diff --git a/doc/pub/week48/html/._week48-bs006.html b/doc/pub/week48/html/._week48-bs006.html
index c599020e7..f6c904588 100644
--- a/doc/pub/week48/html/._week48-bs006.html
+++ b/doc/pub/week48/html/._week48-bs006.html
@@ -266,9 +266,9 @@ MathJax.Hub.Config({
 <h2 id="___sec5" class="anchor">The equations </h2>
 
 <p>
-Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with \( x_i \) and \( y_i \) as variables). We define a vector \( \boldsymbol{x}_i=[x_i,y_i] \) and have
+Suppose we define a polynomial transformation of degree two only. We define a vector \( \boldsymbol{x}_i=[x_i,y_i] \) and have
 $$
-\boldsymbol{z}_i = \phi(\boldsymbol{x}_i)^T =\left(1, \sqrt{2}x_i, \sqrt{2}y_i, x_i^2, y_i^2, \sqrt{2}x_iy_i\right).
+\boldsymbol{z}_i^T = \phi(\boldsymbol{x}_i)^T =\left[1, \sqrt{2}x_i, \sqrt{2}y_i, x_i^2, y_i^2, \sqrt{2}x_iy_i\right].
 $$
 
 <p>
@@ -290,7 +290,7 @@ $$
 
 For the above example, the kernel reads
 $$
-K(\boldsymbol{x}_i,\boldsymbol{x}_j)=[1, \sqrt{2}x_i, \sqrt{2}y_i, x_i^2, y_i^2, \sqrt{2}x_iy_i]^T\begin{bmatrix} 1\\ \sqrt{2}x_j \\ \sqrt{2}y_j \\ x_j^2\\ y_i^2\\ \sqrt{2}x_jy_j \end{bmatrix}=1+2x_ix_j+2y_iy_j+(x_ix_j)^2+(y_iy_j)^2+2x_ix_jy_iy_j.
+K(\boldsymbol{x}_i,\boldsymbol{x}_j)=[1, \sqrt{2}x_i, \sqrt{2}y_i, x_i^2, y_i^2, \sqrt{2}x_iy_i]\begin{bmatrix} 1\\ \sqrt{2}x_j \\ \sqrt{2}y_j \\ x_j^2\\ y_i^2\\ \sqrt{2}x_jy_j \end{bmatrix}=1+2x_ix_j+2y_iy_j+(x_ix_j)^2+(y_iy_j)^2+2x_ix_jy_iy_j.
 $$
 
 <p>
diff --git a/doc/pub/week48/html/week48-reveal.html b/doc/pub/week48/html/week48-reveal.html
index 836f6f418..2d65f4154 100644
--- a/doc/pub/week48/html/week48-reveal.html
+++ b/doc/pub/week48/html/week48-reveal.html
@@ -294,10 +294,10 @@ plt.show()
 <h2 id="___sec5">The equations </h2>
 
 <p>
-Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with \( x_i \) and \( y_i \) as variables). We define a vector \( \boldsymbol{x}_i=[x_i,y_i] \) and have
+Suppose we define a polynomial transformation of degree two only. We define a vector \( \boldsymbol{x}_i=[x_i,y_i] \) and have
 <p>&nbsp;<br>
 $$
-\boldsymbol{z}_i = \phi(\boldsymbol{x}_i)^T =\left(1, \sqrt{2}x_i, \sqrt{2}y_i, x_i^2, y_i^2, \sqrt{2}x_iy_i\right).
+\boldsymbol{z}_i^T = \phi(\boldsymbol{x}_i)^T =\left[1, \sqrt{2}x_i, \sqrt{2}y_i, x_i^2, y_i^2, \sqrt{2}x_iy_i\right].
 $$
 <p>&nbsp;<br>
 
@@ -327,7 +327,7 @@ $$
 For the above example, the kernel reads
 <p>&nbsp;<br>
 $$
-K(\boldsymbol{x}_i,\boldsymbol{x}_j)=[1, \sqrt{2}x_i, \sqrt{2}y_i, x_i^2, y_i^2, \sqrt{2}x_iy_i]^T\begin{bmatrix} 1\\ \sqrt{2}x_j \\ \sqrt{2}y_j \\ x_j^2\\ y_i^2\\ \sqrt{2}x_jy_j \end{bmatrix}=1+2x_ix_j+2y_iy_j+(x_ix_j)^2+(y_iy_j)^2+2x_ix_jy_iy_j.
+K(\boldsymbol{x}_i,\boldsymbol{x}_j)=[1, \sqrt{2}x_i, \sqrt{2}y_i, x_i^2, y_i^2, \sqrt{2}x_iy_i]\begin{bmatrix} 1\\ \sqrt{2}x_j \\ \sqrt{2}y_j \\ x_j^2\\ y_i^2\\ \sqrt{2}x_jy_j \end{bmatrix}=1+2x_ix_j+2y_iy_j+(x_ix_j)^2+(y_iy_j)^2+2x_ix_jy_iy_j.
 $$
 <p>&nbsp;<br>
 
diff --git a/doc/pub/week48/html/week48-solarized.html b/doc/pub/week48/html/week48-solarized.html
index ac04f953f..a342cc370 100644
--- a/doc/pub/week48/html/week48-solarized.html
+++ b/doc/pub/week48/html/week48-solarized.html
@@ -346,9 +346,9 @@ plt.show()
 <h2 id="___sec5">The equations </h2>
 
 <p>
-Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with \( x_i \) and \( y_i \) as variables). We define a vector \( \boldsymbol{x}_i=[x_i,y_i] \) and have
+Suppose we define a polynomial transformation of degree two only. We define a vector \( \boldsymbol{x}_i=[x_i,y_i] \) and have
 $$
-\boldsymbol{z}_i = \phi(\boldsymbol{x}_i)^T =\left(1, \sqrt{2}x_i, \sqrt{2}y_i, x_i^2, y_i^2, \sqrt{2}x_iy_i\right).
+\boldsymbol{z}_i^T = \phi(\boldsymbol{x}_i)^T =\left[1, \sqrt{2}x_i, \sqrt{2}y_i, x_i^2, y_i^2, \sqrt{2}x_iy_i\right].
 $$
 
 <p>
@@ -370,7 +370,7 @@ $$
 
 For the above example, the kernel reads
 $$
-K(\boldsymbol{x}_i,\boldsymbol{x}_j)=[1, \sqrt{2}x_i, \sqrt{2}y_i, x_i^2, y_i^2, \sqrt{2}x_iy_i]^T\begin{bmatrix} 1\\ \sqrt{2}x_j \\ \sqrt{2}y_j \\ x_j^2\\ y_i^2\\ \sqrt{2}x_jy_j \end{bmatrix}=1+2x_ix_j+2y_iy_j+(x_ix_j)^2+(y_iy_j)^2+2x_ix_jy_iy_j.
+K(\boldsymbol{x}_i,\boldsymbol{x}_j)=[1, \sqrt{2}x_i, \sqrt{2}y_i, x_i^2, y_i^2, \sqrt{2}x_iy_i]\begin{bmatrix} 1\\ \sqrt{2}x_j \\ \sqrt{2}y_j \\ x_j^2\\ y_i^2\\ \sqrt{2}x_jy_j \end{bmatrix}=1+2x_ix_j+2y_iy_j+(x_ix_j)^2+(y_iy_j)^2+2x_ix_jy_iy_j.
 $$
 
 <p>
diff --git a/doc/pub/week48/html/week48.html b/doc/pub/week48/html/week48.html
index 473b9f93d..199b62748 100644
--- a/doc/pub/week48/html/week48.html
+++ b/doc/pub/week48/html/week48.html
@@ -351,9 +351,9 @@ plt<span style="color: #666666">.</span>show()
 <h2 id="___sec5">The equations </h2>
 
 <p>
-Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with \( x_i \) and \( y_i \) as variables). We define a vector \( \boldsymbol{x}_i=[x_i,y_i] \) and have
+Suppose we define a polynomial transformation of degree two only. We define a vector \( \boldsymbol{x}_i=[x_i,y_i] \) and have
 $$
-\boldsymbol{z}_i = \phi(\boldsymbol{x}_i)^T =\left(1, \sqrt{2}x_i, \sqrt{2}y_i, x_i^2, y_i^2, \sqrt{2}x_iy_i\right).
+\boldsymbol{z}_i^T = \phi(\boldsymbol{x}_i)^T =\left[1, \sqrt{2}x_i, \sqrt{2}y_i, x_i^2, y_i^2, \sqrt{2}x_iy_i\right].
 $$
 
 <p>
@@ -375,7 +375,7 @@ $$
 
 For the above example, the kernel reads
 $$
-K(\boldsymbol{x}_i,\boldsymbol{x}_j)=[1, \sqrt{2}x_i, \sqrt{2}y_i, x_i^2, y_i^2, \sqrt{2}x_iy_i]^T\begin{bmatrix} 1\\ \sqrt{2}x_j \\ \sqrt{2}y_j \\ x_j^2\\ y_i^2\\ \sqrt{2}x_jy_j \end{bmatrix}=1+2x_ix_j+2y_iy_j+(x_ix_j)^2+(y_iy_j)^2+2x_ix_jy_iy_j.
+K(\boldsymbol{x}_i,\boldsymbol{x}_j)=[1, \sqrt{2}x_i, \sqrt{2}y_i, x_i^2, y_i^2, \sqrt{2}x_iy_i]\begin{bmatrix} 1\\ \sqrt{2}x_j \\ \sqrt{2}y_j \\ x_j^2\\ y_i^2\\ \sqrt{2}x_jy_j \end{bmatrix}=1+2x_ix_j+2y_iy_j+(x_ix_j)^2+(y_iy_j)^2+2x_ix_jy_iy_j.
 $$
 
 <p>
diff --git a/doc/pub/week48/ipynb/ipynb-week48-src.tar.gz b/doc/pub/week48/ipynb/ipynb-week48-src.tar.gz
index 5befe5395..93c9e0356 100644
Binary files a/doc/pub/week48/ipynb/ipynb-week48-src.tar.gz and b/doc/pub/week48/ipynb/ipynb-week48-src.tar.gz differ
diff --git a/doc/pub/week48/ipynb/week48.ipynb b/doc/pub/week48/ipynb/week48.ipynb
index 9b7a9299d..600b2ed39 100644
--- a/doc/pub/week48/ipynb/week48.ipynb
+++ b/doc/pub/week48/ipynb/week48.ipynb
@@ -138,7 +138,7 @@
    "source": [
     "## The equations\n",
     "\n",
-    "Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with $x_i$ and $y_i$ as variables). We define a vector $\\boldsymbol{x}_i=[x_i,y_i]$ and have"
+    "Suppose we define a polynomial transformation of degree two only. We define a vector $\\boldsymbol{x}_i=[x_i,y_i]$ and have"
    ]
   },
   {
@@ -146,7 +146,7 @@
    "metadata": {},
    "source": [
     "$$\n",
-    "\\boldsymbol{z}_i = \\phi(\\boldsymbol{x}_i)^T =\\left(1, \\sqrt{2}x_i, \\sqrt{2}y_i, x_i^2, y_i^2, \\sqrt{2}x_iy_i\\right).\n",
+    "\\boldsymbol{z}_i^T = \\phi(\\boldsymbol{x}_i)^T =\\left[1, \\sqrt{2}x_i, \\sqrt{2}y_i, x_i^2, y_i^2, \\sqrt{2}x_iy_i\\right].\n",
     "$$"
    ]
   },
@@ -211,7 +211,7 @@
    "metadata": {},
    "source": [
     "$$\n",
-    "K(\\boldsymbol{x}_i,\\boldsymbol{x}_j)=[1, \\sqrt{2}x_i, \\sqrt{2}y_i, x_i^2, y_i^2, \\sqrt{2}x_iy_i]^T\\begin{bmatrix} 1\\\\ \\sqrt{2}x_j \\\\ \\sqrt{2}y_j \\\\ x_j^2\\\\ y_i^2\\\\ \\sqrt{2}x_jy_j \\end{bmatrix}=1+2x_ix_j+2y_iy_j+(x_ix_j)^2+(y_iy_j)^2+2x_ix_jy_iy_j.\n",
+    "K(\\boldsymbol{x}_i,\\boldsymbol{x}_j)=[1, \\sqrt{2}x_i, \\sqrt{2}y_i, x_i^2, y_i^2, \\sqrt{2}x_iy_i]\\begin{bmatrix} 1\\\\ \\sqrt{2}x_j \\\\ \\sqrt{2}y_j \\\\ x_j^2\\\\ y_i^2\\\\ \\sqrt{2}x_jy_j \\end{bmatrix}=1+2x_ix_j+2y_iy_j+(x_ix_j)^2+(y_iy_j)^2+2x_ix_jy_iy_j.\n",
     "$$"
    ]
   },
diff --git a/doc/src/week48/week48.do.txt b/doc/src/week48/week48.do.txt
index af6fef55b..44c5dd487 100644
--- a/doc/src/week48/week48.do.txt
+++ b/doc/src/week48/week48.do.txt
@@ -119,10 +119,10 @@ plt.show()
 !split
 ===== The equations =====
 
-Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with $x_i$ and $y_i$ as variables). We define a vector $\bm{x}_i=[x_i,y_i]$ and have
+Suppose we define a polynomial transformation of degree two only. We define a vector $\bm{x}_i=[x_i,y_i]$ and have
 !bt
 \[
-\bm{z}_i = \phi(\bm{x}_i)^T =\left(1, \sqrt{2}x_i, \sqrt{2}y_i, x_i^2, y_i^2, \sqrt{2}x_iy_i\right).
+\bm{z}_i^T = \phi(\bm{x}_i)^T =\left[1, \sqrt{2}x_i, \sqrt{2}y_i, x_i^2, y_i^2, \sqrt{2}x_iy_i\right].
 \]
 !et
 
@@ -148,7 +148,7 @@ K(\bm{x}_i,\bm{x}_j)=\bm{z}_i^T\bm{z}_j= \phi(\bm{x}_i)^T\phi(\bm{x}_j).
 For the above example, the kernel reads
 !bt
 \[
-K(\bm{x}_i,\bm{x}_j)=[1, \sqrt{2}x_i, \sqrt{2}y_i, x_i^2, y_i^2, \sqrt{2}x_iy_i]^T\begin{bmatrix} 1\\ \sqrt{2}x_j \\ \sqrt{2}y_j \\ x_j^2\\ y_i^2\\ \sqrt{2}x_jy_j \end{bmatrix}=1+2x_ix_j+2y_iy_j+(x_ix_j)^2+(y_iy_j)^2+2x_ix_jy_iy_j.
+K(\bm{x}_i,\bm{x}_j)=[1, \sqrt{2}x_i, \sqrt{2}y_i, x_i^2, y_i^2, \sqrt{2}x_iy_i]\begin{bmatrix} 1\\ \sqrt{2}x_j \\ \sqrt{2}y_j \\ x_j^2\\ y_i^2\\ \sqrt{2}x_jy_j \end{bmatrix}=1+2x_ix_j+2y_iy_j+(x_ix_j)^2+(y_iy_j)^2+2x_ix_jy_iy_j.
 \]
 !et