sections: fix things here and there

slides/sections/1.md

@@ -24,19 +24,9 @@ $H_0$: sample following Landau PDF
 The Landau and Moyal PDFs are really similar. Historically, the latter was
 utilized as an approximation of the former.
 
-:::: {.columns}
-:::  {.column width=33%}
-  ![](images/moyal-photo.jpg){height=130pt}
-:::
-
-:::  {.column width=33%}
-  ![](images/mondau-photo.jpg){height=130pt}
-:::
-
-:::  {.column width=33%}
-  ![](images/landau-photo.jpg){height=130pt}
-:::
-::::
+\includegraphics<1>[height=5.5cm]{images/moyal-photo.jpg}
+\includegraphics<2>[height=5.5cm]{images/mondau-photo.jpg}
+\includegraphics<3>[height=5.5cm]{images/landau-photo.jpg}
 
 
 ## Two similar distributions

slides/sections/2.md

@@ -40,7 +40,7 @@
     m = F^{-1}\left(\frac{1}{2}\right)
    $$
 
-  - Numerical integration or QDF is needed
+  -  PDF Numerical integration up to $1/2$ or QDF is needed
 :::
 
 ::::

slides/sections/3.md

@@ -38,21 +38,7 @@ $$
 
 . . .
 
-With the change of variable $z = e^{-\frac{y}{2}}/\sqrt{2}$:
-$$
-  F_M(x) =
-  \frac{-2 \sqrt{2}}{\sqrt{2 \pi}} \int\limits_{+ \infty}^{f(x)} dz \, e^{- z^2}
-  \with f(x) = \frac{e^{- \frac{x}{2}}}{\sqrt{2}}
-$$
-
-
-## Moyal CDF
-
-Remembering the error function
-$$
-  \text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x dy \, e^{-y^2}
-$$
-one finally gets:
+after a bit of math, one finally gets:
 $$
   F_M(x) = 1 - \text{erf} \left( \frac{e^{- \frac{x}{2}}}{\sqrt{2}} \right)
 $$
@@ -125,7 +111,7 @@ $$
 
 We need to compute the maximum value:
 $$
-  M(\mu) = \frac{1}{\sqrt{2 \pi e}} \thus M(x_{\pm}) = \frac{1}{\sqrt{8 \pi e}}
+  M(\mu_M\ex) = \frac{1}{\sqrt{2 \pi e}} \thus M(x_{\pm}) = \frac{1}{\sqrt{8 \pi e}}
 $$
 
 . . .

slides/sections/4.md

@@ -75,7 +75,7 @@ How to estimate sample median, mode and FWHM?
 ::: incremental
 1. Find the smallest interval containing half points
 2. Repeat on the new interval (called modal)
-3. If the interval has less than three points, take average
+3. If the interval has less than two points, take average
 :::
 \End{block}
 

slides/sections/6.md

@@ -127,11 +127,11 @@ $$
   \draw [thick, cyclamen]          (8.5,-0.2) -- (8.5,0.2);
   \node [below right, cyclamen] at (8.5,-0.2) {$a_{j+4}$};
   % notes
-  \node [below] at (1,-1)   {0};
-  \node [below] at (2,-1)   {0};
-  \node [below] at (5.2,-1) {1};
-  \node [below] at (6,-1)   {1};
-  \node [below] at (8.5,-1) {1};
+  \node [below] at (1,-1)   {1};
+  \node [below] at (2,-1)   {1};
+  \node [below] at (5.2,-1) {0};
+  \node [below] at (6,-1)   {0};
+  \node [below] at (8.5,-1) {0};
   \draw [thick, ->] (1,-0.5)   -- (1,-1);
   \draw [thick, ->] (2,-0.5)   -- (2,-1);
   \draw [thick, ->] (5.2,-0.5) -- (5.2,-1);
@@ -162,7 +162,7 @@ $$
 If $a_j$ uniformly distributed, for the CLT:
 $$
   \sum_j \zeta_j (u) \hence
-    G \left( \frac{r}{2}, \frac{r}{4} \right)
+    G \left( \frac{r}{2}, \frac{\sqrt{r}}{2} \right)
   \thus \vartheta (u) \hence
     G \left( 0, 1 \right)
 $$
@@ -195,7 +195,7 @@ $$
 
 ## Trapani test
 
-If $\mu_k \ne + \infty \hence \left\{ a_j \right\}$ are not uniformly distributed
+If $\mu_k < + \infty \hence \left\{ a_j \right\}$ are not uniformly distributed
 
 \vspace{20pt}
 

slides/sections/7.md

@@ -22,18 +22,14 @@ A $M(x)$ similar to $L(x)$ can be found by imposing:
 
 - equal mode
 $$
-  \mu_M\ex = \mu_L\ex \approx −0.22278298...
+  \mu_M\ex = \mu_L\ex \thus \mu \approx −0.22278298...
 $$
 
 . . .
 
 - equal width
 $$
-  w_M\ex = w_L\ex = \sigma \cdot a
-$$
-
-$$
-  \implies \sigma_M \approx 1.1191486...
+  w_M\ex = w_L\ex = \sigma \cdot a \thus \sigma \approx 1.1191486...
 $$
 
 

slides/sections/8.md

@@ -59,7 +59,7 @@ $$
 \setbeamercovered{transparent}
 
 
-## Landau sample results
+## L sample results
 
 \begin{center}
 \begin{tabular}{rcccc}
@@ -79,7 +79,7 @@ $$
 \end{center}
 
 
-## Moyal sample results
+## M sample results
 
 \begin{center}
 \begin{tabular}{rcccc}

slides/sections/9.md

@@ -2,16 +2,18 @@
 
 ## Summary
 
-- All five tests properly work for a high number of points ($N = 50000$);
+- All six tests properly work for a high number of points ($N = 50000$);
 - Properties comparison:
-  - The median estimation decreases in significe as $N$ decreases;
-  - The KDE for FWHM is the least stable test (not working for $N \leq 1000$);
-  - the mode estimation swtill properly works for very few points ($N = 200$);
+  - Median estimation decreases in significance as $N$ decreases;
+  - KDE for FWHM is the least stable test (not working for $N \leq 1000$);
+  - HSM swtill properly works for very few points ($N = 200$);
 - KS still properly works for very few points ($N = 200$);
-- The Trapani test decreases in significe as $N$ decreases.
+- Trapani test (less informative) decreases in significance as $N$ decreases.
 
 
 ## Any questions? {.standout}
 
+Any questions?
+
 
 ## Bibliograph