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The hyperbolastic functions, also known as hyperbolastic growth models, are mathematical functions that are used in medical statistical modeling. These models were originally developed to capture the growth dynamics of multicellular tumor spheres, and were introduced in 2005 by Mohammad Tabatabai, David Williams, and Zoran Bursac. [ 1 ]
The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic functions that does not involve complex numbers. The graph of the function a cosh( x / a ) is the catenary , the curve formed by a uniform flexible chain, hanging freely between two fixed points under uniform gravity.
The hyperbolic distribution is a continuous probability distribution characterized by the logarithm of the probability density function being a hyperbola. Thus the distribution decreases exponentially, which is more slowly than the normal distribution. It is therefore suitable to model phenomena where numerically large values are more probable ...
Another example of hyperbolic growth can be found in queueing theory: the average waiting time of randomly arriving customers grows hyperbolically as a function of the average load ratio of the server. The singularity in this case occurs when the average amount of work arriving to the server equals the server's processing capacity.
The following is a list of integrals (anti-derivative functions) of hyperbolic functions. For a complete list of integral functions, see list of integrals. In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration.
Derivative of the function is defined by the formula: ′ + + + The following conditions are keeping the function limited on y-axes: a ≤ c, b ≤ d.. A family of recurrence-generated parametric Soboleva modified hyperbolic tangent activation functions (NPSMHTAF, FPSMHTAF) was studied with parameters a = c and b = d. [9]
The distance function for the Beltrami–Klein model is a Cayley–Klein metric.Given two distinct points p and q in the open unit ball, the unique straight line connecting them intersects the boundary at two ideal points, a and b, label them so that the points are, in order, a, p, q, b, so that | aq | > | ap | and | pb | > | qb |.
For the hyperbolic model using g(D), the discount for a week from now is () =, which is the same as for f in the exponential model, while the incremental discount for an additional week after a delay of D weeks is not the same: (+) = + From this one can see that the two models of discounting are the same "now"; this is the reason for the choice ...