Continuous approximation

Finally, we assume that the fields 4>, p, and u vary slowly on the length scale of the lattice constant (the size of the molecules) and introduce continuous approximation for the thermodynamical-potential density. In the lattice model the only interactions between the amphiphiles are the steric repulsions provided by the lattice structure. The lattice structure does not allow for changes of the orientation of surfactant for distances smaller than the lattice constant. To assure similar property within the mesoscopic description, we add to the grand-thermodynamical potential a term propor-tional to (V u) - -(V x u) [15], so that the correlation length for the orientational order is equal to the size of the molecules. 

An engineer typically strives to treat discrete variables as continuous even at the cost of achieving a suboptimal solution when the continuous variable is rounded off. Consider the variation of the cost of insulation of various thickness as shown in Figure El.l. Although insulation is only available in 0.5-in. increments, continuous approximation for the thickness can be used to facilitate the solution to this optimization-problem. 

Critical Point. A point where two phases, which are continually approximating each other, become identical and form but one phase. With a liquid in equilibrium with its vapor, the critical point is such a combination of temperature and pressure that the specific volumes of the liquid and its vapor are identical and there is no distinction between the two states (Ref 2, pp 188-89)... 

Up to now we have been considering defect diffusion in continuous approximation, despite the fact that crystalline lattice discreteness was explicitly taken into account defining the initial distribution for geminate pairs. Note, however, that such continuous diffusion approximation is valid only asymptotically when defects (particles) before recombination made large number of hops (see Kotomin and Doktorov [50]). This condition could be violated for recombination of very close defects which can happen in several hops. The lattice statement of the annihilation kinetics has been discussed in detail by Schroder et al. [3, 4, 83], Dederichs and Deutz [34]. Let us consider here just the most important points of this problem. 

Now the effective recombination (annihilation) radius could be defined similarly to that in the continuous approximation... 

A theoretical analysis of the experimental kinetics for Vk centres in KC1-Tl, as well as for self-trapped holes in a-Al203 and Na-salt of DNA, is presented in [55]. The fitting of theory to the experimental curves is shown in Fig. 4.4. Partial agreement of theory and experiment observed in the particular case of Vk centres was attributed to the violation of the continuous approximation in the diffusion description. This point is discussed in detail below in Section 4.3. Note in conclusion that the fact of the observation of prolonged increase in recombination intensity itself demonstrated slow mobility of defects. In the case of pure irradiated crystals, it is a strong... 

Abstract. Calculations of the non-linear wave functions of electrons in single wall carbon nanotubes have been carried out by the quantum field theory method namely the second quantization method. Hubbard model of electron states in carbon nanotubes has been used. Based on Heisenberg equation for second quantization operators and the continual approximation the non-linear equations like non-linear Schroedinger equations have been obtained. Runge-Kutt method of the solution of non-linear equations has been used. Numerical results of the equation solutions have been represented as function graphics and phase portraits. The main conclusions and possible applications of non-linear wave functions have been discussed. 

After applying the continual approximation and restricting only two terms in above expansions the motion equations became following ... 

Figure 5. (a, top) Polarization as a function of the distance from one surface. The solution of the discrete approach (eq 40, circles) and its analytical interpolation (eq 41, line 1) are compared to the solution obtained via the continuous approximation (eq 31, line 2). (b, bottom) Interaction energy, as a function of separation distance, for the discrete approach and for the continuous approximation. 

Assuming that the average dipole moment remains constant in a layer regardless ofthe tilt angle, one obtains for the interaction coefficients the values listed in Table 1. Additionally, we computed the decay lengths for the parameter values given in section II.4 Ai from the continuous approximation, eq 27 A2 from the continuous approximation which treats the polarization of one layer as an average over its two sublayers, eq 30 and Ag from... 

An analogous result is valid for continuous approximations of r when up winding is performed by the streamline upwinding Petrov-Galerkin method (SUPG) [104]. The same is true for finite element methods based on a quadrangular mesh [105]. 

D. Saadri, Finite element approximation of viscoelastic fluid flow existence of approximate solutions and error bounds. Continuous approximation of the stress, SIAM J. Numer. Anal., 31 (1994) 362-377. 

Relation (7) presents an unrealistic infinite first derivative at t = 0, due to the continuous approximation made in the analytical treatment, and further refinements were proposed to avoid this defect. For example, under the continuous approximation, Bendler and Yaris have performed an arbitrary truncation in the mode analysis The expression for the OACF is then ... 

The limitation in equation (2.3.36) that vectors rm+i and cannot coincide with f m, r m becomes unimportant in the continuous approximation containing integrals instead of cell sums. The above mentioned changes in the DFs due to recombination correspond exactly to those done by Dettmann [82]. In the case of the A-I-B -> B reaction one must omit the last... 

Now we see the consequences of (size)-extensivity for the first time. The last term, which depends on <2), arises from the second term on the right in Eq. [29c] after multiplying on the left by and using the fourth-order energy formula. This term plays a critical role in distinguishing MBPT from Cl. If we continued approximating the CID coefficients by higher order perturbation theory, we would have... 

---