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In the following, we check the mean square continuity of J 3 ( t ) on the fixed time t ≠ t k ( k = 1, 2, … ).
On the other hand, as t = t k, it is easy to check that J 1 ( t ), J 2 ( t ), J 3 ( t ) are continuous in mean square on the fixed time t = t k.
For a fixed time t > 0, it is easy to check that J 1 ( t ), J 2 ( t ), J 4 ( t ) are continuous in mean square on the fixed time t ≠ t k, for k = 1, 2, … .
Consequently, when t ≠ t k ( k = 1, 2, … ) in (10), it is easy to find that Q i → 0 as r → 0 for i = 1, …, 4, and so π ( x i ) ( t ) is continuous on the fixed time t ≠ t k ( k = 1, 2, … ).
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Note that the convergence constant C̃ also depends on the exact solution and the fixed time T.
We have shown with the diversification of Chondrostoma, that the pure-birth model with variable rates can be easily adapted to test for specific hypotheses, running the analysis on fixed time frames defined for example on the basis of geological events or climate changes.
Along this idea and omitting the terms when tends to zero, one observes that if we let tend to zero, on any fixed time interval the solution of (2.14) will tend to the solution (with the same initial condition) of the following "averaged" system (see [21] for more details), for any, (3.3).
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The paper culminates in a proof of the existence of oscillatory multidimensional shocks on a fixed time interval independent of the wavelength ε as ε→0.
Participants in the study were, on a fixed time interval, seen by a research nurse or specialized rheumatology nurse after their regular rheumatologist consultation.
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CEO of Professional Science Editing for Scientists @ prosciediting.com