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For a $C^3$-function $f:R^n \to R$, $x \mapsto f(x)$ I have the Taylor series $$ f(x+y)=f(x) + \nabla f (x)^T y + \frac{1}{2} y^T (\nabla^2 f(x)) y + \sum_{|j|=3} \frac{D^j f(z)}{j!} y^{j}$$ with multi-index $j$ and $z$ a point on the line segment between $x$ and $x+y$ But let's just look at $T_3=\sum_{|j|=3} \frac{D^j f(z)}{j!} y^{j}$. The author of the paper I am reading states $T_3$ as follows $$ T'_3= \frac{1}{6} f^{(3)}(z) y \otimes y \otimes y$$

Q: Is this some known shorthand-notation of $T_3$? If not, what does it mean? Has somebody seen the notation $\otimes$ in context of Taylor expansion?

EDIT: Maybe something about notation. multi index notation, $\nabla f(x)$ is the gradient and $\nabla^2 f(x)$ the Hessian at $x$.

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Where did you see this? –  J. M. Sep 1 '11 at 15:28
@J. M. formula (3.1) from the paper $L=f$, $z= \theta_k^{+}$, $x=\hat{\theta_k}$, $y= \Delta_k$ citeseer.ist.psu.edu/viewdoc/… –  Johannes L Sep 1 '11 at 15:32
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