Tangent functional

The simulated free surface of liquid water is relatively stable for several nanoseconds [68-72] because of the strong hydrogen bonds formed by liquid water. The density decrease near the interface is smooth it is possible to describe it by a hyperbolic tangent function [70]. The width of the interface, measured by the distance between the positions where the density equals 90% and 10% of the bulk density, is about 5 A at room temperature [70,71]. The left side of Fig. 3 shows a typical density profile of the free interface for the TIP4P water model [73]. 

Fig. 2.9.12 Phase shift maps measured as the difference in the signal phases with and without current pulses. The rf, field gradient and current pulse program are shown in Figure 2.9.2. The phase shift maps must be unwrapped as the original evaluation yields phase angles only in the principal range between —7t/2 and +7r/2 as a consequence of the 7r periodicity of the tangent function formed as the quotient of the imaginary and real signal...

The first work on pKa determination by zone electrophoresis using paper strips was described by Waldron-Edward in 1965 (15). Also, Kiso et al. in 1968 showed the relationship between pH, mobility, and p/C, using a hyperbolic tangent function (16). Unfortunately, these methods had not been widely accepted because of the manual operation and lower reproducibility of the paper electrophoresis format. The automated capillary electrophoresis (CE) instrument allows rapid and accurate pKa determination. Beckers et al. showed that thermodynamic pATt, (pATf) and absolute ionic mobility values of several monovalent weak acids were determined accurately using effective mobility and activity at two pH points (17). Cai et al. reported pKa values of two monovalent weak bases and p-aminobenzoic acid (18). Cleveland et al. established the thermodynamic pKa determination method using nonlinear regression analysis for monovalent compounds (19). We derived the general equation and applied it to multivalent compounds (20). Until then, there were many reports on pKa determination by CE for cephalosporins (21), sulfonated azo-dyes (22), ropinirole and its impurities (23), cyto-kinins (24), and so on. 

Before training the net, the transfer functions of the neurons must be established. Here, different assays can be made (as detailed in the previous sections), but most often the hyperbolic tangent function tansig function in Table 5.1) is selected for the hidden layer. We set the linear transfer function purelin in Table 5.1) for the output layer. In all cases the output function was the identity function i.e. no further operations were made on the net signal given by the transfer function). 

Functions having a property f(x a) =f x) are known as periodic functions with a period a, and are said to be many-to-one functions. In the examples given above, the period for the sine and cosine functions is 2n, while that for the tangent function is n. 

Figure 2.17 Plots of the trigonometric functions sine (dot-dash line), cos 0 (full line), and tane (dashed line) for -2n s, 0 < 2n. The principal branch of each function is shown by the thick lines. The dotted vertical lines at odd multiples of n 2 indicate the points of discontinuity in the tangent function at these values of e...

The tangent function, tanx = sinx/cosx, shown in Figure 3.3, is interesting because it exhibits infinite discontinuities whenever x passes through an odd multiple of f. 

Figure 1.8 shows the variation of the dimensionless axial velocity on the flow axis ((y/R = 0) along the radial direction. This variation can be approximately represented by the hyperbolic tangent function below ... 

If the arctan (inverse tangent) function is defined to give values between +90° and —90°, we would have control experiment is done with r = 0, and the control field map is subtracted from the field map generated with the delay r. 

Other sigmoidal functions, such as the hyperbolic tangent function, are also commonly used. Finally, Radial Basis Function neural networks, to be described later, use a symmetric function, typically a Gaussian function. 

The neurons weight all inputs and provide an output via the activation function. The complexity of the neural networks used will be determined by the number of nodes in the hidden layer (2,3,5 or 7). The activation applied in this application is a hyperbolic tangent function. In mathematical terms, the output of neuron j is defined by n With yj output of neuron j... 

Figure 2. Temporal evolutions of M(t). U = 0.69. The horizontal line represents the canonical equilibrium value of M. On each curve, two short vertical lines are marked. The first and the second ones are at the end of Stage I and II, respectively. Solid curves are hyperbolic tangent functions (5). [Reproduced with permission from Y. Y. Yamaguchi, Phys. Rev. E 68, 066210 (2003). Copyright 2004 by the American Physical Society.]...

A theoretical prediction of fn/m, the upper curve in Fig. 3, is obtained by fitting the magnetization M(t) as hyperbolic tangent function,... 

From the numerical results of Cp(t x), Fig. 10b, we determine the values of three parameters Cp(0 x), fcorr(x), and (3(x) at some value of x by using the least-squares method. The discrete values of the parameters are not enough to reproduce da (t)jdt accurately, and then we approximate the parameters by hyperbolic tangent functions as follows ... 

The hyperbolic tangent functions are in good agreement with numerical results, as shown in Fig. 14. To confirm the validity of the approximation, we reproduced... 

The translational order parameter permits an estimate of the width of the interfaces. The 10-90 width is defined to be the length over which a specific interfacial order parameter changes from 10% to 90% of the bulk solid value. We have estimated the 10-90 widths of the interfaces using a fit by a simple hyperbolic tangent function, used frequently in earlier studies [17]. In the case of the mass-density profile, the translational order parameter may be extracted from a fitting procedure,... 

The signs in Eq. (11) are governed by the same convention as for Eq. (8b). It has been also shown [49] that in the neighborhood of Tc the interfacial width w is related to the correlation length calculated at coexistence conditions w= 2 b=2 (( >1)=2 (( >2). In practice the hyperbolic tangent function turns out to be also a very good approximate form in the case of NA NB and -dependent y. 

Hyperbolic Tangent Function. This is a form of sigmoid function but it produces values in the range [—1, - -1] instead of [0,1]... 

If the counterion distribution function P is known, the condensed counterion fraction can be characterized in the following geometric way Eqs. 18 and 19 show that P viewed as a function of ln(r) is merely a shifted tangent function with its center of symmetry at (ln(RM) fe). Since, however, tan"(0)... 

Fig. 1. The convex functional F has a unique tangent functional L in the point n0.

Theorem 8. The functional FL has a unique tangent functional for every E-V-density and nowhere else. Moreover the tangent functional at an E-V-density n can be identified with — v where v is the potential that generates this density. 

Let us now suppose that n0 is not an E-V-density. Let us further suppose that there is a tangent functional at n0, i.e., that for some v equation (205) and therefore also... 

---