Parabolic function

In the case of drug design, it may be desirable to use parabolic functions in place of linear functions. The descriptor for an ideal drug candidate often has an optimum value. Drug activity will decrease when the value is either larger or smaller than optimum. This functional form is described by a parabola, not a linear relationship. 

When electrons traverse an alloy rather than a pure metal, tire scattering of electrons is different at tire ion core of each chemical species and so the conductivity reflects a mixture of the effects due to each species. In a series of copper alloys it was found that the resistance, which is the reciprocal of the conductivity, is a parabolic function of tire concentration of the major element... 

This section contains a brief review of the molecular version of Marcus theory, as developed by Warshel [81]. The free energy surface for an electron transfer reaction is shown schematically in Eigure 1, where R represents the reactants and A, P represents the products D and A , and the reaction coordinate X is the degree of polarization of the solvent. The subscript o for R and P denotes the equilibrium values of R and P, while P is the Eranck-Condon state on the P-surface. The activation free energy, AG, can be calculated from Marcus theory by Eq. (4). This relation is based on the assumption that the free energy is a parabolic function of the polarization coordinate. Eor self-exchange transfer reactions, we need only X to calculate AG, because AG° = 0. Moreover, we can write... 

In order to find the relation between Eq. (122) and the Marcus theory, we employ the linear response approximation. In this case, the free energies Fj )(i =1)2) for the donor and acceptor become a parabolic function of as... 

Thus, the potential reaches a maximum at the sphere s center and then decreases as a parabolic function. A completely different behavior is observed outside the sphere. This simple problem allows one to demonstrate again that the potential obeys Poisson s equation. Consider the potential at the point p of an arbitrary body, Fig. 1.12a, assuming that the density may change from point to point. Let us mentally draw a spherical surface around the point p. If its radius is sufficiently small, we can suppose that this sphere is homogeneous. The potential at the point p can be written as... 

Thus, inside the layer the potential is a parabolic function and, in particular, at the boundary we have ... 

We have demonstrated that as in the case of the spherical mass the attraction field increases linearly approaching the spheroid surface and it is equal to zero at the center. At the same time, the potential has a maximum at this point and then decreases gradually as a parabolic function and reaches a minimum on the surface of the spheroid. 

Thus, the pressure has a maximum at the center and the decreases as a parabolic function and it is equal to zero at the pole. Next, consider the distribution of pressure in the channel A, where both the attraction and centrifugal forces act on any particle. Inasmuch as a difference of a pressure at terminal points of both channels is the same and a >, it is natural to assume that the attraction field in the channel A is smaller and suppose that the correction factor is equal to the ratio of axes, bja. Correspondingly, a condition of equilibrium is... 

If AU( 1) has Gaussian fluctuations, this means that the free energy is also a parabolic function of AU(1) itself. AU( 1) is known as the energy gap [52-54] we will denote it by p... 

The idea is to do simulations of the system before and after the proton binding i.e., to simulate the reactant and product states. With the assumption of linear response, these provide all the information needed to compute AA. Indeed, the free energy to introduce a fractional charge Aq into the reactant state (0 < A < 1) is a parabolic function of Aq, which can be written ... 

Laminar flow (LF) is also a form of tubular flow, and is the flow model for an LFR. It is described in Section 2.5. LF occurs at low Reynolds numbers, and is characterized by a lack of mixing in both axial and radial directions. As a consequence, fluid properties vary in both directions. There is a distribution of residence times, since the fluid velocity varies as a parabolic function of radial position. 

Harris et al. also employed a less-known CCK procedure, which Meehl and Yonce (1996) named SQUABAC, but the authors referred to as the Parabolic Function Method. Two SQUABAC analyses were performed, one with the PCL-R total score as the input variable and criminal recidivism as the output variable, and another with adult criminal history and recidivism as input and output variables, respectively. Recidivism history was paired with the two potential taxon indicators because it is a conceptually related but distinct variable. It is expected to be a valid indicator of the taxon, but it is not redundant with other indicators, thus nuisance correlations should not be a problem. 

To a first approximation, the capacity of the diffuse layer in Eqn. 5-4 may be represented by an inverse parabolic function of the diffuse layer potential hp with its tnifiimum at the potential of zero charge, (4ohp = 0) the minimum capacity is given by Eqn. 5-5 ... 

As shown in Fig. 5-21, the reduction of x p reduces the thickness of the compact layer, the magnitude of which increases as an inverse-parabolic function of the interfacial charge (Fig. 5-22). 

Fignre 2.41 indicates that the net peak current is a parabolic function of the electrode kinetic parameter. This is illnstrated in Fig. 2.43. With respect to the electrochemical reversibility of the electrode reaction, approximately three distinct regions can be identified. The reaction is totally irreversible for log(ca) < — 2 and reversible for log(ft)) > 2. Within this interval, the reaction is qnasireversible. The parabolic dependence of the net peak cnrrent on the logarithm of the kinetic parameter asso-... 

Sandorf, 1980 Whitney, 1985 Whitney and Browning, 1985). According to the classical beam theory, the shear stress distribution along the thickness of the specimen is a parabolic function that is symmetrical about the neutral axis where it is at its maximum and decreases toward zero at the compressive and tensile faces. In reality, however, the stress field is dominated by the stress concentration near the loading nose, which completely destroys the parabolic shear distribution used to calculate the apparent ILSS, as illustrated in Fig 3.18. The stress concentration is even more pronounced with a smaller radius of the loading nose (Cui and Wisnom, 1992) and for non-linear materials displaying substantial plastic deformation, such as Kevlar fiber-epoxy matrix composites (Davidovitz et al., 1984 Fisher et al., 1986), which require an elasto-plastic analysis (Fisher and Marom, 1984) to interpret the experimental results properly. 

The bonded interactions are almost always modeled with harmonic (parabolic) functions which practice is acceptable close to equilibrium. For non-bonded interactions, the van der Waals part is modeled with inverse power terms in the interatomic distances, 12, or occa-... 

Contrary to the practical results reviewed above, statistics from correlation work revealed a serious deficiency in the accuracy with which Phase I Equations 3 and 4 predicted -for the Phase II dataset r for Equation 3 predictions for the 103 compound Phase II data was only 0.45 r for Equation 4 predictions for the Phase II dataset was only 0.44. An analysis of the residuals for the Phase II dataset [Potency(observed)-Potency(predicted by Phase I models)] immediately Identified the source of the problem of the 26 Phase II compounds having DICARB >4, 17 had potency for adult observed more than one log unit better than predicted 15 had egg potency observed more than one log unit better than predicted. As schematically shown in Figure 2B, the parabolic functions for DICARB for the Phase I models underpredict at values of DICARB extrapolated beyond those represented in the Phase I dataset. 

There is no doubt that the DICARB function must describe an optimum near 2 (mite) or 3 (egg) carbons. It is also clear that the exponential aspect of a parabolic function exaggerates the detrimental aspects of extreme values of DICARB away from an optimum. The solution to the problem is the creation of transforms of DICARB according to the following expressions ... 

Figure 2. Comparison of linear vs. quadratic (parabolic) function, and planar vs. quadratic surface functions.

This equation describes the velocity of a fluid element as a parabolic function of its radial distance from the center of the tube (see Fig. 4.5b). 

Assume that the maximum of u is at the boundary x = —N at to- Then according to the Hopf theorem for parabolic functions [34a,b],... 

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