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We find the components of the shift vector by solving the equations ruling the shift vector.
We then substitute this expression for the shift vector components to the momentum equation.
With introducing the inner shift vector, the lattice vector is expressed as: (1).
The advection over the coordinates can be counteracted by setting the shift vector appropriately, i.e. (boldsymbol{beta}=-boldsymbol{v}).
The mean shift vector given in (14) can now be computed in the joint space as (19).
By running this algorithm the kernel is recursively moved from to according to the mean shift vector.
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These shift vectors were each applied to the perfectly matching anatomical image to offset it from the activity image.
In a second step, the symmetries identified are applied to all atomic positions in the supercell, to then search for translation symmetries (i.e., shift vectors).
Considering the correspondences between the HREM image and the atomic positions, analysis of the boundaries indicates that they are next-nearest neighbor antiphase boundaries (APB), with two possible shift vectors.
In addition, to verify the robustness of the conclusions against varying magnitudes and directions of shifts of the anatomical image relative to the PET, ten random directions were generated in 3D and used to obtain shift vectors with a magnitude of 1, 2, 4, and 6 mm (40 vectors in total).
Without admitting the spatial shifts, the assertion of the theorem is false: given a minimizer (rho_{0}) and a sequence of shift vectors ((a_{n} e_{3} )in{mathbb{R}}^{3}), the functional F is translation invariant, that is, (F(Trho)=F(rho)).
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com