Mark/handouts
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Mark 2023-04-28 15:33:41 -07:00
parent 8774a3ba59
commit ea24c69100
3 changed files with 6 additions and 6 deletions
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2
Advanced/Lattices/main.tex
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@ -1,7 +1,7 @@
% use [nosolutions] flag to hide solutions. % use [nosolutions] flag to hide solutions.
% use [solutions] flag to show solutions. % use [solutions] flag to show solutions.
\documentclass[ \documentclass[
solutions, nosolutions,
singlenumbering singlenumbering
]{../../resources/ormc_handout} ]{../../resources/ormc_handout}

8
Advanced/Lattices/parts/0 intro.tex
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@ -10,9 +10,9 @@ Draw $\mathbb{Z}^2$.
\definition{} \definition{}
We say a set of vectors $\{v_1, v_2, ..., v_n\}$ \textit{generates} $\mathbb{Z}^n$ if every lattice point can be written uniquely as We say a set of vectors $\{v_1, v_2, ..., v_n\}$ \textit{generates} $\mathbb{Z}^n$ if every lattice point can be written uniquely as
$$ $$
a_1v_1 + a_2v_2 + ... a_nv_n a_1v_1 + a_2v_2 + ... + a_nv_n
$$ $$
for integer coeficcients $a_i$. for integer coefficients $a_i$.
\problem{} \problem{}
Which of the following generate $\mathbb{Z}^3$? Which of the following generate $\mathbb{Z}^3$?
@ -37,13 +37,13 @@ Find a set of vectors that generates $\mathbb{Z}^n$.
\vfill \vfill
\pagebreak \pagebreak
\problem{} \definition{}
A \textit{fundamental region} of a lattice is the parallelepiped spanned by a generating set. The exact shape of this region depends on the generating set we use. A \textit{fundamental region} of a lattice is the parallelepiped spanned by a generating set. The exact shape of this region depends on the generating set we use.
\vfill \vfill
\problem{} \problem{}
Draw two fundamental reions of $\mathbb{Z}^2$ using two different generating sets. Verify that their volumes are the same. Draw two fundamental regions of $\mathbb{Z}^2$ using two different generating sets. Verify that their volumes are the same.
\vfill \vfill
\pagebreak \pagebreak

2
Advanced/Lattices/parts/1 minkowski.tex
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@ -31,7 +31,7 @@ The following picture gives the idea for the proof of Blichfeldt's theorem. Expl
\problem{} \problem{}
Let $X$ be a region $X$ of volume $k$. How many integral points must $X$ contain after a translation? Let $X$ be a region $\in \mathbb{R}^2$ of volume $k$. How many integral points must $X$ contain after a translation?
\vfill \vfill