Sentence examples for by the hyperbolic functions from inspiring English sources

The travelling wave solutions are described by the hyperbolic functions, the trigonometric functions and the rational functions.

The obtained solutions have been articulated by the hyperbolic functions, trigonometric functions and rational functions with arbitrary constants.

Equations (29), (30), (32), and (33), which are expressed by the hyperbolic functions, are a kind of kink-type envelope solitary solutions.

With the aid of solutions of the elliptic-like equation, more explicit traveling wave solutions expressed by the hyperbolic functions, trigonometric functions and rational functions are found out.

({{u}_{j}} ( x,t ) ) ((j=5,6,7,8 )), which are expressed by the hyperbolic functions, are a kind of kink-type envelope solitary solutions.

However, empirical studies in humans and non-human animals reported that delay discounting is better described by the hyperbolic function (see Appendix II for mathematical characteristics of hyperbolic discounting) [ 1- 10]: V D = A/(1+k d D) with the same notations as in Equation 1. Again, a larger k d value corresponds to more rapid discounting.

The probe intensity measured in the microarray experiment is given to a good approximation by the hyperbolic function [ 18, 19, 24- 27] (3) I p P = Mx X p P,S × w p P,S + X p P,N × w p P,N 1 + X p P,S + X p P,N + O with p ≡ g,k where M is the maximum intensity upon complete saturation of the probes and O is the optical background.

Expressing the hyperbolic functions by exponentials, one easily finds begin{aligned} Delta Y T) = U_{mathrm{a}} left( 1 - {hbox {e}}^{-T/varTheta }right) end{aligned} (99 to a very good approximation.

The hyperbolic function proposed by Abbo Sloan was employed not only to approach the Mohr Coulomb criterion but also to express the plastic potential function.

The parameter a influences the shape of the hyperbolic function F q) describing fertilization success.

The bipolar sigmoidal activation function is normally modeled by the hyperbolic tangent function [28, 30, 33] which cannot be easily implemented in the FPGA since it consists of infinite exponential series [29].