Logarithms powers

It is interesting to note that in a study by A. P. Mirabel and A. S. Monin, written after Ya.B. s paper, the expression for the turbulent diffusion coefficient in a two-dimensional turbulent field differs from the one obtained by Ya.B. by a factor which is equal to some logarithmic power of the ratio of the mixing scale to the energy-supply scale. 

Equation (4.4) clearly shows the relative effects of various properties on the rate of nuclei formation. Temperature and supersaturation are the two parameters that can be adjusted to affect nucleation rate. Since most soluble systems exhibit some temperature dependence of the saturation concenU ation (phase boundary), these two parameters are not always independent. For example, raising temperature enhances nucleation kinetics (3rd powerX but usually causes a decrease in supersaturation (square of logarithm power). Thus, the effects of temperature on nucleation are not always easy to predict since they depend on the relative changes of these two terms. 

Fig. 9. a Double-logarithmic power dependence of the population densities, Ni and N2, of levels 1 and 2 in Fig. 5 calculated from Eq. (10) using the same parameters as in Fig. 6 c, where power is taken as G. The dashed lines are the limiting slopes from Eqs. (14) and (15). b Double-logarithmic power dependence of the level 1 depopulation rate ratio between ETU (rate = 2 Wetu i) and linear downconversion (rate = kiNi). The horizontal dashed line indicates equal rates for the two. c Time-evolution of N2 following termination of a cw beam at f = 0 for the high, medium, and low powers indicated in (b), plotted on linear axes. The dashed lines are the limiting behaviors from Eqs. (22) and (24)... 

Ad = 9 nm). The so/id line corresponds to the classical model of fast aggregation, where size dependence of the aggregate diffusion coefficient is neglected. Dashed lines are results of approximation by the power law (s oc t, = 1), or logarithmic power law (s o< (logt ). ... 

With logarithmic, power, or exponential models, one may modify the data by considering the logarithm of one or both of the variables instead of the original data, in order to reduce a nonlinear model to a linear one. This method is sometimes called the transformed least squares criterion. Another criterion, namely the Chebyshev... 

The neutron instrumentation system consists of a neutron measurement and control system and a safety protection system It is designed to measure the neutron flux not only during the reactor operation but also during the reactor shut-down period to have necessary information for reactor operation and safety protection The neutron measurement and control system consists of two start-up channels and two linear power monitoring channels The safety protection system consists of two logarithmic power monitoring channels and two power level safety channels... 

Figure 3 Feature relevance. The weight parameters for every component in the input vector multiplied with the standard deviation for that component are plotted. This is a measure of the significance of this feature (in this case, the logarithm of the power in a small frequency region.)...

Furthermore, one may need to employ data transformation. For example, sometimes it might be a good idea to use the logarithms of variables instead of the variables themselves. Alternatively, one may take the square roots, or, in contrast, raise variables to the nth power. However, genuine data transformation techniques involve far more sophisticated algorithms. As examples, we shall later consider Fast Fourier Transform (FFT), Wavelet Transform and Singular Value Decomposition (SVD). 

As can be seen in Figure 5-17, some search fields (e.g., POW [= Power]) do not need any input in the search mask this means that all entries with any content of those Helds are retrieved. However, other fields always demand an input. In case the input is omitted (for example for the decadic logarithm of the partition coefficient), a corresponding error message results. Since the PCB are more soluble in the organic phase, the input of that Field is restricted to positive values. 

Definition of Logarithm. The logarithm x of the number N to the base b is the exponent of the power to which b must be raised to give N. That is,... 

The logarithm of a power of a number is equal to the logarithm of the base multiplied by the exponent of the power thus,... 

Many other mathematical operations are commonly used in analytical chemistry, including powers, roots, and logarithms. Equations for the propagation of uncertainty for some of these functions are shown in Table 4.9. 

Although two of the mechanisms presented above yield the same power dependence on t, it appears possible to eliminate certain mechanisms by experimentally testing the development of 9 with time. A strategy for this is suggested by Eq. (4.28). Taking the logarithm of both sides of that equation gives... 

Ruths and Granick [95] have studied the self-adhesion of several monolayers and adsorbed polymers onto mica. For loose-packed monolayers, the adhesion, in excess of a constant value observed at low rate, increased as a power law with the square root of the separation rate. In the case of adsorbed diblocks, the excess adhesion increased linearly with logarithmic separation rate. The time effects were ascribed to interdigitation and interdiffusion between the contacting layers. 

Zetmer-logarithmus, m. logarithm to the base 10, common logarithm, -potenz, /. power of ten tenth power, -stein, m. hollow concrete block, -stelle, /. decimal place, -system, n. 

The logarithm of a positive number N is the power to which the base (10 or e) must be raised to produce N. So, x = log N means that e = N, and x = log,(,N means that lO = N. Logarithms to the base 10, frequently used in numerical computation, are called common or denary logarithms, and those to base e, used in theoretical work, are called natural logarithms and frequently notated as In. In either case,... 

Legitimate operations on equations include addition of any quantity to both sides, multiplication by any quantity of both sides (unless this would result in division by zero), raising both sides to any positive power (if is used for even roots) and taking the logarithm or the trigonometric functions of both sides. 

The logarithm of a number n to the base m is defined as the power to which m must be raised to give the number n. Thus... 

The digits that appear before (to the left of) the decimal point specify the value of n, that is, the power of 10 involved in the expression. In that sense, they are not experimentally significant. In contrast, the digits that appear after (to the right of) the decimal point specify the value of the logarithm of C the number of such digits reflects the uncertainty in C. Thus,... 

Logarithm The exponent that indicates the power to which a number is raised to produce a given number. Thus, as an example, 1000 to the base of 10 is 3. This type of mathematics is used extensively in computer software. 

At low values of the Reynolds number, less than about 10, a laminar or viscous zone exists and the slope of the power curve on logarithmic coordinates is — 1, which is typical of most viscous flows. This region, which is characterised by slow mixing at both macro-arid micro-levels, is where the majority of the highly viscous (Newtonian as well as non-Newtonian) liquids are processed. 

The natural logarithm of a number x, denoted In x, is the power to which the number e = 2.718.. . must be raised to equal x. Thus, In 10.0 = 2.303, signifying that e2.30.3 j q q y le value 0f e may Seem a peculiar choice, but it occurs naturally in a number of mathematical expressions, and its use simplifies many formulas. Common and natural logarithms are related by the expression... 

In Figure 4 the logarithm of the observed ion intensities was plotted as a function of the logarithm of the pressure in the collision chamber. As the intensity of a product ion of a certain order increases proportionally to the same power of the pressure, the curves in the diagram corresponding to primary, secondary, and tertiary ions are represented by straight lines of slopes equal to 1, 2, and 3, respectively. Measurements were performed with 11 incident ions with different recombina-... 

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