www/teaching/discrete20/index.html | 26 ++---
www/teaching/graphs2022/index.html | 189 ++++++++++++++++++++++++++++++++++-
www/teaching/index.md | 4 +-
www/teaching/matrices2019/index.html | 6 +-
5 files changed, 224 insertions(+), 32 deletions(-)
diff --git a/app/Main.hs b/app/Main.hs
index bce890b..ec508d4 100644
--- a/app/Main.hs
+++ b/app/Main.hs
@@ -36,6 +36,9 @@ main = do
match "data/licenses.yaml" $ compile $ yamlCompiler @(AliasMap StrictText License_)
match "data/journals.yaml" $ compile $ yamlCompiler @(AliasMap StrictText Journal_)
+ -- load any snippets
+ match "**/_*.html" $ compile getResourceBody
+
match "assets/**" $ do
route $ dropParentRoute 1
compile copyFileCompiler
@@ -83,6 +86,7 @@ main = do
] -- Remove me from the author list
}
getResourceBody >>= applyAsTemplate (mkPaperContext (personCtxt "data/people.yaml" <> journalCtxt "data/journals.yaml") fixedPapers) >>= applyTemplate defaultTemplate baseCtxt
+
match "www/teaching/index.md" $ do
route $ dropParentRoute 1 `composeRoutes` setExtension ".html"
-- compile $ myPandocCompiler >>= applyAsTemplate baseCtxt >>= applyTemplate defaultTemplate baseCtxt
@@ -92,18 +96,27 @@ main = do
route $ dropParentRoute 1 `composeRoutes` setExtension ".html"
compile $ getResourceBody >>= applyAsTemplate baseCtxt >>= myRenderPandoc >>= applyTemplate defaultTemplate baseCtxt
- match "www/contact/*.html" $ do
- route $ dropParentRoute 1
- compile $ getResourceBody >>= applyAsTemplate baseCtxt >>= applyTemplate defaultTemplate baseCtxt
+ -- match "www/contact/*.html" $ do
+ -- route $ dropParentRoute 1
+ -- compile $ getResourceBody >>= applyAsTemplate baseCtxt >>= applyTemplate defaultTemplate baseCtxt
+ --
+ -- match "www/code/*.html" $ do
+ -- route $ dropParentRoute 1
+ -- compile $ getResourceBody >>= applyAsTemplate baseCtxt >>= applyTemplate defaultTemplate baseCtxt
+ --
+ -- match "www/credits/*.html" $ do
+ -- route $ dropParentRoute 1
+ -- compile $ myPandocCompiler >>= applyTemplate defaultTemplate baseCtxt
- match "www/code/*.html" $ do
+ -- compile any other html
+ match "www/**/*.html" $ do
route $ dropParentRoute 1
- compile $ getResourceBody >>= applyAsTemplate baseCtxt >>= applyTemplate defaultTemplate baseCtxt
+ compile $ getResourceBody >>= applyTemplate defaultTemplate baseCtxt
- match "www/credits/*.html" $ do
- route $ dropParentRoute 1
- compile $ myPandocCompiler >>= applyTemplate defaultTemplate baseCtxt
- pure ()
+ -- just copy anything else hanging around
+ match "www/**" $ do
+ route idRoute
+ compile copyFileCompiler
myPandocCompiler :: Compiler (Item String)
diff --git a/www/teaching/discrete20/index.html b/www/teaching/discrete20/index.html
index 127993c..d8bda32 100644
--- a/www/teaching/discrete20/index.html
+++ b/www/teaching/discrete20/index.html
@@ -72,19 +72,19 @@ I type these up quickly, so please let me know if you find any mistakes!
Recitation 2 — 01/21/20
Recitation 3 — 01/21/20: I didn't have time to type detailed notes this week, but here's a list of what we covered:
- - If $$H_n:=\sum_{k=1}^n{1\over k}$$, then $$H_n\approx\int_1^n{dx\over x}=\log n$$. ($$H_n$$ is known as the $$n$$th harmonic number)
+
- If $H_n:=\sum_{k=1}^n{1\over k}$, then $H_n\approx\int_1^n{dx\over x}=\log n$. ($H_n$ is known as the $n$th harmonic number)
- - More precisely, we can bound $$\log(n+1)\leq H_n\leq\log n + 1$$.
- - Since $${1\over x}$$ is convex, the lower bound can be improved to $$H_n\geq\log n + {1\over 2}+{1\over 2n}$$ by using trapezoids instead of rectangles.
+ - More precisely, we can bound $\log(n+1)\leq H_n\leq\log n + 1$.
+ - Since ${1\over x}$ is convex, the lower bound can be improved to $H_n\geq\log n + {1\over 2}+{1\over 2n}$ by using trapezoids instead of rectangles.
- - If $$d(n)$$ denotes the number of divisors of $$n$$, then $$\text{ave}_{k\in[n]}d(k)\approx\log n$$ (see Recitation 2).
- - Using the same ideas, for any $$d>-1$$, we have $$\sum_{k=1}^n k^d\approx {n^{d+1}\over d+1}$$.
+ - If $d(n)$ denotes the number of divisors of $n$, then $\text{ave}_{k\in[n]}d(k)\approx\log n$ (see Recitation 2).
+ - Using the same ideas, for any $d>-1$, we have $\sum_{k=1}^n k^d\approx {n^{d+1}\over d+1}$.
Recitation 4 — 02/04/20
- - We did not cover this in class, but these notes include proof that the number of derangements of $$[n]$$ is precisely the closest integer to $${n!\over e}$$, a fact that was mentioned in Recitation 2.
+ - We did not cover this in class, but these notes include proof that the number of derangements of $[n]$ is precisely the closest integer to ${n!\over e}$, a fact that was mentioned in Recitation 2.
Recitation 5 — 02/11/20 (Review day)
@@ -105,16 +105,16 @@ I type these up quickly, so please let me know if you find any mistakes!
- I made a remark in recitation about how Claim 1 in the notes changes if we require only pairwise independence instead of mutual independence.
There was enough curiosity about this that I wrote up a proof.
- This proof can be modified to show that if $$A_1,\dots,A_n\subseteq\Omega$$ are nontrivial and have the property that any set of at most $$k$$ of them is independent, then $$|\Omega|\geq\sum_{j=0}^{\lfloor k/2\rfloor}{n\choose j}$$.
+ This proof can be modified to show that if $A_1,\dots,A_n\subseteq\Omega$ are nontrivial and have the property that any set of at most $k$ of them is independent, then $|\Omega|\geq\sum_{j=0}^{\lfloor k/2\rfloor}{n\choose j}$.
- The proof here probably appears to be very ad hoc; this is not actually the case.
- Once we learn about random variables (which are simply the elements of $$\mathbb{R}^{\Omega}$$) and expectations, it will be clear that $$\langle X,Y\rangle=\operatorname{\mathbb{E}}XY$$ is a natural inner product on $$\mathbb{R}^{\Omega}$$.
- This is all that's going on in the proof, using the random variables $$X_i=\mathbf{1}_{A_i}-\operatorname{\mathbb{E}}\mathbf{1}_{A_i}$$ and $$X_0=1$$.
+ Once we learn about random variables (which are simply the elements of $\mathbb{R}^{\Omega}$) and expectations, it will be clear that $\langle X,Y\rangle=\operatorname{\mathbb{E}}XY$ is a natural inner product on $\mathbb{R}^{\Omega}$.
+ This is all that's going on in the proof, using the random variables $X_i=\mathbf{1}_{A_i}-\operatorname{\mathbb{E}}\mathbf{1}_{A_i}$ and $X_0=1$.
In fact, the proof is much cleaner when phrased this way.
- - With this terminology, in order to prove the fact about $$k$$-wise independent events mentioned above, first define $$X_i=\mathbf{1}_{A_i}-\operatorname{\mathbb{E}}\mathbf{1}_{A_i}$$ for $$i\in[n]$$.
- Then, for a set $$I\subseteq[n]$$ with $$|I|\leq k/2$$, define $$Y_I=\prod_{i\in I}X_i$$.
- With just a bit of work, it can be shown that the $$Y_I$$'s are non-zero and pairwise orthogonal.
- If $$k$$ is odd, then the bound can be improved slightly by considering also $$Z_I=X_n\prod_{i\in I}X_i$$ for $$I\in{[n-1]\choose\lfloor k/2\rfloor}$$.
+ - With this terminology, in order to prove the fact about $k$-wise independent events mentioned above, first define $X_i=\mathbf{1}_{A_i}-\operatorname{\mathbb{E}}\mathbf{1}_{A_i}$ for $i\in[n]$.
+ Then, for a set $I\subseteq[n]$ with $|I|\leq k/2$, define $Y_I=\prod_{i\in I}X_i$.
+ With just a bit of work, it can be shown that the $Y_I$'s are non-zero and pairwise orthogonal.
+ If $k$ is odd, then the bound can be improved slightly by considering also $Z_I=X_n\prod_{i\in I}X_i$ for $I\in{[n-1]\choose\lfloor k/2\rfloor}$.
diff --git a/www/teaching/graphs2022/index.html b/www/teaching/graphs2022/index.html
index e72f84c..78be034 100644
--- a/www/teaching/graphs2022/index.html
+++ b/www/teaching/graphs2022/index.html
@@ -2,7 +2,6 @@
title: MATH 314 (Spring 2022)
title_short: Graphs
math: true
-
---
MATH 314: Graph Theory
@@ -35,7 +34,7 @@ This will help me ensure I don't miss any submissions.
-If you wish to write your assignments in $$\LaTeX$$, here's a basic homework template you can use.
+If you wish to write your assignments in $\LaTeX$, here's a basic homework template you can use.
A note on pictures and proofs
@@ -162,7 +161,7 @@ While sketches can be very helpful in understanding a problem and reasoning thro
| Mar 29 |
- Matchings, vertex- and edge-covers, $$\alpha,\alpha',\beta,\beta'$$, Kőnig's Theorem, extra notes |
+ Matchings, vertex- and edge-covers, $\alpha,\alpha',\beta,\beta'$, Kőnig's Theorem, extra notes |
HW8 (.tex) |
@@ -180,7 +179,7 @@ While sketches can be very helpful in understanding a problem and reasoning thro
| Apr 7 |
- $$\chi(G)$$ vs $$\chi(\overline{G})$$, Discussion session #4, extra notes |
+ $\chi(G)$ vs $\chi(\overline{G})$, Discussion session #4, extra notes |
DS4 |
@@ -248,4 +247,184 @@ While sketches can be very helpful in understanding a problem and reasoning thro
-$snippet(replaceFileName(path,"_notation.html"))$
+Quick notation
+
+This list will grow throughout the semester.
+
+
+
+ | Notation |
+ Meaning |
+
+
+ | $[n]$ |
+ $\{1,\dots,n\}$ (Note: $[0]=\varnothing$) |
+
+
+ | $2^X$ where $X$ is a set |
+ power-set of $X$ (nb: if $X$ is finite, then $|2^X|=2^{|X|}$) |
+
+
+ | ${X\choose k}$ where $X$ is a set |
+ $\{S\in 2^X:|S|=k\}$ (nb: if $X$ is finite, then $|{X\choose k}|={|X|\choose k}$) |
+
+
+ | $A\sqcup B$ |
+ disjoint union of sets $A$ and $B$ (Note: this is identical to the standard union of sets, but we use it to enforce the idea that $A$ and $B$ are disjoint for the sake of brevity. That is to say, $A\sqcup B$ makes sense iff $A$ and $B$ are disjoint. If $A$ and $B$ aren't disjoint, but one still wants to treat them as such when taking their "union", there's a notion known as the co-product of sets, denoted by $A\amalg B$, but we won't use that in this class.) |
+
+
+ | $V(G)$ |
+ vertex set of $G$ |
+
+
+ | $E(G)$ |
+ edge set of $G$ |
+
+
+ | $K_n$ |
+ clique on $n$ vertices |
+
+
+ | $C_n$ |
+ cycle on $n$ vertices |
+
+
+ | $P_n$ |
+ path on $n$ vertices |
+
+
+ | $K_{m,n}$ |
+ complete bipartite graph with parts of sizes $m$ and $n$ |
+
+
+ | $K_{n_1,n_2,\dots,n_t}$ |
+ complete $t$-partite graph with parts of sizes $n_1,n_2,\dots,n_t$ |
+
+
+ | $\overline{G}$ |
+ complement of $G$ |
+
+
+ | $G-v$ where $v$ is a vertex of $G$ |
+ remove the vertex $v$ along with any incident edges from $G$ |
+
+
+ | $G-e$ where $e$ is an edge of $G$ |
+ remove the edge $e$ from $G$, but keep all vertices |
+
+
+ | $G+e$ where $e$ is an edge not in $G$ |
+ add the edge $e$ to $G$ |
+
+
+ | $G-U$ where $U$ is either a set of vertices or a set of edges of $G$ |
+ If $U$ is a set of vertices, delete each of them from $G$ along with any incident edges.
+ If $U$ is a set of edges, delete each of them from $G$ but keep all vertices.
+ This notation is somewhat ambiguous since an edge $e$ is a set of vertices, so it conflicts with the earlier notation of $G-e$...
+ In other words, context is necessary when using this notation.
+ |
+
+ | $G[X]$ where $X\subseteq V(G)$ |
+ subgraph of $G$ induced by $X$ (Note: $G[X]=G-(V(G)\setminus X)$) |
+
+
+ | $L(G)$ |
+ the line graph of $G$ |
+
+
+ | $G\cup H$ or $G\sqcup H$ |
+ disjoint union of $G$ and $H$ (Note: I will always use $\sqcup$ since it reinforces the disjointness) |
+
+
+ | $G+H$ or $G\vee H$ |
+ join of $G$ and $H$ |
+
+
+ | $G\mathbin\square H$ or $G\times H$ |
+ Cartesian product of $G$ and $H$ (Note: Usually $\times$ denotes a different graph product known as the categorical product, so be careful when reading other texts. I will always use $\square$.) |
+
+
+ | $d(u,v)$ |
+ distance between vertices $u$ and $v$ |
+
+
+ | $N(v)$ |
+ neighborhood of $v$ |
+
+
+ | $\deg v$ |
+ degree of the vertex $v$ |
+
+
+ | $\Delta(G)$ |
+ maximum degree of $G$ |
+
+
+ | $\delta(G)$ |
+ minimum degree of $G$ |
+
+
+ | $\deg^+ v$ or $\operatorname{od}v$ |
+ out-degree of the vertex $v$ in a digraph (Note: I will never use $\operatorname{od}$ since that's just dumb) |
+
+
+ | $\deg^- v$ or $\operatorname{id}v$ |
+ in-degree of the vertex $v$ in a digraph (Note: I will never use $\operatorname{id}$ since that's just dumb) |
+
+
+ | $G\cong H$ |
+ $G$ and $H$ are isomorphic graphs |
+
+
+ | $\operatorname{Aut}(G)$ |
+ automorphism group of $G$ |
+
+
+ | $\kappa(G)$ |
+ vertex-connectivity of $G$ |
+
+
+ | $\lambda(G)$ |
+ edge-connectivity of $G$ |
+
+
+ | $\alpha(G)$ |
+ independence number of $G$ |
+
+
+ | $\omega(G)$ |
+ clique number of $G$ |
+
+
+ | $\alpha'(G)$ |
+ matching number/edge-independence number of $G$, i.e. the largest matching in $G$ |
+
+
+ | $\beta(G)$ |
+ vertex-cover number of $G$, i.e. min number of vertices that touch all edges |
+
+
+ | $\beta'(G)$ |
+ edge-cover number of $G$, i.e. min number of edges that touch all vertices |
+
+
+ | $\chi(G)$ |
+ chromatic number of $G$ |
+
+
+ | $\chi'(G)$ |
+ edge-chromatic number/chromatic index of $G$, i.e. chromatic number of $L(G)$ |
+
+
+ | $R(m,n)$ |
+ The Ramsey number of $K_m$ vs $K_n$, i.e. the smallest integer $N$ such that any red,blue-coloring of $E(K_N)$ contains either a red copy of $K_m$ or a blue copy of $K_n$ |
+
+
+
+
+This is a general notational trend in graph theory.
+If, say, $\zeta(G)$ is some parameter of $G$ which is defined based mainly on the vertices of $G$ (what exactly this means varies from case-to-case), then generally $\zeta'(G)$ is the analogous parameter of $G$ defined based mainly on the edges of $G$ (again, what exactly this means varies from case-to-case).
+Sometimes $\zeta'(G)=\zeta(L(G))$, e.g. $\alpha$ vs $\alpha'$, but not always, e.g. $\beta$ vs $\beta'$.
+Also, this trend is broken often enough, e.g. $\kappa$ vs $\lambda$, but it is common enough to point out.
+
+
diff --git a/www/teaching/index.md b/www/teaching/index.md
index 029d503..71a7c69 100644
--- a/www/teaching/index.md
+++ b/www/teaching/index.md
@@ -59,7 +59,7 @@ Fall 2020
Spring 2020
-- **[21-228: Discrete Mathematics](/teaching/discrete20)** (TA for Tom Bohman)
+- **[21-228: Discrete Mathematics](/teaching/discrete20)** (TA for $personA("Tom Bohman")$)
Summer 2019
@@ -87,7 +87,7 @@ Summer 2016
Spring 2016
-- **[21-228: Discrete Mathematics](/teaching/discrete2016**) (TA for $personA("Tom Bohman")$)
+- **[21-228: Discrete Mathematics](/teaching/discrete2016)** (TA for $personA("Tom Bohman")$)
Fall 2015
diff --git a/www/teaching/matrices2019/index.html b/www/teaching/matrices2019/index.html
index 3541d94..52f94c0 100644
--- a/www/teaching/matrices2019/index.html
+++ b/www/teaching/matrices2019/index.html
@@ -15,8 +15,8 @@ math: true
6201 Wean Hall
MWF 12:00–2:00 p.m.
Textbooks
- Linear Algebra by Jim Hefferon (freely available under the terms of the $license("GFDL")$ or the $license("CC BY-SA 2.5")$)
- A First Course in Linear Algebra by Robert A. Beezer (freely available under the terms of the $license("GFDL")$)
+ Linear Algebra by Jim Hefferon (freely available under the terms of the GFDL or the CC BY-SA 2.5)
+ A First Course in Linear Algebra by Robert A. Beezer (freely available under the terms of the GFDL)
@@ -126,7 +126,7 @@ Solutions will be posted after the lecture that day.
| July 23 |
- Orthogonal matrices, isometries of $$\mathbb{R}^n$$ |
+ Orthogonal matrices, isometries of $\mathbb{R}^n$ |
HW7 |
--
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