The logarithmic derivative of a weakly second order probability measureμon a general vector spaceXis an example of an operator fromXintoL2 which preserves the pairing ofXandX′ under the inner product ofL2.
The data processing during the course of a typical experiment is illustrated in Fig. 1, for the case of a weakly first-order transition, a transition with a small latent heat and considerable pre-transitional contributions, as commonly observed in liquid crystals [2].
The scheme is an upwind second-order extrapolation with simple local limiters, and it is weakly second-order accurate and satisfies maximum principles.
Our first-principles calculations from [ 16], which agree quite well with simulations (see caption to Fig. 7), apply strictly only to the second-order case at low J and near the critical point, but we do not expect them to become meaningless when the transition is weakly first-order.
In Figure 6, we give the comparison among the numerical solution of system (1.3) (left), the weakly nonlinear first order approximation of the solution (medium) and the weakly nonlinear fifth order approximation of the solution (right) under subcritical case.
Figure 8 gives the comparison among the numerical solution of system (1.3) (left), the weakly nonlinear first order approximation of the solution with stable equilibrium ((A_{1infty},0)) (medium) and the weakly nonlinear first order approximation of the solution with stable equilibrium ((0, A_{2infty})) (right) under double and non-resonant eigenvalue case.
In Figure 4, we compare between the numerical solution of system (1.3) and the weakly nonlinear first order approximation of the solution under supercritical circumstance.
We give the comparison between the numerical solution of system (1.3) (left) and the weakly nonlinear first order approximation of the solution (right) with the only stable state (R^= 0, 1.9391)) under double and resonant eigenvalue case.
We give the comparison between the numerical solution of system (1.3) (left) and the weakly nonlinear first order approximation of the solution (right) with the stable state (H_{1}^=(2.2682, -0.0559)) in the double and resonant eigenvalue case.
Figure 9 gives the comparison between the numerical solution of system (1.3) (left) and the weakly nonlinear first order approximation of the solution (right) in the double and non-resonant eigenvalue case.
