Revert "refactor(thesis): introduce commands for the natural and real numbers"

This reverts commit 2ce3ae45 but reintroduces the actually desired changes.

src/internship_results/main.tex

@@ -93,9 +93,9 @@
   The task we eventually aim to solve is:
   \begin{task}
     Let \(X = (X_j)_{j=1, \hdots, N}\) the input quantities of a
-    measurement function \(f \colon \R^N \to \R \colon x \mapsto f
+    measurement function \(f \colon \mathbb R^N \to \mathbb R \colon x \mapsto f
     (x) = y\), which takes the form of an already trained, deep neural network.
-    Let \((x_j)_{j=1, \hdots, N} \in \R^N\) the best estimates representing
+    Let \((x_j)_{j=1, \hdots, N} \in \mathbb R^N\) the best estimates representing
     the input quantities and \(u(x_j), j=1, \ldots, N\) the associated standard
     uncertainties (\cite[paragraph 3.18]{jcgm_evaluation_2008}).
     Propagate the \(x_j\) and \(u(x_j), j=1, \ldots, N\) through
@@ -123,15 +123,15 @@
 
   As starting point we consider an already trained, fully-connected multi-layer
   perceptron (MLP) with input vector \(x = (x_j)_{j=1, \hdots, N}\) comprised of
-  the input quantities, any number \(l \in \N\) of layers and a possibly
+  the input quantities, any number \(l \in \mathbb N\) of layers and a possibly
   vector-valued output quantity \(Y\) of the last layer.
-  Each layer consists of \(n^{(i)} \in \N\) neurons, \(i= 0, 1, \hdots, l\)
+  Each layer consists of \(n^{(i)} \in \mathbb N\) neurons, \(i= 0, 1, \hdots, l\)
   where \(n^{(0)} = N\).
   We denote for \(i = 1, \hdots, l\) the \(i\)th layer's linear transformations
-  \(z^{(i)} \colon \R^{n^{(i-1)}} \to \R^{n^{(i)}}\) as
+  \(z^{(i)} \colon \mathbb R^{n^{(i-1)}} \to \mathbb R^{n^{(i)}}\) as
   \[z^{(i)}(X) = W^{(i)} \cdot x + b^{(i)}\]
-  with weights \(W^{(i)} \in \R^{n^{(i)} \times n^{(i-1)}}\) and biases \({b^{
-    (i)}} \in \R^{n^{(i)}}\).
+  with weights \(W^{(i)} \in \mathbb R^{n^{(i)} \times n^{(i-1)}}\) and biases \({b^{
+    (i)}} \in \mathbb R^{n^{(i)}}\).
   As activation functions we consider
   \begin{itemize}
     \item the so-called \textit{rectifier}
@@ -144,7 +144,7 @@
   \textit{rectifier}:
   \begin{gather}
     \tag{rectifier}
-    g(z) = \max\{0, z\}, \quad z \in \R.
+    g(z) = \max\{0, z\}, \quad z \in \mathbb R\label{eq:rectifier}, \text{ and}
   \end{gather}
   We define the first derivative of \(g\) at \(0\) to be \(g'(0) \coloneqq 1\).
   This results in \(g \in C^\infty([0, \infty))\) and \(g \in C^\infty((-\infty, 0)

src/thesis/Thesis_408230.tex

@@ -219,8 +219,8 @@ in neural network inputs through the networks in a GUM-compliant way.
 The task we eventually aim to solve is:
 \begin{task}[Uncertainty propagation]
   Let \(X = (X_j)_{j=1, \hdots, N}\) the input quantities of a measurement function
-  \(f \colon \R^N \to \R \colon x \mapsto f (x) = y\), which takes the
-  form of an already trained, deep neural network.
+  \(f \colon \R^N \to \R \colon x \mapsto f (x) = y\), which takes the form of an
+  already trained, deep neural network.
   Let \((x_j)_{j=1, \hdots, N} \in \R^N\) the best estimates representing the
   input quantities and \(u(x_j), j=1, \ldots, N\) the associated standard
   uncertainties (\cite[paragraph 3 .18]{jcgm_evaluation_2008}).
@@ -248,8 +248,8 @@ method's inputs and result.
 As starting point we consider an already trained, fully-connected multi-layer
 perceptron (MLP) with input vector \(x^{(0)} = \left( x_k^{(0)} \right)_{k=1, \hdots,
 n^{(0)}}\) comprised of the estimates of the input quantities, any number \(l \in
-\N\) of layers and a possibly vector-valued estimate $y$ of the output
-quantity in the last layer.
+\N\) of layers and a possibly vector-valued estimate $y$ of the output quantity in
+the last layer.
 Each layer consists of \(n^{(i)} \in \N\) neurons, \(i= 0, 1, \hdots, l\).
 We denote for \(i = 1, \hdots, l\) the \(i\)th layer's linear transformations
 \(z^{(i)} \colon \R^{n^{(i-1)}} \to \R^{n^{(i)}}\) as
@@ -304,7 +304,7 @@ with weights \(W^{(i)} \in \R^{n^{(i)} \times n^{(i-1)}}\) and biases \({b^{(i)}
 \begin{gather}
   \label{eq:sigmoid}
   \tag{sigmoid}
-  \sigma(x) = \frac{e^x}{1 + e^x} = \frac{1}{1 + e^{-x}}, \quad x \in \R.
+  \sigma(x) = \frac{e^x}{1 + e^x} = \frac{1}{1 + e^{-x}}.
 \end{gather}
 
 \textit{First derivative of sigmoid}:
@@ -371,8 +371,7 @@ referred to as the
 \begin{gather}
   \label{eq:parametric_softplus}
   \tag{parametric softplus}
-  \softplus(x) = \frac{\ln \left( 1 + e^{nx} \right)}{n}, \quad n \in
-  \R^+.
+  \softplus(x) = \frac{\ln \left( 1 + e^{nx} \right)}{n}, \quad n \in \R^+.
 \end{gather}
 
 Its further investigation is justified by the well-known fact, that in the limit
@@ -393,8 +392,7 @@ transition from $n \to \infty$ one obtains the most wide spread activation funct
 
 \textit{(parametric) softplus first derivative}:
 \begin{gather}
-  \odv{\softplus}{x} (x) = \frac{1}{1 + e^{-nx}} = \sigma(nx), \quad n \in
-  \R^+.
+  \odv{\softplus}{x} (x) = \frac{1}{1 + e^{-nx}} = \sigma(nx), \quad n \in \R^+.
   \label{eq:parametric-softplus-first-derivative}
 \end{gather}