Take the Δ j operator and add the term ( S j u m, ∇ ) Δ j u m on both sides of (2) to have ∂ ∂ t Δ j u m + ( S j u m, ∇ ) Δ j u m = ( S j u m, ∇ ) Δ j u m − Δ j ( u m, ∇ ) u m − Δ j ∇ p m. Consider the trajectory flow { X j m ( x, t ) } along S j u m defined by the solutions of the ordinary differential equations { ∂ ∂ t X j m ( x, t ) = ( S j u m ) ( X j m ( x, t ), t ), X j m ( x, 0 ) = x.
This trajectory flows forward from ((h_{min}(I),s_{max})) under (3) with EP active, say to ((h_{mathit {EP}},s_{mathit {EP}})), and then continues forward under (3) with ER active from ((h_{mathit {EP}},s_{max})) (Fig. 12).
In one region, the trajectories flow to the stable fixed point (i.e., rest potential) quickly without firing; in the other region, the trajectories follow a large excursion (i.e., action potential) before returning back to the stable fixed point.
We then use the derived quasi-potential as a measure of "elevation" to quantitatively map the epigenetic landscape, on which trajectories flow "downhill" from any location.
It is rather a Liapunov function of the dynamical system represented by the gene network, along which trajectories flow monotonically "downhill" towards the steady states of the network [ 41].
The change in the quasi-potential, Δ V q, can be rewritten from Eq. 4 as: (5) For positive increments in time Δ t, Δ V q is thus always negative along an evolving trajectory, ensuring that trajectories flow "downhill" along a putative "quasi-potential surface".
An analysis of the particle trajectories (flows) provides a rational way to create chaotic (or good) mixing.
Emphasis is given to the shoreline trajectory and flow velocity modelling.
This paper focused on the trajectory and flow patterns of submerged horizontal gas jet in stagnant ambient liquid.
The non-invasive positron emission particle tracking (PEPT) technique was used to observe and quantify particle trajectory, solids flow pattern, solids velocity, and solids circulation frequency.
This study demonstrates that hazardous air pollutants from industrial accidents can successfully be identified and traced back to their emission sources by applying a timely and comprehensive air-monitoring campaign and back trajectory air flow models.
