master > master: handschrift - woche 15 aktualisiert

- 2 Beweise für Sem 9.2a)
- Lsg für Sem 9.2b)+c)

.gitignore

@@ -28,6 +28,10 @@
 !/templates
 !/templates/template*
 
+!/notes/
+!/notes/week*.pdf
+!/notes/README.md
+
 !/src
 !/src/**/
 !/src/**/*.py

README.md

@@ -1,6 +1,7 @@
 # Mathematik für Physiker I, WiSe 2022-23 #
 
-Der Inhalt dieses Repository ist keineswegs verpflichtend, sondern dient dem Zweck, „Handnotizen“ anzubieten.
+Der Inhalt dieses Repository ist keineswegs verpflichtend,
+sondern dient dem Zweck, „Handnotizen“ anzubieten (siehe insbesondere den [./notes/](notes/) Ordner).
 Es besteht hier kein Anspruch auf Vollständigkeit.
 Es werden hier keine offiziellen Lösungen für die Übungsblätter gespeichert.
 
@@ -11,6 +12,7 @@ findet man ausschließlich auf der **Moodle**-Seite.
 
 ```text
 .
+├── ./notes     # Handschrift aus ÜG
 ├── ./notebooks # Python/Jupyter Notebooks
 │   ├── ...
 │   └── ... *.ipynb Dateien

justfile

@@ -121,6 +121,17 @@ dist:
     @just build
     @just build-documentation
 
+# exports notebook to format: html, markdown, pdf, latex.
+export name format="pdf" theme="light":
+    @{{PYTHON}} -m jupyter nbconvert \
+        --allow-chromium-download \
+        --HTMLExporter.theme={{theme}} \
+        --TemplateExporter.exclude_input=false \
+        --to {{format}} \
+        --output-dir notes \
+        --ouput "notebook-{{name}}.{{format}}" \
+        notebooks/{{name}}.ipynb
+
 # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 # TARGETS: run
 # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

notebooks/week-10.ipynb

@@ -0,0 +1,265 @@
+{
+ "cells": [
+  {
+   "attachments": {},
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Woche 10 #\n",
+    "\n",
+    "Diese Wochen beschäftigen wir uns mit Matrizen und Vektoren.\n",
+    "In diesem Notebook rechnen wir ein paar (Teil)aufgaben aus dem ÜB."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "'''IMPORTS'''\n",
+    "import os;\n",
+    "import sys; \n",
+    "\n",
+    "# NOTE: need this to force jupyter to reload imports:\n",
+    "for key in list(sys.modules.keys()):\n",
+    "    if key.startswith('src.'):\n",
+    "        del sys.modules[key];\n",
+    "\n",
+    "os.chdir(os.path.dirname(_dh[0]));\n",
+    "sys.path.insert(0, os.getcwd());\n",
+    "\n",
+    "from src.thirdparty.maths import *;\n",
+    "from src.thirdparty.misc import *;\n",
+    "from src.thirdparty.render import *;\n",
+    "from src.maths.diagrams import *;\n",
+    "\n",
+    "np.random.seed(8007253); # zur Wiederholbarkeit"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "'''BEISPIEL-SIGNAL'''\n",
+    "n = 100;\n",
+    "u = np.zeros((n,), dtype=complex);\n",
+    "u[8] = -0.7\n",
+    "u[58] = 1j * sqrt(2);\n",
+    "\n",
+    "display_signal(u, 'Beispiel eines Vektors in $\\mathbb{C}^{100}$ als Signal über $\\{0,1,2,\\ldots,99\\}$ dargestellt');"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "'''SEMINARAUFGABE 10.4 a)'''\n",
+    "n = 3;\n",
+    "t = linspace(0, n-1, n);\n",
+    "i, j = np.meshgrid(t, t);\n",
+    "theta = 2*pi * i * j / n;\n",
+    "\n",
+    "# definiere die Matrix F_n + die Vektoren u_nk\n",
+    "F_n = (1/sqrt(n)) * exp(-1j * theta);\n",
+    "u_n = [ (1/sqrt(n)) * exp(1j * theta[:, k]) for k in range(n) ];\n",
+    "\n",
+    "print(dedent(\n",
+    "    f'''\n",
+    "    Matrix:\n",
+    "\n",
+    "    F_{n} = \\n{np.round(F_n, 3)}\n",
+    "    '''\n",
+    "));\n",
+    "print('');\n",
+    "\n",
+    "# NOTE: Programmiersprachen können nicht exakt berechnen.\n",
+    "# Darum entstehen winzige Fehler, die sich durch Runden entfernen lassen.\n",
+    "print('Matrix-Vektor-Produkte:');\n",
+    "for k in range(n):\n",
+    "    result = F_n @ u_n[k];\n",
+    "    print(dedent(\n",
+    "        f'''\n",
+    "\n",
+    "              u_{{n {k}}} = {np.round(u_n[k], 3)}\n",
+    "        F_n · u_{{n {k}}} = {np.round(result, 3)}\n",
+    "        '''\n",
+    "    ));"
+   ]
+  },
+  {
+   "attachments": {},
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Seminaraufgabe 10.4 b) ##\n",
+    "\n",
+    "Für $i\\in\\{0,1,\\ldots,n-1\\}$ ist die $i$-te von $\\mathcal{F}_{n}\\,u_{n,k}$\n",
+    "\n",
+    "$$\n",
+    "    \\begin{array}{rcl}\n",
+    "    (\\mathcal{F}_{n}\\,u_{n,k})_{i}\n",
+    "    &= &\\displaystyle\\sum_{j=0}^{n-1}\n",
+    "        (\\mathcal{F}_{n})_{ij}\\,(u_{n,k})_{j}\\\\\n",
+    "    &= &\\displaystyle\\sum_{j=0}^{n-1}\n",
+    "        \\tfrac{1}{\\sqrt{n}}\n",
+    "        \\exp(-\\imath\\tfrac{2\\pi\\,ij}{n})\n",
+    "        \\cdot\n",
+    "        \\tfrac{1}{\\sqrt{n}}\n",
+    "        \\exp(\\imath\\tfrac{2\\pi\\,kj}{n})\\\\\n",
+    "    &= &\\frac{1}{n}\\displaystyle\\sum_{j=0}^{n-1}\n",
+    "        \\underbrace{\n",
+    "            \\exp(\\imath\\tfrac{2\\pi\\,(k-i)j}{n})\n",
+    "        }_{= r^{j}}\\\\\n",
+    "    \\end{array}\n",
+    "$$\n",
+    "\n",
+    "wobei $r := \\exp(\\imath\\tfrac{2\\pi\\,(k-i)}{n})$.\n",
+    "Wir haben, dass\n",
+    "    $r = 1$\n",
+    "    $\\Leftrightarrow$\n",
+    "        $\\tfrac{k-i}{n} \\in \\mathbb{Z}$\n",
+    "    $\\Leftrightarrow$\n",
+    "        $i = k$,\n",
+    "da $0\\leq i,k \\leq n-1$.\n",
+    "<br>\n",
+    "Es gilt außerdem $r^{n} = \\exp(\\imath 2\\pi\\,(k-i)) = 1$.\n",
+    "<br>\n",
+    "Aus der o.s. Berechnung erhält man also\n",
+    "\n",
+    "$$\n",
+    "    \\begin{array}{rcl}\n",
+    "    (\\mathcal{F}_{n}\\,u_{n,k})_{i}\n",
+    "    &= &\\tfrac{1}{n}\\displaystyle\\sum_{j=0}^{n-1}r^{j}\\\\\n",
+    "    &= &\\tfrac{1}{n}\n",
+    "        \\left\\{\n",
+    "        \\begin{array}{lcl}\n",
+    "            \\displaystyle\\sum_{j=0}^{n-1}1 &: &i = k\\\\\n",
+    "            \\dfrac{1 - r^{n}}{1 - r} &: &\\text{sonst}\\\\\n",
+    "        \\end{array}\n",
+    "        \\right.\\\\\n",
+    "    &= &\\tfrac{1}{n}\n",
+    "        \\left\\{\n",
+    "        \\begin{array}{lcl}\n",
+    "            n &: &i = k\\\\\n",
+    "            \\dfrac{1 - 1}{1 - r} &: &\\text{sonst}\\\\\n",
+    "        \\end{array}\n",
+    "        \\right.\\\\\n",
+    "    &= &\\left\\{\n",
+    "        \\begin{array}{lcl}\n",
+    "            1 &: &i = k\\\\\n",
+    "            0 &: &\\text{sonst}\\\\\n",
+    "        \\end{array}\n",
+    "        \\right.\\\\\n",
+    "    &= &\\delta_{ik}.\\\\\n",
+    "    \\end{array}\n",
+    "$$\n",
+    "\n",
+    "Darum $\\mathcal{F}_{n}\\,u_{n,k}\n",
+    "    = \\left(\\begin{matrix}\n",
+    "        0\\\\\n",
+    "        0\\\\\n",
+    "        \\vdots\\\\\n",
+    "        1\\\\\n",
+    "        \\vdots\\\\\n",
+    "        0\\\\\n",
+    "    \\end{matrix}\\right)\n",
+    "    \\begin{matrix}\n",
+    "        \\\\\n",
+    "        \\\\\n",
+    "        \\\\\n",
+    "        \\leftarrow k\\\\\n",
+    "        \\\\\n",
+    "        \\\\\n",
+    "    \\end{matrix}\n",
+    "    = \\mathbf{e}_{k}$.\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "'''SEMINARAUFGABE 10.4 d)'''\n",
+    "n = 4;\n",
+    "t = linspace(0, n-1, n);\n",
+    "i, j = np.meshgrid(t, t);\n",
+    "theta = 2*pi * i * j / n;\n",
+    "\n",
+    "# definiere die Matrix F_n + die Vektoren u_nk\n",
+    "F_n = (1/sqrt(n)) * exp(-1j * theta);\n",
+    "u_n = [ (1/sqrt(n)) * exp(1j * theta[:, k]) for k in range(n) ];\n",
+    "\n",
+    "# Eingabe des Vektors:\n",
+    "F_mal_u = np.asarray([100, -3, 0, -3]); # = F_n @ u\n",
+    "\n",
+    "'''\n",
+    "NOTE: Folgende Koeffizienten wurden falsch erraten.\n",
+    "TODO: Wie lauten die richtigen Werte? -> Aufgabe!\n",
+    "'''\n",
+    "c = [10,10,40,50];\n",
+    "\n",
+    "# Aufschreibung des Signals »bzgl. der harmonischen Basis«:\n",
+    "# u = c[0]u_n[0] + c[1]u_n[1] + c[2]u_n[2] + c[3]u_n[3]\n",
+    "u_guess = sum([ c[k] * u_n[k] for k in range(n) ]);\n",
+    "\n",
+    "# Anzeige der Lösung:\n",
+    "display_signal(u_guess, 'Geschätztes Signal');\n",
+    "\n",
+    "# Verifizierung der Lösung:\n",
+    "display_latex(\n",
+    "    f'''\n",
+    "    $$\n",
+    "    \\\\begin{{array}}{{rcl}}\n",
+    "        \\\\%\\\\text{{error}}\n",
+    "            &\\colonequals\n",
+    "                &\\\\dfrac{{\\\\|u_{{\\\\text{{guess}}}} - u\\\\|}}{{\\\\|u\\\\|}}\\\\\\\\\n",
+    "            &=\n",
+    "                &\\\\dfrac{{\n",
+    "                    \\\\|\\\\mathcal{{F}}_{{n}}\\\\,(u_{{\\\\text{{guess}}}} - u)\\\\|\n",
+    "                }}{{\n",
+    "                    \\\\|\\\\mathcal{{F}}_{{n}}\\\\,u\\\\|\n",
+    "                }}\n",
+    "            \\\\quad\\\\text{{\\\\ldots wieso?? [Stichwort: unitär!]}}\\\\\\\\\n",
+    "            &= &\\\\dfrac{{\n",
+    "                    \\\\|\\\\mathcal{{F}}_{{n}}\\\\,u_{{\\\\text{{guess}}}} - \\\\mathcal{{F}}_{{n}}\\\\,u\\\\|\n",
+    "                }}{{\n",
+    "                    \\\\|\\\\mathcal{{F}}_{{n}}\\\\,u\\\\|\n",
+    "                }}\\\\\\\\\n",
+    "            &= &{linalg.norm(F_n @ u_guess - F_mal_u)/linalg.norm(F_mal_u):.2%}\n",
+    "            \\\\%\n",
+    "            \\\\\\\\\n",
+    "    \\\\end{{array}}\n",
+    "    $$\n",
+    "    '''\n",
+    ");\n"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.8"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}

notes/README.md

@@ -0,0 +1,9 @@
+# Handnotizen #
+
+Diese Notizen sollen _nicht_ als Musterlösungen verstanden werden.
+
+Während der ÜG interagieren wir mit dem Stoff + den Aufgaben.
+Wir besprechen **Konzepte** und **Ansätze**.
+Diese Schrift ist also eine Abbildung dieser Diskussionen und Präsentationen.
+Teilweise werden sie nur für die Leute Sinn ergeben, die daran teilgenommen haben und eigene Notizen machen.
+Die Dokumente sind nicht vollständig, da einiges während der Stunden im Raum und an der Tafel erörtert wird.

requirements.txt

@@ -1,4 +1,4 @@
-pip>=22.3
+pip>=22.3.1
 wheel>=0.37.1
 
 # jupyter
@@ -16,7 +16,7 @@ codetiming>=1.3.0
 
 # testing + dev
 coverage[toml]>=6.4
-pytest>=7.1.1
+pytest>=7.2.0
 pytest-asyncio>=0.18.3
 pytest-cov>=3.0.0
 pytest-lazy-fixture>=0.6.3
@@ -33,7 +33,6 @@ lorem>=0.1.1
 safetywrap>=1.5.0
 typing>=3.7.4.3
 nptyping>=2.1.1
-typing-extensions>=3.10.0.2
 
 # config
 python-dotenv>=0.20.0

src/maths/diagrams/__init__.py

@@ -6,6 +6,7 @@
 # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
 from src.maths.diagrams.sets import *;
+from src.maths.diagrams.signal import *;
 
 # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 # EXPORTS
@@ -14,4 +15,5 @@ from src.maths.diagrams.sets import *;
 __all__ = [
     'Function',
     'Functions',
+    'display_signal',
 ];

src/maths/diagrams/signal.py

@@ -0,0 +1,48 @@
+#!/usr/bin/env python3
+# -*- coding: utf-8 -*-
+
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+# IMPORTS
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+from __future__ import annotations;
+
+from src.thirdparty.code import *;
+from src.thirdparty.types import *;
+from src.thirdparty.maths import *;
+from src.thirdparty.plots import *;
+
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+# EXPORTS
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+__all__ = [
+    'display_signal',
+];
+
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+# CONSTANTS / VARIABLES
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+def display_signal(
+    u: np.ndarray,
+    title: str,
+    max_labels = 10,
+):
+    n = len(u);
+    if n <= max_labels:
+        x_axis = list(range(n));
+    else:
+        skip = int(n/max_labels);
+        x_axis = [ k*skip for k in range(max_labels) ];
+
+    fig, axs = mplot.subplots(1, 1, constrained_layout=True);
+    mplot.title(title);
+    mplot.xlabel('»Zeit« [=Indizes]');
+    mplot.ylabel('Werte [=Einträge]');
+    axs.stem(linspace(0,n-1,n), np.real(u), label='reeller Teil', linefmt='red');
+    axs.stem(linspace(0,n-1,n) + 0.05, np.imag(u), label='imaginärer Teil', linefmt='lime');
+    mplot.legend();
+    axs.set_xticks(x_axis, labels=[ str(index) for index in x_axis]);
+    mplot.show();
+    return;

src/thirdparty/maths.py

@@ -6,10 +6,17 @@
 # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
 from fractions import Fraction;
+from numpy import cos;
+from numpy import exp;
+from numpy import linalg;
+from numpy import linspace;
+from numpy import pi;
+from numpy import sin;
+from numpy import sqrt;
+from typing import TypeVar;
 import math;
 import numpy as np;
 import random;
-from typing import TypeVar;
 
 # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 # MODIFICATIONS
@@ -39,8 +46,15 @@ def sample(
 
 __all__ = [
     'Fraction',
+    'cos',
+    'exp',
+    'linalg',
+    'linspace',
     'math',
     'np',
+    'pi',
     'random',
     'sample',
+    'sin',
+    'sqrt',
 ];

src/thirdparty/render.py

@@ -6,24 +6,57 @@
 # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
 from IPython.display import Latex;
-from IPython.display import display_latex;
-from IPython.display import display_png;
-from IPython.display import display_markdown;
 from IPython.display import display;
+from IPython.display import display_png;
+from IPython.display import display as display_text_ipython;
+from IPython.display import display_latex as display_latex_ipython;
+from IPython.display import display_markdown as display_md_ipython;
 import ipywidgets as widgets;
 # from array_to_latex import to_ltx as array_to_latex; # <- has issues
 from qiskit.visualization import array_to_latex;
 
+from functools import wraps;
+from typing import Callable;
+from typing import TypeVar;
+
+from src.thirdparty.misc import dedent;
+
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+# MODIFICATIONS
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+T1 = TypeVar('T1');
+T2 = TypeVar('T2');
+
+def display_with_dedent(pre_process: Callable[[str], T1] = lambda x: x):
+    '''
+    Returns a decorator that modifies string -> string methods
+    '''
+    def dec(method: Callable[[T1], T2]) -> Callable[[str], T2]:
+        '''
+        Performs method but dedents first.
+        '''
+        @wraps(method)
+        def wrapped_method(text: str) -> T2:
+            return method(pre_process(dedent(text)));
+        return wrapped_method;
+
+    return dec;
+
+display_latex = display_with_dedent(Latex)(display_latex_ipython);
+display_text = display_with_dedent()(display_text_ipython);
+display_markdown = display_with_dedent()(display_md_ipython);
+
 # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 # EXPORTS
 # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
 __all__ = [
+    'Latex',
     'array_to_latex',
+    'display',
     'display_latex',
     'display_markdown',
     'display_png',
-    'display',
-    'Latex',
     'widgets',
 ];

src/thirdparty/types.py

@@ -38,8 +38,8 @@ from typing import Generic;
 from typing import Optional;
 from typing import Type;
 from typing import TypeVar;
-from typing_extensions import Concatenate;
+from typing import Concatenate;
-from typing_extensions import ParamSpec;
+from typing import ParamSpec;
 
 # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 # EXPORTS