misc

documents/eaz/eaz.tex

@@ -223,7 +223,35 @@ Will man die Elementarteiler einer Matrix $M$ berechnen, so gilt:
 \end{itemize}
 
 \section*{Weiteres}
-In alten Klausuren begegnen uns desöfteren Ringe der Form $Z[\sqrt{d}]$. 
+Finden von Zerlegungen von Elementen im Ring $\mathbb{Z}[\sqrt{d}] := \Set{a + b \sqrt{d} | a, b \in \mathbb{Z}}$:
+
+Es sei
+\begin{align*}
+	|\cdot|:\mathbb{Z}[\sqrt{d}] &\rightarrow \mathbb{R}_0^+\\
+	|x + y \cdot \sqrt{d}| &:= \sqrt{x^2 + y^2 |d|}
+\end{align*}
+
+und
+
+\begin{align*}
+	N:   \mathbb{Z}[\sqrt{d}] &\rightarrow \mathbb{N}_0\\
+	N(z) &:= |z|^2
+\end{align*}
+und die Konjugation:
+\[\overline{x+ y \sqrt{d}} = \begin{cases}x - y \sqrt{d} &\text{, falls } d < 0\\x + y \sqrt{d} &\text{, falls } x \geq 0\end{cases}\]
+
+Die Konjugation ist multiplikativ:
+\[\overline{wz} = \overline{w} \cdot \overline{z}\]
+
+Außerdem gilt:
+\begin{align*}
+  N(\pi) &= x^2+y^2 |d|\\
+         &= (x+y\sqrt{d}) \cdot (x+\frac{d}{|d|}y\sqrt{d})\\
+		 &= \pi \cdot \overline{\pi}
+\end{align*}
+
+
+In alten Klausuren begegnen uns desöfteren Ringe der Form . 
 In diesem Zusammenhang begegnet uns die Normabbildung. 
 (Ein Beispiel, das in der Vorlesung gesehen wurde, waren die gauß'schen Zahlen.)
 

source-code/Pseudocode/SolveLinearCongruences/Makefile

@@ -0,0 +1,36 @@
+SOURCE = SolveLinearCongruences
+DELAY = 80
+DENSITY = 300
+WIDTH = 500
+
+make:
+	pdflatex $(SOURCE).tex -output-format=pdf
+	pdflatex $(SOURCE).tex -output-format=pdf
+	make clean
+
+clean:
+	rm -rf  $(TARGET) *.class *.html *.log *.aux *.data *.gnuplot
+
+gif:
+	pdfcrop $(SOURCE).pdf
+	convert -verbose -delay $(DELAY) -loop 0 -density $(DENSITY) $(SOURCE)-crop.pdf $(SOURCE).gif
+	make clean
+
+png:
+	make
+	make svg
+	inkscape $(SOURCE).svg -w $(WIDTH) --export-png=$(SOURCE).png
+
+transparentGif:
+	convert $(SOURCE).pdf -transparent white result.gif
+	make clean
+
+svg:
+	make
+	#inkscape $(SOURCE).pdf --export-plain-svg=$(SOURCE).svg
+	pdf2svg $(SOURCE).pdf $(SOURCE).svg
+	# Necessary, as pdf2svg does not always create valid svgs:
+	inkscape $(SOURCE).svg --export-plain-svg=$(SOURCE).svg
+	rsvg-convert -a -w $(WIDTH) -f svg $(SOURCE).svg -o $(SOURCE)2.svg
+	inkscape $(SOURCE)2.svg --export-plain-svg=$(SOURCE).svg
+	rm $(SOURCE)2.svg

source-code/Pseudocode/SolveLinearCongruences/Readme.md

@@ -0,0 +1,3 @@
+Compiled example
+----------------
+![Example](SolveLinearCongruences.png)

source-code/Pseudocode/SolveLinearCongruences/SolveLinearCongruences.tex

@@ -0,0 +1,46 @@
+\documentclass{article}
+\usepackage[pdftex,active,tightpage]{preview}
+\setlength\PreviewBorder{2mm}
+
+\usepackage[utf8]{inputenc} % this is needed for umlauts
+\usepackage[ngerman]{babel} % this is needed for umlauts
+\usepackage[T1]{fontenc}    % this is needed for correct output of umlauts in pdf
+\usepackage{amssymb,amsmath,amsfonts} % nice math rendering
+\usepackage{braket} % needed for \Set
+\usepackage{algorithm,algpseudocode}
+
+\usepackage{tikz}
+\usetikzlibrary{decorations.pathreplacing,calc}
+\newcommand{\tikzmark}[1]{\tikz[overlay,remember picture] \node (#1) {};}
+\newcommand*{\AddNote}[4]{%
+    \begin{tikzpicture}[overlay, remember picture]
+        \draw [decoration={brace,amplitude=0.5em},decorate,very thick]
+            ($(#3)!(#1.north)!($(#3)-(0,1)$)$) --  
+            ($(#3)!(#2.south)!($(#3)-(0,1)$)$)
+                node [align=center, text width=2.5cm, pos=0.5, anchor=west] {#4};
+    \end{tikzpicture}
+}%
+
+\begin{document}
+\begin{preview}
+    \begin{algorithm}[H]
+        \begin{algorithmic}
+            \Require $R \in \mathbb{Z}^n, P \in (\mathbb{N}_{\geq 1})^n, n \in \mathbb{N}_{\geq 1}$, where \\
+					$R$ is a vector with all rests $r_i$ and\\
+					$P$ is a vector with all modulos $p_i$ such that\\
+					($x \equiv r_i \mod p_i$) and $\left(i \neq j \Rightarrow \Call{gcd}{p_i, p_j} = 1 \right)$
+			\\
+			\State $M \gets \prod_{p \in P} p$
+
+			\For{$i \in \{1, \dots, n\}$}
+				\State $M_i \gets \frac{M}{p_i} $
+				\State $y_i \gets \Call{getMultiplicativeInverse}{M_i, R_i}$
+			\EndFor
+            \\
+            \State \Return $(\sum_{i=1}^n R_i y_i M_i, M)$
+        \end{algorithmic}
+    \caption{Solve a system of linear congruences}
+    \label{alg:solveCongruences}
+    \end{algorithm}
+\end{preview}
+\end{document}

source-code/Pseudocode/SolveLinearCongruences/solveLinearCongruences.py

@@ -0,0 +1,49 @@
+#!/usr/bin/env python
+# -*- coding: utf-8 -*-
+
+def ExtendedEuclideanAlgorithm(a, b):
+	"""
+		Calculates gcd(a,b) and a linear combination such that
+		gcd(a,b) = a*x + b*y
+
+		As a side effect:
+		If gcd(a,b) = 1 = a*x + b*y
+		Then x is multiplicative inverse of a modulo b.
+	"""
+	aO, bO = a, b
+
+	x=lasty=0
+	y=lastx=1
+	while (b!=0):
+		q= a/b
+		a, b = b, a%b
+		x, lastx = lastx-q*x, x
+		y, lasty = lasty-q*y, y
+
+	return {
+		"x": lastx,
+		"y": lasty,
+		"gcd": aO * lastx + bO * lasty
+	}
+
+def solveLinearCongruenceEquations(rests, modulos):
+	"""
+	Solve a system of linear congruences.
+
+	>>> solveLinearCongruenceEquations([4, 12, 14], [19, 37, 43])
+	{'congruence class': 22804, 'modulo': 30229}
+	"""
+	assert len(rests) == len(modulos)
+	x = 0
+	M = reduce(lambda x, y: x*y, modulos)
+
+	for mi, resti in zip(modulos, rests):
+		Mi = M / mi
+		s = ExtendedEuclideanAlgorithm(Mi, mi)["x"]
+		e = s * Mi
+		x += resti * e
+	return {"congruence class": ((x % M) + M) % M, "modulo": M}
+
+if __name__ == "__main__":
+	import doctest
+	doctest.testmod()