# What do the two (double) vertical bars mean in this equation?

Quoting the paper...

"The range difference equation between BR (R1) and transmitter (T) and any other receiver (Rj) has the following form:"

$$c \cdot \Delta t_{ij} = \Delta d_{ij} = \left\|\vec{r_T} - \vec{r_{R1}} \right\| - \left\|\vec{r_T} - \vec{r_{Rj}} \right\|.$$

Writing $\lVert \vec{x} \rVert$ for a vector $\vec{x}$ is a standard notation for the norm of $\vec{x}$. The question is which is the norm in question (a usual choice is the Euclidean norm (or "$2$-norm"), the notion of distance we usually deal with in the "real" 3D world).
After finding the "article" you mention [1], where the equation you mention is Eq. (1), it is clear from reading the subsequent equations and paragraph (especially the one leading to Eq.(4), "The least squares solution is in this case:") that the norm considered is indeed the usual Euclidean norm $$\lVert \vec{x}\rVert = \sqrt{ \sum_{i} \vec{x}_i^2 }$$