Then ((Sigmatimes mathbb {R}^{n}, langle,rangle_{Sigma} +langle,rangle_{mathrm{can}}, e^{-f},dnu _{Sigma}otimes,dnu _{mathbb {R}}^{n}, kappa)) is a non-trivial gradient Ricci soliton.
However, the number of tungsten shield layers plays a trivial role on the temperature gradient between the ALN powder sources and the crucible top.
However, handling the gradient data is not trivial because NS and EW differences of Swarm measurements only provide information about the horizontal and not on the vertical gradient.
If the IR-flux has a broad gradient, the heating rate is trivial for timescales of a few hours.
Ji et al. [16] improve the tracking accuracy by adding an l 2 norm regularization on the trivial coefficients and use an accelerated proximal gradient approach for solving the minimization problem which has achieved both tracking accuracy and computational efficiency.
The last point to study (and it is not trivial at all) is the fact that every gradient flow of (mathcal {C}) (w.r.t. the (L^2) distance) is also an EDE gradient flow of (mathcal {E}) for the (W_2) distance.
The core of the criticism is that it is not trivial to establish a stable, noise-resistant and accurate gradient spanning sufficiently relevant distances over the embryo, solely through production, degradation and diffusion of a morphogen alone [ 19, 21- 28].
In the above parental care example, the evolutionary dynamics governed by the selection gradients (Equations 19, 20, 22, and 23) eventually reach the trivial equilibrium phenotypes of maximum care for sons and daughters [ sm zm, zf) = sf zm, zf) = 1].
Discretisation of the horizontal pressure gradient force in layered ocean models is a challenging task, with non-trivial interactions between the thermodynamics of the fluid and the geometry of the layers often leading to numerical difficulties.
Notice that when ͌ q = ͌ T, the response between the temperature gradient and the heat flux is instantaneous and Equation 1 reduces to Fourier law except for a trivial shift in the time scale.
Non-trivial implications of the pH-gradient driven micro-capillary transport are aptly emphasized, so as to offer significant physical insights on the adopted strategy as a guiding principle for facilitating capillary filling processes by inducing a modulation in the effective interfacial energy.
