Hyperbolic cosine Differentiation

One expects that the differential capacity of a semiconductor/electrnlyte interface due to the space charge inside an intrinsic semiconductor will vary in a hyperbolic cosine manner with the potential. Such a variation is shown in Fig. 6.127. 

This function Is plotted in fig. 3.6. Because of the hyperbolic cosine dependency and because both 2U"e proportional to there is some similarity with the differential capacitance, fig. 3.5. The physical background Is that both quantities depend In a similar way to screening. Writing AG as... 

The forces can be obtained from this analysis by simple differentiation, since the sums are absolutely convergent. Although the form in Eq. 69 has a much better convergence than the original form in Eq. 64, its main advantage is a linear computation time with respect to the number of particles N. To see this, the equation has to be rewritten using the addition theorems for the cosine and the hyperbolic cosine. First, one calculates the eight terms... 

The coefficients in the above series in if/p(r) alternate between the hyperbolic sine and cosine value of the uniform contribution, i/r0(z). In contrast to a full linear treatment, which is the usual procedure followed, the 0(if/p) term here does not vanish. As if/0(z) is large we must regard it as satisfying the nonlinear, ordinary differential form of the PB equation,... 

Equation 3-56 is a linear, homogeneous, second-order differential equation with constant coefficients. A fundamental theory of differential equations states (hat such an equation has two linearly independent solution functions, and its general solution is the linear combination of those two solution functions. A careful examination of the differential equation reveals that subtracting a constant multiple of the solution function 0 from its second derivative yields zero. Thus we conclude that the function 0 and its second derivative must be constant multiples of each other. The only functions whose derivatives are constant multiples of the functions themselves are the exponential functions (or a linear combination of exponential functions such as sine and cosine hyperbolic functions). Therefore, the solution functions of the differential equation above are the exponential functions e or or constant multiples of them. This can be verified by direct substitution. For example, the second derivative of e is and its substitution into Eq. 3-56... 

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. 

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