diff --git a/doc/pub/week43/html/week43-bs.html b/doc/pub/week43/html/week43-bs.html
index e5f8d9fd6..0b593b5bf 100644
--- a/doc/pub/week43/html/week43-bs.html
+++ b/doc/pub/week43/html/week43-bs.html
@@ -1458,10 +1458,11 @@ predicted<span style="color: #666666">=</span>np<span style="color: #666666">.</
 trainScore <span style="color: #666666">=</span> model<span style="color: #666666">.</span>evaluate(trainX, trainY, verbose<span style="color: #666666">=0</span>)
 <span style="color: #008000">print</span>(trainScore)
 
-index <span style="color: #666666">=</span> df<span style="color: #666666">.</span>index<span style="color: #666666">.</span>values
-plt<span style="color: #666666">.</span>plot(index,df)
-plt<span style="color: #666666">.</span>plot(index,predicted)
-plt<span style="color: #666666">.</span>axvline(df<span style="color: #666666">.</span>index[Tp], c<span style="color: #666666">=</span><span style="color: #BA2121">&quot;r&quot;</span>)
+
+df <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(x)
+pred <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(predicted)
+plt<span style="color: #666666">.</span>plot(df,c<span style="color: #666666">=</span><span style="color: #BA2121">&quot;b&quot;</span>)
+plt<span style="color: #666666">.</span>plot(pred,c<span style="color: #666666">=</span><span style="color: #BA2121">&quot;r&quot;</span>)
 plt<span style="color: #666666">.</span>show()
 </pre>
 </div>
@@ -1544,24 +1545,24 @@ y_tot <span style="color: #666666">=</span> np<span style="color: #666666">.</sp
 <p>The way the recurrent neural networks are trained in this program
 differs from how machine learning algorithms are usually trained.
 Typically a machine learning algorithm is trained by learning the
-relationship between the x data and the y data.  In this program, the
+relationship between the \( x \) data and the \( y \) data.  In this program, the
 recurrent neural network will be trained to recognize the relationship
-in a sequence of y values.  This is type of data formatting is
-typically used time series forcasting, but it can also be used in any
+in a sequence of \( y \) values.  This is type of data formatting is
+typically used for time series forecasting, but it can also be used in any
 extrapolation (time series forecasting is just a specific type of
 extrapolation along the time axis).  This method of data formatting
-does not use the x data and assumes that the y data are evenly spaced.
+does not use the \( x \) data and assumes that the \( y \) data are evenly spaced.
 </p>
 
 <p>For a standard machine learning algorithm, the training data has the
-form of (x,y) so the machine learning algorithm learns to assiciate a
-y value with a given x value.  This is useful when the test data has x
+form of \( (x,y) \) so the machine learning algorithm learns to associate a
+\( y \) value with a given \( x \) value.  This is useful when the test data has \( x \)
 values within the same range as the training data.  However, for this
 application, the x values of the test data are outside of the x values
 of the training data and the traditional method of training a machine
 learning algorithm does not work as well.  For this reason, the
 recurrent neural network is trained on sequences of y values of the
-form ((y1, y2), y3), so that the network is concerned with learning
+form \( ((y1, y2), y3) \), so that the network is concerned with learning
 the pattern of the y data and not the relation between the x and y
 data.  As long as the pattern of y data outside of the training region
 stays relatively stable compared to what was inside the training
diff --git a/doc/pub/week43/html/week43-reveal.html b/doc/pub/week43/html/week43-reveal.html
index f91554b01..39bb53990 100644
--- a/doc/pub/week43/html/week43-reveal.html
+++ b/doc/pub/week43/html/week43-reveal.html
@@ -1417,10 +1417,11 @@ predicted=np.concatenate((trainPredict,testPredict),axis=<span style="color: #B4
 trainScore = model.evaluate(trainX, trainY, verbose=<span style="color: #B452CD">0</span>)
 <span style="color: #658b00">print</span>(trainScore)
 
-index = df.index.values
-plt.plot(index,df)
-plt.plot(index,predicted)
-plt.axvline(df.index[Tp], c=<span style="color: #CD5555">&quot;r&quot;</span>)
+
+df = pd.DataFrame(x)
+pred = pd.DataFrame(predicted)
+plt.plot(df,c=<span style="color: #CD5555">&quot;b&quot;</span>)
+plt.plot(pred,c=<span style="color: #CD5555">&quot;r&quot;</span>)
 plt.show()
 </pre>
 </div>
@@ -1503,24 +1504,24 @@ y_tot = np.array([-<span style="color: #B452CD">0.03077640549</span>, -<span sty
 <p>The way the recurrent neural networks are trained in this program
 differs from how machine learning algorithms are usually trained.
 Typically a machine learning algorithm is trained by learning the
-relationship between the x data and the y data.  In this program, the
+relationship between the \( x \) data and the \( y \) data.  In this program, the
 recurrent neural network will be trained to recognize the relationship
-in a sequence of y values.  This is type of data formatting is
-typically used time series forcasting, but it can also be used in any
+in a sequence of \( y \) values.  This is type of data formatting is
+typically used for time series forecasting, but it can also be used in any
 extrapolation (time series forecasting is just a specific type of
 extrapolation along the time axis).  This method of data formatting
-does not use the x data and assumes that the y data are evenly spaced.
+does not use the \( x \) data and assumes that the \( y \) data are evenly spaced.
 </p>
 
 <p>For a standard machine learning algorithm, the training data has the
-form of (x,y) so the machine learning algorithm learns to assiciate a
-y value with a given x value.  This is useful when the test data has x
+form of \( (x,y) \) so the machine learning algorithm learns to associate a
+\( y \) value with a given \( x \) value.  This is useful when the test data has \( x \)
 values within the same range as the training data.  However, for this
 application, the x values of the test data are outside of the x values
 of the training data and the traditional method of training a machine
 learning algorithm does not work as well.  For this reason, the
 recurrent neural network is trained on sequences of y values of the
-form ((y1, y2), y3), so that the network is concerned with learning
+form \( ((y1, y2), y3) \), so that the network is concerned with learning
 the pattern of the y data and not the relation between the x and y
 data.  As long as the pattern of y data outside of the training region
 stays relatively stable compared to what was inside the training
diff --git a/doc/pub/week43/html/week43-solarized.html b/doc/pub/week43/html/week43-solarized.html
index c6440f2f5..1b6ae0059 100644
--- a/doc/pub/week43/html/week43-solarized.html
+++ b/doc/pub/week43/html/week43-solarized.html
@@ -1393,10 +1393,11 @@ predicted=np.concatenate((trainPredict,testPredict),axis=<span style="color: #B4
 trainScore = model.evaluate(trainX, trainY, verbose=<span style="color: #B452CD">0</span>)
 <span style="color: #658b00">print</span>(trainScore)
 
-index = df.index.values
-plt.plot(index,df)
-plt.plot(index,predicted)
-plt.axvline(df.index[Tp], c=<span style="color: #CD5555">&quot;r&quot;</span>)
+
+df = pd.DataFrame(x)
+pred = pd.DataFrame(predicted)
+plt.plot(df,c=<span style="color: #CD5555">&quot;b&quot;</span>)
+plt.plot(pred,c=<span style="color: #CD5555">&quot;r&quot;</span>)
 plt.show()
 </pre>
 </div>
@@ -1479,24 +1480,24 @@ y_tot = np.array([-<span style="color: #B452CD">0.03077640549</span>, -<span sty
 <p>The way the recurrent neural networks are trained in this program
 differs from how machine learning algorithms are usually trained.
 Typically a machine learning algorithm is trained by learning the
-relationship between the x data and the y data.  In this program, the
+relationship between the \( x \) data and the \( y \) data.  In this program, the
 recurrent neural network will be trained to recognize the relationship
-in a sequence of y values.  This is type of data formatting is
-typically used time series forcasting, but it can also be used in any
+in a sequence of \( y \) values.  This is type of data formatting is
+typically used for time series forecasting, but it can also be used in any
 extrapolation (time series forecasting is just a specific type of
 extrapolation along the time axis).  This method of data formatting
-does not use the x data and assumes that the y data are evenly spaced.
+does not use the \( x \) data and assumes that the \( y \) data are evenly spaced.
 </p>
 
 <p>For a standard machine learning algorithm, the training data has the
-form of (x,y) so the machine learning algorithm learns to assiciate a
-y value with a given x value.  This is useful when the test data has x
+form of \( (x,y) \) so the machine learning algorithm learns to associate a
+\( y \) value with a given \( x \) value.  This is useful when the test data has \( x \)
 values within the same range as the training data.  However, for this
 application, the x values of the test data are outside of the x values
 of the training data and the traditional method of training a machine
 learning algorithm does not work as well.  For this reason, the
 recurrent neural network is trained on sequences of y values of the
-form ((y1, y2), y3), so that the network is concerned with learning
+form \( ((y1, y2), y3) \), so that the network is concerned with learning
 the pattern of the y data and not the relation between the x and y
 data.  As long as the pattern of y data outside of the training region
 stays relatively stable compared to what was inside the training
diff --git a/doc/pub/week43/html/week43.html b/doc/pub/week43/html/week43.html
index 4efcd90a5..1d41c36b0 100644
--- a/doc/pub/week43/html/week43.html
+++ b/doc/pub/week43/html/week43.html
@@ -1470,10 +1470,11 @@ predicted<span style="color: #666666">=</span>np<span style="color: #666666">.</
 trainScore <span style="color: #666666">=</span> model<span style="color: #666666">.</span>evaluate(trainX, trainY, verbose<span style="color: #666666">=0</span>)
 <span style="color: #008000">print</span>(trainScore)
 
-index <span style="color: #666666">=</span> df<span style="color: #666666">.</span>index<span style="color: #666666">.</span>values
-plt<span style="color: #666666">.</span>plot(index,df)
-plt<span style="color: #666666">.</span>plot(index,predicted)
-plt<span style="color: #666666">.</span>axvline(df<span style="color: #666666">.</span>index[Tp], c<span style="color: #666666">=</span><span style="color: #BA2121">&quot;r&quot;</span>)
+
+df <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(x)
+pred <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(predicted)
+plt<span style="color: #666666">.</span>plot(df,c<span style="color: #666666">=</span><span style="color: #BA2121">&quot;b&quot;</span>)
+plt<span style="color: #666666">.</span>plot(pred,c<span style="color: #666666">=</span><span style="color: #BA2121">&quot;r&quot;</span>)
 plt<span style="color: #666666">.</span>show()
 </pre>
 </div>
@@ -1556,24 +1557,24 @@ y_tot <span style="color: #666666">=</span> np<span style="color: #666666">.</sp
 <p>The way the recurrent neural networks are trained in this program
 differs from how machine learning algorithms are usually trained.
 Typically a machine learning algorithm is trained by learning the
-relationship between the x data and the y data.  In this program, the
+relationship between the \( x \) data and the \( y \) data.  In this program, the
 recurrent neural network will be trained to recognize the relationship
-in a sequence of y values.  This is type of data formatting is
-typically used time series forcasting, but it can also be used in any
+in a sequence of \( y \) values.  This is type of data formatting is
+typically used for time series forecasting, but it can also be used in any
 extrapolation (time series forecasting is just a specific type of
 extrapolation along the time axis).  This method of data formatting
-does not use the x data and assumes that the y data are evenly spaced.
+does not use the \( x \) data and assumes that the \( y \) data are evenly spaced.
 </p>
 
 <p>For a standard machine learning algorithm, the training data has the
-form of (x,y) so the machine learning algorithm learns to assiciate a
-y value with a given x value.  This is useful when the test data has x
+form of \( (x,y) \) so the machine learning algorithm learns to associate a
+\( y \) value with a given \( x \) value.  This is useful when the test data has \( x \)
 values within the same range as the training data.  However, for this
 application, the x values of the test data are outside of the x values
 of the training data and the traditional method of training a machine
 learning algorithm does not work as well.  For this reason, the
 recurrent neural network is trained on sequences of y values of the
-form ((y1, y2), y3), so that the network is concerned with learning
+form \( ((y1, y2), y3) \), so that the network is concerned with learning
 the pattern of the y data and not the relation between the x and y
 data.  As long as the pattern of y data outside of the training region
 stays relatively stable compared to what was inside the training
diff --git a/doc/pub/week43/ipynb/ipynb-week43-src.tar.gz b/doc/pub/week43/ipynb/ipynb-week43-src.tar.gz
index 38921dd10..e21ce151e 100644
Binary files a/doc/pub/week43/ipynb/ipynb-week43-src.tar.gz and b/doc/pub/week43/ipynb/ipynb-week43-src.tar.gz differ
diff --git a/doc/pub/week43/ipynb/week43.ipynb b/doc/pub/week43/ipynb/week43.ipynb
index d8df74187..28ecc2260 100644
--- a/doc/pub/week43/ipynb/week43.ipynb
+++ b/doc/pub/week43/ipynb/week43.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "id": "c5d39e0b",
+   "id": "015c71dd",
    "metadata": {
     "editable": true
    },
@@ -14,7 +14,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6459c579",
+   "id": "f6766299",
    "metadata": {
     "editable": true
    },
@@ -29,7 +29,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "70ebd784",
+   "id": "709ed995",
    "metadata": {
     "editable": true
    },
@@ -59,7 +59,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ad740247",
+   "id": "1e56ca1d",
    "metadata": {
     "editable": true
    },
@@ -85,7 +85,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9293049f",
+   "id": "fc35bfe0",
    "metadata": {
     "editable": true
    },
@@ -112,7 +112,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "67a1fb47",
+   "id": "6f622ae2",
    "metadata": {
     "editable": true
    },
@@ -134,7 +134,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5e4646d6",
+   "id": "52600837",
    "metadata": {
     "editable": true
    },
@@ -152,7 +152,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9a829df6",
+   "id": "df1d6c72",
    "metadata": {
     "editable": true
    },
@@ -182,7 +182,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3e53ba7c",
+   "id": "dc7cac96",
    "metadata": {
     "editable": true
    },
@@ -212,7 +212,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "09e08f91",
+   "id": "04e87f6b",
    "metadata": {
     "editable": true
    },
@@ -252,7 +252,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7c8a5acf",
+   "id": "929941f5",
    "metadata": {
     "editable": true
    },
@@ -281,7 +281,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "12071d8a",
+   "id": "ed9841e7",
    "metadata": {
     "editable": true
    },
@@ -303,7 +303,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c1faf1e4",
+   "id": "4830435d",
    "metadata": {
     "editable": true
    },
@@ -332,7 +332,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "39016bcb",
+   "id": "36474e75",
    "metadata": {
     "editable": true
    },
@@ -349,7 +349,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "945544d3",
+   "id": "f103c962",
    "metadata": {
     "editable": true
    },
@@ -369,7 +369,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "48a2b80c",
+   "id": "f02729ce",
    "metadata": {
     "editable": true
    },
@@ -381,7 +381,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cfd14636",
+   "id": "06532fa8",
    "metadata": {
     "editable": true
    },
@@ -393,7 +393,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "66afa266",
+   "id": "8c39c4cf",
    "metadata": {
     "editable": true
    },
@@ -405,7 +405,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "96f87f9a",
+   "id": "09f8b99c",
    "metadata": {
     "editable": true
    },
@@ -415,7 +415,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f070283e",
+   "id": "ca999e60",
    "metadata": {
     "editable": true
    },
@@ -427,7 +427,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9d3db171",
+   "id": "59cf79b2",
    "metadata": {
     "editable": true
    },
@@ -439,7 +439,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "33f6af14",
+   "id": "d12c26ff",
    "metadata": {
     "editable": true
    },
@@ -454,7 +454,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "02a99cff",
+   "id": "ca17291c",
    "metadata": {
     "editable": true
    },
@@ -466,7 +466,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2c6a4e0c",
+   "id": "a58c56d5",
    "metadata": {
     "editable": true
    },
@@ -476,7 +476,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "08df12bf",
+   "id": "aed190d3",
    "metadata": {
     "editable": true
    },
@@ -488,7 +488,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2d082fe8",
+   "id": "8845e923",
    "metadata": {
     "editable": true
    },
@@ -498,7 +498,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "19f66253",
+   "id": "f1499a4a",
    "metadata": {
     "editable": true
    },
@@ -510,7 +510,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "06e842d7",
+   "id": "cba9bbaf",
    "metadata": {
     "editable": true
    },
@@ -523,7 +523,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "60a4a134",
+   "id": "5aa1c718",
    "metadata": {
     "editable": true
    },
@@ -542,7 +542,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "17cb3b7c",
+   "id": "5123c6a7",
    "metadata": {
     "editable": true
    },
@@ -554,7 +554,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5c318d22",
+   "id": "65073f67",
    "metadata": {
     "editable": true
    },
@@ -566,7 +566,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "35ec45df",
+   "id": "f28c326d",
    "metadata": {
     "editable": true
    },
@@ -576,7 +576,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2b9c44d7",
+   "id": "94999d37",
    "metadata": {
     "editable": true
    },
@@ -588,7 +588,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "70a7d1dc",
+   "id": "3075e47e",
    "metadata": {
     "editable": true
    },
@@ -598,7 +598,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a29e240c",
+   "id": "2aa3debf",
    "metadata": {
     "editable": true
    },
@@ -612,7 +612,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bf4a8589",
+   "id": "4caa7831",
    "metadata": {
     "editable": true
    },
@@ -630,7 +630,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "325858c8",
+   "id": "06f850ab",
    "metadata": {
     "editable": true
    },
@@ -641,7 +641,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "dad45e98",
+   "id": "255aafbb",
    "metadata": {
     "editable": true
    },
@@ -659,7 +659,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "633095c0",
+   "id": "3044a689",
    "metadata": {
     "editable": true
    },
@@ -673,7 +673,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0499b303",
+   "id": "178b17ce",
    "metadata": {
     "editable": true
    },
@@ -687,7 +687,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9f7145ad",
+   "id": "850e9d3b",
    "metadata": {
     "editable": true
    },
@@ -699,7 +699,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c8f1867e",
+   "id": "ffbcd60b",
    "metadata": {
     "editable": true
    },
@@ -709,7 +709,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c86dd048",
+   "id": "4bf03246",
    "metadata": {
     "editable": true
    },
@@ -721,7 +721,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "428656bc",
+   "id": "93e29686",
    "metadata": {
     "editable": true
    },
@@ -731,7 +731,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c70e9d1e",
+   "id": "307c5715",
    "metadata": {
     "editable": true
    },
@@ -743,7 +743,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7b9b0f19",
+   "id": "221ee4d7",
    "metadata": {
     "editable": true
    },
@@ -755,7 +755,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ddc85d9b",
+   "id": "c76cfb13",
    "metadata": {
     "editable": true
    },
@@ -782,7 +782,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "004f94f5",
+   "id": "b0887074",
    "metadata": {
     "editable": true
    },
@@ -794,7 +794,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ecbfa56d",
+   "id": "1ac4e0cc",
    "metadata": {
     "editable": true
    },
@@ -804,7 +804,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ac1647b2",
+   "id": "4d7d6a58",
    "metadata": {
     "editable": true
    },
@@ -830,7 +830,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d43e606a",
+   "id": "1f566b66",
    "metadata": {
     "editable": true
    },
@@ -842,7 +842,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8cce1a2a",
+   "id": "79c5d13d",
    "metadata": {
     "editable": true
    },
@@ -861,7 +861,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c44045c5",
+   "id": "ad7cd27d",
    "metadata": {
     "editable": true
    },
@@ -874,7 +874,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "59a365da",
+   "id": "a281e971",
    "metadata": {
     "editable": true
    },
@@ -886,7 +886,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "34ae8a74",
+   "id": "c9c48d1f",
    "metadata": {
     "editable": true
    },
@@ -907,7 +907,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c67263a9",
+   "id": "197bbf2c",
    "metadata": {
     "editable": true
    },
@@ -928,7 +928,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "795378a9",
+   "id": "36f5a961",
    "metadata": {
     "editable": true
    },
@@ -953,7 +953,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "65b4ac52",
+   "id": "8435603c",
    "metadata": {
     "editable": true
    },
@@ -972,7 +972,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "caf13811",
+   "id": "acbb9c32",
    "metadata": {
     "editable": true
    },
@@ -983,7 +983,7 @@
   {
    "cell_type": "code",
    "execution_count": 1,
-   "id": "f8a921f2",
+   "id": "9c489f64",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1036,7 +1036,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "642845d2",
+   "id": "eb5ef469",
    "metadata": {
     "editable": true
    },
@@ -1047,7 +1047,7 @@
   {
    "cell_type": "code",
    "execution_count": 2,
-   "id": "615d9546",
+   "id": "0b887229",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1080,7 +1080,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "878a352e",
+   "id": "469f5d63",
    "metadata": {
     "editable": true
    },
@@ -1091,7 +1091,7 @@
   {
    "cell_type": "code",
    "execution_count": 3,
-   "id": "05878dc8",
+   "id": "71260f77",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1128,7 +1128,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "538e069f",
+   "id": "ca362756",
    "metadata": {
     "editable": true
    },
@@ -1139,7 +1139,7 @@
   {
    "cell_type": "code",
    "execution_count": 4,
-   "id": "4c91e0c7",
+   "id": "6b40bb7e",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1166,7 +1166,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4708fdf8",
+   "id": "d1b6d608",
    "metadata": {
     "editable": true
    },
@@ -1177,7 +1177,7 @@
   {
    "cell_type": "code",
    "execution_count": 5,
-   "id": "dc871d8f",
+   "id": "5932ad38",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1218,7 +1218,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4cc87ad5",
+   "id": "3a4c3443",
    "metadata": {
     "editable": true
    },
@@ -1234,7 +1234,7 @@
   {
    "cell_type": "code",
    "execution_count": 6,
-   "id": "6a075034",
+   "id": "97da0d0d",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1255,7 +1255,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b48d3436",
+   "id": "8bf7aa7d",
    "metadata": {
     "editable": true
    },
@@ -1268,7 +1268,7 @@
   {
    "cell_type": "code",
    "execution_count": 7,
-   "id": "82256673",
+   "id": "c534f0c4",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1292,7 +1292,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2de5f864",
+   "id": "02e8c931",
    "metadata": {
     "editable": true
    },
@@ -1307,7 +1307,7 @@
   {
    "cell_type": "code",
    "execution_count": 8,
-   "id": "34cd91dc",
+   "id": "4c9fe84a",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1328,7 +1328,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c51270ac",
+   "id": "d52c1474",
    "metadata": {
     "editable": true
    },
@@ -1338,7 +1338,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f7680dfc",
+   "id": "03873aca",
    "metadata": {
     "editable": true
    },
@@ -1357,7 +1357,7 @@
   {
    "cell_type": "code",
    "execution_count": 9,
-   "id": "b923f7c4",
+   "id": "ea66d800",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1373,7 +1373,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8107d910",
+   "id": "0d8b6bb4",
    "metadata": {
     "editable": true
    },
@@ -1383,7 +1383,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d6436c68",
+   "id": "f2aaa69f",
    "metadata": {
     "editable": true
    },
@@ -1394,7 +1394,7 @@
   {
    "cell_type": "code",
    "execution_count": 10,
-   "id": "6b1d461a",
+   "id": "b52ca319",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1410,7 +1410,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a9853275",
+   "id": "933d427a",
    "metadata": {
     "editable": true
    },
@@ -1421,7 +1421,7 @@
   {
    "cell_type": "code",
    "execution_count": 11,
-   "id": "3641ff80",
+   "id": "935e7588",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1440,7 +1440,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4e8bb911",
+   "id": "03da3848",
    "metadata": {
     "editable": true
    },
@@ -1467,7 +1467,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d10dd84c",
+   "id": "b0716086",
    "metadata": {
     "editable": true
    },
@@ -1479,7 +1479,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6791aa19",
+   "id": "99c49087",
    "metadata": {
     "editable": true
    },
@@ -1490,7 +1490,7 @@
   {
    "cell_type": "code",
    "execution_count": 12,
-   "id": "1a9bea34",
+   "id": "4898244d",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1557,16 +1557,17 @@
     "trainScore = model.evaluate(trainX, trainY, verbose=0)\n",
     "print(trainScore)\n",
     "\n",
-    "index = df.index.values\n",
-    "plt.plot(index,df)\n",
-    "plt.plot(index,predicted)\n",
-    "plt.axvline(df.index[Tp], c=\"r\")\n",
+    "\n",
+    "df = pd.DataFrame(x)\n",
+    "pred = pd.DataFrame(predicted)\n",
+    "plt.plot(df,c=\"b\")\n",
+    "plt.plot(pred,c=\"r\")\n",
     "plt.show()"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "7e222e09",
+   "id": "2829f486",
    "metadata": {
     "editable": true
    },
@@ -1582,7 +1583,7 @@
   {
    "cell_type": "code",
    "execution_count": 13,
-   "id": "fa1b4b28",
+   "id": "23baff06",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1621,7 +1622,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c6237960",
+   "id": "1b32a005",
    "metadata": {
     "editable": true
    },
@@ -1631,23 +1632,23 @@
     "The way the recurrent neural networks are trained in this program\n",
     "differs from how machine learning algorithms are usually trained.\n",
     "Typically a machine learning algorithm is trained by learning the\n",
-    "relationship between the x data and the y data.  In this program, the\n",
+    "relationship between the $x$ data and the $y$ data.  In this program, the\n",
     "recurrent neural network will be trained to recognize the relationship\n",
-    "in a sequence of y values.  This is type of data formatting is\n",
-    "typically used time series forcasting, but it can also be used in any\n",
+    "in a sequence of $y$ values.  This is type of data formatting is\n",
+    "typically used for time series forecasting, but it can also be used in any\n",
     "extrapolation (time series forecasting is just a specific type of\n",
     "extrapolation along the time axis).  This method of data formatting\n",
-    "does not use the x data and assumes that the y data are evenly spaced.\n",
+    "does not use the $x$ data and assumes that the $y$ data are evenly spaced.\n",
     "\n",
     "For a standard machine learning algorithm, the training data has the\n",
-    "form of (x,y) so the machine learning algorithm learns to assiciate a\n",
-    "y value with a given x value.  This is useful when the test data has x\n",
+    "form of $(x,y)$ so the machine learning algorithm learns to associate a\n",
+    "$y$ value with a given $x$ value.  This is useful when the test data has $x$\n",
     "values within the same range as the training data.  However, for this\n",
     "application, the x values of the test data are outside of the x values\n",
     "of the training data and the traditional method of training a machine\n",
     "learning algorithm does not work as well.  For this reason, the\n",
     "recurrent neural network is trained on sequences of y values of the\n",
-    "form ((y1, y2), y3), so that the network is concerned with learning\n",
+    "form $((y1, y2), y3)$, so that the network is concerned with learning\n",
     "the pattern of the y data and not the relation between the x and y\n",
     "data.  As long as the pattern of y data outside of the training region\n",
     "stays relatively stable compared to what was inside the training\n",
@@ -1665,7 +1666,7 @@
   {
    "cell_type": "code",
    "execution_count": 14,
-   "id": "777176d4",
+   "id": "815087a3",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1748,7 +1749,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e5b3cd18",
+   "id": "0f248200",
    "metadata": {
     "editable": true
    },
@@ -1759,7 +1760,7 @@
   {
    "cell_type": "code",
    "execution_count": 15,
-   "id": "8bd0ef49",
+   "id": "2ff698a1",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1861,7 +1862,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ca50ecee",
+   "id": "f745c9eb",
    "metadata": {
     "editable": true
    },
@@ -1883,7 +1884,7 @@
   {
    "cell_type": "code",
    "execution_count": 16,
-   "id": "0f56a065",
+   "id": "b2464d8e",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1980,7 +1981,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "231486ad",
+   "id": "950bc163",
    "metadata": {
     "editable": true
    },
@@ -2006,7 +2007,7 @@
   {
    "cell_type": "code",
    "execution_count": 17,
-   "id": "346c01ec",
+   "id": "0c41fb42",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2207,7 +2208,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1e855715",
+   "id": "2088e26d",
    "metadata": {
     "editable": true
    },
@@ -2231,7 +2232,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "74c691f8",
+   "id": "ce21c422",
    "metadata": {
     "editable": true
    },
@@ -2251,7 +2252,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "20b0f4e4",
+   "id": "14ba0435",
    "metadata": {
     "editable": true
    },
@@ -2269,7 +2270,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "60295fd3",
+   "id": "9359ee17",
    "metadata": {
     "editable": true
    },
@@ -2284,7 +2285,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "06f276d9",
+   "id": "056aade8",
    "metadata": {
     "editable": true
    },
@@ -2302,7 +2303,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ac42c4db",
+   "id": "54e238a3",
    "metadata": {
     "editable": true
    },
@@ -2315,7 +2316,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "04c1b0fe",
+   "id": "0a67f20d",
    "metadata": {
     "editable": true
    },
@@ -2333,7 +2334,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "91abaad4",
+   "id": "9a944e06",
    "metadata": {
     "editable": true
    },
@@ -2344,7 +2345,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "18714b71",
+   "id": "d0bc68db",
    "metadata": {
     "editable": true
    },
@@ -2362,7 +2363,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6b83109a",
+   "id": "89f93ab0",
    "metadata": {
     "editable": true
    },
@@ -2389,7 +2390,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f4623825",
+   "id": "8dd6eb78",
    "metadata": {
     "editable": true
    },
@@ -2401,7 +2402,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "97e20104",
+   "id": "b459ce4f",
    "metadata": {
     "editable": true
    },
@@ -2420,7 +2421,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a11dc809",
+   "id": "d11f3e23",
    "metadata": {
     "editable": true
    },
@@ -2430,7 +2431,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4e1c6fc8",
+   "id": "647e4631",
    "metadata": {
     "editable": true
    },
@@ -2450,7 +2451,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6f811ebb",
+   "id": "1a161dc2",
    "metadata": {
     "editable": true
    },
@@ -2462,7 +2463,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "609dbebf",
+   "id": "46db414f",
    "metadata": {
     "editable": true
    },
@@ -2480,7 +2481,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b8e9dc97",
+   "id": "0e1bd4d2",
    "metadata": {
     "editable": true
    },
@@ -2492,7 +2493,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "50adaf2b",
+   "id": "4488b9c5",
    "metadata": {
     "editable": true
    },
@@ -2513,7 +2514,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f7f89a95",
+   "id": "6c7b3ea3",
    "metadata": {
     "editable": true
    },
@@ -2530,7 +2531,7 @@
   {
    "cell_type": "code",
    "execution_count": 18,
-   "id": "e9dae945",
+   "id": "844a2f34",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2548,7 +2549,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8469c1ae",
+   "id": "f999ed2a",
    "metadata": {
     "editable": true
    },
@@ -2559,7 +2560,7 @@
   {
    "cell_type": "code",
    "execution_count": 19,
-   "id": "4f06f394",
+   "id": "1de68c0c",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2585,7 +2586,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9c5cd3cf",
+   "id": "f4e25799",
    "metadata": {
     "editable": true
    },
@@ -2598,7 +2599,7 @@
   {
    "cell_type": "code",
    "execution_count": 20,
-   "id": "db7ee861",
+   "id": "d8a97605",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2611,7 +2612,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "290ed024",
+   "id": "90284486",
    "metadata": {
     "editable": true
    },
@@ -2630,7 +2631,7 @@
   {
    "cell_type": "code",
    "execution_count": 21,
-   "id": "86f144d3",
+   "id": "288b7876",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2698,7 +2699,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8bf7ab88",
+   "id": "6f46d25e",
    "metadata": {
     "editable": true
    },
@@ -2711,7 +2712,7 @@
   {
    "cell_type": "code",
    "execution_count": 22,
-   "id": "ee8291f7",
+   "id": "37b3054d",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2751,7 +2752,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fffb3d71",
+   "id": "551d62e0",
    "metadata": {
     "editable": true
    },
@@ -2763,7 +2764,7 @@
   {
    "cell_type": "code",
    "execution_count": 23,
-   "id": "a78aeff3",
+   "id": "b2514895",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2777,7 +2778,7 @@
   {
    "cell_type": "code",
    "execution_count": 24,
-   "id": "b6841971",
+   "id": "65242556",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2790,7 +2791,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7a0e935e",
+   "id": "c264a585",
    "metadata": {
     "editable": true
    },
@@ -2801,7 +2802,7 @@
   {
    "cell_type": "code",
    "execution_count": 25,
-   "id": "4d955ef1",
+   "id": "bcd6d1d5",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2815,7 +2816,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1f02c4d5",
+   "id": "cfcb27cc",
    "metadata": {
     "editable": true
    },
@@ -2830,7 +2831,7 @@
   {
    "cell_type": "code",
    "execution_count": 26,
-   "id": "4f2c7e3b",
+   "id": "81a0d347",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2846,7 +2847,7 @@
   {
    "cell_type": "code",
    "execution_count": 27,
-   "id": "3e283626",
+   "id": "55e5c0a6",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2863,7 +2864,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f24a66e8",
+   "id": "0488b19f",
    "metadata": {
     "editable": true
    },
@@ -2875,7 +2876,7 @@
   {
    "cell_type": "code",
    "execution_count": 28,
-   "id": "fe431bbc",
+   "id": "9b4e0734",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2889,7 +2890,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "08bedbc8",
+   "id": "e7a01f04",
    "metadata": {
     "editable": true
    },
@@ -2905,7 +2906,7 @@
   {
    "cell_type": "code",
    "execution_count": 29,
-   "id": "dcbbfe02",
+   "id": "faa0f98f",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2939,7 +2940,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "642f178c",
+   "id": "6b1854b0",
    "metadata": {
     "editable": true
    },
@@ -2952,7 +2953,7 @@
   {
    "cell_type": "code",
    "execution_count": 30,
-   "id": "8eb8f078",
+   "id": "b9a84b77",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2977,7 +2978,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "185100ba",
+   "id": "8b996be3",
    "metadata": {
     "editable": true
    },
@@ -2991,7 +2992,7 @@
   {
    "cell_type": "code",
    "execution_count": 31,
-   "id": "6a256e74",
+   "id": "c5047657",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3009,7 +3010,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bd97b51d",
+   "id": "5ab3e5e0",
    "metadata": {
     "editable": true
    },
@@ -3020,7 +3021,7 @@
   {
    "cell_type": "code",
    "execution_count": 32,
-   "id": "42ddd910",
+   "id": "48d4afd1",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3060,7 +3061,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cab3288a",
+   "id": "0caa2763",
    "metadata": {
     "editable": true
    },
@@ -3072,7 +3073,7 @@
   {
    "cell_type": "code",
    "execution_count": 33,
-   "id": "e3e4e46b",
+   "id": "b0e73261",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3084,7 +3085,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fa628eb4",
+   "id": "95eca1ec",
    "metadata": {
     "editable": true
    },
@@ -3098,7 +3099,7 @@
   {
    "cell_type": "code",
    "execution_count": 34,
-   "id": "a29d7a36",
+   "id": "30cbdd71",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3115,7 +3116,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c97bb950",
+   "id": "b72caa6a",
    "metadata": {
     "editable": true
    },
@@ -3129,7 +3130,7 @@
   {
    "cell_type": "code",
    "execution_count": 35,
-   "id": "75c02c4f",
+   "id": "f1ecd7ef",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3146,7 +3147,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7002de8e",
+   "id": "99f348cf",
    "metadata": {
     "editable": true
    },
@@ -3162,7 +3163,7 @@
   {
    "cell_type": "code",
    "execution_count": 36,
-   "id": "7fafeb29",
+   "id": "1596f7d8",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3190,7 +3191,7 @@
   {
    "cell_type": "code",
    "execution_count": 37,
-   "id": "965eee54",
+   "id": "c271a2ed",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3213,7 +3214,7 @@
   {
    "cell_type": "code",
    "execution_count": 38,
-   "id": "08a29dab",
+   "id": "ed9d5fdd",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3226,7 +3227,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "889fcecb",
+   "id": "e602d023",
    "metadata": {
     "editable": true
    },
@@ -3242,7 +3243,7 @@
   {
    "cell_type": "code",
    "execution_count": 39,
-   "id": "437bc288",
+   "id": "0dad14ee",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3269,7 +3270,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ccbd2be7",
+   "id": "eb90f46b",
    "metadata": {
     "editable": true
    },
@@ -3284,7 +3285,7 @@
   {
    "cell_type": "code",
    "execution_count": 40,
-   "id": "61d645f2",
+   "id": "c7f1007b",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3300,7 +3301,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e098db36",
+   "id": "f51dd218",
    "metadata": {
     "editable": true
    },
@@ -3311,7 +3312,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "94429655",
+   "id": "4646c9d7",
    "metadata": {
     "editable": true
    },
@@ -3329,7 +3330,7 @@
   {
    "cell_type": "code",
    "execution_count": 41,
-   "id": "2af9e60d",
+   "id": "d7d4467e",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3347,7 +3348,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "26be8697",
+   "id": "b95c647c",
    "metadata": {
     "editable": true
    },
@@ -3358,7 +3359,7 @@
   {
    "cell_type": "code",
    "execution_count": 42,
-   "id": "b6534afb",
+   "id": "50648dfb",
    "metadata": {
     "collapsed": false,
     "editable": true
diff --git a/doc/src/week43/week43.do.txt b/doc/src/week43/week43.do.txt
index 1d68fa871..61669cac7 100644
--- a/doc/src/week43/week43.do.txt
+++ b/doc/src/week43/week43.do.txt
@@ -921,11 +921,13 @@ predicted=np.concatenate((trainPredict,testPredict),axis=0)
 trainScore = model.evaluate(trainX, trainY, verbose=0)
 print(trainScore)
 
-index = df.index.values
-plt.plot(index,df)
-plt.plot(index,predicted)
-plt.axvline(df.index[Tp], c="r")
+
+df = pd.DataFrame(x)
+pred = pd.DataFrame(predicted)
+plt.plot(df,c="b")
+plt.plot(pred,c="r")
 plt.show()
+
 !ec
 
 
@@ -976,23 +978,23 @@ y_tot = np.array([-0.03077640549, -0.08336233266, -0.1446729567, -0.2116753732,
 The way the recurrent neural networks are trained in this program
 differs from how machine learning algorithms are usually trained.
 Typically a machine learning algorithm is trained by learning the
-relationship between the x data and the y data.  In this program, the
+relationship between the $x$ data and the $y$ data.  In this program, the
 recurrent neural network will be trained to recognize the relationship
-in a sequence of y values.  This is type of data formatting is
-typically used time series forcasting, but it can also be used in any
+in a sequence of $y$ values.  This is type of data formatting is
+typically used for time series forecasting, but it can also be used in any
 extrapolation (time series forecasting is just a specific type of
 extrapolation along the time axis).  This method of data formatting
-does not use the x data and assumes that the y data are evenly spaced.
+does not use the $x$ data and assumes that the $y$ data are evenly spaced.
 
 For a standard machine learning algorithm, the training data has the
-form of (x,y) so the machine learning algorithm learns to assiciate a
-y value with a given x value.  This is useful when the test data has x
+form of $(x,y)$ so the machine learning algorithm learns to associate a
+$y$ value with a given $x$ value.  This is useful when the test data has $x$
 values within the same range as the training data.  However, for this
 application, the x values of the test data are outside of the x values
 of the training data and the traditional method of training a machine
 learning algorithm does not work as well.  For this reason, the
 recurrent neural network is trained on sequences of y values of the
-form ((y1, y2), y3), so that the network is concerned with learning
+form $((y1, y2), y3)$, so that the network is concerned with learning
 the pattern of the y data and not the relation between the x and y
 data.  As long as the pattern of y data outside of the training region
 stays relatively stable compared to what was inside the training