Power law function

An empirical approach can also be used to relate growth kinetics to supersaturation with a power-law function of the form... 

In ion-exchange chromatography (lEC), the mobile phase modulator is typically a salt in aqueous solution, and the stationary phase is an ion-exchanger. For ddnte conditions, the solute retention faclor is commonly found to be a power-law function of the salt uormahty [cf. Eq. (16-27) for ion-exchange equilibrium]. 

Figure 13 Center-of-mass mean-square displacements computed from MD simulations at 323 K. (a) DPPC motion in the plane of a lipid bilayer averaged over 10 ps (b) DPPC motion in the plane of a lipid bilayer averaged over 100 ps (c) comparison of the DPPC m-plane mean-square displacement to linear and power law functions of time (d) comparison of the center-of-mass mean-square displacement from an MD simulation of liquid tetradecane to a linear function of time.

Variable Anemometer Height Option - The anemometer height is used in adjusting the wind speed to stack height wind speed for cavity calculations based on the following power law function ... 

Assuming that the melt viscosity is a power law function of the rate of shear, calculate the percentage difference in the shear stresses given by the two methods of measurement at the rate of shear obtained in the cone and plate experiment. 

A core constituent of BST is to represent all metabolic rate equations as power law functions. Using the power-law formalism, each reaction rate is written as a product... 

For power-law functions the (scaled) elasticities do not depend on the substrate concentration, that is, unlike Michaelis Menten rate equations, power-law functions will not saturate for increasing substrate concentration. 

The mean wind speed is often represented as a power-law function of height which may be expressed as... 

These two functions have been superimposed on the data from Fig. 3.22, and the composite is presented in Fig. 3.23. The zero shear and power law functions as mentioned above are often used in flow simulations. 

It is a common practice to describe mass dependent isotope fractionation processes by a single linear curve on a three-isotope-plot (Matsuhisa et al. 1978). The resulting straight lines are referred to as terrestrial mass fractionation lines and deviations from it are used as indicating nonmass-dependent isotope effects. The three-isotope-plot is based on the approximation of a power law function to linear format. To describe how far a sample plots off the mass-dependent fractionation line, a new term has been introduced A 0, A Mg, A S, etc. Several definitions of A have been introduced in the literature, which have been discussed by Assonov and Bren-ninkmeijer (2005). The simplest definition is given by ... 

Case i) is important close to the melting temperature, while case iii) dominates at lower temperatures. Under Eq. (1), Eqs. (2) and (3) are both linear in h (the same is true in the bulk diffusion case). Consequently, an initially sinusoidal modulation remains sinusoidal during relaxation. The amplitude A t) of modulation decays exponentially with time t. with a time constant, x which grows as a power-law function of the wavelength L,... 

Rheologically, the flow of many non-Newtonian materials can be characterized by a time-independent power law function (sometimes referred to as the Ostwald-deWaele equation)... 

The power-law formalism is a mathematical language or representation with a structure consisting of ordinary nonlinear differential equations whose elements are products of power-law functions. The power-law formalism meets two of the most important criteria for judging the appropriateness of a kinetic representation for complex biological systems the degree to which the formalism is systematically structured, which is related to the issue of mathematical tractability, and the degree to which actual systems in nature conform to the formalism, which is related to the issue of accuracy. 

Conversely, the relationship (7.2) expresses a time-scale invariance (selfsimilarity or fractal scaling property) of the power-law function. Mathematically, it has the same structure as (1.7), defining the capacity dimension dc of a fractal object. Thus, a is the capacity dimension of the profiles following the power-law form that obeys the fundamental property of a fractal self-similarity. A fractal decay process is therefore one for which the rate of decay decreases by some exact proportion for some chosen proportional increase in time the self-similarity requirement is fulfilled whenever the exact proportion, a, remains unchanged, independent of the moment of the segment of the data set selected to measure the proportionality constant. 

Fractals in electrochemistry — Figure. A von Koch curve of Df = 1.5. Note that no characteristic length of the structures can be identified -this is associated with the fact that the size-distribution of the features of the curves is a power-law function... 

In both cases the time (or frequency) functions become power-law functions of time (or frequency) indicating that the measured current or impedance involve no characteristic time (or frequency) with fractals - which have no characteristic sizes. 

Fig. 10. Dependence of relative permeability of gas phase on the volume fraction of gas phase for dry and partially water-saturated porous media of various morphology, fitted by a power-law function and compared with the Carman-Kozeny equation (from Kohout et al, in press).

A useful (also extreme) counterpart to the also idealized linear geometry is fractal geometry which plays a key role in many non-linear processes.280 281 If one measures the length of a fractal interface with different scales, it can be seen that it increases with decreasing scale since more and more details are included. The number which counts how often the scale e is to be applied to measure the fractal object, is not inversely proportional to ebut to a power law function of e with the exponent d being characteristic for the self-similarity of the structure d is called the Hausdorff-dimension. Diffusion limited aggregation is a process that typically leads to fractal structures.283 That this is a nonlinear process follows from the complete neglect of the back-reaction. The impedance of the tree-like metal in Fig. 76 synthesized by electrolysis does not only look like a fractal, it also shows the impedance behavior expected for a fractal electrode.284... 

The rheograms can often be described over a fairly wide range by a power-law function (Figure C4-12). This means that you get a straight line in a logarithmic graph. The slope... 

Plottingy versusx on logarithmic axes is equivalent to plotting In y versus In jr on rectangular axes. If the plot is linear in either case, x and y are related by a power law function, y = ax. ... 

Rather than a clear distinction between colloids and suspended material, separated by a 0.2 pm filtration, a continuum of particle radii exists between the smallest and the largest particles in water. Generally, the particle size distribution follows a power-law function in the form... 

Among natural populations, two major deviations from the ideal pattern are ubiquitous (Pielou, 1977 Lerman, 1979). The first deviation consists of populations with excessive proportions of young units (e.g., planktonic larval stages) because their destruction rate is very high, but chances for survival improve considerably with maturation (type II in Figure 2). The mathematical formalism for such populations (e.g., Lerman, 1979) is a power-law function. 

The mass-transfer coefficient in each film is expected to depend upon molecular diffusivity, and this behavior often is represented by a power-law function k . For two-film theory, n = 1 as discussed above [(Eq. (15-62)]. Subsequent theories introduced by Higbie [Trans. AIChE, 31, p. 365 (1935)] and by Dankwerts [Ind. Eng. Chem., 43, pp. 1460-1467 (1951)] allow for surface renewal or penetration of the stagnant film. These theories indicate a 0.5 power-law relationship. Numerous models have been developed since then where 0.5 < n < 1.0 the results depend upon such things as whether the dispersed drop is treated as a rigid sphere, as a sphere with internal circulation, or as oscillating drops. These theories are discussed by Skelland [ Tnterphase Mass Transfer, Chap. 2 in Science and Practice of Liquid-Liquid Extraction, vol. 1, Thornton, ed. (Oxford, 1992)]. 

Figure 2. Differential Flux versus Energy. The differential flux from the NE cap of SN1006 is shown. Data are grouped in four bins of energy. The differential flux is fitted with a pure power-law function. The statistical errors and those on the energy reconstruction are included. A 30% systematic error is not included. Further spectral studies are limited by the small statistics.

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