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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.
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]
For example, when a = 0, then (b,c) is a point on the standard hyperbola. More generally, there is a hypersurface in M(2,R) of hyperbolic units, any one of which serves in a basis to represent the split-complex numbers as a subring of M(2,R).
The hyperbolic quaternions form a nonassociative ring; the failure of associativity in this algebra curtails the facility of this algebra in transformation theory. . Nevertheless, this algebra put a focus on analytical kinematics by suggesting a mathematical model: When one selects a unit vector r in the hyperbolic quaternions, th