Boundary condition truly

There are two issues that are relevant here. First, the adsorbates in a supercell calculation necessarily have a long-range pattern that is repeated exactly as the supercell is repeated. With periodic boundary conditions, it is impossible to model any kind of truly random arrangement of adsorbates. The good news is that in nature, it happens that adsorbates on crystal surfaces often do exhibit long-range ordering, especially at low temperatures, so it is possible for calculations to imitate real systems in many cases. 

The boundary conditions, while not exact, are close to those of Example 2.10. They would be truly analogous if we make the following transformation ... 

The solution at a given point depends only partly on the initial and boundary conditions, depending on the wave speed a. As illustrated in the lower right-hand panel of Fig. 3.14, the domain of influence slopes away from the point. The higher the sound speed, the shallower becomes the slope. For a truly incompressible fluid, where the sound speed in infinite, the slope approaches zero and characteristics become essentially parabolic. 

Apropos of the analogy noted by Batchelor between vortex velocity and a magnetic field, it should be noted that for the realization of truly steady turbulence a supply of mechanical energy is necessary. The supply of energy comes about either through nonpotential volume forces or through the motion of the surfaces bounding the fluid. With these factors taken into consideration, the set of equations and boundary conditions for a vortex does... 

The conditions on velocity at the outer edge" of the boundary layer arc a little more difficult to define because there is really an interaction between the boundary layer flow and the outer inviscid flow, i.e., because of the reduction in velocity near the surface, the outer flow is somewhat different from that in truly inviscid flow over the surface. However, as discussed above, in many cases, this effect can be ignored because the boundary layer remains thin and in such cases the inviscid flow over the surface considered is calculated and the value that this solution gives for the velocity on the surface at any position is used as the boundary condition at the outer edge" of the boundary layer, i.e., if ui(jc) is the surface velocity distribution given by the solution for inviscid flow over the surface then the boundary coofhtion on the boundary layer solution is... 

Single-phase simulations in relatively small boxes (20 -30 sites) were performed after the determination of the surfactant/silica liquid crystal composition to generate model materials for adsorption simulation. The advantage of generating such materials is that they have truly periodic boundary conditions in the three directions. In contrast, extracted model materials obtained directly from the interfacial simulation will have periodic boundary conditions in two directions, but in the third direction (z-direction) there are two liquid-crystal/dilute phase interfaces. 

In the confined fluid problem, in the absence of time-dependent shearing boundary conditions, we needed a massive thermostating scheme to stabilize the shear flow. In this scheme, each degree of freedom represented in Eqs. [202] is attached to a separate thermostat, to make the p,- of every particle truly peculiar with respect to the flow profile. The use of one global thermostat for... 

The rate model represented by equation 6.9 is particularly interesting since it will be shown shortly that a similar function arises from a consideration of formal diffusion theory. Therefore, providing it is established by experiment that the pseudo rate constant is truly constant over the range of experimental boundary conditions employed a, b, Xg) it remains perfectly valid to equate its value to appropriate mass transfer parameters required by diffusion theory. 

Resonances are not truly bound states, but they are interpreted as metastable states. Because of the boundary conditions of resonances, the problem is not Hermitian even if the Hamiltonian is (that is, for square integrable eigenfunctions). Resonances are characterized by complex eigenvalues (complex poles of the scattering amplitude)... 

We must be aware that equilibrium calculations and log-concentration diagrams of systems involving heterogeneous equilibria may only provide us with the boundary conditions of the system rather than the situation that truly exists. Thus equilibrium calculations involving solids in waters and wastewaters are generally less representative of the true situation than acid-base equilibrium calculations for the following reasons. 

Graphical presentation of redox equilibria, like the graphical treatment of acid-base, complexation, and precipitation equilibria is helpful in understanding complicated problems and in obtaining approximate solutions to equilibrium questions. For redox systems in natural waters the equilibrium condition is truly a boundary condition. In many cases, natural systems are not at equilibrium from a redox standpoint. The diagrams usually present an idea of what is possible, not necessarily of the existing or imminent situation. The graphical presentations of redox equilibria are seldom simple because redox reactions usually involve... 

The stresses and displacements at the boundaries of the deformable domain (applied or enforced) dictate the boundary conditions, which effectively determine the particular solution to the PDFs. If the problem is truly three-dimensional (because geometry and/or loads vary in all three directions) then numerical solutions are needed for all but the most elementary example cases. FFA is the de facto choice for such scenarios and commercial codes abound ABAQUS and ANSYS (software programs) are arguably the most common. An example of a simple geometry that is nevertheless a truly three-dimensional problem is a film capping an elliptical hole, that is, an elhptical plate or membrane. 

For moderately dense coulombic systems the use of Ewald potentials— that is, of truly periodic boundary conditions in the energy calculation—seems to have become almost routine in recent studies. It is also proposed for dipole systems. There seem, however, to be theoretical arguments for examining more critically the consequences of this approximation. These theoretical questions concern the physical realism of the approximation, and may be divided into those of long-range behavior and those of short-range behavior of the truly periodic model ... 

State where one electron is bound and the other free. Worse yet, the channel states in scattering theory that need to be constructed can only be identified by an asymptotic boundary condition, a region where a finite Hilbert basis representation will be inaccurate. What is needed is a method which relies on information about the wave function in the target region to accurately identify and separate processes which are only truly separable in the asymptotic region. This is the interpretation problem. 

Input boundary condition At the inflow boundary, a theoretical particle velocity profile is fed along the vertical input boundary. The exact velocity profile of a truly nonlinear wave under the given condition is not known a priori. Therefore, the best theoretical wave profile is input along the input boundary. Since the fully nonlinear free-surface condition is applied in the computational domain, the input wave immediately takes the feature of fully nonlinear waves. Any unnecessary spurious waves inside the domain is accounted for and corrected. 

Cases (c) and (d) both represent truly ferroelectric behavior, which can be found in smectic C phases of chiral molecules only if the helix is suppressed by the boundary conditions at the liquid crystal glass interfaces. Because it is nearly impossible to align columnar phases uniformly with rubbed or otherwise anisotropic polymer surfaces (the switching is generally studied in polydomain or sheared samples), and because the ferroelectric behavior remains present in thick cells, surface induction of ferroelectricity can be excluded. It must be assumed that the two-dimensional column lattice suppresses helix formation in these materials. 

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