Tanh function

From any real input, this provides an output in the range 0,1. The Tanh function (Figure 2.23) has a similar shape, but an output that covers the range -1, +1. Both functions are differentiable in closed form, which confers a speed advantage during training, a topic we turn to now. 

The coth and tanh functions play an important role in the modelling of the magnetic behaviour of transition metal complexes. 

Both expressions are identical. This is easily seen when remembering the definition of the tanh function ... 

These two figures show the profile twice because for computational reasons (viz. to introduce periodic boundary conditions) it is easier to work with a slab of the liquid under consideration. As expected, the transition layer is thicker at higher temperatures. The weight densities are readily converted into number densities. It appeared that the profiles could be well fitted with a tanh function, so that an effective thickness may be derived, say tgQ jg- See fig. 2.13. 

Equation (2) describes a paramagnetic to ferromagnetic phase transition at = NppoAlk. A similar calculation for atoms with J > 1/2 can be performed by summing over each of the U + 1) magnetization states. The result is that the tanh function in Equation (2) is replaced by the Brillouin function (see any textbook of solid state physics, e.g. Kittel 1976 or Crangle 1977). 

When Lhe argumenL of the tanh function is small, this function may be replaced by the argument. Under these circumstances the potential profile is given by... 

Instead of performing the normal mode analysis we have used a more approximate method to take the qr- -coordinates into account. For the Cl - - CH4/CD4 reactions wc have in some work used a tanh-function in the breaking bond to interpolate between the saddle point and the product asymptote to get both the reaction thermicity and AfA" consistent with the ah initio calculations[18]. In addition, if the effective potential energy surface of the system is modeled by the semiempirical London-Eyring-Polanyi-Sato (LEPS) function, the correction is made directly in the Morse parameters for the two reactive bonds by adjusting the Sato parameters) , 19]. 

Note that Fernandez et al. fitted their results to an error fiinction profile, so their reported interfacial width of 20 A needs to be multiplied by y/2 i to be compared with the width of the tanh function used by Russell s group. 

It is evident that in such reconsideration that it is sufficient to take into account thermodynamic probabilities when the choice of the host grain is made, the probability of choosing one of the two grains should be proportional to the tanh function, as in the Potts model. The probability of choosing the host grain by any atom at the GB is determined by the local curvature of GB. In our experiments, the surface of A1 grains looked flat. So, from here we take into account only the curvature of the GB joint with the surface, but not the curvature of the surface... 

Accordingly, the velocity of GB lateral motion is also proportional to the tanh function... 

A schematic of potentials in a cell with the dead spot on the anode side is shown in Figure 5.38. The spot is characterized by much lower eleetrochemical activity. Let Rs be the spot radius and r be the radial coordinate from the spot center (Figure 5.38). Lower electrochemical activity means reduction of the HOR exchange current density jhy inside the spot. To describe this reduction, one can use the smooth tanh function... 

Toxvaerd minimizes the free energy of a system with a planar surface, with respect to the parameters of a trial function for p(z), and subject to the constraints of fixed gas and liquid densities, p( o°), and of fixed number of molecules, which is achieved by a suitable condition on the position of the equimolar dividing surface. His trial function for p(z) comprises a constant term, Kp +P ). md two tanh functions with arguments [2A, g(z-5)], where A is used for zb, and where i = (Ar -A, )ln2. Clearly there is a discontinuity in the derivative p (z) at z h, but this is imperceptible in his published graphs. From the optimum profile and the minimized free energy he calculates the surface tension as the surface excess free energy with respect to the equimolar dividing surface. 

The characteristic width of the polaron is determined by lp=V21s and M=0.623 Ip [39], where 1 is the soliton width. The soliton width is estimated to ls=3 by fitting a tanh function to the optimized order parameter (see Ref. 26 and below). This length is very close to the value obtained from ab inlto, extended basis set Hartree-Fock calculations for a single soliton in the presence of a screened sodium counterion [40],... 

For 5 = 0, we need only take the limit 8 —> 0 in Eq. [238] for 8 < 0 (the low surface charge density case), we put 8 —> i8 and by analytic continuation obtain the tanh function. The form of the integral chosen is appropriate for highly charged cylindrical polyelectrolytes such as DNA. We then have the solution for the RMP within the PB cell model with no added salt... 

The hyperbolic tangent (tanh) function, with a sigmoid shape, is the most used transfer function for artificial neural networks. The corresponding kernel has the formula ... 

The RBF kernel (experiments 16-24), with ACp between 0.96 and 0.97, has better calibration statistics than the linear kernel, but its performance in prediction only equals that of the linear SVM. Although many tests were performed for the neural kernel (experiments 25-51), the prediction statistics are low, with ACp between 0.64 and 0.88. This result is surprising, because the tanh function gives very good results in neural networks. Even the training statistics are low for the neural kernel, with AC, between 0.68 and 0.89... 

The ANN cross-validation results show that using more than one hidden neuron results in decreased predictive power. Because it is very time consuming, the LOO procedure was not tested with ANN. Therefore, as the best model for the neural network QSAR, we selected the one with a single hidden neuron and with tanh functions for both hidden and output neurons tcai = 0.934, RMSEcai = 0.38 fL5%o=0.906, 5% =0.820, RMSEl5%o = 0-44 and... 

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