Fractional power exponent

We should mention here that the recent extended conductivity and viscosity measurements have revealed a fractional power exponent (i.e., CiJ ... 

Check your calculation of Qc. (Errors in evaluating fractions with exponents are common.) The value of Kc is less than one, so yon wonld expect the numerator to be less than the denominator. This is difficnlt to evaluate without a calculator, however, because of the powers. 

Rational numbers are numbers that can be written as fractions (and decimals and repeating decimals). Similarly, numbers raised to rational exponents are numbers raised to fractional powers ... 

The established concepts predict some features of the Payne effect, that are independent of the specific types of filler. These features are in good agreement with experimental studies. For example, the Kraus-exponent m of the G drop with increasing deformation is entirely determined by the structure of the cluster network [58, 59]. Another example is the scaling relation at Eq. (70) predicting a specific power law behavior of the elastic modulus as a function of the filler volume fraction. The exponent reflects the characteristic structure of the fractal heterogeneity of the CCA-cluster network. 

In Section 5.2.2 it was shown that at large q the intensity I(q) of scattering from a sphere decays as q A, from a thin disk as q 2, and from a thin rod as q l. The power-law exponent at large q is therefore seen to be related to the dimensionality of the scattering object. There are, however, many cases in which the intensity varies as an unexpected or even fractional power of q. In the case of a Gaussian model of a polymer chain, the intensity was seen to decrease as q 2 even though a chain obviously is a three-dimensional object. The inverse power-law exponents that differ from 1, 2, or 4 can be explained in terms of the concept of a fractal. 

The dynamic holdup depends mainly on the particle size and the flow rate and physical properties of the liquid. For laminar flow, the average film thickness is predicted to vary with, as in flow down a wetted-wall column or an inclined plane. In experiments with water in a string-of-spheres column, where the entire surface was wetted, the holdup did agree with theory [28]. For randomly packed beds, the dynamic holdup usually varies with a fractional power of the flow rate, but the reported exponents range from 0.3 to 0.8, and occasionally agreement with the 1/3 power predicted by theory may be fortuitous. 

As this depends on the fractional power 9/4 of the concentration, it is impossible to reach this result no matter how higher-order terms in the perturbational calculation are obtained. It has an exponent higher by 1 /4 than the exponent of the second virial term. Around the overlap concentration where the volume fraction is numerically 0 10", this discrepancy cannot be neglected. 

Related to the preceding is the classification with respect to oidei. In the power law rate equation / = /cC C, the exponent to which any particular reactant concentration is raised is called the order p or q with respect to that substance, and the sum of the exponents p + q is the order of the reaction. At times the order is identical with the molecularity, but there are many reactions with experimental orders of zero or fractions or negative numbers. Complex reactions may not conform to any power law. Thus, there are reactions of ... 

The MW dependences of the normalized chain relaxation times in melts of linear and branched samples are compared in Fig. 12. Both can be represented by scaling power laws, but with remarkably different scaling exponents. For the melts of linear chains, the exponent 3.39 is observed close to the typical value of 3.4 for such systems. In contrast, for the fractions of the branched polymer, the exponent is considerably lower (2.61). It is interesting to note that the value of the normalized chain relaxation time for the feed polymer with the broad M WD fits nicely into the data for the fractions with narrow MWDs. This seems to indicate that conclusions can also be drawn from a series of hyperbranched polymers with broad MWDs. 

The optimum diameter obtained from equations 5.14 and 5.15 should remain valid with time. The cost of piping depends on the cost power and the two costs appear in the equation as a ratio raised to a small fractional exponent. 

In our previous analysis, the dependence of the fraction of predicted folds on rank for the 30 top folds was described by an exponential function, with the top-ranking fold, the P-loop being overrepresented (Wolf et al, 1999). With the improved resolution reported here, which allowed the extension of the plot to a greater number of folds, the data do not fit an exponent (not shown). By contrast, the power law accommodates all the folds (Fig. 4). 

When an exponent is a fraction, the denominator of this fractional exponent means the root of the base number, and he numerator means a raise of the base to that power ... 

For a number with a fractional exponent, the numerator of the exponent tells you the power to raise the number to, and the denominator of the exponent tells you the root you take. 

This equation reveals that when measurements for fractal objects or processes are carried out at various resolutions, the log-log plot of the measured characteristic 9 (oj) against the scale oj is linear. Such simple power laws, which abound in nature, are in fact self-similar if oj is rescaled (multiplied by a constant), then 9 (oj) is still proportional to oja, albeit with a different constant of proportionality. As we will see in the rest of this book, power laws, with integer or fractional exponents, are one of the most abundant sources of self-similarity characterizing heterogeneous media or behaviors. 

The proportionality factor k is called the experimental rate constant with catalytic reactions, this constant is frequently a complex quantity which may be a product of rate constants of several steps or may include equilibrium constants of the fast steps. The exponents m,n,... in the power-law equations may be any fraction or small integer (positive, negative or zero). The constants K, in the denominator of the equations of type 7 are often but not always related to adsorption equilibrium constants. While they have to be evaluated from the experimental data, the values of exponents a, b, a, / and d arc derived from the assumed mechanism in the case of model rate equations. In empirical rate equations these constants can attain any value (fractional or small integer, usually positive) and... 

Accordingly, we expect a power law behavior G,0 (O/Op)3 5 of the small strain elastic modulus for 0>0. Thereby, the exponent (3+df [j)/(3—df)w3.5 reflects the characteristic structure of the fractal heterogeneity of the filler network, i.e., the CCA-clusters. The strong dependency of G 0 on the solid fraction Op of primary aggregates reflects the effect of structure on the storage modulus. 

Tc. The two power-law exponents are not independent but depend on a single parameter, the so-called critical exponent X, which is specific for a given interaction potential (e.g., hard spheres). Actually, the interaction potential enters the MCT equations only indirectly via the structure factor S(q), which fixes the nonlinear coupling in the generalized oscillator equation. It is important to note that the MCT exponents are not universal in contrast to those of second-order phase transitions. In the case of hard spheres, for example, S(q) can be calculated via the Percus-Yevick approximation [26], and the full time and -dependence of < >(q. f) were obtained. As an example, Fig. 10 shows the susceptibility spectra of the hard-sphere system at a particular q. Note that temperature cannot be defined in the hard-sphere system instead, the packing fraction cp is used as a parameter. Above the critical packing fraction 0), which corresponds to T < Tc in systems where T exists, the a-process is absent (frozen) and only the fast dynamics is present. At cp < tpc the a-peak and the concomitant susceptibility minimum shift to lower frequencies with increasing cp, so that the closer cp is to the critical value fast dynamics can be identified (curve c in Fig. 10). 

Mechanical Properties The percolation approach was also employed to model the tensile strength of tablets [47,48]. A critical tablet density was here understood as a minimal solid fraction needed to build a network of relevant contact points spanning the entire tablet. A rising tablet density led to a power law increase of the tensile strength showing an universal exponent Tf = 2.7. 

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