Asymptotic approximation defined

The curves 1 in Figs. 4.6a and b show the functions Fr and FA calculated by formulae (4.3.35) and (4.3.38) for the case of normal molecular orientations (e Oz) and plotted versus the argument AQ/( +AQ). The dimensionless argument and functions of this kind normalized with respect to the sum of the resonance and the band widths were introduced so as to depict their behavior in both limiting cases, ACl rj and Af2 77. The deviation of the solid lines from the dotted ones indicates to which degree the one-parameter approximation defined by Eq. (4.3.38) differs from the realistic dispersion law. As seen, this approximation shows excellent adequacy, but for the region AQ r/, where the asymptotic behavior of the approximation (4.3.38) and Eq. (4.3.35) are as follows ... 

We can use the asymptotic approximation for the LC gain plot as we did for the RC filter. However, the problem with trying to do the same with the phase of the LC is that there will be a very large error, more so if the Q (defined in the figure) becomes very large. If so, we can get a very abrupt phase shift of 180° close to the resonant frequency. This sudden phase shift in fact can become a real problem in a power supply, since it can induce conditional stability (discussed later). Therefore, a certain amount of damping helps from the standpoint of phase and possible conditional stability. 

The task is to find the asymptotic approximation of the solution as e- 0 uniformly over a long time interval 0main goal. The higher order corrections of 0(e) will be defined in order to identify both the secular terms and the time intervals of asymptotic fitness. 

It is necessary to note the essential physical difference between the system (9) and its asymptotic approximation at <5 0 that is the equation (14). The system (9) at finite values S describes the physical waves and is suitable to comparison with experiments. The asymptotic equation (14) simulates the mathematical waves of unbounded length and infinitesimal amplitude and is not included parameters connecting with experimental conditions. This circumstance defines the preference of (9) before (14). It is important to note that mathematical model for nonlinear waves is reduced to single equation in the limiting case (5 0 only. That model includes a system of two equations for finite S values. 

For At -> O the probability density defined in (17) can be asymptotically approximated for the discrete case as... 

So the MCT does not provide an analytical form of the memory function, as other theoretical models do, but it defines a general hierarchical way of building it. Clearly this definition of the memory introduces a self-coupling phenomena in the correlation d5mamics. These coupled equations, (2.47) and (2.48), can be solved using a few asymptotic approximations. The solution of these equations provides an analytical description of the density dynamics, called asymptotic results. 

In practice most distribution cuiwes are not symmetrical about the median but are inherently skewed. The effect of an advertising campaign is usually to increase the rate of sales in the early years. It may also increase the level of mature demand for the product, but this mature demand must be asymptotic to a finite upper Emit of sales c. Such a cui ve is positively skewed since xm — xd) < x(j —x ). This situation can often be approximated by the Gompertz cui ve defined by Eq. (9-96) ... 

In the impact approximation (tc = 0) this equation is identical to Eq. (1.21), angular momentum relaxation is exponential at any times and t = tj. In the non-Markovian approach there is always a difference between asymptotic decay time t and angular momentum correlation time tj defined in Eq. (1.74). In integral (memory function) theory Rotc is equal to 1/t j whereas in differential theory it is 1/t. We shall see that the difference between non-Markovian theories is not only in times but also in long-time relaxation kinetics, especially in dense media. 

The modulus defined by eqn. (10) then has the advantage that the asymptotes to t (0) are approximately coincident for a variety of particle shapes and reaction orders, with the specific exception of a zero-order reaction (n = 0), for which t = 1 when 0 < 1 and 77 = 1/0 when 0 > 1. The curve of 77 as a function of 0 is thus quite general for practical catalyst pellets. Figure 2 illustrates the form of For 0 > 3, it is found that 77 = 1/0 to an accuracy within 0.5%, while the approximation is within 3.5% for 0 > 2. The errors involved in using the generalised curve to estimate 77 are probably no greater than the errors perpetrated by estimating values of parameters in the Thiele modulus. 

The concentration and temperature Tg will, for example, be conditions of reactant concentration and temperature in the bulk gas at some point within a catalytic reactor. Because both c g and Tg will vary with position in a reactor in which there is significant conversion, eqns. (1) and (15) have to be coupled with equations describing the reactor environment (see Sect. 6) for the purpose of commerical reactor design. Because of the nonlinearity of the equations, the problem can only be solved in this form by numerical techniques [5, 6]. However, an approximation may be made which gives an asymptotically exact solution [7] or, alternatively, the exponential function of temperature may be expanded to give equations which can be solved analytically [8, 9]. A convenient solution to the problem may be presented in the form of families of curves for the effectiveness factor as a function of the Thiele modulus. Figure 3 shows these curves for the case of a first-order irreversible reaction occurring in spherical catalyst particles. Two additional independent dimensionless paramters are introduced into the problem and these are defined as... 

The systems considered here are isothermal and at mechanical equilibrium but open to exchanges of matter. Hydrodynamic motion such as convection are not considered. Inside the volume V of Fig. 8, N chemical species may react and diffuse. The exchanges of matter with the environment are controlled through the boundary conditions maintained on the surface S. It should be emphasized that the consideration of a bounded medium is essential. In an unbounded medium, chemical reactions and diffusion are not coupled in the same way and the convergence in time toward a well-defined and asymptotic state is generally not ensured. Conversely, some regimes that exist in an unbounded medium can only be transient in bounded systems. We approximate diffusion by Fick s law, although this simplification is not essential. As a result, the concentration of chemicals Xt (i = 1,2,..., r with r sN) will obey equations of the form... 

DFT calculations [17,331,332] of 02/Pt(l 11) find the two different molecular precursors defined in the discussions above and observed by STM [153,326]. However, DFT calculations of 02 dissociation on surfaces are problematical and two different standard approximations for the exchange correlation functional (PW91 and RPBE) get somewhat different results. The DFT calculations are in only fair agreement with experiments. Most troubling is that all DFT barriers to dissociation from both molecular precursors are significant, 0.6-0.9 eV relative to the 02+ Pt(lll) asymptote. This is qualitatively inconsistent with experiment, which shows that thermal dissociation is energetically favored over desorption. 

Solution to the nondimensional axisymmetric stagnation-flow problem is plotted in Fig. 6.3. Since the viscous boundary layer merges asymptotically into the inviscid potential flow, there is not a distinct edge of the boundary layer. By convention, the boundary-layer thickness is defined as the point at which the radial velocity comes to 99% of its potential-flow value. From Fig. 6.3 it is apparent that the boundary-layer thickness S is approximately z 2. In addition to the boundary-layer thickness, a displacement thickness can be defined. The displacement thickness is the distance that the potential-flow field appears to be displaced from the surface due to the viscous boundary layer. If there were no viscous boundary layer (i.e., the inviscid flow persisted right to the surface), then the axial velocity profile would have a constant slope du/dz = —2. As shown in Fig. 6.3, projecting the constant axial-velocity slope to the surface obtains an intercept of u = 0 at approximately z = 0.55. Since the inviscid flow would have to come to zero velocity at the surface, z = 0.55 is the distance that the potential flow is displaced due to the viscous boundary layer. Otherwise, the potential flow is unaltered by the boundary layer. 

Remark. Equation (7.14) is strictly correct in the sense that it is the first term of a mathematically well-defined asymptotic expansion in 1/y. To estimate whether it is also a good approximation we require that y 1 P(1 differential operator on the left-hand side of (7.9) has eigenvalues of order 1, the zero eigenvalue having been extracted by the solubility condition. Thus P(1 is comparable with the right-hand side, i.e., of order dP(0 /dx — /P(0. Our requirement amounts therefore to... 

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. 

The wavenumbers kn are defined in Equation (2.52) and the potential coupling matrix elements are given in (3.6) with Vj being the interaction potential, i.e., Equation (12.1) without the asymptotic oscillator potential vBc(r) Figure 12.2 shows the diagonal matrix elements Voo(-R) and Vn(.R). As a result of the weak dependence of V/ on r, they are at all distances R vertically displaced by approximately the energy difference between the two vibrational states, i.e., they run roughly parallel. 

Derived from linear approximation of the equations (3.37), the equilibrium correlation function (4.29), defines two conformation relaxation times r+ and r for every mode. The largest relaxation times have appeared to be unrealistically large for strongly entangled systems, which is connected with absence of effect of local anisotropy of mobility. To improve the situation, one can use the complete set of equations (3.37) with local anisotropy of mobility. It is convenient, first, to obtain asymptotic (for the systems of long macromolecules) estimates of relaxation times, using the reptation-tube model. 

Caprani et al. [104], defining the cut-off frequency as the intersection of the low and high-frequency asymptotes, as indicated on Fig. 10.18, have given an approximate method to deduce the size of active sites on a partially blocked electrode from the ratio of the two cut-off frequencies ... 

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