fix doc typos

Author: shanemcq18

docs/source/api/operators.ipynb

@@ -1122,7 +1122,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Affine parametric operators are instantiated with a function $\\boldsymbol{theta}_{\\ell}(\\mu) = [~\\theta_{ell}^{(0)}(\\bfmu)~~\\cdots~~\\theta_{ell}^{(A_{\\ell}-1)}(\\bfmu)~]\\trp$ for the affine expansion coefficients, the number of terms $A_{\\ell}$ in the expansion, and with or without the operator matrices $\\Ohat_{\\ell}^{(1)},\\ldots,\\Ohat_{\\ell}^{(A_{\\ell})}$."
+    "Affine parametric operators are instantiated with a function $\\boldsymbol{\\theta}_{\\ell}(\\mu) = [~\\theta_{\\ell}^{(0)}(\\bfmu)~~\\cdots~~\\theta_{\\ell}^{(A_{\\ell}-1)}(\\bfmu)~]\\trp$ for the affine expansion coefficients, the number of terms $A_{\\ell}$ in the expansion, and with or without the operator matrices $\\Ohat_{\\ell}^{(1)},\\ldots,\\Ohat_{\\ell}^{(A_{\\ell})}$."
    ]
   },
   {

docs/source/tutorials/basics.ipynb

@@ -79,7 +79,6 @@
    "metadata": {},
    "outputs": [],
    "source": [
-    "import time\n",
     "import numpy as np\n",
     "import pandas as pd\n",
     "import scipy.sparse\n",

docs/source/tutorials/inputs.ipynb

@@ -99,7 +99,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "In this section a ROM is trained using data collected for a single choice of the input function $u(t)$."
+    "In this section a ROM is trained using data collected for a single choice of the input function $u(t).$"
    ]
   },
   {
@@ -113,7 +113,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "As in the last tutorial, we use a centered finite difference approximation for the spatial derivative to arrive at a finite-dimensional system of ordinary differential equations.\n",
+    "As in the last tutorial, we use a centered finite difference approximation for the spatial derivative to arrive at a system of $n$ ordinary differential equations.\n",
     "This time, due to the nonzero boundary conditions, the system takes the form\n",
     "\n",
     "$$\n",
@@ -124,7 +124,7 @@
     "\\end{aligned}\n",
     "$$ (eq_inputs_fom)\n",
     "\n",
-    "where $\\A\\in\\RR^{n\\times n}$ and $\\B\\in\\RR^{n}$.\n",
+    "where $\\q:\\RR\\to\\RR^n$, $\\A\\in\\RR^{n\\times n}$, and $\\B\\in\\RR^{n}$.\n",
     "The system {eq}`eq_inputs_fom` is the _full-order model_ (FOM), which we will use to generate training data for the time domain $[0, T'] \\subset [0, T]$."
    ]
   },
@@ -226,7 +226,7 @@
     "\n",
     "# Construct the temporal domain.\n",
     "T = 1\n",
-    "K = T * 10**3 + 1\n",
+    "K = 10**3 + 1\n",
     "t_all = np.linspace(0, T, K)\n",
     "dt = t_all[1] - t_all[0]\n",
     "\n",