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This question is motivated by the consideration of linear control systems in Brunovsky normal form. The idea is that you have $m$ smooth functions with unconstrained dynamics, and the control input for the $i$th function $z^i(t)$ is the $r_i$th derivative of the function. This leads to the control system \begin{align*} \dot{z}^i_k & = z^i_{k+1}, \qquad 0 \leq k \leq r_i - 2, \ \ 1 \leq i \leq m, \\ \dot{z}^i_{r_i-1} & = u^i, \qquad 1 \leq i \leq m, \end{align*} where $u^1, \ldots, u^m$ are the controls.

This ODE system naturally lives on a space that is obtained from $J^1(\mathbb{R}, \mathbb{R}^m)$ with its standard contact structure by prolonging each of the contact forms $$ \theta^i_0 = dz^i_0 - z^i_1\, dt $$ $(r_i-1)$ times. So here's the question: If all the $r_i$ are equal to the same value $r$, the underlying manifold of the resulting prolongation is just the $r$th-order jet space, denoted $J^r(\mathbb{R}, \mathbb{R}^m)$. But is there a standard name/notation for this space when the orders $r_i$ are not all the same?

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