master > master: notebook woche 2

justfile

@@ -68,6 +68,7 @@ _generate-models path name:
         --encoding "UTF-8" \
         --disable-timestamp \
         --use-schema-description \
+        --set-default-enum-member \
         --allow-population-by-field-name \
         --snake-case-field \
         --strict-nullable \

notebooks/week-0.ipynb

@@ -1,67 +0,0 @@
-{
- "cells": [
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "# Woche 0 - 10.-16. Oktober 2022 #\n",
-    "\n",
-    "Herzlich willkommen zur 0. Woche!"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "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.render import *;\n",
-    "\n",
-    "np.random.seed(8007253); # zur Wiederholbarkeit"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "A = np.random.randint(-100, 100, size=(4, 7));\n",
-    "display(array_to_latex(A, precision=3));"
-   ]
-  }
- ],
- "metadata": {
-  "kernelspec": {
-   "display_name": "Python 3.10.6 64-bit",
-   "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": "3.10.6"
-  }
- },
- "nbformat": 4,
- "nbformat_minor": 2
-}

notebooks/week-2.ipynb

@@ -0,0 +1,100 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Woche 2 - 17.-23. Oktober 2022 #\n",
+    "\n",
+    "Herzlich willkommen zur 2. Woche!"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "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.render import *;\n",
+    "\n",
+    "from src.maths.diagrams import *;\n",
+    "from src.maths.sets import *;\n",
+    "\n",
+    "np.random.seed(8007253); # zur Wiederholbarkeit"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "A = np.random.randint(-100, 100, size=(4, 7));\n",
+    "\n",
+    "X = ['a', 'b', 'c', 'd', 'e'];\n",
+    "Y = [2, 3, 5, 7, 11, 13, 17];\n",
+    "Z = ['α', 'β', 'γ', 'δ'];\n",
+    "f = [('a', 5), ('b', 5), ('c', 17), ('d', 2)]\n",
+    "g = [(2, 'γ'), (3, 'α'), (5, 'β'), (7, 'δ'), (11, 'β'), (13, 'γ'), (17, 'β')];\n",
+    "comp = Functions(\n",
+    "    Function(name=('$f$', 'X', 'Y'), domain=X, codomain=Y, fct=f),\n",
+    "    Function(name=('$g$', 'Y', 'Z'), domain=Y, codomain=Z, fct=g),\n",
+    ");\n",
+    "comp.draw(show_labels=True);"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "A = np.random.randint(-100, 100, size=(4, 7));\n",
+    "\n",
+    "X = randomset_alphabet(low=2, high=5);\n",
+    "Y = randomset_integers(low=2, high=5);\n",
+    "Z = randomset_greek(low=2, high=5);\n",
+    "f = random_function(X, Y);\n",
+    "g = random_function(Y, Z);\n",
+    "comp = Functions(\n",
+    "    Function(name=('$f$', 'X', 'Y'), domain=X, codomain=Y, fct=f),\n",
+    "    Function(name=('$g$', 'Y', 'Z'), domain=Y, codomain=Z, fct=g),\n",
+    ");\n",
+    "comp.draw(show_labels=False);"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3.10.8 64-bit",
+   "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
+}

requirements.txt

@@ -40,7 +40,7 @@ datamodel-code-generator>=0.12.0
 # maths
 fraction>=2.2.0
 numpy>=1.22.4
-matplotlib>=3.5.1
+matplotlib>=3.6.1
 
 # tables, data frames
 pandas>=1.4.2

src/maths/diagrams/__init__.py

@@ -0,0 +1,17 @@
+#!/usr/bin/env python3
+# -*- coding: utf-8 -*-
+
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+# IMPORTS
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+from src.maths.diagrams.sets import *;
+
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+# EXPORTS
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+__all__ = [
+    'Function',
+    'Functions',
+];

src/maths/diagrams/sets.py

@@ -0,0 +1,205 @@
+#!/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__ = [
+    'Function',
+    'Functions',
+];
+
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+# CONSTANTS / VARIABLES
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+T1 = TypeVar('T1');
+T2 = TypeVar('T2');
+
+SCALE  = (1., 4.);
+OFFSET = (3., 0.);
+MARGIN = 0.1;
+N_RESOLUTION = 100;
+ANNOTATE_OFFSET = (0, 10);
+FONTSIZE_PTS = 10;
+FONTSIZE_FCT = 14;
+FONTSIZE_SETS = 14;
+
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+# Classes
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+@dataclass
+class Function(Generic[T1,T2]):
+    name: tuple[str, str, str] = field();
+    domain: list[T1] = field();
+    codomain: list[T2] = field();
+    fct: list[tuple[T1,T2]] = field();
+
+    @property
+    def range(self) -> list[T2]:
+        return [y for x, y in self.fct];
+
+    @property
+    def indexes(self) -> list[tuple[int, int]]:
+        # prevent repeated computation:
+        if not hasattr(self, '_indexes'):
+            self._indexes = [
+                (self.domain.index(x), self.codomain.index(y))
+                for x, y in self.fct
+            ];
+        return getattr(self, '_indexes');
+
+    def draw(self) -> Figure:
+        return Functions(self).draw();
+
+class Functions:
+    fcts: list[Function];
+
+    def __init__(self, *f: Function):
+        self.fcts = list(f);
+
+    def draw(self, show_labels: bool = True) -> Figure:
+        N = len(self.fcts);
+        obj = mplot.subplots(1, 1, constrained_layout=True);
+        fig: Figure = obj[0];
+        axs: Axes = obj[1];
+        axs.tick_params(axis='both', which='both', left=False, right=False, top=False, bottom=False, labelbottom=False, labelleft=False);
+        mplot.title('');
+        mplot.xlabel('');
+        mplot.ylabel('');
+        mplot.margins(x=MARGIN, y=MARGIN);
+
+        origin = np.asarray((0., 0.));
+        offset = np.asarray(OFFSET);
+
+        p_set = oval(nr_points=N_RESOLUTION, scale=SCALE, centre=origin);
+        for k in range(N+1):
+            axs.plot(p_set[:, 0] + k*offset[0], p_set[:, 1] + k*offset[1], label='', color='blue');
+
+        p_domain = [];
+        p_codomain = [];
+        comp_range = [];
+
+        anchors = [
+            [
+                # function name
+                origin + (k + 0.5)*offset + (0, -1.1*SCALE[1]),
+                # sets
+                origin + k * offset + (0, 1.1 * SCALE[1]),
+                origin + (k + 1) * offset + (0, 1.1 * SCALE[1]),
+                # arrow start -> end
+                origin + (k + 0.05) * offset + (0, -1.1 * SCALE[1]),
+                origin + (k + 1 - 0.05) * offset + (0, -1.1 * SCALE[1]),
+
+            ]
+            for k in range(N)
+        ];
+
+        for k, f in enumerate(self.fcts):
+            if k == 0:
+                comp_range = f.domain;
+                p_domain = random_points(nr_points=len(f.domain), scale=SCALE, centre=origin + k*offset);
+            else:
+                p_domain = p_codomain;
+            p_codomain = random_points(nr_points=len(f.codomain), scale=SCALE, centre=origin + (k+1)*offset);
+            # range of composition so far:
+            comp_range_next = [y for x, y in f.fct if x in comp_range];
+
+            if k == 0:
+                axs.scatter(p_domain[:, 0], p_domain[:, 1], label='', color='black', marker='o');
+                if show_labels:
+                    for i, p in enumerate(p_domain):
+                        x_name = f.domain[i];
+                        axs.annotate(text=f'{x_name}', xy = p, textcoords='offset points', xytext=ANNOTATE_OFFSET, ha='center', size=FONTSIZE_PTS);
+
+            for j, p in enumerate(p_codomain):
+                y = f.codomain[j];
+                marker = 'o' if (y in comp_range_next) else 'x';
+                axs.scatter([p[0]], [p[1]], label='', color='black', marker=marker);
+                y_name = f.codomain[j];
+                if show_labels:
+                    axs.annotate(text=f'{y_name}', xy=p, textcoords='offset points', xytext=ANNOTATE_OFFSET, ha='center', size=FONTSIZE_PTS);
+
+            for i, j in f.indexes:
+                p = p_domain[i];
+                q = p_codomain[j];
+                x = f.domain[i];
+                if k == 0 or (x in comp_range):
+                    axs.plot([p[0], q[0]], [p[1], q[1]], label='', color='g', linewidth=2);
+                else:
+                    axs.plot([p[0], q[0]], [p[1], q[1]], label='', color='g', linestyle='--', linewidth=1);
+
+            anchor = anchors[k];
+            fct_name, X_name, Y_name = f.name;
+            axs.annotate(text=f'{fct_name}', xy=anchor[0], ha='center', size=FONTSIZE_FCT);
+            if k == 0:
+                axs.annotate(text=f'{X_name}', xy=anchor[1], ha='center', size=FONTSIZE_FCT);
+            axs.annotate(text=f'{Y_name}', xy=anchor[2], ha='center', size=FONTSIZE_FCT);
+            axs.add_patch(FancyArrowPatch(
+                anchor[3], anchor[4],
+                connectionstyle = 'arc3,rad=0.5',
+                arrowstyle      = 'Simple, tail_width=0.5, head_width=4, head_length=8',
+                color           = 'black',
+            ));
+
+            # update range of composition:
+            comp_range = comp_range_next;
+
+        return fig;
+
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+# AUXILIARY
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+def oval(
+    nr_points: int,
+    scale:     tuple[float, float] = (1., 1.),
+    centre:    tuple[float, float] = (0., 0.),
+) -> NDArray[Shape['*, 2'], Float]:
+    theta = np.linspace(start=0, stop=2*np.pi, num=nr_points, endpoint=True);
+    P = np.zeros(shape=(nr_points, 2), dtype=float);
+    P[:, 0] = centre[0] + scale[0] * np.cos(theta);
+    P[:, 1] = centre[1] + scale[1] * np.sin(theta);
+    P[-1, :] = P[0, :];
+    return P;
+
+
+
+def random_points(
+    nr_points: int,
+    scale:     tuple[float, float] = (1., 1.),
+    centre:    tuple[float, float] = (0., 0.),
+    force:     bool = False,
+    tol:       float = 0.2,
+) -> NDArray[Shape['*, 2'], Float]:
+    theta = np.linspace(start=0, stop=2*np.pi, num=nr_points, endpoint=False);
+    r_min = 0.25;
+    r_max = 1;
+    while True:
+        u = np.random.random(size=(nr_points,));
+        u_max = max(u);
+        if u_max == 0.:
+            continue;
+        if force:
+            u = np.minimum((1 + tol) * u / u_max, 1);
+        else:
+            u = (1 - tol) * u / u_max;
+        break;
+    r = r_min + (r_max - r_min) * u;
+    P = np.zeros(shape=(nr_points, 2), dtype=float);
+    P[:, 0] = centre[0] + scale[0] * r * np.cos(theta);
+    P[:, 1] = centre[1] + scale[1] * r * np.sin(theta);
+    return P;

src/maths/sets/__init__.py

@@ -0,0 +1,19 @@
+#!/usr/bin/env python3
+# -*- coding: utf-8 -*-
+
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+# IMPORTS
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+from src.maths.sets.random import *;
+
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+# EXPORTS
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+__all__ = [
+    'random_function',
+    'randomset_alphabet',
+    'randomset_greek',
+    'randomset_integers',
+];

src/maths/sets/random.py

@@ -0,0 +1,62 @@
+#!/usr/bin/env python3
+# -*- coding: utf-8 -*-
+
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+# IMPORTS
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+from __future__ import annotations;
+
+from src.thirdparty.types import *;
+from src.thirdparty.maths import *;
+
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+# EXPORTS
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+__all__ = [
+    'randomset_integers',
+    'randomset_alphabet',
+    'randomset_greek',
+    'random_function',
+];
+
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+# CONSTANTS / VARIABLES
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ALPHA = 'abcdefghijklmnopqrstuvwxyz';
+GREEK = 'αβγδεζηθικλμνξοπρςτυφχψω';
+T1 = TypeVar('T1');
+T2 = TypeVar('T2');
+
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+# METHODS
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+def randomset_integers(low: int, high: int) -> list[int]:
+    N = random.randint(low, high);
+    return list(range(1, N+1));
+
+def randomset_alphabet(low: int, high: int) -> list[int]:
+    N = random.randint(low, high);
+    return list([a for k, a in enumerate(ALPHA) if k < N]);
+
+def randomset_greek(low: int, high: int) -> list[int]:
+    N = random.randint(low, high);
+    return list([a for k, a in enumerate(GREEK) if k < N]);
+
+def random_function(
+    X: list[T1],
+    Y: list[T2],
+    injective: Optional[bool] = None,
+    surjective: Optional[bool] = None,
+) -> list[tuple[T1, T2]]:
+    # TODO: add feature to force injectivity/surjectivity, if possible.
+    # m = len(X);
+    # n = len(Y);
+    # if m > n:
+    #     injective = False;
+    # if m < n:
+    #     surjective = False;
+    return [ (x, random.choice(Y)) for x in X ];

src/thirdparty/misc.py

@@ -7,9 +7,62 @@
 
 from datetime import datetime;
 from datetime import timedelta;
+from functools import wraps;
 import lorem;
 import re;
-from textwrap import dedent;
+from textwrap import dedent as textwrap_dedent;
+from typing import Callable;
+from typing import TypeVar;
+
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+# MODIFICATIONS
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+def prestrip(first: bool = True, last: bool = True, all: bool = False):
+    '''
+    Returns a decorator that modifies string -> string methods
+    '''
+    T = TypeVar('T');
+    def dec(method: Callable[[str], T]) -> Callable[[str], T]:
+        '''
+        Performs method but first strips initial/final (empty) lines.
+        '''
+        @wraps(method)
+        def wrapped_method(text: str) -> T:
+            lines = re.split(pattern=r'\n', string=text);
+            if all:
+                if first:
+                    while len(lines) > 0 and lines[0].strip() == '':
+                        lines = lines[1:];
+                if last:
+                    while len(lines) > 0 and lines[-1].strip() == '':
+                        lines = lines[:-1];
+            else:
+                if first:
+                    lines = lines[1:];
+                if last:
+                    lines = lines[:-1];
+            text = '\n'.join(lines);
+            return method(text);
+        return wrapped_method;
+    return dec;
+
+@prestrip(all=False)
+def dedent(text: str) -> str:
+    '''
+    Remove any common leading whitespace from every line in `text`.
+
+    This can be used to make triple-quoted strings line up with the left
+    edge of the display, while still presenting them in the source code
+    in indented form.
+
+    Note that tabs and spaces are both treated as whitespace, but they
+    are not equal: the lines "  hello" and "\\thello" are
+    considered to have no common leading whitespace.
+
+    Entirely blank lines are normalized to a newline character.
+    '''
+    return textwrap_dedent(text);
 
 # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 # EXPORTS

src/thirdparty/plots.py

@@ -0,0 +1,24 @@
+#!/usr/bin/env python3
+# -*- coding: utf-8 -*-
+
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+# IMPORTS
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+from matplotlib import pyplot as mplot;
+from matplotlib import colors as mcolours;
+from matplotlib.figure import Figure;
+from matplotlib.axes import Axes;
+from matplotlib.patches import FancyArrowPatch;
+
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+# EXPORTS
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+__all__ = [
+    'mplot',
+    'mcolours',
+    'Figure',
+    'Axes',
+    'FancyArrowPatch',
+];