diff --git a/blueprint/src/Toric.bib b/blueprint/src/Toric.bib index d2e29f5..e69de29 100644 --- a/blueprint/src/Toric.bib +++ b/blueprint/src/Toric.bib @@ -1,30 +0,0 @@ -@book{Milne_2017, - place={Cambridge}, - series={Cambridge Studies in Advanced Mathematics}, - title={Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field}, - publisher={Cambridge University Press}, - author={Milne, J. S.}, - year={2017}, - collection={Cambridge Studies in Advanced Mathematics} -} - -@book{Cox_2011, - place={Providence}, - series={Graduate Studies in Mathematics}, - title={Toric Varieties}, - publisher={American Mathematical Society}, - author={Cox, David A. and Little, John B. and Schenck, H. K.}, - year={2011}, - collection={Graduate Studies in Mathematics} -} - -@book{Oda_1988, - author = {Oda, Tadao}, - keywords = {torus embeddings; convex figures in real affine spaces; complex analytic spaces; holomorphic maps; birational geometry; subdivisions of fans; Integral convex polytopes; toric projective varieties; holomorphic differential forms}, - language = {eng}, - location = {Berlin [u.a.]}, - publisher = {Springer}, - title = {Convex bodies and algebraic geometry}, - url = {http://eudml.org/doc/203658}, - year = {1988}, -} diff --git a/blueprint/src/chapters/1-1-over-cat.tex b/blueprint/src/chapters/1-1-over-cat.tex deleted file mode 100644 index 8e3c668..0000000 --- a/blueprint/src/chapters/1-1-over-cat.tex +++ /dev/null @@ -1,51 +0,0 @@ -\section{Over category} - - -\begin{proposition}[Sliced adjoint functors] - \label{0-slice-adj} - \uses{} -% \lean{CategoryTheory.Over.postAdjunctionRight} - \leanok - \mathlibok - - If $a : F \vdash G$ is an adjunction between $F : C \to D$ and $G : D \to C$ and $X : C$, then there is an adjunction between $F / X : C / X \to D / F(X)$ and $G / X : D / F(X) \to C / X$. -\end{proposition} -\begin{proof} - \uses{} - \leanok - - See https://ncatlab.org/nlab/show/sliced+adjoint+functors+--+section. -\end{proof} - - -\begin{proposition}[Limit-preserving functors lift to over categories] - \label{0-over-lim} - \uses{} -% \lean{CategoryTheory.Limits.PreservesLimitsOfShape.overPost, CategoryTheory.Limits.PreservesLimitsOfSize.overPost} - \leanok - - Let $J$ be a shape (i.e. a category). Let $\widetilde J$ denote the category which is the same as $J$, but has an extra object $*$ which is terminal. - If $F : C \to D$ is a functor preserving limits of shape $\widetilde J$, then the obvious functor $C / X \to D / F(X)$ preserves limits of shape $J$. -\end{proposition} -\begin{proof} - \uses{} - \leanok - - Extend a functor $K\colon J \to C / X$ to a functor $\widetilde K\colon \widetilde J \to C$, by letting $\widetilde K (*) = X$. -\end{proof} - - -\begin{proposition}[Essential image of a sliced functor] - \label{0-ess-image-over} - \uses{} -% \lean{CategoryTheory.Functor.essImage_overPost} - \leanok - - If $F : C \to D$ is a full functor between cartesian-monoidal categories, then $F / X : C / X \hom D / F(X)$ has the same essential image as $F$. -\end{proposition} -\begin{proof} - \uses{} - \leanok - - Transfer all diagrams. -\end{proof} diff --git a/blueprint/src/chapters/2-1-tensor-product.tex b/blueprint/src/chapters/2-1-tensor-product.tex deleted file mode 100644 index c9f9ce0..0000000 --- a/blueprint/src/chapters/2-1-tensor-product.tex +++ /dev/null @@ -1,32 +0,0 @@ -\section{Tensor Product} - - -\begin{lemma}[The tensor product of linearly independent families] - \label{0-tensor-lin-indep} - \uses{} -% \lean{LinearIndependent.tmul_of_isDomain} - \leanok - \mathlibok - - Let $R$ be a domain and $M, N$ two $R$-semimodules. - If $f$ and $g$ are linearly independent families of points in $M$ and $N$, then $(i, j) \mapsto f i \ox g j$ is a linearly independent family of points in $M \ox N$. -\end{lemma} -\begin{proof} - \uses{} - \leanok - We will prove the equivalent statement: - - Let $P, Q$ be two free $R$ modules, $f : P \to M$ and $g : Q \to N$ be two $R$-linear injective maps. - Then $f \ox g : P \ox_R Q \to M \ox_R N$ is injective. - - Let $K$ be the field of fractions of $R$. - - The map - \[ P \ox_R Q \to (K \ox_R P) \ox_R (K \ox_R Q) = (K \ox_R P) \ox_K (K \ox_R Q) \] - is injective because $R \to K$ is injective and all the modules involved are flat. - The map - \[ (K \ox_R P) \ox_K (K \ox_R Q) \to (K \ox_R M) \ox_K (K \ox_R N) \] - is injective because all the modules involved are $K$-flat (as $K$ is a field). - - $P \ox_R Q \to M \ox_R N$ is now a factor of the composition of the two injections above, and is thus is injective. -\end{proof} diff --git a/blueprint/src/chapters/Algorithms/Flow/Basic.tex b/blueprint/src/chapters/Algorithms/Flow/Basic.tex new file mode 100644 index 0000000..e69de29 diff --git a/blueprint/src/chapters/Algorithms/MST/Basic.tex b/blueprint/src/chapters/Algorithms/MST/Basic.tex new file mode 100644 index 0000000..e69de29 diff --git a/blueprint/src/chapters/Algorithms/SCC/Basic.tex b/blueprint/src/chapters/Algorithms/SCC/Basic.tex new file mode 100644 index 0000000..e69de29 diff --git a/blueprint/src/chapters/Algorithms/Search/Basic.tex b/blueprint/src/chapters/Algorithms/Search/Basic.tex new file mode 100644 index 0000000..e69de29 diff --git a/blueprint/src/chapters/Algorithms/ShortestPath/Basic.tex b/blueprint/src/chapters/Algorithms/ShortestPath/Basic.tex new file mode 100644 index 0000000..e69de29 diff --git a/blueprint/src/chapters/DataStructures/InverseAckermann.tex b/blueprint/src/chapters/DataStructures/InverseAckermann.tex new file mode 100644 index 0000000..a02a8bc --- /dev/null +++ b/blueprint/src/chapters/DataStructures/InverseAckermann.tex @@ -0,0 +1 @@ +\section{Inverse Ackermann} diff --git a/blueprint/src/chapters/DataStructures/UnionFind/Blueprint.tex b/blueprint/src/chapters/DataStructures/UnionFind/Blueprint.tex new file mode 100644 index 0000000..e69de29 diff --git a/blueprint/src/chapters/Graph/Basic.tex b/blueprint/src/chapters/Graph/Basic.tex new file mode 100644 index 0000000..6cfd41b --- /dev/null +++ b/blueprint/src/chapters/Graph/Basic.tex @@ -0,0 +1,18 @@ +\section{Basic} + +\begin{definition} + An \textit{(undirected) graph} is a pair $G = (V,E)$ consisting of a set $V$ + and a multi-set $E$ of unordered pairs of elements in $V$. + Elements of $V$ are called \textit{vertices} and elements of $E$ are called \textit{edges}. + For two vertices $u,v \in V$, we say $u$ and $v$ are \textit{adjacent} if $\{u,v\} \in E$. + We call an undirected graph $G$ \textit{simple} if all edges are distinct (i.e., $E$ is a set), + and each edge is a pair of distinct vertices. +\end{definition} + +\begin{definition} + A \textit{directed graph} is a pair $G = (V,E)$ consisting of a set $V$ + and a multi-set $E$ of ordered pairs of elements in $V$. + Elements of $V$ are called \textit{vertices} and elements of $E$ are called \textit{edges}. + We call a directed graph $G$ \textit{simple} if all edges are distinct and + each edge is a pair of distinct vertices. +\end{definition} diff --git a/blueprint/src/chapters/Theory/Coloring/Basic.tex b/blueprint/src/chapters/Theory/Coloring/Basic.tex new file mode 100644 index 0000000..e69de29 diff --git a/blueprint/src/chapters/Theory/Connectivity/Basic.tex b/blueprint/src/chapters/Theory/Connectivity/Basic.tex new file mode 100644 index 0000000..e69de29 diff --git a/blueprint/src/chapters/Theory/Embeddings/Basic.tex b/blueprint/src/chapters/Theory/Embeddings/Basic.tex new file mode 100644 index 0000000..e69de29 diff --git a/blueprint/src/chapters/Theory/Matching/Basic.tex b/blueprint/src/chapters/Theory/Matching/Basic.tex new file mode 100644 index 0000000..e69de29 diff --git a/blueprint/src/chapters/Theory/Minors/Basic.tex b/blueprint/src/chapters/Theory/Minors/Basic.tex new file mode 100644 index 0000000..e69de29 diff --git a/blueprint/src/chapters/Theory/Spectral/Basic.tex b/blueprint/src/chapters/Theory/Spectral/Basic.tex new file mode 100644 index 0000000..e69de29 diff --git a/blueprint/src/chapters/Theory/Trees/Basic.tex b/blueprint/src/chapters/Theory/Trees/Basic.tex new file mode 100644 index 0000000..e69de29 diff --git a/blueprint/src/chapters/Theory/Walks/Basic.tex b/blueprint/src/chapters/Theory/Walks/Basic.tex new file mode 100644 index 0000000..e69de29 diff --git a/blueprint/src/content.tex b/blueprint/src/content.tex index 51fb511..895590c 100644 --- a/blueprint/src/content.tex +++ b/blueprint/src/content.tex @@ -6,13 +6,44 @@ % the current file can be a simple sequence of \input. Otherwise It % can start with a \section or \chapter for instance. +\tableofcontents -\chapter{Introduction} +\chapter{Graph} +\input{chapters/Graph/Basic.tex} -\input{chapters/1-1-over-cat.tex} +\chapter{Algorithms} +\section{Flow} +\input{chapters/Algorithms/Flow/Basic.tex} +\section{MST} +\input{chapters/Algorithms/MST/Basic.tex} +\section{SCC} +\input{chapters/Algorithms/SCC/Basic.tex} +\section{Search} +\input{chapters/Algorithms/Search/Basic.tex} +\section{Shortest Path} +\input{chapters/Algorithms/ShortestPath/Basic.tex} -\chapter{Preliminaries} +\chapter{Data Structures} +\input{chapters/DataStructures/InverseAckermann.tex} +\section{Union Find} +\input{chapters/DataStructures/UnionFind/Blueprint.tex} + +\chapter{Theory} +\section{Coloring} +\input{chapters/Theory/Coloring/Basic.tex} +\section{Connectivity} +\input{chapters/Theory/Connectivity/Basic.tex} +\section{Embeddings} +\input{chapters/Theory/Embeddings/Basic.tex} +\section{Matching} +\input{chapters/Theory/Matching/Basic.tex} +\section{Minors} +\input{chapters/Theory/Minors/Basic.tex} +\section{Spectral} +\input{chapters/Theory/Spectral/Basic.tex} +\section{Trees} +\input{chapters/Theory/Trees/Basic.tex} +\section{Walks} +\input{chapters/Theory/Walks/Basic.tex} -\input{chapters/2-1-tensor-product.tex} - \input{chapters/biblio}