Bell-shaped function

The choice of the specific orthogonal polynomial is determined by the convergence. If the signal to be approximated is a bell-shaped function, it is evident to use a polynomial derived from the Gauss function, i.e. one of the so-called classical polynomials, the Hermite polynomial. Widely used is the Chebyschev polynomial one of the special features of this polynomial is that the error will be spread evenly over the whole interval. 

The rate constants for oxidation of alkanes and alkenes with peroxynitrous acid in aqueous-gas phase are bell-shaped functions of the volume ratio between the liquid and gas phases. The kinetics of the generation of OH radicals and its importance in understanding the mechanism of lipid membrane oxidation has been stressed.236... 

The general shape of the cross sections as functions of E, n, and j resembles the initial distribution function in the electronic ground state. If the parent molecule is initially in its lowest vibrational state, the partial cross section behaves qualitatively like a three-dimensional bell-shape function of E,n, and j. If the photodissociation starts from an excited vibrational state, a(E, n,j) will exhibit undulations which reflect the nodal structures of the corresponding bound-state wavefunction [more of this in Chapter 13 see also Shapiro (1981) and Child and Shapiro (1983)]. 

Most of the observations of the M.I.R. actually involve thermal e.t. processes Fig. 11 gives an illustration of the rate constant versus energy plot of what appears to be a complete bell-shaped function, but this consists really of two distinct parts [25 b] the photo-induced forward process is restricted to a normal region, because of the restricted range of free energies and similarly, the thermal back e.t. shows... 

The classical (or semiclassical) equation for the rate constant of e.t. in the Marcus-Hush theory is fundamentally an Arrhenius-Eyring transition state equation, which leads to two quite different temperature effects. The preexponential factor implies only the usual square-root dependence related to the activation entropy so that the major temperature effect resides in the exponential term. The quadratic relationship of the activation energy and the reaction free energy then leads to the prediction that the influence of the temperature on the rate constant should go through a minimum when AG is zero, and then should increase as AG° becomes either more negative, or more positive (Fig. 12). In a quantitative formulation, the derivative dk/dT is expected to follow a bell-shaped function [83]. 

Figures 5.4 and 5.5 summarize results of a recent study of P. versicolor laccase electrochemistry based on cyclic and rotating disk voltammetry [60]. Figure 5.4 shows unequivocally that this laccase is voltammetrically active and gives a kinetically controlled, unpromoted four-electron peak at edge-plane pyrolytic graphite. Electrochemical reduction of 02 catalyzed by an immobilized laccase monolayer is close to reversible, and unrestricted by mass transport. The electrocatalysis follows, moreover, a Michaelis-Menten pattern (Fig. 5.5). Finally, there is a characteristic bell-shaped functional pH-profile with a pronounced maximum at pH 3.1.

Gaussian function A highly useful function named after mathematician Carl Friedrich Gauss. The familiar bell-shaped function is symmetric and has the property that its integral is 1. In statistics, a Gaussian distribution is called a normal distribution and has the familiar parameters mean ((j.) and standard deviation (a) ... 

At each th projection point, the ath atom contribution to the yth descriptor is obtained by a Lorentzian function L (but can be any bell-shaped function) ... 

AC-EO displacement becomes zero as the non-dimensional frequency O goes to zero or infinity, which represents a bell shaped function with both ends approaching zero. 

An alternative approach to analyze the solutions of the filament equation was developed by Menon and Gottwald (2005) and Cox and Gottwald (2006) based on a non-perturbative technique. In order to find an approximate solution close to the bifurcation point or for large Da a test function with a few free parameters is chosen, that is suitable to provide a good approximation for the solution in some particular regimes. For example, close to the bifurcation point the solution is a bell-shaped function and can be approximated by a test function of the form... 

It should be pointed out that Greilf proposed that the thermostability of freeze-dried protein products could be a bell-shape funetion of the residual moisture eontent [26]. In other words, both overdrying and underdrying are detrimental. Hsu et al. also coneluded that the generally accepted concept the drier, the better is not neeessarily appropriate for their product [27]. It is noted that their data were generated from excipient-free proteins. In this study, however, our results did not demonstrate a bell-shape function for the crystalline matrix formulation. 

At room temperature and 1 atm the two peaks are separated by about 29.5 cm" and have line-widths of about 13 and 10 cm for and R2, respectively. The most widely known bell-shaped functions, Gaussian and Lorentzian, can be adjusted to represent the major features of the R lines, but they fail to represent adequately the outer wings, the... 

The mathematical formulation for the representation of the near-fault ground velocity pulses proposed by Mavroeidis and Papageorgiou (2003) is the product of a harmonic oscillation with a bell-shaped function. That is ... 

---