Solution Uniqueness

The non-covalent bond length dL, molecular size dn, molecular separation doo. and the mass density p can be obtained by solving the equation with any one of these parameters as a known input [30], 

38 Molecular Size, Separation, Packing Order, and Density 

The straightforward yet simple solution has thus resolved the seemingly independent geometry and dimension uncertainties of water and ice with the following 

Focusing on the statistic mean of the complete set of all the factors and their cooperativity involved is reliably more revealing than on the instantaneous accuracy of the individual parameter once at a point of time for the strongly correlated and fluctuating water system. 

The size, separation, structural order, and mass density of molecules packing in water and ice are correlated, which is independent of the structural phases of water and ice or the probing conditions. 

---

Apply the simplex rules to minimize /for the formulation. Is the solution unique ... 

To make the solution unique it is not sufficient to fix the initial distribution P(y, 0) in (L, R), but one also needs suitable boundary conditions at L and R, expressing the physical nature of these boundaries. The particle has to know what to do when it arrives at a wall. We have already encountered the condition P = 0, which expresses absorption, and the reflecting boundary condition... 

One essential question is, How many variables must be specified to obtain a solution unique to the phase equilibrium calculation It is possible to have an infinite number of solutions to a problem if too few variables are specified—or no solution if too many variables are specified. One answer to this question is provided by Gibbs Phase Rule (Gibbs, 1928, p. 96), simply stated for nonreacting systems by the equation ... 

To choose the proper solutions uniquely, Pk o),Qk Q)> k = 1,2 must be subject to the following boundary conditions ... 

As the active species is formed on the electrode surface at an interface between solid and solution, unique characteristics in reactivity can occur that are useful to the organic chemist. 

In contrast to the previous example, the state of this system is given by two variables its current angle x and angular velocity x. (Think of it this way we heed the initial values of both X and x to determine the solution uniquely. For example, if we knew only x, we wouldn t know which way the pendulum was swinging.) Because two variables are needed to specify the state, the pendulum belongs in the n = 2 column of Figure 1. .3.1. Moreover, the system is nonlinear, as discussed in the previous section. Hence the pendulum is in the lower, nonlinear half of the n=2 column. 

The coefficients in this equation depend on the approximations used for the gradients and cell face values of the pressure correction. The part which stems from the velocity correction is identical to that for the incompressible case. The second part depends on the approximation used for the term which corresponds to the convective contribution to the conservation equations. The presence of convective terms in the compressible pressure equation makes the solution unique. 

Thus, for the two-dimensional vector function IIoZ only one condition (3.12) is given. This condition does not specify the solution uniquely. To find IloZ and other II terms, we demand that they decay to zero as... 

Detection No unique solution Unique solution Unique solution... 

As in the liquid, evaluating the two additional terms exactly is as difficult as the original problem. However, because of their physical interpretation it is possible to provide simple yet realistic approximations for them. In practice, one sometimes solves the equations of motion not only for the molecular degrees of freedom but also for those surface atoms to which they are directly coupled. Only the rest of the solid is averaged over. This structureless, pillow-like description of the environment has enabled the method of classical trajectories to be applied not only to reactions at the surface but also in solution. Unique to the surface is the need to allow for electron-hole pair excitations. 

For particular. 4 and b, if a solution x e 91 exists, when is it the only solution (uniqueness of solution)... 

---