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To solve this difficulty, we had to assume that the function p is regular enough; this allows us, via an integration by parts to increase the regularity of the nonlinear term N ( t ) at the expense of Lipschitz regularity on p. Step 1. Proof of ( 21 ) and ( 22 ).
In the far southwest of the shield, and especially in the northern areas, precipitation is sufficient to support a considerable vegetation and is regular enough for streams to flow seasonally.
The "molecular clock" is not strictly clock-like, however it is regular enough that substitutions do relate to time.
In other words, whether the weak solution is regular enough to define the trace on the boundary is not so important.
Both algorithms use Floyd Warshall method, have near optimal complexity in terms of the total number of operations, while their matrix-based structure is regular enough to allow for efficient parallel implementation on the GPUs.
Regarding the regularity conditions in Definition 4.1, we note that (widetilde {F}) will be regular enough if and only if F is regular enough (because exp(X t)) is a sufficiently regular functional).
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Theorem 4.1 implies (C^{1,1}) regularity away from (Gamma ^1) in the case the coefficients (g_i) are regular enough to provide (C^{1,1}) solutions away from the free boundary, i.e. Theorem 1.2.
If the changes are regular enough, bacteria can respond ahead of time.
If the coefficients (g_i) are regular enough to provide (C^{1,1}) solutions up to the boundary this is encoded in Assumption C then we obtain full (C^{1,1}) regularity.
Lastly we point out that if the coefficients (g_i) are regular enough to provide (C^{1,1}) solutions at points where the gradient does not vanish, then we obtain full interior (C^{1,1}) regularity.
The unique selling point for retailers is that, while some power shoppers might download those stores' apps, Shopkick offers a "shopping mall" style experience where several stores can get their products and brands in front of shoppers who may not be regular enough to use their own bespoke apps.
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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