Numerical results

The results discussed here apply to an infinite, periodic chain. Identical results are obtained for infinite, linear chains, except that the centre-of-mass momentum, K, is replaced by the pseudo-momentum (3j = jTr/ Nu + l)d. Replacing K by /3j, eqn (E.7) indicates that the exciton energies scale as (j/Nu) in the large Af limit. 

The analysis for this limit is very similar to that of the weak-coupling limit, except that now the hardcore repulsion imposes the boundary condition that V (O) = 0 on all the solutions. Thus, degenerate pairs of even and odd solutions are found by matching (r) with tp —r) as r — 0. So, setting the molecular-orbital 

In this appendix we use the effective-particle exciton models introduced in Chapter 6 to calculate transition dipole moments. These results are summarized in Chapter 8. 

In the weak-coupling limit a general excited state is of the form, 

We now use these equations to calculate the matrix elements for the transitions. 

McMillan has obtained numerical solutions to Eqs. [20] through [23] for several sets of the potential parameters 8 and a. In his first paper he did not yet realize the need for the U(r) term in the potential, Eq. [12], so that he was treating the case of 5 = 0. In his subsequent paper he included the missing term. These two sets of results are now examined. 

When 6 = 0, the term in r drops out of the potential, Eq. [18], and the theory now involves only the order parameters rj and a. The translational order of the smectic-A phase is then described only by the mixed-order parameter a. Fig. 3 shows the temperature depen-  

The collection of phase transition temperatures obtained from the model with 5 = 0 and for numerous values of a is summarized in the phase diagram shown in Fig. 4a. These results are to be compared with the schematic phase diagram shown in Fig. 4b this diagram is representative of many real systems and displays the features discussed in Section 3. Note that while there is qualitative agreement between the model and experiment, there are numerous significant discrepancies in the overall behavior of the phase transition lines. 

In his second paper McMillan included all the terms in the potential, Eq. [18] and studied the model for several values of 5 and a. He was particularly interested in two substances, cholesteryl nonanoate and cholesteryl myristate. The nonanoate calculations were made using a 0.41 5 = 0 and 0.65 for the myristate, a 0.45 was chosen with 5 = 0 and 0.65. With 5 = 0.65 the temperature dependences of the order parameters for both values of a were similar in appearance to those shown in Fig. 3b. In both examples rj looked like the rj of Fig. 3b while r and a resembled the a of Fig. 3b. The model thus displayed successive first-order phase changes—smectic-A to cholesteric fol- 

The statistical thermodynamics is based on the variational principle. Guided by Eqs. [18] and [19], the following variational form was chosen for the single molecule distribution function (of the i -th molecule)  

In order to study the performance of each of the rules, we will determine the average costs for some examples. The size of the examples is small, so we can compute the costs of each production rule. We will consider the following rules  

WW(//) the Wagner-Whitin-like production rale with H periods  

We have considered two sets of simple examples. In the first set we have a maximum lead time A =4, penalty costs p=3, holding costs h = l and a binary demand dii= -dio=d for i=l.4. For some different values of d and s we have the following results  

In the second set of examples we have a maximum lead time N=2, penalty costs p s3, holding costs h = l and a Poisson-distributed demand with parameter 1 for both lead times. For three different vdues of the set-up costs s the results are given in Table 4.2. 

From these examples one might get the impression that the SM-rule is always better than the optimal (x,r)-rule. This however, is not the case if in the second set of examples we take a Poisson-distributed demand with parameter 1.01 and set-up costs r=9 then the average costs of the SM-mle are 5.9308, whereas the average costs of the optimal (x,r)-rule are 5.9082. 

An acceptable experimental value at the present time is (see Table 6.1) 

The agreement between the more accurately calculated values, as represented by the Monte Carlo and consistent P approximations in Table 12.1, and the experimental result is seen to be quite good. 

As an example of the results which may be obtained by use of the different approximations consider the following reactor system  

Reflector Spherical shell of pure water, 10.56 cm thick 

For the analysis of this system the lethargy scale was divide into 29 fast groups plus a thermal group corresponding to 183°F. The spatial integration was carried out using intervals of 0.754 cm which yielded 16 space points in the core and 14 in the reflector. The results of this computation are given in Table 12.2. 

In this section we consider typical examples. They cover all possible cases that could be encountered during the regression of binary VLE data. Illustration of the methods is done with the Trebble-Bishnoi (Trebble and Bishnoi, 1988) EoS with quadratic mixing rules and temperature-independent interaction parameters. It is noted, however, that the methods are not restricted to any particular EoS/mixing rule. 

The anode side of the cell is grounded and, hence, = 0. Solution of Equation 5.217, subject to Equation 5.218, is obtained by varying the only free parameter in this problem, the carbon phase potential of the cathode 4 . Eor simplicity, 5 in Equations 5.207 and 5.213 is taken to be the same. 

The steep gradient of d induces large proton current in the membrane along the z-axis. To support this current, on the anode side of the D/R interface, a virtual fuel cell forms. 

Eigure 5.32 shows the same as in Eigure 5.31 curves for the hundred times smaller oxygen-limiting current density on the anode side (yjj = 0.02 A cm ). This mimics the situation when in the R-domain oxygen penetrates to the anode side through the membrane (crossover). 

Variation of the thickness s of the concentration transition region in Equations 5.207 and 5.213 does not affect the peak value in Eigure 5.31d. Parameter s only scales the width of this peak. At smaller 5, the peak width is lower, while larger s increases it. 

Note that the model above describes the step in O approximately. The calculation of the product with O, resulting from Equation 5.217, shows that the left- 

For relatively large capillary numbers, Ca, the film thickness Hq becomes thick enough so that the effect of the disjoining pressure can be neglected. At the same time. A 0.643, B 2.79 (these values were obtained in Reference 23) and Ao 1 - 0, 643 OCaf -311. 

The dimensionless thickness of the film remaining on the fiber as a function 

FIGURE 3.19 Dimensionless film thickness //g = 1% a on the capillary number, Ca. (1) without disjoining pressure, according to Equation 3.251 and (2) with consideration of the disjoining pressure, according to Equation 3.238. Experimental point from Reference 29. 

Let us detennine the critical velocity, Ca characteristic of the transitional velocity from small to large capillary numbers as the point of intersection of the lines In = const and In 1.33 + 2/3 In Ca. In our case, InCa = 1Z85. The critical velocity obtained from the experimental data in Reference 29 is equal to In Ca, = -13.26, that is, close to the above theoretical prediction. 

---

In the standard method, the metal enclosure (called the air chamber) used to hold the hydrocarbon vapors is immersed in water before the test, then drained but not dried. This mode of operation, often designated as the wet bomb" is stipulated for all materials that are exclusively petroleum. But if the fuels contain alcohols or other organic products soluble in water, the apparatus must be dried in order that the vapors are not absorbed by the water on the walls. This technique is called the dry bomb" it results in RVP values higher by about 100 mbar for some oxygenated motor fuels. When examining the numerical results, it is thus important to know the technique employed. In any case, the dry bomb method is preferred. 

In this section, two illustrative numerical results, obtained by means of the described reconstruction algorithm, are presented. Input data are calculated in the frequency range of 26 to 38 GHz using matrix formulas [8], describing the reflection of a normally incident plane wave from the multilayered half-space. 

Figure C3.5.6 compares the result of this ansatz to the numerical result from the Wiener-Kliintchine theorem. They agree well and the ansatz exliibits the expected exponential energy-gap law (VER rate decreases exponentially with Q). The ansatz was used to detennine the VER rate with no quantum correction Q= 1), with the Bader-Beme hannonic correction [61] and with a correction based [83, M] on Egelstaff s method [62]. The Egelstaff corrected results were within a factor of five of experiment, whereas other corrections were off by orders of magnitude. This calculation represents the present state of the art in computing VER rates in such difficult systems, inasmuch as the authors used only a model potential and no adjustable parameters. However the ansatz procedure is clearly not extendible to polyatomic molecules or to diatomic molecules in polyatomic solvents.

After the assembling of the stochastic matrix Pd we have to solve the associated non-selfadjoint eigenvalue problem. Our present numerical results have been computed using the code speig by Radke AND S0RENSEN in Matlab,... 

These results allows us to connect the observed hysteresis to the conformational changes in the NA molecule and consider it not as a macroscopic phenomenon like capillary hysteresis, but as natural property of the NA-water system. Our experimental and numerical results are in agreement with the data of other authors [13], [12], [14]. 

Fig. 1. The dependence of the stable stationary values of the adsorption and conformational variables on the control parameter, Xe. a-total adsorption per the mole of the nucleotides, b-the probability of finding of an arbitrary NA unit in the A form, c-the probability of finding of an arbitrary NA unit in the B-form. Param-(ders values used to obtain numerical results Vmi = 3,nL = 15.4, = 3.24,6° =...

Numerical Experiment We now present numerical results for non-zero Vi and V2- In particular, we take... 

IXDCf is faster than MINDO/3, MNDO, AMI, and PM3 and, unlike C XDO, can deal with spin effects. It is a particularly appealing choice for UHF calculations on open-shell molecules. It is also available for mixed mode calculations (see the previous section ). IXDO shares the speed and storage advantages of C XDO and is also more accurate. Although it is preferred for numerical results, it loses some of the simplicity and inierpretability of C XDO. 

Although Vickers and DPH microhardness tests should yield the same numerical results on a given material, such is not always the case. Much of the observed variance may be a function of differences ia the volume of sample material displaced by the macro and micro iadentations. 

The numerical results obtained by using either Eq. (14-31) or Eq. (14-33) are identical. Thus, the two equations may be used interchangeably as the need arises. 

The numerical results for the case of two counterions of equal valence where a resin bead, initially partially saturated with A, is completely converted to the B form, is expressed by ... 

This equation has the same form of that obtained for solid diffusion control with D,j replaced by the equivalent concentration-dependent diffusivity = pDpj/[ pn]Ki l - /i,//i)) ]. Numerical results for the case of adsorption on an initially clean particle are given in Fig. 16-18 for different values of A = = 1 - R. The upt e curves become... 

The pumped-discharge case is generally more difficult to solve because of the uncertainty in deahng with negative numerical results. As a final answer, a negative value could indicate that the pump has completely emptied the tank however, as an intermediate value, it could mean that it is not a true solution. A simple check is to try a different initial estimate and see if the intermediate negative results disappear. 

There is a very good correspondence between the analytical and numerical results for temperature and velocity. The airflow rates differ, however, with a factor of 1.46 at a height of 2 m, whereas the correspondence at 4 m is very good. 

FIGURE 10.71 The ratio of the horizontal component of the fluid velocity to its local maximum u/ u, as a function of t), at x = 0.75 m for the experimental results, the numerical results obtained using the commercial package for the offset and equivalent wall jet models, and the original and modified Verhoff empirical formulae. 

The ordinary differential equations for f and C now form a fifth-order system which can be solved using a standard NAG library routine. The results are shown in Fig. 10.73. This figure also shows the numerical results for concentration obtained using a full numerical approach, and there is reasonable agreement between the two. 

At the corner (b, 2ho) of the region, five stress conditions apparently govern the behavior. However, the problem would be overspecified if all five conditions were imposed at the same time. Rather, three are specified and, subsequently, the remaining two are seen to be automatically satisfied thereby acting as a built-in verification of the numerical results. Numerical experimentation revealed that the choice of the three conditions is immaterial the remaining two are always satisfied. 

One of the first solutions to the problem of stresses around an elliptical hole in an infinite anisotropic plate was given by Lekhnitskii [6-7]. A more recent and comprehensive summary of the problem and many others is Savin s monograph [6-8]. Numerous results by Lekhnitskii are shown in his books [6-9 and 6-10]. Two special cases are of particular interest. 

Pagano presented numerical results for several laminates made of a high-modulus graphite-epoxy composite material with... 

M. Lozada-Cassou, E. Diaz-Herrera. Three-point extension for hypernetted chain and other integral equation theories numerical results. J Chem Phys 92 1194-1210, 1990. 

Numerical results for the some model polydisperse systems have been reported in Refs. 81-83. It has been shown that the effect of increasing polydispersity on the number-number distribution function is that the structure decreases with increasing polydispersity. This pattern is common for the behavior of two- and three-dimensional polydisperse fluids [81] and also for three-dimensional quenched-annealed systems [83]. 

For adsorbates out of local equilibrium, an analytic approach to the kinetic lattice gas model is a powerful theoretical tool by which, in addition to numerical results, explicit formulas can be obtained to elucidate the underlying physics. This allows one to extract simplified pictures of and approximations to complicated processes, as shown above with precursor-mediated adsorption as an example. This task of theory is increasingly overlooked with the trend to using cheaper computer power for numerical simulations. Unfortunately, many of the simulations of adsorbate kinetics are based on unnecessarily oversimplified assumptions (for example, constant sticking coefficients, constant prefactors etc.) which rarely are spelled out because the physics has been introduced in terms of a set of computational instructions rather than formulating the theory rigorously, e.g., based on a master equation. 

The distribution function of the vectors normal to the surfaces,/(x), for the direction of the magnetic field B, in accord with the directions of the crystallographic axis (100) for the P, D, G surfaces, is presented in Fig. 6. The histograms for the P, D, G are practically the same, as they should be the differences between the histograms are of the order of a line width. The accuracy of the numerical results can be judged by comparing the histograms obtained in our calculation with the analytically calculated distribution function for the P, D, G surfaces [29]. The sohd line in Fig. 6(a) represents the result of analytical calculations [35]. 

A. Numerical results profiles, center of gravity, and capacitance... 

---