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This is a notation I see in page 8 of Guy Barles and Espen R. Jakobsen, namely $$ \partial_t^{\beta_0}D^{\beta'}\phi(x,t) $$ where $\phi: \mathbb{R}^n\times[0,T]\longrightarrow \mathbb{R}$ is smooth, $\beta_0\in \mathbb{N}$ , $\beta'=(\beta'_i)_i \in \mathbb{N}^n$.

I want to know the precise definition of the notation.

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It means one differentiates the function $\beta_0$ times with respect to $t$ and $\beta'_i$ times with respect to each $x_i$. –  Did Aug 3 '11 at 8:11
    
@Piau Thanks a lot. It makes so much sense now –  user14169 Aug 3 '11 at 8:49
    
Now I'm curious, why are you reading a paper on differential equations without knowing that notation? –  Jonas Teuwen Aug 3 '11 at 17:00
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The notation means that one differentiates the function $\beta_0$ times with respect to $t$ and $\beta'_i$ times with respect to each $x_i$.

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