On the Approximation of the Cut and Step Functions by Logistic and Gompertz Functions

Abstract

Abstract-We study the uniform approximation of the sigmoid cut function by smooth sigmoid functions such as the logistic and the Gompertz functions. The limiting case of the interval-valued step function is discussed using Hausdorff metric. Various expressions for the error estimates of the corresponding uniform and Hausdorff approximations are obtained. Numerical examples are presented using CAS MATHEMATICA.

Convergence in Orlicz spaces by means of the multivariate max-product neural network operators of the Kantorovich type and applications

“…1, left) and hyperbolic tangent sigmoidal functions (see e.g. [16,17,19,41]), defined respectively by: Such kind of sigmoidal functions are the most used in the applications by means of neural networks, since their smoothness make them very suitable for the implementation of learning algorithms, such as the well-known back-propagation algorithm, see e.g. [8,35,36,38,43,44,49,51].…”

Building reaction kinetic models for amiloid fibril growth

“…Reaction scheme (18) describes the reaction mechanism between an enzyme E with a single active site and a substrate S, forming reversibly an enzyme-substrate complex C, which then yields irreversibly a product P . Reaction scheme (18) says that during the transition of the substrate S into product P the enzyme E bounds the substrate into a complex C having specific properties different than the properties of the free enzyme and thus being necessarily considered as a separate substance.…”

“…and applying the mass action law to Henri's reaction scheme (18) we obtain the following system of ODEs:…”

“…In the above reaction network (30 the transition of the aggregated monomer P into fibril follows the Henri enzyme kinetic mechanism (18). Applying the mass action law we obtain: dm/dt = −k + f m + k − c, df /dt = −k + f m + (k − + k c )c + k c p − k f p + 2k p c p , dc/dt = k + f m − (k − + k c )c, dp/dt = k c c − k f p + k c p , dc p /dt = k f p − (k + k p )c p .…”

Pointwise and uniform approximation by multivariate neural network operators of the max-product type

“…These functions are characterized by unsymmetrical growth, hence they fail to satisfy condition (Σ1), see e.g. [20,21,33]. Now, in order to conclude this section and to illustrate the performance of the above approximation processes, we give a numerical example of a concrete approximation that can be achieved by the above operators.…”