diff --git a/01-g-h-filter.ipynb b/01-g-h-filter.ipynb
index 51f3bb7..e59bb37 100644
--- a/01-g-h-filter.ipynb
+++ b/01-g-h-filter.ipynb
@@ -798,7 +798,7 @@
     "weights = [158.0, 164.2, 160.3, 159.9, 162.1, 164.6, \n",
     "           169.6, 167.4, 166.4, 171.0, 171.2, 172.6]\n",
     "\n",
-    "time_step = 1.0 # day\n",
+    "time_step = 1.0  # day\n",
     "scale_factor = 4.0/10\n",
     "\n",
     "def predict_using_gain_guess(weight, gain_rate, do_print=True, sim_rate=0): \n",
@@ -824,6 +824,7 @@
     "\n",
     "        # plot results\n",
     "        gh.plot_gh_results(weights, estimates, predictions, sim_rate)\n",
+    "\n",
     "initial_guess = 160.\n",
     "predict_using_gain_guess(weight=initial_guess, gain_rate=1)     "
    ]
@@ -2482,7 +2483,7 @@
    "source": [
     "import time\n",
     "with interactive_plot():\n",
-    "    for x in range(2,6):\n",
+    "    for x in range(2, 6):\n",
     "        plt.plot([0, 1], [1, x])\n",
     "        plt.gcf().canvas.draw()\n",
     "        time.sleep(0.5)"
@@ -3337,7 +3338,7 @@
    ],
    "source": [
     "weight = 160.  # initial guess\n",
-    "gain_rate = -1.0 # initial guess\n",
+    "gain_rate = -1.0  # initial guess\n",
     "\n",
     "time_step = 1.\n",
     "weight_scale = 4./10\n",
@@ -3373,7 +3374,7 @@
     "```python\n",
     "gain_rate = gain_rate\n",
     "``` \n",
-    "This obviously has no effect, and can be removed. I wrote this to emphasize that in the prediction step you need to predict next value for all variables, both `weight` and `gain_rate`. In this case we are assuming that the gain does not vary, but when we generalize this algorithm we will remove that assumption. "
+    "This obviously has no effect, and can be removed. I wrote this to emphasize that in the prediction step you need to predict the next value for all variables, both `weight` and `gain_rate`. In this case we are assuming that the gain does not vary, but when we generalize this algorithm we will remove that assumption. "
    ]
   },
   {
@@ -3957,15 +3958,15 @@
     "    x_est = x0\n",
     "    results = []\n",
     "    for z in data:\n",
-    "        #prediction step\n",
+    "        # prediction step\n",
     "        x_pred = x_est + (dx*dt)\n",
     "        dx = dx\n",
     "\n",
     "        # update step\n",
     "        residual = z - x_pred\n",
-    "        dx = dx    + h * (residual) / dt\n",
-    "        x_est  = x_pred + g * residual     \n",
-    "        results.append(x_est)  \n",
+    "        dx = dx + h * (residual) / dt\n",
+    "        x_est = x_pred + g * residual\n",
+    "        results.append(x_est)\n",
     "    return np.array(results)\n",
     "\n",
     "book_plots.plot_track([0, 11], [160, 172], label='Actual weight')\n",
@@ -3977,7 +3978,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Choice of g and h"
+    "## Choice of $g$ and $h$"
    ]
   },
   {
@@ -4283,14 +4284,14 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Exercise: Varying g"
+    "## Exercise: Varying $g$"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Now let's look at the effect of varying g. Before you perform this exercise, recall that g is the scale factor for choosing between the measurement and prediction. What do you think the effect of a large value of g will be? A small value? \n",
+    "Now let's look at the effect of varying $g$. Before you perform this exercise, recall that $g$ is the scale factor for choosing between the measurement and prediction. What do you think the effect of a large value of $g$ will be? A small value?\n",
     "\n",
     "Now, let the `noise_factor=50` and `dx=5`. Plot the results of $g = 0.1\\mbox{, } 0.4,\\mbox{ and } 0.8$."
    ]
@@ -4344,7 +4345,7 @@
     "    book_plots.plot_filter(data2, label='g=0.4', marker='v', c='C1')\n",
     "    book_plots.plot_filter(data3, label='g=0.8', c='C2')\n",
     "    plt.legend(loc=4)\n",
-    "    book_plots.set_limits([20,40], [50, 250])"
+    "    book_plots.set_limits([20, 40], [50, 250])"
    ]
   },
   {
@@ -4393,7 +4394,7 @@
    "source": [
     "Here we can see the effects of ignoring the signal. We not only filter out noise, but legitimate changes in the signal as well. \n",
     "\n",
-    "Maybe we need a 'Goldilocks' filter, where is not too large, not too small, but just right? Well, not exactly. As alluded to earlier, different filters choose g and h in different ways depending on the mathematical properties of the problem. For example, the Benedict-Bordner filter was invented to minimize the transient error in this example, where $\\dot{x}$ makes a step jump. We will not discuss this filter in this book, but here are two plots chosen with different allowable pairs of g and h. This filter design minimizes transient errors for step jumps in $\\dot{x}$ at the cost of not being optimal for other types of changes in $\\dot{x}$."
+    "Maybe we need a 'Goldilocks' filter, where $g$ is not too large, not too small, but just right? Well, not exactly. As alluded to earlier, different filters choose $g$ and $h$ in different ways depending on the mathematical properties of the problem. For example, the Benedict-Bordner filter was invented to minimize the transient error in this example, where $\\dot{x}$ makes a step jump. We will not discuss this filter in this book, but here are two plots chosen with different allowable pairs of $g$ and $h$. This filter design minimizes transient errors for step jumps in $\\dot{x}$ at the cost of not being optimal for other types of changes in $\\dot{x}$."
    ]
   },
   {
@@ -4429,14 +4430,14 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Varying h"
+    "## Varying $h$"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Now let's leave g unchanged and investigate the effect of modifying h. We know that h affects how much we favor the measurement of $\\dot{x}$ vs our prediction. But what does this *mean*? If our signal is changing a lot (quickly relative to the time step of our filter), then a large $h$ will cause us to react to those transient changes rapidly. A smaller $h$ will cause us to react more slowly.\n",
+    "Now let's leave $g$ unchanged and investigate the effect of modifying $h$. We know that $h$ affects how much we favor the measurement of $\\dot{x}$ vs our prediction. But what does this *mean*? If our signal is changing a lot (quickly relative to the time step of our filter), then a large $h$ will cause us to react to those transient changes rapidly. A smaller $h$ will cause us to react more slowly.\n",
     "\n",
     "We will look at three examples. We have a noiseless measurement that slowly goes from 0 to 1 in 50 steps. Our first filter uses a nearly correct initial value for $\\dot{x}$ and a small $h$. You can see from the output that the filter output is very close to the signal. The second filter uses the very incorrect guess of $\\dot{x}=2$. Here we see the filter 'ringing' until it settles down and finds the signal. The third filter uses the same conditions but it now sets $h=0.5$. If you look at the amplitude of the ringing you can see that it is much smaller than in the second chart, but the frequency is greater. It also settles down a bit quicker than the second filter, though not by much."
    ]