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The simulations displayed this degeneracy in the direction of the emergent ripples (i.e. the x- and y- components changed between positive and negative with successive simulations), but both components were shown to be equal in magnitude, as expected analytically.
First was the inclusion of solid guidance on what was expected analytically: at the start of the project, students were led through a visual and physiological analysis of one artwork.
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As part of your studies, you'll be expected to write analytically about art, and may also be asked to give presentations to your classmates.
Under linear evolution, it can be seen that a peak exists close to the analytically expected result for linear stability, denoted by the black vertical line.
The white dashed line in Figure 4(a) indicates the light-cone according to the analytically expected group-velocity and is in excellent agreement with the numerical calculations.
Actually, such wide variation can be analytically expected: it has been shown that a system composed of first order reactions that is both 'open' (i.e. including protein synthesis from source, hence violating conservation of mass) and has an initial Poisson distribution will remain a Poisson distribution at any time t > 0 [ 27].
First, we validated the expected (theoretical) performance gains analytically.
We consider problems which require backward induction with analytically intractable expected utility integrals at each stage.
Engineers can use this paper to analytically translate expected permeability changes with pressure, to productivity changes with pressure differences.
The average results from the centrifuge test, pull-in voltage test, and transit time test are all within approximately 20% of the expected values that are analytically calculated.
In order to demonstrate that the "intrinsic risk aversion of selection" arose from correlations between growth rate fluctuations in different generations, we computed analytically the expected population sizes after igen generations.
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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