diff --git a/.gitignore b/.gitignore
index 1f88218..2af6c0b 100644
--- a/.gitignore
+++ b/.gitignore
@@ -53,3 +53,4 @@ rsconnect/
 /.quarto/
 **/*.quarto_ipynb
 _freeze/
+.positai
diff --git a/scripts/plot_non_normal_X.qmd b/scripts/plot_non_normal_X.qmd
index 3790668..2c7a9f7 100644
--- a/scripts/plot_non_normal_X.qmd
+++ b/scripts/plot_non_normal_X.qmd
@@ -24,38 +24,26 @@ library(latex2exp)
 
 ```
 
-## Setup 
-We consider the matrix $QQ^\top$ and look at the smallest eigenvalue, i.e. the
-smallest non-zero singular value of $Q$.
+## Setup
 
-The matrix $Q$ is given by
-$$
-Q_{ik} = \int_{\frac{k}{K}}^{\frac{k+1}{K}} p_a(u| X_i) \, du
-$$
-with
-$$
-p_a(u|X) = \frac{f_v(F_a^{-1}(u) - a^\top X)}{f_a(F_a^{-1}(u))}
-$$
-In this document we plot different the smallest eigenvalue in dependence of the
-parameter $a$ with different "ratios" of the parameters $n$ and
-$$
-k = \lfloor n^\alpha \rfloor
-$$
+We consider the matrix $QQ^\top$ and look at the smallest eigenvalue, i.e. the smallest non-zero singular value of $Q$.
 
-with $\alpha = 0.1, 0.2, \dots 0.5$. The data matrix $X$ is a random matrix with
-i.i.d. distributed entries. We consider $x_{ij} \sim U[0,1]$ and $x_{ij} \sim Exp(\lambda)$.
+The matrix $Q$ is given by $$Q_{ik} = \int_{\frac{k}{K}}^{\frac{k+1}{K}} p_a(u| X_i) \, du$$ with $$p_a(u|X) = \frac{f_v(F_a^{-1}(u) - a^\top X)}{f_a(F_a^{-1}(u))}$$ In this document we plot different the smallest eigenvalue in dependence of the parameter $a$ with different "ratios" of the parameters $n$ and $$k = \lfloor n^\alpha \rfloor$$
+
+with $\alpha = 0.1, 0.2, \dots 0.5$. The data matrix $X$ is a random matrix with i.i.d. distributed entries. We consider $x_{ij} \sim U[0,1]$ and $x_{ij} \sim Exp(\lambda)$.
 
 ## Exponential distribution
+
 ```{r k = n^alpha data generation, rate = 1}
 #| cache: true
 #| echo: false
 #| collapse: true
-ns <- seq(100, 1000, 100)
+ns <- seq(100, 10000, 100)
 as <- seq(0, 20, 2)
 alphas <-  seq(0.1, 0.5, 0.1)
 
 set.seed(100)
-results <- data.frame(dim_n = integer(),
+results01 <- data.frame(dim_n = integer(),
                       dim_k = integer(),
                       param_a = double(),
                       param_alpha = double(),
@@ -79,14 +67,15 @@ for (a in as) {
       ssv <- compute_minmax_sv(Q)[["smallest_singular_value"]]
       
       current_res <- data.frame(dim_n = n, dim_k = K, param_a = a, param_alpha=alphas[j], ssv =ssv)
-      results <- rbind(results, current_res)
+      results01 <- rbind(results01, current_res)
     }
   }
 }
 ```
 
 ```{r k = n^alpha plotting, rate = 1}
-results |>
+# plot the results
+results01 |>
   filter(param_a %in% c(0, 10, 20)) |>
   mutate(param_a = as.factor(param_a),
          param_alpha = as.factor(param_alpha)) |>
@@ -105,17 +94,16 @@ results |>
        shape=latex2exp::TeX("$\\alpha$"))
 ```
 
-
 ```{r k = n^alpha data generation, rate = 3}
 #| cache: true
 #| echo: false
 #| collapse: true
-ns <- seq(100, 1000, 100)
+ns <- seq(100, 10000, 100)
 as <- seq(0, 20, 2)
 alphas <-  seq(0.1, 0.5, 0.1)
 
 set.seed(100)
-results <- data.frame(dim_n = integer(),
+results02 <- data.frame(dim_n = integer(),
                       dim_k = integer(),
                       param_a = double(),
                       param_alpha = double(),
@@ -139,14 +127,14 @@ for (a in as) {
       ssv <- compute_minmax_sv(Q)[["smallest_singular_value"]]
       
       current_res <- data.frame(dim_n = n, dim_k = K, param_a = a, param_alpha=alphas[j], ssv =ssv)
-      results <- rbind(results, current_res)
+      results02 <- rbind(results02, current_res)
     }
   }
 }
 ```
 
 ```{r k = n^alpha plotting, rate = 3}
-results |>
+results02 |>
   filter(param_a %in% c(0, 10, 20)) |>
   mutate(param_a = as.factor(param_a),
          param_alpha = as.factor(param_alpha)) |>
@@ -164,18 +152,19 @@ results |>
        colour=latex2exp::TeX("$a$"),
        shape=latex2exp::TeX("$\\alpha$"))
 ```
+
 For $a = 0$ the smallest singular value is very close to zero.
 
 ```{r k = n^alpha data generation, rate = 5}
 #| cache: true
 #| echo: false
 #| collapse: true
-ns <- seq(100, 1000, 100)
+ns <- seq(100, 10000, 100)
 as <- seq(0, 20, 2)
 alphas <-  seq(0.1, 0.5, 0.1)
 
 set.seed(100)
-results <- data.frame(dim_n = integer(),
+results03 <- data.frame(dim_n = integer(),
                       dim_k = integer(),
                       param_a = double(),
                       param_alpha = double(),
@@ -199,14 +188,14 @@ for (a in as) {
       ssv <- compute_minmax_sv(Q)[["smallest_singular_value"]]
       
       current_res <- data.frame(dim_n = n, dim_k = K, param_a = a, param_alpha=alphas[j], ssv =ssv)
-      results <- rbind(results, current_res)
+      results03 <- rbind(results03, current_res)
     }
   }
 }
 ```
 
 ```{r k = n^alpha plotting, rate = 5}
-results |>
+results03 |>
   filter(param_a %in% c(0, 10, 20)) |>
   mutate(param_a = as.factor(param_a),
          param_alpha = as.factor(param_alpha)) |>
@@ -224,21 +213,21 @@ results |>
        colour=latex2exp::TeX("$a$"),
        shape=latex2exp::TeX("$\\alpha$"))
 ```
-Why is here a perfect match for $\alpha = 0.1$ and $a = 20$ to the square function?
-The difference is of the order of $10^{-11}$!
 
+Why is here a perfect match for $\alpha = 0.1$ and $a = 20$ to the square function? The difference is of the order of $10^{-11}$!
 
 ## Uniform distribution
+
 ```{r k = n^alpha data generation, U[0,1]}
 #| cache: true
 #| echo: false
 #| collapse: true
-ns <- seq(100, 1000, 100)
+ns <- seq(100, 10000, 100)
 as <- seq(0, 20, 2)
 alphas <-  seq(0.1, 0.5, 0.1)
 
 set.seed(100)
-results <- data.frame(dim_n = integer(),
+results04 <- data.frame(dim_n = integer(),
                       dim_k = integer(),
                       param_a = double(),
                       param_alpha = double(),
@@ -262,14 +251,14 @@ for (a in as) {
       ssv <- compute_minmax_sv(Q)[["smallest_singular_value"]]
       
       current_res <- data.frame(dim_n = n, dim_k = K, param_a = a, param_alpha=alphas[j], ssv =ssv)
-      results <- rbind(results, current_res)
+      results04 <- rbind(results04, current_res)
     }
   }
 }
 ```
 
 ```{r k = n^alpha plotting, U[0,1]}
-results |>
+results04 |>
   filter(param_a %in% c(0, 10, 20)) |>
   mutate(param_a = as.factor(param_a),
          param_alpha = as.factor(param_alpha)) |>
@@ -287,19 +276,19 @@ results |>
        colour=latex2exp::TeX("$a$"),
        shape=latex2exp::TeX("$\\alpha$"))
 ```
-Here we have the same effect for $\alpha = 0.1$ and $a = 20$.
 
+Here we have the same effect for $\alpha = 0.1$ and $a = 20$.
 
 ```{r k = n^alpha data generation, U[0,2]}
 #| cache: true
 #| echo: false
 #| collapse: true
-ns <- seq(100, 1000, 100)
+ns <- seq(100, 10000, 100)
 as <- seq(0, 20, 2)
 alphas <-  seq(0.1, 0.5, 0.1)
 
 set.seed(100)
-results <- data.frame(dim_n = integer(),
+results05 <- data.frame(dim_n = integer(),
                       dim_k = integer(),
                       param_a = double(),
                       param_alpha = double(),
@@ -323,14 +312,14 @@ for (a in as) {
       ssv <- compute_minmax_sv(Q)[["smallest_singular_value"]]
       
       current_res <- data.frame(dim_n = n, dim_k = K, param_a = a, param_alpha=alphas[j], ssv =ssv)
-      results <- rbind(results, current_res)
+      results05 <- rbind(results05, current_res)
     }
   }
 }
 ```
 
 ```{r k = n^alpha plotting, U[0,2]}
-results |>
+results05 |>
   filter(param_a %in% c(0, 10, 20)) |>
   mutate(param_a = as.factor(param_a),
          param_alpha = as.factor(param_alpha)) |>
@@ -348,16 +337,17 @@ results |>
        colour=latex2exp::TeX("$a$"),
        shape=latex2exp::TeX("$\\alpha$"))
 ```
+
 ```{r k = n^alpha data generation, N(0,1)}
 #| cache: true
 #| echo: false
 #| collapse: true
-ns <- seq(100, 5000, 100)
+ns <- seq(100, 10000, 100)
 as <- seq(0, 20, 2)
 alphas <-  seq(0.1, 0.5, 0.1)
 
 set.seed(100)
-results <- data.frame(dim_n = integer(),
+results06 <- data.frame(dim_n = integer(),
                       dim_k = integer(),
                       param_a = double(),
                       param_alpha = double(),
@@ -382,14 +372,14 @@ for (a in as) {
       ssv <- compute_minmax_sv(Q)[["smallest_singular_value"]]
       
       current_res <- data.frame(dim_n = n, dim_k = K, param_a = a, param_alpha=alphas[j], ssv =ssv)
-      results <- rbind(results, current_res)
+      results06 <- rbind(results06, current_res)
     }
   }
 }
 ```
 
 ```{r k = n^alpha plotting, U[0,2]}
-results |>
+results06 |>
   filter(param_a %in% c(0, 10, 20)) |>
   mutate(param_a = as.factor(param_a),
          param_alpha = as.factor(param_alpha)) |>
@@ -403,13 +393,13 @@ results |>
   labs(x=latex2exp::TeX("$n$"),
        y=latex2exp::TeX("Smallest singular value of $Q$"),
        title=latex2exp::TeX("Smallest singular value of $Q$ with respect to $a$."),
-       subtitle = latex2exp::TeX(("Hyperparameter $k = n^{\\alpha}$. Black line is $\\sqrt{n}$, and $X \\sim N(0,1) $")),
+       subtitle = latex2exp::TeX(("Hyperparameter $k = n^{\\alpha}$. Black line is $\\sqrt{n}$, and $X \\sim N(0,1) $, use ceil function instead of floor for rounding.")),
        colour=latex2exp::TeX("$a$"),
        shape=latex2exp::TeX("$\\alpha$"))
 ```
 
 ```{r k = n^alpha plotting, U[0,2]}
-results |>
+results06 |>
   filter(param_a %in% c(0, 10, 20)) |>
   mutate(param_a = as.factor(param_a),
          param_alpha = as.factor(param_alpha)) |>
@@ -426,4 +416,9 @@ results |>
        subtitle = latex2exp::TeX(("Hyperparameter $k = n^{\\alpha}$. Black line is $\\sqrt{n}$, and $X \\sim N(0,1) $")),
        colour=latex2exp::TeX("$a$"),
        shape=latex2exp::TeX("$\\alpha$"))
-```
\ No newline at end of file
+```
+
+```{r}
+results <- list(results01, results02, results03, results04, results05, results06)
+save(results, file="results.RData")
+```