Some Numerical Examples

We explore here some numerical aspects of the binding isotherm for this model. The examples worked out in the present section will serve as reference cases for the forthcoming sections of this chapter. We assume that U, the binding energy, is given, hence q = exp(-jSf/) will be constant. We define a new activity of the ligand by 

The eventual saturation cannot be shown on this scale and will be demonstrated 

On the same scale of Fig. 3.2 it is difficult to demonstrate the case of positive cooperativity, i.e., with 5 1. The curves in these cases will be too close to the ordinate. 

FIGURE 3.4. Binding isotherms [Eq. (3.1.37)] for various values of 5 I. Positive cooperativity. 

It is sometimes convenient to plot the isotherm 0(lnx), Fig. 3.3. Here we can demonstrate both positive and negative cooperativity on the same plot. Also, in the case of 5 1 we can see the intermediate saturation at 0 = 3, followed by the eventual 

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A new one-dimensional mierowave imaging approaeh based on suecessive reeonstruetion of dielectrie interfaees is described. The reconstruction is obtained using the complex reflection coefficient data collected over some standard waveguide band. The problem is considered in terms of the optical path length to ensure better convergence of the iterative procedure. Then, the reverse coordinate transformation to the final profile is applied. The method is valid for highly contrasted discontinuous profiles and shows low sensitivity to the practical measurement error. Some numerical examples are presented. 

Some numerical examples will illustrate the discussed cases. Means and standard deviations are given with superfluous significant figures to allow recalculation and comparison. 

When experimental E(tr) data are known, the time delay r is the value of tr when E(tr) - 0. The moments also are obtained from the data. Then values of the other parameters, n and 0, can be deduced from the given equations. Some numerical examples are tabulated. 

We have arrived at the point where further understanding of scattering and absorption by a sphere is difficult to acquire without some numerical examples. What is needed now is some flesh to cover the dry bones of the formal theory we should like to know how the various observable quantities vary with the size and optical properties of the sphere and the nature of the surrounding medium. To do so the first step is to obtain explicit expressions for the scattering coefficients an and bn. 

Previous theoretical treatments of the transition between the helicel and random forms of the desoxyribose nucleic acid (DNA) molecule are extended to Include formally the explicit consideration of the dissociation into two separate chains and the consideration of the effects of the.ends of the chains, An approximate form for the fraction of the base pairs that are bonded is obtained in terms of two parameters, a stability constant for base pairing and a constant representing the interaction of adjacent base pairs. The matrix method of statistical mechanics proves to be adaptable to this problem. Some numerical examples are worked out for very long molecules, for which case it is found that the effect of concentration is small. 

Some numerical examples are given. For a semi-infinite copper melt initially at the fusion temperature, losing heat with an over-all heat transfer coefficient of 0.5 B.t.u./(hr.)(ft.2)(°F.) to the surroundings at ambient temperature, after 4 hr. 771 = 0.98, and the estimated thickness of solidified copper is 44 in. with a 12% error. A second example is a steel sheet subjected to a slowly flowing stream of very hot gas, such that a uniform heat flux of 105 B.t.u./(hr,)(ft.2)(°F.) is imposed at the surface with negligible motion of the melt. After 200 sec., 771 = 0.68, and the melt thickness is estimated to be 1.26 in., with a possible error of 8.6%. 

Before proceeding further, let us use some numerical examples to illustrate the determination of the probability of locating an electron in a certain volume element in space. The ground state wavefunction of the hydrogen atom is... 

Some numerical examples are shown in Fig. 7a. The nonexponential part of 4>v,b(t) for short times is readily seen and leads to a delayed onset of the exponential decay. The latter represents the Markovian limit (straight line in the semi-log plot). A time interval of approximately 2 x rc is required to establish the asymptotic dephasing rate 1/T2. The purely exponential case, rc = 0, is fictitious and shown in the figure only for comparison. 

We shall now illustrate the apphcation of equations (4a) and (9) by some numerical examples. The numerical values of the constants /3, y, etc., are generally small in practice, so that the corresponding terms in the equation are relatively inconsiderable, especially when the heat of reaction Qp is great. In the following calculation we shall assume as a first approximation that the... 

Here, some numerical examples will be presented from Amovilli et al. [41]. As far as possible, bearing in mind the limitations of the model, the examples are designed for conditions which can be achieved in laboratory experiments. However only non-degenerate plasmas will be considered, this then implying the constraint that the ionic number density n,- satisfies... 

In 49 the authors have presented a simple technique that allows to limit the error growth in the Ion-term (long-time) numerical integration of perturbed multi-dimensional oscillators, while using highly efficient and accurate special multistep codes. The author has studied theoretically their behaviour. The new technique has been illustrated with some numerical examples, including a case with non-resonant frequencies. 

In this chapter, the robust PCA via PP techniques is used for chemical data treatment, for this method assembles some excellent performance characteristics of some other robust methods. As a key point of the technique, the GSA algorithm is used as an optimization procedure to guarantee the global optimum. The theory and algorithm of PP PCA together with GSA are described and the necessity and the applicability of robust PCA is demonstrated by some numeric examples. 

In this section, we illustrate the theory by some numerical examples. Each one involves a simulated spectrum and the corresponding two-dimensional plot. We confine ourselves to the energy range below both series limits, where two-dimensional plots are appropriate. 

Table 3.8 gives semiempirical expressions for the direct spin-spin part of several effective spin-spin constants and some numerical examples for valence states of homonuclear molecules where the n orbital is assumed to be an antibonding orbital. For Rydberg states, the contribution of the Rydberg n orbital can be neglected. 

The expressions for i , were collected in Table 11.5 (for S S a different treatment will be used). Some numerical examples are included in Table 11.6. As the individual spins are added to yield the total spin... 

Table 4. Rules of thumb some numerical examples...

For some numerical examples, see Ben-Naim (1990,1992). See also Ragone and Colonna (1994) for a reassessment of the solvation and conditional solvation for the problem of protein folding. 

P is thus dependent on the conversion (p-100). This is illustrated by some numerical examples in Table 4.1. 

Neave, H. R. 1981. Elementary Statistics Tables, Routledge, London. (Good statistical tables are required by all users of statistics. This set is strongly recommended because the tables are accompanied by useful explanatory notes and some numerical examples.)... 

The style of the book is mainly didactic, emphasizing the methods rather than elaborating on specific examples. Occasionally, some numerical examples are presented in the form of a table or a figure. These are included as illustrations only, to give an idea of the order of magnitude of certain quantities. The numerical values are cited in the same units as they appeared in the literature. No attempt was made to use a unified set of units throughout the book. 

Chapter 7 introduces for the first time the basic fluid dynamics principles of packed columns for liquid/liquid extraction. The previously mentioned SBD model for gas/liquid systems is transferable to liquid/liquid systems. The method used to calculate the gas velocity at flooding point of the disperse and continuous phases will be explained by means of some numerical examples. 

These van der Waals forces between colloid particles or surfaces are very strong, as shown in Figure 6.10, where some numerical examples are provided. We can see that the van der Waals forces between macroscopic particles are large and not only when the bodies arc in contact ... 

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