Logarithmic derivative defined

In order to describe the critical behavior of the galvanomagnetic properties quantitatively, we now define the logarithmic derivatives Xa(p)> Xr(p) °f the conductivity and of HalTs coefficient as... 

For finite nucleus models with a well-defined nuclear size parameter R, beyond which the nuclear charge density is exactly( ) zero and the nuclear potential is exactly( ) given as —Z/r (r > R), the energy shifts can be obtained directly from the matching condition for the logarithmic derivatives L (r) = / r)/P [r) of the radial functions in the inner... 

Letting i/ label a chosen energy [30,73], the logarithmic derivative, is already defined (eq. 36), and for the energy derivatives a similar function is introduced ... 

The LMTO method as defined in this section may be regarded as an LCAO formalism in which the muffin-tin potential, rather than the atomic potential, defines the set of basis functions used to construct the trial functions of the variational procedure. Consequently, all overlap integrals can be expressed in terms of the logarithmic derivative parameters, and the muffin-tin Hamiltonian can be solved to any accuracy. 

To do this, we use the Rayleigh-Ritz variational principle in connection with the radial trial function of arbitrary logarithmic derivative D at the sphere boundary defined by the linear combination... 

Such effects can at least be minimized by using a procedure [273, 314] defining via the discrete logarithmic derivative of (3.16)... 

The metabolic pools defined in the denominator of the nutrient-response equation can be viewed as intermediate metabolites. Thus, the logarithmic derivative of each metabolic pool with respect to nutrient intake provides an estimate of the sensitivity of the pool to nutrient intake. Figure 7 shows... 

A local formulation of the self-interaction-corrected (LSIC) energy functionals has been proposed and tested by Luders et al. (2005). This local formulation has increased the functionality of the SIC methodology as presented in Section 3.5. The LSIC method relies on the observation that a localized state may be recognized by the phase shift, tj , defined by the logarithmic derivative ... 

Now we make use of the finite range of the potential U(R). At internuclear distances i > a we will consider the potential to be zero so that the asymptotic form Eqn. (10.40) is the relevant solution to the Schrodinger equation. At shorter distances we assume a well-defined potential form with known solutions TZi for Eqn. (10.30). At the boundary R = a we apply joining conditions to the interior and exterior solutions. Smooth joining requires that the solutions and their derivatives match at the boundary R = a. A convenient way to express this matching is the logarithmic derivative... 

The characteristic behavior of the temperature dependence of conductivity can be understood in detail by defining the reduced activation energy (W) as the logarithmic derivative of the temperature dependence of conductivity, i.e., W = d(lna)/d( nT) [3,... 

To explicitly describe the characteristic behavior of p(7), we define the reduced activation energy W as the logarithmic derivative of p(7) [71,72],... 

The last quantity that we discuss is the mean repulsive force / exerted on the wall. For a single chain this is defined taking the derivative of the logarithm of the chain partition function with respect to the position of the wall (in the —z direction). In the case of a semi-infinite system exposed to a dilute solution of polymer chains at polymer density one can equate the pressure on the wall to the pressure in the bulk which is simply given by the ideal gas law The conclusion then is that [74]... 

In these equations, pIQ is defined as the negative logarithm of IQ. The equivalent expression, derived from Equation (5.25), for a weak base would be... 

The half-life (tl/2) is defined as the time required for the concentration of a reactant to fall to one-half of its initial value, whereas the lifetime is defined as the time it takes for the reactant concentration to fall to /e of its initial value (e is the base of natural logarithms, 2.718). Both tl/2 and r are directly related to the rate constant and to the concentrations of any other reactants involved in the reactions. These relationships are given in general form in Table 5.2 for first-, second-, and third-order reactions and are derived in Box 5.1. 

In general, the right-hand side of eq. (29) will be multiplied by a function depending on p and the derivatives of order less than or equal to i of the natural logarithm of E(w). It is convenient to define a new variable z by... 

Rather than write hydronium ion concentrations in molarity, it s more convenient to express them on a logarithmic scale known as the pH scale. The term pH is derived from the French puissance d hydrogene ("power of hydrogen") and refers to the power of 10 (the exponent) used to express the molar H30+ concentration. The pH of a solution is defined as the negative base-10 logarithm (log) of the molar hydronium ion concentration ... 

In Equation 6.35, o is called the Hammett constant and is defined as the logarithmic ratio of the equilibrium constants of the parent compound and derivative, K() and K, according to ... 

The potency of compounds derived from concentration response assays is expressed most commonly as IC50 or EC50 defined as the compound concentration that produces half maximum response. A common model is the four-parameter Hill-slope equation (Table 14.1). A three-parameter model can be used if a maximum or minimum asymptote is not available because compound potency falls outside the concentration range. One recommendation is to fit the logarithm (loglO) of IC50 or EC50 instead of the untransformed concentration because the concentration response errors are normally... 

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