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The partial derivatives required by the learning algorithms are derived using the adjoint model approach, adapted to the architecture of the networks being used.
The figures in black boxes are derived using the replicating portfolio technique.
Approximate system equations are derived using the assumed mode method.
The exact solutions are derived using the state space approach.
Stress strain relations for a solid polymer are derived using the laws of thermodynamics.
Its detailed features are derived using the computational fluid dynamics-discrete element method (CFD-DEM).
Member stiffness coefficients accounting for rotational stiffness degradation are derived using the modified moment distribution method.
Acceptable daily intakes (ADIs) are derived using the considerable data that are available for APIs.
All mathematical expressions in the paper are derived using the first principles.
The parameters of the distribution model are derived using the method of linear moments.
The governing equations and the corresponding boundary conditions are derived using the Hamilton's principle.
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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