Part of solution verification consists in comparing computed output with analytic solutions (so called "benchmark solutions").
In particular, the so called homoclinics solutions, which are defined as nontrivial solutions x such that lim t → ± ∞ x ( t ) = 0, are studied.
The main motivation for computing the symmetries of a differential equation is the search for the so called invariant solutions.
The point we are interested in now is that u can be recovered via convolution with the so called fundamental solution G ( x, y ) ≈ | x - y | 2 - n if n ≥ 3 - log | x - y | if n = 2 (2)and this means that the following representation formula holds: u ( x ) = ∫ R n G ( x, y ) d μ ( y ).
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Moreover, Cosso and Russo (2016) introduced the so called strong-viscosity solution for semilinear PPDEs as the limit of approximating classical solutions.
In the same direction, a considerable attention has been directed in recent literature to the so called revenge problem: typical solutions, say, of the Liar paradox, rely on notions that, if expressible in the object language, give rise to new versions of the paradox.
When divalent cations are absent from the extracellular solution, so called divalent free (DVF) condition, CatSper is able to conduct monovalent ions, such as K+.
The glycosides are saponins, so called because they form soapy solutions and have other surface active (e.g., hemolytic) properties.
For each assembly function a manual and an automated solution or so called "functional module" is generated.
