Hyperbolic tangent function

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]. 

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). 

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 ... 

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]... 

Radial density profiles perpendicular to the fiber axis can be fitted to a hyperbolic tangent function. Equation (1). For fibers with diameters in the range 5-8 nm, the correlation lengths, are about 0.6 nm, which is close to the value obtained with the models of the free-standing thin films. The end beads are enriched in the surface region, as was also the case with the free-standing thin films. The anisotropy of the chord vectors, as assessed by the order parameter, S, is also similar to the result obtained with the free-standing thin films. 

Activation function Every neuron has its own activation function and generally only two activation functions are used in a particular NN. Neurons in the input layer use the identity function as the activation function. That is, the output of an input neuron equals its input. The activation functions of hidden and output layers can be differentiable and non-linear in theory. Several well-behaved (bounded, monotonically increasing and differentiable) activation functions are commonly used in practice, including (1) the sigmoid function f X) = (1 + exp(-A)) (2) the hyperbolic tangent function f X) = (exp(A) - exp(-A))/ (exp(A) + exp(-A)) (3) the sine or cosine function f(X) = sin(A) or f X) = cos(A) (4) the linear function f X) = X (5) the radial basis function. Among them, the sigmoid function is the most popular, while the radial basis function is only used for radial basis function networks. 

The specific sugar production rate in the chloroplast ql thus depends on the photon flux density Ip. This relation can be described by the following model based on the hyperbolic tangent function, the model ofjassby and Platt (1976)... 

The hyperbolic tangent function is composed of exponential components and can also be written differently ... 

Mean-field theories of the surface tension of polymer solutions have been developed using the Cahn square gradient approach for interfacial properties of solutions and mixtures both for attractive and for repulsive air/liquid interfaces (Cahn and Hilliard 1958), in a way analogous to the treatment of surface segregation in polymer blends given in section 5.1. For situations in which a surface excess was formed, the volume fraction profile was a hyperbolic cotangent, whereas repulsive profiles were described by hyperbolic tangent functions. Values of the surface tension of semi-dilute solutions of polyst)n ene in toluene (a depletion layer) and polydimethyl siloxane in toluene (an attractive interface, a surface excess formed) were well described by this theory. 

Therefore, one may fit the time-dependence of lum to a hyperbolic tangent function to determine kd. Note from the denominator of eq. (39), that, for this system, the total luminescence decay is described as a biexponential, and it is sometimes easier to determine the individual diastereomeric decay constants from a biexponential analysis. 

Optimal control of a batch distillation column consists in the determination of the suitable reflux policy with respect to a particular objective function (e.g. profit) and set of constraints. In the purpose of the present work, the optimisation problem is defined with an operating time objective function and purity constraints set on the recovery ratio (90%) and on the propylene glycol final purity (80% molar). Different basis fimctions have been adopted for the control vector parameterisation of the problem piecewise constant and linear, hyperbolic tangent function. Optimal reflux profiles are determined with the final conditions of the previous optimal reactions as initial conditions. The optimal profiles of the resultant distillations are presented on figure 2. 

The last equality follows from the definition of q in Equation (10.51). A concise way to express this relationship is through use of the hyperbolic tangent function. 

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