# 3. ConventionsΒΆ

We use the notation \(a \triangleq b\) to denote that \(a\) is equal to \(b\) by definition.

We denote by \(\Re\) the set of real numbers, by \(\mathbb Z\) the set of integers and by \(\mathbb N \triangleq \{0, \, 1, \, \ldots \}\) the set of natural numbers. The set \(\mathbb N_+\) is defined as \(\mathbb N_+ \triangleq \{1, \, 2, \, \ldots \}\).

\(f(\cdot)\) denotes a function where \((\cdot)\) stands for the undesignated variables. \(f(x)\) denotes the value of \(f(\cdot)\) at the point \(x\). \(f\colon A \rightarrow B\) indicates that the domain of \(f(\cdot)\) is in the space \(A\) and its range in the space \(B\).

We say that a function \(f \colon \Re^n \to \Re\) is once continuously differentiable if \(f(\cdot)\) is defined on \(\Re^n\), and if \(f(\cdot)\) has a continuous derivative on \(\Re^n\).

For \(f \colon \Re \to \Re\), we denote by \(f^{(N)}(\cdot)\) its \(N\)-th derivative.

For \(f \colon \Re \to \Re\) and \(t \in \Re\), we denote by \(f(t^-) \triangleq \lim_{s \uparrow t} f(s)\) the limit from below.

For \(s \in \Re\), we define the ceiling function as \(\lceil s \rceil \triangleq \arg \min\{ k \in \mathbb Z \ | \ s \le k \}\).

For \(t_R \in \Re\) and \(t_I \in \mathbb N\), we write \(t^+ \triangleq (t_R, \, t_I)^+\) for the right limit at \(t\). It holds that \((t_R, \, t_I)^+ \Leftrightarrow (\lim_{\epsilon \to 0} (t_R+\epsilon), t_{I_{max}})\), where \(I_{max}\) is the largest occurring integer of superdense time. Similarly, we write \(\mathbin{^-t}\) for the left limit at \(t\), for which it holds that \(\mathbin{^-(t_R, \, t_I)} \Leftrightarrow (\lim_{\epsilon \to 0} (t_R-\epsilon), 0)\).

We write a requirement

*shall*be met if it must be fulfilled. If the feature that implements a shall requirement is not in the final system, then the system does not meet this requirement. We write a requirment*should*be met if it is not critical to the system working, but is still desirable.