Linear scaling relations

Deterministic fractals usually have a linear scaling relation between the two mctrices (Le. dmin = 1), but there exists exceptions to this rule, see Rrf. [21] for an example. 

Other investigations using the charge-dipole model include the study of enhanced polarizability of aromatic molecules placed between two silver clusters [32], polarization of fullerenes [28], fullerene clusters [29], silicon clusters [31], and so on. Polarizability of molecular clusters [24] has also been calculated using the dipole interaction model and it would be interesting to investigate the applicability of simple linear scaling relations of polarizability as a function of the cluster size, as obtained by ab initio calculation [56],... 

Now that we understand the origin of the slope in the linear scaling relations, we can focus our attention on the intercept Unfortunately, the intercept cannot be as easily predicted as the slope, but there are two important aspects that need to be mentioned. Using the bond order conservation method to estimate the scaling slope, a single reference calculation can be sufficient to predict adsorption energies on any other catalyst surface. Assume that we... 

Linear scaling relations for complex adsorbates, (a) Parity plot of CH c-NH adsorption energies predicted from scaling and the DFT reference calculations. (b) Linear scaling of C2H4 and C2H6 with methyl (CH3) from F. Studt et al.. Science, 320, 1320-1322 reprinted with permission from AAAS. ... 

To convince ourselves that, for most reactions, both graphs contain essentially the same information, we will derive a mathematical conversion of one graph into the other. We start by postulating that the initial and final state of the reaction is described by linear scaling relations that depend on the descriptors Ej and E2. 

The molar mass dependence of the intrinsic viscosity of rigid chain polymers cannot be described by a simple scaling relation in the form of Equation (36) with molar mass independent of K and a. over a broad molar mass range. Starting from the worm-like chain model, Bohdanecky proposed [29] the linearizing equation... 

The intermediates of an MFEP calculation are usually defined using Hamiltonians related to those of system 0 (J%) and 1 p j). This is generally achieved with a parameter-scaling approach (see Sect. 2.6). Linear scaling is the simplest form... 

Abundances of various types of object at redshifts of order 2 to 3 are summarized in Fig. 12.12, which displays a clear relation with linear scale, or perhaps... 

The data of Rowley and Steiner are shown graphically in Figure 4.5, with k plotted on a logarithmic scale (equivalent to lnifc on a linear scale) against lOOO/T. According to equation 3.1-7, the result should be a linear relation, with a slope of — EA/R and an intercept (not indicated in Figure 4.5) of In A. The values of EA and A obtained by Rowley and Steiner in this way are 115,000 J mol-1 and 3.0 X 107 L mol-1 s 1, respectively. 

In order to estimate the degree of branching, we need information to establish a baseline concerning the behavior of linear PVAc in teinns of the scaling relations R = k M, R, h] = M , etc., which are essentially available in the... 

The wide range in the hydrogen-ion concentrations of aqueous solution makes it difficult to plot these values on a linear scale. As a convenience, we use a logarithmic scale introduced many years ago. Hydrogen-ion concentrations are represented by "pH" and hydroxide-ion concentrations by pOH , defined by the relations... 

Flory-type free energy calculations show that the root mean square end-to-end distance of a polyelectrolyte increases linearly with the chain length at infinite dilution and without added salt [40]. Using the above perturbation theory, scaling relations at finite densities are obtained. The influence of the interaction with other polymer chains, counterions, and added salt is captured in the Debye screening length xT1. 

From all theoretical and experimental results one may conclude that there is no simple scaling relation over a long period of time. The detailed coarsening mechanisms, which are attributed to the intrinsical non-linearity of the phase separation process, determine the exponent a. The time dependence of a reflects cross-over among different coarsening processes. 

For simplicity one searches for a material in which the system provides a natural linear scale for hotness levels. It has been known for three centuries that certain gases such as He, Ar, H2, and the like closely approximate to the relation PV... 

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