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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.
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 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 ...
The model hyperbolic equation is the wave equation. In one spatial dimension, this is ∂ 2 u ∂ t 2 = c 2 ∂ 2 u ∂ x 2 {\displaystyle {\frac {\partial ^{2}u}{\partial t^{2}}}=c^{2}{\frac {\partial ^{2}u}{\partial x^{2}}}} The equation has the property that, if u and its first time derivative are arbitrarily specified initial data on the ...
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]
Depending on the type of censoring, the maximum likelihood function technique along with an appropriate log-likelihood function may be used to estimate the model parameters. If the sample consists of right censored data and the model to use is Hypertabastic proportional hazards model, then, the proportional hazards log-likelihood function is
A hyperbola is an open curve with two branches, the intersection of a plane with both halves of a double cone.The plane does not have to be parallel to the axis of the cone; the hyperbola will be symmetrical in any case.