notes: fix a few small issues in ex-6

notes/sections/6.md

@@ -114,7 +114,7 @@ implemented for discrete arrays of numbers, such as histograms or vectors:
   (f * g)(x)
     &= \int \limits_{- \infty}^{+ \infty} dy f(y) (R \, g)(y-x)      \\
     &= \int \limits_{- \infty}^{+ \infty} dy f(y) (T_x \, R \, g)(y) \\
-    &= (f, T_x \, R \, g)(y)
+    &= (f, T_x \, R \, g)
 \end{align*}
 
 where:
@@ -368,7 +368,7 @@ a known $P(x | \xi)$, @eq:first can be used to calculate an estimate for
 $Q (\xi | x)$. Then, taking the hint provided by @eq:second, an improved
 estimate for $f(\xi)$ can be generated, using the observed sample {$x_i$} to
 give an approximation for $\phi$.  
-Thus, if $f^t$ is the $t-th$ estimate, the next is given by:
+Thus, if $f^t$ is the $t$-th estimate, the next is given by:
 $$
   f^{t + 1}(\xi) = \int dx \, \phi(x) Q^t(\xi | x)
   \with
@@ -544,7 +544,7 @@ merely a fact of floating-points precision) and the best result is obtained for
 $r = 2$, meaning that the convergence of the RL algorithm is really fast and
 this is due to the fact that the histogram was only slightly modified.  
 In @fig:rless-0.5, the curve starts to flatten at about 10 rounds, whereas in
-@fig:rless-1 a minimum occurs around \num{5e3} rounds, meaning that, whit such
+@fig:rless-1 a minimum occurs around \num{5e3} rounds, meaning that, with such
 a large kernel, the convergence is very slow, even if the best results are
 close to the one found for $\sigma = 0.5$.
 The following $r$s were chosen as the most fitting:
@@ -625,7 +625,7 @@ follows.
 
 For each bin, once the convolved histogram was computed, a value $v_N$ was
 randomly sampled from a Gaussian distribution with standard deviation
-$\sigma_N$, and the value $v_n \cdot b$ was added to the bin itself, where $b$
+$\sigma_N$, and the value $v_N \cdot b$ was added to the bin itself, where $b$
 is the count of the bin. An example with $\sigma_N = 0.05$ of the new
 histogram is shown in @fig:noisy.  
 The following three values of $\sigma_N$ were tested: