Curvature exponential function

One of the main criticisms of cubic and polynomial functions is that they produce forward rate curves that exhibit unrealistic properties at the long end, usually a steep fall or rise in the curve. A method proposed by Vasicek and Fong (1982) avoids this feature, and produces smoother forward curves. Their approach characterises the discount function as exponential in shape, which is why splines, being polynomials, do not provide a good fit to the discount function, as they have a different curvature to exponential functions. Vasicek and Fong instead propose a transform to the argument T of the discount function v(T). This transform is given by... 

Note that the distance decay for large separations of Eq. [348] is that of a screened exponential function rather than the faster (and correct) decay of a screened Coulomb. Our derivation is clearly limited to small KdHq for large separations the effect of surface curvature becomes apparent as nonparallel deviations of the field lines for two spheres leads to a faster decay than does... 

The series is a linear model with no accounting for curvature. A plot of this surface is shown in Fig. A.2. Notice that the series is a quite good approximation to the true response surface near the evaluation point and up to around x, y = 0.8, 0.8. At this point, the function begins to fail because the true function begins to grow exponentially whereas the linear approximation remains linear in its growth. Figure A.2 presents a third-order approximation... 

Summarizing the model results, we find that despite many simplifications the model of Fig. 12b provides a detailed description of the lifetime and recombination resistance, quantities that can be measured as a function of the voltage in a DSC. The main feature is that the parabola in the exponential of the Marcus model electron transfer rate between the semiconductor surface and redox acceptor in the electrolyte translates in curvatures of Tn and R ec- These quantities have been reported in hundreds of publications, but usually a linear behavior in semilogarithmic plot is observed, as reported in representative measurements of Figs. 4, 5, and 14. The absence of curvature could be evidence of a large reorganization energy in the DSC with redox couples, and the value X = leV is often used [213]. 

For pure fluids, it is most common to represent the saturated vapor and saturated liquid transport properties as simple polynomial functions in temperature, although polynomials in density or pressure could also be used. Exponential expansions may be preferable in the case of viscosity (Bmsh 1962 Schwen Puhl 1988). For mixtures, the analogous correlation of transport properties along dew curves or bubble curves can be similarly regressed. In the case of thermal conductivity, it is necessary to add a divergent term to account for the steep curvature due to critical enhancement as the critical point is approached. Thus, a reasonable form for a transport property. 

Several major generalizations are needed to discuss vibrational and NLO spectra. The CH units represented by M in Fig. 6.1 have vibrational degrees of freedom, and the PA backbone is planar rather than one-dimensional. The amplitude mode (AM) formalism developed by Horovitz and coworkers [18,19] extends the SSH model to several coupled q = 0 modes, as discussed in Section II.C, with the partitioning taken from experiment. The second term in the Taylor expansion of t(R) appears in force fields, as already recognized by Coulson and Longuet-Higgins [20]. While the form of t (R) need not be specified in advance, wave function overlap is usually taken to be exponential, and this fixes the curvature without additional parameters as... 

Our analysis will examine whether either of these classes of model describes experiment. While a power law and a stretched exponential both can represent a narrow range of measurements to within experimental error, on a log-log plot a power law is always a straight line, while a stretched exponential is always a smooth curve of nonzero curvature. Neither form can fit well data that are described well by the other form, except in the sense that in real measurements with experimental scatter a data set that is descrihed well by either function is tangentially approximated over a narrow region hy the other function. 

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