Cubic model

Besides the void fraction a very important parameter is the size of the particles. Using a simple cubic model for packing spherical balls, which represent the particles, we get the following expression for the void fraction... 

Fit the same data in Problem 2.11 using a quadratic fit. Repeat for a cubic model (y = c0 + cxx + C2X2 + C3JC3). Plot the data and the curves. 

Laviron has studied an especially interesting class of nitro compounds containing a second basic site, e.g. 4-nitropyridine (14)33. Even two-dimensional representations such as those encountered earlier (Schemes 2-4) are inadequate to represent this mechanistically very complex situation. Laviron showed, however, that the electrochemical conversion of 14 to the corresponding ArN(OH)2 species can be satisfactorily explained in terms of a modified so-called bi-cubic diagram (Figure 1). Note that the each of the front and rear planes of the bi-cubic model consists of a seven-component reaction diagram analogous to that of... 

Figure 4.16 Simplex lattice design for a special cubic model with ten...

With the design used it is not possible to construct a complete third order model. The most complex third order model which can be used contains 38 terms. This includes third order blending effects (the so called special cubic model terms [10], e.g. d m c) and temperature and relative humidity dependent blending effects (e.g. d m t and e t h). 

CALCULATED REGRESSION COEFFICIENTS AND MODEL VALIDATION CRITERIA OF THE SPECIAL CUBIC MODEL (EQUATION (3)) FOR THE LIQUID-LIQUID EXTRACTION OF 9 SULPHONAMIDES WITH... 

FIG. 1.13 Spherical and cubic model particles with crystalline or amorphous microstructure (a) spherical zinc sulfide particles (transmission electron microscopy, TEM, see Section 1.6a.2a) x-ray diffraction studies show that the microstructure of these particles is crystalline (b) cubic lead sulfide particles (scanning electron microscopy, SEM, see Section 1.6a.2a) (c) amorphous spherical particles of manganese (II) phosphate (TEM) and (d) crystalline cubic cadmium carbonate particles (SEM). (Reprinted with permission of Matijevic 1993.)... 

We now take the simple cubic model above and allow for the competitive adsorption and desorption of a second species Q. Thus the model becomes... 

With this identification, the stable stationary-state behaviour (found for the cubic model with 1 < A < 4) corresponds to oscillations for which each amplitude is exactly the same as the previous one, i.e. to period-1 oscillatory behaviour. The first bifurcation (A = 4 above) would then give an oscillation with one large and one smaller peak, i.e. a period-2 waveform. The period doubling then continues in the same general way as described above. The B-Z reaction (chapter 14) shows a very convincing sequence, reproducing the Feigenbaum number within experimental error. 

Figure 17. An idealized representation of a cubic model. Restriction imposed on a ball trapped within a cube with different faces lacking. In models 1-10 various faces of the cube are absent. Such an absence increases the freedom the trapped ball experiences as one moves from model 10 to 2. The similarity between this situation and a guest within a restricted space is highlighted.

FIGURE 8.8 Examples of simplex lattices for (a) linear, (b) quadratic, (c) full cubic, and (d) special cubic models. 

The quasi-cubic model approximation can be invoked to calculate the crystal-field splitting (ACf) and spin-orbit coupling (Aso) parameters from Eq [76, 159,160]... 

A useful approach to the wurtzite valence band structure is the quasi-cubic model of J.J. Hop field [3], In this framework, the relative energies of the valence band maxima are ... 

A more detailed k.p description of the valence band dispersion, in particular beyond the quasi cubic model, can be found in [1,4,5,7,8,11,12], The valence bandedge separations are then given by ... 

The shift of the A line in the epilayers has been connected with the variation of the lattice parameters of GaN [1,11,12], The shift of this line was also measured in samples subjected to hydrostatic pressure (see Datareview A3.1). Combination of all these data permits one to obtain the whole series of excitonic deformation potentials [6,16], Two sets of data are available which are consistent with each other and are given in TABLE 1. The discrepancies between them are linked to the differences in the values of the stiflhess coefficients of GaN used by the authors. Gil and Alemu [6] in their work subsequent to the work of Shan et al [16] used data not available when Shan et al calculated their values. The notations are the same and are linked to the relationship with the quasi cubic model of Pikus and Bir [17], Deformation potentials as and a6 have been obtained by Alemu et al [8] who studied the anisotropy of the optical response in the growth plane of GaN epilayers orthorhombically distorted by growth on A-plane sapphire. For a detailed presentation of the theoretical values of deformation potentials of GaN we refer the reader to Suzuki and Uenoyama [20] who took the old values of the stiflhess coefficients of GaN [21]. 

Our objective is to generate response surfaces which will allow the prediction of resolution at any point within the regions ABC and BCD. The response surface is described mathematically by the special cubic model shown in equation 2. 

With 68.0% of the volume of the cell being occupied by the two atoms, the fraction of free space in this structure is 32.0%. Therefore, it represents a more efficient packing model than does the simple cubic model. Also, each atom is surrounded by eight nearest neighbors in the bcc structure, and there are two atoms per unit cell, which means it is preferable to the simple cubic structure on this basis also. 

In a polyhedral foam the liquid is distributed between films and borders and for that reason the structure coefficient B depends not only on foam expansion ratio but also on the liquid distribution between the elements of the liquid phase (borders and films). Manegold [5] has obtained B = 1.5 for a cubic model of foam cells, assuming that from the six films (cube faces) only four contribute to the conductivity. He has also obtained an experimental value for B close to the calculated one, studying a foam from a 2% solution of Nekal BX. Bikerman [7] has discussed another flat cell model in which a raw of cubes (bubbles) is shifted to 1/2 of the edge length and the value obtained was B = 2.25. A more detailed analysis of this model [45,46] gives value for B = 1.5, just as in Manegold s model. 

Somewhat more complex designs have been used in the literature and software has been made commercially available. They require seven or ten chromatograms (Fig. 6.27) obtained with the experimental conditions of Table 6.14. The respective models are called the reduced or special cubic model and the complete cubic model. 

Fig. 3.7. Cubic model of a redox-linlced proton pump. OX and RED denote a redox centre in the oxidised and reduced state. The bar marked M or C next to OX and RED indicates an acidic group, the function of which is linked to the redox centre. M and C mean that the group is connected protonically either with the aqueous matrix or cytoplasmic phases, respectively. When the group is protonated the bar is supplemented with H. Left and right faces of the cube separate states in electronic and protonic contact with the input and output sides of the transducer, respectively. Allowed transitions between these are indicated by thick arrows. Dotted lines denote forbidden transitions. If the latter gain significant probability relative to allowed transitions proton transport becomes decoupled from electron transfer (so-called slipping ). (From Ref. 8.)...

Fig. 19 a-c. Structure of a liquid foam and direction of electrical current, a cubical model b Bickerman s model h c Chistyakov and Chemina model ... 

If one imagines the structure of a liquid foam as a system of cubical gas bubbles (Fig. 19) and that the electrical current is directed upward, the horizontal walls of bubbles (perpendicular to the current direction) do not participate in electroconductivity and 2 of die 6 walls of each cube do not contribute to conduction. Then we have kIk = 4/6 K = 2/3 K, i.e. Manegold s formula is applicable. On the other hand, this equation coincides with Wagner s and Odolevsky s equations, which is to be expected since both these relations are also based on a cubical model of the disperse system. 

The use of other geometrical models for the analytical treatment of the relation Y = f(5, d) does not markedly affect the form of Eqs. (20) and (22) but only alters the value of the factor in the numerator of Eq. (22) For the spherical model discussed above, this factor is tc = 3.14. For other models it may vary from 3.0 to 3.3. In contrast to the spherical model, other models involve greater discrepancies between the experimental and theoretical results. For example, when using a cubic model for PSB plastic foam having a volumetric weight of 90 + 0.5 kg/m, the discrepancy is 18% as compared with 9% in a spherical model. 

The special cubic model describes a eertain third-order eurvature in the response surface ... 

Figure 11. Prediction of stereochemistry of reduction for 2-alkyl cyclohexanones by Jones cubic model. The direction of hydride delivery from NADH is indicated by the...

With any such algorithm it is necessary to specify some tolerance value below which any peaks are assumed to arise from noise in the data. The choice of window width for the quadratic differentiating function and the number of points about the observed inflection to fit the cubic model are selected by the user. These factors depend on the resolution of the recorded spectrum and the shape of the bands present. Results using a IS-point quadratic differentiating convolution function and a nine-point cubic fitting equation are illustrated in Figure 6. 

Figure 6 Results of a peak picking algorithm. At x = 80, the first derivative spectrum crosses zero and the second derivative is negative. A 9-point cubic least-squares fit is applied about this point to derive the coefficients of the cubic model. The peak position (dytdx = 0) is calculated as occurring at x = 80.3...

Table 3 shows results of recorded fluorescence emission intensity as a function of concentration of quinine sulphate in acidic solutions. These data are plotted in Figure 3 with regression lines calculated from least squares estimated lines for a linear model, a quadratic model and a cubic model. The correlation for each fitted model with the experimental data is also given. It is obvious by visual inspection that the straight line represents a poor estimate of the association between the data despite the apparently high value of the correlation coefficient. The observed lack of fit may be due to random errors in the measured dependent variable or due to the incorrect use of a linear model. The latter is the more likely cause of error in the present case. This is confirmed by examining the differences between the model values and the actual results. Figure 4. With the linear model, the residuals exhibit a distinct pattern as a function of concentration. They are not randomly distributed as would be the case if a more appropriate model was employed, e.g. the quadratic function. 

The exercise can be repeated for the fitted cubic model, and the ANOVA table and siuns of squares decomposition are shown in Tables 8 and 9 respectively. In this case, the F-statistic for the cubic term (F= 5.8) is not significant at the 5% level. The cubic term is not required and we can conclude that the quadratic model is sufficient to describe the analytical data accurately, a result which agrees with visual inspection of the line. Figure 3(b). 

---