Table 1 Table of Notations
Notation | Description | Units |
|---|---|---|
N | number of runners in the race | |
i | runner’s identification number (i∈{1,…,N}) | |
L | course’s length | m |
t | time in seconds (race starts at t = 0) | s |
x | vector of the positions of all runners at time t (\(\mathbf{x}:=(x_{1}(t),\ldots ,x_{N}(t)\)) | \(m^{N}\) |
x | position on the course | m |
\(x_{0,i}\) | starting position of runner i (\(x_{0,i}\leq 0\)) | m |
s(x) | slope of the course at position x | |
w(x) | width of the course at position x | m |
\(m_{i}\) | deviation of the speed of runner i per unit of slope | ms−1 |
d | length (along the course) of the vital space | m |
\(\delta _{t}\) | time step | s |
\(G_{i}(t,\mathbf{x})\) | set of runners in the vital space of runner i at time t | |
\(D_{i}(t,\mathbf{x})\) | number of runners per unit of area in the vital space | |
\(v^{G}_{i}(t,\mathbf{x})\) | average speed of the 5 slowest runners of \(G_{i}(t,\mathbf{x})\) | ms−1 |
\(v_{0,i}\) | speed of runner i at departure | ms−1 |
\(\bar{v}_{i}\) | average speed of runner i (fixed) | ms−1 |
\(\bar{v}^{w_{i}}_{0}\) | maximum speed at departure for runners in the wave of runner i | ms−1 |
\(v^{p}_{i}(x)\) | speed profile of runner i evaluated at position x | ms−1 |
\(v_{i}^{l}(t,\mathbf{x})\) | runner i speed component due to density at time t | ms−1 |
\(\bar{d}^{w}_{i}\) | departure time delay for runner i (depends on wave and starting position) | s |
\(\rho _{i}(t,\mathbf{x})\) | density weight for runner i at time t |