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Exact(18)
An analytical expression which includes all the elements that influence the process was used.
An analytical expression which describes the relative solubility S/Sw as a function of the MOS amount, v, was found.
An analytical expression, which reflects the relationship between the member length errors and pretension deviations, is deduced.
Indeed, an analytical expression which gives the optimal IBO maximizing the PA efficiency with respect to any EVM constraint is provided.
An analytical expression which gives the optimal input back-off (IBO) maximizing the PA efficiency with respect to any EVM constraint is provided.
The analytical expression, which gives the optimal IBO for an EVM constraint given by Corollary 5, and the approximated one, given by Corollary 6, are also plotted.
Similar(42)
The analytical expressions, which determine the total displacement of the sphere's centers and the corresponding tangential force, are derived.
The veracity of the analysis has been confirmed with the numerical calculations based on the analytical expressions, which have demonstrated excellent agreement with the simulation results.
Simple analytical expressions which define profiles of feed concentration along the channel and expressions for characteristic length of feed consumption are obtained.
With second-harmonic boundary and initial conditions of excitation, second-harmonic analytical expressions, which are applicable to quantitative analysis, have been derived.
For the equilibrium model, the concentration-constant-patterns and concentration-proportional-patterns may be described by using the Riemann invariants Rk and Hi invariant analytical expressions which are found.
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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