Probabilistic estimators of Lagrangian shape biases: Universal relations and physical insights

Open Access

Table A.2

Notation for tensor operations.

Compact Notation	Explicit Einstein Sum
AB	A_{i₁i₂α₁...α_n}B_{i₁i₂β₁...β_m}
$A_{\cdot}^{(N)} B$ $\[\mathbf{A}~^{(N)}_{~~~\cdot}\mathbf{B}\]$	A_{i₁ i₂...i_Nα₁...α_n} B_{i₁i₂...i_Nβ₁...β_m}
A ⊗ B	A_{i₁i₂...i_n} B_{j₁j₂...j_n}
$\frac{\partial A}{\partial B}$ $\[\frac{\partial \mathbf{A}}{\partial \mathbf{B}}\]$	$\frac{\partial A_{i_{1} i_{2} \dots i_{n}}}{\partial B_{j_{1} j_{2} \dots j_{m}}}$ $\[\frac{\partial A_{i_{1} i_{2} \cdots i_{n}}}{\partial B_{j_{1} j_{2} \cdots j_{m}}}\]$
∥A∥²	A_{i₁,i₂,...,i_n} A_{i₁,i₂,...,i_n}

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