Chapter-06.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 6 - Expressions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib widget\n",
    "from sympy import *\n",
    "init_printing(use_latex=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 6.1 - The Expression Tree"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 6.1.1 - How Sympy represents Expressions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = Symbol(\"x\", real=True)\n",
    "expr1 = x**2 + x * cos(x) - 1 / x\n",
    "expr1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "sr = srepr(expr1)\n",
    "sr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr_rubuilt = S(sr)\n",
    "expr_rubuilt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy_utils import render_tree"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "render_tree(expr1, \"expr1\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr2 = x**(x / cos(x))\n",
    "display(expr2)\n",
    "render_tree(expr2, \"expr2\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr3 = x**2 / exp(3 * x)\n",
    "display(expr3)\n",
    "render_tree(expr3, \"expr3\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr4 = exp(-x + 4 * x**2 - 2)\n",
    "display(expr4)\n",
    "render_tree(expr4, \"expr4\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 6.1.2 - The Basic and Expr classes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "type(expr1), type(expr2), type(expr3), type(expr4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import inspect"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "inspect.getmro(Mul)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "inspect.getmro(Add)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "inspect.getmro(exp)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "for expr in [expr1, expr2, expr3, expr4]:\n",
    "    print(isinstance(expr, Basic), isinstance(expr, Expr))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 6.1.3 - Expression Manipulation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = Symbol(\"x\")\n",
    "expr1 = x**2 + x * cos(x) - 1 / x\n",
    "a1 = expr1.args\n",
    "a1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a1[0].args"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "display(x.args, Integer(2).args)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr1.atoms()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr1.free_symbols"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a, b = symbols(\"a, b\")\n",
    "i = Integral(x, (x, a, b))\n",
    "display(i, i.free_symbols)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr1.func, expr2.func, expr3.func, expr4.func"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a1 = expr1.args\n",
    "expr1.func(*a1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a1 = list(expr1.args)\n",
    "a1[0] = a1[0] / x\n",
    "r = expr1.func(*a1)\n",
    "r, expr1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 6.1.4 - Walking the Expression Tree"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = Symbol(\"x\")\n",
    "expr = x**2 + 2 * x + 1 / cos(x)\n",
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "render_tree(expr, \"expr\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "for arg in preorder_traversal(expr):\n",
    "    print(arg)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "for arg in postorder_traversal(expr):\n",
    "    print(arg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 6.1.5 - About the ordering of the terms"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = cos(x) - sin(x)\n",
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr.args"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a, p, L = symbols(\"alpha, p, L\")\n",
    "expr = p * L * a\n",
    "display(expr, expr.args)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = x**2 + 2 * x + 1 \n",
    "display(expr.args, expr)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 6.1.6 - Expression Evaluation and the class UnevaluatedExpr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x, y = symbols(\"x y\", real=True)\n",
    "expr1 = x + 1 + 2 * x + 2 + 4 * x * x\n",
    "expr2 = 2 * (x + y)\n",
    "display(expr1, expr2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "with evaluate(False):\n",
    "    expr1a = x + 1 + 2 * x + 2 + 4 * x * x\n",
    "    expr2a = 2 * (x + y)\n",
    "expr1b = Add(x, 1, 2 * x, 2, Mul(4, x, x, evaluate=False), evaluate=False)\n",
    "expr2b = Mul(2, x + y, evaluate=False)\n",
    "expr3a = expr1a + expr2a\n",
    "display(expr1a, expr1b, expr2a, expr2b, expr3a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr3 = S(\"x + 1 + 2 * x + 2 + 4 * x * x\", evaluate=False)\n",
    "expr3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "UE = UnevaluatedExpr\n",
    "expr4 = UE(x) + UE(1) + 2 * x + 2 + 4 * x * UE(x)\n",
    "expr4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr5 = 2.6 * x / 2\n",
    "expr6 = UE(2.6) * x / 2\n",
    "expr5, expr6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr5 * 6, expr6 * 4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr6.doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr6.is_commutative"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 6.2 - Expression Comparison – Equality Testing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 6.2.1 - Structural Equality Testing with =="
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = Symbol(\"x\")\n",
    "expr1 = x**2 + 2 * x + 1\n",
    "expr2 = (x + 1)**2\n",
    "display(expr1, expr2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr1 == expr2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "S(5) / 2 == S(2.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 6.2.2 - Equality Testing with the method simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr1 - expr2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "(expr1 - expr2).simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 6.2.3 - Equality Testing with the method equals()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr1.equals(expr2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 6.3 - Polynomials"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 6.3.1 - Symbolic expressions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x, y = symbols(\"x, y\")\n",
    "e1 = (x + 1) * (6 * x**2 - 3 * x + 4)\n",
    "e2 = e1.expand()\n",
    "display(e1, e2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "e3 = x**2*y + x**2 + x*y + y - 1\n",
    "e3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 6.3.2 - The Poly class"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = Poly(e1)\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = f.as_expr()\n",
    "gens = f.gens\n",
    "domain = f.domain\n",
    "display(expr, gens, domain)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "e4 = x**2 + 2 * sqrt(2) * x + 2\n",
    "g = Poly(e4)\n",
    "g"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "g = Poly(e4, extension=sqrt(2))\n",
    "g"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import inspect\n",
    "inspect.getmro(Poly)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "g.args"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "h = f * g\n",
    "h"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "h.eval(x, 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 6.3.3 - Polynomial Rings"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = symbols(\"x\")\n",
    "K = ZZ[x]\n",
    "print(type(K))\n",
    "K"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr1 = x**2 + 2 * x + 1\n",
    "p1 = K.from_sympy(expr1)\n",
    "print(type(p1))\n",
    "p1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "p1.to_dict()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy.polys.monomials import monomial_count\n",
    "x, y = symbols(\"x, y\")\n",
    "num_indeterminates = 2\n",
    "total_degree = 3\n",
    "num_monomials = monomial_count(num_indeterminates, total_degree)\n",
    "monomials = list(itermonomials([x, y], total_degree))\n",
    "display(num_monomials, monomials)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "K2 = ZZ[x, y]\n",
    "expr2 = 1 + x*y + x**2 * y + y**3\n",
    "p2 = K2.from_sympy(expr2)\n",
    "print(p2)\n",
    "p2.to_dict()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "p3 = K.from_sympy(x - 1)\n",
    "p4 = (p1 * p3)**2\n",
    "print(p1)\n",
    "print(p3)\n",
    "print(p4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "p5 = K.from_sympy(x**2 + 2 * x / 3 + 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(p4)\n",
    "p4.evaluate(0, 4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "xr = K(x)\n",
    "print(xr)\n",
    "print(type(xr))\n",
    "p4.evaluate(xr, 4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "xr = K2(x)\n",
    "yr = K2(y)\n",
    "print(p2)\n",
    "p2.evaluate([(xr, 2), (yr, 3)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = K.to_sympy(p4)\n",
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = x**2 + 2 * sqrt(2) * x + 2\n",
    "K = QQ.algebraic_field(sqrt(2))[x]\n",
    "print(K)\n",
    "p = K.from_sympy(expr)\n",
    "p"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 6.3.4 - Factorization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = x**2 - 4\n",
    "expr.factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "p = Poly(expr)\n",
    "p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "coeff, factors = factor_list(p)\n",
    "display(coeff, factors)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "new_expr = prod([f.as_expr()**e for (f, e) in factors]) * coeff\n",
    "new_expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr2 = x**2*y + x**2 + x*y + y - 1\n",
    "display(expr2, expr2.factor(x), expr2.factor(y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr3 = x**2 + 2 * sqrt(2) * x + 2\n",
    "expr3.factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Poly(expr3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr3.factor(extension=sqrt(2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Poly(expr3, x, extension=sqrt(2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr4 = x**2 + 1\n",
    "display(expr4, expr4.factor(), expr4.factor(extension=I))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = symbols(\"a\")\n",
    "expr5 = x**2 - a\n",
    "display(expr5, expr5.factor())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr5.factor(extension=sqrt(a))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "_roots = solve(expr5, x)\n",
    "res = LC(expr5, x) * prod(x - r for r in _roots)\n",
    "res"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 6.4 - Parsing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = sympify(\"(x * cos(y)**2 + x * sin(y)**2) / 2\")\n",
    "fs = list(expr.free_symbols)\n",
    "display(expr, fs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 6.4.1 - From string to SymPy Expressions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "xpos = symbols(\"x\", positive=True)\n",
    "s = \"sqrt(x**2 + 2*x + 1)\"\n",
    "expr1 = parse_expr(s)\n",
    "expr2 = parse_expr(s, local_dict={\"x\": xpos})\n",
    "display(expr1.factor(), expr2.factor())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "s = \"R*I-V\"\n",
    "expr = parse_expr(s)\n",
    "print(type(I))\n",
    "print(expr.has(I))\n",
    "display(expr.free_symbols)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "i = symbols(\"I\")\n",
    "expr = parse_expr(s, local_dict={\"I\": i})\n",
    "print(\"expr contains the imaginary unit:\", expr.has(I))\n",
    "print(\"expr contains the current:\", expr.has(i))\n",
    "display(expr.free_symbols)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy.parsing.sympy_parser import T\n",
    "print(T)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in range(T.N):\n",
    "    help(T[i][0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "s = \"y = x^2 + 2x + 1\"\n",
    "eq = parse_expr(s, transformations=\"all\")\n",
    "print(type(eq))\n",
    "display(eq)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "eq = parse_expr(s, transformations=T[:5, 6, 8, 9])\n",
    "eq"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 6.4.2 - From Latex Code to SymPy Expressions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy.parsing.latex import parse_latex\n",
    "s = r\"\\frac{x \\sin^{2}{\\left(y \\right)}}{2} + \\frac{x \\cos^{2}{\\left(y \\right)}}{2}\"\n",
    "expr = parse_latex(s)\n",
    "fs = list(expr.free_symbols)\n",
    "display(expr, fs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "y, x = fs\n",
    "expr.collect(x).simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 6.5 - Advanced Topics"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[Click here to open the associated notebook](Chapter-06-Advanced-Topics.ipynb)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "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": "3.12.3"
  },
  "toc": {
   "base_numbering": 1,
   "nav_menu": {},
   "number_sections": true,
   "sideBar": true,
   "skip_h1_title": false,
   "title_cell": "Table of Contents",
   "title_sidebar": "Contents",
   "toc_cell": false,
   "toc_position": {},
   "toc_section_display": true,
   "toc_window_display": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}