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improved formulas a bit

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hannes.kuchelmeister
2020-01-30 10:09:12 +01:00
parent cc0e224357
commit 14d864ee35
1 changed files with 6 additions and 6 deletions
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25_Outline/outline.tex
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@@ -74,12 +74,12 @@ The system has one main way to be used as defined in \autoref{table:simulation_p
Set of variables:
\begin{equation}
V = \{V_1, \dotsc, V_m\}
V = \{v_1, \dotsc, v_m\}
\end{equation}
Set of their domains of values:
\begin{equation}
D = \{D_1, \dotsc, D_m\}
D = \{D_1, \dotsc, D_m\}, \qquad where \ D_i = \{d_1, \dotsc, d_o\}
\end{equation}
Set of users:
@@ -90,7 +90,7 @@ Set of users:
Users utility for a domain value of a variable:
\begin{equation}
\begin{split}
u_i(d_j),\ \text{where}\ & d_j \in D_j,\\
u_i(d_j), \qquad \text{where}\ & d_j \in D_j,\\
& 1 <= j <= m, \\
& 1 <= i <= n
\end{split}
@@ -99,18 +99,18 @@ Users utility for a domain value of a variable:
User preferences:
\begin{gather}
P = \{ P_1, \dotsc, P_n\},\ \text{where} \\
P_i = \{(d,\ u_i(d)) \ | \ \forall D_i \in D \ \forall d \in D_i\} \notag
P_i = \{(d,\ u_i(d)) \ | \ \forall d \in D_i,\ i=1,\dotsc,m \} \notag
\end{gather}
Configuration state:
\begin{equation}
S = \{ (v,\ d) \ |\ v \in V_i \ \land \ d \in D_i \}
S = \{ (v_i,\ d) \ |\ v_i \in V \ \land \ d \in D_i,\ i=1,\dotsc,m \}
\end{equation}
Finished configuration state:
\begin{equation}
\begin{split}
S_F \subset S,\ where \ & \forall v \in V_i (\exists (v, d) \in S_F : d \in D_i) \\
S_F \subset S,\ where \ & \forall v_i \in V (\exists (v_i, d) \in S_F : d \in D_i) \\
& \land is\_valid(S_F)
\end{split}
\end{equation}